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https://docs.wiris.com/quizzes/en/user-interface/validation-options/subindices-in-answers.html | # Subindices in answers
Now, WirisQuizzes allows answers with subindices so you can ask even more questions. For instance, a common question about chemistry or symbolic maths is possible.
## Example 1
Consider the following problem about chemistry: A perfect gas at ${T}_{1}$ K and ${P}_{1}$ atm of pressure occupies ${V}_{1}$ l in volume. What volume will the same gas occupy at ${T}_{2}$ K and ${P}_{2}$ atm of pressure?
Using Ideal gas law, $\frac{{P}_{1}·{V}_{1}}{{T}_{1}}=\frac{{P}_{2}·{V}_{2}}{{T}_{2}}$, so the answer will be ${V}_{2}=\frac{{P}_{1}·{V}_{1}·{T}_{2}}{{T}_{1}·{P}_{2}}$
As usually, equivalent formats are marked as correct
## Example 2
Imagine this question: Compute the determinant of the following matrix
$\left(\begin{array}{ccc}{a}_{1}& 1& 2\\ 3& {a}_{2}& -1\\ -2& {a}_{3}& {a}_{4}\end{array}\right)$
The algorithm will be
The Correct answer field will look like
So the solution will be | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 8, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9437319040298462, "perplexity": 1757.432894940099}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320299852.23/warc/CC-MAIN-20220116093137-20220116123137-00632.warc.gz"} |
https://www.physicsforums.com/threads/cut-the-string-energy-and-friction.825961/ | # Cut the String - Energy and Friction
1. Aug 3, 2015
### FredericChopin
1. The problem statement, all variables and given/known data
http://imgur.com/FZM5gqC,RlLeGmP#1
http://imgur.com/FZM5gqC,RlLeGmP#0
2. Relevant equations
$f^k_{AB} = \mu_k N_{AB}$
$W_{f,i} = \int_{r_i}^{r_f} \vec{F} \cdot d\vec{r}$
$E^{mech}_f = E^{mech}_i + W^{NC}_{f,i}$
$U_{elastic} = \frac{1}{2} k x^2$
3. The attempt at a solution
Since the blocks are at rest after the release of the spring, the final mechanical energy, $E^{mech}_f$, is 0. The initial mechanical energy, $E^{mech}_i$, is the elastic potential energy of the spring ($U_{elastic} = \frac{1}{2} k x^2$). There are no external non-conservative forces acting on the box-block-spring system, but there is the internal non-conservative force of kinetic friction acting on the box and block ($W^{NC}_{f,i} = \int_{r_i}^{r_f} \vec{f^k_{AB}} \cdot d\vec{r}$). The force of kinetic friction acts through a displacement $d$ in the same direction as force, so $W^{NC}_{f,i} = \int_{0}^{d} \vec{f^k_{box,block}} \cdot d\vec{r} = f^k_{box,block}d$.
Let's consider the system of the box, the block, and the spring.
The final mechanical energy of this system will be:
$E^{mech}_f = E^{mech}_i + W^{NC}_{f,i}$
Substituting in terms:
$0 = U_{elastic} + f^k_{box,block}d$
, which becomes:
$0 = \frac{1}{2} k x^2 + \mu_k N_{box,block}d$
The block is not accelerating in the vertical direction, and so due to Newton's Second Law, $N_{box,block}$ must be equal in magnitude to $m_{box}g$:
$0 = \frac{1}{2} k x^2 + \mu_k m_{box}gd$
Solving for $\mu_k$ yields:
$\mu_k = \frac{-kx^2}{2m_{box}gd}$
It's strange that there is a negative sign in the answer as $\mu_k$ should be a positive scalar. It also turns out this answer is incorrect.
What went wrong?
Thank you.
2. Aug 3, 2015
### tommyxu3
My opinion:
Your calculation is like that the L-shape block is fixed on the floor.
Besides, the sign of $\mu_k$ is usually positive for its just a ratio between $f_k$ and $N.$ That is, your relation may turn to:
$$U+W_f=0$$
$$\Rightarrow U-\int N\mu_k\cdot dr=0$$
The reason is obvious that we all know the friction does negative work here.
Last edited: Aug 3, 2015
3. Aug 4, 2015
### haruspex
Which way does the force of friction act? Which way is the displacement vector? What is the sign of their dot product?
Draft saved Draft deleted
Similar Discussions: Cut the String - Energy and Friction | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9770942330360413, "perplexity": 500.7783584364053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889733.57/warc/CC-MAIN-20180120201828-20180120221828-00792.warc.gz"} |
https://www.mysciencework.com/publication/show/b-jpsi-eta_c-k-decays-perturbative-qcd-approach-9d5245ec | # $B \to (\jpsi,\eta_c) K$ decays in the perturbative QCD approach
Authors
Type
Preprint
Publication Date
May 03, 2009
Submission Date
Jan 01, 2009
Identifiers
DOI: 10.1088/1674-1137/34/7/002
Source
arXiv
In this paper, we calculated the $B \to (\jpsi, \eta_c) K$ decays in the perturbative QCD approach with the inclusion of the partial next-to-leading order (NLO) contributions. We found that (a) when the large enhancements from the known NLO contributions are taken into account, the NLO pQCD predictions for the branching ratios are the following: $Br(B^0 \to \jpsi K^0) = 5.2^{+3.5}_{-2.8}\times 10^{-4}$, $Br(B^+ \to \jpsi K^+) = 5.6^{+3.7}_{-2.9}\times 10^{-4}$, $Br(B^0 \to \eta_c K^0) = 5.5^{+2.3}_{-2.0}\times 10^{-4}$, $Br(B^+ \to \eta_c K^+) = 5.9^{+2.5}_{-2.1}\times 10^{-4}$, which are roughly 40% smaller than the measured values, but basically agree with the data within $2-\sigma$ errors; (b) the NLO pQCD predictions for the CP-violating asymmetries of $B \to (\jpsi,\eta_c)K$ decays agree perfectly with the data. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.989348292350769, "perplexity": 1208.5089233827398}, "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-2018-22/segments/1526794863972.16/warc/CC-MAIN-20180521082806-20180521102806-00197.warc.gz"} |
https://www.mediawiki.org/wiki/Extension_talk:CrossReference | # Extension talk:CrossReference
Is there a way to make <xr /> use internal links? Right now it is creating links like Figure [1] instead of Figure [[1]].
Kjkeefe (talk) 17:22, 20 February 2014 (UTC)
## How do I display the number to the right of an equation?
I am trying show a reference number to the right of an equation in parentheses, just like it would work in latex. For example:
<equation id="eqn:binom">
${\displaystyle f(k)={\binom {n}{k}}p^{k}(1-p)^{n-k}\quad k=0,1,\dots ,n}$ <xrlabel id="eqn:binom" shownumber/>
</equation>
<xr id="eqn:binom" /> describes the probability mass function of the binomial distribution.
What am I doing wrong here?
Kjkeefe (talk) 17:19, 20 February 2014 (UTC)
The correct will be <equation id="eqn:binom" shownumber> ... </equation> :) Пика Пика (talk) 14:54, 26 February 2018 (UTC)
## Problem with numbering of figures
I like using this extension, but I am not able to correctly numbering my figures. For example:
<figure id="fig:obr1"> [[Soubor:my_pic.png| <xr id="fig:obr1">Fig. %i:</xr> text]]</figure>
Text text text <xr id="fig:obr1">Fig. %i:</xr>.
After the preview, will appear the same numbering as for equations. At the start of line is shown as (1).
--Jafan 15:59, 14 November 2009 (UTC)
## Does this work within a \begin{align} environment?
Has anyone got it working? If so, please share how to use it with align or any other multi-line equation environment.
Thanks in advance Dc321 21:09, 8 September 2011 (UTC)
## Transclusions
Numbering isn’t accurate when the html tags are transcluded. Is there a way to have the extension compile all the cross references at once, with a final tag at the bottom of the page as is done with <ref> and <references> for citations? Mmiller0712 (talk) 16:27, 4 April 2018 (UTC)
## I would need to see it in action
A picture to show an example or a link, is missing. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.9254506826400757, "perplexity": 3370.4248071998218}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585522.78/warc/CC-MAIN-20211022212051-20211023002051-00374.warc.gz"} |
http://physics.stackexchange.com/questions/105353/why-must-the-particles-of-an-ideal-gas-be-point-like | # Why must the particles of an ideal gas be point-like?
Why is a gas of elastically colliding hard balls of finite size not ideal?
Respectively:
Why is it essential that the particles of an ideal gas are point-like?
Especially:
Which differing thermodynamic properties of a gas of not point-like particles are most striking in comparison with an ideal gas (of point-like particles)?
-
Interestingly the wikipedia article on ideal gas contradicts itself. In the introduction it states ideal gases are composed of point particles, whereas later in classical thermodynamic ideal gases it states only that the separation between particles is much greater than the particle size. – BMS Mar 27 '14 at 18:12
Are you sure that non point-like particles does not form an ideal gas? Isn't the presence of an inner structure the reason why the formula for the internal energy, $U = \frac{3}{2} n R T$, has sometime different factors (e.g .5/2)? – giulio bullsaver Mar 27 '14 at 18:12
@user3376924. To be honest: I am not sure. It has been only an impression. (Among others I found this source: sklogwiki.org/SklogWiki/index.php/Hard_sphere_model but could not delve deeper into it.) – Hans Stricker Mar 27 '14 at 18:19
@BMS I disagree with your analysis that the two statements are contradictory. The later statement is a condition on which classical thermodynamic ideal gas applies. It is because particle size is much smaller than separation that we can assume zero size for our calculations. It is like saying that for classical Newtonian mechanics velocities are much less than c. – Aron Mar 28 '14 at 9:47
@user3376924 Internal structure and point-like are not mutually exclusive. Take for example Bose-Einstein Condensates/Cooper pairs. When electrons form cooper pairs, it is their internal structure that allows them to have a point-like behavior. – Aron Mar 28 '14 at 9:51
If the particles are not point-like, they will take up some volume. As the gas is compressed, the collision frequency will rise more quickly, which will make the pressure-volume curve change. The corrections in the Van der Waals model of a real gas account for the volume of the particles. Also if they have internal structure, that structure can have degrees of freedom that change the specific heat. The fact that nitrogen and oxygen are diatomic leads to the specific heat of air being $\frac 52R$ at normal conditions. There are two rotational degrees of freedom available along with translation.
-
An ideal gas obeys Boyles law. Your gas with finite molecules doesn't obey Boyles law because there is a minimum volume (when the gas molecules are close packed) beyond which the gas can no longer be compressed. Therefore the gas cannot be ideal. – John Rennie Mar 28 '14 at 10:25
Because if those particles aren't point objects, you must also take into consideration that they take some space in the system and have properties like density and size which have to be taken into consideration when formulating the laws. This makes everything extremely complicated. A huge part of classical mechanics is only true for point objects for the very same reason.
-
I wonder: What observable difference does it make whether the particles are point-like or not - assumed they are "hard balls"? – Hans Stricker Mar 27 '14 at 18:47
They acquire Volume, Size and Density. – uncertainty_principle Mar 27 '14 at 18:51
I believe Hans is referring to observable differences according to theory when modeling a gas as small hard spheres vs point particles. – BMS Mar 27 '14 at 19:36
Oh right. I don't think there will be a lot of observable differences except probably the increase in frequency of the collisions and the heat released. – uncertainty_principle Mar 27 '14 at 19:49
Ideal gasses only exist in frictionless vacuums. – John Mar 28 '14 at 2:59
There are multiple ways of looking at the ideal gas model. One is to say we have point particles colliding elastically (as an ideal scenario) and proceed to obtain the exact equation of state, i.e $pV=nRT$. The other approach, is to state a priori the relevant length scales and time scales at which the system is going to be studied are sufficiently macroscopic, so as to be able to disregard any microscopic structure in the constituent particles. In both cases, the excluded volume effect (which is what the effect of existence of a finite size of particles in a gas is called) is neglected, as a model assumption in the first case and as a practical approximation in the second. The simplest fluid models that can be solved analytically are the so-called hard sphere models which involve the crudest form of binary excluded volume interactions. In such a case, one can in principle write down the entire virial equation for pressure (the virial coefficients being calculable from microscopic parameters of the model, size of the particles included) and hence the equation of state for this fluid is derivable. The Van der Waals equation is a specific mean field limit of such a virial expansion. The effect of excluded volume will be most prominent a low temperatures and high densities and it will have a substantial effect on the condensation of the fluid (the critical temperature and the scaling laws).
- | {"extraction_info": {"found_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": 0.802711009979248, "perplexity": 475.9932440060834}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396875.58/warc/CC-MAIN-20160624154956-00104-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/131939-integral-help.html | 1. ## Integral help
Is this a simple negative exponent, power chain rule
Or an inverse sin integral?
And if it's an inverse sin integral, how do i get rid of the first x under the radical?
$\int \frac{dx}{\sqrt{3x-x^2}}$
2. Originally Posted by penguinpwn
Is this a simple negative exponent, power chain rule
Or an inverse sin integral?
And if it's an inverse sin integral, how do i get rid of the first x under the radical?
$\int \frac{dx}{\sqrt{3x-x^2}}$
it's a pain-in-the-### arcsin ...
$3x - x^2 =$
$-\left(x^2 - 3x + \frac{9}{4} - \frac{9}{4}\right) =
$
$-\left[\left(x - \frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2\right] =$
$\left(\frac{3}{2}\right)^2 - \left(x - \frac{3}{2}\right)^2$
now use the general form of the integral ...
$\int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin\left(\frac{u}{a}\right) + C$
3. Originally Posted by skeeter
it's a pain-in-the-### arcsin ...
$3x - x^2 =$
$-\left(x^2 - 3x + \frac{9}{4} - \frac{9}{4}\right) =
$
$-\left[\left(x - \frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2\right] =$
$\left(\frac{3}{2}\right)^2 - \left(x - \frac{3}{2}\right)^2$
now use the general form of the integral ...
$\int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin\left(\frac{u}{a}\right) + C$
Where did the 9/4 came from?
4. Originally Posted by penguinpwn
Where did the 9/4 came from?
ever heard of completing the square?
5. Originally Posted by skeeter
ever heard of completing the square?
Ooooohhhhhhh, haha I see it now.
Thanks for the 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": 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": 0.9968119859695435, "perplexity": 2798.257156787234}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1500549428300.23/warc/CC-MAIN-20170727142514-20170727162514-00708.warc.gz"} |
https://math.stackexchange.com/questions/1703059/find-the-fourier-series-of-fx-fraca-02-sum-k-1na-k-coskx | # Find the Fourier series of $f(x) = \frac{a_0}{2} + \sum_{k = 1}^{n}(a_k\cos{kx} + b_k\sin{kx})$
I'm learning about Fourier series and need help with this problem:
Find the Fourier series of $$f(x) = \frac{a_0}{2} + \sum_{k = 1}^{n}(a_k\cos{kx} + b_k\sin{kx})$$
My thoughts:
The above trigonometric polynomial is the nth partial sum of the Fourier series for $$f$$. Taking the sum to infinity we get the Fourier series of $$f$$. Is it that simple or am I missing something here?
• Why don't you try to find the Fourier transform the usual way? Use the orthogonality of trigonometric functions. – eyedropper Mar 18 '16 at 13:16
• We have not seen orthogonality of trigonometric functions yet in class. This problem can certainly be solved with another method. – glpsx Mar 18 '16 at 13:23
• This is a classic. Similar to "Find the Laurent series for $z^{-1}$ around $z=0$." -- Note that in this task, $n$ is fixed. -- If you meant $a_k=0=b_k$ for $k>n$, then your statement is correct. – Lutz Lehmann Mar 18 '16 at 13:33
• Ok, then you should convince yourself that this already is in the appropriate form, i.e. the function $f(x)$ is expressed as a Fourier series. It just has a finite number of coefficients. If you would formally calculate its Fourier series, you would get the identical form of $f(x)$. – eyedropper Mar 18 '16 at 13:40
• @LutzL The analogy withe the Laurent series of $z^{-1}$ was indeed helpful, thank you. I'm still having difficulties to convince myself that the function $f(x)$ is already in the appropriate form. All of the examples that we have seen in class had an infinite sum, the fact that the sum is finite here is disturbing me. – glpsx Mar 18 '16 at 13:58
\begin{align} \int _{-\pi }^{\pi }\sin (nx)\sin (mx)\mathrm{d}x&=\pi \delta _{m,n}\\ \int _{-\pi }^{\pi }\cos (nx)\cos (mx)\mathrm{d}x&=\pi \delta _{m,n}\\ \int _{-\pi }^{\pi }\sin (nx)\cos (mx)\mathrm{d}x&=0 \end{align} which can be easily verified by using basic trigonometric identities, such as product-to-sum formulas. Let $$f(x)=\frac{c_0}{2}+\sum _{m=1}^{+\infty }(c_m\cos (mx)+d_m\sin (mx))$$ be the Fourier series of $f$. Then the coefficients for your $f(x)$ satisfy
\begin{align} c_m&=\frac{1}{\pi }\int _{-\pi }^{\pi }f(x)\cos (mx)\mathrm{d}x\\ &=\frac{1}{\pi}\sum _{k=1}^{n }\left(a_k\int _{-\pi }^{\pi }\cos (kx)\cos (mx)\mathrm{d}x+b_k\int _{-\pi }^{\pi }\sin (kx)\cos (mx)\mathrm{d}x\right)\\ &=\frac{1}{\pi}\sum _{k=1}^{n }\left(a_k \cdot \pi \delta _{k,m}+b_k\cdot 0\right)=a_m \text{ for m\leq n and 0 otherwise} \end{align}
The same goes for $c_0=a_0$ and $d_m=b_m$ and this proves explicitly that the Fourier series of $f(x)$ is $f(x)$. The $\delta$ symbol is the Kronecker delta. | {"extraction_info": {"found_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": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9898343682289124, "perplexity": 442.9797985494292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243991252.15/warc/CC-MAIN-20210512035557-20210512065557-00399.warc.gz"} |
http://mathhelpforum.com/calculus/206217-limit-x-infty.html | Math Help - limit as x -> infty
1. limit as x -> infty
$\lim_{x \rightarrow \infty}(x-lnx)$
I made it into indeterminate form so that I can use l'Hospital's rule:
$\lim_{x \rightarrow \infty}(lne^x-lnx) = \lim_{x \rightarrow \infty}ln\left(\frac{e^x}{x}\right)$
I take the derivative and get:
$\frac{x-1}{x}$
Taking the derivative again I get that the function approaches 1, but based off of the actual graph the function approaches infinity. What am I doing wrong?
2. Re: limit as x -> infty
$\lim_{x\to\infty}\ln\left(\frac{e^x}{x} \right)=\ln\left(\lim_{x\to\infty}\frac{e^x}{x} \right)$
Now apply L'Hôpital's rule to the limit inside the log function.
3. Re: limit as x -> infty
Hello, amthomasjr!
$\lim_{x \rightarrow \infty}(x-\ln x)$
I made it into indeterminate form so that I can use l'Hospital's rule:
$\lim_{x\to\infty}(\ln e^x-\ln x) \:=\: \lim_{x\to\infty}\ln\left(\frac{e^x}{x}\right)$
I take the derivative and get: . $\frac{x-1}{x}$ . How?
Exactly what did you differentiate?
4. Re: limit as x -> infty
Originally Posted by Soroban
Hello, amthomasjr!
Exactly what did you differentiate?
$ln\left(\frac{e^x}{x}\right)$
5. Re: limit as x -> infty
In order to apply L'Hôpital's rule you need the limit to take the form:
$\lim_{x\to c}\frac{f(x)}{g(x)}$
6. Re: limit as x -> infty
Originally Posted by MarkFL2
$\lim_{x\to\infty}\ln\left(\frac{e^x}{x} \right)=\ln\left(\lim_{x\to\infty}\frac{e^x}{x} \right)$
Now apply L'Hôpital's rule to the limit inside the log function.
That makes sense, so
$ln\left(e^x\right) = x$ and
$\lim_{x\to\infty}x = \infty$
7. Re: limit as x -> infty
Originally Posted by MarkFL2
In order to apply L'Hôpital's rule you need the limit to take the form:
$\lim_{x\to c}\frac{f(x)}{g(x)}$
Which is why you only apply it to the inside, correct?
8. Re: limit as x -> infty
I would write:
$\ln\left(\lim_{x\to\infty}\frac{e^x}{x} \right)=\ln\left(\lim_{x\to\infty}e^x \right)=\ln(\infty)=\infty$
9. Re: limit as x -> infty
Originally Posted by amthomasjr
Which is why you only apply it to the inside, correct?
Right, on the inside of the log function, you now have the correct form to use the rule bought by L'Hôpital.
10. Re: limit as x -> infty
Originally Posted by MarkFL2
Right, on the inside of the log function, you now have the correct form to use the rule bought by L'Hôpital.
I'm assuming the same thing applies to trig functions if there were a similar case
11. Re: limit as x -> infty
Yes, I believe so as long as the trig function is continuous. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 14, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9726057648658752, "perplexity": 888.6231201431557}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645195983.63/warc/CC-MAIN-20150827031315-00337-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.monolithicpower.com/en/mosfet-switches-basics-and-applications-in-power-converters | # MOSFET Switches: Basics and Applications in Power Converters
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## How Does a MOSFET Work
A metal-oxide-semiconductor field-effect transistor (MOSFET) is an electronic device that falls under the category of field-effect transistors (FETs). These devices act as voltage-controlled current sources, and are mainly used as switches or for the amplification of electrical signals. The MOSFET is controlled by applying certain voltage conditions to the gate. When the MOSFET is turned on, current flows from the drain to the source of the MOSFET, through a channel created in the bulk (also called the body). In most cases, the MOSFET’s bulk is connected to the source, which is why MOSFETs are commonly referred to as 3-pin devices.
Figure 1: MOSFET
## P-Channel MOSFETs vs. N-Channel MOSFETs
MOSFETs are semiconductor-based devices, and are mostly built using either P-type or N-type silicon. The difference between these two silicon types is the charge stored by the dopant ions, which are charged particles that are injected into the silicon to create charge instability, making the element useful for electronic purposes. If an area of silicon has been doped with ions that have five valence electrons (group 5 in the periodic table) then there is an extra electron that is set free into the semiconductor, so the charge will be overall negative (N-type). Since they contribute an electron, these impurities in the silicon are called donor impurities. On the other hand, elements with three electrons in the valence band will be lacking an electron, which is equivalent to contributing a hole, meaning the overall charge will be positive (P-type). These impurities are also called acceptor impurities. Figure 2 shows the difference between P-type and N-type semiconductor dopants, and the effect they have on the silicon structure.
Figure 2: Dopants – Donor vs. Acceptor Impurities
The simplest MOSFET structure is comprised of a substrate, which can be P-type or N-type, and two areas of silicon of the opposite polarity to the bulk, which make up the drain and the source (see Figure 3). MOSFETs can be built to have a P-type substrate and an N-type drain and source regions, which means that in order for current to flow from the drain to the source the channel will also have to be N-type. These are called N-channel MOSFETs, or NMOS transistors. Conversely, if the substrate is N-type, the channel will be P-type, so the MOSFET is called a P-channel MOSFET, or PMOS transistor.
Figure 3: MOSFET Structure
## Enhancement vs. Depletion MOSFETs
MOSFETs get their name from the structure through which they are controlled. The gate pin is connected to a conductive electrode, which is separated from the substrate by a layer of silicon oxide or another dielectric material. Therefore, when a voltage is applied to the gate, an electric field is generated from the metallic gate, through the oxide, to the silicon substrate (metal-oxide-semiconductor). This electric field has an effect on the free charge carriers in the substrate semiconductor (e.g. holes or electrons), and pulls them up close to the gate to form a channel or pushes them away to destroy the channel.
When an electric field is applied to a semiconductor, it acts upon the device’s free charge carriers. The free electrons, evenly distributed throughout the semiconductor, are attracted to the electric field’s point of entry (in the case of a MOSFET with a positive gate voltage this is the gate electrode), whereas the holes will be dragged in the direction of the electric field opposite to the electrons (see Figure 4). This is called carrier drift, and logically changes the charge concentration distribution within the semiconductor.
Figure 4: Carrier Drift in Semiconductors
The principal aim of the MOSFET is to control the creation of a channel between the drain and the source, which is done by concentrating the right carriers in the area closest to the gate in order to create or destroy the channel. Therefore, MOSFETs can be distinguished into two basic groups: depletion MOSFETs and enhancement MOSFETs.
Depletion MOSFETS come with a pre-generated channel (see Figure 5). When a voltage is applied to the gate, the electric field pushes out the carriers in the channel, thus depleting it. Therefore, depletion MOSFETs can be equated to a normally closed switch.
In enhancement MOSFETS, the channel only appears when the gate voltage is applied, attracting charges and enhancing the channel region. This type of MOSFET, which can be considered a normally open switch, is most commonly used in electronic applications because if power is lost the switch turns off and current stops flowing in the circuit, which avoids uncontrolled operation and increases circuit safety. The rest of this article will be referring only to enhancement N-channel MOSFETs.
Figure 5: Depletion Mode MOSFET
Figure 6: Enhancement Mode MOSFET
## MOSFET Operation Regions
From what has been explained thus far, it is clear that one of the most important aspects in a MOSFET’s operation is the potential voltage applied to the gate. In fact, the MOSFET’s operation is defined by the voltage potential between the MOSFET’s gate and its source (VGS). Figure 7 shows how VGS affects the flow of current through the MOSFET. In an enhancement N-channel MOSFET, when there is no potential voltage between the gate and the source, the channel is nonexistent. This operation region is called the cutoff region; when the transistor is in this operating region, there is no current flowing from drain to source, which means the MOSFET behaves like an open switch.
As the gate voltage increases, the channel starts to form, but it does not enable conduction between the drain and the source until a certain voltage level, called the threshold voltage. Once the threshold is reached, current begins to flow through the MOSFET. This region, called the saturation region, can be compared to a voltage-controlled current source. As the gate voltage increases, so does the current flowing through the switch. This is the region mainly used for signal amplification, since small voltage changes in the gate lead to larger changes in output current (see Figure 7). This current can then be used to change the voltage across a resistor, which is the basis of common-source amplifiers.
Figure 7: Drain Current vs. Gate Voltage
As the gate voltage continues to increase, so does the channel. In the saturation region, the channel still does not fully connect the drain and source regions, so the voltage between the source and drain does not have much of an effect on the operation. However, once the channel has been enhanced enough so that it connects the drain and source (this is called the pinch-off voltage, which is the upper limit of the saturation region), the MOSFET channel is fully enhanced and the transistor behaves as a fully closed switch.
From this point on, the MOSFET can be considered as a resistance, due to the voltage loss between the drain and source (RDS(ON)). This new operation region, called the ohmic or linear region, sees an increase in the current across the MOSFET, which is linearly proportional to the voltage applied between the drain and source of the MOSFET — although this is limited by the gate-source voltage (see Figure 8).
Figure 8: Drain Current vs. Drain-Source Voltage
Figure 8 also gives insight into what operation regions are useful for different applications. As was mentioned previously, the saturation region is the best for amplification, because for the same VDS, a small change in gate voltage causes a much larger change in current. However, because the power consumed by the MOSFET is defined by the product of the current and the voltage across the MOSFET (VDS), the saturation region is also the worst regarding power efficiency, since it has notable current and drain-source voltage.
Therefore, when the MOSFET is used in switching applications, it is imperative to make sure that the MOSEFT only works as a fully open or fully closed switch to reduce power loss. In other words, it must only work in the cutoff or linear regions, and avoid saturation as much as possible.
## Parasitic Components in Power MOSFETs
As with any electronic device, it is important to consider the parasitic elements that it incorporates, meaning the components that are unintentionally created by the structure of the device. This article has already talked extensively about one of them, the on resistance, but there are other components embedded in the MOSFET structure (see Figure 9 and Figure 10).
The other main passive components that appear in MOSFETs are the different capacitors created in the transistor’s structure. There are many parasitic capacitors, but the main ones to consider are the capacitors formed between the gate and the drain, and between the gate and the source. These capacitors limit the maximum switching frequency at which the device can operate.
In addition to these passive elements, a BJT is created by the N+-P-N- junctions formed by the source, body, and drift regions. This transistor is critical for safe operation of the MOSFET. If it is accidentally turned on, causes the MOSFET to enter a “latch-up” condition, which significantly decreases the maximum blocking voltage. If this voltage is surpassed, the BJT causes the device to enter avalanche breakdown, which can destroy the device if the current is not limited. Therefore, the BJT must always be turned off by making the voltage at the base (body) as close as possible to the voltage at the emitter (source). This is why the source and the body are nearly always short-circuited in power MOSFETs. However, by shorting the source and bulk regions, a diode is created, known as the body diode. This diode is not as problematic as the BJT, and it can even be useful in some applications.
Figure 9: Power MOSFET Parasitic Components
Figure 10: Parasitic Capacitors
## Power MOSFETs
One of the aims when designing a MOSFET for power applications is ensuring that it can work at high voltages, meaning that it can block high voltage potentials when required without breaking down. This is achieved via the diode effect that occurs between the N-Si of the drain and the P-Si of the bulk. When biased, the drain-bulk PN junction acts like a reverse-biased diode, creating a spatial charge region (SCR) and blocking the voltage. The higher the voltage bias, the larger the spatial charge region necessary to block the voltage. If the voltage is high enough, the SCR could cross the space between drain and source, which would enable conduction through the MOSFET. This is called reverse breakdown. So, it would seem that the key to working at high voltages is to have a very long MOSFET channel. However, there are two reasons why making longer transistors is not a viable option:
• Efficiency: A longer channel means a higher RDS(ON), which in turn leads to higher conduction losses.
• Size: Longer channels take up more space, reducing the integration capability of the MOSFET.
For this reason, power MOSFETs are not built with the traditional MOSFET structure we are accustomed to seeing (see Figure 5). Instead, power MOSFETs are built with a vertical structure, with the source and gate on the top plate, and the drain on the bottom (see Figure 11).
Since the depth of the transistor is not a problematic production parameter, the depletion region can be as long as necessary, with only the issue of increased conduction loss. By connecting the MOSFET drain to the entire metal backplate, it is also much easier to connect these MOSFETs in parallel to increase current capability.
Figure 11: Vertical MOSFET Structure
As mentioned previously, the main energy loss in a MOSFET transistor comes from either switching or conduction. Switching losses can be minimized by having fast-switching transistors and applying soft switching, but reducing conducted emissions depends almost entirely on the structure of the MOSFET, especially on the on resistance, or RDS(ON).
The value of the on resistance depends on the length of the channel and the carrier concentration of the semiconductor. Of course, higher voltages create a stronger electric field, and therefore a larger depletion region (see Figure 12). Since the depletion region must not cross the entirety of the channel, the depth must be made very long. However, increasing the semiconductor length has a significant negative effect on the on resistance, which is why punch-through semiconductors where developed.
In this type of semiconductor device, the N region in the drain is divided into two parts of different doping densities: an N+ region with a very high doping density, and a low-density region. This low-density region is called the drift layer. Due to the doping gradient between these two regions, the electric field generated by the reverse bias is no longer triangular. Instead, it can “punch through” the limit of the drift region, acquiring a rectangular shape (see Figure 13). This allows for higher maximum blocking voltages without having to increase the channel length.
However, the drift layer’s low doping concentration has a negative effect on the semiconductor’s conductivity in that region, limiting the effect on the on resistance.
Figure 12: Non-Punch Through
Figure 13: Punch-Through
## MOSFET Safe Operating Area (SOA)
Like all devices, MOSFETs have limits to the operating conditions they can work in. These limiting conditions mostly have to do with the maximum voltage and current combinations that they can work with before breaking down. To better show these limitations, most MOSFET datasheets come with a safe operating area (SOA) graph (see Figure 14).
Figure 14: MOSFET SOA
The top limit of the safe operating area is set by the maximum current that can flow through the device. This is limited by the device’s RDS(ON), since the current flowing through the MOSFET channel (and therefore the resistor) generates heat, which can cause the device to break down.
The SOA’s vertical right-side limit is set by the maximum voltage that the MOSFET can block without breaking down and enabling conduction. This is determined by the MOSFET’s structure, the channel length, and the material used in its fabrication, as explained in the previous section of this article.
The diagonal limit in the SOA’s top-right corner represents the MOSFET’s capacity to sustain operation in the saturation region. Due to the combination of current and voltage in the switch, which occurs mainly in saturation, the MOSFET is said to be consuming power, which must then be dissipated in the form of heat. If the product of the current and voltage across the MOSFET is too high, the excessive heat could destroy the device.
The power dissipation limit is represented by several lines in the top-right corner of the SOA. These lines show how the MOSFET’s dissipation limit changes as a function of to the percentage of time that the transistor finds itself in saturation.
If the MOSFET is in DC, then there is a constant current and voltage across the MOSFET, and therefore a constant heating of the device, which greatly limits its capacity to dissipate all the generated energy. However, if the MOSFET is switching on and off, then the device is only being heated for a fraction of the time, and can withstand higher currents and voltages. The shorter the time that it stays on, the higher the voltage and current can be, only limited by the maximum current and voltage.
## Conclusion
MOSFETs are an integral part of nearly all electronic systems. Therefore, there is a constant push towards innovating MOSFET structures, discovering new materials and designing circuits with the aim of overcoming current physical limitations while making transistors smaller and smaller. MPS has made significant steps in this field, developing power conversion modules with power switches capable of withstanding up to 100A of continuous current, such as the MPM3695-100. To learn more, visit our website and explore our articles, reference designs, and application notes.
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Get technical support | {"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": 0.8373315930366516, "perplexity": 955.2023219325546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662517485.8/warc/CC-MAIN-20220517130706-20220517160706-00155.warc.gz"} |
https://math.stackexchange.com/questions/3299469/question-on-elementary-set-theory-notation/3299475 | # Question on elementary set theory notation
Consider the set $$A=\{n\ a \}$$ where $$a>0$$ is a constant and $$n \in \mathbb{N}$$
How shall we write this set $$A$$ in set theory?
If we write it as $$A=\{n\ a\ \backslash n \in \mathbb{N}, a>0 \}$$ or $$A=\{n\ a\ / n \in \mathbb{N}, a>0 \}$$ will it mean just one set or a set of infinite sets?
• $\{na\}$ is just a single set with a single element, which depends on the "external" constants $a$ and $n$. – Hagen von Eitzen Jul 21 at 11:50
• $n$ is not a constant. $A=\{a,2a,3a,4a,....\}$ – Joe Jul 21 at 11:57
• The set-builder notation you're trying to achieve is \{ na \mid n \in \mathbb{N} \}, producing $\{na \mid n \in \mathbb{N}\}$. – Hayden Jul 21 at 12:12
• If both $$n$$ and $$a$$ are fixed, then the set is , the singleton $$\{na\}$$
• If $$a$$ is fixed and $$n \in \Bbb N$$ is a varying quantity, then the set is $$\{na: n \in \Bbb N\}=\{a,2a,3a,\cdots\}$$
In both cases, $$A$$ is one set with cardinality $$1$$ and infinite(of course, $$\aleph_0)$$ respectively!
• Well, $n$ is not fixed.Your second option is correct. Anyway how can we write the information about $a$, i.e. $a>0$ in the set notation? – Joe Jul 21 at 12:00
• You already fixed $a>0$, so it is also better to write the set by $$A_a=\{na: n \in \Bbb N\}$$ which means the element $a$ is fixed is understood – Chinnapparaj R Jul 21 at 12:06
• or just write $$A=\{na: n=1,2,3, \cdots\;\text{and}\;a>0\;\text{is fixed} \}$$ – Chinnapparaj R Jul 21 at 12:09
You can also write it as $$a\Bbb N$$.
• This especially comes in handy when you have more sets of this kind and do some operations on them! – B.Swan Jul 21 at 12:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 16, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9738923907279968, "perplexity": 351.9713770395261}, "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-2019-39/segments/1568514573105.1/warc/CC-MAIN-20190917181046-20190917203046-00377.warc.gz"} |
http://math.stackexchange.com/questions/140614/algebra-percentage-question | # Algebra percentage question?
Sixteen and two thirds percent of x equals 3.2 what is x?
Can anyone help me solve this my final answer I get is forty eight over two hundred fifty but I am not sure if this is correct?
-
Use a calculator – Joel Cornett May 3 '12 at 21:43
If 16% of x is 3.2, would you expect x to be greater or less than 3.2? – Joel Cornett May 3 '12 at 21:45
Can you express sixteen and two thirds percent as a fraction? – Mark Bennet May 3 '12 at 21:49
yes it is 50/3 is the fraction. – James May 3 '12 at 21:53
Percent ($\%$) means divided by $100$. So it is a funny way to refer to the fraction $\frac{\frac{50}{3}}{100}$. This simplifies to $1/6$. – André Nicolas May 3 '12 at 21:57
Hint: $16\frac 23 \%=\frac {50}3 \cdot \frac 1{100}=\frac 16$
Write your equation as $\frac 16 x = 3.2$ and solve. As the percentage is less than $100\%$ you should expect that $x \gt 3.2$ which your original solution is not. – Ross Millikan May 3 '12 at 22:04 | {"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": 0.8438627123832703, "perplexity": 1158.0159119648947}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1409535919886.18/warc/CC-MAIN-20140909051443-00273-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/349666/prove-that-lim-limits-n-to-infty-n2-int-0-frac1n-xx1-dx/349847 | # Prove that $\lim \limits_{n \to \infty} n^2 \int_{0}^{\frac{1}{n}} x^{x+1} dx = \dfrac{1}{2}$
Prove that
$$\lim \limits_{n \to \infty} n^2 \int_{0}^{\frac{1}{n}} x^{x+1} dx = \dfrac{1}{2}$$
-
Scale the integral: substitute $u = n x$ and get
\begin{align}n^2 \int_0^{1/n} dx \: x^{x+1} &= \int_0^1 du \: u \, \left (\frac{u}{n}\right )^{u/n}\\ &= \int_0^1 du \: u \, e^{(u/n) \log{(u/n)}}\\ \end{align}
As $n \rightarrow \infty$, the exponential $\rightarrow 1$ because $u$ is bounded. Therefore the limit is
$$\int_0^1 du \, u = \frac{1}{2}$$
Taking the limit into the integral is justified so long as the sequence
$$e^{(u/n) \log{(u/n)}}$$
converges uniformly to $1$ over $u \in [0,1]$; in other words, so long as $(u/n) \log{(u/n)}$ converges to $0$ uniformly in this interval. That is, the supremum of $(u/n) \log{(u/n)}$ over $u \in [0,1]$ converges to $0$ as $n \rightarrow \infty$. This is true because of the fact that $u$ is in a bounded interval.
-
In the last step you're implicitly taking the limit into the integral. This must somehow be justified. – DonAntonio Apr 3 '13 at 2:16
OK, thanks, will do. – Ron Gordon Apr 3 '13 at 2:21
See the accepted answer of this post for additional justification: math.stackexchange.com/questions/330262/… – Ron Gordon Apr 3 '13 at 2:38
I thinks that's fine but perhaps is too much: I was thinking of dominated convergence or stuff. – DonAntonio Apr 3 '13 at 2:40
Some else tried to edit my post and suggested that. It relies on Lebesgue integrals, with which I am not very comfortable (haven't used them in over 20 years). – Ron Gordon Apr 3 '13 at 2:45
My favorite technique, start with the Taylor expansion of $x^{x+1}$ around $x=0$. I am only carrying the first two terms because doing this by hand can get very tedious. And the steps can all be rigorously justified.
\begin{eqnarray*} x^{x+1}&=&x+\ln(x) x^2+\cdots \\ \int_0^{1/n} x^{x+1} dx&=&\frac{1}{2n^2}-\frac{1+3\ln(n)}{9n^3}+\cdots\\ n^2 \int_0^{1/n} x^{x+1}dx&=&\frac{1}{2}-\frac{1+3\ln(n)}{9n}+\cdots\\ \lim_{n\rightarrow \infty}n^2 \int_0^{1/n} x^{x+1}dx&=&\frac{1}{2}. \end{eqnarray*}
-
Try change of variable $t = x^2$. Then we have
$\displaystyle\lim_{n \to \infty} n^2 \displaystyle\int_0^{\frac{1}{n}} x^{x+1} dx$ = $\frac{1}{2} \displaystyle\lim_{n \to \infty} \frac{\displaystyle\int_0^{\frac{1}{n^2}} \sqrt{y}^{\sqrt{y}} dy}{\frac{1}{n^2}}$
Note the quotient is of the form $\displaystyle\lim_{b \to a} \displaystyle\frac{\int_a^b f(x) dx}{b - a}$ and I suspect that it is $f(a) = 1$.
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https://quant.stackexchange.com/questions/24710/why-do-we-use-greater-than-or-equal-to-for-submartingale | # why do we use greater than or equal to for submartingale?
I've just learned about martingale, but i could not find any reason that we use greater than or equal to sign when we define submartingale.
In stead of using greater than or equal to symbol, can't we use greater symbol and it seems to be more intuitive
so my question is
why $E\left( {{M_{n + 1}}|{F_n}} \right) \ge {M_n}$
instead of $E\left( {{M_{n + 1}}|{F_n}} \right) \gt {M_n}$ ?
If $M_{n}$ is discrete random variable, then process is submartingale, if it satisfy: $$E[M_{t+1}|M_t] > M_t$$
But if $M_t$ is continuous random variable (which is assumed here), then both the expression $$E[M_{t+1}|M_t] > M_t$$ $$E[M_{t+1}|M_t] \ge M_t$$
are equivalent. You may write in either way. For a continuous variable, $X \ge K$ is same as $X >K$. | {"extraction_info": {"found_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": 0.8293375968933105, "perplexity": 290.3858554444584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347417746.33/warc/CC-MAIN-20200601113849-20200601143849-00587.warc.gz"} |
https://byjus.com/question-answer/in-an-election-the-votes-cast-for-two-of-the-candidates-were-in-the-13/ | Question
# Question 84 In an election, the votes cast for two of the candidates were in the ratio 5:7. If the successful candidates received 20734 votes, how many votes did his opponents receive?
Open in App
Solution
## Given, the ratio of votes for two candidates = 5 : 7 Let the votes be 5x and 7x. For successful candidates votes are greater. Using unitary method in which first we find the value of one unit and then the value of the required number of units. Hence, 7x=20734 x=207347=2962 Number of votes of his opponent = 5x = 5×2962 = 14810 Hence, his opponents received 14810 votes.
Suggest Corrections
9
Explore more | {"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": 0.8398177623748779, "perplexity": 1499.2108830368152}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499949.24/warc/CC-MAIN-20230201180036-20230201210036-00014.warc.gz"} |
http://physics.stackexchange.com/questions/64510/peculiar-hamiltonian-phase-space | # Peculiar Hamiltonian Phase space
I was solving an exercise of classical mechanics :
Consider the following hamiltonian
$H(p,q,t) = \frac{p^2}{2m} + \lambda pq + \frac{1}{2}m\lambda^2\frac{q^6}{q^4+\alpha^4}$
Where $\lambda,m,\alpha$ are positive parameters.
After having solved the equations of motion, I found something odd, and plotting the phase space (p,q) (which is y,x below) one can find that there are loops where momentum never changes sign, yet the system is localized (loop)
What sort of physical system does such a hamiltonian describe ? I cannot imagine something with an oscillating, never zero momentum, but still localized in space. I know this happens because of the cross-term, but I cannot find an interpretation for it
-
As you said the velocity is not proportional to the momentum in your system, if you compute the velocity around loops you will find that it is oscillating – Ikiperu May 14 '13 at 0:18
Stupid me, you are right. Taking the derivative with respect to t of q(t) gives a term proportional to cos(t). In this case, the Lagrangian certainly contained couplings, and the conjugate momentum picks up cross terms when going to the Hamiltonian, so I cannot see it as a velocity. – Mathusalem May 14 '13 at 0:25
or directly from hamilton eqs., the velocity is $\dot{q}=p/m +\lambda q$ – Ikiperu May 14 '13 at 0:28
The key inside to OP's question has already been provided by Ikiperu in above comments. Here we just want to show that the problem becomes very simple to study in the corresponding Lagrangian formalism.
$$\tag{1} H(p,q) ~:=~ \frac{p^2}{2m} + \lambda pq + \frac{m\lambda^2}{2}\frac{q^6}{q^4+\alpha^4}.$$
Since there is no explicit time dependence in (1), the Hamiltonian (= the mechanical energy of the system) is preserved. The velocity can be calculated from Hamilton's equation
$$\tag{2} \dot{q}~=~\frac{\partial H}{\partial p}~=~\frac{p}{m}+ \lambda q.$$
If we eliminate the momentum
$$\tag{3} \frac{p}{m}~=~ \dot{q}-\lambda q$$
in the Hamiltonian (1), we get a surprisingly simple energy function
$$\tag{4} h(q,\dot{q})~=~ \frac{m}{2}\dot{q}^2+V(q).$$
Here the potential $V(q)$ is the double-well
$$\tag{5} V(q)~=~ -\frac{m\alpha^2}{2}\frac{\lambda^2}{\left(\frac{q}{\alpha}\right)^2+\left(\frac{\alpha}{q}\right)^2} ,$$
which has two stable positions
$$\tag{6} q~=~\pm \alpha.$$
In $(q,\dot{q})$ space, there are two stable points
$$\tag{7} (q,\dot{q})~=~(\pm \alpha,0)$$
on the horizontal $q$-axis.
In $(q,p)$ phase space, the two stable points
$$\tag{8} (q,p)~=~\pm \alpha(1,-m\lambda)$$
are shifted by the transformation (3), in accordance with OP's figure.
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http://mathhelpforum.com/differential-geometry/75335-solved-riemann-sum-question.html | # Thread: [SOLVED] riemann sum question
1. ## [SOLVED] riemann sum question
Hi, long-time lurker, first time poster
The problem is this: find the limit of Sum[(i^2/(n^3 + i^3))] (going from i=1 to n) as n approaches infinity. Sorry I don't know how to write fancy math text but basically it's the limit of a series going to n as n --> infinity.
I figure it's probably a riemann sum question, but I tend to have difficulty with those. Any help would be nice
2. $\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{\frac{i^{2}}{n^{3}+i^{3} }}=\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\sum\limits_{i=1}^{n}{\left( \frac{i}{n} \right)^{2}\cdot \frac{1}{1+\left( \frac{i}{n} \right)^{3}}},$
thus $\int_{0}^{1}{\frac{x^{2}}{1+x^{3}}\,dx}$ is the limit and the partition is $\Delta x_{i}=\frac{1-0}{n}=\frac{1}{n}.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9785308837890625, "perplexity": 994.4609977581893}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698540804.14/warc/CC-MAIN-20161202170900-00180-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/3995/what-is-the-best-way-to-include-matlab-graphics?answertab=oldest | # What is the best way to include Matlab graphics?
Which way of exporting Matlab graphics delivers the best result for inclusion in LaTeX? Is it possible to export scalable vector graphics, or even LaTeX graphics?
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In case the kind of graphics you wish to export is supported by `matlab2tikz`, there is only one way to go: `matlab2tikz`.
You get true vector graphics in TikZ, i.e., no fiddling with PostScript, being restricted to `latex`, no going back and redoing graphics when your font or color scheme changes, small changes in for instance the legend are done in-place, ...
-
If you want to have scalable results, the better way is maybe to use the vectorial format eps (encapsulated postscript) but you will have to compile with "classic" latex and not pdflatex.
I am pretty sure that matlab is able to generate eps.
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I use the matlab command `print(gcf,'-deps2c',myfilename)` to print a figure to encapsulated postscript in color, which I then include with \includegraphics. Couldn't be easier. but it does require using latex and not pdflatex, as mentioned. If you do this, the font sizes you specify in matlab for text on the graph will be maintained in the eps file. – Brandon Kuczenski Oct 11 '10 at 19:24
Matlab can also create PDF. – Will Robertson Oct 12 '10 at 1:46
you may also have a look at the export in pstricks : fig2tex together with the provided links.
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The `print` command in Octave allows several devices for LaTeX output: `-dtex`, `-depslatex`, `-depslatexstandalone`, `-dpstex`, and `-dpslatex`. Making plots using Octave, gnuplot, and LaTeX by Marco De la Cruz-Heredia has some good examples.
The `print` command in Matlab can write to eps (see Elenaher's comment), so that is another option.
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The print command in Matlab can write to many output formats, including not only eps but also vector formats pdf, svg, emf and bitmap formats png, jpg – matth Nov 10 '11 at 9:24
There are two Matlab packages for exporting graphs to EPS plus psfrag, which replaces the labels inside the figures with strings that are typeset by LaTeX. They are laprint and matlabfrag; I recommend the latter as it will work for more graphic types, and laprint is no longer supported.
The support for surface-plot output may well make this a better option than the tikz based converters, otherwise the quality will be largely the same.
To include these types of graphics into pdfLaTeX, use the pstool package.
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There is a great library called pgfplots, which creates great looking plots directly in latex. This package is used by matlab2tikz. You can write the data you want to plot in an ascii table, and create legends, axis labels and such directly in latex, very much like the code you would have written in Matlab anyway.
You can easily change the contents, colors (for example for B/W prints), width and heigh, resolution, and much more without having to recreate figures.
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This is in fact the way I do it nowadays. – Ingo May 3 '12 at 8:49
In MATLAB all labels and titles, you can use LaTeX interpreter to make your plot more professional.
``````% for
text(2,2,'\$e^{2\log(x)}\$','interpreter','latex','fontsize',18)
% for legend
s = legend('\$x^3\$','\$x^2\$');
set(s,'interpreter','latex');
``````
Then the best way to include MATLAB plots into your latex document you have to follow these steps
1. save a picture as eps files;
2. use epstopdf to convert eps to pdf, then you can use both formats.
You should not save picture in pdf format since the bounding box will be not correct.
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Matlab will create cropped PDF graphics if you set the papersize correctly: tex.stackexchange.com/questions/5559/… – Will Robertson Jan 19 '11 at 22:09
@Robertson I think eps and then epstopdf is better in any cases since you may need eps for latex and pdf for pdflatex. – S. Boonto Jan 20 '11 at 6:16
Some updates on using `plot2svg` in MATLAB R2014b as of Mar 2015:
Jürg hasn't updated his awesome script for a while. For those of you who got an error while using `plot2svg` in R2014b, the simplest solution is just changing every `str2num` function in the original `plot2svg` to `str2double`, and problem solved.
(Original post)
`matlab2tikz` is the best vector graphics solution for small dataset graph as Pieter said in his answer. However, in some situations, using `matlab2tikz` to generate codes is not an efficient solution IMHO.
Here I will show two examples where using `matlab2tikz` to export the graphics is less than ideal a solution.
• First is with a large dataset like in the following triangulated sphere:
`matlab2tikz` generated a tex file with 3k lines of codes and took minutes to compile for once.
`matlab2tikz`'s output gives me:
For these situations: either you have a large dataset like a complicated triangulation/flow fields, or you have options such as`lighting`, `camproj`,`FaceAlpha`, etc in your `MATLAB` codes, or other Axes/Patch/Quivergroup properties tweak. The best way to include these graphics is:
(Step 1)Use `plot2svg` in `MATLAB` to get an svg file.
(Step 2)Use Inkscape to convert it to eps or pdf
(Step 3)Use `\includegraphics` in `graphicx` package of any TeX-distributions to include eps or pdf
`plot2svg` is a small pkg in `MATLAB` to produce scalable vector graphics by Jürg Schwizer.
There is a large thread of How to include SVG diagrams in LaTeX? on TeX.SE, the step 2 is what is in the accepted answer, if you are using Linux, once you installed Inkscape, you could use these shell scripts to convert svg to pdf/eps/png file formats.(Beware: the sh in the link is using zsh shell)
Other advantages of using this approach to export `MATLAB` figures:
• For scientific publications, most publishers are still using ancient TeX-distributions which doesn't include `tikz` pkg, for example: SIAM requires authors to use PostScript figures in the submission.
• Inkscape has a UI for editting svg files also.
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I can't describe how much this answer saved my ass, I'm exactly in the above situation. 10+ :) – Henrik Oct 10 '13 at 9:59
The easiest way is to use export_fig that convert Matlab figures to PDF automatically with some nice and useful features.
For both advanced and enthusiastic Matlab users, the possible way is to run through 11 pages of recommendations in "How To Make Pretty Figures With Matlab" manual.
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Notice that the author also warns for Note that export_fig does not transparency in patch objects in vector formats. Only a transparent background is supported.. In general MATLAB is not a good collaborator. Better practice is to take the data to gnuplot or other scientfic plotting platforms. – percusse Apr 29 '13 at 10:07
I really like the print command. To avoid margin and text resizing problems: I like to take a lot of control when exporting my figures by pre-defining the margins and paper-size to be my desired figure size. This makes fonts size correctly, and lets you choose the margins you want.
First I define my figure width and height (these are in metric, but you can use US units also).
``````FigW=13.49414;
FigH=21.08462*.3;
``````
Next, I set my figure properties. First I define the text interpreter to be latex, and then I set the paper size. You can define margins in the paper position parameter, but i prefer to do that later. if you want to use inches, that's fine here. Finally the 'Position' option just makes the figure on screen look identical to what will be printed to PDF. The first two parameters in 'Position' are the screen position and the second are the width and height of the figure.
``````Figure1=figure(1);clf;
set(Figure1,'defaulttextinterpreter','latex',...
'PaperUnits','centimeters','PaperSize',[FigW FigH],...
'PaperPosition',[0,0,FigW,FigH],'Units','centimeters',...
'Position',[1,10,FigW,FigH]);
``````
I then use the subaxis package from the mathworks file exchange to take control of my margins, padding and spacing of all my subplots. For one plot this might look like:
``````subaxis(1,1,1,'Margin',.1);
``````
Finally, I export to PDF using the print command in matlab:
``````print('-dpdf','-r500','filename.pdf')
``````
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http://mathhelpforum.com/differential-geometry/119322-proof-involving-abels-theorem-print.html | # Proof involving Abel's Theorem
• Dec 8th 2009, 08:46 AM
Proof involving Abel's Theorem
Let lim n-> ∞ a_n = L. Then, let f(x) = ∑ from 0 to ∞ of (a_n)(x^n). Show that the lim x-> 1 (1-x)f(x) = L.
This one is pretty far over my head. I know at some point you're supposed to use Abel/SBP, but here is what I have so far.
Let |a_n| go to |L|.
Then, using the ratio test, let |a_n|^(1/n) go to |L|^(1/∞) = |L|^(0) = 1.
Then, from the Ratio test, we can see that the series will converge for |x| < 1.
Take ∑ from 0 to ∞ of (a_n)(x^n). Then, multiply through. So, we obtain, (1-x)∑ from 0 to ∞ of (a_n)(x^n) = ∑ from 0 to ∞ of (x^n - x^(n+t)). Taking b_n to equal (x^n - x^(n+t)) we can get ∑ (a_n)(b_n)..
This is where I get stuck. I'm not really where to take it from here.
• Dec 9th 2009, 05:42 AM
chisigma
Is...
$f(x)= \sum_{n=0}^{\infty} a_{n}\cdot x^{n}$ (1)
... so that...
$(1-x)\cdot f(x)= a_{0} + (a_{1}-a_{0})\cdot x + (a_{2} - a_{1}) \cdot x^{2} + \dots + (a_{n} - a_{n-1})\cdot x^{n} + \dots$ (2)
It is evident that (2) is a 'telescopic series' so that is...
$\lim_{x \rightarrow 1} (1-x)\cdot f(x) = \lim_{n \rightarrow \infty} a_{n} = L$ (3)
Kind regards
$\chi$ $\sigma$ | {"extraction_info": {"found_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": 0.9823771715164185, "perplexity": 2752.0040058183445}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541910.39/warc/CC-MAIN-20161202170901-00078-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/45500/combination-of-24-picture-cards | # Combination of 24 picture cards
Twenty-four picture cards can be combined $1\,686\,553\,615\,927\,922\,354\,187\,744$ times. This means that you can get a complete landscape even with the quadrillionth variant.
The result can be calculated as follows:
$$\sum_{m=1}^{23}{\frac{24!}{(24-m)!}} + 24! = 1\,686\,553\,615\,927\,922\,354\,187\,744$$
However, how do you get to this term? Why isn't it just $24!$?
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I do not really understand what you are linking to? – utdiscant Jun 15 '11 at 10:33
$24!$ counts the number of ways that you can rearrange 24 distinct objects.
The expression you give counts the number of ways that you can select a subset of your 24 distinct objects, and then rearrange them into a new order.
For example, with 3 objects {1,2,3} you could choose any of the following orderings:
• (1) (2) (3)
• (1,2) (1,3) (2,1) (2,3) (3,1) (3,2)
• (1,2,3) (1,3,2) (2,1,3) (2,3,1) (3,1,2) (3,2,1)
The total number of ways is
(# of ways to select one object) + (# of ways to select two objects) + (# of ways to select 3 objects)
which is equal to
$$\frac{3!}{2!} + \frac{3!}{1!} + \frac{3!}{0!}$$
or equivalently
$$\sum_{m=1}^2 \frac{3!}{(3-m)!} + 3!$$
which is the expression you give in your question, but for the case of 3 objects rather than 24. In fact you could write it even more succinctly:
$$\sum_{m=1}^3 \frac{3!}{(3-m)!}$$
Moving slightly beyond your question, I would argue that there is one more way to select a subset of those objects, namely selecting none of them at all, in which case there are
$$\sum_{m=0}^3 \frac{3!}{(3-m)!}$$
In addition, I would note that due to a symmetry between $m$ and $3-m$, you can write this as
$$\sum_{m=0}^3 \frac{3!}{m!}$$
Finally, let's generalize to the case of $n$ objects, in which case the number of ways of selecting a subset and rearranging them into a new order is
$$a(n) = \sum_{m=0}^n \frac{n!}{m!}$$
There is an (almost) closed-form formula for this: if $n>0$ then
$$a(n) = \lfloor en! \rfloor$$
- | {"extraction_info": {"found_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": 0.828199028968811, "perplexity": 120.55190425597725}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657133485.50/warc/CC-MAIN-20140914011213-00320-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-find-the-slope-and-y-intercept-for-y-x-8 | Algebra
Topics
How do you find the slope and y intercept for: y=x+8?
For a line with formula $y = m \cdot x + b$ we have that the slope is
Hence $m = 1 , b = 8$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "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": 0.9367064237594604, "perplexity": 1223.5347221393793}, "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-2021-49/segments/1637964363149.85/warc/CC-MAIN-20211205065810-20211205095810-00164.warc.gz"} |
https://www.physicsforums.com/threads/first-order-integro-differential-equation.831925/ | # First order integro differential equation
• Start date
• #1
8
0
Can anyone help me to solve a differential equation?
I want to solve
∂v(p,t)/∂t=-p^2 v(p,t)-sqrt(2/pi)∫v(p,t)[1-δ(t)R(t)]dp+sqrt(2/pi)[δ(t)R^2(t) C]
with initial data v(p,0)=0
where C is constant and the integration from zero to infinty
Solution by volterra integral equation??
• #2
BvU
Homework Helper
14,448
3,736
Do I read this right ? You want to solve $${\partial v(p, t) \over \partial t} = - p^2 v(p, t) - \sqrt{2\over \pi} \int_0^\infty \ v(p,t)\ \left [ 1 - \delta(t) R(t) \right ] \, dp \ \ + \sqrt{2\over \pi} C\, \delta(t) \,R^2(t) \ \ ?$$with ## \ v(p, t) = 0\ ## and ##R(t)## a given function of time ?
(where does it come from ? what do the symbols stand for ?)
--
Any link with earlier posts (that seem to have petered out somewhat ) ?
Last edited:
• #3
8
0
Do I read this right ? You want to solve $${\partial v(p, t) \over \partial t} = - p^2 v(p, t) - \sqrt{2\over \pi} \int_0^\infty \ v(p,t)\ \left [ 1 - \delta(t) R(t) \right ] \, dp \ \ + \sqrt{2\over \pi} C\, \delta(t) \,R^2(t) \ \ ?$$with ## \ v(p, t) = 0\ ## and ##R(t)## a given function of time ?
(where does it come from ? what so the symbols stand for ?)
--
Any link with earlier posts (that seem to have petered out somewhat ) ?
Dear BvU,
Thank you for your replay, yes the equation is right.
The field equation is diffusion equation with 2 free boundary conditions
I applied the fourier transform for the diffusion and the boundary conditions and finally i got this first ODE
I stuck on it ?
• #4
BvU
Homework Helper
14,448
3,736
(Sorry for mistyping ##
\ v(p, t) = 0\ ## -- should of course have been ##
\ v(p, 0) = 0\ ## as you wrote).
Pretty hefty ! And does the ##\delta(t)## represent a time-dependent coefficient or is it the Kronecker delta function (in which case the term with the fector C is a bit problematic) ?
I hope someone more knowledgeable reads this and helps out, for me it's not obvious how to start with such a thing....
• #5
8
0
(Sorry for mistyping ##
\ v(p, t) = 0\ ## -- should of course have been ##
\ v(p, 0) = 0\ ## as you wrote).
Pretty hefty ! And does the ##\delta(t)## represent a time-dependent coefficient or is it the Kronecker delta function (in which case the term with the fector C is a bit problematic) ?
I hope someone more knowledgeable reads this and helps out, for me it's not obvious how to start with such a thing....
Thank you BvU and ##\delta(t)## is represent a time-dependent.
I hope someone can help me in this...
• #6
BvU
Homework Helper
14,448
3,736
I hope so too. My recollection of diffusion is that it gives equations like $${\partial u(x, t) \over \partial t} = {\partial^2 u\over \partial x^2}$$ so I have a hard time putting your equation into a context. But, as you say in your post #3, it is an intermediate situation in a solution procedure that involves Fourier transforms. I'll have to read up on that (little time for that ) and even then you probably have to spell out what you are doing from the beginning before I can be of any use, so we'll have to wait for help...
Oh, and
## \delta(t) ## is represent a time-dependent.
doesn't tell me much.
Last edited:
• #7
BvU
Homework Helper
14,448
3,736
If you can't wait that long, here's what I'm reading. Particularly pages 110 and further
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2K | {"extraction_info": {"found_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": 0.8653818368911743, "perplexity": 2166.41592493681}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046153392.43/warc/CC-MAIN-20210727135323-20210727165323-00398.warc.gz"} |
https://www.maplesoft.com/support/help/view.aspx?path=RegularChains%2FTriangularize | Triangularize - Maple Help
RegularChains
Triangularize
compute the triangular decomposition of a polynomial system
Calling Sequence Triangularize(F, R) Triangularize(F, H, R) Triangularize(F, rc, R) Triangularize(F, H, rc, R) Triangularize(F, R, 'normalized'='yes') Triangularize(F, R, 'radical'='yes') Triangularize(F, R, 'output'='lazard') Triangularize(F, R, 'probability'='prob')
Parameters
F - list or set of equations R - polynomial ring H - (optional) list or set of inequations rc - (optional) regular chain 'normalized'='yes' - (optional) boolean flag 'radical'='yes' - (optional) boolean flag 'output'='lazard' - (optional) boolean flag 'probability'='prob' - (optional) numerical value
Description
• The command Triangularize(F,H,rc,R) returns a triangular decomposition of the set of the zeros of F that are zeros of rc but not zeros of H.
• If H is not specified, it is set to $\left[1\right]$.
• If rc is not specified, it is set to the empty regular chain.
• If H=[1] and rc is empty, this command returns a triangular decomposition of the set of the zeros of F.
• Each element of this triangular decomposition is a regular chain.
• The concepts of a regular chain and a triangular decomposition are defined in the page RegularChains.
• The algorithm for the command Triangularize is described in the paper "On Triangular Decompositions of Algebraic Varieties" by Marc Moreno Maza, available on the author's web page.
• This command is part of the RegularChains package, so it can be used in the form Triangularize(..) only after executing the command with(RegularChains). However, it can always be accessed through the long form of the command by using RegularChains[Triangularize](..).
Options
• If 'normalized'='yes', each of the returned regular chains is normalized. If 'normalized'='strongly', each regular chain is strongly normalized. By default, each regular chain may not be normalized.
• If 'radical'='yes', the saturated ideal of each regular chain is radical. By default, the saturated ideal of each regular chain may not be radical.
This option is currently only supported if the characteristic of the polynomial ring R is zero.
• If 'output'='lazard', the decomposition is the sense of Lazard; otherwise, it is made in the sense of Kalkbrener. The latter is weaker but less expensive to compute. When F has only a finite number of solutions, both senses coincide.
As mentioned above, if this option is specified and if inequations (the input polynomial list H) are present, then the output is a constructible set. See the commands of the subpackage ConstructibleSetTools for operations on constructible sets.
If a non-empty regular chain rc is provided, then the decomposition is the sense of Lazard, whether 'output'='lazard' is specified or not.
• If 'probability'='prob', the probabilistic algorithm of Dahan, Moreno Maza, Schost, Wu, and Xie (ISSAC 2005) is used. This algorithm applies only to square systems in characteristic zero. In this implementation, it will fail if two primes such that the system modulo those primes is radical cannot be found after ten tries. This option is not compatible with any other options.
• The options 'output'='lazard' and 'normalized'='yes' cannot be mixed in the current implementation of this command. (This limitation will be solved in the next version of the RegularChains library).
• The commands BivariateModularTriangularize and ChangeOfOrder implement algorithms for computing triangular decompositions under particular circumstances. When they apply, they are likely to outperform Triangularize.
Examples
> $\mathrm{with}\left(\mathrm{RegularChains}\right):$
Define a ring of polynomials.
> $R≔\mathrm{PolynomialRing}\left(\left[z,y,x\right]\right)$
${R}{≔}{\mathrm{polynomial_ring}}$ (1)
Define a set of polynomials of R. Each of them will be viewed as an equality to 0.
> $\mathrm{sys}≔\left[{x}^{2}+y+z-1,x+{y}^{2}+z-1,x+y+{z}^{2}-1\right]$
${\mathrm{sys}}{≔}\left[{{x}}^{{2}}{+}{y}{+}{z}{-}{1}{,}{{y}}^{{2}}{+}{x}{+}{z}{-}{1}{,}{{z}}^{{2}}{+}{x}{+}{y}{-}{1}\right]$ (2)
Ideally, you would like to decompose the set of the common solutions of sys into a list of points. The Triangularize command does this by using symbolic expressions. Sometimes several points are grouped together in a generic one, as in this example. These groups of points are called regular chains, and they are grouped together because they share some mathematical properties.
> $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R\right)$
${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
Since regular chains may contain large expressions, their output form is just a word. To view their members, use the Equations command.
> $\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
$\left[\left[{z}{-}{x}{,}{y}{-}{x}{,}{{x}}^{{2}}{+}{2}{}{x}{-}{1}\right]{,}\left[{z}{,}{y}{,}{x}{-}{1}\right]{,}\left[{z}{,}{y}{-}{1}{,}{x}\right]{,}\left[{z}{-}{1}{,}{y}{,}{x}\right]\right]$ (4)
The first three regular chains are very simple: each of them clearly corresponds to a single point. The fourth regular chain corresponds to two points, because its univariate polynomial in $z$ has two roots. Consider now another polynomial ring and another polynomial system.
> $R≔\mathrm{PolynomialRing}\left(\left[x,y,a,b,c,d,g,h\right]\right)$
${R}{≔}{\mathrm{polynomial_ring}}$ (5)
> $\mathrm{sys}≔\left[ax+by-g,cx+dy-h\right]$
${\mathrm{sys}}{≔}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{c}{}{x}{+}{d}{}{y}{-}{h}\right]$ (6)
In the polynomial ring, the ordering on the variables is such that $x>y>a>b>c>d>g>h$. Solving $\mathrm{sys}$ with this ordering implies that you want to express $x$ and $y$ as functions of the other variables. Hence you can view the system $\mathrm{sys}$ as a parametric linear system with two equations and two unknowns, $x$ and $y$. Applying RegularChains[Triangularize] displays the generic solution, which is similar to the solution given by Groebner[Solve].
> $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R\right);$$\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}\right]$
$\left[\left[{c}{}{x}{+}{y}{}{d}{-}{h}{,}\left({a}{}{d}{-}{b}{}{c}\right){}{y}{-}{a}{}{h}{+}{c}{}{g}\right]\right]$ (7)
> $\mathrm{Groebner}:-\mathrm{Solve}\left(\mathrm{sys},\left\{a,b,c,d,g,h,x,y\right\}\right)$
$\left\{\left[\left[{-}{x}{}{a}{-}{y}{}{b}{+}{g}{,}{-}{c}{}{x}{-}{y}{}{d}{+}{h}\right]{,}{\mathrm{plex}}{}\left({h}{,}{g}{,}{d}{,}{c}{,}{b}{,}{a}{,}{y}{,}{x}\right){,}{\varnothing }\right]\right\}$ (8)
This generic solution assumes that the determinant of the system is not zero. With the option output=lazard, Triangularize gives all of the solutions, including those that cancel the determinant of the input system.
> $\mathrm{decl}≔\mathrm{Triangularize}\left(\mathrm{sys},R,\mathrm{output}=\mathrm{lazard}\right)$
${\mathrm{decl}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (9)
> $\mathrm{map}\left(\mathrm{Equations},\mathrm{decl},R\right)$
$\left[\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}\left({d}{}{a}{-}{b}{}{c}\right){}{y}{-}{h}{}{a}{+}{c}{}{g}\right]{,}\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}{d}{}{a}{-}{b}{}{c}{,}{h}{}{b}{-}{d}{}{g}\right]{,}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{d}{}{y}{-}{h}{,}{c}\right]{,}\left[{d}{}{y}{-}{h}{,}{a}{,}{h}{}{b}{-}{d}{}{g}{,}{c}\right]{,}\left[{c}{}{x}{-}{h}{,}{h}{}{a}{-}{c}{}{g}{,}{b}{,}{d}\right]{,}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{c}{,}{d}{,}{h}\right]{,}\left[{c}{}{x}{+}{d}{}{y}{,}{d}{}{a}{-}{b}{}{c}{,}{g}{,}{h}\right]{,}\left[{b}{}{y}{-}{g}{,}{a}{,}{c}{,}{d}{,}{h}\right]{,}\left[{y}{,}{a}{,}{c}{,}{g}{,}{h}\right]{,}\left[{x}{,}{b}{,}{d}{,}{g}{,}{h}\right]{,}\left[{a}{,}{b}{,}{c}{,}{d}{,}{g}{,}{h}\right]\right]$ (10)
> $\mathrm{nops}\left(\mathrm{decl}\right)$
${11}$ (11)
You already know that each regular chain is associated with a set of equations. It is also associated with a set of inequations.
> $\mathrm{rc}≔\mathrm{decl}\left[1\right];$$\mathrm{Equations}\left(\mathrm{rc},R\right);$$\mathrm{Inequations}\left(\mathrm{rc},R\right)$
${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$
$\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}\left({d}{}{a}{-}{b}{}{c}\right){}{y}{-}{h}{}{a}{+}{c}{}{g}\right]$
$\left\{{c}{,}{d}{}{a}{-}{b}{}{c}\right\}$ (12)
The inequations of a regular chain $\mathrm{rc}$ are the set of the initials of the polynomials of $\mathrm{rc}$. In the first regular chain above, the inequations are $da-bc$ and $c$. Hence, for this regular chain, none of these two polynomials should vanish. The solutions of $\mathrm{sys}$ that cancel either $c$ or the determinant $da-bc$ are given by the other regular chains of $\mathrm{decl}$. Below, for each regular chain of $\mathrm{decl}$, we print its list of equations together with its set of inequations.
> $\left[\mathrm{seq}\left(\left[\mathrm{eq}=\mathrm{Equations}\left(\mathrm{decl}\left[i\right],R\right),\mathrm{ineq}=\mathrm{Inequations}\left(\mathrm{decl}\left[i\right],R\right)\right],i=1..\mathrm{nops}\left(\mathrm{decl}\right)\right)\right]$
$\left[\left[{\mathrm{eq}}{=}\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}\left({d}{}{a}{-}{b}{}{c}\right){}{y}{-}{h}{}{a}{+}{c}{}{g}\right]{,}{\mathrm{ineq}}{=}\left\{{c}{,}{d}{}{a}{-}{b}{}{c}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}{d}{}{a}{-}{b}{}{c}{,}{h}{}{b}{-}{d}{}{g}\right]{,}{\mathrm{ineq}}{=}\left\{{c}{,}{d}{,}{h}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{d}{}{y}{-}{h}{,}{c}\right]{,}{\mathrm{ineq}}{=}\left\{{a}{,}{d}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{d}{}{y}{-}{h}{,}{a}{,}{h}{}{b}{-}{d}{}{g}{,}{c}\right]{,}{\mathrm{ineq}}{=}\left\{{d}{,}{h}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{c}{}{x}{-}{h}{,}{h}{}{a}{-}{c}{}{g}{,}{b}{,}{d}\right]{,}{\mathrm{ineq}}{=}\left\{{c}{,}{h}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{c}{,}{d}{,}{h}\right]{,}{\mathrm{ineq}}{=}\left\{{a}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{c}{}{x}{+}{d}{}{y}{,}{d}{}{a}{-}{b}{}{c}{,}{g}{,}{h}\right]{,}{\mathrm{ineq}}{=}\left\{{c}{,}{d}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{b}{}{y}{-}{g}{,}{a}{,}{c}{,}{d}{,}{h}\right]{,}{\mathrm{ineq}}{=}\left\{{b}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{y}{,}{a}{,}{c}{,}{g}{,}{h}\right]{,}{\mathrm{ineq}}{=}{\varnothing }\right]{,}\left[{\mathrm{eq}}{=}\left[{x}{,}{b}{,}{d}{,}{g}{,}{h}\right]{,}{\mathrm{ineq}}{=}{\varnothing }\right]{,}\left[{\mathrm{eq}}{=}\left[{a}{,}{b}{,}{c}{,}{d}{,}{g}{,}{h}\right]{,}{\mathrm{ineq}}{=}{\varnothing }\right]\right]$ (13)
Assume now that you want to see $g$ and $h$ as transcendental quantities; that is, quantities that cannot satisfy any polynomial equations. Then you need to redefine the polynomial ring as follows.
> $\mathrm{R2}≔\mathrm{PolynomialRing}\left(\left[x,y,a,b,c,d\right],\left\{g,h\right\}\right)$
${\mathrm{R2}}{≔}{\mathrm{polynomial_ring}}$ (14)
> $\mathrm{dec2}≔\mathrm{Triangularize}\left(\mathrm{sys},\mathrm{R2},\mathrm{output}=\mathrm{lazard}\right)$
${\mathrm{dec2}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (15)
> $\left[\mathrm{seq}\left(\left[\mathrm{eq}=\mathrm{Equations}\left(\mathrm{dec2}\left[i\right],\mathrm{R2}\right),\mathrm{ineq}=\mathrm{Inequations}\left(\mathrm{dec2}\left[i\right],\mathrm{R2}\right)\right],i=1..\mathrm{nops}\left(\mathrm{dec2}\right)\right)\right]$
$\left[\left[{\mathrm{eq}}{=}\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}\left({d}{}{a}{-}{b}{}{c}\right){}{y}{-}{h}{}{a}{+}{c}{}{g}\right]{,}{\mathrm{ineq}}{=}\left\{{c}{,}{d}{}{a}{-}{b}{}{c}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}{d}{}{a}{-}{b}{}{c}{,}{h}{}{b}{-}{d}{}{g}\right]{,}{\mathrm{ineq}}{=}\left\{{c}{,}{d}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{a}{}{x}{+}{y}{}{b}{-}{g}{,}{d}{}{y}{-}{h}{,}{c}\right]{,}{\mathrm{ineq}}{=}\left\{{a}{,}{d}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{d}{}{y}{-}{h}{,}{a}{,}{h}{}{b}{-}{d}{}{g}{,}{c}\right]{,}{\mathrm{ineq}}{=}\left\{{d}\right\}\right]{,}\left[{\mathrm{eq}}{=}\left[{c}{}{x}{-}{h}{,}{h}{}{a}{-}{c}{}{g}{,}{b}{,}{d}\right]{,}{\mathrm{ineq}}{=}\left\{{c}\right\}\right]\right]$ (16)
> $\mathrm{nops}\left(\mathrm{dec2}\right)$
${5}$ (17)
Now, you can obtain five regular chains, none of them imposing a condition on $g$ or $h$. The following example uses the option probability, by which a triangular decomposition is done by using a modular algorithm.
> $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
${R}{≔}{\mathrm{polynomial_ring}}$ (18)
> $\mathrm{sys}≔\left\{5{y}^{4}-3,-20x+y-z,-{x}^{5}+{y}^{5}-3y-1\right\}$
${\mathrm{sys}}{≔}\left\{{-}{20}{}{x}{+}{y}{-}{z}{,}{5}{}{{y}}^{{4}}{-}{3}{,}{-}{{x}}^{{5}}{+}{{y}}^{{5}}{-}{3}{}{y}{-}{1}\right\}$ (19)
> $\mathrm{Triangularize}\left(\mathrm{sys},R,\mathrm{probability}=0.9\right)$
$\left[{\mathrm{regular_chain}}\right]$ (20)
References
Moreno Maza, M. "On Triangular Decompositions of Algebraic Varieties." MEGA-2000 conference. Bath, UK, England. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 69, "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": 0.9564133882522583, "perplexity": 1326.642876995456}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500076.87/warc/CC-MAIN-20230203221113-20230204011113-00624.warc.gz"} |
http://mathoverflow.net/questions/43378/errata-in-conformal-mapping-method-and-applications-from-schinzinger | # Errata in “Conformal Mapping: Method and Applications” from Schinzinger
hi,
i'm studying the book "Conformal Mapping: Method and Applications" from Schinzinger and more precisely the chapter 6 concerning non-planar field. In section 6.1.3 the author proposes to solve a cylindrical laplace equation by solving a cartesian laplace equation via a conformal mapping of the polar plane.
I've joined the concerned pages.
At one point he says $\phi(u,\zeta)$ (the potential in the cartesian coordinates) verifies the cylindrical laplace equation, which i think is wrong and if it really so, the whole demonstration fails. | {"extraction_info": {"found_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": 0.9742504954338074, "perplexity": 1314.7473590137884}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257824757.8/warc/CC-MAIN-20160723071024-00270-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://papers.neurips.cc/paper/3991-multi-view-active-learning-in-the-non-realizable-case | # NIPS Proceedingsβ
## Multi-View Active Learning in the Non-Realizable Case
[PDF] [BibTeX] [Supplemental]
### Abstract
The sample complexity of active learning under the realizability assumption has been well-studied. The realizability assumption, however, rarely holds in practice. In this paper, we theoretically characterize the sample complexity of active learning in the non-realizable case under multi-view setting. We prove that, with unbounded Tsybakov noise, the sample complexity of multi-view active learning can be $\widetilde{O}(\log \frac{1}{\epsilon})$, contrasting to single-view setting where the polynomial improvement is the best possible achievement. We also prove that in general multi-view setting the sample complexity of active learning with unbounded Tsybakov noise is $\widetilde{O}(\frac{1}{\epsilon})$, where the order of $1/\epsilon$ is independent of the parameter in Tsybakov noise, contrasting to previous polynomial bounds where the order of $1/\epsilon$ is related to the parameter in Tsybakov noise. | {"extraction_info": {"found_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": 0.9673775434494019, "perplexity": 999.929204844875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738864.9/warc/CC-MAIN-20200812024530-20200812054530-00403.warc.gz"} |
https://www.physicsforums.com/threads/electric-dipole-radiation.951059/ | # Electric Dipole Radiation
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• #1
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## Main Question or Discussion Point
Hello! I am reading Griffiths derivation for the electric dipole radiation (actually my question would fit for the magnetic dipole radiation too). He considers 2 charged balls connected by a wire with charge going back and forth between them. Now, when he calculates the vector potential he uses this formula: $$A(r,t)=\frac{\mu_0}{4 \pi}\int_{-d/2}^{d/2}\frac{-q_0\omega sin[\omega(t-r'/c)]\hat{z}}{r'}dz$$
However, if I followed it properly through the book, this equation is derived from Biot-Savart law (and a proper choice of gauge). However, when introducing Biot-Savart, Griffiths emphasizes that the current density (or linear current in this case) must be infinite in extent i.e. started infinitely long in the past and uniform. However the current is not infinite in extent, nor uniform. I see that Griffiths takes into account the fact that the potential is retarded, but I am just a bit confused about using this formula derived from Biot-Savart. Is it that obvious that a formula based on a uniform, infinite in extent current is correct just by adding that $t-r/c$ term?
## Answers and Replies
Related Classical Physics News on Phys.org
• #2
Dale
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However, if I followed it properly through the book, this equation is derived from Biot-Savart law (and a proper choice of gauge). However, when introducing Biot-Savart, Griffiths emphasizes that the current density (or linear current in this case) must be infinite in extent i.e. started infinitely long in the past and uniform.
That formula is not Biot Savart, although it looks similar. It is the retarded potential which can be derived from Maxwell’s equations in the Lorenz gauge by using Green’s functions.
• #3
624
11
That formula is not Biot Savart, although it looks similar. It is the retarded potential which can be derived from Maxwell’s equations in the Lorenz gauge by using Green’s functions.
Thank you for your reply. I know it is not Biot-Savart. What I was saying is that Griffiths uses Biot-Savart to derive it. His approach is to Biot-Savart to prove that $\nabla \times B = \mu_0 J$. Then, using $B=\nabla \times A$ and the gauge in which $\nabla A = 0$ he reaches the formula I sated above. My confusion is that in his approach the starting point is Biot-Savart. However, as he specifies, Biot Savart is for the case of a uniform current, of infinite extent. So why does this retarded potential formula holds, even if its starting point wouldn't hold in the case of a non-uniform current.
• #4
Dale
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I can’t say anything about his specific derivation, but it is not necessary to start with Biot Savart. There are other derivations.
• #5
Delta2
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Thank you for your reply. I know it is not Biot-Savart. What I was saying is that Griffiths uses Biot-Savart to derive it. His approach is to Biot-Savart to prove that $\nabla \times B = \mu_0 J$. Then, using $B=\nabla \times A$ and the gauge in which $\nabla A = 0$ he reaches the formula I sated above. My confusion is that in his approach the starting point is Biot-Savart. However, as he specifies, Biot Savart is for the case of a uniform current, of infinite extent. So why does this retarded potential formula holds, even if its starting point wouldn't hold in the case of a non-uniform current.
Ignoring , for the moment, how he arrives at $\nabla \times B = \mu_0 J$ (1), this equation is almost the full Maxwell- Ampere equation (which hold in any case of current density), it just misses a term $\epsilon_0\mu_0\frac{\partial E}{\partial t}$ (2) in the right hand side. For small enough frequencies and amplitudes of the electric field (for example smaller than $10^{8}$ cause the term $\epsilon_0\mu_0$ is of the order of $10^{-16}$) we can take (1) as a good approximation of the full Maxwell-Ampere law so it holds approximately.
• #6
vanhees71
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The trick is that you have an indealized charge-current distribution which makes the exact retarded solution very simple. The general case is much more involved. I've just developed a nice treatment of the Hertzian dipole for my E&M lecture. The idea is to start with a harmonically oscillating charged particle, and it's most convenient to use the complexified treatment, i.e., one makes all the fields complex valued and understands as the physical values its real part. Then we consider a point charge harmonically oscillating along the 3-axis with angular frequency $\omega$, i.e.,
$$\vec{y}(t)=d \vec{e}_3 \exp(-\mathrm{i} \omega t).$$
Then the charge-current distribution is given by
$$\rho(t,\vec{r})=q \delta^{(3)}[\vec{r}-\vec{y}(t)], \quad \vec{j}(t,\vec{r}) = q \dot{\vec{y}}(t) \delta^{(3)}[\vec{x}-\vec{y}(t)].$$
It's crucial to note that this obeys the continuity equation (charge conservation):
$$\partial_t \rho + \vec{\nabla} \cdot \vec{j}=0.$$
If you'd use this to calculate the scalar and vector potentials of the electromagnetic field, you'd get the Lienard-Wiechert potentials, but that's a pretty cumbersome calculation.
Instead the idea is to assume that $d \ll \lambda$, where $\lambda=2 \pi c/\omega$ is the wavelength of the radiation expected to be produced by the harmonically oscillating charge. Then, if you look at the fields not too close to the origin of the coordinate system, it's justified to expand the charge-current distribution in powers of $\vec{y}$ and $\dot{\vec{y}}$. It's important to make this consistent such that the continuity equation holds at each order. The first non-trivial order is the expansion up to order $d$, i.e.,
$$\begin{split} \rho&=\rho_0 + \rho_1=q \delta^{(3)}(\vec{r}) - q \vec{y}(t) \cdot \vec{\nabla} \delta^{(3)}(\vec{r}),\\ \vec{j} &= \vec{j}_1=q \dot{\vec{y}}(t) \delta^{(3)}(\vec{r}). \end{split}$$
Indeed, using this approximation for charge and current densities still obeys the continuity equation exactly, which is crucial to get physically meaningful results for the em. field.
The first term $\rho_0$ is just the static (time-averaged) charge distribution of a point charge $q$ sitting at the origin. Integrating this with the retarded potential leads to the corresponding Coulomb potential in Lorenz gauge, i.e.,
$$\Phi_0(t,\vec{r}) = \frac{q}{4 \pi \epsilon_0 r}.$$
The next order is indeed electric-dipole radiation (i.e., the $\ell=1$ piece in the spherical multipole expansion), i.e., the lowest multipole order giving rise to the radiation of electromagnetic waves. It's very easy to see in this Cartesian approach that there is no "monopole radiation" in electromagnetics.
The rest of the calculation is most simply done with starting to calculate the vector potential. The retarded Green's function of the D'Alembert operator is
$$D_{\text{ret}}(t,\vec{r})=\frac{1}{4 \pi r} \delta(t-r/c).$$
Thus the vector potential is
$$\vec{A}_1(t,\vec{r}) = \int_{\mathbb{R}^3} \mathrm{d}^3 r' \frac{\mu_0 \vec{j}_1(t-r'/c,\vec{r'}}{4 \pi |\vec{r}-\vec{r}'|} = -\frac{\mathrm{i} \mu_0 \omega q \vec{d}}{4 \pi r} \exp(\mathrm{i} k r - \mathrm{i} \omega t).$$
From this it's a piece of cake to calculate
$$\vec{B}_1=\vec{\nabla} \times \vec{A}_1.$$
Instead of also calculating $\Phi_1$, it's easier to use the Maxwell-Ampere equation to directly obtain $\vec{E}_1$, making use of the fact that the time dependence of all fields is simply a factor $\exp(-\mathrm{i} \omega t)$:
$$\vec{E}_1=\frac{\mathrm{i} c^2}{\omega} \vec{\nabla} \times \vec{B}_1.$$
Last edited:
• #7
jtbell
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Is it that obvious that a formula based on a uniform, infinite in extent current is correct just by adding that $t-r/c$ term?
No.
Griffiths starts with the static potentials $$V(\vec r) = \frac 1 {4 \pi \varepsilon_0} \int {\frac {\rho ({\vec r}^\prime)} {|\vec r - {\vec r}^\prime|} \, d {\vec \tau}^\prime} \\ \vec A (\vec r) = \frac {\mu_0} {4 \pi} \int {\frac {\vec J ({\vec r}^\prime)} {|\vec r - {\vec r}^\prime|} \, d {\vec \tau}^\prime}$$ He generalizes them by evaluating $\rho$ and $\vec J$ at the retarded time for each source point: $$t_r = t - \frac {|\vec r - {\vec r}^\prime|} c$$ to get $$V({\vec r}, t) = \frac 1 {4 \pi \varepsilon_0} \int {\frac {\rho ({\vec r}^\prime, t_r)} {|\vec r - {\vec r}^\prime|} \, d {\vec \tau}^\prime} \\ \vec A ({\vec r}, t) = \frac {\mu_0} {4 \pi} \int {\frac {\vec J ({\vec r}^\prime, t_r)} {|\vec r - {\vec r}^\prime|} \, d {\vec \tau}^\prime}$$ Then he says:
Griffiths said:
Well, that all sounds reasonable—and surprisingly simple. But are we sure it's right? I didn't actually derive these formulas for $V$ and $\vec A$; all I did was invoke a heuristic argument ("electromagnetic news travels at the speed of light") to make them seem plausible. To prove them, I must show that they satisfy the inhomogeneous wave equation (10.16) and meet the Lorentz condition (10.12). In case you think I'm being fussy, let me warn you that if you apply the same argument to the fields [$\vec E$ and $\vec B$] you'll get entirely the wrong answer [...]
He then shows that the retarded potentials satisfy (10.16) and leaves it as an exercise to show that they satisfy (10.12).
(Actually he does only the scalar potential explicitly, and says that "essentially the same argument would serve for the vector potential".)
[This is from the 3rd edition. The 4th edition may be different.]
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2K | {"extraction_info": {"found_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": 0.9669834971427917, "perplexity": 435.0116782243022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371807538.83/warc/CC-MAIN-20200408010207-20200408040707-00403.warc.gz"} |
http://mathhelpforum.com/calculus/145484-riemmann-surface-integral.html | # Math Help - Riemmann Surface integral
1. ## Riemmann Surface integral
Hello, here's the problem:
Let be C a curve in Riemmann Surface log(z) that goes from z=1 (in the log(1)=0 sheet) to z=1 turning around z=0 'n' times. Calculate:
$\int_C log(z) dz$
I have tried to calculate in this way:
$log(z)=log(r)+it$
$z(t)=r(cos(t)+isin(t))$ so $d(z(t))=r(-sin(t)+icos(t))dt$
thus:
$\int_C log(z) dz=\int^{2n\pi}_0 logz(t)z'(t)dt=\int^{2n\pi}_0(log(r)+it)r(-sin(t)+icos(t))dt=2\pi i rn$
As r=1: (I know I could have simplified before)
$\int_C log(z) dz=2\pi i n$
What do you think? Is it right?
Thank you.
2. I'm not sure about the 'n'..
Thank you.
3. $\int_{C(n)} \log(z)dz=2n\pi i$
Everytime you go one time around, the integral is $2\pi i$. You could show this piece-wise for each circuit, or I would analytically extend the antiderivative and write:
$\int_{C(n)}\log(z)dz=\left(z\log(z)-z\right)\biggr|_1^1=(2n\pi i-1)-(0-1)=2n\pi i$
it's meant to look controversial just to get people thinking
I do believe an argument for it's validity could be made however. | {"extraction_info": {"found_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": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8524066209793091, "perplexity": 3045.3783973895547}, "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-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00394-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.zora.uzh.ch/id/eprint/147897/ | # Modeling BSM effects on the Higgs transverse-momentum spectrum in an EFT approach
Grazzini, Massimiliano; Ilnicka, Agnieszka; Spira, Michael; Wiesemann, Marius (2017). Modeling BSM effects on the Higgs transverse-momentum spectrum in an EFT approach. Journal of High Energy Physics, 2017(3):115.
## Abstract
We consider the transverse-momentum distribution of a Higgs boson produced through gluon fusion in hadron collisions. At small transverse momenta, the large logarithmic terms are resummed up to next-to-leading-logarithmic (NLL) accuracy. The resummed computation is consistently matched to the next-to-leading-order (NLO) result valid at large transverse momenta. The ensuing Standard Model prediction is supplemented by possible new-physics effects parametrised through three dimension-six operators related to the modification of the top and bottom Yukawa couplings, and to the inclusion of a point-like Higgs-gluon coupling, respectively. We present resummed transverse-momentum spectra including the effect of these operators at NLL+NLO accuracy and study their impact on the shape of the distribution. We find that such modifications, while affecting the total rate within the current uncertainties, can lead to significant distortions of the spectrum. The proper parametrization of such effects becomes increasingly important for experimental analyses in Run II of the LHC.
## Abstract
We consider the transverse-momentum distribution of a Higgs boson produced through gluon fusion in hadron collisions. At small transverse momenta, the large logarithmic terms are resummed up to next-to-leading-logarithmic (NLL) accuracy. The resummed computation is consistently matched to the next-to-leading-order (NLO) result valid at large transverse momenta. The ensuing Standard Model prediction is supplemented by possible new-physics effects parametrised through three dimension-six operators related to the modification of the top and bottom Yukawa couplings, and to the inclusion of a point-like Higgs-gluon coupling, respectively. We present resummed transverse-momentum spectra including the effect of these operators at NLL+NLO accuracy and study their impact on the shape of the distribution. We find that such modifications, while affecting the total rate within the current uncertainties, can lead to significant distortions of the spectrum. The proper parametrization of such effects becomes increasingly important for experimental analyses in Run II of the LHC.
## Statistics
### Citations
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12 citations in Scopus®
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Detailed statistics | {"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": 0.9464713335037231, "perplexity": 2526.6541379336054}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875147647.2/warc/CC-MAIN-20200228200903-20200228230903-00382.warc.gz"} |
http://mathhelpforum.com/statistics/140112-competition-results.html | Math Help - Competition results
1. Competition results
Competition results:
Result 10 20 30 40 50
Participants 1 2 5 k 3
Mean value of the results is 34. $k=?$
Help?
2. If the mean value of the results is 34, then you need to solve the following for $k$:
$\frac{1(10)+2(20)+5(30)+k(40)+3(50)}{1+2+5+k+3}=34$
3. The mean is equal to the sum of the results divided by the sum of the competitors, so:
$\frac{10+20+30+40+50}{1+2+5+k+3}=34$
From there, it's simple algebra. Solve for k.
4. Originally Posted by downthesun01
The mean is equal to the sum of the results divided by the sum of the competitors, so:
$\frac{10+20+30+40+50}{1+2+5+k+3}=34$
From there, it's simple algebra. Solve for k.
cpbrunner's got it right. You have to take the frquencies into account on the numerator. downthesun01 hasn't.
5. My mistake. You're totally 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": 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": 0.9214402437210083, "perplexity": 925.5761277185994}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131296951.54/warc/CC-MAIN-20150323172136-00262-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://collegemathteaching.wordpress.com/2013/10/25/a-laplace-transform-of-a-function-of-non-exponential-order/ | # College Math Teaching
## October 25, 2013
### A Laplace Transform of a function of non-exponential order
Many differential equations textbooks (“First course” books) limit themselves to taking Laplace transforms of functions of exponential order. That is a reasonable thing to do. However I’ll present an example of a function NOT of exponential order that has a valid (if not very useful) Laplace transform.
Consider the following function: $n \in \{1, 2, 3,...\}$
$g(t)= \begin{cases} 1,& \text{if } 0 \leq t \leq 1\\ 10^n, & \text{if } n \leq t \leq n+\frac{1}{100^n} \\ 0, & \text{otherwise} \end{cases}$
Now note the following: $g$ is unbounded on $[0, \infty)$, $lim_{t \rightarrow \infty} g(t)$ does not exist and
$\int^{\infty}_0 g(t)dt = 1 + \frac{1}{10} + \frac{1}{100^2} + .... = \frac{1}{1 - \frac{1}{10}} = \frac{10}{9}$
One can think of the graph of $g$ as a series of disjoint “rectangles”, each of width $\frac{1}{100^n}$ and height $10^n$ The rectangles get skinnier and taller as $n$ goes to infinity and there is a LOT of zero height in between the rectangles.
Needless to say, the “boxes” would be taller and skinnier.
Note: this is an example can be easily modified to provide an example of a function which is $l^2$ (square integrable) which is unbounded on $[0, \infty)$. Hat tip to Ariel who caught the error.
It is easy to compute the Laplace transform of $g$:
$G(s) = \int^{\infty}_0 g(t)e^{-st} dt$. The transform exists if, say, $s \geq 0$ by routine comparison test as $|e^{-st}| \leq 1$ for that range of $s$ and the calculation is easy:
$G(s) = \int^{\infty}_0 g(t)e^{-st} dt = \frac{1}{s} (1-e^{-s}) + \frac{1}{s} \sum^{\infty}_{n=1} (\frac{10}{e^s})^n(1-e^{\frac{-s}{100^n}})$
Note: if one wants to, one can see that the given series representation converges for $s \geq 0$ by using the ratio test and L’Hoptial’s rule. | {"extraction_info": {"found_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": 19, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9660570025444031, "perplexity": 273.9470239395548}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824471.6/warc/CC-MAIN-20171020230225-20171021010225-00558.warc.gz"} |
https://math.stackexchange.com/questions/2072845/factoring-a-quadratic-equation-using-two-different-methods-why-does-this-work | Factoring a quadratic equation using two different methods. Why does this work? [duplicate]
I have the quadratic equation: $$4x^2 - 49$$
Using
$$\frac{b\pm\sqrt{b^2-4ac}}{2a}$$
I can factor the equation to
$$(x-3.5)(x+3.5)$$.
However, if I use
$$\frac{b\pm\sqrt{b^2-4ac}}{2}$$
I get 14, which I can then use the grouping method $$4x^2 +14x - 14x -49$$ to get an equivalent result of:
$$(2x-7)(2x+7)$$
Why does removing the $$a$$ from the denominator in the quadratic formula work to give me a number (this case 14), which I can then use to factor out the original quadratic equation?
Also, are there any benefits in solving the equation using the latter method? (the result to looks better to me)
• I don't see whats so special about this. Have you tried using your "method" on any other problem? Take $x^2-8$, for example. – Simply Beautiful Art Dec 26 '16 at 20:11
• Not my method and I'm not saying it's special. Just wondering why it works. – B.Vanjorek Dec 26 '16 at 20:13
• "which I can then use the grouping method" What is "the grouping method" and where did you learn it? – fleablood Dec 27 '16 at 19:44
You got lucky actually. It just so happens that in general, a problem of the form $ax^2-b$ can be factored via grouping if $a$ and $b$ are perfect squares, and it further happens that the value needed to perform the grouping process was given. In particular, doing this will apparently result in the correct grouping coefficients if $a=4$.
This is too long for a comment and I don't know what is the "grouping method" but note that the quadratic formula
$$x_{1,2} = \frac{b \pm \sqrt{b^2 - 4ac}}{2a}$$
gives you the roots $x_{1,2}$ of $ax^2 + bx + c = 0$. When you factor the expression, it is not enough to know the roots but you also need to know $a$. Thus, using the "first" method, you discovered that the roots are $\pm 3.5$ but this does not mean you can factor $4x^2 - 49$ as $(x - 3.5)(x - 3.5)$!
If we open the brackets, we see
$$(x - 3.5)(x + 3.5) = x^2 - (3.5)^2 = x^2 - \left( \frac{7}{2} \right)^2 = x^2 - \frac{49}{4}$$
so we don't get the same expression. But if we multiply by $a = 4$ both sides, we get the correct factorization
$$4x^2 - 49 = 4(x - 3.5)(x + 3.5) = 2(x-3.5)2(x+3.5) = (2x - 7)(2x + 7).$$
Back to completing the square:
\begin{align*} ax^2+bx+c &= a\left( x+\frac{b}{2a} \right)^2- \left( \frac{b^2-4ac}{4a} \right) \\ &= a\left[ \left( x+\frac{b}{2a} \right)^2- \left( \frac{b^2-4ac}{4a^2} \right) \right] \\ &= a\left(x+\frac{b+\sqrt{b^2-4ac}}{2a} \right) \left(x+\frac{b-\sqrt{b^2-4ac}}{2a} \right) \end{align*}
Perhaps this simplification is worth investigating? \begin{align} ax^2+bx+c&=\frac{1}{a}(a^2x^2+abx+ac)\\ &=\frac{1}{a}((ax)^2+b(ax)+ac)\\ &=\frac{1}{a}(t^2+bt+ac)\\ \end{align} Where $t = ax$
If $ax^2 + bx + c = 0;a \ne 0$ has solutions $x_1,x_2 = \frac{b \pm \sqrt{b^2 - 4ac}}{2a}$ and $v_1, v_2 = \frac{b \pm \sqrt{b^2 - 4ac}}{2} = ax_1, ax_2$
then $(x -x_1)(x - x_2)=x^2 + \frac bax + \frac ca = 0 \iff a(x- x_1)(x-x_2) = ax^2 + bx + c = 0 \iff (x - \frac {v_1}a)(x - \frac{v_2}a) =(x -x_1)(x - x_2)= 0 \iff (ax - v_1)(ax-v_2) = a^2(x - x_1)(x-x_2) = a^2x^2 + bax + ca = 0$
And $(\sqrt{a}x - \sqrt{a}x_1)(\sqrt{a}x - \sqrt{a}x_2) = (\sqrt{a}x - \frac{v_1}{\sqrt{a}})(\sqrt{a}x - \frac{v_2}{\sqrt{a}}) = a^2 + bx + c = 0 \iff etc.$.
If I knew what the "grouping method" was supposed to be (I've never heard of it) I'd be able to answer specifically but the above should be ... exhaustive and exhausting.
No you didn't get lucky what you found is to two methods for factoring quadratic polymomials that are a consequence of the $\color{red}{\text{the ac method}}$.
$\mathbf{THEOREM}$. Suppose that $a\ne 0,b$ and $c$ are relatively prime integers and $ax^2 + bx + c$ factors over the set of rational numbers. Then there exists integers $u$ and $v$ such that
• $x^2 + bx + ac = (x-u)(x-v)$
• $ax^2 + bx + c$ can be factored by grouping as $ax^2 + bx + c = (ax^2 - ux) + (-vx + c)$
• The roots of $ax^2 + bx + c$ are $\dfrac ua$ and $\dfrac va$.
• $ax^2 + bx + c$ can be factored by simplifying $\dfrac{(ax-u)}{\gcd(a,u)}\;\dfrac{(ax-v)}{\gcd(a,v)}$
• $\gcd(a,u)\gcd(a,v) = a$
$\mathbf{EXAMPLE}$. $ax^2 + bx + c = 10x^2 -3x - 18$.
$x^2 + bx + ac = x^2 - 3x - 180 = (x - 15)(x + 12) = (x-u)(x-v)$
\begin{align} ax^2 + bx + c &= (ax^2 - ux) + (-vx + c) \\ &= (10x^2 - 15x) + (12x - 18) \\ &= 5x(2x - 3) + 6(2x - 3) \\ &= (5x + 6)(2x - 3) \end{align}
\begin{align} ax^2 + bx + c &= \dfrac{(ax-u)}{\gcd(a,u)}\dfrac{(ax-v)}{\gcd(a,v)} \\ &= \dfrac{(10x-15)}{5} \dfrac{(10x+12)}{2}\\ &= (2x-3)(5x+6) \\ \end{align}
$\mathbf{PROOF}$.
We assume that $\gcd(a,b,c) = 1$ and that $ax^2 + bx + c$ has two real rational roots.
The roots of $x^2 + bx + ac$ are $u,v = \dfrac{-b \pm \sqrt{b^2-4ac}}{2}$. So the roots of $ax^2 + bx + c$ are $\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}=\dfrac ua, \dfrac va$.
The next part is ugly, but I couldn't think of any other way to present it.
We need to reduce the fractions $\dfrac ua$ and $\dfrac va$. Let $s = \gcd(a,u)$ and let $t = \gcd(a,v)$
Then $\dfrac ua = \dfrac{u'}{s'}$ and $\dfrac va = \dfrac{v'}{t'}$ where $u' = \dfrac us,\; s' = \dfrac as,\; v' = \dfrac vt,\; t' = \dfrac at$. Then we must have $$ax^2 + bx + c = a\left( x - \dfrac{u'}{s'} \right) \left( x - \dfrac{v'}{t'} \right) = ax^2 - a\left( \dfrac{u'}{s'} + \dfrac{v'}{t'}\right)x +\dfrac{au'v'}{s't'}.$$
Because $c = \dfrac{au'v'}{s't'}$ is an integer, then $s't'$ must be a divisor of $a$. Lets say $a=s't'a'$. So \begin{align} ax^2 + bx + c &= s't'a'x^2 - s't'a'\left( \dfrac{u'}{s'} + \dfrac{v'}{t'}\right)x +\dfrac{s't'a'u'v'}{s't'} \\ &= s't'a'x^2 - a'(t'u' +s'v')x +a'u'v'\\ \end{align}
Comparing coefficients, we must have \begin{align} a &= s't'a' \\ b &= -a'( t'u' + s'v') \\ c &= a'u'v' \end{align} It follows that $a' \mid \gcd(a,b,c) = 1$. So $a'=1$ Hence \begin{align} a &= s't' \\ b &= -t'u' - s'v' \\ c &= u'v' \\ u &= t'u' \\ v &= s'v' \\ \end{align}
As a consequence, we find that $s = \dfrac{a}{s'} = t'$ and $t = \dfrac{a}{t'} = s'$
We end up with \begin{align} s &= \gcd(a,u) \\ t &= \gcd(a,v) \\ a &= st \\ b &= -u - v \\ c &= u'v' \\ u &= su' \\ v &= tv' \\ \end{align}
If you replace $b$ with $-u-v$, then you find
\begin{align} ax^2 + bx + c &= ax^2 - (u+v)x + c\\ &= stx^2 - (su'+tv')x + u'v'\\ &= (stx^2 - su'x) - (tv'x - u'v') \\ &= sx(tx - u') - v'(tx - u') \\ &= (sx - v')(tx - u') \\ \end{align}
Which agrees with the roots being $\dfrac{u'}{t} = \dfrac ua$ and $\dfrac{v'}{s} = \dfrac va$.
We also note that $\dfrac{(ax-u)}{\gcd(a,u)}\;\dfrac{(ax-v)}{\gcd(a,v)} = \dfrac{(stx-su')}{s}\;\dfrac{(stx-tv')}{t} = (tx-u')(sx-v')$ | {"extraction_info": {"found_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": 8, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000100135803223, "perplexity": 507.2396659710722}, "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/1606141187753.32/warc/CC-MAIN-20201126084625-20201126114625-00072.warc.gz"} |
http://clay6.com/qa/11481/a-body-is-placed-on-rough-inclined-plane-of-inclination-theta-as-angle-thet | Browse Questions
# A body is placed on rough inclined plane of inclination $\theta$. As angle $\theta$ is increased from $0^{\circ}$ to $90^{\circ}$ the contact force between the block and plane
a) remains constant b) first remains constant then decreases c) first decreases then increases d) first increases then decreases
Till $\theta < \tan ^{-1} \mu$ contact force is constant and gravitational force are equal and opposite.
Once the body starts sliding
$F_c=\sqrt {(mg \cos \theta)^2+(\mu \;mg\; \cos \theta)^2}$
$\quad= mg \cos \theta \sqrt {1+\mu ^2}$
So as $\theta$ increases $\cos \theta$ decreases.
So contact force decreases.
Hence b is the correct answer.
edited Mar 21, 2014
+1 vote | {"extraction_info": {"found_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": 0.8692174553871155, "perplexity": 1617.7312899172612}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1480698543170.25/warc/CC-MAIN-20161202170903-00469-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/83315-proving-field-not-algebraically-closed.html | # Math Help - Proving a field is not algebraically closed
1. ## Proving a field is not algebraically closed
Hi,
My problem is this: Prove that if p is a prime, then the field Zp is not algebraically closed. I know that using Fermat's little theorem will help but I can't see how it's not closed. Can anybody help please?
Thanks
2. Using Fermat's little theorem we have x^(p-1)-1=0for any x doesn't equal zero,so the polynomial x^(p-1)-2(suppose p>2)doesn't have the zero points inZp
if p=2,we can find x^2+x+1 hasn't the zero points in Z2.
So field Zp is not algebraically closed.
Originally Posted by Zinners
Hi,
My problem is this: Prove that if p is a prime, then the field Zp is not algebraically closed. I know that using Fermat's little theorem will help but I can't see how it's not closed. Can anybody help please?
Thanks
3. Originally Posted by Zinners
Hi,
My problem is this: Prove that if p is a prime, then the field Zp is not algebraically closed. I know that using Fermat's little theorem will help but I can't see how it's not closed. Can anybody help please?
Thanks
In general let $F$ be a finite field with $q$ elements.
Define $f(x) = x^q - x +1$, we know that $a^{q-1}=1 \implies a^q - a = 0$ for all $a\in F^{\times}$.
Therefore, $f(x) = x^q - x + 1$ always has no zero. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8865850567817688, "perplexity": 325.89899488155027}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1419447563403.84/warc/CC-MAIN-20141224185923-00075-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/105330/equilateral-triangle-with-integer-coordinates/105387 | # equilateral triangle with integer coordinates
Is it possible to construct an equilateral triangle with coordinates on a grid of integers?
I think the answer is no, but how can I prove this?
I started with a triangle with coordinates (0,0) (a,b) and (c,d). Equating the size of the 3 sides, I get
$a^{2}+b^{2}=c^{2}+d^{2}=2ab+2cd$
How should I continue?
I see there are solutions based on the fact that the angle between two edges can not be 60°. Is it possible to have a solution based on the fact that the length of the edges can not be the same?
-
This deals with a general version of your question. In particular, since $\tan\frac{\pi}{3}$ isn't rational, you can't have lattice points as the corners of an equilateral triangle. – Ｊ. Ｍ. Feb 3 '12 at 15:02
possible duplicate of Which internal angles can a lattice polygon have? – joriki Feb 3 '12 at 15:09
You've accepted an answer which I believe is incorrect. Please see my comment under the answer, and unaccept it in case you agree, as the checkmark will otherwise mislead others. – joriki Feb 3 '12 at 15:23
$(a-c)^{2}+(b-d)^{2}=a^{2}+b^{2}+c^{2}+d^{2}-2ab-2cd=2a^{2}+2b^{2}-2ab-2cd$ this implies $a^{2}+b^{2}=2ab+2cd$ – wnvl Feb 3 '12 at 15:25
a) If you edit your post such that a correct earlier comment by someone else now appears wrong, please indicate this clearly (e.g. by adding "Edit" or the like). b) It's still wrong, since you've shown that $a^{2}+b^{2}=2ab+2cd$, whereas the post now says $a^{2}+b^{2}=2ac+2bd$. – joriki Feb 3 '12 at 15:30
Let the vertices of our triangle be $(0,0)$, $(a,b)$, and $(c,d)$, where $a$, $b$, $c$, and $d$ are integers. If all edge lengths are the same, then $$a^2+b^2=c^2+d^2=(a-c)^2+(b-d)^2.$$ Minor manipulation turns this into $$a^2+b^2=c^2+d^2=2ac+2bd.$$
Now we use my favourite identity, which was known more than a millenium ago in India, and even earlier by Diophantus, and so has often been called the Fermat Identity: $$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2.\qquad\qquad(\ast)$$ This identity can be easily verified by expanding both sides, or more conceptually by noting that the norm of the product of two complex numbers is the product of the norms.
Let $N=a^2+b^2=c^2+d^2=**2(ac+bd)**$. Then $ac+bd=N/2$. The identity $(\ast)$ now gives $$N^2=\frac{N^2}{4}+(ad-bc)^2$$ or equivalently $$3N^2=4(ad-bc)^2.$$ This is impossible, since $3$ times the perfect square $N^2$ cannot be a square unless $N=0$, which gives a very tiny triangle.
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A cute proof of the fermat identity: If $z = a+ib$ and $w = c + id$, then $(zz') (ww') = (zw)(zw)'$ where $x'$ is the conjugate of $x$. – Aryabhata Feb 3 '12 at 21:05
It is possible, but you need three dimensions in order to do it.
Consider $\bigtriangleup v_{1}v_{2}v_{3}$ with:
$v_{1}=(1,0,0)$
$v_{2}=(0,1,0)$
$v_{3}=(0,0,1)$
For $a,b\in{1,2,3}$, $a\neq b$, $d(v_{a},v_{b})=\sqrt{2}$, therefore the triangle is equilateral. It is not possible (as other answers indicate) to have an equilateral triangle with integer coordinates for the vertices in a two dimensional square lattice (a grid is just a 2d lattice).
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Solution 1 (by me): Assume WLOG that two of the points are $(0,0), (m,n), m,n \in \mathbb{Q}$. Then the third point is $(m/2 - n \sqrt{3} / 2, n/2 + m \sqrt{3}/2)$, which is not a rational point.
Solution 2 (by a friend): The determinant formula for area is rational, so if the all three points are rational points, then the area of the triangle is also rational, so whereas the area of an equilateral triangle with side length s is $\frac{s^2 \sqrt{3}}{4}$, which is irrational since $s^2$ is an integer.
Note that the above solutions both generalize from integer points to rational points.
You can also use Pick's theorem for integer points.
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You can start like this. Without loss of generality, let the three points be $(0,0), (0,a)$ and $(b,a/2)$. You can do this because you can always rotate and translate the axis to get these points. Now for all the points to be integral, you need $a$ to be even. This is the first constraint. Secondly, from the basic trigonometry,
$$\tan \theta = \frac{2b}{a} = \sqrt{3}.$$ From this, you get $b = a\sqrt{3}/2$, which is irrational.
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You can't isometrically map any three lattice points to points of this form. The lengths of all sides may be irrational, whereas the length of the side from $(0,0)$ to $(0,a)$ is always an integer. – joriki Feb 3 '12 at 15:17
Ohh yes! Thanks for pointing it out. – Jalaj Feb 3 '12 at 22:43 | {"extraction_info": {"found_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": 0.9300987720489502, "perplexity": 235.44737089834373}, "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-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00132-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://planetmath.org/fundamentaltheoremofspacecurves | # fundamental theorem of space curves
## Informal summary.
The curvature and torsion of a space curve are invariant with respect to Euclidean motions. Conversely, a given space curve is determined up to a Euclidean motion, by its curvature and torsion, expressed as functions of the arclength.
## Theorem.
Let $\boldsymbol{\gamma}:I\to\mathbb{R}$ be a regular, parameterized space curve, without points of inflection. Let $\kappa(t),\tau(t)$ be the corresponding curvature and torsion functions. Let $T:\mathbb{R}^{3}\to\mathbb{R}^{3}$ be a Euclidean isometry. The curvature and torsion of the transformed curve $T(\boldsymbol{\gamma}(t))$ are given by $\kappa(t)$ and $\tau(t)$, respectively.
Conversely, let $\kappa,\tau:I\to\mathbb{R}$ be continuous functions, defined on an interval $I\subset\mathbb{R}$, and suppose that $\kappa(t)$ never vanishes. Then, there exists an arclength parameterization $\boldsymbol{\gamma}:I\to\mathbb{R}$ of a regular, oriented space curve, without points of inflection, such that $\kappa(t)$ and $\tau(t)$ are the corresponding curvature and torsion functions. If $\hat{\boldsymbol{\gamma}}:I\to\mathbb{R}$ is another such space curve, then there exists a Euclidean isometry $T:\mathbb{R}^{3}\to\mathbb{R}^{3}$ such that $\hat{\boldsymbol{\gamma}}(t)=T(\boldsymbol{\gamma}(t)).$
Title fundamental theorem of space curves FundamentalTheoremOfSpaceCurves 2013-03-22 13:23:28 2013-03-22 13:23:28 rmilson (146) rmilson (146) 5 rmilson (146) Theorem msc 53A04 SpaceCurve | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 15, "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": 0.9913254976272583, "perplexity": 429.6305833521334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1568514570830.42/warc/CC-MAIN-20190915072355-20190915094355-00259.warc.gz"} |
https://support.xmatters.com/hc/en-us/articles/360044689852--The-system-cannot-find-the-specified-path-error-when-running-the-wspasswd-command | Submit a Request
# "The system cannot find the specified path" error when running the ./wspasswd command
## Question
While setting up an integration using the Integration Agent, we get an error at the point where you are required to run:
`c:\xmatters\integrationagent-5.3.0\bin>iapassword.bat --new "password" --file c:\xmatters\integrationagent-5.3.0\conf/.wspasswd`
The error says "The system cannot find the specified path".
## Environment
All versions of xMatters, Integration Agent 5.3.0
The error is being caused because of the lack of a suitable JRE on the system.
Note: you can either replace this `jre` directory or the contents.
You should then be able to successfully run the following command:
`C:\xmatters\integrationagent-5.3.0\bin>iapassword.bat --new "password" --file c:\xmatters\integrationagent-5.3.0\conf/.wspasswd` | {"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": 0.8554345965385437, "perplexity": 2085.7728775001597}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046153897.89/warc/CC-MAIN-20210729203133-20210729233133-00440.warc.gz"} |
http://meta.math.stackexchange.com/questions/8336/are-questions-of-the-form-has-this-ever-been-studied-appropriate | # Are questions of the form “has this ever been studied?” appropriate?
Are requests of the form "has this ever been studied, and if so, may I please have a reference?" considered appropriate?
But let me be more specific.
So firstly, there's the point-set or "classical" approach to topology, which concerns itself with ordered pairs $(X,\tau)$ called topological spaces.
Then there's the pointless approach to topology, which concerns itself with lattices $(\tau,\wedge,\vee)$ called frames (in which finite meets distribute over arbitrary joins.)
I'm interested in a concept halfway between the two. We might call it "the classical approach, but with lattices." Rather than $(X,\tau)$, we concern ourselves $(P,\tau),$ where $P$ is a lattice that is isomorphic to a powerset lattice, and $\tau$ is a subset of $P$ that is closed with respect to arbitrary joins etc.
The motivation for this idea is as follows: we may be able to weaken the requirement that $P$ needs to be isomorphic to a powerset, and still be able to develop classical topology just fine.
So my question is, has this idea been studied before, and if so, may I please have reference recommended?
-
One problem with this type of question is that they are semi-answerable: if the object has been studied, one can point to a reference, but there is no way to show that the subject has not been studied. It may have been studied, for example, and found to be useless so nothing was written as a result. – Mariano Suárez-Alvarez Jan 28 at 17:41
@Mariano This comes to mind... ;-) – Michael Greinecker Jan 28 at 20:07
@MichaelGreinecker, well, that is somewhat different. There can't be a standard notion of something for which one cannot give references! :-) – Mariano Suárez-Alvarez Jan 28 at 20:38
Ask away, but now, ask it at math.stackexchange, the main site! – amWhy Feb 1 at 18:05
In my opinion, this kind of question is fine. I ask fellow mathematicians this kind of thing all the time and, when the answer is positive, it's very useful.
Questioners should understand, though, that if the true answer is "no" then the question will probably never be answered.
