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https://www.quimicaorganica.org/en/sulfur-phosphorus-and-silicon/1048-synthesis-13-ditians-react-umpolung.html | The 1,3-dithianes allow to change the polarity of the carbonyl carbon of the aldehydes by subtraction of the acid hydrogen, obtaining an organolithic capable of attacking a wide variety of electrophiles.
The initial carbonyl, with positive polarity on carbon, changes in the umpolung reaction to a carboanion. The sulphurs of 1,3-dithiane are vital in stabilizing the negative charge, and the reaction with the oxygenated equivalent is not viable.
Methanal is the most versatile aldehyde in these reactions as it has two hydrogens that can be replaced by electrophiles. | {"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": 0.8054589629173279, "perplexity": 3299.9698782058067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296948871.42/warc/CC-MAIN-20230328201715-20230328231715-00404.warc.gz"} |
https://rupress.org/jcb/article/104/4/1035/47937/Phase-dynamics-at-microtubule-ends-the-coexistence | The length dynamics both of microtubule-associated protein (MAP)-rich and MAP-depleted bovine brain microtubules were examined at polymer mass steady state. In both preparations, the microtubules exhibited length redistributions shortly after polymer mass steady state was attained. With time, however, both populations relaxed to a state in which no further changes in length distributions could be detected. Shearing the microtubules or diluting the microtubule suspensions transiently increased the extent to which microtubule length redistributions occurred, but again the microtubules relaxed to a state in which changes in the polymer length distributions were not detected. Under steady-state conditions of constant polymer mass and stable microtubule length distribution, both MAP-rich and MAP-depleted microtubules exhibited behavior consistent with treadmilling. MAPs strongly suppressed the magnitude of length redistributions and the steady-state treadmilling rates. These data indicate that the inherent tendency of microtubules in vitro is to relax to a steady state in which net changes in the microtubule length distributions are zero. If the basis of the observed length redistributions is the spontaneous loss and regain of GTP-tubulin ("GTP caps") at microtubule ends, then in order to account for stable length distributions the microtubule ends must reside in the capped state far longer than in the uncapped state, and uncapped microtubule ends must be rapidly recapped. The data suggest that microtubules in cells may have an inherent tendency to remain in the polymerized state, and that microtubule disassembly must be induced actively.
This content is only available as a PDF. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9298332929611206, "perplexity": 4398.628671627489}, "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-2021-31/segments/1627046154302.46/warc/CC-MAIN-20210802012641-20210802042641-00650.warc.gz"} |
https://quics.umd.edu/research/publications?page=1&s=title&%3Bf%5Bag%5D=N&%3Bf%5Bauthor%5D=1470&f%5Bauthor%5D=207&o=asc | # Publications
Export 89 results:
Author [ Title] Type Year
Filters: Author is A. M. Childs [Clear All Filters]
H
, Hamiltonian simulation with nearly optimal dependence on all parameters, Proceedings of the 56th IEEE Symposium on Foundations of Computer Science, pp. 792-809, 2015.
L
, Levinson's theorem for graphs, Journal of Mathematical Physics, vol. 52, no. 8, p. 082102, 2011.
, Levinson's theorem for graphs II, Journal of Mathematical Physics, vol. 53, no. 10, p. 102207, 2012.
, The limitations of nice mutually unbiased bases, Journal of Algebraic Combinatorics, vol. 25, no. 2, pp. 111 - 123, 2007.
, Locality and digital quantum simulation of power-law interactions, Phys. Rev. X 9, 031006, vol. 9, no. 031006, 2019.
, Lower bounds on the complexity of simulating quantum gates, Physical Review A, vol. 68, no. 5, 2003.
M
, Momentum switches, Quantum Information and Computation, vol. 15, no. 7-8, pp. 601-621, 2015.
N
, Nearly optimal lattice simulation by product formulas, Phys. Rev. Lett. , vol. 123, no. 050503, 2019.
, Non-interactive classical verification of quantum computation, Theory of Cryptography Conference (TCC), vol. Lecture Notes in Computer Science 12552, pp. 153-180, 2020.
O
, Optimal quantum algorithm for polynomial interpolation, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), vol. 55, p. 16:1--16:13, 2016.
, Optimal state discrimination and unstructured search in nonlinear quantum mechanics, Physical Review A, vol. 93, no. 2, p. 022314, 2016.
P
, Product Formulas for Exponentials of Commutators, Journal of Mathematical Physics, vol. 54, no. 6, p. 062202, 2013.
Q
, Quantum algorithm for linear differential equations with exponentially improved dependence on precision, Communications in Mathematical Physics, vol. 356, no. 3, pp. 1057-1081, 2017.
, Quantum algorithm for multivariate polynomial interpolation, Proceedings of The Royal Society A, vol. 474, no. 2209, 2018.
, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision, SIAM Journal on Computing, vol. 46, no. 6, pp. 1920-1950, 2017. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8336398601531982, "perplexity": 570.1887467545506}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703565541.79/warc/CC-MAIN-20210125092143-20210125122143-00018.warc.gz"} |
http://math.stackexchange.com/questions/618782/help-me-understanding-logic-behind-limits-of-recurence-relations | # Help me understanding logic behind limits of recurence relations
I was trying to understand how limits of recurence relations are working. I have one.
$$a_0 = \dfrac32 ,\ a_{n+1} = \frac{3}{4-a_n}$$
So, from what i know, if this recurence relation has a limit, it have to be bounded and monotonous. To check if it's bounded i have to calculate $$\lim_{x \rightarrow \infty} \frac{3}{4-x}$$ and it goes to 0, so it's bounded.
Now to check if it's monotonous i have to check if $$a_{n+1} - a_n$$ is monotonous.
$$a_{n+1} - a_n = \frac{3}{4-a_n} - a_n = - \frac{(a_n-3)(a_n-1)}{(a_n-4)}$$ This expression is monotonous ( decreasing ) but only starting at x = 5. Is it enough to say that this is monotonous?
If it is. We know that $$\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} a_{n+1}$$
So we calculate it $$L = \frac{3}{4-L}$$ and end up with $$L^2 - 4L + 3 = 0 \rightarrow (L-1)(L-3) = 0$$ We know that the limit of this recurence relation can be 1 or 3. On our classes there always were some examples that had only 1 possible limit to choose. In this example we have decreasing sequence and $a_0 = \dfrac32$ then one possible limit is 1.
I was starting to check it... What if $a_0 = \dfrac12$? What if $a_0 = 5$ or $a_0 = 2$ it turns out that, no matter what starting value we have, sequence always goes to the same limit ( at least in this example ). Is it true for all recurence relations? You can check using wolframalpha clicking here and just manipulate with starting value. Please help me with this and explain those weird things. I'd be so thankful!
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You have some conceptual mistakes right from the beginning.
So, from what i know, if this recurence relation has a limit, it have to be bounded and monotonous.
No, you have that the wrong way around. Every sequence that is bounded and monotonic has a limit, but there are sequences that have limits without being monotonic, such as $$2, \frac{1}{2}, \frac{5}{4}, \frac{7}{8}, \frac{17}{16}, \frac{31}{32}, \frac{65}{64}, \frac{127}{128}$$ (which is generated by the recurrence $x_{n+1}=\frac{3-x}{2}$ and has limit $1$, but alternately increases and decreases).
(The word is "monotonic", not "monotonous", by the way).
Now to check if it's monotonous i have to check if $a_{n+1}−a_n$ is monotonous.
Um, no, checking whether the sequence is monotonic is not the same as checking whether the sequence of first differences are monotonic.
Instead of trying to concoct a single test for montonoicity, it is better to think of it as asking two questions: Is is increasing? Is it decreasing?
In this case, you should be able to prove that if $a_n$ is between $1$ and $3$, then $a_{n+1}$ is smaller than $a_n$ and still between $1$ and $3$. So this case will continue holding forever, and the sequence is decreasing (which is one of the ways it can be monotonic). The same proof shows it is bounded.
(Why $1$ and $3$? Because those are the fixed points of the functions you're iterating, and I know from experience that the behavior of iterated functions change near such points -- effectively I have seen informally that the limit is probably going to be $1$, and I'm now trying to construct a proof that my hunch is right, not trying to feign stupidity and approach it with no hunches).
If you graph the function $y=\frac{3}{4-x}$ it is possible that the only proof you need here is some handwaving that says "for $1<x<3$ this graph is below the line $y=x$ and the function value is always in $[1,3]$".
Your computation of the fixed points now shows that the only limit that's consistent with being decreasing and staying between $1$ and $3$ is $1$, so $1$ must be the limit.
I was starting to check it... What if $a_0=1/2$? What if $a_0=5$ or $a_0=2$ it turns out that, no matter what starting value we have, sequence always goes to the same limit ( at least in this example ). Is it true for all recurence relations?
Even in this example, if $a_0=3$, then $a_n=3$ for all $n$, and therefore the limit is $3$. But otherwise, if $a_0\ne 4$ such that you avoid dividing by zero, the sequence does tend to $1$ no matter where you start it. To convince yourself of this you need to prove
• If $a_n<1$ then from that point on the sequence will increase monotonically towards $1$ (but never become greater than $1$).
• If $a_n>4$ then the next term in the sequence will be negative, and then we're in the previous case.
• If $3<a_n<4$, then the $a_n$ will get progressively further away from $3$ until you reach one that is larger than $4$, and then we're in the previous case.
There are also functions where you can end up with a sequence that grows without bound (such as $a_{n+1}=a_n^2$ if $a_0>1$), or sequences that stay bounded but don't tend to a limit -- such as $a_{n+1}=4a_n(a_n-1)$ which famously exhibits chaotic behavior when $a_0\in(0,1)$.
There are also recurrences that have several different attractive limits, depending on where you start them, such as $a_{n+1}=a_n+\sin a_n$.
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Hey man, thanks for correcting me and nice explaination about those starting values. In fact i have one more doubt about proving monotonoicity. You said that if $1<a_n<3$ then $1<a_{n+1}<a_n<3$, what about if we have 3 fixed points? Which to pick? I'm so confused about it... Sometimes we can do $\frac{a_{n+1}}{a_n} < 1$ to prove it's decreasing, right? But sometimes, like in this case, we end up with some fixed points. Please explain what to do, like to 5 years old, confused kid. I'd appreciate it so much. – Chris Dec 26 '13 at 15:24
@Chris: In general what I'd do is chop the x-axis into intervals with endpoints at each fixed point and whenever the function to iterate has a discontinuity (such as $x=4$ in this case). Then there's some hope (it's not quite a certainty) that we can prove that the function has "similar" behavior within any one of the intervals. So the first one to investigate would be the one that $a_0$ is in. – Henning Makholm Dec 26 '13 at 15:36
@Chris: Yes, sometimes the ratio $a_{n+1}/a_n$ is an easier way to prove motonicity. It depends on the exact function you're investigating which one is easiest -- sometimes there's more than one way that works; at other times only one of them does. And sometimes (hopefully not in homework problems) there's nothing that seems to work at all. The best general advice I can give is try a variety of approaches and hope one of them will work. After some time you may get enough experience to have an idea what to try first depending on how the functions look. – Henning Makholm Dec 26 '13 at 15:39
Okay, thanks a lot :-) – Chris Dec 26 '13 at 15:43
To understand the behaviour of a sequence defined by $u_{n+1}=f(u_n)$ with given $u_0$ sometimes it helps to draw the graphs of $y=x$ and $y=f(x)$ on the same picture. Then it becomes possible to "draw" the sequence $u_n$. For example, when $f(x)=\sqrt{x}$ and $u_0=1/5$ the picture looks like this:
The main trick is that once $u_{n+1}$ is drawn, it is easy to find a point $u_{n+1}$ on the horisontal axis by means of "reflection" using the graph $y=x$. There are some more details on this on the page where I have have found this image: http://www.fmaths.com/recursivethinking/lesson.php
By the way, this example also demonstrates that the limit can depend on $u_0$. Namely, $$\lim_{n\to \infty}u_n=\begin{cases}0, & u_0=0 \\ 1, & u_0\in(0,1], \\ +\infty, & u_0>1\end{cases}$$
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I agree that drawing a graph is very helpful for this kind of problems. But beware that the function in this drawing is qualitatively different from the one in the question! – Henning Makholm Dec 26 '13 at 15:50
@HenningMakholm: Yes, it is different, but the same method works (try "plot[x, 3/(4-x), {x,-5,5},{y,-5,5}]" in WolframAlpha and then argue the same way). I have suggested this drawing only because it better explains the method, and it was the only drawing of this kind I found on the web ;) – tour Dec 26 '13 at 16:19
You have most of your analysis right. Here is the complete discussion:
The two possible limits are $L=1$ and $L=3$ as you correctly point out.
Secondly, $$a_{n+1}-a_n = \frac{a_n^2 -4 a_n +3}{4 - a_n}$$
Between the two possible limits the numerator is negative, and outside the two possible limits it is positive. So we can make the following general observations
If $a_n < 1$ then $a_{n+1} > a_n$ (since numerator is $>0$, denominator $>0$)
If $1 < a_n < 3$ then $a_{n+1} <a_n$ (since numerator is $<0$, denominator $>0$)
If $3 < a_n < 4$ then $a_{n+1} > a_n$ (since numerator is $>0$, denominator $>0$)
If $4 < a_n$ then $a_{n+1} < 0$
So if you start to the left of $3$, $a_n \to 1$
If you start between $3$ and $4$, $a_n \to 4$ where the recurrence blows up, or you may go past $4$ in which case you down to next case.
If you start to the right of $4$, next number is less than zero and you back to the first case, i.e. $a_n \to 1$
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You are missing (-) before polynomial. :-) Thanks for your analysis. – Chris Dec 26 '13 at 15:33
oops! I need coffee badly :-) – user44197 Dec 26 '13 at 15:35
Whoops, i think i need coffee, not you, you just reversed dominator instead of putting (-) in front of whole fraction :-D forgive me. – Chris Dec 26 '13 at 15:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9268877506256104, "perplexity": 216.19992724000102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1408500836108.12/warc/CC-MAIN-20140820021356-00090-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/90167-linked-probabilities.html | # Math Help - Linked probabilities
Hi everyone,
My question seems to be easy, but I can't solve it...
Here are three random variables :
- X ~ N(0,sigma_x) : normal distribtution with mean = 0 and variance = sigma_x²
- Y ~ N(0,sigma_y)
- C ~ N(0,1)
How can the following probability be calculated :
Pr { X+C<a et Y+C<b } ??? (with a and b two real numbers)
The aim is to find a result depending on a, b, sigma_x and sigma_y.
Have a nice day,
2. Tiens, t'es Français ?
Are the three variables independent ? (looks like it is, but I prefer to be sure)
If it is, then you know the joint probability density function of (X,Y,C) (product of the density functions)
Now find the region associated to $\{(x,y,c)\in\mathbb{R}^3 ~:~ x+c. By "find", I mean get the boundaries for x,y,c so that you can write the triple integral : $\iiint_D f_X(x)f_Y(y)f_C(c) ~dx~dy~dc$
The idea is to get a result without integrals.
Because, after, I will have to find a and b such as :
Pr { X+C<a et Y+C<b } = 0.9
I tried hard to calculate the integral that you defined in your post, but without any success.
Do you have any idea?
P.S.: oui, je suis français
4. Originally Posted by guigui1024
The idea is to get a result without integrals.
Because, after, I will have to find a and b such as :
Pr { X+C<a et Y+C<b } = 0.9
I tried hard to calculate the integral that you defined in your post, but without any success.
Do you have any idea?
P.S.: oui, je suis français
It is not possible to get an exact value...
You'll have to make approximations, or use a z-table : http://www.science.mcmaster.ca/psych...e/z-table2.jpg
I'm a bit busy to make the computations, but even so, I think it would be impossible to find a and b. Or at least you may be able to find a constraint...
Maybe you can make the transformation (x,y,c) -> (u,v,w)=(x+c,y+c,c) to see more clearly the boundaries.
I don't know :s
Hello,
Trying to make the transformation you mentionned, I find a ugly piece of calculation which is a kind of :
$
\int^{c=\infty}_{c=-\infty}\int^{x=a-c}_{x=-\infty}\int^{y=b-c}_{y=-\infty}\exp(-(x^2+y^2+c^2))dxdydc
$
I can't solve it... I sent a post in the calculation forum.
Do you have any idea? I mean, without going throug integration, or to calculate easily this integrate...?
Have a nice day,
P.S.: yes, I know the z-table, but it seems that I can't use it directly in my case
6. [1] From the definition of X,Y,C, it seems they are independant.
[2]Theorem:
The distribution of muti-dimensional random variable $\left(X_1,X_2,\ldots,X_n\right)$ is norm distribution ,if and only if
any linear combination of $X_1,X_2,\ldots X_n$
$L_1X_1+L_2X_2+\ldots L_nX_n$
is one-dimensional norm distribution.
[3]According to theorem, it's easy to verify the $\left(X+C,Y+C\right)$ is the norm distribution.
[4]The coefficients between the two random variables $cov\left(X+C,Y+C\right)$ is easy to handle.
[5]Based on the above statements, it seem easy to deal with a, b | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9639171361923218, "perplexity": 2397.4539774356927}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1433195035316.21/warc/CC-MAIN-20150601214355-00002-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://byjus.com/ohms-law-formula/ | # Ohm's Law Formula
Ohms law says that the current running through the conductor is directly proportional to the potential difference across its extremities as long as the temperature and other physical conditions are constant.
## Formula of Ohm’s Law
Ohms law formula is articulated as
V=IR
Where
Voltage is V and is measured in Volts,
The current flowing through the conductor is I and it represented in amperes,
the resistance is R and is measured in ohms
Ohm’s law formula (potential difference formula) is made use of to calculate the Resistance, Current, and Voltage in any given circuit if any of the two quantities are given.
### Ohm’s Law Solved Examples
Underneath are some numerical on Ohm’s law which might be useful for you.
Problem 1: A potential difference of 10 V is applied across a conductor whose resistance is 25 ω. Calculate the current flowing through it?
Given: Potential difference V = 10 V,
Resistance R = 25 ω,
V=I/R
V=10/25
V=0.4volts
Problem 2: If a conductor resistance is 50 Ω and the current passing through the is 5 A. Calculate the potential difference?
Current I = 5 A,
Resistance R = 50 ω,
Potential difference V = IR
= 5× 50
V = 250 volts
For more such valuable equations and formulas stay tuned with BYJU’S!!
#### 1 Comment
1. raymond timothy
You are giving the best thank you | {"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": 0.8671541810035706, "perplexity": 1365.246377210875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652663035797.93/warc/CC-MAIN-20220529011010-20220529041010-00231.warc.gz"} |
https://www.arxiv-vanity.com/papers/hep-th/9402115/ | 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.
\oddsidemargin
= 0.9 in
SU–ITP–94–3
hep-th/9402115
February 13, 1994
TOPOLOGICAL DEFECTS
AS SEEDS FOR ETERNAL INFLATION
Andrei Linde111On leave from: Lebedev Physical Institute, Moscow, Russia. E-mail: , Department of Physics, Stanford University, Stanford, CA 94305-4060, USA
Dmitri Linde222E-mail: California Institute of Technology, Pasadena, CA 91125, USA
ABSTRACT
We investigate the global structure of inflationary universe both by analytical methods and by computer simulations of stochastic processes in the early Universe. We show that the global structure of inflationary universe depends crucially on the mechanism of inflation. In the simplest models of chaotic inflation with the effective potentials or the Universe looks like a sea of thermalized phase, surrounding permanently self-reproducing inflationary domains. On the other hand, in the theories where inflation may occur near a local extremum of the effective potential corresponding to a metastable state, the Universe looks like de Sitter space surrounding islands of thermalized phase. A similar picture appears even if the state is unstable but the effective potential has a discrete symmetry, e.g. the symmetry . In this case the Universe becomes divided into domains containing different phases ( or ). These domains will be separated from each other by domain walls. However, unlike ordinary domain walls often discussed in the literature, these domain walls will inflate, and their thickness will exponentially grow. In the theories with continuous symmetries inflation will generate exponentially expanding strings and monopoles surrounded by thermalized phase. Inflating topological defects will be stable, and they will unceasingly produce new inflating topological defects. This means that topological defects may play a role of indestructible seeds for eternal inflation.
## 1 Introduction
Inflationary cosmology is gradually changing our point of view on the global structure of the Universe [1]. One of the most radical changes has occurred when it was realized that in many versions of inflationary theory the process of inflation never ends. Originally this statement was shown to be correct for the old inflationary scenario [2], and for the new inflationary scenario [3, 4]. The main idea is that the field near the top of the effective potential does not move. Therefore if the Universe expands fast enough, there always will be enough space where the field stays at the top (or occasionally jumps back to the top), and inflation continues. However, this conclusion did not attract much attention. Old inflation did not work anyway, and new inflation was also extremely problematic. It was plagued by the problem of initial conditions, and all its semi-realistic versions looked very complicated and not very natural [1].
Chaotic inflation scenario [5] has brought two surprises. First of all, it was realized that inflation can occur even if there was no thermal equilibrium in the early Universe, and even if the effective potential does not have any maximum at all, or if its maximum is not sufficiently flat. In particular, chaotic inflation scenario can be realized in the theories with potentials , , , and . But the most surprising realization was that inflation in these theories also goes on without end. Due to quantum fluctuations the scalar field in some parts of the Universe perpetually climbs to higher and higher values of its potential energy , until it approaches the Planck density . The existence of this regime may seem counterintuitive. Indeed, the probability that the field jumps up all the time is very small. However, those rare domains where it happens continue growing exponentially, much faster than the domains with small . This scenario was called “eternal inflation” [6].
An important feature of this scenario was the existence of domains where the field may jump for a long time not far away from the Planck density. In these domains the Hubble constant is extremely large, . This induces strong perturbations in all other scalar fields, which eventually leads to division of the Universe into exponentially large domains filled with matter with all possible types of symmetry breaking [1], and maybe even with different types of compactification of space-time [7]. This provides a physical justification of the weak anthropic principle. Under certain conditions, eternally inflating universe enters a stationary regime, where the probability to find domains with given properties does not depend on time [8]. This is a considerable deviation of inflationary cosmology from the standard big bang paradigm. A detailed discussion of this scenario was given recently in [9].
In the simplest versions of chaotic inflation scenario describing only one scalar field the Universe looks like a sea of a thermalized phase, surrounding islands of inflating space [9]. A considerably different picture appears in the old inflationary theory, as well as in those versions of new inflation where the field can stay near the top of the effective potential for a long time, being in a kind of metastable state. For example, if the probability of formation of bubbles of the new phase in the old inflationary universe scenario is sufficiently small, the distance between previously generated bubbles grows exponentially before any new bubbles appear. Thus, the new bubbles appear far away from the old ones. In such a situation the bubbles of a new phase do not percolate; they always remain surrounded by de Sitter space [10]. As similar conclusion was reached in [11] concerning the structure of the Universe in the new inflationary universe scenario. The authors performed a computer simulation of inflation and of quantum fluctuations in a simple theory with a potential which looked like a step function. It was equal to some positive constant for , and it was equal to zero for . This potential mimics many properties of realistic potentials used in the new inflation scenario. However, an important feature of this potential was its absolute flatness near , which effectively made the scalar field near metastable. The conclusion of ref. [11] was that in the new inflationary universe scenario the Universe also consists of islands of thermalized phase surrounded by de Sitter space.
In the present paper we will report the results of our investigation of the global structure of the Universe in the theories with potentials of the type . Inflation in such models may occur in two different regimes. If it begins at , all consequences will be the same as in the simple model . This means that the inflationary domains will look like islands surrounded by the thermalized phase.
On the other hand, for , inflation may occur near as well, as in the new inflationary universe scenario. We will argue that the global structure of the Universe in this case will depend on the properties of the theory. If we consider a theory of a real scalar field with a discrete symmetry , the Universe will consist of islands of thermalized phase with . However, if the scalar field has more than one component, for example, if it is a complex field , then the situation will be different. The Universe will be filled by the thermalized phase containing inflating strings. In the -symmetric theory where the scalar field is a vector , thermalized phase will surround inflating monopoles. This means that topological defects may play an extremely important role in formation of the global structure of the Universe.
Investigation of this issue should help us to obtain a better understanding of a very interesting piece of physics which was missed in our previous studies of new inflation. Until very recently all experts in inflationary theory believed that primordial monopoles produced during inflation in the new inflationary scenario were effectively pointlike objects, which did not inflate themselves. For example, in the first version of this scenario based on the Coleman-Weinberg theory [12] the Hubble constant during inflation was of the order GeV, which is five orders of magnitude smaller than the mass of the -boson GeV. The size of a monopole estimated by is five orders of magnitude smaller than the curvature of the Universe given by the size of the horizon . It seemed obvious that such monopoles simply could not know that the Universe is curved.
This conclusion finds an independent confirmation in the calculation of the probability of spontaneous creation of monopoles during inflation. According to [13], this probability is suppressed by a factor of , where is the monopole mass. In the model discussed above this factor is given by , which is negligibly small. This result is rather general. In all (or almost all) realistic models of inflation the Hubble constant at the end of inflation is smaller than GeV [1]. Meanwhile most of the superheavy topological defects that may have interesting cosmological consequences appear in the theories with the scale of spontaneous symmetry breaking GeV, which is at least two orders of magnitude greater than . The probability of creation of such topological defects by the mechanism described in [13] is extremely small. Even if there were no barrier for production of such objects, their density would have been suppressed by a factor [14].333Superheavy topological defects can be created, however, during inflationary phase transitions [15].
Despite all these considerations, in the present paper (see also [16], [17]) we will show that in those theories where inflation is possible near a local maximum of the effective potential, topological defects expand exponentially and can be copiously produced during inflation. The main reason can be explained as follows. In the cores of topological defects the scalar field always corresponds to the maximum of effective potential. When inflation begins, it makes the field almost homogeneous. This provides ideal conditions for inflation inside topological defects. These conditions remain satisfied inside the topological defects even after inflation finishes outside of them. Moreover, as we are going to argue, each such topological defect will create many other inflating topological defects. We have called such configurations fractal topological defects [16].
The paper is organized in the following way. In Section 2 we will give a description of inflation of domain walls at the level of classical theory. In Section 3 we will briefly describe this process with an account of quantum fluctuations, and present the results of our computer simulations of this process. In Section 4 we will describe similar processes in the case of inflating strings and monopoles. In Section 5 we will consider the problem of initial conditions for inflation near a local maximum of . In the concluding Section 6 we will discuss our main results.
## 2 Inflating domain walls
To explain the basic idea of our work, we will begin with a discussion of inflating domain walls. The Lagrangian of the simplest model where such walls may appear is given by
L=12(∂μϕ)2−λ4(ϕ2−m2λ)2 . (1)
Here is a real scalar field. Symmetry breaking in this model leads to formation of domains with , where . These domains are divided by domain walls which interpolate between the two minima. Neglecting gravitational effects, one can easily obtain a solution for a static domain wall in the plane:
ϕ=η tanh(√λ2 ηx) . (2)
For small our neglect of gravitational effects is reasonable. However, the situation becomes more complicated if becomes comparable to the Planck mass . The potential energy density in the center of the wall (2) at is equal to , the gradient energy is also equal to . This energy density remains almost constant at , and then it rapidly decreases. Gravitational effects can be neglected if the Schwarzschild radius corresponding to the distribution of matter with energy density and radius is much smaller than . Here . This condition implies that gravitational effects can be neglected for . In the opposite case,
η \vbox\hbox to 0.0pt{>}\lower 5.0pt\vbox{\hbox{∼}} 32πMP , (3)
gravitational effects can be very significant. A similar conclusion is valid for other topological defects as well. For example, recently it was shown that magnetic monopoles in the theory with the scale of spontaneous symmetry breaking form Reissner-Nordström black holes [18].
Now let us look at it from the point of view of inflationary theory. Inflation occurs at in the model (1) if the curvature of the effective potential at is much smaller than , where is the Hubble constant supported by the effective potential [1]. This gives , which leads to the condition almost exactly coinciding with (3): .
This coincidence by itself does not mean that domain walls and monopoles in the theories with will inflate. Indeed, inflation occurs only if the energy density is dominated by the vacuum energy. As we have seen, for the wall (2) this was not the case: gradient energy density for the solution (2) near is equal to the potential energy density. However, this is correct only after inflation and only if gravitational effects are not taken into account.
At the initial stages of inflation the field is equal to zero. Even if originally there were any gradients of this field, they rapidly become exponentially small. Each time new perturbations with the amplitude and the wavelength are produced, but their gradient energy density is always much smaller than for [9]. Therefore originally the vacuum energy inside the walls dominated its gradient energy, and walls could easily expand. The reason why we did not understand this before is the same as the reason why we thought that the interior of the bubbles of the new phase cannot expand: We thought that the bubble walls during inflation were thin from the very beginning. Then we understood that this was wrong, and the new inflationary scenario was proposed. Here we encounter the same situation. Domain walls, just as the bubble walls, originally were thick, and they were exponentially expanding.
## 3 Self-reproduction of the Universe and fractal structure of domain walls
Previous description was purely classical. Meanwhile quantum fluctuations play extremely important role in this scenario.
The wavelengths of quantum fluctuations of the scalar field grow exponentially in the expanding Universe. When the wavelength of any particular fluctuation becomes greater than , this fluctuation stops oscillating, and its amplitude freezes at some nonzero value because of the large friction term in the equation of motion of the field . The amplitude of this fluctuation then remains almost unchanged for a very long time, whereas its wavelength grows exponentially. Therefore, the appearance of such a frozen fluctuation is equivalent to the appearance of a classical field that does not vanish after averaging over macroscopic intervals of space and time.
One can visualize fluctuations generated during the typical time as sinusoidal waves with average amplitude
δϕ=H2π . (4)
and with a wavelength . Since phases of each wave are random, the sum of all waves at a given point fluctuates and experiences Brownian jumps in all directions. As a result, the values of the scalar field in different points become different from each other, and the corresponding variance grows as
⟨ϕ2⟩=H34π2 t , (5)
which means that dispersion grows as .
In general, the Hubble constant strongly depends on the value of the scalar field . However, we consider the case when inflation occurs near a local maximum of the effective potential at . This gives , and the average amplitude of fluctuations generated during the time is given by
δϕ=√λ6πη2MP=m2√6πλMP . (6)
These perturbations appear in the background of classically moving field , which grows each time by
Δϕ=V′(ϕ)3H2=ϕλM2P2πm2 . (7)
Comparison of these two quantities shows that for
ϕ<ϕ∗≡m4√2M2Pλ√λ . (8)
If from the very beginning the scalar field was sufficiently small, , then the quantum jumps of the field could always return the field back to even smaller values of . The field jumps back only in a half of domains with . However, this is quite enough since each typical time interval the total volume of such domains grows approximately times [4].
Let us consider fluctuations near in a more detailed way. Suppose that after the typical time quantum fluctuations of the scalar field pushed it away from , and it acquired a positive value inside a domain of a size . During the next period of time the original domain grows in size times, its volume grows times. Therefore it becomes divided into domains of a size of the horizon . Evolution of the field inside each of them occurs independently of the processes in the other domains (no-hair theorem for de Sitter space). In each of these domains the scalar field with a probability may jump back, or it may jump in the same direction. However, these jumps will occur on the scale which is times smaller than the length scale of the previous fluctuation. In average those points which originally jumped to positive will remain positive, and the value of the field at these points will grow.
Suppose now that we paint white domains with positive , paint grey domains with negative , and black – the boundary between these domains. Then after the first step the domain will consist of two parts, one is homogeneously white, and another is homogeneously grey. After the second interval the size of each domain will grow times. The white domain after expansion will contain some grey islands inside it, and the grey domain will contain some white islands. These domains will be separated by black domain walls corresponding to . Only at the domain walls does the Universe return to its state . Outside the walls the field always moves down to the minima of its effective potential. After a while, the Universe becomes divided into white and grey islands separated from each other by black domain walls. These domain walls still continue expand exponentially. Therefore qualitatively the picture we obtain is very similar to the one which emerges in the old inflationary scenario: The islands of thermalized phase are surrounded de Sitter space. However, the physical reason for this picture is somewhat different.
If the field is in a metastable state, or if it is in a state of equilibrium for a certain sufficiently large range of its values, then the bubbles of the new phase always appear surrounded by the old phase. If the decay rate of the old phase is small enough, thermalized phase will be always surrounded by de Sitter space, even if the field can roll only in one direction from its original position. On the other hand, the main reason for the existence of the domain structure of the Universe in the model under consideration is the possibility of the field falling down in two different directions from the maximum of the effective potential . In our model this was achieved due to the discrete symmetry of the effective potential. We should emphasize, however, that in fact we do not need exact or even approximate symmetry. The same conclusions will remain valid for any one-component scalar field which has a potential with a sufficiently flat local maximum. This maximum can be at any point . The flatness condition reads .
As we have mentioned already, the jumps of the field in our model can occasionally change its sign and create grey domains inside white surroundings. Simultaneously this forms new inflating domain walls. These new walls will be formed only in those places where the scalar field is sufficiently small for the jumps with the change of the sign of the field to be possible. Therefore the new walls will be created predominantly near the old ones (where ), thus forming a fractal domain wall structure.
As a part of our investigation, we made a series of computer simulations of this process in a two-dimensional slice of the Universe. All calculations were performed in comoving coordinates, which did not change during the expansion of the Universe. In such coordinates, expansion of the Universe results in an exponential shrinking of wavelengths of perturbations. We represent perturbations as sinusoidal waves in a two-dimensional universe,
δϕ(x,y)=H√u√2π⋅sin(H eHt(xcosθn+ysinθn+αn)) . (9)
Here is some small parameter which controls the time between two consequent steps of our simulations, and are random numbers. Equation (9) follows from the corresponding equations of our paper [9] in the case . This is a very good approximation for describing inflation near the top of the effective potential. It fails in the thermalized regions, but thermalized regions do not influence geometry of exponentially expanding part of the Universe at a distance greater than from the boundary between these regions [10]. This condition was satisfied during our simulations. Therefore we expect that our simulations correctly represent the behavior of the field not too far away from the top of the effective potential. This is all we need. One should not take too seriously the distribution of the field in the regions with in our figures. Fortunately, this distribution for is of no interest for us.
We performed our calculations using the grids containing and points. At each step of our calculations we added to the previous distribution of the field the wave (9) and also took into account the classical drift of the field by
Δϕ(x,y)=−uV′(ϕ)3H2 . (10)
A more detailed description of our method can be found in [9]. Here we will just briefly describe our results, which are shown in Fig. 1.
As we mentioned above, we paint black the regions corresponding to the domain walls. However, in our figures we included into the definition of a domain wall all points where . Thus, the points in white and grey area (, and ) practically never change their color, since at the amplitude of quantum jumps typically is much smaller than the classical drift . Therefore one can consider these regions as the regions containing thermalized matter. We begin our simulations in a domain of a typical size with a field . As we see, after a few steps white and grey islands appear inside the black area, Fig. 1a. Then new and new islands become formed, Figs. 1b–1d. The fractal structure of the domain walls is obvious from these simulations. These simulations are similar to those performed in an important paper by Aryal and Vilenkin [11]. The method used in the present work allows us to reveal the physical nature of the exponentially expanding phase. This phase corresponds to expanding domain walls dividing regions filled by different phases. Note that there are no black walls which separate white regions from white regions. (Such walls would exist in the old inflationary scenario.) This demonstrates an important role topological defects can play in inflationary cosmology: they can determine the global structure of the Universe. This suggests also that in the models with different types of topological defects, the global structure of the Universe may look different.
## 4 Inflating strings and monopoles
We will consider now more complicated models where instead of a discrete symmetry we have a continuous symmetry. For example, instead of the model (1) describing a real scalar field one can consider a model
L=∂μϕ∗∂μϕ−λ(ϕ∗ϕ−η22)2 , (11)
where is a complex scalar field, . Spontaneous breaking of the symmetry in this theory may produce global cosmic strings. Each string contains a line with . Outside this line the absolute value of the field increases and asymptotically approaches the limiting value . This string will be topologically stable if the isotopic vector rotates by when the point takes a closed path around the string.
The (global) monopole solutions for the first time appear in the theory with symmetry,
L=12(∂μ→ϕ)2−λ4(→ϕ2−η2)2 , (12)
where is a vector . The simplest monopole configuration contains a point with surrounded by the scalar field . Asymptotically this field approaches regime with .
The basic feature of all topological defects including strings and monopoles is the existence of the points where . Effective potential has an extremum at , and if the curvature of the effective potential is smaller than , space around the points with will expand exponentially, just as in the domain wall case considered above.
Now we can add gauge fields. We begin with the Higgs model, which is a direct generalization of the model (11):
L=Dμϕ∗Dμϕ−14FμνFμν−λ(ϕ∗ϕ−η22)2 . (13)
Here is a covariant derivative of the scalar field, which in this case is given by . In this model strings of the scalar field contain magnetic flux . This flux is localized near the center of the string with , for the reason that the vector field becomes heavy at large , see e.g. [19]. However, if inflation takes place inside the string, then the field becomes vanishingly small not only at the central line with , but even exponentially far away from it. In such a situation the flux of magnetic field will not be confined near the center of the string. The thickness of the flux will grow together with the growth of the Universe. Since the total flux of magnetic field inside the string is conserved, its strength will decrease exponentially, and very soon its effect on the string expansion will become negligibly small. Therefore vector fields will not prevent inflation of strings.
The final step is to consider magnetic monopoles. With this purpose one can add non-Abelian gauge fields to the symmetric theory (12):
L=12|Dμ→ϕ|2−14FaμνFaμν−λ4(→ϕ2−η2)2 . (14)
Global monopoles of the theory (11) become magnetic monopoles in the theory (13). They also have in the center. Vector fields in the center of the monopole are massless (). During inflation these fields exponentially decrease, and therefore they do not affect inflation of the monopoles.
We should emphasize that even though the field around the monopole during inflation is very small, its topological charge is well defined, it cannot change and it cannot annihilate with the charge of other monopoles as soon as the radius of the monopole becomes greater than . However, an opposite process is possible. Just as domain walls can be easily produced by quantum fluctuations near other inflating domain walls, pairs of monopoles can be produced in the vicinity of an inflationary monopole. The distance between these monopoles grow exponentially, but the new monopoles will appear in the vicinity of each of them. We will show how it happens using computer simulations of this process.
Note that in the simple models discussed above inflation of monopoles occurs only if spontaneous symmetry breaking is extremely strong, . However, this is not a necessary condition. Our arguments remain valid for all models where the curvature of the effective potential near is smaller than the Hubble constant supported by . This condition is satisfied by all models which were originally proposed for the realization of the new inflationary universe scenario. In particular, the monopoles in the Coleman-Weinberg theory also should expand exponentially. The reason why we thought that this is impossible was explained in the Introduction: The Hubble constant during inflation in the Coleman-Weinberg theory is much smaller than the mass of the vector field , which is usually related to the size of the monopole. However, this argument is misleading. The effective mass of the vector field GeV can determine effective size of the monopole only after inflation. Effective mass of the vector field is always equal to zero in the center of the monopole. Once inflation begins in a domain of a size around the center of the monopole, it expels vectors fields away from the center and does not allow them to penetrate back as far as inflation continues.
Of course, one may argue that there is no much reason to consider inflation generated by magnetic monopoles. If the inflaton field is not a gauge singlet, the density perturbations produced after inflation typically are too large [1]. However, there may be many different stages of inflation, and the last one can be driven by a different mechanism. The main problem is how to obtain good initial conditions for the first stage of inflation and (if possible) how to make it eternal. Here topological defects may be of some help.
An interesting feature of this scenario is that inflation of monopoles is eternal for purely classical (topological) reasons [16, 17]. There is the only way for a monopole to stop inflating. Even though we have estimated the amplitude of quantum fluctuations around the monopole to be very small, eventually at some moment this amplitude may appear to be much larger than its typical value . The probability of large jumps of the scalar field is exponentially small [9], but small probabilities can accumulate when we are speaking about eternity. If the gradients of the classical field become sufficiently large because of the large fluctuation , the monopole may stop inflating. However, the probability of this event is much smaller than the probability of the monopole pair creation in the vicinity of an expanding monopole. Therefore quantum fluctuations which may kill inflation of the monopole simultaneously create many new inflationary monopoles. Moreover, even if quantum fluctuation can terminate inflation of a monopole, they certainly cannot do the same for inflating strings and domain walls.
We have performed computer simulations illustrating some of these issues. Our simulations were two-dimensional, and analogs of the monopoles were the centers of the strings in the model (11). The centers of the monopoles should correspond to the points where . There are three series of figures in our simulations. Fig. 2 shows the distribution of potential energy density in the two-dimensional Universe. In the beginning potential energy density is equal to in the whole domain of initial size . After a few steps of expansion the surface shown in Fig. 2a, bends a little, but still the value of the effective potential does not differ much from . (The box (x,y,V) is not shown in this figure.) Later it decreases everywhere except for some points where it remains equal to . These points are the peaks of the mountains surrounded by the thermalized phase in Fig. 2. In the beginning we see just a few such mountains, Fig. 2b, but then they split and form new mountains separated from others by the thermalized phase, Figs. 2c, 2d. Note that all these mountains have equal height. It is instructive to compare this picture with a typical distribution obtained in the chaotic inflation scenario with the potential , Fig. 3. In this case mountains are also separated by the thermalized phase, but their height can be as large as .
Our calculations have been performed with several different sets of parameters. The results shown in Fig. 2 correspond to , , . Inflationary condition is satisfied for these values of parameters. Of course, these parameters are far from their values in realistic models. Still they should give us a qualitatively correct picture of the process.
It is very tempting to identify the peaks of the mountains shown at Fig. 2 with monopoles. However, most of the mountains correspond to topologically trivial field configurations. Moreover, most of them do not even have in the center. Indeed, the only condition which is necessary for the self-reproduction of inflationary domains with large is that the absolute value of the field should be smaller than (8). This means that at the peaks of the mountains is somewhere in the interval . Typically this means that on the peaks of the mountains is very close to , but it may be slightly different from . Thus one should not overemphasize the role of topological defects in the eternal process of self-reproduction of the Universe. This process can occur without topological defects as well. Still the possibility of exponential expansion and self-reproduction of topological defects adds some new dimension to this theory.
The field should make many jumps back from to . Consequently, the number of the monopoles produced due to these jumps will be suppressed by a factor . Monopoles will be copiously produced in this scenario, but only near the points where the field is sufficiently small, , in particular, near other monopoles.
In order to identify those mountains which correspond to monopoles we performed another series of computer simulations. We used color to show the direction of the vector in the isotopic space. Namely, we used white color if this vector was looking in the direction (i.e. positive and vanishing ), and then we gradually increased the level of darkness as the vector rotated by the angle approaching . The point for obvious reasons corresponds to a discontinuity; the color is either white or black depending on the way we approach it. This discontinuity does not imply existence of any physical singularity. However, this color map allows us to identify the monopoles as the points where the boundary lines between black and white end in a grey area.
Fig. 4 shows the distribution of the direction of the vector using this color map. As we can see, monopoles are created in this process, and their distribution indeed looks like a fractal, which becomes more and more complicated in the course of time. (For the attentive reader: there are eight monopoles in Fig. 4a and thirteen monopoles in Fig. 4b.) However, if we impose these pictures on the distribution of the energy density , we will see that some mountains correspond to monopoles, and some do not, see Fig. 5. The stage of the process shown in Fig. 5c corresponds to the field distribution in Fig. 4a. The first monopole can be seen in the upper right part of Fig. 5a.
As we already mentioned, the centers of inflationary domains in Fig. 2 do not form walls surrounding the thermalized phase. On the contrary, inflating domains are surrounded by the thermalized phase. The reason for this behavior in the simplest versions of chaotic inflation scenario can be easily understood. Nothing prevents the field at each particular point to roll down to the minimum of the effective potential. Only very rarely the field jumps against the classical flow down. Those rare points where this happens form the peaks of mountains in Fig. 3. After a sufficiently large time these peaks become surrounded by the thermalized phase.
As we already mentioned, in the situation where the state is metastable with a sufficiently large lifetime we would encounter an opposite regime. Independently of all topological considerations we would obtain islands of thermalized phase surrounded by de Sitter space. Is there any strict boundary between these two regimes? Is it possible that topological defects will prevent rolling of the field down to the minimum of in a considerable part of space even in the situations where the state is unstable?
One can get some insight by a more detailed investigation of the shape of domain walls (generalization to monopoles is straightforward) by using a slight extension of the method of ref. [17]. Let us assume that the field initially is very small, , and its configuration is sufficiently smooth. Here is a comoving coordinate of the point we consider. We will assume also that near the center of a domain wall one can write in the first approximation as , where is some small constant. In this case the amplitude of the scalar field at each particular point will grow exponentially [1],
ϕ(x,t)≈cxexp(m2t3H) , (15)
Let us write this equation in terms of the physical distance :
ϕ≈cXexp[−(H−m23H)t] . (16)
This equation means that at a physical distance from the center of the topological defect the value of the field does not change in time. In other words, inflation stretches the domain wall without changing its shape at small . However, the Universe stretches domain walls in the -direction more slowly that it stretches itself. In the comoving coordinates the thickness of the wall exponentially decrease. Indeed, one can easily see that the value of the field (15) does not change for
x∼exp(−m2t3H) . (17)
This makes it easier to understand the difference between the topological structure of the Universe in the old inflation scenario and in the new one. In the old inflation scenario de Sitter phase decays due to spontaneous appearance of holes inside it, which leads to formation of islands of thermalized phase surrounded by de Sitter space. In our scenario the state is unstable, and all space has a tendency to go to the thermalized phase. Inflation still continues near the regions with , but the comoving size of these regions exponentially decreases in some directions. In the case of domain walls this is not very important; they surround domains of thermalized phase for topological reasons.444The possibility of percolation of thermalized domains in a three-dimensional space remains an open question [10, 20]. On the other hand, shrinking (in the comoving coordinates) strings and monopoles gradually become surrounded by the thermalized phase. This picture is consistent with the results of our calculations.
Note, however, that our last argument was based on the assumption that the effective potential is quadratic near . Meanwhile in the Coleman-Weinberg model the effective potential near the maximum looks like . Behavior of domain walls in this theory is more complicated. At small the field decreases more slowly. This changes the shape of the domain wall, making it more flat near , which more closely resembles the situation in the old inflation scenario. On the other hand, one can argue that due to quantum fluctuations the field spends most of the time at greater than , and therefore in average is rolls down at least as fast as the field in the theory (1) with . Therefore we expect that our conclusions concerning the global structure of the Universe will remain qualitatively correct for the theories with the effective potentials . However, this subject clearly requires further investigation.
## 5 The problem of initial conditions
The possibility of inflation of topological defects can lead to some improvement with the problem of initial conditions in the models where inflation occurs near a local maximum of . Initially the models of that type were introduced in the context of the new inflationary universe scenario [12]. The basic assumption of old and new inflation was that inflation begins in a state of thermal equilibrium at . This idea was not particularly successful, and no realistic versions of new inflation were suggested so far. Still it is possible for inflation to begin at in the context of chaotic inflation scenario if for some reason the scalar field appears near the top of the effective potential inside a domain of a size greater than . But is it possible to achieve it in a natural way?
In order to analyse this question let us imagine that we are witnessing the moment of the Universe creation (“Planck time”), when the first domain of classical space-time with the Planck energy density emerged from the space-time foam. It seems extremely unlikely that this first domain is infinite from the very beginning. In this case we would face the horizon problem: How was it possible for the same event (the appearance of matter with the Planck density) to be correlated in infinitely many causally disconnected domains?
The only natural length scale in general relativity theory is the Planck length . Therefore the most natural assumption is that the initial domain has the Planck length.555 Of course, this size might be much larger if there was a preceding stage of evolution of the Universe, for example something like stringy pre-inflation [21]. This possibility is extremely interesting, but its discussion is outside the scope of the present paper. If inflation of this domain does not begin immediately after that, there is a good chance that such a domain will momentarily collapse within the time . This is definitely the case if this domain locally looks like a part of a closed universe, but even if the domain looks like a part of an open universe of a size immersed into space-time foam, the only obvious way for it to avoid collapse and to evolve into a large homogeneous universe would be to begin inflation instantaneously.
This is not a problem at all for the simplest versions of chaotic inflation, where inflation can easily begin at [1]. However, in all models where inflation occurs near the top of the effective potential, the value of appears to be at least ten orders of magnitude smaller than the Planck density, and typically it is even much smaller than that. Inflation in such models can begin only at a much later stage of the evolution of the Universe, at a time . The size of initial domain of inflationary universe at that time should be greater than . Suppose for simplicity that the Universe from the very beginning was dominated by ultrarelativistic matter. Then its scale factor expanded as , where is the energy density at the pre-inflationary stage. Therefore at the Planck time the size of the part of the Universe which later evolved into inflationary domain was not , but somewhat smaller: . This whole scenario can work only if at the Planck time the domain of this size was sufficiently homogeneous, . However, at the Planck time this domain consisted of domains of a Planck size, and energy density in each of them was absolutely uncorrelated with the energy density in other domains. Therefore a priori one could expect changes of density when going from one causally disconnected parts of the Universe of a size to another. Simple combinatorial analysis suggests that the probability of formation of a reasonably homogeneous part of the Universe of a size at the Planck time is suppressed by the exponential factor
P∼exp(−CM3PV3/4(0)) , (18)
where . To get a numerical estimate, one can take . This gives . For the original Coleman-Weinberg model this number is even much smaller. Note that this estimate is very similar to the estimate of the probability of a direct quantum creation of inflationary universe with the vacuum energy density [22],
P∼exp(−3M4P8V(ϕ)) , (19)
which gives even smaller value of the probability of inflation at than eq. (18). Meanwhile this equation tells us that there is no suppression of probability of chaotic inflation with .
One of the differences between these two estimates is that eq. (18) still does not guarantee that the homogeneous part of the Universe will inflate. Inflation begins near the local maximum of the effective potential only if the field in this domain appears in a state with , and is sufficiently homogeneous. Meanwhile in the theory (1) the field initially can take any value in the interval from to . In realistic models with this means that the typical initial value of would be of the order of . Then it will participate in inflation and roll down to , just as in the simplest versions of chaotic inflation scenario. The probability to obtain a domain containing homogeneous field (assuming that ) will be even smaller than . At this stage topology may help [17]. Once we have a sufficiently large and homogeneous domain, it is most probable that the field in the model (1) will take both positive and negative values in its different parts. Consequently, there will be domain walls. Since in this model domain walls can be stretched by inflation, they will be even more easily stretched at the pre-inflationary stage, because at that time the Hubble constant was even greater. This naturally creates good conditions for inflation inside the domain wall. However, equations (18) and (19) clearly indicate that the probability to obtain inflation beginning at large is much better.
Does this mean that we should abandon the idea of chaotic inflation near the local maximum of effective potential? In our opinion, this would be incorrect. First of all, it might happen that in a future theory of elementary particles inflation cannot occur anywhere else except for a local maximum of . Still it will be much better than no inflation at all. On the other hand, there exist several different ways to create good conditions for inflation at . The simplest way is to add to the theory some heavy field with a simple effective potential for which inflation may begin at . This stage of inflation initiated by the field will force the field to jump to the top of the effective potential at least in some part of the Universe. Initially the part of the volume of the Universe where the field stays at the top of the effective potential will be relatively small, but later these regions will become increasingly important, since they will eternally inflate [9].
Another way is to consider potentials of the new inflationary type in the context of the Brans-Dicke theory. In this case the Planck mass depends on the value of the Brans-Dicke field , and the condition can be satisfied at the local maximum of [23, 24].
There is also another interesting possibility [25, 9]. The wave function of the Universe should describe all possible initial conditions and all possible outcomes. However, we are interested only in the conditional probability to obtain particular observational data under an obvious but very nontrivial condition of our own existence. There may be many branches of the wave function of the Universe which may seem natural from the point of view of initial conditions, but most of them describe the Universe where intelligent observers cannot live. In our calculation of the probability (18) we simply counted all trajectories, even those which correspond to “virtual” universes collapsing within the Planck time. But why should we count them? Perhaps we should see where most of the observers can live, and we should call the corresponding trajectories “typical”. There will be many problems with such approach, in particular the problem of introducing a proper measure on the set of all such trajectories [24]. However, it seems plausible that with any reasonable choice of measure the trajectories corresponding to eternal inflation will always win being compared to the trajectories which do not possess this property.
The only real problem appears if we should compare many different possibilities corresponding to different realizations of the eternal inflation scenario. In this case one should take into account that it is much more difficult for inflation to begin at than at .
## 6 Discussion
When we began this investigation, our main purpose was to study the difference between the global structure of the Universe in the models of two different classes: those models where inflation occurs near a local maximum of the effective potential and those models where inflation begins at large , outside the equilibrium. However, during our work we recognized that some other important features of inflation in the models of the first class theories have not been properly analysed. For more than ten years we knew that inflation solves the primordial monopole problem, but we did not know that monopoles and other topological defects can inflate.
Now it appears that under certain conditions they do inflate, and their inflation never ends [16, 17]. According to this scenario, our part of the Universe could be formed from what initially was an interior of an inflating topological defect. The first attempt to investigate this question was made in our paper [9], where we have shown that in accordance to the most natural realization of the “natural inflation” scenario [26] we should live in the remnants of an inflating domain wall. Now we understand that this situation is much more general.
The structure of space-time near inflating topological defects is very complicated; it should be studied by the methods developed in [27] for description of a bubble of de Sitter space immersed into vacuum with vanishing energy density. Depending on initial conditions, many possible configurations may appear. For example, an inflating monopole may look from outside like a small magnetically charged Reissner-Nordström black hole [18]. However, it will contain a part of exponentially expanding space inside it. This will be a wormhole configuration similar to those studied in [27][30].
At the quantum level the situation becomes even more interesting. Fluctuations of the field near the center of a monopole are strong enough to create new regions of space with , some of which will become monopoles. After a while, the distance between these monopoles becomes exponentially large, so that they cannot annihilate. This process of monopole-antimonopole pair creation produces a fractal structure consisting of monopoles created in the vicinity of other monopoles.
One of the original motivations for the development of inflationary cosmology was a desire to get rid of primordial magnetic monopoles and dangerous domain walls produced in the theories with spontaneous breaking of discrete symmetries. For a long time topological defects and inflation were opposed to each other as two almost incompatible sources of density perturbations in the early Universe. Now we see that the interplay between inflationary theory and the theory of topological defects can be very constructive. According to our scenario, inflation can produce inflating topological defects which in their turn can serve as seeds for eternal inflation.
We are very grateful to Victor Berezin, Valery Frolov, Renata Kallosh, Arthur Mezhlumian, Igor Tkachev and especially to Alex Vilenkin for valuable discussions. This work was supported in part by NSF grant PHY-8612280.
## References
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• [16] A.D. Linde, Monopoles as Big as a Universe, Stanford University preprint SU-ITP-94-2 (1994), astro-ph/9402031.
• [17] A. Vilenkin, Topological Inflation, MIT preprint, hep-th/9402085.
• [18] A. Frieman and C.T. Hill, SLAC-PUB-4283, 1987 (unpublished); K. Lee, V.P. Nair and E.J. Weinberg, Phys. Rev. D 45, 2751 (1992); M.E. Ortiz, Phys. Rev. D 45, R2586 (1992); P. Breitenlohner, P. Forgács and D. Maison, Nucl. Phys. B 383, 357 (1992).
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• [22] A.D. Linde, JETP 60, 211 (1984); Lett. Nuovo Cim. 39, 401 (1984);
A. Vilenkin, Phys. Rev. D 30, 549 (1984).
• [23] A.D. Linde, Phys. Rev. D 49, 748 (1994).
• [24] J. García–Bellido, A.D. Linde and D.A. Linde, Fluctuations of the Gravitational Constant in the Inflationary Brans-Dicke Cosmology, Stanford University preprint SU-ITP-93-29 (1993), astro-ph/9312039.
• [25] M. Mijić, Phys. Rev. D 42, 2469 (1990); Int. J. Mod. Phys. A6, 2685 (1991).
• [26] K. Freese, J.A. Frieman and A.V. Olinto Phys. Rev. Lett. 65, 3233 (1990); F.C. Adams, J.R. Bond, K. Freese, J.A. Frieman and A.V. Olinto Phys. Rev. D47, 426 (1993).
• [27] V.A. Berezin, V.A. Kuzmin, I.I. Tkachev Phys. Lett. 120 B, 91 (1983); Phys. Rev. D 36, 2919 (1987); A. Aurilia, G. Denardo, F. Legovini, and E. Spalucci, Nucl. Phys. B252, 523 (1985); P. Laguna-Gastillo, and R. A. Matzner, Phys. Rev. D 34, 2913 (1986); S.K. Blau, E.I. Guendelman, and A.H. Guth, Phys. Rev. D 35, 1747 (1987).
• [28] M.A. Markov and V.P. Frolov, Teor. Mat. Fiz. 3, 3 (1970).
• [29] K. Sato, H. Kodama, M. Sasaki and K. Maeda, Phys. Lett. B108, 35 (1982).
• [30] V.A. Berezin, V.A. Kuzmin and I.I. Tkachev Jetp Lett. 41, 547 (1985) (Pisma Zh. Eksp. Teor. Fiz. 41, 446 (1985)); Sov. Phys. JETP 66, 654 (1987) (Zh. Eksp. Teor. Fiz. 93, 1159 (1987); Phys. Lett. B210, 64 (1988).
## Figure Captions
Fig. 1
The domain structure of the Universe in the theory (1) with spontaneously broken discrete symmetry .
Fig. 2
Energy density distribution during inflation in the theory (11).
Fig. 3
Energy density distribution during inflation in the simplest chaotic inflation model with the effective potential .
Fig. 4
Distribution of the field during inflation in the theory (11).
Fig. 5
These figures show simultaneously the energy density of the field in the theory (11), and its direction in the isotopic space. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8370577692985535, "perplexity": 460.9857096836704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400206133.46/warc/CC-MAIN-20200922125920-20200922155920-00097.warc.gz"} |
http://cpr-condmat-other.blogspot.com/2012/09/12093235-wlodek-zawadzki.html | ## Electron dynamics in crystalline semiconductors [PDF]
Electron dynamics in crystalline semiconductors is described by distinguishing between an instantaneous velocity related to electron's momentum and an average velocity related to its quasi-momentum in a periodic potential. It is shown that the electron velocity used in the theory of electron transport and free-carrier optics is the average electron velocity, not the instantaneous velocity. An effective mass of charge carriers in solids is considered and it is demonstrated that, in contrast to the "acceleration" mass introduced in textbooks, it is a "velocity" mass relating carrier velocity to its quasi-momentum that is a much more useful physical quantity. Among other advantages, the velocity mass is a scalar for spherical but nonparabolic energy bands $\epsilon(k)$, whereas the acceleration mass is not a scalar. Important applications of the velocity mass are indicated. A two-band ${\bm k}\cdot {\bm \hp}$ model is introduced as the simplest example of a band structure that still keeps track of the periodic lattice potential. It is remarked that the two-band model, adequately describing narrow-gap semiconductors (including zero-gap graphene), strongly resembles the special theory of relativity. Instructive examples of the "semi-relativistic" analogy are given. The presentation has both scientific and pedagogical aspects. | {"extraction_info": {"found_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": 0.9523324370384216, "perplexity": 800.9577476896753}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320685.63/warc/CC-MAIN-20170626064746-20170626084746-00017.warc.gz"} |
https://www.palmsens.com/knowledgebase-article/the-cottrell-experiment-and-diffusion-limitation-double-layer/ | # The Cottrell Experiment and Diffusion Limitation 3/3 – Electrochemical Double Layer
This chapter is the final chapter of the series ‘The Cottrell experiment and diffusion limitation ’. In this chapter the electrochemical double layer and its features are discussed.
## Electrochemical double layer
As soon as an electrode surface is charged, due to a potentiostat or its Nernst potential, an electric field is created. Charged particles will move in this field. Ions of the other charge than the electrodes charge will accumulate directly at the electrode, forming a layer of ions.
Assuming that the electrode is positive charged, anions will accumulate. Attracted by these anions, kations will be attracted and form another loose layer on top of the first layer. The first layer is known as the outer Helmholtz plane. The charges accumulating in the metal surface are the inner Helmholtz plane (see Figure 2.1). The layer of ions and the electrode act like a capacitor and this has impact on most electrochemical techniques.
This is only a rudimentary description of the electrochemical double layer; there are more sophisticated models, but for many electrochemical experiments the simple model will suffice.
Figure 2.1 | Scheme of the electrochemical double layer
## Migration
Up to now it was assumed that the electrochemical active species Red is transported by diffusion or convection only, but there is one more way for mass transport: migration. Migration is mass transport due to an electric field. If Red is negatively charged, the positive potential of the electrode will attract the Red-ions.
Why are complete models and observations done without taking migration into consideration? Usually an electrochemical measurement is done in a well-conducting solution. Since the current flows from the working to the counter electrode through the solution, a high solution resistance will make the measured current smaller and increase the Ohmic drop (i.e. the difference in the potential applied between the reference electrode and the working electrode and the potential that the working electrode is feeling, see also Ohmic drop).
To reduce the resistance of a solution, an electrochemical inert support electrolyte is added. Often the buffer itself in a pH buffered solution is sufficient or a salt with a high solubility, for example KCl, NaCl, NaSO4, NaNO3, is added.
If the support electrolyte has a high concentration compared to the investigated species, in this example Red, the electric field will be compensated by the ions of the support electrolyte and almost only these will migrate. A rule of thumb is that the support electrolyte should have a hundred times higher concentration. Since this effect is suppressed so easily, migration is often negligible.
Another effect is the capacitive charging current or short capacitive current. As mentioned the electrochemical double layers acts as capacitor. Capacitors store charge.
A simple capacitor is the plate capacitor. It comprises two conducting parallel plates that are not in contact with each other. If a power source is connected to the plates, a current flows that is exponentially decaying until it is insignificant. A current flows because one plate is charged negative and the other positive. The separation of charges means current flows. At some point the plates cannot store more charge and the current stops flowing. The current decays over time according to
Equation 2.1
EC is the charging potential or voltage, I0 is the starting current, R is the resistance of the circuit around the capacitor, t the time and C the capacity of the capacitor. The capacity is a property of the capacitor and is defined as the charge Q that can be stored per applied potential E or as equation
Equation 2.2
Usually U is used for voltage, but since these equations need to be transferred to electrochemical experiments, it is useful to start with the potential E instead of the voltage U. These two are not synonymous but in this context it is fine to exchange them.
## Properties of the electrochemical double layer
If it is assumed that the electrochemical double layer behaves exactly like a plate capacitor, the two equations 2.1 and 2.2 show three important facts:
1. The capacitive current decays exponentially with the time t. The higher resistance R and capacity C are, the slower it will be decaying. The product of resistance R and capacity C is often called the time constant τ.
2. The charge Q that can be stored is proportional to the applied potential. Every time the charge Q that can be stored changes a current I flows until the charge Q is adjusted. The charge Q that can be stored changes, if the potential E is changing. This is expressed in the equation:
Equation 2.3
3. In equation 2.2 it is shown implicitly and in 2.3 explicitly that the higher the capacity C is, the more capacitive current will flow if the potential changes.
Usually electrochemists are interested in the Faraday current, that is the current caused by an electrochemical reaction; the capacitive current, caused by physics, is an unwanted side effect (see also Capacitive current).
What does this mean for measurements? If the potential of the electrode is changed, for example during a potential step, a current will flow that has no chemical but only a physical meaning. This current decays exponential with t, while the Faraday current decays with t. This means that the capacitive current decays much faster than the Faraday current (see Figure 2.2). The higher the capacity C the higher the capacitive current. The capacity C for a plate capacitor can be calculated with
Equation 2.4
where ε0 is the electric field constant, εr is the relative permittivity of the medium between the plates, d is the distance between the two plates and A is the surface area of the two plates.
In conclusion, most factors influencing the capacity cannot be altered in an electrochemical experiment. The constant ε0 cannot be changed. The distance d and the relative permittivity εr can only be changed by changing the solution, because d is defined by the distance between inner and outer Helmholtz plane (see Figure 2.1). The area A is influenced by the surface roughness. The rougher a surface the higher its area. If a reusable electrode is used, a proper polishing that leads to a smooth surface can reduce capacitive current drastically.
Figure 2.2 | Scheme of the capacitive and Faraday current through time
The electrochemical double layer
The electrochemical double layer acts as a capacitor and every change in the potential of the electrode will induce a capacitive charging current that is caused by physics not by a chemical reaction. This current decays exponentially.
The Cottrell Experiment – The Cottrell ExperimentThe Cottrell Experiment – PDF | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9062525629997253, "perplexity": 898.7933465157967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154032.75/warc/CC-MAIN-20210730220317-20210731010317-00358.warc.gz"} |
http://tex.stackexchange.com/questions/94334/xetex-and-fedora-cant-use-otf-fonts | # XeTeX and Fedora: can't use OTF fonts
When trying to compile a document with XeLaTeX, I'm getting the following error:
``````!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
! fontspec error: "font-not-found"
!
! The font "Linux Libertine Slanted O" cannot be found.
!
! See the fontspec documentation for further information.
!
! For immediate help type H <return>.
!...............................................
``````
Now, I've got the font files installed, `/usr/share/texlive/texmf-dist/fonts/opentype/public/libertineotf` contains the font files that I need.
Fontconfig doesn't see those files, however. I tried adding the directory with `fc-cache` and it reports having added 30 new fonts, but I still can't see them when running `fc-list`.
Now it all kinda works well on Linux Mint, I'm on Fedora now, and it somehow doesn't work anymore.
What am I doing wrong?
I'm using Fedora 18, with everything latex related installed from the official repos. `This is XeTeX, Version 3.1415926-2.5-0.9999 (TeX Live 2013/dev)` is what xelatex reports. Fontconfig is the one installed that came with Fedora 18.
-
you should tell us which version of fedora you're (now) using. the banner that xetex produces when you start could help, too. things may be different if you're using a “too old” version of xetex, or of fontconfig. – wasteofspace Jan 18 '13 at 10:17
@wasteofspace ok, I added the information you asked me to provide. – polemon Jan 18 '13 at 10:40
In section 4.2 (p6) of the `fontspec` manual, it says:
4.6 By file name
XETEX and LuaTEX also allow fonts to be loaded by file name instead of font name. When you have a very large collection of fonts, you will sometimes not wish to have them all installed in your system’s font directories. In this case, it is more convenient to load them from a different location on your disk. This technique is also necessary in XETEX when loading OpenType fonts that are present within your TEX distribution, such as /usr/local/texlive/2010/texmf-dist/fonts/opentype/ public. Fonts in such locations are visible to XETEX but cannot be loaded by font name, only file name; LuaTEX does not have this restriction.
This means that you cannot simply use
``````\setmainfont{Linux Libertine O}
``````
with XeLaTeX unless the file is seen by your system.
In order to use TeX distribution fonts with XeLaTeX, you need to load them by hand which gets very tedious.
Your best alternative is to use the `libertine` TeX package (`\usepackage{libertine}`).
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it //is// the libertineotf package! I have installed this package, and you can even tell by the directory, it is installed. But still, fontconfig doesn't see it. It worked that way on Linux Mint, though – polemon Jan 18 '13 at 11:52
fontconfig would not see it and the `libertineotf` package would not be using it either. When you say that it is the `libertineotf` package, what do you mean? are you actually loading it in your tex file is `\usepackage{libertineoft}`. At the point an MWE would be highly advantageous as we abviously do not have all the information. – ArTourter Jan 18 '13 at 12:22
Ahh, I was a bit confused here: The distro has a package called <code>libterineotf</code>, I didn't try the latex package yet. – polemon Jan 18 '13 at 12:59
The most recent `libertine` package also provides OpenType support. If you have that package (5.3.0 is the latest version) you should use the `libertine` package, not `libertineotf` (but `libertineotf` will tell you that as well). – Silex Jan 18 '13 at 13:58
@Silex I must admit I am losing track with these:-) it used to be `libertine`, then they split it into `libertine` and `libertineotf`, and now you are telling me they have merged again?? Or am I getting it wrong? – ArTourter Jan 18 '13 at 14:04 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.926727831363678, "perplexity": 3103.6616011391807}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1410657121288.75/warc/CC-MAIN-20140914011201-00228-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://fr.maplesoft.com/support/help/maple/view.aspx?path=geom3d/form&L=F | geom3d - Maple Programming Help
Home : Support : Online Help : Mathematics : Geometry : 3-D Euclidean : geom3d/form
geom3d
form
returns the form of the geometric object
Calling Sequence form(obj)
Parameters
obj - geometric object
Description
• The routine returns the form of the given object supported by the geom3d package; FAIL otherwise.
• The command with(geom3d,form) allows the use of the abbreviated form of this command.
Examples
> $\mathrm{with}\left(\mathrm{geom3d}\right):$
define point A with coordinates 1, 2, 3.
> $\mathrm{point}\left(A,1,2,3\right)$
${A}$ (1)
> $\mathrm{form}\left(A\right)$
${\mathrm{point3d}}$ (2)
Define a dodecahedron with center (0,0,0), radius of circum-sphere 1
> $\mathrm{dodecahedron}\left(d,\mathrm{point}\left(o,0,0,0\right),1\right)$
${d}$ (3)
> $\mathrm{form}\left(d\right)$
${\mathrm{dodecahedron3d}}$ (4) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "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": 0.9884852170944214, "perplexity": 3740.4597831992323}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1596439736972.79/warc/CC-MAIN-20200806151047-20200806181047-00546.warc.gz"} |
https://www.lessonplanet.com/teachers/estimate-solutions | # Estimate Solutions
In this estimate solutions instructional activity, students solve and complete 10 different multiple choice problems. First, they determine the number that is reasonable for each word problem described. Then, students find the best estimate for a given amount of items. | {"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": 0.9018201231956482, "perplexity": 1490.0844090288806}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549426133.16/warc/CC-MAIN-20170726102230-20170726122230-00542.warc.gz"} |
https://www.physicsforums.com/threads/friction-on-an-inclined-plane.373242/ | Friction on an Inclined Plane
1. Jan 27, 2010
mmalone11
1. The problem statement, all variables and given/known data
The diagram shows a 5kg block of lead released from rest at the top of an incline. The block has a speed of 6 m/s when it reaches the bottom. The angle between the slope and the ground is 40° and the slope is 10 m long.
a) What is its PE at the top?
b) What is its KE at the bottom?
c) What is the work done by friction?
d) What must be the coefficient of friction?
2. Relevant equations
I am having trouble finding the friction. Once i find the friction, i know how to find the coefficient force of friciton using Ffr=mu*Fn.
3. The attempt at a solution
For part a first I found what the height was by doing 10(sin(40))=Height and got 6.43 m. Than i plugged that into PE=mgh... 5(9.8)(6.43) ... getting 315.07J ...
For part b I used KE=1/2(m)(v2) ... 1/2(5)(62) ... getting 90J ...
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. Jan 27, 2010
rl.bhat
What is the component of the weight normal to the surface? That will be fn.
What is the component of the weight along the surface?
In the absence of the friction, what will be the KE at the bottom?
Difference in the KE = fr*d, where d is the distance moved by the block.
Last edited: Jan 27, 2010 | {"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": 0.9117364287376404, "perplexity": 758.8860635453665}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647758.97/warc/CC-MAIN-20180322014514-20180322034514-00374.warc.gz"} |
https://mathhelpboards.com/threads/complex-integral.306/ | [SOLVED]Complex integral
dwsmith
Well-known member
Feb 1, 2012
1,673
$\displaystyle \int_0^1 \frac{2t+i}{t^2+it+1} dt = \int_0^1 \left(\frac{t}{2} + \frac{i}{4} + \frac{5/4}{2t+i}\right) dt = \frac{1}{4} + \frac{5}{8} \ln\left(\sqrt{5}\right) + i\left(\frac{1}{4} + \frac{5}{8}\tan^{-1}\left(\frac{1}{2}\right)\right)$
Is this correct?
Last edited:
Ackbach
Indicium Physicus
Staff member
Jan 26, 2012
4,197
$\displaystyle \int_0^1 \frac{2t+i}{t^2+it+1} dt = \int_0^1 \left(\frac{t}{2} + \frac{i}{4} + \frac{5/4}{2t+i}\right) dt = \frac{1}{4} + \frac{5}{8} \ln\left(\sqrt{5}\right) + i\left(\frac{1}{4} + \tan^{-1}(-2)\right)$
Is this correct?
I don't think it is. Note that the numerator is the derivative of the denominator. What does that suggest?
dwsmith
Well-known member
Feb 1, 2012
1,673
I don't think it is. Note that the numerator is the derivative of the denominator. What does that suggest?
U-sub isn't defined for Complex integrals because any closed path would be zero. A counter example is 1/z around the unit circle which isn't 0 and is closed path.
Random Variable
Well-known member
MHB Math Helper
Jan 31, 2012
253
What Ackbach suggests is correct because you're integrating with respect to a real variable ($t$), in which case $i$ is just a constant.
Ackbach
Indicium Physicus
Staff member
Jan 26, 2012
4,197
U-sub isn't defined for Complex integrals because any closed path would be zero. A counter example is 1/z around the unit circle which isn't 0 and is closed path.
What Ackbach suggests is correct because you're integrating with respect to a real variable ($t$), in which case $i$ is just a constant.
Also, the denominator is nonzero on the integration path.
dwsmith
Well-known member
Feb 1, 2012
1,673
What Ackbach suggests is correct because you're integrating with respect to a real variable ($t$), in which case $i$ is just a constant.
I think I understand what my professor means. If we substitute,
$\int_{\gamma}\frac{1}{z}dz = \int_0^{2\pi}\frac{ie^{it}}{e^{it}}dt$
The numerator is the derivative of the denominator so substitutions is viable here as well.
$\int_1^1\frac{du}{u}=0\neq 2\pi i$.
Random Variable
Well-known member
MHB Math Helper
Jan 31, 2012
253
EDIT: Erased a bunch of nonsesne
Last edited:
Random Variable
Well-known member
MHB Math Helper
Jan 31, 2012
253
Ignore my previous post. It contains a bit of nonsense.
$u = e^{it}$ for $t$ from $0$t o $2 \pi$ is the unit circle.
So $\displaystyle \int_{\gamma}\frac{1}{z}dz = \int_0^{2\pi}\frac{ie^{it}}{e^{it}}dt {\color {red} \ne } \int_{1}^{1}\frac{1}{u}du$.
But rather $\displaystyle \int_{\gamma}\frac{1}{z}dz = \int_0^{2\pi}\frac{ie^{it}}{e^{it}}dt = \int_{\gamma}\frac{1}{u}du$. And we're backing to where we started.
dwsmith
Well-known member
Feb 1, 2012
1,673
Ignore my previous post. It contains a bit of nonsense.
$u = e^{it}$ for $t$ from $0$t o $2 \pi$ is the unit circle.
So $\displaystyle \int_{\gamma}\frac{1}{z}dz = \int_0^{2\pi}\frac{ie^{it}}{e^{it}}dt {\color {red} \ne } \int_{1}^{1}\frac{1}{u}du$.
But rather $\displaystyle \int_{\gamma}\frac{1}{z}dz = \int_0^{2\pi}\frac{ie^{it}}{e^{it}}dt = \int_{\gamma}\frac{1}{u}du$. And we're backing to where we started.
I just multiplied by the conjugate and obtained the answer without the use of substitution.
Random Variable
Well-known member
MHB Math Helper
Jan 31, 2012
253
So you said that $\displaystyle \int_{0}^{1} \frac{2t +i}{t^{2}+it+1} \ dt \ \frac{t^{2}-it+1}{t^2-it+1} = \int_{0}^{1} \frac{2t^{3}+3t}{t^4+3t^{2}+1} \ dt + i \int_{0}^{1} \frac{1-t^{2}}{t^{4}+3t^{2}+1} \ dt$?
The first integral is easy, but the second integral looks fairly nasty. But that is a valid approach.
But I would still make that substitution to get $\displaystyle \int \frac{2t +i}{t^{2}+it+1} \ dt = \int \frac{1}{u} \ du = \ln u + C = \ln(t^{2}+it+1) + C$
Then $\displaystyle \int_{0}^{1} \frac{2t +i}{t^{2}+it+1} \ dt = \ln(1^{2}+i(1)+1) - \ln(0^{2}+i(0)+1) = \ln(2+i) - \ln(1) = \ln(2+i)$ which is the answer that WolframAlpha gives
http://www.wolframalpha.com/input/?i=integrate+(2t%2Bi)%2F(t^2%2Bit%2B1)+from+0+to+1
dwsmith
Well-known member
Feb 1, 2012
1,673
So you said that $\displaystyle \int_{0}^{1} \frac{2t +i}{t^{2}+it+1} \ dt \ \frac{t^{2}-it+1}{t^2-it+1} = \int_{0}^{1} \frac{2t^{3}+3t}{t^4+3t^{2}+1} \ dt + i \int_{0}^{1} \frac{1-t^{2}}{t^{4}+3t^{2}+1} \ dt$?
The first integral is easy, but the second integral looks fairly nasty. But that is a valid approach.
But I would still make that substitution to get $\displaystyle \int \frac{2t +i}{t^{2}+it+1} \ dt = \int \frac{1}{u} \ du = \ln u + C = \ln(t^{2}+it+1) + C$
Then $\displaystyle \int_{0}^{1} \frac{2t +i}{t^{2}+it+1} \ dt = \ln(1^{2}+i(1)+1) - \ln(0^{2}+i(0)+1) = \ln(2+i) - \ln(1) = \ln(2+i)$ which is the answer that WolframAlpha gives
http://www.wolframalpha.com/input/?i=integrate+(2t+i)/(t^2+it+1)+from+0+to+1
Yup that is what I did.
Since u-sub isn't defined for complex integrals, we can't use it even though it may work in some cases.
Random Variable
Well-known member
MHB Math Helper
Jan 31, 2012
253
Can you quote where in your textbook such a claim is made?
dwsmith
Well-known member
Feb 1, 2012
1,673
Can you quote where in your textbook such a claim is made?
It is made by Richard Foote.
Random Variable
Well-known member
MHB Math Helper
Jan 31, 2012
253
I do believe you misinterpreted what he said. Because to evaluate every complex integral by breaking it into it's real and imaginary parts will become ridiculously time-consuming.
dwsmith
Well-known member
Feb 1, 2012
1,673
I do believe you misinterpreted what he said. Because to evaluate every complex integral by breaking it into it's real and imaginary parts will become ridiculously time-consuming.
I don't do that every time. He wanted us to appreciate Cauchy's Integral Formula, Residue Theory, etc. So we were doing these integrals the hard way. I didn't misinterpret. He had a had a brief discussion about it on Friday when he noticed that some students were using it. That is when he gave the counter example of all closed curves will evaluate to 0 when we know that isn't the case.
Random Variable
Well-known member
MHB Math Helper
Jan 31, 2012
253
But your assertion that making the substitution $u =e^{it}$ would mean that $\displaystyle \int_{0}^{2 \pi} \frac{i e^{it}}{e^{it}} \ dt = \int_{1}^{1} \frac{1}{u} \ du$ is false.
That substitution, as I already stated, would mean that $\displaystyle \int_{0}^{2 \pi} \frac{i e^{it}}{e^{it}} \ dt = \int_{\gamma} \frac{1}{u} \ du$
dwsmith
Well-known member
Feb 1, 2012
1,673
But your assertion that making the substitution $u =e^{it}$ would mean that $\displaystyle \int_{0}^{2 \pi} \frac{i e^{it}}{e^{it}} \ dt = \int_{1}^{1} \frac{1}{u} \ du$ is false.
That substitution, as I already stated, would mean that $\displaystyle \int_{0}^{2 \pi} \frac{i e^{it}}{e^{it}} \ dt = \int_{\gamma} \frac{1}{u} \ du$
If you do a change of bounds, you get 1 and 1. e^0 = 1 and e^{2\pi i} = 1
Random Variable
Well-known member
MHB Math Helper
Jan 31, 2012
253
We made the substitution $u = e^{it}$, and the limits are from $t=0$ to $t= 2 \pi$. Isn't that the definition of the unit circle? | {"extraction_info": {"found_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": 0.96928870677948, "perplexity": 1028.8179169550986}, "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-2020-50/segments/1606141182794.28/warc/CC-MAIN-20201125125427-20201125155427-00143.warc.gz"} |
https://stacks.math.columbia.edu/tag/04TD | ## 92.14 2-Fibre products of algebraic stacks
The $2$-category of algebraic stacks has products and $2$-fibre products. The first lemma is really a special case of Lemma 92.14.3 but its proof is slightly easier.
Lemma 92.14.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}$, $\mathcal{Y}$ be algebraic stacks over $S$. Then $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ is an algebraic stack, and is a product in the $2$-category of algebraic stacks over $S$.
Proof. An object of $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ over $T$ is just a pair $(x, y)$ where $x$ is an object of $\mathcal{X}_ T$ and $y$ is an object of $\mathcal{Y}_ T$. Hence it is immediate from the definitions that $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ is a stack in groupoids. If $(x, y)$ and $(x', y')$ are two objects of $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ over $T$, then
$\mathit{Isom}((x, y), (x', y')) = \mathit{Isom}(x, x') \times \mathit{Isom}(y, y').$
Hence it follows from the equivalences in Lemma 92.10.11 and the fact that the category of algebraic spaces has products that the diagonal of $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ is representable by algebraic spaces. Finally, suppose that $U, V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, and let $x, y$ be surjective smooth morphisms $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$, $y : (\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$. Note that
$(\mathit{Sch}/U \times _ S V)_{fppf} = (\mathit{Sch}/U)_{fppf} \times _{(\mathit{Sch}/S)_{fppf}} (\mathit{Sch}/V)_{fppf}.$
The object $(\text{pr}_ U^*x, \text{pr}_ V^*y)$ of $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ over $(\mathit{Sch}/U \times _ S V)_{fppf}$ thus defines a $1$-morphism
$(\mathit{Sch}/U \times _ S V)_{fppf} \longrightarrow \mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$
which is the composition of base changes of $x$ and $y$, hence is surjective and smooth, see Lemmas 92.10.6 and 92.10.5. We conclude that $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ is indeed an algebraic stack. We omit the verification that it really is a product. $\square$
Lemma 92.14.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{Z}$ be a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ whose diagonal is representable by algebraic spaces. Let $\mathcal{X}$, $\mathcal{Y}$ be algebraic stacks over $S$. Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of stacks in groupoids. Then the $2$-fibre product $\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}$ is an algebraic stack.
Proof. We have to check conditions (1), (2), and (3) of Definition 92.12.1. The first condition follows from Stacks, Lemma 8.5.6.
The second condition we have to check is that the $\mathit{Isom}$-sheaves are representable by algebraic spaces. To do this, suppose that $T$ is a scheme over $S$, and $u, v$ are objects of $(\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y})_ T$. By our construction of $2$-fibre products (which goes all the way back to Categories, Lemma 4.32.3) we may write $u = (x, y, \alpha )$ and $v = (x', y', \alpha ')$. Here $\alpha : f(x) \to g(y)$ and similarly for $\alpha '$. Then it is clear that
$\xymatrix{ \mathit{Isom}(u, v) \ar[d] \ar[rr] & & \mathit{Isom}(y, y') \ar[d]^{\phi \mapsto g(\phi ) \circ \alpha } \\ \mathit{Isom}(x, x') \ar[rr]^-{\psi \mapsto \alpha ' \circ f(\psi )} & & \mathit{Isom}(f(x), g(y')) }$
is a cartesian diagram of sheaves on $(\mathit{Sch}/T)_{fppf}$. Since by assumption the sheaves $\mathit{Isom}(y, y')$, $\mathit{Isom}(x, x')$, $\mathit{Isom}(f(x), g(y'))$ are algebraic spaces (see Lemma 92.10.11) we see that $\mathit{Isom}(u, v)$ is an algebraic space.
Let $U, V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, and let $x, y$ be surjective smooth morphisms $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$, $y : (\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$. Consider the morphism
$(\mathit{Sch}/U)_{fppf} \times _{f \circ x, \mathcal{Z}, g \circ y} (\mathit{Sch}/V)_{fppf} \longrightarrow \mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}.$
As the diagonal of $\mathcal{Z}$ is representable by algebraic spaces the source of this arrow is representable by an algebraic space $F$, see Lemma 92.10.11. Moreover, the morphism is the composition of base changes of $x$ and $y$, hence surjective and smooth, see Lemmas 92.10.6 and 92.10.5. Choosing a scheme $W$ and a surjective étale morphism $W \to F$ we see that the composition of the displayed $1$-morphism with the corresponding $1$-morphism
$(\mathit{Sch}/W)_{fppf} \longrightarrow (\mathit{Sch}/U)_{fppf} \times _{f \circ x, \mathcal{Z}, g \circ y} (\mathit{Sch}/V)_{fppf}$
is surjective and smooth which proves the last condition. $\square$
Lemma 92.14.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be algebraic stacks over $S$. Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of algebraic stacks. Then the $2$-fibre product $\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}$ is an algebraic stack. It is also the $2$-fibre product in the $2$-category of algebraic stacks over $(\mathit{Sch}/S)_{fppf}$.
Proof. The fact that $\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}$ is an algebraic stack follows from the stronger Lemma 92.14.2. The fact that $\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}$ is a $2$-fibre product in the $2$-category of algebraic stacks over $S$ follows formally from the fact that the $2$-category of algebraic stacks over $S$ is a full sub $2$-category of the $2$-category of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9960119128227234, "perplexity": 133.93950803136727}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585302.56/warc/CC-MAIN-20211020024111-20211020054111-00107.warc.gz"} |
http://math.stackexchange.com/questions/18301/proving-that-the-given-two-integrals-are-equal | # Proving that the given two integrals are equal
I am stuck up with this simple problem.
If $\alpha \cdot \beta = \pi$, then show that $$\sqrt{\alpha}\int\limits_{0}^{\infty} \frac{e^{-x^{2}}}{\cosh{\alpha{x}}} \ \textrm{dx} = \sqrt{\beta} \int\limits_{0}^{\infty} \frac{e^{-x^{2}}}{\cosh{\beta{x}}} \ \textrm{dx}$$
I tried replacing $\cosh{x} = \frac{e^{x}+e^{-x}}{2}$ and tried doing some manipulations, but it's of no use. Seems to be a clever problem. Moreover since we have $\alpha \cdot \beta = \pi$, we get $\sqrt{\alpha} = \frac{\sqrt{\pi}}{\sqrt{\beta}}$, but the Beta factor is in the numerator, which bewilders me.
-
You mean $\sqrt{\pi}$ in the last identity. – AD. Jan 20 '11 at 14:16
@AD: Yes, sorry it's a typo – anonymous Jan 20 '11 at 14:32
This was shown by Hardy back in 1903/1904.
A mention of it can be found here: Quarterly Journal Of Pure And Applied Mathematics, Volume 35, Page 203, which is somewhere in the middle of a long paper.
Here is a snapshot in case that link does not work:
Note, the integral is slightly different, but I suppose it won't be too hard to convert it into the form you have.
(Edit:) Since the journal itself has no reliable electronic copies, and the proof is actually somewhat more involved then just the excerpt shown above, I'll give a quick description of the proof that Hardy provided.
• First is the concept of reciprocal functions of the first and second kind introduced by Cauchy. Two functions $\phi$ and $\psi$ defined on the positive real line is called reciprocal functions of the first kind if $$\phi(y) = \sqrt{\frac{2}{\pi}} \int_0^\infty \cos(y x) \psi(x) dx$$ and also the same formula with $\phi$ and $\psi$ swapped. They are called reciprocal functions of the second kind if the $\cos$ in the formula above is replaced by $\sin$. Cauchy gave several examples of each, and also examples of functions which are their own reciprocal function of the first kind (but not for the second), and proved that those functions have the following property: whenever $\alpha \beta = \pi$ $$\sqrt\alpha \left( \frac12 \phi(0) + \phi(\alpha) + \phi(2\alpha) + \cdots\right) = \sqrt\beta \left(\frac12 \psi(0) + \psi(\beta) + \psi(2\beta) + \cdots \right)$$
• In the article linked above, Hardy proved the following two facts (among others).
1. The function $f(x) = e^{x^2/2}\int_x^\infty e^{-t^2/2}dt$ is its own reciprocal function of the second kind. (That proof is about 3 pages long, condensed in the typical Hardy fashion.)
2. If $\phi$ and $\psi$ are reciprocal functions of the second kind, the following summation formula (analogue of the one above for functions of the first kind) holds: when $\lambda \mu = 2\pi$, one has $$\sqrt\lambda \sum_0^\infty (-1)^n \phi\left( (n + \frac12)\lambda\right) = \sqrt\mu \sum_0^\infty (-1)^n \psi\left( (n+\frac12)\mu\right)$$ This expression being the one termed equation (9) in the screenshot above.
• Hardy provided two proves of the formula asked about above in the question. The first proof proceeds by giving the series expansion $$\int_0^\infty \frac{e^{-\alpha x^2}}{\cosh \pi x} dx = \frac{2}{\pi} \sum (-1)^n F\left( (n + \frac12)\alpha\right)$$ where $$F(x) = \sqrt\pi e^{x^2}\int_x^\infty e^{-t^2}dt$$ and using equation (9) above. The second proof is shown in section 10 in the image above: he obtained a different series expansion of the expression we want on the left hand side, which can be shown to be term by term equal to the first series expansion of the expression on the right hand side, avoiding the need to invoke equation (9).
-
Hum, so $F$ is a modified version of erfc? He seems to be using some property of $F$ that I don't know how to prove. Have you any idea how Hardy got from that last displayed equation to his conclusion? – Willie Wong Jan 20 '11 at 19:42
@willie: Check out the earlier page (202) of the link I gave and the top of page 203, which got cut off from the snapshot. – Aryabhata Jan 20 '11 at 19:48
unfortunately Google does not want to show me the book. (No preview available it says.) I'll drop by the library tomorrow to see if I can find a hard copy. – Willie Wong Jan 20 '11 at 19:52
@Willie: Thanks for taking the time to elaborate! – Aryabhata Jan 21 '11 at 17:45
No problem. It was mostly for my own benefit: the journal's publication date is awfully close to the line beyond which the library won't lend it out off-premises anymore. – Willie Wong Jan 21 '11 at 18:58 | {"extraction_info": {"found_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": 0.8703141212463379, "perplexity": 284.2514456483592}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115860277.59/warc/CC-MAIN-20150124161100-00024-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/a-student-of-weight-800-n-rides-a-steadily-rotating-ferris-wheel-the-student-sits-upright--q3649242 | A student of weight 800 N rides a steadily rotating Ferris wheel (the student sits upright). At the highest point, the magnitude of the normal force on the student from the seat is 700 N. a)what is the magnitude of the normal force at the lowest point? b)if the wheel's speed is doubled, what is the magnitude FN at the highest point? | {"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": 0.9751744270324707, "perplexity": 514.6193099013847}, "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-2013-20/segments/1368704117624/warc/CC-MAIN-20130516113517-00089-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://pdfkul.com/electroweak-d-waves_5b7730a1097c47d8608b4567.html | Physics Letters B 526 (2002) 90–96 www.elsevier.com/locate/npe
Electroweak d-waves Xavier Calmet, Harald Fritzsch Ludwig-Maximilians-University Munich, Sektion Physik, Theresienstraße 37, D-80333 Munich, Germany Received 4 April 2001; received in revised form 7 December 2001; accepted 8 December 2001 Editor: R. Gatto
Abstract We consider phenomenological implications of a model recently proposed for the electroweak interactions based on a SU(2)L confining theory. We concentrate on the production of excited states of the electroweak bosons at future colliders and we consider their contribution to the reaction W + + W − → W + + W − . We expect large deviations from the standard model in the TeV region. 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The aim of this work is to investigate the phenomenological implications of a model recently proposed for the electroweak interactions based on a SU(2)L confining theory [1]. We have shown in [1] that the minimal sector of the model is identical to the standard model, indeed the standard model can be rewritten using gauge invariant fields representing bound states. Once the gauge is fixed, one obtains exactly the standard model, and finds that the fermions couple with the same strength to the electroweak bosons as in the standard model. The confinement mechanism in our model cannot be identical to that of QCD, due to the weak coupling involved. As stressed by ’t Hooft, vortices could be responsible for the confinement [2]. The duality described in [1] allows to make computation in the bosonic sector of the theory, in particular, to compute the Higgs boson mass [3], but it does not allow to
E-mail address: [email protected] (X. Calmet).
compute the masses of the fermions. The same mechanism as in the standard model, namely, the Yukawa coupling, is used to obtain the smallness of the fermion masses compared to the scale of the theory. We shall concentrate on orbital and radial excitations of the electroweak bosons which are expected if the duality presented in [1] breaks down and if Nature is described by the confinement phase. It was emphasized in Ref. [4] that models of a similar class imply different search strategies for the Higgs boson than those usually adopted when searching for the standard model, supersymmetric or fermiophobic Higgs bosons. In our model the left-handed particles appear as bound states of fundamental, unobservable fermions fL and qL and a scalar h. These particles transform as doublets under SU(2)L . Besides this, qL is a triplet under SU(3)c . We can then identify the following physical left-handed fermions, Higgs boson and electroweak bosons: Neutrino:
¯ νL → hl;
Electron:
eL → hl;
0370-2693/02/\$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 1 4 7 1 - X
X. Calmet, H. Fritzsch / Physics Letters B 526 (2002) 90–96
Higgs particle:
¯ φ → hh,
(s-wave);
√ − 2 ij = (Dµ h)i (Dν h)j g2 F 2 † + (Dν h)i (Dµ h)j
W 3 boson:
¯ W 3 → hh
(p-wave);
≈ bµ3 bν+ + bν3 bµ+ ,
W − boson:
W − → hh
(p-wave);
W + boson:
W + → (hh)†
(p-wave);
Up type quark:
¯ uL → hq;
+ Dµν
Down type quark: dL → hq;
(1)
assuming a SU(2)L confinement. The right-handed particles are those of the standard model. In our approach the electroweak bosons appear as excited states of the Higgs boson. It was shown in Ref. [1] that the minimal sector of the model, i.e., the sector containing only the particles predicted by the standard model, is identical to the standard model [5] if one chooses the unitary gauge, we call this property duality. This is done by fixing the gauge and performing a 1/F expansion, where F ∼ = 492 GeV is the scale of the theory. In this model new particles corresponding to exotic particles like leptoquarks can be introduced. But, they do not obey to this expansion, and the duality cannot be applied to describe their properties. Forces between two fermions can be very much different than those between a fermion and a scalar or between two scalars. If leptoquarks do exist, their mass scale is presumably very high. Of particular interest are radially excited versions of the Higgs boson H ∗ and of the electroweak bosons W 3∗ and W ±∗ . As described in [1], the most promising candidates for energies available at the LHC or at future linear colliders are the excited states of the Higgs boson and of the electroweak bosons. Especially the orbital excitation, i.e., the spin 2 d-waves 3 , D − and D + , of the electroweak bosons have a Dµν µν µν well defined 1/F expansion (we use the unitary gauge: h = (h(1) + F, 0), 3 = Dµν
− Dµν
2 (Dµ h)† (Dν h) + (Dν h)† (Dµ h) 2 2 g F
≈ bν3 bµ3 + bν+ bµ− + bµ+ bν− , √ − 2 ij = 2 2 (Dµ h)i (Dν h)j + (Dν h)i (Dµ h)j g F ≈ bµ3 bν− + bν3 bµ− ,
91
(2)
where Dµ is the covariant derivative, bµa , a = {3, +, −} are the gauge fields and g the coupling constant corresponding to the gauge group SU(2)L . Although the masses and the couplings of these electroweak d-waves to other particles are fixed by the dynamics of the model, it is difficult to determine these parameters. In analogy to Quantum Chromodynamics, it is expected that these d-waves couple with a reasonable strength to the corresponding p-waves, the electroweak bosons. In the following, we assume in accordance with the duality property, that the d-waves only couple to the electroweak bosons and not to the photon, Higgs boson or the fermions.
2. Production of the electroweak d-waves The cross sections and decay width of d-waves predicted in a variety of composite models were considered in Ref. [6]. Here we shall consider different effective couplings of our electroweak d-waves that are more suitable for the model proposed in [1]. If their masses are of the order of the scale of the theory, they will be accessible at the LHC. Of particular interest is the neutral electroweak d-wave because it is expected to couple to the W ± electroweak bosons. This particle can thus be produced by the fusion of two electroweak bosons at the LHC or at linear colliders. We shall use the formalism developed by van Dam and Veltman [7] for massive d-waves to compute the 3 into W + W − . We use the decay width of the Dµν following relation: 5
i i eµν (p)eαβ (p)
i=1
1 = (δµα δνβ + δµβ δνα − δµν δαβ ) 2 pν pβ 1 pµ pα pν pα + δµα 2 + δνβ 2 + δµβ 2 2 mD mD mD pµ pβ + δνα 2 mD
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X. Calmet, H. Fritzsch / Physics Letters B 526 (2002) 90–96
pα pβ 1 2 1 pµ pν δµν − 2 δαβ − + 3 2 2 mD m2D
(3)
i of the d-wave. for the sum over the polarizations eµν In the notation of Ref. [7] the sum over the polarizations of the W ± is given by 3
eµi (p)eνi (p) = δµν +
i=1
pµ pν m2W
,
(4)
where δµν is the Euclidean metric. Averaging over the polarizations of the d-wave, we obtain Γ D3 → W + W − 2 gD 4 2 (xW − 4) = (5) 1− 1920mD π xW with xW = (mD /mW )2 , where mD is the mass of the d-wave and gD is a dimensionfull coupling constant with dim[gD ] = GeV. A dimensionless coupling constant is obtained by a redefinition of the coupling constant gD → mD g¯ D . We shall discuss plausible numerical inputs in Section 3. Assuming that the Z-boson couples with the same strength to the d-wave as the W bosons, we can approximate the decay width into Z bosons in the following way Γ D 3 → ZZ 2 gD 4 2 (xZ − 4) 1− = 3840mD π xZ 1 ≈ Γ D3 → W + W − (6) 2 with xZ = (mD /mZ )2 . The Breit–Wigner resonance cross section for the reaction W + + W − → D 3 thus reads (see, e.g., [8]) (res)
σW + +W − →D 3 =
(tot) 10π m2D ΓD Γ (D 3 → W + W − ) , (tot)2 q2 (m2D − s)2 + m2D ΓD (7)
where q 2 = (s − 4m2W )/4 and ΓD(tot) ≈ 3/2Γ (D 3 → W + W − ) is the total decay width of the neutral d-wave. Due to the background, the W bosons might be difficult to observe. But, if the electroweak d-waves states are produced we expect an excess of Z-bosons compared to the standard model expectation. Note that the Z bosons are easier to observe.
As we shall see in the next section, the neutral d-waves give a sizable contribution to the reaction W + + W − → W + + W −. 3. The reaction W + + W − → W + + W − A considerable attention has been paid to the scattering of electroweak bosons since this represents a stringent test of the gauge structure of the standard model. In particular, the reaction W + + W − → W + + W − is known to be of prime interest. If the Higgs boson is heavier than 1 TeV, the electroweak bosons will start to interact strongly [9]. This reaction has been studied in the framework of the standard model in Ref. [10] and the one loop corrections were considered in [11] and are known to be sizable. For the sake of this Letter the tree level diagrams are sufficient to show that the contribution of the neutral electroweak d-wave will have a considerable impact to that reaction and cannot be overlooked in forthcoming experiments. As described in [12] (see also Ref. [10]) the W ’s emitted by the beam particles are dominantly longitudinally polarized if the following relations are fulfilled: m2W m2W W s at an e+ e− collider, and m2W m2W W sq q¯ s at a hadron collider, and we shall only consider the especially interesting reaction WL+ + WL− → WL+ + WL− as described in [10]. In the standard model, this reaction is a test of the gauge structure of the theory [13]. The Feynman graphs contributing in the standard model to this reaction can be found in Figs. 1, 2. The amplitudes corresponding to these graphs are [10] 2 1 2 2 2 ig xs β 3 − β 2 cos θ, 16 2 1 s3 AsZ = − ig 2 (1 − x) β 2 3 − β 2 cos θ, 16 s − ξZ Asγ = −
1 2 s3 ig x 32 t 2 × β 4 − 2β 2 + β 4 + β 2 4 − 10β 2 + β 4 cos θ + 2 − 11β 2 + 10β 4 cos2 θ + β 2 cos3 θ ,
At γ = −
At Z = −
1 2 s3 ig (1 − x) 32 t − ξZ
X. Calmet, H. Fritzsch / Physics Letters B 526 (2002) 90–96
(a)
(b)
93
(c)
Fig. 1. (a) Photon and Z-boson in the s-channel; (b) photon and Z-boson in the t-chanel; (c) four W vertex.
(a)
(b)
Fig. 2. (a) Higgs boson in the s-channel; (b) Higgs boson in the t-channel.
× β 2 4 − 2β 2 + β 4 + β 2 4 − 10β 2 + β 4 cos θ + 2 − 11β 2 + 10β 4 cos2 θ + β 2 cos3 θ , 1 A4 = − ig 2 s 2 1 + 2β 2 − 6β 2 cos θ − cos2 θ , 16 (1 + β 2 )2 1 AsH = − ig 2 s 2 , 16 s − ξH + iγH (β 2 − cos θ )2 1 At H = − ig 2 s 2 (8) , 16 t − ξH + iγH where x = sin2 θW , ξZ = (1 − x)−1 = m2Z /m2W , ξH = √ m2H /m2W , γH = mH ΓH /m2W and β = 1 − 4/s. The variables s and t are scaled with respect to m2W . The scattering angle is θ , t = −1/2sβ 2 (1 − cos θ ). These notations are the same as those introduced in Ref. [10]. The standard model amplitude is thus Asum SM = Asγ + AsZ + At γ + At Z + A4 + AsH + At H .
(9)
In the high energy limit, one observes the cancellation of the leading powers in s and finds [10]
t 1 2 s sum ASM ≈ ig ξZ 1 + + (10) + ξH − iγH 2 t s
for the sum of these amplitudes. The cross section with the angular cut −z0 < cos θ < z0 is then 1 σ= 16πs 2 β 2
t+ sum 2 A dt
(11)
t−
in dimensionless units, t± = (2 − s/2)(1 ∓ z0 ). The excitations of the Higgs and electroweak bosons also contribute via the s- and t-channel. The amplitudes corresponding to the contribution of a radially excited Higgs boson (H ∗ ) of mass mH ∗ and decay width ΓH ∗ to this reaction are AsH ∗ = −
1 2 2 (1 + β 2 )2 ig ∗ s , 16 H s − ξH ∗ + iγH ∗
At H ∗ = −
1 2 2 (β 2 − cos θ )2 ig ∗ s , 16 H t − ξH ∗ + iγH ∗
(12)
where ξH ∗ = m2H ∗ /m2W , γH ∗ = mH ∗ ΓH ∗ /m2W and gH ∗ is the strength of the coupling between two W bosons and the H ∗ scalar particle. We shall now consider the contribution of the radially (W 3∗ ) and orbitally (D µν ) excited neutral Z boson. The amplitudes for the W 3∗ can be at once deduced from those of the standard model contribution
94
X. Calmet, H. Fritzsch / Physics Letters B 526 (2002) 90–96
of the Z boson s3 1 2 igW 3∗ 16 s − ξW 3∗ + iγW 3∗ 2 2 × β 3 − β 2 cos θ,
AsW 3∗ = −
1 2 s3 At W 3∗ = − igW 3∗ 32 t − ξW 3∗ + iγW 3∗ × β 2 4 − 2β 2 + β 4 + β 2 4 − 10β 2 + β 4 cos θ + 2 − 11β 2 + 10β 4 cos2 θ + β 2 cos3 θ (13) where ξW 3∗ = m2W 3∗ /m2W , γW 3∗ = mW 3∗ ΓW 3∗ /m2W and gW 3∗ is the strength of the coupling between two W bosons and the W 3∗ boson. The orbitally excited Z boson (D µν ) is a d-wave, and its propagation is thus described by a propagator corresponding to a massive spin 2 particle. The propagator of a massive spin two particle is as follows (see Ref. [7]): 1 1 2 g g g + g g − g Γµνρσ = µρ νσ µσ νρ µν ρσ 3 p2 − m2D 2 (14) and we assume that the vertex W +µ W −ν Dµν is of the form igD . We obtain the following amplitudes for the s- and t-channel exchange AsD =
−1 2 m2D s2 igD 2 , 48 mW s − ξD + iγD × 2β 4 + 3 cos2 θ − 2β 2 − 1 ,
p p2 p3 p4
As 1
m2D
1 1 µ (p1 ) ν (p2 ) s − ξD + iγD 2 2 × gµρ gνσ + gµσ gνρ − gµν gρσ 3 ∗ρ ∗σ × (p3 ) (p4 )
2 = −igD
m2W
(17)
and p p2 p3 p4
At 1
m2D
1 1 µ (p1 ) ρ (p2 ) t − ξD + iγD 2 2 × gµρ gνσ + gµσ gνρ − gµν gρσ 3 × ∗ν (p3 ) ∗σ (p4 ),
2 = −igD
m2W
(18)
where pi stands for the polarization and also using the following relations (−p, 0, 0, E) (0, −1, ±i, 0) µ , , 1 (±) = √ mW 2 (−p, 0, 0, −E) (0, 1, ±i, 0) µ µ 2 (0) = , 2 (±) = √ , mW 2 (p, −E sin θ, 0, −E cos θ ) ∗µ , 3 (0) = mW (0, − cos θ, ∓i, sin θ ) ∗µ 3 (±) = √ , 2 (p, E sin θ, 0, E cos θ ) ∗µ , 4 (0) = mW (0, cos θ, ∓i, − sin θ ) ∗µ 4 (±) = (19) √ 2 µ
1 (0) =
(15)
s2 −1 2 m2D igD 2 96 mW t − ξD + iγD 4 × 4β + 6β 2 + 3 + 10β 2 cos θ + 1 cos θ 2 . (16) Since there is a pole in the t channel whose origin is the photon exchange, one has to impose cuts on the cross sections. For the numerical evaluation of the cross section, we impose a cut of 10◦ , which is the cut chosen in Ref. [11]. The spin of the particle can be determined from the angular distribution of the cross section. We have neglected the decay width of the Z boson and that of the Higgs boson since we assume that the energy of the process is such that no Z boson At D =
or Higgs resonance appear. For numerical estimates, we took mH = 100 GeV. We have considered only the reaction involving longitudinally polarized W . The amplitudes for different polarizations for the standard model can be found in the literature [10]. The amplitudes for a H ∗ or a W 3∗ can be deduced from the standard model calculations by replacing the masses, the decay widths and the coupling constants. Those for the neutral d-wave can be easily calculated using
valid in the center of mass system
where E is the energy of the W bosons, p = E 2 − m2W is their momentum and θ is the scattering angle.
X. Calmet, H. Fritzsch / Physics Letters B 526 (2002) 90–96
95
4. Discussion The differential decay widths for the reaction WL+ WL− → WL+ + WL− can be found in Fig. 3 for the reaction involving the neutral d-wave, Fig. 4 for that involving the W 3∗ spin 1 boson and Fig. 5 for that involving the H ∗ scalar. The particles W 3∗ and H ∗ are assumed to couple, in a first approximation, only to the W ’s. This allows to compute their decay rates using standard model formulas. As mentioned previously it is not an easy task to predict the mass spectrum of the model, thus we assumed, for numerical illustration, three different masses: 350, 500 and 800 GeV. The coupling constants are assumed to sizable (see Figs. 5, 6). If the cross sections are extrapolated to very high energies, the unitarity is violated. However, as expected in any substructure models, it will be restored by bound states effects. It is very instructive to plot the ratio of the differential cross section involving new physics to the standard model differential cross section. We have done so for the neutral d-wave (Ref. [9]). It is obvious from this picture that any deviation from the standard model, even at high energy will manifest it-self already in a deviation from one for that ratio. Already at an energy
Fig. 3. Dimensionless cross section of the reaction WL+ WL− → WL+ + WL− including the d-wave. The solid line is the standard model cross section, the dotted line corresponds to a d-wave of mass 350 GeV, with Γ = 4.38 GeV and g¯ W 3∗ = 0.8g, the long dashed line to a d-wave of mass 500 GeV, with Γ = 27.49 GeV and g¯ W 3∗ = 0.7g and the dot-dashed line to a d-wave of mass 800 GeV, with Γ = 251.03 GeV and g¯ W 3∗ = 0.6g.
Fig. 4. Dimensionless cross section of the reaction WL+ WL− → WL+ + WL− including the W 3∗ boson. The solid line is the standard model cross section, the dotted line corresponds to a W 3∗ boson of mass 350 GeV, with Γ = 66.2 GeV and g¯ W 3∗ = 0.8 sin2 θW g, the long dashed line to a W 3∗ boson of mass 500 GeV, with Γ = 266.8 GeV and g¯W 3∗ = 0.7 sin2 θW g and the dot-dashed line to a W 3∗ boson of mass 800 GeV, with Γ = 1795.5 GeV and g¯ W 3∗ = 0.6 sin2 θW g.
Fig. 5. Dimensionless cross section of the reaction WL+ WL− → WL+ + WL− including the H ∗ boson. The solid line is the standard model cross section, the dotted line corresponds to a H ∗ boson of mass 350 GeV, with Γ = 6.72 GeV and g¯ H ∗ = 0.8g, the long dashed line to a H ∗ boson of mass 500 GeV, with Γ = 17.6 GeV and g¯ H ∗ = 0.7g and the dot-dashed line to a W 3∗ boson of mass 800 GeV, with Γ = 58.25 GeV and g¯ H ∗ = 0.6g.
96
X. Calmet, H. Fritzsch / Physics Letters B 526 (2002) 90–96
of in the reaction WL+ + WL− → WL+ + WL− , thus any new particle contributing to that reaction will have a large impact already at energies well below the mass of this new particle. This reaction is thus not only of prime interest if the Higgs boson is heavy but should also be studied if the Higgs boson was light.
Acknowledgements We should like to thank P. Bambade, G.L. Kane, A. Leike, Z. Xing, V.I. Zakharov and P. Zerwas for useful discussions.
Fig. 6. Ratio of the cross section for the of the reaction involving the d-wave to the standard model cross section for different values of the d-wave mass and different coupling constant. The dotted line corresponds to a d-wave of mass 350 GeV, with Γ = 4.38 GeV and g¯W 3∗ = 0.8g, the long dashed line to a d-wave of mass 500 GeV, with Γ = 27.49 GeV and g¯ W 3∗ = 0.7g and the dot-dashed line to a d-wave of mass 800 GeV, with Γ = 251.03 GeV and g¯ W 3∗ = 0.6g.
which is low compared to the mass of the new particle, i.e., well bellow the resonance, one observes a deviation from unity. Nevertheless the calculation of the full reaction, e.g., e+ e− → W + W − ν ν¯ involves the convolution of the cross section of the reaction W + W − → W + + W − with functions describing the radiative emission of the W ’s from the fermions. When this integral is performed some sensitivity is lost. Nevertheless the effects are expected to be so large that they cannot be overlooked. The reaction will allow to test a mass range of a few TeV’s so that even if the new particles are too massive to be produced on-shell, their effects will be noticeable at future colliders.
5. Conclusions We have discussed the production of a neutral d-wave D 3 at the LHC or at a linear collider. If the mass of this particle is of the order of the scale of the theory, i.e., 300 GeV, it can be produced at these colliders. We have also shown that this particle as well as radial excitations of the Higgs boson and Z boson would spoil the cancellation of the leading powers in s
References [1] X. Calmet, H. Fritzsch, Phys. Lett. B 496 (2000) 161, hepph/0008243. [2] G. ’t Hooft, hep-th/9812204; See, also, G. ’t Hooft, in: G. ’t Hooft et al. (Eds.), Recent Developments in Gauge Theories, Cargesè 1979, Plenum, New York, 1980, Lecture II, p. 117. [3] X. Calmet, H. Fritzsch, hep-ph/0107085, to appear in Phys. Lett. B. [4] X. Calmet, H. Fritzsch, Phys. Lett. B 496 (2000) 190, hepph/0008252. [5] S.L. Glashow, Nucl. Phys. 22 (1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264. [6] P. Chiappetta, J.L. Kneur, S. Larbi, S. Narison, Phys. Lett. B 193 (1987) 346; J.L. Kneur, S. Larbi, S. Narison, Phys. Lett. B 194 (1987) 147. [7] H. van Dam, M. Veltman, Nucl. Phys. B 22 (1970) 397; V.I. Zakharov, JETP Lett. 12 (1970) 312. [8] J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model, Cambridge Univ. Press, Cambridge, 1992, p. 540. [9] D.A. Dicus, V.S. Mathur, Phys. Rev. D 7 (1973) 3111; M. Veltman, Acta Phys. Pol. B8 (1977) 475; See also, M.S. Chanowitz, M.K. Gaillard, Phys. Lett. B 142 (1984) 85; M.S. Chanowitz, M.K. Gaillard, Nucl. Phys. B 261 (1985) 379. [10] M.J. Duncan, G.L. Kane, W.W. Repko, Nucl. Phys. B 272 (1986) 517. [11] A. Denner, T. Hahn, Nucl. Phys. B 525 (1998) 27, hepph/9711302. [12] J.F. Gunion, H.E. Haber, G.L. Kane, S. Dawson, The Higgs Hunter’s Guide, SCIPP-89/13. [13] C.H. Llewellyn Smith, Phys. Lett. B 46 (1973) 233; J.M. Cornwall, D.N. Levin, G. Tiktopoulos, Phys. Rev. Lett. 30 (1973) 1268; J.M. Cornwall, D.N. Levin, G. Tiktopoulos, Phys. Rev. D 10 (1974) 1145.
Electroweak d-waves
Received 4 April 2001; received in revised form 7 December 2001; accepted 8 December 2001. Editor: R. Gatto ... [4] that models of a similar class im- ply different ..... York, 1980, Lecture II, p. 117. ... Hunter's Guide, SCIPP-89/13. [13] C.H. ...
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pdf-1462\electroweak-interactions-by-luciano-maiani.pdf ... | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9833118319511414, "perplexity": 2911.484640136761}, "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-2020-10/segments/1581875149238.97/warc/CC-MAIN-20200229114448-20200229144448-00554.warc.gz"} |
https://www.qalaxia.com/questions/Dividing-circle-into-geometrically-different-thirds | D
#### Dividing circle into geometrically different thirds
45 viewed last edited 1 year ago
Deepika Visala
0
A question that I was recently thinking of and couldn't work out an answer to was to determine a general solution as to the size of the shapes needed to make a circle divided into equal size parts using this shape: http://imgur.com/a/0Z5t0 where the white dot in the middle is the centre of the circle. Thanks :)
Krishna
1
Given in the question The size of the shapes needed to make a circle divided into equal size parts using this shape: We have to determine the three equal parts. So we need to imagine chords on the circle that divides the circle area into three equal parts. The area of the each part is must be \frac{1}{3}. First, we have to find out the area of the one part under the chord. Remaining are similar to this. We don,t have a direct formula for the area under the chord. but we know the formula of 1) The area of a sector in terms of \Theta. A = \frac{r^2 ( \theta)}{2} 2)The area of the isosceles triangle within the circular sector. A = \frac{r^2 * \sin (\theta)}{2} See the figure: chord area( \frac{\pi * r^2}{3}) = (the area of the sector) - (the area of the isosceles triangle ) \frac{\pi * r^2}{3} = \frac{r^2 * \theta}{2} - \frac{r^2 * \sin (\theta)}{2} \frac{\pi * r^2}{3} = \frac{r^2 }{2}(\theta -\sin (\theta) 2(\pi ) = 3*(\theta -\sin (\theta) f(\theta) = 3*\theta -3\sin (\theta) - 2* \pi Clearly, must be between zero and \pi radians, so using those as the initial boundaries, I quickly divided the range of possible values of in half and zeroed in on the approximate value of that answers your question. f(0) = -2(3.14) f(3.14) = 3(3.14) - 3 sin (3.14) - 2 (3.14) = 3.13522 f(2.6053) = 3(2.6053) - 3sin (2.6053) - 2(3.14) = 0.0003657 verify for the numbers in between the 0 to 3.14 \theta = 2.6053 a circular segment subtended by an angle of 2.6053 radians has an area of one third of the area of the circle itself. The length of the chord can be calculated given the radius r and this angle \theta using simple triangle.
Mahesh Godavarti
0
Do you have a Geometric method that accomplishes this?
Vivekanand Vellanki
1
In the attached figure, FE is parallel to AB. Let, the angle OAD be \theta degrees. It is also known that \theta < 30 since if angle OAD is 30, the area of the sector AOC will be 1/3rd the area of the circle. The area of ADC is given to the 1/3 area of the circle. Assuming unit circle, Area of ADC = Area of sector AOC + area of triangle AOD. Area of sector AOC = \pi\frac{90+\theta}{360} To find area of triangle AOD. OD=\cos{\theta} and AD=\sin{\theta} Area of triangle AOD = \frac{\cos{\theta}\sin{\theta}}{2} This gives the equation: \pi\frac{90+\theta}{360} + \frac{\cos{\theta}\sin{\theta}}{2} = \pi/3 \frac{\cos{\theta}\sin{\theta}}{2} = \pi(1/3 - \frac{90+\theta}{360}) \frac{\cos{\theta}\sin{\theta}}{2} = \pi\frac{30 - \theta}{360} \cos{\theta}\sin{\theta} = \pi\frac{30 - \theta}{180}
Mahesh Godavarti
0
Do you have a geometric construction for this?
Sangeetha Pulapaka
0
Draw a circle of any radius and mark the center of the circle as O. Mark some random point on the circumference of the circle and name it as A. Join OA. Keep the protractor on line OA and measure 120 ^\circand mark it as point B. Now keep the protractor on line OB and measure 120 ^\circ again, this time mark the next point as C.The circle is divided this way into three equal parts. Join the points ABC to get an equilateral triangle with angles 60 ^\circ each.This means that all the three sides are of equal length. | {"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": 0.906154215335846, "perplexity": 1077.2301019325782}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578532882.36/warc/CC-MAIN-20190421195929-20190421221929-00433.warc.gz"} |
https://en.wikibooks.org/wiki/Trigonometry/Selected_Angles_Reference | # Trigonometry/Selected Angles Reference
Note: Some values in the table are given in forms that include a radical in the denominator — this is done both to simplify recognition of reciprocal pairs and because the form given in the table is simpler in some sense. Note also that all absolute values of trigonometric functions for remarkable points that are listed in this table are contained in the first quadrant (from 0 to 90° or ${\displaystyle {\frac {\pi }{2}}}$ radians, inclusive); all others are deduced by simple symmetries with the horizontal or vertical axis, or by swapping axes (on the trigonometric circle) so that one trigonometric function is also swapped with its co-function.
${\displaystyle \theta }$ (positive) ${\displaystyle \sin(\theta )}$ ${\displaystyle \cos(\theta )}$ ${\displaystyle \tan(\theta )}$ ${\displaystyle \cot(\theta )}$ ${\displaystyle \sec(\theta )}$ ${\displaystyle \csc(\theta )}$ ${\displaystyle \theta }$ (negative)
${\displaystyle 0^{\circ }}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ not
defined
${\displaystyle 1}$ not
defined
${\displaystyle -360^{\circ }}$ ${\displaystyle -2\pi }$
${\displaystyle 15^{\circ }}$ ${\displaystyle {\frac {\pi }{12}}}$ ${\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}$ ${\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}$ ${\displaystyle 2-{\sqrt {3}}}$ ${\displaystyle 2+{\sqrt {3}}}$ ${\displaystyle {\sqrt {6}}-{\sqrt {2}}}$ ${\displaystyle {\sqrt {6}}+{\sqrt {2}}}$ ${\displaystyle -345^{\circ }}$ ${\displaystyle -{\frac {13\pi }{12}}}$
${\displaystyle 22.5^{\circ }}$ ${\displaystyle {\frac {\pi }{8}}}$ ${\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$ ${\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$ ${\displaystyle {\sqrt {2}}-1}$ ${\displaystyle {\sqrt {2}}+1}$ ${\displaystyle {\sqrt {4-2{\sqrt {2}}}}}$ ${\displaystyle {\sqrt {4+2{\sqrt {2}}}}}$ ${\displaystyle -337.5^{\circ }}$ ${\displaystyle -{\frac {15\pi }{8}}}$
${\displaystyle 30^{\circ }}$ ${\displaystyle {\frac {\pi }{6}}}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle {\frac {\sqrt {3}}{2}}}$ ${\displaystyle {\frac {1}{\sqrt {3}}}}$ ${\displaystyle {\sqrt {3}}}$ ${\displaystyle {\frac {2}{\sqrt {3}}}}$ ${\displaystyle 2}$ ${\displaystyle -330^{\circ }}$ ${\displaystyle -{\frac {11\pi }{6}}}$
${\displaystyle 45^{\circ }}$ ${\displaystyle {\frac {\pi }{4}}}$ ${\displaystyle {\frac {1}{\sqrt {2}}}}$ ${\displaystyle 1}$ ${\displaystyle {\sqrt {2}}}$ ${\displaystyle -315^{\circ }}$ ${\displaystyle -{\frac {7\pi }{4}}}$
${\displaystyle 60^{\circ }}$ ${\displaystyle {\frac {\pi }{3}}}$ ${\displaystyle {\frac {\sqrt {3}}{2}}}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle {\sqrt {3}}}$ ${\displaystyle {\frac {1}{\sqrt {3}}}}$ ${\displaystyle 2}$ ${\displaystyle {\frac {2}{\sqrt {3}}}}$ ${\displaystyle -300^{\circ }}$ ${\displaystyle -{\frac {5\pi }{3}}}$
${\displaystyle 67.5^{\circ }}$ ${\displaystyle {\frac {3\pi }{8}}}$ ${\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$ ${\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$ ${\displaystyle {\sqrt {2}}+1}$ ${\displaystyle {\sqrt {2}}-1}$ ${\displaystyle {\sqrt {4+2{\sqrt {2}}}}}$ ${\displaystyle {\sqrt {4-2{\sqrt {2}}}}}$ ${\displaystyle -292.5^{\circ }}$ ${\displaystyle -{\frac {11\pi }{8}}}$
${\displaystyle 75^{\circ }}$ ${\displaystyle {\frac {5\pi }{12}}}$ ${\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}$ ${\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}$ ${\displaystyle 2+{\sqrt {3}}}$ ${\displaystyle 2-{\sqrt {3}}}$ ${\displaystyle {\sqrt {6}}+{\sqrt {2}}}$ ${\displaystyle {\sqrt {6}}-{\sqrt {2}}}$ ${\displaystyle -285^{\circ }}$ ${\displaystyle -{\frac {19\pi }{12}}}$
${\displaystyle 90^{\circ }}$ ${\displaystyle {\frac {\pi }{2}}}$ ${\displaystyle 1}$ ${\displaystyle 0}$ not
defined
${\displaystyle 0}$ not
defined
${\displaystyle 1}$ ${\displaystyle -270^{\circ }}$ ${\displaystyle -{\frac {3\pi }{2}}}$
${\displaystyle 105^{\circ }}$ ${\displaystyle {\frac {7\pi }{12}}}$ ${\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}$ ${\displaystyle -{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}$ ${\displaystyle -2-{\sqrt {3}}}$ ${\displaystyle -2+{\sqrt {3}}}$ ${\displaystyle -{\sqrt {6}}-{\sqrt {2}}}$ ${\displaystyle {\sqrt {6}}-{\sqrt {2}}}$ ${\displaystyle -255^{\circ }}$ ${\displaystyle -{\frac {17\pi }{12}}}$
${\displaystyle 112.5^{\circ }}$ ${\displaystyle {\frac {5\pi }{8}}}$ ${\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$ ${\displaystyle -{\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$ ${\displaystyle -{\sqrt {2}}-1}$ ${\displaystyle -{\sqrt {2}}+1}$ ${\displaystyle -{\sqrt {4+2{\sqrt {2}}}}}$ ${\displaystyle {\sqrt {4-2{\sqrt {2}}}}}$ ${\displaystyle -247.5^{\circ }}$ ${\displaystyle -{\frac {11\pi }{8}}}$
${\displaystyle 120^{\circ }}$ ${\displaystyle {\frac {2\pi }{3}}}$ ${\displaystyle {\frac {\sqrt {3}}{2}}}$ ${\displaystyle -{\frac {1}{2}}}$ ${\displaystyle -{\sqrt {3}}}$ ${\displaystyle -{\frac {1}{\sqrt {3}}}}$ ${\displaystyle -2}$ ${\displaystyle {\frac {2}{\sqrt {3}}}}$ ${\displaystyle -240^{\circ }}$ ${\displaystyle -{\frac {4\pi }{3}}}$
${\displaystyle 135^{\circ }}$ ${\displaystyle {\frac {3\pi }{4}}}$ ${\displaystyle {\frac {1}{\sqrt {2}}}}$ ${\displaystyle -{\frac {1}{\sqrt {2}}}}$ ${\displaystyle -1}$ ${\displaystyle -{\sqrt {2}}}$ ${\displaystyle {\sqrt {2}}}$ ${\displaystyle -225^{\circ }}$ ${\displaystyle -{\frac {5\pi }{4}}}$
${\displaystyle 150^{\circ }}$ ${\displaystyle {\frac {5\pi }{6}}}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle -{\frac {\sqrt {3}}{2}}}$ ${\displaystyle -{\frac {1}{\sqrt {3}}}}$ ${\displaystyle -{\sqrt {3}}}$ ${\displaystyle -{\frac {2}{\sqrt {3}}}}$ ${\displaystyle 2}$ ${\displaystyle -210^{\circ }}$ ${\displaystyle -{\frac {7\pi }{6}}}$
${\displaystyle 157.5^{\circ }}$ ${\displaystyle {\frac {7\pi }{8}}}$ ${\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$ ${\displaystyle -{\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$ ${\displaystyle -{\sqrt {2}}+1}$ ${\displaystyle -{\sqrt {2}}-1}$ ${\displaystyle -{\sqrt {4-2{\sqrt {2}}}}}$ ${\displaystyle {\sqrt {4+2{\sqrt {2}}}}}$ ${\displaystyle -202.5^{\circ }}$ ${\displaystyle -{\frac {9\pi }{8}}}$
${\displaystyle 165^{\circ }}$ ${\displaystyle {\frac {11\pi }{12}}}$ ${\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}$ ${\displaystyle -{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}$ ${\displaystyle -2+{\sqrt {3}}}$ ${\displaystyle -2-{\sqrt {3}}}$ ${\displaystyle -{\sqrt {6}}+{\sqrt {2}}}$ ${\displaystyle {\sqrt {6}}+{\sqrt {2}}}$ ${\displaystyle -195^{\circ }}$ ${\displaystyle -{\frac {13\pi }{12}}}$
${\displaystyle 180^{\circ }}$ ${\displaystyle \pi }$ ${\displaystyle 0}$ ${\displaystyle -1}$ ${\displaystyle 0}$ not
defined
${\displaystyle -1}$ not
defined
${\displaystyle -180^{\circ }}$ ${\displaystyle -\pi }$
${\displaystyle 195^{\circ }}$ ${\displaystyle {\frac {13\pi }{12}}}$ ${\displaystyle -{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}$ ${\displaystyle -{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}$ ${\displaystyle 2-{\sqrt {3}}}$ ${\displaystyle 2+{\sqrt {3}}}$ ${\displaystyle -{\sqrt {6}}+{\sqrt {2}}}$ ${\displaystyle -{\sqrt {6}}-{\sqrt {2}}}$ ${\displaystyle -165^{\circ }}$ ${\displaystyle -{\frac {11\pi }{12}}}$
${\displaystyle 202.5^{\circ }}$ ${\displaystyle {\frac {9\pi }{8}}}$ ${\displaystyle -{\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$ ${\displaystyle -{\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$ ${\displaystyle {\sqrt {2}}-1}$ ${\displaystyle {\sqrt {2}}+1}$ ${\displaystyle -{\sqrt {4-2{\sqrt {2}}}}}$ ${\displaystyle -{\sqrt {4+2{\sqrt {2}}}}}$ ${\displaystyle -157.5^{\circ }}$ ${\displaystyle -{\frac {7\pi }{8}}}$
${\displaystyle 210^{\circ }}$ ${\displaystyle {\frac {7\pi }{6}}}$ ${\displaystyle -{\frac {1}{2}}}$ ${\displaystyle -{\frac {\sqrt {3}}{2}}}$ ${\displaystyle {\frac {1}{\sqrt {3}}}}$ ${\displaystyle {\sqrt {3}}}$ ${\displaystyle -{\frac {2}{\sqrt {3}}}}$ ${\displaystyle -2}$ ${\displaystyle -150^{\circ }}$ ${\displaystyle -{\frac {5\pi }{6}}}$
${\displaystyle 225^{\circ }}$ ${\displaystyle {\frac {5\pi }{4}}}$ ${\displaystyle -{\frac {1}{\sqrt {2}}}}$ ${\displaystyle 1}$ ${\displaystyle -{\sqrt {2}}}$ ${\displaystyle -135^{\circ }}$ ${\displaystyle -{\frac {3\pi }{4}}}$
${\displaystyle 240^{\circ }}$ ${\displaystyle {\frac {4\pi }{3}}}$ ${\displaystyle -{\frac {\sqrt {3}}{2}}}$ ${\displaystyle -{\frac {1}{2}}}$ ${\displaystyle {\sqrt {3}}}$ ${\displaystyle {\frac {1}{\sqrt {3}}}}$ ${\displaystyle -2}$ ${\displaystyle -{\frac {2}{\sqrt {3}}}}$ ${\displaystyle -120^{\circ }}$ ${\displaystyle -{\frac {2\pi }{3}}}$
${\displaystyle 247.5^{\circ }}$ ${\displaystyle {\frac {11\pi }{8}}}$ ${\displaystyle -{\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$ ${\displaystyle -{\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$ ${\displaystyle {\sqrt {2}}+1}$ ${\displaystyle {\sqrt {2}}-1}$ ${\displaystyle -{\sqrt {4+2{\sqrt {2}}}}}$ ${\displaystyle -{\sqrt {4-2{\sqrt {2}}}}}$ ${\displaystyle -112.5^{\circ }}$ ${\displaystyle -{\frac {5\pi }{8}}}$
${\displaystyle 255^{\circ }}$ ${\displaystyle {\frac {17\pi }{12}}}$ ${\displaystyle -{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}$ ${\displaystyle -{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}$ ${\displaystyle 2+{\sqrt {3}}}$ ${\displaystyle 2-{\sqrt {3}}}$ ${\displaystyle -{\sqrt {6}}-{\sqrt {2}}}$ ${\displaystyle -{\sqrt {6}}+{\sqrt {2}}}$ ${\displaystyle -105^{\circ }}$ ${\displaystyle -{\frac {7\pi }{12}}}$
${\displaystyle 270^{\circ }}$ ${\displaystyle {\frac {3\pi }{2}}}$ ${\displaystyle -1}$ ${\displaystyle 0}$ not
defined
${\displaystyle 0}$ not
defined
${\displaystyle -1}$ ${\displaystyle -90^{\circ }}$ ${\displaystyle -{\frac {\pi }{2}}}$
${\displaystyle 285^{\circ }}$ ${\displaystyle {\frac {19\pi }{12}}}$ ${\displaystyle -{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}$ ${\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}$ ${\displaystyle -2-{\sqrt {3}}}$ ${\displaystyle -2+{\sqrt {3}}}$ ${\displaystyle {\sqrt {6}}+{\sqrt {2}}}$ ${\displaystyle -{\sqrt {6}}+{\sqrt {2}}}$ ${\displaystyle -75^{\circ }}$ ${\displaystyle -{\frac {5\pi }{12}}}$
${\displaystyle 292.5^{\circ }}$ ${\displaystyle {\frac {11\pi }{8}}}$ ${\displaystyle -{\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$ ${\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$ ${\displaystyle -{\sqrt {2}}-1}$ ${\displaystyle -{\sqrt {2}}+1}$ ${\displaystyle {\sqrt {4+2{\sqrt {2}}}}}$ ${\displaystyle -{\sqrt {4-2{\sqrt {2}}}}}$ ${\displaystyle -67.5^{\circ }}$ ${\displaystyle -{\frac {3\pi }{8}}}$
${\displaystyle 300^{\circ }}$ ${\displaystyle {\frac {5\pi }{3}}}$ ${\displaystyle -{\frac {\sqrt {3}}{2}}}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle -{\sqrt {3}}}$ ${\displaystyle -{\frac {1}{\sqrt {3}}}}$ ${\displaystyle 2}$ ${\displaystyle -{\frac {2}{\sqrt {3}}}}$ ${\displaystyle -60^{\circ }}$ ${\displaystyle -{\frac {\pi }{3}}}$
${\displaystyle 315^{\circ }}$ ${\displaystyle {\frac {7\pi }{4}}}$ ${\displaystyle -{\frac {1}{\sqrt {2}}}}$ ${\displaystyle {\frac {1}{\sqrt {2}}}}$ ${\displaystyle -1}$ ${\displaystyle {\sqrt {2}}}$ ${\displaystyle -{\sqrt {2}}}$ ${\displaystyle -45^{\circ }}$ ${\displaystyle -{\frac {\pi }{4}}}$
${\displaystyle 330^{\circ }}$ ${\displaystyle {\frac {11\pi }{6}}}$ ${\displaystyle -{\frac {1}{2}}}$ ${\displaystyle {\frac {\sqrt {3}}{2}}}$ ${\displaystyle -{\frac {1}{\sqrt {3}}}}$ ${\displaystyle -{\sqrt {3}}}$ ${\displaystyle {\frac {2}{\sqrt {3}}}}$ ${\displaystyle -2}$ ${\displaystyle -30^{\circ }}$ ${\displaystyle -{\frac {\pi }{6}}}$
${\displaystyle 337.5^{\circ }}$ ${\displaystyle {\frac {15\pi }{8}}}$ ${\displaystyle -{\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$ ${\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$ ${\displaystyle -{\sqrt {2}}+1}$ ${\displaystyle -{\sqrt {2}}-1}$ ${\displaystyle {\sqrt {4-2{\sqrt {2}}}}}$ ${\displaystyle -{\sqrt {4+2{\sqrt {2}}}}}$ ${\displaystyle -22.5^{\circ }}$ ${\displaystyle -{\frac {\pi }{8}}}$
${\displaystyle 345^{\circ }}$ ${\displaystyle {\frac {13\pi }{12}}}$ ${\displaystyle -{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}$ ${\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}$ ${\displaystyle -2+{\sqrt {3}}}$ ${\displaystyle -2-{\sqrt {3}}}$ ${\displaystyle {\sqrt {6}}-{\sqrt {2}}}$ ${\displaystyle -{\sqrt {6}}-{\sqrt {2}}}$ ${\displaystyle -15^{\circ }}$ ${\displaystyle -{\frac {\pi }{12}}}$
${\displaystyle 360^{\circ }}$ ${\displaystyle 2\pi }$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ not
defined
${\displaystyle 1}$ not
defined
${\displaystyle 0^{\circ }}$ ${\displaystyle 0}$
Notice that for certain values of ${\displaystyle x}$ , the tangent, cotangent, secant, and cosecant functions are undefined. This is because these functions are defined as ${\displaystyle {\frac {\sin(x)}{\cos(x)}}}$ , ${\displaystyle {\frac {\cos(x)}{\sin(x)}}}$ , ${\displaystyle {\frac {1}{\cos(x)}}}$ , and ${\displaystyle {\frac {1}{\sin(x)}}}$ , respectively. Since an expression is undefined if it contains division by 0, the functions are therefore undefined at angle measures where the denominator (the sine or cosine of ${\displaystyle x}$ , depending on the trigonometric function) is equal to 0. For example, the tangent function for 90º (${\displaystyle \pi /2\,}$ radians) is equivalent to ${\displaystyle {\frac {\sin \left({\frac {\pi }{2}}\right)}{\cos \left({\frac {\pi }{2}}\right)}}}$ , or ${\displaystyle {\frac {1}{0}}}$ , which is an undefined value. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 330, "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": 0.985552191734314, "perplexity": 29.055888865830248}, "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/1495463608665.33/warc/CC-MAIN-20170526124958-20170526144958-00027.warc.gz"} |
https://www.physicsforums.com/threads/a-plane-a-pulley-and-two-boxes.22525/ | # A Plane, A pulley and two boxes
1. Apr 25, 2004
### Divergent13
Hi everyone here is the picture of the this plane/friction problem.
The question asks what are the minimum and maximum values of m1 in the figure to keep the system from accelerating...
Take µk = µs = 0.50
------------------------
Ok when drawing my free body diagrams I have come up with this method of solving the problem... tell me if you agree.
The force of friction can either be up or down the slope, if m2 = 0 or sufficiently small, then m1 would tend to slide down the plane. so Ffr would be directed Up the incline.
We know that newtons second law for the y direction (i chose my xy coordinate axes along the plane-- IE horizontal x being the plane) shows that
Fnormal - m1*g*cos(30) = m1*ay = 0
since theres no y motion Fnormal = m1*g*cos(30)
Now for the x motion... For the first case (smallest m1) f = ma shows that
m1*g*sin(30) - Ftension - Ffr = m1*ax <---- x direction
since we want ax to be 0, we can solve Ftension since thats related to m2.
Since Ffr can be AT MOST µs * Fnormal= µs*m1*g*cos 30 the minumum value m2 can have to prevent motion (ax = 0) is (after dividing by g)
m2 = m1 * sin(30) - µs*m1*cos(theta).
And then finding the max value wouldnt be much more difficult from there since we already set up our equations..
Am I correct here? If you are willing can someone work it --- what range do you get for the mass?
Thanks for you help. Mechanics gets soooo tough!
2. Apr 25, 2004
### Staff: Mentor
Some of your thinking seems OK, but you are getting a little mixed up. First of all, your diagram shows m2 (the hanging mass) as being fixed at 5 kg, so I don't know why you are solving for m2!
The smallest value of m1 would just prevent it from sliding up the plane. Taking up the plane as positive, the forces on m1 (I just call it m) are:
-mg sin(30) -μmg cos(30) + T = ma = 0
(note that T must equal 5g)
You can solve this for the minimum value of m.
The maximum value of m1 would just prevent it from sliding down the plane. The forces on m in this case are:
-mg sin(30) +μmg cos(30) + T = ma = 0
You can solve this for the maximum value of m.
The difference between the two cases, as I'm sure you realize, is that the friction acts in different directions.
3. Apr 25, 2004
### Divergent13
Interesting, I see why the tension is 5g, and I get a minimum value of 5.35 kg which seems appropriate... but my max value is nearly 75kg!!!
Is that correct? Or is there something else that we should consider for max value...
4. Apr 25, 2004
### UrbanXrisis
I think that's correct. Know that 75 kg also causes more friction.
5. Apr 25, 2004
### Divergent13
Thanks urban... but wow-- that is a huge difference. | {"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": 0.8351992964744568, "perplexity": 1429.0234795852925}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719416.57/warc/CC-MAIN-20161020183839-00467-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/2237-find-missing-factor.html | # Math Help - find the missing factor
1. ## find the missing factor
how do i do this problem
find the missing factor 7m2 times (?) = 35m8 -28m5 -42m2
2. Originally Posted by jim1174
how do i do this problem
find the missing factor 7m2 times (?) = 35m8 -28m5 -42m2
I understand the problem as,
$7m^2 (?)=35m^8-28m^5-42m^2$
On the right side you get $35m^8$ thus, which number when multiplied by $7m^2$ gives $35m^8$? The answer is $5m^6$. Now on the right side you also have $-28m^5$ now which number when multiplied by $7m^2$ gives $-28m^5$? The answer is $-4m^3$. Now on the right side you also have $-42m^2$ now which number multiplied by $7m^2$ gives $-42m^2$? The answer is $-6$ Thus, you have,
$7m^2(5m^6-4m^3-6)=35m^8-28m^5-42m^2$
$5m^6-4m^3-42$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.952134370803833, "perplexity": 717.7680600352451}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1410657116650.48/warc/CC-MAIN-20140914011156-00195-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://mathonline.wikidot.com/the-order-of-a-mod-m | The Order of a (mod m)
# The Order of a (mod m)
Recall that by Euler's theorem that if $(a, m) = 1$, then $a^{\phi (m)} \equiv 1 \pmod {m}$. However, we will now be interested in the congruence $a^t \equiv 1 \pmod {m}$ for $1 ≤ t ≤ \phi (m)$. The smallest such positive t that satisfies this congruence is called the order of a (mod m) or alternatively the exponent to which a belongs (mod m).
Notice that if $(a, m) = 1$, then the least residues of the integers $a, a^2, a^3, ...$ (mod m) are all relatively prime to m. We know that there are $\phi (m)$ least residues that are relatively prime to m, but there are infinitely many powers of a. Hence it follows that for some j ≠ k and j > k then:
(1)
\begin{align} a^j \equiv a^k \pmod m \end{align}
Since $(a, m) = 1$, it thus follows that we can divide both sides by a^k to obtain:
(2)
\begin{align} a^{j - k} \equiv 1 \pmod m \end{align}
In fact there are infinitely many powers of a that satisfy this congruence. For example since $a^{t + k\phi (m)} = a^t (a^k)^{\phi m}$, then for any positive integer k:
(3)
\begin{align} a^{t + k\phi (m)} \equiv a^t (a^{k})^{\phi (m)} \equiv a^t (1) \equiv 1 \pmod m \end{align}
Let's create a table for $a^n \mod {13}$. We will let a = 1, a = 2, …, a = 12, and calculate a (mod m), a2 (mod m), …, a^12 (mod m).
a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12
1 1 1 1 1 1 1 1 1 1 1 1
2 4 8 3 6 12 11 9 5 10 7 1
3 9 1 3 9 1 3 9 1 3 9 1
4 3 12 9 10 1 4 3 12 9 10 1
5 12 8 1 5 12 8 1 5 12 8 1
6 10 8 9 2 12 7 3 5 4 11 1
7 10 5 9 11 12 6 3 8 4 2 1
8 12 5 1 8 12 5 1 8 12 5 1
9 3 1 9 3 1 9 3 1 9 3 1
10 9 12 3 4 1 10 9 12 3 4 1
11 4 5 3 7 12 2 9 8 10 6 1
12 1 12 1 12 1 12 1 12 1 12 1
a Order of a (mod 13)
1 1
12 2
3, 9 3
5, 8 4
4, 10 6
2, 6, 7, 11 12
Notice that the only possible orders for a are 1, 2, 3, 4, 6, and 12. Interestingly enough, these are all divisors of 12. So how is 12 related to 13? Well we know that 13 is a prime, so $\phi (13) = 12$.
# Theorem 1: If (a, m) = 1 and a has order t (mod m), then an ≡ 1 (mod m) if and only if n is a multiple of t.
• Proof: Suppose that $n = tq$, that is, n is a multiple of t. It thus follows that $a^n \equiv a^{tq} \equiv (a^t)^q \equiv (1)^q \equiv 1 \pmod m$. Hence if $a^n \equiv 1 \pmod m$, then n must be a multiple of t.
• Proof (Converse): Suppose that $a^n \equiv 1 \pmod m$. Since t is the order of a (mod m), n ≥ t. Hence we can rewrite n as $n = qt + r$ by the division algorithm. We thus obtain that: $a^n \equiv a^{qt + r} \equiv (a^t)^qa^r \equiv (1)^q a^r \equiv a^r \pmod m$. But r = 0 since 0 ≤ r < t since the order of a (mod m) is t. Hence n = qt.
# Theorem 2: If (a, m) = 1 and a has order t (mod m), then t | Φ(m).
• Proof: We know that from theorem 1 that $\phi (m)$ must be a multiple of t since by Euler's theorem $a^{\phi (m)} \equiv 1 \pmod m$. Hence t | Φ(m).
We can verify from the original example that all of the orders of a (mod 13) were 1, 2, 3, 4, 6, and 12. All of these divide $\phi (13) = 12$. 1 | 12, 2 | 12, 3 | 12, 4 | 12, 6 | 12, and 12 | 12.
# Theorem 3: If p and q are both odd primes, and q | ap - 1, then either q | a - 1 or q = 2kp + 1.
• Proof: By the definition of a congruence, we know that $a^p \equiv 1 \pmod q$ since q | ap - 1. The order of a (mod q) must be a divisor of p by theorem 1, but there are only two divisors of p, namely 1 and p. Suppose that the order of a (mod q) is 1. Hence $a^1 \equiv 1 \pmod q$, so then q | a - 1. Suppose that the order of a (mod q) is p. By theorem 2, p must divided $\phi (q) = q - 1$. Hence p | q - 1, or rather $np = q - 1$ or $q = np + 1$. The lefthand side of this equation is odd, hence, the righthand side of this equation must also be odd. Since p is an odd prime and 1 is odd, then n must be even. Hence n = 2k for k ≥ 1. Hence $q = 2kp + 1$.
## Example 1
Verify theorem 3 with q = 3 and a = 4, and p = 5.
We must first verify that q | ap - 1. In fact, this is true. 3 | 45 - 1, or rather 3 | 1023. So either 3 | 4 - 1 or 3 = 2k(5) + 1. Clearly 3 | 3, so this is true, and 3 = 10k + 1 does not work since $k \in \mathbb{Z}$ and 10k + 1 > 3.
# Theorem 4: If t is the order of a (mod m), then ar ≡ as (mod m) if and only if r ≡ s (mod t).
• Proof: Suppose that r ≥ s. Hence with the congruence $a^r \equiv a^s \pmod m$, then $a^{r-s} \equiv 1 \pmod m$. We know that t is the order of a (mod m). By theorem 1, r-s must be a multiple of t, that is t | r - s. Hence $r \equiv s \pmod t$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9567480087280273, "perplexity": 145.34133565929554}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1502886105961.34/warc/CC-MAIN-20170820015021-20170820035021-00256.warc.gz"} |
https://phys.libretexts.org/Courses/Joliet_Junior_College/Sandbox/Sandbox_stuff_-_guide/Getting_started_in_the_sandbox | $$\require{cancel}$$
# Getting started in the sandbox
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http://tex.stackexchange.com/questions/94379/how-to-uppercase-heavily-customised-sections-sectstyfontspec | # How to uppercase heavily customised sections (sectsty+fontspec)?
I am trying to customise the aspect of a scrbook to look like a book I have.
In my case, sections should be formatted as follows: the font face should be Conques Demi, the font size 26 pt, the font colour a particular shade of dark red (CMYK: 0% 87% 87% 50%) with a 1 pt thick line rule of the same colour that is large the width of a column, the label should be in upper case and aligned to right.
Since I have not managed to realise how to draw coloured line rules with KOMA scripts, I tried to abstain that results with other packages. If you know how to get the exact result that I described above, please feel free to show me how in KOMA! Thanks in advance!
Anyway I managed to mimic the font face, size and colour, the rule and the alignment, but I started to experience problems when I tried to make the label uppercase. Note that the problem persists even if I remove any reference to the fontspec package and all its commands. Consider the following minimal working example:
\documentclass[twocolumn]{scrbook}
\usepackage{xcolor}
\usepackage{sectsty}
\usepackage{fontspec}
\setcounter{secnumdepth}{0}
\definecolor{darkred}{cmyk}{0.0,0.87,0.87,0.50}
\newfontfamily\myfont[]{Conques}
\sectionfont{\raggedleft\myfont\Huge\color{darkred}\sectionrule{0pt}{0pt}{-2pt}{1pt}} %
\begin{document}
\mainmatter
\section{A Section} % (fold)
\label{sec:a_section}
Lorem ipsum dolor sit amet, consectetur adipisicing elit.
% section a_section (end)
\end{document}
As other answers (to simpler cases) suggest, in order to uppercase the section's label, I am expected to add \uppercase to the end of the \sectionfont{...} command. If I do so, however, I am warned that a { is missing and that \begin{document} is closed by \end{DOCUMENT}. If I open the resulting .pdf, all the document is uppercase except for an additional A Section the appear just below the first occurrence and the rule is gone.
This is due to the fact, I think, that \uppercase is a parametric command while here it is used as an "absolute" statement. In my opinion, in fact, the second occurrence of A Section is due to \uppercase messing with the definition of the section's headings for the table of content, which in turn becomes empty.
According to my researches, there is no such "absolute" \uppercase equivalent. Some answers suggest to use \MakeUppercase instead as in some cases it is more "respectful" of the surrounding environment. If I do so, however, I get the following result: the string [0pt][r] at the beginning of an empty line with a rule that is finishing out of the vertical column border; the proper section's label appears below, not uppercased, with no rule, but correctly moved to the right. Even if this might suggest to reorder the commands within \sectionfont{...}, no permutation actually works. I have even tried with the textcase package and its \MakeTextUppercase command as well, with no luck at all.
Can someone please tell me where I am wrong and how to fix the issue?
-
– Claudio Fiandrino Jan 18 '13 at 14:15
As far as I know, sectsty doesn't allow to put at the end a command with argument; it's titlesec that has this feature. However, also the KoMa-Script classes should have some facility for this job, without the need of external packages. – egreg Jan 18 '13 at 14:23
It's a pity we're not allowed to use KOMA's own setup commands. It'd be rather easy, then. – cgnieder Jan 18 '13 at 14:25
My main concern in regards of KOMA commands is that I have found no evidence that they allow to define line rules like \sectionrule{0pt}{0pt}{-2pt}{1pt}. Do you know if they indeed do allow that? I'm editing my question to include this possibility. – Stefano Bragaglia Jan 18 '13 at 14:53
If KOMA's \setkomafont{section}{<decl>} can be used then the task is rather easy since the last command in there may have an argument. KOMA doesn't have a \sectionrule but one can define something similar using LaTeX's \rule[<raise>]{<width>}{<thickness>} command.
So maybe something like the following:
\documentclass[twocolumn]{scrbook}
% choose the font:
\usepackage{fontspec}
% I don't have Conques' installed, so I use another font
% I also added a LetterSpace' greater than zero as all-caps
% words _always_ should be spaced out a little
\newfontfamily\sectionfont[LetterSpace=2]{Linux Libertine O}
% define the color:
\usepackage{xcolor}
\definecolor{darkred}{cmyk}{0.0,0.87,0.87,0.50}
\colorlet{section}{darkred}
% the section layout:
\newcommand\sectionrule{%
\makebox[0pt][l]{\rule[-.25ex]{\linewidth}{1pt}}}
\newcommand\sectionformat[1]{%
\sectionfont\Huge\color{section}%
\sectionrule
\hfill\MakeUppercase{#1}}
\setkomafont{section}{\sectionformat}
\setcounter{secnumdepth}{0}
\begin{document}
\mainmatter
\section{A Section}\label{sec:a_section}
Lorem ipsum dolor sit amet, consectetur adipisicing elit.
\end{document}
-
\makebox[0pt][l]{\rule...} would be better than skipping back by \linewidth. However this would not work for multiline section titles. – egreg Jan 18 '13 at 16:25
@egreg indeed, thanks! – cgnieder Jan 18 '13 at 16:26
You might instead define a new command such as \newcommand\printsectiontitle[1]{\MakeUppercase{#1}\makebox[0pt][r]{\rule...}} and use \printsectiontitle as the last command in \setkomafont{section} – egreg Jan 18 '13 at 16:31
@egreg Yes, that would be better and if it was my document I'd probably define some \sectionformat{<arg>} command and used it as only argument to \setkomafont. I consider that fine-tuning which I haven't done in my answer. Maybe I should, though... – cgnieder Jan 18 '13 at 16:38
Dear cgnieder and egreg, thanks for both the solution and insightful suggestions to my problem! Now it works... and also reasonably well! I'm doing more experiments but I consider the problem solved. The above short discussion about best practices is very interesting as well, thanks for sharing! Regards. – Stefano Bragaglia Jan 18 '13 at 16:51 | {"extraction_info": {"found_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": 0.9688860177993774, "perplexity": 1593.7775544981043}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1410657137906.42/warc/CC-MAIN-20140914011217-00319-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://arxiv.org/list/hep-ph/new | # High Energy Physics - Phenomenology
## New submissions
[ total of 45 entries: 1-45 ]
[ showing up to 2000 entries per page: fewer | more ]
### New submissions for Thu, 16 Aug 18
[1]
Title: Dispersion relation for hadronic light-by-light scattering: pion pole
Comments: 55 pages, 16 figures, result for the space-like pion transition form factor attached as ancillary material
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex); High Energy Physics - Lattice (hep-lat); Nuclear Theory (nucl-th)
The pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon $(g-2)_\mu$ is fully determined by the doubly-virtual pion transition form factor. Although this crucial input quantity is, in principle, directly accessible in experiment, a complete measurement covering all kinematic regions relevant for $(g-2)_\mu$ is not realistic in the foreseeable future. Here, we report in detail on a reconstruction from available data, both space- and time-like, using a dispersive representation that accounts for all the low-lying singularities, reproduces the correct high- and low-energy limits, and proves convenient for the evaluation of the $(g-2)_\mu$ loop integral. We concentrate on the systematics of the fit to $e^+e^-\to 3\pi$ data, which are key in constraining the isoscalar dependence, as well as the matching to the asymptotic limits. In particular, we provide a detailed account of the pion transition form factor at low energies in the time- and space-like region, including the error estimates underlying our final result for the pion-pole contribution, $a_\mu^{\pi^0\text{-pole}}=62.6^{+3.0}_{-2.5}\times 10^{-11}$, and demonstrate how forthcoming singly-virtual measurements will further reduce its uncertainty.
[2]
Title: LHC luminosity and energy upgrades confront natural supersymmetry models
Comments: 19 pages with 8 .png figures
Subjects: High Energy Physics - Phenomenology (hep-ph)
The electroweak fine-tuning measure Delta(EW) allows for correlated SUSY soft terms as are expected in any ultra-violet complete theory. Requiring no less than 3% electroweak fine-tuning implies upper bounds of about 360~GeV on all higgsinos, while top squarks are lighter than ~3 TeV and gluinos are bounded by ~ 6-9 TeV. We examine the reach for SUSY of the planned high luminosity (HL: 3 ab^{-1} at 14 TeV) and the proposed high energy (HE: 15 ab^{-1} at 27 TeV) upgrades of the LHC via four LHC collider search channels relevant for natural SUSY: 1. gluino pair production followed by gluino decay to third generation (s)quarks, 2. top-squark pair production followed by decay to third generation quarks and light higgsinos, 3. neutral higgsino pair production with QCD jet radiation (resulting in monojet events with soft dileptons), and 4. wino pair production followed by decay to light higgsinos leading to same-sign diboson production. We confront our reach results with upper limits on superpartner masses in four natural SUSY models: natural gravity-mediation via the 1. two- and 2. three-extra-parameter non-universal Higgs models, 3. natural mini-landscape models with generalized mirage mediation and 4. natural anomaly-mediation. We find that while the HL-LHC can probe considerable portions of natural SUSY parameter space in all these models, the HE-LHC will decisively cover the entire natural SUSY parameter space with better than 3% fine-tuning.
[3]
Title: Dark matter bound states via emission of two scalar mediators
Comments: 35 pages, 7 figures, 2 tables
Subjects: High Energy Physics - Phenomenology (hep-ph)
If dark matter (DM) couples to a force carrier that is much lighter than itself, then it may form bound states in the early universe and inside haloes today. This occurs typically with emission of a force mediator that dissipates the binding energy. While bound-state formation via vector emission is known to be very efficient, the radiative capture via scalar emission is thought to require larger couplings, and therefore be relevant to more limited parameter space. Indeed, for particle-antiparticle or identical-particle pairs, the lowest order s- and p-wave contributions to the capture with emission of one scalar mediator cancel. We compute the cross-section for bound-state formation with emission of two scalar mediators, and show that the lowest order s-wave contribution does not cancel. The corresponding cross-section is of lower order in the trilinear DM-DM-mediator coupling than the cap- ture via emission of one scalar, and even than the capture via emission of one vector boson in theories with vector mediators. For pairs of annihilating particles, this process is also of lower order than the annihilation into mediators, provided that the relative velocity is sufficiently low. We showcase the phenomenological importance of bound-state formation via two-scalar emission by computing its effect on the relic density of self-conjugate DM, and discuss its potential implications for models of symmetric and asymmetric DM.
[4]
Title: Extended scalar sectors, effective operators and observed data
Subjects: High Energy Physics - Phenomenology (hep-ph)
The available data on the 125 GeV scalar $h$ is analysed to explore the room for new physics in the electroweak symmetry breaking sector. The first part of the study is model-independent, with $h$ couplings to standard model particles scaled by quantities that are taken to be free parameters. At the same time, the additional loop contributions to $h \rightarrow \gamma\gamma$ and $h \rightarrow Z\gamma$, mediated by charged scalar contributions in the extended scalar sector, are treated in terms of gauge-invariant effective operators. Having justified this approach for cases where the concerned scalar masses are a little above the $Z$-boson mass, we fit the existing data to obtain marginalized 1$\sigma$ and 2$\sigma$ regions in the space of the coefficients of such effective operators, where the limit on the $h \rightarrow Z\gamma$ branching ratio is used as a constraint. The correlation between, say, the gluon fusion and vector-boson fusion channels, as reflected in a non-diagonal covariance matrix, is taken into account. After thus obtaining model-independent fits, the allowed values of the coefficients are translated into permissible regions of the parameter spaces of several specific models. In this spirit we constrain four different types of two Higgs doublet models, and also models with one or two $Y = 2$ scalar triplets, taking into account the correlatedness of the scale factors in $h$-interactions and the various couplings of charged Higgs states in each extended scenario.
[5]
Title: Multiparticle Production at Mid-Rapidity in the Color-Glass Condensate
Subjects: High Energy Physics - Phenomenology (hep-ph); Nuclear Theory (nucl-th)
In this paper, we compute a number of cross sections for the production of multiple particles at mid-rapidity in the semi-dilute / dense regime of the color-glass condensate (CGC) effective field theory. In particular, we present new results for the production of two quark-antiquark pairs (whether the same or different flavors) and for the production of one quark-antiquark pair and a gluon. We also demonstrate the existence of a simple mapping which transforms the cross section to produce a quark-antiquark pair into the corresponding cross section to produce a gluon, which we use to obtain various results and to cross-check them against the literature. We also discuss hadronization effects in the heavy flavor sector, writing explicit expressions for the production of various combinations of $D$ and $\bar D$ mesons, $J/\psi$ mesons, and light hadrons. The various multiparticle cross sections presented here contain a wealth of information and can be used to study heavy flavor production, charge-dependent correlations, and "collective" flow phenomena arising from initial-state dynamics.
[6]
Title: Non-Global and Clustering Effects for Groomed Multi-Prong Jet Shapes
Authors: Duff Neill
Subjects: High Energy Physics - Phenomenology (hep-ph)
We present a resummation valid to next-to leading logarithmic accuracy of the non-global and clustering effects in groomed (with modified mass drop tagger) multi-pronged observables. These effects are universal in the sense that they depend only on the flavor structure of the $1\to 2$ splitting forming the multi-pronged subjets and the opening angle of the splitting, being insensitive to the underlying hard process or underlying event. The differential spectra with and without the non-global and clustering effects are presented, and the change in the spectra is found to be small.
[7]
Title: Multi-component dark matter from a hidden gauged SU(3)
Comments: 20 pages, 5 figures, 5 tables
Subjects: High Energy Physics - Phenomenology (hep-ph)
We study Dark Matter (DM) phenomenology with multiple DM species consisting of both scalar and vector DM particles. More specifically, we study the Hidden Gauged SU(3) model of Arcadi {\it et al}. Before proceeding to the Hidden Gauged SU(3) model, we study the relic abundances of simplified multi-species DM scenarios to gain some insights when multiple species and interactions are included. In the Hidden Gauged SU(3) model, because of the large parameter space, we restrict ourselves to three representative benchmark points, each with multiple DM species. The relic densities for the benchmark points were found using a program developed to solve the coupled Boltzmann equations for an arbitrary number of interacting DM species with two particles in the final state. For each case, we varied the mass of the DM particles and then found the value of the dark SU(3) gauge coupling that gives the correct relic density. We found that in some regions of the parameter space, the DM would be difficult to observe in direct detection experiments while easier to observe in indirect detection experiments and vice versa, so that complementary measurements could help pinpoint the details of the Hidden Gauged SU(3) model. Important to this, is that even for moderate changes in input parameter values, the relic density of some species can change by orders of magnitude resulting in large changes in the observability of multi-species DM by direct or indirect detection.
[8]
Title: Energy evolution and the Bose-Einstein enhancement for double parton densities
Authors: E. Gotsman (Tel Aviv U.), E. Levin (Tel Aviv U./UTFSM)
Comments: 19pp. 7 figures in pdf files
Subjects: High Energy Physics - Phenomenology (hep-ph)
In this paper we found that the Bose-Einstein enhancement generates the strong correlations, which increase with energy in the BFKL evolution. This increase leads to the double parton densities ( $\Phi$), that are much larger than the product of the single parton densities ($\phi$). However, numerically, it turns out that the ratio $\Phi/\phi^2 \propto \Lb 1/x\Rb^{\delta_2}$ with $\delta_2 \sim \bas/\Lb N^2_c - 1\Rb^{2/3}\,\,\ll\,\,1$ and we do not expect a large correction for the accessible range of energies. However, for $N_c=3$ it tuns out that $\delta_2 = 0.07 \Delta_{\rm BFKL}$ where $\Delta_{\rm BFKL}$ is the intercept of the BFKL Pomeron and we can anticipate an substantial increase for the range of rapidities $Y \sim 20$.It is shown that all $1/(N^2_c -1)$ corrections to the double parton densities stem from the Bose-Einstein enhancement.
[9]
Authors: J. Haidenbauer
Subjects: High Energy Physics - Phenomenology (hep-ph); Nuclear Theory (nucl-th)
Two-particle momentum correlation functions as measured in heavy ion collisions or in high-energetic proton-proton collisions are studied. Special emphasis is put on systems like $\Lambda\Lambda$ or $K^-p$ where effects from the coupling to other channels could be relevant. In both cases other channels open at relatively low momenta or are already open at the reaction threshold. To have a solid basis, realistic coupled-channel interactions for $\Lambda\Lambda-\Xi N-\Lambda\Sigma-\Sigma\Sigma$ and $\pi\Lambda-\pi\Sigma-\bar KN$ are utilized in the actual calculations. It is found that the opening of the $\Xi N$ channel leaves a trace in the $\Lambda\Lambda$ correlation function that could be detectable in experiments. Should the proposed $H$-dibaryon be located close to or below the $\Xi N$ it will have a very pronounced effect. The presence of open channels in systems like $\Xi^- p$ or $K^-p$ does influence the correlation functions significantly at low momenta and will certainly complicate any dedicated analysis.
[10]
Title: Test of semi-local duality in a large $N_C$ framework
Subjects: High Energy Physics - Phenomenology (hep-ph); Nuclear Theory (nucl-th)
In this paper we test the semi-local duality based on the method of Ref.[1] for calculating final-state interactions at varying number of colors ($N_C$). We compute the amplitudes by dispersion relations that respect analyticity and coupled channel unitarity, as well as accurately describing experiment. The $N_C$ dependence of the $\pi\pi\to\pi\pi$ scattering amplitudes is obtained by comparing these amplitudes to the one of chiral perturbation theory. The semi-local duality is investigated by varying $N_C$. Our results show that the semi-local duality is not violated when $N_C$ is large. At large $N_C$, the contributions of the $f_2(1270)$, the $f_0(980)$ and the $f_0(1370)$ cancel that of the $\rho(770)$ in the finite energy sum rules, while the $f_0(500)$ has almost no effect. This gives further credit to the method developed in Ref.[1] for investigating the $N_C$ dependence of hadron-hadron scattering with final-state interactions. This study is also helpful to understand the structure of the scalar mesons.
[11]
Title: $μ-τ$ Reflection Symmetry and Its Explicit Breaking for Leptogenesis in a Minimal Seesaw Model
Authors: Newton Nath
Subjects: High Energy Physics - Phenomenology (hep-ph)
The minimal seesaw framework, embroiling the Dirac neutrino mass matrix $M_D$ and the Majorana neutrino mass matrix $M_R$, is quite successful to explain the current global-fit results of neutrino oscillation data. In this context, we consider most predictive forms of $M_D$ and $M_R$ with two simple parameters. Considering these matrices, we obtain the low energy neutrino mass matrix under type-I seesaw formalism which obeys $\mu-\tau$ reflection symmetry and predicts $\theta_{23} = \pi/4$ and $\delta = \pm \pi/2$. In the given set-up, we also evaluate the Baryon Asymmetry of the Universe (BAU) through successful leptogenesis and find that perturbation of $\mathcal{O}(10^{-2})$ leads to the observed BAU and breaks exactness of the symmetry. Moreover, we also perform various correlation studies among different parameters in the framework of broken symmetry. Finally, we add a remark that the concerned model is consistent with the current global-fit data only for the normal mass ordering.
[12]
Title: Quantifying jet modifications with substructure
Comments: Proceedings of the XXVIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018), 13-19 May 2018, Venice, Italy
Subjects: High Energy Physics - Phenomenology (hep-ph); Nuclear Theory (nucl-th)
The striking suppression and modification patterns that are observed in jet observables measured in heavy-ion collisions with respect to the proton-proton baseline have the potential to constrain the spatio-temporal branching process of energetic partons in a dense QCD medium. The mechanism of jet energy loss is intricately associated with medium resolution of jet substructure fluctuations. This naturally affects the behavior of the suppression of jets at high-pT, inducing an explicit dependence on jet scales. In this contribution, we review recent work on using the insight from multi-parton quenching to calculate leading-logarithmic corrections to the single-inclusive jet spectrum, and discuss its impact on a wide range of observables, including jet substructure.
[13]
Title: Chiral symmetry breaking for fermions charged under large Lie groups
Subjects: High Energy Physics - Phenomenology (hep-ph)
We reexamine the dynamical generation of mass for fermions charged under various Lie groups with equal charge and mass at a high Grand Unification scale, extending the Renormalization Group Equations in the perturbative regime to two-loops and matching to the Dyson-Schwinger Equations in the strong coupling regime.
[14]
Title: Hypothesis about semi-weak interaction and experiments with solar neutrinos. II. Deuteron disintegration by neutral currents
Subjects: High Energy Physics - Phenomenology (hep-ph); Solar and Stellar Astrophysics (astro-ph.SR); High Energy Physics - Experiment (hep-ex); Nuclear Theory (nucl-th)
The present work provides one more evidence of that the solar neutrino problem has an elegant solution based on the hypothesis about the existence of a new, semi-weak, interaction. The analysis of the deuteron disintegration by neutral currents of solar neutrinos, generated by both the electroweak and semi-weak interactions, is fulfilled. A good agreement between the theoretical and experimental results for this process is obtained, which is in harmony with the conclusions of the first part of the work on the other four observed processes with solar neutrinos.
[15]
Title: Five-Particle Phase-Space Integrals in QCD
Comments: 11 pages, 2 tables; contribution to the proceedings of Loops and Legs in Quantum Field Theory, 29 April 2018 - 04 May 2018, St. Goar, Germany; based on arXiv:1803.09084
Subjects: High Energy Physics - Phenomenology (hep-ph)
We present analytical expressions for the 31 five-particle phase-space master integrals in massless QCD as an $\epsilon$-series with coefficients being multiple zeta values of weight up to 12. In addition, we provide a computer code for the Monte-Carlo integration in higher dimensions, based on the RAMBO algorithm, that has been used to numerically cross-check the obtained results in 4, 6, and 8 dimensions.
[16]
Title: Dulaity and charged pion condensation in chirally asymmetric dense quark matter in the framework of an NJL$_2$ model
Comments: Talk given at XXXI-th International Workshop on High Energy Physics "Critical points in the modern particle physics", July 5-7, 2017, in Protvino, Moscow region, Russia; the same results were also presented at XQCD 2017 in Pisa 26-28 June 2017
Journal-ref: International Journal of Modern Physics: Conf. Ser. Vol. 47, 1860093 (2018); Int.\ J.\ Mod.\ Phys.\ Conf.\ Ser.\ {\bf 47} (2018) 1860093
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th)
In this talk we present investigation of the phase structure of a (1+1)-dimensional quark model with four-quark interaction and in the presence of baryon ($\mu_B$), isospin ($\mu_I$) and chiral isospin ($\mu_{I5}$) chemical potentials. Spatially homogeneous and inhomogeneous (chiral density wave (for chiral condensate) and single wave (for charged pion condensate)) condensates are considered. It is established that in the large-$N_c$ limit ($N_c$ is the number of colored quarks) there exists a duality correspondence between the chiral symmetry breaking phase and the charged pion condensation one. The primary conclusion of this investigation is the fact that chiral isospin chemical potential generates charged pion condensation with non-zero baryon density in dense quark matter. Moreover, it is shown that inhomogeneous charged PC phase with nonzero baryon density is induced in the model by arbitrary small values of the chemical potential $\mu_{I5}$ (for a rather large region of $\mu_B$ and $\mu_I$).
[17]
Title: Electromagnetic spectral function and dilepton rate in a hot magnetized QCD medium
Subjects: High Energy Physics - Phenomenology (hep-ph); Nuclear Theory (nucl-th)
The dilepton production rate in hot QCD medium is studied within a effective description of the medium in the presence of magnetic field. This could be done by obtaining the one-loop self energy of photon due to the effective (quasi-) quark loop at finite temperature under an arbitrary external magnetic field while employing the real time formalism of Thermal Field Theory. The effective quarks and gluons encode hot QCD medium effective in terms of their respective effective fugacities. The magnetic field enters in the form of landau level quantization, in the matter sector (quarks, antiquarks). The full Schwinger proper time propagator including all the Landau levels is considered for the quasi quarks while calculating the photon self energy. The electromagnetic Debye screening (in terms of the self-energy) has seen to be influenced both by the hot QCD medium effects and magnetic field. Analogous results are also obtained from the semi classical transport theory. The imaginary part of the photon self energy function is obtained from the discontinuities of the self energy across the Unitary cuts which are also present at zero magnetic field and the Landau cuts which are purely due to the magnetic field. The dilepton production rate is then obtained in terms of the product of electromagnetic spectral functions due to quark loop and lepton loop. The modifications of both the quarks/antiquarks as well as leptons in presence of an arbitrary external magnetic field have been considered in the formalism. Significant enhancement of the low invariant mass dileptons due the appearance of the Landau cuts in the electromagnetic spectral function at finite external magnetic field has been observed. A substantial enhancement of dilepton rate is also found when the EOS effects are considered through the effective quarks/antiquraks.
### Cross-lists for Thu, 16 Aug 18
[18] arXiv:1808.04051 (cross-list from astro-ph.HE) [pdf, other]
Title: Meaningful Details: The value of adding baseline dependence to the Neutrino-Dark Matter Effect
Subjects: High Energy Astrophysical Phenomena (astro-ph.HE); High Energy Physics - Phenomenology (hep-ph)
The possibility of a dark matter effect on the flavour spectra of astronomical neutrinos was explored by De Salas, Lineros, and T\'ortola using a simplified version of the standard neutrino mixing formula which excised the baseline dependency. In this work, we report results calculated with the full formula (i.e. including the baseline dependency) and employing two different bases for the dark-matter neutrino interactions: the flavour basis, as for the weak interaction in the Standard Model, and the mass basis, which is predicted by certain non-Standard Models (specifically Scotogenic models). It was found that including baseline dependency dramatically increased the explorable parameter space. The two bases yield substantially different oscillation patterns, which suggests that the observation of astronomical neutrinos can conceivably shed light on the neutrino mass generation mechanism. Additionally, we presented results for the blazar TXS 0506+056, the first identified source of high-energy astrophysical neutrinos. It was found that a source like TXS 0506+056 can put significant constraints on neutrino-dark matter interactions.
[19] arXiv:1808.04827 (cross-list from hep-th) [pdf, other]
Title: Anyonic particle-vortex statistics and the nature of dense quark matter
Subjects: High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph); Nuclear Theory (nucl-th)
We show that $\mathbb{Z}_3$-valued particle-vortex braiding phases are present in high density quark matter. Certain mesonic and baryonic excitations, in the presence of a superfluid vortex, have orbital angular momentum quantized in units of $\hbar/3$. Such non-local topological features can distinguish phases whose realizations of global symmetries, as probed by local order parameters, are identical. If $\mathbb{Z}_3$ braiding phases and angular momentum fractionalization are absent in lower density hadronic matter, as is widely expected, then the quark matter and hadronic matter regimes of dense QCD must be separated by at least one phase transition.
[20] arXiv:1808.05134 (cross-list from gr-qc) [pdf, other]
Title: Preinflationary dynamics in loop quantum cosmology: Monodromy Potential
Comments: revtex4, seven figures and five tables
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th)
In this article we explore the pre-inflationary background dynamics of an FLRW universe sourced by a scalar field with monodromy potential in LQC framework. In particular we calculate the number of e-folds, $N_{inf}$, produced during the slowly rolling phase of the inflation and find out the critical value of the ratio of the kinetic to potential energy, $r_w^c$, at the quantum bounce that is required to produce $N_{inf}\simeq 60.$ Two different monodromy potentials, namely, linear and quadratic with a modulation term are investigated to this effect. The effects on the value of $N_{inf}$ due to parameters associated with the strength, decay constant and the phase factor of the modulation term are calculated. In addition to this we present the qualitative picture of the background dynamics by carrying out a dynamical system analysis. We produce the phase portraits and carry out a detailed linear stability analysis of the finite fixed points, if any, for each of the potentials.
