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http://chemwiki.ucdavis.edu/index.php?title=Physical_Chemistry/Quantum_Mechanics/Quantum_Theory/Trapped_Particles/Particle_in_a_1-dimensional_box&revision=66 | More
# Particle in a 1-dimensional box
Version as of 17:10, 19 May 2013
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A particle in a 1-dimensional box is a fundamental quantum mechanical approximation describing the translational motion of a single particle confined inside an infinitely deep well from which it cannot escape.
### Introduction
The particle in a box problem is a common application of a quantum mechanical model to a simplified system consisting of a particle moving horizontally within an infinitely deep well from which it cannot escape. The solutions to the problem give possible values of E and
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that the particle can possess. E represents allowed energy values and
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is a wavefunction, which when squared gives us the probability of locating the particle at a certain position within the box at a given energy level.
To solve the problem for a particle in a 1-dimensional box, we must follow our Big, Big recipe for Quantum Mechanics:
1. Define the Potential Energy, V
2. Solve the Schrödinger Equation
3. Define the wavefunction
4. Define the allowed energies
### Step 1: Define the Potential Energy V
The potential energy is 0 inside the box (V=0 for 0<x<L) and goes to infinity at the walls of the box (V=∞ for x<0 or x>L). We assume the walls have infinite potential energy to ensure that the particle has zero probability of being at the walls or outside the box. Doing so significantly simplifies our later mathematical calculations as we employ these boundary conditions when solving the Schrödinger Equation.
### Step 2: Solve the Schrödinger Equation
The time-independent Schrödinger equation for a particle of mass m moving in one direction with energy E is
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where
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is the reduced Planck Constant and is equal to {{ math.formula("\\hbar = 1{\\dfrac{h}{2\\pi}}") }}
m is the mass of the particle
{{ math.formula("\\psi(x)") }} is the stationary time-independent wavefunction
V(x) is the potential energy as a function of position
E is the energy, a real number
This equation can be modified for a particle of mass m free to move parallel to the x-axis with zero potential energy (V = 0 everywhere) resulting in the quantum mechanical description of free motion in one dimension:
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This equation has been well studied and gives a general solution of:
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where A, B, and k are constants
### Step 3: Define the wavefunction
The solution to the Schrodinger equation we found above is the general solution for a 1-dimensional system, we now need to apply our boundary conditions to find the solution to our particular system.
The probability to find the particle at x=0 or x=L is zero, remember? So, when x=0 sin(0)=0 and cos(0)=1; therefore, B must equal 0 to fulfill this boundary condition giving:
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We can now solve for our constants (A and k) systematically to define the wavefunction.
Solving for k
Differentiate the wavefunction with respect to x:
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Since
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, then
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If we then solve for k by comparing with the Schrödinger equation above, we find:
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Now we plug k into our wavefunction:
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Solving for A
To determine A, we have to apply the boundary conditions again. Remember that the probability of finding a particle at x = 0 or x = L is zero.
When x = L:
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This is only true when
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, where n = 1,2,3…
Plugging this back in gives us:
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To determine A, remember that the total probability of finding the particle is 100% or 1. When we find the probability and set it equal to 1, we are normalizing the wavefunction.
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For our system, the normalization looks like:
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Using the solution for this integral from an integral table, we find our normalization constant, A:
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Which results in the normalized wavefunction for a particle in a 1-dimensional box:
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### Step 4: Determine the Allowed Energies
Solving for E results in the allowed energies for a particle in a box:
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This is a very important result; it tells us that
1. The energy of a particle is quantized
2. The lowest possible energy is NOT zero
This is also consistent with the Heisenberg Uncertainty Principle: if the particle had zero energy, we would know where it was in both space and time.