- | {"extraction_info": {"found_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": 0.8329654932022095, "perplexity": 397.38631803749246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164022208/warc/CC-MAIN-20131204133342-00002-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://demo.sgmapps.com/phonex-container-vvxlkj/0aa19b-convex-hull-simplices | denoting the vertices, then the boundary If the convex hull lies in a flat (affine subspace) of dimension d', the output will comprise a list of d'-tuples, the vertices of the convex hull relative to that flat. Bei einer großen Anzahl von Punkten möchte ich herausfinden, ob die Punkte in der konvexen Hülle der Punktwolke liegen. (2) The Delaunay triangulation contains O(#n#^(#d#/2)) simplices. + ( The Delaunay triangulation contains O(n ⌈d / 2⌉) simplices. If P is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the n-parallelotope is the image of the unit n-hypercube by the linear isomorphism that sends the canonical basis of Simplicial complexes are used to define a certain kind of homology called simplicial homology. 1 A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. ! ) 1 Properties: (1) The union of all simplices in the triangulation is the convex hull of the points. 2 j 1 R 1 ) Ich habe eine Punktwolke von Koordinaten in numpy. where the {\displaystyle 1\leq i\leq n} {\displaystyle \partial \sigma } = ! ] -1 denotes no neighbor. p {\displaystyle (v_{0},\ v_{1},\ v_{2},\ldots v_{n})} , ; and the fact that the angle subtended through the center of the simplex by any two vertices is 1 {\displaystyle \,(p_{i})_{i}} ) ( ) or {3,3} and so on. Thus, if we denote one positively oriented affine simplex as, with the / ∙ Empty 2 and 3-simplices and hollow 2-polytope. The following assertions hold: If P is the unit n-hypercube, then the union of the n-simplexes formed by the convex hull of each n-path is P, and these simplexes are congruent and pairwise non-overlapping. When n is odd, the condition means that exactly one of the diagonal blocks is 1 × 1, equal to −1, and acts upon a non-zero entry of v; while the remaining diagonal blocks, say Q1, ..., Q(n − 1) / 2, are 2 × 2, there is an equality of sets, and each diagonal block acts upon a pair of entries of v which are not both zero. : Δ Face and facet can have different meanings when describing types of simplices in a simplicial complex; see simplical complex for more detail. n 1 Similar hyperplane equations for the Delaunay triangulation correspond to the convex hull facets on the corresponding N+1 dimensional paraboloid. 2 R Alternatively, the volume can be computed by an iterated integral, whose successive integrands are , {\displaystyle v_{n}} The Hasse diagram of the face lattice of an n-simplex is isomorphic to the graph of the (n + 1)-hypercube's edges, with the hypercube's vertices mapping to each of the n-simplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). ( simplices (ndarray of ints, shape (nfacet, ndim)) Indices of points forming the simplical facets of the convex hull. , ) : Abstract This paper deals with the following question concerning the volume In each of the following de nitions of d-simplices, d-cubes, and d-cross-polytopes we give both a V- and an H-presentation. = 1 X Simplices Definition 1. d ) {\displaystyle a_{i}} if joggle: return ConvexHull(qhull_data, qhull_options="QJ i").simplices else: return ConvexHull(qhull_data, qhull_options="Qt i").simplices """ if joggle: return ConvexHull(qhull_data, qhull_options="QJ i").simplices else: return ConvexHull(qhull_data, qhull_options="Qt i").simplices x {\displaystyle \mathbb {R} ^{n}} as can be seen by multiplying the previous formula by xn+1, to get the volume under the n-simplex as a function of its vertex distance x from the origin, differentiating with respect to x, at The data type is derived from Convex_hull_d via the lifting map. | Δ To carry this out, first observe that for any orthogonal matrix Q, there is a choice of basis in which Q is a block diagonal matrix, where each Qi is orthogonal and either 2 × 2 or 1 × 1. 0 The convex hull of a given set $${\displaystyle X}$$ may be defined as A set of points in a Euclidean space is defined to be convex if it contains the line segments connecting each pair of its points. + / v … The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form (α/n, ..., α/n) for some real number α. { method. In the study moduli spaces of spherical minimal immersions, in [22,23] the author intro-duced a sequence of measures of symmetry {σm}m≥1 associated to a convex body K ⊂ En (of dimension n) with a specified interior point O ∈ intK.Themth measure of symmetry σm is defined as follows. As the convex hull is unique, so is the triangulation, assuming all facets of the convex hull are simplices. ( In probability theory, the points of the standard n-simplex in (n + 1)-space form the space of possible probability distributions on a finite set consisting of n+1 possible outcomes. 1 2 , call a list of vertices . The convex hull of any nonempty subset of the n + 1 points that define an n-simplex is called a face of the simplex. Convex Hull. every simplex. The kth neighbor is opposite to the kth vertex. The kth neighbor is opposite to the kth vertex. 1 ) Solving this equation shows that there are two choices for the additional vertex: Either of these, together with the standard basis vectors, yields a regular n-simplex. . n n − with. For a point x in d-dimensional space let lift(x) be its lifting to the unit paraboloid of revolution. The contour of the obtained polygon is … ball. Throughout this article, simplices are n -simplices in R n exclusively, i.e., those polytopes formed by the convex hull of (n +1) affine independent points in R n (the This is the simplex used in the simplex method, which is based at the origin, and locally models a vertex on a polytope with n facets. ( Returns: List of simplices of the Convex Hull. """ It can be shown that the following is true: {\displaystyle \partial } There are several sets of equations that can be written down and used for this purpose. ( We call S the underlying point set and $$d$$ or dim the dimension of the underlying space. x -1 denotes no neighbor. {\displaystyle (n-1)} n : Um politopo convexo pode ser decomposto em um complexo simplicial, ou união de simplicial, satisfazendo certas propriedades. ! π [ use.random. n-paths and … of σ is the chain. for 2-D are guaranteed to be in counterclockwise order: (ndarray of double, shape (npoints, ndim)) Coordinates of input points. The simplex Δn lies in the affine hyperplane obtained by removing the restriction ti ≥ 0 in the above definition. i This correspondence is an affine homeomorphism. … plot (player50471. v + Convex Hulls, Convex Polyhedra, and Simplices Definition 6. i R 1 n Proposition 10.1. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. while the interior corresponds to the inequalities becoming strict (increasing sequences). n 1 1 2 x 3. Every n-simplex is an n-dimensional manifold with corners. n / ≤ 1 Raised when Qhull encounters an error condition, such as The n + 1 vertices of the standard n-simplex are the points ei ∈ Rn+1, where, There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v0, ..., vn) given by. The simplexes in a chain need not be unique; they may occur with multiplicity. , , {\displaystyle (0,{\frac {1}{n}},\dots ,{\frac {1}{n}})} It turns out that CH(v 0;:::;v k)= n w2Rn:9l 0;:::;l k 2R s.t. In Ziegler's Lectures on Polytopes (7th printing), on page 8, it is said that "the convex hull of any set of points that are in general position in $\mathbb{R}^d$ is a simplicial polytope", where "simplicial polytope" is defined slightly above as a "polytope, all of whose proper faces are simplices" (Ziegler uses "polytope" to mean "convex polytope"). Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … 0 void O {\displaystyle \mathbf {R} ^{n}} ≤ (where the n-simplex side length is 1), and normalizing by the length e log n n Throughout this article, simplices are n-simplices in Rn exclusively, i.e., those polytopes formed by the convex hull of (n +1) affine independent points in Rn (the vertices). , from which the dihedral angles are calculated. 0 [9] Projecting onto the simplex is computationally similar to projecting onto the n Chapter Ten - Convex Sets, Simplices, and All That Definition. Convex hull facets also define a hyperplane equation: Wie kann man effizient herausfinden, ob ein Punkt in der konvexen Hülle einer Punktwolke liegt? 1 Δ ] n If TRUE and the input is of class Hypervolume, sets boundaries based on the @RandomPoints slot; otherwise uses @Data. of / elements of the symmetric group divides the n-cube into More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map {\displaystyle \mathbf {R} ^{n}} } it is the formula for the volume of an n-parallelotope. The output tuples represent the facets of the convex hull of the input set. ∂ A convex polytope can be decomposed into a simplicial complex, or union of simplices, satisfying certain properties. 1 v R The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the map operation (by definition of a map). Suppose that P ˆRn is the union of finitely many simplices T (not necessarily of the same dimension). Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. to and ). assemble into one cosimplicial object , between the origin and the simplex in Rn+1) is, The volume of a regular n-simplex with unit side length is. d + In particular, an empty d-simplex is the convex hull of d+1affinely independent integer points and not containing other integer points. , Suppose that v 0;:::;v k 2Rn. [10] A more symmetric way to write it is, | x The convex hull of fv 0;:::;v kg is the smallest convex set containing v 0;:::;v k. It is denotedCH(v 0;:::;v k). for details. + . A different rescaling produces a simplex that is inscribed in a unit hypersphere. … , (ndarray of ints, shape (nfacet, ndim)) Indices of neighbor facets for each facet. We could also have directly used the vertices of the hull, which , For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X. , 1 {\displaystyle e_{1},\ldots ,e_{n}} 0 here correspond to successive coordinates being equal, The union of all simplices in the triangulation is the convex hull of the points. ) , [ … ∂ As previously, this implies that the volume of a simplex coming from a n-path is: Conversely, given an n-simplex , (ndarray of ints, shape (nvertices,)) Indices of points forming the vertices of the convex hull. (so there are n! CGAL::Convex_hull_d Definition. ) i 1 (ndarray of double, shape (nfacet, ndim+1)) [normal, offset] forming the hyperplane equation of the facet (see, (ndarray of int, shape (ncoplanar, 3)) Indices of coplanar points and the corresponding indices of the nearest facets and nearest vertex indices. {\displaystyle 1\leq i\leq n} with possibly negative entries, the closest point More generally, there is a canonical map from the standard where ( 1. n complexity via median-finding algorithms. n ) A vector subspace of Rn is a subset which is closed under (finite) linear combinations. R 1 i {\displaystyle 1/n!} v ( R n / Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen vertices, forms a regular simplex. R simplices (ndarray of ints, shape (nfacet, ndim)) Indices of points forming the simplical facets of the convex hull. {\displaystyle p_{i}} … ≤ Coplanar points are input points which were. . 2 Denote the basis vectors of Rn by e1 through en. n ( To create a convex hull, we need to build it from a list of coordinates. In general, the number of m-faces is equal to the binomial coefficient $${\displaystyle {\tbinom {n+1}{m+1}}}$$. e on the simplex has coordinates, where ( {\displaystyle A_{1}\ldots A_{n}} n ( + Gemeinschaften (8) Booking - 10% Rabatt python numpy convex-hull. f ⋯ v , x {\displaystyle \arccos(-1/n)} ) The 0-faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1-faces are called the edges, the (n − 1)-faces are called the facets, and the sole n-face is the whole n-simplex itself. {\displaystyle O(n\log n)} 1 , y, 'o') #Loop through each of the hull's simplices for simplex in hull. n of the increment, to a topological space X is frequently referred to as a singular n-simplex. n n 1 i . [12] In particular, the volume of such a simplex is. Thenthe trianglewith the vertices a, b and c can be introducedas the set abc ={αa+βb+γc: α,β,γ∈[0,1], α+β+γ=1}. ( ( (ndarray of ints, shape (nfacet, ndim)) Indices of points forming the simplical facets of the convex hull. the ring of regular functions on the algebraic n-simplex (for any ring n 1 are the integers denoting orientation and multiplicity. Pastebin is a website where you can store text online for a set period of time. ∂ This simplex is inscribed in a hypersphere of radius That is, the kth vertex of the simplex is assigned too have the kth probability of the (n+1)-tuple as its barycentric coefficient. , , © Copyright 2008-2009, The Scipy community. x {\displaystyle (0,{\frac {1}{n}},\dots ,{\frac {1}{n}})} 3 n ( does not depend on the permutation). Δ det : 1 {\displaystyle \Delta } These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is − {\displaystyle {\sqrt {2(n+1)/n}}} simplices: #Draw a black line between each plt. Additional options to pass to Qhull. … , ( e {\displaystyle \mathbf {R} ^{n}} -simplex is the softmax function, or normalized exponential function; this generalizes the standard logistic function. − 2 ρ So r t the points according to increasing x-coordinate. Example. , Given a permutation … , v 1 + n 0 , }\det {\begin{pmatrix}v_{0}&v_{1}&\cdots &v_{n}\\1&1&\cdots &1\end{pmatrix}}\right|}, Another common way of computing the volume of the simplex is via the Cayley–Menger determinant. v For example, when n = 4, one possible matrix is, Applying this to the vector (1, 0, 1, 0) results in the simplex whose vertices are, each of which has distance √5 from the others. M n This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes. . = {\displaystyle \ell _{1}} , one has: where ρ is a chain. , In mathematics, the convex hull or convex envelope or convex closure of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. / n {\displaystyle ({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}})} ℓ An instance C of type Convex_hull_d is the convex hull of a multi-set S of points in d-dimensional space.We call S the underlying point set and d or dim the dimension of the underlying space. A continuous map It is also possible to directly write down a particular regular n-simplex in Rn which can then be translated, rotated, and scaled as desired. of v , along the normal vector. n {\displaystyle R[\Delta ^{n}]} It is the smallest convex set that contains X. − Pastebin.com is the number one paste tool since 2002. v Finally, the formula at the beginning of this section is obtained by observing that, From this formula, it follows immediately that the volume under a standard n-simplex (i.e. v + {\displaystyle \Delta _{n}(R)=\operatorname {Spec} (R[\Delta ^{n}])} {\displaystyle (v_{0},e_{1},\ldots ,e_{n})} The running time is O(n 2) in the worst case and O(nlog n) for most inputs. )[15], Since classical algebraic geometry allows to talk about polynomial equations, but not inequalities, the algebraic standard n-simplex is commonly defined as the subset of affine (n + 1)-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). σ 0 ) That is. From this one can see that the H- ... convex hull of d+1 a nely independent points as a d-simplex, since any two such polytopes are equivalent with … {\displaystyle \mathbf {R} ^{n}} This results in the simplex whose vertices are: for One proof is to inductively build a triangulation of P. If P is the convex hull of vertices { v 1, …, v n } and P k is the convex hull of { v 1, …, v k } such that a triangulation of P k is given, construct a triangulation of P k + 1 by taking the simplices formed by v k + 1 and the faces of P k that are "visible" from v k + 1. One way to write down a regular n-simplex in Rn is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. 0 n Convex Hull A convex hull is the smallest polygon that covers all of the given points. {\displaystyle R[\Delta ^{\bullet }]} {\displaystyle v_{0},\ v_{1},\ldots ,v_{n}} 1 { v R ) Since all simplices are self-dual, they can form a series of compounds; In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. / R The dimension of the convex hull of V is the dimension of the affine space of V. Simplex. (in the category of schemes resp. + 1 i CONVEX_HULL takes as argument a list of points and returns the (planar embedded) surface graph H of the convex hull of L. The algorithm is based on an incremental space sweep. If TRUE, prints diagnostic progress messages. 2 The input is a list of points, and the output is a list of facets of the convex hull of the points, each facet presented as a list of its vertices. Option “Qt” is always enabled. a For 2-D convex hulls, the vertices are in counterclockwise order. It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero: Likewise, the boundary of the boundary of a chain is zero: . (3) Thus the triangle abc is the convex hull of the vertices set {a,b,c}. i This is an n × n orthogonal matrix Q such that Qn + 1 = I is the identity matrix, but no lower power of Q is. PDF | On Jan 1, 2008, Á. G. Horváth published Maximal convex hull of connecting simplices. Suppose S is a subset of a real linear space. Spec . {\displaystyle f\colon \mathbb {R} ^{n}\rightarrow M} By adding an additional vertex, these become a face of a regular n-simplex. 1 {\displaystyle n!} Considering the parallelotope constructed from , which is the facet opposite the orthogonal corner. = If some of the simplexes occur in the set more than once, these are prefixed with an integer count. A convex body in Rn is a compact convex set with non-empty interior. … I would like to generate a convexhull (from the scipy package) and convert it to a mesh (for a viewer library). Unit hypersphere by orthogonal matrices [ 12 ] in particular, the vertices are: for 1 ≤ i n. D-Simplices, d-cubes, and the input is of interest you can store text online for a point x d-dimensional! 2 from the others highly symmetric way to write it is, 1... Used for this purpose its vertices are: for 1 ≤ i ≤ {. In Rn+1 ) is, | 1 n 12 ] in particular, the vertices of the polygon. Of higher Chow groups d+1affinely independent integer points a_ { i } } does not depend on the corresponding dimensional. Is equivalent to an n-ball all that definition p i { \displaystyle }! Vertices of the convex hull of the triangle notation volume of a real linear space matrix... ( ).These examples are extracted from open source projects toutes les faces de l'enveloppe convexe est unique, triangulation. Both the summation convention for denoting the set, and the boundary operator {! By orthogonal matrices not call the add_points method from a __del__ destructor divide the problem of convex! Convex sets, simplices, and the boundary operator ∂ { \displaystyle 1\leq i\leq }... That can be given unit side length hulls, convex Polyhedra, and simplices definition 6 need ResearchGate. ( disjoint except for boundaries ), showing that this simplex has volume 1 / n v 0 ;:! ) is the smallest polygon that covers all of the same dimension ) ≥. K 2Rn __del__ destructor not a simplex a is a compact convex set of... Hull into finding the upper convex hull of the vertices set { a, B and c be non-collinearpointsin plane... Ensure that each newly chosen vertex, together with convex hull simplices previously chosen vertices forms. Underlying point set and \ ( d\ ) or dim the dimension of S.The data type is derived Convex_hull_d! _ { i } \max\ { p_ { i } } does not depend on the permutation ) Find... Punktwolke liegen shape ( nvertices, ) ) Indices of points which has distance 2 from the.. Otherwise uses @ data this is done, its vertices called the barycentric coordinates of a regular simplex n! To the kth vertex, c } kann man effizient herausfinden, ob ein Punkt der! S of points forming the simplical facets of the points according to increasing.... Further memory allocation not a simplex that is inscribed in a combinatorial fashion ) simplices become a face a! 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Input is of class Hypervolume, sets boundaries based on the corresponding N+1 dimensional paraboloid of time (! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.888795018196106, "perplexity": 1134.513180207815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243990449.41/warc/CC-MAIN-20210514091252-20210514121252-00119.warc.gz"} |
https://mathoverflow.net/questions/410001/explicit-estimates-for-nt-chi-not-nt-chint-overline-chi | # Explicit estimates for $N(T,\chi)$ (not $N(T,\chi)+N(T,\overline{\chi})$)
Let $$N(T,\chi)$$ denote the number of zeros of $$L(s,\chi)$$ with imaginary part between $$0$$ and $$T$$, with any zero with imaginary part equal to $$T$$ or to $$0$$ (not that the latter kind really exists) counting as half a zero. Here I am following the convention in Montgomery-Vaughan, rather than that in part of the literature, where $$N(T,\chi)$$ means what I would call $$N(T,\chi) + N(T,\overline{\chi})$$.
The explicit literature generally (McCurley, Trudgian, Bennett-Martin-O'Bryant-Rechnitzer...) generally bounds $$N(T,\chi) + N(T,\overline{\chi})$$. The question is: what kind of explicit bounds we can extract from their proofs for $$N(T,\chi)$$?
The first step is easy: we can express $$N(T,\chi)$$ as $$\text{main term} + S(T,\chi)-S(0,\chi)$$, as in Montgomery-Vaughan, Thm. 14.5, where $$S(T,\chi) = \frac{1}{\pi} \arg L(1/2+iT,\chi)$$. One would then decompose $$S(T,\chi)-S(0,\chi) = \frac{1}{\pi} \left(\arg L(\sigma+i T,\chi)|_{\sigma=\sigma_0}^{1/2} + \arg L(\sigma_0+i t,\chi)|_{t=0}^T + \arg L(\sigma,\chi)|_{\sigma=1/2}^{\sigma_0}\right)$$ for some $$\sigma_0>1$$ of our choice. The literature gives the bound $$2 \log \zeta(\sigma_0)$$ on $$\left|\arg L(\sigma_0+it)|_{t=-T}^T\right|$$. The reason is a mystery to me -- it is obvious that $$2 \sum_p \arcsin p^{-\sigma}$$ is a tighter upper bound on $$\left|\arg L(\sigma_0+it)|_{t=-T}^T\right|$$ (and it is easy to compute). I do not know how to do better than $$2 \sum_p \arcsin p^{-\sigma}$$ as an upper bound on $$\left|\arg L(\sigma_0+it)|_{t=0}^T\right|$$, and suspect one cannot, in general, as $$t$$ and $$\chi$$ could conspire.
The bulk of the explicit literature deals with bounding $$\arg L(\sigma+i T,\chi)|_{\sigma=\sigma_0}^{1/2}$$. Is there a better bound on $$\arg L(\sigma,\chi)|_{\sigma=1/2}^{\sigma_0}$$ than what one would get just by setting $$T=0$$?
• Important self-correction: I should have said $\arcsin$, not $\arctan$. You still get a tighter upper bound. Dec 6, 2021 at 7:48
• And yes, by the linear independence of $\pi$ and $\log 2, \log 3,\dotsc,\log p$ over $\mathbb{Q}$, $t$ and $\chi$ can conspire, and so the bound is tight, for $t$ and $\chi$ unbounded. (As @juan points out below - use Kronecker's theorem.) Dec 6, 2021 at 7:50
For the first question about $$2\log\zeta(\sigma_0)|$$, I think the reasoning is this:
For $$\sigma>2$$ we have $$|L(s,\chi)-1|\le \sum_{n=2}^\infty\frac{1}{n^\sigma}=\zeta(\sigma)-1<1.$$ Hence $$\log L(s,\chi)$$ can be defined by the power series $$\log L(s,\chi)=-\sum_{k=1}^\infty \frac{1}{k}(1-L(s,\chi))^k$$ In particular $$|\Im \log L(2+it,\chi)|<\pi/2$$ and coincide with $$\arg(L(2+it,\chi))$$. Also this is equal to $$\log L(2+it)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\log n}\frac{\chi(n)}{n^{2+it}}$$ It follows that $$|\arg L(2+it)|\le |\log L(2+it)|\le \sum_{n=2}^\infty \frac{\Lambda(n)}{\log n}\frac{1}{n^{2}}=\log\zeta(2).$$ Now it follows that $$|\arg L(2+it,\chi)-\arg L(2,\chi)|\le 2\log\zeta(2).$$
• Oh, I know how to derive that - I was just pointing out that (a) one can easily do better, (b) I don't know how to get rid of the factor of $2$ (and one most likely can't). Dec 4, 2021 at 17:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 36, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9740306735038757, "perplexity": 163.496341285755}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500074.73/warc/CC-MAIN-20230203185547-20230203215547-00277.warc.gz"} |
https://www.physicsforums.com/threads/probability-of-independent-events.800900/ | # Probability of Independent Events
1. Mar 2, 2015
### _N3WTON_
1. The problem statement, all variables and given/known data
Consider three events A1, A2, and A3, and let pi = P(Ai), for i = 1, 2, 3.
a) Express the probability that at least one of these three events occurs in terms of the pi ’s.
b) Express the probability that at least two of the events occur.
c) Suppose that A1, A2 and A3 are independent events. Verify that P(A1|A2 ∩ A3) = P(A1).
2. Relevant equations
3. The attempt at a solution
For part c I had no trouble obtaining a solution:
P(A1|A2 ∩ A3) = P[A1 ∩ (A2 ∩ A3)] P(A2 ∩ A3)
By independence, we have P[A1 ∩ (A2 ∩ A3)] = P(A1)P(A2)P(A3) and P(A2 ∩ A3) = P(A2)P(A3), and the result follows. However, I am having some trouble with the first two. The answer given by the instructor for a is:
1 − (1 − p1)(1 − p2)(1 − p3)
and for b is:
p1p2(1 − p3) + p1(1 − p2)p3 + (1 − p1)p2p3 + p1p2p3. I am not trying to dispute these answers, I am just having trouble understanding where they come from. Specifically, I do not understand the expression (1-p1), I assume that this is the probability that either p2 or p3 occur, but I'm not sure why. I was hoping somebody could give me some sort of explanation. Thanks.
2. Mar 2, 2015
### Orodruin
Staff Emeritus
1-pi is the probability of Ai not occuring.
3. Mar 2, 2015
### _N3WTON_
Ok, thanks...so then the expression p1p2(1-p3) would be the probability that either p1 or p2 occur?
4. Mar 2, 2015
### Orodruin
Staff Emeritus
It is the probability that A1 and A2 occur and that A3 does not - assuming the events are independent.
5. Mar 2, 2015
### _N3WTON_
ok, thank you
6. Mar 2, 2015
### Ray Vickson
The way the problem is written, independence does not enter until part (c), so (presumably), A1, A2, A3 need not be independent in parts (a) and (b). In that case you do not have enough information to do parts (a) and (b); you would need to know also the probabilities $P(A_1 \cap A_2)$, $P(A_1 \cap A_3$, $P(A_2 \cap A_3$ and $P(A_1 \cap A_2 \cap A_3)$. If you did know these, the inclusion-exclusion principle would allow you to compute P{at least one }. A less well-known extension of the inclusion-exclusion principle would allow for calculation of P{exactly 1 occurs}, P{exactly 2 occur}, P{at least 2 occur}, etc. See, eg.,
http://www.tricki.org/article/To_co...nts_occur_use_generalized_inclusion-exclusion
If the events ARE independent, the inclusion-exclusion probabilities simplify a lot, and you can obtain explicit results. Below, let $p_{i} = P(A_i)$, $p_{ij} = P(A_i \cap A_j)$ and $p_{123} = P(A_1 \cap A_2 \cap A_3)$ (with no independence assumptions).
The instructor's answer for (a) comes from the fact that
$$P(\text{at least one occurs}) = P(A_1 \cup A_2 \cup A_3)\\ = p_1 + p_2 + p_2 - p_{12} - p_{13} - p_{23} + p_{123} .$$
Assuming indepedence, we can go further, as we would then have
$$P(\text{at least one occurs}) = p_1 + p_2 + p_3 - p_1 p_2 - p_1 p_3 - p_2 p_3 + p_1 p_2 p_3 \\ = 1 - (1-p_1)(1 - p_2)(1 - p_3)$$
BTW: even "pairwise independence" is not enough: it is possible to have $p_{12} = p_1 p_2$, $p_{13} = p_1 p_3$ and $p_{23} = p_2 p_3$, but $p_{123} \neq p_1 p_2 p_3$.
7. Mar 2, 2015
### _N3WTON_
Just to be clear, you simplification assuming independence comes from the fact that:
$P(A_{i1} \cap A_{i2} \cap ... \cap A_{ik}) = P(A_{i1}) * P(A_{i2}) *...*P(A_{ik})$ ?
8. Mar 2, 2015
Yes.
9. Mar 2, 2015
### _N3WTON_
Ok fantastic, thank you for the explanation!
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https://logan-thomas.com/2020/12/19/something-about-gamma/ | In this level-1 post, we take a closer look at the gamma function, a generalization of the factorial, and see how it makes sense even when it blows up.
###### Introduction
For a blog called Singular Points, I have gone quite a while without mentioning a singularity (unless you count a black hole). We are here now to change that! Recently, I came across a fact about a special function while working on a physics project. For an integer $n \geq 0$,
$\displaystyle\lim_{x\rightarrow-n}\left[-\frac{\Gamma^2(x)}{\Gamma'(x)}\right]=\frac{(-1)^n}{n!}$
This fact caught me a little bit by surprise, in that it reminded me of an idea that I had years ago that didn’t work… until now. It involves the Gamma function ($\Gamma$), which is a generalization of the factorial function, so it isn’t surprising to see a factorial show up, but this particular one struck me. In this post, we will try to reproduce the discovery that made me realize that this must be true, but reorganized in a way that hopefully makes more sense than how I originally came to it. We’ll start by talking about the factorial and why it’s important. Then, we will introduce the Gamma function and discuss why it is so important. Finally, we will talk about how to extract this special fact about this special function and why it just had to be this way.
###### Factorials
The factorial is a simple operation, defined on the non-negative integers. It is the number of ways or arranging a number of objects in a line. For example, if we had 3 frisbees, and we wanted to rank how cool they looked, there would be $3!=6$ different rankings. This is because we start with 3 choices for first place, then for each choice of first place, we can pick one of two frisbees for second place (no frisbee could rank first and second, that would be ludicrous). And then finally, whichever frisbee is left gets third place. That makes $3\times2\times1=6$ different rankings. In general, we can define the factorial of $n$ as being the product of all the whole numbers up to $n$.
The observant among you would have noticed that I said the factorial is defined for non-negative numbers, which includes 0. That doesn’t quite fit into the definition about multiplying whole numbers. Let’s think about ordering objects then. How many ways are there to order 0 objects? Well, the only thing to do is nothing, and many mathematicians would say that means there is only one way to order 0 things: by not doing anything. This doesn’t fly with everyone though, some people don’t want to count not doing anything as an ordering (though there are good mathematical reasons why one would want to say that). Instead, I will convince these naysayers with the ultimate method: recursion.
Recursion is a way of defining a sequence from a starting value(s). There is a special algorithm to go from the starting value to the next and so on. In the case of the factorial, we can reason things out like so. $n!$ is the product of the first $n$ whole numbers, which just so happens to contain the product of the first $n-1$ whole numbers. In our frisbee example, $3! = 3\times2\times1$ contains the product of the first 2 whole numbers, which is also called $2!$. In general, we can write
$n! = n\times(n-1)!\text{ , with }1!=1$.
This definition is equivalent to the previous definition for all whole numbers, but it also gives an answer for other numbers too, namely 0. By using the recursion relation backwards,
$(n-1)! = \frac{n!}{n}$,
we can see that for $n=1$, this gives $0! = 1$. That settles it!
We could try to go even further back and define the factorial of a negative number, but you run into trouble pretty quickly.
$(-1)! = \frac{0!}{0} = \frac{1}{0}$
We could just throw our hands up and say all is lost, but let’s try something else. If we just keep going, we can spot a pattern (even though it isn’t technically defined).
$(-2)! = \frac{(-1)!}{-1} = \frac{1}{0}\times\frac{-1}{1}$
$(-3)! = \frac{(-2)!}{-2} = \frac{1}{0}\times\frac{1}{2!}$
$(-4)! = \frac{(-3)!}{-3} = \frac{1}{0}\times\frac{-1}{3!}$
We can see the pattern forming, and indeed if we continue this out to some negative integer $-n$, then we will see
$(-n)! = \frac{1}{0}\times\frac{(-1)^{n-1}}{(n-1)!}$.
So the factorial of a negative integer is not defined, but there is this hidden structure behind it, which we will come back to later.
###### The Gamma Function
All the way back in the 1720’s, mathematicians were interested to know whether there was a meaningful way to extend the factorial to include non-integers. The problem was solved by Leonhard Euler, and his solution was the gamma function, $\Gamma(x)$. The gamma function matches the factorial everywhere that it is defined, but actually exists for all complex numbers, except for non-positive integers (where the recurrence relation failed before). Unfortunately the gamma function can be a little confusing, because $\Gamma(n+1)=n!$. So it isn’t exactly the factorial, it is shifted over by one. Here is a graph of the gamma function:
The gamma function has familiar properties that it inherits from the factorial, like the recurrence relation, $\Gamma(n+1) = n\Gamma(n)$, but it satisfies this property for non-integer $n$ as well. It isn’t the only way to extend the factorial, though it is the extension that appears the most in all the mathematics I have ever seen. There is a notable extension that doesn’t blow up anywhere, which is called Hadamard’s gamma function, $H(x)$. It can be defined in terms of Euler’s gamma function as
$H(x) = \frac{1}{\Gamma(1-x)}\cdot\frac{d}{dx}\ln\left[\frac{\Gamma\left(\frac{1}{2}-\frac{x}{2}\right)}{\Gamma\left(1-\frac{x}{2}\right)}\right]$.
Hadamard’s gamma function is well-defined for all negative numbers (and in fact all complex numbers). Here is a graph of Hadamard’s gamma function:
There are other extensions of the factorial (infinitely many, with varying properties). The Hadamard’s gamma function and some others are explained here, for example.
###### Pulling Out Singularities
We want to extract the singular bit of the Gamma function and see what is left over. There are plenty of different ways of discarding singularities, so we would like to pick one that suits our intuition for what is happening. From the recursion relation that defines the factorial, we can see the structure of what we are dealing with: it looks like a singular bit multiplying a regular bit.
In calculus, we have this notion that a differentiable function looks like a straight line if you look close enough. Similarly, and without going into too much detail, the singularities of the Gamma function ‘look like’ the singularity of the function $1/x$. Using our intuition for how the gamma function behaves near a singularity, we can think of this as being like $f(x)/x$ near $x=0$. Our goal is to get rid of the factor of $1/x$ without disturbing the underlying function $f(x)$. The easy thing may be to just multiply everything by $x$, which will work. However, if we do it another way, then we can make the connection with the equation we started with at the beginning of this post.
Notice that the function $F(x)=1/x$ has a particular relationship with its derivative, $F'(x)=-1/x^2$:
$-\frac{F(x)^2}{F'(x)} = 1$.
This relationship allows us to ‘cancel out’ the singularity of the function using its derivative. This trick doesn’t just work with $1/x$, but you can show that it also holds for any inverse power $G(x)=1/x^p$, with some minor changes:
$(-p)^{p}\,\frac{G(x)^{p+1}}{G'(x)^{p}} = 1$.
For $p=1$, this matches the previous equation, and already this should remind us of the equation we started with. But that equation isn’t just an inverse power, it looks more like an inverse power times a function. So what happens to something like that under this operation? Let’s take $F(x) = f(x)/x$ and put it through the grinder. We find that
$-\frac{F(x)^2}{F'(x)} = \frac{f(x)}{1-xf'(x)/f(x)}$.
It’s close, but not quite $f(x)$. However, we don’t really care about the function, but rather it’s limit as you approach the singularity, i.e. $x\rightarrow0$. In this limit, the denominator goes to 1, or at least I couldn’t find a counterexample to that fact (and really we only need this to hold for the Gamma function, which it does). So the whole expression goes to $f(0)$.
###### The Gamma Function Knows
Let’s consolidate the discussion so far. First, we had the factorial recurrence that, when iterated backwards, immediately ran into problems, but formally gave us
$(-k)! = \frac{1}{0}\times\frac{(-1)^{k-1}}{(k-1)!}$.
Then, we introduced the Gamma function, which also obeys the (shifted) recurrence relation, and therefore has Singular Points corresponding to these negative factorials. Finally, we showed that there is a simple operation involving derivatives that strips the $x^{-p}$ behavior from a function and leaves whatever is left. If we combine these facts with the knowledge that $\Gamma$ behaves like $1/x$ at Singular Points, then we should see
$\displaystyle\lim_{x\rightarrow-n}\left[-\frac{\Gamma^2(x)}{\Gamma'(x)}\right]=\frac{(-1)^n}{n!}$,
which is indeed the case as we saw at the beginning of the post! This has a special name in complex analysis: the residue, though this is not the typical way of computing it. If you have ever seen this method of computing the residue before, let me know! | {"extraction_info": {"found_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": 51, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.931340754032135, "perplexity": 190.95199144705157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296943471.24/warc/CC-MAIN-20230320083513-20230320113513-00443.warc.gz"} |
http://tex.stackexchange.com/questions/2811/is-there-any-reason-to-compile-to-dvi-rather-than-pdf-these-days?answertab=active | # Is there any reason to compile to DVI rather than PDF these days?
I appreciate that in the past, `latex` was faster than `pdflatex`, but computer speeds being what they are nowadays, I can't see any difference in how quickly documents compile...
So, given that the end product I want is the PDF, are there good reasons not to always compile to pdf?
-
Closely related question: tex.stackexchange.com/questions/349/… – provides many good reasons for using pdflatex instead of latex (and one good reason for using latex instead of pdflatex). – Jukka Suomela Sep 6 '10 at 21:25
Unless you are a heavy PSTricks user, I think the answer is no, you can always go to pdf directly. Actually, in modern TeX distributions, latex is pdflatex in dvi mode; the underlying engine is the same.
-
as a side note: and tikz/pgf which works well with pdflatex and is a kind of substitute for pstricks, thus there is probably no reason to use pstricks today – maxschlepzig Sep 6 '10 at 20:52
@Khaled: There are two overlapping, but basically different layout implementations in Pdftex. Pdftex came from the work Hàn Thế Thành did towards his Phd thesis on microptypography, and it isn't a conservative change to Tex's layout: microtypography only works with the PDF layout engine, not the DVI engine. Pdftex keeps track of state associated with Pdf generation, whilst the DVI engine just injects raw Postscript code into PS specials. They're quite different. – Charles Stewart Sep 7 '10 at 8:20
@Khaled: I'm less sure of what I've been saying... Some details though: the microtype package says that font expansion, not surprisingly, doesn't work with the DVI engine/pathway. There's stuff to do with page geometry that Pdftex has to keep track of to generate PDFs which, to be conservative over Tex, it should not bother with in DVI mode. – Charles Stewart Sep 7 '10 at 12:41
@Khaled: Be very careful with your conclusion that, since `latex` and `pdflatex` are symlinks to the same command, they must be the same. This is simply not true: a program can check how it was invoked (i.e. via which symlink/command name) and act accordingly. Many applications do that, and so does `pdftex` – clearly at least for the purpose of loading the LaTeX macro packages when invoked via `[pdf]latex`. (But you could still be right.) – Konrad Rudolph Sep 8 '10 at 8:43
xelatex compiles to PDF and can handle PSTricks (and eps,png,jpg,...). – Dean Serenevy Jun 2 '11 at 13:01
There is another reason: PDF files produced with latex and dvipdfmx are much smaller than those produced with pdftex. The reason is that dvipdfmx embeds fonts as CFF (Compact Font Format). For short texts the difference is big. You can achive the same (and more, such as image compression) by using pdfsizeopt. I always use dvipdfmx instead of pdflatex. A short text that has 73 KB with pdflatex has only 9 KB with latex and dvipdfmx.
-
Some of the journals I use require figures to be submitted separately in eps or tiff format, so that they can deal with not just the latex work, but also the archiving of the figures in higher resolution for the journal website. I don't know why they prefer eps and tiff to pdf, but they do, and they are "the boss".
-
Some humanities journals don't even accept submissions in LaTeX, so consider yourself lucky that the journals you use are savvy enough to even think of specifying what image formats to use... – Seamus Sep 8 '10 at 12:34
One reason I can think of: xdvi is supposed to support reverse search (a.k.a. inverse search). I.e. you click in xdvi on some text and directly jump to the corresponding location in the text-editor.
I never tried it, but it sounds neat.
It seems that some xdvi clones support that feature for dvi files, too.
I guess that current PDF viewer/pdflatex combinations does not support reverse search ...
-
See the question tex.stackexchange.com/questions/2006/… . SumatraPDF is said to support it. – Roman Plášil Sep 7 '10 at 20:03
TeXWorks, Evince and Okular also support reverse search, if you are using a sufficiently recent engine with synctex support (all the popular three engine have synctex support for a while now). – Khaled Hosny Sep 8 '10 at 2:08
@Khaled: Cool, do you have some nice link in your back pocket that show how to use synctex and reverse search? – Johan Sep 8 '10 at 5:31
@Johan: You need to activate synctex for your document by either passing `--synctex=1` at command line or the primitive `\synctex=1` in your document, how to reverse search now depends on your viewer/editor (in TeXWorks you press ctrl while clicking the target either in source or pdf). – Khaled Hosny Sep 8 '10 at 9:54
I must try that :) – Johan Sep 8 '10 at 15:17
There is a reason: `pdflatex`'s lack of support for eps figures. If you have a program that only outputs eps graphics and wants to include them in your document, the best alternative is to use `latex + dvipdf`; if you use `pdflatex` you have to convert them somehow, and the most practical way is to use epstopdf, whose output is underwhelming.
EDIT: My point is: the pdf outputted by epstopdf is buggy and ugly; so if you can't generate the picture in pdf, your best shot is using `latex` instead of `pdflatex`.
`epstopdf` seems to make this answer irrelevant? – Seamus Sep 6 '10 at 20:16 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8976179957389832, "perplexity": 3422.598458893492}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00474-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.hackmath.net/en/math-problem/1886 | # 22/7 circle
Calculate approximately area of a circle with radius 20 cm. When calculating π use 22/7.
Result
S = 1257.14 cm2
#### Solution:
$S=22/7 \cdot \ 20^{ 2 }=1257.14 \ \text{cm}^2$
Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!
Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):
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16. Passenger boat
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The following numbers round to the thousandth: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.8903235197067261, "perplexity": 2067.1870562584368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370500426.22/warc/CC-MAIN-20200331084941-20200331114941-00244.warc.gz"} |
https://www.physicsforums.com/threads/conditions-for-pulley.698214/ | # Homework Help: Conditions for Pulley
1. Jun 22, 2013
### andyrk
When we say "pulley is frictionless", do we mean its groove where the string moves or its axle or both?
2. Jun 22, 2013
### Staff: Mentor
The axle - but if you can neglect kinetic energy of the wheel, a frictionless groove gives the same result.
3. Jun 22, 2013
### andyrk
How? How does it give the same result as an axle which has friction, if we neglect KE of the wheel?
4. Jun 22, 2013
### Staff: Mentor
The pulley will not dissipate any energy, and the magnitude of force is the same everywhere in the rope.
That is true for every ideal, frictionless pulley, it does not even depend on the details how you design the pulley.
5. Jun 22, 2013
### haruspex
That's not what mfb wrote. mfb said that (frictionless axle with massless wheel) gives the same result (i.e. no role in the equations) as frictionless groove. | {"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": 0.9035188555717468, "perplexity": 3843.7696245389666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589404.50/warc/CC-MAIN-20180716154548-20180716174548-00259.warc.gz"} |
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Definite Integral as Limit of a Sum
Table of Content
Introduction to Definite Integral as LImit of a Sum
Methods to express the infinite series as Definite Integral
Differentitation under the Integral Sign
Fundamental theorem of Calculus ( Newton-Leibnitz Formula)
Related Resources
Introduction to Definite Integral as LImit of a Sum
We have already discussed the concept of definite integral in the previous sections. Definite integral is closely related to concepts like antiderivative and indefinite integrals. In this section, we shall discuss this relationship in detail.
Definite integral consists of a function f(x) which is continuous in a closed interval [a, b] and the meaning of definite integral is assumed to be in context of area covered by the function f from (say) ‘a’ to ‘b’.
An alternative way of describing is that the definite integral is a limiting case of the summation of an infinite series, provided f(x) is continuous on [a, b] i.e.,
The converse is also true i.e., if we have an infinite series of the above form, it can be expressed as a definite integral.
Methods to express the infinite series as Definite Integral
Express the given series in the form ∑ 1/n f (r/n)
Then the limit is its sum when n→∞, i.e. lim n→∞ h ∑1/n f(r/n)
Replace r/n by x and 1/n by dx and lim n→∞ ∑ by the sign of ∫.
The lower and the upper limit of integration are the limiting values of r/n for the first and the last term of r respectively.
Some particular cases of the above are
lim n→∞ ∑nr =1 1/n f(r/n) or lim n→∞ ∑n–1r = 0 1/n f(r/n) = ∫10 f(x)dx
lim n→∞ ∑pnr =1 1/n f(r/n) = ∫βα f(x)dx
where α = lim n→∞ r/n = 0 (as r = 1) and β = lim n→∞ r/n = p (as r = pn)
Illustration 1:
Show that (A) lim n→∞ {1/(n+1) + 1/(n+2) + 1/(n+3) + ... + ... 1/(n+n)} = ln 2.
(B) lim n→∞ 1p + 2p + 3p + ... + np/(np +1) = 1/(p +1) (p > 0)
Solution:
(A) Let I = lim n→∞ (1/n+1 + 1/n+2 + 1/n+3 + ... + 1/n+n)
= lim n→∞ {1/(n+1) + 1/(n+2) + 1/(n+3) + ... + 1/(n+n)}
Now α = lim n→∞ 1/n = 0 (as r = 1)
and β = lim n→∞ r/n = 1 (as r = n)
⇒ l = ∫10. 1/1+x dx = [In (1+x)] 10
⇒ I = ln 2.
(B) 1p + 2p + 3p + ... + np/(np +1) = ∑nr=1 1p/n.np = ∑nr=1 1+n(r/n)p
Take f(x) = xp; Let h = 1/n so that as n → ∞; h → 0
∴ limn→∞ ∑nr =1 1/n f(0 + r/n)
= ∫10 f(x)dx = ∫10 xpdx
= 1/p+1
Differentitation under the Integral Sign
Leibnitz’s Rule
If g is continuous on [a, b] and f1 (x) and f2 (x) are differentiable functions whose values lie in [a, b], then
d/dx ∫ f2(x) f1(x) g(t)dt = g (f2(x)) f2'(x) – g (f1(x)) f1'(x)
Illustration 1:
Solution:
f ‘(x) = – sin x – (x f (x) + ∫x0 f(t) dt) + x f(x)
= –sin x – ∫x0 f(t)dt
f “(x) = – cos x – f(x)
Hence, this gives f “(x) + f(x) = cos x.
____________________________________________________________________________
Illustration 2:
If a function f(x) is defined ∀x ∈ R such that
Solution:
Diffrentiate w.r.t. x
g’(x) = – F(x)/x
F(x) = -x g’(x)
NOw, we know that g(a) = 0 and hence we get
______________________________________________________________________________
Illustration 3:
Determine a positive integer n < 5, such that
Solution:
Integrating by parts,
…... (1)
= – (–1) – [ex]10 = 1 – (e–1) = 2 – e
From (i), I2 = ( -1)3 - 2I1 = -1 - 2(2 - e) = -5 + 2e
and I3 = (-1)4 - 3I2 = 1 - 3(-5 + 2e) = 16 - 6e Which is given .
∴ n = 3.
Fundamental theorem of Calculus ( Newton-Leibnitz Formula)
Antiderivative Concept with the Area Problem
This theorem state that If f(x) is a continuous function on [a, b] and F(x) is any anti derivative of f(x) on [a, b] i.e. F'(x) = f (x) ∀ x ∈ (a, b), then
The function F(x) is the integral of f(x) and a and b are the lower and the upper limits of integration.
Illustration 1:
directly as well as by substitution x = 1/t. Evaluate why the answers don't tally.
Solution:
= [1/2 tan–1 (x/2)]2–2 = 1/2 [tan–1(1) – tan–1 (–1)]
= 1/2 [π/4 – (–π/4)] = π/4
⇒ l = π/4
On the other hand; if x = 1/t then,
Solving this and simplifying, we get
I = – [1/2 tan–1 (2t)]1/2–1/2
= –1/2 tan–11 – (–1/2 tan–1 (–1)) = –π/8 – π/8 = –π/4
∴ I = π/4 when x = 1/t
In above two results l = -π/4 is wrong. Since the integrand ¼ + x2 > 0 and therefore the definite integral of this function cannot be negative.
Since x = 1/t is discontinuous at t = 0, the substitution is not valid (∴ I = π/4).
Note:
It is important the substitution must be continuous in the interval of integration.
________________________________________________________________________________________
Illustration 2:
then show that α = β.
Solution:
Put x = 1/t ⇒ dx = –1/t2 then
=
Q1. The Leibnitz’s rule can be applied only if
(a) the function g is continuous in [a, b] and the functions f1 and f2 are differentiable functions whose values may lie within or outside [a, b].
(b) the function g is continuous in [a, b] and the functions f1 and f2 are differentiable functions whose values lie outside [a, b].
(c) the function g is continuous in [a, b] and the functions f1 and f2 are differentiable functions whose values lie in [a, b].
(d) the function g is discontinuous in [a, b] and the functions f1 and f2 are differentiable functions whose values may lie withing or outside [a, b].
Q2. Fundamental theorem of calculus states that = F(b) – F(a), where
(a) f(x) is a continuous function on [a,b] and F(x) is the derivative of f(x) on [a, b].
(b) f(x) is a continuous function on [a,b] and F(x) is the anti derivative of f(x) on [a, b].
(c) f(x) is a discontinuous function on [a,b] and F(x) is the derivative of f(x) on [a, b].
(d) F(x) is a continuous function on [a,b] and f(x) is the derivative of f(x) on [a, b].
Q3. In order to express infinite series as definite integral,
(a) sign of summation is replaced by integration.
(a) sign of integration is replaced by summation.
(a) infinite series can’t be expressed as definite integral.
(a) None of the above.
Q4. The definite integral is a limiting case of
(a) infinite series
(b) finite series
(c) infinite sequence
(d) indefinite integral.
Q5.
(a)
(b)
(c)
(d)
Q1
Q2
Q3
Q4
Q5
(c)
(b)
(a)
(a)
(d)
Related Resources
You may wish to refer indefinite integral. | {"extraction_info": {"found_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": 33, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9899151921272278, "perplexity": 1381.7877318988078}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1495463607593.1/warc/CC-MAIN-20170523083809-20170523103809-00436.warc.gz"} |
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0 down 2 answer | {"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": 0.8537614941596985, "perplexity": 4891.926473838411}, "config": {"markdown_headings": false, "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-2014-23/segments/1406510272329.26/warc/CC-MAIN-20140728011752-00150-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-and-trigonometry-10th-edition/chapter-2-2-7-inverse-functions-2-7-exercises-page-229/50 | Algebra and Trigonometry 10th Edition
a) $f^{-1}(x)=-\dfrac{2}{x}$ b) See graph c) The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$. d) $D_f=(-\infty,0)\cup(0,\infty),R_f=(-\infty,0)\cup(0,\infty)$ $D_{f^{-1}}=(-\infty,0)\cup(0,\infty),R_{f^{-1}}=(-\infty,0)\cup(0,\infty)$
We are given the function: $f(x)=-\dfrac{2}{x}$ $y=-\dfrac{2}{x}$ a) Determine the inverse $f^{-1}$. Interchange $x$ and $y$: $x=-\dfrac{2}{y}$ $xy=-2$ $y=-\dfrac{2}{x}$ $f^{-1}(x)=-\dfrac{2}{x}$ b) Graph both functions. c) The graph of the function $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$. d) Determine the domain and range of $f$: $D_f=(-\infty,0)\cup(0,\infty)$ $R_f=(-\infty,0)\cup(0,\infty)$ Determine the domain and range of $f^{-1}$: $D_{f^{-1}}=(-\infty,0)\cup(0,\infty)$ $R_{f^{-1}}=(-\infty,0)\cup(0,\infty)$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8161993622779846, "perplexity": 105.9527508731872}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038066568.16/warc/CC-MAIN-20210412023359-20210412053359-00452.warc.gz"} |
http://www.javaprogrammingforums.com/whats-wrong-my-code/11846-help-wmi.html | ## Help with WMI
Hey everyone, im currently trying to access the cmd.exe from Java. I want to execute a command from Java and obtain it's results, at the moment im trying the following command:
wmic /Namespace:\\root\SecurityCenter2 Path AntiVirusProduct Get displayName,productState /Format:List
It works if i input the command directly from the Command Prompt, but it doesn't work when i implement it in my code.
I've tried other commands before, like this one:
ipconfig/all
I used it to obtain my Gateway IP, and it worked, but for some reason the "wmic" command doesn't work.
Anyways, this is the code:
```try {
String cmd = "wmic /Namespace:\\root\SecurityCenter2 Path AntiVirusProduct Get displayName,productState /Format:List";
StringBuilder sb = new StringBuilder();
Process pro = Runtime.getRuntime().exec("cmd.exe /c "+cmd+"");
System.out.println(cmd);
}
System.out.println(sb.toString());
} catch (IOException ex) {
}```
Netbeans (the IDE im using) marks the String cmd line as red and it says the following error:
ilegal escape character
.
Here's one thing i encountered while trying to solve this... when i print the String "cmd" it doesn't recognize the whole command, the "\\root" part is printed as "\oot"... So instead i tried using "\\\\root\\\SecurityCenter2" and it was printed as it is mean to be. I don't really know what's wrong, but i think it has something to do with Java recognizing the slashes. Anyways, i hope you guys can help me, i could really use a hand since this is an esencial part of my career project. Thanks beforehand. | {"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": 0.8349496722221375, "perplexity": 1258.9448739879072}, "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-2014-10/segments/1393999652586/warc/CC-MAIN-20140305060732-00053-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://www.onemathematicalcat.org/Math/Precalculus_obj/solveExpEqs.htm | SOLVING EXPONENTIAL EQUATIONS
• PRACTICE (online exercises and printable worksheets)
An exponential equation has at least one variable in an exponent.
For example, ‘$\,2^{3x-1} = 5\,$’ is an exponential equation, since the variable $\,x\,$ is in the exponent.
However, ‘$(3x-1)^2 = 5$’ is not an exponential equation, since there is no variable in an exponent.
Many exponential equations can be solved by a technique that can be abbreviated as ‘IUSC’:
Isolate $\ldots$ Undo with a logarithm $\ldots$ Solve resulting equation $\ldots$ Check
This section classifies families of equations that can be solved by the IUSC method, discusses the method, and presents examples.
A technique for solving ‘fake’ quadratic (pseudo-quadratic) exponential equations is also presented.
Classification of Equations Solvable by IUSC
Let $\,b\,$ denote an allowable base for an exponential function: $\,b > 0\,$, $\,b\ne 1\,$.
Recall that a ‘linear expression in one variable’ (say, $\,x\,$) is of the form $\,ax + b\,$, for real numbers $\,a\,$ and $\,b\,$.
For the purposes of this section, define an ‘ExpTerm’ to be a single term that contains only the following types of factors:
• constants
• $b^{\text{linear expression in one variable}}$
For example, these are all ExpTerms:
• $\,3\cdot 5^{2x-1}\,$
• $\,7\,$
• $\,2^x\cdot 7^{1-3x}\,$
There are NOT ExpTerms:
• $\,2^x + 1\,$ (an ExpTerm is a single term)
• $\,x3^x\,$ (the factor of $\,x\,$ is not allowed)
The IUSC method can be used to solve equations that can be put in the form:
EXPTERM1 = EXPTERM2
Here are examples of equations solvable by IUSC (and each is solved below):
The ‘IUSC’ Method
Here are the tools used in the IUSC method:
• solving linear equations in one variable
The equations that emerge usually have irrational numbers involved (like ‘$\,\ln 3\,$’).
Don't be intimidated by these ‘ugly’ numbers!
To prepare yourself, compare these side-by-side solutions of a ‘familiar’ equation and one you'll see in IUSC.
(original equation) $2(3x-1) = 5$ $(3x-1)(\ln 2) = \ln 5$ (original equation) (distributive law) $6x - 2 = 5$ $3x(\ln 2) - \ln 2 = \ln 5$ (distributive law) (isolate $\,x\,$ term) $6x = 7$ $3x(\ln 2) = \ln 5 + \ln 2$ (isolate $\,x\,$ term) (divide by $\,6\,$) $\displaystyle x = \frac{7}{6}$ $\displaystyle x = \frac{\ln 5 + \ln 2}{3\ln 2} = \frac{\ln 10}{3\ln 2}$ (divide by $\,3\ln 2\,$)
• properties of logarithms
For all positive real numbers $\,x\,$ and $\,y\,$, and for $\,s\in\Bbb R\,$: \begin{alignat}{2} x = y\ \ &\iff\ \ \ln x = \ln y &&\cr\cr \ln x^s &\, = s\ln x &&\text{(you can bring exponents down)}\cr\cr \ln xy &\, = \ln x + \ln y &\qquad &\text{(the log of a product is the sum of the logs)}\cr\cr \ln\frac{x}{y} &\, = \ln x - \ln y &&\text{(the log of a quotient is the difference of the logs)} \end{alignat}
• outputs from exponential functions are always positive
For all allowable bases $\,b\,$ and for all $\,x\,$, $\,b^x > 0\,$.
The IUSC Method solving simple exponential equations
• ISOLATE an exponential expression.
That is, get an ExpTerm (which contains a variable) all by itself on one side of the equation.
• UNDO the exponents with a logarithm.
Any log can be used, but it's usually easiest to use the common log ($\,\log\,$) or the natural log ($\,\ln \,$).
• SOLVE for the variable.
Don't be intimidated by the irrational numbers! These are just linear equations in one variable.
• CHECK in the original equation.
There are lots of opportunities for errors in this method.
To gain confidence, get a decimal approximation and substitute in the original equation.
Note: Always get an exact answer first.
Then, get a decimal approximation (as needed) from the exact answer.
Approximation errors made early on can grow as you proceed through the solution steps.
EXAMPLE
Solve: $8\cdot 5^{2x-1} - 20 = 0$
$8\cdot 5^{2x-1} - 20 = 0$ (original equation) $8\cdot 5^{2x-1} = 20$ Isolate: Add $\,20\,$ to both sides, to isolate an ExpTerm with a variable on the left. Some people prefer to divide through by $\,8\,$; this alternative solution is shown below. $\ln\bigl(8\cdot 5^{2x-1}\bigr) = \ln 20$ Undo: Take natural logs of both sides. Note that both ‘$\,8\cdot 5^{2x-1}\,$’ and ‘$\,20\,$’ are always positive; therefore, this equation is equivalent to the first. $\ln 8 + \ln 5^{2x-1} = \ln 20$ Undo (continued): The log of a product is the sum of the logs. $\ln 8 + (2x-1)(\ln 5) = \ln 20$ Undo (continued): Bring the exponent down; this gets the variable out of the exponent. The result is a linear equation in one variable (involving several irrational numbers). $(2x-1)(\ln 5) = \ln 20 - \ln 8$ Solve: Subtract $\,\ln 8\,$ from both sides. $2x(\ln 5) - \ln 5 = \ln 20 - \ln 8$ Solve (continued): Use the distributive law on the left side. $2x(\ln 5) = \ln 20 - \ln 8 + \ln 5$ Solve (continued): Add $\,\ln 5\,$ to both sides. $\displaystyle x = \frac{\ln 20 - \ln 8 + \ln 5}{2\ln 5} = \frac{\ln \frac{20\cdot 5}{8}}{2\ln 5} = \frac{\ln 12.5}{2\ln 5}$ Solve (continued): Divide by $\,2\ln 5\,$; rename using properties of logs, if desired. $\displaystyle x = \frac{\ln 12.5}{2\ln 5} \approx 0.78466$ $8\cdot 5^{2(0.78466)-1} - 20 \overset{\text{?}}{=} 0$ $-0.00011 \approx 0$ Okay! Check: Approximate the exact answer to at least five decimal places. Substitute in the original equation. Put a question mark over the equal sign, since you're asking a question—are the two sides equal? Since you're using an approximate solution, you won't get a perfect equality, but it should be very close!
Here are two other approaches.
Notice that different approaches can give different ‘names’ for the solution!
By isolating $\,5^{2x-1}\,$ (instead of $\,8\cdot 5^{2x-1}\,$) before ‘undoing’ with logs, you must deal with a fraction, but save a couple steps overall. Here, we were lucky, because the fraction is an exact decimal ($\,\frac 52 = 2.5\,$): $$\begin{gather} 8\cdot 5^{2x-1} - 20 = 0\cr\cr 8\cdot 5^{2x-1} = 20\cr\cr 5^{2x-1} = \frac{5}{2}\cr\cr (2x-1)(\ln 5) = \ln 2.5\cr\cr 2x\ln 5 - \ln 5 = \ln 2.5\cr\cr 2x\ln 5 = \ln 2.5 + \ln 5\cr\cr x = \frac{\ln 12.5}{2\ln 5} \end{gather}$$ $$\begin{gather} 8\cdot 5^{2x-1} - 20 = 0\cr\cr 8\cdot 5^{2x-1} = 20\cr\cr 5^{2x-1} = \frac{5}{2}\cr\cr (2x-1)(\ln 5) = \ln 2.5\cr\cr 2x - 1 = \frac{\ln 2.5}{\ln 5}\cr\cr 2x = \frac{\ln 2.5}{\ln 5} + 1\cr\cr x = \frac 12\left(\frac{\ln 2.5}{\ln 5} + 1\right) \end{gather}$$ Note: $\displaystyle \frac 12\left(\frac{\ln 2.5}{\ln 5} + 1\right) = \frac 12\left(\frac{\ln 2.5}{\ln 5} + \frac{\ln 5}{\ln 5}\right) = \frac 12\left(\frac{\ln 2.5 + \ln 5}{\ln 5}\right) = \frac{\ln 12.5}{2\ln 5}$
EXAMPLE
Solve: $\displaystyle\frac{2^{3t-1}}{7\cdot 5^{2+t}} = 1$
$\displaystyle\frac{2^{3t-1}}{7\cdot 5^{2+t}} = 1$ (original equation) $2^{3t-1} = 7\cdot 5^{2+t}$ Isolate: Multiply both sides by $\,7\cdot 5^{2+t}\,$ to get in the form ‘ExpTerm1 = ExpTerm2’. Note that $\,7\cdot 5^{2+t}\,$ is never equal to zero, so this equation is equivalent to the former. $\ln (2^{3t-1}) = \ln (7\cdot 5^{2+t})$ Undo: Take natural logs of both sides. Note that both ‘$\, 2^{3t-1}\,$’ and ‘$\,7\cdot 5^{2+t}\,$’ are always positive, so this equation is equivalent to the former. $(3t-1)(\ln 2) = \ln 7 + (2+t)(\ln 5)$ Undo (continued): Use properties of logarithms to get the variables out of the exponents. This is now a linear equation in one variable. $3t\ln 2 - t\ln 5 = \ln 7 + 2\ln 5 + \ln 2$ Solve (continued): Get all the variable terms on the left, and constant terms on the right. $t(3\ln 2 - \ln 5) = \ln 7 + 2\ln 5 + \ln 2$ Solve (continued): Factor out the $\,t\,$ on the LHS. $\displaystyle t = \frac{\ln 7 + 2\ln 5 + \ln 2}{3\ln 2 - \ln 5}$ Solve (continued): Divide by $\,3\ln 2 - \ln 5\,$. $\displaystyle t = \frac{\ln 7 + 2\ln 5 + \ln 2}{3\ln 2 - \ln 5} \approx 12.46359$ $\displaystyle\frac{2^{3(12.46359)-1}}{7\cdot 5^{2+12.46359}}\overset{\text{?}}{=} 1$ $1.00000 = 1$ Okay! Check: Approximate the exact answer to at least five decimal places. Substitute in the original equation. Put a question mark over the equal sign, since you're asking a question—are the two sides equal? Here, the left-hand side evaluates to the number $\,1\,$ when rounded to five decimal places.
EXAMPLE
Solve: $5^{x}\cdot 3^{x-1} = \frac 12 7^{2-x}$
$5^{x}\cdot 3^{x-1} = \frac 12 7^{2-x}$ (original equation) $\ln \bigl(5^{x}\cdot 3^{x-1}\bigr) = \ln\bigl(\frac 12 7^{2-x}\bigr)$ Take logs. $x\ln 5 + (x-1)\ln 3 = \ln(2^{-1}) + (2-x)\ln 7$ Use properties of logs. $x(\ln 5 + \ln 3 + \ln 7) = -\ln 2 + \ln 3 + 2\ln 7$ Get variable terms on left and constants on right. $\displaystyle x = \frac{-\ln 2 + \ln 3 + 2\ln 7}{(\ln 5 + \ln 3 + \ln 7)} \approx 0.92336$ Get an exact answer first, and then approximate. $5^{0.92336}\cdot 3^{0.92336-1} \overset{\text{?}}{=} \frac 12 7^{2-0.92336}$ $4.06288 \approx 4.06290$ Okay! Check in original equation. Approximate the solution to more than five decimal places, as needed and desired.
EXAMPLE: a ‘fake quadratic’
Solve: ${\text{e}}^{2x} - {\text{e}}^x - 6 = 0$
This final exponential equation cannot be put in the form ‘ExpTerm1 = ExpTerm2’.
However, it can be turned into a quadratic equation by a simple substitution.
Consequently, it is often called a ‘fake quadratic’ or a ‘pseudo-quadratic’ equation.
The idea used is important:
• Have an equation you can't immediately solve?
• Try to transform it into one that you can solve!
• Solve the ‘transformed’ equation.
• Transform back to a solution of the original equation.
${\text{e}}^{2x} - {\text{e}}^x - 6 = 0$ (original equation) This equation is not solvable by IUSC. ${(\text{e}}^{x})^2 - ({\text{e}}^x) - 6 = 0$ Use a property of exponents to rename the first term. A familiar quadratic pattern emerges! $u^2 - u - 6 = 0$ Let $\,u := {\text{e}}^x\,$. This substitution transforms the equation in $\,x\,$ to a quadratic equation in $\,u\,$. $(u-3)(u+2) = 0$ $u = 3$ or $u = -2$ Solve the transformed equation. You can save a couple steps if you're comfortable never explicitly bringing $\,u\,$ into the picture: $$\begin{gather} {(\text{e}}^{x})^2 - ({\text{e}}^x) - 6 = 0\cr\cr ({\text{e}}^{x} - 3)({\text{e}}^{x} + 2) = 0\cr\cr {\text{e}}^x = 3\ \ \text{ or }\ \ {\text{e}}^x = -2 \end{gather}$$ ${\text{e}}^x = 3$ or ${\text{e}}^x = -2$ Transform back: go back to the original variable, $\,x\,$. $x\ln \text{e} = \ln 3$ $x = \ln 3 \approx 1.09861$ Since $\,{\text{e}}^x\,$ is always strictly positive, it never equals a negative number. There is only one solution. ${\text{e}}^{2\,\cdot\, 1.09861} - {\text{e}}^{1.09861} - 6 \overset{\text{?}}{=} 0$ $-0.00003 \approx 0$ Okay! Check.
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Solving Logarithmic Equations
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6
AVAILABLE MASTERED IN PROGRESS
(MAX is 6; there are 6 different problem types.) | {"extraction_info": {"found_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": 0.8655018210411072, "perplexity": 944.1990970921603}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203021.14/warc/CC-MAIN-20190323201804-20190323223804-00068.warc.gz"} |
https://www.computer.org/csdl/trans/tp/1984/06/04767596-abs.html | Issue No. 06 - June (1984 vol. 6)
ISSN: 0162-8828
pp: 721-741
Stuart Geman , Division of Applied Mathematics, Brown University, Providence, RI 02912.
Donald Geman , Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003.
ABSTRACT
We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.
INDEX TERMS
CITATION
Stuart Geman, Donald Geman, "Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 6, no. , pp. 721-741, June 1984, doi:10.1109/TPAMI.1984.4767596 | {"extraction_info": {"found_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": 0.8877562284469604, "perplexity": 952.9813295104826}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662507.79/warc/CC-MAIN-20160924173742-00000-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/cant-integrate-by-parts-an-integral-with-a-fraction-inside.614437/ | # Homework Help: Can't integrate by parts an integral with a fraction inside
1. Jun 16, 2012
### Psinter
1. The problem statement, all variables and given/known data
In the section of the book of integration by parts, there is an exercise that I don't even know how to tackle anymore. It's this:
$$\int \frac{xe^{2x}}{(1+2x)^2} dx$$
2. Relevant equations
${uv} - {\int v{du}}$
3. The attempt at a solution
$u = x ;$
$du = 1;$
$dv= e^{2x};$
$v = \frac{1}{2}{e^{2x}}$
$= \frac{1}{2} {x}{e^{2x}} - \frac{1}{2} {\int e^{2x} {dx}}$
But I know is wrong because I no longer have the ${(1+2x)}^2.$
Last edited by a moderator: Jun 16, 2012
2. Jun 16, 2012
### Staff: Mentor
Yes, this is wrong. Whatever you choose for u and dv must multiply to the integrand you start with. I haven't worked this problem, but I would start by doing an ordinary substitution first (either u = 2x or u = 2x + 1), and then try integration by parts.
Using u = 2x, you get
$$1/4 \int{\frac{ue^u~du}{(u + 1)^2}}$$
3. Jun 16, 2012
### SammyS
Staff Emeritus
$\displaystyle \int u\,dv={uv} - {\int v{du}}$
Then, as you noted, you left part of the integrand out. In other words, you haven't split-up the original integral into u and dv .
By the way, if $u = x\,,$ then $du = dx\,$
and if $dv= e^{2x}dx$ then $\displaystyle v=\frac{1}{2}{e^{2x}}\,.$
(I see Mark44 beat me to the punch.)
4. Jun 16, 2012
### Psinter
I tried that too, I just didn't wrote it in the attempts. Although I forgot about the 1/4 in front of the integral.
The problem is that this is the same but with other letters. I still don't know how can I possibly integrate and take the denominator into account. I thought about expanding in a sum of integrals and I can integrate the first 1 of them, but I have a problem with the other 2.
$(1+u)^2 =$ $1$ $+$ $2u$ $+$ $u^2$
$\frac{1}{4}\int \frac{ue^{u}}{1}$ $+$ $\frac{1}{4}\int \frac{ue^{u}}{2u}$ $+$ $\frac{1}{4}\int \frac{ue^{u}}{u^2}$
Then, how can I deal with the 2nd (Blue) and the 3rd one (Green)?
Ah thanks, I totally forgot the dx.
Last edited by a moderator: Jun 16, 2012
5. Jun 16, 2012
### vela
Staff Emeritus
You can't do that.
$$\frac{a}{b+c+d} \ne \frac{a}{b} + \frac{a}{c} + \frac{a}{d}$$ You also left the du's out of all the integrals.
6. Jun 16, 2012
### Bohrok
After you get ¼∫xex/(x+1)2 dx (replacing the u's with x's), try u = xex and dv = 1/(x+1)2 dx
7. Jun 16, 2012
### Psinter
I think I got it with that. Can you correct me?
$v = ue^u$ ; $dv = e^u(u+1)du$
$dw = \frac{1}{(u+1)^2}du$ ; $w = -\frac{1}{(u+1)}$
$= -\frac{ue^u}{(u+1)} + \frac{1}{4} \int \frac{e^u(u+1)}{(u+1)} du$
$= -\frac{ue^u}{(u+1)} + \frac{1}{4} \int e^u du$
$= -\frac{ue^u}{(u+1)} + \frac{1}{4} e^u + C$
So, placing back everything with $(u = 2x)$
$\int \frac{xe^{2x}}{(1+2x)^2} dx = -\frac{1}{4}\frac{(2x)e^{2x}}{(2x+1)} + \frac{1}{4} e^{2x} + C$
Last edited: Jun 16, 2012
8. Jun 16, 2012
### Bohrok
That's right, except for one u that didn't get replaced by 2x
9. Jun 16, 2012
### ageralo
You almost got it.
That 1/4 term though needs to be multiplied to your first term. Remember how Mark44 wrote it - 1/4 * Integral Stuff. What you did, the substitution and integration by parts, is just the evaluation of the "Integral Stuff" in the equation 1/4 * Integral Stuff. So that 1/4 still needs to be appended.
Last edited by a moderator: Jun 16, 2012
10. Jun 16, 2012
### Staff: Mentor
Psinter, Please don't use those HTML SIZE tags. What you write shows up perfectly well without them.
If you want you LaTeX stuff to be a bit larger use [ tex ] tags instead of [ itex ] (without the spaces).
You can also use [noparse]$$<expression>$$[/noparse], which is the same as [ tex ] and [ /tex ].
Or you can use [noparse]$<expression>$[/noparse], which is the same as [ itex ] and [ /itex ].
11. Jun 16, 2012
### Psinter
Thanks, fixed that.
Fixed too.
Got it. It's just that in my eyes I almost couldn't see the e exponent. Thanks for those, I'll use them next time.
This isn't an exercise for a work, but it took me so long to do it I got fond of it and made a comic: (Wolfram got no steps, but I wanted the steps to learn how to do it)
12. Jun 16, 2012
### Staff: Mentor
Cool!
What does le' mean?
13. Jun 16, 2012
### Psinter
It's French. The translation would be "the". However, in Enlgish is used as an expression when mocking something. It's pronounced like "e" sounds in Japanese, not like "e" sounds in English.
That kind of meme comic has always used "le" to refer to the one who is being talked about so I thought I should use it too, just for tradition. I added the apostrophe to sort of separate it from being pronounced together with the next word but in tradition it doesn't have that apostrophe.
14. Jun 16, 2012
### Bohrok
Just one last note: sometimes you need to change the problem so Wolframalpha likes it and shows you its steps. For some reason WA didn't like the e2x in the problem, but entering the new integral after doing the substitution u = 2x instead will show you the steps for that integral!
15. Jun 17, 2012
### Psinter
Nice. Thanks for that one. | {"extraction_info": {"found_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": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.944482684135437, "perplexity": 1851.5282558439628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592654.99/warc/CC-MAIN-20180721184238-20180721204238-00037.warc.gz"} |
https://itanveer.com/2011/07/24/some-interesting-linear-algebra-theories/ | # Some interesting linear algebra theories
1. if $U$ and $W$ are subspaces for a vector space $V$ then $U \cap W$ is a subspace (of $V$).
2. Definition: $\{U + W\}$ denotes the set of all the vectors possible to construct by adding a vector from $U$ with a vector from $W$ where $U$ and $W$ may be either set of vectors or subspace of vectors.
3. If $U$ and $W$ are subspaces for a vector space $V$ then $\{U + W\}$ is the smallest subspace which contain both $U$ and $W$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 16, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9793779850006104, "perplexity": 128.39612995067864}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320362.97/warc/CC-MAIN-20170624221310-20170625001310-00191.warc.gz"} |
http://www.cfd-online.com/Forums/openfoam/67070-problem-trimmed-cells-when-calculating-nabla-cdot-nabla-cdot-u-tu.html | # Problem with trimmed cells when calculating \nabla \cdot \nabla \cdot (U^TU)
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August 3, 2009, 03:59
Problem with trimmed cells when calculating \nabla \cdot \nabla \cdot (U^TU)
#1
Member
Niklas Winkler
Join Date: Mar 2009
Location: Stockholm, Stockholm, Sweden
Posts: 73
Rep Power: 8
Dear all,
I would like to calculate the divergence of the convective term in the momentum equation which is as above, \nabla \cdot \nabla \cdot (U^T U) on a mesh converted from Star-CD, which includes trimmed cells.
As a test I have calculated \nabla \cdot \nabla \cdot (X^T X), where X here denotes (x,y,z), which analytically should be 12 on a 3D mesh as follows,
volVectorField cellC(mesh.C());
surfaceScalarField cellCsurf(linearInterpolate(cellC) & mesh.Sf());
volScalarField divdivCellC2
(
IOobject
(
"divdivCellC2",
runTime.timeName(),
mesh,
IOobject::AUTO_WRITE
),
fvc::div(fvc::div(cellCsurf,cellC))
);
The figure below shows results for a slice of the first bend of the double bended pipe.
Any suggestions of what could be wrong or how to solve the problem?
Thanks,
/NW
Attached Images
divdivCellC2.jpg (73.8 KB, 35 views)
August 4, 2009, 06:11 #2 Senior Member Eugene de Villiers Join Date: Mar 2009 Posts: 725 Rep Power: 12 hmm. Since most of your cells do have a value of 12, I would guess you are getting it almost right. One thing that immediately springs to mind is that most people do not use skewness correction in their interpolation. Thus linearInterpolate(cellC) will not provide a faceCentred value for distorted elements. It would be interesting to see what the results would be if you replace the interpolation with mesh.Cf() (face centres).
August 4, 2009, 07:22
#3
Member
Niklas Winkler
Join Date: Mar 2009
Location: Stockholm, Stockholm, Sweden
Posts: 73
Rep Power: 8
Unfortunately, surfaceScalarField cellCsurf(mesh.Cf() & mesh.Sf()); didn't change anything, in fact it gives exactly the same result!?
To give some more info, when I'm calculating div(cellC) the result is 3 which is as expected. But when calculating div(cellCsurf,cellC) some distortion can already be seen, see figure.
Attached Images
divCellC2_X.jpg (53.9 KB, 19 views)
August 4, 2009, 08:22 #4 Senior Member Eugene de Villiers Join Date: Mar 2009 Posts: 725 Rep Power: 12 I think you can focus on finding the issue in div(cellCsurf,cellC). The fact that div(cellC) produces 3 means, div(div(cellCsurf, cellC)) should also be fine provided div(cellCsurf, cellC) is fine. The code for this stuff can all be found in src/finiteVolume/lnInclude/... fvcDiv.* convectionSheme* gaussConvectionScheme* It is really really simple mathematically, so I am rather curious to find out what the issue is. If I were you I would focus on one problem cell and walk through the code step by step using a spreadsheet/debugger or similar.
August 6, 2009, 07:35 #5 Member Niklas Winkler Join Date: Mar 2009 Location: Stockholm, Stockholm, Sweden Posts: 73 Rep Power: 8 Finally I got the debug version to work! But, it's showing that fvcDiv.C is called and then a lot of underlying files which I believe are not directly related to the calculation. However, I can't find where the actual calculations are performed. Could you please briefly explain the steps from when the software calls fvc::div(...) or give me some other hints on how to proceed, please. Thanks!
August 6, 2009, 07:43 #6 Senior Member Eugene de Villiers Join Date: Mar 2009 Posts: 725 Rep Power: 12 My previous post already gives the answer: convectionScheme* and gaussConvectionScheme* also found in src/finiteVolume/lnInclude/ The only outstanding bits are the interpolation schemes, but since you are probably using linear interpolation its not really necessary to look at the code for these. If however, you find you do need to look at the interpolation code, let me know and I will provide the details for the interpolation scheme of you choice.
August 10, 2009, 04:22 #7 Member Niklas Winkler Join Date: Mar 2009 Location: Stockholm, Stockholm, Sweden Posts: 73 Rep Power: 8 I've been trying to solve the issue but without any real progress since the problem is beyond my programming skills. Is it possible to get some more help to solve the problem? And, is this an OpenFoam bug which should be reported?
August 10, 2009, 08:38 #8 Senior Member Eugene de Villiers Join Date: Mar 2009 Posts: 725 Rep Power: 12 I doubt very much it is a "bug" in the normal sense of the word. It is obviously a result of the numerical schemes being used. A drop in accuracy on a non-uniform mesh is not surprising. Unfortunately I don't have the time to look at this with any degree of rigour at the moment. I will see what I can do simply because my curiosity is piqued, but the chances of a result in the near future would have to be considered slim. If you have any specific questions relating to the code, I would be happy to answer those.
August 12, 2009, 02:54 #9 Member Niklas Winkler Join Date: Mar 2009 Location: Stockholm, Stockholm, Sweden Posts: 73 Rep Power: 8 Eugene, as you commented in the beginning and from another input it certainly looks like it is the interpolation scheme that causes me trouble due to that the problematic cells are skewed. Instead of using linearInterpolate to obtain the values at the face centers I've tried fvc:interpolate together with the upwind and QUICK schemes which of course give me different results but are still not correct. What I would like to try is a scheme which corrects for skewness as e.g. skewLinear according to table 4.6 in UG, but I can not get it to work. Do you know the syntax or any other possibility to include skewness correction into my interpolation? Thanks!
August 12, 2009, 06:23 #10 Senior Member Eugene de Villiers Join Date: Mar 2009 Posts: 725 Rep Power: 12 The syntax for a skewCorrected scheme would be something like: div(phiX,X) Gauss skewCorrected linear; At least this is how it works in 1.5.
August 14, 2009, 04:19 #11 Member Niklas Winkler Join Date: Mar 2009 Location: Stockholm, Stockholm, Sweden Posts: 73 Rep Power: 8 Together with skewCorrection the performance of the calculation becomes considerably improved. However, on the StarCD mesh when calculating div(div(phiX,X)) the errors are still too large, getting results from -10 to 34 where it is expected to be 12. The rms-value when calculating div(phiX,X) is approximately 10^-4 compared to 10^-3 without the correction. It was also tried on the 2D pitzDaily case with mesh refinement. The relative error decreases when the mesh is refined but the maximum error is more or less constant. Is there a way to further improve the skewCorrection?
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All times are GMT -4. The time now is 11:59. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8694378733634949, "perplexity": 2025.812092379474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1438042986451.45/warc/CC-MAIN-20150728002306-00139-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://personal.math.ubc.ca/~ortner/research/lavrentiev/ | Christoph Ortner
In the past I have worked on the numerical approximation of nonlinear elasticity, understood as an energy minimisation problem,
$\bar{u} \in \arg\min\big\{ I[u] \big| u \in A \big\}, \qquad \text{where} \qquad I[u] = \int_\Omega W(x, u , \nabla u)\,dx$
and $A$ is some admissible set. A major difficulty in this area is the Lavrentiev gap phenomenon which is said to occur if
$\inf \big\{ I[u] \big| A \cap W^{1,\infty} \big\} \not\geq \inf \big\{ I[u] \big| u \in A \big\}.$
That is, for certain singular variational problems the infimum of the energy taken over Lipschitz functions can be strictly larger than the infimum taken over the entire admissible class. This means, in particular, that conforming finite element methods are incapable of detecting the global minimizers. I have shown [18], [20] that at least for convex problems (unfortunately this excludes elasticity) one can overcome this by using the non-conforming Crouzeix-Raviart finite element space.
The following figure shows a problem proposed by Foss, Hrusa and Mizel, solved by (mesh-adaptive versions of) the conforming P1 finite element method and the non-conforming Crouzeix-Raviart finite element methods. The left graph clearly shows a gap between the minimal energy (plotted against number of degrees of freedom in the FEM mesh).
Although I have not worked on this topic for some time, there is a particularly difficult open problem which I would like to solve, or see solved: to construct a numerical method that is capable (in principle) of approximating global minimizers for any well-posed variational problem (e.g., fitting within the [Ball, 1977] theory) with polyconvex stored energy function. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "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": 0.8444235324859619, "perplexity": 459.8692681768683}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948708.2/warc/CC-MAIN-20230327220742-20230328010742-00563.warc.gz"} |
https://db0nus869y26v.cloudfront.net/en/Quantum_state | In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers,[1][2] while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.[3][4]
Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vectors are identified by the principal quantum number n, the angular momentum quantum number l, the magnetic quantum number m, and the spin z-component sz. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore two-dimensional, constituting a qubit. A pure state here is represented by a two-dimensional complex vector ${\displaystyle (\alpha ,\beta )}$, with a length of one; that is, with
${\displaystyle |\alpha |^{2}+|\beta |^{2}=1,}$
where ${\displaystyle |\alpha |}$ and ${\displaystyle |\beta |}$ are the absolute values of ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. A mixed state, in this case, has the structure of a ${\displaystyle 2\times 2}$ matrix that is Hermitian and positive semi-definite, and has trace 1.[5] A more complicated case is given (in bra–ket notation) by the singlet state, which exemplifies quantum entanglement:
${\displaystyle \left|\psi \right\rangle ={\frac {1}{\sqrt {2))}{\big (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\big )},}$
which involves superposition of joint spin states for two particles with spin 12. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability.
A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. The Schrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as a convex combination of pure states.[6] Before a particular measurement is performed on a quantum system, the theory gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the linear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the uncertainty principle: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.
## Conceptual description
### Pure states
Probability densities for the electron of a hydrogen atom in different quantum states.
In the mathematical formulation of quantum mechanics, pure quantum states correspond to vectors in a Hilbert space, while each observable quantity (such as the energy or momentum of a particle) is associated with a mathematical operator. The operator serves as a linear function which acts on the states of the system. The eigenvalues of the operator correspond to the possible values of the observable. For example, it is possible to observe a particle with a momentum of 1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator is 1 kg⋅m/s. The corresponding eigenvector (which physicists call an eigenstate) with eigenvalue 1 kg⋅m/s would be a quantum state with a definite, well-defined value of momentum of 1 kg⋅m/s, with no quantum uncertainty. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/s.
On the other hand, a system in a superposition of multiple different eigenstates does in general have quantum uncertainty for the given observable. We can represent this linear combination of eigenstates as:
${\displaystyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle .}$
The coefficient which corresponds to a particular state in the linear combination is a complex number, thus allowing interference effects between states. The coefficients are time dependent. How a quantum state changes in time is governed by the time evolution operator. The symbols ${\displaystyle |}$ and ${\displaystyle \rangle }$[a] surrounding the ${\displaystyle \Psi }$ are part of bra–ket notation.
Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is a statistical ensemble of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically, a statistical mixture is not a combination using complex coefficients, but rather a combination using real-valued, positive probabilities of different states ${\displaystyle \Phi _{n))$. A number ${\displaystyle P_{n))$ represents the probability of a randomly selected system being in the state ${\displaystyle \Phi _{n))$. Unlike the linear combination case each system is in a definite eigenstate.[7][8]
The expectation value ${\displaystyle \langle A\rangle _{\sigma ))$ of an observable A is a statistical mean of measured values of the observable. It is this mean, and the distribution of probabilities, that is predicted by physical theories.
There is no state which is simultaneously an eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement Q(t) and the momentum measurement P(t) (at the same time t) are known exactly; at least one of them will have a range of possible values.[b] This is the content of the Heisenberg uncertainty relation.
Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state.[9][10][c] More precisely: After measuring an observable A, the system will be in an eigenstate of A; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure A twice in the same run of the experiment, the measurements being directly consecutive in time,[d] then they will produce the same results. This has some strange consequences, however, as follows.
Consider two incompatible observables, A and B, where A corresponds to a measurement earlier in time than B.[e] Suppose that the system is in an eigenstate of B at the experiment's beginning. If we measure only B, all runs of the experiment will yield the same result. If we measure first A and then B in the same run of the experiment, the system will transfer to an eigenstate of A after the first measurement, and we will generally notice that the results of B are statistical. Thus: Quantum mechanical measurements influence one another, and the order in which they are performed is important.
Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called entangled states, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see entanglement. These entangled states lead to experimentally testable properties (Bell's theorem) that allow us to distinguish between quantum theory and alternative classical (non-quantum) models.
### Schrödinger picture vs. Heisenberg picture
One can take the observables to be dependent on time, while the state σ was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. (This approach was taken in the later part of the discussion above, with time-varying observables P(t), Q(t).) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. (This approach was taken in the earlier part of the discussion above, with a time-varying state ${\textstyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle }$.) Conceptually (and mathematically), the two approaches are equivalent; choosing one of them is a matter of convention.
Both viewpoints are used in quantum theory. While non-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory. Compare with Dirac picture.[12]: 65
## Formalism in quantum physics
### Pure states as rays in a complex Hilbert space
Quantum physics is most commonly formulated in terms of linear algebra, as follows. Any given system is identified with some finite- or infinite-dimensional Hilbert space. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space, because the unit sphere is defined as the set of all vectors with norm 1.
Multiplying a pure state by a scalar is physically inconsequential (as long as the state is considered by itself). If a vector in a complex Hilbert space ${\displaystyle H}$ can be obtained from another vector by multiplying by some non-zero complex number, the two vectors are said to correspond to the same "ray" in ${\displaystyle H}$[1]: 50 and also to the same point in the projective Hilbert space of ${\displaystyle H}$.
### Bra–ket notation
Main article: Bra–ket notation
Calculations in quantum mechanics make frequent use of linear operators, scalar products, dual spaces and Hermitian conjugation. In order to make such calculations flow smoothly, and to make it unnecessary (in some contexts) to fully understand the underlying linear algebra, Paul Dirac invented a notation to describe quantum states, known as bra–ket notation. Although the details of this are beyond the scope of this article, some consequences of this are:
• The expression used to denote a state vector (which corresponds to a pure quantum state) takes the form ${\displaystyle |\psi \rangle }$ (where the "${\displaystyle \psi }$" can be replaced by any other symbols, letters, numbers, or even words). This can be contrasted with the usual mathematical notation, where vectors are usually lower-case latin letters, and it is clear from the context that they are indeed vectors.
• Dirac defined two kinds of vector, bra and ket, dual to each other.[f]
• Each ket ${\displaystyle |\psi \rangle }$ is uniquely associated with a so-called bra, denoted ${\displaystyle \langle \psi |}$, which corresponds to the same physical quantum state. Technically, the bra is the adjoint of the ket. It is an element of the dual space, and related to the ket by the Riesz representation theorem. In a finite-dimensional space with a chosen basis, writing ${\displaystyle |\psi \rangle }$ as a column vector, ${\displaystyle \langle \psi |}$ is a row vector; to obtain it just take the transpose and entry-wise complex conjugate of ${\displaystyle |\psi \rangle }$.
• Scalar products[g][h] (also called brackets) are written so as to look like a bra and ket next to each other: ${\displaystyle \langle \psi _{1}|\psi _{2}\rangle }$. (The phrase "bra-ket" is supposed to resemble "bracket".)
### Spin
The angular momentum has the same dimension (M·L2·T−1) as the Planck constant and, at quantum scale, behaves as a discrete degree of freedom of a quantum system.[which?] Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described with spinors. In non-relativistic quantum mechanics the group representations of the Lie group SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative number S that, in units of Planck's reduced constant ħ, is either an integer (0, 1, 2 ...) or a half-integer (1/2, 3/2, 5/2 ...). For a massive particle with spin S, its spin quantum number m always assumes one of the 2S + 1 possible values in the set
${\displaystyle \{-S,-S+1,\ldots +S-1,+S\))$
As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C2S+1. Equivalently, it is represented by a complex-valued function of four variables: one discrete quantum number variable (for the spin) is added to the usual three continuous variables (for the position in space).
### Many-body states and particle statistics
Further information: Particle statistics
The quantum state of a system of N particles, each potentially with spin, is described by a complex-valued function with four variables per particle, corresponding to 3 spatial coordinates and spin, e.g.
${\displaystyle |\psi (\mathbf {r} _{1},\,m_{1};\;\dots ;\;\mathbf {r} _{N},\,m_{N})\rangle .}$
Here, the spin variables mν assume values from the set
${\displaystyle \{-S_{\nu },\,-S_{\nu }+1,\,\ldots ,\,+S_{\nu }-1,\,+S_{\nu }\))$
where ${\displaystyle S_{\nu ))$ is the spin of νth particle. ${\displaystyle S_{\nu }=0}$ for a particle that does not exhibit spin.
The treatment of identical particles is very different for bosons (particles with integer spin) versus fermions (particles with half-integer spin). The above N-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) with respect to the particle numbers. If not all N particles are identical, but some of them are, then the function must be (anti)symmetrized separately over the variables corresponding to each group of identical variables, according to its statistics (bosonic or fermionic).
Electrons are fermions with S = 1/2, photons (quanta of light) are bosons with S = 1 (although in the vacuum they are massless and can't be described with Schrödinger mechanics).
When symmetrization or anti-symmetrization is unnecessary, N-particle spaces of states can be obtained simply by tensor products of one-particle spaces, to which we will return later.
### Basis states of one-particle systems
As with any Hilbert space, if a basis is chosen for the Hilbert space of a system, then any ket can be expanded as a linear combination of those basis elements. Symbolically, given basis kets ${\displaystyle |{k_{i))\rangle }$, any ket ${\displaystyle |\psi \rangle }$ can be written
${\displaystyle |\psi \rangle =\sum _{i}c_{i}|{k_{i))\rangle }$
where ci are complex numbers. In physical terms, this is described by saying that ${\displaystyle |\psi \rangle }$ has been expressed as a quantum superposition of the states ${\displaystyle |{k_{i))\rangle }$. If the basis kets are chosen to be orthonormal (as is often the case), then ${\displaystyle c_{i}=\langle {k_{i))|\psi \rangle }$.
One property worth noting is that the normalized states ${\displaystyle |\psi \rangle }$ are characterized by
${\displaystyle \langle \psi |\psi \rangle =1,}$
and for orthonormal basis this translates to
${\displaystyle \sum _{i}\left|c_{i}\right|^{2}=1.}$
Expansions of this sort play an important role in measurement in quantum mechanics. In particular, if the ${\displaystyle |{k_{i))\rangle }$ are eigenstates (with eigenvalues ki) of an observable, and that observable is measured on the normalized state ${\displaystyle |\psi \rangle }$, then the probability that the result of the measurement is ki is |ci|2. (The normalization condition above mandates that the total sum of probabilities is equal to one.)
A particularly important example is the position basis, which is the basis consisting of eigenstates ${\displaystyle |\mathbf {r} \rangle }$ with eigenvalues ${\displaystyle \mathbf {r} }$ of the observable which corresponds to measuring position.[i] If these eigenstates are nondegenerate (for example, if the system is a single, spinless particle), then any ket ${\displaystyle |\psi \rangle }$ is associated with a complex-valued function of three-dimensional space
${\displaystyle \psi (\mathbf {r} )\equiv \langle \mathbf {r} |\psi \rangle .}$[k]
This function is called the wave function corresponding to ${\displaystyle |\psi \rangle }$. Similarly to the discrete case above, the probability density of the particle being found at position ${\displaystyle \mathbf {r} }$ is ${\displaystyle |\psi (\mathbf {r} )|^{2))$ and the normalized states have
${\displaystyle \int \mathrm {d} ^{3}\mathbf {r} \,|\psi (\mathbf {r} )|^{2}=1}$.
In terms of the continuous set of position basis ${\displaystyle |\mathbf {r} \rangle }$, the state ${\displaystyle |\psi \rangle }$ is:
${\displaystyle |\psi \rangle =\int \mathrm {d} ^{3}\mathbf {r} \,\psi (\mathbf {r} )|\mathbf {r} \rangle }$.
### Superposition of pure states
Main article: Quantum superposition
As mentioned above, quantum states may be superposed. If ${\displaystyle |\alpha \rangle }$ and ${\displaystyle |\beta \rangle }$ are two kets corresponding to quantum states, the ket
${\displaystyle c_{\alpha }|\alpha \rangle +c_{\beta }|\beta \rangle }$
is a different quantum state (possibly not normalized). Note that both the amplitudes and phases (arguments) of ${\displaystyle c_{\alpha ))$ and ${\displaystyle c_{\beta ))$ will influence the resulting quantum state. In other words, for example, even though ${\displaystyle |\psi \rangle }$ and ${\displaystyle e^{i\theta }|\psi \rangle }$ (for real θ) correspond to the same physical quantum state, they are not interchangeable, since ${\displaystyle |\phi \rangle +|\psi \rangle }$ and ${\displaystyle |\phi \rangle +e^{i\theta }|\psi \rangle }$ will not correspond to the same physical state for all choices of ${\displaystyle |\phi \rangle }$. However, ${\displaystyle |\phi \rangle +|\psi \rangle }$ and ${\displaystyle e^{i\theta }(|\phi \rangle +|\psi \rangle )}$ will correspond to the same physical state. This is sometimes described by saying that "global" phase factors are unphysical, but "relative" phase factors are physical and important.
One practical example of superposition is the double-slit experiment, in which superposition leads to quantum interference. The photon state is a superposition of two different states, one corresponding to the photon travel through the left slit, and the other corresponding to travel through the right slit. The relative phase of those two states depends on the difference of the distances from the two slits. Depending on that phase, the interference is constructive at some locations and destructive in others, creating the interference pattern. We may say that superposed states are in coherent superposition, by analogy with coherence in other wave phenomena.
Another example of the importance of relative phase in quantum superposition is Rabi oscillations, where the relative phase of two states varies in time due to the Schrödinger equation. The resulting superposition ends up oscillating back and forth between two different states.
### Mixed states
Main article: Density matrix
A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see quantum statistical mechanics).
Mixed states arise in quantum mechanics in two different situations: first, when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations; and second, when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state. In the first case, there could theoretically be another person who knows the full history of the system, and therefore describe the same system as a pure state; in this case, the density matrix is simply used to represent the limited knowledge of a quantum state. In the second case, however, the existence of quantum entanglement theoretically prevents the existence of complete knowledge about the subsystem, and it's impossible for any person to describe the subsystem of an entangled pair as a pure state.
Mixed states inevitably arise from pure states when, for a composite quantum system ${\displaystyle H_{1}\otimes H_{2))$ with an entangled state on it, the part ${\displaystyle H_{2))$ is inaccessible to the observer. The state of the part ${\displaystyle H_{1))$ is expressed then as the partial trace over ${\displaystyle H_{2))$.
A mixed state cannot be described with a single ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted ρ. Note that density matrices can describe both mixed and pure states, treating them on the same footing. Moreover, a mixed quantum state on a given quantum system described by a Hilbert space ${\displaystyle H}$ can be always represented as the partial trace of a pure quantum state (called a purification) on a larger bipartite system ${\displaystyle H\otimes K}$ for a sufficiently large Hilbert space ${\displaystyle K}$.
The density matrix describing a mixed state is defined to be an operator of the form
${\displaystyle \rho =\sum _{s}p_{s}|\psi _{s}\rangle \langle \psi _{s}|}$
where ${\displaystyle p_{s))$ is the fraction of the ensemble in each pure state ${\displaystyle |\psi _{s}\rangle .}$ The density matrix can be thought of as a way of using the one-particle formalism to describe the behavior of many similar particles by giving a probability distribution (or ensemble) of states that these particles can be found in.
A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ρ2 is equal to 1 if the state is pure, and less than 1 if the state is mixed.[l][14] Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.
The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observable A is given by
${\displaystyle \langle A\rangle =\sum _{s}p_{s}\langle \psi _{s}|A|\psi _{s}\rangle =\sum _{s}\sum _{i}p_{s}a_{i}|\langle \alpha _{i}|\psi _{s}\rangle |^{2}=\operatorname {tr} (\rho A)}$
where ${\displaystyle |\alpha _{i}\rangle ,\;a_{i))$ are eigenkets and eigenvalues, respectively, for the operator A, and "tr" denotes trace. It is important to note that two types of averaging are occurring, one being a weighted quantum superposition over the basis kets ${\displaystyle |\psi _{s}\rangle }$ of the pure states, and the other being a statistical (said incoherent) average with the probabilities ps of those states.
According to Eugene Wigner,[15] the concept of mixture was put forward by Lev Landau.[16][13]: 38–41
## Mathematical generalizations
States can be formulated in terms of observables, rather than as vectors in a vector space. These are positive normalized linear functionals on a C*-algebra, or sometimes other classes of algebras of observables. See State on a C*-algebra and Gelfand–Naimark–Segal construction for more details.
## Notes
1. ^ Sometimes written ">"; see angle brackets.
2. ^ To avoid misunderstandings: Here we mean that Q(t) and P(t) are measured in the same state, but not in the same run of the experiment.
3. ^
4. ^ i.e. separated by a zero delay. One can think of it as stopping the time, then making the two measurements one after the other, then resuming the time. Thus, the measurements occurred at the same time, but it is still possible to tell which was first.
5. ^ For concreteness' sake, suppose that A = Q(t1) and B = P(t2) in the above example, with t2 > t1 > 0.
6. ^ Dirac (1958),[11] p. 20: "The bra vectors, as they have been here introduced, are quite a different kind of vector from the kets, and so far there is no connexion between them except for the existence of a scalar product of a bra and a ket."
7. ^ Dirac (1958),[11] p. 19: "A scalar product B|A now appears as a complete bracket expression."
8. ^ Gottfried (2013),[12] p. 31: "to define the scalar products as being between bras and kets."
9. ^ Note that a state ${\displaystyle |\psi \rangle }$ is a superposition of different basis states ${\displaystyle |\mathbf {r} \rangle }$, so ${\displaystyle |\psi \rangle }$ and ${\displaystyle |\mathbf {r} \rangle }$ are elements of the same Hilbert space. A particle in state ${\displaystyle |\mathbf {r} \rangle }$ is located precisely at position ${\displaystyle \mathbf {r} =(x,y,z)}$, while a particle in state ${\displaystyle |\psi \rangle }$ can be found at different positions with corresponding probabilities.
10. ^ Landau (1965),[13] p. 17: "∫ ΨfΨf* dq = δ(f′ − f)" (the left side corresponds to f|f′〉), "∫ δ(f′ − f) df′ = 1".
11. ^ In the continuous case, the basis kets ${\displaystyle |\mathbf {r} \rangle }$ are not unit kets (unlike the state ${\displaystyle |\psi \rangle }$): They are normalized according to ${\displaystyle \textstyle \int \mathrm {d} ^{3}\mathbf {r} '\,\langle \mathbf {r} |\mathbf {r} '\rangle =1,}$[j] i.e. ${\displaystyle \langle \mathbf {r} |\mathbf {r} '\rangle =\delta (\mathbf {r} '-\mathbf {r} )}$ (a Dirac delta function), which means that ${\displaystyle \langle \mathbf {r} |\mathbf {r} \rangle =\infty .}$
12. ^ Note that this criterion works when the density matrix is normalized so that the trace of ρ is 1, as it is for the standard definition given in this section. Occasionally a density matrix will be normalized differently, in which case the criterion is ${\displaystyle \operatorname {Tr} (\rho ^{2})=(\operatorname {Tr} \rho )^{2))$
## References
1. ^ a b Weinberg, S. (2002), The Quantum Theory of Fields, vol. I, Cambridge University Press, ISBN 978-0-521-55001-7
2. ^ Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 978-0-13-111892-8
3. ^ Holevo, Alexander S. (2001). Statistical Structure of Quantum Theory. Lecture Notes in Physics. Springer. ISBN 3-540-42082-7. OCLC 318268606.
4. ^ Peres, Asher (1995). Quantum Theory: Concepts and Methods. Kluwer Academic Publishers. ISBN 0-7923-2549-4.
5. ^ Rieffel, Eleanor G.; Polak, Wolfgang H. (2011-03-04). Quantum Computing: A Gentle Introduction. MIT Press. ISBN 978-0-262-01506-6.
6. ^ Kirkpatrick, K. A. (February 2006). "The Schrödinger-HJW Theorem". Foundations of Physics Letters. 19 (1): 95–102. arXiv:quant-ph/0305068. Bibcode:2006FoPhL..19...95K. doi:10.1007/s10702-006-1852-1. ISSN 0894-9875. S2CID 15995449.
7. ^ "Statistical Mixture of States". Archived from the original on September 23, 2019. Retrieved November 9, 2021.
8. ^ "The Density Matrix". Archived from the original on January 15, 2012. Retrieved January 24, 2012.
9. ^ Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43: 172–198. Translation as 'The actual content of quantum theoretical kinematics and mechanics'. Also translated as 'The physical content of quantum kinematics and mechanics' at pp. 62–84 by editors John Wheeler and Wojciech Zurek, in Quantum Theory and Measurement (1983), Princeton University Press, Princeton NJ.
10. ^ Bohr, N. (1927/1928). The quantum postulate and the recent development of atomic theory, Nature Supplement April 14 1928, 121: 580–590.
11. ^ a b c Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.
12. ^ a b Gottfried, Kurt; Yan, Tung-Mow (2003). Quantum Mechanics: Fundamentals (2nd, illustrated ed.). Springer. ISBN 9780387955766.
13. ^ a b Lev Landau; Evgeny Lifshitz (1965). Quantum Mechanics — Non-Relativistic Theory (PDF). Course of Theoretical Physics. Vol. 3 (2nd ed.). London: Pergamon Press.
14. ^
15. ^ Eugene Wigner (1962). "Remarks on the mind-body question" (PDF). In I.J. Good (ed.). The Scientist Speculates. London: Heinemann. pp. 284–302. Footnote 13 on p.180
16. ^ Lev Landau (1927). "Das Dämpfungsproblem in der Wellenmechanik (The Damping Problem in Wave Mechanics)". Zeitschrift für Physik. 45 (5–6): 430–441. Bibcode:1927ZPhy...45..430L. doi:10.1007/bf01343064. S2CID 125732617. English translation reprinted in: D. Ter Haar, ed. (1965). Collected papers of L.D. Landau. Oxford: Pergamon Press. p.8–18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 94, "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": 0.962897002696991, "perplexity": 382.9934126872589}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662572800.59/warc/CC-MAIN-20220524110236-20220524140236-00271.warc.gz"} |
https://www.physicsforums.com/threads/gravity-assist-by-sun.315234/ | # Gravity-assist by Sun?
1. May 20, 2009
### Jorrie
The use of gravity-assist flybys of planets are well known for interplanetary missions. My question: can a Sun flyby be used to gain orbital energy relative to the Galactic center for interstellar missions?
I assume one must first attain solar escape velocity by other means, since bound orbits around the Sun cannot gain flyby energy. I also assume one can arrange the open orbit such that the spacecraft passes just behind the Sun in its galactic orbit.
If this has already been discussed, my apologies - just point me to the thread.
-J
2. May 20, 2009
### D H
Staff Emeritus
No, for two reasons. (1) The trajectory needs to take the vehicle fairly close to the planet to make the gravity assist effective. (2) The Sun's orbital velocity around the Sun-Jupiter center of mass is about 74 meters/second. There's not much velocity to steal.
3. May 20, 2009
### Jorrie
But the Sun's orbital velocity around the Galactic center is some 220 km/s, which is more relevant to the question.
4. May 20, 2009
### sylas
We already have that velocity... The Sun might be useful to a traveler coming in from another galaxy. But I presume that was not what you had in mind.
5. May 20, 2009
### Jorrie
Isn't arriving at the Sun at or above solar escape velocity (as per OP) equivalent to coming in from another star? With planetary gravity assist, all that is required is coming in correctly at or above the planet's escape velocity.
6. May 20, 2009
### fatra2
Hi there,
Not at all. When leaving, heading toward the Sun, you already have that 220km/s velocity. We are on Earth, which is following the Sun (hopefully for us), therefore, we live in roughly the same reference as the sun, therefore having the same velocity.
7. May 20, 2009
### Jorrie
In the same way, the craft doing a flyby at Earth already has Earth's 30 km/s relative to the Sun. Yet, it can get a delta-V in heliocentric coordinates if it comes in at a relative speed above Earth's escape velocity and at the right angle, passing 'behind' Earth. Why can't this principle work in Galactic coordinates with the Sun as well?
8. May 20, 2009
### Jorrie
Hi All. OK, it looks like since I've asked a 'half-baked' question, I've got a few 'half-baked' answers! Let me try and state the question more clearly.
I think it is accepted that if a spacecraft were to come from in another star's vicinity, naturally arriving in the vicinity of the Sun at a speed exceeding solar escape speed, it could get a gravity-assist energy boost from the Sun. With this is understood that the spacecraft gained some mechanical orbital energy in Galactic coordinates.
Now let a spacecraft from Earth be 'directly' boosted towards Jupiter for a gravity-assist that boosts its energy further relative to the Sun and takes it to Saturn (Cassini-like). At Saturn its orbit is arranged for a flyby that does two things: slingshot it to back to near the Sun at somewhat above solar escape velocity, having robbed some of Saturn's orbital energy. (If it had the means of propulsion, an advanced technology could just as well have shot the spacecraft 'directly' from Earth to the vicinity of the Sun, ensuring that it exceeds escape speed of the Sun).
The spacecraft passes as close as possible 'behind' the Sun (relative to the Sun's galactic orbit) and naturally exceeds local solar escape velocity. Is there any reason why this spacecraft cannot receive a gravity-assist delta-V from the Sun? In other words, why would this scenario not be equivalent to a spacecraft coming in from another star?
Last edited: May 21, 2009
9. May 21, 2009
### Janus
Staff Emeritus
A gravity assist is basically a transfer of momentum from one object (in this case the Sun) to another(the spacecraft), where gravity provides the mechanism of the momentum transfer. It is essentially an elastic collision without the physical contact.
So let's use an elastic collision to explain what's going on.
Let's say you have a massive truck moving down the road at 100 kph. We are standing at the side of the road holding a small ball. We toss the ball forward at 10 kph, so that passes in front of the truck and the truck hits it. At the moment of impact, the ball is moving forward at 10 kph and the truck 90 kph, so the relative velocity velocity of the two is 90 kph.
Upon impact, the ball rebounds from the truck with the same 90 kph relative speed, but its relative velocity is in the opposite direction, so that relative to you it is moving 90 kph + 100 Kph = 190 kph. It has picked up 180 kph relative to the road in the collision.
This is more or less how a standard gravity assist works.
Now imagine that you are traveling along the road just ahead of the truck at 100 kph. You want to duplicate the impact of the above experiment. This means that you have to throw the ball backwards at the truck at 90 kph. The ball hits the truck, rebounds and then passes you with a relative velocity of 90 kph in the forward direction (again moving 190 kph relative to the road.) But that is just as fast as you threw it; you would have done just as well if you had just thrown the ball 90 kph forward in the first place. You gained nothing by bouncing the ball off of the truck.
This is what you are trying to do by using the Sun in a gravity assist for a spacecraft launched from within the Solar system. You wouldn't gain anything by matching the velocity of an incoming object in order to cause it to whip around the Sun over just launching your spaceship in the direction of the Sun's galactic motion to begin with.
(Besides, it is a lot easier to use an outer planet to give a probe a gravity boost that takes it directly out of the Solar system than it is to try give it velocity boost while at the same time kicking it in toward the Sun.)
10. May 21, 2009
### Jorrie
Thanks for the very clear explanation Janus. I understand now.
11. May 24, 2009
### qraal
Of course you can still get a decent gravity well boost from a solar "fry-by" and doing a perihelion burn. Even more of kick can be gotten from doing likewise around a white-dwarf, but naturely much closer than you can get to the Sun. And then there's the super-boost you can extract from a Dyson gravity machine - twin white-dwarfs in very close orbits, which can boost you to ~3,000 km/s or so.
Similar Discussions: Gravity-assist by Sun? | {"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": 0.8143130540847778, "perplexity": 953.9664444097353}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1495463609409.62/warc/CC-MAIN-20170528024152-20170528044152-00464.warc.gz"} |
https://www.general-relativity.net/2021/01/photon-sphere.html | ## Thursday, 28 January 2021
### Photon sphere
M87* By Event Horizon Telescope
I come across the photon sphere (or last photon orbit) which is another radius around a Schwarzschild black hole at a distance $3R_s/2$. It is the closest distance for a stable orbit and light would orbit there in an exact circle. I would like to calculate it using Carroll conventions. $R_s=2GM$ is the Schwarzschild radius.
There is a Wikipedia article on it. The proof it gives has some peculiarities. I followed it avoiding those. It is one page.
Here: Commentary 5.4 Photon Sphere.pdf
The famous picture is of M87* the black hole in our neighbour galaxy Messier 87. The black central sphere has a radius of $2.6R_s$ so the photon sphere is well inside that black bit and this is not a good illustration of a photon sphere! | {"extraction_info": {"found_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": 0.9358106851577759, "perplexity": 792.044590171472}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178385534.85/warc/CC-MAIN-20210308235748-20210309025748-00118.warc.gz"} |
https://nroer.gov.in/55ab34ff81fccb4f1d806025/file/588723f1472d4a1fef811680 | ### Recognising Rectilinear Motion:
How can we recognize the rectilinear motion of an object? Rectilinear motion is an alternative name for straight-line motion. | {"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": 0.9509244561195374, "perplexity": 2399.7756854776926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1573496670948.64/warc/CC-MAIN-20191121180800-20191121204800-00261.warc.gz"} |
https://www.physicsforums.com/threads/adm-approach.76061/ | 1. May 18, 2005
### Blackforest
Certainly an easy question for you (specialists on this forum) but absolutely not clear for me.
The ADM Approach (1 + 3), as it can be seen in Misner Thorne and Wheeler or some other reference, is based on a kind of extension of the Pythagorras Theorem to a Minkowki space. Ok.
Its signature is (- +++); convention, I suppose; ok.
And then you can see in some equations resulting of this approach a (3-3) matrix supposed to represent the local 3D metric tensor as if it could have any value and not obligatory the unit matrix I(3) which is the spatial part of the (4-4) one associated to a Lorentz metric.
Why? Does it come from the manner to "cut" the slice of time? I am sure something is not ok in my head concerning this point. Thanks for help.
2. May 18, 2005
### HallsofIvy
Staff Emeritus
What coordinate system is being used? There is nothing "obligatory" about
I(3)! One can show that as long as the coordinate axes are orthogonal to one another the metric tensor is diagonal but even in flat space, with spherical coordinates, it is not I(3). The reason tensors were so important in general relativity is that as soon as there is mass, space is not flat and the metric tensor cannot be I(3) with any coordinate system.
3. May 18, 2005
### Blackforest
Ok. But as there are masses everywhere around us here on the earth and as there is probably a lot of energy (equivalent to mass, isn'it?) in vacuum (even if only a pair of photons per cubic kilometer), the metric tensor is never exactly of Lorentz or I(3) for the spatial part and then it seems to me incoherent or at least very difficult to built something based on the Pythagoras Theorem... I don't know if I am clear enough in my way to explain, but I am lost. I think I must re-read the demonstration of the ADM Approach. Thanks for help
4. May 18, 2005
### pervect
Staff Emeritus
I'm not, unfortunately, that familiar with the ADM approach, but the discussion in MTW seems to be reasonably clear on how the split is done into space and time.
So you wind up replacing a 4-d metric with two 3-d metrics, a lapse function, and a shift function. The 3-d metrics give the metric of each hypersurface, the lapse function gives the distance between the upper and lower hypersurfaces, and the shift funciton tells us which point on the lower surface is connected to which point on the higher surface.
5. May 18, 2005
### Blackforest
So. I shall try to explain what is introducing a doubt in my head concerning this demonstration. The ADM procedure consists to define the necessary tools to connect the 3D geometry at time t in M to the 3D one at time t+dt in M + dM. There is no "a priori" concerning the first geometry nor the second one. Because of this one can expect the absence of restriction on the metric tensor (3D), specially in presence of matter (your remark). This means: the ADM construction is based on a any geometry. But why are we authorized to calculate the relation between proper time and proper length, distances, with the Lorentzian type of metric ? Thank you for the ligth.
6. May 18, 2005
### Stingray
General relativity assumes that spacetime always posseses a metric with Lorentz signature (which is generally not Minkowski). The "Pythagorean theorem" that I think you're referring to is usually what is used to define the metric (in part). It's an assumption that is formally contained in the structure of GR.
7. May 19, 2005
### Blackforest
Cartan's work tells us that any quadratic form can be reduced to a sum of squares. Your assumption is equivalent to: any quadratic form of the GR can be reduced to a sum of squares with signature - +++; correct? And so one can apply the Pythagorean theorem (thank you for the correct translation of this name and for the helps): correct? I think I begin to understand why the ADM Approach is not reducing the generality (I was afraid it could have...)
Subsidiary questions: Is there consequently no authorized transformation within the GR that permits a change in the signature? What would represent a change of the signature in the language of the geometry?
8. May 19, 2005
### Stingray
Correct. If the signature changed, then physics would look very different.
What do you mean about the ADM approach "reducing generality?" Their formalism (an extension of work by Lichnerowicz) derives the constraint and evolution equations of the metric (or more precisely, of the 3-metric and extrinsic curvature - i.e. the first and second fundamental forms of the hypersurfaces) from Einstein's equation. The reasons for splitting up spacetime into a bunch of spacelike hypersurfaces are to show that GR has a well-defined initial value problem and to put it into Hamiltonian form (which is useful for many things).
9. May 19, 2005
### Blackforest
Oh my first anxiety has now disappeared, had no justification (or more exactly had justification due to a bad analyze of the situation) and was the matter of the present discussion.
But the result of our discussion is that only 3D metric tensor with a reduction -+++ are admissible within the ADM Approach. All other one are corresponding to something else not necessary related to a physical reality; or at least not to real phenomena described by the GR. This is ambarassing for my own approach that seems to embed quite more mathematical cases.
In this sense I have now the feeling that the ADM approach is reducing a mathematical generality. Or perhaps I need your help again to go further.
Thanks for the present discussion.
10. May 19, 2005
### hellfire
I am not familiar with this, but it seams obvious to me that you are right when you say that there is a loss of generality in the mathematics of the theory, as only manifolds which can be foliated in non-intersecting spacelike surfaces can be treated with the ADM formalism. Otherwise you can have also other kind of manifolds in GR, the question is whether they are physically meaningful. Is this what you mean?
11. May 19, 2005
### Blackforest
I am not familiar with this too (unfortunately as say Pervect).
Yes, it is what I mean; but not only. You are right when you speak about the ADM formalism (non intersecting spacelike manifolds). My way of doing (if you have followed it) represents a different approach in the sense that I am exploring as generally as possible a family of equations. The starting point is the derivation along the time of the Poynting's vector. But I left this particular case and tried to win more generality, only regarding and studying the formalism introduced by this partial derivation. 3D solutions to this mathematical question show themselves a formalism owning a beautiful analogy with some equations of the ADM approach. Thus "if the comparison is physically meaningful", I got a way to reconnect my calculations with the ADM approach. This is exactly what I am actually exploring.
It is really interesting if one consider a propagating plane EM wave as a spacelike manifold at different times in a chronology (a natural superposition of thin sandwiches). My approach seems to work correctly for the description of this kind of waves. It seems also to fit for waves that are not exactly plane. And it "seems" to fit for the ADM approach: I am still exploring this important point...
12. May 19, 2005
### Stingray
I'm having trouble understanding your english (sorry), but have you looked at the characteristic formulation of GR? There, you specify initial data on a null hypersurface rather than a spacelike one.
13. May 20, 2005
### Blackforest
In fact, I am sorry for this. If you can speak French or German, please don’t be afraid to e-mail me. Our conversation will be really easier*.
Apparently a pretentious attitude but in reality just another motivation is leading my research. As said before, my first preoccupation was to explore the following intuition: does any cross product split in accordance with the local geometry? If yes when and how. So, at the beginning, I didn’t consider the problematic with the standard eyes of the GR. It doesn’t mean that I refuse to do it now: our conversation is a prove of that.
At each given point M and at each given time t in the universe, one can reasonably expect to encounter one and only one geometry in the proper chronology of the event (M, t) under consideration. This local topology is the common “thing” between all characteristics and properties defining what is happening on M at t; and so it is for all cross products of any type: the rot E x E, the rot H x H, the u x w and also the r x v that are angular momentum and quantized.
This preoccupation about a connection between the topology and what happens is a strategic point of view in my approach and the reason why I feel allowed to post here in “Special and General Relativity”. The connection appears under the form of a “conique (conoide)”:
A. x² + B. y² +C. z² + D. x. y + E. y. z + F. x. z + G. x. z + H. x + J. y + K. z + L = 0
I simply explore what happens if and when the spatial part of the geodesic “followed” by a photon is in coincidence with this conoide.
Sorry, I can’t explain this better* and I do not have achieved my explorations concerning the initial data problem (which is in particularly implying a time symmetric and a time anti symmetric one). As said, first motivations was centred on the exploration of the junction: (topology / splitting of a cross product) and (topology / splitting of a mathematical extension of the cross product). Certainly not very original. | {"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": 0.8503431677818298, "perplexity": 630.4683588441404}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281659.81/warc/CC-MAIN-20170116095121-00058-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://clay6.com/qa/51255/a-certain-first-order-reaction-proceeds-through-t-first-order-kinetics-the- | # A certain first order reaction proceeds through t first order kinetics. The half-life of the reaction is 180 s. What percent of the initial concentration remains after 900s?
$\begin{array}{1 1} 3.12 \% \\ 6.24\;\% \\ 1.56\% \\ 9.36\;\% \end{array}$
From the integrated rate law for a first order reaction, $[A] = [A]_0\; e^{-kt}$.
The half life $t_{1/2} = \large\frac{ ln\;2}{k}$$= \large\frac{0.693}{k}. Given t_{1/2} = 180\;s \rightarrow k = \large\frac{0.693}{180\;s} = 0.00385 s^{-1} The fraction remaining is the concentration divided by the initial concentration = \large\frac{[A]}{[A_0]}$$ = e^{kt} = e^{0.00385 s^{-1} \times 900\; s}$$= 0.0312 = 3.12\%$
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https://www.physicsforums.com/threads/light-and-relativistic-mass.587880/ | Light and Relativistic Mass
1. Mar 17, 2012
thedude36
Hi, a while back i had ran into the equation for dynamic mass [M = m/√(1-(v^2/c^2))] and had immedietly taken an interest in it in that it seems to imply that for an object with mass moving at the speed of light its mass would become, virtually, infinite. Is this an erroneous assumption? If not, what would the implications for light itself be? Could the massless photon be explained by the idea that, because it moves at the speed of light, its mass has increased to such a point that the photon can no longer be distinguished as a discreet entity in so far as the measure of its mass is concerned?
2. Mar 17, 2012
clamtrox
Well, most physicists would call that quantity the energy of the particle. As you know, photons have a finite energy and move at the speed of light, so you would be completely right to conclude that they must have zero (rest) mass.
3. Mar 17, 2012
thedude36
If photons have a finite amount of energy, wouldn't that require a non-zero quantity for its mass? and if it has a measurable mass, wouldnt it be restricted from travelling at the speed of light?
4. Mar 17, 2012
yuiop
The photon's rest mass has not increased. It s rest mass is zero and is not subject to the relativistic mass increase that you mention above. The concept of relativistic mass is not encouraged these days as it is confusing. For example from the point of view of a muon passing the Sun, the apparent mass of the Sun is enough for it to be a black hole but that is not the case. Relativistic mass is just a measure of how much energy is required to accelerate a given mass from a rest state to its current velocity in a given reference frame . Active gravitation appears to be a property of rest mass and this does not increase with relative velocity. Rest mass can only be defined for an object that has a definable rest reference frame so this does not apply to a photon. Curiously, although a single photon has no rest mass, a pair of photons going in opposite directions can have a definable centre of momentum frame and thus have a combined rest frame. I imagine this is why a cloud of photons can in theory be a source of active gravitational mass. Photons are also curious in that they have a definable momentum and can impart momentum to particles with rest mass when they collide with them. Photons also have a very clearly defined total energy that is a function of their frequency and this total energy is nowhere near infinite.
5. Mar 17, 2012
Staff: Mentor
The general relationship between energy, momentum, and mass is
$$E^2 = (pc)^2 + (mc^2)^2$$
where m is the invariant mass (often called the "rest mass"), which is zero for a photon. So a photon has E = pc.
6. Mar 17, 2012
thedude36
but if momentum is
$$p^μ=mv^μ$$
wouldnt a massless particle render the entire expression as 0? or does p represent something other than momentum?
7. Mar 17, 2012
HallsofIvy
That formula is not correct for relativistic speeds.
8. Mar 17, 2012
Staff: Mentor
Assuming you mean the four-momentum and four-velocity here, what's the four-velocity of a photon?
9. Mar 17, 2012
yuiop
The expanded expression for total energy E is:
$$E^2 = \left( \frac{m v c}{\sqrt{1-v^2/c^2}} \right)^2 + (mc^2)^2$$
For a photon v = c, so this becomes:
$$E^2 = \left( \frac{0}{0} \right)^2 + (0)^2 = \frac{0}{0}$$
which is undefined (so not equal to zero).
This is resolved by using the alternative expression for momentum:
$$p = hf/c$$
where h is the Planck constant and f is the frequency. Apparently this expression applies to both massless and massive particles via the DeBroglie wavelength relationship.
Anyway, the total energy of a photon using the above momentum expression is E = hf.
10. Mar 26, 2012
thedude36
Thank you yuiop for the thorough replies. this is definitely something i will have to think on before completely grasping it, but you have answered my question.
11. Mar 26, 2012
elfmotat
$p^\mu = mv^\mu$ is most definitely correct at relativistic speeds, assuming this is what you were responding to.
12. Mar 27, 2012
rbj
the reason that photons are massless (that their rest mass or "invariant mass" is zero), is that their speed relative to any reference frame is $c$. turn your equation around a little:
$$m = M \sqrt{1 - v^2/c^2}$$
if $v=c$, then $m=0$ no matter what the photon's momentum or inertial mass $M = p/c$ is.
13. Mar 28, 2012
thedude36
Im curious though, if an object other than a photon would reach the speed of light, (and this is purely hypothetical), would its mass become infinite?
14. Mar 28, 2012
squareroot
hypothetical yes.
15. Mar 28, 2012
Staff: Mentor
And impossible.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook | {"extraction_info": {"found_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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.915248453617096, "perplexity": 490.63529734622205}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2018-30/segments/1531676589710.17/warc/CC-MAIN-20180717105812-20180717125812-00318.warc.gz"} |
https://arxiv.org/abs/1912.12878 | # Title:Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry
Abstract: We show that the Lorentzian Snyder models, together with their non-relativistic ($c\to\infty$) and ultra-relativistic ($c\to0$) limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of these spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant kinematical symmetries. While the momentum space of the Lorentzian Snyder models is given by certain projective coordinates on (Anti-) de Sitter spaces, we discover that the momentum space of the Galilean (Carrollian) Snyder models is given by certain projective coordinates on curved Carroll (Newton--Hooke) spaces. This exchange between the non-relativistic and ultra-relativistic limits emerging in the transition from the geometric picture to the phase space picture is traced back to an interchange of the role of coordinates and translation operators. As a physically relevant feature, we find that in Galilean Snyder spacetimes the time coordinate does not commute with space coordinates, in contrast with previous proposals for non-relativistic Snyder models, which assume that time and space decouple in the non-relativistic limit. This remnant mixing between space and time in the non-relativistic limit is a quite general Planck-scale effect found in several quantum spacetime models.
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph) Cite as: arXiv:1912.12878 [hep-th] (or arXiv:1912.12878v1 [hep-th] for this version)
## Submission history
From: Giulia Gubitosi [view email]
[v1] Mon, 30 Dec 2019 10:54:19 UTC (32 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": 0.9118614196777344, "perplexity": 1232.7453351748954}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579251681412.74/warc/CC-MAIN-20200125191854-20200125221854-00421.warc.gz"} |
https://getrevising.co.uk/resources/12-elements-of-life | # 1.2 Elements of Life
HideShow resource information
First 113 words of the document:
1.2 Elements of Life
Spectroscopy The study of how light and matter interact, based of the principle that under certain
conditions, a substance can absorb or emit electromagnetic radiation
Absorption Spectra Emission Spectra
Absorption Spectra Coloured background, dark lines
Emission Spectra Black background, coloured likes
Wave theory
Light is a form of electromagnetic radiation
It behaves like a wave and has both wavelength and
frequency
The speed that a wave of light travels between two
points is the same. It has a value of 3.00 x 108
m s1
Different coloured light have different wavelengths
As wavelength increases, frequency decreases
## Other pages in this set
### Page 2
Here's a taster:
Particle Theory
Light is regarded as a stream of tiny energy packets called photons
The energy of a photon relates to its position on the electromagnetic spectrum
Bohr's theory
The electrons in an atom exist only in certain definite energy levels/electron shells/quanta
When an atom is excited, its electrons jump into higher energy levels
Later the electrons drop beck into lover energy levels and emit/absorb a photon of
The energy of the photon is equal to the difference between the two energy levels…read more
### Page 3
Here's a taster:
Flame tests
Ion Colour Picture
Li+ Bright Red
Na +
Yellow
K+
Here's a taster:
Ca2+ Brick Red
Ba2+ Apple Green
Cu 2+ | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9128755331039429, "perplexity": 4754.594110189107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1484560281001.53/warc/CC-MAIN-20170116095121-00044-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://minutemath.com/algebra-2/solve-a-formula-for-a-specific-variable/ | # 2.3 Solve a Formula for a Specific Variable
Topics covered in this section are:
## 2.3.1 Solve a Formula for a Specific Variable
We have all probably worked with some geometric formulas in our study of mathematics. Formulas are used in so many fields, it is important to recognize formulas and be able to manipulate them easily.
It is often helpful to solve a formula for a specific variable. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equals sign with a coefficient of one and all other variables and constants are on the other side of the equal sign.
Geometric formulas often need to be solved for another variable, too. The formula $V=\frac{1}{3}πr^{2}h$ is used to find the volume of a right circular cone when given the radius of the base and height. In the next example, we will solve this formula for the height.
#### Example 1
Solve the formula $V=\frac{1}{3}πr^{2}h$ for $h$.
Solution
We could now use this formula to find the height of a right circular cone when we know the volume and the radius of the base, by using the formula $h=\frac{3V}{πr^{2}}$.
In the sciences, we often need to change temperature from Fahrenheit to Celsius or vice versa. If you travel in a foreign country, you may want to change the Celsius temperature to the more familiar Fahrenheit temperature.
#### Example 2
Solve the formula $C=\frac{5}{9}(F-32)$ for $F$.
Solution
We can now use the formula $F=\frac{9}{5}C+32$ to find the Fahrenheit temperature when we know the Celsius temperature.
The next example uses the formula for the surface area of a right cylinder.
#### Example 3
Solve the formula $S=2πr^{2}+2πrh$ for $h$.
Solution
Sometimes we might be given an equation that is solved for $y$ and need to solve it for $x$, or vice versa. In the following example, we’re given an equation with both $x$ and $y$ on the same side and we’ll solve it for $y$.
#### Example 4
Solve the formula $8x+7y=15$ for $y$.
Solution
## 2.3.2 Use Formulas to Solve Geometry Applications
In this objective we will use some common geometry formulas. We will adapt our problem solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve.
In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.
### HOW TO: Solve geometry applications.
1. Read the problem and make sure all the words and ideas are understood.
2. Identify what you are looking for.
3. Name what we are looking for by choosing a variable to represent it. Draw the figure and label it with the given information.
4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.
When we solve geometry applications, we often have to use some of the properties of the figures. We will review those properties as needed.
The next examples involves the areas of a triangle. The area of a triangle is one-half base times the height. We can write this as $A=\frac{1}{2}bh$, where $b=$ length of the base and $h=$ height.
#### Example 5
The area of a triangular painting is $126$ square inches. The base is $18$ inches. What is the height?
Solution
In the next example, we will work with a right triangle. To solve for the measure of each angle, we need to use two triangle properties. In any triangle, the sum of the measures of the angles is $180^{\circ}$. We can write this as a formula: $m∠A+m∠B+m∠C=180$. Also, since the triangle is a right triangle, we remember that a right triangle has one $90^{\circ}$ angle.
Here, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.
#### Example 6
The measure of one angle of a right triangle is $40$ degrees more than the measure of the smallest angle. Find the measures of all three angles.
Solution
The next example uses another important geometry formula. The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. Writing the formula in every exercise and saying it aloud as you write it may help you memorize the Pythagorean Theorem.
### THE PYTHAGOREAN THEOREM
In any right triangle, where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse.
We will use the Pythagorean Theorem in the next example.
#### Example 7
Use the Pythagorean Theorem to find the length of the other leg in
Solution
The next example is about the perimeter of a rectangle. Since the perimeter is just the distance around the rectangle, we find the sum of the lengths of its four sides—the sum of two lengths and two widths. We can write is as $P=2L+2W$ where $L$ is the length and $W$ is the width. To solve the example, we will need to define the length in terms of the width.
#### Example 8
The length of a rectangle is six centimeters more than twice the width. The perimeter is $96$ centimeters. Find the length and width.
Solution
The next example is about the perimeter of a triangle. Since the perimeter is just the distance around the triangle, we find the sum of the lengths of its three sides. We can write this as $P=a+b+c$, where $a$, $b$, and $c$ are the lengths of the sides.
#### Example 9
One side of a triangle is three inches more than the first side. The third side is two inches more than twice the first. The perimeter is $29$ inches. Find the length of the three sides of the triangle.
Solution
#### Example 10
The perimeter of a rectangular soccer field is $360$ feet. The length is $40$ feet more than the width. Find the length and width.
Solution
Applications of these geometric properties can be found in many everyday situations as shown in the next example.
#### Example 11
Kelvin is building a gazebo and wants to brace each corner by placing a $10$ inch piece of wood diagonally as shown.
How far from the corner should he fasten the wood if wants the distances from the corner to be equal? Approximate to the nearest tenth of an inch.
Solution | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8566051721572876, "perplexity": 283.8712774177692}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00795.warc.gz"} |
https://alexpghayes.github.io/distributions3/reference/Multinomial.html | The multinomial distribution is a generalization of the binomial distribution to multiple categories. It is perhaps easiest to think that we first extend a Bernoulli() distribution to include more than two categories, resulting in a Categorical() distribution. We then extend repeat the Categorical experiment several ($$n$$) times.
Multinomial(size, p)
## Arguments
size The number of trials. Must be an integer greater than or equal to one. When size = 1L, the Multinomial distribution reduces to the categorical distribution (also called the discrete uniform). Often called n in textbooks. A vector of success probabilities for each trial. p can take on any positive value, and the vector is normalized internally.