### Replacements for Thu, 16 Aug 18
[21] arXiv:1502.03262 (replaced) [pdf, ps, other]
Title: Hypothesis about semi-weak interaction and experiments with solar neutrinos
Subjects: High Energy Physics - Phenomenology (hep-ph); Solar and Stellar Astrophysics (astro-ph.SR); High Energy Physics - Experiment (hep-ex); Nuclear Theory (nucl-th)
[22] arXiv:1702.02875 (replaced) [pdf, other]
Title: The pressure of a weakly magnetized hot and dense deconfined QCD matter in one-loop Hard-Thermal-Loop perturbation theory
Comments: 44 pages, 7 figures; Revised substantially from the previous version; One new author inducted; Title is little changed;
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Lattice (hep-lat); Nuclear Theory (nucl-th)
[23] arXiv:1708.08462 (replaced) [pdf, other]
Title: Nucleon Axial Radius and Muonic Hydrogen - A New Analysis and Review
Comments: 36 pages, 7 figures. v2: improved discussion of radiative corrections with reduced uncertainty; updated values for V_{ud} and g_A; references added; version to appear in Rept. Prog. Phys. v3: included missing sentence in g_A discussion on page 12 to make arxiv and published versions identical
Journal-ref: Rept.Prog.Phys. 81 (2018) no.9, 096301
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Lattice (hep-lat); Nuclear Experiment (nucl-ex); Nuclear Theory (nucl-th)
[24] arXiv:1708.09514 (replaced) [pdf]
Title: A hypothetical effect of the Maxwell-Proca electromagnetic stresses on galaxy rotation curves
Subjects: Astrophysics of Galaxies (astro-ph.GA); High Energy Physics - Phenomenology (hep-ph); Plasma Physics (physics.plasm-ph)
[25] arXiv:1709.09193 (replaced) [pdf, ps, other]
Title: New tracker models of dark energy
Comments: 24 pages, 16 figures, matches published version in JCAP
Journal-ref: JCAP 08 (2018) 009
Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th)
[26] arXiv:1712.05366 (replaced) [pdf, other]
Title: Non-universal Z' from SO(10) GUTs with vector-like family and the origin of neutrino masses
Comments: Revised version, published in NPB. New material, general conclusions unchanged. 30 pages, 4 figures, 2 tables
Subjects: High Energy Physics - Phenomenology (hep-ph)
[27] arXiv:1712.07160 (replaced) [pdf, other]
Title: Collider phenomenology of Hidden Valley mediators of spin 0 or 1/2 with semivisible jets
Comments: 40 pages, 11 figures, published version
Subjects: High Energy Physics - Phenomenology (hep-ph)
[28] arXiv:1801.08791 (replaced) [pdf, ps, other]
Title: Strong decays of higher charmonium states into open-charm meson pairs
Comments: 20 pages, 2 figure, and 9 tables, published version
Journal-ref: Phys. Rev. D 98, 016010 (2018)
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex)
[29] arXiv:1803.01636 (replaced) [pdf, ps, other]
Title: The first $Δ(27)$ flavor 3-3-1 model with low scale seesaw mechanism
Comments: 13 pages, 4 figures. Revised version. Text improved
Subjects: High Energy Physics - Phenomenology (hep-ph)
[30] arXiv:1803.02745 (replaced) [pdf, other]
Title: Distribution of primordial black holes and 21cm signature
Subjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph)
[31] arXiv:1803.07045 (replaced) [pdf, other]
Title: Non-global logarithms in jet and isolation cone cross sections
Comments: 39 pages, 13 figures. v2: journal version with new result (4.18) for narrow isolation cones
Subjects: High Energy Physics - Phenomenology (hep-ph)
[32] arXiv:1804.07407 (replaced) [pdf, other]
Title: NLO QCD corrections to SM-EFT dilepton and electroweak Higgs boson production, matched to parton shower in POWHEG
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex)
[33] arXiv:1804.11018 (replaced) [pdf, ps, other]
Title: DAMA annual modulation from electron recoils
Authors: R. Foot
Subjects: High Energy Physics - Phenomenology (hep-ph); Astrophysics of Galaxies (astro-ph.GA)
[34] arXiv:1805.05030 (replaced) [pdf, other]
Title: Breaking flavor democracy with symmetric perturbations
Comments: 13 pages, 2 figures, 2 tables. Version to appear in PLB
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex)
[35] arXiv:1805.09024 (replaced) [pdf, other]
Title: Dynamically integrated transport approach for heavy-ion collisions at high baryon density
Comments: 8 pages, 8 figures, new figures added; published in Physical Review C
Journal-ref: Phys. Rev. C 98, 024909 (2018)
Subjects: Nuclear Theory (nucl-th); High Energy Physics - Phenomenology (hep-ph); Nuclear Experiment (nucl-ex)
[36] arXiv:1805.11322 (replaced) [pdf, other]
Title: Neutralino/chargino pair production at NLO+NLL with resummation-improved PDFs for LHC Run II
Authors: J. Fiaschi, M. Klasen
Comments: 14 pages, 5 figures, 4 tables
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex)
[37] arXiv:1806.00660 (replaced) [pdf, ps, other]
Title: Flavourful Axion Phenomenology
Comments: v1: 27 pages. v2: version to appear in JHEP. 31 pages. Extended discussion on axion basis, updated mixing bounds
Subjects: High Energy Physics - Phenomenology (hep-ph)
[38] arXiv:1806.04447 (replaced) [pdf, ps, other]
Title: The doubly charmed pseudoscalar tetraquarks $T_{cc;\bar{s} \bar{s}}^{++}$ and $T_{cc;\bar{d} \bar{s}}^{++}$
Comments: 9 Pages, 4 Figures and 1 Table
Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex); High Energy Physics - Lattice (hep-lat)
[39] arXiv:1806.06324 (replaced) [pdf, other]
Title: Some diagrams and basic formalism to the low energy kaon-hyperon interaction
Subjects: High Energy Physics - Phenomenology (hep-ph)
[40] arXiv:1806.09367 (replaced) [pdf, other]
Title: Formation of Relativistic Axion Stars
Subjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th)
[41] arXiv:1806.10107 (replaced) [pdf, ps, other]
Title: Neutron Dark Matter Decays
Comments: 22 pages, 8 figures, the paper is substantially revised. We propose a gauge invariant quantum field theory model with SU_L(2)\times U_R(1) \times U_R'(1)\times U''_L(1) symmetry for the UV completion of the effective (n\chi \ell \bar{\ell}) interaction
Subjects: High Energy Physics - Phenomenology (hep-ph); General Relativity and Quantum Cosmology (gr-qc); Nuclear Experiment (nucl-ex)
[42] arXiv:1807.02211 (replaced) [pdf, other]
Title: Scalar resonance in a top partner model
Subjects: High Energy Physics - Phenomenology (hep-ph)
[43] arXiv:1807.03801 (replaced) [pdf, other]
Title: Displaced Vertices from Hidden Glue | {"extraction_info": {"found_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": 0.9262011051177979, "perplexity": 2282.257180643032}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1534221210559.6/warc/CC-MAIN-20180816074040-20180816094040-00175.warc.gz"} |
https://discuss.pytorch.org/t/usage-of-cross-entropy-loss/14841 | # Usage of cross entropy loss
Is cross entropy loss good for multi-label classification or for binary-class classification?
Please also tell how to use it?
``````criterion = nn.CrossEntropyLoss().cuda()
loss = criterion(input,tgt)
``````
Tried above, but got error
TypeError: FloatClassNLLCriterion_updateOutput received an invalid combination of arguments - got (int, torch.FloatTensor, torch.FloatTensor, torch.FloatTensor, bool, NoneType, torch.FloatTensor), but expected (int state, torch.FloatTensor input, torch.LongTensor target, torch.FloatTensor output, bool sizeAverage, [torch.FloatTensor weights or None], torch.FloatTensor total_weight)
2 Likes
Have a look at the documentation of CrossEntropyLoss.
It states:
It is useful when training a classification problem with C classes.
The error message gives you a hint, that some types are wrong.
You should pass the target as a `LongTensor`.
Try changing the `tgt` to:
``tgt = torch.autograd.Variable(torch.LongTensor(3).random_(5))``
3 Likes
How then it is showing multi-class classification in this case as 3 can be assumed as number of examples and 5 can be number of classes.
Each input row can be interpreted as probability to map to that corresponding class?
What if I have multiple classes? How to write that as vector?
I’m confused a bit. Do you mean multiclass classification or multi-label classification?
`CrossEntropyLoss` is used for multiclass classification, i.e. predict one of several classes for each example.
For multi-label classification, there are some losses like MultiLabelMarginLoss.
4 Likes
Sorry, I meant multi-label classification.
Can you tell how can I define the accuracy function for above problem?
My label vector has ones at the classes which are there in the feature.
Sorry, I haven’t used `MultiLabelMarginLoss` yet and would have to get familiar with it, before posting a wrong approach.
However, for multi-label classification, you could use a `sigmoid` in your last layer and feed it to `BCELoss`:
``````x = Variable(torch.randn(10, 3))
output = F.sigmoid(x)
target = Variable(torch.Tensor(10, 3).random_(2))
criterion = nn.BCELoss(reduce=False)
loss = criterion(output, target)
print(loss)
``````
3 Likes
reduce = False causing error?
TypeError: init() got an unexpected keyword argument ‘reduce’
What can be the cause and its need?
I just used it, so that you can see the loss of each sample instead of the mean or sum. You can safely skip this argument.
However, which Pytorch version are you using? I would suggest to update it, since the newer versions have some nice features and bug fixes.
0.1.12_1
Thanks a lot. @ptrblck
Oh yeah, you should definitely update
You can find the install instructions on Pytorch.org.
1 Like
Can you suggest how can I write the accuracy function for multilabel classification?
You could use the hamming loss or “hamming score”:
``````
target = torch.FloatTensor([[0, 1, 0],
[1, 1, 1],
[0 ,0 ,0]])
pred = torch.FloatTensor([[0, 1, 1],
[1, 1, 1],
[0 ,1 ,0]])
hamming_score = 1 - (target != pred).sum() / float(target.nelement())
``````
Scikit provides other metrics like Jaccard similarity coefficient.
Would this work for you?
2 Likes
How shall I saturate my outputs? Basically my outputs are some figures which are like prob of lying in that label if I do a softmax.
But after softmax how shall I do thresholding to assign them to 0 or 1?
Please feel free to question back if I am not clear.
I would threshold the `output` to get the predictions. There might be some “accuracy” metrics for probabilities which I’m not aware of, though.
I suppose you are using `sigmoid` instead of `softmax`.
In sigmoid, can we use 0.5 as the thresholding?
Sure, you could also tune it to favor some classes, if that’s important in your use case.
Do you know what is normally done in such cases?
Go for `0.5` and see if your score is good enough.
If you have an imbalanced dataset, I also compute the confusion matrix and sometimes the Cohens Kappa.
1 Like
So I have an image of size 9x50 as:
tensor([[ 0.4115, -0.6465, -1.6343, 0.6694, -0.8929, 0.7482, -0.6784, -1.2556,
-0.9919, 0.7736, -1.3033, -1.4822, 1.6883, 1.3857, -0.4635, -0.4117,
0.1361, 1.2751, 1.5286, -1.0493, 0.4839, -2.1620, -1.4373, -0.3013,
0.5121, 0.7913, 0.7924, -0.7720, -0.3467, 1.1353, 0.5904, -1.8757,
0.5789, -2.0829, 1.2716, -0.2533, -0.6339, 0.5726, -0.1584, 1.2937,
-0.6060, -0.7181, -1.1443, 0.1927, 0.0326, -1.3743, -0.5325, 0.7743,
-1.0776, 0.5832],
[-0.4022, -0.0806, 0.6202, 1.4176, -0.0325, 0.2146, 0.4789, 0.2615,
-1.9354, -0.9925, -1.3699, 1.4623, 1.1422, 0.4273, 0.7865, 0.4704,
0.7516, -0.8715, -0.7594, -0.3551, 0.6217, 1.5333, -1.7359, 0.7198,
-0.4480, 0.4198, 0.5431, 0.2605, -0.5880, -0.3684, 0.5031, -1.3644,
0.3791, 0.4395, -0.0098, -0.3250, -1.9895, 0.5293, 0.5274, 1.5332,
1.0197, -1.1839, 0.2819, 1.7081, 0.1653, 0.3076, -1.0679, -0.5644,
2.5712, -0.6777],
[ 0.3608, 0.7212, -1.5474, 1.0859, -0.5586, 1.3594, -1.2196, -1.5036,
0.8116, 0.6708, 0.9988, -0.7967, -0.7120, 0.5176, -1.9599, 0.2420,
0.0513, -1.1133, 0.6954, -0.4826, -1.5786, 0.1810, 0.7230, 0.4276,
-0.2598, 0.4369, -0.3106, -0.0446, 1.1185, -0.7355, 0.0219, -0.0619,
0.0329, 0.1079, 0.2461, 0.7204, 1.0873, -1.1423, 0.0986, -0.6493,
1.1245, -0.8159, 1.3520, -0.8926, 0.4020, 1.0555, -1.1234, -0.0147,
1.3508, 0.6182],
[-0.7430, 0.5251, -0.6153, -0.0003, -0.6046, 1.1388, -0.7799, -1.9012,
-0.4144, -0.0861, 0.0823, 0.6609, 1.0585, -0.5026, -0.1830, 0.8965,
0.1796, 0.7578, 0.2869, -1.3962, -1.7420, 1.7718, 1.6606, 0.5634,
-0.1225, 0.5426, -2.1004, 0.0133, 0.7839, 1.8201, -0.0306, 0.2149,
-0.2372, -0.3642, 0.3713, 0.1301, -0.2877, -0.4470, -0.1347, -1.3249,
0.6950, 0.0947, -0.0682, -0.3107, -0.5063, -0.1554, 0.5312, 0.2986,
-0.7677, 0.9213],
[ 1.1000, -0.6128, -0.6937, -0.9583, 0.2561, -0.0408, 0.5273, -0.1111,
-0.3420, 0.9789, -0.5763, -0.3564, -1.1349, 0.2419, 1.0597, 1.3880,
0.3580, -1.2515, -0.1734, 0.2403, -0.2600, 0.4373, -0.5632, 0.5021,
1.9840, -0.5519, -1.5868, 1.2105, 1.0267, 1.4813, -1.5021, 1.6625,
-0.9624, 1.4024, 2.0388, 0.0238, 0.3076, 1.6528, -0.4595, -0.7159,
-0.8997, -1.8804, -1.1647, 1.8108, -1.4731, -1.1084, 0.5496, 1.5376,
0.1698, -0.4175],
[-0.7766, -2.0425, -0.8977, 0.0425, 1.8165, -0.6411, 0.1768, -0.7219,
-0.4880, -0.4142, 0.7928, -0.5951, 1.1639, -0.0928, -0.3169, 1.5937,
-1.1871, 0.2590, -0.1274, 0.1017, -1.0488, 0.1753, 0.5793, 0.1125,
-0.4837, -1.4312, 0.0187, -0.6604, -0.3871, 1.6479, -1.4328, 0.9142,
0.0699, 0.8660, 1.0728, -0.8291, 1.0222, 0.1272, -0.5531, 0.8532,
0.5304, 0.4040, -0.7247, 0.1954, -0.2499, 0.9694, 0.8410, -0.1247,
-1.5646, -1.3319],
[-0.2229, -1.6662, 0.5105, -0.2770, 0.3966, 1.0326, 0.9928, -0.4494,
0.6234, 0.4386, -0.6726, 1.1923, 1.1223, -0.5312, 0.2890, -0.8353,
-1.3872, -0.2604, 1.7785, 0.2281, 0.8691, 0.8132, -0.0213, -1.0649,
0.3980, -0.2038, 1.5023, 0.3054, 0.8736, 1.8556, -1.3965, 1.0579,
-0.0868, -0.3515, -1.2344, -0.2689, 1.1425, 0.1928, 1.0721, -1.5331,
-0.2131, 1.0340, 1.6211, 0.2218, 1.7555, 0.3581, 2.6108, -0.1747,
0.1864, 0.0211],
[-2.1773, 0.4278, 0.2847, 0.4405, 0.9457, -0.1819, -0.3713, 1.0402,
-0.9497, -0.0645, 0.1729, -0.6848, 0.2156, -0.0078, 0.3848, -0.4249,
1.2975, -0.4167, 0.0660, 1.6326, -0.4543, 0.7339, 0.6010, 0.8946,
1.2881, -1.0936, 1.1421, -0.5225, 0.1843, -1.0033, 0.1155, -0.4692,
1.5356, 0.1045, -1.0899, 2.0136, 1.7887, 2.1656, -1.2265, -0.0519,
0.0472, 0.2626, -0.5554, 1.6628, -1.0357, 0.4898, 1.1277, 0.0699,
0.4967, 0.8722],
[ 0.7352, -0.6486, -0.6952, 2.6622, 0.2339, -0.0961, 1.8036, -0.3650,
-1.2539, -0.0111, 0.6007, -0.3418, -0.9551, 0.3020, 1.3864, -0.0676,
0.8362, 0.1694, -0.5506, 0.4202, 0.2058, -0.9739, 1.5484, -1.0143,
-0.7052, -0.2831, 0.7834, -3.0195, 0.3679, 0.9377, 0.0174, -1.4630,
-0.4082, -0.5332, 0.8701, -0.5404, 1.8485, -1.9600, -0.1757, -0.1020,
-0.1524, -1.8317, -0.0961, 1.1949, -1.2083, 1.7236, -0.2691, 0.9958,
and it has labels of size 9x1 like so,
[3, 3, 7, 0, 8, 3, 3, 3, 1],
between 0-8 (no one hot encoding)
which means each row of 50 dimensions in this image has a label. To summarize,
image
9x50 --> 9x1
and, we have a batch size of 128.
So, 128x9x50 maps to 128x9.
Since, each image has 9 labels, can we apply 2d crossentropy? Am not sure how to use that though.
OR , do you suggest some other loss function? What about the accuracy calculation? I guess I may not be able to use hamming distance since am not using one-hot-encoded label vectors. my labels are class indexes
Regards
I tried to use crossentropy loss for video generation but it does not work.
The input dimension is the same as target dimension but crossentropy loss expects the target to be of 1-lower dimension. How should I fix that? | {"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": 0.8114995956420898, "perplexity": 1885.2265544459897}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662512229.26/warc/CC-MAIN-20220516172745-20220516202745-00090.warc.gz"} |
https://www.physicsforums.com/threads/properties-of-sound.307676/ | # Properties of Sound
1. Apr 15, 2009
### whitehorsey
1. Carol drops a stone into a mine shaft 122.5 m deep. How soon after she drops the stone does she hear it hit the bottom of the shaft?
2. d= vt
3. d=vt
t = d/v
= (122.5/343)2
= .7143s
I'm not sure if i multiply by 2 or divide by 2 or just not put a 2 there, but i multiplied because i think the sound hits the bottom then goes back up to where Carol is. So that would double the time?
2. Apr 15, 2009
### rl.bhat
t = d/v is the time taken by the sound from bottom of the mine to reach Carol. But you have not taken the time required by the stone to travel from Carol to bottom of the mine.
3. Apr 15, 2009
### whitehorsey
so i add (122.5/343) to the 0.7143s?
4. Apr 15, 2009
### rl.bhat
No. That is not the time. Use kinematics equation to find the time of fall of the stone.
5. Apr 15, 2009
### whitehorsey
so i use this eq. d =vit + 1/2at2?
getting t2 = 122.5/.5(9.8)
= 5 s
thus all togther would be 5 + .7143 = 5.7143 s
6. Apr 16, 2009
### 3.211k
If time is distance divided by velocity, what does the "2" signify when you multiply it in the third line here?
7. Apr 16, 2009
### rl.bhat
No. It should be 5s + 0.35715s
8. Apr 19, 2009
### whitehorsey
oh i see thank you!!
Similar Discussions: Properties of Sound | {"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": 0.8195230960845947, "perplexity": 3968.5297637909202}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934803848.60/warc/CC-MAIN-20171117170336-20171117190336-00009.warc.gz"} |
https://www.emathhelp.net/pt/notes/pre-algebra/irrational-numbers/nth-root/ | # Nth Root
Similarly to square root and cube root, we can define nth root.
Nth root of a number $b$ is such number $a$, that $a^n=b$.
Notation for the nth root is the following: $\color{purple}{\sqrt[n]{b}}$.
Square root is the 2nd root, which is $\sqrt{b}$ (we just don't write 2).
Cube root is 3rd root, which is ${\sqrt[{{3}}]{{{b}}}}$.
Fourth root is ${\sqrt[{{4}}]{{{b}}}}$.
And so forth.
Nth root symbol $\color{purple}{\sqrt[n]{\phantom{0}}}$ is a radical with a small $n$.
This is done to emphasize the fact, that we are looking for a number, that when raised to nth power) will give original number.
Let's go through a couple of examples.
Example 1. Find ${\sqrt[{{4}}]{{{2401}}}}$.
It is known that ${{7}}^{{4}}={2401}$, so ${\sqrt[{{4}}]{{{2401}}}}={7}$.
If ${n}$ is odd, then we can find nth root of negative number as well.
Example 2. Find ${\sqrt[{{5}}]{{-{32}}}}$.
Since ${{\left(-{2}\right)}}^{{5}}=-{32}$, then ${\sqrt[{{5}}]{{-{32}}}}=-{2}$.
Is it possible, to find nth root of a number if ${n}$ is even?
Example 3. Find ${\sqrt[{{8}}]{{-{256}}}}$.
What is ${\sqrt[{{8}}]{{-{256}}}}$. It is such number ${a}$, that ${{a}}^{{8}}=-{256}$.
But can we really find such number, that when raised to even power, will give negative number?
Answer is no, because any number, raised to even power will give positive number.
For, example ${{\left(-{3}\right)}}^{{4}}={\left(-{3}\right)}\cdot{\left(-{3}\right)}\cdot{\left(-{3}\right)}\cdot{\left(-{3}\right)}={81}$ (there is even number of minuses, so they just disappear).
As a result, we have the following useful fact.
Nth root of negative number (if ${n}$ is even) doesn't exist.
Finally, notice that nth root undoes raising to nth power (if we take nth root of a number and then raise the result to nth power, we will get back to the original number), and vice versa (with a slight modification):
${\color{red}{{{{\left({\sqrt[{{n}}]{{{b}}}}\right)}}^{{n}}={b}}}}$
${\color{green}{{{\sqrt[{{n}}]{{{{b}}^{{n}}}}}={b}}}}$, if ${n}$ is odd
${\color{purple}{{{\sqrt[{{n}}]{{{{b}}^{{n}}}}}={\left|{b}\right|}}}}$, if ${n}$ is even
Did you notice absolute value? Why is that?
Because raising any number to even number will give positive number, and taking nth root of the result will also give positive number. Absolute value guarantees, that we wil have positive number.
We could write, that ${{\left({\sqrt[{{n}}]{{{b}}}}\right)}}^{{n}}={\left|{b}\right|}$, but that is not necessary, because if ${n}$ is even, ${b}$ is already non-negative number (n-th root of negative number doesn't exist, if ${n}$ is even).
Let's go through a couple of example.
Example 1. ${{\left({\sqrt[{{4}}]{{{15}}}}\right)}}^{{4}}={15}$.
Example 2. ${{\left({\sqrt[{{3}}]{{-{2}}}}\right)}}^{{3}}=-{2}$.
Example 3. ${{\left({\sqrt[{{8}}]{{-{5}}}}\right)}}^{{8}}$ doesn't exist.
Example 4. ${\sqrt[{{3}}]{{{{\left({\color{red}{{-{0.2}}}}\right)}}^{{3}}}}}={\sqrt[{{3}}]{{-{0.008}}}}={\color{red}{{-{0.2}}}}$.
Example 5. ${\sqrt[{{10}}]{{{{\left({\color{red}{{-{2}}}}\right)}}^{{10}}}}}={\sqrt[{{10}}]{{{1024}}}}={2}={\left|{\color{red}{{-{2}}}}\right|}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8739842176437378, "perplexity": 535.462073941965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662546071.13/warc/CC-MAIN-20220522190453-20220522220453-00376.warc.gz"} |
http://mathoverflow.net/questions/133666/finding-primes-using-eulers-sum-of-divisors-recurrence-relation | Finding primes using Euler's sum of divisors recurrence relation
Euler came up with following recurrence relation for the sum of divisors (refer to http://arxiv.org/abs/math/0411587) $$\sigma(n) = \sigma(n−1) + \sigma(n−2) − \sigma(n−5) − \sigma(n−7) \dots$$
Since $\sigma(p) = p+1$, where $p$ is a prime number, we can use the recurrence relation to verify if a number is prime. I'm wondering how efficient it is to use this method to find a prime, or more specifically to build up all the primes up to a number. It seems pretty fast to add a few numbers, especially the numbers subtracted in the recurrence relation are increasing quadratically?
(I've also asked this question on math.stackexchange: http://math.stackexchange.com/questions/419059/eulers-sum-of-divisors-recurrence-relation)
-
We would need a lot of values $\sigma(j)$ for larger $j$. Try to do it , say, for $n=2^{9941}-1$ with your computer. – Dietrich Burde Jun 13 '13 at 18:02
Then I suppose the sieve methods are a lot more efficient? – JoeS Jun 13 '13 at 18:05
Yes, the general number field sieve will be much better. – Dietrich Burde Jun 13 '13 at 18:12
If this formula would have been efficient to find primes, then no doubt Euler would have known about it -- even if there was no computer at the time. – François Brunault Jun 13 '13 at 21:25
Kayu, "add a few numbers"? Sure, but first you have to calculate those numbers. You can't calculate $\sigma(n-1)$ without factoring $n-1$, and factoring $n-1$ is likely to be harder than using, say, APR to test $n$ for primality, so even calculating the first number in your sum is likely to be highly inefficient as part of a primality test. – Gerry Myerson Jun 14 '13 at 3:47
Perhaps a summary of the comments is useful. Using Euler's formula for $\sigma(n)$ to test $n$ for primality certainly is not efficient. The values $\sigma(n-1),\sigma(n-2),\sigma(n-5),\cdots$ are hard to compute.
More precisely, in general computing $\sigma(n)$ is equivalent to factoring $n$ in the following sense:
1. Given the factorization of $n$, $\sigma(n)$ can be computed in polynomial time. (This follows immediately from the formula for $\sigma(n)$, given a prime factorization of $n$).
2. Given $\sigma(n)$, we can produce the factorization of $n$ in random polynomial time. (This has been proved by Bach, Miller, Shallit 1986).
The AKS-primality test shows that cheking primality can be done in polynomial time. About the growth of $\sigma(n)$, it holds $\lim \sup_{n\to \infty}\frac{\sigma(n)}{n\log (\log(n))}=e^{\gamma}$, and assuming the Riemann hypothesis, there is the estimate (Robin) $\sigma(n)< e^{\gamma}n\log(\log(n))$ for all $n\ge 5041$.
-
I agree with everything that's been said about Euler's formula not being a practical way of testing for primality, but it occurs to me there might be special numbers for which it could be useful. Indulge me in a laborious "proof" that $n=82$ is not a prime.
Euler's formula in this case says
\begin{align} \sigma(82) = &\sigma(81) + \sigma(80) - \sigma(77) - \sigma(75) + \sigma(70) + \sigma(67) - \sigma(60) - \sigma(56)+\cr &\sigma(47) + \sigma(42) - \sigma(31) -\sigma(25)+\sigma(12)+\sigma(5)\cr \end{align}
As it happens, "most" of the numbers inside the $\sigma$s on the right hand side factor into "small" primes, where by "small" I mean up to $11$. In particular, we have \begin{align} \sigma(81)&=\sigma(3^4)=121\cr \sigma(80)&=\sigma(2^4)\sigma(5)=186\cr \sigma(77)&=\sigma(7)\sigma(11)=96\cr \sigma(75)&=\sigma(3)\sigma(5^2)=124\cr \sigma(70)&=\sigma(2)\sigma(5)\sigma(7)=144\cr \sigma(60)&=\sigma(2^2)\sigma(3)\sigma(5)=168\cr \sigma(56)&=\sigma(2^3)\sigma(7)=120\cr \sigma(42)&=\sigma(2)\sigma(3)\sigma(7)=96\cr \sigma(25)&=\sigma(5^2)=31\cr \sigma(12)&=\sigma(2^2)\sigma(3)=28\cr \sigma(5)&=6\cr \end{align}
When you add and subtract all this stuff up, you have
$$\sigma(82)=42+\sigma(67)+\sigma(47)-\sigma(31)$$
Now we're not allowing ourselves to know that $67$, $47$, and $31$ are primes, but we do know that $\sigma(n)\ge n+1$ for all $n$. Therefore we have
$$\sigma(82)\ge 42+ 68 + 48 - 32 -\text{stuff} = 126-\text{stuff},$$
where "$\text{stuff}$" is the sum of the divisors (if any) of $31$ other than $1$ and $31$. These divisors must come in pairs $d,31/d$, with $d$ odd and less than $\sqrt{31}$. Thus
$$\text{stuff} \le 3+{31\over3}+5+{31\over5} = 24.5333\ldots,$$
and hence
$$\sigma(82) \gt 126-24=112 \gt 83,$$
from which we can conclude that $82$ is not prime.
The obvious drawback to such a "proof" is that it requires an awful lot of computation: There are always $O(\sqrt n)$ terms to deal with, so its only advantage over trial-and-error division is that you can hope to avoid doing trial divisions by "large" primes. It also requires a considerable amount of luck -- in this case we were left subtracting the $\sigma$ of only one number that was "too big" to factor, and even it was fairly small. But still, there might be some rare values of $n$ for which one can deduce something nontrivial from Euler's formula without ever dividing by anything other than "small" primes.
-
That was very fun to read. Thanks! – JoeS Jun 15 '13 at 1:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9090426564216614, "perplexity": 362.7206079925378}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049279525.46/warc/CC-MAIN-20160524002119-00173-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://madhavamathcompetition.com/2018/05/21/solution-to-a-nice-analysis-question-for-rmo-practice/ | # Solution to a “nice analysis question for RMO practice”
The question from a previous blog is re-written here for your convenience.
Question:
How farthest from the edge of a table can a deck of playing cards be stably overhung if the cards are stacked on top of one another? And, how many of them will be overhanging completely away from the edge of the table?
Solution:
The figure below shows how two and three cards can be stacked so that the mass of cards is equal on either side of the vertical line passing through the corner of table’s edge in order to just balance them under gravity:
the set of first two cards are arranged as follows (the horizontal lines represents the cards):
$xxxxxxxxxxxxxxxx\line(5,0){170}$
$\line(5,0){150}xxxxxxxxxxxxxxxxxxx$
the set of three cards are arranged as follows:
$xxxxxxxxxxxxxxxxxxxxxx\line(5,0){150}$
$xxxxxxxxxxxxx\line(5,0){150}$
$\line(5,0){150}xxxxxxxxxxxxxxxxxxxxxx$
We can see that the length of the overhand is a harmonic series of even numbers multiplied by the length of one card, L.
Overhand distance is $(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \ldots + \frac{1}{52 \times 2})L$ for 52 cards.
It may be noted that the series if continued to infinity leads to $H_{\infty}^{E}$.
That is, $H_{\infty}^{E}=\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \ldots$
This series is known to diverge as proved below:
First consider, $H_{\infty}=1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}+ \ldots$, which is, greater than
$1+ \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}+ \ldots$, which is greater than $1+ \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \ldots$. Hence, $H_{\infty}$ diverges as we go on adding 1/2 indefinitely.
Now, let $H_{E}=\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \ldots = \frac{1}{2}(1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots)=\frac{1}{2}H_{\infty}$
Since $H_{\infty}$ diverges, $H_{E}$also diverges.
Hence, the “overhang series” also diverges.
This means that the cards can be stacked indefinitely and the overhang distance can reach infinity. However, this will happen very slowly as shown in the table below:
$\begin{array}{cc} n^{E} & H_{n}^{E}\\ 2 & 0.5 \\ 10 & 1.46 \\ 100 & 2.59 \\ 1000 & 3.74 \\ 10000 & 4.89 \\ 100000 & 6.05 \end{array}$
Computing the number of cards that completely overhang off the table needs information about the overhang distance for different number of cards. As shown below in the figure, four cards are required to have one card completely away from the edge of the table. This is because $(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8}=1.0417 >1)$.
(the set of four cards are arranged as follows:)
$xxxxxxxxxxxxxxxxxxxxxxxxxxxx\line(5,0){150}$
$xxxxxxxxxxxxxxxx\line(5,0){150}$
$xxxxxxxxxx\line(5,0){150}$
$xxxxx\line(5,0){150}$
We can see that the length of the overhang is a harmonic series of even numbers multiplied by the length of one card, L:
Overhang distance = $(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \ldots + \frac{1}{52 \times 2})L$ for 52 cards
It may be noted that the series if continued to infinity, leads to $H_{\infty}^{E}$
$H_{\infty}^{E} = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \ldots$
This series is known to diverge. This means that the cards can be stacked indefinitely and the overhang can reach infinity. However, this will happen very slowly as shown in the table below:
$\begin{array}{cc} n^{E} & H_{\infty}^{E} \\ 2 & 0.5 \\ 10 & 1.46 \\ 100 & 2.59 \\ 1000 & 3.74 \\ 10000 & 4.89 \\ 100000 & 6.05 \end{array}$
Computing the number of cards that completely overhang off the table needs information about the overhang distance for different numbers of cards. As shown in the above schematic figures of cards with overhangs, four cards are required to have one card completely away from the edge of the table. This is because
$(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8})=1.0417>1$
For the second card to overhang completely, leaving the first card (and hence one half) that is already completely overhung, it is now necessary that
$(\frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \ldots + \frac{1}{2n})>1$, or
$(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \ldots + \frac{1}{2n} )>1+ \frac{1}{2}$
where n needs to be found out. By generating some more data, we can find the value of n to be 11.
For third overhanging card, we need
$(\frac{1}{6} + \frac{1}{8} + \ldots + \frac{1}{2n})>1$ or
$(\frac{1}{2} + \frac{1}{4} + \frac{1}{6}+\frac{1}{8}+ \ldots + \frac{1}{2n})>1+\frac{1}{2} + \frac{1}{4}$
Thus, for m completely overhanging cards, we find n such that $H_{2n}^{E} > 1+ H_{2(m-1)}^{E}$
The table below shows these values wherein we see an approximate pattern of arithmetic progression by 7.
$\begin{array}{cccc} m & n & m & n \\ 1 & 4 & 11 & 78 \\ 2 & 11 & 12 & 85 \\ 3 & 19 & 13 & 92 \\ 4 & 26 & 14 & 100 \\ 5 & 33 & 15 & 107 \\ 6 & 41 & 16 & 115 \\ 7 & 48 & 17 & 122 \\ 8 & 55 & 18 & 129 \\ 9 & 63 & 19 & 137 \\ 10 & 70 & 20 & 144 \end{array}$
By examining the pattern in the table, we can get a simple rule to estimate the number of completely overhanging number of cards m, with an error of utmost one, for n cards stacked.
$m = round(\frac{n}{7.4})=round(\frac{10n}{74})$.
Reference:
Popular Problems and Puzzles in Mathematics by Asok Kumar Mallik, IISc Press, Foundation Books.
Hope you enjoyed the detailed analysis…
More later,
Nalin Pithwa
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https://brilliant.org/problems/physics-19/ | # An algebra problem by Ankit vijay
Algebra Level 1
If the sum of the first n terms of an AP is given by $${ S }_{ n }={ n }^{ 2 }+3n$$, then the first term of the AP is :
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https://math.stackexchange.com/questions/555698/how-to-prove-continuity-in-multivariable-functions | # how to prove continuity in multivariable functions?
$$\frac{x^3y}{x^4+y^2},etc.,$$ in these multi-variable functions its easy to prove discontinuity by giving counterexamples but for proving continuity are there any tricks?
using $\epsilon$ definitions seems to so tougher in multivariable functions compared to single variable calculus
• say at which point u can use epsilon definition also – user106301 Nov 8 '13 at 1:40
You're interested in continuity at the point of $$\left ( 0,0 \right )$$.
Use the squeeze theorem $$\left| \frac{x^3y}{x^4+y^2} \right| \le \left| \frac{x^3y}{2x^2y} \right| = \frac{1}{2}\left|x \right| \to 0$$ for $$\left ( x,y \right) \to \left ( 0,0 \right)$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9445143938064575, "perplexity": 806.0334201030294}, "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-2021-17/segments/1618038098638.52/warc/CC-MAIN-20210417011815-20210417041815-00472.warc.gz"} |
http://theblogreaders.com/flipkart-placement-paper-for-freshers-part-2/ | Sunday, June 24th, 2018
# Flipkart Placement paper For Freshers Part-2
Be First!
by February 22, 2016
11. 13 sheeps and 9 pigs were bought for Rs. 1291.85. If the average price of a sheep be Rs. 74. What is the average price of a pig?
12. A batsman in his 18th innings makes a score of 150 runs and there by increasing his Average by 6. Find his average after 18th innings?
13. Find the H.C.F of 777 and 1147?
14. The L.C.M of two numbers is 2310 and their H.C.F is 30. If one number is 210, the other number is?
15. The average of 50 numbers is 38. If two numbers namely 45 and 55 are discarded, The average of remaining numbers is?
16. Divide 50 in two parts so that the sum of reciprocals is (1/12), the numbers are?
17. Five years ago the average age of a family of 3 members was 27 years. A child has Been born, due to which the average age of the family is 25 years today. What is the Present age of the child?
18. In a class of 20 students in an examination in Mathematics 2 students scored 100 Marks each, 3 get zero each and the average of the rest was 40. What is the average Of the whole class?
19. The greatest number, which can divide 432, 534 and 398 leaving the same remainder 7 in each. Required number is the H.C.F of?
20. A man traveled a certain distance at the rate of 15 miles an hour and came back at the rate of 10 miles an hour. What is his average speed ?
Answer: 12 miles an hour
(90) | {"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": 0.8655238747596741, "perplexity": 654.4518967950387}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267865995.86/warc/CC-MAIN-20180624005242-20180624025242-00425.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/16160/what-does-the-identity-operator-represent-when-computing-langle-varphii-otime/16161#16161 | # What does the identity operator represent when computing $\langle\varphi|I\otimes Z|\varphi\rangle$?
Consider a single qubit state $$|\varphi\rangle$$ and a hamiltonian $$H = Z$$. Evaluating $$\langle \varphi | H | \varphi \rangle$$ corresponds to a measurement of $$|\varphi\rangle$$ in the computational basis. This is easy to generalise to $$n$$ qubits and any Hamiltonian expressed as a tensor product of Pauli matrices $$X, Z, Y$$.
My question is: what to do when the Hamiltonian is, for example, $$H = I \otimes Z$$. What does $$\langle \varphi | H | \varphi \rangle$$ mean in this case?
When you have $$\langle \varphi | I \otimes Z | \varphi \rangle$$ It means you are calculating the expectation of the operator $$I \otimes Z$$ with respect to some state $$|\varphi \rangle$$. Since $$I$$ is on the first qubit, we would not need to do anything there, no need to do any rotation or even measurement. The eigenspace is decomposed into two halves depending on the second qubit. That is, the states $$\{ |00\rangle, |10\rangle \}$$ belong to the $$+1$$ eigenspace, and the states $$\{| 01\rangle, |11\rangle \}$$ belong to the $$-1$$ eigenspace. Notice how only the second qubit value matter.
Similarly, if we have $$\langle Z \otimes I \rangle$$ instead, then we would measure the first qubit and leave the second qubit alone. In this case, the states $$\{ |00\rangle, |01\rangle \}$$ belong to the $$+1$$ eigenspace, and the states $$\{| 10\rangle, |11\rangle \}$$ belong to the $$-1$$ eigenspace.
If we have $$\langle I \otimes X \rangle$$ then we would do a Hadamard rotation before measuring the second qubit and leave the first qubit alone. | {"extraction_info": {"found_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": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9742187857627869, "perplexity": 124.80839673355253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363189.92/warc/CC-MAIN-20211205130619-20211205160619-00425.warc.gz"} |
https://mathzsolution.com/how-can-one-prove-that-e/ | # How can one prove that e<πe<\pi?
This question is inspired by another one, asking to prove that something approximately equal to $1.2$ is bigger than something approximately equal to $0.9$. The numerical answer to this question was (expectedly) downvoted, though in my opinion it is the most reasonable approach to this kind of problems (${\tiny \text{which I personally find completely useless}}$).
My question will consist of 2 parts:
1. Prove (without calculator) that $e<\pi$;
2. Explain what do we learn from the proof/what makes this problem interesting.
Edit: Existing answers only confirm my point of view about various weird inequalitites. Fortunately there is $3$ between $e$ and $\pi$, otherwise the things would be very boring.
Inscribe a regular hexagon in a circle of radius $1$. Since a straight line is the shortest distance between two points the circumference of the circle is longer than the circumference of the hexagon. We take the definition of $\pi$ as half the circumference of the unit circle.
Putting all this together we obtain $2\pi \gt 6$ or $\pi \gt 3$
We take $e$ as the sum $1+1+\frac 12+\frac 1{3!}+\cdots$ which converges absolutely and which, after the first three terms, is term by term less than the sum $1+1+\frac 12+\frac 1{2^2}+\cdots$ since the later terms in the second sum are obtained by dividing the previous term by $2$, and in the first sum by $n\gt 2$ (crudely for $n\ge 3$ we have $n!\gt 2^{n-1}$).
Summing the geometric series we have $e\lt 3 \lt\pi$.
What do we learn - well how easy it is to make an estimate depends on the definition. The geometric definition of $\pi$ lends itself to a good enough estimate. There are different ways of defining $e$ too, but the sum offers a range of possibilities for estimating, particularly as the terms decrease very quickly. But the geometric definition for $\pi$ requires assumed knowledge about a straight line as the shortest distance between two points, which seems obvious - yet conceals the trickiness of defining the length of a curve - so this looks simpler than it is. | {"extraction_info": {"found_math": true, "script_math_tex": 22, "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": 0.9385321140289307, "perplexity": 112.51668832686406}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337531.3/warc/CC-MAIN-20221005011205-20221005041205-00157.warc.gz"} |
http://media.nips.cc/nipsbooks/nipspapers/paper_files/nips30/reviews/2880.html | NIPS 2017
Mon Dec 4th through Sat the 9th, 2017 at Long Beach Convention Center
Paper ID: 2880 Do Deep Neural Networks Suffer from Crowding?
### Reviewer 1
The paper study the effect of crowding, under particular conditions, on the object recognition performance of deep neural networks. The authors considered different DNN architectures, in particular Convnets and eccentricity-dependent networks, both with different pooling strategies. For the convnets they tried: (i) no pooling (60-54-48-42), (ii) progressive pooling (60-27-11-1), and (iii) at the end pooling (60-54-48-1) and trained with minibatches of 128 images and 32 feature channels. The eccentricity-dependent network constructs a multiscale representation where smaller sections are sample densely at a high resolution and larger sections are sampled with at a lower resolution. The configurations for this type of network were: (i) at the beginning (11-1-1-1-1), (ii) progressively (11-7-5-3-1), and (iii) at the end (11-11-11-11-1). Models were trained on even MNIST digits with/without flankers of different datasets and embedded in a dataset of places. From the results it can be seen that when the models are trained with isolated objects, the recognition rate falls when flankers are added to the image. This is specially true with flankers that are similar to the objects to be recognized. The eccentricity dependent network has better results when the object to recognize is in the center of the image. This is an interesting paper that is testing the robustness of DNNs and helps to explain some degradation in performance when different background objects are shown in testing images.
### Reviewer 2
Crowding is an effect in human vision, in which objects that can be recognized in isolation can no longer be recognized in the presence of nearby objects, even though there is no occlusion. This paper tries to answer the question whether deep neural network suffers from the crowding effect as well as the relationship of crowding to various types of configurations such as pooling, eccentricity, target-flanker similarity, flanker configurations etc. I think this paper is a good attempt of understand crowding of neural networks, but the execution of this work needs major improvements. 1. Most of the result shown in this paper are well expected. For example, neural network could suffer from crowding, and training the network with the same type of flankers can improve the performance. 2. Some of the statements are not well supported by the experiments. For example, line 168 "thus in order for a model to be truly robust to all kinds of clutter, it needs to be trained with all possible target-flanker configurations". This is not well justified, since the experiments are conducted based on 120x and 240x spacing only. 3. The study is purely empirical and the settings studied are quite limited in their scope. For example, it is stated that "we investigate pooling in particular, because some computational models of crowding have suggested that feature integration may partly be the cause of this phenomenon in humans". However, neural networks are very different from human vision. The effect of crowding may be well based on other aspects of the network rather than the pooling operation. 4. I do not see utilities of the proposed method. Towards the end of this paper, "our results suggest that the eccentricity -dependent model coupled with a system for selecting eye fixation location would give benefit of low sample complexity in training". This is a conjecture rather than conclusion.
### Reviewer 3
This paper studies if crowding, a visual effect suffered by human visual systems, happens to deep neural network as well. The paper systematically analyzes the performance difference when (1) clutter/flankers is present; (2) the similarity and proximity to the target; (3) when different architectures of the network is used. Pros: There are very few papers to study if various visual perceptual phenomenon exists in deep neural nets, or in vision algorithms in general. This paper studies the effect of crowding in DNN/DCNN image classification problem, and presents some interesting results which seems to suggest similar effect exists in DNN because of pooling layers merges nearby responses. And this is related to the theories of crowding in humans, which is also interesting. The paper also suggests what we should not do prematurely pooling if when designing architectures. In my opinion such papers should be encouraged. Cons: My main criticism to the paper is that it solely studies crowding in the context of image classification. However, if crowding is studied in the context of object detection, where the task is localize the object and recognize its category, the effect may be significantly lessened. For example, R-CNN proposes high saliency regions where the object might be and perform classification on that masked region. Because targets are usually centered in such proposed region and background clutters are excluded from the proposed region, the accuracy can potentially be much higher. After all, the extent to which crowding is present in DNN depends a lot on the chosen architecture. And the architecture in this paper is very primitive compare to what researchers consider state-of-the-art these days, and the accuracy of the MNIST tasks reported by the paper are way lower than what most researchers would expect from a practical system. For example, [1] performs digit OCR which has much more clutters but with very high accuracy. It is not obvious architectures like that also suffer from crowding. Suggestion: The paper is overall easy to follow. I feel the experimental setup can be more clear if some more example images (non-cropped, like the ones in the Fig1 of supplementary material). Overall, this paper has an interesting topic and is a nice read. The conclusions are not too strong because it uses simplistic architecture/datasets. But I think it is nonetheless this is a helpful paper to (re-)generate some interest on drawing relation between theories of human visual recognition and neural nets. [1] Goodfellow, Ian J., et al. "Multi-digit number recognition from street view imagery using deep convolutional neural networks." arXiv preprint arXiv:1312.6082 (2013). | {"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": 0.8095982074737549, "perplexity": 1012.1881942580882}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1550247481994.42/warc/CC-MAIN-20190217132048-20190217154048-00267.warc.gz"} |
http://theoreticalatlas.wordpress.com/2009/07/02/conference-makkaifest-09-models-logic-higher-categories/ | It’s taken me a while to write this up, since I’ve been in the process of moving house – packing and unpacking and all the rest. However, a bit over a week ago, I was in Montreal, attending MakkaiFest ’09 at the Centre de Recherches Mathematiques at the University of Montréal (and a pre-conference workshop hosted at McGill, which I’m including in the talks I mention here). This was in honour of the 70th birthday of Mihaly (Michael) Makkai, of McGill University. Makkai has done a lot of important foundational work in logic, model theory, and category theory, and a great many of the talks were from former students who’d gone on and been inspired by him, so one got sense of the range of things he’s worked on through his life.
The broad picture of Makkai’s work was explained to us by J.P. Marquis, from the Philosophy department at U of M. He is interested in philosophy of mathematics, and described Makkai’s project by contrast with the program of axiomatization of the early 20th century, along the lines suggested by Hilbert. This program provided a formal language for concrete structures – the problem, which category theory is part of a solution to, is to do the same for abstract structures. Contrast, for instance, the concrete description of a group $G$ as a (particular) set with some (particular) operation, with the abstract definition of a group object in a category. Makkai’s work in categorical logic, said Marquis, is about formalizing the process of abstraction that example illustrates.
Model Theory/Logic
This matter – of the relation between abstract theories and concrete models of the theories – is really what model theory is about, and this is one of the major areas Makkai has worked on. Roughly, a theory is most basically a schema with symbols for types, members of types, and some function symbols – and a collection of sentences built using these symbols (usually generated from some axioms by rules of logical inference). A model is (intuitively), an interpretation of the terms: a way of assigning concrete data to the symbols – say, a symbol for a type is assigned the set of all entities of that type, and a function symbol is assigned an actual function between sets, and so on – making all propositions true. A morphism of models is a map that preserves all the properties of the model that can be stated using first order logic.
This is an older way to say things – Victor Harnik gave an expository talk called “Model Theory vs. Categorical Logic” in which he compared two ways of adding an equivalence relation to a theory. The model theory way (invented by Shelah) involves taking the theory (list of sentences) $T$ and extending it to a new theory $T^{eq}$. This has, for instance, some new types – if we had a type for “element of group”, for example, we might then get a new type “equivalence class of elements of group”, and so on. Now, this extension is “tight” in the sense that the categories of all models of $T$ and of $T^{eq}$ are equivalent (by a forgetful functor $Mod(T^{eq}) \rightarrow Mod(T)$) – but one can prove new theorems in the extended theory. To make this clear, he described work (due to Makkai and Reyes) about pretopos completion. Here, one has the concept of a “Boolean logical category” – $Set$ is an example, as is, for any theory, a certain category whose objects are the formulas of the theory. This is related to Lawvere theories (see below). There are logical functors between such categories – functors into $Set$ are models, but there are also logical functors between theories. The point is that a theory $T$ embeds into $T^{eq}$ (abusing notation here – these are now the boolean logical categories). Then the point is that $T^{eq}$ arises as a kind of completion of $T$ – namely, it’s a boolean pretopos (not just category). Moreover, it has some nice universal properties, making this point of view a bit more natural than the model-theoretic construction.
Bradd Hart’s talk, “Conceptual Completeness for Cantinuous Logic”, was a bit over my head, but made some use of this kind of extension of a theory to $T^{eq}$. The basic point seems to be to add some kind of continuous structure to logic. One example comes from a metric structure – defining a metric space of terms, where the metric function $d(x,y)$ is some sum $\sum_n \phi_n (x,y)$, where the $\phi_n$ are formulas with two variables, either true or false – where true gives a $0$, and false gives a $1$ in this sum. This defines a distance from $x$ to $y$ associated to the given list of formulas $\phi_n$. A continuous logic is one with a structure like this. The business about equivalence relations arises if we say two things are equivalent when the distance between them is 0 – this leads to a concept of completion, and again there’s a notion that the categories of models are equivalent (though proving it here involves some notion of approximating terms to arbitrary epsilon, which doesn’t appear in standard logic).
Anand Pillay gave a talk which used model theory to describe some properties of the free group on n generators. This involved a “theory of the free group” which applies to any free group, and regard each such group as a model of the theory – in fact a submodel of some large model, and using model-theoretic methods to examine “stability” properties, in some sense which amounts to a notion of defining “generic” subsets of the group.
Logic and Higher Categories
A number of talks specifically addressed the ground where logic meets higher dimensional categories, since Makkai has worked with both.
In one talk, Robert Paré described a way of thinking about first-order theories as examples of “double Lawvere theories”. Lawvere’s way of formalizing “theories and models” was to say that the theory is a category itself (which has just the objects needed to describe the kind of structure it’s a theory of) – and a model is a functor into $Sets$ (or some other category – a model of the theory of groups in topological spaces, say, is a topological group). For example, the theory of groups includes an object $G$ and powers of it, multiplication and inverse maps, and expresses the axioms by the fact that certain diagrams commute. A model is a functor $M : Th(Grp) \rightarrow Sets$, assigning to the “group object” a set of elements, which then get the group structure from the maps. Instead of a category, this uses a double category. There are two kinds of morphisms – horizontal and vertical – and these are used to represent two kinds of symbols: function symbols, and relation symbols. (For example, one can talk about the theory of an ordered field – so one needs symbols for multiplication and addition and so forth, but also for the order relation $\leq$). Then a model of such a theory is a double functor into the double category whose objects are sets, and whose horizontal and vertical morphisms are respectively functions and relations.
André Joyal gave a talk about the first order logic of higher structures. He started by commenting on some fields which began life close together, and are now gradually re-merging: logic and category theory; category theory and homotopy theory (via higher categories); homotopy theory and algebraic geometry. The higher categories Joyal was thinking of are quasicategories, or “$( \infty, 1)$-categories, which are simplicial sets satisfying a weak version of a horn-filling condition (the “strict” version of this, a Kan complex, includes as example $N(C)$, the nerve of a category $C$ – there’s an n-simplex for each sequence of n composable morphisms, whose other edges are the various composites, and whose faces are “compositors”, “associators”, and so on – which for $N(C)$ are identities). The point of this is that one can reproduce most of category theory for quasicategories – in particular, he mentioned limits and colimits, factorization systems, pretoposes, and model theory.
Moving to quasicategories on one side of the parallel between category theory and logic has a corresponding move on the other side – on the logic side, one aspect is that the usual notion of a language is replaced by what’s called Martin-Löf type theory. This, in fact, was the subject of Michael Warren’s talk, “Martin-Löf complexes” (I reported on a similar talk he gave at Octoberfest last year). The idea here is to start by defining a globular set, given a theory and type $A$ – a complex whose n-cells have two faces, of dimension (n-1). The 0-cells are just terms of some type $A$. The 1-cells are terms of types like $\underline{A}(a,b)$, where $a$ and $b$ are variables of type $A$ – the type has an interpretation as a proposition that $a=b$ “extensionally” (i.e. not via a proof – but as for instance when two programs with non-equivalent code happen to always produce the same output). This kind of operation can be repeated to give higher cells, like $\underline{A(a,b)}(f,g)$, and so on. Given a globular set $G$, one gets a theory by an adjoint construction. Putting the two together, one has a monad on the category of globular sets – algebras for the monad are Martin-Löf complexes. Throwing in syntactic rules to truncate higher cells (I suppose by declaring all cells to be identities) gives n-truncated versions of these complexes, $MLC_n$. Then there is some interesting homotopy theory, in that the category of n-truncated Martin-Löf complexes is expected to be a model for homotopy n-types. For example, $MLC_0$ is equivalent to $Sets$, and there is an adjunction (in fact, a Quillen equivalence – that is, a kind of “homotopy” equivalence) between $MLC_1$ and $Gpd$.
Category Theory/Higher Categories
There were a number of talks that just dealt with categories – including higher categories – in their own right. Makkai has worked, for example, on computads, which were touched on by Marek Zawadowski in one of his two talks (one in the pre-conference workshop, the other in the conference). The first was about categories of “many-to-one shapes”, which are important to computads – these are a notion of higher-category, where every cell takes many “input” faces to one “output” face. Zawadowski described a “shape” of an n-cell as an initial object in a certain category built from the category of computads with specified faces. Then there’s a category of shapes, and an abstract description of “shape” in terms of a graded tensor theory (graded for dimension, and tensor because there’s a notion of composition, I believe). Zawadowski’s second talk, “Opetopic Sets in Lax Monoidal Fibrations”, dealt with a similar topic from a different point of view. A lax monoidal fibration (LMF) is a kind of gadget for dealing with multi-level structures (categories, multicategories, quasicategories, etc). There’s a lot of stuff here I didn’t entirely follow, but just to illustrate: categories arise as LMF, by the fibration $cod : Set^{B} \rightarrow Set$, where $B$ is the category with two objects $M, O$, and two arrows from $M$ to $O$. An object in the functor category $Set^{B}$ consists of a “set of morphisms and set of objects” with maps – making this a category involves the monoidal structure, and how composition is defined, and the real point is that this is quite general machinery.
Joachim Lambek and Gonzalo Reyez, both longtime collaborators and friends of Makkai, also both gave talks that touched on physics and categories, though in very different ways. Lambek talked about the “Lorentz category” and its appearance in special relativity. This involves a reformulation of SR in terms of biquaternions: like complex numbers, these are of the form $u + iv$, but $u$ and $v$ are quaternions. They have various conjugation operations, and the geometry of SR can be described in terms of their algebra (just as, say, rotations in 3D can be described in terms of quaternions). The Lorentz category is a way of organizing this – its two objects correspond to “unconjugated” and “conjugated” states.
Gonzalo Reyez gave a derivation of General Relativity in the context of synthetic differential geometry. The substance of this derivation is not so different from the usual one, but with one exception. Einstein’s field equations can be derived in terms of the motions of small regions full of of freely falling test particles – synthetic differential geometry makes it possible to do the same analysis using infinitesimals rigorously all the way through. The basic point here is that in SDG one replaces the real line as usually conceived, with a “real line with infinitesimals” (think of the ring $\mathbb{R}[\epsilon]/\langle \epsilon^2 \rangle$, which is like the reals, but has the infinitesimal $\epsilon$, whose square is zero).
Among other talks: John Power talked about the correspondence between Lawvere theories in universal algebra and finitary tree monads on sets – and asked about what happens to the left hand side of this correspondence when we replace “sets” with other categories on the righ hand side. Jeff Egger talked about measure theory from a categorical point of view – namely, the correspondence of NCG between C*-algebras and “noncommutative” topological spaces, and between W*-algebras and “noncommutative” measure spaces, thought of in terms of locales. Hongde Hu talked about the “codensity theorem”, and a way to classify certain kinds of categories – he commented on how it was inspired by Makkai’s approach to mathematics: 1) Find new proofs of old theorems, (2) standardize the concepts used in them, and (3) prove new theorems with those concepts. Fred Linton gave a talk describing Heath’s “V-space”, which is a half-plane with a funny topology whose open sets are “V” shapes, and described how the topos of locally finite sheaves over it has surprising properties having to do with nonexistence of global sections. Manoush Sadrzadeh, whom I met recently at CQC (see the bottom of the previous post) was again talking about linguistics using monoidal categories – she described some rules for “clitic movement” and changes in word order, and what these rules look like in categorical terms.
Other
A few other talks are a little harder for me to fit into the broad classification above. There was Charles Steinhorn’s talk about ordered “o-minimal” structures, which touched on a bit of economics – essentially, a lot of economics is based on the assumption that preference orders can be made into real-valued functions, but in fact in many cases one has (variants on) “lexicographic order”, involving ranked priorities. He talked about how typically one has a space of possibilities which can be cut up into cells, with one sort of order in each cell. There was Julia Knight, talking about computable structures of “high Scott rank” – in particular, this is about infinite structures that can still be dealt with computably – for example, infinitary logical formulas involving an infinite number of “OR” statements where all the terms being joined are of some common form. This ends up with an analysis of certain infinite trees. Hal Kierstead gave a talk about Ramsey theory which I found notable because it used the kind of construction based on a game: to prove that any colouring of a graph (or hypergraph) has some property, one devises a game where one player tries to build a graph, and the other tries to colour it, and proves a winning strategy for one player. Finally, Michael Barr gave a talk about a duality between certain categories of modules over commutative rings.
All in all, an interesting conference, with plenty of food for thought. | {"extraction_info": {"found_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": 54, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8651952743530273, "perplexity": 495.21538623323795}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802775225.37/warc/CC-MAIN-20141217075255-00163-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://arxiv.org/abs/1712.03652 | hep-ph
# Title:DAMPE excess from decaying right-handed neutrino dark matter
Abstract: The flux of high-energy cosmic-ray electrons plus positrons recently measured by the DArk Matter Particle Explorer (DAMPE) exhibits a tentative peak excess at an energy of around $1.4$ TeV. In this paper, we consider the minimal gauged $U(1)_{B-L}$ model with a right-handed neutrino (RHN) dark matter (DM) and interpret the DAMPE peak with a late-time decay of the RHN DM into $e^\pm W^\mp$. We find that a DM lifetime $\tau_{DM} \sim 10^{28}$ s can fit the DAMPE peak with a DM mass $m_{DM}=3$ TeV. This favored lifetime is close to the current bound on it by Fermi-LAT, our decaying RHN DM can be tested once the measurement of cosmic gamma ray flux is improved. The RHN DM communicates with the Standard Model particles through the $U(1)_{B-L}$ gauge boson ($Z^\prime$ boson), and its thermal relic abundance is controlled by only three free parameters: $m_{DM}$, the $U(1)_{B-L}$ gauge coupling ($\alpha_{BL}$), and the $Z^\prime$ boson mass ($m_{Z^\prime}$). For $m_{DM}=3$ TeV, the rest of the parameters are restricted to be $m_{Z^\prime}\simeq 6$ TeV and $0.00807 \leq \alpha_{BL} \leq 0.0149$, in order to reproduce the observed DM relic density and to avoid the Landau pole for the running $\alpha_{BL}$ below the Planck scale. This allowed region will be tested by the search for a $Z^\prime$ boson resonance at the future Large Hadron Collider.
Comments: 13 pages, 2 figures, the final journal version Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Astrophysical Phenomena (astro-ph.HE) DOI: 10.1142/S0217732318501572 Report number: EPHOU-17-016 Cite as: arXiv:1712.03652 [hep-ph] (or arXiv:1712.03652v3 [hep-ph] for this version)
## Submission history
From: Osamu Seto [view email]
[v1] Mon, 11 Dec 2017 06:20:29 UTC (96 KB)
[v2] Fri, 15 Dec 2017 05:56:38 UTC (96 KB)
[v3] Fri, 31 Aug 2018 07:54:01 UTC (112 KB) | {"extraction_info": {"found_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": 0.9112856984138489, "perplexity": 2571.418358496844}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232255071.27/warc/CC-MAIN-20190519161546-20190519183546-00171.warc.gz"} |
https://sunrisenews.co/the-international-space-station-is-all-set-for-the-creation-of-the-coldest-environment/ | Undoubtedly, space is a cold place. However, the scientists of the International Space Station are thinking to make it a bit colder, though temporarily. The researcher’s crew is working on the strategy to power the Cold Atom Laboratory. The Cold Atom Laboratory is a small device which is used to plunge the atoms into temperature to shoot absolute zero.
According to NBC March, the research experiment will be concentrating on the particle movement at the temperatures that are just above the absolute zero values. The perfect zero temperature is the theoretical temperature in which there is no movement. The experiment will be focusing on the concepts of a quirk of quantum mechanics.
The Chinese satellite from the early period of the year 2017, made the use of a strange quirk known as quantum entanglement. The role of this quantum entanglement is to send an unhackable message at a distance of approximately seven hundred and fifty miles on our Planet Blue.
The Quantum movement targets explicitly two particles to be in unison. However, these twin particles are separated by a considerable distance between them. Any interaction with either of the particle will produce a similar reaction in the other particle. When carefully observed the two particles are same, they exist in two different places at one time.
The research analysis infers that when the temperature of the particles is lowered, it will favour scientists to observe the behaviours of these two particles. The International Space Station has a low-gravity environment. It is this low-gravity environment that makes it favourable for the scientists to conclude similar testing. It will, in turn, help the researchers to work on the chilled particle clouds before their breakdown.
Frank is a writer and editor from Edmonton, Canada. He enjoys writing about every topic under the sun, but he has a particular interest in science and technology - as well as the odd guilty pleasure movie! | {"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": 0.8233740925788879, "perplexity": 715.4608569948457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610703520883.15/warc/CC-MAIN-20210120120242-20210120150242-00413.warc.gz"} |
http://math.stackexchange.com/questions/132768/proof-of-equivalence | # Proof of equivalence?
How do I prove that if two numbers $a$ and $N$ are co-prime, then in the equation:
$$ax ≡ ay \pmod N$$
necessarily $x ≡ y \pmod N$
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(Standard boilerplate): What have you tried? Is this homework? Are you working from a certain course/textbook on number theory? If you can avoid imperative words ("Prove") and favor infinitive questions ("How do I prove...?"), you might find a more fulfilling response from other users! – The Chaz 2.0 Apr 17 '12 at 1:23
Thanks for the advice! I'll keep it in mind next time. This isn't homework, just a bit of reading I am doing. – user26649 Apr 17 '12 at 1:36
$ax ≡ ay$ $(mod$ $N$) implies that $ax = ay + pN$ where $p \in \mathbb{Z}$. Then by subtracting $ay$ from both sides, we see that $ax - ay = a(x-y) = pN$. $a$ divides the left hand side of the equation, so it also must divide $pN$. But because $a \mid pN$ and $\gcd(a, N) = 1$, it must be the case that $a \mid p$. So there exists an integer $m$ such that $am = p$.
Then going back to $ax = ay + pN$, we can rewrite it as $ax = ay + (am)N$. If we divide the equation by $a$, we get $x = y + mN$. So we get $x \equiv y$ $(mod$ $N$)
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You completely lost me after you arrived at a(x-y)= pN. Could you please explain the rest with a bit more detail? – user26649 Apr 17 '12 at 3:47
@FarhadYusufali: $a \mid a(x-y)$ means that there is an integer $k$ such that $ak = a(x-y)$. Do you see what $k$ should be? Since $a(x-y) = pN$, by substitution we see that $a \mid pN$. Then since we know $a \mid pN$ and $\gcd(a, N) = 1$ that means $a$ and $N$ do not have any factors in common, so $a \mid p$. – Student Apr 17 '12 at 15:34
Awesome thanks! – user26649 Apr 17 '12 at 15:53
$ax \equiv ay \mod N \implies N | (ax - ay) \implies N|a(x-y)$
But $N$ doesn't divide $a$, so $N | x-y \implies x \equiv y \mod N$
Here, I used that if $(c,d) = 1$, then $c | de \implies c | e$. If that's not immediately obvious, or known, try to prove that first.
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Hint $\rm\ (a,n) = 1,\ n\:|\:az\:\Rightarrow\:n\:|\:az,nz\:\Rightarrow\:n\:|\:(az,nz) = (a,n)z = z.\:$ Now put $\rm\:z = x-y.$
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Hint: If $\gcd(a,b)=1$ then there are $x,y\in\mathbb Z$ so that $ax+by=1$.
- | {"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": 0.932206928730011, "perplexity": 217.4848934636518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375095874.61/warc/CC-MAIN-20150627031815-00131-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://www.nag.com/numeric/CL/nagdoc_cl24/html/S/s22bec.html | s Chapter Contents
s Chapter Introduction
NAG Library Manual
# NAG Library Function Documentnag_specfun_2f1_real (s22bec)
## 1 Purpose
nag_specfun_2f1_real (s22bec) returns a value for the Gauss hypergeometric function ${}_{2}F_{1}\left(a,b;c;x\right)$ for real parameters $a,b$ and $c$, and real argument $x$.
## 2 Specification
#include #include
double nag_specfun_2f1_real (double a, double b, double c, double x, NagError *fail)
## 3 Description
nag_specfun_2f1_real (s22bec) returns a value for the Gauss hypergeometric function ${}_{2}F_{1}\left(a,b;c;x\right)$ for real parameters $a$, $b$ and $c$, and for real argument $x$.
The associated function nag_specfun_2f1_real_scaled (s22bfc) performs the same operations, but returns ${}_{2}F_{1}\left(a,b;c;x\right)$ in the scaled form ${}_{2}F_{1}\left(a,b;c;x\right)={f}_{\mathrm{fr}}×{2}^{{f}_{\mathrm{sc}}}$ to allow calculations to be performed when ${}_{2}F_{1}\left(a,b;c;x\right)$ is not representable as a single working precision number. It also accepts the parameters $a$, $b$ and $c$ as summations of an integer and a decimal fraction, giving higher accuracy when any are close to an integer.
The Gauss hypergeometric function is a solution to the hypergeometric differential equation,
$x1-x d2 f dx2 + c-a+b+1x d f dx - a b f = 0 .$ (1)
For $\left|x\right|<1$, it may be defined by the Gauss series,
$F1 2 a,b;c;x = ∑ s=0 ∞ as bs cs s! xs = 1+ ab c x + aa+1 bb+1 cc+1 2! x2 + ⋯ ,$ (2)
where ${\left(a\right)}_{s}=1\left(a\right)\left(a+1\right)\left(a+2\right)\dots \left(a+s-1\right)$ is the rising factorial of $a$. ${}_{2}F_{1}\left(a,b;c;x\right)$ is undefined for $c=0$ or $c$ a negative integer.
For $\left|x\right|<1$, the series is absolutely convergent and ${}_{2}F_{1}\left(a,b;c;x\right)$ is finite.
For $x<1$, linear transformations of the form,
$F1 2 a,b;c;x = C1 a1,b1,c1,x1 F1 2 a1, b1 ;c1;x1 + C2 a2,b2,c2,x2 F1 2 a2, b2 ;c2;x2$ (3)
exist, where ${x}_{1}$, ${x}_{2}\in \left(0,1\right]$. ${C}_{1}$ and ${C}_{2}$ are real valued functions of the parameters and argument, typically involving products of gamma functions. When these are degenerate, finite limiting cases exist. Hence for $x<0$, ${}_{2}F_{1}\left(a,b;c;x\right)$ is defined by analytic continuation, and for $x<1$, ${}_{2}F_{1}\left(a,b;c;x\right)$ is real and finite.
For $x=1$, the following apply:
• If $c>a+b$, ${}_{2}F_{1}\left(a,b;c;1\right)=\frac{\Gamma \left(c\right)\Gamma \left(c-a-b\right)}{\Gamma \left(c-a\right)\Gamma \left(c-b\right)}$, and hence is finite. Solutions also exist for the degenerate cases where $c-a$ or $c-b$ are negative integers or zero.
• If $c\le a+b$, ${}_{2}F_{1}\left(a,b;c;1\right)$ is infinite, and the sign of ${}_{2}F_{1}\left(a,b;c;1\right)$ is determinable as $x$ approaches $1$ from below.
In the complex plane, the principal branch of ${}_{2}F_{1}\left(a,b;c;z\right)$ is taken along the real axis from $x=1.0$ increasing. ${}_{2}F_{1}\left(a,b;c;z\right)$ is multivalued along this branch, and for real parameters $a,b$ and $c$ is typically not real valued. As such, this function will not compute a solution when $x>1$.
The solution strategy used by this function is primarily dependent upon the value of the argument $x$. Once trivial cases and the case $x=1.0$ are eliminated, this proceeds as follows.
For $0, sets of safe parameters $\left\{{\alpha }_{i,j};{\beta }_{i,j};{\zeta }_{i,j};{\chi }_{j}\left|1\le j\le 2\right|;1\le i\le 4\right\}$ are determined, such that the values of ${}_{2}F_{1}\left({a}_{j},{b}_{j};{c}_{j};{x}_{j}\right)$ required for an appropriate transformation of the type (3) may be calculated either directly or using recurrence relations from the solutions of ${}_{2}F_{1}\left({\alpha }_{i,j},{\beta }_{i,j};{\zeta }_{i,j};{\chi }_{j}\right)$. If $c$ is positive, then only transformations with ${C}_{2}=0.0$ will be used, implying only ${}_{2}F_{1}\left({a}_{1},{b}_{1};{c}_{1};{x}_{1}\right)$ will be required, with the transformed argument ${x}_{1}=x$. If $c$ is negative, in some cases a transformation with ${C}_{2}\ne 0.0$ will be used, with the argument ${x}_{2}=1.0-x$. The function then cycles through these sets until acceptable solutions are generated. If no computation produces an accurate answer, the least inaccurate answer is selected to complete the computation. See Section 7.
For $0.5, an identical approach is first used with the argument $x$. Should this fail, a linear transformation resulting in both transformed arguments satisfying ${x}_{j}=1.0-x$ is employed, and the above strategy for $0 is utilized on both components. Further transformations in these sub-computations are however limited to single terms with no argument transformation.
For $x<0$, a linear transformation mapping the argument $x$ to the interval $\left(0,0.5\right]$ is first employed. The strategy for $0 is then used on each component, including possible further two term transforms. To avoid some degenerate cases, a transform mapping the argument $x$ to $\left[0.5,1\right)$ may also be used.
In addition to the above restrictions on $c$ and $x$, an artificial bound, arbnd, is placed on the magnitudes of $a,b,c$ and $x$ to minimize the occurrence of overflow in internal calculations, particularly those involving real to integer conversions. $\mathit{arbnd}=0.0001×{I}_{\mathrm{max}}$, where ${I}_{\mathrm{max}}$ is the largest machine integer (see nag_max_integer (X02BBC)). It should however not be assumed that this function will produce accurate answers for all values of $a,b,c$ and $x$ satisfying this criterion.
This function also tests for non-finite values of the parameters and argument on entry, and assigns non-finite values upon completion if appropriate. See Section 9 and Chapter x07.
Please consult the NIST Digital Library of Mathematical Functions or the companion (2010) for a detailed discussion of the Gauss hypergeometric function including special cases, transformations, relations and asymptotic approximations.
## 4 References
NIST Handbook of Mathematical Functions (2010) (eds F W J Olver, D W Lozier, R F Boisvert, C W Clark) Cambridge University Press
Pearson J (2009) Computation of hypergeometric functions MSc Dissertation, Mathematical Institute, University of Oxford
## 5 Arguments
On entry: the parameter $a$.
Constraint: $\left|{\mathbf{a}}\right|\le \mathit{arbnd}$.
2: bdoubleInput
On entry: the parameter $b$.
Constraint: $\left|{\mathbf{b}}\right|\le \mathit{arbnd}$.
3: cdoubleInput
On entry: the parameter $c$.
Constraints:
• $\left|{\mathbf{c}}\right|\le \mathit{arbnd}$;
• ${\mathbf{c}}\ne 0,-1,-2,\dots$.
4: xdoubleInput
On entry: the argument $x$.
Constraint: $-\mathit{arbnd}<{\mathbf{x}}\le 1$.
5: failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
## 6 Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_CANNOT_CALCULATE
An internal calculation has resulted in an undefined result.
NE_COMPLEX
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
In general, ${}_{2}F_{1}\left(a,b;c;x\right)$ is not real valued when $x>1$.
NE_INFINITE
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$, $c=⟨\mathit{\text{value}}⟩$, $a+b=⟨\mathit{\text{value}}⟩$.
${}_{2}F_{1}\left(a,b;c;1\right)$ is infinite in the case $c\le a+b$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_OVERFLOW
Overflow occurred in a subcalculation of ${}_{2}F_{1}\left(a,b;c;x\right)$. The result may or may not be infinite.
NE_REAL
On entry, ${\mathbf{c}}=⟨\mathit{\text{value}}⟩$.
${}_{2}F_{1}\left(a,b;c;x\right)$ is undefined when $c$ is zero or a negative integer.
NE_REAL_RANGE_CONS
On entry, a does not satisfy $\left|{\mathbf{a}}\right|\le \mathit{arbnd}=⟨\mathit{\text{value}}⟩$.
On entry, b does not satisfy $\left|{\mathbf{b}}\right|\le \mathit{arbnd}=⟨\mathit{\text{value}}⟩$.
On entry, c does not satisfy $\left|{\mathbf{c}}\right|\le \mathit{arbnd}=⟨\mathit{\text{value}}⟩$.
On entry, x does not satisfy $\left|{\mathbf{x}}\right|\le \mathit{arbnd}=⟨\mathit{\text{value}}⟩$.
NE_TOTAL_PRECISION_LOSS
All approximations have completed, and the final residual estimate indicates no accuracy can be guaranteed.
$\text{Relative residual}=⟨\mathit{\text{value}}⟩$.
NW_OVERFLOW_WARN
On completion, overflow occurred in the evaluation of ${}_{2}F_{1}\left(a,b;c;x\right)$.
NW_SOME_PRECISION_LOSS
All approximations have completed, and the final residual estimate indicates some precision may have been lost.
$\text{Relative residual}=⟨\mathit{\text{value}}⟩$.
NW_UNDERFLOW_WARN
Underflow occurred during the evaluation of ${}_{2}F_{1}\left(a,b;c;x\right)$. The returned value may be inaccurate.
## 7 Accuracy
In general, if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the value of ${}_{2}F_{1}\left(a,b;c;x\right)$ may be assumed accurate, with the possible loss of one or two decimal places. Assuming the result does not under or overflow, an error estimate $\mathit{res}$ is made internally using equation (1). If the magnitude of $\mathit{res}$ is sufficiently large, a different fail.code will be returned. Specifically,
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NW_UNDERFLOW_WARN $\mathit{res}\le 1000\epsilon$ ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_SOME_PRECISION_LOSS $1000\epsilon <\mathit{res}\le 0.1$ ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TOTAL_PRECISION_LOSS $\mathit{res}>0.1$
where $\epsilon$ is the machine precision as returned by nag_machine_precision (X02AJC).
A further estimate of the residual can be constructed using equation (1), and the differential identity,
$d F 1 2 a,b; c;x dx = ab c F 1 2 a+1,b+1; c+1;x d2 F 1 2 a,b; c;x dx2 = aa+1 bb+1 cc+1 F 1 2 a+2,b+2; c+2;x$ (4)
This estimate is however dependent upon the error involved in approximating ${}_{2}F_{1}\left(a+1,b+1;c+1;x\right)$ and ${}_{2}F_{1}\left(a+2,b+2;c+2;x\right)$.
Furthermore, the accuracy of the solution, and the error estimate, can be dependent upon the accuracy of the decimal fraction of the input parameters $a$ and $b$. For example, if $c={c}_{i}+{c}_{r}=100+\text{1.0e−6}$, then on a machine with $16$ decimal digits of precision, the internal calculation of ${c}_{r}$ will only be accurate to $8$ decimal places. This can subsequently pollute the final solution by several decimal places without affecting the residual estimate as greatly. Should you require higher accuracy in such regions, then you should use nag_specfun_2f1_real_scaled (s22bfc), which requires you to supply the correct decimal fraction.
## 8 Parallelism and Performance
Not applicable.
nag_specfun_2f1_real (s22bec) returns non-finite values when appropriate. See Chapter x07 for more information on the definitions of non-finite values.
Should a non-finite value be returned, this will be indicated in the value of fail, as detailed in the following cases.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, or ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TOTAL_PRECISION_LOSSNW_SOME_PRECISION_LOSS or NW_UNDERFLOW_WARN, a finite value will have been returned with an approximate accuracy as detailed in Section 7.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INFINITE then ${}_{2}F_{1}\left(a,b;c;x\right)$ is infinite, and a signed infinity will have been returned. The sign of the infinity should be correct when taking the limit as $x$ approaches $1$ from below.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_OVERFLOW_WARN then upon completion, $\left|{}_{2}F_{1}\left(a,b;c;x\right)\right|>{R}_{\mathrm{max}}$, where ${R}_{\mathrm{max}}$ is the largest machine number given by nag_real_largest_number (X02ALC), and hence is too large to be representable. The result will be returned as a signed infinity. The sign should be correct.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_OVERFLOW then overflow occurred during a subcalculation of ${}_{2}F_{1}\left(a,b;c;x\right)$. A signed infinity will have been returned, however there is no guarantee that this is representative of either the magnitude or the sign of ${}_{2}F_{1}\left(a,b;c;x\right)$.
For all other error exits, nag_specfun_2f1_real (s22bec) will return a signalling NaN (see nag_create_nan (x07bbc)).
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CANNOT_CALCULATE then an internal computation produced an undefined result. This may occur when two terms overflow with opposite signs, and the result is dependent upon their summation for example.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL then $c$ is too close to a negative integer or zero on entry, and ${}_{2}F_{1}\left(a,b;c;x\right)$ is considered undefined. Note, this will also be the case when $c$ is a negative integer, and a (possibly trivial) linear transformation of the form (3) would result in either:
(i) all ${c}_{j}$ not being negative integers, (ii) for any ${c}_{j}$ which remain as negative integers, one of the corresponding parameters ${a}_{j}$ or ${b}_{j}$ is a negative integer of magnitude less than ${c}_{j}$.
In the first case, the transformation coefficients ${C}_{j}\left({a}_{j},{b}_{j},{c}_{j},{x}_{j}\right)$ are typically either infinite or undefined, preventing a solution being constructed. In the second case, the series (2) will terminate before the degenerate term, resulting in a polynomial of fixed degree, and hence potentially a finite solution.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_RANGE_CONS then no computation will have been performed. The actual solution may however be finite.
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_COMPLEX indicates $x>1$. Hence the requested solution is on the boundary of the principal branch of ${}_{2}F_{1}\left(a,b;c;x\right)$, and hence is multivalued, typically with a non-zero imaginary component. It is however strictly finite.
## 10 Example
This example evaluates ${}_{2}F_{1}\left(a,b;c;x\right)$ at a fixed set of parameters $a,b$ and $c$, and for several values for the argument $x$.
### 10.1 Program Text
Program Text (s22bece.c)
None.
### 10.3 Program Results
Program Results (s22bece.r) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 162, "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": 0.9938381314277649, "perplexity": 1008.7666432094529}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1492917121153.91/warc/CC-MAIN-20170423031201-00279-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/308440/find-and-example-of-two-elements-a-b-in-a-finite-group-g-such-that-a-b | # Find and example of two elements $a,b$ in a finite group $G$ such that $|a| = |b| = 2, a \ne b$ and $|ab|$ is odd.
Find and example of two elements $a,b$ in a finite group $G$ such that $|a| = |b| = 2, a \ne b$ and $|ab|$ is odd. Any ideas as to how I would go about finding it?
Thanks
-
For this to happen it is necessary that $a$ and $b$ do not commute for otherwise $(ab)^2=abab=aabb=a^2b^2=1$. Have you tried the smallest non-abelian group that you know? – Jyrki Lahtonen Feb 19 '13 at 21:14
Any ideas as to how I would go about finding [blah]? The most general answer to this question, which covers a vast expanse of territory in many forms of endeavor: pick stuff repeatedly and test it; explore. – anon Feb 19 '13 at 21:20
Get to know as many examples as you can swallow and digest. You undoubtedly saw the group had to be nonabelian, so you should try every one of the (I hope several) nonabelian groups you know. – Lubin Feb 19 '13 at 21:56
In $S_3$ the multiplication of $(a,b)$ and $(b,c)$ is $(a,b,c)$
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There is a very general example you should know about, that of dihedral groups. A dihedral group has order $2n$, for any $n \ge 2$, and it is generated by two elements of order $2$, whose product has order $n$.
Probably the simplest way to see these groups is as a group of bijective maps on $\mathbf{Z}_{n}$, $$a : x \mapsto -x, \qquad b: x \mapsto -x-1.$$ (Coefficients are written as integers, but meant to be in $\mathbf{Z}_{n}$.) These are clearly elements of order $2$, while because of $$a \circ b(x) = (a(b(x)) = -(-x - 1) = x + 1$$ $ab = a \circ b$ has order $n$.
(Geometrically, such a group is the group of congruences of a regular $n$-gon.)
If you do the same thing over $\mathbf{Z}$, you find that $ab$ has infinite order. | {"extraction_info": {"found_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": 0.8039976358413696, "perplexity": 174.79357216850173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1394011237144/warc/CC-MAIN-20140305092037-00071-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/182589-closest-point-mmn14-4a.html | # Math Help - closest point mmn14 4a
1. ## closest point mmn14 4a
on the surface $x^2y^2z^2=1$
find a point on this plane where it is the closest to (0,0,0)
i know we should find a function f(x,y) which is the distance between the point on the suface oand (0,0,0)
how to find this f(x,y)
?
2. Originally Posted by transgalactic
on the surface $x^2y^2z^2=1$
find a point on this plane where it is the closest to (0,0,0)
i know we should find a function f(x,y) which is the distance between the point on the suface oand (0,0,0)
how to find this f(x,y)
?
Minimise
$d^2=x^2+y^2+z^2$
subject to
$x^2y^2z^2=1$
Lagrange multiplier problem, now show us what you have done or explain where you are having difficulties.
CB
3. Originally Posted by CaptainBlack
Minimise
$d^2=x^2+y^2+z^2$
subject to
$x^2y^2z^2=1$
Lagrange multiplier problem, now show us what you have done or explain where you are having difficulties.
CB
ok
$z=g(x,y)=\sqrt{\frac{1}{x^2y^2}}$
$d^2=x^2+y^2+z^2$
$f(x,y)=d^2=x^2+y^2+(\sqrt{\frac{1}{x^2y^2}})^2$
can i use the d^2 formula
and find its minimal points instead of the "d" formula
?
4. Originally Posted by transgalactic
ok
$z=g(x,y)=\sqrt{\frac{1}{x^2y^2}}$
$d^2=x^2+y^2+z^2$
$f(x,y)=d^2=x^2+y^2+(\sqrt{\frac{1}{x^2y^2}})^2$
can i use the d^2 formula
and find its minimal points instead of the "d" formula
?
d is a minimum if and only if d^2 is a minimum when d is a distance.
CB | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8514506816864014, "perplexity": 1128.2907188981708}, "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-2015-35/segments/1440645315227.83/warc/CC-MAIN-20150827031515-00215-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.isid.ac.in/~statmath/index.php?module=Preprint&Action=ViewAbs&PreprintId=159 | # Publications and Preprints
Effect of truncation on large deviations for heavy-tailed random vectors
by
Arijit Chakrabarty
This paper studies the effect of truncation on the large deviations behavior of the partial sum of a triangular array coming from a truncated power law model. Each row of the triangular array consists of i.i.d. random vectors, whose distribution matches a power law on a ball of radius going to infinity, and outside that it has a light-tailed modification. The random vectors are assumed to be $\mathbb{R}^d$-valued. It turns out that there are two regimes depending on the growth rate of the truncating threshold, so that in one regime, much of the heavy tailedness is retained, while in the other regime, the same is lost.
isid/ms/2011/14 [fulltext] | {"extraction_info": {"found_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": 0.9462161064147949, "perplexity": 379.32594889563075}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316549.78/warc/CC-MAIN-20190821220456-20190822002456-00410.warc.gz"} |
http://mathhelpforum.com/pre-calculus/173623-finding-coordinates-print.html | # Finding coordinates
• March 6th 2011, 06:54 AM
Veronica1999
Finding coordinates
Let F = (3,2). There is a point P on the y-axis for which the distance from P to the X-axis equals the distance PF. Find the coordinates of P.
Hints pls.
• March 6th 2011, 07:06 AM
Plato
Quote:
Originally Posted by Veronica1999
Let F = (3,2). There is a point P on the y-axis for which the distance from P to the X-axis equals the distance PF. Find the coordinates of P.
Let $P:(0,b)$ then $3^2+(b-2)^2=b^2$. | {"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": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8029724359512329, "perplexity": 874.6530479420437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246639414.6/warc/CC-MAIN-20150417045719-00288-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://search.datacite.org/works/10.3204/DESY-PROC-2012-02/239 | ### Search for the rare decays $B \rightarrow \mu^+\mu^-$ with the CMS detector.
Franco Simonetto
I summarize here the results of a search for the rare decays $B^0 \rightarrow \mu^+\mu^-$ and $B^0_s\rightarrow \mu^+\mu^-$, based on a sample of data collected by the CMS detector from $pp$ collisions at $\sqrt{s} =7$ \TeV, corresponding to an integrated luminosity of 4.9 \invfb . No excess of events over the expected background is observed. The resulting upper limits on the branching fractions are \mbox{${\cal B}(B_s^0 \rightarrow \mu^+ \mu^- ) < 7.7 \times 10^{-9}$} and... | {"extraction_info": {"found_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": 0.9985224604606628, "perplexity": 638.6976868166745}, "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-2018-34/segments/1534221210615.10/warc/CC-MAIN-20180816093631-20180816113631-00236.warc.gz"} |
https://gea.esac.esa.int/archive/documentation/GDR2/Catalogue_consolidation/chap_cu9val_cu9val/sec_cu9val_942/ssec_cu9val_942_dupl.html | # 10.2.2 Duplicated sources
The detection system on-board Gaia may sometimes generate more than one detection for a given source. The on-ground cross match process will then have to decide if two simultaneous detections represent two sources or merely repeated detections of the same single source. With dozens of thousands of millions detections, chances are that some small fraction of the single sources end up with part of their observations labelled with one source identifier, and the rest with a different source identifier. When this happens, the various pipelines produce two sets of independent results each with their own source identifier. We will of course expect, that the two sets are essentially identical. In the final catalogue only one solution is kept and the other removed, and the published source carries a flag saying it was a ‘duplicated source’. In deciding which solution to keep, priority is given to the one with better astrometric results. Before removing the poorer solution, this set of duplicated sources from the ‘pre-DR2’ Catalogue provided an attractive opportunity for the catalogue validation. Some results are presented in Arenou et al. (2018) and others in Section 10.2.5, typically pointing to a certain underestimation of the errors.
First, however, we must decide when to consider that a close source-pair is a duplicate. From simple considerations on the image size, data acquisition, and the present state of the on-ground processing, sources can hardly be expected to be reliably resolved if they are separated by less than 0.3 ${}^{\prime\prime}$, and even that is an optimistic limit as long as no dedicated data processing for binaries is activated.
Figure 10.6 shows the relation between separation and magnitude difference for sources in a dense test field in a processing step immediately before duplicates were removed. The sign of the magnitude difference depends, for processing reasons, on which source had the highest declination and can be considered random. We notice several characteristics: a concentration at small separations (below 0.05 ${}^{\prime\prime}$) and magnitude differences; a continuation up to 0.4 ${}^{\prime\prime}$ separation with small magnitude differences; a widening up to 0.7 ${}^{\prime\prime}$; and a more gradual widening at higher separations. A tentative interpretation is that the concentration at very small separations then represents the well-behaved, single sources. The, relatively sparse, continuation to 0.4 ${}^{\prime\prime}$ is a mixture of genuine duplicate sources and actual binaries; the widening up to 0.7 ${}^{\prime\prime}$ show the gradual decrease of acquisition conflicts; and the final widening demonstrates the continued decrease of conflicts and the increasing ability to deal with magnitude contrasts. Typical acquisition windows are 0.7 ${}^{\prime\prime}$ by 2.1 ${}^{\prime\prime}$, so two sources closer than 0.7 ${}^{\prime\prime}$ will always be in conflict.
Minor conflicts, affecting a small part of the acquisition windows, are accepted in Gaia DR2, and for pairs with similar magnitudes either of them may be detected as the brighter in a given transit and get a complete window. Even so, the closeness of a neighbour of similar magnitude will be disturbing. All in all it was decided to define duplicate sources for Gaia DR2 as pairs closer than 0.4 ${}^{\prime\prime}$ at the reference epoch. The vast majority of these pairs are genuine duplicates and only a minor fraction are binaries.
Figure 10.7 shows the separations for source pairs identified as duplicates. Because of the way duplicates involving more than two sources are handled, separations may occasionally exceed 0.4 ${}^{\prime\prime}$. The left panel shows a strong peak in the centre with a short tail that gradually becomes more rectangular and end in a hint of a diagonal cross. As shown in the right panel, the more exotic behaviour only occurs for cases where the full astrometric solution had to be abandoned. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 11, "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": 0.8365585207939148, "perplexity": 1153.006938047693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1534221209040.29/warc/CC-MAIN-20180814131141-20180814151141-00433.warc.gz"} |
https://www.physicsforums.com/threads/differential-equations-pop-growth.255024/ | Differential equations- Pop Growth
1. Sep 9, 2008
dm59
dy/dt=(.5+sint)y/5 what is time t when the population has doubled?
2. Sep 9, 2008
sutupidmath
$$\frac{dy}{dt}=\frac{(.5+sint)y}{5}$$ Well, you first need to solve this one, and come up with a function that predicts population at any point in time t. Do you know how to solve this diff, eq?
$$\frac{dy}{dt}=\frac{(.5+sint)y}{5}=>\frac{dy}{y}=\frac{1}{5}(.5+sint)dt$$
Now integrate on both sides, and solve for y. I guess Y=f(t) or sth like that.
SO there is another info. that you have been provided with, Y(t)=2Yo, where Yo is your initial population. Have you been provided with initial population?or with another info. on this problem?
3. Sep 9, 2008
Dick
You don't need the initial population. The ode is linear. But you do need to write down the solution, as stupidmath points out. The general solution will have the form C*f(t). So you just need to solve C*f(t)=2*C*f(0) for t. You don't need to know C. It cancels.
4. Sep 9, 2008
dm59
Got it!
Thank both of you so much for your help.
Similar Discussions: Differential equations- Pop Growth | {"extraction_info": {"found_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": 0.9008103609085083, "perplexity": 1051.7908961136334}, "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-2017-13/segments/1490218189316.33/warc/CC-MAIN-20170322212949-00368-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://www.act-math-practice.com/multiple-solutions/ | # Multiple Solutions
### Problems with Multiple Solutions for ACT Math
Problems with multiple solutions on the ACT math exam are questions that give you an equation and then ask you how many solutions there are for the equation provided.
These types of questions will usually appear on the algebra section of the test.
Normally, the questions will involve an equation that has a squared number and an integer, although you may also see other types of equations, such as the one in example 3 below.
You will need to consider both positive and negative numbers as potential solutions.
### Multiple Solutions – Practice Questions
Look at these examples.
Example 1: How many solutions exist for the following equation?
x2 + 8 = 0
A. 0
B. 1
C. 2
D. 4
E. 8
Remember that any real number squared will always equal a positive number.
Since 8 is added to the first value x2, the result will always be 8 or greater.
In other words, since x2 is always a positive number, the result of the equation would never be 0.
So there are zero solutions for this equation.
Example 2: How many solutions exist for the following equation?
x2 − 9 = 0
F. 0
G. 1
H. 2
I. 4
J. 8
As mentioned above, any real number squared will always equal a positive number.
Since 9 is subtracted from x2, x2 needs to be equal to 9.
Both 3 and −3 solve the equation.
So there are two solutions for this equation.
Example 3: How many solutions exist for the following equation?
2(x + 5) = 14
A. 0
B. 1
C. 2
D. Infinite
E. Cannot be determined
Solve the problem as you normally would.
2(x + 5) = 14
2x + 10 = 14
2x = 4
x = 2
Then ask yourself if any other solutions are possible.
A negative number or zero would result in the incorrect value on the left side of the equation.
So, there is only one solution to this problem.
In conclusion, the answers for questions like these are that there might be 1, 2, 3, or infinite solutions to the problem provided.
Go back to the Algebra Questions.
Have a look at our other Free Preparation Materials. | {"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": 0.8484756946563721, "perplexity": 431.18161412571686}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1516084890187.52/warc/CC-MAIN-20180121040927-20180121060927-00211.warc.gz"} |
http://launchloop.com/FilamentGravity | # Gravity Field of a Filament Perpendicular to a Galaxy
Integrate the gravitational acceleration from an object at radius r toward a line mass perpendicular to its orbit.
Assume gravitational constant G , radius r , line density { \large \rho } , and perpendicular axis z . Calculate the radial gravitational acceleration a .
Assume a mass element dm = { \large \rho } dz at position z . The distance between the object and the mass element is the hypotenuse h = \sqrt{ r^2 + z^2 } . The diagonal gravitational acceleration towards dm is
Eq 1: da_d = G { \large \rho } dz / h^2 dz ~ = ~ G { \large \rho } / ( r^2 + z^2 ) dz ~ ~ ~ ~ ~ in the diagonal direction
This is force is diagonal to the center of the particle's orbit; however, we are only concerned with the force component towards the center in the radial direction, because there is equal mass in the plus and minus z direction. Hence, we only care about the "cosine" component in the radial direction,
Eq 2: cos( ) = r / \sqrt{ r^2 + z^2 }
So the force component in the r direction is:
Eq 3: da_r = cos() G { \large \rho } r / ( r^2 + z^2 ) ~ dz ~ = ~ G { \large \rho } r / ( r^2 + z^2 )^{3/2} ~ dz
The total acceleration is the integral of this between -\infty and +\infty :
Eq 4: a_r = {\Large \int_{-\infty}^{+\infty}} ~ da_d = {\Large \int_{-\infty}^{+\infty}} ~ G { \large \rho } r / ( r^2 + z^2 )^{3/2} ~ dz
I am a lazy fellow, so I will normalize z to units of r with z' = z / r or z = z' r :
Eq 5: a_d = {\Large \int_{-\infty}^{+\infty}} ~ G { \large \rho } r / ( r^2 + (z' r)^2 )^{3/2} ~ r ~ dz'
This allows us to pull everything out of the integral besides z' :
Eq 6: a_d = G { \large \rho } {\Large \int_{-\infty}^{+\infty}} ~ r / ( r^2 ( 1 + {z'}^2 ))^{3/2} ~ r ~ dz'
Eq 7: a_d = G { \large \rho } {\Large \int_{-\infty}^{+\infty}} ~ r / ( r^3 ( 1 + {z'}^2)^{3/2} ) ~ r ~ dz'
Eq 8: a_d = G { \large \rho } ( r^2 / r^3 ) {\Large \int_{-\infty}^{+\infty}}~ 1 / ( 1 + {z'}^2)^{3/2} ~ dz' ~ move the constant r out
Eq 9: a_d = ( G { \large \rho } / r ) {\Large \int_{-\infty}^{+\infty}} ~ 1 / ( 1 + {z'}^2)^{3/2} ~ dz'
I cheated and looked the integral up on Wolfram alpha:
{\Large \int_{-\infty}^{+\infty}} 1 / ( 1 + {z'}^2 )^{3/2} dz' = 2 ~ ~ ~ so:
Eq 10: a_d = 2 G { \large \rho } / r ~ ~ ~ the acceleration from an infinite line mass is proportional to 1 / r
Units check: ( m ~ s^{-2} ) = ( m^3 kg^{-1} s^{-2} ) ( kg ~ m^{-1} ) ( m^{-1} ) = ( m ~ s^{-2} ) ~ ~ ~ ... they match!
Am I supposed to put Q.E.D. here?
A reminder: for a general NON-Keplerian orbit, v^2 / r = a = 2 G { \large \rho } / r ~ so ~ v^2 = 2 G { \large \rho } = constant.
As to what "holds the filaments up" relative to a galaxy over gigayears, I have no clue. They are actually there (hot and ionized and vaguely detectable in bulk), so nature is smarter than I am. We might expect the material close to the galaxy to fall into the central black hole, leaving a void; that will reduce the effect for the inner galactic material more than the outer material, so this may reduce inner rotation velocities relative to outer rotation velocities.
And I have no idea whether the filaments are actually massive enough to affect galactic rotation, but they seem to be massive enough to account for a large fraction of the "dark matter" the cosmologists are looking for. Perhaps all of it, if "dark energy" is an observational error because the SN1a "standard candle" claim is untrue.
Are the filaments perpendicular to rotation? If not, they may tilt themselves and the galaxy until they are. Keep in mind that we have NOT detected filaments associated with our own galaxy, we just detect them statistically between millions of pairs of other distant galaxies. We only see the radial rotation of other galaxies, we are imbedded in ours, and can only infer whole-Milky-Way behavior from the observed behavior of others.
FilamentGravity (last edited 2017-11-01 19:36:34 by KeithLofstrom) | {"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": 0.9354036450386047, "perplexity": 1882.098326560449}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934805894.15/warc/CC-MAIN-20171120013853-20171120033853-00772.warc.gz"} |
https://proceedings.mlr.press/v14/mohan11a.html | # Web-Search Ranking with Initialized Gradient Boosted Regression Trees
Ananth Mohan, Zheng Chen, Kilian Weinberger
Proceedings of the Learning to Rank Challenge, PMLR 14:77-89, 2011.
#### Abstract
In May 2010 Yahoo! Inc. hosted the Learning to Rank Challenge. This paper summarizes the approach by the highly placed team Washington University in St. Louis. We investigate Random Forests (RF) as a low-cost alternative algorithm to Gradient Boosted Regression Trees (GBRT) (the de facto standard of web-search ranking). We demonstrate that it yields surprisingly accurate ranking results – comparable to or better than GBRT. We combine the two algorithms by first learning a ranking function with RF and using it as initialization for GBRT. We refer to this setting as iGBRT. Following a recent discussion by ?, we show that the results of iGBRT can be improved upon even further when the web-search ranking task is cast as classification instead of regression. We provide an upper bound of the Expected Reciprocal Rank (?) in terms of classification error and demonstrate that iGBRT outperforms GBRT and RF on the Microsoft Learning to Rank and Yahoo Ranking Competition data sets with surprising consistency. | {"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": 0.9046947956085205, "perplexity": 2457.91915406596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363135.71/warc/CC-MAIN-20211205035505-20211205065505-00431.warc.gz"} |
https://proofwiki.org/wiki/Definition:Symmetry_Group_of_Square | # Definition:Symmetry Group of Square
## Group Example
Let $\mathcal S = ABCD$ be a square.
The various symmetry mappings of $\mathcal S$ are:
The identity mapping $e$
The rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ counterclockwise respectively about the center of $\mathcal S$.
The reflections $t_x$ and $t_y$ are reflections about the $x$ and $y$ axis respectively.
The reflection $t_{AC}$ is a reflection about the diagonal through vertices $A$ and $C$.
The reflection $t_{BD}$ is a reflection about the diagonal through vertices $B$ and $D$.
This group is known as the symmetry group of the square.
### Cayley Table
The Cayley table of the symmetry group of the square can be written:
$\begin{array}{c|cccccc} & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\ \hline e & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\ r & r & r^2 & r^3 & e & t_{AC} & t_{BD} & t_y & t_x \\ r^2 & r^2 & r^3 & e & r & t_y & t_x & t_{BD} & t_{AC} \\ r^3 & r^3 & e & r & r^2 & t_{BD} & t_{AC} & t_x & t_y \\ t_x & t_x & t_{BD} & t_y & t_{AC} & e & r^2 & r^3 & r \\ t_y & t_y & t_{AC} & t_x & t_{BD} & r^2 & e & r & r^3 \\ t_{AC} & t_{AC} & t_x & t_{BD} & t_y & r & r^3 & e & r^2 \\ t_{BD} & t_{BD} & t_y & t_{AC} & t_x & r^3 & r & r^2 & e\\ \end{array}$
## Also known as
The symmetry group of the square is also known as:
the dihedral group of order $8$ and denoted $D_4$
the octic group.
Some sources denote $D_4$ as ${D_4}^*$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9744685888290405, "perplexity": 418.42375443885317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107894426.63/warc/CC-MAIN-20201027170516-20201027200516-00015.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-7-section-7-5-systems-of-inequalities-exercise-set-page-863/68 | ## Precalculus (6th Edition) Blitzer
Step 1. Based on the given conditions, we can write the first inequality as $x^2+y^2\leq25$ shown with the red shaded area. Step 2. The second inequality can be written as $x+2y\geq5$ shown with the blue shaded area in the figure. Step 3. The overlapping area (appears in purple) shows the solution for both inequalities. | {"extraction_info": {"found_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": 0.9044925570487976, "perplexity": 311.3567218213921}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243988753.91/warc/CC-MAIN-20210506083716-20210506113716-00376.warc.gz"} |
https://www.expertsmind.com/library/find-the-final-kinetic-energy-of-first-glider-5543467.aspx | ### Find the final kinetic energy of first glider
Assignment Help Physics
##### Reference no: EM13543467
A 0.300 kg} glider is moving to the right on a frictionless, horizontal air track with a speed of 0.900 m/s} when it makes a head-on collision with a stationary 0.150kg} glider.
Part A
Find the magnitude of the final velocity of first glider if the collision is elastic.
v = m/s}
Part C
Find the magnitude of the final velocity of second glider if the collision is elastic.
v = m/s}
Part E
Find the final kinetic energy of first glider.
K = J}
Part F
Find the final kinetic energy of second glider.
K = J}
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All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd | {"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": 0.8180446624755859, "perplexity": 755.5467274876165}, "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-2023-14/segments/1679296949387.98/warc/CC-MAIN-20230330194843-20230330224843-00037.warc.gz"} |
https://math.stackexchange.com/questions/2003239/how-to-do-a-unitary-diagonalization-of-a-normal-matrix | # How to do a unitary diagonalization of a normal matrix?
It is easy to diagonalize a normal matrix such that $D = P^{-1} A P$ by simply putting all the orthogonal eigenvectors as columns for $P$. But I spent hours trying a unitary diagonalization of the following Hermitian (and therefore Normal) matrix: $$A = \begin{bmatrix} 0 & i & 1 \\ -i & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}$$ such that $D = U^*AU$. I know that by definition every normal matrix is unitarily diagonalizable. The eigenvalues of this matrix are $\{ 0, -\sqrt{2}, \sqrt{2} \}$.
What did not work but was my most promising try, was to scale down the eigenvectors by their norm so the matrix $P$ became orthonormal. The result does not give me the diagonal matrix with the desired eigenvalues though. Also, Google search did not yield a single nicely explained way to do a unitary transform of a normal matrix. The only document that I believe to try to explain it is here, although it does not show clearly how to construct $U$.
• Note that the Matlab expression U' returns the conjugate transpose of U for matrices U with complex entries and that U.' returns the nonconjugate transpose. I'm guessing that your promising try was correct, but your verification failed. – DCarter Nov 7 '16 at 15:33
• Nearly! My try was not correct since I didn't normalize properly. – Flaudre Nov 8 '16 at 3:17
The eigenvalues of $A$ are $0, \sqrt{2}, -\sqrt{2}$. These eigenvalues correspond to the eigenvectors $$\begin{bmatrix} 0\\ i\\ 1 \end{bmatrix},\quad \begin{bmatrix} \sqrt{2}\\ -i\\ 1 \end{bmatrix},\quad \begin{bmatrix} -\sqrt{2}\\ -i\\ 1 \end{bmatrix},$$ respectively. You will observe that the eigenvectors are orthogonal with respect to the standard inner product on $\mathbb{C}^n$. Normalizing the eigenvectors gives the unitary matrix $$U = \begin{bmatrix} 0 & 1/\sqrt{2} & -1/\sqrt{2}\\ i/\sqrt{2} & -i/2 & -i/2\\ 1/\sqrt{2} & 1/2 & 1/2 \end{bmatrix}$$ that diagonalizes $A$ to $D = \operatorname{diag}(0,\sqrt{2},-\sqrt{2})$.
• Absolutely. I failed to orthonormalize properly with respect to the standard inner product in $\mathbb{C}^n$ as you showed. Thanks! – Flaudre Nov 8 '16 at 3:22
• Is any diagonalizable matrix can be in the form $U D {U}^{H}$ or there are special requirements for that? – Royi Aug 25 '17 at 11:15
• Only the normal matrices are unitarily diagonalizable. A matrix $A$ is normal if $A^*A = AA^*$. – K. Miller Aug 25 '17 at 13:15 | {"extraction_info": {"found_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": 0.9609130024909973, "perplexity": 291.9030398334877}, "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-2019-39/segments/1568514574532.44/warc/CC-MAIN-20190921145904-20190921171904-00486.warc.gz"} |
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[61] Jürgen Richter and Bernd Sturmfels. On the topology and geometric construction of oriented matroids and convex polytopes . Trans. Amer. Math. Soc. 325 (1991) 389-412. MR 994170. Abstract, references, and article information View Article: PDF This article is available free of charge [62] Günter M. Ziegler. Binary supersolvable matroids and modular constructions . Proc. Amer. Math. Soc. 113 (1991) 817-829. MR 1068134. Abstract, references, and article information View Article: PDF This article is available free of charge [63] Sergey Yuzvinsky. Cohomology of local sheaves on arrangement lattices . Proc. Amer. Math. Soc. 112 (1991) 1207-1217. MR 1062840. Abstract, references, and article information View Article: PDF This article is available free of charge [64] Zuowei Shen. Dimension of certain kernel spaces of linear operators . Proc. Amer. Math. Soc. 112 (1991) 381-390. MR 1065091. Abstract, references, and article information View Article: PDF This article is available free of charge [65] Günter M. Ziegler. Matroid representations and free arrangements . Trans. Amer. Math. Soc. 320 (1990) 525-541. MR 986703. Abstract, references, and article information View Article: PDF This article is available free of charge [66] François Jaeger. Tutte polynomials and bicycle dimension of ternary matroids . Proc. Amer. Math. Soc. 107 (1989) 17-25. MR 979049. Abstract, references, and article information View Article: PDF This article is available free of charge [67] Geoff Whittle. A generalisation of the matroid lift construction . Trans. Amer. Math. Soc. 316 (1989) 141-159. MR 957084. Abstract, references, and article information View Article: PDF This article is available free of charge [68] Gary Gordon and Elizabeth McMahon. A greedoid polynomial which distinguishes rooted arborescences . Proc. Amer. Math. Soc. 107 (1989) 287-298. MR 967486. 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Abstract, references, and article information View Article: PDF This article is available free of charge [73] Bernd Sturmfels. On the decidability of Diophantine problems in combinatorial geometry. Bull. Amer. Math. Soc. 17 (1987) 121-124. MR 888886. Abstract, references, and article information View Article: PDF [74] James G. Oxley. The binary matroids with no $4$-wheel minor . Trans. Amer. Math. Soc. 301 (1987) 63-75. MR 879563. Abstract, references, and article information View Article: PDF This article is available free of charge [75] James G. Oxley. On nonbinary $3$-connected matroids . Trans. Amer. Math. Soc. 300 (1987) 663-679. MR 876471. Abstract, references, and article information View Article: PDF This article is available free of charge [76] Bernt Lindström. A reduction of algebraic representations of matroids . Proc. Amer. Math. Soc. 100 (1987) 388-389. MR 884485. 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Abstract, references, and article information View Article: PDF This article is available free of charge [81] Curtis Greene and Thomas Zaslavsky. On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs . Trans. Amer. Math. Soc. 280 (1983) 97-126. MR 712251. Abstract, references, and article information View Article: PDF This article is available free of charge [82] J. Kahn and J. P. S. Kung. Varieties of combinatorial geometries . Trans. Amer. Math. Soc. 271 (1982) 485-499. MR 654846. Abstract, references, and article information View Article: PDF This article is available free of charge [83] James G. Oxley. On connectivity in matroids and graphs . Trans. Amer. Math. Soc. 265 (1981) 47-58. MR 607106. Abstract, references, and article information View Article: PDF This article is available free of charge [84] Jeff Kahn and Joseph P. S. Kung. Varieties and universal models in the theory of combinatorial geometries. Bull. Amer. Math. Soc. 3 (1980) 857-858. MR 578380. Abstract, references, and article information View Article: PDF [85] George Purdy. Triangles in arrangements of lines. II . Proc. Amer. Math. Soc. 79 (1980) 77-81. MR 560588. Abstract, references, and article information View Article: PDF This article is available free of charge [86] Hien Q. Nguyen. Weak cuts of combinatorial geometries . Trans. Amer. Math. Soc. 250 (1979) 247-262. MR 530054. Abstract, references, and article information View Article: PDF This article is available free of charge [87] Joseph P. S. Kung. Alternating basis exchanges in matroids . Proc. Amer. Math. Soc. 71 (1978) 355-358. MR 500521. Abstract, references, and article information View Article: PDF This article is available free of charge [88] Hien Quang Nguyen. Semimodular functions and combinatorial geometries . Trans. Amer. Math. Soc. 238 (1978) 355-383. MR 0491269. 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Results: 61 to 90 of 106 found Go to page: 1 2 3 4 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9112143516540527, "perplexity": 1994.1728121894541}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934804724.3/warc/CC-MAIN-20171118094746-20171118114746-00679.warc.gz"} |
https://www.physicsforums.com/threads/linear-velocity-of-a-spring-with-mass.561777/ | # Linear velocity of a spring with mass
1. Dec 20, 2011
### tomerb
1. The problem statement, all variables and given/known data
why does the velocity of an small spring element will be in linear proportion to the distance from the fixed end?
2. Relevant equations
v(x)=$\frac{x}{l}$V$_{0}$
Thank you very much,
Tomer
2. Dec 21, 2011
### tomerb
I would like to add my attemp (although its probably way too far from the right direction):
the general force equation for any coordinate of a mass spring with mass M attached to it is (I think):
L - lenght of loose spring
z$_{0}$ - the lenght from the fixed wall
Z - the coordinate of the small mass element.
m- mass of the spring
M - mass attached to the spring
(M+m($\frac{L-z_{0}}{L}$))$\ddot{Z}$=-$\frac{L}{z_{0}}$k(Z-z$_{0}$)
if z$_{0}$ will be L then the equation will be the "normal" equation for mass M attached to a fixed spring.
from this differential equation i've got the general velocity depends on z$_{0}$.
as you can see, this is probably not the right way to approach this question - way too complicated..
thanks, again. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.981508195400238, "perplexity": 862.5862931716097}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1550247482478.14/warc/CC-MAIN-20190217192932-20190217214932-00351.warc.gz"} |
https://www.coursehero.com/file/17588569/MTH405-Quiz-Hints/ | MTH405 Quiz (Hints) - Quiz(Hints MTH-405A(1 Why `p with |x|p:= X |xi |p is a normed linear space for 1 p <[5 points i=1(2 Give an example of a normed
# MTH405 Quiz (Hints) - Quiz(Hints MTH-405A(1 Why `p with...
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Quiz (Hints) MTH-405A (1) Why p with || x || p := X i =1 | x i | p ! is a normed linear space for 1 p < ? [5 points] (2) Give an example of a normed linear space X with two norms on it which are not equivalent. Justify your answer. [5 points] Done in class. (3) Show that the range of a bounded linear map may not be always closed. [5 points] Hint: consider the function F : as F ( x n ) = x n n . This is a linear map whose range is not closed because consider the set of sequences y n := (1 , 1 2 , 1 3 , · · · , 1 n , 0 , · · · ). Each of these y n ’s are in the range set because the inverse image of this is the set of sequences z n = (1 , 2 , 3 , · · · , n, 0 , · · · ). Notice that the full sequence ( 1 n ) is the limit of the y n in the sup-norm. So suppose range were a closed subspace, then the inverse image of this full sequence should be in and certainly the inverse image is the full sequence ( n ) / . | {"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": 0.9625754952430725, "perplexity": 411.16761111176265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243989012.26/warc/CC-MAIN-20210509183309-20210509213309-00633.warc.gz"} |
http://eprints.maths.ox.ac.uk/41/ | # A numerical study of the Schrödinger-Newton equations
Harrison, R. (2001) A numerical study of the Schrödinger-Newton equations. PhD thesis, University of Oxford.
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## Abstract
The Schrödinger-Newton (S-N) equations were proposed by Penrose [18] as a model for gravitational collapse of the wave-function. The potential in the Schrödinger equation is the gravity due to the density of , where is the wave-function. As with normal Quantum Mechanics the probability, momentum and angular momentum are conserved. We first consider the spherically symmetric case, here the stationary solutions have been found numerically by Moroz et al [15] and Jones et al [3]. The ground state which has the lowest energy has no zeros. The higher states are such that the th state has zeros. We consider the linear stability problem for the stationary states, which we numerically solve using spectral methods. The ground state is linearly stable since it has only imaginary eigenvalues. The higher states are linearly unstable having imaginary eigenvalues except for quadruples of complex eigenvalues for the th state, where a quadruple consists of . Next we consider the nonlinear evolution, using a method involving an iteration to calculate the potential at the next time step and Crank-Nicolson to evolve the Schrödinger equation. To absorb scatter we use a sponge factor which reduces the reflection back from the outer boundary condition and we show that the numerical evolution converges for different mesh sizes and time steps. Evolution of the ground state shows it is stable and added perturbations oscillate at frequencies determined by the linear perturbation theory. The higher states are shown to be unstable, emitting scatter and leaving a rescaled ground state. The rate at which they decay is controlled by the complex eigenvalues of the linear perturbation. Next we consider adding another dimension in two different ways: by considering the axisymmetric case and the 2-D equations. The stationary solutions are found. We modify the evolution method and find that the higher states are unstable. In 2-D case we consider rigidly rotationing solutions and show they exist and are unstable.
Item Type: Thesis (PhD) O - Z > Partial differential equationsO - Z > Quantum theoryH - N > Numerical analysis Mathematical Physics GroupOxford Centre for Industrial and Applied Mathematics 41 Eprints Administrator 10 Mar 2004 29 May 2015 18:15
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# Accelerated sea ice loss in the Wandel Sea points to a change in the Arctic’s Last Ice Area
## Abstract
The Arctic Ocean’s Wandel Sea is the easternmost sector of the Last Ice Area, where thick, old sea ice is expected to endure longer than elsewhere. Nevertheless, in August 2020 the area experienced record-low sea ice concentration. Here we use satellite data and sea ice model experiments to determine what caused this record sea ice minimum. In our simulations there was a multi-year sea-ice thinning trend due to climate change. Natural climate variability expressed as wind-forced ice advection and subsequent melt added to this trend. In spring 2020, the Wandel Sea had a mixture of both thin and—unusual for recent years—thick ice, but this thick ice was not sufficiently widespread to prevent the summer sea ice concentration minimum. With continued thinning, more frequent low summer sea ice events are expected. We suggest that the Last Ice Area, an important refuge for ice-dependent species, is less resilient to warming than previously thought.
## Introduction
In August 2020, as the German Icebreaker Polarstern transited northward as part of the MOSAiC experiment, it followed a route across the Wandel Sea (WS) north of Greenland (Fig. 1a). This area is normally marked by compact, thick multiyear ice created by cold temperatures and onshore winds and currents1 that stretches from the WS southwestward along the northern coast of the Canadian Arctic Archipelago. This area has been predicted to retain multiyear ice much longer than elsewhere2,3 as the climate warms, and is therefore referred to as the “Last Ice Area”4 (LIA, Fig. S1a). In fact, predictions of when sea ice-free summers in the Arctic will occur usually assume an area threshold of 1 × 106 km2, a number that implicitly accounts for the LIA3,5. The LIA is considered to be a last refuge for ice-associated Arctic marine mammals, such as polar bears (Ursus maritimus), ice-dependent seals such as ringed seals (Pusa hispida) and bearded seals (Erignathus barbatus), and walrus (Odobendus rosmarus) throughout the 21st century6,7. The LIA is also important for ivory gulls that breed in north Greenland8.
The first sign of change in the LIA emerged during the spring of 2018, when a large polynya formed in response to anomalous northward winds which drove sea ice away from the coast9,10. These winds were so strong that model simulations indicated that the polynya would have developed even with the thicker ice that had been present there several decades ago11. This serves as a reminder that not all notable events encountered in the Arctic are the result of climate change and much is yet to be discovered and understood.
But what about the summer of 2020? How unusual were the sea ice conditions that helped the Polarstern on its northward journey? Was this simply an example of interannual variability or did climate change contribute significantly to these ice conditions? And most importantly, is the summer of 2020 an early warning that the WS may not be as resilient to climate change as we expect? Here we use satellite data and sea ice model experiments to diagnose the mechanisms responsible for the summer 2020 record minimum and to put them into historical context. We also speculate about potential future changes in this area. We examine the interplay of weather- and climate change-driven changes in sea ice conditions and hypothesize about the potential short- and long-term implications for sea ice-associated species. Our focus is on the WS in the eastern LIA, although the implications of this study may be relevant to the entire LIA region.
## Results and discussion
### Observed summer sea ice conditions
The Polarstern’s route was guided by satellite images showing extensive areas of open water and sea ice concentration (SIC) as low as 70% at 87N (Figs. 1a, S1b). We define our WS study area by 81.5°N–85°N, 10°W–50°W, the same area where we saw signs of change in February 201810. Daily 2020 WS SIC drops below the 5th percentile of the 1979–2020 time series on July 25 and stays there almost until the end of August (Fig. 1b). August 14, 2020 constitutes a record low 52% SIC minimum (Fig. 1c). Several earlier years (e.g., 1985: 57%, 1990: 67%, and 1991: 62%) also show significant low SIC minima, although none as low as 2020.
### Summer 2020 sea ice mass budget
Summer sea ice loss in any particular area happens in response to ice advection (i.e., dynamics) and ice melt (i.e., thermodynamics). Thus, to fully understand the causes of the summer 2020 WS sea ice loss event, we must examine these processes separately and in detail. To do this, we here use the Pan-arctic Ice Ocean Modeling and Assimilation System (PIOMAS; see “Methods”). For a time period Δt, a change in sea ice thickness (SIT) Δh is described by Δh/Δt = Fadv + Fprod, where “ice advection” Fadv is a short-hand term used here for SIT flux convergence. Positive (negative) ice advection indicates ice gain (loss). The quantity Fprod is net ice production defined by Fprod = Fatm-ice + Fbot where Fatm-ice is surface ice melt due to atmospheric heating, and Fbot is ice melt on the bottom and lateral floe surfaces due to ocean-ice heat flux (e.g. ref. 12).
First, we discuss ice loss from dynamics. Ice advection anomalies dominate monthly variability from 1979 to 2020, but with no long-term trend (Fig. 2a). Anomalous ice advection for June, July, and August (JJA) 2020 is substantially negative (i.e., SIT loss), although several earlier years also have anomalies of similar magnitude. July and August ice motion anomalies show ice being driven northwestward away from the WS (Fig. 2b). Daily Fadv variability over the summer in 2020 (Fig. 2c) shows several strong negative events in early June, and mostly negative values for much of July through mid-August (see Fig. S2b for anomalies). Many, but not all of these Fadv extrema correspond to peaks in surface stress as a result of strong wind events (Fig. S2a). The temporal evolution of cumulative SIT anomaly closely tracks the cumulative sea ice advection anomaly (Fig. 2d), underlining the importance of dynamics on SIT in the WS.
Next, we discuss ice loss from thermodynamics. In contrast to ice advection, Fig. 2a shows that 2020 JJA ice production is a record low value of −0.3 m/month (i.e., SIT loss). In fact, all years since 2016 show negative JJA ice production anomalies. Figure 3a shows that in 2020, Fatm-ice is generally the largest contributor to ice melt until mid-July, when Fbot becomes dominant as the ice recedes and more solar radiation enters the ocean. In order to better understand the physics of Fbot, we write the ocean heat budget12 for the upper 60 m (i.e., the layer that includes both the summer and winter mixed layer depths) of the WS region as:
$$\frac{\Delta H}{\Delta t}={F}_{{{\rm{atm}}}-{{\rm{ocn}}}}+{F}_{{{\rm{bot}}}}+{F}_{{{\rm{ocndyn}}}}$$
(1)
where ΔH is the change in ocean heat content over a time period Δt (here one day), Fatm-ocn the net atmospheric heat flux at the ocean surface either in open water or under the ice cover (the latter referring to penetrating solar shortwave radiative flux), Fbot the ocean-forced melting, and Focdyn the sum of all oceanic dynamical heat fluxes at the lateral and bottom boundaries including convection and lateral and vertical advection and diffusion (see “Methods”). The dominant term in summertime Fatm-ocn is solar shortwave radiation13 which contributes to both bottom melt Fbot and upper-ocean warming ΔH.
Figure 3a shows how the terms in Eq. (1) vary over the summer of 2020 in the WS and how they interact to create the 2020 SIC anomaly. An increase in Fatm-ocn starting in mid June increases ocean heat content H and results in warmer subsurface ocean temperatures that form a Near-Surface Temperature Maximum (NSTM)12 (Fig. 3b). This heat mixes upward during periods of high surface stress, cooling the NSTM and melting ice (most notably during August 9–16 when the upper part of the NSTM above the temperature maximum loses heat). We note that Focndyn is generally near zero, meaning that net ocean dynamics plays only a secondary role in ocean warming and bottom ice melt. This near-zero value is actually a sum of substantial warming via lateral heat flux convergence that is balanced by cooling via downwelling (see “Methods”). Total ice thickness during the time from July 25 and August 16th when we track the NSTM drops from about 3 m to 1.5 m (Fig. S3b)
It is important to recognize that ice advection and ice melt are not independent of each other. Negative advection (i.e., thickness divergence) creates thin ice and open water which then allows for more absorbed solar radiation, which in turn enhances melt: the classic ice-albedo feedback. This is evident in Fig. 3a and S2b, where Fatm-ocn increases during periods of negative advection (e.g. during August 9–12). This is a key result of our study: we find significant ice-albedo feedback in the WS, an area of usually thick, compact ice that traditionally should not sustain such a process14.
The above analysis shows that the development of the record SIC anomaly in the WS was in large part driven by ice advection whose effect was amplified by anomalous melt. The fact that dynamics, specifically sea ice advection, in this area displays large interannual variability with no statistically significant trend suggests that low ice concentrations and low thickness during August 2020 may have been predominantly driven by dynamics. For such events, climate change attribution is often aided by separating the thermodynamic component which often more clearly shows a climate change signal15. In our case, we consider the long-term thinning of the ice cover to be dominated by thermodynamics16 analogous to the sea surface temperatures identified in (ref. 15). Therefore, we next take a closer look at the sea ice conditions in spring 2020 (what we will call “initial conditions” for the events later that summer) relative to past years.
### Initial conditions and the role of the ice thickness distribution
During spring 2020, ice accumulated in the WS (Fig. 4a, b) in response to anomalous advection (mostly in February; Fig. 4c, d). As a result, ice thickness was near its 1979–2020 mean value by June 1 according to PIOMAS; Fig. 2c), and actually thicker than in recent years (2011–2019) as confirmed by the combined CryoSat-2/SMOS satellite product (Fig. 4b)12. We might then wonder how this initial spring state evolved into the low SICs observed in August. As shown next, mean SIT does not tell the entire story: we must look at the sub-grid scale distribution of SIT which describes how much ice in each model grid cell is relatively thin or thick (see “Methods”).
First, we consider how the SIT distribution changed over 2020, specifically from January through August 2020 (Fig. 5a). We see how thin ice in January transformed (via advection, Fig. 4c) to a distribution with significant thick ice while still retaining large amounts of thin ice. This substantial thick ice component is unusual in the context of recent years and is corroborated by observations that show large amounts of old ice in 2020 that are not evident in recent years (Fig. 5b).
Next, we consider the bigger picture i.e. longer-term changes in the SIT distribution over the period 1979–2020. During this time, the fraction of ice mass in PIOMAS’s thicker ice categories decreased, while the fraction in the thin ice categories increased (Fig. 5c), in keeping with the general thinning trend observed in the Arctic17,18 as well as in the LIA10. Thus, in recent years, thinner ice is the norm (Fig. 5d), which could lead to low summer SICs (such as seen in 2020) via reduced ice strength, increased susceptibility to advection, and enhanced shortwave absorption by the ocean and thus anomalous melt. In fact, one might ask, why haven’t other recent years also shown low summer SICs? The following section attempts to answer this question, looking specifically at the roles of springtime initial conditions vs. summer atmospheric forcing.
### Model experiments: initial SIT vs. summer weather
Two model experiment ensembles (see “Methods” for further details) are analyzed in this section and compared to the historical simulation “HIST” previously described. First, we ask what role did the particular sea ice conditions found at the end of spring 2020 play in the extreme sea ice loss over the following summer? Experiment “INIT” addresses this question by starting 42 separate simulations using June 1 sea ice conditions from each of the years 1979–2020, while always using the atmospheric forcing only from 2020. We next ask, what role did the particular atmospheric forcings (wind, heat fluxes, etc.) found during summer 2020 play in the sea ice loss of that year? Experiment “ATMOS” addresses this question by starting 42 separate simulations all with June 1, 2020 sea ice conditions but then forcing the ice-ocean model with the atmospheric forcing from each summer of 1979–2020.
SIC for the HIST simulation is below the median of the INIT ensemble throughout the summer, dropping below the 10th percentile by the middle of June and staying there until the end of August (Fig. 6a). This shows that the spring initial conditions were key in producing the anomalous summer conditions in 2020. However, recent years with similar amounts of thin ice to 2020 (i.e., 2018 and 2019) also show low August SIC under 2020 atmospheric forcing. This means that if 2020 summer atmospheric conditions had occurred in 2018 or 2019, the WS would have seen similarly low ice concentrations as in 2020. It also suggests that the thicker and older ice present at the start of the summer of 2020 did little to prevent the record SIC anomaly in August.
The ATMOS ensemble (Fig. 6b) indicates that the impact of 2020 atmospheric forcing on that summer’s SIC anomalies was minimal until late July, but then this forcing did indeed become important. HIST SIC drops below the 10th percentile of the ATMOS ensemble by the end of July, staying there through the end of August.
These ensemble experiments underline the importance of both spring sea ice and summer atmospheric forcing to August SIC. In summary, we find that: Spring ice conditions were mostly responsible for the summer SIC anomaly through the end of July, while the atmosphere was mainly responsible for driving SIC to a record low during August. Partitioning the impact of 2020 spring initial sea ice conditions vs. summer atmospheric forcing on the sea ice anomaly at the time of the WS sea ice minimum on August 14 (see “Methods”) attributes ~20% to the initial conditions while ~80% is the due to the atmospheric forcing. Assuming that the increasing presence over the past 40 years of thin ice and open water at the beginning of the melt season (Fig. 5c) is primarily driven by climate change and that the summer atmospheric conditions were due to internal variability, the above 20/80 partitioning provides an approximate measure of the contributions of climate change and internal variability to the 2020 event (see further discussion in “Methods”).
Is this 20% climate change signal significant or not? Extreme local events such as storms, heat waves, or floods are almost always dominated by dynamics driven by internal climate variability15. For example, flooding in New York City in response to Superstorm Sandy was on the order of 20% more extreme owing to long-term sea level rise17,18, a signal of the same magnitude we have detected in the present study. Using this example as a scale, we conclude that climate change was indeed a significant contributor to sea ice loss during the summer of 2020 in the WS.
### Atmospheric variability in context
To put 2020 WS sea ice advection into a larger scale context, we consider here the fundamental modes of Arctic atmospheric variability, i.e., the Arctic Oscillation (AO), the Arctic Dipole Mode (ADA), and the Barents Oscillation (BO)19,20. Each of these correspond to the principal components (PCs) of empirical orthogonal functions computed from monthly mean sea level pressure fields north of 30°N (Figs. 7 and S4). During January–March 2020, when sea ice was advected into the Wandel Sea, sea level pressure over the Arctic was low, with a sea level pressure pattern similar to that found in 2017 when the Beaufort Gyre reversed21,22. The resulting onshore ice motion contributed to anomalously thick ice north of Greenland. At this time, the AO and ADA were both very high (the AO was in fact a record), a situation not found in any other year over the 41-year time series. Interestingly, summer 2020 conditions show the opposite, with ice motion westward away from the WS and the AO and ADA near record negative values. It seems clear that the anomalous 2020 WS wind forcing was associated with anomalous large-scale surface wind patterns.
### Discussion and conclusions
While primarily driven by unusual weather, climate change in the form of thinning sea ice contributed significantly to the record low August 2020 SIC in the WS. Several advection events, some relatively early in the melt season, transported sea ice out of the region and allowed the accumulation of heat from the absorption of solar radiation in the ocean. This heat was mixed upward and contributed to rapid melt during high wind events, notably between August 9 and 16. Ocean-forced melting in this area that is traditionally covered by thick, compact ice is a key finding of this study. However, in some ways this should not be surprising given that this area (like most areas of the Arctic) has experienced a long-term thinning trend (Fig. 5c). Given the long-term thinning trend and strong interannual variability in atmospheric forcing, it seems reasonable to expect that summer sea ice conditions in the WS will likely become more variable in the future. In fact, mean SIT at the start of summer in 2018 and 2019 was even thinner than in 2020, which implies that with 2020-type atmospheric forcing, we might have seen even lower August SICs in those years, relative to that observed in 2020. In other words, SIC in the WS is now poised to plunge to low summer values, given the right atmospheric forcing.
#### So, is the LIA in trouble?
The WS is a key part of the LIA, one that has recently experienced anomalous conditions9,10. We have shown that climate change-associated thinning ice in this region is a prerequisite for the record low ice concentrations seen in August 2020. Further, the unusually high SIT at the start of 2020 suggests that a temporary replenishment of sea ice from other parts of the Arctic may do little to protect this area from eventual sea ice loss. Recent work indicates that while western and eastern sectors of the LIA have distinct physics11, they both are experiencing long-term sea ice thinning and thus are both vulnerable to the processes discussed in this study. Our work suggests a re-examination of climate model simulations in this area, since most do not predict summer 2020-level low SICs until several decades or more into the future. Given that most climate models presently feature a subgrid-scale thickness distribution23 its evolution over time in those models should be a focal point of future investigations (i.e., rather than simply focusing on grid-cell mean thickness). Coupled model simulations where atmosphere and ocean conditions are nudged to 2020 conditions would provide useful insights into the capabilities of the current generation of climate models to replicate our results. In addition, our results should be replicated with ice-ocean models using different resolutions and physics.
While the WS is only one part of the LIA, our results should give us pause when making assumptions about the persistence and resilience of summer sea ice in the LIA. Currently, little is known about marine mammal densities and biological productivity in the WS and the broader LIA. Recent studies indicate there may be some transient benefits for polar bears in areas transitioning from thick multi-year ice to thinner first year ice24,25, as biological productivity in the system increases26. However, this is largely the case in shallow water <300 m in depth and it is unclear if this will occur in multi-year ice regions elsewhere. The assumption that the LIA will be available as a refuge over the next century is inherently linked to projections about species’ population status, because for some species the LIA will be the last remaining summer sea ice habitat e.g. ref. 27. It is critical that future work quantify the resilience of this area for conservation and management of ice-dependent mammals under climate change.
## Methods
### Model and model configuration details
PIOMAS consists of coupled sea ice and ocean model components28. The sea ice model is a multi-category thickness and enthalpy distribution sea ice model which employs a teardrop viscous plastic rheology29, a mechanical redistribution function for ice ridging30,31 and a LSR (line successive relaxation) dynamics solver32. The model features 12 ice thickness categories covering ice up to 28 m thick. Sea ice volume per unit area h provides an “effective sea ice thickness” which includes open water (or leads) and ice of varying thicknesses. Unless otherwise noted, we refer to this quantity as sea ice thickness. The sea ice model is coupled with the Parallel Ocean Program model developed at the Los Alamos National Laboratory33. The PIOMAS model domain is based on a curvilinear grid with the north pole of the grid displaced into Greenland. It covers the area north of 49°N and is one-way nested into a similar, but global, ice-ocean model34. The average resolution of the model is 30 km but features its highest resolution in the Wandel Sea, with grid cell sizes on the order of 15 × 30 km. Vertical model resolution is 5 m in the upper 30 m, and less than 10 m at depths down to 100 m35, a resolution that has been shown sufficient to provide a realistic representation of upper ocean heat fluxes and the NSTM12.
PIOMAS is capable of assimilating satellite sea ice concentration data using an optimal interpolation approach36 either over the whole ice-covered area or only near ice edge. In our run HIST, satellite ice concentrations are assimilated only near the ice edge (defined as 0.15 ice concentration). This means that no assimilation is conducted in the areas where both model and satellite ice concentrations are above 0.15. If the observed ice edge exceeds the model ice edge, then sea ice is added to the thinnest sea ice thickness category and sea surface temperature (SST) is set to the freezing point. If the model ice edge exceeds observations, excess ice is removed in all thickness categories proportionally. This ice-edge assimilation approach forces the simulated ice edge close to observations, while preventing satellite-derived ice concentrations (which can be biased low during the summer e.g. ref. 37) from inaccurately correcting model ice concentrations in the interior of the ice pack. Ice concentrations used for assimilation are from the Hadley Centre (HadISST v1)38 for 1979-2006 and from the NSIDC near real time product39 for 2007 to present.
PIOMAS also assimilates SST40, using observations provided in the NCAR/NCEP reanalysis (see below for atmospheric forcing) which in turn are derived from NOAA’s OISSTv2.1 data set41. SST assimilation is only conducted in the open water areas, not in the ice-covered areas to avoid introducing an additional heat source into the sea ice budget42. For this study, we also conducted a number of sensitivity simulations in which no assimilation of ice concentration and SST is performed (see below).
Daily mean NCEP/NCAR reanalysis data43 are used as atmospheric forcing, i.e., 10-m surface winds, 2-m surface air temperature, specific humidity, precipitation, evaporation, downwelling longwave radiation, sea level pressure, and cloud fraction. Cloud fraction is used to calculate downwelling shortwave radiation following Parkinson and Kellogg44. Precipitation less evaporation is calculated from precipitation and latent heat fluxes provided by the reanalysis model and specified at monthly time resolution to allow the calculation of snow depth over sea ice and input of fresh water into the ocean. There is no explicit representation of melt-ponds in this version of PIOMAS. River runoff into the model domain is specified from climatology45. Because of the uncertainty of net precipitation and river runoff, the surface ocean salinity is restored to a salinity climatology46 with a 3-year restoring constant. Surface atmospheric momentum and turbulent heat fluxes are calculated using a surface layer model47 that is part of the PIOMAS framework. Additional model information can be found in in Zhang and Rothrock28. PIOMAS has undergone substantial validation23,48,49,50,51,52 and has been shown to simulate sea ice thickness with error statistics similar to the uncertainty of the observations52. Validation results for ocean profiles for the WS are shown in S5.
### Sea ice mass and upper ocean heat budgets
Components of the sea ice mass and upper ocean heat budgets are computed directly from model output and residuals. Fprod is calculated as Fprod= Δh/Δt – Fadv and Fbot=FprodFatm-ice. All heat entering the uppermost ocean grid cell is used to melt ice until SIT = 0; however, subsurface shortwave radiation penetration and attenuation are allowed, which can warm the ocean below the uppermost grid cell. Focndyn over the upper 60 m (Eq. (1)) can be partitioned into Focndyn=Focnadv + Fdiff + Fconvect where Focnadv, Fdiff, and Fconvect are heat exchanges between the upper 60 m of the WS and the adjacent ocean via horizontal and vertical advection, horizonal and vertical diffusion, and vertical convection, respectively. Focnadv is calculated directly from model ocean temperatures and velocities, and the sum of Fdiff + Fconvect found as a residual, i.e., Fdiff + Fconvect= ΔH/ΔtFatm-ocnFbotFocnadv where ΔH/Δt is calculated directly from model temperature profiles. We find that Fdiff + Fconvect is negligible, meaning that horizontal and vertical advection terms (more formally, heat flux convergence) dominate. This is illustrated in Fig. S6, which shows a strong ocean warming within ~100 km of the north Greenland coast owing to lateral heat flux convergence. This is nearly exactly balanced (not shown) by the vertical fluxes, i.e., downwelling, in keeping with previous results53. Finally, by comparing the heat budget for summer 2020 simulations with and without data assimilation (i.e., HIST vs. INIT), we find that this numerical effect produces only a negligible heat flux term and so is neglected here (it might be larger in other regions or over a longer time period of simulation). All ice mass and ocean heat budget terms are presented in units of meters of ice melt, assuming an ice density of 917 kg m−3 and latent heat of sea ice fusion of 3.293 × 105 J/kg.
### HIST, INIT, and ATMOS Runs
The single HIST simulation uses data assimilation for the entire simulation period and is the basis for our analysis except for the sensitivity experiments described next. The INIT and ATMOS ensemble runs turn off the assimilation after May 31, 2020. For the JJA period of comparison, differences between the HIST run (which includes assimilation) and the equivalent members from the following ensemble runs (which do not include assimilation) are negligible.
The INIT and ATMOS ensembles allow a partitioning of the proximate causes of the 2020 sea ice anomaly into those driven by the initial spring conditions (sea ice and ocean) and those related to the evolution of the atmosphere (winds, temperature, radiation, humidity) over the summer. To compute the relative contribution, we calculate spatially averaged SIC and SIT differences between the INIT and ATMOS ensemble medians and the HIST median at the time of the observed and simulated WS SIC sea ice minimum (August 14). The ensemble median here represents normal conditions as the reference to which conditions (sea ice for INIT, atmosphere for ATMOS) being tested are compared. The difference from HIST is considered the contribution of the respective 2020 condition, initial ice thickness for “INIT” and atmosphere for “ATMOS.” This difference in SIC (SIT) is 6.3% (0.35 m) for INIT and 31% (1.5 m) for ATMOS. Adding these differences yields a total SIC response of 37.3%, and with respective fractions for “INIT and ATMOS” yields a 17% (6.3%/37.3%) role of initial conditions and a 83% (31%/37.3%) role for the atmosphere. The impact on SIT is slightly higher with respective contributions of 19% and 81%.
This partitioning can be used as a measure of the relative impacts of climate change and internal variability. Loosely following the framework of Trenberth at al. we assume atmospheric variability is governed by internal variability, and initial (i.e., spring) sea ice conditions to be driven by long-term climate change. Therefore the 20%/80% partitioning provides an approximate measure of the contributions of climate change and internal variability on the 2020 event. This separation is not perfect because atmospheric warming appears to be playing a role as evident in the fact that ATMOS ensemble members 2018/2019 both yield ice concentrations well below the 1979–2020 mean/median. The assumption that initial ice conditions are entirely due to climate change is also not entirely correct either, since internal variability also plays a role in sea ice conditions16. Nevertheless, our experiments clearly show that the climate signal of thinning sea ice exerts an impact on the magnitude of internally driven extreme events in the WS. Moreover, the fact that dynamic thickening of WS spring sea ice conditions (likely the result of internal variability) did little to improve the resilience of sea ice later in the summer provides an indication that climate change-driven thinning will likely influence future events.
### Model uncertainties
As noted, PIOMAS has undergone substantial validation with respect to sea ice thickness, volume23,48,50,51,52 and motion54. A measure of the uncertainty of ice-mass budget terms can be obtained from a recent study55 that compared monthly advection and ice production terms from PIOMAS with another numerical model and estimates derived from satellite observations. Mass budget terms from the three different sources are highly correlated and provide confidence that the relationship of budget terms is correct even if their magnitudes may have error. In addition, our INIT and ATMOS model simulations incur additional uncertainties due to the lack of a direct feedback between the atmosphere and ice-ocean system. However, this problem is less severe in the summer season which is our focus here, because summer thermal contrasts are small between the marine surface and the atmosphere. Future experiments with coupled models that allow for a “replay” of observed variability will be needed to verify this.
## Data availability
NOAA/NSIDC-Climate Data Record (CDR) Ice Concentrations. For 1979–2019 monthly and daily observed SIC, we use the NSIDC-CDR data set (Meier et al. 2017, Peng et al. 2013, https://doi.org/10.7265/N59P2ZTG), specifically the “goddard_merged_seaice_conc” algorithm which is based on a merging of the NASA Team and NASA Bootstrap algorithms56. For 2020, we use the seaice_conc_cdr variable from the Near Real Time Version of the NSIDC-CDR57 https://doi.org/10.7265/N5FF3QJ6. This variable is derived using the identical algorithm as the “goddard_merged_seaice_conc” and therefore should provide a consistent record. Prior to 1987 the data are from SMMR and thus are only provided every other day and with a polar data gap (i.e., “pole hole”) that straddles the northern edge of our domain. We, therefore, set ice concentrations to 100% in this area; the SIC minimum series is not affected by this filling.
ERA-5 forcing data. ERA-5 data were downloaded from the Copernicus Data System (CDS) https://doi.org/10.24381/cds.adbb2d47 and from NCAR-CISL RDA https://doi.org/10.5065/D6X34W69.
Artist Sea Ice Concentrations (ASI). The ARTIST sea ice algorithm (ASI)58 provides daily high-resolution ice concentration data derived from the 89 Ghz passive microwave channels on SSM/I and SSM/IS and from the AMSR-(E/2) sensors on board of Aqua and Global Change Observation Mission (GCOM). Data sets were downloaded from the University of Bremen, https://seaice.uni-bremen.de/data/amsr2/asi_daygrid_swath/n3125. This data set has higher resolution relative to the NSIDC-CDR SIC’s, but it only covers the period 2001-present. It thus complements the NSIDC-CDR SIC data set which offers the full 41-year record.
NSIDC Ice Age. This product provides the age of sea ice up to 15–16 years old. The age of sea ice is computed by tracking the motion of contiguous areas of sea ice from observed ice motion data derived by blending sea ice motion derived from passive microwave, visible band AVHRR data and in-situ observations59. https://doi.org/10.5067/UTAV7490FEPB.
AWI Cryos/SMOS. The Alfred Wegener Institute provides weekly and monthly products of sea ice thickness derived from the ESA CryoSat-2 radar altimeter merged with ice thickness derived from the SMOS L-band passive microwave instrument60. CS-2 provides a more accurate measurement for thicker ice, while SMOS provides ice thickness data for sea ice up to 1 m thickness. Neither product is available from May through September when increases in snow water content make retrievals of thickness too uncertain to be useful. Data were downloaded from ftp.awi.de:/sea_ice/product/cryosat2_smos/v203/nh.
Cape Morris-Jessup Wind Data. Wind measurements for Cape-Morris-Jessup are from Cappelen, J. Weather observations from Greenland 1958–2017. Observation data with description., (Danish Meteorological Institute, 2018). Data are available from https://confluence.govcloud.dk/display/FDAPI/About+meteorological+observations.
Observed Ocean Profiles. Validation Ocean Profiles shown in Figure S5 are from the Word Ocean Data Base https://www.ncei.noaa.gov/products/world-ocean-database61.
PIOMAS model output. Available from http://psc.apl.uw.edu/research/projects/arctic-sea-ice-volume-anomaly/data/model_grid.
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## Acknowledgements
A.S. was supported by NSF grant OPP-1744587, NASA grants 80NSSC20K1253 and NNX15AG68G. J.Z. was supported by NSF grants PLR-1603259, PLR-1602985, and NNA-1927785, and NASA grant NNX17AD27G. M.S. was supported by NSF grants OPP-1751363 and PLR-1602521, NASA grant NNX16AK43G, NOAA grant NA15OAR4320063-AM170, and ONR grant N00014-17-1-2545. K.M. was supported by the Natural Sciences and Engineering Research Council of Canada. Additional support was provided by the World Wildlife Fund (Canada) under grant (G-1122-035-00-I) and NASA grant 80NSSC20K1361. The authors thank W. Ermold and B. Cohen for assistance with ocean hydrographic data analysis. We acknowledge the service of all the data providers.
## Author information
Authors
### Contributions
This work had critical contributions from all authors in developing the ideas, the analysis, as well as writing and review. A.J.S. took the lead in writing and conducted the analysis of model simulations. M.S. led the ocean heat and ice mass budget analysis and edited the drafts. G.W.K.M. conducted the analysis of model and satellite data to show the influence of the sea ice thickness distribution. J.Z. conducted model experiments and computed ocean heat budget components. K.L. provided the perspective on the implications for marine mammals.
### Corresponding author
Correspondence to Axel J. Schweiger.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Communications Earth & Environment thanks the anonymous reviewers for their contribution to the peer review of this work. Primary handling editors: Clare Davis. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Schweiger, A.J., Steele, M., Zhang, J. et al. Accelerated sea ice loss in the Wandel Sea points to a change in the Arctic’s Last Ice Area. Commun Earth Environ 2, 122 (2021). https://doi.org/10.1038/s43247-021-00197-5
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• ### Rare events in the Arctic
• James E. Overland
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https://worldwidescience.org/topicpages/t/two-dimensional+lattice+gas.html | #### Sample records for two-dimensional lattice gas
1. One-dimensional lattices topologically equivalent to two-dimensional lattices within the context of the lattice gas model
Science.gov (United States)
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2016-10-01
Continuum partial differential equations are obtained from a set of discrete stochastic evolution equations of both non-Markovian and Markovian processes and applied to the diffusion within the context of the lattice gas model. A procedure allowing to construct one-dimensional lattices that are topologically equivalent to two-dimensional lattices is described in detail in the case of a rectangular lattice. This example shows the general features that possess the procedure and extensions are also suggested in order to provide a wider insight in the present approach.
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Science.gov (United States)
Gotcheva, Violeta; Wang, Albert T J; Teitel, S
2004-06-18
A continuous time Monte Carlo lattice gas dynamics is developed to model driven steady states of vortices in two dimensional superconducting networks. Dramatic differences are found when compared to a simpler Metropolis dynamics. Subtle finite size effects are found at low temperature, with a moving smectic that becomes unstable to an anisotropic liquid on sufficiently large length scales.
3. One-dimensional lattices topologically equivalent to two-dimensional lattices within the context of the lattice gas model. II The triangular lattice
Science.gov (United States)
Costanza, E. F.; Costanza, G.
2016-12-01
Continuum partial differential equations are obtained from a set of discrete stochastic evolution equations of both non-Markovian and Markovian processes and applied to the diffusion within the context of the lattice gas model. A procedure allowing to construct one-dimensional lattices that are topologically equivalent to two-dimensional lattices is described in detail in the case of a triangular lattice. This example shows the general features that possess the procedure and extensions are also suggested in order to provide a wider insight in the present approach.
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Science.gov (United States)
Costanza, E. F.; Costanza, G.
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Budantsev, M. V., E-mail: [email protected]; Lavrov, R. A.; Pogosov, A. G.; Zhdanov, E. Yu.; Pokhabov, D. A. [Russian Academy of Sciences, Rzhanov Institute of Semiconductor Physics, Siberian Branch (Russian Federation)
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Institute of Scientific and Technical Information of China (English)
罗孟波; 陈庆虎; 许祝安; 焦正宽
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8. Two-Dimensional Toda-Heisenberg Lattice
Directory of Open Access Journals (Sweden)
2013-06-01
Full Text Available We consider a nonlinear model that is a combination of the anisotropic two-dimensional classical Heisenberg and Toda-like lattices. In the framework of the Hirota direct approach, we present the field equations of this model as a bilinear system, which is closely related to the Ablowitz-Ladik hierarchy, and derive its N-soliton solutions.
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CERN Document Server
Ye, F; Hu, B; Panoiu, N C
2010-01-01
We present a theoretical study of plasmonic lattice solitons (PLSs) formed in two-dimensional (2D) arrays of metallic nanowires embedded into a nonlinear medium with Kerr nonlinearity. We analyze two classes of 2D PLSs families, namely, fundamental and vortical PLSs in both focusing and defocusing media. Their existence, stability, and subwavelength spatial confinement are studied in detai
10. Two Dimensional Lattice Boltzmann Method for Cavity Flow Simulation
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Panjit MUSIK
2004-01-01
Full Text Available This paper presents a simulation of incompressible viscous flow within a two-dimensional square cavity. The objective is to develop a method originated from Lattice Gas (cellular Automata (LGA, which utilises discrete lattice as well as discrete time and can be parallelised easily. Lattice Boltzmann Method (LBM, known as discrete Lattice kinetics which provide an alternative for solving the Navier–Stokes equations and are generally used for fluid simulation, is chosen for the study. A specific two-dimensional nine-velocity square Lattice model (D2Q9 Model is used in the simulation with the velocity at the top of the cavity kept fixed. LBM is an efficient method for reproducing the dynamics of cavity flow and the results which are comparable to those of previous work.
11. Spatiotemporal surface solitons in two-dimensional photonic lattices.
Science.gov (United States)
Mihalache, Dumitru; Mazilu, Dumitru; Lederer, Falk; Kivshar, Yuri S
2007-11-01
We analyze spatiotemporal light localization in truncated two-dimensional photonic lattices and demonstrate the existence of two-dimensional surface light bullets localized in the lattice corners or the edges. We study the families of the spatiotemporal surface solitons and their properties such as bistability and compare them with the modes located deep inside the photonic lattice.
12. Spatiotemporal dissipative solitons in two-dimensional photonic lattices.
Science.gov (United States)
Mihalache, Dumitru; Mazilu, Dumitru; Lederer, Falk; Kivshar, Yuri S
2008-11-01
We analyze spatiotemporal dissipative solitons in two-dimensional photonic lattices in the presence of gain and loss. In the framework of the continuous-discrete cubic-quintic Ginzburg-Landau model, we demonstrate the existence of novel classes of two-dimensional spatiotemporal dissipative lattice solitons, which also include surface solitons located in the corners or at the edges of the truncated two-dimensional photonic lattice. We find the domains of existence and stability of such spatiotemporal dissipative solitons in the relevant parameter space, for both on-site and intersite lattice solitons. We show that the on-site solitons are stable in the whole domain of their existence, whereas most of the intersite solitons are unstable. We describe the scenarios of the instability-induced dynamics of dissipative solitons in two-dimensional lattices.
13. Two dimensional axisymmetric smooth lattice Ricci flow
CERN Document Server
Brewin, Leo
2015-01-01
A lattice based method will be presented for numerical investigations of Ricci flow. The method will be applied to the particular case of 2-dimensional axially symmetric initial data on manifolds with S^2 topology. Results will be presented that show that the method works well and agrees with results obtained using contemporary finite difference methods.
14. Two-dimensional lattice Boltzmann model for magnetohydrodynamics.
Science.gov (United States)
Schaffenberger, Werner; Hanslmeier, Arnold
2002-10-01
We present a lattice Boltzmann model for the simulation of two-dimensional magnetohydro dynamic (MHD) flows. The model is an extension of a hydrodynamic lattice Boltzman model with 9 velocities on a square lattice resulting in a model with 17 velocities. Earlier lattice Boltzmann models for two-dimensional MHD used a bidirectional streaming rule. However, the use of such a bidirectional streaming rule is not necessary. In our model, the standard streaming rule is used, allowing smaller viscosities. To control the viscosity and the resistivity independently, a matrix collision operator is used. The model is then applied to the Hartmann flow, giving reasonable results.
15. Pseudo-two-dimensional random dimer lattices
Energy Technology Data Exchange (ETDEWEB)
Naether, U., E-mail: [email protected] [Instituto de Ciencia de Materiales de Aragón and Departamento de Física de la Materia Condensada, CSIC – Universidad de Zaragoza, 50009 Zaragoza (Spain); Mejía-Cortés, C.; Vicencio, R.A. [Departamento de Física and MSI – Nucleus for Advanced Optics, Center for Optics and Photonics (CEFOP), Facultad de Ciencias, Universidad de Chile, Santiago (Chile)
2015-06-05
We study the long-time wave transport in correlated and uncorrelated disordered 2D arrays. When a separation of dimensions is applied to the model, we find that the previously predicted 1D random dimer phenomenology also appears in so-called pseudo-2D arrays. Therefore, a threshold behavior is observed in terms of the effective size for eigenmodes, as well as in long-time dynamics. A minimum system size is required to observe this threshold, which is very important when considering a possible experimental realization. For the long-time evolution, we find that for correlated lattices a super-diffusive long-range transport is observed. For completely uncorrelated disorder 2D transport becomes sub-diffusive within the localization length and for random binary pseudo-2D arrays localization is observed.
16. Existence and Stability of Two-Dimensional Compact-Like Discrete Breathers in Discrete Two-Dimensional Monatomic Square Lattices
Institute of Scientific and Technical Information of China (English)
XU Quan; TIAN Qiang
2007-01-01
Two-dimensional compact-like discrete breathers in discrete two-dimensional monatomic square lattices are investigated by discussing a generafized discrete two-dimensional monatomic model.It is proven that the twodimensional compact-like discrete breathers exist not only in two-dimensional soft Ф4 potentials but also in hard two-dimensional Ф4 potentials and pure two-dimensional K4 lattices.The measurements of the two-dimensional compact-like discrete breather cores in soft and hard two-dimensional Ф4 potential are determined by coupling parameter K4,while those in pure two-dimensional K4 lattices have no coupling with parameter K4.The stabilities of the two-dimensional compact-like discrete breathers correlate closely to the coupling parameter K4 and the boundary condition of lattices.
17. Logarithmic divergent thermal conductivity in two-dimensional nonlinear lattices.
Science.gov (United States)
Wang, Lei; Hu, Bambi; Li, Baowen
2012-10-01
Heat conduction in three two-dimensional (2D) momentum-conserving nonlinear lattices are numerically calculated via both nonequilibrium heat-bath and equilibrium Green-Kubo algorithms. It is expected by mainstream theories that heat conduction in such 2D lattices is divergent and the thermal conductivity κ increases with lattice length N logarithmically. Our simulations for the purely quartic lattice firmly confirm it. However, very robust finite-size effects are observed in the calculations for the other two lattices, which well explain some existing studies and imply the extreme difficulties in observing their true asymptotic behaviors with affordable computation resources.
18. Dielectric-barrier discharges in two-dimensional lattice potentials
CERN Document Server
Sinclair, Josiah
2011-01-01
We use a pin-grid electrode to introduce a corrugated electrical potential into a planar dielectric-barrier discharge (DBD) system, so that the amplitude of the applied electric field has the profile of a two-dimensional square lattice. The lattice potential provides a template for the spatial distribution of plasma filaments in the system and has pronounced effects on the patterns that can form. The positions at which filaments become localized within the lattice unit cell vary with the width of the discharge gap. The patterns that appear when filaments either overfill or under-fill the lattice are reminiscent of those observed in other physical systems involving 2d lattices. We suggest that the connection between lattice-driven DBDs and other areas of physics may benefit from the further development of models that treat plasma filaments as interacting particles.
19. Two-Dimensional Breather Lattice Solutions and Compact-Like Discrete Breathers and Their Stability in Discrete Two-Dimensional Monatomic β-FPU Lattice
Institute of Scientific and Technical Information of China (English)
XU Quan; TIAN Qiang
2009-01-01
We restrict our attention to the discrete two-dimensional monatomic β-FPU lattice. We look for twodimensional breather lattice solutions and two-dimensional compact-like discrete breathers by using trying method and analyze their stability by using Aubry's linearly stable theory. We obtain the conditions of existence and stability of two-dimensional breather lattice solutions and two-dimensional compact-like discrete breathers in the discrete twodimensional monatomic β-FPU lattice.
20. Stress Wave Propagation in Two-dimensional Buckyball Lattice
Science.gov (United States)
Xu, Jun; Zheng, Bowen
2016-11-01
Orderly arrayed granular crystals exhibit extraordinary capability to tune stress wave propagation. Granular system of higher dimension renders many more stress wave patterns, showing its great potential for physical and engineering applications. At nanoscale, one-dimensionally arranged buckyball (C60) system has shown the ability to support solitary wave. In this paper, stress wave behaviors of two-dimensional buckyball (C60) lattice are investigated based on square close packing and hexagonal close packing. We show that the square close packed system supports highly directional Nesterenko solitary waves along initially excited chains and hexagonal close packed system tends to distribute the impulse and dissipates impact exponentially. Results of numerical calculations based on a two-dimensional nonlinear spring model are in a good agreement with the results of molecular dynamics simulations. This work enhances the understanding of wave properties and allows manipulations of nanoscale lattice and novel design of shock mitigation and nanoscale energy harvesting devices.
1. Topological states in two-dimensional hexagon lattice bilayers
Science.gov (United States)
Zhang, Ming-Ming; Xu, Lei; Zhang, Jun
2016-10-01
We investigate the topological states of the two-dimensional hexagon lattice bilayer. The system exhibits a quantum valley Hall (QVH) state when the interlayer interaction t⊥ is smaller than the nearest neighbor hopping energy t, and then translates to a trivial band insulator state when t⊥ / t > 1. Interestingly, the system is found to be a single-edge QVH state with t⊥ / t = 1. The topological phase transition also can be presented via changing bias voltage and sublattice potential in the system. The QVH states have different edge modes carrying valley current but no net charge current. The bias voltage and external electric field can be tuned easily in experiments, so the present results will provide potential application in valleytronics based on the two-dimensional hexagon lattice.
2. Vibrational Properties of a Two-Dimensional Silica Kagome Lattice.
Science.gov (United States)
Björkman, Torbjörn; Skakalova, Viera; Kurasch, Simon; Kaiser, Ute; Meyer, Jannik C; Smet, Jurgen H; Krasheninnikov, Arkady V
2016-12-27
Kagome lattices are structures possessing fascinating magnetic and vibrational properties, but in spite of a large body of theoretical work, experimental realizations and investigations of their dynamics are scarce. Using a combination of Raman spectroscopy and density functional theory calculations, we study the vibrational properties of two-dimensional silica (2D-SiO2), which has a kagome lattice structure. We identify the signatures of crystalline and amorphous 2D-SiO2 structures in Raman spectra and show that, at finite temperatures, the stability of 2D-SiO2 lattice is strongly influenced by phonon-phonon interaction. Our results not only provide insights into the vibrational properties of 2D-SiO2 and kagome lattices in general but also suggest a quick nondestructive method to detect 2D-SiO2.
3. Electronic Transmission Properties of Two-Dimensional Quasi-Lattice
Institute of Scientific and Technical Information of China (English)
侯志林; 傅秀军; 刘有延
2002-01-01
In the framework of the tight binding model, the electronic transmission properties of two-dimensional Penrose lattices with free boundary conditions are studied using the generalized eigenfunction method (Phys. Rev. B 60(1999)13444). The electronic transmission coefficients for Penrose lattices with different sizes and widths are calculated, and the result shows strong energy dependence because of the quasiperiodic structure and quantum coherent effect. Around the Fermi level E = 0, there is an energy region with zero transmission amplitudes,which suggests that the studied systems are insulating. The spatial distributions of several typical electronic states with different transmission coefficients are plotted to display the propagation process.
4. Quantum computing via defect states in two-dimensional antidot lattices.
Science.gov (United States)
Flindt, Christian; Mortensen, Niels Asger; Jauho, Antti-Pekka
2005-12-01
We propose a new structure suitable for quantum computing in a solid-state environment: designed defect states in antidot lattices superimposed on a two-dimensional electron gas at a semiconductor heterostructure. State manipulation can be obtained with gate control. Model calculations indicate that it is feasible to fabricate structures whose energy level structure is robust against thermal dephasing.
5. Two-dimensional chiral topological superconductivity in Shiba lattices
Science.gov (United States)
Li, Jian; Neupert, Titus; Wang, Zhijun; MacDonald, A. H.; Yazdani, A.; Bernevig, B. Andrei
2016-07-01
The chiral p-wave superconductor is the archetypal example of a state of matter that supports non-Abelian anyons, a highly desired type of exotic quasiparticle. With this, it is foundational for the distant goal of building a topological quantum computer. While some candidate materials for bulk chiral superconductors exist, they are subject of an ongoing debate about their actual paring state. Here we propose an alternative route to chiral superconductivity, consisting of the surface of an ordinary superconductor decorated with a two-dimensional lattice of magnetic impurities. We furthermore identify a promising experimental platform to realize this proposal.
6. The characters of nonlinear vibration in the two-dimensional discrete monoatomic lattice
Institute of Scientific and Technical Information of China (English)
XU Quan; TIAN Qiang
2005-01-01
The two-dimensional discrete monoatomic lattice is analyzed. Taking nearest-neighbor interaction into account, the characters of the nonlinear vibration in two-dimensional discrete monoatomic lattice are described by the two-dimensional cubic nonlinear Schrodinger equation. Considering the quartic nonlinear potential, the two-dimensional discrete-soliton trains and the solutions perturbed by the neck mode are presented.
7. Many body localization in two dimensional square and triangular lattices
CERN Document Server
Gonzalez-Garcia, L; Paredes, R
2016-01-01
Ultracold interacting Bose atoms placed in disordered two dimensional optical lattices with square and triangular symmetries are found to be localized above a certain disorder strength amplitude. From a Gross-Pitaevskii mean analysis we determine the localization length as a function of the disorder strength and investigate the energy spectrum in terms of the disorder magnitude. We found that the localization length is observed to decrease faster in triangular geometries than in square ones. In the presence of a harmonic confinement localization is observed at the center of the trap. The analysis of the energy spectrum reveals that discrete energy levels acquire a finite width that is always smaller than the distance among energy levels.
8. Compact triplexer in two-dimensional hexagonal lattice photonic crystals
Institute of Scientific and Technical Information of China (English)
Hongliang Ren; Jianping Ma; Hao Wen; Yali Qin; Zhefu Wu; Weisheng Hu; Chun Jiang; Yaohui Jin
2011-01-01
We design a contpact triplexer based on two-dimensional (2D) hexagonal lattice photonic crystals (PCs). A folded directional coupler (FDC) is introduced in the triplexer beside the point-defect micro-cavities and line-defect waveguides. Because of the reflection feedback of the FDC, high channel drop efficiency can be realized and a compact size with the order of micrometers can be maintained. The proposed device is analyzed using the plane wave expansion method, and its transmission characteristics are calculated using the finites-difference time-domain method. The footprint of the triplexer is about 12× 9 μm, and its extinction ratios are less than -20 dB for 1310 nm, approximately -20 dB for 1490 nm, and under -4O dB for 1550 nm, making it a potentially essential device ii future fiber-to-the-home networks.%@@ We design a compact triplexer based on two-dimensional (2D) hexagonal lattice photonic crystals (PCs).A folded directional coupler (FDC) is introduced in the triplexer beside the point-defect micro-cavities and line-defect waveguides.Because of the reflection feedback of the FDC, high channel drop efficiency can be realized and a compact size with the order of micrometers can be maintained.The proposed device is analyzed using the plane wave expansion method, and its transmission characteristics are calculated using the finite-difference time-domain method.The footprint of the triplexer is about 12×9 μm, and its extinction ratios are less than -20 dB for 1310 nm, approximately -20 dB for 1490 nm, and under -40 dB for 1550 nm, making it a potentially essential device in future fiber-to-the-home networks.
9. Two-dimensional discrete gap breathers in a two-dimensional discrete diatomic Klein-Gordon lattice
Institute of Scientific and Technical Information of China (English)
XU Quan; QIANG Tian
2009-01-01
We study the existence and stability of two-dimensional discrete breathers in a two-dimensional discrete diatomic Klein-Gordon lattice consisting of alternating light and heavy atoms, with nearest-neighbor harmonic coupling.Localized solutions to the corresponding nonlinear differential equations with frequencies inside the gap of the linear wave spectrum, i.e. two-dimensional gap breathers, are investigated numerically. The numerical results of the corresponding algebraic equations demonstrate the possibility of the existence of two-dimensional gap breathers with three types of symmetries, i.e., symmetric, twin-antisymmetric and single-antisymmetric. Their stability depends on the nonlinear on-site potential (soft or hard), the interaction potential (attractive or repulsive)and the center of the two-dimensional gap breather (on a light or a heavy atom).
10. Configurational entropy of a set of dipoles placed on a two-dimensional lattice
Science.gov (United States)
Dammig Quiña, P. L.; Irurzun, I. M.; Mola, E. E.
2017-01-01
In the present work we calculate the configurational entropy of an arbitrary number of dipoles placed on a square lattice. We use a quasi-two-dimensional (Q2D) space to capture the main features determining the occupation statistics of this system. We show that our result is in agreement with both, lattice-gas predictions at low coverages and the exact value derived in the close-packed limit as well. Therefore our equation provides a substantial improvement to the most recent calculations based on semiempirical models and Monte Carlo simulations.
11. On some classes of two-dimensional local models in discrete two-dimensional monatomic FPU lattice with cubic and quartic potential
Institute of Scientific and Technical Information of China (English)
Xu Quan; Tian Qiang
2009-01-01
This paper discusses the two-dimensional discrete monatomic Fermi-Pasta-Ulam lattice, by using the method of multiple-scale and the quasi-discreteness approach. By taking into account the interaction between the atoms in the lattice and their nearest neighbours, it obtains some classes of two-dimensional local models as follows: two-dimensional bright and dark discrete soliton trains, two-dimensional bright and dark line discrete breathers, and two-dimensional bright and dark discrete breather.
12. Larkin-Ovchinnikov phases in two-dimensional square lattices
Science.gov (United States)
Baarsma, J. E.; Törmä, P.
2016-10-01
We consider a two-component gas of fermions in optical lattices in the presence of a population imbalance within a mean-field theory. We study phase transitions from a normal gas of unpaired fermions to a superfluid phase of Bose-condensed Cooper pairs. The possibility of Cooper pairs with a nonzero centre-of-mass momentum is included, which corresponds to a so-called Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state. We find that for population-imbalanced systems such states can form the ground state. The FF and LO state are compared and it is shown that actually the LO state is energetically more favourable. We complete the mean-field phase diagram for the LO phase and show that it is qualitatively in excellent agreement with recent diagrammatic Monte Carlo calculations. Subsequently, we calculate the atomic density modulations in the LO phase.
13. Quantum magnetotransport in a modulated two-dimensional electron gas
Science.gov (United States)
Park, Tae-ik; Gumbs, Godfrey
1997-09-01
Quantum mechanical calculations of the magnetotransport coefficients of a modulated two-dimensional electron gas in a perpendicular magnetic field are presented using the Kubo method. The model modulation potential used is such that the effect of the steepness of the potential and its strength on the band part of the longitudinal resistivity ρxxand the Hall resistivity ρxycould be studied. In the extreme limit of a very steep potential, a two-dimensional square array of antidots is simulated. Impurity scattering is included in the self-consistent t-matrix approximation. The results show that for a strong lateral superlattice potential, ρxyis quenched in the low magnetic field regime and as the magnetic field increases there is a large negative Hall resistivity. The intensity of this negative peak is suppressed as the strength of the modulation potential is decreased. It is also shown that the height of the negative peak depends on the steepness of the potential. The longitudinal resistivity also has some interesting features. There are Aharonov-Bohm oscillations and a double peak structure which depends on both the strength of the modulation potential as well as its slope. The numerical results show that the position and intensity of the lower peak is not very sensitive to a change in the strength of the lattice potential or its steepness. However, the upper peak is greatly reduced when the lattice potential is diminished in strength. The double peak feature in ρxxand the negative peak and quenching of the Hall effect at low magnetic fields have been observed experimentally for antidots in both the quasiclassical and quantum regimes.
14. Decoherence in a Landau Quantized Two Dimensional Electron Gas
Directory of Open Access Journals (Sweden)
McGill Stephen A.
2013-03-01
Full Text Available We have studied the dynamics of a high mobility two-dimensional electron gas as a function of temperature. The presence of satellite reflections in the sample and magnet can be modeled in the time-domain.
15. Critical phenomena in the majority voter model on two-dimensional regular lattices.
Science.gov (United States)
Acuña-Lara, Ana L; Sastre, Francisco; Vargas-Arriola, José Raúl
2014-05-01
In this work we studied the critical behavior of the critical point as a function of the number of nearest neighbors on two-dimensional regular lattices. We performed numerical simulations on triangular, hexagonal, and bilayer square lattices. Using standard finite-size scaling theory we found that all cases fall in the two-dimensional Ising model universality class, but that the critical point value for the bilayer lattice does not follow the regular tendency that the Ising model shows.
16. Forensic potential of comprehensive two-dimensional gas chromatography
NARCIS (Netherlands)
Sampat, A.; Lopatka, M.; Sjerps, M.; Vivo-Truyols, G.; Schoenmakers, P.; van Asten, A.
2016-01-01
In this study, the application of comprehensive two-dimensional (2D) gas chromatography (GC × GC) in forensic science is reviewed. The peer-reviewed publications on the forensic use of GC × GC and 2D gas chromatography with mass spectrometric detection (GC × GC-MS) have been studied in detail, not o
17. Forensic potential of comprehensive two-dimensional gas chromatography
NARCIS (Netherlands)
Sampat, A.; Lopatka, M.; Sjerps, M.; Vivo-Truyols, G.; Schoenmakers, P.; van Asten, A.
2016-01-01
In this study, the application of comprehensive two-dimensional (2D) gas chromatography (GC × GC) in forensic science is reviewed. The peer-reviewed publications on the forensic use of GC × GC and 2D gas chromatography with mass spectrometric detection (GC × GC-MS) have been studied in detail, not o
18. Two-Dimensional Lattice Gravity as a Spin System
CERN Document Server
Beirl, W; Riedler, J
1994-01-01
Quantum gravity is studied in the path integral formulation applying the Regge calculus. Restricting the quadratic link lengths of the originally triangular lattice the path integral can be transformed to the partition function of a spin system with higher couplings on a Kagome lattice. Various measures acting as external field are considered. Extensions to matter fields and higher dimensions are discussed.
19. Thermal diode from two-dimensional asymmetrical Ising lattices.
Science.gov (United States)
Wang, Lei; Li, Baowen
2011-06-01
Two-dimensional asymmetrical Ising models consisting of two weakly coupled dissimilar segments, coupled to heat baths with different temperatures at the two ends, are studied by Monte Carlo simulations. The heat rectifying effect, namely asymmetric heat conduction, is clearly observed. The underlying mechanisms are the different temperature dependencies of thermal conductivity κ at two dissimilar segments and the match (mismatch) of flipping frequencies of the interface spins.
20. Spin dynamics in a two-dimensional quantum gas
DEFF Research Database (Denmark)
Pedersen, Poul Lindholm; Gajdacz, Miroslav; Deuretzbacher, Frank
2014-01-01
We have investigated spin dynamics in a two-dimensional quantum gas. Through spin-changing collisions, two clouds with opposite spin orientations are spontaneously created in a Bose-Einstein condensate. After ballistic expansion, both clouds acquire ring-shaped density distributions with superimp......We have investigated spin dynamics in a two-dimensional quantum gas. Through spin-changing collisions, two clouds with opposite spin orientations are spontaneously created in a Bose-Einstein condensate. After ballistic expansion, both clouds acquire ring-shaped density distributions...
1. Optical properties of two-dimensional magnetoelectric point scattering lattices
DEFF Research Database (Denmark)
Hansen, Per Lunnemann; Sersic, Ivana; Koenderink, A. Femius
2013-01-01
of split ring resonators and provide a quantitative comparison of measured and calculated transmission spectra at normal incidence as a function of lattice density, showing excellent agreement. We further show angle-dependent transmission calculations for circularly polarized light and compare...
2. Zero sound in a two-dimensional dipolar Fermi gas
NARCIS (Netherlands)
Lu, Z.K.; Matveenko, S.I.; Shlyapnikov, G.V.
2013-01-01
We study zero sound in a weakly interacting two-dimensional (2D) gas of single-component fermionic dipoles (polar molecules or atoms with a large magnetic moment) tilted with respect to the plane of their translational motion. It is shown that the propagation of zero sound is provided by both mean-f
3. Quantum holographic encoding in a two-dimensional electron gas
Energy Technology Data Exchange (ETDEWEB)
Moon, Christopher
2010-05-26
The advent of bottom-up atomic manipulation heralded a new horizon for attainable information density, as it allowed a bit of information to be represented by a single atom. The discrete spacing between atoms in condensed matter has thus set a rigid limit on the maximum possible information density. While modern technologies are still far from this scale, all theoretical downscaling of devices terminates at this spatial limit. Here, however, we break this barrier with electronic quantum encoding scaled to subatomic densities. We use atomic manipulation to first construct open nanostructures - 'molecular holograms' - which in turn concentrate information into a medium free of lattice constraints: the quantum states of a two-dimensional degenerate Fermi gas of electrons. The information embedded in the holograms is transcoded at even smaller length scales into an atomically uniform area of a copper surface, where it is densely projected into both two spatial degrees of freedom and a third holographic dimension mapped to energy. In analogy to optical volume holography, this requires precise amplitude and phase engineering of electron wavefunctions to assemble pages of information volumetrically. This data is read out by mapping the energy-resolved electron density of states with a scanning tunnelling microscope. As the projection and readout are both extremely near-field, and because we use native quantum states rather than an external beam, we are not limited by lensing or collimation and can create electronically projected objects with features as small as {approx}0.3 nm. These techniques reach unprecedented densities exceeding 20 bits/nm{sup 2} and place tens of bits into a single fermionic state.
4. Two-dimensional Chern semimetals on the Lieb lattice
Science.gov (United States)
Palumbo, Giandomenico; Meichanetzidis, Konstantinos
2015-12-01
In this work we propose a simple model that supports Chern semimetals. These gapless topological phases share several properties with the Chern insulators like a well-defined Chern number associated with each band, topologically protected edge states and topological phase transitions that occur when the bands touch each, with linear dispersion around the contact points. The tight-binding model, defined on the Lieb lattice with intra-unit-cell and suitable nearest-neighbor hopping terms between three different species of spinless fermions, supports a single Dirac-like point. The dispersion relation around this point is fully relativistic and the 3 ×3 matrices in the corresponding effective Hamiltonian satisfy the Duffin-Kemmer-Petiau algebra. We show the robustness of the topologically protected edge states by employing the entanglement spectrum. Moreover, we prove that the Chern number of the lowest band is robust with respect to weak disorder. For its simplicity, our model can be naturally implemented in real physical systems like cold atoms in optical lattices.
5. Two-dimensional lattice gauge theories with superconducting quantum circuits
Energy Technology Data Exchange (ETDEWEB)
Marcos, D., E-mail: [email protected] [Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck (Austria); Widmer, P. [Albert Einstein Center, Institute for Theoretical Physics, Bern University, CH-3012, Bern (Switzerland); Rico, E. [IPCMS (UMR 7504) and ISIS (UMR 7006), University of Strasbourg and CNRS, 67000 Strasbourg (France); Hafezi, M. [Joint Quantum Institute, NIST/University of Maryland, College Park 20742 (United States); Department of Electrical Engineering and Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742 (United States); Rabl, P. [Institute of Atomic and Subatomic Physics, TU Wien, Stadionallee 2, 1020 Wien (Austria); Wiese, U.-J. [Albert Einstein Center, Institute for Theoretical Physics, Bern University, CH-3012, Bern (Switzerland); Zoller, P. [Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck (Austria); Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck (Austria)
2014-12-15
A quantum simulator of U(1) lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.
6. Negative Dispersion of Lattice Waves in a Two-Dimensional Yukawa System
Institute of Scientific and Technical Information of China (English)
刘艳红; 刘斌; 杨思泽; 王龙
2002-01-01
Collective motion modes existing in a two-dimensional Yukawa system are investigated by molecular dynamics simulation. The dispersion relations of transverse and longitudinal lattice waves obtained for hexagonal lattice are in agreement with the theoretical results. The negative dispersion of the parallel longitudinal wave is demonstrated by the simulation, and is explained by a physical model.