### What does all this mean?
The wavefunction for a particle in a box at the n=1 and n=2 energy levels look like this:
The probability of finding a particle a certain spot in the box is determined by squaring Psi. The probability distribution for a particle in a box at the n=1 and n=2 energy levels look like this:
### Important Facts to Learn from the Particle in the Box
• The energy of a particle is quantized
• The lowest possible energy for a particle is NOT zero (even at 0 K)
• The square of the wavefunction is related to the probability of finding the particle in a specific position
• The probability changes with increasing energy of the particle and depends on where in the box you look
• In classical physics, the probability of finding the particle is independent of the energy and the same at all points in the box
### Questions
1. Draw the wave function for a particle in a box at the n = 4 energy level.
2. Draw the probability distribution for a particle in a box at the n = 3 energy level.
3. What is the probability of locating a particle of mass m between x = L/4 and x = L/2 in a 1-D box of length L? Assume the particle is in the n=1 energy state.
4. Calculate the electronic transition energy of acetylaldehyde (the stuff that gives you a hangover) using the particle in a box model. Assume that aspirin is a box of length 300 pm that contains 4 electrons.
5. Suggest where along the box the n=1 to n=2 electronic transition would most likely take place.
Provides a live quantum mechanical simulation of the particle in a box model and allows you to visualize the solutions to the Schrödinger Equation: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9681729674339294, "perplexity": 902.3741571958429}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368698150793/warc/CC-MAIN-20130516095550-00062-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/electric-potential-problem.154926/ | # Electric potential problem
1. Feb 6, 2007
### debwaldy
1. The problem statement, all variables and given/known data
derive an expression for the electrical potential at a distance x measured along the axis from the centre of a circular ring of radius R on which a charge Q is uniformly distributed.hence derive an expression for the electric field strength at this point
2. Relevant equations
to be honest im not too sure where to go after:
V= -⌠E.dl
any guidance at all would be much appreciated!!!thanks
3. The attempt at a solution
2. Feb 6, 2007
### denverdoc
It looks like a start, what if the "ring" were simply two charges, each of 1/2Q, and -r,0 and r,0 relative to a test charge at 0,x. What would that look like?
3. Feb 6, 2007
### Mr.4
This simply involves a lil integration.
Q is distributed uniformly throughout the ring. This linear charge denstity on it would be lambda= Q/2pi*R.
Let dV due to each infinitesimally small element (dl) on the ring = (1/4pi*epsilon)*(lambda*dl/(x^2+R^2)^1/2.
Then just integrate from 0 to 2pi*R.
4. Feb 6, 2007
### debwaldy
oh i see, so its ok to treat them as 2 separate point charges and then sum the electric potentials at the end.
doing this i got an expression:
V= - Q/4*pi*ε(x^2 + R^2)^1/2
and E= dV/dx
E = - [Qx]/[4*pi*ε(x^2 + r^2)^3/2]
5. Feb 6, 2007
### denverdoc
no, what i meant was look at it first as 2 point charges, then 4, then an infinite number spread around the ring, but Mr 4 points the way I was hinting at in post above directly.
6. Feb 6, 2007
### Mr.4
Quite so. I think your logic would be more useful in some questions. Thanks for that denverdoc.
7. Feb 6, 2007
### denverdoc
Still getting the hang of helping without doing the work, ie trying to help posters conceptualize w/o telling them how to pursue directly. Sometimes I think I just add to the confusion:grumpy:
J
8. Feb 7, 2007
### Mr.4
Nope. I get you loud and clear! Maybe its coz I'm just as confusing!
Similar Discussions: Electric potential 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": 0.9042513370513916, "perplexity": 2059.193343542757}, "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/1516084886952.14/warc/CC-MAIN-20180117173312-20180117193312-00772.warc.gz"} |
https://www.clutchprep.com/chemistry/practice-problems/143386/determine-whether-a-given-substance-is-a-mixture-a-homogeneous-mixture-a-heterog-2 | # Problem: A blood sample contains several different types of cells, each of which contains different combinations of specific ions and molecules. The cells are suspended in water, and the water also contains many molecules and ions. Which term or terms could be used to describe this sample of blood?Matter can be classified into several categories. All substances are either pure chemical substances or mixtures. Mixtures can be separated into their component parts through physical means, whereas pure chemical substances require chemical or nuclear reactions to separate them.Pure chemical substances can be further divided into elements or compounds.Mixtures can be either heterogeneous or homogeneous. Homogeneous mixtures are also called solutions
###### Problem Details
A blood sample contains several different types of cells, each of which contains different combinations of specific ions and molecules. The cells are suspended in water, and the water also contains many molecules and ions. Which term or terms could be used to describe this sample of blood?