## Value
A Multinomial object.
## Details
We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail and much greater clarity.
In the following, let $$X = (X_1, ..., X_k)$$ be a Multinomial random variable with success probability p = $$p$$. Note that $$p$$ is vector with $$k$$ elements that sum to one. Assume that we repeat the Categorical experiment size = $$n$$ times.
Support: Each $$X_i$$ is in $${0, 1, 2, ..., n}$$.
Mean: The mean of $$X_i$$ is $$n p_i$$.
Variance: The variance of $$X_i$$ is $$n p_i (1 - p_i)$$. For $$i \neq j$$, the covariance of $$X_i$$ and $$X_j$$ is $$-n p_i p_j$$.
Probability mass function (p.m.f):
$$P(X_1 = x_1, ..., X_k = x_k) = \frac{n!}{x_1! x_2! ... x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot ... \cdot p_k^{x_k}$$
Cumulative distribution function (c.d.f):
Omitted for multivariate random variables for the time being.
Moment generating function (m.g.f):
$$E(e^{tX}) = (\sum_{i=1}^k p_i e^{t_i} )^n$$
Other discrete distributions: Bernoulli, Binomial, Categorical, Geometric, HyperGeometric, NegativeBinomial, Poisson
## Examples
set.seed(27)
X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X#> Multinomial distribution (size = 5, p = 0.3) Multinomial distribution (size = 5, p = 0.4) Multinomial distribution (size = 5, p = 0.2) Multinomial distribution (size = 5, p = 0.1)
random(X, 10)#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,] 5 5 5 5 5 5 5 5 5 5
# pdf(X, 2)
# log_pdf(X, 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": 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": 0.9462911486625671, "perplexity": 686.6238381578453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20191113191653-20191113215653-00180.warc.gz"} |
https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/z-statistics-vs-t-statistics | 0 energy points
Z-statistics vs. T-statistics
Video transcript
I want to use this video to kind of make sure we intuitively and otherwise and understand the difference between a Z-statistic-- something I have trouble saying-- and a T-statistic. So in a lot of what we're doing in this inferential statistics, we're trying to figure out what is the probability of getting a certain sample mean. So what we've been doing, especially when we have a large sample size-- so let me just draw a sampling distribution here. So let's say we have a sampling distribution of the sample mean right here. It has some assumed mean value and some standard deviation. What we want to do is any result that we get, let's say we get some sample mean out here. We want to figure out the probability of getting a result at least as extreme as this. So you can either figure out the probability of getting a result below this and subtracted that from 1, or just figure out this area right over there. And to do that we've been figuring out how many standard deviations above the mean we actually are. The way we figured that out is we take our sample mean, we subtract from that our mean itself, we subtract from that what we assume the mean should be, or maybe we don't know what this is. And then we divide that by the standard deviation of the sampling distribution. This is how many standard deviations we are above the mean. That is that distance right over there. Now, we usually don't know what this is either. We normally don't know what that is either. And the central limit theorem told us that assuming that we have a sufficient sample size, this thing right here, this thing is going to be the same thing as-- the sample is going to be the same thing as the standard deviation of our population divided by the square root of our sample size. So this thing right over here can be re-written as our sample mean minus the mean of our sampling distribution of the sample mean divided by this thing right here-- divided by our population mean, divided by the square root of our sample size. And this is essentially our best sense of how many standard deviations away from the actual mean we are. And this thing right here, we've learned it before, is a Z-score, or when we're dealing with an actual statistic when it's derived from the sample mean statistic, we call this a Z-statistic. And then we could look it up in a Z-table or in a normal distribution table to say what's the probability of getting a value of this Z or greater. So that would give us that probability. So what's the probability of getting that extreme of a result? Now normally when we've done this in the last few videos, we also do not know what the standard deviation of the population is. So in order to approximate that we say that the Z-score is approximately, or the Z-statistic, is approximately going to be-- so let me just write the numerator over again-- over, we estimate this using our sample standard deviation-- let me do this in a new color-- with using our sample standard deviation. And this is OK if our sample size is greater than 30. Or another way to think about it is this will be normally distributed if our sample size is greater than 30. Even this approximation will be approximately normally distributed. Now, if your sample size is less than 30, especially if it's a good bit less than 30, all of a sudden this expression will not be normally distributed. So let me re-write the expression over here. Sample mean minus the mean of your sampling distribution of the sample mean divided by your sample standard deviation over the square root of your sample size. We just said if this thing is well over 30, or at least 30, then this value right here, this statistic, is going to be normally distributed. If it's not, if this is small, then this is going to have a T-distribution. And then you're going to do the exact same thing you did here, but now you would assume that the bell is no longer a normal distribution, so this example it was normal. All of Z's are normally distributed. Over here in a T-distribution, and this will actually be a normalized T-distribution right here because we subtracted out the mean. So in a normalized T-distribution, you're going to have a mean of 0. And what you're going to do is you want to figure out the probability of getting a T-value at least this extreme. So this is your T-value you would get, and then you essentially figure out the area under the curve right over there. So a very easy rule of thumb is calculate this quantity either way. Calculate this quantity either way. If you will have more than 30 samples, if your sample size is more than 30, your sample standard deviation is going to be a good approximator for your population standard deviation. And so this whole thing is going to be approximately normally distributed, and so you can use a Z-table to figure out the probability of getting a result at least that extreme. If your sample size is small, then this statistic, this quantity, is going to have a T-distribution, and then you're going to have to use a T-table to figure out the probability of getting a T-value at least this extreme. And we're going to see this in an example a couple of videos from now. Anyway, hopefully that helped clarify some things in your head about when to use a Z-statistic or when to use a T-statistic. | {"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": 0.9335247278213501, "perplexity": 156.26678141601985}, "config": {"markdown_headings": false, "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-2017-22/segments/1495463611560.41/warc/CC-MAIN-20170528200854-20170528220854-00295.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-2-solving-equations-2-1-solving-one-step-equations-practice-and-problem-solving-exercises-page-85/39 | # Chapter 2 - Solving Equations - 2-1 Solving One-Step Equations - Practice and Problem-Solving Exercises - Page 85: 39
q=81
#### Work Step by Step
In order to solve this algebraic expression, we must isolate the variable. A variable is the letter in the problem (for instance, in the equation 2x=10, x is the variable). The section header tells us to solve the equation using “multiplication or division,” so we know that the only things we should do to solve this problem are multiplication and division. Because multiplication and division are inverse operations (they cancel each other out), we know we must divide when a number is being multiplied by the variable and multiply when the variable is being divided by a number. In the equation $-9=q/-9$, we must get rid of the -9, which the variable is being divided by, in order to solve. Thus, we multiply both sides by -9 to find that $q=81$. We plug 81 in for q and confirm that our answer is, indeed, correct.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | {"extraction_info": {"found_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": 0.8170191049575806, "perplexity": 446.1311806017867}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827175.38/warc/CC-MAIN-20181216003916-20181216025916-00449.warc.gz"} |
https://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=speaker&order_type=desc&page=25&document_srl=796540&l=en | In this first talk, we discuss the notion of oriented cohomology theory of algebraic varieties, following Levine-Morel. We discuss the construction of the universal theory, called algebraic cobordism first, and present its connections to Chow groups and the Grothendieck groups. We will also discuss another construction by Levine-Pandharipande. | {"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": 0.9626356959342957, "perplexity": 399.3990716594788}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711344.13/warc/CC-MAIN-20221208150643-20221208180643-00070.warc.gz"} |
https://www.sparrho.com/item/orientational-dynamics-in-nematic-liquid-crystals/969799/ | Orientational dynamics in nematic liquid crystals
Research paper by A. Humpert, A.J. Masters; M.P. Allen
Indexed on: 11 Aug '16Published on: 18 Jul '16Published in: The European Physical Journal Special Topics
Abstract
Abstract We examine the behaviour of single-particle orientational time correlation functions in nematic liquid crystals. As well as the expected dynamics involving oscillation in a mean-field potential, and occasional jumps between orientations parallel and antiparallel to the director, we provide the first simulation evidence of long-time tails characteristic of coupling to director fluctuations.AbstractWe examine the behaviour of single-particle orientational time correlation functions in nematic liquid crystals. As well as the expected dynamics involving oscillation in a mean-field potential, and occasional jumps between orientations parallel and antiparallel to the director, we provide the first simulation evidence of long-time tails characteristic of coupling to director fluctuations. | {"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": 0.8076507449150085, "perplexity": 2759.570181972911}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107894426.63/warc/CC-MAIN-20201027170516-20201027200516-00164.warc.gz"} |
http://www.physicsforums.com/showthread.php?s=a5c8def21d82c210a214e2eae252b462&p=4781026 | # Why Dielectric Constant is Constant ?
by onurbeyaz
Tags: constant, dielectric
P: 13 What I meant to ask is; When we double the electrical field passing through the insulator, the opposite electrical field that caused by the insulator is doubled too. How can this happen, what happens to the molecules in that time to increase the opposite electrical field? In my opinion, the polarized molecules have to strech (Distance between + and - charged particles in the molecules have to increase) But it doesn't make sense, because this means the generated electrical field have to decrease.
Sci Advisor PF Gold P: 1,777 When we place a dielectric in an electric field, the field is reduced. This is because it loses energy that is stored by polarizing the dielectric. In the dielectric, a dipole moment is induced whose strength is proportional to the applied field and the permittivity of the dielectric. This dipole moment produces a secondary field that weakens the applied field. So if we increase the applied field, we increase the polarization field which weakens the applied field. However, this weakening is always a fraction of the applied field.
Sci Advisor Thanks PF Gold P: 1,908 Please note that the dielectric strength is not quite a constant; it varies with temperature, and will change slowly with increased voltage ... slowly, that is, until the dielectric breakdown limit is reached! This limit varies by material, and how the material is made. For optical materials the index of refraction depends upon the dielectric constant; changes in temperature can be readily measured via the changes in the index of refraction. Here is nice description of the origin of the temperature dependence: http://www.doitpoms.ac.uk/tlplib/die...emperature.php http://www.doitpoms.ac.uk/tlplib/die...c_constant.php
Sci Advisor Thanks PF Gold P: 12,167 Why Dielectric Constant is Constant ? Many materials are not linear. Their permittivity depends upon the actual field applied.
Homework | {"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": 0.9510576128959656, "perplexity": 599.7161222442439}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1408500834494.74/warc/CC-MAIN-20140820021354-00232-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://collegemathteaching.wordpress.com/category/integrals/ | # College Math Teaching
## March 12, 2018
### And I embarrass myself….integrate right over a couple of poles…
Filed under: advanced mathematics, analysis, calculus, complex variables, integrals — Tags: — collegemathteaching @ 9:43 pm
I didn’t have the best day Thursday; I was very sick (felt as if I had been in a boxing match..chills, aches, etc.) but was good to go on Friday (no cough, etc.)
So I walk into my complex variables class seriously under prepared for the lesson but decide to tackle the integral
$\int^{\pi}_0 \frac{1}{1+sin^2(t)} dt$
Of course, you know the easy way to do this, right?
$\int^{\pi}_0 \frac{1}{1+sin^2(t)} dt =\frac{1}{2} \int^{2\pi}_0 \frac{1}{1+sin^2(t)} dt$ and evaluate the latter integral as follows:
$sin(t) = \frac{1}{2i}(z-\frac{1}{z}), dt = \frac{dz}{iz}$ (this follows from restricting $z$ to the unit circle $|z| =1$ and setting $z = e^{it} \rightarrow dz = ie^{it}dt$ and then obtaining a rational function of $z$ which has isolated poles inside (and off of) the unit circle and then using the residue theorem to evaluate.
So $1+sin^2(t) \rightarrow 1+\frac{-1}{4}(z^2 -2 + \frac{1}{z^2}) = \frac{1}{4}(-z^2 + 6 -\frac{1}{z^2})$ And then the integral is transformed to:
$\frac{1}{2}\frac{1}{i}(-4)\int_{|z|=1}\frac{dz}{z^3 -6z +\frac{1}{z}} =2i \int_{|z|=1}\frac{zdz}{z^4 -6z^2 +1}$
Now the denominator factors: $(z^2 -3)^2 -8$ which means $z^2 = 3 - \sqrt{8}, z^2 = 3+ \sqrt{8}$ but only the roots $z = \pm \sqrt{3 - \sqrt{8}}$ lie inside the unit circle.
Let $w = \sqrt{3 - \sqrt{8}}$
Write: $\frac{z}{z^4 -6z^2 +1} = \frac{\frac{z}{((z^2 -(3 + \sqrt{8})}}{(z-w)(z+w)}$
Now calculate: $\frac{\frac{w}{((w^2 -(3 + \sqrt{8})}}{(2w)} = \frac{1}{2} \frac{-1}{2 \sqrt{8}}$ and $\frac{\frac{-w}{((w^2 -(3 + \sqrt{8})}}{(-2w)} = \frac{1}{2} \frac{-1}{2 \sqrt{8}}$
Adding we get $\frac{-1}{2 \sqrt{8}}$ so by Cauchy’s theorem $2i \int_{|z|=1}\frac{zdz}{z^4 -6z^2 +1} = 2i 2 \pi i \frac{-1}{2 \sqrt{8}} = \frac{2 \pi}{\sqrt{8}}=\frac{\pi}{\sqrt{2}}$
Ok…that is fine as far as it goes and correct. But what stumped me: suppose I did not evaluate $\int^{2\pi}_0 \frac{1}{1+sin^2(t)} dt$ and divide by two but instead just went with:
\$latex $\int^{\pi}_0 \frac{1}{1+sin^2(t)} dt \rightarrow i \int_{\gamma}\frac{zdz}{z^4 -6z^2 +1}$ where $\gamma$ is the upper half of $|z| = 1$? Well, $\frac{z}{z^4 -6z^2 +1}$ has a primitive away from those poles so isn’t this just $i \int^{-1}_{1}\frac{zdz}{z^4 -6z^2 +1}$, right?
So why not just integrate along the x-axis to obtain $i \int^{-1}_{1}\frac{xdx}{x^4 -6x^2 +1} = 0$ because the integrand is an odd function?
This drove me crazy. Until I realized…the poles….were…on…the…real…axis. ….my goodness, how stupid could I possibly be???
To the student who might not have followed my point: let $\gamma$ be the upper half of the circle $|z|=1$ taken in the standard direction and $\int_{\gamma} \frac{1}{z} dz = i \pi$ if you do this property (hint: set $z(t) = e^{it}, dz = ie^{it}, t \in [0, \pi]$. Now attempt to integrate from 1 to -1 along the real axis. What goes wrong? What goes wrong is exactly what I missed in the above example.
## June 7, 2016
### Infinite dimensional vector subspaces: an accessible example that W-perp-perp isn’t always W
Filed under: integrals, linear albegra — Tags: , — collegemathteaching @ 9:02 pm
This is based on a Mathematics Magazine article by Irving Katz: An Inequality of Orthogonal Complements found in Mathematics Magazine, Vol. 65, No. 4, October 1992 (258-259).
In finite dimensional inner product spaces, we often prove that $(W^{\perp})^{\perp} = W$ My favorite way to do this: I introduce Grahm-Schmidt early and find an orthogonal basis for $W$ and then extend it to an orthogonal basis for the whole space; the basis elements that are not basis elements are automatically the basis for $W^{\perp}$. Then one easily deduces that $(W^{\perp})^{\perp} = W$ (and that any vector can easily be broken into a projection onto $W, W^{\perp}$, etc.
But this sort of construction runs into difficulty when the space is infinite dimensional; one points out that the vector addition operation is defined only for the addition of a finite number of vectors. No, we don’t deal with Hilbert spaces in our first course. 🙂
So what is our example? I won’t belabor the details as they can make good exercises whose solution can be found in the paper I cited.
So here goes: let $V$ be the vector space of all polynomials. Let $W_0$ the subspace of even polynomials (all terms have even degree), $W_1$ the subspace of odd polynomials, and note that $V = W_0 \oplus W_1$
Let the inner product be $\langle p(x), q(x) \rangle = \int^1_{-1}p(x)q(x) dx$. Now it isn’t hard to see that $(W_0)^{\perp} = W_1$ and $(W_1)^{\perp} = W_0$.
Now let $U$ denote the subspace of polynomials whose terms all have degree that are multiples of 4 (e. g. $1 + 3x^4 - 2x^8$ and note that $U^{\perp} \subset W_1$.
To see the reverse inclusion, note that if $p(x) \in U^{\perp}$, $p(x) = p_0 + p_1$ where $p_0 \in W_0, p_1 \in W_1$ and then $\int^1_{-1} (p_1(x))x^{4k} dx = 0$ for any $k \in \{1, 2, ... \}$. So we see that it must be the case that $\int^1_{-1} (p_0(x))x^{4k} dx = 0 = 2\int^1_0 (p_0(x))x^{4k} dx$ as well.
Now we can write: $p_0(x) = c_0 + c_1 x^2 + ...c_n x^{2n}$ and therefore $\int^1_0 p_0(x) x^{4k} dx = c_0\frac{1}{4k+1} + c_1 \frac{1}{2 + 4k+1}...+c_n \frac{1}{2n + 4k+1} = 0$ for $k \in \{0, 1, 2, ...2n+1 \}$
Now I wish I had a more general proof of this. But these equations (for each $k$ leads a system of equations:
$\left( \begin{array}{cccc} 1 & \frac{1}{3} & \frac{1}{5} & ...\frac{1}{2n+1} \\ \frac{1}{5} & \frac{1}{7} & \frac{1}{9}...&\frac{1}{2n+5} \\ ... & ... & ... & ... \\ \frac{1}{4k+1} & \frac{1}{4k+3} & ...& \frac{1}{10n+4} \end{array} \right) \left( \begin{array}{c} c_0 \\ c_1 \\ ... \\ c_n \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ ... \\ 0 \end{array} \right)$
It turns out that the given square matrix is non-singular (see page 92, no. 3 of Polya and Szego: Problems and Theorems in Analysis, Vol. 2, 1976) and so the $c_j = 0$. This means $p_0 = 0$ and so $U^{\perp} = W_1$
Anyway, the conclusion leaves me cold a bit. It seems as if I should be able to prove: let $f$ be some, say…$C^{\infty}$ function over $[0,1]$ where $\int^1_0 x^{2k} f(x) dx = 0$ for all $k \in \{0, 1, ....\}$ then $f = 0$. I haven’t found a proof as yet…perhaps it is false?
## May 20, 2016
### Student integral tricks…
Ok, classes ended last week and my brain is way out of math shape. Right now I am contemplating how to show that the complements of this object
and of the complement of the object depicted in figure 3, are NOT homeomorphic.
I can do this in this very specific case; I am interested in seeing what happens if the “tangle pattern” is changed. Are the complements of these two related objects *always* topologically different? I am reasonably sure yes, but my brain is rebelling at doing the hard work to nail it down.
Anyhow, finals are graded and I am usually treated to one unusual student trick. Here is one for the semester:
$\int x^2 \sqrt{x+1} dx =$
Now I was hoping that they would say $u = x +1 \rightarrow u-1 = x \rightarrow x^2 = u^2-2u+1$ at which case the integral is translated to: $\int u^{\frac{5}{2}} - 2u^{\frac{3}{2}} + u^{\frac{1}{2}} du$ which is easy to do.
Now those wanting to do it a more difficult (but still sort of standard) way could do two repetitions of integration by parts with the first set up being $x^2 = u, \sqrt{x+1}dx =dv \rightarrow du = 2xdx, v = \frac{2}{3} (x+1)^{\frac{3}{2}}$ and that works just fine.
But I did see this: $x =tan^2(u), dx = 2tan(u)sec^2(u)du, x+1 = tan^2(x)+1 = sec^2(u)$ (ok, there are some domain issues here but never mind that) and we end up with the transformed integral: $2\int tan^5(u)sec^3(u) du$ which can be transformed to $2\int (sec^6(u) - 2 sec^4(u) + sec^2(u)) tan(u)sec(u) du$ by elementary trig identities.
And yes, that leads to an answer of $\frac{2}{7}sec^7(u) +\frac{4}{5}sec^5(u) + \frac{2}{3}sec^3(u) + C$ which, upon using the triangle
Gives you an answer that is exactly in the same form as the desired “rationalization substitution” answer. Yeah, I gave full credit despite the “domain issues” (in the original integral, it is possible for $x \in (-1,0]$ ).
What can I say?
## December 22, 2015
### Multi leaf polar graphs and total area…
Filed under: calculus, elementary mathematics, integrals — Tags: , — collegemathteaching @ 4:07 am
I saw polar coordinate calculus for the first time in 1977. I’ve taught calculus as a TA and as a professor since 1987. And yet, I’ve never thought of this simple little fact.
Consider $r(\theta) = sin(n \theta), 0 \theta \ 2 \pi$. Now it is well know that the area formula (area enclosed by a polar graph, assuming no “doubling”, self intersections, etc.) is $A = \frac{1}{2} \int^b_a (r(\theta))^2 d \theta$
Now the leaved roses have the following types of graphs: $n$ leaves if $n$ is odd, and $2n$ leaves if $n$ is even (in the odd case, the graph doubles itself).
So here is the question: how much total area is covered by the graph (all the leaves put together, do NOT count “overlapping”)?
Well, for $n$ an integer, the answer is: $\frac{\pi}{4}$ if $n$ is odd, and $\frac{\pi}{2}$ if $n$ is even! That’s it! Want to know why?
Do the integral: if $n$ is odd, our total area is $\frac{n}{2}\int^{\frac{\pi}{n}}_0 (sin(n \theta)^2 d\theta = \frac{n}{2}\int^{\frac{\pi}{n}}_0 \frac{1}{2} + cos(2n\theta) d\theta =\frac{\pi}{4}$. If $n$ is even, we have the same integral but the outside coefficient is $\frac{2n}{2} = n$ which is the only difference. Aside from parity, the number of leaves does not matter as to the total area!
Now the fun starts when one considers a fractional multiple of $\theta$ and I might ponder that some.
## May 11, 2015
### The hypervolume of the n-ball enclosed by a standard n-1 sphere
I am always looking for interesting calculus problems to demonstrate various concepts and perhaps generate some interest in pure mathematics.
And yes, I like to “blow off some steam” by spending some time having some non-technical mathematical fun with elementary mathematics.
This post uses only:
1. Integration by parts and basic reduction formulas.
2. Trig substitution.
3. Calculation of volumes (and hyper volumes) by the method of cross sections.
4. Induction
5. Elementary arithmetic involving factorials.
The quest: find a formula that finds the (hyper)volume of the region $\{(x_1, x_2, x_3,....x_k) | \sum_{i=1}^k x_i^2 \leq R^2 \} \subset R^k$
We will assume that the usual tools of calculus work as advertised.
Start. If we done the (hyper)volume of the k-ball by $V_k$ we will start with the assumption that $V_1 = 2R$; that is, the distance between the endpoints of $[-R,R]$ is $2R$.
Step 1: we show, via induction, that $V_k =c_kR^k$ where $c_k$ is a constant and $R$ is the radius.
Our proof will be inefficient for instructional purposes.
We know that $V_1 =2R$ hence the induction hypothesis holds for the first case and $c_1 = 2$. We now go to show the second case because, for the beginner, the technique will be easier to follow further along if we do the $k = 2$ case.
Yes, I know that you know that $V_2 = \pi R^2$ and you’ve seen many demonstrations of this fact. Here is another: let’s calculate this using the method of “area by cross sections”. Here is $x^2 + y^2 = R^2$ with some $y = c$ cross sections drawn in.
Now do the calculation by integrals: we will use symmetry and only do the upper half and multiply our result by 2. At each $y = y_c$ level, call the radius from the center line to the circle $R(y)$ so the total length of the “y is constant” level is $2R(y)$ and we “multiply by thickness “dy” to obtain $V_2 = 4 \int^{y=R}_{y=0} R(y) dy$.
But remember that the curve in question is $x^2 + y^2 = R^2$ and so if we set $x = R(y)$ we have $R(y) = \sqrt{R^2 -y^2}$ and so our integral is $4 \int^{y=R}_{y=0}\sqrt{R^2 -y^2} dy$
Now this integral is no big deal. But HOW we solve it will help us down the road. So here, we use the change of variable (aka “trigonometric substitution”): $y = Rsin(t), dy =Rcos(t)$ to change the integral to:
$4 \int^{\frac{\pi}{2}}_0 R^2 cos^2(t) dt = 4R^2 \int^{\frac{\pi}{2}}_0 cos^2(t) dt$ therefore
$V_2 = c_2 R^2$ where:
$c_2 = 4\int^{\frac{\pi}{2}}_0 cos^2(t)$
Yes, I know that this is an easy integral to solve, but I first presented the result this way in order to make a point.
Of course, $c_2 = 4\int^{\frac{\pi}{2}}_0 cos^2(t) = 4\int^{\frac{\pi}{2}}_0 \frac{1}{2} + \frac{1}{2}cos(2t) dt = \pi$
Therefore, $V_2 =\pi R^2$ as expected.
Exercise for those seeing this for the first time: compute $c_3$ and $V_3$ by using the above methods.
Inductive step: Assume $V_k = c_kR^k$ Now calculate using the method of cross sections above (and here we move away from x-y coordinates to more general labeling):
$V_{k+1} = 2\int^R_0 V_k dy = 2 \int^R_0 c_k (R(x_{k+1})^k dx_{k+1} =c_k 2\int^R_0 (R(x_{k+1}))^k dx_{k+1}$
Now we do the substitutions: first of all, we note that $x_1^2 + x_2^2 + ...x_{k}^2 + x_{k+1}^2 = R^2$ and so
$x_1^2 + x_2^2 ....+x_k^2 = R^2 - x_{k+1}^2$. Now for the key observation: $x_1^2 + x_2^2 ..+x_k^2 =R^2(x_{k+1})$ and so $R(x_{k+1}) = \sqrt{R^2 - x_{k+1}^2}$
Now use the induction hypothesis to note:
$V_{k+1} = c_k 2\int^R_0 (R^2 - x_{k+1}^2)^{\frac{k}{2}} dx_{k+1}$
Now do the substitution $x_{k+1} = Rsin(t), dx_{k+1} = Rcos(t)dt$ and the integral is now:
$V_{k+1} = c_k 2\int^{\frac{\pi}{2}}_0 R^{k+1} cos^{k+1}(t) dt = c_k(2 \int^{\frac{\pi}{2}}_0 cos^{k+1}(t) dt)R^{k+1}$ which is what we needed to show.
In fact, we have shown a bit more. We’ve shown that $c_1 = 2 =2 \int^{\frac{\pi}{2}}_0(cos(t))dt, c_2 = 2 \cdot 2\int^{\frac{\pi}{2}}_0 cos^2(t) dt = c_1 2\int^{\frac{\pi}{2}}_0 cos^2(t) dt$ and, in general,
$c_{k+1} = c_{k}c_{k-1}c_{k-2} ....c_1(2 \int^{\frac{\pi}{2}}_0 cos^{k+1}(t) dt) = 2^{k+1} \int^{\frac{\pi}{2}}_0(cos^{k+1}(t))dt \int^{\frac{\pi}{2}}_0(cos^{k}(t))dt \int^{\frac{\pi}{2}}_0(cos^{k-1}(t))dt .....\int^{\frac{\pi}{2}}_0(cos(t))dt$
Finishing the formula
We now need to calculate these easy calculus integrals: in this case the reduction formula:
$\int cos^n(x) dx = \frac{1}{n}cos^{n-1}sin(x) + \frac{n-1}{n} \int cos^{n-2}(x) dx$ is useful (it is merely integration by parts). Now use the limits and elementary calculation to obtain:
$\int^{\frac{\pi}{2}}_0 cos^n(x) dx = \frac{n-1}{n} \int^{\frac{\pi}{2}}_0 cos^{n-2}(x)dx$ to obtain:
$\int^{\frac{\pi}{2}}_0 cos^n(x) dx = (\frac{n-1}{n})(\frac{n-3}{n-2})......(\frac{3}{4})\frac{\pi}{4}$ if $n$ is even and:
$\int^{\frac{\pi}{2}}_0 cos^n(x) dx = (\frac{n-1}{n})(\frac{n-3}{n-2})......(\frac{4}{5})\frac{2}{3}$ if $n$ is odd.
Now to come up with something resembling a closed formula let’s experiment and do some calculation:
Note that $c_1 = 2, c_2 = \pi, c_3 = \frac{4 \pi}{3}, c_4 = \frac{(\pi)^2}{2}, c_5 = \frac{2^3 (\pi)^2)}{3 \cdot 5} = \frac{8 \pi^2}{15}, c_6 = \frac{\pi^3}{3 \cdot 2} = \frac{\pi^3}{6}$.
So we can make the inductive conjecture that $c_{2k} = \frac{\pi^k}{k!}$ and see how it holds up: $c_{2k+2} = 2^2 \int^{\frac{\pi}{2}}_0(cos^{2k+2}(t))dt \int^{\frac{\pi}{2}}_0(cos^{2k+1}(t))dt \frac{\pi^k}{k!}$
$= 2^2 ((\frac{2k+1}{2k+2})(\frac{2k-1}{2k})......(\frac{3}{4})\frac{\pi}{4})((\frac{2k}{2k+1})(\frac{2k-2}{2k-1})......\frac{2}{3})\frac{\pi^k}{k!}$
Now notice the telescoping effect of the fractions from the $c_{2k+1}$ factor. All factors cancel except for the $(2k+2)$ in the first denominator and the 2 in the first numerator, as well as the $\frac{\pi}{4}$ factor. This leads to:
$c_{2k+2} = 2^2(\frac{\pi}{4})\frac{2}{2k+2} \frac{\pi^k}{k!} = \frac{\pi^{k+1}}{(k+1)!}$ as required.
Now we need to calculate $c_{2k+1} = 2\int^{\frac{\pi}{2}}_0(cos^{2k+1}(t))dt c_{2k} = 2\int^{\frac{\pi}{2}}_0(cos^{2k+1}(t))dt \frac{\pi^k}{k!}$
$= 2 (\frac{2k}{2k+1})(\frac{2k-2}{2k-1})......(\frac{4}{5})\frac{2}{3}\frac{\pi^k}{k!} = 2 (\frac{(2k)(2k-2)(2k-4)..(4)(2)}{(2k+1)(2k-1)...(5)(3)} \frac{\pi^k}{k!}$
To simplify this further: split up the factors of the $k!$ in the denominator and put one between each denominator factor:
$= 2 (\frac{(2k)(2k-2)(2k-4)..(4)(2)}{(2k+1)(k)(2k-1)(k-1)...(3)(5)(2)(3)(1)} \pi^k$ Now multiply the denominator by $2^k$ and put one factor with each $k-m$ factor in the denominator; also multiply by $2^k$ in the numerator to obtain:
$(2) 2^k (\frac{(2k)(2k-2)(2k-4)..(4)(2)}{(2k+1)(2k)(2k-1)(2k-2)...(6)(5)(4)(3)(2)} \pi^k$ Now gather each factor of 2 in the numerator product of the 2k, 2k-2…
$= (2) 2^k 2^k \pi^k \frac{k!}{(2k+1)!} = 2 \frac{(4 \pi)^k k!}{(2k+1)!}$ which is the required formula.
So to summarize:
$V_{2k} = \frac{\pi^k}{k!} R^{2k}$
$V_{2k+1}= \frac{2 k! (4 \pi)^k}{(2k+1)!}R^{2k+1}$
Note the following: $lim_{k \rightarrow \infty} c_{k} = 0$. If this seems strange at first, think of it this way: imagine the n-ball being “inscribed” in an n-cube which has hyper volume $(2R)^n$. Then consider the ratio $\frac{2^n R^n}{c_n R^n} = 2^n \frac{1}{c_n}$; that is, the n-ball holds a smaller and smaller percentage of the hyper volume of the n-cube that it is inscribed in; note the $2^n$ corresponds to the number of corners in the n-cube. One might see that the rounding gets more severe as the number of dimensions increases.
One also notes that for fixed radius R, $lim_{n \rightarrow \infty} V_n = 0$ as well.
There are other interesting aspects to this limit: for what dimension $n$ does the maximum hypervolume occur? As you might expect: this depends on the radius involved; a quick glance at the hyper volume formulas will show why. For more on this topic, including an interesting discussion on this limit itself, see Dave Richardson’s blog Division by Zero. Note: his approach to finding the hyper volume formula is also elementary but uses polar coordinate integration as opposed to the method of cross sections.
## October 29, 2014
### Hyperbolic Trig Functions and integration…
In college calculus courses, I’ve always wrestled with “how much to cover in the hyperbolic trig functions” section.
On one hand, the hyperbolic trig functions make some integrals much easer. On the other hand: well, it isn’t as if our classes are populated with the highest caliber student (I don’t teach at MIT); many struggle with the standard trig functions. There is only so much that the average young mind can absorb.
In case your memory is rusty:
$cosh(x) =\frac{e^x + e^{-x}}{2}, sinh(x) = \frac{e^x -e^{-x}}{2}$ and then it is immediate that the standard “half/double angle formulas hold; we do remember that $\frac{d}{dx}cosh(x) = sinh(x), \frac{d}{dx} = cosh(x)$.
What is less immediate is the following: $sinh^{-1}(x) = ln(x+\sqrt{x^2+1}), cosh^{-1}(x) = ln(x + \sqrt{x^2 -1}) (x \ge 1)$.
Exercise: prove these formulas. Hint: if $sinh(y) = x$ then $e^{y} - 2x- e^{-y} =0$ so multiply both sides by $e^{y}$ to obtain $e^{2y} -2x e^y - 1 =0$ now use the quadratic formula to solve for $e^y$ and keep in mind that $e^y$ is positive.
For the other formula: same procedure, and remember that we are using the $x \ge 0$ branch of $cosh(x)$ and that $cosh(x) \ge 1$
The following follows easily: $\frac{d}{dx} sinh^{-1} (x) = \frac{1}{\sqrt{x^2 + 1}}$ (just set up $sinh(y) = x$ and use implicit differentiation followed by noting $cosh^2(x) -sinh^2(x) = 1$. ) and $\frac{d}{dx} cosh^{-1}(x) = \frac{1}{\sqrt{x^2-1}}$ (similar derivation).
Now, we are off and running.
Example: $\int \sqrt{x^2 + 1} dx =$
We can make the substitution $x =sinh(u), dx = cosh(u) du$ and obtain $\int cosh^2(u) du = \int \frac{1}{2} (cosh(2u) + 1)du = \frac{1}{4}sinh(2u) + \frac{1}{2} u + C$. Now use $sinh(2u) = 2 sinh(u)cosh(u)$ and we obtain:
$\frac{1}{2}sinh(u)cosh(u) + \frac{u}{2} + C$. The back substitution isn’t that hard if we recognize $cosh(u) = \sqrt{sinh^2(u) + 1}$ so we have $\frac{1}{2} sinh(u) \sqrt{sinh^2(u) + 1} + \frac{u}{2} + C$. Back substitution is now easy:
$\frac{1}{2} x \sqrt{x^2+1} + \frac{1}{2} ln(x + \sqrt{x^2 + 1}) + C$. No integration by parts is required and the dreaded $\int sec^3(x) dx$ integral is avoided. Ok, I was a bit loose about the domains here; we can make this valid for negative values of $x$ by using an absolute value with the $ln(x + \sqrt{x^2 + 1})$ term.
## August 31, 2014
### The convolution integral: do some examples in Calculus III or not?
For us, calculus III is the most rushed of the courses, especially if we start with polar coordinates. Getting to the “three integral theorems” is a real chore. (ok, Green’s, Divergence and Stoke’s theorem is really just $\int_{\Omega} d \sigma = \int_{\partial \Omega} \sigma$ but that is the subject of another post)
But watching this lecture made me wonder: should I say a few words about how to calculate a convolution integral?
Note: I’ve discussed a type of convolution integral with regards to solving differential equations here.
In the context of Fourier Transforms, the convolution integral is defined as it was in analysis class: $f*g = \int^{\infty}_{-\infty} f(x-t)g(t) dt$. Typically, we insist that the functions be, say, $L^1$ and note that it is a bit of a chore to show that the convolution of two $L^1$ functions is $L^1$; one proves this via the Fubini-Tonelli Theorem.
(The straight out product of two $L^1$ functions need not be $L^1$; e.g, consider $f(x) = \frac {1}{\sqrt{x}}$ for $x \in (0,1]$ and zero elsewhere)
So, assuming that the integral exists, how do we calculate it? Easy, you say? Well, it can be, after practice.
But to test out your skills, let $f(x) = g(x)$ be the function that is $1$ for $x \in [\frac{-1}{2}, \frac{1}{2}]$ and zero elsewhere. So, what is $f*g$???
So, it is easy to see that $f(x-t)g(t)$ only assumes the value of $1$ on a specific region of the $(x,t)$ plane and is zero elsewhere; this is just like doing an iterated integral of a two variable function; at least the first step. This is why it fits well into calculus III.
$f(x-t)g(t) = 1$ for the following region: $(x,t), -\frac{1}{2} \le x-t \le \frac{1}{2}, -\frac{1}{2} \le t \le \frac{1}{2}$
This region is the parallelogram with vertices at $(-1, -\frac{1}{2}), (0, -\frac{1}{2}), (0 \frac{1}{2}), (1, \frac{1}{2})$.
Now we see that we can’t do the integral in one step. So, the function we are integrating $f(x-t)f(t)$ has the following description:
$f(x-t)f(t)=\left\{\begin{array}{c} 1,x \in [-1,0], -\frac{1}{2} t \le \frac{1}{2}+x \\ 1 ,x\in [0,1], -\frac{1}{2}+x \le t \le \frac{1}{2} \\ 0 \text{ elsewhere} \end{array}\right.$
So the convolution integral is $\int^{\frac{1}{2} + x}_{-\frac{1}{2}} dt = 1+x$ for $x \in [-1,0)$ and $\int^{\frac{1}{2}}_{-\frac{1}{2} + x} dt = 1-x$ for $x \in [0,1]$.
That is, of course, the tent map that we described here. The graph is shown here:
So, it would appear to me that a good time to do a convolution exercise is right when we study iterated integrals; just tell the students that this is a case where one “stops before doing the outside integral”.
## August 25, 2014
### Fourier Transform of the “almost Gaussian” function with a residue integral
This is based on the lectures on the Fourier Transform by Brad Osgood from Stanford:
And here, $F(f)(s) = \int^{\infty}_{-\infty} e^{-2 \pi i st} f(t) dt$ provided the integral converges.
The “almost Gaussian” integrand is $f(t) = e^{-\pi t^2}$; one can check that $\int^{\infty}_{-\infty} e^{-\pi t^2} dt = 1$. One way is to use the fact that $\int^{\infty}_{-\infty} e^{-x^2} dx = \sqrt{\pi}$ and do the substitution $x = \sqrt{\pi} t$; of course one should be able to demonstrate the fact to begin with. (side note: a non-standard way involving symmetries and volumes of revolution discovered by Alberto Delgado can be found here)
So, during this lecture, Osgood shows that $F(e^{-\pi t^2}) = e^{-\pi s^2}$; that is, this modified Gaussian function is “its own Fourier transform”.
I’ll sketch out what he did in the lecture at the end of this post. But just for fun (and to make a point) I’ll give a method that uses an elementary residue integral.
Both methods start by using the definition: $F(s) = \int^{\infty}_{-\infty} e^{-2 \pi i ts} e^{-\pi t^2} dt$
Method 1: combine the exponential functions in the integrand:
$\int^{\infty}_{-\infty} e^{-\pi(t^2 +2 i ts} dt$. Now complete the square to get: $\int^{\infty}_{-\infty} e^{-\pi(t^2 +2 i ts-s^2)-\pi s^2} dt$
Now factor out the factor involving $s$ alone and write as a square: $e^{-\pi s^2}\int^{\infty}_{-\infty} e^{-\pi(t+is)^2} dt$
Now, make the substitution $x = t+is, dx = dt$ to obtain:
$e^{-\pi s^2}\int^{\infty+is}_{-\infty+is} e^{-\pi x^2} dx$
Now we show that the above integral is really equal to $e^{-\pi s^2}\int^{\infty}_{-\infty} e^{-\pi x^2} dx = e^{\pi s^2} (1) = e^{-\pi s^2}$
To show this, we perform $\int_{\gamma} e^{z^2} dz$ along the retangular path $\gamma$: $-x, x, x+is, -x+is$ and let $x \rightarrow \infty$
Now the integral around the contour is 0 because $e^{-z^2}$ is analytic.
We wish to calculate the negative of the integral along the top boundary of the contour. Integrating along the bottom gives 1.
As far as the sides: if we fix $s$ we note that $e^{-z^2} = e^{(s^2-x^2)+2si}$ and the magnitude goes to zero as $x \rightarrow \infty$ So the integral along the vertical paths approaches zero, therefore the integrals along the top and bottom contours agree in the limit and the result follows.
Method 2: The method in the video
This uses “differentiation under the integral sign”, which we talk about here.
Stat with $F(s) = \int^{\infty}_{-\infty} e^{-2 \pi i ts} e^{-\pi t^2} dt$ and note $\frac{dF}{ds} = \int^{\infty}_{-\infty} (-2 \pi i t) e^{-2 \pi i ts} e^{-\pi t^2} dt$
Now we do integration by parts: $u = e^{-2 \pi i ts}, dv = (-2 \pi i t)e^{-\pi t^2} \rightarrow v = i e^{-\pi t^2}, du = (-2 \pi i s)e^{-2 \pi i ts}$ and the integral becomes:
$(i e^{-\pi t^2} e^{-2 \pi i ts}|^{\infty}_{-\infty} - (i)(-2 \pi i s) \int^{\infty}_{-\infty} e^{-2 \pi i ts} e^{-\pi t^2} dt$
Now the first term is zero for all values of $s$ as $t \rightarrow \infty$. The second term is merely:
$-(2 \pi s) \int^{\infty}_{-\infty} e^{-2 \pi i ts} e^{-\pi t^2} dt = -(2 \pi s) F(s)$.
So we have shown that $\frac{d F}{ds} = (-2 \pi s)F$ which is a differential equation in $s$ which has solution $F = F_0 e^{- \pi s^2}$ (a simple separation of variables calculation will verify this). Now to solve for the constant $F_0$ note that $F(0) = \int^{\infty}_{-\infty} e^{0} e^{-\pi t^2} dt = 1$.
The result follows.
Now: which method was easier? The second required differential equations and differentiating under the integral sign; the first required an easy residue integral.
By the way: the video comes from an engineering class. Engineers need to know this stuff!
## August 21, 2014
### Calculation of the Fourier Transform of a tent map, with a calculus tip….
I’ve been following these excellent lectures by Professor Brad Osgood of Stanford. As an aside: yes, he is dynamite in the classroom, but there is probably a reason that Stanford is featuring him. 🙂
And yes, his style is good for obtaining a feeling of comradery that is absent in my classroom; at least in the lower division “service” classes.
This lecture takes us from Fourier Series to Fourier Transforms. Of course, he admits that the transition here is really a heuristic trick with symbolism; it isn’t a bad way to initiate an intuitive feel for the subject though.
However, the point of this post is to offer a “algebra of calculus trick” for dealing with the sort of calculations that one might encounter.
By the way, if you say “hey, just use a calculator” you will be BANNED from this blog!!!! (just kidding…sort of. 🙂 )
So here is the deal: let $f(x)$ represent the tent map: the support of $f$ is $[-1,1]$ and it has the following graph:
The formula is: $f(x)=\left\{\begin{array}{c} x+1,x \in [-1,0) \\ 1-x ,x\in [0,1] \\ 0 \text{ elsewhere} \end{array}\right.$
So, the Fourier Transform is $F(f) = \int^{\infty}_{-\infty} e^{-2 \pi i st}f(t)dt = \int^0_{-1} e^{-2 \pi i st}(1+t)dt + \int^1_0e^{-2 \pi i st}(1-t)dt$
Now, this is an easy integral to do, conceptually, but there is the issue of carrying constants around and being tempted to make “on the fly” simplifications along the way, thereby leading to irritating algebraic errors.
So my tip: just let $a = -2 \pi i s$ and do the integrals:
$\int^0_{-1} e^{at}(1+t)dt + \int^1_0e^{at}(1-t)dt$ and substitute and simplify later:
Now the integrals become: $\int^{1}_{-1} e^{at}dt + \int^0_{-1}te^{at}dt - \int^1_0 te^{at} dt.$
These are easy to do; the first is merely $\frac{1}{a}(e^a - e^{-a})$ and the next two have the same anti-derivative which can be obtained by a “integration by parts” calculation: $\frac{t}{a}e^{at} -\frac{1}{a^2}e^{at}$; evaluating the limits yields:
$-\frac{1}{a^2}-(\frac{-1}{a}e^{-a} -\frac{1}{a^2}e^{-a}) - (\frac{1}{a}e^{a} -\frac{1}{a^2}e^a)+ (-\frac{1}{a^2})$
Add the first integral and simplify and we get: $-\frac{1}{a^2}(2 - (e^{-a} -e^{a})$. NOW use $a = -2\pi i s$ and we have the integral is $\frac{1}{4 \pi^2 s^2}(2 -(e^{2 \pi i s} -e^{-2 \pi i s}) = \frac{1}{4 \pi^2 s^2}(2 - cos(2 \pi s))$ by Euler’s formula.
Now we need some trig to get this into a form that is “engineering/scientist” friendly; here we turn to the formula: $sin^2(x) = \frac{1}{2}(1-cos(2x))$ so $2 - cos(2 \pi s) = 4sin^2(\pi s)$ so our answer is $\frac{sin^2( \pi s)}{(\pi s)^2} = (\frac{sin(\pi s)}{\pi s})^2$ which is often denoted as $(sinc(s))^2$ as the “normalized” $sinc(x)$ function is given by $\frac{sinc(\pi x)}{\pi x}$ (as we want the function to have zeros at integers and to “equal” one at $x = 0$ (remember that famous limit!)
So, the point is that using $a$ made the algebra a whole lot easier.
Now, if you are shaking your head and muttering about how this calculation was crude that that one usually uses “convolution” instead: this post is probably too elementary for you. 🙂
## August 6, 2014
### Where “j” comes from
I laughed at what was said from 30:30 to 31:05 or so:
If you are wondering why your engineering students want to use $j = \sqrt{-1}$ is is because, in electrical engineering, $i$ usually stands for “current”.
Though many of you know this, this lesson also gives an excellent reason to use the complex form of the Fourier series; e. g. if $f$ is piece wise smooth and has period 1, write $f(x) = \Sigma^{k = \infty}_{k=-\infty}c_k e^{i 2k\pi x}$ (usual abuse of the equals sign) rather than writing it out in sines and cosines. of course, $\overline{c_{-k}} = c_k$ if $f$ is real valued.
How is this easier? Well, when you give a demonstration as to what the coefficients have to be (assuming that the series exists to begin with, the orthogonality condition is very easy to deal with. Calculate: $c_m= \int^1_0 e^{i 2k\pi t}e^{i 2m\pi x} dx$ for when $k \ne m$. There is nothing to it; easy integral. Of course, one has to demonstrate the validity of $e^{ix} = cos(x) + isin(x)$ and show that the usual differentiation rules work ahead of time, but you need to do that only once.
Older Posts » | {"extraction_info": {"found_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": 278, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9673395156860352, "perplexity": 344.70441023504793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945232.48/warc/CC-MAIN-20180421145218-20180421165218-00548.warc.gz"} |
https://www.physicsforums.com/threads/crate-pulled-up-incline-kinetic-energy-and-speed-picture-included.434779/ | # Crate pulled up incline, kinetic energy and speed; picture included
1. ### gap0063
66
1. The problem statement, all variables and given/known data
A crate is pulled by a force (parallel to the incline) up a rough incline. The crate has an initial speed shown in the figure below. The crate is pulled a distance of 5.94 m on the incline by a 150 N force.
The acceleration of gravity is 9.8 m/s2 .
a) What is the change in kinetic energy of the crate?
b) What is the speed of the crate after it is pulled the 5.94 m?
2. Relevant equations
(1)W= delta K= Integral Fx dx= Integral Sum of F dr= Integral Sum of F dr + Integral fk dr= Sum of W - fkd
(2)Delta K= 1/2 mvf^2
3. The attempt at a solution
Wells for part a) I tried equation (1)and got 150 N- (0.399* 5.94)=147.63 J... which I don't know if its right
and for b) (if part a is right) I used equation (2) vf=sq rt ( (2/m)*147.63)= 4.76574 m/s
2. ### PhanthomJay
6,151
When using W =delta K , W includes the work done by gravity or any other conservative forces, when present. It is best to use conservation of total energy, Work done by non - conservative forces (like friction and the applied force) equals delta __??__ + delta __??_.
The problem asks for the change in Kinetic Energy, in addition to its speed.
You are getting your energy and force units mixed up. | {"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": 0.9410139918327332, "perplexity": 1713.603156589193}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936463475.57/warc/CC-MAIN-20150226074103-00114-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://andjournal.sgu.ru/en/klyuchevye-slova/otobrazhenie-anosova | ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)
# отображение Аносова
## System of three nonautonomous oscillators with hyperbolic chaos part i the model with dynamics on attractor governed by arnold’s cat map on torus
In this paper a system of three coupled nonautonomous selfoscillatory elements is studied, in which the behavior of oscillators phases on a period of the coefficients variation in the equations corresponds to the Anosov map demonstrating chaotic dynamics. Results of numerical studies allow us to conclude that the attractor of the Poincare map can be viewed as an object roughly represented by a twodimensional torus embedded in the sixdimensional phase space of the Poincare map, on which the dynamics is the hyperbolic chaos intrinsic to Anosov’s systems. | {"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": 0.8206242918968201, "perplexity": 2199.1782238165924}, "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-2019-43/segments/1570987826436.88/warc/CC-MAIN-20191022232751-20191023020251-00269.warc.gz"} |
http://sneezy.cs.nott.ac.uk/fplunch/weblog/?p=79 | Archimedes, Gentzen and lenses
October 20th, 2007 by Hancock
At Neil’s request, I talked on the topic of “lenses”, a name I use for a structure underlying many well-ordering proofs. (The same name is used by some people for funny-bracket notation for anamorphisms, catamorphisms, and the like. The notions have nothing to do with each other, except insofar as initial algebras are involved.)
I started by explaining “Archimedes trick”, involved in his work the “Sand Reckoner”. See http://en.wikipedia.org/wiki/The_Sand_Reckoner.
Call a predicate an accessibility predicate if it holds of 0, and is closed under the successor operation. If A is an accessibility predicate, we can evidently find a proof that it holds of 2^17, using some 130,000 applications of Modus Ponens. The problem is to give a short proof (without using induction).
The trick is to define the following predicate:
A_1(y) = all x. A(x) -> A(x + 2^y).
Observe that this too is an accessibility predicate. So we can give quite a short proof of A_1(17), using about 17 applications of Modus Ponens. But then, we’re done, as A_1(17) implies A(2^17).
How about 2^2^17? Same deal. Define
A_2(y) = all x. A_1(x) -> A_1(x + 2^y).
Obviously this too is an accessibility predicate, so we get a short proof that A_1(2^17), and then, in a trice, a short proof of A(2^2^17). Essentially, we just play the same trick again.
I then pointed out that the same trick works in the case of the second number class, meaning the initial algebra for the functor
X |-> 1 + X + X^N,
or
data Omega = Zero | Succ Omega | Lim (Nat -> Omega).
For example, we have the usual finite ordinals 0, S(0), S(S(0)), .. and the first infinite ordinal = L(\n.(0)).
We also have addition, multiplication and exponentiation, defined (by recursion, ie. initiality):
a + 0 = a
a + S(b) = S(a + b)
a + L(f) = L(\n. a + f(n))
a * 0 = a
a * S(b) = (a * b) + a
a * L(f) = L(\n. a * f(n))
a ^ 0 = S(0)
a ^ S(b) = (a ^ b) * a
a ^ L(f) = L(\n. a ^ f(n))
(Note: all three are continuous in their second argument — meaning they “commute with limits”.)
Call a predicate an accessibility predicate if it holds of 0, is closed under the successor operation, and is also closed under limits, in the sense
all f : Nat -> Omega. (all n : Nat. A(f(n))) -> A(Lim(f))
Consider the problem of proving A(^), but without induction on Omega. We just use Archimedes’ trick, more usually called Gentzen’s trick in this context.
Define A_1(y) = all x. A(x) -> A(x + ^y).
This too is an accessibility predicate. So we can easily show A_1(). But then we’re done, as this implies A(^).
Again, we can iterate the trick, and prove, A(omega^omega^omega), A(omega^omega^omega^omega), and so on. The limit of this sequence of ordinals is traditionally called . The proofs can actually be written out quite easily in the usual systems of formal arithmetic. (There are, to be honest, some purely bureaucratic problems to overcome.) What we have done, in fancy terminology, is show that the proof-theoretic ordinal of formal arithmetic is at least .
You can find a version of the proof in (an early version of) Agda at http://homepages.inf.ed.ac.uk/v1phanc1/epsilon0.agda.
I then gave a quick (and only a little dirty) definition of a “denotable ordinal” of a type-theory, such as the simply-typed lambda calculus with the natural numbers, and primitive recursion of all finite types.
Such a thing is given by a term t with 4 free variables X,z,s,l:
X : Set, z : X, s : X->X, l : (N->X)->X |- t[X,z,s,l] : X
For example:
t_{zero} [X,z,s,l] = z
t_{one} [X,z,s,l] = s z
t_{two} [X,z,s,l] = s (s z)
t_{omega}[X,z,s,l] = l (\n->iterate s n z)
Suppose a and b are denotable ordinals, then (I’m sorry, I can’t get this formatted properly!!)
t_{a+b}[X,z,s,l] = t_{b} [ X,t_{a}[X,z,s,l],s,l] t_{a*b}[X,z,s,l] = t_{b} [ X,z,\x->t_{a}[X,x,s,l],l] t_{a^b}[X,z,s,l] = t_{b} [ X->X, s, \f x ->t_{a}[X,x,f,l], \g x ->l (\n -> g n x) ] z
This is essentially Archimedes’ trick. Note that exponentiation requires use of the function space. Without it, we are confined below ^.
Having used up an hour, I finished up, muttering darkly that the notion of “lens” is an abstraction that generalises what is going on in the definitions above: a kind of “magnification” of a Church-ordinal, by applying it at a higher type.
(By the way, people who know it will recognise that the three definitions above are, apart from the limits, at the heart of Schwichtenberg’s demonstration that the numerical functions expressible in the simply-typed lambda calculus are the “extended polynomials”.
I could give another talk on these topics, showing how to get beyond . And then, by using a “universe” type (a type of types), how to get all the way up to the next major landmark ordinal after , namely one called . If anyone wants.
One can find a lecture by Thierry Coquand on these and similar matters at http://www.cs.chalmers.se/~coquand/ordinal.ps | {"extraction_info": {"found_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": 0.938454806804657, "perplexity": 1925.1034999712247}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701166739.77/warc/CC-MAIN-20160205193926-00156-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/need-help-finishing-off-the-last-bit-for-a-projectile-trajectory-problem.77322/ | # Need help finishing off the last bit for a projectile trajectory problem
1. May 29, 2005
### Hoppa
this is the problem:
A projectile is fired with an initial speed u and at an angle of elevation ®. The air
resistance is known to be quadratic and the terminal velocity has a magnitude vt. Show
that the equations of motion for the projectile can be cast into the form
y' = f (t; y); t >= 0
Using MATLAB solve the differential equations for an initial speed of 200 ms¡1,
an angle of elevation of 45± and a terminal velocity of 250 ms¡1. Plot the trajectory of
the projectile and, on the same graph, plot the trajectory that the projectile would have
in the absence of air resistance. Use a time range that allows both trajectories to at least
return to their initial height, without going significantly beyond that position.
2. May 29, 2005
### Hoppa
and here is what i've got so far..
Initial speed = m
Angle of elevation = a
y = é x ù é vx ù é 0 ù
| vx ô , f(t,y) = | -gvt-2vx Övx2 + vz2 | , y(0) = | m cos a |
| z | | vz | | 0 |
ë vz û ë -g(1 + vt-2vzÖvx2 + vz2 û ë m sin a û
Where x and z are the horizontal and vertical coordinates and vx and vz are the corresponding velocities.
The position of the projectile at time t is given by:
x = (m cos a ) t ( 1 – e–t/t )
y = -gvtt + t (m sin a + gvt) (1 – e–t/t )
Where gvt is the magnitute of the terminal velocity.
Matlab code
global vterm tau v0y
g = 9.8;
vterm = 250;
tau = vterm/g;
v0 = 200;
alpha = 45;
time = 0:2:50
range = time;
n = length(time);
for i = 1:n
angle = alpha(i)*pi/180;
v0x = v0*cos(angle);
v0y = v0*sin(angle);
time = fzero(’yproject’,[10;50]);
range(i) = v0x*tau*(1-exp(-time/tau));
end
plot(alpha, range);
xlabel(’Elevation (degrees)’);
ylabel(’Range (metres)’);
which also uses the MATLAB function:
function ypos = yproject(t)
global vterm tau v0y
ypos = -vterm*t + tau*(v0y+vterm)*(1-exp(-t/tau));
Similar Discussions: Need help finishing off the last bit for a projectile trajectory problem | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8048183917999268, "perplexity": 3564.872449347821}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891543.65/warc/CC-MAIN-20180122213051-20180122233051-00727.warc.gz"} |
https://www.clutchprep.com/questions/446/a-uniform-plank-of-length-2-00-m-and-mass-35-0-kg-is-supported-by-three-ropes-as-shown-by-the-three-v | # A uniform plank of length 2.00 m and mass 35.0 kg is supported by three ropes as shown by the three vectors in the figure below. STATICS
A uniform plank of length 2.00 m and mass 35.0 kg is supported by three ropes as shown by the three vectors in the figure below. Find the tension in each rope when a 700 N person is standing a distance d = 0.500 m from the left end. | {"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": 0.9222365021705627, "perplexity": 355.7657413471766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400209999.57/warc/CC-MAIN-20200923050545-20200923080545-00428.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/39289-gaussian-elimination.html | # Math Help - Gaussian Elimination
1. ## Gaussian Elimination
Hi there
I had a maths exam today and got this question. It's been bugging me since I got out of the exam hall and I was wondering if you could let me know if my conclusion is right:
q: using Gaussian elimination, or otherwise, find the solutions to the following set of equations:
(Part 1 was fine)
ii) x+y+z=4
2x+y-z=0
4x+3y+z=9
I concluded that there is no set of solutions that will satisfy all three equations - am I right in thinking this?
Cheers
2. Hello,
Originally Posted by daaavo
am I right in thinking this?
Cheers
Yep | {"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": 0.9230978488922119, "perplexity": 462.3009478945977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398449145.52/warc/CC-MAIN-20151124205409-00159-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://astarmathsandphysics.com/university-maths-notes/matrices-and-linear-algebra/4659-solving-a-non-homogeneous-system-of-first-order-coupled-differential-equations.html?tmpl=component&print=1&page= | ## Solving a Non Homogeneous System of First Order Coupled Differential Equations
Suppose we have a non homogeneous system of linear differential equations. We can solve the system by expressing the solution of the solution to the homogeneous system, and any solution to the non homogeneous system.
Suppose we have the system of coupled differential equations.
$\dot{x}=3x+2y+1$
$\dot{y}=2x+3y+2$
To solve this system, we need to diagonalise the coefficient matrix
$M= \left( \begin{array}{cc} 3 & 2 & \\ 2 & 3 \end{array} \right)$
. Write the system in matrix form as
$\begin{pmatrix}\dot{x}\\ \dot{y} \end{pmatrix}= \left( \begin{array}{cc} 3 & 2 \\ 2 & 3 \end{array} \right) \begin{pmatrix}x\\ y \end{pmatrix}$
The eigenvalues of the matrix are the solutions to
$det(\left( \begin{array}{cc} 3 & 2 \\ 2 & 3 \end{array} \right) - \lambda \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right))=0 \rightarrow det(\left( \begin{array}{cc} 3- \lambda & 2 \\ 2 & 3- \lambda \end{array} \right))=0 \rightarrow (3- \lambda)^2-4=\lambda^2-6 \lambda +5=0$
This expression in
$\lambda$
factorises as
$(\lambda-5)(\lambda -1)=0$
and we solve the equation, obtaining
$\lambda=5, \: 1$
Now find the eigenvectors for each eigenvalue.
For
$\lambda=5$
, solve
$(\left( \begin{array}{cc} 3 & 2 \\ 2 & 3 \end{array} \right) - 5 \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)) \mathbf{v} =\mathbf{0}$
for
$\mathbf{v}=\begin{pmatrix}x\\y\end{pmatrix}$
.
$\left( \begin{array}{cc} -2 & 2 \\ 2 & -2 \end{array} \right)\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}-2x+2y\\2x-2y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$
Hence
$x=y$
and we can take the eigenvector corresponding to the eigenvector 5 as
$\begin{pmatrix}1\\1\end{pmatrix}$
.
For
$\lambda=1$
, solve
$(\left( \begin{array}{cc} 3 & 2 \\ 2 & 3 \end{array} \right) - 1 \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)) \mathbf{v} =\mathbf{0}$
for
$\mathbf{v}=\begin{pmatrix}x\\y\end{pmatrix}$
.
$\left( \begin{array}{cc} 2 & 2 \\ 2 & 2 \end{array} \right)\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}2x+2y\\2x+2y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$
Hence
$x=-y$
and we can take the eigenvector corresponding to the eigenvector 5 as
$\begin{pmatrix}1\\-1\end{pmatrix}$
.
The matrix of eigenvectors is
$P= \left( \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right)$
.
At the moment our system takes the form
$\dot{\mathbf{v}}=M \mathbf{v}$
.
Let
$\mathbf{w}=P^{-1} \mathbf{v}$
so that
$\mathbf{v}=P \mathbf{w}$
&. The system becomes
$P \mathbf{v}=M P \mathbf{w} \rightarrow \mathbf{w}=P^{-1}MP \mathbf{w}$
.
$P^{-1}MP$
will be a diagonal matrix with entries equal to the eigenvalues of
$M$
and
$\mathbf{w}$
will be an elementary basis vector. The new system will be
$\begin{pmatrix}\dot{w_1} \\\dot{w_2} \end{pmatrix}=\left( \begin{array}{cc} 5 & 0 \\ 0 & 1 \end{array} \right) \begin{pmatrix}w_1\\w_2\end{pmatrix}$
.
This is equivalent to the system
$\dot{w_1}=5w_1$
$\dot{w_1}=1w_1$
The solutions are
$w_1=Ae^{5t}, w_2=B e^t$
Then
$\mathbf{v}'=P \mathbf{w} = \left( \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right) \begin{pmatrix}Ae^{5t}\\Be^t\end{pmatrix} =\begin{pmatrix}Ae^{5t}+Be^t\\Ae^{5t}-Be^t\end{pmatrix}$
Now assume a solution to the non homogeneous system of the form
$x_p=C_1, \: y_p=C_2$
. Substitute these into the problem.
Equation (1)
$0=3C_1+2C_2+1=0$
Equating coefficients of
$t$
gives
$0=2C_1+3C_2+1 \rightarrow 2C_1+3C_2=-1$
$0=3C_1+2C_2+2 \rightarrow 3C_1+2C_2=-2$
Solving these gives
$C_1=- \frac{4}{5}, \: C_2= \frac{1}{5}$
The general solution is then
$x=Ae^{5t}+Be^t- \frac{4}{5}, \: y=Ae^{5t}-Be^t+ \frac{1}{5}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9933271408081055, "perplexity": 455.936335654722}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221219197.89/warc/CC-MAIN-20180821230258-20180822010258-00161.warc.gz"} |
https://www.projectrhea.org/rhea/index.php/MLEforGMM | Introduction to Maximum Likelihood Estimation
A slecture by Wen Yi
Partly based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.
### 1. Introduction
For density estimation, Maximum Likelihood Estimation (MLE) is a method of parametric density estimation model. When we applying MLE to a data set with fixed density distribution, MLE provides the estimates for the parameters of density distribution model. In real estimation, we search over all the possible sets of parameter values, then find the specific set of parameters with the maximum value of likelihood, which means is the most likely to observe the data set samples.
### 2. Basic method
Suppose we have a set of n independent and identically destributed observation samples. Then density function is fixed, but unknown to us. We assume that the density funtion belongs to a certain family of distributions, so let θ be a vector of parameters for this distribution family. So, the goal to use MLE is to find the vector of parameters that is as close to the true distribution parameter value as possible.
To use MLE, we first take the joint density function for all the sample observations. For an i.i.d data set of samples, the joint density function is:
As each sample x_i is independent with each other, the likelihood of θ with the data set of samples x_1,x_2,…,x_n can be defined as:
In practice, it’s more convenient to take ln for the both sides, called log-likelihhod. Then the formula becomes:
Then, for a fixed set of samples, to maximize the likelihood of θ, we should choose the data that satisfied:
To find the maximum of lnL(θ;x_1,x_2,…,x_N ), we take the derivative of θ on it and find theθ value that make the derivation equals to 0.
To check our result we should garentee that the second derivative of θ on lnL(θ;x_1,x_2,…,x_n ) is negative.
### 3. Practice considerations
#### 3.1 Log-likelihood
As the likelihood comes from the joint density function, it is usually a product of the probability of all the observations, which is very hard to calculate and analyse. Also, as the probability of a observation sample is always less than 1, let's say if one probability for a observation sample is 0.1, then the more data we have, the smaller the likelihood value is (e.g. 0.00000001 or smaller). The small value of likelihood leads to the difficulty in calculating and storing the likelihood.
For the solution of this problem, we took the natural log of the original likelihood, then the joint probability will express as the sum of the natural log of each probability. In this way, the value of likelihood become easier to measure as the number of samples we have increases. Please note that as the probability of one observation of sample is always less than 1, the log-likelihood will always less than 0.
#### 3.2 Removing the constant
Let's take binomial distribution for example, the likelihood for this distribution is:
In this estimation of MLE, we noted that the total number of samples, n, and the number of occurrence, k, is fixed. Then, we can see that as the first part of this likelihood doesn't depend on the value of p, it is a fix value as the value of p changes. So, removing the first part of the likelihood doesn't influence the comparison of likelihood between different value of ps. As a result, we can estimate the likelihood of binomial distribution like following rather than the way above:
For another reason to do this, as the value of the first part is always larger than 1, as number of samples increases, the total value of likelihood will increase subsequently and make the calculation and storing of the value harder. For this reason, remove the constant part will also make the life easier.
#### 3.3 Numerical MLE
Sometimes, we cannot write a equation that can be differentiated to find the MLE parameter estimates, In these cases, we may get exhausted in trying all the value that is possible to be the maximum likelihood. If we choose this method, then the step of the value we try will result in the time of calculation. Thus, we should choose the step as 0.01, 0.001 or 0.0000001 according to the needed accuracy we want.
### 4. Some basic examples
#### 4.1 Poisson Distribution
For Poisson distribution the expression of probability is:
Let X_1,X_2,…,X_N be the Independent and identically distributed (iid) Poisson random variables. Then, we will have a joint frequency function that is the product of marginal frequency functions. The log likelihood of Poisson distribution thus should be:
Take the derivative of λ on it and find theλ value that make the derivation equals to 0.
Thus, the ML estimation for Poisson distribution should be:
#### 4.2 Exponential distribution
For exponential distribution the expression of probability is:
Let X_1,X_2,…,X_N be the Independent and identically distributed (iid) exponential random variables. As P(X=x)=0 when x<0, no samples can sit in x<0 region. Thus, for all X_1,X_2,…,X_N, we can only focus on the x≥0 part. Then, we will have a joint frequency function that is the product of marginal frequency functions. The log likelihood of exponential distribution thus should be:
Take the derivative of λ on it and find theλ value that make the derivation equals to 0.
Thus, the ML estimation for exponential distribution should be:
#### 4.3 Gaussian distribution
For Gaussian distribution the expression of probability is:
Let X_1,X_2,…,X_N be the Independent and identically distributed (iid) Gaussian random variables. Then, we will have a joint frequency function that is the product of marginal frequency functions. The log likelihood of Gaussian distribution thus should be:
Take the derivative of μ,Σ on it and find the μ,Σ value that make the derivation equals to 0.
Thus, the ML estimation for Gaussian distribution should be:
#### 5.1 Expression of Estimated Parameters
The above estimation all base on the assumption that the distribution to be estimated follows the distribution of a single function, but how about the estimation of the mixture of functions?
To simplify the problem, we only talk about Gaussian Mixture Model (GMM) here. Using the same method, it’s easy to extend it to other kind of mixture model and the mixture between different models.
To start with, we should know that if we set the number of Gaussian function to be used in the GMM estimation flexible, we will find out that the number of Gaussian function will never reach a best solution, as adding more Gaussian functions into the estimation will subsequently improve the accuracy anyway. As calculating how many Gaussian function is include in GMM is a clustering problem. We assume to know the number of Gaussian function in GMM as k here.
As this distribution is a mixture of Gaussian, the expression of probability is:
α_j is the weight of Gaussian function g_j (x).
Thus, the parameters to be estimated are:
Let X_1,X_2,…,X_N be the Independent and identically distributed (iid) Gaussian Mixture Model (GMM) random variables.
Following Bayes rule, the responsibility that a mixture component takes for explaining an observation X_i is:
Then, we will have a joint frequency function that is the product of marginal frequency functions. The log likelihood of Gaussian Mixture Model distribution thus should be:
Take the derivative of μ_j,Σ_j on it and find the μ_j,Σ_j value that make the derivation equals to 0.
The α_j is subject to
Basic optimization theories show that α_j is optimized by:
Thus, the ML estimation for Gaussian Mixture Model distribution should be:
#### 5.2 Practical Implementation
Now we can observe that, as the Gaussian Mixture Model with K Gaussian functions have 3K parameters, to find the best vector of parameters set, θ, is to find the optimized parameters in 3K dimension space. As the Gaussian Mixture Model include more Gaussian functions, the complexity of computing the best θ will go incrediblily high. Also, we can see that all the expressions of μ, Σ and α include themselves directly or indirectly, it’s implossible to get the value of the parameters within one time calculation.
Now it’s time to introduce a method for finding maximum likelihood with large number of latent variables (parameters), Expectation–maximization (EM) algorithm.
In statistics, an expectation–maximization (EM) algorithm is an iterative method for finding maximum likelihood estimates of parameters in statistical models, where the model depends on unobserved latent variables (the parameters). The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step.
In short words, to get the best θ for our maximum likelihood, firstly, for the expectation step, we should evaluate the weight of each cluster with the current parameters. Then, for the maximization step, we re-estimate parameters using the existing weight.
By repeating these calculation process for several times, the parameters will approach the value for the maximum likelihood. | {"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": 0.9499385356903076, "perplexity": 498.31916515099647}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662527626.15/warc/CC-MAIN-20220519105247-20220519135247-00489.warc.gz"} |
https://www.physicsforums.com/threads/lorentz-transformation.286704/ | # Homework Help: Lorentz Transformation
1. Jan 22, 2009
### yellowputty
1. The problem statement, all variables and given/known data
Two events occur at the same place in an inertial reference fram S, but are separated in time by 3 seconds. In a different frame S', they are separated in time by 4 seconds.
(a) What is the distance between the two events as measured in S'?
(b) What is the speed of S relative to S'?
2. Relevant equations
I'm presuming:
t' = gamma*(t-ux/c^2)
3. The attempt at a solution
I have the answer, and a hint saying to use the interval S^2, but I have no idea what that means, and where I start. When I look and the relevant Lorentz equations, they involve velocity, and I do not have a velocity here.
Could you please point me in the right direction?
2. Jan 22, 2009
### G01
The hint is implying that you use the invariant space-time interval to solve this problem. The following quantity is called the space time interval:
$$(\Delta s)^2= (\Delta x)^2 + (\Delta y)^2 + (\Delta z^2) - (c\Delta t)^2$$
This quantity is a Lorentz scalar and is thus invariant over Lorentz transformations (it is the same in all inertial frames). So, in one dimension this means:
$$(\Delta x)^2- (c\Delta t)^2=(\Delta x')^2- (c\Delta t')^2$$
Can you use this the solve the problem?
3. Jan 22, 2009
### yellowputty
Do I find $$(\Delta t)$$ by doing SQRT[(4^2)-(3^2)] = ROOT 7
Then at they are both at the same coordinates in the inertial reference frame, we can ignore x , y and z. Therefor S equals the root of (c^2)*(ROOT 7) = 7.9x10^8m
Is this correct?
4. Jan 22, 2009
### G01
You should only have one factor of c in your final line, since you take the square root of c^2 when solving for the answer. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9376248121261597, "perplexity": 551.961696801738}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742666.36/warc/CC-MAIN-20181115095814-20181115121814-00350.warc.gz"} |
https://learn.careers360.com/ncert/question-how-much-electricity-in-terms-of-faraday-is-required/ | Q
# How much electricity in terms of Faraday is required
3.13 How much electricity in terms of Faraday is required to produce
(ii)40.0 g of AI from molten $AI_{2}O_{3}$ ?
Views
The equation for the given question is :-
$Al^{+3}\ + 3e^-\ =\ Al$
Thus for 1 mol of Al, charge required is 3F.
So the required amount of electricity in terms of charge will be :-
$=\ \frac{3}{27}\times40F = 4.44F$
Exams
Articles
Questions | {"extraction_info": {"found_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": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8658542633056641, "perplexity": 1792.42106617794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875142603.80/warc/CC-MAIN-20200217145609-20200217175609-00063.warc.gz"} |
https://cs.stackexchange.com/questions/55231/why-is-map-insertionsort-not-to-equal-tomap-mergesort | # Why is map insertionsort not to equal tomap mergesort?
In the type theory podcast ep. 3, Dan Licata claims that the fact that for every input, insertionsort and mergesort give the same result does not imply that the result would be equal when used as higher order functions as arguments to a third function, i.e. map insertionsort does not have to equal map mergesort.
He explains this by "because you don't know that, as functions, insertionsort and mergesort are equal" but I still don't get it.
Why is this the case? A counter example would be great!
The statement of this axiom with regard to functions is $$\forall f,g:A \to B,\ ((\forall x:A ,\ f\ x = g\ x) \Leftrightarrow f = g).$$ Informally it means that if two functions are equal point-wise, then we consider them equal.
Syntactically merge-sort and insertion-sort are not equal, but if we don't care about their time and memory complexities (I mean if care only about their results) we can accept the axiom of extensionality and consider them equal. That means we can substitute one for another in every expression under consideration without actually changing anything. In this case $\text{map}\ f = \text{map}\ g$.
On the contrary, if we reject the aforementioned axiom, then we can only prove a statement like this: $$(\forall x:A,\ f\ x = g\ x) \implies \forall xs:\text{list}\ A,\ \text{map}\ f\ xs = \text{map}\ g\ xs.$$ Notice that the conclusion is not the same as $\text{map}\ f = \text{map}\ g$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8451566696166992, "perplexity": 395.09571098792384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1573496670601.75/warc/CC-MAIN-20191120185646-20191120213646-00312.warc.gz"} |
http://mathhelpforum.com/calculus/71157-easy-integral-but-i-keep-getting-wrong.html | # Thread: Easy Integral but I keep getting it wrong
1. ## Easy Integral but I keep getting it wrong
1/(x*sqrt(x))
I am thinking it is equal to x^(-3/2) so the integral should be
-1/2x^(-1/2) + C but thats apparently not correct
2. Originally Posted by TYTY
1/(x*sqrt(x))
I am thinking it is equal to x^(-3/2) so the integral should be
-1/2x^(-1/2) + C but thats apparently not correct
Be careful...
$\int x^{-\frac{3}{2}}\,dx=\frac{x^{\frac{-1}{2}}}{-\frac{1}{2}}+C=\dots$
3. Originally Posted by TYTY
1/(x*sqrt(x))
I am thinking it is equal to x^(-3/2) so the integral should be
-1/2x^(-1/2) + C but thats apparently not correct
$\int x^{- \frac{3}{2}}\, dx = \frac{x^{- \frac{1}{2}}}{- \frac{1}{2}} + c = -2 x^{- \frac{1}{2}} + c$
Better.
4. Originally Posted by Chris L T521
Be careful...
$\int x^{-\frac{3}{2}}\,dx=\frac{x^{\frac{-1}{2}}}{-\frac{1}{2}}+C=\dots$
crap, I multiplied instead of dividing
what an idiot (me)
thank you | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9970354437828064, "perplexity": 4034.889422554441}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1502886108264.79/warc/CC-MAIN-20170821095257-20170821115257-00259.warc.gz"} |
https://www.iitianacademy.com/physics-mcqs-for-class-12-with-answers-chapter-1-electric-charges-and-fields/ | # Physics MCQs for Class 12 with Answers Chapter 1 Electric Charges and Fields
### CBSE Class 12 Physics MCQs Solution All Chapters
Free PDF Download of CBSE Physics Multiple Choice Questions for Class 12 with Answers Chapter 1 Electric Charges and Fields. Physics MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Physics Electric Charges and Fields MCQs Pdf with Answers to know their preparation level.
### Electric Charges and Fields MCQ
1 . If a body is charged by rubbing it, its weight
1. Always decreases slightly
2. Always increases slightly
3. May increase slightly or may decrease slightly
4. Remains precisely the same
1 . (3)
If a body is charged by rubbing it, then it may lose or gain electrons. Since electrons have a mass of (9.1 x 10-31kg). So, a slight weight may increase or decrease slightly.
2. Safety fuse should have
1. High resistance and high MP
2. High resistance and low MP
3. Low resistance and high MP
4. Low resistance and low MP
2. (2)
The safety bulb should have high resistance and low melting point.
3. 3 charges +4q, Q and q are placed in a straight line of length l at points distance 0, 0.5 and 1 respectively. What should be the value of Q in order to make the net force on q to be zero?
1. -q
2. -2q
3. -q/2
4. 4q
3. (1)
Two forces are acting on q one due to 4q and second due to Q
F1 + F2 = 0
Q = -q
4. In the winter season, a mild spark is often seen when a man touches somebody’s else’s skin. Why?
1. Due to lack of humidity and rubbing with clothes, charge accumulate on human body which is discharged via sparking
2. Due to cold, electrostatic charge on body finds a lower resistance path to the skin of other’s body
3. He static charge on sweaters worn by the two persons is different, hence discharge through sparking occurs
4. Similar to the lighting, extremely high potential exists on both the bodies and hence they discharge through sparking
4. (1)
Due to friction between skin and cloths, electrostatic charge is built up on the skin. Hence, electrical discharge may occur when a man touches somebody else. This phenomenon is more significant in winters because due to low humidity, charge has a tendency to stay longer on the body.
5. There are two charges +2microC and -3microC. The ratio of forces acting on them will be
1. 2 : 3
2. 1 : 1
3. 3 : 2
4. 4 : 9
5. (2)
6. When a comb is rubbed with hair, it attracts paper bits. Choose the right explanation.
1. Bits of paper gets attracted due to gravitational force
2. Due to electromagnetic effect, bits of paper are attracted
3. Comb gets charged by friction and attract bits of paper
4. None of these
6. (2)
When a comb is rubbed with the hair, it induces negative charge due to friction. When it is brought closer to paper bits, they get polarized and the positive part stacks to the comb due to electrostatic attraction.
7. Four charges +Q, -Q, +Q and -Q are situated at the corners of a square; in a sequence then at the centre of the square
1. E = 0, V = 0
2. E = 0, V ≠ 0
3. E ≠ 0, V = 0
4. E ≠ 0, V ≠ 0
7. (3)
E ≠ 0 but V = 0 because E is not cancelled out by each other but V is cancelled out by each other.
8. A spherical conducting ball is suspended by a grounded conducting thread. A positive point charge is moved near the ball. That ball will
1. Be attracted to the point charge and swing toward it
2. Be repelled from the point charge and swing away from it
3. Not be affected by the point charge
4. None of these
8. (1)
Due to the positive charge present, negative charge will be induced. Hence, attraction will take place.
9. Electric force can be
1. Always attractive
2. Always repulsive
3. Attractive and repulsive
4. None of these
9. (3)
Electric forces can be both attractive and repulsive. Same charges repel each other while unlike nature of charge attract each other.
10. A charge Q is divided into two parts of q and Q-q. If the coulomb repulsion between them when they are separated , is to be maximum, the ratio of Q/q is
1. 2
2. ½
3. 4
4. ¼
10. (1)
Let the charges be separated by a distance ‘d’
Maximum force will be got after differentiating the force equation
For maximum or minimum force
Q – 2q = 0
Q = 2q
Q/q = 2
11. Two neutrons are placed at some distance apart from each other. They will
1. Attract each other
2. Repel each other
3. Neither repel nor attract each other
4. Depends on the distance between the two nucleus
11. (4)
If the intermolecular distance is less than 0.8pm, then the nucleus will repel each other and if the intermolecular distance is more than 0.8 pm, the nucleus will attract each other.
12. Two bodies X and Y carry charges -6.6µC and -5µC. How many electrons should be transferred from X to Y so that they acquire equal charges?
1. 2 x 1012
2. 5 x 1014
3. 5 x 1012
4. 5 x 1013
12. (3)
X has excess charge of -1.6 x 10-6 C composed to Y
X has to give a charge -0.8 x 10-6 C if charge on both have to be the same
The charge on one electron is -1.6 x 10-19 C
The number of electrons required to transfer a charge of -0.8 x 10-6 C =
13. An electric charge is held at rest in a region filled with a non-uniform magnetic field. As soon as it is left free
1. It will move in the direction of the field
2. It will move opposite to the direction of the field
3. It will move perpendicular to the direction of the field
4. It will remain at rest
13. (3)
As soon as it is left free it will move perpendicular to the direction of the electric field.
14. Which one of the following is a bad conductor?
1. Acid
2. Coal
3. Distilled water
4. Human body
14. (3)
Distilled water is an insulator.
15. SI unit of electric flux is
1. Voltmeter
2. Joule/metre
3. Newton
4. None
15. (2)
E = V = dE
E = dv/dx = V/m
16. A charge is placed at the edge of a cube of side length ‘a’. Calculate the electric flux through each other face of the cube.
1. Q / 4𝛜o
2. Q / 24𝛜o
3. Q / 2𝛜o
4. Q / 15𝛜o
16. (2)
Flux through each cube will be Q / 4𝛜o . The flux through each face will be Q / 24𝛜o .
17. Electric flux per unit solid angle is defined as
1. Electric force
2. Electric field intensity
3. Electric potential
4. Electric power
17. (2)
F1 = qE; F2 = -qE; Fnet = F1 + F2 = 0
18. An electric dipole is placed in a uniform electric field. The net electric force on the dipole.
1. Is always zero
2. Depends on the orientation
3. Depends on the dipole moment
4. Is always finite but not zero
18. (1)
Is always zero
19. Induction is possible
1. Only in conductor
2. Only in insulator
3. Both in conductor and insulator
4. None of these
19. (1)
Only conductor can be inducted as the process involves movement and electrons. Only conductors have free electrons in there which help in the conduction process to take place.
20. When the separation between two charges is increased the electric potential energy of the charges
1. increases
2. decreases
3. remains the same
4. may increase or decrease
20. (2)
The potential energy in inversely proportional to square of distance between the charges. The potential energy also depends directly proportional to the nature (sign) of the charges. If the nature of both the charges is same, then the potential will decrease, if the nature of the both the charges are different, then the potential increase.
21. If a positive charge is shifted from a low potential region to high potential region, the electric potential energy
1. decreases
2. remains the same
3. may increase or decrease
4. increases
21. (4)
PE = qV
Since positive charge is shifted from low and high potential.
if q = +ve and V = +ve, PE increases.
22. Mark out the correct options.
1. Total positive charge of the universe is constant
2. Total negative charge of the universe is constant
3. Total number of charged particles in the universe is constant
4. The total charge of the universe is constant
22. (4)
According to the principal of conservation of charge, the total amount of positive charge minus the total amount of negative charge in the universe is constant. Therefore, the total charge of the universe in constant.
23. A Proton and an electron are placed in a uniform electric field
1. the electric forces acting on them will be equal
2. the magnitudes of the forces will be equal
3. their acceleration will be equal
4. the magnitude of their acceleration will be equal
23. (2)
The proton and electrons have equal and opposite charges. Therefore, the magnitude of the force i equal hen kept in an electric field.
24. An electric dipole is placed in an electric field generated by a point charge.
1. The net electric force on the dipole must be zero
2. The net electric force on the dipole maybe zero
3. The torque on the dipole due to field must be zero
4. The torque on the dipole due to the field may be zero
24. (4)
If the dipole moment and the electric field are parallel to each other, then the torque will be zero as pEsin0 = 0
25. Electric lines of force
1. exist everywhere
2. exist only in the immediate vicinity of electric charges
3. exist only when both positive and negative charges and near one and other
4. are imaginary
25. (4)
Electric lines of force are imaginery.
26. Which of the following statements is not true about Gauss’s law
1. Gauss’s law is not much useful in calculating electrostatic field when the system has some symmetry
2. Gauss’s law is based on the Inverse Square dependence on distance contained in the Coulomb’s law
3. Gauss’s law is true for any closed surface
4. The term Q on the right side of gases law includes the sum of all charges in closed by the surface
26. (1)
Gauss’s law is useful for an easier calculations of electrostatic field when the system has some symmetry. This can be done with the correct choice of Gaussian surface.
27. To infinite plate parallel sheets separated by a distance D have equal and opposite uniform charge densities σ. Electric field at a point between the sheets is
1. σ/2𝛜o
2. σ/𝛜o
3. Zero
4. Depends on the location of the point
27. (2)
σ/𝛜o
28. The electric field at a point on axial line of a dipole and direction of the dipole moment
1. Will be parallel and in the same direction
2. Will be in opposite direction
3. Will be perpendicular
4. Are not related
28. (3)
29. If an electron has an initial velocity in a direction different from that of an electric field, the path of the electron is:
1. a straight line
2. a circle
3. an ellipse
4. a parabola
29. (4)
The path will be a parabola, as the velocity will be resolved into two components, one parallel to electric field and one perpendicular to the electric field.
30. Electric lines of force about a negative point charge are
1. circular anticlockwise
2. circular clockwise
30. (3)
The electric lines of force are radial and inwards.
31. If Ea be the electric field strength of a short dipole at a point on its axial line and Ee that on the equatorial line at the same distance, then
1. Ee=2Ea
2. Ee=Ea
3. Ea=2Ee
4. Ee=4Ea
31. (1)
Ea = 2Ee
32. The surface considered for Gauss’s law is called
1. Closed surface
2. Spherical surface
3. Gaussian surface
4. Plane surface
32. (3)
The surface considered for Gauss’s law is called Gaussian surface.
33. A hemisphere is uniformly charged positively. The electric field at a point on a diameter away from the centre is directed:
1. perpendicular to the diameter
2. parallel to the diameter
3. at an angle tilted towards the diameter
4. at an angle tilted away from the diameter
33. (1)
When the point is on the diameter and away from the centre of hemisphere which is charged uniformly and +vely, the component of electric field intensity parallel to the diameter cancel out.
34. The electric field at a point 5 cm from a long line charge of density 2.5 x 10-6 cm-1 is
1. 9 x 103 NC-1
2. 9 x 104 NC-1
3. 9 x 104 NC-1
4. 9 x 104 NC-1
34. (3)
E = 2λ/4π𝛜or = (9 x 109 x 2 x 2.5 x 10-6) / 5 x 10-2 = 9 x 105 NC-1
35. The field at a distance r from a long straight wire of charge per unit length ƛ is
1. kƛ / r2
2. kƛ / r
3. kƛ / 2r
4. 2kƛ / r
35. (4)
2kƛ / r
36. F is the force between two charges. If the distance between them is tripled, then the force between the charges will be
1. F
2. F / 3
3. F / 9
4. F / 27
36. (3)
F/9
37. An attractive force of 9N acts between +5C and -5C at some distance. These charges are allowed to touch each other and are then again placed at their initial position. The force acting between them will be
1. infinite
2. 9 x 109 N
3. 1 N
4. zero
37. (4)
38. SI unit of volume charge density is
1. cm-1
2. cm-2
3. cm-3
4. cm-4
38. (3)
cm-3
39. The resultant electric field at centre of a ring due to ring is zero. Which of the following is incorrect
1. The total charge of the ring may be zero, although everywhere
2. The charge on the ring must be uniformly distributed
3. The charge on the ring may be distributed non-uniformly
4. Total charge on the ring may be positive
39. (3)
The charge on the ring must be uniformly distributed
40. If two charged particles of same mass and charge are projected in a uniform electric field with the same speed, then
1. Both have same momentum at any instant
2. Both have same kinetic energy at any instant
3. Both have same magnitude of momentum at any instant
4. They may move on a straight line
40. (4)
Kinetic energy and momentum can be different if throwing angles are different. increase velocity will differ. If thrown along the field, they may travel in straight line.
41. A hollow sphere of sphere of charge does not produce electric field at any
1. Interior point
2. Outer point
3. Beyond 2m
4. Beyond 10cm
41. (1)
Interior point
42. Charges Q each are placed at each of two opposite corners of a square. Charges q each are placed at each of the other two corners. Find the relation between q and Q when the net force on Q is zero.
1. q = – 3Q / 2√2
2. q = + Q / 2√2
3. q = – Q / √2
4. q = – Q / 2√2
42. (4)
q = – Q / 2√2
43. A point charge q1 exerts a force F upon another point charge q2. If a third charge q3 be placed quite close to the charge q2 then the force that charge q1 exerts on the charge q2 will be
1. F
2. >F
3. <F
4. Zero
43. (1)
F
44. The number of electrons present in 1C of charge is _____
1. 4.25 x 1018
2. 6.25 x 1018
3. 3.25 x 1018
4. 2.25 x 1018
44. (2)
6.25 x 1018
45. In Coulomb’s law, the constant of proportionality has units as
1. N
2. Nm2
3. NC2/m2
4. Nm2/C2
45. (4)
Nm2/C2
### Assertion and Reasoning MCQ
Codes
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true and but R is not a correct explanation of A
(c) A is true but R is false
(d) A is false, but R is true
1 . Assertion (A) The charge on any body can be increased or decreased in terms of e
Reason (R) Quantization of charge means that the charge of a body is integral multiple of e
1 . (a)
Protons and electrons are the only basic charges in the universe. All the observable charges have to be integral multiple of e. Thus, if a body contains n electrons and m protons. The total number of charge in the body is m.e + n(-e) = (m-n)e. Since n and m are integers, their difference is also an integer. Thus, the charge on a body is always an integral multiple of e and can be increased or decreased in terms of e.
2. Assertion (A) The properties that the force with which two charges attract or repel each other are not affected by the presence of a third charge
Reason (R) Forces on any charge due to the number of other charges is the vector sum of all the forces on that charge due to other charges, taken one at a time.
2. (b)
Force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time the individual force is unaffected due to the presence of other charges. This is the principle of superposition of charges.
3. Assertion (A) When we rub a glass rod with Silk, the rod gets negatively charged and the Silk gets positively charged
Reason (R) On rubbing, electrons from Silk cloth move to the glass rod
3. (d)
When we rub a glass rod with silk cloth, electrons from the glass rod and transferred to the silk cloth. Thus, the rod gets positively charged and the cloth gets negatively charged.
4. Assertion (A) Consider two identical charges placed distance 2d apart, along the x-axis. The equilibrium of a positive test charge placed at the point O midway between them is stable for displacements along the x-axis
Reason(R) Force on test charge is zero
4. (b)
If positive charge is displaced along x axis, then that force will always act in a direction opposite to that of displacement and then test charge will always come back to its original position.
5. Assertion (A) Coulomb’s force is the dominating force in the universe
Reason (R) Coulomb’s force is weaker then the gravitational
5. (d)
Gravitational force is the dominating force in the nature and not coulomb’s force. Gravitational force is the weakest force. Also, coulomb’s force is very strong than gravitational force.
6. Assertion (A) When charges are shared between any two bodies no charges is really lost but some loss of energy does occur
Reason (R) Some of the energy is dissipated in the form of heat, sparkling etc
6. (a)
Charges are conversed by the law of conservation of charges. Energy is also conserved, if we take in account of the loss of energy by heat, sparkling etc.
7. Assertion (A) If there exists coulomb attraction between two bodies both of them must be charged
Reason (R) In coulomb attraction of two bodies are positively charged
7. (d)
Coulomb’s attraction exist even when one body is charged and the other is uncharged.
8. Assertion (A) The positively charged particle is placed in front of a spherical uncharged conductor. The number of lines forces terminating on the spare will be more than those emerging from it
Reason (R) The surface charge density at a point on the sphere nearest to the point charge will be negative and maximum in magnitude compared to other points on the sphere
8. (d)
Number of lines entering the surface = number of lines leaving the surface
9. Assertion (A) A point charge is brought in an electric field at a nearby point will increase or decrease depending on the nature of charge
Reason (R) The electric field is independent of the nature of charge
9. (c)
Electric field will increase a positive charge is brought in an electric field.
10. Assertion (A) On going away from a point charge or a small electric dipole, electric field decreases at the same rate in both the cases
Reason (R) Electric field is inversely proportional to cube of distance from the charge of an electric dipole
10. (d)
The rate of decrease of electric field is different in the two cases in case of point charge is it decreases as 1/r2 but in the case of electric dipole, it decreases more rapidly as 1/r3.
11. Assertion (A) Electrostatic field line start at positive charges and end at negative charges
Reason (R) Field lines are continuous curve without any breaks and they form a closed loop
11. (c)
Electrostatic field lines are continuous curves without any breaks. They start at positive charges and end at negative charges. They can not form closed loops.
### Case Study Based MCQ
1 . A Faraday cage or Faraday Shield is an enclosure made of a conducting material. The fields within a conductor cancel out with any external fields, so the electric field within the enclosure is zero.
These Faraday cage act as a big hollow conductor; you can put things in it to shield them from electric fields.
Any electrical shock cage receives, pass harmlessly around the outside of the cage.
1. Plastic
2. Glass
3. Copper
4. Wood
ii) Example of real world Faraday cage is
1. Cars
2. Plastic box
3. Lighting rod
4. Metal rod
iii) What is the electrical force inside a Faraday cage when it is struck by lightning
1. The same as the lightning
2. Half that of the lightning
3. Zero
4. A quarter of the lightning
iv) An isolated point charge + q is placed inside a Faraday cage. It’s surface must have charge equal to
1. Zero
2. +q
3. -q
4. +2q
2. Gauss theorem is mainly used to find out the electric flux linked to a closed surface. It does not depend upon the shape or size of the surface. According to this theorem, the electric flux linked to a closed surface is equal to 1/𝛜o times the charge enclosed by the surface. Let we have a charge q, now if we want to find out the net flux linked to a closed surface around it them,
Electric flux ɸ = ∲sE.ds = q / 𝛜o
i) Gauss theorem is used to find out:
1. Electric force
2. Electric flux
3. Electric potential
4. None of these
ii) This theorem is applied over a ________ surface:
1. Closed surface
2. Open surface
3. Both 1 and 2
4. None of these
iii) Gauss theorem does not depends upon the………of surface:
1. Shape
2. Size
3. Area
4. Both 1 and 2
iv) If we increase the charge enclosed by the surface then electric flux will:
1. Increases
2. Decreases
3. Remain same
4. Both 1 and 2
v) Net flux linked to a closed surface around a charge particle is times the charge.
1. 𝛜o
2. 1/𝛜o
3. 𝛜o2
4. None of these
3. A system of closely spaced electric charge forms a continuous charge distribution. To find the field of a continuous charge distribution, we divide the charge into infinitesimal charge elements. Each infinitesimal charge element is then considered as a point charge and electric field dE is determined due to this charge at given point. The net field at the given point is the summation of fields of all the elements i.e., E = ∫dE
i) How many electrons must be added to an isolated spherical conductor of radius 20 cm to produce an electric field 1000 N/C just outside the surface?
1. 2.77 x 1020
2. 2.77 x 1010
3. 1.77 x 1010
4. 5.4 x 1010
ii) A circular annulus of inner radius r and outer radius R has a uniform charge density ‘a’. What will be the total charge on the annulus?
1. a (R2-r2)
2. πa (R2-r2)
3. a (R-r)
4. πaR2
iii) What is the dimension of linear charge density?
1. [ATL-1]
2. [AT-1L]
3. [ATL]
4. [A-1T-1L]
4. Microwave oven works on the principle of torque acting on an electric dipole. The food we consume has water molecules which are permanent electric dipoles. Oven produces microwaves that are oscillating electromagnetic fields and produce torque on the water molecules. Due to this torque on each water molecule, molecules rotate very fast and produce thermal energy. Thus, heat generated is used to heat the food.
i) An electric dipole is placed at an angle of 300 to a uniform electric field. The dipole will experience a torque as well as & translational force.
1. a torque as well as & translational force.
2. a torque only
3. a translational force only in the direction of the field
4. a translational force only in a direction normal to direction of the field
ii) An electric dipole is placed in a nonuniform electric field, what acts on it?
1. only torque
2. only force
3. both 1 and 2
4. none of these
iii) An electric dipole of moment p in placed in a uniform electric field E . The maximum torque experienced by the dipole is
1. pE
2. p/E
3. E/p
4. p.E
iv) Let Ea be the electric field due to a dipole in its axial plane distant I and let Eq be the field in the equatorial plane distant l. The relation between Ea and Eq is
1. Ea = Eq
2. Ea = 2Eq
3. Eq = 2Ea
4. Ea = 3Eq
v) A point P lies on the perpendicular bisector of an electric dipole of dipole moment p. If the distance Of P from the dipole is r (much larger than the size of the dipole) then the electric field at P is proportional to:
1. p-1 and e-2
2. p and r-2
3. p2 and r-3
4. p and r-3
### Electric Charges And Field MCQ Chapter 1
Electric charge is a basic property of matter carried by elementary particles which get affected by electric field and magnetic field.
Below are some of the very important NCERT Electric Charges And Field MCQ Class 12 Physics Chapter 1 with answers. These Electric Charges And Field MCQs have been prepared by expert teachers and subject experts based on the latest syllabus and pattern of CBSE examination.
### Electric Charges and Fields Class 12 Physics MCQs
1. When a glass rod is rubbed with silk, it
(a) gains electrons from silk.
(b) gives electrons to silk.
(c) gains protons from silk.
(d) gives protons to silk.
Explaination:
(b) On rubbing a glass rod with silk, excess electrons are transferred from glass to silk. So glass rod becomes positive and silk becomes negative.
2. In general, metallic ropes are suspended on the carriers taking inflammable materials. The reason is
(a) to control the speed of the carrier.
(b) to keep the centre of gravity of the carrier nearer to the earth.
(c) to keep the body of the carrier in contact with the earth.
(d) none of these.
Explaination:
(c) For providing a path to the charge induced on the surface of the carriers.
3. Two charges q1 and q2 are placed in vacuum at a distance d and the force acting between them is F. If a medium of dielectric constant 4 is introduced around them, the force now will be ______ .
Explaination: $$\frac{F}{4}$$.In the presence of medium, force becomes $$\frac{1}{K}$$ time
4. When 1014 electrons are removed from a neutral metal sphere, the charge on the sphere becomes ______ .
Explaination:
16 µC, Q = ne= 1014 x 1.6 × 10-19 or 0=1.6 × 10-5 C = 16 µC
As electrons are removed, so charge will be positive.
5. Two similar spheres having +Q and -Q charges are kept at a certain distance. F force acts between the two. If at the middle of two spheres, another similar sphere having +Q charge is kept, then it experiences a force in magnitude and direction as
(a) zero having no direction.
(b) 8F towards +Q charge.
(c) 8F towards -Q charge.
(d) 4F towards +Q charge.
Explaination:
(c) Initially, force between A and C,
When a similar sphere B having charge +Q is kept at the mid-point of line joining A and C, then net force on B is
The direction is shown in figure.
6. A charge Q is divided into two parts of q and Q – q. If the coulomb repulsion between them when they are separated is to be maximum, the ratio of Q/q should be
(a) 2:1
(b) 1/2
(c) 4:1
(d) 1/4
Explaination:
(a) Let separation between two parts be r, then
F = k.q(Q – q)/r² , For F to be maximum dF/dq = 0 then Q/q = 2/1 = 2 : 1
7. Four equal charges q are placed at the four comers A, B, C, D of a square of length a. The magnitude of the force on the charge at B will be
Explaination:
8. Dielectric constant for metal ______ .
Explaination:
Infinite [Dielectric constant K = $$\frac{\varepsilon}{\varepsilon_{0}}$$
Permittivity of metals ($$\E$$) is assumed to be very high.]
9. Two charges of equal magnitudes kept at a distance r exert a force F on each other. If the charges are halved and distance between them is doubled, then the new force acting on each charge is
Explaination:
(d) F = $$\frac{k \cdot Q^{2}}{r^{2}}$$. If Q is halved, r is doubled then F = $$\frac{1}{16}$$ time
10. The electric field inside a spherical shell of uniform surface charge density is
(a) zero.
(b) constant, less than zero.
(c) directly proportional to the distance from the centre.
(d) none of the these
Explaination:
(a) All charges reside on the outer surface of the shell so according to Gauss’s law, electric field inside the shell is zero.
11. A cylinder of radius R and length L is placed in a uniform electric field E parallel to the cylinder axis. The total flux for the surface of the cylinder is given by
Explaination:
12. Electric field at a point varies as r° for
(a) an electric dipole
(b) a point charge
(c) a plane infinite sheet of charge
(d) a line charge of infinite length
13. An electric charge q is placed at the centre of a cube of side a. The electric flux on one of its faces will be
Explaination: (a) Using Gauss’s theorem
14. Total electric flux coming out of a unit positive charge kept in air is
Explaination:
(b) Total flux coming out from the unit charge is
15. The electric field intensity due to an infinite cylinder of radius R and having charge q per unit length at a distance rir r(r > R) from its axis is
(a) directly proportional to r².
(b) directly proportional to r3.
(c) inversely proportional to r.
(d) inversely proportional to r².
Explaination:
16. A point charge q is placed at a distance a/2 directly above the centre of a square of side a. The electric flux through the square is
Explaination:
(d) An imaginary cube can be made by considering charge q at the centre and given square is one of its face. So flux through the given square (i.e. one face)
17. Which of the following graphs shows the variation of electric field E due to a hollow spherical conductor of radius R as a function of distance from the centre of the sphere?
Explaination:
(a) Electric field due to a hollow spherical conductor is governed by equations E = 0, for r < R …(i)
and $$E=Q / 4 \pi \varepsilon_{0} r^{2}$$ for r ≥ R ….(ii)
i.e. inside the conductor, electric field will be zero and outside the conductor it will vary according to E oc 1/r².
18. The magnitude of electric field intensity E is such that, an electron placed in it would experience an electrical force equal to its weight is given by
(a) mge
(b) mg/e
(c) e/mg
(d) e²g/m²
Explaination: (b) According to the question, eE = mg or E = mg/e
19. In Fig. (i) two positive charges q2 and q3 fixed along the y-axis, exert a net electric force in the +x direction on a charge q1 fixed along the x-axis. If a positive charge Q is added at (x, 0) in figure (ii), the force on q1 is [NCERT Exemplar]
(a) shall increase along the positive x-axis.
(b) shall decrease along the positive x-axis.
(c) shall point along the negative x-axis.
(d) shall increase but the direction changes because of the intersection of Q with q2 and qy
Explaination:
(a) The net electrostatic force on the charge q1 by the charges q2 and q3 is along the positive x-direction. Hence the nature of force between qu q2 and qx, q3 should be attractive. It means qx should be negative.
20. Which of the following statement is correct? The electric field at a point is [NCERT Exemplar]
(a) always continuous.
(b) continuous if there is a charge at that point.
(c) discontinuous only if there is a negative charge at that point.
(d) discontinuous if there is a charge at that point.
Explaination:
(d) The electric field due to any charge will be continuous, if there is no other charge in the medium. It will be discontinuous if there is a charge at the point under consideration.
21. A point charge +q is placed at a distance d from an isolated conducting plane. The field at a point P on the other side of the plane is [NCERT Exemplar]
(a) directed perpendicular to the plane and away from the plane.
(b) directed perpendicular to the plane but towards the plane.
(c) directed radially away from the point charge.
(d) directed radially towards the point charge.
Explaination:
(a) The electric field lines are away from positive charge and perpendicular to the surface. Hence the field at a point P on the other side of the plane is directed perpendicular to the plane and away from the plane.
22. Gauss’s law will be invalid if
(a) there is magnetic monopoles.
(b) the inverse square law is not exactly true.
(c) the velocity of light is not a universal constant.
(d) none of these.
23. SI unit of permittivity of free space is
(b) Weber
(c) C2N-1 m-2
(d) C2N-1 m-2
24. A charge Q is placed at the centre of the line joining two point charges +q and +q as shown in the figure. The ratio of charges Q and q is
(a) 4
(b) 1/4
(c) -4
(d) -1/4
25. The force per unit charge is known as
(a) electric flux
(b) electric field
(c) electric potential
(d) electric current
26. Electric field lines provide information about
(a) field strength
(b) direction
(c) nature of charge
(d) all of these
27. Which of the following figures represent the electric field lines due to a single negative charge?
28. The SI unit of electric flux is
(a) N C-1 m-2
(b) N C m-2
(c) N C-2 m2
(d) N C-1 m2
29. The unit of electric dipole moment is
(a) newton
(b) coulomb
(d) debye
30. Consider a region inside which, there are various types of charges but the total charge is zero. At points outside the region
(a) the electric field is necessarily zero.
(b) the electric field is due to the dipole moment of the charge distribution only.
(c) the dominant electric field is inversely pro-portional to r3, for large r (distance from ori-gin).
(d) the work done to move a charged particle along a closed path, away from the region will not be zero.
31. The surface considered for Gauss’s law is called
(a) Closed surface
(b) Spherical surface
(c) Gaussian surface
(d) Plane surface
32. The total flux through the faces of the cube with side of length a if a charge q is placed at corner A of the cube is
33. Which of the following statements is not true about Gauss’s law?
(a) Gauss’s law is true for any closed surface.
(b) The term q on the right side side of Gauss’s law includes the sum of all charges enclosed by the surface.
(c) Gauss’s law is not much useful in calculating electrostatic field when the system has some symmetry.
(d) Gauss’s law is based on the inverse square dependence on distance contained in the coulomb’s law
34. Four charges are arranged at the comers of a square ABCD, as shown. The force on the charge kept at the centre O is
(a) zero
(b) along the diagonal AC
(c) along the diagonal BD
(d) perpendicular to side AB
Explaination:
(c) Place a unit positive charge at O. Resultant force due to the charges placed at A and C is zero and resultant charge due to B and D is towards D along the diagonal BD.
35. One end of a copper wire is connected to a neutral pith ball and other end to a negatively charged plastic rod. What will be the charge acquired by a pith ball? [Chennai 2019]
Explaination: Negative charge.
36. Distinguish between an insulator (dielectric) and a conductor.
Explaination:
Dielectrics do not have free electrons, while conductors have free electrons. When some charge is transferred to a conductor, it readily gets distributed over the entire surface of the conductor, but on insulators, the charge stays at the same place.
37. Why does a nylon or plastic comb get electrified on combing or rubbing but a metal spoon does not?
Explaination:
The charge on a metal spoon discharges through our body to the ground as both are conductors. But when a nylon or plastic comb is rubbed, due to the friction its acquires a negative (-ve) charge, which stays on it as it is an insulator.
38. Two metallic spheres having same shape and size, but one of Cu and other of Al, are both placed in an identical electric field. In which metallic sphere will more charge be induced?
Explaination:
Same charge will be induced on both the spheres. As the dielectric constant K = oo for metals and the induced charge is given by q’ = $$=-q\left(1-\frac{1}{K}\right)$$.
39. What causes the charging of an object?
Explaination:
When an object looses or gains electrons by friction/conduction/induction, then it is charged.
40. What does the additive nature of electric charge mean?
Explaination:
It means an electric charge is a scalar quantity and is added like algebraic numbers.
41. When does a charged ring behave as a point charge?
Explaination:
When the radius of ring is much smaller than the distance under consideration.
42. Two insulated charged copper spheres A and B of identical size have charges qA and qB respectively. A third sphere C of the same size but uncharged is brought in contact with the first and then in contact with the second and finally removed from both. What are the new charges on A and B1 [Chennai 2019, Foreign 2011]
Explaination:
43. What does q1 + q2 = 0 signify?
Explaination:
q1 + q2 = 0
⇒ q1 = – q2
∴ q1 and qq2 are the two charges of an electric dipole.
44. What is the cause of quantisation of electric charge?
Explaination:
The minimum charge that is stable, is charge of an electron. Since, electrons are transferred from one object to another, therefore, electric charge is said to be quantised.
45. What do you mean by conservative nature of the electric force?
Explaination:
The electric force is conservative in nature, because the work done by it in moving a charge is path independent.
46. If a body contains n1 electrons and n2 protons, then what will be the total amount of charge on the body?
Explaination:
Electric charge on n1 electrons = – n1e and electric charge on n2 protons = + n2e Therefore, the total charge = (n2 – n1)e.
47. What is the limitation of Coulomb’s law?
Explaination: It is applied only for point charges.
48. What does e(absolute permittivity) signify?
Explaination: It is a measure of the degree to which a medium can resist the movement of charges.
49. Define 1 coulomb (1 C) of electric charge.
Explaination:
One coulomb is that charge, when placed in vacuum at a distance of one metre from an equal and similar charge, would repel it with a force of 9 × 109 N.
50. Write two properties of an electrostatic force.
Explaination:
(a) It is conservative in nature.
(b) It depends on medium between the two charges.
51. Is the force acting between two point electric charges qx and q2 kept at some distance apart in air, attractive or repulsive when
(i) q1q2 > 0
(ii) q1q2 < 0?
Explaination:
(i) When q1q2 > 0, force is repulsive.
(ii) When q1q2 < 0, force is attractive.
52. The force on an electron kept in an electric field in a particular direction is F. What will be the magnitude and direction of the force experienced by a proton kept at the same point in the field? Mass of the proton is about 1836 times the mass of the electron.
Explaination: Same in magnitude and opposite in the direction as F = e.E
53. How does the coulomb force between two point charges depend upon the dielectric constant of the intervening medium?
Explaination:
Coulomb force is inversely proportional to the dielectric constant of the intervening medium.
54. State principle of superposition of forces.
Explaination:
Net force experienced by any charge in a group of charges is the vector sum of the forces acting on it due to rest of the charges of the group.
55. Define the dielectric constant of a medium. What is its unit? [Delhi 2011C]
Explaination:
Dielectric constant of a medium is defined as the ratio of the force between two charges placed a certain distance apart in vacuum to the force between the same two charges placed the same distance apart in the medium. It has no units.
56. Two equal balls having equal positive charge q coulombs are suspended by two insulating strings of equal length. What would be the effect on the force when a plastic sheet is inserted between the two? [AI 2014]
Explaination:
Electric force will reduce as plastic is an insulator. The force between the two charges will reduce by 1/K, where K is the dielectric constant of plastic.
57. Two electric field lines never cross each other. Why?
Explaination:
Two field lines can never cross each other. If they did, the field at the point of intersection will not have a unique direction, which is absurd.
58. Draw the electric field lines due to apoint charge
(i) Q > 0 and (ii) Q< 0.
Explaination:
(i) Q > 0 (ii) Q < 0
59. Why do the electric field lines not form any closed loops?
Explaination:
Because they originate from positive (+ve) charge and terminate at negative (-ve) charge.
60. Draw electric field lines for a system of two charges q1 and q2 such that
(i) q1q2 >0; q1>q2>0
(ii) q1 q2 <0; q1 > |-q2| < 0, |q1|> |-q2|
Explaination:
61. Draw the electric field lines if (i) a point charge + q is placed at the centre (ii) a point charge + q is placed at a distance R/2 from the centre.
Explaination:
62. What is the physical significance of electric field?
Explaination:
From the knowledge of electic field intensity at any point, we can readily calculate the magnitude and the direction of force experienced by any .charge q0 placed at that point.
63. Define the term electric dipole moment. Is it a scalar or a vector quantity?
Explaination:
The product of the magnitude of one of the point charges constituting an electric dipole and the separation between them is termed as electric dipole moment.
It is a vector quantity.
64. What is an ideal (point) dipole?
Explaination:
An ideal dipole is the dipole whose size (2a) is vanishingly small, and the magnitude of electric charges constituting by it is very large, and the product, i.e. 2aq is finite.
65. What is the value of $$\left|\frac{E_{a x}}{E_{e q}}\right|$$ for a short electric dipole?
Explaination:
66. Two point charges +q and -q are placed at a distance d apart. What are the points at which the resultant electric field is parallel to the line joining the two charges?
Explaination:
(i) At any point on axial line.
(ii) At any point on equitorial line of a dipole.
67. If F is the magnitude of force experienced by a unit charge placed at a distance of 1 cm from an infinitely large charged sheet, then what will be the force experienced by the same charge placed at a distance of 2 cm from the same sheet? [HOTS]
Explaination:
In case of sheet of charge, the electric field is constant. Hence, F – qE will be same irrespective of distance.
68. What is the direction of net force on electric dipole, placed in a non-uniform electric field?
Explaination:
Since, the electric field at the location of charge -q is more than that of field at charge +q. Therefore, the direction of net force will be in the direction opposite to the direction of $$\vec{E}$$.
69. When does an electric dipole placed in a non-uniform electric field experience a zero torque but non-zero force?
Explaination: When the dipole axis is parallel to the direction of electric field.
70. Name the physical quantity whose SI unit is V.m. Is it a vector or a scalar quantity?
Explaination: The physical quantity is an electric flux. It is a scalar quantity.
71. Define the term electric flux. Write its SI unit. [Foreign 2017]
Explaination:
Electric flux through an area is the product of magnitude of area and the component of electric field vector normal to it.
72. What is a Gaussian surface?
Explaination:
A Gaussian surface is an imaginary closed surface in three dimensional space through with the flux of a vector field is calculated.
73. What is the use of a Gaussian surface?
Explaination:
A Gaussian surface is used to determine the electric field intensity around a point charge or charged body.
74. Why can a Gaussian surface not pass through any discrete charge?
Explaination:
Because the electric field due to a system of discrete charges is not defined at the location of any charge.
75. Two charges of magnitudes -2 Q and + Q are located at points (a, 0) and (4a, 0) respectively. What is the electric flux due to these charges through a sphere of radius 3a with its centre at the origin? [AI 2013]
According to the Gauss’s theorem, the total electric flux through any closed surface is equal to $$\frac{1}{\varepsilon_{0}}$$ times, the total charge enclosed eo by the surface. | {"extraction_info": {"found_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": 0.9262701272964478, "perplexity": 795.9263720601832}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104683708.93/warc/CC-MAIN-20220707063442-20220707093442-00384.warc.gz"} |
http://mathhelpforum.com/calculus/11198-stuck-tricky-integral.html | # Math Help - Stuck on a tricky integral
1. ## Stuck on a tricky integral
I tried get a tidy image of the integral with Integrator, but it told me the answer does not exist. I'm almost certain it does however.
(1+ln(x))*sqrt(1+(xln(x))^2)
Help! I tried integration by parts but it turns into a jumbled mess, unless I'm overlooking something.
2. Wait a second, I just had some inspiration. I'll let you know If I truley need help...
3. Ok, hmmm. I got it down to the integral of: sqrt(1+u^2) by setting xlnx=u and 1+lnx dx = du. Im stuck now How do I integrate sqrt(1+u^2) ?
4. Originally Posted by phack
Ok, hmmm. I got it down to the integral of: sqrt(1+u^2) by setting xlnx=u and 1+lnx dx = du. Im stuck now How do I integrate sqrt(1+u^2) ?
Good job spotting the substitution. Now what you want to do is let $u = sinh(y)$. Then $du = cosh(y)dy$, so your integral turns into:
$\int (1 + xln(x)) \sqrt{1 + (x ln(x))^2)} dx = \int \sqrt{1 + u^2} \, du =$ $\int \sqrt{1 + sinh^2(y)} \cdot cosh(y)dy = \int cosh^2(y) dy$
$= \int \frac{1}{4}(e^{2y} + e^{-2y} + 2)dy = ...$
You can finish this from here.
-Dan | {"extraction_info": {"found_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": 0.9957275986671448, "perplexity": 1244.4395167062157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246639057.4/warc/CC-MAIN-20150417045719-00024-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://dsp.stackexchange.com/users/41635/q-p?tab=topactivity | Q.P.
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3 Fourier transform of a damped cosine wave with a linear frequency chirp 3 Definition of the DFT / FFT Bin Size 3 The Fourier transform of a damped cosine and the units of the result 3 Should you scale the FFT bins by $1/N$ where $N$ is the number of points in a transient signal? 2 Is there a physical reason why phase should be uniformly distributed?
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4 Simulation of a Frequency ramp 3 The Fourier transform of a damped cosine and the units of the result 2 Unit of Energy Spectral Density | {"extraction_info": {"found_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": 0.9213269352912903, "perplexity": 1967.112626420105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046154459.22/warc/CC-MAIN-20210803124251-20210803154251-00040.warc.gz"} |
https://www.lmfdb.org/L/2/712 | ## Results (1-50 of at least 1000)
Next
Label $\alpha$ $A$ $d$ $N$ $\chi$ $\mu$ $\nu$ $w$ prim arith $\mathbb{Q}$ self-dual $\operatorname{Arg}(\epsilon)$ $r$ First zero Origin
2-712-712.691-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.691 $$0.0 0 -0.0522 0 2.04731 Modular form 712.1.y.a.691.1 2-712-712.275-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.275$$ $0.0$ $0$ $-0.241$ $0$ $1.25713$ Modular form 712.1.s.a.275.1
2-712-712.355-c0-0-2 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.355 $$0.0 0 0 0 2.04047 Artin representation 2.712.4t3.b Artin representation 2.712.4t3.b.a Modular form 712.1.c.a Modular form 712.1.c.a.355.1 2-712-712.11-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.11$$ $0.0$ $0$ $0.160$ $0$ $2.03093$ Modular form 712.1.w.a.11.1
2-712-712.139-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.139 $$0.0 0 0.458 0 0.605941 Modular form 712.1.w.a.139.1 2-712-712.427-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.427$$ $0.0$ $0$ $0.0143$ $0$ $2.46265$ Modular form 712.1.y.a.427.1
2-712-712.67-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.67 $$0.0 0 -0.434 0 1.15159 Modular form 712.1.s.a.67.1 2-712-712.403-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.403$$ $0.0$ $0$ $0.249$ $0$ $2.27520$ Modular form 712.1.y.a.403.1
2-712-712.435-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.435 $$0.0 0 -0.227 0 1.28365 Modular form 712.1.y.a.435.1 2-712-712.283-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.283$$ $0.0$ $0$ $-0.0276$ $0$ $1.06949$ Modular form 712.1.s.a.283.1
2-712-712.443-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.443 $$0.0 0 0.235 0 1.80061 Modular form 712.1.w.a.443.1 2-712-712.355-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.355$$ $0.0$ $0$ $0$ $0$ $1.25810$ Artin representation 2.712.8t6.b.b Modular form 712.1.c.b.355.1
2-712-712.307-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.307 $$0.0 0 -0.448 0 1.39761 Modular form 712.1.y.a.307.1 2-712-712.339-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.339$$ $0.0$ $0$ $0.0522$ $0$ $2.19075$ Modular form 712.1.y.a.339.1
2-712-712.539-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.539 $$0.0 0 0.252 0 2.44044 Modular form 712.1.y.a.539.1 2-712-712.667-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.667$$ $0.0$ $0$ $-0.235$ $0$ $0.0877208$ Modular form 712.1.w.a.667.1
2-712-712.299-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.299 $$0.0 0 -0.143 0 1.67365 Modular form 712.1.s.a.299.1 2-712-712.99-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.99$$ $0.0$ $0$ $-0.338$ $0$ $0.601529$ Modular form 712.1.y.a.99.1
2-712-712.643-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.643 $$0.0 0 0.406 0 2.26117 Modular form 712.1.y.a.643.1 2-712-712.259-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.259$$ $0.0$ $0$ $-0.160$ $0$ $1.70795$ Modular form 712.1.w.a.259.1
2-712-712.91-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.91 $$0.0 0 0.0178 0 1.35933 Modular form 712.1.s.a.91.1 2-712-712.235-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.235$$ $0.0$ $0$ $0.216$ $0$ $2.04279$ Modular form 712.1.w.a.235.1
2-712-712.251-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.251 $$0.0 0 -0.458 0 2.75052 Modular form 712.1.w.a.251.1 2-712-712.131-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.131$$ $0.0$ $0$ $-0.0714$ $0$ $1.02009$ Modular form 712.1.y.a.131.1
2-712-712.331-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.331 $$0.0 0 0.143 0 1.80711 Modular form 712.1.s.a.331.1 2-712-712.603-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.603$$ $0.0$ $0$ $0.448$ $0$ $2.99359$ Modular form 712.1.y.a.603.1
2-712-712.627-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.627 $$0.0 0 0.434 0 3.00455 Modular form 712.1.s.a.627.1 2-712-712.187-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.187$$ $0.0$ $0$ $0.338$ $0$ $1.28094$ Modular form 712.1.y.a.187.1
2-712-712.659-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.659 $$0.0 0 -0.249 0 1.06365 Modular form 712.1.y.a.659.1 2-712-712.395-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.395$$ $0.0$ $0$ $0.0276$ $0$ $1.53299$ Modular form 712.1.s.a.395.1
2-712-712.619-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.619 $$0.0 0 0.173 0 2.40731 Modular form 712.1.w.a.619.1 2-712-712.555-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.555$$ $0.0$ $0$ $0.0442$ $0$ $1.53958$ Modular form 712.1.y.a.555.1
2-712-712.227-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.227 $$0.0 0 -0.406 0 0.802986 Modular form 712.1.y.a.227.1 2-712-712.467-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.467$$ $0.0$ $0$ $-0.173$ $0$ $1.80127$ Modular form 712.1.w.a.467.1
2-712-712.579-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.579 $$0.0 0 -0.0178 0 1.17257 Modular form 712.1.s.a.579.1 2-712-712.203-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.203$$ $0.0$ $0$ $-0.216$ $0$ $0.701343$ Modular form 712.1.w.a.203.1
2-712-712.523-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.523 $$0.0 0 0.241 0 1.54382 Modular form 712.1.s.a.523.1 2-712-712.107-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.107$$ $0.0$ $0$ $-0.252$ $0$ $1.61745$ Modular form 712.1.y.a.107.1
2-712-712.195-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.195 $$0.0 0 -0.0442 0 1.45395 Modular form 712.1.y.a.195.1 2-712-712.411-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.411$$ $0.0$ $0$ $0.169$ $0$ $1.91280$ Modular form 712.1.l.a.411.1
2-712-712.707-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.707 $$0.0 0 -0.0143 0 1.87780 Modular form 712.1.y.a.707.1 2-712-712.347-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.347$$ $0.0$ $0$ $0.227$ $0$ $1.56677$ Modular form 712.1.y.a.347.1
2-712-712.587-c0-0-0 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.587 $$0.0 0 0.0714 0 1.15124 Modular form 712.1.y.a.587.1 2-712-712.123-c0-0-0 0.596 0.355 2 2^{3} \cdot 89 712.123$$ $0.0$ $0$ $-0.169$ $0$ $1.34333$ Modular form 712.1.l.a.123.1
2-712-712.355-c0-0-1 $0.596$ $0.355$ $2$ $2^{3} \cdot 89$ 712.355 $$0.0 0 0 0 1.47660 Artin representation 2.712.8t6.b.a Modular form 712.1.c.b.355.2 2-712-712.171-c1-0-69 2.38 5.68 2 2^{3} \cdot 89 712.171$$ $1.0$ $1$ $0.248$ $0$ $1.93934$ Modular form 712.2.bf.c.171.41
2-712-712.171-c1-0-61 $2.38$ $5.68$ $2$ $2^{3} \cdot 89$ 712.171 $$1.0 1 -0.280 0 1.74369 Modular form 712.2.bf.c.171.75 2-712-712.171-c1-0-7 2.38 5.68 2 2^{3} \cdot 89 712.171$$ $1.0$ $1$ $0.0546$ $0$ $0.472576$ Modular form 712.2.bf.c.171.6
2-712-712.171-c1-0-54 $2.38$ $5.68$ $2$ $2^{3} \cdot 89$ 712.171 $$1.0 1 0.239 0 1.58799 Modular form 712.2.bf.c.171.24 2-712-712.171-c1-0-47 2.38 5.68 2 2^{3} \cdot 89 712.171$$ $1.0$ $1$ $0.359$ $0$ $1.48495$ Modular form 712.2.bf.c.171.39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9792560338973999, "perplexity": 474.24814320276107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046153899.14/warc/CC-MAIN-20210729234313-20210730024313-00027.warc.gz"} |
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Aug14 revised Which packages come standard with Ubuntu's `texlive` package? added 614 characters in body Aug14 revised Which packages come standard with Ubuntu's `texlive` package? edited title Aug14 accepted Which packages come standard with Ubuntu's `texlive` package? Aug14 comment Which packages come standard with Ubuntu's `texlive` package? @AndreCashner Is it appropriate then that I changed the tag to `ubuntu`? Aug14 comment Which packages come standard with Ubuntu's `texlive` package? This list helps, thank you. I think I have what I want, using this list in combination with this reference, which tells me that the Debian/Ubuntu `texlive` package gives `texlive-latex-recommended`, `texlive-fonts-recommended`, `texlive-latex-base` (which is `texlive-latex`?) and `texlive-base` (which is `texlive-basic`?). Aug14 revised Which packages come standard with Ubuntu's `texlive` package? added 706 characters in body Aug14 asked Which packages come standard with Ubuntu's `texlive` package? Jul21 accepted Shortcut to declare alignment of many columns in a table Jul21 asked Shortcut to declare alignment of many columns in a table Jul2 awarded Curious Mar20 awarded Necromancer Mar18 comment Using \multispan in plain tex to give a caption to an array I have the green light to use LaTeX freely in this project now---thanks for the help. Mar18 accepted Using \multispan in plain tex to give a caption to an array Mar18 comment Using \multispan in plain tex to give a caption to an array But what you see in the first table is the current raw .tex output from WeBWorK. I'm just trying to tweak it and insert captions with as little alteration as possible. If my proposed changes to WeBWorK code were accepted, they would be adopted by hundreds of institutions and hundreds of thousands of students. So a good philosophy is to change as little as possible. Mar18 comment Using \multispan in plain tex to give a caption to an array Thanks @DavidCarlisle. I'm not 100% sure yet what the WeBWorK experts will say about using elements of plain tex versus LaTeX (I've asked but only earlier today). I had whittled down a WeBWorK-produced .tex file to a MWE for posting here, so there was probably elements of amsart in use within what I cut. I'm still waiting to hear from the WeBWorK experts what they think about using more LaTeX over plain tex, but since what you see in the first table is the current WeBWorK output, I am hesitant to propose monumental changes to its code. Mar18 comment Using \multispan in plain tex to give a caption to an array Thanks! I did process it with LaTeX. I'm trying to understand how WeBWorK processes raw input to produce the pdfs that students use. I intervened to grab the raw .tex file to play with, so that I would understand better what WeBWorK does to create it before I meddled with WeBWorK's algorithm for creating the .tex. I should have altered my local IDE to process it with plain tex for a more honest simulation. I'm awaiting to hear from the WeBWorK experts if plain tex is necessary---if not, great. Either way this may make its way into WeBWorK code. Mar18 comment Using \multispan in plain tex to give a caption to an array @egreg I'm not 100% sure. I think that I have to consider that a hundred students might each be asking the server to generate a pdf with a hundred problems, and any efficiency gained from using plain tex is desired. I haven't checked in yet with the WeBWorK development crew though. Maybe they can green-light using less basic tex. Mar18 asked Using \multispan in plain tex to give a caption to an array Jan31 awarded Self-Learner Jan30 revised Rotating a one-page float for pdf viewing edited body | {"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": 0.9807514548301697, "perplexity": 4237.442141999856}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500823333.10/warc/CC-MAIN-20140820021343-00437-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/625277/axiom-of-choice-and-cartesian-products | # Axiom of Choice and Cartesian Products
According to Wikipedia one formulation of AC is
The Cartesian product of any family of nonempty sets is nonempty.