7. Two-dimensional ion trap lattice on a microchip for quantum simulation
CERN Document Server
Sterling, R C; Weidt, S; Lake, K; Srinivasan, P; Webster, S C; Kraft, M; Hensinger, W K
2013-01-01
Using a controllable quantum system it is possible to simulate other highly complex quantum systems efficiently overcoming an in-principle limitation of classical computing. Trapped ions constitute such a highly controllable quantum system. So far, no dedicated architectures for the simulation of two-dimensional spin lattices using trapped ions in radio-frequency ion traps have been produced, limiting the possibility of carrying out such quantum simulations on a large scale. We report the operation of a two-dimensional ion trap lattice integrated in a microchip capable of implementing quantum simulations of two-dimensional spin lattices. Our device provides a scalable microfabricated architecture for trapping such ion lattices with coupling strengths between neighbouring ions sufficient to provide a powerful platform for the implementation of quantum simulations. In order to realize this device we developed a specialist fabrication process that allows for the application of very large voltages. We fabricated ...
8. Quantum search on the two-dimensional lattice using the staggered model with Hamiltonians
Science.gov (United States)
Portugal, R.; Fernandes, T. D.
2017-04-01
Quantum search on the two-dimensional lattice with one marked vertex and cyclic boundary conditions is an important problem in the context of quantum algorithms with an interesting unfolding. It avails to test the ability of quantum walk models to provide efficient algorithms from the theoretical side and means to implement quantum walks in laboratories from the practical side. In this paper, we rigorously prove that the recent-proposed staggered quantum walk model provides an efficient quantum search on the two-dimensional lattice, if the reflection operators associated with the graph tessellations are used as Hamiltonians, which is an important theoretical result for validating the staggered model with Hamiltonians. Numerical results show that on the two-dimensional lattice staggered models without Hamiltonians are not as efficient as the one described in this paper and are, in fact, as slow as classical random-walk-based algorithms.
9. Bloch oscillations and Zener tunneling in two-dimensional photonic lattices.
Science.gov (United States)
Trompeter, Henrike; Krolikowski, Wieslaw; Neshev, Dragomir N; Desyatnikov, Anton S; Sukhorukov, Andrey A; Kivshar, Yuri S; Pertsch, Thomas; Peschel, Ulf; Lederer, Falk
2006-02-10
We report on the first experimental observation of photonic Bloch oscillations and Zener tunneling in two-dimensional periodic systems. We study the propagation of an optical beam in a square lattice superimposed on a refractive index ramp. We observe oscillations of the beam inside the first Brilloin zone and tunneling of light from the first to the higher-order bands of the lattice band gap spectrum.
10. Tensor renormalization group approach to two-dimensional classical lattice models.
Science.gov (United States)
Levin, Michael; Nave, Cody P
2007-09-21
We describe a simple real space renormalization group technique for two-dimensional classical lattice models. The approach is similar in spirit to block spin methods, but at the same time it is fundamentally based on the theory of quantum entanglement. In this sense, the technique can be thought of as a classical analogue of the density matrix renormalization group method. We demonstrate the method - which we call the tensor renormalization group method - by computing the magnetization of the triangular lattice Ising model.
11. Topological phase transitions driven by next-nearest-neighbor hopping in two-dimensional lattices
NARCIS (Netherlands)
Beugeling, W.; Everts, J.C.; de Morais Smith, C.
2012-01-01
For two-dimensional lattices in a tight-binding description, the intrinsic spin-orbit coupling, acting as a complex next-nearest-neighbor hopping, opens gaps that exhibit the quantum spin Hall effect. In this paper, we study the effect of a real next-nearest-neighbor hopping term on the band structu
12. Dipolar fermions in a two-dimensional lattice at non-zero temperature
DEFF Research Database (Denmark)
Larsen, Anne-Louise G.; Bruun, Georg
2012-01-01
We examine density-ordered and superfluid phases of fermionic dipoles in a two-dimensional square lattice at nonzero temperature. The critical temperature of the density-ordered phases is determined and is shown to be proportional to the coupling strength for strong coupling. We calculate...
13. Lattice Methods for Pricing American Strangles with Two-Dimensional Stochastic Volatility Models
Directory of Open Access Journals (Sweden)
Xuemei Gao
2014-01-01
Full Text Available The aim of this paper is to extend the lattice method proposed by Ritchken and Trevor (1999 for pricing American options with one-dimensional stochastic volatility models to the two-dimensional cases with strangle payoff. This proposed method is compared with the least square Monte-Carlo method via numerical examples.
14. Hofstadter butterfly evolution in the space of two-dimensional Bravais lattices
Science.gov (United States)
Yılmaz, F.; Oktel, M. Ö.
2017-06-01
The self-similar energy spectrum of a particle in a periodic potential under a magnetic field, known as the Hofstadter butterfly, is determined by the lattice geometry as well as the external field. Recent realizations of artificial gauge fields and adjustable optical lattices in cold-atom experiments necessitate the consideration of these self-similar spectra for the most general two-dimensional lattice. In a previous work [F. Yılmaz et al., Phys. Rev. A 91, 063628 (2015), 10.1103/PhysRevA.91.063628], we investigated the evolution of the spectrum for an experimentally realized lattice which was tuned by changing the unit-cell structure but keeping the square Bravais lattice fixed. We now consider all possible Bravais lattices in two dimensions and investigate the structure of the Hofstadter butterfly as the lattice is deformed between lattices with different point-symmetry groups. We model the optical lattice with a sinusoidal real-space potential and obtain the tight-binding model for any lattice geometry by calculating the Wannier functions. We introduce the magnetic field via Peierls substitution and numerically calculate the energy spectrum. The transition between the two most symmetric lattices, i.e., the triangular and the square lattices, displays the importance of bipartite symmetry featuring deformation as well as closing of some of the major energy gaps. The transitions from the square to rectangular lattice and from the triangular to centered rectangular lattices are analyzed in terms of coupling of one-dimensional chains. We calculate the Chern numbers of the major gaps and Chern number transfer between bands during the transitions. We use gap Chern numbers to identify distinct topological regions in the space of Bravais lattices.
15. Extension of the approximate two-dimensional electron gas formulation
Science.gov (United States)
Pierret, R. F.
1985-07-01
The functional two-dimensional electron gas (2DEG) formalism employed in the analysis of modulation-doped field-effect transistors is extended to properly account for the bulk charge and to more accurately model sub- and near-threshold behavior. The implemented changes basically transform the functional formulation from an above-threshold formalism for lightly doped structures to one of additional utility which automatically approaches expected limits under widely divergent conditions. Sample computations of the surface carrier concentration, relevant energy level positionings, and the semiconductor depletion width as a function of surface potential and doping are also presented and examined. These computations exhibit the general utility of the extended theory and provide an indirect evaluation of the standard two-level 2DEG theory.
16. Polarons and molecules in a two-dimensional Fermi gas
DEFF Research Database (Denmark)
Zöllner, Sascha; Bruun, Georg Morten; Pethick, C. J.
2011-01-01
We study an impurity atom in a two-dimensional Fermi gas using variational wave functions for (i) an impurity dressed by particle-hole excitations (polaron) and (ii) a dimer consisting of the impurity and a majority atom. In contrast to three dimensions, where similar calculations predict a sharp...... transition to a dimer state with increasing interspecies attraction, we show that the polaron Ansatz always gives a lower energy. However, the exact solution for a heavy impurity reveals that both a two-body bound state and distortions of the Fermi sea are crucial. This reflects the importance of particle......-hole pairs in lower dimensions and makes simple variational calculations unreliable. We show that the energy of an impurity gives important information about its dressing cloud, for which both Ansätze give inaccurate results....
17. Diffusion in the two-dimensional nonoverlapping Lorentz gas
Science.gov (United States)
James, Corinne P.; Evans, Glenn T.
1987-10-01
The self-diffusion coefficient, velocity autocorrelation function, and distribution of collision times for a two-dimensional nonoverlapping Lorentz gas were calculated using molecular dynamics simulation. The systems studied covered a range of densities, from a packing fraction (πNr2/L2) of 0.01 to 0.8. Self-diffusion coefficients were found to agree to all densities with kinetic theory predictions [A. Weijland and J. M. J. van Leeuwen, Physica 38, 35 (1968)] if the radial distribution function (rdf) was taken into account. The density dependence of the decay of the velocity autocorrelation function was qualitatively different from that predicted by kinetic theory. The distribution of collision times was nearly exponential for all but the highest density studied.
18. Two-dimensional Talbot self-imaging via Electromagnetically induced lattice
Science.gov (United States)
Wen, Feng; Wang, Wei; Ahmed, Irfan; Wang, Hongxing; Zhang, Yiqi; Zhang, Yanpeng; Mahesar, Abdul Rasheed; Xiao, Min
2017-02-01
We propose a lensless optical method for imaging two-dimensional ultra-cold atoms (or molecules) in which the image can be non-locally observed by coincidence recording of entangled photon pairs. In particular, we focus on the transverse and longitudinal resolutions of images under various scanning methods. In addition, the role of the induced nonmaterial lattice on the image contrast is investigated. Our work shows a non-destructive and lensless way to image ultra-cold atoms or molecules that can be further used for two-dimensional atomic super-resolution optical testing and sub-wavelength lithography.
19. Tuning of band gaps for a two-dimensional piezoelectric phononic crystal with a rectangular lattice
Institute of Scientific and Technical Information of China (English)
Yize Wang; Fengming Li; Yuesheng Wang; Kikuo Kishimoto; Wenhu Huang
2009-01-01
In this paper, the elastic wave propagation in a two-dimensional piezoelectric phononic crystal is studied by considering the mechanic-electric coupling. The gener-alized eigenvalue equation is obtained by the relation of the mechanic and electric fields as well as the Bloch-Floquet the-orem. The band structures of both the in-plane and anti-plane modes are calculated for a rectangular lattice by the plane-wave expansion method. The effects of the lattice constant ratio and the piezoelectricity with different filling fractions are analyzed. The results show that the largest gap width is not always obtained for a square lattice. In some situations, a rectangular lattice may generate larger gaps. The band gap characteristics are influenced obviously by the piezoelectric-ity with the larger lattice constant ratios and the filling frac-tions.
20. Periodic, quasiperiodic, and chaotic breathers in two-dimensional discrete β-Fermi-Pasta-Ulam lattice
Institute of Scientific and Technical Information of China (English)
Xu Quan; Tian Qiang
2013-01-01
Using numerical method,we investigate whether periodic,quasiperiodic,and chaotic breathers are supported by the two-dimensional discrete Fermi-Pasta-Ulam (FPU) lattice with linear dispersion term.The spatial profile and time evolution of the two-dimensional discrete β-FPU lattice are segregated by the method of separation of variables,and the numerical simulations suggest that the discrete breathers (DBs) are supported by the system.By introducing a periodic interaction into the linear interaction between the atoms,we achieve the coupling of two incommensurate frequencies for a single DB,and the numerical simulations suggest that the quasiperiodic and chaotic breathers are supported by the system,too.
1. Non-classical photon correlation in a two-dimensional photonic lattice
CERN Document Server
Gao, Jun; Lin, Xiao-Feng; Jiao, Zhi-Qiang; Feng, Zhen; Zhou, Zheng; Gao, Zhen-Wei; Xu, Xiao-Yun; Chen, Yuan; Tang, Hao; Jin, Xian-Min
2016-01-01
Quantum interference and quantum correlation, as two main features of quantum optics, play an essential role in quantum information applications, such as multi-particle quantum walk and boson sampling. While many experimental demonstrations have been done in one-dimensional waveguide arrays, it remains unexplored in higher dimensions due to tight requirement of manipulating and detecting photons in large-scale. Here, we experimentally observe non-classical correlation of two identical photons in a fully coupled two-dimensional structure, i.e. photonic lattice manufactured by three-dimensional femtosecond laser writing. Photon interference consists of 36 Hong-Ou-Mandel interference and 9 bunching. The overlap between measured and simulated distribution is up to $0.890\\pm0.001$. Clear photon correlation is observed in the two-dimensional photonic lattice. Combining with controllably engineered disorder, our results open new perspectives towards large-scale implementation of quantum simulation on integrated phot...
2. Tunable band topology reflected by fractional quantum Hall States in two-dimensional lattices.
Science.gov (United States)
Wang, Dong; Liu, Zhao; Cao, Junpeng; Fan, Heng
2013-11-01
Two-dimensional lattice models subjected to an external effective magnetic field can form nontrivial band topologies characterized by nonzero integer band Chern numbers. In this Letter, we investigate such a lattice model originating from the Hofstadter model and demonstrate that the band topology transitions can be realized by simply introducing tunable longer-range hopping. The rich phase diagram of band Chern numbers is obtained for the simple rational flux density and a classification of phases is presented. In the presence of interactions, the existence of fractional quantum Hall states in both |C| = 1 and |C| > 1 bands is confirmed, which can reflect the band topologies in different phases. In contrast, when our model reduces to a one-dimensional lattice, the ground states are crucially different from fractional quantum Hall states. Our results may provide insights into the study of new fractional quantum Hall states and experimental realizations of various topological phases in optical lattices.
3. Two-Dimensional Photonic Band-Gap Defect Modes with Deformed Lattice
Institute of Scientific and Technical Information of China (English)
CAI Xiang-Hua; ZHENG Wan-Hua; MA Xiao-Tao; REN Gang; XIA Jian-Bai
2005-01-01
@@ A numerical study of the defect modes in two-dimensional photonic crystals with deformed triangular lattice is presented by using the supercell method and the finite-difference time-domain method We find the stretch or shrink of the lattice can bring the change not only on the frequencies of the defect modes but also on their magnetic field distributions. We obtain the separation of the doubly degenerate dipole modes with the change of the lattice and find that both the stretch and the shrink of the lattice can make the dipole modes separate large enough to realize the single-mode emission. These results may be advantageous to the manufacture of photonic crystal lasers and provide a new way to realize the single-mode operation in photonic crystal lasers.
4. The mean field study of phase transitions in two dimensional Kagome lattice under local anisotropy
Directory of Open Access Journals (Sweden)
S. Mortezapour
2007-06-01
Full Text Available In this work we investigated the critical properties of the anti-ferromagnetic XY model on a two dimensional Kagome lattice under single-ion easy-axes anisotropy. Employing the mean field theory, we found that this model shows a second order phase transition from disordered to all-in all-out state for any value of anisotropy.
5. Light-Induced Hofstadter's Butterfly Spectrum of Ultracold Atoms on the Two-Dimensional Kagome Lattice
Institute of Scientific and Technical Information of China (English)
HOU Jing-Min
2009-01-01
We investigate the energy spectrum of ultracold atoms on the two-dimensional Kagome optical lattice under an effective magnetic field,which can be realized with laser beams.We derive the generalized Harper's equations from the Schr(o)dinger equation.The energy spectrum with a fractal band structure is obtained by numerically solving the generalized Harper's equations.We analyze the properties of the Hofstadter's butterfly spectrum and discuss its observability.
6. Quantum Monte Carlo simulation of a two-dimensional Majorana lattice model
Science.gov (United States)
Hayata, Tomoya; Yamamoto, Arata
2017-07-01
We study interacting Majorana fermions in two dimensions as a low-energy effective model of a vortex lattice in two-dimensional time-reversal-invariant topological superconductors. For that purpose, we implement ab initio quantum Monte Carlo simulation to the Majorana fermion system in which the path-integral measure is given by a semipositive Pfaffian. We discuss spontaneous breaking of time-reversal symmetry at finite temperatures.
7. Closed-form evaluation of two-dimensional static lattice sums
Science.gov (United States)
Yakubovich, S.; Drygas, P.; Mityushev, V.
2016-11-01
Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sums S2 (for conductivity) and T2 (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between S2 and T2 for different lattices are established. In particular, the value S2=π for the square and hexagonal arrays is discussed and T2=π/2 for the hexagonal is deduced.
8. Competitive irreversible random one-, two-, three-, . . . point adsorption on two-dimensional lattices
Science.gov (United States)
Evans, J. W.; Nord, R. S.
1985-02-01
An analytic treatment of competitive, irreversible (immobile) random one-, two-, three-, . . . point adsorption (or monomer, dimer, trimer, . . . filling) on infinite, uniform two-dimensional lattices is provided by applying previously developed truncation schemes to the hierarchial form of the appropriate master equations. The behavior of these processes for two competing species is displayed by plotting families of filling trajectories'' in the partial-coverage plane for various ratios of adsorption rates. The time or coverage dependence of various subconfiguration probabilities can also be analyzed. For processes where no one-point (monomer) adsorption occurs, the lattice cannot fill completely; accurate estimates of the total (and partial) saturation coverages can be obtained.
9. Fundamental and vortex solitons in a two-dimensional optical lattice
CERN Document Server
Yang, J; Yang, Jianke; Musslimani, Ziad
2003-01-01
Fundamental and vortex solitons in a two-dimensional optically induced waveguide array are reported. In the strong localization regime, the fundamental soliton is largely confined to one lattice site, while the vortex state comprises of four fundamental modes superimposed in a square configuration with a phase structure that is topologically equivalent to the conventional vortex. However, in the weak localization regime, both the fundamental and vortex solitons spread over many lattice sites. We further show that fundamental and vortex solitons are stable against small perturbations in the strong localization regime.
10. Dynamic Effective Medium Theory for Two-Dimensional Non-Magnetic Metamaterial Lattices using Multipole Expansion
CERN Document Server
Chremmos, Ioannis; Giamalaki, Melpomeni; Yannopapas, Vassilios; Paspalakis, Emmanuel
2014-01-01
We present a formulation for deriving effective medium properties of infinitely periodic two-dimensional metamaterial lattice structures beyond the static and quasi-static limits. We utilize the multipole expansions, where the polarization currents associated with the supported Bloch modes are expressed via the electric dipole, magnetic dipole, and electric quadrupole moments per unit length. We then propose a method to calculate the Bloch modes based on the lattice geometry and individual unit element structure. The results revert to well-known formulas in the quasistatic limit and are useful for the homogenization of nanorod-type metamaterials which are frequently used in optical applications.
11. Heteroepitaxial growth modes with dislocations in a two-dimensional elastic lattice model
Science.gov (United States)
Katsuno, Hiroyasu; Uwaha, Makio; Saito, Yukio
2008-11-01
We study equilibrium shapes of adsorbate crystals by allowing a possibility of dislocations on an elastic substrate in a two-dimensional lattice model. The ground state energy is calculated numerically with the use of an elastic lattice Green's function. From the equilibrium shapes determined for various coverages, we infer the growth mode. As the misfit parameter increases, the growth mode changes from the Frank-van der Merwe (FM) to the Stranski-Krastanov (SK), further to the FM with dislocations for a parameter range of ordinary semiconductor materials. Conceivable growth modes such as the SK with dislocations appear in a parameter range between the SK and the FM with dislocations.
12. Boundaries determine the formation energies of lattice defects in two-dimensional buckled materials
Science.gov (United States)
Jain, Sandeep K.; Juričić, Vladimir; Barkema, Gerard T.
2016-07-01
Lattice defects are inevitably present in two-dimensional materials, with direct implications on their physical and chemical properties. We show that the formation energy of a lattice defect in buckled two-dimensional crystals is not uniquely defined as it takes different values for different boundary conditions even in the thermodynamic limit, as opposed to their perfectly planar counterparts. Also, the approach to the thermodynamic limit follows a different scaling: inversely proportional to the logarithm of the system size for buckled materials, rather than the usual power-law approach. In graphene samples of ˜1000 atoms, different boundary conditions can cause differences exceeding 10 eV. Besides presenting numerical evidence in simulations, we show that the universal features in this behavior can be understood with simple bead-spring models. Fundamentally, our findings imply that it is necessary to specify the boundary conditions for the energy of the lattice defects in the buckled two-dimensional crystals to be uniquely defined, and this may explain the lack of agreement in the reported values of formation energies in graphene. We argue that boundary conditions may also have an impact on other physical observables such as the melting temperature.
13. Designing artificial two dimensional electron lattice on metal surface: a Kagome-like lattice as an example.
Science.gov (United States)
Li, Shuai; Qiu, Wen-Xuan; Gao, Jin-Hua
2016-07-07
Recently, a new kind of artificial two dimensional (2D) electron lattice on the nanoscale, i.e. molecular graphene, has drawn a lot of interest, where the metal surface electrons are transformed into a honeycomb lattice via absorbing a molecular lattice on the metal surface [Gomes et al., Nature, 2012, 438, 306; Wang et al., Phys. Rev. Lett., 2014, 113, 196803]. In this work, we theoretically demonstrate that this technique can be readily used to build other complex 2D electron lattices on a metal surface, which are of high interest in the field of condensed matter physics. The main challenge to build a complex 2D electron lattice is that this is a quantum antidot system, where the absorbed molecule normally exerts a repulsive potential on the surface electrons. Thus, there is no straightforward corresponding relation between the molecular lattice pattern and the desired 2D lattice of surface electrons. Here, we give an interesting example about the Kagome lattice, which has exotic correlated electronic states. We design a special molecular pattern and show that this molecular lattice can transform the surface electrons into a Kagome-like lattice. The numerical simulation is conducted using a Cu(111) surface and CO molecules. We first estimate the effective parameters of the Cu/CO system by fitting experimental data of the molecular graphene. Then, we calculate the corresponding energy bands and LDOS of the surface electrons in the presence of the proposed molecular lattice. Finally, we interpret the numerical results by the tight binding model of the Kagome lattice. We hope that our work can stimulate further theoretical and experimental interest in this novel artificial 2D electron lattice system.
14. Theoretical and numerical investigation of HF elastic wave propagation in two-dimensional periodic beam lattices
Science.gov (United States)
Tie, B.; Tian, B. Y.; Aubry, D.
2013-12-01
The elastic wave propagation phenomena in two-dimensional periodic beam lattices are studied by using the Bloch wave transform. The numerical modeling is applied to the hexagonal and the rectangular beam lattices, in which, both the in-plane (with respect to the lattice plane) and out-of-plane waves are considered. The dispersion relations are obtained by calculating the Bloch eigenfrequencies and eigenmodes. The frequency bandgaps are observed and the influence of the elastic and geometric properties of the primitive cell on the bandgaps is studied. By analyzing the phase and the group velocities of the Bloch wave modes, the anisotropic behaviors and the dispersive characteristics of the hexagonal beam lattice with respect to the wave propagation are highlighted in high frequency domains. One important result presented herein is the comparison between the first Bloch wave modes to the membrane and bending/transverse shear wave modes of the classical equivalent homogenized orthotropic plate model of the hexagonal beam lattice. It is shown that, in low frequency ranges, the homogenized plate model can correctly represent both the in-plane and out-of-plane dynamic behaviors of the beam lattice, its frequency validity domain can be precisely evaluated thanks to the Bloch modal analysis. As another important and original result, we have highlighted the existence of the retropropagating Bloch wave modes with a negative group velocity, and of the corresponding "retro-propagating" frequency bands.
15. Quantum State Transfer in a Two-dimensional Regular Spin Lattice of Triangular Shape
CERN Document Server
Miki, Hiroshi; Vinet, Luc; Zhedanov, Alexei
2012-01-01
Quantum state transfer in a triangular domain of a two-dimensional, equally-spaced, spin lat- tice with non-homogeneous nearest-neighbor couplings is analyzed. An exact solution of the one- excitation dynamics is provided in terms of 2-variable Krawtchouk orthogonal polynomials that have been recently defined. The probability amplitude for an excitation to transit from one site to another is given. For some values of the parameters, perfect transfer is shown to take place from the apex of the lattice to the boundary hypotenuse.
16. High applicability of two-dimensional phosphorous in Kagome lattice predicted from first-principles calculations
OpenAIRE
Peng-Jen Chen; Horng-Tay Jeng
2016-01-01
A new semiconducting phase of two-dimensional phosphorous in the Kagome lattice is proposed from first-principles calculations. The band gaps of the monolayer (ML) and bulk Kagome phosphorous (Kagome-P) are 2.00 and 1.11 eV, respectively. The magnitude of the band gap is tunable by applying the in-plane strain and/or changing the number of stacking layers. High optical absorption coefficients at the visible light region are predicted for multilayer Kagome-P, indicating potential applications ...
17. Hamiltonian dynamics of the two-dimensional lattice {phi}{sup 4} model
Energy Technology Data Exchange (ETDEWEB)
Caiani, Lando [Scuola Internazionale Superiore di Studi Avanzati (SISSA/ISAS), Trieste (Italy); Casetti, Lapo [Istituto Nazionale di Fisica della Materia (INFM), Unita di Ricerca del Politecnico di Torino, Dipartimento di Fisica, Politecnico di Torino, Turin (Italy); Pettini, Marco [Osservatorio Astrofisico di Arcetri, Florence (Italy)
1998-04-17
The Hamiltonian dynamics of the classical {phi}{sup 4} model on a two-dimensional square lattice is investigated by means of numerical simulations. The macroscopic observables are computed as time averages. The results clearly reveal the presence of the continuous phase transition at a finite energy density and are consistent both qualitatively and quantitatively with the predictions of equilibrium statistical mechanics. The Hamiltonian microscopic dynamics also exhibits critical slowing down close to the transition. Moreover, the relationship between chaos and the phase transition is considered, and interpreted in the light of a geometrization of dynamics. (author)
18. Restoration of supersymmetry in two-dimensional SYM with sixteen supercharges on the lattice
CERN Document Server
Giguère, Eric
2015-01-01
We perform lattice simulations of two-dimensional supersymmetric Yang-Mills theory with sixteen supercharges with a lattice action which has two exact supercharges (Sugino lattice action). According to the gauge/gravity duality, the theory at finite temperature is expected to be well described by the corresponding black 1-branes, at low temperature in the large N limit. We aim to confirm the duality conjecture by comparing the lattice results with the theoretical predictions obtained in the gravity side. In this article, at the beginning of this study, we examine the supersymmetric Ward-Takahashi identity to test whether the lattice action reproduces the correct continuum theory. Numerical results of the SUSY WTI strongly suggest us that any cut-off effects, which break supersymmetry, disappear in the continuum limit. In addition, we study the issue of degenerate vacua and find that the admissiblilty condition or any other constraints of the link fields which guarantee the unique vacuum are not always needed.
19. Restoration of supersymmetry in two-dimensional SYM with sixteen supercharges on the lattice
Energy Technology Data Exchange (ETDEWEB)
Giguère, Eric [Department of Physics, University of Hokkaido,Sapporo, Hokkaido 060-0810 (Japan); Kadoh, Daisuke [KEK Theory Center, High Energy Accelerator Research Organization (KEK),Tsukuba, Ibaraki 305-0801 (Japan)
2015-05-18
We perform lattice simulations of two-dimensional supersymmetric Yang-Mills theory with sixteen supercharges with a lattice action which has two exact supercharges (Sugino lattice action). According to the gauge/gravity duality, the theory at finite temperature is expected to be well described by the corresponding black 1-branes, at low temperature in the large N limit. We aim to confirm the duality conjecture by comparing the lattice results with the theoretical predictions obtained in the gravity side. In this article, at the beginning of this study, we examine the supersymmetric Ward-Takahashi identity to test whether the lattice action reproduces the correct continuum theory. Numerical results of the SUSY WTI strongly suggest us that any cut-off effects, which break supersymmetry, disappear in the continuum limit. In addition, we study the issue of degenerate vacua and find that the admissiblilty condition or any other constraints of the link fields which guarantee the unique vacuum are not always needed.
20. Hydration of an apolar solute in a two-dimensional waterlike lattice fluid.
Science.gov (United States)
Buzano, C; De Stefanis, E; Pretti, M
2005-05-01
In a previous work, we investigated a two-dimensional lattice-fluid model, displaying some waterlike thermodynamic anomalies. The model, defined on a triangular lattice, is now extended to aqueous solutions with apolar species. Water molecules are of the "Mercedes Benz" type, i.e., they possess a D3 (equilateral triangle) symmetry, with three equivalent bonding arms. Bond formation depends both on orientation and local density. The insertion of inert molecules displays typical signatures of hydrophobic hydration: large positive transfer free energy, large negative transfer entropy (at low temperature), strong temperature dependence of the transfer enthalpy and entropy, i.e., large (positive) transfer heat capacity. Model properties are derived by a generalized first order approximation on a triangle cluster.
1. Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice
CERN Document Server
Casini, Horacio
2014-01-01
We study entanglement entropy (EE) for a Maxwell field in 2+1 dimensions. We do numerical calculations in two dimensional lattices. This gives a concrete example of the general results of our recent work on entropy for lattice gauge fields using an algebraic approach. To evaluate the entropies we extend the standard calculation methods for the entropy of Gaussian states in canonical commutation algebras to the more general case of algebras with center and arbitrary numerical commutators. We find that while the entropy depends on the details of the algebra choice, mutual information has a well defined continuum limit. We study several universal terms for the entropy of the Maxwell field and compare with the case of a massless scalar field. We find some interesting new phenomena: An "evanescent" logarithmically divergent term in the entropy with topological coefficient which does not have any correspondence with ultraviolet entanglement in the universal quantities, and a non standard way in which strong subaddi...
2. Competitive irreversible random one-, two-, three-,. point adsorption on two-dimensional lattices
Energy Technology Data Exchange (ETDEWEB)
Evans, J.W.; Nord, R.S.
1985-02-15
An analytic treatment of competitive, irreversible (immobile) random one-, two-, three-, . . . point adsorption (or monomer, dimer, trimer, . . . filling) on infinite, uniform two-dimensional lattices is provided by applying previously developed truncation schemes to the hierarchial form of the appropriate master equations. The behavior of these processes for two competing species is displayed by plotting families of ''filling trajectories'' in the partial-coverage plane for various ratios of adsorption rates. The time or coverage dependence of various subconfiguration probabilities can also be analyzed. For processes where no one-point (monomer) adsorption occurs, the lattice cannot fill completely; accurate estimates of the total (and partial) saturation coverages can be obtained.
3. Spin superconductivity in the frustrated two-dimensional antiferromagnet in the square lattice
Science.gov (United States)
Lima, L. S.
2017-02-01
We use the SU(2) Schwinger boson formalism to study the spin transport in the two-dimensional S = 1 / 2 frustrated Heisenberg antiferromagnet in a square lattice, considering the second-neighbors interactions in the diagonal. We have obtained a spin superfluid behavior for the spin transport to this system similar to obtained recently to the triangular lattice. We consider an antiferromagnetic inter-chain coupling on the diagonal, J2 > 0 , and the nearest-neighbor coupling antiferromagnetic J1 > 0 . We also have in the critical temperature T0, where the correlation length ξ → 0 , that the system suffers a transition from an ordered ground state to a disordered ground state.
4. Vector meson masses in two-dimensional SU(NC) lattice gauge theory with massive quarks
Institute of Scientific and Technical Information of China (English)
JIANG Jun-Qin
2008-01-01
Using an improved lattice Hamiltonian with massive Wilson quarks a variational method is applied to study the dependence of the vector meson mass Mv on the quark mass m and the Wilson parameter r in in the scaling window 1 ≤ 1/g2 ≤ 2, Mv/g is approximately linear in m, but Mv/g obviously does not depend on r (this differs from the quark condensate). Particularly for m → 0 our numerical results agree very well with Bhattacharya's analytical strong coupling result in the continuum, and the value of ((e)Mv/(e)m) |mm=0 in two-dimensional SU(NC) lattice gauge theory is very close to that in Schwinger model.
5. Two-dimensional-lattice spin models with long-range antiferromagnetic interactions
Science.gov (United States)
Romano, S.
1991-10-01
We consider a classical system, consisting of m-component unit vectors (m=2,3), associated with a two-dimensional lattice \\{uk||k∈openZ2\\} and interacting via translationally and rotationally invariant antiferromagnetic pair potentials of the long-range form W=Wjk=ɛ||xj-xk||-puj.uk, p>2, where ɛ is a positive quantity, setting energy and temperature scales (i.e., T*=kBT/ɛ), and xk are the coordinates of the lattice sites. A spin-wave approach predicts orientational disorder (in the thermodynamic limit) at all finite temperatures and for all p>2 this agrees with available rigorous results for p>=4, whereas no such theorems are known in the literature when 22.
6. Piezoelectricity in two-dimensional materials: Comparative study between lattice dynamics and ab initio calculations
Science.gov (United States)
Michel, K. H.; ćakır, D.; Sevik, C.; Peeters, F. M.
2017-03-01
The elastic constant C11 and piezoelectric stress constant e1 ,11 of two-dimensional (2D) dielectric materials comprising h-BN, 2 H -MoS2 , and other transition-metal dichalcogenides and dioxides are calculated using lattice dynamical theory. The results are compared with corresponding quantities obtained with ab initio calculations. We identify the difference between clamped-ion and relaxed-ion contributions with the dependence on inner strains which are due to the relative displacements of the ions in the unit cell. Lattice dynamics allows us to express the inner-strain contributions in terms of microscopic quantities such as effective ionic charges and optoacoustical couplings, which allows us to clarify differences in the piezoelectric behavior between h-BN and MoS2. Trends in the different microscopic quantities as functions of atomic composition are discussed.
7. Conical wave propagation and diffraction in two-dimensional hexagonally packed granular lattices.
Science.gov (United States)
Chong, C; Kevrekidis, P G; Ablowitz, M J; Ma, Yi-Ping
2016-01-01
Linear and nonlinear mechanisms for conical wave propagation in two-dimensional lattices are explored in the realm of phononic crystals. As a prototypical example, a statically compressed granular lattice of spherical particles arranged in a hexagonal packing configuration is analyzed. Upon identifying the dispersion relation of the underlying linear problem, the resulting diffraction properties are considered. Analysis both via a heuristic argument for the linear propagation of a wave packet and via asymptotic analysis leading to the derivation of a Dirac system suggests the occurrence of conical diffraction. This analysis is valid for strong precompression, i.e., near the linear regime. For weak precompression, conical wave propagation is still possible, but the resulting expanding circular wave front is of a nonoscillatory nature, resulting from the complex interplay among the discreteness, nonlinearity, and geometry of the packing. The transition between these two types of propagation is explored.
8. Identification of the dynamics of a two-dimensional grid structure using least square lattice filters
Science.gov (United States)
Montgomery, R. C.; Sundararajan, N.
1984-01-01
The basic theory of least square lattice filters and their use in identification of structural dynamics systems is summarized. Thereafter, this theory is applied to a two-dimensional grid structure made of overlapping bars. Previously, this theory has been applied to an integral beam. System identification results are presented for both simulated and experimental tests and they are compared with those predicted using finite element modelling. The lattice filtering approach works well for simulated data based on finite element modelling. However, considerable discrepancy exists between estimates obtained from experimental data and the finite element analysis. It is believed that this discrepancy is the result of inadequacies in the finite element modelling to represent the damped motion of the laboratory apparatus.
9. Suppression of photothermal convection of microparticles in two dimensional nanoplasmonic optical lattice
Science.gov (United States)
Chen, Yi-Chung; Yossifon, Gilad; Yang, Ya-Tang
2016-11-01
Photothermal convection has been a major obstacle for stable particle trapping in plasmonic optical tweezer at high optical power. Here, we demonstrate a strategy to suppress the plasmonic photothermal convection by using vanishingly small thermal expansion coefficient of water at low temperature. A simple square nanoplasmonic array is illuminated with a loosely Gaussian beam to produce a two dimensional optical lattice for trapping of micro particles. We observe stable particle trapping due to near-field optical gradient forces at elevated optical power at low temperature. In contrast, for the same optical power at room temperature, the particles are convected away from the center of the optical lattice without their accumulation. This technique will greatly increase usable optical power and enhance the trapping capability of plasmonic optical tweezer.
10. An improvement of the lattice theory of dislocation for a two-dimensional triangular crystal
Institute of Scientific and Technical Information of China (English)
Wang Shao-Feng
2005-01-01
The structure of dislocation in a two-dimensional triangular crystal has been studied theoretically on the basis of atomic interaction and lattice statics. The theory presented in this paper is an improvement to that published previously.Within a reasonable interaction approximation, a new dislocation equation is obtained, which remedies a fault existing in the lattice theory of dislocation. A better simplification of non-diagonal terms of the kernel is given. The solution of the new dislocation equation asymptotically becomes the same as that obtained in the elastic theory, and agrees with experimental data. It is found that the solution is formally identical with that proposed phenomenologically by Foreman et al, where the parameter can be chosen freely, but cannot uniquely determined from theory. Indeed, if the parameter in the expression of the solution is selected suitably, the expression can be well applied to describe the fine structure of the dislocation.
11. Dual geometric worm algorithm for two-dimensional discrete classical lattice models
Science.gov (United States)
Hitchcock, Peter; Sørensen, Erik S.; Alet, Fabien
2004-07-01
We present a dual geometrical worm algorithm for two-dimensional Ising models. The existence of such dual algorithms was first pointed out by Prokof’ev and Svistunov [N. Prokof’ev and B. Svistunov, Phys. Rev. Lett. 87, 160601 (2001)]. The algorithm is defined on the dual lattice and is formulated in terms of bond variables and can therefore be generalized to other two-dimensional models that can be formulated in terms of bond variables. We also discuss two related algorithms formulated on the direct lattice, applicable in any dimension. These latter algorithms turn out to be less efficient but of considerable intrinsic interest. We show how such algorithms quite generally can be “directed” by minimizing the probability for the worms to erase themselves. Explicit proofs of detailed balance are given for all the algorithms. In terms of computational efficiency the dual geometrical worm algorithm is comparable to well known cluster algorithms such as the Swendsen-Wang and Wolff algorithms, however, it is quite different in structure and allows for a very simple and efficient implementation. The dual algorithm also allows for a very elegant way of calculating the domain wall free energy.
12. Long-range ferrimagnetic order in a two-dimensional supramolecular Kondo lattice
Science.gov (United States)
Girovsky, Jan; Nowakowski, Jan; Ali, Md. Ehesan; Baljozovic, Milos; Rossmann, Harald R.; Nijs, Thomas; Aeby, Elise A.; Nowakowska, Sylwia; Siewert, Dorota; Srivastava, Gitika; Wäckerlin, Christian; Dreiser, Jan; Decurtins, Silvio; Liu, Shi-Xia; Oppeneer, Peter M.; Jung, Thomas A.; Ballav, Nirmalya
2017-05-01
Realization of long-range magnetic order in surface-supported two-dimensional systems has been challenging, mainly due to the competition between fundamental magnetic interactions as the short-range Kondo effect and spin-stabilizing magnetic exchange interactions. Spin-bearing molecules on conducting substrates represent a rich platform to investigate the interplay of these fundamental magnetic interactions. Here we demonstrate the direct observation of long-range ferrimagnetic order emerging in a two-dimensional supramolecular Kondo lattice. The lattice consists of paramagnetic hexadeca-fluorinated iron phthalocyanine (FeFPc) and manganese phthalocyanine (MnPc) molecules co-assembled into a checkerboard pattern on single-crystalline Au(111) substrates. Remarkably, the remanent magnetic moments are oriented in the out-of-plane direction with significant contribution from orbital moments. First-principles calculations reveal that the FeFPc-MnPc antiferromagnetic nearest-neighbour coupling is mediated by the Ruderman-Kittel-Kasuya-Yosida exchange interaction via the Au substrate electronic states. Our findings suggest the use of molecular frameworks to engineer novel low-dimensional magnetically ordered materials and their application in molecular quantum devices.
13. Long-range ferrimagnetic order in a two-dimensional supramolecular Kondo lattice.
Science.gov (United States)
Girovsky, Jan; Nowakowski, Jan; Ali, Md Ehesan; Baljozovic, Milos; Rossmann, Harald R; Nijs, Thomas; Aeby, Elise A; Nowakowska, Sylwia; Siewert, Dorota; Srivastava, Gitika; Wäckerlin, Christian; Dreiser, Jan; Decurtins, Silvio; Liu, Shi-Xia; Oppeneer, Peter M; Jung, Thomas A; Ballav, Nirmalya
2017-05-22
Realization of long-range magnetic order in surface-supported two-dimensional systems has been challenging, mainly due to the competition between fundamental magnetic interactions as the short-range Kondo effect and spin-stabilizing magnetic exchange interactions. Spin-bearing molecules on conducting substrates represent a rich platform to investigate the interplay of these fundamental magnetic interactions. Here we demonstrate the direct observation of long-range ferrimagnetic order emerging in a two-dimensional supramolecular Kondo lattice. The lattice consists of paramagnetic hexadeca-fluorinated iron phthalocyanine (FeFPc) and manganese phthalocyanine (MnPc) molecules co-assembled into a checkerboard pattern on single-crystalline Au(111) substrates. Remarkably, the remanent magnetic moments are oriented in the out-of-plane direction with significant contribution from orbital moments. First-principles calculations reveal that the FeFPc-MnPc antiferromagnetic nearest-neighbour coupling is mediated by the Ruderman-Kittel-Kasuya-Yosida exchange interaction via the Au substrate electronic states. Our findings suggest the use of molecular frameworks to engineer novel low-dimensional magnetically ordered materials and their application in molecular quantum devices.
14. Quantum spin liquid and magnetic order in a two-dimensional nonsymmorphic lattice: Considering the distorted kagome lattice of volborthite
Science.gov (United States)
Chern, Li Ern; Hwang, Kyusung; Mizoguchi, Tomonari; Huh, Yejin; Kim, Yong Baek
2017-07-01
The Kagome-lattice-based material, volborthite, Cu3V2O7(OH) 2.2 H2O , has been considered as a promising platform for discovery of unusual quantum ground states due to the frustrated nature of spin interaction. We explore possible quantum spin liquid and magnetically ordered phases in a two-dimensional nonsymmorphic lattice, which is described by the plane group p 2 g g , consistent with the spatial anisotropy of the spin model derived from density functional theory (DFT) for volborthite. Using the projective symmetry group (PSG) analysis and Schwinger boson mean field theory, we classify possible spin liquid phases with bosonic spinons and investigate magnetically ordered phases connected to such states. It is shown, in general, that only translationally invariant mean field spin liquid ansatzes are allowed in two-dimensional nonsymmorphic lattices. We study the mean field phase diagram of the DFT-derived spin model and find that possible quantum spin liquid phases are connected to two types of magnetically ordered phases, a coplanar incommensurate (q ,0 ) spiral order as the ground state and a closely competing coplanar commensurate (π ,π ) spin density wave order. In addition, periodicity enhancement of the two-spinon continuum, a consequence of symmetry fractionalization, is found in the spin liquid state connected to the (π ,π ) spin density wave order. We discuss relevance of these results to recent and future experiments on volborthite.
15. Dipolar matter-wave solitons in two-dimensional anisotropic discrete lattices
Science.gov (United States)
Chen, Huaiyu; Liu, Yan; Zhang, Qiang; Shi, Yuhan; Pang, Wei; Li, Yongyao
2016-05-01
We numerically demonstrate two-dimensional (2D) matter-wave solitons in the disk-shaped dipolar Bose-Einstein condensates (BECs) trapped in strongly anisotropic optical lattices (OLs) in a disk's plane. The considered OLs are square lattices which can be formed by interfering two pairs of plane waves with different intensities. The hopping rates of the condensates between two adjacent lattices in the orthogonal directions are different, which gives rise to a linearly anisotropic system. We find that when the polarized orientation of the dipoles is parallel to disk's plane with the same direction, the combined effects of the linearly anisotropy and the nonlocal nonlinear anisotropy strongly influence the formations, as well as the dynamics of the lattice solitons. Particularly, the isotropy-pattern solitons (IPSs) are found when these combined effects reach a balance. Motion, collision, and rotation of the IPSs are also studied in detail by means of systematic simulations. We further find that these IPSs can move freely in the 2D anisotropic discrete system, hence giving rise to an anisotropic effective mass. Four types of collisions between the IPSs are identified. By rotating an external magnetic field up to a critical angular velocity, the IPSs can still remain localized and play as a breather. Finally, the influences from the combined effects between the linear and the nonlocal nonlinear anisotropy with consideration of the contact and/or local nonlinearity are discussed too.
16. Bandgaps and directional properties of two-dimensional square beam-like zigzag lattices
Energy Technology Data Exchange (ETDEWEB)
Wang, Yan-Feng; Wang, Yue-Sheng, E-mail: [email protected] [Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044 (China); Zhang, Chuanzeng [Department of Civil Engineering, University of Siegen, Siegen 57068 (Germany)
2014-12-15
In this paper we propose four kinds of two-dimensional square beam-like zigzag lattice structures and study their bandgaps and directional propagation of elastic waves. The band structures are calculated by using the finite element method. Both the in-plane and out-of-plane waves are investigated simultaneously via the three-dimensional Euler beam elements. The mechanism of the bandgap generation is analyzed by studying the vibration modes at the bandgap edges. The effects of the geometry parameters of the xy- and z-zigzag lattices on the bandgaps are investigated and discussed. Multiple complete bandgaps are found owing to the separation of the degeneracy by introducing bending arms. The bandgaps are sensitive to the geometry parameters of the periodic systems. The deformed displacement fields of the harmonic responses of a finite lattice structure subjected to harmonic loads at different positions are illustrated to show the directional wave propagation. An extension of the proposed concept to the hexagonal lattices is also presented. The research work in this paper is relevant to the practical design of cellular structures with enhanced vibro-acoustics performance.
17. Current-injection in a ballastic multiterminal superconductor/two-dimensional electron gas Josephson junction
NARCIS (Netherlands)
Schäpers, Th.; Guzenko, V.A.; Müller, R.P.; Golubov, A.A.; Brinkman, A.; Crecelius, G.; Kaluza, A.; Lüth, H.
2003-01-01
We study the suppression of the critical current in a multi-terminal superconductor/two-dimensional electron gas/superconductor Josephson junction by means of hot carrier injection. As a superconductor Nb is used, while the two-dimensional electron gas is located in a strained InGaAs/InP heterostruc
18. Magnetic properties of two dimensional silicon carbide triangular nanoflakes-based kagome lattices
Energy Technology Data Exchange (ETDEWEB)
Li Xiaowei [Peking University, Center for Applied Physics and Technology, College of Engineering (China); Zhou Jian [Peking University, Department of Materials Science and Engineering (China); Wang Qian, E-mail: [email protected] [Peking University, Center for Applied Physics and Technology, College of Engineering (China); Jena, Puru [Virginia Commonwealth University, Department of Physics (United States)
2012-08-15
Two-dimensional (2D) magnetic kagome lattices are constructed using silicon carbide triangular nanoflakes (SiC-TNFs). Two types of structures with alternating Si and C atoms are studied: the first one is constructed using the C-edged SiC-TNFs as the building blocks and C atoms as the linkers of kagome sites (TNF{sub N}-C-TNF{sub N}) while the second one is composed of the Si-edged SiC-TNFs with Si atoms as linkers (TNF{sub N}-Si-TNF{sub N}). Using density functional theory-based calculations, we show that the fully relaxed TNF{sub N}-C-TNF{sub N} retains the morphology of regular kagome lattice and is ferromagnetism. On the other hand, the TNF{sub N}-Si-TNF{sub N} structure is deformed and antiferromagnetic. However, the ground state of TNF{sub N}-Si-TNF{sub N} structure can be transformed from the antiferromagnetic to ferromagnetic state by applying tensile strain. Monte Carlo simulations indicate that the SiC-TNFs-based kagome lattices can be ferromagnetic at room temperature.
19. Realization and Characterization of a Curved Two-dimensional Electron Gas
Science.gov (United States)
Shaji, Nakul; Deneke, Christoph
2005-03-01
Using the built-in strain from lattice mismatch between Al0.33Ga0.67As and In0.2Ga0.8As as a bending force, a strip of two-dimensional electron gas (2DEG) in an AlxGa1-xAs/GaAs/AlxGa1-xAs heterostructure is curved into a tube when released from the substrate by wet etching. A variety of mesoscopic quantum devices can be defined in such curved 2DEG structures. This technology opens the door for investigating geometry-dependent electron transport under non-uniform magnetic fields. We have defined Hall bar patterns from a sheet of 2DEG using both optical and electron-beam lithography. The sample characterization under an external magnetic field will be discussed.
20. Two-dimensional gas of massless Dirac fermions in graphene.
Science.gov (United States)
Novoselov, K S; Geim, A K; Morozov, S V; Jiang, D; Katsnelson, M I; Grigorieva, I V; Dubonos, S V; Firsov, A A
2005-11-10
Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry. The ideas underlying quantum electrodynamics also influence the theory of condensed matter, but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrödinger equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Dirac's (relativistic) equation. The charge carriers in graphene mimic relativistic particles with zero rest mass and have an effective 'speed of light' c* approximately 10(6) m s(-1). Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions. In particular we have observed the following: first, graphene's conductivity never falls below a minimum value corresponding to the quantum unit of conductance, even when concentrations of charge carriers tend to zero; second, the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; and third, the cyclotron mass m(c) of massless carriers in graphene is described by E = m(c)c*2. This two-dimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a bench-top experiment.
1. Pythagoras's theorem on a two-dimensional lattice from a natural' Dirac operator and Connes's distance formula
Science.gov (United States)
Dai, Jian; Song, Xing-Chang
2001-07-01
One of the key ingredients of Connes's noncommutative geometry is a generalized Dirac operator which induces a metric (Connes's distance) on the pure state space. We generalize such a Dirac operator devised by Dimakis et al, whose Connes distance recovers the linear distance on an one-dimensional lattice, to the two-dimensional case. This Dirac operator has the local eigenvalue property and induces a Euclidean distance on this two-dimensional lattice, which is referred to as natural'. This kind of Dirac operator can be easily generalized into any higher-dimensional lattices.
2. Analysis of Photonic Band Gaps in a Two-Dimensional Triangular Lattice with Superconducting Hollow Rods
Science.gov (United States)
Diaz-Valencia, B. F.; Calero, J. M.
2017-02-01
In this work, we use the plane wave expansion method to calculate photonic band structures in two-dimensional photonic crystals which consist of high-temperature superconducting hollow rods arranged in a triangular lattice. The variation of the photonic band structure with respect to both, the inner radius and the system temperature, is studied, taking into account temperatures below the critical temperature of the superconductor in the low frequencies regime and assuming E polarization of the incident light. Permittivity contrast and nontrivial geometry of the hollow rods lead to the appearance of new band gaps as compared with the case of solid cylinders. Such band gaps can be modulated by means of the inner radius and system temperature.
3. Two-dimensional novel optical lattices with multi-well traps for cold atoms or molecules
Institute of Scientific and Technical Information of China (English)
Junfa Lu; Xianming Ji; Jianping Yin
2006-01-01
We propose some new schemes to constitute two-dimensional (2D) array of multi-well optical dipole traps for cold atoms (or molecules) by using an optical system consisting of a binary π-phase grating and a 2D array of rectangle microlens. We calculate the intensity distribution of each optical well in 2D array of multi-well traps and its geometric parameters and so on. The proposed 2D array of multi-well traps can be used to form novel 2D optical lattices with cold atoms (or molecules), and form various novel optical crystals with cold atoms (or molecules), or to perform quantum computing and quantum information processing on an atom chip, even to realize an array of all-optical multi-well atomic (or molecular) BoseEinstein condensates (BECs) on an all-optical integrated atom (or molecule) chip.
4. Pair formation in Fermi systems with population imbalance in one- and two-dimensional optical lattices
Science.gov (United States)
Batrouni, George
2011-03-01
I will discuss pairing in fermionic systems in one- and two-dimensional optical lattices with population imbalance. This will be done in the context of the attractive fermionic Hubbard model using the Stochastic Green Function algorithm in d=1 while for d=2 we use Determinant Quantum Monte Carlo. This is the first exact QMC study examining the effects of finite temperature which is very important in experiments on ultra-cold atoms. Our results show that, in the ground state, the dominant pairing mechanism is at nonzero center of mass momentum, i.e. FFLO. I will then discuss the effect of finite temperature in the uniform and confined systems and present finite temperature phase diagrams. The numerical results will be compared with experiments. With M. J. Wolak (CQT, National University of Singapore) and V. G. Rousseau (Department of Physics and Astronomy, Louisiana State University).
5. Duality and Fisher zeros in the two-dimensional Potts model on a square lattice.
Science.gov (United States)
Astorino, Marco; Canfora, Fabrizio
2010-05-01
A phenomenological approach to the ferromagnetic two-dimensional (2D) Potts model on square lattice is proposed. Our goal is to present a simple functional form that obeys the known properties possessed by the free energy of the q-state Potts model. The duality symmetry of the 2D Potts model together with the known results on its critical exponent α allows us to fix consistently the details of the proposed expression for the free energy. The agreement of the analytic ansatz with numerical data in the q=3 case is very good at high and low temperatures as well as at the critical point. It is shown that the q>4 cases naturally fit into the same scheme and that one should also expect a good agreement with numerical data. The limiting q=4 case is shortly discussed.
6. Nonlinear localized modes in dipolar Bose–Einstein condensates in two-dimensional optical lattices
Energy Technology Data Exchange (ETDEWEB)
Rojas-Rojas, Santiago, E-mail: [email protected] [Center for Optics and Photonics and MSI-Nucleus on Advanced Optics, Universidad de Concepción, Casilla 160-C, Concepción (Chile); Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción (Chile); Naether, Uta [Instituto de Ciencia de Materiales de Aragón and Departamento de Física de la Materia Condensada, CSIC-Universidad de Zaragoza, 50009 Zaragoza (Spain); Delgado, Aldo [Center for Optics and Photonics and MSI-Nucleus on Advanced Optics, Universidad de Concepción, Casilla 160-C, Concepción (Chile); Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción (Chile); Vicencio, Rodrigo A. [Center for Optics and Photonics and MSI-Nucleus on Advanced Optics, Universidad de Concepción, Casilla 160-C, Concepción (Chile); Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago (Chile)
2016-09-16
Highlights: • We study discrete two-dimensional breathers in dipolar Bose–Einstein Condensates. • Important differences in the properties of three fundamental modes are found. • Norm threshold for existence of 2D breathers varies with dipolar interaction. • The Effective Potential Method is implemented for stability analysis. • Uncommon mobility of 2D discrete solitons is observed. - Abstract: We analyze the existence and properties of discrete localized excitations in a Bose–Einstein condensate loaded into a periodic two-dimensional optical lattice, when a dipolar interaction between atoms is present. The dependence of the Number of Atoms (Norm) on the energy of solutions is studied, along with their stability. Two important features of the system are shown, namely, the absence of the Norm threshold required for localized solutions to exist in finite 2D systems, and the existence of regions in the parameter space where two fundamental solutions are simultaneously unstable. This feature enables mobility of localized solutions, which is an uncommon feature in 2D discrete nonlinear systems. With attractive dipolar interaction, a non-trivial behavior of the Norm dependence is obtained, which is well described by an analytical model.
7. Creating tuneable microwave media from a two-dimensional lattice of re-entrant posts
Science.gov (United States)
Goryachev, Maxim; Tobar, Michael E.
2015-11-01
The potential capabilities of resonators based on two dimensional arrays of re-entrant posts is demonstrated. Such posts may be regarded as magnetically coupled lumped element microwave harmonic oscillators, arranged in a 2D lattices structure, which is enclosed in a 3D cavity. By arranging these elements in certain 2D patterns, we demonstrate how to achieve certain requirements with respect to field localisation and device spectra. Special attention is paid to symmetries of the lattices, mechanical tuning, design of areas of high localisation of magnetic energy; this in turn creates unique discrete mode spectra. We demonstrate analogies between systems designed on the proposed platform and well known physical phenomena such as polarisation, frustration, and Whispering Gallery Modes. The mechanical tunability of the cavity with multiple posts is analysed, and its consequences to optomechanical applications is calculated. One particular application to quantum memory is demonstrated with a cavity design consisting of separate resonators analogous to discrete Fabry-Pérot resonators. Finally, we propose a generalised approach to a microwave system design based on the concept of Programmable Cavity Arrays.
8. Ground State and Collective Modes of Magnetic Dipoles Fixed on Two-Dimensional Lattice Sites
Science.gov (United States)
Feldmann, John; Kalman, Gabor; Hartmann, Peter; Rosenberg, Marlene
2006-10-01
In complex (dusty) plasmas the grains may be endowed with intrinsic dipole moments. We present here our results of theoretical calculations accompanied by and Molecular Dynamics simulation findings on the ground state configuration and on the collective modes mode spectrum of a system of magnetic dipoles, interacting via the magnetic dipole pair-dipole potential, fixed on two-dimensional (2D) lattice sites. In particular, we We study a family of lattices that can be characterized by two parameters: (parallelogram)---the aspect ratio, c/a, and the rhombic angle, phi. The The new collective modes of in the system associated with the dipole-dipole interaction are the angular oscillations (or wobbling) of the direction of the dipoles about their equilibrium configurations. We identify in-plane and out-of-plane modes and display their dispersions. Orders of magnitudes of the parameters of the system relevant to possible future experiments will be discussed. JD Feldmann, G J Kalman and M Rosenberg, J. Phys. A: Math. Gen. 39 (2006) 4549-4553
9. Creating tuneable microwave media from a two-dimensional lattice of re-entrant posts
Energy Technology Data Exchange (ETDEWEB)
Goryachev, Maxim; Tobar, Michael E. [ARC Centre of Excellence for Engineered Quantum Systems, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009 (Australia)
2015-11-28
The potential capabilities of resonators based on two dimensional arrays of re-entrant posts is demonstrated. Such posts may be regarded as magnetically coupled lumped element microwave harmonic oscillators, arranged in a 2D lattices structure, which is enclosed in a 3D cavity. By arranging these elements in certain 2D patterns, we demonstrate how to achieve certain requirements with respect to field localisation and device spectra. Special attention is paid to symmetries of the lattices, mechanical tuning, design of areas of high localisation of magnetic energy; this in turn creates unique discrete mode spectra. We demonstrate analogies between systems designed on the proposed platform and well known physical phenomena such as polarisation, frustration, and Whispering Gallery Modes. The mechanical tunability of the cavity with multiple posts is analysed, and its consequences to optomechanical applications is calculated. One particular application to quantum memory is demonstrated with a cavity design consisting of separate resonators analogous to discrete Fabry–Pérot resonators. Finally, we propose a generalised approach to a microwave system design based on the concept of Programmable Cavity Arrays.
10. Robustness and breakup of the spiral wave in a two-dimensional lattice network of neurons
Institute of Scientific and Technical Information of China (English)
2010-01-01
The robustness and breakup of spiral wave in a two-dimensional lattice networks of neurons are investigated. The effect of small- world type connection is often simplified with local regular connection and the long-range connection with certain probability. The network effect on the development of spiral wave can be better described by local regular connection and changeable long-range connection probability than fixed long-range connection probability because the long-range probability could be changeable in realistic biological system. The effect from the changeable probability for long-range connection is simplified by multiplicative noise. At first, a stable rotating spiral wave is developed by using appropriate initial values, parameters and no-flux boundary conditions, and then the effect of networks is investigated. Extensive numerical studies show that spiral wave keeps its alive and robust when the intensity of multiplicative noise is below a certain threshold, otherwise, the breakup of spiral wave occurs. A statistical factor of synchronization in two-dimensional array is defined to study the phase transition of spiral wave by checking the membrane potentials of all neurons corresponding to the critical parameters(the intensity of noise or forcing current)in the curve for factor of synchronization. The Hindmarsh-Rose model is investigated, the Hodgkin-Huxley neuron model in the presence of the channel noise is also studied to check the model independence of our conclusions. And it is found that breakup of spiral wave is easier to be induced by the multiplicative noise in presence of channel noise.
11. Periodic, Quasiperiodic and Chaotic Discrete Breathers in a Parametrical Driven Two-Dimensional Discrete Klein-Gordon Lattice
Institute of Scientific and Technical Information of China (English)
XU Quan; TIAN Qiang; LUO Jun
2009-01-01
@@ We study a two-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the two-dimensional Klein-Gordon lattice with hard on-site potential. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers and chaotic discrete breathers by changing the amplitude of the driver.
12. Theory of a Nearly Two-Dimensional Dipolar Bose Gas
Science.gov (United States)
2016-05-11
optical data storage, and quantum computing. Today , BECs can be made with a variety of atomic species, including Chromium (Cr) [8] and the rare-earth...out many of these experiments. However, the experi- ments involved atoms that possess negligible dipole moments (like the alkali atoms ). Today , there...gases of bosonic atoms at ultracold, but finite temperatures. Under these circumstances, the gas can undergo a phase transition to a purely quantum
13. Dipolar quantum electrodynamics of the two-dimensional electron gas
Science.gov (United States)
Todorov, Yanko
2015-03-01
Similarly to a previous work on the homogeneous electron gas [Y. Todorov, Phys. Rev. B 89, 075115 (2014), 10.1103/PhysRevB.89.075115], we apply the Power-Zienau-Wooley (PZW) formulation of the quantum electrodynamics to the case of an electron gas quantum confined by one-dimensional potential. We provide a microscopic description of all collective plasmon modes of the gas, oscillating both along and perpendicular to the direction of quantum confinement. Furthermore, we study the interaction of the collective modes with a photonic structure, planar metallic waveguide, by using the full expansion of the electromagnetic field into normal modes. We show how the boundary conditions for the electromagnetic field influence both the transverse light-matter coupling and the longitudinal particle-particle interactions. The PZW descriptions appear thus as a convenient tool to study semiconductor quantum optics in geometries where quantum-confined particles interact with strongly confined electromagnetic fields in microresonators, such as the ones used to achieve the ultrastrong light-matter coupling regime.
14. Optical generation of a spatially variant two-dimensional lattice structure by using a phase only spatial light modulator
CERN Document Server
Kumar, Manish
2016-01-01
We propose a simple and straightforward method to generate a spatially variant lattice structures by optical interference lithography method. Using this method, it is possible to independently vary the orientation and period of the two-dimensional lattice. The method consists of two steps which are: numerical synthesis of corresponding phase mask by employing a two-dimensional integrated gradient calculations and experimental implementation of synthesized phase mask by making use of a phase only spatial light modulator in an optical 4f Fourier filtering setup. As a working example, we provide the experimental fabrication of a spatially variant square lattice structure which has the possibility to guide a Gaussian beam through a 90{\\deg} bend by photonic crystal self-collimation phenomena. The method is digitally reconfigurable, is completely scalable and could be extended to other kind of lattices as well.
15. Optical generation of a spatially variant two-dimensional lattice structure by using a phase only spatial light modulator
Energy Technology Data Exchange (ETDEWEB)
Kumar, Manish, E-mail: [email protected]; Joseph, Joby, E-mail: [email protected] [Photonics Research Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016 (India)
2014-08-04
We propose a simple and straightforward method to generate spatially variant lattice structures by optical interference lithography method. Using this method, it is possible to independently vary the orientation and period of the two-dimensional lattice. The method consists of two steps which are: numerical synthesis of corresponding phase mask by employing a two-dimensional integrated gradient calculations and experimental implementation of synthesized phase mask by making use of a phase only spatial light modulator in an optical 4f Fourier filtering setup. As a working example, we provide the experimental fabrication of a spatially variant square lattice structure which has the possibility to guide a Gaussian beam through a 90° bend by photonic crystal self-collimation phenomena. The method is digitally reconfigurable, is completely scalable, and could be extended to other kind of lattices as well.
16. Negative refraction and imaging of acoustic waves in a two-dimensional square chiral lattice structure
Science.gov (United States)
Zhao, Sheng-Dong; Wang, Yue-Sheng
2016-05-01
The negative refraction behavior and imaging effect for acoustic waves in a kind of two-dimensional square chiral lattice structure are studied in this paper. The unit cell of the proposed structure consists of four zigzag arms connected through a thin circular ring at the central part. The relation of the symmetry of the unit cell and the negative refraction phenomenon is investigated. Using the finite element method, we calculate the band structures and the equi-frequency surfaces of the system, and confirm the frequency range where the negative refraction is present. Due to the rotational symmetry of the unit cell, a phase difference is induced to the waves propagating from a point source through the structure to the other side. The phase difference is related to the width of the structure and the frequency of the source, so we can get a tunable deviated imaging. This kind of phenomenon is also demonstrated by the numerical simulation of two Gaussian beams that are symmetrical about the interface normal with the same incident angle, and the different negative refractive indexes are presented. Based on this special performance, a double-functional mirror-symmetrical slab is proposed for realizing acoustic focusing and beam separation.
17. The sequence d(CGGCGGCCGC) self-assembles into a two dimensional rhombic DNA lattice
Energy Technology Data Exchange (ETDEWEB)
Venkadesh, S.; Mandal, P.K. [CAS in Crystallography and Biophysics, University of Madras, Chennai 600 025 (India); Gautham, N., E-mail: [email protected] [CAS in Crystallography and Biophysics, University of Madras, Chennai 600 025 (India)
2011-04-15
Highlights: {yields} This is the first crystal structure of a four-way junction with sticky ends. {yields} Four junction structures bind to each other and form a rhombic cavity. {yields} Each rhombus binds to others to form 'infinite' 2D tiles. {yields} This is an example of bottom-up fabrication of a DNA nano-lattice. -- Abstract: We report here the crystal structure of the partially self-complementary decameric sequence d(CGGCGGCCGC), which self assembles to form a four-way junction with sticky ends. Each junction binds to four others through Watson-Crick base pairing at the sticky ends to form a rhombic structure. The rhombuses bind to each other and form two dimensional tiles. The tiles stack to form the crystal. The crystal diffracted in the space group P1 to a resolution of 2.5 A. The junction has the anti-parallel stacked-X conformation like other junction structures, though the formation of the rhombic net noticeably alters the details of the junction geometry.
18. High applicability of two-dimensional phosphorous in Kagome lattice predicted from first-principles calculations.
Science.gov (United States)
Chen, Peng-Jen; Jeng, Horng-Tay
2016-03-16
A new semiconducting phase of two-dimensional phosphorous in the Kagome lattice is proposed from first-principles calculations. The band gaps of the monolayer (ML) and bulk Kagome phosphorous (Kagome-P) are 2.00 and 1.11 eV, respectively. The magnitude of the band gap is tunable by applying the in-plane strain and/or changing the number of stacking layers. High optical absorption coefficients at the visible light region are predicted for multilayer Kagome-P, indicating potential applications for solar cell devices. The nearly dispersionless top valence band of the ML Kagome-P with high density of states at the Fermi level leads to superconductivity with Tc of ~9 K under the optimal hole doping concentration. We also propose that the Kagome-P can be fabricated through the manipulation of the substrate-induced strain during the process of the sample growth. Our work demonstrates the high applicability of the Kagome-P in the fields of electronics, photovoltaics, and superconductivity.
19. Non-equilibrium relaxation in a two-dimensional stochastic lattice Lotka-Volterra model
Science.gov (United States)
Chen, Sheng; Täuber, Uwe C.
We employ Monte Carlo simulations to study a stochastic Lotka-Volterra model on a two-dimensional square lattice with periodic boundary conditions. There are stable states when the predators and prey coexist. If the local prey carrying capacity is finite, there emerges an extinction threshold for the predator population at a critical value of the predation rate. We investigate the non-equilibrium relaxation of the predator density in the vicinity of this critical point. The expected power law dependence between the relaxation time and predation rate is observed (critical slowing down). The numerically determined associated critical exponents are in accord with the directed percolation universality class. Following a sudden predation rate change to its critical value, one observes critical aging for the predator density autocorrelation function with a universal scaling exponent. This aging scaling signature of the absorbing state phase transition emerges at significantly earlier times than stationary critical power laws, and could thus serve as an advanced indicator of the population's proximity to its extinction threshold. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-09ER46613.
20. Screening phase transitions in two-dimensional Coulomb gas
Energy Technology Data Exchange (ETDEWEB)
Gallavotti, G.; Nicolo, F.
1984-07-01
Infrared properties of a Coulomb gas in two dimensions and with fixed ultraviolet cutoff are studied. The existence of infinitely many thresholds Tu = 1/Ke 1/8 pi (1-1/zu)sup-1 in the interval of temperatures 1/Ke1/8 pi, 1/4 pi, where K is the Boltzmann constant and e = /e/ is the charge of the positive particle, is proved. Such thresholds are conjectured to reflect a sequence of transitions from a pure multipole phase (the Koesterlitz-Thouless region) to the plasma phase via an infinite number of intermediate phases. Mathematically the free energy becomes more and more differentiable as a function of the activity lambda, near lambda = 0, as the temperature decreases.
1. Ferroelectric control of two dimensional electron gas in oxide heterointerface
Science.gov (United States)
Thanh, Tra Vu; Chen, Jhih-Wei; Yeh, Chao-Hui; Chen, Yi-Chun; Wu, Chung-Lin; Lin, Jiunn Yuan; Chu, Ying-Hao
2012-02-01
Oxide heterointerfaces are emerging as one of the most exciting materials systems in condensed-matter science. One remarkable example is the LaAlO3 /SrTiO3 (LAO/STO) interface, a model system in which a highly mobile electron gas forms between two band insulators. Our study to manipulate the conductivity at this interface by using ferroeletricity of Pb(Zr,Ti)O3. Our transport data strongly suggests that down polarization direction depletes the conducting interface of LAO/STO. After switching the polarization direction (up), it becomes accumulation. In addition, our experiments show there is obvious the band structure changed by cross-sectional scanning tunneling microscopy and combining with X-ray photoelectron spectroscopy (XPS) measurements. The transport properties are measured to build up the connection between macroscopic properties and local electronic structures that have been applied to study this structure. Controlling the conductivity of this oxide interface suggests that this technique may not only extend more generally to other oxide systems but also open much potential to ferroelectric field effect transistors.
2. Surface Reconstruction-Induced Coincidence Lattice Formation Between Two-Dimensionally Bonded Materials and a Three-Dimensionally Bonded Substrate
NARCIS (Netherlands)
Boschker, Jos E.; Momand, Jamo; Bragaglia, Valeria; Wang, Ruining; Perumal, Karthick; Giussani, Alessandro; Kooi, Bart J.; Riechert, Henning; Calarco, Raffaella
2014-01-01
Sb2Te3 films are used for studying the epitaxial registry between two-dimensionally bonded (2D) materials and three-dimensional bonded (3D) substrates. In contrast to the growth of 3D materials, it is found that the formation of coincidence lattices between Sb2Te3 and Si(111) depends on the geometry
3. Fabrication of deep-profile Al-doped ZnO one- and two-dimensional lattices as plasmonic elements
DEFF Research Database (Denmark)
Jensen, Flemming; Shkondin, Evgeniy; Takayama, Osamu
2016-01-01
In this work, we report on fabrication of deep-profile one- and two-dimensional lattices made from Al-doped ZnO (AZO). AZO is considered as an alternative plasmonic material having the real part of the permittivity negative in the near infrared range. The exact position of the plasma frequency...
4. Photonic band structures of two-dimensional photonic crystals with deformed lattices
Institute of Scientific and Technical Information of China (English)
Cai Xiang-Hua; Zheng Wan-Hua; Ma Xiao-Tao; Ren Gang; Xia Jian-Bai
2005-01-01
Using the plane-wave expansion method, we have calculated and analysed the changes of photonic band structures arising from two kinds of deformed lattices, including the stretching and shrinking of lattices. The square lattice with square air holes and the triangular lattice with circular air holes are both studied. Calculated results show that the change of lattice size in some special ranges can enlarge the band gap, which depends strongly on the filling factor of air holes in photonic crystals; and besides, the asymmetric band edges will appear with the broken symmetry of lattices.
5. The Bargmann type reduction for some Lax integrable two-dimensional generalization of the relativistic Toda lattice
Directory of Open Access Journals (Sweden)
O. Ye. Hentosh
2016-01-01
Full Text Available The possibility of applying the method of reducing upon finite-dimensional invariant subspaces, generated by the eigenvalues of the associated spectral problem, to some two-dimensional generalization of the relativistic Toda lattice with the triple matrix Lax type linearization is investigated. The Hamiltonian property and Lax-Liouville integrability of the vector fields, given by this system, on the invariant subspace related with the Bargmann type reduction are found out.
6. Absolute band gaps of a two-dimensional triangular-lattice dielectric photonic crystal with different shapes
Institute of Scientific and Technical Information of China (English)
2010-01-01
Absolute band gaps of a two-dimensional triangular-lattice photonic crystal are calculated with the finite-difference time-domain method in this paper.Through calculating the photonic band structures of the triangular-lattice photonic crystal consisting of Ge rods immersed in air with different shapes,it is found that a large absolute band gap of 0.098 (2c/a) can be obtained for the structures with hollow triangular Ge rods immersed in air,corresponding to 19.8% of the middle frequency.The influence of the different factors on the width of the absolute band gaps is also discussed.
7. Direct Measurement of the Band Structure of a Buried Two-Dimensional Electron Gas
DEFF Research Database (Denmark)
Miwa, Jill; Hofmann, Philip; Simmons, Michelle Y.;
2013-01-01
We directly measure the band structure of a buried two dimensional electron gas (2DEG) using angle resolved photoemission spectroscopy. The buried 2DEG forms 2 nm beneath the surface of p-type silicon, because of a dense delta-type layer of phosphorus n-type dopants which have been placed there. ...
8. Model for ballistic spin-transport in ferromagnet/two-dimensional electron gas/ferromagnet structures
NARCIS (Netherlands)
Schapers, T; Nitta, J; Heersche, HB; Takayanagi, H
2002-01-01
The spin dependent conductance of a ferromagnet/two-dimensional electron gas ferromagnet structure is theoretically examined in the ballistic transport regime. It is shown that the spin signal can be improved considerably by making use of the spin filtering effect of a barrier at the ferromagnet two
9. Tunable secondary dimension selectivity in comprehensive two-dimensional gas chromatography
NARCIS (Netherlands)
J. Mommers; G. Pluimakers; J. Knooren; T. Dutriez; S. van der Wal
2013-01-01
In this paper two tunable two-dimensional gas chromatography setups are compared and described in which the secondary dimension consists of two different capillary columns coupled in series. In the first setup the selectivity of the second dimension can be tuned by adjusting the effective column len
10. Collective modes of a quasi-two-dimensional Bose condensate in large gas parameter regime
Indian Academy of Sciences (India)
S R Mishra; S P Ram; Arup Banerjee
2007-06-01
We have theoretically studied the collective modes of a quasi-two-dimensional (Q2D) Bose condensate in the large gas parameter regime by using a formalism which treats the interaction energy beyond the mean-field approximation. The results show that incorporation of this higher order term leads to significant modifications in the mode frequencies.
11. Comprehensive two-dimensional gas chromatography for the analysis of organohalogenated micro-contaminants
NARCIS (Netherlands)
Korytar, P.; Haglund, P.; Boer, de J.; Brinkman, U.A.Th.
2006-01-01
We explain the principles of comprehensive two-dimensional gas chromatography (GC × GC), and discuss key instrumental aspects - with emphasis on column combinations and mass spectrometric detection. As the main item of interest, we review the potential of GC × GC for the analysis of organohalogenate
12. Spin and charge transport in a gated two dimensional electron gas
NARCIS (Netherlands)
Lerescu, Alexandru Ionut
2007-01-01
The work presented in this thesis is centered around the idea of how one can inject, transport and detect the electron's spin in a two dimensional electron gas (a semiconductor heterostructure). Metal based spintronic devices have been established to be the easy way to implement spintronic concepts
13. Model of two-dimensional electron gas formation at ferroelectric interfaces
Energy Technology Data Exchange (ETDEWEB)
Aguado-Puente, P.; Bristowe, N. C.; Yin, B.; Shirasawa, R.; Ghosez, Philippe; Littlewood, P. B.; Artacho, Emilio
2015-07-01
The formation of a two-dimensional electron gas at oxide interfaces as a consequence of polar discontinuities has generated an enormous amount of activity due to the variety of interesting effects it gives rise to. Here, we study under what circumstances similar processes can also take place underneath ferroelectric thin films. We use a simple Landau model to demonstrate that in the absence of extrinsic screening mechanisms, a monodomain phase can be stabilized in ferroelectric films by means of an electronic reconstruction. Unlike in the LaAlO3/SrTiO3 heterostructure, the emergence with thickness of the free charge at the interface is discontinuous. This prediction is confirmed by performing first-principles simulations of free-standing slabs of PbTiO3. The model is also used to predict the response of the system to an applied electric field, demonstrating that the two-dimensional electron gas can be switched on and off discontinuously and in a nonvolatile fashion. Furthermore, the reversal of the polarization can be used to switch between a two-dimensional electron gas and a two-dimensional hole gas, which should, in principle, have very different transport properties. We discuss the possible formation of polarization domains and how such configuration competes with the spontaneous accumulation of free charge at the interfaces.
14. Comprehensive two-dimensional gas chromatography for the analysis of organohalogenated micro-contaminants
NARCIS (Netherlands)
Korytar, P.; Haglund, P.; Boer, de J.; Brinkman, U.A.Th.
2006-01-01
We explain the principles of comprehensive two-dimensional gas chromatography (GC × GC), and discuss key instrumental aspects - with emphasis on column combinations and mass spectrometric detection. As the main item of interest, we review the potential of GC × GC for the analysis of organohalogenate
15. Coherent electron focusing with quantum point contacts in a two-dimensional electron gas
NARCIS (Netherlands)
Houten, H. van; Beenakker, C.W.J.; Williamson, J.G.; Broekaart, M.E.I.; Loosdrecht, P.H.M. van; Wees, B.J. van; Mooij, J.E.; Foxon, C.T.; Harris, J.J.
1989-01-01
Transverse electron focusing in a two-dimensional electron gas is investigated experimentally and theoretically for the first time. A split Schottky gate on top of a GaAs-AlxGa1–xAs heterostructure defines two point contacts of variable width, which are used as injector and collector of ballistic el
16. Quantitative analysis of target components by comprehensive two-dimensional gas chromatography
NARCIS (Netherlands)
Mispelaar, V.G. van; Tas, A.C.; Smilde, A.K.; Schoenmakers, P.J.; Asten, A.C. van
2003-01-01
Quantitative analysis using comprehensive two-dimensional (2D) gas chromatography (GC) is still rarely reported. This is largely due to a lack of suitable software. The objective of the present study is to generate quantitative results from a large GC x GC data set, consisting of 32 chromatograms. I
17. A Robust Thermal Modulator for Comprehensive Two-Dimensional Gas Chromatography
NARCIS (Netherlands)
Geus, de H.J.; Boer, de J.
1999-01-01
In comprehensive two dimensional gas chromatography (GCxGC), two capillary columns are connected in series through an interface known as a 'thermal modulator'. This device transforms effluent from the first capillary column into a series of sharp injection-like chemical pulses suitable for high-spee
18. Spin-polarized transport in a two-dimensional electron gas with interdigital-ferromagnetic contacts
DEFF Research Database (Denmark)
Hu, C.-M.; Nitta, Junsaku; Jensen, Ane
2001-01-01
Ferromagnetic contacts on a high-mobility, two-dimensional electron gas (2DEG) in a narrow gap semiconductor with strong spin-orbit interaction are used to investigate spin-polarized electron transport. We demonstrate the use of magnetized contacts to preferentially inject and detect specific spin...
19. Thermodynamics of Two-Dimensional Electron Gas in a Magnetic Field
Directory of Open Access Journals (Sweden)
V. I. Nizhankovskii
2011-01-01
Full Text Available Change of the chemical potential of electrons in a GaAs-AlGa1−As heterojunction was measured in magnetic fields up to 6.5 T at several temperatures from 2.17 to 12.3 K. A thermodynamic equation of state of two-dimensional electron gas well describes the experimental results.
20. Universal relations for the two-dimensional spin-1/2 Fermi gas with contact interactions
DEFF Research Database (Denmark)
Valiente, Manuel; Zinner, Nikolaj Thomas; Mølmer, Klaus
2011-01-01
We present universal relations for a two-dimensional Fermi gas with pairwise contact interactions. The derivation of these relations is made possible by obtaining the explicit form of a generalized function—selector—in the momentum representation. The selector implements the short-distance bounda...
1. Pixel-based analysis of comprehensive two-dimensional gas chromatograms (color plots) of petroleum
DEFF Research Database (Denmark)
Furbo, Søren; Hansen, Asger B.; Skov, Thomas;
2014-01-01
We demonstrate how to process comprehensive two-dimensional gas chromatograms (GC × GC chromatograms) to remove nonsample information (artifacts), including background and retention time shifts. We also demonstrate how this, combined with further reduction of the influence of irrelevant informati...
2. Dispersion of guided modes in two-dimensional split ring lattices
CERN Document Server
Lunnemann, Per
2014-01-01
We present a semi-analytical point-dipole method that uses Ewald lattice summation to find the dispersion relation of guided plasmonic and bianisotropic modes in metasurfaces composed of 2D periodic lattices of arbitrarily strongly scattering magneto-electric dipole scatterers. This method takes into account all retarded electrodynamic interactions as well as radiation damping selfconsistently. As illustration we analyze the dispersion of plasmon nanorod lattices, and of 2D split ring resonator lattices. Plasmon nanorod lattices support transverse and longitudinal in-plane electric modes. Scatterers that have an in-plane electric and out-of-plane magnetic polarizability, but without intrinsic magnetoelectric coupling, result in two bands that are mixtures of the bands of electric-only and magnetic-only lattices. Thereby bianisotropy through mutual coupling, in absence of building-block bianisotropy, is evident. Once strong bi-anisotropy is included in each building block, the Bloch modes become even more stro...
3. Generation and enumeration of compact conformations on the two-dimensional triangular and three-dimensional fcc lattices
Science.gov (United States)
Peto, Myron; Sen, Taner Z.; Jernigan, Robert L.; Kloczkowski, Andrzej
2007-07-01
We enumerated all compact conformations within simple geometries on the two-dimensional (2D) triangular and three-dimensional (3D) face centered cubic (fcc) lattice. These compact conformations correspond mathematically to Hamiltonian paths and Hamiltonian circuits and are frequently used as simple models of proteins. The shapes that were studied for the 2D triangular lattice included m ×n parallelograms, regular equilateral triangles, and various hexagons. On the 3D fcc lattice we generated conformations for a limited class of skewed parallelepipeds. Symmetries of the shape were exploited to reduce the number of conformations. We compared surface to volume ratios against protein length for compact conformations on the 3D cubic lattice and for a selected set of real proteins. We also show preliminary work in extending the transfer matrix method, previously developed by us for the 2D square and the 3D cubic lattices, to the 2D triangular lattice. The transfer matrix method offers a superior way of generating all conformations within a given geometry on a lattice by completely avoiding attrition and reducing this highly complicated geometrical problem to a simple algebraic problem of matrix multiplication.
4. Generation and enumeration of compact conformations on the two-dimensional triangular and three-dimensional fcc lattices.
Science.gov (United States)
Peto, Myron; Sen, Taner Z; Jernigan, Robert L; Kloczkowski, Andrzej
2007-07-28
We enumerated all compact conformations within simple geometries on the two-dimensional (2D) triangular and three-dimensional (3D) face centered cubic (fcc) lattice. These compact conformations correspond mathematically to Hamiltonian paths and Hamiltonian circuits and are frequently used as simple models of proteins. The shapes that were studied for the 2D triangular lattice included mxn parallelograms, regular equilateral triangles, and various hexagons. On the 3D fcc lattice we generated conformations for a limited class of skewed parallelepipeds. Symmetries of the shape were exploited to reduce the number of conformations. We compared surface to volume ratios against protein length for compact conformations on the 3D cubic lattice and for a selected set of real proteins. We also show preliminary work in extending the transfer matrix method, previously developed by us for the 2D square and the 3D cubic lattices, to the 2D triangular lattice. The transfer matrix method offers a superior way of generating all conformations within a given geometry on a lattice by completely avoiding attrition and reducing this highly complicated geometrical problem to a simple algebraic problem of matrix multiplication.
5. Cluster algorithm for two-dimensional U(1) lattice gauge theory
Science.gov (United States)
Sinclair, R.
1992-03-01
We use gauge fixing to rewrite the two-dimensional U(1) pure gauge model with Wilson action and periodic boundary conditions as a nonfrustrated XY model on a closed chain. The Wolff single-cluster algorithm is then applied, eliminating critical slowing down of topological modes and Polyakov loops.
6. From spin flip excitations to the spin susceptibility enhancement of a two-dimensional electron gas.
Science.gov (United States)
Perez, F; Aku-leh, C; Richards, D; Jusserand, B; Smith, L C; Wolverson, D; Karczewski, G
2007-07-13
The g-factor enhancement of the spin-polarized two-dimensional electron gas was measured directly over a wide range of spin polarizations, using spin flip resonant Raman scattering spectroscopy on two-dimensional electron gases embedded in Cd(1-x)Mn(x)Te semimagnetic quantum wells. At zero Raman transferred momentum, the single-particle spin flip excitation, energy Z*, coexists in the Raman spectrum with the spin flip wave of energy Z, the bare giant Zeeman splitting. We compare the measured g-factor enhancement with recent spin-susceptibility enhancement theories and deduce the spin-polarization dependence of the mass renormalization.
7. Universal relations for the two-dimensional spin-1/2 Fermi gas with contact interactions
Energy Technology Data Exchange (ETDEWEB)
Valiente, Manuel; Zinner, Nikolaj T.; Moelmer, Klaus [Lundbeck Foundation Theoretical Center for Quantum System Research, Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C (Denmark)
2011-12-15
We present universal relations for a two-dimensional Fermi gas with pairwise contact interactions. The derivation of these relations is made possible by obtaining the explicit form of a generalized function--selector--in the momentum representation. The selector implements the short-distance boundary condition between two fermions in a straightforward manner and leads to simple derivations of the universal relations, in the spirit of Tan's original method for the three-dimensional gas.
8. Nonlinear two-dimensional terahertz photon echo and rotational spectroscopy in the gas phase
CERN Document Server
Lu, Jian; Hwang, Harold Y; Ofori-Okai, Benjamin K; Fleischer, Sharly; Nelson, Keith A
2016-01-01
Ultrafast two-dimensional spectroscopy utilizes correlated multiple light-matter interactions for retrieving dynamic features that may otherwise be hidden under the linear spectrum. Its extension to the terahertz regime of the electromagnetic spectrum, where a rich variety of material degrees of freedom reside, remains an experimental challenge. Here we report ultrafast two-dimensional terahertz spectroscopy of gas-phase molecular rotors at room temperature. Using time-delayed terahertz pulse pairs, we observe photon echoes and other nonlinear signals resulting from molecular dipole orientation induced by three terahertz field-dipole interactions. The nonlinear time-domain orientation signals are mapped into the frequency domain in two-dimensional rotational spectra which reveal J-state-resolved nonlinear rotational dynamics. The approach enables direct observation of correlated rotational transitions and may reveal rotational coupling and relaxation pathways in the ground electronic and vibrational state.
9. Design, fabrication, and characterization of lightweight and broadband microwave absorbing structure reinforced by two dimensional composite lattice
Science.gov (United States)
Chen, Mingji; Pei, Yongmao; Fang, Daining
2012-07-01
Microwave absorbing structures (MASs) reinforced by two dimensional (2D) composite lattice elements have been designed and fabricated. The density of these MASs is lower than 0.5 g/cm3. Experimental measurements show that the sandwich structure with glass fiber reinforced composite (GFRC) lattice core can serve as a broadband MAS with its reflectivity below -10 dB over the frequency range of 4-18 GHz. The low permittivity GFRC is indicated to be the proper material for both the structural element of the core and the transparent face sheet. Calculations by the periodic moment method (PMM) demonstrate that the 2D Kagome lattice performs better for microwave absorbing than the square one at relatively low frequencies. The volume fraction and cell size of the structural element are also revealed to be key factors for microwave absorbing performance.
10. Analysis of two-dimensional photonic band gap structure with a rhombus lattice
Institute of Scientific and Technical Information of China (English)
Limei Qi; Ziqiang Yang; Xi Gao; Zheng Liang
2008-01-01
@@ The relative band gap for a rhombus lattice photonic crystal is studied by plane wave expansion method and high frequency structure simulator (HFSS) simulation. General wave vectors in the first Briliouin zone are derived. The relative band gap as a function of air-filling factor and background material is investigated, respectively, and the nature of photonic band gap for different lattice angles is analyzed by the distribution of electric energy. These results would provide theoretical instruction for designing optical integrated devices using photonic crystal with a rhombus lattice.
11. Properties of the two-dimensional spin-1/2 Heisenberg model on a honeycomb lattice with interlayer coupling
Directory of Open Access Journals (Sweden)
U. Löw
2009-01-01
Full Text Available The magnetic properties of the two-dimensional S=1/2 (quantum antiferromagnetic Heisenberg model on a honeycomb lattice with and without interlayer coupling are studied by means of a continuous Euclidean time Quantum Monte-Carlo algorithm. The internal energy, the magnetic susceptibility and the staggered magnetization are determined in the full temperature range. For the two-dimensional system the ground-state energy/bond is found to be E0hc=-0.36303(13, and the zero temperature staggered magnetization mst=0.2681(8. For coupled planes of honeycomb systems a phase transition from an ordered phase to a disordered phase is found at T/J=0.695(10.
12. Two-Dimensional Anharmonic Crystal Lattices: Solitons, Solectrons, and Electric Conduction
OpenAIRE
Velarde, Manuel G.; Ebeling, Werner; Chetverikov, Alexander P.
2011-01-01
Reported here are salient features of soliton-mediated electron transport in anharmonic crystal lattices.After recalling how an electron-soliton bound state (solectron) can be formed we comment on consequences like electron surfing on a sound wave and balistic transport, possible percolation in 2d lattices, and a novel form of electron pairing with strongly correlated electrons both in real space and momentum space.
13. Percolation in spatial evolutionary prisoner's dilemma game on two-dimensional lattices
Science.gov (United States)
Choi, Woosik; Yook, Soon-Hyung; Kim, Yup
2015-11-01
We study the spatial evolutionary prisoner's dilemma game with updates of imitation max on triangular, hexagonal, and square lattices. We use the weak prisoner's dilemma game with a single parameter b . Due to the competition between the temptation value b and the coordination number z of the base lattice, a greater variety of percolation properties is expected to occur on the lattice with the larger z . From the numerical analysis, we find six different regimes on the triangular lattice (z =6 ). Regardless of the initial densities of cooperators and defectors, cooperators always percolate in the steady state in two regimes for small b . In these two regimes, defectors do not percolate. In two regimes for the intermediate value of b , both cooperators and defectors undergo percolation transitions. The defector always percolates in two regimes for large b . On the hexagonal lattice (z =3 ), there exist two distinctive regimes. For small b , both the cooperators and the defectors undergo percolation transitions while only defectors always percolate for large b . On the square lattice (z =4 ), there exist three regimes. Combining with the finite-size scaling analyses, we show that all the observed percolation transitions belong to the universality class of the random percolation. We also show how the detailed growth mechanism of cooperator and defector clusters decides each regime.
14. Tunable spin wave spectra in two-dimensional Ni{sub 80}Fe{sub 20} antidot lattices with varying lattice symmetry
Energy Technology Data Exchange (ETDEWEB)
Mandal, R.; Barman, S.; Saha, S.; Barman, A., E-mail: [email protected] [Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 098 (India); Otani, Y. [CEMS-RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198 (Japan); Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581 (Japan)
2015-08-07
Ferromagnetic antidot lattices are important systems for magnetic data storage and magnonic devices, and understanding their magnetization dynamics by varying their structural parameters is an important problems in magnetism. Here, we investigate the variation in spin wave spectrum in two-dimensional nanoscale Ni{sub 80}Fe{sub 20} antidot lattices with lattice symmetry. By varying the bias magnetic field values in a broadband ferromagnetic resonance spectrometer, we observed a stark variation in the spin wave spectrum with the variation of lattice symmetry. The simulated mode profiles showed further difference in the spatial nature of the modes between different lattices. While for square and rectangular lattices extended modes are observed in addition to standing spin wave modes, all modes in the hexagonal, honeycomb, and octagonal lattices are either localized or standing waves. In addition, the honeycomb and octagonal lattices showed two different types of modes confined within the honeycomb (octagonal) units and between two such consecutive units. Simulated internal magnetic fields confirm the origin of such a wide variation in the frequency and spatial nature of the spin wave modes. The tunability of spin waves with the variation of lattice symmetry is important for the design of future magnetic data storage and magnonic devices.
15. The focusing effect of electromagnetic waves in two-dimensional photonic crystals with gradually varying lattice constant
Directory of Open Access Journals (Sweden)
F Bakhshi Garmi
2016-02-01
Full Text Available In this paper we studied the focusing effect of electromagnetic wave in the two-dimensional graded photonic crystal consisting of Silicon rods in the air background with gradually varying lattice constant. The results showed that graded photonic crystal can focus wide beams on a narrow area at frequencies near the lower edge of the band gap, where equal frequency contours are not concave. For calculation of photonic band structure and equal frequency contours, we have used plane wave expansion method and revised plane wave expansion method, respectively. The calculation of the electric and magnetic fields was performed by finite difference time domain method.
16. Fission-gas release at extended burnups: effect of two-dimensional heat transfer
Energy Technology Data Exchange (ETDEWEB)
Tayal, M. [Atomic Energy of Canada Limited, Mississauga, Ontario (Canada); Yu, S.D. [Ryerson Polytechnic Univ., Toronto, Ontario (Canada); Lau, J.H.K
2000-09-01
To better simulate the performance of high-burnup CANDU fuel, a two-dimensional model for heat transfer between the pellet and the sheath has been added to the computer code ELESTRES. The model covers four relative orientations of the pellet and the sheath and their impacts on heat transfer and fission-gas release. The predictions of the code were compared to a database of 27 experimental irradiations involving extended burnups and normal burnups. The calculated values of fission gas release matched the measurements to an average of 94%. Thus, the two-dimensional heat transfer model increases the versatility of the ELESTRES code to better simulate fuels at normal as well as at extended burnups. (author)
17. [Determination of aromatics in light petroleum products by comprehensive two-dimensional gas chromatography].
Science.gov (United States)
Li, Yanyan
2006-07-01
In recent years, comprehensive two-dimensional gas chromatography (GC x GC) have been used widely, and the applications of this technique to many fields have already been reported. In the standard method of oil analysis, the concentrations of aromatics and naphthalene hydrocarbons in light petroleum products must be detected by more than two methods. Mono-aromatics, di-aromatics etc. in light petroleum products were detected only by comprehensive two-dimensional gas chromatography. After the proper selection of column system and optimization of chromatographic conditions, the method can achieve the group separations of paraffins, olefins, naphthenes, aromatics with 1 to 2 rings and some target components in light petroleum products with good reproducibility and good precision. The recoveries of standard compounds were 89.5% - 106.1%, and the relative standard deviations of repeatedly detecting the components were all lower than 5.8%. It took only 30 min to finish a determination.
18. Bloch waves in an arbitrary two-dimensional lattice of subwavelength Dirichlet scatterers
CERN Document Server
Schnitzer, Ory
2016-01-01
We study waves governed by the planar Helmholtz equation, propagating in an infinite lattice of subwavelength Dirichlet scatterers, the periodicity being comparable to the wavelength. Applying the method of matched asymptotic expansions, the scatterers are effectively replaced by asymptotic point constraints. The resulting coarse-grained Bloch-wave dispersion problem is solved by a generalised Fourier series, whose singular asymptotics in the vicinities of scatterers yield the dispersion relation governing modes that are strongly perturbed from plane-wave solutions existing in the absence of the scatterers; there are also empty-lattice waves that are only weakly perturbed. Characterising the latter is useful in interpreting and potentially designing the dispersion diagrams of such lattices. The method presented, that simplifies and expands on Krynkin & McIver [Waves Random Complex, 19 347 2009], could be applied in the future to study more sophisticated designs entailing resonant subwavelength elements di...
19. Auxetic two-dimensional lattice with Poisson's Ratio arbitrarily close to -1
CERN Document Server
Cabras, L
2014-01-01
In this paper we propose a new lattice structure having macroscopic Poisson's ratio arbitrarily close to the stability limit -1. We tested experimentally the effective Poisson's ratio of the micro-structured medium; the uniaxial test has been performed on a thermoplastic lattice produced with a 3d printing technology. A theoretical analysis of the effective properties has been performed and the expression of the macroscopic constitutive properties is given in full analytical form as a function of the constitutive properties of the elements of the lattice and on the geometry of the microstructure. The analysis has been performed on three micro-geometry leading to an isotropic behaviour for the cases of three-fold and six-fold symmetry and to a cubic behaviour for the case of four-fold symmetry.
20. Magnetoresistance of a two-dimensional electron gas in a random magnetic field
DEFF Research Database (Denmark)
Smith, Anders; Taboryski, Rafael Jozef; Hansen, Luise Theil
1994-01-01
We report magnetoresistance measurements on a two-dimensional electron gas made from a high-mobility GaAs/AlxGa1-xAs heterostructure, where the externally applied magnetic field was expelled from regions of the semiconductor by means of superconducting lead grains randomly distributed on the surf...... on the surface of the sample. A theoretical explanation in excellent agreement with the experiment is given within the framework of the semiclassical Boltzmann equation. © 1994 The American Physical Society...
1. A New Class of Resonances at the Edge of the Two Dimensional Electron Gas
OpenAIRE
Zhitenev, N. B.; Brodsky, M; Ashoori, R. C.; Melloch, M. R.
1996-01-01
We measure the frequency dependent capacitance of a gate covering the edge and part of a two-dimensional electron gas in the quantum Hall regime. In applying a positive gate bias, we create a metallic puddle under the gate surrounded by an insulating region. Charging of the puddle occurs via electron tunneling from a metallic edge channel. Analysis of the data allows direct extraction of this tunneling conductance. Novel conductance resonances appear as a function of gate bias. Samples with g...
2. Interaction of a Surface Acoustic Wave with a Two-dimensional Electron Gas
Institute of Scientific and Technical Information of China (English)
YANG Shi-Jie; ZHAO Hu; YU Yue
2005-01-01
When a surface acoustic wave (SAW) propagates on the surface of a GaAs semiconductor, coupling between electrons in the two-dimensional electron gas beneath the interface and the elastic host crystal through piezoelectric interaction will attenuate the SAW. The coupling coefficient is calculated for the SAW propagating along an arbitrary direction. It is found that the coupling strength is strongly dependent on the propagating direction. When the SAW propagates along the [011] direction, the coupling becomes quite weak.
3. Interaction-induced huge magnetoresistance in a high mobility two-dimensional electron gas
Energy Technology Data Exchange (ETDEWEB)
Bockhorn, L.; Haug, R. J. [Institut für Festkörperphysik, Leibniz Universität Hannover, D-30167 Hannover (Germany); Gornyi, I. V. [Institut für Nanotechnologie, Karlsruher Institut of Technology, D-76021 Karlsruhe (Germany); Schuh, D. [Institut für Experimentelle und Angewandte Physik, Universität Regensburg, D-93053 Regensburg (Germany); Wegscheider, W. [ETH Zürich (Switzerland)
2013-12-04
A strong negative magnetoresistance is observed in a high-mobility two-dimensional electron gas in a GaAs/Al{sub 0.3}Ga{sub 0.7}As quantum well. We discuss that the negative magnetoresistance consists of a small peak induced by a combination of two types of disorder and a huge magnetoresistance explained by the interaction correction to the conductivity for mixed disorder.
4. Average site perimeter of directed animals on the two-dimensional lattices
CERN Document Server
Bacher, Axel
2009-01-01
We introduce new combinatorial (bijective) methods that enable us to compute the average value of three parameters of directed animals of a given area, including the site perimeter. Our results cover directed animals of any one-line source on the square lattice and its bounded variants, and we give counterparts for most of them in the triangular lattices. We thus prove conjectures by Conway and Le Borgne. The techniques used are based on Viennot's correspondence between directed animals and heaps of pieces (or elements of a partially commutative monoid).
5. Guide modes in photonic crystal heterostructures composed of rotating non-circular air cylinders in two-dimensional lattices
CERN Document Server
Zhou Yun Song; Wang Fu He
2003-01-01
We investigate the properties of guide modes localized at the interfaces of photonic crystal (PC) heterostructures which are composed of two semi-infinite two-dimensional PCs consisting of non-circular air cylinders with different rotating angles embedded in a homogeneous host dielectric. Photonic band gap structures are calculated with the use of the plane-wave expansion method in combination with a supercell technique. We consider various configurations, for instance, rectangular (square) lattice-rectangular (square) air cylinders, and different rotating angles of the cylinders in the lattices on either side of the interface of a heterostructure. We find that the absolute gap width and the number of guide modes strongly depend on geometric and physical parameters of the heterostructures. It is anticipated that the guide modes in such heterostructures can be engineered by adjusting parameters.
6. Normally attracting manifolds and periodic behavior in one-dimensional and two-dimensional coupled map lattices
Science.gov (United States)
Giberti, Claudio; Vernia, Cecilia
1994-12-01
We consider diffusively coupled logistic maps in one- and two-dimensional lattices. We investigate periodic behaviors as the coupling parameter varies, i.e., existence and bifurcations of some periodic orbits with the largest domain of attraction. Similarity and differences between the two lattices are shown. For small coupling the periodic behavior appears to be characterized by a number of periodic orbits structured in such a way to give rise to distinct, reverse period-doubling sequences. For intermediate values of the coupling a prominent role in the dynamics is played by the presence of normally attracting manifolds that contain periodic orbits. The dynamics on these manifolds is very weakly hyperbolic, which implies long transients. A detailed investigation allows the understanding of the mechanism of their formation. A complex bifurcation is found which causes an attracting manifold to become unstable.
7. Periodic, quasiperiodic and chaotic discrete breathers in a parametrical driven two-dimensional discrete diatomic Klein-Gordon lattice
Institute of Scientific and Technical Information of China (English)
Xu Quan; Tian Qiang
2009-01-01
We study a two-dimensional (2D) diatomic lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers (DBs) can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the 2D discrete diatomic Klein-Gordon lattice with hard and soft on-site potentials. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers (QDBs) and chaotic discrete breathers (CDBs) by changing the amplitude of the driver. But the DBs and QDBs with symmetric and anti-symmetric profiles that are centered at a heavy atom are more stable than at a light atom, because the frequencies of the DBs and QDBs centered at a heavy atom are lower than those centered at a light atom.
8. Noise Threshold for a Fault-Tolerant Two-Dimensional Lattice Architecture
CERN Document Server
Svore, K M; Terhal, B M; Svore, Krysta M.; Vincenzo, David P. Di; Terhal, Barbara M.
2006-01-01
We consider a model of quantum computation in which the set of operations is limited to nearest-neighbor interactions on a 2D lattice. We model movement of qubits with noisy SWAP operations. For this architecture we design a fault-tolerant coding scheme using the concatenated [[7,1,3
9. Programming Self-Assembly of DNA Origami Honeycomb Two-Dimensional Lattices and Plasmonic Metamaterials.
Science.gov (United States)
Wang, Pengfei; Gaitanaros, Stavros; Lee, Seungwoo; Bathe, Mark; Shih, William M; Ke, Yonggang
2016-06-22
Scaffolded DNA origami has proven to be a versatile method for generating functional nanostructures with prescribed sub-100 nm shapes. Programming DNA-origami tiles to form large-scale 2D lattices that span hundreds of nanometers to the micrometer scale could provide an enabling platform for diverse applications ranging from metamaterials to surface-based biophysical assays. Toward this end, here we design a family of hexagonal DNA-origami tiles using computer-aided design and demonstrate successful self-assembly of micrometer-scale 2D honeycomb lattices and tubes by controlling their geometric and mechanical properties including their interconnecting strands. Our results offer insight into programmed self-assembly of low-defect supra-molecular DNA-origami 2D lattices and tubes. In addition, we demonstrate that these DNA-origami hexagon tiles and honeycomb lattices are versatile platforms for assembling optical metamaterials via programmable spatial arrangement of gold nanoparticles (AuNPs) into cluster and superlattice geometries.
10. Design of a Photonic-Crystal Channel-Drop Filter Based on the Two-Dimensional Triangular-Lattice Hole Structure
Institute of Scientific and Technical Information of China (English)
Kyu; Hwan; Hwang; G.; Hugh; Song; Chanmook; Lim; Soan; Kim; Kyung-Won; Chun; Mahn; Yong; Park
2003-01-01
A channel-drop filter has been designed based on the two-dimensional triangular-lattice hole photonic-crystal structure, which consists of two line defects and two point defects, by a two-dimensional finite-difference time-domain simulation.
11. Two-dimensional lattice solitons in polariton condensates with spin-orbit coupling
CERN Document Server
Kartashov, Yaroslav V
2016-01-01
We study two-dimensional fundamental and vortex solitons in polariton condensates with spin-orbit coupling and Zeeman splitting evolving in square arrays of microcavity pillars. Due to repulsive excitonic nonlinearity such states are encountered in finite gaps in the spectrum of the periodic array. Spin-orbit coupling between two polarization components stemming from TE-TM energy splitting of the cavity photons acting together with Zeeman splitting lifts the degeneracy between vortex solitons with opposite topological charges and makes their density profiles different for a fixed energy. This results in formation of four distinct families of vortex solitons with topological charges m=+-1, all of which can be stable. At the same time, only two stable families of fundamental gap solitons characterized by domination of different polarization components are encountered.
12. Phase diagram of the two-dimensional O(3) model from dual lattice simulations
CERN Document Server
Bruckmann, Falk; Kloiber, Thomas; Sulejmanpasic, Tin
2016-01-01
We have simulated the asymptotically free two-dimensional O(3) model at nonzero chemical potential using the model's dual representation. We first demonstrate how the latter solves the sign (complex action) problem. The system displays a crossover at nonzero temperature, while at zero temperature it undergoes a quantum phase transition when mu reaches the particle mass (generated dynamically similar to QCD). The density follows a square root behavior universal for repulsive bosons in one spatial dimension. We have also measured the spin stiffness, known to be sensitive to the spatial correlation length, using different scaling trajectories to zero temperature and infinite size. It points to a dynamical critical exponent z=2. Comparisons to thermodynamic Bethe ansaetze are shown as well.
13. Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension
Science.gov (United States)
Lin, Qian; Xiao, Meng; Yuan, Luqi; Fan, Shanhui
2016-12-01
Weyl points, as a signature of 3D topological states, have been extensively studied in condensed matter systems. Recently, the physics of Weyl points has also been explored in electromagnetic structures such as photonic crystals and metamaterials. These structures typically have complex three-dimensional geometries, which limits the potential for exploring Weyl point physics in on-chip integrated systems. Here we show that Weyl point physics emerges in a system of two-dimensional arrays of resonators undergoing dynamic modulation of refractive index. In addition, the phase of modulation can be controlled to explore Weyl points under different symmetries. Furthermore, unlike static structures, in this system the non-trivial topology of the Weyl point manifests in terms of surface state arcs in the synthetic space that exhibit one-way frequency conversion. Our system therefore provides a versatile platform to explore and exploit Weyl point physics on chip.
14. Unconventional phases in quantum spin and pseudospin systems in two dimensional and three dimensional lattices
Science.gov (United States)
Xu, Cenke
Several examples of quantum spin systems and pseudo spin systems have been studied, and unconventional states of matters and phase transitions have been realized in all these systems under consideration. In the p +/- ip superconductor Josephson lattice and the p--band cold atomic system trapped in optical lattices, novel phases which behave similarly to 1+1 dimensional systems are realized, despite the fact that the real physical systems are in two or three dimensional spaces. For instance, by employing a spin-wave analysis together with a new duality transformation, we establish the existence and stability of a novel gapless "critical phase", which we refer to as a "bond algebraic liquid". This novel critical phase is analogous to the 1+1 dimensional algebraic boson liquid phase. The reason for the novel physics is that there is a quasilocal gauge symmetry in the effective low energy Hamiltonian. In a spin-1 system on the kagome lattice, and a hard-core boson system on the honeycomb lattice, the low energy physics is controlled by two components of compact U(1) gauge symmetries that emerge at low energy. Making use of the confinement nature of the 2+1 dimensional compact gauge theories and the powerful duality between gauge theories and height field theories, the crystalline phase diagrams are studied for both systems, and the transitions to other phases are also considered. These phase diagrams might be accessible in strongly correlated materials, or atomic systems in optical lattices. A novel quantum ground state of matter is realized in a bosonic model on three dimensional fcc lattice with emergent low energy excitations. The novel phase obtained is a stable gapless boson liquid phase, with algebraic boson density correlations. The stability of this phase is protected against the instanton effect and superfluidity by self-duality and large gauge symmetries on both sides of the duality. The gapless collective excitations of this phase closely resemble the
15. On Berezinskii-Kosterlitz-Thouless phase transition and universal breathing mode in two dimensional photon gas
OpenAIRE
Vyas, Vivek M.; Panigrahi, Prasanta. K.; Banerji, J.
2013-01-01
A system of two dimensional photon gas has recently been realized experimentally. It is pointed out that this setup can be used to observe a universal breathing mode of photon gas. It is shown that a modification in the experimental setup would open up a possibility of observing the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in such a system. It is shown that the universal jump in the superfluid density of light in the output channel can be used as an unambiguous signature for the...
16. On Berezinskii-Kosterlitz-Thouless phase transition and universal breathing mode in two dimensional photon gas
CERN Document Server
Vyas, Vivek M; Banerji, J
2013-01-01
A system of two dimensional photon gas has recently been realized experimentally. It is pointed out that this setup can be used to observe a universal breathing mode of photon gas. It is shown that a modification in the experimental setup would open up a possibility of observing the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in such a system. It is shown that the universal jump in the superfluid density of light in the output channel can be used as an unambiguous signature for the experimental verification of the BKT transition.
17. Cromatografia gasosa bidimensional abrangente (GC × GC Comprehensive two-dimensional gas chromatography (GC × GC
Directory of Open Access Journals (Sweden)
Marcio Pozzobon Pedroso
2009-01-01
Full Text Available This paper presents the fundamental principles, instrumentation and selected applications of comprehensive two-dimensional gas chromatography (GC × GC. In this technique, introduced in 1991, two capillary columns are coupled and proper modulating interfaces continuously collect the eluate from the first column, transferring it to the second column. The result is a geometric increment in the chromatographic resolution, ensuring separation of extremely complex mixtures in time periods shorter or comparable to those of analysis using conventional gas chromatography and with better detectabilities and sensitivities.
18. Anisotropic ordering in a two-temperature lattice gas
DEFF Research Database (Denmark)
Szolnoki, Attila; Szabó, György; Mouritsen, Ole G.
1997-01-01
We consider a two-dimensional lattice gas model with repulsive nearest- and next-nearest-neighbor interactions that evolves in time according to anisotropic Kawasaki dynamics. The hopping of particles along the principal directions is governed by two heat baths at different temperatures T-x and T...
19. Fabrication of deep-profile Al-doped ZnO one- and two-dimensional lattices as plasmonic elements
Science.gov (United States)
Jensen, Flemming; Shkondin, Evgeniy; Takayama, Osamu; Larsen, Pernille V.; Mar, Mikkel D.; Malureanu, Radu; Lavrinenko, Andrei V.
2016-09-01
In this work, we report on fabrication of deep-profile one- and two-dimensional lattices made from Al-doped ZnO (AZO). AZO is considered as an alternative plasmonic material having the real part of the permittivity negative in the near infrared range. The exact position of the plasma frequency of AZO is doping concentration dependent, allowing for tuning possibilities. In addition, the thickness of the AZO film also affects its material properties. Physical vapor deposition techniques typically applied for AZO coating do not enable deep profiling of a plasmonic structure. Using the atomic layer deposition technique, a highly conformal deposition method, allows us to fabricate high-aspect ratio structures such as one-dimensional lattices with a period of 400 nm and size of the lamina of 200 nm in width and 3 μm in depth. Thus, our structures have an aspect ratio of 1:15 and are homogeneous on areas of 2×2 cm2 and more. We also produce two-dimensional arrays of circular nanopillars with similar dimensions. Instead of nanopillars hollow tubes with a wall thickness on demand from 20 nm up to a complete fill can be fabricated.
20. Qualitative and quantitative analysis of vetiver essential oils by comprehensive two-dimensional gas chromatography and comprehensive two-dimensional gas chromatography/mass spectrometry.
Science.gov (United States)
Filippi, Jean-Jacques; Belhassen, Emilie; Baldovini, Nicolas; Brevard, Hugues; Meierhenrich, Uwe J
2013-05-03
Vetiver essential oils (VEO) are important raw ingredients used in perfume industry, entering the formula of numerous modern fragrances. Vetiver oils are considered to be among the most complex essential oils, resulting most of the time in highly coeluted chromatograms whatever the analytical technique. In this context, conventional gas chromatography has failed to provide a routine tool for the accurate qualitative and quantitative analysis of their constituents. Applying comprehensive two-dimensional gas chromatography techniques (GC×GC-FID/MS) afforded the mean to separate efficiently vetiver oil constituents in order to identify them in a more reliable way. Moreover, this is the first time that a complete true quantitation of each constituent is carried out on such complex oils by means of internal calibration. Finally, we have studied the influence of the injection mode on the determined chemical composition, and showed that several alcohols underwent dehydration under defined chromatographic conditions (splitless mode) usually recommended for quantitation purposes. Copyright © 2013 Elsevier B.V. All rights reserved.
1. General model for a entanglement-enhanced composed quantum game on a two-dimensional lattice
CERN Document Server
2013-01-01
We introduce a method of analyzing entanglement enhanced quantum games on regular lattices of agents. Our method is valid for setups with periodic and non-periodic boundary conditions. To demonstrate our approach we study two different types games, namely the prisoner's dilemma game and a cooperative Parrondo's game. In both cases we obtain results showing, that entanglement is a crucial resource necessary for the agents to achieve positive capital gain.
2. Three-dimensional vortex solitons in quasi-two-dimensional lattices.
Science.gov (United States)
Leblond, Hervé; Malomed, Boris A; Mihalache, Dumitru
2007-08-01
We consider the three-dimensional (3D) Gross-Pitaevskii or nonlinear Schrödinger equation with a quasi-2D square-lattice potential (which corresponds to the optical lattice trapping a self-attractive Bose-Einstein condensate, or, in some approximation, to a photonic-crystal fiber, in terms of nonlinear optics). Stable 3D solitons, with embedded vorticity S=1 and 2, are found by means of the variational approximation and in a numerical form. They are built, basically, as sets of four fundamental solitons forming a rhombus, with phase shifts piS2 between adjacent sites, and an empty site in the middle. The results demonstrate two species of stable 3D solitons, which were not studied before, viz., localized vortices ("spinning light bullets," in terms of optics) with S>1 , and vortex solitons (with any S not equal 0 ) supported by a lattice in the 3D space. Typical scenarios of instability development (collapse or decay) of unstable localized vortices are identified too.
3. Observation of a pairing pseudogap in a two-dimensional Fermi gas.
Science.gov (United States)
Feld, Michael; Fröhlich, Bernd; Vogt, Enrico; Koschorreck, Marco; Köhl, Michael
2011-11-30
Pairing of fermions is ubiquitous in nature, underlying many phenomena. Examples include superconductivity, superfluidity of (3)He, the anomalous rotation of neutron stars, and the crossover between Bose-Einstein condensation of dimers and the BCS (Bardeen, Cooper and Schrieffer) regime in strongly interacting Fermi gases. When confined to two dimensions, interacting many-body systems show even more subtle effects, many of which are not understood at a fundamental level. Most striking is the (as yet unexplained) phenomenon of high-temperature superconductivity in copper oxides, which is intimately related to the two-dimensional geometry of the crystal structure. In particular, it is not understood how the many-body pairing is established at high temperature, and whether it precedes superconductivity. Here we report the observation of a many-body pairing gap above the superfluid transition temperature in a harmonically trapped, two-dimensional atomic Fermi gas in the regime of strong coupling. Our measurements of the spectral function of the gas are performed using momentum-resolved photoemission spectroscopy, analogous to angle-resolved photoemission spectroscopy in the solid state. Our observations mark a significant step in the emulation of layered two-dimensional strongly correlated superconductors using ultracold atomic gases.
4. Dispersion of guided modes in two-dimensional split ring lattices
DEFF Research Database (Denmark)
Hansen, Per Lunnemann; Koenderink, A. Femius
2014-01-01
-plane electric modes. Scatterers that have an in-plane electric and out-of-plane magnetic polarizability, but without intrinsic magnetoelectric coupling, result in two bands that are mixtures of the bands of electric-only and magnetic-only lattices. Thereby, bianisotropy through mutual coupling, in absence...... of building-block bianisotropy, is evident. Once strong bianisotropy is included in each building block, the Bloch modes become even more strongly magnetoelectric. Our results are important to understand spatial dispersion and bianisotropy of metasurface and metamaterial designs....
5. Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities
Science.gov (United States)
El-Jallal, Said; Oudich, Mourad; Pennec, Yan; Djafari-Rouhani, Bahram; Laude, Vincent; Beugnot, Jean-Charles; Martínez, Alejandro; Escalante, José María; Makhoute, Abdelkader
2013-11-01
We theoretically investigate phonon-photon interaction in cavities created in a phoxonic crystal slab constituted by a two-dimensional (2D) square array of holes in a silicon membrane. The structure without defects provides 2D band gaps for both electromagnetic and elastic waves. We consider two types of cavities, namely, an L3 cavity (a row of three holes is removed) and a cross-shape cavity, which both possess highly confined phononic and photonic localized modes suitable for enhancing their interaction. In our theoretical study, we take into account two mechanisms that contribute to optomechanical interaction, namely, the photoelastic and the interface motion effects. We show that, depending on the considered pair of photonic and phononic modes, the two mechanisms can have similar or very different magnitudes, and their contributions can be either in or out of phase. We find out that only acoustic modes with a specific symmetry are allowed to couple with photonic cavity modes. The coupling strength is quantified by two different methods. In the first method, we compute a direct estimation of coupling rates by overlap integrals, while in the second one, we analyze the temporal modulation of the resonant photonic frequency by the phonon-induced acoustic vibrational motion during one acoustic period. Interestingly, we obtain high optomechanical interaction, with the coupling rate reaching more than 2.4 MHz for some specific phonon-photon pairs.
6. Critical Behavior of Spatial Evolutionary Game with Altruistic to Spiteful Preferences on Two-Dimensional Lattices
Science.gov (United States)
Yang, Bo; Li, Xiao-Teng; Chen, Wei; Liu, Jian; Chen, Xiao-Song
2016-10-01
Self-questioning mechanism which is similar to single spin-flip of Ising model in statistical physics is introduced into spatial evolutionary game model. We propose a game model with altruistic to spiteful preferences via weighted sums of own and opponent's payoffs. This game model can be transformed into Ising model with an external field. Both interaction between spins and the external field are determined by the elements of payoff matrix and the preference parameter. In the case of perfect rationality at zero social temperature, this game model has three different phases which are entirely cooperative phase, entirely non-cooperative phase and mixed phase. In the investigations of the game model with Monte Carlo simulation, two paths of payoff and preference parameters are taken. In one path, the system undergoes a discontinuous transition from cooperative phase to non-cooperative phase with the change of preference parameter. In another path, two continuous transitions appear one after another when system changes from cooperative phase to non-cooperative phase with the prefenrence parameter. The critical exponents v, β, and γ of two continuous phase transitions are estimated by the finite-size scaling analysis. Both continuous phase transitions have the same critical exponents and they belong to the same universality class as the two-dimensional Ising model. Supported by the National Natural Science Foundation of China under Grant Nos. 11121403 and 11504384
7. First-order phase transitions in repulsive rigid k-mers on two-dimensional lattices
Science.gov (United States)
Pasinetti, P. M.; Romá, F.; Ramirez-Pastor, A. J.
2012-02-01
In a previous paper [F. Romá, A. J. Ramirez-Pastor, and J. L. Riccardo, Phys. Rev. B 72, 035444 (2005)], the critical behavior of repulsive rigid rods of length k (k-mers) on a square lattice at half coverage has been studied by using Monte Carlo (MC) simulations. The obtained results indicated that (1) the phase transition occurring in the system is a second-order phase transition for all adsorbate sizes k; and (2) the universality class of the transition changes from 2D Ising-type for monomers (k = 1) to an unknown universality class for k ≥ 2. In the present work, we revisit our previous results together with further numerical evidences, resulting from new extensive MC simulations based on an efficient exchange algorithm and using high-performance computational capabilities. In contrast to our previous conclusions (1) and (2), the new numerical calculations clearly support the occurrence of a first-order phase transition for k ≥ 2. In addition, a similar scenario was found for k-mers adsorbed on the triangular lattice at coverage k/(2k+1).
8. Kinetic Theory of a Confined Quasi-Two-Dimensional Gas of Hard Spheres
Directory of Open Access Journals (Sweden)
J. Javier Brey
2017-02-01
Full Text Available The dynamics of a system of hard spheres enclosed between two parallel plates separated a distance smaller than two particle diameters is described at the level of kinetic theory. The interest focuses on the behavior of the quasi-two-dimensional fluid seen when looking at the system from above or below. In the first part, a collisional model for the effective two-dimensional dynamics is analyzed. Although it is able to describe quite well the homogeneous evolution observed in the experiments, it is shown that it fails to predict the existence of non-equilibrium phase transitions, and in particular, the bimodal regime exhibited by the real system. A critical revision analysis of the model is presented , and as a starting point to get a more accurate description, the Boltzmann equation for the quasi-two-dimensional gas has been derived. In the elastic case, the solutions of the equation verify an H-theorem implying a monotonic tendency to a non-uniform steady state. As an example of application of the kinetic equation, here the evolution equations for the vertical and horizontal temperatures of the system are derived in the homogeneous approximation, and the results compared with molecular dynamics simulation results.
9. Two-dimensional lattice Boltzmann model for compressible flows with high Mach number
Science.gov (United States)
Gan, Yanbiao; Xu, Aiguo; Zhang, Guangcai; Yu, Xijun; Li, Yingjun
2008-03-01
In this paper we present an improved lattice Boltzmann model for compressible Navier-Stokes system with high Mach number. The model is composed of three components: (i) the discrete-velocity-model by M. Watari and M. Tsutahara [Phys. Rev. E 67 (2003) 036306], (ii) a modified Lax-Wendroff finite difference scheme where reasonable dissipation and dispersion are naturally included, (iii) artificial viscosity. The improved model is convenient to compromise the high accuracy and stability. The included dispersion term can effectively reduce the numerical oscillation at discontinuity. The added artificial viscosity helps the scheme to satisfy the von Neumann stability condition. Shock tubes and shock reflections are used to validate the new scheme. In our numerical tests the Mach numbers are successfully increased up to 20 or higher. The flexibility of the new model makes it suitable for tracking shock waves with high accuracy and for investigating nonlinear nonequilibrium complex systems.
10. From Discreteness to Continuity: Dislocation Equation for Two-Dimensional Triangular Lattice
Institute of Scientific and Technical Information of China (English)
WANG Shao-Feng
2007-01-01
@@ A systematic method from the discreteness to the continuity is presented for the dislocation equation of the triangular lattice. A modification of the Peierls equation has been derived strictly. The modified equation includes the higher order corrections of the discrete effect which are important for the core structure of dislocation. It is observed that the modified equation possesses a universal form which is model-independent except the factors.The factors, which depend on the detail of the model, are related to the derivatives of the kernel at its zero point in the wave-vector space. The results open a way to deal with the complicated models because what one needs to do is to investigate the behaviour near the zero point of the kernel in the wave-vector space instead of calculating the kernel completely.
11. Spectroscopy of Dipolar Fermions in Layered Two-Dimensional and Three-Dimensional Lattices
Science.gov (United States)
2011-09-06
term. We approximate Vαβ(j,ρ) by Vαβ(j,ρ) ≈ γαβU(j,ρ) (8) and Vsf by Vsf(j,ρ) ≈ ηU(j,ρ), (9) with U(j,ρ) = δ0j 1 ρ3 + A + (1 − δ0j ) 1 − 3( ( jdl )2 ρ2... jdl )2 ) [ρ2 + ( jdl )2]3/2 , (10) where γαβ = dααdββ/(4π 0), η = | ∫ dzw∗1(z)w2(z)|2d212/(4π 0), and dl is the lattice spacing, where A ∼ 3, with...3( ( jdl )2 ρ2+( jdl )2 ) [ρ2 + ( jdl )2]3/2 ] = 2π3/2 √ m β n (γ12 − γ11 + η) (thermal), (28) 033608-5 HAZZARD, GORSHKOV, AND REY PHYSICAL REVIEW A 84
12. Magnetoelectronic transport of the two-dimensional electron gas in CdSe single quantum wells
Indian Academy of Sciences (India)
P K Ghosh; A Ghosal; D Chattopadhyay
2009-02-01
Hall mobility and magnetoresistance coefficient for the two-dimensional (2D) electron transport parallel to the heterojunction interfaces in a single quantum well of CdSe are calculated with a numerical iterative technique in the framework of Fermi–Dirac statistics. Lattice scatterings due to polar-mode longitudinal optic (LO) phonons, and acoustic phonons via deformation potential and piezoelectric couplings, are considered together with background and remote ionized impurity interactions. The parallel mode of piezoelectric scattering is found to contribute more than the perpendicular mode. We observe that the Hall mobility decreases with increasing temperature but increases with increasing channel width. The magnetoresistance coefficient is found to decrease with increasing temperature and increase with increasing magnetic field in the classical region.
13. Two-dimensional transition metal dichalcogenides with a hexagonal lattice: Room-temperature quantum spin Hall insulators
Science.gov (United States)
Ma, Yandong; Kou, Liangzhi; Li, Xiao; Dai, Ying; Heine, Thomas
2016-01-01
So far, several transition metal dichalcogenide (TMDC)-based two-dimensional (2D) topological insulators (TIs) have been discovered, all of them based on a tetragonal lattice. However, in 2D crystals, the hexagonal rather than the tetragonal symmetry is the most common motif. Here, based on first principles calculations, we propose a class of stable 2D TMDCs of composition MX2(M =Mo ,W ;X =S ,Se ,Te ) with a hexagonal lattice. They are all in the same stability range as other 2D TMDC allotropes that have been demonstrated experimentally, and they are identified to be practical 2D TIs with large band gaps ranging from 41 to 198 meV, making them suitable for applications at room temperature. Besides, in contrast to tetragonal 2D TMDCs, their hexagonal lattice will greatly facilitate the integration of theses novel TI state van der Waals crystals with other hexagonal or honeycomb materials and thus provide a route for 2D material-based devices for wider nanoelectronic and spintronic applications. The nontrivial band gaps of both WS e2 and WT e2 2D crystals are 198 meV, which are larger than that in any previously reported TMDC-based TIs. These large band gaps entirely stem from the strong spin orbit coupling strength within the d orbitals of Mo/W atoms near the Fermi level. Our findings broaden the scientific and technological impact of both 2D TIs and TMDCs.
14. Metal Oxide Gas Sensor Drift Compensation Using a Two-Dimensional Classifier Ensemble
Directory of Open Access Journals (Sweden)
Hang Liu
2015-04-01
Full Text Available Sensor drift is the most challenging problem in gas sensing at present. We propose a novel two-dimensional classifier ensemble strategy to solve the gas discrimination problem, regardless of the gas concentration, with high accuracy over extended periods of time. This strategy is appropriate for multi-class classifiers that consist of combinations of pairwise classifiers, such as support vector machines. We compare the performance of the strategy with those of competing methods in an experiment based on a public dataset that was compiled over a period of three years. The experimental results demonstrate that the two-dimensional ensemble outperforms the other methods considered. Furthermore, we propose a pre-aging process inspired by that applied to the sensors to improve the stability of the classifier ensemble. The experimental results demonstrate that the weight of each multi-class classifier model in the ensemble remains fairly static before and after the addition of new classifier models to the ensemble, when a pre-aging procedure is applied.
15. Piezoelectric Electromechanical Coupling in Nanomechanical Resonators with a Two-Dimensional Electron Gas
Science.gov (United States)
Shevyrin, A. A.; Pogosov, A. G.; Bakarov, A. K.; Shklyaev, A. A.
2016-07-01
The electrical response of a two-dimensional electron gas to vibrations of a nanomechanical cantilever containing it is studied. Vibrations of perpendicularly oriented cantilevers are experimentally shown to oppositely change the conductivity near their bases. This indicates the piezoelectric nature of electromechanical coupling. A physical model is developed, which quantitatively explains the experiment. It shows that the main origin of the conductivity change is a rapid change in the mechanical stress on the boundary between suspended and nonsuspended areas, rather than the stress itself.
16. Electromechanical coupling in suspended nanomechanical resonators with a two-dimensional electron gas
Science.gov (United States)
Shevyrin, A. A.; Pogosov, A. G.; Bakarov, A. K.; Shklyaev, A. A.
2017-06-01
A physical model describing the piezoelectric-effect-mediated influence of bending of a thin suspended cantilever with a two-dimensional electron gas on the conductivity is proposed. The model shows that the conductivity change is almost entirely caused by the rapid change in mechanical stress near the boundary of suspended and non-suspended areas, rather than by the stress itself. An experiment confirming that the electromechanical coupling is associated with the piezoelectric effect is performed. The experimentally measured conductance sensitivity to the cantilever’s vibrations agree with the developed physical model.
17. Ultra-low-temperature cooling of two-dimensional electron gas
Science.gov (United States)
Xia, J. S.; Adams, E. D.; Shvarts, V.; Pan, W.; Stormer, H. L.; Tsui, D. C.
2000-05-01
A new design has been used for cooling GaAs/Al xGa 1- xAs sample to ultra-low-temperatures. The sample, with electrical contacts directly soldered to the sintered silver powder heat exchangers, was immersed in liquid 3He, which was cooled by a PrNI 5 nuclear refrigerator. The data analysis shows that the two-dimensional electron gas (2DEG) was cooled to 4.0 mK at the refrigerator base temperature Tb of 2.0 mK. The design with heat exchanger cooling is applicable to any ultra-low-temperature transport measurements of 2DEG system.
18. Spin injection into a two-dimensional electron gas using inter-digital-ferromagnetic contacts
DEFF Research Database (Denmark)
Hu, C.M.; Nitta, J.; Jensen, Ane;
2002-01-01
We present a model that describes the spin injection across a single interface with two electrodes. The spin-injection rate across a typical hybrid junction made of ferromagnet (FM) and a two-dimensional electron gas (2DEG) is found at the percentage level. We perforin spin-injection-detection ex......-injection-detection experiment on devices with two ferromagnetic contacts on a 2DEG confined in an InAs quantum well. A spin-injection rate of 4.5% is estimated from the measured magnetoresistance....
19. Quantum spin-glass transition in the two-dimensional electron gas
Indian Academy of Sciences (India)
Subir Sachdev
2002-02-01
We discuss the possibility of spin-glass order in the vicinity of the unexpected metallic state of the two-dimensional electron gas in zero applied magnetic field. An average ferromagnetic moment may also be present, and the spin-glass order then resides in the plane orthogonal to the ferromagnetic moment. We argue that a quantum transition involving the destruction of the spin-glass order in an applied in-plane magnetic field offers a natural explanation of some features of recent magnetoconductance measurements. We present a quantum field theory for such a transition and compute its mean field properties.
20. Electrically Detected Magnetic Resonance of Neutral Donors Interacting with a Two-Dimensional Electron Gas
Energy Technology Data Exchange (ETDEWEB)
Lo, C. C.; Lang, V.; George, R. E.; Morton, J. J. L.; Tyryshkin, A. M.; Lyon, A.; Bokor, J.; Schenkel, T.
2011-04-20
We have measured the electrically detected magnetic resonance of donor-doped silicon field-effect transistors in resonant X- (9.7 GHz) and W-band (94 GHz) microwave cavities. The two-dimensional electron gas (2DEG) resonance signal increases by two orders of magnitude from X- to W-band, while the donor resonance signals are enhanced by over one order of magnitude. Bolometric effects and spin-dependent scattering are inconsistent with the observations. We propose that polarization transfer from the donor to the 2DEG is the main mechanism giving rise to the spin resonance signals.
1. Photon-assisted spin transport in a two-dimensional electron gas
OpenAIRE
Fistul, M. V.; Efetov, K. B.
2007-01-01
We study spin-dependent transport in a two-dimensional electron gas subject to an external step-like potential $V(x)$ and irradiated by an electromagnetic field (EF). In the absence of EF the electronic spectrum splits into spin sub-bands originating from the "Rashba" spin-orbit coupling. We show that the resonant interaction of propagating electrons with the component EF parallel to the barrier induces a \\textit{% non-equilibrium dynamic gap} $(2\\Delta_{R})$ between the spin sub-bands. Exist...
2. Thermodynamic magnetization of two-dimensional electron gas measured over wide range of densities
OpenAIRE
Reznikov, M.; Kuntsevich, A. Yu.; Teneh, N.; Pudalov, V. M.
2011-01-01
We report measurements of dm/dn in Si MOSFET, where m is the magnetization of the two-dimensional electron gas and n is its density. We extended the density range of measurements from well in the metallic to deep in the insulating region. The paper discusses in detail the conditions under which this extension is justified, as well as the corrections one should make to extract dm/dn properly. At low temperatures, dm/dn was found to be strongly nonlinear already in weak magnetic fields, on a sc...
3. Extraordinary waves in two dimensional electron gas with separate spin evolution and Coulomb exchange interaction
CERN Document Server
Andreev, Pavel A
2016-01-01
Hydrodynamics analysis of waves in two-dimensional degenerate electron gas with the account of separate spin evolution is presented. The transverse electric field is included along with the longitudinal electric field. The Coulomb exchange interaction is included in the analysis. In contrast with the three-dimensional plasma-like mediums the contribution of the transverse electric field is small. We show the decrease of frequency of both the extraordinary (Langmuir) wave and the spin-electron acoustic wave due to the exchange interaction. Moreover, spin-electron acoustic wave has negative dispersion at the relatively large spin-polarization. Corresponding dispersion dependencies are presented and analyzed.
4. Resonant Peak Splitting for Ballistic Conductance in Two-Dimensional Electron Gas Under Electromagnetic Modulation
Institute of Scientific and Technical Information of China (English)
WANG Ru-Zhi; YAN Xiao-Hong
2000-01-01
By developing a transfer-matrix method, the resonant peaks splitting of ballistic conductance are investigated into the two-dimensional electron gas system with both electric and magnetic modulations of nanoscale periods. It is found that there exists the n-fold resonant peak splitting for ballistic conductance through n perpendicular magnetic barriers to n electric barriers. With a combination of m magnetic barriers and n electric barriers by increasing the amplitude of electric field, the folds of the splitting would shift from m － 1 to n － 1.
5. The two-dimensional alternative binary L-J system: liquid-gas phase diagram
Institute of Scientific and Technical Information of China (English)
张陟; 陈立溁
2003-01-01
A two-dimensional (2D) binary system without considering the Lennard-Jones (L-J) potential has been studied by using the Collins model. In this paper, we introduce the L-J potential into the 2D binary system and consider the existence of the holes that are called the "molecular fraction". The liquid-gas phase diagram of the 2D alternative binary L-J system is obtained. The results are quite analogous to the behaviour of 3D substances.
6. Diffusivity and weak clustering in a quasi-two-dimensional granular gas.
Science.gov (United States)
Perera-Burgos, J A; Pérez-Ángel, G; Nahmad-Molinari, Y
2010-11-01
We present results from a detailed simulation of a quasi-two-dimensional dissipative granular gas, kept in a noncondensed steady state via vertical shaking over a rough substrate. This gas shows a weak power-law decay in the tails of its pair distribution functions, indicating clustering. This clustering depends monotonically on the dissipation coefficient and disappears when the sphere-sphere collisions are conservative. Clustering is also sensitive to the packing fraction. This gas also displays the standard nonequilibrium characteristics of similar systems, including non-Maxwellian velocity distributions. The diffusion coefficients are calculated over all the conditions of the simulations, and it is found that diluted gases are more diffusive for smaller restitution coefficients.
7. Response Functions for the Two-Dimensional Ultracold Fermi Gas: Dynamical BCS Theory and Beyond
Science.gov (United States)
Vitali, Ettore; Shi, Hao; Qin, Mingpu; Zhang, Shiwei
2017-08-01
Response functions are central objects in physics. They provide crucial information about the behavior of physical systems, and they can be directly compared with scattering experiments involving particles such as neutrons or photons. Calculations of such functions starting from the many-body Hamiltonian of a physical system are challenging and extremely valuable. In this paper, we focus on the two-dimensional (2D) ultracold Fermi atomic gas which has been realized experimentally. We present an application of the dynamical BCS theory to obtain response functions for different regimes of interaction strengths in the 2D gas with zero-range attractive interaction. We also discuss auxiliary-field quantum Monte Carlo (AFQMC) methods for the calculation of imaginary time correlations in these dilute Fermi gas systems. Illustrative results are given and comparisons are made between AFQMC and dynamical BCS theory results to assess the accuracy of the latter.
8. Shear viscosity and spin-diffusion coefficient of a two-dimensional Fermi gas
DEFF Research Database (Denmark)
Bruun, Georg
2012-01-01
Using kinetic theory, we calculate the shear viscosity and the spin-diffusion coefficient as well as the associated relaxation times for a two-component Fermi gas in two dimensions, as a function of temperature, coupling strength, polarization, and mass ratio of the two components. It is demonstr......Using kinetic theory, we calculate the shear viscosity and the spin-diffusion coefficient as well as the associated relaxation times for a two-component Fermi gas in two dimensions, as a function of temperature, coupling strength, polarization, and mass ratio of the two components....... It is demonstrated that the minimum value of the viscosity decreases with the mass ratio, since Fermi blocking becomes less efficient. We furthermore analyze recent experimental results for the quadrupole mode of a two-dimensional gas in terms of viscous damping, obtaining a qualitative agreement using no fitting...
9. Device for two-dimensional gas-phase separation and characterization of ion mixtures
Science.gov (United States)
Tang, Keqi; Shvartsburg, Alexandre A.; Smith, Richard D.
2006-12-12
The present invention relates to a device for separation and characterization of gas-phase ions. The device incorporates an ion source, a field asymmetric waveform ion mobility spectrometry (FAIMS) analyzer, an ion mobility spectrometry (IMS) drift tube, and an ion detector. In one aspect of the invention, FAIMS operating voltages are electrically floated on top of the IMS drift voltage. In the other aspect, the FAIMS/IMS interface is implemented employing an electrodynamic ion funnel, including in particular an hourglass ion funnel. The present invention improves the efficiency (peak capacity) and sensitivity of gas-phase separations; the online FAIMS/IMS coupling creates a fundamentally novel two-dimensional gas-phase separation technology with high peak capacity, specificity, and exceptional throughput.
10. Pythagoras's theorem on a two-dimensional lattice from a 'natural' Dirac operator and Connes's distance formula
Energy Technology Data Exchange (ETDEWEB)
Dai Jian [Theory Group, Department of Physics, Peking University, Beijing (China)]. E-mail: [email protected]; Song Xingchang [Theory Group, Department of Physics, Peking University, Beijing (China)]. E-mail: [email protected]
2001-07-13
One of the key ingredients of Connes's noncommutative geometry is a generalized Dirac operator which induces a metric (Connes's distance) on the pure state space. We generalize such a Dirac operator devised by Dimakis et al, whose Connes distance recovers the linear distance on an one-dimensional lattice, to the two-dimensional case. This Dirac operator has the local eigenvalue property and induces a Euclidean distance on this two-dimensional lattice, which is referred to as 'natural'. This kind of Dirac operator can be easily generalized into any higher-dimensional lattices. (author)
11. Finite-Difference Lattice Boltzmann Scheme for High-Speed Compressible Flow: Two-Dimensional Case
Science.gov (United States)
Gan, Yan-Biao; Xu, Ai-Guo; Zhang, Guang-Cai; Zhang, Ping; Zhang, Lei; Li, Ying-Jun
2008-07-01
Lattice Boltzmann (LB) modeling of high-speed compressible flows has long been attempted by various authors. One common weakness of most of previous models is the instability problem when the Mach number of the flow is large. In this paper we present a finite-difference LB model, which works for flows with flexible ratios of specific heats and a wide range of Mach number, from 0 to 30 or higher. Besides the discrete-velocity-model by Watari [Physica A 382 (2007) 502], a modified Lax Wendroff finite difference scheme and an artificial viscosity are introduced. The combination of the finite-difference scheme and the adding of artificial viscosity must find a balance of numerical stability versus accuracy. The proposed model is validated by recovering results of some well-known benchmark tests: shock tubes and shock reflections. The new model may be used to track shock waves and/or to study the non-equilibrium procedure in the transition between the regular and Mach reflections of shock waves, etc.
12. Magnon Breakdown in a Two Dimensional Triangular Lattice Heisenberg Antiferromagnet of Multiferroic LuMnO3
Science.gov (United States)
Oh, Joosung; Le, Manh Duc; Jeong, Jaehong; Lee, Jung-hyun; Woo, Hyungje; Song, Wan-Young; Perring, T. G.; Buyers, W. J. L.; Cheong, S.-W.; Park, Je-Geun
2013-12-01
The breakdown of magnons, the quasiparticles of magnetic systems, has rarely been seen. By using an inelastic neutron scattering technique, we report the observation of spontaneous magnon decay in multiferroic LuMnO3, a simple two dimensional Heisenberg triangular lattice antiferromagnet, with large spin S=2. The origin of this rare phenomenon lies in the nonvanishing cubic interaction between magnons in the spin Hamiltonian arising from the noncollinear 120° spin structure. We observed all three key features of the nonlinear effects as theoretically predicted: a rotonlike minimum, a flat mode, and a linewidth broadening, in our inelastic neutron scattering measurements of single crystal LuMnO3. Our results show that quasiparticles in a system hitherto thought of as “classical” can indeed break down.
13. Magnon breakdown in a two dimensional triangular lattice Heisenberg antiferromagnet of multiferroic LuMnO3.
Science.gov (United States)
Oh, Joosung; Le, Manh Duc; Jeong, Jaehong; Lee, Jung-hyun; Woo, Hyungje; Song, Wan-Young; Perring, T G; Buyers, W J L; Cheong, S-W; Park, Je-Geun
2013-12-20
The breakdown of magnons, the quasiparticles of magnetic systems, has rarely been seen. By using an inelastic neutron scattering technique, we report the observation of spontaneous magnon decay in multiferroic LuMnO3, a simple two dimensional Heisenberg triangular lattice antiferromagnet, with large spin S=2. The origin of this rare phenomenon lies in the nonvanishing cubic interaction between magnons in the spin Hamiltonian arising from the noncollinear 120° spin structure. We observed all three key features of the nonlinear effects as theoretically predicted: a rotonlike minimum, a flat mode, and a linewidth broadening, in our inelastic neutron scattering measurements of single crystal LuMnO3. Our results show that quasiparticles in a system hitherto thought of as "classical" can indeed break down.
14. Floquet-Weyl and Floquet-topological-insulator phases in a stacked two-dimensional ring-network lattice
CERN Document Server
Ochiai, Tetsuyuki
2016-01-01
We show the presence of Floquet-Weyl and Floquet-topological-insulator phases in a stacked two-dimensional ring-network lattice. The Weyl points in the three-dimensional Brillouin zone and Fermi-arc surface states are clearly demonstrated in the quasienergy spectrum of the system in the Weyl phase. In addition, chiral surface states coexist in this phase. The Floquet-topological-insulator phase is characterized by the winding number of two in the reflection matrices of the semi-infinite system and resulting two gapless surface states in the quasienergy g ap of the bulk. The phase diagram of the system is derived in the two-parameter space of hopping S-matrices among the rings. We also discuss a possible optical realization of the system together with the introduction of synthetic gauge fields.
15. Localization of a two-component Bose–Einstein condensate in a two-dimensional bichromatic optical lattice
Energy Technology Data Exchange (ETDEWEB)
Xi, Kui-Tian, E-mail: [email protected]; Li, Jinbin, E-mail: [email protected]; Shi, Da-Ning, E-mail: [email protected]
2014-03-01
We consider a weakly interacting two-component Bose–Einstein condensate (BEC) in a two-dimensional (2D) quasi-periodic bichromatic optical lattice (BOL). The problem is studied by means of split-step Crank–Nicolson method. The effects of weak intra- and inter-component interactions on localization of a two-component BEC are investigated. It is shown that in the quasi-2D regime, due to the enhanced disorder, there is no symmetry breaking like that in the one-dimensional (1D) case under a sine-typed potential, while configurations of density profiles are also quite different from that in the 1D case. By modulating interactions, the interplay of disorder and weak repulsive or attractive interactions is studied in detail. The cases with sine- and cosine-typed potentials acting on components 1 and 2 respectively are also discussed.
16. Identification of the dynamics of a two-dimensional grid structure using least square lattice filters. [for large space structures
Science.gov (United States)
Montgomery, R. C.; Sundararajan, N.
1984-01-01
It is doubtful whether the dynamics of large space structures (LSS) can be predicted well enough for control system design applications. Hence, dynamic modeling from on-orbit measurements followed by a modification of the control system is of interest, taking into account the utilization of adaptive control concepts. The present paper is concerned with the model determination phase of the adaptive control problem. Using spectral decoupling to determine mode shapes, mode frequency and damping data can be obtained with the aid of an equation error parameter identification method. This method employs a second-order auto-regressive moving average (ARMA) model to represent the natural mode amplitudes. The discussed procedure involves an extension of the application of the least square lattice filter in system identification to a nonintegral, two-dimensional grid structure made of overlapping bars.
17. Floquet-Weyl and Floquet-topological-insulator phases in a stacked two-dimensional ring-network lattice
Science.gov (United States)
Ochiai, Tetsuyuki
2016-10-01
We show the presence of Floquet-Weyl and Floquet-topological-insulator phases in a stacked two-dimensional ring-network lattice. The Weyl points in the three-dimensional Brillouin zone and Fermi-arc surface states are clearly demonstrated in the quasienergy spectrum of the system in the Floquet-Weyl phase. In addition, chiral surface states coexist in this phase. The Floquet-topological-insulator phase is characterized by the winding number of two in the reflection matrices of the semi-infinite system and resulting two gapless surface states in the quasienergy gap of the bulk. The phase diagram of the system is derived in the two-parameter space of hopping S-matrices among the rings. We also discuss a possible optical realization of the system together with the introduction of synthetic gauge fields.
18. Deformation of Two-Dimensional Nonuniform-Membrane Red Blood Cells Simulated by a Lattice Boltzmann Model
Institute of Scientific and Technical Information of China (English)
LI Hua-Bing; JIN Li; QIU Bing
2008-01-01
To study two-dimensional red blood cells deforming in a shear flow with the membrane nonuniform on the rigidity and mass, the membrane is discretized into equilength segments. The fluid inside and outside the red blood cell is simulated by the D2Q9 lattice Boltzmann model and the hydrodynamic forces exerted on the membrane from the inner and outer of the red blood cell are calculated by a stress-integration method. Through the global deviation from the curvature of uniform-membrane, we find that when the membrane is nonuniform on the rigidity, the deviation first decreases with the time increases and implies that the terminal profile of the red blood cell is static. To a red blood cell with the mass nonuniform on the membrane, the deviation becomes more large, and the mass distribution affects the profile of the two sides of the flattened red blood cell in a shear flow.
19. A novel two-dimensional MgB6 crystal: metal-layer stabilized boron kagome lattice.
Science.gov (United States)
Xie, Sheng-Yi; Li, Xian-Bin; Tian, Wei Quan; Chen, Nian-Ke; Wang, Yeliang; Zhang, Shengbai; Sun, Hong-Bo
2015-01-14
Based on first-principles calculations, we designed for the first time a boron-kagome-based two-dimensional MgB6 crystal, in which two boron kagome layers sandwich a triangular magnesium layer. The two-dimensional lattice is metallic with several bands across the Fermi level, and among them a Dirac point appears at the K point of the first Brillouin zone. This metal-stabilized boron kagome system displays electron-phonon coupling, with a superconductivity critical transition temperature of 4.7 K, and thus it is another possible superconducting Mg-B compound besides MgB2. Furthermore, the proposed 2D MgB6 can also be used for hydrogen storage after decoration with Ca. Up to five H2 molecules can be attracted by one Ca with an average binding energy of 0.225 eV. The unique properties of 2D MgB6 will spur broad interest in nanoscience and technology.
20. Phase correlations and quasicondensate in a two-dimensional ultracold Fermi gas
Energy Technology Data Exchange (ETDEWEB)
Tempere, J., E-mail: [email protected] [Theory of Quantum and Complex Systems, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen (Belgium); Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138 (United States); Klimin, S.N. [Theory of Quantum and Complex Systems, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen (Belgium)
2015-02-15
The interplay between dimensionality, coherence and interaction in superfluid Fermi gases is analyzed by the phase correlation function of the field of fermionic pairs. We calculate this phase correlation function for a two-dimensional superfluid Fermi gas with s-wave interactions within the Gaussian pair fluctuation formalism. The spatial behavior of the correlation function is shown to exhibit a rapid (exponential) decay at short distances and a characteristic algebraic decay at large distances, with an exponent matching that expected from the Berezinskii–Kosterlitz–Thouless theory of 2D Bose superfluids. We conclude that the Gaussian pair fluctuation approximation is able to capture the physics of quasi-long-range order in two-dimensional Fermi gases. - Highlights: • The phase correlation functions for an ultracold Fermi gas in 2D are calculated. • The decay of the correlation functions is algebraic at long distances. • The Gaussian pair fluctuation approach is shown to capture the quasicondensate physics in 2D Fermi gases.
1. Bubble dynamics in a two-dimensional gas-solid fluidized bed
Institute of Scientific and Technical Information of China (English)
2007-01-01
Related referential studies on gas-solid two-phase flows were briefly reviewed. Bubble ascending in a two-dimensional (2D) gas-solid fluidized bed was studied both experimentally and numerically. A modified continuum model expressed in the conservation form was used in numerical simulation. Solid-phase pressure was modeled via local sound speed; gas-phase turbulence was described by the K-ε two-equation model. The modified implicit multiphase formulation (IMF) scheme was used to solve the model equations in 2D Cartesian/cylindrical coordinates. The bubble ascending velocity and particle motion in the 2D fluidized bed were measured using the photochromic dye activation (PDA) technique, which was based on UV light activation of particles impregnated with the dye. Effects of bed height and superficial gas velocity on bubble formation and ascent were investigated numerically. The numerically obtained bubble ascending velocities were compared with experimental measurements. Gas bubble in jetting gas-solids fluidized bed was also simulated numerically.
2. Properties of two-dimensional electron gas containing self-organized quantum antidots
Science.gov (United States)
Vasilyev, Yu.; Suchalkin, S.; Zundel, M.; Heisenberg, D.; Eberl, K.; von Klitzing, K.
1999-11-01
A nonuniform two-dimensional electron gas in a heterojunction with inserted self-organized electrically inactive dots (acting as antidots) has been fabricated by molecular-beam epitaxy of AlGaAs/AlInAs/GaAs layer sequences. Transport measurements give the ratio of the transport mobility to the quantum mobility less than four, which suggests that the dominant scattering at low magnetic fields is the short-range scattering from the lateral potential of the antidots. Far-infrared cyclotron resonance (CR) spectra show an absorption mode as narrow as 0.5 cm-1 at high magnetic fields associated with the high-mobility electron gas formed between the antidot islands and confined in the lateral directions. The confinement energy of 14 cm-1 is derived from the CR spectra.
3. Two-dimensional intra-band solitons in lattice potentials with local defects and self-focusing nonlinearity
CERN Document Server
Zeng, Jianhua
2013-01-01
It is commonly known that stable bright solitons in periodic potentials, which represent gratings in photonics/plasmonics, or optical lattices in quantum gases, exist either in the spectral semi-infinite gap (SIG) or in finite bandgaps. Using numerical methods, we demonstrate that, under the action of the cubic self-focusing nonlinearity, defects in the form of "holes" in two-dimensional (2D) lattices support continuous families of 2D solitons \\textit{embedded} into the first two Bloch bands of the respective linear spectrum, where solitons normally do not exist. The two families of the \\textit{embedded defect solitons} (EDSs) are found to be continuously linked by the branch of \\textit{gap defect solitons} (GDSs) populating the first finite bandgap. Further, the EDS branch traversing the first band links the GDS family with the branch of regular defect-supported solitons populating the SIG. Thus, we construct a continuous chain of regular, embedded, and gap-mode solitons ("superfamily") threading the entire ...
4. Two-dimensional O(3) model at nonzero density: From dual lattice simulations to repulsive bosons
Science.gov (United States)
Bruckmann, Falk; Gattringer, Christof; Kloiber, Thomas; Sulejmanpasic, Tin
2016-12-01
We discuss the thermodynamics of the O(3) nonlinear sigma model in 1 +1 dimensions at nonzero chemical potential (equivalent to a magnetic field). In its conventional field theory representation the model suffers from a sign problem. By dualizing the model, we are able to fully access the nonzero density regime of an asymptotically free theory with dynamical mass gap at arbitrary chemical potential values. We find a quantum phase transition at zero temperature where as a function of the chemical potential the density assumes a nonzero value. Measuring the spin stiffness we present evidence for a corresponding dynamical critical exponent z close to 2. The low energy O(3) model is conjectured to be described by a massive boson triplet with repulsive interactions. We confirm the universal square-root behavior expected for such a system at low density (and temperature) and compare our data to the results of Bethe Ansatz solutions of the relativistic and nonrelativistic one-dimensional Bose gas. We also comment on a potential Berezinskii-Kosterlitz-Thouless transition at nonzero density.
5. Direct observation of two dimensional trace gas distributions with an airborne Imaging DOAS instrument
Directory of Open Access Journals (Sweden)
K.-P. Heue
2008-11-01
Full Text Available In many investigations of tropospheric chemistry information about the two dimensional distribution of trace gases on a small scale (e.g. tens to hundreds of metres is highly desirable. An airborne instrument based on imaging Differential Optical Absorption Spectroscopy has been built to map the two dimensional distribution of a series of relevant trace gases including NO2, HCHO, C2H2O2, H2O, O4, SO2, and BrO on a scale of 100 m.
Here we report on the first tests of the novel aircraft instrument over the industrialised South African Highveld, where large variations in NO2 column densities in the immediate vicinity of several sources e.g. power plants or steel works, were measured. The observed patterns in the trace gas distribution are interpreted with respect to flux estimates, and it is seen that the fine resolution of the measurements allows separate sources in close proximity to one another to be distinguished.
6. Retention modelling of polychlorinated biphenyls in comprehensive two-dimensional gas chromatography.
Science.gov (United States)
D'Archivio, Angelo Antonio; Incani, Angela; Ruggieri, Fabrizio
2011-01-01
In this paper, we use a quantitative structure-retention relationship (QSRR) method to predict the retention times of polychlorinated biphenyls (PCBs) in comprehensive two-dimensional gas chromatography (GC×GC). We analyse the GC×GC retention data taken from the literature by comparing predictive capability of different regression methods. The various models are generated using 70 out of 209 PCB congeners in the calibration stage, while their predictive performance is evaluated on the remaining 139 compounds. The two-dimensional chromatogram is initially estimated by separately modelling retention times of PCBs in the first and in the second column ((1) t (R) and (2) t (R), respectively). In particular, multilinear regression (MLR) combined with genetic algorithm (GA) variable selection is performed to extract two small subsets of predictors for (1) t (R) and (2) t (R) from a large set of theoretical molecular descriptors provided by the popular software Dragon, which after removal of highly correlated or almost constant variables consists of 237 structure-related quantities. Based on GA-MLR analysis, a four-dimensional and a five-dimensional relationship modelling (1) t (R) and (2) t (R), respectively, are identified. Single-response partial least square (PLS-1) regression is alternatively applied to independently model (1) t (R) and (2) t (R) without the need for preliminary GA variable selection. Further, we explore the possibility of predicting the two-dimensional chromatogram of PCBs in a single calibration procedure by using a two-response PLS (PLS-2) model or a feed-forward artificial neural network (ANN) with two output neurons. In the first case, regression is carried out on the full set of 237 descriptors, while the variables previously selected by GA-MLR are initially considered as ANN inputs and subjected to a sensitivity analysis to remove the redundant ones. Results show PLS-1 regression exhibits a noticeably better descriptive and predictive
7. A two-dimensional position sensitive gas chamber with scanned charge transfer readout
Science.gov (United States)
Gómez, F.; Iglesias, A.; Lobato, R.; Mosquera, J.; Pardo, J.; Pena, J.; Pazos, A.; Pombar, M.; Rodríguez, A.
2003-10-01
We have constructed and tested a two-dimensional position sensitive parallel-plate gas ionization chamber with scanned charge transfer readout. The scan readout method described here is based on the development of a new position-dependent charge transfer technique. It has been implemented by using gate strips perpendicularly oriented to the collector strips. This solution reduces considerably the number of electronic readout channels needed to cover large detector areas. The use of a 25 μm thick kapton etched circuit allows high charge transfer efficiency with a low gating voltage, consequently needing a very simple commutating circuit. The present prototype covers 8×8 cm2 with a pixel size of 1.27×1.27 mm2. Depending on the intended use and beam characteristics a smaller effective pixel is feasible and larger active areas are possible. This detector can be used for X-ray or other continuous beam intensity profile monitoring.
8. Hydrodynamics for a model of a confined quasi-two-dimensional granular gas.
Science.gov (United States)
Brey, J Javier; Buzón, V; Maynar, P; García de Soria, M I
2015-05-01
The hydrodynamic equations for a model of a confined quasi-two-dimensional gas of smooth inelastic hard spheres are derived from the Boltzmann equation for the model, using a generalization of the Chapman-Enskog method. The heat and momentum fluxes are calculated to Navier-Stokes order, and the associated transport coefficients are explicitly determined as functions of the coefficient of normal restitution and the velocity parameter involved in the definition of the model. Also an Euler transport term contributing to the energy transport equation is considered. This term arises from the gradient expansion of the rate of change of the temperature due to the inelasticity of collisions, and it vanishes for elastic systems. The hydrodynamic equations are particularized for the relevant case of a system in the homogeneous steady state. The relationship with previous works is analyzed.
9. Modulation techniques and applications in comprehensive two-dimensional gas chromatography (GC x GC)
Energy Technology Data Exchange (ETDEWEB)
Pursch, Matthias [Dow Deutschland GmbH and Co. OHG, Analytical Sciences, 77836 Rheinmuenster (Germany); Sun, Kefu; Winniford, Bill; Weber, Andy [Dow Chemical Company, Analytical Sciences, Freeport, TX 77541 (United States); Cortes, Hernan; McCabe, Terry [Dow Chemical Company, Analytical Sciences, Midland MI 48667 (United States); Luong, Jim [Dow Canada, Analytical Sciences, Fort Saskatchewan (Canada)
2002-07-01
More than a decade after Phillips' first published work this article reviews recent developments in comprehensive two-dimensional gas chromatography (GC x GC). Special attention is devoted to the further development and diversity of modulation devices. These include heated sweepers, cryofocused modulators, and a variety of diaphragm valve-switching strategies. It is demonstrated that all modulation approaches can be very well suited to GC x GC, depending on the particular application. Diaphragm-valve modulation is very powerful for volatile organic compounds. Slotted heater and cryofocused modulation are preferred for samples that contain non-volatile components. Applications ranging from petroleum to environmental and biological samples are illustrated. Extension of the technique to GC x GC-mass spectrometry (MS) is also discussed and trends for future research activity are pointed out. (orig.)
10. Gas-kinetic numerical schemes for one- and two-dimensional inner flows
Institute of Scientific and Technical Information of China (English)
Zhi-hui LI; Lin BI; Zhi-gong TANG
2009-01-01
Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics of the gas-kinetic numerical algorithm for Boltzmann model equation.The peculiar flow phenomena and mechanism from various flow regimes are revealed in the numerical simulations of the unsteady Sod shock-tube problems and the two-dimensional channel flows with different Knudsen numbers.The numerical remainder-effects of the difference schemes are investigated and analyzed based on the computed results.The ways of improving the computational efficiency of the gaskinetic numerical method and the computing principles of difference discretization are discussed.
11. Automated multivariate analysis of comprehensive two-dimensional gas chromatograms of petroleum
DEFF Research Database (Denmark)
Skov, Søren Furbo
Petroleum is an economically and industrially important resource. Crude oil must be refined before use to ensure suitable properties of the product. Among the processes used in this refining is distillation and desulfurization. In order to optimize these processes, it is essential to understand...... them. Comprehensive two-dimensional gas chromatography (GCGC) is a method for analyzing the volatile parts of a sample. It can separate hundreds or thousands of compounds based on their boiling point, polarity and polarizability. This makes it ideally suited for petroleum analysis. The number...... impossible to find it. For a special class of models, multi-way models, unique solutions often exist, meaning that the underlying phenomena can be found. I have tested this class of models on GCGC data from petroleum and conclude that more work is needed before they can be automated. I demonstrate how...
12. Cavity quantum electrodynamics with many-body states of a two-dimensional electron gas.
Science.gov (United States)
Smolka, Stephan; Wuester, Wolf; Haupt, Florian; Faelt, Stefan; Wegscheider, Werner; Imamoglu, Ataç
2014-10-17
Light-matter interaction has played a central role in understanding as well as engineering new states of matter. Reversible coupling of excitons and photons enabled groundbreaking results in condensation and superfluidity of nonequilibrium quasiparticles with a photonic component. We investigated such cavity-polaritons in the presence of a high-mobility two-dimensional electron gas, exhibiting strongly correlated phases. When the cavity was on resonance with the Fermi level, we observed previously unknown many-body physics associated with a dynamical hole-scattering potential. In finite magnetic fields, polaritons show distinct signatures of integer and fractional quantum Hall ground states. Our results lay the groundwork for probing nonequilibrium dynamics of quantum Hall states and exploiting the electron density dependence of polariton splitting so as to obtain ultrastrong optical nonlinearities.
13. Fermi liquid-to-Bose condensate crossover in a two-dimensional ultracold gas experiment
Science.gov (United States)
Barmashova, T. V.; Mart'yanov, K. A.; Makhalov, V. B.; Turlapov, A. V.
2016-02-01
By controling interparticle interactions, it is possible to transform a fermionic system into a bosonic system and vice versa, while preserving quantum degeneracy. Evidence of such a transformation may be found by monitoring the pressure and interference. The Fermi pressure is an indication of the fermion?ic character of a system, while the interference implies a nonzero order parameter and Bose condensation. Lowering from three to two spatial dimensions introduces new physics and makes the system more difficult to describe due to the increased fluctuations and the reduced applicability of mean field methods. An experiment with a two-dimensional ultracold atomic gas shows a crossover between the Bose and Fermi limits, as evident from the value of pressure and from the interference pattern, and provides data to test models of 2D Fermi and Bose systems, including the most-difficult-to-model strongly coupled systems.
14. Terahertz Radiation Heterodyne Detector Using Two-Dimensional Electron Gas in a GaN Heterostructure
Science.gov (United States)
Karasik, Boris S.; Gill, John J.; Mehdi, Imran; Crawford, Timothy J.; Sergeev, Andrei V.; Mitin, Vladimir V.
2012-01-01
High-resolution submillimeter/terahertz spectroscopy is important for studying atmospheric and interstellar molecular gaseous species. It typically uses heterodyne receivers where an unknown (weak) signal is mixed with a strong signal from the local oscillator (LO) operating at a slightly different frequency. The non-linear mixer devices for this frequency range are unique and are not off-the-shelf commercial products. Three types of THz mixers are commonly used: Schottky diode, superconducting hot-electron bolometer (HEB), and superconductor-insulation-superconductor (SIS) junction. A HEB mixer based on the two-dimensional electron gas (2DEG) formed at the interface of two slightly dissimilar semiconductors was developed. This mixer can operate at temperatures between 100 and 300 K, and thus can be used with just passive radiative cooling available even on small spacecraft.
15. Hidden long-range order in a two-dimensional spin-orbit coupled Bose gas
CERN Document Server
Su, Shih-Wei; Gou, Shih-Chuan; Liao, Renyuan; Fialko, Oleksandr; Brand, Joachim
2016-01-01
A two-dimensional spin-orbit coupled Bose gas is shown to simultaneously possess quasi and true long-range orders in the total and relative phases, respectively. The total phase undergoes a conventional Berenzinskii- Kosterlitz-Thouless transition, where an quasi long-range order is expected. Additionally, the relative phase undergoes an Ising-type transition building up true long-range order, which is induced by the anisotropic spin- orbit coupling. Based on the Bogoliubov approach, expressions for the total- and relative-phase fluctuations are derived analytically for the low temperature regime. Numerical simulations of the stochastic projected Gross- Pitaevskii equation give a good agreement with the analytical predictions.
16. Optimization of micro-strip gas chamber as two-dimensional neutron detector using gadolinium converter
Energy Technology Data Exchange (ETDEWEB)
Masaoka, Sei; Nakamura, Tatsuya; Yamagishi, Hideshi; Soyama, Kazuhiko [Japan Atomic Energy Research Inst., Tokai, Ibaraki (Japan). Tokai Research Establishment
2002-02-01
A micro-strip gas chamber (MSGC) has been developing as a two-dimensional position sensitive neutron detector for neutron scattering experiments using high-intensity pulsed-neutron source in a high-intensity proton accelerator facility. MSGC is required for the high count rate, high detective efficiency, high positional resolution, stabilization and covering large area. Our purpose in this paper is to verify the proper of Gadolinium as MSGC converter. First, the basic property of Gadolinium converter was examined by simple experiments using a zero-dimensional neutron detector on the purpose of deriving the detective efficiency. Second, the optimization of the arrangement of a capillary plate in MSGC has been done by simulation on the MSGC using Gadolinium converter. As a result of that, it has been proved that Gadolinium can be theoretically used as a converter of MSGC. (author)
17. Elastic wave band gaps tuned by configuring radii of rods in two-dimensional phononic crystals with a hybrid square-like lattice
Science.gov (United States)
Liu, Rongqiang; Zhao, Haojiang; Zhang, Yingying; Guo, Honghwei; Deng, Zongquan
2015-12-01
The plane wave expansion (PWE) method is used to calculate the band gaps of two-dimensional (2D) phononic crystals (PCs) with a hybrid square-like (HSL) lattice. Band structures of both XY-mode and Z-mode are calculated. Numerical results show that the band gaps between any two bands could be maximized by altering the radius ratio of the inclusions at different positions. By comparing with square lattice and bathroom lattice, the HSL lattice is more efficient in creating larger gaps.
18. Tunable spin wave dynamics in two-dimensional Ni{sub 80}Fe{sub 20} nanodot lattices by varying dot shape
Energy Technology Data Exchange (ETDEWEB)
Mahato, Bipul Kumar; Rana, Bivas; Kumar, Dheeraj; Barman, Saswati; Barman, Anjan, E-mail: [email protected] [Thematic Unit of Excellence on Nanodevice Technology, Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700098 (India); Sugimoto, Satoshi [Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581 (Japan); Otani, YoshiChika [Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581 (Japan); CEMS-RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198 (Japan)
2014-07-07
We demonstrate tunable spin wave spectrum in two-dimensional Ni{sub 80}Fe{sub 20} nanodot lattices by varying dot shape. A single collective mode in elliptical dot lattices transforms into three distinct modes for the half-elliptical, rectangular, and diamond dot lattices, albeit with different peak frequencies and intensities. A drastic change is observed for the triangular dots, where eight modes covering a broad band are observed. Using micromagnetic simulations, we characterized the modes as different localized, extended, and quantized modes, whose frequencies and spatial profiles are determined by a combination of internal field profiles within the nanodots and the stray magnetic field within the lattice.
19. Thermalization of a two-dimensional photon gas in a polymeric host matrix
Science.gov (United States)
Schmitt, Julian; Damm, Tobias; Vewinger, Frank; Weitz, Martin; Klaers, Jan
2012-07-01
We investigate thermodynamic properties of a two-dimensional photon gas confined by a dye-filled optical microcavity. A thermally equilibrated state of the photon gas is achieved by radiative coupling to a heat bath that is realized with dye molecules embedded in a polymer at room temperature. The chemical potential of the gas is freely adjustable. The optical microcavity consisting of two curved mirrors induces both a non-vanishing effective photon mass and a harmonic trapping potential for the photons. While previous experiments of our group have used liquid dye solutions, the measurements described here are based on dye molecules incorporated into a polymer host matrix. The solid state material allows a simplified operation of the experimental scheme. We furthermore describe studies of fluorescence properties of dye-doped polymers, and verify the applicability of Kennard-Stepanov theory in this system. In the future, dye-based solid state systems hold promise for the realization of single-mode light sources in thermal equilibrium based on Bose-Einstein condensation of photons, as well as for solar energy concentrators.
20. Shear viscosity of a two-dimensional emulsion of drops using a multiple-relaxation-time-step lattice Boltzmann method.
Science.gov (United States)
Halliday, I; Xu, X; Burgin, K
2017-02-01
An extended Benzi-Dellar lattice Boltzmann equation scheme [R. Benzi, S. Succi, and M. Vergassola, Europhys. Lett. 13, 727 (1990)EULEEJ0295-507510.1209/0295-5075/13/8/010; R. Benzi, S. Succi, and M. Vergassola, Phys. Rep. 222, 145 (1992)PRPLCM0370-157310.1016/0370-1573(92)90090-M; P. J. Dellar, Phys. Rev. E 65, 036309 (2002)1063-651X10.1103/PhysRevE.65.036309] is developed and applied to the problem of confirming, at low Re and drop fluid concentration, c, the variation of effective shear viscosity, η_{eff}=η_{1}[1+f(η_{1},η_{2})c], with respect to c for a sheared, two-dimensional, initially crystalline emulsion [here η_{1} (η_{2}) is the fluid (drop fluid) shear viscosity]. Data obtained with our enhanced multicomponent lattice Boltzmann method, using average shear stress and hydrodynamic dissipation, agree well once appropriate corrections to Landau's volume average shear stress [L. Landau and E. M. Lifshitz, Fluid Mechanics, 6th ed. (Pergamon, London, 1966)] are applied. Simulation results also confirm the expected form for f(η_{i},η_{2}), and they provide a reasonable estimate of its parameters. Most significantly, perhaps, the generality of our data supports the validity of Taylor's disputed simplification [G. I. Taylor, Proc. R. Soc. London, Ser. A 138, 133 (1932)1364-502110.1098/rspa.1932.0175] to reduce the effect of one hydrodynamic boundary condition (on the continuity of the normal contraction of stress) to an assumption that interfacial tension is sufficiently strong to maintain a spherical drop shape.
1. Shear viscosity of a two-dimensional emulsion of drops using a multiple-relaxation-time-step lattice Boltzmann method
Science.gov (United States)
Halliday, I.; Xu, X.; Burgin, K.
2017-02-01
An extended Benzi-Dellar lattice Boltzmann equation scheme [R. Benzi, S. Succi, and M. Vergassola, Europhys. Lett. 13, 727 (1990), 10.1209/0295-5075/13/8/010; R. Benzi, S. Succi, and M. Vergassola, Phys. Rep. 222, 145 (1992), 10.1016/0370-1573(92)90090-M; P. J. Dellar, Phys. Rev. E 65, 036309 (2002), 10.1103/PhysRevE.65.036309] is developed and applied to the problem of confirming, at low Re and drop fluid concentration, c , the variation of effective shear viscosity, ηeff=η1[1 +f (η1,η2) c ] , with respect to c for a sheared, two-dimensional, initially crystalline emulsion [here η1 (η2) is the fluid (drop fluid) shear viscosity]. Data obtained with our enhanced multicomponent lattice Boltzmann method, using average shear stress and hydrodynamic dissipation, agree well once appropriate corrections to Landau's volume average shear stress [L. Landau and E. M. Lifshitz, Fluid Mechanics, 6th ed. (Pergamon, London, 1966)] are applied. Simulation results also confirm the expected form for f (ηi,η2) , and they provide a reasonable estimate of its parameters. Most significantly, perhaps, the generality of our data supports the validity of Taylor's disputed simplification [G. I. Taylor, Proc. R. Soc. London, Ser. A 138, 133 (1932), 10.1098/rspa.1932.0175] to reduce the effect of one hydrodynamic boundary condition (on the continuity of the normal contraction of stress) to an assumption that interfacial tension is sufficiently strong to maintain a spherical drop shape.
2. Synthetic gauge field and pseudospin-orbit interaction in a stacked two-dimensional ring-network lattice
Science.gov (United States)
Ochiai, Tetsuyuki
2017-02-01
We study the effects of a synthetic gauge field and pseudospin-orbit interaction in a stacked two-dimensional ring-network model. The model was introduced to simulate light propagation in the corresponding ring-resonator lattice, and is thus completely bosonic. Without these two items, the model exhibits Floquet-Weyl and Floquet-topological-insulator phases with topologically gapless and gapped band structures, respectively. The synthetic magnetic field implemented in the model results in a three-dimensional Hofstadter-butterfly-type spectrum in a photonic platform. The resulting gaps are characterized by the winding number of relevant S-matrices together with the Chern number of the bulk bands. The pseudospin-orbit interaction is defined as the mixing term between two pseudospin degrees of freedom in the rings, namely, the clockwise and counter-clockwise modes. It destroys the Floquet-topological-insulator phases, while the Floquet-Weyl phase with multiple Weyl points can be preserved by breaking the space-inversion symmetry. Implementing both the synthetic gauge field and pseudospin-orbit interaction requires a certain nonreciprocity.
3. Phase transitions in a two-dimensional antiferromagnetic Potts model on a triangular lattice with next-nearest neighbor interactions
Energy Technology Data Exchange (ETDEWEB)
Babaev, A. B., E-mail: [email protected]; Magomedov, M. A.; Murtazaev, A. K. [Russian Academy of Sciences, Amirkhanov Institute of Physics, Dagestan Scientific Center (Russian Federation); Kassan-Ogly, F. A.; Proshkin, A. I. [Russian Academy of Sciences, Institute of Metal Physics, Ural Branch (Russian Federation)
2016-02-15
Phase transitions (PTs) and frustrations in two-dimensional structures described by a three-vertex antiferromagnetic Potts model on a triangular lattice are investigated by the Monte Carlo method with regard to nearest and next-nearest neighbors with interaction constants J{sub 1} and J{sub 2}, respectively. PTs in these models are analyzed for the ratio r = J{sub 2}/J{sub 1} of next-nearest to nearest exchange interaction constants in the interval |r| = 0–1.0. On the basis of the analysis of the low-temperature entropy, the density of states function of the system, and the fourth-order Binder cumulants, it is shown that a Potts model with interaction constants J{sub 1} < 0 and J{sub 2} < 0 exhibits a first-order PT in the range of 0 ⩽ r < 0.2, whereas, in the interval 0.2 ⩽ r ⩽ 1.0, frustrations arise in the system. At the same time, for J{sub 1} > 0 and J{sub 2} < 0, frustrations arise in the range 0.5 < |r| < 1.0, while, in the interval 0 ⩽ |r| ⩽ 1/3, the model exhibits a second-order PT.
4. Bouncing, rolling, energy flows, and cluster formation in a two-dimensional vibrated granular gas
Science.gov (United States)
Pérez-Ángel, Gabriel; Nahmad-Molinari, Yuri
2011-10-01
We study the formation of crystalline clusters for a two-dimensional (2D) sinusoidally vibrated granular gas, with maximum vertical acceleration smaller than gravity, using fully 3D simulations. It is found that this phenomenon arises from the spontaneous segregation of the granulate into two dynamical modes: one of grains that bounce in synchrony with the motion of the sustaining plate (“bouncers”) and another of grains that cease to bounce and simply rolls on the plate, without ever loosing contact with it (“rollers”). These two dynamical categories are quite robust with respect to perturbations. The populations for bouncers and rollers depend on the preparation of the granulate and can be made to take arbitrary values in all the range of accelerations where both dynamical modes are present. It is found that the dynamical mode with the largest population coalesces in clusters under the influence of the other mode, whose grains act as a higher pressure gas that compresses the clusters. In this way it is possible to produce clusters of rollers or clusters of bouncers. A gas made of grains from only one dynamical class shows only weak density fluctuations. When the occupation fractions for both modes are similar, one observes segregation and clusters of both types. The clustering of the gas is monitored using both the average coordination number and the local hexatic order parameter ψ6. Energy flows in the plane are monitored, and it is shown that roller-bouncer collisions increase horizontal kinetic energy, while all other types of collisions reduce this energy. We find that friction with the substrate is the main sink of horizontal energy for these granular gases.