Matter can be classified into several categories. All substances are either pure chemical substances or mixtures.
Mixtures can be separated into their component parts through physical means, whereas pure chemical substances require chemical or nuclear reactions to separate them.
Pure chemical substances can be further divided into elements or compounds.
Mixtures can be either heterogeneous or homogeneous. Homogeneous mixtures are also called solutions | {"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": 0.8173271417617798, "perplexity": 1005.8552308964199}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107890586.57/warc/CC-MAIN-20201026061044-20201026091044-00716.warc.gz"} |
https://mathhelpboards.com/threads/analogy-between-long-division-and-polynomial-long-division.6476/ | # [SOLVED]Analogy Between Long Division and Polynomial Long Division
• #1
#### Ackbach
##### Indicium Physicus
Staff member
Jan 26, 2012
4,202
Hopefully, anyone who has studied polynomial long division understands the link between it and regular long division. If you divide $58$ into $302985$, you could follow the usual long division procedure and obtain the answer. Alternatively, if you divide $x+4$ into $x^{3}+2x^{2}+6x+7$, you could use the exact same math to perform this division. However, polynomial long division offers one feature that I am not sure has its analogy in regular long division: minus signs. I can also divide $x-4$ into $x^{3}+2x^{2}+6x+7$ or $x+4$ into $x^{3}-2x^{2}+6x-7$. My question is this: is there an analogy in regular long division for minus signs in polynomial long division?
#### Klaas van Aarsen
##### MHB Seeker
Staff member
Mar 5, 2012
9,593
Hopefully, anyone who has studied polynomial long division understands the link between it and regular long division. If you divide $58$ into $302985$, you could follow the usual long division procedure and obtain the answer. Alternatively, if you divide $x+4$ into $x^{3}+2x^{2}+6x+7$, you could use the exact same math to perform this division. However, polynomial long division offers one feature that I am not sure has its analogy in regular long division: minus signs. I can also divide $x-4$ into $x^{3}+2x^{2}+6x+7$ or $x+4$ into $x^{3}-2x^{2}+6x-7$. My question is this: is there an analogy in regular long division for minus signs in polynomial long division?
When you do a long division with regular numbers, you are converting the result to decimal form: $\sum a_i 10^i$.
By convention all $a_i$ are between 0 and 9.
We don't have to do it that way though.
We can also use a different base than 10, say 16.
Or we can choose that all $a_i$ should be between -5 and +4.
That way you'll get a different representation of the result, similar to doing polynomial division.
Note that if you define x=10 in polynomial division, and also constrain the factors to be between 0 and 9, you get regular long division.
#### agentmulder
##### Active member
Feb 9, 2012
33
If you allow me to lose the place value in the answer then yes , i can make make it happen.
$\ \ \$ 6000 - 900 + 100 + 20 + 5 - 1
_________ ________ ________ ________ ________ _____
58 | 302985
-(348000)
________________ ___________ ________ ___
-45015
-(-52200)
______________ ______________ ______
7185
-(5800)
____ __________ ____________ ____
1385
-(1160)
_______ ____________ ___________ ______
225
-(290)
____ _______ _____________ ______ _____
-(65)
-(-58)
_________ _______________ ________
-7
You can stop now if you're bored or you can continue for as long as you like , what i have done is make a guess at each step until the subtraction produced a number that is obviously a remainder , -7. It doesn't matter if i go 'over' or 'under' at any step AND it doesn't matter by how much i am in error , the rules of arithmetic (and algebra indirectly) gaurantee the result is true.