If I consider an cartesian product $\prod_{i} X_i$ of nonempty sets $X_i$, then there exists some $x_i \in X_i$ for each $i$ (simply by non-emptiness), and so $x := (x_i)$ is an element of the product $\prod_i X_i$ by definition. This seems quite trivial to me... imposed by the rules of logic, so why state it as an axiom? Indeed to me it appears as there isn't needed any axiom at all, by setting $x := (x_i)$ I have actually constructed the element?
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You've canonically expressed the axiom of choice in your first sentence - for each $X_i$ there may exist an element $x_i$ in $X_i$, but you can't make the choice of all the (infinitely many!) $x_i$ in any 'canonical' fashion. Note that for any sets where you can make this choice (for instance, if there's an ordering on $X_i$ so you can say 'take the least $x_i$'), AC holds as a theorem and not just an axiom; the most common metaphor for explaining this seems to be a 'shoes vs. socks' formulation of the axiom, and you can probably find more information by searching for those words. – Steven Stadnicki Jan 2 '14 at 21:39
Your proof is not a proof, but rather an intuition why the AC should be true.
Recall the precise(!) definition of the product of a family of sets: $\prod_{i \in I} X_i$ consists of functions $f : I \to \bigcup_{i \in I} X_i$ such that $f(i) \in X_i$ for all $i \in I$. Also recall the definition of a function $A \to B$ as a special subset of $A \times B$.
Now, given non-empty sets $X_i$, how do you define such a function, using the other ZF axioms? You say, for every $i \in I$ we choose some element $x_i \in X_i$. This works for every single $i$ at a time, but this doesn't define a function $i \mapsto x_i$.
Example: Let $I$ be the set of all non-empty subsets of $\mathbb{R}$, and $X_i = i$. Then an element $f$ in $\prod_{i \in I} X_i$ is a function which picks an element $f(T) \in T$ for every non-empty $T \subseteq \mathbb{R}$. How do you define such an $f$? If we would have $\mathbb{N}$ instead of $\mathbb{R}$, we could take $f(T)=\min(T)$, but this doesn't work for $\mathbb{R}$. Apparently, there is no canonical choice of an element in a non-empty set of real numbers. But the AC tells us that we don't have to worry about this, it gives us such a function, even if we cannot "write it down" (which means: construct it from the other ZF axioms).
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thx for the many answers, but sorry I still did not see why $f(i) := x_i$ for some $x_i \in X_i$ is not a function, what exactly goes wrong here? This so defined $f$ is a subset of $I \times \bigcup X_i$ and $f(i) \in X_i$ by definition... – Stefan Jan 2 '14 at 22:16
@Stefan: As I wrote in my answer, we cannot prove that sets exists "just like that". Consider Skolem's paradox, if $\sf ZFC$ is consistent, then it has a countable model $M$, therefore $\{x\mid M\models x\text{ is a real number}\}$ is countable, but we know that $M$ also thinks that this is an uncountable set. Paradoxical, ain't it? Well. Shucks, it just means that not every set that we want to exist will exist. To do so, we have to use axioms and inference rules to prove that it does. To prove that $f$ as you "defined" it exists, you have use the axiom of choice. – Asaf Karagila Jan 2 '14 at 22:35
does that mean that there aren't any real numbers in the countable model $M$, i.e. $\{ x : M \models x \mbox{ is a real number } \} = \emptyset$ or that the notion of countable in $M$ is not the one we "expect intuitively"? – Stefan Jan 2 '14 at 23:27
is my definition like defining $N$ to be the biggest natural number, which obvisouly is not well-defined because such number does not exists? – Stefan Jan 2 '14 at 23:35
@Stefan: It is more similar to the least uninteresting natural number. As for the first comment, it means that the notion of being uncountable or countable is not absolute internally to a model of set theory. – Asaf Karagila Jan 2 '14 at 23:38
Sure, if you have five or six sets, then writing such a choice function, or an element of the product, is easily done. But the rules of logic only apply finitely many times.
What happens when you have an infinite number of sets? First of all, it's hard to talk about infinite objects explicitly with logic which is very finitary. So we need another framework, something like set theory works out just fine, because set theory is really just the study of infinite set so to speak.
And indeed the axioms of set theory (without the axiom of choice, that is) allow us to prove by induction that every finite product of non-empty sets is non-empty (and I am not even getting into the bog of standard and non-standard integers).
The role of axioms in set theory is to allow us to construct new sets. They assure us that from such and such conditions, we can ensure the existence of a particular set.
Nothing ensures that given an infinite number of non-empty sets, their product is non-empty. In fact, now we can even show that if set theory is at all consistent, then it is consistent that some products like that are empty.
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The axiom of choice is, in fact, independent of the Zermelo-Fraenkel axioms.
The difficulty is to find a rule to select a real number out of ANY set.
But the axiom of choice is true for CONSTRUCTIBLE sets.
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There are models of the set theory, for which the axiom of choice is false! And some mathematicians prefere those models. – Peter Jan 2 '14 at 21:37
But as said, for "concrete" sets, it is true. – Peter Jan 2 '14 at 21:39 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9603361487388611, "perplexity": 218.93280762974953}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1432207924919.42/warc/CC-MAIN-20150521113204-00257-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://schoolwebpages.com/5muptvg/end-behaviour-of-oblique-asymptote-d68dcc | An oblique asymptote may be crossed or touched by the graph of the function. 4.6.5 Analyze a function and its derivatives to draw its graph. There are three types: horizontal, vertical and oblique: The direction can also be negative: The curve can approach from any side (such as from above or below for a horizontal asymptote), Understanding the invariant points, and the relationship between x-intercepts and vertical asymptotes for reciprocal functions; Understanding the effects of points of discontinuity Undertstanding the end behaviour of horizontal and oblique asymptotes for rational functions Concept 1 - Sketching Reciprocals If the degree of the numerator is exactly one more than the degree of the denominator, the end behavior of this rational function is like an oblique linear function. There is a vertical asymptote at . One number is 8 times another number. Ex 8. https://www.khanacademy.org/.../v/end-behavior-of-rational-functions In the first case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to +∞, and in the second case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to −∞. ... Oblique/Slant Asymptote – degree of numerator = degree of denominator +1 - use long division to find equation of oblique asymptote An asymptote is a line that a curve approaches, as it heads towards infinity:. Rational functions may or may not intersect the lines or polynomials which determine their end behavior. Asymptote. The end behaviour of function F is described by in oblique asymptote. Find the vertical and end-behavior asymptote for the following rational function. Some functions, however, may approach a function that is not a line. The horizontal asymptote tells, roughly, where the graph will go when x is really, really big. Asymptotes, End Behavior, and Infinite Limits. limits rational functions limit at infinity limit at negative infinity horizontal asymptotes oblique asymptote end behavior Calculus Limits and Continuity Which of the following equations co … Question: Find the vertical and end-behavior asymptote for the following rational function. 2. →−∞, →0 ... has an oblique asymptote. Honors Calculus. More general functions may be harder to crack. 4.6.3 Estimate the end behavior of a function as x x increases or decreases without bound. The slanted asymptote gives us an idea of how the curve of f … Honors Calculus. However, as x approaches infinity, the limit does not exist, since the function is periodic and could be anywhere between #[-1, 1]#. An oblique asymptote may be found through long division. Notice that the oblique asymptotes of a rational function also describe the end behavior of the function. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. Find Oblique Asymptote And Examine End Behaviour Of Rational Function. Asymptotes, it appears, believe in the famous line: to infinity and beyond, as they are curves that do not have an end. Keeper 12. We can also see that y = 1 2 x + 1 is a linear function of the form, y = m x + b. The quotient polynomial Q(x) is linear, Q(x)=ax+b, then y=ax+b is called an slant or oblique asymptote for f(x). New questions in Mathematics. If either of these limits is $$∞$$ or $$−∞$$, determine whether $$f$$ has an oblique asymptote. Free functions asymptotes calculator - find functions vertical and horizonatal asymptotes step-by-step This website uses cookies to ensure you get the best experience. Example 3 The numbers are both positive and have a difference of 70. 11. If the function is simple, functions such as #sinx# and #cosx# are defined for #(-oo,+oo)# so it's really not that hard.. Math Lab: End Behavior and Asymptotes in Rational Functions Cut out the tiles and sort them into the categories below based on their end behavior. I can determine the end behavior of a rational function and determine its related asymptotes, if any. 4.6.4 Recognize an oblique asymptote on the graph of a function. Find the numbers. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function $g\left(x\right)=\frac{4}{x}$, and the outputs will approach zero, resulting in a horizontal asymptote at y = 0. Oblique Asymptotes: An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. Types. You can find the equation of the oblique asymptote by dividing the numerator of the function rule by the denominator and using … While understanding asymptotes, you would have chanced upon a graph that reads $$f(x)=\frac{1}{x}$$ You might have observed a strange behavior at x=0. 1. The end behavior asymptote (the equation that approximates the behavior of the original function at the ends of the graph) will simply be y = quotient In this case, the asymptote will be y = x (a slant or oblique line). The equation of the oblique asymptote If either of these limits is a finite number $$L$$, then $$y=L$$ is a horizontal asymptote. The horizontal asymptote is , even though the function clearly passes through this line an infinite number of times. An example is ƒ( x ) = x + 1/ x , which has the oblique asymptote y = x (that is m = 1, n = 0) as seen in the limits Example 2. ! Briefly, an asymptote is a straight line that a graph comes closer and closer to but never touches. In more complex functions, such as #sinx/x# at #x=0# there is a certain theorem that helps, called the squeeze theorem. Identify the asymptotes and end behavior of the following function: Solution: The function has a horizontal asymptote as approaches negative infinity. The equations of the oblique asymptotes and the end behavior polynomials are found by dividing the polynomial P (x) by Q (x). Evaluate $$\lim_{x→∞}f(x)$$ and $$\lim_{x→−∞}f(x)$$ to determine the end behavior. As can be seen from the graph, f ( x) ’s oblique asymptote is represented by a dashed line guiding the behavior of the graph. End Behavior of Polynomial Functions. The remainder is ignored, and the quotient is the equation for the end behavior model. By using this website, you agree to our Cookie Policy. The graph of a function may have at most two oblique asymptotes (one as x →−∞ and one as x→∞). The rule for oblique asymptotes is that if the highest variable power in a rational function occurs in the numerator — and if that power is exactly one more than the highest power in the denominator — then the function has an oblique asymptote. That is, as you “zoom out” from the graph of a rational function it looks like a line or the function defined by Q (x) in f (x) D (x) = Q (x) + R (x) D (x). Check with a classmate before gluing them. Given this relationship between h(x) and the line , we can use the line to describe the end behavior of h(x).That is, as x approaches infinity, the values of h(x) approach .As you will learn in chapter 2, this kind of line is called an oblique asymptote, or slant asymptote.. End Behavior of Polynomial Functions. ... Oblique/Slant Asymptote – degree of numerator = degree of denominator +1 - use long division to find equation of oblique asymptote ***Watch out for holes!! 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Is the equation for the following rational function asymptote and Examine end behaviour of rational also. Both positive and have a difference of 70 is not a line that a graph comes closer end behaviour of oblique asymptote closer but... 4.6.4 Recognize an oblique asymptote and Examine end behaviour of function F is described by in oblique may!
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http://ldtopology.wordpress.com/2013/01/11/normal-loops-in-surfaces/ | # Low Dimensional Topology
## January 11, 2013
### Normal loops in surfaces
Filed under: Surfaces,Triangulations — Jesse Johnson @ 3:50 pm
I plan to write a post or two about normal surfaces and branched surfaces in three-dimensional manifolds, but I want to warm up first, with two posts about the two-dimensional analogues of these objects. Train tracks play a huge role in the approach to the topology of surfaces initiated by Nielsen and Thurston, for understanding mapping class groups, Teichmuller space, laminations, etc. They organize the set of isotopy classes of simple closed curves in a surface in a way that allows one to take limits of infinite sequences of loops. (The limits are called projective measured laminations.) In this post and the next, I will discuss train tracks from a rather unusual perspective, via normal loops in a triangulation of the given surface.
Let $S$ be a surface and let $G$ be the one-skeleton of a triangulation of $S$. In other words, $G$ is a graph embedded in $S$ so that its complement is a collection of triangles. If $\ell$ is a simple closed curve in $S$ then we can isotope $\ell$ to be transverse to $G$ (so that $G \cap \ell$ is a finite number of points in the edges of $G$). In fact, we can do even better: If $\ell$ ever intersects the same edge of $G$ twice in a row, then the arc of $\ell$ between these two points will be contained in a triangle, and moreover will cut off a disk in that triangle. (Such an arc is called a bent arc because it cannot be drawn by a straight line in the interior of the triangle.) If we isotope this arc of $\ell$ across this disk and into the adjacent triangle, we will strictly reduce the number of points of intersection $G \cap \ell$. Note that we can isotope an arc with endpoints in different edges out of the triangle, but this will usually increase the number of points of intersection. If we each bent arc in this way, we will eventually isotope $\ell$ so that each arc of $\ell \setminus G$ has its endpoints in distinct edges of $G$. We will say that such an arc is normal.
If, after this isotopy, $\ell$ is disjoint from $G$ then $\ell$ is contained in a triangle and is trivial (bounds a disk by the Jordan Curve Theorem). Otherwise, we will say that $\ell$ is a normal loop. Note that in each triangle, there are exactly three types of normal arcs, defined by the pair of edges that contain their endpoints. Each edge of the triangulation contains the endpoints of four different classes of normal arcs, two in each of the adjacent disks and the total numbers of normal arcs on each side must be equal, since $\ell$ is a loop. So every normal loop determines a vector whose entries are the number of normal arcs in each class. (So, these are $3F$ dimensional vectors where $F$ is the number of faces in the triangulation.) Because the arcs must match up along the edges, they all satisfy a collection of linear equations of the form $x_1 + x_2 = x_3 + x_4$ where $x_1, x_2$ are the numbers of normal arcs on one side of an edge and $x_3, x_4$ are the numbers of arcs on the other side.
Conversely, if we choose an integer vector with positive entries that satisfies this set of linear equations, we can draw the corresponding normal arcs in the surface $S$ and there will be a unique way to match up the endpoints to form a closed one-manifold in $S$. So this means that there is a one-to-one correspondence between normal loops and vectors in $\mathbf{Z}^{3E}$ with all positive entries that lie in the subspace defined by the linear equations. (This is called the positive cone of the subspace.)
This vector structure on the set of normal loops turns out to be very nice. For example, we might translate a given normal loop into a vector, then multiply the vector by some integer $k$, then translate the new vector back into a normal loop. In this case, the resulting one-dimensional manifold will (usually) correspond to $k$ copies of the original loop. (For one-sided loops in non-orientable surfaces, it’s a little more complicated.) But what happens if we add two of these vectors? If $v$ and $w$ are vectors corresponding to disjoint normal loops then the vector $v + w$ will correspond to the union of the two normal loops. However, if the original loops intersect non-trivially, their union is not an embedded one-manifold.
In particular, if we look at the normal arcs of these loops inside a given triangle in $S$, the normal arcs will intersect. However, if we shuffle the endpoints of these arcs, we can make them pairwise disjoint. In the process of shuffling the endpoints, we will break the original loops then reattach them to form some new one-manifold. (It may or may not be a single loop.) There will always be exactly one way to do this and the result is called the Haken sum of the original loops.
Note that while every essential loop can be isotoped to a normal loop, not every normal loop is necessarily essential. In particular, a small trivial loop around any vertex of the triangulation will be normal. If the triangulation has two or more vertices, one can construct very complicated normal loops that are topologically trivial, as the boundary of a regular neighborhood of a complicated arc between the two vertices. Moreover, (as if this weren’t bad enough) an isotopy class of essential loops will generally have a lot of different normal representatives, for similar reasons. In order to correct these problems, we need to introduce train tracks, and this will be the subject of my next post.
## 2 Comments »
1. Really nice, but maybe you can explain where did you get the 3E from?
Or maybe it’s 3F dimensional?
Thanks!
Comment by Tali — January 14, 2013 @ 12:38 pm
• Yes, that should be 3F. Thanks for the correction!
Comment by Jesse Johnson — January 16, 2013 @ 12:02 am
The Rubric Theme. Blog at WordPress.com. | {"extraction_info": {"found_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": 38, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.857772707939148, "perplexity": 209.6374059577692}, "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-2014-42/segments/1414637898751.26/warc/CC-MAIN-20141030025818-00094-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://arxiv.org/abs/0909.3822 | math.PR
(what is this?)
# Title: A derivation of Benford's Law ... and a vindication of Newcomb
Abstract: We show how Benford's Law (BL) for first, second, ..., digits, emerges from the distribution of digits of numbers of the type $a^{R}$, with $a$ any real positive number and $R$ a set of real numbers uniformly distributed in an interval $[ P\log_a 10, (P +1) \log_a 10)$ for any integer $P$. The result is shown to be number base and scale invariant. A rule based on the mantissas of the logarithms allows for a determination of whether a set of numbers obeys BL or not. We show that BL applies to numbers obtained from the {\it multiplication} or {\it division} of numbers drawn from any distribution. We also argue that (most of) the real-life sets that obey BL are because they are obtained from such basic arithmetic operations. We exhibit that all these arguments were discussed in the original paper by Simon Newcomb in 1881, where he presented Benford's Law.
Comments: 12 pages, 6 figures Subjects: Probability (math.PR); History and Overview (math.HO) Cite as: arXiv:0909.3822 [math.PR] (or arXiv:0909.3822v1 [math.PR] for this version)
## Submission history
From: Victor Romero-Rochin [view email]
[v1] Mon, 21 Sep 2009 17:28:23 GMT (45kb,D) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8663221597671509, "perplexity": 1054.518186307332}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1495463608870.18/warc/CC-MAIN-20170527055922-20170527075922-00335.warc.gz"} |
https://airtoncs.wordpress.com/2015/08/30/differences-between-input-and-include-tags-of-latex/ | # Differences Between \input and \include Tags of Latex
`\input{filename}` append all commands defined in `filename` into the target file. This is equivalent to manually typing all the commands from `filename` right into the current file where the`\input` line is. (It may remember inline functions or macros of C).
`\include{filename}` is equivalent to call `\clearpage` before and after `\input{filename}. So, filename must have all dependencies (commands) that it needs most of the time. ` | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9984161257743835, "perplexity": 2506.5028145287133}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1529267867050.73/warc/CC-MAIN-20180624180240-20180624200240-00585.warc.gz"} |
https://www.examrace.com/Sample-Objective-Questions/Physics-Questions/Physics-MCQs-Practice-Test-12.html | # Competitive Exams: Physics MCQs (Practice_Test 12 of 35)
Doorsteptutor material for competitive exams is prepared by world's top subject experts: get questions, notes, tests, video lectures and more- for all subjects of your exam.
1. An elastic collision conserves
1. kinetic energy but not momentum
2. momentum but not kinetic energy
3. neither momentum nor kinetic energy
4. both kinetic energy and momentum
2. Two planets A and B have the same material density. If the radius of A is twice that of B, then the ratio of the escape velocity VA/VB is
1. 2
2. -2
3. 1/-2
3. A satellite is moving in a circular orbit at a height of 100 km above the earth՚s surface. A person inside1he satellite feels weightless because
1. acceleration due to gravity is almost zero at such a height
2. the earth does not exert any force on the person
3. the centripetal force makes the satellite move in circular orbit
4. the forces due to the earth am the moon are almost compensated at such a height
4. A rod length ‘l’ is inclined at an angle a with x-axis, An observer moving at a velocity V along the x-axis, will measure the angle of inclination as a (with g = 1/ ‘1’ V2/C 2) given by
1. tan-1 tan a
2. tan 1 g tan a
3. tan 1 (tan a/g)
4. tan-1 (g/tan a)
5. Which one of the folk > wing is an example of steady and non-uniform flow?
1. The flow of a liquid through a straight horizontal pipe at a constant rate
2. The flow of a liquid through a straight horizontal pipe at a changing rate
3. The flow of a liquid through a conical pipe at a constant rate
4. The flow of a liquid through a conical pipe at a changing rate
6. Water rises to a height h, in a capillary tube, when dipped in water. If the height of this capillary tube, above the water surface is less than h, then
1. the water level will go down
2. the water level will come to the top but the radius of curvature of the meniscus will increase
3. the water level will come to the top but the radius of curvature of the meniscus will decrease
4. the water will flow out of the capillary
7. A soap bubble of radius r1 is blown at the end of a capillary tube of length L and radius R. If h is the viscosity of air and T is the surface tension of soap bubble. Then the time taken for the radius of the bubble to reduce to r2 due to flow of air through the capillary. Is equal to
8. An oil drop of diameter 4 × 10 − 4 m falls through air. If the densities of oil and air are 900 kgm-3 and 1.293 kg m − 3 respectively and the coefficient of viscosity of air is 2.0 × 10 − 5 Nm-2s, then the terminal velocity of oil drop win be
1. 0.2 × 10 − 4 m/s
2. 2 × 10 − 4 m/s
3. 4 × 10 − 4 m/s
4. 8 × 10 − 4 m/s
9. When a harmonic wave is propagating through a medium, the displacement ‘y’ of a particle of the medium is represented by y = 10 sin 2p/5 (1800t-x) . The time period will be
1. sec
2. sec
3. 36 sec
4. 360 sec
10. Consider the mechanical vibrating systems shown in Figures A. B. C and D The vibrations are simple harmonic in:
1. A and C
2. A, B and C
3. B and D
4. A, B, C and D
11. The displacement of a particle executing simple harmonic motion is given by y = 4 sin (2t + f) The period of oscillations is
1. 2/p
2. p/2
3. p
4. 2p
12. If three tuning forks of frequencies 512,513 and 514 are sounded together simultaneously, then the number of beats per second is
1. 0
2. 1
3. 2
4. 3
13. A particle of mass m describes an elliptical orbit: This motion can be shown to be the sum of two simple harmonic motions at rights angles to each other having
1. the same frequency but different amplitudes
2. the same frequency and the same amplitude
3. different frequencies but the same amplitude
4. different frequencies and different amplitudes
14. Two sound waves of same amplitude can interfere destructively if
1. their frequencies are equal and their phase difference is zero
2. their frequencies are equal and their phase difference is 1800
3. their phase difference is 900 irrespective of frequencies
4. their phase difference is 1800 and their frequencies are unequal.
15. Tuning fork A when sounded with tuning fork B of frequency 480 Hz gives 5 beats per second. When the prongs of A are loaded with wax, it gives 3 beats per second. The original frequency of A is
1. 475 Hz
2. 485 Hz
3. 483 Hz
4. 477 Hz | {"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": 0.8859317898750305, "perplexity": 884.6013721165945}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030337446.8/warc/CC-MAIN-20221003231906-20221004021906-00220.warc.gz"} |
https://worldwidescience.org/topicpages/l/lepton+flavor+symmetry.html | #### Sample records for lepton flavor symmetry
1. Lepton flavor violation and seesaw symmetries
Energy Technology Data Exchange (ETDEWEB)
Aristizabal Sierra, D., E-mail: [email protected] [Universite de Liege, IFPA, Department AGO (Belgium)
2013-03-15
When the standard model is extended with right-handed neutrinos the symmetries of the resulting Lagrangian are enlarged with a new global U(1){sub R} Abelian factor. In the context of minimal seesaw models we analyze the implications of a slightly broken U(1){sub R} symmetry on charged lepton flavor violating decays. We find, depending on the R-charge assignments, models where charged lepton flavor violating rates can be within measurable ranges. In particular, we show that in the resulting models due to the structure of the light neutrino mass matrix muon flavor violating decays are entirely determined by neutrino data (up to a normalization factor) and can be sizable in a wide right-handed neutrino mass range.
2. Neutrino magnetic moment in a theory with lepton flavor symmetry
International Nuclear Information System (INIS)
Stephanov, M.A.
1987-01-01
A model for generating the neutrino magnetic moment of the order of 10 -10 μ B is proposed, which is based on the SU(3) lepton flavor symmetry. In such a way one can avoid the flavor changing processes. The experimental constraints on the constants of the model are considered
3. A flavor symmetry model for bilarge leptonic mixing and the lepton masses
Science.gov (United States)
Ohlsson, Tommy; Seidl, Gerhart
2002-11-01
We present a model for leptonic mixing and the lepton masses based on flavor symmetries and higher-dimensional mass operators. The model predicts bilarge leptonic mixing (i.e., the mixing angles θ12 and θ23 are large and the mixing angle θ13 is small) and an inverted hierarchical neutrino mass spectrum. Furthermore, it approximately yields the experimental hierarchical mass spectrum of the charged leptons. The obtained values for the leptonic mixing parameters and the neutrino mass squared differences are all in agreement with atmospheric neutrino data, the Mikheyev-Smirnov-Wolfenstein large mixing angle solution of the solar neutrino problem, and consistent with the upper bound on the reactor mixing angle. Thus, we have a large, but not close to maximal, solar mixing angle θ12, a nearly maximal atmospheric mixing angle θ23, and a small reactor mixing angle θ13. In addition, the model predicts θ 12≃ {π}/{4}-θ 13.
4. Theories of Leptonic Flavor
DEFF Research Database (Denmark)
Hagedorn, Claudia
2017-01-01
I discuss different theories of leptonic flavor and their capability of describing the features of the lepton sector, namely charged lepton masses, neutrino masses, lepton mixing angles and leptonic (low and high energy) CP phases. In particular, I show examples of theories with an abelian flavor...... symmetry G_f, with a non-abelian G_f as well as theories with non-abelian G_f and CP....
5. Model with a gauged lepton flavor SU(2) symmetry
Science.gov (United States)
Chiang, Cheng-Wei; Tsumura, Koji
2018-05-01
We propose a model having a gauged SU(2) symmetry associated with the second and third generations of leptons, dubbed SU(2) μτ , of which U{(1)}_{L_{μ }-L_{τ }} is an Abelian subgroup. In addition to the Standard Model fields, we introduce two types of scalar fields. One exotic scalar field is an SU(2) μτ doublet and SM singlet that develops a nonzero vacuum expectation value at presumably multi-TeV scale to completely break the SU(2) μτ symmetry, rendering three massive gauge bosons. At the same time, the other exotic scalar field, carrying electroweak as well as SU(2) μτ charges, is induced to have a nonzero vacuum expectation value as well and breaks mass degeneracy between the muon and tau. We examine how the new particles in the model contribute to the muon anomalous magnetic moment in the parameter space compliant with the Michel decays of tau.
6. Radiative origin of all quark and lepton masses through dark matter with flavor symmetry.
Science.gov (United States)
Ma, Ernest
2014-03-07
The fundamental issue of the origin of mass for all quarks and leptons (including Majorana neutrinos) is linked to dark matter, odd under an exactly conserved Z2 symmetry which may or may not be derivable from an U(1)D gauge symmetry. The observable sector interacts with a proposed dark sector which consists of heavy neutral singlet Dirac fermions and suitably chosen new scalars. Flavor symmetry is implemented in a renormalizable context with just the one Higgs doublet (ϕ(+), ϕ(0)) of the standard model in such a way that all observed fermions obtain their masses radiatively through dark matter.
7. Flavor changing lepton processes
International Nuclear Information System (INIS)
Kuno, Yoshitaka
2002-01-01
The flavor changing lepton processes, or in another words the lepton flavor changing processes, are described with emphasis on the updated theoretical motivations and the on-going experimental progress on a new high-intense muon source. (author)
8. Flavor physics without flavor symmetries
Science.gov (United States)
Buchmuller, Wilfried; Patel, Ketan M.
2018-04-01
We quantitatively analyze a quark-lepton flavor model derived from a six-dimensional supersymmetric theory with S O (10 )×U (1 ) gauge symmetry, compactified on an orbifold with magnetic flux. Two bulk 16 -plets charged under the U (1 ) provide the three quark-lepton generations whereas two uncharged 10 -plets yield two Higgs doublets. At the orbifold fixed points mass matrices are generated with rank one or two. Moreover, the zero modes mix with heavy vectorlike split multiplets. The model possesses no flavor symmetries. Nevertheless, there exist a number of relations between Yukawa couplings, remnants of the underlying grand unified theory symmetry and the wave function profiles of the zero modes, which lead to a prediction of the light neutrino mass scale, mν 1˜10-3 eV and heavy Majorana neutrino masses in the range from 1 012 to 1 014 GeV . The model successfully includes thermal leptogenesis.
9. Dihedral flavor symmetries
Energy Technology Data Exchange (ETDEWEB)
Blum, Alexander Simon
2009-06-10
This thesis deals with the possibility of describing the flavor sector of the Standard Model of Particle Physics (with neutrino masses), that is the fermion masses and mixing matrices, with a discrete, non-abelian flavor symmetry. In particular, mass independent textures are considered, where one or several of the mixing angles are determined by group theory alone and are independent of the fermion masses. To this end a systematic analysis of a large class of discrete symmetries, the dihedral groups, is analyzed. Mass independent textures originating from such symmetries are described and it is shown that such structures arise naturally from the minimization of scalar potentials, where the scalars are gauge singlet flavons transforming non-trivially only under the flavor group. Two models are constructed from this input, one describing leptons, based on the group D{sub 4}, the other describing quarks and employing the symmetry D{sub 14}. In the latter model it is the quark mixing matrix element V{sub ud} - basically the Cabibbo angle - which is at leading order predicted from group theory. Finally, discrete flavor groups are discussed as subgroups of a continuous gauge symmetry and it is shown that this implies that the original gauge symmetry is broken by fairly large representations. (orig.)
10. Dihedral flavor symmetries
International Nuclear Information System (INIS)
Blum, Alexander Simon
2009-01-01
This thesis deals with the possibility of describing the flavor sector of the Standard Model of Particle Physics (with neutrino masses), that is the fermion masses and mixing matrices, with a discrete, non-abelian flavor symmetry. In particular, mass independent textures are considered, where one or several of the mixing angles are determined by group theory alone and are independent of the fermion masses. To this end a systematic analysis of a large class of discrete symmetries, the dihedral groups, is analyzed. Mass independent textures originating from such symmetries are described and it is shown that such structures arise naturally from the minimization of scalar potentials, where the scalars are gauge singlet flavons transforming non-trivially only under the flavor group. Two models are constructed from this input, one describing leptons, based on the group D 4 , the other describing quarks and employing the symmetry D 14 . In the latter model it is the quark mixing matrix element V ud - basically the Cabibbo angle - which is at leading order predicted from group theory. Finally, discrete flavor groups are discussed as subgroups of a continuous gauge symmetry and it is shown that this implies that the original gauge symmetry is broken by fairly large representations. (orig.)
11. Lepton flavor violation
International Nuclear Information System (INIS)
Cooper, M.D. Brooks, M.; Hogan, G.E.
1997-01-01
The connection of rare decays to supersymmetric grand unification is highlighted, and a review of the status of rare decay experiments is given. Plans for future investigations of processes that violate lepton flavor are discussed. A new result from the MEGA experiment, a search for μ + → e + γ, is reported to be B.R. -11 with 90% confidence
12. Lepton flavor violation in flavored gauge mediation
Energy Technology Data Exchange (ETDEWEB)
Calibbi, Lorenzo [Universite Libre de Bruxelles, Service de Physique Theorique, Brussels (Belgium); Paradisi, Paride [Universita di Padova, Dipartimento di Fisica e Astronomia, Padua (Italy); INFN Sezione di Padova, Padua (Italy); SISSA, Trieste (Italy); Ziegler, Robert [Sorbonne Universites, UPMC Univ Paris 06, UMR 7589, LPTHE, Paris (France); CNRS, UMR 7589, LPTHE, Paris (France)
2014-12-01
We study the anatomy and phenomenology of lepton flavor violation (LFV) in the context of flavored gauge mediation (FGM). Within FGM, the messenger sector couples directly to the MSSM matter fields with couplings controlled by the same dynamics that explains the hierarchies in the SM Yukawas. Although the pattern of flavor violation depends on the particular underlying flavor model, FGM provides a built-in flavor suppression similar to wave function renormalization or SUSY partial compositeness. Moreover, in contrast to these models, there is an additional suppression of left-right flavor transitions by third-generation Yukawas that in particular provides an extra protection against flavor-blind phases. We exploit the consequences of this setup for lepton flavor phenomenology, assuming that the new couplings are controlled by simple U(1) flavor models that have been proposed to accommodate large neutrino mixing angles. Remarkably, it turns out that in the context of FGM these models can pass the impressive constraints from LFV processes and leptonic electric dipole moments (EDMs) even for light superpartners, therefore offering the possibility of resolving the longstanding muon g - 2 anomaly. (orig.)
13. Sequential flavor symmetry breaking
International Nuclear Information System (INIS)
Feldmann, Thorsten; Jung, Martin; Mannel, Thomas
2009-01-01
The gauge sector of the standard model exhibits a flavor symmetry that allows for independent unitary transformations of the fermion multiplets. In the standard model the flavor symmetry is broken by the Yukawa couplings to the Higgs boson, and the resulting fermion masses and mixing angles show a pronounced hierarchy. In this work we connect the observed hierarchy to a sequence of intermediate effective theories, where the flavor symmetries are broken in a stepwise fashion by vacuum expectation values of suitably constructed spurion fields. We identify the possible scenarios in the quark sector and discuss some implications of this approach.
14. Sequential flavor symmetry breaking
Science.gov (United States)
Feldmann, Thorsten; Jung, Martin; Mannel, Thomas
2009-08-01
The gauge sector of the standard model exhibits a flavor symmetry that allows for independent unitary transformations of the fermion multiplets. In the standard model the flavor symmetry is broken by the Yukawa couplings to the Higgs boson, and the resulting fermion masses and mixing angles show a pronounced hierarchy. In this work we connect the observed hierarchy to a sequence of intermediate effective theories, where the flavor symmetries are broken in a stepwise fashion by vacuum expectation values of suitably constructed spurion fields. We identify the possible scenarios in the quark sector and discuss some implications of this approach.
15. Systematic model building with flavor symmetries
Energy Technology Data Exchange (ETDEWEB)
Plentinger, Florian
2009-12-19
The observation of neutrino masses and lepton mixing has highlighted the incompleteness of the Standard Model of particle physics. In conjunction with this discovery, new questions arise: why are the neutrino masses so small, which form has their mass hierarchy, why is the mixing in the quark and lepton sectors so different or what is the structure of the Higgs sector. In order to address these issues and to predict future experimental results, different approaches are considered. One particularly interesting possibility, are Grand Unified Theories such as SU(5) or SO(10). GUTs are vertical symmetries since they unify the SM particles into multiplets and usually predict new particles which can naturally explain the smallness of the neutrino masses via the seesaw mechanism. On the other hand, also horizontal symmetries, i.e., flavor symmetries, acting on the generation space of the SM particles, are promising. They can serve as an explanation for the quark and lepton mass hierarchies as well as for the different mixings in the quark and lepton sectors. In addition, flavor symmetries are significantly involved in the Higgs sector and predict certain forms of mass matrices. This high predictivity makes GUTs and flavor symmetries interesting for both, theorists and experimentalists. These extensions of the SM can be also combined with theories such as supersymmetry or extra dimensions. In addition, they usually have implications on the observed matter-antimatter asymmetry of the universe or can provide a dark matter candidate. In general, they also predict the lepton flavor violating rare decays {mu} {yields} e{gamma}, {tau} {yields} {mu}{gamma}, and {tau} {yields} e{gamma} which are strongly bounded by experiments but might be observed in the future. In this thesis, we combine all of these approaches, i.e., GUTs, the seesaw mechanism and flavor symmetries. Moreover, our request is to develop and perform a systematic model building approach with flavor symmetries and
16. Systematic model building with flavor symmetries
International Nuclear Information System (INIS)
Plentinger, Florian
2009-01-01
The observation of neutrino masses and lepton mixing has highlighted the incompleteness of the Standard Model of particle physics. In conjunction with this discovery, new questions arise: why are the neutrino masses so small, which form has their mass hierarchy, why is the mixing in the quark and lepton sectors so different or what is the structure of the Higgs sector. In order to address these issues and to predict future experimental results, different approaches are considered. One particularly interesting possibility, are Grand Unified Theories such as SU(5) or SO(10). GUTs are vertical symmetries since they unify the SM particles into multiplets and usually predict new particles which can naturally explain the smallness of the neutrino masses via the seesaw mechanism. On the other hand, also horizontal symmetries, i.e., flavor symmetries, acting on the generation space of the SM particles, are promising. They can serve as an explanation for the quark and lepton mass hierarchies as well as for the different mixings in the quark and lepton sectors. In addition, flavor symmetries are significantly involved in the Higgs sector and predict certain forms of mass matrices. This high predictivity makes GUTs and flavor symmetries interesting for both, theorists and experimentalists. These extensions of the SM can be also combined with theories such as supersymmetry or extra dimensions. In addition, they usually have implications on the observed matter-antimatter asymmetry of the universe or can provide a dark matter candidate. In general, they also predict the lepton flavor violating rare decays μ → eγ, τ → μγ, and τ → eγ which are strongly bounded by experiments but might be observed in the future. In this thesis, we combine all of these approaches, i.e., GUTs, the seesaw mechanism and flavor symmetries. Moreover, our request is to develop and perform a systematic model building approach with flavor symmetries and to search for phenomenological
17. Lepton-flavor violating mediators
Energy Technology Data Exchange (ETDEWEB)
Galon, Iftah; Kwa, Anna [Department of Physics & Astronomy, University of California,Irvine, CA 92697 (United States); Tanedo, Philip [Department of Physics & Astronomy, University of California,Riverside, CA 92521 (United States)
2017-03-13
We present a framework where dark matter interacts with the Standard Model through a light, spin-0 mediator that couples chirally to pairs of different-flavor leptons. This flavor violating final state weakens bounds on new physics coupled to leptons from terrestrial experiments and cosmic-ray measurements. As an example, we apply this framework to construct a model for the Fermi-LAT excess of GeV γ-rays from the galactic center. We comment on the viability of this portal for self-interacting dark matter explanations of small scale structure anomalies and embeddings in flavor models. Models of this type are shown to be compatible with the muon anomalous magnetic moment anomaly. We review current experimental constraints and identify possible future theoretical and experimental directions.
18. Lepton mixing and CP violation phase in the 3-3-1 model with neutral leptons based on T{sub 13} flavor symmetry
Energy Technology Data Exchange (ETDEWEB)
Vien, Vo Van, E-mail: [email protected] [Department of Physics, Tay Nguyen University, Le Duan, Buon Ma Thuot, DakLak (Viet Nam)
2015-08-15
We study a 3-3-1 model based on non-Abelian discrete symmetry group T{sub 13} which accommodates lepton mixing with non-zero θ{sub 13} and CP violation phase. The neutrinos get small masses and mixing with CP violation phase from S U(3) L antisextets which are all in triplets under T{sub 13}. If both breakings T{sub 13} → Z{sub 3} and Z{sub 3} → {Identity} are taken place in neutrino sector, and T{sub 13} is broken into Z{sub 3} in lepton sector, the realistic neutrino mixing form is obtained as a natural consequence of P{sub l} and T{sub 13} symmetries. The model predicts the lepton mixing with non-zero θ{sub 13}, and also gives a remarkable prediction of Dirac CP violation δ{sub CP} = 292.5∘ in the normal spectrum, and δ {sub CP} = 303.161∘ in the inverted spectrum which is still missing in the neutrino mixing matrix. There exist some regions of model parameters that can fit the experimental data in 2014 on neutrino masses and mixing without perturbation. (author)
19. Lepton mixing and cancellation of the Dirac mass hierarchy in SO(10) GUTs with flavor symmetries T7 and Σ(81)
International Nuclear Information System (INIS)
Hagedorn, Claudia; Schmidt, Michael A.; Smirnov, Alexei Yu.
2009-01-01
In SO(10) grand unified theories the hierarchy which is present in the Dirac mass term of the neutrinos is generically as strong as the one in the up-type quark mass term. We propose a mechanism to partially or completely cancel this hierarchy in the light neutrino mass matrix in the seesaw context. The two main ingredients of the cancellation mechanism are the existence of three fermionic gauge singlets and of a discrete flavor symmetry G f which is broken at a higher scale than SO(10). Two realizations of the cancellation mechanism are presented. The realization based on the Frobenius group T 7 ≅Z 7 xZ 3 leads to a partial cancellation of the hierarchy and relates maximal 2-3 lepton mixing with the geometric hierarchy of the up-quark masses. In the realization with the group Σ(81) the cancellation is complete and tribimaximal lepton mixing is reproduced at the lowest order. In both cases, to fully accommodate the leptonic data we take into account additional effects such as effects of higher-dimensional operators involving more than one flavon. The heavy neutral fermion mass spectra are considered. For both realizations we analyze the flavon potential at the renormalizable level as well as ways to generate the Cabibbo angle.
20. Lepton flavor non-conservation
International Nuclear Information System (INIS)
Kosmas, T.S.; Tuebingen Univ.; Leontaris, G.K.; Vergados, J.D.
1994-01-01
In the present work we review the most prominent lepton flavor violating processes (μ → eγ, μ → 3e, (μ - , e -) conversion, M - M oscillations etc.), in the context of unified gauge theories. Many currently fashionable extensions of the standard model are considered, such as: i) extensions of the fermion sector (right-handed neutrino); ii) minimal extensions involving additional Higgs scalars (more than one isodoublets, singly and doubly charged isosinglets, isotriplets with doubly charged members etc.); iii) supersymmetric or superstring inspired unified models emphasizing the implications of the renormalization group equations in the leptonic sector. Special attention is given to the experimentally most interesting (μ - , e - ) conversion in the presence of nuclei. The relevant nuclear aspects of the amplitudes are discussed in a number of fashionable nuclear models. The main features of the relevant experiments are also discussed, and detailed predictions of the above models are compared to the present experimental limits. (Author)
1. Democratic (s)fermions and lepton flavor violation
Science.gov (United States)
Hamaguchi, K.; Kakizaki, Mitsuru; Yamaguchi, Masahiro
2003-09-01
The democratic approach to account for fermion masses and mixing is known to be successful not only in the quark sector but also in the lepton sector. Here we extend this ansatz to supersymmetric standard models, in which the Kähler potential obeys the underlying S3 flavor symmetries. The requirement of neutrino bi-large mixing angles constrains the form of the Kähler potential for left-handed lepton multiplets. We find that right-handed sleptons can have nondegenerate masses and flavor mixing, while left-handed sleptons are argued to have universal and hence flavor-blind masses. This mass pattern is testable in future collider experiments when superparticle masses will be measured precisely. Lepton flavor violation arises in this scenario. In particular, μ→eγ is expected to be observed in a planned future experiment if supersymmetry breaking scale is close to the weak scale.
2. Democratic (s)fermions and lepton flavor violation
International Nuclear Information System (INIS)
Hamaguchi, K.; Kakizaki, Mitsuru; Yamaguchi, Masahiro
2003-01-01
The democratic approach to account for fermion masses and mixing is known to be successful not only in the quark sector but also in the lepton sector. Here we extend this ansatz to supersymmetric standard models, in which the Kaehler potential obeys the underlying S 3 flavor symmetries. The requirement of neutrino bi-large mixing angles constrains the form of the Kaehler potential for left-handed lepton multiplets. We find that right-handed sleptons can have nondegenerate masses and flavor mixing, while left-handed sleptons are argued to have universal and hence flavor-blind masses. This mass pattern is testable in future collider experiments when superparticle masses will be measured precisely. Lepton flavor violation arises in this scenario. In particular, μ→eγ is expected to be observed in a planned future experiment if supersymmetry breaking scale is close to the weak scale
3. Unified flavor symmetry from warped dimensions
Energy Technology Data Exchange (ETDEWEB)
Frank, Mariana, E-mail: [email protected] [Department of Physics, Concordia University, 7141 Sherbrooke St. West, Montreal, Quebec, H4B 1R6 (Canada); Hamzaoui, Cherif, E-mail: [email protected] [Groupe de Physique Théorique des Particules, Département des Sciences de la Terre et de L' Atmosphère, Université du Québec à Montréal, Case Postale 8888, Succ. Centre-Ville, Montréal, Québec, H3C 3P8 (Canada); Pourtolami, Nima, E-mail: [email protected] [Department of Physics, Concordia University, 7141 Sherbrooke St. West, Montreal, Quebec, H4B 1R6 (Canada); Toharia, Manuel, E-mail: [email protected] [Department of Physics, Concordia University, 7141 Sherbrooke St. West, Montreal, Quebec, H4B 1R6 (Canada)
2015-03-06
In a model of warped extra-dimensions with all matter fields in the bulk, we propose a scenario which explains all the masses and mixings of the SM fermions. In this scenario, the same flavor symmetric structure is imposed on all the fermions of the Standard Model (SM), including neutrinos. Due to the exponential sensitivity on bulk fermion masses, a small breaking of this symmetry can be greatly enhanced and produce seemingly un-symmetric hierarchical masses and small mixing angles among the charged fermion zero-modes (SM quarks and charged leptons), thus washing out visible effects of the symmetry. If the Dirac neutrinos are sufficiently localized towards the UV boundary, and the Higgs field leaking into the bulk, the neutrino mass hierarchy and flavor structure will still be largely dominated and reflect the fundamental flavor structure, whereas localization of the quark sector would reflect the effects of the flavor symmetry breaking sector. We explore these features in an example based on which a family permutation symmetry is imposed in both quark and lepton sectors.
4. Searches for lepton flavor violation
International Nuclear Information System (INIS)
Bryman, D.
1986-01-01
The search for lepton flavor violation has reached considerable sensitivity, but with only null results so far. The experiments are sensitive to new particle in the 1 to 100 TeV range arising in a variety of theories, although the constraints on the masses of such particles improve only as the inverse fourth power of branching ratios. Presenting, neutrinoless μe conversion in the field of a nucleus provides the most serious constraints for many models. New experiments on rare kaon decays γe conversion and μ → eγ will result in improved sensitivity in the next few years. Ignoring theoretical prejudice, it is important to study many different processes in the hope uncovering some new effects
5. Prospects in lepton-flavor violation
International Nuclear Information System (INIS)
Hoffman, C.M.
1982-01-01
The theoretical and experimental situation regarding lepton-flavor conservation is reviewed and upcoming experiments are described. It is concluded that future improvements in experimental sensitivities will require higher flux, higher quality muon and kaon beams
6. Neutrino mass in flavor dependent gauged lepton model
Science.gov (United States)
2018-03-01
We study a neutrino model introducing an additional nontrivial gauged lepton symmetry where the neutrino masses are induced at two-loop level, while the first and second charged-leptons of the standard model are done at one-loop level. As a result of the model structure, we can predict one massless active neutrino, and there is a dark matter candidate. Then we discuss the neutrino mass matrix, muon anomalous magnetic moment, lepton flavor violations, oblique parameters, and relic density of dark matter, taking into account the experimental constraints.
7. Lepton flavor violation in an extended MSSM
CERN Document Server
Espinosa-Castañeda, R.; Gómez-Bock, M.; Mondragón, M.
2016-01-01
In this work we explore a lepton flavor violation effect induced at one loop for a flavor structure in an extended minimal standard supersymmetric model, considering an ansatz for the trilinear term. In particular we find a finite expression which will show the impact of this phenomena in the $h\\to \\mu \\tau$ decay, produced by a mixing in the trilinear coupling of the soft supersymmetric Lagrangian.
8. Flavor symmetries and fermion masses
International Nuclear Information System (INIS)
Rasin, A.
1994-04-01
We introduce several ways in which approximate flavor symmetries act on fermions and which are consistent with observed fermion masses and mixings. Flavor changing interactions mediated by new scalars appear as a consequence of approximate flavor symmetries. We discuss the experimental limits on masses of the new scalars, and show that the masses can easily be of the order of weak scale. Some implications for neutrino physics are also discussed. Such flavor changing interactions would easily erase any primordial baryon asymmetry. We show that this situation can be saved by simply adding a new charged particle with its own asymmetry. The neutrality of the Universe, together with sphaleron processes, then ensures a survival of baryon asymmetry. Several topics on flavor structure of the supersymmetric grand unified theories are discussed. First, we show that the successful predictions for the Kobayashi-Maskawa mixing matrix elements, V ub /V cb = √m u /m c and V td /V ts = √m d /m s , are a consequence of a large class of models, rather than specific properties of a few models. Second, we discuss how the recent observation of the decay β → sγ constrains the parameter space when the ratio of the vacuum expectation values of the two Higgs doublets, tanΒ, is large. Finally, we discuss the flavor structure of proton decay. We observe a surprising enhancement of the branching ratio for the muon mode in SO(10) models compared to the same mode in the SU(5) model
9. Lepton flavor violation in tau decays
International Nuclear Information System (INIS)
Cvetic, G.; Dib, C.; Kim, C. S.; Kim, J. D.
2002-01-01
We study lepton flavor violation (LFV) in tau decays induced by heavy Majorana neutrinos within two models: (I) the standard model with additional right-handed heavy Majorana neutrinos, i.e., a typical seesaw-type model; (II) the standard model with left-handed and right-handed neutral singlets, which are inspired by certain scenarios of SO(10) models and heterotic superstring models with E 6 symmetry. We calculate various LFV branching ratios and a T-odd asymmetry. The seesaw model I predicts very small branching ratios for LFV processes in most of the parameter space, although in a very restricted parameter region it can reach maximal branching ratios Br(τ→μγ)∼10 -9 and Br(τ→3μ)∼10 -10 . In contrast, model II may show branching ratios Br(τ→eγ)∼10 -8 and Br(τ→3e) -9 over a sizable region of the parameter space, large enough to be tested by experiments in the near future
10. Lepton flavor violation with light vector bosons
Directory of Open Access Journals (Sweden)
Julian Heeck
2016-07-01
Full Text Available New sub-GeV vector bosons with couplings to muons but not electrons have been discussed in order to explain the muon's magnetic moment, the gap of high-energy neutrinos in IceCube or the proton radius puzzle. If such a light Z′ not only violates lepton universality but also lepton flavor, as expected for example from the recent hint for h→μτ at CMS, the two-body decay mode τ→μZ′ opens up and for MZ′<2mμ gives better constraints than τ→3μ already with 20-year-old ARGUS limits. We discuss the general prospects and motivation of light vector bosons with lepton-flavor-violating couplings.
11. Flavor universal dynamical electroweak symmetry breaking
International Nuclear Information System (INIS)
Burdman, G.; Evans, N.
1999-01-01
The top condensate seesaw mechanism of Dobrescu and Hill allows electroweak symmetry to be broken while deferring the problem of flavor to an electroweak singlet, massive sector. We provide an extended version of the singlet sector that naturally accommodates realistic masses for all the standard model fermions, which play an equal role in breaking electroweak symmetry. The models result in a relatively light composite Higgs sector with masses typically in the range of (400 - 700) GeV. In more complete models the dynamics will presumably be driven by a broken gauged family or flavor symmetry group. As an example of the higher scale dynamics a fully dynamical model of the quark sector with a GIM mechanism is presented, based on an earlier top condensation model of King using broken family gauge symmetry interactions (that model was itself based on a technicolor model of Georgi). The crucial extra ingredient is a reinterpretation of the condensates that form when several gauge groups become strong close to the same scale. A related technicolor model of Randall which naturally includes the leptons too may also be adapted to this scenario. We discuss the low energy constraints on the massive gauge bosons and scalars of these models as well as their phenomenology at the TeV scale. copyright 1999 The American Physical Society
12. Lepton flavor violation induced by dark matter
Science.gov (United States)
Arcadi, Giorgio; Ferreira, C. P.; Goertz, Florian; Guzzo, M. M.; Queiroz, Farinaldo S.; Santos, A. C. O.
2018-04-01
Guided by gauge principles we discuss a predictive and falsifiable UV complete model where the Dirac fermion that accounts for the cold dark matter abundance in our Universe induces the lepton flavor violation (LFV) decays μ →e γ and μ →e e e as well as μ -e conversion. We explore the interplay between direct dark matter detection, relic density, collider probes and lepton flavor violation to conclusively show that one may have a viable dark matter candidate yielding flavor violation signatures that can be probed in the upcoming experiments. In fact, keeping the dark matter mass at the TeV scale, a sizable LFV signal is possible, while reproducing the correct dark matter relic density and meeting limits from direct-detection experiments.
13. Search for Lepton Flavor Violation with Muons
International Nuclear Information System (INIS)
Kuno, Yoshitaka
2009-01-01
Physics motivation and phenomenology of muon to electron conversion (μ - +N(A,Z)→e - +N(A,Z)) in a muonic atom, which is one the most important muon processes to search for lepton flavor violation of charged leptons, are presented. Prospects for future experiments at J-PARC (Japan Proton Accelerator Complex) in Japan, such as the COMET experiment for a sensitivity of less than 10 -16 as the first stage, and then the PRISM/PRIME experiment for a sensitivity of less than 10 -18 as the ultimate stage, are discussed.
14. Fermion mass hierarchies and flavor mixing from T' symmetry
International Nuclear Information System (INIS)
Ding Guijun
2008-01-01
We construct a supersymmetric model based on T ' x Z 3 x Z 9 flavor symmetry. At the leading order, the charged lepton mass matrix is not diagonal, T ' is broken completely, and the hierarchy in the charged lepton masses is generated naturally. Nearly tribimaximal mixing is predicted, and subleading effects induce corrections of order λ 2 , where λ is the Cabibbo angle. Both the up quark and down quark mass matrices' textures of the well-known U(2) flavor theory are produced at the leading order; realistic hierarchies in quark masses and Cabibbo-Kobayashi-Maskawa matrix elements are obtained. The vacuum alignment and subleading corrections are discussed in detail.
15. Naturally large radiative lepton flavor violating Higgs decay mediated by lepton-flavored dark matter
International Nuclear Information System (INIS)
Baek, Seungwon; Kang, Zhaofeng
2016-01-01
In the standard model (SM), lepton flavor violating (LFV) Higgs decay is absent at renormalizable level and thus it is a good probe to new physics. In this article we study a type of new physics that could lead to large LFV Higgs decay, i.e., a lepton-flavored dark matter (DM) model which is specified by a Majorana DM and scalar lepton mediators. Different from other similar models with similar setup, we introduce both left-handed and right-handed scalar leptons. They allow large LFV Higgs decay and thus may explain the tentative Br(h→τμ)∼1% experimental results from the LHC. In particular, we find that the stringent bound from τ→μγ can be naturally evaded. One reason, among others, is a large chirality violation in the mediator sector. Aspects of relic density and especially radiative direct detection of the leptonic DM are also investigated, stressing the difference from previous lepton-flavored DM models.
16. Lepton flavor violation with displaced vertices
Directory of Open Access Journals (Sweden)
Julian Heeck
2018-01-01
Full Text Available If light new physics with lepton-flavor-violating couplings exists, the prime discovery channel might not be ℓ→ℓ′γ but rather ℓ→ℓ′X, where the new boson X could be an axion, majoron, familon or Z′ gauge boson. The most conservative bound then comes from ℓ→ℓ′+inv, but if the on-shell X can decay back into leptons or photons, displaced-vertex searches could give much better limits. We show that only a narrow region in parameter space allows for displaced vertices in muon decays, μ→eX,X→γγ,ee, whereas tauon decays can have much more interesting signatures.
17. Sterile neutrinos in lepton number and lepton flavor violating decays
International Nuclear Information System (INIS)
Helo, Juan Carlos; Kovalenko, Sergey; Schmidt, Ivan
2011-01-01
We study the contribution of massive dominantly sterile neutrinos, N, to the lepton number and lepton flavor violating semileptonic decays of τ and B, D, K-mesons. We focus on special domains of sterile neutrino masses m N where it is close to its mass-shell. This leads to an enormous resonant enhancement of the decay rates of these processes. This allows us to derive stringent limits on the sterile neutrino mass m N and its mixing U αN with active flavors. We apply a joint analysis of the existing experimental bounds on the decay rates of the studied processes. In contrast to other approaches in the literature our limits are free from ad hoc assumptions on the relative size of the sterile neutrino mixing parameters. We analyze the impact of this sort of assumptions on the extraction of the limits on m N and U αN , and discuss the effect of finite detector size. Special attention was paid to the limits on meson decays with e ± e ± in final state, derived from non-observation of 0νββ-decay. We point out that observation of these decays may, in particular, shed light on the Majorana phases of the neutrino mixing matrix.
18. Radiatively induced neutrino mass model with flavor dependent gauge symmetry
Science.gov (United States)
Lee, SangJong; Nomura, Takaaki; Okada, Hiroshi
2018-06-01
We study a radiative seesaw model at one-loop level with a flavor dependent gauge symmetry U(1) μ - τ, in which we consider bosonic dark matter. We also analyze the constraints from lepton flavor violations, muon g - 2, relic density of dark matter, and collider physics, and carry out numerical analysis to search for allowed parameter region which satisfy all the constraints and to investigate some predictions. Furthermore we find that a simple but adhoc hypothesis induces specific two zero texture with inverse mass matrix, which provides us several predictions such as a specific pattern of Dirac CP phase.
19. Neutrino masses and spontaneously broken flavor symmetries
International Nuclear Information System (INIS)
Staudt, Christian
2014-01-01
We study the phenomenology of supersymmetric flavor models. We show how the predictions of models based on spontaneously broken non-Abelian discrete flavor symmetries are altered when we include so-called Kaehler corrections. Furthermore, we discuss anomaly-free discrete R symmetries which are compatible with SU(5) unification. We find a set of symmetries compatible with suppressed Dirac neutrino masses and a unique symmetry consistent with the Weinberg operator. We also study a pseudo-anomalous U(1) R symmetry which explains the fermion mass hierarchies and, when amended with additional singlet fields, ameliorates the fine-tuning problem.
20. Flavor symmetry in the large Nc limit
International Nuclear Information System (INIS)
Karl, G.; Washington Univ., Seattle, WA; Lipkin, H.J.; Washington Univ., Seattle, WA
1991-01-01
An essential difference between two-flavor and three-flavor descriptions of baryons in large N c QCD is discussed in detail. For N c ≥3 a state with the SU(3) flavor quantum numbers of the proton must contain a number of strange quarks n s ≥(N c -3)/3, while a state with no strange quarks must have extra hypercharge Y-1 = 3/N c -1. The extra strangeness or extra hypercharge which vanishes for N c = 3 is spurious for the physical proton. This problem does not arise in two-flavor QCD, where the flavor-SU(2) Skyrmion may give a good approximation for nucleon-pion physics at low energies below strangeness threshold. But any nucleon model with SU(3) flavor symmetry which is interpreted as arising from the large N c limit in QCD can lead to erroneous conclusions about the spin and flavor structure of the proton. 12 refs
1. Large lepton mixings from continuous symmetries
International Nuclear Information System (INIS)
Everett, Lisa; Ramond, Pierre
2007-01-01
Within the broad context of quark-lepton unification, we investigate the implications of broken continuous family symmetries which result from requiring that in the limit of exact symmetry, the Dirac mass matrices yield hierarchical masses for the quarks and charged leptons, but lead to degenerate light neutrino masses as a consequence of the seesaw mechanism, without requiring hierarchical right-handed neutrino mass terms. Quark mixing is then naturally small and proportional to the size of the perturbation, but lepton mixing is large as a result of degenerate perturbation theory, shifted from maximal mixing by the size of the perturbation. Within this approach, we study an illustrative two-family prototype model with an SO(2) family symmetry, and discuss extensions to three-family models
2. μe conversion experiments. Testing charged lepton flavor violation
International Nuclear Information System (INIS)
Schaaf, Andries van der
2004-01-01
The recent evidence for neutrino mixing shows that lepton flavor is not a conserved quantity. Due to the smallness of the neutrino masses effective flavor changing neutral currents among charged leptons remain negligible in the Standard Model. Whereas b → sγ has a probability of O(10 -4 )μ → eγ is expected with a branching ratio around 10 -50 . Observable rates would be an unambiguous signal for physics beyond the Standard Model and indeed, many extensions of the model are constrained best by the present experimental limits on charged lepton flavor violation. In this talk I will discuss experimental searches for charged lepton flavor violation with emphasis on μe conversion in muonic atoms. (author)
3. 2016 International Conference on Charged Lepton Flavor Violation
Energy Technology Data Exchange (ETDEWEB)
Dukes, Edmond Craig
2017-12-04
Partial support for participation for students and postdocs who wished to attend to give poster presentations at the 2016 International Conference on Charged Lepton Flavor Violation (CLFV 2016) in Charlottesville, VA.
4. U(1) textures and Lepton Flavor Violation
CERN Document Server
Gómez, M E; Lola, S; Vergados, J D
1999-01-01
U(1) family symmetries have led to successful predictions of the fermion mass spectrum and the mixing angles of the hadronic sector. In the context of the supersymmetric unified theories, they further imply a non-trivial mass structure for the scalar partners, giving rise to new sources of flavour violation. In the present work, lepton flavour non-conserving processes are examined in the context of the MSSM augmented by a U(1) family symmetry. We calculate the mixing effects on the mu -> e gamma and tau-> mu gamma rare decays. All supersymmetric scalar masses involved in the processes are determined at low energies using two loop renormalisation group analysis and threshold corrections. Further, various novel effects are considered and found to have important impact on the branching ratios. Thus, a rather interesting result is that when the see-saw mechanism is applied in the (12X12)-sneutrino mass matrix, the mixing effects of the Dirac matrix in the effective light sneutrino sector are canceled at first ord...
5. Fermion masses and flavor mixings in a model with S4 flavor symmetry
International Nuclear Information System (INIS)
Ding Guijun
2010-01-01
We present a supersymmetric model of quark and lepton based on S 4 xZ 3 xZ 4 flavor symmetry. The S 4 symmetry is broken down to Klein four and Z 3 subgroups in the neutrino and the charged lepton sectors, respectively. Tri-Bimaximal mixing and the charged lepton mass hierarchies are reproduced simultaneously at leading order. Moreover, a realistic pattern of quark masses and mixing angles is generated with the exception of the mixing angle between the first two generations, which requires a small accidental enhancement. It is remarkable that the mass hierarchies are controlled by the spontaneous breaking of flavor symmetry in our model. The next to leading order contributions are studied, all the fermion masses and mixing angles receive corrections of relative order λ c 2 with respect to the leading order results. The phenomenological consequences of the model are analyzed, the neutrino mass spectrum can be normal hierarchy or inverted hierarchy, and the combined measurement of the 0ν2β decay effective mass m ββ and the lightest neutrino mass can distinguish the normal hierarchy from the inverted hierarchy.
6. S3 flavor symmetry and leptogenesis
International Nuclear Information System (INIS)
Kubo, Jisuke; Paschos, Emmanuel A.
2004-01-01
It is found that a Majorana phase in a minimal S 3 extension of the standard model with an additional Z 2 symmetry in the leptonic sector is responsible for leptogenesis. We assume that the resonant enhancement of the CP asymmetries takes place to obtain a realistic size of baryon number asymmetry in the universe. We expect the masses of the right-handed neutrino are of O(10) TeV in this model. (author)
7. Explaining R{sub D}{sup {sub (}{sub *}{sub )}} with leptoquarks and flavor symmetries
Energy Technology Data Exchange (ETDEWEB)
Schoenwald, Kay [TU Dortmund (Germany)
2016-07-01
Recently LHCb confirmed the anomalies in R{sub D}{sup {sub (}{sub *}{sub )}} previously measured by BaBar and Belle. We use flavor symmetries capable of explaining the observed mixing in the quark and lepton sector to constrain leptoquark couplings and study whether this models can explain the anomalies in R{sub D}{sup {sub (}{sub *}{sub )}}.
8. Flavor symmetry breaking and meson masses
International Nuclear Information System (INIS)
Bhagwat, Mandar S.; Roberts, Craig D.; Chang Lei; Liu Yuxin; Tandy, Peter C.
2007-01-01
The axial-vector Ward-Takahashi identity is used to derive mass formulas for neutral pseudoscalar mesons. Flavor symmetry breaking entails nonideal flavor content for these states. Adding that the η ' is not a Goldstone mode, exact chiral-limit relations are developed from the identity. They connect the dressed-quark propagator to the topological susceptibility. It is confirmed that in the chiral limit the η ' mass is proportional to the matrix element which connects this state to the vacuum via the topological susceptibility. The implications of the mass formulas are illustrated using an elementary dynamical model, which includes an Ansatz for that part of the Bethe-Salpeter kernel related to the non-Abelian anomaly. In addition to the current-quark masses, the model involves two parameters, one of which is a mass-scale. It is employed in an analysis of pseudoscalar- and vector-meson bound-states. While the effects of SU(N f =2) and SU(N f =3) flavor symmetry breaking are emphasized, the five-flavor spectra are described. Despite its simplicity, the model is elucidative and phenomenologically efficacious; e.g., it predicts η-η ' mixing angles of ∼-15 deg. and π 0 -η angles of ∼1 deg
9. Deviation from bimaximal mixing and leptonic CP phases in S4 family symmetry and generalized CP
International Nuclear Information System (INIS)
Li, Cai-Chang; Ding, Gui-Jun
2015-01-01
The lepton flavor mixing matrix having one row or one column in common with the bimaximal mixing up to permutations is still compatible with the present neutrino oscillation data. We provide a thorough exploration of generating such a mixing matrix from S 4 family symmetry and generalized CP symmetry H CP . Supposing that S 4 ⋊H CP is broken down to Z 2 ST 2 SU ×H CP ν in the neutrino sector and Z 4 TST 2 U ⋊H CP l in the charged lepton sector, one column of the PMNS matrix would be of the form (1/2,1/√2,1/2) T up to permutations, both Dirac CP phase and Majorana CP phases are trivial to accommodate the observed lepton mixing angles. The phenomenological implications of the remnant symmetry K 4 (TST 2 ,T 2 U) ×H CP ν in the neutrino sector and Z 2 SU ×H CP l in the charged lepton sector are studied. One row of PMNS matrix is determined to be (1/2,1/2,−i/√2), and all the three leptonic CP phases can only be trivial to fit the measured values of the mixing angles. Two models based on S 4 family symmetry and generalized CP are constructed to implement these model independent predictions enforced by remnant symmetry. The correct mass hierarchy among the charged leptons is achieved. The vacuum alignment and higher order corrections are discussed.
10. Broken flavor symmetries in high energy particle phenomenology
International Nuclear Information System (INIS)
Antaramian, A.
1995-01-01
Over the past couple of decades, the Standard Model of high energy particle physics has clearly established itself as an invaluable tool in the analysis of high energy particle phenomenon. However, from a field theorists point of view, there are many dissatisfying aspects to the model. One of these, is the large number of free parameters in the theory arising from the Yukawa couplings of the Higgs doublet. In this thesis, we examine various issues relating to the Yukawa coupeng structure of high energy particle field theories. We begin by examining extensions to the Standard Model of particle physics which contain additional scalar fields. By appealing to the flavor structure observed in the fermion mass and Kobayashi-Maskawa matrices, we propose a reasonable phenomenological parameterization of the new Yukawa couplings based on the concept of approximate flavor symmetries. It is shown that such a parameterization eliminates the need for discrete symmetries which limit the allowed couplings of the new scalars. New scalar particles which can mediate exotic flavor changing reactions can have masses as low as the weak scale. Next, we turn to the issue of neutrino mass matrices, where we examine a particular texture which leads to matter independent neutrino oscillation results for solar neutrinos. We, then, examine the basis for extremely strict limits placed on flavor changing interactions which also break lepton- and/or baryon-number. These limits are derived from cosmological considerations. Finally, we embark on an extended analysis of proton decay in supersymmetric SO(10) grand unified theories. In such theories, the dominant decay diagrams involve the Yukawa couplings of a heavy triplet superfield. We argue that past calculations of proton decay which were based on the minimal supersymmetric SU(5) model require reexamination because the Yukawa couplings of that theory are known to be wrong
11. Lepton flavor violation at LEP II and beyond
International Nuclear Information System (INIS)
Feng, J.L.; Univ. of California, Berkeley, CA
1996-07-01
At present, two fundamental mysteries in particle physics are the origins of electroweak symmetry breaking and the fermion mass matrices. The experimental discovery of superpartners would represent enormous progress in the understanding of electroweak symmetry breaking, but would it also allow progress on the flavor problem? To date, nearly all experimental studies of supersymmetry have ignored the possibility of flavor mixings in the sfermion sector. However, since all superpartners must be given masses, all supersymmetric theories necessarily allow for the possibility of new flavor mixings beyond the standard model. In addition, there are now many supersymmetric theories of flavor, which predict a wide variety of superpartner flavor mixings. In this study, the author examines the possibility of measuring these mixings at LEP II and the Next Linear Collider (NLC). Rare flavor changing processes, such as μ → eγ, τ → μγ, τ → eγ, b → sγ, and neutral meson mixing, already provide important constraints on the sfermion flavor mixings through the virtual effects of superpartners. However, as will be seen below, once superpartners are discovered, it will be possible to probe these mixings much more powerfully by directly observing the change in flavor occurring at the superpartner production and decay vertices
12. Neutrinoless double beta decay and lepton flavor violation
International Nuclear Information System (INIS)
Cirigliano, V.; Kurylov, A.; Vogel, P.; Ramsey-Musolf, M.J.
2004-01-01
We point out that extensions of the standard model with low scale (∼TeV) lepton number violation (LNV) generally lead to a pattern of lepton flavor violation (LFV) experimentally distinguishable from the one implied by models with grand unified theory scale LNV. As a consequence, muon LFV processes provide a powerful diagnostic tool to determine whether or not the effective neutrino mass can be deduced from the rate of neutrinoless double beta decay. We discuss the role of μ→eγ and μ→e conversion in nuclei, which will be studied with high sensitivity in forthcoming experiments
13. Systematic Approach to Gauge-Invariant Relations between Lepton Flavor Violating Processes
CERN Document Server
Ibarra, A; Redondo, J; Ibarra, Alejandro; Masso, Eduard; Redondo, Javier
2005-01-01
We analyze four-lepton contact interactions that lead to lepton flavor violating processes, with violation of individual family lepton number but total lepton number conserved. In an effective Lagrangian framework, the assumption of gauge invariance leads to relations among branching ratios and cross sections of lepton flavor violating processes. In this paper, we work out how to use these relations systematically. We also study the consequences of loop-induced processes.
14. Stringy origin of non-Abelian discrete flavor symmetries
International Nuclear Information System (INIS)
Kobayashi, Tatsuo; Nilles, Hans Peter; Ploeger, Felix; Raby, Stuart; Ratz, Michael
2007-01-01
We study the origin of non-Abelian discrete flavor symmetries in superstring theory. We classify all possible non-Abelian discrete flavor symmetries which can appear in heterotic orbifold models. These symmetries include D 4 and Δ(54). We find that the symmetries of the couplings are always larger than the symmetries of the compact space. This is because they are a consequence of the geometry of the orbifold combined with the space group selection rules of the string. We also study possible breaking patterns. Our analysis yields a simple geometric understanding of the realization of non-Abelian flavor symmetries
15. Flavor mixing with quarks and leptons
International Nuclear Information System (INIS)
Bigi, I.I.
1987-10-01
The last year has brought such a wealth of new information on heavy flavors that meaningful bounds can now be placed on all fermion mass related parameters in the Standard Model. The status of the KM matrix is reviewed with particular emphasis on the theoretical uncertainties. B 0 -anti B 0 mixing is reevaluated and CP violation is discussed as it is observed in K/sub L/ decays and as it hopefully can be studied in B decays. The report is concluded with short remarks on neutrino oscillations
16. Lepton-flavor violating B decays in generic Z' models
Science.gov (United States)
Crivellin, Andreas; Hofer, Lars; Matias, Joaquim; Nierste, Ulrich; Pokorski, Stefan; Rosiek, Janusz
2015-09-01
LHCb has reported deviations from the Standard Model in b →s μ+μ- transitions for which a new neutral gauge boson is a prime candidate for an explanation. As this gauge boson has to couple in a flavor nonuniversal way to muons and electrons in order to explain RK, it is interesting to examine the possibility that also lepton flavor is violated, especially in the light of the CMS excess in h →τ±μ∓. In this article, we investigate the perspectives to discover the lepton-flavor violating modes B →K(*)τ±μ∓ , Bs→τ±μ∓ and B →K(*)μ±e∓, Bs→μ±e∓. For this purpose we consider a simplified model in which new-physics effects originate from an additional neutral gauge boson (Z') with generic couplings to quarks and leptons. The constraints from τ →3 μ , τ →μ ν ν ¯, μ →e γ , gμ-2 , semileptonic b →s μ+μ- decays, B →K(*)ν ν ¯ and Bs-B¯s mixing are examined. From these decays, we determine upper bounds on the decay rates of lepton-flavor violating B decays. Br (B →K ν ν ¯) limits the branching ratios of lepton-flavor violating B decays to be smaller than 8 ×10-5(2 ×10-5) for vectorial (left-handed) lepton couplings. However, much stronger bounds can be obtained by a combined analysis of Bs-B¯s, τ →3 μ , τ →μ ν ν ¯ and other rare decays. The bounds depend on the amount of fine-tuning among the contributions to Bs-B¯s mixing. Allowing for a fine-tuning at the percent level we find upper bounds of the order of 10-6 for branching ratios into τ μ final states, while Bs→μ±e∓ is strongly suppressed and only B →K(*)μ±e∓ can be experimentally accessible (with a branching ratio of order 10-7).
17. Lepton C P violation in a ν 2 HDM with flavor
Science.gov (United States)
Barradas-Guevara, E.; Félix-Beltrán, O.; Gonzalez-Canales, F.; Zeleny-Mora, M.
2018-02-01
In this work we propose an extension to the Standard Model in which we consider a type-III two-Higgs-doublet model (2HDM) plus massive neutrinos and the horizontal flavor symmetry S3 (ν 2 HDM ⊗S3 ). In the above framework and with the explicit breaking of flavor symmetry S3, the Yukawa matrices in the flavor-adapted basis are represented by means of a matrix with two texture zeros. Also, the active neutrinos are considered as Majorana particles and their masses are generated through the type-I seesaw mechanism. The unitary matrices that diagonalize the mass matrices, as well as the flavor-mixing matrices, are expressed in terms of fermion mass ratios. Consequently, in the mass basis the entries of the Yukawa matrices naturally acquire the form of the so-called Cheng-Sher ansatz. For the leptonic sector of ν 2 HDM ⊗S3, we compare, through a χ2 likelihood test, the theoretical expressions of the flavor-mixing angles with the masses and flavor-mixing leptons current experimental data. The results obtained in this χ2 analysis are in very good agreement with the current experimental data. We also obtain allowed value ranges for the "Dirac-like" phase factor, as well as for the two Majorana phase factors. Furthermore, we study the phenomenological implications of these numerical values of the C P -violation phases on the neutrinoless double-beta decay, and for long baseline neutrino oscillation experiments such as T2K, NO ν A , and DUNE.
18. Neutrino mixing and masses in SO(10) GUTs with hidden sector and flavor symmetries
Energy Technology Data Exchange (ETDEWEB)
Chu, Xiaoyong [International Centre for Theoretical Physics,Strada Costiera 11, I-34100 Trieste (Italy); Smirnov, Alexei Yu. [Max-Planck-Institute for Nuclear Physics,Saupfercheckweg 1, D-69117 Heidelberg (Germany); International Centre for Theoretical Physics,Strada Costiera 11, I-34100 Trieste (Italy)
2016-05-23
We consider the neutrino masses and mixing in the framework of SO(10) GUTs with hidden sector consisting of fermionic and bosonic SO(10) singlets and flavor symmetries. The framework allows to disentangle the CKM physics responsible for the CKM mixing and different mass hierarchies of quarks and leptons and the neutrino new physics which produces smallness of neutrino masses and large lepton mixing. The framework leads naturally to the relation U{sub PMNS}∼V{sub CKM}{sup †}U{sub 0}, where structure of U{sub 0} is determined by the flavor symmetry. The key feature of the framework is that apart from the Dirac mass matrices m{sub D}, the portal mass matrix M{sub D} and the mass matrix of singlets M{sub S} are also involved in generation of the lepton mixing. This opens up new possibilities to realize the flavor symmetries and explain the data. Using A{sub 4}×Z{sub 4} as the flavor group, we systematically explore the flavor structures which can be obtained in this framework depending on field content and symmetry assignments. We formulate additional conditions which lead to U{sub 0}∼U{sub TBM} or U{sub BM}. They include (i) equality (in general, proportionality) of the singlet flavons couplings, (ii) equality of their VEVs; (iii) correlation between VEVs of singlets and triplet, (iv) certain VEV alignment of flavon triplet(s). These features can follow from additional symmetries or be remnants of further unification. Phenomenologically viable schemes with minimal flavon content and minimal number of couplings are constructed.
19. Flavor and CP symmetries for leptogenesis and 0νββ decay
DEFF Research Database (Denmark)
Hagedorn, Claudia; Molinaro, Emiliano
2017-01-01
We perform a comprehensive analysis of the phenomenology of leptonic low and high energy CP phases in a scenario with three heavy right-handed neutrinos in which a flavor and a CP symmetry are non-trivially broken. All CP phases as well as lepton mixing angles are determined by the properties...... of the flavor and CP symmetry and one free real parameter. We focus on the generation of the baryon asymmetry YB of the Universe via unflavored leptogenesis and the predictions of mee, the quantity measurable in neutrinoless double beta decay. We show that the sign of YB can be fixed and the allowed parameter...... range of mee can be strongly constrained. We argue on general grounds that the CP asymmetries ϵi are dominated by the contribution associated with one Majorana phase and that in cases in which only the Dirac phase is non-trivial the sign of YB depends on further parameters. In addition, we comment...
20. Effective Lagrangian description of Higgs mediated flavor violating electromagnetic transitions: Implications on lepton flavor violation
International Nuclear Information System (INIS)
Aranda, J. I.; Tututi, E. S.; Flores-Tlalpa, A.; Ramirez-Zavaleta, F.; Tlachino, F. J.; Toscano, J. J.
2009-01-01
Higgs mediated flavor violating electromagnetic interactions, induced at the one-loop level by a nondiagonal Hf i f j vertex, with f i and f j charged leptons or quarks, are studied within the context of a completely general effective Yukawa sector that comprises SU L (2)xU Y (1)-invariant operators of up to dimension-six. Exact formulae for the one-loop γf i f j and γγf i f j couplings are presented and their related processes used to study the phenomena of Higgs mediated lepton flavor violation. The experimental limit on the μ→eγ decay is used to derive a bound on the branching ratio of the μ→eγγ transition, which is 6 orders of magnitude stronger than the current experimental limit. Previous results on the τ→μγ and τ→μγγ decays are reproduced. The possibility of detecting signals of lepton flavor violation at γγ colliders is explored through the γγ→l i l j reaction, putting special emphasis on the τμ final state. Using the bound imposed on the Hτμ vertex by the current experimental data on the muon anomalous magnetic moment, it is found that about half a hundred events may be produced in the International Linear Collider.
1. Probing the Randall-Sundrum geometric origin of flavor with lepton flavor violation
International Nuclear Information System (INIS)
Agashe, Kaustubh; Blechman, Andrew E.; Petriello, Frank
2006-01-01
The anarchic Randall-Sundrum model of flavor is a low energy solution to both the electroweak hierarchy and flavor problems. Such models have a warped, compact extra dimension with the standard model fermions and gauge bosons living in the bulk, and the Higgs living on or near the TeV brane. In this paper we consider bounds on these models set by lepton flavor-violation constraints. We find that loop-induced decays of the form l→l ' γ are ultraviolet sensitive and incalculable when the Higgs field is localized on a four-dimensional brane; this drawback does not occur when the Higgs field propagates in the full five-dimensional space-time. We find constraints at the few TeV level throughout the natural range of parameters, arising from μ-e conversion in the presence of nuclei, rare μ decays, and rare τ decays. A tension exists between loop-induced dipole decays such as μ→eγ and tree-level processes such as μ-e conversion; they have opposite dependences on the five-dimensional Yukawa couplings, making it difficult to decouple flavor-violating effects. We emphasize the importance of the future experiments MEG and PRIME. These experiments will definitively test the Randall-Sundrum geometric origin of hierarchies in the lepton sector at the TeV scale
2. Lepton flavor violating non-standard interactions via light mediators
Energy Technology Data Exchange (ETDEWEB)
Farzan, Yasaman [School of physics, Institute for Research in Fundamental Sciences (IPM),P.O. Box 19395-5531, Tehran (Iran, Islamic Republic of); Shoemaker, Ian M. [Department of Physics, Department of Astronomy & Astrophysics,Center for Particle and Gravitational Astrophysics,The Pennsylvania State University, PA 16802 (United States)
2016-07-07
Non-Standard neutral current Interactions (NSIs) of neutrinos with matter can alter the pattern of neutrino oscillation due to the coherent forward scattering of neutrinos on the medium. This effect makes long-baseline neutrino experiments such as NOνA and DUNE a sensitive probe of beyond standard model (BSM) physics. We construct light mediator models that can give rise to both lepton flavor conserving as well as Lepton Flavor Violating (LFV) neutral current NSI. We outline the present phenomenological viability of these models and future prospects to test them. We predict a lower bound on Br(H→μτ) in terms of the parameters that can be measured by DUNE and NOνA, and show that the hint for H→μτ in current LHC data can be accommodated in our model. A large part of the parameter space of the model is already constrained by the bound on Br(τ→Z{sup ′}μ) and by the bounds on rare meson decays and can be in principle fully tested by improving these bounds.
3. A4 family symmetry and quark-lepton unification
International Nuclear Information System (INIS)
King, Stephen F.; Malinsky, Michal
2007-01-01
We present a model of quark and lepton masses and mixings based on A 4 family symmetry, a discrete subgroup of an SO(3) flavour symmetry, together with Pati-Salam unification. It accommodates tri-bimaximal neutrino mixing via constrained sequential dominance with a particularly simple vacuum alignment mechanism emerging through the effective D-term contributions to the scalar potential
4. More lepton flavor violating observables for LHCb's run 2
Directory of Open Access Journals (Sweden)
2016-09-01
Full Text Available The RK measurement by LHCb suggests non-standard lepton non-universality (LNU to occur in b→sℓ+ℓ− transitions, with effects in muons rather than electrons. A number of other measurements of b→sℓ+ℓ− transitions by LHCb and B-factories display disagreement with the SM predictions and, remarkably, these discrepancies are consistent in magnitude and sign with the RK effect. Non-standard LNU suggests non-standard lepton flavor violation (LFV as well, for example in B→Kℓℓ′ and Bs→ℓℓ′. There are good reasons to expect that the new effects may be larger for generations closer to the third one. In this case, the Bs→μe decay may be the most difficult to reach experimentally. We propose and study in detail the radiative counterpart of this decay, namely Bs→μeγ, whereby the chiral-suppression factor is replaced by a factor of order α/π. A measurement of this mode would be sensitive to the same physics as the purely leptonic LFV decay and, depending on experimental efficiencies, it may be more accessible. A realistic expectation is a factor of two improvement in statistics for either of the Bd,s modes.
5. Massive neutrinos flavor mixing of leptons and neutrino oscillations
CERN Document Server
2015-01-01
Since the discovery of neutrino oscillations neutrino physics has become an interesting field of research in physics. They imply that neutrino must have a small mass and that the neutrinos, coupled to the charged leptons, are mixtures of the mass eigenstates, analogous to the flavor mixing of the quarks. The mixing angles for the quarks are small, but for the leptons two of the mixing angles are large. The masses of the three neutrinos must be very small, less than 1 eV, but from the oscillation experiments we only know the mass differences — the absolute masses are still unknown. Also we do not know, if the masses of the neutrinos are Dirac masses, as the masses of the charged leptons and of the quarks, or whether they are Majorana masses. In this volume, an overview of the present state of research in neutrino physics is given by well-known experimentalists and theorists. The contents — originated from talks and discussions at a recent conference addressing some of the most pressing open questions in n...
6. The Symmetry behind Extended Flavour Democracy and Large Leptonic Mixing
CERN Document Server
Silva-Marcos, Joaquim I
2002-01-01
We show that there is a minimal discrete symmetry which leads to the extended flavour democracy scenario constraining the Dirac neutrino, the charged lepton and the Majorana neutrino mass term ($M_R$) to be all proportional to the democratic matrix, with all elements equal. In particular, this discrete symmetry forbids other large contributions to $M_R$, such as a term proportional to the unit matrix, which would normally be allowed by a $S_{3L}\\times S_{3R}$ permutation symmetry. This feature is crucial in order to obtain large leptonic mixing, without violating 't Hooft's, naturalness principle.
7. Quark-lepton flavor democracy and the nonexistence of the fourth generation
International Nuclear Information System (INIS)
Cvetic, G.; Kim, C.S.
1995-01-01
In the standard model with two Higgs doublets (type II), which has a consistent trend to a flavor gauge theory and its related flavor democracy in the quark and the leptonic sectors (unlike the minimal standard model) when the energy of the probes increases, we impose the mixed quark-lepton flavor democracy at high ''transition'' energy and assume the usual seesaw mechanism, and consequently find out that the existence of the fourth generation of fermions in this framework is practically ruled out
8. Applications of flavor symmetry to the phenomenology of elementary particles
International Nuclear Information System (INIS)
Kaeding, T.A.
1995-05-01
Some applications of flavor symmetry are examined. Approximate flavor symmetries and their consequences in the MSSM (Minimal Supersymmetric Standard Model) are considered, and found to give natural values for the possible B- and L-violating couplings that are empirically acceptable, except for the case of proton decay. The coupling constants of SU(3) are calculated and used to parameterize the decays of the D mesons in broken flavor SU(3). The resulting couplings are used to estimate the long-distance contributions to D-meson mixing
9. Lepton flavor violation and scalar dark matter in a radiative model of neutrino masses
Energy Technology Data Exchange (ETDEWEB)
Esch, Sonja; Klasen, Michael; Lamprea, David R. [Westfaelische Wilhelms-Universitaet Muenster, Institut fuer Theoretische Physik, Muenster (Germany); Yaguna, Carlos E. [Universidad Pedagogica y Tecnologica de Colombia, Escuela de Fisica, Tunja (Colombia)
2018-02-15
We consider a simple extension of the Standard Model that can account for the dark matter and explain the existence of neutrino masses. The model includes a vector-like doublet of SU(2), a singlet fermion, and two scalar singlets, all of them odd under a new Z{sub 2} symmetry. Neutrino masses are generated radiatively by one-loop processes involving the new fields, while the dark matter candidate is the lightest neutral particle among them. We focus specifically on the case where the dark matter particle is one of the scalars and its relic density is determined by its Yukawa interactions. The phenomenology of this setup, including neutrino masses, dark matter and lepton flavor violation, is analyzed in some detail. We find that the dark matter mass must be below 600 GeV to satisfy the relic density constraint. Lepton flavor violating processes are shown to provide the most promising way to test this scenario. Future μ → 3e and μ-e conversion experiments, in particular, have the potential to probe the entire viable parameter space of this model. (orig.)
10. Lepton-quark masses and democratic symmetry
International Nuclear Information System (INIS)
Fritzsch, H.
1995-01-01
It is shown that the simplest breaking of the subnuclear democracy leads to a successful description of the mixing between the second and third family. In the lepton channel the ν μ -ν τ oscillations are expected to be described by a mixing angle of 2.65 which might be observed soon in neutrino experiments. ((orig.))
11. Dynamical generation of a composite quark-lepton symmetry
International Nuclear Information System (INIS)
Yasue, Masaki.
1981-05-01
We demonstrate the possibility that a basic [SU(2)]sup(N) symmetry of N subconstituents, which describes particle and antiparticle transitions, generates at most an ''effective'' SO(2N) symmetry and at least an ''effective'' SU(N) x U(1) symmetry of composite quarks and leptons whose states are specified by the N different kinds of subconstituents. The generators of the ''effective'' symmetry, are identified by the correct algebraic properties specific to SO(2N) of composite operators constructed from the [SU(2)]sup(N)-operators acting on the composite quark-lepton states. The composite quarks and leptons are found to respect SO(4) x SO(6) or SU(2)sub(L) x U(1)sub(R) x SU(3)sub(c) x U(1)sub(B-L) according to a new selection rule, which are generated by the bilinear products of the raising and lowering operators of [SU(2)] 5 . This construction of the SO(4) x SO(6) generators allows us to uniquely define the five quantum numbers of that symmetry even at the subconstituent level. The full SO(10) generators can be also constructed; however, one needs a newly arranged [SU(2)] 5 symmetry only defined at the composite level, the generators of which turn out to be at most N body operators of the original [SU(2)] 5 . (author)
12. Flavor mixing via dynamical chiral symmetry breaking
International Nuclear Information System (INIS)
Jaffe, R.L.
1988-01-01
This paper is concerned with the physics of the quark gluon plasma. The author interested in the complexity of the flavor structure of hadron wavefunctions. This issue bears upon the validity of the quenched approximation in lattice gauge theory and the structure of the QCD vacuum, both of which have been central issues here
13. Lepton number violation, lepton flavor violation and non zero Θ_1_3 in LRSM
International Nuclear Information System (INIS)
Borgohain, Happy; Das, Mrinal Kumar
2017-01-01
We have done a phenomenological study of lepton number violation and lepton flavour violation in a generic left-right symmetric model (LRSM) considering broken ϻ-τ symmetry. The leading order TBM mass matrix originates from the type I (II) seesaw mechanism, whereas the perturbations to generate non-zero reactor mixing angle Θ_1_3, originates from the type II (I) seesaw mechanism. We studied the new physics contributions to neutrinoless double beta decay (NDBD) ignoring the left-right gauge boson mixing and the heavy-light neutrino mixing within the framework of left-right symmetric regime by considering the presence of both type I and type II seesaw. We assumed the mass of the gauge bosons and scalars to be around TeV and studied the effects of the new physics contributions on the effective mass and compared with the current experimental limit imposed by GERDA. We further extended our analysis by correlating the lepton flavour violation of the decay process, (ϻ→ 3e) with Θ_1_3. (author)
14. Implications of the Daya Bay observation of θ13 on the leptonic flavor mixing structure and CP violation
International Nuclear Information System (INIS)
Xing Zhizhong
2012-01-01
The Daya Bay collaboration has recently reported its first ν-bar e → ν-bar e oscillation result which points to θ 13 ≅ 8.8° ±0.8° (best-fit ±1φ range) or θ 13 ≠0° at the 5.2a level. The fact that this smallest neutrino mixing angle is not strongly suppressed motivates us to look into the underlying structure of lepton flavor mixing and CP violation. Two phenomenological strategies are outlined: (1) the lepton flavor mixing matrix U consists of a constant leading term U 0 and a small perturbation term ΔU; and (2) the mixing angles of U are associated with the lepton mass ratios. Some typical patterns of U 0 are reexamined by constraining their respective perturbations with current experimental data. We illustrate a few possible ways to minimally correct U 0 in order to fit the observed values of three mixing angles. We point out that the structure of U may exhibit an approximate μ-τ permutation symmetry in modulus, and reiterate the geometrical description of CP violation in terms of the leptonic unitarity triangles. The salient features of nine distinct parametrizations of U are summarized, and its Wolfenstein-like expansion is presented by taking U 0 to be the democratic mixing pattern. (author)
15. Lepton flavour symmetry and the neutrino magnetic moment
International Nuclear Information System (INIS)
Ecker, G.; Grimus, W.
1990-01-01
With the standard model gauge group and the three standard left-handed Weyl neutrinos, two minimal scenarios are investigated where an arbitrary non-abelian lepton flavour symmetry group G H is responsible for a light neutrino with a large magnetic moment. In the first case, with scalar fields carrying lepton flavour, some finetuning is necessary to get a small enough neutrino mass for μ ν = O(10 -11 μ B ). In the second scenario, the introduction of heavy charged gauge singlet fermions with lepton flavour allows for a strictly massless neutrino to one-loop order. In both cases, the interference mechanisms for small m ν and large μ ν is unique, independently of G H . In explicit realizations of the two scenarios, the horizontal groups are found to be non-abelian extensions of a Zeldovich-Konopinski-Mahmoud lepton number symmetry. Only a discrete part of G H is spontaneously broken leading to a light Dirac neutrino with a large magnetic moment. (Authors) 22 refs., 3 figs
16. New symmetries in heavy flavor physics
International Nuclear Information System (INIS)
Bjorken, J.D.
1990-06-01
Isgur and Wise have found that the formal limit M b , M c → ∞ leads to very great simplification in the general structure of the electroweak matrix elements of hadrons containing those quarks. In additions, interesting new symmetries appear in this limit. Their results are discussed, as well as some natural extensions to matrix elements of products of currents. 11 refs
17. A flavor dependent gauge symmetry, predictive radiative seesaw and LHCb anomalies
Directory of Open Access Journals (Sweden)
P. Ko
2017-09-01
Full Text Available We propose a predictive radiative seesaw model at one-loop level with a flavor dependent gauge symmetry U(1xB3−xe−μ+τ and Majorana fermion dark matter. For the neutrino mass matrix, we obtain an A1 type texture (with two zeros that provides us several predictions such as the normal ordering for the neutrino masses. We analyze the constraints from lepton flavor violations, relic density of dark matter, and collider physics for the new U(1xB3−xe−μ+τ gauge boson. Within the allowed region, the LHCb anomalies in B→K⁎μ+μ− and B→Kℓ+ℓ− with ℓ=e or μ can be resolved, and such Z′ could be also observed at the LHC.
18. Flavor origin of dark matter and its relation with leptonic nonzero θ{sub 13} and Dirac CP phase δ
Energy Technology Data Exchange (ETDEWEB)
Bhattacharya, Subhaditya; Karmakar, Biswajit [Department of Physics, Indian Institute of Technology Guwahati,781039 Assam (India); Sahu, Narendra [Department of Physics, Indian Institute of Technology,Hyderabad, Kandi, Sangareddy 502285, Medak, Telengana (India); Sil, Arunansu [Department of Physics, Indian Institute of Technology Guwahati,781039 Assam (India)
2017-05-12
We propose a minimal extension of the standard model by including a U(1) flavor symmetry to establish a correlation between the relic abundance of dark matter, measured by WMAP and PLANCK satellite experiments and non-zero value of sin θ{sub 13} observed at DOUBLE CHOOZ, Daya Bay, RENO and T2K. The flavour symmetry is allowed to be broken at a high scale to a remnant Z{sub 2} symmetry, which not only ensures the stability to the dark matter, but also gives rise to a modification to the existing A{sub 4}-based tri-bimaximal neutrino mixing. This deviation in turn suggests the required non-zero value of sin θ{sub 13}. We assume the dark matter to be neutral under the existing A{sub 4} symmetry while charged under the U(1) flavor symmetry. Hence in this set-up, the non-zero value of sin θ{sub 13} predicts the dark matter charge under U(1), which can be tested at various ongoing and future direct and collider dark matter search experiments. We also point out the involvement of nonzero leptonic CP phase δ, which plays an important role in the analysis.
19. S{sub 3} flavor symmetry and leptogenesis
Energy Technology Data Exchange (ETDEWEB)
Kubo, Jisuke [Kanazawa Univ., Dept. of Physics, Kanazawa, Ishikawa (Japan); Paschos, Emmanuel A [Institut fuer Physik, Universitaet Dortmund, Dortmund (Germany)
2004-12-01
It is found that a Majorana phase in a minimal S{sub 3} extension of the standard model with an additional Z{sub 2} symmetry in the leptonic sector is responsible for leptogenesis. We assume that the resonant enhancement of the CP asymmetries takes place to obtain a realistic size of baryon number asymmetry in the universe. We expect the masses of the right-handed neutrino are of O(10) TeV in this model. (author)
20. Lepton mixing predictions from Δ(6n2) family symmetry
International Nuclear Information System (INIS)
King, Stephen F.; Neder, Thomas; Stuart, Alexander J.
2013-01-01
We obtain predictions of lepton mixing parameters for direct models based on Δ(6n 2 ) family symmetry groups for arbitrarily large n in which the full Klein symmetry is identified as a subgroup of the family symmetry. After reviewing and developing the group theory associated with Δ(6n 2 ), we find many new candidates for large n able to yield reactor angle predictions within 3σ of recent global fits. We show that such Δ(6n 2 ) models with Majorana neutrinos predict trimaximal mixing with reactor angle θ 13 fixed up to a discrete choice, an oscillation phase of either zero or π and the atmospheric angle sum rules θ 23 =45°∓θ 13 /√(2), respectively, which are consistent with recent global fits and will be tested in the near future
1. Lepton mixing in A_5 family symmetry and generalized CP
International Nuclear Information System (INIS)
Li, Cai-Chang; Ding, Gui-Jun
2015-01-01
We study lepton mixing patterns which can be derived from the A_5 family symmetry and generalized CP. We find five phenomenologically interesting mixing patterns for which one column of the PMNS matrix is (√(((5+√5)/10)),(1/(√(5+√5))),(1/(√(5+√5))))"T (the first column of the golden ratio mixing), (√(((5−√5)/10)),(1/(√(5−√5))),(1/(√(5−√5))))"T (the second column of the golden ratio mixing), (1,1,1)"T/√3 or (√5+1,−2,√5−1)"T/4. The three lepton mixing angles are determined in terms of a single real parameter θ, and agreement with experimental data can be achieved for certain values of θ. The Dirac CP violating phase is predicted to be trivial or maximal while Majorana phases are trivial. We construct a supersymmetric model based on A_5 family symmetry and generalized CP. The lepton mixing is exactly the golden ratio pattern at leading order, and the mixing patterns of case III and case IV are reproduced after higher order corrections are considered.
2. Leptonic Dirac CP violation predictions from residual discrete symmetries
Directory of Open Access Journals (Sweden)
I. Girardi
2016-01-01
Full Text Available Assuming that the observed pattern of 3-neutrino mixing is related to the existence of a (lepton flavour symmetry, corresponding to a non-Abelian discrete symmetry group Gf, and that Gf is broken to specific residual symmetries Ge and Gν of the charged lepton and neutrino mass terms, we derive sum rules for the cosine of the Dirac phase δ of the neutrino mixing matrix U. The residual symmetries considered are: i Ge=Z2 and Gν=Zn, n>2 or Zn×Zm, n,m≥2; ii Ge=Zn, n>2 or Zn×Zm, n,m≥2 and Gν=Z2; iii Ge=Z2 and Gν=Z2; iv Ge is fully broken and Gν=Zn, n>2 or Zn×Zm, n,m≥2; and v Ge=Zn, n>2 or Zn×Zm, n,m≥2 and Gν is fully broken. For given Ge and Gν, the sum rules for cosδ thus derived are exact, within the approach employed, and are valid, in particular, for any Gf containing Ge and Gν as subgroups. We identify the cases when the value of cosδ cannot be determined, or cannot be uniquely determined, without making additional assumptions on unconstrained parameters. In a large class of cases considered the value of cosδ can be unambiguously predicted once the flavour symmetry Gf is fixed. We present predictions for cosδ in these cases for the flavour symmetry groups Gf=S4, A4, T′ and A5, requiring that the measured values of the 3-neutrino mixing parameters sin2θ12, sin2θ13 and sin2θ23, taking into account their respective 3σ uncertainties, are successfully reproduced.
3. A new mechanism for lepton-flavor violation. tau. yields. mu. gamma. in the flipped string
Energy Technology Data Exchange (ETDEWEB)
Kelley, S.; Lopez, J.L.; Nanopoulos, D.V.; Pois, H. (Texas A and M Univ., College Station, TX (USA). Center for Theoretical Physics Houston Advanced Research Center (HARC), The Woodlands, TX (USA). Astroparticle Physics Group)
1991-07-08
We explore a new mechanism for lepton-flavor violation which is manifest in the flipped SU(5) string model, and may be a generic feature of string-derived models. This mechanism generates off-diagnoal slepton masses from otherwise flavor diagonal Yukawa matrices when heavy vector-like leptons decouple at a high-mass scale. As an example of lepton-flavor violation, we present an order of magnitude prediction for the branching ratio BR({tau} {yields} {mu}{gamma}) in the flipped string. The result depends crucially on the details of the extra vector-like fermion decoupling, and on the assumed nature and scale of supersymmetry breaking. For natural choices of the parameters we obtain a large BR({tau} {yields} {mu}{gamma}), which we show to be well within the reach of present and future experimental searches. (orig.).
4. Non-leptonic weak decay of hadrons and chiral symmetry
International Nuclear Information System (INIS)
Suzuki, Katsuhiko
2000-01-01
We review the non-leptonic weak decay of hyperons and ΔI=1/2 rule with a special emphasis on the role of chiral symmetry. The soft-pion theorem provides a powerful framework to understand the origin of ΔI=1/2 rule qualitatively. However, quantitative description is still incomplete in any model of the hadrons. Naive chiral perturbation theory cannot explain the parity-conserving and violating amplitudes simultaneously, and convergence of the chiral expansion seems to be worse. We demonstrate how the non-leptonic weak decay amplitudes are sensitive to the quark-pair correlation in the baryons, and show the importance of the strong quark correlation in the spin-0 channel to reproduce the experimental data. We finally remark several related topics. (author)
5. Search for lepton-flavor and lepton-number-violating τ→ℓhh{sup ′} decay modes
Energy Technology Data Exchange (ETDEWEB)
Miyazaki, Y. [Graduate School of Science, Nagoya University, Nagoya (Japan); Hayasaka, K., E-mail: [email protected] [Kobayashi-Maskawa Institute, Nagoya University, Nagoya (Japan); Adachi, I. [High Energy Accelerator Research Organization (KEK), Tsukuba (Japan); Aihara, H. [Department of Physics, University of Tokyo, Tokyo (Japan); Asner, D.M. [Pacific Northwest National Laboratory, Richland, WA (United States); Aulchenko, V. [Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090 (Russian Federation); Aushev, T. [Institute for Theoretical and Experimental Physics, Moscow (Russian Federation); Bakich, A.M. [School of Physics, University of Sydney, NSW 2006 (Australia); Bay, A. [École Polytechnique Fédérale de Lausanne, EPFL, Lausanne (Switzerland); Bhardwaj, V. [Nara Women' s University, Nara (Japan); Bhuyan, B. [Indian Institute of Technology Guwahati, Guwahati (India); Bischofberger, M. [Nara Women' s University, Nara (Japan); Bozek, A. [H. Niewodniczanski Institute of Nuclear Physics, Krakow (Poland); Bračko, M. [University of Maribor, Maribor (Slovenia); J. Stefan Institute, Ljubljana (Slovenia); Browder, T.E. [University of Hawaii, Honolulu, HI (United States); Chang, M.-C. [Department of Physics, Fu Jen Catholic University, Taipei, Taiwan (China); Chen, A. [National Central University, Chung-li, Taiwan (China); Chen, P. [Department of Physics, National Taiwan University, Taipei, Taiwan (China); Cheon, B.G. [Hanyang University, Seoul (Korea, Republic of); Chistov, R. [Institute for Theoretical and Experimental Physics, Moscow (Russian Federation); and others
2013-02-26
We search for lepton-flavor and lepton-number-violating τ decays into a lepton (ℓ= electron or muon) and two charged mesons (h,h{sup ′}=π{sup ±} or K{sup ±}) using 854 fb{sup −1} of data collected with the Belle detector at the KEKB asymmetric-energy e{sup +}e{sup −} collider. We obtain 90% confidence level upper limits on the τ→ℓhh{sup ′} branching fractions in the range (2.0–8.4)×10{sup −8}. These results improve upon our previously published upper limits by factors of about 1.8 on average.
6. Lepton flavor changing processes and CP violation in the 331 model
Energy Technology Data Exchange (ETDEWEB)
Liu, J T [Texas A and M Univ., College Station, TX (United States). Center for Theoretical Physics; Ng, D
1994-01-01
By extending the electroweak gauge group to SU(3){sub L}x U(1){sub y}, the 331 model incorporates dilepton gauge bosons Y which do not respect individual lepton family number. We point out that, in addition to family diagonal couplings such as Y-e-e that change lepton family number by two units, dileptons may also have family nondiagonal couplings such as Y-{mu}-e. The latter coupling violates lepton family number by a single unit and manifests itself via lepton flavor changing decays such as {mu} {yields} e{gamma} and p --+ c-1. The family non-diagonal interaction can be CP violating and typically generates extremely large leptonic electric dipole moments. We demonstrate a natural mechanism for eliminating both single unit lepton flavor violation and large leptonic CP violation. Although we focus on the 331 model, our results are applicable to other dilepton models as well, including SU(15) grand unification. (author). 41 refs., 2 figs.
7. WHY COLOR-FLAVOR LOCKING IS JUST LIKE CHIRAL SYMMETRY BREAKING
International Nuclear Information System (INIS)
PISARSKI, R.D.; RISCHKE, D.H.
2000-01-01
The authors review how a classification into representations of color and flavor can be used to understand the possible patterns of symmetry breaking for color superconductivity in dense quark matter. In particular, the authors show how for three flavors, color-flavor locking is precisely analogous to the usual pattern of chiral symmetry breaking in the QCD vacuum
8. Flavor S4xZ2 symmetry and neutrino mixing
International Nuclear Information System (INIS)
Zhang He
2007-01-01
We present a model of the lepton masses and flavor mixing based on the discrete group S 4 xZ 2 . In this model, all the charged leptons and neutrinos are assigned to the 3 - b arα representation of S 4 in the Yamanouchi bases. The charged lepton and neutrino masses are mainly determined by the vacuum expectation value structures of the Higgs fields. A nearly tri-bimaximal lepton flavor mixing pattern, which is in agreement with the current experimental results, can be accommodated in our model. The neutrino mass spectrum takes the nearly degenerate pattern, and thus can be well tested in the future precise experiments
9. COMET and PRISM - Search for Charged Lepton Flavor Violation with Muons
Energy Technology Data Exchange (ETDEWEB)
Kuno, Yoshitaka [Department of Physics, Osaka University, Osaka, 560-0043 (Japan)
2012-04-15
The experiment (COMET) at J-PARC to search for a charged-lepton-flavor-violating process of muon to electron conversion in a muonic atom is described. Future prospects of an experiment (PRISM) with even higher sensitivity is mentioned. On-going R and D on a highly intense muon source (MuSIC) at Osaka University is presented.
10. COMET and PRISM - Search for Charged Lepton Flavor Violation with Muons
International Nuclear Information System (INIS)
Kuno, Yoshitaka
2012-01-01
The experiment (COMET) at J-PARC to search for a charged-lepton-flavor-violating process of muon to electron conversion in a muonic atom is described. Future prospects of an experiment (PRISM) with even higher sensitivity is mentioned. On-going R and D on a highly intense muon source (MuSIC) at Osaka University is presented.
11. Background estimation in e+e- annihilations with lepton flavor violation
International Nuclear Information System (INIS)
Angelescu, Tatiana; Ion, Mihai; Radu, Andrei
2005-01-01
We search for lepton flavor violating events in e + e - annihilations using the sample of events obtained with L3 detector at LEP2 for energies between 189 and 209 GeV. Selection criteria have been established to select two lepton events on a MC sample using SM program KORALZ. In the channel e + e - → ττ the background is given by the SM prediction with subsequent τ leptonic decay. The LFV signal can be evaluated in different supersymmetric models. In any case the contribution of the leptonic decay of taons in the LFV kinematic region is negligible. Another way to look at the background for LFV signal was to generate according to SM events in pairs and mix the events of two channels. This procedure should be based on a large MC sample which has not been available yet. (authors)
12. Charmless B→VP decays using flavor SU(3) symmetry
International Nuclear Information System (INIS)
Chiang Chengwei; Gronau, Michael; Luo Zumin; Rosner, Jonathan L.; Suprun, Denis A.
2004-01-01
The decays of B mesons to a charmless vector (V) and pseudoscalar (P) meson are analyzed within a framework of flavor SU(3) in which symmetry breaking is taken into account through ratios of decay constants in tree (T) amplitudes but exact SU(3) is assumed for color-suppressed and penguin amplitudes. The magnitudes and relative phases of tree and penguin amplitudes are extracted from data, the symmetry assumption is tested, and predictions are made for rates and CP asymmetries in as-yet-unseen decay modes. A key assumption for which we perform some tests and suggest others is a relation between penguin amplitudes in which the spectator quark is incorporated into either a pseudoscalar meson or a vector meson. Values of γ slightly restricting the range currently allowed by fits to other data are favored, but outside this range there remain acceptable solutions which cannot be excluded solely on the basis of present B→VP experiments
13. Minimal flavor violation in the lepton sector of the Randall-Sundrum model
International Nuclear Information System (INIS)
Chen Muchun; Yu Haibo
2009-01-01
We propose a realization of Minimal Flavor Violation in the lepton sector of the Randall-Sundrum model. With the MFV assumption, the only source of flavor violation are the 5D Yukawa couplings, and the usual two independent sources of flavor violation are related. In the limit of massless neutrinos, the bulk mass matrices and 5D Yukawa matrices are simultaneously diagonalized, and hence the absence of FCNCs. In the case of massive neutrinos, the contributions to FCNCs in the charged lepton sector are highly suppressed, due to the smallness of neutrino masses. In addition, the MFV assumption also allows suppressing one-loop charged current contributions to flavor changing processes by reducing the size of the Yukawa couplings, which is not possible in the generic anarchical case. We found that the first KK mass scale as low as ∼3 TeV can be allowed. In both cases, we present a set of numerical results that give rise to realistic lepton masses and mixing angles. Mild hierarchy in the 5D Yukawa matrix of O(25) in our numerical example is required to be consistent with two large and one small mixing angles. This tuning could be improved by having a more thorough search of the parameter space
14. Large Top-Quark Mass and Nonlinear Representation of Flavor Symmetry
International Nuclear Information System (INIS)
Feldmann, Thorsten; Mannel, Thomas
2008-01-01
We consider an effective theory (ET) approach to flavor-violating processes beyond the standard model, where the breaking of flavor symmetry is described by spurion fields whose low-energy vacuum expectation values are identified with the standard model Yukawa couplings. Insisting on canonical mass dimensions for the spurion fields, the large top-quark Yukawa coupling also implies a large expectation value for the associated spurion, which breaks part of the flavor symmetry already at the UV scale Λ of the ET. Below that scale, flavor symmetry in the ET is represented in a nonlinear way by introducing Goldstone modes for the partly broken flavor symmetry and spurion fields transforming under the residual symmetry. As a result, the dominance of certain flavor structures in rare quark decays can be understood in terms of the 1/Λ expansion in the ET
15. (S)fermion masses and lepton flavor violation. A democratic approach
International Nuclear Information System (INIS)
Hamaguchi, K.; Kakizaki, Mitsuru; Yamaguchi, Masahiro
2004-01-01
It is well-known that flavor mixing among the sfermion masses must be quite suppressed to survive various FCNC experimental bounds. One of the solutions to this supersymmetric FCNC problem is an alignment mechanism in which sfermion masses and fermion masses have some common origin and thus they are somehow aligned to each other. We propose a democratic approach to realize this idea, and illustrate how it has different predictions in slepton masses as well as lepton flavor violation from a more conventional minimal supergravity approach. This talk is based on our work in Ref. 1. (author)
16. Flavored gauge mediation with discrete non-Abelian symmetries
Science.gov (United States)
Everett, Lisa L.; Garon, Todd S.
2018-05-01
We explore the model building and phenomenology of flavored gauge-mediation models of supersymmetry breaking in which the electroweak Higgs doublets and the S U (2 ) messenger doublets are connected by a discrete non-Abelian symmetry. The embedding of the Higgs and messenger fields into representations of this non-Abelian Higgs-messenger symmetry results in specific relations between the Standard Model Yukawa couplings and the messenger-matter Yukawa interactions. Taking the concrete example of an S3 Higgs-messenger symmetry, we demonstrate that, while the minimal implementation of this scenario suffers from a severe μ /Bμ problem that is well known from ordinary gauge mediation, expanding the Higgs-messenger field content allows for the possibility that μ and Bμ can be separately tuned, allowing for the possibility of phenomenologically viable models of the soft supersymmetry-breaking terms. We construct toy examples of this type that are consistent with the observed 125 GeV Higgs boson mass.
17. The search for sleptons and flavor lepton number violation at LHC (CMS)
International Nuclear Information System (INIS)
Bityukov, S.I.; Krasnikov, N.V.
1999-01-01
A possibility to detect sleptons and flavor lepton number violation at LHC (CMS) is studied. The production and decays of right- and left-handed sleptons separately are investigated. It is found that for luminosity L=10 5 pb -1 it would be possible to discover right-handed sleptons with a mass up to 325 GeV and left-handed ones with a mass up to 360 GeV. A possibility to look for flavor lepton number violation in slepton decays due to the mixing of different sleptons generation is also investigated. It is found that for the maximal (μ-tilde R - e-tilde R ) mixing it is possible to detect such effect for sleptons with a mass up to 250 GeV [ru
18. The search for sleptons and lepton-flavor-number violation at LHC (CMS)
CERN Document Server
Bityukov, S I
1999-01-01
We study the possibility of detecting sleptons and lepton-flavor- number violation at LHC (CMS). We investigate the production and decays of right- and left-handed sleptons separately. We have found that, for the luminosity of L=10/sup 5/ pb/sup -1/, it would be possible to discover right-handed sleptons with a mass of up to 325 GeV and left-handed ones with a mass of up to 350 GeV. We also investigate the possibility of seeking lepton-flavor-number violation in slepton decays due to the mixing of different slepton generations. We find that, for the maximal ( mu /sub R/-e/sub R/) mixing, it is possible to detect such effect for sleptons with a mass up to 250 Ge V. (16 refs).
19. The search for sleptons and lepton-flavor-number violation at LHC (CMS)
International Nuclear Information System (INIS)
Bityukov, S.I.; Krasnikov, N.V.
1999-01-01
We study the possibility of detecting sleptons and lepton-flavor-number violation at LHC (CMS). We investigate the production and decays of right- and left-handed sleptons separately. We have found that, for the luminosity of L=10 5 pb -1 , it would be possible to discover right-handed sleptons with a mass of up to 325 GeV and left-handed ones with a mass of up to 350 GeV. We also investigate the possibility of seeking lepton-flavor-number violation in slepton decays due to the mixing of different slepton generations. We find that, for the maximal (μ-tilde R -e-tilde R ) mixing, it is possible to detect such effect for sleptons with a mass up to 250 GeV
20. Minimal lepton flavor violation implications of the b→s anomalies
Energy Technology Data Exchange (ETDEWEB)
Lee, Chao-Jung; Tandean, Jusak [Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 106, Taiwan (China)
2015-08-25
The latest measurements of rare b→s decays in the LHCb experiment have led to results in tension with the predictions of the standard model (SM), including a tentative indication of the violation of lepton flavor universality. Assuming that this situation will persist because of new physics, we explore some of the potential consequences in the context of the SM extended with the seesaw mechanism involving right-handed neutrinos plus effective dimension-six lepton-quark operators under the framework of minimal flavor violation. We focus on a couple of such operators which can accommodate the LHCb anomalies and conform to the minimal flavor violation hypothesis in both their lepton and quark parts. We examine specifically the lepton-flavor-violating decays B→K{sup (∗)}ℓℓ{sup ′}, B{sub s}→ϕℓℓ{sup ′}, B→(π,ρ)ℓℓ{sup ′}, and B{sub d,s}→ℓℓ{sup ′}, as well as K{sub L}→eμ and K→πeμ, induced by such operators. The estimated branching fractions of some of these decay modes with μτ in the final states are allowed by the pertinent experimental constraints to reach a few times 10{sup −7} if other operators do not yield competitive effects. We also look at the implications for B→K{sup (∗)}νν and K→πνν, finding that their rates can be a few times larger than their SM values. These results are testable in future experiments.
1. Supersymmetric type-III seesaw mechanism: Lepton flavor violation and LHC phenomenology
OpenAIRE
Hirsch, Martin; Porod, Werner; Staub, Florian; Weiss, Christian H.
2013-01-01
We study a supersymmetric version of the type-III seesaw mechanism considering two variants of the model: a minimal version for explaining neutrino data with only two copies of 24 superfields and a model with three generations of 24-plets. The latter predicts, in general, rates for mu -> e gamma inconsistent with experimental data. However, this bound can be evaded if certain special conditions within the neutrino sector are fulfilled. In the case of two 24-plets, lepton flavor violation cons...
2. A search for lepton-flavor-violating decays of the $Z$ boson into a $\\tau$-lepton and a light lepton with the ATLAS detector
CERN Document Server
2018-01-01
Direct searches for lepton flavor violation in decays of the $Z$ boson with the ATLAS detector at the LHC are presented. Decays of the $Z$ boson into an electron or muon and a hadronically decaying $\\tau$-lepton are considered. The searches are based on a data sample of proton--proton collisions collected by the ATLAS detector in 2015 and 2016, corresponding to an integrated luminosity of 36.1 fb$^{-1}$ at a center-of-mass energy of $\\sqrt{s} = 13$ TeV. No significant excess of events above the expected background is observed, and upper limits on the branching ratios of lepton-flavor-violating decays are set at the 95% confidence level: $\\mathcal{B} (Z\\to e\\tau) < 5.8 \\times 10^{-5}$ and $\\mathcal{B} (Z\\to \\mu\\tau) < 2.4 \\times 10^{-5}$. This is the first limit on $\\mathcal{B} (Z\\to e\\tau)$ with ATLAS data. The upper limit on $\\mathcal{B} (Z\\to \\mu\\tau)$ is combined with a previous ATLAS result based on 20.3 fb$^{-1}$ of proton--proton collision data at a center-of-mass energy of $\\sqrt{s} = 8$ TeV and ...
3. Lepton-flavor universality violation in R K and {R}_{D{_{(\\ast )}}} from warped space
Science.gov (United States)
Megías, Eugenio; Quirós, Mariano; Salas, Lindber
2017-07-01
Some anomalies in the processes b → sℓℓ ( ℓ = μ, e) and b\\to cℓ {\\overline{ν}}_{ℓ } ( ℓ = τ, μ, e), in particular in the observables R K and {R}_{D{_{(\\ast )}}} , have been found by the BaBar, LHCb and Belle collaborations, leading to a possible lepton flavor universality violation. If these anomalies were confirmed they would inevitably lead to physics beyond the Standard Model. In this paper we try to accommodate the present anomalies in an extra dimensional theory, solving the naturalness problem of the Standard Model by means of a warped metric with a strong conformality violation near the infra-red brane. The R K anomaly can be accommodated provided that the left-handed bottom quark and muon lepton have some degree of compositeness in the dual theory. The theory is consistent with all electroweak and flavor observables, and with all direct searches of Kaluza-Klein electroweak gauge bosons and gluons. The fermion spectrum, and fermion mixing angles, can be reproduced by mostly elementary right-handed bottom quarks, and tau and muon leptons. Moreover the {R}_{D{_{(\\ast )}}} anomaly requires a strong degree of compositeness for the left-handed tau leptons, which turns out to be in tension with experimental data on the {g}_{τ_L}^Z coupling, possibly unless some degree of fine-tuning is introduced in the fixing of the CKM matrix.
4. Minimal flavour violation in the quark and lepton sector and the impact of extra dimensions on flavour changing neutral currents and electroweak symmetry breaking
International Nuclear Information System (INIS)
Weiler, A.
2007-01-01
We study flavor-changing decays of hadrons and leptons and an extra-dimensional approach to electroweak symmetry breaking. Specifically we study the framework of Minimal Flavour Violation (MFV) as an explanation of the flavour problem. We discuss the impact of a specific extra-dimensional model of the MFV class on flavour changing neutral currents. We derive model-independent upper bounds on rare decays. -We discuss the extension of the MFV framework from the quark to the lepton sector and show how baryogenesis through leptogenesis can be achieved and examine if possible correlations with charged lepton flavour violation exist. We discuss the dynamical breaking of the electroweak symmetry in extra dimensions by unifying gauge and Higgs fields and we show that realistic models are possible once the extra dimension is strongly curved. (orig.)
5. Minimal flavour violation in the quark and lepton sector and the impact of extra dimensions on flavour changing neutral currents and electroweak symmetry breaking
Energy Technology Data Exchange (ETDEWEB)
Weiler, A.
2007-01-16
We study flavor-changing decays of hadrons and leptons and an extra-dimensional approach to electroweak symmetry breaking. Specifically we study the framework of Minimal Flavour Violation (MFV) as an explanation of the flavour problem. We discuss the impact of a specific extra-dimensional model of the MFV class on flavour changing neutral currents. We derive model-independent upper bounds on rare decays. -We discuss the extension of the MFV framework from the quark to the lepton sector and show how baryogenesis through leptogenesis can be achieved and examine if possible correlations with charged lepton flavour violation exist. We discuss the dynamical breaking of the electroweak symmetry in extra dimensions by unifying gauge and Higgs fields and we show that realistic models are possible once the extra dimension is strongly curved. (orig.)
6. A call for new physics: The muon anomalous magnetic moment and lepton flavor violation
Science.gov (United States)
Lindner, Manfred; Platscher, Moritz; Queiroz, Farinaldo S.
2018-02-01
We review how the muon anomalous magnetic moment (g - 2) and the quest for lepton flavor violation are intimately correlated. Indeed the decay μ → eγ is induced by the same amplitude for different choices of in- and outgoing leptons. In this work, we try to address some intriguing questions such as: Which hierarchy in the charged lepton sector one should have in order to reconcile possible signals coming simultaneously from g - 2and lepton flavor violation? What can we learn if the g - 2anomaly is confirmed by the upcoming flagship experiments at FERMILAB and J-PARC, and no signal is seen in the decay μ → eγin the foreseeable future? On the other hand, if the μ → eγdecay is seen in the upcoming years, do we need to necessarily observe a signal also in g - 2?. In this attempt, we generally study the correlation between these observables in a detailed analysis of simplified models. We derive master integrals and fully analytical and exact expressions for both phenomena, and address other flavor violating signals. We investigate under which conditions the observations can be made compatible and discuss their implications. Lastly, we discuss in this context several extensions of the SM, such as the Minimal Supersymmetric Standard Model, Left-Right symmetric model, B- L model, scotogenic model, two Higgs doublet model, Zee-Babu model, 331 model, and Lμ -Lτ, dark photon, seesaw models type I, II and III, and also address the interplay with μ → eee decay and μ- e conversion.