5. Determination of toxaphene enantiomers by comprehensive two-dimensional gas chromatography with electron-capture detection.
Science.gov (United States)
Bordajandi, Luisa R; Ramos, Lourdes; González, María José
2006-09-01
Comprehensive two-dimensional gas chromatography with micro electron-capture detection (GC x GC-microECD) has been evaluated for the enantioseparation of five chiral toxaphenes typically found in real-life samples (Parlar 26, 32, 40, 44 and 50). From the two enantioselective beta-cyclodextrin-based columns evaluated as first dimension column, BGB-176SE and BGB-172, the latter provided the best results and was further combined with three non-enantioselective columns in the second dimension: HT-8, BPX-50 and Supelcowax-10. The combination BGB-172 x BPX-50 was finally selected because it provided a complete separation among all enantiomers. A satisfactory repeatability and reproducibility of the retention times in both the first and the second dimension were observed for all target compounds (RSDs below 0.8%, n = 4). Linear responses in the tested range of 10-200 pg/microl and limits of detection in the range of 2-6 pg/microl were obtained. The repeatability and reproducibility at a concentration of 100 pg/microl, evaluated as the RSDs calculated for the enantiomeric fraction (EF), was better than 11% (n = 4) in all instances. The feasibility of the method developed for real-life analyses was illustrated by the determination of the enantiomeric ratios and concentration levels of the test compounds in four commercial fish oil samples. These results were compared to those obtained by heart-cut multidimensional gas chromatography using the same enantioselective column.
6. Thermalization of a two-dimensional photonic gas in a white wall' photon box
Science.gov (United States)
Klaers, Jan; Vewinger, Frank; Weitz, Martin
2010-07-01
Bose-Einstein condensation, the macroscopic accumulation of bosonic particles in the energetic ground state below a critical temperature, has been demonstrated in several physical systems. The perhaps best known example of a bosonic gas, blackbody radiation, however exhibits no Bose-Einstein condensation at low temperatures. Instead of collectively occupying the lowest energy mode, the photons disappear in the cavity walls when the temperature is lowered-corresponding to a vanishing chemical potential. Here we report on evidence for a thermalized two-dimensional photon gas with a freely adjustable chemical potential. Our experiment is based on a dye-filled optical microresonator, acting as a white wall' box for photons. Thermalization is achieved in a photon-number-conserving way by photon scattering off the dye molecules, and the cavity mirrors provide both an effective photon mass and a confining potential-key prerequisites for the Bose-Einstein condensation of photons. As a striking example of the unusual system properties, we demonstrate a yet unobserved light concentration effect into the centre of the confining potential, an effect with prospects for increasing the efficiency of diffuse solar light collection.
7. Observation of Spin Coulomb Drag in a Two-Dimensional Electron Gas
Energy Technology Data Exchange (ETDEWEB)
Weber, C.P.
2011-08-19
An electron propagating through a solid carries spin angular momentum in addition to its mass and charge. Of late there has been considerable interest in developing electronic devices based on the transport of spin, which offer potential advantages in dissipation, size, and speed over charge-based devices. However, these advantages bring with them additional complexity. Because each electron carries a single, fixed value (-e) of charge, the electrical current carried by a gas of electrons is simply proportional to its total momentum. A fundamental consequence is that the charge current is not affected by interactions that conserve total momentum, notably collisions among the electrons themselves. In contrast, the electron's spin along a given spatial direction can take on two values, {+-} {h_bar}/2 (conventionally {up_arrow}, {down_arrow}), so that the spin current and momentum need not be proportional. Although the transport of spin polarization is not protected by momentum conservation, it has been widely assumed that, like the charge current, spin current is unaffected by electron-electron (e-e) interactions. Here we demonstrate experimentally not only that this assumption is invalid, but that over a broad range of temperature and electron density, the flow of spin polarization in a two-dimensional gas of electrons is controlled by the rate of e-e collisions.
8. A laterally averaged two-dimensional simulation of unsteady supersaturated total dissolved gas in deep reservoir
Institute of Scientific and Technical Information of China (English)
FENG Jing-jie; LI Ran; YANG Hui-xia; LI Jia
2013-01-01
Elevated levels of the Total Dissolved Gas (TDG) may be reached downstream of dams,leading to increased incidences of gas bubble diseases in fish.The supersaturated TDG dissipates and transports more slowly in reservoirs than in natural rivers because of the greater depth and the lower turbulence,which endangers the fish more seriously.With consideration of the topographical characteristics of a deep reservoir,a laterally averaged two-dimensional unsteady TDG model for deep reservoir is proposed.The dissipation process of the TDG inside the waterbody and the mass transfer through the free surface are separately modeled with different functions in the model.Hydrodynamics equations are solved coupling with those of water temperature and density.The TDG concentration is calculated based on the density current field.A good agreement is found in the simulation of the Dachaoshan Reservoir between the simulation results and the field data of the hydrodynamics parameters and the TDG distribution in the vertical direction and their unsteady evolution with time.The hydrodynamics parameters,the temperature and the TDG concentration are analyzed based on the simulation results.This study demonstrates that the model can be used to predict the evolutions of hydrodynamics parameters,the temperature and the TDG distribution in a deep reservoir with unsteady inflow and outflow.The results can be used in the study of the mitigation measures of the supersaturated TDG.
9. Density of states in a two-dimensional electron gas: Impurity bands and band tails
Science.gov (United States)
Gold, A.; Serre, J.; Ghazali, A.
1988-03-01
We calculate the density of states of a two-dimensional electron gas in the presence of charged impurities within Klauder's best multiple-scattering approach. The silicon metal-oxide-semiconductor (MOS) system with impurities at the interface is studied in detail. The finite extension of the electron wave function into the bulk is included as well as various dependences of the density of states on the electron, the depletion, and the impurity densities. The transition from an impurity band at low impurity concentration to a band tail at high impurity concentration is found to take place at a certain impurity concentration. If the screening parameter of the electron gas is decreased, the impurity band shifts to lower energy. For low impurity density we find excited impurity bands. Our theory at least qualitatively explains conductivity and infrared-absorption experiments on impurity bands in sodium-doped MOS systems and deep band tails in the gap observed for high doping levels in these systems.
10. High-Throughput Design of Two-Dimensional Electron Gas Systems Based on Polar/Nonpolar Perovskite Oxide Heterostructures
Science.gov (United States)
Yang, Kesong; Nazir, Safdar; Behtash, Maziar; Cheng, Jianli
2016-10-01
The two-dimensional electron gas (2DEG) formed at the interface between two insulating oxides such as LaAlO3 and SrTiO3 (STO) is of fundamental and practical interest because of its novel interfacial conductivity and its promising applications in next-generation nanoelectronic devices. Here we show that a group of combinatorial descriptors that characterize the polar character, lattice mismatch, band gap, and the band alignment between the perovskite-oxide-based band insulators and the STO substrate, can be introduced to realize a high-throughput (HT) design of SrTiO3-based 2DEG systems from perovskite oxide quantum database. Equipped with these combinatorial descriptors, we have carried out a HT screening of all the polar perovskite compounds, uncovering 42 compounds of potential interests. Of these, Al-, Ga-, Sc-, and Ta-based compounds can form a 2DEG with STO, while In-based compounds exhibit a strain-induced strong polarization when deposited on STO substrate. In particular, the Ta-based compounds can form 2DEG with potentially high electron mobility at (TaO2)+/(SrO)0 interface. Our approach, by defining materials descriptors solely based on the bulk materials properties, and by relying on the perovskite-oriented quantum materials repository, opens new avenues for the discovery of perovskite-oxide-based functional interface materials in a HT fashion.
11. Broadband terahertz radiation from a biased two-dimensional electron gas in an AlGaN/GaN heterostructure
Science.gov (United States)
Zhongxin, Zheng; Jiandong, Sun; Yu, Zhou; Zhipeng, Zhang; Hua, Qin
2015-10-01
The broadband terahertz (THz) emission from drifting two-dimensional electron gas (2DEG) in an AlGaN/GaN heterostructure at 6 K is reported. The devices are designed as THz plasmon emitters according to the Smith-Purcell effect and the ‘shallow water’ plasma instability mechanism in 2DEG. Plasmon excitation is excluded since no signature of electron-density dependent plasmon mode is observed. Instead, the observed THz emission is found to come from the heated lattice and/or the hot electrons. Simulated emission spectra of hot electrons taking into account the THz absorption in air and Fabry-Pérot interference agree well with the experiment. It is confirmed that a blackbody-like THz emission will inevitably be encountered in similar devices driven by a strong in-plane electric field. A conclusion is drawn that a more elaborate device design is required to achieve efficient plasmon excitation and THz emission. Project supported by the National Basic Research Program of China (No. G2009CB929303), the National Natural Science Foundation of China (No. 61271157), the China Postdoctoral Science Foundation (No. 2014M551678), and the Jiangsu Planned Projects for Postdoctoral Research Funds (No. 1301054B).
12. Macroporous polymer monoliths as second dimension columns in comprehensive two-dimensional gas chromatography: a feasibility study
NARCIS (Netherlands)
D. Peroni; R.J. Vonk; W. van Egmond; H.-G. Janssen
2012-01-01
When the typical column combinations are used, comprehensive two-dimensional gas chromatography (GC × GC) suffers from the impossibility to operate both dimensions at their optimum carrier gas velocities at the same time. This as a result of the flow mismatch caused by the different dimensions of th
13. Partition function for the two-dimensional square lattice Ising model in a non-zero magnetic field-A heuristic analysis
OpenAIRE
2008-01-01
The exact partition function of the two-dimensional nearest neighbour Ising model pertaining to square lattices is derived for N sites in the case of a non-vanishing magnetic field.When the magnetic field is zero,the partition functions estimated from the present analysis are identical with those arising from Onsager's exact solution.
14. Effect of long range spatial correlations on the lifetime statistics of an emitter in a two-dimensional disordered lattice
CERN Document Server
de Sousa, N; García-Martín, A; Froufe-Pérez, L S; Marqués, M I
2014-01-01
The effect of spatial correlations on the Purcell effect in a bidimensional dispersion of resonant nanoparticles is analyzed. We perform extensive calculations of the fluorescence decay rate of a point emitter embedded in a system of nanoparticles statistically distributed according to a sim- ple 2D lattice-gas model near the critical point. For short range correlations (high temperature thermalization) the Purcell factors present a non-Gaussian long-tailed statistics which evolves to- wards a bimodal distribution as approaching the critical point where the spatial correlation length diverges. Our results suggest long range correlations as a possible origin of the large fluctuations of experimental decay rates in disordered metal films.
15. Soliton assisted control of source to drain electron transport along natural channels - crystallographic axes - in two-dimensional triangular crystal lattices
Science.gov (United States)
Chetverikov, A. P.; Ebeling, W.; Velarde, M. G.
2016-09-01
We present computational evidence of the possibility of fast, supersonic or subsonic, nearly loss-free ballistic-like transport of electrons bound to lattice solitons (a form of electron surfing on acoustic waves) along crystallographic axes in two-dimensional anharmonic crystal lattices. First we study the structural changes a soliton creates in the lattice and the time lapse of recovery of the lattice. Then we study the behavior of one electron in the polarization field of one and two solitons with crossing pathways with suitably monitored delay. We show how an electron surfing on a lattice soliton may switch to surf on the second soliton and hence changing accordingly the direction of its path. Finally we discuss the possibility to control the way an excess electron proceeds from a source at a border of the lattice to a selected drain at another border by following appropriate straight pathways on crystallographic axes.
16. Pressure Tuning of First Dimension Columns in Comprehensive Two-Dimensional Gas Chromatography.
Science.gov (United States)
Sharif, Khan M; Kulsing, Chadin; Marriott, Philip J
2016-09-20
The experimental approach and mechanism of pressure tuning (PT) are introduced for the first stage of a comprehensive two-dimensional gas chromatography (GC × GC) separation. The PT-GC × GC system incorporates a first dimension ((1)D) coupled column ensemble comprising a pair of (1)D columns ((1)D1 and (1)D2) connected via a microfluidic splitter device, allowing variable decompression of carrier gas across each (1)D column, and a conventional (2)D narrow bore column. By variation of junction pressure between the (1)D1 and (1)D2 columns, tunable total (1)D retentions of analytes are readily derived. Separations of a standard mixture comprising a number of different chemical classes (including alkanes, monoaromatics, alcohols, aldehydes, ketones, and esters) and Australian tea tree oil (TTO) were studied as practical examples of the PT-GC × GC system application. This illustrated the change of analyte retention time with experimental conditions depending on void time and retention on the different columns. In addition to void time change, variation of carrier gas relative decompression in the (1)D ensemble leads to tunable contribution of the (1)D1/(1)D2 columns that changes apparent polarity and selectivity of the ensemble. The resulting changes in (1)D elution order further altered elution temperature and thus retention of each analyte on the (2)D column in temperature-programmed GC × GC. 2D orthogonality measurements were then conducted to evaluate overall separation performance under application of different (1)D junction pressure. As a result, distribution and selectivity of particular target compounds, monoterpenes, sesquiterpenes, and oxygenated terpenes in 2D space, and thus orthogonality, could be adequately tuned. This indicates the potential of PT-GC × GC to be applicable for practical sample separation and provides a general approach to tune selectivity of target compounds.
17. Doppler Velocimetry of Current Driven Spin Helices in a Two-Dimensional Electron Gas
Science.gov (United States)
Yang, Luyi
Spins in semiconductors provide a pathway towards the development of spin-based electronics. The appeal of spin logic devices lies in the fact that the spin current is even under time reversal symmetry, yielding non-dissipative coupling to the electric field. To exploit the energy-saving potential of spin current it is essential to be able to control it. While recent demonstrations of electrical-gate control in spin-transistor configurations show great promise, operation at room temperature remains elusive. Further progress requires a deeper understanding of the propagation of spin polarization, particularly in the high mobility semiconductors used for devices. This thesis presents the demonstration and application of a powerful new optical technique, Doppler spin velocimetry, for probing the motion of spin polarization at the level of 1 nm on a picosecond time scale. We discuss experiments in which this technique is used to measure the motion of spin helices in high mobility n-GaAs quantum wells as a function of temperature, in-plane electric field, and photoinduced spin polarization amplitude. We find that the spin helix velocity changes sign as a function of wave vector and is zero at the wave vector that yields the largest spin lifetime. This observation is quite striking, but can be explained by the random walk model that we have developed. We discover that coherent spin precession within a propagating spin density wave is lost at temperatures near 150 K. This finding is critical to understanding why room temperature operation of devices based on electrical gate control of spin current has so far remained elusive. We report that, at all temperatures, electron spin polarization co-propagates with the high-mobility electron sea, even when this requires an unusual form of separation of spin density from photoinjected electron density. Furthermore, although the spin packet co-propagates with the two-dimensional electron gas, spin diffusion is strongly suppressed
18. Quantum and thermal phase transitions in a bosonic atom-molecule mixture in a two-dimensional optical lattice
Science.gov (United States)
de Forges de Parny, L.; Rousseau, V. G.
2017-01-01
We study the ground state and the thermal phase diagram of a two-species Bose-Hubbard model, with U(1 ) ×Z2 symmetry, describing atoms and molecules on a two-dimensional optical lattice interacting via a Feshbach resonance. Using quantum Monte Carlo simulations and mean-field theory, we show that the conversion between the two species, coherently coupling the atomic and molecular states, has a crucial impact on the Mott-superfluid transition and stabilizes an insulating phase with a gap controlled by the conversion term—the Feshbach insulator—instead of a standard Mott-insulating phase. Depending on the detuning between atoms and molecules, this model exhibits three phases: the Feshbach insulator, a molecular condensate coexisting with noncondensed atoms, and a mixed atomic-molecular condensate. Employing finite-size scaling analysis, we observe three-dimensional (3D) X Y (3D Ising) transition when U(1 ) (Z2) symmetry is broken, whereas the transition is first order when both U(1 ) and Z2 symmetries are spontaneously broken. The finite-temperature phase diagram is also discussed. The thermal disappearance of the molecular superfluid leads to a Berezinskii-Kosterlitz-Thouless transition with unusual universal jump in the superfluid density. The loss of the quasi-long-range coherence of the mixed atomic and molecular superfluid is more subtle since only atoms exhibit conventional Berezinskii-Kosterlitz-Thouless criticality. We also observe a signal compatible with a classical first-order transition between the mixed superfluid and the normal Bose liquid at low temperature.
19. Strongly anomalous non-thermal fixed point in a quenched two-dimensional Bose gas
CERN Document Server
Karl, Markus
2016-01-01
Universal scaling behavior in the relaxation dynamics of an isolated two-dimensional Bose gas is studied by means of semi-classical stochastic simulations of the Gross-Pitaevskii model. The system is quenched far out of equilibrium by imprinting vortex defects into an otherwise phase-coherent condensate. A strongly anomalous non-thermal fixed point is identified, associated with a slowed decay of the defects in the case that the dissipative coupling to the thermal background noise is suppressed. At this fixed point, a large anomalous exponent $\\eta \\simeq -3$ and, related to this, a large dynamical exponent $z \\simeq 5$ are identified. The corresponding power-law decay is found to be consistent with three-vortex-collision induced loss. The article discusses these aspects of non-thermal fixed points in the context of phase-ordering kinetics and coarsening dynamics, thus relating phenomenological and analytical approaches to classifying far-from-equilibrium scaling dynamics with each other. In particular, a clo...
20. Strongly anomalous non-thermal fixed point in a quenched two-dimensional Bose gas
Science.gov (United States)
Karl, Markus; Gasenzer, Thomas
2017-09-01
Universal scaling behavior in the relaxation dynamics of an isolated two-dimensional Bose gas is studied by means of semi-classical stochastic simulations of the Gross–Pitaevskii model. The system is quenched far out of equilibrium by imprinting vortex defects into an otherwise phase-coherent condensate. A strongly anomalous non-thermal fixed point is identified, associated with a slowed decay of the defects in the case that the dissipative coupling to the thermal background noise is suppressed. At this fixed point, a large anomalous exponent η ≃ -3 and, related to this, a large dynamical exponent z≃ 5 are identified. The corresponding power-law decay is found to be consistent with three-vortex-collision induced loss. The article discusses these aspects of non-thermal fixed points in the context of phase-ordering kinetics and coarsening dynamics, thus relating phenomenological and analytical approaches to classifying far-from-equilibrium scaling dynamics with each other. In particular, a close connection between the anomalous scaling exponent η, introduced in a quantum-field theoretic approach, and conservation-law induced scaling in classical phase-ordering kinetics is revealed. Moreover, the relation to superfluid turbulence as well as to driven stationary systems is discussed.
1. A two-dimensional position sensitive gas chamber with scanned charge transfer readout
Energy Technology Data Exchange (ETDEWEB)
Gomez, F. E-mail: [email protected]; Iglesias, A.; Lobato, R.; Mosquera, J.; Pardo, J.; Pena, J.; Pazos, A.; Pombar, M.; Rodriguez, A
2003-10-21
We have constructed and tested a two-dimensional position sensitive parallel-plate gas ionization chamber with scanned charge transfer readout. The scan readout method described here is based on the development of a new position-dependent charge transfer technique. It has been implemented by using gate strips perpendicularly oriented to the collector strips. This solution reduces considerably the number of electronic readout channels needed to cover large detector areas. The use of a 25 {mu}m thick kapton etched circuit allows high charge transfer efficiency with a low gating voltage, consequently needing a very simple commutating circuit. The present prototype covers 8x8 cm{sup 2} with a pixel size of 1.27x1.27 mm{sup 2}. Depending on the intended use and beam characteristics a smaller effective pixel is feasible and larger active areas are possible. This detector can be used for X-ray or other continuous beam intensity profile monitoring.
2. Forensic profiling of sassafras oils based on comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Schäffer, M; Gröger, T; Pütz, M; Zimmermann, R
2013-06-10
Safrole, the main compound in the essential oil of several plants of the Laurel family (Lauraceae), and its secondary product piperonylmethylketone are the predominantly used precursors for the illicit synthesis of 3,4-methylenedioxymethamphetamine (MDMA) which is, in turn, the most common active ingredient in Ecstasy tablets. Analytical methods with adequate capacity to identify links and origin of precursors, such as safrole, provide valuable information for drug-related police intelligence. Authentic sassafras oil samples from police seizures were subjected to comparative analysis based on their chemical profiles obtained by comprehensive two-dimensional gas chromatography time-of-flight mass spectrometry (GC × GC-TOFMS). The enhanced separation power and increased sensitivity of GC × GC allowed for the detection of minor compounds present in the essential oils which were of particular interest in case of very pure samples whose impurity profiles were not very pronounced. Discrimination of such samples was still possible even in the absence of characteristic main compounds.
3. Comprehensive two-dimensional gas chromatographic separations with a temperature programmed microfabricated thermal modulator.
Science.gov (United States)
Collin, William R; Nuñovero, Nicolas; Paul, Dibyadeep; Kurabayashi, Katsuo; Zellers, Edward T
2016-04-29
Comprehensive two-dimensional gas chromatography (GC×GC) with a temperature-programmed microfabricated thermal modulator (μTM) is demonstrated. The 0.78 cm(2), 2-stage μTM chip with integrated heaters and a PDMS coated microchannel was placed in thermal contact with a solid-state thermoelectric cooler and mounted on top of a bench scale GC. It was fluidically coupled through heated interconnects to an upstream first-dimension ((1)D) PDMS-coated capillary column and a downstream uncoated capillary or second-dimension ((2)D) PEG-coated capillary. A mixture of n-alkanes C6-C10 was separated isothermally and the full-width-at-half-maximum (fwhm) values of the modulated peaks were assessed as a function of the computer-controlled minimum and maximum stage temperatures of μTM, Tmin and Tmax, respectively. With Tmin and Tmax fixed at -25 and 100°C, respectively, modulated peaks of C6 and C7 had fwhm valuesthermal modulator. Replacing the PDMS phase in the μTM with a trigonal-tricationic room temperature ionic liquid eliminated the bleed observed with the PDMS, but also reduced the capacity for several test compounds. Regardless, the demonstrated capability to independently temperature program this low resource μTM enhances its versatility and its promise for use in bench-scale GC×GC systems.
4. Superfluidity and relaxation dynamics of a laser-stirred two-dimensional Bose gas
Science.gov (United States)
Singh, Vijay Pal; Weitenberg, Christof; Dalibard, Jean; Mathey, Ludwig
2017-04-01
We investigate the superfluid behavior of a two-dimensional (2D) Bose gas of 87Rb atoms using classical field dynamics. In the experiment by R. Desbuquois et al. [Nat. Phys. 8, 645 (2012), 10.1038/nphys2378], a 2D quasicondensate in a trap is stirred with a blue-detuned laser beam along a circular path around the trap center. Here, we study this experiment from a theoretical perspective. The heating induced by stirring increases rapidly above a velocity vc, which we define as the critical velocity. We identify the superfluid, the crossover, and the thermal regime by a finite, a sharply decreasing, and a vanishing critical velocity, respectively. We demonstrate that the onset of heating occurs due to the creation of vortex-antivortex pairs. A direct comparison of our numerical results to the experimental ones shows a good agreement, if a systematic shift of the critical phase-space density is included. We relate this shift to the absence of thermal equilibrium between the condensate and the thermal wings, which were used in the experiment to extract the temperature. We expand on this observation by studying the full relaxation dynamics between the condensate and the thermal cloud.
5. Analysis of siloxanes in hydrocarbon mixtures using comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Ghosh, Abhijit; Seeley, Stacy K; Nartker, Steven R; Seeley, John V
2014-09-19
A comprehensive two-dimensional gas chromatography (GC×GC) method for separating siloxanes from hydrocarbons has been developed using a systematic process. First, the retention indices of a set of siloxanes and a set of hydrocarbons were determined on 6 different stationary phases. The retention indices were then used to model GC×GC separation on 15 different stationary phase pairs. The SPB-Octyl×DB-1 pair was predicted to provide the best separation of the siloxanes from the hydrocarbons. The efficacy of this stationary phase pair was experimentally tested by performing a GC×GC analysis of gasoline spiked with siloxanes and by analyzing biogas obtained from a local wastewater treatment facility. The model predictions agreed well with the experimental results. The SPB-Octyl×DB-1 stationary phase pair constrained the hydrocarbons to a narrow range of secondary retention times and fully isolated the siloxanes from the hydrocarbon band. The resulting GC×GC method allows siloxanes to be resolved from complex mixtures of hydrocarbons without requiring the use of a selective detector.
6. Electrical transport of an AlGaN/GaN two-dimensional electron gas
Energy Technology Data Exchange (ETDEWEB)
Saxler, A.; Debray, P.; Perrin, R. [and others
2000-07-01
An Al{sub x}Ga{sub 1{minus}x}N/GaN two-dimensional electron gas structure with x = 0.13 deposited by molecular beam epitaxy on a GaN layer grown by organometallic vapor phase epitaxy on a sapphire substrate was characterized. Hall effect measurements gave a sheet electron concentration of 5.1x10{sup 12} cm{sup {minus}2} and a mobility of 1.9 x 10{sup 4} cm{sup 2}/Vs at 10 K. Mobility spectrum analysis showed single-carrier transport and negligible parallel conduction at low temperatures. The sheet carrier concentrations determined from Shubnikov-de Haas magnetoresistance oscillations were in good agreement with the Hall data. The electron effective mass was determined to be 0.21 {+-} 0.006 m{sub 0} based on the temperature dependence of the amplitude of Shubnikov-de Haas oscillations. The quantum lifetime was about one-fifth of the transport lifetime of 2.3 x 10{sup {minus}12} s.
7. Analysis of oxidised heavy paraffininc products by high temperature comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Potgieter, H; Bekker, R; Beigley, J; Rohwer, E
2017-08-04
Heavy petroleum fractions are produced during crude and synthetic crude oil refining processes and they need to be upgraded to useable products to increase their market value. Usually these fractions are upgraded to fuel products by hydrocracking, hydroisomerization and hydrogenation processes. These fractions are also upgraded to other high value commercial products like lubricant oils and waxes by distillation, hydrogenation, and oxidation and/or blending. Oxidation of hydrogenated heavy paraffinic fractions produces high value products that contain a variety of oxygenates and the characterization of these heavy oxygenates is very important for the control of oxidation processes. Traditionally titrimetric procedures are used to monitor oxygenate formation, however, these titrimetric procedures are tedious and lack selectivity toward specific oxygenate classes in complex matrices. Comprehensive two-dimensional gas chromatography (GC×GC) is a way of increasing peak capacity for the comprehensive analysis of complex samples. Other groups have used HT-GC×GC to extend the carbon number range attainable by GC×GC and have optimised HT-GC×GC parameters for the separation of aromatics, nitrogen-containing compounds as well as sulphur-containing compounds in heavy petroleum fractions. HT-GC×GC column combinations for the separation of oxygenates in oxidised heavy paraffinic fractions are optimised in this study. The advantages of the HT-GC×GC method in the monitoring of the oxidation reactions of heavy paraffinic fraction samples are illustrated. Copyright © 2017 Elsevier B.V. All rights reserved.
8. Noble gas adsorption in two-dimensional zeolites: a combined experimental and density functional theory study
Science.gov (United States)
Wang, Mengen; Zhong, Jianqiang; Boscoboinik, Jorge Anibal; Lu, Deyu
Zeolites are important industrial catalysts with porous three-dimensional structures. The catalytically active sites are located inside the pores, thus rendering them inaccessible for surface science measurements. We synthesized a two-dimensional (2D) zeolite model system, consisting of an (alumino)silicate bilayer weakly bound to a Ru (0001) surface. The 2D zeolite is suitable for surface science studies; it allows a detailed characterization of the atomic structure of the active site and interrogation of the model system during the catalytic reaction. As an initial step, we use Ar adsorption to obtain a better understanding of the atomic structure of the 2D zeolite. In addition, atomic level studies of rare gas adsorption and separation by zeolite are important for its potential application in nuclear waste sequestration. Experimental studies found that Ar atoms can be trapped inside the 2D-zeolite, raising an interesting question on whether Ar atoms are trapped inside the hexagonal prism nano-cages or at the interface between the (alumino)silicate bilayer and Ru(0001), or both. DFT calculations using van der Waals density functionals were carried out to determine the preferred Ar adsorption sites and the corresponding adsorption energies. This research used resources of the Center for Functional Nanomaterials, which is a U.S. DOE Office of Science Facility, at Brookhaven National Laboratory under Contract No. DE-SC0012704.
9. Comprehensive two-dimensional gas chromatography for determination of the terpenes profile of blue honeysuckle berries.
Science.gov (United States)
Kupska, Magdalena; Chmiel, Tomasz; Jędrkiewicz, Renata; Wardencki, Waldemar; Namieśnik, Jacek
2014-01-01
Terpenes are the main group of secondary metabolites, which play essential role in human. The establishment of the terpenes profile of berries of different blue honeysuckle cultivars was achieved by headspace solid-phase microextraction coupled with comprehensive two-dimensional gas chromatography with time-of-flight mass spectrometry (HS-SPME/GC×GC-TOFMS). The berries were found to contain 44 terpenes identified by GC×GC-TOFMS. From these, 10 were previously reported in blueberries. According to their chemical structure, the compounds were organised in different groups: monoterpene hydrocarbons and monoterpene oxygen-containing compounds (oxides, alcohols, aldehydes, and ketones). Positive identification of some of the compounds was performed using authentic standards, while tentative identification of the compounds was based on deconvoluted mass spectra and comparison of linear retention indices (LRI) with literature values. The major components of volatile fraction were monoterpenes, such as eucalyptol, linalool and p-cymene. Furthermore, quantitative analysis showed that eucalyptol was the most abundant bioactive terpene in analysed berries (12.4-418.2 μg/L).
10. Magnetic oscillations in two-dimensional Dirac systems and Shear viscosity and spin diffusion in a two-dimensional Fermi gas
NARCIS (Netherlands)
Küppersbusch, C.S.
2015-01-01
In the first part of the thesis I derive a full quantitative formula which describes the amplitude and frequency of magnetic oscillations in two-dimensional Dirac systems. The investigations are on the basis of graphene, but they generally also hold for other two-dimensional Dirac systems. Starting
11. DISPLAY-2: a two-dimensional shallow layer model for dense gas dispersion including complex features.
Science.gov (United States)
Venetsanos, A G; Bartzis, J G; Würtz, J; Papailiou, D D
2003-04-25
A two-dimensional shallow layer model has been developed to predict dense gas dispersion, under realistic conditions, including complex features such as two-phase releases, obstacles and inclined ground. The model attempts to predict the time and space evolution of the cloud formed after a release of a two-phase pollutant into the atmosphere. The air-pollutant mixture is assumed ideal. The cloud evolution is described mathematically through the Cartesian, two-dimensional, shallow layer conservation equations for mixture mass, mixture momentum in two horizontal directions, total pollutant mass fraction (vapor and liquid) and mixture internal energy. Liquid mass fraction is obtained assuming phase equilibrium. Account is taken in the conservation equations for liquid slip and eventual liquid rainout through the ground. Entrainment of ambient air is modeled via an entrainment velocity model, which takes into account the effects of ground friction, ground heat transfer and relative motion between cloud and surrounding atmosphere. The model additionally accounts for thin obstacles effects in three ways. First a stepwise description of the obstacle is generated, following the grid cell faces, taking into account the corresponding area blockage. Then obstacle drag on the passing cloud is modeled by adding flow resistance terms in the momentum equations. Finally the effect of extra vorticity generation and entrainment enhancement behind obstacles is modeled by adding locally into the entrainment formula without obstacles, a characteristic velocity scale defined from the obstacle pressure drop and the local cloud height.The present model predictions have been compared against theoretical results for constant volume and constant flux gravity currents. It was found that deviations of the predicted cloud footprint area change with time from the theoretical were acceptably small, if one models the frictional forces between cloud and ambient air, neglecting the Richardson
12. Programmed automation of modulator cold jet flow for comprehensive two-dimensional gas chromatographic analysis of vacuum gas oils.
Science.gov (United States)
Rathbun, Wayne
2007-01-01
A method is described for automating the regulation of cold jet flow of a comprehensive two-dimensional gas chromatograph (GCxGC) configured with flame ionization detection. This new capability enables the routine automated separation, identification, and quantitation of hydrocarbon types in petroleum fractions extending into the vacuum gas oil (VGO) range (IBP-540 degrees C). Chromatographic data acquisition software is programmed to precisely change the rate of flow from the cold jet of a nitrogen cooled loop modulator of a GCxGC instrument during sample analysis. This provides for the proper modulation of sample compounds across a wider boiling range. The boiling point distribution of the GCxGC separation is shown to be consistent with high temperature simulated distillation results indicating recovery of higher boiling semi-volatile VGO sample components. GCxGC configured with time-of-flight mass spectrometry is used to determine the molecular identity of individual sample components and boundaries of different molecular types.
13. The Characteristics Method Applied to Stationary Two-Dimensional and Rotationally Symmetrical Gas Flows
Science.gov (United States)
Pfeiffer, F.; Meyer-Koenig, W.
1949-01-01
By means of characteristics theory, formulas for the numerical treatment of stationary compressible supersonic flows for the two-dimensional and rotationally symmetrical cases have been obtained from their differential equations.
14. A Lattice-Gas Model of Microemulsions
CERN Document Server
Boghosian, B M; Emerton, A N; Boghosian, Bruce M.; Coveney, Peter V.; Emerton, Andrew N.
1995-01-01
We develop a lattice gas model for the nonequilibrium dynamics of microemulsions. Our model is based on the immiscible lattice gas of Rothman and Keller, which we reformulate using a microscopic, particulate description so as to permit generalisation to more complicated interactions, and on the prescription of Chan and Liang for introducing such interparticle interactions into lattice gas dynamics. We present the results of simulations to demonstrate that our model exhibits the correct phenomenology, and we contrast it with both equilibrium lattice models of microemulsions, and to other lattice gas models.
15. Effect of a Two-Dimensional Periodic Dielectric Background on Complete Photonic Band Gap in Complex Square Lattices
Institute of Scientific and Technical Information of China (English)
ZHANG Yan; SHI Jun-Jie
2008-01-01
A two-dimensional photonic crystal model with a periodic square dielectric background is proposed.The photonic band modulation effects due to the two-dimensional periodic background are investigated jn detail.It is found that periodic modulation of the dielectric background greatly alters photonic band structures,especially for the Epolarization modes.The number,width and position of the photonic band gaps sensitively depend on the dielectric constants of the two-dimensional periodic background.Complete band gaps are found,and the dependence of the widths of these gaps on the structural and material parameters of the two alternating rods/holes is studied.
16. Amino acid analysis by using comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Mayadunne, Renuka; Nguyen, Thuy-Tien; Marriott, Philip J
2005-06-01
The separation characteristics of alkylchloroformate-derivatised amino acids (AAs) by using comprehensive two-dimensional gas chromatography (GCxGC) is reported. The use of a low-polarity/polar column set did not provide as good a separation performance as that achieved with a polar/non-polar column set, where the latter appeared to provide less correlation over the separation space. The degree of component correlation in each column set was estimated by using the correlation coefficient (r(2); for (1)t(R) and (2)t(R) data) with the low-polarity/polar and polar/low-polarity sets returning correlation coefficients of 0.86, and 0.00 respectively, under the respective conditions employed for the experiments. The 1.5-m non-polar (2)D column (0.1-mm ID; 0.1-mum film thickness) gave peak halfwidths of the order of 50-80 ms. Linearity of detection was good, over a three order of magnitude concentration range, with typical lower detection limit of ca. 0.01 mg L(-1), compared with 0.5 mg L(-1) for normal GC operation with splitless injection. The method was demonstrated for analysis of AAs in a range of food and beverage products, including wine, beer and honey. The major AA in these samples was proline. The Heineken beer sample had a relatively more complex and more abundant AA content compared with the other beer sample. The wine and honey samples also gave a range of AA compounds. Repetition of the sample preparation/analysis procedure for the honey sample gave acceptable reproducibility for individual AAs.
17. Allergic asthma exhaled breath metabolome: a challenge for comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Caldeira, M; Perestrelo, R; Barros, A S; Bilelo, M J; Morête, A; Câmara, J S; Rocha, S M
2012-09-07
Allergic asthma represents an important public health issue, most common in the paediatric population, characterized by airway inflammation that may lead to changes in volatiles secreted via the lungs. Thus, exhaled breath has potential to be a matrix with relevant metabolomic information to characterize this disease. Progress in biochemistry, health sciences and related areas depends on instrumental advances, and a high throughput and sensitive equipment such as comprehensive two-dimensional gas chromatography-time of flight mass spectrometry (GC×GC-ToFMS) was considered. GC×GC-ToFMS application in the analysis of the exhaled breath of 32 children with allergic asthma, from which 10 had also allergic rhinitis, and 27 control children allowed the identification of several hundreds of compounds belonging to different chemical families. Multivariate analysis, using Partial Least Squares-Discriminant Analysis in tandem with Monte Carlo Cross Validation was performed to assess the predictive power and to help the interpretation of recovered compounds possibly linked to oxidative stress, inflammation processes or other cellular processes that may characterize asthma. The results suggest that the model is robust, considering the high classification rate, sensitivity, and specificity. A pattern of six compounds belonging to the alkanes characterized the asthmatic population: nonane, 2,2,4,6,6-pentamethylheptane, decane, 3,6-dimethyldecane, dodecane, and tetradecane. To explore future clinical applications, and considering the future role of molecular-based methodologies, a compound set was established to rapid access of information from exhaled breath, reducing the time of data processing, and thus, becoming more expedite method for the clinical purposes.
18. Thermodynamic-based retention time predictions of endogenous steroids in comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Silva, Aline C A; Ebrahimi-Najafadabi, Heshmatollah; McGinitie, Teague M; Casilli, Alessandro; Pereira, Henrique M G; Aquino Neto, Francisco R; Harynuk, James J
2015-05-01
This work evaluates the application of a thermodynamic model to comprehensive two-dimensional gas chromatography (GC × GC) coupled with time-of-flight mass spectrometry for anabolic agent investigation. Doping control deals with hundreds of drugs that are prohibited in sports. Drug discovery in biological matrices is a challenging task that requires powerful tools when one is faced with the rapidly changing designer drug landscape. In this work, a thermodynamic model developed for the prediction of both primary and secondary retention times in GC × GC has been applied to trimethylsilylated hydroxyl (O-TMS)- and methoxime-trimethylsilylated carbonyl (MO-TMS)-derivatized endogenous steroids. This model was previously demonstrated on a pneumatically modulated GC × GC system, and is applied for the first time to a thermally modulated GC × GC system. Preliminary one-dimensional experiments allowed the calculation of thermodynamic parameters (ΔH, ΔS, and ΔC p ) which were successfully applied for the prediction of the analytes' interactions with the stationary phases of both the first-dimension column and the second-dimension column. The model was able to predict both first-dimension and second-dimension retention times with high accuracy compared with the GC × GC experimental measurements. Maximum differences of -8.22 s in the first dimension and 0.4 s in the second dimension were encountered for the O-TMS derivatives of 11β-hydroxyandrosterone and 11-ketoetiocholanolone, respectively. For the MO-TMS derivatives, the largest discrepancies were from testosterone (9.65 ) for the first-dimension retention times and 11-keto-etiocholanolone (0.4 s) for the second-dimension retention times.
19. Reentrant resistance and giant Andreev back scattering in a two-dimensional electron gas coupled to superconductors
NARCIS (Netherlands)
den Hartog, Sander; Wees, B.J. van; Nazarov, Yu.V.; Klapwijk, T.M.; Borghs, G.
1998-01-01
We first present the bias-voltage dependence of the superconducting phase-dependent reduction in the differential resistance of a disordered T-shaped two-dimensional electron gas (2DEG) coupled to two superconductors. This reduction exhibits a reentrant behavior, since it first increases upon loweri
20. Attempt to unravel the composition of toxaphene by comprehensive two-dimensional gas chromatography with selective detection
NARCIS (Netherlands)
Korytar, P.; Stee, van L.L.P.; Leonards, P.E.G.; Boer, de J.; Brinkman, U.A.Th.
2003-01-01
Comprehensive two-dimensional gas chromatography (GCxGC) coupled with micro electron-capture and time-of-flight mass spectrometric (TOF-MS) detection has been used to analyse technical toxaphene. An HP-1xHT-8 column combination yielded highly structured chromatograms and revealed a complex mixture o
1. Four-terminal magnetoresistance of a two-dimensional electron-gas constriction in the ballistic regime
NARCIS (Netherlands)
Houten, H. van; Beenakker, C.W.J.; Loosdrecht, P.H.M. van; Thornton, T.J.; Ahmed, H.; Pepper, M.; Foxon, C.T.; Harris, J.J.
1988-01-01
A novel negative magnetoresistance effect is found in four-terminal measurements of the voltage drop across a short constriction of variable width in a high-mobility two-dimensional electron gas. The effect is interpreted as the suppression by a magnetic field of the geometrical constriction resista
2. Classification of highly similar crude oils using data sets from comprehensive two-dimensional gas chromatography and multivariate techniques
NARCIS (Netherlands)
Mispelaar, V.G. van; Smilde, A.K.; Noord, O.E. de; Blomberg, J.; Schoenmakers, P.J.
2005-01-01
Comprehensive two-dimensional gas chromatography (GC × GC) has proven to be an extremely powerful separation technique for the analysis of complex volatile mixtures. This separation power can be used to discriminate between highly similar samples. In this article we will describe the use of GC × GC
3. Integration of complementary circuits and two-dimensional electron gas in a Si/SiGe heterostructure
Science.gov (United States)
Lu, T. M.; Lee, C.-H.; Tsui, D. C.; Liu, C. W.
2010-06-01
We have realized complementary devices on an undoped Si/SiGe substrate where both two-dimensional electrons and holes can be induced capacitively. The design of the heterostructure and the fabrication process are reported. Magnetotransport measurements show that the induced two-dimensional electron gas exhibits the quantum Hall effect characteristics. A p-channel field-effect transistor is characterized and the operation of an inverter is demonstrated. The proof-of-principle experiment shows the feasibility of integrating complementary logic circuits with quantum devices.
4. Partition Function for Two-Dimensional Nearest Neighbour Ising Model in a Non-Zero Magnetic Field for a Square Lattice of 16 Sites
OpenAIRE
2007-01-01
An explicit expression for the partition function of two-dimensional nearest neighbour Ising models in the presence of a magnetic field is derived by a systematic enumeration of all the spin configurations pertaining to a square lattice of sixteen sites. The critical temperature is shown to be in excellent agreement with the reported values while the corresponding dimensionless magnetic field is obtained as 0.004.
5. A two-dimensional model for gas mixing in the upper dilute zone of a circulating fluidized bed
Energy Technology Data Exchange (ETDEWEB)
Kruse, M.; Schoenfelder, H.; Werther, J. [Technical University of Hamburg-Harburg, Hamburg (Germany)
1995-10-01
A two-dimensional two-phase flow model for gas/solid flow and gas mixing in the upper zone of a circulating fluidized bed is described. Continuous functions are used to describe variations of local flow parameters horizontally and vertically. Numerical values of dispersion parameters and interfacial mass transfer coefficients are derived from the results of tracer gas mixing experiments. There is good agreement between calculated and measured tracer gas profiles in the upper dilute zone of the circulating fluidized bed. The model is applicable to calculation of chemical reactions in CFB risers. 37 refs., 26 figs., 3 tabs.
6. Doppler Velocimetry of Current Driven Spin Helices in a Two-Dimensional Electron Gas
Energy Technology Data Exchange (ETDEWEB)
Yang, Luyi [Univ. of California, Berkeley, CA (United States)
2013-05-17
Spins in semiconductors provide a pathway towards the development of spin-based electronics. The appeal of spin logic devices lies in the fact that the spin current is even under time reversal symmetry, yielding non-dissipative coupling to the electric field. To exploit the energy-saving potential of spin current it is essential to be able to control it. While recent demonstrations of electrical-gate control in spin-transistor configurations show great promise, operation at room temperature remains elusive. Further progress requires a deeper understanding of the propagation of spin polarization, particularly in the high mobility semiconductors used for devices. This dissertation presents the demonstration and application of a powerful new optical technique, Doppler spin velocimetry, for probing the motion of spin polarization at the level of 1 nm on a picosecond time scale. We discuss experiments in which this technique is used to measure the motion of spin helices in high mobility n-GaAs quantum wells as a function of temperature, in-plane electric field, and photoinduced spin polarization amplitude. We find that the spin helix velocity changes sign as a function of wave vector and is zero at the wave vector that yields the largest spin lifetime. This observation is quite striking, but can be explained by the random walk model that we have developed. We discover that coherent spin precession within a propagating spin density wave is lost at temperatures near 150 K. This finding is critical to understanding why room temperature operation of devices based on electrical gate control of spin current has so far remained elusive. We report that, at all temperatures, electron spin polarization co-propagates with the high-mobility electron sea, even when this requires an unusual form of separation of spin density from photoinjected electron density. Furthermore, although the spin packet co-propagates with the two-dimensional electron gas, spin diffusion is strongly
7. Three-body recombination in a quasi-two-dimensional quantum gas
Science.gov (United States)
Huang, Bo; Zenesini, Alessandro; Grimm, Rudolf
2016-05-01
Quantum three-body recombination in three-dimensional systems is influenced by a series of weakly bound trimers known as Efimov states, which are induced by short-range interactions and exhibit a discrete scaling symmetry. On the other hand, two-dimensional systems with contact interactions are characterized by continuous scale invariance and support no Efimov physics. This raises questions about the behaviour of three-body recombination in the transition from three to two dimensions. We use ultracold caesium atoms trapped in anisotropic potentials formed by a pair of counter-propagating laser beams to experimentally investigate three-body recombination in quasi-two-dimensional systems with tunable confinement and tunable interactions. In our recent experiments, we observed a smooth transition of the three-body recombination rate coefficient from a three-dimensional to a deeply quasi-two-dimensional system. A comparison between the results obtained near two Feshbach resonances indicates a universal behaviour of three-body recombination in the quasi-two-dimensional regime. Austrian Science Fund FWF within project P23106.
8. Complete elution of vacuum gas oil resins by comprehensive high-temperature two-dimensional gas chromatography.
Science.gov (United States)
Boursier, Laure; Souchon, Vincent; Dartiguelongue, Cyril; Ponthus, Jérémie; Courtiade, Marion; Thiébaut, Didier
2013-03-08
The development of efficient conversion processes requires extended knowledge on vacuum gas oils (VGOs). Among these processes, hydrocracking is certainly one of the best suited to meet the increasing demand on high quality diesel fuels. Most of refractory and inhibiting compounds towards hydrocracking and especially nitrogen containing compounds are contained in a fraction of the VGO called the resin fraction, which corresponds to the most polar fraction of a VGO obtained by liquid chromatography (LC) fractionation on a silica column. However, the lack of resolution observed through existing analytical methods does not allow a detailed characterization of these fractions. A recent study showed that comprehensive high temperature two-dimensional gas chromatography (HT-GC×GC) methods could be optimized in order to elute heavy compounds. This method was implemented for the analysis of VGO resin fractions and complete elution was reached. Firstly, the method was validated through repeatability, accuracy, linearity and response factors calculations. Four VGO resin fractions were analyzed and their HT-GC×GC simulated distillation curves were compared to their GC simulated distillation (GC-SimDist) curves. This comparison showed that the method allows complete elution of most of the analyzed VGO resin fractions. However, a detailed characterization of these fractions is not yet obtained due to the very large number of heteroatomic and aromatic species that a flame ionization detector can detect. Current work aims at increasing the selectivity of GC×GC by using heteroatom selective detectors in order to improve the characterization of such products.
9. Terahertz magneto-optical spectroscopy of a two-dimensional hole gas
Energy Technology Data Exchange (ETDEWEB)
Kamaraju, N., E-mail: [email protected]; Taylor, A. J.; Prasankumar, R. P., E-mail: [email protected] [Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States); Pan, W.; Reno, J. [Sandia National Laboratories, Albuquerque, New Mexico 87123 (United States); Ekenberg, U. [Semiconsultants, Brunnsgrnd 12, SE-18773 Täby (Sweden); Gvozdić, D. M. [School of Electrical Engineering, University of Belgrade, Belgrade 11120 (Serbia); Boubanga-Tombet, S. [Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States); Research Institute of Electrical Communication, Tohoku University, 2-1-1 Katahira, Aoba-Ku, Sendai (Japan); Upadhya, P. C. [Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States); Laboratory for Electro-Optics Systems, Indian Space Research Organization, Bangalore 560058 (India)
2015-01-19
Two-dimensional hole gases (2DHGs) have attracted recent attention for their unique quantum physics and potential applications in areas including spintronics and quantum computing. However, their properties remain relatively unexplored, motivating the use of different techniques to study them. We used terahertz magneto-optical spectroscopy to investigate the cyclotron resonance frequency in a high mobility 2DHG, revealing a nonlinear dependence on the applied magnetic field. This is shown to be due to the complex non-parabolic valence band structure of the 2DHG, as verified by multiband Landau level calculations. We also find that impurity scattering dominates cyclotron resonance decay in the 2DHG, in contrast with the dominance of superradiant damping in two-dimensional electron gases. Our results shed light on the properties of 2DHGs, motivating further studies of these unique 2D nanosystems.
10. Collective modes of a two-dimensional spin-1/2 Fermi gas in a harmonic trap
DEFF Research Database (Denmark)
Baur, Stefan; Vogt, Enrico; Köhl, Michael
2013-01-01
We derive analytical expressions for the frequency and damping of the lowest collective modes of a two-dimensional Fermi gas using kinetic theory. For strong coupling, we furthermore show that pairing correlations overcompensate the effects of Pauli blocking on the collision rate for a large rang...... the experimental bounds results in a damping of the breathing mode which is comparable to what is observed, even for a scale-invariant system....
11. Two-Dimensional Gas Chromatography-Mass Spectrometry to Determine Composition of the Pro ducts of Waste Tire Pyrolysis
OpenAIRE
Gertsiuk, M.M.; Kovalchuk, T.; Kapral, K.; Lysychenko, G.V.
2014-01-01
The method of two-dimensional gas chromatography cou pled with mass-spectrometry detection was used for determination of pyrolysis liquid — a mixture of pyrolysis products of waste tires. 6500 organic compounds have been identified: the saturated, unsaturated, aromatic hydrocarbons, the derivatives of thiophene, cyclic aminocompounds. By its composition pyrolytic liquid is close to the diesel fuel and can be used as the alternative fuel.
12. Two-Dimensional Gas Chromatography-Mass Spectrometry to Determine Composition of the Pro ducts of Waste Tire Pyrolysis
Directory of Open Access Journals (Sweden)
Gertsiuk, M.M.
2014-03-01
Full Text Available The method of two-dimensional gas chromatography cou pled with mass-spectrometry detection was used for determination of pyrolysis liquid — a mixture of pyrolysis products of waste tires. 6500 organic compounds have been identified: the saturated, unsaturated, aromatic hydrocarbons, the derivatives of thiophene, cyclic aminocompounds. By its composition pyrolytic liquid is close to the diesel fuel and can be used as the alternative fuel.
13. Mixed valence as a necessary criteria for quasi-two dimensional electron gas in oxide hetero-interfaces
Science.gov (United States)
Singh, Vijeta; Pulikkotil, J. J.
2017-02-01
The origin of quasi-two dimensional electron gas at the interface of polar-nonpolar oxide hetero-structure, such as LaAlO3/SrTiO3, is debated over electronic reconstruction and defects/disorder models. Common to these models is the partial valence transformation of substrate Ti ions from its equilibrium 4 + state to an itinerant 3 + state. Given that the Hf ions have a lower ionization potential than Ti due to the 4 f orbital screening, one would expect a hetero-interface conductivity in the polar-nonpolar LaAlO3/SrHfO3 system as well. However, our first principles calculations show the converse. Unlike the Ti3+ -Ti4+ valence transition which occur at a nominal energy cost, the barrier energy associated with its isoelectronic Hf3+ -Hf4+ counterpart is very high, hence suppressing the formation of quasi-two dimensional electron gas at LaAlO3/SrHfO3 hetero-interface. These calculations, therefore, emphasize on the propensity of mixed valence at the interface as a necessary condition for an oxide hetero-structure to exihibit quasi two-dimensional electron gas.
14. Peierls-Nabarro energy surfaces and directional mobility of discrete solitons in two-dimensional saturable nonlinear Schr\\"odinger lattices
CERN Document Server
Naether, Uta; Johansson, Magnus
2010-01-01
We address the problem of directional mobility of discrete solitons in two-dimensional rectangular lattices, in the framework of a discrete nonlinear Schr\\"odinger model with saturable on-site nonlinearity. A numerical constrained Newton-Raphson method is used to calculate two-dimensional Peierls-Nabarro energy surfaces, which describe a pseudopotential landscape for the slow mobility of coherent localized excitations, corresponding to continuous phase-space trajectories passing close to stationary modes. Investigating the two-parameter space of the model through independent variations of the nonlinearity constant and the power, we show how parameter regimes and directions of good mobility are connected to existence of smooth surfaces connecting the stationary states. In particular, directions where solutions can move with minimum radiation can be predicted from flatter parts of the surfaces. For such mobile solutions, slight perturbations in the transverse direction yield additional transverse oscillations w...
15. Exact two-body solutions and quantum defect theory of two-dimensional dipolar quantum gas
Science.gov (United States)
Jie, Jianwen; Qi, Ran
2016-10-01
In this paper, we provide the two-body exact solutions of the two-dimensional (2D) Schrödinger equation with isotropic +/- 1/{r}3 interactions. An analytic quantum defect theory is constructed based on these solutions and it is applied to investigate the scattering properties as well as two-body bound states of an ultracold polar molecules confined in a quasi-2D geometry. Interestingly, we find that for the attractive case, the scattering resonance happens simultaneously in all partial waves, which has not been observed in other systems. The effect of this feature on the scattering phase shift across such resonances is also illustrated.
16. Pseudogap Phenomena Near the BKT Transition of a Two-Dimensional Ultracold Fermi Gas in the Crossover Region
Science.gov (United States)
Matsumoto, M.; Hanai, R.; Inotani, D.; Ohashi, Y.
2017-06-01
We investigate strong-coupling properties of a two-dimensional ultracold Fermi gas in the normal phase. In the three-dimensional case, it has been shown that the so-called pseudogap phenomena can be well described by a (non-self-consistent) T-matrix approximation (TMA). In the two-dimensional case, while this strong-coupling theory can explain the pseudogap phenomenon in the strong-coupling regime, it unphysically gives large pseudogap size in the crossover region, as well as in the weak-coupling regime. We show that this difficulty can be overcome when one improves TMA to include higher-order pairing fluctuations within the framework of a self-consistent T-matrix approximation (SCTMA). The essence of this improvement is also explained. Since the observation of the BKT transition has recently been reported in a two-dimensional ^6{Li} Fermi gas, our results would be useful for the study of strong-coupling physics associated with this quasi-long-range order.
17. Determination of the Critical Exponents for the Isotropic-Nematic Phase Transition in a System of Long Rods on Two-dimensional Lattices: Universality of the Transition
OpenAIRE
Matoz-Fernandez, D. A.; Linares, D. H.; Ramirez-Pastor, A. J.
2007-01-01
Monte Carlo simulations and finite-size scaling analysis have been carried out to study the critical behavior and universality for the isotropic-nematic phase transition in a system of long straight rigid rods of length $k$ ($k$-mers) on two-dimensional lattices. The nematic phase, characterized by a big domain of parallel $k$-mers, is separated from the isotropic state by a continuous transition occurring at a finite density. The determination of the critical exponents, along with the behavi...
18. Similarity between the superconductivity in the graphene with the spin transport in the two-dimensional antiferromagnet in the honeycomb lattice
Science.gov (United States)
Lima, L. S.
2017-02-01
We have used the Dirac's massless quasi-particles together with the Kubo's formula to study the spin transport by electrons in the graphene monolayer. We have calculated the electric conductivity and verified the behavior of the AC and DC currents of this system, that is a relativistic electron plasma. Our results show that the AC conductivity tends to infinity in the limit ω → 0 , similar to the behavior obtained for the spin transport in the two-dimensional frustrated antiferromagnet in the honeycomb lattice. We have made a diagrammatic expansion for the Green's function and we have not gotten significative change in the results.
19. Bound states in optical absorption of semiconductor quantum wells containing a two-dimensional electron Gas
Science.gov (United States)
Huard; Cox; Saminadayar; Arnoult; Tatarenko
2000-01-01
The dependence of the optical absorption spectrum of a semiconductor quantum well on two-dimensional electron concentration n(e) is studied using CdTe samples. The trion peak (X-) seen at low n(e) evolves smoothly into the Fermi edge singularity at high n(e). The exciton peak (X) moves off to high energy, weakens, and disappears. The X,X- splitting is linear in n(e) and closely equal to the Fermi energy plus the trion binding energy. For Cd0.998Mn0.002Te quantum wells in a magnetic field, the X,X- splitting reflects unequal Fermi energies for M = +/-1/2 electrons. The data are explained by Hawrylak's theory of the many-body optical response including spin effects.
20. Strongly anisotropic spin-orbit splitting in a two-dimensional electron gas
DEFF Research Database (Denmark)
Michiardi, Matteo; Bianchi, Marco; Dendzik, Maciej
2015-01-01
Near-surface two-dimensional electron gases on the topological insulator Bi$_2$Te$_2$Se are induced by electron doping and studied by angle-resolved photoemission spectroscopy. A pronounced spin-orbit splitting is observed for these states. The $k$-dependent splitting is strongly anisotropic...... Rashba Hamiltonian. However, a $\\mathbf{k} \\cdot \\mathbf{p}$ model that includes the possibility of band structure anisotropy as well as both isotropic and anisotropic third order Rashba splitting can explain the results. The isotropic third order contribution to the Rashba Hamiltonian is found...... to be negative, reducing the energy splitting at high $k$. The interplay of band structure, higher order Rashba effect and tuneable doping offers the opportunity to engineer not only the size of the spin-orbit splitting but also its direction....
1. [Characterization of aromatic hydrocarbons in heavy gas oil using comprehensive two-dimensional gas chromatography coupled to time-of-flight mass spectrometry].
Science.gov (United States)
Guo, Kun; Zhou, Jian; Liu, Zelong
2012-02-01
An analytical method for separating and identifying the aromatic hydrocarbons in heavy gas oil using comprehensive two-dimensional gas chromatography (GC x GC) coupled to time-of-flight mass spectrometry (TOF MS) was established. The two-dimensional distribution by ring number of the aromatic hydrocarbons was obtained. Besides phenanthrene and methyl-phenanthrene, many other polycyclic aromatic hydrocarbons (PAHs) such as pyrene and benzo [a] anthracene were identified by using the retention times, standard mass spectra or literature reports. The method was successfully applied to the hydrotreating process of heavy gas oil and the hydrotreated products of phenanthrene, pyrene were identified. This method provided technical support for the characterization of aromatic hydrocarbons in heavy gas oil and the investigation of hydrogenation mechanism of polycyclic aromatic hydrocarbons. Compared with the conventional method, gas chromatography coupled to mass spectrometry (GC-MS), the GC x GC-TOF MS method illustrated the obvious advantages for heavy gas oil analysis.
2. Evolution of the vortex state in the BCS-BEC crossover of a quasi two-dimensional superfluid Fermi gas
Science.gov (United States)
Luo, Xuebing; Zhou, Kezhao; Zhang, Zhidong
2016-11-01
We use the path-integral formalism to investigate the vortex properties of a quasi-two dimensional (2D) Fermi superfluid system trapped in an optical lattice potential. Within the framework of mean-field theory, the cooper pair density, the atom number density, and the vortex core size are calculated from weakly interacting BCS regime to strongly coupled while weakly interacting BEC regime. Numerical results show that the atoms gradually penetrate into the vortex core as the system evolves from BEC to BCS regime. Meanwhile, the presence of the optical lattice allows us to analyze the vortex properties in the crossover from three-dimensional (3D) to 2D case. Furthermore, using a simple re-normalization procedure, we find that the two-body bound state exists only when the interaction is stronger than a critical one denoted by G c which is obtained as a function of the lattice potential’s parameter. Finally, we investigate the vortex core size and find that it grows with increasing interaction strength. In particular, by analyzing the behavior of the vortex core size in both BCS and BEC regimes, we find that the vortex core size behaves quite differently for positive and negative chemical potentials. Project supported by the National Natural Science Foundation of China (Grant Nos. 51331006, 51590883, and 11204321) and the Project of Chinese Academy of Sciences (Grant No. KJZD-EW-M05-3).
3. First-principles design of a half-filled flat band of the kagome lattice in two-dimensional metal-organic frameworks
Science.gov (United States)
Yamada, Masahiko G.; Soejima, Tomohiro; Tsuji, Naoto; Hirai, Daisuke; Dincǎ, Mircea; Aoki, Hideo
2016-08-01
We design from first principles a type of two-dimensional metal-organic framework (MOF) using phenalenyl-based ligands to exhibit a half-filled flat band of the kagome lattice, which is one of a family of lattices that show Lieb-Mielke-Tasaki's flat-band ferromagnetism. Among various MOFs, we find that trans-Au-THTAP (THTAP=trihydroxytriaminophenalenyl) has such an ideal band structure, where the Fermi energy is adjusted right at the flat band due to unpaired electrons of radical phenalenyl. The spin-orbit coupling opens a band gap giving a nonzero Chern number to the nearly flat band, as confirmed by the presence of the edge states in first-principles calculations and by fitting to the tight-binding model. This is a novel and realistic example of a system in which a nearly flat band is both ferromagnetic and topologically nontrivial.
4. Synthetic gauge field and pseudospin-orbit interaction in a stacked two-dimensional ring network lattice
CERN Document Server
Ochiai, Tetsuyuki
2016-01-01
Synthetic gauge field and pseudospin-orbit interaction are implemented in the stacked two-dimensional ring network model proposed by the present author. The model was introduced to simulate light propagation in the corresponding ring-resonator network, and is thus completely bosonic. Without these two items, the system exhibits Floquet-Weyl and Floquet-topological-insulator phases with topologically gapless and gapped band structures, respectively. The synthetic magnetic field implemented in the model results in a three-dimensional Hofstadter-butterfly-type spectrum in a photonic platform. The resulting gaps are characterization by the winding number of relevant S-matrices together with the Chern number of the bulk bands. The pseudospin-orbit interaction is defined as the mixing term between two pseudospin degrees of freedom in the rings, namely, the clockwise and counter-clockwise modes in the rings. It destroys the Floquet-topological-insulator phases, while the Floquet-Weyl phase with multiple Weyl points ...
5. Aerosol assisted fabrication of two dimensional ZnO island arrays and honeycomb patterns with identical lattice structures
Directory of Open Access Journals (Sweden)
Mitsuhiro Numata
2010-11-01
Full Text Available Two dimensional island arrays and honeycomb patterns consisting of ZnO nanocrystal clusters were fabricated on predefined TiO2 seed patterns prepared by vacuum free, aerosol assisted wet-chemical synthesis. The TiO2 seed patterns were prepared by applying an aerosol of a water soluble titanium complex on hexagonally close-packed polystyrene bead arrays for different lengths of time. Scanning electron microscopy revealed that a dot array grows into a honeycomb shape as increasing amounts of the precursor were deposited. ZnO nucleation on substrates with a dot array and honeycomb patterns resulted in the formation of two discrete patterns with contrasting fill fractions of the materials.
6. A new model for two-dimensional numerical simulation of pseudo-2D gas-solids fluidized beds
Energy Technology Data Exchange (ETDEWEB)
Li, Tingwen; Zhang, Yongmin
2013-10-11
Pseudo-two dimensional (pseudo-2D) fluidized beds, for which the thickness of the system is much smaller than the other two dimensions, is widely used to perform fundamental studies on bubble behavior, solids mixing, or clustering phenomenon in different gas-solids fluidization systems. The abundant data from such experimental systems are very useful for numerical model development and validation. However, it has been reported that two-dimensional (2D) computational fluid dynamic (CFD) simulations of pseudo-2D gas-solids fluidized beds usually predict poor quantitative agreement with the experimental data, especially for the solids velocity field. In this paper, a new model is proposed to improve the 2D numerical simulations of pseudo-2D gas-solids fluidized beds by properly accounting for the frictional effect of the front and back walls. Two previously reported pseudo-2D experimental systems were simulated with this model. Compared to the traditional 2D simulations, significant improvements in the numerical predictions have been observed and the predicted results are in better agreement with the available experimental data.
7. Optical probing of the metal-to-insulator transition in a two-dimensional high-mobility electron gas
Energy Technology Data Exchange (ETDEWEB)
Dionigi, F; Rossella, F; Bellani, V [Dipartimento di Fisica ' A Volta' and CNISM, Universita degli Studi di Pavia, 27100 Pavia (Italy); Amado, M [GISC and Departamento de Fisica de Materiales, Universidad Complutense, 28040 Madrid (Spain); Diez, E [Laboratorio de Bajas Temperaturas, Universidad de Salamanca, 37008 Salamanca (Spain); Kowalik, K [Laboratoire National des Champs Magnetiques Intenses, CNRS, 38042 Grenoble (France); Biasiol, G [Istituto Officina dei Materiali CNR, Laboratorio TASC, 34149 Trieste (Italy); Sorba, L, E-mail: [email protected] [NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, 56126 Pisa (Italy)
2011-06-15
We study the quantum Hall liquid and the metal-insulator transition in a high-mobility two-dimensional electron gas, by means of photoluminescence and magnetotransport measurements. In the integer and fractional regime at {nu}>1/3, by analyzing the emission energy dispersion we probe the magneto-Coulomb screening and the hidden symmetry of the electron liquid. In the fractional regime above {nu}=1/3, the system undergoes metal-to-insulator transition, and in the insulating phase the dispersion becomes linear with evidence of an increased renormalized mass.
8. Hole-Hole Interaction Effect in the Conductance of the Two-Dimensional Hole Gas in the Ballistic Regime
OpenAIRE
Proskuryakov, Y. Y.; Savchenko, A. K.; Safonov, S. S.; Pepper, M; Simmons, M.Y.; Ritchie, D. A.
2001-01-01
On a high mobility two-dimensional hole gas (2DHG) in a GaAs/GaAlAs heterostructure we study the interaction correction to the Drude conductivity in the ballistic regime, $k_BT\\tau /\\hbar$ $>1$. It is shown that the 'metallic' behaviour of the resistivity ($d\\rho /dT>0$) of the low-density 2DHG is caused by hole-hole interaction effect in this regime. We find that the temperature dependence of the conductivity and the parallel-field magnetoresistance are in agreement with this description, a...
9. The effect of the lateral interactions on the critical behavior of long straight rigid rods on two-dimensional lattices
Directory of Open Access Journals (Sweden)
A. J. Ramirez-Pastor
2009-09-01
Full Text Available Using Monte Carlo simulations and finite-size scaling analysis, the critical behavior of attractive rigid rods of length $k$ ($k$-mers on square lattices at intermediate density has been studied. A nematic phase, characterized by a big domain of parallel $k$-mers, was found. This ordered phase is separated from the isotropic state by a continuous transition occurring at a intermediate density $heta_c$, which increases linearly with the magnitude of the lateral interactions.
10. Tunable all-angle negative refraction and photonic band gaps in two-dimensional plasma photonic crystals with square-like Archimedean lattices
Energy Technology Data Exchange (ETDEWEB)
Zhang, Hai-Feng, E-mail: [email protected], E-mail: [email protected] [Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing Univ. Aeronaut. Astronaut.), Ministry of Education, Nanjing University of Aeronautics and Astronautics, Nanjing 210016 (China); Nanjing Artillery Academy, Nanjing 211132 (China); Liu, Shao-Bin, E-mail: [email protected], E-mail: [email protected]; Jiang, Yu-Chi [Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing Univ. Aeronaut. Astronaut.), Ministry of Education, Nanjing University of Aeronautics and Astronautics, Nanjing 210016 (China)
2014-09-15
In this paper, the tunable all-angle negative refraction and photonic band gaps (PBGs) in two types of two-dimensional (2D) plasma photonic crystals (PPCs) composed of homogeneous plasma and dielectric (GaAs) with square-like Archimedean lattices (ladybug and bathroom lattices) for TM wave are theoretically investigated based on a modified plane wave expansion method. The type-1 structure is dielectric rods immersed in the plasma background, and the complementary structure is named as type-2 PPCs. Theoretical simulations demonstrate that the both types of PPCs with square-like Archimedean lattices have some advantages in obtaining the higher cut-off frequency, the larger PBGs, more number of PBGs, and the relative bandwidths compared to the conventional square lattices as the filling factor or radius of inserted rods is same. The influences of plasma frequency and radius of inserted rod on the properties of PBGs for both types of PPCs also are discussed in detail. The calculated results show that PBGs can be manipulated by the parameters as mentioned above. The possibilities of all-angle negative refraction in such two types of PPCs at low bands also are discussed. Our calculations reveal that the all-angle negative phenomena can be observed in the first two TM bands, and the frequency range of all-angle negative refraction can be tuned by changing plasma frequency. Those properties can be used to design the optical switching and sensor.
11. Tunable all-angle negative refraction and photonic band gaps in two-dimensional plasma photonic crystals with square-like Archimedean lattices
Science.gov (United States)
Zhang, Hai-Feng; Liu, Shao-Bin; Jiang, Yu-Chi
2014-09-01
In this paper, the tunable all-angle negative refraction and photonic band gaps (PBGs) in two types of two-dimensional (2D) plasma photonic crystals (PPCs) composed of homogeneous plasma and dielectric (GaAs) with square-like Archimedean lattices (ladybug and bathroom lattices) for TM wave are theoretically investigated based on a modified plane wave expansion method. The type-1 structure is dielectric rods immersed in the plasma background, and the complementary structure is named as type-2 PPCs. Theoretical simulations demonstrate that the both types of PPCs with square-like Archimedean lattices have some advantages in obtaining the higher cut-off frequency, the larger PBGs, more number of PBGs, and the relative bandwidths compared to the conventional square lattices as the filling factor or radius of inserted rods is same. The influences of plasma frequency and radius of inserted rod on the properties of PBGs for both types of PPCs also are discussed in detail. The calculated results show that PBGs can be manipulated by the parameters as mentioned above. The possibilities of all-angle negative refraction in such two types of PPCs at low bands also are discussed. Our calculations reveal that the all-angle negative phenomena can be observed in the first two TM bands, and the frequency range of all-angle negative refraction can be tuned by changing plasma frequency. Those properties can be used to design the optical switching and sensor.
12. Local membrane length conservation in two-dimensional vesicle simulation using a multicomponent lattice Boltzmann equation method.
Science.gov (United States)
Halliday, I; Lishchuk, S V; Spencer, T J; Pontrelli, G; Evans, P C
2016-08-01
We present a method for applying a class of velocity-dependent forces within a multicomponent lattice Boltzmann equation simulation that is designed to recover continuum regime incompressible hydrodynamics. This method is applied to the problem, in two dimensions, of constraining to uniformity the tangential velocity of a vesicle membrane implemented within a recent multicomponent lattice Boltzmann simulation method, which avoids the use of Lagrangian boundary tracers. The constraint of uniform tangential velocity is carried by an additional contribution to an immersed boundary force, which we derive here from physical arguments. The result of this enhanced immersed boundary force is to apply a physically appropriate boundary condition at the interface between separated lattice fluids, defined as that region over which the phase-field varies most rapidly. Data from this enhanced vesicle boundary method are in agreement with other data obtained using related methods [e.g., T. Krüger, S. Frijters, F. Günther, B. Kaoui, and J. Harting, Eur. Phys. J. 222, 177 (2013)10.1140/epjst/e2013-01834-y] and underscore the importance of a correct vesicle membrane condition.
13. General velocity, pressure, and initial condition for two-dimensional and three-dimensional lattice Boltzmann simulations
Science.gov (United States)
Mohammadipour, Omid Reza; Niazmand, Hamid; Succi, Sauro
2017-03-01
In this paper, an alternative approach to implement initial and boundary conditions in the lattice Boltzmann method is presented. The main idea is to approximate the nonequilibrium component of distribution functions as a third-order power series in the lattice velocities and formulate a procedure to determine boundary node distributions by using fluid variables, consistent with such an expansion. The velocity shift associated with the body force effects is included in this scheme, along with an approximation to determine the mass density in complex geometries. Different strategies based on the present scheme are developed to implement velocity and pressure conditions for arbitrarily shaped boundaries, using the D2Q9, D3Q15, D3Q19 and D3Q27 lattices, in two and three space dimensions, respectively. The proposed treatment is tested against several well-established problems, showing second-order spatial accuracy and often improved behavior as compared to various existing methods, with no appreciable computational overhead.
14. Magnetocapacitance oscillations and thermoelectric effect in a two-dimensional electron gas irradiated by microwaves
Science.gov (United States)
Levin, A. D.; Gusev, G. M.; Raichev, O. E.; Momtaz, Z. S.; Bakarov, A. K.
2016-07-01
To study the influence of microwave irradiation on two-dimensional electrons, we apply a method based on capacitance measurements in GaAs quantum well samples where the gate covers a central part of the layer. We find that the capacitance oscillations at high magnetic fields, caused by the oscillations of thermodynamic density of states, are not essentially modified by microwaves. However, in the region of fields below 1 T, we observe another set of oscillations, with the period and the phase identical to those of microwave-induced resistance oscillations. The phenomenon of microwave-induced capacitance oscillations is explained in terms of violation of the Einstein relation between conductivity and the diffusion coefficient in the presence of microwaves, which leads to a dependence of the capacitor charging on the anomalous conductivity. We also observe microwave-induced oscillations in the capacitive response to periodic variations of external heating. These oscillations appear due to the thermoelectric effect and are in antiphase with microwave-induced resistance oscillations because of the Corbino-like geometry of our experimental setup.
15. Observation of mesoscopic crystalline structures in a two-dimensional Rydberg gas
CERN Document Server
Schauß, Peter; Endres, Manuel; Fukuhara, Takeshi; Hild, Sebastian; Omran, Ahmed; Pohl, Thomas; Gross, Christian; Kuhr, Stefan; Bloch, Immanuel
2012-01-01
The ability to control and tune interactions in ultracold atomic gases has paved the way towards the realization of new phases of matter. Whereas experiments have so far achieved a high degree of control over short-ranged interactions, the realization of long-range interactions would open up a whole new realm of many-body physics and has become a central focus of research. Rydberg atoms are very well-suited to achieve this goal, as the van der Waals forces between them are many orders of magnitude larger than for ground state atoms. Consequently, the mere laser excitation of ultracold gases can cause strongly correlated many-body states to emerge directly when atoms are transferred to Rydberg states. A key example are quantum crystals, composed of coherent superpositions of different spatially ordered configurations of collective excitations. Here we report on the direct measurement of strong correlations in a laser excited two-dimensional atomic Mott insulator using high-resolution, in-situ Rydberg atom imag...
16. Two-Dimensional Riemann Solver for Euler Equations of Gas Dynamics
Science.gov (United States)
Brio, M.; Zakharian, A. R.; Webb, G. M.
2001-02-01
We construct a Riemann solver based on two-dimensional linear wave contributions to the numerical flux that generalizes the one-dimensional method due to Roe (1981, J. Comput. Phys.43, 157). The solver is based on a multistate Riemann problem and is suitable for arbitrary triangular grids or any other finite volume tessellations of the plane. We present numerical examples illustrating the performance of the method using both first- and second-order-accurate numerical solutions. The numerical flux contributions are due to one-dimensional waves and multidimensional waves originating from the corners of the computational cell. Under appropriate CFL restrictions, the contributions of one-dimensional waves dominate the flux, which explains good performance of dimensionally split solvers in practice. The multidimensional flux corrections increase the accuracy and stability, allowing a larger time step. The improvements are more pronounced on a coarse mesh and for large CFL numbers. For the second-order method, the improvements can be comparable to the improvements resulting from a less diffusive limiter.
17. Coupled two-dimensional edge plasma and neutral gas modeling of tokamak scrape-off-layers
Energy Technology Data Exchange (ETDEWEB)
Maingi, R. [North Carolina State Univ., Raleigh, NC (United States)
1992-08-01
The objective of this study is to devise a detailed description of the tokamak scrape-off-layer (SOL), which includes the best available models of both the plasma and neutral species and the strong coupling between the two in many SOL regimes. A good estimate of both particle flux and heat flux profiles at the limiter/divertor target plates is desired. Peak heat flux is one of the limiting factors in determining the survival probability of plasma-facing-components at high power levels. Plate particle flux affects the neutral flux to the pump, which determines the particle exhaust rate. A technique which couples a two-dimensional (2-D) plasma and a 2-D neutral transport code has been developed (coupled code technique), but this procedure requires large amounts of computer time. Relevant physics has been added to an existing two-neutral-species model which takes the SOL plasma/neutral coupling into account in a simple manner (molecular physics model), and this model is compared with the coupled code technique mentioned above. The molecular physics model is benchmarked against experimental data from a divertor tokamak (DIII-D), and a similar model (single-species model) is benchmarked against data from a pump-limiter tokamak (Tore Supra). The models are then used to examine two key issues: free-streaming-limits (ion energy conduction and momentum flux) and the effects of the non-orthogonal geometry of magnetic flux surfaces and target plates on edge plasma parameter profiles.
18. The effect of surface roughness on rarefied gas flows by lattice Boltzmann method
Institute of Scientific and Technical Information of China (English)
Liu Chao-Feng; Ni Yu-Shan
2008-01-01
This paper studies the roughness effect combining with effects of rarefaction and compressibility by a lattice Boltzmann model for rarefied gas flows at high Knudsen numbers. By discussing the effect of the tangential momentum accommodation coefficient on the rough boundary condition, the lattice Boltzmann simulations of nitrogen and helium flows are performed in a two-dimensional microchannel with rough boundaries. The surface roughness effects in the microchannel on the velocity field, the mass flow rate and the friction coefficient are studied and analysed. Numerical results for the two gases in micro scale show different characteristics from macroscopic flows and demonstrate the feasibility of the lattice Boltzmann model in rarefied gas dynamics.
19. Ballistic and diffusive dynamics in a two-dimensional ideal gas of macroscopic chaotic Faraday waves.
Science.gov (United States)
Welch, Kyle J; Hastings-Hauss, Isaac; Parthasarathy, Raghuveer; Corwin, Eric I
2014-04-01
We have constructed a macroscopic driven system of chaotic Faraday waves whose statistical mechanics, we find, are surprisingly simple, mimicking those of a thermal gas. We use real-time tracking of a single floating probe, energy equipartition, and the Stokes-Einstein relation to define and measure a pseudotemperature and diffusion constant and then self-consistently determine a coefficient of viscous friction for a test particle in this pseudothermal gas. Because of its simplicity, this system can serve as a model for direct experimental investigation of nonequilibrium statistical mechanics, much as the ideal gas epitomizes equilibrium statistical mechanics.
20. A two-dimensional Fermi gas in the BEC-BCS crossover
Energy Technology Data Exchange (ETDEWEB)
Ries, Martin Gerhard
2016-01-21
This thesis reports on the preparation of a 2D Fermi gas in the BEC-BCS crossover and the observation of the BKT transition into a quasi long-range ordered superfluid phase. The pair momentum distribution of the gas is probed by means of a matter-wave focusing technique which relies on time-of-flight evolution in a weak harmonic potential. This distribution holds the coherence properties of the gas. The quasi long-range ordered phase manifests itself as a sharp low-momentum peak. The temperature where it forms is identified as the transition temperature. By tuning the temperature and the interaction strength, the phase diagram of the 2D Fermi gas in the BEC-BCS crossover is mapped out. The phase coherence is investigated in a self-interference experiment. Furthermore, algebraic decay of correlations is observed in the trap average of the first order correlation function, which is obtained from the Fourier transform of the pair momentum distribution. This is in qualitative agreement with predictions of homogeneous theory for the superfluid phase in a 2D gas. The presented results provide a foundation for future experimental and theoretical studies of strongly correlated 2D Fermi gases. They might thus help to elucidate complex systems such as the electron gas in high-T{sub c} superconductors.
1. Periodic ordering of clusters and stripes in a two-dimensional lattice model. II. Results of Monte Carlo simulation.
Science.gov (United States)
Almarza, N G; Pȩkalski, J; Ciach, A
2014-04-28
The triangular lattice model with nearest-neighbor attraction and third-neighbor repulsion, introduced by Pȩkalski, Ciach, and Almarza [J. Chem. Phys. 140, 114701 (2014)] is studied by Monte Carlo simulation. Introduction of appropriate order parameters allowed us to construct a phase diagram, where different phases with patterns made of clusters, bubbles or stripes are thermodynamically stable. We observe, in particular, two distinct lamellar phases-the less ordered one with global orientational order and the more ordered one with both orientational and translational order. Our results concern spontaneous pattern formation on solid surfaces, fluid interfaces or membranes that is driven by competing interactions between adsorbing particles or molecules.
2. Two Dimensional Incommensurate Spin Excitations and Lattice Fluctuations in La2 - x Bax CuO4
Science.gov (United States)
Wagman, J. J.; Carlo, J. P.; van Gastel, G.; Zhao, Y.; Kallin, A. B.; Mazurek, E.; Dabkowska, H. A.; Savicii, A.; Granroth, G. E.; Yamani, Z.; Tun, Z.; Gaulin, B. D.
2013-03-01
'Hour-glass' shaped dispersions of antiferromagnetic (AF) spin fluctuations are a robust feature common to many high temperature superconductors. In 214 cuprates, these phenomena are well known to display a strong dependence on the concentration of holes that are introduced into the copper oxide planes by doping. The incommensurability (IC) of the two dimensional magnetic order in this system is sensitive to hole concentration. Here, we present a series of neutron scattering measurements on single crystals of La2 - x Bax CuO4 (LBCO), with 0 . 035 <= x <= 0 . 095 , a doping range that spans the transition from diagonal to parallel IC ordering wavevectors, and from non-superconducting to superconducting ground states. Our measurements map out the evolution of the spin excitations for energies below ~ 50 meV, and focus on an enhancement in the scattered intensity centered in the 17-20 meV at the AF IC positions. This regime corresponds to the approximate crossing of very dispersive spin excitations and weakly dispersive low lying optic phonons in LBCO. NSERC, Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy
3. Analysis of cave atmospheres by comprehensive two-dimensional gas chromatography (GC×GC with flame ionization detection (FID
Directory of Open Access Journals (Sweden)
Ryan C. Blase
2015-03-01
Full Text Available In this paper, we describe a simple method for sampling, pre-concentrating, and separating volatile and semi-volatile components from two different cave atmospheres. Sampling is performed by capturing a volume of cave atmosphere in a Tedlar bag or Suma canister for sample storage and transport back to the laboratory. Loading a portion of the sample on a multi-bed sorption trap allows for sample pre-concentration prior to separation and detection of components on a comprehensive two-dimensional gas chromatograph (GC×GC. Comparison of two Texas caves reveals the power of comprehensive two-dimensional gas chromatography (GC×GC for volatile separation and detection, and to our knowledge marks the first use of GC×GC for the analysis of cave atmospheres. Analysis of the results revealed 138 and 146 chromatographic signals over an S/N threshold of 500 and direct comparison of the two samples revealed 50 identical chromatographic signals. This study is a first step toward demonstrating the ability of GC×GC to separate the complex volatiles and semi-volatiles in the cave atmosphere as a fingerprinting tool.
4. Comprehensive two-dimensional gas chromatography for characterizing mineral oils in foods and distinguishing them from synthetic hydrocarbons.
Science.gov (United States)
Biedermann, Maurus; Grob, Koni
2015-01-02
Many foods are contaminated by hydrocarbons of mineral oil or synthetic origin. High performance liquid chromatography on-line coupled with gas chromatography and flame ionization detection (HPLC-GC-FID) is a powerful tool for the quantitative determination, but it would often be desirable to obtain more information about the type of hydrocarbons in order to identify the source of the contamination and specify pertinent legislation. Comprehensive two-dimensional gas chromatography (GC×GC) is shown to produce plots distinguishing mineral oil saturated hydrocarbons (MOSH) from polymer oligomeric saturated hydrocarbons (POSH) and characterizing the degree of raffination of a mineral oil. The first dimension separation occurred on a phenyl methyl polysiloxane, the second on a dimethyl polysiloxane. Mass spectrometry (MS) was used for identification, FID for quantitative determination. This shows the substantial advances in chromatography to characterize complex hydrocarbon mixtures even as contaminants in food.
5. Spin-orbital exchange of strongly interacting fermions in the p band of a two-dimensional optical lattice.
Science.gov (United States)
Zhou, Zhenyu; Zhao, Erhai; Liu, W Vincent
2015-03-13
Mott insulators with both spin and orbital degeneracy are pertinent to a large number of transition metal oxides. The intertwined spin and orbital fluctuations can lead to rather exotic phases such as quantum spin-orbital liquids. Here, we consider two-component (spin 1/2) fermionic atoms with strong repulsive interactions on the p band of the optical square lattice. We derive the spin-orbital exchange for quarter filling of the p band when the density fluctuations are suppressed, and show that it frustrates the development of long-range spin order. Exact diagonalization indicates a spin-disordered ground state with ferro-orbital order. The system dynamically decouples into individual Heisenberg spin chains, each realizing a Luttinger liquid accessible at higher temperatures compared to atoms confined to the s band.
6. Quantum anomaly, universal relations, and breathing mode of a two-dimensional Fermi gas.
Science.gov (United States)
Hofmann, Johannes
2012-05-01
In this Letter, we show that the classical SO(2,1) symmetry of a harmonically trapped Fermi gas in two dimensions is broken by quantum effects. The anomalous correction to the symmetry algebra is given by a two-body operator that is well known as the contact. Taking into account this modification, we are able to derive the virial theorem for the system and a universal relation for the pressure of a homogeneous gas. The existence of an undamped breathing mode is associated with the classical symmetry. We provide an estimate for the anomalous frequency shift of this oscillation at zero temperature and compare the result with a recent experiment by [E. Vogt et al., Phys. Rev. Lett. 108, 070404 (2012)]. Discrepancies are attributed to finite temperature effects.
7. Direct Imaging of Charge Density Modulation in Switchable Two-Dimensional Electron Gas at the Oxide Hetero-Interfaces by Using Electron Bean Inline Holography
Science.gov (United States)
2015-08-16
SUPPLEMENTARY NOTES 14. ABSTRACT The recent discovery of a two-dimensional electron gas (2DEG) at the interface between insulating perovskite ...3/10/2015 Abstract The recent discovery of a two-dimensional electron gas (2DEG) at the interface between insulating perovskite oxides SrTiO3...associated charge distributions in semiconductor materials, and therefore regarded as the only tool that can completely visualize the spatial
8. Two Dimensional Spin-Polarized Electron Gas at the Oxide Interfaces
OpenAIRE
Nanda, B. R. K.; Satpathy, S.
2008-01-01
The formation of a novel spin-polarized 2D electron gas at the LaMnO$_3$ monolayer embedded in SrMnO$_3$ is predicted from the first-principles density-functional calculations. The La (d) electrons become confined in the direction normal to the interface in the potential well of the La layer, serving as a positively-charged layer of electron donors. These electrons mediate a ferromagnetic alignment of the Mn t$_{2g}$ spins near the interface via the Anderson-Hasegawa double exchange and becom...
9. An accurate predictor-corrector HOC solver for the two dimensional Riemann problem of gas dynamics
Science.gov (United States)
Gogoi, Bidyut B.
2016-10-01
The work in the present manuscript is concerned with the simulation of twodimensional (2D) Riemann problem of gas dynamics. We extend our recently developed higher order compact (HOC) method from one-dimensional (1D) to 2D solver and simulate the problem on a square geometry with different initial conditions. The method is fourth order accurate in space and second order accurate in time. We then compare our results with the available benchmark results. The comparison shows an excellent agreement of our results with the existing ones in the literature. Being a finite difference solver, it is quite straight-forward and simple.
10. Quenching Plasma Waves in Two Dimensional Electron Gas by a Femtosecond Laser Pulse
Science.gov (United States)
Shur, Michael; Rudin, Sergey; Greg Rupper Collaboration; Andrey Muraviev Collaboration
Plasmonic detectors of terahertz (THz) radiation using the plasma wave excitation in 2D electron gas are capable of detecting ultra short THz pulses. To study the plasma wave propagation and decay, we used femtosecond laser pulses to quench the plasma waves excited by a short THz pulse. The femtosecond laser pulse generates a large concentration of the electron-hole pairs effectively shorting the 2D electron gas channel and dramatically increasing the channel conductance. Immediately after the application of the femtosecond laser pulse, the equivalent circuit of the device reduces to the source and drain contact resistances connected by a short. The total response charge is equal to the integral of the current induced by the THz pulse from the moment of the THz pulse application to the moment of the femtosecond laser pulse application. This current is determined by the plasma wave rectification. Registering the charge as a function of the time delay between the THz and laser pulses allowed us to follow the plasmonic wave decay. We observed the decaying oscillations in a sample with a partially gated channel. The decay depends on the gate bias and reflects the interplay between the gated and ungated plasmons in the device channel. Army Research Office.
11. Two dimensional numerical simulation of gas discharges: comparison between particle-in-cell and FCT techniques
Energy Technology Data Exchange (ETDEWEB)
Soria-Hoyo, C; Castellanos, A [Departamento de Electronica y Electromagnetismo, Facultad de Fisica, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla (Spain); Pontiga, F [Departamento de Fisica Aplicada II, EUAT, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla (Spain)], E-mail: [email protected]
2008-10-21
Two different numerical techniques have been applied to the numerical integration of equations modelling gas discharges: a finite-difference flux corrected transport (FD-FCT) technique and a particle-in-cell (PIC) technique. The PIC technique here implemented has been specifically designed for the simulation of 2D electrical discharges using cylindrical coordinates. The development and propagation of a streamer between two parallel electrodes has been used as a convenient test to compare the performance of both techniques. In particular, the phase velocity of the cathode directed streamer has been used to check the internal consistency of the numerical simulations. The results obtained from the two techniques are in reasonable agreement with each other, and both techniques have proved their ability to follow the high gradients of charge density and electric field present in this type of problems. Moreover, the streamer velocities predicted by the simulation are in accordance with the typical experimental values.
12. A high density two-dimensional electron gas in an oxide heterostructure on Si (001
Directory of Open Access Journals (Sweden)
E. N. Jin
2014-11-01
Full Text Available We present the growth and characterization of layered heterostructures comprised of LaTiO3 and SrTiO3 epitaxially grown on Si (001. Magnetotransport measurements show that the sheet carrier densities of the heterostructures scale with the number of LaTiO3/SrTiO3 interfaces, consistent with the presence of an interfacial 2-dimensional electron gas (2DEG at each interface. Sheet carrier densities of 8.9 × 1014 cm−2 per interface are observed. Integration of such high density oxide 2DEGs on silicon provides a bridge between the exceptional properties and functionalities of oxide 2DEGs and microelectronic technologies.
13. Development and Design of a Single-Stage Cryogenic Modulator for Comprehensive Two-Dimensional Gas Chromatography.
Science.gov (United States)
Mostafa, Ahmed; Górecki, Tadeusz
2016-05-17
A new liquid nitrogen-based single-stage cryogenic modulator was developed and characterized. In addition, a dedicated liquid nitrogen delivery system was developed. A well-defined restriction placed inside a deactivated fused silica capillary was used to increase the cooling surface area and provide very efficient trapping. At the same time, it enabled modulation of the carrier gas flow owing to changes in gas viscosity with temperature. Gas flow is almost unimpeded at the trapping temperature but reduced to nearly zero at the desorption temperature, which prevents analyte breakthrough. Peak widths for n-alkanes of 30-40 ms at half height were obtained. Most importantly, even the solvent peak could be modulated, which is not feasible with any commercially available thermal modulator. Evaluation of the newly developed system in two-dimensional gas chromatography (GC × GC) separations of some real samples such as regular gasoline and diesel fuel showed that the analytical performance of this single-stage modulator is fully competitive to those of the more complicated dual-stage modulators.
14. Comprehensive two-dimensional gas chromatography combined to multivariate data analysis for detection of disease-resistant clones of Eucalyptus.
Science.gov (United States)
Hantao, Leandro Wang; Toledo, Bruna Regina; Ribeiro, Fabiana Alves de Lima; Pizetta, Marilia; Pierozzi, Caroline Geraldi; Furtado, Edson Luiz; Augusto, Fabio
2013-11-15
In this paper it is reported the use of the chromatographic profiles from volatile fractions of plant clones - in this case, hybrids of Eucalyptus grandis×Eucalyptus urophylla - to determine specimens susceptible to rust disease. The analytes were isolated by headspace solid phase microextraction (HS-SPME) and analyzed by comprehensive two-dimensional gas chromatography combined to fast quadrupole mass spectrometry (GC×GC-qMS). Parallel Factor Analysis (PARAFAC) was employed for estimate the correlation between the chromatographic profiles and resistance against Eucalyptus rust, after preliminary variable selection performed by Fisher ratio analysis. The proposed method allowed the differentiation between susceptible and non-susceptible clones and determination of three resistance biomarkers. This approach can be a valuable alternative for the otherwise time-consuming and labor-intensive methods commonly used.
15. Liquid-Gated High Mobility and Quantum Oscillation of the Two-Dimensional Electron Gas at an Oxide Interface.
Science.gov (United States)
Zeng, Shengwei; Lü, Weiming; Huang, Zhen; Liu, Zhiqi; Han, Kun; Gopinadhan, Kalon; Li, Changjian; Guo, Rui; Zhou, Wenxiong; Ma, Haijiao Harsan; Jian, Linke; Venkatesan, Thirumalai; Ariando
2016-04-26
Electric field effect in electronic double layer transistor (EDLT) configuration with ionic liquids as the dielectric materials is a powerful means of exploring various properties in different materials. Here, we demonstrate the modulation of electrical transport properties and extremely high mobility of two-dimensional electron gas at LaAlO3/SrTiO3 (LAO/STO) interface through ionic liquid-assisted electric field effect. With a change of the gate voltages, the depletion of charge carrier and the resultant enhancement of electron mobility up to 19 380 cm(2)/(V s) are realized, leading to quantum oscillations of the conductivity at the LAO/STO interface. The present results suggest that high-mobility oxide interfaces, which exhibit quantum phenomena, could be obtained by ionic liquid-assisted field effect.
16. Effect of In Composition on Two-Dimensional Electron Gas in Wurtzite AlGaN/InGaN Heterostructures
Institute of Scientific and Technical Information of China (English)
KIM Bong-Hwan; PARK Seoung-Hwan; LEE Jung-Hee; MOON Yong-Tae
2010-01-01
@@ The effect of In composition on two-dimensional electron gas in wurtzite AIGaN/InGaN heterostructures is theoretically investigated.The sheet carrier density is shown to increase nearly linearly with In mole fraction x,due to the increase in the polarization charge at the AlGaN/InGaN interface.The electron sheet density is enhanced with the doping in the AlGaN layer.The sheet carrier density is as high as 3.7 × 1013 cm-2 at the donor density of 10 x 1018 cm-3 for the HEMT structure with x = 0.3.The contribution of additional donor density on the electron sheet density is nearly independent of the In mole fraction.
17. Hole-hole interaction effect in the conductance of the two-dimensional hole gas in the ballistic regime.
Science.gov (United States)
Proskuryakov, Y Y; Savchenko, A K; Safonov, S S; Pepper, M; Simmons, M Y; Ritchie, D A
2002-08-12
On a high-mobility two-dimensional hole gas (2DHG) in a GaAs/GaAlAs heterostructure we study the interaction correction to the Drude conductivity in the ballistic regime, k(B)Ttau/ variant Planck's over 2pi >1. It is shown that the "metallic" behavior of the resistivity (drho/dT>0) of the low-density 2DHG is caused by the hole-hole interaction effect in this regime. We find that the temperature dependence of the conductivity and the parallel-field magnetoresistance are in agreement with this description, and determine the Fermi-liquid interaction constant Fsigma0 which controls the sign of drho/dT.
18. Development of an ultra-fast data-acquisition system for a two-dimensional microstrip gas chamber.
Science.gov (United States)
Ochi, A; Tanimori, T; Nishi, Y; Aoki, S; Nishi, Y
1998-05-01
A high-performance data-acquisition system has been developed in order to obtain time-resolved sequential images from a two-dimensional microstrip gas chamber (MSGC). This was achieved using fully digital processing with a synchronized pipeline method. Complex logical circuits for processing large numbers of signals are mounted on a small number of complex programmable logic devices. The system is operated with a 10 MHz synchronous clock, and has the capability of handling more than 3 x 10(6) counts s(-1) for asynchronous events. The system was examined using a 5 x 5 cm MSGC and the recently developed 10 x 10 cm MSGC (1024 outputs); the anticipated performances were achieved.
19. Bound states of a negative test charge due to many-body effects in the two-dimensional electron gas
Science.gov (United States)
Ghazali, A.; Gold, A.
1995-12-01
Bound states of a negative test electron in the low-density regime of the two-dimensional electron gas are obtained when many-body effects (exchange and correlation) are incorporated in the screening function via the local-field correction. Using the Green's-function method and a variational method we determine the energies and the wave functions of the ground state and the excited states as functions of the electron density. For high electron density no bound state is found. Below a critical density the number and the energy of bound states increase with decreasing electron density. The ground state is described by the wave function ψ2s~r exp(-r/α).
20. Shock wave structure in a lattice gas
Science.gov (United States)
Broadwell, James E.; Han, Donghee
2007-05-01
The motion and structure of shock and expansion waves in a simple particle system, a lattice gas and cellular automaton, are determined in an exact computation. Shock wave solutions, also exact, of a continuum description, a model Boltzmann equation, are compared with the lattice results. The comparison demonstrates that, as proved by Caprino et al. ["A derivation of the Broadwell equation," Commun. Math. Phys. 135, 443 (1991)] only when the lattice processes are stochastic is the model Boltzmann description accurate. In the strongest shock wave, the velocity distribution function is the bimodal function proposed by Mott-Smith.
1. Determining indicator toxaphene congeners in soil using comprehensive two-dimensional gas chromatography-tandem mass spectrometry.
Science.gov (United States)
Zhu, Shuai; Gao, Lirong; Zheng, Minghui; Liu, Huimin; Zhang, Bing; Liu, Lidan; Wang, Yiwen
2014-01-01
Toxaphene, which is a broad spectrum chlorinated pesticide, is a complex mixture of several hundred congeners, mainly polychlorinated bornanes. Quantifying toxaphene in environmental samples is difficult because of its complexity, and because each congener has a different response factor. Toxaphene chromatograms acquired using one-dimensional gas chromatography (1DGC) show that this technique cannot be used to separate all of the toxaphene congeners. We developed and validated a sensitive and quantitative method for determining three indicator toxaphene congeners in soil using an isotope dilution/comprehensive two-dimensional gas chromatography-tandem mass spectrometry (GC × GC-MS). The samples were extracted using accelerated solvent extraction, and then the extracts were purified using silica gel columns. (13)C₁₀-labeled Parlar 26 and 50 were used as internal standards and (13)C₁₀-labeled Parlar 62 was used as an injection standard. The sample extraction and purification treatments and the GC × GC-MS parameters were optimized. Subsequently the samples were determined by GC × GC-MS. The limits of detection for Parlar 26, 50, and 62 were 0.6 pg/g, 0.4 pg/g, and 1.0 pg/g (S/N=3), respectively, and the calibration curves had good linear correlations between 50 and 1000 μg/L (r(2)>0.99). Comprehensive two-dimensional GC gave substantial improvements over one-dimensional GC in the toxaphene analysis. We analyzed soil samples containing trace quantities of toxaphene to demonstrate that the developed method could be used to analyze toxaphene in environmental samples.
2. Interplay between Rashba spin-orbit coupling and adiabatic rotation in a two-dimensional Fermi gas
Science.gov (United States)
Doko, E.; Subaşı, A. L.; Iskin, M.
2017-01-01
We explore the trap profiles of a two-dimensional atomic Fermi gas in the presence of a Rashba spin-orbit coupling and under an adiabatic rotation. We first consider a noninteracting gas and show that the competition between the effects of Rashba coupling on the local density of single-particle states and the Coriolis effects caused by rotation gives rise to a characteristic ring-shaped density profile that survives at experimentally accessible temperatures. Furthermore, Rashba splitting of the Landau levels gives the density profiles a ziggurat shape in the rapid-rotation limit. We then consider an interacting gas under the BCS mean-field approximation for local pairing, and study the pair-breaking mechanism that is induced by the Coriolis effects on superfluidity, where we calculate the critical rotation frequencies both for the onset of pair breaking and for the complete destruction of superfluidity in the system. In particular, by comparing the results of a fully-quantum-mechanical Bogoliubov-de Gennes approach with those of a semiclassical local-density approximation, we construct extensive phase diagrams for a wide range of parameter regimes in the trap where the aforementioned competition may, e.g., favor an outer normal edge that is completely phase separated from the central superfluid core by vacuum.
3. Simple stochastic lattice gas automaton model for formation of river networks
Science.gov (United States)
Yan, Guangwu; Zhang, Jianying; Wang, Huimin; Guo, Li
2008-12-01
A stochastic lattice gas automata model for formation of river networks is proposed. The model is based on two-dimensional lattice gas automata with three fundamental principles at each node. The water source is regarded as a fixed point where a drop of water drips every time step. This system can be treated as a memory network: the probability of water moving along a direction relies on the history of the channel segment along which water drops have moved. Last, we find that the width of the river channel and the number of channels with this width meet a scaling law when the system reaches a critical status.
4. Dynamic structure factor in single- and two-species thermal GBL lattice gas
Science.gov (United States)
Dubbeldam, D.; Hoekstra, A. G.; Sloot, P. M. A.
2000-07-01
The two-dimensional 19-bits GBL lattice gas model conserves energy in a non-trivial way, allowing temperature, temperature gradients, and heat conduction. We describe the thermodynamics of the model, its equilibrium properties, and confirm the change of sound speed with energy density at fixed density with simulation results. The sound speed, the sound damping, and the thermal diffusivity are extracted from the dynamic structure factor and shown for various energy densities at fixed density. We have extended the 19 bits GBL model with multiple-species (miscible fluid model) and have measured the dynamic structure factor for this two-component thermal lattice gas model.
5. Chemical characterization of bio-oils using comprehensive two-dimensional gas chromatography with time-of-flight mass spectrometry.
Science.gov (United States)
Tessarolo, Nathalia S; dos Santos, Luciana R M; Silva, Raphael S F; Azevedo, Débora A
2013-03-01
The liquid product obtained via the biomass flash pyrolysis is commonly called bio-oil or pyrolysis oil. Bio-oils can be used as sources for chemicals or as fuels, primarily in mixtures or emulsions with fossil fuels. A detailed chemical characterization of bio-oil is necessary to determine its potential uses. Such characterization demands a powerful analytical technique such as comprehensive two-dimensional gas chromatography coupled with time-of-flight mass spectrometry (GC×GC-TOFMS). Limited chemical information can be obtained from conventional gas chromatography coupled mass spectrometry (GC-MS) because of the large number of compounds and coelutions. Thus, GC×GC-TOFMS was used for the individual identification of bio-oil components from two samples prepared via the flash pyrolysis of empty palm fruit bunch and pine wood chips. To the best of our knowledge, few papers have reported comprehensive two-dimensional gas chromatography (GC×GC) for bio-oil analysis. Many classes of compounds such as phenols, benzenediols, cyclopentenones, furanones, indanones and alkylpyridines were identified. Several coelutions present in the GC-MS were resolved using GC×GC-TOFMS. Many peaks were detected for the samples by GC-MS (~166 and 129), but 631 and 857 were detected by GC×GC-TOFMS, respectively. The GC×GC-TOFMS analyses indicated that the major classes of components (analytes>0.5% relative area) in the two bio-oil samples are ketones, cyclopentenones, furanones, furans, phenols, benzenediols, methoxy- and dimethoxy-phenols and sugars. In addition, esters, aldehydes and pyridines were found for sample obtained from empty palm fruit bunch, while alcohols and cyclopentanediones were found in sample prepared from pine wood chips indicating different composition profiles due to the biomass sources. The elucidation of the composition of empty fruit bunch and pine wood chips bio-oils indicates that these oils are suitable for the production of value-added chemicals. The
6. A numerical theory of lattice gas and lattice Boltzmann methods in the computation of solutions to nonlinear advective-diffusive systems
Energy Technology Data Exchange (ETDEWEB)
Elton, A.B.H.
1990-09-24
A numerical theory for the massively parallel lattice gas and lattice Boltzmann methods for computing solutions to nonlinear advective-diffusive systems is introduced. The convergence theory is based on consistency and stability arguments that are supported by the discrete Chapman-Enskog expansion (for consistency) and conditions of monotonicity (in establishing stability). The theory is applied to four lattice methods: Two of the methods are for some two-dimensional nonlinear diffusion equations. One of the methods is for the one-dimensional lattice method for the one-dimensional viscous Burgers equation. And one of the methods is for a two-dimensional nonlinear advection-diffusion equation. Convergence is formally proven in the L{sub 1}-norm for the first three methods, revealing that they are second-order, conservative, conditionally monotone finite difference methods. Computational results which support the theory for lattice methods are presented. In addition, a domain decomposition strategy using mesh refinement techniques is presented for lattice gas and lattice Boltzmann methods. The strategy allows concentration of computational resources on regions of high activity. Computational evidence is reported for the strategy applied to the lattice gas method for the one-dimensional viscous Burgers equation. 72 refs., 19 figs., 28 tabs.
7. Smart multi-channel two-dimensional micro-gas chromatography for rapid workplace hazardous volatile organic compounds measurement.
Science.gov (United States)
Liu, Jing; Seo, Jung Hwan; Li, Yubo; Chen, Di; Kurabayashi, Katsuo; Fan, Xudong
2013-03-07
We developed a novel smart multi-channel two-dimensional (2-D) micro-gas chromatography (μGC) architecture that shows promise to significantly improve 2-D μGC performance. In the smart μGC design, a non-destructive on-column gas detector and a flow routing system are installed between the first dimensional separation column and multiple second dimensional separation columns. The effluent from the first dimensional column is monitored in real-time and decision is then made to route the effluent to one of the second dimensional columns for further separation. As compared to the conventional 2-D μGC, the greatest benefit of the smart multi-channel 2-D μGC architecture is the enhanced separation capability of the second dimensional column and hence the overall 2-D GC performance. All the second dimensional columns are independent of each other, and their coating, length, flow rate and temperature can be customized for best separation results. In particular, there is no more constraint on the upper limit of the second dimensional column length and separation time in our architecture. Such flexibility is critical when long second dimensional separation is needed for optimal gas analysis. In addition, the smart μGC is advantageous in terms of elimination of the power intensive thermal modulator, higher peak amplitude enhancement, simplified 2-D chromatogram re-construction and potential scalability to higher dimensional separation. In this paper, we first constructed a complete smart 1 × 2 channel 2-D μGC system, along with an algorithm for automated control/operation of the system. We then characterized and optimized this μGC system, and finally employed it in two important applications that highlight its uniqueness and advantages, i.e., analysis of 31 workplace hazardous volatile organic compounds, and rapid detection and identification of target gas analytes from interference background.
8. Lipidic ionic liquid stationary phases for the separation of aliphatic hydrocarbons by comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Nan, He; Zhang, Cheng; O'Brien, Richard A; Benchea, Adela; Davis, James H; Anderson, Jared L
2017-01-20
Lipidic ionic liquids (ILs) possessing long alkyl chains as well as low melting points have the potential to provide unique selectivity as well as wide operating ranges when used as stationary phases in gas chromatography. In this study, a total of eleven lipidic ILs containing various structural features (i.e., double bonds, linear thioether chains, and cyclopropanyl groups) were examined as stationary phases in comprehensive two dimensional gas chromatography (GC×GC) for the separation of nonpolar analytes in kerosene. N-alkyl-N'-methyl-imidazolium-based ILs containing different alkyl side chains were used as model structures to investigate the effects of alkyl moieties with different structural features on the selectivities and operating temperature ranges of the IL-based stationary phases. Compared to a homologous series of ILs containing saturated side chains, lipidic ILs exhibit improved selectivity toward the aliphatic hydrocarbons in kerosene. The palmitoleyl IL provided the highest selectivity compared to all other lipidic ILs as well as the commercial SUPELCOWAX 10 column. The linoleyl IL containing two double bonds within the alkyl side chain showed the lowest chromatographic selectivity. The lipidic IL possessing a cyclopropanyl group within the alkyl moiety exhibited the highest thermal stability. The Abraham solvation parameter model was used to evaluate the solvation properties of the lipidic ILs. This study provides the first comprehensive examination into the relation between lipidic IL structure and the resulting solvation characteristics. Furthermore, these results establish a basis for applying lipidic ILs as stationary phases for solute specific separations in GC×GC.
9. Using comprehensive two-dimensional gas chromatography to explore the geochemistry of the Santa Barbara oil seeps
Energy Technology Data Exchange (ETDEWEB)
Reddy, Christopher; Nelson, Robert
2013-03-27
The development of comprehensive two-dimensional gas chromatography (GC x GC) has expanded the analytical window for studying complex mixtures like oil. Compared to traditional gas chromatography, this technology separates and resolves at least an order of magnitude more compounds, has a much larger signal to noise ratio, and sorts compounds based on their chemical class; hence, providing highly refined inventories of petroleum hydrocarbons in geochemical samples that was previously unattainable. In addition to the increased resolution afforded by GC x GC, the resulting chromatograms have been used to estimate the liquid vapor pressures, aqueous solubilities, octanol-water partition coefficients, and vaporization enthalpies of petroleum hydrocarbons. With these relationships, powerful and incisive analyses of phase-transfer processes affecting petroleum hydrocarbon mixtures in the environment are available. For example, GC x GC retention data has been used to quantitatively deconvolve the effects of phase transfer processes such as water washing and evaporation. In short, the positive attributes of GC x GC-analysis have led to a methodology that has revolutionized the analysis of petroleum hydrocarbons. Overall, this research has opened numerous fields of study on the biogeochemical "genetics" (referred to as petroleomics) of petroleum samples in both subsurface and surface environments. Furthermore, these new findings have already been applied to the behavior of oil at other seeps as well, for petroleum exploration and oil spill studies.
10. Comprehensive two-dimensional gas chromatography coupled with fast sulphur-chemiluminescence detection: implications of detector electronics.
Science.gov (United States)
Blomberg, Jan; Riemersma, Toby; van Zuijlen, Manfred; Chaabani, Hassan
2004-09-24
Within the petrochemical industry, there has been a growing interest in methods capable of providing detailed information on the distribution of sulphur-containing compounds in various product streams, going down to the level of separating and quantifying individual sulphur species. Since no single capillary gas chromatographic column is able to perform this separation, a refuge to multi-dimensional separation techniques has to be taken. In this respect, comprehensive two-dimensional gas chromatography (GC x GC) coupled with sulphur chemiluminescence detection (SCD) has shown to be highly promising. It has been suggested, however, that the detector volume of an SCD restricts its potential to keep up with the fast second-dimension separations of contemporary GC x GC. In this paper, we will demonstrate that the lack of speed of the SCD does not originate from its physical dimensions, but is largely determined by the speed of the electronics used. Additionally, some typical examples will be presented to illustrate the potential of GC x GC coupled with fast SCD.
11. Two-dimensional time-dependent modelling of fume formation in a pulsed gas metal arc welding process
Science.gov (United States)
Boselli, M.; Colombo, V.; Ghedini, E.; Gherardi, M.; Sanibondi, P.
2013-06-01
Fume formation in a pulsed gas metal arc welding (GMAW) process is investigated by coupling a time-dependent axi-symmetric two-dimensional model, which takes into account both droplet detachment and production of metal vapour, with a model for fume formation and transport based on the method of moments for the solution of the aerosol general dynamic equation. We report simulative results of a pulsed process (peak current = 350 A, background current 30 A, period = 9 ms) for a 1 mm diameter iron wire, with Ar shielding gas. Results showed that metal vapour production occurs mainly at the wire tip, whereas fume formation is concentrated in the fringes of the arc in the spatial region close to the workpiece, where metal vapours are transported by convection. The proposed modelling approach allows time-dependent tracking of fumes also in plasma processes where temperature-time variations occur faster than nanoparticle transport from the nucleation region to the surrounding atmosphere, as is the case for most pulsed GMAW processes.
12. Analysis of special surfactants by comprehensive two-dimensional gas chromatography coupled to time-of-flight mass spectrometry.
Science.gov (United States)
Wulf, Volker; Wienand, Nils; Wirtz, Michaela; Kling, Hans-Willi; Gäb, Siegmar; Schmitz, Oliver J
2010-01-29
Multidimensional gas-chromatographic analyses of olesochemically based nonionic, anionic and several cationic surfactants in industrial cleaners are demonstrated. Comprehensive two-dimensional gas chromatography coupled with time-of-flight mass spectrometry allows the simultaneous determination of fatty alcohols, fatty alcohol sulphates and alkyl polyglucosides. In addition, the determination of fatty alcohol ethoxylates up to C(10)EO(8) (highest degree of ethoxylation) and C(18)EO(5) (longest C-chain at an ethoxylation degree of five) and the analysis of fatty alcohol alkoxylates that contain ethoxy (EO) and propoxy (PO) groups could be realized. Because of decomposition in the injector and a weak EI-fragmentation, cationic surfactants such as alkyl benzyl dimethyl ammonium chloride could also be identified by their characteristic fragments. Thermogravimetric analyses confirmed that the temperature in a normal GC injector is not high enough to cause thermal decomposition of esterquats. However, we could demonstrate that a modified silylation procedure forms decomposition products of esterquats in the GC injector which are detectable by GCxGC-(TOF)MS and allows the identification of such GC-atypical analytes.
13. Multisite Interactions in Lattice-Gas Models
Science.gov (United States)
Einstein, T. L.; Sathiyanarayanan, R.
For detailed applications of lattice-gas models to surface systems, multisite interactions often play at least as significant a role as interactions between pairs of adatoms that are separated by a few lattice spacings. We recall that trio (3-adatom, non-pairwise) interactions do not inevitably create phase boundary asymmetries about half coverage. We discuss a sophisticated application to an experimental system and describe refinements in extracting lattice-gas energies from calculations of total energies of several different ordered overlayers. We describe how lateral relaxations complicate matters when there is direct interaction between the adatoms, an issue that is important when examining the angular dependence of step line tensions. We discuss the connector model as an alternative viewpoint and close with a brief account of recent work on organic molecule overlayers.
14. Gas-Sensing Devices Based on Zn-Doped NiO Two-Dimensional Grainy Films with Fast Response and Recovery for Ammonia Molecule Detection.
Science.gov (United States)
Wang, Jian; Wei, Xiaowei; Wangyang, Peihua
2015-12-01
Zn-doped NiO two-dimensional grainy films on glass substrates are shown to be an ammonia-sensing material with excellent comprehensive performance, which could real-time detect and monitor ammonia (NH3) in the surrounding environment. The morphology and structure analysis indicated that the as-fabricated semiconductor films were composed of particles with diameters ranging from 80 to 160 nm, and each particle was composed of small crystalline grain with a narrow size about 20 nm, which was the face-centered cubic single crystal structure. X-ray diffraction peaks shifted toward lower angle, and the size of the lattice increased compared with undoped NiO, which demonstrated that zinc ions have been successfully doped into the NiO host structure. Simultaneously, we systematically investigated the gas-sensing properties of the Zn-doped NiO sensors for NH3 detection at room temperature. The sensor based on doped NiO sensing films gave four to nine times faster response and four to six times faster recovery speeds than those of sensor with undoped NiO films, which is important for the NiO sensor practical applications. Moreover, we found that the doped NiO sensors owned outstanding selectivity toward ammonia.
15. Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed: A hard-sphere approach.
NARCIS (Netherlands)
Hoomans, B.P.B.; Kuipers, J.A.M.; Briels, Willem J.; van Swaaij, Willibrordus Petrus Maria
1996-01-01
A discrete particle model of a gas-fluidised bed has been developed and in this the two-dimensional motion of the individual, spherical particles was directly calculated from the forces acting on them, accounting for the interaction between the particles and the interstitial gas phase. Our collision
16. Self-similarity of phase-space networks of frustrated spin models and lattice gas models
Science.gov (United States)
Peng, Yi; Wang, Feng; Han, Yilong
2013-03-01
We studied the self-similar properties of the phase-spaces of two frustrated spin models and two lattice gas models. The frustrated spin models included (1) the anti-ferromagnetic Ising model on a two-dimensional triangular lattice (1a) at the ground states and (1b) above the ground states and (2) the six-vertex model. The two lattice gas models were (3) the one-dimensional lattice gas model and (4) the two-dimensional lattice gas model. The phase spaces were mapped to networks so that the fractal analysis of complex networks could be applied, i.e. the box-covering method and the cluster-growth method. These phase spaces, in turn, establish new classes of networks with unique self-similar properties. Models 1a, 2, and 3 with long-range power-law correlations in real space exhibit fractal phase spaces, while models 1b and 4 with short-range exponential correlations in real space exhibit nonfractal phase spaces. This behavior agrees with one of untested assumptions in Tsallis nonextensive statistics. Hong Kong GRC grants 601208 and 601911
17. Diffusive description of lattice gas models
DEFF Research Database (Denmark)
Fiig, T.; Jensen, H.J.
1993-01-01
in time. We have numerically investigated the power spectrum of the density fluctuations, the lifetime distribution, and the spatial correlation function. We discuss the appropriate Langevin-like diffusion equation which can reproduce our numerical findings. Our conclusion is that the deterministic...... lattice gases are described by a diffusion equation without any bulk noise. The open lattice gas exhibits a crossover behavior as the probability for introducing particles at the edge of the system becomes small. The power spectrum changes from a 1/f to a 1/f2 spectrum. The diffusive description, proven...
18. Determination of disease biomarkers in Eucalyptus by comprehensive two-dimensional gas chromatography and multivariate data analysis.
Science.gov (United States)
Hantao, Leandro Wang; Aleme, Helga Gabriela; Passador, Martha Maria; Furtado, Edson Luiz; Ribeiro, Fabiana Alves de Lima; Poppi, Ronei Jesus; Augusto, Fabio
2013-03-01
In this paper is reported the use of the chromatographic profiles of volatiles to determine disease markers in plants - in this case, leaves of Eucalyptus globulus contaminated by the necrotroph fungus Teratosphaeria nubilosa. The volatile fraction was isolated by headspace solid phase microextraction (HS-SPME) and analyzed by comprehensive two-dimensional gas chromatography-fast quadrupole mass spectrometry (GC×GC-qMS). For the correlation between the metabolic profile described by the chromatograms and the presence of the infection, unfolded-partial least squares discriminant analysis (U-PLS-DA) with orthogonal signal correction (OSC) were employed. The proposed method was checked to be independent of factors such as the age of the harvested plants. The manipulation of the mathematical model obtained also resulted in graphic representations similar to real chromatograms, which allowed the tentative identification of more than 40 compounds potentially useful as disease biomarkers for this plant/pathogen pair. The proposed methodology can be considered as highly reliable, since the diagnosis is based on the whole chromatographic profile rather than in the detection of a single analyte.
19. Mineral oil in human tissues, part II: characterization of the accumulated hydrocarbons by comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Biedermann, Maurus; Barp, Laura; Kornauth, Christoph; Würger, Tanja; Rudas, Margaretha; Reiner, Angelika; Concin, Nicole; Grob, Koni
2015-02-15
Mineral oil hydrocarbons are by far the largest contaminant in the human body. Their composition differs from that in the mineral oils humans are exposed to, and varies also between different tissues of the same individual. Using the presently best technique for characterizing the composition of mineral oil hydrocarbons, comprehensive two-dimensional gas chromatography (GC×GC), the hydrocarbons in human tissues were compared to those of various mineral oils. This provided information about the strongly accumulated species and might give hints on the flow path through the human body. The selectivity of accumulation is probably also of interest for the risk assessment of synthetic hydrocarbons (polyolefins). GC×GC grouped the MOSH into classes of n-alkanes, paraffins with a low degree of branching, multibranched paraffins and naphthenes (alkylated cyclic hydrocarbons) with 1-4 rings. Metabolic elimination was observed for constituents of all these classes, but was selective within each class. The MOSH in the subcutaneous abdominal fat tissues and the mesenteric lymph nodes (MLN) had almost the same composition and included the distinct signals observed in mineral oil, though in reduced amounts relative to the cloud of unresolved hydrocarbons. The MOSH in the liver and the spleen were different from those in the MLN and fat tissue, but again with largely identical composition for a given individual. Virtually all constituents forming distinct signals were eliminated, leaving an unresolved residue of highly isomerized hydrocarbons.
20. The low-temperature mobility of two-dimensional electron gas in AlGaN/GaN heterostructures
Institute of Scientific and Technical Information of China (English)
Zhang Jin-Feng; Mao Wei; Zhang Jin-Cheng; Hao Yue
2008-01-01
To reveal the internal physics of the low-temperature mobility of two-dimensional electron gas (2DEG) in AlGaN/GaN heterostructures, we present a theoretical study of the strong dependence of 2DEG mobility on Al content and thickness of AlGaN barrier layer. The theoretical results are compared with one of the highest measured of 2DEG mobility reported for AlGaN/GaN heterostructures. The 2DEG mobility is modelled as a combined effect of the scattering mechanisms including acoustic deformation-potential, piezoelectric, ionized background donor, surface donor, dislocation, alloy disorder and interface roughness scattering. The analyses of the individual scattering processes show that the dominant scattering mechanisms are the alloy disorder scattering and the interface roughness scattering at low temperatures. The variation of 2DEG mobility with the barrier layer parameters results mainly from the change of 2DEG density and distribution. It is suggested that in AlGaN/GaN samples with a high Al content or a thick AlGaN layer, the interface roughness scattering may restrict the 2DEG mobility significantly, for the AlGan/GaN interface roughness increases due to the stress accumulation in AlGaN layer.
1. The mobility of two-dimensional electron gas in AlGaN/GaN heterostructures with varied Al content
Institute of Scientific and Technical Information of China (English)
ZHANG JinFeng; HAO Yue; ZHANG JinCheng; NI JinYu
2008-01-01
The mobility of the two-dimensional electron gas (2DEG) in AlGaN/GaN hetero-structures changes significantly with AI content in the AlGaN barrier layer, while few mechanism analyses focus on it. Theoretical calculation and analysis of the 2DEG mobility in AlGaN/GaN heterostructures with varied Al content are carried out based on the recently reported experimental data. The 2DEG mobility is modeled analytically as the total effects of the scattering mechanisms including acoustic deformation-potential, piezoelectric, polar optic phonon, alloy disorder, interface roughness, dislocation and remote modulation doping scattering. We show that the increase of the 2DEG density, caused by the ascension of the Al content in the barrier layer, is a dominant factor that leads to the changes of the individual scat-tering processes. The change of the 2DEG mobility with Al content are mainly de-termined by the interface roughness scattering and the alloy disorder scattering at 77 K, and the polar optic phonon scattering and the interface roughness scattering at the room temperature. The calculated function of the interface roughness pa-rameters on the Al content shows that the stress caused AlGaN/GaN interface degradation at higher Al content is an important factor in the limitation of the in-terface roughness scattering on the 2DEG mobility in AlGaN/GaN heterostructures with high Al content.
2. Characterization of the Clostridium difficile volatile metabolome using comprehensive two-dimensional gas chromatography time-of-flight mass spectrometry.
Science.gov (United States)
Rees, Christiaan A; Shen, Aimee; Hill, Jane E
2016-12-15
Clostridium difficile is a bacterial pathogen capable of causing life-threatening infections of the gastrointestinal tract characterized by severe diarrhea. Exposure to certain classes of antibiotics, advanced age, and prolonged hospitalizations are known risk factors for infection by this organism. Anecdotally, healthcare providers have reported that they can smell C. difficile infections in their patients, and several studies have suggested that there may indeed be an olfactory signal associated with C. difficile-associated diarrhea. In this study, we sought to characterize the volatile molecules produced by an epidemic strain of C. difficile (R20291) using headspace solid-phase microextraction (HS-SPME) followed by two-dimensional gas chromatography time-of-flight mass spectrometry (GC×GC-TOFMS). We report on a set of 77 volatile compounds, of which 59 have not previously been associated with C. difficile growth in vitro. Amongst these reported compounds, we detect both straight-chain and branched-chain carboxylic acids, as well as p-cresol, which have been the primary foci of C. difficile volatile metabolomic studies to-date. We additionally report on novel sulfur-containing and carbonyl-containing molecules that have not previously been reported for C. difficile. With the identification of these novel C. difficile-associated volatile compounds, we demonstrate the superior resolution and sensitivity of GC×GC-TOFMS relative to traditional GC-MS. Copyright © 2016 Elsevier B.V. All rights reserved.
3. Tuning the conductivity threshold and carrier density of two-dimensional electron gas at oxide interfaces through interface engineering
Directory of Open Access Journals (Sweden)
H. J. Harsan Ma
2015-08-01
Full Text Available The two-dimensional electron gas (2DEG formed at the perovskite oxides heterostructures is of great interest because of its potential applications in oxides electronics and nanoscale multifunctional devices. A canonical example is the 2DEG at the interface between a polar oxide LaAlO3 (LAO and non-polar SrTiO3 (STO. Here, the LAO polar oxide can be regarded as the modulating or doping layer and is expected to define the electronic properties of 2DEG at the LAO/STO interface. However, to practically implement the 2DEG in electronics and device design, desired properties such as tunable 2D carrier density are necessary. Here, we report the tuning of conductivity threshold, carrier density and electronic properties of 2DEG in LAO/STO heterostructures by insertion of a La0.5Sr0.5TiO3 (LSTO layer of varying thicknesses, and thus modulating the amount of polarization of the oxide over layers. Our experimental result shows an enhancement of carrier density up to a value of about five times higher than that observed at the LAO/STO interface. A complete thickness dependent metal-insulator phase diagram is obtained by varying the thickness of LAO and LSTO providing an estimate for the critical thickness needed for the metallic phase. The observations are discussed in terms of electronic reconstruction induced by polar oxides.
4. Two-Dimensional Electron Gas at SrTiO3-Based Oxide Heterostructures via Atomic Layer Deposition
Directory of Open Access Journals (Sweden)
Sang Woon Lee
2016-01-01
Full Text Available Two-dimensional electron gas (2DEG at an oxide interface has been attracting considerable attention for physics research and nanoelectronic applications. Early studies reported the formation of 2DEG at semiconductor interfaces (e.g., AlGaAs/GaAs heterostructures with interesting electrical properties such as high electron mobility. Besides 2DEG formation at semiconductor junctions, 2DEG was realized at the interface of an oxide heterostructure such as the LaAlO3/SrTiO3 (LAO/STO heterojunction. The origin of 2DEG was attributed to the well-known “polar catastrophe” mechanism in oxide heterostructures, which consist of an epitaxial LAO layer on a single crystalline STO substrate among proposed mechanisms. Recently, it was reported that the creation of 2DEG was achieved using the atomic layer deposition (ALD technique, which opens new functionality of ALD in emerging nanoelectronics. This review is focused on the origin of 2DEG at oxide heterostructures using the ALD process. In particular, it addresses the origin of 2DEG at oxide interfaces based on an alternative mechanism (i.e., oxygen vacancies.
5. A comparison of the transport properties of bilayer graphene,monolayer graphene, and two-dimensional electron gas
Institute of Scientific and Technical Information of China (English)
Sun Li-Feng; Dong Li-Min; Wu Zhi-Fang; Fang Chao
2013-01-01
we studied and compared the transport properties of charge carriers in bilayer graphene,monolayer graphene,and the conventional semiconductors (the two-dimensional electron gas (2DEG)).It is elucidated that the normal incidence transmission in the bilayer graphene is identical to that in the 2DEG but totally different from that in the monolayer graphene.However,resonant peaks appear in the non-normal incidence transmission profile for a high barrier in the bilayer graphene,which do not occur in the 2DEG.Furthermore,there are tunneling and forbidden regions in the transmission spectrum for each material,and the division of the two regions has been given in the work.The tunneling region covers a wide range of the incident energy for the two graphene systems,but only exists under specific conditions for the 2DEG.The counterparts of the transmission in the conductance profile are also given for the three materials,which may be used as high-performance devices based on the bilayer graphene.
6. Characterization of incense smoke by solid phase microextraction—Comprehensive two-dimensional gas chromatography (GC×GC)
Science.gov (United States)
Tran, Tin C.; Marriott, Philip J.
Comprehensive two-dimensional gas chromatography in tandem with flame ionization detection (GC×GC-FID) was used for the qualitative fingerprint characterisation of four different types of powdered incense headspace (H/S), and incense smoke. Volatile organic compounds (VOCs) in the incense powder and smoke were extracted by using solid phase microextraction (SPME) with a polydimethylsiloxane/divinylbenzene (PDMS/DVB) 65 μm fiber. Low-polarity/polar, and polar/non-polar phase combinations were tested to contrast the GC×GC separation of components in these two column sets. A total of 324 compounds were tentatively identified, with more than 100 compounds in incense powders and more than 200 compounds in the incense smoke, by using GC coupled to quadrupole mass spectrometric detection. Identification required at least 90% match with the NIST library; otherwise they were considered as unidentified. The smoke stream comprised compounds originating from the incense powder, and combustion products such as PAH, N-heterocyclics, and furans. However, GC×GC was able to separate many more volatile compounds (possibly hundreds more) present in the complex smoke samples, many of which cannot be separated by conventional 1D-GC; this is a direct consequence of the high-resolution power of GC×GC. GC×GC fingerprint comparison of powder H/S with smoke allows facile subtraction of the former from the latter to assist identification of compounds generated from burning incense.
7. Andreev reflection and bound state formation in a ballistic two-dimensional electron gas probed by a quantum point contact
Science.gov (United States)
Irie, Hiroshi; Todt, Clemens; Kumada, Norio; Harada, Yuichi; Sugiyama, Hiroki; Akazaki, Tatsushi; Muraki, Koji
2016-10-01
We study coherent transport and bound state formation of Bogoliubov quasiparticles in a high-mobility I n0.75G a0.25As two-dimensional electron gas (2DEG) coupled to a superconducting Nb electrode by means of a quantum point contact (QPC) as a tunable single-mode probe. Below the superconducting critical temperature of Nb, the QPC shows a single-channel conductance greater than the conductance quantum 2 e2/h at zero bias, which indicates the presence of Andreev-reflected quasiparticles, time-reversed states of the injected electron, returning back through the QPC. The marked sensitivity of the conductance enhancement to voltage bias and perpendicular magnetic field suggests a mechanism analogous to reflectionless tunneling—a hallmark of phase-coherent transport, with the boundary of the 2DEG cavity playing the role of scatterers. When the QPC transmission is reduced to the tunneling regime, the differential conductance vs bias voltage probes the single-particle density of states in the proximity area. Measured conductance spectra show a double peak within the superconducting gap of Nb, demonstrating the formation of Andreev bound states in the 2DEG. Both of these results, obtained in the open and closed geometries, underpin the coherent nature of quasiparticles, i.e., phase-coherent Andreev reflection at the InGaAs/Nb interface and coherent propagation in the ballistic 2DEG.
8. Durability-enhanced two-dimensional hole gas of C-H diamond surface for complementary power inverter applications
Science.gov (United States)
Kawarada, Hiroshi; Yamada, Tetsuya; Xu, Dechen; Tsuboi, Hidetoshi; Kitabayashi, Yuya; Matsumura, Daisuke; Shibata, Masanobu; Kudo, Takuya; Inaba, Masafumi; Hiraiwa, Atsushi
2017-02-01
Complementary power field effect transistors (FETs) based on wide bandgap materials not only provide high-voltage switching capability with the reduction of on-resistance and switching losses, but also enable a smart inverter system by the dramatic simplification of external circuits. However, p-channel power FETs with equivalent performance to those of n-channel FETs are not obtained in any wide bandgap material other than diamond. Here we show that a breakdown voltage of more than 1600 V has been obtained in a diamond metal-oxide-semiconductor (MOS) FET with a p-channel based on a two-dimensional hole gas (2DHG). Atomic layer deposited (ALD) Al2O3 induces the 2DHG ubiquitously on a hydrogen-terminated (C-H) diamond surface and also acts as both gate insulator and passivation layer. The high voltage performance is equivalent to that of state-of-the-art SiC planar n-channel FETs and AlGaN/GaN FETs. The drain current density in the on-state is also comparable to that of these two FETs with similar device size and VB.
9. Attempt to unravel the composition of toxaphene by comprehensive two-dimensional gas chromatography with selective detection.
Science.gov (United States)
Korytár, P; van Stee, L L P; Leonards, P E G; de Boer, J; Brinkman, U A Th
2003-04-25
Comprehensive two-dimensional gas chromatography (GC x GC) coupled with micro electron-capture and time-of-flight mass spectrometric (TOF-MS) detection has been used to analyse technical toxaphene. An HP-1 x HT-8 column combination yielded highly structured chromatograms and revealed a complex mixture of over 1000 compounds what is significantly higher number than in any study before. The analysis of a mixture of 23 individual congeners and TOF-MS evaluation of technical toxaphene showed that the chromatogram is structured according to the number of chlorine substituents in a molecule. The nature of the compounds (bornane and camphene) does not appear to have any influence. The sum of the peak areas of all congeners in each group was calculated using laboratory-written software; based on these results, the composition of technical toxaphene as a function of the number of chlorine substituents was provisionally calculated and was found that hepta- and octachlorinated compounds represents 75% of the total toxaphene area.
10. Screened test-charge - test-charge interaction in the two-dimensional electron gas: bound states
Science.gov (United States)
Gold, A.; Ghazali, A.
1997-08-01
We study the test-charge - test-charge interaction when screening effects of a two-dimensional electron gas are taken into account. The Schrödinger equation is solved in the momentum space by diagonalizing the corresponding matrix and the results are compared with variational calculations. For two positive (or negative) test-charges bound states are obtained for low electron densities when many-body effects are incorporated in the screening function. For a density larger than a critical density, 0953-8984/9/32/011/img5 (0953-8984/9/32/011/img6 is the Wigner - Seitz parameter), no bound states are found. Below the critical density, 0953-8984/9/32/011/img7, the number of bound states and their energy increase with decreasing density and the ground-state binding energy saturates near 0953-8984/9/32/011/img8. Finite-width effects for quantum wells are also discussed. We present new results for bound states between a positive and a negative test charge and we discuss effects of exchange and correlation on the binding energies.
11. Quantification of real thermal, catalytic, and hydrodeoxygenated bio-oils via comprehensive two-dimensional gas chromatography with mass spectrometry.
Science.gov (United States)
Silva, Raquel V S; Tessarolo, Nathalia S; Pereira, Vinícius B; Ximenes, Vitor L; Mendes, Fábio L; de Almeida, Marlon B B; Azevedo, Débora A
2017-03-01
The elucidation of bio-oil composition is important to evaluate the processes of biomass conversion and its upgrading, and to suggest the proper use for each sample. Comprehensive two-dimensional gas chromatography with time-of-flight mass spectrometry (GC×GC-TOFMS) is a widely applied analytical approach for bio-oil investigation due to the higher separation and resolution capacity from this technique. This work addresses the issue of analytical performance to assess the comprehensive characterization of real bio-oil samples via GC×GC-TOFMS. The approach was applied to the individual quantification of compounds of real thermal (PWT), catalytic process (CPO), and hydrodeoxygenation process (HDO) bio-oils. Quantification was performed with reliability using the analytical curves of oxygenated and hydrocarbon standards as well as the deuterated internal standards. The limit of quantification was set at 1ngµL(-1) for major standards, except for hexanoic acid, which was set at 5ngµL(-1). The GC×GC-TOFMS method provided good precision (bio-oil samples. Sugars, furans, and alcohols appear as the major constituents of the PWT, CPO, and HDO samples, respectively. In order to obtain bio-oils with better quality, the catalytic pyrolysis process may be a better option than hydrogenation due to the effective reduction of oxygenated compound concentrations and the lower cost of the process, when hydrogen is not required to promote deoxygenation in the catalytic pyrolysis process.
12. Durability-enhanced two-dimensional hole gas of C-H diamond surface for complementary power inverter applications
Science.gov (United States)
Kawarada, Hiroshi; Yamada, Tetsuya; Xu, Dechen; Tsuboi, Hidetoshi; Kitabayashi, Yuya; Matsumura, Daisuke; Shibata, Masanobu; Kudo, Takuya; Inaba, Masafumi; Hiraiwa, Atsushi
2017-01-01
Complementary power field effect transistors (FETs) based on wide bandgap materials not only provide high-voltage switching capability with the reduction of on-resistance and switching losses, but also enable a smart inverter system by the dramatic simplification of external circuits. However, p-channel power FETs with equivalent performance to those of n-channel FETs are not obtained in any wide bandgap material other than diamond. Here we show that a breakdown voltage of more than 1600 V has been obtained in a diamond metal-oxide-semiconductor (MOS) FET with a p-channel based on a two-dimensional hole gas (2DHG). Atomic layer deposited (ALD) Al2O3 induces the 2DHG ubiquitously on a hydrogen-terminated (C-H) diamond surface and also acts as both gate insulator and passivation layer. The high voltage performance is equivalent to that of state-of-the-art SiC planar n-channel FETs and AlGaN/GaN FETs. The drain current density in the on-state is also comparable to that of these two FETs with similar device size and VB. PMID:28218234
13. High-temperature two-dimensional gas chromatography of hydrocarbons up to nC60 for analysis of vacuum gas oils.
Science.gov (United States)
Dutriez, Thomas; Courtiade, Marion; Thiébaut, Didier; Dulot, Hugues; Bertoncini, Fabrice; Vial, Jérôme; Hennion, Marie-Claire
2009-04-03
In a tense energetic context, the characterization of heavy petroleum fractions becomes essential. Conventional comprehensive two-dimensional gas chromatography (2D-GC or GCxGC) is widely used for middle distillates analysis, but only a few applications are devoted to these heavier fractions. In this paper, it is shown how the optimization of GCxGC separation allowed the determination of suitable high-temperature (HT) conditions, adjusting column properties and operating conditions. 2D separations were evaluated using 2D separation criteria and a new concept of 2D asymmetry (As(2D)). New HT conditions allowed the extension of GCxGC range of applications to heavier hydrocarbons, up to nC(60). A first application of high-temperature two-dimensional gas chromatography (HT-2D-GC) to a full vacuum gas oil (VGO) feed stock is described. Comparisons with other standardized methods illustrate the high potential of HT-2D-GC for heavy fractions analysis.
14. Lattice gas hydrodynamics: Theory and simulations
Energy Technology Data Exchange (ETDEWEB)
Hasslacher, B.
1993-01-01
The first successful application of a microscopic analogy to create a skeleton cellular automaton and analyze it with statistical mechanical tools, was the work of Frisch, Hasslacher and Pomeau on the Navier-Stokes equation in two and three dimensions. This has become a very large research area with lattice gas models and methods being used for both fundamental investigations into the foundations of statistical mechanics and a large number of diverse applications. This present research was devoted to enlarging the fundamental scope of lattice gas models and proved quite successful. Since the beginning of this proposal, cellular automata have been constructed for statistical mechanical models, fluids, diffusion and shock systems in fundamental investigations. In applied areas, there are now excellent lattice gas models for complex flows through porous media, chemical reaction and combustion dynamics, multiphase flow systems, and fluid mixtures with natural boundaries. With extended cellular fluid models, one can do problems with arbitrary pairwise potentials. Recently, these have been applied to such problems as non-newtonian or polymeric liquids and a mixture of immiscible fluids passing through fractal or spongelike media in two and three dimensions. This proposal has contributed to and enlarged the scope of this work.
15. Detection of an extended human volatome with comprehensive two-dimensional gas chromatography time-of-flight mass spectrometry.
Directory of Open Access Journals (Sweden)
Michael Phillips
Full Text Available BACKGROUND: Comprehensive two-dimensional gas chromatography coupled with time-of-flight mass spectrometry (GCxGC-TOF MS has been proposed as a powerful new tool for multidimensional analysis of complex chemical mixtures. We investigated GCxGC-TOF MS as a new method for identifying volatile organic compounds (VOCs in normal human breath. METHODS: Samples of alveolar breath VOCs and ambient room air VOC were collected with a breath collection apparatus (BCA onto separate sorbent traps from 34 normal healthy volunteers (mean age = 40 yr, SD = 17 yr, male/female = 19/15. VOCs were separated on two serial capillary columns separated by a cryogenic modulator, and detected with TOF MS. The first and second dimension columns were non-polar and polar respectively. RESULTS: BCA collection combined with GC×GC-TOF MS analysis identified approximately 2000 different VOCs in samples of human breath, many of which have not been previously reported. The 50 VOCs with the highest alveolar gradients (abundance in breath minus abundance in ambient room air mostly comprised benzene derivatives, acetone, methylated derivatives of alkanes, and isoprene. CONCLUSIONS: Collection and analysis of breath VOCs with the BCA-GC×GC-TOF MS system extended the size of the detectable human volatile metabolome, the volatome, by an order of magnitude compared to previous reports employing one-dimensional GC-MS. The size of the human volatome has been under-estimated in the past due to coelution of VOCs in one-dimensional GC analytical systems.
16. Negative differential conductivity induced current instability in two-dimensional electron gas system in high magnetic fields
Science.gov (United States)
Lee, Ching-Ping; Komiyama, Susumu; Chen, Jeng-Chung
2015-03-01
High mobility two-dimensional electron gas (2DEG) formed in the interface of a GaAs/AlGaAs hetero-structure in high magnetic field (B) exhibits interring nonlinear response either under microwave radiation or to a dc electric field (E). It is general believed that this kind nonlinear behavior is closely related to the occurrence of negative-differential conductance (NDC) in the presence of strong B and E. We observe a new type NDC state driven by a direct current above a threshold value (Ith) applied to a 2DEG as a function of B at relatively high temperatures (T). A current instability is observed in 2DEG system at high B ~6-8 T and at high T ~ 20- 30 K while the applied current is over Ith. The longitudinal voltage Vxx shows sub-linear behavior with the increase of I. As the current exceed Ith, Vxx suddenly drops a ΔVxx and becomes irregular associated with the appearance of hysteresis with sweeping I. We find that Ith increases with the increase of B and of T; meanwhile, ΔVxx is larger at higher B but lower T. Data analysis suggest that the onset of voltage fluctuation can be described by a NDC model proposed by Kurosawa et al. in 1976. The general behaviors of T and B dependence of current instability are analog to those recently reported at lower both T and B. This consistence suggests the same genuine mechanism of NDC phenomena observed in 2DEG system.
17. Quantitative analysis of essential oils in perfume using multivariate curve resolution combined with comprehensive two-dimensional gas chromatography.
Science.gov (United States)
de Godoy, Luiz Antonio Fonseca; Hantao, Leandro Wang; Pedroso, Marcio Pozzobon; Poppi, Ronei Jesus; Augusto, Fabio
2011-08-05
The use of multivariate curve resolution (MCR) to build multivariate quantitative models using data obtained from comprehensive two-dimensional gas chromatography with flame ionization detection (GC×GC-FID) is presented and evaluated. The MCR algorithm presents some important features, such as second order advantage and the recovery of the instrumental response for each pure component after optimization by an alternating least squares (ALS) procedure. A model to quantify the essential oil of rosemary was built using a calibration set containing only known concentrations of the essential oil and cereal alcohol as solvent. A calibration curve correlating the concentration of the essential oil of rosemary and the instrumental response obtained from the MCR-ALS algorithm was obtained, and this calibration model was applied to predict the concentration of the oil in complex samples (mixtures of the essential oil, pineapple essence and commercial perfume). The values of the root mean square error of prediction (RMSEP) and of the root mean square error of the percentage deviation (RMSPD) obtained were 0.4% (v/v) and 7.2%, respectively. Additionally, a second model was built and used to evaluate the accuracy of the method. A model to quantify the essential oil of lemon grass was built and its concentration was predicted in the validation set and real perfume samples. The RMSEP and RMSPD obtained were 0.5% (v/v) and 6.9%, respectively, and the concentration of the essential oil of lemon grass in perfume agreed to the value informed by the manufacturer. The result indicates that the MCR algorithm is adequate to resolve the target chromatogram from the complex sample and to build multivariate models of GC×GC-FID data. Copyright © 2011 Elsevier B.V. All rights reserved.
18. High-Throughput Computational Design of Advanced Functional Materials: Topological Insulators and Two-Dimensional Electron Gas Systems
Science.gov (United States)
Yang, Kesong
As a rapidly growing area of materials science, high-throughput (HT) computational materials design is playing a crucial role in accelerating the discovery and development of novel functional materials. In this presentation, I will first introduce the strategy of HT computational materials design, and take the HT discovery of topological insulators (TIs) as a practical example to show the usage of such an approach. Topological insulators are one of the most studied classes of novel materials because of their great potential for applications ranging from spintronics to quantum computers. Here I will show that, by defining a reliable and accessible descriptor, which represents the topological robustness or feasibility of the candidate, and by searching the quantum materials repository aflowlib.org, we have automatically discovered 28 TIs (some of them already known) in five different symmetry families. Next, I will talk about our recent research work on the HT computational design of the perovskite-based two-dimensional electron gas (2DEG) systems. The 2DEG formed on the perovskite oxide heterostructure (HS) has potential applications in next-generation nanoelectronic devices. In order to achieve practical implementation of the 2DEG in the device design, desired physical properties such as high charge carrier density and mobility are necessary. Here I show that, using the same strategy with the HT discovery of TIs, by introducing a series of combinatorial descriptors, we have successfully identified a series of candidate 2DEG systems based on the perovskite oxides. This work provides another exemplar of applying HT computational design approach for the discovery of advanced functional materials.
19. Enantiomeric separation and quantification of ephedrine-type alkaloids in herbal materials by comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Wang, Min; Marriott, Philip J; Chan, Wing-Hong; Lee, Albert W M; Huie, Carmen W
2006-04-21
The separation of ephedrine-type alkaloids and their enantiomers in raw herbs and commercial herbal products was investigated by carrying out enantioselective separation in the first-dimension column (containing beta-cyclodextrin as the chiral selector) of a comprehensive two-dimensional gas chromatography (GC x GC) system, whereas a polar polyethylene glycol capillary column was used for separation in the second dimension. Naturally occurring ephedrine-type alkaloids and their synthetic analogues (enantiomeric counterparts) were adequately resolved from each other, as well as from potential interference species in the sample matrix using GC x GC, whereas single column GC analysis was unable to separate all the alkaloids of interest. Detection limits in the order of 0.1-1.3 microg/mL and linearity of calibration with R(2)>or=0.999 over approximately the range of 0.5-100 microg/mL for the quantitative determination of various ephedrine-type alkaloids were obtained. The commercial herbal products tested contained mostly (-)-ephedrine, (+)-pseudoephedrine, (-)-N-methylephedrine and (-)-norephedrine, with concentrations in the range of 40-2100, 0-1,300, 15-300 and 0-30 microg/g of the product, respectively, and repeatability of analysis was generally in the range of 1-5%. The present GCxGC method is effective and useful for the determination of the dosage levels of the principle ephedrine-type alkaloids in commercial health supplements and complex raw herb formulations, as well the differentiation of ephedrine-containing products that were derived from natural plant or synthetic sources, e.g., simply by visualizing the presence or absence of the enantiomeric pairs of (+/-) ephedrine and (+/-)-N-methylephedrine in the GC x GC chromatograms.
20. Hydrocarbon Specificity During Aerobic oil Biodegradation Revealed in Marine Microcosms With the use of Comprehensive, Two-Dimensional Gas Chromatography.
Science.gov (United States)
Wardlaw, G. D.; Reddy, C. M.; Nelson, R. K.; Valentine, D. L.
2008-12-01
In 2003 the National Research Council reported more than 380 million gallons of oil is emitted into the ocean each year from natural seepage and as a result of anthropogenic activities. Many of the hydrocarbons making up this oil are persistent and toxic to marine life. Petroleum emitted into biologically sensitive areas can lead to environmental stress and ecosystem collapse. As a result many studies and a substantial amount of resources have been devoted to creating efficient and effective remediation tools and developing a better understanding of natural hydrocarbon weathering processes occurring in marine environments. The goal of this study is to elucidate patterns and extent of aerobic hydrocarbon degradation in marine sediments. In order to assess the specific molecular transformations occurring in petroleum emitted into oxic marine environments, we prepared microcosm experiments using sediments and seawater collected from the natural oil seeps offshore Coal Oil Point, California. Petroleum recovered from Platform Holly in the Santa Barbara Channel, was added to a sediment-seawater mixture and the microcosm bottles were allowed to incubate under aerobic conditions for slightly more than 100 days. Comprehensive, two-dimensional gas chromatography was employed in this study to quantify changes in the concentrations of individual hydrocarbon compounds because of the increased resolution and resolving power provided with this robust analytical method. We show significant hydrocarbon mass loss due to aerobic biodegradation for hundreds of tracked compounds in the microcosm bottles. The results shown here provide quantitative evidence for broad-scale metabolic specificity during aerobic hydrocarbon degradation in surface and shallow subsurface marine sediments.
1. A data acquisition system for two-dimensional position sensitive micropattern gas detectors with delay-line readout
Energy Technology Data Exchange (ETDEWEB)
Hanu, A.R., E-mail: [email protected] [Department of Medical Physics and Applied Radiation Sciences, McMaster University, Hamilton, Ontario, Canada L8S 4K1 (Canada); NASA Goddard Space Flight Center, Greenbelt, MD 20771 (United States); Prestwich, W.V.; Byun, S.H. [Department of Medical Physics and Applied Radiation Sciences, McMaster University, Hamilton, Ontario, Canada L8S 4K1 (Canada)
2015-04-21
We present a data acquisition (DAQ) system for two-dimensional position sensitive micropattern gas detectors using the delay-line method for readout. The DAQ system consists of a field programmable gate array (FPGA) as the main data processor and our time-to-digital (TDC) mezzanine card for making time measurements. We developed the TDC mezzanine card around the Acam TDC-GPX ASIC and it features four independent stop channels referenced to a common start, a typical timing resolution of ~81 ps, and a 17-bit measurement range, and is compliant with the VITA 57.1 standard. For our DAQ system, we have chosen the Xilinx SP601 development kit which features a single Spartan 6 FPGA, 128 MB of DDR2 memory, and a serial USB interface for communication. Output images consist of 1024×1024 square pixels, where each pixel has a 32-bit depth and corresponds to a time difference of 162 ps relative to its neighbours. When configured for a 250 ns acquisition window, the DAQ can resolve periodic event rates up to 1.8×10{sup 6} Hz without any loses and will report a maximum event rate of 6.11×10{sup 5} Hz for events whose arrival times follow Poisson statistics. The integral and differential non-linearities have also been measured and are better than 0.1% and 1.5%, respectively. Unlike commercial units, our DAQ system implements the delay-line image reconstruction algorithm entirely in hardware and is particularly attractive for its modularity, low cost, ease of integration, excellent linearity, and high throughput rate.
2. Analysis of sex pheromone gland content of individual Symmetrischema tangolias by means of direct gland introduction into a two-dimensional gas chromatograph
NARCIS (Netherlands)
Griepink, F.C.; Drijfhout, F.P.; Beek, van T.A.; Visser, H.J.; Groot, de C.P.G.M.
2000-01-01
The amounts and ratios of the four constituents of the sex pheromone gland of the moth Symmetrischema tangolias were measured during a 24-hr dark–light cycle. A new approach was followed that involved the direct introduction of sex pheromone glands into the liner of a two-dimensional gas chromatogra
3. Characterisation of volatile components of Pinotage wines using comprehensive two-dimensional gas chromatography coupled to time-of-flight mass spectrometry (GC x GC–TOFMS)
NARCIS (Netherlands)
Weldegergis, B.T.; Villiers, de A.; McNeish, C.; Seethapathy, S.; Mostafa, A.; Górecki, T.; Crouch, A.M.
2011-01-01
As part of the ongoing research into the chemical composition of the uniquely South African wine cultivar Pinotage, the volatile composition of nine young wines of this cultivar was investigated using comprehensive two-dimensional gas chromatography (GC × GC) in combination with time-of-flight mass
4. Characterisation of volatile components of Pinotage wines using comprehensive two-dimensional gas chromatography coupled to time-of-flight mass spectrometry (GC x GC–TOFMS)
NARCIS (Netherlands)
Weldegergis, B.T.; Villiers, de A.; McNeish, C.; Seethapathy, S.; Mostafa, A.; Górecki, T.; Crouch, A.M.
2011-01-01
As part of the ongoing research into the chemical composition of the uniquely South African wine cultivar Pinotage, the volatile composition of nine young wines of this cultivar was investigated using comprehensive two-dimensional gas chromatography (GC × GC) in combination with time-of-flight mass
5. Effects of anisotropy and magnetic fields on the specific heat of a quasi-two-dimensional Boltzmann gas in an elliptical parabolic quantum dot
Institute of Scientific and Technical Information of China (English)
Zhai Zhi-Yuan; Li Yu-Qi; Pan Xiao-Yin
2012-01-01
We investigate the effects due to anisotropy and magnetic field interaction for a quasi-two-dimensional Boltzmann gas in an elliptical parabolic quantum dot.The specific heat is studied with varying temperature,anisotropy,and magnetic field strength.The cases without and with the inclusion of the spin Zeeman interaction are considered.
6. Zigzag antiferromagnetic ground state with anisotropic correlation lengths in the quasi-two-dimensional honeycomb lattice compound N a2C o2Te O6
Science.gov (United States)
Bera, A. K.; Yusuf, S. M.; Kumar, Amit; Ritter, C.
2017-03-01
The crystal structure, magnetic ground state, and the temperature-dependent microscopic spin-spin correlations of the frustrated honeycomb lattice antiferromagnet N a2C o2Te O6 have been investigated by powder neutron diffraction. A long-range antiferromagnetic (AFM) ordering has been found below TN˜24.8 K . The magnetic ground state, determined to be zigzag antiferromagnetic and characterized by a propagation vector k =(1 /2 0 0 ) , occurs due to the competing exchange interactions up to third-nearest neighbors within the honeycomb lattice. The exceptional existence of a limited magnetic correlation length along the c axis (perpendicular to the honeycomb layers in the a b planes) has been found even at 1.8 K, well below the TN˜24.8 K . The observed limited correlation along the c axis is explained by the disorder distribution of the Na ions within the intermediate layers between honeycomb planes. The reduced ordered moments mCo (1 )=2.77 (3 ) μB/C o2 + and mCo (2 )=2.45 (2 ) μB/C o2 + at 1.8 K reflect the persistence of spin fluctuations in the ordered state. Above TN˜24.8 K , the presence of short-range magnetic correlations, manifested by broad diffuse magnetic peaks in the diffraction patterns, has been found. Reverse Monte Carlo analysis of the experimental diffuse magnetic scattering data reveals that the spin correlations are mainly confined within the two-dimensional honeycomb layers (a b plane) with a correlation length of ˜12 Å at 25 K. The nature of the spin arrangements is found to be similar in both the short-range and long-range ordered magnetic states. This implies that the short-range correlation grows with decreasing temperature and leads to the zigzag AFM ordering at T ≤TN . The present study provides a comprehensive picture of the magnetic correlations over the temperature range above and below the TN and their relation to the crystal structure. The role of intermediate soft Na layers on the magnetic coupling between honeycomb planes is
7. AlGaAs/GaAs two-dimensional electron gas structures studied by photo reflectance spectroscopy
Energy Technology Data Exchange (ETDEWEB)
Guillen Cervantes, A; Rivera Alvarez, Z; Hernandez, F; Huerta, J. [Instituto Politecnico Nacional, Mexico, D.F. (Mexico); Mendez Garcia, V. H.; Lastras Martinez, A.; Zamora, L.; Saucedo, N. [Universidad Autonoma de San Luis Potosi, San Luis Potosi (Mexico); Melendez Lira, M; Lopez, M [Instituto Politecnico Nacional, Mexico, D.F. (Mexico)
2001-12-01
Al{sub x} Ga{sub 1}-x As/GaAs two-dimensional electron gas (2-DEG) heterostructures were fabricated by molecular beam epitaxy in three different laboratories. The samples were characterized by room temperature Photo reflectance (PR) spectroscopy and measurements at 77 K. Internal electric fields were detected by the presence of Franz-Keldysh (FK) oscillations in the PR spectra. >From a FK analysis we obtained the GaAs band-gap energy and the built-in electric field strength in each sample. On the other hand, in the energy region corresponding to Al{sub x} Ga{sub 1}-x As a broad PR signal was registered typical of a highly doped material. Using the third derivative theory we obtained the Al{sub x} Ga{sub 1}-x As band-gap energy, and from this value the Al concentration in the samples. Results showed that the sample with highest electron mobility exhibited the lowest internal electric field strength. [Spanish] Se fabricaron heteroestructuras del tipo Al{sub x} Ga{sub 1}-x As/GaAs con un gas de electrones en dos dimensiones por medio de epitaxia de haces moleculares en tres laboratorios diferentes. Las muestras se caracterizaron por fotorreflectancia (FR) a temperatura ambiente y por mediciones del efecto mayor a 77 K. Campos electricos internos se detectaron por la presencia de oscilaciones Franz-Leldysh (FK) en los espectros de FR. Del analisis de las oscilaciones FK obtuvimos la energia de la brecha prohibida del GaAs y la intensidad del campo electrico interno en cada muestra. Por otra parte, en la region de energia correspondiente al Al{sub x} Ga{sub 1}-x As observamos una senal de FR ancha, tipica de un material altamente impurificado. Usando la teoria de la tercera derivada, obtuvimos el valor de la brecha de energia del Al{sub x}Ga{sub 1}-xAs, y de este valor la concentracion de Al en las muestras. Los resultados mostraron que la estructura con el valor de movilidad electronica mas alto tiene la intensidad de campo electrico interno mas baja.
8. Improved electrical properties of the two-dimensional electron gas in AlGaN/GaN heterostructures using high temperature AlN interlayers
Institute of Scientific and Technical Information of China (English)
2010-01-01
The electrical properties of two-dimensional electron gas (2DEG) in AlGaN/GaN heterostructures using high temperature (HT) AlN interlayers (ITs) grown on c-plane sapphire substrate by metal organic chemical vapor deposition (MOCVD) have been investigated.It is found that the electrical properties (electron mobility and sheet carrier density) are improved compared with those in the conventional AlGaN/GaN heterostructures without HT AlN ITs,and the improved 2DEG properties result in the reduction of the sheet resistance.The results from high resolution X-ray diffraction (HRXRD) and Raman spectroscopy measurements show that HT AlN ITs increase the in-plane compressive strain in the upper GaN layer,which enhances the piezoelectric polarization in it and consequently causes increasing of 2DEG density at the AlGaN/GaN interface.Meanwhile,the compressive strain induced by HT AlN ITs leads to a less tensile strain in AlGaN barrier layer and causes positive and negative effects on the sheet carrier density of 2DEG,which counteract each other.The HT AlN ITs reduce the lattice mismatch between the GaN and AlGaN layers and smooth the interface between them,thus increasing the electric mobility of 2DEG by weakening the alloy-related interface roughness and scattering.In addition,the surface morphology of AlGaN/GaN heterostructures is improved by the insertion of HT AlN ITs.The reason for the improved properties is discussed in this paper.
9. The high density phase of the k-NN hard core lattice gas model
Science.gov (United States)
Nath, Trisha; Rajesh, R.
2016-07-01
The k-NN hard core lattice gas model on a square lattice, in which the first k next nearest neighbor sites of a particle are excluded from being occupied by another particle, is the lattice version of the hard disc model in two dimensional continuum. It has been conjectured that the lattice model, like its continuum counterpart, will show multiple entropy-driven transitions with increasing density if the high density phase has columnar or striped order. Here, we determine the nature of the phase at full packing for k up to 820 302 . We show that there are only eighteen values of k, all less than k = 4134, that show columnar order, while the others show solid-like sublattice order.
10. Organophosphorus pesticide and ester analysis by using comprehensive two-dimensional gas chromatography with flame photometric detection
Energy Technology Data Exchange (ETDEWEB)
Liu, Xiangping; Li, Dengkun; Li, Jiequan [Nanjing Centre for Disease Control and Prevention, Zizhulin Street, Gulou 210003, Nanjing (China); Rose, Gavin [Department of Environment and Primary Industries, Macleod Centre, Ernest Jones Drive, Macleod, Vic 3085 (Australia); Marriott, Philip J., E-mail: [email protected] [Australian Centre for Research on Separation Science, School of Chemistry, Monash University, Wellington Road, Clayton 3800 (Australia)
2013-12-15
Highlights: • GC × GC-FPD(P-mode) was applied to detection of 37 phosphorus (P)-containing compounds. • The method improves resolution of P-compounds that coelute in the first dimension. • P-compounds are analyzed with excellent sensitivity supported by cryogenic modulation. • The FPD(P-mode) selectivity allows analysis in high hydrocarbon (H/C) matrix. • Soil samples and spiked chemical weapon compounds in H/C matrix are readily screened. -- Abstract: Thirty-seven phosphorus (P)-containing compounds comprising organophosphorus pesticides and organophosphate esters were analyzed by using comprehensive two-dimensional gas chromatography with flame photometric detection in P mode (GC × GC-FPD(P)), with a non-polar/moderately polar column set. A suitable modulation temperature and period was chosen based on experimental observation. A number of co-eluting peak pairs on the {sup 1}D column were well separated in 2D space. Excellent FPD(P) detection selectivity, responding to compounds containing the P atom, produces clear 2D GC × GC plots with little interference from complex hydrocarbon matrices. Limits of detection (LOD) were within the range of 0.0021–0.048 μmol L{sup −1}, and linear calibration correlation coefficients (R{sup 2}) for all 37 P-compounds were at least 0.998. The P-compounds were spiked in 2% diesel and good reproducibility for their response areas and retention times was obtained. Spiked recoveries were 88%–157% for 5 μg L{sup −1} and 80%–138% for 10 μg L{sup −1} spiked levels. Both {sup 1}t{sub R} and {sup 2}t{sub R} shifts were noted when the content of diesel was in excess of 5% in the matrix. Soil samples were analyzed by using the developed method; some P-compounds were positively detected. In general, this study shows that GC × GC-FPD(P) is an accurate, sensitive and simple method for P-compound analysis in complicated environmental samples.
11. Multidimensional gas chromatography for the detailed PIONA analysis of heavy naphtha: hyphenation of an olefin trap to comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Vendeuvre, Colombe; Bertoncini, Fabrice; Espinat, Didier; Thiébaut, Didier; Hennion, Marie-Claire
2005-10-07
A multidimensional method providing the composition of a heavy naphtha in paraffins, isoparaffins, olefins, naphthenes, and aromatics (PIONA) in the C8-C14 range is presented. The analytical set-up consists in a silver modified silica olefin trap on-line coupled to comprehensive two-dimensional gas chromatography (GC x GC). In this configuration, hydrocarbons are separated, in gaseous state, in two fractions, saturate and unsaturate, each fraction being subsequently analysed by GC or by GC x GC. The resolution between saturates and olefins was found to be improved compared to a single GC x GC run. The characterisation of the olefin trap highlights the benefits and the limits related to the use of that stationary phase as a double bond selective fractionation medium.
12. Lattice-Gas Automata for the Problem Of Kinetic Theory of Gas During Free Expansion
Science.gov (United States)
Khotimah, Siti Nurul; Arif, Idam; Liong, The Houw
The lattice-gas method has been applied to solve the problem of kinetic theory of gas in the Gay-Lussac-Joule experiment. Numerical experiments for a two-dimensional gas were carried out to determine the number of molecules in one vessel (Nr), the ratio between the mean square values of the components of molecule velocity (/line{vx2}//line{v_y^2}), and the change in internal energy (ΔU) as a function of time during free expansion. These experiments were repeated for different sizes of an aperture in the partition between the two vessels. After puncturing the partition, the curve for the particle number in one vessel shows a damped oscillation for about half of the total number. The oscillations do not vanish after a sampling over different initial configurations. The system is in nonequilibrium due to the pressure equilibration, and here the flow is actually compressible. The equilibration time (in time steps) decreases with decreased size of aperture in the partition. For very small apertures (equal or less than 9{√{3}}/{2} lattice units), the number of molecules in one vessel changes with time in a smooth way until it reaches half of the total number; their curves obey the analytical solution for quasi-static processes. The calculations on /line{vx2}//line{v_y^2} and ΔU also support the results that the equilibration time decreases with decreased size of aperture in the partition.
13. Vacuum ultraviolet absorption spectroscopy in combination with comprehensive two-dimensional gas chromatography for the monitoring of volatile organic compounds in breath gas: A feasibility study.
Science.gov (United States)
Gruber, Beate; Groeger, Thomas; Harrison, Dale; Zimmermann, Ralf
2016-09-16
Vacuum ultraviolet (VUV) absorption spectroscopy was recently introduced as a new detection system for one, as well as comprehensive two-dimensional gas chromatography (GC×GC) and successfully applied to the analysis of various analytes in several matrices. In this study, its suitability for the analysis of breath metabolites was investigated and the impact of a finite volume of the absorption cell and makeup gas pressure was evaluated for volatile analytes in terms of sensitivity and chromatographic resolution. A commercial available VUV absorption spectrometer was coupled to GC×GC and applied to the analysis of highly polar volatile organic compounds (VOCs). Breath gas samples were acquired by needle trap micro extraction (NTME) during a glucose challenge and analysed by the applied technique. Regarding qualitative and quantitative information, the VGA-100 is compatible with common GC×GC detection systems like FID and even TOFMS. Average peak widths of 300ms and LODs in the lower ng range were achieved using GC×GC-VUV. Especially small oxygenated breath metabolites show intense and characteristic absorption patterns in the VUV region. Challenge responsive VOCs could be identified and monitored during a glucose challenge. The new VUV detection technology might especially be of benefit for applications in clinical research.
14. Determination of diamondoids in crude oils using gas purge microsyringe extraction with comprehensive two dimensional gas chromatography-time-of-flight mass spectrometry.
Science.gov (United States)
Zhang, Wanfeng; Zhu, Shukui; Pang, Liling; Gao, Xuanbo; Zhu, Gang-Tian; Li, Donghao
2016-12-23
Based on a homemade device, gas purge microsyringe extraction (GP-MSE) of crude oil samples was developed for the first time. As a simple, fast, low-cost, sensitive and solvent-saving technique, GP-MSE provides some outstanding advantages over the widely used sample preparation methods for crude oils such as column chromatography (ASTM D2549). Several parameters affecting extraction efficiency were optimized, including extraction temperature, extraction time, extraction solvent, condensing temperature and purge gas flow rate. With the optimized GP-MSE conditions, several real crude oil samples were extracted, and trace diamondoids were determined using comprehensive two dimensional gas chromatography-time-of-flight mass spectrometry (GC×GC-TOFMS). In total, more than 100 diamondoids were detected and 27 marker compounds were identified and quantified accurately. The limits of detection (LODs, S/N=3) were less than 0.08μg/L for all diamondoids. The relative standard deviation (RSD) was below 8%, ranging from 1.1 to 7.6%. The linearity of the developed method was in the range of 0.5-100.0μg/L with correlation coefficients (R(2)) more than 0.996. The recoveries obtained at spiking 50μg/L were between 81 and 108% for diamondoids in crude oil samples. The developed method can also be extended to the analysis of other components in crude oils and other complex matrices.
15. Terahertz signal detection in a short gate length field-effect transistor with a two-dimensional electron gas
Energy Technology Data Exchange (ETDEWEB)
Vostokov, N. V., E-mail: [email protected]; Shashkin, V. I. [Institute for Physics of Microstructures of the Russian Academy of Sciences, 603950 Nizhny Novgorod, Russia and N. I. Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod (Russian Federation)
2015-11-28
We consider the problem of non-resonant detection of terahertz signals in a short gate length field-effect transistor having a two-dimensional electron channel with zero external bias between the source and the drain. The channel resistance, gate-channel capacitance, and quadratic nonlinearity parameter of the transistor during detection as a function of the gate bias voltage are studied. Characteristics of detection of the transistor connected in an antenna with real impedance are analyzed. The consideration is based on both a simple one-dimensional model of the transistor and allowance for the two-dimensional distribution of the electric field in the transistor structure. The results given by the different models are discussed.
16. Measurements and analysis of Hall effect of a two dimensional electron gas in the close proximity of a superconducting YBa2Cu3O(7 - x) film
Science.gov (United States)
Tseng, M. Z.; Jiang, W. N.; Hu, E. L.
1994-09-01
A direct integration of YBa2Cu3O(7 - x) and a two dimensional electron gas Hall probe was made possible through the use of a MgO buffer layer. We demonstrate the use of this structure for the measurements of the magnetization hysteresis of a superconducting YBa2Cu3O(7 - x) thin film, and we make an estimate of the sensitivity and resolution that can be achieved with this probe structure. The close proximity of the YBa2Cu3O(7 - x) to the two dimensional electron gas (approximately 1700 A) allows sensitive measurements of interactions between the two; more importantly, closer superconductor-semiconductor spacing can be achieved without severe compromise of the component material quality.
17. Comparison of column phase configurations for comprehensive two dimensional gas chromatographic analysis of crude oil and bitumen
Energy Technology Data Exchange (ETDEWEB)
Tran, T.C.; Harynuk, J.; Marriott, P. [RMIT University, Melbourne (Australia). Dept. of Applied Chemistry; Logan, G.A.; Grosjean, E. [Geoscience Australia, Canberra (Australia); Ryan, D. [Charles Sturt University, Wagga Wagga (Australia). School of Science and Technology
2006-09-15
An inverted phase (polar to non-polar) column set has been compared with a non-polar to polar column set for the GC x GC separation of petroleum hydrocarbons. This column configuration is shown to provide greatly enhanced resolution for less polar compounds and makes greater use of the two dimensional separation space. It improves resolution of a greater number of components within one analysis and offers new possibilities for crude oil fingerprinting. (Author)
18. [Characterization of compounds in crude oils by gas purge micro-syringe extraction coupled to comprehensive two-dimensional gas chromatography].
Science.gov (United States)
Tong, Ting; Zhang, Wanfeng; Li, Donghao; Zhao, Jinhua; Chang, Zhenyang; Gao, Xuanbo; Dai, Wei; He, Sheng; Zhu, Shukui
2014-10-01
A novel sample pretreatment method, gas purge micro-syringe extraction (GP- MSE), coupled to comprehensive two-dimensional gas chromatography/time-of-flight mass spectrometry (GC x GC/TOFMS) has been developed for the characterization of volatile and semi-volatile compounds in crude oils. In the sample pretreatment process, the analytes were carried to the microsyringe barrel by inert gas, and at the same time, trapped by an organic solvent. The whole process of extraction takes less than 10 min, and only 20 μL of organic solvent was needed. Using two custom standard solutions containing alkanes and polycyclic aromatic hydrocarbons (PAHs), the influences of the extraction conditions were investigated. The optimized conditions were as follows: 5 mg crude oil, 20 μL hexane (extraction solvent), extraction for 3 min at 300 °C, condensation temperature set at -2 °C, gas flow rate set at 2 mL/min. Under the optimized conditions, a real crude oil sample was extracted and then analyzed in detail. It showed that the proposed method was very effective in simultaneously analyzing the normal and branched alkanes, cycloalkanes, aromatic hydrocarbons, and biomarkers of crude oil such as steranes and terpanes. The recoveries obtained ranged from 82.0% to 107.3% and the detection limits ranged from 34 to 93 μg/L. The correlation coefficients (R2) were more than 0.99. The relative standard deviations (RSDs, n = 5) for all the analytes were below 10%. The results indicate that the proposed method is suitable for the characterization of volatile and semi-volatile compounds in crude oils with easy operation, high sensitivity and efficiency.
19. Elucidation of the aroma compositions of Zhenjiang aromatic vinegar using comprehensive two dimensional gas chromatography coupled to time-of-flight mass spectrometry and gas chromatography-olfactometry.
Science.gov (United States)
Zhou, Zhilei; Liu, Shuangping; Kong, Xiangwei; Ji, Zhongwei; Han, Xiao; Wu, Jianfeng; Mao, Jian
2017-03-03
In this work, a method to characterize the aroma compounds of Zhenjiang aromatic vinegar (ZAV) was developed using comprehensive two dimensional gas chromatography (GC×GC) coupled with time-of-flight mass spectrometry (TOFMS) and gas chromatography olfactometry (GC-O). The column combination was optimized and good separation was achieved. Structured chromatograms of furans and pyrazines were obtained and discussed. A total of 360 compounds were tentatively identified based on mass spectrum match factors, structured chromatogram and linear retention indices comparison. The most abundant class in number was ketones. A large number of esters, furans and derivatives, aldehydes and alcohols were also detected. The odor-active components were identified by comparison of the reported odor of the identified compounds with the odor of corresponding GC-O region. The odorants of methanethiol, 2-methyl-propanal, 2-methyl-butanal/3-methyl-butanal, octanal, 1-octen-3-one, dimethyl trisulfide, trimethyl-pyrazine, acetic acid, 3-(methylthio)-propanal, furfural, benzeneacetaldehyde, 3-methyl-butanoic acid/2-methyl-butanoic acid and phenethyl acetate were suspected to be the most potent. About half of them were identified as significant aroma constituents in ZAV for the first time. Their contribution to specific sensory attribute of ZAJ was also studied. The results indicated that the presented method is suitable for characterization of ZAV aroma constituents. This study also enriches our knowledge on the components and aroma of ZAV.
20. Chemical characterization of aromatic compounds in extra heavy gas oil by comprehensive two-dimensional gas chromatography coupled to time-of-flight mass spectrometry.
Science.gov (United States)
Avila, Bárbara M F; Pereira, Ricardo; Gomes, Alexandre O; Azevedo, Débora A
2011-05-27
Comprehensive two-dimensional gas chromatography coupled to time-of-flight mass spectrometry (GC×GC-TOFMS) was used for the characterization of aromatic compounds present in extra heavy gas oil (EHGO) from Brazil. Individual identification of EHGO compounds was successfully achieved in addition to group-type separation on the chromatographic plane. Many aromatic hydrocarbons, especially polycyclic aromatic hydrocarbons and sulfur compounds, were detected and identified, such as chrysenes, phenanthrenes, perylenes, benzonaphthothiophenes and alkylbenzonaphthothiophenes. In addition, triaromatic steroids, methyl-triaromatic steroids, tetrahydrochrysenes and tetraromatic pentacyclic compounds were present in the EHGO aromatic fractions. Considering the roof-tile effect observed for many of these compound classes and the high number of individual compounds identified, GC×GC-TOFMS is an excellent technique to characterize the molecular composition of the aromatic fraction from EHGO samples. Moreover, data processing allowed the quantification of aromatic compounds, in class and individually, using external standards. EHGO data were obtained in μgg(-1), e.g., benzo[a]pyrene were in the range 351 to 1164μgg(-1). Thus, GC×GC-TOFMS was successfully applied in EHGO quantitative analysis.
1. Determination of aromatic sulphur compounds in heavy gas oil by using (low-)flow modulated comprehensive two-dimensional gas chromatography-triple quadrupole mass spectrometry.