$6000 - 900 + 100 + 20 + 5 - 1 - \frac{7}{58} = 5224 - \frac{7}{58} = 302985 ÷ 58$
#### agentmulder
##### Active member
Feb 9, 2012
33
An easier example that doesn't use too many numbers and operations would be to do 9 ÷ 7
$\ \ \$ 2 - 1
$\ \ \$ _______
7 | 9
-(14)
__________
-5
-(-7)
______
2
So the answer is $\ \ 2 - 1 + \frac{2}{7} = 1 + \frac{2}{7}$
Note that we can stop after the first 'guess' of 2 and we would get
$2 - \frac{5}{7}$
which is the correct answer. So at every step the algorithm produces a correct answer ... in my previous post as well ... it must be , otherwise we would not get the correct answer in the end because each subsequent calculation depends on the previous calculation , but some answers are more useful than others , depending on what you like or are looking for a particular representation of the same correct answer.
• #5
#### Ackbach
##### Indicium Physicus
Staff member
Jan 26, 2012
4,202
When you do a long division with regular numbers, you are converting the result to decimal form: $\sum a_i 10^i$.
By convention all $a_i$ are between 0 and 9.
We don't have to do it that way though.
We can also use a different base than 10, say 16.
Or we can choose that all $a_i$ should be between -5 and +4.
That way you'll get a different representation of the result, similar to doing polynomial division.
Note that if you define x=10 in polynomial division, and also constrain the factors to be between 0 and 9, you get regular long division.
If you allow me to lose the place value in the answer then yes , i can make make it happen.
$\ \ \$ 6000 - 900 + 100 + 20 + 5 - 1
_________ ________ ________ ________ ________ _____
58 | 302985
-(348000)
________________ ___________ ________ ___
-45015
-(-52200)
______________ ______________ ______
7185
-(5800)
____ __________ ____________ ____
1385
-(1160)
_______ ____________ ___________ ______
225
-(290)
____ _______ _____________ ______ _____
-(65)
-(-58)
_________ _______________ ________
-7
You can stop now if you're bored or you can continue for as long as you like , what i have done is make a guess at each step until the subtraction produced a number that is obviously a remainder , -7. It doesn't matter if i go 'over' or 'under' at any step AND it doesn't matter by how much i am in error , the rules of arithmetic (and algebra indirectly) gaurantee the result is true.
$6000 - 900 + 100 + 20 + 5 - 1 - \frac{7}{58} = 5224 - \frac{7}{58} = 302985 ÷ 58$
An easier example that doesn't use too many numbers and operations would be to do 9 ÷ 7
$\ \ \$ 2 - 1
$\ \ \$ _______
7 | 9
-(14)
__________
-5
-(-7)
______
2
So the answer is $\ \ 2 - 1 + \frac{2}{7} = 1 + \frac{2}{7}$
Note that we can stop after the first 'guess' of 2 and we would get
$2 - \frac{5}{7}$
which is the correct answer. So at every step the algorithm produces a correct answer ... in my previous post as well ... it must be , otherwise we would not get the correct answer in the end because each subsequent calculation depends on the previous calculation , but some answers are more useful than others , depending on what you like or are looking for a particular representation of the same correct answer.
Excellent! So the analogy was always there under my very nose; but because regular long division is hardly ever taught this way (ever taught this way?), it would not be obvious unless you go looking for it. This relieves the tension in my mind between the two; it seemed to me that the analogy had to be exact! You really are doing the same math.
#### agentmulder
##### Active member
Feb 9, 2012
33
You can also follow 'THEIR' rules about keeping place value and positive remainders until you decide to break the rules , Kobayashi Maru style.