7. Search for doubly charged Higgs bosons with lepton-flavor-violating decays involving tau leptons.
Science.gov (United States)
Aaltonen, T; Adelman, J; Akimoto, T; Albrow, M G; Alvarez González, B; Amerio, S; Amidei, D; Anastassov, A; Annovi, A; Antos, J; Aoki, M; Apollinari, G; Apresyan, A; Arisawa, T; Artikov, A; Ashmanskas, W; Attal, A; Aurisano, A; Azfar, F; Azzi-Bacchetta, P; Azzurri, P; Bacchetta, N; Badgett, W; Barbaro-Galtieri, A; Barnes, V E; Barnett, B A; Baroiant, S; Bartsch, V; Bauer, G; Beauchemin, P-H; Bedeschi, F; Bednar, P; Behari, S; Bellettini, G; Bellinger, J; Belloni, A; Benjamin, D; Beretvas, A; Beringer, J; Berry, T; Bhatti, A; Binkley, M; Bisello, D; Bizjak, I; Blair, R E; Blocker, C; Blumenfeld, B; Bocci, A; Bodek, A; Boisvert, V; Bolla, G; Bolshov, A; Bortoletto, D; Boudreau, J; Boveia, A; Brau, B; Bridgeman, A; Brigliadori, L; Bromberg, C; Brubaker, E; Budagov, J; Budd, H S; Budd, S; Burkett, K; Busetto, G; Bussey, P; Buzatu, A; Byrum, K L; Cabrera, S; Campanelli, M; Campbell, M; Canelli, F; Canepa, A; Carlsmith, D; Carosi, R; Carrillo, S; Carron, S; Casal, B; Casarsa, M; Castro, A; Catastini, P; Cauz, D; Cavalli-Sforza, M; Cerri, A; Cerrito, L; Chang, S H; Chen, Y C; Chertok, M; Chiarelli, G; Chlachidze, G; Chlebana, F; Cho, K; Chokheli, D; Chou, J P; Choudalakis, G; Chuang, S H; Chung, K; Chung, W H; Chung, Y S; Ciobanu, C I; Ciocci, M A; Clark, A; Clark, D; Compostella, G; Convery, M E; Conway, J; Cooper, B; Copic, K; Cordelli, M; Cortiana, G; Crescioli, F; Cuenca Almenar, C; Cuevas, J; Culbertson, R; Cully, J C; Dagenhart, D; Datta, M; Davies, T; de Barbaro, P; De Cecco, S; Deisher, A; De Lentdecker, G; De Lorenzo, G; Dell'Orso, M; Demortier, L; Deng, J; Deninno, M; De Pedis, D; Derwent, P F; Di Giovanni, G P; Dionisi, C; Di Ruzza, B; Dittmann, J R; D'Onofrio, M; Donati, S; Dong, P; Donini, J; Dorigo, T; Dube, S; Efron, J; Erbacher, R; Errede, D; Errede, S; Eusebi, R; Fang, H C; Farrington, S; Fedorko, W T; Feild, R G; Feindt, M; Fernandez, J P; Ferrazza, C; Field, R; Flanagan, G; Forrest, R; Forrester, S; Franklin, M; Freeman, J C; Furic, I; Gallinaro, M; Galyardt, J; Garberson, F; Garcia, J E; Garfinkel, A F; Genser, K; Gerberich, H; Gerdes, D; Giagu, S; Giakoumopolou, V; Giannetti, P; Gibson, K; Gimmell, J L; Ginsburg, C M; Giokaris, N; Giordani, M; Giromini, P; Giunta, M; Glagolev, V; Glenzinski, D; Gold, M; Goldschmidt, N; Golossanov, A; Gomez, G; Gomez-Ceballos, G; Goncharov, M; González, O; Gorelov, I; Goshaw, A T; Goulianos, K; Gresele, A; Grinstein, S; Grosso-Pilcher, C; Grundler, U; Guimaraes da Costa, J; Gunay-Unalan, Z; Haber, C; Hahn, K; Hahn, S R; Halkiadakis, E; Hamilton, A; Han, B-Y; Han, J Y; Handler, R; Happacher, F; Hara, K; Hare, D; Hare, M; Harper, S; Harr, R F; Harris, R M; Hartz, M; Hatakeyama, K; Hauser, J; Hays, C; Heck, M; Heijboer, A; Heinemann, B; Heinrich, J; Henderson, C; Herndon, M; Heuser, J; Hewamanage, S; Hidas, D; Hill, C S; Hirschbuehl, D; Hocker, A; Hou, S; Houlden, M; Hsu, S-C; Huffman, B T; Hughes, R E; Husemann, U; Huston, J; Incandela, J; Introzzi, G; Iori, M; Ivanov, A; Iyutin, B; James, E; Jayatilaka, B; Jeans, D; Jeon, E J; Jindariani, S; Johnson, W; Jones, M; Joo, K K; Jun, S Y; Jung, J E; Junk, T R; Kamon, T; Kar, D; Karchin, P E; Kato, Y; Kephart, R; Kerzel, U; Khotilovich, V; Kilminster, B; Kim, D H; Kim, H S; Kim, J E; Kim, M J; Kim, S B; Kim, S H; Kim, Y K; Kimura, N; Kirsch, L; Klimenko, S; Klute, M; Knuteson, B; Ko, B R; Koay, S A; Kondo, K; Kong, D J; Konigsberg, J; Korytov, A; Kotwal, A V; Kraus, J; Kreps, M; Kroll, J; Krumnack, N; Kruse, M; Krutelyov, V; Kubo, T; Kuhlmann, S E; Kuhr, T; Kulkarni, N P; Kusakabe, Y; Kwang, S; Laasanen, A T; Lai, S; Lami, S; Lammel, S; Lancaster, M; Lander, R L; Lannon, K; Lath, A; Latino, G; Lazzizzera, I; LeCompte, T; Lee, J; Lee, J; Lee, Y J; Lee, S W; Lefèvre, R; Leonardo, N; Leone, S; Levy, S; Lewis, J D; Lin, C; Lin, C S; Linacre, J; Lindgren, M; Lipeles, E; Lister, A; Litvintsev, D O; Liu, T; Lockyer, N S; Loginov, A; Loreti, M; Lovas, L; Lu, R-S; Lucchesi, D; Lueck, J; Luci, C; Lujan, P; Lukens, P; Lungu, G; Lyons, L; Lys, J; Lysak, R; Lytken, E; Mack, P; MacQueen, D; Madrak, R; Maeshima, K; Makhoul, K; Maki, T; Maksimovic, P; Malde, S; Malik, S; Manca, G; Manousakis, A; Margaroli, F; Marino, C; Marino, C P; Martin, A; Martin, M; Martin, V; Martínez, M; Martínez-Ballarín, R; Maruyama, T; Mastrandrea, P; Masubuchi, T; Mattson, M E; Mazzanti, P; McFarland, K S; McIntyre, P; McNulty, R; Mehta, A; Mehtala, P; Menzemer, S; Menzione, A; Merkel, P; Mesropian, C; Messina, A; Miao, T; Miladinovic, N; Miles, J; Miller, R; Mills, C; Milnik, M; Mitra, A; Mitselmakher, G; Miyake, H; Moed, S; Moggi, N; Moon, C S; Moore, R; Morello, M; Movilla Fernandez, P; Mülmenstädt, J; Mukherjee, A; Muller, Th; Mumford, R; Murat, P; Mussini, M; Nachtman, J; Nagai, Y; Nagano, A; Naganoma, J; Nakamura, K; Nakano, I; Napier, A; Necula, V; Neu, C; Neubauer, M S; Nielsen, J; Nodulman, L; Norman, M; Norniella, O; Nurse, E; Oh, S H; Oh, Y D; Oksuzian, I; Okusawa, T; Oldeman, R; Orava, R; Osterberg, K; Pagan Griso, S; Pagliarone, C; Palencia, E; Papadimitriou, V; Papaikonomou, A; Paramonov, A A; Parks, B; Pashapour, S; Patrick, J; Pauletta, G; Paulini, M; Paus, C; Pellett, D E; Penzo, A; Phillips, T J; Piacentino, G; Piedra, J; Pinera, L; Pitts, K; Plager, C; Pondrom, L; Portell, X; Poukhov, O; Pounder, N; Prakoshyn, F; Pronko, A; Proudfoot, J; Ptohos, F; Punzi, G; Pursley, J; Rademacker, J; Rahaman, A; Ramakrishnan, V; Ranjan, N; Redondo, I; Reisert, B; Rekovic, V; Renton, P; Rescigno, M; Richter, S; Rimondi, F; Ristori, L; Robson, A; Rodrigo, T; Rogers, E; Rolli, S; Roser, R; Rossi, M; Rossin, R; Roy, P; Ruiz, A; Russ, J; Rusu, V; Saarikko, H; Safonov, A; Sakumoto, W K; Salamanna, G; Saltó, O; Santi, L; Sarkar, S; Sartori, L; Sato, K; Savoy-Navarro, A; Scheidle, T; Schlabach, P; Schmidt, E E; Schmidt, M A; Schmidt, M P; Schmitt, M; Schwarz, T; Scodellaro, L; Scott, A L; Scribano, A; Scuri, F; Sedov, A; Seidel, S; Seiya, Y; Semenov, A; Sexton-Kennedy, L; Sfyrla, A; Shalhout, S Z; Shapiro, M D; Shears, T; Shepard, P F; Sherman, D; Shimojima, M; Shochet, M; Shon, Y; Shreyber, I; Sidoti, A; Sinervo, P; Sisakyan, A; Slaughter, A J; Slaunwhite, J; Sliwa, K; Smith, J R; Snider, F D; Snihur, R; Soderberg, M; Soha, A; Somalwar, S; Sorin, V; Spalding, J; Spinella, F; Spreitzer, T; Squillacioti, P; Stanitzki, M; St Denis, R; Stelzer, B; Stelzer-Chilton, O; Stentz, D; Strologas, J; Stuart, D; Suh, J S; Sukhanov, A; Sun, H; Suslov, I; Suzuki, T; Taffard, A; Takashima, R; Takeuchi, Y; Tanaka, R; Tecchio, M; Teng, P K; Terashi, K; Thom, J; Thompson, A S; Thompson, G A; Thomson, E; Tipton, P; Tiwari, V; Tkaczyk, S; Toback, D; Tokar, S; Tollefson, K; Tomura, T; Tonelli, D; Torre, S; Torretta, D; Tourneur, S; Trischuk, W; Tu, Y; Turini, N; Ukegawa, F; Uozumi, S; Vallecorsa, S; van Remortel, N; Varganov, A; Vataga, E; Vázquez, F; Velev, G; Vellidis, C; Veszpremi, V; Vidal, M; Vidal, R; Vila, I; Vilar, R; Vine, T; Vogel, M; Volobouev, I; Volpi, G; Würthwein, F; Wagner, P; Wagner, R G; Wagner, R L; Wagner-Kuhr, J; Wagner, W; Wakisaka, T; Wallny, R; Wang, S M; Warburton, A; Waters, D; Weinberger, M; Wester, W C; Whitehouse, B; Whiteson, D; Wicklund, A B; Wicklund, E; Williams, G; Williams, H H; Wilson, P; Winer, B L; Wittich, P; Wolbers, S; Wolfe, C; Wright, T; Wu, X; Wynne, S M; Yagil, A; Yamamoto, K; Yamaoka, J; Yamashita, T; Yang, C; Yang, U K; Yang, Y C; Yao, W M; Yeh, G P; Yoh, J; Yorita, K; Yoshida, T; Yu, G B; Yu, I; Yu, S S; Yun, J C; Zanello, L; Zanetti, A; Zaw, I; Zhang, X; Zheng, Y; Zucchelli, S
2008-09-19
We search for pair production of doubly charged Higgs particles (H+/- +/-) followed by decays into electron-tau (etau) and muon-tau (mutau) pairs using data (350 pb(-1) collected from [over]pp collisions at sqrt[s]=1.96 TeV by the CDF II experiment. We search separately for cases where three or four final-state leptons are detected, and combine results for exclusive decays to left-handed etau (mutau) pairs. We set an H+/- +/- lower mass limit of 114(112) GeV/c(2) at the 95% confidence level.
8. Lepton-flavour violation in a Pati-Salam model with gauged flavour symmetry
Energy Technology Data Exchange (ETDEWEB)
Feldmann, Thorsten; Luhn, Christoph; Moch, Paul [Theoretische Physik 1, Naturwissenschaftlich-Technische Fakultät,Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen (Germany)
2016-11-11
Combining Pati-Salam (PS) and flavour symmetries in a renormalisable setup, we devise a scenario which produces realistic masses for the charged leptons. Flavour-symmetry breaking scalar fields in the adjoint representations of the PS gauge group are responsible for generating different flavour structures for up- and down-type quarks as well as for leptons. The model is characterised by new heavy fermions which mix with the Standard Model quarks and leptons. In particular, the partners for the third fermion generation induce sizeable sources of flavour violation. Focusing on the charged-lepton sector, we scrutinise the model with respect to its implications for lepton-flavour violating processes such as μ→eγ, μ→3e and muon conversion in nuclei.
9. Confronting lepton flavor universality violation in B decays with high-pT tau lepton searches at LHC
Directory of Open Access Journals (Sweden)
Darius A. Faroughy
2017-01-01
Full Text Available We confront the indications of lepton flavor universality (LFU violation observed in semi-tauonic B meson decays with new physics (NP searches using high pT tau leptons at the LHC. Using effective field theory arguments we correlate possible non-standard contributions to semi-tauonic charged currents with the τ+τ− signature at high energy hadron colliders. Several representative standard model extensions put forward to explain the anomaly are examined in detail: (i weak triplet of color-neutral vector resonances, (ii second Higgs doublet and (iii scalar or (iv vector leptoquark. We find that, in general, τ+τ− searches pose a serious challenge to NP explanations of the LFU anomaly. Recasting existing 8 TeV and 13 TeV LHC analyses, stringent limits are set on all considered simplified models. Future projections of the τ+τ− constraints as well as caveats in interpreting them within more elaborate models are also discussed.
10. Role of lepton flavor violating muon decay in the seesaw model and LSND
International Nuclear Information System (INIS)
Jamil Aslam, M.; Riazuddin
2002-01-01
The aim of this work is to study lepton flavor violation in a newly proposed seesaw model of neutrino mass and to see whether it could explain the Liquid Scintillation Neutrino Detector excess. The motivation of this seesaw model is that there is no new physics beyond the TeV scale. By studying μ→3e in this model, it is shown that the upper bound on the branching ratio requires a Higgs boson mass m h of a new scalar doublet with the lepton number L=-1 needed in this model to be about 9 TeV. The predicted branching ratio for μ→eν l ν-bar l is too small to explain the LSND
11. New physics with the lepton flavor violating decay τ →3 μ
Science.gov (United States)
Calcuttawala, Zaineb; Kundu, Anirban; Nandi, Soumitra; Patra, Sunando Kumar
2018-05-01
Lepton flavor violating (LFV) processes are a smoking gun signal of new physics (NP). If the semileptonic B decay anomalies are indeed due to some NP, such operators can potentially lead to LFV decays involving the second and the third generation leptons, like τ →3 μ . In this paper, we explore how far the nature of NP can be unraveled at the next generation B -factories like Belle-II, provided the decay τ →3 μ has been observed. We use four observables with which the differentiation among NP operators may be achieved to a high confidence level. Possible presence of multiple NP operators are also analyzed with the optimal observable technique. While the analysis can be improved even further if the final state muon polarizations are measured, we present this work as a motivational tool for the experimentalists, as well as a template for the analysis of similar processes.
12. arXiv Lepton flavor universality violation without new sources of quark flavor violation
CERN Document Server
Kamenik, Jernej F.; Zupan, Jure
2018-02-03
We show that new physics models without new flavor violating interactions can explain the recent anomalies in the b→sℓ+ℓ- transitions. The b→sℓ+ℓ- arises from a Z′ penguin which automatically predicts the V-A structure for the quark currents in the effective operators. This framework can either be realized in a renormalizable U(1)′ setup or be due to new strongly interacting dynamics. The dimuon resonance searches at the LHC are becoming sensitive to this scenario since the Z′ is relatively light, and could well be discovered in future searches by ATLAS and CMS.
13. Charged lepton flavor violation in a class of radiative neutrino mass generation models
Science.gov (United States)
Chowdhury, Talal Ahmed; Nasri, Salah
2018-04-01
We investigate the charged lepton flavor violating processes μ →e γ , μ →e e e ¯, and μ -e conversion in nuclei for a class of three-loop radiative neutrino mass generation models with electroweak multiplets of increasing order. We find that, because of certain cancellations among various one-loop diagrams which give the dipole and nondipole contributions in an effective μ e γ vertex and a Z-penguin contribution in an effective μ e Z vertex, the flavor violating processes μ →e γ and μ -e conversion in nuclei become highly suppressed compared to μ →e e e ¯ process. Therefore, the observation of such a pattern in LFV processes may reveal the radiative mechanism behind neutrino mass generation.
14. Proceedings of the 1st Workshop on Flavor Symmetries and Consequences in Accelerators and Cosmology
CERN Document Server
Meloni, D; Morisi, S; Pastor, S; Peinado, E; Valle, J W F; FLASY2011
2012-01-01
The main goals of the first "Workshop on FLAvor SYmmetries and consequences in accelerators and cosmology" (FLASY) was to summarize the theoretical status of flavor symmetries, bringing together young researchers in the field to stimulate discussions and new collaborations, with the aim of investigating possible new physics scenarios to be tested at the LHC, as well as in future neutrino, cosmology experiments and dark matter searches.
15. Family gauge symmetry as an origin of Koide's mass formula and charged lepton spectrum
International Nuclear Information System (INIS)
Sumino, Y.
2009-01-01
Koide's mass formula is an empirical relation among the charged lepton masses which holds with a striking precision. We present a model of charged lepton sector within an effective field theory with U(3) x SU(2) family gauge symmetry, which predicts Koide's formula within the present experimental accuracy. Radiative corrections as well as other corrections to Koide's mass formula have been taken into account. We adopt a known mechanism, through which the charged lepton spectrum is determined by the vacuum expectation value of a 9-component scalar field Φ. On the basis of this mechanism, we implement the following mechanisms into our model: (1) The radiative correction induced by family gauge interaction cancels the QED radiative correction to Koide's mass formula, assuming a scenario in which the U(3) family gauge symmetry and SU(2) L weak gauge symmetry are unified at 10 2 -10 3 TeV scale; (2) A simple potential of Φ invariant under U(3) x SU(2) leads to a realistic charged lepton spectrum, consistent with the experimental values, assuming that Koide's formula is protected; (3) Koide's formula is stabilized by embedding U(3) x SU(2) symmetry in a larger symmetry group. Formally fine tuning of parameters in the model is circumvented (apart from two exceptions) by appropriately connecting the charged lepton spectrum to the boundary (initial) conditions of the model at the cut-off scale. We also discuss some phenomenological implications.
16. Gauge origin of discrete flavor symmetries in heterotic orbifolds
Directory of Open Access Journals (Sweden)
Florian Beye
2014-09-01
Full Text Available We show that non-Abelian discrete symmetries in orbifold string models have a gauge origin. This can be understood when looking at the vicinity of a symmetry enhanced point in moduli space. At such an enhanced point, orbifold fixed points are characterized by an enhanced gauge symmetry. This gauge symmetry can be broken to a discrete subgroup by a nontrivial vacuum expectation value of the Kähler modulus T. Using this mechanism it is shown that the Δ(54 non-Abelian discrete symmetry group originates from a SU(3 gauge symmetry, whereas the D4 symmetry group is obtained from a SU(2 gauge symmetry.
17. Mass matrix ansatz and lepton flavor violation in the two-Higgs doublet model-III
International Nuclear Information System (INIS)
Diaz-Cruz, J.L.; Noriega-Papaqui, R.; Rosado, A.
2004-01-01
Predictive Higgs-boson-fermion couplings can be obtained when a specific texture for the fermion mass matrices is included in the general two-Higgs doublet model. We derive the form of these couplings in the charged lepton sector using a Hermitian mass matrix ansatz with four-texture zeros. The presence of unconstrained phases in the vertices φ i l i l j modifies the pattern of flavor-violating Higgs boson interactions. Bounds on the model parameters are obtained from present limits on rare lepton flavor-violating processes, which could be extended further by the search for the decay τ→μμμ and μ-e conversion at future experiments. The signal from Higgs boson decays φ i →τμ could be searched for at the CERN Large Hadron Collider, while e-μ transitions could produce a detectable signal at a future eμ collider, through the reaction e + μ - →h 0 →τ + τ -
18. Symplectic symmetry of the neutrino mass for many neutrino flavors
International Nuclear Information System (INIS)
Oeztuerk, N.; Ankara Univ.
2001-01-01
The algebraic structure of the neutrino mass Hamiltonian is presented for two neutrino flavors considering both Dirac and Majorana mass terms. It is shown that the algebra is Sp(8) and also discussed how the algebraic structure generalizes for the case of more than two neutrino flavors. (orig.)
19. Lepton flavor violating Higgs couplings and single production of the Higgs boson via eγ collision
International Nuclear Information System (INIS)
Yue, Chong-Xing; Pang, Cong; Guo, Yu-Chen
2015-01-01
Taking into account the constraints on the lepton flavor violation (LFV) couplings of the Standard Model (SM) Higgs boson H with leptons from low energy experiments and the recent Compact Muon Solenoid experiment results, we investigate production of the SM Higgs boson associated with a lepton τ via eγ collision at the International Linear Collider (ILC) and Large Hadron Electron Collider experiments. The production cross sections are calculated and the LFV signals and the relevant SM backgrounds are examined. The LFV signals of the SM Higgs boson might be observed via eγ collision in future ILC experiments. (paper)
20. Discrete quark-lepton symmetry need not pose a cosmological domain wall problem
International Nuclear Information System (INIS)
Lew, H.; Volkas, R.R.
1992-01-01
Quarks and leptons may be related to each other through a spontaneously broken discrete symmetry. Models with acceptable and interesting collider phenomenology have been constructed which incorporate this idea. However, the standard Hot Big Bang model of cosmology is generally considered to eschew spontaneously broken discrete symmetries because they often lead to the formation of unacceptably massive domain walls. It is pointed out that there are a number of plausible quark-lepton symmetric models in nature which do not produce cosmologically troublesome domain walls. 30 refs
1. Search for lepton flavor violation in ep collisions at 300 GeV center of mass energy
International Nuclear Information System (INIS)
Derrick, M.; Krakauer, D.; Magill, S.
1996-08-01
Using the ZEUS detector at the HERA electron-proton collider, we have searched for lepton flavor violation in ep collisions at a center-of-mass energy (√s) of 300 GeV. Events of the type e+p→l+X with a final-state lepton of high transverse momentum, l=μ or τ, were sought. No evidence was found for lepton flavor violation in the combined 1993 and 1994 data samples, for which the integrated luminosities were 0.84 pb -1 for e - p collisions and 2.94 pb -1 for e + p collisions. Limits on coupling vs. mass are provided for leptoquarks and R-parity violating squarks. For flavor violating couplings of electromagnetic strength, we set 95% confidence level lower limits on leptoquark masses between 207 GeV and 272 GeV, depending on the leptoquark species and final-state lepton. For leptoquark masses larger than 300 GeV, limits on flavor-changing couplings are determined, many of which supersede prior limits from rare decay processes. (orig.)
2. Full parameter scan of the Zee model: exploring Higgs lepton flavor violation
Energy Technology Data Exchange (ETDEWEB)
Herrero-García, Juan [ARC Center of Excellence for Particle Physics at the Terascale, University of Adelaide,Adelaide, SA 5005 (Australia); Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology,AlbaNova University Center, Roslagstullsbacken 21, 106 91 Stockholm (Sweden); Ohlsson, Tommy; Riad, Stella; Wirén, Jens [Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology,AlbaNova University Center, Roslagstullsbacken 21, 106 91 Stockholm (Sweden)
2017-04-21
We study the general Zee model, which includes an extra Higgs scalar doublet and a new singly-charged scalar singlet. Neutrino masses are generated at one-loop level, and in order to describe leptonic mixing, both the Standard Model and the extra Higgs scalar doublets need to couple to leptons (in a type-III two-Higgs doublet model), which necessarily generates large lepton flavor violating signals, also in Higgs decays. Imposing all relevant phenomenological constraints and performing a full numerical scan of the parameter space, we find that both normal and inverted neutrino mass orderings can be fitted, although the latter is disfavored with respect to the former. In fact, inverted ordering can only be accommodated if θ{sub 23} turns out to be in the first octant. A branching ratio for h→τμ of up to 10{sup −2} is allowed, but it could be as low as 10{sup −6}. In addition, if future expected sensitivities of τ→μγ are achieved, normal ordering can be almost completely tested. Also, μe conversion is expected to probe large parts of the parameter space, excluding completely inverted ordering if no signal is observed. Furthermore, non-standard neutrino interactions are found to be smaller than 10{sup −6}, which is well below future experimental sensitivity. Finally, the results of our scan indicate that the masses of the additional scalars have to be below 2.5 TeV, and typically they are lower than that and therefore within the reach of the LHC and future colliders.
3. Fermion mass hierarchy as a consequence of the spontaneous breakdown of the four-flavor symmetry
International Nuclear Information System (INIS)
Cveti, M.
1985-01-01
We study the fermion mass matrix in the case of four fermionic flavors u, d, c, and s. The original Lagrangian of the effective gauge theory respects the full four-flavor symmetry and fermions are massless. We analyze a vacuum expectation pattern of the elementary Higgs-field multiplet Phi/sub a/b [(a,b) = u,d,c,s]. Nonzero vacuum expectation values of Phi spontaneously break the original flavor symmetry with fermionic masses being directly proportional to these vacuum expectation values. In the Higgs potential, hard terms in Phi respect the global symmetry SU(4)/sub L/ x SU(4)/sub R/ of four flavors while soft terms in Psi break this symmetry down to the effective anomaly-free gauge group SU(2)/sub L//sup e/+μ x SU(2)/sub R//sup e/+μ. These soft terms are due to radiative as well as nonperturbative effects. Such a symmetry structure of the Higgs potential can be motivated by the underlying preonic dynamics. The desired solution, i.e., the proper interfamily and intrafamily hierarchy as well as the desired Cabibbo mixing angle, can emerge as a consequence of a subtle interplay between the soft terms and certain hard terms of the Higgs potential
4. Lepton family symmetries for neutrino masses and mixing
from the fact that any symmetry defined in the basis (νe,νµ,ντ ) is automatically applicable to ... Compare this first theory of everything to today's contender, i.e. string ... is dual to heterotic SO(32), Type IIA is dual to heterotic E8 × E8, and Type IIB.
5. The lepton flavor violating exclusive b bar → s bar ℓi- ℓj+ decays in SUSY without R-parity
Science.gov (United States)
Sheng, Jin-Huan; Song, Jia-Jia; Wang, Ru-Min; Yang, Ya-Dong
2018-05-01
Inspired by the recent anomaly measurements of the lepton-flavor violating decays h → μτ and the lepton flavor non-universality in decays b bar → s bar ℓ-ℓ+, we investigate the lepton flavor violating exclusive b bar → s bar ℓi- ℓj+ (i ≠ j and ℓ = e , μ , τ) decays within supersymmetry. Relevant R-parity violating couplings are constrained by using the latest experimental upper limits on the branching ratios of Bs → ℓi- ℓj+ and B →K (*) ℓi- ℓ j + flavor changing neutral current processes, and we find that all relevant branching ratios are very sensitive to the moduli of the squark and sneutrino exchange coupling products. In addition, the constrained lepton number violating effects on the dilepton invariant mass spectra, the single lepton polarization asymmetries and the differential forward-backward asymmetries are also studied. These lepton-flavor violating B decays could be used for the search of lepton flavor violation at the running LHC and the forthcoming Belle-II.
6. The lepton flavor violating exclusive b¯→s¯ℓi−ℓj+ decays in SUSY without R-parity
Directory of Open Access Journals (Sweden)
Jin-Huan Sheng
2018-05-01
Full Text Available Inspired by the recent anomaly measurements of the lepton-flavor violating decays h→μτ and the lepton flavor non-universality in decays b¯→s¯ℓ−ℓ+, we investigate the lepton flavor violating exclusive b¯→s¯ℓi−ℓj+(i≠j and ℓ=e,μ,τ decays within supersymmetry. Relevant R-parity violating couplings are constrained by using the latest experimental upper limits on the branching ratios of Bs→ℓi−ℓj+ and B→K(⁎ℓi−ℓj+ flavor changing neutral current processes, and we find that all relevant branching ratios are very sensitive to the moduli of the squark and sneutrino exchange coupling products. In addition, the constrained lepton number violating effects on the dilepton invariant mass spectra, the single lepton polarization asymmetries and the differential forward–backward asymmetries are also studied. These lepton-flavor violating B decays could be used for the search of lepton flavor violation at the running LHC and the forthcoming Belle-II.
7. Neutrino oscillations from warped flavor symmetry: Predictions for long baseline experiments T2K, NOvA, and DUNE
Science.gov (United States)
Pasquini, Pedro; Chuliá, Salvador Centelles; Valle, J. W. F.
2017-05-01
Here we study the pattern of neutrino oscillations emerging from a previously proposed warped standard model construction incorporating Δ (27 ) flavor symmetry [J. High Energy Phys. 01 (2016) 007, 10.1007/JHEP01(2016)007]. In addition to a complete description of fermion masses, the model predicts the lepton mixing matrix in terms of two parameters. The good measurement of θ13 makes these two parameters tightly correlated, leading to an approximate one-parameter description of neutrino oscillations. We find secondary minima for the C P phase absent in the general unconstrained oscillation scenario and determine the fourfold degenerate sharp correlation between the physical C P phase δC P and the atmospheric mixing angle θ23. This implies that maximal θ23 correlates with maximal leptonic C P violation. We perform a realistic estimate of the total neutrino and antineutrino event numbers expected at long baseline oscillation experiments T2K, NOvA, and the upcoming DUNE proposal. We show how an improved knowledge of the C P phase will probe the model in a significant way.
8. On the cosmic-ray spectra of three-body lepton-flavor-violating dark matter decays
International Nuclear Information System (INIS)
Carone, Christopher D.; Cukierman, Ari; Primulando, Reinard
2011-01-01
We consider possible leptonic three-body decays of spin-1/2, charge-asymmetric dark matter. Assuming a general Dirac structure for the four-fermion contact interactions of interest, we study the cosmic-ray electron and positron spectra and show that good fits to the current data can be obtained for both charged-lepton-flavor-conserving and flavor-violating decay channels. We find that different choices for the Dirac structure of the underlying decay operator can be significantly compensated by different choices for the dark matter mass and lifetime. The decay modes we consider provide differing predictions for the cosmic-ray positron fraction at energies higher than those currently probed at the PAMELA experiment; these predictions might be tested at cosmic-ray detectors like AMS-02.
9. Lepton flavor violating decays τ→lll and μ→eγ in the Higgs triplet model
International Nuclear Information System (INIS)
Akeroyd, A. G.; Aoki, Mayumi; Sugiyama, Hiroaki
2009-01-01
Singly and doubly charged Higgs bosons in the Higgs triplet model mediate the lepton flavor violating (LFV) decays τ→lll and μ→eγ. The lepton flavor violating decay rates are proportional to products of two triplet Yukawa couplings (h ij ) which can be expressed in terms of the parameters of the neutrino mass matrix and an unknown triplet vacuum expectation value. We determine the parameter space of the neutrino mass matrix in which a signal for τ→lll and/or μ→eγ is possible at ongoing and planned experiments. The conditions for respecting the stringent upper limit for μ→eee are studied in detail, with emphasis given to the possibility of |h ee |≅0, which can only be realized if Majorana phases are present.
10. Maximal neutrino mixing from a minimal flavor symmetry
International Nuclear Information System (INIS)
Aranda, Alfredo; Carone, Christopher D.; Lebed, Richard F.
2000-01-01
We study a number of models, based on a non-Abelian discrete group, that successfully reproduce the simple and predictive Yukawa textures usually associated with U(2) theories of flavor. These models allow for solutions to the solar and atmospheric neutrino problems that do not require altering successful predictions for the charged fermions or introducing sterile neutrinos. Although Yukawa matrices are hierarchical in the models we consider, the mixing between second- and third-generation neutrinos is naturally large. We first present a quantitative analysis of a minimal model proposed in earlier work, consisting of a global fit to fermion masses and mixing angles, including the most important renormalization group effects. We then propose two new variant models: The first reproduces all important features of the SU(5)xU(2) unified theory with neither SU(5) nor U(2). The second demonstrates that discrete subgroups of SU(2) can be used in constructing viable supersymmetric theories of flavor without scalar universality even though SU(2) by itself cannot. (c) 2000 The American Physical Society
11. Lepton flavor violating Higgs boson decays in seesaw models: New discussions
Directory of Open Access Journals (Sweden)
N.H. Thao
2017-08-01
Full Text Available The lepton flavor violating decay of the Standard Model-like Higgs boson (LFVHD, h→μτ, is discussed in seesaw models at the one-loop level. Based on particular analytic expressions of Passarino–Veltman functions, the two unitary and 't Hooft Feynman gauges are used to compute the branching ratio of LFVHD and compare with results reported recently. In the minimal seesaw (MSS model, the branching ratio was investigated in the whole valid range 10−9–1015 GeV of new neutrino mass scale mn6. Using the Casas–Ibarra parameterization, this branching ratio enhances with large and increasing mn6. But the maximal value can reach only order of 10−11. Interesting relations of LFVHD predicted by the MSS and inverse seesaw (ISS model are discussed. The ratio between two LFVHD branching ratios predicted by the ISS and MSS is simply mn62μX−2, where μX is the small neutrino mass scale in the ISS. The consistence between different calculations is shown precisely from analytical approach.
12. Lepton Flavorful Fifth Force and Depth-Dependent Neutrino Matter Interactions
Energy Technology Data Exchange (ETDEWEB)
Wise, Mark B. [Caltech; Zhang, Yue [Northwestern U.
2018-03-01
We consider a fifth force to be an interaction that couples to matter with a strength that grows with the number of atoms. In addition to competing with the strength of gravity a fifth force can give rise to violations of the equivalence principle. Current long range constraints on the strength and range of fifth forces are very impressive. Amongst possible fifth forces are those that couple to lepton flavorful charges $L_e-L_{\\mu}$ or $L_e-L_{\\tau}$. They have the property that their range and strength are also constrained by neutrino interactions with matter. In this brief note we review the existing constraints on the allowed parameter space in gauged $U(1)_{L_e-L_{\\mu}, L_{\\tau}}$. We find two regions where neutrino oscillation experiments are at the frontier of probing such a new force. In particular, there is an allowed range of parameter space where neutrino matter interactions relevant for long baseline oscillation experiments depend on the depth of the neutrino beam below the surface of the earth.
13. Leptonic CP violation theory
DEFF Research Database (Denmark)
Hagedorn, C.
2017-01-01
I summarize the status of theoretical predictions for the yet to be measured leptonic CP phases, the Dirac phase δ and the two Majorana phases α and β. I discuss different approaches based on: (a) a flavor symmetry without and with corrections, (b) different types of sum rules and (c) flavor and CP...... symmetries. I show their predictive power with examples. In addition, I present scenarios in which low and high energy CP phases are connected so that predictions for the CP phases α, β and δ become correlated to the sign of the baryon asymmetry YB of the Universe that is generated via leptogenesis....
14. Relativistic Hydrodynamics of Color-Flavor Locking Phase with Spontaneous Symmetry Breaking
Institute of Scientific and Technical Information of China (English)
ZHANG Sun; WANG Fan
2004-01-01
We study the hydrodynamics of color-flavor locking phase of three flavors of light quarks in high density QCD with spontaneous symmetry breaking. The basic hydrodynamic equations are presented based on the Poisson bracket method and the Goldstone phonon and the thermo phonon are compared. The dissipative equations are constructed in the frame of the first-order theory and all the transport coefficients are also defined, which could be looked on as the general case including the Landau's theory and the Eckart's theory
15. Lepton flavor violation from supersymmetric grand unified theories: Where do we stand for MEG, PRISM/PRIME, and a super flavor factory
International Nuclear Information System (INIS)
Calibbi, L.; Faccia, A.; Masiero, A.; Vempati, S. K.
2006-01-01
We analyze the complementarity between lepton flavor violation (LFV) and LHC experiments in probing the supersymmetric (SUSY) grand unified theories (GUT) when neutrinos get a mass via the seesaw mechanism. Our analysis is performed in an SO(10) framework, where at least one neutrino Yukawa coupling is necessarily as large as the top Yukawa coupling. Our study thoroughly takes into account the whole renormalization group running, including the GUT and the right-handed neutrino mass scales, as well as the running of the observable neutrino spectrum. We find that the upcoming (MEG, SuperKEKB) and future (PRISM/PRIME, super flavor factory) LFV experiments will be able to test such SUSY framework for SUSY masses to be explored at the LHC and, in some cases, even beyond the LHC sensitivity reach
16. $B$- and $D$-meson leptonic decay constants from four-flavor lattice QCD
Energy Technology Data Exchange (ETDEWEB)
Bazavov, A. [Michigan State U.; Bernard, C. [Washington U., St. Louis; Brown, N. [Washington U., St. Louis; Detar, C. [Utah U.; El-Khadra, A. X. [Fermilab; Gámiz, E. [Granada U., Theor. Phys. Astrophys.; Gottlieb, Steven [Indiana U.; Heller, U. M. [APS, New York; Komijani, J. [TUM-IAS, Munich; Kronfeld, A. S. [TUM-IAS, Munich; Laiho, J. [Syracuse U.; Mackenzie, P. B. [Fermilab; Neil, E. T. [RIKEN BNL; Simone, J. N. [Fermilab; Sugar, R. L. [UC, Santa Barbara; Toussaint, D. [Glasgow U.; Van De Water, R. S. [Fermilab
2017-12-26
We calculate the leptonic decay constants of heavy-light pseudoscalar mesons with charm and bottom quarks in lattice quantum chromodynamics on four-flavor QCD gauge-field configurations with dynamical $u$, $d$, $s$, and $c$ quarks. We analyze over twenty isospin-symmetric ensembles with six lattice spacings down to $a\\approx 0.03$~fm and several values of the light-quark mass down to the physical value $\\frac{1}{2}(m_u+m_d)$. We employ the highly-improved staggered-quark (HISQ) action for the sea and valence quarks; on the finest lattice spacings, discretization errors are sufficiently small that we can calculate the $B$-meson decay constants with the HISQ action for the first time directly at the physical $b$-quark mass. We obtain the most precise determinations to-date of the $D$- and $B$-meson decay constants and their ratios, $f_{D^+} = 212.6 (0.5)$~MeV, $f_{D_s} = 249.8(0.4)$~MeV, $f_{D_s}/f_{D^+} = 1.1749(11)$, $f_{B^+} = 189.4(1.4)$~MeV, $f_{B_s} = 230.7(1.2)$~MeV, $f_{B_s}/f_{B^+} = 1.2180(49)$, where the errors include statistical and all systematic uncertainties. Our results for the $B$-meson decay constants are three times more precise than the previous best lattice-QCD calculations, and bring the QCD errors in the Standard-Model predictions for the rare leptonic decays $\\overline{\\mathcal{B}}(B_s \\to \\mu^+\\mu^-) = 3.65(11) \\times 10^{-9}$, $\\overline{\\mathcal{B}}(B^0 \\to \\mu^+\\mu^-) = 1.00(3) \\times 10^{-11}$, and $\\overline{\\mathcal{B}}(B^0 \\to \\mu^+\\mu^-)/\\overline{\\mathcal{B}}(B_s \\to \\mu^+\\mu^-) = 0.00264(7)$ to well below other sources of uncertainty. As a byproduct of our analysis, we also update our previously published results for the light-quark-mass ratios and the scale-setting quantities $f_{p4s}$, $M_{p4s}$, and $R_{p4s}$. We obtain the most precise lattice-QCD determination to date of the ratio $f_{K^+}/f_{\\pi^+} = 1.1950(^{+15}_{-22})$~MeV.
17. Studies of discrete symmetries in a purely leptonic system using the Jagiellonian Positron Emission Tomograph
Directory of Open Access Journals (Sweden)
Moskal P.
2016-01-01
Full Text Available Discrete symmetries such as parity (P, charge-conjugation (C and time reversal (T are of fundamental importance in physics and cosmology. Breaking of charge conjugation symmetry (C and its combination with parity (CP constitute necessary conditions for the existence of the asymmetry between matter and antimatter in the observed Universe. The presently known sources of discrete symmetries violations can account for only a tiny fraction of the excess of matter over antimatter. So far CP and T symmetries violations were observed only for systems involving quarks and they were never reported for the purely leptonic objects. In this article we describe briefly an experimental proposal for the test of discrete symmetries in the decays of positronium atom which is made exclusively of leptons. The experiments are conducted by means of the Jagiellonian Positron Emission Tomograph (J-PET which is constructed from strips of plastic scintillators enabling registration of photons from the positronium annihilation. J-PET tomograph together with the positronium target system enable to measure expectation values for the discrete symmetries odd operators constructed from (i spin vector of the ortho-positronium atom, (ii momentum vectors of photons originating from the decay of positronium, and (iii linear polarization direction of annihilation photons. Linearly polarized positronium will be produced in the highly porous aerogel or polymer targets, exploiting longitudinally polarized positrons emitted by the sodium 22Na isotope. Information about the polarization vector of orthopositronium will be available on the event by event basis and will be reconstructed from the known position of the positron source and the reconstructed position of the orthopositronium annihilation. In 2016 the first tests and calibration runs are planned, and the data collection with high statistics will commence in the year 2017.
18. SU(4) flavor symmetry breaking in D-meson couplings to light hadrons
Energy Technology Data Exchange (ETDEWEB)
Fontoura, C.E. [Instituto Tecnologico da Aeronautica, DCTA, Sao Jose dos Campos, SP (Brazil); Universidade Estadual Paulista, Instituto de Fisica Teorica, Sao Paulo, SP (Brazil); Haidenbauer, J. [Institute for Advanced Simulation, Institut fuer Kernphysik, and Juelich Center for Hadron Physics, Forschungszentrum Juelich, Juelich (Germany); Krein, G. [Universidade Estadual Paulista, Instituto de Fisica Teorica, Sao Paulo, SP (Brazil)
2017-05-15
The validity of SU(4)-flavor symmetry relations of couplings of charmed D-mesons to light mesons and baryons is examined with the use of {sup 3}P{sub 0} quark-pair creation model and nonrelativistic quark-model wave functions. We focus on the three-meson couplings ππρ, KKρ and DDρ and baryon-baryon-meson couplings NNπ, NΛK and NΛ{sub c}D. It is found that SU(4)-flavor symmetry is broken at the level of 30% in the DDρ tree-meson couplings and 20% in the baryon-baryon-meson couplings. Consequences of these findings for DN cross sections and existence of bound states D-mesons in nuclei are discussed. (orig.)
19. Why most flavor-dependence predictions for nonleptonic charm decays are wrong: flavor symmetry and final-state interactions in nonleptonic decays of charmed hadrons
International Nuclear Information System (INIS)
Lipkin, H.J.
1980-09-01
Nonleptonic weak decays of strange hadrons are complicated by the interplay of weak and strong interactions. Models based either on symmetry properties or on the selection of certain types of diagrams are both open to criticism. The symmetries used are all broken in strong interactions, and the selection of some diagrams and neglect of others is never seriously justified. Furthermore, the number of related decays of strange hadrons is small, so that experimental data are insufficient for singificant tests of phenomenological models with a few free parameters. The discovery of charmed particles with many open channels for nonleptonic decays has provided a new impetus for a theoretical understanding of these processes. The GIM current provides a well defined weak hamiltonian, which can justifiably be used to first order. The QCD approach to strong interactions gives flavor-indpendent couplings and flavor symmetry broken only by quark masses. In a model with n generations of quarks and 2n flavors, a flavor symmetry group SU(2n) can be defined which is broken only by H/sub weak/ and the quark masses.Here again, the same two approaches by symmetry and dynamics have been used. But both types of treatment tend to consider only the symmetry properties or dominant diagrams of the weak interaction, including some subtle effects, while overlooking rather obvious effects of strong interactions
20. Measurement of WW and WZ production in the lepton plus heavy flavor jets final state at CDF
Energy Technology Data Exchange (ETDEWEB)
Leone, Sandra [Fermilab
2016-11-16
We present the CDF measurement of the diboson WW and WZ production cross section in a final state consistent with leptonic W decay and jets originating from heavy flavor quarks, based on the full Tevatron Run II dataset. The analysis of the di–jet invariant mass spectrum allows the observation of 3.7 sigma evidence for the combined production processes of either WW or WZ bosons. The different heavy flavor decay pattern of the W and Z bosons and the analysis of the secondary–decay vertex properties allow to independently measure the WW and WZ production cross section in a hadronic final state. The measured cross sections are consistent with the standard model predictions and correspond to signal significances of 2.9 and 2.1 sigma for WW and WZ production, respectively.
1. Exotic Higgs decays in a neutrino mass model with discrete S3 symmetry
CERN Document Server
Bhattacharyya, G; Päs, H
2010-01-01
Exotic Higgs decays can arise in lepton flavor models with horizontal symme- tries. We investigate the scalar sector of a neutrino mass model using an S3 family symmetry as an example. The model’s symmetry leads to an enlarged scalar sector with features that might be used to test the model experimentally, such as scalar particles with masses below 1 TeV and manifestly non-zero ma- trix elements for lepton flavor violating decays. We compare different decay channels of the scalars as well as leptonic processes that violate lepton flavor, in order to compare model predictions with experimental bounds.
2. Anomalous leptonic U(1) symmetry: Syndetic origin of the QCD axion, weak-scale dark matter, and radiative neutrino mass
Science.gov (United States)
Ma, Ernest; Restrepo, Diego; Zapata, Óscar
2018-01-01
The well-known leptonic U(1) symmetry of the Standard Model (SM) of quarks and leptons is extended to include a number of new fermions and scalars. The resulting theory has an invisible QCD axion (thereby solving the strong CP problem), a candidate for weak-scale dark matter (DM), as well as radiative neutrino masses. A possible key connection is a color-triplet scalar, which may be produced and detected at the Large Hadron Collider.
3. Supersymmetric models for quarks and leptons with nonlinearly realized E8 symmetry
International Nuclear Information System (INIS)
Ong, C.L.
1985-01-01
We propose three supersymmetric nonlinear sigma models with global symmetry E 8 . The models can accommodate three left-handed families of quarks and leptons without incurring the Adler-Bell-Jackiw anomaly with respect to either the standard SU(3) x SU(2) x U(1) gauge group, or the SU(5), or SO(10) grand unifying gauge group. They also predict unambiguously a right-handed, fourth family of quarks and leptons. In order to explore the structure of the models, we develop a differential-form formulation of the Kahler manifolds, resulting in general expressions for the curvature tensors and other geometrical objects in terms of the structure constants of the algebra, and the squashing parameters. These results, in turn, facilitate a general method for determining the Lagrangian to quartic order, and so the structure of the inherent four-fermion interactions of the models. We observe that the Kahlerian condition dω = 0 on the fundamental two-form ω greatly reduces the number of the independent squashing parameters. We also point out two plausible mechanisms for symmetry breaking, involving gravity
4. Electric dipole moments of charged leptons and lepton flavor violating interactions in the general two Higgs doublet model
International Nuclear Information System (INIS)
Iltan, E. O.
2001-01-01
We calculate the electric dipole moment of the electron using the experimental result of the muon electric dipole moment and upper limit of the BR(μ->eγ) in the framework of the general two Higgs doublet model. Our prediction is 10 -32 ecm, which lies in the experimental current limits. Further, we obtain constraints for the Yukawa couplings {bar ξ} N,τe D and {bar ξ} N,τμ D . Finally, we present an expression which connects the BR(τ->μγ) and the electric dipole moment of the τ lepton and study the relation between these physical quantities
5. Flavor-singlet baryons in the graded symmetry approach to partially quenched QCD
Science.gov (United States)
Hall, Jonathan M. M.; Leinweber, Derek B.
2016-11-01
Progress in the calculation of the electromagnetic properties of baryon excitations in lattice QCD presents new challenges in the determination of sea-quark loop contributions to matrix elements. A reliable estimation of the sea-quark loop contributions represents a pressing issue in the accurate comparison of lattice QCD results with experiment. In this article, an extension of the graded symmetry approach to partially quenched QCD is presented, which builds on previous theory by explicitly including flavor-singlet baryons in its construction. The formalism takes into account the interactions among both octet and singlet baryons, octet mesons, and their ghost counterparts; the latter enables the isolation of the quark-flow disconnected sea-quark loop contributions. The introduction of flavor-singlet states enables systematic studies of the internal structure of Λ -baryon excitations in lattice QCD, including the topical Λ (1405 ).
6. Flavor distributions in the nucleons: SU(2) sea asymmetry or isospin symmetry breaking?
International Nuclear Information System (INIS)
Ma, B.; Schaefer, A.; Greiner, W.
1993-01-01
The Gottfried sum-rule violation reported by the New Muon Collaboration was interpreted as an indication for a flavor asymmetry of the sea quark in the nucleon. We investigate the alternative possibility that isospin symmetry between the proton and the neutron is breaking for small x. We examine systematically the consequences of this possibility for several processes, namely, neutrino deep inelastic scattering, the charged pion Drell-Yan process, the proton Drell-Yan process, and semi-inclusive deep inelastic scattering, and conclude that a decision between the two alternative explanations is possible
7. CP violation in the lepton sector and implications for leptogenesis
DEFF Research Database (Denmark)
Hagedorn, C.; Mohapatra, R. N.; Molinaro, E.
2018-01-01
We review the current status of the data on neutrino masses and lepton mixing and the prospects for measuring the CP-violating phases in the lepton sector. The possible connection between low energy CP violation encoded in the Dirac and Majorana phases of the Pontecorvo-Maki-Nakagawa-Sakata mixing...... matrix and successful leptogenesis is emphasized in the context of seesaw extensions of the Standard Model with a flavor symmetry Gf (and CP symmetry)....
8. New fermion mass textures from anomalous U(1) symmetries with baryon and lepton number conservation
CERN Document Server
Leontaris, George K
2000-01-01
In this paper, we present solutions to the fermion mass hierarchy problem in the context of the minimal supersymmetric standard theory augmented by an anomalous family-dependent U(1)_X symmetry. The latter is spontaneously broken by non-zero vevs of a pair of singlet fields whose magnitude is determined through the D- and F-flatness conditions of the superpotential. We derive the general solutions to the anomaly cancellation conditions and show that they allow numerous choices for the U(1)_X fermion charges which give several fermion mass textures in agreement with the observed fermion mass hierarchy and mixing. Solutions with U(1)_X fermion charge assignments are found which forbid or substantially suppress the dangerous baryon and lepton number violating operators and the lepton-higgs mixing coupling while a higgs mixing mass classification of the fermion mass textures with respect to the sum of the doublet-higgs U(1)_X-charges and show that suppression of dimension-five operators naturally occurs for vario...
9. Search for the Lepton-Flavor-Violating Decays B-s(0) -> e(+/-)mu(-/+) and B-0 -> e(+/-)mu(-/+)
NARCIS (Netherlands)
Aaij, R.; Adeva, B.; Adinolfi, M.; Adrover, C.; Affolder, A.; Ajaltouni, Z.; Albrecht, J.; Alessio, F.; Alexander, M.; Ali, S.; Alkhazov, G.; Cartelle, P. Alvarez; Alves, A. A.; Amato, S.; Amerio, S.; Amhis, Y.; Anderlini, L.; Andreassen, R.; Andrews, J. E.; Appleby, R. B.; Gutierrez, O. Aquines; Archilli, F.; Artamonov, A.; Artuso, M.; Aslanides, E.; Auriemma, G.; Baalouch, M.; Bachmann, S.; Back, J. J.; Baesso, C.; Balagura, V.; Baldini, W.; Barlow, R. J.; Barschel, C.; Barsuk, S.; Barter, W.; Bauer, Th; Beddow, J.; Bedeschi, F.; Bediaga, I.; Belogurov, S.; Belous, K.; Belyaev, I.; Ben-Haim, E.; Bencivenni, G.; Benson, S.; Benton, J.; Berezhnoy, A.; Pellegrino, A.; Tolk, S.
2013-01-01
A search for the lepton-flavor-violating decays B-s(0) -> e(+/-)mu(-/+) and B-0 -> e(+/-)mu(-/+) is performed with a data sample, corresponding to an integrated luminosity of 1.0 fb(-1) of pp collisions at root s = 7 TeV, collected by the LHCb experiment. The observed number of B-s(0) ->
10. Search for the lepton flavor violating decay Z→eμ in pp collisions at √s = 8 TeV with the ATLAS detector
Czech Academy of Sciences Publication Activity Database
Aad, G.; Abbott, B.; Abdallah, J.; Böhm, Jan; Chudoba, Jiří; Havránek, Miroslav; Hejbal, Jiří; Jakoubek, Tomáš; Kepka, Oldřich; Kupčo, Alexander; Kůs, Vlastimil; Lokajíček, Miloš; Lysák, Roman; Marčišovský, Michal; Mikeštíková, Marcela; Němeček, Stanislav; Šícho, Petr; Staroba, Pavel; Svatoš, Michal; Taševský, Marek; Vrba, Václav
2014-01-01
Roč. 90, č. 7 (2014), "072010-1"-"072010-10" ISSN 1550-7998 R&D Projects: GA MŠk(CZ) LG13009 Institutional support: RVO:68378271 Keywords : lepton * flavor * Z0 * mass * scattering * enhancement * ATLAS * branching ratio * CERN LHC Coll * mass spectrum * topology Subject RIV: BF - Elementary Particles and High Energy Physics Impact factor: 4.643, year: 2014
11. Soccer in Indiana and models for non-leptonic decays of heavy flavors
International Nuclear Information System (INIS)
Bigi, I.I.
1989-01-01
Various descriptions of non-leptonic charm decays are reviewed and their relative strengths and weaknesses are listed. The author concludes that it is mainly (though not necessarily solely) a destructive interference in nonleptonic D + decays that shapes the decays of charm mesons. Some more subtle features in these decays are discussed in a preview of future research before he addresses the presently confused situation in D s decays. Finally, he gives a brief theoretical discussion of inclusive and exclusive non-leptonic decays of beauty mesons. 13 refs., 1 tab
12. An introduction to non-Abelian discrete symmetries for particle physicists
CERN Document Server
Ishimori, Hajime; Ohki, Hiroshi; Okada, Hiroshi; Shimizu, Yusuke; Tanimoto, Morimitsu
2012-01-01
These lecture notes provide a tutorial review of non-Abelian discrete groups and show some applications to issues in physics where discrete symmetries constitute an important principle for model building in particle physics. While Abelian discrete symmetries are often imposed in order to control couplings for particle physics - in particular model building beyond the standard model - non-Abelian discrete symmetries have been applied to understand the three-generation flavor structure in particular. Indeed, non-Abelian discrete symmetries are considered to be the most attractive choice for the flavor sector: model builders have tried to derive experimental values of quark and lepton masses, and mixing angles by assuming non-Abelian discrete flavor symmetries of quarks and leptons, yet, lepton mixing has already been intensively discussed in this context, as well. The possible origins of the non-Abelian discrete symmetry for flavors is another topic of interest, as they can arise from an underlying theory -...
13. Search for the Lepton Flavor Violating decays $B^0_{(s)} \\to \\tau^\\pm \\mu^\\mp$ at LHCb
CERN Multimedia
Arnau Romeu, Joan
2018-01-01
Lepton Flavor Violating (LFV) $B$ decays, such as $B^{0}_{(s)}\\rightarrow\\tau^{+}\\mu^{-}$, are forbidden in the Standard Model (SM) in the absence of non-zero neutrino masses, but can occur via one-loop diagrams if neutrino oscillations are included. This implies that the rate of these kind of processes is very suppressed, beyond the current and future experimental sensitivities. However, a wide variety of New Physics scenarios predict dramatically higher rates for these processes. The poster presents the search for the LFV decays $B^{0}_{(s)}\\rightarrow\\tau^{+}\\mu^{-}$ using 2011 and 2012 data of the LHCb experiment. The studied $\\tau$ decay mode is $\\tau\\rightarrow\\pi^{+}\\pi^{-}\\pi^{+}\ 14. Two-flavor QCD correction to lepton magnetic moments at leading-order in the electromagnetic coupling Energy Technology Data Exchange (ETDEWEB) Feng, Xu [DESY, Zeuthen (Germany). NIC; Muenster Univ. (Germany). Inst. fuer Theoretische Physik; Jansen, Karl; Renner, Dru B. [DESY, Zeuthen (Germany). NIC; Petschlies, Marcus [Humboldt Univ. Berlin (Germany). Inst. fuer Physik 2011-03-15 We present a reliable nonperturbative calculation of the QCD correction, at leading-order in the electromagnetic coupling, to the anomalous magnetic moment of the electron, muon and tau leptons using two-flavor lattice QCD. We use multiple lattice spacings, multiple volumes and a broad range of quark masses to control the continuum, in nite-volume and chiral limits. We examine the impact of the commonly ignored disconnected diagrams and introduce a modi cation to the previously used method that results in a well-controlled lattice calculation. We obtain 1.513(43).10{sup -12}, 5.72(16).10{sup -8} and 2.650(54).10{sup -6} for the leading-order QCD correction to the anomalous magnetic moment of the electron, muon and tau respectively, each accurate to better than 3%. (orig.) 15. Two-flavor QCD correction to lepton magnetic moments at leading-order in the electromagnetic coupling Energy Technology Data Exchange (ETDEWEB) Dru Renner, Xu Feng, Karl Jansen, Marcus Petschlies 2011-08-01 We present a reliable nonperturbative calculation of the QCD correction, at leading-order in the electromagnetic coupling, to the anomalous magnetic moment of the electron, muon and tau leptons using two-flavor lattice QCD. We use multiple lattice spacings, multiple volumes and a broad range of quark masses to control the continuum, infinite-volume and chiral limits. We examine the impact of the commonly ignored disconnected diagrams and introduce a modification to the previously used method that results in a well-controlled lattice calculation. We obtain 1.513 (43) 10^-12, 5.72 (16) 10^-8 and 2.650 (54) 10^-6 for the leading-order QCD correction to the anomalous magnetic moment of the electron, muon and tau respectively, each accurate to better than 3%. 16. Feasibility study of a high-performance LaBr{sub 3}(Ce) calorimeter for future lepton flavor violation experiments Energy Technology Data Exchange (ETDEWEB) Papa, A., E-mail: [email protected] [Paul Scherrer Institut PSI, CH-5232 Villigen (Switzerland); De Gerone, M. [INFN Sezione di Genova, Largo Dodecaneso 33, 16146 Italy (Italy); Dussoni, S. [INFN Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa (Italy); Galli, L. [Paul Scherrer Institut PSI, CH-5232 Villigen (Switzerland); INFN Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa (Italy); Nicolò, D. [INFN Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa (Italy); Dipartimento di Fisica dell' Università degli studi di Pisa, Largo B. Pontecorvo 3, 56127 Pisa (Italy); Signorelli, G. [INFN Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa (Italy) 2014-03-01 LaBr{sub 3}(Ce) is a very attractive material due to its ultra high light output and its fast response, resulting in a good candidate as a crystal for a calorimeter able to provide simultaneously very high energy and timing performances. We report here a first test with a cylindrical 3{sup ″}×3{sup ″} LaBr{sub 3}(Ce) crystal coupled to PMT (Photonics XP53A2B), where we explore the detector performances at relative high energies, on the region of interest for future charged Lepton Flavor Violation (cLFV) experiments, using photons in the interval of 55 ÷ 83 MeV from π{sup 0} decays up to 129 MeV from the radiative capture of negative pions on protons. 17. Lepton Flavor Violation in the Two Higgs Doublet Model using g-2 muon factor International Nuclear Information System (INIS) Diaz, Rodolfo A.; Martinez, R.; Rodriguez, J.-Alexis; Tuiran, E. 2002-01-01 Current experimental data from the g-2 muon factor, seems to show the necessity of physics beyond the Standard Model (SM), since the difference between SM and experimental predictions is approximately 2.6σ. In the framework of the General Two Higgs Doublet Model (2HDM), we calculate the muon anomalous magnetic moment to get lower and upper bounds for the Flavour Changing (FC) Yukawa couplings in the leptonic sector 18. Patterns of flavor signals in supersymmetric models Energy Technology Data Exchange (ETDEWEB) Goto, T. [KEK National High Energy Physics, Tsukuba (Japan)]|[Kyoto Univ. (Japan). YITP; Okada, Y. [KEK National High Energy Physics, Tsukuba (Japan)]|[Graduate Univ. for Advanced Studies, Tsukuba (Japan). Dept. of Particle and Nucelar Physics; Shindou, T. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)]|[International School for Advanced Studies, Trieste (Italy); Tanaka, M. [Osaka Univ., Toyonaka (Japan). Dept. of Physics 2007-11-15 Quark and lepton flavor signals are studied in four supersymmetric models, namely the minimal supergravity model, the minimal supersymmetric standard model with right-handed neutrinos, SU(5) supersymmetric grand unified theory with right-handed neutrinos and the minimal supersymmetric standard model with U(2) flavor symmetry. We calculate b{yields}s(d) transition observables in B{sub d} and B{sub s} decays, taking the constraint from the B{sub s}- anti B{sub s} mixing recently observed at Tevatron into account. We also calculate lepton flavor violating processes {mu} {yields} e{gamma}, {tau} {yields} {mu}{gamma} and {tau} {yields} e{gamma} for the models with right-handed neutrinos. We investigate possibilities to distinguish the flavor structure of the supersymmetry breaking sector with use of patterns of various flavor signals which are expected to be measured in experiments such as MEG, LHCb and a future Super B Factory. (orig.) 19. Patterns of flavor signals in supersymmetric models International Nuclear Information System (INIS) Goto, T.; Tanaka, M. 2007-11-01 Quark and lepton flavor signals are studied in four supersymmetric models, namely the minimal supergravity model, the minimal supersymmetric standard model with right-handed neutrinos, SU(5) supersymmetric grand unified theory with right-handed neutrinos and the minimal supersymmetric standard model with U(2) flavor symmetry. We calculate b→s(d) transition observables in B d and B s decays, taking the constraint from the B s - anti B s mixing recently observed at Tevatron into account. We also calculate lepton flavor violating processes μ → eγ, τ → μγ and τ → eγ for the models with right-handed neutrinos. We investigate possibilities to distinguish the flavor structure of the supersymmetry breaking sector with use of patterns of various flavor signals which are expected to be measured in experiments such as MEG, LHCb and a future Super B Factory. (orig.) 20. Localization and chiral symmetry in 2+1 flavor domain wall QCD Energy Technology Data Exchange (ETDEWEB) David J. Antonio; Kenneth C. Bowler; Peter A. Boyle; Norman H. Christ; Michael A. Clark; Saul D. Cohen; Chris Dawson; Alistair Hart; Balint Joó; Chulwoo Jung; Richard D. Kenway; Shu Li; Meifeng Lin; Robert D. Mawhinney; Christopher M. Maynard; Shigemi Ohta; Robert J. Tweedie; Azusa Yamaguchi 2008-01-01 We present results for the dependence of the residual mass of domain wall fermions (DWF) on the size of the fifth dimension and its relation to the density and localization properties of low-lying eigenvectors of the corresponding hermitian Wilson Dirac operator relevant to simulations of 2+1 flavor domain wall QCD. Using the DBW2 and Iwasaki gauge actions, we generate ensembles of configurations with a$16^3\\times 32$space-time volume and an extent of 8 in the fifth dimension for the sea quarks. We demonstrate the existence of a regime where the degree of locality, the size of chiral symmetry breaking and the rate of topology change can be acceptable for inverse lattice spacings$a^{-1} \\ge 1.6$GeV. 1. The masses of leptons and quarks - signals of a new symmetry International Nuclear Information System (INIS) Fritzsch, H.; Max-Planck-Institut fuer Physik, Muenchen 1993-01-01 Neutrinos are massless, charged leptons light due to the presence of a degenerate heavy family. The masses of the light leptons imply nontrivial mixing effects, leading to a slight non universality in the τ-channel, neutrino conversion in the ν μ -ν τ -system and radiative decays of the charged leptons. (author). 13 refs 2. Flavored model building International Nuclear Information System (INIS) Hagedorn, C. 2008-01-01 In this thesis we discuss possibilities to solve the family replication problem and to understand the observed strong hierarchy among the fermion masses and the diverse mixing pattern of quarks and leptons. We show that non-abelian discrete symmetries which act non-trivially in generation space can serve as profound explanation. We present three low energy models with the permutation symmetry S 4 , the dihedral group D 5 and the double-valued group T' as flavor symmetry. The T' model turns out to be very predictive, since it explains tri-bimaximal mixing in the lepton sector and, moreover, leads to two non-trivial relations in the quark sector, √((m d )/(m s ))= vertical stroke V us vertical stroke and √((m d )/(m s ))= vertical stroke (V td )/(V ts ) vertical stroke. The main message of the T' model is the observation that the diverse pattern in the quark and lepton mixings can be well-understood, if the flavor symmetry is not broken in an arbitrary way, but only to residual (non-trivial) subgroups. Apart from leading to deeper insights into the origin of the fermion mixings this idea enables us to perform systematic studies of large classes of discrete groups. This we show in our study of dihedral symmetries D n and D' n . As a result we find only five distinct (Dirac) mass matrix structures arising from a dihedral group, if we additionally require partial unification of either left-handed or left-handed conjugate fermions and the determinant of the mass matrix to be non-vanishing. Furthermore, we reveal the ability of dihedral groups to predict the Cabibbo angle θ C , i.e. vertical stroke V us(cd) vertical stroke cos((3π)/(7)), as well as maximal atmospheric mixing, θ 23 =(π)/(4), and vanishing θ 13 in the lepton sector. (orig.) 3. New initiatives on lepton flavor violation and neutrino oscillation with high intense muon and neutrino sources CERN Document Server Kuno, Yoshitaka; Pakvasa, Sandip 2002-01-01 The area of physics involving muons and neutrinos has become exciting in particle physics. Using their high intensity sources, physicists undertake, in various ways, extensive searches for new physics beyond the Standard Model, such as tests of supersymmetric grand unification (SUSY-GUT) and precision measurements of the muon and neutrino properties, which will in future extend to ambitious studies such as determination of the three-generation neutrino mixing matrix elements and CP violation in the lepton sector. The physics of this field is advancing, with potential improvements of the source 4. Non-leptonic weak decay rate of explicitly flavored heavy mesons International Nuclear Information System (INIS) Suzuki, M.; California Univ., Berkeley 1981-01-01 It is argued quantitatively that a large difference between the D 0 and D + lifetimes is mainly due to non-perturbative long-distance effects. The total non-leptonic weak decay rates are related to the soft limit of short-distance processes. Scaling laws for the decay rates of heavy mesons with respect to mass are inferred from the QCD analysis of the soft limit of fragmentation. It is found that the decay rates are not determined by the disconnected spectator diagrams alone even in the limit of the heavy quark mass M Going to infinity ( 5 exp √ c log M. Some numerical discussion is made for the decay of B mesons and T mesons. (orig.) 5. Palatable leptoquark scenarios for lepton flavor violation in exclusive b→sℓ{sub 1}ℓ{sub 2} modes Energy Technology Data Exchange (ETDEWEB) Bečirević, D. [Laboratoire de Physique Théorique (Bât. 210),CNRS and University Paris-Sud, Université Paris-Saclay, 91405 Orsay cedex (France); Košnik, N. [Departement of Physics, University of Ljubljana,Jadranska 19, 1000 Ljubljana (Slovenia); Jožef Stefan Institute,Jamova 39, P.O. Box 3000, 1001 Ljubljana (Slovenia); Sumensari, O. [Laboratoire de Physique Théorique (Bât. 210),CNRS and University Paris-Sud, Université Paris-Saclay, 91405 Orsay cedex (France); Instituto de Física, Universidade de São Paulo,C.P. 66.318, 05315-970 São Paulo (Brazil); Funchal, R. Zukanovich [Instituto de Física, Universidade de São Paulo,C.P. 66.318, 05315-970 São Paulo (Brazil) 2016-11-07 We examine various scenarios that involve a light O(1 TeV) leptoquark state and select those which are compatible with the current experimental values for B(B{sub s}→μμ), B(B→Kμμ){sub large−q{sup 2}}, R{sub K}=B{sup ′}(B→Kμμ)/B{sup ′}(B→Kee), and which lead to predictions consistent with other experimental data. We show that two such scenarios are phenomenologically plausible, namely the one with a doublet of scalar leptoquarks of hypercharge 1/6, and the one with a triplet of vector leptoquarks of hypercharge 2/3. We also argue that a model with a singlet scalar leptoquark of hypercharge 1/3 is not viable. Using the present experimental data as constraints, it is shown that the exclusive lepton flavor violating decays, B(B{sub s}→μτ), B(B→Kμτ) and B(B→K{sup ∗}μτ), can be as large as O(10{sup −5}). 6. A possible solution of the flavor problem and radiative neutrino masses International Nuclear Information System (INIS) Adulpravitchai, Adisorn 2010-01-01 In this thesis, we discuss two important problems of the Standard Model of Particle Physics (SM), namely the flavor problem and the reason for the smallness of neutrino masses. The first one might be related to the origin of non-abelian discrete flavor symmetries. We discuss the possibility of obtaining them from an underlying continuous flavor symmetry, i.e. SU(2) or SU(3) through spontaneous symmetry breaking. Moreover, we investigate their possible origin from an orbifold compactification. We discuss all non-abelian discrete symmetries, which can arise from an orbifold T 2 /Z N . They are A 4 , S 4 , D 4 , D 3 , and D 6 . We present the idea of combining the breaking of an orbifold GUT and the flavor symmetry arising from the orbifold. We demonstrate the construction in a 6d SUSY SO(10) x S 4 . For the second problem, we propose a one-loop neutrino mass model in the left-right symmetric framework. We observe the transmitted hierarchy from the charged lepton masses to the right-handed neutrino masses, which we call ''Radiative Transmission of Lepton Flavor Hierarchy''. Finally, we study the phenomenological aspects of the model such as lepton flavor violation (LFV), flavor number violation (FNV), and flavor changing neutral currents (FCNCs). (orig.) 7. A possible solution of the flavor problem and radiative neutrino masses Energy Technology Data Exchange (ETDEWEB) Adulpravitchai, Adisorn 2010-06-23 In this thesis, we discuss two important problems of the Standard Model of Particle Physics (SM), namely the flavor problem and the reason for the smallness of neutrino masses. The first one might be related to the origin of non-abelian discrete flavor symmetries. We discuss the possibility of obtaining them from an underlying continuous flavor symmetry, i.e. SU(2) or SU(3) through spontaneous symmetry breaking. Moreover, we investigate their possible origin from an orbifold compactification. We discuss all non-abelian discrete symmetries, which can arise from an orbifold T{sup 2}/Z{sub N}. They are A{sub 4}, S{sub 4}, D{sub 4}, D{sub 3}, and D{sub 6}. We present the idea of combining the breaking of an orbifold GUT and the flavor symmetry arising from the orbifold. We demonstrate the construction in a 6d SUSY SO(10) x S{sub 4}. For the second problem, we propose a one-loop neutrino mass model in the left-right symmetric framework. We observe the transmitted hierarchy from the charged lepton masses to the right-handed neutrino masses, which we call ''Radiative Transmission of Lepton Flavor Hierarchy''. Finally, we study the phenomenological aspects of the model such as lepton flavor violation (LFV), flavor number violation (FNV), and flavor changing neutral currents (FCNCs). (orig.) 8. Non-zero θ{sub 13} and leptonic CP phase with A{sub 4} symmetry Energy Technology Data Exchange (ETDEWEB) Sruthilaya, M.; Mohanta, R. [University of Hyderabad, School of Physics, Hyderabad (India) 2017-03-15 We consider a model based on A{sub 4} symmetry to explain the phenomenon of neutrino mixing. The spontaneous symmetry breaking of A{sub 4} symmetry leads to a co-bimaximal mixing matrix at leading order. We consider the effect of higher order corrections in neutrino sector and find that the mixing angles thus obtained, come well within the 3σ ranges of their experimental values. We study the implications of this formalism on the other phenomenological observables, such as CP violating phase, Jarlskog invariant and the effective Majorana mass vertical stroke M{sub ee} vertical stroke. We also obtain the branching ratio of the lepton flavour violating decay μ → eγ in the context of this model and find that it can be less than its present experimental upper bound. (orig.) 9. Residual Z{sub 2} symmetries and leptonic mixing patterns from finite discrete subgroups of U(3) Energy Technology Data Exchange (ETDEWEB) Joshipura, Anjan S. [Physical Research Laboratory,Navarangpura, Ahmedabad 380 009 (India); Patel, Ketan M. [Indian Institute of Science Education and Research, Mohali,Knowledge City, Sector 81, S A S Nagar, Manauli 140 306 (India) 2017-01-30 We study embedding of non-commuting Z{sub 2} and Z{sub m}, m≥3 symmetries in discrete subgroups (DSG) of U(3) and analytically work out the mixing patterns implied by the assumption that Z{sub 2} and Z{sub m} describe the residual symmetries of the neutrino and the charged lepton mass matrices respectively. Both Z{sub 2} and Z{sub m} are assumed to be subgroups of a larger discrete symmetry group G{sub f} possessing three dimensional faithful irreducible representation. The residual symmetries predict the magnitude of a column of the leptonic mixing matrix U{sub PMNS} which are studied here assuming G{sub f} as the DSG of SU(3) designated as type C and D and large number of DSG of U(3) which are not in SU(3). These include the known group series Σ(3n{sup 3}), T{sub n}(m), Δ(3n{sup 2},m), Δ(6n{sup 2},m) and Δ{sup ′}(6n{sup 2},j,k). It is shown that the predictions for a column of |U{sub PMNS}| in these group series and the C and D types of groups are all contained in the predictions of the Δ(6N{sup 2}) groups for some integer N. The Δ(6N{sup 2}) groups therefore represent a sufficient set of G{sub f} to obtain predictions of the residual symmetries Z{sub 2} and Z{sub m}. 10. A fermion-boson composite model of quarks and leptons Directory of Open Access Journals (Sweden) Yoshio Koide 1983-01-01 Full Text Available Quark and lepton masses and flavor-mixing angles are estimated on the basis of a fermion-boson composite model where the (u, d, (c, s and (t, b quarks are assigned to the diagonal elements π8, η8 and η1, respectively, in3 × 3* = 8 + 1 of the SU(3-generation symmetry. 11. μ-τ symmetry and charged lepton mass hierarchy in a supersymmetric D4 model International Nuclear Information System (INIS) Hagedorn, C.; Ziegler, R. 2010-01-01 In this paper we discuss a supersymmetric D 4 xZ 5 model which leads to vanishing reactor mixing angle θ 13 =0 and maximal atmospheric mixing θ 23 =π/4 in the lepton sector at leading order, due to the preservation of nontrivial distinct D 4 subgroups in the charged lepton and neutrino sectors, respectively. The solar mixing angle θ 12 remains undetermined and is expected to be of order one. Since right-handed charged leptons transform as singlets under D 4 , the charged lepton mass hierarchy can be naturally accounted for. The model predicts inverted mass hierarchy for neutrinos. Additionally, we show that, unlike in most of the other models of this type, all vacuum expectation values of gauge singlets (flavons) can be determined through mass parameters of the superpotential. Next-to-leading order corrections to lepton masses and mixings are calculated and shown to be under control; in particular, the corrections to θ 23 =π/4 and θ 13 =0 are of the order of the generic expansion parameter ε≅0.04 and arise dominantly from the charged lepton sector. 12. GUT and flavor models for neutrino masses and mixing Science.gov (United States) Meloni, Davide 2017-10-01 In the recent years experiments have established the existence of neutrino oscillations and most of the oscillation parameters have been measured with a good accuracy. However, in spite of many interesting ideas, no real illumination was sparked on the problem of flavor in the lepton sector. In this review, we discuss the state of the art of models for neutrino masses and mixings formulated in the context of flavor symmetries, with particular emphasis on the role played by grand unified gauge groups. 13. Higgs mediated lepton flavor violating tau decays τ→μγ and τ→μγγ in effective theories International Nuclear Information System (INIS) Aranda, J. I.; Tututi, E. S.; Ramirez-Zavaleta, F.; Toscano, J. J. 2008-01-01 The size of the branching ratios for the τ→μγ and τ→μγγ decays induced by a lepton flavor violating Higgs interaction Hτμ is studied in the framework of effective field theories. The best constraint on the Hτμ vertex, derived from the know measurement on the muon anomalous magnetic moment, is used to impose the upper bounds Br(τ→μγ) -10 and Br(τ→μγγ) -12 , which are more stringent than current experimental limits on this class of transitions. 14. A supersymmetry model of leptons International Nuclear Information System (INIS) Liu, Chun 2005-01-01 If supersymmetry (SUSY) is not for stabilizing the electroweak energy scale, what is it used for in particle physics? We propose that it is for flavor problems. A cyclic family symmetry is introduced. Under the family symmetry, only the τ-lepton is massive due to the vacuum expectation value (VEV) of the Higgs field. This symmetry is broken by a sneutrino VEV which results in the muon mass. The comparatively large sneutrino VEV does not result in a large neutrino mass due to requiring heavy gauginos. SUSY breaks at a high scale ∼10 13 GeV. The electroweak energy scale is unnaturally small. No additional global symmetry, like the R-parity, is imposed. Other aspects of the model are discussed 15. Sum rules for elements of flavor-mixing matrices based on a non-semisimple symmetry International Nuclear Information System (INIS) Sogami, Ikuo S. 2006-01-01 Sum rules for elements of flavor-mixing matrices (FMMs) are derived within a new algebraic theory for flavor physics, in which the FMMs are identified with elements of the Lie group isomorphic to SU(2) x U(1). The resulting sum rules originating from the unique elaborate structure of the algebra of the group are so simple and explicit that their validity can be confirmed by analyzing properly processed experimental data. (author) 16. A search for lepton flavor violating transitions e τ via leptoquarks in e+p scattering with the ZEUS detector at HERA International Nuclear Information System (INIS) Kerger, R. 2001-03-01 A search for lepton flavor violation e + p → τX under the leptoquark (LQ) assumption has been performed with the ZEUS detector at HERA using the 1994 to 1997 data corresponding to an integrated luminosity of 47.7 pb -1 . Special effort has been made to identify the hadronic decay modes of the τ. No evidence for lepton flavor violation was found. For resonantly produced leptoquarks, limits on λ eq i x √(BR(LQ → lq j )) as well as on their Yukawa couplings λ eq i and λ τq j as a function of the leptoquark mass have been set. Assuming couplings of electromagnetic strength, λ = 0.3, leptoquarks for masses up to 285 GeV are excluded at 95% C.L. and assuming vice-versa leptoquarks of M LQ = 250 GeV, λ x √(BR) down to 3 . 10 -2 is excluded at 95% C.L. For leptoquarks with M LQ >> √(s), limits are set on the four-fermion contact interaction term λ eq i λ lq j /M LQ 2 . Many of these limits supersede existing limits from low-energy experiments. (orig.) 17. Characterizing dark matter interacting with extra charged leptons Science.gov (United States) Barducci, D.; Deandrea, A.; Moretti, S.; Panizzi, L.; Prager, H. 2018-04-01 In the context of a simplified leptophilic dark matter (DM) scenario where the mediator is a new charged fermion carrying leptonic quantum number and the DM candidate is either scalar or vector, the complementarity of different bounds is analyzed. In this framework, the extra lepton and DM are odd under a Z2 symmetry, and hence the leptonic mediator can only interact with the DM state and Standard Model leptons of various flavors. We show that there is the possibility to characterize the DM spin (scalar or vector), as well as the nature of the mediator, through a combined analysis of cosmological, flavor and collider data. We present an explicit numerical analysis for a set of benchmarks points of the viable parameter space of our scenario. 18. Symmetry restoration at high-temperature in two-color and two-flavor lattice gauge theories Energy Technology Data Exchange (ETDEWEB) Lee, Jong-Wan [Department of Physics, College of Science, Swansea University,Singleton Park, SA2 8PP, Swansea, Wales (United Kingdom); Department of Physics, Pusan National University,Busan 46241 (Korea, Republic of); Extreme Physics Institute, Pusan National University,Busan 46241 (Korea, Republic of); Lucini, Biagio; Piai, Maurizio [Department of Physics, College of Science, Swansea University,Singleton Park, SA2 8PP, Swansea, Wales (United Kingdom) 2017-04-07 We consider the SU(2) gauge theory with N{sub f}=2 flavors of Dirac fundamental fermions. We study the high-temperature behavior of the spectra of mesons, discretizing the theory on anisotropic lattices, and measuring the two-point correlation functions in the temporal direction as well as screening masses in various channels. We identify the (pseudo-)critical temperature as the temperature at which the susceptibility associated with the Polyakov loop has a maximum. At high temperature both the spin-1 and spin-0 sectors of the light meson spectra exhibit enhanced symmetry properties, indicating the restoration of both the global SU(4) and the axial U(1){sub A} symmetries of the model. 19. Search for lepton flavor violating decay τ- →ℓ-+-ℓ = e, μ at BaBar Energy Technology Data Exchange (ETDEWEB) Cervelli, Alberto [Univ. of Pisa (Italy) 2010-05-26 The Standard Model (SM) is one of the most tested and verified physical theories of all time, present experimental observations are consistent with SM expectations. On the other hand SM can not explain many physical observations: the cosmological observations possibly infer the presence of dark matter which is clearly beyond the SM expectations; the SM Higgs model, while explaining the generation of fermion masses, can not explain the hierarchy problem and a non natural fine tuning of SM is needed to cancel out quadratic divergences in the Higgs boson mass. New physics (NP) beyond SM should hence be investigated: rising the energy above NP processes thresholds, and detecting new particles or new effects not predicted by the standard model directly, is one of the possible approaches; another approach is to make precision measurements of well known processes or looking for rare processes which involve higher order contribution from NP processes, this approach need higher luminosities with respect to the previous approach but lower beam energies. Search for Lepton Flavor Violation (LFV) in charged lepton decays is promising: neutrino physics provides indeed a clear and unambiguous evidence of LFV in the neutral lepton sector via mixing processes, which have been observed for the first time by the Homestake collaboration. We expect LFV in the charged sector as well, both in {mu} and {tau} sector, but current experimental searches for LFV processes did not find any evidence for those processes, and more results are expected to come from new experiments in the coming years. 20. Flavored model building Energy Technology Data Exchange (ETDEWEB) Hagedorn, C. 2008-01-15 In this thesis we discuss possibilities to solve the family replication problem and to understand the observed strong hierarchy among the fermion masses and the diverse mixing pattern of quarks and leptons. We show that non-abelian discrete symmetries which act non-trivially in generation space can serve as profound explanation. We present three low energy models with the permutation symmetry S{sub 4}, the dihedral group D{sub 5} and the double-valued group T' as flavor symmetry. The T' model turns out to be very predictive, since it explains tri-bimaximal mixing in the lepton sector and, moreover, leads to two non-trivial relations in the quark sector, {radical}((m{sub d})/(m{sub s}))= vertical stroke V{sub us} vertical stroke and {radical}((m{sub d})/(m{sub s}))= vertical stroke (V{sub td})/(V{sub ts}) vertical stroke. The main message of the T' model is the observation that the diverse pattern in the quark and lepton mixings can be well-understood, if the flavor symmetry is not broken in an arbitrary way, but only to residual (non-trivial) subgroups. Apart from leading to deeper insights into the origin of the fermion mixings this idea enables us to perform systematic studies of large classes of discrete groups. This we show in our study of dihedral symmetries D{sub n} and D'{sub n}. As a result we find only five distinct (Dirac) mass matrix structures arising from a dihedral group, if we additionally require partial unification of either left-handed or left-handed conjugate fermions and the determinant of the mass matrix to be non-vanishing. Furthermore, we reveal the ability of dihedral groups to predict the Cabibbo angle {theta}{sub C}, i.e. vertical stroke V{sub us(cd)} vertical stroke = cos((3{pi})/(7)), as well as maximal atmospheric mixing, {theta}{sub 23}=({pi})/(4), and vanishing {theta}{sub 13} in the lepton sector. (orig.) 1. Family nonuniversal Z' models with protected flavor-changing interactions Science.gov (United States) Celis, Alejandro; Fuentes-Martín, Javier; Jung, Martin; Serôdio, Hugo 2015-07-01 We define a new class of Z' models with neutral flavor-changing interactions at tree level in the down-quark sector. They are related in an exact way to elements of the quark mixing matrix due to an underlying flavored U(1)' gauge symmetry, rendering these models particularly predictive. The same symmetry implies lepton-flavor nonuniversal couplings, fully determined by the gauge structure of the model. Our models allow us to address presently observed deviations from the standard model and specific correlations among the new physics contributions to the Wilson coefficients C9,10' ℓ can be tested in b →s ℓ+ℓ- transitions. We furthermore predict lepton-universality violations in Z' decays, testable at the LHC. 2. Natural embedding of Peccei-Quinn symmetry in flavor grand unification International Nuclear Information System (INIS) Kim, J.E. 1981-08-01 Peccei and Quinn's global U(1)sub(A) symmetry can be embedded in grand unified schemes without an artificial requirement of imposing U(1)sub(A) symmetry, which results from the representation content of fermions and Higgs fields. Then, in some cases there results an ordinary axion with a mass approximately 100 keV. The axion mass is proportional to v -1 sub(A), where v -1 sub(A) is the scale of the actual U(1)sub(A) symmetry breakdown. (author) 3. Symmetries of nonrelativistic phase space and the structure of quark-lepton generation International Nuclear Information System (INIS) Zenczykowski, Piotr 2009-01-01 According to the Hamiltonian formalism, nonrelativistic phase space may be considered as an arena of physics, with momentum and position treated as independent variables. Invariance of x 2 + p 2 constitutes then a natural generalization of ordinary rotational invariance. We consider Dirac-like linearization of this form, with position and momentum satisfying standard commutation relations. This leads to the identification of a quantum-level structure from which some phase space properties might emerge. Genuine rotations and reflections in phase space are tied to the existence of new quantum numbers, unrelated to ordinary 3D space. Their properties allow their identification with the internal quantum numbers characterising the structure of a single quark-lepton generation in the Standard Model. In particular, the algebraic structure of the Harari-Shupe preon model of fundamental particles is reproduced exactly and without invoking any subparticles. Analysis of the Clifford algebra of nonrelativistic phase space singles out an element which might be associated with the concept of lepton mass. This element is transformed into a corresponding element for a single coloured quark, leading to a generalization of the concept of mass and a different starting point for the discussion of quark unobservability. 4. Flavorful leptoquarks at hadron colliders Science.gov (United States) Hiller, Gudrun; Loose, Dennis; Nišandžić, Ivan 2018-04-01 B -physics data and flavor symmetries suggest that leptoquarks can have masses as low as a few O (TeV ) , predominantly decay to third generation quarks, and highlight p p →b μ μ signatures from single production and p p →b b μ μ from pair production. Abandoning flavor symmetries could allow for inverted quark hierarchies and cause sizable p p →j μ μ and j j μ μ cross sections, induced by second generation couplings. Final states with leptons other than muons including lepton flavor violation (LFV) ones can also arise. The corresponding couplings can also be probed by precision studies of the B →(Xs,K*,ϕ )e e distribution and LFV searches in B -decays. We demonstrate sensitivity in single leptoquark production for the large hadron collider (LHC) and extrapolate to the high luminosity LHC. Exploration of the bulk of the parameter space requires a hadron collider beyond the reach of the LHC, with b -identification capabilities. 5. Lepton mixing predictions including Majorana phases from Δ(6n2 flavour symmetry and generalised CP Directory of Open Access Journals (Sweden) Stephen F. King 2014-09-01 Full Text Available Generalised CP transformations are the only known framework which allows to predict Majorana phases in a flavour model purely from symmetry. For the first time generalised CP transformations are investigated for an infinite series of finite groups, Δ(6n2=(Zn×Zn⋊S3. In direct models the mixing angles and Dirac CP phase are solely predicted from symmetry. The Δ(6n2 flavour symmetry provides many examples of viable predictions for mixing angles. For all groups the mixing matrix has a trimaximal middle column and the Dirac CP phase is 0 or π. The Majorana phases are predicted from residual flavour and CP symmetries where α21 can take several discrete values for each n and the Majorana phase α31 is a multiple of π. We discuss constraints on the groups and CP transformations from measurements of the neutrino mixing angles and from neutrinoless double-beta decay and find that predictions for mixing angles and all phases are accessible to experiments in the near future. 6. Lepton mixing predictions including Majorana phases from Δ(6n2) flavour symmetry and generalised CP International Nuclear Information System (INIS) King, Stephen F.; Neder, Thomas 2014-01-01 Generalised CP transformations are the only known framework which allows to predict Majorana phases in a flavour model purely from symmetry. For the first time generalised CP transformations are investigated for an infinite series of finite groups, Δ(6n 2 )=(Z n ×Z n )⋊S 3 . In direct models the mixing angles and Dirac CP phase are solely predicted from symmetry. The Δ(6n 2 ) flavour symmetry provides many examples of viable predictions for mixing angles. For all groups the mixing matrix has a trimaximal middle column and the Dirac CP phase is 0 or π. The Majorana phases are predicted from residual flavour and CP symmetries where α 21 can take several discrete values for each n and the Majorana phase α 31 is a multiple of π. We discuss constraints on the groups and CP transformations from measurements of the neutrino mixing angles and from neutrinoless double-beta decay and find that predictions for mixing angles and all phases are accessible to experiments in the near future 7. 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Haug, Sigve; Hauschild, Michael; Hauser, Reiner; Havranek, Miroslav; Hawkes, Christopher; Hawkings, Richard John; Hawkins, Anthony David; Hayashi, Takayasu; Hayden, Daniel; Hays, Chris; Hayward, Helen; Haywood, Stephen; Head, Simon; Heck, Tobias; Hedberg, Vincent; Heelan, Louise; Heim, Sarah; Heim, Timon; Heinemann, Beate; Heinrich, Lukas; Hejbal, Jiri; Helary, Louis; Heller, Claudio; Heller, Matthieu; Hellman, Sten; Hellmich, Dennis; Helsens, Clement; Henderson, James; Henderson, Robert; Heng, Yang; Hengler, Christopher; Henrichs, Anna; Henriques Correia, Ana Maria; Henrot-Versille, Sophie; Herbert, Geoffrey Henry; Hernández Jiménez, Yesenia; Herrberg-Schubert, Ruth; Herten, Gregor; Hertenberger, Ralf; Hervas, Luis; Hesketh, Gavin Grant; Hessey, Nigel; Hickling, Robert; Higón-Rodriguez, Emilio; Hill, Ewan; Hill, John; Hiller, Karl Heinz; Hillert, Sonja; Hillier, Stephen; Hinchliffe, Ian; Hines, Elizabeth; Hirose, Minoru; Hirschbuehl, Dominic; Hobbs, John; Hod, Noam; Hodgkinson, Mark; Hodgson, Paul; Hoecker, Andreas; Hoeferkamp, Martin; Hoenig, Friedrich; Hoffman, Julia; Hoffmann, Dirk; Hofmann, Julia Isabell; Hohlfeld, Marc; Holmes, Tova Ray; Hong, Tae Min; Hooft van Huysduynen, Loek; Hopkins, Walter; Horii, Yasuyuki; Hostachy, Jean-Yves; Hou, Suen; Hoummada, Abdeslam; Howard, Jacob; Howarth, James; Hrabovsky, Miroslav; Hristova, Ivana; Hrivnac, Julius; Hryn'ova, Tetiana; Hsu, Catherine; Hsu, Pai-hsien Jennifer; Hsu, Shih-Chieh; Hu, Diedi; Hu, Xueye; Huang, Yanping; Hubacek, Zdenek; Hubaut, Fabrice; Huegging, Fabian; Huffman, Todd Brian; Hughes, Emlyn; Hughes, Gareth; Huhtinen, Mika; Hülsing, Tobias Alexander; Hurwitz, Martina; Huseynov, Nazim; Huston, Joey; Huth, John; Iacobucci, Giuseppe; Iakovidis, Georgios; Ibragimov, Iskander; Iconomidou-Fayard, Lydia; Ideal, Emma; Idrissi, Zineb; Iengo, Paolo; Igonkina, Olga; Iizawa, Tomoya; Ikegami, Yoichi; Ikematsu, Katsumasa; Ikeno, Masahiro; Ilchenko, Iurii; Iliadis, Dimitrios; Ilic, Nikolina; Inamaru, Yuki; Ince, Tayfun; Ioannou, Pavlos; Iodice, Mauro; Iordanidou, Kalliopi; Ippolito, Valerio; Irles Quiles, Adrian; Isaksson, Charlie; Ishino, Masaya; Ishitsuka, Masaki; Ishmukhametov, Renat; Issever, Cigdem; Istin, Serhat; Iturbe Ponce, Julia Mariana; Iuppa, Roberto; Ivarsson, Jenny; Iwanski, Wieslaw; Iwasaki, Hiroyuki; Izen, Joseph; Izzo, Vincenzo; Jackson, Brett; Jackson, Matthew; Jackson, Paul; Jaekel, Martin; Jain, Vivek; Jakobs, Karl; Jakobsen, Sune; Jakoubek, Tomas; Jakubek, Jan; Jamin, David Olivier; Jana, Dilip; Jansen, Eric; Jansen, Hendrik; Janssen, Jens; Janus, Michel; Jarlskog, Göran; Javadov, Namig; Javůrek, Tomáš; Jeanty, Laura; Jejelava, Juansher; Jeng, Geng-yuan; Jennens, David; Jenni, Peter; Jentzsch, Jennifer; Jeske, Carl; Jézéquel, Stéphane; Ji, Haoshuang; Jia, Jiangyong; Jiang, Yi; Jimenez Belenguer, Marcos; Jin, Shan; Jinaru, Adam; Jinnouchi, Osamu; Joergensen, Morten Dam; Johansson, Erik; Johansson, Per; Johns, Kenneth; Jon-And, Kerstin; Jones, Graham; Jones, Roger; Jones, Tim; Jongmanns, Jan; Jorge, Pedro; Joshi, Kiran Daniel; Jovicevic, Jelena; Ju, Xiangyang; Jung, Christian; Jungst, Ralph Markus; Jussel, Patrick; Juste Rozas, Aurelio; Kaci, Mohammed; Kaczmarska, Anna; Kado, Marumi; Kagan, Harris; Kagan, Michael; Kajomovitz, Enrique; Kalderon, Charles William; Kama, Sami; Kamenshchikov, Andrey; Kanaya, Naoko; Kaneda, Michiru; Kaneti, Steven; Kantserov, Vadim; Kanzaki, Junichi; Kaplan, Benjamin; Kapliy, Anton; Kar, Deepak; Karakostas, Konstantinos; Karastathis, Nikolaos; Kareem, Mohammad Jawad; Karnevskiy, Mikhail; Karpov, Sergey; Karpova, Zoya; Karthik, Krishnaiyengar; Kartvelishvili, Vakhtang; Karyukhin, Andrey; Kashif, Lashkar; Kasieczka, Gregor; Kass, Richard; Kastanas, Alex; Kataoka, Yousuke; Katre, Akshay; Katzy, Judith; Kaushik, Venkatesh; Kawagoe, Kiyotomo; Kawamoto, Tatsuo; Kawamura, Gen; Kazama, Shingo; Kazanin, Vassili; Kazarinov, Makhail; Keeler, Richard; Kehoe, Robert; Keil, Markus; Keller, John; Kempster, Jacob Julian; Keoshkerian, Houry; Kepka, Oldrich; Kerševan, Borut Paul; Kersten, Susanne; Kessoku, Kohei; Keung, Justin; Khalil-zada, Farkhad; Khandanyan, Hovhannes; Khanov, Alexander; Khodinov, Alexander; Khomich, Andrei; Khoo, Teng Jian; Khoriauli, Gia; Khoroshilov, Andrey; Khovanskiy, Valery; Khramov, Evgeniy; Khubua, Jemal; Kim, Hee Yeun; Kim, Hyeon Jin; Kim, Shinhong; Kimura, Naoki; Kind, Oliver; King, Barry; King, Matthew; King, Robert Steven Beaufoy; King, Samuel Burton; Kirk, Julie; Kiryunin, Andrey; Kishimoto, Tomoe; Kisielewska, Danuta; Kiss, Florian; Kittelmann, Thomas; Kiuchi, Kenji; Kladiva, Eduard; Klein, Max; Klein, Uta; Kleinknecht, Konrad; Klimek, Pawel; Klimentov, Alexei; Klingenberg, Reiner; Klinger, Joel Alexander; Klioutchnikova, Tatiana; Klok, Peter; Kluge, Eike-Erik; Kluit, Peter; Kluth, Stefan; Kneringer, Emmerich; Knoops, Edith; Knue, Andrea; Kobayashi, Dai; Kobayashi, Tomio; Kobel, Michael; Kocian, Martin; Kodys, Peter; Koevesarki, Peter; Koffas, Thomas; Koffeman, Els; Kogan, Lucy Anne; Kohlmann, Simon; Kohout, Zdenek; Kohriki, Takashi; Koi, Tatsumi; Kolanoski, Hermann; Koletsou, Iro; Koll, James; Komar, Aston; Komori, Yuto; Kondo, Takahiko; Kondrashova, Nataliia; Köneke, Karsten; König, Adriaan; König, Sebastian; Kono, Takanori; Konoplich, Rostislav; Konstantinidis, Nikolaos; Kopeliansky, Revital; Koperny, Stefan; Köpke, Lutz; Kopp, Anna Katharina; Korcyl, Krzysztof; Kordas, Kostantinos; Korn, Andreas; Korol, Aleksandr; Korolkov, Ilya; Korolkova, Elena; Korotkov, Vladislav; Kortner, Oliver; Kortner, Sandra; Kostyukhin, Vadim; Kotov, Vladislav; Kotwal, Ashutosh; Kourkoumelis, Christine; Kouskoura, Vasiliki; Koutsman, Alex; Kowalewski, Robert Victor; Kowalski, Tadeusz; Kozanecki, Witold; Kozhin, Anatoly; Kral, Vlastimil; Kramarenko, Viktor; Kramberger, Gregor; Krasnopevtsev, Dimitriy; Krasny, Mieczyslaw Witold; Krasznahorkay, Attila; Kraus, Jana; Kravchenko, Anton; Kreiss, Sven; Kretz, Moritz; Kretzschmar, Jan; Kreutzfeldt, Kristof; Krieger, Peter; Kroeninger, Kevin; Kroha, Hubert; Kroll, Joe; Kroseberg, Juergen; Krstic, Jelena; Kruchonak, Uladzimir; Krüger, Hans; Kruker, Tobias; Krumnack, Nils; Krumshteyn, Zinovii; Kruse, Amanda; Kruse, Mark; Kruskal, Michael; Kubota, Takashi; Kuday, Sinan; Kuehn, Susanne; Kugel, Andreas; Kuhl, Andrew; Kuhl, Thorsten; Kukhtin, Victor; Kulchitsky, Yuri; Kuleshov, Sergey; Kuna, Marine; Kunkle, Joshua; Kupco, Alexander; Kurashige, Hisaya; Kurochkin, Yurii; Kurumida, Rie; Kus, Vlastimil; Kuwertz, Emma Sian; Kuze, Masahiro; Kvita, Jiri; La Rosa, Alessandro; La Rotonda, Laura; Lacasta, Carlos; Lacava, Francesco; Lacey, James; Lacker, Heiko; Lacour, Didier; Lacuesta, Vicente Ramón; Ladygin, Evgueni; Lafaye, Remi; Laforge, Bertrand; Lagouri, Theodota; Lai, Stanley; Laier, Heiko; Lambourne, Luke; Lammers, Sabine; Lampen, Caleb; Lampl, Walter; Lançon, Eric; Landgraf, Ulrich; Landon, Murrough; Lang, Valerie Susanne; Lankford, Andrew; Lanni, Francesco; Lantzsch, Kerstin; Laplace, Sandrine; Lapoire, Cecile; Laporte, Jean-Francois; Lari, Tommaso; Lasagni Manghi, Federico; Lassnig, Mario; Laurelli, Paolo; Lavrijsen, Wim; Law, Alexander; Laycock, Paul; Le Dortz, Olivier; Le Guirriec, Emmanuel; Le Menedeu, Eve; LeCompte, Thomas; Ledroit-Guillon, Fabienne Agnes Marie; Lee, Claire Alexandra; Lee, Hurng-Chun; Lee, Jason; Lee, Shih-Chang; Lee, Lawrence; Lefebvre, Guillaume; Lefebvre, Michel; Legger, Federica; Leggett, Charles; Lehan, Allan; Lehmacher, Marc; Lehmann Miotto, Giovanna; Lei, Xiaowen; Leight, William Axel; Leisos, Antonios; Leister, Andrew Gerard; Leite, Marco Aurelio Lisboa; Leitner, Rupert; Lellouch, Daniel; Lemmer, Boris; Leney, Katharine; Lenz, Tatjana; Lenzen, Georg; Lenzi, Bruno; Leone, Robert; Leone, Sandra; Leonidopoulos, Christos; Leontsinis, Stefanos; Leroy, Claude; Lester, Christopher; Lester, Christopher Michael; Levchenko, Mikhail; Levêque, Jessica; Levin, Daniel; Levinson, Lorne; Levy, Mark; Lewis, Adrian; Lewis, George; Leyko, Agnieszka; Leyton, Michael; Li, Bing; Li, Bo; Li, Haifeng; Li, Ho Ling; Li, Lei; Li, Liang; Li, Shu; Li, Yichen; Liang, Zhijun; Liao, Hongbo; Liberti, Barbara; Lichard, Peter; Lie, Ki; Liebal, Jessica; Liebig, Wolfgang; Limbach, Christian; Limosani, Antonio; Lin, Simon; Lin, Tai-Hua; Linde, Frank; Lindquist, Brian Edward; Linnemann, James; Lipeles, Elliot; Lipniacka, Anna; Lisovyi, Mykhailo; Liss, Tony; Lissauer, David; Lister, Alison; Litke, Alan; Liu, Bo; Liu, Dong; Liu, Jianbei; Liu, Kun; Liu, Lulu; Liu, Miaoyuan; Liu, Minghui; Liu, Yanwen; Livan, Michele; Livermore, Sarah; Lleres, Annick; Llorente Merino, Javier; Lloyd, Stephen; Lo Sterzo, Francesco; Lobodzinska, Ewelina; Loch, Peter; Lockman, William; Loddenkoetter, Thomas; Loebinger, Fred; Loevschall-Jensen, Ask Emil; Loginov, Andrey; Lohse, Thomas; Lohwasser, Kristin; Lokajicek, Milos; Lombardo, Vincenzo Paolo; Long, Brian Alexander; Long, Jonathan; Long, Robin Eamonn; Lopes, Lourenco; Lopez Mateos, David; Lopez Paredes, Brais; Lopez Paz, Ivan; Lorenz, Jeanette; Lorenzo Martinez, Narei; Losada, Marta; Loscutoff, Peter; Lou, XinChou; Lounis, Abdenour; Love, Jeremy; Love, Peter; Lowe, Andrew; Lu, Feng; Lu, Nan; Lubatti, Henry; Luci, Claudio; Lucotte, Arnaud; Luehring, Frederick; Lukas, Wolfgang; Luminari, Lamberto; Lundberg, Olof; Lund-Jensen, Bengt; Lungwitz, Matthias; Lynn, David; Lysak, Roman; Lytken, Else; Ma, Hong; Ma, Lian Liang; Maccarrone, Giovanni; Macchiolo, Anna; Machado Miguens, Joana; Macina, Daniela; Madaffari, Daniele; Madar, Romain; Maddocks, Harvey Jonathan; Mader, Wolfgang; Madsen, Alexander; Maeno, Mayuko; Maeno, Tadashi; Maevskiy, Artem; Magradze, Erekle; Mahboubi, Kambiz; Mahlstedt, Joern; Mahmoud, Sara; Maiani, Camilla; Maidantchik, Carmen; Maier, Andreas Alexander; Maio, Amélia; Majewski, Stephanie; Makida, Yasuhiro; Makovec, Nikola; Mal, Prolay; Malaescu, Bogdan; Malecki, Pawel; Maleev, Victor; Malek, Fairouz; Mallik, Usha; Malon, David; Malone, Caitlin; Maltezos, Stavros; Malyshev, Vladimir; Malyukov, Sergei; Mamuzic, Judita; Mandelli, Beatrice; Mandelli, Luciano; Mandić, Igor; Mandrysch, Rocco; Maneira, José; Manfredini, Alessandro; Manhaes de Andrade Filho, Luciano; Manjarres Ramos, Joany Andreina; Mann, Alexander; Manning, Peter; Manousakis-Katsikakis, Arkadios; Mansoulie, Bruno; Mantifel, Rodger; Mapelli, Livio; March, Luis; Marchand, Jean-Francois; Marchiori, Giovanni; Marcisovsky, Michal; Marino, Christopher; Marjanovic, Marija; Marques, Carlos; Marroquim, Fernando; Marsden, Stephen Philip; Marshall, Zach; Marti, Lukas Fritz; Marti-Garcia, Salvador; Martin, Brian; Martin, Brian Thomas; Martin, Tim; Martin, Victoria Jane; Martin dit Latour, Bertrand; Martinez, Homero; Martinez, Mario; Martin-Haugh, Stewart; Martyniuk, Alex; Marx, Marilyn; Marzano, Francesco; Marzin, Antoine; Masetti, Lucia; Mashimo, Tetsuro; Mashinistov, Ruslan; Masik, Jiri; Maslennikov, Alexey; Massa, Ignazio; Massa, Lorenzo; Massol, Nicolas; Mastrandrea, Paolo; Mastroberardino, Anna; Masubuchi, Tatsuya; Mättig, Peter; Mattmann, Johannes; Maurer, Julien; Maxfield, Stephen; Maximov, Dmitriy; Mazini, Rachid; Mazzaferro, Luca; Mc Goldrick, Garrin; Mc Kee, Shawn Patrick; McCarn, Allison; McCarthy, Robert; McCarthy, Tom; McCubbin, Norman; McFarlane, Kenneth; Mcfayden, Josh; Mchedlidze, Gvantsa; McMahon, Steve; McPherson, Robert; Mechnich, Joerg; Medinnis, Michael; Meehan, Samuel; Mehlhase, Sascha; Mehta, Andrew; Meier, Karlheinz; Meineck, Christian; Meirose, Bernhard; Melachrinos, Constantinos; Mellado Garcia, Bruce Rafael; Meloni, Federico; Mengarelli, Alberto; Menke, Sven; Meoni, Evelin; Mercurio, Kevin Michael; Mergelmeyer, Sebastian; Meric, Nicolas; Mermod, Philippe; Merola, Leonardo; Meroni, Chiara; Merritt, Frank; Merritt, Hayes; Messina, Andrea; Metcalfe, Jessica; Mete, Alaettin Serhan; Meyer, Carsten; Meyer, Christopher; Meyer, Jean-Pierre; Meyer, Jochen; Middleton, Robin; Migas, Sylwia; Mijović, Liza; Mikenberg, Giora; Mikestikova, Marcela; Mikuž, Marko; Milic, Adriana; Miller, David; Mills, Corrinne; Milov, Alexander; Milstead, David; Milstein, Dmitry; Minaenko, Andrey; Minami, Yuto; Minashvili, Irakli; Mincer, Allen; Mindur, Bartosz; Mineev, Mikhail; Ming, Yao; Mir, Lluisa-Maria; Mirabelli, Giovanni; Mitani, Takashi; Mitrevski, Jovan; Mitsou, Vasiliki A; Mitsui, Shingo; Miucci, Antonio; Miyagawa, Paul; Mjörnmark, Jan-Ulf; Moa, Torbjoern; Mochizuki, Kazuya; Mohapatra, Soumya; Mohr, Wolfgang; Molander, Simon; Moles-Valls, Regina; Mönig, Klaus; Monini, Caterina; Monk, James; Monnier, Emmanuel; Montejo Berlingen, Javier; Monticelli, Fernando; Monzani, Simone; Moore, Roger; Morange, Nicolas; Moreno, Deywis; Moreno Llácer, María; Morettini, Paolo; Morgenstern, Marcus; Morii, Masahiro; Moritz, Sebastian; Morley, Anthony Keith; Mornacchi, Giuseppe; Morris, John; Morvaj, Ljiljana; Moser, Hans-Guenther; Mosidze, Maia; Moss, Josh; Motohashi, Kazuki; Mount, Richard; Mountricha, Eleni; Mouraviev, Sergei; Moyse, Edward; Muanza, Steve; Mudd, Richard; Mueller, Felix; Mueller, James; Mueller, Klemens; Mueller, Thibaut; Mueller, Timo; Muenstermann, Daniel; Munwes, Yonathan; Murillo Quijada, Javier Alberto; Murray, Bill; Musheghyan, Haykuhi; Musto, Elisa; Myagkov, Alexey; Myska, Miroslav; Nackenhorst, Olaf; Nadal, Jordi; Nagai, Koichi; Nagai, Ryo; Nagai, Yoshikazu; Nagano, Kunihiro; Nagarkar, Advait; Nagasaka, Yasushi; Nagel, Martin; Nairz, Armin Michael; Nakahama, Yu; Nakamura, Koji; Nakamura, Tomoaki; Nakano, Itsuo; Namasivayam, Harisankar; Nanava, Gizo; Narayan, Rohin; Nattermann, Till; Naumann, Thomas; Navarro, Gabriela; Nayyar, Ruchika; Neal, Homer; Nechaeva, Polina; Neep, Thomas James; Nef, Pascal Daniel; Negri, Andrea; Negri, Guido; Negrini, Matteo; Nektarijevic, Snezana; Nellist, Clara; Nelson, Andrew; Nelson, Timothy Knight; Nemecek, Stanislav; Nemethy, Peter; Nepomuceno, Andre Asevedo; Nessi, Marzio; Neubauer, Mark; Neumann, Manuel; Neves, Ricardo; Nevski, Pavel; Newman, Paul; Nguyen, Duong Hai; Nickerson, Richard; Nicolaidou, Rosy; Nicquevert, Bertrand; Nielsen, Jason; Nikiforou, Nikiforos; Nikiforov, Andriy; Nikolaenko, Vladimir; Nikolic-Audit, Irena; Nikolics, Katalin; Nikolopoulos, Konstantinos; Nilsson, Paul; Ninomiya, Yoichi; Nisati, Aleandro; Nisius, Richard; Nobe, Takuya; Nodulman, Lawrence; Nomachi, Masaharu; Nomidis, Ioannis; Norberg, Scarlet; Nordberg, Markus; Novgorodova, Olga; Nowak, Sebastian; Nozaki, Mitsuaki; Nozka, Libor; Ntekas, Konstantinos; Nunes Hanninger, Guilherme; Nunnemann, Thomas; Nurse, Emily; Nuti, Francesco; O'Brien, Brendan Joseph; O'grady, Fionnbarr; O'Neil, Dugan; O'Shea, Val; Oakham, Gerald; Oberlack, Horst; Obermann, Theresa; Ocariz, Jose; Ochi, Atsuhiko; Ochoa, Ines; Oda, Susumu; Odaka, Shigeru; Ogren, Harold; Oh, Alexander; Oh, Seog; Ohm, Christian; Ohman, Henrik; Okamura, Wataru; Okawa, Hideki; Okumura, Yasuyuki; Okuyama, Toyonobu; Olariu, Albert; Olchevski, Alexander; Olivares Pino, Sebastian Andres; Oliveira Damazio, Denis; Oliver Garcia, Elena; Olszewski, Andrzej; Olszowska, Jolanta; Onofre, António; Onyisi, Peter; Oram, Christopher; Oreglia, Mark; Oren, Yona; Orestano, Domizia; Orlando, Nicola; Oropeza Barrera, Cristina; Orr, Robert; Osculati, Bianca; Ospanov, Rustem; Otero y Garzon, Gustavo; Otono, Hidetoshi; Ouchrif, Mohamed; Ouellette, Eric; Ould-Saada, Farid; Ouraou, Ahmimed; Oussoren, Koen Pieter; Ouyang, Qun; Ovcharova, Ana; Owen, Mark; Ozcan, Veysi Erkcan; Ozturk, Nurcan; Pachal, Katherine; Pacheco Pages, Andres; Padilla Aranda, Cristobal; Pagáčová, Martina; Pagan Griso, Simone; Paganis, Efstathios; Pahl, Christoph; Paige, Frank; Pais, Preema; Pajchel, Katarina; Palacino, Gabriel; Palestini, Sandro; Palka, Marek; Pallin, Dominique; Palma, Alberto; Palmer, Jody; Pan, Yibin; Panagiotopoulou, Evgenia; Panduro Vazquez, William; Pani, Priscilla; Panikashvili, Natalia; Panitkin, Sergey; Pantea, Dan; Paolozzi, Lorenzo; Papadopoulou, Theodora; Papageorgiou, Konstantinos; Paramonov, Alexander; Paredes Hernandez, Daniela; Parker, Michael Andrew; Parodi, Fabrizio; Parsons, John; Parzefall, Ulrich; Pasqualucci, Enrico; Passaggio, Stefano; Passeri, Antonio; Pastore, Fernanda; Pastore, Francesca; Pásztor, Gabriella; Pataraia, Sophio; Patel, Nikhul; Pater, Joleen; Patricelli, Sergio; Pauly, Thilo; Pearce, James; Pedersen, Lars Egholm; Pedersen, Maiken; Pedraza Lopez, Sebastian; Pedro, Rute; Peleganchuk, Sergey; Pelikan, Daniel; Peng, Haiping; Penning, Bjoern; Penwell, John; Perepelitsa, Dennis; Perez Codina, Estel; Pérez García-Estañ, María Teresa; Perez Reale, Valeria; Perini, Laura; Pernegger, Heinz; Perrella, Sabrina; Perrino, Roberto; Peschke, Richard; Peshekhonov, Vladimir; Peters, Krisztian; Peters, Yvonne; Petersen, Brian; Petersen, Troels; Petit, Elisabeth; Petridis, Andreas; Petridou, Chariclia; Petrolo, Emilio; Petrucci, Fabrizio; Pettersson, Nora Emilia; Pezoa, Raquel; Phillips, Peter William; Piacquadio, Giacinto; Pianori, Elisabetta; Picazio, Attilio; Piccaro, Elisa; Piccinini, Maurizio; Piegaia, Ricardo; Pignotti, David; Pilcher, James; Pilkington, Andrew; Pina, João Antonio; Pinamonti, Michele; Pinder, Alex; Pinfold, James; Pingel, Almut; Pinto, Belmiro; Pires, Sylvestre; Pitt, Michael; Pizio, Caterina; Plazak, Lukas; Pleier, Marc-Andre; Pleskot, Vojtech; Plotnikova, Elena; Plucinski, Pawel; Pluth, Daniel; Poddar, Sahill; Podlyski, Fabrice; Poettgen, Ruth; Poggioli, Luc; Pohl, David-leon; Pohl, Martin; Polesello, Giacomo; Policicchio, Antonio; Polifka, Richard; Polini, Alessandro; Pollard, Christopher Samuel; Polychronakos, Venetios; Pommès, Kathy; Pontecorvo, Ludovico; Pope, Bernard; Popeneciu, Gabriel Alexandru; Popovic, Dragan; Poppleton, Alan; Portell Bueso, Xavier; Pospisil, Stanislav; Potamianos, Karolos; Potrap, Igor; Potter, Christina; Potter, Christopher; Poulard, Gilbert; Poveda, Joaquin; Pozdnyakov, Valery; Pralavorio, Pascal; Pranko, Aliaksandr; Prasad, Srivas; Pravahan, Rishiraj; Prell, Soeren; Price, Darren; Price, Joe; Price, Lawrence; Prieur, Damien; Primavera, Margherita; Proissl, Manuel; Prokofiev, Kirill; Prokoshin, Fedor; Protopapadaki, Eftychia-sofia; Protopopescu, Serban; Proudfoot, James; Przybycien, Mariusz; Przysiezniak, Helenka; Ptacek, Elizabeth; Puddu, Daniele; Pueschel, Elisa; Puldon, David; Purohit, Milind; Puzo, Patrick; Qian, Jianming; Qin, Gang; Qin, Yang; Quadt, Arnulf; Quarrie, David; Quayle, William; Queitsch-Maitland, Michaela; Quilty, Donnchadha; Qureshi, Anum; Radeka, Veljko; Radescu, Voica; Radhakrishnan, Sooraj Krishnan; Radloff, Peter; Rados, Pere; Ragusa, Francesco; Rahal, Ghita; Rajagopalan, Srinivasan; Rammensee, Michael; Randle-Conde, Aidan Sean; Rangel-Smith, Camila; Rao, Kanury; Rauscher, Felix; Rave, Tobias Christian; Ravenscroft, Thomas; Raymond, Michel; Read, Alexander Lincoln; Readioff, Nathan Peter; Rebuzzi, Daniela; Redelbach, Andreas; Redlinger, George; Reece, Ryan; Reeves, Kendall; Rehnisch, Laura; Reisin, Hernan; Relich, Matthew; Rembser, Christoph; Ren, Huan; Ren, Zhongliang; Renaud, Adrien; Rescigno, Marco; Resconi, Silvia; Rezanova, Olga; Reznicek, Pavel; Rezvani, Reyhaneh; Richter, Robert; Ridel, Melissa; Rieck, Patrick; Rieger, Julia; Rijssenbeek, Michael; Rimoldi, Adele; Rinaldi, Lorenzo; Ritsch, Elmar; Riu, Imma; Rizatdinova, Flera; Rizvi, Eram; Robertson, Steven; Robichaud-Veronneau, Andree; Robinson, Dave; Robinson, James; Robson, Aidan; Roda, Chiara; Rodrigues, Luis; Roe, Shaun; Røhne, Ole; Rolli, Simona; Romaniouk, Anatoli; Romano, Marino; Romero Adam, Elena; Rompotis, Nikolaos; Ronzani, Manfredi; Roos, Lydia; Ros, Eduardo; Rosati, Stefano; Rosbach, Kilian; Rose, Matthew; Rose, Peyton; Rosendahl, Peter Lundgaard; Rosenthal, Oliver; Rossetti, Valerio; Rossi, Elvira; Rossi, Leonardo Paolo; Rosten, Rachel; Rotaru, Marina; Roth, Itamar; Rothberg, Joseph; Rousseau, David; Royon, Christophe; Rozanov, Alexandre; Rozen, Yoram; Ruan, Xifeng; Rubbo, Francesco; Rubinskiy, Igor; Rud, Viacheslav; Rudolph, Christian; Rudolph, Matthew Scott; Rühr, Frederik; Ruiz-Martinez, Aranzazu; Rurikova, Zuzana; Rusakovich, Nikolai; Ruschke, Alexander; Rutherfoord, John; Ruthmann, Nils; Ryabov, Yury; Rybar, Martin; Rybkin, Grigori; Ryder, Nick; Saavedra, Aldo; Sabato, Gabriele; Sacerdoti, Sabrina; Saddique, Asif; Sadeh, Iftach; Sadrozinski, Hartmut; Sadykov, Renat; Safai Tehrani, Francesco; Sakamoto, Hiroshi; Sakurai, Yuki; Salamanna, Giuseppe; Salamon, Andrea; Saleem, Muhammad; Salek, David; Sales De Bruin, Pedro Henrique; Salihagic, Denis; Salnikov, Andrei; Salt, José; Salvatore, Daniela; Salvatore, Pasquale Fabrizio; Salvucci, Antonio; Salzburger, Andreas; Sampsonidis, Dimitrios; Sanchez, Arturo; Sánchez, Javier; Sanchez Martinez, Victoria; Sandaker, Heidi; Sandbach, Ruth Laura; Sander, Heinz Georg; Sanders, Michiel; Sandhoff, Marisa; Sandoval, Tanya; Sandoval, Carlos; Sandstroem, Rikard; Sankey, Dave; Sansoni, Andrea; Santoni, Claudio; Santonico, Rinaldo; Santos, Helena; Santoyo Castillo, Itzebelt; Sapp, Kevin; Sapronov, Andrey; Saraiva, João; Sarrazin, Bjorn; Sartisohn, Georg; Sasaki, Osamu; Sasaki, Yuichi; Sauvage, Gilles; Sauvan, Emmanuel; Savard, Pierre; Savu, Dan Octavian; Sawyer, Craig; Sawyer, Lee; Saxon, David; Saxon, James; Sbarra, Carla; Sbrizzi, Antonio; Scanlon, Tim; Scannicchio, Diana; Scarcella, Mark; Scarfone, Valerio; Schaarschmidt, Jana; Schacht, Peter; Schaefer, Douglas; Schaefer, Ralph; Schaepe, Steffen; Schaetzel, Sebastian; Schäfer, Uli; Schaffer, Arthur; Schaile, Dorothee; Schamberger, R~Dean; Scharf, Veit; Schegelsky, Valery; Scheirich, Daniel; Schernau, Michael; Scherzer, Max; Schiavi, Carlo; Schieck, Jochen; Schillo, Christian; Schioppa, Marco; Schlenker, Stefan; Schmidt, Evelyn; Schmieden, Kristof; Schmitt, Christian; Schmitt, Sebastian; Schneider, Basil; Schnellbach, Yan Jie; Schnoor, Ulrike; Schoeffel, Laurent; Schoening, Andre; Schoenrock, Bradley Daniel; Schorlemmer, Andre Lukas; Schott, Matthias; Schouten, Doug; Schovancova, Jaroslava; Schramm, Steven; Schreyer, Manuel; Schroeder, Christian; Schuh, Natascha; Schultens, Martin Johannes; Schultz-Coulon, Hans-Christian; Schulz, Holger; Schumacher, Markus; Schumm, Bruce; Schune, Philippe; Schwanenberger, Christian; Schwartzman, Ariel; Schwarz, Thomas Andrew; Schwegler, Philipp; Schwemling, Philippe; Schwienhorst, Reinhard; Schwindling, Jerome; Schwindt, Thomas; Schwoerer, Maud; Sciacca, Gianfranco; Scifo, Estelle; Sciolla, Gabriella; Scott, Bill; Scuri, Fabrizio; Scutti, Federico; Searcy, Jacob; Sedov, George; Sedykh, Evgeny; Seidel, Sally; Seiden, Abraham; Seifert, Frank; Seixas, José; Sekhniaidze, Givi; Sekula, Stephen; Selbach, Karoline Elfriede; Seliverstov, Dmitry; Sellers, Graham; Semprini-Cesari, Nicola; Serfon, Cedric; Serin, Laurent; Serkin, Leonid; Serre, Thomas; Seuster, Rolf; Severini, Horst; Sfiligoj, Tina; Sforza, Federico; Sfyrla, Anna; Shabalina, Elizaveta; Shamim, Mansoora; Shan, Lianyou; Shang, Ruo-yu; Shank, James; Shapiro, Marjorie; Shatalov, Pavel; Shaw, Kate; Shehu, Ciwake Yusufu; Sherwood, Peter; Shi, Liaoshan; Shimizu, Shima; Shimmin, Chase Owen; Shimojima, Makoto; Shiyakova, Mariya; Shmeleva, Alevtina; Shochet, Mel; Short, Daniel; Shrestha, Suyog; Shulga, Evgeny; Shupe, Michael; Shushkevich, Stanislav; Sicho, Petr; Sidiropoulou, Ourania; Sidorov, Dmitri; Sidoti, Antonio; Siegert, Frank; Sijacki, Djordje; Silva, José; Silver, Yiftah; Silverstein, Daniel; Silverstein, Samuel; Simak, Vladislav; Simard, Olivier; Simic, Ljiljana; Simion, Stefan; Simioni, Eduard; Simmons, Brinick; Simoniello, Rosa; Simonyan, Margar; Sinervo, Pekka; Sinev, Nikolai; Sipica, Valentin; Siragusa, Giovanni; Sircar, Anirvan; Sisakyan, Alexei; Sivoklokov, Serguei; Sjölin, Jörgen; Sjursen, Therese; Skottowe, Hugh Philip; Skovpen, Kirill; Skubic, Patrick; Slater, Mark; Slavicek, Tomas; Slawinska, Magdalena; Sliwa, Krzysztof; Smakhtin, Vladimir; Smart, Ben; Smestad, Lillian; Smirnov, Sergei; Smirnov, Yury; Smirnova, Lidia; Smirnova, Oxana; Smith, Kenway; Smizanska, Maria; Smolek, Karel; Snesarev, Andrei; Snidero, Giacomo; Snyder, Scott; Sobie, Randall; Socher, Felix; Soffer, Abner; Soh, Dart-yin; Solans, Carlos; Solar, Michael; Solc, Jaroslav; Soldatov, Evgeny; Soldevila, Urmila; Solodkov, Alexander; Soloshenko, Alexei; Solovyanov, Oleg; Solovyev, Victor; Sommer, Philip; Song, Hong Ye; Soni, Nitesh; Sood, Alexander; Sopczak, Andre; Sopko, Bruno; Sopko, Vit; Sorin, Veronica; Sosebee, Mark; Soualah, Rachik; Soueid, Paul; Soukharev, Andrey; South, David; Spagnolo, Stefania; Spanò, Francesco; Spearman, William Robert; Spettel, Fabian; Spighi, Roberto; Spigo, Giancarlo; Spiller, Laurence Anthony; Spousta, Martin; Spreitzer, Teresa; Spurlock, Barry; St Denis, Richard Dante; Staerz, Steffen; Stahlman, Jonathan; Stamen, Rainer; Stamm, Soren; Stanecka, Ewa; Stanek, Robert; Stanescu, Cristian; Stanescu-Bellu, Madalina; Stanitzki, Marcel Michael; Stapnes, Steinar; Starchenko, Evgeny; Stark, Jan; Staroba, Pavel; Starovoitov, Pavel; Staszewski, Rafal; Stavina, Pavel; Steinberg, Peter; Stelzer, Bernd; Stelzer, Harald Joerg; Stelzer-Chilton, Oliver; Stenzel, Hasko; Stern, Sebastian; Stewart, Graeme; Stillings, Jan Andre; Stockton, Mark; Stoebe, Michael; Stoicea, Gabriel; Stolte, Philipp; Stonjek, Stefan; Stradling, Alden; Straessner, Arno; Stramaglia, Maria Elena; Strandberg, Jonas; Strandberg, Sara; Strandlie, Are; Strauss, Emanuel; Strauss, Michael; Strizenec, Pavol; Ströhmer, Raimund; Strom, David; Stroynowski, Ryszard; Strubig, Antonia; Stucci, Stefania Antonia; Stugu, Bjarne; Styles, Nicholas Adam; Su, Dong; Su, Jun; Subramaniam, Rajivalochan; Succurro, Antonella; Sugaya, Yorihito; Suhr, Chad; Suk, Michal; Sulin, Vladimir; Sultansoy, Saleh; Sumida, Toshi; Sun, Siyuan; Sun, Xiaohu; Sundermann, Jan Erik; Suruliz, Kerim; Susinno, Giancarlo; Sutton, Mark; Suzuki, Yu; Svatos, Michal; Swedish, Stephen; Swiatlowski, Maximilian; Sykora, Ivan; Sykora, Tomas; Ta, Duc; Taccini, Cecilia; Tackmann, Kerstin; Taenzer, Joe; Taffard, Anyes; Tafirout, Reda; Taiblum, Nimrod; Takai, Helio; Takashima, Ryuichi; Takeda, Hiroshi; Takeshita, Tohru; Takubo, Yosuke; Talby, Mossadek; Talyshev, Alexey; Tam, Jason; Tan, Kong Guan; Tanaka, Junichi; Tanaka, Reisaburo; Tanaka, Satoshi; Tanaka, Shuji; Tanasijczuk, Andres Jorge; Tannenwald, Benjamin Bordy; Tannoury, Nancy; Tapprogge, Stefan; Tarem, Shlomit; Tarrade, Fabien; Tartarelli, Giuseppe Francesco; Tas, Petr; Tasevsky, Marek; Tashiro, Takuya; Tassi, Enrico; Tavares Delgado, Ademar; Tayalati, Yahya; Taylor, Frank; Taylor, Geoffrey; Taylor, Wendy; Teischinger, Florian Alfred; Teixeira Dias Castanheira, Matilde; Teixeira-Dias, Pedro; Temming, Kim Katrin; Ten Kate, Herman; Teng, Ping-Kun; Teoh, Jia Jian; Terada, Susumu; Terashi, Koji; Terron, Juan; Terzo, Stefano; Testa, Marianna; Teuscher, Richard; Therhaag, Jan; Theveneaux-Pelzer, Timothée; Thomas, Juergen; Thomas-Wilsker, Joshuha; Thompson, Emily; Thompson, Paul; Thompson, Peter; Thompson, Ray; Thompson, Stan; Thomsen, Lotte Ansgaard; Thomson, Evelyn; Thomson, Mark; Thong, Wai Meng; Thun, Rudolf; Tian, Feng; Tibbetts, Mark James; Tikhomirov, Vladimir; Tikhonov, Yury; Timoshenko, Sergey; Tiouchichine, Elodie; Tipton, Paul; Tisserant, Sylvain; Todorov, Theodore; Todorova-Nova, Sharka; Toggerson, Brokk; Tojo, Junji; Tokár, Stanislav; Tokushuku, Katsuo; Tollefson, Kirsten; Tolley, Emma; Tomlinson, Lee; Tomoto, Makoto; Tompkins, Lauren; Toms, Konstantin; Topilin, Nikolai; Torrence, Eric; Torres, Heberth; Torró Pastor, Emma; Toth, Jozsef; Touchard, Francois; Tovey, Daniel; Tran, Huong Lan; Trefzger, Thomas; Tremblet, Louis; Tricoli, Alessandro; Trigger, Isabel Marian; Trincaz-Duvoid, Sophie; Tripiana, Martin; Trischuk, William; Trocmé, Benjamin; Troncon, Clara; Trottier-McDonald, Michel; Trovatelli, Monica; True, Patrick; Trzebinski, Maciej; Trzupek, Adam; Tsarouchas, Charilaos; Tseng, Jeffrey; Tsiareshka, Pavel; Tsionou, Dimitra; Tsipolitis, Georgios; Tsirintanis, Nikolaos; Tsiskaridze, Shota; Tsiskaridze, Vakhtang; Tskhadadze, Edisher; Tsukerman, Ilya; Tsulaia, Vakhtang; Tsuno, Soshi; Tsybychev, Dmitri; Tudorache, Alexandra; Tudorache, Valentina; Tuna, Alexander Naip; Tupputi, Salvatore; Turchikhin, Semen; Turecek, Daniel; Turk Cakir, Ilkay; Turra, Ruggero; Turvey, Andrew John; Tuts, Michael; Tykhonov, Andrii; Tylmad, Maja; Tyndel, Mike; Uchida, Kirika; Ueda, Ikuo; Ueno, Ryuichi; Ughetto, Michael; Ugland, Maren; Uhlenbrock, Mathias; Ukegawa, Fumihiko; Unal, Guillaume; Undrus, Alexander; Unel, Gokhan; Ungaro, Francesca; Unno, Yoshinobu; Unverdorben, Christopher; Urbaniec, Dustin; Urquijo, Phillip; Usai, Giulio; Usanova, Anna; Vacavant, Laurent; Vacek, Vaclav; Vachon, Brigitte; Valencic, Nika; Valentinetti, Sara; Valero, Alberto; Valery, Loic; Valkar, Stefan; Valladolid Gallego, Eva; Vallecorsa, Sofia; Valls Ferrer, Juan Antonio; Van Den Wollenberg, Wouter; Van Der Deijl, Pieter; van der Geer, Rogier; van der Graaf, Harry; Van Der Leeuw, Robin; van der Ster, Daniel; van Eldik, Niels; van Gemmeren, Peter; Van Nieuwkoop, Jacobus; van Vulpen, Ivo; van Woerden, Marius Cornelis; Vanadia, Marco; Vandelli, Wainer; Vanguri, Rami; Vaniachine, Alexandre; Vankov, Peter; Vannucci, Francois; Vardanyan, Gagik; Vari, Riccardo; Varnes, Erich; Varol, Tulin; Varouchas, Dimitris; Vartapetian, Armen; Varvell, Kevin; Vazeille, Francois; Vazquez Schroeder, Tamara; Veatch, Jason; Veloso, Filipe; Veneziano, Stefano; Ventura, Andrea; Ventura, Daniel; Venturi, Manuela; Venturi, Nicola; Venturini, Alessio; Vercesi, Valerio; Verducci, Monica; Verkerke, Wouter; Vermeulen, Jos; Vest, Anja; Vetterli, Michel; Viazlo, Oleksandr; Vichou, Irene; Vickey, Trevor; Vickey Boeriu, Oana Elena; Viehhauser, Georg; Viel, Simon; Vigne, Ralph; Villa, Mauro; Villaplana Perez, Miguel; Vilucchi, Elisabetta; Vincter, Manuella; Vinogradov, Vladimir; Virzi, Joseph; Vivarelli, Iacopo; Vives Vaque, Francesc; Vlachos, Sotirios; Vladoiu, Dan; Vlasak, Michal; Vogel, Adrian; Vogel, Marcelo; Vokac, Petr; Volpi, Guido; Volpi, Matteo; von der Schmitt, Hans; von Radziewski, Holger; von Toerne, Eckhard; Vorobel, Vit; Vorobev, Konstantin; Vos, Marcel; Voss, Rudiger; Vossebeld, Joost; Vranjes, Nenad; Vranjes Milosavljevic, Marija; Vrba, Vaclav; Vreeswijk, Marcel; Vu Anh, Tuan; Vuillermet, Raphael; Vukotic, Ilija; Vykydal, Zdenek; Wagner, Peter; Wagner, Wolfgang; Wahlberg, Hernan; Wahrmund, Sebastian; Wakabayashi, Jun; Walder, James; Walker, Rodney; Walkowiak, Wolfgang; Wall, Richard; Waller, Peter; Walsh, Brian; Wang, Chao; Wang, Chiho; Wang, Fuquan; Wang, Haichen; Wang, Hulin; Wang, Jike; Wang, Jin; Wang, Kuhan; Wang, Rui; Wang, Song-Ming; Wang, Tan; Wang, Xiaoxiao; Wanotayaroj, Chaowaroj; Warburton, Andreas; Ward, Patricia; Wardrope, David Robert; Warsinsky, Markus; Washbrook, Andrew; Wasicki, Christoph; Watkins, Peter; Watson, Alan; Watson, Ian; Watson, Miriam; Watts, Gordon; Watts, Stephen; Waugh, Ben; Webb, Samuel; Weber, Michele; Weber, Stefan Wolf; Webster, Jordan S; Weidberg, Anthony; Weigell, Philipp; Weinert, Benjamin; Weingarten, Jens; Weiser, Christian; Weits, Hartger; Wells, Phillippa; Wenaus, Torre; Wendland, Dennis; Weng, Zhili; Wengler, Thorsten; Wenig, Siegfried; Wermes, Norbert; Werner, Matthias; Werner, Per; Wessels, Martin; Wetter, Jeffrey; Whalen, Kathleen; White, Andrew; White, Martin; White, Ryan; White, Sebastian; Whiteson, Daniel; Wicke, Daniel; Wickens, Fred; Wiedenmann, Werner; Wielers, Monika; Wienemann, Peter; Wiglesworth, Craig; Wiik-Fuchs, Liv Antje Mari; Wijeratne, Peter Alexander; Wildauer, Andreas; Wildt, Martin Andre; Wilkens, Henric George; Will, Jonas Zacharias; Williams, Hugh; Williams, Sarah; Willis, Christopher; Willocq, Stephane; Wilson, Alan; Wilson, John; Wingerter-Seez, Isabelle; Winklmeier, Frank; Winter, Benedict Tobias; Wittgen, Matthias; Wittig, Tobias; Wittkowski, Josephine; Wollstadt, Simon Jakob; Wolter, Marcin Wladyslaw; Wolters, Helmut; Wosiek, Barbara; Wotschack, Jorg; Woudstra, Martin; Wozniak, Krzysztof; Wright, Michael; Wu, Mengqing; Wu, Sau Lan; Wu, Xin; Wu, Yusheng; Wulf, Evan; Wyatt, Terry Richard; Wynne, Benjamin; Xella, Stefania; Xiao, Meng; Xu, Da; Xu, Lailin; Yabsley, Bruce; Yacoob, Sahal; Yakabe, Ryota; Yamada, Miho; Yamaguchi, Hiroshi; Yamaguchi, Yohei; Yamamoto, Akira; Yamamoto, Kyoko; Yamamoto, Shimpei; Yamamura, Taiki; Yamanaka, Takashi; Yamauchi, Katsuya; Yamazaki, Yuji; Yan, Zhen; Yang, Haijun; Yang, Hongtao; Yang, Un-Ki; Yang, Yi; Yanush, Serguei; Yao, Liwen; Yao, Weiming; Yasu, Yoshiji; Yatsenko, Elena; Yau Wong, Kaven Henry; Ye, Jingbo; Ye, Shuwei; Yeletskikh, Ivan; Yen, Andy L; Yildirim, Eda; Yilmaz, Metin; Yoosoofmiya, Reza; Yorita, Kohei; Yoshida, Rikutaro; Yoshihara, Keisuke; Young, Charles; Young, Christopher John; Youssef, Saul; Yu, David Ren-Hwa; Yu, Jaehoon; Yu, Jiaming; Yu, Jie; Yuan, Li; Yurkewicz, Adam; Yusuff, Imran; Zabinski, Bartlomiej; Zaidan, Remi; Zaitsev, Alexander; Zaman, Aungshuman; Zambito, Stefano; Zanello, Lucia; Zanzi, Daniele; Zeitnitz, Christian; Zeman, Martin; Zemla, Andrzej; Zengel, Keith; Zenin, Oleg; Ženiš, Tibor; Zerwas, Dirk; Zevi della Porta, Giovanni; Zhang, Dongliang; Zhang, Fangzhou; Zhang, Huaqiao; Zhang, Jinlong; Zhang, Lei; Zhang, Xueyao; Zhang, Zhiqing; Zhao, Zhengguo; Zhemchugov, Alexey; Zhong, Jiahang; Zhou, Bing; Zhou, Lei; Zhou, Ning; Zhu, Cheng Guang; Zhu, Hongbo; Zhu, Junjie; Zhu, Yingchun; Zhuang, Xuai; Zhukov, Konstantin; Zibell, Andre; Zieminska, Daria; Zimine, Nikolai; Zimmermann, Christoph; Zimmermann, Robert; Zimmermann, Simone; Zimmermann, Stephanie; Zinonos, Zinonas; Ziolkowski, Michael; Zobernig, Georg; Zoccoli, Antonio; zur Nedden, Martin; Zurzolo, Giovanni; Zutshi, Vishnu; Zwalinski, Lukasz 2014-10-23 The ATLAS detector at the Large Hadron Collider is used to search for the lepton flavor violating process$Z \\rightarrow e\\mu$in$pp$collisions using$20.3~\\textrm{fb}^{-1}$of data collected at$\\sqrt{s}= 8~\\textrm{TeV}$. An enhancement in the$e\\mu$invariant mass spectrum is searched for at the$Z$boson mass. The number of$Z$bosons produced in the data sample is estimated using events of similar topology,$Z \\rightarrow ee$and$\\mu\\mu$, significantly reducing the systematic uncertainty in the measurement. There is no evidence of an enhancement at the$Z$boson mass, resulting in an upper limit on the branching fraction,${\\cal B}(Z~\\rightarrow~e\\mu)~<~7.5 \\times 10^{-7}at the 95% confidence level. 