Science.gov (United States)
Franchina, Flavio Antonio; Machado, Maria Elisabete; Tranchida, Peter Quinto; Zini, Cláudia Alcaraz; Caramão, Elina Bastos; Mondello, Luigi
2015-03-27
The present research is focused on the development of a flow-modulated comprehensive two-dimensional gas chromatography-triple quadrupole mass spectrometry (FM GC × GC-MS/MS) method for the determination of classes of aromatic organic sulphur compounds (benzothiophenes, dibenzothiophenes, and benzonaphthothiophene) in heavy gas oil (HGO). The MS/MS instrument was used to provide both full-scan and multiple-reaction-monitoring (MRM) data. Linear retention index (LRI) ranges were used to define the MRM windows for each chemical class. Calibration solutions (internal standard: 1-fluoronaphthalene) were prepared by using an HGO sample, depleted of S compounds. Calibration information was also derived for the thiophene class (along with MRM and LRI data), even though such constituents were not present in the HGO. Linearity was satisfactory over the analyzed concentration range (1-100 mg/L); intra-day precision for the lowest calibration point was always below 17%. Accuracy was also satisfactory, with a maximum percentage error of 3.5% (absolute value) found among the S classes subjected to (semi-)quantification. The highest limit of quantification was calculated to be 299 μg/L (for the C1-benzothiophene class), while the lowest was 21 μg/L (for the C4-benzothiophene class).
2. Ultra resolution chemical fingerprinting of dense non-aqueous phase liquids from manufactured gas plants by reversed phase comprehensive two-dimensional gas chromatography.
Science.gov (United States)
McGregor, Laura A; Gauchotte-Lindsay, Caroline; Daéid, Niamh Nic; Thomas, Russell; Daly, Paddy; Kalin, Robert M
2011-07-22
Ultra resolution chemical fingerprinting of dense non-aqueous phase liquids (DNAPLs) from former manufactured gas plants (FMGPs) was investigated using comprehensive two-dimensional gas chromatography coupled with time of flight mass spectrometry (GC×GC TOFMS). Reversed phase GC×GC (i.e. a polar primary column coupled to a non-polar secondary column) was found to significantly improve the separation of polycyclic aromatic hydrocarbons (PAHs) and their alkylated homologues. Sample extraction and cleanup was performed simultaneously using accelerated solvent extraction (ASE), with recovery rates between 76% and 97%, allowing fast, efficient extraction with minimal solvent consumption. Principal component analysis (PCA) of the GC×GC data was performed in an attempt to differentiate between twelve DNAPLs based on their chemical composition. Correlations were discovered between DNAPL composition and historic manufacturing processes used at different FMGP sites. Traditional chemical fingerprinting methods generally follow a tiered approach with sample analysis on several different instruments. We propose ultra resolution chemical fingerprinting as a fast, accurate and precise method of obtaining more chemical information than traditional tiered approaches while using only a single analytical technique.
3. Random-lattice models and simulation algorithms for the phase equilibria in two-dimensional condensed systems of particles with coupled internal and translational degrees of freedom
DEFF Research Database (Denmark)
Nielsen, Morten; Miao, Ling; Ipsen, John Hjorth;
1996-01-01
In this work we concentrate on phase equilibria in two-dimensional condensed systems of particles where both translational and internal degrees of freedom are present and coupled through microscopic interactions, with a focus on the manner of the macroscopic coupling between the two types...
4. Strongly interacting two-dimensional Dirac fermions
NARCIS (Netherlands)
Lim, L.K.; Lazarides, A.; Hemmerich, Andreas; de Morais Smith, C.
2009-01-01
We show how strongly interacting two-dimensional Dirac fermions can be realized with ultracold atoms in a two-dimensional optical square lattice with an experimentally realistic, inherent gauge field, which breaks time reversal and inversion symmetries. We find remarkable phenomena in a temperature
5. Terahertz optical-Hall effect characterization of two-dimensional electron gas properties in AlGaN/GaN high electron mobility transistor structures
Science.gov (United States)
Schöche, S.; Shi, Junxia; Boosalis, A.; Kühne, P.; Herzinger, C. M.; Woollam, J. A.; Schaff, W. J.; Eastman, L. F.; Schubert, M.; Hofmann, T.
2011-02-01
The free-charge carrier mobility, sheet density, and effective mass of a two-dimensional electron gas are exemplarily determined in the spectral range from 640 GHz to 1 THz in a AlGaN/GaN heterostructure using the optical-Hall effect at room temperature. Complementary midinfrared spectroscopic ellipsometry measurements are performed for analysis of heterostructure constituents layer thickness, phonon mode, and free-charge carrier parameters. The electron effective mass is determined to be (0.22±0.04)m0. The high-frequency sheet density and carrier mobility parameters are in good agreement with results from dc electrical Hall effect measurements, indicative for frequency-independent carrier scattering mechanisms of the two-dimensional carrier distribution.
6. Determination of scale-invariant equations of state without fitting parameters: application to the two-dimensional Bose gas across the Berezinskii-Kosterlitz-Thouless transition.
Science.gov (United States)
Desbuquois, Rémi; Yefsah, Tarik; Chomaz, Lauriane; Weitenberg, Christof; Corman, Laura; Nascimbène, Sylvain; Dalibard, Jean
2014-07-11
We present a general "fit-free" method for measuring the equation of state (EoS) of a scale-invariant gas. This method, which is inspired from the procedure introduced by Ku et al. [Science 335, 563 (2012)] for the unitary three-dimensional Fermi gas, provides a general formalism which can be readily applied to any quantum gas in a known trapping potential, in the frame of the local density approximation. We implement this method on a weakly interacting two-dimensional Bose gas across the Berezinskii-Kosterlitz-Thouless transition and determine its EoS with unprecedented accuracy in the critical region. Our measurements provide an important experimental benchmark for classical-field approaches which are believed to accurately describe quantum systems in the weakly interacting but nonperturbative regime.
7. Density-functional theory of a lattice-gas model with vapour, liquid, and solid phases
OpenAIRE
Prestipino, S.; Giaquinta, P. V.
2003-01-01
We use the classical version of the density-functional theory in the weighted-density approximation to build up the entire phase diagram and the interface structure of a two-dimensional lattice-gas model which is known, from previous studies, to possess three stable phases -- solid, liquid, and vapour. Following the common practice, the attractive part of the potential is treated in a mean-field-like fashion, although with different prescriptions for the solid and the fluid phases. It turns o...
8. A high-mobility two-dimensional electron gas at the spinel/perovskite interface of γ-Al2O3/SrTiO3
DEFF Research Database (Denmark)
Chen, Yunzhong; Bovet, N.; Trier, Felix
2013-01-01
The discovery of two-dimensional electron gases at the heterointerface between two insulating perovskite-type oxides, such as LaAlO3 and SrTiO3, provides opportunities for a new generation of all-oxide electronic devices. Key challenges remain for achieving interfacial electron mobilities much be...... confined within a layer of 0.9 nm in proximity to the interface. Our findings pave the way for studies of mesoscopic physics with complex oxides and design of high-mobility all-oxide electronic devices.......The discovery of two-dimensional electron gases at the heterointerface between two insulating perovskite-type oxides, such as LaAlO3 and SrTiO3, provides opportunities for a new generation of all-oxide electronic devices. Key challenges remain for achieving interfacial electron mobilities much...... beyond the current value of approximately 1,000 cm2V-1 s-1 (at low temperatures). Here we create a new type of two-dimensional electron gas at the heterointerface between SrTiO3 and a spinel g-Al2O3 epitaxial film with compatible oxygen ions sublattices. Electron mobilities more than one order...
9. Offline coupling of high-speed counter-current chromatography and gas chromatography/mass spectrometry generates a two-dimensional plot of toxaphene components.
Science.gov (United States)
Kapp, Thomas; Vetter, Walter
2009-11-20
High-speed counter-current chromatography (HSCCC), a separation technique based solely on the partitioning of solutes between two immiscible liquid phases, was applied for the fractionation of technical toxaphene, an organochlorine pesticide which consists of a complex mixture of structurally closely related compounds. A solvent system (n-hexane/methanol/water 34:24:1, v/v/v) was developed which allowed to separate compounds of technical toxaphene (CTTs) with excellent retention of the stationary phase (S(f) = 88%). Subsequent analysis of all HSCCC fractions by gas chromatography coupled to electron-capture negative ion mass spectrometry (GC/ECNI-MS) provided a wealth of information regarding separation characteristics of HSCCC and the composition of technical toxaphene. The visualization of the large amount of data obtained from the offline two-dimensional HSCCC-GC/ECNI-MS experiment was facilitated by the creation of a two-dimensional (2D) contour plot. The contour plot not only provided an excellent overview of the HSCCC separation progress, it also illustrated the differences in selectivity between HSCCC and GC. The results of this proof-of-concept study showed that the 2D chromatographic approach involving HSCCC facilitated the separation of CTTs that coelute in unidimensional GC. Furthermore, the creation of 2D contour plots may provide a useful means of enhancing data visualization for other offline two-dimensional separations.
10. Evolution and structure of the plasma of current sheets forming in two-dimensional magnetic fields with a null line at low initial gas ionization and their interpretation
Science.gov (United States)
Ostrovskaya, G. V.; Frank, A. G.
2012-04-01
An analysis of the experimental data obtained by holographic interferometry in our work [1] makes it possible to explain most of the observed specific features of the structure and evolution of the plasma sheets developing in a two-dimensional magnetic field with a null line in a plasma with a low initial degree of ionization (≈10-4). The following two processes are shown to play a key role here: additional gas ionization in an electric field and the peculiarities of plasma dynamics in a current sheet expanding in time.
11. Onset of quantum criticality in the topological-to-nematic transition in a two-dimensional electron gas at filling factor ν =5 /2
Science.gov (United States)
Schreiber, K. A.; Samkharadze, N.; Gardner, G. C.; Biswas, Rudro R.; Manfra, M. J.; Csáthy, G. A.
2017-07-01
Under hydrostatic pressure, the ground state of a two-dimensional electron gas at ν =5 /2 changes from a fractional quantum Hall state to the stripe phase. By measuring the energy gap of the fractional quantum Hall state and of the onset temperature of the stripe phase, we mapped out a phase diagram of these competing phases in the pressure-temperature plane. Our data highlight the dichotomy of two descriptions of the half-filled Landau level near the quantum critical point: one based on electrons and another on composite fermions.
12. Acoustic phonon-limited resistivity of spin-orbit coupled two-dimensional electron gas: the deformation potential and piezoelectric scattering.
Science.gov (United States)
Biswas, Tutul; Ghosh, Tarun Kanti
2013-01-23
We study the interaction between electron and acoustic phonons in a Rashba spin-orbit coupled two-dimensional electron gas using Boltzmann transport theory. Both the deformation potential and piezoelectric scattering mechanisms are considered in the Bloch-Grüneisen (BG) regime as well as in the equipartition (EP) regime. The effect of the Rashba spin-orbit interaction on the temperature dependence of the resistivity in the BG and EP regimes is discussed. We find that the effective exponent of the temperature dependence of the resistivity in the BG regime decreases due to spin-orbit coupling.
13. Enantioselective comprehensive two-dimensional gas chromatography. A route to elucidate the authenticity and origin of Rosa damascena Miller essential oils.
Science.gov (United States)
Krupčík, Ján; Gorovenko, Roman; Špánik, Ivan; Sandra, Pat; Armstrong, Daniel W
2015-10-01
The analysis of Bulgarian and Turkish Rosa damascena Miller essential oils was performed by flow-modulated comprehensive two-dimensional gas chromatography using simultaneous detection of the second column effluent by flame ionization and quadrupole mass spectrometric detection. Enantioselective separations were obtained by running the samples on 2,3-di-O-ethyl-6-O-tert-butyldimethylsilyl-β-cyclodextrin column as the first column and on polyethylene glycol as the second column. The determination of enantiomeric or diastereomeric excess of some terpenoic solutes is a possible route for quality or authenticity control as well as for the elucidation of the country of origin.
14. Suppression of the two-dimensional electron gas in LaGaO3/SrTiO3 by cation intermixing
KAUST Repository
Nazir, S.
2013-12-03
Cation intermixing at the n-type polar LaGaO 3 /SrTiO 3 (001) interface is investigated by first principles calculations. Ti"Ga, Sr"La, and SrTi"LaGa intermixing are studied in comparison to each other, with a focus on the interface stability. We demonstrate in which cases intermixing is energetically favorable as compared to a clean interface. A depopulation of the Ti 3d xy orbitals under cation intermixing is found, reflecting a complete suppression of the two-dimensional electron gas present at the clean interface.
15. Giant spin splitting of the two-dimensional electron gas at the surface of SrTiO3
Science.gov (United States)
Santander-Syro, A. F.; Fortuna, F.; Bareille, C.; Rödel, T. C.; Landolt, G.; Plumb, N. C.; Dil, J. H.; Radović, M.
2014-12-01
Two-dimensional electron gases (2DEGs) forming at the interfaces of transition metal oxides exhibit a range of properties, including tunable insulator-superconductor-metal transitions, large magnetoresistance, coexisting ferromagnetism and superconductivity, and a spin splitting of a few meV (refs , ). Strontium titanate (SrTiO3), the cornerstone of such oxide-based electronics, is a transparent, non-magnetic, wide-bandgap insulator in the bulk, and has recently been found to host a surface 2DEG (refs , , , ). The most strongly confined carriers within this 2DEG comprise two subbands, separated by an energy gap of 90 meV and forming concentric circular Fermi surfaces. Using spin- and angle-resolved photoemission spectroscopy (SARPES), we show that the electron spins in these subbands have opposite chiralities. Although the Rashba effect might be expected to give rise to such spin textures, the giant splitting of almost 100 meV at the Fermi level is far larger than anticipated. Moreover, in contrast to a simple Rashba system, the spin-polarized subbands are non-degenerate at the Brillouin zone centre. This degeneracy can be lifted by time-reversal symmetry breaking, implying the possible existence of magnetic order. These results show that confined electronic states at oxide surfaces can be endowed with novel, non-trivial properties that are both theoretically challenging to anticipate and promising for technological applications.
16. Absence of localization in disordered two-dimensional electron gas at weak magnetic field and strong spin-orbit coupling
Science.gov (United States)
Su, Ying; Wang, C.; Avishai, Y.; Meir, Yigal; Wang, X. R.
2016-09-01
The one-parameter scaling theory of localization predicts that all states in a disordered two-dimensional system with broken time reversal symmetry are localized even in the presence of strong spin-orbit coupling. While at constant strong magnetic fields this paradigm fails (recall the quantum Hall effect), it is believed to hold at weak magnetic fields. Here we explore the nature of quantum states at weak magnetic field and strongly fluctuating spin-orbit coupling, employing highly accurate numerical procedure based on level spacing distribution and transfer matrix technique combined with one parameter finite-size scaling hypothesis. Remarkably, the metallic phase, (known to exist at zero magnetic field), persists also at finite (albeit weak) magnetic fields, and eventually crosses over into a critical phase, which has already been confirmed at high magnetic fields. A schematic phase diagram drawn in the energy-magnetic field plane elucidates the occurrence of localized, metallic and critical phases. In addition, it is shown that nearest-level statistics is determined solely by the symmetry parameter β and follows the Wigner surmise irrespective of whether states are metallic or critical.
17. Conductivity of the two-dimensional electron gas at LaAlO3/SrTiO3 interface
Science.gov (United States)
Kirichenko, E. V.; Stephanovich, V. A.; Dugaev, V. K.
2017-02-01
We propose an analytical theory of metallic conductivity in the two-dimensional (2D) LaAlO3/SrTiO3 (LAO/STO) interface. For that we consider the electron-phonon interaction at the interface. The electronic part is taken from our previous work [Phys. Chem. Chem. Phys. 18, 2104 (2016), 10.1039/C5CP06627A], considering the conditions for the interfacial charge carrier (electron or hole) to become itinerant. The second ingredient deals with the atomic oscillations localized near the interface and decaying rapidly at its both sides, which can be regarded as 2D acoustic phonons. The dispersion of such phonons depends on the characteristics of phonon spectra of LAO and STO. Calculating the corresponding scattering rate by Fermi's golden rule, we show that the resulting resistivity (i.e., inverse conductivity) has typical metallic character, growing linearly with temperature and tending to zero (without defects forming so-called residual resistivity) at T →0 . The results of our calculations are in agreement with available experimental data.
18. Critical-like behavior in a lattice gas model
CERN Document Server
Wieloch, A; Lukasik, J; Pawlowski, P; Pietrzak, T; Trautmann, W
2010-01-01
ALADIN multifragmentation data show features characteristic of a critical behavior, which are very well reproduced by a bond percolation model. This suggests, in the context of the lattice gas model, that fragments are formed at nearly normal nuclear densities and temperatures corresponding to the Kertesz line. Calculations performed with a lattice gas model have shown that similarly good reproduction of the data can also be achieved at lower densities, particularly in the liquid-gas coexistence region.
19. Qualitative analysis of Copaifera oleoresin using comprehensive two-dimensional gas chromatography and gas chromatography with classical and cold electron ionisation mass spectrometry.
Science.gov (United States)
Wong, Yong Foo; Uekane, Thais M; Rezende, Claudia M; Bizzo, Humberto R; Marriott, Philip J
2016-12-16
Improved separation of both sesquiterpenes and diterpenic acids in Copaifera multijuga Hayne oleoresin, is demonstrated by using comprehensive two-dimensional gas chromatography (GC×GC) coupled to accurate mass time-of-flight mass spectrometry (accTOFMS). GC×GC separation employs polar phases (including ionic liquid phases) as the first dimension ((1)D) column, combined with a lower polarity (2)D phase. Elution temperatures (Te) of diterpenic acids (in methyl ester form, DAME) increased as the (1)D McReynolds' polarity value of the column phase decreased. Since Te of sesquiterpene hydrocarbons decreased with increased polarity, the very polar SLB-IL111 (1)D phase leads to excessive peak broadening in the (2)D apolar phase due to increased second dimension retention ((2)tR). The combination of SLB-IL59 with a nonpolar column phase was selected, providing reasonable separation and low Te for sesquiterpenes and DAME, compared to other tested column sets, without excessive (2)tR. Identities of DAME were aided by both soft (30eV) electron ionisation (EI) accurate mass TOFMS analysis and supersonic molecular beam ionisation (cold EI) TOFMS, both which providing less fragmentation and increased relative abundance of molecular ions. The inter-relation between EI energies, emission current, signal-to-noise and mass error for the accurate mass measurement of DAME are reported. These approaches can be used as a basis for conducting of GC×GC with soft EI accurate mass measurement of terpenes, particularly for unknown phytochemicals.
20. Detailed compositional characterization of plastic waste pyrolysis oil by comprehensive two-dimensional gas-chromatography coupled to multiple detectors.
Science.gov (United States)
Toraman, Hilal E; Dijkmans, Thomas; Djokic, Marko R; Van Geem, Kevin M; Marin, Guy B
2014-09-12
The detailed compositional characterization of plastic waste pyrolysis oil was performed with comprehensive two-dimensional GC (GC×GC) coupled to four different detectors: a flame ionization detector (FID), a sulfur chemiluminescence detector (SCD), a nitrogen chemiluminescence detector (NCD) and a time of flight mass spectrometer (TOF-MS). The performances of different column combinations were assessed in normal i.e. apolar/mid-polar and reversed configurations for the GC×GC-NCD and GC×GC-SCD analyses. The information obtained from the four detectors and the use of internal standards, i.e. 3-chlorothiophene for the FID and the SCD and 2-chloropyridine for the NCD analysis, enabled the identification and quantification of the pyrolysis oil in terms of both group type and carbon number: hydrocarbon groups (n-paraffins, iso-paraffins, olefins and naphthenes, monoaromatics, naphthenoaromatics, diaromatics, naphthenodiaromatics, triaromatics, naphthenotriaromatics and tetra-aromatics), nitrogen (nitriles, pyridines, quinolines, indole, caprolactam, etc.), sulfur (thiols/sulfides, thiophenes/disulfides, benzothiophenes, dibenzothiophenes, etc.) and oxygen containing compounds (ketones, phenols, aldehydes, ethers, etc.). Quantification of trace impurities is illustrated for indole and caprolactam. The analyzed pyrolysis oil included a significant amount of nitrogen containing compounds (6.4wt%) and to a lesser extent sulfur containing compounds (0.6wt%). These nitrogen and sulfur containing compounds described approximately 80% of the total peak volume for respectively the NCD and SCD analysis. TOF-MS indicated the presence of the oxygen containing compounds. However only a part of the oxygen containing compounds (2.5wt%) was identified because of their low concentrations and possible overlap with the complex hydrocarbon matrix as no selective detector or preparative separation for oxygen compounds was used.
1. Kinetics of diffuesion-controlled oxygen ordering in a lattic-gas model of YBa2Cu3O7-δ
DEFF Research Database (Denmark)
Andersen, Jørgen Vitting; Bohr, Henrik; Mouritsen, Ole G.
1990-01-01
Nonequilibrium properties of oxygen ordering in high-Tc superconductors of the Y-Ba-Cu-O type are studied via computer simulation of an anisotropic two-dimensional lattice-gas model in which the ordering processes are controlled by diffusion across the sample edges. With a view to designing optimal...
2. Exact diagonalization study of the spin-1 two-dimensional J{sub 1}–J{sub 3} Heisenberg model on a triangular lattice
Energy Technology Data Exchange (ETDEWEB)
Rubin, P., E-mail: [email protected]; Sherman, A.
2014-11-07
The spin-1 Heisenberg model on a triangular lattice with the ferromagnetic nearest-neighbor and antiferromagnetic third-nearest-neighbor exchange interactions, J{sub 1}=−(1−p)J and J{sub 2}=pJ, J>0(0≤p≤1), is studied with the use of the SPINPACK code. This model is applicable for the description of the magnetic properties of NiGa{sub 2}S{sub 4}. The ground, low-lying excited state energies and spin-spin correlation functions have been found for lattices with N=16 and N=20 sites with the periodic boundary conditions. These results are in qualitative agreement with earlier authors' results obtained with Mori's projection operator technique. - Highlights: • The S=1J{sub 1}–J{sub 3} Heisenberg model on a triangular lattice is studied. • The ferromagnetic nearest and AF 3rd-nearest-neighbor couplings are considered. • The exact diagonalization study of finite lattices was done. • The SPINPACK code using Lanczos' method is used for calculations. • The obtained results are in agreement with those obtained by Mori's approach.
3. Two-dimensional gas chromatography and trilinear partial least squares for the quantitative analysis of aromatic and naphthene content in naphtha.
Science.gov (United States)
Prazen, B J; Johnson, K J; Weber, A; Synovec, R E
2001-12-01
Quantitative analysis of naphtha samples is demonstrated using comprehensive two-dimensional gas chromatography (GC x GC) and chemometrics. This work is aimed at providing a GC system for the quantitative and qualitative analysis of complex process streams for process monitoring and control. The high-speed GC x GC analysis of naphtha is accomplished through short GC columns, high carrier gas velocities, and partial chromatographic peak resolution followed by multivariate quantitative analysis. Six min GC x GC separations are analyzed with trilinear partial least squares (tri-PLS) to predict the aromatic and naphthene (cycloalkanes) content of naphtha samples. The 6-min GC x GC separation time is over 16 times faster than a single-GC-column standard method in which a single-column separation resolves the aromatic and naphthene compounds in naphtha and predicts the aromatic and naphthene percent concentrations through addition of the resolved signals. Acceptable quantitative precision is provided by GC x GC/tri-PLS.
4. Identification and quantification of alkene-based drilling fluids in crude oils by comprehensive two-dimensional gas chromatography with flame ionization detection.
Science.gov (United States)
Reddy, Christopher M; Nelson, Robert K; Sylva, Sean P; Xu, Li; Peacock, Emily A; Raghuraman, Bhavani; Mullins, Oliver C
2007-04-27
Comprehensive two-dimensional gas chromatography with flame ionization detection (GC x GC-FID) was used to measure alkene-based drilling fluids in crude oils. Compared to one-dimensional gas chromatography, GC x GC-FID is more robust for detecting alkenes due to the increased resolution afforded by second dimension separations. Using GC x GC-FID to analyze four oil samples from one reservoir contaminated with the same drilling fluid, C(15), C(16), C(17), C(18) and C(20) alkenes were identified. The drilling fluid that contaminated these samples also differed from another commercially obtained fluid, which only contained C(16) and C(18) alkenes. These results should motivate the petroleum industry to consider GC x GC-FID for measuring drilling fluids.
5. Flow-modulated comprehensive two-dimensional gas chromatography combined with a vacuum ultraviolet detector for the analysis of complex mixtures.
Science.gov (United States)
Zoccali, Mariosimone; Schug, Kevin A; Walsh, Phillip; Smuts, Jonathan; Mondello, Luigi
2017-05-12
The present paper is focused on the use of a vacuum ultraviolet absorption spectrometer (VUV) for gas chromatography (GC), within the context of flow modulated comprehensive two-dimensional gas chromatography (FM GC×GC). The features of the VUV detector were evaluated through the analysis of petrochemical and fatty acids samples. Besides responding in a predictable fashion via Beer's law principles, the detector provides additional spectroscopic information for qualitative analysis. Virtually all chemical species absorb and have unique gas phase absorption features in the 120-240nm wavelength range monitored. The VUV detector can acquire up to 90 full range absorption spectra per second, allowing its coupling with comprehensive two-dimensional gas chromatography. This recent form of detection can address specific limitations related to mass spectrometry (e.g., identification of isobaric and isomeric species with very similar mass spectra or labile chemical compounds), and it is also able to deconvolute co-eluting peaks. Moreover, it is possible to exploit a pseudo-absolute quantitation of analytes based on pre-recorded absorption cross-sections for target analytes, without the need for traditional calibration. Using this and the other features of the detector, particular attention was devoted to the suitability of the FM GC×GC-VUV system toward qualitative and quantitative analysis of bio-diesel fuel and different kinds of fatty acids. Satisfactory results were obtained in terms of tailing factor (1.1), asymmetry factor (1.1), and similarity (average value 97%), for the FAMEs mixtures analysis. Copyright © 2017 Elsevier B.V. All rights reserved.
6. Ba8CoNb6O24 : A spin-1/2 triangular-lattice Heisenberg antiferromagnet in the two-dimensional limit
Science.gov (United States)
Rawl, R.; Ge, L.; Agrawal, H.; Kamiya, Y.; Dela Cruz, C. R.; Butch, N. P.; Sun, X. F.; Lee, M.; Choi, E. S.; Oitmaa, J.; Batista, C. D.; Mourigal, M.; Zhou, H. D.; Ma, J.
2017-02-01
The perovskite Ba8CoNb6O24 comprises equilateral effective spin-1/2 Co2 + triangular layers separated by six nonmagnetic layers. Susceptibility, specific heat, and neutron scattering measurements combined with high-temperature series expansions and spin-wave calculations confirm that Ba8CoNb6O24 is basically a two-dimensional magnet with no detectable spin anisotropy and no long-range magnetic ordering down to 0.06 K. In other words, Ba8CoNb6O24 is very close to be a realization of the paradigmatic spin-1/2 triangular Heisenberg model, which is not expected to exhibit symmetry breaking at finite temperatures according to the Mermin and Wagner theorem.
7. Description and results of a two-dimensional lattice physics code benchmark for the Canadian Pressure Tube Supercritical Water-cooled Reactor (PT-SCWR)
Energy Technology Data Exchange (ETDEWEB)
Hummel, D.W.; Langton, S.E.; Ball, M.R.; Novog, D.R.; Buijs, A., E-mail: [email protected] [McMaster Univ., Hamilton, Ontario (Canada)
2013-07-01
Discrepancies have been observed among a number of recent reactor physics studies in support of the PT-SCWR pre-conceptual design, including differences in lattice-level predictions of infinite neutron multiplication factor, coolant void reactivity, and radial power profile. As a first step to resolving these discrepancies, a lattice-level benchmark problem was designed based on the 78-element plutonium-thorium PT-SCWR fuel design under a set of prescribed local conditions. This benchmark problem was modeled with a suite of both deterministic and Monte Carlo neutron transport codes. The results of these models are presented here as the basis of a code-to-code comparison. (author)
8. Investigating the sign problem for two-dimensional $\\mathcal{N}=(2,2)$ and $\\mathcal{N}=(8,8)$ lattice super Yang--Mills theories
CERN Document Server
Galvez, Richard; Joseph, Anosh; Mehta, Dhagash
2012-01-01
Recently there has been some controversy in the literature concerning the existence of a fermion sign problem in the $\\mathcal{N}=(2,2)$ supersymmetric Yang--Mills (SYM) theories on the lattice. In this work, we address this issue by conducting Monte Carlo simulations not only for $\\mathcal{N}=(2,2)$ but also for $\\mathcal{N}=(8,8)$ SYM in two dimensions for the U(N) theories with N=2, using the new ideas derived from topological twisting followed by geometric discretization. Our results from simulations provide the evidence that these theories do {\\it not} suffer from a sign problem as the continuum limit is approached. These results thus boost confidence that these new lattice formulations can be used successfully to explore the nonperturbative aspects of the four-dimensional $\\mathcal{N}=4$ SYM theory.
9. A two-dimensional microscale model of gas exchange during photosynthesis in maize (Zea mays L.) leaves.
Science.gov (United States)
Retta, Moges; Ho, Quang Tri; Yin, Xinyou; Verboven, Pieter; Berghuijs, Herman N C; Struik, Paul C; Nicolaï, Bart M
2016-05-01
CO2 exchange in leaves of maize (Zea mays L.) was examined using a microscale model of combined gas diffusion and C4 photosynthesis kinetics at the leaf tissue level. Based on a generalized scheme of photosynthesis in NADP-malic enzyme type C4 plants, the model accounted for CO2 diffusion in a leaf tissue, CO2 hydration and assimilation in mesophyll cells, CO2 release from decarboxylation of C4 acids, CO2 fixation in bundle sheath cells and CO2 retro-diffusion from bundle sheath cells. The transport equations were solved over a realistic 2-D geometry of the Kranz anatomy obtained from light microscopy images. The predicted responses of photosynthesis rate to changes in ambient CO2 and irradiance compared well with those obtained from gas exchange measurements. A sensitivity analysis showed that the CO2 permeability of the mesophyll-bundle sheath and airspace-mesophyll interfaces strongly affected the rate of photosynthesis and bundle sheath conductance. Carbonic anhydrase influenced the rate of photosynthesis, especially at low intercellular CO2 levels. In addition, the suberin layer at the exposed surface of the bundle sheath cells was found beneficial in reducing the retro-diffusion. The model may serve as a tool to investigate CO2 diffusion further in relation to the Kranz anatomy in C4 plants.
10. Tailoring the Two Dimensional Electron Gas at Polar ABO3/SrTiO3 Interfaces for Oxide Electronics
Science.gov (United States)
Li, Changjian; Liu, Zhiqi; Lü, Weiming; Wang, Xiao Renshaw; Annadi, Anil; Huang, Zhen; Zeng, Shengwei; Ariando; Venkatesan, T.
2015-08-01
The 2D electron gas at the polar/non-polar oxide interface has become an important platform for several novel oxide electronic devices. In this paper, the transport properties of a wide range of polar perovskite oxide ABO3/SrTiO3 (STO) interfaces, where ABO3 includes LaAlO3, PrAlO3, NdAlO3, NdGaO3 and LaGaO3 in both crystalline and amorphous forms, were investigated. A robust 4 unit cell (uc) critical thickness for metal insulator transition was observed for crystalline polar layer/STO interface while the critical thickness for amorphous ones was strongly dependent on the B site atom and its oxygen affinity. For the crystalline interfaces, a sharp transition to the metallic state (i.e. polarization catastrophe induced 2D electron gas only) occurs at a growth temperature of 515 °C which corresponds to a critical relative crystallinity of ~70 ± 10% of the LaAlO3 overlayer. This temperature is generally lower than the metal silicide formation temperature and thus offers a route to integrate oxide heterojunction based devices on silicon.
11. Kinetic view of chirped optical lattice gas heating
Science.gov (United States)
Graul, J. S.; Gimelshein, S. F.; Lilly, T. C.
2014-12-01
With a focus on optical lattice gas heating, direct simulation Monte Carlo was used to investigate the interaction between molecular nitrogen, argon and methane, initially at 300 K and 0.8 atm, with pulsed, chirped optical lattices. Created from two 700 mJ, 532 nm, flattop laser pulses, the optical lattice parameters simulated are based on published optical lattice-based experiments, to ensure that pulse energies and durations do not exceed published optical breakdown (ionization) thresholds. Resultant translational gas temperatures, as well as induced bulk velocities, were used quantify energy and momentum deposition. To maximize available gas temperature changes achieved using the technique, laser pulses were linearly chirped to produce lattice velocities able to more effectively facilitate energy deposition throughout the pulse duration. From the initial conditions, the maximum, end pulse axial translational temperature obtained in nitrogen was approximately 755 K, at a lattice velocity of 400 m/s linearly chirped at 25 Gm/s2 over the 40 ns pulse duration. To date, this stands as the single largest, numerically-predicted temperature change from optical lattice gas heating under the numerical integration of real world energy and laser-based limitations.
12. Effects of antidot shape on the spin wave spectra of two-dimensional Ni{sub 80}Fe{sub 20} antidot lattices
Energy Technology Data Exchange (ETDEWEB)
Mandal, Ruma; Laha, Pinaki; Das, Kaustuv; Saha, Susmita; Barman, Saswati; Raychaudhuri, A. K.; Barman, Anjan, E-mail: [email protected] [Thematic Unit of Excellence on Nanodevice Technology, Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 098 (India)
2013-12-23
We show that the optically induced spin wave spectra of nanoscale Ni{sub 80}Fe{sub 20} (permalloy) antidot lattices can be tuned by changing the antidot shape. The spin wave spectra also show an anisotropy with the variation of the in-plane bias field orientation. Analyses show this is due to various quantized and extended modes, whose nature changes with the antidot shape and bias field orientation as a result of the variation of the internal magnetic field profile. The observed variation and anisotropy in the spin waves with the internal and external parameters are important for their applications in magnonic devices.
13. Quantification of trace O-containing compounds in GTL process samples via Fischer-Tropsch reaction by comprehensive two-dimensional gas chromatography/mass spectrometry.
Science.gov (United States)
Fernandes, Daniella R; Pereira, Vinícius B; Stelzer, Karen T; Gomes, Alexandre O; Neto, Francisco R Aquino; Azevedo, Débora A
2015-11-01
Comprehensive two-dimensional gas chromatography coupled to time-of-flight mass spectrometry (GC×GC-TOFMS) was successfully applied to eight real Brazilian Fischer-Tropsch (FT) product samples for the quantitative analysis of O-containing compounds. It not only allowed identifying and quantifying simultaneously a large number of O-containing compounds but also resolved many co-eluting components, such as carboxylic acids, which co-elute in one-dimensional gas chromatography. The homologous series of alcohols and carboxylic acids as trimethylsilyl derivatives were detected and identified at trace levels. The absolute quantification of each compound was accomplished with reliability using analytical curves. Linear alcohols (from C5 to C19), branched alcohols (C6-C13) and carboxylic acids (C4 to C12) were obtained in the range of 1.58 mg g(-1) to 14.75 mg g(-1), 0.51 mg g(-1) to 1.12 mg g(-1) and 0.21 mg g(-1) to 1.63 mg g(-1) of FT product samples, respectively. GC×GC-TOFMS provided a linear range (from 0.3 ng µL(-1) to 10 ng µL(-1)), good precision (gas-to-liquid technologies from natural gas and guide the choice of an FT conversion process that generates clean products with higher added value.
14. Comprehensive two-dimensional gas chromatography in combination with rapid scanning quadrupole mass spectrometry in perfume analysis.
Science.gov (United States)
Mondello, Luigi; Casillia, Alessandro; Tranchida, Peter Quinto; Dugo, Giovanni; Dugo, Paola
2005-03-04
Single column gas chromatography (GC) in combination with a flame ionization detector (FID) and/or a mass spectrometer is routinely employed in the determination of perfume profiles. The latter are to be considered medium to highly complex matrices and, as such, can only be partially separated even on long capillaries. Inevitably, several monodimensional peaks are the result of two or more overlapping components, often hindering reliable identification and quantitation. The present investigation is based on the use of a comprehensive GC (GC x GC) method, in vacuum outlet conditions, for the near to complete resolution of a complex perfume sample. A rapid scanning quadrupole mass spectrometry (qMS) system, employed for the assignment of GC x GC peaks, supplied high quality mass spectra. The validity of the three-dimensional (3D) GC x GC-qMS application was measured and compared to that of GC-qMS analysis on the same matrix. Peak identification, in all applications, was achieved through MS spectra library matching and the interactive use of linear retention indices (LRI).
15. Cooperative Reformable Channel System with Unique Recognition of Small Gas Molecules in a two-dimensional ZIF-membrane
Science.gov (United States)
Motevalli, Benyamin; Taherifar, Neda; Liu, Zhe
We report a cooperative reformable channel system in a coordination porous polymer, named as ZIF-L. Three types of local flexible ligands coexist in the crystal structure of this polymer, resulting in ultra-flexibility. The reformable channel is able to regulate permeation of a nonspherical guest molecule, such as N2 or CO2, based on its longer molecular dimension, which is in a striking contrast to conventional molecular sieves that regulate the shorter cross-sectional dimension of the guest molecules. Our density functional theory (DFT) calculations reveal that the guest molecule induces dynamic motion of the flexible ligands, leading to the channel reformation, and then the guest molecule reorientates itself to fit in the reformed channel. Such a unique induced fit-in'' mechanism causes the gas molecule to pass through 6 membered-ring windows in the c- crystal direction of ZIF-L with its longer axis parallel to the window plane. Our experimental permeance of N2 through the ZIF-L membranes is about three times greater than that of CO2, supporting the DFT simulation predictions.
16. In Silico Modeling of Hundred Thousand Experiments for Effective Selection of Ionic Liquid Phase Combinations in Comprehensive Two-Dimensional Gas Chromatography.
Science.gov (United States)
Nolvachai, Yada; Kulsing, Chadin; Marriott, Philip J
2016-02-16
The selection of the best column sets is one of the most tedious processes in comprehensive two-dimensional gas chromatography (GC × GC) where a multitude of choices of column sets could be employed for an individual sample analysis. We demonstrate analyte/stationary phase dependent selection approaches based on the linear solvation energy relationship (LSER), which is a reliable concept for the study of interaction mechanisms and retention prediction with a large database pool of columns and compounds. Good correlations between our predicted results, with experimental results reported in the literature, were obtained. The developed approaches were applied to the simulation of 157 920 individual experiments in GC × GC, focusing on the application of 30 nonionic liquid and 111 ionic liquid (IL) stationary phases for separation of some example sets of model compounds present in practical samples. The best column sets for each sample separation could then be extracted according to maximizing orthogonality, which estimates the quality of separation.
17. Two-dimensional electron gas in the regime of strong light-matter coupling: Dynamical conductivity and all-optical measurements of Rashba and Dresselhaus coupling
Science.gov (United States)
Yudin, Dmitry; Shelykh, Ivan A.
2016-10-01
A nonperturbative interaction of an electronic system with a laser field can substantially modify its physical properties. In particular, in two-dimensional (2D) materials with a lack of inversion symmetry, the achievement of a regime of strong light-matter coupling allows direct optical tuning of the strength of the Rashba spin-orbit interaction (SOI). Capitalizing on these results, we build a theory of the dynamical conductivity of a 2D electron gas with both Rashba and Dresselhaus SOIs coupled to an off-resonant high-frequency electromagnetic wave. We argue that strong light-matter coupling modifies qualitatively the dispersion of the electrons and can be used as a powerful tool to probe and manipulate the coupling strengths and adjust the frequency range where optical conductivity is essentially nonzero.
18. Comprehensive two-dimensional gas chromatography for enhanced analysis of naphthas: new column combination involving permethylated cyclodextrin in the second dimension.
Science.gov (United States)
Adam, Frédérick; Vendeuvre, Colombe; Bertoncini, Fabrice; Thiébaut, Didier; Espinat, Didier; Hennion, Marie-Claire
2008-01-18
A new column association using comprehensive two-dimensional gas chromatography for the detailed molecular analysis of hydrocarbon mixtures is reported in this paper. In order to compare the impact of two different secondary columns, a novel column combination relying on a GC x 2GC system was used. This system is based on a non-polar first column (PONA) combined with both a permethylated beta-cyclodextrin (beta-Dex 120) stationary phase and a polysilphenylensiloxane (BPX 50) in the second dimension. Compared to BPX 50 stationary phase, the implementation of beta-cyclodextrin columns as the second dimension was found to improve the resolution between paraffins and naphthenes in the naphtha range but not in the middle distillate range. Attempts to improve the results and to understand the interaction mechanism remained unsuccessful. Therefore, the benefits of the beta-Dex 120-column are only demonstrated on heavy naphtha cut for the quantitation of hydrocarbons.
19. Comprehensive two-dimensional gas chromatography coupled with time-of-flight mass spectrometry reveals the correlation between chemical compounds in Japanese sake and its organoleptic properties.
Science.gov (United States)
Takahashi, Kei; Kabashima, Fumie; Tsuchiya, Fumihiko
2016-03-01
Japanese sake is a traditional alcoholic beverage composed of a wide variety of metabolites, which give it many types of tastes and flavors. Previously, we have reported that medium-chain fatty acids contribute to a fatty odor in sake (Takahashi, K., et al., J. Agric. Food Chem., 62, 8478-8485, 2014). In this study, we have reanalyzed the data obtained using two-dimensional gas chromatography coupled with time-of-flight mass spectrometry. The relationship between the chemical components in sake and specific organoleptic properties such as off-flavor and quality has been explored. This led to the identification of the type of chemical compounds present and an assessment of the numerous candidate compounds that correlate with such organoleptic properties in sake. This research provides important fundamental knowledge for the sake-brewing industry. Copyright © 2015 The Society for Biotechnology, Japan. Published by Elsevier B.V. All rights reserved.
20. Spin beam splitter based on Goos-Haenchen shifts in two-dimensional electron gas modulated by ferromagnetic and Schottky metal stripes
Energy Technology Data Exchange (ETDEWEB)
Lu, Mao-Wang; Huang, Xin-Hong; Zhang, Gui-Lin; Chen, Sai-Yan [College of Science, Guilin University of Technology, Guilin 541004 (China)
2012-11-15
We present a theoretical study on the spin-dependent Goos-Haenchen (GH) effect in a two-dimensional electron gas modulated by ferromagnetic and Schottky metal (SM) stripes. The GH shifts for spin electron beams across this device are calculated with the help of the stationary phase method. It is shown that the GH shift of spin-up beam is significantly different from that of spin-down beam, i.e., this device shows up a considerable spin polarization effect in GH shifts of electron beams. It also is shown that both magnitude and sign of spin polarization of GH shifts are closely related to the stripe width, the magnetic strength and the gated voltage under SM stripe. These interesting properties not only provide an effective method of spin injection for spintronics application, but also give rise to a tunable spin beam splitter. (Copyright copyright 2012 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
1. Detailed analysis of petroleum hydrocarbon attenuation in biopiles by high-performance liquid chromatography followed by comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Mao, Debin; Lookman, Richard; Van De Weghe, Hendrik; Van Look, Dirk; Vanermen, Guido; De Brucker, Nicole; Diels, Ludo
2009-02-27
Enhanced bioremediation of petroleum hydrocarbons in two biopiles was quantified by high-performance liquid chromatography (HPLC) followed by comprehensive two-dimensional gas chromatography (GCXGC). The attenuation of 34 defined hydrocarbon classes was calculated by HPLC-GCXGC analysis of representative biopile samples at start-up and after 18 weeks of biopile operation. In general, a-cyclic alkanes were most efficiently removed from the biopiles, followed by monoaromatic hydrocarbons. Cycloalkanes and polycyclic aromatic hydrocarbons (PAHs) were more resistant to degradation. A-cyclic biomarkers farnesane, trimethyl-C13, norpristane, pristane and phytane dropped to only about 10% of their initial concentrations. On the other hand, C29-C31 hopane concentrations remained almost unaltered after 18 weeks of biopile operation, confirming their resistance to biodegradation. They are thus reliable indicators to estimate attenuation potential of petroleum hydrocarbons in biopile processed soils.
2. Quantum transport in two dimensional electron gas/p-wave superconductor junction with Rashba spin–orbit coupling at the interface and in the normal layer
Energy Technology Data Exchange (ETDEWEB)
Mohammadkhani, R., E-mail: [email protected]; Hassanloo, Gh.
2014-11-01
We have studied the tunneling conductance of a clean two dimensional electron gas/p- wave superconductor junction with Rashba spin–orbit coupling (RSOC) which is present in the normal layer and at the interface. Using the extended Blonder–Tinkham–Klapwijk formalism we have found that the subgap conductance peaks are shifted to a nonzero bias by RSOC at the interface which are the same as Ref. [1]. It is shown that for low insulating barrier and in the absence of the interface RSOC, the tunneling conductance decreases within energy gap with increasing of the RSOC in the normal layer while for high insulating barrier it enhances by increase of the RSOC. We have also shown that the RSOC inside the normal cannot affect the location of the subgap conductance peaks shifted by the interface RSOC.
3. Analysis of Salvinorin A in plants, water, and urine using solid-phase microextraction-comprehensive two-dimensional gas chromatography-time of flight mass spectrometry.
Science.gov (United States)
Barnes, Brian B; Snow, Nicholas H
2012-02-24
Salvinorin A, a psychoactive hallucinogen, and related compounds, were analyzed in plants, water, and urine using liquid-liquid extraction (LLE), solid-phase microextraction (SPME) and comprehensive two-dimensional gas chromatography-time of flight mass spectrometry (GC×GC-ToFMS). A semi-qualitative study of the extraction of Salvinorin A and analogs from Salvia divinorum plants by LLE showed ppb levels of Salvinorin A and several analogs in the leaves and stems of S. divinorum plants, much lower than expected. Quantitative analysis of Salvinorin A spiked into water and urine showed much better figures of merit for SPME than LLE, with limit of detection of about 5 ng/mL, linear range from 8 to 500 ng/mL and precision about ±10% for the SPME-based analyses using external standard quantitation. GC×GC-ToFMS was especially effective in separating the peaks of interest from matrix and chromatographic interferences.
4. Transport properties of the two-dimensional electron gas in GaN/AlGaN heterostructures grown by ammonia molecular-beam epitaxy
Energy Technology Data Exchange (ETDEWEB)
Pogosov, A.G.; Budantsev, M.V.; Lavrov, R.A.; Mansurov, V.G.; Nikitin, A.Yu.; Preobrazhenskii, V.V.; Zhuravlev, K.S. [Institute of Semiconductor Physics, 13 Lavrentiev Avenue, 630090 Novosibirsk (Russian Federation)
2006-07-15
Transport properties of the two-dimensional electron gas in AlGaN/GaN heterostructures grown by ammonia molecular-beam epitaxy are experimentally investigated. Conventional Hall and Shubnikov-de Haas measurements as well as investigations of quantum transport phenomena are reported. It is found that negative magnetoresistance (NMR) caused by weak localization demonstrates an unusual behavior at low temperature (1.8 K). The observed NMR cannot be described by the ordinary theory of quantum corrections to conductivity based on a single phase breaking time {tau}{sub {phi}}. The anomalous NMR behavior can be explained by the presence of two occupied quantum subbands, characterized by their own phase breaking times {tau} {sub {phi}} {sub 1} and {tau} {sub {phi}} {sub 2}. (copyright 2006 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) (orig.)
5. Donor-Like Surface Traps on Two-Dimensional Electron Gas and Current Collapse of AlGaN/GaN HEMTs
Science.gov (United States)
Yu, Chen-hui; Luo, Qing-zhou; Luo, Xiang-dong; Liu, Pei-sheng
2013-01-01
The effect of donor-like surface traps on two-dimensional electron gas (2DEG) and drain current collapse of AlGaN/GaN high electron mobility transistors (HEMTs) has been investigated in detail. The depletion of 2DEG by the donor-like surface states is shown. The drain current collapse is found to be more sensitive to the addition of positive surface charges. Surface trap states with higher energy levels result in weaker current collapse and faster collapse process. By adopting an optimized backside doping scheme, the electron density of 2DEG has been improved greatly and the current collapse has been greatly eliminated. These results give reference to the improvement in device performance of AlGaN/GaN HEMTs. PMID:24348195
6. Two-dimensional electron gas generated by La-doping at SrTiO3(001 surface: A first-principles study
Directory of Open Access Journals (Sweden)
Yun Li
2013-06-01
Full Text Available We carried out first-principles calculations to study the electronic properties of SrO-terminated and TiO2-terminated SrTiO3(001 surfaces with La-doping at the surface. We find that an intrinsic lower-lying state at the SrO-terminated surface can accommodate a two-dimensional electron gas (2DEG. By introducing La-doping at the SrO-terminated surface the energy position of the surface state and the 2DEG density can be tuned by changing the doping concentration. The higher the La-doping concentration, the lower the lower-lying state and the higher the 2DEG density. This 2DEG has a small effective mass and hopefully shows a high mobility.
7. Patterning the two dimensional electron gas at the LaAlO3/SrTiO3 interface by structured Al capping
Science.gov (United States)
Zhou, Y.; Wang, P.; Luan, Z. Z.; Shi, Y. J.; Jiang, S. W.; Ding, H. F.; Wu, D.
2017-04-01
We demonstrate an approach for patterning a quasi-two dimensional electron gas (q-2DEG) at the interface of LaAlO3 (LAO) and SrTiO3 (STO) utilizing a structured Al capping layer. The capping of Al enables the formation of q-2DEG at the interface of 1-3 unit cells (uc) of LAO on STO, which was originally insulating before capping. The properties of the q-2DEG induced by the Al capping layer are essentially the same as those of q-2DEG without Al. Therefore, we can pattern q-2DEG by simply patterning the Al film on LAO (2 or 3 uc)/STO using a one-step liftoff process. Our approach circumvents the difficulty of direct patterning of oxide materials and provides a simple and robust patterning method for future device applications based on complex oxide interfaces.
8. Predictive Control over Charge Density in the Two-Dimensional Electron Gas at the Polar-Nonpolar NdTiO3/SrTiO3 Interface
Science.gov (United States)
Xu, Peng; Ayino, Yilikal; Cheng, Christopher; Pribiag, Vlad S.; Comes, Ryan B.; Sushko, Peter V.; Chambers, Scott A.; Jalan, Bharat
2016-09-01
Through systematic control of the Nd concentration, we show that the carrier density of the two-dimensional electron gas (2DEG) in SrTiO3/NdTiO3/SrTiO3(001 ) can be modulated over a wide range. We also demonstrate that the NdTiO3 in heterojunctions without a SrTiO3 cap is degraded by oxygen absorption from air, resulting in the immobilization of donor electrons that could otherwise contribute to the 2DEG. This system is, thus, an ideal model to understand and control the insulator-to-metal transition in a 2DEG based on both environmental conditions and film-growth processing parameters.
9. Photoluminescence Investigation of Two-Dimensional Electron Gas in an Undoped AlxGa1-xN/GaN Heterostructure
Institute of Scientific and Technical Information of China (English)
HAN Xiu-Xun; WU Jie-Jun; LI Jie-Min; CONG Guang-Wei; LIU Xiang-Lin; ZHU Qin-Sheng; WANG Zhan-Guo
2005-01-01
@@ Low-temperature photoluminescence measurement is performed on an undoped Alx Ga1-xN/GaN heterostructure. Temperature-dependent Hall mobility confirms the formation of two-dimensional electron gas (2DEG) near the heterointerface. A weak photoluminescence (PL) peak with the energy of ～79meV lower than the free exciton (FE) emission of bulk GaN is related to the radiative recombination between electrons confined in the triangular well and the holes near the flat-band region of GaN. Its identification is supported by the solution of coupled one-dimensional Poisson and Schrodinger equations. When the temperature increases, the red shift of the 2DEG related emission peak is slower than that of the FE peak. The enhanced screening effect coming from the increasing 2DEG concentration and the varying electron distribution at two lowest subbands as a function of temperature account for such behaviour.
10. Subband Structure of a Two-Dimensional Electron Gas Formed at the Polar Surface of the Strong Spin-Orbit Perovskite KTaO3
Energy Technology Data Exchange (ETDEWEB)
King, P.D.C.
2012-03-01
We demonstrate the formation of a two-dimensional electron gas (2DEG) at the (100) surface of the 5d transition-metal oxide KTaO{sub 3}. From angle-resolved photoemission, we find that quantum confinement lifts the orbital degeneracy of the bulk band structure and leads to a 2DEG composed of ladders of subband states of both light and heavy carriers. Despite the strong spin-orbit coupling, we find no experimental signatures of a Rashba spin splitting, which has important implications for the interpretation of transport measurements in both KTaO{sub 3}- and SrTiO{sub 3}-based 2DEGs. The polar nature of the KTaO{sub 3}(100) surface appears to help mediate formation of the 2DEG as compared to non-polar SrTiO{sub 3}(100).
11. Two-Dimensional MHD Numerical Simulations of Magnetic Reconnection Triggered by A Supernova Shock in Interstellar Medium, Generation of X-Ray Gas in Galaxy
CERN Document Server
Tanuma, S; Kudoh, T; Shibata, K; Tanuma, Syuniti; Yokoyama, Takaaki; Kudoh, Takahiro; Shibata, Kazunari
2001-01-01
We examine the magnetic reconnection triggered by a supernova (or a point explosion) in interstellar medium, by performing two-dimensional resistive magnetohydrodynamic (MHD) numerical simulations with high spatial resolution. We found that the magnetic reconnection starts long after a supernova shock (fast-mode MHD shock) passes a current sheet. The current sheet evolves as follows: (i) Tearing-mode instability is excited by the supernova shock, and the current sheet becomes thin in its nonlinear stage. (ii) The current-sheet thinning is saturated when the current-sheet thickness becomes comparable to that of Sweet-Parker current sheet. After that, Sweet-Parker type reconnection starts, and the current-sheet length increases. (iii) Secondary tearing-mode instability'' occurs in the thin Sweet-Parker current sheet. (iv) As a result, further current-sheet thinning occurs and anomalous resistivity sets in, because gas density decreases in the current sheet. Petschek type reconnection starts and heats interste...
12. Quantitative analysis of biodiesel in blends of biodiesel and conventional diesel by comprehensive two-dimensional gas chromatography and multivariate curve resolution.
Science.gov (United States)
Mogollon, Noroska Gabriela Salazar; Ribeiro, Fabiana Alves de Lima; Lopez, Monica Mamian; Hantao, Leandro Wang; Poppi, Ronei Jesus; Augusto, Fabio
2013-09-24
In this paper, a method to determine the composition of blends of biodiesel with mineral diesel (BXX) by multivariate curve resolution with Alternating Least Squares (MRC-ALS) combined to comprehensive two-dimensional gas chromatography with Flame Ionization Detection (GC×GC-FID) is presented. Chromatographic profiles of BXX blends produced with biodiesels from different sources were used as input data. An initial evaluation carried out after multiway principal component analysis (MPCA) was used to reveal regions of the chromatograms were the signal was likely to be dependent on the concentration of biodiesel, regardless its vegetable source. After this preliminary step MCR-ALS modeling was carried out only using relevant parts of the chromatograms. The resulting procedure was able to predict accurately the concentration of biodiesel in the BXX samples regardless of its origin.
13. Comprehensive two-dimensional gas chromatography - time-of-flight mass spectrometry and simultaneous electron capture detection/nitrogen phosphorous detection for incense analysis
Science.gov (United States)
Tran, Tin C.; Marriott, Philip J.
This study reports comprehensive two-dimensional gas chromatography hyphenated to time-of-flight mass spectrometry detection (GC × GC/TOFMS) for characterisation and identification of components generated by four different types of powdered incense headspace (H/S) and incense smoke. GC × GC/TOFMS allowed simultaneous separation and identification of compounds emitted into the atmosphere as a result of combustion of incense powder. The smoke stream comprised compounds originating from the incense powder, and combustion products such as saturated and unsaturated hydrocarbons, essential oil type compounds, nitromusks, fatty acid methyl esters (FAMEs), polycyclic aromatic hydrocarbons (PAHs, which possibly include oxygenated and nitrated PAH), N-heterocyclics, pyrans and furans, which were detected and tentatively identified by GC × GC/TOFMS. GC × GC-electron capture detector/nitrogen phosphorous detector (ECD/NPD) potentially offers the prospect of providing selective chemical compositional information of incense powder and smoke, such as nitrogen-containing (N-containing) and halogenated compounds. Results of GC×GC-ECD/NPD showed that both incense powder and smoke generated emission of N-containing and halogenated compounds. A significant number of halogenated and N-containing compounds were emitted during the incomplete combustion of incense. However, one further objective of this paper is to demonstrate the capacity of comprehensive two-dimensional gas chromatography coupled to specific and/or selective detectors such as those used in this study (GC × GC-ECD/NPD) for the detection of particular classes of compounds such as N-containing and halogenated compounds at trace level concentrations in complex smoke samples.
14. Application of a quantitative structure retention relationship approach for the prediction of the two-dimensional gas chromatography retention times of polycyclic aromatic sulfur heterocycle compounds.
Science.gov (United States)
Gieleciak, Rafal; Hager, Darcy; Heshka, Nicole E
2016-03-11
Information on the sulfur classes present in petroleum is a key factor in determining the value of refined products and processing behavior in the refinery. A large part of the sulfur present is included in polycyclic aromatic sulfur heterocycles (PASHs), which in turn are difficult to desulfurize. Furthermore, some PASHs are potentially more mutagenic and carcinogenic than polycyclic aromatic hydrocarbons, PAHs. All of this calls for improved methods for the identification and quantification of individual sulfur species. Recent advances in analytical techniques such as comprehensive two-dimensional gas chromatography (GC×GC) have enabled the identification of many individual sulfur species. However, full identification of individual components, particularly in virgin oil fractions, is still out of reach as standards for numerous compounds are unavailable. In this work, a method for accurately predicting retention times in GC×GC using a QSRR (quantitative structure retention relationship) method was very helpful for the identification of individual sulfur compounds. Retention times for 89 saturated, aromatic, and polyaromatic sulfur-containing heterocyclic compounds were determined using two-dimensional gas chromatography. These retention data were correlated with molecular descriptors generated with CODESSA software. Two independent QSRR relationships were derived for the primary as well as the secondary retention characteristics. The predictive ability of the relationships was tested by using both independent sets of compounds and a cross-validation technique. When the corresponding chemical standards are unavailable, the equations developed for predicting retention times can be used to identify unknown chromatographic peaks by matching their retention times with those of sulfur compounds of known molecular structure.
15. Simulating high Reynolds number flow in two-dimensional lid-driven cavity by multi-relaxation-time lattice Boltzmann method
Institute of Scientific and Technical Information of China (English)
Chai Zhen-Hua; Shi Bao-Chang; Zheng Lin
2006-01-01
By coupling the non-equilibrium extrapolation scheme for boundary condition with the multi-relaxation-time lattice Boltzmann method, this paper finds that the stability of the multi-relaxation-time model can be improved greatly, especially on simulating high Reynolds number (Re) flow. As a discovery, the super-stability analysed by Lallemand and Luo is verified and the complex structure of the cavity flow is also exhibited in our numerical simulation when Re is high enough. To the best knowledge of the authors, the maximum of Re which has been investigated by direct numerical simulation is only around 50 000 in the literature; however, this paper can readily extend the maximum to 1000 000 with the above combination.
16. Self-consistent Bogoliubov-de Gennes theory of the vortex lattice state in a two-dimensional strongly type-II superconductor at high magnetic fields
Science.gov (United States)
Zhuravlev, Vladimir; Duan, Wenye; Maniv, Tsofar
2017-01-01
A self-consistent Bogoliubov-de Gennes theory of the vortex lattice state in a 2D strong type-II superconductor at high magnetic fields reveals a novel quantum mixed state around the semiclassical Hc 2, characterized by a well-defined Landau-Bloch band structure in the quasiparticle spectrum and suppressed order-parameter amplitude, which sharply crossover into the well-known semiclassical (Helfand-Werthamer) results upon decreasing magnetic field. Application to the 2D superconducting state observed recently on the surface of the topological insulator Sb2Te3 accounts well for the experimental data, revealing a strong type-II superconductor, with unusually low carrier density and very small cyclotron mass, which can be realized only in the strong coupling superconductor limit.
17. Disordered phase of the two-dimensional J{sub 1}–J{sub 2} Heisenberg antiferromagnet with S=1 on a square lattice
Energy Technology Data Exchange (ETDEWEB)
Pires, A.S.T. [Departamento de Fisica, Universidade Federal de Minas Gerais, Belo Horizonte, MG, CP 702, 30123-970 (Brazil); Lapa, R.S., E-mail: [email protected] [Departamento de Fisica, Universidade Federal de Minas Gerais, Belo Horizonte, MG, CP 702, 30123-970 (Brazil)
2013-08-15
We study the disordered phase of the J{sub 1}–J{sub 2} frustrated Heisenberg antiferromagnet with spin S=1 on a square lattice using a SU(3) Schwinger boson representation. In the approximation of Bose condensation and in a mean-field treatment of the four-operator terms, we calculate the gap as a function of J{sub 1}/J{sub 2}, and the quadrupole static structure factor at zero temperature. Our results indicate the existence of a nematic state in the paramagnetic phase. - Highlights: ► The disordered phase is studied. ► The energy gap in the paramagnetic phase is calculated. ► The static spin structure factor is calculated. ► The static quadrupole structure factor is calculated. ► The ground state energy is calculated.
18. Two-dimensional radiation MHD modeling assessment of designs for argon gas puff distributions for future experiments on the refurbished Z machine
Science.gov (United States)
Thornhill, J. W.; Giuliani, J. L.; Chong, Y. K.; Velikovich, A. L.; Dasgupta, A.; Apruzese, J. P.; Jones, B.; Ampleford, D. J.; Coverdale, C. A.; Jennings, C. A.; Waisman, E. M.; Lamppa, D. C.; McKenney, J. L.; Cuneo, M. E.; Krishnan, M.; Coleman, P. L.; Madden, R. E.; Elliott, K. W.
2012-09-01
Argon Z-pinch experiments are to be performed on the refurbished Z machine (which we will refer to as ZR here in order to distinguish between pre-refurbishment Z) at Sandia National Laboratories with a new 8 cm diameter double-annulus gas puff nozzle constructed by Alameda Applied Sciences Corporation (AASC). The gas exits the nozzle from an outer and inner annulus and a central jet. The amount of gas present in each region can be varied. Here a two-dimensional radiation MHD (2DRMHD) model, MACH2-TCRE, with tabular collisional radiative equilibrium atomic kinetics is used to theoretically investigate stability and K-shell emission properties of several measured (interferometry) initial gas distributions emanating from this new nozzle. Of particular interest is to facilitate that the distributions employed in future experiments have stability and K-shell emission properties that are at least as good as the Titan nozzle generated distribution that was successfully fielded in earlier experiments on the Z machine before it underwent refurbishment. The model incorporates a self-consistent calculation for non-local thermodynamic equilibrium kinetics and ray-trace based radiation transport. This level of detail is necessary in order to model opacity effects, non-local radiation effects, and the high temperature state of K-shell emitting Z-pinch loads. Comparisons of radiation properties and stability of measured AASC gas profiles are made with that of the distribution used in the pre-refurbished Z experiments. Based on these comparisons, an optimal K-shell emission producing initial gas distribution is determined from among the AASC nozzle measured distributions and predictions are made for K-shell yields attainable from future ZR experiments.
19. Two-Dimensional Numerical Simulations of Ultrasound in Liquids with Gas Bubble Agglomerates: Examples of Bubbly-Liquid-Type Acoustic Metamaterials (BLAMMs).
Science.gov (United States)
Vanhille, Christian
2017-01-17
This work deals with a theoretical analysis about the possibility of using linear and nonlinear acoustic properties to modify ultrasound by adding gas bubbles of determined sizes in a liquid. We use a two-dimensional numerical model to evaluate the effect that one and several monodisperse bubble populations confined in restricted areas of a liquid have on ultrasound by calculating their nonlinear interaction. The filtering of an input ultrasonic pulse performed by a net of bubbly-liquid cells is analyzed. The generation of a low-frequency component from a single cell impinged by a two-frequency harmonic wave is also studied. These effects rely on the particular dispersive character of attenuation and nonlinearity of such bubbly fluids, which can be extremely high near bubble resonance. They allow us to observe how gas bubbles can change acoustic signals. Variations of the bubbly medium parameters induce alterations of the effects undergone by ultrasound. Results suggest that acoustic signals can be manipulated by bubbles. This capacity to achieve the modification and control of sound with oscillating gas bubbles introduces the concept of bubbly-liquid-based acoustic metamaterials (BLAMMs).
20. A hybrid wavelet-based adaptive immersed boundary finite-difference lattice Boltzmann method for two-dimensional fluid-structure interaction
Science.gov (United States)
Cui, Xiongwei; Yao, Xiongliang; Wang, Zhikai; Liu, Minghao
2017-03-01
A second generation wavelet-based adaptive finite-difference Lattice Boltzmann method (FD-LBM) is developed in this paper. In this approach, the adaptive wavelet collocation method (AWCM) is firstly, to the best of our knowledge, incorporated into the FD-LBM. According to the grid refinement criterion based on the wavelet amplitudes of density distribution functions, an adaptive sparse grid is generated by the omission and addition of collocation points. On the sparse grid, the finite differences are used to approximate the derivatives. To eliminate the special treatments in using the FD-based derivative approximation near boundaries, the immersed boundary method (IBM) is also introduced into FD-LBM. By using the adaptive technique, the adaptive code requires much less grid points as compared to the uniform-mesh code. As a consequence, the computational efficiency can be improved. To justify the proposed method, a series of test cases, including fixed boundary cases and moving boundary cases, are invested. A good agreement between the present results and the data in previous literatures is obtained, which demonstrates the accuracy and effectiveness of the present AWCM-IB-LBM.
1. Study the two dimensional triangular lattice photonic crystal band gap and coupling characters%二维三角形光子晶体带隙与耦合特性研究
Institute of Scientific and Technical Information of China (English)
李未; 陈小玲
2011-01-01
利用二维三角晶格介质柱光子晶体TE偏振的禁带与介质柱半径的变化关系,分析了二维光子晶体的带隙分布及斜边耦合特性.结果表明,光子禁带的大小受到构成光子晶体的介电材料的空间排列分布以及介质柱半径大小的影响;束缚在光子晶体中的光波可以在波导和谐振腔中进行传输,达到选择输出光波的目的.%The paper study the relation between two dimensional triangular lattice photonic crystal band gap for TE polarizationand dielectric cylinder radius, and study distribution of two dimensional photonic crystal defect state. Results show, the photonic crystal band gaps were distributed by dielectric material space distribution and medium size of the radius; Tied in the photon crystals of light waves can transmission in waveguides and resonator cavity to select the output of light waves.
2. Superconductivity in the two-dimensional electron gas induced by high-energy optical phonon mode and large polarization of the SrTiO3 substrate
Science.gov (United States)
Rosenstein, Baruch; Shapiro, B. Ya.; Shapiro, I.; Li, Dingping
2016-07-01
Pairing in one-atomic-layer-thick two-dimensional electron gas (2DEG) by a single flat band of high-energy longitudinal optical phonons is considered. The polar dielectric SrTiO3 (STO) exhibits such an energetic phonon mode and the 2DEG is created both when one unit cell FeSe layer is grown on its (100 ) surface and on the interface with another dielectric like LaAlO3 (LAO). We obtain a quantitative description of both systems solving the gap equation for Tc for arbitrary Fermi energy ɛF, electron-phonon coupling λ , and the phonon frequency Ω , and direct (random-phase approximation) electron-electron repulsion strength α . The focus is on the intermediate region between the adiabatic, ɛF>>Ω , and the nonadiabatic, ɛF<<Ω , regimes. The high-temperature superconductivity in single-unit-cell FeSe/STO is possible due to a combination of three factors: high-longitudinal-optical phonon frequency, large electron-phonon coupling λ ˜0.5 , and huge dielectric constant of the substrate suppression the Coulomb repulsion. It is shown that very low density electron gas in the interfaces is still capable of generating superconductivity of the order of 0.1 K in LAO/STO.
3. Using comprehensive two-dimensional gas chromatography for the analysis of oxygenates in middle distillates I. Determination of the nature of biodiesels blend in diesel fuel.
Science.gov (United States)
Adam, Frédérick; Bertoncini, Fabrice; Coupard, Vincent; Charon, Nadège; Thiébaut, Didier; Espinat, Didier; Hennion, Marie-Claire
2008-04-04
In the current energetic context (increasing consumption of vehicle fuels, greenhouse gas emission etc.) government policies lead to mandatory introduction in fossil fuels of fuels resulting from renewable sources of energy such as biomass. Blending of fatty acid alkyl esters from vegetable oils (also known as biodiesel) with conventional diesel fuel is one of the solutions technologically available; B5 blends (up to 5%w/w esters in fossil fuel) are marketed over Europe. Therefore, for quality control as well as for forensic reasons, it is of major importance to monitor the biodiesel origin (i.e. the fatty acid ester distribution) and its content when it is blend with petroleum diesel. This paper reports a comprehensive two-dimensional gas chromatography (GC x GC) method that was developed for the individual quantitation of fatty acid esters in middle distillates matrices. Several first and the second dimension columns have been investigated and their performances to achieve (i) a group type separation of hydrocarbons and (ii) individual identification and quantitation of fatty acid ester blend with diesel are reported and discussed. Finally, comparison of quantitative GC x GC results with reference methods demonstrates the benefits of GC x GC approach which enables fast and reliable individual quantitation of fatty acid esters in one single run. Results show that under developed chromatographic conditions, quantitative group type analysis of hydrocarbons is also possible, meaning that simultaneous quantification of hydrocarbons and fatty acid esters can be achieved in one single run.
4. Mechanism to generate a two-dimensional electron gas at the surface of the charge-ordered semiconductor BaBiO3.
Science.gov (United States)
Vildosola, Verónica; Güller, Francisco; Llois, Ana María
2013-05-17
In this Letter, we find by means of first-principles calculations a new physical mechanism to generate a two-dimensional electron gas, namely, the breaking of charge ordering at the surface of a charge-ordered semiconductor due to the incomplete oxygen environment of the surface ions. The emergence of the 2D gas is independent of the presence of oxygen vacancies or polar discontinuities; this is a self-doping effect. This mechanism might apply to many charge-ordered systems, in particular, we study the case of BaBiO(3)(001). Our calculations show that the outer layer of the Bi-terminated simulated surface turns more cubiclike and metallic while the inner layers remain in the insulating monoclinic state that the system present in the bulk form. On the other hand, the metallization does not occur for the Ba termination, a fact that makes this system appealing for nanostructuring. Finally, in view of the bulk properties of this material under doping, this particular finding sets another possible route for future exploration: the potential scenario of 2D superconductivity at the BaBiO(3) surface.
5. A one-step method for priority compounds of concern in tar from former industrial sites: trimethylsilyl derivatisation with comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Gauchotte-Lindsay, C; Richards, P; McGregor, L A; Thomas, R; Kalin, R M
2012-08-31
A dense non-aqueous phase liquid sample formed by release of coal tar into the environment was derivatised by trimethylsilylation using the reagent N,O-bis(trimethylsilyl)trifluoroacetamide (BSTFA) and extracted in hexane using accelerated solvent extraction. This procedure enables comprehensive extraction of an extensive suite of organic compounds from tar, which has not previously been described. Comprehensive two dimensional gas chromatography coupled to time of flight mass spectrometry (GC×GC-TOFMS) was used for the analysis of the sample for concurrent evaluation of -OH functional group-containing compounds along with aliphatics, polycyclic aromatic hydrocarbons and other typical tar compounds normally determined via classic gas chromatography. Using statistically designed experiments, a range of conditions were tested for complete recovery of four different surrogates. The robustness and repeatability of the optimised derivatisation/extraction method was demonstrated. Finally, more than a hundred and fifty derivatised compounds were identified using mass spectra elucidation and GC×GC logical order of elution. Copyright © 2012 Elsevier B.V. All rights reserved.
6. Complex Saddles in Two-dimensional Gauge Theory
CERN Document Server
Buividovich, P V; Valgushev, S N
2015-01-01
We study numerically the saddle point structure of two-dimensional (2D) lattice gauge theory, represented by the Gross-Witten-Wadia unitary matrix model. The saddle points are in general complex-valued, even though the original integration variables and action are real. We confirm the trans-series/instanton gas structure in the weak-coupling phase, and identify a new complex-saddle interpretation of non-perturbative effects in the strong-coupling phase. In both phases, eigenvalue tunneling refers to eigenvalues moving off the real interval, into the complex plane, and the weak-to-strong coupling phase transition is driven by saddle condensation.
7. A Vacuum Ultraviolet Absorption Array Spectrometer as a Selective Detector for Comprehensive Two-Dimensional Gas Chromatography: Concept and First Results.
Science.gov (United States)
Gröger, Thomas; Gruber, Beate; Harrison, Dale; Saraji-Bozorgzad, Mohammad; Mthembu, Makhosazana; Sutherland, Aimée; Zimmermann, Ralf
2016-03-15
Fast and selective detectors are very interesting for comprehensive two-dimensional gas chromatography (GC × GC). This is particularly true if the detector system can provide additional spectroscopic information on the compound structure and/or functionality. Other than mass spectrometry (MS), only optical spectroscopic detectors are able to provide selective spectral information. However, until present the application of optical spectroscopy technologies as universal detectors for GC × GC has been restricted mainly due to physical limitations such as insufficient acquisition speed or high detection limits. A recently developed simultaneous-detection spectrometer working in the vacuum ultraviolet (VUV) region of 125-240 nm overcomes these limitations and meets all the criteria of a universal detector for GC × GC. Peak shape and chromatographic resolution is preserved and unique spectral information, complementary to mass spectrometry data, is gained. The power of this detector is quickly recognized as it has the ability to discriminate between isomeric compounds or difficult to separate structurally related isobaric species; thus, it provides additional selectivity. A further promising feature of this detector is the data analysis concept of spectral filtering, which is accomplished by targeting special electronic transitions that allows for a fast screening of GC × GC chromatograms for designated compound classes.
8. Quantitative and qualitative analysis of hemicellulose, cellulose and lignin bio-oils by comprehensive two-dimensional gas chromatography with time-of-flight mass spectrometry.
Science.gov (United States)
Michailof, Chrysoula; Sfetsas, Themistoklis; Stefanidis, Stylianos; Kalogiannis, Konstantinos; Theodoridis, Georgios; Lappas, Angelos
2014-11-21
Thermal and catalytic pyrolysis are efficient processes for the transformation of biomass to bio-oil, a liquid energy carrier and a general source of chemicals. The elucidation of the bio-oil's composition is essential for a rational design of both its production and utilization process. However, the complex composition of bio-oils hinders their complete qualitative and quantitative analysis, and conventional chromatographic techniques lack the necessary separation power. Two-dimensional gas chromatography with time-of-flight mass spectrometry (GC×GC-ToFMS) is considered a suitable technique for bio-oil analysis due to its increased separation and resolution capacity. This work presents the tentative qualitative and quantitative analysis of bio-oils resulting from the thermal and catalytic pyrolysis of standard xylan, cellulose, lignin and their mixture by GC×GC-ToFMS. Emphasis is placed on the development of the quantitative method using phenol-d6 as internal standard. During the method development, a standard solution of 39 compounds was used for the determination of the respective Relative Response Factors (RRF) employing statistical methods, ANOVA and WLSLR, for verification of the data. The developed method was applied to the above mentioned bio-oils and their detailed analysis is presented. The different compounds produced and their diverse concentration allows for an elucidation of the pyrolysis mechanism and highlight the effect of the catalyst.
9. Analysis of alkyl phosphates in petroleum samples by comprehensive two-dimensional gas chromatography with nitrogen phosphorus detection and post-column Deans switching.
Science.gov (United States)
Nizio, Katie D; Harynuk, James J
2012-08-24
Alkyl phosphate based gellants used as viscosity builders for fracturing fluids used in the process of hydraulic fracturing have been implicated in numerous refinery-fouling incidents in North America. In response, industry developed an inductively coupled plasma optical emission spectroscopy (ICP-OES) based method for the analysis of total volatile phosphorus in distillate fractions of crude oil; however, this method is plagued by poor precision and a high limit of detection (0.5±1μg phosphorus mL(-1)). Furthermore this method cannot provide speciation information, which is critical for developing an understanding of the challenge of alkyl phosphates at a molecular level. An approach using comprehensive two-dimensional gas chromatography with nitrogen phosphorus detection (GC×GC-NPD) and post-column Deans switching is presented. This method provides qualitative and quantitative profiles of alkyl phosphates in industrial petroleum samples with increased precision and at levels comparable to or below those achievable by ICP-OES. A recovery study in a fracturing fluid sample and a profiling study of alkyl phosphates in four recovered fracturing fluid/crude oil mixtures (flowback) are also presented.
10. In-plane anisotropy in two-dimensional electron gas at LaAlO3/SrTiO3(110) interface
Science.gov (United States)
Sheng-Chun, Shen; Yan-Peng, Hong; Cheng-Jian, Li; Hong-Xia, Xue; Xin-Xin, Wang; Jia-Cai, Nie
2016-07-01
A systematic study of the two-dimensional electron gas at LaAlO3/SrTiO3(110) interface reveals an anisotropy along two specific directions, [001] and . The anisotropy becomes distinct for the interface prepared under high oxygen pressure with low carrier density. Angular dependence of magnetoresistance shows that the electron confinement is stronger along the direction. Gate-tunable magnetoresistance reveals a clear in-plane anisotropy of the spin-orbit coupling, and the spin relaxation mechanism along both directions belongs to D’yakonov-Perel’ (DP) scenario. Moreover, in-plane anisotropic superconductivity is observed for the sample with high carrier density, the superconducting transition temperature is lower but the upper critical field is higher along the direction. This in-plane anisotropy could be ascribed to the anisotropic band structure along the two crystallographic directions. Project supported by the Ministry of Science and Technology of China (Grant Nos. 2013CB921701, 2013CBA01603, and 2014CB920903), the National Natural Science Foundation of China (Grant Nos. 10974019, 51172029, 91121012, 11422430, 11374035, 11474022, and 11474024), the Program for New Century Excellent Talents in the University of the Ministry of Education of China (Grant No. NCET-13-0054), and the Beijing Higher Education Young Elite Teacher Project, China (Grant No. YETP0238).
11. A peaklet-based generic strategy for the untargeted analysis of comprehensive two-dimensional gas chromatography mass spectrometry data sets.
Science.gov (United States)
Egert, Björn; Weinert, Christoph H; Kulling, Sabine E
2015-07-31
Comprehensive two-dimensional gas chromatography mass spectrometry (GC×GC-MS) is a well-established key technology in analytical chemistry and increasingly used in the field of untargeted metabolomics. However, automated processing of large GC×GC-MS data sets is still a major bottleneck in untargeted, large-scale metabolomics. For this reason we introduce a novel peaklet-based alignment strategy. The algorithm is capable of an untargeted deterministic alignment exploiting a density based clustering procedure within a time constrained similarity matrix. Exploiting minimal (1)D and (2)D retention time shifts between peak modulations, the alignment is done without the need for peak merging which also eliminates the need for linear or nonlinear retention time correction procedures. The approach is validated in detail using data of urine samples from a large human metabolomics study. The data was acquired by a Shimadzu GCMS-QP2010 Ultra GC×GC-qMS system and consists of 512 runs, including 312 study samples and 178 quality control sample injections, measured within a time period of 22 days. The final result table consisted of 313 analytes, each of these being detectable in at least 75% of the study samples. In summary, we present an automated, reliable and fully transparent workflow for the analysis of large GC×GC-qMS metabolomics data sets.
12. Identifying important structural features of ionic liquid stationary phases for the selective separation of nonpolar analytes by comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Zhang, Cheng; Ingram, Isaiah C; Hantao, Leandro W; Anderson, Jared L
2015-03-20
A series of dicationic ionic liquid (IL)-based stationary phases were evaluated as secondary columns in comprehensive two-dimensional gas chromatography (GC×GC) for the separation of aliphatic hydrocarbons from kerosene. In order to understand the role that structural features of ILs play on the selectivity of nonpolar analytes, the solvation parameter model was used to probe the solvation properties of the IL-based stationary phases. It was observed that room temperature ILs containing long free alkyl side chain substituents and long linker chains between the two cations possess less cohesive forces and exhibited the highest resolution of aliphatic hydrocarbons. The anion component of the IL did not contribute significantly to the overall separation, as similar selectivities toward aliphatic hydrocarbons were observed when examining ILs with identical cations and different anions. In an attempt to further examine the separation capabilities of the IL-based GC stationary phases, columns of the best performing stationary phases were prepared with higher film thickness and resulted in enhanced selectivity of aliphatic hydrocarbons.
13. Validated Method for the Quantification of Buprenorphine in Postmortem Blood Using Solid-Phase Extraction and Two-Dimensional Gas Chromatography-Mass Spectrometry.
Science.gov (United States)
Nahar, Limon Khatun; Andrews, Rebecca; Paterson, Sue
2015-09-01
A highly sensitive and fully validated method was developed for the quantification of buprenorphine in postmortem blood. After a two-step protein precipitation process using acetonitrile, buprenorphine was purified using mixed-mode (C8/cation exchange) solid-phase extraction cartridges. Endogenous water-soluble compounds and lipids were removed from the cartridges before the samples were eluted, concentrated and derivatized using N-methyl-N-trimethylsilyltrifluoroacetamide. The samples were analyzed using two-dimensional gas chromatography-mass spectrometry (2D GC-MS) in selective ion-monitoring mode. A low polarity Rxi(®)-5MS (30 m × 0.25 mm I.D. × 0.25 µm) was used as the primary column and the secondary column was a mid-polarity Rxi(®) -17Sil MS (15 m × 0.32 mm I.D. × 0.25 µm). The assay was linear from 1.0 to 50.0 ng/mL (r(2) > 0.99; n = 6). Intraday (n = 6) and interday (n = 9) imprecisions (percentage relative standard deviation, % RSD) were selective with no interference from endogenous compounds or from 62 commonly encountered drugs. To prove method applicability to forensic postmortem cases, 14 authentic postmortem blood samples were analyzed.
14. Solid phase microextraction-comprehensive two-dimensional gas chromatography-time-of-flight mass spectrometry for the analysis of honey volatiles.
Science.gov (United States)
Cajka, Tomás; Hajslová, Jana; Cochran, Jack; Holadová, Katerina; Klimánková, Eva
2007-03-01
Head-space solid phase microextration (SPME), followed by comprehensive two-dimensional gas chromatography-time-of-flight mass spectrometry (GCxGC-TOFMS), has been implemented for the analysis of honey volatiles, with emphasis on the optimal selection of SPME fibre and the first- and second-dimension GC capillaries. From seven SPME fibres investigated, a divinylbenzene/Carboxen/polydimethylsiloxane (DVB/CAR/PDMS) 50/30 microm fibre provided the best sorption capacity and the broadest range of volatiles extracted from the headspace of a mixed honey sample. A combination of DB-5ms x SUPELCOWAX 10 columns enabled the best resolution of sample components compared to the other two tested column configurations. Employing this powerful analytical strategy led to the identification of 164 volatile compounds present in a honey mixture during a 19-min GC run. Combination of this simple and inexpensive SPME-based sampling/concentration technique with the advanced separation/identification approach represented by GCxGC-TOFMS allows a rapid and comprehensive examination of the honey volatiles profile. In this way, the laboratory sample throughput can be increased significantly and, at the same time, the risk of erroneous identification, which cannot be avoided in one-dimensional GC separation, is minimised.
15. The quantification of short-chain chlorinated paraffins in sediment samples using comprehensive two-dimensional gas chromatography with μECD detection.
Science.gov (United States)
Muscalu, Alina M; Morse, Dave; Reiner, Eric J; Górecki, Tadeusz
2017-03-01
The analysis of persistent organic pollutants in environmental samples is a challenge due to the very large number of compounds with varying chemical and physical properties. Chlorinated paraffins (CPs) are complex mixtures of chlorinated n-alkanes with varying chain lengths (C10 to C30) and degree of chlorination (30 to 70% by weight). Their physical-chemical properties make these compounds persistent in the environment and able to bioaccumulate in living organisms. Comprehensive two-dimensional gas chromatography (GC × GC) coupled with micro-electron capture detection (μECD) was used to separate and quantify short-chain chlorinated paraffins (SCCP) in sediment samples. Distinct ordered bands were observed in the GC × GC chromatograms pointing to group separation. Using the Classification function of the ChromaTOF software, summary tables were generated to determine total area counts to set up multilevel-calibration curves for different technical mixes. Fortified sediment samples were analyzed by GC × GC-μECD with minimal extraction and cleanup. Recoveries ranged from 120 to 130%. To further validate the proposed method for the analysis of SCCPs, the laboratory participated in interlaboratory studies for the analysis of standards and sediment samples. The results showed recoveries between 75 and 95% and z-score values <2, demonstrating that the method is suitable for the analysis of SCCPs in soil/sediment samples. Graphical abstract Quantification of SCCPs by 2D-GC-μECD.
16. Two-dimensional gas chromatography/mass spectrometry, physical property modeling and automated production of component maps to assess the weathering of pollutants.
Science.gov (United States)
Antle, Patrick M; Zeigler, Christian D; Livitz, Dimitri G; Robbat, Albert
2014-10-17
Local conditions influence how pollutants will weather in subsurface environments and sediment, and many of the processes that comprise environmental weathering are dependent upon these substances' physical and chemical properties. For example, the effects of dissolution, evaporation, and organic phase partitioning can be related to the aqueous solubility (SW), vapor pressure (VP), and octanol-water partition coefficient (KOW), respectively. This study outlines a novel approach for estimating these physical properties from comprehensive two-dimensional gas chromatography-mass spectrometry (GC×GC/MS) retention index-based polyparameter linear free energy relationships (LFERs). Key to robust correlation between GC measurements and physical properties is the accurate and precise generation of retention indices. Our model, which employs isovolatility curves to calculate retention indices, provides improved retention measurement accuracy for families of homologous compounds and leads to better estimates of their physical properties. Results indicate that the physical property estimates produced from this approach have the same error on a logarithmic-linear scale as previous researchers' log-log estimates, yielding a markedly improved model. The model was embedded into a new software program, allowing for automated determination of these properties from a single GC×GC analysis with minimal model training and parameter input. This process produces component maps that can be used to discern the mechanism and progression of how a particular site weathers due to dissolution, organic phase partitioning, and evaporation into the surrounding environment.
17. Carrier dynamics of optical emission from two-dimensional electron gas in undoped AlGaN/GaN single heterojunctions
Energy Technology Data Exchange (ETDEWEB)
Kwack, H.S.; Cho, Y.H. [Department of Physics and Institute for Basic Science Research, Chungbuk National University, Cheongju 361-763 (Korea); Kim, G.H. [School of Information and Communication Engineering, Sungkyunkwan University, Suwon 440-746 (Korea); Park, M.R.; Youn, D.H.; Bae, S.B.; Lee, K.S. [Basic Research Laboratory, Electronics and Telecommunications Research Institute, Daejeon 305-350 (Korea); Lee, J.H.; Lee, J.H. [Department of Electric and Electronic Engineering, Kyungpook National University, Taegu 702-701 (Korea)
2006-06-15
The structural and optical properties of undoped AlGaN/GaN single heterojunctions (HJs) were studied by means of high-resolution x-ray diffraction, photoluminescence (PL), cathodoluminescence (CL), and time-resolved PL spectroscopy. An additional two-dimensional electron gas (2DEG)-related PL and CL emission appeared at about 40 meV below the GaN band-edge emission energy and persisted up to about 100 K, while this peak disappeared when the top AlGaN layer was removed by reactive ion etching. Depth-resolved CL spectra reveal the presence of a 2DEG at the heterointerface. The additional PL and CL emission below the GaN band-edge emission is attributed to the recombination between photogenerated holes and electrons confined at 2DEG states in the triangular-shaped interface potential. For the 2DEG emission, we observed an about 50-ps delayed rise time than the GaN and AlGaN emissions by using time-resolved PL, indicating effective carrier transfer from the GaN flatband and AlGaN regions to the heterointerface. From the results, we explained the optical properties and carrier recombination dynamics of 2DEG, GaN, and AlGaN emissions in undoped AlGaN/GaN single HJs. (copyright 2006 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) (orig.)
18. Carrier dynamics of optical emission from two-dimensional electron gas in undoped AlGaN/GaN single heterojunctions
Science.gov (United States)
Kwack, H. S.; Cho, Y. H.; Kim, G. H.; Park, M. R.; Youn, D. H.; Bae, S. B.; Lee, K.-S.; Lee, J. H.; Lee, J. H.
2006-06-01
The strucutral and optical properties of undoped AlGaN/GaN single heterojunctions (HJs) were studied by means of high-resolution x-ray diffraction, photoluminescence (PL), cathodoluminescence (CL), and time-resolved PL spectroscopy. An additional two-dimensional electron gas (2DEG)-related PL and CL emission appeared at about 40 meV below the GaN band-edge emission energy and persisted up to about 100 K, while this peak disappeared when the top AlGaN layer was removed by reactive ion etching. Depth-resolved CL spectra reveal the presence of a 2DEG at the heterointerface. The additional PL and CL emission below the GaN band-edge emission is attributed to the recombination between photogenerated holes and electrons confined at 2DEG states in the triangular-shaped interface potential. For the 2DEG emission, we observed an about 50-ps delayed rise time than the GaN and AlGaN emissions by using time-resolved PL, indicating effective carrier transfer from the GaN flatband and AlGaN regions to the heterointerface. From the results, we explained the optical properties and carrier recombination dynamics of 2DEG, GaN, and AlGaN emissions in undoped AlGaN/GaN single HJs.
19. Fano resonance in the nonadiabatically pumped shot noise of a time-dependent quantum well in a two-dimensional electron gas and graphene
Energy Technology Data Exchange (ETDEWEB)
Zhu, Rui, E-mail: [email protected]; Dai, Jiao-Hua [Department of Physics, South China University of Technology, Guangzhou 510641 (China); Guo, Yong [Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084 (China); Collaborative Innovation Center of Quantum Matter, Beijing (China)
2015-04-28
Interference between different quantum paths can generate Fano resonance. One of the examples is transport through a quasibound state driven by a time-dependent scattering potential. Previously it is found that Fano resonance occurs as a result of energy matching in one-dimensional systems. In this work, we demonstrate that when transverse motion is present, Fano resonance occurs precisely at the wavevector matching situation. Using the Floquet scattering theory, we considered the transport properties of a nonadiabatic time-dependent well both in a two-dimensional electron gas and monolayer graphene structure. Dispersion of the quasibound state of a static quantum well is obtained with transverse motion present. We found that Fano resonance occurs when the wavevector in the transport direction of one of the Floquet sidebands is exactly identical to that of the quasibound state in the well at equilibrium and follows the dispersion pattern of the latter. To observe the Fano resonance phenomenon in the transmission spectrum, we also considered the pumped shot noise properties when time and spatial symmetry secures vanishing current in the considered configuration. Prominent Fano resonance is found in the differential pumped shot noise with respect to the reservoir Fermi energy.
20. Comparison between theoretical and experimental results for energy states of two-dimensional electron gas in pseudomorphically strained InAs high-electron-mobility transistors
Science.gov (United States)
Nishio, Yui; Tange, Takahiro; Hirayama, Naomi; Iida, Tsutomu; Takanashi, Yoshifumi
2014-01-01
The energy states of a two-dimensional electron gas (2DEG) in high-electron-mobility transistors with a pseudomorphically strained InAs channel (PHEMTs) were analyzed rigorously using a recently established theory that takes into account the nonparabolicity of the conduction band of the channel layer. The sheet density of the 2DEG in InxGa1-xAs-PHEMTs and the drain I-V characteristics of those devices were calculated theoretically and compared with the density and characteristics obtained experimentally. Not only the calculated threshold voltage (VTH) but also the calculated transconductance agreed fairly well with the corresponding values obtained experimentally. When the effects of the compositions of the InxGa1-xAs subchannel layer in the composite channel and the channel layer on energy states of 2DEG were investigated in order to establish a guiding principle for a design of the channel structure in PHEMTs, it was found that VTH is determined by the effective conduction-band offset energy ΔEC between the InAlAs barrier and the channel layers.
1. On-line combination of high performance liquid chromatography with comprehensive two-dimensional gas chromatography-triple quadrupole mass spectrometry: a proof of principle study.
Science.gov (United States)
Zoccali, Mariosimone; Tranchida, Peter Quinto; Mondello, Luigi
2015-02-03
The present contribution is focused on the on-line combination of high performance liquid chromatography (HPLC), cryogenically modulated comprehensive two-dimensional gas chromatography (GC × GC), and triple quadrupole mass spectrometry (QqQ MS), generating a very powerful unified separation-science tool. The instrument can be used in seven different combinations ranging from one-dimensional HPLC with a photodiode array detector to on-line LC × GC × GC/QqQ MS. The main focus of the present research is directed to the LC-GC × GC/QqQ MS configuration, with its analytical potential shown in a proof-of-principle study involving a very complex sample, namely, coal tar. Specifically, a normal-phase LC process enabled the separation of three classes of coal tar compounds: (1) nonaromatic hydrocarbons; (2) unsaturated compounds (with and without S); (3) oxygenated constituents. The HPLC fractions were transferred to the GC × GC instrument via a syringe-based interface mounted on an autosampler. Each fraction was subjected to a specific programmed temperature vaporizer GC × GC/QqQ MS untargeted or targeted analysis. For example, the coal tar S-containing compounds were pinpointed through multiple-reaction-monitoring analysis, while full-scan information was attained for the oxygenated constituents.
2. Comprehensive magnetotransport characterization of two dimensional electron gas in AlGaN/GaN high electron mobility transistor structures leading to the assessment of interface roughness
Energy Technology Data Exchange (ETDEWEB)
Mishra, Manna Kumari [Solid State Physics Laboratory, Lucknow Road, Timarpur, Delhi-110054 (India); Netaji Subhas Institute of Technology, Dwarka, New Delhi-110078 (India); Sharma, Rajesh K., E-mail: [email protected]; Manchanda, Rachna; Bag, Rajesh K.; Muralidharan, Rangarajan [Solid State Physics Laboratory, Lucknow Road, Timarpur, Delhi-110054 (India); Thakur, Om Prakash [Netaji Subhas Institute of Technology, Dwarka, New Delhi-110078 (India)
2014-09-15
Magnetotransport in two distinct AlGaN/GaN HEMT structures grown by Molecular Beam Epitaxy (MBE) on Fe-doped templates is investigated using Shubnikov de-Haas Oscillations in the temperature range of 1.8–6 K and multicarrier fitting in the temperature range of 1.8–300 K. The temperature dependence of the two dimensional electron gas mobility is extracted from simultaneous multicarrier fitting of transverse and longitudinal resistivity as a function of magnetic field and the data is utilized to estimate contribution of interface roughness to the mobility and the corresponding transport lifetime. The quantum scattering time obtained from the analysis of Shubnikov de Haas Oscillations in transverse magnetoresistance along with the transport lifetime time were used to estimate interface roughness amplitude and lateral correlation length. The results indicate that the insertion of AlN over layer deposited prior to the growth of GaN base layer on Fe doped GaN templates for forming HEMT structures reduced the parallel conduction but resulted in an increase in interface roughness.
3. Comprehensive magnetotransport characterization of two dimensional electron gas in AlGaN/GaN high electron mobility transistor structures leading to the assessment of interface roughness
Directory of Open Access Journals (Sweden)
Manna Kumari Mishra
2014-09-01
Full Text Available Magnetotransport in two distinct AlGaN/GaN HEMT structures grown by Molecular Beam Epitaxy (MBE on Fe-doped templates is investigated using Shubnikov de-Haas Oscillations in the temperature range of 1.8–6 K and multicarrier fitting in the temperature range of 1.8–300 K. The temperature dependence of the two dimensional electron gas mobility is extracted from simultaneous multicarrier fitting of transverse and longitudinal resistivity as a function of magnetic field and the data is utilized to estimate contribution of interface roughness to the mobility and the corresponding transport lifetime. The quantum scattering time obtained from the analysis of Shubnikov de Haas Oscillations in transverse magnetoresistance along with the transport lifetime time were used to estimate interface roughness amplitude and lateral correlation length. The results indicate that the insertion of AlN over layer deposited prior to the growth of GaN base layer on Fe doped GaN templates for forming HEMT structures reduced the parallel conduction but resulted in an increase in interface roughness.
4. Giant tunneling electroresistance induced by ferroelectrically switchable two-dimensional electron gas at nonpolar BaTiO3/SrTiO3 interface
Science.gov (United States)
Wu, Qingyun; Shen, Lei; Yang, Ming; Zhou, Jun; Chen, Jingsheng; Feng, Yuan Ping
2016-10-01
Using first-principles calculations, we investigate the tunneling electroresistance (TER) of ferroelectric tunnel junctions [Pt /BaTiO3(BTO)/SrTiO3(STO )/Pt ]. It is found that the TER of Pt/BTO/STO/Pt junctions can be greatly increased with increasing thickness of STO layers. The underlying physics of this giant TER is the switchable two-dimensional electron gas (2DEG) at a nonpolar BTO/STO interface induced by the ferroelectric polarization. Our calculations show that when the ferroelectric polarization is pointing from BTO to STO, a 2DEG forms at the interface and acts as bridge for electrons to tunnel through the junctions. Nevertheless, there is no 2DEG at the interface under the opposite direction of the ferroelectric polarization, which results in a large tunnel resistance. More importantly, this ferroelectrically switchable 2DEG leads to a low resistance area product for Pt/BTO/STO/Pt junctions, which offers good compatibility with other components in an integrated circuit and is highly desired for industrial applications.
5. Application of comprehensive two-dimensional gas chromatography with mass spectrometric detection for the analysis of selected drug residues in wastewater and surface water
Institute of Scientific and Technical Information of China (English)
Petr Lacina; Ludmila Mravcová; Milada Vávrová
2013-01-01
Pharmaceutical residues have become tightly controlled environmental contaminants in recent years,due to their increasing concentration in environmental components.This is mainly caused by their high level of production and everyday consumption.Therefore there is a need to apply new and sufficiently sensitive analytical methods,which can detect the presence of these contaminants even in very low concentrations.This study is focused on the application of a reliable analytical method for the analysis of 10 selected drug residues,mainly from the group of non-steroidal anti-iaffammatory drugs (salicylic acid,acetylsalicylic acid,clofibric acid,ibuprofen,acetaminophen,caffeine,naproxen,mefenamic acid,ketoprofen,and dicofenac),in wastewaters and surface waters.This analytical method is based on solid phase extraction,derivatization by N-methyl-N-(trimethylsilyl)trifluoroacetamide (MSTFA) and finally analysis by comprehensive two-dimensional gas chromatography with Time-of-Flight mass spectrometric detection (GCxGC-TOF MS).Detection limits ranged from 0.18 to 5 ng/L depending on the compound and selected matrix.The method was successfully applied for detection of the presence of selected pharmaceuticals in the Svratka River and in wastewater from the wastewater treatment plant in Brno-Modrice,Czech Republic.The concentration of pharmaceuticals varied from one to several hundreds of ng/L in surface water and from one to several tens of μg/L in wastewater.
6. Analysis of honeybush tea (Cyclopia spp.) volatiles by comprehensive two-dimensional gas chromatography using a single-stage thermal modulator.
Science.gov (United States)
Ntlhokwe, Gaalebalwe; Tredoux, Andreas G J; Górecki, Tadeusz; Edwards, Matthew; Vestner, Jochen; Muller, Magdalena; Erasmus, Lené; Joubert, Elizabeth; Christel Cronje, J; de Villiers, André
2017-07-01
The applicability of comprehensive two-dimensional gas chromatography (GC×GC) using a single-stage thermal modulator was explored for the analysis of honeybush tea (Cyclopia spp.) volatile compounds. Headspace solid phase micro-extraction (HS-SPME) was used in combination with GC×GC separation on a non-polar × polar column set with flame ionisation (FID) detection for the analysis of fermented Cyclopia maculata, Cyclopia subternata and Cyclopia genistoides tea infusions of a single harvest season. Method optimisation entailed evaluation of the effects of several experimental parameters on the performance of the modulator, the choice of columns in both dimensions, as well as the HS-SPME extraction fibre. Eighty-four volatile compounds were identified by co-injection of reference standards. Principal component analysis (PCA) showed clear differentiation between the species based on their volatile profiles. Due to the highly reproducible separations obtained using the single-stage thermal modulator, multivariate data analysis was simplified. The results demonstrate both the complexity of honeybush volatile profiles and the potential of GC×GC separation in combination with suitable data analysis techniques for the investigation of the relationship between sensory properties and volatile composition of these products. The developed method therefore offers a fast and inexpensive methodology for the profiling of honeybush tea volatiles. Graphical abstract Surface plot obtained for the GC×GC-FID analysis of honeybush tea volatiles.
7. Study of phase separation using liquid-gas model of lattice-gas cellular automata
Energy Technology Data Exchange (ETDEWEB)
Ebihara, Kenichi; Watanabe, Tadashi; Kaburaki, Hideo [Japan Atomic Energy Research Inst., Tokai, Ibaraki (Japan). Tokai Research Establishment
1997-07-01
This report describes the study of phase separation by the liquid gas model of lattice gas cellular automata. The lattice gas cellular automaton is one model for simulating fluid phenomena which was proposed by Frisch, Hasslacher and Pomeau in 1986. In 1990, Appert and Zaleski added a new long-range interaction to lattice gas cellular automata to construct a model, the liquid-gas model, which could simulate phase separation using lattice-gas cellular automata. Gerits et al formulated the liquid-gas model mathematically using the theory of statistical dynamics in 1993 and explained the mechanism of phase separation in the liquid-gas model using the equation of state. At first this report explains the FHP model of lattice gas cellular automata and derives fluid dynamics equations such as the equation of continuity and the Navier-Stokes equation. Then the equation of state for the liquid-gas model which was derived by Gerits et al is modified by adding the interactions which were proposed by Appert but not considered by Gerits et al. The modified equation of state is verified by the computer simulation using the liquid gas model. The relation between phase separation and the equation of state is discussed. (author)
8. Characterisation of dense non-aqueous phase liquids of coal tar using comprehensive two-dimensional gas chromatography coupled with time of flight mass spectrometry.
Science.gov (United States)
Gauchotte-Lindsay, Caroline; McGregor, Laura; Richards, Phil; Kerr, Stephanie; Glenn, Aliyssa; Thomas, Russell; Kalin, Robert
2013-04-01
Comprehensive two-dimensional gas chromatography (GCxGC) is a recently developed analytical technique in which two capillary columns with different stationary phases are placed in series enabling planar resolution of the analytes. The resolution power of GCxGC is one order of magnitude higher than that of one dimension gas chromatography. Because of its high resolution capacity, the use of GCxGC for complex environmental samples such as crude oils, petroleum derivatives and polychlorinated biphenyls mixtures has rapidly grown in recent years. We developed a one-step method for the forensic analysis of coal tar dense non-aqueous phase liquids (DNAPLs) from former manufactured gas plant (FMGP) sites. Coal tar is the by-product of the gasification of coal for heating and lighting and it is composed of thousands of organic and inorganic compounds. Before the boom of natural gases and oils, most towns and cities had one or several manufactured gas plants that have, in many cases, left a devastating environmental print due to coal tar contamination. The fate of coal tar DNAPLs, which can persist in the environment for more than a hundred years, is therefore of crucial interest. The presented analytical method consists of a unique clean-up/ extraction stage by pressurized liquid extraction and a single analysis of its organic chemical composition using GCxGC coupled with time of flight mass spectrometry (TOFMS). The chemical fingerprinting is further improved by derivatisation by N,O-bis(trimethylsilyl)trifluoroacetamide (BSTFA) of the tar compounds containing -OH functions such as alcohols and carboxylic acids. We present here how, using the logical order of elution in GCxGC-TOFMS system, 1) the identification of never before observed -OH containing compounds is possible and 2) the isomeric selectivity of an oxidation reaction on a DNAPL sample can be revealed. Using samples collected at various FMGP sites, we demonstrate how this GCxGC method enables the simultaneous
9. Two-dimensional calculus
CERN Document Server
Osserman, Robert
2011-01-01
The basic component of several-variable calculus, two-dimensional calculus is vital to mastery of the broader field. This extensive treatment of the subject offers the advantage of a thorough integration of linear algebra and materials, which aids readers in the development of geometric intuition. An introductory chapter presents background information on vectors in the plane, plane curves, and functions of two variables. Subsequent chapters address differentiation, transformations, and integration. Each chapter concludes with problem sets, and answers to selected exercises appear at the end o
10. Two dimensional vernier
Science.gov (United States)
Juday, Richard D. (Inventor)
1992-01-01
A two-dimensional vernier scale is disclosed utilizing a cartesian grid on one plate member with a polar grid on an overlying transparent plate member. The polar grid has multiple concentric circles at a fractional spacing of the spacing of the cartesian grid lines. By locating the center of the polar grid on a location on the cartesian grid, interpolation can be made of both the X and Y fractional relationship to the cartesian grid by noting which circles coincide with a cartesian grid line for the X and Y direction.
11. Damage spreading in a driven lattice gas model
Science.gov (United States)
Rubio Puzzo, M. Leticia; Saracco, Gustavo P.; Albano, Ezequiel V.
2013-06-01
We studied damage spreading in a Driven Lattice Gas (DLG) model as a function of the temperature T, the magnitude of the external driving field E, and the lattice size. The DLG model undergoes an order-disorder second-order phase transition at the critical temperature Tc(E), such that the ordered phase is characterized by high-density strips running along the direction of the applied field; while in the disordered phase one has a lattice-gas-like behavior. It is found that the damage always spreads for all the investigated temperatures and reaches a saturation value D that depends only on T. D increases for TTc(E=∞) and is free of finite-size effects. This behavior can be explained as due to the existence of interfaces between the high-density strips and the lattice-gas-like phase whose roughness depends on T. Also, we investigated damage spreading for a range of finite fields as a function of T, finding a behavior similar to that of the case with E=∞.
12. Fick's Law in a Random Lattice Lorentz Gas
Science.gov (United States)
Lefevere, Raphaël
2015-06-01
We provide a proof that the stationary macroscopic current of particles in a random lattice Lorentz gas satisfies Fick's law when connected to particles reservoirs. We consider a box on a d + 1-dimensional lattice and when , we show that under a diffusive rescaling of space and time, the probability of finding a current different from its stationary value is exponentially small in time. Its stationary value is given by the conductivity times the difference of chemical potentials of the reservoirs. The proof is based on the fact that in a high dimension, random walks have a small probability of making loops or intersecting each other when starting sufficiently far apart.
13. Crosslinked structurally-tuned polymeric ionic liquids as stationary phases for the analysis of hydrocarbons in kerosene and diesel fuels by comprehensive two-dimensional gas chromatography.
Science.gov (United States)
Zhang, Cheng; Park, Rodney A; Anderson, Jared L
2016-04-01
Structurally-tuned ionic liquids (ILs) have been previously applied as the second dimension column in comprehensive two-dimensional gas chromatography (GC×GC) and have demonstrated high selectivity in the separation of individual aliphatic hydrocarbons from other aliphatic hydrocarbons. However, the maximum operating temperatures of these stationary phases limit the separation of analytes with high boiling points. In order to address this issue, a series of polymeric ionic liquid (PIL)-based stationary phases were prepared in this study using imidazolium-based IL monomers via in-column free radical polymerization. The IL monomers were functionalized with long alkyl chain substituents to provide the needed selectivity for the separation of aliphatic hydrocarbons. Columns were prepared with different film thicknesses to identify the best performing stationary phase for the separation of kerosene. The bis[(trifluoromethyl)sulfonyl]imide ([NTf2](-))-based PIL stationary phase with larger film thickness (0.28μm) exhibited higher selectivity for aliphatic hydrocarbons and showed a maximum allowable operating temperature of 300°C. PIL-based stationary phases containing varied amount of IL-based crosslinker were prepared to study the effect of the crosslinker on the selectivity and thermal stability of the resulting stationary phase. The optimal resolution of aliphatic hydrocarbons was achieved when 50% (w/w) of crosslinker was incorporated into the PIL-based stationary phase. The resulting stationary phase exhibited good selectivity for different groups of aliphatic hydrocarbons even after being conditioned at 325°C. Finally, the crosslinked PIL-based stationary phase was compared with SUPELCOWAX 10 and DB-17 columns for the separation of aliphatic hydrocarbons in diesel fuel. Better resolution of aliphatic hydrocarbons was obtained when employing the crosslinked PIL-based stationary phase as the second dimension column.
14. Impurity Profiling of a Chemical Weapon Precursor for Possible Forensic Signatures by Comprehensive Two-Dimensional Gas Chromatography/Mass Spectrometry and Chemometrics
Energy Technology Data Exchange (ETDEWEB)
Hoggard, Jamin C.; Wahl, Jon H.; Synovec, Robert E.; Mong, Gary M.; Fraga, Carlos G.
2010-01-15
In this work we present the feasibility of using analytical chemical and chemometric methodologies to reveal and exploit the organic impurity profiles from commercial dimethyl methylphosphonate (DMMP) samples to illustrate the type of forensic information that may be obtained from chemical-attack evidence. Using DMMP as a model compound for a toxicant that may be used in a chemical attack, we used comprehensive two-dimensional gas chromatography mass spectrometric detection (GC × GC-TOFMS) to detect and identify trace organic impurities in six samples of commercially acquired DMMP. The GC x GC-TOFMS data were analyzed to produce impurity profiles for all six DMMP samples using 29 analyte impurities. The use of PARAFAC for the mathematical resolution of overlap GC x GC peaks ensured clean spectra for the identification of many of the detected analytes by spectral library matching. The use of statistical pairwise comparison revealed that there were trace impurities that were quantitatively similar and different among five of the six DMMP samples. Two of the DMMP samples were revealed to have identical impurity profiles by this approach. The use of nonnegative matrix factorization proved that there were five distinct DMMP sample types as illustrated by the clustering of the multiple DMMP analyses into 5 distinct clusters in the scores plots. The two indistinguishable DMMP samples were confirmed by their chemical supplier to be from the same bulk source. Sample information from the other chemical suppliers supported that the other five DMMP samples were likely from different bulk sources. These results demonstrate that the matching of synthesized products from the same source is possible using impurity profiling. In addition, the identified impurities common to all six DMMP samples provide strong evidence that basic route information can be obtained from impurity profiles. In addition, impurities that may be unique to the sole bulk manufacturer of DMMP were found in
15. Partial least squares analysis of rocket propulsion fuel data using diaphragm valve-based comprehensive two-dimensional gas chromatography coupled with flame ionization detection.
Science.gov (United States)
Freye, Chris E; Fitz, Brian D; Billingsley, Matthew C; Synovec, Robert E
2016-06-01
The chemical composition and several physical properties of RP-1 fuels were studied using comprehensive two-dimensional (2D) gas chromatography (GC×GC) coupled with flame ionization detection (FID). A "reversed column" GC×GC configuration was implemented with a RTX-wax column on the first dimension ((1)D), and a RTX-1 as the second dimension ((2)D). Modulation was achieved using a high temperature diaphragm valve mounted directly in the oven. Using leave-one-out cross-validation (LOOCV), the summed GC×GC-FID signal of three compound-class selective 2D regions (alkanes, cycloalkanes, and aromatics) was regressed against previously measured ASTM derived values for these compound classes, yielding root mean square errors of cross validation (RMSECV) of 0.855, 0.734, and 0.530mass%, respectively. For comparison, using partial least squares (PLS) analysis with LOOCV, the GC×GC-FID signal of the entire 2D separations was regressed against the same ASTM values, yielding a linear trend for the three compound classes (alkanes, cycloalkanes, and aromatics), yielding RMSECV values of 1.52, 2.76, and 0.945 mass%, respectively. Additionally, a more detailed PLS analysis was undertaken of the compounds classes (n-alkanes, iso-alkanes, mono-, di-, and tri-cycloalkanes, and aromatics), and of physical properties previously determined by ASTM methods (such as net heat of combustion, hydrogen content, density, kinematic viscosity, sustained boiling temperature and vapor rise temperature). Results from these PLS studies using the relatively simple to use and inexpensive GC×GC-FID instrumental platform are compared to previously reported results using the GC×GC-TOFMS instrumental platform.
16. Effect of {sup 60}Co gamma-irradiation on two-dimensional electron gas transport and device characteristics of AlGaN/GaN HEMTs
Energy Technology Data Exchange (ETDEWEB)
Umana-Membreno, G.A.; Dell, J.M.; Parish, G.; Nener, B.D.; Faraone, L. [School of Electrical, Electronic and Computer Engineering, The University ofWestern Australia, Crawley WA 6009 (Australia); Ventury, R.; Mishra, U.K. [Dept. of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106 (United States)
2005-05-01
The effect of {sup 60}Co gamma-irradiation on Al{sub 0.33}Ga{sub 0.67}N/GaN high-electron-mobility transistors (HEMTs) has been investigated using DC and geometrical magnetoresistance measurements. The devices studied were of similar epitaxial structure, yet differed in the doping levels of the Al{sub 0.33}Ga{sub 0.67}N barrier layer: (A) nominally undoped and (B) Si-doped with SiN{sub x} passivation. Exposure to cumulative gamma-ray doses up to 20 Mrad(Si) is shown to induce significant changes in drain-current level, threshold voltage and gate leakage current level. Analysis of magnetoresistance characteristics measured at 80 K indicated that irradiation induced an increase in two-dimensional electron-gas (2DEG) density, which leads to negative threshold voltage shifts observable in the drain current versus gate voltage characteristics, attributable to the introduction of defect centers in the Al{sub 0.33}Ga{sub 0.67}N layer and/or at the gate-AlGaN interface. The 2DEG mobility-concentration profiles are shown to remain approximately unchanged for doses up to 20 Mrad(Si). Device failure, evidenced as loss of gate control over the channel and/or excessive gate leakage, occurred after exposure to 30 Mrad(Si) for device A, whereas sample B failed after 20 Mrad(Si) total dose due to failure of half of the gate contact. Degradation of gate and source/drain contacts characteristics with total dose appears to limit the tolerance of the studied HEMTs to {sup 60}Co gamma-irradiation. (copyright 2005 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) (orig.)
17. Separation and screening of short-chain chlorinated paraffins in environmental samples using comprehensive two-dimensional gas chromatography with micro electron capture detection.
Science.gov (United States)
Xia, Dan; Gao, Lirong; Zhu, Shuai; Zheng, Minghui
2014-11-01
Short-chain chlorinated paraffins (SCCPs) are highly complex technical mixtures with thousands of isomers and numerous homologs. They are classified as priority candidate persistent organic pollutants under the Stockholm Convention for their persistence, bioaccumulation, and toxicity. Analyzing SCCPs is challenging because of the complexity of the mixtures. Chromatograms of SCCPs acquired using one-dimensional (1D) gas chromatography (GC) contain a large characteristic "peak" with a broad and unresolved profile. Comprehensive two-dimensional GC (GC×GC) shows excellent potential for separating complex mixtures. In this study, GC×GC coupled with micro electron capture detection (μECD) was used to separate and screen SCCPs. The chromatographic parameters, including the GC column types, oven temperature program, and modulation period, were systematically optimized. The SCCP congeners were separated into groups using a DM-1 column connected to a BPX-50 column. The SCCP congeners in technical mixtures were separated according to the number of chlorine substituents for a given carbon chain length and according to the number of carbon atoms plus chlorine atoms for different carbon chain lengths. A fish tissue sample was analyzed to illustrate the feasibility of the GC×GC-μECD method in analyzing biological samples. Over 1,500 compounds were identified in the fish extract, significantly more than were identified using 1D GC. The detection limits for five selected SCCP congeners were between 1 and 5 pg/L using the GC×GC method, and these were significantly lower than those achieved using 1D GC. This method is a good choice for analysis of SCCPs in environmental samples, exhibiting good separation and good sensitivity.
18. Implementation of comprehensive two-dimensional gas chromatography-time-of-flight mass spectrometry for the simultaneous determination of halogenated contaminants and polycyclic aromatic hydrocarbons in fish
Energy Technology Data Exchange (ETDEWEB)
Kalachova, Kamila; Pulkrabova, Jana; Cajka, Tomas; Drabova, Lucie; Hajslova, Jana [Institute of Chemical Technology, Prague (Czech Republic). Department of Food Chemistry and Analysis, Faculty of Food and Biochemical Technology
2012-07-15
In the presented study, comprehensive two-dimensional gas chromatography coupled to time-of-flight mass spectrometry (GC x GC-TOFMS) was shown to be a powerful tool for the simultaneous determination of various groups of contaminants including 18 polychlorinated biphenyls (PCBs), seven polybrominated diphenyl ethers (PBDEs), and 16 polycyclic aromatic hydrocarbons (PAHs). Since different groups of analytes (traditionally analyzed separately) were included into one instrumental method, significant time savings were achieved. Following the development of an integrated sample preparation procedure for an effective and rapid isolation of several groups of contaminants from fish tissue, the GC x GC-TOFMS instrumental method was optimized to obtain the best chromatographic resolution and low quantification limits (LOQs) of all target analytes in a complex mixture. Using large-volume programmable temperature vaporization, the following LOQs were achieved - PCBs, 0.01-0.25 {mu}g/kg; PBDEs, 0.025-5 {mu}g/kg; PAHs 0.025-0.5 {mu}g/kg. Furthermore, several capillary column combinations (BPX5, BPX50, and Rxi-17Sil-ms in the first dimension and BPX5, BPX50, Rt-LC35, and HT8 in the second dimension) were tested during the experiments, and the optimal separation of all target analytes even of critical groups of PAHs (group (a): benz[a]anthracene, cyclopenta[cd]pyrene and chrysene; group (b): benzo[b]fluoranthene, benzo[j]fluoranthene and benzo[k]fluoranthene; group (c): dibenz[ah]anthracene, indeno[1,2,3-cd]pyrene and benzo[ghi]perylene) was observed on BPX5 x BPX50 column setup. Moreover, since the determination of target analytes was performed using TOFMS detector, further identification of other non-target compounds in real life samples was also feasible. (orig.)
19. Congener-specific carbon isotopic analysis of technical PCB and PCN mixtures using two-dimensional gas chromatography-isotope ratio mass spectrometry.
Science.gov (United States)
Horii, Yuichi; Kannan, Kurunthachalam; Petrick, Gert; Gamo, Toshitaka; Falandysz, Jerzy; Yamashita, Nobuyoshi
2005-06-01
Analysis of stable carbon isotope fractionation is a useful method to study the sources and fate of anthropogenic organic contaminants such as polychlorinated biphenyls (PCBs) in the environment. To evaluate the utility of carbon isotopes, determination of isotopic ratios of 13C/12C in source materials, for example, technical PCB preparations, is needed. In this study, we determined delta13C values of 31 chlorobiphenyl (CB) congeners in 18 technical PCB preparations and 15 chloronaphthalene (CN) congeners in 6 polychlorinated naphthalene preparations using two-dimensional gas chromatography-combustion furnace-isotope ratio mass spectrometry (2DGC-C-IRMS). Development of 2DGC-IRMS enabled improved resolution and sensitivity of compound-specific carbon isotope analysis (CSIA) of CB or CN congeners. Delta13C values of PCB congeners ranged from -34.4 (Delors) to -22.0/1000 (Sovol). Analogous PCB preparations with similar chlorine content, but different geographical origin, had different delta13C values. PCB preparations from Eastern European countries--Delors, Sovol, Trichlorodiphenyl, and Chlorofen--had distinct delta13C values. PCB mixtures showed increased 13C depletion with increasing chlorine content. Delta13C values for individual CB congeners varied depending on the degree of chlorination in technical mixtures. Delta13C values of CN congeners in Halowaxes ranged from -26.3 to -21.7/1000 and these values are within the ranges observed for PCBs. This study establishes the range of delta13C values in technical PCB and PCN preparations, which may prove to be useful in the determination of sources of these compounds in the environment. This is the first study to employ 2DGC-IRMS analysis of delta13C values in technical PCB and PCN preparations.
20. Energetic, spatial, and momentum character of the electronic structure at a buried interface: The two-dimensional electron gas between two metal oxides
Science.gov (United States)
Nemšák, S.; Conti, G.; Gray, A. X.; Palsson, G. K.; Conlon, C.; Eiteneer, D.; Keqi, A.; Rattanachata, A.; Saw, A. Y.; Bostwick, A.; Moreschini, L.; Rotenberg, E.; Strocov, V. N.; Kobayashi, M.; Schmitt, T.; Stolte, W.; Ueda, S.; Kobayashi, K.; Gloskovskii, A.; Drube, W.; Jackson, C. A.; Moetakef, P.; Janotti, A.; Bjaalie, L.; Himmetoglu, B.; Van de Walle, C. G.; Borek, S.; Minar, J.; Braun, J.; Ebert, H.; Plucinski, L.; Kortright, J. B.; Schneider, C. M.; Balents, L.; de Groot, F. M. F.; Stemmer, S.; Fadley, C. S.
2016-06-01
The interfaces between two condensed phases often exhibit emergent physical properties that can lead to new physics and novel device applications and are the subject of intense study in many disciplines. We here apply experimental and theoretical techniques to the characterization of one such interesting interface system: the two-dimensional electron gas (2DEG) formed in multilayers consisting of SrTi O3 (STO) and GdTi O3 (GTO). This system has been the subject of multiple studies recently and shown to exhibit very high carrier charge densities and ferromagnetic effects, among other intriguing properties. We have studied a 2DEG-forming multilayer of the form [6unit cells (u .c .) STO /3 u .c .of GTO ] 20 using a unique array of photoemission techniques including soft and hard x-ray excitation, soft x-ray angle-resolved photoemission, core-level spectroscopy, resonant excitation, and standing-wave effects, as well as theoretical calculations of the electronic structure at several levels and of the actual photoemission process. Standing-wave measurements below and above a strong resonance have been exploited as a powerful method for studying the 2DEG depth distribution. We have thus characterized the spatial and momentum properties of this 2DEG in detail, determining via depth-distribution measurements that it is spread throughout the 6 u.c. layer of STO and measuring the momentum dispersion of its states. The experimental results are supported in several ways by theory, leading to a much more complete picture of the nature of this 2DEG and suggesting that oxygen vacancies are not the origin of it. Similar multitechnique photoemission studies of such states at buried interfaces, combined with comparable theory, will be a very fruitful future approach for exploring and modifying the fascinating world of buried-interface physics and chemistry. | {"extraction_info": {"found_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": 0.8204216957092285, "perplexity": 3548.971653679587}, "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-2018-09/segments/1518891811794.67/warc/CC-MAIN-20180218062032-20180218082032-00251.warc.gz"} |
https://infoscience.epfl.ch/record/141672 | Infoscience
Journal article
# Si-O-Si bond-angle distribution in vitreous silica from first-principles Si-29 NMR analysis
The correlation between Si-29 chemical shifts and Si-O-Si bond angles in SiO2 is determined within density-functional theory for the full range of angles present in vitreous silica. This relation closely reproduces measured shifts of crystalline polymorphs. The knowledge of the correlation allows us to reliably extract from the experimental NMR spectrum the mean (151 degrees) and the standard deviation (11 degrees) of the Si-O-Si angular distribution of vitreous silica. In particular, we show that the Mozzi-Warren Si-O-Si angular distribution is not consistent with the NMR data. This analysis illustrates the potential of our approach for structural determinations of silicate glasses. | {"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": 0.8274826407432556, "perplexity": 4272.223123367545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1487501171781.5/warc/CC-MAIN-20170219104611-00496-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://dataspace.princeton.edu/jspui/handle/88435/dsp01np193c633 | Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01np193c633
Title: Parallel electron force balance and the L-H transition Contributors: Stoltzfus-Dueck, T.U. S. Department of Energy contract number DE-AC02-09CH11466 Keywords: L-H transitionTokamakReynolds stressZonal flows Issue Date: May-2016 Publisher: Princeton Plasma Physics Laboratory, Princeton University Related Publication: Physics of Plasmas, Vol. 23, p. 054505 (May 2016) Abstract: In one popular description of the L-H transition, energy transfer to the mean flows directly depletes turbulence fluctuation energy, resulting in suppression of the turbulence and a corresponding transport bifurcation. However, electron parallel force balance couples nonzonal velocity fluctuations with electron pressure fluctuations on rapid timescales, comparable with the electron transit time. For this reason, energy in the nonzonal velocity stays in a fairly fixed ratio to the free energy in electron density fluctuations, at least for frequency scales much slower than electron transit. In order for direct depletion of the energy in turbulent fluctuations to cause the L-H transition, energy transfer via Reynolds stress must therefore drain enough energy to significantly reduce the sum of the free energy in nonzonal velocities and electron pressure fluctuations. At low k, the electron thermal free energy is much larger than the energy in nonzonal velocities, posing a stark challenge for this model of the L-H transition. URI: http://arks.princeton.edu/ark:/88435/dsp01np193c633 Referenced By: http://dx.doi.org/10.1063/1.4951015 Appears in Collections: NSTX-U
Files in This Item:
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https://gilkalai.wordpress.com/tag/borsuk/ | Borsuk’s Conjecture
Karol Borsuk conjectured in 1933 that every bounded set in $R^d$ can be covered by $d+1$ sets of smaller diameter. Jeff Kahn and I found a counterexample in 1993. It is based on the Frankl-Wilson theorem.
Let $\cal G$ be the set of $\pm 1$ vectors of length $n$. Suppose that $n=4p$ and $p$ is a prime, as the conditions of Frankl-Wilson theorem require. Let ${\cal G'} = \{(1/\sqrt n)x:x \in {\cal G}\}$. All vectors in ${\cal G}'$ are unit vectors.
Consider the set $X=\{x \otimes x: x \in {\cal G}'\}$. $X$ is a subset of $R^{n^2}$.
Remark: If $x=(x_1,x_2,\dots,x_n)$, regard $x\otimes x$ as the $n$ by $n$ matrix with entries $(x_ix_j)$.
It is easy to verify that:
Claim: $ = ^2$.
It follows that all vectors in $X$ are unit vectors, and that the inner product between every two of them is nonnegative. The diameter of $X$ is therefore $\sqrt 2$. (Here we use the fact that the square of the distance between two unit vectors $x$ and $y$ is 2 minus twice their inner product.)
Suppose that $Y \subset X$ has a smaller diameter. Write $Y=\{x \otimes x: x \in {\cal F}\}$ for some subset $\cal F$ of $\cal G$. This means that $Y$ (and hence also $\cal F$) does not contain two orthogonal vectors and therefore by the Frankl-Wilson theorem
$|{\cal F}| \le U=4({{n} \choose {0}}+{{n}\choose {1}}+\dots+{{n}\choose{p-1}})$.
It follows that the number of sets of smaller diameter needed to cover $X$ is at least $2^n / U$. This clearly refutes Borsuk’s conjecture for large enough $n$. Sababa.
Let me explain in a few more words Continue reading | {"extraction_info": {"found_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": 33, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.993916928768158, "perplexity": 132.56926452915775}, "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-2015-32/segments/1438042982502.13/warc/CC-MAIN-20150728002302-00061-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/153721/is-gravitational-chern-simons-action-topological-or-not/153774 | # Is gravitational Chern-Simons action “topological” or not?
Here are the 2+1D gravitational Chern-Simons action of the connection $\Gamma$ or spin-connection:
$$S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{a}$$
$$S=\int\omega\wedge\mathrm{d}\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega \tag{b}$$
A usual Chern-Simons theory of 1-form gauge field is said to be topological, since $S=\int A\wedge\mathrm{d}A + \frac{2}{3}A\wedge A \wedge A$ does not depend on the spacetime metric.
(1) Are (a) and (b) topological or not?
(2) Do (a) and (b) they depend on the spacetime metric (the action including the integrand)?
(3) Do we have topological gravitational Chern-Simons theory then? Then, what do questions (1) and (2) mean in this context of being topological?
Classically they are clearly topological. The metric does not appear, and you don't need a metric for integration on manifolds to make sense. Now in dimension 3 you can cast the Einstein-Hilbert action into a Chern-Simons theory as you say. The connection takes it values in the Lie algebra of the Poincare group.
In higher dimensions you need to use higher invariant polynomials, remember you need the integration to make sense. In this way you can get higher dimensional theories and this includes the Einstein-Hilbert action, but also with higher curvature terms.
There is no experimental evidence for the inclusion of this higher curvature terms in gravity. They are however, interesting from a non-perturbative quantum gravity perspective using the renormalisation group flow and the asymptotic safety.
Now, in perturbation quantisation of the Chern-Simons theory you do need a metric to define the path integrals. Witten in 1989 did this [1]. You get a expressions that do depend on this choice of metric, but he then showed how to make this all metric independent by adding another term.
References [1] Edward Witten, Quantum Field Theory and the Jones Polynomialm 121 (3) (1989) 351–399.
Whilst the question is not a resource request, I would recommend Edward Witten's paper on the topic published in 1988, titled, 2+1 Dimensional Gravity as a Soluble System. In the paper, Witten shows:
• $2+1$ dimensional gravity with or without $\Lambda$ is soluble classically and at the quantum level
• $2+1$ dimensional gravity is related to a Yang-Mills theory with only a Chern-Simons term
• At the quantum level, such a theory has a vanishing beta function
Witten also discusses other routes, such as the relation to representations of the Virasoro algebra which is related to conformal field theory and string theory. Finally, to answer your question directly, if we interpret the fields as gauge fields, yes, the action is a topological invariant, at least classically.
The gravitational Chern-Simons action is topological, yes. The gauge connection encodes the field of gravity and since it is being integrated over, the result does not depend on a metric. (In the expressions you write maybe the vielbein contribution is missing? Or maybe you mean to have absorbed it in the notation.) Notice that it's just the usual Chern-Simons term which may be written down for many gauge groups, here specialized to the the Poincaré group or an AdS groups.
What one needs to know to understand what's going on here is this:
1. The Einstein-Hilbert action functional always has a first-order formulation in terms of vielbeing and spin connections, which are nothing but the componentes of a 1-form with values in the Poincaré Lie algebra. More precisely, the field of gravity may always be written as a Cartan connection for the inclusion of the Lorentz group into the Poincaré group.
2. Now when one writes down this first-order version of the Einstein Hilbert action in 3-dimensions then a little miracle happens: it turs out to be equal to the Chern-Simons action functional with that gauge group. See at Chern-Simons gravity.
I think the statement in Witten's paper, "Quantum Field Theory and the Jones Polynomial" saying the term is topological in the sense it is indeed independent of metric. However, he also mentioned, in order to make sense of this integration, i.e. make it to be a number, you need to choose a trivialization of tangent bundle, i.e. choose a framing. The tricky thing is the action is not invariant under twisting of framing. In this sense, the gravitational Chern-Simons is in fact a topological invariants of 3-manifold with chosen framing.
Well, if you want a truly topological invariants, you can choose a proper coefficient to make it independent of framing. E.g., in the paper you mentioned, if you choose $c=24$, the partition function will be independent of framing. | {"extraction_info": {"found_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": 0.9359186291694641, "perplexity": 274.0633710606536}, "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-2021-04/segments/1610703529179.46/warc/CC-MAIN-20210122082356-20210122112356-00344.warc.gz"} |
https://www.physicsforums.com/threads/really-basic-stuff-exponents.146703/ | # Really Basic Stuff: Exponents
1. Dec 4, 2006
### Swapnil
I always get confused when you get things like:
$${(-4)}^{\frac{3}{2}}$$ or $$8^{\frac{2}{3}}$$.
Which operation do you do first and why?
Do you take the square/squareroot first and then take the cube/cuberoot or what?
2. Dec 4, 2006
For $$(-4)^{\frac{3}{2}}$$ take the square root first and then cube it.
For $$(8)^{\frac{2}{3}}$$ take the cube root first and then square it.
Last edited: Dec 4, 2006
3. Dec 4, 2006
### Hurkyl
Staff Emeritus
To compute it directly, you do neither. You merely raise -4 to the 3/2 power, and raise 8 to the 2/3 power.
If you want to compute it in some other fashion, you have to invoke some extra knowledge. For example, if a > 0, you have a theorem1 that says
(ab)c = abc,
which would allow you to split this single exponent operation into two separate exponent operations. If you were able to do so, then it's clear what you would do first.
1: This theorem is for real arithmetic; things get messier when you are doing complex arithmetic
4. Dec 5, 2006
### Staff: Mentor
courtrigrad showed you the way we would do it -- perform the simplest operation first. Whichever makes the most sense if there is a simple operation to do in your head, do that first and deal with the rest second. So in the first one, taking the square root of the perfect square 4 is easiest (you have to deal with the imaginary part of sqrt(-4), but that's no big deal). In the second one, the cube root of 8 is easiest to do first in your head. In other examples, look for stuff you can do first for an even answer, then deal with the rest.
5. Dec 5, 2006
### Swapnil
But chosing which operation to do first actually gives you a different answer! Look at this:
$$(-4)^\frac{3}{2} = \Big[(-4)^\frac{1}{2}\Big]^3 = (2i)^3 = -8i$$
but then you can also do
$$(-4)^\frac{3}{2} = \Big[(-4)^3\Big]^\frac{1}{2} = (-64)^\frac{1}{2} = 8i$$
See what I mean?
Last edited: Dec 5, 2006
6. Dec 5, 2006
### d_leet
This is because the square root is not single valued, notice that both 2i and -2i are square roots of -4.
7. Dec 5, 2006
### Swapnil
But don't we always take the positive square root when we do these types of things. You know, taking only the principle square root...
8. Dec 5, 2006
### d_leet
Positive doesn't have much meaning when dealing with complex numbers, but yes I believe we usually do take the principle root.
9. Dec 5, 2006
### D H
Staff Emeritus
Two points:
1. you didn't read Hurkyl's note, repeated below with critical part in bold.
2. (-4)^(1/2) has two solutions, as does (-64)^1/2.
10. Dec 6, 2006
### Swapnil
I did read it. But the theorem doesn't help me when I have a negative base. And all he said about complex aritmetic was that things get messy. That didn't really helped me.
I see. But why this impartiality. Why is it that be consider both positive and negative complex roots but only consider positive real roots (i.e. the principle square root)?
Last edited: Dec 6, 2006
11. Dec 6, 2006
### Hurkyl
Staff Emeritus
The point of my post was "invoke other knowledge" -- use theorems and other facts you know about arithmetic in order to clarify the situation. (definitions are always a good start) The one theorem I put in my post was just an example of the process.
12. Dec 6, 2006
### Swapnil
I see, but I didn't meant any disrespect when I said that btw.
You know, now that I think about it why would it be messier for complex numbers? Doesn't that same theorem work for negative bases too?
Last edited: Dec 6, 2006
13. Dec 7, 2006
### Hurkyl
Staff Emeritus
My postings so far have been based on the assumption that you know the definition of the complex exponential,
$$a^b = \exp(b \log a) = e^{\Re (b \log a)} ( \cos (\Im (b \log a)) + i \sin (\Im (b \log a))$$
(and the principal value of $a^b$ is found by using the principal value of the compex logarithm)
Was that correct?
14. Dec 7, 2006
### Swapnil
What was what correct?
15. Dec 8, 2006
### Hurkyl
Staff Emeritus
My assumption that you knew the definition of complex exponentiation. I guess that's irrelevant now since I just told you. (Though you would need to know the complex logarithm to use it)
16. Dec 18, 2006
### Swapnil
Yes it was correct.
So if I want to evaluate $$(-4)^\frac{3}{2}$$ then
$$(-4)^\frac{3}{2} = \exp(\frac{3}{2}\log(-4)) = \exp( \Re (\frac{3}{2} \log(-4)) ) ( \cos (\Im (\frac{3}{2} \log(-4))) + i \sin (\Im (\frac{3}{2} \log(-4)))$$
$$= \exp( \Re (\frac{3}{2} (\log(4)+i\pi)) ) ( \cos (\Im (\frac{3}{2}(\log(4)+i\pi))) + i \sin (\Im (\frac{3}{2}(\log(4)+i\pi)))$$
$$= \exp(\frac{3}{2}\log(4)) ( \cos(\frac{3}{2}\pi) + i \sin (\frac{3}{2}\pi) )$$
$$= 4^\frac{3}{2}(-i)$$
$$= -8i$$
what happened to the solution $$+8i$$?
Last edited: Dec 18, 2006
17. Dec 18, 2006
### Hurkyl
Staff Emeritus
Well, it looks like you were taking the principal value of the logarithm. (You should capitalize Log when you do that) Thus, you got the principal value of the exponential.
18. Dec 19, 2006
### Swapnil
...or I can just use the fact that
$$4^\frac{3}{2} = \pm 8$$,
right?
19. Dec 19, 2006
### tehno
that's right.
Similar Discussions: Really Basic Stuff: Exponents | {"extraction_info": {"found_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": 0.9067801833152771, "perplexity": 1469.7596959270556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424559.25/warc/CC-MAIN-20170723122722-20170723142722-00516.warc.gz"} |
https://proofwiki.org/wiki/Definition:Rhombus | (Redirected from Definition:Rhombus)
## Definition
A rhombus is a parallelogram whose sides are all the same length.
Its angles may or may not all be equal.
## Also known as
A rhombus is also known as a rhomb.
Particularly when it is oriented so that the longer diagonal is a vertical line, a rhombus is often referred to as a diamond.
## Euclid's Definitions
In the words of Euclid:
Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia. | {"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": 0.9395926594734192, "perplexity": 528.20552493279}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1575540490743.16/warc/CC-MAIN-20191206173152-20191206201152-00511.warc.gz"} |
https://scicomp.stackexchange.com/questions/25510/does-the-box-covering-algorithm-work-also-for-directed-graphs | # Does the box-covering algorithm work also for directed graphs?
According to this article from Wikipedia, the box-covering algorithm calculates the fractal dimension of a graph. The algorithm is based on the concept of distance between nodes; see for example the sentence:
A box consists of nodes separated by a distance $l < l_B$.
The distance between nodes can be defined also for directed graphs, so I think the algorithm should work also in that case. However, on the Internet, I cannot find any explicit statement about the possibility to use this algorithm for directed graphs. | {"extraction_info": {"found_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": 0.9753590226173401, "perplexity": 183.06398894944243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585199.76/warc/CC-MAIN-20211018062819-20211018092819-00151.warc.gz"} |
https://math.stackexchange.com/questions/2252048/why-is-sqrtx-1-extraneous | # Why is $\sqrt{x} =- 1$ extraneous?
An equation is extraneous if (at least to my understanding) it has no valid solutions. The example my math teacher used was $\sqrt{x}=-1$, citing this proof
$$\sqrt{x}=-1 \\ x=1$$
They then stated that $\sqrt{1} \ne -1$, and therefor the equation is extraneous. While this wasn't initially a problem, I seemed to accept it for some reason, I now realize that $\sqrt{1} = \pm 1$, and therefor -1 is a square root of 1, so why is the initial equation extraneous?
Am I missing something major here or was my math teacher wrong?
• The function $\sqrt{x}$, when used in the real numbers, is defined as the non-negative solution $y$ to $x=y^2$. Since $-1$ is not non-negative, $\sqrt{x}=-1$ won't ever happen in the real numbers. – vrugtehagel Apr 25 '17 at 20:24
• it is $(\pm 1)^2 =1$ and not $\sqrt 1=\pm 1$ – Surb Apr 25 '17 at 20:25
• We say solutions are extraneous. – Sean Roberson Apr 25 '17 at 20:27
• It is a matter of convention. The usual convention when dealing with the real numbers is that $\sqrt{}$ denotes the nonnegative square root. On the other hand, in the context of complex numbers $\sqrt{}$ is often taken to be a multivalued function. – Robert Israel Apr 25 '17 at 20:31
• Note that the square root of a number's square is equal to the absolute value of that number. That is why, when you have $x^2=1$, the answer is $x=\pm 1$. But when you square a number's square root, the result is always that number, and not its absolute value. – Franklin Pezzuti Dyer Apr 25 '17 at 20:32
We talk about an extraneous or spurious solution when we solve an algebraic equation by raising both sides to some power. The map $x\mapsto x^n$ is not injective, hence once the original problem has been reduced to finding the roots of some polynomial, it is not granted that every root of such polynomial is indeed a solution of the original equation. For instance $$\sqrt{x-1} = 7-x \tag{1}$$ implies $$x-1 = (x-7)^2 \tag{2}$$ and $$p(x) = (x-7)^2-(x-1) = x^2-15x+50 = 0 \tag{3}$$ but while $x=5$ is an actual solution of $(1)$, the other root of $p(x)$, i.e. $x=10$, is a spurious solution, because it fulfills $(2)$ but not $(1)$.
It is enough to recall that the very definition of the square root function over the real numbers:
Def. $\sqrt{x}$ is the only non-negative real number whose square equals $x$.
In particular the maximal domain of the square root function is the set of non-negative real numbers, and over such set the square root function is non-negative. So $\sqrt{1}=1$, not $\pm 1$.
Over the set of complex numbers, for any $z\neq 0$ there are two opposite numbers whose squares equal $z$: in such context we write $\sqrt{z}=\pm w$ by meaning that both $w$ and $-w$ are roots of the polynomial $q(t)=t^2-z$, i.e. we regard $\sqrt{\cdot}$ as a multi-valued function: not a function, strictly speaking.
I now realize that $\sqrt{1} = \pm 1$
You are mistaken. The square root symbol $\sqrt{\phantom0}$ denotes the positive square root function. Therefore $\sqrt1 = 1 \neq -1.$
A point of confusion is that we often look at equations such as $$x^2 = 4$$ and observe that they have two solutions, $x = \pm 2.$ But this does not say that $\sqrt4$ is "equal" to $\pm 2$; the actual solution is $$x = \pm \sqrt4,$$ where $x$ is an unknown. The $\pm$ sign tells us that $x$ might be $2$ or might be $-2.$ We need the $\pm$ sign in front of $\sqrt4$ to tell us that, because $\sqrt4$ by itself is always $2,$ never $-2.$
Basically, the symbol $\sqrt{x}$ means that, if $x$ is a non-negative real number, then $\sqrt{x}$ is the single non-negative real number which becomes $x$ when you square it. This is because $\sqrt{x}$ is more useful when it's a function, so each $x$ input needs to have at most one output.
However, when you use the square root to solve an equation, for example $x^2 = 1$, you need to recognize that undoing a square means you could have started with a positive or a negative number, because squaring either will make it positive. So, we say $x = \pm 1$ in this case, but $\sqrt{1}$ is still just $1$.
• "the single real number that become x when you square it"? – operatorerror Apr 25 '17 at 20:42
• @EthanBolker: Thanks for catching that, I said it in my head but it didn't come out of my fingers! – AlexanderJ93 Apr 25 '17 at 20:43
• It should still be "$\sqrt{x}$ is the non-negative real number which..." – Christopher.L Apr 25 '17 at 20:44
• @qbert: I simply mean that it is unique (there is only one) and that $\sqrt{x}^2= \sqrt{x}\sqrt{x} = x$. – AlexanderJ93 Apr 25 '17 at 20:44
• @MartinArgerami : Sorry, yes, I deleted that comment now, since AlexanderJ93 had just updated his answer, and I had not catched that yet. – Christopher.L Apr 25 '17 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": 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": 0.8567978739738464, "perplexity": 174.99577590498302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107878921.41/warc/CC-MAIN-20201022053410-20201022083410-00468.warc.gz"} |
https://physics.stackexchange.com/questions/28197/cy-moduli-fields | # CY moduli fields
When one does string compactification on a Calabi-Yau 3-fold. The parameters in Kähler moduli and complex moduli gives the scalar fields in 4-dimensions. It is claimed that the Kähler potentials of the CY moduli space gives the kinetic terms of the scalar fields in 4d. Could anyone let me know why?
I know that a consistent coupling of a SUSY multiplet containing scalars with supergravity requires the scalar kinetic term comes from a Kähler potential. But I am not sure why precisely this Kähler potential coincides with the one for CY moduli spaces in the case of string compactification. Could anyone explain it to me? Thanks!
• Unfortunately I don't know the answer, but I've tried to share this question around and hopefully someone else who does will stumble upon it. Welcome to Physics Stack Exchange! – David Z May 12 '12 at 23:44
In order to describe physics of 4 dimensional space-time starting from 10 dimensional space, we consider that the extra 6 dimensional space has very tiny size (about the Planck length). This is called the comopactification in string theory. Then the symmetry of theory requires this internal space to be Ricci flat: $$R_{IJ}=0$$ where $I$ and $J$ run from 0 to 6.
Since C-Y manifolds have the structure of the complex manifolds, it is appropriate to use the indices of the complex manifolds, i.e. $i,j=1,2,3$. The metric on C-Y manifolds is of type (1,1) $G_{i,\bar j}$ and we obtain the closed (1,1)-form from the metric $$\omega =\sqrt{-1}G_{i\bar j} dz^i\wedge d\bar z^{\bar j} .$$ And there exists nowhere vanishing holomorphic 3-form $\Omega_{klm}$ since the Ricci tensor vanishes.
In what follows we consider the moduli of C-Y manifolds. If the geometric object can be continuously deformed with its geometric properties preserved, we call the parameter of this deformation moduli. Suppose that the metric $G_{IJ}(y)$ is given on a C-Y manifold $M$ where $y$ is the local coordinate of $M$. And assume that the metric change $G_{IJ}+g_{IJ}$ and this new metric also gives RIcci flat. Then taking the first order of the deformation of metric, we have $$\Delta_6 g_{IJ}(y)=0$$ where $\Delta_6$ is a 6 dimensional Laplacian. Thus, the deformation of the metric which preserves the definition of C-Y manifolds is given by the eigenfuntion of the Laplacian with its eigenvalue zero. In general, the eigenvalues of $-\Delta_6$ take zero and positive values: $$-\Delta_6 f_{IJ}^\alpha(y)=m_\alpha^2 f_{IJ}^\alpha(y), \,\,\,\alpha=1,2,\cdots.$$ Let us consider the case that the deformation of metric $g_{IJ}$ is of type (1,1), $\delta g_{i\bar j}$, and of type (2,0) , $\delta g_{i j}$, respectively. Generally, the metric on a K\"ahler manifold is of type (1,1) and $g_{i\bar j}$ gives the deformation preserving this type. This is called the deformation of K\"ahler structure. The deformation of K\"ahler structure is described by a solution of the equation $$\Delta_6 \omega_{i j}(y)=0,$$ i.e. it is given by a harmonic (1,1)-form. The number of a harmonic (1,1)-form is provided by the Hodge number $h_{1,1}$ of the manifold $M$. On the other hand, the type (2,0) deformation of metric implies the deformation of complex structure on a complex manifold. Using the complex conjugate $\bar \Omega$ of $\Omega$ , we obtain $$\chi_{i\bar j \bar k} \equiv \delta g_{ij} G^{j\bar k} \bar \Omega_{\bar k\bar l\bar m}.$$ Therefore the deformation of complex structure is described by harmonic (1,2)-form and the degree of freedom of the deformation becomes the Hodge number $h_{1,2}=h_{2,1}$. As we see, C-Y manifolds have two kinds of the deformation parameters, the K\"ahler parameters and the parameters of complex structure, which are called moduli, and the K\"ahler parameters correspond to the degree of the change of the size and the parameters of complex structure correspond to that of the deformation of the shape. The metric of the complex structure moduli space is $$G_{\alpha\bar\beta}^{\rm mod}=-\frac{i\int\chi_\alpha\wedge\bar\chi_{\bar \beta}}{i\int\Omega\wedge\bar\Omega}$$ Recalling that the metric $G^{\rm mod}_{\alpha\bar\beta}$ of the complex structure moduli space can be obtained from the K\"ahler potential $\cal K$ $$G_{\alpha\bar\beta}^{\rm mod}=\partial_{\alpha}\partial_{\bar\beta}{\cal K} \ ,$$ one finds that the K\"ahler potential can be written as $${\cal K}=-\log\int\left(i\int \Omega\wedge\bar\Omega \right) \ .$$
Let us choose the basis $C_a$ $(a=1,\cdots, h_{1,1})$ of the 4-cycle as the duals of the harmonic (1,1)-form $\omega^a\equiv\omega^a_{i\bar j}d\!z^i\wedge d\! z^{\bar j}$ $(a=1,\cdots, h_{1,1})$. Then we can expand $$\ast C+\sqrt{-1}\omega=\sum t_a \omega^a$$ where $\omega$ is the K\"ahler form and $C$ is 4-th antisymmetric tensor field which is the partner of the gravitational field. Then the coefficients of the expansion is given by $$t_a =\int _{C_a} (C+\sqrt{-1}\ast\omega).$$ These are the parameters of the complexified K\"ahler moduli.
Similarly, let us choose the basis $A_a$ and $B_a$ $(a=0,1,\cdots, h_{1,2})$ of 3-cycle so that the intersection numbers satisfy $A_a\cap B_b=\delta_{ab}, A_a\cap A_b=B_a\cap B_b=0$. In this case, it is known that the parameters are taken as the moduli of the deformation of complex structure $$z_a=\int_{A_a} \Omega, \,\,\, a=a,\cdots, h_{1,2}.$$ And also it is know that the integral of $\Omega$ over the cycle $B_a$ can be written $$\frac{\partial F}{\partial z_a}=\int _{B_a} \Omega,\,\,\, a=a,\cdots, h_{1,2}$$ where $F$ is a holomorphic function of $z_a$ and is called the prepotential.
Under the compactification of a C-Y manifold fields of 10 dimensional theory can be expanded by the eigenfunctions of the 6 dimensional Laplacian $$f_{IJ}(x,y)=\sum_\alpha \phi^\alpha(x)f_{IJ}^\alpha(y).$$ Then the wave function of 10 dimensional theory reduces to the 4 dimensional field equation: \begin{eqnarray} &&\Delta_{10}f_{IJ}(x,y)=(\Delta_{4}+\Delta_{6})\sum_\alpha\phi^\alpha(x)f_{IJ}^\alpha(y)=0 \ && \rightarrow (\Delta_{4}-m_\alpha^2) \phi^\alpha(x)=0. \end{eqnarray} Hence the scalar field $\phi_\alpha$ with mass $m_\alpha$ shows up in 4 dimensional space corresponding to the eigenvalue $m_\alpha^2$ of the six dimensional Laplacian. Especially, the mass of the scalar field corresponding to the moduli of the manifold becomes zero and the massless scalar particle, moduli particle shows up in the four dimensional effective theory. Then the expectation value of the vacuum corresponding to the parameter $\{t_a, z_a\}$. Since the nonzero eigenvalue $m_\alpha$ of the Laplacian is inversely proportional to the square of the size of the space $M$ and acquire very heavy mass, we can neglect them.
${\rm \bf Note\ Added}$: Once Type II string theory is compactified on a Calabi-Yau manifold $M$, the 4d low-energy effective theory is described by ${\cal N}=2$ supergravity. The field contents of ${\cal N}=2$ supergravity consist of the Weyl (gravity) multiplet, vector multiplets and hypermultiplets. The effective actions for vector multiplets and hypermultiplets are described by non-linear sigma model with target spaces, the vector multiplet moduli space and the hypermultiplet moduli space respectively. In particular, the kinetic terms of the scalars $\phi^i$ in the vector multiplets and the hypermultiplets can be written as $$\int_{M_4}d^4x\sqrt{g}G_{ij}\partial_\mu \phi^i \partial^\mu \phi^j +\cdots$$ where $G_{ij}$ is the metric of the moduli space. In type IIA compactications, the vector multiplet moduli space coincide with the complex K\"ahler moduli and the hypermultiplet moduli space is the complex structure moduli space. In type IIB, vice-versa. In type IIB compactifications, the low energy effective action of the vector multiplets are dictated by the prepotential $F$ due to supersymmetry. Due to mirror symmetry we can restrict our attention to one of the two type II theories.
• This is very nice +1, but it doesn't answer the question, which is about the Kahler potential of the scalars, the field dependence of the kinetic terms, or the derivative terms in the equation of motion. It's not asking why the moduli correspond to massless fields, nor is it asking what gives rise to the superpotential for the massive case. The question is entirely restricted to the massless scalars parametrizing the Kahler moduli--- it is asking why the kinetic term of these fields coincides with the Kahler potential of the Moduli space. Maybe its only true in SUGRA approximation. – Ron Maimon May 14 '12 at 1:35
• Thanks for the answering Satoshi. But indeed my question is why the kinetic term of these fields coincides with the Kahler metric of the CY moduli space. – color May 15 '12 at 10:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 15, "x-ck12": 0, "texerror": 0, "math_score": 0.9792804718017578, "perplexity": 179.52830787696217}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027322170.99/warc/CC-MAIN-20190825021120-20190825043120-00175.warc.gz"} |
http://nbviewer.jupyter.org/github/gpeyre/numerical-tours/blob/master/matlab/segmentation_4_convex_segmentation.ipynb | # Convex Region-Based Image Segmentation¶
Important: Please read the installation page for details about how to install the toolboxes. $\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\enscond}[2]{\lbrace #1, #2 \rbrace}$ $\newcommand{\pd}[2]{ \frac{ \partial #1}{\partial #2} }$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\umax}[1]{\underset{#1}{\max}\;}$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\uargmin}[1]{\underset{#1}{argmin}\;}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\diag}[1]{{diag}\left( #1 \right)}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\qarrq}{ \quad \Longrightarrow \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\Zz}{\mathcal{Z}}$ $\newcommand{\Ww}{\mathcal{W}}$ $\newcommand{\Vv}{\mathcal{V}}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\NN}{\mathcal{N}}$ $\newcommand{\Hh}{\mathcal{H}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\Ee}{\mathcal{E}}$ $\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Gg}{\mathcal{G}}$ $\newcommand{\Ss}{\mathcal{S}}$ $\newcommand{\Pp}{\mathcal{P}}$ $\newcommand{\Ff}{\mathcal{F}}$ $\newcommand{\Xx}{\mathcal{X}}$ $\newcommand{\Mm}{\mathcal{M}}$ $\newcommand{\Ii}{\mathcal{I}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Ll}{\mathcal{L}}$ $\newcommand{\Tt}{\mathcal{T}}$ $\newcommand{\si}{\sigma}$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$
This numerical tour explores a convex relaxation of the piecewise constant Mumford-Shah. This relaxation is exact, and leads to a global solution to the segmentation problem. It can be solved using proximal splitting scheme, and we propose to use here the Douglas-Rachford algorithm. Of independent interest is the introduction of auxiliary gradient variables that enables the use of purely primal splitting schemes.
Special thanks to Jalal Fadili for telling me about the "auxiliary variable" trick (i.e. adding the variable $u=\nabla f$), that allows one to solve TV regularization without the need to use primal-dual schemes.
In [2]:
addpath('toolbox_signal')
## Binary Segmentation ¶
We consider some input image $I(x) \in \RR^d$ ($d=1$ for grayscale image and $d=3$ for color images). Given weights $w_0$ and $w_1$ computed from $I$, where $N$ is the number of pixel, the goal is to find a region $\Om$ that minimize $$\umin{\Om} \int_\Om w_0(x) d x + \int_{\Om^c} w_1(x) d x + \la \abs{\partial \Om},$$ where $\abs{\partial \Om}$ is the perimeter of $\Om$.
To perform region based piecewise-constant segmentation, we assume we know the target values $c_0, c_1 \in \RR^d$ for the inside/outside of the segmented domain, and use quadratic weigths $$\text{for } i=0,1, \quad w_i(x) = \norm{I(x)-c_i}^2.$$ We use these weights in all the remaining part of this tour.
In the special case $\la=0$, no regularization is performed, and the optimal set $$is obtained by a simple thresholding$$ \Om = \enscond{x}{ w_0(x) \leq w_1(x). } $$We load the image I. In [3]: name = 'hibiscus'; n = 256; I = rescale( load_image(name,n) ); Display it. In [4]: clf; imageplot(I); Take as target colors red and green. In [5]: c0 = [1;0;0]; c1 = [0;1;0]; Exercise 1 Compute w_0 and w_1. Compute and display the segmentation when \la=0. In [6]: exo1() In [7]: %% Insert your code here. Define w=w_0-w_1. In [8]: w = w0-w1; ## Convex Discrete Formulation¶ If one represents \Om using its indicator function f$$ f(x) = \chi_\Om(x) = \choice{ 1 \qifq x \in \Om. \\ 0 \quad \text{otherwise}, } $$this problem is re-casted equivalently$$ \umin{ f(x) \in \{0,1\} } \dotp{f}{w} + \la \norm{f}_{\text{TV}}, $$where w=w_0-w_1 and \norm{f}_{\text{TV}} is the total variation pseudo-norm, that is equal to \abs{\partial \Om} for binary indicator f=\chi_\Om. Here the inner product is the canonical one \dotp{f}{w}=\int f w. The variational problem is discretized on a grid of N=n \times n pixels, and we define the total variation pseudo-norm, for f \in \RR^N, as$$ \norm{f}_{\text{TV}} = \norm{\nabla f}_1 \qwhereq \norm{u}_1 = \sum_{i=1}^N \norm{u_i}, $$when u=(u_i)_{i=1}^N \in \RR^{N \times 2}, u_i \in \RR^2 is a vector field. We use a finite difference gradient operator$$ (\nabla f)_i = (f_{i+\de_1}-f_i, f_{i+\de_2}-f_i) \in \RR^2, $$(we assume the pixels are indexed on a 2-D grid) where \de_1=(1,0) and \de_2=(0,1). We use periodic boundary conditions for simplicity. In [9]: options.bound = 'per'; Grad = @(x)grad(x,options); Div = @(x)div(x,options); The inner product in the objective is discretized using the canonical inner product in \RR^N$$ \dotp{f}{w} = \sum_{i=1}^N f_i w_i . $$To obtain a convex program, one replaces the binary constraint f_i \in \{0,1\} by a box constraint f_i \in [0,1] . This defines the folowing finite dimensional convex problem$$ \umin{ f \in [0,1]^N } \dotp{f}{w} + \la \norm{\nabla f}_{1}. $$One can prove that this relaxation is exact, meaning that the minimizer f, when it is unique, is binary, f \in \{0,1\}^N. It means that \Om such that f=\chi_\Om actually solves the original segmentation problem. See for instance: Tony F. Chan, Selim Esedoglu, and Mila Nikolova. Algorithms for finding global minimizers of image segmentation and denoising models SIAM J. Appl. Math., 66(5):1632-1648, 2006. It is possible to generalize this convexification method to the segmentation problem with more than 2 partitions. See for instance: Antonin Chambolle, Daniel Cremers, Thomas Pock, A convex approach to minimal partitions, Preprint hal-00630947, 2011. To solve this problem using primal proximal splitting scheme, we introduce an auxiliary variable u=\nabla f, and write the optimization problem as$$ \umin{z=(f,u) \in \Zz = \RR^N \times \RR^{N \times 2} } F(z) + G(z) \qwhereq \choice{ F(f,u) = \dotp{f}{w} + \iota_{[0,1]^N}(f) + \la \norm{u}_1, \\ G(f,u) = \iota_{\Cc}(f,u), } $$where here we included the constraints using indicator functions$$ \iota_{A}(z) = \choice{ 0 \qifq z \in A, \\ +\infty \quad \text{otherwise}. } $$The constraint linking f to u is$$ \Cc = \enscond{z = (f,u) \in \Zz}{ u=\nabla f }. $$## Douglas-Rachford Algorithm¶ To minimize the segmentation energy, we will make use of proximal splitting scheme. These scheme are adapted to solve structured non-smooth optimization problem. They basically replace the traditional gradient-descent step (that is not available because neither F nor G are smooth functionals) by proximal mappings, defined as$$ \text{Prox}_{\gamma F}(z) = \uargmin{y} \frac{1}{2}\norm{z-y}^2 + \ga F(y) $$(the same definition applies also for G). The Douglas-Rachford (DR) algorithm is an iterative scheme to minimize functionals of the form$$ \umin{z} F(z) + G(z) $$where F and G are convex functions for which one is able to comptue the proximal mappings \text{Prox}_{\gamma F} and \text{Prox}_{\gamma G} . The important point is that F and G do not need to be smooth. One onely needs then to be "proximable". A DR iteration reads$$ \tilde z_{k+1} = \pa{1-\frac{\mu}{2}} \tilde z_k + \frac{\mu}{2} \text{rPox}_{\gamma G}( \text{rProx}_{\gamma F}(\tilde z_k) ) \qandq z_{k+1} = \text{Prox}_{\gamma F}(\tilde z_{k+1},) $$We have use the following shortcuts:$$ \text{rProx}_{\gamma F}(z) = 2\text{Prox}_{\gamma F}(z)-z $$It is of course possible to inter-change the roles of F and G, which defines another set of iterations. One can show that for any value of \gamma>0, any 0 < \mu < 2 , and any \tilde z_0, z_k \rightarrow z^\star which is a minimizer of F+G. Please note that it is actually z_k that converges, and not \tilde z_k. To learn more about this algorithm, you can read: Proximal Splitting Methods in Signal Processing, Patrick L. Combettes and Jean-Christophe Pesquet, in: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, New York: Springer-Verlag, 2010. ## Proximal Operator of G¶ The proximal mapping of G is the orthogonal projection on the convex set G:$$ (\tilde f, \tilde u) = \text{Prox}_{\ga G}(f,u) = \text{Proj}_\Cc(f,u). $$It can be computed by solving a linear system of equations since$$ \tilde u = \nabla \tilde f \qwhereq \tilde f = (\text{Id}_N - \Delta)^{-1}(f-\text{div}(u)). $$Here, by convention, \Delta=\text{div} \circ \nabla and \text{div}=-\nabla^*. Since we use periodic boundary conditions for the gradient operator, it is possible to solve this linear system in O(N \log(N)) operations using the FFT algorithm. Note that a similar method can be used with non-periodic Neumann condition (this requires to extend by symmetry the image). One indeed has$$ \forall \om=(\om_1,\om_2), \quad \hat {\tilde f}(\om) = \frac{\hat g(\om)}{K(\om)} \qwhereq K(\om) = 1+4\sin\pa{\frac{\pi \om_1}{n}}^2+4\sin\pa{\frac{\pi \om_2}{n}}^2, $$where g = f-\text{div}(u) and where \hat g is the 2-D discrete Fourier transform of an image g. Compute K(\om). In [10]: [X Y] = meshgrid(0:n-1, 0:n-1); K = 1 + 4*sin(X*pi/n).^2 + 4*sin(Y*pi/n).^2; Define Proj_\Cc. In [11]: Replicate = @(z)deal(z, Grad(z)); ProjC = @(f,u)Replicate( real( ifft2( fft2( f - Div(u) ) ./ K ) ) ); One has \text{Prox}_{\ga G} = \text{Proj}_{\Cc}, whatever the value of \ga. In [12]: ProxG = @(f,u,gamma)ProjC(f,u); ## Proximal Operator of F¶ Recall that the function F(f,u) is actully a separable sum of a function that only depends on f and a function that depends only on u:$$ F(f,u) = F_0(f) + \la \norm{u}_1 \qwhereq F_0(f) = \dotp{f}{w} + \iota_{[0,1]^N}(f) $$The proximal operator of F reads$$ \text{Prox}_{\ga F}(f,u) = ( \text{Prox}_{\ga F_0 }(f), \text{Prox}_{\ga \la \norm{\cdot}_1 }(u) ). $$Define the value of \la>0 (you can change this value). In [13]: lambda = .1; The proximal operator of F_0 is obtained by using the projection on the box constraint$$ \text{Prox}_{\ga F_0 }(f) = \text{Proj}_{[0,1]^N}( f - \ga w ) \qwhereq \text{Proj}_{[0,1]^N}(g) = \max(0,\min(1, g)). $$In [14]: ProxF0 = @(f,gamma)max(0, min(1, f-gamma*w) ); The proximal operator of the \ell^1-\ell^2 norm \norm{\cdot}_1 is a soft thresholding of the amplitude of the vector field:$$ \text{Prox}_{\ga \norm{\cdot}_1}(u)_i = \max\pa{ 0, \frac{\ga}{\norm{u_i}} } u_i. $$In [15]: amplitude = @(u)repmat( sqrt( sum(u.^2, 3) ), [1 1 2]); ProxL1 = @(u,gamma)max(0,1-gamma./max(1e-9, amplitude(u))) .* u; Define the proximal operator of F. In [16]: ProxF = @(f,u,gamma)deal( ProxF0(f,gamma), ProxL1(u,gamma*lambda) ); ## Douglas-Rachford for Convex Segmentation¶ Set the value of \mu and \gamma. You might consider using your own value to speed up the convergence. In [17]: mu = 1; gamma = 1; Number of iterations. In [18]: niter = 800; Exercise 2 Implement the DR iterative algorithm on |niter| iterations. Keep track of the evolution of the minimized energy$$ E(f) = \dotp{w}{f} + \la \norm{\nabla f}_1 during the iterations. Remark: to speedup the convergence, you can use a "clever" initialization.
In [19]:
exo2()
In [20]:
%% Insert your code here.
Display the result image $f$ at convergence. Note that $f$ is almost binary.
In [21]:
clf;
imageplot(f);
Exercise 3
Test with different value of the $\lambda$ parameter.
In [24]:
exo3()
In [23]:
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http://math.stackexchange.com/questions/657131/find-the-last-two-digits-of-999 | # Find the last two digits of 9^(9^9) [duplicate]
I want to find the last two digits of $9^{9^9}$, that is $9$ raised to the power $9^9$. I tried using Euler's theorem but I can't make anything of it.
As always, I ask only for a minor hint, not a solution.
-
## marked as duplicate by Brad, M Turgeon, Tomás, Ivo Terek, Carl MummertJul 30 at 15:45
you tried math.stackexchange.com/questions/116548/…?? does it help somehow? – Praphulla Koushik Jan 30 at 11:34
– lab bhattacharjee Jan 30 at 18:10
Last $2$ digit essentially implies $\pmod{10^2}$
Method $1:$ $$9^{(9^9)}=(10-1)^{(9^9)}=\left[(-1)(1-10)\right]^{(9^9)}=-(1-10)^{(9^9)}\text{ as }9^9\text{ is odd}$$
Now uisng Binomial Theorem , $\displaystyle(1-10)^{(9^9)}=1-\binom{9^9}110^1\pmod{100}\equiv1-10\cdot9^9$
Again, $\displaystyle9^9=(10-1)^9\equiv-1\pmod{10}$
$\displaystyle\implies10\cdot9^9\equiv10(-1)\pmod{100}\equiv-10$
So,$$9^{(9^9)}\equiv-[1-(-10)]\equiv-11\pmod{100}\equiv89$$
as $9^{(9^9)}>0$ as any positive integer $n\pmod{100}$ lies $\in[0,100-1]$
Method $2:$ Alternatively, using Carmichael's theorem (which is generally more useful than Euler's totient theorem when the modulus is composite (why?))
We have $\displaystyle\lambda(100)=20\implies9^{(9^9)}\equiv9^{(9^9\pmod{20})}\pmod{100}$
Now, $\displaystyle9^2=81\equiv1\pmod{20}\implies 9^9\equiv(9^2)^4\cdot9\equiv9\pmod{20}$
So, $\displaystyle\lambda(100)=20\implies9^{(9^9)}\equiv9^9\pmod{100}$
Method $\displaystyle2A:$ Now, $\displaystyle9^9=(10-1)^9\equiv\binom9110^1-1\pmod{100}\equiv90-1$
Method $\displaystyle2B:$
$\displaystyle9^9=(3^2)^9=3^{18}\equiv3^{-2}\pmod{100}$ as $\displaystyle\lambda(100)=20\implies3^{20}\equiv1\pmod{100}$
Now, $\displaystyle3^{-2}\equiv\frac19\pmod{100}\equiv\frac{-99}9$ as $99\equiv-1\pmod{100}\iff-99\equiv1$
$\displaystyle\implies3^{-2}\equiv-11\pmod{100}\equiv100-11$ as $9^{(9^9)}>0$ as any positive integer $n\pmod{100}$ lies $\in[0,100-1]$
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I asked for a hint, but this is good too ( awesome to be honest, just not fun since I like to try to do it myself :D I tried with BT and (10-1) but got stuck.. Thanks ;) – Transcendental Jan 31 at 18:29
@Irrational, sorry I could not determine where to stop before the end – lab bhattacharjee Feb 1 at 8:20
The last two digits of $9^n$ form a repetition cycle of length $10$, starting at $n=0$. So your question is equivalent to finding the remainder of $n\mod10$, where $n=9^9$, which shouldn't be too hard, given that $9=10-1$, so $9^9\mod10=(-1)^9\mod10=-1$, so we are looking for the last element of the afore-mentioned cycle (the first being $1$, for $n=0$), which is $89$.
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Hint : Reduce the exponent $9^9$ modulo 40
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[This ought to be a comment on @Lucian’s answer, but I don’t have the 50 requisite points yet for that. So, here it is as answer.]
Although, mathematically, answers like lab bhattacharjee’s are preferable, because of their generality, for some problems, it’s worth getting your fingers dirty with actual digits and looking at how the power series plays out—what empirical scientists call “making friends with your data”.
In the case of powers of 9, the cycle of digits is very pretty. For units, they just alternate between 1 (for even powers of 9) and 9 (for odd powers). In the tens column, the digits are always even, hitting the lowest first (0), then the highest (8), then the next lowest (2), then the next highest (6), then landing on (4), before reversing the whole pattern:
08264 46280 08264 46280 08264 46280 ...
where the $2n+1$-th zero arises whenever the power of 9 equals 0 (mod 10).
Obviously, then, $9^{9^9} = 9^{81}$ ends in $89$, as $81 \equiv 1$ (mod $10$) and $81 \equiv 1$ (mod $2$).
As said, this way of looking at things has its limitations, but, in this instance, it happens to be rewarding to see what’s going on under the bonnet.
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https://research-explorer.app.ist.ac.at/record/7014 | # Non-polynomial worst-case analysis of recursive programs
K. Chatterjee, H. Fu, A.K. Goharshady, ACM Transactions on Programming Languages and Systems 41 (2019).
Journal Article | Published | English
Scopus indexed
Department
Abstract
We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of nonrecursive programs. First, we apply ranking functions to recursion, resulting in measure functions. We show that measure functions provide a sound and complete approach to prove worst-case bounds of non-deterministic recursive programs. Our second contribution is the synthesis of measure functions in nonpolynomial forms. We show that non-polynomial measure functions with logarithm and exponentiation can be synthesized through abstraction of logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem using linear programming. While previous methods obtain worst-case polynomial bounds, our approach can synthesize bounds of the form $\mathcal{O}(n\log n)$ as well as $\mathcal{O}(n^r)$ where $r$ is not an integer. We present experimental results to demonstrate that our approach can obtain efficiently worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the divide-and-conquer algorithm for the Closest-Pair problem, where we obtain $\mathcal{O}(n \log n)$ worst-case bound, and (ii) Karatsuba's algorithm for polynomial multiplication and Strassen's algorithm for matrix multiplication, where we obtain $\mathcal{O}(n^r)$ bound such that $r$ is not an integer and close to the best-known bounds for the respective algorithms.
Publishing Year
Date Published
2019-10-01
Journal Title
ACM Transactions on Programming Languages and Systems
Volume
41
Issue
4
Article Number
20
IST-REx-ID
### Cite this
Chatterjee K, Fu H, Goharshady AK. Non-polynomial worst-case analysis of recursive programs. ACM Transactions on Programming Languages and Systems. 2019;41(4). doi:10.1145/3339984
Chatterjee, K., Fu, H., & Goharshady, A. K. (2019). Non-polynomial worst-case analysis of recursive programs. ACM Transactions on Programming Languages and Systems. ACM. https://doi.org/10.1145/3339984
Chatterjee, Krishnendu, Hongfei Fu, and Amir Kafshdar Goharshady. “Non-Polynomial Worst-Case Analysis of Recursive Programs.” ACM Transactions on Programming Languages and Systems. ACM, 2019. https://doi.org/10.1145/3339984.
K. Chatterjee, H. Fu, and A. K. Goharshady, “Non-polynomial worst-case analysis of recursive programs,” ACM Transactions on Programming Languages and Systems, vol. 41, no. 4. ACM, 2019.
Chatterjee K, Fu H, Goharshady AK. 2019. Non-polynomial worst-case analysis of recursive programs. ACM Transactions on Programming Languages and Systems. 41(4), 20.
Chatterjee, Krishnendu, et al. “Non-Polynomial Worst-Case Analysis of Recursive Programs.” ACM Transactions on Programming Languages and Systems, vol. 41, no. 4, 20, ACM, 2019, doi:10.1145/3339984.
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arXiv 1705.00317 | {"extraction_info": {"found_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": 0.9487713575363159, "perplexity": 2667.616972967556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610704799711.94/warc/CC-MAIN-20210126073722-20210126103722-00236.warc.gz"} |
http://tex.stackexchange.com/questions/68860/increase-spacing-after-minipage?answertab=oldest | # Increase spacing after minipage
Hello i am using a line of code to draw 2 equations next to eachother using \minipage{}:
\begin{minipage}{0.5\textwidth}
\small
$$\label{e3} \begin{split} \frac{\mathrm d x'}{\mathrm d t} &= \frac{\mathrm d x}{\mathrm d t} - \frac{\mathrm d} {\mathrm d t}(ut)\\ \frac{\mathrm d x'}{\mathrm d t} &= \frac{\mathrm d x}{\mathrm d t} - u \, \frac{\mathrm d}{\mathrm d t}(t)\\ \frac{\mathrm d x'}{\mathrm d t} &= \frac{\mathrm d x}{\mathrm d t} - u \, \frac{\mathrm d t}{\mathrm d t}\\ v' &= v - u \end{split}$$
\end{minipage}
\begin{minipage}{0.5\textwidth}
\small
$$\label{e4} \begin{split} \frac{\mathrm d x}{\mathrm d t} &= \frac{\mathrm d x'}{\mathrm d t} + \frac{\mathrm d}{\mathrm d t}(ut)\\ \frac{\mathrm d x}{\mathrm d t} &= \frac{\mathrm d x'}{\mathrm d t} + u \, \frac{\mathrm d}{\mathrm d t}(t)\\ \frac{\mathrm d x}{\mathrm d t} &= \frac{\mathrm d x'}{\mathrm d t} + u \, \frac{\mathrm d t}{\mathrm d t}\\ v &= v' + u \end{split}$$
\end{minipage}
After compiling, using PDFLaTeX i get a resulting document which looks like this:
What can i do to increase vertical spacing after minipage, which is very small? I would also like to know, how can i take care of an annoying \small comand which i have to put inside every \minipage{} for my equations to look smaller? Can it be done in preamble?
Thank you.
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I've updated my answer with a possible solution. – Gonzalo Medina Aug 28 '12 at 0:12
Below there's one possible solution; the key ideas were:
1. I used the solution given in How to keep a constant baselineskip when using minipages (or \parboxes)? to guarantee spacing after the minipages.
2. Before the minipages I used \smallskip\nointerlinespacing.
3. I defined a newenvironment sminipage (to apply \small inside minipage); the optional argument (set by default to t) controls the alignment of the minipage and the mandatory argument sets the width of the minipage.