1156432 ÷ 7
I'm going to follow the rules by putting 7 into 11 once then 7 into 45 6 times then 7 into 36 5 times , then break the rules and start 'guessing'
So far i've got this ,
$\ \ \ \ \ \$ 165' + 200 + 5
$\ \ \$ __________ ________
7 | 1156432
$\$ -(1155000)
$\ \$ ________ ________ ___
$\ \ \ \ \ \ \ \ \ \$ 1432
$\ \ \ \ \ \ \ \$ -(1400)
$\ \$ ________ ________ ________
$\ \ \ \ \ \ \ \ \ \ \ \ \ \$ 32
$\ \ \ \ \ \ \ \ \ \ \$ -(35)
$\ \$ ________ ________ ________
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ -3
That 165 represents 165000 because i'm 'honoring' place value , now start the guesswork and break the rule , any way you like ... i make a 'guess' that 7 will go into 1432 200 times then subtract 1400 and then 'guess' 7 will go into 32 5 times so that i can break the other rule about 'positive remainders'
Puting it all together gives $165000 + 200 + 5 - \frac{3}{7} = 165205 - \frac{3}{7}$
I could have made different 'guesses' , more wildly inaccurate 'guesses' , the procedure would have been longer but if do my arithmetic correctly , every step is a valid representation of the answer.
We could put a mark on 165 to let us know we're breaking the rule after that point and make a note
165' = 165000
as i have done above. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8398934602737427, "perplexity": 829.6850171808982}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585265.67/warc/CC-MAIN-20211019105138-20211019135138-00083.warc.gz"} |
https://thosgood.com/blog/2020/03/29/simplicial-chern-weil.html | Simplicial Chern-Weil theory
29th March, 2020
This week just gone I uploaded two preprints to the arXiv:
Both have been extracted from my PhD thesis (which I’ve just sent off to the referees) and contain about 90% of the main mathematical content of my thesis, but with about 90% fewer inane footnotes and digressions. There are also a few appendices in my thesis which explain the background of some of the subjects in a bit more detail, which I think are quite nice, but I’ll probably turn the good ones into blog posts at some point in the coming months.
So what are these two papers about? And why is it split into two parts?
The main goal
In 1980, H.I. Green wrote their thesis “Chern classes for coherent sheaves”, which, as the title so succinctly implies, gave a construction of Chern classes of coherent (analytic) sheaves. What was special about this construction, however, was that it used simplicial methods to mirror the classical Chern-Weil construction of characteristic classes, which shouldn’t really be possible, at a first glance.
As a brief recap, for anybody who doesn’t know what some of these words mean:
• The idea of Chern-Weil theory is that you take a connection on a vector bundle, calculate its curvature, and then take the trace, to end up with something that you can show really is a characteristic class.
• Connections are things from differential geometry and Lie theory that have a lot of interpretations, so just go check out some other source to learn about these (these notes are a really nice introduction).
• Characteristic classes are classes in cohomology that “describe” principal G-bundles, and common examples include Chern classes, Stiefel-Whitney classes, Pontryagin classes, and Euler classes (with each one corresponding to a certain choice of G, roughly). For homotopy/category theorists, the most familiar construction is probably that where you look at the generators of the cohomology of BG and then pull them back along the unique map to X corresponding to your principal G-bundle.
• Coherent sheaves are a type of particularly nice sheaf, and are somehow analogous to finitely-generated modules. In general, algebraic geometers like to study a space X by studying the category of sheaves on X, and coherent sheaves are some of the most well behaved. Another very nice type of sheaf is the locally free sheaf, which is somehow equivalent to a vector bundle, or principal \mathrm{GL}_n-bundle.
• The distinction between coherent analytic sheaves and coherent algebraic sheaves is the distinction between our ringed spaces: analytic geometry means we study a complex-analytic manifold with its \mathbb{C}^n-induced subspace topology, and look at holomorphic functions on it; algebraic geometry means we study a complex-analytic manifold with the Zariski topology, and look at regular functions on it (and this is where the language of schemes normally comes into play).
So the problem is that Chern-Weil theory doesn’t really work in the setting that Green was interested in: holomorphic vector bundles on complex-analytic manifolds rarely have holomorphic connections (in fact, the first Chern class is exactly the obstruction towards admitting a global holomorphic connection, and so the moment something has a non-zero first Chern class, we can’t use this method to calculate the Chern classes!). Even worse, coherent analytic sheaves do not, in general, admit resolutions by vector bundles (i.e. locally free sheaves), unlike coherent algebraic sheaves, which always do admit such resolutions.