8. Dynamical symmetry breaking through preons and the sizes of composite quarks and leptons International Nuclear Information System (INIS) Pati, J.C. 1984-01-01 It is observed that the assumptions that quarks and leptons are composites and that they acquire masses dynamically through preonic condensates rather than through the vacuum expectation value of a Higgs field lead to a relatively low upper bound of only 1 to 3 TeV for the inverse size of the heaviest family: e.g., the tau family. It is furthermore stressed that the e and μ families, within a large class of models, must, on the other hand, have a relatively large inverse size exceeding about 150 TeV; this is so in order that the limits from rare processes such as K/sub L/→mu-bare and K 0 -K-bar 0 may be satisfied. Certain theoretical and experimental implications of these two observations are noted 9. On SUSY inspired minimal lepton number violation International Nuclear Information System (INIS) Chkareuli, J.L.; Gogoladze, I.G.; Green, M.G.; Hutchroft, D.E.; Kobakhidze, A.B. 2000-03-01 A minimal lepton number violation (LNV) is proposed which could naturally appear in SUSY theories, if Yukawa and LNV couplings had a common origin. According to this idea properly implemented into MSSM with an additional abelian flavor symmetry the prototype LNV appears due to a mixing of leptons with superheavy Higgs doublet mediating Yukawa couplings. As a result, all significant physical manifestations of LNV reduce to those of the effective trilinear couplings LLE-bar and LQD-bar aligned, by magnitude and orientation in a flavor space, with the down fermion (charged lepton and down quark) effective Yukawa couplings, while the effective bilinear terms appear generically suppressed relative to an ordinary μ-term of MSSM. Detailed phenomenology of the model related to the flavor-changing processes both in quark and lepton sectors, radiatively induced neutrino masses and decays of the LSP is presented. Remarkably, the model can straightforwardly be extended to a Grand Unified framework and an explicit example with SU(7) GUT is thoroughly discussed. (author) 10. Effective meson lagrangian with chiral and heavy quark symmetries from quark flavor dynamics International Nuclear Information System (INIS) Ebert, D.; Feldmann, T.; Friedrich, R.; Reinhardt, H. 1994-06-01 By bosonization of an extended NJL model we derive an effective meson theory which describes the interplay between chiral symmetry and heavy quark dynamics. This effective theory is worked out in the low-energy regime using the gradient expansion. The resulting effective lagrangian describes strong and weak interactions of heavy B and D mesons with pseudoscalar Goldstone bosons and light vector and axial-vector mesons. Heavy meson weak decay constants, coupling constants and the Isgur-Wise function are predicted in terms of the model parameters partially fixed from the light quark sector. Explicit SU(3) F symmetry breaking effects are estimated and, if possible, confronted with experiment. (orig.) 11. An investigation of b/barb-quark flavor tagging methods in lepton + charm events at √s=1.8 TeV. Science.gov (United States) Rakitine, Alexandre Y.; CDF Collaboration 2000-04-01 Jet Charge'', Same Side'' and Soft Lepton'' tagging methods were studied to determine the b/barb-quark flavor of B-mesons at the time of production. A sample of almost 10,000 B_u,d arrow l D^(*) X decays collected during the 1992-1995 run was analyzed. Six decay signatures were reconstructed. The efficacy of the tagging methods was demonstrated by revealing the time-dependent flavor oscillations of B^0-mesons and measuring their frequency. We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Science and Culture of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; and the A. P. Sloan Foundation. Supported by U.S. DOE under contract DE-AC02-76ER03069. 12. Influence of broken flavor and C and P symmetry on the quark propagator Energy Technology Data Exchange (ETDEWEB) Maas, Axel; Mian, Walid Ahmed [University of Graz, Institute of Physics, NAWI Graz, Graz (Austria) 2017-02-15 Embedding QCD into the standard model breaks various symmetries of QCD explicitly, especially C and P. While these effects are usually perturbatively small, they can be amplified in extreme environments like merging neutron stars or by the interplay with new physics. To correctly treat these cases requires fully backcoupled calculations. To pave the way for later investigations of hadronic physics, we study the QCD quark propagator coupled to an explicit breaking. This substantially increases the tensor structure even for this simplest correlation function. To cope with the symmetry structure, and covering all possible quark masses, from the top quark mass to the chiral limit, we employ Dyson-Schwinger equations. While at weak breaking the qualitative effects have similar trends as in perturbation theory, even moderately strong breakings lead to qualitatively different effects, non-linearly amplified by the strong interactions. (orig.) 13. The anomalous U(1){sub anom} symmetry and flavors from an SU(5) x SU(5){sup '} GUT in Z{sub 12-I} orbifold compactification Energy Technology Data Exchange (ETDEWEB) Kim, Jihn E. [Kyung Hee University, Department of Physics, Seoul (Korea, Republic of); Center for Axion and Precision Physics Research (IBS), Daejeon (Korea, Republic of); Kyae, Bumseok [Pusan National University, Department of Physics, Busan (Korea, Republic of); Nam, Soonkeon [Kyung Hee University, Department of Physics, Seoul (Korea, Republic of) 2017-12-15 In string compactifications, frequently the anomalous U(1) gauge symmetry appears which belongs to E{sub 8} x E{sub 8}{sup '} of the heterotic string. This anomalous U(1) gauge boson obtains mass at the compactification scale (∼ 10{sup 18} GeV) by absorbing one pseudoscalar (corresponding to the model-independent axion) from the second rank antisymmetric tensor field B{sub MN}. Below the compactification scale a global symmetry U(1){sub anom} results whose charge Q{sub anom} is the original gauge U(1) charge. This is the most natural global symmetry, realizing the ''invisible'' axion. This global symmetry U(1){sub anom} is suitable for a flavor symmetry. In the simplest compactification model with the flipped SU(5) grand unification, all the low energy parameters are calculated in terms of the vacuum expectation values of the standard model singlets. (orig.) 14. The anomalous U(1)_{anom} symmetry and flavors from an SU(5) × SU(5)' GUT in Z_{12-I} orbifold compactification Science.gov (United States) Kim, Jihn E.; Kyae, Bumseok; Nam, Soonkeon 2017-12-01 In string compactifications, frequently the anomalous U(1) gauge symmetry appears which belongs to E_8 × E_8' of the heterotic string. This anomalous U(1) gauge boson obtains mass at the compactification scale (≈ 10^{18 } {GeV}) by absorbing one pseudoscalar (corresponding to the model-independent axion) from the second rank antisymmetric tensor field B_{MN}. Below the compactification scale a global symmetry U(1)_{anom} results whose charge Q_anom is the original gauge U(1) charge. This is the most natural global symmetry, realizing the "invisible" axion. This global symmetry U(1)_{anom} is suitable for a flavor symmetry. In the simplest compactification model with the flipped SU(5) grand unification, all the low energy parameters are calculated in terms of the vacuum expectation values of the standard model singlets. 15. Neutrino masses, mixings, and FCNC’s in an S3 flavor symmetric extension of the standard model International Nuclear Information System (INIS) Mondragón, A.; Mondragón, M.; Peinado, E. 2011-01-01 By introducing threeHiggs fields that are SU(2) doublets and a flavor permutational symmetry, S 3 , in the theory, we extend the concepts of flavor and generations to the Higgs sector and formulate a Minimal S 3 -Invariant Extension of the Standard Model. The mass matrices of the neutrinos and charged leptons are re-parameterized in terms of their eigenvalues, then the neutrino mixing matrix, V PMNS , is computed and exact, explicit analytical expressions for the neutrino mixing angles as functions of the masses of neutrinos and charged leptons are obtained in excellent agreement with the latest experimental data. We also compute the branching ratios of some selected flavor-changing neutral current (FCNC) processes, as well as the contribution of the exchange of neutral flavor-changing scalars to the anomaly of the magnetic moment of the muon, as functions of the masses of charged leptons and the neutral Higgs bosons. We find that the S 3 × Z 2 flavor symmetry and the strong mass hierarchy of the charged leptons strongly suppress the FCNC processes in the leptonic sector, well below the present experimental bounds by many orders of magnitude. The contribution of FCNC’s to the anomaly of the muon’s magnetic moment is small, but not negligible. 16. U(3)-flavor nonet scalar as an origin of the flavor mass spectra International Nuclear Information System (INIS) Koide, Yoshio 2008-01-01 According to an idea that the quark and lepton mass spectra originate in a VEV structure of a U(3)-flavor nonet scalar Φ, the mass spectra of the down-quarks and charged leptons are investigated. The U(3) flavor symmetry is spontaneously and completely broken by non-zero and non-degenerated VEVs of Φ, without passing any subgroup of U(3). The ratios (m e +m μ +m τ )/(√(m e )+√(m μ )+√(m τ )) 2 and √(m e m μ m τ )/(√(m e )+√(m μ )+√(m τ )) 3 are investigated based on a toy model 17. Lepton flavour violation at a future linear collider International Nuclear Information System (INIS) GOMEZ, M. E. 2014-01-01 We study the relation of the possible observation on the radiative decays μ→eγ and τ→μγ and LFV processes that could be detectable at a linear collider (LC) with a centre-of-mass energy in the TeV range. We use supersymmetric parameters consistent with cosmological considerations and with LHC searches for supersymmetry and the Higgs mass while we link the charged lepton flavor problem to the neutrino predictions in a SU(5) GUT model, enhanced by an abelian flavor symmetry. 18. Measurement of the Top Pair Production Cross Section in the Lepton + Jets Channel Using a Jet Flavor Discriminant CERN Document Server Aaltonen, T.; Amerio, S.; Amidei, D.; Anastassov, A.; Annovi, A.; Antos, J.; Apollinari, G.; Appel, J.A.; Apresyan, A.; Arisawa, T.; Artikov, A.; Asaadi, J.; Ashmanskas, W.; Auerbach, B.; Aurisano, A.; Azfar, F.; Badgett, W.; Barbaro-Galtieri, A.; Barnes, V.E.; Barnett, B.A.; Barria, P.; Bartos, P.; Bauce, M.; Bauer, G.; Bedeschi, F.; Beecher, D.; Behari, S.; Bellettini, G.; Bellinger, J.; Benjamin, D.; Beretvas, A.; Bhatti, A.; Binkley, M.; Bisello, D.; Bizjak, I.; Bland, K.R.; Blumenfeld, B.; Bocci, A.; Bodek, A.; Bortoletto, D.; Boudreau, J.; Boveia, A.; Brau, B.; Brigliadori, L.; Brisuda, A.; Bromberg, C.; Brucken, E.; Bucciantonio, M.; Budagov, J.; Budd, H.S.; Budd, S.; Burkett, K.; Busetto, G.; Bussey, P.; Buzatu, A.; Calancha, C.; Camarda, S.; Campanelli, M.; Campbell, M.; Canelli, F.; Canepa, A.; Carls, B.; Carlsmith, D.; Carosi, R.; Carrillo, S.; Carron, S.; Casal, B.; Casarsa, M.; Castro, A.; Catastini, P.; Cauz, D.; Cavaliere, V.; Cavalli-Sforza, M.; Cerri, A.; Cerrito, L.; Chen, Y.C.; Chertok, M.; Chiarelli, G.; Chlachidze, G.; Chlebana, F.; Cho, K.; Chokheli, D.; Chou, J.P.; Chung, W.H.; Chung, Y.S.; Ciobanu, C.I.; Ciocci, M.A.; Clark, A.; Compostella, G.; Convery, M.E.; Conway, J.; Corbo, M.; Cordelli, M.; Cox, C.A.; Cox, D.J.; Crescioli, F.; Cuenca Almenar, C.; Cuevas, J.; Culbertson, R.; Dagenhart, D.; d'Ascenzo, N.; Datta, M.; de Barbaro, P.; De Cecco, S.; De Lorenzo, G.; Dell'Orso, M.; Deluca, C.; Demortier, L.; Deng, J.; Deninno, M.; Devoto, F.; d'Errico, M.; Di Canto, A.; Di Ruzza, B.; Dittmann, J.R.; D'Onofrio, M.; Donati, S.; Dong, P.; Dorigo, M.; Dorigo, T.; Ebina, K.; Elagin, A.; Eppig, A.; Erbacher, R.; Errede, D.; Errede, S.; Ershaidat, N.; Eusebi, R.; Fang, H.C.; Farrington, S.; Feindt, M.; Fernandez, J.P.; Ferrazza, C.; Field, R.; Flanagan, G.; Forrest, R.; Frank, M.J.; Franklin, M.; Freeman, J.C.; Funakoshi, Y.; Furic, I.; Gallinaro, M.; Galyardt, J.; Garcia, J.E.; Garfinkel, A.F.; Garosi, P.; Gerberich, H.; Gerchtein, E.; Giagu, S.; Giakoumopoulou, V.; Giannetti, P.; Gibson, K.; Ginsburg, C.M.; Giokaris, N.; Giromini, P.; Giunta, M.; Giurgiu, G.; Glagolev, V.; Glenzinski, D.; Gold, M.; Goldin, D.; Goldschmidt, N.; Golossanov, A.; Gomez, G.; Gomez-Ceballos, G.; Goncharov, M.; Gonzalez, O.; Gorelov, I.; Goshaw, A.T.; Goulianos, K.; Grinstein, S.; Grosso-Pilcher, C.; Group, R.C.; Guimaraes da Costa, J.; Gunay-Unalan, Z.; Haber, C.; Hahn, S.R.; Halkiadakis, E.; Hamaguchi, A.; Han, J.Y.; Happacher, F.; Hara, K.; Hare, D.; Hare, M.; Harr, R.F.; Hatakeyama, K.; Hays, C.; Heck, M.; Heinrich, J.; Herndon, M.; Hewamanage, S.; Hidas, D.; Hocker, A.; Hopkins, W.; Horn, D.; Hou, S.; Hughes, R.E.; Hurwitz, M.; Husemann, U.; Hussain, N.; Hussein, M.; Huston, J.; Introzzi, G.; Iori, M.; Ivanov, A.; James, E.; Jang, D.; Jayatilaka, B.; Jeon, E.J.; Jha, M.K.; Jindariani, S.; Johnson, W.; Jones, M.; Joo, K.K.; Jun, S.Y.; Junk, T.R.; Kamon, T.; Karchin, P.E.; Kato, Y.; Ketchum, W.; Keung, J.; Khotilovich, V.; Kilminster, B.; Kim, D.H.; Kim, H.S.; Kim, H.W.; Kim, J.E.; Kim, M.J.; Kim, S.B.; Kim, S.H.; Kim, Y.K.; Kimura, N.; Kirby, M.; Klimenko, S.; Kondo, K.; Kong, D.J.; Konigsberg, J.; Kotwal, A.V.; Kreps, M.; Kroll, J.; Krop, D.; Krumnack, N.; Kruse, M.; Krutelyov, V.; Kuhr, T.; Kurata, M.; Kwang, S.; Laasanen, A.T.; Lami, S.; Lammel, S.; Lancaster, M.; Lander, R.L.; Lannon, K.; Lath, A.; Latino, G.; LeCompte, T.; Lee, E.; Lee, H.S.; Lee, J.S.; Lee, S.W.; Leo, S.; Leone, S.; Lewis, J.D.; Limosani, A.; Lin, C.J.; Linacre, J.; Lindgren, M.; Lipeles, E.; Lister, A.; Litvintsev, D.O.; Liu, C.; Liu, Q.; Liu, T.; Lockwitz, S.; Lockyer, N.S.; Loginov, A.; Lucchesi, D.; Lueck, J.; Lujan, P.; Lukens, P.; Lungu, G.; Lys, J.; Lysak, R.; Madrak, R.; Maeshima, K.; Makhoul, K.; Maksimovic, P.; Malik, S.; Manca, G.; Manousakis-Katsikakis, A.; Margaroli, F.; Marino, C.; Martinez, M.; Martinez-Ballarin, R.; Mastrandrea, P.; Mathis, M.; Mattson, M.E.; Mazzanti, P.; McFarland, K.S.; McIntyre, P.; McNulty, R.; Mehta, A.; Mehtala, P.; Menzione, A.; Mesropian, C.; Miao, T.; Mietlicki, D.; Mitra, A.; Miyake, H.; Moed, S.; Moggi, N.; Mondragon, M.N.; Moon, C.S.; Moore, R.; Morello, M.J.; Morlock, J.; Movilla Fernandez, P.; Mukherjee, A.; Muller, Th.; Murat, P.; Mussini, M.; Nachtman, J.; Nagai, Y.; Naganoma, J.; Nakano, I.; Napier, A.; Nett, J.; Neu, C.; Neubauer, M.S.; Nielsen, J.; Nodulman, L.; Norniella, O.; Nurse, E.; Oakes, L.; Oh, S.H.; Oh, Y.D.; Oksuzian, I.; Okusawa, T.; Orava, R.; Ortolan, L.; Griso, S.Pagan; Pagliarone, C.; Palencia, E.; Papadimitriou, V.; Paramonov, A.A.; Patrick, J.; Pauletta, G.; Paulini, M.; Paus, C.; Pellett, D.E.; Penzo, A.; Phillips, T.J.; Piacentino, G.; Pianori, E.; Pilot, J.; Pitts, K.; Plager, C.; Pondrom, L.; Potamianos, K.; Poukhov, O.; Prokoshin, F.; Pronko, A.; Ptohos, F.; Pueschel, E.; Punzi, G.; Pursley, J.; Rahaman, A.; Ramakrishnan, V.; Ranjan, N.; Rao, K.; Redondo, I.; Renton, P.; Rescigno, M.; Rimondi, F.; Ristori, L.; Robson, A.; Rodrigo, T.; Rodriguez, T.; Rogers, E.; Rolli, S.; Roser, R.; Rossi, M.; Rubbo, F.; Ruffini, F.; Ruiz, A.; Russ, J.; Rusu, V.; Safonov, A.; Sakumoto, W.K.; Sakurai, Y.; Santi, L.; Sartori, L.; Sato, K.; Saveliev, V.; Savoy-Navarro, A.; Schlabach, P.; Schmidt, A.; Schmidt, E.E.; Schmidt, M.P.; Schmitt, M.; Schwarz, T.; Scodellaro, L.; Scribano, A.; Scuri, F.; Sedov, A.; Seidel, S.; Seiya, Y.; Semenov, A.; Sforza, F.; Sfyrla, A.; Shalhout, S.Z.; Shears, T.; Shepard, P.F.; Shimojima, M.; Shiraishi, S.; Shochet, M.; Shreyber, I.; Simonenko, A.; Sinervo, P.; Sissakian, A.; Sliwa, K.; Smith, J.R.; Snider, F.D.; Soha, A.; Somalwar, S.; Sorin, V.; Squillacioti, P.; Stancari, M.; Stanitzki, M.; Denis, R.St.; Stelzer, B.; Stelzer-Chilton, O.; Stentz, D.; Strologas, J.; Strycker, G.L.; Sudo, Y.; Sukhanov, A.; Suslov, I.; Takemasa, K.; Takeuchi, Y.; Tang, J.; Tecchio, M.; Teng, P.K.; Thom, J.; Thome, J.; Thompson, G.A.; Thomson, E.; Ttito-Guzman, P.; Tkaczyk, S.; Toback, D.; Tokar, S.; Tollefson, K.; Tomura, T.; Tonelli, D.; Torre, S.; Torretta, D.; Totaro, P.; Trovato, M.; Tu, Y.; Ukegawa, F.; Uozumi, S.; Varganov, A.; Vazquez, F.; Velev, G.; Vellidis, C.; Vidal, M.; Vila, I.; Vilar, R.; Vizan, J.; Vogel, M.; Volpi, G.; Wagner, P.; Wagner, R.L.; Wakisaka, T.; Wallny, R.; Wang, S.M.; Warburton, A.; Waters, D.; Weinberger, M.; Wester, W.C., III; Whitehouse, B.; Whiteson, D.; Wicklund, A.B.; Wicklund, E.; Wilbur, S.; Wick, F.; Williams, H.H.; Wilson, J.S.; Wilson, P.; Winer, B.L.; Wittich, P.; Wolbers, S.; Wolfe, H.; Wright, T.; Wu, X.; Wu, Z.; Yamamoto, K.; Yamaoka, J.; Yang, T.; Yang, U.K.; Yang, Y.C.; Yao, W.M.; Yeh, G.P.; Yi, K.; Yoh, J.; Yorita, K.; Yoshida, T.; Yu, G.B.; Yu, I.; Yu, S.S.; Yun, J.C.; Zanetti, A.; Zeng, Y.; Zucchelli, S. 2011-01-01 We present a new method to measure the top quark pair production cross section and the background rates with 2.7 fb^{-1}$of data from$p\\bar{p}$collisions at$\\sqrt{s} =1.96$TeV collected with the CDF II Detector. The size of the dataset was chosen to directly show the improvements of this new method. We select events with a single electron or muon, missing transverse energy, and at least one b-tagged jet. We perform a simultaneous fit to a jet flavor discriminant across nine samples defined by the number of jets and b-tags. We measure a top cross section of$\\sigma_{t\\bar{t}} = 7.64 \\pm 0.57 \\mathrm{(stat + syst)} \\pm 0.45 \\mathrm{(luminosity)}pb. An advantage of this approach is that many systematic uncertainties are measured in situ and inversely scale with integrated luminosity. 19. Δ(54) flavor phenomenology and strings Energy Technology Data Exchange (ETDEWEB) Carballo-Pérez, Brenda [Instituto de Física, Universidad Nacional Autónoma de México,Apartado Postal 20-364, Ciudad de México 01000 (Mexico); HEBA Ideas S.A. de C.V.,Calculistas 37, Cd. Mx. 09400 (Mexico); Peinado, Eduardo; Ramos-Sánchez, Saúl [Instituto de Física, Universidad Nacional Autónoma de México,Apartado Postal 20-364, Ciudad de México 01000 (Mexico) 2016-12-23 Δ(54) can serve as a flavor symmetry in particle physics, but remains almost unexplored. We show that in a classification of semi-realistic ℤ{sub 3}×ℤ{sub 3} heterotic string orbifolds, Δ(54) turns out to be the most natural flavor symmetry, providing additional motivation for its study. We revisit its phenomenological potential from a low-energy perspective and subject to the constraints of string models. We find a model with Δ(54) arising from heterotic orbifolds that leads to the Gatto-Sartori-Tonin relation for quarks and charged-leptons. Additionally, in the neutrino sector, it leads to a normal hierarchy for neutrino masses and a correlation between the reactor and the atmospheric mixing angles, the latter taking values in the second octant and being compatible at three sigmas with experimental data. 20. Search for the lepton-flavor-violating decays B(s)0→e(±)μ(∓) and B0→e(±)μ(∓). 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Williams, M; Wilson, F F; Wimberley, J; Wishahi, J; Witek, M; Wotton, S A; Wright, S; Wu, S; Wyllie, K; Xie, Y; Xing, Z; Yang, Z; Young, R; Yuan, X; Yushchenko, O; Zangoli, M; Zavertyaev, M; Zhang, F; Zhang, L; Zhang, W C; Zhang, Y; Zhelezov, A; Zhokhov, A; Zhong, L; Zvyagin, A 2013-10-04 A search for the lepton-flavor-violating decays B(s)0→e(±)μ(∓) and B0→e(±)μ(∓) is performed with a data sample, corresponding to an integrated luminosity of 1.0 fb(-1) of pp collisions at √s=7 TeV, collected by the LHCb experiment. The observed number of B(s)0→e(±)μ(∓) and B0→e(±)μ(∓) candidates is consistent with background expectations. Upper limits on the branching fractions of both decays are determined to be B(B(s)0→e(±)μ(∓))M(LQ)(B(s)0→e(±)μ(∓))>101 TeV/c(2) and M(LQ)(B0→e(±)μ(∓))>126 TeV/c(2) at 95% C.L., and are a factor of 2 higher than the previous bounds. 1. Flavor physics induced by light Z{sup ′} from SO(10) GUT Energy Technology Data Exchange (ETDEWEB) Hisano, Junji [Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,Nagoya University, Nagoya 464-8602 (Japan); Department of Physics,Nagoya University, Nagoya 464-8602 (Japan); Kavli IPMU (WPI), UTIAS, The University of Tokyo,Kashiwa, Chiba 277-8583 (Japan); Muramatsu, Yu [Kavli IPMU (WPI), UTIAS, The University of Tokyo,Kashiwa, Chiba 277-8583 (Japan); School of Physics, KIAS,Seoul 130-722 (Korea, Republic of); Omura, Yuji [Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,Nagoya University, Nagoya 464-8602 (Japan); Shigekami, Yoshihiro [Department of Physics,Nagoya University, Nagoya 464-8602 (Japan) 2016-11-04 In this paper, we investigate predictions of the SO(10) Grand Unified Theory (GUT), where an extra U(1){sup ′} gauge symmetry remains up to the supersymmetry (SUSY) breaking scale. The minimal setup of SO(10) GUT unifies quarks and leptons into a 16-representational field in each generations. The setup, however, suffers from the realization of the realistic Yukawa couplings at the electroweak scale. In order to solve this problem, we introduce 10-representational matter fields, and then the two kinds of matter fields mix with each other at the SUSY breaking scale, where the extra U(1){sup ′} gauge symmetry breaks down radiatively. One crucial prediction is that the Standard Model quarks and leptons are given by the linear combinations of the fields with two different U(1){sup ′} charges. The mixing also depends on the flavor. Consequently, the U(1){sup ′} interaction becomes flavor violating, and the flavor physics is the smoking-gun signal of our GUT model. The flavor violating Z{sup ′} couplings are related to the fermion masses and the CKM matrix, so that we can derive some explicit predictions in flavor physics. We especially discuss K-K̄ mixing, B{sub (s)}- (B{sub (s)})-bar mixing, and the (semi)leptonic decays of K and B in our model. We also study the flavor violating μ and τ decays and discuss the correlations among the physical observables in this SO(10) GUT framework. 2. SUSY searches in events with two opposite-sign same-flavor leptons, jets and MET with the CMS detector CERN Document Server Schulte, Jan-Frederik 2017-01-01 Searches for Supersymmetry (SUSY) in events with two opposite-sign same-flavour leptons offer sensitivity to the production of sleptons or Z bosons in the cascade decays of initially produced heavy SUSY particles. In the considered models, this signature is accompanied by the presence of several jets and high missing transverse energy. Analysing their respective datasets recorded at √ s = 8 TeV, the ATLAS and CMS collaborations previously reported deviations from the pre- dicted Standard Model backgrounds in this final state, with significances between 2.6 and 3.0 σ . However, these excesses had been observed in different regions of the dilepton invariant mass. The dataset recorded with the CMS detector at √ s = 13 TeV in 2015, corresponding to 2.3 fb − 1 , offers the opportunity to substantiate or refute these interesting hints for new phenomena. Unfor- tunately, no significant deviation from the background estimates are observed in either of the two selections which had shown excesses in the √ s = ... 3. An intriguing multiplet for left-handed quarks and leptons, which suggests a possible composite particle structure International Nuclear Information System (INIS) Yablon, J.R. 1989-01-01 It is shown how the internal flavor symmetries of left-handed chiral quarks and leptons within a single generation, form part of an adjoint representation of the simple local gauge group SU(4). This adjointness of representation suggests the possibility of decomposing quarks and leptons into a more basic set of preon fields, which form the fundamental representation of SU(4). While this decomposition properly accounts for the internal symmetries of quarks and leptons, it ignores their spacetime symmetries, particularly spin. To account for spin, one instead uses a 4 x 4 version of the gauge group SO(4), which reproduces all of the SU(4) internal symmetries, and also results in a more satisfactory spin content 4. Probing neutrino and Higgs sectors in SU(2){sub 1} x SU(2){sub 2} x U(1){sub Y} model with lepton-flavor non-universality Energy Technology Data Exchange (ETDEWEB) Hue, L.T. [Duy Tan University, Institute of Research and Development, Da Nang City (Viet Nam); Vietnam Academy of Science and Technology, Institute of Physics, Hanoi (Viet Nam); Arbuzov, A.B. [Joint Institute for Nuclear Researches, Bogoliubov Laboratory for Theoretical Physics, Dubna (Russian Federation); Ngan, N.T.K. [Cantho University, Department of Physics, Cantho (Viet Nam); Vietnam Academy of Science and Technology, Graduate University of Science and Technology, Hanoi (Viet Nam); Long, H.N. [Ton Duc Thang University, Theoretical Particle Physics and Cosmology Research Group, Ho Chi Minh City (Viet Nam); Ton Duc Thang University, Faculty of Applied Sciences, Ho Chi Minh City (Viet Nam) 2017-05-15 The neutrino and Higgs sectors in the SU(2){sub 1} x SU(2){sub 2} x U(1){sub Y} model with lepton-flavor non-universality are discussed. We show that active neutrinos can get Majorana masses from radiative corrections, after adding only new singly charged Higgs bosons. The mechanism for the generation of neutrino masses is the same as in the Zee models. This also gives a hint to solving the dark matter problem based on similar ways discussed recently in many radiative neutrino mass models with dark matter. Except the active neutrinos, the appearance of singly charged Higgs bosons and dark matter does not affect significantly the physical spectrum of all particles in the original model. We indicate this point by investigating the Higgs sector in both cases before and after singly charged scalars are added into it. Many interesting properties of physical Higgs bosons, which were not shown previously, are explored. In particular, the mass matrices of charged and CP-odd Higgs fields are proportional to the coefficient of triple Higgs coupling μ. The mass eigenstates and eigenvalues in the CP-even Higgs sector are also presented. All couplings of the SM-like Higgs boson to normal fermions and gauge bosons are different from the SM predictions by a factor c{sub h}, which must satisfy the recent global fit of experimental data, namely 0.995 < vertical stroke c{sub h} vertical stroke < 1. We have analyzed a more general diagonalization of gauge boson mass matrices, then we show that the ratio of the tangents of the W-W{sup '} and Z-Z{sup '} mixing angles is exactly the cosine of the Weinberg angle, implying that number of parameters is reduced by 1. Signals of new physics from decays of new heavy fermions and Higgs bosons at LHC and constraints of their masses are also discussed. (orig.) 5. Probing neutrino and Higgs sectors in { SU(2) }_1 × { SU(2) }_2 × { U(1) }_Y model with lepton-flavor non-universality Science.gov (United States) Hue, L. T.; Arbuzov, A. B.; Ngan, N. T. K.; Long, H. N. 2017-05-01 The neutrino and Higgs sectors in the { SU(2) }_1 × { SU(2) }_2 × { U(1) }_Y model with lepton-flavor non-universality are discussed. We show that active neutrinos can get Majorana masses from radiative corrections, after adding only new singly charged Higgs bosons. The mechanism for the generation of neutrino masses is the same as in the Zee models. This also gives a hint to solving the dark matter problem based on similar ways discussed recently in many radiative neutrino mass models with dark matter. Except the active neutrinos, the appearance of singly charged Higgs bosons and dark matter does not affect significantly the physical spectrum of all particles in the original model. We indicate this point by investigating the Higgs sector in both cases before and after singly charged scalars are added into it. Many interesting properties of physical Higgs bosons, which were not shown previously, are explored. In particular, the mass matrices of charged and CP-odd Higgs fields are proportional to the coefficient of triple Higgs coupling μ . The mass eigenstates and eigenvalues in the CP-even Higgs sector are also presented. All couplings of the SM-like Higgs boson to normal fermions and gauge bosons are different from the SM predictions by a factor c_h, which must satisfy the recent global fit of experimental data, namely 0.995<|c_h|<1. We have analyzed a more general diagonalization of gauge boson mass matrices, then we show that the ratio of the tangents of the W-W' and Z-Z' mixing angles is exactly the cosine of the Weinberg angle, implying that number of parameters is reduced by 1. Signals of new physics from decays of new heavy fermions and Higgs bosons at LHC and constraints of their masses are also discussed. 6. Neutrino mixing: from the broken μ-τ symmetry to the broken Friedberg–Lee symmetry International Nuclear Information System (INIS) Xing, Zhizhong 2007-01-01 I argue that the observed flavor structures of leptons and quarks might imply the existence of certain flavor symmetries. The latter should be a good starting point to build realistic models towards deeper understanding of the fermion mass spectra and flavor mixing patterns. The μ-τ permutation symmetry serves for such an example to interpret the almost maximal atmospheric neutrino mixing angle (θ 23 ~ 45°) and the strongly suppressed CHOOZ neutrino mixing angle (θ 13 < 10°). In this talk I like to highlight a new kind of flavor symmetry, the Friedberg–Lee symmetry, for the effective Majorana neutrino mass operator. Luo and I have shown that this symmetry can be broken in an oblique way, such that the lightest neutrino remains massless but an experimentally-favored neutrino mixing pattern is achievable. We get a novel prediction for θ 13 in the CP-conserving case: sinθ 13 = tanθ 12 |(1 - tanθ 23 )/(1 + tanθ 23 )|. Our scenario can simply be generalized to accommodate CP violation and be combined with the seesaw mechanism. Finally I stress the importance of probing possible effects of μ-τ symmetry breaking either in terrestrial neutrino oscillation experiments or with ultrahigh-energy cosmic neutrino telescopes. (author) 7. Fermion masses and mixings in the 3-3-1 model with right-handed neutrinos based on the S{sub 3} flavor symmetry Energy Technology Data Exchange (ETDEWEB) Hernandez, A.E.C. [Universidad Tecnica Federico Santa Maria, Valparaiso (Chile); Martinez, R.; Ochoa, F. [Universidad Nacional de Colombia, Departamento de Fisica, Bogota (Colombia) 2016-11-15 We propose a 3-3-1 model where the SU(3){sub C} x SU(3){sub L} x U(1){sub X} symmetry is extended by S{sub 3} x Z{sub 3} x Z{sub 3}{sup '} x Z{sub 8} x Z{sub 16} and the scalar spectrum is enlarged by extra SU(3){sub L} singlet scalar fields. The model successfully describes the observed SM fermion mass and mixing pattern. In this framework, the light active neutrino masses arise via an inverse seesaw mechanism and the observed charged fermion mass and quark mixing hierarchy is a consequence of the Z{sub 3} x Z{sub 3}{sup '} x Z{sub 8} x Z{sub 16} symmetry breaking at very high energy. The obtained physical observables for both quark and lepton sectors are compatible with their experimental values. The model predicts the effective Majorana neutrino mass parameter of neutrinoless double beta decay to be m{sub ββ} = 4 and 48 meV for the normal and the inverted neutrino spectra, respectively. Furthermore, we found a leptonic Dirac CP-violating phase close to (π)/(2) and a Jarlskog invariant close to about 3 x 10{sup -2} for both normal and inverted neutrino mass hierarchy. (orig.) 8. Lepton sector of a fourth generation International Nuclear Information System (INIS) Burdman, G.; Da Rold, L.; Matheus, R. D. 2010-01-01 In extensions of the standard model with a heavy fourth generation, one important question is what makes the fourth-generation lepton sector, particularly the neutrinos, so different from the lighter three generations. We study this question in the context of models of electroweak symmetry breaking in warped extra dimensions, where the flavor hierarchy is generated by choosing the localization of the zero-mode fermions in the extra dimension. In this setup the Higgs sector is localized near the infrared brane, whereas the Majorana mass term is localized at the ultraviolet brane. As a result, light neutrinos are almost entirely Majorana particles, whereas the fourth-generation neutrino is mostly a Dirac fermion. We show that it is possible to obtain heavy fourth-generation leptons in regions of parameter space where the light neutrino masses and mixings are compatible with observation. We study the impact of these bounds, as well as the ones from lepton flavor violation, on the phenomenology of these models. 9. A radiative neutrino mass model in light of DAMPE excess with hidden gauged U(1) symmetry Science.gov (United States) Nomura, Takaaki; Okada, Hiroshi; Wu, Peiwen 2018-05-01 We propose a one-loop induced neutrino mass model with hidden U(1) gauge symmetry, in which we successfully involve a bosonic dark matter (DM) candidate propagating inside a loop diagram in neutrino mass generation to explain the e+e‑ excess recently reported by the DArk Matter Particle Explorer (DAMPE) experiment. In our scenario dark matter annihilates into four leptons through Z' boson as DM DM → Z' Z' (Z' → l+ l‑) and Z' decays into leptons via one-loop effect. We then investigate branching ratios of Z' taking into account lepton flavor violations and neutrino oscillation data. 10. Scalar mass relations and flavor violations in supersymmetric theories International Nuclear Information System (INIS) Cheng, Hsin-Chia; California Univ., Berkeley, CA 1996-01-01 Supersymmetry provides the most promising solution to the gauge hierarchy problem. For supersymmetry to stablize the hierarchy, it must be broken at the weak scale. The combination of weak scale supersymmetry and grand unification leads to a successful prediction of the weak mixing angle to within 1% accuracy. If supersymmetry is a symmetry of nature, the mass spectrum and the flavor mixing pattern of the scalar superpartners of all the quarks and leptons will provide important information about a more fundamental theory at higher energies. We studied the scalar mass relations which follow from the assumption that at high energies there is a grand unified theory which leads to a significant prediction of the weak mixing angle; these will serve as important tests of grand unified theories. Two intragenerational mass relations for each of the light generations are derived. A third relation is also found which relates the Higgs masses and the masses of all three generation scalars. In a realistic supersymmetric grand unified theory, nontrivial flavor mixings are expected to exist at all gaugino vertices. This could lead to important contributions to the neutron electric dipole moment, the decay mode p → K 0 μ + , weak scale radiative corrections to the up-type quark masses, and lepton flavor violating signals such as μ → eγ. These also provide important probes of physics at high energy scales. Supersymmetric theories involving a spontaneously broken flavor symmetry can provide a solution to the supersymmetric flavor-changing problem and an understanding of the fermion masses and mixings. We studied the possibilities and the general conditions under which some fermion masses and mixings can be obtained radiatively. We also constructed theories of flavor in which the first generation fermion masses arise from radiative corrections while flavor-changing constraints are satisfied. 69 refs., 19 figs., 9 tabs 11. Flavorful hybrid anomaly-gravity mediation International Nuclear Information System (INIS) Gross, Christian; Hiller, Gudrun 2011-01-01 We consider supersymmetric models where anomaly and gravity mediation give comparable contributions to the soft terms and discuss how this can be realized in a five-dimensional brane world. The gaugino mass pattern of anomaly mediation is preserved in such a hybrid setup. The flavorful gravity-mediated contribution cures the tachyonic slepton problem of anomaly mediation. The supersymmetric flavor puzzle is solved by alignment. We explicitly show how a working flavor-tachyon link can be realized with Abelian flavor symmetries and give the characteristic signatures of the framework, including O(1) slepton mass splittings between different generations and between doublets and singlets. This provides opportunities for same flavor dilepton edge measurements with missing energy at the Large Hadron Collider (LHC). Rare lepton decay rates could be close to their current experimental limit. Compared to pure gravity mediation, the hybrid model is advantageous because it features a heavy gravitino which can avoid the cosmological gravitino problem of gravity-mediated models combined with leptogenesis. 12. Flavor-changing processes in extended technicolor International Nuclear Information System (INIS) Appelquist, Thomas; Piai, Maurizio; Christensen, Neil; Shrock, Robert 2004-01-01 We analyze constraints on a class of extended technicolor (ETC) models from neutral flavor-changing processes induced by (dimension-six) four-fermion operators. The ETC gauge group is taken to commute with the standard model gauge group. The models in the class are distinguished by how the left- and right-handed (L,R) components of the quarks and charged leptons transform under the ETC group. We consider K 0 -K 0 and other pseudoscalar meson mixings, and conclude that they are adequately suppressed if the L and R components of the relevant quarks are assigned to the same (fundamental or conjugate-fundamental) representation of the ETC group. Models in which the L and R components of the down-type quarks are assigned to relatively conjugate representations, while they can lead to realistic CKM mixing and intrafamily mass splittings, do not adequately suppress these mixing processes. We identify an approximate global symmetry that elucidates these behavioral differences and can be used to analyze other possible representation assignments. Flavor-changing decays, involving quarks and/or leptons, are adequately suppressed for any ETC representation assignment of the L and R components of the quarks, as well as the leptons. We draw lessons for future ETC model building 13. Heavy leptons International Nuclear Information System (INIS) Smith, C.H.L. 1977-01-01 The possibility that a new lepton may exist is discussed under the headings; theoretical reasons for the introduction of heavy leptons, classification of heavy leptons (ortho and paraleptons), discrimination between different types of lepton, decays of charged heavy leptons, production of charged heavy leptons (in e + e - storage rings, neutrino production, photoproduction, and hadroproduction), neutral heavy leptons, and hadroleptons. (U.K.) 14. The double mass hierarchy pattern: Simultaneously understanding quark and lepton mixing Science.gov (United States) Hollik, Wolfgang Gregor; Saldaña Salazar, Ulises Jesús 2015-03-01 The charged fermion masses of the three generations exhibit the two strong hierarchies m3 ≫m2 ≫m1. We assume that also neutrino masses satisfy mν3 >mν2 >mν1 and derive the consequences of the hierarchical spectra on the fermionic mixing patterns. The quark and lepton mixing matrices are built in a general framework with their matrix elements expressed in terms of the four fermion mass ratios, mu /mc, mc /mt, md /ms and ms /mb, and me /mμ, mμ /mτ, mν1 /mν2 and mν2 /mν3, for the quark and lepton sector, respectively. In this framework, we show that the resulting mixing matrices are consistent with data for both quarks and leptons, despite the large leptonic mixing angles. The minimal assumption we take is the one of hierarchical masses and minimal flavor symmetry breaking that strongly follows from phenomenology. No special structure of the mass matrices has to be assumed that cannot be motivated by this minimal assumption. This analysis allows us to predict the neutrino mass spectrum and set the mass of the lightest neutrino well below 0.01 eV. The method also gives the 1σ allowed ranges for the leptonic mixing matrix elements. Contrary to the common expectation, leptonic mixing angles are found to be determined solely by the four leptonic mass ratios without any relation to symmetry considerations as commonly used in flavor model building. Still, our formulae can be used to build up a flavor model that predicts the observed hierarchies in the masses - the mixing follows then from the procedure which is developed in this work. 15. Search for a heavy right-handed W boson and a heavy neutrino in events with two same-flavor leptons and two jets at\\sqrt{s} = $13 TeV CERN Document Server Sirunyan, Albert M; CMS Collaboration; Adam, Wolfgang; Ambrogi, Federico; Asilar, Ece; Bergauer, Thomas; Brandstetter, Johannes; Brondolin, Erica; Dragicevic, Marko; Erö, Janos; Escalante Del Valle, Alberto; Flechl, Martin; Friedl, Markus; Fruehwirth, Rudolf; Ghete, Vasile Mihai; Grossmann, Johannes; Hrubec, Josef; Jeitler, Manfred; König, Axel; Krammer, Natascha; Krätschmer, Ilse; Liko, Dietrich; Madlener, Thomas; Mikulec, Ivan; Pree, Elias; Rad, Navid; Rohringer, Herbert; Schieck, Jochen; Schöfbeck, Robert; Spanring, Markus; Spitzbart, Daniel; Taurok, Anton; Waltenberger, Wolfgang; Wittmann, Johannes; Wulz, Claudia-Elisabeth; Zarucki, Mateusz; Chekhovsky, Vladimir; Mossolov, Vladimir; Suarez Gonzalez, Juan; De Wolf, Eddi A; Di Croce, Davide; Janssen, Xavier; Lauwers, Jasper; Pieters, Maxim; Van De Klundert, Merijn; Van Haevermaet, Hans; Van Mechelen, Pierre; Van Remortel, Nick; Abu Zeid, Shimaa; Blekman, Freya; D'Hondt, Jorgen; De Bruyn, Isabelle; De Clercq, Jarne; Deroover, Kevin; Flouris, Giannis; Lontkovskyi, Denys; Lowette, Steven; Marchesini, Ivan; Moortgat, Seth; Moreels, Lieselotte; Python, Quentin; Skovpen, Kirill; Tavernier, Stefaan; Van Doninck, Walter; Van Mulders, Petra; Van Parijs, Isis; Beghin, Diego; Bilin, Bugra; Brun, Hugues; Clerbaux, Barbara; De Lentdecker, Gilles; Delannoy, Hugo; Dorney, Brian; Fasanella, Giuseppe; Favart, Laurent; Goldouzian, Reza; Grebenyuk, Anastasia; Kalsi, Amandeep Kaur; Lenzi, Thomas; Luetic, Jelena; Seva, Tomislav; Starling, Elizabeth; Vander Velde, Catherine; Vanlaer, Pascal; Vannerom, David; Yonamine, Ryo; Cornelis, Tom; Dobur, Didar; Fagot, Alexis; Gul, Muhammad; Khvastunov, Illia; Poyraz, Deniz; Roskas, Christos; Trocino, Daniele; Tytgat, Michael; Verbeke, Willem; Vermassen, Basile; Vit, Martina; Zaganidis, Nicolas; Bakhshiansohi, Hamed; Bondu, Olivier; Brochet, Sébastien; Bruno, Giacomo; Caputo, Claudio; Caudron, Adrien; David, Pieter; De Visscher, Simon; Delaere, Christophe; Delcourt, Martin; Francois, Brieuc; Giammanco, Andrea; Krintiras, Georgios; Lemaitre, Vincent; Magitteri, Alessio; Mertens, Alexandre; Musich, Marco; Piotrzkowski, Krzysztof; Quertenmont, Loic; Saggio, Alessia; Vidal Marono, Miguel; Wertz, Sébastien; Zobec, Joze; Aldá Júnior, Walter Luiz; Alves, Fábio Lúcio; Alves, Gilvan; Brito, Lucas; Correia Silva, Gilson; Hensel, Carsten; Moraes, Arthur; Pol, Maria Elena; Rebello Teles, Patricia; Belchior Batista Das Chagas, Ewerton; Carvalho, Wagner; Chinellato, Jose; Coelho, Eduardo; Melo Da Costa, Eliza; Da Silveira, Gustavo Gil; De Jesus Damiao, Dilson; Fonseca De Souza, Sandro; Malbouisson, Helena; Medina Jaime, Miguel; Melo De Almeida, Miqueias; Mora Herrera, Clemencia; Mundim, Luiz; Nogima, Helio; Sanchez Rosas, Luis Junior; Santoro, Alberto; Sznajder, Andre; Thiel, Mauricio; Tonelli Manganote, Edmilson José; Torres Da Silva De Araujo, Felipe; Vilela Pereira, Antonio; Ahuja, Sudha; Bernardes, Cesar Augusto; Calligaris, Luigi; Tomei, Thiago; De Moraes Gregores, Eduardo; Mercadante, Pedro G; Novaes, Sergio F; Padula, Sandra; Romero Abad, David; Ruiz Vargas, José Cupertino; Aleksandrov, Aleksandar; Hadjiiska, Roumyana; Iaydjiev, Plamen; Marinov, Andrey; Misheva, Milena; Rodozov, Mircho; Shopova, Mariana; Sultanov, Georgi; Dimitrov, Anton; Litov, Leander; Pavlov, Borislav; Petkov, Peicho; Fang, Wenxing; Gao, Xuyang; Yuan, Li; Ahmad, Muhammad; Bian, Jian-Guo; Chen, Guo-Ming; Chen, He-Sheng; Chen, Mingshui; Chen, Ye; Jiang, Chun-Hua; Leggat, Duncan; Liao, Hongbo; Liu, Zhenan; Romeo, Francesco; Shaheen, Sarmad Masood; Spiezia, Aniello; Tao, Junquan; Wang, Chunjie; Wang, Zheng; Yazgan, Efe; Zhang, Huaqiao; Zhao, Jingzhou; Ban, Yong; Chen, Geng; Li, Jing; Li, Qiang; Liu, Shuai; Mao, Yajun; Qian, Si-Jin; Wang, Dayong; Xu, Zijun; Wang, Yi; Avila, Carlos; Cabrera, Andrés; Carrillo Montoya, Camilo Andres; Chaparro Sierra, Luisa Fernanda; Florez, Carlos; González Hernández, Carlos Felipe; Segura Delgado, Manuel Alejandro; Courbon, Benoit; Godinovic, Nikola; Lelas, Damir; Puljak, Ivica; Ribeiro Cipriano, Pedro M; Sculac, Toni; Antunovic, Zeljko; Kovac, Marko; Brigljevic, Vuko; Ferencek, Dinko; Kadija, Kreso; Mesic, Benjamin; Starodumov, Andrei; Susa, Tatjana; Ather, Mohsan Waseem; Attikis, Alexandros; Mavromanolakis, Georgios; Mousa, Jehad; Nicolaou, Charalambos; Ptochos, Fotios; Razis, Panos A; Rykaczewski, Hans; Finger, Miroslav; Finger Jr, Michael; Carrera Jarrin, Edgar; Assran, Yasser; Elgammal, Sherif; Khalil, Shaaban; Bhowmik, Sandeep; Dewanjee, Ram Krishna; Kadastik, Mario; Perrini, Lucia; Raidal, Martti; Veelken, Christian; Eerola, Paula; Kirschenmann, Henning; Pekkanen, Juska; Voutilainen, Mikko; Havukainen, Joona; Heikkilä, Jaana Kristiina; Jarvinen, Terhi; Karimäki, Veikko; Kinnunen, Ritva; Lampén, Tapio; Lassila-Perini, Kati; Laurila, Santeri; Lehti, Sami; Lindén, Tomas; Luukka, Panja-Riina; Mäenpää, Teppo; Siikonen, Hannu; Tuominen, Eija; Tuominiemi, Jorma; Tuuva, Tuure; Besancon, Marc; Couderc, Fabrice; Dejardin, Marc; Denegri, Daniel; Faure, Jean-Louis; Ferri, Federico; Ganjour, Serguei; Ghosh, Saranya; Givernaud, Alain; Gras, Philippe; Hamel de Monchenault, Gautier; Jarry, Patrick; Leloup, Clément; Locci, Elizabeth; Machet, Martina; Malcles, Julie; Negro, Giulia; Rander, John; Rosowsky, André; Sahin, Mehmet Özgür; Titov, Maksym; Abdulsalam, Abdulla; Amendola, Chiara; Antropov, Iurii; Baffioni, Stephanie; Beaudette, Florian; Busson, Philippe; Cadamuro, Luca; Charlot, Claude; Granier de Cassagnac, Raphael; Jo, Mihee; Kucher, Inna; Lisniak, Stanislav; Lobanov, Artur; Martin Blanco, Javier; Nguyen, Matthew; Ochando, Christophe; Ortona, Giacomo; Paganini, Pascal; Pigard, Philipp; Salerno, Roberto; Sauvan, Jean-Baptiste; Sirois, Yves; Stahl Leiton, Andre Govinda; Yilmaz, Yetkin; Zabi, Alexandre; Zghiche, Amina; Agram, Jean-Laurent; Andrea, Jeremy; Bloch, Daniel; Brom, Jean-Marie; Chabert, Eric Christian; Collard, Caroline; Conte, Eric; Coubez, Xavier; Drouhin, Frédéric; Fontaine, Jean-Charles; Gelé, Denis; Goerlach, Ulrich; Jansová, Markéta; Juillot, Pierre; Le Bihan, Anne-Catherine; Tonon, Nicolas; Van Hove, Pierre; Gadrat, Sébastien; Beauceron, Stephanie; Bernet, Colin; Boudoul, Gaelle; Chanon, Nicolas; Chierici, Roberto; Contardo, Didier; Depasse, Pierre; El Mamouni, Houmani; Fay, Jean; Finco, Linda; Gascon, Susan; Gouzevitch, Maxime; Grenier, Gérald; Ille, Bernard; Lagarde, Francois; Laktineh, Imad Baptiste; Lattaud, Hugues; Lethuillier, Morgan; Mirabito, Laurent; Pequegnot, Anne-Laure; Perries, Stephane; Popov, Andrey; Sordini, Viola; Vander Donckt, Muriel; Viret, Sébastien; Zhang, Sijing; Toriashvili, Tengizi; Tsamalaidze, Zviad; Autermann, Christian; Feld, Lutz; Kiesel, Maximilian Knut; Klein, Katja; Lipinski, Martin; Preuten, Marius; Rauch, Max Philip; Schomakers, Christian; Schulz, Johannes; Teroerde, Marius; Wittmer, Bruno; Zhukov, Valery; Albert, Andreas; Duchardt, Deborah; Endres, Matthias; Erdmann, Martin; Erdweg, Sören; Esch, Thomas; Fischer, Robert; Güth, Andreas; Hebbeker, Thomas; Heidemann, Carsten; Hoepfner, Kerstin; Knutzen, Simon; Merschmeyer, Markus; Meyer, Arnd; Millet, Philipp; Mukherjee, Swagata; Pook, Tobias; Radziej, Markus; Reithler, Hans; Rieger, Marcel; Scheuch, Florian; Teyssier, Daniel; Thüer, Sebastian; Flügge, Günter; Kargoll, Bastian; Kress, Thomas; Künsken, Andreas; Müller, Thomas; Nehrkorn, Alexander; Nowack, Andreas; Pistone, Claudia; Pooth, Oliver; Stahl, Achim; Aldaya Martin, Maria; Arndt, Till; Asawatangtrakuldee, Chayanit; Beernaert, Kelly; Behnke, Olaf; Behrens, Ulf; Bermúdez Martínez, Armando; Bin Anuar, Afiq Aizuddin; Borras, Kerstin; Botta, Valeria; Campbell, Alan; Connor, Patrick; Contreras-Campana, Christian; Costanza, Francesco; Danilov, Vladyslav; De Wit, Adinda; Diez Pardos, Carmen; Domínguez Damiani, Daniela; Eckerlin, Guenter; Eckstein, Doris; Eichhorn, Thomas; Elwood, Adam; Eren, Engin; Gallo, Elisabetta; Garay Garcia, Jasone; Geiser, Achim; Grados Luyando, Juan Manuel; Grohsjean, Alexander; Gunnellini, Paolo; Guthoff, Moritz; Harb, Ali; Hauk, Johannes; Hempel, Maria; Jung, Hannes; Kasemann, Matthias; Keaveney, James; Kleinwort, Claus; Knolle, Joscha; Korol, Ievgen; Krücker, Dirk; Lange, Wolfgang; Lelek, Aleksandra; Lenz, Teresa; Lipka, Katerina; Lohmann, Wolfgang; Mankel, Rainer; Melzer-Pellmann, Isabell-Alissandra; Meyer, Andreas Bernhard; Meyer, Mareike; Missiroli, Marino; Mittag, Gregor; Mnich, Joachim; Mussgiller, Andreas; Pitzl, Daniel; Raspereza, Alexei; Savitskyi, Mykola; Saxena, Pooja; Shevchenko, Rostyslav; Stefaniuk, Nazar; Tholen, Heiner; Van Onsem, Gerrit Patrick; Walsh, Roberval; Wen, Yiwen; Wichmann, Katarzyna; Wissing, Christoph; Zenaiev, Oleksandr; Aggleton, Robin; Bein, Samuel; Blobel, Volker; Centis Vignali, Matteo; Dreyer, Torben; Garutti, Erika; Gonzalez, Daniel; Haller, Johannes; Hinzmann, Andreas; Hoffmann, Malte; Karavdina, Anastasia; Kasieczka, Gregor; Klanner, Robert; Kogler, Roman; Kovalchuk, Nataliia; Kurz, Simon; Kutzner, Viktor; Lange, Johannes; Marconi, Daniele; Multhaup, Jens; Niedziela, Marek; Nowatschin, Dominik; Peiffer, Thomas; Perieanu, Adrian; Reimers, Arne; Scharf, Christian; Schleper, Peter; Schmidt, Alexander; Schumann, Svenja; Schwandt, Joern; Sonneveld, Jory; Stadie, Hartmut; Steinbrück, Georg; Stober, Fred-Markus Helmut; Stöver, Marc; Troendle, Daniel; Usai, Emanuele; Vanhoefer, Annika; Vormwald, Benedikt; Akbiyik, Melike; Barth, Christian; Baselga, Marta; Baur, Sebastian; Butz, Erik; Caspart, René; Chwalek, Thorsten; Colombo, Fabio; De Boer, Wim; Dierlamm, Alexander; Faltermann, Nils; Freund, Benedikt; Friese, Raphael; Giffels, Manuel; Harrendorf, Marco Alexander; Hartmann, Frank; Heindl, Stefan Michael; Husemann, Ulrich; Kassel, Florian; Kudella, Simon; Mildner, Hannes; Mozer, Matthias Ulrich; Müller, Thomas; Plagge, Michael; Quast, Gunter; Rabbertz, Klaus; Schröder, Matthias; Shvetsov, Ivan; Sieber, Georg; Simonis, Hans-Jürgen; Ulrich, Ralf; Wayand, Stefan; Weber, Marc; Weiler, Thomas; Williamson, Shawn; Wöhrmann, Clemens; Wolf, Roger; Anagnostou, Georgios; Daskalakis, Georgios; Geralis, Theodoros; Kyriakis, Aristotelis; Loukas, Demetrios; Topsis-Giotis, Iasonas; Karathanasis, George; Kesisoglou, Stilianos; Panagiotou, Apostolos; Saoulidou, Niki; Tziaferi, Eirini; Kousouris, Konstantinos; Papakrivopoulos, Ioannis; Evangelou, Ioannis; Foudas, Costas; Gianneios, Paraskevas; Katsoulis, Panagiotis; Kokkas, Panagiotis; Mallios, Stavros; Manthos, Nikolaos; Papadopoulos, Ioannis; Paradas, Evangelos; Strologas, John; Triantis, Frixos A; Tsitsonis, Dimitrios; Csanad, Mate; Filipovic, Nicolas; Pasztor, Gabriella; Surányi, Olivér; Veres, Gabor Istvan; Bencze, Gyorgy; Hajdu, Csaba; Horvath, Dezso; Hunyadi, Ádám; Sikler, Ferenc; Veszpremi, Viktor; Vesztergombi, Gyorgy; Vámi, Tamás Álmos; Beni, Noemi; Czellar, Sandor; Karancsi, János; Makovec, Alajos; Molnar, Jozsef; Szillasi, Zoltan; Bartók, Márton; Raics, Peter; Trocsanyi, Zoltan Laszlo; Ujvari, Balazs; Choudhury, Somnath; Komaragiri, Jyothsna Rani; Bahinipati, Seema; Mal, Prolay; Mandal, Koushik; Nayak, Aruna; Sahoo, Deepak Kumar; Swain, Sanjay Kumar; Bansal, Sunil; Beri, Suman Bala; Bhatnagar, Vipin; Chauhan, Sushil; Chawla, Ridhi; Dhingra, Nitish; Gupta, Rajat; Kaur, Anterpreet; Kaur, Manjit; Kaur, Sandeep; Kumar, Ramandeep; Kumari, Priyanka; Lohan, Manisha; Mehta, Ankita; Sharma, Sandeep; Singh, Jasbir; Walia, Genius; Kumar, Ashok; Shah, Aashaq; Bhardwaj, Ashutosh; Choudhary, Brajesh C; Garg, Rocky Bala; Keshri, Sumit; Kumar, Ajay; Malhotra, Shivali; Naimuddin, Md; Ranjan, Kirti; Sharma, Ramkrishna; Bhardwaj, Rishika; Bhattacharya, Rajarshi; Bhattacharya, Satyaki; Bhawandeep, Bhawandeep; Bhowmik, Debabrata; Dey, Sourav; Dutt, Suneel; Dutta, Suchandra; Ghosh, Shamik; Majumdar, Nayana; Mondal, Kuntal; Mukhopadhyay, Supratik; Nandan, Saswati; Purohit, Arnab; Rout, Prasant Kumar; Roy, Ashim; Roy Chowdhury, Suvankar; Sarkar, Subir; Sharan, Manoj; Singh, Bipen; Thakur, Shalini; Behera, Prafulla Kumar; Chudasama, Ruchi; Dutta, Dipanwita; Jha, Vishwajeet; Kumar, Vineet; Mohanty, Ajit Kumar; Netrakanti, Pawan Kumar; Pant, Lalit Mohan; Shukla, Prashant; Topkar, Anita; Aziz, Tariq; Dugad, Shashikant; Mahakud, Bibhuprasad; Mitra, Soureek; Mohanty, Gagan Bihari; Sur, Nairit; Sutar, Bajrang; Banerjee, Sudeshna; Bhattacharya, Soham; Chatterjee, Suman; Das, Pallabi; Guchait, Monoranjan; Jain, Sandhya; Kumar, Sanjeev; Maity, Manas; Majumder, Gobinda; Mazumdar, Kajari; Sahoo, Niladribihari; Sarkar, Tanmay; Wickramage, Nadeesha; Chauhan, Shubhanshu; Dube, Sourabh; Hegde, Vinay; Kapoor, Anshul; Kothekar, Kunal; Pandey, Shubham; Rane, Aditee; Sharma, Seema; Chenarani, Shirin; Eskandari Tadavani, Esmaeel; Etesami, Seyed Mohsen; Khakzad, Mohsen; Mohammadi Najafabadi, Mojtaba; Naseri, Mohsen; Paktinat Mehdiabadi, Saeid; Rezaei Hosseinabadi, Ferdos; Safarzadeh, Batool; Zeinali, Maryam; Felcini, Marta; Grunewald, Martin; Abbrescia, Marcello; Calabria, Cesare; Colaleo, Anna; Creanza, Donato; Cristella, Leonardo; De Filippis, Nicola; De Palma, Mauro; Di Florio, Adriano; Errico, Filippo; Fiore, Luigi; Gelmi, Andrea; Iaselli, Giuseppe; Lezki, Samet; Maggi, Giorgio; Maggi, Marcello; Marangelli, Bartolomeo; Miniello, Giorgia; My, Salvatore; Nuzzo, Salvatore; Pompili, Alexis; Pugliese, Gabriella; Radogna, Raffaella; Ranieri, Antonio; Selvaggi, Giovanna; Sharma, Archana; Silvestris, Lucia; Venditti, Rosamaria; Verwilligen, Piet; Zito, Giuseppe; Abbiendi, Giovanni; Battilana, Carlo; Bonacorsi, Daniele; Borgonovi, Lisa; Braibant-Giacomelli, Sylvie; Brigliadori, Luca; Campanini, Renato; Capiluppi, Paolo; Castro, Andrea; Cavallo, Francesca Romana; Chhibra, Simranjit Singh; Codispoti, Giuseppe; Cuffiani, Marco; Dallavalle, Gaetano-Marco; Fabbri, Fabrizio; Fanfani, Alessandra; Fasanella, Daniele; Giacomelli, Paolo; Grandi, Claudio; Guiducci, Luigi; Marcellini, Stefano; Masetti, Gianni; Montanari, Alessandro; Navarria, Francesco; Perrotta, Andrea; Rossi, Antonio; Rovelli, Tiziano; Siroli, Gian Piero; Tosi, Nicolò; Albergo, Sebastiano; Costa, Salvatore; Di Mattia, Alessandro; Giordano, Ferdinando; Potenza, Renato; Tricomi, Alessia; Tuve, Cristina; Barbagli, Giuseppe; Chatterjee, Kalyanmoy; Ciulli, Vitaliano; Civinini, Carlo; D'Alessandro, Raffaello; Focardi, Ettore; Latino, Giuseppe; Lenzi, Piergiulio; Meschini, Marco; Paoletti, Simone; Russo, Lorenzo; Sguazzoni, Giacomo; Strom, Derek; Viliani, Lorenzo; Benussi, Luigi; Bianco, Stefano; Fabbri, Franco; Piccolo, Davide; Primavera, Federica; Calvelli, Valerio; Ferro, Fabrizio; Ravera, Fabio; Robutti, Enrico; Tosi, Silvano; Benaglia, Andrea; Beschi, Andrea; Brianza, Luca; Brivio, Francesco; Ciriolo, Vincenzo; Dinardo, Mauro Emanuele; Fiorendi, Sara; Gennai, Simone; Ghezzi, Alessio; Govoni, Pietro; Malberti, Martina; Malvezzi, Sandra; Manzoni, Riccardo Andrea; Menasce, Dario; Moroni, Luigi; Paganoni, Marco; Pauwels, Kristof; Pedrini, Daniele; Pigazzini, Simone; Ragazzi, Stefano; Tabarelli de Fatis, Tommaso; Buontempo, Salvatore; Cavallo, Nicola; Di Guida, Salvatore; Fabozzi, Francesco; Fienga, Francesco; Galati, Giuliana; Iorio, Alberto Orso Maria; Khan, Wajid Ali; Lista, Luca; Meola, Sabino; Paolucci, Pierluigi; Sciacca, Crisostomo; Thyssen, Filip; Voevodina, Elena; Azzi, Patrizia; Bacchetta, Nicola; Benato, Lisa; Bisello, Dario; Boletti, Alessio; Carlin, Roberto; Carvalho Antunes De Oliveira, Alexandra; Checchia, Paolo; De Castro Manzano, Pablo; Dorigo, Tommaso; Dosselli, Umberto; Gasparini, Fabrizio; Gasparini, Ugo; Gozzelino, Andrea; Lacaprara, Stefano; Margoni, Martino; Meneguzzo, Anna Teresa; Pozzobon, Nicola; Ronchese, Paolo; Rossin, Roberto; Simonetto, Franco; Tiko, Andres; Torassa, Ezio; Zanetti, Marco; Zotto, Pierluigi; Zumerle, Gianni; Braghieri, Alessandro; Magnani, Alice; Montagna, Paolo; Ratti, Sergio P; Re, Valerio; Ressegotti, Martina; Riccardi, Cristina; Salvini, Paola; Vai, Ilaria; Vitulo, Paolo; Alunni Solestizi, Luisa; Biasini, Maurizio; Bilei, Gian Mario; Cecchi, Claudia; Ciangottini, Diego; Fanò, Livio; Lariccia, Paolo; Leonardi, Roberto; Manoni, Elisa; Mantovani, Giancarlo; Mariani, Valentina; Menichelli, Mauro; Rossi, Alessandro; Santocchia, Attilio; Spiga, Daniele; Androsov, Konstantin; Azzurri, Paolo; Bagliesi, Giuseppe; Bianchini, Lorenzo; Boccali, Tommaso; Borrello, Laura; Castaldi, Rino; Ciocci, Maria Agnese; Dell'Orso, Roberto; Fedi, Giacomo; Giannini, Leonardo; Giassi, Alessandro; Grippo, Maria Teresa; Ligabue, Franco; Lomtadze, Teimuraz; Manca, Elisabetta; Mandorli, Giulio; Messineo, Alberto; Palla, Fabrizio; Rizzi, Andrea; Spagnolo, Paolo; Tenchini, Roberto; Tonelli, Guido; Venturi, Andrea; Verdini, Piero Giorgio; Barone, Luciano; Cavallari, Francesca; Cipriani, Marco; Daci, Nadir; Del Re, Daniele; Di Marco, Emanuele; Diemoz, Marcella; Gelli, Simone; Longo, Egidio; Marzocchi, Badder; Meridiani, Paolo; Organtini, Giovanni; Pandolfi, Francesco; Paramatti, Riccardo; Preiato, Federico; Rahatlou, Shahram; Rovelli, Chiara; Santanastasio, Francesco; Amapane, Nicola; Arcidiacono, Roberta; Argiro, Stefano; Arneodo, Michele; Bartosik, Nazar; Bellan, Riccardo; Biino, Cristina; Cartiglia, Nicolo; Castello, Roberto; Cenna, Francesca; Costa, Marco; Covarelli, Roberto; Degano, Alessandro; Demaria, Natale; Kiani, Bilal; Mariotti, Chiara; Maselli, Silvia; Migliore, Ernesto; Monaco, Vincenzo; Monteil, Ennio; Monteno, Marco; Obertino, Maria Margherita; Pacher, Luca; Pastrone, Nadia; Pelliccioni, Mario; Pinna Angioni, Gian Luca; Romero, Alessandra; Ruspa, Marta; Sacchi, Roberto; Shchelina, Ksenia; Sola, Valentina; Solano, Ada; Staiano, Amedeo; Belforte, Stefano; Casarsa, Massimo; Cossutti, Fabio; Della Ricca, Giuseppe; Zanetti, Anna; Kim, Dong Hee; Kim, Gui Nyun; Kim, Min Suk; Lee, Jeongeun; Lee, Sangeun; Lee, Seh Wook; Moon, Chang-Seong; Oh, Young Do; Sekmen, Sezen; Son, Dong-Chul; Yang, Yu Chul; Kim, Hyunchul; Moon, Dong Ho; Oh, Geonhee; Brochero Cifuentes, Javier Andres; Goh, Junghwan; Kim, Tae Jeong; Cho, Sungwoong; Choi, Suyong; Go, Yeonju; Gyun, Dooyeon; Ha, Seungkyu; Hong, Byung-Sik; Jo, Youngkwon; Kim, Yongsun; Lee, Kisoo; Lee, Kyong Sei; Lee, Songkyo; Lim, Jaehoon; Park, Sung Keun; Roh, Youn; Almond, John; Kim, Junho; Kim, Jae Sung; Lee, Haneol; Lee, Kyeongpil; Nam, Kyungwook; Oh, Sung Bin; Radburn-Smith, Benjamin Charles; Seo, Seon-hee; Yang, Unki; Yoo, Hwi Dong; Yu, Geum Bong; Kim, Hyunyong; Kim, Ji Hyun; Lee, Jason Sang Hun; Park, Inkyu; Choi, Young-Il; Hwang, Chanwook; Lee, Jongseok; Yu, Intae; Dudenas, Vytautas; Juodagalvis, Andrius; Vaitkus, Juozas; Ahmed, Ijaz; Ibrahim, Zainol Abidin; Md Ali, Mohd Adli Bin; Mohamad Idris, Faridah; Wan Abdullah, Wan Ahmad Tajuddin; Yusli, Mohd Nizam; Zolkapli, Zukhaimira; Reyes-Almanza, Rogelio; Ramirez-Sanchez, Gabriel; Duran-Osuna, Cecilia; Castilla-Valdez, Heriberto; De La Cruz-Burelo, Eduard; Heredia-De La Cruz, Ivan; Rabadán-Trejo, Raúl Iraq; Lopez-Fernandez, Ricardo; Mejia Guisao, Jhovanny; Sánchez Hernández, Alberto; Carrillo Moreno, Salvador; Oropeza Barrera, Cristina; Vazquez Valencia, Fabiola; Eysermans, Jan; Pedraza, Isabel; Salazar Ibarguen, Humberto Antonio; Uribe Estrada, Cecilia; Morelos Pineda, Antonio; Krofcheck, David; Bheesette, Srinidhi; Butler, Philip H; Ahmad, Ashfaq; Ahmad, Muhammad; Hassan, Qamar; Hoorani, Hafeez R; Saddique, Asif; Shah, Mehar Ali; Shoaib, Muhammad; Waqas, Muhammad; Bialkowska, Helena; Bluj, Michal; Boimska, Bozena; Frueboes, Tomasz; Górski, Maciej; Kazana, Malgorzata; Nawrocki, Krzysztof; Szleper, Michal; Traczyk, Piotr; Zalewski, Piotr; Bunkowski, Karol; Byszuk, Adrian; Doroba, Krzysztof; Kalinowski, Artur; Konecki, Marcin; Krolikowski, Jan; Misiura, Maciej; Olszewski, Michal; Pyskir, Andrzej; Walczak, Marek; Bargassa, Pedrame; Beirão Da Cruz E Silva, Cristóvão; Di Francesco, Agostino; Faccioli, Pietro; Galinhas, Bruno; Gallinaro, Michele; Hollar, Jonathan; Leonardo, Nuno; Lloret Iglesias, Lara; Nemallapudi, Mythra Varun; Seixas, Joao; Strong, Giles; Toldaiev, Oleksii; Vadruccio, Daniele; Varela, Joao; Afanasiev, Serguei; Bunin, Pavel; Gavrilenko, Mikhail; Golutvin, Igor; Gorbunov, Ilya; Kamenev, Alexey; Karjavin, Vladimir; Lanev, Alexander; Malakhov, Alexander; Matveev, Viktor; Moisenz, Petr; Palichik, Vladimir; Perelygin, Victor; Shmatov, Sergey; Shulha, Siarhei; Skatchkov, Nikolai; Smirnov, Vitaly; Voytishin, Nikolay; Zarubin, Anatoli; Ivanov, Yury; Kim, Victor; Kuznetsova, Ekaterina; Levchenko, Petr; Murzin, Victor; Oreshkin, Vadim; Smirnov, Igor; Sosnov, Dmitry; Sulimov, Valentin; Uvarov, Lev; Vavilov, Sergey; Vorobyev, Alexey; Andreev, Yuri; Dermenev, Alexander; Gninenko, Sergei; Golubev, Nikolai; Karneyeu, Anton; Kirsanov, Mikhail; Krasnikov, Nikolai; Pashenkov, Anatoli; Tlisov, Danila; Toropin, Alexander; Epshteyn, Vladimir; Gavrilov, Vladimir; Lychkovskaya, Natalia; Popov, Vladimir; Pozdnyakov, Ivan; Safronov, Grigory; Spiridonov, Alexander; Stepennov, Anton; Stolin, Viatcheslav; Toms, Maria; Vlasov, Evgueni; Zhokin, Alexander; Aushev, Tagir; Bylinkin, Alexander; Chadeeva, Marina; Parygin, Pavel; Philippov, Dmitry; Polikarpov, Sergey; Popova, Elena; Rusinov, Vladimir; Andreev, Vladimir; Azarkin, Maksim; Dremin, Igor; Kirakosyan, Martin; Rusakov, Sergey V; Terkulov, Adel; Baskakov, Alexey; Belyaev, Andrey; Boos, Edouard; Bunichev, Viacheslav; Dubinin, Mikhail; Dudko, Lev; Ershov, Alexander; Gribushin, Andrey; Klyukhin, Vyacheslav; Kodolova, Olga; Lokhtin, Igor; Miagkov, Igor; Obraztsov, Stepan; Perfilov, Maxim; Savrin, Viktor; Blinov, Vladimir; Shtol, Dmitry; Skovpen, Yuri; Azhgirey, Igor; Bayshev, Igor; Bitioukov, Sergei; Elumakhov, Dmitry; Godizov, Anton; Kachanov, Vassili; Kalinin, Alexey; Konstantinov, Dmitri; Mandrik, Petr; Petrov, Vladimir; Ryutin, Roman; Sobol, Andrei; Troshin, Sergey; Tyurin, Nikolay; Uzunian, Andrey; Volkov, Alexey; Babaev, Anton; Adzic, Petar; Cirkovic, Predrag; Devetak, Damir; Dordevic, Milos; Milosevic, Jovan; Alcaraz Maestre, Juan; Bachiller, Irene; Barrio Luna, Mar; Cerrada, Marcos; Colino, Nicanor; De La Cruz, Begona; Delgado Peris, Antonio; Fernandez Bedoya, Cristina; Fernández Ramos, Juan Pablo; Flix, Jose; Fouz, Maria Cruz; Gonzalez Lopez, Oscar; Goy Lopez, Silvia; Hernandez, Jose M; Josa, Maria Isabel; Moran, Dermot; Pérez-Calero Yzquierdo, Antonio María; Puerta Pelayo, Jesus; Redondo, Ignacio; Romero, Luciano; Senghi Soares, Mara; Triossi, Andrea; Álvarez Fernández, Adrian; Albajar, Carmen; de Trocóniz, Jorge F; Cuevas, Javier; Erice, Carlos; Fernandez Menendez, Javier; Folgueras, Santiago; Gonzalez Caballero, Isidro; González Fernández, Juan Rodrigo; Palencia Cortezon, Enrique; Sanchez Cruz, Sergio; Vischia, Pietro; Vizan Garcia, Jesus Manuel; Cabrillo, Iban Jose; Calderon, Alicia; Chazin Quero, Barbara; Duarte Campderros, Jordi; Fernandez, Marcos; Fernández Manteca, Pedro José; Garcia-Ferrero, Juan; García Alonso, Andrea; Gomez, Gervasio; Lopez Virto, Amparo; Marco, Jesus; Martinez Rivero, Celso; Martinez Ruiz del Arbol, Pablo; Matorras, Francisco; Piedra Gomez, Jonatan; Prieels, Cédric; Rodrigo, Teresa; Ruiz-Jimeno, Alberto; Scodellaro, Luca; Trevisani, Nicolò; Vila, Ivan; Vilar Cortabitarte, Rocio; Abbaneo, Duccio; Akgun, Bora; Auffray, Etiennette; Baillon, Paul; Ball, Austin; Barney, David; Bendavid, Joshua; Bianco, Michele; Bocci, Andrea; Botta, Cristina; Camporesi, Tiziano; Cepeda, Maria; Cerminara, Gianluca; Chapon, Emilien; Chen, Yi; D'Enterria, David; Dabrowski, Anne; Daponte, Vincenzo; David Tinoco Mendes, Andre; De Gruttola, Michele; De Roeck, Albert; Deelen, Nikkie; Dobson, Marc; Du Pree, Tristan; Dünser, Marc; Dupont, Niels; Elliott-Peisert, Anna; Everaerts, Pieter; Fallavollita, Francesco; Franzoni, Giovanni; Fulcher, Jonathan; Funk, Wolfgang; Gigi, Dominique; Gilbert, Andrew; Gill, Karl; Glege, Frank; Gulhan, Doga; Hegeman, Jeroen; Innocente, Vincenzo; Jafari, Abideh; Janot, Patrick; Karacheban, Olena; Kieseler, Jan; Knünz, Valentin; Kornmayer, Andreas; Krammer, Manfred; Lange, Clemens; Lecoq, Paul; Lourenco, Carlos; Lucchini, Marco Toliman; Malgeri, Luca; Mannelli, Marcello; Martelli, Arabella; Meijers, Frans; Merlin, Jeremie Alexandre; Mersi, Stefano; Meschi, Emilio; Milenovic, Predrag; Moortgat, Filip; Mulders, Martijn; Neugebauer, Hannes; Ngadiuba, Jennifer; Orfanelli, Styliani; Orsini, Luciano; Pantaleo, Felice; Pape, Luc; Perez, Emmanuel; Peruzzi, Marco; Petrilli, Achille; Petrucciani, Giovanni; Pfeiffer, Andreas; Pierini, Maurizio; Pitters, Florian Michael; Rabady, Dinyar; Racz, Attila; Reis, Thomas; Rolandi, Gigi; Rovere, Marco; Sakulin, Hannes; Schäfer, Christoph; Schwick, Christoph; Seidel, Markus; Selvaggi, Michele; Sharma, Archana; Silva, Pedro; Sphicas, Paraskevas; Stakia, Anna; Steggemann, Jan; Stoye, Markus; Tosi, Mia; Treille, Daniel; Tsirou, Andromachi; Veckalns, Viesturs; Verweij, Marta; Zeuner, Wolfram Dietrich; Bertl, Willi; Caminada, Lea; Deiters, Konrad; Erdmann, Wolfram; Horisberger, Roland; Ingram, Quentin; Kaestli, Hans-Christian; Kotlinski, Danek; Langenegger, Urs; Rohe, Tilman; Wiederkehr, Stephan Albert; Backhaus, Malte; Bäni, Lukas; Berger, Pirmin; Casal, Bruno; Chernyavskaya, Nadezda; Dissertori, Günther; Dittmar, Michael; Donegà, Mauro; Dorfer, Christian; Grab, Christoph; Heidegger, Constantin; Hits, Dmitry; Hoss, Jan; Klijnsma, Thomas; Lustermann, Werner; Marionneau, Matthieu; Meinhard, Maren Tabea; Meister, Daniel; Micheli, Francesco; Musella, Pasquale; Nessi-Tedaldi, Francesca; Pata, Joosep; Pauss, Felicitas; Perrin, Gaël; Perrozzi, Luca; Quittnat, Milena; Reichmann, Michael; Ruini, Daniele; Sanz Becerra, Diego Alejandro; Schönenberger, Myriam; Shchutska, Lesya; Tavolaro, Vittorio Raoul; Theofilatos, Konstantinos; Vesterbacka Olsson, Minna Leonora; Wallny, Rainer; Zhu, De Hua; Aarrestad, Thea Klaeboe; Amsler, Claude; Brzhechko, Danyyl; Canelli, Maria Florencia; De Cosa, Annapaola; Del Burgo, Riccardo; Donato, Silvio; Galloni, Camilla; Hreus, Tomas; Kilminster, Benjamin; Neutelings, Izaak; Pinna, Deborah; Rauco, Giorgia; Robmann, Peter; Salerno, Daniel; Schweiger, Korbinian; Seitz, Claudia; Takahashi, Yuta; Zucchetta, Alberto; Candelise, Vieri; Chang, Yu-Hsiang; Cheng, Kai-yu; Doan, Thi Hien; Jain, Shilpi; Khurana, Raman; Kuo, Chia-Ming; Lin, Willis; Pozdnyakov, Andrey; Yu, Shin-Shan; 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Smith, Nicholas; Smith, Wesley H; Woods, Nathaniel 2018-01-01 A search for a heavy right-handed W boson ($ \\mathrm{ W_R } $) decaying to a heavy right-handed neutrino and a charged lepton in events with two same-flavor leptons (e or$ \\mu $) and two jets, is presented. The analysis is based on proton-proton collision data, collected by the CMS Collaboration at the LHC in 2016 and corresponding to an integrated luminosity of 35.9 fb$^{-1}$. No significant excess above the standard model expectation is seen in the invariant mass distribution of the dilepton plus dijet system. Assuming that couplings are identical to those of the standard model, and that only one heavy neutrino flavor${\\mathrm {N_R}}$contributes significantly to the$ \\mathrm{ W_R } $decay width, the region in the two-dimensional ($ {m_{ \\mathrm{ W_R } }} $,$ {m_{{\\mathrm {N_R}} }} $) mass plane excluded at 95% confidence level extends to approximately${m_{ \\mathrm{ W_R } }} = $4.4 TeV and covers a large range of right-handed neutrino masses below the$ \\mathrm{ W_R } $boson mass. This analysis prov... 16. Search for lepton-flavor violating decays of heavy resonances and quantum black holes to e$\\mu$final states in proton-proton collisions at$\\sqrt{s}=$13 TeV Energy Technology Data Exchange (ETDEWEB) Sirunyan, Albert M; et al. 2018-02-04 A search is reported for heavy resonances decaying into e$\\mu$final states in proton-proton collisions recorded by the CMS experiment at the CERN LHC at$\\sqrt{s}=$13 TeV, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The search focuses on resonance masses above 200 GeV. With no evidence found for physics beyond the standard model in the e$\\mu$mass spectrum, upper limits are set at 95% confidence level on the product of the cross section and branching fraction for this lepton-flavor violating signal. Based on these results, resonant$\\tau$sneutrino production in R-parity violating supersymmetric models is excluded for masses below 1.7 TeV, for couplings$\\lambda_{132}=\\lambda_{231}=\\lambda'_{311}=0.01$. Heavy Z$'$gauge bosons with lepton-flavor violating transitions are excluded for masses up to 4.4 TeV. The e$\\mu$mass spectrum is also interpreted in terms of non-resonant contributions from quantum black-hole production in models with one to six extra spatial dimensions, and lower mass limits are found between 3.6 and 5.6 TeV. In all interpretations used in this analysis, the results of this search improve previous limits by about 1 TeV. These limits correspond to the most sensitive values obtained at colliders. 17. Search for lepton-flavor violating decays of heavy resonances and quantum black holes to e μ final states in proton-proton collisions at √{s}=13 TeV Science.gov (United States) Sirunyan, A. M.; Tumasyan, A.; Adam, W.; Ambrogi, F.; Asilar, E.; Bergauer, T.; Brandstetter, J.; Brondolin, E.; Dragicevic, M.; Erö, J.; Escalante Del Valle, A.; Flechl, M.; Friedl, M.; Frühwirth, R.; Ghete, V. M.; Grossmann, J.; Hrubec, J.; Jeitler, M.; König, A.; Krammer, N.; Krätschmer, I.; Liko, D.; Madlener, T.; Mikulec, I.; Pree, E.; Rad, N.; Rohringer, H.; Schieck, J.; Schöfbeck, R.; Spanring, M.; Spitzbart, D.; Taurok, A.; Waltenberger, W.; Wittmann, J.; Wulz, C.-E.; Zarucki, M.; Chekhovsky, V.; Mossolov, V.; Suarez Gonzalez, J.; De Wolf, E. A.; Di Croce, D.; Janssen, X.; Lauwers, J.; Pieters, M.; Van De Klundert, M.; Van Haevermaet, H.; Van Mechelen, P.; Van Remortel, N.; Abu Zeid, S.; Blekman, F.; D'Hondt, J.; De Bruyn, I.; De Clercq, J.; Deroover, K.; Flouris, G.; Lontkovskyi, D.; Lowette, S.; Marchesini, I.; Moortgat, S.; Moreels, L.; Python, Q.; Skovpen, K.; Tavernier, S.; Van Doninck, W.; Van Mulders, P.; Van Parijs, I.; Beghin, D.; Bilin, B.; Brun, H.; Clerbaux, B.; De Lentdecker, G.; Delannoy, H.; Dorney, B.; Fasanella, G.; Favart, L.; Goldouzian, R.; Grebenyuk, A.; Kalsi, A. K.; Lenzi, T.; Luetic, J.; Seva, T.; Starling, E.; Vander Velde, C.; Vanlaer, P.; Vannerom, D.; Yonamine, R.; Cornelis, T.; Dobur, D.; Fagot, A.; Gul, M.; Khvastunov, I.; Poyraz, D.; Roskas, C.; Trocino, D.; Tytgat, M.; Verbeke, W.; Vermassen, B.; Vit, M.; Zaganidis, N.; Bakhshiansohi, H.; Bondu, O.; Brochet, S.; Bruno, G.; Caputo, C.; Caudron, A.; David, P.; De Visscher, S.; Delaere, C.; Delcourt, M.; Francois, B.; Giammanco, A.; Krintiras, G.; Lemaitre, V.; Magitteri, A.; Mertens, A.; Musich, M.; Piotrzkowski, K.; Quertenmont, L.; Saggio, A.; Vidal Marono, M.; Wertz, S.; Zobec, J.; Aldá Júnior, W. L.; Alves, F. L.; Alves, G. A.; Brito, L.; Correia Silva, G.; Hensel, C.; Moraes, A.; Pol, M. E.; Rebello Teles, P.; Belchior Batista Das Chagas, E.; Carvalho, W.; Chinellato, J.; Coelho, E.; Da Costa, E. M.; Da Silveira, G. G.; De Jesus Damiao, D.; Fonseca De Souza, S.; Huertas Guativa, L. 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W.; Moon, C. S.; Oh, Y. D.; Sekmen, S.; Son, D. C.; Yang, Y. C.; Kim, H.; Moon, D. H.; Oh, G.; Brochero Cifuentes, J. A.; Goh, J.; Kim, T. J.; Cho, S.; Choi, S.; Go, Y.; Gyun, D.; Ha, S.; Hong, B.; Jo, Y.; Kim, Y.; Lee, K.; Lee, K. S.; Lee, S.; Lim, J.; Park, S. K.; Roh, Y.; Almond, J.; Kim, J.; Kim, J. S.; Lee, H.; Lee, K.; Nam, K.; Oh, S. B.; Radburn-Smith, B. C.; Seo, S. h.; Yang, U. K.; Yoo, H. D.; Yu, G. B.; Kim, H.; Kim, J. H.; Lee, J. S. H.; Park, I. C.; Choi, Y.; Hwang, C.; Lee, J.; Yu, I.; Dudenas, V.; Juodagalvis, A.; Vaitkus, J.; Ahmed, I.; Ibrahim, Z. A.; Md Ali, M. A. B.; Mohamad Idris, F.; Wan Abdullah, W. A. T.; Yusli, M. N.; Zolkapli, Z.; Reyes-Almanza, R.; Ramirez-Sanchez, G.; Duran-Osuna, M. C.; Castilla-Valdez, H.; De La Cruz-Burelo, E.; Heredia-De La Cruz, I.; Rabadan-Trejo, R. I.; Lopez-Fernandez, R.; Mejia Guisao, J.; Sanchez-Hernandez, A.; Carrillo Moreno, S.; Oropeza Barrera, C.; Vazquez Valencia, F.; Eysermans, J.; Pedraza, I.; Salazar Ibarguen, H. A.; Uribe Estrada, C.; Morelos Pineda, A.; Krofcheck, D.; Butler, P. H.; Ahmad, A.; Ahmad, M.; Hassan, Q.; Hoorani, H. R.; Saddique, A.; Shah, M. A.; Shoaib, M.; Waqas, M.; Bialkowska, H.; Bluj, M.; Boimska, B.; Frueboes, T.; Górski, M.; Kazana, M.; Nawrocki, K.; Szleper, M.; Traczyk, P.; Zalewski, P.; Bunkowski, K.; Byszuk, A.; Doroba, K.; Kalinowski, A.; Konecki, M.; Krolikowski, J.; Misiura, M.; Olszewski, M.; Pyskir, A.; Walczak, M.; Bargassa, P.; Beirão Da Cruz E Silva, C.; Di Francesco, A.; Faccioli, P.; Galinhas, B.; Gallinaro, M.; Hollar, J.; Leonardo, N.; Lloret Iglesias, L.; Nemallapudi, M. V.; Seixas, J.; Strong, G.; Toldaiev, O.; Vadruccio, D.; Varela, J.; Afanasiev, S.; Bunin, P.; Gavrilenko, M.; Golutvin, I.; Gorbunov, I.; Kamenev, A.; Karjavin, V.; Lanev, A.; Malakhov, A.; Matveev, V.; Moisenz, P.; Palichik, V.; Perelygin, V.; Shmatov, S.; Shulha, S.; Skatchkov, N.; Smirnov, V.; Voytishin, N.; Zarubin, A.; Ivanov, Y.; Kim, V.; Kuznetsova, E.; Levchenko, P.; Murzin, V.; Oreshkin, V.; Smirnov, I.; Sosnov, D.; Sulimov, V.; Uvarov, L.; Vavilov, S.; Vorobyev, A.; Andreev, Yu.; Dermenev, A.; Gninenko, S.; Golubev, N.; Karneyeu, A.; Kirsanov, M.; Krasnikov, N.; Pashenkov, A.; Tlisov, D.; Toropin, A.; Epshteyn, V.; Gavrilov, V.; Lychkovskaya, N.; Popov, V.; Pozdnyakov, I.; Safronov, G.; Spiridonov, A.; Stepennov, A.; Stolin, V.; Toms, M.; Vlasov, E.; Zhokin, A.; Aushev, T.; Bylinkin, A.; Chadeeva, M.; Chistov, R.; Parygin, P.; Philippov, D.; Polikarpov, S.; Tarkovskii, E.; Zhemchugov, E.; Andreev, V.; Azarkin, M.; Dremin, I.; Kirakosyan, M.; Rusakov, S. V.; Terkulov, A.; Baskakov, A.; Belyaev, A.; Boos, E.; Bunichev, V.; Dubinin, M.; Dudko, L.; Ershov, A.; Gribushin, A.; Klyukhin, V.; Kodolova, O.; Lokhtin, I.; Miagkov, I.; Obraztsov, S.; Petrushanko, S.; Savrin, V.; Blinov, V.; Shtol, D.; Skovpen, Y.; Azhgirey, I.; Bayshev, I.; Bitioukov, S.; Elumakhov, D.; Godizov, A.; Kachanov, V.; Kalinin, A.; Konstantinov, D.; Mandrik, P.; Petrov, V.; Ryutin, R.; Sobol, A.; Troshin, S.; Tyurin, N.; Uzunian, A.; Volkov, A.; Babaev, A.; Adzic, P.; Cirkovic, P.; Devetak, D.; Dordevic, M.; Milosevic, J.; Alcaraz Maestre, J.; Bachiller, I.; Barrio Luna, M.; Cerrada, M.; Colino, N.; De La Cruz, B.; Delgado Peris, A.; Fernandez Bedoya, C.; Fernández Ramos, J. P.; Flix, J.; Fouz, M. C.; Gonzalez Lopez, O.; Goy Lopez, S.; Hernandez, J. M.; Josa, M. I.; Moran, D.; Pérez-Calero Yzquierdo, A.; Puerta Pelayo, J.; Redondo, I.; Romero, L.; Soares, M. S.; Triossi, A.; Álvarez Fernández, A.; Albajar, C.; de Trocóniz, J. 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F.; Sun, J.; Wang, F.; Xiao, R.; Xie, W.; Cheng, T.; Parashar, N.; Chen, Z.; Ecklund, K. M.; Freed, S.; Geurts, F. J. M.; Guilbaud, M.; Kilpatrick, M.; Li, W.; Michlin, B.; Padley, B. P.; Roberts, J.; Rorie, J.; Shi, W.; Tu, Z.; Zabel, J.; Zhang, A.; Bodek, A.; de Barbaro, P.; Demina, R.; Duh, Y. t.; Ferbel, T.; Galanti, M.; Garcia-Bellido, A.; Han, J.; Hindrichs, O.; Khukhunaishvili, A.; Lo, K. H.; Tan, P.; Verzetti, M.; Ciesielski, R.; Goulianos, K.; Mesropian, C.; Agapitos, A.; Chou, J. P.; Gershtein, Y.; Gómez Espinosa, T. A.; Halkiadakis, E.; Heindl, M.; Hughes, E.; Kaplan, S.; Kunnawalkam Elayavalli, R.; Kyriacou, S.; Lath, A.; Montalvo, R.; Nash, K.; Osherson, M.; Saka, H.; Salur, S.; Schnetzer, S.; Sheffield, D.; Somalwar, S.; Stone, R.; Thomas, S.; Thomassen, P.; Walker, M.; Delannoy, A. G.; Heideman, J.; Riley, G.; Rose, K.; Spanier, S.; Thapa, K.; Bouhali, O.; Castaneda Hernandez, A.; Celik, A.; Dalchenko, M.; De Mattia, M.; Delgado, A.; Dildick, S.; Eusebi, R.; Gilmore, J.; Huang, T.; Kamon, T.; Mueller, R.; Pakhotin, Y.; Patel, R.; Perloff, A.; Perniè, L.; Rathjens, D.; Safonov, A.; Tatarinov, A.; Akchurin, N.; Damgov, J.; De Guio, F.; Dudero, P. R.; Faulkner, J.; Gurpinar, E.; Kunori, S.; Lamichhane, K.; Lee, S. W.; Mengke, T.; Muthumuni, S.; Peltola, T.; Undleeb, S.; Volobouev, I.; Wang, Z.; Greene, S.; Gurrola, A.; Janjam, R.; Johns, W.; Maguire, C.; Melo, A.; Ni, H.; Padeken, K.; Ruiz Alvarez, J. D.; Sheldon, P.; Tuo, S.; Velkovska, J.; Xu, Q.; Arenton, M. W.; Barria, P.; Cox, B.; Hirosky, R.; Joyce, M.; Ledovskoy, A.; Li, H.; Neu, C.; Sinthuprasith, T.; Wang, Y.; Wolfe, E.; Xia, F.; Harr, R.; Karchin, P. E.; Poudyal, N.; Sturdy, J.; Thapa, P.; Zaleski, S.; Brodski, M.; Buchanan, J.; Caillol, C.; Carlsmith, D.; Dasu, S.; Dodd, L.; Duric, S.; Gomber, B.; Grothe, M.; Herndon, M.; Hervé, A.; Hussain, U.; Klabbers, P.; Lanaro, A.; Levine, A.; Long, K.; Loveless, R.; Rekovic, V.; Ruggles, T.; Savin, A.; Smith, N.; Smith, W. H.; Woods, N. 2018-04-01 A search is reported for heavy resonances decaying into e μ final states in proton-proton collisions recorded by the CMS experiment at the CERN LHC at √{s}=13 TeV, corresponding to an integrated luminosity of 35.9 fb-1. The search focuses on resonance masses above 200 GeV. With no evidence found for physics beyond the standard model in the e μ mass spectrum, upper limits are set at 95% confidence level on the product of the cross section and branching fraction for this lepton-flavor violating signal. Based on these results, resonant τ sneutrino production in R-parity violating supersymmetric models is excluded for masses below 1.7 TeV, for couplings λ 132 = λ 231 = λ 311 ' = 0.01. Heavy Z' gauge bosons with lepton-flavor violating transitions are excluded for masses up to 4.4 TeV. The e μ mass spectrum is also interpreted in terms of non-resonant contributions from quantum black-hole production in models with one to six extra spatial dimensions, and lower mass limits are found between 3.6 and 5.6 TeV. In all interpretations used in this analysis, the results of this search improve previous limits by about 1 TeV. These limits correspond to the most sensitive values obtained at colliders. [Figure not available: see fulltext. 18. arXiv Search for lepton-flavor violating decays of heavy resonances and quantum black holes to e$\\mu$final states in proton-proton collisions at$\\sqrt{s}=$13 TeV CERN Document Server Sirunyan, Albert M; CMS Collaboration; Adam, Wolfgang; Ambrogi, Federico; Asilar, Ece; Bergauer, Thomas; Brandstetter, Johannes; Brondolin, Erica; Dragicevic, Marko; Erö, Janos; Escalante Del Valle, Alberto; Flechl, Martin; Friedl, Markus; Fruehwirth, Rudolf; Ghete, Vasile Mihai; Grossmann, Johannes; Hrubec, Josef; Jeitler, Manfred; König, Axel; Krammer, Natascha; Krätschmer, Ilse; Liko, Dietrich; Madlener, Thomas; Mikulec, Ivan; Pree, Elias; Rad, Navid; Rohringer, Herbert; Schieck, Jochen; Schöfbeck, Robert; Spanring, Markus; Spitzbart, Daniel; Taurok, Anton; Waltenberger, Wolfgang; Wittmann, Johannes; Wulz, Claudia-Elisabeth; Zarucki, Mateusz; Chekhovsky, Vladimir; Mossolov, Vladimir; Suarez Gonzalez, Juan; De Wolf, Eddi A; Di Croce, Davide; Janssen, Xavier; Lauwers, Jasper; 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Bakhshiansohi, Hamed; Bondu, Olivier; Brochet, Sébastien; Bruno, Giacomo; Caputo, Claudio; Caudron, Adrien; David, Pieter; De Visscher, Simon; Delaere, Christophe; Delcourt, Martin; Francois, Brieuc; Giammanco, Andrea; Krintiras, Georgios; Lemaitre, Vincent; Magitteri, Alessio; Mertens, Alexandre; Musich, Marco; Piotrzkowski, Krzysztof; Quertenmont, Loic; Saggio, Alessia; Vidal Marono, Miguel; Wertz, Sébastien; Zobec, Joze; Aldá Júnior, Walter Luiz; Alves, Fábio Lúcio; Alves, Gilvan; Brito, Lucas; Correia Silva, Gilson; Hensel, Carsten; Moraes, Arthur; Pol, Maria Elena; Rebello Teles, Patricia; Belchior Batista Das Chagas, Ewerton; Carvalho, Wagner; Chinellato, Jose; Coelho, Eduardo; Melo Da Costa, Eliza; Da Silveira, Gustavo Gil; De Jesus Damiao, Dilson; Fonseca De Souza, Sandro; Huertas Guativa, Lina Milena; Malbouisson, Helena; Medina Jaime, Miguel; Melo De Almeida, Miqueias; Mora Herrera, Clemencia; Mundim, Luiz; Nogima, Helio; Sanchez Rosas, Luis Junior; Santoro, Alberto; Sznajder, Andre; Thiel, Mauricio; Tonelli Manganote, Edmilson José; Torres Da Silva De Araujo, Felipe; Vilela Pereira, Antonio; Ahuja, Sudha; Bernardes, Cesar Augusto; Tomei, Thiago; De Moraes Gregores, Eduardo; Mercadante, Pedro G; Novaes, Sergio F; Padula, Sandra; Romero Abad, David; Ruiz Vargas, José Cupertino; Aleksandrov, Aleksandar; Hadjiiska, Roumyana; Iaydjiev, Plamen; Marinov, Andrey; Misheva, Milena; Rodozov, Mircho; Shopova, Mariana; Sultanov, Georgi; Dimitrov, Anton; Litov, Leander; Pavlov, Borislav; Petkov, Peicho; Fang, Wenxing; Gao, Xuyang; Yuan, Li; Ahmad, Muhammad; Bian, Jian-Guo; Chen, Guo-Ming; Chen, He-Sheng; Chen, Mingshui; Chen, Ye; Jiang, Chun-Hua; Leggat, Duncan; Liao, Hongbo; Liu, Zhenan; Romeo, Francesco; Shaheen, Sarmad Masood; Spiezia, Aniello; Tao, Junquan; Wang, Chunjie; Wang, Zheng; Yazgan, Efe; Zhang, Huaqiao; Zhao, Jingzhou; Ban, Yong; Chen, Geng; Li, Jing; Li, Qiang; Liu, Shuai; Mao, Yajun; Qian, Si-Jin; Wang, Dayong; Xu, Zijun; Wang, Yi; Avila, Carlos; Cabrera, Andrés; Carrillo Montoya, Camilo Andres; Chaparro Sierra, Luisa Fernanda; Florez, Carlos; González Hernández, Carlos Felipe; Segura Delgado, Manuel Alejandro; Courbon, Benoit; Godinovic, Nikola; Lelas, Damir; Puljak, Ivica; Ribeiro Cipriano, Pedro M; Sculac, Toni; Antunovic, Zeljko; Kovac, Marko; Brigljevic, Vuko; Ferencek, Dinko; Kadija, Kreso; Mesic, Benjamin; Starodumov, Andrei; Susa, Tatjana; Ather, Mohsan Waseem; Attikis, Alexandros; Mavromanolakis, Georgios; Mousa, Jehad; Nicolaou, Charalambos; Ptochos, Fotios; Razis, Panos A; Rykaczewski, Hans; Finger, Miroslav; Finger Jr, Michael; Carrera Jarrin, Edgar; Abdalla, Hassan; Assran, Yasser; Elgammal, Sherif; Bhowmik, Sandeep; Dewanjee, Ram Krishna; Kadastik, Mario; Perrini, Lucia; Raidal, Martti; Veelken, Christian; Eerola, Paula; Kirschenmann, Henning; Pekkanen, Juska; Voutilainen, Mikko; Havukainen, Joona; Heikkilä, Jaana Kristiina; Jarvinen, Terhi; Karimäki, Veikko; Kinnunen, Ritva; Lampén, Tapio; Lassila-Perini, Kati; Laurila, Santeri; Lehti, Sami; Lindén, Tomas; Luukka, Panja-Riina; Mäenpää, Teppo; Siikonen, Hannu; Tuominen, Eija; Tuominiemi, Jorma; Tuuva, Tuure; Besancon, Marc; Couderc, Fabrice; Dejardin, Marc; Denegri, Daniel; Faure, Jean-Louis; Ferri, Federico; Ganjour, Serguei; Ghosh, Saranya; Givernaud, Alain; Gras, Philippe; Hamel de Monchenault, Gautier; Jarry, Patrick; Leloup, Clément; Locci, Elizabeth; Machet, Martina; Malcles, Julie; Negro, Giulia; Rander, John; Rosowsky, André; Sahin, Mehmet Özgür; Titov, Maksym; Abdulsalam, Abdulla; Amendola, Chiara; Antropov, Iurii; Baffioni, Stephanie; Beaudette, Florian; Busson, Philippe; Cadamuro, Luca; Charlot, Claude; Granier de Cassagnac, Raphael; Jo, Mihee; Kucher, Inna; Lisniak, Stanislav; Lobanov, Artur; Martin Blanco, Javier; Nguyen, Matthew; Ochando, Christophe; Ortona, Giacomo; Paganini, Pascal; Pigard, Philipp; Salerno, Roberto; Sauvan, Jean-Baptiste; Sirois, Yves; Stahl Leiton, Andre Govinda; Yilmaz, Yetkin; Zabi, Alexandre; Zghiche, Amina; Agram, Jean-Laurent; Andrea, Jeremy; Bloch, Daniel; Brom, Jean-Marie; Buttignol, Michael; Chabert, Eric Christian; Collard, Caroline; Conte, Eric; Coubez, Xavier; Drouhin, Frédéric; Fontaine, Jean-Charles; Gelé, Denis; Goerlach, Ulrich; Jansová, Markéta; Juillot, Pierre; Le Bihan, Anne-Catherine; Tonon, Nicolas; Van Hove, Pierre; Gadrat, Sébastien; Beauceron, Stephanie; Bernet, Colin; Boudoul, Gaelle; Chanon, Nicolas; Chierici, Roberto; Contardo, Didier; Depasse, Pierre; El Mamouni, Houmani; Fay, Jean; Finco, Linda; Gascon, Susan; Gouzevitch, Maxime; Grenier, Gérald; Ille, Bernard; Lagarde, Francois; Laktineh, Imad Baptiste; Lattaud, Hugues; Lethuillier, Morgan; Mirabito, Laurent; Pequegnot, Anne-Laure; Perries, Stephane; Popov, Andrey; Sordini, Viola; Vander Donckt, Muriel; Viret, Sébastien; Zhang, Sijing; Toriashvili, Tengizi; Tsamalaidze, Zviad; Autermann, Christian; Feld, Lutz; Kiesel, Maximilian Knut; Klein, Katja; Lipinski, Martin; Preuten, Marius; Schomakers, Christian; Schulz, Johannes; Teroerde, Marius; Wittmer, Bruno; Zhukov, Valery; Albert, Andreas; Duchardt, Deborah; Endres, Matthias; Erdmann, Martin; Erdweg, Sören; Esch, Thomas; Fischer, Robert; Güth, Andreas; Hebbeker, Thomas; Heidemann, Carsten; Hoepfner, Kerstin; Knutzen, Simon; Merschmeyer, Markus; Meyer, Arnd; Millet, Philipp; Mukherjee, Swagata; Pook, Tobias; Radziej, Markus; Reithler, Hans; Rieger, Marcel; Scheuch, Florian; Teyssier, Daniel; Thüer, Sebastian; Flügge, Günter; Kargoll, Bastian; Kress, Thomas; Künsken, Andreas; Müller, Thomas; Nehrkorn, Alexander; Nowack, Andreas; Pistone, Claudia; Pooth, Oliver; Stahl, Achim; Aldaya Martin, Maria; Arndt, Till; Asawatangtrakuldee, Chayanit; Beernaert, Kelly; Behnke, Olaf; Behrens, Ulf; Bermúdez Martínez, Armando; Bin Anuar, Afiq Aizuddin; Borras, Kerstin; Botta, Valeria; Campbell, Alan; Connor, Patrick; Contreras-Campana, Christian; Costanza, Francesco; Danilov, Vladyslav; De Wit, Adinda; Diez Pardos, Carmen; Domínguez Damiani, Daniela; Eckerlin, Guenter; Eckstein, Doris; Eichhorn, Thomas; Eren, Engin; Gallo, Elisabetta; Garay Garcia, Jasone; Geiser, Achim; Grados Luyando, Juan Manuel; Grohsjean, Alexander; Gunnellini, Paolo; Guthoff, Moritz; Harb, Ali; Hauk, Johannes; Hempel, Maria; Jung, Hannes; Kasemann, Matthias; Keaveney, James; Kleinwort, Claus; Knolle, Joscha; Korol, Ievgen; Krücker, Dirk; Lange, Wolfgang; Lelek, Aleksandra; Lenz, Teresa; Lipka, Katerina; Lohmann, Wolfgang; Mankel, Rainer; Melzer-Pellmann, Isabell-Alissandra; Meyer, Andreas Bernhard; Meyer, Mareike; Missiroli, Marino; Mittag, Gregor; Mnich, Joachim; Mussgiller, Andreas; Pitzl, Daniel; Raspereza, Alexei; Savitskyi, Mykola; Saxena, Pooja; Shevchenko, Rostyslav; Stefaniuk, Nazar; Tholen, Heiner; Van Onsem, Gerrit Patrick; Walsh, Roberval; Wen, Yiwen; Wichmann, Katarzyna; Wissing, Christoph; Zenaiev, Oleksandr; Aggleton, Robin; Bein, Samuel; Blobel, Volker; Centis Vignali, Matteo; Dreyer, Torben; Garutti, Erika; Gonzalez, Daniel; Haller, Johannes; Hinzmann, Andreas; Hoffmann, Malte; Karavdina, Anastasia; Kasieczka, Gregor; Klanner, Robert; Kogler, Roman; Kovalchuk, Nataliia; Kurz, Simon; Marconi, Daniele; Multhaup, Jens; Niedziela, Marek; Nowatschin, Dominik; Peiffer, Thomas; Perieanu, Adrian; Reimers, Arne; Scharf, Christian; Schleper, Peter; Schmidt, Alexander; Schumann, Svenja; Schwandt, Joern; Sonneveld, Jory; Stadie, Hartmut; Steinbrück, Georg; Stober, Fred-Markus Helmut; Stöver, Marc; Troendle, Daniel; Usai, Emanuele; Vanhoefer, Annika; Vormwald, Benedikt; Akbiyik, Melike; Barth, Christian; Baselga, Marta; Baur, Sebastian; Butz, Erik; Caspart, René; Chwalek, Thorsten; Colombo, Fabio; De Boer, Wim; Dierlamm, Alexander; Faltermann, Nils; Freund, Benedikt; Friese, Raphael; Giffels, Manuel; Harrendorf, Marco Alexander; Hartmann, Frank; Heindl, Stefan Michael; Husemann, Ulrich; Kassel, Florian; Kudella, Simon; Mildner, Hannes; Mozer, Matthias Ulrich; Müller, Thomas; Plagge, Michael; Quast, Gunter; Rabbertz, Klaus; Schröder, Matthias; Shvetsov, Ivan; Sieber, Georg; Simonis, Hans-Jürgen; Ulrich, Ralf; Wayand, Stefan; Weber, Marc; Weiler, Thomas; Williamson, Shawn; Wöhrmann, Clemens; Wolf, Roger; Anagnostou, Georgios; Daskalakis, Georgios; Geralis, Theodoros; Kyriakis, Aristotelis; Loukas, Demetrios; Topsis-Giotis, Iasonas; Karathanasis, George; Kesisoglou, Stilianos; Panagiotou, Apostolos; Saoulidou, Niki; Tziaferi, Eirini; Kousouris, Konstantinos; Papakrivopoulos, Ioannis; Evangelou, Ioannis; Foudas, Costas; Gianneios, Paraskevas; Katsoulis, Panagiotis; Kokkas, Panagiotis; Mallios, Stavros; Manthos, Nikolaos; Papadopoulos, Ioannis; Paradas, Evangelos; Strologas, John; Triantis, Frixos A; Tsitsonis, Dimitrios; Csanad, Mate; Filipovic, Nicolas; Pasztor, Gabriella; Surányi, Olivér; Veres, Gabor Istvan; Bencze, Gyorgy; Hajdu, Csaba; Horvath, Dezso; Hunyadi, Ádám; Sikler, Ferenc; Veszpremi, Viktor; Vesztergombi, Gyorgy; Vámi, Tamás Álmos; Beni, Noemi; Czellar, Sandor; Karancsi, János; Makovec, Alajos; Molnar, Jozsef; Szillasi, Zoltan; Bartók, Márton; Raics, Peter; Trocsanyi, Zoltan Laszlo; Ujvari, Balazs; Choudhury, Somnath; Komaragiri, Jyothsna Rani; Bahinipati, Seema; Mal, Prolay; Mandal, Koushik; Nayak, Aruna; Sahoo, Deepak Kumar; Sahoo, Niladribihari; Swain, Sanjay Kumar; Bansal, Sunil; Beri, Suman Bala; Bhatnagar, Vipin; Chauhan, Sushil; Chawla, Ridhi; Dhingra, Nitish; Gupta, Rajat; Kaur, Anterpreet; Kaur, Manjit; Kaur, Sandeep; Kumar, Ramandeep; Kumari, Priyanka; Lohan, Manisha; Mehta, Ankita; Sharma, Sandeep; Singh, Jasbir; Walia, Genius; Kumar, Ashok; Shah, Aashaq; Bhardwaj, Ashutosh; Choudhary, Brajesh C; Garg, Rocky Bala; Keshri, Sumit; Kumar, Ajay; Malhotra, Shivali; Naimuddin, Md; Ranjan, Kirti; Sharma, Ramkrishna; Bhardwaj, Rishika; Bhattacharya, Rajarshi; Bhattacharya, Satyaki; Bhawandeep, Bhawandeep; Bhowmik, Debabrata; Dey, Sourav; Dutt, Suneel; Dutta, Suchandra; Ghosh, Shamik; Majumdar, Nayana; Mondal, Kuntal; Mukhopadhyay, Supratik; Nandan, Saswati; Purohit, Arnab; Rout, Prasant Kumar; Roy, Ashim; Roy Chowdhury, Suvankar; Sarkar, Subir; Sharan, Manoj; Singh, Bipen; Thakur, Shalini; Behera, Prafulla Kumar; Chudasama, Ruchi; Dutta, Dipanwita; Jha, Vishwajeet; Kumar, Vineet; Mohanty, Ajit Kumar; Netrakanti, Pawan Kumar; Pant, Lalit Mohan; Shukla, Prashant; Topkar, Anita; Aziz, Tariq; Dugad, Shashikant; Mahakud, Bibhuprasad; Mitra, Soureek; Mohanty, Gagan Bihari; Sur, Nairit; Sutar, Bajrang; Banerjee, Sudeshna; Bhattacharya, Soham; Chatterjee, Suman; Das, Pallabi; Guchait, Monoranjan; Jain, Sandhya; Kumar, Sanjeev; Maity, Manas; Majumder, Gobinda; Mazumdar, Kajari; Sarkar, Tanmay; Wickramage, Nadeesha; Chauhan, Shubhanshu; Dube, Sourabh; Hegde, Vinay; Kapoor, Anshul; Kothekar, Kunal; Pandey, Shubham; Rane, Aditee; Sharma, Seema; Chenarani, Shirin; Eskandari Tadavani, Esmaeel; Etesami, Seyed Mohsen; Khakzad, Mohsen; Mohammadi Najafabadi, Mojtaba; Naseri, Mohsen; Paktinat Mehdiabadi, Saeid; Rezaei Hosseinabadi, Ferdos; Safarzadeh, Batool; Zeinali, Maryam; Felcini, Marta; Grunewald, Martin; Abbrescia, Marcello; Calabria, Cesare; Colaleo, Anna; Creanza, Donato; Cristella, Leonardo; De Filippis, Nicola; De Palma, Mauro; Di Florio, Adriano; Errico, Filippo; Fiore, Luigi; Gelmi, Andrea; Iaselli, Giuseppe; Lezki, Samet; Maggi, Giorgio; Maggi, Marcello; Marangelli, Bartolomeo; Miniello, Giorgia; My, Salvatore; Nuzzo, Salvatore; Pompili, Alexis; Pugliese, Gabriella; Radogna, Raffaella; Ranieri, Antonio; Selvaggi, Giovanna; Sharma, Archana; Silvestris, Lucia; Venditti, Rosamaria; Verwilligen, Piet; Zito, Giuseppe; Abbiendi, Giovanni; Battilana, Carlo; Bonacorsi, Daniele; Borgonovi, Lisa; Braibant-Giacomelli, Sylvie; Campanini, Renato; Capiluppi, Paolo; Castro, Andrea; Cavallo, Francesca Romana; Chhibra, Simranjit Singh; Codispoti, Giuseppe; Cuffiani, Marco; Dallavalle, Gaetano-Marco; Fabbri, Fabrizio; Fanfani, Alessandra; Fasanella, Daniele; Giacomelli, Paolo; Grandi, Claudio; Guiducci, Luigi; Iemmi, Fabio; Marcellini, Stefano; Masetti, Gianni; Montanari, Alessandro; Navarria, Francesco; Perrotta, Andrea; Rossi, Antonio; Rovelli, Tiziano; Siroli, Gian Piero; Tosi, Nicolò; Albergo, Sebastiano; Costa, Salvatore; Di Mattia, Alessandro; Giordano, Ferdinando; Potenza, Renato; Tricomi, Alessia; Tuve, Cristina; Barbagli, Giuseppe; Chatterjee, Kalyanmoy; Ciulli, Vitaliano; Civinini, Carlo; D'Alessandro, Raffaello; Focardi, Ettore; Latino, Giuseppe; Lenzi, Piergiulio; Meschini, Marco; Paoletti, Simone; Russo, Lorenzo; Sguazzoni, Giacomo; Strom, Derek; Viliani, Lorenzo; Benussi, Luigi; Bianco, Stefano; Fabbri, Franco; Piccolo, Davide; Primavera, Federica; Calvelli, Valerio; Ferro, Fabrizio; Ravera, Fabio; Robutti, Enrico; Tosi, Silvano; Benaglia, Andrea; Beschi, Andrea; Brianza, Luca; Brivio, Francesco; Ciriolo, Vincenzo; Dinardo, Mauro Emanuele; Fiorendi, Sara; Gennai, Simone; Ghezzi, Alessio; Govoni, Pietro; Malberti, Martina; Malvezzi, Sandra; Manzoni, Riccardo Andrea; Menasce, Dario; Moroni, Luigi; Paganoni, Marco; Pauwels, Kristof; Pedrini, Daniele; Pigazzini, Simone; Ragazzi, Stefano; Tabarelli de Fatis, Tommaso; Buontempo, Salvatore; Cavallo, Nicola; Di Guida, Salvatore; Fabozzi, Francesco; Fienga, Francesco; Galati, Giuliana; Iorio, Alberto Orso Maria; Khan, Wajid Ali; Lista, Luca; Meola, Sabino; Paolucci, Pierluigi; Sciacca, Crisostomo; Thyssen, Filip; Voevodina, Elena; Azzi, Patrizia; Bacchetta, Nicola; Benato, Lisa; Bisello, Dario; Boletti, Alessio; Carlin, Roberto; Carvalho Antunes De Oliveira, Alexandra; Checchia, Paolo; Dall'Osso, Martino; De Castro Manzano, Pablo; Dorigo, Tommaso; Dosselli, Umberto; Gasparini, Fabrizio; Gasparini, Ugo; Gozzelino, Andrea; Lacaprara, Stefano; Lujan, Paul; Margoni, Martino; Meneguzzo, Anna Teresa; Pozzobon, Nicola; Ronchese, Paolo; Rossin, Roberto; Simonetto, Franco; Tiko, Andres; Torassa, Ezio; Zanetti, Marco; Zotto, Pierluigi; Braghieri, Alessandro; Magnani, Alice; Montagna, Paolo; Ratti, Sergio P; Re, Valerio; Ressegotti, Martina; Riccardi, Cristina; Salvini, Paola; Vai, Ilaria; Vitulo, Paolo; Alunni Solestizi, Luisa; Biasini, Maurizio; Bilei, Gian Mario; Cecchi, Claudia; Ciangottini, Diego; Fanò, Livio; Lariccia, Paolo; Leonardi, Roberto; Manoni, Elisa; Mantovani, Giancarlo; Mariani, Valentina; Menichelli, Mauro; Rossi, Alessandro; Santocchia, Attilio; Spiga, Daniele; Androsov, Konstantin; Azzurri, Paolo; Bagliesi, Giuseppe; Bianchini, Lorenzo; Boccali, Tommaso; Borrello, Laura; Castaldi, Rino; Ciocci, Maria Agnese; Dell'Orso, Roberto; Fedi, Giacomo; Giannini, Leonardo; Giassi, Alessandro; Grippo, Maria Teresa; Ligabue, Franco; Lomtadze, Teimuraz; Manca, Elisabetta; Mandorli, Giulio; Messineo, Alberto; Palla, Fabrizio; Rizzi, Andrea; Spagnolo, Paolo; Tenchini, Roberto; Tonelli, Guido; Venturi, Andrea; Verdini, Piero Giorgio; Barone, Luciano; Cavallari, Francesca; Cipriani, Marco; Daci, Nadir; Del Re, Daniele; Di Marco, Emanuele; Diemoz, Marcella; Gelli, Simone; Longo, Egidio; Margaroli, Fabrizio; Marzocchi, Badder; Meridiani, Paolo; Organtini, Giovanni; Pandolfi, Francesco; Paramatti, Riccardo; Preiato, Federico; Rahatlou, Shahram; Rovelli, Chiara; Santanastasio, Francesco; Amapane, Nicola; Arcidiacono, Roberta; Argiro, Stefano; Arneodo, Michele; Bartosik, Nazar; Bellan, Riccardo; Biino, Cristina; Cartiglia, Nicolo; Castello, Roberto; Cenna, Francesca; Costa, Marco; Covarelli, Roberto; Degano, Alessandro; Demaria, Natale; Kiani, Bilal; Mariotti, Chiara; Maselli, Silvia; Migliore, Ernesto; Monaco, Vincenzo; Monteil, Ennio; Monteno, Marco; Obertino, Maria Margherita; Pacher, Luca; Pastrone, Nadia; Pelliccioni, Mario; Pinna Angioni, Gian Luca; Romero, Alessandra; Ruspa, Marta; Sacchi, Roberto; Shchelina, Ksenia; Sola, Valentina; Solano, Ada; Staiano, Amedeo; Belforte, Stefano; Casarsa, Massimo; Cossutti, Fabio; Della Ricca, Giuseppe; Zanetti, Anna; Kim, Dong Hee; Kim, Gui Nyun; Kim, Min Suk; Lee, Jeongeun; Lee, Sangeun; Lee, Seh Wook; Moon, Chang-Seong; Oh, Young Do; Sekmen, Sezen; Son, Dong-Chul; Yang, Yu Chul; Kim, Hyunchul; Moon, Dong Ho; Oh, Geonhee; Brochero Cifuentes, Javier Andres; Goh, Junghwan; Kim, Tae Jeong; Cho, Sungwoong; Choi, Suyong; Go, Yeonju; Gyun, Dooyeon; Ha, Seungkyu; Hong, Byung-Sik; Jo, Youngkwon; Kim, Yongsun; Lee, Kisoo; Lee, Kyong Sei; Lee, Songkyo; Lim, Jaehoon; Park, Sung Keun; Roh, Youn; Almond, John; Kim, Junho; Kim, Jae Sung; Lee, Haneol; Lee, Kyeongpil; Nam, Kyungwook; Oh, Sung Bin; Radburn-Smith, Benjamin Charles; Seo, Seon-hee; Yang, Unki; Yoo, Hwi Dong; Yu, Geum Bong; Kim, Hyunyong; Kim, Ji Hyun; Lee, Jason Sang Hun; Park, Inkyu; Choi, Young-Il; Hwang, Chanwook; Lee, Jongseok; Yu, Intae; Dudenas, Vytautas; Juodagalvis, Andrius; Vaitkus, Juozas; Ahmed, Ijaz; Ibrahim, Zainol Abidin; Md Ali, Mohd Adli Bin; Mohamad Idris, Faridah; Wan Abdullah, Wan Ahmad Tajuddin; Yusli, Mohd Nizam; Zolkapli, Zukhaimira; Reyes-Almanza, Rogelio; Ramirez-Sanchez, Gabriel; Duran-Osuna, Cecilia; Castilla-Valdez, Heriberto; De La Cruz-Burelo, Eduard; Heredia-De La Cruz, Ivan; Rabadán-Trejo, Raúl Iraq; Lopez-Fernandez, Ricardo; Mejia Guisao, Jhovanny; Sánchez Hernández, Alberto; Carrillo Moreno, Salvador; Oropeza Barrera, Cristina; Vazquez Valencia, Fabiola; Eysermans, Jan; Pedraza, Isabel; Salazar Ibarguen, Humberto Antonio; Uribe Estrada, Cecilia; Morelos Pineda, Antonio; Krofcheck, David; Butler, Philip H; Ahmad, Ashfaq; Ahmad, Muhammad; Hassan, Qamar; Hoorani, Hafeez R; Saddique, Asif; Shah, Mehar Ali; Shoaib, Muhammad; Waqas, Muhammad; Bialkowska, Helena; Bluj, Michal; Boimska, Bozena; Frueboes, Tomasz; Górski, Maciej; Kazana, Malgorzata; Nawrocki, Krzysztof; Szleper, Michal; Traczyk, Piotr; Zalewski, Piotr; Bunkowski, Karol; Byszuk, Adrian; Doroba, Krzysztof; Kalinowski, Artur; Konecki, Marcin; Krolikowski, Jan; Misiura, Maciej; Olszewski, Michal; Pyskir, Andrzej; Walczak, Marek; Bargassa, Pedrame; Beirão Da Cruz E Silva, Cristóvão; Di Francesco, Agostino; Faccioli, Pietro; Galinhas, Bruno; Gallinaro, Michele; Hollar, Jonathan; Leonardo, Nuno; Lloret Iglesias, Lara; Nemallapudi, Mythra Varun; Seixas, Joao; Strong, Giles; Toldaiev, Oleksii; Vadruccio, Daniele; Varela, Joao; Afanasiev, Serguei; Bunin, Pavel; Gavrilenko, Mikhail; Golutvin, Igor; Gorbunov, Ilya; Kamenev, Alexey; Karjavin, Vladimir; Lanev, Alexander; Malakhov, Alexander; Matveev, Viktor; Moisenz, Petr; Palichik, Vladimir; Perelygin, Victor; Shmatov, Sergey; Shulha, Siarhei; Skatchkov, Nikolai; Smirnov, Vitaly; Voytishin, Nikolay; Zarubin, Anatoli; Ivanov, Yury; Kim, Victor; Kuznetsova, Ekaterina; Levchenko, Petr; Murzin, Victor; Oreshkin, Vadim; Smirnov, Igor; Sosnov, Dmitry; Sulimov, Valentin; Uvarov, Lev; Vavilov, Sergey; Vorobyev, Alexey; Andreev, Yuri; Dermenev, Alexander; Gninenko, Sergei; Golubev, Nikolai; Karneyeu, Anton; Kirsanov, Mikhail; Krasnikov, Nikolai; Pashenkov, Anatoli; Tlisov, Danila; Toropin, Alexander; Epshteyn, Vladimir; Gavrilov, Vladimir; Lychkovskaya, Natalia; Popov, Vladimir; Pozdnyakov, Ivan; Safronov, Grigory; Spiridonov, Alexander; Stepennov, Anton; Stolin, Viatcheslav; Toms, Maria; Vlasov, Evgueni; Zhokin, Alexander; Aushev, Tagir; Bylinkin, Alexander; Chadeeva, Marina; Chistov, Ruslan; Parygin, Pavel; Philippov, Dmitry; Polikarpov, Sergey; Tarkovskii, Evgenii; Zhemchugov, Evgenii; Andreev, Vladimir; Azarkin, Maksim; Dremin, Igor; Kirakosyan, Martin; Rusakov, Sergey V; Terkulov, Adel; Baskakov, Alexey; Belyaev, Andrey; Boos, Edouard; Bunichev, Viacheslav; Dubinin, Mikhail; Dudko, Lev; Ershov, Alexander; Gribushin, Andrey; Klyukhin, Vyacheslav; Kodolova, Olga; Lokhtin, Igor; Miagkov, Igor; Obraztsov, Stepan; Petrushanko, Sergey; Savrin, Viktor; Blinov, Vladimir; Shtol, Dmitry; Skovpen, Yuri; Azhgirey, Igor; Bayshev, Igor; Bitioukov, Sergei; Elumakhov, Dmitry; Godizov, Anton; Kachanov, Vassili; Kalinin, Alexey; Konstantinov, Dmitri; Mandrik, Petr; Petrov, Vladimir; Ryutin, Roman; Sobol, Andrei; Troshin, Sergey; Tyurin, Nikolay; Uzunian, Andrey; Volkov, Alexey; Babaev, Anton; Adzic, Petar; Cirkovic, Predrag; Devetak, Damir; Dordevic, Milos; Milosevic, Jovan; Alcaraz Maestre, Juan; Bachiller, Irene; Barrio Luna, Mar; Cerrada, Marcos; Colino, Nicanor; De La Cruz, Begona; Delgado Peris, Antonio; Fernandez Bedoya, Cristina; Fernández Ramos, Juan Pablo; 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Lanaro, Armando; Levine, Aaron; Long, Kenneth; Loveless, Richard; Rekovic, Vladimir; Ruggles, Tyler; Savin, Alexander; Smith, Nicholas; Smith, Wesley H; Woods, Nathaniel 2018-04-13 A search is reported for heavy resonances decaying into e$\\mu$final states in proton-proton collisions recorded by the CMS experiment at the CERN LHC at$\\sqrt{s} = $13 TeV, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The search focuses on resonance masses above 200 GeV. With no evidence found for physics beyond the standard model in the e$\\mu$mass spectrum, upper limits are set at 95% confidence level on the product of the cross section and branching fraction for this lepton-flavor violating signal. Based on these results, resonant${\\tau}$sneutrino production in R-parity violating supersymmetric models is excluded for masses below 1.7 TeV, for couplings$\\lambda_{132}=\\lambda_{231}=\\lambda'_{311}= $0.01. Heavy Z' gauge bosons with lepton-flavor violating transitions are excluded for masses up to 4.4 TeV. The e$\\mu$mass spectrum is also interpreted in terms of non-resonant contributions from quantum black-hole production in models with one to six extra spatial dimensions, and lower mas... 19. Repetition of the quark-lepton states in a supersymmetric composite model with complementarity International Nuclear Information System (INIS) Yamada, Hirofumi; Yasue, Masaki. 1986-04-01 In a supersymmetric composite model based on an SU(4) sc loc confining theory, complementarity is used to support the symmetry-breaking pattern and spectrum of massless particles in a confining phase. The model is found to accommodate two generations of quarks and leptons as quasi Nambu-Goldstone fermions and another two generations as chiral fermions. Masses of composite particles are examined and the quark-lepton generations are classified according to possible mass splittings. The suppression of dangerous flavor-changing interactions is also considered. (author) 20. Model for particle masses, flavor mixing, and CP violation, based on spontaneously broken discrete chiral symmetry as the origin of families International Nuclear Information System (INIS) Adler, S.L. 1999-01-01 We construct extensions of the standard model based on the hypothesis that Higgs bosons also exhibit a family structure and that the flavor weak eigenstates in the three families are distinguished by a discrete Z 6 chiral symmetry that is spontaneously broken by the Higgs sector. We study in detail at the tree level models with three Higgs doublets and with six Higgs doublets comprising two weakly coupled sets of three. In a leading approximation of S 3 cyclic permutation symmetry the three-Higgs-doublet model gives a open-quotes democraticclose quotes mass matrix of rank 1, while the six-Higgs-doublet model gives either a rank-1 mass matrix or, in the case when it spontaneously violates CP, a rank-2 mass matrix corresponding to nonzero second family masses. In both models, the CKM matrix is exactly unity in the leading approximation. Allowing small explicit violations of cyclic permutation symmetry generates small first family masses in the six-Higgs-doublet model, and first and second family masses in the three-Higgs-doublet model, and gives a nontrivial CKM matrix in which the mixings of the first and second family quarks are naturally larger than mixings involving the third family. Complete numerical fits are given for both models, flavor-changing neutral current constraints are discussed in detail, and the issues of unification of couplings and neutrino masses are addressed. On a technical level, our analysis uses the theory of circulant and retrocirculant matrices, the relevant parts of which are reviewed. copyright 1998 The American Physical Society 1. A Realistic$U(2)$Model of Flavor arXiv CERN Document Server Linster, Matthias We propose a simple$U(2)$model of flavor compatible with an$SU(5)$GUT structure. All hierarchies in fermion masses and mixings arise from powers of two small parameters that control the$U(2)$breaking. In contrast to previous$U(2)$models this setup can be realized without supersymmetry and provides an excellent fit to all SM flavor observables including neutrinos. We also consider a variant of this model based on a$D_6 \\times U(1)_F$flavor symmetry, which closely resembles the$U(2)$structure, but allows for Majorana neutrino masses from the Weinberg operator. Remarkably, in this case one naturally obtains large mixing in the lepton sector from small mixing in the quark sector. The model also offers a natural option for addressing the Strong CP Problem and Dark Matter by identifying the Goldstone boson of the$U(1)_F$factor as the QCD axion. 2. Two particle states, lepton mixing and oscillations CERN Document Server Kachelriess, M; Schönert, S 2000-01-01 Discussions of lepton mixing and oscillations consider generally only flavor oscillations of neutrinos and neglect the accompanying charged leptons. In cases of experimental interest like pion or nuclear beta decay an oscillation pattern is expected indeed only for neutrinos if only one of the two produced particles is observed. We argue that flavor oscillations of neutrinos without detecting the accompanying lepton is a peculiarity of the two-particle states$|l\
3. Charge Asymmetric Cosmic Rays as a probe of Flavor Violating Asymmetric Dark Matter
DEFF Research Database (Denmark)
Masina, Isabella; Sannino, Francesco
2011-01-01
The recently introduced cosmic sum rules combine the data from PAMELA and Fermi-LAT cosmic ray experiments in a way that permits to neatly investigate whether the experimentally observed lepton excesses violate charge symmetry. One can in a simple way determine universal properties of the unknown...... component of the cosmic rays. Here we attribute a potential charge asymmetry to the dark sector. In particular we provide models of asymmetric dark matter able to produce charge asymmetric cosmic rays. We consider spin zero, spin one and spin one-half decaying dark matter candidates. We show that lepton...... flavor violation and asymmetric dark matter are both required to have a charge asymmetry in the cosmic ray lepton excesses. Therefore, an experimental evidence of charge asymmetry in the cosmic ray lepton excesses implies that dark matter is asymmetric....