I also made some other modifications to improve your code:
1. I removed the spurious blank space after the first \end{minipage}.
2. I defined a \Pder command to facilitate the writing of the partial derivatives.
The idea was to obtain (approximately) the same spacing around the equations inside the minipage as the one used for regular equations not inside a minipage (I added a regular equation environment at the end just for comparison purposes):
\documentclass{article}
\usepackage{amsmath}
\usepackage[nopar]{lipsum}% just to generate text for the example
\newcommand\Pder[2]{%
\frac{\mathrm{d}#1}{\mathrm{d}#2}}
\newenvironment{sminipage}[2][t]
{\minipage[t]{#2}\small}
{\endminipage}
\begin{document}
\lipsum*[4]\par\smallskip\nointerlineskip
\noindent\begin{sminipage}[t]{0.5\textwidth}
$$\label{e3} \begin{split} \Pder{x'}{t} &= \Pder{x}{t} - \Pder{\phantom{x}}{t} (ut) \\ \Pder{x'}{t} &= \Pder{x}{t} - u\,\Pder{\phantom{x}}{t} (t) \\ \Pder{x'}{t} &= \Pder{x}{t} - u\,\Pder{t}{t} \\ v' &= v - u \end{split}$$
\end{sminipage}%
\begin{sminipage}{0.5\textwidth}
$$\label{e4} \begin{split} \Pder{x}{t} &= \Pder{x'}{t} - \Pder{\phantom{x}}{t} (ut) \\ \Pder{x}{t} &= \Pder{x'}{t} - u\,\Pder{\phantom{x}}{t} (t) \\ \Pder{x}{t} &= \Pder{x'}{t} - u\,\Pder{t}{t} \\ v &= v' + u \end{split}$$\null
\par\xdef\tpd{\the\prevdepth}
\end{sminipage}
\prevdepth\tpd
\noindent\lipsum[2]
$$a=b$$
\lipsum[4]
\end{document}
![enter image description here][1]
I defined now (as was requested in a comment) a new environment eqmpage which basically is a top aligned minipage with constant width of \linewidth which automates all the preparations mentioned above:
\documentclass{article}
\usepackage{amsmath}
\usepackage[nopar]{lipsum}% just to generate text for the example
\newcommand\Pder[2]{%
\frac{\mathrm{d}#1}{\mathrm{d}#2}}
\newenvironment{sminipage}[2][t]
{\minipage[t]{#2}\small}
{\endminipage}
\newenvironment{eqmpage}
{\par\smallskip\nointerlineskip%
\noindent\minipage[t]{\textwidth}}
{\par\xdef\tpd{\the\prevdepth}\endminipage\par\prevdepth\tpd}
\begin{document}
\lipsum*[4]
\begin{eqmpage}
\begin{sminipage}[t]{0.5\textwidth}
$$\label{e3} \begin{split} \Pder{x'}{t} &= \Pder{x}{t} - \Pder{\phantom{x}}{t} (ut) \\ \Pder{x'}{t} &= \Pder{x}{t} - u\,\Pder{\phantom{x}}{t} (t) \\ \Pder{x'}{t} &= \Pder{x}{t} - u\,\Pder{t}{t} \\ v' &= v - u \end{split}$$
\end{sminipage}%
\begin{sminipage}{0.5\textwidth}
$$\label{e4} \begin{split} \Pder{x}{t} &= \Pder{x'}{t} - \Pder{\phantom{x}}{t} (ut) \\ \Pder{x}{t} &= \Pder{x'}{t} - u\,\Pder{\phantom{x}}{t} (t) \\ \Pder{x}{t} &= \Pder{x'}{t} - u\,\Pder{t}{t} \\ v &= v' + u \end{split}$$\null
\end{sminipage}
\end{eqmpage}
\noindent\lipsum[2]
$$a=b$$
\lipsum[4]
\end{document}
-
But this still has to have \small written in each minipage environment every time which is part of the problem of the OP. From what I understand, OP likes to have this thing taken care of in the preamble. – hpesoj626 Aug 27 '12 at 6:47
@hpesoj626 I've updated my answer. – Gonzalo Medina Aug 27 '12 at 11:59
Nice. I am trying to learn how to write packages and this example helps a lot. Thanks. – hpesoj626 Aug 28 '12 at 0:16
I love the idea about partial derivatives TY! – 71GA Aug 28 '12 at 17:03
@71GA You're welcome! I'm glad I could help. – Gonzalo Medina Aug 28 '12 at 17:10
You can make use of the command \bigskip or \medskip.
If you will always use the same size, you can define
\def\bmp{\begin{minipage}{0.48\linewidth}\small}
\def\emp{\end{minipage}\smallskip}
for begin and end. Note the use of 48% for the width. You can change. You just need to put some \hfill between them.
\bmp \emp
\hfill
\bmp \emp
% this paragraph is important
text here....
-
For some reason this doesnt work. Makes absolutely no change to my document. – 71GA Aug 26 '12 at 19:57
@71GA: Write \bmp$$...$$\emp instead of \begin{minipage}$$...$$\end{minipage} – hpesoj626 Aug 26 '12 at 23:43
@Sigur: Maybe editing the answer so that the modified MWE with your solution is shown will help. – hpesoj626 Aug 27 '12 at 0:10
I know. The problem is that you should use 2 consecutive, to produce the 2 minipage side by side. But the \par does not allow this. I am thinking about. – Sigur Aug 27 '12 at 11:32
I have upvoted @Sigur's answer. But you might also want to look at genmpage package. It adds additional options to your minipage environment and you can also write the options in the preamble.
To set your minipage text to small, put the following into your preamble:
\usepackage{genmpage}
\setkeys{GenMP}{resetfont,fsize=small,inner=s}
Then write your minipage environment as you usually do and you can use \medskip and \bigskip as suggested by @Sigur.
-
Nice package. Thanks. – Sigur Aug 26 '12 at 15:00
@Sigur: You are welcome. – hpesoj626 Aug 26 '12 at 15:28 | {"extraction_info": {"found_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": 10, "x-ck12": 0, "texerror": 0, "math_score": 0.9613690972328186, "perplexity": 2656.4097988346894}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510274979.56/warc/CC-MAIN-20140728011754-00324-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/491877/what-is-the-probability-distribution-of-a-minibatch-of-data | # What is the probability distribution of a minibatch of data?
Suppose there are numbers $$\{1, \ldots, 10\}$$
You pick one at random, call it $$i$$
Then $$i$$ is a Uniform random variable (https://en.wikipedia.org/wiki/Discrete_uniform_distribution), $$i \sim U\{1, 10\}$$. Then you use it for SGD by taking the gradient of the $$i$$th loss.
What about minibatch $$B$$?
In this case you are sampling several numbers from $$\{1, \ldots, 10\}$$ without replacement. Suppose you are assuming a batch size of $$2$$.
The probability of sampling the first one is $$1/10$$, the second one is $$1/9$$.
Then the random vector $$B$$ follows some joint distribution. Does anyone know how to model this joint distribution? I feel like this is something super obvious (thinking in the line of products of uniform distribution - not sure how to express this mathematically).
By symmetry, $$B$$ must be uniformly distributed over the set of subsets of the original dataset whose size is equal to the batch size.
For example, if the batch size is 2, then $$B$$ can take on any value in the set $$\{\{1,2\}, \{1,3\}, \{1,4\}, ..., \{8,9\}\} = \{S: S \in \mathcal{P}(\mathcal{D}), \hspace{0.3em} |S| = 2 \}\,$$ where $$\mathcal{P}$$ is the power set function and $$\mathcal{D}$$ is the full dataset. Since there's no reason why, say, the batch $$\{2, 7\}$$ should be more likely than the batch $$\{4, 5\}$$, we can conclude that $$B$$ is uniformly distributed over the set of batches above. | {"extraction_info": {"found_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": 19, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.957805335521698, "perplexity": 109.65753340159378}, "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/1620243991488.53/warc/CC-MAIN-20210515223209-20210516013209-00606.warc.gz"} |
https://www.physicsforums.com/threads/the-true-quantum.19091/ | The True Quantum
1. Apr 12, 2004
Antonio Lao
What is the true quantum? The quantization of matter, energy, charge, angular momentum (spin), scalar and vector field and many other forms of quantization to date can all be approximation to the true quantum.
The true quantum is the square of the energy. This is a quantum of time (with two directions) and quantum of double-spin. The true quantum is a scalar field like the Higgs field.
2. Apr 13, 2004
Antonio Lao
In a way, the complete understanding of quantum entanglement is intimately related to the true quantum.
This understanding of entanglement is presently beyond the reach of current physics. For us (creatures of reality), like what Einstein's demanded, our understanding depends on some element of reality. But John Bell and all the experiments on entanglement have shown, these elements of reality simply did not materialize at this time.
3. Apr 13, 2004
ahrkron
Staff Emeritus
There's no such a thing as "the true quantum". There seems to be a "quantum principle" in all of physics, in the sense that the measurement process, by which we acquire information from experimental setups, behaves according to QM principles. However, the "quantum" in QM is a property, not a noun.
4. Apr 13, 2004
Antonio Lao
It is not a noun because it is still not found. Once located, it's ghostly appearance will eventually disappear. At the moment, the quantum is intimately linked to the wave function of QM. And it's measuement requires the probability amplitude which is the square absolute value of the wave function.
My independent research strongly indicated to me that the "true" quantum of reality is the square of energy and the internal structure is that of two Hopf rings.
5. Apr 13, 2004
ahrkron
Staff Emeritus
In many circumstances, QM systems do allow for continuous spectra. For instance, the energy of a free electron can have any value. If there was such a thing as "the true quantum" as a fundamental element of reality, every system would be quantized, instead of only bound states (as is the case).
6. Apr 13, 2004
ahrkron
Staff Emeritus
Energy is a number. As such, it can be squared, and the result of such operation is also a number; hence, you are saying that the "true" quantum is only a number, not a physical entity.
7. Apr 13, 2004
Antonio Lao
In a simplistic way, you are correct to say that the "true' quantum is a number. But in reality, it's a matrix. But once the operations of addition and multiplication are done on the matrices, numbers are produced. Addition produces numbers for values of electric charge and multiplication produces values for mass.
8. Apr 13, 2004
Antonio Lao
A first hint that leads me to suspect the "trueness" of quantum of energy squared is th following relativistic energy equation:
$$E^2 = c^2 p^2 + m^2c^4$$
Dirac used this to propose the existence of antimatter. Further, he used the same square root form of the equation for the concept of spin. But the quantization of spin is only an approximation to the "trueness" of the quantum. A deeper and more subtle symmetry can be found in the double spins of two Hopf rings. But this implies reality as being one-dimensional instead of 4-dim (3 of space and one of time) as we are used to believe.
Last edited: Apr 13, 2004
9. Apr 13, 2004
ahrkron
Staff Emeritus
Just in a very loose sense. He used the operation version of the equation, and defined a type of "number" (that turned out to be representable as a matrix) that was able to solve it.
10. Apr 13, 2004
Antonio Lao
Dirac's matrices contain 0,1, -1, and i as elements. The matrices I use only contains 1, and -1. I was able to calculate the mass ratio of proton to electron within a percent of the experimental value.
11. Apr 13, 2004
ahrkron
Staff Emeritus
A number, or a matrix, are representations of physical quantities, not physical entities themselves. The "true quantum" would only be a concept, and not able to be the building block of anything else.
Sure
This is an absurd generalization. Addition of electrical charges produces an electrical charge. Addition of masses produces a value of mass. As of multiplication, you need to specify what magnitudes you are multiplying.
12. Apr 13, 2004
ahrkron
Staff Emeritus
1. Are "your matrices" supposed to have the same function as that of Dirac matrices? what commutation relations do your matrices satisfy? what is their relation to Dirac's eqn? If they have only real entries, they cannot possibly have the same role, which makes the comparison pointless.
2. What elements go into the calculation? I'm not asking for the full thing; only for a summary of the main ideas.
13. Apr 13, 2004
Antonio Lao
The matrices always commute. The matrices are symmetrical and square with alternate elements of 1 and -1. They are basically Hadamard matrices. What is done to these matrices as operators operating on themselves to generate numerical values for charge and mass.
14. Apr 13, 2004
ahrkron
Staff Emeritus
If they always commute, they cannot be used to solve Dirac's eqn. comparing them with Dirac matrices is then mixing apples and oranges.
15. Apr 13, 2004
Antonio Lao
They are not used to solve any equation. it's only the elements that are different.
The Pauli's and Dirac's matrices contains 1, -1, 0, and i. Hadamard matrices contain only 1 and -1.
These matrices are all singular. Their determinants are zero. At first, I tried to fit these matrices into a group. But the multiplication operation does not satisfy the group property of possessing an inverse and also successive matrix addition or matrix multiplication produced scalar factors that cannot be part of the group. These matrices more or less formed an algebraic ring that are Abelian group of matrix addition and semi-group of matrix multiplication.
16. Apr 13, 2004
Antonio Lao
If one can assume the existence of infinitesimal distances and forces r1, F1, and r2, F2 then the square of energy E is given by:
$$E^2 = r_i \times F_i \cdot r_j \times F_j$$
where i=1 and j=2.
Expanded by Langrange's identity give
$$E^2 = (r_i \cdot r_j)(F_i \cdot F_j) - (r_i \cdot F_j)(r_j \cdot F_i)$$
17. Apr 13, 2004
Antonio Lao
The square of energy can also be equally given by
$$E^2 = r_i \times F_i \cdot F_j \times r_j$$
and when expanded give
$$E^2 = (r_i \cdot F_j)(r_j \cdot F_i) - (r_i \cdot r_j)(F_i \cdot F_j)$$
which one of these two forms of E^2 happens more often in reality can be determined by probability theory. One form represents the kinetic energy (or kinetic mass) and the other represents the potential energy (or potential mass). | {"extraction_info": {"found_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": 0.8863527178764343, "perplexity": 923.5527233786202}, "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-2017-04/segments/1484560280825.87/warc/CC-MAIN-20170116095120-00325-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://www.ck12.org/book/CK-12-Algebra-I-Concepts/r1/section/11.10/ | <meta http-equiv="refresh" content="1; url=/nojavascript/"> Applications Using the Pythagorean Theorem | CK-12 Foundation
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# 11.10: Applications Using the Pythagorean Theorem
Created by: CK-12
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Practice Applications Using the Pythagorean Theorem
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What if a fireman needed to rescue a cat from a tree? The cat is 40 feet up from the ground. The fireman places a ladder 30 feet away from the base of the tree? How tall does the ladder need to be for him to reach the cat? After completing this Concept, you'll be able to solve real-world applications like this one using the Pythagorean Theorem and its converse.
### Guidance
The Pythagorean Theorem and its converse have many applications for finding lengths and distances.
#### Example A
Maria has a rectangular cookie sheet that measures $10 \ inches \times 14 \ inches$ . Find the length of the diagonal of the cookie sheet.
Solution
Draw a sketch:
Define variables: Let $c =$ length of the diagonal.
Write a formula: Use the Pythagorean Theorem: $a^2+b^2=c^2$
Solve the equation:
$10^2+14^2 &= c^2\\100+196 &= c^2\\c^2 = 296 & \Rightarrow c=\sqrt{296} \Rightarrow c=2 \sqrt{74} \ \text{or} \ c = 17.2 \ inches$
Check: $10^2+14^2=100+196=296$ and $c^2=17.2^2=296$ . The solution checks out.
#### Example B
Find the area of the shaded region in the following diagram:
Solution
Draw the diagonal of the square in the figure:
Notice that the diagonal of the square is also the diameter of the circle.
Define variables: Let $c =$ diameter of the circle.
Write the formula: Use the Pythagorean Theorem: $a^2+b^2=c^2$ .
Solve the equation:
$2^2+2^2 &= c^2\\4+4 &= c^2\\c^2 = 8 & \Rightarrow c=\sqrt{8} \Rightarrow c=2 \sqrt{2}$
The diameter of the circle is $2 \sqrt{2}$ , therefore the radius $R=\sqrt{2}$ .
Area of a circle formula: $A=\pi \cdot R^2=\pi \left(\sqrt{2}\right)^2=2 \pi$ .
The area of the shaded region is therefore $2 \pi - 4 = 2.28$ .
#### Example C
In a right triangle, one leg is twice as long as the other and the perimeter is 28. What are the measures of the sides of the triangle?
Solution
Make a sketch and define variables:
Let: $a =$ length of the short leg
$2a =$ length of the long leg
$c =$ length of the hypotenuse
Write formulas:
The sides of the triangle are related in two different ways.
The perimeter is 28, so $a+2a+c=28 \Rightarrow 3a+c=28$
The triangle is a right triangle, so the measures of the sides must satisfy the Pythagorean Theorem:
$&\qquad \qquad a^2+(2a)^2 = c^2 \Rightarrow a^2+4a^2=c^2 \Rightarrow 5a^2=c^2\\&\text{or} \qquad \quad c = a\sqrt{5}=2.236a$
Solve the equation:
Plug the value of $c$ we just obtained into the perimeter equation: $3a+c=28$
$3a+2.236a=28 \Rightarrow 5.236a=28 \Rightarrow a=5.35$
The short leg is: $a = 5.35$
The long leg is: $2a = 10.70$
The hypotenuse is: $c = 11.95$
Check: The legs of the triangle should satisfy the Pythagorean Theorem:
$a^2+b^2=5.35^2+10.70^2=143.1, c^2=11.95^2=142.80$ . The results are approximately the same.
The perimeter of the triangle should be 28:
$a+b+c=5.35+10.70+11.95=28$ The answer checks out.
Watch this video for help with the Examples above.
### Vocabulary
• The Pythagorean Theorem is a statement of how the lengths of the sides of a right triangle are related to each other. A right triangle is one that contains a 90 degree angle. The side of the triangle opposite the 90 degree angle is called the hypotenuse and the sides of the triangle adjacent to the 90 degree angle are called the legs .
• If we let $a$ and $b$ represent the legs of the right triangle and $c$ represent the hypotenuse then the Pythagorean Theorem can be stated as:
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. That is: $a^2+b^2=c^2$ .
### Guided Practice
Mike is loading a moving van by walking up a ramp. The ramp is 10 feet long and the bed of the van is 2.5 feet above the ground. How far does the ramp extend past the back of the van?
Solution
Make a sketch:
Define variables: Let $x =$ how far the ramp extends past the back of the van.
Write a formula: Use the Pythagorean Theorem: $x^2+2.5^2=10^2$
Solve the equation:
$x^2+6.25 &= 100\\x^2 &= 93.5\\x &= \sqrt{93.5} = 9.7 \ ft$ Check by plugging the result in the Pythagorean Theorem:
$9.7^2+2.5^2=94.09+6.25=100.34 \approx 100$ . So the ramp is 10 feet long. The answer checks out.
### Practice
1. In order to make a ramp that is $3ft$ high and covers $4ft$ of ground, how long must the ramp be?
2. A regulation baseball diamond is a square with 90 feet between bases. How far is second base from home plate?
3. Emanuel has a cardboard box that measures $20 \ cm \ \text{long} \ \times \ 10 \ cm \ \text{wide} \ \times \ 8 \ cm \ \text{deep}$ .
1. What is the length of the diagonal across the bottom of the box?
2. What is the length of the diagonal from a bottom corner to the opposite top corner?
4. Samuel places a ladder against his house. The base of the ladder is 6 feet from the house and the ladder is 10 feet long.
1. How high above the ground does the ladder touch the wall of the house?
2. If the edge of the roof is 10 feet off the ground and sticks out 1.5 feet beyond the wall, how far is it from the edge of the roof to the top of the ladder?
5. Find the area of the triangle below if the area of a triangle is defined as $A=\frac{1}{2} \ base \times height$ :
6. Instead of walking along the two sides of a rectangular field, Mario decided to cut across the diagonal. He thus saves a distance that is half of the long side of the field.
1. Find the length of the long side of the field given that the short side is 123 feet.
2. Find the length of the diagonal.
7. Marcus sails due north and Sandra sails due east from the same starting point. In two hours Marcus’ boat is 35 miles from the starting point and Sandra’s boat is 28 miles from the starting point.
1. How far are the boats from each other?
2. Sandra then sails 21 miles due north while Marcus stays put. How far is Sandra from the original starting point?
3. How far is Sandra from Marcus now?
8. Determine the area of the circle below. (Hint: the hypotenuse of the triangle is the diameter of the circle.)
9. A rectangle's length is $1in$ longer than its width and if the diagonal has a length of $29in$ , what are the lengths of the sides of the rectangle?
10. For an isosceles triangle with sides of the length given, find the length of hypotenuse:
1. 1
2. 2
3. 3
4. $n$
Oct 01, 2012
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https://engineering.stackexchange.com/questions/17774/isentropic-flow-and-conservation-of-mass/17782 | # Isentropic Flow and Conservation of Mass
I have some doubt on fluid mechanics that need some confirmation. I am on the section that discusses about compressible flow and the effect of variation of flow cross section area.
It stated that that an isentropic flow would obey conservation of mass and what I wonder about is that any chance that an isentropic flow would disobey conversation of mass?
What made me wondering this was I was considering the mass flow rate of isentropic flow in converging-diverging nozzle. If my understanding towards isentropic flow is not wrong, the mass flow rate of an isentropic flow would always be constant no matter of what position it is in the nozzle.
• If it does not obey conservation of mass then where is that mass difference stored in the condi nozzle? Oh conversation is not conservation, autocorrect / predictive type tend to have a limited vocabulary and guess the nearest word... – Solar Mike Oct 27 '17 at 18:20 | {"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": 0.8780779242515564, "perplexity": 731.7985799052111}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487586390.4/warc/CC-MAIN-20210612193058-20210612223058-00169.warc.gz"} |
http://math.stackexchange.com/questions/79535/real-analysis-on-cauchy-sequences | # Real analysis on Cauchy sequences?
Cauchy sequences of rationals can be used to model the reals.
Has anyone actually tried though to develop a theory of real analysis explicitly in this model?
Some definitions seem like they would be straightforward to convert into this model, but I am not sure about others.
I guess we probably wouldn't gain anything, but it would be interesting to see.
Does someone have a reference maybe?
-
I wonder why you would want to do this? Since you have to take equivalence classes of the Cauchy sequences, which essentially gives you the reals, there's nothing to gain here. I mean, the reals are equivalence classes of Cauchy sequences, just with a more compact notation. – pki Nov 6 '11 at 15:11
Yes, I know that. The only thing that might be gained is insight. It just seemed interesting to me. No other reason. – Tim Seguine Nov 6 '11 at 16:49
"Some definitions seem like they would be straightforward to convert into this model" - do you have any examples? – Srivatsan Nov 6 '11 at 16:58
@Srivatsan Like for example, I could define the exponential function of a Cauchy sequence as $\exp{\{x_n\}}_{n\in\mathbb{N}}:=\{\sum_{k=0}^{n}\frac{x_n^k}{k!}\}_{n\in\mathbb{N}}$ – Tim Seguine Nov 6 '11 at 17:40
@pki The reason I originally came up with the idea was for discretizing algorithms. I thought maybe by manipulating the series directly, one might find explicit representations for the series that converges to the solution. Seems probably impractical now, but the idea is still intriguing to me. – Tim Seguine Nov 6 '11 at 18:04 | {"extraction_info": {"found_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": 0.9092604517936707, "perplexity": 575.4551234168673}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00596-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://worldwidescience.org/topicpages/l/lattice+guage+theory.html | #### Sample records for lattice guage theory
1. Cognitive Learning Theories as One Effective Foundation of Second Lan-guage Teaching and Learning
CHENG Li
2016-01-01
Cognitive theory is based on the work of psychologists and psycholinguists working on the internal factors of individu-al’s mind. As language learning is a complex mental process therefore the focus of cognitive learning theories is the learning process in learners’minds. This paper aims to show that cognitive learning theories are the theoretical foundation of second lan-guage teaching and learning according to the comparison of theories between Anderson and McLaughlin. The comparing result indicates that cognitive learning theories play a significant role in second language teaching and learning but the learners’exter-nal factors cannot be neglected either.
2. Lattice theory
Donnellan, Thomas; Maxwell, E A; Plumpton, C
1968-01-01
Lattice Theory presents an elementary account of a significant branch of contemporary mathematics concerning lattice theory. This book discusses the unusual features, which include the presentation and exploitation of partitions of a finite set. Organized into six chapters, this book begins with an overview of the concept of several topics, including sets in general, the relations and operations, the relation of equivalence, and the relation of congruence. This text then defines the relation of partial order and then partially ordered sets, including chains. Other chapters examine the properti
3. Automated Lattice Perturbation Theory
Monahan, Christopher
2014-11-01
I review recent developments in automated lattice perturbation theory. Starting with an overview of lattice perturbation theory, I focus on the three automation packages currently "on the market": HiPPy/HPsrc, Pastor and PhySyCAl. I highlight some recent applications of these methods, particularly in B physics. In the final section I briefly discuss the related, but distinct, approach of numerical stochastic perturbation theory.
4. Lattice Gerbe Theory
Lipstein, Arthur E
2014-01-01
We formulate the theory of a 2-form gauge field on a Euclidean spacetime lattice. In this approach, the fundamental degrees of freedom live on the faces of the lattice, and the action can be constructed from the sum over Wilson surfaces associated with each fundamental cube of the lattice. If we take the gauge group to be $U(1)$, the theory reduces to the well-known abelian gerbe theory in the continuum limit. We also propose a very simple and natural non-abelian generalization with gauge group $U(N) \\times U(N)$, which gives rise to $U(N)$ Yang-Mills theory upon dimensional reduction. Formulating the theory on a lattice has several other advantages. In particular, it is possible to compute many observables, such as the expectation value of Wilson surfaces, analytically at strong coupling and numerically for any value of the coupling.
5. Lattice gauge theories
Weisz, Peter; Majumdar, Pushan
2012-03-01
Lattice gauge theory is a formulation of quantum field theory with gauge symmetries on a space-time lattice. This formulation is particularly suitable for describing hadronic phenomena. In this article we review the present status of lattice QCD. We outline some of the computational methods, discuss some phenomenological applications and a variety of non-perturbative topics. The list of references is severely incomplete, the ones we have included are text books or reviews and a few subjectively selected papers. Kronfeld and Quigg (2010) supply a reasonably comprehensive set of QCD references. We apologize for the fact that have not covered many important topics such as QCD at finite density and heavy quark effective theory adequately, and mention some of them only in the last section "In Brief". These topics should be considered in further Scholarpedia articles.
6. Digital lattice gauge theories
Zohar, Erez; Reznik, Benni; Cirac, J Ignacio
2016-01-01
We propose a general scheme for a digital construction of lattice gauge theories with dynamical fermions. In this method, the four-body interactions arising in models with $2+1$ dimensions and higher, are obtained stroboscopically, through a sequence of two-body interactions with ancillary degrees of freedom. This yields stronger interactions than the ones obtained through pertubative methods, as typically done in previous proposals, and removes an important bottleneck in the road towards experimental realizations. The scheme applies to generic gauge theories with Lie or finite symmetry groups, both Abelian and non-Abelian. As a concrete example, we present the construction of a digital quantum simulator for a $\\mathbb{Z}_{3}$ lattice gauge theory with dynamical fermionic matter in $2+1$ dimensions, using ultracold atoms in optical lattices, involving three atomic species, representing the matter, gauge and auxiliary degrees of freedom, that are separated in three different layers. By moving the ancilla atoms...
7. Digital lattice gauge theories
Zohar, Erez; Farace, Alessandro; Reznik, Benni; Cirac, J. Ignacio
2017-02-01
We propose a general scheme for a digital construction of lattice gauge theories with dynamical fermions. In this method, the four-body interactions arising in models with 2 +1 dimensions and higher are obtained stroboscopically, through a sequence of two-body interactions with ancillary degrees of freedom. This yields stronger interactions than the ones obtained through perturbative methods, as typically done in previous proposals, and removes an important bottleneck in the road towards experimental realizations. The scheme applies to generic gauge theories with Lie or finite symmetry groups, both Abelian and non-Abelian. As a concrete example, we present the construction of a digital quantum simulator for a Z3 lattice gauge theory with dynamical fermionic matter in 2 +1 dimensions, using ultracold atoms in optical lattices, involving three atomic species, representing the matter, gauge, and auxiliary degrees of freedom, that are separated in three different layers. By moving the ancilla atoms with a proper sequence of steps, we show how we can obtain the desired evolution in a clean, controlled way.
8. London relation and fluxoid quantization for monopole currents in U(1) lattice gauge theory
Singh, Vandana; Browne, Dana A; 10.1103/PhysRevD.47.1715
2009-01-01
We explore the analogy between quark confinement and the Meissner effect in superconductors. We measure the response of color-magnetic "supercurrents" from Dirac magnetic monopoles to the presence of a static quark-antiquark pair in four dimensional U(1) lattice gauge theory. Our results indicate that in the confined phase these currents screen the color-electric flux due to the quarks in an electric analogy of the Meisner effect. We show that U(1) lattice guage theory obeys both a dual London equation and an electric fluxoid quantization condition.
9. Introduction to lattice gauge theory
Gupta, R.
The lattice formulation of Quantum Field Theory (QFT) can be exploited in many ways. We can derive the lattice Feynman rules and carry out weak coupling perturbation expansions. The lattice then serves as a manifestly gauge invariant regularization scheme, albeit one that is more complicated than standard continuum schemes. Strong coupling expansions: these give us useful qualitative information, but unfortunately no hard numbers. The lattice theory is amenable to numerical simulations by which one calculates the long distance properties of a strongly interacting theory from first principles. The observables are measured as a function of the bare coupling g and a gauge invariant cut-off approx. = 1/alpha, where alpha is the lattice spacing. The continuum (physical) behavior is recovered in the limit alpha yields 0, at which point the lattice artifacts go to zero. This is the more powerful use of lattice formulation, so in these lectures the author focuses on setting up the theory for the purpose of numerical simulations to get hard numbers. The numerical techniques used in Lattice Gauge Theories have their roots in statistical mechanics, so it is important to develop an intuition for the interconnection between quantum mechanics and statistical mechanics.
10. Technicolor and Lattice Gauge Theory
Chivukula, R Sekhar
2010-01-01
Technicolor and other theories of dynamical electroweak symmetry breaking invoke chiral symmetry breaking triggered by strong gauge-dynamics, analogous to that found in QCD, to explain the observed W, Z, and fermion masses. In this talk we describe why a realistic theory of dynamical electroweak symmetry breaking must, relative to QCD, produce an enhanced fermion condensate. We quantify the degree to which the technicolor condensate must be enhanced in order to yield the observed quark masses, and still be consistent with phenomenological constraints on flavor-changing neutral-currents. Lattice studies of technicolor and related theories provide the only way to demonstrate that such enhancements are possible and, hopefully, to discover viable candidate models. We comment briefly on the current status of non-perturbative investigations of dynamical electroweak symmetry breaking, and provide a "wish-list" of phenomenologically-relevant properties that are important to calculate in these theories
11. Effective Field Theories and Lattice QCD
Bernard, C
2015-01-01
I describe some of the many connections between lattice QCD and effective field theories, focusing in particular on chiral effective theory, and, to a lesser extent, Symanzik effective theory. I first discuss the ways in which effective theories have enabled and supported lattice QCD calculations. Particular attention is paid to the inclusion of discretization errors, for a variety of lattice QCD actions, into chiral effective theory. Several other examples of the usefulness of chiral perturbation theory, including the encoding of partial quenching and of twisted boundary conditions, are also described. In the second part of the talk, I turn to results from lattice QCD for the low energy constants of the two- and three-flavor chiral theories. I concentrate here on mesonic quantities, but the dependence of the nucleon mass on the pion mass is also discussed. Finally I describe some recent preliminary lattice QCD calculations by the MILC Collaboration relating to the three-flavor chiral limit.
12. Introduction to Vortex Lattice Theory
Santiago Pinzón
2015-10-01
Full Text Available Panel methods have been widely used in industry and are well established since the 1970s for aerodynamic analysis and computation. The Vortex Lattice Panel Method presented in this study comes across a sophisticated method that provides a quick solution time, allows rapid changes in geometry and suits well for aerodynamic analysis. The aerospace industry is highly competitive in design efficiency, and perhaps one of the most important factors on airplane design and engineering today is multidisciplinary optimization. Any cost reduction method in the design cycle of a product becomes vital in the success of its outcome. The subsequent sections of this article will further explain in depth the theory behind the vortex lattice method, and the reason behind its selection as the method for aerodynamic analysis during preliminary design work and computation within the aerospace industry. This article is analytic in nature, and its main objective is to present a mathematical summary of this widely used computational method in aerodynamics.
13. Lattice theory special topics and applications
Wehrung, Friedrich
2014-01-01
George Grätzer's Lattice Theory: Foundation is his third book on lattice theory (General Lattice Theory, 1978, second edition, 1998). In 2009, Grätzer considered updating the second edition to reflect some exciting and deep developments. He soon realized that to lay the foundation, to survey the contemporary field, to pose research problems, would require more than one volume and more than one person. So Lattice Theory: Foundation provided the foundation. Now we complete this project with Lattice Theory: Special Topics and Applications, written by a distinguished group of experts, to cover some of the vast areas not in Foundation. This first volume is divided into three parts. Part I. Topology and Lattices includes two chapters by Klaus Keimel, Jimmie Lawson and Ales Pultr, Jiri Sichler. Part II. Special Classes of Finite Lattices comprises four chapters by Gabor Czedli, George Grätzer and Joseph P. S. Kung. Part III. Congruence Lattices of Infinite Lattices and Beyond includes four chapters by Friedrich W...
14. Lattice gauge theories and Monte Carlo simulations
Rebbi, Claudio
1983-01-01
This volume is the most up-to-date review on Lattice Gauge Theories and Monte Carlo Simulations. It consists of two parts. Part one is an introductory lecture on the lattice gauge theories in general, Monte Carlo techniques and on the results to date. Part two consists of important original papers in this field. These selected reprints involve the following: Lattice Gauge Theories, General Formalism and Expansion Techniques, Monte Carlo Simulations. Phase Structures, Observables in Pure Gauge Theories, Systems with Bosonic Matter Fields, Simulation of Systems with Fermions.
15. Working Group Report: Lattice Field Theory
Blum, T.; et al.,
2013-10-22
This is the report of the Computing Frontier working group on Lattice Field Theory prepared for the proceedings of the 2013 Community Summer Study ("Snowmass"). We present the future computing needs and plans of the U.S. lattice gauge theory community and argue that continued support of the U.S. (and worldwide) lattice-QCD effort is essential to fully capitalize on the enormous investment in the high-energy physics experimental program. We first summarize the dramatic progress of numerical lattice-QCD simulations in the past decade, with some emphasis on calculations carried out under the auspices of the U.S. Lattice-QCD Collaboration, and describe a broad program of lattice-QCD calculations that will be relevant for future experiments at the intensity and energy frontiers. We then present details of the computational hardware and software resources needed to undertake these calculations.
16. A classification of 2-dim Lattice Theory
Kieburg, Mario; Zafeiropoulos, Savvas
2013-01-01
A unified classification and analysis is presented of two dimensional Dirac operators of QCD-like theories in the continuum as well as in a naive lattice discretization. Thereby we consider the quenched theory in the strong coupling limit. We do not only consider the case of a lattice which has an even number of lattice sites in both directions and is thus equivalent to the case of staggered fermions. We also study lattices with one or both directions with an odd parity to understand the general mechanism of changing the universality class via a discretization. Furthermore we identify the corresponding random matrix ensembles sharing the global symmetries of these QCD-like theories. Despite the Mermin-Wagner-Coleman theorem we find good agreement of lattice data with our random matrix predictions.
17. Modified $U(1)$ lattice gauge theory towards realistic lattice QED
Bornyakov, V G; Müller-Preussker, M
1992-01-01
We study properties of the compact $~4D~$ $U(1)$ lattice gauge theory with monopoles {\\it removed}. Employing Monte Carlo simulations we calculate correlators of scalar, vector and tensor operators at zero and nonzero momenta $~\\vec{p}~$. We confirm that the theory without monopoles has no phase transition, at least, in the interval $~0 < \\beta \\leq 2~$. There the photon becomes massless and fits the lattice free field theory dispersion relation very well. The energies of the $~0^{++}~$, $~1^{+-}~$ and $~2^{++}~$ states show a rather weak dependence on the coupling in the interval of $~\\beta~$ investigated, and their ratios are practically constant. We show also a further modification of the theory suppressing the negative plaquettes to improve drastically the overlap with the lowest states (at least, for $~J=1$).
18. Lattice gauge theory for QCD
DeGrand, T. [Univ. of Colorado, Boulder, CO (United States). Dept. of Physics
1997-06-01
These lectures provide an introduction to lattice methods for nonperturbative studies of Quantum Chromodynamics. Lecture 1: Basic techniques for QCD and results for hadron spectroscopy using the simplest discretizations; lecture 2: Improved actions--what they are and how well they work; lecture 3: SLAC physics from the lattice-structure functions, the mass of the glueball, heavy quarks and {alpha}{sub s} (M{sub z}), and B-{anti B} mixing. 67 refs., 36 figs.
19. Lattice methods and effective field theory
Nicholson, Amy N
2016-01-01
Lattice field theory is a non-perturbative tool for studying properties of strongly interacting field theories, which is particularly amenable to numerical calculations and has quantifiable systematic errors. In these lectures we apply these techniques to nuclear Effective Field Theory (EFT), a non-relativistic theory for nuclei involving the nucleons as the basic degrees of freedom. The lattice formulation of [1,2] for so-called pionless EFT is discussed in detail, with portions of code included to aid the reader in code development. Systematic and statistical uncertainties of these methods are discussed at length, and extensions beyond pionless EFT are introduced in the final Section.
20. Quantum Finite Elements for Lattice Field Theory
Brower, Richard C; Gasbarro, Andrew; Raben, Timothy; Tan, Chung-I; Weinberg, Evan
2016-01-01
Viable non-perturbative methods for lattice quantum field theories on curved manifolds are difficult. By adapting features from the traditional finite element methods (FEM) and Regge Calculus, a new simplicial lattice Quantum Finite Element (QFE) Lagrangian is constructed for fields on a smooth Riemann manifold. To reach the continuum limit additional counter terms must be constructed to cancel the ultraviolet distortions. This is tested by the comparison of phi 4-th theory at the Wilson-Fisher fixed point with the exact Ising (c =1/2) CFT on a 2D Riemann sphere. The Dirac equation is also constructed on a simplicial lattice approximation to a Riemann manifold by introducing a lattice vierbein and spin connection on each link. Convergence of the QFE Dirac equation is tested against the exact solution for the 2D Riemann sphere. Future directions and applications to Conformal Field Theories are suggested.
1. 二语习得动态系统理论综述%Review of the Theory of Dynamic System on Second Lan―guage Acquisition
王一凡
2015-01-01
二语习得理论研究范式众多,每种范式都从不同的角度和侧面对二语习得研究做出解释,动态系统理论便是其中一个.本文尝试对动态系统理论的理论框架、研究发展及其在二语习得领域的贡献和不足进行简单的介绍,为进一步的研究提供参考.%There are many kinds of paradigms about theories' re-search on second language acquisition (SLA), and each paradigm explains the SLA from different aspects and angels. The theory of dynamic system is one of them. This thesis tries to make a brief introduction on its theory framework, research development and its contribution and disadvantage in the field of SLA.
2. Quiver gauge theories and integrable lattice models
Yagi, Junya
2015-01-01
We discuss connections between certain classes of supersymmetric quiver gauge theories and integrable lattice models from the point of view of topological quantum field theories (TQFTs). The relevant classes include 4d $\\mathcal{N} = 1$ theories known as brane box and brane tilling models, 3d $\\mathcal{N} = 2$ and 2d $\\mathcal{N} = (2,2)$ theories obtained from them by compactification, and 2d $\\mathcal{N} = (0,2)$ theories closely related to these theories. We argue that their supersymmetric indices carry structures of TQFTs equipped with line operators, and as a consequence, are equal to the partition functions of lattice models. The integrability of these models follows from the existence of extra dimension in the TQFTs, which emerges after the theories are embedded in M-theory. The Yang-Baxter equation expresses the invariance of supersymmetric indices under Seiberg duality and its lower-dimensional analogs.
3. Quiver gauge theories and integrable lattice models
Yagi, Junya [International School for Advanced Studies (SISSA),via Bonomea 265, 34136 Trieste (Italy); INFN - Sezione di Trieste,via Valerio 2, 34149 Trieste (Italy)
2015-10-09
We discuss connections between certain classes of supersymmetric quiver gauge theories and integrable lattice models from the point of view of topological quantum field theories (TQFTs). The relevant classes include 4d N=1 theories known as brane box and brane tilling models, 3d N=2 and 2d N=(2,2) theories obtained from them by compactification, and 2d N=(0,2) theories closely related to these theories. We argue that their supersymmetric indices carry structures of TQFTs equipped with line operators, and as a consequence, are equal to the partition functions of lattice models. The integrability of these models follows from the existence of extra dimension in the TQFTs, which emerges after the theories are embedded in M-theory. The Yang-Baxter equation expresses the invariance of supersymmetric indices under Seiberg duality and its lower-dimensional analogs.
4. Lattice Gauge Theories and Spin Models
Mathur, Manu
2016-01-01
The Wegner $Z_2$ gauge theory-$Z_2$ Ising spin model duality in $(2+1)$ dimensions is revisited and derived through a series of canonical transformations. These $Z_2$ results are directly generalized to SU(N) lattice gauge theory in $(2+1)$ dimensions to obtain a dual SU(N) spin model in terms of the SU(N) magnetic fields and electric scalar potentials. The gauge-spin duality naturally leads to a new gauge invariant disorder operator for SU(N) lattice gauge theory. A variational ground state of the dual SU(2) spin model with only nearest neighbour interactions is constructed to analyze SU(2) lattice gauge theory.
5. Multiphase lattice Boltzmann methods theory and application
Huang, Haibo; Lu, Xiyun
2015-01-01
Theory and Application of Multiphase Lattice Boltzmann Methods presents a comprehensive review of all popular multiphase Lattice Boltzmann Methods developed thus far and is aimed at researchers and practitioners within relevant Earth Science disciplines as well as Petroleum, Chemical, Mechanical and Geological Engineering. Clearly structured throughout, this book will be an invaluable reference on the current state of all popular multiphase Lattice Boltzmann Methods (LBMs). The advantages and disadvantages of each model are presented in an accessible manner to enable the reader to choose the
6. Topological Charge of Lattice Abelian Gauge Theory
Fujiwara, T; Wu, K
2001-01-01
Configuration space of abelian gauge theory on a periodic lattice becomes topologically disconnected by excising exceptional gauge field configurations. It is possible to define a U(1) bundle from the nonexceptional link variables by a smooth interpolation of the transition functions. The lattice analogue of Chern character obtained by a cohomological technique based on the noncommutative differential calculus is shown to give a topological charge related to the topological winding number of the U(1) bundle.
7. Nuclear effective field theory on the lattice
Krebs, H; Epelbaum, E; Lee, D; ner, Ulf-G Mei\\ss
2008-01-01
In the low-energy region far below the chiral symmetry breaking scale (which is of the order of 1 GeV) chiral perturbation theory provides a model-independent approach for quantitative description of nuclear processes. In the two- and more-nucleon sector perturbation theory is applicable only at the level of an effective potential which serves as input in the corresponding dynamical equation. To deal with the resulting many-body problem we put chiral effective field theory (EFT) on the lattice. Here we present the results of our lattice EFT study up to next-to-next-to-leading order in the chiral expansion. Accurate description of two-nucleon phase-shifts and ground state energy ratio of dilute neutron matter up to corrections of higher orders shows that lattice EFT is a promising tool for a quantitative description of low-energy few- and many-body systems.
8. Chiral Perturbation Theory With Lattice Regularization
Ouimet, P P A
2005-01-01
In this work, alternative methods to regularize chiral perturbation theory are discussed. First, Long Distance Regularization will be considered in the presence of the decuplet of the lightest spin 32 baryons for several different observables. This serves motivation and introduction to the use of the lattice regulator for chiral perturbation theory. The mesonic, baryonic and anomalous sectors of chiral perturbation theory will be formulated on a lattice of space time points. The consistency of the lattice as a regulator will be discussed in the context of the meson and baryon masses. Order a effects will also be discussed for the baryon masses, sigma terms and magnetic moments. The work will close with an attempt to derive an effective Wess-Zumino-Witten Lagrangian for Wilson fermions at non-zero a. Following this discussion, there will be a proposal for a phenomenologically useful WZW Lagrangian at non-zero a.
9. Chiral perturbation theory for lattice QCD
Baer, Oliver
2010-07-21
The formulation of chiral perturbation theory (ChPT) for lattice Quantum Chromodynamics (QCD) is reviewed. We start with brief summaries of ChPT for continuum QCD as well as the Symanzik effective theory for lattice QCD. We then review the formulation of ChPT for lattice QCD. After an additional chapter on partial quenching and mixed action theories various concrete applications are discussed: Wilson ChPT, staggered ChPT and Wilson ChPT with a twisted mass term. The remaining chapters deal with the epsilon regime with Wilson fermions and selected results in mixed action ChPT. Finally, the formulation of heavy vector meson ChPT with Wilson fermions is discussed. (orig.)
10. A theory of latticed plates and shells
Pshenichnon, Gi
1993-01-01
The book presents the theory of latticed shells as continual systems and describes its applications. It analyses the problems of statics, stability and dynamics. Generally, a classical rod deformation theory is applied. However, in some instances, more precise theories which particularly consider geometrical and physical nonlinearity are employed. A new effective method for solving general boundary value problems and its application for numerical and analytical solutions of mathematical physics and reticulated shell theory problems is described. A new method of solving the shell theory's nonli
11. Lattice gaugefixing and other optics in lattice gauge theory
Yee, Ken
1992-06-01
We present results from four projects. In the first, quark and gluon propagators and effective masses and {Delta}I = 1/2 Rule operator matching coefficients are computed numerically in gaugefixed lattice QCD. In the second, the same quantities are evaluated analytically in the strong coupling, N {yields} {infinity} limit. In the third project, the Schwinger model is studied in covariant gauges, where we show that the effective electron mass varies with the gauge parameter and that longitudinal gaugefixing ambiguities affect operator product expansion coefficients (analogous to {Delta}I = 1/2 Rule matching coefficients) determined by matching gauge variant matrix elements. However, we find that matching coefficients even if shifted by the unphysical modes are {xi} invariant. In the fourth project, we show that the strong coupling parallelogram lattice Schwinger model as a different thermodynamic limit than the weak coupling continuum limit. As a function of lattice skewness angle these models span the {Delta} = {minus}1 critical line of 6-vertex models which, in turn, have been identified as c = 1 conformal field theories.
12. Lattice gaugefixing and other optics in lattice gauge theory
Yee, Ken.
1992-06-01
We present results from four projects. In the first, quark and gluon propagators and effective masses and {Delta}I = 1/2 Rule operator matching coefficients are computed numerically in gaugefixed lattice QCD. In the second, the same quantities are evaluated analytically in the strong coupling, N {yields} {infinity} limit. In the third project, the Schwinger model is studied in covariant gauges, where we show that the effective electron mass varies with the gauge parameter and that longitudinal gaugefixing ambiguities affect operator product expansion coefficients (analogous to {Delta}I = 1/2 Rule matching coefficients) determined by matching gauge variant matrix elements. However, we find that matching coefficients even if shifted by the unphysical modes are {xi} invariant. In the fourth project, we show that the strong coupling parallelogram lattice Schwinger model as a different thermodynamic limit than the weak coupling continuum limit. As a function of lattice skewness angle these models span the {Delta} = {minus}1 critical line of 6-vertex models which, in turn, have been identified as c = 1 conformal field theories.
13. Recent advances in lattice gauge theories
R V Gavai
2000-04-01
Recent progress in the field of lattice gauge theories is briefly reviewed for a nonspecialist audience. While the emphasis is on the latest and more definitive results that have emerged prior to this symposium, an effort has been made to provide them with minimal technicalities.
14. Integrable Lattice Models From Gauge Theory
Witten, Edward
2016-01-01
These notes provide an introduction to recent work by Kevin Costello in which integrable lattice models of classical statistical mechanics in two dimensions are understood in terms of quantum gauge theory in four dimensions. This construction will be compared to the more familiar relationship between quantum knot invariants in three dimensions and Chern-Simons gauge theory. (Based on a Whittaker Colloquium at the University of Edinburgh and a lecture at Strings 2016 in Beijing.)
15. Large N lattice gauge theory
Narayanan, Rajamani
2008-01-01
Wilson loops in large N gauge theory exhibit a weak to strong coupling transition as the loop is dilated. A multiplicative matrix model captures the universal behavior associated with this transition. A universal scaling function is obtained in a double scaling limit. Numerical studies show that both large N QCD in three dimensions and the SU(N) principal chiral model in two dimensions are in the same universality class.
16. Gauge theories and integrable lattice models
Witten, Edward
1989-08-01
Investigations of new knot polynomials discovered in the last few years have shown them to be intimately connected with soluble models of two dimensional lattice statistical mechanics. In this paper, these results, which in time may illuminate the whole question of why integrable lattice models exist, are reconsidered from the point of view of three dimensional gauge theory. Expectation values of Wilson lines in three dimensional Chern-Simons gauge theories can be computed by evaluating the partition functions of certain lattice models on finite graphs obtained by projecting the Wilson lines to the plane. The models in question — previously considered in both the knot theory and statistical mechanics — are IRF models in which the local Boltzmann weights are the matrix elements of braiding matrices in rational conformal field theories. These matrix elements, in turn, can be presented in three dimensional gauge theory in terms of the expectation value of a certain tetrahedral configuration of Wilson lines. This representation makes manifest a surprising symmetry of the braiding matrix elements in conformal field theory.
17. Noncompact lattice formulation of gauge theories
Friedberg, R; Pang, Y; Ren, H C
1995-01-01
We expand the gauge field in terms of a suitably constructed complete set of Bloch wave functions, each labeled by a band designation \\,n\\, and a wave number \\,\\vec K\\, restricted to the Brillouin zone. A noncompact formulation of lattice QCD (or QED) can be derived by restricting the expansion only to the \\,0^{th}-band (\\,n = 0\\,) functions, which are simple continuum interpolations of discrete values associated with sites or links on a lattice. The exact continuum theory can be reached through the inclusion of all \\,n = 0\\, and \\,n \
18. National Computational Infrastructure for Lattice Gauge Theory
Brower, Richard C.
2014-04-15
SciDAC-2 Project The Secret Life of Quarks: National Computational Infrastructure for Lattice Gauge Theory, from March 15, 2011 through March 14, 2012. The objective of this project is to construct the software needed to study quantum chromodynamics (QCD), the theory of the strong interactions of sub-atomic physics, and other strongly coupled gauge field theories anticipated to be of importance in the energy regime made accessible by the Large Hadron Collider (LHC). It builds upon the successful efforts of the SciDAC-1 project National Computational Infrastructure for Lattice Gauge Theory, in which a QCD Applications Programming Interface (QCD API) was developed that enables lattice gauge theorists to make effective use of a wide variety of massively parallel computers. This project serves the entire USQCD Collaboration, which consists of nearly all the high energy and nuclear physicists in the United States engaged in the numerical study of QCD and related strongly interacting quantum field theories. All software developed in it is publicly available, and can be downloaded from a link on the USQCD Collaboration web site, or directly from the github repositories with entrance linke http://usqcd-software.github.io
19. Lattice Theories with Nonlinearly Realized Chiral Symmetry
Chandrasekharan, S; Steffen, F D; Wiese, U J
2003-01-01
We present the lattice formulation of effective Lagrangians in which chiral symmetry is realized nonlinearly on the fermion fields. In this framework both the Wilson term removing unphysical doubler fermions and the fermion mass term do not break chiral symmetry. Our lattice formulation allows us to address non-perturbative questions in effective theories of baryons interacting with pions and in models involving constitutent quarks interacting with pions and gluons. With the presented methods, a system containing a non-zero density of static baryons interacting with pions can be studied on the lattice without encountering a complex action problem. This might lead to new insights into the phase diagram of strongly interacting matter at non-zero chemical potential.
20. Chiral symmetry and lattice gauge theory
Creutz, M
1994-01-01
I review the problem of formulating chiral symmetry in lattice gauge theory. I discuss recent approaches involving an infinite tower of additional heavy states to absorb Fermion doublers. For hadronic physics this provides a natural scheme for taking quark masses to zero without requiring a precise tuning of parameters. A mirror Fermion variation provides a possible way of extending the picture to chirally coupled light Fermions. Talk presented at "Quark Confinement and the Hadron Spectrum," Como, Italy, 20-24 June 1994.
1. Cosmological phase transitions from lattice field theory
Jansen, Karl [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC
2011-11-22
In this proceedings contribution we discuss the fate of the electroweak and the quantum chromodynamics phase transitions relevant for the early stage of the universe at non-zero temperature. These phase transitions are related to the Higgs mechanism and the breaking of chiral symmetry, respectively. We will review that non-perturbative lattice field theory simulations show that these phase transitions actually do not occur in nature and that physical observables show a completely smooth behaviour as a function of the temperature.
2. Lattice gauge theories and Monte Carlo algorithms
Creutz, M. (Brookhaven National Lab., Upton, NY (USA). Physics Dept.)
1989-07-01
Lattice gauge theory has become the primary tool for non-perturbative calculations in quantum field theory. These lectures review some of the foundations of this subject. The first lecture reviews the basic definition of the theory in terms of invariant integrals over group elements on lattice bonds. The lattice represents an ultraviolet cutoff, and renormalization group arguments show how the bare coupling must be varied to obtain the continuum limit. Expansions in the inverse of the coupling constant demonstrate quark confinement in the strong coupling limit. The second lecture turns to numerical simulation, which has become an important approach to calculating hadronic properties. Here I discuss the basic algorithms for obtaining appropriately weighted gauge field configurations. The third lecture turns to algorithms for treating fermionic fields, which still require considerably more computer time than needed for purely bosonic simulations. Some particularly promising recent approaches are based on global accept-reject steps and should display a rather favorable dependence of computer time on the system volume. (orig.).
3. Global anomalies in Chiral Lattice Gauge Theory
Bär, Oliver; Campos, Isabel
As first realized by Witten an SU(2) gauge theory coupled to a single Weyl fermion suffers from a global anomaly. This problem is addressed here in the context of the recent developments on chiral gauge theories on the lattice. We find Witten's anomaly manifests in the impossibility of defining globally a fermion measure that reproduces the proper continuum limit. Moreover, following Witten's original argument, we check numerically the crossing of the lowest eigenvalues of Neuberger's operator along a path connecting two gauge fields that differ by a topologically non-trivial gauge transformation.
4. Jarzynski's theorem for lattice gauge theory
Caselle, Michele; Nada, Alessandro; Panero, Marco; Toniato, Arianna
2016-01-01
Jarzynski's theorem is a well-known equality in statistical mechanics, which relates fluctuations in the work performed during a non-equilibrium transformation of a system, to the free-energy difference between two equilibrium states. In this article, we extend Jarzynski's theorem to lattice gauge theory, and present examples of applications for two challenging computational problems, namely the calculation of interface free energies and the determination of the equation of state. We conclude with a discussion of further applications of interest in QCD and in other strongly coupled gauge theories, in particular for the Schroedinger functional and for simulations at finite density using reweighting techniques.
5. The ergodic theory of lattice subgroups
Gorodnik, Alexander
2010-01-01
The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean
6. Quantum Holonomy Theory, Lattice-Independent Formulation
Aastrup, Johannes
2016-01-01
Quantum holonomy theory is a candidate for a non-perturbative theory of quantum gravity coupled to fermions. The theory is based on the QHD-algebra, which essentially encodes how local degrees of freedom are moved on a three-dimensional manifold. In this paper we continue the development of the theory by providing a lattice-independent formulation. We first define a Dirac type operator over a configuration space of Ashtekar connections and use it to formulate a graded version of the QHD-algebra. Next we formulate necessary conditions for a state to exist on this algebra and use the GNS construction to build a kinematical Hilbert space. Finally we find that operators, that correspond to the Dirac and gravitational Hamiltonians in a semi-classical limit, are background independent.
7. Lattice gauge theories and spin models
Mathur, Manu; Sreeraj, T. P.
2016-10-01
The Wegner Z2 gauge theory-Z2 Ising spin model duality in (2 +1 ) dimensions is revisited and derived through a series of canonical transformations. The Kramers-Wannier duality is similarly obtained. The Wegner Z2 gauge-spin duality is directly generalized to SU(N) lattice gauge theory in (2 +1 ) dimensions to obtain the SU(N) spin model in terms of the SU(N) magnetic fields and their conjugate SU(N) electric scalar potentials. The exact and complete solutions of the Z2, U(1), SU(N) Gauss law constraints in terms of the corresponding spin or dual potential operators are given. The gauge-spin duality naturally leads to a new gauge invariant magnetic disorder operator for SU(N) lattice gauge theory which produces a magnetic vortex on the plaquette. A variational ground state of the SU(2) spin model with nearest neighbor interactions is constructed to analyze SU(2) gauge theory.
8. Rough Set Theory over Fuzzy Lattices
Guilong Liu
2006-01-01
Rough set theory, proposed by Pawlak in 1982, is a tool for dealing with uncertainty and vagueness aspects of knowledge model. The main idea of roug h sets corresponds to the lower and upper approximations based on equivalence relations. This paper studies the rough set and its extension. In our talk, we present a linear algebra approach to rough set and its extension, give an equivalent definition of the lower and upper approximations of rough set based on the characteristic function of sets, and then we explain the lower and upper approximations as the colinear map and linear map of sets, respectively. Finally, we define the rough sets over fuzzy lattices, which cover the rough set and fuzzy rough set, and the independent axiomatic systems are constructed to characterize the lower and upper approximations of rough set over fuzzy lattices, respectively, based on inner and outer products. The axiomatic systems unify the axiomization of Pawlak's rough sets and fuzzy rough sets.
9. Fractional Quantum Field Theory: From Lattice to Continuum
Vasily E. Tarasov
2014-01-01
Full Text Available An approach to formulate fractional field theories on unbounded lattice space-time is suggested. A fractional-order analog of the lattice quantum field theories is considered. Lattice analogs of the fractional-order 4-dimensional differential operators are proposed. We prove that continuum limit of the suggested lattice field theory gives a fractional field theory for the continuum 4-dimensional space-time. The fractional field equations, which are derived from equations for lattice space-time with long-range properties of power-law type, contain the Riesz type derivatives on noninteger orders with respect to space-time coordinates.
10. Ultracold Quantum Gases and Lattice Systems: Quantum Simulation of Lattice Gauge Theories
Wiese, U -J
2013-01-01
Abelian and non-Abelian gauge theories are of central importance in many areas of physics. In condensed matter physics, Abelian U(1) lattice gauge theories arise in the description of certain quantum spin liquids. In quantum information theory, Kitaev's toric code is a Z(2) lattice gauge theory. In particle physics, Quantum Chromodynamics (QCD), the non-Abelian SU(3) gauge theory of the strong interactions between quarks and gluons, is non-perturbatively regularized on a lattice. Quantum link models extend the concept of lattice gauge theories beyond the Wilson formulation, and are well suited for both digital and analog quantum simulation using ultracold atomic gases in optical lattices. Since quantum simulators do not suffer from the notorious sign problem, they open the door to studies of the real-time evolution of strongly coupled quantum systems, which are impossible with classical simulation methods. A plethora of interesting lattice gauge theories suggests itself for quantum simulation, which should al...
11. Entanglement in Weakly Coupled Lattice Gauge Theories
2015-01-01
We present a direct lattice gauge theory computation that, without using dualities, demonstrates that the entanglement entropy of Yang-Mills theories with arbitrary gauge group $G$ contains a generic logarithmic term at sufficiently weak coupling $e$. In two spatial dimensions, for a region of linear size $r$, this term equals $\\frac{1}{2} \\dim(G) \\log\\left(e^2 r\\right)$ and it dominates the universal part of the entanglement entropy. Such logarithmic terms arise from the entanglement of the softest mode in the entangling region with the environment. For Maxwell theory in two spatial dimensions, our results agree with those obtained by dualizing to a compact scalar with spontaneous symmetry breaking.
12. Observable currents in lattice field theories
Zapata, José A
2016-01-01
Observable currents are spacetime local objects that induce physical observables when integrated on an auxiliary codimension one surface. Since the resulting observables are independent of local deformations of the integration surface, the currents themselves carry most of the information about the induced physical observables. I study observable currents in a multisymplectic framework for Lagrangian field theory over discrete spacetime. A weak version of observable currents preserves many of their properties, while inducing a family of observables capable of separating points in the space of physically distinct solutions. A Poisson bracket gives the space of observable currents the structure of a Lie algebra. Peierls bracket for bulk observables gives an algebra homomorphism mapping equivalence classes of bulk observables to weak observable currents. The study covers scalar fields, nonlinear sigma models and gauge theories (including gauge theory formulations of general relativity) on the lattice. Even when ...
13. Entanglement of Distillation for Lattice Gauge Theories
Van Acoleyen, Karel; Bultinck, Nick; Haegeman, Jutho; Marien, Michael; Scholz, Volkher B.; Verstraete, Frank
2016-09-01
We study the entanglement structure of lattice gauge theories from the local operational point of view, and, similar to Soni and Trivedi [J. High Energy Phys. 1 (2016) 1], we show that the usual entanglement entropy for a spatial bipartition can be written as the sum of an undistillable gauge part and of another part corresponding to the local operations and classical communication distillable entanglement, which is obtained by depolarizing the local superselection sectors. We demonstrate that the distillable entanglement is zero for pure Abelian gauge theories at zero gauge coupling, while it is in general nonzero for the non-Abelian case. We also consider gauge theories with matter, and show in a perturbative approach how area laws—including a topological correction—emerge for the distillable entanglement. Finally, we also discuss the entanglement entropy of gauge fixed states and show that it has no relation to the physical distillable entropy.
14. Statistical mechanics approach to lattice field theory
Amador, Arturo; Olaussen, Kåre
2016-01-01
The mean spherical approximation (MSA) is a closure relation for pair correlation functions (two-point functions) in statistical physics. It can be applied to a wide range of systems, is computationally fairly inexpensive, and when properly applied and interpreted lead to rather good results. In this paper we promote its applicability to euclidean quantum field theories formulated on a lattice, by demonstrating how it can be used to locate the critical lines of a class of multi-component bosonic models. The MSA has the potential to handle models lacking a positive definite integration measure, which therefore are difficult to investigate by Monte-Carlo simulations.
15. Lattice Gauge Field Theory and Prismatic Sets
Akyar, Bedia; Dupont, Johan Louis
as and in particular the latter we use to study lattice gauge theory in the sense of Phillips and Stone. Thus for a Lie group and a set of parallel transport functions defining the transition over faces of the simplices, we define a classifying map from the prismatic star to a prismatic version of the classifying......We study prismatic sets analogously to simplicial sets except that realization involves prisms, i.e., products of simplices rather than just simplices. Particular examples are the prismatic subdivision of a simplicial set and the prismatic star of . Both have the same homotopy type...
16. Parallel supercomputers for lattice gauge theory.
Brown, F R; Christ, N H
1988-03-18
During the past 10 years, particle physicists have increasingly employed numerical simulation to answer fundamental theoretical questions about the properties of quarks and gluons. The enormous computer resources required by quantum chromodynamic calculations have inspired the design and construction of very powerful, highly parallel, dedicated computers optimized for this work. This article gives a brief description of the numerical structure and current status of these large-scale lattice gauge theory calculations, with emphasis on the computational demands they make. The architecture, present state, and potential of these special-purpose supercomputers is described. It is argued that a numerical solution of low energy quantum chromodynamics may well be achieved by these machines.
17. Axial Anomaly in Lattice Abelian Gauge Theory in Arbitrary Dimensions
Fujiwara, T; Wu, K; Fujiwara, Takanori; Suzuki, Hiroshi; Wu, Ke
1999-01-01
Axial anomaly of lattice abelian gauge theory in hyper-cubic regular lattice in arbitrary even dimensions is investigated by applying the method of exterior differential calculus. The topological invariance, gauge invariance and locality of the axial anomaly determine the explicit form of the topological part. The anomaly is obtained up to a multiplicative constant for finite lattice spacing and can be interpreted as the Chern character of the abelian lattice gauge theory.
18. Matrix product states for lattice field theories
Banuls, M.C.; Cirac, J.I. [Max-Planck-Institut fuer Quantenoptik (MPQ), Garching (Germany); Cichy, K. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Poznan Univ. (Poland). Faculty of Physics; Jansen, K. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Saito, H. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Tsukuba Univ., Ibaraki (Japan). Graduate School of Pure and Applied Sciences
2013-10-15
The term Tensor Network States (TNS) refers to a number of families of states that represent different ansaetze for the efficient description of the state of a quantum many-body system. Matrix Product States (MPS) are one particular case of TNS, and have become the most precise tool for the numerical study of one dimensional quantum many-body systems, as the basis of the Density Matrix Renormalization Group method. Lattice Gauge Theories (LGT), in their Hamiltonian version, offer a challenging scenario for these techniques. While the dimensions and sizes of the systems amenable to TNS studies are still far from those achievable by 4-dimensional LGT tools, Tensor Networks can be readily used for problems which more standard techniques, such as Markov chain Monte Carlo simulations, cannot easily tackle. Examples of such problems are the presence of a chemical potential or out-of-equilibrium dynamics. We have explored the performance of Matrix Product States in the case of the Schwinger model, as a widely used testbench for lattice techniques. Using finite-size, open boundary MPS, we are able to determine the low energy states of the model in a fully non-perturbativemanner. The precision achieved by the method allows for accurate finite size and continuum limit extrapolations of the ground state energy, but also of the chiral condensate and the mass gaps, thus showing the feasibility of these techniques for gauge theory problems.
19. Wilson loop expectations in $SU(N)$ lattice gauge theory
Jafarov, Jafar
2016-01-01
This article gives a rigorous formulation and proof of the $1/N$ expansion for Wilson loop expectations in strongly coupled $SU(N)$ lattice gauge theory in any dimension. The coefficients of the expansion are represented as absolutely convergent sums over trajectories in a string theory on the lattice, establishing a kind of gauge-string duality. Moreover, it is shown that in large $N$ limit, calculations in $SU(N)$ lattice gauge theory with coupling strength $2\\beta$ corresponds to those in $SO(N)$ lattice gauge theory with coupling strength $\\beta$ when $|\\beta|$ is sufficiently small.
20. CERN Theory Institute: Future directions in lattice gauge theory
2010-01-01
The main goal of the Institute is to bring together researchers in lattice gauge theory and in its applications to phenomenology to discuss interesting future directions of research. The focus will be on new ideas rather than on the latest computation of the usual quantities. The aim is to identify calculations in QCD, flavour physics, other strongly-interacting theories, etc. which are of high physics interest, and to clarify the theoretical and technical difficulties which, at present, prevent us from carrying them out.
1. From lattice gauge theories to hydrogen atoms
Manu Mathur
2015-10-01
Full Text Available We construct canonical transformations to obtain a complete and most economical realization of the physical Hilbert space Hp of pure SU(22+1 lattice gauge theory in terms of Wigner coupled Hilbert spaces of hydrogen atoms. One hydrogen atom is assigned to every plaquette of the lattice. A complete orthonormal description of the Wilson loop basis in Hp is obtained by all possible angular momentum Wigner couplings of hydrogen atom energy eigenstates |n l m〉 describing electric fluxes on the loops. The SU(2 gauge invariance implies that the total angular momenta of all hydrogen atoms vanish. The canonical transformations also enable us to rewrite the Kogut–Susskind Hamiltonian in terms of fundamental Wilson loop operators and their conjugate electric fields. The resulting loop Hamiltonian has a global SU(2 invariance and a simple weak coupling (g2→0 continuum limit. The canonical transformations leading to the loop Hamiltonian are valid for any SU(N. The ideas and techniques can also be extended to higher dimension.
2. Attribute reduction theory and approach to concept lattice
ZHANG Wenxiu; WEI Ling; QI Jianjun
2005-01-01
The theory of the concept lattice is an efficient tool for knowledge representation and knowledge discovery, and is applied to many fields successfully. One focus of knowledge discovery is knowledge reduction. This paper proposes the theory of attribute reduction in the concept lattice, which extends the theory of the concept lattice. In this paper, the judgment theorems of consistent sets are examined, and the discernibility matrix of a formal context is introduced, by which we present an approach to attribute reduction in the concept lattice. The characteristics of three types of attributes are analyzed.
3. Topics in lattice QCD and effective field theory
Buchoff, Michael I.
Quantum Chromodynamics (QCD) is the fundamental theory that governs hadronic physics. However, due to its non-perturbative nature at low-energy/long distances, QCD calculations are difficult. The only method for performing these calculations is through lattice QCD. These computationally intensive calculations approximate continuum physics with a discretized lattice in order to extract hadronic phenomena from first principles. However, as in any approximation, there are multiple systematic errors between lattice QCD calculation and actual hardronic phenomena. Developing analytic formulae describing the systematic errors due to the discrete lattice spacings is the main focus of this work. To account for these systematic effects in terms of hadronic interactions, effective field theory proves to be useful. Effective field theory (EFT) provides a formalism for categorizing low-energy effects of a high-energy fundamental theory as long as there is a significant separation in scales. An example of this is in chiral perturbation theory (chiPT), where the low-energy effects of QCD are contained in a mesonic theory whose applicability is a result of a pion mass smaller than the chiral breaking scale. In a similar way, lattice chiPT accounts for the low-energy effects of lattice QCD, where a small lattice spacing acts the same way as the quark mass. In this work, the basics of this process are outlined, and multiple original calculations are presented: effective field theory for anisotropic lattices, I=2 pipi scattering for isotropic, anisotropic, and twisted mass lattices. Additionally, a combination of effective field theory and an isospin chemical potential on the lattice is proposed to extract several computationally difficult scattering parameters. Lastly, recently proposed local, chiral lattice actions are analyzed in the framework of effective field theory, which illuminates various challenges in simulating such actions.
4. Topics in Effective Field Theory for Lattice QCD
Walker-Loud, A
2006-01-01
In this work, we extend and apply effective field theory techniques to systematically understand a subset of lattice artifacts which pollute the lattice correlation functions for a few processes of physical interest. Where possible, we compare to existing lattice QCD calculations. In particular, we extend the heavy baryon Lagrangian to the next order in partially quenched chiral perturbation theory and use it to compute the masses of the lightest spin-1/2 and spin-3/2 baryons to next-to-next-to leading order. We then construct the twisted mass chiral Lagrangian for baryons and apply it to compute the lattice spacing corrections to the baryon masses simulated with twisted mass lattice QCD. We extend computations of the nucleon electromagnetic structure to account for finite volume effects, as these observables are particularly sensitive to the finite extent of the lattice. We resolve subtle peculiarities for lattice QCD simulations of polarizabilities and we show that using background field techniques, one can...
5. Lattice field theory applications in high energy physics
Gottlieb, Steven
2016-10-01
Lattice gauge theory was formulated by Kenneth Wilson in 1974. In the ensuing decades, improvements in actions, algorithms, and computers have enabled tremendous progress in QCD, to the point where lattice calculations can yield sub-percent level precision for some quantities. Beyond QCD, lattice methods are being used to explore possible beyond the standard model (BSM) theories of dynamical symmetry breaking and supersymmetry. We survey progress in extracting information about the parameters of the standard model by confronting lattice calculations with experimental results and searching for evidence of BSM effects.
6. Lattice field theory applications in high energy physics
Gottlieb, Steven
2016-01-01
Lattice gauge theory was formulated by Kenneth Wilson in 1974. In the ensuing decades, improvements in actions, algorithms, and computers have enabled tremendous progress in QCD, to the point where lattice calculations can yield sub-percent level precision for some quantities. Beyond QCD, lattice methods are being used to explore possible beyond the standard model (BSM) theories of dynamical symmetry breaking and supersymmetry. We survey progress in extracting information about the parameters of the standard model by confronting lattice calculations with experimental results and searching for evidence of BSM effects.
7. Matrix Product States for Lattice Field Theories
Bañuls, Mari Carmen; Cirac, J Ignacio; Jansen, Karl; Saito, Hana
2013-01-01
The term Tensor Network States (TNS) refers to a number of families of states that represent different ans\\"atze for the efficient description of the state of a quantum many-body system. Matrix Product States (MPS) are one particular case of TNS, and have become the most precise tool for the numerical study of one dimensional quantum many-body systems, as the basis of the Density Matrix Renormalization Group method. Lattice Gauge Theories (LGT), in their Hamiltonian version, offer a challenging scenario for these techniques. While the dimensions and sizes of the systems amenable to TNS studies are still far from those achievable by 4-dimensional LGT tools, Tensor Networks can be readily used for problems which more standard techniques, such as Markov chain Monte Carlo simulations, cannot easily tackle. Examples of such problems are the presence of a chemical potential or out-of-equilibrium dynamics. We have explored the performance of Matrix Product States in the case of the Schwinger model, as a widely used ...
8. Introduction to lattice theory with computer science applications
Garg, Vijay K
2015-01-01
A computational perspective on partial order and lattice theory, focusing on algorithms and their applications This book provides a uniform treatment of the theory and applications of lattice theory. The applications covered include tracking dependency in distributed systems, combinatorics, detecting global predicates in distributed systems, set families, and integer partitions. The book presents algorithmic proofs of theorems whenever possible. These proofs are written in the calculational style advocated by Dijkstra, with arguments explicitly spelled out step by step. The author's intent
9. Applications Of Chiral Perturbation Theory To Lattice Qcd
Van de Water, R S
2005-01-01
Quantum chromodynamics (QCD) is the fundamental theory that describes the interaction of quarks and gluons. Thus, in principle, one should be able to calculate all properties of hadrons from the QCD Lagrangian. It turns out, however, that such calculations can only be performed numerically on a computer using the nonperturbative method of lattice QCD, in which QCD is simulated on a discrete spacetime grid. Because lattice simulations use unphysically heavy quark masses (for computational reasons), lattice results must be connected to the real world using expressions calculated in chiral perturbation theory (χPT), the low-energy effective theory of QCD. Moreover, because real spacetime is continuous, they must be extrapolated to the continuum using an extension of χPT that includes lattice discretization effects, such as staggered χPT. This thesis is organized as follows. We motivate the need for lattice QCD and present the basic methodology in Chapter 1. We describe a common approximat...
10. Effective field theory of interactions on the lattice
Valiente, Manuel; Zinner, Nikolaj T.
2015-01-01
We consider renormalization of effective field theory interactions by discretizing the continuum on a tight-binding lattice. After studying the one-dimensional problem, we address s-wave collisions in three dimensions and relate the bare lattice coupling constants to the continuum coupling consta...
11. Heavy-quarkonium potential with input from lattice gauge theory
Serenone, Willian Matioli
2014-01-01
In this dissertation we study potential models incorporating a nonperturbative propagator obtained from lattice simulations of a pure gauge theory. Initially we review general aspects of gauge theories, the principles of the lattice formulation of quantum chromodynamics (QCD) and some properties of heavy quarkonia, i.e. bound states of a heavy quark and its antiquark. As an illustration of Monte Carlo simulations of lattice models, we present applications in the case of the harmonic oscillator and SU(2) gauge theory. We then study the effect of using a gluon propagator from lattice simulations of pure SU(2) theory as an input in a potential model for the description of quarkonium, in the case of bottomonium and charmonium. We use, in both cases, a numerical approach to evaluate masses of quarkonium states. The resulting spectra are compared to calculations using the Coulomb plus linear (or Cornell) potential.
12. Comparison of SO(3) and SU(2) lattice gauge theory
De Forcrand, Philippe; Forcrand, Philippe de; Jahn, Oliver
2003-01-01
The Villain form of SO(3) lattice gauge theory is studied and compared to Wilson's SU(2) theory. The topological invariants in SO(3) which correspond to twisted boundary conditions in SU(2) are discussed and lattice observables are introduced for them. An apparent SO(3) phase with negative adjoint Polyakov loop is explained in terms of these observables. The electric twist free energy, an order parameter for the confinement-deconfinement transition, is measured in both theories to calibrate the temperature. The results indicate that lattices with about 700^4 sites or larger will be needed to study the SO(3) confined phase. Alternative actions are discussed and an analytic path connecting SO(3) and SU(2) lattice gauge theory at weak coupling is exhibited. The relevance for confinement of the centre of the gauge group is discussed.
13. Effective Field Theory of Interactions on the Lattice
Valiente, Manuel; Zinner, Nikolaj Thomas
2015-12-01
We consider renormalization of effective field theory interactions by discretizing the continuum on a tight-binding lattice. After studying the one-dimensional problem, we address s-wave collisions in three dimensions and relate the bare lattice coupling constants to the continuum coupling constants. Our method constitutes a very simple avenue for the systematic renormalization in effective field theory, and is especially useful as the number of interaction parameters increases.
14. Proton–proton fusion in lattice effective field theory
Gautam Rupak
2015-02-01
Full Text Available The proton–proton fusion rate is calculated at low energy in a lattice effective field theory (EFT formulation. The strong and the Coulomb interactions are treated non-perturbatively at leading order in the EFT. The lattice results are shown to accurately describe the low energy cross section within the validity of the theory at energies relevant to solar physics. In prior works in the literature, Coulomb effects were generally not included in non-perturbative lattice calculations. Work presented here is of general interest in nuclear lattice EFT calculations that involve Coulomb effects at low energy. It complements recent developments of the adiabatic projection method for lattice calculations of nuclear reactions.
15. Testing chiral effective theory with quenched lattice QCD
Giusti, Leonardo; Necco, S; Peña, C; Wennekers, J; Wittig, H
2008-01-01
We investigate two-point correlation functions of left-handed currents computed in quenched lattice QCD with the Neuberger-Dirac operator. We consider two lattice spacings a~0.09,0.12 fm and two different lattice extents L~ 1.5, 2.0 fm; quark masses span both the p- and the epsilon-regimes. We compare the results with the predictions of quenched chiral perturbation theory, with the purpose of testing to what extent the effective theory reproduces quenched QCD at low energy. In the p-regime we test volume and quark mass dependence of the pseudoscalar decay constant and mass; in the epsilon-regime, we investigate volume and topology dependence of the correlators. While the leading order behaviour predicted by the effective theory is very well reproduced by the lattice data in the range of parameters that we explored, our numerical data are not precise enough to test next-to-leading order effects.
16. Testing chiral effective theory with quenched lattice QCD
Giusti, L.; Hernández, P.; Necco, S.; Pena, C.; Wennekers, J.; Wittig, H.
2008-05-01
We investigate two-point correlation functions of left-handed currents computed in quenched lattice QCD with the Neuberger-Dirac operator. We consider two lattice spacings a simeq 0.09,0.12 fm and two different lattice extents L simeq 1.5,2.0 fm; quark masses span both the p- and the epsilon-regimes. We compare the results with the predictions of quenched chiral perturbation theory, with the purpose of testing to what extent the effective theory reproduces quenched QCD at low energy. In the p-regime we test volume and quark mass dependence of the pseudoscalar decay constant and mass; in the epsilon-regime, we investigate volume and topology dependence of the correlators. While the leading order behaviour predicted by the effective theory is very well reproduced by the lattice data in the range of parameters that we explored, our numerical data are not precise enough to test next-to-leading order effects.
17. Vortex lattice theory: A linear algebra approach
Chamoun, George C.
Vortex lattices are prevalent in a large class of physical settings that are characterized by different mathematical models. We present a coherent and generalized Hamiltonian fluid mechanics-based formulation that reduces all vortex lattices into a classic problem in linear algebra for a non-normal matrix A. Via Singular Value Decomposition (SVD), the solution lies in the null space of the matrix (i.e., we require nullity( A) > 0) as well as the distribution of its singular values. We demonstrate that this approach provides a good model for various types of vortex lattices, and makes it possible to extract a rich amount of information on them. The contributions of this thesis can be classified into four main points. The first is asymmetric equilibria. A 'Brownian ratchet' construct was used which converged to asymmetric equilibria via a random walk scheme that utilized the smallest singular value of A. Distances between configurations and equilibria were measured using the Frobenius norm ||·||F and 2-norm ||·||2, and conclusions were made on the density of equilibria within the general configuration space. The second contribution used Shannon Entropy, which we interpret as a scalar measure of the robustness, or likelihood of lattices to occur in a physical setting. Third, an analytic model was produced for vortex street patterns on the sphere by using SVD in conjunction with expressions for the center of vorticity vector and angular velocity. Equilibrium curves within the configuration space were presented as a function of the geometry, and pole vortices were shown to have a critical role in the formation and destruction of vortex streets. The fourth contribution entailed a more complete perspective of the streamline topology of vortex streets, linking the bifurcations to critical points on the equilibrium curves.
18. Lattice gas hydrodynamics: Theory and simulations
Hasslacher, B.
1993-01-01
The first successful application of a microscopic analogy to create a skeleton cellular automaton and analyze it with statistical mechanical tools, was the work of Frisch, Hasslacher and Pomeau on the Navier-Stokes equation in two and three dimensions. This has become a very large research area with lattice gas models and methods being used for both fundamental investigations into the foundations of statistical mechanics and a large number of diverse applications. This present research was devoted to enlarging the fundamental scope of lattice gas models and proved quite successful. Since the beginning of this proposal, cellular automata have been constructed for statistical mechanical models, fluids, diffusion and shock systems in fundamental investigations. In applied areas, there are now excellent lattice gas models for complex flows through porous media, chemical reaction and combustion dynamics, multiphase flow systems, and fluid mixtures with natural boundaries. With extended cellular fluid models, one can do problems with arbitrary pairwise potentials. Recently, these have been applied to such problems as non-newtonian or polymeric liquids and a mixture of immiscible fluids passing through fractal or spongelike media in two and three dimensions. This proposal has contributed to and enlarged the scope of this work.
19. Lattice theory of nonequilibrium fermion production
Gelfand, Daniil
2014-07-22
In this thesis we investigate non-equilibrium production of fermionic particles using modern lattice techniques. The presented applications range from preheating after inflation in the early Universe cosmology to pre-thermalization dynamics in heavy-ion collisions as well as pair production and string breaking in a lower-dimensional model of quantum chromodynamics. Strong enhancement of fermion production in the presence of overoccupied bosons is observed in scalar models undergoing instabilities. Both parametric resonance and tachyonic instability are considered as scenarios for preheating after inflation. The qualitative and quantitative features of the resulting fermion distribution are found to depend largely on an effective coupling parameter. In order to simulate fermions in three spatial dimensions we apply a stochastic low-cost lattice algorithm, which we verify by comparison with an exact lattice approach and with a functional method based on a coupling expansion. In the massive Schwinger model, we analyse the creation of fermion/anti-fermion pairs from homogeneous and inhomogeneous electric fields and observe string formation between charges. As a follow-up we study the dynamics of string breaking and establish a two-stage process, consisting of the initial particle production followed by subsequent charge separation and screening. In quantum chromodynamics, our focus lies on the properties of the quark sector during turbulent bosonic energy cascade as well as on the isotropization of quarks and gluons starting from different initial conditions.
20. Perfect Lattice Perturbation Theory A Study of the Anharmonic Oscillator
Bietenholz, W
1999-01-01
As an application of perfect lattice perturbation theory, we construct an O(\\lambda) perfect lattice action for the anharmonic oscillator analytically in momentum space. In coordinate space we obtain a set of 2-spin and 4-spin couplings \\propto \\lambda, which we evaluate for various masses. These couplings never involve variables separated by more than two lattice spacings. The O(\\lambda) perfect action is simulated and compared to the standard action. We discuss the improvement for the first two energy gaps \\Delta E_1, \\Delta E_2 and for the scaling quantity \\Delta E_2 / \\Delta E1 in different regimes of the interaction parameter, and of the correlation length.
1. Automated Methods in Chiral Perturbation Theory on the Lattice
Borasoy, B; Krebs, H; Lewis, R; Borasoy, Bugra; Hippel, Georg M. von; Krebs, Hermann; Lewis, Randy
2005-01-01
We present a method to automatically derive the Feynman rules for mesonic chiral perturbation theory with a lattice regulator. The Feynman rules can be output both in a human-readable format and in a form suitable for an automated numerical evaluation of lattice Feynman diagrams. The automated method significantly simplifies working with improved or extended actions. Some applications to the study of finite-volume effects will be presented.
2. Quantum study of Foldy-Wouthuysen-Tani theory on lattice
Liu Da-Qing
2007-01-01
We study here a quantum version of Foldy-Wouthuysen-Tani (FWT) transformation and compare the similarities and differences between the quantum and the classic FWT theories. Then the improvement of action on lattice is discussed. The result shows that it is not necessary to improve the covariant difference along the time direction on lattice. Finally we discuss briefly the structure of physical vacuum and give a model independent of field condensate.
3. Microscopic theory of photonic band gaps in optical lattices
Samoylova, M; Bachelard, R; Courteille, Ph W
2013-01-01
We propose a microscopic model to describe the scattering of light by atoms in optical lattices. The model is shown to efficiently capture Bragg scattering, spontaneous emission and photonic band gaps. A connection to the transfer matrix formalism is established in the limit of a one-dimensional optical lattice, and we find the two theories to yield results in good agreement. The advantage of the microscopic model is, however, that it suits better for studies of finite-size and disorder effects.
4. Topics in effective field theory as applied to lattice QCD
Smigielski, Brian
This thesis focuses on understanding aspects of hadronic physics using numerical and analytic computations which comprise the research fields of Lattice QCD and Effective Field Theories. Lattice QCD is a numerical approximation to QCD that is computed within a finite spacetime volume, a finite lattice spacing, and unphysically large values of the quark mass used to limit computational run time. Because Lattice QCD calculations are implemented with these constraints, it becomes necessary to understand how these constraints influence the physics if we are to extract physical observables. This requires the use and matching of an effective field theory for mesons and baryons which are the fundamental degrees of freedom of the effective field theory Lagrangian. We consider pion and nucleon interactions in Chapter 3 when computational demands force the use of small, spacetime lattices, and extract the axial charge of the nucleon. In Chapters 4 and 5 we examine systems of up to twelve particles of single species, pions or kaons, and mixed species systems of pions and kaons. From these systems we learn about the scattering lengths and three-body forces of these particles. These multi-particle systems also allow one to understand the behavior of finite density systems on the lattice. Lastly in Chapter 6, we examine parton distributions of the pion for a nonzero change in the pion's momentum. These are known as generalized parton distributions and reveal information regarding the valence quarks within a particular hadron. Before the advent of QCD, however, these particles were also known as partons.
5. Recent progress in lattice supersymmetry: from lattice gauge theory to black holes
2016-01-01
Supersymmetry (SUSY) is a fascinating topic in theoretical physics, because of its unique and counterintuitive properties. It is expected to emerge as new physics beyond the standard model, and it is also a building block for supergravity and superstring theory. A number of exact results obtained via SUSY theories provide insights into field theory. However, the dynamics of many SUSY theories are not yet fully understood, and numerical study of SUSY theories through lattice simulations is promising as regards furthering this understanding. In this paper, I overview the current status of lattice SUSY by discussing its development in chronological order, and by reviewing some simple models. In addition, I discuss the numerical verification of gauge/gravity duality, which is one of the recent significant developments in this field.
6. Long-range interactions in lattice field theory
Rabin, J.M.
1981-06-01
Lattice quantum field theories containing fermions can be formulated in a chirally invariant way provided long-range interactions are introduced. It is established that in weak-coupling perturbation theory such a lattice theory is renormalizable when the corresponding continuum theory is, and that the continuum theory is indeed recovered in the perturbative continuum limit. In the strong-coupling limit of these theories one is led to study an effective Hamiltonian describing a Heisenberg antiferromagnet with long-range interactions. Block-spin renormalization group methods are used to find a critical rate of falloff of the interactions, approximately as inverse distance squared, which separates a nearest-neighbor-antiferromagnetic phase from a phase displaying identifiable long-range effects. A duality-type symmetry is present in some block-spin calculations.
7. Automata theory based on complete residuated lattice-valued logic
邱道文
2001-01-01
This paper establishes a fundamental framework of automata theory based on complete residuated lattice-valued logic. First it deals with how to extend the transition relation of states and particularly presents a characterization of residuated lattice by fuzzy automata (called valued automata).After that fuzzy subautomata (called valued subautomata), successor and source operators are proposed and their basic properties as well as the equivalent relation among them are discussed, from which it follows that the two fuzzy operators are exactly fuzzy closure operators. Finally an L bifuzzy topological characterization of valued automata is presented, so a more generalized fuzzy automata theory is built.
8. Heavy dense QCD from a 3d effective lattice theory
Glesaaen, Jonas; Philipsen, Owe
2015-01-01
The cold and dense regime of the QCD phase diagram is to this day inaccessible to first principle lattice calculations owing to the sign problem. Here we present progress of an ongoing effort to probe this particularly difficult regime utilising a dimensionally reduced effective lattice theory with a significantly reduced sign problem. The effective theory is derived by combined character and hopping expansion and is valid for heavy quarks near the continuum. We show an extension of the effective theory to order $u^5\\kappa^8$ in the cold regime. A linked cluster expansion is applied to the effective theory resulting in a consistent mechanism for handling the effective theory fully analytically. The new results are consistent with the ones from simulations confirming the viability of analytic methods. Finally we resum the analytical result which doubles the convergence region of the expansion.
9. Lattice Field Theory Study of Magnetic Catalysis in Graphene
DeTar, Carleton; Zafeiropoulos, Savvas
2016-01-01
We discuss the simulation of the low-energy effective field theory (EFT) for graphene in the presence of an external magnetic field. Our fully nonperturbative calculation uses methods of lattice gauge theory to study the theory using a hybrid Monte Carlo approach. We investigate the phenomenon of magnetic catalysis in the context of graphene by studying the chiral condensate which is the order parameter characterizing the spontaneous breaking of chiral symmetry. In the EFT, the symmetry breaking pattern is given by $U(4) \\to U(2) \\times U(2)$. We also comment on the difficulty, in this lattice formalism, of studying the time-reversal-odd condensate characterizing the ground state in the presence of a magnetic field. Finally, we study the mass spectrum of the theory, in particular the Nambu-Goldstone (NG) mode as well as the Dirac quasiparticle, which is predicted to obtain a dynamical mass.
10. Fermion frontiers in vector lattice gauge theories: Proceedings. Volume 8
NONE
1998-11-01
The inclusion of fermions into simulations of lattice gauge theories is very difficult both theoretically and numerically. With the presence of Teraflops-scale computers for lattice gauge theory, the authors wanted a forum to discuss new approaches to lattice fermions. The workshop concentrated on approaches which are ripe for study on such large machines. Although lattice chiral fermions are vitally important to understand, there is not technique at hand which is viable on these Teraflops-scale machines for real-world problems. The discussion was therefore focused on recent developments and future prospects for QCD-like theories. For the well-known fermion formulations, the Aoki phase in Wilson fermions, novelties of U{sub A}(1) symmetry and the {eta}{prime} for staggered fermions and new approaches for simulating the determinant for Wilson fermions were discussed. The newer domain-wall fermion formulation was reviewed, with numerical results given by many speakers. The fermion proposal of Friedberg, Lee and Pang was introduced. They also were able to compare and contrast the dependence of QCD and QCD-like SUSY theories on the number of quark flavors. These proceedings consist of several transparencies and a summary page from each speaker. This should serve to outline the major points made in each talk.
11. High precision simulation techniques for lattice field theory
Wolff, U
1993-01-01
An overview is given over the recently developed and now widely used Monte Carlo algorithms with reduced or eliminated critical slowing down. The basic techniques are overrelaxation, cluster algorithms and multigrid methods. With these tools one is able to probe much closer than before the universal continuum behavior of field theories on the lattice.
12. Gauge-fixing approach to lattice chiral gauge theories
Bock, W; Shamir, Y; Bock, Wolfgang; Golterman, Maarten F.L.; Shamir, Yigal
1998-01-01
We review the status of our recent work on the gauge-fixing approach to lattice chiral gauge theories. New numerical results in the reduced version of a model with a U(1) gauge symmetry are presented which strongly indicate that the factorization of the correlation functions of the left-handed neutral and right-handed charged fermion fields, which we established before in perturbation theory, holds also nonperturbatively.
13. Radial Quantization for Conformal Field Theories on the Lattice
Brower, Richard C; Neuberger, Herbert
2012-01-01
We consider radial quantization for conformal quantum field theory with a lattice regulator. A Euclidean field theory on $\\mathbb R^D$ is mapped to a cylindrical manifold, $\\mathbb R\\times \\mathbb S^{D-1}$, whose length is logarithmic in scale separation. To test the approach, we apply this to the 3D Ising model and compute $\\eta$ for the first $Z_2$ odd primary operator.
14. Dynamical Systems On Weighted Lattices: General Theory
Maragos, Petros
2016-01-01
In this work a theory is developed for unifying large classes of nonlinear discrete-time dy- namical systems obeying a superposition of a weighted maximum or minimum type. The state vectors and input-output signals evolve on nonlinear spaces which we call complete weighted lat- tices and include as special cases the nonlinear vector spaces of minimax algebra. Their algebraic structure has a polygonal geometry. Some of the special cases unified include max-plus, max- product, and probabilistic...
15. The Alternation Hierarchy for the Theory of µ-lattices
Santocanale, Luigi
2002-01-01
independent of φ. In this paper we give a proof that the alternation hierarchy for the theory of µ-lattices is strict, meaning that such a constant does not exist if µ-term are built up from the basic lattice operations and are interpreted as expected. The proof relies on the explicit characterization of free......The alternation hierarchy problem asks whether every µ-term φ, that is, a term built up also using a least fixed point constructor as well as a greatest fixed point constructor, is equivalent to a µ-term where the number of nested fixed points of a different type is bounded by a constant...
16. Multigrid methods for propagators in lattice gauge theories
Kalkreuter, T
1994-01-01
Multigrid methods were invented for the solution of discretized partial differential equations in ordered systems. The slowness of traditional algorithms is overcome by updates on various length scales. In this article we discuss generalizations of multigrid methods for disordered systems, in particular for propagators in lattice gauge theories. A discretized nonabelian gauge theory can be formulated as a system of statistical mechanics where the gauge field degrees of freedom are SU(N) matrices on the links of the lattice. These SU(N) matrices appear as random coefficients in Dirac equations. We aim at finding an efficient method by which one can solve Dirac equations without critical slowing down. If this could be achieved, Monte Carlo simulations of Quantum Chromodynamics (the theory of the strong interaction) would be accelerated considerably. In principle, however, the methods discussed can be used in arbitrary space-time dimension and for arbitrary gauge group. Moreover, there are applications in multig...
17. Independent Plaquette Trial Action for 4-Dimensional Lattice Gauge Theory
LIU Jin-Ming
2001-01-01
Based on the explicit expressions of the plaquette formulations, the independent plaquette trial action for 4-dimensional lattice gauge theory is introduced. As an example, the mean plaquette energy EP for the SU(2) lattice gauge theory is calculated by using action variational approach with the independent trial action. The results are in good agreement with the Monte Carlo results in the strong coupling and the crossover region, and the curve is smooth in the whole region, which show that 4-dimensional SU(2) theory has only a single, confining phase. The unwanted discontinuity of EP given by the single link trial action, which is used in the earlier variational calculations has been avoided.
18. Renormalisation and off-shell improvement in lattice perturbation theory
Capitani, S; Horsley, R; Perlt, H; Rakow, P E L; Schierholz, G; Schiller, A
2001-01-01
We discuss the improvement of flavour non-singlet point and one-link lattice quark operators, which describe the quark currents and the first moment of the DIS structure functions respectively. Suitable bases of improved operators are given, and the corresponding renormalisation factors and improvement coefficients are calculated in one-loop lattice perturbation theory, using the Sheikholeslami-Wohlert (clover) action. To this order we achieve off-shell improvement by eliminating the effect of contact terms. We use massive fermions, and our calculations are done keeping all terms up to first order in the lattice spacing, for arbitrary m^2/p^2, in a general covariant gauge. We also compare clover fermions with fermions satisfying the Ginsparg-Wilson relation, and show how to remove O(a) effects off-shell in this case too, and how this is in many aspects simpler than for clover fermions. Finally, tadpole improvement is also considered.
19. Tadpole-improved SU(2) lattice gauge theory
Shakespeare, N H; Shakespeare, Norman H.; Trottier, Howard D.
1999-01-01
A comprehensive analysis of tadpole-improved SU(2) lattice gauge theory is made. Simulations are done on isotropic and anisotropic lattices, with and without improvement. Two tadpole renormalization schemes are employed, one using average plaquettes, the other using mean links in Landau gauge. Simulations are done with spatial lattice spacings $a_s$ in the range of about 0.1--0.4 fm. Results are presented for the static quark potential, the renormalized lattice anisotropy $a_t/a_s$ (where $a_t$ is the temporal'' lattice spacing), and for the scalar and tensor glueball masses. Tadpole improvement significantly reduces discretization errors in the static quark potential and in the scalar glueball mass, and results in very little renormalization of the bare anisotropy that is input to the action. We also find that tadpole improvement using mean links in Landau gauge results in smaller discretization errors in the scalar glueball mass (as well as in the static quark potential), compared to when average plaquett...
20. Light-cone Wilson loop in classical lattice gauge theory
Laine, M
2013-01-01
The transverse broadening of an energetic jet passing through a non-Abelian plasma is believed to be described by the thermal expectation value of a light-cone Wilson loop. In this exploratory study, we measure the light-cone Wilson loop with classical lattice gauge theory simulations. We observe, as suggested by previous studies, that there are strong interactions already at short transverse distances, which may lead to more efficient jet quenching than in leading-order perturbation theory. We also verify that the asymptotics of the Wilson loop do not change qualitatively when crossing the light cone, which supports arguments in the literature that infrared contributions to jet quenching can be studied with dimensionally reduced simulations in the space-like domain. Finally we speculate on possibilities for full four-dimensional lattice studies of the same observable, perhaps by employing shifted boundary conditions in order to simulate ensembles boosted by an imaginary velocity.
1. Chiral effective theory with a light scalar and lattice QCD
Soto, J., E-mail: [email protected] [Departament d' Estructura i Constituents de la Materia, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia (Spain); Institut de Ciencies del Cosmos, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia (Spain); Talavera, P., E-mail: [email protected] [Institut de Ciencies del Cosmos, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia (Spain); Departament de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Comte Urgell 187, E-08036 Barcelona (Spain); Tarrus, J., E-mail: [email protected] [Departament d' Estructura i Constituents de la Materia, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia (Spain); Institut de Ciencies del Cosmos, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia (Spain)
2013-01-21
We extend the usual chiral perturbation theory framework ({chi}PT) to allow the inclusion of a light dynamical isosinglet scalar. Using lattice QCD results, and a few phenomenological inputs, we explore the parameter space of the effective theory. We discuss the S-wave pion-pion scattering lengths, extract the average value of the two light quark masses and evaluate the impact of the dynamical singlet field in the low-energy constants l{sup Macron }{sub 1}, l{sup Macron }{sub 3} and l{sup Macron }{sub 4} of {chi}PT. We also show how to extract the mass and width of the sigma resonance from chiral extrapolations of lattice QCD data.
2. Thick vortices in SU(2) lattice gauge theory
Cheluvaraja, Srinath
2004-01-01
Three dimensional SU(2) lattice gauge theory is studied after eliminating thin monopoles and the smallest thick monopoles. Kinematically this constraint allows the formation of thick vortex loops which produce Z(2) fluctuations at longer length scales. The thick vortex loops are identified in a three dimensional simulation. A condensate of thick vortices persists even after the thin vortices have all disappeared. The thick vortices decouple at a slightly lower temperature (higher beta) than t...
3. Uncertainties of Euclidean Time Extrapolation in Lattice Effective Field Theory
Lähde, Timo A; Krebs, Hermann; Lee, Dean; Meißner, Ulf-G; Rupak, Gautam
2014-01-01
Extrapolations in Euclidean time form a central part of Nuclear Lattice Effective Field Theory (NLEFT) calculations using the Projection Monte Carlo method, as the sign problem in many cases prevents simulations at large Euclidean time. We review the next-to-next-to-leading order NLEFT results for the alpha nuclei up to $^{28}$Si, with emphasis on the Euclidean time extrapolations, their expected accuracy and potential pitfalls. We also discuss possible avenues for improving the reliability of Euclidean time extrapolations in NLEFT.
4. Fusion basis for lattice gauge theory and loop quantum gravity
Delcamp, Clement; Dittrich, Bianca; Riello, Aldo
2017-02-01
We introduce a new basis for the gauge-invariant Hilbert space of lattice gauge theory and loop quantum gravity in (2 + 1) dimensions, the fusion basis. In doing so, we shift the focus from the original lattice (or spin-network) structure directly to that of the magnetic (curvature) and electric (torsion) excitations themselves. These excitations are classified by the irreducible representations of the Drinfel'd double of the gauge group, and can be readily "fused" together by studying the tensor product of such representations. We will also describe in detail the ribbon operators that create and measure these excitations and make the quasi-local structure of the observable algebra explicit. Since the fusion basis allows for both magnetic and electric excitations from the onset, it turns out to be a precious tool for studying the large scale structure and coarse-graining flow of lattice gauge theories and loop quantum gravity. This is in neat contrast with the widely used spin-network basis, in which it is much more complicated to account for electric excitations, i.e. for Gauß constraint violations, emerging at larger scales. Moreover, since the fusion basis comes equipped with a hierarchical structure, it readily provides the language to design states with sophisticated multi-scale structures. Another way to employ this hierarchical structure is to encode a notion of subsystems for lattice gauge theories and (2 + 1) gravity coupled to point particles. In a follow-up work, we have exploited this notion to provide a new definition of entanglement entropy for these theories.
5. Comparing lattice Dirac operators with Random Matrix Theory
Farchioni, F; Lang, C B
2000-01-01
We study the eigenvalue spectrum of different lattice Dirac operators (staggered, fixed point, overlap) and discuss their dependence on the topological sectors. Although the model is 2D (the Schwinger model with massless fermions) our observations indicate possible problems in 4D applications. In particular misidentification of the smallest eigenvalues due to non-identification of the topological sector may hinder successful comparison with Random Matrix Theory (RMT).
6. Effective theory of interacting fermions in shaken square optical lattices
Keleş, Ahmet; Zhao, Erhai; Liu, W. Vincent
2017-06-01
We develop a theory of weakly interacting fermionic atoms in shaken optical lattices based on the orbital mixing in the presence of time-periodic modulations. Specifically, we focus on fermionic atoms in a circularly shaken square lattice with near-resonance frequencies, i.e., tuned close to the energy separation between the s band and the p bands. First, we derive a time-independent four-band effective Hamiltonian in the noninteracting limit. Diagonalization of the effective Hamiltonian yields a quasienergy spectrum consistent with the full numerical Floquet solution that includes all higher bands. In particular, we find that the hybridized s band develops multiple minima and therefore nontrivial Fermi surfaces at different fillings. We then obtain the effective interactions for atoms in the hybridized s band analytically and show that they acquire momentum dependence on the Fermi surface even though the bare interaction is contactlike. We apply the theory to find the phase diagram of fermions with weak attractive interactions and demonstrate that the pairing symmetry is s +d wave. Our theory is valid for a range of shaking frequencies near resonance, and it can be generalized to other phases of interacting fermions in shaken lattices.
7. Applications of Jarzynski's relation in lattice gauge theories
Nada, Alessandro; Costagliola, Gianluca; Panero, Marco; Toniato, Arianna
2016-01-01
Jarzynski's equality is a well-known result in statistical mechanics, relating free-energy differences between equilibrium ensembles with fluctuations in the work performed during non-equilibrium transformations from one ensemble to the other. In this work, an extension of this relation to lattice gauge theory will be presented, along with numerical results for the $\\mathbb{Z}_2$ gauge model in three dimensions and for the equation of state in $\\mathrm{SU}(2)$ Yang-Mills theory in four dimensions. Then, further applications will be discussed, in particular for the Schr\\"odinger functional and for the study of QCD in strong magnetic fields.
8. Approaches to the sign problem in lattice field theory
Gattringer, Christof
2016-01-01
Quantum field theories (QFTs) at finite densities of matter generically involve complex actions. Standard Monte-Carlo simulations based upon importance sampling, which have been producing quantitative first principle results in particle physics for almost fourty years, cannot be applied in this case. Various strategies to overcome this so-called Sign Problem or Complex Action Problem were proposed during the last thirty years. We here review the sign problem in lattice field theories, focussing on two more recent methods: Dualization to world-line type of representations and the density-of-states approach.
9. Approaches to the sign problem in lattice field theory
Gattringer, Christof; Langfeld, Kurt
2016-08-01
Quantum field theories (QFTs) at finite densities of matter generically involve complex actions. Standard Monte Carlo simulations based upon importance sampling, which have been producing quantitative first principle results in particle physics for almost forty years, cannot be applied in this case. Various strategies to overcome this so-called sign problem or complex action problem were proposed during the last thirty years. We here review the sign problem in lattice field theories, focusing on two more recent methods: dualization to worldline type of representations and the density-of-states approach.
10. Theory of Lattice Strain for Materials Undergoing Plastic Deformation
Karato, S.
2008-12-01
Radial x-ray diffraction is used to probe physical properties of materials including elastic and plastic properties. The theory used behind such an practice is the one developed by Singh (1993) in which the relation between lattice strain and elastic constants and macroscopic stress is derived. In this theory, the variation of inferred stress with the crystallographic planes, (hkl), is due to the elastic anisotropy. However, recent experimental studies showed that in many cases, the variation of stress with (hkl) far exceeds the value expected from this theory. I have developed a modified theory to rectify this problem with Singh's theory. In Singh's theory, the stress distribution in a polycrystalline material is treated only either unrelaxed or relaxed state. The role of plastic deformation is included only to the extent that plastic flow influences this stress state. Such an assumption corresponds to a Voigt model behavior, which is not an appropriate model at high temperatures where continuing plastic flow occurs with concurrent microscopic equilibrium, elastic deformation. This is a Maxwell model type behavior, and my model provides a stress analysis in a Maxwell material with anisotropic and non-linear power-law rheology. In this theory, the lattice strain corresponding to an imposed macroscopic strain-rate is calculated by three steps: (i) conversion of macroscopic strain-rate to macroscopic stress, (ii) conversion of macroscopic stress to microscopic stress at individual grains, and (iii) calculation of microscopic strain due to microscopic stress. The first step involves anisotropy in macroscopic viscosity that depends on anisotropy in crystal plasticity and lattice-preferred orientation. The second step involves anisotropic crystal plasticity and finally the third step involves elastic crystal anisotropy. In most cases, the influence of LPO is weak and in such a case, the lattice strain depends on (hkl) due to the anisotropy in both elastic and plastic
11. N=1 supersymmetric Yang-Mills theory on the lattice
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.
12. Numerical methods for the sign problem in Lattice Field Theory
Bongiovanni, Lorenzo
2016-01-01
The great majority of algorithms employed in the study of lattice field theory are based on Monte Carlo's importance sampling method, i.e. on probability interpretation of the Boltzmann weight. Unfortunately in many theories of interest one cannot associated a real and positive weight to every configuration, that is because their action is explicitly complex or because the weight is multiplied by some non positive term. In this cases one says that the theory on the lattice is affected by the sign problem. An outstanding example of sign problem preventing a quantum field theory to be studied, is QCD at finite chemical potential. Whenever the sign problem is present, standard Monte Carlo methods are problematic to apply and, in general, new approaches are needed to explore the phase diagram of the complex theory. Here we will review three of the main candidate methods to deal with the sign problem, namely complex Langevin dynamics, Lefschetz thimbles and density of states method. We will first study complex Lan...
13. Kitaev Lattice Models as a Hopf Algebra Gauge Theory
Meusburger, Catherine
2017-07-01
We prove that Kitaev's lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern-Simons theory for the Drinfeld double D( H). This shows that Kitaev models are a special case of the older and more general combinatorial models. This equivalence is an analogue of the relation between Turaev-Viro and Reshetikhin-Turaev TQFTs and relates them to the quantisation of moduli spaces of flat connections. We show that the topological invariants of the two models, the algebra of operators acting on the protected space of the Kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism, are isomorphic. This is established in a gauge theoretical picture, in which both models appear as Hopf algebra valued lattice gauge theories. We first prove that the triangle operators of a Kitaev model form a module algebra over a Hopf algebra of gauge transformations and that this module algebra is isomorphic to the lattice algebra in the combinatorial formalism. Both algebras can be viewed as the algebra of functions on gauge fields in a Hopf algebra gauge theory. The isomorphism between them induces an algebra isomorphism between their subalgebras of invariants, which are interpreted as gauge invariant functions or observables. It also relates the curvatures in the two models, which are given as holonomies around the faces of the lattice. This yields an isomorphism between the subalgebras obtained by projecting out curvatures, which can be viewed as the algebras of functions on flat gauge fields and are the topological invariants of the two models.
14. On the definition of entanglement entropy in lattice gauge theories
Aoki, Sinya; Iritani, Takumi; Nozaki, Masahiro; Numasawa, Tokiro; Shiba, Noburo; Tasaki, Hal
2015-06-01
We focus on the issue of proper definition of entanglement entropy in lattice gauge theories, and examine a naive definition where gauge invariant states are viewed as elements of an extended Hilbert space which contains gauge non-invariant states as well. Working in the extended Hilbert space, we can define entanglement entropy associated with an arbitrary subset of links, not only for abelian but also for non-abelian theories. We then derive the associated replica formula. We also discuss the issue of gauge invariance of the entanglement entropy. In the Z N gauge theories in arbitrary space dimensions, we show that all the standard properties of the entanglement entropy, e.g. the strong subadditivity, hold in our definition. We study the entanglement entropy for special states, including the topological states for the Z N gauge theories in arbitrary dimensions. We discuss relations of our definition to other proposals.
15. On the definition of entanglement entropy in lattice gauge theories
Aoki, Sinya; Nozaki, Masahiro; Numasawa, Tokiro; Shiba, Noburo; Tasaki, Hal
2015-01-01
We focus on the issue of proper definition of entanglement entropy in lattice gauge theories, and examine a naive definition where gauge invariant states are viewed as elements of an extended Hilbert space which contain gauge non-invariant states as well. Working in the extended Hilbert space, we can define entanglement entropy associated with an arbitrary subset of links, not only for abelian but also for non-abelian theories. We then derive the associated replica formula. We also discuss the issue of gauge invariance of the entanglement entropy. In the $Z_N$ gauge theories in arbitrary space dimensions, we show that all the standard properties of the entanglement entropy, e.g. the strong subadditivity, hold in our definition. We study the entanglement entropy for special states, including the topological states for the $Z_N$ gauge theories in arbitrary dimensions. We discuss relations of our definition to other proposals.
16. Fusion basis for lattice gauge theory and loop quantum gravity
Delcamp, Clement; Riello, Aldo
2016-01-01
We introduce a new basis for the gauge--invariant Hilbert space of lattice gauge theory and loop quantum gravity in $(2+1)$ dimensions, the fusion basis. In doing so, we shift the focus from the original lattice (or spin--network) structure directly to that of the magnetic (curvature) and electric (torsion) excitations themselves. These excitations are classified by the irreducible representations of the Drinfel'd double of the gauge group, and can be readily "fused" together by studying the tensor product of such representations. We will also describe in detail the ribbon operators that create and measure these excitations and make the quasi--local structure of the observable algebra explicit. Since the fusion basis allows for both magnetic and electric excitations from the onset, it turns out to be a precious tool for studying the large scale structure and coarse--graining flow of lattice gauge theories and loop quantum gravity. This is in neat contrast with the widely used spin--network basis, in which it ...
17. Prepotential formulation of SU(3) lattice gauge theory
Anishetty, Ramesh [Institute of Mathematical Sciences, CIT-Campus, Taramani, Chennai 600 113 (India); Mathur, Manu; Raychowdhury, Indrakshi [S N Bose, National Centre for Basic Sciences, JD Block, Sector III, Salt Lake City, Kolkata 700 098 (India)], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected]
2010-01-22
The SU(3) lattice gauge theory is reformulated in terms of SU(3) prepotential harmonic oscillators. This reformulation has enlarged SU(3)xU(1)xU(1) gauge invariance under which the prepotential operators transform like matter fields. The Hilbert space of SU(3) lattice gauge theory is shown to be equivalent to the Hilbert space of the prepotential formulation satisfying certain color invariant Sp(2, R) constraints. The SU(3) irreducible prepotential operators which solve these Sp(2, R) constraints are used to construct SU(3) gauge invariant Hilbert spaces at every lattice site in terms of SU(3) gauge invariant vertex operators. The electric fields and the link operators are reconstructed in terms of these SU(3) irreducible prepotential operators. We show that all the SU(3) Mandelstam constraints become local and take a very simple form within this approach. We also discuss the construction of all possible linearly independent SU(3) loop states which solve the Mandelstam constraints. The techniques can be easily generalized to SU(N)
18. Chiral effective theory with a light scalar and lattice QCD
Soto, J; Tarrús, J
2011-01-01
We extend the usual chiral perturbation theory framework ($\\chi$PT) to allow the inclusion of a light dynamical isosinglet scalar. Using lattice QCD results, and a few phenomenological inputs, we explore the parameter space of the effective theory. The extended theory collects already at LO the ball park contribution to the pion mass and decay constant, thus achieving an accuracy that is comparable to the one of the standard $\\chi$PT at NLO results. We check explicitly that radiative corrections do not spoil this behavior and keep the theory stable under mild variations of the parameters. The parameter sets that are compatible with the current mass and width of the sigma resonance turn out to reproduce the experimental values of the S-wave pion-pion scattering lengths very accurately. We also extract the average value of the two light quark--masses and evaluate the impact of the dynamical singlet field in the low--energy constants $\\bar{l}_3$ and $\\bar{l}_4$ of $\\chi$PT. We emphasize that more accurate lattic...
19. Real Representation in Chiral Gauge Theories on the Lattice
Suzuki, H
2000-01-01
The Weyl fermion belonging to the real representation of the gauge group provides a simple illustrative example for L\\"uscher's gauge-invariant lattice formulation of chiral gauge theories. We can explicitly construct the fermion integration measure globally over the gauge-field configuration space in the arbitrary topological sector; there is no global obstruction corresponding to the Witten anomaly. It is shown that this Weyl formulation is equivalent to a lattice formulation based on the Majorana (left--right-symmetric) fermion, in which the fermion partition function is given by the Pfaffian with a definite sign, up to physically irrelevant contact terms. This observation suggests a natural relative normalization of the fermion measure in different topological sectors for the Weyl fermion belonging to the complex representation.
20. Lattice Effective Field Theory for Medium-Mass Nuclei
Lähde, Timo A; Krebs, Hermann; Lee, Dean; Meißner, Ulf-G; Rupak, Gautam
2014-01-01
We extend Nuclear Lattice Effective Field Theory (NLEFT) to the regime of medium-mass nuclei, and describe a method which allows us to greatly decrease the uncertainties due to extrapolation at large Euclidean time. We present results for the ground states of alpha nuclei from $^4$He to $^{28}$Si, calculated up to next-to-next-to-leading order (NNLO) in the EFT expansion. We discuss systematic errors associated with the momentum-cutoff scale and the truncation of the EFT expansion. While the long-term objectives of NLEFT are a decrease in the lattice spacing and the inclusion of higher-order contributions, we show that the missing physics at NNLO can be approximated by an effective four-nucleon interaction.
1. 31st International Symposium on Lattice Field Theory
2013-01-01
The annual lattice symposium brings together a global community of researchers from theoretical particle physics and beyond, who employ numerical and computational methods to study the properties of strongly interacting physical systems, above all Quantum Chromodynamics (QCD), the theory describing the interactions of quarks and gluons. Topics include studies of the spectrum and structure of hadrons, lattice studies of matter under extreme conditions, hadronic contributions to weak decay amplitudes, as well as recent developments in simulation algorithms and computer hardware. The 2013 conference in Mainz was attended by over 500 participants from all over the globe, making it the biggest in this series so far. This proceedings volume is dedicated to the memory of Nobel Laureate Kenneth G. Wilson (June 8, 1936 - June 15, 2013).
2. 31st International Symposium on Lattice Field Theory
The annual lattice symposium brings together a global community of researchers from theoretical particle physics and beyond, who employ numerical and computational methods to study the properties of strongly interacting physical systems, above all Quantum Chromodynamics (QCD), the theory describing the interactions of quarks and gluons. Topics include studies of the spectrum and structure of hadrons, lattice studies of matter under extreme conditions, hadronic contributions to weak decay amplitudes, as well as recent developments in simulation algorithms and computer hardware. The 2013 conference in Mainz was attended by over 500 participants from all over the globe, making it the biggest in this series so far. This proceedings volume is dedicated to the memory of Nobel Laureate Kenneth G. Wilson (June 8, 1936 - June 15, 2013).
3. Latfield2: A c++ library for classical lattice field theory
David, Daverio; Bevis, Neil
2015-01-01
latfield2 is a C++ library designed to simplify writing parallel codes for solving partial differen- tial equations, developed for application to classical field theories in particle physics and cosmology. It is a significant rewrite of the latfield framework, moving from a slab domain decomposition to a rod decomposition, where the last two dimension of the lattice are scattered into a two dimensional process grid. Parallelism is implemented using the Message Passing Interface (MPI) standard, and hidden in the basic objects of grid-based simulations: Lattice, Site and Field. It comes with an integrated parallel fast Fourier transform, and I/O server class permitting computation to continue during the writing of large files to disk. latfield2 has been used for production runs on tens of thousands of processor elements, and is expected to be scalable to hundreds of thousands.
4. The XXV International Symposium on Lattice Field Theory
Bali; Braun, Gunnar; Gattring, Vladimir; Göckeler, Christof; Schäfer, Meinulf; Weisz, Andreas; Wettig, Peter; Tilo
Lattice 2007, the XXV International Symposium on Lattice Field Theory, was held from July 30 to August 4, 2007 at the University of Regensburg, Germany. The scientific program contained 24 plenary session talks and 338 parallel session contributions (talks and posters). The conference topics included: algorithms and machines; applications beyond QCD; chiral symmetry; hadron spectroscopy; hadron structure; nonzero temperature and density; standard model parameters and renormalization; theoretical developments; vacuum structure and confinement; weak decays and matrix elements. We gratefully acknowledge financial support by the following companies and institutions, which was essential for the success of the conference: Bull, Eurotech, IBM, Intel, Sun, DESY, GSI, FZ Jülich, Vielberth Foundation, Kneitinger.Editorial Board:Gunnar Bali, Vladimir Braun, Christof Gattringer (chairman), Meinulf Göckeler, Andreas Schäfer, Peter Weisz, Tilo Wettig
5. Simulating thimble regularization of lattice quantum field theories
Di Renzo, Francesco
2016-01-01
Monte Carlo simulations of lattice quantum field theories on Lefschetz thimbles are non trivial. We discuss a new Monte Carlo algorithm based on the idea of computing contributions to the functional integral which come from complete flow lines. The latter are the steepest ascent paths attached to critical points, i.e. the basic building blocks of thimbles. The measure to sample is thus dictated by the contribution of complete flow lines to the partition function. The algorithm is based on a heat bath sampling of the gaussian approximation of the thimble: this defines the proposals for a Metropolis-like accept/reject step. The effectiveness of the algorithm has been tested on a few models, e.g. the chiral random matrix model. We also discuss thimble regularization of gauge theories, and in particular the successfull application to 0+1 dimensional QCD and the status and prospects for Yang-Mills theories.
6. A Formulation of Lattice Gauge Theories for Quantum Simulations
Zohar, Erez
2014-01-01
We examine the Kogut-Susskind formulation of lattice gauge theories under the light of fermionic and bosonic degrees of freedom that provide a description useful to the development of quantum simulators of gauge invariant models. We consider both discrete and continuous gauge groups and adopt a realistic multi-component Fock space for the definition of matter degrees of freedom. In particular, we express the Hamiltonian of the gauge theory and the Gauss law in terms of Fock operators. The gauge fields are described in two different bases, based on either group elements or group representations. This formulation allows for a natural scheme to achieve a consistent truncation of the Hilbert space for continuous groups, and provides helpful tools to study the connections of gauge theories with topological quantum double and string-net models for discrete groups. Several examples, including the case of the discrete $D_3$ gauge group, are presented.
7. On a lattice-independent formulation of quantum holonomy theory
Aastrup, Johannes; Møller Grimstrup, Jesper
2016-11-01
Quantum holonomy theory is a candidate for a non-perturbative theory of quantum gravity coupled to fermions. The theory is based on the {{QHD}}(M)-algebra, which essentially encodes how matter degrees of freedom are moved on a three-dimensional manifold. In this paper we commence the development of a lattice-independent formulation. We first introduce a flow-dependent version of the {{QHD}}(M)-algebra and formulate necessary conditions for a state to exist hereon. We then use the GNS construction to build a kinematical Hilbert space. Finally, we find that operators, that correspond to the Dirac and gravitational Hamiltonians in a semi-classical limit, are background independent.
8. Symanzik improvement of the gradient flow in lattice gauge theories
Ramos, Alberto [PH-TH, CERN, Geneva (Switzerland); Sint, Stefan [Trinity College Dublin, School of Mathematics, Dublin (Ireland)
2016-01-15
We apply the Symanzik improvement programme to the 4 + 1-dimensional local re-formulation of the gradient flow in pure SU(N) lattice gauge theories. We show that the classical nature of the flow equation allows one to eliminate all cutoff effects at O(a{sup 2}), which originate either from the discretised gradient flow equation or from the gradient flow observable. All the remaining O(a{sup 2}) effects can be understood in terms of local counterterms at the zero flow-time boundary. We classify these counterterms and provide a complete set as required for on-shell improvement. Compared to the 4-dimensional pure gauge theory only a single additional counterterm is required, which corresponds to a modified initial condition for the flow equation. A consistency test in perturbation theory is passed and allows one to determine all counterterm coefficients to lowest non-trivial order in the coupling. (orig.)
9. Phase diagrams of exceptional and supersymmetric lattice gauge theories
Wellegehausen, Bjoern-Hendrik
2012-07-10
In this work different strongly-coupled gauge theories with and without fundamental matter have been studied on the lattice with an emphasis on the confinement problem and the QCD phase diagram at nonvanishing net baryon density as well as on possible supersymmetric extensions of the standard model of particle physics. In gauge theories with a non-trivial centre symmetry, as for instance SU(3)-Yang-Mills theory, confinement is intimately related to the centre of the gauge group, and the Polyakov loop serves as an order parameter for confinement. In QCD, this centre symmetry is explicitly broken by quarks in the fundamental representation of the gauge group. But still quarks and gluons are confined in mesons, baryons and glueballs at low temperatures and small densities, suggesting that centre symmetry is not responsible for the phenomenon of confinement. Therefore it is interesting to study pure gauge theories without centre symmetry. In this work this has been done by replacing the gauge group SU(3) of the strong interaction with the exceptional Lie group G{sub 2}, that has a trivial centre. To investigate G{sub 2} gauge theory on the lattice, a new and highly efficient update algorithm has been developed, based on a local HMC algorithm. Employing this algorithm, the proposed and already investigated first order phase transition from a confined to a deconfined phase has been confirmed, showing that indeed a first order phase transition without symmetry breaking or an order parameter is possible. In this context, also the deconfinement phase transition of the exceptional Lie groups F4 and E6 in three spacetime dimensions has been studied. It has been shown that both theories also possess a first order phase transition.
10. Thermodynamics and reference scale of SU(3) gauge theory from gradient flow on fine lattices
Kitazawa, Masakiyo; Hatsuda, Tetsuo; Iritani, Takumi; Itou, Etsuko; Suzuki, Hiroshi
2015-01-01
We study the parametrization of lattice spacing and thermodynamics of SU(3) gauge theory on the basis of the Yang-Mills gradient flow on fine lattices. The lattice spacing of the Wilson gauge action is determined over a wide range $6.3\\le\\beta\\le7.5$ with high accuracy. The measurements of the flow time and lattice spacing dependences of the expectation values of the energy-momentum tensor are performed on fine lattices.
11. Fracton topological order, generalized lattice gauge theory, and duality
Vijay, Sagar; Haah, Jeongwan; Fu, Liang
2016-12-01
We introduce a generalization of conventional lattice gauge theory to describe fracton topological phases, which are characterized by immobile, pointlike topological excitations, and subextensive topological degeneracy. We demonstrate a duality between fracton topological order and interacting spin systems with symmetries along extensive, lower-dimensional subsystems, which may be used to systematically search for and characterize fracton topological phases. Commutative algebra and elementary algebraic geometry provide an effective mathematical tool set for our results. Our work paves the way for identifying possible material realizations of fracton topological phases.
12. Lattice Gauge Theory and the Origin of Mass
Kronfeld, Andreas S.
2013-08-01
Most of the mass of everyday objects resides in atomic nuclei/ the total of the electrons' mass adds up to less than one part in a thousand. The nuclei are composed of nucleons---protons and neutrons---whose nuclear binding energy, though tremendous on a human scale, is small compared to their rest energy. The nucleons are, in turn, composites of massless gluons and nearly massless quarks. It is the energy of these confined objects, via $M=E/c^2$, that is responsible for everyday mass. This article discusses the physics of this mechanism and the role of lattice gauge theory in establishing its connection to quantum chromodynamics.
13. The Hoyle state in nuclear lattice effective field theory
Timo A Lähde; Evgeny Epelbaum; Hermann Krebs; Dean Lee; Ulf-G Meißner; Gautam Rupak
2014-11-01
We review the calculation of the Hoyle state of 12C in nuclear lattice effective field theory (NLEFT) and its anthropic implications in the nucleosynthesis of 12C and 16O in red giant stars. We also analyse the extension of NLEFT to the regime of medium-mass nuclei, with emphasis on the determination of the ground-state energies of the nuclei 16O, 20Ne, 24Mg, and 28Si by Euclidean time projection. Finally, we discuss recent NLEFT results for the spectrum, electromagnetic properties, and α-cluster structure of 16O.
14. Variational Calculation in SU(3) Lattice Gauge Theory
YANG Chun; ZHANG Qi-Ren; GAO Chun-Yuan
2001-01-01
Using the Hamiltonian lattice gauge theory, we perform some variational calculations to obtain the ground-state energy of SU(3) gauge field and scalar (0++) glueball mass. The agreement of our data with the strong and weak expansion results in the corresponding limits indicates that this method can provide us with reliable information in the most interesting medium region. The trial wavefunction used in our variational method is also proven to be a good first approximation of the ground-state of the SU(3) gauge field. Upgrading this function according to correlations of adjacent plaquettes may mean better results.
15. CONDITIONAL FACTORIZATION BASED ON LATTICE THEORY FOR -INTEGERS
Zheng Yonghui; Zhu Yuefei
2008-01-01
In this paper, the integer N = pkq is called a -integer, if p and q are odd primes with almost the same size and k is a positive integer. Such integers were previously proposed for various cryptographic applications. The conditional factorization based on lattice theory for n-bit -integers is considered, and there is an algorithm in time polynomial in n to factor these integers if the least significant |(2k-1)n/(3k-1)(k-1)| bits of p are given.
16. Deconstruction and other approaches to supersymmetric lattice field theories
Giedt, J
2006-01-01
This report contains both a review of recent approaches to supersymmetric lattice field theories and some new results on the deconstruction approach. The essential reason for the complex phase problem of the fermion determinant is shown to be derivative interactions that are not present in the continuum. These irrelevant operators violate the self-conjugacy of the fermion action that is present in the continuum. It is explained why this complex phase problem does not disappear in the continuum limit. The fermion determinant suppression of various branches of the classical moduli space is explored, and found to be supportive of previous claims regarding the continuum limit.
17. Lattice Gauge Theory and the Origin of Mass
Kronfeld, Andreas S
2012-01-01
Most of the mass of everyday objects resides in atomic nuclei; the total of the electrons' mass adds up to less than one part in a thousand. The nuclei are composed of nucleons---protons and neutrons---whose nuclear binding energy, though tremendous on a human scale, is small compared to their rest energy. The nucleons are, in turn, composites of massless gluons and nearly massless quarks. It is the energy of these confined objects, via $M=E/c^2$, that is responsible for everyday mass. This article discusses the physics of this mechanism and the role of lattice gauge theory in establishing its connection to quantum chromodynamics.
18. Lattice gauge theory on the Intel parallel scientific computer
Gottlieb, S. (Department of Physics, Indiana University, Bloomington, IN (USA))
1990-08-01
Intel Scientific Computers (ISC) has just started producing its third general of parallel computer, the iPSC/860. Based on the i860 chip that has a peak performance of 80 Mflops and with a current maximum of 128 nodes, this computer should achieve speeds in excess of those obtainable on conventional vector supercomputers. The hardware, software and computing techniques appropriate for lattice gauge theory calculations are described. The differences between a staggered fermion conjugate gradient program written under CANOPY and for the iPSC are detailed.
19. Flux-tubes in three-dimensional lattice gauge theories
Trottier, H D; Trottier, Howard D.
1993-01-01
Flux-tubes in different representations of SU(2) and U(1) lattice gauge theories in three dimensions are measured. Wilson loops generate heavy quark-antiquark'' pairs in fundamental ($j=1/2$), adjoint ($j=1$), and quartet ($j=3/2$) representations of SU(2). The first direct lattice measurements of the flux-tube cross-section ${\\cal A}_j$ as a function of representation are made. It is found that ${\\cal A}_j \\approx {\\rm constant}$, to about 10\\%. Results are consistent with a connection between the string tension $\\sigma_j$ and ${\\cal A}_j$ suggested by a simplified flux-tube model, $\\sigma_j = g^2 j(j+1) / (2 {\\cal A}_j)$ [$g$ is the gauge coupling], given that $\\sigma_j$ scales like the Casimir $j(j+1)$, as observed in previous lattice studies in both three and four dimensions. The results can discriminate among phenomenological models of the physics underlying confinement. Flux-tubes for singly- and doubly-charged Wilson loops in compact QED$_3$ are also measured. It is found that the string tension scal...
20. Lattice regularization of gauge theories without loss of chiral symmetry
't Hooft, Gerardus
1994-01-01
Abstract: A lattice regularization procedure for gauge theories is proposed in which fermions are given a special treatment such that all chiral flavor symmetries that are free of Adler-Bell-Jackiw anomalies are kept intact. There is no doubling of fermionic degrees of freedom. A price paid for this feature is that the number of fermionic degrees of freedom per unit cell is still infinite, although finiteness of the complete functional integrals can be proven (details are outlined in an Appendix). Therefore, although perhaps of limited usefulness for numerical simulations, our scheme can be applied for studying aspects such as analytic convergence questions, spontaneous symmetry breakdown and baryon number violation in non-Abelian gauge theories.
1. Two-dimensional lattice gauge theories with superconducting quantum circuits
Marcos, D., E-mail: [email protected] [Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck (Austria); Widmer, P. [Albert Einstein Center, Institute for Theoretical Physics, Bern University, CH-3012, Bern (Switzerland); Rico, E. [IPCMS (UMR 7504) and ISIS (UMR 7006), University of Strasbourg and CNRS, 67000 Strasbourg (France); Hafezi, M. [Joint Quantum Institute, NIST/University of Maryland, College Park 20742 (United States); Department of Electrical Engineering and Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742 (United States); Rabl, P. [Institute of Atomic and Subatomic Physics, TU Wien, Stadionallee 2, 1020 Wien (Austria); Wiese, U.-J. [Albert Einstein Center, Institute for Theoretical Physics, Bern University, CH-3012, Bern (Switzerland); Zoller, P. [Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck (Austria); Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck (Austria)
2014-12-15
A quantum simulator of U(1) lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.
2. Lattice cluster theory for dense, thin polymer films.
Freed, Karl F
2015-04-07
While the application of the lattice cluster theory (LCT) to study the miscibility of polymer blends has greatly expanded our understanding of the monomer scale molecular details influencing miscibility, the corresponding theory for inhomogeneous systems has not yet emerged because of considerable technical difficulties and much greater complexity. Here, we present a general formulation enabling the extension of the LCT to describe the thermodynamic properties of dense, thin polymer films using a high dimension, high temperature expansion. Whereas the leading order of the LCT for bulk polymer systems is essentially simple Flory-Huggins theory, the highly non-trivial leading order inhomogeneous LCT (ILCT) for a film with L layers already involves the numerical solution of 3(L - 1) coupled, highly nonlinear equations for the various density profiles in the film. The new theory incorporates the essential "transport" constraints of Helfand and focuses on the strict imposition of excluded volume constraints, appropriate to dense polymer systems, rather than the maintenance of chain connectivity as appropriate for lower densities and as implemented in self-consistent theories of polymer adsorption at interfaces. The ILCT is illustrated by presenting examples of the computed profiles of the density, the parallel and perpendicular bonds, and the chain ends for free standing and supported films as a function of average film density, chain length, temperature, interaction with support, and chain stiffness. The results generally agree with expected general trends.
3. Report of the Snowmass 2013 Computing Frontier working group on Lattice Field Theory -- Lattice field theory for the energy and intensity frontiers: Scientific goals and computing needs
Blum, T; Holmgren, D; Brower, R; Catterall, S; Christ, N; Kronfeld, A; Kuti, J; Mackenzie, P; Neil, E T; Sharpe, S R; Sugar, R
2013-01-01
This is the report of the Computing Frontier working group on Lattice Field Theory prepared for the proceedings of the 2013 Community Summer Study ("Snowmass"). We present the future computing needs and plans of the U.S. lattice gauge theory community and argue that continued support of the U.S. (and worldwide) lattice-QCD effort is essential to fully capitalize on the enormous investment in the high-energy physics experimental program. We first summarize the dramatic progress of numerical lattice-QCD simulations in the past decade, with some emphasis on calculations carried out under the auspices of the U.S. Lattice-QCD Collaboration, and describe a broad program of lattice-QCD calculations that will be relevant for future experiments at the intensity and energy frontiers. We then present details of the computational hardware and software resources needed to undertake these calculations.
4. The Alternation Hierarchy for the Theory of µ-lattices
Santocanale, Luigi
2002-01-01
The alternation hierarchy problem asks whether every µ-term φ, that is, a term built up also using a least fixed point constructor as well as a greatest fixed point constructor, is equivalent to a µ-term where the number of nested fixed points of a different type is bounded by a constant independ......The alternation hierarchy problem asks whether every µ-term φ, that is, a term built up also using a least fixed point constructor as well as a greatest fixed point constructor, is equivalent to a µ-term where the number of nested fixed points of a different type is bounded by a constant...... independent of φ. In this paper we give a proof that the alternation hierarchy for the theory of µ-lattices is strict, meaning that such a constant does not exist if µ-term are built up from the basic lattice operations and are interpreted as expected. The proof relies on the explicit characterization of free...
5. Non-commutative Differential Calculus and the Axial Anomaly in Abelian Lattice Gauge Theories
Fujiwara, T; Wu, K; Fujiwara, Takanori; Suzuki, Hiroshi; Wu, Ke
2000-01-01
The axial anomaly in lattice gauge theories has topological nature when the Dirac operator satisfies the Ginsparg-Wilson relation. We study the axial anomaly in Abelian gauge theories on an infinite hypercubic lattice by utilizing cohomological techniques. The crucial tool in our approach is the non-commutative differential calculus (NCDC) which validates the Leibniz rule of exterior derivatives on the lattice. The topological nature of the Chern character'' on the lattice becomes manifest with NCDC. Our result provides an algebraic proof of Lüscher's theorem for a four-dimensional lattice and its generalization to arbitrary dimensions.
6. Theory of vortex-lattice melting in a one-dimensional optical lattice
Snoek, M.; Stoof, H.T.C.
2006-01-01
We investigate quantum and temperature fluctuations of a vortex lattice in a one-dimensional optical lattice. We discuss in particular the Bloch bands of the Tkachenko modes and calculate the correlation function of the vortex positions along the direction of the optical lattice. Because of the
7. Universal dimer-dimer scattering in lattice effective field theory
Elhatisari, Serdar; Lee, Dean; Meißner, Ulf-G; Rupak, Gautam
2016-01-01
We consider two-component fermions with short-range interactions and large scattering length. This system has universal properties that are realized in many different fields including atomic, nuclear and particle physics. In the limit of large fermion-fermion scattering length $a_\\mathrm{ff}$ and zero range interaction, all properties of the system scale proportionally with the only length scale $a_\\mathrm{ff}$. We consider the case where there are bound dimers and calculate the scattering phase shifts for the two-dimer system near threshold using lattice effective field theory. From the scattering phase shifts, we extract the universal dimer-dimer scattering length $a_\\mathrm{dd}/a_\\mathrm{ff}=0.645(89)$ and effective range $r_\\mathrm{dd}/a_\\mathrm{ff}=-0.413(79)$.
8. Lattice gauge theory of three dimensional Thirring model
Kim, S; Kim, Seyong; Kim, Yoonbai
1999-01-01
Three dimensional Thirring model with N four-component Dirac fermions, reformulated as a lattice gauge theory, is studied by computer simulation. According to an 8^{3} data and preliminary 16^3 data, chiral symmetry is found to be spontaneously broken for N=2,\\;4 and 6. N=2 data exhibits long tail of the non-vanishing chiral condensate into weak coupling region, and N=6 case shows phase separation between the strong coupling region and the weak coupling region. Although the comparison between 8^3 data and 16^3 data shows large finite volume effects, an existence of the critical fermion flavor number N_{{\\rm cr}} (2
9. Cluster density functional theory for lattice models based on the theory of Möbius functions
Lafuente, Luis; Cuesta, José A.
2005-08-01
Rosenfeld's fundamental-measure theory for lattice models is given a rigorous formulation in terms of the theory of Möbius functions of partially ordered sets. The free-energy density functional is expressed as an expansion in a finite set of lattice clusters. This set is endowed with a partial order, so that the coefficients of the cluster expansion are connected to its Möbius function. Because of this, it is rigorously proven that a unique such expansion exists for any lattice model. The low-density analysis of the free-energy functional motivates a redefinition of the basic clusters (zero-dimensional cavities) which guarantees a correct zero-density limit of the pair and triplet direct correlation functions. This new definition extends Rosenfeld's theory to lattice models with any kind of short-range interaction (repulsive or attractive, hard or soft, one or multicomponent ...). Finally, a proof is given that these functionals have a consistent dimensional reduction, i.e. the functional for dimension d' can be obtained from that for dimension d (d' < d) if the latter is evaluated at a density profile confined to a d'-dimensional subset.
10. Cluster density functional theory for lattice models based on the theory of Moebius functions
Lafuente, Luis; Cuesta, Jose A [Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matematicas, Universidad Carlos III de Madrid, 28911 Leganes, Madrid (Spain)
2005-08-26
Rosenfeld's fundamental-measure theory for lattice models is given a rigorous formulation in terms of the theory of Moebius functions of partially ordered sets. The free-energy density functional is expressed as an expansion in a finite set of lattice clusters. This set is endowed with a partial order, so that the coefficients of the cluster expansion are connected to its Moebius function. Because of this, it is rigorously proven that a unique such expansion exists for any lattice model. The low-density analysis of the free-energy functional motivates a redefinition of the basic clusters (zero-dimensional cavities) which guarantees a correct zero-density limit of the pair and triplet direct correlation functions. This new definition extends Rosenfeld's theory to lattice models with any kind of short-range interaction (repulsive or attractive, hard or soft, one or multicomponent ...). Finally, a proof is given that these functionals have a consistent dimensional reduction, i.e. the functional for dimension d' can be obtained from that for dimension d (d' < d) if the latter is evaluated at a density profile confined to a d'-dimensional subset.
11. Monopoles and Confinement in U(1) Lattice Gauge Theory
Copeland, Timothy John
Available from UMI in association with The British Library. Requires signed TDF. Confinement in U(1) gauge theory is investigated, with particular emphasis on the role of monopoles. Starting from the work of Polyakov, the theoretical aspects are considered first, in some detail. This leads to the conclusion that the conventional techniques for analysing Monte Carlo data may not be adequate, and motivates the development of an alternative interpretation based on the theoretical insight gained. This takes more account of the expected physical properties of the theory, and does not assume beforehand that one type of behaviour (perturbative, or monopole driven) dominates. It is found that better fits to the Monte Carlo data can be achieved this way than by using the conventional methods, although different string tensions are found. The small distance behaviour is found to be best explained in terms of Coulomb effects, rather than the Luscher vibrating string picture sometimes used before. Perturbative calculations are made of Wilson loops on lattices of different shapes, and some comparisons with Monte Carlo data are made. Comments are made on the significance of these results for four dimensions, and for SU(2) and SU(3).
12. Lattice Field Theory with the Sign Problem and the Maximum Entropy Method
Masahiro Imachi
2007-02-01
Full Text Available Although numerical simulation in lattice field theory is one of the most effective tools to study non-perturbative properties of field theories, it faces serious obstacles coming from the sign problem in some theories such as finite density QCD and lattice field theory with the θ term. We reconsider this problem from the point of view of the maximum entropy method.
13. Supersymmetric gauge theories on the lattice: Pfaffian phases and the Neuberger 0/0 problem
Mehta, Dhagash; Galvez, Richard; Joseph, Anosh
2011-01-01
Recently a class of supersymmetric gauge theories have been successfully implemented on the lattice. However, there has been an ongoing debate on whether lattice versions of some of these theories suffer from a sign problem, with independent simulations for the ${\\cal N} = (2, 2)$ supersymmetric Yang-Mills theories in two dimensions yielding seemingly contradictory results. Here, we address this issue from an interesting theoretical point of view. We conjecture that the sign problem observed in some of the simulations is related to the so called Neuberger 0/0 problem, which arises in ordinary non-supersymmetric lattice gauge theories, and prevents the realization of Becchi-Rouet-Stora-Tyutin symmetry on the lattice. After discussing why we expect a sign problem in certain classes of supersymmetric lattice gauge theories far from the continuum limit, we argue that these problems can be evaded by use of a non-compact parametrization of the gauge link fields.
14. Common Misconceptions about the Dynamical Theory of Crystal Lattices: Cauchy Relations, Lattice Potentials and Infinite Crystals
Elcoro, Luis; Etxebarria, Jesus
2011-01-01
The requirement of rotational invariance for lattice potential energies is investigated. Starting from this condition, it is shown that the Cauchy relations for the elastic constants are fulfilled if the lattice potential is built from pair interactions or when the first-neighbour approximation is adopted. This is seldom recognized in widely used…
15. Polyakov line actions from SU(3) lattice gauge theory with dynamical fermions via relative weights
Höllwieser, Roman
2016-01-01
We extract an effective Polyakov line action from an underlying SU(3) lattice gauge theory with dynamical fermions via the relative weights method. The center-symmetry breaking terms in the effective theory are fit to a form suggested by effective action of heavy-dense quarks, and the effective action is solved at finite chemical potential by a mean field approach. We show results for a small sample of lattice couplings, lattice actions, and lattice extensions in the time direction. We find in some instances that the long-range couplings in the effective action are very important to the phase structure, and that these couplings are responsible for long-lived metastable states in the effective theory. Only one of these states corresponds to the underlying lattice gauge theory.
16. Exact Maps in Density Functional Theory for Lattice Models
Dimitrov, Tanja; Fuks, Johanna I; Rubio, Angel
2015-01-01
In the present work, we employ exact diagonalization for model systems on a real-space lattice to explicitly construct the exact density-to-potential and for the first time the exact density-to-wavefunction map that underly the Hohenberg-Kohn theorem in density functional theory. Having the explicit wavefunction-to- density map at hand, we are able to construct arbitrary observables as functionals of the ground-state density. We analyze the density-to-potential map as the distance between the fragments of a system increases and the correlation in the system grows. We observe a feature that gradually develops in the density-to-potential map as well as in the density-to-wavefunction map. This feature is inherited by arbitrary expectation values as functional of the ground-state density. We explicitly show the excited-state energies, the excited-state densities, and the correlation entropy as functionals of the ground-state density. All of them show this exact feature that sharpens as the coupling of the fragmen...
17. Lattice cluster theory for polymer melts with specific interactions
Xu, Wen-Sheng, E-mail: [email protected] [James Franck Institute, The University of Chicago, Chicago, Illinois 60637 (United States); Freed, Karl F., E-mail: [email protected] [James Franck Institute, The University of Chicago, Chicago, Illinois 60637 (United States); Department of Chemistry, The University of Chicago, Chicago, Illinois 60637 (United States)
2014-07-28
Despite the long-recognized fact that chemical structure and specific interactions greatly influence the thermodynamic properties of polymer systems, a predictive molecular theory that enables systematically addressing the role of chemical structure and specific interactions has been slow to develop even for polymer melts. While the lattice cluster theory (LCT) provides a powerful vehicle for understanding the influence of various molecular factors, such as monomer structure, on the thermodynamic properties of polymer melts and blends, the application of the LCT has heretofore been limited to the use of the simplest polymer model in which all united atom groups within the monomers of a species interact with a common monomer averaged van der Waals energy. Thus, the description of a compressible polymer melt involves a single van der Waals energy. As a first step towards developing more realistic descriptions to aid in the analysis of experimental data and the design of new materials, the LCT is extended here to treat models of polymer melts in which the backbone and side groups have different interaction strengths, so three energy parameters are present, namely, backbone-backbone, side group-side group, and backbone-side group interaction energies. Because of the great algebraic complexity of this extension, we retain maximal simplicity within this class of models by further specializing this initial study to models of polymer melts comprising chains with poly(n-α-olefin) structures where only the end segments on the side chains may have different, specific van der Waals interaction energies with the other united atom groups. An analytical expression for the LCT Helmholtz free energy is derived for the new model. Illustrative calculations are presented to demonstrate the degree to which the thermodynamic properties of polymer melts can be controlled by specific interactions.
18. Universality and the approach to the continuum limit in lattice gauge theory
De Divitiis, G M; Guagnelli, M; Lüscher, Martin; Petronzio, Roberto; Sommer, Rainer; Weisz, P; Wolff, U; de Divitiis, G; Frezzotti, R; Guagnelli, M; Luescher, M; Petronzio, R; Sommer, R; Weisz, P; Wolff, U
1995-01-01
The universality of the continuum limit and the applicability of renormalized perturbation theory are tested in the SU(2) lattice gauge theory by computing two different non-perturbatively defined running couplings over a large range of energies. The lattice data (which were generated on the powerful APE computers at Rome II and DESY) are extrapolated to the continuum limit by simulating sequences of lattices with decreasing spacings. Our results confirm the expected universality at all energies to a precision of a few percent. We find, however, that perturbation theory must be used with care when matching different renormalized couplings at high energies.
19. SU(2) lattice gauge theory at non-zero temperature with fixed holonomy boundary condition
Ilgenfritz, E M; Müller-Preussker, M; Veselov, A I
2001-01-01
We study SU(2) lattice gauge theory at $T>0$ in a finite box with fixed holonomy value at the spatial boundary. We search for (approximate) classical solutions of the lattice field equations and find in particular the dissociated calorons recently discussed by van Baal and collaborators.
20. Density Functional Theory for General Hard-Core Lattice Gases
Lafuente, Luis; Cuesta, José A.
2004-09-01
We put forward a general procedure to obtain an approximate free-energy density functional for any hard-core lattice gas, regardless of the shape of the particles, the underlying lattice, or the dimension of the system. The procedure is conceptually very simple and recovers effortlessly previous results for some particular systems. Also, the obtained density functionals belong to the class of fundamental measure functionals and, therefore, are always consistent through dimensional reduction. We discuss possible extensions of this method to account for attractive lattice models.
1. Exact maps in density functional theory for lattice models
Dimitrov, Tanja; Appel, Heiko; Fuks, Johanna I.; Rubio, Angel
2016-08-01
In the present work, we employ exact diagonalization for model systems on a real-space lattice to explicitly construct the exact density-to-potential and graphically illustrate the complete exact density-to-wavefunction map that underly the Hohenberg-Kohn theorem in density functional theory. Having the explicit wavefunction-to-density map at hand, we are able to construct arbitrary observables as functionals of the ground-state density. We analyze the density-to-potential map as the distance between the fragments of a system increases and the correlation in the system grows. We observe a feature that gradually develops in the density-to-potential map as well as in the density-to-wavefunction map. This feature is inherited by arbitrary expectation values as functional of the ground-state density. We explicitly show the excited-state energies, the excited-state densities, and the correlation entropy as functionals of the ground-state density. All of them show this exact feature that sharpens as the coupling of the fragments decreases and the correlation grows. We denominate this feature as intra-system steepening and discuss how it relates to the well-known inter-system derivative discontinuity. The inter-system derivative discontinuity is an exact concept for coupled subsystems with degenerate ground state. However, the coupling between subsystems as in charge transfer processes can lift the degeneracy. An important conclusion is that for such systems with a near-degenerate ground state, the corresponding cut along the particle number N of the exact density functionals is differentiable with a well-defined gradient near integer particle number.
2. Chiral Effective Theory Methods and their Application to the Structure of Hadrons from Lattice QCD
Shanahan, P E
2016-01-01
For many years chiral effective theory (ChEFT) has enabled and supported lattice QCD calculations of hadron observables by allowing systematic effects from unphysical lattice parameters to be controlled. In the modern era of precision lattice simulations approaching the physical point, ChEFT techniques remain valuable tools. In this review we discuss the modern uses of ChEFT applied to lattice studies of hadron structure in the context of recent determinations of important and topical quantities. We consider muon g-2, strangeness in the nucleon, the proton radius, nucleon polarizabilities, and sigma terms relevant to the prediction of dark-matter-hadron interaction cross-sections, among others.
3. Chiral effective theory methods and their application to the structure of hadrons from lattice QCD
Shanahan, P. E.
2016-12-01
For many years chiral effective theory (ChEFT) has enabled and supported lattice QCD calculations of hadron observables by allowing systematic effects from unphysical lattice parameters to be controlled. In the modern era of precision lattice simulations approaching the physical point, ChEFT techniques remain valuable tools. In this review we discuss the modern uses of ChEFT applied to lattice studies of hadron structure in the context of recent determinations of important and topical quantities. We consider muon g-2, strangeness in the nucleon, the proton radius, nucleon polarizabilities, and sigma terms relevant to the prediction of dark-matter-hadron interaction cross-sections, among others.
4. Digital Quantum Simulation of Z2 Lattice Gauge Theories with Dynamical Fermionic Matter
Zohar, Erez; Farace, Alessandro; Reznik, Benni; Cirac, J. Ignacio
2017-02-01
We propose a scheme for digital quantum simulation of lattice gauge theories with dynamical fermions. Using a layered optical lattice with ancilla atoms that can move and interact with the other atoms (simulating the physical degrees of freedom), we obtain a stroboscopic dynamics which yields the four-body plaquette interactions, arising in models with (2 +1 ) and higher dimensions, without the use of perturbation theory. As an example we show how to simulate a Z2 model in (2 +1 ) dimensions.
5. Classical light dispersion theory in a regular lattice
Marino, M.; Carati, A.; Galgani, L.
2007-04-01
We study the dynamics of an infinite regular lattice of classical charged oscillators. Each individual oscillator is described as a point particle subject to a harmonic restoring potential, to the retarded electromagnetic field generated by all the other particles, and to the radiation reaction expressed according to the Lorentz-Dirac equation. Exact normal mode solutions, describing the propagation of plane electromagnetic waves through the lattice, are obtained for the complete linearized system of infinitely many oscillators. At variance with all the available results, our method is valid for any values of the frequency, or of the ratio between wavelength and lattice parameter. A remarkable feature is that the proper inclusion of radiation reaction in the dynamics of the individual oscillators does not give rise to any extinction coefficient for the global normal modes of the lattice. The dispersion relations resulting from our solution are numerically studied for the case of a simple cubic lattice. New predictions are obtained in this way about the behavior of the crystal at frequencies near the proper oscillation frequency of the dipoles.
6. Chern-Simons theory on a lattice and a new description of 3-manifolds invariants
Buffenoir, E
1995-01-01
A new approach to the quantization of Chern-Simons theory has been developed in recent papers of the author. It uses a "simulation" of the moduli space of flat connections modulo the gauge group which reveals to be related to a lattice gauge theory based on a quantum group. After a generalization of the formalism of q-deformed gauge theory to the case of root of unity, we compute explicitely the correlation functions associated to Wilson loops (and more generally to graphs) on a surface with punctures, which are the interesting quantity in the study of moduli space. We then give a new description of Chern-Simons three manifolds invariants based on a description in terms of the mapping class group of a surface. At last we introduce a three dimensional lattice gauge theory based on a quantum group which is a lattice regularization of Chern-Simons theory.
7. Chaos, scaling and existence of a continuum limit in classical non-Abelian lattice gauge theory
Nielsen, H.B. [Niels Bohr Inst., Kobenhavn (Denmark); Rugh, H.H. [Univ. of Warwick, Coventry (United Kingdom); Rugh, S.E. [Los Alamos National Lab., NM (United States)
1996-12-31
We discuss space-time chaos and scaling properties for classical non-Abelian gauge fields discretized on a spatial lattice. We emphasize that there is a {open_quote}no go{close_quotes} for simulating the original continuum classical gauge fields over a long time span since there is a never ending dynamical cascading towards the ultraviolet. We note that the temporal chaotic properties of the original continuum gauge fields and the lattice gauge system have entirely different scaling properties thereby emphasizing that they are entirely different dynamical systems which have only very little in common. Considered as a statistical system in its own right the lattice gauge system in a situation where it has reached equilibrium comes closest to what could be termed a {open_quotes}continuum limit{close_quotes} in the limit of very small energies (weak non-linearities). We discuss the lattice system both in the limit for small energies and in the limit of high energies where we show that there is a saturation of the temporal chaos as a pure lattice artifact. Our discussion focuses not only on the temporal correlations but to a large extent also on the spatial correlations in the lattice system. We argue that various conclusions of physics have been based on monitoring the non-Abelian lattice system in regimes where the fields are correlated over few lattice units only. This is further evidenced by comparison with results for Abelian lattice gauge theory. How the real time simulations of the classical lattice gauge theory may reach contact with the real time evolution of (semi-classical aspects of) the quantum gauge theory (e.g. Q.C.D.) is left an important question to be further examined.
8. Generating Quadrilateral and Circular Lattices in KP Theory
Doliwa, A; Alonso, L M; Doliwa, Adam; Manas, Manuel; Alonso, Luis Martinez
1998-01-01
The bilinear equations of the $N$-component KP and BKP hierarchies and a corresponding extended Miwa transformation allow us to generate quadrilateral and circular lattices from conjugate and orthogonal nets, respectively. The main geometrical objects are expressed in terms of Baker functions.
9. Hamiltonian Effective Field Theory Study of the N^{*}(1535) Resonance in Lattice QCD.
Liu, Zhan-Wei; Kamleh, Waseem; Leinweber, Derek B; Stokes, Finn M; Thomas, Anthony W; Wu, Jia-Jun
2016-02-26
Drawing on experimental data for baryon resonances, Hamiltonian effective field theory (HEFT) is used to predict the positions of the finite-volume energy levels to be observed in lattice QCD simulations of the lowest-lying J^{P}=1/2^{-} nucleon excitation. In the initial analysis, the phenomenological parameters of the Hamiltonian model are constrained by experiment and the finite-volume eigenstate energies are a prediction of the model. The agreement between HEFT predictions and lattice QCD results obtained on volumes with spatial lengths of 2 and 3 fm is excellent. These lattice results also admit a more conventional analysis where the low-energy coefficients are constrained by lattice QCD results, enabling a determination of resonance properties from lattice QCD itself. Finally, the role and importance of various components of the Hamiltonian model are examined.
10. Dualization of non-abelian lattice gauge theory with Abelian Color Cycles (ACC)
Marchis, Carlotta
2016-01-01
We discuss a new approach to strong coupling expansion and dual representations for non-abelian lattice gauge theories. The Wilson gauge action is decomposed into a sum over "abelian color cycles" (ACC), which are loops around plaquettes visiting different colors at the corners. ACCs are complex numbers and thus commute such that a dual representation of a non-abelian theory can be obtained as in the abelian case. We apply the ACC approach to SU(2) and SU(3) lattice gauge theory and exactly rewrite the two partition sums in a strong coupling series where all gauge integrals are known in closed form.
11. Density functional theory for nearest-neighbor exclusion lattice gases in two and three dimensions
Lafuente, Luis; Cuesta, José A.
2003-12-01
To speak about fundamental measure theory obliges us to mention dimensional crossover. This feature, inherent to the systems themselves, was incorporated in the theory almost from the beginning. Although at first it was thought to be a consistency check for the theory, it rapidly became its fundamental pillar, thus becoming the only density functional theory which possesses such a property. It is straightforward that dimensional crossover connects, for instance, the parallel hard cube system (three dimensional) with that of squares (two dimensional) and rods (one dimensional). We show here that there are many more connections which can be established in this way. Through them we deduce from the functional for parallel hard (hyper)cubes in the simple (hyper)cubic lattice the corresponding functionals for the nearest-neighbor exclusion lattice gases in the square, triangular, simple cubic, face-centered-cubic, and body-centered-cubic lattices. As an application, the bulk phase diagram for all these systems is obtained.
12. Efficient implementation of the Monte Carlo method for lattice gauge theory calculations on the floating point systems FPS-164
Moriarty, K.J.M. (Royal Holloway Coll., Englefield Green (UK). Dept. of Mathematics); Blackshaw, J.E. (Floating Point Systems UK Ltd., Bracknell)
1983-04-01
The computer program calculates the average action per plaquette for SU(6)/Z/sub 6/ lattice gauge theory. By considering quantum field theory on a space-time lattice, the ultraviolet divergences of the theory are regulated through the finite lattice spacing. The continuum theory results can be obtained by a renormalization group procedure. Making use of the FPS Mathematics Library (MATHLIB), we are able to generate an efficient code for the Monte Carlo algorithm for lattice gauge theory calculations which compares favourably with the performance of the CDC 7600.
13. Transient response of lattice structures based on exact member theory
Anderson, Melvin S.
1989-01-01
The computer program BUNVIS-RG, which treats vibration and buckling of lattice structures using exact member stiffness matrices, has been extended to calculate the exact modal mass and stiffness quantities that can be used in a conventional transient response analysis based on modes. The exact nature of the development allows inclusion of local member response without introduction of any interior member nodes. Results are given for several problems in which significant interaction between local and global response occurs.
14. Ab initio nuclear structure from lattice effective field theory
Lee, Dean [Department of Physics, North Carolina State University, Raleigh NC 27695 (United States)
2014-11-11
This proceedings article reviews recent results by the Nuclear Lattice EFT Collaboration on an excited state of the {sup 12}C nucleus known as the Hoyle state. The Hoyle state plays a key role in the production of carbon via the triple-alpha reaction in red giant stars. We discuss the structure of low-lying states of {sup 12}C as well as the dependence of the triple-alpha reaction on the masses of the light quarks.
15. Freedom and confinement in lattice Yang-Mills theories. A case for divorce
Colangelo, P.; Cosmai, L.; Pellicoro, M.; Preparata, G.
1986-03-01
We present evidence that nonperturbative effects in lattice gauge theories do not obey at small coupling constant (large ..beta..) asymptotic scaling, but they rather behave as suggested by a recent result in continuum Yang-Mills theories. We also discuss the possible impact of these results on our understanding of QCD.
16. Vortex free energies in SO(3) and SU(2) lattice gauge theory
De Forcrand, Philippe; Forcrand, Philippe de; Jahn, Oliver
2003-01-01
Lattice gauge theories with gauge groups SO(3) and SU(2) are compared. The free energy of electric twist, an order parameter for the confinement-deconfinement transition which does not rely on centre-symmetry breaking, is measured in both theories. The results are used to calibrate the scale in SO(3).
17. Lattice gas hydrodynamics: Theory and simulations. Final report
Hasslacher, B.
1993-05-01
The first successful application of a microscopic analogy to create a skeleton cellular automaton and analyze it with statistical mechanical tools, was the work of Frisch, Hasslacher and Pomeau on the Navier-Stokes equation in two and three dimensions. This has become a very large research area with lattice gas models and methods being used for both fundamental investigations into the foundations of statistical mechanics and a large number of diverse applications. This present research was devoted to enlarging the fundamental scope of lattice gas models and proved quite successful. Since the beginning of this proposal, cellular automata have been constructed for statistical mechanical models, fluids, diffusion and shock systems in fundamental investigations. In applied areas, there are now excellent lattice gas models for complex flows through porous media, chemical reaction and combustion dynamics, multiphase flow systems, and fluid mixtures with natural boundaries. With extended cellular fluid models, one can do problems with arbitrary pairwise potentials. Recently, these have been applied to such problems as non-newtonian or polymeric liquids and a mixture of immiscible fluids passing through fractal or spongelike media in two and three dimensions. This proposal has contributed to and enlarged the scope of this work.
18. Low Energy Continuum and Lattice Effective Field Theories
Elhatisari, Serdar
In this thesis we investigate several constraints and their impacts on the short-range potentials in the low-energy limits of quantum mechanics.We also present lattice Monte Carlo calculations using the adiabatic projection method. In the first part we consider the constraints of causality and unitarity for the low-energy interactions of particles. We generalize Wigner's causality bound to the case of non-vanishing partial-wave mixing. Specifically we analyze the system of the low-energy interactions between protons and neutrons. We derive a general theorem that non-vanishing partial-wave mixing cannot be reproduced with zero-range interactions without violating causality or unitarity. We also analyze low-energy scattering for systems with arbitrary short-range interactions plus an attractive 1/ralpha tail for alpha ≥ 2. In particular, we focus on the case of alpha = 6 and we derive the constraints of causality and unitarity also for these systems and find that the van derWaals length scale dominates over parameters characterizing the short-distance physics of the interaction. This separation of scales suggests a separate universality class for physics characterizing interactions with an attractive 1{r6 tail. We argue that a similar universality class exists for any attractive potential 1/ralpha for alpha ≥ 2. In the second part of the thesis we present lattice Monte Carlo calculations of fermion-dimer scattering in the limit of zero-range interactions using the adiabatic projection method. The adiabatic projection method uses a set of initial cluster states and Euclidean time projection to give a systematically improvable description of the low-lying scattering cluster states in a finite volume. We use Luscher's finite-volume relations to determine the s-wave, p-wave, and d-wave phase shifts. For comparison, we also compute exact lattice results using Lanczos iteration and continuum results using the Skorniakov-Ter-Martirosian equation. For our Monte Carlo
19. Gauge Invariant Effective Action in Abelian Chiral Gauge Theory on the Lattice
Suzuki, H
1999-01-01
Lüscher's recent formulation of Abelian chiral gauge theories on the lattice, in the vacuum (or perturbative) sector in infinite lattice volume, is re-interpreted in terms of the lattice covariant regularization. The gauge invariance of the effective action and the integrability of the gauge current in anomaly-free cases become transparent then. The real part of the effective action is simply one-half of that of the Dirac fermion and, when the Dirac operator has proper properties in the continuum limit, the imaginary part in the continuum limit reproduces the $\\eta$-invariant.}
20. Lattice heavy quark effective theory and the isgur-wise function
Hashimoto, S
1996-01-01
We compute the Isgur-Wise function using heavy quark effective theory formulated on the lattice. The non-relativistic kinetic energy term of the heavy quark is included to the action as well as terms remaining in the infinite quark mass limit. The classical velocity of the heavy quark is renormalized on the lattice and we determine the renormalized velocity non-perturbatively using the energy-momentum dispersion relation. The slope parameter of the Isgur-Wise function at zero recoil is obtained at \\beta=6.0 on a 24^3\\times 48 lattice for three values of m_{Q}.
1. Maximum-Likelihood Approach to Topological Charge Fluctuations in Lattice Gauge Theory
Brower, R C; Fleming, G T; Lin, M F; Neil, E T; Osborn, J C; Rebbi, C; Rinaldi, E; Schaich, D; Schroeder, C; Voronov, G; Vranas, P; Weinberg, E; Witzel, O
2014-01-01
We present a novel technique for the determination of the topological susceptibility (related to the variance of the distribution of global topological charge) from lattice gauge theory simulations, based on maximum-likelihood analysis of the Markov-chain Monte Carlo time series. This technique is expected to be particularly useful in situations where relatively few tunneling events are observed. Restriction to a lattice subvolume on which topological charge is not quantized is explored, and may lead to further improvement when the global topology is poorly sampled. We test our proposed method on a set of lattice data, and compare it to traditional methods.
2. Geometric approach to chaos in the classical dynamics of Abelian lattice gauge theory
Casetti, Lapo [Istituto Nazionale per la Fisica della Materia (INFM), Unita di Ricerca del Politecnico di Torino, Dipartimento di Fisica, Politecnico di Torino, Turin (Italy); Gatto, Raoul [Departement de Physique Theorique, Universite de Geneve, Geneva (Switzerland); Pettini, Marco [Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, Florence (Italy)
1999-04-23
A Riemannian geometrization of dynamics is used to study chaoticity in the classical Hamiltonian dynamics of a U(1) lattice gauge theory. This approach allows one to obtain analytical estimates of the largest Lyapunov exponent in terms of time averages of geometric quantities. These estimates are compared with the results of numerical simulations, and turn out to be very close to the values extrapolated for very large lattice sizes even when the geometric quantities are computed using small lattices. The scaling of the Lyapunov exponent {lambda} with the energy density {epsilon} is found to be well described by the law {lambda}{proportional_to}{epsilon}{sup 2}. (author)
3. Geometric approach to chaos in the classical dynamics of abelian lattice gauge theory
Casetti, L; Pettini, M; Casetti, Lapo; Gatto, Raoul; Pettini, Marco
1998-01-01
A Riemannian geometrization of dynamics is used to study chaoticity in the classical Hamiltonian dynamics of a U(1) lattice gauge theory. This approach allows one to obtain analytical estimates of the largest Lyapunov exponent in terms of time averages of geometric quantities. These estimates are compared with the results of numerical simulations, and turn out to be very close to the values extrapolated for very large lattice sizes even when the geometric quantities are computed using small lattices. The scaling of the Lyapunov exponent with the energy density is found to be well described by a quadratic power law.
4. Parallel implementation of a lattice-gauge-theory code: studying quark confinement on PC clusters
Cucchieri, A; Travieso, G; Taurines, A R; Cucchieri, Attilio; Mendes, Tereza; Travieso, Gonzalo; Taurines, Andre R.
2003-01-01
We consider the implementation of a parallel Monte Carlo code for high-performance simulations on PC clusters with MPI. We carry out tests of speedup and efficiency. The code is used for numerical simulations of pure SU(2) lattice gauge theory at very large lattice volumes, in order to study the infrared behavior of gluon and ghost propagators. This problem is directly related to the confinement of quarks and gluons in the physics of strong interactions.
5. apeNEXT: A multi-TFlops Computer for Simulations in Lattice Gauge Theory
Bodin, F; Cabibbo, Nicola; Carlo, F D; De Pietri, R; Renzo, F D; Kaldass, H; Lonardo, A; Lukyanov, M; De Luca, S; Micheli, J; Morénas, V; Pène, O; Pleiter, D; Paschedag, N; Rapuano, F; Sartori, L; Schifano, F; Simma, H; Tripiccione, R; Vicini, P; Boucaud, Ph.
2003-01-01
We present the APE (Array Processor Experiment) project for the development of dedicated parallel computers for numerical simulations in lattice gauge theories. While APEmille is a production machine in today's physics simulations at various sites in Europe, a new machine, apeNEXT, is currently being developed to provide multi-Tflops computing performance. Like previous APE machines, the new supercomputer is largely custom designed and specifically optimized for simulations of Lattice QCD.
6. Recent developments in neutron-proton scattering with Lattice Effective Field Theory
Alarcón, Jose Manuel
2015-01-01
In this contribution, we show some recent progress in the study of neutron-proton scattering with Nuclear Lattice Effective Field Theory (NLEFT). We present preliminary studies of both, the uncertainties in the $np$ phase shifts extracted with NLEFT, and the lattice spacing dependence in the transfer matrix formalism. Such investigations have not been performed before in the literature, and will be relevant for Monte Carlo simulations of nuclear structure with NLEFT.
7. Topological susceptibility in lattice Yang-Mills theory with open boundary condition
Chowdhury, Abhishek; Harindranath, A. [Theory Division, Saha Institute of Nuclear Physics,1/AF Bidhan Nagar, Kolkata 700064 (India); Maiti, Jyotirmoy [Department of Physics, Barasat Government College,10 KNC Road, Barasat, Kolkata 700124 (India); Majumdar, Pushan [Department of Theoretical Physics, Indian Association for the Cultivation of Science,Kolkata 700032 (India)
2014-02-11
We find that using open boundary condition in the temporal direction can yield the expected value of the topological susceptibility in lattice SU(3) Yang-Mills theory. As a further check, we show that the result agrees with numerical simulations employing the periodic boundary condition. Our results support the preferability of the open boundary condition over the periodic boundary condition as the former allows for computation at smaller lattice spacings needed for continuum extrapolation at a lower computational cost.
8. A numerical theory of lattice gas and lattice Boltzmann methods in the computation of solutions to nonlinear advective-diffusive systems
Elton, A.B.H.
1990-09-24
A numerical theory for the massively parallel lattice gas and lattice Boltzmann methods for computing solutions to nonlinear advective-diffusive systems is introduced. The convergence theory is based on consistency and stability arguments that are supported by the discrete Chapman-Enskog expansion (for consistency) and conditions of monotonicity (in establishing stability). The theory is applied to four lattice methods: Two of the methods are for some two-dimensional nonlinear diffusion equations. One of the methods is for the one-dimensional lattice method for the one-dimensional viscous Burgers equation. And one of the methods is for a two-dimensional nonlinear advection-diffusion equation. Convergence is formally proven in the L{sub 1}-norm for the first three methods, revealing that they are second-order, conservative, conditionally monotone finite difference methods. Computational results which support the theory for lattice methods are presented. In addition, a domain decomposition strategy using mesh refinement techniques is presented for lattice gas and lattice Boltzmann methods. The strategy allows concentration of computational resources on regions of high activity. Computational evidence is reported for the strategy applied to the lattice gas method for the one-dimensional viscous Burgers equation. 72 refs., 19 figs., 28 tabs.
9. Nonequilibrium Dynamical Mean-Field Theory for Bosonic Lattice Models
2015-01-01
We develop the nonequilibrium extension of bosonic dynamical mean-field theory and a Nambu real-time strong-coupling perturbative impurity solver. In contrast to Gutzwiller mean-field theory and strong-coupling perturbative approaches, nonequilibrium bosonic dynamical mean-field theory captures not only dynamical transitions but also damping and thermalization effects at finite temperature. We apply the formalism to quenches in the Bose-Hubbard model, starting from both the normal and the Bos...
10. Mean field theory for Lyapunov exponents and KS entropy in Lorentz lattice gases
Ernst, M H; Nix, R; Jacobs, D; Ernst, M H; Dorfman, J R; Nix, R; Jacobs, D
1994-01-01
automata lattice gases are useful systems for systematically exploring the connections between non-equilibrium statistical mechanics and dynamical systems theory. Here the chaotic properties of a Lorentz lattice gas are studied analytically and by means of computer simulations. The escape-rates, Lyapunov exponents, and KS entropies are estimated for a one- dimensional example using a mean field theory. The results are compared with simulations for a range of densities and scattering parameters of the lattice gas. The computer results show a distribution of values for the dynamical quantities with average values that are in good agreement with the mean field theory and consistent with the escape-rate formalism for the coefficient of diffusion.
11. First-principles theories for anharmonic lattice vibrations.
Hirata, So; Keçeli, Murat; Yagi, Kiyoshi
2010-07-21
Size-extensive generalizations of the vibrational self-consistent field (VSCF), vibrational Moller-Plesset perturbation (VMP), and vibrational coupled-cluster (VCC) methods are made to anharmonic lattice vibrations of extended periodic systems on the basis of a quartic force field (QFF) in delocalized normal coordinates. Copious terms in the formalisms of VSCF that have nonphysical size dependence are identified algebraically and eliminated, leading to compact and strictly size-extensive equations. This "quartic" VSCF method (qVSCF) thus defined has no contributions from cubic force constants and alters only the transition energies of the underlying harmonic-oscillator reference from a subset of quartic force constants. It also provides a way to evaluate an anharmonic correction to the lattice structure due to cubic force constants of a certain type. The second-order VMP and VCC methods in the QFF based on the qVSCF reference are shown to account for anharmonic effects due to all cubic and quartic force constants in a size-extensive fashion. These methods can be readily extended to a higher-order truncated Taylor expansion of a potential energy surface in normal coordinates. An algebraic proof of the lack of size-extensivity in the vibrational configuration-interaction method is also presented.
12. Dense baryonic matter in strong coupling lattice gauge theory
Bringoltz, B
2004-01-01
We investigate the strong coupling limit of lattice QCD in the Hamiltonian formulation for systems with non-zero baryon density. In leading order the Hamiltonian looks like an antiferromagnet that is invariant under global U(N_f)xU(N_f) and local SU(N_c). Physically it describes meson dynamics with a fixed background of baryon density. We study this Hamiltonian with several baryon number distributions, and concentrate on the global symmetries of the ground state and on the properties of low lying excitations. In particular, for uniform non-zero baryon density we write the partition function as a path integral that is tractable in the limit of large N_c. We find that the ground state spontaneously breaks chiral symmetry as well as discrete lattice rotations in a way that depends on N_f and the density. The low energy excitations include type I and type II Goldstone bosons. The energies of the latter are of order 1/N_c, and are quadratic in momentum. Bosons of either type can develop anisotropic dispersion rela...
13. Dual of 3-dimensional pure SU(2) Lattice Gauge Theory and the Ponzano-Regge Model
Anishetty, R; Sharatchandra, H S; Mathur, M; Anishetty, Ramesh; Cheluvaraja, Srinath; Mathur, Manu
1993-01-01
By carrying out character expansion and integration over all link variables, the partition function of 3-dimensional pure SU(2) lattice gauge theory is rewritten in terms of 6j symbols. The result is Ponzano-Regge model of 3-dimensional gravity with a term that explicitly breaks general coordinate invariance. Conversely, we show that dual of Ponzano-Regge model is an SU(2) lattice gauge theory where all plaquette variables are constrained to the identity matrix and therefore the model needs no further regularization. Our techniques are applicable to other models with non-abelian symmetries in any dimension and provide duality transform for the partition function.
14. Third and higher order NFPA twisted constructions of conformal field theories from lattices
Montague, P.S. [Cambridge Univ. (United Kingdom). Dept. of Applied Mathematics and Theoretical Physics (DAMTP)
1995-05-08
We investigate orbifold constructions of conformal field theories from lattices by no-fixed-point automorphisms (NFPAs) Z{sub p} for p prime, p>2, concentrating on the case p=3. Explicit expressions are given for most of the relevant vertex operators, and we consider the locality relations necessary for these to define a consistent conformal field theory. A relation to constructions of lattices from codes, analogous to that found in earlier work in the p=2 case which led to a generalisation of the triality structure of the Monster module, is also demonstrated. ((orig.)).
15. Third and higher order NFPA twisted constructions of conformal field theories from lattices
Montague, P S
1995-01-01
We investigate orbifold constructions of conformal field theories from lattices by no-fixed-point automorphisms (NFPA's) Z_p for p prime, p>2 concentrating on the case p=3. Explicit expressions are given for most of the relevant vertex operators, and we consider the locality relations necessary for these to define a consistent conformal field theory. A relation to constructions of lattices from codes, analogous to that found in earlier work in the p=2 case which led to a generalisation of the triality structure of the Monster module, is also demonstrated.