So there are two problems with trying to do Chern-Weil theory on coherent analytic sheaves:
1. global holomorphic connections rarely exist, and
2. coherent analytic sheaves are rarely resolved by locally free sheaves.
Green managed to solve both of these problems by using everybody’s favourite abstraction of triangles: simplices!
Part I and Part II
First of all: what have I actually done, except read Green’s thesis many times over the last three-and-a-bit years? Well, although Green gives a very explicit construction, he doesn’t really say too much about why it works, or how to generalise it to get an abstract framework, and these are the two main points that I’ve tried to work on.
There are two papers because they have different goals: Part I aims to introduce some definitions about what a simplicial connection should really be, as well as proving that it satisfies a bunch of nice properties, and that Green’s construction is a specific example of this slightly more abstract notion; Part II, on the other hand, is a lot more hands-on, and applies the abstract nonsense of Part I to show that Chern-Weil theory ‘works’, as well as showing some calculations (done ‘manually’, without Green’s construction) that are proved to explicitly agree with what Green gives (but that also contain some extra data: they are lifts of closed elements in the de Rham complex (i.e. those which Green constructs) to closed elements in the Čech-de Rham bicomplex).
In a sense, Part I deals with the formalities of the construction, and Part II shows how it works on vector bundles, which, justified by some things in Part I, is enough to show us how it works for coherent analytic sheaves.
Simplicial connections
Building simplicial connections is somehow quite easy, once you get past all the fiddly notation. Indeed, they are basically connections on each simplicial degree of a locally free sheaf on the nerve that satisfy some sort of gluing condition. Of course, we need to define what a ‘locally free sheaf on the nerve’ is, and we do that too — these things have previously been called simplicial sheaves, but since they are not simplicial objects in the category of sheaves (I mean, if they were to be anything, they would be _co_simplicial for a start), I was pretty careful to give them a less misleading (albeit much less catchy) name.
Then we can define some particularly nice families simplicial connections, namely admissible simplicial connections. These are defined so that we can apply (‘generalised’) invariant polynomials (such as the trace) to their curvatures and get something well defined.
Finally, there is an even nicer type of simplicial connection, which are those that are generated in degree zero. We show that, for sufficiently nice complexes of locally free sheaves on the nerve, being generated in degree zero does indeed imply admissibility. This is nice because simplicial connections that are generated in degree zero are really so much nicer to work with than arbitrary admissible ones.
Simplicial resolutions
Green used objects called holomorphic twisting resolutions, pioneered by Toledo and Tong, as well as one of his supervisors (O’Brian), to construct resolutions of coherent analytic sheaves by locally free sheaves on the nerve. This can be seen almost as a strictification result: twisting cochains (the things that make up a twisting resolution) are the data of a bunch of homotopies that hold up to higher homotopies that themselves hold up to higher homotopies… and so on; what Green constructs from such a thing is actually a nice complex of ‘simplicial sheaves’ that satisfies a bunch of really useful properties.
What is shown in my thesis (and the Part I preprint) is that we can lift everything to the level of (\infty,1)-categories (presented here by relative categories and complete Segal spaces, following the work of Rezk and Barwick & Kan). This is pretty fiddly work, since Green’s construction is somehow not functorial, but we can get around this. We also need to be super careful by what we mean when we say ‘a complex of coherent sheaves’, since, in the algebraic world, this can mean one of two equivalent things: a (cochain) complex in the category of coherent sheaves, or a complex in the category of sheaves such that its internal cohomology (i.e. the cohomology of the cochain complex) consists of coherent sheaves. In the analytic world, however, these two things are not necessarily equivalent: we know that they are in low dimensions, but have no general results (as far as I can tell). Annoyingly, Green’s construction works on complexes of actually coherent sheaves, but the category that is really of interest (even in the analytic world) is the category of complexes with coherent cohomology. But we can get around this by using some clever argument found in Toën and Vezzosi’s HAG II, amongst other technical results.