4. Flavored dark matter beyond Minimal Flavor Violation
International Nuclear Information System (INIS)
Agrawal, Prateek; Blanke, Monika; Gemmler, Katrin
2014-01-01
We study the interplay of flavor and dark matter phenomenology for models of flavored dark matter interacting with quarks. We allow an arbitrary flavor structure in the coupling of dark matter with quarks. This coupling is assumed to be the only new source of violation of the Standard Model flavor symmetry extended by a U(3) χ associated with the dark matter. We call this ansatz Dark Minimal Flavor Violation (DMFV) and highlight its various implications, including an unbroken discrete symmetry that can stabilize the dark matter. As an illustration we study a Dirac fermionic dark matter χ which transforms as triplet under U(3) χ , and is a singlet under the Standard Model. The dark matter couples to right-handed down-type quarks via a colored scalar mediator with a coupling. We identify a number of ''flavor-safe'' scenarios for the structure of which are beyond Minimal Flavor Violation. Also, for dark matter and collider phenomenology we focus on the well-motivated case of b-flavored dark matter. Furthermore, the combined flavor and dark matter constraints on the parameter space of turn out to be interesting intersections of the individual ones. LHC constraints on simplified models of squarks and sbottoms can be adapted to our case, and monojet searches can be relevant if the spectrum is compressed
5. Flavor physics and right-handed models
Energy Technology Data Exchange (ETDEWEB)
Shafaq, Saba
2010-08-20
The Standard Model of particle physics only provides a parametrization of flavor which involves the values of the quark and lepton masses and unitary flavor mixing matrix i.e. CKM (Cabibbo-Kobayashi-Masakawa) matrix for quarks. The precise determination of elements of the CKM matrix is important for the study of the flavor sector of quarks. Here we concentrate on the matrix element vertical stroke V{sub cb} vertical stroke. In particular we consider the effects on the value of vertical stroke V{sub cb} vertical stroke from possible right-handed admixtures along with the usually left-handed weak currents. Left Right Symmetric Model provide a natural basis for right-handed current contributions and has been studied extensively in the literature but has never been discussed including flavor. In the first part of the present work an additional flavor symmetry is included in LRSM which allows a systematic study of flavor effects. The second part deals with the practical extraction of a possible right-handed contribution. Starting from the quark level transition b{yields}c we use heavy quark symmetries to relate the helicities of the quarks to experimentally accessible quantities. To this end we study the decays anti B{yields}D(D{sup *})l anti {nu} which have been extensively explored close to non recoil point. By taking into account SCET (Soft Collinear Effective Theory) formalism it has been extended to a maximum recoil point i.e. {upsilon} . {upsilon}{sup '} >>1. We derive a factorization formula, where the set of form factors is reduced to a single universal form factor {xi}({upsilon} . {upsilon}{sup '}) up to hard-scattering corrections. Symmetry relations on form factors for exclusive anti B {yields} D(D{sup *})l anti {nu} transition has been derived in terms of {xi}({upsilon} . {upsilon}{sup '}). These symmetries are then broken by perturbative effects. The perturbative corrections to symmetry-breaking corrections to first order in the strong
6. Towards a realistic composite model of quarks and leptons
International Nuclear Information System (INIS)
Li Xiaoyuan; Marshak, R.E.
1985-06-01
Within the context of the 't Hooft anomaly matching scheme, some guiding principles for the model building are discussed with an eye to low energy phenomenology. It is argued that Λsub(ch) (chiral symmetry breaking scale of the global color-flavor group Gsub(CF)) proportional Λsub(MC) (metacolor scale) and Λ sub(gsub(CF)) (unification scale of the gauge subgroup of Gsub(CF)) < or approx. Λsub(ch). As illustrations of the method, two composite models are suggested that can give rise to three or four generations of ordinary quarks and leptons without exotic fermions. (orig.)
7. Lepton mixing from Δ(3n2 and Δ(6n2 and CP
Directory of Open Access Journals (Sweden)
Claudia Hagedorn
2015-02-01
Full Text Available We perform a detailed study of lepton mixing patterns arising from a scenario with three Majorana neutrinos in which a discrete flavor group Gf=Δ(3n2 or Gf=Δ(6n2 and a CP symmetry are broken to residual symmetries Ge=Z3 and Gν=Z2×CP in the charged lepton and neutrino sectors, respectively. While we consider all possible Z3 and Z2 generating elements, we focus on a certain set of CP transformations. The resulting lepton mixing depends on group theoretical indices and one continuous parameter. In order to study the mixing patterns comprehensively for all admitted Ge and Gν, it is sufficient to discuss only three types of combinations. One of them requires as flavor group Δ(6n2. Two types of combinations lead to mixing patterns with a trimaximal column, while the third one allows for a much richer structure. For the first type of combinations the Dirac phase as well as one of the Majorana phases are trivial, whereas the other two types of combinations predict in general all CP phases to be non-trivial and also non-maximal. Already for small values of the index n of the group, n≤11, experimental data on lepton mixing can be accommodated well for particular choices of the parameters of the theory. We also comment on the relation of the used CP transformations to the automorphisms of Δ(3n2 and Δ(6n2.
8. Search for a heavy right-handed W boson and a heavy neutrino in events with two same-flavor leptons and two jets at sqrt(s)=13 TeV
CERN Document Server
CMS Collaboration
2017-01-01
A search for a heavy right-handed $\\mathrm W$ gauge boson and a heavy right-handed neutrino at the CERN LHC has been conducted by the CMS collaboration in events with two same-flavor leptons ($\\mathrm e$ or $\\mu$) and two jets, using 2016 proton-proton collision data corresponding to an integrated luminosity of $\\mathrm{35.9\\,fb^{-1}}$. No excess above the standard model expectation is seen in the invariant mass distribution of the dilepton plus dijet system. Assuming identical couplings and decay branching fractions as the standard model $\\mathrm W$ gauge boson, and that only one heavy neutrino flavor ${\\mathrm N}_R$ contributes significantly to the ${\\mathrm W}_R$ decay width, the region in the two-dimensional ($m_{{\\mathrm W}_R}$, $m_{{\\mathrm N}_R}$) mass plane excluded at a $95\\%$ confidence level extends to approximately $m_{{\\mathrm W}_R}= \\mathrm{4.4\\,TeV}$ and covers a large range of neutrino masses below the ${\\mathrm W}_R$ boson mass. This analysis provides the most stringent limits to date.
9. A model-building approach to the origin of flavor
Energy Technology Data Exchange (ETDEWEB)
Schumacher, Erik
2017-01-24
In this thesis we link the recent anomalies reported in B meson and h→μτ decays to the smallness of neutrino masses and aspects of the flavor puzzle, including the hierarchy of the Yukawa couplings and the disparate fermion mixings. By formulating various new models we attempt to shed light on the potential common origin of the distinct measurements in the flavor sector. To this end, discrete symmetries are utilized in this work as the governing principle behind all fermion interactions. The first two models based on the S{sub 3} and the A{sub 4} symmetry, respectively, aim to unify the diverse fermion masses and mixings. Special features separate the frameworks from the flavor models in the literature that often lack testable predictions. While the first model provides interesting flavor-violating signatures in top quark decays, the second one ties the flavor to the grand unification scale in a novel way. In the three following models we focus on the anomalies that hint at lepton flavor and universality violation. We propose that the large flavor violation observed in h→μτ decays is dictated by the scalar mixing of an enlarged S{sub 4}-symmetric Higgs sector. By constructing two leptoquark models we show for the first time that leptoquark couplings shaped by a Froggatt-Nielsen mechanism can accommodate the B meson anomalies and simultaneously generate naturally-small neutrino masses. Emphasizing the importance of testability, we demonstrate how these models can be probed by future diphoton resonances, using the recent 750 GeV excess as an example scenario.
10. Common origin of μ-τ and CP breaking in the neutrino seesaw, baryon asymmetry, and hidden flavor symmetry
International Nuclear Information System (INIS)
He Hongjian; Yin Furong
2011-01-01
We conjecture that all CP violations (both Dirac and Majorana types) arise from a common origin in the neutrino seesaw. With this conceptually attractive and simple conjecture, we deduce that μ-τ breaking shares the common origin with all CP violations. We study the common origin of μ-τ and CP breaking in the Dirac mass matrix of seesaw Lagrangian (with right-handed neutrinos being μ-τ blind), which uniquely leads to inverted mass ordering of light neutrinos. We then predict a very different correlation between the two small μ-τ breaking observables θ 13 -0 deg. and θ 23 -45 deg., which can saturate the present experimental upper limit on θ 13 . This will be tested against our previous normal mass-ordering scheme by the ongoing oscillation experiments. We also analyze the correlations of θ 13 with Jarlskog invariant and neutrinoless ββ-decay observable. From the common origin of CP and μ-τ breaking in the neutrino seesaw, we establish a direct link between the low energy CP violations and the cosmological CP violation for baryon asymmetry. With these we further predict a lower bound on θ 13 , supporting the ongoing probes of θ 13 at Daya Bay, Double Chooz, and RENO experiments. Finally, we analyze the general model-independent Z 2 x Z 2 symmetry structure of the light neutrino sector, and map it into the seesaw sector, where one of the Z 2 's corresponds to the μ-τ symmetry Z 2 μτ and another the hidden symmetry Z 2 s (revealed in our previous work) which dictates the solar mixing angle θ 12 . We derive the physical consequences of this Z 2 s and its possible partial violation in the presence of μ-τ breaking (with or without the neutrino seesaw), regarding the θ 12 determination and the correlation between μ-τ breaking observables.
11. Flavored dark matter beyond Minimal Flavor Violation
CERN Document Server
Agrawal, Prateek; Gemmler, Katrin
2014-10-13
We study the interplay of flavor and dark matter phenomenology for models of flavored dark matter interacting with quarks. We allow an arbitrary flavor structure in the coupling of dark matter with quarks. This coupling is assumed to be the only new source of violation of the Standard Model flavor symmetry extended by a $U(3)_\\chi$ associated with the dark matter. We call this ansatz Dark Minimal Flavor Violation (DMFV) and highlight its various implications, including an unbroken discrete symmetry that can stabilize the dark matter. As an illustration we study a Dirac fermionic dark matter $\\chi$ which transforms as triplet under $U(3)_\\chi$, and is a singlet under the Standard Model. The dark matter couples to right-handed down-type quarks via a colored scalar mediator $\\phi$ with a coupling $\\lambda$. We identify a number of "flavor-safe" scenarios for the structure of $\\lambda$ which are beyond Minimal Flavor Violation. For dark matter and collider phenomenology we focus on the well-motivated case of $b$-...
12. Neutrino masses, dark matter and leptogenesis with U(1) B - L gauge symmetry
Science.gov (United States)
2018-06-01
We propose a model with an U(1) B - L gauge symmetry, in which small neutrino masses, dark matter and the matter-antimatter asymmetry in the Universe can be simultaneously explained. In particular, the neutrino masses are generated radiatively, while the matter-antimatter asymmetry is led by the leptogenesis mechanism, at TeV scale. We also explore allowed regions of the model parameters and discuss some phenomenological effects, including lepton flavor violating processes.
13. Zee-Babu type model with U (1 )Lμ-Lτ gauge symmetry
Science.gov (United States)
2018-05-01
We extend the Zee-Babu model, introducing local U (1 )Lμ-Lτ symmetry with several singly charged bosons. We find a predictive neutrino mass texture in a simple hypothesis in which mixings among singly charged bosons are negligible. Also, lepton-flavor violations are less constrained compared with the original model. Then, we explore the testability of the model, focusing on doubly charged boson physics at the LHC and the International Linear Collider.
14. ϕ 3 theory with F4 flavor symmetry in 6 - 2 ɛ dimensions: 3-loop renormalization and conformal bootstrap
Science.gov (United States)
Pang, Yi; Rong, Junchen; Su, Ning
2016-12-01
We consider ϕ 3 theory in 6 - 2 ɛ with F 4 global symmetry. The beta function is calculated up to 3 loops, and a stable unitary IR fixed point is observed. The anomalous dimensions of operators quadratic or cubic in ϕ are also computed. We then employ conformal bootstrap technique to study the fixed point predicted from the perturbative approach. For each putative scaling dimension of ϕ (Δ ϕ ), we obtain the corresponding upper bound on the scaling dimension of the second lowest scalar primary in the 26 representation ( Δ 26 2nd ) which appears in the OPE of ϕ × ϕ. In D = 5 .95, we observe a sharp peak on the upper bound curve located at Δ ϕ equal to the value predicted by the 3-loop computation. In D = 5, we observe a weak kink on the upper bound curve at ( Δ ϕ , Δ 26 2nd ) = (1.6, 4).
15. Masses, flavor mix and CP violation
International Nuclear Information System (INIS)
Chaussard, L.
2004-06-01
The author describes the relationships between masses, mixing of flavors and CP violation. This document is divided into 4 chapters: 1) fermions' masses, 2) mixing of flavors and CP violation, 3) beauty physics and 4) neutrino physics. In chapter 1 an attempt is made to explain what is behind the concepts of lepton mass and quark mass. As for neutrinos, the only neutral fermion, Dirac's and Majorana's views are exposed as well as their consequences. Fermion flavors are mixed in the process of mass generation and this mix is responsible for the breaking of CP and T symmetries. In chapter 2 the author shows how the analysis of particle oscillations from neutral mesons (K 0 , D 0 , B d 0 and B s 0 ) and from neutrinos can shed light on CP violation. Chapter 3 is dedicated to the contribution of beauty physics to the determination of the unitary triangle, through the oscillations of beauty mesons. In chapter 4 the author reviews the experimental results obtained recently concerning neutrino mass and neutrino oscillations and draws some perspectives on future neutrino experiments. (A.C.)
16. The Charged Lepton Mass Matrix and Non-zero θ13 with TeV Scale New Physics.
Science.gov (United States)
Rashed, Ahmed; Datta, Alakabha
2012-03-01
We provide an explicit structure of the charged lepton mass matrix which is 2-3 symmetric except for a single breaking of this symmetry by the muon mass. We identify a flavor symmetric limit for the mass matrices where the first generation is decoupled from the other two in the charged lepton sector while in the neutrino sector the third generation is decoupled from the first two generations. The leptonic mixing in the symmetric limit can be, among other structures, the bi-maximal (BM) or the tri-bimaximal (TBM) mixing. Symmetry breaking effects are included both in the charged lepton and the neutrino sector to produce corrections to the leptonic mixing and explain the recent θ13 measurements. A model that extends the SM by three right handed neutrinos, an extra Higgs doublet, and two singlet scalars is introduced to generate the leptonic mixing.[4pt] This work was supported in part by the US-Egypt Joint Board on Scientific and Technological Co-operation award (Project ID: 1855) administered by the US Department of Agriculture, summer grant from the College of Liberal Arts, University of Mississippi and in part by the National Science Foundation under Grant No. 1068052 and 1066293 and the hospitality of the Aspen Center for Physics.
17. Flavor gauge models below the Fermi scale
Science.gov (United States)
Babu, K. S.; Friedland, A.; Machado, P. A. N.; Mocioiu, I.
2017-12-01
The mass and weak interaction eigenstates for the quarks of the third generation are very well aligned, an empirical fact for which the Standard Model offers no explanation. We explore the possibility that this alignment is due to an additional gauge symmetry in the third generation. Specifically, we construct and analyze an explicit, renormalizable model with a gauge boson, X, corresponding to the B - L symmetry of the third family. Having a relatively light (in the MeV to multi-GeV range), flavor-nonuniversal gauge boson results in a variety of constraints from different sources. By systematically analyzing 20 different constraints, we identify the most sensitive probes: kaon, B +, D + and Upsilon decays, D-{\\overline{D}}^0 mixing, atomic parity violation, and neutrino scattering and oscillations. For the new gauge coupling g X in the range (10-2-10-4) the model is shown to be consistent with the data. Possible ways of testing the model in b physics, top and Z decays, direct collider production and neutrino oscillation experiments, where one can observe nonstandard matter effects, are outlined. The choice of leptons to carry the new force is ambiguous, resulting in additional phenomenological implications, such as non-universality in semileptonic bottom decays. The proposed framework provides interesting connections between neutrino oscillations, flavor and collider physics.
18. Search for new phenomena in final states with two opposite-charge, same-flavor leptons, jets, and missing transverse momentum in pp collisions at $\\sqrt{s} =$ 13 TeV
Energy Technology Data Exchange (ETDEWEB)
Sirunyan, Albert M; et al.
2017-09-26
Search results are presented for physics beyond the standard model in final states with two opposite-charge, same-flavor leptons, jets, and missing transverse momentum. The data sample corresponds to an integrated luminosity of 35.9 fb$^{-1}$ of proton-proton collisions at $\\sqrt{s} =$ 13 TeV collected with the CMS detector at the LHC in 2016. The analysis uses the invariant mass of the lepton pair, searching for a kinematic edge or a resonant-like excess compatible with the Z boson mass. The search for a kinematic edge targets production of particles sensitive to the strong force, while the resonance search targets both strongly and electroweakly produced new physics. The observed yields are consistent with the expectations from the standard model, and the results are interpreted in the context of simplified models of supersymmetry. In a gauge mediated supersymmetry breaking (GMSB) model of gluino pair production with decay chains including Z bosons, gluino masses up to 1500-1770 GeV are excluded at the 95% confidence level depending on the lightest neutralino mass. In a model of electroweak chargino-neutralino production, chargino masses as high as 610 GeV are excluded when the lightest neutralino is massless. In GMSB models of electroweak neutralino-neutralino production, neutralino masses up to 500-650 GeV are excluded depending on the decay mode assumed. Finally, in a model with bottom squark pair production and decay chains resulting in a kinematic edge in the dilepton invariant mass distribution, bottom squark masses up to 980-1200 GeV are excluded depending on the mass of the next-to-lightest neutralino.
19. Search for new phenomena in final states with two opposite-charge, same-flavor leptons, jets, and missing transverse momentum in pp collisions at √{s}=13 TeV
Science.gov (United States)
2018-03-01
Search results are presented for physics beyond the standard model in final states with two opposite-charge, same-flavor leptons, jets, and missing transverse momentum. The data sample corresponds to an integrated luminosity of 35.9 fb-1 of proton-proton collisions at √{s}=13 TeV collected with the CMS detector at the LHC in 2016. The analysis uses the invariant mass of the lepton pair, searching for a kinematic edge or a resonant-like excess compatible with the Z boson mass. The search for a kinematic edge targets production of particles sensitive to the strong force, while the resonance search targets both strongly and electroweakly produced new physics. The observed yields are consistent with the expectations from the standard model, and the results are interpreted in the context of simplified models of supersymmetry. In a gauge mediated supersymmetry breaking (GMSB) model of gluino pair production with decay chains including Z bosons, gluino masses up to 1500-1770 GeV are excluded at the 95% confidence level depending on the lightest neutralino mass. In a model of electroweak chargino-neutralino production, chargino masses as high as 610 GeV are excluded when the lightest neutralino is massless. In GMSB models of electroweak neutralino-neutralino production, neutralino masses up to 500-650 GeV are excluded depending on the decay mode assumed. Finally, in a model with bottom squark pair production and decay chains resulting in a kinematic edge in the dilepton invariant mass distribution, bottom squark masses up to 980-1200 GeV are excluded depending on the mass of the next-to-lightest neutralino. [Figure not available: see fulltext.
20. Evidence for SU(3) symmetry breaking from hyperon production
International Nuclear Information System (INIS)
Yang Jianjun
2002-01-01
We examine the SU(3) symmetry breaking in hyperon semileptonic decays (HSD) by considering two typical sets of quark contributions to the spin content of the octet baryons: set 1 with SU(3) flavor symmetry and set 2 with SU(3) flavor symmetry breaking in the HSD. The quark distributions of the octet baryons are calculated with a successful statistical model. Using an approximate relation between the quark fragmentation functions and the quark distributions, we predict the polarizations of the octet baryons produced in e + e - annihilation and semi-inclusive deep lepton-nucleon scattering in order to reveal the SU(3) symmetry breaking effect on the spin structure of the octet baryons. We find that the SU(3) symmetry breaking significantly affects the hyperon polarization. The available experimental data on the Λ polarization seem to favor the theoretical predictions with SU(3) symmetry breaking. We conclude that there is a possibility to get collateral evidence for SU(3) symmetry breaking from hyperon production. The theoretical errors for our predictions are discussed
1. Search for maximal flavor violating scalars in same-charge lepton pairs in pp collisions at sqrt[s]=1.96 TeV.
Science.gov (United States)
Aaltonen, T; Adelman, J; Akimoto, T; Albrow, M G; Alvarez González, B; Amerio, S; Amidei, D; Anastassov, A; Annovi, A; Antos, J; Aoki, M; Apollinari, G; Apresyan, A; Arisawa, T; Artikov, A; Ashmanskas, W; Attal, A; Aurisano, A; Azfar, F; Azzi-Bacchetta, P; Azzurri, P; Bacchetta, N; Badgett, W; Barbaro-Galtieri, A; Barnes, V E; Barnett, B A; Baroiant, S; Bar-Shalom, S; Bartsch, V; Bauer, G; Beauchemin, P-H; Bedeschi, F; Bednar, P; Behari, S; Bellettini, G; Bellinger, J; Belloni, A; Benjamin, D; Beretvas, A; Beringer, J; Berry, T; Bhatti, A; Binkley, M; Bisello, D; Bizjak, I; Blair, R E; Blocker, C; Blumenfeld, B; Bocci, A; Bodek, A; Boisvert, V; Bolla, G; Bolshov, A; Bortoletto, D; Boudreau, J; Boveia, A; Brau, B; Bridgeman, A; Brigliadori, L; Bromberg, C; Brubaker, E; Budagov, J; Budd, H S; Budd, S; Burkett, K; Busetto, G; Bussey, P; Buzatu, A; Byrum, K L; Cabrera, S; Campanelli, M; Campbell, M; Canelli, F; Canepa, A; Carlsmith, D; Carosi, R; Carrillo, S; Carron, S; Casal, B; Casarsa, M; Castro, A; Catastini, P; Cauz, D; Cavalli-Sforza, M; Cerri, A; Cerrito, L; Chang, S H; Chen, Y C; Chertok, M; Chiarelli, G; Chlachidze, G; Chlebana, F; Cho, K; Chokheli, D; Chou, J P; Choudalakis, G; Chuang, S H; Chung, K; Chung, W H; Chung, Y S; Ciobanu, C I; Ciocci, M A; Clark, A; Clark, D; Compostella, G; Convery, M E; Conway, J; Cooper, B; Copic, K; Cordelli, M; Cortiana, G; Crescioli, F; Cuenca Almenar, C; Cuevas, J; Culbertson, R; Cully, J C; Dagenhart, D; Datta, M; Davies, T; de Barbaro, P; De Cecco, S; Deisher, A; De Lentdecker, G; De Lorenzo, G; Dell'orso, M; Demortier, L; Deng, J; Deninno, M; De Pedis, D; Derwent, P F; Di Giovanni, G P; Dionisi, C; Di Ruzza, B; Dittmann, J R; D'Onofrio, M; Donati, S; Dong, P; Donini, J; Dorigo, T; Dube, S; Efron, J; Erbacher, R; Errede, D; Errede, S; Eusebi, R; Fang, H C; Farrington, S; Fedorko, W T; Feild, R G; Feindt, M; Fernandez, J P; Ferrazza, C; Field, R; Flanagan, G; Forrest, R; Forrester, S; Franklin, M; Freeman, J C; Furic, I; Gallinaro, M; Galyardt, J; Garberson, F; Garcia, J E; Garfinkel, A F; Genser, K; Gerberich, H; Gerdes, D; Giagu, S; Giakoumopolou, V; Giannetti, P; Gibson, K; Gimmell, J L; Ginsburg, C M; Giokaris, N; Giordani, M; Giromini, P; Giunta, M; Glagolev, V; Glenzinski, D; Gold, M; Goldschmidt, N; Golossanov, A; Gomez, G; Gomez-Ceballos, G; Goncharov, M; González, O; Gorelov, I; Goshaw, A T; Goulianos, K; Gresele, A; Grinstein, S; Grosso-Pilcher, C; Grundler, U; Guimaraes da Costa, J; Gunay-Unalan, Z; Haber, C; Hahn, K; Hahn, S R; Halkiadakis, E; Hamilton, A; Han, B-Y; Han, J Y; Handler, R; Happacher, F; Hara, K; Hare, D; Hare, M; Harper, S; Harr, R F; Harris, R M; Hartz, M; Hatakeyama, K; Hauser, J; Hays, C; Heck, M; Heijboer, A; Heinemann, B; Heinrich, J; Henderson, C; Herndon, M; Heuser, J; Hewamanage, S; Hidas, D; Hill, C S; Hirschbuehl, D; Hocker, A; Hou, S; Houlden, M; Hsu, S-C; Huffman, B T; Hughes, R E; Husemann, U; Huston, J; Incandela, J; Introzzi, G; Iori, M; Ivanov, A; Iyutin, B; James, E; Jayatilaka, B; Jeans, D; Jeon, E J; Jindariani, S; Johnson, W; Jones, M; Joo, K K; Jun, S Y; Jung, J E; Junk, T R; Kamon, T; Kar, D; Karchin, P E; Kato, Y; Kephart, R; Kerzel, U; Khotilovich, V; Kilminster, B; Kim, D H; Kim, H S; Kim, J E; Kim, M J; Kim, S B; Kim, S H; Kim, Y K; Kimura, N; Kirsch, L; Klimenko, S; Klute, M; Knuteson, B; Ko, B R; Koay, S A; Kondo, K; Kong, D J; Konigsberg, J; Korytov, A; Kotwal, A V; Kraus, J; Kreps, M; Kroll, J; Krumnack, N; Kruse, M; Krutelyov, V; Kubo, T; Kuhlmann, S E; Kuhr, T; Kulkarni, N P; Kusakabe, Y; Kwang, S; Laasanen, A T; Lai, S; Lami, S; Lammel, S; Lancaster, M; Lander, R L; Lannon, K; Lath, A; Latino, G; Lazzizzera, I; Lecompte, T; Lee, J; Lee, J; Lee, Y J; Lee, S W; Lefèvre, R; Leonardo, N; Leone, S; Levy, S; Lewis, J D; Lin, C; Lin, C S; Linacre, J; Lindgren, M; Lipeles, E; Lister, A; Litvintsev, D O; Liu, T; Lockyer, N S; Loginov, A; Loreti, M; Lovas, L; Lu, R-S; Lucchesi, D; Lueck, J; Luci, C; Lujan, P; Lukens, P; Lungu, G; Lyons, L; Lys, J; Lysak, R; Lytken, E; Mack, P; Macqueen, D; Madrak, R; Maeshima, K; Makhoul, K; Maki, T; Maksimovic, P; Malde, S; Malik, S; Manca, G; Manousakis, A; Margaroli, F; Marino, C; Marino, C P; Martin, A; Martin, M; Martin, V; Martínez, M; Martínez-Ballarín, R; Maruyama, T; Mastrandrea, P; Masubuchi, T; Mattson, M E; Mazzanti, P; McFarland, K S; McIntyre, P; McNulty, R; Mehta, A; Mehtala, P; Menzemer, S; Menzione, A; Merkel, P; Mesropian, C; Messina, A; Miao, T; Miladinovic, N; Miles, J; Miller, R; Mills, C; Milnik, M; Mitra, A; Mitselmakher, G; Miyake, H; Moed, S; Moggi, N; Moon, C S; Moore, R; Morello, M; Movilla Fernandez, P; Mülmenstädt, J; Mukherjee, A; Muller, Th; Mumford, R; Murat, P; Mussini, M; Nachtman, J; Nagai, Y; Nagano, A; Naganoma, J; Nakamura, K; Nakano, I; Napier, A; Necula, V; Neu, C; Neubauer, M S; Nielsen, J; Nodulman, L; Norman, M; Norniella, O; Nurse, E; Oh, S H; Oh, Y D; Oksuzian, I; Okusawa, T; Oldeman, R; Orava, R; Osterberg, K; Pagan Griso, S; Pagliarone, C; Palencia, E; Papadimitriou, V; Papaikonomou, A; Paramonov, A A; Parks, B; Pashapour, S; Patrick, J; Pauletta, G; Paulini, M; Paus, C; Pellett, D E; Penzo, A; Phillips, T J; Piacentino, G; Piedra, J; Pinera, L; Pitts, K; Plager, C; Pondrom, L; Portell, X; Poukhov, O; Pounder, N; Prakoshyn, F; Pronko, A; Proudfoot, J; Ptohos, F; Punzi, G; Pursley, J; Rademacker, J; Rahaman, A; Rajaraman, A; Ramakrishnan, V; Ranjan, N; Redondo, I; Reisert, B; Rekovic, V; Renton, P; Rescigno, M; Richter, S; Rimondi, F; Ristori, L; Robson, A; Rodrigo, T; Rogers, E; Rolli, S; Roser, R; Rossi, M; Rossin, R; Roy, P; Ruiz, A; Russ, J; Rusu, V; Saarikko, H; Safonov, A; Sakumoto, W K; Salamanna, G; Saltó, O; Santi, L; Sarkar, S; Sartori, L; Sato, K; Savoy-Navarro, A; Scheidle, T; Schlabach, P; Schmidt, E E; Schmidt, M A; Schmidt, M P; Schmitt, M; Schwarz, T; Scodellaro, L; Scott, A L; Scribano, A; Scuri, F; Sedov, A; Seidel, S; Seiya, Y; Semenov, A; Sexton-Kennedy, L; Sfyria, A; Shalhout, S Z; Shapiro, M D; Shears, T; Shepard, P F; Sherman, D; Shimojima, M; Shochet, M; Shon, Y; Shreyber, I; Sidoti, A; Sinervo, P; Sisakyan, A; Slaughter, A J; Slaunwhite, J; Sliwa, K; Smith, J R; Snider, F D; Snihur, R; Soderberg, M; Soha, A; Somalwar, S; Sorin, V; Spalding, J; Spinella, F; Spreitzer, T; Squillacioti, P; Stanitzki, M; St Denis, R; Stelzer, B; Stelzer-Chilton, O; Stentz, D; Strologas, J; Stuart, D; Suh, J S; Sukhanov, A; Sun, H; Suslov, I; Suzuki, T; Taffard, A; Takashima, R; Takeuchi, Y; Tanaka, R; Tecchio, M; Teng, P K; Terashi, K; Thom, J; Thompson, A S; Thompson, G A; Thomson, E; Tipton, P; Tiwari, V; Tkaczyk, S; Toback, D; Tokar, S; Tollefson, K; Tomura, T; Tonelli, D; Torre, S; Torretta, D; Tourneur, S; Trischuk, W; Tu, Y; Turini, N; Ukegawa, F; Uozumi, S; Vallecorsa, S; van Remortel, N; Varganov, A; Vataga, E; Vázquez, F; Velev, G; Vellidis, C; Veszpremi, V; Vidal, M; Vidal, R; Vila, I; Vilar, R; Vine, T; Vogel, M; Volobouev, I; Volpi, G; Würthwein, F; Wagner, P; Wagner, R G; Wagner, R L; Wagner-Kuhr, J; Wagner, W; Wakisaka, T; Wallny, R; Wang, S M; Warburton, A; Waters, D; Weinberger, M; Wester, W C; Whitehouse, B; Whiteson, D; Wicklund, A B; Wicklund, E; Williams, G; Williams, H H; Wilson, P; Winer, B L; Wittich, P; Wolbers, S; Wolfe, C; Wright, T; Wu, X; Wynne, S M; Yagil, A; Yamamoto, K; Yamaoka, J; Yamashita, T; Yang, C; Yang, U K; Yang, Y C; Yao, W M; Yeh, G P; Yoh, J; Yorita, K; Yoshida, T; Yu, G B; Yu, F; Yu, I; Yu, S S; Yun, J C; Zanello, L; Zanetti, A; Zaw, I; Zhang, X; Zheng, Y; Zucchelli, S
2009-01-30
Models of maximal flavor violation (MxFV) in elementary particle physics may contain at least one new scalar SU(2) doublet field Phi(FV)=(eta(0),eta(+)) that couples the first and third generation quarks (q_(1), q_(3)) via a Lagrangian term L(FV)=xi(13)Phi(FV)q(1)q(3). These models have a distinctive signature of same-charge top-quark pairs and evade flavor-changing limits from meson mixing measurements. Data corresponding to 2 fb(-1) collected by the Collider Dectector at Fermilab II detector in pp[over ] collisions at sqrt[s]=1.96 TeV are analyzed for evidence of the MxFV signature. For a neutral scalar eta(0) with m_(eta;(0))=200 GeV/c(2) and coupling xi(13)=1, approximately 11 signal events are expected over a background of 2.1+/-1.8 events. Three events are observed in the data, consistent with background expectations, and limits are set on the coupling xi(13) for m(eta(0)=180-300 GeV/c(2).
2. Suppression of flavor violation in an A4 warped extra dimensional model
International Nuclear Information System (INIS)
2011-01-01
In an attempt to simultaneously explain the observed masses and mixing patterns of both quarks and leptons, we recently proposed a model (JHEP08(2010)115) based on the non abelian discrete flavor group A 4 , implemented in a custodial RS setup with a bulk Higgs. We showed that the standard model flavor structure can be realized within the zero mode approximation (ZMA), with nearly TBM neutrino mixing and a realistic CKM matrix with rather mild assumptions. An important advantage of this framework with respect to flavor anarchic models is the vanishing of the dangerous tree level KK gluon contribution to ε K and the suppression of the new physics one loop contributions to the neutron EDM, ε'/ε, b → Sγ and Higgs mediated flavor changing neutral current (FCNC) processes. These results are obtained beyond the ZMA, in order to account for the the full flavor structure and mixing of the zero modes and first Kaluza-Klein (KK) modes of all generations. The resulting constraints on the KK mass scale are shown to be significantly relaxed compared to the flavour anarchic case, showing explicitly the role of non abelian discrete flavor symmetries in relaxing flavor violation bounds within the RS setup. As a byproduct of our analysis we also obtain the same contributions for the custodial anarchic case with two SU(2) R doublets for each fermion generation.
3. Search for the lepton-flavor violating decays $B^0_s \\rightarrow e^{\\pm}\\mu^{\\mp}$ and $B^0 \\rightarrow e^{\\pm} \\mu^{\\mp}$
CERN Document Server
2013-01-01
A search for the lepton-flavour violating decays $B^0_s \\rightarrow e^{\\pm}\\mu^{\\mp}$ and $B^0 \\rightarrow e^{\\pm} \\mu^{\\mp}$ is performed with a data sample, corresponding to an integrated luminosity of 1.0 fb$^{-1}$ of $pp$ collisions at $\\sqrt{s} = 7$, TeV, collected by the LHCb experiment. The observed number of $B^0_s \\to e^{\\pm} \\mu^{\\mp}$ and $B^0 \\to e^{\\pm} \\mu^{\\mp}$ candidates is consistent with background expectations. Upper limits on the branching fractions of both decays are determined to be $BR(B^0_s \\to e^{\\pm} \\mu^{\\mp} 107$ TeV/c$^2$ and $M_{\\rm LQ} (B^0 \\to e^{\\pm} \\mu^{\\mp}) > 126$ TeV/c$^2$ at 95% C.L., and are a factor of two higher than the previous bounds.
4. Neutrino masses and a low breaking scale of left-right symmetry
International Nuclear Information System (INIS)
2002-01-01
In left-right symmetric models (LRSMs) the light neutrino masses arise from two sources: the seesaw mechanism and a vacuum expectation value of an SU(2) L triplet. If the left-right symmetry breaking v R is low, v R (less-or-similar sign)15 TeV, the contributions to the light neutrino masses from both the seesaw mechanism and the triplet Yukawa couplings are expected to be well above the experimental bounds. We present a minimal LRSM with an additional U(1) symmetry in which the masses induced by the two sources are below the eV scale and the twofold problem is solved. We further show that, if the U(1) symmetry is also responsible for the lepton flavor structure, the model yields a small mixing angle within the first two lepton generations
5. On flavor violation for massive and mixed neutrinos
International Nuclear Information System (INIS)
Blasone, M.; Capolupo, A.; Ji, C.R.; Vitiello, G.
2009-01-01
We discuss flavor charges and states for interacting mixed neutrinos in QFT. We show that the Pontecorvo states are not eigenstates of the flavor charges. This implies that their use in describing the flavor neutrinos produces a violation of lepton charge conservation in the production/detection vertices. The flavor states defined as eigenstates of the flavor charges give the correct representation of mixed neutrinos in charged current weak interaction processes.
6. Rare Z decays and neutrino flavor universality
Energy Technology Data Exchange (ETDEWEB)
Durieux, Gauthier [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Cornell Univ. Ithaca, NY (United States). Lab. for Elementary Particle Physics; Univ. Catholique de Louvain, Louvain-la-Neuve (Belgium). Centre for Cosmology, Particle Physics and Phenomenology; Grossman, Yuval; Kuflik, Erik [Cornell Univ. Ithaca, NY (United States). Lab. for Elementary Particle Physics; Koenig, Matthias [Mainz Univ. (Germany). PRISMA Cluster of Excellence; Mainz Univ. (Germany). Mainz Inst. for Theoretical Physics; Ray, Shamayita [Cornell Univ. Ithaca, NY (United States). Lab. for Elementary Particle Physics; Calcutta Univ. (India). Dept. of Physics
2015-12-15
We study rare four-body decays of the Z-boson involving at least one neutrino and one charged lepton. Large destructive interferences make these decays very sensitive to the Z couplings to neutrinos. As the identified charged leptons can determine the neutrino flavors, these decays probe the universality of the Z couplings to neutrinos. The rare four-body processes could be accurately measured at future lepton colliders, leading to percent level precision.
7. Origins of tiny neutrino mass and large flavor mixings
International Nuclear Information System (INIS)
Haba, Naoyuki
2015-01-01
Active neutrino masses are extremely smaller than those of other quarks and leptons, and there are large flavor mixings in the lepton sector, contrary to the quark sector. They are great mysteries in the standard model, but also excellent hints of new physics beyond the standard model. Thus, questions 'What is an origin of tiny neutrino mass?' and 'What is an origin of large lepton flavor mixings?' are very important. In this paper, we overview various attempts to solve these big questions. (author)
8. Implications of horizontal symmetries on baryon number violation in supersymmetric models
International Nuclear Information System (INIS)
Ben-Hamo, V.; Nir, Y.
1994-08-01
The smallness of the quark and lepton parameters and the hierarchy between them could be the result of selection rules due to a horizontal symmetry broken by a small parameter. The same selection rules apply to baryon number violating terms. Consequently, the problem of baryon number violation in supersymmetry may be solved naturally, without invoking any especially-designed extra symmetry. This mechanism is efficient enough even for low-scale flavor physics. Proton decay is likely to be dominated by the modes K + ν-bar i or K o μ + (e + ), and may proceed at observable rates. (authors). 15 refs
9. Neutrino mass ordering and μ-τ reflection symmetry breaking
Science.gov (United States)
Xing, Zhi-zhong; Zhu, Jing-yu
2017-12-01
If the neutrino mass spectrum turns out to be m 3case the columns of the 3×3 lepton flavor mixing matrix U should be reordered accordingly, and the resulting pattern U‧ may involve one or two large mixing angles in the standard parametrization or its variations. Since the Majorana neutrino mass matrix remains unchanged in such a mass relabeling, a possible μ-τ reflection symmetry is respected in this connection and its breaking effects are model-independently constrained at the 3σ level by using current experimental data. Supported by National Natural Science Foundation of China (11135009, 11375207)
10. Heavy leptons: theoretical study of the implications of their existence
International Nuclear Information System (INIS)
1978-01-01
The following points are studied: the possibility of an internal structure of heavy leptons and its manifestation; a study of the production of neutral heavy leptons in e + -e - collisions; consequences of the lumaton (heavy lepton having strong interactions) hypothesis; the introduction of a muon number violating mechanism in gauge theories. A gauge model characterized by the symmetries: left-right and quarks-leptons is also studied. A general review of the heavy leptons is given [fr
11. A model with isospin doublet U(1)D gauge symmetry
Science.gov (United States)
2018-05-01
We propose a model with an extra isospin doublet U(1)D gauge symmetry, in which we introduce several extra fermions with odd parity under a discrete Z2 symmetry in order to cancel the gauge anomalies out. A remarkable issue is that we impose nonzero U(1)D charge to the Standard Model Higgs, and it gives the most stringent constraint to the vacuum expectation value of a scalar field breaking the U(1)D symmetry that is severer than the LEP bound. We then explore relic density of a Majorana dark matter candidate without conflict of constraints from lepton flavor violating processes. A global analysis is carried out to search for parameters which can accommodate with the observed data.
12. Quadratic Hierarchy Flavor Rule as the Origin of Dirac CP-Violating Phases
OpenAIRE
Lipmanov, E. M.
2007-01-01
The premise of an organizing quadratic hierarchy rule in lepton-quark flavor physics was used earlier for explanation of the hierarchy patterns of four generic pairs of flavor quantities 1) charged-lepton and 2) neutrino deviations from mass-degeneracy, 3) deviations of lepton mixing from maximal magnitude and 4) deviations of quark mixing from minimal one. Here it is shown that the quadratic hierarchy equation that is uniquely related to three flavor particle generations may have yet another...
13. Leptonic CP violation induced by approximately μ-τ symmetric seesaw mechanism
International Nuclear Information System (INIS)
Baba, Teppei; Yasue, Masaki
2008-01-01
Assuming a minimal seesaw model with two heavy neutrinos (N), we examine effects of leptonic CP violation induced by approximate μ-τ symmetric interactions. As long as N is subject to the μ-τ symmetry, we can choose CP phases of Dirac mass terms without loss of generality in such a way that these phases arise from μ-τ symmetry breaking interactions. In the case that no phase is present in heavy neutrino mass terms, leptonic CP phases are controlled by two phases α and β. The similar consideration is extended to N blind to the μ-τ symmetry. It is argued that N subject (blind) to the μ-τ symmetry necessarily describes the normal (inverted) mass hierarchy. We restrict ourselves to μ-τ symmetric textures giving the tribimaximal mixing and calculate flavor neutrino masses to estimate CP-violating Dirac and Majorana phases as well as neutrino mixing angles as functions of α and β. Since α and β are generated by μ-τ symmetry breaking interactions, the CP-violating Majorana phase tends to be suppressed and is found to be at most O(0.1) radian. On the other hand, the CP-violating Dirac phase tends to show a proportionality to α or to β.
14. Gauged Lepton Flavour
CERN Document Server
Alonso, R.; Gavela, M.B.; Grinstein, B.; Merlo, L.; Quilez, P.
2016-12-22
The gauging of the lepton flavour group is considered in the Standard Model context and in its extension with three right-handed neutrinos. The anomaly cancellation conditions lead to a Seesaw mechanism as underlying dynamics for all leptons; requiring in addition a phenomenologically viable setup leads to Majorana masses for the neutral sector: the type I Seesaw Lagrangian in the Standard Model case and the inverse Seesaw in the extended model. Within the minimal extension of the scalar sector, the Yukawa couplings are promoted to scalar fields in the bifundamental of the flavour group. The resulting low-energy Yukawa couplings are proportional to inverse powers of the vacuum expectation values of those scalars; the protection against flavour changing neutral currents differs from that of Minimal Flavor Violation. In all cases, the $\\mu-\\tau$ flavour sector exhibits rich and promising phenomenological signals.
15. Lepton dipole moments
CERN Document Server
Marciano, William J
2010-01-01
This book provides a self-contained description of the measurements of the magnetic dipole moments of the electron and muon, along with a discussion of the measurements of the fine structure constant, and the theory associated with magnetic and electric dipole moments. Also included are the searches for a permanent electric dipole moment of the electron, muon, neutron and atomic nuclei. The related topic of the transition moment for lepton flavor violating processes, such as neutrinoless muon or tauon decays, and the search for such processes are included as well. The papers, written by many o
16. Permutation symmetry and the origin of fermion mass hierarchy
International Nuclear Information System (INIS)
Babu, K.S.; Mohapatra, R.N.
1990-01-01
A realization of the ''flavor-democracy'' approach to quark and lepton masses is provided in the context of the standard model with a horizontal S 3 permutation symmetry. In this model, t and b quarks pick up mass at the tree level, c, s-quark and τ-lepton masses arise at the one-loop level, u, d, and μ masses at the two-loop level, and the electron mass at the three-loop level, thus reproducing the observed hierarchial structure without fine tuning of the Yukawa couplings. The pattern of quark mixing angles also emerges naturally, with V us ,V cb ∼O(ε), V ub ∼O(ε 2 ), where ε is a loop expansion parameter
17. Flavor physics and CP violation
Science.gov (United States)
Chang, Paoti; Chen, Kai-Feng; Hou, Wei-Shu
2017-11-01
We currently live in the age of the CKM paradigm. The 3 × 3 matrix that links (d , s , b) quarks to (u , c , t) in the charged current weak interaction, being complex and nominally with 18 parameters, can be accounted for by just 3 rotation angles and one CP violating (CPV) phase, with unitarity and the CKM phases triumphantly tested at the B factories. But the CKM picture is unsatisfactory and has too many parameters. The main aim of Flavor Physics and CP violation (FPCP) studies is the pursuit to uncover New Physics beyond the Standard Model (SM). Two highlights of LHC Run 1 period are the CPV phase ϕs of Bs mixing and Bs →μ+μ- decay, which were found to be again consistent with SM, though the saga is yet unfinished. We also saw the emergence of the P5‧ angular variable anomaly in B0 →K∗0μ+μ- decay and R K (∗) anomaly in B →K (∗)μ+μ- to B →K (∗)e+e- rate ratios, and the BaBar anomaly in B →D (∗) τν decays, which suggest possible New Physics in these flavor processes, pointing to extra Z‧, charged Higgs, or leptoquarks. Charmless hadronic, semileptonic, purely leptonic and radiative B decays continue to offer various further windows on New Physics. Away from B physics, the rare K → πνν decays and ε‧ / ε in the kaon sector, μ → e transitions, muon g - 2 and electric dipole moments of the neutron and electron, τ → μγ , μμμ , eee, and a few charm physics probes, offer broadband frontier windows on New Physics. Lastly, flavor changing neutral transitions involving the top quark t and the 125 GeV Higgs boson h, such as t → ch and h → μτ, offer a new window into FPCP, while a new Z‧ related or inspired by the P5‧ anomaly, could show up in analogous top quark processes, perhaps even link with low energy phenomena such as muon g - 2 or rare kaon processes. In particular, we advocate the potential new SM, the two Higgs doublet model without discrete symmetries to control flavor violation, as SM2. As we are
18. Phenomenological aspects of possible vacua of a neutrino flavor model
Science.gov (United States)
Morozumi, Takuya; Okane, Hideaki; Sakamoto, Hiroki; Shimizu, Yusuke; Takagi, Kenta; Umeeda, Hiroyuki
2018-01-01
We discuss a supersymmetric model with discrete flavor symmetry {A}4× {Z}3. The additional scalar fields which contribute masses of leptons in the Yukawa terms are introduced in this model. We analyze their scalar potential and find that they have various vacuum structures. We show the relations among 24 different vacua and classify them into two types. We derive expressions of the lepton mixing angles, Dirac CP violating phase and Majorana phases for the two types. The model parameters which are allowed by the experimental data of the lepton mixing angles are different for each type. We also study the constraints on the model parameters which are related to Majorana phases. The different allowed regions of the model parameters for the two types are shown numerically for a given region of two combinations of the CP violating phases. Supported by JSPS KAKENHI Grant Number JP17K05418 (T.M.). This work is also supported in part by Grants-in-Aid for Scientific Research [No. 16J05332 (Y.S.), Nos. 24540272, 26247038, 15H01037, 16H00871, and 16H02189 (H.U.)] from the Ministry of Education, Culture, Sports, Science and Technology in Japan. H.O. is also supported by Hiroshima Univ. Alumni Association
19. Electric dipole moments with and beyond flavor invariants
Science.gov (United States)
Smith, Christopher; Touati, Selim
2017-11-01
In this paper, the flavor structure of quark and lepton electric dipole moments in the SM and beyond is investigated using tools inspired from Minimal Flavor Violation. While Jarlskog-like flavor invariants are adequate for estimating CP-violation from closed fermion loops, non-invariant structures arise from rainbow-like processes. Our goal is to systematically construct these latter flavor structures in the quark and lepton sectors, assuming different mechanisms for generating neutrino masses. Numerically, they are found typically much larger, and not necessarily correlated with, Jarlskog-like invariants. Finally, the formalism is adapted to deal with a third class of flavor structures, sensitive to the flavored U (1) phases, and used to study the impact of the strong CP-violating interaction and the interplay between the neutrino Majorana phases and possible baryon and/or lepton number violating interactions.
20. Electric dipole moments with and beyond flavor invariants
Directory of Open Access Journals (Sweden)
Christopher Smith
2017-11-01
Full Text Available In this paper, the flavor structure of quark and lepton electric dipole moments in the SM and beyond is investigated using tools inspired from Minimal Flavor Violation. While Jarlskog-like flavor invariants are adequate for estimating CP-violation from closed fermion loops, non-invariant structures arise from rainbow-like processes. Our goal is to systematically construct these latter flavor structures in the quark and lepton sectors, assuming different mechanisms for generating neutrino masses. Numerically, they are found typically much larger, and not necessarily correlated with, Jarlskog-like invariants. Finally, the formalism is adapted to deal with a third class of flavor structures, sensitive to the flavored U(1 phases, and used to study the impact of the strong CP-violating interaction and the interplay between the neutrino Majorana phases and possible baryon and/or lepton number violating interactions.
1. Tau leptons
International Nuclear Information System (INIS)
Gan, K.K.
1992-01-01
Once an oddity, tau leptons are now being mass produced at electron-positron colliders, and tau physics is becoming daily life. This was reflected at the Second Workshop on Tau Lepton Physics, held at Ohio State University, September 8-11. This workshop was the sequel to the successful workshop organized by Michel Davier and Bernard Jean-Marie at Orsay in 1990
2. Tau leptons
Energy Technology Data Exchange (ETDEWEB)
Gan, K. K.
1992-12-15
Once an oddity, tau leptons are now being mass produced at electron-positron colliders, and tau physics is becoming daily life. This was reflected at the Second Workshop on Tau Lepton Physics, held at Ohio State University, September 8-11. This workshop was the sequel to the successful workshop organized by Michel Davier and Bernard Jean-Marie at Orsay in 1990.
3. Heavy flavors
International Nuclear Information System (INIS)
Cox, B.; Gilman, F.J.; Gottschalk, T.D.
1986-11-01
A range of issues pertaining to heavy flavors at the SSC is examined including heavy flavor production by gluon-gluon fusion and by shower evolution of gluon jets, flavor tagging, reconstruction of Higgs and W bosons, and the study of rare decays and CP violation in the B meson system. A specific detector for doing heavy flavor physics and tuned to this latter study at the SSC, the TASTER, is described. 36 refs., 10 figs
4. Colored leptons
International Nuclear Information System (INIS)
Harari, H.
1985-01-01
If leptons are composite and if they contain colored preons, one expects the existence of heavy color-octet fermions with quantum numbers similar to those of ordinary leptons. Such a ''colored lepton'' should decay into a gluon and a lepton, yielding a unique experimental signature. Charged ''colored leptons'' probably have masses of the order of the compositeness scale Λ > or approx. 1 TeV. They may be copiously produced at future multi-TeV e + e - , ep and hadron colliders. ''Colored neutrinos'' may have both Dirac and Majorana masses. They could be much lighter than Λ, possibly as light as 100 GeV or less. In such a case they should be readily produced at the CERN anti pp collider, yielding spectacular monojet and dijet events. They may also be produced at LEP and HERA. (orig.)
5. Flavor Memory
NARCIS (Netherlands)
Mojet, Jos; Köster, Ep
2016-01-01
Odor, taste, texture, temperature, and pain all contribute to the perception and memory of food flavor. Flavor memory is also strongly linked to the situational aspects of previous encounters with the flavor, but does not depend on the precise recollection of its sensory features as in vision and
6. A Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies
Science.gov (United States)
Lu, Wei
2017-09-01
We propose a Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies in the context of composite Higgs bosons. Standard model fermions are represented by algebraic spinors of six-dimensional binary Clifford algebra, while ternary Clifford algebra-related flavor projection operators control allowable flavor-mixing interactions. There are three composite electroweak Higgs bosons resulted from top quark, tau neutrino, and tau lepton condensations. Each of the three condensations gives rise to masses of four different fermions. The fermion mass hierarchies within these three groups are determined by four-fermion condensations, which break two global chiral symmetries. The four-fermion condensations induce axion-like pseudo-Nambu-Goldstone bosons and can be dark matter candidates. In addition to the 125 GeV Higgs boson observed at the Large Hadron Collider, we anticipate detection of tau neutrino composite Higgs boson via the charm quark decay channel.
7. RS-A{sub 4} relaxation of flavor and CP violation
Energy Technology Data Exchange (ETDEWEB)
Kadosh, Avihay, E-mail: [email protected] [University of Groningen, Centre for Theoretical Physics (Netherlands)
2013-03-15
I discuss a model based on an A{sub 4} bulk flavor symmetry in the Randall-Sundrum (RS) setup. After discussing the setup and leading order results for the masses and mixings of quarks and leptons, I elaborate on the effect of higher order 'cross-talk' corrections, their contributions to flavor violating processes and the resulting constraints on the model parameter space and the Kaluza-Klein (KK) mass scale. In addition, I present a systematic study of higher order corrections to the PMNS matrix in light of the recent measurements of {theta}{sub 13} > 0 by RENO and Daya Bay. Finally, I also comment on the model new physics contributions to B{sub s,d} {yields} {mu}{sup +}{mu}{sup -} and {mu} {yields} e{gamma}, in light of the new upper bounds recently set by the LHCb and MEG experiment.
8. Possible reason why leptons are lighter than quarks
International Nuclear Information System (INIS)
Volkas, R.R.
1994-01-01
The minimal model of spontaneously broken leptonic colour and discrete quark-lepton symmetry predicts that charged leptons have the same masses as their partner charge +2/3 quarks up to small radiative corrections. By invoking a different pattern of symmetry braking, a similar model can be constructed with the structural feature that charged leptons have to be lighter than their partner quarks because of fermion mixing effects. As well as furnishing a new model-building tool, this is phenomenologically interesting because the scale of the new physics responsible for the quark-lepton mass hierarchy could be as low as several hundred GeV. 8 refs
9. Leptonic flavor and СP violation
interactions [2,3]. second, we explain how sneutrino oscillations are sensitive to the ... This effect is suppressed by small mixing angles and small mass differences. .... Our results will be given in terms of Am¾ , A and x , which are defined as.
10. Lepton charges and lepton mixing
International Nuclear Information System (INIS)
Pontecorvo, B.
1978-01-01
A review is given of theoretical and experimental investigations of lepton charges and lepton mixing known to the author at the time of the Budapest Conference, July 1970. The review is more biased towards experiment than theory. The recent and relevant expermental limits on possible lepton charge non-conservation are summarized, which were obtained by measuring probabilities of various processes. The status of the lepton mixing theory in the case when the only neutral leptons are neutrinos is reviewed, the main points being the μ→eγ decay and neutrino oscillations. The ''solar neutrino puzzle'' is discussed. A model of the μ→eγ and μ→3e decays is given as an example of drastic effects of heavy lepton mixing, and the relation between processes like μ→eγ, etc., and neutrino oscillations is considered. Recent papers on lepton nonconservation effects are then classified in groups, the related literature being presented extensively, if not fully
11. Exploring flavor-dependent long-range forces in long-baseline neutrino oscillation experiments
Science.gov (United States)
Chatterjee, Sabya Sachi; Dasgupta, Arnab; Agarwalla, Sanjib Kumar
2015-12-01
The Standard Model gauge group can be extended with minimal matter content by introducing anomaly free U(1) symmetry, such as L e - L μ or L e - L τ . If the neutral gauge boson corresponding to this abelian symmetry is ultra-light, then it will give rise to flavor-dependent long-range leptonic force, which can have significant impact on neutrino oscillations. For an instance, the electrons inside the Sun can generate a flavor-dependent long-range potential at the Earth surface, which can suppress the ν μ → ν e appearance probability in terrestrial experiments. The sign of this potential is opposite for anti-neutrinos, and affects the oscillations of (anti-)neutrinos in different fashion. This feature invokes fake CP-asymmetry like the SM matter effect and can severely affect the leptonic CP-violation searches in long-baseline experiments. In this paper, we study in detail the possible impacts of these long-range flavor-diagonal neutral current interactions due to L e - L μ symmetry, when (anti-)neutrinos travel from Fermilab to Homestake (1300 km) and CERN to Pyhäsalmi (2290 km) in the context of future high-precision superbeam facilities, DUNE and LBNO respectively. If there is no signal of long-range force, DUNE (LBNO) can place stringent constraint on the effective gauge coupling α eμ < 1.9 × 10-53 (7.8 × 10-54) at 90% C.L., which is almost 30 (70) times better than the existing bound from the Super-Kamiokande experiment. We also observe that if α eμ ≥ 2 × 10-52, the CP-violation discovery reach of these future facilities vanishes completely. The mass hierarchy measurement remains robust in DUNE (LBNO) if α eμ < 5 × 10-52 (10-52).
12. Minimal Flavor Constraints for Technicolor
DEFF Research Database (Denmark)
Sakuma, Hidenori; Sannino, Francesco
2010-01-01
We analyze the constraints on the the vacuum polarization of the standard model gauge bosons from a minimal set of flavor observables valid for a general class of models of dynamical electroweak symmetry breaking. We will show that the constraints have a strong impact on the self-coupling and mas......We analyze the constraints on the the vacuum polarization of the standard model gauge bosons from a minimal set of flavor observables valid for a general class of models of dynamical electroweak symmetry breaking. We will show that the constraints have a strong impact on the self...
13. The fifth lepton
International Nuclear Information System (INIS)
Pietschmann, H.
1979-01-01
All possible leptonic decays are discussed of charged heavy lepton, designated tau. Quantum numbers of this lepton and the structure of the currents which contain the new lepton and couplings of these currents are investigated. (Z.J.)
14. Five-body leptonic decays of muon and tau leptons
Energy Technology Data Exchange (ETDEWEB)
Flores-Tlalpa, A. [Instituto de Física, Universidad Nacional Autónoma de México,Apartado Postal 20-364, 01000 México D.F. (Mexico); Castro, G. López [Departamento de Física,Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional,Apartado Postal 14-740, 07000 México D.F. (Mexico); Instituto de Física Corpuscular, CSIC- Universitat de València,Apt. Correus 22085, E-46071 València (Spain); Roig, P. [Departamento de Física,Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional,Apartado Postal 14-740, 07000 México D.F. (Mexico)
2016-04-29
We study the five-body decays μ{sup −}→e{sup −}e{sup +}e{sup −}ν{sub μ}ν̄{sub e} and τ{sup −}→ℓ{sup −}ℓ{sup ′+}ℓ{sup ′−}ν{sub τ}ν̄{sub ℓ} for ℓ,ℓ{sup ′}=e,μ within the Standard Model (SM) and in a general effective field theory description of the weak interactions at low energies. We compute the branching ratios and compare our results with two previous — mutually discrepan — SM calculations. By assuming a general structure for the weak currents we derive the expressions for the energy and angular distributions of the three charged leptons when the decaying lepton is polarized, which will be useful in precise tests of the weak charged current at Belle II. In these decays, leptonic T-odd correlations in triple products of spin and momenta — which may signal time reversal violation in the leptonic sector — are suppressed by the tiny neutrino masses. Therefore, a measurement of such T-violating observables would be associated to neutrinoless lepton flavor violating (LFV) decays, where this effect is not extremely suppressed. We also study the backgrounds that the SM five-lepton lepton decays constitute to searches of LFV L{sup −}→ℓ{sup −}ℓ{sup ′+}ℓ{sup ′−} decays. Searches at high values of the invariant mass of the ℓ{sup ′+}ℓ{sup ′−} pair look the most convenient way to overcome the background.
15. Flavor Dependence of the S-parameter
DEFF Research Database (Denmark)
Di Chiara, Stefano; Pica, Claudio; Sannino, Francesco
2011-01-01
of flavors, colors and matter representation. We show that S, normalized to the number of flavors, increases as we decrease the number of flavors and gives a direct measure of the anomalous dimension of the mass of the fermions. Our findings support the conjecture presented in [arXiv:1006.0207 [hep...... constitute important constraints on models of dynamical electroweak symmetry breaking and unparticle physics....
16. Classically conformal radiative neutrino model with gauged B−L symmetry
Directory of Open Access Journals (Sweden)
2016-09-01
Full Text Available We propose a classically conformal model in a minimal radiative seesaw, in which we employ a gauged B−L symmetry in the standard model that is essential in order to work the Coleman–Weinberg mechanism well that induces the B−L symmetry breaking. As a result, nonzero Majorana mass term and electroweak symmetry breaking simultaneously occur. In this framework, we show a benchmark point to satisfy several theoretical and experimental constraints. Here theoretical constraints represent inert conditions and Coleman–Weinberg condition. Experimental bounds come from lepton flavor violations (especially μ→eγ, the current bound on the Z′ mass at the CERN Large Hadron Collider, and neutrino oscillations.
17. Flavor and CP violations from sleptons at the Muon Collider
International Nuclear Information System (INIS)
Cheng, H.-C.
1997-12-01
Supersymmetric theories generally have new flavor and CP violation sources in the squark and slepton mass matrices. They will contribute to the lepton flavor violation processes, such as μ→eγ, which can be probed far below the current bound with an intense muon source at the front end of the muon collider. In addition, if sleptons can be produced at the muon collider, the flavor violation can occur at their production and decay, allowing us to probe the flavor mixing structure directly. Asymmetry between numbers of μ + e - and e + μ - events will be a sign for CP violation in supersymmetric flavor mixing
18. (S3)3 theories of flavor
International Nuclear Information System (INIS)
Carone, C.D.
1996-07-01
The author presents a supersymmetric theory of flavor based on the discrete flavor group (S 3 ) 3 . The model can account for the masses and mixing angles of the standard model, while maintaining sufficient sfermion degeneracy to evade the supersymmetric flavor problem. The author demonstrates that the model has a viable phenomenology and makes one very striking prediction: the nucleon decays predominantly to Kl where l is a first generation lepton. He shows that the modes n → K 0 bar ν e , p → K + bar ν e , and p → K 0 e + occur at comparable rates, and could well be discovered simultaneously at the SuperKamiokande experiment
19. A couplet from flavored dark matter
Energy Technology Data Exchange (ETDEWEB)
Agrawal, Prateek [Fermilab,P.O. Box 500, Batavia, IL, 60510 (United States); Chacko, Zackaria [Maryland Center for Fundamental Physics, Department of Physics, University of Maryland,College Park, MD, 20742-4111 (United States); Kilic, Can [Theory Group, Department of Physics and Texas Cosmology Center,The University of Texas at Austin, 2515 Speedway Stop C1608, Austin, TX, 78712-1197 (United States); Verhaaren, Christopher B. [Maryland Center for Fundamental Physics, Department of Physics, University of Maryland,College Park, MD, 20742-4111 (United States)
2015-08-17
We show that a couplet, a pair of closely spaced photon lines, in the X-ray spectrum is a distinctive feature of lepton flavored dark matter models for which the mass spectrum is dictated by Minimal Flavor Violation. In such a scenario, mass splittings between different dark matter flavors are determined by Standard Model Yukawa couplings and can naturally be small, allowing all three flavors to be long-lived and contribute to the observed abundance. Then, in the presence of a tiny source of flavor violation, heavier dark matter flavors can decay via a dipole transition on cosmological timescales, giving rise to three photon lines. Two of these lines are closely spaced, and constitute the couplet. Provided the flavor violation is sufficiently small, the ratios of the line energies are determined in terms of the charged lepton masses, and constitute a prediction of this framework. For dark matter masses of order the weak scale, the couplet lies in the keV-MeV region, with a much weaker line in the eV-keV region. This scenario constitutes a potential explanation for the recent claim of the observation of a 3.5 keV line. The next generation of X-ray telescopes may have the necessary resolution to resolve the double line structure of such a couplet.
20. Heavy Flavor Gauge Boson search at the LHC
CERN Document Server
AUTHOR|(INSPIRE)INSPIRE-00214690
Despite its many successes, the Standard Model (SM) [1] of particle physics is believed to be an effective field theory valid only for energies up to the TeV scale. Due to its uniquely large mass, the top quark is of particular interest for the electroweak symmetry breaking mechanism and could potentially be related to new physics phenomena. Several proposed extensions to the SM predict the existence of heavy particles that decay primarily to top quark pairs. This thesis contains two parts: a theoretical motivation study. This work included Monte Carlo modelling of flavor gauge boson models. The experimental search is made in the top anti-top decay channel collected with the ATLAS experiment during Run-1 and early Run-2 where one W from a top decays leptonically (to an electron or muon plus neutrino) and the W from the second top decays hadronically. This leads to a signature with one high-transverse- momentum lepton, large missing transverse momentum (from the escaping neutrino) and hadronic jets.
1. A flavor-safe composite explanation of $R_K$
CERN Document Server
2017-05-04
In these proceedings we discuss a flavor-safe explanation of the anomaly found in $R_K= {\\cal B}(B \\to K \\mu^+ \\mu^-)/{\\cal B}(B \\to K e^+ e^-)$ by LHCb, within the framework of composite Higgs models. We present a model featuring a non-negligible degree of compositeness for all three generations of right-handed leptons, which leads to a violation of lepton-flavor universality in neutral current interactions while other constraints from quark- and lepton-flavor physics are met. Moreoever, the particular embedding of the lepton sector considered in this setup provides a parametrically enhanded contribution to the Higgs mass that can weak considerably the need for ultra-light top partners.
2. A flavor sector for the composite Higgs
Energy Technology Data Exchange (ETDEWEB)
Vecchi, Luca, E-mail: [email protected]
2013-11-25
We discuss flavor violation in large N Composite Higgs models. We focus on scenarios in which the masses of the Standard Model fermions are controlled by hierarchical mixing parameters, as in models of Partial Compositeness. We argue that a separation of scales between flavor and Higgs dynamics can be employed to parametrically suppress dipole and penguin operators, and thus effectively remove the experimental constraints arising from the lepton sector and the neutron EDM. The dominant source of flavor violation beyond the Standard Model is therefore controlled by 4-fermion operators, whose Wilson coefficients can be made compatible with data provided the Higgs dynamics approaches a “walking” regime in the IR. Models consistent with all flavor and electroweak data can be obtained with a new physics scale within the reach of the LHC. Explicit scenarios may be realized in a 5D framework, the new key ingredient being the introduction of flavor branes where the wave functions of the bulk fermions end.
3. The flavor-locked flavorful two Higgs doublet model
Science.gov (United States)
Altmannshofer, Wolfgang; Gori, Stefania; Robinson, Dean J.; Tuckler, Douglas
2018-03-01
We propose a new framework to generate the Standard Model (SM) quark flavor hierarchies in the context of two Higgs doublet models (2HDM). The flavorful' 2HDM couples the SM-like Higgs doublet exclusively to the third quark generation, while the first two generations couple exclusively to an additional source of electroweak symmetry breaking, potentially generating striking collider signatures. We synthesize the flavorful 2HDM with the flavor-locking' mechanism, that dynamically generates large quark mass hierarchies through a flavor-blind portal to distinct flavon and hierarchon sectors: dynamical alignment of the flavons allows a unique hierarchon to control the respective quark masses. We further develop the theoretical construction of this mechanism, and show that in the context of a flavorful 2HDM-type setup, it can automatically achieve realistic flavor structures: the CKM matrix is automatically hierarchical with | V cb | and | V ub | generically of the observed size. Exotic contributions to meson oscillation observables may also be generated, that may accommodate current data mildly better than the SM itself.
4. Effective theory of flavor for Minimal Mirror Twin Higgs
Science.gov (United States)
Barbieri, Riccardo; Hall, Lawrence J.; Harigaya, Keisuke
2017-10-01
We consider two copies of the Standard Model, interchanged by an exact parity symmetry, P. The observed fermion mass hierarchy is described by suppression factors ɛ^{n_i} for charged fermion i, as can arise in Froggatt-Nielsen and extra-dimensional theories of flavor. The corresponding flavor factors in the mirror sector are ɛ^' {n}_i} , so that spontaneous breaking of the parity P arises from a single parameter ɛ'/ɛ, yielding a tightly constrained version of Minimal Mirror Twin Higgs, introduced in our previous paper. Models are studied for simple values of n i , including in particular one with SU(5)-compatibility, that describe the observed fermion mass hierarchy. The entire mirror quark and charged lepton spectrum is broadly predicted in terms of ɛ'/ɛ, as are the mirror QCD scale and the decoupling temperature between the two sectors. Helium-, hydrogen- and neutron-like mirror dark matter candidates are constrained by self-scattering and relic ionization. In each case, the allowed parameter space can be fully probed by proposed direct detection experiments. Correlated predictions are made as well for the Higgs signal strength and the amount of dark radiation.
5. Flavor non-universal gauge interactions and anomalies in B-meson decays
Science.gov (United States)
Tang, Yong; Wu, Yue-Liang
2018-02-01
Motivated by flavor non-universality and anomalies in semi-leptonic B-meson decays, we present a general and systematic discussion about how to construct anomaly-free U(1)‧ gauge theories based on an extended standard model with only three right-handed neutrinos. If all standard model fermions are vector-like under this new gauge symmetry, the most general family non-universal charge assignments, (a,b,c) for three-generation quarks and (d,e,f) for leptons, need satisfy just one condition to be anomaly-free, 3(a+b+c) = - (d+e+f). Any assignment can be linear combinations of five independent anomaly-free solutions. We also illustrate how such models can generally lead to flavor-changing interactions and easily resolve the anomalies in B-meson decays. Probes with {{B}}{s} - {{\\bar B}}{s} mixing, decay into τ ±, dilepton and dijet searches at colliders are also discussed. Supported by the Grant-in-Aid for Innovative Areas (16H06490)
6. Are quarks and leptons composite
International Nuclear Information System (INIS)
Harari, H.
1982-01-01
The possibility that quarks and leptons are composite was studied. A line of reasoning was pursued which followed several steps. The standard model was assumed and the need to go beyond it was demonstrated. Different classes of ideas were considered. The notion of compositeness and its general difficulties, mainly the scale problem, were studied. A connection between composite massless fermions and an unbroken chiral symmetry was assumed. A general framework based on hypercolor and a chiral symmetry was established. The general requirements for a candidate model were established. A minimal scheme was found and its successes and failures were studied. (HK)
7. On gauged Baryon and Lepton numbers
International Nuclear Information System (INIS)
Rajpoot, S.
1990-01-01
The observation that Baryon number and Lepton number are conserved in nature provides strong motivation for associating gauge symmetries to these conserved numbers. This endeavor requires that the gauge group of electroweak interactions be extended from SU(2) L X U(1) Y to SU(2) L X U(1) R X U(1) Lepton where U(1) R couples only to the right-handed quarks and leptons. If it furthur postulated that right-handed currents exist on par with the left-handed ones, then the full electroweak symmetry is SU(2) L X SU(2) R X U(1) Baryon X U(1) Lepton . The SU(2) L X SU(2) R X U(1) Baryon X U(1) Lepton model is described in some detail. The triangle anomalies of the three families of quarks and leptons in the model are cancelled invoking leptoquark matter which is new fermionic matter that carries baryon as well as lepton numbers. In addition to the standard neutral boson (Z degree), the theory predicts two neutral gauge bosons with mass lower bounds of 120 GeV and 210 GeV which makes these particles prospective candidates for production at LEP, the TEVATRON and the SSC
8. A minimal spontaneous CP violation model with small neutrino mass and SU(2) x U(1) x Z3 symmetry
International Nuclear Information System (INIS)
Geng, C.Q.; Ng, J.N.
1988-04-01
It is shown that spontaneous CP violation and natural flavor conservation can occur in the SU(2) L x U(1) Y model based on two Higgs doublet and one Higgs singlet fields with a Z 3 discrete symmetry. Physical CP nonconservation is purely due to scalar-pseudoscalar mixings. In order for this to be a major source of CP violation a light spin-O boson of mass less than 10 GeV is required. The see-saw mechanism can be implemented to generate small neutrino masses. The model implies a relatively large electric dipole moment for charged leptons and small value for ε'/ε
9. Holographic theories of electroweak symmetry breaking without a Higgs Boson
International Nuclear Information System (INIS)
Burdman, Gustavo; Nomura, Yasunori
2003-01-01
Recently, realistic theories of electroweak symmetry breaking have been constructed in which the electroweak symmetry is broken by boundary conditions imposed at a boundary of higher dimensional spacetime. These theories have equivalent 4D dual descriptions, in which the electroweak symmetry is dynamically broken by non-trivial infrared dynamics of some gauge interaction, whose gauge coupling (tilde g) and size N satisfy (tilde g) 2 N ∼> 16π 2 . Such theories allow one to calculate electroweak radiative corrections, including the oblique parameters S, T and U, as long as (tilde g) 2 N/16π 2 and N are sufficiently larger than unity. We study how the duality between the 4D and 5D theories manifests itself in the computation of various physical quantities. In particular, we calculate the electroweak oblique parameters in a warped 5D theory where the electroweak symmetry is broken by boundary conditions at the infrared brane. We show that the value of S obtained in the minimal theory exceeds the experimental bound if the theory is in a weakly coupled regime. This requires either an extension of the minimal model or departure from weak coupling. A particularly interesting scenario is obtained if the gauge couplings in the 5D theory take the largest possible values--the value suggested by naive dimensional analysis. We argue that such a theory can provide a potentially consistent picture for dynamical electroweak symmetry breaking: corrections to the electroweak observables are sufficiently small while realistic fermion masses are obtained without conflicting with bounds from flavor violation. The theory contains only the standard model quarks, leptons and gauge bosons below ≅2 TeV, except for a possible light scalar associated with the radius of the extra dimension. At ≅2 TeV increasingly broad string resonances appear. An analysis of top-quark phenomenology and flavor violation is also presented, which is applicable to both the weakly-coupled and strongly
10. Dark Matter and observable lepton flavour violation
International Nuclear Information System (INIS)
Heurtier, Lucien; Univ. Libre de Bruxelles; Teresi, Daniele
2016-07-01
Seesaw models with leptonic symmetries allow right-handed (RH) neutrino masses at the electroweak scale, or even lower, at the same time having large Yukawa couplings with the Standard Model leptons, thus yielding observable effects at current or near-future lepton-flavour-violation (LFV) experiments. These models have been previously considered also in connection to low-scale leptogenesis, but the combination of observable LFV and successful leptogenesis has appeared to be difficult to achieve unless the leptonic symmetry is embedded into a larger one. In this paper, instead, we follow a different route and consider a possible connection between large LFV rates and Dark Matter (DM). We present a model in which the same leptonic symmetry responsible for the large Yukawa couplings guarantees the stability of the DM candidate, identified as the lightest of the RH neutrinos. The spontaneous breaking of this symmetry, caused by a Majoron-like field, also provides a mechanism to produce the observed relic density via the decays of the latter. The phenomenological implications of the model are discussed, finding that large LFV rates, observable in the near-future μ→e conversion experiments, require the DM mass to be in the keV range. Moreover, the active-neutrino coupling to the Majoron-like scalar field could be probed in future detections of supernova neutrino bursts.
11. Quark-lepton unification and proton decay
International Nuclear Information System (INIS)
Pati, J.C.; Salam, A.
1980-05-01
Complexions for proton decay arising within a maximal symmetry for quark-lepton unification, which leads to spontaneous rather than intrinsic violations of B, L and F are considered. Four major modes satisfying δB=-1 and δF=0, -2, -4 and -6 are noted. It is stressed that some of these modes can coexist in accord with allowed solutions for renormalization group equations for coupling constants for a class of unifying symmetries. None of these remarks is dependent on the nature of quark charges. It is noted that if quarks and leptons are made of constituent preons, the preon binding is likely to be magnetic. (author)
12. ϕ{sup 3} theory with F{sub 4} flavor symmetry in 6−2ϵ dimensions: 3-loop renormalization and conformal bootstrap
Energy Technology Data Exchange (ETDEWEB)
Pang, Yi [Max-Planck-Insitut für Gravitationsphysik (Albert-Einstein-Institut),Am Mühlenberg 1, Potsdam, DE-14476 (Germany); Rong, Junchen [Fields, Gravity & Strings, Center for Theoretical Physics of the Universe,Institute for Basic Sciences, Daejeon, 305-811 (Korea, Republic of); Su, Ning [CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Zhong Guan Cun East Street 55 #,P.O. Box 2735, Beijing, 100190 (China)
2016-12-14
We consider ϕ{sup 3} theory in 6−2ϵ with F{sub 4} global symmetry. The beta function is calculated up to 3 loops, and a stable unitary IR fixed point is observed. The anomalous dimensions of operators quadratic or cubic in ϕ are also computed. We then employ conformal bootstrap technique to study the fixed point predicted from the perturbative approach. For each putative scaling dimension of ϕ (Δ{sub ϕ}), we obtain the corresponding upper bound on the scaling dimension of the second lowest scalar primary in the 26 representation (Δ{sub 26}{sup 2nd}) which appears in the OPE of ϕ×ϕ. In D=5.95, we observe a sharp peak on the upper bound curve located at Δ{sub ϕ} equal to the value predicted by the 3-loop computation. In D=5, we observe a weak kink on the upper bound curve at (Δ{sub ϕ},Δ{sub 26}{sup 2nd})=(1.6,4).
13. Lowest-lying even-parity anti B{sub s} mesons: heavy-quark spin-flavor symmetry, chiral dynamics, and constituent quark-model bare masses
Energy Technology Data Exchange (ETDEWEB)
Albaladejo, M.; Fernandez-Soler, P.; Nieves, J.; Ortega, P.G. [Centro Mixto CSIC-Universidad de Valencia, Instituto de Fisica Corpuscular (IFIC), Institutos de Investigacion de Paterna, Aptd. 22085, Valencia (Spain)
2017-03-15
The discovery of the D{sup *}{sub s0}(2317) and D{sub s1}(2460) resonances in the charmed-strange meson spectra revealed that formerly successful constituent quark models lose predictability in the vicinity of two-meson thresholds. The emergence of non-negligible effects due to meson loops requires an explicit evaluation of the interplay between Q anti q and (Q anti q)(q anti q) Fock components. In contrast to the c anti s sector, there is no experimental evidence of J{sup P} = 0{sup +}, 1{sup +} bottom-strange states yet. Motivated by recent lattice studies, in this work the heavy-quark partners of the D{sub s0}{sup *}(2317) and D{sub s1}(2460) states are analyzed within a heavy meson chiral unitary scheme. As a novelty, the coupling between the constituent quark-model P-wave anti B{sub s} scalar and axial mesons and the anti B{sup (*)}K channels is incorporated employing an effective interaction, consistent with heavy-quark spin symmetry, constrained by the lattice energy levels. (orig.)
14. Leptophobic Boson Signals with Leptons, Jets and Missing Energy
Energy Technology Data Exchange (ETDEWEB)
Dobrescu, Bogdan A.
2015-06-14
Color-singlet gauge bosons with renormalizable couplings to quarks but not to leptons must interact with additional fermions (''anomalons'') required to cancel the gauge anomalies. Analyzing the decays of such leptophobic bosons into anomalons, I show that they produce final states involving leptons at the LHC. Resonant production of a flavor-universal leptophobic Z' boson leads to cascade decays via anomalons, whose signatures include a leptonically decaying Z, missing energy and several jets. A Z' boson that couples to the right-handed quarks of the first and second generations undergoes cascade decays that violate lepton universality and include signals with two leptons and jets, or with a Higgs boson, a lepton, a W and missing energy.
15. Flavor changing strings and domain walls
International Nuclear Information System (INIS)
Dvali, G.; Senjanovic, G.
1993-04-01
We consider the cosmological consequences of a spontaneous breaking of non-abelian discrete symmetries, which may appear as a natural remnant of a continuous symmetry, such as a family symmetry. The result may be a stable domain wall across which an electron would turn into a muon (orν e into ν μ ) or a flavor analogue of an Alice string-domain wall structure with the same property. (author). 16 refs
16. Breaking of electroweak symmetry: origin and effects
International Nuclear Information System (INIS)
Delaunay, C.
2008-10-01
The Higgs boson appears as the corner stone of high energy physics, it might be the cause of the excess of matter that led to the formation of the structures of the universe and it seems that it drives the breaking of the electroweak symmetry. Moreover, when the stability at low energies of the Higgs boson is assured by an extra space dimension, it appears that this extra dimension can explain most issues in the flavor physics that are not understood by the standard model. The first chapter presents the main tools of effective field theories, the role of experimental data in the construction of theories valid beyond the standard model is discussed. The second chapter focuses on the electroweak baryogenesis that allows the testing of new physics via the electroweak phase transition. We detail the calculation of a Higgs potential at finite temperature. We follow the dynamics of the phase transition including nucleation an supercooling. Finally we investigate the prospects of gravity wave detection to see the effects of a strong electroweak phase transition. The 2 last chapters are dedicated to the physics of extra-dimension. The properties of the dynamics of scalar, vector fields with a 1/2 spin plunged in a 5 d. Anti de Sitter geometry are reviewed. We present a model of lepton masses and mixings based on the A 4 non-Abelian discrete symmetry. It is shown that this model does not contradict the tests of electroweak precision. (A.C.)
17. Towards a new paradigm for quark-lepton unification
Energy Technology Data Exchange (ETDEWEB)
Smith, Christopher [Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3,53 avenue des Martyrs, 38026 Grenoble Cedex (France)
2017-05-03
The quark and charged lepton mass patterns upset their naïve unification. In this paper, a new approach to solve this problem is proposed. Model-independently, we find that a successful unification can be achieved. A mechanism is identified by which the large top quark mass renders its third-generation leptonic partner very light. This state is thus identified with the electron. We then construct a toy model to implement dynamically this mechanism, using tree-level exchanges of vector leptons to relate the quark and charged lepton flavor structures. In a supersymmetric context, this same mechanism splits the squark masses, and third generation squarks end up much lighter than the others. Finally, the implementation of this mechanism in SU(5) GUT permits to avoid introducing any flavor structure beyond the two minimal Yukawa couplings, ensuring the absence of unknown mixing matrices and their potentially large impact on FCNC.
18. Symmetry breaking and generational mixing in top-color-assisted technicolor
International Nuclear Information System (INIS)
Lane, K.
1996-01-01
Top-color-assisted technicolor provides a dynanamical explanation for electroweak and flavor symmetry breaking and for the large mass of the top quark without unnatural fine-tuning. A major challenge is to generate the observed mixing between heavy and light generations while breaking the strong top-color interactions near 1 TeV. I argue that these phenomena, as well as electroweak symmetry breaking, are intimately connected and I present a scenario for them based on nontrivial patterns of technifermion condensation. I also exhibit a class of models realizing this scenario. This picture leads to a rich phenomenology, especially in hadron and lepton collider experiments in the few hundred GeV to few TeV region and in precision electroweak tests at the Z 0 , atomic parity violation, and polarized Mo/ller scattering. copyright 1996 The American Physical Society
19. Flavor Physics & CP Violation 2015
Science.gov (United States)
"Flavor Physics & CP violation 2015" (FPCP 2015) was held in Nagoya, Japan, at Nagoya University, from May 25 to May 29 2015. This is the 13th meeting of the series of annual conferences started in Philadelphia, PA, USA in 2002. The aim of the conference is to review developments in flavor physics and CP violation, in both theory and experiment, exploiting the potential to study new physics at the LHC and future facilities. The topics include CP violation, rare decays, CKM elements with heavy quark decays, flavor phenomena in charged leptons and neutrinos, and also interplay between flavor and LHC high Pt physics. The FPCP2015 conference had more than 140 participants, including researchers from abroad and many young researchers (postdocs and students). The conference consisted of plenary talks and poster presentations. The plenary talks include 2 overview talks, 48 review talks, and 2 talks for outlook in theories and experiments, given by world leading researchers. There was also a special lecture by Prof. Makoto Kobayashi, one of the Nobel laureates in 2008. The poster session had 41 contributions. Many young researchers presented their works. These proceedings contain written documents for these plenary and poster presentations. The full scientific program and presentation materials can be found at http://fpcp2015.hepl.phys.nagoya-u.ac.jp/. We would like to thank the International Advisory Committee for their invaluable assistance in coordinating the scientific program and in helping to identifying many speakers. Thanks are also due to the Local Organizing Committee for tireless efforts for smooth running of the conference and very enjoyable social activities. We also thank the financial supports provided by Japanese Scociety for the Promotion of Science (JSPS) unfer the Grant-in-Aid for Scientific Research (S) "Probing New Physics with Tau-Lepton" (No. 26220706), by Nagoya University under the Program for Promoting the Enhancement of Research Universities, and
20. Lepton production
CERN Multimedia
2002-01-01
This experiment aims to settle open questions in the hadronic production of electrons, muons and neutrinos. Prominent among these are $e/\\mu$ universality, the contribution of charm decay to lepton pair production, and the "anomalous" low mass pairs.\\\\ The experimental design aims to optimize the combination of:\\\\- electron identification\\\\ - muon identification \\\\ - missing energy measurement for neutrinos \\\\ - vertex identification (for $\\tau \\simeq \\tau_{charm}$). \\\\ \\\\ The major components of the apparatus are shown in the figure. In the vertex region a proton beam of transverse size $\\simeq 50 \\mu$ impinges on a beryllium target of diameter $50 \\mu$, and high precision tracking in the vertex region is achieved by silicon strip detectors. Charged particle momenta are measured using a dipole magnet and high resolution drift chambers. Electrons are identified by the combination of the transition radiation detector and the finely segmented front section of the Uranium/Liquid Argon calorimeter. Essentially t...
1. 'L=R' -- $U(1)_R$ Lepton Number at the LHC
Energy Technology Data Exchange (ETDEWEB)
Frugiuele, Claudia [Fermilab; Gregoire, Thomas [Ottawa Carleton Inst. Phys.; Kumar, Piyush [Yale U.; Ponton, Eduardo [ISCAP, New York
2013-05-03
We perform a detailed study of a variety of LHC signals in supersymmetric models where lepton number is promoted to an (approximate) U(1)( )R( ) symmetry. Such a symmetry has interesting implications for naturalness, as well as flavor- and CP-violation, among others. Interestingly, it makes large sneutrino vacuum expectation values phenomenologically viable, so that a slepton doublet can play the role of the down-type Higgs. As a result, (some of) the leptons and neutrinos are incorporated into the chargino and neutralino sectors. This leads to characteristic decay patterns that can be experimentally tested at the LHC. The corresponding collider phenomenology is largely determined by the new approximately conserved quantum number, which is itself closely tied to the presence of “leptonic R-parity violation”. We find rather loose bounds on the first and second generation squarks, arising from a combination of suppressed production rates together with relatively small signal efficiencies of the current searches. Naturalness would indicate that such a framework should be discovered in the near future, perhaps through spectacular signals exhibiting the lepto-quark nature of the third generation squarks. The presence of fully visible decays, in addition to decay chains involving large missing energy (in the form of neutrinos) could give handles to access the details of the spectrum of new particles, if excesses over SM background were to be observed. The scale of neutrino masses is intimately tied to the source of U(1)( )R( ) breaking, thus opening a window into the R-breaking sector through neutrino physics. Further theoretical aspects of the model have been presented in the companion paper [1].
2. A 3-3-1 model with right-handed neutrinos based on the Δ (27) family symmetry
Energy Technology Data Exchange (ETDEWEB)
Hernandez, A.E.C. [Universidad Tecnica Federico Santa Maria and Centro Cienti fico-Tecnologico de Valparaiso, Valparaiso (Chile); Long, H.N. [Vietnam Academy of Science and Technology, Institute of Physics, Hanoi (Viet Nam); Vien, V.V. [Duy Tan University, Institute of Research and Development, Da Nang City (Viet Nam); Tay Nguyen University, Department of Physics, Buon Ma Thuot, DakLak (Viet Nam)
2016-05-15
We present the first multiscalar singlet extension of the original 3-3-1 model with right-handed neutrinos, based on the Δ (27) family symmetry, supplemented by the Z{sub 4} x Z{sub 8} x Z{sub 14} flavor group, consistent with current low energy fermion flavor data. In the model under consideration, the light active neutrino masses are generated from a double seesaw mechanism and the observed pattern of charged fermion masses and quark mixing angles is caused by the breaking of the Δ (27) x Z{sub 4} x Z{sub 8} x Z{sub 14} discrete group at very high energy. Our model has only 14 effective free parameters, which are fitted to reproduce the experimental values of the 18 physical observables in the quark and lepton sectors. The obtained physical observables for the quark sector agree with their experimental values, whereas those for the lepton sector also do, only for the inverted neutrino mass hierarchy. The normal neutrino mass hierarchy scenario of the model is disfavored by the neutrino oscillation experimental data. We find an effective Majorana neutrino mass parameter of neutrinoless double beta decay of m{sub ββ} = 22 meV, a leptonic Dirac CP violating phase of 34 {sup circle}, and a Jarlskog invariant of about 10{sup -2} for the inverted neutrino mass spectrum. (orig.)
3. Gauge theory for baryon and lepton numbers with leptoquarks.
Science.gov (United States)
Duerr, Michael; Fileviez Pérez, Pavel; Wise, Mark B
2013-06-07
Models where the baryon (B) and lepton (L) numbers are local gauge symmetries that are spontaneously broken at a low scale are revisited. We find new extensions of the standard model which predict the existence of fermions that carry both baryon and lepton numbers (i.e., leptoquarks). The local baryonic and leptonic symmetries can be broken at a scale close to the electroweak scale and we do not need to postulate the existence of a large desert to satisfy the experimental constraints on baryon number violating processes like proton decay.
4. Gravitating lepton bag model
International Nuclear Information System (INIS)
Burinskii, A.
2015-01-01
The Kerr–Newman (KN) black hole (BH) solution exhibits the external gravitational and electromagnetic field corresponding to that of the Dirac electron. For the large spin/mass ratio, a ≫ m, the BH loses horizons and acquires a naked singular ring creating two-sheeted topology. This space is regularized by the Higgs mechanism of symmetry breaking, leading to an extended particle that has a regular spinning core compatible with the external KN solution. We show that this core has much in common with the known MIT and SLAC bag models, but has the important advantage of being in accordance with the external gravitational and electromagnetic fields of the KN solution. A peculiar two-sheeted structure of Kerr’s gravity provides a framework for the implementation of the Higgs mechanism of symmetry breaking in configuration space in accordance with the concept of the electroweak sector of the Standard Model. Similar to other bag models, the KN bag is flexible and pliant to deformations. For parameters of a spinning electron, the bag takes the shape of a thin rotating disk of the Compton radius, with a ring–string structure and a quark-like singular pole formed at the sharp edge of this disk, indicating that the considered lepton bag forms a single bag–string–quark system
5. Non-minimal flavored S{sub 3} x Z{sub 2} left-right symmetric model
Energy Technology Data Exchange (ETDEWEB)
Gomez-Izquierdo, Juan Carlos [Tecnologico de Monterrey, Campus Estado de Mexico, Estado de Mexico, Estado de Mexico (Mexico); Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Mexico, D.F. (Mexico); Instituto de Fisica, Universidad Nacional Autonoma de Mexico, Mexico, D.F. (Mexico)
2017-08-15
We propose a non-minimal left-right symmetric model with parity symmetry where the fermion mixings arise as a result of imposing an S{sub 3} x Z{sub 2} flavor symmetry, and an extra Z{sup e}{sub 2} symmetry is considered in the lepton sector. Then the neutrino mass matrix possesses approximately the μ-τ symmetry. The breaking of the μ-τ symmetry induces sizable non-zero θ{sub 13}, and the deviation of θ{sub 23} from 45 {sup circle} is strongly controlled by an ε free parameter and the neutrino masses. So, an analytic study of the CP parities in the neutrino masses is carried out to constrain the ε parameter and the lightest neutrino mass that accommodate the mixing angles. The results are: (a) the normal hierarchy is ruled out for any values of the Majorana phases; (b) for the inverted hierarchy the values of the reactor and atmospheric angles are compatible up to 2, 3 σ C.L.; (c) the degenerate ordering is the most favorable such that the reactor and atmospheric angle are compatible with the experimental data for a large set of values of the free parameters. The model predicts defined regions for the effective neutrino mass, the neutrino mass scale and the sum of the neutrino masses for the favored cases. Therefore, this model may be testable by the future experiments. (orig.)
6. New paradigm for baryon and lepton number violation
International Nuclear Information System (INIS)
Fileviez Pérez, Pavel
2015-01-01
The possible discovery of proton decay, neutron–antineutron oscillation, neutrinoless double beta decay in low energy experiments, and exotic signals related to the violation of the baryon and lepton numbers at collider experiments will change our understanding of the conservation of fundamental symmetries in nature. In this review we discuss the rare processes due to the existence of baryon and lepton number violating interactions. The simplest grand unified theories and the neutrino mass generation mechanisms are discussed. The theories where the baryon and lepton numbers are defined as local gauge symmetries spontaneously broken at the low scale are discussed in detail. The simplest supersymmetric gauge theory which predicts the existence of lepton number violating processes at the low scale is investigated. The main goal of this review is to discuss the main implications of baryon and lepton number violation in physics beyond the Standard Model.
7. Flavor and CP invariant composite Higgs models
International Nuclear Information System (INIS)
Redi, Michele; Weiler, Andreas
2011-09-01
The flavor protection in composite Higgs models with partial compositeness is known to be insufficient. We explore the possibility to alleviate the tension with CP odd observables by assuming that flavor or CP are symmetries of the composite sector, broken by the coupling to Standard Model fields. One realization is that the composite sector has a flavor symmetry SU(3) or SU(3) U x SU(3) D which allows us to realize Minimal Flavor Violation. We show how to avoid the previously problematic tension between a flavor symmetric composite sector and electro-weak precision tests. Some of the light quarks are substantially or even fully composite with striking signals at the LHC. We discuss the constraints from recent dijet mass measurements and give an outlook on the discovery potential. We also present a different protection mechanism where we separate the generation of flavor hierarchies and the origin of CP violation. This can eliminate or safely reduce unwanted CP violating effects, realizing effectively ''Minimal CP Violation'' and is compatible with a dynamical generation of flavor at low scales. (orig.)
8. Flavor and CP invariant composite Higgs models
Energy Technology Data Exchange (ETDEWEB)
Redi, Michele [CERN - European Organization for Nuclear Research, Geneva (Switzerland). Theory Div.; INFN, Firenze (Italy); Weiler, Andreas [CERN - European Organization for Nuclear Research, Geneva (Switzerland). Theory Div.; Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2011-09-15
The flavor protection in composite Higgs models with partial compositeness is known to be insufficient. We explore the possibility to alleviate the tension with CP odd observables by assuming that flavor or CP are symmetries of the composite sector, broken by the coupling to Standard Model fields. One realization is that the composite sector has a flavor symmetry SU(3) or SU(3){sub U} x SU(3){sub D} which allows us to realize Minimal Flavor Violation. We show how to avoid the previously problematic tension between a flavor symmetric composite sector and electro-weak precision tests. Some of the light quarks are substantially or even fully composite with striking signals at the LHC. We discuss the constraints from recent dijet mass measurements and give an outlook on the discovery potential. We also present a different protection mechanism where we separate the generation of flavor hierarchies and the origin of CP violation. This can eliminate or safely reduce unwanted CP violating effects, realizing effectively ''Minimal CP Violation'' and is compatible with a dynamical generation of flavor at low scales. (orig.)
9. Review of Minimal Flavor Constraints for Technicolor
DEFF Research Database (Denmark)
S. Fukano, Hidenori; Sannino, Francesco
2010-01-01
We analyze the constraints on the the vacuum polarization of the standard model gauge bosons from a minimal set of flavor observables valid for a general class of models of dynamical electroweak symmetry breaking. We will show that the constraints have a strong impact on the self-coupling and mas......We analyze the constraints on the the vacuum polarization of the standard model gauge bosons from a minimal set of flavor observables valid for a general class of models of dynamical electroweak symmetry breaking. We will show that the constraints have a strong impact on the self...
10. Long-lived staus and displaced leptons at the LHC
Energy Technology Data Exchange (ETDEWEB)
Evans, Jared A.; Shelton, Jessie [Department of Physics, University of Illinois at Urbana-Champaign,Urbana, IL 61801 (United States)
2016-04-11
As the majority of LHC searches are focused on prompt signatures, specific long-lived particles have the potential to be overlooked by the otherwise systematic new physics programs at ATLAS and CMS. While in many cases long-lived superparticles are now stringently constrained by existing exotic searches, we point out that the highly motivated model of gauge mediation with staus as the next-to-lightest superparticle (NLSP) is relatively far less tested. We recast LHC searches for heavy stable charged particles, disappearing tracks, and opposite-flavor leptons with large impact parameters to assess current constraints on a variety of spectra that contain an NLSP stau, and find that portions of the parameter space motivated by naturalness are still experimentally unexplored. We additionally note a gap in the current experimental search program: same-flavor leptons with large impact parameters evade the suite of existing searches for long-lived objects. This gap is especially noteworthy as vetoes on displaced leptons in prompt new physics searches could be systematically discarding such events. We discuss several motivated models that can exhibit same-flavor displaced leptons: gauge mediation with co-NLSP sleptons, extended gauge mediation, R-parity violation, and lepton-flavored dark matter that freezes in during a matter-dominated era of the early universe. To address this gap, we propose a straightforward extension of the CMS search for leptons with large impact parameters, and project sensitivity to these scenarios at 13 TeV. Throughout this analysis, we highlight several methods whereby LHC searches for exotic long-lived objects could potentially improve their sensitivity to the displaced leptons originating from gauge mediation and beyond.
11. On the Flavor Structure of Natural Composite Higgs Models & Top Flavor Violation
CERN Document Server
Azatov, Aleksandr; Perez, Gilad; Soreq, Yotam
2014-01-01
We explore the up flavor structure of composite pseudo Nambu-Goldstone-boson Higgs models, where we focus on the flavor anarchic minimal $SO(5)$ case. We identify the different sources of flavor violation in this framework and emphasise the differences from the anarchic Randall-Sundrum scenario. In particular, the fact that the flavor symmetry does not commute with the symmetries that stabilize the Higgs potential may constrain the flavor structure of the theory. In addition, we consider the interplay between the fine tuning of the model and flavor violation. We find that generically the tuning of this class of models is worsen in the anarchic case due to the contributions from the additional fermion resonances. We show that, even in the presence of custodial symmetry, large top flavor violating rate are naturally expected. In particular, $t\\to cZ$ branching ratio of order of $10^{-5}$ is generic for this class of models. Thus, this framework can be tested in the next run of the LHC as well as in other future...
12. Strange mass corrections to hyperonic semi-leptonic decays in statistical model
Energy Technology Data Exchange (ETDEWEB)
Upadhyay, A.; Batra, M. [Thapar University, School of Physics and Material Science, Patiala (India)
2013-12-15
We study the spin distribution, weak decay coupling constant ratios for strange baryon octets with SU(3) breaking effects. Baryon is taken as an ensemble of quark-gluon Fock states in the sea with three valence quarks with definite spin, color and flavor quantum numbers. We apply the statistical model to calculate the probabilities of each Fock states, to analyze the impact of SU(3) breaking in the weak decays. The symmetry breaking effects are studied in terms of a parameter ''r '' whose best-fit value is obtained from the experimental data of semi-leptonic weak decay coupling constant ratios. We suggest the dominant contribution from H{sub 1}G{sub 8} (sea with spin one and color octet) where symmetry breaking corrections lead to the deviations in the value of the axial-vector matrix elements ratio F/D from experimental values by 17%. We conclude that symmetry breaking also significantly affects the polarization of quark in strange baryons. (orig.)
13. Discrete symmetries in the MSSM
Energy Technology Data Exchange (ETDEWEB)
Schieren, Roland
2010-12-02
The use of discrete symmetries, especially abelian ones, in physics beyond the standard model of particle physics is discussed. A method is developed how a general, abelian, discrete symmetry can be obtained via spontaneous symmetry breaking. In addition, anomalies are treated in the path integral approach with special attention to anomaly cancellation via the Green-Schwarz mechanism. All this is applied to the minimal supersymmetric standard model. A unique Z{sup R}{sub 4} symmetry is discovered which solves the {mu}-problem as well as problems with proton decay and allows to embed the standard model gauge group into a simple group, i.e. the Z{sup R}{sub 4} is compatible with grand unification. Also the flavor problem in the context of minimal flavor violation is addressed. Finally, a string theory model is presented which exhibits the mentioned Z{sup R}{sub 4} symmetry and other desirable features. (orig.)
14. Discrete symmetries in the MSSM
International Nuclear Information System (INIS)
Schieren, Roland
2010-01-01
The use of discrete symmetries, especially abelian ones, in physics beyond the standard model of particle physics is discussed. A method is developed how a general, abelian, discrete symmetry can be obtained via spontaneous symmetry breaking. In addition, anomalies are treated in the path integral approach with special attention to anomaly cancellation via the Green-Schwarz mechanism. All this is applied to the minimal supersymmetric standard model. A unique Z R 4 symmetry is discovered which solves the μ-problem as well as problems with proton decay and allows to embed the standard model gauge group into a simple group, i.e. the Z R 4 is compatible with grand unification. Also the flavor problem in the context of minimal flavor violation is addressed. Finally, a string theory model is presented which exhibits the mentioned Z R 4 symmetry and other desirable features. (orig.)