16. Digital quantum simulation of $\\mathbb{Z}_2$ lattice gauge theories with dynamical fermionic matter
Zohar, Erez; Reznik, Benni; Cirac, J Ignacio
2016-01-01
We propose a scheme for digital quantum simulation of lattice gauge theories with dynamical fermions. Using a layered optical lattice with ancilla atoms that can move and interact with the other atoms (simulating the physical degrees of freedom), we obtain a stroboscopic dynamics which yields the four-body plaquette interactions, arising in models with $2+1$ and higher dimensions, without the use of perturbation theory. As an example we show how to simulate a $\\mathbb{Z}_2$ model in $2+1$ dimensions.
17. Volume scaling of Dirac eigenvalues in SU(3) lattice gauge theory with color sextet fermions
DeGrand, Thomas
2009-01-01
I observe a rough volume-dependent scaling of the low eigenvalues of a chiral Dirac operator in lattice studies of SU(3) lattice gauge theory with two flavors of color sextet fermions, in its weak-coupling phase. The mean value of the ith eigenvalue scales with the simulation volume V=L^4 as L^p ~zeta_i, where zeta_i is a volume-independent constant. The exponent p is about 1.4. A possible explanation for this phenomenon is that p is the leading relevant exponent associated with the fermion mass dependence of correlation functions in a theory whose zero-mass limit is conformal.
18. Discriminating between two reformulations of SU(3) Yang-Mills theory on a lattice
Shibata, Akihiro [Computing Research Center, High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 (Japan); 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)
2016-01-22
In order to investigate quark confinement, we give a new reformulation of the SU (N) Yang-Mills theory on a lattice and present the results of the numerical simulations of the SU (3) Yang-Mills theory on a lattice. 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.
19. Casimir effect on the lattice: U(1) gauge theory in two spatial dimensions
Chernodub, M N; Molochkov, A V
2016-01-01
We propose a general numerical method to study the Casimir effect in lattice gauge theories. We illustrate the method by calculating the energy density of zero-point fluctuations around two parallel wires of finite static permittivity in Abelian gauge theory in two spatial dimensions. We discuss various subtle issues related to the lattice formulation of the problem and show how they can successfully be resolved. Finally, we calculate the Casimir potential between the wires of a fixed permittivity, extrapolate our results to the limit of ideally conducting wires and demonstrate excellent agreement with a known theoretical result.
20. Casimir effect on the lattice: U(1) gauge theory in two spatial dimensions
Chernodub, M. N.; Goy, V. A.; Molochkov, A. V.
2016-11-01
We propose a general numerical method to study the Casimir effect in lattice gauge theories. We illustrate the method by calculating the energy density of zero-point fluctuations around two parallel wires of finite static permittivity in Abelian gauge theory in two spatial dimensions. We discuss various subtle issues related to the lattice formulation of the problem and show how they can successfully be resolved. Finally, we calculate the Casimir potential between the wires of a fixed permittivity, extrapolate our results to the limit of ideally conducting wires and demonstrate excellent agreement with a known theoretical result.
1. Application of graph theory to the statistical thermodynamics of lattice polymers. I. Elements of theory and test for dimers
Brazhnik, Olga D.; Freed, Karl F.
1996-07-01
The lattice cluster theory (LCT) is extended to enable inclusion of longer range correlation contributions to the partition function of lattice model polymers in the athermal limit. A diagrammatic technique represents the expansion of the partition function in powers of the inverse lattice coordination number. Graph theory is applied to sort, classify, and evaluate the numerous diagrams appearing in higher orders. New general theorems are proven that provide a significant reduction in the computational labor required to evaluate the contributions from higher order correlations. The new algorithm efficiently generates the correction to the Flory mean field approximation from as many as eight sterically interacting bonds. While the new results contain the essential ingredients for treating a system of flexible chains with arbitrary lengths and concentrations, the complexity of our new algorithm motivates us to test the theory here for the simplest case of a system of lattice dimers by comparison to the dimer packing entropies from the work of Gaunt. This comparison demonstrates that the eight bond LCT is exact through order φ5 for dimers in one through three dimensions, where φ is the volume fraction of dimers. A subsequent work will use the contracted diagrams, derived and tested here, to treat the packing entropy for a system of flexible N-mers at a volume fraction of φ on hypercubic lattices.
2. Matrix product states for Hamiltonian lattice gauge theories
Buyens, Boye; Haegeman, Jutho; Verstraete, Frank
2014-01-01
Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems. The matrix product states (MPS) are one particular case of TNS and are used for the simulation of 1+1 dimensional systems. In [1] we considered the MPS formalism for the simulation of the Hamiltonian lattice gauge formulation of 1+1 dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model. We deduced the ground state and lowest lying excitations. Furthermore, we performed a full quantum real-time simulation for a quench with a uniform background electric field. In this proceeding we continue our work on the Schwinger model. We demonstrate the advantage of working with gauge invariant MPS by comparing with MPS simulations on the full Hilbert space, that includes numerous non-physical gauge variant states. Furthermore, we compute the chiral condensate and recover the predicted UV-divergent behavior.
3. Lattice Simulations for Light Nuclei: Chiral Effective Field Theory at Leading Order
Borasoy, B; Krebs, H; Lee, D; Meißner, Ulf G; Borasoy, Bugra; Epelbaum, Evgeny; Krebs, Hermann; Lee, Dean; Mei{\\ss}ner, Ulf-G.
2006-01-01
We discuss lattice simulations of light nuclei at leading order in chiral effective field theory. Using lattice pion fields and auxiliary fields, we include the physics of instantaneous one-pion exchange and the leading-order S-wave contact interactions. We also consider higher-derivative contact interactions which adjust the S-wave scattering amplitude at higher momenta. By construction our lattice path integral is positive definite in the limit of exact Wigner SU(4) symmetry for any even number of nucleons. This SU(4) positivity and the approximate SU(4) symmetry of the low-energy interactions play an important role in suppressing sign and phase oscillations in Monte Carlo simulations. We assess the computational scaling of the lattice algorithm for light nuclei with up to eight nucleons and analyze in detail calculations of the deuteron, triton, and helium-4.
4. What are the Confining Field Configurations of Strong-Coupling Lattice Gauge Theory?
Faber, M; Olejník, S
2000-01-01
Starting from the strong-coupling SU(2) Wilson action in D=3 dimensions, we derive an effective, semi-local action on a lattice of spacing L times the spacing of the original lattice. It is shown that beyond the adjoint color-screening distance, i.e. for $L \\ge 5$, thin center vortices are stable saddlepoints of the corresponding effective action. Since the entropy of these stable objects exceeds their energy, center vortices percolate throughout the lattice, and confine color charge in half-integer representations of the SU(2) gauge group. This result contradicts the folklore that confinement in strong-coupling lattice gauge theory, for D>2 dimensions, is simply due to plaquette disorder, as is the case in D=2 dimensions. It also demonstrates explicitly how the emergence and stability of center vortices is related to the existence of color screening by gluon fields.
5. Nonequilibrium Dynamical Mean-Field Theory for Bosonic Lattice Models
Strand, Hugo U. R.; Eckstein, Martin; Werner, Philipp
2015-01-01
We develop the nonequilibrium extension of bosonic dynamical mean-field theory and a Nambu real-time strong-coupling perturbative impurity solver. In contrast to Gutzwiller mean-field theory and strong-coupling perturbative approaches, nonequilibrium bosonic dynamical mean-field theory captures not only dynamical transitions but also damping and thermalization effects at finite temperature. We apply the formalism to quenches in the Bose-Hubbard model, starting from both the normal and the Bose-condensed phases. Depending on the parameter regime, one observes qualitatively different dynamical properties, such as rapid thermalization, trapping in metastable superfluid or normal states, as well as long-lived or strongly damped amplitude oscillations. We summarize our results in nonequilibrium "phase diagrams" that map out the different dynamical regimes.
6. Chiral effective field theory on the lattice at next-to-leading order
Borasoy, Bugra; Krebs, Hermann; Lee, Dean; Meißner, Ulf-G
2007-01-01
We study nucleon-nucleon scattering on the lattice at next-to-leading order in chiral effective field theory. We determine phase shifts and mixing angles from the properties of two-nucleon standing waves induced by a hard spherical wall in the center-of-mass frame. At fixed lattice spacing we test model independence of the low-energy effective theory by computing next-to-leading-order corrections for two different leading-order lattice actions. The first leading-order action includes instantaneous one-pion exchange and same-site contact interactions. The second leading-order action includes instantaneous one-pion exchange and Gaussian-smeared interactions. We find that in each case the results at next-to-leading order are accurate up to corrections expected at higher order.
7. Leech Lattice Extension of the Non-linear Schrodinger Equation Theory of Einstein spaces
Chapline, George
2015-01-01
Although the nonlinear Schrodinger equation description of Einstein spaces has provided insights into how quantum mechanics might modify the classical general relativistic description of space-time, an exact quantum description of space-times with matter has remained elusive. In this note we outline how the nonlinear Schrodinger equation theory of Einstein spaces might be generalized to include matter by transplanting the theory to the 25+1 dimensional Lorentzian Leech lattice. Remarkably when a hexagonal section of the Leech lattice is set aside as the stage for the nonlinear Schrodinger equation, the discrete automorphism group of the complex Leech lattice with one complex direction fixed can be lifted to continuous Lie group symmetries. In this setting the wave function becomes an 11x11 complex matrix which represents matter degrees of freedom consisting of a 2-form abelian gauge field and vector nonabelian SU(3)xE6 gauge fields together with their supersymmetric partners. The lagrangian field equations fo...
8. Topological Objects And Confinement In Non-abelian Lattice Gauge Theory
Tucker, W W
2005-01-01
We use lattice methods to study the connection between topological objects and the confining potential in SU(2) and SU(3) Yang-Mills theories. We use Monte Carlo techniques, generating and performing measurements on sample configurations of SU(2) and SU(3) gauge fields. We isolate topological objects, specifically Abelian monopoles and center vortices, in these configurations. We then measure the contribution to the string tension from these objects, and compare the results to “full” measurements made on the original configurations. In addition we investigate the effects of gauge ambiguities (Gribov effects) and cooling on these sets of measurements. For the case of SU(2) lattice gauge theory, our results from monopoles agree with full values but are somewhat lower when gauge ambiguities are taken into account. The situation is not stable under cooling. When we carry out analogous procedures on sample SU(3) lattice configurations, we find disagreement between full SU(3) values and those fr...
9. Apparently non-invariant terms of nonlinear sigma models in lattice perturbation theory
Harada, Koji; Kubo, Hirofumi; Yamamoto, Yuki
2009-01-01
Apparently non-invariant terms (ANTs) which appear in loop diagrams for nonlinear sigma models (NLSs) are revisited in lattice perturbation theory. The calculations have been done mostly with dimensional regularization so far. In order to establish that the existence of ANTs is independent of the regularization scheme, and of the potential ambiguities in the definition of the Jacobian of the change of integration variables from group elements to "pion" fields, we employ lattice regularization, in which everything (including the Jacobian) is well-defined. We show explicitly that lattice perturbation theory produces ANTs in the four-point functions of the "pion" fields at one-loop and the Jacobian does not play an important role in generating ANTs.
10. Decorated tensor network renormalization for lattice gauge theories and spin foam models
Dittrich, Bianca; Mizera, Sebastian; Steinhaus, Sebastian
2016-05-01
Tensor network techniques have proved to be powerful tools that can be employed to explore the large scale dynamics of lattice systems. Nonetheless, the redundancy of degrees of freedom in lattice gauge theories (and related models) poses a challenge for standard tensor network algorithms. We accommodate for such systems by introducing an additional structure decorating the tensor network. This allows to explicitly preserve the gauge symmetry of the system under coarse graining and straightforwardly interpret the fixed point tensors. We propose and test (for models with finite Abelian groups) a coarse graining algorithm for lattice gauge theories based on decorated tensor networks. We also point out that decorated tensor networks are applicable to other models as well, where they provide the advantage to give immediate access to certain expectation values and correlation functions.
11. Z{sub c}(3900): confronting theory and lattice simulations
Albaladejo, Miguel; Fernandez-Soler, Pedro; Nieves, Juan [Instituto de Fisica Corpuscular (IFIC), Centro Mixto CSIC-Universidad de Valencia, Institutos de Investigacion de Paterna, Valencia (Spain)
2016-10-15
We consider a recent T-matrix analysis by Albaladejo et al. (Phys Lett B 755:337, 2016), which accounts for the J/ψπ and D{sup *} anti D coupled-channels dynamics, and which successfully describes the experimental information concerning the recently discovered Z{sub c}(3900){sup ±}. Within such scheme, the data can be similarly well described in two different scenarios, where Z{sub c}(3900) is either a resonance or a virtual state. To shed light into the nature of this state, we apply this formalism in a finite box with the aim of comparing with recent Lattice QCD (LQCD) simulations. We see that the energy levels obtained for both scenarios agree well with those obtained in the single-volume LQCD simulation reported in Prelovsek et al. (Phys Rev D 91:014504, 2015), thus making it difficult to disentangle the two possibilities. We also study the volume dependence of the energy levels obtained with our formalism and suggest that LQCD simulations performed at several volumes could help in discerning the actual nature of the intriguing Z{sub c}(3900) state. (orig.)
12. The QCD Abacus A New Formulation for Lattice Gauge Theories
Brower, R C
1998-01-01
A quantum Hamiltonian is constructed for SU(3) lattice QCD entirely from color triplet Fermions --- the standard quarks and a new Fermionic constituent'' of the gluon we call rishons''. The quarks are represented by Dirac spinors on each site and the gauge fields by rishon-antirishon bilinears on each link which together with the local gauge transforms are the generators of an SU(6) algebra. The effective Lagrangian for the path integral lives in $R^4 \\times S^1$ Euclidean space with a compact fifth time'' of circumference ($\\beta$) and non-Abelian charge ($e^2$) both of which carry dimensions of length. For large $\\beta$, it is conjectured that continuum QCD is reached and that the dimensionless ratio $g^2 = e^2/\\beta$ becomes the QCD gauge coupling. The quarks are introduced as Kaplan chiral Fermions at either end of the finite slab in fifth time. This talk will emphasize the gauge and algebraic structure of the rishon or link Fermions and the special properties that may lead to fast discrete dynamics...
13. Z_c(3900): confronting theory and lattice simulations
Albaladejo, Miguel; Fernandez-Soler, Pedro; Nieves, Juan
2016-10-01
We consider a recent T-matrix analysis by Albaladejo et al. (Phys Lett B 755:337, 2016), which accounts for the J/ψ π and D^*bar{D} coupled-channels dynamics, and which successfully describes the experimental information concerning the recently discovered Z_c(3900)^± . Within such scheme, the data can be similarly well described in two different scenarios, where Z_c(3900) is either a resonance or a virtual state. To shed light into the nature of this state, we apply this formalism in a finite box with the aim of comparing with recent Lattice QCD (LQCD) simulations. We see that the energy levels obtained for both scenarios agree well with those obtained in the single-volume LQCD simulation reported in Prelovsek et al. (Phys Rev D 91:014504, 2015), thus making it difficult to disentangle the two possibilities. We also study the volume dependence of the energy levels obtained with our formalism and suggest that LQCD simulations performed at several volumes could help in discerning the actual nature of the intriguing Z_c(3900) state.
14. $Z_c(3900)$: Confronting theory and lattice simulations
Albaladejo, M; Fernandez-Soler, P
2016-01-01
We consider a recent $T$-matrix analysis by Albaladejo {\\it et al.}, [Phys.\\ Lett.\\ B {\\bf 755}, 337 (2016)] which accounts for the $J/\\psi\\pi$ and $D^\\ast\\bar{D}$ coupled--channels dynamics, and that successfully describes the experimental information concerning the recently discovered $Z_c(3900)^\\pm$. Within such scheme, the data can be similarly well described in two different scenarios, where the $Z_c(3900)$ is either a resonance or a virtual state. To shed light into the nature of this state, we apply this formalism in a finite box with the aim of comparing with recent Lattice QCD (LQCD) simulations. We see that the energy levels obtained for both scenarios agree well with those obtained in the single-volume LQCD simulation reported in Prelovsek {\\it et al.} [Phys.\\ Rev.\\ D {\\bf 91}, 014504 (2015)], making thus difficult to disentangle between both possibilities. We also study the volume dependence of the energy levels obtained with our formalism, and suggest that LQCD simulations performed at several vo...
15. Structure of flux tube in SU(2) lattice gauge theory
Shiba, H
1994-01-01
The structure of the flux tube is studied in SU(2) QCD from the standpoint of the abelian projection theory. It is shown that the flux distributions of the orthogonal electric field and the magnetic field are produced by the effect that the abelian monopoles in the maximally abelian (MA) gauge are expelled from the string region.
16. Lattice gauge theory simulations in the quantum information era
Dalmonte, M.; Montangero, S.
2016-07-01
The many-body problem is ubiquitous in the theoretical description of physical phenomena, ranging from the behaviour of elementary particles to the physics of electrons in solids. Most of our understanding of many-body systems comes from analysing the symmetric properties of Hamiltonian and states: the most striking examples are gauge theories such as quantum electrodynamics, where a local symmetry strongly constrains the microscopic dynamics. The physics of such gauge theories is relevant for the understanding of a diverse set of systems, including frustrated quantum magnets and the collective dynamics of elementary particles within the standard model. In the last few years, several approaches have been put forward to tackle the complex dynamics of gauge theories using quantum information concepts. In particular, quantum simulation platforms have been put forward for the realisation of synthetic gauge theories, and novel classical simulation algorithms based on quantum information concepts have been formulated. In this review, we present an introduction to these approaches, illustrating the basics concepts and highlighting the connections between apparently very different fields, and report the recent developments in this new thriving field of research.
17. A lattice model for the second $\\mathbb{Z}_{3}$ parafermionic field theory
Estienne, Benoit
2008-01-01
The second $\\mathbb{Z}_{3}$ parafermionic conformal theories are associated with the coset construction $\\frac{SU(2)_{k}\\times SU(2)_{4}}{SU(2)_{k+4}}$. Solid-on-solid integrable lattice models obtained by fusion of the model based on level-1 representation of the affine algebra $B_1^{(1)}$ have a critical point described by these conformal theories. Explicit values for the Boltzmann weights are derived for these models, and it is shown that the Boltzmann weights can be made positive for a particular value of the spectral parameter, opening a way to eventual numerical simulations of these conformal field theories. Away from criticality, these lattice models describe an integrable, massive perturbation of the parafermionic conformal theory by the relevant field $\\Psi_{-2/3}^{\\dagger}D_{1,3}$.
18. Attribute reduction theory of concept lattice based on decision formal contexts
WEI Ling; QI diandun; ZHANG WenXiu
2008-01-01
The theory of concept lattices is an efficient tool for knowledge representation and knowledge discovery,and is applied to many fields successfully.One focus of knowledge discovery is knowledge reduction.Based on the reduction theory of classical formal context,this paper proposes the definition of decision formal context and its reduction theory,which extends the reduction theory of concept lattices.In this paper,strong consistence and weak consistence of decision formal context are defined respectively.For strongly consistent decision formal context,the judgment theorems of consistent sets are examined,and approaches to reduc-tion are given.For weakly consistent decision formal context,implication mapping is defined,and its reduction is studied.Finally,the relation between reducts of weakly consistent decision formal context and reducts of implication mapping is discussed.
19. An improvement of the lattice theory of dislocation for a two-dimensional triangular crystal
Wang Shao-Feng
2005-01-01
The structure of dislocation in a two-dimensional triangular crystal has been studied theoretically on the basis of atomic interaction and lattice statics. The theory presented in this paper is an improvement to that published previously.Within a reasonable interaction approximation, a new dislocation equation is obtained, which remedies a fault existing in the lattice theory of dislocation. A better simplification of non-diagonal terms of the kernel is given. The solution of the new dislocation equation asymptotically becomes the same as that obtained in the elastic theory, and agrees with experimental data. It is found that the solution is formally identical with that proposed phenomenologically by Foreman et al, where the parameter can be chosen freely, but cannot uniquely determined from theory. Indeed, if the parameter in the expression of the solution is selected suitably, the expression can be well applied to describe the fine structure of the dislocation.
20. Lattice calculations for A=3,4,6,12 nuclei using chiral effective field theory
Epelbaum, Evgeny; Lee, Dean; Meißner, Ulf-G
2010-01-01
We present lattice calculations for the ground state energies of tritium, helium-3, helium-4, lithium-6, and carbon-12 nuclei. Our results were previously summarized in a letter publication. This paper provides full details of the calculations. We include isospin-breaking, Coulomb effects, and interactions up to next-to-next-to-leading order in chiral effective field theory.
1. Lattice effective field theory for nuclei from A = 4 to A = 28
Lähde, Timo A; Krebs, Hermann; Lee, Dean; Meißner, Ulf-G; Rupak, Gautam
2013-01-01
We present an overview of the extension of Nuclear Lattice Effective Field Theory simulations to the regime of medium-mass nuclei. We focus on the determination of the ground-state energies of the alpha nuclei $^{16}$O, $^{20}$Ne, $^{24}$Mg and $^{28}$Si by means of Euclidean time projection.
2. Lattice effective field theory calculations for A = 3,4,6,12 nuclei
Epelbaum, Evgeny; Lee, Dean; Meißner, Ulf-G
2009-01-01
We present lattice results for the ground state energies of tritium, helium-3, helium-4, lithium-6, and carbon-12 nuclei. Our analysis includes isospin-breaking, Coulomb effects, and interactions up to next-to-next-to-leading order in chiral effective field theory.
3. Doubled Lattice Chern-Simons-Yang-Mills Theories with Discrete Gauge Group
Caspar, Stephan; Olesen, Therkel Z; Vlasii, Nadiia D; Wiese, Uwe-Jens
2016-01-01
We construct doubled lattice Chern-Simons-Yang-Mills theories with discrete gauge group $G$ in the Hamiltonian formulation. Here, these theories are considered on a square spatial lattice and the fundamental degrees of freedom are defined on pairs of links from the direct lattice and its dual, respectively. This provides a natural lattice construction for topologically-massive gauge theories, which are invariant under parity and time-reversal symmetry. After defining the building blocks of the doubled theories, paying special attention to the realization of gauge transformations on quantum states, we examine the dynamics in the group space of a single cross, which is spanned by a single link and its dual. The dynamics is governed by the single-cross electric Hamiltonian and admits a simple quantum mechanical analogy to the problem of a charged particle moving on a discrete space affected by an abstract electromagnetic potential. Such a particle might accumulate a phase shift equivalent to an Aharonov-Bohm pha...
4. Mathematical Derivation of Chiral Anomaly in Lattice Gauge Theory with Wilson's Action
Hattori, T G; Hattori, Tetsuya; Watanabe, Hiroshi
1998-01-01
Chiral U(1) anomaly is derived with mathematical rigor for a Euclidean fermion coupled to a smooth external U(1) gauge field on an even dimensional torus as a continuum limit of lattice regularized fermion field theory with the Wilson term in the action. The present work rigorously proves for the first time that the Wilson term correctly reproduces the chiral anomaly.
5. National Computational Infrastructure for Lattice Gauge Theory SciDAC-2 Closeout Report Indiana University Component
Gottlieb, Steven Arthur [Indiana University; DeTar, Carleton [University of Utah; Tousaint, Doug [University of Arizona
2014-07-24
This is the closeout report for the Indiana University portion of the National Computational Infrastructure for Lattice Gauge Theory project supported by the United States Department of Energy under the SciDAC program. It includes information about activities at Indian University, the University of Arizona, and the University of Utah, as those three universities coordinated their activities.
6. The Gribov ambiguity for maximal abelian and center gauges in SU(2) lattice gauge theory
Stack, John D.; Tucker, William W
2001-03-01
We present results for the fundamental string tension in SU(2) lattice gauge theory after projection to maximal abelian and direct maximal center gauges. We generate 20 Gribov copies/configuration. Abelian and center projected string tensions slowly decrease as higher values of the gauge functionals are reached.
7. The Spiral and the Lattice: Changes in Cognitive Learning Theory with Implications for Art Education.
Efland, Arthur D.
1995-01-01
Contrasts recent views of learning and cognition with cognitive learning theories of the late 1950s and early 1960s. Maintains that Jerome Bruner's spiral curriculum approach, still valuable, is not sufficient to explain cognitive development. Proposes a lattice-like cognitive development structure, inviting differing paths of exploration. (CFR)
8. Libraries and Development Environments for Monte Carlo Simulations of Lattice Gauge Theories on Parallel Computers
Decker, K. M.; Jayewardena, C.; Rehmann, R.
We describe the library lgtlib, and lgttool, the corresponding development environment for Monte Carlo simulations of lattice gauge theory on multiprocessor vector computers with shared memory. We explain why distributed memory parallel processor (DMPP) architectures are particularly appealing for compute-intensive scientific applications, and introduce the design of a general application and program development environment system for scientific applications on DMPP architectures.
9. Phase structure of (2+1)d strongly coupled lattice gauge theories
Strouthos, C G
2003-01-01
We study the chiral phase transition in (2+1)d strongly coupled U(N) lattice gauge theories with staggered fermions. We show with high precision simulations performed directly in the chiral limit that these models undergo a Berezinski-Kosterlitz-Thouless (BKT) transition. We also show that this universality class is unaffected even in the large N limit.
10. From Doubled Chern-Simons-Maxwell Lattice Gauge Theory to Extensions of the Toric Code
Olesen, T Z; Wiese, U -J
2015-01-01
We regularize compact and non-compact Abelian Chern-Simons-Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group $\\mathbb{R}$, each local Hilbert space is analogous to the one of a charged "particle" moving in the link-pair group space $\\mathbb{R}^2$ in a constant "magnetic" background field. In the compact case, the link-pair group space is a torus $U(1)^2$ threaded by $k$ units of quantized "magnetic" flux, with $k$ being the level of the Chern-Simons theory. The holonomies of the torus $U(1)^2$ give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry...
11. The Spiral and the Lattice: Changes in Cognitive Learning Theory with Implications for Art Education.
Efland, Arthur D.
1995-01-01
Contrasts recent views of learning and cognition with cognitive learning theories of the late 1950s and early 1960s. Maintains that Jerome Bruner's spiral curriculum approach, still valuable, is not sufficient to explain cognitive development. Proposes a lattice-like cognitive development structure, inviting differing paths of exploration. (CFR)
12. On the Path Integral Loop Representation of (2+1) Lattice Non-Abelian Theory
Aroca, J M; Gambini, R
1998-01-01
A gauge invariant Hamiltonian representation for SU(2) in terms of a spin network basis is introduced. The vectors of the spin network basis are independent and the electric part of the Hamiltonian is diagonal in this representation. The corresponding path integral for SU(2) lattice gauge theory is expressed as a sum over colored surfaces, i.e. only involving the $j_p$ attached to the lattice plaquettes. This surfaces may be interpreted as the world sheets of the spin networks In 2+1 dimensions, this can be accomplished by working in a lattice dual to a tetrahedral lattice constructed on a face centered cubic Bravais lattice. On such a lattice, the integral of gauge variables over boundaries or singular lines - which now always bound three coloured surfaces - only contributes when four singular lines intersect at one vertex and can be explicitly computed producing a 6-j or Racah symbol. We performed a strong coupling expansion for the free energy. The convergence of the series expansions is quite different fr...
13. Bloch Waves in Minimal Landau Gauge and the Infinite-Volume Limit of Lattice Gauge Theory
Cucchieri, Attilio; Mendes, Tereza
2017-05-01
By exploiting the similarity between Bloch's theorem for electrons in crystalline solids and the problem of Landau gauge fixing in Yang-Mills theory on a "replicated" lattice, we show that large-volume results can be reproduced by simulations performed on much smaller lattices. This approach, proposed by Zwanziger [Nucl. Phys. B412, 657 (1994), 10.1016/0550-3213(94)90396-4], corresponds to taking the infinite-volume limit for Landau-gauge field configurations in two steps: first for the gauge transformation alone, while keeping the lattice volume finite, and second for the gauge-field configuration itself. The solutions to the gauge-fixing condition are then given in terms of Bloch waves. Applying the method to data from Monte Carlo simulations of pure SU(2) gauge theory in two and three space-time dimensions, we are able to evaluate the Landau-gauge gluon propagator for lattices of linear extent up to 16 times larger than that of the simulated lattice. This approach is reminiscent of the Fisher-Ruelle construction of the thermodynamic limit in classical statistical mechanics.
14. Phase structure of pure SU(3) lattice gauge theory in 5-dimensions
Itou, Etsuko; Nakamoto, Norihiro
2014-01-01
We investigate the nonperturbative phase structure of five-dimensional SU(3) pure Yang-Mills theory on the lattice. We perform numerical simulations using the Wilson plaquette gauge action on an anisotropic lattice with a four-dimensional lattice spacing (a4) and with an independent value in the fifth dimension (a5). We investigate both cases of a4 > a5 and a4 < a5. The Polyakov loops in the fourth and the fifth directions are observed, and we find that there are four possible phases for the anisotropic five-dimensional quenched QCD theory on the lattice. We determine the critical values of the lattice bare coupling and the anisotropic parameter for each phase transition. Furthermore, we find that there is novel meta-stable vacuum, where the global gauge symmetry would be spontaneously broken. It appears only in the phase where the center symmetry in four dimensions is preserved while the symmetry in the fifth dimension is spontaneously broken.
15. Lattice model of linear telechelic polymer melts. I. Inclusion of chain semiflexibility in the lattice cluster theory
Xu, Wen-Sheng, E-mail: [email protected] [James Franck Institute, The University of Chicago, Chicago, Illinois 60637 (United States); Freed, Karl F., E-mail: [email protected] [James Franck Institute, The University of Chicago, Chicago, Illinois 60637 (United States); Department of Chemistry, The University of Chicago, Chicago, Illinois 60637 (United States)
2015-07-14
The lattice cluster theory (LCT) for the thermodynamics of polymer systems has recently been reformulated to treat strongly interacting self-assembling polymers composed of fully flexible linear telechelic chains [J. Dudowicz and K. F. Freed, J. Chem. Phys. 136, 064902 (2012)]. Here, we further extend the LCT for linear telechelic polymer melts to include a description of chain semiflexibility, which is treated by introducing a bending energy penalty whenever a pair of consecutive bonds from a single chain lies along orthogonal directions. An analytical expression for the Helmholtz free energy is derived for the model of semiflexible linear telechelic polymer melts. The extension provides a theoretical tool for investigating the influence of chain stiffness on the thermodynamics of self-assembling telechelic polymers, and for further exploring the influence of self-assembly on glass formation in such systems.
16. Direct evidence for a Coulombic phase in monopole-suppressed SU(2) lattice gauge theory
2013-11-21
Further evidence is presented for the existence of a non-confining phase at weak coupling in SU(2) lattice gauge theory. Using Monte Carlo simulations with the standard Wilson action, gauge-invariant SO(3)–Z2 monopoles, which are strong-coupling lattice artifacts, have been seen to undergo a percolation transition exactly at the phase transition previously seen using Coulomb gauge methods, with an infinite lattice critical point near β=3.2. The theory with both Z2 vortices and monopoles and SO(3)–Z2 monopoles eliminated is simulated in the strong-coupling (β=0) limit on lattices up to 60{sup 4}. Here, as in the high-β phase of the Wilson-action theory, finite size scaling shows it spontaneously breaks the remnant symmetry left over after Coulomb gauge fixing. Such a symmetry breaking precludes the potential from having a linear term. The monopole restriction appears to prevent the transition to a confining phase at any β. Direct measurement of the instantaneous Coulomb potential shows a Coulombic form with moderately running coupling possibly approaching an infrared fixed point of α∼1.4. The Coulomb potential is measured to 50 lattice spacings and 2 fm. A short-distance fit to the 2-loop perturbative potential is used to set the scale. High precision at such long distances is made possible through the use of open boundary conditions, which was previously found to cut random and systematic errors of the Coulomb gauge fixing procedure dramatically. The Coulomb potential agrees with the gauge-invariant interquark potential measured with smeared Wilson loops on periodic lattices as far as the latter can be practically measured with similar statistics data.
17. Direct evidence for a Coulombic phase in monopole-suppressed SU(2) lattice gauge theory
2013-11-01
Further evidence is presented for the existence of a non-confining phase at weak coupling in SU(2) lattice gauge theory. Using Monte Carlo simulations with the standard Wilson action, gauge-invariant SO(3)-Z2 monopoles, which are strong-coupling lattice artifacts, have been seen to undergo a percolation transition exactly at the phase transition previously seen using Coulomb gauge methods, with an infinite lattice critical point near β=3.2. The theory with both Z2 vortices and monopoles and SO(3)-Z2 monopoles eliminated is simulated in the strong-coupling (β=0) limit on lattices up to 604. Here, as in the high-β phase of the Wilson-action theory, finite size scaling shows it spontaneously breaks the remnant symmetry left over after Coulomb gauge fixing. Such a symmetry breaking precludes the potential from having a linear term. The monopole restriction appears to prevent the transition to a confining phase at any β. Direct measurement of the instantaneous Coulomb potential shows a Coulombic form with moderately running coupling possibly approaching an infrared fixed point of α˜1.4. The Coulomb potential is measured to 50 lattice spacings and 2 fm. A short-distance fit to the 2-loop perturbative potential is used to set the scale. High precision at such long distances is made possible through the use of open boundary conditions, which was previously found to cut random and systematic errors of the Coulomb gauge fixing procedure dramatically. The Coulomb potential agrees with the gauge-invariant interquark potential measured with smeared Wilson loops on periodic lattices as far as the latter can be practically measured with similar statistics data.
18. Non-Abelian Lattice Gauge Theories in Superconducting Circuits
Mezzacapo, A; Sabín, C; Egusquiza, I L; Lamata, L; Solano, E
2015-01-01
We propose a digital quantum simulator of non-Abelian pure-gauge models with a superconducting circuit setup. Within the framework of quantum link models, we build a minimal instance of a pure $SU(2)$ gauge theory, using triangular plaquettes involving geometric frustration. This realization is the least demanding, in terms of quantum simulation resources, of a non-Abelian gauge dynamics. We present two superconducting architectures that can host the quantum simulation, estimating the requirements needed to run possible experiments. The proposal establishes a path to the experimental simulation of non-Abelian physics with solid-state quantum platforms.
19. Lattice gauge theory and gluon color-confinement in curved spacetime
Villegas, Kristian Hauser
2014-01-01
The lattice gauge theory for curved spacetime is formulated. A discretized action is derived for both gluon and quark fields which reduces to the generally covariant form in the continuum limit. Using the Wilson action, it is shown analytically that for a general curved spacetime background, two propagating gluons are always color-confined. The fermion-doubling problem is discussed in the specific case of Friedman-Robertson-Walker metric. Lastly, we discussed possible future numerical implementation of lattice QCD in curved spacetime.
20. Decorated tensor network renormalization for lattice gauge theories and spin foam models
Dittrich, Bianca; Steinhaus, Sebastian
2014-01-01
Tensor network techniques have proved to be powerful tools that can be employed to explore the large scale dynamics of lattice systems. Nonetheless, the redundancy of degrees of freedom in lattice gauge theories (and related models) poses a challenge for standard tensor network algorithms. We accommodate for such systems by introducing an additional structure decorating the tensor network. This allows to explicitly preserve the gauge symmetry of the system under coarse graining and straightforwardly interpret the fixed point tensors. Using this novel information encoded in the decoration might eventually lead to new methods incorporating both analytical and numerical techniques.
1. Renormalisation of composite operators in lattice perturbation theory with clover fermions. Non-forward matrix elements
Goeckeler, M.; Schaefer, A. [Regensburg Univ. (Germany). Inst. fuer Physik 1 - Theoretische Physik; Horsley, R. [Edinburgh Univ. (United Kingdom). School of Physics; Perlt, H.; Schiller, A. [Leipzig Univ. (Germany). Inst. fuer Theoretische Physik; Rakow, P.E.L. [Liverpool Univ. (United Kingdom). Theoretical Physics Division, Dept. of Mathematical Sciences; Schierholz, G. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC]|[Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2006-06-15
We consider the renormalisation of lattice QCD operators with one and two covariant derivatives related to the first and second moments of generalised parton distributions and meson distribution amplitudes. Employing the clover fermion action we calculate their non-forward quark matrix elements in one-loop lattice perturbation theory. For some representations of the hypercubic group commonly used in simulations we determine the sets of all possible mixing operators and compute the matrices of renormalisation factors in one-loop approximation. We describe how tadpole improvement is applied to the results. (Orig.)
2. Two loop computation of a running coupling in lattice Yang-Mills theory
Narayanan, R A; Narayanan, Rajamani; Wolff, Ulli
1995-01-01
We compute the two loop coefficient in the relation between the lattice bare coupling and the running coupling defined through the Schroedinger functional for the case of pure SU(2) gauge theory. This result is needed as one computational component to relate the latter to the MSbar-coupling, and it allows us to implement O(a) improvement of the Schroedinger functional to two-loop order. In addition, the two-loop beta-function is verified in a perturbative computation on the lattice, and the behavior of an improved bare coupling is investigated beyond one loop.
3. Low-lying Dirac operator eigenvalues, lattice effects and random matrix theory
Heller, Urs M
2011-01-01
Recently, random matrix theory predictions for the distribution of low-lying Dirac operator eigenvalues have been extended to include lattice effects for both staggered and Wilson fermions. We computed low-lying eigenvalues for the Hermitian Wilson-Dirac operator and for improved staggered fermions on several quenched ensembles with size $\\approx 1.5$ fm. Comparisons to the expectations from RMT with lattice effects included are made. Wilson RMT describes our Wilson data nicely. For improved staggered fermions we find strong indications that taste breaking effects on the low-lying spectrum disappear in the continuum limit, as expected from staggered RMT.
4. Lattice QCD in the {epsilon}-regime and random matrix theory
Giusti, L.; Luescher, M. [CERN, Geneva (Switzerland); Weisz, P. [Max-Planck-Institut fuer Physik, Muenchen (Germany); Wittig, H. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2003-11-01
In the {epsilon}-regime of QCD the main features of the spectrum of the low-lying eigenvalues of the (euclidean) Dirac operator are expected to be described by a certain universality class of random matrix models. In particular, the latter predict the joint statistical distribution of the individual eigenvalues in any topological sector of the theory. We compare some of these predictions with high-precision numerical data obtained from quenched lattice QCD for a range of lattice spacings and volumes. While no complete matching is observed, the results agree with theoretical expectations at volumes larger than about 5 fm{sup 4}. (orig.)
5. Lattice QCD in the {epsilon}-regime and random matrix theory
Giusti, Leonardo; Luescher, Martin [CERN, Theory Division, Geneva (Switzerland)]. E-mail addresses: [email protected]; [email protected]; Weisz, Peter [Max-Planck-Institut fuer Physik, Munich (Germany)]. E-mail: [email protected]; Wittig, Hartmut [DESY, Theory Group, Hamburg (Germany)]. E-mail: [email protected]
2003-11-01
In the {epsilon}-regime of QCD the main features of the spectrum of the low-lying eigenvalues of the (euclidean) Dirac operator are expected to be described by a certain universality class of random matrix models. In particular, the latter predict the joint statistical distribution of the individual eigenvalues in any topological sector of the theory. We compare some of these predictions with high-precision numerical data obtained from quenched lattice QCD for a range of lattice spacings and volumes. While no complete matching is observed, the results agree with theoretical expectations at volumes larger than about 5 fm{sup 4}. (author)
6. Non-renormalization theorem in a lattice supersymmetric theory and the cyclic Leibniz rule
Kato, Mitsuhiro; So, Hiroto
2016-01-01
N=4 supersymmetric quantum mechanical model is formulated on the lattice. Two supercharges, among four, are exactly conserved with the help of the cyclic Leibniz rule without spoiling the locality. In use of the cohomological argument, any possible local terms of the effective action are classified into two categories which we call type-I and type-II, analogous to the D- and F-terms in the supersymmetric field theories. We prove non-renormalization theorem on the type-II terms which include mass and interaction terms with keeping a lattice constant finite, while type-I terms such as the kinetic terms have nontrivial quantum corrections.
7. Direct evidence for a Coulombic phase in monopole-suppressed SU(2) lattice gauge theory
2013-01-01
Further evidence is presented for the existence of a non-confining phase at weak coupling in SU(2) lattice gauge theory. Using Monte Carlo simulations with the standard Wilson action, gauge-invariant SO(3)-Z2 monopoles, which are strong-coupling lattice artifacts, have been seen to undergo a percolation transition exactly at the phase transition previously seen using Coulomb-gauge methods, with an infinite lattice critical point near $\\beta = 3.2$. The theory with both Z2 vortices and monopoles and SO(3)-Z2 monopoles eliminated is simulated in the strong coupling ($\\beta = 0$) limit on lattices up to $60^4$. Here, as in the high-$\\beta$ phase of the Wilson action theory, finite size scaling shows it spontaneously breaks the remnant symmetry left over after Coulomb gauge fixing. Such a symmetry breaking precludes the potential from having a linear term. The monopole restriction appears to prevent the transition to a confining phase at any $\\beta$. Direct measurement of the instantaneous Coulomb potential shows...
8. Entanglement entropy for pure gauge theories in 1+1 dimensions using the lattice regularization
Aoki, Sinya; Nagata, Keitaro
2016-01-01
We study the entanglement entropy (EE) for pure gauge theories in 1+1 dimensions with the lattice regularization. Using the definition of the EE for lattice gauge theories proposed in a previous paper [1] (S. Aoki, T. Iritani, M. Nozaki, T. Numasawa, N. Shiba and H. Tasaki, JHEP 1506 (2015) 187), we calculate the EE for arbitrary pure as well as mixed states in terms of eigenstates of the transfer matrix in 1+1 dimensional lattice gauge theory. We find that the EE of an arbitrary pure state does not depend on the lattice spacing, thus giving the EE in the continuum limit, and show that the EE for an arbitrary pure state is independent of the real (Minkowski) time evolution. We also explicitly demonstrate the dependence of EE on the gauge fixing at the boundaries between two subspaces, which was pointed out for general cases in the paper [1]. In addition, we calculate the EE at zero as well as finite temperature by the replica method, and show that our result in the continuum limit corresponds to the result ob...
9. The potential of the effective Polyakov line action from the underlying lattice gauge theory
Greensite, Jeff
2012-01-01
I adapt a numerical method, previously applied to investigate the Yang-Mills vacuum wavefunctional, to the problem of extracting the effective Polyakov line action from SU(N) lattice gauge theories, with or without matter fields. The method can be used to find the variation of the effective Polyakov line action along any trajectory in field configuration space; this information is sufficient to determine the potential term in the action, and strongly constrains the possible form of the kinetic term. The technique is illustrated for both pure and gauge-Higgs SU(2) lattice gauge theory at finite temperature. A surprise, in the pure gauge theory, is that the potential of the corresponding Polyakov line action contains a non-analytic (yet center-symmetric) term proportional to |P|^3, where P is the trace of the Polyakov line at a given point, in addition to the expected analytic terms proportional to even powers of P.
10. Theory of the Lattice Boltzmann Equation: Symmetry properties of Discrete Velocity Sets
Rubinstein, Robert; Luo, Li-Shi
2007-01-01
In the lattice Boltzmann equation, continuous particle velocity space is replaced by a finite dimensional discrete set. The number of linearly independent velocity moments in a lattice Boltzmann model cannot exceed the number of discrete velocities. Thus, finite dimensionality introduces linear dependencies among the moments that do not exist in the exact continuous theory. Given a discrete velocity set, it is important to know to exactly what order moments are free of these dependencies. Elementary group theory is applied to the solution of this problem. It is found that by decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group, it becomes relatively straightforward to assess the behavior of moments in the theory. The construction of some standard two- and three-dimensional models is reviewed from this viewpoint, and procedures for constructing some new higher dimensional models are suggested.
11. Graph Theoretic Lattice Mining Based on Formal Concept Analysis (FCA Theory for Text Mining
Hasni Hassan
2017-08-01
Full Text Available The growth of the semantic web has fueled the need to search for information based on the understanding of the intent of the searcher, coupled with the contextual meaning of the keywords supplied by the searcher. The common solution to enhance the searching process includes the deployment of formal concept analysis (FCA theory to extract concepts from a set of text with the use of corresponding domain ontology. However, creating a domain ontology or cross-platform ontology is a tedious and time consuming process that requires validation from domain experts. Therefore, this study proposed an alternative solution called Lattice Mining (LM that utilizes FCA theory and graph theory. This is because the process of matching a query to related documents is similar to the process of graph matching if both the query and the documents are represented using graphs. This study adopted the idea of FCA in the determination of the concepts based on texts and deployed the lattice diagrams obtained from an FCA tool for further analysis using graph theory. The LM technique employed in this study utilized the adjacency matrices obtained from the lattice outputs and performed a distance measure technique to calculate the similarity between two graphs. The process was realized successively via the implementation of three algorithms called the Relatedness Algorithm (RA, the Adjacency Matrix Algorithm (AMA and the Concept-Based Lattice Mining (CBLM Algorithm. A similarity measure between FCA output lattices yielded promising results based on the ranking of the trace values from the matrices. Recognizing the potential of this method, future work includes refinement in the steps of the CBLM algorithm for a more efficient implementation of the process.
12. Doubled lattice Chern-Simons-Yang-Mills theories with discrete gauge group
Caspar, S.; Mesterházy, D.; Olesen, T. Z.; Vlasii, N. D.; Wiese, U.-J.
2016-11-01
We construct doubled lattice Chern-Simons-Yang-Mills theories with discrete gauge group G in the Hamiltonian formulation. Here, these theories are considered on a square spatial lattice and the fundamental degrees of freedom are defined on pairs of links from the direct lattice and its dual, respectively. This provides a natural lattice construction for topologically-massive gauge theories, which are invariant under parity and time-reversal symmetry. After defining the building blocks of the doubled theories, paying special attention to the realization of gauge transformations on quantum states, we examine the dynamics in the group space of a single cross, which is spanned by a single link and its dual. The dynamics is governed by the single-cross electric Hamiltonian and admits a simple quantum mechanical analogy to the problem of a charged particle moving on a discrete space affected by an abstract electromagnetic potential. Such a particle might accumulate a phase shift equivalent to an Aharonov-Bohm phase, which is manifested in the doubled theory in terms of a nontrivial ground-state degeneracy on a single cross. We discuss several examples of these doubled theories with different gauge groups including the cyclic group Z(k) ⊂ U(1) , the symmetric group S3 ⊂ O(2) , the binary dihedral (or quaternion) group D¯2 ⊂ SU(2) , and the finite group Δ(27) ⊂ SU(3) . In each case the spectrum of the single-cross electric Hamiltonian is determined exactly. We examine the nature of the low-lying excited states in the full Hilbert space, and emphasize the role of the center symmetry for the confinement of charges. Whether the investigated doubled models admit a non-Abelian topological state which allows for fault-tolerant quantum computation will be addressed in a future publication.
13. Fundamental measure theory for lattice fluids with hard-core interactions
Lafuente, Luis; Cuesta, José A.
2002-11-01
We present the extension of Rosenfeld's fundamental measure theory to lattice models by constructing a density functional for d-dimensional mixtures of parallel hard hypercubes on a simple hypercubic lattice. The one-dimensional case is exactly solvable and two cases must be distinguished: all the species with the same length parity (additive mixture), and arbitrary length parity (nonadditive mixture). To the best of our knowledge, this is the first time that the latter case has been considered. Based on the one-dimensional exact functional form, we propose the extension to higher dimensions by generalizing the zero-dimensional cavity method to lattice models. This assures the functional will have correct dimensional crossovers to any lower dimension, including the exact zero-dimensional limit. Some applications of the functional to particular systems are also shown.
14. Numerical study of renormalization group flows of nuclear effective field theory without pions on a lattice
Harada, Koji; Yahiro, Masanobu
2016-01-01
We formulate the next-to-leading order nuclear effective field theory without pions in the two-nucleon sector on a spatial lattice, and investigate nonperturbative renormalization group flows in the strong coupling region by diagonalizing the Hamiltonian numerically. The cutoff (proportional to the inverse of the lattice constant) dependence of the coupling constants is obtained by changing the lattice constant with the binding energy and the asymptotic normalization constant for the groundstate being fixed. We argue that the critical line can be obtained by looking at the finite-size dependence of the groundstate energy. We determine the relevant operator and locate the nontrivial fixed point, as well as the physical flow line corresponding to the deuteron in the two-dimensional plane of dimensionless coupling constants. It turns out that the location of the nontrivial fixed point is very close to the one obtained by the corresponding analytic calculation, but the relevant operator is quite different.
15. Generalized Courant-Snyder theory for charged-particle dynamics in general focusing lattices.
Qin, Hong; Davidson, Ronald C; Chung, Moses; Burby, Joshua W
2013-09-06
The Courant-Snyder (CS) theory for one degree of freedom is generalized to the case of coupled transverse dynamics in general linear focusing lattices with quadrupole, skew-quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation of beam energy. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D sympletic rotation. The envelope equation, the transfer matrix, and the CS invariant of the original CS theory all have their counterparts, with remarkably similar expressions, in the generalized theory.
16. On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity
Delcamp, Clement; Riello, Aldo
2016-01-01
Entanglement entropy is a valuable tool for characterizing the correlation structure of quantum field theories. When applied to gauge theories, subtleties arise which prevent the factorization of the Hilbert space underlying the notion of entanglement entropy. Borrowing techniques from extended topological field theories, we introduce a new definition of entanglement entropy for both Abelian and non--Abelian gauge theories. Being based on the notion of excitations, it provides a completely relational way of defining regions. Therefore, it naturally applies to background independent theories, e.g. gravity, by circumventing the difficulty of specifying the position of the entangling surface. We relate our construction to earlier proposals and argue that it brings these closer to each other. In particular, it yields the non--Abelian analogue of the magnetic centre choice', as obtained through an extended--Hilbert--space method, but applied to the recently introduced fusion basis for 3D lattice gauge theories. W...
17. Lattice study for conformal windows of SU(2) and SU(3) gauge theories with fundamental fermions
Huang, Cynthia Y -H; Lin, C.-J. David; Ogawa, Kenji; Ohki, Hiroshi; Ramos, Alberto; Rinaldi, Enrico
2016-01-01
We present our investigation of SU(2) gauge theory with 8 flavours, and SU(3) gauge theory with 12 flavours. For the SU(2) case, at strong bare coupling, $\\beta \\lesssim 1.45$, the distribution of the lowest eigenvalue of the Dirac operator can be described by chiral random matrix theory for the Gaussian symplectic ensemble. Our preliminary result indicates that the chiral phase transition in this theory is of bulk nature. For the SU(3) theory, we use high-precision lattice data to perform the step-scaling study of the coupling, $g_{{\\rm GF}}$, in the Gradient Flow scheme. We carefully examine the reliability of the continuum extrapolation in the analysis, and conclude that the scaling behaviour of this SU(3) theory is not governed by possible infrared conformality at $g_{{\\rm GF}}^{2} \\lesssim 6$.
18. Lattice study for conformal windows of SU(2) and SU(3) gauge theories with fundamental fermions
Huang, Cynthia Y.-H.; Lin, C.-J. David; Ogawa, Kenji; Ohki, Hiroshi; Ramos, Alberto; Rinaldi, Enrico
2015-10-30
We present our investigation of SU(2) gauge theory with 8 flavours, and SU(3) gauge theory with 12 flavours. For the SU(2) case, at strong bare coupling, $\\beta \\lesssim 1.45$, the distribution of the lowest eigenvalue of the Dirac operator can be described by chiral random matrix theory for the Gaussian symplectic ensemble. Our preliminary result indicates that the chiral phase transition in this theory is of bulk nature. For the SU(3) theory, we use high-precision lattice data to perform the step-scaling study of the coupling, $g_{{\\rm GF}}$, in the Gradient Flow scheme. We carefully examine the reliability of the continuum extrapolation in the analysis, and conclude that the scaling behaviour of this SU(3) theory is not governed by possible infrared conformality at $g_{{\\rm GF}}^{2} \\lesssim 6$.
19. Chern-Simons theory for Heisenberg spins on the Kagome Lattice
Kumar, Krishna; Sun, Kai; Fradkin, Eduardo
2015-03-01
We study the problem of Heisenberg spins on the frustrated Kagome lattice using a 2D Jordan-Wigner transformation that maps the spins (hard-core bosons) onto a system of (interacting) fermions coupled to a Chern-Simons gauge field. This mapping requires us to define a discretized version of the Chern-Simons term on the lattice. Using a recently developed result on how to define Chern-Simons theories on a class of planar lattices, we can consistently study spin models beyond the mean-field level and include the effects of fluctuations, which are generally strong in frustrated systems. Here, we apply these results to study magnetization plateau type states on the Kagome lattice in the regime of XY anisotropy. We find that the 1/3 and 2/3 magnetization plateaus are chiral spin liquid states equivalent to a primary Laughlin fractional quantum Hall state of bosons with (spin) Hall conductivity 1/2 1/4 π and semionic excitations. The 5/9 plateau is a chiral spin liquid equivalent to the first Jain descendant. We also consider the spin-1/2 Heisenberg model on the Kagome lattice with a chirality-breaking term on the triangular plaquettes. This situation also leads to a primary Laughlin bosonic fractional quantum Hall type state with filling fraction 1 / 2 .
20. Lattice simulations of QCD-like theories at finite baryon density
Scior, Philipp Friedrich
2016-07-13
The exploration of the phase diagram of quantum chromodynamics (QCD) is of great importance to describe e.g. the properties of neutron stars or heavy-ion collisions. Due to the sign problem of lattice QCD at finite chemical potential we need effective theories to study QCD at finite density. Here, we use a three-dimensional Polyakov-loop theory to study the phase diagrams of QCD-like theories. In particular, we investigate the heavy quark limit of the QCD-like theories where the effective theory can be derived from the full theory by a combined strong coupling and hopping expansion. This expansion can be systematically improved order by order. Since there is no sign problem for the QCD-like theories we consider, we can compare our results to data from lattice calculations of the full theories to make qualitative and quantitative statements of the effective theory's validity. We start by deriving the effective theory up to next-to-next-to leading-order, in particular for two-color and G{sub 2}-QCD where replace the three colors in QCD with only two colors or respectively replace the gauge group SU(3) of QCD with G{sub 2}. We will then apply the effective theory at finite temperature mainly to test the theory and the implementation but also to make some predictions for the deconfinement phase transition in G{sub 2} Yang-Mills theory. Finally, we turn our attention to the cold and dense regime of the phase diagram where we observe a sharp increase of the baryon density with the quark chemical potential μ, when μ reaches half the diquark mass. At vanishing temperature this is expected to happen in a quantum phase transition with Bose-Einstein-condensation of diquarks. In contrast to the liquid-gas transition in QCD, the phase transition to the Bose-Einstein condensate is continuous. We find evidence that the effective theories for heavy quarks are able to describe the qualitative difference between first and second order phase transitions. For even higher μ we
1. Beyond Flory theory: Distribution functions for interacting lattice trees
Rosa, Angelo; Everaers, Ralf
2017-01-01
While Flory theories [J. Isaacson and T. C. Lubensky, J. Physique Lett. 41, 469 (1980), 10.1051/jphyslet:019800041019046900; M. Daoud and J. F. Joanny, J. Physique 42, 1359 (1981), 10.1051/jphys:0198100420100135900; A. M. Gutin et al., Macromolecules 26, 1293 (1993), 10.1021/ma00058a016] provide an extremely useful framework for understanding the behavior of interacting, randomly branching polymers, the approach is inherently limited. Here we use a combination of scaling arguments and computer simulations to go beyond a Gaussian description. We analyze distribution functions for a wide variety of quantities characterizing the tree connectivities and conformations for the four different statistical ensembles, which we have studied numerically in [A. Rosa and R. Everaers, J. Phys. A: Math. Theor. 49, 345001 (2016), 10.1088/1751-8113/49/34/345001 and J. Chem. Phys. 145, 164906 (2016), 10.1063/1.4965827]: (a) ideal randomly branching polymers, (b) 2 d and 3 d melts of interacting randomly branching polymers, (c) 3 d self-avoiding trees with annealed connectivity, and (d) 3 d self-avoiding trees with quenched ideal connectivity. In particular, we investigate the distributions (i) pN(n ) of the weight, n , of branches cut from trees of mass N by severing randomly chosen bonds; (ii) pN(l ) of the contour distances, l , between monomers; (iii) pN(r ⃗) of spatial distances, r ⃗, between monomers, and (iv) pN(r ⃗|l ) of the end-to-end distance of paths of length l . Data for different tree sizes superimpose, when expressed as functions of suitably rescaled observables x ⃗=r ⃗/√{ } or x =l / . In particular, we observe a generalized Kramers relation for the branch weight distributions (i) and find that all the other distributions (ii-iv) are of Redner-des Cloizeaux type, q (x ⃗) =C |x| θexp(-(K|x |) t) . We propose a coherent framework, including generalized Fisher-Pincus relations, relating most of the RdC exponents to each other and to the contact and Flory
2. Two topics in nonperturbative lattice field theories: The U(1) quantum link model and perfect actions for scalar theories
Tsapalis, Antonios S.
This thesis deals with two topics in lattice field theories. In the first part we discuss aspects of renormalization group flow and non-perturbative improvement of actions for scalar theories regularized on a lattice. We construct a perfect action, an action which is free of lattice artifacts, for a given theory. It is shown how a good approximation to the perfect action-referred to as classically perfect-can be constructed based on a well-defined blocking scheme for the O(3) non-linear σ-model. We study the O(N) non- linear σ-model in the large-N limit and derive analytically its perfect action. This action is applied to the O(3) model on a square lattice. The Wolff cluster algorithm is used to simulate numerically the system. We perform scaling tests and discuss the scaling properties of the large- N inspired perfect action as opposed to the standard and the classically perfect action. In the second part we present a new formulation for a quantum field theory with Abelian gauge symmetry. A Hamiltonian is constructed on a four-dimensional Euclidean space-time lattice which is invariant under local transformations. The model is formulated as a 5- dimensional path integral of discrete variables. We argue that dimensional reduction will allow us to study the behavior of the standard compact U(1) gauge theory in 4-d. Based on the idea of the loop- cluster algorithm for quantum spins, we present the construction of a flux-cluster algorithm for the U(1) quantum link model for the spin-1/2 quantization of the electric flux. It is shown how improved estimators for Wilson loop expectation values can be defined. This is important because the Wilson loops are traditionally used to identify confining and Coulomb phases in gauge theories. Our study indicates that the spin-1/2 U(1) quantum link model is strongly coupled for all bare coupling values we examined. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)
3. Novel Approach to Super Yang-Mills Theory on Lattice - Exact fermionic symmetry and "Ichimatsu" pattern -
Itoh, K; Sawanaka, H; So, H; Ukita, N
2003-01-01
We present a lattice theory with an exact fermionic symmetry, which mixes the link and the fermionic variables. The staggered fermionic variables may be reconstructed into a Majorana fermion in the continuum limit. The gauge action has a novel structure. Though it is the ordinary plaquette action, two different couplings are assigned in the Ichimatsu pattern'' or the checkered pattern. In the naive continuum limit, the fermionic symmetry survives as a continuum (or an $O(a^0)$) symmetry. The transformation of the fermion is proportional to the field strength multiplied by the difference of the two gauge couplings in this limit. This work is an extension of our recently proposed cell model toward the realization of supersymmetric Yang-Mills theory on lattice.
4. Heavy Quark Thermalization in Classical Lattice Gauge Theory Lessons for Strongly-Coupled QCD
Laine, Mikko; Philipsen, Owe; Tassler, Marcus
2009-01-01
Thermalization of a heavy quark near rest is controlled by the correlator of two electric fields along a temporal Wilson line. We address this correlator within real-time, classical lattice Yang-Mills theory, and elaborate on the analogies that exist with the dynamics of hot QCD. In the weak-coupling limit, it can be shown analytically that the dynamics on the two sides are closely related to each other. For intermediate couplings, we carry out non-perturbative simulations within the classical theory, showing that the leading term in the weak-coupling expansion significantly underestimates the heavy quark thermalization rate. Our analytic and numerical results also yield a general understanding concerning the overall shape of the spectral function corresponding to the electric field correlator, which may be helpful in subsequent efforts to reconstruct it from Euclidean lattice Monte Carlo simulations.
5. The Nc dependencies of baryon masses: Analysis with Lattice QCD and Effective Theory
Calle Cordon, Alvaro C. [JLAB; DeGrand, Thomas A. [University of Colorado; Goity, Jose L. [JLAB
2014-07-01
Baryon masses at varying values of Nc and light quark masses are studied with Lattice QCD and the results are analyzed in a low energy effective theory based on a combined framework of the 1/Nc and Heavy Baryon Chiral Perturbation Theory expansions. Lattice QCD results for Nc=3, 5 and 7 obtained in quenched calculations, as well as results for unquenched calculations for Nc=3, are used for the analysis. The results are consistent with a previous analysis of Nc=3 LQCD results, and in addition permit the determination of sub-leading in 1/Nc effects in the spin-flavor singlet component of the baryon masses as well as in the hyperfine splittings.
6. Lattice gluon and ghost propagators and the strong coupling in pure SU(3) Yang-Mills theory: Finite lattice spacing and volume effects
Duarte, Anthony G.; Oliveira, Orlando; Silva, Paulo J.
2016-07-01
The dependence of the Landau gauge two-point gluon and ghost correlation functions on the lattice spacing and on the physical volume are investigated for pure SU(3) Yang-Mills theory in four dimensions using lattice simulations. We present data from very large lattices up to 1284 and for two lattice spacings 0.10 fm and 0.06 fm corresponding to volumes of ˜(13 fm )4 and ˜(8 fm )4 , respectively. Our results show that, for sufficiently large physical volumes, both propagators have a mild dependence on the lattice volume. On the other hand, the gluon and ghost propagators change with the lattice spacing a in the infrared region, with the gluon propagator having a stronger dependence on a compared to the ghost propagator. In what concerns the strong coupling constant αs(p2), as defined from gluon and ghost two-point functions, the simulations show a sizeable dependence on the lattice spacing for the infrared region and for momenta up to ˜1 GeV .
7. Lattice Gluon and Ghost Propagators, and the Strong Coupling in Pure SU(3) Yang-Mills Theory: Finite Lattice Spacing and Volume Effects
Duarte, Anthony G; Silva, Paulo J
2016-01-01
The dependence of the Landau gauge two point gluon and ghost correlation functions on the lattice spacing and on the physical volume are investigated for pure SU(3) Yang-Mills theory in four dimensions using lattice simulations. We present data from very large lattices up to $128^4$ and for two lattice spacings $0.10$ fm and $0.06$ fm corresponding to volumes of $\\sim$ (13 fm)$^4$ and $\\sim$ (8 fm)$^4$, respectively. Our results show that, for sufficiently large physical volumes, both propagators have a mild dependence on the lattice volume. On the other hand, the gluon and ghost propagators change with the lattice spacing $a$ in the infrared region, with the gluon propagator having a stronger dependence on $a$ compared to the ghost propagator. In what concerns the strong coupling constant $\\alpha_s (p^2)$, as defined from gluon and ghost two point functions, the simulations show a sizeable dependence on the lattice spacing for the infrared region and for momenta up to $\\sim 1$ GeV.
8. Z2 gauge theory for valence bond solids on the kagome lattice
Hwang, Kyusung; Huh, Yejin; Kim, Yong Baek
We present an effective Z2 gauge theory that captures various competing phases in spin-1/2 kagome lattice antiferromagnets: the topological Z2 spin liquid (SL) phase, and the 12-site and 36- site valence bond solid (VBS) phases. Our effective theory is a generalization of the recent Z2 gauge theory proposed for SL phases by Wan and Tchernyshyov. In particular, we investigate possible VBS phases that arise from vison condensations in the SL. In addition to the 12-site and 36-site VBS phases, there exists 6-site VBS that is closely related to the symmetry-breaking valence bond modulation patterns observed in the recent density matrix renormalization group simulations. We find that our results have remarkable consistency with a previous study using a different Z2 gauge theory. Motivated by the lattice geometry in the recently reported vanadium oxyfluoride kagome antiferromagnet, our gauge theory is extended to incorporate lowered symmetry by inequivalent up- and down-triangles. We investigate effects of this anisotropy on the 12-site, 36-site, and 6-site VBS phases. Particularly, interesting dimer melting effects are found in the 36-site VBS. We discuss the implications of our findings and also compare the results with a different type of Z2 gauge theory used in previous studies.
9. From lattice BF gauge theory to area-angle Regge calculus
Bonzom, Valentin
2009-01-01
We consider Riemannian 4d BF lattice gauge theory, on a triangulation of spacetime. Introducing the simplicity constraints which turn BF theory into simplicial gravity, some geometric quantities of Regge calculus, areas, and 3d and 4d dihedral angles, are identified. The parallel transport conditions are taken care of to ensure a consistent gluing of simplices. We show that these gluing relations, together with the simplicity constraints, contain the constraints of area-angle Regge calculus in a simple way, via the group structure of the underlying BF gauge theory. This provides a precise road from constrained BF theory to area-angle Regge calculus. Doing so, a framework combining variables of lattice BF theory and Regge calculus is built. The action takes a form {\\it \\a la Regge} and includes the contribution of the Immirzi parameter. In the absence of simplicity constraints, the standard spin foam model for BF theory is recovered. Insertions of local observables are investigated, leading to Casimir inserti...
10. θ dependence of the vacuum energy in SU(3) gauge theory from the lattice
Giusti, Leonardo; Petrarca, Silvano; Taglienti, Bruno
2007-11-01
We report on a precise computation of the topological charge distribution in the SU(3) Yang-Mills theory. It is carried out on the lattice with high statistics Monte Carlo simulations by employing the definition of the topological charge suggested by Neuberger’s fermions. We observe significant deviations from a Gaussian distribution. Our results disfavor the θ behavior of the vacuum energy predicted by dilute instanton models, while they are compatible with the expectation from the large Nc expansion.
11. Theta dependence of the vacuum energy in the SU(3) gauge theory from the lattice
Giusti, Leonardo; Taglienti, B
2007-01-01
We report on a precise computation of the topological charge distribution in the SU(3) Yang--Mills theory. It is carried out on the lattice with high statistics Monte Carlo simulations by employing the definition of the topological charge suggested by Neuberger's fermions. We observe significant deviations from a Gaussian distribution. Our results disfavour the theta behaviour of the vacuum energy predicted by instanton models, while they are compatible with the expectation from the large Nc expansion.
12. String Representation of the Abelian Higgs Theory and Aharonov-Bohm Effect on the Lattice
Polikarpov, M I; Zubkov, M A
1993-01-01
The partition function of the $4D$ lattice Abelian Higgs theory is represented as the sum over world sheets of Nielsen--Olesen strings. The creation and annihilation operators of the strings are constructed. The topological long--range interaction of the strings and charged particles is shown to exist; it is proportional to the linking number of the string world sheet and particle world trajectory.
13. Higgs-Yukawa model in chirally-invariant lattice field theory
Bulava, John [CERN, Geneva (Switzerland). Physics Department; Gerhold, Philipp; Kallarackal, Jim; Nagy, Attila [Humboldt Univ. Berlin (Germany). Inst. fuer Physik; Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Jansen, Karl [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Knippschild, Bastian [National Taiwan Univ., Taipei (China). Dept. of Physics; Lin, C.J. David [National Chiao-Tung Univ., Hsinchu (China). Inst. of Physics; National Centre for Theoretical Sciences, Hsinchu (China). Div. of Physics; Nagai, Kei-Ichi [Nagoya Univ., Nagoya, Aichi (Japan). Kobayashi-Maskawa Institute; Ogawa, Kenji [Chung-Yuan Christian Univ., Chung-Li (China). Dept. of Physics
2012-10-15
Non-perturbative numerical lattice studies of the Higgs-Yukawa sector of the standard model with exact chiral symmetry are reviewed. In particular, we discuss bounds on the Higgs boson mass at the standard model top quark mass, and in the presence of heavy fermions. We present a comprehensive study of the phase structure of the theory at weak and very strong values of the Yukawa coupling as well as at non-zero temperature.
14. Density of states techniques for lattice field theories using the functional fit approach (FFA)
Gattringer, Christof; Lehmann, Alexander; Törek, Pascal
2015-01-01
We discuss a variant of density of states (DoS) techniques for lattice field theories, the so-called "functional fit approach" (FFA). The DoS FFA is based on a density of states rho(x) which is parameterized on small intervals of the argument x of rho(x). On these intervals restricted Monte Carlo simulations with an additional Boltzmann factor exp(lambda x) allow to determine rho(x) very precisely by obtaining its parameters from fitting the Monte Carlo data to a known function of lambda. We describe the method in detail and show its applicability in four different systems, three of which have a complex action problem: The SU(3) spin model with a chemical potential, U(1) lattice gauge theory, the Z(3) spin model with chemical potential, and 2-dimensional U(1) lattice gauge theory with a topological term. In all cases we compare to reference calculations, which partly were done in a dual formulation where the complex action problem is absent. In all four cases we find a very encouraging performance of the DoS ...
15. Area-preserving diffeomorphisms in gauge theory on a non-commutative plane. A lattice study
Bietenholz, W. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Bigarini, A. [Univ. degli Studi di Perugia (Italy). Dipt. di Fisica]|[INFN, Sezione di Perugia (Italy)]|[Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik; Torrielli, A. [Massachusetts Institute of Technology, Cambridge, MA (United States). Center for Theoretical Physics, Lab. for Nuclear Sciences
2007-06-15
We consider Yang-Mills theory with the U(1) gauge group on a non-commutative plane. Perturbatively it was observed that the invariance of this theory under area-preserving diffeomorphisms (APDs) breaks down to a rigid subgroup SL(2,R). Here we present explicit results for the APD symmetry breaking at finite gauge coupling and finite non-commutativity. They are based on lattice simulations and measurements of Wilson loops with the same area but with a variety of different shapes. Our results confirm the expected loss of invariance under APDs. Moreover, they strongly suggest that non-perturbatively the SL(2,R) symmetry does not persist either. (orig.)
16. A first look at quasi-Monte Carlo for lattice field theory problems
Jansen, K; Nube, A; Griewank, A; Mueller-Preussker, M
2012-01-01
In this project we initiate an investigation of the applicability of Quasi-Monte Carlo methods to lattice field theories in order to improve the asymptotic error behavior of observables for such theories. In most cases the error of an observable calculated by averaging over random observations generated from an ordinary Monte Carlo simulation behaves like 1/sqrt(N), where N is the number of observations. By means of Quasi-Monte Carlo methods it is possible to improve this behavior for certain problems to up to 1/N. We adapted and applied this approach to simple systems like the quantum harmonic and anharmonic oscillator and verified an improved error scaling.
17. New lessons from the nucleon mass, lattice QCD and heavy baryon chiral perturbation theory
Walker-Loud, A
2008-01-01
I will review heavy baryon chiral perturbation theory for the nucleon delta degrees of freedom and then examine the recent dynamical lattice calculations of the nucleon mass from the BMW, ETM, JLQCD, LHP, MILC, NPLQCD, PACS-CS, QCDSF/UKQCD and RBC/UKQCD Collaborations. Performing the chiral extrapolations of these results, one finds remarkable agreement with the physical nucleon mass, from each lattice data set. However, a careful examination of the lattice data and the resulting extrapolation functions reveals some unexpected results, serving to highlight the significant challenges in performing chiral extrapolations of baryon quantities. All the N_f=2+1 dynamical results can be quantitatively described by theoretically unmotivated fit function linear in the pion mass with m_pi ~ 750 -190 MeV. When extrapolated to the physical point, the results are in striking agreement with the physical nucleon mass. I will argue that knowledge of each lattice datum of the nucleon mass is required at the 1-2% level, includ...
18. Coulomb-gauge ghost and gluon propagators in SU(3) lattice Yang-Mills theory
Nakagawa, Y.; Voigt, A.; Ilgenfritz, E.-M.; Müller-Preussker, M.; Nakamura, A.; Saito, T.; Sternbeck, A.; Toki, H.
2009-06-01
We study the momentum dependence of the ghost propagator and of the space and time components of the gluon propagator at equal time in pure SU(3) lattice Coulomb-gauge theory carrying out a joint analysis of data collected independently at the Research Center for Nuclear Physics, Osaka and Humboldt University, Berlin. We focus on the scaling behavior of these propagators at β=5.8,…,6.2 and apply a matching technique to relate the data for the different lattice cutoffs. Thereby, lattice artifacts are found to be rather strong for both instantaneous gluon propagators at a large momentum. As a byproduct we obtain the respective lattice scale dependences a(β) for the transversal gluon and the ghost propagator which indeed run faster with β than two-loop running, but slightly slower than what is known from the Necco-Sommer analysis of the heavy quark potential. The abnormal a(β) dependence as determined from the instantaneous time-time gluon propagator, D44, remains a problem, though. The role of residual gauge-fixing influencing D44 is discussed.
19. Coulomb-gauge ghost and gluon propagators in SU(3) lattice Yang-Mills theory
Nakagawa, Y; Ilgenfritz, E -M; Müller-Preussker, M; Nakamura, A; Saitô, T; Sternbeck, A; Toki, H
2009-01-01
We study the momentum dependence of the ghost propagator and of the space and time components of the gluon propagator at equal time in pure SU(3) lattice Coulomb gauge theory carrying out a joint analysis of data collected independently at RCNP Osaka and Humboldt University Berlin. We focus on the scaling behavior of these propagators at beta=5.8,...,6.2 and apply a matching technique to relate the data for the different lattice cutoffs. Thereby, lattice artifacts are found to be rather strong for both instantaneous gluon propagators at large momentum. As a byproduct we obtain the respective lattice scale dependences a(beta) for the transversal gluon and the ghost propagator which indeed run faster with beta than two-loop running, but slightly slower than what is known from the Necco-Sommer analysis of the heavy quark potential. The abnormal a(beta) dependence as determined from the instantaneous time-time gluon propagator, D_{44}, remains a problem, though. The role of residual gauge-fixing influencing D_{44...
20. Four-loop lattice-regularized vacuum energy density of the three-dimensional SU(3) + adjoint Higgs theory
Di Renzo, F; Schröder, Y; Torrero, C
2008-01-01
The pressure of QCD admits at high temperatures a factorization into purely perturbative contributions from hard'' thermal momenta, and slowly convergent as well as non-perturbative contributions from soft'' thermal momenta. The latter can be related to various effective gluon condensates in a dimensionally reduced effective field theory, and measured there through lattice simulations. Practical measurements of one of the relevant condensates have suffered, however, from difficulties in extrapolating convincingly to the continuum limit. In order to gain insight on this problem, we employ Numerical Stochastic Perturbation Theory to estimate the problematic condensate up to 4-loop order in lattice perturbation theory. Our results seem to confirm the presence of large'' disretization effects, going like $a\\ln(1/a)$, where $a$ is the lattice spacing. For definite conclusions, however, it would be helpful to repeat the corresponding part of our study with standard lattice perturbation theory techniques.
1. Real-time dynamics of lattice gauge theories with a few-qubit quantum computer
Martinez, E A; Schindler, P; Nigg, D; Erhard, A; Heyl, M; Hauke, P; Dalmonte, M; Monz, T; Zoller, P; Blatt, R
2016-01-01
Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. In the spirit of Feynman's vision of a quantum simulator, this has recently stimulated theoretical effort to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented. Here we report the first experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realising 1+1-dimensional quantum electrodynamics (Schwinger model) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism, describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron-posi...
2. Effective lattice Polyakov loop theory vs. full SU(3) Yang-Mills at finite temperature
Bergner, Georg; Philipsen, Owe
2013-01-01
A three-dimensional effective theory of Polyakov loops has recently been derived from Wilson's Yang-Mills lattice action by means of a strong coupling expansion. It is valid in the confined phase up to the deconfinement phase transition, for which it predicts the correct order and gives quantitative estimates for the critical coupling. In this work we study its predictive power for further observables like correlation functions and the equation of state. We find that the effective theory correctly reproduces qualitative features and symmetries of the full theory as the continuum is approached. Regarding quantitative predictions, we identify two classes of observables by numerical comparison as well as analytic calculations: correlation functions and their associated mass scales cannot be described accurately from a truncated effective theory, due to its inherently non-local nature involving long-range couplings. On the other hand, phase transitions and bulk thermodynamic quantities are accurately reproduced b...
3. The properties of nuclear matter with lattice $NN$ potential in relativistic Brueckner-Hartree-Fock theory
Hu, Jinniu; Shen, Hong
2016-01-01
We study the properties of nuclear matter with lattice nucleon-nucleon ($NN$) potential in the relativistic Brueckner-Hartree-Fock (RBHF) theory. To use this potential in such a microscopic many-body theory, we firstly have to construct a one-boson-exchange potential (OBEP) based on the latest lattice $NN$ potential. Three mesons, pion, $\\sigma$ meson, and $\\omega$ meson, are considered. Their coupling constants and cut-off momenta are determined by fitting the on-shell behaviors and phase shifts of the lattice force, respectively. Therefore, we obtain two parameter sets of the OBEP potential (named as LOBEP1 and LOBEP2) with these two fitting ways. We calculate the properties of symmetric and pure neutron matter with LOBEP1 and LOBEP2. In non-relativistic Brueckner-Hartree-Fock case, the binding energies of symmetric nuclear matter are around $-3$ and $-5$ MeV at saturation densities, while it becomes $-8$ and $-12$ MeV in relativistic framework with $^1S_0,~^3S_1,$ and $^3D_1$ channels using our two paramet...
4. Lattice thermal expansion and anisotropic displacements in -sulfur from diffraction experiments and first-principles theory.
George, Janine; Deringer, Volker L; Wang, Ai; Müller, Paul; Englert, Ulli; Dronskowski, Richard
2016-12-21
Thermal properties of solid-state materials are a fundamental topic of study with important practical implications. For example, anisotropic displacement parameters (ADPs) are routinely used in physics, chemistry, and crystallography to quantify the thermal motion of atoms in crystals. ADPs are commonly derived from diffraction experiments, but recent developments have also enabled their first-principles prediction using periodic density-functional theory (DFT). Here, we combine experiments and dispersion-corrected DFT to quantify lattice thermal expansion and ADPs in crystalline α-sulfur (S8), a prototypical elemental solid that is controlled by the interplay of covalent and van der Waals interactions. We begin by reporting on single-crystal and powder X-ray diffraction measurements that provide new and improved reference data from 10 K up to room temperature. We then use several popular dispersion-corrected DFT methods to predict vibrational and thermal properties of α-sulfur, including the anisotropic lattice thermal expansion. Hereafter, ADPs are derived in the commonly used harmonic approximation (in the computed zero-Kelvin structure) and also in the quasi-harmonic approximation (QHA) which takes the predicted lattice thermal expansion into account. At the PPBE+D3(BJ) level, the QHA leads to excellent agreement with experiments. Finally, more general implications of this study for theory and experiment are discussed.
5. The relation between random matrix theory, chiral perturbation theory and lattice-QCD; Die Beziehungen zwischen Random-Matrix-Theorie, chiraler Stoerungstheorie und Gitter-QCD
Hehl, H.
2002-07-01
This thesis has studied the range of validity of the chiral random matrix theory in QCD on the example of the quenched staggered Dirac operator. The eigenvalues of this operator in the neighbourhood of zero are essential for the understanding of the spontaneous breaking of the chiral symmetry and the phase transition connected with this. The phase transition cannot be understood in the framework of perturbation theory, so that the formulation of QCD on the lattice has been chosen as the only non-perturbative approach. In order to circumvent both the problem of the fermion doubling and to study chiral properties on the lattice with acceptable numerical effort, quenched Kogut-Susskind fermions have been applied. The corresponding Dirac operator can be completely diagonalized by the Lanczos procedure of Cullum and Willoughby. Monte carlo simulations on hypercubic lattice have been performed and the Dirac operators of very much configurations diagonalized at different lattice lengths and coupling constants. The eigenvalue correlations on the microscopic scale are completely described by the chiral random matrix theory for the topological sector zero, which has been studied by means of the distribution of the smallest eigenvalue, the microscopic spectral density and the corresponding 2-point correlation function. The found universal behaviour shows, that on the scale of the lowest eigenvalue only completely general properties of the theory are important, but not the full dynamics. In order to determine the energy scale, from which the chiral random matrix theory losses its validity, - the Thouless energy - with the scalar susceptibilities observables have been analyzed, which are because of their spectral mass dependence sensitive on this. For each combination of the lattice parameter so the deviation point has been identified.
6. Tricritical points in a compact $U(1)$ lattice gauge theory at strong coupling
De, Asit K
2016-01-01
Pure compact $U(1)$ lattice gauge theory exhibits a phase transition at gauge coupling $g \\sim {\\cal{O}}(1)$ separating a familiar weak coupling Coulomb phase, having free massless photons, from a strong coupling phase. However, the phase transition was found to be of first order, ruling out any non-trivial theory resulting from a continuum limit from the strong coupling side. In this work, a compact $U(1)$ lattice gauge theory is studied with addition of a dimension-two mass counter-term and a higher derivative (HD) term that ensures a unique vacuum and produces a covariant gauge-fixing term in the naive continuum limit. For a reasonably large coefficient of the HD term, now there exists a continuous transition from a regular ordered phase to a spatially modulated ordered phase which breaks Euclidean rotational symmetry. For weak gauge couplings, a continuum limit from the regular ordered phase results in a familiar theory consisting of free massless photons. For strong gauge couplings with $g\\ge {\\cal{O}}(1... 7. Lattice density functional for colloid-polymer mixtures: Comparison of two fundamental measure theories Cuesta, José A.; Lafuente, Luis; Schmidt, Matthias 2005-09-01 We consider a binary mixture of colloid and polymer particles with positions on a simple cubic lattice. Colloids exclude both colloids and polymers from nearest neighbor sites. Polymers are treated as effective particles that are mutually noninteracting, but exclude colloids from neighboring sites; this is a discrete version of the (continuum) Asakura-Oosawa-Vrij model. Two alternative density functionals are proposed and compared in detail. The first is based on multioccupancy in the zero-dimensional limit of the bare model, analogous to the corresponding continuum theory that reproduces the bulk fluid free energy of free volume theory. The second is based on mapping the polymers onto a multicomponent mixture of polymer clusters that are shown to behave as hard cores; the corresponding property of the extended model in strong confinement permits direct treatment with lattice fundamental measure theory. Both theories predict the same topology for the phase diagram with a continuous fluid-fcc freezing transition at low polymer fugacity and, upon crossing a tricritical point, a first-order freezing transition for high polymer fugacities with rapidly broadening density jump. 8. New approach to the Dirac spectral density in lattice gauge theory applications Fodor, Zoltan; Kuti, Julius; Mondal, Santanu; Nogradi, Daniel; Wong, Chik Him 2016-01-01 We report tests and results from a new approach to the spectral density and the mode number distribution of the Dirac operator in lattice gauge theories. The algorithm generates the spectral density of the lattice Dirac operator as a continuous function over all scales of the complete eigenvalue spectrum. This is distinct from an earlier method where the integrated spectral density (mode number) was calculated efficiently for some preselected fixed range of the integration. The new algorithm allows global studies like the chiral condensate from the Dirac spectrum at any scale including the cutoff-dependent IR and UV range of the spectrum. Physics applications include the scale-dependent mass anomalous dimension, spectral representation of composite fermion operators, and the crossover transition from the$\\epsilon$-regime of Random Matrix Theory to the p-regime in chiral perturbation theory. We present thorough tests of the algorithm in the 2-flavor sextet SU(3) gauge theory that we continue to pursue for its... 9. Continuum formulation of the Scheutjens-Fleer lattice statistical theory for homopolymer adsorption from solution Mavrantzas, Vlasis G.; Beris, Antony N.; Leermakers, Frans; Fleer, Gerard J. 2005-11-01 Homopolymer adsorption from a dilute solution on an interacting (attractive) surface under static equilibrium conditions is studied in the framework of a Hamiltonian model. The model makes use of the density of chain ends n1,e and utilizes the concept of the propagator G describing conformational probabilities to locally define the polymer segment density or volume fraction φ; both n1,e and φ enter into the expression for the system free energy. The propagator G obeys the Edwards diffusion equation for walks in a self-consistent potential field. The equilibrium distribution of chain ends and, consequently, of chain conformational probabilities is found by minimizing the system free energy. This results in a set of model equations that constitute the exact continuum-space analog of the Scheutjens-Fleer (SF) lattice statistical theory for the adsorption of interacting chains. Since for distances too close to the surface the continuum formulation breaks down, the continuum model is here employed to describe the probability of chain configurations only for distances z greater than 2l, where l denotes the segment length, from the surface; instead, for distances z ⩽2l, the SF lattice model is utilized. Through this novel formulation, the lattice solution at z =2l provides the boundary condition for the continuum model. The resulting hybrid (lattice for distances z ⩽2l, continuum for distances z >2l) model is solved numerically through an efficient implementation of the pseudospectral collocation method. Representative results obtained with the new model and a direct application of the SF lattice model are extensively compared with each other and, in all cases studied, are found to be practically identical. 10. Shear viscosity to relaxation time ratio in SU(3) lattice gauge theory Kohno, Yasuhiro; Kitazawa, Masakiyo 2011-01-01 We evaluate the ratio of the shear viscosity to the relaxation time of the shear flux above but near the critical temperature$T_c$in SU(3) gauge theory on the lattice. The ratio is related to Kubo's canonical correlation of the energy-momentum tensor in Euclidean space with the relaxation time approximation and an appropriate regularization. Using this relation, the ratio is evaluated by direct measurements of the Euclidean observables on the lattice. We obtained the ratio with reasonable statistics for the range of temperature$1.3T_c \\lesssim T \\lesssim 4T_c$. We also found that the characteristic speed of the transverse plane wave in gluon media is almost constant,$v \\simeq 0.5$, for$T \\gtrsim 1.5T_c$, which is compatible with the causality in the second order dissipative hydrodynamics. 11. Using the Dempster-Shafer theory of evidence with a revised lattice structure for activity recognition. Liao, Jing; Bi, Yaxin; Nugent, Chris 2011-01-01 This paper explores a sensor fusion method applied within smart homes used for the purposes of monitoring human activities in addition to managing uncertainty in sensor-based readings. A three-layer lattice structure has been proposed, which can be used to combine the mass functions derived from sensors along with sensor context. The proposed model can be used to infer activities. Following evaluation of the proposed methodology it has been demonstrated that the Dempster-Shafer theory of evidence can incorporate the uncertainty derived from the sensor errors and the sensor context and subsequently infer the activity using the proposed lattice structure. The results from this study show that this method can detect a toileting activity within a smart home environment with an accuracy of 88.2%. 12. Vector meson masses in two-dimensional SU(NC) lattice gauge theory with massive quarks JIANG Jun-Qin 2008-01-01 Using an improved lattice Hamiltonian with massive Wilson quarks a variational method is applied to study the dependence of the vector meson mass Mv on the quark mass m and the Wilson parameter r in in the scaling window 1 ≤ 1/g2 ≤ 2, Mv/g is approximately linear in m, but Mv/g obviously does not depend on r (this differs from the quark condensate). Particularly for m → 0 our numerical results agree very well with Bhattacharya's analytical strong coupling result in the continuum, and the value of ((e)Mv/(e)m) |mm=0 in two-dimensional SU(NC) lattice gauge theory is very close to that in Schwinger model. 13. Flux tubes and their interaction in U(1) lattice gauge theory Zach, M P; Skála, P; Zach, Martin; Faber, Manfried; Skala, Peter 1997-01-01 We investigate singly and doubly charged flux tubes in U(1) lattice gauge theory. By simulating the dually transformed path integral we are able to consider large flux tube lengths, low temperatures, and multiply charged systems without loss of numerical precision. We simulate flux tubes between static sources as well as periodically closed flux tubes, calculating flux tube profiles, the total field energy and the free energy. Our main results are that the string tension in both three and four dimensions scales proportionally to the charge -- which is in contrast to previous lattice results -- and that in four-dimensional U(1) there is an attractive interaction between flux tubes for beta approaching the phase transition. 14. On the chiral limit in lattice gauge theories with Wilson fermions Hoferichter, A; Müller-Preussker, M 1995-01-01 The chiral limit ~\\kappa \\simeq \\kappa_c(\\beta)~ in lattice gauge theories with Wilson fermions and problems related to near--to--zero ('exceptional') eigenvalues of the fermionic matrix are studied. For this purpose we employ compact lattice QED in the confinement phase. A new estimator ~\\mpr_{\\pi}~ for the calculation of the pseudoscalar mass ~m_{\\pi}~ is proposed which does not suffer from 'divergent' contributions at \\kappa \\simeq \\kappa_c(\\beta). We conclude that the main contribution to the pion mass comes from larger modes, and 'exceptional' eigenvalues play {\\it no} physical role. The behaviour of the subtracted chiral condensate ~\\langle \\psb \\psi \\rangle_{subt}~ near ~\\kappa_c(\\beta)~ is determined. We observe a comparatively large value of ~\\langle \\psb \\psi \\rangle_{subt} \\cdot Z_P^{-1}~, which could be interpreted as a possible effect of the quenched approximation. 15. Compact U(1) lattice gauge-Higgs theory with monopole suppression Krishnan, B; Mitrjushkin, V K; Müller-Preussker, M; Krishnan, Balasubramanian 1996-01-01 We investigate a model of a U(1)-Higgs theory on the lattice with compact gauge fields but completely suppressed (elementary) monopoles. We study the model at two values of the quartic Higgs self-coupling, a strong coupling, \\lambda = 3.0, and a weak coupling, \\lambda=0.01. We map out the phase diagrams and find that the monopole suppression eliminated the confined phase of the standard lattice model at strong gauge coupling. We perform a detailed analysis of the static potential and study the mass spectrum in the Coulomb and Higgs phases for three values of the gauge coupling. We also probe the existence of a scalar bosonium to the extent that our data allow and conclude that further investigations are required in the Coulomb phase. 16. Charmless chiral perturbation theory for N_f=2+1+1 twisted mass lattice QCD Bar, Oliver 2014-01-01 The chiral Lagrangian describing the low-energy behavior of N_f=2+1+1 twisted mass lattice QCD is constructed through O(a^2). In contrast to existing results the effects of a heavy charm quark are consistently removed, leaving behind a charmless 3-flavor Lagrangian. This Lagrangian is used to compute the pion and kaon masses to one loop in a regime where the pion mass splitting is large and taken as a leading order effect. In comparison with continuum chiral perturbation theory additional chiral logarithms are present in the results. In particular, chiral logarithms involving the neutral pion mass appear. These predict rather large finite volume corrections in the kaon mass which roughly account for the finite volume effects observed in lattice data. 17. Generalized Courant-Snyder theory for coupled transverse dynamics of charged particles in electromagnetic focusing lattices Hong Qin 2009-06-01 Full Text Available The Courant-Snyder theory gives a complete description of the uncoupled transverse dynamics of charged particles in electromagnetic focusing lattices. In this paper, the Courant-Snyder theory is generalized to the case of coupled transverse dynamics with two degrees of freedom. The generalized theory has the same structure as the original Courant-Snyder theory for one degree of freedom. The four basic components of the original Courant-Snyder theory, i.e., the envelope equation, phase advance, transfer matrix, and the Courant-Snyder invariant, all have their counterparts, with remarkably similar expressions, in the generalized theory presented here. In the generalized theory, the envelope function is generalized into an envelope matrix, and the envelope equation becomes a matrix envelope equation with matrix operations that are noncommutative. The generalized theory gives a new parametrization of the 4D symplectic transfer matrix that has the same structure as the parametrization of the 2D symplectic transfer matrix in the original Courant-Snyder theory. All of the parameters used in the generalized Courant-Snyder theory correspond to physical quantities of importance, and this parametrization can provide a valuable framework for accelerator design and particle simulation studies. A time-dependent canonical transformation is used to develop the generalized Courant-Snyder theory. Applications of the new theory to strongly and weakly coupled dynamics are given. It is shown that the stability of coupled dynamics can be determined by the generalized phase advance developed. Two stability criteria are given, which recover the known results about sum and difference resonances in the weakly coupled limit. 18. Kinetic theory of correlated fluids: from dynamic density functional to Lattice Boltzmann methods. Marconi, Umberto Marini Bettolo; Melchionna, Simone 2009-07-07 Using methods of kinetic theory and liquid state theory we propose a description of the nonequilibrium behavior of molecular fluids, which takes into account their microscopic structure and thermodynamic properties. The present work represents an alternative to the recent dynamic density functional theory, which can only deal with colloidal fluids and is not apt to describe the hydrodynamic behavior of a molecular fluid. The method is based on a suitable modification of the Boltzmann transport equation for the phase space distribution and provides a detailed description of the local structure of the fluid and its transport coefficients. Finally, we propose a practical scheme to solve numerically and efficiently the resulting kinetic equation by employing a discretization procedure analogous to the one used in the Lattice Boltzmann method. 19. Thermal imaginary part of a real-time static potential from classical lattice gauge theory simulations Laine, M; Tassler, M 2007-01-01 Recently, a finite-temperature real-time static potential has been introduced via a Schr\\"odinger-type equation satisfied by a certain heavy quarkonium Green's function. Furthermore, it has been pointed out that it possesses an imaginary part, which induces a finite width for the tip of the quarkonium peak in the thermal dilepton production rate. The imaginary part originates from Landau-damping of low-frequency gauge fields, which are essentially classical due to their high occupation number. Here we show how the imaginary part can be measured with classical lattice gauge theory simulations, accounting non-perturbatively for the infrared sector of finite-temperature field theory. We demonstrate that a non-vanishing imaginary part indeed exists non-perturbatively; and that its value agrees semi-quantitatively with that predicted by Hard Loop resummed perturbation theory. 20. 建构主义学习理论对外语课堂教师提问策略的启发%The Enlightenment of Constructivism Learning Theory to the Strategies of Teachers' Questioning in Foreign Lan-guage Class 赵秋英 2014-01-01 教师提问作为教师话语的一部分，在外语教与学过程中起着重要的作用，然而其在实际课堂中存在很多问题，导致课堂教学质量未能达到预期的效果。本文着重从建构主义学习理论出发，结合教师提问存在的问题，从教学和学习的角度来探究外语教学课堂中教师提问的策略。%As apart of teacher talk, teachers' questioning plays an important role in the process of foreign language teaching and learning. However, in real teaching environment,it has lots of problems, failing to achieve the expected teaching result. Based on constructivism learning theory, this paper analyzes the prob-lems of teachers' questioning, and most importantly, explores the strategies of teachers' questioning in foreign language teaching class from the angel of teaching and learning. 1. Real-time dynamics of lattice gauge theories with a few-qubit quantum computer Martinez, Esteban A.; Muschik, Christine A.; Schindler, Philipp; Nigg, Daniel; Erhard, Alexander; Heyl, Markus; Hauke, Philipp; Dalmonte, Marcello; Monz, Thomas; Zoller, Peter; Blatt, Rainer 2016-06-01 Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. This has recently stimulated theoretical effort, using Feynman’s idea of a quantum simulator, to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented. Here we report the experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realizing (1 + 1)-dimensional quantum electrodynamics (the Schwinger model) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism, describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron-positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields in favour of exotic long-range interactions, which can be directly and efficiently implemented on an ion trap architecture. We explore the Schwinger mechanism of particle-antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulation of high-energy theories using atomic physics experiments—the long-term intention is to extend this approach to real-time quantum simulations of non-Abelian lattice gauge theories. 2. Real-time dynamics of lattice gauge theories with a few-qubit quantum computer. Martinez, Esteban A; Muschik, Christine A; Schindler, Philipp; Nigg, Daniel; Erhard, Alexander; Heyl, Markus; Hauke, Philipp; Dalmonte, Marcello; Monz, Thomas; Zoller, Peter; Blatt, Rainer 2016-06-23 Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. This has recently stimulated theoretical effort, using Feynman's idea of a quantum simulator, to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented. Here we report the experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realizing (1 + 1)-dimensional quantum electrodynamics (the Schwinger model) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism, describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron-positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields in favour of exotic long-range interactions, which can be directly and efficiently implemented on an ion trap architecture. We explore the Schwinger mechanism of particle-antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulation of high-energy theories using atomic physics experiments-the long-term intention is to extend this approach to real-time quantum simulations of non-Abelian lattice gauge theories. 3. SU(3) gauge theory with four degenerate fundamental fermions on the lattice Aoki, Yasumichi; Bennett, Ed; Kurachi, Masafumi; Maskawa, Toshihide; Miura, Kohtaroh; Nagai, Kei-ichi; Ohki, Hiroshi; Rinaldi, Enrico; Shibata, Akihiro; Yamawaki, Koichi; Yamazaki, Takeshi 2015-01-01 As a part of the project studying large$N_f$QCD, the LatKMI Collaboration has been investigating the SU(3) gauge theory with four fundamental fermions (four-flavor QCD). The main purpose of studying four-flavor QCD is to provide a qualitative comparison to$N_f= 8$,$12$,$16$QCD; however, a quantitative comparison to real-world QCD is also interesting. To make such comparisons more meaningful, it is desirable to use the same kind of lattice action consistently, so that qualitative difference of different theories are less affected by artifacts of lattice discretization. Here, we adopt the highly-improved staggered quark action with the tree-level Symanzik gauge action (HISQ/tree), which is exactly the same as the setup for our simulations for$SU(3)$gauge theories with$N_f=8$,$12$and$16$fundamental fermions~\\cite{Aoki:2013xza, Aoki:2012eq, Aoki:2014oma}. In the next section, we show the fermion mass dependence of$F_\\pi$,$\\langle\\bar{\\psi}\\psi\\rangle$,$M_\\pi$,$M_\\rho$,$M_N$and their chiral extr... 4. Theory of supersymmetry protected'' topological phases of isostatic lattices and highly frustrated magnets Lawler, Michael I generalize the theory of phonon topological band structures of isostatic lattices to highly frustrated antiferromagnets. I achieve this with a discovery of a many-body supersymmetry (SUSY) in the phonon problem of balls and springs which also applies to geometrically frustrated magnets. The Witten index of the SUSY model, when restricted to the single body problem (meaningful for linearized phonons), is then shown to be the Calladine-Kane-Lubensky index of mechanical structures that forms the cornerstone of the phonon topological band structure theory. Spontaneous supersymmetry breaking'' is then identified as the need to gap all modes in the bulk to create the topological state. The many-body SUSY formulation shows that the topology is not restricted to a band structure problem but extends to systems of coupled bosons and fermions that are in principle also realizable in solid state systems. The analogus supersymmetry of the magnon problem turns out to be particularly useful for highly frustrated magnets with the kagome family of antiferromagnets an analog of topological isostatic lattices. Thus, a solid state realization of the theory of phonon topological band structure may be found in highly frustrated magnets. However, our results show that this topology is protected not 5. Transport theory with self-consistent confinement related to the lattice data Bozek, P; Hüfner, J 1998-01-01 The space-time development of a quark-gluon plasma is calculated from a Vlasov equation for the distribution function of quasiparticles with medium dependent masses. At each space-time point the masses are calculated selfconsistently from a gap equation, whose form is determined by the requirement that in thermal equilibrium and for a range of temperatures the energy density of the quasi-particle system is identical to the one from lattice calculations . The numerical solutions of the Vlasov equation display confinement. Relations to effective theories like that by Friedberg Lee and Nambu Jona-Lasinio are established. 6. Density-functional theory of a lattice-gas model with vapour, liquid, and solid phases Prestipino, S.; Giaquinta, P. V. 2003-01-01 We use the classical version of the density-functional theory in the weighted-density approximation to build up the entire phase diagram and the interface structure of a two-dimensional lattice-gas model which is known, from previous studies, to possess three stable phases -- solid, liquid, and vapour. Following the common practice, the attractive part of the potential is treated in a mean-field-like fashion, although with different prescriptions for the solid and the fluid phases. It turns o... 7. Quantum electrodynamical time-dependent density functional theory for many-electron systems on a lattice Farzanehpour, Mehdi; Tokatly, Ilya; Nano-Bio Spectroscopy Group; ETSF Scientific Development Centre Team 2015-03-01 We present a rigorous formulation of the time-dependent density functional theory for interacting lattice electrons strongly coupled to cavity photons. We start with an example of one particle on a Hubbard dimer coupled to a single photonic mode, which is equivalent to the single mode spin-boson model or the quantum Rabi model. For this system we prove that the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate, provided the initial state and the density satisfy a set of well defined conditions. Then we generalize the formalism to many interacting electrons on a lattice coupled to multiple photonic modes and prove the general mapping theorem. We also show that for a system evolving from the ground state of a lattice Hamiltonian any density with a continuous second time derivative is locally v-representable. Spanish Ministry of Economy and Competitiveness (Grant No. FIS2013-46159-C3-1-P), Grupos Consolidados UPV/EHU del Gobierno Vasco (Grant No. IT578-13), COST Actions CM1204 (XLIC) and MP1306 (EUSpec). 8. Fortran code for SU(3) lattice gauge theory with and without MPI checkerboard parallelization Berg, Bernd A.; Wu, Hao 2012-10-01 We document plain Fortran and Fortran MPI checkerboard code for Markov chain Monte Carlo simulations of pure SU(3) lattice gauge theory with the Wilson action in D dimensions. The Fortran code uses periodic boundary conditions and is suitable for pedagogical purposes and small scale simulations. For the Fortran MPI code two geometries are covered: the usual torus with periodic boundary conditions and the double-layered torus as defined in the paper. Parallel computing is performed on checkerboards of sublattices, which partition the full lattice in one, two, and so on, up to D directions (depending on the parameters set). For updating, the Cabibbo-Marinari heatbath algorithm is used. We present validations and test runs of the code. Performance is reported for a number of currently used Fortran compilers and, when applicable, MPI versions. For the parallelized code, performance is studied as a function of the number of processors. Program summary Program title: STMC2LSU3MPI Catalogue identifier: AEMJ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEMJ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 26666 No. of bytes in distributed program, including test data, etc.: 233126 Distribution format: tar.gz Programming language: Fortran 77 compatible with the use of Fortran 90/95 compilers, in part with MPI extensions. Computer: Any capable of compiling and executing Fortran 77 or Fortran 90/95, when needed with MPI extensions. Operating system: Red Hat Enterprise Linux Server 6.1 with OpenMPI + pgf77 11.8-0, Centos 5.3 with OpenMPI + gfortran 4.1.2, Cray XT4 with MPICH2 + pgf90 11.2-0. Has the code been vectorised or parallelized?: Yes, parallelized using MPI extensions. Number of processors used: 2 to 11664 RAM: 200 Mega bytes per process. Classification: 11 9. Hamiltonian Study of Improved$U(1)_{2+1}Lattice Gauge Theory Loan, M; Hamer, C; Loan, Mushtaq; Byrnes, Tim; Hamer, Chris 2003-01-01 Monte Carlo results are presented, in the Hamiltonian limit, for the string tension and antisymmetric mass gap for U(1) lattice gauge theory in (2+1) dimensions, using mean-field improved anisotropic Wilson action, are presented. Evidence of scaling in the string tension and antisymmetric mass gap is observed in the weak coupling regime of the theory. The results are compared to previous simulation data using the standard Wilson action and we find that a more accurate determination of the string tension and scalar glueball masses has been achieved. The scaling behaviour observed is in good agreement with the results from other numerical calculations. Finally comparisons are made with previous estimates obtained in the Hamiltonian limit by various other studies. 10. Energy-momentum tensor on the lattice: non-perturbative renormalization in Yang--Mills theory Giusti, Leonardo 2015-01-01 We construct an energy-momentum tensor on the lattice which satisfies the appropriate Ward Identities (WIs) and has the right trace anomaly in the continuum limit. It is defined by imposing suitable WIs associated to the Poincare invariance of the continuum theory. These relations come forth when the length of the box in the temporal direction is finite, and they take a particularly simple form if the coordinate and the periodicity axes are not aligned. We implement the method for the SU(3) Yang--Mills theory discretized with the standard Wilson action in presence of shifted boundary conditions in the (short) temporal direction. By carrying out extensive numerical simulations, the renormalization constants of the traceless components of the tensor are determined with a precision of roughly half a percent for values of the bare coupling constant in the range 0<= g^2_0<=1. 11. A first look at Quasi-Monte Carlo for lattice field theory problems Jansen, K. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Leovey, H.; Griewank, A. [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Mathematik; Nube, A. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik; Mueller-Preussker, M. [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik 2012-11-15 In this project we initiate an investigation of the applicability of Quasi-Monte Carlo methods to lattice field theories in order to improve the asymptotic error behavior of observables for such theories. In most cases the error of an observable calculated by averaging over random observations generated from an ordinary Monte Carlo simulation behaves like N{sup -1/2}, where N is the number of observations. By means of Quasi-Monte Carlo methods it is possible to improve this behavior for certain problems to up to N{sup -1}. We adapted and applied this approach to simple systems like the quantum harmonic and anharmonic oscillator and verified an improved error scaling. 12. On the generalized eigenvalue method for energies and matrix elements in lattice field theory Blossier, Benoit; von Hippel, Georg; Mendes, Tereza; Sommer, Rainer 2009-01-01 We discuss the generalized eigenvalue problem for computing energies and matrix elements in lattice gauge theory, including effective theories such as HQET. It is analyzed how the extracted effective energies and matrix elements converge when the time separations are made large. This suggests a particularly efficient application of the method for which we can prove that corrections vanish asymptotically as\\exp(-(E_{N+1}-E_n) t)$. The gap$E_{N+1}-E_n$can be made large by increasing the number$N$of interpolating fields in the correlation matrix. We also show how excited state matrix elements can be extracted such that contaminations from all other states disappear exponentially in time. As a demonstration we present numerical results for the extraction of ground state and excited B-meson masses and decay constants in static approximation and to order$1/m_b$in HQET. 13. Twenty-first Century Lattice Gauge Theory: Results from the QCD Lagrangian Kronfeld, Andreas S.; /Fermilab 2012-03-01 Quantum chromodynamics (QCD) reduces the strong interactions, in all their variety, to an elegant nonabelian gauge theory. It clearly and elegantly explains hadrons at short distances, which has led to its universal acceptance. Since its advent, however, many of its long-distance, emergent properties have been believed to be true, without having been demonstrated to be true. This paper reviews a variety of results in this regime that have been established with lattice gauge theory, directly from the QCD Lagrangian. This body of work sheds light on the origin of hadron masses, its interplay with dynamical symmetry breaking, as well as on other intriguing features such as the phase structure of QCD. In addition, nonperturbative QCD is quantitatively important to many aspects of particle physics (especially the quark flavor sector), nuclear physics, and astrophysics. This review also surveys some of the most interesting connections to those subjects. 14. Radiative contribution to the effective potential in composite Higgs models from lattice gauge theory DeGrand, Thomas; Golterman, Maarten; Jay, William I.; Neil, Ethan T.; Shamir, Yigal; Svetitsky, Benjamin 2016-09-01 We develop methods to calculate the electroweak gauge boson contribution to the effective Higgs potential in the context of composite Higgs models, using lattice gauge theory. The calculation is analogous to that of the electromagnetic mass splitting of the pion multiplet in QCD. We discuss technical details of carrying out this calculation, including modeling of the momentum and fermion-mass dependence of the underlying current-current correlation function, direct integration of the correlation function over momentum, and fits based on the minimal-hadron approximation. We show results of a numerical study using valence overlap fermions, carried out in an SU(4) gauge theory with two flavors of Dirac fermions in the two-index antisymmetric representation. 15. Entropic lattice Boltzmann model for gas dynamics: Theory, boundary conditions, and implementation. Frapolli, N; Chikatamarla, S S; Karlin, I V 2016-06-01 We present in detail the recently introduced entropic lattice Boltzmann model for compressible flows [N. Frapolli et al., Phys. Rev. E 92, 061301(R) (2015)PLEEE81539-375510.1103/PhysRevE.92.061301]. The model is capable of simulating a wide range of laminar and turbulent flows, from thermal and weakly compressible flows to transonic and supersonic flows. The theory behind the construction of the model is laid out and its thermohydrodynamic limit is discussed. Based on this theory and the hydrodynamic limit thereof, we also construct the boundary conditions necessary for the simulation of solid walls. We present the inlet and outlet boundary conditions as well as no-slip and free-slip boundary conditions. Details necessary for the implementation of the compressible lattice Boltzmann model are also reported. Finally, simulations of compressible flows are presented, including two-dimensional supersonic and transonic flows around a diamond and a NACA airfoil, the simulation of the Schardin problem, and the three-dimensional simulation of the supersonic flow around a conical geometry. 16. Automated lattice perturbation theory in the Schroedinger functional. Implementation and applications in HQET Hesse, Dirk 2012-07-13 The author developed the pastor software package for automated lattice perturbation theory calculations in the Schroedinger functional scheme. The pastor code consists of two building blocks, dealing with the generation of Feynman rules and Feynman diagrams respectively. Accepting a rather generic class of lattice gauge and fermion actions, passed to the code in a symbolic form as input, a low level part of pastor will generate Feynman rules to an arbitrary order in the bare coupling with a trivial or an Abelian background field. The second, high level part of pastor is a code generator whose output relies on the vertex generator. It writes programs that evaluate Feynman diagrams for a class of Schroedinger functional observables up to one loop order automatically, the relevant O(a) improvement terms are taken into account. We will describe the algorithms used for implementation of both parts of the code in detail, and provide cross checks with perturbative and non-perturbative data to demonstrate the correctness of our code. We demonstrate the usefulness of the pastor package through various applications taken from the matching process of heavy quark effective theory with quantum chromodynamics. We have e.g. completed a one loop analysis for new candidates for matching observables timely and with rather small effort, highlighting two advantages of an automated software setup. The results that were obtained so far will be useful as a guideline for further non-perturbative studies. 17. Lattice study on QCD-like theory with exact center symmetry Iritani, Takumi; Misumi, Tatsuhiro 2015-01-01 We investigate QCD-like theory with exact center symmetry, with emphasis on the finite-temperature phase transition concerning center and chiral symmetries. On the lattice, we formulate center symmetric$SU(3)$gauge theory with three fundamental Wilson quarks by twisting quark boundary conditions in a compact direction ($Z_3$-QCD model). We calculate the expectation value of Polyakov loop and the chiral condensate as a function of temperature on 16^3 x 4 and 20^3 x 4 lattices along the line of constant physics realizing$m_{PS}/m_{V}=0.70$. We find out the first-order center phase transition, where the hysteresis of the magnitude of Polyakov loop exists depending on thermalization processes. We show that chiral condensate decreases around the critical temperature in a similar way to that of the standard three-flavor QCD, as it has the hysteresis in the same range as that of Polyakov loop. We also show that the flavor symmetry breaking due to the twisted boundary condition gets qualitatively manifest in the h... 18. Parent Hamiltonians for lattice Halperin states from free-boson conformal field theories Anna Hackenbroich 2017-03-01 Full Text Available We introduce a family of many-body quantum states that describe interacting spin one-half hard-core particles with bosonic or fermionic statistics on arbitrary one- and two-dimensional lattices. The wave functions at lattice filling fraction ν=2/(2m+1 are derived from deformations of the Wess–Zumino–Witten model su(31 and are related to the (m+1,m+1,m Halperin fractional quantum Hall states. We derive long-range SU(2 invariant parent Hamiltonians for these states which in two dimensions are chiral t–J–V models with additional three-body interaction terms. In one dimension we obtain a generalisation to open chains of a periodic inverse-square t–J–V model proposed in [25]. We observe that the gapless low-energy spectrum of this model and its open-boundary generalisation can be described by rapidity sets with the same generalised Pauli exclusion principle. A two-component compactified free boson conformal field theory is identified as the low-energy effective theory for the periodic inverse-square t–J–V model. 19. Finite Size Scaling and the Universality Class of SU(2) Lattice Gauge Theory Staniford-Chen, Stuart Gresley For a system near a second order phase transition, the correlation length becomes extremely large. This gives rise to much interesting physics such as the existence of critical exponents and the division of physical theories into universality classes. SU(2) lattice gauge theory has such a phase transition at finite temperature and it has been persuasively argued in the literature that it should be in the same universality class as the Ising model in a space with dimensionality one less than the gauge theory. This is in the sense that the effective theory for the SU(2) Wilson lines is universal with the Ising model. This prediction has been checked for d = 3 + 1 SU(2) by comparing the critical exponents, and those checks appear to confirm it to the modest accuracy currently available. However, the theory of finite size scaling predicts a very rich set of objects which should be the same across universality classes. For example, the shape of the graph of various observables against temperature near the transition is universal. Not only that, but whole collections of probability distributions as a function of temperature can be given a scaling form and the shape of this object is universal. I develop a methodology for comparing such sets of distributions. This gives a two dimensional surface for each theory which can then be used in comparisons. I then use this approach and compare the surface for the order parameter in SU(2) with that in phi^4. The visual similarity is very striking. I perform a semi-quantitative error analysis which does not reveal significant differences between the two surfaces. This strengthens the idea that the SU(2) effective line theory is in the Ising universality class. I conclude by discussing the advantages and disadvantages of the method used here. 20. Study of compact U(1) flux tubes in 3+1 dimensions in lattice gauge theory using GPU's Amado, André; Cardoso, Marco; Bicudo, Pedro 2012-01-01 We utilize Polyakov loop correlations to study (3+1)D compact U(1) flux tubes and the static electron-positron potential in lattice gauge theory. By using field operators it is possible in U(1) lattice gauge theory to probe directly the electric and magnetic fields. In order to improve the signal-to-noise ratio in the confinement phase, we apply the L\\"uscher-Weiss multilevel algorithm. Our code is written in CUDA, and we run it in NVIDIA FERMI generation GPU's, in order to achieve the necessary performance for our computations. 1. Canonical transformations and loop formulation of SU(N) lattice gauge theories Mathur, Manu; Sreeraj, T. P. 2015-12-01 We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop and string flux operators along with their canonically conjugate loop and string electric fields. The canonical relations between the initial SU(N) link operators and the final SU(N) loop and string operators, consistent with SU(N) gauge transformations, are explicitly constructed over the entire lattice. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space Hp. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the (1 /g2 ) magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the (g2 ) electric field terms describe all their interactions. In the weak coupling (g2→0 ) continuum limit the SU(N) loop dynamics is described by SU(N) spin Hamiltonian with nearest neighbor interactions. In the simplest SU(2) case, where the canonical transformations map the SU(2) loop Hilbert space into the Hilbert spaces of hydrogen atoms, we analyze the special role of the hydrogen atom dynamical symmetry group S O (4 ,2 ) in the loop dynamics and the spectrum. A simple tensor network ansatz in the SU(2) gauge invariant hydrogen atom loop basis is discussed. 2. Canonical Transformations and Loop Formulation of SU(N) Lattice Gauge Theories Mathur, Manu 2015-01-01 We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop & string flux operators along with their canonically conjugate loop & string electric fields. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space${\\cal H}^p$. The canonical relations between the initial SU(N) link operators and the final SU(N) loop & string operators over the entire lattice are worked out in a self consistent manner. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the$(1/g^2)$magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the$(g^2)$electric field terms describe all their interactions. In the weak coupling ($g^2 \\rightarrow 0$) continuum limit the SU(N) loop dynamics is described b... 3. Lattice-Gas Automata for the Problem Of Kinetic Theory of Gas During Free Expansion Khotimah, Siti Nurul; Arif, Idam; Liong, The Houw The lattice-gas method has been applied to solve the problem of kinetic theory of gas in the Gay-Lussac-Joule experiment. Numerical experiments for a two-dimensional gas were carried out to determine the number of molecules in one vessel (Nr), the ratio between the mean square values of the components of molecule velocity (/line{vx2}//line{v_y^2}), and the change in internal energy (ΔU) as a function of time during free expansion. These experiments were repeated for different sizes of an aperture in the partition between the two vessels. After puncturing the partition, the curve for the particle number in one vessel shows a damped oscillation for about half of the total number. The oscillations do not vanish after a sampling over different initial configurations. The system is in nonequilibrium due to the pressure equilibration, and here the flow is actually compressible. The equilibration time (in time steps) decreases with decreased size of aperture in the partition. For very small apertures (equal or less than 9{√{3}}/{2} lattice units), the number of molecules in one vessel changes with time in a smooth way until it reaches half of the total number; their curves obey the analytical solution for quasi-static processes. The calculations on /line{vx2}//line{v_y^2} and ΔU also support the results that the equilibration time decreases with decreased size of aperture in the partition. 4. Accurate determination of lattice parameters based on Niggli reduced cell theory by using digitized electron diffraction micrograph. Yang, Yi; Cai, Canying; Lin, Jianguo; Gong, Lunjun; Yang, Qibin 2017-05-01 In this paper, we used Niggli reduced cell theory to determine lattice constants of a micro/nano crystal by using electron diffraction patterns. The Niggli reduced cell method enhanced the accuracy of lattice constant measurement obviously, because the lengths and the angles of lattice vectors of a primitive cell can be measured directly on the electron micrographs instead of a double tilt holder. With the aid of digitized algorithm and least square optimization by using three digitized micrographs, a valid reciprocal Niggli reduced cell number can be obtained. Thus a reciprocal and real Bravais lattices are acquired. The results of three examples, i.e., Mg4Zn7, an unknown phase (Precipitate phase in nickel-base superalloy) and Ba4Ti13O30 showed that the maximum errors are 1.6% for lengths and are 0.3% for angles. 5. Effective-Field Theory for Kinetic Ising Model on Honeycomb Lattice SHI Xiao-Ling; WEI Guo-Zhu 2009-01-01 As an analytical method, the effective-field theory (EFT) is used to study the dynamical response of the kinetic Ising model in the presence of a sinusoidal oscillating field. The effective-field equations of motion of the average magnetization are given for the honeycomb lattice (Z = 3). The Liapunov exponent A is calculated for discussing the stability of the magnetization and it is used to determine the phase boundary. In the field amplitude ho / Z J-temperature T/ Z J plane, the phase boundary separating the dynamic ordered and the disordered phase has been drawn. In contrast to previous analytical results that predicted a tricritical point separating a dynamic phase boundary line of continuous and discontinuous transitions, we find that the transition is always continuous. There is inconsistency between our results and previous analytical restdts, because they do not introduce sufficiently strong fluctuations. 6. Optimizing the performance of Lattice Gauge Theory simulations with Streaming SIMD extensions Srinivasan, Shyam 2013-01-01 Two factors, which affect simulation quality are the amount of computing power and implementation. The Streaming SIMD (single instruction multiple data) extensions (SSE) present a technique for influencing both by exploiting the processor's parallel functionalism. In this paper, we show how SSE improves performance of lattice gauge theory simulations. We identified two significant trends through an analysis of data from various runs. The speed-ups were higher for single precision than double precision floating point numbers. Notably, though the use of SSE significantly improved simulation time, it did not deliver the theoretical maximum. There are a number of reasons for this: architectural constraints imposed by the FSB speed, the spatial and temporal patterns of data retrieval, ratio of computational to non-computational instructions, and the need to interleave miscellaneous instructions with computational instructions. We present a model for analyzing the SSE performance, which could help factor in the bot... 7. Field Theory Simulations on a Fuzzy Sphere - an Alternative to the Lattice Medina, J; Hofheinz, F; O'Connor, D; Medina, Julieta; Bietenholz, Wolfgang; Hofheinz, Frank; Connor, Denjoe O' 2005-01-01 We explore a new way to simulate quantum field theory, without introducing a spatial lattice. As a pilot study we apply this method to the 3d \\lambda \\phi^4 model. The regularisation consists of a fuzzy sphere with radius R for the two spatial directions, plus a discrete Euclidean time. The fuzzy sphere approximates the algebra of functions of the sphere with a matrix algebra, and the scalar field is represented by a Hermitian N x N matrix at each time site. We evaluate the phase diagram, where we find a disordered phase and an ordered regime, which splits into phases of uniform and non-uniform order. We discuss the behaviour of the model in different limits of large N and R, which lead to a commutative or to a non-commutative \\lambda \\phi^4 model in flat space. 8. Finite-representation approximation of lattice gauge theories at the continuum limit with tensor networks Buyens, Boye; Montangero, Simone; Haegeman, Jutho; Verstraete, Frank; Van Acoleyen, Karel 2017-05-01 It has been established that matrix product states can be used to compute the ground state and single-particle excitations and their properties of lattice gauge theories at the continuum limit. However, by construction, in this formalism the Hilbert space of the gauge fields is truncated to a finite number of irreducible representations of the gauge group. We investigate quantitatively the influence of the truncation of the infinite number of representations in the Schwinger model, one-flavor QED2 , with a uniform electric background field. We compute the two-site reduced density matrix of the ground state and the weight of each of the representations. We find that this weight decays exponentially with the quadratic Casimir invariant of the representation which justifies the approach of truncating the Hilbert space of the gauge fields. Finally, we compute the single-particle spectrum of the model as a function of the electric background field. 9. The spin-temperature theory of dynamic nuclear polarization and nuclear spin-lattice relaxation Byvik, C. E.; Wollan, D. S. 1974-01-01 A detailed derivation of the equations governing dynamic nuclear polarization (DNP) and nuclear spin lattice relaxation by use of the spin temperature theory has been carried to second order in a perturbation expansion of the density matrix. Nuclear spin diffusion in the rapid diffusion limit and the effects of the coupling of the electron dipole-dipole reservoir (EDDR) with the nuclear spins are incorporated. The complete expression for the dynamic nuclear polarization has been derived and then examined in detail for the limit of well resolved solid effect transitions. Exactly at the solid effect transition peaks, the conventional solid-effect DNP results are obtained, but with EDDR effects on the nuclear relaxation and DNP leakage factor included. Explicit EDDR contributions to DNP are discussed, and a new DNP effect is predicted. 10. Pion Structure in Qcd: from Theory to Lattice to Experimental Data Bakulev, A. P.; Mikhailov, S. V.; Pimikov, A. V.; Stefanis, N. G. We describe the present status of the pion distribution amplitude (DA) as it originates from several sources: (i) a nonperturbative approach based on QCD sum rules with nonlocal condensates, (ii) an O(as) QCD analysis of the CLEO data on Fgg*p(Q2) with asymptotic and renormalon models for higher twists and (iii) recent high-precision lattice QCD calculations of the second moment of the pion DA. We show predictions for the pion electromagnetic form factor, obtained in analytic QCD perturbation theory, and compare it with the JLab data on Fp(Q2). We also discuss in this context an improved model for nonlocal condensates in QCD and show its consequences for the pion DA and the gg*p transition form factor. We include a brief analysis of meson-induced massive lepton (muon) Drell-Yan production for the process p-Nm+m-X, considering both an unpolarized nucleon target and longitudinally polarized protons. 11. Phase transitions in strongly coupled 3d Z(N) lattice gauge theories at finite temperature Borisenko, O; Cortese, G; Fiore, R; Gravina, M; Papa, A; Surzhikov, I 2012-01-01 We perform an analytical and numerical study of the phase transitions in three-dimensional Z(N) lattice gauge theories at finite temperature for N>4. In the strong coupling limit these models are equivalent to a generalized version of the vector Potts models in two dimensions, where Polyakov loops play the role of Z(N) spins. The effective couplings of these two-dimensional spin models are calculated explicitly. It is argued that the effective spin models have two phase transitions of BKT type. This is confirmed by large-scale Monte Carlo simulations. Using a cluster algorithm we locate the position of the critical points and study the critical behavior across both phase transitions in details. In particular, we determine various critical indices, compute the helicity modulus, the average action and the specific heat. A scaling formula for the critical points with N is proposed. 12. In-medium quarkonium properties from a lattice QCD based effective field theory Kim, Seyong; Petreczky, Peter; Rothkopf, Alexander 2016-12-01 In order to understand the experimental data on heavy quarkonium production in heavy ion collisions at RHIC and LHC it is necessary (though not sufficient) to pinpoint the properties of heavy Q Q ‾ bound states in the deconfined quark-gluon plasma, including their dissolution. Here we present recent results on the temperature dependence of bottomonium and charmonium correlators, as well as their spectral functions in a lattice QCD based effective field theory called NRQCD, surveying temperatures close to the crossover transition 140MeV < T < 249MeV. The spectra are reconstructed based on a novel Bayesian prescription, whose systematic uncertainties are assessed. We present indications for sequential melting of different quarkonium species with respect to their vacuum binding energies and give estimates on the survival of S-wave and P-wave ground states. 13. In-medium quarkonium properties from a lattice QCD based effective field theory Kim, Seyong; Rothkopf, Alexander 2015-01-01 In order to understand the experimental data on heavy quarkonium production in heavy ion collisions at RHIC and LHC it is necessary (though not sufficient) to pinpoint the properties of heavy$Q\\bar{Q}$bound states in the deconfined quark-gluon plasma, including their dissolution. Here we present recent results on the temperature dependence of bottomonium and charmonium correlators, as well as their spectral functions in a lattice QCD based effective field theory called NRQCD, surveying temperatures close to the crossover transition$140 {\\rm MeV} < T< 249 {\\rm MeV}$. The spectra are reconstructed based on a novel Bayesian prescription, whose systematic uncertainties are assessed. We present indications for sequential melting of different quarkonium species with respect to their vacuum binding energies and give estimates on the survival of S-wave and P-wave ground states. 14. A Candidate for Solvable Large N Lattice Gauge Theory in D>2 Dubin, A Yu 1999-01-01 I propose a class of D\\geq{2} lattice SU(N) gauge theories dual to certain vector models endowed with the local [U(N)]^{D} conjugation-invariance and Z_{N} gauge symmetry. In the latter models, both the partitition function and Wilson loop observables depend nontrivially only on the eigenvalues of the link-variables. Therefore, the vector-model facilitates a master-field representation of the large N loop-averages in the corresponding induced gauge system. As for the partitition function, in the limit N->{infinity} it is reduced to the 2Dth power of an effective one-matrix eigenvalue-model which makes the associated phase structure accessible. In particular a simple scaling-condition, that ensures the proper continuum limit of the induced gauge theory, is proposed. We also derive a closed expression for the large N average of a generic nonself-intersecting Wilson loop in the D=2 theory defined on an arbitrary 2d surface. 15. Statistical mechanics and field theory. [Path integrals, lattices, pseudofree vertex model Samuel, S.A. 1979-05-01 Field theory methods are applied to statistical mechanics. Statistical systems are related to fermionic-like field theories through a path integral representation. Considered are the Ising model, the free-fermion model, and close-packed dimer problems on various lattices. Graphical calculational techniques are developed. They are powerful and yield a simple procedure to compute the vacuum expectation value of an arbitrary product of Ising spin variables. From a field theorist's point of view, this is the simplest most logical derivation of the Ising model partition function and correlation functions. This work promises to open a new area of physics research when the methods are used to approximate unsolved problems. By the above methods a new model named the 128 pseudo-free vertex model is solved. Statistical mechanics intuition is applied to field theories. It is shown that certain relativistic field theories are equivalent to classical interacting gases. Using this analogy many results are obtained, particularly for the Sine-Gordon field theory. Quark confinement is considered. Although not a proof of confinement, a logical, esthetic, and simple picture is presented of how confinement works. A key ingredient is the insight gained by using an analog statistical system consisting of a gas of macromolecules. This analogy allows the computation of Wilson loops in the presence of topological vortices and when symmetry breakdown occurs in the topological quantum number. Topological symmetry breakdown calculations are placed on approximately the same level of rigor as instanton calculations. The picture of confinement that emerges is similar to the dual Meissner type advocated by Mandelstam. Before topological symmetry breakdown, QCD has monopoles bound linearly together by three topological strings. Topological symmetry breakdown corresponds to a new phase where these monopoles are liberated. It is these liberated monopoles that confine quarks. 64 references. 16. Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory Hong Qin 2014-04-01 Full Text Available The dynamics of charged particles in general linear focusing lattices with quadrupole, skew-quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation of beam energy is parametrized using a generalized Courant-Snyder (CS theory, which extends the original CS theory for one degree of freedom to higher dimensions. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D symplectic rotation, or a U(2 element. The 1D envelope equation, also known as the Ermakov-Milne-Pinney equation in quantum mechanics, is generalized to an envelope matrix equation in higher dimensions. Other components of the original CS theory, such as the transfer matrix, Twiss functions, and CS invariant (also known as the Lewis invariant all have their counterparts, with remarkably similar expressions, in the generalized theory. The gauge group structure of the generalized theory is analyzed. By fixing the gauge freedom with a desired symmetry, the generalized CS parametrization assumes the form of the modified Iwasawa decomposition, whose importance in phase space optics and phase space quantum mechanics has been recently realized. This gauge fixing also symmetrizes the generalized envelope equation and expresses the theory using only the generalized Twiss function β. The generalized phase advance completely determines the spectral and structural stability properties of a general focusing lattice. For structural stability, the generalized CS theory enables application of the Krein-Moser theory to greatly simplify the stability analysis. The generalized CS theory provides an effective tool to study coupled dynamics and to discover more optimized lattice designs in the larger parameter space of general focusing lattices. 17. Contrastive Analysis of Impact by Affective Factors on Adult Foreign Lan-guage Learning and Child Language Development 田小丽 2016-01-01 Affective factors are the meaning of"decisive essence"in foreign language learning. Adult foreign language learning is much influenced by affective factors than children language development in the process of language development. In this pa-per, on the basis of a Krashen's"Affective Filter Hypothesis", the writer analyses the reasons, and analyses adult foreign lan-guage learning and child language development affect differently from four main affective factors:motivation, self-confidence, anxiety and empathy. 18. Lattice simulation of a center symmetric three dimensional effective theory for SU(2) Yang-Mills 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.) 19. SO(3) vs. SU(2) Yang-Mills theory on the lattice: an investigation at non-zero temperature Barresi, A; Müller-Preussker, M 2003-01-01 The adjoint SU(2) lattice gauge theory in 3+1 dimensions with the Wilson plaquette action modified by a Z(2) monopole suppression term is reinvestigated with special emphasis on the existence of a finite-temperature phase transition decoupling from the well-known bulk transitions. 20. Third-Order Approximation of 0++ Glueball Mass and Wavefunction of (2 + 1)-Dimensional SU(3) Lattice Gauge Theory LI Jie-Ming; CHEN Qi-Zhou; GUO Shuo-Hong 2001-01-01 The random phase approximation is applied to the coupled-cluster expansions of lattice gauge theory (LGT). Using this method, wavefunctions are approximated by linear combination of graphs consisting of only one connected Wilson loop. We study the excited state energy and wavefunction in (2 + 1)-D SU(3) LGT up to thc third order. The glueballmass shows a good scaling behavior. 1. Lattice Study of the Extent of the Conformal Window in Two-Color Yang-Mills Theory Voronov, Gennady 2013-01-01 We perform a lattice calculation of the Schr\\"odinger functional running coupling in SU(2) Yang-Mills theory with six massless Wilson fermions in the fundamental representation. The aim of this work is to determine whether the above theory has an infrared fixed point. Due to sensitivity of the$SF$renormalized coupling to the tuning of the fermion bare mass we were unable to reliably extract the running coupling for stronger bare couplings. 2. Semiclassical theory of the magnetization process of the triangular lattice Heisenberg model Coletta, Tommaso; Tóth, Tamás A.; Penc, Karlo; Mila, Frédéric 2016-08-01 Motivated by the numerous examples of 1/3 magnetization plateaux in the triangular-lattice Heisenberg antiferromagnet with spins ranging from 1/2 to 5/2, we revisit the semiclassical calculation of the magnetization curve of that model, with the aim of coming up with a simple method that allows one to calculate the full magnetization curve and not just the critical fields of the 1/3 plateau. We show that it is actually possible to calculate the magnetization curve including the first quantum corrections and the appearance of the 1/3 plateau entirely within linear spin-wave theory, with predictions for the critical fields that agree to order 1 /S with those derived a long time ago on the basis of arguments that required going beyond linear spin-wave theory. This calculation relies on the central observation that there is a kink in the semiclassical energy at the field where the classical ground state is the collinear up-up-down structure and that this kink gives rise to a locally linear behavior of the energy with the field when all semiclassical ground states are compared to each other for all fields. The magnetization curves calculated in this way for spin 1/2, 1, and 5/2 are shown to be in good agreement with available experimental data. 3. Parameters of heavy quark effective theory from N{sub f}=2 lattice QCD Blossier, Benoit [CNRS, Orsay (France). LPT; Paris-11 Univ., 91 - Orsay (France); Della Morte, Michele [Mainz Univ. (Germany). Inst. fuer Kernphysik; Fritzsch, Patrick [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik; Garron, Nicolas [Edinburgh Univ. (United Kingdom). School of Physics and Astronomy; Heitger, Jochen [Muenster Univ. (Germany). Inst. fuer Theoretische Physik 1; Simma, Hubert; Sommer, Rainer [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Tantalo, Nazario [Rome-3 Univ. (Italy). Dipt. di Fisica; INFN, Sezione di Roma (Italy) 2012-07-15 We report on a non-perturbative determination of the parameters of the lattice Heavy Quark Effective Theory (HQET) Lagrangian and of the time component of the heavy-light axial-vector current with N{sub f} = 2 flavors of massless dynamical quarks. The effective theory is considered at the 1/m{sub h} order, and the heavy mass m{sub h} covers a range from slightly above the charm to beyond the beauty region. These HQET parameters are needed to compute, for example, the b-quark mass, the heavy-light spectrum and decay constants in the static approximation and to order 1/m{sub h} in HQET. The determination of the parameters is done non-perturbatively. The computation reported in this paper uses the plaquette gauge action and two different static actions for the heavy quark described by HQET. For the light-quark action we choose non-perturbatively O(a)-improved Wilson fermions. 4. Phase Structure of lattice$SU(2) x U_{S}(1)$three-dimensional Gauge Theory Farakos, K; McNeill, D 1999-01-01 We discuss a phase diagram for a relativistic SU(2) x U_{S}(1) lattice gauge theory, with emphasis on the formation of a parity-invariant chiral condensate, in the case when the$U_{S}(1)$field is infinitely coupled, and the SU(2) field is moved away from infinite coupling by means of a strong-coupling expansion. We provide analytical arguments on the existence of (and partially derive) a critical line in coupling space, separating the phase of broken SU(2) symmetry from that where the symmetry is unbroken. We review uncoventional (Kosterlitz-Thouless type) superconducting properties of the model, upon coupling it to external electromagnetic potentials. We discuss the rôle of instantons of the unbroken subgroup U(1) of SU(2), in eventually destroying superconductivity under certain circumstances. The model may have applications to the theory of high-temperature superconductivity. In particular, we argue that in the regime of the couplings leading to the broken SU(2) phase, the model may provide an explanati... 5. Smooth Gauge Strings and D > 2 Lattice Yang-Mills Theories Dubin, A Yu 2000-01-01 Employing the nonabelian duality transformation \\cite{Dub2}, I derive theGauge String representation of certain D>2 lattice Yang-Mills theories in theSC phase. With the judicious choice of the actions, in D>2 our constructiongeneralizes the Gross-Taylor stringy reformulation of the continuous YM_{2} ona 2d manifold. Using the Twisted Eguchi-Kawai model as an example, we developethe algorithm to determine the weights w[\\tilde{M}] for connected YM-fluxworldsheets$\\tilde{M}$immersed, \\tilde{M}->T, into a given 2d cell-complex T.Owing to the invariance of w[\\tilde{M}] under a continuous group ofarea-preserving worldsheet homeomorphisms, the weights {w[\\tilde{M}]} can bereadily used to define the theory of the smooth YM-fluxes which unambiguouslyrefers to a particular continuous YM_{D} system. I argue that the latter YM_{D}models (with a finite ultraviolet cut-off \\Lambda) for sufficiently largevalues of the coupling constant(s) are reproduced, to all orders in 1/N, by thesmooth Gauge String thus associated. The... 6. Lattice chiral effective field theory with three-body interactions at next-to-next-to-leading order Epelbaum, Evgeny; Lee, Dean; Meißner, Ulf-G 2009-01-01 We consider low-energy nucleons at next-to-next-to-leading order in lattice chiral effective field theory. Three-body interactions first appear at this order, and we discuss several methods for determining three-body interaction coefficients on the lattice. We compute the energy of the triton and low-energy neutron-deuteron scattering phase shifts in the spin-doublet and spin-quartet channels using Luescher's finite volume method. In the four-nucleon system we calculate the energy of the alpha particle using auxiliary fields and projection Monte Carlo. 7. Path Integral Monte Carlo Approach to the U(1) Lattice Gauge Theory in (2+1) Dimensions Loan, M; Sloggett, C; Hamer, C; Loan, Mushtaq; Brunner, Michael; Sloggett, Clare; Hamer, Chris 2003-01-01 Path Integral Monte Carlo simulations have been performed for U(1) lattice gauge theory in (2+1) dimensions on anisotropic lattices. We extract the static quark potential, the string tension and the low-lying "glueball" spectrum. The Euclidean string tension and mass gap decrease exponentially at weak coupling in excellent agreement with the predictions of Polyakov and G{\\" o}pfert and Mack, but their magnitudes are five times bigger than predicted. Extrapolations are made to the extreme anisotropic or Hamiltonian limit, and comparisons are made with previous estimates obtained in the Hamiltonian formulation. 8. SU(2) lattice gauge theory in 2+1 dimensions: critical couplings from twisted boundary conditions and universality Edwards, Sam 2009-01-01 We present a precision determination of the critical coupling beta_c for the deconfinement transition in pure SU(2) gauge theory in 2+1 dimensions. This is possible from universality, by intersecting the center vortex free energy as a function of the lattice coupling beta with the exactly known value of the interface free energy in the 2D Ising model at criticality. Results for lattices with different numbers of sites N_t along the Euclidean time direction are used to determine how beta varies with temperature for a given N_t around the deconfinement transition. 9. Flux tube widening in compact U (1) lattice gauge theory computed at T < Tc with the multilevel method and GPUs Amado, A; Bicudo, P 2013-01-01 We utilize Polyakov loop correlations to study d=3+1 compact U (1) flux tubes and the static electron-positron potential in lattice gauge theory. With the plaquette field operator, in U(1) lattice gauge theory, we probe directly the components of the electric and magnetic fields. In order to improve the signal-to-noise ratio in the confinement phase, we apply the L\\"uscher-Weiss multilevel algorithm. Our code is written in CUDA, and we run it in NVIDIA FERMI generation GPUs, in order to achieve the necessary efficiency for our computations. We measure in detail the quantum widening of the flux tube, as a function of the intercharge distance and at different finite temperatures T < Tc . Our results are compatible with the Effective String Theory. 10. Looijenga's weighted projective space, Tate's algorithm and Mordell-Weil Lattice in F-theory and heterotic string theory Mizoguchi, Shun'ya 2016-01-01 It is now well known that the moduli space of a vector bundle for heterotic string compactifications to four dimensions is parameterized by a set of sections of a weighted projective space bundle of a particular kind, known as Looijenga's weighted projective space bundle. We show that the requisite weighted projective spaces and the Weierstrass equations describing the spectral covers for gauge groups E_N (N=4,...,8) and SU(n+1) (n=1,2,3) can be obtained systematically by a series of blowing-up procedures according to Tate's algorithm, thereby the sections of correct line bundles claimed to arise by Looijenga's theorem can be automatically obtained. They are nothing but the four-dimensional analogue of the set of independent polynomials in the six-dimensional F-theory parameterizing the complex structure, which is further confirmed in the constructions of D_4, A_5, D_6 and E_3 bundles. We also explain why we can obtain them in this way by using the structure theorem of the Mordell-Weil lattice, which is also ... 11. Dilute neutron matter on the lattice at next-to-leading order in chiral effective field theory Borasoy, Bugra; Krebs, Hermann; Lee, Dean; Meißner, Ulf-G 2007-01-01 We discuss lattice simulations of the ground state of dilute neutron matter at next-to-leading order in chiral effective field theory. In a previous paper the coefficients of the next-to-leading-order lattice action were determined by matching nucleon-nucleon scattering data for momenta up to the pion mass. Here the same lattice action is used to simulate the ground state of up to 12 neutrons in a periodic cube using Monte Carlo. We explore the density range from 2% to 8% of normal nuclear density and analyze the ground state energy as an expansion about the unitarity limit with corrections due to finite scattering length, effective range, and P-wave interactions. 12. Program package for multicanonical simulations of U(1) lattice gauge theory-Second version Bazavov, Alexei; Berg, Bernd A. 2013-03-01 A new version STMCMUCA_V1_1 of our program package is available. It eliminates compatibility problems of our Fortran 77 code, originally developed for the g77 compiler, with Fortran 90 and 95 compilers. New version program summaryProgram title: STMC_U1MUCA_v1_1 Catalogue identifier: AEET_v1_1 Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html Programming language: Fortran 77 compatible with Fortran 90 and 95 Computers: Any capable of compiling and executing Fortran code Operating systems: Any capable of compiling and executing Fortran code RAM: 10 MB and up depending on lattice size used No. of lines in distributed program, including test data, etc.: 15059 No. of bytes in distributed program, including test data, etc.: 215733 Keywords: Markov chain Monte Carlo, multicanonical, Wang-Landau recursion, Fortran, lattice gauge theory, U(1) gauge group, phase transitions of continuous systems Classification: 11.5 Catalogue identifier of previous version: AEET_v1_0 Journal Reference of previous version: Computer Physics Communications 180 (2009) 2339-2347 Does the new version supersede the previous version?: Yes Nature of problem: Efficient Markov chain Monte Carlo simulation of U(1) lattice gauge theory (or other continuous systems) close to its phase transition. Measurements and analysis of the action per plaquette, the specific heat, Polyakov loops and their structure factors. Solution method: Multicanonical simulations with an initial Wang-Landau recursion to determine suitable weight factors. Reweighting to physical values using logarithmic coding and calculating jackknife error bars. Reasons for the new version: The previous version was developed for the g77 compiler Fortran 77 version. Compiler errors were encountered with Fortran 90 and Fortran 95 compilers (specified below). Summary of revisions: epsilon=one/10**10 is replaced by epsilon/10.0D10 in the parameter statements of the subroutines u1_bmha.f, u1_mucabmha.f, u1wl 13. A Unified Theory of Non-Ideal Gas Lattice Boltzmann Models Luo, Li-Shi 1998-01-01 A non-ideal gas lattice Boltzmann model is directly derived, in an a priori fashion, from the Enskog equation for dense gases. The model is rigorously obtained by a systematic procedure to discretize the Enskog equation (in the presence of an external force) in both phase space and time. The lattice Boltzmann model derived here is thermodynamically consistent and is free of the defects which exist in previous lattice Boltzmann models for non-ideal gases. The existing lattice Boltzmann models for non-ideal gases are analyzed and compared with the model derived here. 14. Hamiltonian effective field theory study of the$\\mathbf{N^*(1440)}$resonance in lattice QCD Liu, Zhan-Wei; Leinweber, Derek B; Stokes, Finn M; Thomas, Anthony W; Wu, Jia-Jun 2016-01-01 We examine the phase shifts and inelasticities associated with the$N^*(1440)$Roper resonance and connect these infinite-volume observables to the finite-volume spectrum of lattice QCD using Hamiltonian effective field theory. We explore three hypotheses for the structure of the Roper resonance. In the first scenario, the Roper is postulated to have a triquark-like bare or core component with a mass exceeding the resonance mass. This component mixes with attractive virtual meson-baryon contributions, including the$\\pi N$,$\\pi\\Delta$, and$\\sigma N$channels, to reproduce the observed pole position. In the second hypothesis, the Roper resonance is dynamically generated purely from the meson-baryon channels. However, given the presence of a bare state associated with the ground state nucleon, we proceed to consider a third scenario incorporating the presence of this low-lying basis state. All three hypotheses are able to describe the scattering data well. However, the first hypothesis predicts a low-lying st... 15. Research in Lattice Gauge Theory and in the Phenomenology of Neutrinos and Dark Matter Meurice, Yannick L [Univ. of Iowa, Iowa City, IA (United States); Reno, Mary Hall [Univ. of Iowa, Iowa City, IA (United States) 2016-06-23 Research in theoretical elementary particle physics was performed by the PI Yannick Meurice and co-PI Mary Hall Reno. New techniques designed for precision calculations of strong interaction physics were developed using the tensor renormalization group method. Large-scale Monte Carlo simulations with dynamical quarks were performed for candidate models for Higgs compositeness. Ab-initio lattice gauge theory calculations of semileptonic decays of B-mesons observed in collider experiments and relevant to test the validity of the standard model were performed with the Fermilab/MILC collaboration. The phenomenology of strong interaction physics was applied to new predictions for physics processes in accelerator physics experiments and to cosmic ray production and interactions. A research focus has been on heavy quark production and their decays to neutrinos. The heavy quark contributions to atmospheric neutrino and muon fluxes have been evaluated, as have the neutrino fluxes from accelerator beams incident on heavy targets. Results are applicable to current and future particle physics experiments and to astrophysical neutrino detectors such as the IceCube Neutrino Observatory. 16. Unified approach to thermodynamic Bethe ansatz and finite size corrections for lattice models and field theories Destri, C 1994-01-01 We present a unified approach to the Thermodynamic Bethe Ansatz (TBA) for magnetic chains and field theories that includes the finite size (and zero temperature) calculations for lattice BA models. In all cases, the free energy follows by quadratures from the solution of a {\\bf single} non-linear integral equation (NLIE). [A system of NLIE appears for nested BA]. We derive the NLIE for: a) the six-vertex model with twisted boundary conditions; b) the XXZ chain in an external magnetic field h_z and c) the sine-Gordon-massive Thirring model (sG-mT) in a periodic box of size \\b \\equiv 1/T using the light-cone approach. This NLIE is solved by iteration in one regime (high T in the XXZ chain and low T in the sG-mT model). In the opposite (conformal) regime, the leading behaviors are obtained in closed form. Higher corrections can be derived from the Riemann-Hilbert form of the NLIE that we present. 17. Application of perturbation theory to lattice calculations based on method of cyclic characteristics Assawaroongruengchot, Monchai Perturbation theory is a technique used for the estimation of changes in performance functionals, such as linear reaction rate ratio and eigenvalue affected by small variations in reactor core compositions. Here the algorithm of perturbation theory is developed for the multigroup integral neutron transport problems in 2D fuel assemblies with isotropic scattering. The integral transport equation is used in the perturbative formulation because it represents the interconnecting neutronic systems of the lattice assemblies via the tracking lines. When the integral neutron transport equation is used in the formulation, one needs to solve the resulting integral transport equations for the flux importance and generalized flux importance functions. The relationship between the generalized flux importance and generalized source importance functions is defined in order to transform the generalized flux importance transport equations into the integro-differential equations for the generalized adjoints. Next we develop the adjoint and generalized adjoint transport solution algorithms based on the method of cyclic characteristics (MOCC) in DRAGON code. In the MOCC method, the adjoint characteristics equations associated with a cyclic tracking line are formulated in such a way that a closed form for the adjoint angular function can be obtained. The MOCC method then requires only one cycle of scanning over the cyclic tracking lines in each spatial iteration. We also show that the source importance function by CP method is mathematically equivalent to the adjoint function by MOCC method. In order to speed up the MOCC solution algorithm, a group-reduction and group-splitting techniques based on the structure of the adjoint scattering matrix are implemented. A combined forward flux/adjoint function iteration scheme, based on the group-splitting technique and the common use of a large number of variables storing tracking-line data and exponential values, is proposed to reduce the 18. An O(a) modified lattice set-up of the Schr\\"odinger functional in SU(3) gauge theory Pérez-Rubio, Paula; Takeda, Shinji 2011-01-01 The set-up of the QCD Schr\\"odinger functional (SF) on the lattice with staggered quarks requires an even number of points$L/a$in the spatial directions, while the Euclidean time extent of the lattice,$T/a$, must be odd. Identifying a unique renormalisation scale,$L=T$, is then only possible up to O($a$) lattice artefacts. In this article we study such lattices in the pure SU(3) gauge theory, where we can also compare to the standard set-up. We consider the SF coupling as obtained from the variation of an SU(3) Abelian and spatially constant background field. The O($a$) lattice artefacts can be cancelled by the existing O($a$) boundary counterterm. However, its coefficient,$\\ct$, differs at the tree-level from its standard value, so that one first needs to re-determine the induced background gauge field. The perturbative one-loop correction to the coupling allows to determine$\\ct$to one-loop order. A few numerical simulations serve to demonstrate that residual cutoff effects in the step scaling functio... 19. Parent Hamiltonians for lattice Halperin states from free-boson conformal field theories Hackenbroich, Anna 2016-01-01 We introduce a family of many-body quantum states that describe interacting spin one-half hard-core particles with bosonic or fermionic statistics on arbitrary one- and two-dimensional lattices. The wave functions at lattice filling fraction$\
20. General point dipole theory for periodic metasurfaces: magnetoelectric scattering lattices coupled to planar photonic structures
Chen, Yuntian
2015-01-01
We study semi-analytically the light emission and absorption properties of arbitrary stratified photonic structures with embedded two-dimensional magnetoelectric point scattering lattices, as used in recent plasmon-enhanced LEDs and solar cells. By employing dyadic Green's function for the layered structure in combination with Ewald lattice summation to deal with the particle lattice, we develop an efficient method to study the coupling between planar 2D scattering lattices of plasmonic, or metamaterial point particles, coupled to layered structures. Using the array scanning method' we deal with localized sources. Firstly, we apply our method to light emission enhancement of dipole emitters in slab waveguides, mediated by plasmonic lattices. We benchmark the array scanning method against a reciprocity-based approach to find that the calculated radiative rate enhancement in k-space below the light cone shows excellent agreement. Secondly, we apply our method to study absorption-enhancement in thin-film solar ...
1. Ground state energy of dilute neutron matter at next-to-leading order in lattice chiral effective field theory
Epelbaum, Evgeny; Lee, Dean; Meißner, Ulf-G
2008-01-01
We present lattice calculations for the ground state energy of dilute neutron matter at next-to-leading order in chiral effective field theory. This study follows a series of recent papers on low-energy nuclear physics using chiral effective field theory on the lattice. In this work we introduce an improved spin- and isospin-projected leading-order action which allows for a perturbative treatment of corrections at next-to-leading order and smaller estimated errors. Using auxiliary fields and Euclidean-time projection Monte Carlo, we compute the ground state of 8, 12, and 16 neutrons in a periodic cube, covering a density range from 2% to 10% of normal nuclear density.
2. Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory
Qin, Hong; Burby, J W; Chung, Moses
2015-01-01
The dynamics of charged particles in general linear focusing lattices with quadrupole, skew-quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation of beam energy is parameterized using a generalized Courant-Snyder (CS) theory, which extends the original CS theory for one degree of freedom to higher dimensions. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D symplectic rotation, or an U(2) element. The 1D envelope equation, also known as the Ermakov-Milne-Pinney equation in quantum mechanics, is generalized to an envelope matrix equation in higher dimensions. Other components of the original CS theory, such as the transfer matrix, Twiss functions, and CS invariant (also known as the Lewis invariant) all have their counterparts, with remarkably similar expressions, in the generalized theory. The gauge group structure of the generalized theory is analyzed. By fixing the gauge freedom with a desired symmetr...
3. Four-loop lattice-regularized vacuum energy density of the three-dimensional SU(3) + adjoint Higgs theory
Di Renzo, F.; M. Laine; Y. Schroder(Bielefeld U.); Torrero, C.
2008-01-01
The pressure of QCD admits at high temperatures a factorization into purely perturbative contributions from "hard" thermal momenta, and slowly convergent as well as non-perturbative contributions from "soft" thermal momenta. The latter can be related to various effective gluon condensates in a dimensionally reduced effective field theory, and measured there through lattice simulations. Practical measurements of one of the relevant condensates have suffered, however, from difficulties in extra...
4. A novel approach for computing glueball masses and matrix elements in Yang-Mills theories on the lattice
Della Morte, Michele; Giusti, Leonardo
2011-05-01
We make use of the global symmetries of the Yang-Mills theory on the lattice to design a new computational strategy for extracting glueball masses and matrix elements which achieves an exponential reduction of the statistical error with respect to standard techniques. By generalizing our previous work on the parity symmetry, the partition function of the theory is decomposed into a sum of path integrals each giving the contribution from multiplets of states with fixed quantum numbers associated to parity, charge conjugation, translations, rotations and central conjugations Z N 3. Ratios of path integrals and correlation functions can then be computed with a multi-level Monte Carlo integration scheme whose numerical cost, at a fixed statistical precision and at asymptotically large times, increases power-like with the time extent of the lattice. The strategy is implemented for the SU(3) Yang-Mills theory, and a full-fledged computation of the mass and multiplicity of the lightest glueball with vacuum quantum numbers is carried out at a lattice spacing of 0.17 fm.
5. Lattice parameters and electronic structure of BeMgZnO quaternary solid solutions: Experiment and theory
Toporkov, M.; Demchenko, D. O.; Zolnai, Z.; Volk, J.; Avrutin, V.; Morkoç, H.; Özgür, Ü.
2016-03-01
BexMgyZn1-x-yO semiconductor solid solutions are attractive for UV optoelectronics and electronic devices owing to their wide bandgap and capability of lattice-matching to ZnO. In this work, a combined experimental and theoretical study of lattice parameters, bandgaps, and underlying electronic properties, such as changes in band edge wavefunctions in BexMgyZn1-x-yO thin films, is carried out. Theoretical ab initio calculations predicting structural and electronic properties for the whole compositional range of materials are compared with experimental measurements from samples grown by plasma assisted molecular beam epitaxy on (0001) sapphire substrates. The measured a and c lattice parameters for the quaternary alloys BexMgyZn1-x with x = 0-0.19 and y = 0-0.52 are within 1%-2% of those calculated using generalized gradient approximation to the density functional theory. Additionally, composition independent ternary BeZnO and MgZnO bowing parameters were determined for a and c lattice parameters and the bandgap. The electronic properties were calculated using exchange tuned Heyd-Scuseria-Ernzerhof hybrid functional. The measured optical bandgaps of the quaternary alloys are in good agreement with those predicted by the theory. Strong localization of band edge wavefunctions near oxygen atoms for BeMgZnO alloy in comparison to the bulk ZnO is consistent with large Be-related bandgap bowing of BeZnO and BeMgZnO (6.94 eV). The results in aggregate show that precise control over lattice parameters by tuning the quaternary composition would allow strain control in BexMgyZn1-x-yO/ZnO heterostructures with possibility to achieve both compressive and tensile strain, where the latter supports formation of two-dimensional electron gas at the interface.
6. Topology in SU(2) lattice gauge theory and parallelization of functional magnetic resonance imaging
Solbrig, Stefan
2008-07-01
In this thesis, I discuss topological properties of quenched SU(2) lattice gauge fields. In particular, clusters of topological charge density exhibit a power-law. The exponent of that power-law can be used to validate models for lattice gauge fields. Instead of working with fixed cutoffs of the topological charge density, using the notion of a ''watermark'' is more convenient. Furthermore, I discuss how a parallel computer, originally designed for lattice gauge field simulations, can be used for functional magnetic resonance imaging. Multi parameter fits can be parallelized to achieve almost real-time evaluation of fMRI data. (orig.)
7. Defects in higher-dimensional quantum field theory. Relations to AdS/CFT-correspondence and Kondo lattices
Schmidt, R.
2007-03-15
The present work is addressed to defects and boundaries in quantum field theory considering the application to AdS/CFT correspondence. We examine interactions of fermions with spins localised on these boundaries. Therefore, an algebra method is emphasised adding reflection and transmission terms to the canonical quantisation prescription. This method has already been applied to bosons in two space-time dimensions before. We show the possibilities of such reflection-transmission algebras in two, three, and four dimensions. We compare with models of solid state physics as well as with the conformal field theory approach to the Kondo effect. Furthermore, we discuss ansatzes of extensions to lattice structures. (orig.)
8. Calculation of mixture critical diagrams using an equation of state based on the lattice fluid theory
S. Mattedi
2000-12-01
Full Text Available A modified form of the Hicks and Young algorithm was used with the Mattedi-Tavares-Castier lattice equation of state (MTC lattice EOS to calculate critical points of binary mixtures that exhibit several types of critical behavior. Several qualitative aspects of the critical curves, such as maxima and minima in critical pressure, and minima in critical temperature, could be predicted using the MTC lattice EOS. These results were in agreement with experimental information available in the literature, illustrating the flexibility of the functional form of the MTC lattice EOS. We observed however that the MTC lattice EOS failed to predict maxima in pressure for two of the studied systems: ethane + ethanol and methane + n-hexane. We also observed that the agreement between the calculated and experimental critical properties was at most semi-quantitative in some examples. Despite these limitations, in many ways similar to those of other EOS in common use when applied to critical point calculations, we can conclude that the MTC lattice EOS has the ability to predict several types of critical curves of complex shape.
9. Lattice thermal expansion and anisotropic displacements in 𝜶-sulfur from diffraction experiments and first-principles theory
George, Janine; Deringer, Volker L.; Wang, Ai; Müller, Paul; Englert, Ulli; Dronskowski, Richard
2016-12-01
Thermal properties of solid-state materials are a fundamental topic of study with important practical implications. For example, anisotropic displacement parameters (ADPs) are routinely used in physics, chemistry, and crystallography to quantify the thermal motion of atoms in crystals. ADPs are commonly derived from diffraction experiments, but recent developments have also enabled their first-principles prediction using periodic density-functional theory (DFT). Here, we combine experiments and dispersion-corrected DFT to quantify lattice thermal expansion and ADPs in crystalline α-sulfur (S8), a prototypical elemental solid that is controlled by the interplay of covalent and van der Waals interactions. We begin by reporting on single-crystal and powder X-ray diffraction measurements that provide new and improved reference data from 10 K up to room temperature. We then use several popular dispersion-corrected DFT methods to predict vibrational and thermal properties of α-sulfur, including the anisotropic lattice thermal expansion. Hereafter, ADPs are derived in the commonly used harmonic approximation (in the computed zero-Kelvin structure) and also in the quasi-harmonic approximation (QHA) which takes the predicted lattice thermal expansion into account. At the PPBE+D3(BJ) level, the QHA leads to excellent agreement with experiments. Finally, more general implications of this study for theory and experiment are discussed.
10. Theory of the lattice dynamics of model crystals containing screw dislocations
Glass, N. E.
1976-08-01
A theoretical study of the lattice dynamics of a simple cubic model-crystal is made. The perturbation matrix of a single screw dislocation is determined and is used with the perfect lattice Green function to find four secular equations for the frequencies altered by the dislocation. The solutions yield, depending on the model parameters, up to four separate bands of optic localized-modes across the Brillouin zone. No shifts in the perfect lattice acoustical bands are found. The frequencies of the dislocation-induced localized modes are well separated from the frequencies of the perfect lattice modes and should present no difficulty in being distinguished experimentally. The Green function of the lattice containing many parallel screw dislocations is determined by following the method in use for point defects. With this imperfect-lattice Green function, the neutron cross-section for coherent one-phonon inelastic scattering by the dislocation localized-modes is obtained. Using model parameters corresponding to simple metals, the numerical evaluation yields cross-sections on the borderline of present capabilities for experimental detection and indicates the desirability of an experimental test-search. The most important parameter is found to be the ratio of the longitudinal (lambda) to the transverse (..mu..) force constants. As lambda:..mu.. increases, the localized-mode branches separate, the many-dislocation effects become noticeable, and the cross-section for inelastic scattering by the localized-modes rises. Crystals undergoing transverse mode softening, in which lambda:..mu.. grows as ..mu.. tends toward zero, may be useful in the experimental detection of dislocation-induced lattice modes.
11. GINZBURG-LANDAU THEORY AND VORTEX LATTICE OF HIGH-TEMPERATURE SUPERCONDUCTORS
ZHOU SHI-PING
2001-01-01
The thermodynamics of the vortex lattice of high-temperature superconductors has been studied by solving the generalized Ginzburg-Landau equations derived microscopically. Our numerical simulation indicates that the structure of the vortex lattice is oblique at the temperature far away from the transition temperature Tc, where the mixed s-dx2-ya state is expected to have the lowest energy. Whereas, very close to Tc, the dx2-ya wave is slightly lower energetically, and a triangular vortex lattice recovers. The coexistence and the coupling between the s and d waves would account for the unusual dynamic behaviours such as the upward curvature of the upper critical field curve Hc2(T), as observed in dc magnetization measurements on single-crystal YBa2Cu307 samples.
12. Dynamic Effective Medium Theory for Two-Dimensional Non-Magnetic Metamaterial Lattices using Multipole Expansion
Chremmos, Ioannis; Giamalaki, Melpomeni; Yannopapas, Vassilios; Paspalakis, Emmanuel
2014-01-01
We present a formulation for deriving effective medium properties of infinitely periodic two-dimensional metamaterial lattice structures beyond the static and quasi-static limits. We utilize the multipole expansions, where the polarization currents associated with the supported Bloch modes are expressed via the electric dipole, magnetic dipole, and electric quadrupole moments per unit length. We then propose a method to calculate the Bloch modes based on the lattice geometry and individual unit element structure. The results revert to well-known formulas in the quasistatic limit and are useful for the homogenization of nanorod-type metamaterials which are frequently used in optical applications.
13. Chiral extension of lattice field theory with Ginsparg-Wilson fermions
Lim, Kyung-Taek
In 1994, Brower, Shen and Tan proposed "chirally extended QCD" (or XQCD), and current research extends this method to incorporate fermions obeying Ginsparg-Wilson relation, e.g. Overlap fermion. The hope in this research is that the XQCD can overcome the difficulty in standard lattice approach associated with small quark mass by adding explicit fields while maintaining chiral symmetry on the lattice, and that the XQCD has desired continuum limit. I show that the 4-d Yukawa Overlap XQCD fermion action can be derived from the standard 5-d domain-wall action. I also present study on the imaginary part of the determinant of the coset XQCD Dirac operator.
14. Composite (Goldstone) Higgs Dynamics on the Lattice: Spectrum of SU(2) Gauge Theory with two Fundamental Fermions
Arthur, Rudy; Hansen, Martin; Hietanen, Ari; Lewis, Randy; Pica, Claudio; Sannino, Francesco
2014-01-01
We study the meson spectrum of the SU(2) gauge theory with two Wilson fermions in the fundamental representation. The theory unifies both Technicolor and composite Goldstone Boson Higgs models of electroweak symmetry breaking. We have calculated the masses of the lightest spin one vector and axial vector mesons. In addition, we have also obtained preliminary results for the mass of the lightest scalar (singlet) meson state. The simulations have been done with multiple masses and two different lattice spacings for chiral and continuum extrapolations. The spin one meson masses set lower limits for accelerator experiments, whereas the scalar meson will mix with a pGB of the theory and produce two scalar states. The lighter of the states is the 125 GeV Higgs boson, and the heavier would be a new yet unobserved scalar state.
15. A novel approach for computing glueball masses and matrix elements in Yang-Mills theories on the lattice
Della Morte, Michele
2011-01-01
We make use of the global symmetries of the Yang-Mills theory on the lattice to design a new computational strategy for extracting glueball masses and matrix elements which achieves an exponential reduction of the statistical error with respect to standard techniques. By generalizing our previous work on the parity symmetry, the partition function of the theory is decomposed into a sum of path integrals each giving the contribution from multiplets of states with fixed quantum numbers associated to parity, charge conjugation, translations, rotations and central conjugations Z_N^3. Ratios of path integrals and correlation functions can then be computed with a multi-level Monte Carlo integration scheme whose numerical cost, at a fixed statistical precision and at asymptotically large times, increases power-like with the time extent of the lattice. The strategy is implemented for the SU(3) Yang--Mills theory, and a full-fledged computation of the mass and multiplicity of the lightest glueball with vacuum quantum ...
16. Renormalization of two-loop diagrams in scalar lattice field theory
Borasoy, B
2006-01-01
We present a method to calculate to very high precision the coefficients of the divergences occuring in two-loop diagrams for a massive scalar field on the lattice. The approach is based on coordinate space techniques and extensive use of the precisely known Green's function.
17. Anatomy of isolated monopole in Abelian projection od SU(2) lattice gauge theory
Belavin, V A; Veselov, A I
2001-01-01
The structure of the isolated static monopolies in the maximum Abelian projection of the SU(2) gluodynamics on the lattice studied. The standard parametrization of the coupling matrix was used by determining the maximum Abelian projection of the R functional maximization relative to all scale transformations. The monopole radius R approx = 0.06 fm is evaluated
18. APPLICATION OF PARAMETRIC DERIVATION METHOD TO THE CALCULATION OF PEIERLS ENERGY AND PEIERLS STRESS IN LATTICE THEORY
Xiaozhi Wu; Shaofeng Wang
2007-01-01
Applying the parametric derivation method, Peierls energy and Peierls stress are calculated with a non-sinusoidal force law in the lattice theory, while the results obtained by the power-series expansion according to sinusoidal law can be deduced as a limiting case of nonsinusoidal law. The simplified expressions of Peierls energy and Peierls stress are obtained for the limit of wide and narrow. Peierls energy and Peierls stress decrease monotonically with the factor of modification of force law. Present results can be used expediently for prediction of the correct order of magnitude of Peierls stress for materials.
19. Individual complex Dirac eigenvalue distributions from random matrix theory and comparison to quenched lattice QCD with a quark chemical potential.
Akemann, G; Bloch, J; Shifrin, L; Wettig, T
2008-01-25
We analyze how individual eigenvalues of the QCD Dirac operator at nonzero quark chemical potential are distributed in the complex plane. Exact and approximate analytical results for both quenched and unquenched distributions are derived from non-Hermitian random matrix theory. When comparing these to quenched lattice QCD spectra close to the origin, excellent agreement is found for zero and nonzero topology at several values of the quark chemical potential. Our analytical results are also applicable to other physical systems in the same symmetry class.
20. Lattice parameters and electronic structure of BeMgZnO quaternary solid solutions: Experiment and theory
Toporkov, M.; Avrutin, V.; Morkoç, H.; Özgür, Ü. [Department of Electrical and Computer Engineering, Virginia Commonwealth University, Richmond, Virginia 23284 (United States); Demchenko, D. O. [Department of Physics, Virginia Commonwealth University, Richmond, Virginia 23284 (United States); Zolnai, Z. [MTA EK Institute of Technical Physics and Materials Science, Budapest (Hungary); Volk, J. [Department of Electrical and Computer Engineering, Virginia Commonwealth University, Richmond, Virginia 23284 (United States); MTA EK Institute of Technical Physics and Materials Science, Budapest (Hungary)
2016-03-07
Be{sub x}Mg{sub y}Zn{sub 1−x−y}O semiconductor solid solutions are attractive for UV optoelectronics and electronic devices owing to their wide bandgap and capability of lattice-matching to ZnO. In this work, a combined experimental and theoretical study of lattice parameters, bandgaps, and underlying electronic properties, such as changes in band edge wavefunctions in Be{sub x}Mg{sub y}Zn{sub 1−x−y}O thin films, is carried out. Theoretical ab initio calculations predicting structural and electronic properties for the whole compositional range of materials are compared with experimental measurements from samples grown by plasma assisted molecular beam epitaxy on (0001) sapphire substrates. The measured a and c lattice parameters for the quaternary alloys Be{sub x}Mg{sub y}Zn{sub 1−x} with x = 0−0.19 and y = 0–0.52 are within 1%–2% of those calculated using generalized gradient approximation to the density functional theory. Additionally, composition independent ternary BeZnO and MgZnO bowing parameters were determined for a and c lattice parameters and the bandgap. The electronic properties were calculated using exchange tuned Heyd-Scuseria-Ernzerhof hybrid functional. The measured optical bandgaps of the quaternary alloys are in good agreement with those predicted by the theory. Strong localization of band edge wavefunctions near oxygen atoms for BeMgZnO alloy in comparison to the bulk ZnO is consistent with large Be-related bandgap bowing of BeZnO and BeMgZnO (6.94 eV). The results in aggregate show that precise control over lattice parameters by tuning the quaternary composition would allow strain control in Be{sub x}Mg{sub y}Zn{sub 1−x−y}O/ZnO heterostructures with possibility to achieve both compressive and tensile strain, where the latter supports formation of two-dimensional electron gas at the interface.
1. Critical adsorbing properties in slits predicted by tradi-tional polymer adsorption theories on Ising lattice
LIU Meitang; MU Bozhong
2005-01-01
The critical adsorbing properties in slits and three-dimension (3D) phase transitions can be predicted by either Freed theory or Flory-Huggins theory. The mean field approximation in Flory-Huggins theory may cause apparent system errors, from which one can observe two-dimension (2D) phase transitions although it is not true. Monte Carlo simulation has demonstrated that Freed theory is more suitable for predicting adsorbing properties of fluids in slits than Flory-Huggins theory. It was found that from Freed theory prediction multilevel adsorption occurs in slits and the spreading pressure curves exhibit binodal points.
2. Noether-type theory for discrete mechanico-electrical dynamical systems with nonregular lattices
2010-01-01
We investigate Noether symmetries and conservation laws of the discrete mechanico-electrical systems with nonregular lattices.The operators of discrete transformation and discrete differentiation to the right and left are introduced for the systems.Based on the invariance of discrete Hamilton action on nonregular lattices of the systems with the dissipation forces under the infinitesimal transformations with respect to the time,generalized coordinates and generalized charge quantities,we work out the discrete analog of the generalized variational formula.From this formula we derive the discrete analog of generalized Noether-type identity,and then we present the generalized quasi-extremal equations and properties of these equations for the systems.We also obtain the discrete analog of Noether-type conserved laws and the discrete analog of generalized Noether theorems for the systems.Finally we use an example to illustrate these results.
3. Geometrical Lattice models for N=2 supersymmetric theories in two dimensions
Saleur, H
1992-01-01
We introduce in this paper two dimensional lattice models whose continuum limit belongs to the $N=2$ series. The first kind of model is integrable and obtained through a geometrical reformulation, generalizing results known in the $k=1$ case, of the $\\Gamma_{k}$ vertex models (based on the quantum algebra $U_{q}sl(2)$ and representation of spin $j=k/2$). We demonstrate in particular that at the $N=2$ point, the free energy of the $\\Gamma_{k}$ vertex model can be obtained exactly by counting arguments, without any Bethe ansatz computation, and we exhibit lattice operators that reproduce the chiral ring. The second class of models is more adequately described in the language of twisted $N=2$ supersymmetry, and consists of an infinite series of multicritical polymer points, which should lead to experimental realizations. It turns out that the exponents $\ 4. Dynamic mean field theory for lattice gas models of fluids confined in porous materials: higher order theory based on the Bethe-Peierls and path probability method approximations. Edison, John R; Monson, Peter A 2014-07-14 Recently we have developed a dynamic mean field theory (DMFT) for lattice gas models of fluids in porous materials [P. A. Monson, J. Chem. Phys. 128(8), 084701 (2008)]. The theory can be used to describe the relaxation processes in the approach to equilibrium or metastable states for fluids in pores and is especially useful for studying system exhibiting adsorption/desorption hysteresis. In this paper we discuss the extension of the theory to higher order by means of the path probability method (PPM) of Kikuchi and co-workers. We show that this leads to a treatment of the dynamics that is consistent with thermodynamics coming from the Bethe-Peierls or Quasi-Chemical approximation for the equilibrium or metastable equilibrium states of the lattice model. We compare the results from the PPM with those from DMFT and from dynamic Monte Carlo simulations. We find that the predictions from PPM are qualitatively similar to those from DMFT but give somewhat improved quantitative accuracy, in part due to the superior treatment of the underlying thermodynamics. This comes at the cost of greater computational expense associated with the larger number of equations that must be solved. 5. The theoretical analysis of the lattice hydrodynamic models for traffic flow theory Ge, H. X.; Cheng, R. J.; Lei, L. 2010-07-01 The lattice hydrodynamic model is not only a simplified version of the macroscopic hydrodynamic model, but also connected with the microscopic car following model closely. The modified Korteweg-de Vries (mKdV) equation related to the density wave in a congested traffic region has been derived near the critical point since Nagatani first proposed it. But the Korteweg-de Vries (KdV) equation near the neutral stability line has not been studied, which has been investigated in detail for the car following model. We devote ourselves to obtaining the KdV equation from the original lattice hydrodynamic models and the KdV soliton solution to describe the traffic jam. Especially, we obtain the general soliton solution of the KdV equation and the mKdV equation. We review several lattice hydrodynamic models, which were proposed recently. We compare the modified models and carry out some analysis. Numerical simulations are conducted to demonstrate the nonlinear analysis results. 6. Molecular structure stability of short-chain chlorinated paraffins (SCCPs): Evidence from lattice compatibility and Simha-Somcynsky theories Yumak, A.; Boubaker, K.; Petkova, P.; Yahsi, U. 2015-10-01 In is known that short-chain chlorinated paraffins (SCCPs) are highly complex technical mixtures of polychlorinated n-alkanes with single chlorine content. Due to their physical properties (viscosity, flame resistance) they are used in many different applications, such as lubricant additives, metal processing, leather fat-liquoring, plastics softening, PVC plasticizing and flame retardants in paints, adhesives and sealants. SCCPs are studied here in terms of processing-linked molecular structure stability, under Simha and Somcynsky-EOS theory calculations and elements from Simha-Somcynsky-related Lattice Compatibility Theory. Analyses were carried out on 1-chloropropane, 2-chloropropane, 1-chlorobutane, 2-chlorobutane, 1-chloro 2-methylane, and 2-chloro 2-methylane as (SCCPs) universal representatives. This paper gives evidence to this stability and reviews the current state of knowledge and highlights the need for further research in order to improve future (SCCPs) monitoring efforts. 7. Lattice Regularization and Symmetries Hasenfratz, Peter; Von Allmen, R; Allmen, Reto von; Hasenfratz, Peter; Niedermayer, Ferenc 2006-01-01 Finding the relation between the symmetry transformations in the continuum and on the lattice might be a nontrivial task as illustrated by the history of chiral symmetry. Lattice actions induced by a renormalization group procedure inherit all symmetries of the continuum theory. We give a general procedure which gives the corresponding symmetry transformations on the lattice. 8. The exact decomposition of gauge variables in lattice Yang-Mills theory Shibata, Akihiro; Kondo, Kei-Ichi; Shinohara, Toru 2010-07-01 In this Letter, we consider lattice versions of the decomposition of the Yang-Mills field a la Cho-Faddeev-Niemi, which was extended by Kondo, Shinohara and Murakami in the continuum formulation. For the SU (N) gauge group, we propose a set of defining equations for specifying the decomposition of the gauge link variable and solve them exactly without using the ansatz adopted in the previous studies for SU (2) and SU (3). As a result, we obtain the general form of the decomposition for SU (N) gauge link variables and confirm the previous results obtained for SU (2) and SU (3). 9. Numerical study of tree-level improved lattice gradient flows in pure Yang-Mills theory Kamata, Norihiko 2016-01-01 We study several types of tree-level improvement in the Yang-Mills gradient flow method in order to reduce the lattice discretization errors in line with a reference [Fodor et al., arXiv:1406.0827]. The tree-level$\\mathcal{O}(a^2)$improvement can be achieved in a simple manner, where an appropriate weighted average is computed between two definitions of the action density$\\langle E(t)\\rangle$measured at every flow time$t$. We further develop the idea of achieving the tree-level$\\mathcal{O}(a^4)$improvement. For testing our proposal, we present numerical results of$\\langle E(t)\\rangle$obtained on gauge configurations generated with the Wilson and Iwasaki gauge actions at three lattice spacings ($a\\approx 0.1, 0.07$and 0.05 fm). Our results show that tree-level improved flows significantly eliminate the discretization corrections in the relatively small-$t$regime. To demonstrate the feasibility of our proposal, we also study the scaling behavior of the dimensionless combinations of the$\\Lambda_{\\ove...