In the end, we can construct an equivalence of (\infty,1)-categories that reassures us that, yes, indeed, coherent sheaves (meaning complexes with coherent cohomology) are indeed equivalent to sufficiently nice complexes of locally free sheaves on the nerve endowed with sufficiently nice simplicial connections, and so we really can apply Chern-Weil.
The rest
There are some other results and points of view in my thesis that I’d like to talk about, but I figured that the shorter I keep this post, the more chance of people actually reading it! Anyway, it’s nice to take a break from this subject for a little while, and spend some time thinking about another side project… | {"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": 0.8662975430488586, "perplexity": 503.96764187595363}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057403.84/warc/CC-MAIN-20210922223752-20210923013752-00442.warc.gz"} |
http://www.thefullwiki.org/Proportionality_(mathematics) | # Proportionality (mathematics): Wikis
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
# Encyclopedia
y is directly proportional to x.
In mathematics, two quantities are said to be proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio.
Proportion also refers to the equality of two ratios.
## Symbol
The mathematical symbol '∝' is used to indicate that two values are proportional. For example, A ∝ B.
In Unicode this is symbol U+221D..
## Direct proportionality
Given two variables x and y, y is (directly) proportional to x (x and y vary directly, or x and y are in direct variation) if there is a non-zero constant k such that
$y = kx.\,$
The relation is often denoted
$y \propto x$
or, alternatively,
$y \sim x$
and the constant ratio
$k = y/x\,$
is called the proportionality constant or constant of proportionality.
### Examples
• If an object travels at a constant speed, then the distance traveled is proportional to the time spent traveling, with the speed being the constant of proportionality.
• On a map drawn to scale, the distance between any two points on the map is proportional to the distance between the two locations that the points represent, with the constant of proportionality being the scale of the map.
### Properties
Since
$y = kx\,$
is equivalent to
$x = \left(\frac{1}{k}\right)y,$
it follows that if y is proportional to x, with (nonzero) proportionality constant k, then x is also proportional to y with proportionality constant 1/k.
If y is proportional to x, then the graph of y as a function of x will be a straight line passing through the origin with the slope of the line equal to the constant of proportionality: it corresponds to linear growth.
## Inverse proportionality
As noted in the definition above, two proportional variables are sometimes said to be directly proportional. This is done so as to contrast direct proportionality with inverse proportionality.
Two variables are inversely proportional (or varying inversely, or in inverse variation, or in inverse proportion or reciprocal proportion) if one of the variables is directly proportional with the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant. It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that
$y = {k \over x}$
The constant can be found by multiplying the original x variable and the original y variable.
Basically, the concept of inverse proportion means that as the absolute value or magnitude of one variable gets bigger, the absolute value or magnitude of another gets smaller, such that their product (the constant of proportionality) is always the same.
For example, the time taken for a journey is inversely proportional to the speed of travel; the time needed to dig a hole is (approximately) inversely proportional to the number of people digging.
The graph of two variables varying inversely on the Cartesian coordinate plane is a hyperbola. The product of the X and Y values of each point on the curve will equal the constant of proportionality (k). Since k can never equal zero, the graph will never cross either axis.
## Hyperbolic coordinates
The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that locates a point on a ray and the constant of inverse proportionality that locates a point on a hyperbola.