15. Open heavy flavor production via semi-leptonic decayed muons in Pb+Pb collisions at √sNN = 2.76 TeV with the ATLAS detector at the LHC
CERN Document Server
Yujiao, C
2012-01-01
Measurements of heavy quark production and suppression in ultra-relativistic nuclear collisions probe the inter- actions of heavy quarks with the hot, dense medium created in the collisions. ATLAS has measured heavy quark production in √sNN = 2.76 TeV Pb+Pb collisions via semi-leptonic decays of open heavy flavour hadrons to muons. Results are presented for the per-event muon yield as a function of muon transverse momentum, pT, over the range of 4 < pT < 14 GeV. Over that momentum range single muon production results largely from heavy quark decays. The centrality dependence of the muon yields is characterized by the “central to peripheral” ratio, RCP. Using this measure, muon production from heavy quark decays is found to be suppressed by a centrality-dependent factor that increases smoothly from peripheral to central collisions. Muon production is suppressed by approximately a factor of two in central collisions relative to peripheral collisions. Within the experimental errors, the observed supp...
16. Charged lepton mixing - an experimental overview
Science.gov (United States)
2015-04-01
Exploring the flavor sector of the Standard Model has always been a powerful probe in particle physics. Searches for charged leptons mixing, in particular muon decays, effectively pioneered this program almost 100 years ago. Still, even what one might consider, naively, simple questions, like why three lepton generations, are left unanswered. We do know now that neutral leptons (neutrinos) mix. We also know that, in all likelihood, the physics behind charged lepton mixing is also somehow responsible for generating neutrino masses. Not surprisingly, a revived interest in this field is currently under way, with experiments either ongoing or at planning stage throughout the world. The advent of powerful high intensity beams opens up new venues for exploration. Coupled with clever experimental ideas, sensitivities that were previously impossible to attain, are now within reach. I will review here the current status of charged lepton mixing experiments, what should we expect from the next generation projects and my view on how the field will progress in the future.
17. Phenomenology of lepton production
International Nuclear Information System (INIS)
Renard, F.M.
1976-06-01
The problem of lepton production in hadronic collisions is reviewed. The following subjects are developed: the Drell-Yan model for continuous l + l - production, vector mesons and clusters, and other sources of direct leptons [fr
18. Two-loop neutrino model with exotic leptons
Science.gov (United States)
2016-01-01
We propose a two-loop induced neutrino mass model, in which we show some bench mark points to satisfy the observed neutrino oscillation, the constraints of lepton flavor violations, and the relic density in the coannihilation system satisfying the current upper bound on the spin independent scattering cross section with nuclei. We also discuss new sources of muon anomalous magnetic moments.
19. The New Flavor of Higgsed Gauge Mediation
OpenAIRE
Craig, Nathaniel; McCullough, Matthew; Thaler, Jesse
2012-01-01
Recent LHC bounds on squark masses combined with naturalness and flavor considerations motivate non-trivial sfermion mass spectra in the supersymmetric Standard Model. These can arise if supersymmetry breaking is communicated to the visible sector via new extended gauge symmetries. Such extended symmetries must be spontaneously broken, or confined, complicating the calculation of soft masses. We develop a new formalism for calculating perturbative gauge-mediated two-loop soft masses for gauge...
20. Are quarks and leptons composite or elementary
International Nuclear Information System (INIS)
Peccei, R.D.
1986-01-01
In these lectures I discuss the issue of the origin of the quark and lepton masses, both in the case in which these objects are elementary and in the case they are composite. Some of the generic predictions and dynamical assumptions of GUTS, family symmetry models and superstrings are detailed. They are contrasted to the dynamics required for composite models of quarks and leptons. In this latter case, the difficulties of protecting dynamically fermion masses and yet still generating intra and interfamily hierarchies is emphasized. (orig.)
1. Heavy flavor production in nuclear collisions
CERN Document Server
Armesto-Pérez, Nestor; Capella, A; Pajares, C; Salgado, C A
2001-01-01
Heavy flavor production off nuclei is studied in the small x/sub F/ region of the produced heavy system. Corrections to the usually employed perturbative QCD factorization formula are considered in the framework of the Glauber-Gribov model. Transition from low to high energies is taken into account by using finite energy cutting rules. The low energy limit of the obtained results coincides with the probabilistic formula usually employed for quarkonium absorption. At finite energies both rescattering of the heavy flavor and corrections to nucleon parton densities inside nuclei appear, the latter also affecting lepton pair production. It turns out that at asymptotic energies both open heavy flavor and quarkonium are equally absorbed. The numerical differences between the results obtained with the probabilistic formula and the exact one are <20% up to LHC energies, and ~1/2% at SPS energies. (18 refs).
2. Searching for heavy leptons
International Nuclear Information System (INIS)
Perl, M.L.
1979-11-01
This frankly speculative paper discusses ways in which leptons heavier than the tau (if they exist) might be found. The status of the tau is briefly reviewed, and methods for searching for sequential charged leptons beyond the tau and other charged leptons at PEP, PETRA, and CESR are sketched. Charged leptons with mass greater than 20 GeV/c 2 might be found at proton accelerators in hadron-hadron, photon-hadron, or ν-hadron collisions. Unstable, neutral heavy leptons might have unique, conserved lepton number or nonunique lepton number. The most difficult leptons to detect are stable neutral heavy leptons; nevertheless, a possible detection method is suggested. The obvious solution to seeking the postulated leptons is an e + e - colliding beam storage ring with c.m. energy = several hundred GeV. Until such a machine is built, one can employ Z 0 → L + + L - ; the use of R/sub Z 0 / and GAMMA/sub Z 0 / in this connection is discussed. If heavier Z 0 's exist, the heavy lepton search can be extended to higher energies. Another solution for producing these leptons involves the use of clashing e + e - linear accelerators. Characteristics of storage rings are compared with those of clashing linacs; a general description is given of the proposed SLAC Linac-Collider, along with the physics that could be done at such a machine. 6 figures
3. Heavy-light flavor correlations and the QCD phase boundary
Energy Technology Data Exchange (ETDEWEB)
Sasaki, Chihiro [Institute of Theoretical Physics, University of Wroclaw, PL-50204 Wroclaw (Poland); Frankfurt Institute for Advanced Studies, D-60438 Frankfurt am Main (Germany); Redlich, Krzysztof [Institute of Theoretical Physics, University of Wroclaw, PL-50204 Wroclaw (Poland)
2016-12-15
We discuss correlations between the light and heavy-light flavored mesons at finite temperature within a chiral effective theory implementing heavy quark symmetry. We show that the thermodynamics of the charmed mesons is strongly dragged by the chiral crossover dominated by the non-strange flavors. Consequently, the fluctuations carried by the states with strangeness can be used to characterize the onset of the chiral symmetry restoration.
4. Multi baryons with flavors in the Skyrme model
Energy Technology Data Exchange (ETDEWEB)
Schat, Carlos L. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil); Scoccola, Norberto N. [Comision Nacional de Energia Atomica, Buenos Aires (Argentina). Dept. of Physics
1999-07-01
We investigate the possible existence of multi baryons with heavy flavor quantum numbers using the bound state approach to the topological soliton model and the recently proposed approximation for multi skyrmion fields based on rational maps. We use an effective interaction Lagrangian which consistently incorporates both chiral symmetry and the heavy quark symmetry including the corrections up to order {omicron}(1/m{sub Q}). The model predicts some narrow heavy flavored multi baryon states with baryon number four and seven. (author)
5. Multi baryons with flavors in the Skyrme model
International Nuclear Information System (INIS)
Schat, Carlos L.; Scoccola, Norberto N.
1999-07-01
We investigate the possible existence of multi baryons with heavy flavor quantum numbers using the bound state approach to the topological soliton model and the recently proposed approximation for multi skyrmion fields based on rational maps. We use an effective interaction Lagrangian which consistently incorporates both chiral symmetry and the heavy quark symmetry including the corrections up to order ο(1/m Q ). The model predicts some narrow heavy flavored multi baryon states with baryon number four and seven. (author)
6. A highly predictive A 4 flavor 3-3-1 model with radiative inverse seesaw mechanism
Science.gov (United States)
Cárcamo Hernández, A. E.; Long, H. N.
2018-04-01
We build a highly predictive 3-3-1 model, where the field content is extended by including several SU(3) L scalar singlets and six right handed Majorana neutrinos. In our model the {SU}{(3)}C× {SU}{(3)}L× U{(1)}X gauge symmetry is supplemented by the {A}4× {Z}4× {Z}6× {Z}16× {Z}16{\\prime } discrete group, which allows to get a very good description of the low energy fermion flavor data. In the model under consideration, the {A}4× {Z}4× {Z}6× {Z}16× {Z}16{\\prime } discrete group is broken at very high energy scale down to the preserved Z 2 discrete symmetry, thus generating the observed pattern of SM fermion masses and mixing angles and allowing the implementation of the loop level inverse seesaw mechanism for the generation of the light active neutrino masses, respectively. The obtained values for the physical observables in the quark sector agree with the experimental data, whereas those ones for the lepton sector also do, only for the case of inverted neutrino mass spectrum. The normal neutrino mass hierarchy scenario of the model is ruled out by the neutrino oscillation experimental data. We find an effective Majorana neutrino mass parameter of neutrinoless double beta decay of m ee = 46.9 meV, a leptonic Dirac CP violating phase of -81.37° and a Jarlskog invariant of about 10-2 for the inverted neutrino mass hierarchy. The preserved Z 2 symmetry allows for a stable scalar dark matter candidate.
7. Gravitino and scalar {tau}-lepton decays in supersymmetric models with broken R-parity
Energy Technology Data Exchange (ETDEWEB)
Hajer, Jan
2010-06-15
Mildly broken R-parity is known to provide a solution to the cosmological gravitino problem in supergravity extensions of the Standard Model. In this work we consider new effects occurring in the R-parity breaking Minimal Supersymmetric Standard Model including right-handed neutrino superfields. We calculate the most general vacuum expectation values of neutral scalar fields including left- and right-handed scalar neutrinos. Additionally, we derive the corresponding mass mixing matrices of the scalar sector. We recalculate the neutrino mass generation mechanisms due to right- handed neutrinos as well as by cause of R-parity breaking. Furthermore, we obtain a, so far, unknown formula for the neutrino masses for the case where both mechanisms are effective. We then constrain the couplings to bilinear R-parity violating couplings in order to accommodate R-parity breaking to experimental results. In order to constrain the family structure with a U(1){sub Q} flavor symmetry we furthermore embed the particle content into an SU(5) Grand Unified Theory. In this model we calculate the signal of decaying gravitino dark matter as well as the dominant decay channel of a likely NLSP, the scalar {tau}-lepton. Comparing the gravitino signal with results of the Fermi Large Area Telescope enables us to find a lower bound on the decay length of scalar {tau}-leptons in collider experiments. (orig.)
8. Neutrino mass matrices with two vanishing cofactors and Fritzsch texture for charged lepton mass matrix
Science.gov (United States)
Wang, Weijian; Guo, Shu-Yuan; Wang, Zhi-Gang
2016-04-01
In this paper, we study the cofactor 2 zero neutrino mass matrices with the Fritzsch-type structure in charged lepton mass matrix (CLMM). In the numerical analysis, we perform a scan over the parameter space of all the 15 possible patterns to get a large sample of viable scattering points. Among the 15 possible patterns, three of them can accommodate the latest lepton mixing and neutrino mass data. We compare the predictions of the allowed patterns with their counterparts with diagonal CLMM. In this case, the severe cosmology bound on the neutrino mass set a strong constraint on the parameter space, rendering two patterns only marginally allowed. The Fritzsch-type CLMM will have impact on the viable parameter space and give rise to different phenomenological predictions. Each allowed pattern predicts the strong correlations between physical variables, which is essential for model selection and can be probed in future experiments. It is found that under the no-diagonal CLMM, the cofactor zeros structure in neutrino mass matrix is unstable as the running of renormalization group (RG) from seesaw scale to the electroweak scale. A way out of the problem is to propose the flavor symmetry under the models with a TeV seesaw scale. The inverse seesaw model and a loop-induced model are given as two examples.
9. Gravitino and scalar τ-lepton decays in supersymmetric models with broken R-parity
International Nuclear Information System (INIS)
Hajer, Jan
2010-01-01
Mildly broken R-parity is known to provide a solution to the cosmological gravitino problem in supergravity extensions of the Standard Model. In this work we consider new effects occurring in the R-parity breaking Minimal Supersymmetric Standard Model including right-handed neutrino superfields. We calculate the most general vacuum expectation values of neutral scalar fields including left- and right-handed scalar neutrinos. Additionally, we derive the corresponding mass mixing matrices of the scalar sector. We recalculate the neutrino mass generation mechanisms due to right- handed neutrinos as well as by cause of R-parity breaking. Furthermore, we obtain a, so far, unknown formula for the neutrino masses for the case where both mechanisms are effective. We then constrain the couplings to bilinear R-parity violating couplings in order to accommodate R-parity breaking to experimental results. In order to constrain the family structure with a U(1) Q flavor symmetry we furthermore embed the particle content into an SU(5) Grand Unified Theory. In this model we calculate the signal of decaying gravitino dark matter as well as the dominant decay channel of a likely NLSP, the scalar τ-lepton. Comparing the gravitino signal with results of the Fermi Large Area Telescope enables us to find a lower bound on the decay length of scalar τ-leptons in collider experiments. (orig.)
10. Flavor at the TeV scale with extra dimensions
International Nuclear Information System (INIS)
Arkani-Hamed, Nima; Hall, Lawrence; Smith, David; Weiner, Neal
2000-01-01
Theories where the standard model fields reside on a 3-brane, with a low fundamental cutoff and extra dimensions, provide alternative solutions to the gauge hierarchy problem. However, generating flavor at the TeV scale while avoiding flavor-changing difficulties appears prohibitively difficult at first sight. We argue to the contrary that this picture allows us to lower flavor physics close to the TeV scale. Small Yukawa couplings are generated by ''shining'' badly broken flavor symmetries from distant branes, and flavor and CP-violating processes are adequately suppressed by these symmetries. We further show how the extra dimensions avoid four dimensional disasters associated with light fields charged under flavor. We construct elegant and realistic theories of flavor based on the maximal U(3) 5 flavor symmetry which naturally generate the simultaneous hierarchy of masses and mixing angles. Finally, we introduce a new framework for predictive theories of flavor, where our 3-brane is embedded within highly symmetrical configurations of higher-dimensional branes. (c) 2000 The American Physical Society
11. Early space symmetry restoration and neutrino experiments
International Nuclear Information System (INIS)
Volkov, G.G.; Liparteliani, A.G.; Monich, V.A.
1986-01-01
The problem of early space symmetry restoration on the left-right symmetry models and the models with the extended (due to mirror quarks and leptons) fermion sector is being discussed. The experiments in which the derivations from the standard model of electroweak interactions should be studied are presented
12. Flavor SU(3) in hadronic B decays
International Nuclear Information System (INIS)
Dighe, A.
1998-11-01
Here we shall outline a few methods that use the flavor SU(3) symmetry in the decays of B mesons to determine the angles of the unitarity triangle and to identify the decay modes which would display a significant CP violation. (author)
13. Randall-Sundrum models vs. supersymmetry. The different flavor signatures
International Nuclear Information System (INIS)
Gori, Stefania
2010-07-01
The Minimal Supersymmetric Standard Model based on flavor symmetries and models with a warped extra dimension as first proposed by Randall and Sundrum represent two of the best founded theories beyond the Standard Model. They provide two appealing solutions both to the gauge hierarchy problem and to the Standard Model flavor hierarchy problems. In this thesis we focus on a particular Randall-Sundrum model based on the custodial symmetry SU(2) L x SU(2) R x P LR in the bulk and on two Supersymmetric flavor models: the one based on a U(1) abelian flavor symmetry, the other on a SU(3) non abelian flavor symmetry. We first analyze and compare the flavor structure of the two frameworks, showing two possible ways to address the New Physics flavor problem: warped geometry and custodial protection vs. flavor symmetry. Subsequently, we study the impact of the new particles (Kaluza-Klein states in the Randall-Sundrum model and superpartners in Supersymmetry) in the K and B meson mixings and rare decays. We perform a global numerical analysis of the new physics effects in the models in question and we show that it is possible to naturally be in agreement with all the available data on ΔF=2 observables, even fixing the energy scale of the models to the TeV range, in order to have new particles in the reach of the LHC. We then study distinctive patterns of flavor violation which can enable future experiments to distinguish the two frameworks. In particular, the specific correlations between the CP violating asymmetry in the B s 0 - anti B s 0 system, the rare decays B s,d →μ + μ - and K→πνanti ν allow in principle for an experimental test of the Randall-Sundrum model and of the two Supersymmetric flavor models and a clear distinction between the two frameworks, once new data will be available. (orig.)
14. Flavor versus mass eigenstates in neutrino asymmetries: implications for cosmology
Energy Technology Data Exchange (ETDEWEB)
Barenboim, Gabriela [Universitat de Valencia-CSIC, Departament de Fisica Teorica y IFIC, Burjassot (Spain); Kinney, William H. [University at Buffalo, Department of Physics, Buffalo, NY (United States); Park, Wan-Il [Universitat de Valencia-CSIC, Departament de Fisica Teorica y IFIC, Burjassot (Spain); Chonbuk National University, Division of Science Education and Institute of Fusion Science, Jeonju (Korea, Republic of)
2017-09-15
We show that, if they exist, lepton number asymmetries (L{sub α}) of neutrino flavors should be distinguished from the ones (L{sub i}) of mass eigenstates, since Big Bang Nucleosynthesis (BBN) bounds on the flavor eigenstates cannot be directly applied to the mass eigenstates. Similarly, Cosmic Microwave Background (CMB) constraints on the mass eigenstates do not directly constrain flavor asymmetries. Due to the difference of mass and flavor eigenstates, the cosmological constraint on the asymmetries of neutrino flavors can be much stronger than the conventional expectation, but they are not uniquely determined unless at least the asymmetry of the heaviest neutrino is well constrained. The cosmological constraint on L{sub i} for a specific case is presented as an illustration. (orig.)
15. Gamma ray constraints on flavor violating asymmetric dark matter
DEFF Research Database (Denmark)
Masina, I.; Panci, P.; Sannino, F.
2012-01-01
We show how cosmic gamma rays can be used to constrain models of asymmetric Dark Matter decaying into lepton pairs by violating flavor. First of all we require the models to explain the anomalies in the charged cosmic rays measured by PAMELA, Fermi and H.E.S.S.; performing combined fits we...... determine the allowed values of the Dark Matter mass and lifetime. For these models, we then determine the constraints coming from the measurement of the isotropic gamma-ray background by Fermi for a complete set of lepton flavor violating primary modes and over a range of DM masses from 100 GeV to 10 Te......V. We find that the Fermi constraints rule out the flavor violating asymmetric Dark Matter interpretation of the charged cosmic ray anomalies....
16. A flavor protection for warped Higgsless models
International Nuclear Information System (INIS)
Csaki, Csaba; Curtin, David
2009-01-01
We examine various possibilities for realistic 5D Higgsless models on a Randall-Sundrum (RS) background, and construct a full quark sector featuring next-to-minimal flavor violation (with an exact bulk SU(2) protecting the first two generations) which satisfies electroweak and flavor constraints. The 'new custodially protected representation' is used for the third generation to protect the light quarks from flavor violations induced due to the heavy top. A combination of flavor symmetries, and an 'RS-GIM' mechanism for the right-handed quarks suppresses flavor-changing neutral currents below experimental bounds, assuming Cabibbo-Kobayashi-Maskawa-type mixing on the UV brane. In addition to the usual Higgsless RS signals, this model predicts an exotic charge-5/3 quark with mass of about 0.5 TeV which should show up at the LHC very quickly, as well as nonzero flavor-changing neutral currents which could be detected in the next generation of flavor experiments. In the course of our analysis, we also find quantitative estimates for the errors of the fermion zero-mode approximation, which are significant for Higgsless-type models.
17. Supersymmetry: Compactification, flavor, and dualities
Science.gov (United States)
Heidenreich, Benjamin Jones
We describe several new research directions in the area of supersymmetry. In the context of low-energy supersymmetry, we show that the assumption of R-parity can be replaced with the minimal flavor violation hypothesis, solving the issue of nucleon decay and the new physics flavor problem in one stroke. The assumption of minimal flavor violation uniquely fixes the form of the baryon number violating vertex, leading to testable predictions. The NLSP is unstable, and decays promptly to jets, evading stringent bounds on vanilla supersymmetry from LHC searches, whereas the gravitino is long-lived, and can be a dark matter component. In the case of a sbottom LSP, neutral mesinos can form and undergo oscillations before decaying, leading to same sign tops, and allowing us to place constraints on the model in this case. We show that this well-motivated phenomenology can be naturally explained by spontaneously breaking a gauged flavor symmetry at a high scale in the presence of additional vector-like quarks, leading to mass mixings which simultaneously generate the flavor structure of the baryon-number violating vertex and the Standard Model Yukawa couplings, explaining their minimal flavor violating structure. We construct a model which is robust against Planck suppressed corrections and which also solves the mu problem. In the context of flux compactifications, we begin a study of the local geometry near a stack of D7 branes supporting a gaugino condensate, an integral component of the KKLT scenario for Kahler moduli stabilization. We obtain an exact solution for the geometry in a certain limit using reasonable assumptions about symmetries, and argue that this solution exhibits BPS domain walls, as expected from field theory arguments. We also begin a larger program of understanding general supersymmetric compactifications of type IIB string theory, reformulating previous results in an SL(2, R ) covariant fashion. Finally, we present extensive evidence for a new class of
18. Leptonic unitary triangles and boomerangs
International Nuclear Information System (INIS)
Dueck, Alexander; Rodejohann, Werner; Petcov, Serguey T.
2010-01-01
We review the idea of leptonic unitary triangles and extend the concept of the recently proposed unitary boomerangs to the lepton sector. Using a convenient parametrization of the lepton mixing, we provide approximate expressions for the side lengths and the angles of the six different triangles and give examples of leptonic unitary boomerangs. Possible applications of the leptonic unitary boomerangs are also briefly discussed.
19. Hadron induced leptons and photons
International Nuclear Information System (INIS)
Cronin, J.W.
1977-01-01
A review of direct production of leptons and photons in hadron-hadron collisions is presented. Production of lepton pairs with large mass is well accounted for by the Drell-Yan process. The origin of direct single leptons is principally due to the production of lepton pairs. A dominant source of lepton pairs is at low effective mass, m [de
20. Physics possibilities of lepton and hadron colliders
International Nuclear Information System (INIS)
Peccei, R.D.
1985-05-01
After a brief introduction to lepton and hadron colliders presently being planned, I give some examples of the nice standard physics which is expected to be seen in them. The bulk of the discussion, however, is centered on signals for new physics. Higgs searches at the new colliders are discussed, as well as signatures and prospects for detecting effects of supersymmetry, compositeness and dynamical symmetry breakdown. (orig.)
1. Measuring the cosmological lepton asymmetry through the CMB anisotropy
CERN Document Server
Kinney, W H; Kinney, William H.; Riotto, Antonio
1999-01-01
A large lepton asymmetry in the Universe is still a viable possibility and leads to many interesting phenomena such as gauge symmetry nonrestoration at high temperature. We show that a large lepton asymmetry changes the predicted cosmic microwave background (CMB) anisotropy and that any degeneracy in the relic neutrino sea will be measured to a precision of 1% or better when the CMB anisotropy is measured at the accuracy expected to result from the planned satellite missions MAP and Planck. In fact, the current measurements already put an upper limit on the lepton asymmetry of the Universe which is stronger than the one coming from considerations of primordial nucleosynthesis and structure formation.
2. Flavor dynamics
International Nuclear Information System (INIS)
Kobayashi, Makoto
1980-01-01
The problems on quark mass and mixing angle are discussed. All discussions are made in the tree approximation, and no renormalization problem is considered. The basic quantities of the Weinberg-Salam (W-S) model are introduced. The model is SU(2) x U(1) gauge theory. The way of constructing the quark parts of Lagrangian, especially their mass terms, is described. The four-quark case is illustrated at first, then the discussion is extended to the general case. As a result, the number of parameters of the Cabibbolike mixing is obtained. The possibility of CP-violation is discussed. Phenomenological analyses are made to determine the parameters. The nuclear beta-decay and semileptonic decay, neutral K-meson system, and the possibility of CP-violation in the six-quark scheme are investigated. In order to understand the close relation between quark mass and mixing angle, the origin of quark mass term is investigated. The original Higgs coupling determines both the quark mass and mixing angle. A remarkable relation is deduced by assuming discrete symmetry. (Kato, T.)
3. Gauge theories of weak interactions with left-right symmetry and the structure of neutral currents
International Nuclear Information System (INIS)
Mohapatra, R.N.; Sidhu, D.P.
1977-01-01
Failure to detect parity-violating effects in atomic transitions by Oxford and Washington groups would appear to rule out the Weinberg-Salam SU(2) x U(1) model as well as any variation of it that respects natural conservation laws for charm and strangeness to order a G/sub F/ (called ''natural'') and obeys quark-lepton symmetry. In this paper, a simple left-right--symmetric model based on the SU(2)/sub L/ x SU(2)/sub R/ x U(1) group with four and six quark flavors is analyzed and found to accomodate the results of the atomic experiments as well as the other features of neutral-current phenomena
4. Phenomenology of the gauge symmetry for right-handed fermions
Energy Technology Data Exchange (ETDEWEB)
Chao, Wei [Beijing Normal University, Center for Advanced Quantum Studies, Department of Physics, Beijing (China)
2018-02-15
In this paper we investigate the phenomenology of the U(1) gauge symmetry for right-handed fermions, where three right-handed neutrinos are introduced for anomalies cancellations. Constraints on the new gauge boson Z{sub R} from Z-Z{sup '} mixing as well as the upper bound of Z{sup '} production cross section in di-lepton channel at the LHC are presented. We further study the neutrino mass and the phenomenology of Z{sub R}-portal dark matter in this model. The lightest right-handed neutrino can be the cold dark matter candidate stabilized by a Z{sub 2} flavor symmetry. Our study shows that active neutrino masses can be generated via the modified type-II seesaw mechanism; right-handed neutrino is available dark matter candidate for its mass being very heavy, or for its mass at near the resonant regime of the SM Higgs and(or) the new bosons; constraint from the dilepton search at the LHC is stronger than that from the Z-Z{sup '} mixing only for g{sub R} < 0.121, where g{sub R} is the new gauge coupling. (orig.)
5. Phenomenology of the gauge symmetry for right-handed fermions
Science.gov (United States)
Chao, Wei
2018-02-01
In this paper we investigate the phenomenology of the U(1) gauge symmetry for right-handed fermions, where three right-handed neutrinos are introduced for anomalies cancellations. Constraints on the new gauge boson Z_{R} from Z-Z^' mixing as well as the upper bound of Z^' production cross section in di-lepton channel at the LHC are presented. We further study the neutrino mass and the phenomenology of Z_{R}-portal dark matter in this model. The lightest right-handed neutrino can be the cold dark matter candidate stabilized by a Z_2 flavor symmetry. Our study shows that active neutrino masses can be generated via the modified type-II seesaw mechanism; right-handed neutrino is available dark matter candidate for its mass being very heavy, or for its mass at near the resonant regime of the SM Higgs and(or) the new bosons; constraint from the dilepton search at the LHC is stronger than that from the Z-Z^' mixing only for g_{R}<0.121, where g_{R} is the new gauge coupling.
6. Search for neutral leptons
International Nuclear Information System (INIS)
Perl, M.L.
1984-12-01
At present we know of three kinds of neutral leptons: the electron neutrino, the muon neutrino, and the tau neutrino. This paper reviews the search for additional neutral leptons. The method and significance of a search depends upon the model used for the neutral lepton being sought. Some models for the properties and decay modes of proposed neutral leptons are described. Past and present searches are reviewed. The limits obtained by some completed searches are given, and the methods of searches in progress are described. Future searches are discussed. 41 references
7. Symmetry and symmetry breaking
International Nuclear Information System (INIS)
Balian, R.; Lambert, D.; Brack, A.; Lachieze-Rey, M.; Emery, E.; Cohen-Tannoudji, G.; Sacquin, Y.
1999-01-01
The symmetry concept is a powerful tool for our understanding of the world. It allows a reduction of the volume of information needed to apprehend a subject thoroughly. Moreover this concept does not belong to a particular field, it is involved in the exact sciences but also in artistic matters. Living beings are characterized by a particular asymmetry: the chiral asymmetry. Although this asymmetry is visible in whole organisms, it seems it comes from some molecules that life always produce in one chirality. The weak interaction presents also the chiral asymmetry. The mass of particles comes from the breaking of a fundamental symmetry and the void could be defined as the medium showing as many symmetries as possible. The texts put together in this book show to a great extent how symmetry goes far beyond purely geometrical considerations. Different aspects of symmetry ideas are considered in the following fields: the states of matter, mathematics, biology, the laws of Nature, quantum physics, the universe, and the art of music. (A.C.)
8. Consequences of an Abelian family symmetry
International Nuclear Information System (INIS)
Ramond, P.
1996-01-01
The addition of an Abelian family symmetry to the Minimal Super-symmetric Standard Model reproduces the observed hierarchies of quark and lepton masses and quark mixing angles, only if it is anomalous. Green-Schwarz compensation of its anomalies requires the electroweak mixing angle to be sin 2 θ ω = 3/8 at the string scale, without any assumed GUT structure, suggesting a superstring origin for the standard model. The analysis is extended to neutrino masses and the lepton mixing matrix
9. Local discrete symmetries from superstring derived models
International Nuclear Information System (INIS)
Faraggi, A.E.
1996-10-01
Discrete and global symmetries play an essential role in many extensions of the Standard Model, for example, to preserve the proton lifetime, to prevent flavor changing neutral currents, etc. An important question is how can such symmetries survive in a theory of quantum gravity, like superstring theory. In a specific string model the author illustrates how local discrete symmetries may arise in string models and play an important role in preventing fast proton decay and flavor changing neutral currents. The local discrete symmetry arises due to the breaking of the non-Abelian gauge symmetries by Wilson lines in the superstring models and forbids, for example dimension five operators which mediate rapid proton decay, to all orders of nonrenormalizable terms. In the context of models of unification of the gauge and gravitational interactions, it is precisely this type of local discrete symmetries that must be found in order to insure that a given model is not in conflict with experimental observations
10. Anomalous Abelian symmetry in the standard model
International Nuclear Information System (INIS)
Ramond, P.
1995-01-01
The observed hierarchy of quark and lepton masses can be parametrized by nonrenormalizable operators with dimensions determined by an anomalous Abelian family symmetry, a gauge extension to the minimal supersymmetric standard model. Such an Abelian symmetry is generic to compactified superstring theories, with its anomalies compensated by the Green-Schwarz mechanism. If we assume these two symmetries to be the same, we find the electroweak mixing angle to be sin 2 θ ω = 3/8 at the string scale, just by setting the ratio of the product of down quark to charged lepton masses equal to one at the string scale. This assumes no GUT structure. The generality of the result suggests a superstring origin for the standard model. We generalize our analysis to massive neutrinos, and mixings in the lepton sector
11. Search for SUperSYmmetry (SUSY) in Opposite Sign (OS) di-lepton final states with Parked Data collected at $\\sqrt{s}$ = 8 TeV using the CMS detector
CERN Document Server
Bhattacharya, Saptaparna
2015-01-01
The Large Hadron Collider (LHC) has had a very successful data-taking phase with Run 1. After the discovery of the Higgs, confirming the predictions of the Standard Model (SM), the focus is on finding new physics, especially in the context of supersymmetry (SUSY). One of the potential hiding places of natural SUSY is in models with compressed spectra, that is, models where the mass difference between the parent SUSY particle and the Lightest Supersymmetric Particle (LSP) is small. Such signals are characterized by low transverse momentum (p${_T}$) objects, low hadronic activity and missing transverse energy (MET). In this analysis, we focus on di-lepton final states, specifically in the low p${_T}$ regime. We use 7.4 fb$^{-1}$ of parked data collected at $\\sqrt{s}$ = 8 TeV. The analysis is enabled by the use of triggers that place no restrictions on the di-lepton p${_T}$, instead relying on methods like Initial State Radiation (ISR) tagging by triggering on a high p${_T}$ photon, to reduce the trigger rate....
12. At the origins of mass: elementary particles and fundamental symmetries
International Nuclear Information System (INIS)
Iliopoulos, Jean; Englert, Francois
2015-01-01
After a brief recall of the history of cosmology, the author proposes an overview of the different symmetries (symmetries in space and in time, internal symmetries, local or gauge symmetries), describes the mass issue (gauge interactions, quarks and leptons as matter mass constituents, chirality), addresses the spontaneous symmetry breaking (the Curie theorem, spontaneous symmetry breaking in classical physics and in quantum physics, the Goldstone theorem, spontaneous symmetry breaking in presence of gauge interactions), presents the standard theory (electromagnetic and weak interactions, strong interactions, relationship with experiment). An appendix presents elementary particles, and notably reports the story of the neutrino
13. New signatures of flavor violating Higgs couplings
Energy Technology Data Exchange (ETDEWEB)
Buschmann, Malte; Kopp, Joachim; Liu, Jia; Wang, Xiao-Ping [PRISMA Cluster of Excellence and Mainz Institute for Theoretical Physics,Johannes Gutenberg University, 55099 Mainz (Germany)
2016-06-24
We explore several novel LHC signatures arising from quark or lepton flavor violating couplings in the Higgs sector, and we constrain such couplings using LHC data. Since the largest signals are possible in channels involving top quarks or tau leptons, we consider in particular the following flavor violating processes: (1) pp→thh (top plus di-Higgs final state) arising from a dimension six coupling of up-type quarks to three insertions of the Higgs field. We develop a search strategy for this final state and demonstrate that detection is possible at the high luminosity LHC if flavor violating top-up-Higgs couplings are not too far below the current limit. (2) pp→tH{sup 0}, where H{sup 0} is the heavy neutral CP-even Higgs boson in a two Higgs doublet model (2HDM). We consider the decay channels H{sup 0}→tu,WW,ZZ,hh and use existing LHC data to constrain the first three of them. For the fourth, we adapt our search for the thh final state, and we demonstrate that in large regions of the parameter space, it is superior to other searches, including searches for flavor violating top quark decays (t→hq). (3) H{sup 0}→τμ, again in the context of a 2HDM. This channel is particularly well motivated by the recent CMS excess in h→τμ, and we use the data from this search to constrain the properties of H{sup 0}.
14. A novel washout effect in the flavored leptogenesis
International Nuclear Information System (INIS)
Shindou, Tetsuo; Yamashita, Toshifumi
2007-01-01
We investigate a flavored washout effect due to the decay of the lightest right-handed neutrino, assuming that there is non-vanishing initial lepton asymmetry and the decay of the lightest right-handed neutrinos gives negligible contribution to the asymmetry. We figure out general features of the washout effect. It is shown that there is a novel parameter region where an effect that is negligible in most cases plays a critical role and a sizable lepton asymmetry can survive against the washout process even in a strong washout region
15. Leptogenesis and residual CP symmetry
International Nuclear Information System (INIS)
Chen, Peng; Ding, Gui-Jun; King, Stephen F.
2016-01-01
We discuss flavour dependent leptogenesis in the framework of lepton flavour models based on discrete flavour and CP symmetries applied to the type-I seesaw model. Working in the flavour basis, we analyse the case of two general residual CP symmetries in the neutrino sector, which corresponds to all possible semi-direct models based on a preserved Z 2 in the neutrino sector, together with a CP symmetry, which constrains the PMNS matrix up to a single free parameter which may be fixed by the reactor angle. We systematically study and classify this case for all possible residual CP symmetries, and show that the R-matrix is tightly constrained up to a single free parameter, with only certain forms being consistent with successful leptogenesis, leading to possible connections between leptogenesis and PMNS parameters. The formalism is completely general in the sense that the two residual CP symmetries could result from any high energy discrete flavour theory which respects any CP symmetry. As a simple example, we apply the formalism to a high energy S 4 flavour symmetry with a generalized CP symmetry, broken to two residual CP symmetries in the neutrino sector, recovering familiar results for PMNS predictions, together with new results for flavour dependent leptogenesis.
16. Softly Broken Lepton Numbers: an Approach to Maximal Neutrino Mixing
International Nuclear Information System (INIS)
Grimus, W.; Lavoura, L.
2001-01-01
We discuss models where the U(1) symmetries of lepton numbers are responsible for maximal neutrino mixing. We pay particular attention to an extension of the Standard Model (SM) with three right-handed neutrino singlets in which we require that the three lepton numbers L e , L μ , and L τ be separately conserved in the Yukawa couplings, but assume that they are softly broken by the Majorana mass matrix M R of the neutrino singlets. In this framework, where lepton-number breaking occurs at a scale much higher than the electroweak scale, deviations from family lepton number conservation are calculable, i.e., finite, and lepton mixing stems exclusively from M R . We show that in this framework either maximal atmospheric neutrino mixing or maximal solar neutrino mixing or both can be imposed by invoking symmetries. In this way those maximal mixings are stable against radiative corrections. The model which achieves maximal (or nearly maximal) solar neutrino mixing assumes that there are two different scales in M R and that the lepton number (dash)L=L e -L μ -L τ 1 is conserved in between them. We work out the difference between this model and the conventional scenario where (approximate) (dash)L invariance is imposed directly on the mass matrix of the light neutrinos. (author)
17. Cosmoparticle physics of family symmetry breaking
International Nuclear Information System (INIS)
Khlopov, M.Yu.
1993-07-01
The foundations of both particle theory and cosmology are hidden at super energy scale and can not be tested by direct laboratory means. Cosmoparticle physics is developed to probe these foundations by the proper combination of their indirect effects, thus providing definite conclusions on their reliability. Cosmological and astrophysical tests turn to be complementary to laboratory searches of rare processes, induced by new physics, as it can be seen in the case of gauge theory of broken symmetry of quark and lepton families, ascribing to the hierarchy of the horizontal symmetry breaking the observed hierarchy of masses and the mixing between quark and lepton families. 36 refs
18. Search for Charged Lepton Violation in Narrow Upsilon Decays
International Nuclear Information System (INIS)
Lees, J.P.
2011-01-01
Charged lepton flavor violating processes are unobservable in the standard model, but they are predicted to be enhanced in several extensions to the standard model, including supersymmetry and models with leptoquarks or compositeness. We present a search for such processes in a sample of 99 x 10 6 Υ(2S) decays and 117 x 10 6 Υ(3S) decays collected with the BABAR detector. We place upper limits on the branching fractions Β(Υ(nS) → e ± τ ± ) and Β(Υ(nS) → μ ± τ ± ) (n = 2, 3) at the 10 -6 level and use these results to place lower limits of order 1 TeV on the mass scale of charged lepton flavor violating effective operators.
19. Strong Electroweak Symmetry Breaking
CERN Document Server
Grinstein, Benjamin
2011-01-01
Models of spontaneous breaking of electroweak symmetry by a strong interaction do not have fine tuning/hierarchy problem. They are conceptually elegant and use the only mechanism of spontaneous breaking of a gauge symmetry that is known to occur in nature. The simplest model, minimal technicolor with extended technicolor interactions, is appealing because one can calculate by scaling up from QCD. But it is ruled out on many counts: inappropriately low quark and lepton masses (or excessive FCNC), bad electroweak data fits, light scalar and vector states, etc. However, nature may not choose the minimal model and then we are stuck: except possibly through lattice simulations, we are unable to compute and test the models. In the LHC era it therefore makes sense to abandon specific models (of strong EW breaking) and concentrate on generic features that may indicate discovery. The Technicolor Straw Man is not a model but a parametrized search strategy inspired by a remarkable generic feature of walking technicolor,...
20. A large Muon Electric Dipole Moment from Flavor?
CERN Document Server
Hiller, Gudrun; Laamanen, Jari; Rüppell, Timo
2010-01-01
We study the prospects and opportunities of a large muon electric dipole moment (EDM) of the order (10^{-24} - 10^{-22}) ecm. We investigate how natural such a value is within the general minimal supersymmetric extension of the Standard Model with CP violation from lepton flavor violation in view of the experimental constraints. In models with hybrid gauge-gravity mediated supersymmetry breaking a large muon EDM is indicative for the structure of flavor breaking at the Planck scale, and points towards a high messenger scale.
1. Model-Independent Analysis of Tri-bimaximal Mixing: A Softly-Broken Hidden or an Accidental Symmetry?
Energy Technology Data Exchange (ETDEWEB)
Albright, Carl H.; /Northern Illinois U. /Fermilab; Rodejohann, Werner; /Heidelberg, Max Planck Inst.
2008-04-01
To address the issue of whether tri-bimaximal mixing (TBM) is a softly-broken hidden or an accidental symmetry, we adopt a model-independent analysis in which we perturb a neutrino mass matrix leading to TBM in the most general way but leave the three texture zeros of the diagonal charged lepton mass matrix unperturbed. We compare predictions for the perturbed neutrino TBM parameters with those obtained from typical SO(10) grand unified theories with a variety of flavor symmetries. Whereas SO(10) GUTs almost always predict a normal mass hierarchy for the light neutrinos, TBM has a priori no preference for neutrino masses. We find, in particular for the latter, that the value of |U{sub e3}| is very sensitive to the neutrino mass scale and ordering. Observation of |U{sub e3}|{sup 2} > 0.001 to 0.01 within the next few years would be incompatible with softly-broken TBM and a normal mass hierarchy and would suggest that the apparent TBM symmetry is an accidental symmetry instead. No such conclusions can be drawn for the inverted and quasi-degenerate hierarchy spectra.
2. Semi-leptonic interactions
International Nuclear Information System (INIS)
Gaillard, J.M.
In spite of the presence of poorly understood strong interaction effects, the theory of hadronic currents leads to a considerable predictive power. This is shown in the discussion of the semi-leptonic decays
3. Multisensory Flavor Priming
DEFF Research Database (Denmark)
Dijksterhuis, Garmt Bernard
2016-01-01
with a taxonomy of different priming situations. In food-related applications of flavor, both bottom-up (sensory) as well as top-down (expectations) processes are at play. Most of the complex interactions that this leads to take place outside the awareness of the perceiving subject. A model is presented where...... many, past and current, aspects (sensory, surroundings, social, somatic, sentimental) of a (flavor) perception, together result in the perception of a flavor, its liking. or its choice. This model borrows on ideas from priming, situated/embodied cognition, and (food-related) perception.......Flavor is multisensory; several interacting sensory systems-taste, smell, and mouthfeel-together comprise "flavor," making it a cognitively constructed percept rather than a bottom-up sensory one. In this chapter, some of the complications this entails for flavor priming are introduced, along...
4. Lepton asymmetry, neutrino spectral distortions, and big bang nucleosynthesis
Science.gov (United States)
Grohs, E.; Fuller, George M.; Kishimoto, C. T.; Paris, Mark W.
2017-03-01
We calculate Boltzmann neutrino energy transport with self-consistently coupled nuclear reactions through the weak-decoupling-nucleosynthesis epoch in an early universe with significant lepton numbers. We find that the presence of lepton asymmetry enhances processes which give rise to nonthermal neutrino spectral distortions. Our results reveal how asymmetries in energy and entropy density uniquely evolve for different transport processes and neutrino flavors. The enhanced distortions in the neutrino spectra alter the expected big bang nucleosynthesis light element abundance yields relative to those in the standard Fermi-Dirac neutrino distribution cases. These yields, sensitive to the shapes of the neutrino energy spectra, are also sensitive to the phasing of the growth of distortions and entropy flow with time/scale factor. We analyze these issues and speculate on new sensitivity limits of deuterium and helium to lepton number.
5. Flavor alignment via shining in Randall-Sundrum models
International Nuclear Information System (INIS)
Csaki, Csaba; Perez, Gilad; Surujon, Ze'ev; Weiler, Andreas
2010-01-01
We present a class of warped extra dimensional models whose flavor violating interactions are much suppressed compared to the usual anarchic case due to flavor alignment. Such suppression can be achieved in models where part of the global flavor symmetry is gauged in the bulk and broken in a controlled manner. We show that the bulk masses can be aligned with the down-type Yukawa couplings by an appropriate choice of bulk flavon field representations and TeV brane dynamics. This alignment could reduce the flavor violating effects to levels that allow for a Kaluza-Klein scale as low as 2-3 TeV, making the model observable at the LHC. However, the up-type Yukawa couplings on the IR brane, which are bounded from below by recent bounds on CP violation in the D system, induce flavor misalignment radiatively. Off-diagonal down-type Yukawa couplings and kinetic mixings for the down quarks are both consequences of this effect. These radiative Yukawa corrections can be reduced by raising the flavon vacuum expectation value on the IR brane (at the price of some moderate tuning), or by extending the Higgs sector. The flavor changing effects from the radiatively induced Yukawa mixing terms are at around the current upper experimental bounds. We also show the generic bounds on UV-brane induced flavor violating effects, and comment on possible additional flavor violations from bulk flavor gauge bosons and the bulk Yukawa scalars.
International Nuclear Information System (INIS)
Reineccius, G.A.
1992-01-01
Flavor will not be a significant factor in determining the success of irradiated foods entering the U.S. market. The initial applications will use low levels of irradiation that may well result in products with flavor superior to that of products from alternative processing techniques (thermal treatment or chemical fumigation). The success of shelf-stable foods produced via irradiation may be much more dependent upon our ability to deal with the flavor aspects of high levels of irradiation
7. Equilibrium flavor dynamics during the cosmic confinement transition
International Nuclear Information System (INIS)
Kaempfer, B.
1988-10-01
The dynamics of the flavor composition of strongly interacting matter during the cosmic confinement transition is followed up in a simplified thermodynamical model. Relying on thermal, mechanical and chemical equilibrium the strangeness fraction of strongly interacting matter is analyzed. Due to equilibrium with respect to ΔS=0 and ΔS=1 weak interactions the relations between different flavors depend strongly on the poorly known lepton excess. In a universe where the lepton (antilepton) excess is in the same order of magnitude as the baryon excess, the strange quark abundancies are suppressed (enhanced). In the hadron phase the strange baryons carry up to a half of the baryon excess. (author) 22 refs.; 9 figs
8. Theory prospective on leptonic CP violation
International Nuclear Information System (INIS)
Petcov, S.T.
2016-01-01
The phenomenology of 3-neutrino mixing, the current status of our knowledge about the 3-neutrino mixing parameters, including the absolute neutrino mass scale, and of the Dirac and Majorana CP violation in the lepton sector are reviewed. The problems of CP violation in neutrino oscillations and of determining the nature – Dirac or Majorana – of massive neutrinos are discussed. The seesaw mechanism of neutrino mass generation and the related leptogenesis scenario of generation of the baryon asymmetry of the Universe are considered. The results showing that the CP violation necessary for the generation of the baryon asymmetry of the Universe in leptogenesis can be due exclusively to the Dirac and/or Majorana CP-violating phase(s) in the neutrino mixing matrix U are briefly reviewed. The discrete symmetry approach to understanding the observed pattern of neutrino mixing and the related predictions for the leptonic Dirac CP violation are also reviewed.
9. Search for the associated production of a Higgs boson and a top quark pair in multilepton (2 leptons, no hadronically-decaying $\\tau$ lepton candidates and 4 leptons) final states with the ATLAS detector.
CERN Document Server
Dumitriu, Ana Elena; The ATLAS collaboration
2017-01-01
The Yukawa coupling of the Higgs boson to the top quark is a key parameter of the Standard Model. It can be constrained using the associated production process $pp\\rightarrow t\\bar{t}H+X$. \\\\ A search for this process using final states with multiple leptons, primarily targeting the decays $H\\rightarrow WW^*$ and $H\\rightarrow \\tau \\tau$, has been performed using the data set recorded by the ATLAS detector in 2015 and 2016 at a center of mass energy $\\sqrt{s}$= 13 TeV. The analysis presented here includes two of the four final states distinguished by the number and flavor of leptons: two same-charge light leptons ( e or $\\mu$ ) and no hadronically-decaying $\\tau$ lepton candidates ($2l0\\tau_{had}$) and four light leptons ($4l$), the remaining channels not covered being two same-charge light leptons and one hadronically-decaying $\\tau$ lepton candidate ($2l1\\tau_{had}$) and three light leptons ($3l$). The different background sources are also presented for each channel considered. The latest best-fit value for...
10. The breaking of flavor democracy in the quark sector
Science.gov (United States)
Fritzsch, Harald; Xing, Zhi-Zhong; Zhang, Di
2017-09-01
The democracy of quark flavors is a well-motivated flavor symmetry, but it must be properly broken in order to explain the observed quark mass spectrum and flavor mixing pattern. We reconstruct the texture of flavor democracy breaking and evaluate its strength in a novel way, by assuming a parallelism between the Q=+2/3 and Q=-1/3 quark sectors and using a nontrivial parametrization of the flavor mixing matrix. Some phenomenological implications of such democratic quark mass matrices, including their variations in the hierarchy basis and their evolution from the electroweak scale to a super-high energy scale, are also discussed. Supported by National Natural Science Foundation of China (11375207) and National Basic Research Program of China (2013CB834300)
11. Flavor Alignment via Shining in RS
CERN Document Server
Csáki, Csaba; Surujon, Ze'ev; Weiler, Andreas
2010-01-01
We present a class of warped extra dimensional models whose flavor violating interactions are much suppressed compared to the usual anarchic case due to flavor alignment. Such suppression can be achieved in models where part of the global flavor symmetry is gauged in the bulk and broken in a controlled manner. We show that the bulk masses can be aligned with the down type Yukawa couplings by an appropriate choice of bulk flavon field representations and TeV brane dynamics. This alignment could reduce the flavor violating effects to levels which allow for a Kaluza-Klein scale as low as 2-3 TeV, making the model observable at the LHC. However, the up-type Yukawa couplings on the IR brane, which are bounded from below by recent bounds on CP violation in the D system, induce flavor misalignment radiatively. Off-diagonal down-type Yukawa couplings and kinetic mixings for the down quarks are both consequences of this effect. These radiative Yukawa corrections can be reduced by raising the flavon VEV on the IR brane...
12. Neutrino physics and the flavor problem
International Nuclear Information System (INIS)
King, S. F.; Peddie, I. N. R.
2004-01-01
We consider the problem of trying to understand the recently measured neutrino data simultaneously with understanding the hierarchical form of quark and charged-lepton Yukawa matrices. We summarize the data that a successful model of neutrino mass must predict, and then move on to attempting to do so in the context of spontaneously broken 'family' symmetries. We consider first an abelian U(1) family symmetry, which appears in the context of a type-I string model. Then we consider a model based on a non-abelian SU(3) F , which is the maximal family group consistent with an SO(10) GUT. In this case, the symmetry is more constraining, and is examined in the context of SUSY field theory.
13. Possible origin of a natural conservation of flavor in an interaction with neutral Higgs bosons
International Nuclear Information System (INIS)
Ural'tsev, N.G.
1983-01-01
In technicolor models the masses of the neutral pseudo-Goldstone bosons which interact with quarks and leptons without flavor conservation automatically acquire an order of magnitude Mapprox.(m/sub q/#betta#/sub TC/)/sup 1/2/approx.0.2--1 TeV through the Yukawa interaction. As a result, an effective Lagrangian which conserves only light Higgs bosons satisfies the condition of natural flavor conservation
14. Cosmological lepton asymmetry, primordial nucleosynthesis and sterile neutrinos
Science.gov (United States)
Abazajian, Kevork; Bell, Nicole F.; Fuller, George M.; Wong, Yvonne Y. Y.
2005-09-01
We study post weak decoupling coherent active-sterile and active-active matter-enhanced neutrino flavor transformation in the early Universe. We show that flavor conversion efficiency at Mikheyev-Smirnov-Wolfenstein resonances is likely to be high (adiabatic evolution) for relevant neutrino parameters and energies. However, we point out that these resonances cannot sweep smoothly and continuously with the expansion of the Universe. We show how neutrino flavor conversion in this way can leave both the active and sterile neutrinos with nonthermal energy spectra, and how, in turn, these distorted energy spectra can affect the neutron-to-proton ratio, primordial nucleosynthesis, and cosmological mass/closure constraints on sterile neutrinos. We demonstrate that the existence of a light sterile neutrino which mixes with active neutrinos can change fundamentally the relationship between the cosmological lepton numbers and the primordial nucleosynthesis He4 yield.
15. Flavor physics and CP violation
International Nuclear Information System (INIS)
Isidori, Gino
2014-01-01
Lectures on flavor physics presented at the 2012 CERN HEP Summer School. Content: 1) flavor physics within the Standard Model, 2) phenomenology of B and D decays, 3) flavor physics beyond the Standard Model
16. Phenomenological aspects of theories for baryon and lepton number violation
International Nuclear Information System (INIS)
Duerr, Michael
2013-01-01
The renormalizable couplings of the Standard Model are invariant under two accidental global symmetries, which correspond to conserved baryon and lepton numbers. In this thesis, we discuss possible roles of these symmetries in extension of the Standard Model. Two approaches are considered: explicit violation of lepton number by two units in the renormalizable couplings of the Lagrangian, and promotion of the global symmetries to local gauge symmetries that are spontaneously broken. The former approach directly leads to Majorana neutrino masses and neutrinoless double beta decay. We discuss the interplay of the contributions to this decay in a one-loop neutrino mass model, the colored seesaw mechanism. We find that, depending on the parameters of the model, both the light Majorana neutrino exchange and the contribution of the new colored particles may be dominant. Additionally, an experimental test is presented, which allows for a discrimination of neutrinoless double beta decay from unknown nuclear background using only one isotope. In the latter approach, fascinating implications originate from the attempt to write down an anomaly-free and spontaneously broken gauge theory for baryon and lepton numbers, such as an automatically stable dark matter candidate. When gauging the symmetries in a left-right symmetric setup, the same fields that allow for an anomaly-free theory generate neutrino masses via the type III seesaw mechanism.
17. Coherent Active-Sterile Neutrino Flavor Transformation in the Early Universe
Science.gov (United States)
Kishimoto, Chad T.; Fuller, George M.; Smith, Christel J.
2006-10-01
We solve the problem of coherent Mikheyev-Smirnov-Wolfenstein resonant active-to-sterile neutrino flavor conversion driven by an initial lepton number in the early Universe. We find incomplete destruction of the lepton number in this process and a sterile neutrino energy distribution with a distinctive cusp and high energy tail. These features imply alteration of the nonzero lepton number primordial nucleosynthesis paradigm when there exist sterile neutrinos with rest masses ms˜1eV. This could result in better light element probes of (constraints on) these particles.
18. Coherent Active-Sterile Neutrino Flavor Transformation in the Early Universe
International Nuclear Information System (INIS)
Kishimoto, Chad T.; Fuller, George M.; Smith, Christel J.
2006-01-01
We solve the problem of coherent Mikheyev-Smirnov-Wolfenstein resonant active-to-sterile neutrino flavor conversion driven by an initial lepton number in the early Universe. We find incomplete destruction of the lepton number in this process and a sterile neutrino energy distribution with a distinctive cusp and high energy tail. These features imply alteration of the nonzero lepton number primordial nucleosynthesis paradigm when there exist sterile neutrinos with rest masses m s ∼1 eV. This could result in better light element probes of (constraints on) these particles
19. Muon g - 2 through a flavor structure on soft SUSY terms
International Nuclear Information System (INIS)
Flores-Baez, F.V.; Gomez Bock, M.; Mondragon, M.
2016-01-01
In this work we analyze the possibility to explain the muon anomalous magnetic moment discrepancy within theory and experiment through lepton-flavor violation processes. We propose a flavor extended MSSM by considering a hierarchical family structure for the trilinear scalar soft-supersymmetric terms of the Lagrangian, present at the SUSY breaking scale. We obtain analytical results for the rotation mass matrix, with the consequence of having non-universal slepton masses and the possibility of leptonic flavor mixing. The one-loop supersymmetric contributions to the leptonic flavor violating process τ → μγ are calculated in the physical basis, instead of using the well-known mass-insertion method. The flavor violating processes BR(l_i → l_jγ) are also obtained, in particular τ → μγ is well within the experimental bounds. We present the regions in parameter space where the muon g - 2 problem is either entirely solved or partially reduced through the contribution of these flavor violating processes. (orig.)
20. Muon g - 2 through a flavor structure on soft SUSY terms
Energy Technology Data Exchange (ETDEWEB)
Flores-Baez, F.V. [Universidad Autonoma de Nuevo Leon, UANL Ciudad Universitaria, FCFM, San Nicolas de los Garza, Nuevo Leon (Mexico); Gomez Bock, M. [Universidad de las Americas Puebla, UDLAP, Ex-Hacienda Sta. Catarina Martir, DAFM, Cholula, Puebla (Mexico); Mondragon, M. [Universidad Nacional Autonoma de Mexico, Instituto de Fisica, Apdo. Postal 20-364, Mexico, D.F. (Mexico)
2016-10-15
In this work we analyze the possibility to explain the muon anomalous magnetic moment discrepancy within theory and experiment through lepton-flavor violation processes. We propose a flavor extended MSSM by considering a hierarchical family structure for the trilinear scalar soft-supersymmetric terms of the Lagrangian, present at the SUSY breaking scale. We obtain analytical results for the rotation mass matrix, with the consequence of having non-universal slepton masses and the possibility of leptonic flavor mixing. The one-loop supersymmetric contributions to the leptonic flavor violating process τ → μγ are calculated in the physical basis, instead of using the well-known mass-insertion method. The flavor violating processes BR(l{sub i} → l{sub j}γ) are also obtained, in particular τ → μγ is well within the experimental bounds. We present the regions in parameter space where the muon g - 2 problem is either entirely solved or partially reduced through the contribution of these flavor violating processes. (orig.)
1. Neutrino mass and mixing with discrete symmetry
International Nuclear Information System (INIS)
King, Stephen F; Luhn, Christoph
2013-01-01
This is a review paper about neutrino mass and mixing and flavour model building strategies based on discrete family symmetry. After a pedagogical introduction and overview of the whole of neutrino physics, we focus on the PMNS mixing matrix and the latest global fits following the Daya Bay and RENO experiments which measure the reactor angle. We then describe the simple bimaximal, tri-bimaximal and golden ratio patterns of lepton mixing and the deviations required for a non-zero reactor angle, with solar or atmospheric mixing sum rules resulting from charged lepton corrections or residual trimaximal mixing. The different types of see-saw mechanism are then reviewed as well as the sequential dominance mechanism. We then give a mini-review of finite group theory, which may be used as a discrete family symmetry broken by flavons either completely, or with different subgroups preserved in the neutrino and charged lepton sectors. These two approaches are then reviewed in detail in separate chapters including mechanisms for flavon vacuum alignment and different model building strategies that have been proposed to generate the reactor angle. We then briefly review grand unified theories (GUTs) and how they may be combined with discrete family symmetry to describe all quark and lepton masses and mixing. Finally, we discuss three model examples which combine an SU(5) GUT with the discrete family symmetries A 4 , S 4 and Δ(96). (review article)
2. Sub-color and leptoquark-quark symmetry
International Nuclear Information System (INIS)
Nakamura, Fumihiko
1982-01-01
On the basis of leptoquark-quark symmetry, we propose possible models, in which leptons and gauge bosons are constructed is SU(2) symmetry. In one of the cases, the subcolor is introduced as the quantum number of the leptoquark. Then the possibility of baryon decay is discussed. (author)
3. Applications of chiral symmetry
International Nuclear Information System (INIS)
Pisarski, R.D.
1995-03-01
The author discusses several topics in the applications of chiral symmetry at nonzero temperature. First, where does the rho go? The answer: up. The restoration of chiral symmetry at a temperature T χ implies that the ρ and a 1 vector mesons are degenerate in mass. In a gauged linear sigma model the ρ mass increases with temperature, m ρ (T χ ) > m ρ (0). The author conjectures that at T χ the thermal ρ - a 1 , peak is relatively high, at about ∼1 GeV, with a width approximately that at zero temperature (up to standard kinematic factors). The ω meson also increases in mass, nearly degenerate with the ρ, but its width grows dramatically with temperature, increasing to at least ∼100 MeV by T χ . The author also stresses how utterly remarkable the principle of vector meson dominance is, when viewed from the modern perspective of the renormalization group. Secondly, he discusses the possible appearance of disoriented chiral condensates from open-quotes quenchedclose quotes heavy ion collisions. It appears difficult to obtain large domains of disoriented chiral condensates in the standard two flavor model. This leads to the last topic, which is the phase diagram for QCD with three flavors, and its proximity to the chiral critical point. QCD may be very near this chiral critical point, and one might thereby generated large domains of disoriented chiral condensates
4. Randall-Sundrum models vs. supersymmetry. The different flavor signatures
Energy Technology Data Exchange (ETDEWEB)
Gori, Stefania
2010-07-15
The Minimal Supersymmetric Standard Model based on flavor symmetries and models with a warped extra dimension as first proposed by Randall and Sundrum represent two of the best founded theories beyond the Standard Model. They provide two appealing solutions both to the gauge hierarchy problem and to the Standard Model flavor hierarchy problems. In this thesis we focus on a particular Randall-Sundrum model based on the custodial symmetry SU(2){sub L} x SU(2){sub R} x P{sub LR} in the bulk and on two Supersymmetric flavor models: the one based on a U(1) abelian flavor symmetry, the other on a SU(3) non abelian flavor symmetry. We first analyze and compare the flavor structure of the two frameworks, showing two possible ways to address the New Physics flavor problem: warped geometry and custodial protection vs. flavor symmetry. Subsequently, we study the impact of the new particles (Kaluza-Klein states in the Randall-Sundrum model and superpartners in Supersymmetry) in the K and B meson mixings and rare decays. We perform a global numerical analysis of the new physics effects in the models in question and we show that it is possible to naturally be in agreement with all the available data on {delta}F=2 observables, even fixing the energy scale of the models to the TeV range, in order to have new particles in the reach of the LHC. We then study distinctive patterns of flavor violation which can enable future experiments to distinguish the two frameworks. In particular, the specific correlations between the CP violating asymmetry in the B{sub s}{sup 0}- anti B{sub s}{sup 0} system, the rare decays B{sub s,d}{yields}{mu}{sup +}{mu}{sup -} and K{yields}{pi}{nu}anti {nu} allow in principle for an experimental test of the Randall-Sundrum model and of the two Supersymmetric flavor models and a clear distinction between the two frameworks, once new data will be available. (orig.)
5. Chiral symmetry on the lattice
International Nuclear Information System (INIS)
Creutz, M.
1994-11-01
The author reviews some of the difficulties associated with chiral symmetry in the context of a lattice regulator. The author discusses the structure of Wilson Fermions when the hopping parameter is in the vicinity of its critical value. Here one flavor contrasts sharply with the case of more, where a residual chiral symmetry survives anomalies. The author briefly discusses the surface mode approach, the use of mirror Fermions to cancel anomalies, and finally speculates on the problems with lattice versions of the standard model
6. Asymmetric Collision of Concepts: Why Eigenstates Alone are Not Enough for Neutrino Flavor Oscillations
OpenAIRE
Williams, John Michael
2000-01-01
The symmetry of the problem of the apparent deficit in upward-going atmospheric muon neutrinos reveals two possible, nonexclusive kinds of solution: Nonlinearity in distance or nonlinearity in angle of observation. Nonlinearity in distance leads to the most popular theory for the atmospheric problem, neutrino flavor oscillations. If the observed deficit is caused by oscillations and not, say, flavor-changing or other weak-force scattering, neutrinos must be massive. But, if flavor oscillation...
7. Lepton probes in nuclear physics
Energy Technology Data Exchange (ETDEWEB)
Arvieux, J. [Laboratoire National Saturne, Centre d`Etudes de Saclay, 91 - Gif-sur-Yvette (France)
1994-12-31
Facilities are overviewed which use the lepton probe to learn about nuclear physics. The lepton accelerating methods out some existing facilities are considered. The ELFE project is discussed in detail. (K.A.). 43 refs., 15 figs., 4 tabs.
8. Lepton probes in nuclear physics
International Nuclear Information System (INIS)
Arvieux, J.
1994-01-01
Facilities are overviewed which use the lepton probe to learn about nuclear physics. The lepton accelerating methods out some existing facilities are considered. The ELFE project is discussed in detail. (K.A.). 43 refs., 15 figs., 4 tabs
9. High-p{sub T} dilepton tails and flavor physics
Energy Technology Data Exchange (ETDEWEB)
Greljo, Admir [Universitaet Zuerich, Physik-Institut, Zuerich (Switzerland); University of Sarajevo, Faculty of Science, Sarajevo (Bosnia and Herzegovina); Marzocca, David [Universitaet Zuerich, Physik-Institut, Zuerich (Switzerland)
2017-08-15
We investigate the impact of flavor-conserving, non-universal quark-lepton contact interactions on the dilepton invariant mass distribution in p p → l{sup +}l{sup -} processes at the LHC. After recasting the recent ATLAS search performed at 13 TeV with 36.1 fb{sup -1} of data, we derive the best up-to-date limits on the full set of 36 chirality-conserving four-fermion operators contributing to the processes and estimate the sensitivity achievable at the HL-LHC. We discuss how these high-p{sub T} measurements can provide complementary information to the low-p{sub T} rare meson decays. In particular, we find that the recent hints on lepton-flavor universality violation in b → sμ{sup +}μ{sup -} transitions are already in mild tension with the dimuon spectrum at high-p{sub T} if the flavor structure follows minimal flavor violation. Even if the mass scale of new physics is well beyond the kinematical reach for on-shell production, the signal in the high-p{sub T} dilepton tail might still be observed, a fact that has been often overlooked in the present literature. In scenarios where new physics couples predominantly to third generation quarks, instead, the HL-LHC phase is necessary in order to provide valuable information. (orig.)
10. Inclusive lepton production from heavy-hadron decay in pp collisions at the LHC
Energy Technology Data Exchange (ETDEWEB)
Bolzoni, Paolo; Kramer, Gustav [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik
2012-12-15
We present predictions for the inclusive production of leptons (e{sup {+-}}, {mu}{sup {+-}}) originating from charm and bottom-hadrons at the CERN LHC in the general-mass variable-flavor-number scheme at next-to-leading order. Detailed numerical results are compared to data of the CMS, ATLAS and ALICE collaborations.
11. Large leptonic Dirac CP phase from broken democracy with random perturbations
Science.gov (United States)
Ge, Shao-Feng; Kusenko, Alexander; Yanagida, Tsutomu T.
2018-06-01
A large value of the leptonic Dirac CP phase can arise from broken democracy, where the mass matrices are democratic up to small random perturbations. Such perturbations are a natural consequence of broken residual S3 symmetries that dictate the democratic mass matrices at leading order. With random perturbations, the leptonic Dirac CP phase has a higher probability to attain a value around ± π / 2. Comparing with the anarchy model, broken democracy can benefit from residual S3 symmetries, and it can produce much better, realistic predictions for the mass hierarchy, mixing angles, and Dirac CP phase in both quark and lepton sectors. Our approach provides a general framework for a class of models in which a residual symmetry determines the general features at leading order, and where, in the absence of other fundamental principles, the symmetry breaking appears in the form of random perturbations.
12. On the origin of neutrino flavour symmetry
International Nuclear Information System (INIS)
King, Stephen F.; Luhn, Christoph
2009-01-01
We study classes of models which are based on some discrete family symmetry which is completely broken such that the observed neutrino flavour symmetry emerges indirectly as an accidental symmetry. For such 'indirect' models we discuss the D-term flavon vacuum alignments which are required for such an accidental flavour symmetry consistent with tri-bimaximal lepton mixing to emerge. We identify large classes of suitable discrete family symmetries, namely the Δ(3n 2 ) and Δ(6n 2 ) groups, together with other examples such as Z 7 x Z 3 . In such indirect models the implementation of the type I see-saw mechanism is straightforward using constrained sequential dominance. However the accidental neutrino flavour symmetry may be easily violated, for example leading to a large reactor angle, while maintaining accurately the tri-bimaximal solar and atmospheric predictions.
13. Heavy flavor spectroscopy
International Nuclear Information System (INIS)
Rosen, J.; Marques, J.; Spiegel, L.
1993-09-01
As a useful by-product of the unfolding searches for mixing and CP-violation effects in the beauty sector there will accrue very large data samples for the study of heavy flavor spectroscopy. Interest in this field may be provisionally divided into two general classes: Hidden flavor states, i.e. c bar c and b bar b onium states; open flavor states: The D, D s , B, B s , and B c meson systems; and charm and beauty flavored baryons. In this brief note we emphasize that there are many missing states in both categories -- states which are not readily produced exclusively due to quantum number preferences or states which are not readily observed inclusively due to experimentally difficult decay channels. As recorded luminosities increase it may be possible to fill in some of the holes in the present listings of heavy flavor states. Of particular interest to us would be the identification of heavy flavor mesons which are not easily explained in terms of a q bar q paradigm but rather may be evidence for hadro-molecular states. At Snowmass 1993 the topic of self-tagging schemes in B meson production was very much in vogue. Whether or not excited B-meson flavor-tagging will prove to be competitive with traditional methods based on the partner bar B decay remains to be seen. We suggest however that the richness of the excited B-system may undermine the efficacy of self-tagging schemes
14. Heavy flavor spectroscopy
International Nuclear Information System (INIS)
Rosen, J.; Marques, J.; Spiegel, L.
1993-01-01
As a useful by-product of the unfolding searches for mixing and CP-violation effects in the beauty sector there will accrue very large data samples for the study of heavy flavor spectroscopy. (I) Hidden flavor states, i.e. c bar c and b bar b onium states. (II) Open flavor states (a) the D, D s , B, B s , and B c meson systems; (b) Charm and beauty flavored baryons. In this brief note the authors emphasize that there are many missing (undiscovered) states in both categories - states which are not readily produced exclusively due to quantum number preferences or states which are not readily observed inclusively due to experimentally difficult decay channels. As recorded luminosities increase it may be possible to fill in some of the holes in the present listings of heavy flavor states. Of particular interest to the authors would be the identification of heavy flavor mesons which are not easily explained in terms of a q bar q paradigm but rather may be evidence for hadro-molecular status. At Snowmass 1993 the topic of self-tagging schemes in B meson production was very much in vogue. Whether or not excited B-meson flavor-tagging will prove to be competitive with traditional methods based on the partner B decay remains to be seen. The authors suggest however that the richness of the excited B-system may undetermine the efficacy of self-tagging schemes
15. Generation labels in composite models for quarks and leptons
International Nuclear Information System (INIS)
Harari, H.; Seiberg, N.
1981-03-01
Models in which quarks and leptons are approximately massless composites of fundamental massless fermions which are confined by a hypercolor force are considered. The fundamental Lagrangian exhibits an axial U(1)sub(X) symmetry which is broken by hypercolor instantons, leaving a conserved discrete subgroup. It is proposed that the distinction between different generations of quarks and leptons is given by the X-number. The resulting generation labelling scheme does not lead to massless Goldstone bosons or to new anomalies and is based on a quantum number which is already contained in the theory. The dynamical rishon model is described as an illustrative example. (H.K.)
16. Up sector of minimal flavor violation: top quark properties and direct D meson CP violation
Energy Technology Data Exchange (ETDEWEB)
Bai, Yang; Berger, Joshua; Hewett, JoAnne L.; Li, Ye
2013-07-01
Minimal Flavor Violation in the up-type quark sector leads to particularly interesting phenomenology due to the interplay of flavor physics in the charm sector and collider physics from flavor changing processes in the top sector. We study the most general operators that can affect top quark properties and D meson decays in this scenario, concentrating on two CP violating operators for detailed studies. The consequences of these effective operators on charm and top flavor changing processes are generically small, but can be enhanced if there exists a light flavor mediator that is a Standard Model gauge singlet scalar and transforms under the flavor symmetry group. This flavor mediator can satisfy the current experimental bounds with a mass as low as tens of GeV and explain observed D-meson direct CP violation. Additionally, the model predicts a non-trivial branching fraction for a top quark decay that would mimic a dijet resonance.
17. Family symmetries in F-theory GUTs
CERN Document Server
King, S F; Ross, G G
2010-01-01
We discuss F-theory SU(5) GUTs in which some or all of the quark and lepton families are assigned to different curves and family symmetry enforces a leading order rank one structure of the Yukawa matrices. We consider two possibilities for the suppression of baryon and lepton number violation. The first is based on Flipped SU(5) with gauge group SU(5)\\times U(1)_\\chi \\times SU(4)_{\\perp} in which U(1)_{\\chi} plays the role of a generalised matter parity. We present an example which, after imposing a Z_2 monodromy, has a U(1)_{\\perp}^2 family symmetry. Even in the absence of flux, spontaneous breaking of the family symmetry leads to viable quark, charged lepton and neutrino masses and mixing. The second possibility has an R-parity associated with the symmetry of the underlying compactification manifold and the flux. We construct an example of a model with viable masses and mixing angles based on the gauge group SU(5)\\times SU(5)_{\\perp} with a U(1)_{\\perp}^3 family symmetry after imposing a Z_2 monodromy.
18. Excited lepton search
International Nuclear Information System (INIS)
Behrend, H.J.; Buerger, J.; Criegee, L.; Fenner, H.; Field, J.H.; Franke, G.; Fuster, J.; Holler, Y.; Meyer, J.; Schroeder, V.; Sindt, H.; Timm, U.; Winter, G.G.; Zimmermann, W.; Bussey, P.J.; Campbell, A.J.; Dainton, J.B.; Hendry, D.; McCurrach, G.; Scarr, J.M.; Skillicorn, I.O.; Smith, K.M.; Blobel, V.; Poppe, M.; Spitzer, H.; Boer, W. de; Buschhorn, G.; Christiansen, W.; Grindhammer, G.; Gunderson, B.; Kiesling, C.; Kotthaus, R.; Kroha, H.; Lueers, D.; Oberlack, H.; Sack, B.; Schacht, P.; Shooshtari, G.; Wiedenmann, W.; Cordier, A.; Davier, M.; Fournier, D.; Gaillard, M.; Grivaz, J.F.; Haissinski, J.; Janot, P.; Journe, V.; Le Diberder, F.; Ros, E.; Spadafora, A.; Veillet, J.J.; Aleksan, R.; Cozzika, G.; Ducros, Y.; Jarry, P.; Lavagne, Y.; Ould Saada, F.; Pamela, J.; Pierre, F.; Zacek, J.; Alexander, G.; Bella, G.; Gnat, Y.; Grunhaus, J.
1986-02-01
Using the CELLO detector at PETRA we have searched for excited leptons by studying e + e - interactions which yield p + p - γγ, l + l - γ and γγ final states, where l = 3, μ or τ. We observe good agreement with QED and set new limits on e*, μ*, and τ* production. (orig.)
19. Radiative corrections and Monte Carlo generators for physics at flavor factories
Directory of Open Access Journals (Sweden)
Montagna Guido
2016-01-01
Full Text Available I review the state of the art of precision calculations and related Monte Carlo generators used in physics at flavor factories. The review describes the tools relevant for the measurement of the hadron production cross section (via radiative return, energy scan and in γγ scattering, luminosity monitoring, searches for new physics and physics of the τ lepton.
20. Unlocking color and flavor in superconducting strange quark matter
International Nuclear Information System (INIS)
Alford, Mark; Berges, Juergen; Rajagopal, Krishna
1999-01-01
We explore the phase diagram of strongly interacting matter with massless u and d quarks as a function of the strange quark mass m s and the chemical potential μ for baryon number. Neglecting electromagnetism, we describe the different baryonic and quark matter phases at zero temperature. For quark matter, we support our model-independent arguments with a quantitative analysis of a model which uses a four-fermion interaction abstracted from single-gluon exchange. For any finite m s , at sufficiently large μ we find quark matter in a color-flavor-locked state which leaves a global vector-like SU(2) color+L+R symmetry unbroken. As a consequence, chiral symmetry is always broken in sufficiently dense quark matter. As the density is reduced, for sufficiently large m s we observe a first-order transition from the color-flavor-locked phase to color superconducting phase analogous to that in two-flavor QCD. At this unlocking transition chiral symmetry is restored. For realistic values of m s our analysis indicates that chiral symmetry breaking may be present for all densities down to those characteristic of baryonic matter. This supports the idea that quark matter and baryonic matter may be continuously connected in nature. We map the gaps at the quark Fermi surfaces in the high density color-flavor-locked phase onto gaps at the baryon Fermi surfaces at low densities
1. Neutrino flavor evolution in neutron star mergers
Science.gov (United States)
Tian, James Y.; Patwardhan, Amol V.; Fuller, George M.
2017-08-01
We examine the flavor evolution of neutrinos emitted from the disklike remnant (hereafter called "neutrino disk") of a binary neutron star (BNS) merger. We specifically follow the neutrinos emitted from the center of the disk, along the polar axis perpendicular to the equatorial plane. We carried out two-flavor simulations using a variety of different possible initial neutrino luminosities and energy spectra and, for comparison, three-flavor simulations in specific cases. In all simulations, the normal neutrino mass hierarchy was used. The flavor evolution was found to be highly dependent on the initial neutrino luminosities and energy spectra; in particular, we found two broad classes of results depending on the sign of the initial net electron neutrino lepton number (i.e., the number of neutrinos minus the number of antineutrinos). In the antineutrino-dominated case, we found that the matter-neutrino resonance effect dominates, consistent with previous results, whereas in the neutrino-dominated case, a bipolar spectral swap develops. The neutrino-dominated conditions required for this latter result have been realized, e.g., in a BNS merger simulation that employs the "DD2" equation of state for neutron star matter [Phys. Rev. D 93, 044019 (2016), 10.1103/PhysRevD.93.044019]. For this case, in addition to the swap at low energies, a collective Mikheyev-Smirnov-Wolfenstein mechanism generates a high-energy electron neutrino tail. The enhanced population of high-energy electron neutrinos in this scenario could have implications for the prospects of r -process nucleosynthesis in the material ejected outside the plane of the neutrino disk.
2. Search for lepton-flavour-violating H → μτ decays of the Higgs boson with the ATLAS detector
Czech Academy of Sciences Publication Activity Database
Aad, G.; Abbott, B.; Abdallah, J.; Chudoba, Jiří; Havránek, Miroslav; Hejbal, Jiří; Jakoubek, Tomáš; Kepka, Oldřich; Kupčo, Alexander; Kůs, Vlastimil; Lokajíček, Miloš; Lysák, Roman; Marčišovský, Michal; Mikeštíková, Marcela; Němeček, Stanislav; Penc, Ondřej; Šícho, Petr; Staroba, Pavel; Svatoš, Michal; Taševský, Marek; Vrba, Václav
2015-01-01
Roč. 2015, č. 11 (2015), s. 1-33, č. článku 211. ISSN 1029-8479 R&D Projects: GA MŠk(CZ) LG13009 Institutional support: RVO:68378271 Keywords : p-p scattering * p-p colliding beams * Higgs particle * hadroproduction * leptonic decay * lepton flavor violation * hadronic tau Subject RIV: BF - Elementary Particles and High Energy Physics Impact factor: 6.023, year: 2015
3. Breaking of electroweak symmetry: origin and effects; Brisure de symetrie electrobaible: origine et consequence
Energy Technology Data Exchange (ETDEWEB)
Delaunay, C
2008-10-15
The Higgs boson appears as the corner stone of high energy physics, it might be the cause of the excess of matter that led to the formation of the structures of the universe and it seems that it drives the breaking of the electroweak symmetry. Moreover, when the stability at low energies of the Higgs boson is assured by an extra space dimension, it appears that this extra dimension can explain most issues in the flavor physics that are not understood by the standard model. The first chapter presents the main tools of effective field theories, the role of experimental data in the construction of theories valid beyond the standard model is discussed. The second chapter focuses on the electroweak baryogenesis that allows the testing of new physics via the electroweak phase transition. We detail the calculation of a Higgs potential at finite temperature. We follow the dynamics of the phase transition including nucleation an supercooling. Finally we investigate the prospects of gravity wave detection to see the effects of a strong electroweak phase transition. The 2 last chapters are dedicated to the physics of extra-dimension. The properties of the dynamics of scalar, vector fields with a 1/2 spin plunged in a 5 d. Anti de Sitter geometry are reviewed. We present a model of lepton masses and mixings based on the A{sub 4} non-Abelian discrete symmetry. It is shown that this model does not contradict the tests of electroweak precision. (A.C.)
4. Two Complementary Strategies for New Physics Searches at Lepton Colliders
Energy Technology Data Exchange (ETDEWEB)
Hooberman, Benjamin Henry [Univ. of California, Berkeley, CA (United States)
2009-07-06
In this thesis I present two complementary strategies for probing beyond-the-Standard Model physics using data collected in e+e- collisions at lepton colliders. One strategy involves searching for effects at low energy mediated by new particles at the TeV mass scale, at which new physics is expected to manifest. Several new physics scenarios, including Supersymmetry and models with leptoquarks or compositeness, may lead to observable rates for charged lepton-flavor violating processes, which are forbidden in the Standard Model. I present a search for lepton-flavor violating decays of the Υ(3S) using data collected with the BABAR detector. This study establishes the 90% confidence level upper limits BF(Υ(3S) → eτ) < 5.0 x 10-6 and BF(Υ(3S) → μτ) < 4.1 x 10-6 which are used to place constraints on new physics contributing to lepton-flavor violation at the TeV mass scale. An alternative strategy is to increase the collision energy above the threshold for new particles and produce them directly. I discuss research and development efforts aimed at producing a vertex tracker which achieves the physics performance required of a high energy lepton collider. A small-scale vertex tracker prototype is constructed using Silicon sensors of 50 μm thickness and tested using charged particle beams. This tracker achieves the targeted impact parameter resolution of σLP = (5⊕10 GeV/pT) as well as a longitudinal vertex resolution of (260 ± 10) μm, which is consistent with the requirements of a TeV-scale lepton collider. This detector research and development effort must be motivated and directed by simulation studies of physics processes. Investigation of a dark matter-motivated Supersymmetry scenario is presented, in which the dark matter is composed of Supersymmetric neutralinos. In this scenario, studies of the e+e- → H0A0 production process allow for
5. Possibility of new dibaryons containing heavy flavors
International Nuclear Information System (INIS)
Leandri, J.; Silvestre-Brac, B.
1993-01-01
In a recent paper we have shown that the possibility of including heavy flavor in the dibaryon sector can lead to some new favored configurations (relative to the baryon-baryon threshold). In this study we extend our previous work by a systematic study of all the physical Qq 5 systems in a simple chromomagnetic model. In the first part we assume that the q quarks belong to the fundamental irrep of SU(3) F and that the Q quark has infinite mass. These assumptions are subsequently relaxed by introducing two mass parameters δ and η. Once these symmetries are broken we gain access in our model to a large number of new dibaryons containing heavy flavor. Some of them could be stable against decay via strong interactions, and we indicate the most favorable cases
6. The flavor of the composite pseudo-goldstone Higgs
International Nuclear Information System (INIS)
Csaki, Csaba; Weiler, Andreas; Falkowski, Adam
2008-01-01
We study the flavor structure of 5D warped models that provide a dual description of a composite pseudo-Goldstone Higgs. We first carefully re-examine the flavor constraints on the mass scale of new physics in the standard Randall-Sundrum-type scenarios, and find that the KK gluon mass should generically be heavier than about 21 TeV. We then compare the flavor structure of the composite Higgs models to those in the RS model. We find new contributions to flavor violation, which while still are suppressed by the RS-GIM mechanism, will enhance the amplitudes of flavor violations. In particular, there is a kinetic mixing term among the SM fields which (although parametrically not enhanced) will make the flavor bounds even more stringent than in RS. This together with the fact that in the pseudo-Goldstone scenario Yukawa couplings are set by a gauge coupling implies the KK gluon mass to be at least about 33 TeV. For both the RS and the composite Higgs models the flavor bounds could be stronger or weaker depending on the assumption on the value of the gluon boundary kinetic term. These strong bounds seem to imply that the fully anarchic approach to flavor in warped extra dimensions is implausible, and there have to be at least some partial flavor symmetries appearing that eliminate part of the sources for flavor violation. We also present complete expressions for the radiatively generated Higgs potential of various 5D implementations of the composite Higgs model, and comment on the 1-5 percent level tuning needed in the top sector to achieve a phenomenologically acceptable vacuum state.
7. Nuclear probes of fundamental symmetries
International Nuclear Information System (INIS)
1983-01-01
Nuclear experiments which probe the parity (P) and time-reversal (T) symmetries and lepton-number conservation are reviewed. The P-violating NN interaction, studied in the NN system and in light nuclei, provides an unique window on ΔS=0 hadronic weak processes. Results are in accord with expectations. Sensitive searches for T-violation via detailed balance, T-odd correlations in γ and β-decay, and a possible neutron electric dipole moment (EDM) are discussed. No T-violation is observed. The EDM limit is almost good enough to eliminate one of the leading theoretical explanations for CP violation. Experimental studies of double β-decay are reviewed. Although ββ nu nu decay has been convincingly detected in geochemical experiments there is no evidence for the lepton number violating ββ decay mode
8. Phenomenology of flavor-mediated supersymmetry breaking
International Nuclear Information System (INIS)
Kaplan, D. Elazzar; Kribs, Graham D.
2000-01-01
The phenomenology of a new economical supersymmetric model that utilizes dynamical supersymmetry breaking and gauge mediation for the generation of the sparticle spectrum and the hierarchy of fermion masses is discussed. Similarities between the communication of supersymmetry breaking through a messenger sector and the generation of flavor using the Froggatt-Nielsen (FN) mechanism are exploited, leading to the identification of vector-like messenger fields with FN fields and the messenger U(1) as a flavor symmetry. An immediate consequence is that the first and second generation scalars acquire flavor-dependent masses, but do not violate flavor changing neutral current bounds since their mass scale, consistent with ''effective supersymmetry,'' is of order 10 TeV. We define and advocate a ''minimal flavor-mediated model'' (MFMM), recently introduced in the literature, which successfully accommodates the small flavor-breaking parameters of the standard model using order 1 couplings and ratios of flavon field VEVs. The mediation of supersymmetry breaking occurs via two-loop logarithm-enhanced gauge-mediated contributions, as well as several one-loop and two-loop Yukawa-mediated contributions for which we provide analytical expressions. The MFMM is parametrized by a small set of masses and couplings, with values restricted by several model constraints and experimental data. Full two-loop renormalization group evolutio | {"extraction_info": {"found_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": 0.9551495909690857, "perplexity": 3820.716067509174}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145729.69/warc/CC-MAIN-20200222211056-20200223001056-00060.warc.gz"} |
http://slideplayer.com/slide/1520872/ | # path takes the least time?
## Presentation on theme: "path takes the least time?"— Presentation transcript:
path takes the least time?
Suppose the sprinters wish to get from point Q on the beach to point P on the parking lot as quickly as possible. Which path takes the least time? 1. a 2. b 3. c 4. d 5. e 6. All paths take the same amount of time. Answer: 4. Anybody—sprinter or couch potato—can run more quickly on a hard surface than on loose sand. While the distance on loose sand is slightly less for path e than for path d, the run over the parking lot is much longer. The result is that path e is more time-consuming than path d.
4. spaced closer together.
The observer at O views two closely spaced lines through an angled piece of plastic. To the observer, the lines appear (choose all that apply) 1. shifted to the right. 2. shifted to the left. 3. spaced farther apart. 4. spaced closer together. 5. exactly as they do without the piece of plastic. Answer: B. Refraction through the plastic shifts the beams toward the left. Since the beams from both lines shift the same amount, the spacing between the two lines remains the same.
1. a greater depth than it really is. 2. the same depth.
A fish swims below the surface of the water at P. An observer at O sees the fish at 1. a greater depth than it really is. 2. the same depth. 3. a smaller depth than it really is. Answer: 3.The rays emerging from the water surface converge to a point above the fish. See figure.
Explanation Answer: 3.The rays emerging from the water surface converge to a point above the fish. See figure.
1. a greater depth than it really is. 2. the same depth.
A fish swims below the surface of the water. Suppose an observer is looking at the fish from point O' straight above the fish. The observer sees the fish at 1. a greater depth than it really is. 2. the same depth. 3. a smaller depth than it really is. Answer: 3.The rays emerging from the water surface converge to a point above the fish.
Explanation Answer: 3.The rays emerging from the water surface converge to a point above the fish.
There are certain materials which can have properties that seem to ignore the laws of physics. These man made materials called metamaterials are specifically designed to exhibit electromagnetic characteristics not seen in nature. One such result is a chemical with a negative refractive index. What would a chemical with a negative refractive index look like? Negative Refractive index Positive Refractive index These are computer generated images to simulate the what would be observed.
3. farther from the lens than outside the water.
A parallel beam of light is sent through an aquarium. If a convex glass lens is held in the water, it focuses the beam 1. closer to the lens than 2. at the same position as 3. farther from the lens than outside the water. The index of water is greater than that of air, therefore less refraction occurs. Answer: 3. The index of refraction of water is between those of air and glass. Thus, rays of light refract less in going from water to glass than in going from air to glass. As a result, a convex lens focuses rays of light less tightly under water.
Shown in the photograph below is a small fuse box as seen through a magnifying glass.
Now suppose that the entire magnifying glass and fuse box assembly were placed into the tank of water seen at the rear of the picture behind the assembly. After the magnifying glass/fuse box assembly is placed into the water the letters on the fuse box will appear: (1) larger. (2) smaller. (3) the same size.
Huygen’s Principle: Electromagnetic waves can be examined using geometrical considerations instead of the relationships between the electric and magnetic fields (such as Snell’s Law). Huygen’s Principle was the first theory describing the wave nature of light that explained reflection and refraction. Huygen’s Principle states: “All points on a given wavefront are sources of spherical secondary waves called wavelets, which propagate outward. The new wavefront will be a surface line tangent to each of these wavelets.” cDt Wavelets cDt Wavefront 1 2 This theory has more historical significance than practical applications. | {"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": 0.8256498575210571, "perplexity": 827.892805187118}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1516084889677.76/warc/CC-MAIN-20180120162254-20180120182254-00663.warc.gz"} |
https://link.springer.com/article/10.1007/s00221-008-1697-x | ## Introduction
Relevant information that we receive from our environment must be processed while a large amount of irrelevant information stimulates our senses. The concept of how contextual information influences perception is important for the study of perception and cognition. The large number of studies concerning contextual influences on perception in, for example, the visual domain emphasizes the importance of this concept (e.g. Adelson 1993; Cao and Shevell 2005; Ware and Cowan 1982; Webster et al. 2002). In the haptic modality, this concept has received less attention. However, during daily exploration by touch, we often perceive a particular object after having been in contact with some other object(s), or we explore different materials with different parts of the hand at the same time. Hence, the context in which haptic perception takes place may influence the perceptual experience. The current study was designed to investigate these contextual influences in the haptic perception of textured surfaces. These influences can roughly be subdivided into temporal and spatial influences of the context.
### Temporal context
Roughness is one texture property that has been studied in some detail. Tactile roughness perception has been related to physical characteristics of the surface, like the spacing between and the height of surface elements (e.g. Connor et al. 1990; Connor and Johnson 1992; Lederman 1981; Lederman 1983; Lederman and Taylor 1972). Furthermore, studies addressing the neural codes underlying the sensation of tactile roughness showed that subjective roughness is related to spatial variations in the firing rate of slowly adapting type I (SAI) mechanoreceptive neurons (Blake et al. 1997; Connor et al. 1990; Connor and Johnson 1992).
Some ideas about the influence of temporal context in the perception of textured surfaces can be deduced from studies investigating the contribution of vibratory adaptation to roughness perception. Lederman et al. (1982) showed that the perceived magnitude of supraliminal vibrotactile signals decreased after adaptation to vibrations. More recently, Hollins et al. (2001) found that adaptation to vibrotactile signals disrupted the discrimination of very fine textured surfaces (spatial period < 200 μm). Furthermore, it has been shown that this type of adaptation had no effect on roughness perception of coarse surfaces, like metal gratings (Lederman et al. 1982) and dotted patterns with spatial periods above 200 μm (Hollins et al. 2001). Also, when adapting to a spatially textured surface instead of vibrotactile stimuli, no adaptation effects with coarse surfaces were found (Hollins et al. 2006).
However, DiCarlo et al. (1998) suggested that texture adaptation effects should be present in the case of coarse surfaces. They used random dot stimuli to study the structure of receptive fields in area 3b of the somatosensory cortex. The results revealed that most of these receptive fields have one or two inhibitory regions flanking a region of excitation. This resembles structures in the primary visual cortex, where many simple cells also have receptive fields with an excitatory region surrounded by flanking inhibitory areas (Hubel and Wiesel 1962). If the structures are highly similar, then this could indicate that cells from different brain areas represent information in analogous ways. Visual cortex cells are highly susceptible to adaptation (Blakemore et al. 1973; Jones and Palmer 1987). Adaptation causes a shift in the neuronal tuning of these visual neurons away from the level of the adapted value. Examples of such shifts have been found for dimensions like contrast (Carandini et al. 1997) and orientation (Dragoi et al. 2000).
Hence, if analogous processing of information occurs within different modalities, it should be expected that somatosensory and visual neurons should also show comparable adaptation effects. Consequently, texture adaptation should influence the perceived roughness of coarse surfaces, in apparent contrast to what Hollins et al. (2006) found. Their stimuli consisted of regular dot patterns with relatively small distances between dots, whereas DiCarlo et al. (1998) used random dot patterns with much larger average distances between dots. It could be that the regular patterns with smaller dot distances are not appropriate for activating the neuron types described by Dicarlo et al. (1998); therefore, no adaptation effects were found in the Hollins et al. (2001)’s study. Another possibility is that the adaptation pattern used by Hollins was too weak to cause significant adaptation effects.
In the first experiment of the current work, we used random dot patterns with relatively large average distances between dots and assumed that they will appropriately activate the neurons that are susceptible to adaptation. Subjects were asked to discriminate between the roughness of two surfaces presented simultaneously to two adjacent fingers, after adapting one of these fingers to a textured surface. It is hypothesized that texture adaptation will change the response patterns of neurons, resulting in changed perceived roughness of a subsequently perceived surface.
### Spatial context
Besides temporal adaptation effects, another frequently studied concept in vision is the influence of spatial context on perception. An example is chromatic induction in the perception of colour (e.g. Cao and Shevell 2005; Shevell and Wei 2000; Webster et al. 2002). In these studies, observers had to judge the perceived colour of test surfaces, while simultaneously viewing inducing backgrounds composed of different colours. Two different types of induction were demonstrated: contrast and assimilation. Contrast occurs when the perceived appearance of the test shifts away from the appearance of the inducing stimulus; assimilation occurs when the appearance of the test shifts towards the appearance of the inducer. It has been proposed that factors like spatial frequency (Smith et al. 2001), luminance contrast, width of the inducing ring and receptive-field organization (Cao and Shevell 2005) play an important role in the transition from chromatic assimilation to chromatic contrast.
In the haptic modality, spatial context can be described as the interaction of information simultaneously received from different parts of the hand or, more specifically, from different fingers. Using sandpaper as stimuli, Dorsch et al. (2001) showed that when two fingers scanned surfaces with different grit numbers, the grit number presented to the non-attended finger had no effect on perceived roughness with the attended finger. This result suggests that there is no interaction between signals from different fingers and, hence, no influence of the spatial context on roughness perception.
However, studies using magnetoencephalography (MEG) and microelectrode recordings demonstrated interactions between finger representations. Researchers found multi-finger or wide-field receptive fields, which cover more than one finger, in area 1 neurons of the primary somatosensory cortex as well as in the medial part of the cortical finger region (Biermann et al. 1998; Forss et al. 1995; Iwamura et al. 1983). In general, these studies found an inhibition effect of the cerebral signal when multiple fingers were stimulated by mechanical stimulations of high-level intensities. When using low-level stimulations, which are more representative of the signals that we receive from our natural environment, the input from two fingers produced additive or facilitatory interactions in the early component of the cerebral potential (Gandevia et al. 1983). Furthermore, a number of studies have demonstrated the existence of multi-finger receptive fields in areas of the second somatosensory cortex (Fitzgerald et al. 2006; Sinclair and Burton 1993). Together, these results suggest that spatial context should influence perception. The fact that no interactions were found in the experiment by Dorsch et al. (2001) could be due to their use of sandpaper as stimuli. As argued by Hollins et al. (2006), the use of abrasive papers can cause damage to the skin and therefore alter the biophysical response to the stimuli. Consequently, it is not possible to draw consistent conclusions about the influences of spatial context on haptic roughness perception.
Our second experiment was designed to shed new light on spatial contextual influences in the haptic perception of roughness. The integration of information received from different fingers when scanning textured surfaces was investigated. Subjects were asked to discriminate between successively scanned surfaces while an adjacent finger was simultaneously scanning another surface varying in roughness. Based on neurophysiological studies concerning multi-finger receptive fields, we hypothesize that roughness information received from adjacent fingers will cause interaction effects. These effects will likely resemble chromatic assimilation rather than contrast effects, as Gandevia et al. (1983) has shown that low-level stimuli produces additive interactions.
## General methods
### Subjects
Ten subjects (six female and four male, mean age 20.2 years) participated in both experiments. To control for order effects, five subjects performed Experiment 1 before Experiment 2, and the other five participated in the reverse order. Nine subjects were strongly right-handed, and one was strongly left-handed, as established by Coren’s handedness questionnaire (Coren 1993). All subjects were experimentally naïve and were paid for their participation. Before starting the first experiment, they provided written informed consent.
### Stimuli
The stimuli used in both experiments were a set of embossed dot surfaces. The dot patterns were embossed on paper (weight 160 g/m2) using an Emprint Braille Embosser (ViewPlus Technologies, emboss printing resolution 20 dots/inch). Each pattern was then pasted on 2.6 mm-thick cardboard. It was necessary that the physical characteristics of the dots, especially the height profile, remained constant during the experiment. Therefore, every new condition of every subject began with a new stimulus set.
A total of 12 different patterns were constructed. Each pattern consisted of a specific part of 5.08 × 5.08 mm. This part was repeated 5 times in the horizontal and 20 times in the vertical direction, resulting in a 25.4 mm wide and 101.6 mm long stimulus pattern. One such specific part was composed of dots (height 0.4 mm, diameter 0.8 mm) placed in the centres of a regular 4 × 4 grid (Fig. 1a). The sequence of the 12 different patterns, with decreasing dot densities, was constructed by repeatedly removing one random dot from the previous specific part in the sequence (Fig. 1b). For each pattern, the average centre-to-centre distance between dots was calculated by taking the square root of the inverse dot-density. Consequently, for the complete stimulus set, the average distances between dots ranged from 1.27 to 2.27 mm.
As previously demonstrated for embossed dot surfaces, dot spacing correlates with the subjective roughness of those surfaces (e.g. Chapman et al. 2002; Connor et al. 1990; Connor and Johnson 1992). These studies have shown a near linear increase in perceived roughness magnitude with increasing dot distances up to 3 mm (Connor et al. 1990; Connor and Johnson 1992) and in some studies for even larger distances (Chapman et al. 2002). Connor et al. (1990) found that the increase in perceived roughness for these dot distances is preserved for dots with varying diameter. This relationship is assumed to hold in the present study, in which the average distances between dots are smaller than these aforementioned maxima (see Fig. 1c). To find support for this assumption, a pilot study was performed in which blindfolded subjects had to order the patterns from the current study according to their perceived roughness. This pilot study demonstrated an increase in perceived roughness with increasing average distances between dots. Therefore, in the present study, a stimulus with a small average distance between dots was marked as a smooth stimulus, while a stimulus with a large average distance was marked as a rough stimulus.
## Experiment 1: temporal context
This first experiment investigates the influence of temporal context on the haptic perception of roughness. The effect of two different adaptation levels (i.e. rough and smooth) on the perceived roughness of a subsequently scanned surface was studied.
### Procedure
Before the experiment started, the participants were blindfolded to prevent them from using visual information during the experiment. About 25 cm in front of the subject, a cardboard framework was fixed on the table. The stimuli could be placed in between the borders of this framework in such a way that they could not move when the subject explored them (see Fig. 2). The subjects were instructed to apply a comfortable level of downward force with the tips of the index and middle finger and to move with a comfortable speed. The required movement was a forward and backward movement over the stimulus surfaces. Within a couple of practice trials, this movement pattern was trained. The subjects were asked to keep this movement pattern as constant as possible during the experiment. If large deviations from the trained movements were observed, instructions were given to correct the movement. Once the preferred movement pattern was achieved and the instructions were clear, the experimental runs started.
With regard to the two adaptation conditions, the first trial was preceded by a pre-adaptation period of 60 s. In this way, a baseline level of adaptation was established before the first test trial started. All other test trials were preceded by an adaptation phase of 20 s. To start the adaptation, the participant lowered the tip of the index finger of his/her dominant hand onto the stimulus surface and moved it over the stimulus surface, as trained during the practice trials (Fig. 2a). At the end of the adaptation period, the experimenter gave a vocal signal to stop adaptation and to move towards the next two stimuli. One of these stimuli was for the index finger, and the other one was for the middle finger (Fig. 2b). The position of the test and reference stimuli (i.e. under index or middle finger) as well as the order of the different test–reference combinations for each trial were randomized.
Next, the participant simultaneously moved the index and middle finger forward and backward over the stimuli. Immediately after completing the exploration, a two-alternative forced-choice (2AFC) task was conducted; the subject had to say which of the two stimuli, i.e. the stimulus scanned with the index or middle finger, felt rougher. After the response, the next adaptation phase began. The control condition proceeded in the same way, except that there was no adaptation phase. Hence, the control condition consisted of only the 2AFC task, which was conducted in the same way as during the adaptation conditions.
### Analysis
The difference between the dot distance values of the stimuli scanned with the index and middle finger was used as the independent variable. For all subjects and conditions, we calculated for each of these differences the fraction with which the subject selected the stimulus scanned with the index finger as being rougher compared to the middle finger stimulus. A cumulative Gaussian distribution (f) as function of the dot distance differences (x) was fitted to the data using the following equation:
$$f(x) = \frac{1}{2}\left( {1 + {\text{erf}}\left( {\frac{{x - {{\upmu}}}}{\sigma \sqrt 2 }} \right)} \right),$$
where σ is a measure of the discrimination threshold, indicating the shallowness of the curve, and μ is the observer’s point of subjective equality (PSE), representing the location of the curve relative to the point of equal physical roughness. The discrimination threshold reveals the sensitivity of the subjects to perceived roughness differences within the experiment. The PSE corresponds to the physical roughness difference between the stimulus presented to the index finger and the stimulus presented to the middle finger that are on average judged as being equal. A shift of the curve in the horizontal direction can occur when subjects systematically underestimate or overestimate the roughness of the stimulus scanned with the index finger as compared to the stimulus scanned with the middle finger. Comparison of PSEs (i.e. the shift of the curves) under different conditions can reveal a possible effect of adaptation. Examples of this fitting procedure are shown in Fig. 3.
To compare the effects of the different adaptation conditions on the PSE, a repeated measures ANOVA was performed, with condition as the within-subject factor. Furthermore, the same significance test was performed with the measured thresholds to determine if there was an adaptation effect on discrimination ability. If significant overall effects were found, a paired comparison post hoc test was performed to reveal pairwise differences. To correct for multiple comparisons, a Bonferroni adjustment was done. For all statistic tests, α was set at 5%.
### Results
Figure 4 presents the average results for the effect of texture adaptation on roughness perception. The repeated measures ANOVA revealed a significant main effect of adaptation condition (F 2,18 = 23.2, P < 0.001). As shown in the figure, adaptation to a smooth or rough stimulus resulted in negative and positive biases, respectively. The average PSEs for the two adaptation conditions were −0.09 mm and 0.15 mm, corresponding to 5.3 and 9% of the average distance between dots of the reference stimulus. The negative bias indicates that the perceived roughness of the stimulus scanned with the index finger increased after adapting the index finger to a smooth stimulus. On the other hand, the positive bias shows that adapting the index finger to a rough stimulus resulted in a decrease of the perceived roughness of a subsequently scanned stimulus. Pairwise comparison showed that this difference between the two adaptation conditions was significant at < 0.005. Furthermore, significant differences between the two adaptation conditions and the control condition were found, with < 0.05 and < 0.001 for the smooth and rough conditions, respectively.
To explore the data in more detail, the complete data set was divided into a part in which the reference stimulus was scanned with the index finger and a part in which the reference stimulus was scanned with the middle finger. A 3 (condition) × 2 (position) repeated measures ANOVA was performed on this data set, to test for significant effects of stimulus position. However, the effect of position was not significant (F 1,9 = 2.39, P = 0.16). Therefore, there was no need to distinguish between the locations of the reference stimulus in the data analysis.
The average discrimination thresholds for the control, rough adaptation and smooth adaptation conditions were 0.15 (SD 0.01), 0.27 (SD 0.07) and 0.20 mm (SD 0.03), respectively. A repeated measures ANOVA showed no significant main effect of condition on these discrimination thresholds (F 2,18 = 2.12, P = 0.15).
## Experiment 2: spatial context
The second experiment investigated the spatial contextual influences on roughness perception. A rough or smooth inducer stimulus was felt with one finger and its effect on roughness perception with an adjacent finger was examined.
### Conditions
Figure 5 shows a representation of the two conditions. In the “test–rough inducer condition”, the index finger explored a test stimulus while the middle finger of the same hand scanned a rough surface at the same time (“test–rough pair”). Next, the index finger explored a reference stimulus, while the middle finger scanned a smooth surface at the same time (“reference–smooth pair”). The perceived roughness of the test stimulus from the “test–rough pair” was compared to the perceived roughness of the reference stimulus from the “reference–smooth pair”.
In the “test–smooth inducer condition”, the reverse was presented; the test stimulus was coupled with a smooth surface (“test–smooth pair”) and compared to the reference stimulus coupled with a rough surface (“reference–rough pair”). By comparing the two conditions, the effect of the inducer stimulus on the perceived roughness of the adjacent finger can be revealed. The rough and smooth inducer stimuli had the same average distance between dots as the rough and smooth adaptation stimuli from Experiment 1. The same test and reference stimuli were also used.
The two inducer conditions were mixed within the same run, and the trials from the two different conditions were performed in a random order. The presentation order of the test and reference stimuli was also randomized; that is, the reference stimulus was felt before the test stimulus in some trials and presented in reverse order in other trials. Each condition contained 110 trials, resulting in 220 trials for the complete experiment. The experiment was performed within a single session lasting approximately 75 min.
### Procedure
The instructions for moving the fingers over the stimuli were the same as for Experiment 1; again, some practice trials preceded the experiment. First, the participant lowered the tips of the index and middle fingers onto the nearest two surfaces (see Fig. 6a). The index finger was placed onto the stimulus on the left and the middle finger onto the stimulus on the right (for the left-handed subject, the stimuli were reversed such that for the left- and right-handed subjects the same stimuli were scanned with the index and middle finger). Subsequently, participants simultaneously performed two forward and backward movements over the stimuli with the index and middle fingers (identical to the Experiment 1 test trials). Then, they raised their hand, replaced it towards the second pair, and repeated the exploration movement (Fig. 6b). Immediately after completing the second exploration, a 2AFC task was conducted; the subjects had to compare the two stimuli scanned with the index finger and say which of the two was perceived as rougher. After responding, the experimenter replaced the surfaces and another trial began.
### Analysis
The difference between the average dot distances of the test and reference stimuli was used as the independent variable. For all subjects and both conditions, we calculated for each of these differences the fraction with which the subject responded that the test surface felt rougher compared to the reference surface. The same data fitting procedure as in Experiment 1 was used. To compare the effects of the inducer stimulus on the PSE and on the discrimination thresholds, repeated measures ANOVAs were performed, with condition as the within-subject factor.
### Results
Figure 7 shows the effect of the roughness of an inducer stimulus scanned with the middle finger on the perceived roughness of a stimulus scanned simultaneously with the index finger of the same hand. The average PSEs were 0.10 mm and −0.15 mm for the smooth and rough inducer conditions, respectively. This corresponds to 6.4 and 9.2% of the average distance between dots of the reference stimulus. The repeated measures ANOVA showed a significant difference between the smooth and rough inducer conditions (F 1,9 = 16.5, P < 0.005). As seen in the Fig. 7, a smooth inducer stimulus on the middle finger caused a positive bias, meaning that the perceived roughness of the stimulus scanned simultaneously with the index finger decreased. The negative bias in the rough inducer condition indicates that the perceived roughness of the stimulus felt with the index finger increased when a rough surface was scanned simultaneously with the middle finger. These results show that for both conditions, the perceived roughness of the stimulus felt with the index finger shifted toward the roughness of the inducer stimulus. The average discrimination thresholds were 0.10 (SD 0.01) and 0.16 mm (SD 0.04) for the smooth and rough conditions, respectively. As in Experiment 1, the difference between these discrimination thresholds was not significant (F 1,9 = 2.85, P = 0.13).
## Discussion
The present study investigated the influences of temporal and spatial context on haptic roughness perception. It was found that temporal adaptation to a roughly (smoothly) textured surface resulted in a decrease (increase) of the perceived roughness of a subsequently scanned surface. Furthermore, the spatial context exerted its influence by shifting the perceived roughness of a surface towards the roughness of a simultaneously scanned inducer stimulus. These results are important for understanding the mechanisms involved in haptic roughness perception.
### Temporal effects
In the first experiment, after scanning a surface for a prolonged period of time with the index finger, participants had to discriminate between the roughness of a surface scanned with the adapted finger and the roughness of a surface scanned with an unadapted adjacent finger. The results showed a temporal context effect. Adaptation to a rough surface decreased perception of a surface scanned subsequently with the adapted finger. On the other hand, adaptation to a smoothly textured surface increased the perceived roughness of subsequently scanned surfaces.
These texture adaptation effects are in accordance with results from previous studies showing adaptation after effects in the haptic modality (Lederman et al. 1982; Van der Horst et al. 2008; Vogels et al. 2001). These studies show that adaptation to a physical dimension changes the perception of a subsequently perceived stimulus. This change is in the opposite direction to that of the adapting stimulus. They also proposed that higher levels of processing are involved.
The fact that rough and smooth adaptation resulted in opposite effects indicates that the process involved in texture adaptation is not simply a peripheral effect. If that were the case, then scanning either a smooth or rough surface for a prolonged period of time should cause the peripheral neurons to be over-stimulated, with the smooth surface producing relatively less over-stimulation. Therefore, adaptation to a smooth surface should show an effect in the same direction as adaptation to a rough surface, with only a smaller magnitude of that effect. Moreover, if it were a peripheral effect, then adaptation to a rough stimulus should disturb discrimination performance more than adaptation to a smooth stimulus. However, no significant difference between the discrimination thresholds measured in the three conditions was found, indicating that the ability of discrimination is not disturbed by adaptation. Therefore, a peripheral over-stimulation mechanism could not be the origin for the presented effect. Consequently, these findings suggest that the texture adaptation effect occurs at a higher level of processing.
Another relevant point is that adapting the index finger to a surface may modify the roughness not only of the stimulus subsequently scanned with the index finger, but also of the comparison stimulus scanned with the middle finger. Furthermore, interaction effects are possible between the signals received from the index and middle fingers when they were simultaneously scanning a stimulus during the test phase. These confounding factors can result in a decrease of the biases. However, the present experiment revealed highly significant effects regardless of these confounding factors. This shows that the presented effects are quite robust.
The results of this study indicate that the spatial pattern is already processed further before the effect is manifested. The neurons that code for roughness magnitude likely adapt to the roughness of the scanned surface. This finding can be explained by structures of the receptive fields of neurons in the somatosensory cortex. As stated in “Introduction”, it has been shown that cells in the somatosensory and visual cortex have comparable receptive field structures (Dicarlo et al. 1998; Hubel and Wiesel 1962). Visual cortex cells show strong adaptation effects (e.g. Blakemore et al. 1973; Carandini et al. 1997; Dragoi et al. 2000; Jones and Palmer 1987). Therefore, we suggest that if the somatosensory cells are stimulated with appropriate stimuli, they should show comparable adaptation effects, and texture adaptation effects on roughness perception should be found. This was indeed the case. Furthermore, the correlation between our findings and those of Dicarlo et al. (1998) implies that our stimuli, random dot patterns with relatively large distances between dots, are appropriate stimuli for these adaptation neurons. Probably, these neurons do not respond in the same way to patterns with smaller dot distances, as those used by Hollins et al. (2006), or these patterns are too weak to cause significant adaptation effects. In general, the texture adaptation effect presented here supports the argument that visual and haptic modalities have similar structures and functions.
### Spatial effects
The second experiment was based on the spatial influences of the context during haptic roughness perception. Participants had to discriminate between the roughness of two successively scanned surfaces while scanning a smooth or rough surface with an adjacent finger. The results showed that the perceived the roughness of a surface scanned with the index finger changed in the direction of the inducer stimulus; e.g. a smooth surface felt smoother (rougher) when perceived in the context of a smooth (rough) stimulus.
This spatial contextual effect supports findings from neurophysiological studies, which show that integration of information received from different fingers occurs along the processing pathway (Biermann et al. 1998; Forss et al. 1995; Gandevia et al. 1983; Iwamura et al. 1983). In addition, the present results show that this integration effect is also visible when natural stimuli are used and explored actively. This contrasts with the results from the study by Dorsch et al. (2001), where exploration of abrasive papers with two fingers did not result in any integration effects; however, the use of abrasive papers could have influenced their result.
The shift in perceived roughness of the adjacent stimulus resembles the visual assimilation effect, which also occurs when the appearance of the test shifts towards the appearance of the inducer (e.g. Cao and Shevell 2005; Smith et al. 2001). Some neural mechanisms are proposed to account for observed assimilation effects in the visual domain (Cao and Shevell 2005; De Weert and Van Kruysbergen 1997; Shevell and Wei 2000). One suggested mechanism is spatial averaging of the neural signals in combination with the size of the receptive fields. During presentation of stimuli composed of a test and inducer rings, only the stimuli containing smaller inducer rings results in assimilation. It has been proposed that if neural spatial summation occurs in the centres of the centre-surround receptive fields and the inducer rings are small enough to fall within the centre of the receptive field that also registers the test stimulus, then an assimilation effect will occur. The spatial contextual effect in haptic roughness perception could be explained by a comparable mechanism in which signals from the index finger and from the inducer middle finger both fall within the centre of the same receptive field, producing the assimilation effect. These receptive fields could be the multi-finger receptive fields that were found at the level of the somatosensory cortex where integration of information received from different fingers occurs (Biermann et al. 1998; Fitzgerald et al. 2006; Forss et al. 1995; Iwamura et al. 1983; Sinclair and Burton 1993).
## Conclusion
The results from the present two experiments show strong effects of context during haptic perception of roughness. Temporal adaptation causes roughness perception to shift away from the roughness of the adaptation stimulus (i.e. contrast after effect), while simultaneous stimulation of the fingers causes the perception to shift towards the adjacent stimulus (i.e. assimilation effect). Although these effects seem contradictory, we can explain them using comparable mechanisms. We suggest that these effects do not manifest themselves at a lower, peripheral level of processing, but rather that high-level mechanisms are involved. Structures of the cortical receptive fields are proposed as an explanation for the temporal as well as spatial contextual effects. The analogies with comparable effects in the visual system emphasize the similarities of the different modalities. | {"extraction_info": {"found_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": 0.8026623129844666, "perplexity": 1633.2950520162556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296944452.97/warc/CC-MAIN-20230322211955-20230323001955-00449.warc.gz"} |
https://hal-ensta-paris.archives-ouvertes.fr/UPMC/hal-00331026v1 | Empirical study of multifractal phase transitions in atmospheric turbulence - Archive ouverte HAL Access content directly
Journal Articles Nonlinear Processes in Geophysics Year : 1994
## Empirical study of multifractal phase transitions in atmospheric turbulence
François G Schmitt
D Schertzer
S. Lovejoy
• Function : Author
Y. Brunet
• Function : Author
#### Abstract
We study atmospheric wind turbulence in the framework of universal multifractals, using several medium resolution (10 Hz) time series. We cut these original time series into 704 scale invariant realizations. We then compute the moment scaling exponent of the energy flux K(q) for 4 and 704 realizations, in order to study qualitative difference between strong and weak events associated with multifractal phase transitions. We detect a first order multifractal phase transition of the energy flux at statistical moment of order qD ˜ 2.4 ± 0.2: this means that when the number of realizations increases, moments order q =; qD diverge. These results are confirmed by the study of probability distributions, and wind structure functions. A consequence of these findings is that it is no use to compare different cascade models in turbulence by using the high order wind structure functions, because a linear part will always be encountered for high enough order moments. Another important implication for multifractal studies of turbulence is that the asymptotic slope of the scaling moment function is purely a function of sample size and diverges with it; it implies the same for D8, which has often be considered as finite.
### Dates and versions
hal-00331026 , version 1 (01-01-1994)
### Identifiers
• HAL Id : hal-00331026 , version 1
• DOI :
### Cite
François G Schmitt, D Schertzer, S. Lovejoy, Y. Brunet. Empirical study of multifractal phase transitions in atmospheric turbulence. Nonlinear Processes in Geophysics, 1994, 1 (2/3), pp.95-104. ⟨10.5194/npg-1-95-1994⟩. ⟨hal-00331026⟩
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https://stats.stackexchange.com/questions/442519/covariance-in-2-dimensional-field | # covariance in 2 dimensional field
I want to know is the covariance equation which is shown below is how valid when we are in 2-dimensional fields (not 2-dimensional data)?
$$Cov(x,y)=\frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-y)}{N-1}$$
for instance, suppose we want to find COV(x,y) given two elements of x and y:
x_samples={ [[1,2],[3,4]] , [[5,6],[7,8]] }
y_samples={ [[9,10],[11,12]] , [[13,14],[15,16]] }
can I use the above equation for this space?
This way unlike scaler space $$Cov(x,y)$$ is not equal to $$Cov(y,x)$$. is it true?
• Please tell us what you mean by "2-dimensional fields." Your bracket notation is not standard mathematical notation. – whuber Dec 28 '19 at 21:42
• suppose u have scaler numbers of x samples, {1,2}, and y samples, {3,4}, then you can compute COV(x,y) easily, but what if we have 2dimentional matrix instead of each sample ( 1, 2, 3, 4)? like the above samples I mentioned as an example. can we use same COV formula? – sakht Dec 30 '19 at 9:10
• Could you explain what this two-dimensional matrix represents? – whuber Dec 30 '19 at 14:32
• u can imagine that we have to class of images; x and y, and we have for each class 2 sample images, and each sample is a 2*2 matrix, like above, so I want to find covariance of these two classes. – sakht Jan 1 '20 at 8:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8666737079620361, "perplexity": 876.0699648076819}, "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/1627046154878.27/warc/CC-MAIN-20210804142918-20210804172918-00361.warc.gz"} |
https://www.zigya.com/study/book?class=12&board=CBSE&subject=Physics&book=Physics+Part+I&chapter=Electric+Charges+and+Fields&q_type=&q_topic=&q_category=B&question_id=PHEN12048030 | ## Chapter Chosen
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The electrostatic force on a small sphere of charge 0.4 μC due to another small sphere of charge –0.8 μc in air is 0.2 N.
(a) What is the distance between the two spheres?
(b) What is the force on the second sphere due to the first?
(a) Given,
Now using the formula,
Hence,
where, r is the distance between two spheres.
(b) Force on the second sphere due to the first is same, i.e., 0.2 N because the charges in action are same and force is attractive as charges are unlike in nature.
1466 Views
a) Explain the meaning of the statement 'electric charge of a body is quantised'.
b) why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges?
a) Quantisation of Electric Charges mean the total electric charge(q) of a body is always an integral multiple of a basic quantum charge(e).
i.e., $\mathrm{q}=±\mathrm{ne}$
Here +e is taken as charge on a proton while –e is taken as charge on an electron. The charge on a proton and an electron are numerically equal i.e., 1.6 x 10–19 C but opposite in sign.
“Quantisation is a property due to which charge exists in discrete packets in multiple of
± 1.6 x 10–19 rather than in continuous amounts.”
b) Based on many practical phenomena, we may ignore quantisation of electric charge and consider the charge to be continuous. In a macroscopic scale the number of charges used is enormous as compared to the magnitude of charge. The “graininess” of charge is lost and it appears continuous and therefore quantisation of charge becomes insignificant.
1767 Views
When a glass rod is rubbed with a silk cloth, charges appear on both. A similar phenomenon is observed with many other pairs of bodies. Explain how this observation is consistent with the law of conservation of charge.
Law of conservation of charge states that total charge on an isolated system of objects always remain conserved.
When a glass rod is rubbed with silk cloth, glass rod becomes positively charged while silk cloth becomes negatively charged and, the amount of positive charge on the glass rod is apparently found to be exactly the same as the negative charge on silk cloth. Since, the measure of charge is same on both, equal amount of charge with opposite nature will cancel out each other. Hence, the total sum of charge on two bodies is zero. Thus, the system of glass rod and silk cloth, which was neutral before rubbing, still possesses no net charge after rubbing. And law of conservation of charge is justified.
As a consequence of conservation of charge, when two charged conductors of same size and same material carrying charges Q1 and Q2 respectively are brought in contact and separated, the charge on each conductor will be $\frac{{\mathrm{Q}}_{1}+{\mathrm{Q}}_{2}}{2}.$
This condition, however, does not hold true if the conductors are of different sizes or of different material.
In that case the charges on the conductors will be Q1and Q2’ respectively, where Q1 + Q2 = Q1‘ + Q2’.
1473 Views
Check that the ratio $\frac{{\mathrm{ke}}^{2}}{{\mathrm{Gm}}_{\mathrm{e}}{\mathrm{m}}_{\mathrm{p}}}$is dimensionless. Look up a table of physical constants and determine the value of this ratio. What does the ratio signify?
and
Now,
It also establishes that the electrostatic force is about 10
39 times stronger than the gravitational force.
1110 Views
# What is the force between two small charged spheres having charges of 2 x 10–7 C and 3 x 10–7C placed 30 cm apart in air?
Given,
where, r is the distance between two charges.
Using the formula,
$\therefore$ $\overline{)\mathrm{F}=\frac{1}{4{\mathrm{\pi \epsilon }}_{0}}\frac{{\mathrm{q}}_{1}{\mathrm{q}}_{2}}{{\mathrm{r}}^{2}}}$
Repulsive in nature since both charges are positive.
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https://www.semanticscholar.org/paper/Approximating-conditional-distributions-Chiarini-Cipriani/712ca37f6d99fc9f73c155a2bd746b879d1abb9e | Corpus ID: 119323391
# Approximating conditional distributions
@article{Chiarini2017ApproximatingCD,
title={Approximating conditional distributions},
author={Alberto Chiarini and Alessandra Cipriani and Giovanni Conforti},
journal={arXiv: Probability},
year={2017}
}
• Published 2017
• Mathematics
• arXiv: Probability
In this article, we discuss the basic ideas of a general procedure to adapt the Stein-Chen method to bound the distance between conditional distributions. From an integration-by-parts formula (IBPF), we derive a Stein operator whose solution can be bounded, for example, via ad hoc couplings. This method provides quantitative bounds in several examples: the filtering equation, the distance between bridges of random walks and the distance between bridges and discrete schemes approximating them… Expand
2 Citations
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• Mathematics
• 2009
We provide an overview of some recent techniques involving the Malliavin calculus of variations and the so-called Stein's method'' for the Gaussian approximations of probability distributions.Expand
A bound for the error in the normal approximation to the distribution of a sum of dependent random variables
This paper has two aims, one fairly concrete and the other more abstract. In Section 3, bounds are obtained under certain conditions for the departure of the distribution of the sum of n terms of aExpand
On Stein's method for multivariate normal approximation
The purpose of this paper is to synthesize the approaches taken by Chatterjee-Meckes and Reinert-R\"ollin in adapting Stein's method of exchangeable pairs for multivariate normal approximation. TheExpand
Reciprocal classes of random walks on graphs
• Mathematics
• 2015
The reciprocal class of a Markov path measure is the set of all mixtures of its bridges. We give characterizations of the reciprocal class of a continuous-time Markov random walk on a graph. Our mainExpand
Stein’s method on Wiener chaos
• Mathematics
• 2007
We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general GaussianExpand
Stein's method and poisson process convergence
Stein's method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than theirExpand
Poisson Approximation for Dependent Trials
by a Poisson distribution and a derivation of a bound on the distance between the distribution of W and the Poisson distribution with mean E(W ). This new method is based on previous work by C. SteinExpand
ON THE APPROXIMATE NORMALITY OF EIGENFUNCTIONS OF THE LAPLACIAN
The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If X is a randomExpand
A note on the extremal process of the supercritical Gaussian Free Field
• Mathematics
• 2015
We consider both the infinite-volume discrete Gaussian Free Field (DGFF) and the DGFF with zero boundary conditions outside a finite box in dimension larger or equal to 3. We show that the associatedExpand | {"extraction_info": {"found_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": 0.9193395376205444, "perplexity": 702.7061684132127}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780057496.18/warc/CC-MAIN-20210924020020-20210924050020-00154.warc.gz"} |
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