10. On the development of a model predicting the recrystallization texture of aluminum using the Taylor model for rolling textures and the coincidence lattice site theory
T, Morimoto; F, Yoshida; A, Yanagida; J, Yanagimoto
2015-04-01
First, hardening model in f.c.c. metals was formulated with collinear interactions slips, Hirth slips and Lomer-Cottrell slips. Using the Taylor and the Sachs rolling texture prediction model, the residual dislocation densities of cold-rolled commercial pure aluminum were estimated. Then, coincidence site lattice grains were investigated from observed cold rolling texture. Finally, on the basis of oriented nucleation theory and coincidence site lattice theory, the recrystallization texture of commercial pure aluminum after low-temperature annealing was predicted.
11. Diffractive stacks of metamaterial lattices with a complex unit cell: Self-consistent long-range bianisotropic interactions in experiment and theory
Kwadrin, Andrej; Koenderink, A. Femius
2014-01-01
Metasurfaces and metamaterials promise arbitrary rerouting of light using two-dimensional (2D) planar arrangements of electric and magnetic scatterers, respectively, 3D stacks built out of such 2D planes. An important problem is how to self-consistently model the response of these systems in a manner that retains dipole intuition yet does full justice to the self-consistent multiple scattering via near-field and far-field retarded interactions. We set up such a general model for metamaterial lattices of complex 2D unit cells of poly-atomic basis as well as allowing for stacking in a third dimension. In particular, each scatterer is quantified by a magnetoelectric polarizability tensor and Ewald lattice summation deals with all near-field and long-range retarded electric, magnetic, and magnetoelectric couplings self-consistently. We show in theory and experiment that grating diffraction orders of dilute split ring lattices with complex unit cells show a background-free signature of magnetic dipole response. For denser lattices experiment and theory show that complex unit cells can reduce the apparent effect of bianisotropy, i.e., the strong oblique-incidence handed response that was reported for simple split ring lattices. Finally, the method is applied to calculate transmission of finite stacks of lattices. Thereby our simple methodology allows us to trace the emergence of effective material constants when building a 3D metamaterial layer by layer, as well as facilitating the design of metasurfaces.
12. Bond operator theory for the frustrated anisotropic Heisenberg antiferromagnet on a square lattice
Pires, A.S.T., E-mail: [email protected] [Departamento de Fisica, Universidade Federal de Minas Gerais, Belo Horizonte, Cp 702, 30123-970 MG (Brazil)
2012-07-15
The quantum anisotropic antiferromagnetic Heisenberg model with single ion anisotropy, spin S=1 and up to the next-next-nearest neighbor coupling (the J{sub 1}-J{sub 2}-J{sub 3} model) on a square lattice, is studied using the bond-operator formalism in a mean field approximation. The quantum phase transitions at zero temperature are obtained. The model features a complex T=0 phase diagram, whose ordering vector is subject to quantum corrections with respect to the classical limit. The phase diagram shows a quantum paramagnetic phase situated among Neel, spiral and collinear states. - Highlights: Black-Right-Pointing-Pointer The quantum phase transition at zero temperature is studied. Black-Right-Pointing-Pointer The phase diagram up to the next-next-nearest neighbor coupling is calculated. Black-Right-Pointing-Pointer The energy gap is calculated in several regions of the phase diagram.
13. Coupling LAMMPS with Lattice Boltzmann fluid solver: theory, implementation, and applications
Tan, Jifu; Sinno, Talid; Diamond, Scott
2016-11-01
Studying of fluid flow coupled with solid has many applications in biological and engineering problems, e.g., blood cell transport, particulate flow, drug delivery. We present a partitioned approach to solve the coupled Multiphysics problem. The fluid motion is solved by the Lattice Boltzmann method, while the solid displacement and deformation is simulated by Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The coupling is achieved through the immersed boundary method so that the expensive remeshing step is eliminated. The code can model both rigid and deformable solids. The code also shows very good scaling results. It was validated with classic problems such as migration of rigid particles, ellipsoid particle's orbit in shear flow. Examples of the applications in blood flow, drug delivery, platelet adhesion and rupture are also given in the paper. NIH.
14. Geometry of dynamics and phase transitions in classical lattice $\\phi^{4}$ theories
Caiani, L; Clementi, C; Pettini, G; Pettini, M; Gatto, R; Caiani, Lando; Casetti, Lapo; Clementi, Cecilia; Pettini, Giulio; Pettini, Marco; Gatto, Raoul
1998-01-01
We perform a microcanonical study of classical lattice phi^4 field models in 3 dimensions with O(n) symmetries. The Hamiltonian flows associated to these systems that undergo a second order phase transition in the thermodynamic limit are here investigated. The microscopic Hamiltonian dynamics neatly reveals the presence of a phase transition through the time averages of conventional thermodynamical observables. Moreover, peculiar behaviors of the largest Lyapunov exponents at the transition point are observed. A Riemannian geometrization of Hamiltonian dynamics is then used to introduce other relevant observables, that are measured as functions of both energy density and temperature. On the basis of a simple and abstract geometric model, we suggest that the apparently singular behaviour of these geometric observables might probe a major topological change of the manifolds whose geodesics are the natural motions.
15. Lattice oscillator model, scattering theory and a many-body problem
Valiente, Manuel, E-mail: [email protected] [Department of Physics and Astronomy, Lundbeck Foundation Theoretical Center for Quantum System Research, Aarhus University, DK-8000 Aarhus C (Denmark)
2011-11-18
We propose a model for the quantum harmonic oscillator on a discrete lattice which can be written in a supersymmetric form, in contrast with the more direct discretization of the harmonic oscillator. Its ground state is easily found to be annihilated by the annihilation operator defined here, and its excitation spectrum is obtained numerically. We then define an operator whose continuum limit corresponds to an angular momentum in terms of the creation-annihilation operators of our model. Coherent states with the correct continuum limit are also constructed. The versatility of the model is then used to calculate, in a simple way, the generalized position-dependent scattering length for a particle colliding with a single static impurity in a periodic potential and the exact ground state of an interacting many-body problem in a one-dimensional ring. (paper)
16. Density of States FFA analysis of SU(3) lattice gauge theory at a finite density of color sources
Giuliani, Mario; Gattringer, Christof
2017-10-01
We present a Density of States calculation with the Functional Fit Approach (DoS FFA) in SU(3) lattice gauge theory with a finite density of static color sources. The DoS FFA uses a parameterized density of states and determines the parameters of the density by fitting data from restricted Monte Carlo simulations with an analytically known function. We discuss the implementation of DoS FFA and the results for a qualitative picture of the phase diagram in a model which is a further step towards implementing DoS FFA in full QCD. We determine the curvature κ in the μ-T phase diagram and find a value close to the results published for full QCD.
17. Cluster mean-field theory study of J1-J2 Heisenberg model on a square lattice.
Ren, Yong-Zhi; Tong, Ning-Hua; Xie, Xin-Cheng
2014-03-19
We study the spin-1/2 J1-J2 Heisenberg model on a square lattice using the cluster mean-field theory. We find a rapid convergence of phase boundaries with increasing cluster size. By extrapolating the cluster size L to infinity, we obtain accurate phase boundaries J(c1)(2) ≈ 0.42 (between the Néel antiferromagnetic phase and non-magnetic phase), and J(c2)(2) ≈ 0.59 (between non-magnetic phase and the collinear antiferromagnetic phase). Our results support the second-order phase transition at J(c1)(2) and the first-order one at J(c2)(2). For the spin-anisotropic J1-J2 model, we present its finite temperature phase diagram and demonstrate that the non-magnetic state is unstable towards the first-order phase transition under intermediate spin anisotropy.
18. Finite-size scaling tests for spectra in SU(3) lattice gauge theory coupled to 12 fundamental flavor fermions
Degrand, Thomas
2011-12-01
I carry out a finite-size scaling study of the correlation length in SU(3) lattice gauge theory coupled to 12 fundamental flavor fermions, using recent data published by Fodor, Holland, Kuti, Nógradi and Schroeder [Z. Fodor, K. Holland, J. Kuti, D. Nogradi, and C. Schroeder, Phys. Lett. B 703, 348 (2011).PYLBAJ0370-269310.1016/j.physletb.2011.07.037]. I make the assumption that the system is conformal in the zero-mass, infinite volume limit, that scaling is violated by both nonzero fermion mass and by finite volume, and that the scaling function in each channel is determined self-consistently by the data. From several different observables I extract a common exponent for the scaling of the correlation length ξ with the fermion mass mq, ξ˜mq-1/ym with ym˜1.35. Shortcomings of the analysis are discussed.
19. Systematics of flux tubes in the dual Ginzburg-Landau theory and Casimir scaling hypothesis: folklore and lattice facts
Koma, Y
2003-01-01
The ratios between the string tensions sigma sub D of color-electric flux tubes in higher and fundamental SU(3) representations, d sub D ident to sigma sub D /sigma sub 3 , are systematically studied in a Weyl symmetric formulation of the DGL theory. The ratio is found to depend on the Ginzburg-Landau (GL) parameter, kappa ident to m subchi/m sub B , the mass ratio between the monopoles (m subchi) and the masses of the dual gauge bosons (m sub B). While the ratios d sub D follow a simple flux counting rule in the Bogomol'nyi limit, kappa=1.0, systematic deviations appear with increasing kappa due to interactions between the fundamental flux inside a higher representation flux tube. We find that in a type-II dual superconducting vacuum near kappa= 3.0 this leads to a consistent description of the ratios d sub D as observed in lattice QCD simulations. (orig.)
20. Intersections of thick Center Vortices, Dirac Eigenmodes and Fractional Topological Charge in SU(2) Lattice Gauge Theory
Höllwieser, Roman; Heller, Urs M
2011-01-01
Intersections of thick, plane vortices are characterized by the topological charge $|Q|=1/2$. We compare such intersections with the distribution of zeromodes of the Dirac operator in the fundamental and adjoint representation using both the overlap and asqtad staggered fermion formulations in SU(2)-lattice gauge theory. We analyze configurations with four intersections and find that the probability density distribution of fundamental zeromodes in the intersection plane differs significantly from the one obtained analytically in [Phys.\\ Rev.\\ D 66, 85004 (2002)]. The Dirac eigenmodes are clearly sensitive to the traces of the Polyakov (Wilson) lines and do not exactly locate topological charge contributions. Although, the adjoint Dirac operator is able to produce zeromodes for configurations with topological charge $|Q|=1/2$, they do not locate single vortex intersections, as we prove by forming arbitrary linear combinations of these zeromodes - their scalar density peaks at least at two intersection points. ...
1. Testing the Standard Model and Fundamental Symmetries in Nuclear Physics with Lattice QCD and Effective Field Theory
Walker-Loud, Andre [College of William and Mary, Williamsburg, VA (United States)
2016-10-14
The research supported by this grant is aimed at probing the limits of the Standard Model through precision low-energy nuclear physics. The work of the PI (AWL) and additional personnel is to provide theory input needed for a number of potentially high-impact experiments, notably, hadronic parity violation, Dark Matter direct detection and searches for permanent electric dipole moments (EDMs) in nucleons and nuclei. In all these examples, a quantitative understanding of low-energy nuclear physics from the fundamental theory of strong interactions, Quantum Chromo-Dynamics (QCD), is necessary to interpret the experimental results. The main theoretical tools used and developed in this work are the numerical solution to QCD known as lattice QCD (LQCD) and Effective Field Theory (EFT). This grant is supporting a new research program for the PI, and as such, needed to be developed from the ground up. Therefore, the first fiscal year of this grant, 08/01/2014-07/31/2015, has been spent predominantly establishing this new research effort. Very good progress has been made, although, at this time, there are not many publications to show for the effort. After one year, the PI accepted a job at Lawrence Berkeley National Laboratory, so this final report covers just a single year of five years of the grant.
2. Numerical study of tree-level improved lattice gradient flows in pure Yang-Mills theory
Kamata, Norihiko; Sasaki, Shoichi
2017-03-01
We study several types of tree-level improvement in the Yang-Mills gradient flow method in order to reduce the lattice discretization errors in line with Fodor et al. [J. High Energy Phys. 09 (2014) 018., 10.1007/JHEP09(2014)018]. The tree-level O (a2) improvement can be achieved in a simple manner, where an appropriate weighted average is computed between the plaquette and clover-leaf definitions of the action density ⟨E (t )⟩ measured at every flow time t . We further develop the idea of achieving the tree-level O (a4) improvement within a usage of actions consisting of the 1 ×1 plaquette and 1 ×2 planar loop for both the flow and gauge actions. For testing our proposal, we present numerical results for ⟨E (t )⟩ obtained on gauge configurations generated with the Wilson and Iwasaki gauge actions at three lattice spacings (a ≈0.1 ,0.07 , and 0.05 fm). Our results show that tree-level improved flows significantly eliminate the discretization corrections on t2⟨E (t )⟩ in the relatively small-t regime for up to t ≳a2 . To demonstrate the feasibility of our tree-level improvement proposal, we also study the scaling behavior of the dimensionless combinations of the ΛMS ¯ parameter and the new reference scale tX, which is defined through tX2⟨E (tX)⟩=X for the smaller X , e.g., X =0.15 . It is found that √{t0.15 }ΛMS ¯ shows a nearly perfect scaling behavior as a function of a2 regardless of the types of gauge action and flow, after tree-level improvement is achieved up to O (a4) . Further detailed study of the scaling behavior exposes the presence of the remnant O (g2 na2) corrections, which are beyond the tree level. Although our proposal is not enough to eliminate all O (a2) effects, we show that the O (g2 na2) corrections can be well under control even by the simplest tree-level O (a2) improved flow.
3. Isotriplet Dark Matter on the Lattice: SO(4)-gauge theory with two Vector Wilson fermions
Hietanen, Ari; Sannino, Francesco; Søndergaard, Ulrik Ishøj
2012-01-01
We present preliminary results for simulations of SO(4)-gauge theory with two Dirac Wilson fermions transforming according to the vector representation. We map out the phase diagram including the strong coupling bulk phase transition line as well as the zero PCAC-mass line. In addition, we measure the pseudo scalar and vector meson masses, and investigate whether the theory features chiral symmetry breaking. If the theory is used for breaking the electroweak symmetry dynamically it is the orthogonal group equivalent of the Minimal Walking Technicolor model but with the following distinctive features: a] It provides a natural complex weak isotriplet of Goldstone bosons of which the neutral component can be identified with a light composite dark matter state; b] It is expected to break the global symmetry spontaneously; c] It is free from fermionic composite states made by a techniglue and a technifermion.
4. Lattice instability and martensitic transformation in LaAg predicted from first-principles theory
Vaitheeswaran, G.; Kanchana, V.; Zhang, X.
2012-01-01
, calculated using density functional perturbation theory, are in good agreement with available inelastic neutron scattering data. Under pressure, the phonon dispersions develop imaginary frequencies, starting at around 2.3 GPa, in good accordance with the martensitic instability observed above 3.4 GPa...
5. A portable high-quality random number generator for lattice field theory simulations
Lüscher, Martin
1994-01-01
The theory underlying a proposed random number generator for numerical simulations in elementary particle physics and statistical mechanics is discussed. The generator is based on an algorithm introduced by Marsaglia and Zaman, with an important added feature leading to demonstrably good statistical properties. It can be implemented exactly on any computer complying with the IEEE--754 standard for single precision floating point arithmetic.
6. A Portable High-Quality Random Number Generator for Lattice Field Theory Simulations
Luescher, Martin
1993-01-01
The theory underlying a proposed random number generator for numerical simulations in elementary particle physics and statistical mechanics is discussed. The generator is based on an algorithm introduced by Marsaglia and Zaman, with an important added feature leading to demonstrably good statistical properties. It can be implemented exactly on any computer complying with the IEEE--754 standard for single precision floating point arithmetic.
7. Investigating the sign problem for two-dimensional $\\mathcal{N}=(2,2)$ and $\\mathcal{N}=(8,8)$ lattice super Yang--Mills theories
Galvez, Richard; Joseph, Anosh; Mehta, Dhagash
2012-01-01
Recently there has been some controversy in the literature concerning the existence of a fermion sign problem in the $\\mathcal{N}=(2,2)$ supersymmetric Yang--Mills (SYM) theories on the lattice. In this work, we address this issue by conducting Monte Carlo simulations not only for $\\mathcal{N}=(2,2)$ but also for $\\mathcal{N}=(8,8)$ SYM in two dimensions for the U(N) theories with N=2, using the new ideas derived from topological twisting followed by geometric discretization. Our results from simulations provide the evidence that these theories do {\\it not} suffer from a sign problem as the continuum limit is approached. These results thus boost confidence that these new lattice formulations can be used successfully to explore the nonperturbative aspects of the four-dimensional $\\mathcal{N}=4$ SYM theory.
8. The Lattice Compatibility Theory LCT: Physical and Chemical Arguments from the Growth Behavior of Doped Compounds in terms of Bandgap Distortion and Magnetic Effects
K. Boubaker
2013-01-01
Full Text Available Physical and chemical arguments for the recently discussed materials-related Lattice Compatibility Theory are presented. The discussed arguments are based on some differences of Mn ions incorporation kinetics inside some compounds. These differences have been evaluated and quantified in terms of alteration of bandgap edges, magnetic patterns, and Faraday effect.
9. Persistent current and Drude weight of one-dimensional interacting fermions on imperfect ring from current lattice density functional theory
Akande, Akinlolu; Sanvito, Stefano
2016-11-01
We perform a numerical study of interacting one-dimensional Hubbard rings with a single impurity potential and pierced by a magnetic flux. Our calculations are carried out at the level of current lattice density functional theory (CLDFT) for the Hubbard model and compared to known results obtained in the thermodynamical limit from the Bethe ansatz. In particular, we investigate the effects of disorder and Coulomb interaction on the persistent current (PC) and the Drude weight. It is found that CLDFT is able to accurately describe qualitative and quantitative features of these ground state properties in the presence of disorder and electronic interaction. When the impurity potential is switched off, the CLDFT approach describes well the velocity of the Luttinger liquid excitations as a function of both interaction strength and electron filling. Then, when the impurity scattering potential is finite, we find the PC to vanish as {{L}-{{α\\text{B}}-1}} for large L and independent on the strength of the scattering potential, in good agreement with Luttinger liquid theory.
10. p-adic string theories provide lattice Discretization to the ordinary string worldsheet.
Ghoshal, Debashis
2006-10-13
A class of models called p-adic strings is useful in understanding the tachyonic instability of string theory. These are found to be empirically related to the ordinary strings in the p-->1 limit. We propose that these models provide discretization for the string worldsheet and argue that the limit is naturally thought of as a continuum limit in the sense of the renormalization group.
11. A portable high-quality random number generator for lattice field theory simulations
Lüscher, Martin
1994-02-01
The theory underlying a proposed random number generator for numerical simulations in elementary particle physics and statistical mechanics is discussed. The generator is based on an algorithm introduced by Marsaglia and Zaman, with an important added feature leading to demonstrably good statistical properties. It can be implemented exactly on any computer complying with the IEEE-754 standard for single-precision floating-point arithmetic.
12. Dynamic mean field theory for lattice gas models of fluid mixtures confined in mesoporous materials.
Edison, J R; Monson, P A
2013-11-12
We present the extension of dynamic mean field theory (DMFT) for fluids in porous materials (Monson, P. A. J. Chem. Phys. 2008, 128, 084701) to the case of mixtures. The theory can be used to describe the relaxation processes in the approach to equilibrium or metastable equilibrium states for fluids in pores after a change in the bulk pressure or composition. It is especially useful for studying systems where there are capillary condensation or evaporation transitions. Nucleation processes associated with these transitions are emergent features of the theory and can be visualized via the time dependence of the density distribution and composition distribution in the system. For mixtures an important component of the dynamics is relaxation of the composition distribution in the system, especially in the neighborhood of vapor-liquid interfaces. We consider two different types of mixtures, modeling hydrocarbon adsorption in carbon-like slit pores. We first present results on bulk phase equilibria of the mixtures and then the equilibrium (stable/metastable) behavior of these mixtures in a finite slit pore and an inkbottle pore. We then use DMFT to describe the evolution of the density and composition in the pore in the approach to equilibrium after changing the state of the bulk fluid via composition or pressure changes.
13. Perturbation Theory for PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold
Jones, H F
2011-01-01
The $PT$ symmetric potential $V_0[\\cos(2\\pi x/a)+i\\lambda\\sin(2\\pi x/a)]$ has a completely real spectrum for $\\lambda\\le 1$, and begins to develop complex eigenvalues for $\\lambda>1$. At the symmetry-breaking threshold $\\lambda=1$ some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to give rise to a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for an initial wave packet this growth is suppressed, giving instead a constant maximum amplitude. We revisit this problem by developing the perturbation theory further. We verify that the results found by Longhi persist to second order, and with different input wave packets we are able to see the seeds in perturbation theory of the phenomenon of birefringence first discovered by El-Ganainy et al.
14. Scattering theory for lattice operators in dimension $d\\geq 3$
Bellissard, Jean
2011-01-01
This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension $d\\geq 3$ the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in presence of embedded eigenvalues and threshold singularities.
15. Electromagnetic superconductivity of vacuum induced by strong magnetic field: Numerical evidence in lattice gauge theory
Braguta, V.V. [IHEP, Protvino, Moscow region, 142284 (Russian Federation); ITEP, B. Cheremushkinskaya str. 25, Moscow, 117218 (Russian Federation); Buividovich, P.V. [ITEP, B. Cheremushkinskaya str. 25, Moscow, 117218 (Russian Federation); JINR, Joliot-Curie str. 6, Dubna, Moscow region, 141980 (Russian Federation); Institute of Theoretical Physics, University of Regensburg, Universitaetsstrasse 31, D-93053 Regensburg (Germany); Chernodub, M.N., E-mail: [email protected] [CNRS, Laboratoire de Mathematiques et Physique Theorique, Universite Francois-Rabelais Tours, Parc de Grandmont, 37200 Tours (France); Department of Physics and Astronomy, University of Gent, Krijgslaan 281, S9, B-9000 Gent (Belgium); Kotov, A.Yu.; Polikarpov, M.I. [ITEP, B. Cheremushkinskaya str. 25, Moscow, 117218 (Russian Federation); MIPT, Institutskii per. 9, Dolgoprudny, Moscow region, 141700 (Russian Federation)
2012-12-05
Using numerical simulations of quenched SU(2) gauge theory we demonstrate that an external magnetic field leads to spontaneous generation of quark condensates with quantum numbers of electrically charged {rho} mesons if the strength of the magnetic field exceeds the critical value eB{sub c}=0.927(77) GeV{sup 2} or B{sub c}=(1.56{+-}0.13) Dot-Operator 10{sup 16} Tesla. The condensation of the charged {rho} mesons in strong magnetic field is a key feature of the magnetic-field-induced electromagnetic superconductivity of the vacuum.
16. Six-dimensional regularization of chiral gauge theories on a lattice
Fukaya, Hidenori; Yamamoto, Shota; Yamamura, Ryo
2016-01-01
We propose a six-dimensional regularization of four dimensional chiral gauge theories. We consider a massive Dirac fermion in six dimensions with two different operators having domain-wall profiles in the fifth and the sixth directions, respectively. A Weyl fermion appears as a localized mode at the junction of the two domain-walls. In our formulation, the Stora-Zumino chain of the anomaly descent equations, starting from the axial $U(1)$ anomaly in six-dimensions to the gauge anomaly in four-dimensions, is naturally embedded. Moreover, a similar inflow of the global anomalies is found. The anomaly free condition is equivalent to requiring that the axial $U(1)$ anomaly and the parity anomaly are canceled among the six-dimensional Dirac fermions. Putting the gauge field at the four- dimensional junction and extending it to the bulk using the Yang-Mills gradient flow, as recently proposed by Grabowska and Kaplan, we define the four-dimensional path integral of the target chiral gauge theory.
17. Technicolor on the Lattice
Pica, C; Lucini, B; Patella, A; Rago, A
2009-01-01
Technicolor theories provide an elegant mechanism for dynamical electroweak symmetry breaking. We will discuss the use of lattice simulations to study the strongly-interacting dynamics of some of the candidate theories, with matter fields in representations other than the fundamental. To be viable candidates for phenomenology, such theories need to be different from a scaled-up version of QCD, which were ruled out by LEP precision measurements, and represent a challenge for modern lattice computations.
18. Connecting phase transitions between the 3-d O(4) Heisenberg model and 4-d SU(2) lattice gauge theory
2011-01-01
SU(2) lattice gauge theory is extended to a larger coupling space where the coupling parameter for horizontal (spacelike) plaquettes, $\\beta_H$, differs from that for vertical (Euclidean timelike) plaquettes, $\\beta_V$. When $\\beta_H \\rightarrow \\infty$ the system, when in Coulomb Gauge, splits into multiple independent 3-d O(4) Heisenberg models on spacelike hyperlayers. Through consideration of the robustness of the Heisenberg model phase transition to small perturbations, and illustrated by Monte Carlo simulations, it is shown that the ferromagnetic phase transition in this model persists for $\\beta_H < \\infty$. Once it has entered the phase-plane it must continue to another edge due to its symmetry-breaking nature, and therefore must necessarily cross the $\\beta_V = \\beta_H$ line at a finite value. Indeed, a higher-order SU(2) phase transition is found at $\\beta = 3.18 \\pm 0.08$, from a finite-size scaling analysis of the Coulomb gauge magnetization from Monte Carlo simulations, which also yields criti...
19. Abelian color cycles: A new approach to strong coupling expansion and dual representations for non-abelian lattice gauge theory
Gattringer, Christof; Marchis, Carlotta
2017-03-01
We propose a new approach to strong coupling series and dual representations for non-abelian lattice gauge theories using the SU(2) case as an example. The Wilson gauge action is written as a sum over "abelian color cycles" (ACC) which correspond to loops in color space around plaquettes. The ACCs are complex numbers which can be commuted freely such that the strong coupling series and the dual representation can be obtained as in the abelian case. Using a suitable representation of the SU(2) gauge variables we integrate out all original gauge links and identify the constraints for the dual variables in the SU(2) case. We show that the construction can be generalized to the case of SU(2) gauge fields with staggered fermions. The result is a strong coupling series where all gauge integrals are known in closed form and we discuss its applicability for possible dual simulations. The abelian color cycle concept can be generalized to other non-abelian gauge groups such as SU(3).
20. The three-loop $\\beta$-function of SU(N) lattice gauge theories with overlap fermions
Constantinou, M
2007-01-01
We briefly report our calculation of the 2-loop coefficient of the coupling constant renormalization function Z_g in lattice perturbation theory. The quantity under study is defined through g_0 = Z_g g, where g_0 (g) is the bare (renormalized) coupling constant. The 2-loop expression for Z_g can be directly related to the 3-loop bare beta-function beta_L(g_0). Our calculation is performed using overlap fermions and Wilson gluons, and the background field technique has been chosen for convenience. Our results depend explicitly on the number of fermion flavors (N_f) and colors (N). Since the dependence of Z_g on the overlap parameter rho cannot be extracted analytically, we tabulate our results for different values of rho in the allowed range (0
1. Systematics of flux tubes in the dual Ginzburg-Landau theory and Casimir scaling hypothesis: folklore and lattice facts
Koma, Y. [Institute for Theoretical Physics, Kanazawa University, Kanazawa, Ishikawa 920-1192 (Japan); Koma, M. [Research Center for Nuclear Physics (RCNP), Osaka University, Mihogaoka 10-1, Ibaraki, Osaka 567-0047 (Japan)
2003-01-01
The ratios between the string tensions {sigma}{sub D} of color-electric flux tubes in higher and fundamental SU(3) representations, d{sub D} {identical_to}{sigma}{sub D}/{sigma}{sub 3}, are systematically studied in a Weyl symmetric formulation of the DGL theory. The ratio is found to depend on the Ginzburg-Landau (GL) parameter, {kappa}{identical_to}m{sub {chi}}/m{sub B}, the mass ratio between the monopoles (m{sub {chi}}) and the masses of the dual gauge bosons (m{sub B}). While the ratios d{sub D} follow a simple flux counting rule in the Bogomol'nyi limit, {kappa}=1.0, systematic deviations appear with increasing {kappa} due to interactions between the fundamental flux inside a higher representation flux tube. We find that in a type-II dual superconducting vacuum near {kappa}= 3.0 this leads to a consistent description of the ratios d{sub D} as observed in lattice QCD simulations. (orig.)
2. Finite-density phase diagram of a (1+1)-d non-abelian lattice gauge theory with tensor networks
Silvi, Pietro; Dalmonte, Marcello; Tschirsich, Ferdinand; Montangero, Simone
2016-01-01
We investigate the finite-density phase diagram of a non-abelian SU(2) lattice gauge theory, encoding Yang-Mills microscopical dynamics, in (1+1)-dimensions using tensor network methods. We numerically characterise the phase diagram as a function of the filling and of the matter-field coupling, individuating different phases, some of them appearing only at finite densities. At unit filling, we find a meson BCS liquid phase, which at strong coupling undergoes a phase transition to a charge density wave of single-site (spin-0) mesons via spontaneous chiral symmetry breaking. At finite densities, the chiral symmetry is restored almost everywhere, and the meson BCS liquid becomes a simple liquid at strong couplings, with the exception of filling two-thirds, where a charge density wave of mesons spreading over neighbouring sites appears. Finally, we individuate two tri-critical points between the chiral and the two liquid phases which are compatible with a SU(2)$_2$ Wess-Zumino-Novikov-Witten model.
3. Calculation of the (liquid + liquid) equilibrium of solutions of hyperbranched polymers with the lattice-cluster theory combined with an association model
Zeiner, T. [Technical University of Dortmund, Lehrstuhl fuer Fluidverfahrenstechnik, Emil-Figge Str. 70, 44227 Dortmund (Germany); Browarzik, C. [Technical University of Berlin, Institut fuer Prozess- und Verfahrenstechnik (TK7), Fachgebiet Thermodynamik und thermische Verfahrenstechnik, Strasse des 17, Juni 135, 10623 Berlin (Germany); Browarzik, D., E-mail: [email protected] [Martin-Luther University Halle-Wittenberg, Institut fuer Chemie/Physikalische Chemie, 06099 Halle (Germany); Enders, S. [Technical University of Berlin, Institut fuer Prozess- und Verfahrenstechnik (TK7), Fachgebiet Thermodynamik und thermische Verfahrenstechnik, Strasse des 17, Juni 135, 10623 Berlin (Germany)
2011-12-15
Highlights: > The (liquid + liquid) equilibrium of hyperbranched polyester solutions is calculated. > The solvents are n-alkanes, propan-1-ol, and butan-1-ol. > The lattice-cluster theory is combined with a chemical association model. > The solvent molecules are assumed to be linear chains of segments. > The calculations agree reasonably well with the experimental data. - Abstract: The (liquid + liquid) equilibrium of solutions of hyperbranched polyesters is calculated with the lattice-cluster theory (LCT) combined with a chemical association model. The considered solvents are n-alkanes as well as propan-1-ol and butan-1-ol. The structure of the solvents is also considered in the framework of the LCT, assuming the solvent molecules as linear chains of several segments. For polymer solutions with the non-associating n-alkanes only the self association of the hyperbranched polymer molecules has to be considered by the chemical association lattice model (CALM). For the solutions of the type alcohol + hyperbranched polymer additionally the cross association is taken into account by a modified version of the extended chemical association lattice model (ECALM). The association effects are proved to influence strongly the phase equilibrium. Calculating the cloud-point curve and the critical point the polydispersity of the polymer samples is neglected. There is a reasonable agreement of the calculated curves with the experimental data taken from the literature.
4. GNSS Ambiguity Resolution Using the Lattice Theory%基于格论的GNSS模糊度解算
刘经南; 于兴旺; 张小红
2012-01-01
Fast and reliable ambiguity resolution plays a critical role in GNSS real-time precise positioning, however, the computational complexity of finding the optimal integer ambiguity vector directly will be high, because there is strong correction between ambiguities. Thus the lattice theory was introduced in this paper to reduce the correlation for improving the computational efficiency of ambiguity resolution, and the main decorrelation methods currently existed were also analyzed and transformed into the methods of lattice reduction. Then the Bootstrapping success rate was suggested and tested by an experiment to show the performance of these decorrelation methods used for long-baseline ambiguity resolution with multi-frequency GNSS signals.%快速、准确地解算整周模糊度是实现GNSS载波相位实时高精度定位的关键，由于模糊度之间的强相关，基于整数最小二乘估计准则时，需要较长的时间才能搜索出最优的整周模糊度向量。为了提高模糊度的搜索效率，本文在扼要介绍格论的理论框架基础上，引入基于格论的模糊度解算方法，通过格基规约来降低模糊度之间的相关性，从而快速搜索出最优的整数模糊度向量。与此同时，将GNSS领域的主要降相关方法统一到格论框架下，探讨了并建议采用Bootstrapping成功率作为格基规约的性能指标之一。最后实验分析了三频多系统长基线相对定位情况下，不同格基规约可获得的性能。
5. Nuclear lattice simulations
Epelbaum E.
2010-04-01
Full Text Available We review recent progress on nuclear lattice simulations using chiral effective field theory. We discuss lattice results for dilute neutron matter at next-to-leading order, three-body forces at next-to-next-toleading order, isospin-breaking and Coulomb effects, and the binding energy of light nuclei.
6. Functional Fit Approach (FFA) for Density of States method: SU(3) spin system and SU(3) lattice gauge theory with static quarks
Giuliani, Mario
2016-01-01
We apply a recently developed variant of the Density of States (DoS) method, the so-called Functional Fit Approach (FFA) to two different models: the SU(3) spin model and SU(3) lattice gauge theory with static quarks. Both models can be derived from QCD and inherit the complex action problem at finite density. We discuss the implementation of DoS FFA in the two models and compute observables related to the particle density. For the SU(3) spin model we show that the results are in good agreement with the results from a Monte Carlo simulation in the dual formulation, which is free of the complex action problem. For the case of SU(3) lattice gauge theory with static quarks we present first results for the particle number as a function of the coupling for different values of the chemical potential.
7. A Brief Analysis on the Cultivation of Children's Lan-guage Competence%幼儿语言表达能力培养浅析
杨兆梅
2013-01-01
幼儿期是语言发展的一个非常重要和关键的时期，应通过谈话、讲述、故事、游戏等多种教育活动，再通过日常生活中、在游戏活动中，发展幼儿语言表达能力。%Early childhood is an important and crucial period for children's language development, so we should develop their lan-guage competence through taking, narrating, stories and games in daily life and activities.
8. Basis reduction for layered lattices
Torreão Dassen, Erwin
2011-01-01
We develop the theory of layered Euclidean spaces and layered lattices. We present algorithms to compute both Gram-Schmidt and reduced bases in this generalized setting. A layered lattice can be seen as lattices where certain directions have infinite weight. It can also be interpre
9. Lattice of ℤ-module
Futa Yuichi
2016-03-01
Full Text Available In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].
10. Supersymmetric correspondence in spectra on a graph and its line graph: From circuit theory to spoof plasmons on metallic lattices
Nakata, Yosuke; Nakanishi, Toshihiro; Miyamaru, Fumiaki; Takeda, Mitsuo Wada; Kitano, Masao
2016-01-01
We investigate the supersymmetry (SUSY) structures for inductor-capacitor circuit networks on a simple regular graph and its line graph. We show that their eigenspectra must coincide (except, possibly, for the highest eigenfrequency) due to SUSY, which is derived from the topological nature of the circuits. To observe this spectra correspondence in the high-frequency range, we study spoof plasmons on metallic hexagonal and kagom\\'e lattices. The band correspondence between them is predicted by a simulation. Using terahertz time-domain spectroscopy, we demonstrate the band correspondence of fabricated metallic hexagonal and kagom\\'e lattices.
11. Branes and integrable lattice models
Yagi, Junya
2016-01-01
This is a brief review of my work on the correspondence between four-dimensional $\\mathcal{N} = 1$ supersymmetric field theories realized by brane tilings and two-dimensional integrable lattice models. I explain how to construct integrable lattice models from extended operators in partially topological quantum field theories, and elucidate the correspondence as an application of this construction.
12. Lattice implementation of Abelian gauge theories with Chern-Simons number and an axion field arXiv
Figueroa, Daniel G.
Real time evolution of classical gauge fields is relevant for a number of applications in particle physics and cosmology, ranging from the early Universe to dynamics of quark-gluon plasma. We present a lattice formulation of the interaction between a $shift$-symmetric field and some $U(1)$ gauge sector, $a(x)\\tilde{F}_{\\mu\ 13. Exact Lattice Supersymmetry Catterall, Simon; Kaplan, David B.; Unsal, Mithat 2009-03-31 We provide an introduction to recent lattice formulations of supersymmetric theories which are invariant under one or more real supersymmetries at nonzero lattice spacing. These include the especially interesting case of N = 4 SYM in four dimensions. We discuss approaches based both on twisted supersymmetry and orbifold-deconstruction techniques and show their equivalence in the case of gauge theories. The presence of an exact supersymmetry reduces and in some cases eliminates the need for fine tuning to achieve a continuum limit invariant under the full supersymmetry of the target theory. We discuss open problems. 14. A Numerical Study of Spectral Flows of Hermitian Wilson-Dirac Operator and The Index Theorem in Abelian Gauge Theories on Finite Lattices Fujiwara, T 2000-01-01 We investigate the index of the Neuberger's Dirac operator in abelian gauge theories on finite lattices by numerically analyzing the spectrum of the hermitian Wilson-Dirac operator for a continuous family of gauge fields connecting different topological sectors. By clarifying the characteristic structure of the spectrum leading to the index theorem we show that the index coincides to the topological charge for a wide class of gauge field configurations. We also argue that the index can be found exactly for some special but nontrivial configurations in two dimensions by directly analyzing the spectrum. 15. A braided Yang-Baxter Algebra in a Theory of two coupled Lattice Quantum KdV algebraic properties and ABA representations Fioravanti, D; Fioravanti, Davide; Rossi, Marco 2001-01-01 A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in involution which form the Cartan sub-algebra of the braided quantum group. Representations diagonalizing these operators are described through relying on an easy generalization of Algebraic Bethe Ansatz techniques. The conjecture that this monodromy matrix algebra leads, {\\it in the cylinder continuum limit}, to a Perturbed Minimal Conformal Field Theory description is analysed and supported. 16. Lattice Bosons Chakrabarti, J; Bagchi, B; Chakrabarti, Jayprokas; Basu, Asis; Bagchi, Bijon 2000-01-01 Fermions on the lattice have bosonic excitations generated from the underlying periodic background. These, the lattice bosons, arise near the empty band or when the bands are nearly full. They do not depend on the nature of the interactions and exist for any fermion-fermion coupling. We discuss these lattice boson solutions for the Dirac Hamiltonian. 17. Measuring on Lattices Knuth, Kevin H. 2009-12-01 Previous derivations of the sum and product rules of probability theory relied on the algebraic properties of Boolean logic. Here they are derived within a more general framework based on lattice theory. The result is a new foundation of probability theory that encompasses and generalizes both the Cox and Kolmogorov formulations. In this picture probability is a bi-valuation defined on a lattice of statements that quantifies the degree to which one statement implies another. The sum rule is a constraint equation that ensures that valuations are assigned so as to not violate associativity of the lattice join and meet. The product rule is much more interesting in that there are actually two product rules: one is a constraint equation arises from associativity of the direct products of lattices, and the other a constraint equation derived from associativity of changes of context. The generality of this formalism enables one to derive the traditionally assumed condition of additivity in measure theory, as well introduce a general notion of product. To illustrate the generic utility of this novel lattice-theoretic foundation of measure, the sum and product rules are applied to number theory. Further application of these concepts to understand the foundation of quantum mechanics is described in a joint paper in this proceedings. 18. Lattice Theory Properties of Fuzzy Rough Sets%粗糙模糊集的格论性质 刘贵龙 2003-01-01 Let U denote a finite and nonempty set called the universe, and P(U) a power set. Suppose R is an equiva-lence relation on U. Consider the equivalence relation ≈ (X≈Y←→-RX=-R and RX=RY, X,Y, ∈ F(U)) on F(U),the quotient set denoted by F(U)/≈. In this paper we show that F(U)/≈ is a distributive lattice. 19. Critical line of the Φ4 scalar field theory on a 4D cubic lattice in the local potential approximation J.-M. Caillol 2013-01-01 Full Text Available We establish the critical line of the one-component Φ4 (or Landau-Ginzburg model on a simple four dimensional cubic lattice. Our study is performed in the framework of the non-perturbative renormalization group in the local potential approximation with a soft infra-red regulator. The transition is found to be of second order even in the Gaussian limit where first order would be expected according to some recent theoretical predictions. 20. Lattice gradient flow with tree-level$\\mathcal{O}(a^4)$improvement in pure Yang-Mills theory Kamata, Norihiko 2015-01-01 Following a recent paper by Fodor et al. (arXiv:1406.0827), we reexamine several types of tree-level improvements on the flow action with various gauge actions in order to reduce the lattice discretization errors in the Yang-Mills gradient flow method. We propose two types of tree-level,$\\mathcal{O}(a^4)$improved lattice gradient flow including the rectangle term in both the flow and gauge action within the minimal way. We then perform numerical simulations with the simple plaquette gauge action for testing our proposal. Our numerical results of the expectation value of the action density,$\\langle E(t)\\rangle$, show that two$\\mathcal{O}(a^4)$improved flows significantly eliminate the discretization corrections in the small flow time$t$regime. On the other hand, the values of$t^2\\langle E(t)\\rangle$in the large$t$regime, where the lattice spacing dependence of the tree-level term dies out as inverse powers of$t/a^2$, are different between the results given by two optimal flows leading to the same$...
1. Correlation of Critical Loci for Water-Hydrocarbon Binary Systems by EOS Based on the Multi-Fluid Nonrandom Lattice Theory
Hun; Yong; SHIN; Hwayong; KIM; 等
2002-01-01
Quantitative representation of complicated behavior of fluid mixtures in the critical region by any of equation-of-state theories remains as a difficults thermodynamic topics to date.In the present work,a computational efforts were made for representing various types of critical loci of binary water with hydrocarbon systems showing Type Ⅱ and Type Ⅲ phase behavior by an elementary equation of state[called multi-fluid nonrandom lattice fluid EOS(MF-NLF EOS)]based on the lattice statistical mechanical theory.The model EOS requires two molecular parameters which representing molecular size and interaction energy for a pure component and single adjustable interaction energy parameter for binary mixtures.Critical temperature and pressure data were used to obtain molecular size parameter and vapor pressure data were used to obtain interaction energy parameter.The MF-NLF EOS model adapted in the present study correlated quantitatively well the critical loci of various binary water with hydrocarbon systems.
2. Exploration of Variability from the Perspective of Speech Style Theory
郑洁
2013-01-01
From the emergency of the term, interlanguage is one of the key of the language teaching researches. The essay illus⁃trates the variability of interlanguage from the perspective of Speech Style Theory in order to promote the development of lan⁃guage teaching.
3. Investigation of structural and optical properties in Cobalt–Chromium co-doped ZnO thin films within the Lattice Compatibility Theory scope
Mimouni, R.; Boubaker, K., E-mail: [email protected]; Amlouk, M.
2015-03-05
Highlights: • Co/Cr co-doped ZnO thin films were synthesized by a low-cost spray technique. • Optical and morphological properties of the Co/Cr co-doped ZnO system were described. • Lattice Compatibility Theory explains Co preferential incorporation in ZnO lattice. - Abstract: (Co,Cr)-codoped zinc oxide thin films (ZnO:Cr:Co) at different percentages (0%, 1–1%, 1–2%, 2–1%) were deposited on glass substrates using a chemical low-cost spray technique. The effect of Cr and Co concentration on the structural, morphological and optical properties of the ZnO:Cr:Co thin films were investigated by means of X-ray diffraction, optical measurement, contact Atomic Force Microscopy (AFM), and Photoluminescence spectroscopy. The results revealed that all films consist of single phase ZnO and were well crystallized in würtzite phase with the crystallites preferentially oriented towards (0 0 2) direction parallel to c-axis. Also, the co-doping has effective role in the enhancement of the crystallinity and leads to an improvement of roughness of the ZnO films. Doping by chrome and cobalt resulted in a slight decrease in the optical band gap energy of the films. The optical band gap of these films is calculated. The optical absorption spectra show that the absorption mechanism is a direct transition. The UV peak positions for ZnO:Cr:Co samples slightly red shift to the longer wavelength in comparison with the pure ZnO which can be attributed to the change in the acceptor level induced by the substitutional Co{sup 2+} and Cr{sup 3+} and the band-gap narrowing of ZnO with the Cr and Co dopants. The Lattice Compatibility Theory analyses have been applied in order to give original, plausible and founded explanation to the recorded preferential incorporation of cobalt ions within ZnO lattice over chromium.
4. Kenneth Wilson and lattice QCD
Ukawa, Akira
2015-01-01
We discuss the physics and computation of lattice QCD, a space-time lattice formulation of quantum chromodynamics, and Kenneth Wilson's seminal role in its development. We start with the fundamental issue of confinement of quarks in the theory of the strong interactions, and discuss how lattice QCD provides a framework for understanding this phenomenon. A conceptual issue with lattice QCD is a conflict of space-time lattice with chiral symmetry of quarks. We discuss how this problem is resolved. Since lattice QCD is a non-linear quantum dynamical system with infinite degrees of freedom, quantities which are analytically calculable are limited. On the other hand, it provides an ideal case of massively parallel numerical computations. We review the long and distinguished history of parallel-architecture supercomputers designed and built for lattice QCD. We discuss algorithmic developments, in particular the difficulties posed by the fermionic nature of quarks, and their resolution. The triad of efforts toward b...
5. Improved Lattice Radial Quantization
Brower, Richard C; Fleming, George T
2014-01-01
Lattice radial quantization was proposed in a recent paper by Brower, Fleming and Neuberger[1] as a nonperturbative method especially suited to numerically solve Euclidean conformal field theories. The lessons learned from the lattice radial quantization of the 3D Ising model on a longitudinal cylinder with 2D Icosahedral cross-section suggested the need for an improved discretization. We consider here the use of the Finite Element Methods(FEM) to descretize the universally-equivalent $\\phi^4$ Lagrangian on $\\mathbb R \\times \\mathbb S^2$. It is argued that this lattice regularization will approach the exact conformal theory at the Wilson-Fisher fixed point in the continuum. Numerical tests are underway to support this conjecture.
6. Neural Network for Quantum Brain Dynamics: 4D CP$^1$+U(1) Gauge Theory on Lattice and its Phase Structure
Sakane, Shinya; Matsui, Tetsuo
2016-01-01
We consider a system of two-level quantum quasi-spins and gauge bosons put on a 3+1D lattice. As a model of neural network of the brain functions, these spins describe neurons quantum-mechanically, and the gauge bosons describes weights of synaptic connections. It is a generalization of the Hopfield model to a quantum network with dynamical synaptic weights. At the microscopic level, this system becomes a model of quantum brain dynamics proposed by Umezawa et al., where spins and gauge field describe water molecules and photons, respectively. We calculate the phase diagram of this system under quantum and thermal fluctuations, and find that there are three phases; confinement, Coulomb, and Higgs phases. Each phase is classified according to the ability to learn patterns and recall them. By comparing the phase diagram with that of classical networks, we discuss the effect of quantum fluctuations and thermal fluctuations (noises in signal propagations) on the brain functions.
7. Dark matter on the lattice
Lewis, Randy
2014-01-01
Several collaborations have recently performed lattice calculations aimed specifically at dark matter, including work with SU(2), SU(3), SU(4) and SO(4) gauge theories to represent the dark sector. Highlights of these studies are presented here, after a reminder of how lattice calculations in QCD itself are helping with the hunt for dark matter.
8. Lattice cluster theory of associating polymers. II. Enthalpy and entropy of self-assembly and Flory-Huggins interaction parameter χ for solutions of telechelic molecules.
Dudowicz, Jacek; Freed, Karl F; Douglas, Jack F
2012-02-14
The lattice cluster theory for solutions of telechelic polymer chains, developed in paper I, is applied to determine the enthalpy Δh(p) and entropy Δs(p) of self-assembly of linear telechelics and to evaluate the Flory-Huggins (FH) interaction parameter χ governing the phase behavior of these systems. Particular focus is placed on examining how these interaction variables depend on the composition of the solution, temperature, van der Waals and local "sticky" interaction energies, and the length of the individual telechelic chains. The FH interaction parameter χ is found to exhibit an entropy-enthalpy compensation effect between the "entropic" and "enthalpic" portions as either the composition or mass of the telechelic species is varied, providing unique theoretical insights into this commonly reported, yet, enigmatic phenomenon.
9. Phase diagram of a spin-1 magnetic bilayer by cluster variational theory: Exact results for a BEG model on a Bethe lattice with five-fold coordination
Tucker, J. W.; Balcerzak, T.; Gzik, M.; Sukiennicki, A.
1998-09-01
The complete global phase diagram for a magnetic spin-1 bilayer, whose interactions are described by the Blume Emery Griffiths model (BEG), is studied by cluster variational theory within the pair approximation. The results obtained, are also the exact results pertaining to the BEG model on a Bethe lattice having coordination number, z=5. Useful analytic expressions are derived for trajectories in phase space containing the second-order (continuous) phase boundaries. The physical existence of these second-order boundaries, together with the location of the first-order phase boundaries, are determined from a Gibbs free energy analysis. Detailed comparison of the results with those of other workers on this, and closely related systems, is made.
10. Self-consistent Bogoliubov-de Gennes theory of the vortex lattice state in a two-dimensional strongly type-II superconductor at high magnetic fields
Zhuravlev, Vladimir; Duan, Wenye; Maniv, Tsofar
2017-01-01
A self-consistent Bogoliubov-de Gennes theory of the vortex lattice state in a 2D strong type-II superconductor at high magnetic fields reveals a novel quantum mixed state around the semiclassical Hc 2, characterized by a well-defined Landau-Bloch band structure in the quasiparticle spectrum and suppressed order-parameter amplitude, which sharply crossover into the well-known semiclassical (Helfand-Werthamer) results upon decreasing magnetic field. Application to the 2D superconducting state observed recently on the surface of the topological insulator Sb2Te3 accounts well for the experimental data, revealing a strong type-II superconductor, with unusually low carrier density and very small cyclotron mass, which can be realized only in the strong coupling superconductor limit.
11. Non-linear spin wave theory results for the frustrated [Formula: see text] Heisenberg antiferromagnet on a body-centered cubic lattice.
Majumdar, Kingshuk; Datta, Trinanjan
2009-10-07
At zero temperature the sublattice magnetization of the quantum spin- 1/2 Heisenberg antiferromagnet on a body-centered cubic lattice with competing first and second neighbor exchange (J(1) and J(2)) is investigated using the non-linear spin wave theory. The zero temperature phases of the model consist of a two sublattice Néel phase for small J(2) (AF(1)) and a collinear phase at large J(2) (AF(2)). We show that quartic corrections due to spin wave interactions enhance the sublattice magnetization in both the AF(1) and the AF(2) phase. The magnetization corrections are prominent near the classical transition point of the model and in the J(2)>J(1) regime. The ground state energy with quartic interactions is also calculated. It is found that up to quartic corrections the first order phase transition (previously observed in this model) between the AF(1) and the AF(2) phase survives.
Wang, Da-Wei; Zhu, Shi-Yao; Scully, Marlan O
2014-01-01
We show that the timed Dicke states of a collection of three-level atoms can form a tight-binding lattice in the momentum space. This lattice, coined the superradiance lattice (SL), can be constructed based on an electromagnetically induced transparency (EIT) system. For a one-dimensional SL, we need the coupling field of the EIT system to be a standing wave. The detuning between the two components of the standing wave introduces an effective electric field. The quantum behaviours of electrons in lattices, such as Bloch oscillations, Wannier-Stark ladders, Bloch band collapsing and dynamic localization can be observed in the SL. The SL can be extended to two, three and even higher dimensions where no analogous real space lattices exist and new physics are waiting to be explored.
13. Measuring on Lattices
Knuth, Kevin H
2009-01-01
Previous derivations of the sum and product rules of probability theory relied on the algebraic properties of Boolean logic. Here they are derived within a more general framework based on lattice theory. The result is a new foundation of probability theory that encompasses and generalizes both the Cox and Kolmogorov formulations. In this picture probability is a bi-valuation defined on a lattice of statements that quantifies the degree to which one statement implies another. The sum rule is a constraint equation that ensures that valuations are assigned so as to not violate associativity of the lattice join and meet. The product rule is much more interesting in that there are actually two product rules: one is a constraint equation arises from associativity of the direct products of lattices, and the other a constraint equation derived from associativity of changes of context. The generality of this formalism enables one to derive the traditionally assumed condition of additivity in measure theory, as well in...
14. Weakly deformed soliton lattices
Dubrovin, B. (Moskovskij Gosudarstvennyj Univ., Moscow (USSR). Dept. of Mechanics and Mathematics)
1990-12-01
In this lecture the author discusses periodic and quasiperiodic solutions of nonlinear evolution equations of phi{sub t}=K (phi, phi{sub x},..., phi{sup (n)}), the so-called soliton lattices. After introducing the theory of integrable systems of hydrodynamic type he discusses their Hamiltonian formalism, i.e. the theory of Poisson brackets of hydrodynamic type. Then he describes the application of algebraic geometry to the effective integration of such equations. (HSI).
15. Lattice radial quantization by cubature
Neuberger, Herbert
2014-01-01
Basic aspects of a program to put field theories quantized in radial coordinates on the lattice are presented. Only scalar fields are discussed. Simple examples are solved to illustrate the strategy when applied to the 3D Ising model.
16. Large-N theory of the Sp(N) Heisenberg quantum antiferromagnet on an anisotropic triangular lattice
Chung, Chung-Hou; Marston, Brad
2000-03-01
The magnetic properties of the two-dimensional layered organic superconductors κ-(BEDT-TTF)_2X are modeled by a spin-1/2 Heisenberg quantum antiferromagnet on an anisotropic triangular lattice. The phase diagram is ascertained by means of a large-N expansion of the Sp(N) generalization of the physical SU(2) \\cong Sp(1) Heisenberg magnet.(S. Sachdev and N. Reed, Int. J. Mod. Phys. B5), 219 (1991). The phase diagram is presented in the two-dimensional parameter space of J_1/J_2, the ratio of the nearest to next-nearest neighbor Heisenberg exchange, and the ratio nb / N, which sets the strength of the quantum fluctuations. At large nb / N (equivalent to the large-spin limit of the physical SU(2) model) quantum effects are small, the ground states break global Sp(N) spin-rotational symmetry, and exhibit magnetic long-range-order (LRO). At small nb / N, however, quantum fluctuations overwhelm the tendency to order and there is only short-range magnetic order (SRO). The LRO and SRO phases can be further classified into two types depending on the size of the anisotropy: (i) ground states with commensurate, collinear, spin correlations; and (ii) ground states with incommensurate, coplanar, spin correlations. Finite-N corrections due to a Berry's phase term modify the character of the SRO phases, leading in the case of the commensurate state to spin-Peierls order and the confinement of spinons.
17. Lattice QCD: A Brief Introduction
Meyer, H. B.
A general introduction to lattice QCD is given. The reader is assumed to have some basic familiarity with the path integral representation of quantum field theory. Emphasis is placed on showing that the lattice regularization provides a robust conceptual and computational framework within quantum field theory. The goal is to provide a useful overview, with many references pointing to the following chapters and to freely available lecture series for more in-depth treatments of specifics topics.
18. Annealed lattice animal model and Flory theory for the melt of non-concatenated rings: towards the physics of crumpling.
Grosberg, Alexander Y
2014-01-28
A Flory theory is constructed for a long polymer ring in a melt of unknotted and non-concatenated rings. The theory assumes that the ring forms an effective annealed branched object and computes its primitive path. It is shown that the primitive path follows self-avoiding statistics and is characterized by the corresponding Flory exponent of a polymer with excluded volume. Based on that, it is shown that rings in the melt are compact objects with overall size proportional to their length raised to the 1/3 power. Furthermore, the contact probability exponent γcontact is estimated, albeit by a poorly controlled approximation, with the result close to 1.1 consistent with both numerical and experimental data.
19. Ausubel＇s Meaningful Learning Theory and Its Enlightenment to the Teaching Reform
2012-01-01
The article briefly comments Ausubel＇s mearingful learning theory and draws some enlightenment from it. Nowdays we should combine the traditional instruction method and the demonstration method with class software. Ausubel is a famous American educational psychologist, who has made remarkable achievement by using cognitive perspective to study directly the meaningful lan- guage learning activities of human in the practical educational environment.
20. 5D Maximally Supersymmetric Yang-Mills on the Lattice
Joseph, Anosh
2016-01-01
We provide details of the lattice construction of five-dimensional maximally supersymmetric Yang-Mills theory. The lattice theory is supersymmetric, gauge invariant and free from spectrum doublers. Such a supersymmetric lattice formulation is interesting as it can be used for non-perturbative explorations of the five-dimensional theory, which has a known gravitational dual.
1. Holographic Lattices Give the Graviton a Mass
Blake, Mike; Vegh, David
2014-01-01
We discuss the DC conductivity of holographic theories with translational invariance broken by a background lattice. We show that the presence of the lattice induces an effective mass for the graviton via a gravitational version of the Higgs mechanism. This allows us to obtain, at leading order in the lattice strength, an analytic expression for the DC conductivity in terms of the size of the lattice at the horizon. In locally critical theories this leads to a power law resistivity that is in agreement with an earlier field theory analysis of Hartnoll and Hofman.
2. Lattices of dielectric resonators
Trubin, Alexander
2016-01-01
This book provides the analytical theory of complex systems composed of a large number of high-Q dielectric resonators. Spherical and cylindrical dielectric resonators with inferior and also whispering gallery oscillations allocated in various lattices are considered. A new approach to S-matrix parameter calculations based on perturbation theory of Maxwell equations, developed for a number of high-Q dielectric bodies, is introduced. All physical relationships are obtained in analytical form and are suitable for further computations. Essential attention is given to a new unified formalism of the description of scattering processes. The general scattering task for coupled eigen oscillations of the whole system of dielectric resonators is described. The equations for the expansion coefficients are explained in an applicable way. The temporal Green functions for the dielectric resonator are presented. The scattering process of short pulses in dielectric filter structures, dielectric antennas and lattices of d...
3. Lattice Based Tools in Cryptanalysis for Public Key Cryptography
R.Santosh Kumar
2012-03-01
Full Text Available Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Latticereduction has been successfully utilizing in Number Theory, Linear algebra and Cryptology. Not only the existence of lattice based cryptosystems of hard in nature, but also has vulnerabilities by lattice reduction techniques. In this survey paper, we are focusing on point lattices and then describing an introduction to the theoretical and practical aspects of lattice reduction. Finally, we describe the applications of lattice reduction in Number theory, Linear algebra
4. Lattice models of ionic systems
Kobelev, Vladimir; Kolomeisky, Anatoly B.; Fisher, Michael E.
2002-05-01
A theoretical analysis of Coulomb systems on lattices in general dimensions is presented. The thermodynamics is developed using Debye-Hückel theory with ion-pairing and dipole-ion solvation, specific calculations being performed for three-dimensional lattices. As for continuum electrolytes, low-density results for simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices indicate the existence of gas-liquid phase separation. The predicted critical densities have values comparable to those of continuum ionic systems, while the critical temperatures are 60%-70% higher. However, when the possibility of sublattice ordering as well as Debye screening is taken into account systematically, order-disorder transitions and a tricritical point are found on sc and bcc lattices, and gas-liquid coexistence is suppressed. Our results agree with recent Monte Carlo simulations of lattice electrolytes.
5. Quantum Gravity on the Lattice
Hamber, Herbert W
2009-01-01
I review the lattice approach to quantum gravity, and how it relates to the non-trivial ultraviolet fixed point scenario of the continuum theory. After a brief introduction covering the general problem of ultraviolet divergences in gravity and other non-renormalizable theories, I cover the general methods and goals of the lattice approach. An underlying theme is an attempt at establishing connections between the continuum renormalization group results, which are mainly based on diagrammatic perturbation theory, and the recent lattice results, which should apply to the strong gravity regime and are inherently non-perturbative. A second theme in this review is the ever-present natural correspondence between infrared methods of strongly coupled non-abelian gauge theories on the one hand, and the low energy approach to quantum gravity based on the renormalization group and universality of critical behavior on the other. Towards the end of the review I discuss possible observational consequences of path integral q...
6. Density functional theory study of the structural, electronic, lattice dynamical, and thermodynamic properties of Li4SiO4 and its capability for CO2 capture
Duan, Yuhua; Parlinski, K.
2011-01-01
The structural, electronic, lattice dynamical, optical, thermodynamic, and CO{sub 2} capture properties of monoclinic and triclinic phases of Li{sub 4}SiO{sub 4} are investigated by combining density functional theory with phonon lattice dynamics calculations. We found that these two phases have some similarities in their bulk and thermodynamic properties. The calculated bulk modulus and the cohesive energies of these two phases are close to each other. Although both of them are insulators, the monoclinic phase of Li{sub 4}SiO{sub 4} has a direct band gap of 5.24 eV while the triclinic Li{sub 4}SiO{sub 4} phase has an indirect band gap of 4.98 eV. In both phases of Li{sub 4}SiO{sub 4}, the s orbital of O mainly contributes to the lower-energy second valence band (VB{sub 2}) and the p orbitals contribute to the fist valence band (VB{sub 1}) and the conduction bands (CBs). The s orbital of Si mainly contributes to the lower portions of the VB1 and VB{sub 2}, and Si p orbitals mainly contribute to the higher portions of the VB{sub 1} and VB{sub 2}. The s and p orbitals of Li contribute to both VBs and to CBs, and Li p orbitals have a higher contribution than the Li s orbital. There is possibly a phonon soft mode existing in triclinic {gamma}-Li{sub 4}SiO{sub 4}; in the monoclinic Li{sub 4}SiO{sub 4}, there are three phonon soft modes, which correspond to the one type of Li disordered over a few sites. Their LO-TO splitting indicates that both phases of Li{sub 4}SiO{sub 4} are polar anisotropic materials. The calculated infrared absorption spectra for LO and TO modes are different for these two phases of Li{sub 4}SiO{sub 4}. The calculated relationships of the chemical potential versus temperature and CO{sub 2} pressure for reaction of Li{sub 4}SiO{sub 4} with CO{sub 2} shows that Li{sub 4}SiO{sub 4} could be a good candidate for a high-temperature CO{sub 2} sorbent while used for postcombustion capture technology.
7. Chiral Four-Dimensional Heterotic Covariant Lattices
Beye, Florian
2014-01-01
In the covariant lattice formalism, chiral four-dimensional heterotic string vacua are obtained from certain even self-dual lattices which completely decompose into a left-mover and a right-mover lattice. The main purpose of this work is to classify all right-mover lattices that can appear in such a chiral model, and to study the corresponding left-mover lattices using the theory of lattice genera. In particular, the Smith-Minkowski-Siegel mass formula is employed to calculate a lower bound on the number of left-mover lattices. Also, the known relationship between asymmetric orbifolds and covariant lattices is considered in the context of our classification.
8. The quantum spin-1/2 J1-J2 antiferromagnet on a stacked square lattice: a study of effective-field theory in a finite cluster.
Nunes, Wagner A; de Sousa, J Ricardo; Viana, J Roberto; Richter, J
2010-04-14
The ground state phase diagram of the quantum spin-1/2 Heisenberg antiferromagnet in the presence of nearest-neighbor (J(1)) and next-nearest-neighbor (J(2)) interactions (J(1)-J(2) model) on a stacked square lattice, where we introduce an interlayer coupling through nearest-neighbor bonds of strength J(), is studied within the framework of the differential operator technique. The Hamiltonian is solved by effective-field theory in a cluster with N=4 spins (EFT-4). We obtain the sublattice magnetization m(A) for the ordered phases: antiferromagnetic (AF) and collinear (CAF-collinear antiferromagnetic). We propose a functional for the free energy Ψ(μ)(m(μ)) (μ=A, B) to obtain the phase diagram in the λ-α plane, where λ=J()/J(1) and α=J(2)/J(1). Depending on the values of λ and α, we found different ordered states (AF and CAF) and a disordered state (quantum paramagnetic (QP)). For an intermediate region α(1c)(λ) α(2c)(λ), and below λ(1), we have the AF and CAF semi-classically ordered states, respectively. At α=α(1c)(λ) a second-order transition between the AF and QP states occurs and at α=α(2c)(λ) a first-order transition between the AF and CAF phases takes place. The boundaries between these ordered phases merge at the critical end point CEP≡(λ(1), α(c)), where α(c)≈0.56. Above this CEP there is again a direct first-order transition between the AF and CAF phases, with a behavior described by the point α(c) independent of λ ≥ λ(1).
9. Lattice QCD for nuclear physics
Meyer, Harvey
2015-01-01
With ever increasing computational resources and improvements in algorithms, new opportunities are emerging for lattice gauge theory to address key questions in strongly interacting systems, such as nuclear matter. Calculations today use dynamical gauge-field ensembles with degenerate light up/down quarks and the strange quark and it is possible now to consider including charm-quark degrees of freedom in the QCD vacuum. Pion masses and other sources of systematic error, such as finite-volume and discretization effects, are beginning to be quantified systematically. Altogether, an era of precision calculation has begun, and many new observables will be calculated at the new computational facilities. The aim of this set of lectures is to provide graduate students with a grounding in the application of lattice gauge theory methods to strongly interacting systems, and in particular to nuclear physics. A wide variety of topics are covered, including continuum field theory, lattice discretizations, hadron spect...
10. Topological Lattice Actions
Bietenholz, W; Pepe, M; Wiese, U -J
2010-01-01
We consider lattice field theories with topological actions, which are invariant against small deformations of the fields. Some of these actions have infinite barriers separating different topological sectors. Topological actions do not have the correct classical continuum limit and they cannot be treated using perturbation theory, but they still yield the correct quantum continuum limit. To show this, we present analytic studies of the 1-d O(2) and O(3) model, as well as Monte Carlo simulations of the 2-d O(3) model using topological lattice actions. Some topological actions obey and others violate a lattice Schwarz inequality between the action and the topological charge $Q$. Irrespective of this, in the 2-d O(3) model the topological susceptibility $\\chi_t = \\l/V$ is logarithmically divergent in the continuum limit. Still, at non-zero distance the correlator of the topological charge density has a finite continuum limit which is consistent with analytic predictions. Our study shows explicitly that some cla...
11. Dynamical Gauge Fields on Optical Lattices: A Lattice Gauge Theorist Point of View
Meurice, Yannick
2011-01-01
Dynamical gauge fields are essential to capture the short and large distance behavior of gauge theories (confinement, mass gap, chiral symmetry breaking, asymptotic freedom). I propose two possible strategies to use optical lattices to mimic simulations performed in lattice gauge theory. I discuss how new developments in optical lattices could be used to generate local invariance and link composite operators with adjoint quantum numbers that could play a role similar to the link variables used in lattice gauge theory. This is a slightly expanded version of a poster presented at the KITP Conference: Frontiers of Ultracold Atoms and Molecules (Oct 11-15, 2010) that I plan to turn into a more comprehensive tutorial that could be used by members of the optical lattice and lattice gauge theory communities. Suggestions are welcome.
12. Lattice QCD on fine lattices
Schaefer, Stefan [DESY (Germany). Neumann Inst. for Computing
2016-11-01
These configurations are currently in use in many on-going projects carried out by researchers throughout Europe. In particular this data will serve as an essential input into the computation of the coupling constant of QCD, where some of the simulations are still on-going. But also projects computing the masses of hadrons and investigating their structure are underway as well as activities in the physics of heavy quarks. As this initial project of gauge field generation has been successful, it is worthwhile to extend the currently available ensembles with further points in parameter space. These will allow to further study and control systematic effects like the ones introduced by the finite volume, the non-physical quark masses and the finite lattice spacing. In particular certain compromises have still been made in the region where pion masses and lattice spacing are both small. This is because physical pion masses require larger lattices to keep the effects of the finite volume under control. At light pion masses, a precise control of the continuum extrapolation is therefore difficult, but certainly a main goal of future simulations. To reach this goal, algorithmic developments as well as faster hardware will be needed.
13. Lattice QCD for Cosmology
Borsanyi, Sz; Kampert, K H; Katz, S D; Kawanai, T; Kovacs, T G; Mages, S W; Pasztor, A; Pittler, F; Redondo, J; Ringwald, A; Szabo, K K
2016-01-01
We present a full result for the equation of state (EoS) in 2+1+1 (up/down, strange and charm quarks are present) flavour lattice QCD. We extend this analysis and give the equation of state in 2+1+1+1 flavour QCD. In order to describe the evolution of the universe from temperatures several hundreds of GeV to several tens of MeV we also include the known effects of the electroweak theory and give the effective degree of freedoms. As another application of lattice QCD we calculate the topological susceptibility (chi) up to the few GeV temperature region. These two results, EoS and chi, can be used to predict the dark matter axion's mass in the post-inflation scenario and/or give the relationship between the axion's mass and the universal axionic angle, which acts as a initial condition of our universe.
14. Varieties of lattices
Jipsen, Peter
1992-01-01
The study of lattice varieties is a field that has experienced rapid growth in the last 30 years, but many of the interesting and deep results discovered in that period have so far only appeared in research papers. The aim of this monograph is to present the main results about modular and nonmodular varieties, equational bases and the amalgamation property in a uniform way. The first chapter covers preliminaries that make the material accessible to anyone who has had an introductory course in universal algebra. Each subsequent chapter begins with a short historical introduction which sites the original references and then presents the results with complete proofs (in nearly all cases). Numerous diagrams illustrate the beauty of lattice theory and aid in the visualization of many proofs. An extensive index and bibliography also make the monograph a useful reference work.
15. Chiral Fermions on the Lattice
Bietenholz, Wolfgang
2010-01-01
In the last century the non-perturbative regularization of chiral fermions was a long-standing problem. We review how this problem was finally overcome by the formulation of a modified but exact form of chiral symmetry on the lattice. This also provides a sound definition of the topological charge of lattice gauge configurations. We illustrate a variety of applications to QCD in the p-, the epsilon- and the delta-regime, where simulation results can now be related to Random Matrix Theory and Chiral Perturbation Theory. The latter contains Low Energy Constants as free parameters, and we comment on their evaluation from first principles of QCD.
16. Computing iceberg concept lattices with Titanic
Stumme, Gerd; Taouil, Rafik; Bastide, Yves; Pasquier, Nicolas; Lakhal, Lotfi
2002-01-01
International audience; We introduce the notion of iceberg concept lattices and show their use in knowledge discovery in databases. Iceberg lattices are a conceptual clustering method, which is well suited for analyzing very large databases. They also serve as a condensed representation of frequent itemsets, as starting point for computing bases of association rules, and as a visualization method for association rules. Iceberg concept lattices are based on the theory of Formal Concept Analysi...
17. Nuclear Reactions from Lattice QCD
Briceño, Raúl A; Luu, Thomas C
2014-01-01
One of the overarching goals of nuclear physics is to rigorously compute properties of hadronic systems directly from the fundamental theory of strong interactions, Quantum Chromodynamics (QCD). In particular, the hope is to perform reliable calculations of nuclear reactions which will impact our understanding of environments that occur during big bang nucleosynthesis, the evolution of stars and supernovae, and within nuclear reactors and high energy/density facilities. Such calculations, being truly ab initio, would include all two-nucleon and three- nucleon (and higher) interactions in a consistent manner. Currently, lattice QCD provides the only reliable option for performing calculations of some of the low- energy hadronic observables. With the aim of bridging the gap between lattice QCD and nuclear many-body physics, the Institute for Nuclear Theory held a workshop on Nuclear Reactions from Lattice QCD on March 2013. In this review article, we report on the topics discussed in this workshop and the path ...
18. Algebraic Lattices in QFT Renormalization
Borinsky, Michael
2016-07-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
19. Dual Lattice of ℤ-module Lattice
Futa Yuichi
2017-07-01
Full Text Available In this article, we formalize in Mizar [5] the definition of dual lattice and their properties. We formally prove that a set of all dual vectors in a rational lattice has the construction of a lattice. We show that a dual basis can be calculated by elements of an inverse of the Gram Matrix. We also formalize a summation of inner products and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász base reduction algorithm and cryptographic systems with lattice [20], [10] and [19].
20. Robots and lattice automata
2015-01-01
The book gives a comprehensive overview of the state-of-the-art research and engineering in theory and application of Lattice Automata in design and control of autonomous Robots. Automata and robots share the same notional meaning. Automata (originated from the latinization of the Greek word “αυτόματον”) as self-operating autonomous machines invented from ancient years can be easily considered the first steps of robotic-like efforts. Automata are mathematical models of Robots and also they are integral parts of robotic control systems. A Lattice Automaton is a regular array or a collective of finite state machines, or automata. The Automata update their states by the same rules depending on states of their immediate neighbours. In the context of this book, Lattice Automata are used in developing modular reconfigurable robotic systems, path planning and map exploration for robots, as robot controllers, synchronisation of robot collectives, robot vision, parallel robotic actuators. All chapters are...
1. Lattice dynamics of ferromagnetic superconductor UGe2
Satyam Shinde; Prafulla K Jha
2008-11-01
This paper reports the lattice dynamical study of the UGe2 using a lattice dynamical model theory based on pairwise interactions under the framework of the shell model. The calculated phonon dispersion curves and phonon density of states are in good agreement with the measured data.
2. An Application of Linear Algebra over Lattices
M. Hosseinyazdi
2008-03-01
Full Text Available In this paper, first we consider L n as a semimodule over a complete bounded distributive lattice L. Then we define the basic concepts of module theory for L n. After that, we proved many similar theorems in linear algebra for the space L n. An application of linear algebra over lattices for solving linear systems, was given
3. Resummation of Cactus Diagrams in Lattice QCD
Panagopoulos, H
1998-01-01
We show how to perform a resummation, to all orders in perturbation theory, of a certain class of gauge invariant diagrams in Lattice QCD. These diagrams are often largely responsible for lattice artifacts. Our resummation leads to an improved perturbative expansion. Applied to a number of cases of interest, this expansion yields results remarkably close to corresponding nonperturbative estimates.
4. A Study of Teacher Talk in English Classroom Based on SLA Theories
ZHONG Ying
2015-01-01
Teacher talk is of great significance in the classroom organization as well as the process of language acquisition. This pa⁃per explores the teacher talk from the angel of SLA Theories and analyzes the connection between teacher talk and SLA Theories. Thus it gives some suggestions to improve Teacher Talk in English classroom and consequently promotes students ’second lan⁃guage acquisition.
5. Hadron Physics from Lattice QCD
2016-01-01
We sketch the basic ideas of the lattice regularization in Quantum Field Theory, the corresponding Monte Carlo simulations, and applications to Quantum Chromodynamics (QCD). This approach enables the numerical measurement of observables at the non-perturbative level. We comment on selected results, with a focus on hadron masses and the link to Chiral Perturbation Theory. At last we address two outstanding issues: topological freezing and the sign problem.
6. A Lattice Study of the Glueball Spectrum
LIU Chuan
2001-01-01
The glueball spectrum is studied using an improved gluonic action on asymmetric lattices in the pure SU(3) lattice gauge theory. The smallest spatial lattice spacing is about 0.08 fm which makes the extrapolation to the ontinuum limit more reliable. Converting our lattice results to physical units using the scale set by the static quark potential, we obtain the following results for the glueball masses: MG(0++) -＝ 1730(90) MeV for the scalarglueball and MG(2++) ＝ 2400(95) MeV for the tensor glueball.
7. Long-time tails of the velocity autocorrelation function in 2D and 3D lattice gas cellular automata: a test of mode-coupling theory
Hoef, M.A. van der; Frenkel, D.
1990-01-01
We report simulations of the velocity autocorrelation function (VACF) of a tagged particle in two- and three-dimensional lattice-gas cellular automata, using a new technique that is about a million times more efficient than the conventional techniques. The simulations clearly show the algebraic t-D/
8. N=4 supersymmetry on a space-time lattice
Catterall, Simon; Schaich, David; Damgaard, Poul H.
2014-01-01
Maximally supersymmetric Yang–Mills theory in four dimensions can be formulated on a space-time lattice while exactly preserving a single supersymmetry. Here we explore in detail this lattice theory, paying particular attention to its strongly coupled regime. Targeting a theory with gauge group SU...
9. Monte Carlo methods in continuous time for lattice Hamiltonians
Huffman, Emilie
2016-01-01
We solve a variety of sign problems for models in lattice field theory using the Hamiltonian formulation, including Yukawa models and simple lattice gauge theories. The solutions emerge naturally in continuous time and use the dual representation for the bosonic fields. These solutions allow us to construct quantum Monte Carlo methods for these problems. The methods could provide an alternative approach to understanding non-perturbative dynamics of some lattice field theories.
10. Triangle Lattice Green Functions for Vector Fields
Moritz, Brian; Schwalm, William
2000-03-01
The triangle lattice is convenient for modeling fields and fluid flows in two dimensions. Discrete vector field equations are defined through the analogy between differential forms and simplicial homology theory. The basic vector difference operators on the lattice correspond to the graph adjacency matricies of the triangle, honeycomb, and Kagomé lattices. The scalar Green functions for nearest neighbor interactions on the triangle lattice are known in closed form in terms of the complete elliptic integrals. Green functions for vector field operators are obtained explicitly in terms of the known scalar Green functions. The scalar Green functions for the Kagomé lattice are thus written in terms of the Green functions for the triangle lattice and ultimately in closed form. Thus, Green functions for a wide range of vector difference models are reduced to closed form in terms of the complete elliptic integrals.
11. Crystalline Scaling Geometries from Vortex Lattices
Bao, Ning
2013-01-01
We study magnetic geometries with Lifshitz and/or hyperscaling violation exponents (both with a hard wall cutoff in the IR and a smooth black brane horizon) which have a complex scalar field which couples to the magnetic field. The complex scalar is unstable to the production of a vortex lattice in the IR. The lattice is a normalizable mode which is relevant (i.e. grows into the IR.) When one considers linearized backreaction of the lattice on the metric and gauge field, the metric forms a crystalline structure. We analyze the scaling of the free energy, thermodynamic entropy, and entanglement in the lattice phase and find that in the smeared limit, the leading order correction to thermodynamic properties due to the lattice has the scaling behavior of a theory with a hyperscaling violation exponent between 0 and 1, indicating a flow to an effectively lower-dimensional theory in the deep IR.
12. A lattice approach to spinorial quantum gravity
Renteln, Paul; Smolin, Lee
1989-01-01
A new lattice regularization of quantum general relativity based on Ashtekar's reformulation of Hamiltonian general relativity is presented. In this form, quantum states of the gravitational field are represented within the physical Hilbert space of a Kogut-Susskind lattice gauge theory. The gauge field of the theory is a complexified SU(2) connection which is the gravitational connection for left-handed spinor fields. The physical states of the gravitational field are those which are annihilated by additional constraints which correspond to the four constraints of general relativity. Lattice versions of these constraints are constructed. Those corresponding to the three-dimensional diffeomorphism generators move states associated with Wilson loops around on the lattice. The lattice Hamiltonian constraint has a simple form, and a correspondingly simple interpretation: it is an operator which cuts and joins Wilson loops at points of intersection.
13. Phase structure of lattice N=4 super Yang-Mills
Catterall, Simon; Damgaard, Poul H.; DeGrand, Thomas;
2012-01-01
We make a first study of the phase diagram of four-dimensional N = 4 super Yang-Mills theory regulated on a space-time lattice. The lattice formulation we employ is both gauge invariant and retains at all lattice spacings one exactly preserved supersymmetry charge. Our numerical results are consi...
14. Lattice Based Attack on Common Private Exponent RSA
Santosh Kumar Ravva
2012-03-01
Full Text Available Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Lattice reduction has been successfully utilizing in Number Theory, Linear algebra and Cryptology. Not only the existence of lattice based cryptosystems of hard in nature, but also has vulnerabilities by lattice reduction techniques. In this paper, we show that Wieners small private exponent attack, when viewed as a heuristic lattice based attack, is extended to attack many instances of RSA when they have the same small private exponent.
15. Free µ-Lattices
Santocanale, Luigi
2002-01-01
A μ-lattice is a lattice with the property that every unary polynomial has both a least and a greatest fix-point. In this paper we define the quasivariety of μ-lattices and, for a given partially ordered set P, we construct a μ-lattice JP whose elements are equivalence classes of games in a preor...
16. Structural Investigation of Photocatalyst Solid Ag1−xCuxInS2 Quaternary Alloys Sprayed Thin Films Optimized within the Lattice Compatibility Theory (LCT Scope
A. Colantoni
2014-01-01
Full Text Available CuxAg1−xInS2 solid thin films were fabricated through a low-cost process. Particular process-related enhanced properties lead to reaching a minimum of lattice mismatch between absorber and buffer layers within particular solar cell devices. First, copper-less samples X-ray diffraction analysis depicts the presence of AgInS2 ternary compound in chalcopyrite tetragonal phase with privileged (112 peak (d112=1.70 Å according to JCPDS 75-0118 card. Second, when x content increases, we note a shift of the same preferential orientation (112 and its value reaches 1.63 Å corresponding to CuInS2 chalcopyrite tetragonal material according to JCPDS 89-6095 file. Finally, the formation and stability of these quaternaries have been discussed in terms of the lattice compatibility in relation with silver-copper duality within indium disulfide lattice structure. Plausible explanations for the extent and dynamics of copper incorporation inside AgInS2 elaborated ternary matrices have been proposed.
17. Latest results from lattice N=4 supersymmetric Yang--Mills
Schaich, David; Damgaard, Poul H; Giedt, Joel
2016-01-01
We present some of the latest results from our numerical investigations of N=4 supersymmetric Yang--Mills theory formulated on a space-time lattice. Based on a construction that exactly preserves a single supersymmetry at non-zero lattice spacing, we recently developed an improved lattice action that is now being employed in large-scale calculations. Here we update our studies of the static potential using this new action, also applying tree-level lattice perturbation theory to improve the analysis of the potential itself. Considering relatively weak couplings, we obtain results for the Coulomb coefficient that are consistent with continuum perturbation theory.
18. Application of Noncommutative Differential Geometry on Lattice to Anomaly
Fujiwara, T; Wu, K; Fujiwara, Takanori; Suzuki, Hiroshi; Wu, Ke
1999-01-01
The chiral anomaly in lattice abelian gauge theory is investigated by applying the geometric and topological method in noncommutative differential geometry(NCDG). A new kind of double complex and descent equation are proposed on infinite hypercubic lattice in arbitrary even dimensional Euclidean space, in the framework of NCDG. Using the general solutions to proposed descent equation, we derive the chiral anomaly in Abelian lattice gauge theory. The topological origin of anomaly is nothing but the Chern classes in NCDG.
19. Graphene as a hexagonal 2-lattice: Evaluation of the in-plane material constants for the linear theory. A multiscale approach
Sfyris, D., E-mail: [email protected], E-mail: [email protected] [Foundation for Research and Technology, Institute of Chemical Engineering Sciences, Patras (Greece); Koukaras, E. N. [Foundation for Research and Technology, Institute of Chemical Engineering Sciences, Patras (Greece); Department of Physics, University of Patras, Patras (Greece); Pugno, N. [Laboratory of Bio-Inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, Universita' di Trento, via Mesiano, 77, 38123 Trento (Italy); Center for Materials and Microsystems, Fondazione Bruno Kessler, Via Sommarive 18, 38123 Povo (Trento) (Italy); School of Engineering and Materials Science, Queen Mary University of London, Mile End Road, London E1 4NS (United Kingdom); Galiotis, C. [Foundation for Research and Technology, Institute of Chemical Engineering Sciences, Patras (Greece); Department of Chemical Engineering, University of Patras, Patras (Greece)
2015-08-21
Continuum modeling of free-standing graphene monolayer, viewed as a two dimensional 2-lattice, requires specification of the components of the shift vector that acts as an auxiliary variable. If only in-plane motions are considered, the energy depends on an in-plane strain measure and the shift vector. The assumption of geometrical and material linearity leads to quadratic energy terms with respect to the shift vector, the strain tensor, and their combinations. Graphene's hexagonal symmetry reduces the number of independent moduli then to four. We evaluate these four material parameters using molecular calculations and the adaptive intermolecular reactive empirical bond order potential and compare them with standard linear elastic constitutive modeling. The results of our calculations show that the predicted values are in reasonable agreement with those obtained solely from our molecular calculations as well as those from the literature. To the best of our knowledge, this is the first attempt to measure mechanical properties when graphene is modeled as a hexagonal 2-lattice. This work targets at the continuum scale when the insight measurements come from finer scales using atomistic simulations.
20. First Results from Lattice Simulation of the PWMM
Catterall, Simon
2010-01-01
We present results of lattice simulations of the Plane Wave Matrix Model (PWMM). The PWMM is a theory of supersymmetric quantum mechanics that has a well-defined canonical ensemble. We simulate this theory by applying rational hybrid Monte Carlo techniques to a naive lattice action. We examine the strong coupling behaviour of the model focussing on the deconfinement transition. | {"extraction_info": {"found_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": 0.8802808523178101, "perplexity": 1256.1374671767367}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647681.81/warc/CC-MAIN-20180321180325-20180321200325-00274.warc.gz"} |
https://bio.libretexts.org/Courses/City_College_of_San_Francisco/Introduction_to_Genetics/11%3A_Genomics_and_Systems_Biology/11.02%3A_Whole_Genome_Sequencing | # 11.2: Whole Genome Sequencing
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## The need for assembly
Given that the length of a single, individual sequencing read is somewhere between 45bp and 700bp, we are faced with a problem determining the sequence of longer fragments, such as the chromosomes in an entire genome of humans (3 x109 bp). Obviously, we need to break the genome into smaller fragments. There are two different strategies for doing this:
1. clone-by-clone sequencing, which relies on the creation of a physical map first then sequencing, and
2. whole genome shotgun sequencing, which sequences first and does not require a physical map.
## Physical mapping
A physical map is a representation of a genome, comprised of cloned fragments of DNA. The map is therefore made from physical entities (pieces of DNA) rather than abstract concepts such as the linkage frequencies and genes that make up a genetic map (Figure $$\PageIndex{1}$$). It is usually possible to correlate genetic and physical maps, for example by identifying the clone that contains a particular molecular marker. The connection between physical and genetic maps allows the genes underlying particular mutations to be identified through a process call map-based cloning.
To create a physical map, large fragments of the genome are cloned into plasmid vectors, or into larger vectors called bacterial artificial chromosomes (BACs). BACs can contain approximately 100kb fragments. The set of BACs produced in a cloning reaction will be redundant, meaning that different clones will contain DNA from the same part of the genome. Because of this redundancy, it is useful to select the minimum set of clones that represent the entire genome, and to order these clones respective to the sequence of the original chromosome. Note that this is all to be done without knowing the complete sequence of each BAC. Making a physical map may therefore rely on techniques related to Southern blotting: DNA from the ends of one BAC is used as a probe to find clones that contain the same sequence. These clones are then assumed to overlap each other. A set of overlapping clones is called a contig.
## Clone-by-clone sequencing
Physical mapping of cloned sequences was once considered a pre-requisite for genome sequencing. The process would begin by breaking the genome into BAC-sized pieces, arranging these BACs into a map, then breaking each BAC up into a series of smaller clones, which were usually then also mapped. Eventually, a minimum set of smaller clones would be identified, each of which was small enough to be sequenced (Figure $$\PageIndex{8}$$). Because the order of clones relative to the complete chromosome was known prior to sequencing, the resulting sequence information could be easily assembled into one complete chromosome at the end of the project. Clone-by-clone sequencing therefore minimizes the number of sequencing reactions that must be performed, and makes sequence assembly straightforward and reliable. However, a drawback of this strategy is the tedious process of building physical map prior to any sequencing.
## Whole genome shotgun sequencing
This strategy breaks the genome into fragments that are small enough to be sequenced, then reassembles them simply by looking for overlaps in the sequence of each fragment. It avoids the laborious process of making a physical map (Figure $$\PageIndex{2}$$). However, it requires many more sequencing reactions than the clone-by-clone method, because, in the shotgun approach, there is no way to avoid sequencing redundant fragments. There is also a question of the feasibility of assembling complete chromosomes based simply on the sequence overlaps of many small fragments. This is particularly a problem when the size of the fragments is smaller than the length of a repetitive region of DNA. Nevertheless, this method has now been successfully demonstrated in the nearly complete sequencing of many large genomes (rice, human, and many others). It is the current standard methodology.
However, shotgun assemblies are rarely able to complete entire genomes. The human genome, for example, relied on a combination of shotgun sequence and physical mapping to produce contiguous sequence for the length of each arm of each chromosome. Note that because of the highly repetitive nature of centromeric and telomeric DNA, sequencing projects rarely include these heterochromatic, gene poor regions.
## Genome analysis
An assembled genome is a string of millions of A’s,C’s,G’s,T’s. Which of these represent nucleotides that encode proteins, and which of these represent other features of genes and their regulatory elements? The process of genome annotation relies on computers to define features such a start and stop codons, introns, exons, and splice sites. However, few of the predictions made by these programs is entirely accurate, and most must be verified experimentally for any gene of particular importance or interest.
11.2: Whole Genome Sequencing is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by Todd Nickle and Isabelle Barrette-Ng 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": 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": 0.8766175508499146, "perplexity": 1590.764304420853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103624904.34/warc/CC-MAIN-20220629054527-20220629084527-00687.warc.gz"} |
https://www.physicsforums.com/threads/modelling-helium-in-the-joule-kelvin-expansion.364123/ | Homework Help: Modelling helium in the joule kelvin expansion
1. Dec 17, 2009
sheelbe999
How is it best to model Helium gas,
Ideal gas equation?
Dieterici?
Van der Waals?
some other equation?
Also if it is dieterici or van der waals,what the values of a and b, the constants in the equation of state.
2. Dec 17, 2009
ideasrule
It depends on what temperature/pressure the helium is at and how accurate you want your model to be. For helium at atmospheric temperature and pressure, the ideal gas model is already extremely accurate. | {"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": 0.85349041223526, "perplexity": 1950.7170851624935}, "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-2018-26/segments/1529267864364.38/warc/CC-MAIN-20180622065204-20180622085204-00518.warc.gz"} |
https://cstheory.stackexchange.com/questions/39488/on-collapsing-the-exponential-time-hierarchy | # On collapsing the Exponential time hierarchy
1. Define $\Sigma^E_0 = \Pi^E_0=E$,
2. for every $n>0$, define $\Sigma^E_n=NE^{\Sigma^p_{n-1}}$,
3. for every $n>0$, define $\Pi^E_n=CoNE^{\Sigma^p_{n-1}}$.
Define the Exponential time hierarchy by $EH=\bigcup_{i\in\mathbb{N}}\Sigma^E_i$. We know that $\Sigma^p_n=\Pi^p_n$ implies $PH=\Sigma^p_n$. My question is whether something like this is true for $EH$ or not.
Q1. Does $\Sigma^E_n=\Pi^E_n$ imply the collapse of the $EH$ to some level? For example, does $E=NE$ imply $EH=E$?
Q2. If we do not know the answer to the above question, Is there any oracle separation? For example is there any oracle $A$ such that $E^A=NE^A$, but $E^A\not={\Sigma^E_2}^A$? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.997043251991272, "perplexity": 172.5704463643842}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250598726.39/warc/CC-MAIN-20200120110422-20200120134422-00475.warc.gz"} |
https://www.codecogs.com/library/engineering/theory_of_machines/turning-moment-diagrams.php | I have forgotten
• https://me.yahoo.com
# Turning Moment Diagrams
Turning Moment Diagrams with particular reference to Engines and Flywheels.
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## Introduction
In any machine there is at least one point where energy is supplied, and at least one other point from which energy is delivered. In an ideal machine no energy would be lost and these two would be equal. In practice this state of affairs does not exist since it is inevitable that some energy is absorbed in over coming friction at the various joints, couplings and bearings.
The ratio of energy out to energy in is known as the Mechanical Efficiency of the machine.
A flywheel is a mechanical device with a significant moment of inertia used as a storage device for rotational energy.
In addition, over a given interval of time, the kinetic and potential energies of each link will change so that either some of the energy supplied will be absorbed increasing the total energy (Kinetic and potential) of the moving parts, or alternatively the energy supplied will be supplemented by a decrease in the total energy of the moving parts.
It should be stressed that over the time taken for the machine to complete one cycle, the net change of energy for each moving part is nil, since at the end of the cycle each part is in the same position and has the same speed as at the beginning of the cycle. However, during the cycle the input of energy or the load on the machine may vary considerably. In most cases this fluctuation is kept to a minimum by the use of a flywheel.
## Fluctuations Of Energy And Speed
The driving torque produced by a reciprocating engine fluctuates during any one cycle. The manner in which it varies depends upon the type of engine, number of cylinders, the characteristics of the flywheel etc. It can usually be assumed that the resisting torque due to the load is constant, and when > the engine will be accelerating; and vice versa.
If there are complete cycles per minute and the engine speed is Then the power transmitted is
i.e. = The mean height of the turning moment diagram
For any period during which the area cut off on the turning moment diagram represents "excess energy" , which will go to increase the speed of the rotating parts.
where is the moment of inertia of the flywheel and rotating parts and and are the maximum and minimum speeds during one cycle.
Thus,
This can be re-written to include the mean speed as:
(Approximately)
The coefficient of fluctuation of speed is:
This is usually expressed as a percentage variation from the mean. i.e.
In simple cases is given by the area of one "loop" intercepted between and , but for multi-cylinder engines a further analysis is necessary. (See Example 5)
## Disc And Rim Flywheels
The purpose of a flywheel is to absorb energy when the supply of energy to a machine exceeds the requirement, and to provide energy when there is a deficit.
A diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle.
A body when it rotates behaves as if all of its mass were concentrated in a ring at a distance from the axis of rotation. The radius is known as the Radius of Gyration of the body. The product is known as the Moment of Inertia of the body and given the symbol .
For a solid disc of diameter ,
For a ring or rim of diameters and ,
Example:
[imperial]
##### Example - Example 1
Problem
The mean speed of an engine is 250 r.p.m. The maximum fluctuation of energy generated in the engine is 850 ft.lb. and the resisting torque is constant. Determine the moment of inertia of the flywheel required to keep the speed within the range 1% above to 1% below the mean speed. State clearly the units in which the moment of inertia is expressed.
It is desired to reduce the coefficient of speed fluctuation by one fifth by bolting a plain cast-iron ring to the side of the flywheel. The ring is to have an outside diameter of 3ft. 4in. and an inside diameter of 2ft. 8in. and to be made of material which weighs 0.28 lb./cu.in. Find the width of the required ring.
Workings
The coefficient of fluctuation of speed and from equation (1)
Note. For those of you who are not used to the foot slug second system of engineering quantities, a full description can be found on the Codecogs site under References: Engineering: General. However in brief the Slug is that mass which on this planet weighs one pound (lb.)
To reduce the coefficient of speed fluctuation by one-fifth it is necessary to increase by one-quarter. As this is to be done by adding a ring of thickness :
Note. is now measured in lb.in.^2 and the density of the flywheel material has been converted into slugs/cu.in. since the equation demands the mass of the flywheel rather than its weight.
Solution
• The moment of inertia is
• The width is | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 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": 0.9406213760375977, "perplexity": 435.3276568454481}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1573496667333.2/warc/CC-MAIN-20191113191653-20191113215653-00250.warc.gz"} |
https://www.doubleloopmedia.com/dqto42t/1c285f-tangent-secant-formula | Secant Line Definition. A secant and a tangent meet at a 90° angle outside the circle. \\ These inverse functions have the same name but with 'arc' in front.So the inverse of sec is arcsec etc. So x = 40. m \angle x = \frac{1}{2} \left( \overparen{Farc} - \overparen{Narc} \right) In order to find the tangent line at a point, you need to solve for the slope function of a secant line. \\ Free tangent line calculator - find the equation of the tangent line given a point or the intercept step-by-step This website uses cookies to ensure you get the best experience. \\ m \angle x = \frac{1}{2} \left( \overparen{CAH} - \overparen{CH} \right) For the given function, find (a) the equation of the secant line through the points where x has the given values and (b) the equation of the tangent line when x has the first value. The secant function is the reciprocal of the cosine function. = \frac{\class{data-angle-0}{035.92} - \class{data-angle-1}{89.84}}{2} The outer arc is 143º. Your IP: 68.183.188.176 A secant and a tangent meet at a 90° angle outside the circle. This is because secant is defined as. If you look at each theorem, you really only need to remember ONE formula. λ = c / f = wave speed c (m/s) / frequency f (Hz). Sometimes written as asec or sec-1 Remember that this theorem only used the intercepted arcs . Example 1: Find Sec X if Cos x = 3 ⁄ 8. Two secants extend from the same point and intersect the circle as shown in the diagram below. However, the reciprocal functions (secant, cosecant and cotangent) can be helpful in solving trig equations and simplifying trig identities. Secant is the reciprocal of cosine. Secant of a Circle Formula. Introduction In trigonometry, the secant and tangent are two functions, and they have a direct relation between them in square form but their relationship is derived from Pythagorean theorem . m \angle x = \frac{1}{2} (50) The measure of an angle formed by a secant and a Plot of the six trigonometric functions, the unit circle, and a line for the angle θ = 0.7 radians. We wil… the circle. 143 - 63 = 80. Tangent is a special case of a secant where the two points of intersection of a line with a circle coincide. \\ Introduction to the Tangent Function. Secant Line Definition. 12(a + 12) = 102 10 + 12 = a2 10(a + 10) = 122 10(12) = a2 - the answers to estudyassistant.com Finally, we’ll use the term tangent for a line that intersects the circle at just one point. E. Gunter (1624) used the notation "tan", and J. H. Lambert (1770) discovered the continued fraction representation of this function. Before getting stuck into the functions, it helps to give a nameto each side of a right triangle: circle is $$\frac 1 2$$ the difference of the intercepted arcs . In other words, we can say that the lines that intersect the circles exactly in one single point are Tangents. Therefore to find this angle (angle K in Cotangent is the reciprocal of tangent. Then x = [1/2] (143 - 63). A secant line (from the Latin Secare, to cut) connects two ore more points on a curve.. You can find any secant line with the following formula: (f(x + Δx) – f(x))/Δx or lim (f(x + h) – f(x))/h. the circle is half the the difference of the intercepted arcs: In the picture below, the measure of $$\angle x$$ is $$\frac 1 2$$ the difference of the arcs intercepted by the two secants. function in trigonometry. A tangent is a line that touches the parabola at exactly one point. The cotangent function is the reciprocal of the tangent function. = \class{data-angle-outer}{26.96} ^{\circ} The angle formed by the intersection of 2 tangents, 2 secants or 1 tangent and 1 secant outside the circle equals half the The measure of an angle formed by a 2 secants drawn from a point outside The secant function that we are talking about is defined as one of the reciprocal of our basic three functions. Please enable Cookies and reload the page. Secant is Reciprocal of Cos, Sec x = $$\frac{1}{CosX}$$ Examples of Secant Math Formula. In one way, this case seems to differ from the others-- because all circle is included in the intercepted arcs. So, Sec X = 8/3 Solution for For the function f(x) = - 6x, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at x= 3. A secant of a parabola is a line, or line segment, that joins two distinct points on the parabola. The three theorems for the intercepted arcs to the angle of two tangents, two secants or 1 tangent and 1 secant are summarized the examples below), all that you have to do is take the far intercepted arc When we see "arcsec A", we interpret it as "the angle whose secant is A". As we work through this lesson, remember that a chord of a circle is a line segment that has both of its endpoints on the circle. [1/2]⋅80 = 40. The angle formed by the intersection of 2 tangents, 2 secants or 1 tangent and 1 secant outside the circle equals half the difference of the intercepted arcs! $$. Sine, Cosine and Tangent. \\ So, we have cosecant which is the reciprocal of sine, secant which is the reciprocal of cosine, and cotangent is the reciprocal of the tangent function. \overparen{\rm Near} = \class{data-angle-1}{89.84} \\ m \angle x = \frac{1}{2}(140-50) Therefore, its basic formula is: s e c X = H y p o t e n u s e A d j a c e n t S i d e. sec X = \frac {Hypotenuse} {Adjacent Side} secX = Adj acentS ideH ypotenuse.$$ Look up above to see the easy way to remember the formulas. This result is found as Proposition 36 in Book 3 of Euclid's Elements.. Only one of the two circles below includes the intersection of a For every trigonometry function such as sec, there is an inverse function that works in reverse. m \angle x = \frac{1}{2} (205-155) used in this theorem's formula. $$The segment is not tangent to the circle at C. However,$$\frac{1}{2}(115- 45) = 35 $$so the segment intersects point D. (the 115 represents 113 + 2 which is the sum of arc ABC + arc CD),$$ We … (From the Latin tangens "touching", like in the word "tangible".) The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. Given a secant g intersecting the circle at points G 1 and G 2 and a tangent t intersecting the circle at point T and given that g and t intersect at point P, the following equation holds: Tangent and Secant. \\ m \angle x = \frac{1}{2}(90) Notice in particular that sine and tangent are odd functions, being symmetric about the origin, while cosine is an even function, being symmetric about the y-axis. Secant of a Circle Formula If a secant and a tangent of a circle are drawn from a point outside the circle, then; Lengths of the secant × its external segment = (length of the tangent segment… Cross multiplying the equation gives. m \angle x = 25^{\circ} \\ 60 = 210 - \overparen{\rm CH} The following image shows a secant line that connects two points, along with a tangent line (which skims the curve at one point): Since $$\frac{1}{2}(113- 45) \ne 35. Slope; Finding the Equation; Exsecant Function; 1. The inner arc is 63º. Remember that this theorem only makes use of the intercepted arcs. Solution. (From the Latin secare "cut or sever") The line that joins two infinitely close points from a point on the circle is a Tangent. More precisely, a straight line is said to be a tangent of a curve y = f at a point x = c if the line passes through the point on the curve and has slope f', where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space. (Both lines in the picture are tangent to the circle),$$ m \angle x = 45^{\circ} A tangent line just touches a curve at a point, matching the curve's slope there. drawn from a point outside the circle is $$\frac 1 2$$ the the difference of the intercepted arcs . Formula: If a secant and a tangent of a circle are drawn from a point outside the circle, then the product of the lengths of the secant and its external segment equals the square of the length of the tangent segment. What is the measure of x in the picture on the left. At the point of tangency, a tangent is perpendicular to the radius. The average rate of change of a function between two points and the slope between two points are the same thing. Lets take a look at tangent Tangent is defined as sin tan cos x x x Now that we. The length of the hypotenuse, when divided by the length of the adjacent side, will give the secant of the angle in a right triangle. These six trigonometric functions in relation to a right triangle are displayed in the figure. In trigonometry (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). Only Circle 1 on the left is consistent with the formula. You can find any secant line with the following formula: The measure of an angle formed by a two tangents Solution: As Sec X = 1/ Cos X =1/3/8 =8/3. Cloudflare Ray ID: 616960152d4c1924 Example problem: Find the tangent line at a point for f(x) = x 2. Use your knowledge of the theorems on this page to determine at whether point C or point D is where the bottom segment \\ The formula for time is: T (period) = 1 / f (frequency). this formula. What is the measure of $$\overparen{\rm CH}$$? Sine, cosine, secant, and cosecant have period 2π while tangent and cotangent have period π. Identities for negative angles. Tangent to a Circle; Angle Formed by a Tangent and a Chord; Angle Formed by Two Chords; Angle Formed by Tangents and Secants; Segments Formed by Two Chords; Segments Formed by Two Secants; Segments Formed by a Tangent and a Secant; Circle: Equation; Equation of a Tangent Line: Circle; System of Equations: Circle, Line; Circle: Area; Sector: Area (See above.) Since both of the lines are tangents, they touch the circle in one point and therefore they do not 'cut off' any parts of Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. Real World Math Horror Stories from Real encounters. Besides that, we’ll use the term secant for a line segment that has one endpoint outside the circle and intersects the circle at two points. By using this website, you agree to our Cookie Policy. As we know there are six trigonometric functions and out of these, Secant, cotangent, and cosecant are hardly used. Slope; Finding the Equation; Exsecant Function; 1. Consider the circle below. The points labelled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. m \angle x = \frac{1}{2} \left( \overparen{CAH} - \overparen{CH} \right) Defining the tangent function. It was mentioned in 1583 by T. Fincke who introduced the word "tangens" in Latin. Secant Formula The length of the hypotenuse, when divided by the length of the adjacent side, becomes the secant of an angle in a right triangle. $$. Relationship to Tangent-Secant Theorem In the figure above, drag point B around the top until it meets point A. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Diameter of Circle – Secant.$$ ... 2 2 cos sin 1 x x + = and if we also recall the definition of secant in terms of cosine we arrive at, ... A potentially easier way to do this is to think of the minus sign as part of the first function in the product. \\ Therefore to find this angle (angle K in the examples below), all that you have to do is take the far intercepted arc and near the smaller intercepted arc and then divide that number by two! What is the formula of period? Three Functions, but same idea. Suppose line DB is the secant and AB is the tangent of the circle, then the of the secant and the tangent are related as follows: DB/AB = AB/CB. So, we have cosecant which is the reciprocal of sine, secant which is the reciprocal of cosine, and cotangent is the reciprocal of the tangent function. The cosine graph crosses the … In mathematics, the trigonometric functions are a set of functions which relate angles to the sides of a right triangle.There are many trigonometric functions, the 3 most common being sine, cosine, tangent, followed by cotangent, secant and cosecant. The abbreviation of secant is sec. \\ It is written as Sec, and the formula for secant is: The formula for secant theta What is the measure of $$\overparen{\rm CH}$$? When solving right triangles the three main identities are traditionally used. For example, the triangle contains an angle A, and the ratio of the side opposite to … As with tangent and cotangent, the graph of secant has asymptotes. The length of two tangents from a common external point to a circle are equal. difference of the intercepted arcs! Several theorems are related to this because it plays a significant role in geometrical constructionsand proofs. m \angle x = \frac{1}{2} \left( \overparen{Farc} - \overparen{Narc} \right) \\ The abbreviation of cotangent is cot. You may need to download version 2.0 now from the Chrome Web Store. Where n is an integer. \\ A tangent line just touches a curve at a point, matching the curve's slope there. All of the formulas on this page can be thought of in terms of a "far arc" and a "near arc". If Tangents of two circles intersect at a common point is called the internal tangents. only the intercepted arcs count. \\ In order to find the tangent line at a point, you need to solve for the slope function of a secant line. The tangent function is an old mathematical function. Right Triangle. \\ What must be the difference between the measures of the intercepted arcs? Performance & security by Cloudflare, Please complete the security check to access. 30 =\frac{1}{2}(210- \overparen{\rm CH}) That's why we call this the Far Arc Near Arc theorem (sometimes abbreviated Farc - Narc). In other words, is point D tangent to tangent and a secant. Point of tangency is the point where the tangent touches the circle. \\ The last three are called reciprocal trigonometric functions, because they act as the reciprocals of other functions. • \overparen{\rm Far} = \class{data-angle-0}{35.92} A secant line intersects two or more points on a curve. 2 \cdot 30= (210- \overparen{\rm CH}) Pierre de Fermat anticipated the calculus with his approach to finding the tangent line to a given curve. tangent drawn from a point outside the A secant line (from the Latin Secare, to cut) connects two ore more points on a curve.. \angle{Outer} = \frac{\overparen{\rm Far} - \overparen{\rm Near}}{2} The abbreviation of cosecant is csc or cosec. Answer: 2 question Which equation results from applying the secant and tangent segment theorem to the figure? A tangent line is a straight line that touches a function at only one point. The models of this kind are suggested in various references, such as: Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle.. Before getting stuck into the functions, it helps to give a name to each side of a right triangle: Tangent Lines and Secant Lines (This is about lines, you might want the tangent and secant functions). by the pictures below. The secant function that we are talking about is defined as one of the reciprocal of our basic three functions. m \angle x = \frac{1}{2} \left( \overparen{ABC} - \overparen{XYZ} \right) \\ The angle formed outside of the circle is always equal to the the far arc minus the near arc divided by 2. When the equation of continuous curve is used to establish the bond stress–slip model, the values of tangent and secant bond stiffness obtained vary continuously and definitely, which is convenient to be used in finite element analysis. The domain, in other words, is. The following image shows a secant line that connects two points, along with a tangent line (which skims the curve at one point): Another way to prevent getting this page in the future is to use Privacy Pass. If a secant and a tangent of a circle are drawn from a point outside the circle, then; Lengths of the secant × its external segment = (length of the tangent segment) 2. A secant line intersects two or more points on a curve. y=f(x) = x² +x; x= -2, x=2 a. the circle? Sin(θ), Tan(θ), and 1 are the heights to the line starting from the x-axis, while Cos(θ), 1, and Cot(θ) are lengths along the x-axis starting from the origin. Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle. What must be the difference between the measures of the intercepted arcs? Keep in mind that f (x) is also equal to y, and that the slope-intercept formula for a line is y = mx + b where m is equal to the slope, and b is equal to the y intercept of the line. Therefore, the red arcs in the picture below are not and near the smaller intercepted arc and then divide that number by two! More about Secant angles formula. xº: is the angle. Internally. 150^{\circ} = \overparen{\rm CH}$$. The fact that you can take the argument's "minus" sign outside (for sine and tangent) or eliminate it entirely (for cosine) can be helpful when working with complicated expressions. 2 \cdot 30= 2 \cdot \frac{1}{2}(210- \overparen{\rm CH}) Finding tangents to curves is historically an important problem going back to P. Fermat, and is a key motivator for the differential calculus. Secant Line Definition. • Secant Line Definition. What is the value of x? (From the Latin tangens "touching", like in the word "tangible".) Therefore, the red arc in the picture below is not used in m \angle x = \frac{1}{2} \left( \overparen{Farc} - \overparen{Narc} \right) The subtraction of square of tan function from square of secant function equals to one is called the Pythagorean identity of secant and tangent functions. Note: In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. As Secant line = Average Rate of Change = Slope. Length PR = Length PQ How to Find the Tangent of a Circle? Do This (*) Draw a circle and a secant PQ of the circle on a paper as shown below. formed by a tangent and a secant. Interactive simulation the most controversial math riddle ever! Since … \\ If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. intersects the circle. The line is now a tangent to the circle, and PA=PB. 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Book 3 of Euclid 's Elements we call this the Far arc the. | {"extraction_info": {"found_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": 2, "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": 0.9522075653076172, "perplexity": 602.6639276532551}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057830.70/warc/CC-MAIN-20210926053229-20210926083229-00438.warc.gz"} |
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# Given $A=\begin{bmatrix}2 & 4 & 0\\3 & 9 & 6\end{bmatrix}\;and\;B=\begin{bmatrix}1& 4\\2 & 8\\1 & 3\end{bmatrix}.\;Is \;(AB)'=B'A'?$
Toolbox:
• If A_{i,j} be a matrix m*n matrix , then the matrix obtained by interchanging the rows and column of A is called as transpose of A.
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Step1:
By the property of transpose of a matrix, we have $(AB)'=B'A'$
Given:
$A=\begin{bmatrix}2 & 4 & 0\\3 & 9 & 6\end{bmatrix}_{2\times 3}\;\;B=\begin{bmatrix}1& 4\\2 & 8\\1 & 3\end{bmatrix}_{3\times 2}$
$AB=\begin{bmatrix}2 & 4 & 0\\3 & 9 & 6\end{bmatrix}\begin{bmatrix}1& 4\\2 & 8\\1 & 3\end{bmatrix}$
$AB=\begin{bmatrix}2 (1)+4(2)+0(1)& 2(4)+4(8)+0(3)\\3(1)+9(2)+6(1) & 3(4)+9(8)+6(3)\end{bmatrix}$
$\;\;\;=\begin{bmatrix}2 +8+0& 8+32+0\\3+18+6 & 12+72+18\end{bmatrix}$
$\;\;\;=\begin{bmatrix}10& 40\\27 & 102\end{bmatrix}$
$(AB)'=\begin{bmatrix}10& 27\\40 & 102\end{bmatrix}$ [Transpose of a matrix can be obtained by changing the rows and a column.]
Step2:
Let $B'A'$
Given:B=$\begin{bmatrix}1 & 4\\2 & 8\\1 & 3\end{bmatrix}$
$B'=\begin{bmatrix}1 & 2 &1\\4 & 8 & 3\end{bmatrix}$
$A=\begin{bmatrix}2 & 4 & 0\\3 & 9 & 6\end{bmatrix}$
$A'=\begin{bmatrix}2 & 3 \\4 & 9 \\0 & 6\end{bmatrix}$
$B'A'=\begin{bmatrix}1 & 2 & 1\\4 & 8 & 3\end{bmatrix}\begin{bmatrix}2& 3\\4 & 9\\0 & 6\end{bmatrix}$
$\;\;\;=\begin{bmatrix}1 (2)+2(4)+1(0)& 1(3)+2(9)+1(6)\\4(2)+8(4)+3(0) & 4(3)+8(9)+3(6)\end{bmatrix}$
$\;\;\;=\begin{bmatrix}2 +8+0& 3+18+6\\ 8+32+0& 12+72+18\end{bmatrix}$
$\;\;\;=\begin{bmatrix}10& 27\\40 & 102\end{bmatrix}$
$\Rightarrow(AB)'=B'A'$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9314929842948914, "perplexity": 377.59666068109715}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1495463609061.61/warc/CC-MAIN-20170527210417-20170527230417-00200.warc.gz"} |
https://www.vedantu.com/maths/angle-between-a-line-and-a-plane | # Angle Between a Line and a Plane
### Basics to Find the Angle Between Line and Plane:
In geometry, a line and an angle are used with a different meaning. A line in geometry is a set of closely arranged points that extend lengthwise in both the directions. When a set of lines are arranged adjacent to each other, a plane is obtained. A plane is a geometric surface that has two dimensions. The line has only one dimension which is measured in terms of length whereas a plane is a two-dimensional surface measured in terms of length and width. When a line is an incident on a plane, it forms an angle with the plane at the point of contact. This angle is said to be the angle between a line and a plane.
### How to Find the Angle Between Line and Plane?
When a line is inclined to a plane at a certain angle and touches the plane at a particular point, the angle between the line and plane is equivalent to the complement of the angle between the line and the normal to the plane at the point of contact of line and plane.
Consider a line indicated in the above diagram in brown color. Let vector ‘n’ represent the normal drawn to the plane at the point of contact of line and plane. Let the angle between the line and the plane be ‘α’ and the angle between the line and the normal to the plane be ‘β’. The angle between a line and a plane as represented in the figure is equal to the complement of the angle between the line and the normal to the plane. i.e. the angle ‘α’ is equal to the complement of the angle ‘β’. In other words, the value of angle ‘α’ is equal to the value obtained when the value of angle ‘β’ is subtracted from 900.
### How to Find the Angle Between Line and Plane Formula?
Consider a line and a plane in which the line is inclined to the plane at a certain angle. A line is represented by a vector equation as where ‘a’ is the position vector of the initial point of the line and r is the position vector of the endpoint of the line. ‘b’ indicates the direction vector of the line drawn from the initial to the final point. Similarly, the vector equation of a plane is . Let the angle between the line and the normal is ‘θ’ and the angle between the line and the plane is ‘Φ’. The cosine of the angle between the line and the normal is given as:
$\cos \theta = \left| {\frac{{\overrightarrow b {\text{ }}.\overrightarrow {{\text{ }}n} }}{{\left| {\overrightarrow b } \right|.\left| {\overrightarrow n } \right|}}} \right|$
The line drawn normal to the plane is always perpendicular to the plane. The sum of the angle between the line and the plane and the angle between the line and the normal is equal to the right angle. So, the angle between the line and a plane is equal to the complement of the angle between the line and the normal. From the formulas of trigonometric ratios of complementary angles, Cos θ = Sin (90 - θ) = Sin Φ. So, the sine of the angle between the line and a plane is given by the equation for Cos θ as:
$\sin \Phi = \left| {\frac{{\overrightarrow b {\text{ }}.\overrightarrow {{\text{ }}n} }}{{\left| {\overrightarrow b } \right|.\left| {\overrightarrow n } \right|}}} \right|$
So, the angle between a line and a plane is given as:
$\Phi = {\sin ^{ - 1}} = \left| {\frac{{\overrightarrow b {\text{ }}.\overrightarrow {{\text{ }}n} }}{{\left| {\overrightarrow b } \right|.\left| {\overrightarrow n } \right|}}} \right|$
### Angle Between a Line and a Plane Example Problems:
1. Find the angle between line and plane where the line is represented by $r = \frac{{x + 1}}{2} = \frac{{y + 1}}{1} = \frac{z}{2}$and the equation of the plane is x + y - 1 = 0. (Hint: Use the angle between line and plane formula)
Solution:
Let us consider the angle between the line and the plane as Φ.
The given line can be written in vector form as
$\hat r = \left( {i - j} \right) + \lambda \left( {2i + j + 2k} \right)$
The equation of the normal to the plane in vector form is given as:
$\hat r = i + j$
So, b = 2i + 1j + 2k and n = i + j
The sine of the angle between the line and the plane is given as:
$\sin \Phi = \left| {\frac{{\overrightarrow b {\text{ }}.\overrightarrow {{\text{ }}n} }}{{\left| {\overrightarrow b } \right|.\left| {\overrightarrow n } \right|}}} \right|$
Substituting the values of b and n, we get
$\sin \Phi = \left| {\frac{{\left( {2i + 1j + 2k} \right)\left( {i + j - k} \right)}}{{\sqrt {{2^2} + {1^2} + {2^2}} .\sqrt {{1^2} + {1^2} + {0^2}} }}} \right|$
$Sin\phi\;=|\frac{2\;.\;1\;+\;1\;.\;1\;+\;2\;.\;0}{\sqrt{9}\;.\;\sqrt{2}}|$
$Sin\phi\;=|\frac{3}{3\;.\;\sqrt{2}}|$
$Sin\phi\;=|\frac{1}{\sqrt{2}}|$
$\phi\;=Sin^{-1}(\frac{1}{\sqrt{2}})$
$\phi\;=45^{0}$
So, the angle between the line and the plane is 450.
### Fun Facts About Angle Between a Line and a Plane:
• When a line is parallel to a plane or on the plane, the line does not form any angle with the plane and the line is perpendicular to the normal drawn to the plane.
• If a line is perpendicular to any two lines on the same plane, then that line is perpendicular to the plane also. | {"extraction_info": {"found_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": 0.820893406867981, "perplexity": 99.29154784866039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655889877.72/warc/CC-MAIN-20200705215728-20200706005728-00232.warc.gz"} |
http://mathhelpforum.com/number-theory/189117-multiplicative-order.html | 1. ## Multiplicative order
Hi, need a some help with this proof.
Prove the multiplicative order of $(1+n) \ mod \ n^2$ is $n$ for odd prime $n$.
Thank You for any help.
2. ## Re: Multiplicative order
You have to try first. What is the definition of the multiplicative order?
3. ## Re: Multiplicative order
step one: what is $(1 + p)^p \ (mod \ p^2)$? justify your answer.
4. ## Re: Multiplicative order
Originally Posted by Deveno
step one: what is $(1 + p)^p \ (mod \ p^2)$? justify your answer.
I am sorry, I don't know.
I can see that $n^2$ does not divide $(1+n)^n$.
$\frac{(1+n)^n}{n^2}=0$
$(1+n)^n=0$
which is false for positive n.
I need a little more help if you can.
5. ## Re: Multiplicative order
try expanding $(1+p)^p$ using the binomial theorem. which terms don't have $p^2$ in them?
6. ## Re: Multiplicative order
Originally Posted by Deveno
try expanding $(1+p)^p$ using the binomial theorem. which terms don't have $p^2$ in them?
$(p+1)^p= \sum_{k=0}^p \tbinom pk p^k \cdot 1^{p-k} = \sum_{k=0}^p \tbinom pk p^k$
So in the expansion of $(1+p)^p$ the only terms than don't have $p^2$ in them are $1$ and $p$.
So where to from here then?
7. ## Re: Multiplicative order
um, no....
the coefficent of p in the expansion is p:
$(p+1)^p = 1 + p^2 + \frac{p(p-1)}{2}p^2 + \dots$ where all the remaining terms involve higher powers of p.
thus $(p+1)^p = 1 (mod\ p^2)$.
so the multiplicative order of p+1 (mod p^2) has to divide p. what are our choices, given that p is an odd prime?
8. ## Re: Multiplicative order
Originally Posted by Deveno
um, no....
the coefficent of p in the expansion is p:
$(p+1)^p = 1 + p^2 + \frac{p(p-1)}{2}p^2 + \dots$ where all the remaining terms involve higher powers of p.
thus $(p+1)^p = 1 (mod\ p^2)$.
so the multiplicative order of p+1 (mod p^2) has to divide p. what are our choices, given that p is an odd prime?
Oh yeah, slight error on my part. So if the order of p+1 mod p^2 has to divide p and p is an odd prime, then by the definition of a prime, the order of p+1 mod p^2 has to be p.
9. ## Re: Multiplicative order
almost...why isn't the order of p+1, 1 (after all, 1 divides p)?
it's a small detail, and easily answered, but it never hurts to dot the "i's" and cross the "t's".
10. ## Re: Multiplicative order
Originally Posted by Deveno
almost...why isn't the order of p+1, 1 (after all, 1 divides p)?
it's a small detail, and easily answered, but it never hurts to dot the "i's" and cross the "t's".
Yes, of course! That makes perfect sense. Thanks again for your fantastic 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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 22, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9503574967384338, "perplexity": 1401.9359924705452}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1484560280266.9/warc/CC-MAIN-20170116095120-00109-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://twistedone151.wordpress.com/2008/12/15/monday-math-50/ | ## Monday Math 50
Let us consider Laplace’s equation, in two dimensions. In Cartesian coordinates, this is . We can see by immediate inspection that linear functions f(x,y)=ax+by+c satisfy this. We also see that terms of the form axy satisfy the equation.
A useful method for finding general solutions is the separation of variables for partial differential equations technique. Here, we substitute in a function of the form f(x,y)=X(x)Y(y). Then Laplace’s equation becomes:
Dividing both sides by f(x,y)=X(x)Y(y), we have:
Now, we note that is a function of x only, while is a function of y only. By varying x while holding y fixed, and vice versa, we see that the sum of these two can be zero for all x and y if and only if both functions are constants, which sum to zero. Thus, we have , for some constant λ.
Both of these are second order differential equations: has solution , while has solution .
Combining these, we have solutions of the form , and we can sum any of these to produce another solution.
For a more specific example, let us choose the region 0≤x,y≤1 with boundary conditions (Dirichlet) , , , .
Then we have, from the above, solutions of the form , , . To get a solution , we want
,
for a function . For a nonzero solution which obeys , we need to be periodic in x (complex exponential, rather than real), so we need λ<0. Thus, let λ=-ω2. We then have
,
and tells us that , and thus for positive integer n,
Thus, we have λ=-ω2=-n2π2, and . tells us , which means
,
and so
,
and our solutions for these three parts of the boundary are .
Consider the Fourier series for the square wave function with period 2π , which is
Compare to
With for odd n and for even n, we have when y=1 the terms of the Fourier series for . Thus, our last boundary is met, and our solution found, by
.
Here is a graph of the partial series with the first 25 terms:
Now, let us consider polar coordinates, useful for solving Laplace’s equation on the disk. In polar coordinates, the laplacian is
.
Trying a function of the form in Laplace’s equation, we get:
Multiplying that equation by r2 and then dividing by , we have:
As the terms in the first set of parentheses is a function of r only, and the second set holds a function of θ only, these two functions are constants, and so we have , .
The latter equation is easily solved: has general solution . Meanwhile, the second order linear differential equation
is a second order Cauchy-Euler equation, which we solve by substituting a trial solution of the form rm:
for λ≠0. With λ=0, we have the equation: , which, with , gives the first order equation , which is separable:
So we have for λ≠0, and for λ=0.
Note that if our region covers all angles θ, then we have the requirement that must be periodic, with . This tells us first that λ≥0; with λ=ω2, we have . Second, the requirement that means that ω=n, n an integer:
.
Examining our radial solution for λ=ω2=n2, we have for positive n the radial function . The latter term has a singularity at the origin, so if the region over which we wish to solve contains the origin, we must have A2=0 for n>0. For n=0, we have radial function , and to avoid a singularity at the origin, we again set A2=0. Thus, the solutions for Laplace’s equation for a domain in polar coordinates containing the origin (and all angles θ) are of the form , n=0,1,2,3,…
For example, n=0 is the constant function; n=1 gives
; these confirm our early result of linear functions in x and y being solutions. n=2 gives us and , this latter which we also noted earlier.
Suppose we want to solve Laplace’s equation on the unit disk with (Dirichlet) boundary condition . Then we see that we want to combine solutions of the form . As our boundary condition is an odd function of θ, the cosine terms disappear, and we have , plugging in r=1, and noting that our boundary condition is the square wave function we defined earlier: , we put , giving solution
Here is a graph of the partial series with the first 25 terms:
### One Response to “Monday Math 50”
1. Monday Math 51 « Twisted One 151’s Weblog Says:
[…] Math 51 By twistedone151 Last week, we looked at solving Laplace’s equation, in two dimensions. In particular, I demonstrated […] | {"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": 76, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9845212697982788, "perplexity": 502.60807829855924}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607963.70/warc/CC-MAIN-20170525025250-20170525045250-00560.warc.gz"} |
https://www.longevitas.co.uk/site/informationmatrix/overflow.html | ## Overflow
A good general-purpose formula for describing pensioner mortality rates is the logistic function:
q = exp(α) / (1 + exp(α))
where the value of α varies by age. This particular formula arises when using logistic regression, a type of generalised linear model (GLM). Any mathematician looking at the equation above will see that the value of q tends to 1 as α increases. This is demonstrated in Table 1 for various increasing values of α.
Table 1. Evaluation of the logistic function for various values of α.
αLogistic function
50.993307149
100.999954602
150.999999694
200.999999998
251
So far so good: as α increases, the value of the logistic function approaches 1 as expected. At α=25 the logistic function is around 1e13 different from 1, which is so close that the answer is just 1 when shown to nine decimal places in Table 1 above.
Now try this in Microsoft Excel or any other PC calculator. In particular, try the value α=710. Instead of getting the answer 1, you will probably see that the calculation has failed and the result will be something like #NUM!. The problem lies in how computers store real numbers. Machine arithmetic typically dedicates a fixed amount of space to hold an approximation of a real number. In order to keep things compact and relatively fast, this involves various compromises. As a result there is a limit to how large a number can be and still be stored. If a number is larger than this during any intermediate calculation, overflow occurs and the whole calculation fails. This happens even if the final result of the whole formula is well within the computer's arithmetical range. In the case of the double-precision standard being used by Microsoft Excel, any intermediate value which exceeds 10^320 will overflow and the entire calculation will fail.
The solution lies in careful programming. For example, the formula at the top of this post can be re-written as follows:
q = 1 / (1 + exp())
Although this formula is mathematically equivalent to the one at the top, in terms of computer arithmetic it is far more robust for large, positive values of α. For example, if you set α=710 this new formula will yield 1 in Microsoft Excel instead of the #NUM! error previously.
Observant readers will note that this new formula will itself fail when α=-710 and below. The complete solution is therefore to use the formula at the top of the page when α is negative, and to use the second formula when α is positive. This allows evaluation of the logistic function over the entire real line without error.
Assume we have a random variable, $$X$$, with expected value ... Read more | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8334434032440186, "perplexity": 576.6549324270848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794865913.52/warc/CC-MAIN-20180524033910-20180524053910-00193.warc.gz"} |
https://www.albert.io/ie/general-chemistry/equilibrium-pressure-for-a-gas-phase-reaction | Free Version
Difficult
# Equilibrium Pressure for a Gas Phase Reaction
CHEM-L0JAKP
The gases nitrogen (${ N }_{ 2 }$) and oxygen (${ O }_{ 2 }$) are the two most abundant gases in the atmosphere. These two gases can react to form nitric oxide ($NO$) by the following reaction.
$${ N }_{ 2 }(g)+{ O }_{ 2 }(g)⇌2NO(g)$$
What is the partial pressure of $NO$ at equilibrium with ${ N }_{ 2 }$ and ${ O }_{ 2 }$ if ${ P }_{ { N }_{ 2 } }=0.78\text{ atm}$ and ${ P }_{ { O }_{ 2 } }=0.21\text{ atm}$?
$K_{ P }=2.0\times { 10 }^{ -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": 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": 0.8455268740653992, "perplexity": 738.8966187537017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1487501172404.62/warc/CC-MAIN-20170219104612-00153-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/derivation-for-magnification-equation-for-convex-mirrors.612552/ | # Derivation for magnification equation for convex mirrors
1. Jun 8, 2012
### 330730358
Hello everyone, I really run into a problem here.
The magnification equation for mirrors describes such a relation: M=-distance of image/distance of object = height of image/height of object. (M=(-i/o)=h'/h).
I understand how this formula can be proved using a ray diagram for concave mirrors simply by proving similar triangles between the image distance and the object distance. However, it didn't quite work out for the convex mirrors as I worked for a few days trying to identify a similar triangle using the two. All I can get from the diagram is that h'/h= distance from object to the centre of mirror/distance from image to the centre of mirror. Interestingly this gives the same result as i compute h'/h, indicating that this is the correct magnification equation.
I would like to ask the physics experts here to kindly give me some suggestions on how to mathematically solve this derivation here; greatly appreciated, thanks!
2. Jun 9, 2012
### 330730358
Yes, but where are the similar triangles, and how do you proof them?
3. Jun 9, 2012
### Simon Bridge
The important part of this sort fo thing is finding the triangles an important skill.
Lets label the diagram above:
O M I F and C are the points on the horizontal axis for the object, mirror, image, focus and center or curvature. So |MF| = f right?
A and B are the top of the object and the top of the image.
D E and G are the three places the rays touch the vertical (mirror) axis.
Using these - triangle AOF is similar to GMF ... now do you see them? I can see three similar triangles involving point C.
Triangle AOF has height h and base o+f.
To use them, write the formulas you get from considering that the ratios of corresponding sides are equal ... this will get you a set of simultanious equations: eliminate the unwanted variables and rearrange.
4. Jun 9, 2012
### 330730358
Yes, I drew a diagram just like you described, and I did find a similar triangle of AOF and GMF, but pardon my slowness in math; I failed to see how the rest of the derivation goes. So can you please elaborate on the rest of the similar triangles and how the final result is derived?
5. Jun 9, 2012
### Simon Bridge
Well those two give you:$$\frac{h'}{h} = \frac{f}{f+o}$$... or something like that right?
That looks close - but you need to get rid of that pesky f, and you are missing an i ... can you see any other similar triangle sets you can use? Maybe one that lets you express f in terms of i?
6. Jun 9, 2012
### 330730358
I appreciate your patience, but sorry, I can't see the other similar triangle that can help.
On the other hand, the relation ΔAOF≈ΔGMF only gives you AO/GM=MF/OF which is h/GM=(f+o)/f in this case i think we cannot assume GM=h' because if you draw a bigger diagram you can see that GM is slightly bigger than h', the tangenline drawn at M and the incident ray that goes to F, and its reflected parallel ray do not all intercept at point G.
7. Jun 9, 2012
### 330730358
sorry for the unclearness of the picture, my cellphone camera is very crappy.
#### Attached Files:
• ###### moto_0211.jpg
File size:
13.4 KB
Views:
271
8. Jun 10, 2012
### Simon Bridge
That's not an assumption - it has to be the case by geometry. You just made a mistake in your drawing: you have drawn all your rays to the curved line representing the mirror surface - then bending the rays at that surface. This is not how you draw ray diagrams. (Also your horizontal lines are not very parallel... but that's not so important.)
In all the optics you are learning you are using the par-axial approximation ... the mirror surface in that approximation is so small the entire curve would fit inside the thickness of the vertical line.
You should always draw your rays so they bend at the vertical axis - that's why you draw it. Look again at the one I showed you - the rays seem to bounce off the vertical and not the "mirror". This is why. The big curve people like to draw on these diagrams is indicative only and not meant to be part of the analysis. It is actually better to leave the curve off the diagram completely... they'll make more sense.
|GM| has to be h' since it is (by definition) the intersection with the vertical axis with a horizontal line drawn through B and |BI|=h'.
9. Jun 11, 2012
### 330730358
I see, now I can derive the equation. Thank you so much for your help!
10. Jun 12, 2012
### Simon Bridge
No worries - you should find these diagrams easier to understand and use now.
Similar Discussions: Derivation for magnification equation for convex mirrors | {"extraction_info": {"found_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": 0.8804224729537964, "perplexity": 971.2991751166463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812405.3/warc/CC-MAIN-20180219052241-20180219072241-00613.warc.gz"} |
http://link.springer.com/chapter/10.1007%2F978-3-540-70575-8_48 | Chapter
Automata, Languages and Programming
Volume 5125 of the series Lecture Notes in Computer Science pp 587-596
# Understanding the Complexity of Induced Subgraph Isomorphisms
• Yijia ChenAffiliated withBASICS, Department of Computer Science, Shanghai Jiaotong University
• , Marc ThurleyAffiliated withInstitut für Informatik, Humboldt-Universität zu Berlin
• , Mark WeyerAffiliated withInstitut für Informatik, Humboldt-Universität zu Berlin
* Final gross prices may vary according to local VAT.
## Abstract
We study left-hand side restrictions of the induced subgraph isomorphism problem: Fixing a class , for given graphs G and arbitrary H we ask for induced subgraphs of H isomorphic to G.
For the homomorphism problem this kind of restriction has been studied by Grohe and Dalmau, Kolaitis and Vardi for the decision problem and by Dalmau and Jonsson for its counting variant.
We give a dichotomy result for both variants of the induced subgraph isomorphism problem. Under some assumption from parameterized complexity theory, these problems are solvable in polynomial time if and only if contains no arbitrarily large graphs.
All classifications are given by means of parameterized complexity. The results are presented for arbitrary structures of bounded arity which implies, for example, analogous results for directed graphs.
Furthermore, we show that no such dichotomy is possible in the sense of classical complexity. That is, if there are classes such that the induced subgraph isomorphism problem on is neither in nor -complete. This argument may be of independent interest, because it is applicable to various parameterized problems. | {"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": 0.9663305282592773, "perplexity": 987.245767106039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542455.45/warc/CC-MAIN-20161202170902-00076-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://rybu.org/aggregator | # Recent MathOverflow Questions
### Why every affine variety admit $C^\infty$-exhaustion which admit the complex homogeneous Monge-Amphre equation and a non-degenerate condition
Math Overflow Recent Questions - Mon, 06/26/2017 - 10:24
Why we have always the following fact for affine variety
If $M$ is an affine variety of dimension $n$ then there exists an $C^\infty$ exhaustion $\tau: M\to [0,r)$, $0<r\leq \infty$ such that
on $M^*=M\setminus \tau^{-1}\{0\}$ the function $u=\log \tau$ satisfies
1) $(\partial\bar\partial u)^n=0$
2)$\sqrt{-1}\partial\bar\partial u\geq 0$ and $(\sqrt{-1}\partial\bar\partial u)^{n-1}\neq 0$
outside of the ramification divisor
### Partitioning a rectangle into different isosceles triangles
Math Overflow Recent Questions - Mon, 06/26/2017 - 09:48
After all the discussion raised by this old question, I am wondering about a somewhat complementary one:
For any given rectangle, does there exist a finite set of pairwise different isosceles triangles which tile it?
It is easy to tile e.g. a $1\times a$ rectangle for $1<a<2$ by four isosceles triangles, but with two of them being equal. In the case that $a=\sqrt{\frac{5-\sqrt{5}}2}$, we are lucky and can split one of those into two smaller ones, obtaining a tiling into 5 different isosceles triangles (with all occurring angles being multiples of $\frac\pi{10}$). BTW, we can iterate that by splitting the blue triangle again etc., getting tilings of the same rectangle into $k$ different isosceles triangles for all $k\ge5$.
I am quite sure the answer to the initial question is no, and it may even be interesting to restrict it to the following:
For which other rectangles is such a tiling known to exist?
And possibly, it doesn't even make a difference if we allow an infinite set of pairwise different isosceles triangles!
### Sobolev trace theorem on Lipschitz domains
Math Overflow Recent Questions - Mon, 06/26/2017 - 08:31
Supposing that D is a bounded Lipschitz domain (and not smooth) in $\mathbb{R}^d$. From what I know, it is known that the trace operator is well-defined and continuous from $H^s(D)$ to $H^l(\partial D)$ when $l=s-1/2$ and $1/2<s<3/2$. My question is what happens when $s>3/2$, is the above result true? Also I am interested in versions concerning more general spaces like Besov or Tribel-Lizorkin. Any answer or reference will be appreciated.
### Categories whose auto-equivalences are naturally isomorphic to the identity
Math Overflow Recent Questions - Mon, 06/26/2017 - 07:56
Are there any useful characterisation of categories whose auto-equivalences are all naturally isomorphic to the identity? For example, I read in this thread, What are the auto-equivalences of the category of groups? , that the category of groups is one such category. Is there some nice general property that I can use to check whether or not a category admits of auto-equivalences that are not naturally isomorphic to the identity? If not, it'd be useful just to find some more interesting examples of categories with this property.
More specifically, what examples (if any) are there of auto-equivalences that send every object to an isomorphic object but are not naturally isomorphic to the identity? For instance, I know that any auto-equivalence of SETS must send every object to an isomorphic object, but is it true that any such equivalence is naturally isomorphic to the identity? If so, is this a general property of toposes or just a special case?
### Understanding poincare conjecture for high dimensions
Math Overflow Recent Questions - Mon, 06/26/2017 - 06:51
I am a masters student of mathematics. Me and my friends wish to organize a small seminar, with aim of understanding the poincare conjecture.
We do not wish to delve into the case of 3-manifolds, but only understand it for high dimension.
My question is: What resources (articles, books) should we use in order to understand the proof and the specific tools that are used in the proof?
### Share of fortunate people in some pie splitting setting
Math Overflow Recent Questions - Mon, 06/26/2017 - 06:45
(This question is a follow-up on an older one.)
A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the last guest gets all of what is left.)
A guest is said to be fortunate if his share of pie is strictly greater than the average share (which is $1/N$ of the original pie). Let $f(N)$ denote the number of fortunate guests out of the total of $N$ guests. What is the value of $$\lim\sup_{N\to\infty}\frac{f(N)}{N}$$ ?
### Lower Bound for $\sum_{\{i,j\}\subseteq \partial S \\ (g_i-g_j)^2 \le1}{(g_i-g_j)^2}$
Math Overflow Recent Questions - Mon, 06/26/2017 - 05:53
Let $\emptyset \subsetneq S \subsetneq \{1,\cdots,n\}$ be a set with cardinality $s$, and $g\in\mathbb{R}^n$ be a vector such that $$\sum_{\{i,j\}\subseteq \{1,\cdots,n\}}{(g_i-g_j)^2} = s(n-s).$$
Question. Is this true? $$\sum_{\{i,j\}\subseteq \partial S \\(g_i-g_j)^2\le1}{(g_i-g_j)^2} \ge \frac{s(n-s)}{n}$$ where $\partial S$ is the set of all $2$-subsets $\{i,j\}\subseteq \{1,\cdots,n\}$ that exactly one of $i$ or $j$ is in $S$.
### On certain sums involving the reciprocals of primes
Math Overflow Recent Questions - Mon, 06/26/2017 - 05:36
Let $p$ be a prime and $f(x,k)= \sum_{p^{}\leq x} \frac{1}{p^k\log p}$ where $k\geq 1$ is an integer. Is there a known asymptotic expression for $f(x,k)$, even for $k=1$ ?
Motivation: If no such results are known, i'm intending to take this as my Bachelor's thesis problem.
### What is a fat point?
Math Overflow Recent Questions - Mon, 06/26/2017 - 05:27
In our scriptum we're talking about singularities. And there is the term "fat point" (for example of "tangent of fat point") . I cannot find any definition :-/ Has somebody an idea?
### Does the Krylov subspace exponential preserve structure?
Math Overflow Recent Questions - Mon, 06/26/2017 - 05:02
It is possible to approximate the action of a matrix exponential $exp(A)$ on a vector $v$ in the corresponding Krylov subspace, i.e. calculate $exp_{Kr}(A)v$ [Saad].
Some sources claim that it is possible to compute the action of $exp_{Kr}(A)V$, where $V$ is a matrix, preserving symplecticity [Lopez] (however, the algorithm they provide does not produce the results described).
Are there any works showing that the Krylov-type exponential $exp_{Kr}(A)v$ preserves structure (especially, symplecticity) as does $exp(A)v$, when $v$ is a vector? Or, maybe, there is a way to prove it which I do not see?
For example, if $A\in sp(2n)$, then $\Psi := exp(A) \in Sp(2n)$, and for any two vectors $v, w$ holds true $\omega(v,w) = \omega(\Psi v, \Psi w)$, where $\omega(v,w) = v^T J w$ is the symplectic form. Does it holds for Krylov subspace exp?
### Non-smooth version of Stoll theorem
Math Overflow Recent Questions - Mon, 06/26/2017 - 04:07
Stoll in
Stoll, W.: The characterization of strictly parabolic manifolds. Ann. Scuola Norm. Sup. Pisa, VII, 87-154 (1980)
showed that if $M$ is a connected complex manifold of dimension $n$ with $C^\infty$ exhaustion $\tau: M\to [0,r)$, $0<r\leq \infty$ such that
1) $\sqrt{-1}\partial\bar\partial\tau>0$ on $M$
and on $M^*=M\setminus \tau^{-1}\{0\}$ the function $u=\log \tau$ satisfies
2) $(\partial\bar\partial u)^n=0$
3)$\sqrt{-1}\partial\bar\partial u\geq 0$ and $(\sqrt{-1}\partial\bar\partial u)^{n-1}\neq 0$
then $M$ with its Kaehler metric is biholomorphically isometric to the ball of radius $r$ in $\mathbb C^n$ with the Euclidean metric.
I am wondering about the singular case, i.e. if $M$ is a singular complex variety and $\tau$ is not $C^\infty$ then what can we say about $M$?
### Finite group cohomology with roots of unity as coefficients
Math Overflow Recent Questions - Mon, 06/26/2017 - 01:11
Let $G$ be a finite group of order $n$, and let $L := (\mathbb{Q}/\mathbb{Z})^d$ be a (not necessarily trivial) $G$-module (we assume that $d$ is finite).
By a direct limit argument, there must be a finite $G$-module $M$ (in $L$) such that $$H^2(G,M) = H^2(G,L).$$
How small can I take $M$ (in terms of $G$ or even $n$)?
In particular, since $H^2(G,L)$ is $n$-torsion, can I just take $M$ to be the $n$-torsion points of $L$?
### integral transform of Fibonacci polynomials is integral
Math Overflow Recent Questions - Sun, 06/25/2017 - 16:20
The Fibonacci polynomials are defined recursively by $F_0(x)=0, F_1(x)=1$ and $F_n(x)=xF_{n-1}(x)+F_{n-2}(x)$, for $n\geq2$.
While computing certain integrals, I observe the following (numerically) which prompted me to ask:
Question. For $n, k\in\mathbb{N}$, are these always integers? $$\int_0^1F_n(k+nz)\,dz$$
To help clarify, here is a list of the first few polynomials: $$F_2(x)=x, \qquad F_3(x)=x^2+1, \qquad F_4(x)=x^3+2x.$$
### Problem with an integral equation taken from a paper
Math Overflow Recent Questions - Sun, 06/25/2017 - 15:53
I was reading a paper (the 2015 paper by A. Falkowsy and L. Slominski Stochastic Differential Equation with Constraints Driven by Processes with Bounded $p-$variation, page 353, proof of the Lemma 3.1) for my master degree thesis.
The framework is the following: we fix $$f:\Bbb R^d\to\Bbb R^d$$ continous and such that satisfies the linear growth condition $$|f(x)|\le L(1+|x|),\;\;\;x\in\Bbb R^d$$ and if $1<p\le2$ take $$g:\Bbb R^d\to\mathcal M_d(\Bbb R)$$ $\alpha$-Holder continous function, where $p-1<\alpha\le1$ i.e. $$||g(x)-g(y)||\le C_{\alpha}|x-y|^{\alpha},\;\;x,y\in\Bbb R^d$$ where $||A||:=\sup\{|Ax|\;:\;|x|=1\}$ is the usual matrix norm.
Fix then $a:\Bbb R_{\ge0}\to\Bbb R$ and $z:\Bbb R_{\ge0}\to\Bbb R^d$ right continous functions which admit finite left limit (RCFLF for short) such that \begin{align*} V_1(a)_{[0,T]}&:=\sup_{\pi[0,T]}\sum_{j=1}^n|a_{t_j}-a_{t_{j-1}}|<+\infty\\ V_p(z)_{[0,T]}&:=\sup_{\pi[0,T]}\left[\sum_{j=1}^n|z_{t_j}-z_{t_{j-1}}|^p\right]^{1/p}<+\infty\\ \end{align*} where obviously $\pi[0,T]$ denotes the generic subdivision of the closed interval $[0,T]$.
Fix then $l:\Bbb R_{\ge0}\to\Bbb R^d$ another RCFLF function and consider the following integral equation $$x_t=x_0+\int_0^tf(x_{s-})\,da_s+\int_0^tg(x_s)\,dz_s+k_t$$ whose unknown are the RCLFL functions $x,k:\Bbb R_{\ge0}\to\Bbb R^d$, with the constraint given by $x_0\ge l_0$ (inequality taken componentwise); both integral are Riemann-Stieltjes ones.
Now the Lemma of the paper I refer to, says what follows: suppose there exists $b>0$ such that $$\max\left[V_1(a)_{[0,T]},\;V_p(z)_{[0,T]},\;\sup_{t\le T}|l_t|\right]\le b$$ then there exists $\bar C$ depending ONLY on $d, p, \alpha, L, g(0), x_0, b$ such that, if $(x,k)$ is a solution of the integral equation above, then $$\bar V_p(x)_{[0,T]}:=|x_0|+V_p(x)_{[0,T]} \le\bar C\;.$$
I will skip all the detail of the proof, going directly to the core of the problem (otherwise the post would be the longest ever!).
I am able to prove the following inequality: \begin{align*} \bar V_p(x)_{[0,t]} &\le (d+1)\left[|x_0|+LV_1(a)_{[0,t]}(1+V_p(x)_{[0,t]})+DV_p(z)_{[0,t]}(1+V_p(x)_{[0,t]})\right]+db\\ &= (d+1)\left[|x_0|+(LV_1(a)_{[0,t]}+DV_p(z)_{[0,t]})(1+V_p(x)_{[0,t]})\right]+db \end{align*} where $D$ is another absolute constant. Here $t\le T$.
Starting from here, we define $$t_1:=\inf\left\{t>0\;:\;LV_1(a)_{[0,t]}>\frac1{4(d+1)}\;\;\mbox{or}\;\;DV_p(z)_{[0,t]}>\frac1{4(d+1)}\right\}\wedge T$$ from which immediately $$\bar V_p(x)_{[0,t_1[}\le(d+1)|x_0|+\frac12(1+V_p(x)_{[0,t_1[})+db$$ and thus $$\bar V_p(x)_{[0,t_1[}\le2(d+1)|x_0|+1+2db\;\;.$$ Next we accept that $$|\Delta x_{t_1}|=|x_{t_1}-x_{t_1-}|\le(L(1+|x_{t_1-}|)+C_{\alpha}|x_{t_1}|^{\alpha}+||g(0)||+2)b.$$
Then the authors write there exists then $C_1,C_2>0$ depending only on $d, p, \alpha, L, g(0), b$, such that $$\bar V_p(x)_{[0,t_1]}\le C_1+C_2|x_0|.$$ WHY?! I know that $\bar V_p(x)_{[0,t_1]}\le\bar V_p(x)_{[0,t_1[}+|\Delta x_{t_1}|$ but controlling $\Delta x_{t_1}$, terms depending on $x$ appear!!
Then they set $$t_k:=\inf\left\{t>t_{k-1}\;:\;LV_1(a)_{[t_{k-1},t_k]}>\frac1{4(d+1)}\;\;\mbox{or}\;\;DV_p(z)_{[t_{k-1},t_k]}>\frac1{4(d+1)}\right\}\wedge T$$ thus for the same constants $$V_p(x)_{[t_{k-1},t_k]}\le C_1+C_2|x_{t_{k-1}}|\le C_1+C_2\bar V_p(x)_{[0,t_{k-1}]}$$ Set now $m:=\sup\{k\;:\;t_k\le T\}$ and accept that $m$ is finite and absolute (in the sense it doesn't depend on $(x,k)$ but only on the constants already written).
How can I reach the conclusion from here? I am really confused: how can I get rid of the terms depending on $x$ in order to get my absolute estimate?
### An integral involving hyperbolic functions
Math Overflow Recent Questions - Sun, 06/25/2017 - 13:57
I am wondering if it is possible to obtain a closed-form formula for $$f(\alpha) = \frac{1}{{\sqrt{2 \pi } \; \alpha }} \int^\infty_{-\infty} x^2 \cosh(x) \; e^{-\frac{\sinh ^2(x)}{2 \alpha ^2}} .$$ This integral came up when I tried to calculate the second moment of the random variable $$\DeclareMathOperator\arsinh{arsinh} X = \arsinh(Z \cdot \alpha)$$ where $Z \sim \mathcal{N}(0, 1)$ is a normally distributed random variable and $\arsinh$ denotes the inverse hyperbolic sine.
Motivation/Context: The above came up as I was investigating whether $\arsinh$ could be a useful variance-stabilizing, non-saturating activation function for artificial neural networks. For more details, see this Reddit thread and the original research cited there.
### A weak version of high dimensional Abhyankar's conjecture
Math Overflow Recent Questions - Sun, 06/25/2017 - 13:26
I am encountering the following situation which is similar to the Abhyankar's higher dimensional conjecture on étale fundamental groups, but with much stronger assumptions:
Let $S$ be a finitely generated subring of $\mathbb{C}$, let $X$ be a smooth affine variety over $S$, and let $G$ be a finite group such that the following holds. For any large enough prime $p$ and a base change $S\to k$ to an algebraically closed field of characteristic $p$, the variety $X_{k}$ admits a Galois covering with the Galois group $G$. Does this imply that $X_{\mathbb{C}}$ also admits a Galois covering with the group $G?$
I believe the answer is "yes" if $X$ is a curve, or is a complement of divisors with normal crossings in a projective space (by a result of Abhyankar).
Any suggestions or references would be greatly appreciated.
### On "topological" Hopf map eta and its relation to the motivic one
Math Overflow Recent Questions - Sun, 06/25/2017 - 12:21
Morel has defined the motivic Hopf map $\eta$ (in the motivic stable homotopy category $SH(k)$). I suspect that the following facts are valid for it and its topological "cousin"; please correct me if they are false and give me some (nice) references if they are true.
1) For the topological Hopf map we have $\eta^4=0$.
2) The action ot the topological $\eta$ on the values of oriented cohomology theories is zero.
3) If $k$ is the field of complex numbers then the "topological realization" of motivic $\eta$ is the topological Hopf morphism in $SH$ (also denoted by $\eta$?).
### Research-only permanent positions worldwide
Math Overflow Recent Questions - Sat, 06/24/2017 - 12:14
Most academic jobs involve some amount of teaching. Post-docs generally do not, but they are only short-term positions.
Question: in which countries can one obtain a research-only permanent position in mathematics? Please provide a link to a relevant website if possible.
Please mention only one country per answer, and since there is obviously no best answer, this is a community-wiki question.
### Simply connected slices
Math Overflow Recent Questions - Fri, 06/23/2017 - 17:23
Assume $\Omega$ is an open set in $\mathbb R^3$ such that the intersection of $\Omega$ with any horizontal plane is simply connected.
Can you prove that $\Omega$ is simply connected?
(Note that by the definition, simply connected set can not be empty.)
• The proof given by Tom Goodwillie below is done with bare hands. I would prefer to find ready to use tool for answering this and similar questions.
### Nick Katz observation: "the rationality of the zeta function!"
Math Overflow Recent Questions - Fri, 06/23/2017 - 09:55
In the proceedings "Algebraic Geometry - Arcata 1974" edited by R. Hartshorne there is an article by Nick Katz called "$p$-adic $L$-functions via moduli of elliptic curves". He starts by recalling $p$-adic measures. In particular, he characterizes all $p$-adic measures on $\mathbb{Z}_p$, with values in $\mathbb{Z}_p$, which correspond to rational functions in $\mathbb{Z}_p[[T-1]]$ under the Iwasawa isomorphism. Then, on page 488, he makes the following observation:
The measures $\mu_F$ we considered above correspond exactly to the rational functions in $\mathbb{Z}_p[[T-1]]$ (the "rationality of the zeta function"!).
Assuming he is referring to the rationality of the zeta function in the Weil conjectures, my question is:
What is the relation between $p$-adic measures and the rationality of the zeta function of an algebraic variety over a finite field?
The exclamation mark in his claim disturbs me. Is this something obvious?
I have studied Dwork's proof (from his paper and from Koblitz book), but I'm not familiar with Deligne et al proof of the whole conjecture.
Thank you very much. | {"extraction_info": {"found_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": 0.9272109270095825, "perplexity": 392.47763691133235}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128321306.65/warc/CC-MAIN-20170627083142-20170627103142-00183.warc.gz"} |
https://perminc.com/resources/publications/stress-gas-absorptive-effect-coal-densities-laboratory-cbmecbm-processes/ | The Stress and Gas Absorptive Effect on Coal Densities in Laboratory CBM/ECBM Processes
Guo, R. and Kantzas, A.
DOI: 10.2118/2008-142
CIM 2008-142 presented at the 59th Annual Technical Meeting of the Petroleum Society held in Calgary, June 17-19, 2008.
ABSTRACT
Density is an important coal property that determines the potential of gas resources in CBM reservoir. This paper aims to investigate coal density and structure variation during primary CBM and CO2-ECBM experiments. A coal core sample from Alberta Mannville formation with the rank of SubB was used to conduct the core flooding experiments covering the stages of inert gas flow, methane production, methane displacement by CO2 and inert gas flow after CO2 desorption. The x-ray CT experiments were carried out parallel to the core flooding experiment to provide x-ray images of coal core saturated with different gases at different stress conditions. The x-ray techniques were used for visualization and mapping of larger fractures and mineral streaks, as well as identification of flow paths. The coal density and density distribution changed with the gas adsorptive capacity and the stress condition were obtained.
The results show that net stress, gas adsorption capacity, and the production history are all key factors affecting coal core structure, leading coal density and density distribution variations. Hence, the core flow path, which contributes to the coal permeability, changes with those factors during CBM/ECBM processes. The results from this study provide laboratory coal characterization techniques using x-ray imaging analysis.
A full version of this paper is available on OnePetro Online. | {"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": 0.8572084307670593, "perplexity": 3891.8482787299413}, "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-2021-43/segments/1634323588257.34/warc/CC-MAIN-20211028034828-20211028064828-00688.warc.gz"} |
https://www.dorais.org/archives/448/ | # François G. Dorais
## Classically valid theorems of intuitionistic analysis
It is well-known that proofs in intuitionistic logic are more constructive than proofs in classical logic. Indeed, this is what the (informal) Brouwer–Heyting–Kolmogorov (BHK) interpretation leads to believe. Thus, intuitionistic proofs tend to have more information content than classical proofs, a fact that is well exploited in proof mining.
This suggests that looking at intuitionistic proofs of classical results from classical analysis may yield some new insights on the constructive nature of these classical results. However, there is one major hurdle with this program: theorems and axioms of intuitionistic analysis are not necessarily classically valid. Thus, inuitionistic proofs must be inspected very closely before drawing any kind of classical conclusions from them. Often, this is done by only looking at intuitionistic proofs over weak systems of intuitionistic analysis whose axioms are all classically valid. This amounts to doing classical analysis with a very limited set of axioms and deduction rules and it basically prohibits the use of the most important methods and results from intuitionistic analysis. This is not very satisfactory, but there is another way. There is a rich class of statements for which intuitionistic analysis is conservative over some weak subtheory whose axioms are all classically valid. So long as we restrict our attention to statements from this class, we can use methods and results of intuitionistic analysis with impunity and draw classical conclusions from the mere existence of a proof in the weaker subsystem, without going through the pain of finding such a proof.
One important result of intuitionistic analysis that is not classically valid is Brouwer’s Continuity Theorem, which simply says that every function defined on the unit interval is (uniformly) continuous (Brouwer 1923) (translated in van Heijenoort (1967)). The Continuity Theorem becomes especially powerful when combined with the Axiom of Choice, together they imply that if the statement $\forall\xi\exists\zeta A(\xi,\zeta)$ is true, then there must be a continuous function $F$ such that $\forall\xi A(\xi,F(\xi))$. On the one hand, this combination captures the constructive strength of intuitionistic analysis, as it provides a uniform and effective way to obtain witnesses to existential statements. On the other hand, this combination has some disastrous consequences for classically minded mathematicians. For example, the only subsets of $\R$ that have complements are $\emptyset$ and $\R$. Indeed, if $A \subseteq \R$ has a complement then it has a characteristic function by the Axiom of Choice, which must then be continuous by the Continuity Theorem. Since $\R$ is connected, the only continuous $\set{0,1}$-valued functions on $\R$ are the two constant functions. Therefore, $A$ must either be empty or all of $\R$. In particular, it follows that equality for real numbers is undecidable: $\forall\xi(\xi = 0 \lor \xi \neq 0)$ cannot be true since we cannot partition $\R$ into the singleton zero and the set of nonzero real numbers.
There are many ways to formalize all of this. In my paper (Dorais 2011), I look at two related systems intuitionistic analysis and I exploit conservation results by Troelstra (1973) and van Oosten (1990) to draw some classical consequences in subsystems of second-order arithmetic commonly used in reverse mathematics. The main results are of the following form: if $A$ and $B$ satisfy certain syntactic constraints and
is provable in a certain formal system of intuitionistic analysis, then the sequential form
is provable in a corresponding classical subsystem of second-order arithmetic. Results of this kind had already been obtained by Jeff Hirst and Carl Mummert for a different class of intuitionistic systems (Hirst–Mummert 2010). However, their systems only include classically valid axioms, so the fact that both systems I used prove Brouwer’s Continuity Theorem and some form of the Axiom of Choice adds an interesting new twist to this picture.1
$\newcommand{\EL}{\mathsf{EL}}\newcommand{\GC}{\mathsf{GC}}\newcommand{\RCA}{\mathsf{RCA}}\newcommand{\krf}{\mathrel{\mathtt{rf}}}\newcommand{\cat}{\mathop{^\frown}}$The basic intuitionistic system I will use is called $\EL$. The system $\EL$ has two sorts: natural numbers ($\N$) and functions from numbers to numbers ($\N^\N$). I will use roman letters for numbers and greek letters for functions. The details of $\EL$ can be found in (Troelstra 1973) and (Dorais 2011). The axioms of $\EL$ are classically valid. In fact, by adding the law of excluded middle to $\EL$, you obtain a classical system equivalent to the subsystem $\RCA$ of second-order arithmetic. To make this into a full-blown system for intuitionistic analysis, we can add Troelstra’s Generalized Continuity ($\GC$) axiom scheme:
where $B(\xi)$ is restricted to the syntactic class $\mathrm{N}_K$ (defined below) but $A(\xi,\zeta)$ is arbitrary. The notation $\alpha\mathop{\vert}\xi$ is a clever way of coding partial continuous functions $\N^\N\to\N^\N$ due to Kleene that I will describe shortly. Since the class $\mathrm{N}_K$ includes the tautology $0 = 0$, $\EL + \GC$ directly implies the Axiom of Choice: if the set $A_\xi = \set{\zeta \in \N^\N : A(\xi,\zeta)}$ is nonempty for every $\xi \in \N^\N$, then there is an $\alpha$ such that $\alpha\mathop{\vert}\xi$ picks out a single element of $A_\xi$ for every $\xi \in \N^\N$. $\EL + \GC$ also implies Brouwer’s Continuity Theorem: if $A(\xi,\zeta)$ describes the graph of a function $F:\N^\N\to\N^\N$, then there must be an $\alpha$ such that for every $\xi$, $\alpha\mathop{\vert}\xi$ is the unique $\zeta$ such that $A(\xi,\zeta)$. Since $\xi \mapsto \alpha\mathop{\vert}\xi$ is necessarily continuous, the function $F$ must be continuous.
In order to talk about the relevant conservation result for $\EL + \GC$, I need to talk about Kleene’s realizability with functions. The basic idea of realizability with functions is both natural and beautiful. Unfortunately, the beauty is hidden behind a thick wall of coding tricks. Three standard coding tricks are the following:
• Pairs of functions can be coded by alternately meshing their values: given functions $\xi_0,\xi_1$ there is a unique function $\seq{\xi_0,\xi_1}$ such that $\seq{\xi_0,\xi_1}\pi_0 = \xi_0$ and $\seq{\xi_0,\xi_1}\pi_1 = \xi_1$, where $\pi_0 = \lambda n.2n$ and $\pi_1 = \lambda n.2n+1$.
• Number-function pairs can be coded by prefixing the number: given a number $x$ and a function $\xi$ there is a unique function $x\cat\xi$ such that $(x\cat\xi)(0) = x$ and $(x\cat\xi)\sigma = \xi$, where $\sigma = \lambda n.n+1$.
• Every function $\xi$ can be thought of as encoding an infinite list of functions $\xi_n = \lambda m.\xi(2^n(2m+1)-1)$. The next trick is a very clever way of encoding partial continuous functions $\N^\N\to\N^\N$ due to Kleene that I alluded to above. Given a function $\xi$ and a number $\ell$, let $\bar{\xi}\ell$ denote the finite initial segment $\seq{\xi(0),\dots,\xi(\ell-1)}$ coded as a number in some primitive recursive fashion. Kleene’s encoding trick is best thought of as a two-step process.
1. A function $\alpha \in \N^\N$ encodes a partial continuous function $\N^\N\to\N$ as follows: to compute $\alpha(\xi)$ for some $\xi \in \N^\N$, look for the first number $\ell$ (if any) such that $\alpha(\bar{\xi}\ell) \neq 0$ and then set $\alpha(\xi) = \alpha(\bar{\xi}\ell)-1$. Write $\alpha(\xi){\uparrow}$ when no such $\ell$ can be found, and $\alpha(\xi){\downarrow}$ when $\alpha(\xi)$ is defined.
2. Thinking of functions in $\N^\N$ as encoding number-function pairs $n\cat\xi$, the above scheme lets $\alpha \in \N^\N$ encode a partial continuous function $\N\times\N^\N\to\N$ via $\alpha(n\cat\xi)$. By currying, we obtain a partial continuous function $\N^\N\to\N^\N$. Thus, for each $\xi \in \N^\N$, the value $\alpha\mathop{\vert}\xi$ of the partial continuous function $\N^\N\to\N^\N$ coded by $\alpha$ is the function $\lambda n.\alpha(n\cat\xi)$ provided that $\alpha(n\cat\xi){\downarrow}$ for every $n$. Write $\alpha\mathop{\vert}\xi{\uparrow}$ when $\alpha(n\cat\xi){\uparrow}$ for some $n$, and $\alpha\mathop{\vert}\xi{\downarrow}$ when $\alpha\mathop{\vert}\xi$ is defined. This encoding of partial continuous functions is rather dense at first sight, but it is actually quite natural after getting used to it. The $\alpha\mathop{\vert}\xi$ notation may seem strange, but it is actually well motivated. Thinking of $\mathop{\vert}$ as a partial binary operation on the set $\N^\N$, we obtain a partial combinatory algebra known as Kleene’s second algebra. The post-modern approach to realizability goes through realizability topos, which can be constructed on top of any partial combinatory algebra.
Kleene’s realizability with functions is a syntactic transformation of formulas which uses all the tricks above to pack most existential quantifiers into an auxiliary function parameter. The transformation is built by induction on complexity as follows2:
• $\alpha \krf A$ is $A$ for atomic $A$.
• $\alpha \krf (A \land B)$ is $\alpha\pi_0 \krf A \land \alpha\pi_1 \krf B$.
• $\alpha \krf (A \lthen B)$ is $\forall\beta(\beta \krf A \lthen \alpha\mathop{\vert}\beta{\downarrow} \land \alpha\mathop{\vert}\beta \krf B)$.
• $\alpha \krf \forall x A$ is $\forall x(\alpha_x \krf A)$.3
• $\alpha \krf \forall \xi A$ is $\forall \xi(\alpha\mathop{\vert}\xi{\downarrow} \land \alpha\mathop{\vert}\xi \krf A)$.
• $\alpha \krf \exists x A$ is $\alpha\sigma \krf A[x/\alpha(0)]$.
• $\alpha \krf \exists \xi A$ is $\alpha\pi_1 \krf A[\xi/\alpha\pi_0]$. Note that the translated formula $\alpha \krf A$ never involves existential quantifiers, except to say that $\alpha\mathop{\vert}\xi{\downarrow}$ and the scope of this implicit existential quantifiers is quantifier-free. Therefore, $\alpha \krf A$ always belongs to the class of formulas $\mathrm{N}_K$, which is defined as follows:
• If $A$ is quantifier-free, then $A$, $\exists x A$, $\exists\xi A$ are in $\mathrm{N}_K$.
• If $A, B$ are in $\mathrm{N}_K$, then so are $A \land B$, $A \lthen B$, $\forall x A$, $\forall \xi A$. It seems odd to include existential function quantifiers in the first clause but this is harmless since a quantifier-free formula $A$ can only use a finite initial segment of $\xi$ and thus $\exists\xi A$ is easily simulated by an existential number quantifier.
Now that we have described the system $\EL + \GC$ and Kleene’s realizability with functions, we can start tying everything together. The first step is to relate provability in $\EL + \GC$ with Kleene’s realizability with functions.
Troelstra’s Characterization Theorem. For every formula $A$:
1. $\EL + \GC \vdash A \liff \exists\alpha(\alpha \krf A)$.
2. $\EL + \GC \vdash A \IFF \EL \vdash \exists\alpha(\alpha \krf A)$. An important consequence of this characterization of $\krf$ is that $\EL + \GC$ is relatively consistent with $\EL$. In particular, Brouwer’s Continuity Theorem is relatively consistent with the Axiom of Choice so that the above discussion was not meaningless.
The next step is to formulate the relevant conservation result. The class of formulas $\Gamma_K$ is defined as follows:
• Quantifier-free formulas are in $\Gamma_K$.
• If $A, B$ are in $\Gamma_K$ then so are $A \land B$, $\forall x A$, $\forall \xi A$, $\exists x A$, $\exists \xi A$.
• If $A$ is in $\mathrm{N}_K$ and $B$ is in $\Gamma_K$ then $A \lthen B$ is in $\Gamma_K$. Although $\Gamma_K$ is far from including all formulas, it is actually a rather rich fragment which plays well with Kleene’s realizability with functions.
Troelstra’s Conservation Theorem. If $A \in \Gamma_K$ then $\EL \vdash \exists\alpha(\alpha \krf A) \lthen A$.
Combining this with the second part of Troelstra’s Characterization Theorem, we see that if $A \in \Gamma_K$, then
In other words, $\EL + \GC$ is conservative over $\EL$ for formulas in $\Gamma_K$.
What is this all good for? Well, suppose your friend Bertus (a hard-core intuitionist working in $\EL + \GC$) tells you (a hard-core classicist working in $\mathsf{ZFC}$) about his most recent result:
Every $n \times n$ real matrix with nonzero determinant is invertible.
A priory, you have no reason to believe your friend since he also believes in false statements like every function is continuous. After some thought, you realize that $\det(A) \neq 0$ can be formalized as a formula in $\mathrm{N}_K$ and that
can be formalized as a formula in $\Gamma_K$. You can then conclude that Bertus’s result is classically true and, because of $\GC$, that there is a continuous function to compute the inverse of a $n \times n$ real matrix $A$ with nonzero determinant. Not only were you able to meaningfully communicate with your friend Bertus, but you also gained a new classical result that you can call your own!4
The system $\EL + \GC$ is not the only system for which the above scheme goes through. van Oosten (1990) proves a similar characterization result for an intuitionistic system that incorporates the Weak König Lemma. This system also proves Brouwer’s Continuity Theorem. However, since the Weak König Lemma and Brouwer’s Continuity Theorem are incompatible with the Axiom of Choice, van Oosten has to settle for a weaker form of choice that continuously selects a nonempty compact set of witnesses to existential statements instead of a single witness. These results of van Oosten are also exploited in my paper (Dorais 2011).
#### References
1. L. E. J. Brouwer, 1923: Über Definitionsbereiche von Funktionen, Math. Ann. 97, no. 1, 60–75.
2. F. G. Dorais, 2011: Classical consequences of continuous choice principles from intuitionistic analysis, Notre Dame J. Form. Log. 55, no. 1, 25–39.
3. J. van Heijenoort, 1967: From Frege to Godel: A source book in mathematical logic, 1879-1931, Harvard University Press (Cambridge, Mass.).
4. J. Hirst, C. Mummert, 2010: Reverse mathematics and uniformity in proofs without excluded middle, Notre Dame J. Form. Log. 52, no. 2, 149–162.
5. J. van Oosten, 1990: Lifschitz’ realizability, J. Symbolic Logic 55, no. 2, 805–821.
6. A. S. Troelstra, 1973: Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics 344, Springer–Verlag (Berlin).
1. I think that it is probably safe to add certain intuitionistic principles like Brouwer’s Continuity Theorem to the systems considered by Hirst and Mummert (2010), but I have not checked that.
2. There is no clause for disjunction since because I prefer to think of $A \lor B$ as an abbreviation for $\exists x((x = 0 \lthen A) \land (x \neq 0 \lthen B))$. Since equality of numbers is decidable, this interpretation is intuitionistically correct.
3. Troelstra (1973) actually uses $\forall x(\alpha\mathop{\vert}\lambda n.x{\downarrow} \land \alpha\mathop{\vert}\lambda n.x \krf A)$ for this case. This amounts to the same idea except that $\alpha_x$ is always defined whereas $\alpha\mathop{\vert}\lambda n.x$ might not. There does not appear to be any need for the added complexity except to make this case more parallel to the other universal quantification case.
4. Don’t start writing that paper right away: this is not a new result… However, you can now captivate your friends for hours explaining how you discovered the existence of Cramer’s Rule without opening a linear algebra textbook!
Originally posted on by François G. Dorais. To the extent possible under law, François G. Dorais has waived all copyright and neigboring rights to this work. | {"extraction_info": {"found_math": true, "script_math_tex": 180, "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": 0.9679699540138245, "perplexity": 287.79478831626017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780057227.73/warc/CC-MAIN-20210921191451-20210921221451-00023.warc.gz"} |
https://mailman.ntg.nl/pipermail/ntg-context/2007/025720.html | # [NTG-context] question about macro's
Jelle Huisman jelle at jhnet.nl
Thu Jun 28 16:03:57 CEST 2007
Hello Aditya,
> Do you have a blank line between the two lines? If so, see below.
No, I don't have a blank line. (well, I can change my test file, but
that is not the solution I am looking for.)
>> To process the tags I have defined some macro's, like this one:
>>
>> \def\TAGc#1
>> {\dosomething{#1} % the \dosomething has to do with fonts etc.
>> \dosomethingelse}
>>
>> However, when I typeset the document this macro only works with the
>> first word of the line tagged with \TAGc. Is it possible to tell the
>> macro to process the whole line? (changing my source document is no option.)
>>
>
> Have a look at \dowithpargument.
Thank you for this hint and your example. I see in the source browser
that there is also an \dowithwargument, so I think I am looking for
something like \dowithlargument (with l for line). Any idea's how to do
that?
Jelle
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http://mathhelpforum.com/pre-calculus/190357-sequences-series-print.html | # Sequences and Series
• October 14th 2011, 11:05 AM
GrigOrig99
1 Attachment(s)
Sequences and Series
In this case, the text book has included the answer. I have 3^n/2, they have just 3^n. Can someone help me figure out how to factor out the 2? Thank you.
• October 14th 2011, 11:11 AM
e^(i*pi)
Re: Sequences and Series
Quote:
Originally Posted by GrigOrig99
In this case, the text book has included the answer. I have 3^n/2, they have just 3^n. Can someone help me figure out how to factor out the 2? Thank you.
You're very close, it is that last simplification that is wrong - the laws of exponents do not work like that. Use the law that says $a^{b+c} = a^ba^c$ to express $3^{n+1}$ in terms of $3^n$
Spoiler:
$\dfrac{3^{n+1} - 3^n}{2} = \dfrac{3(3^n) - 3^n}{2} = \dfrac{3^n(3-1)}{2} = 3^n$
• October 14th 2011, 11:12 AM
TheChaz
Re: Sequences and Series
$\frac{3^{n+1}}{2} - \frac{3^n}{2} = \frac{3 \cdot 3^n}{2} - \frac{1 \cdot 3^n}{2} = \frac{(3 - 1) \cdot 3^n}{2} = ...$
edit: the biggest spoiler was that someone posted before me!
(Punch)
• October 14th 2011, 12:03 PM
GrigOrig99
Re: Sequences and Series
Thanks for the help guys. I really appreciate it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9270362854003906, "perplexity": 1181.189822829555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1454701157472.18/warc/CC-MAIN-20160205193917-00317-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://getrevising.co.uk/revision-tests/coastal-zones-2-1 | # Coastal Zones 2
HideShow resource information
N J P S O F T E N G I N E E R I N G O M N H R E O I L O Q Y L Y T O M B O L O E B N W H A R D E N G I N E E R I N G V I Y P N F L O N G S H O R E D R I F T V S S K G C T M O J S O Y C G K T Y K K S L J K L J R H O B I B F H C K E A J P O V N Q Y S K W D B E A C H R E P L E N I S H M E N T V B Q L T N L R J S I X A O U E M F W V K J G V Q M F Q E D Q C A U N C V Y G H Q K S E O Y D I T U T T X E N X S A O H R B F R T A T J O G L I X X S E T M S E A W A L L K V M A N A G E D R E T R E A T R L M I Y N K N O H G Y M O L C G B P B Y Y O V R Y H F I R M N G H J H S T A C K E X F F W Y Q T M V A U D F I G E K N E K J V Y L K O U V E D D C R Y G R W B Q V F J B O F D V J W T F W S A Y D O J E C O S Y S T E M C I L E S Q M E K E Y N Q B Q H A L N F G T X X R V Q P O H R N O P O Y X W C P Q X S S M Q N N P I I X E K E V L Q O H I K U C M D Y X W J M E L R Q A D O C O F V H D W V
### Clues
• A high up wall or embankment, made of concrete which reflects the waves back into the sea. (3, 4)
• Dumping or pumping sand from else where onto an eroding shoreline. (5, 13)
• Formed gradually as over time the arch is enlarged by erosion at the base and sides and by weathering processes action on the roof. The roof then collapses leaving a ... (5)
• Involves using artificial structures to control the forces of nature. (4, 11)
• Joins the gap between an island and the mainland. (7)
• Leave nature to take it's course. (7, 7)
• the interrelationship between the living and non living elements of a habitat (9)
• The transportation of sand and pebbles along the coast by waves. (9, 5)
• Timber or rock structures that are built into the sea at right angles to the coast. (6)
• To try and fit in and work with natural coastal processes. (4, 11) | {"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": 0.9278631806373596, "perplexity": 1649.0937689239597}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1487501170614.88/warc/CC-MAIN-20170219104610-00458-ip-10-171-10-108.ec2.internal.warc.gz"} |
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