## Exponential and logarithmic proportionality
A variable y is exponentially proportional to a variable x, if y is directly proportional to the exponential function of x, that is if there exists a non-zero constant k
$y = k a^x.\,$
Likewise, a variable y is logarithmically proportional to a variable x, if y is directly proportional to the logarithm of x, that is if there exists a non-zero constant k
$y = k \log_a (x).\,$
## Experimental determination
To determine experimentally whether two physical quantities are directly proportional, one performs several measurements and plots the resulting data points in a Cartesian coordinate system. If the points lie on or close to a straight line that passes through the origin (0, 0), then the two variables are probably proportional, with the proportionality constant given by the line's slope. | {"extraction_info": {"found_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": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9626601338386536, "perplexity": 307.8717364049561}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1534221210249.41/warc/CC-MAIN-20180815181003-20180815201003-00180.warc.gz"} |
https://www.physicsforums.com/threads/angular-velocity-problem.716826/ | # Angular Velocity Problem
1. Oct 15, 2013
### hoxbug
1. The problem statement, all variables and given/known data
A wheel rotating about a fixed axis with a constant angular acceleration of 2.0 rad/s2 turns through 2.165 revolutions during a 2.0 s time interval. What is the angular velocity at the end of this time interval?
2. Relevant equations
I was using the kinematic Equations
$\theta$f = $\theta$i + $\omega$it + 0.5$\alpha$t2
Am i doing something wrong?
All the answers I get are way larger
I must be doing something obviously wrong but I just cant see it.
2. Oct 15, 2013 | {"extraction_info": {"found_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": 0.9429552555084229, "perplexity": 885.1915228554277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647519.62/warc/CC-MAIN-20180320170119-20180320190119-00395.warc.gz"} |
http://math.stackexchange.com/questions/19815/whats-special-about-the-greatest-common-divisor-of-a-b-and-a-b | # What's special about the greatest common divisor of a + b and a - b?
Problem:
Prove that if gcd( a, b ) = 1, then gcd( a - b, a + b ) is either 1 or 2.
From Bezout's Theorem, I see that am + bn = 1, and a, b are relative primes. However, I could not find a way to link this idea to a - b and a + b. I realized that in order to have gcd( a, b ) = 1, they must not be both even. I played around with some examples (13, 17), ...and I saw it's actually true :( ! Any idea?
Thanks,
Chan
-
+1 for showing some thought and trying something before posting. – Ross Millikan Feb 1 '11 at 4:32
The gcd of $x$ and $y$ divides any linear combination of $x$ and $y$. And any number that divides $r$ and $s$ divides the gcd of $r$ and $s$.
If you add $a+b$ and $a-b$, you get <blank>, so $\mathrm{gcd}(a+b,a-b)$ divides <blank>.
If you subtract $a-b$ from $a+b$, you get <blankity>, so $\mathrm{gcd}(a+b,a-b)$ divides <blankity>.
So $\mathrm{gcd}(a+b,a-b)$ divides $\mathrm{gcd}($<blank>,<blankity>$) =$<blankety-blank>.
(For good measure, assuming the result is true you'll want to come up with examples where you get $1$ and examples where you get $2$, just to convince yourself that the statement you are trying to prove is the best you can do).
-
@Arturo Magidin: Haha, I got it. Much simpler than I thought. Thanks a lot! – Chan Feb 1 '11 at 4:33
By the way, I can't go from gcd( a + b, a - b ) right? since this is inverse error if A then B, does not guarantee B then A. Should I prove it by contradiction instead? – Chan Feb 1 '11 at 4:36
@Chan: I'm not sure what you mean... You cannot begin by assuming that gcd(a+b,a-b) is equal to 1 or to 2, but you do not need to make any assumption. The hints above give you information about what gcd(a+b,a-b) must divide, and you have other information (remember what you are told about $a$ and $b$). All of that together should be sufficient (PEV was more explicit, and you seemed to think it was a great hint). – Arturo Magidin Feb 1 '11 at 4:46
@Arturo Magidin: All I tried to say is that, when proving if A then B. If we assume B is true, then infer A is true -> this is wrong. – Chan Feb 1 '11 at 4:46
Yes. You should never affirm the consequent. If you assume B is true and infer A is true, then you prove $B\rightarrow A$, which may be something interesting, but is not equivalent to $A\rightarrow B$. – Arturo Magidin Feb 1 '11 at 4:47
Note that $d|(a-b)$ and $d|(a+b)$ where $d = \gcd(a-b, a+b)$. So $d$ divides the sum and difference (i.e. $2a$ and $2b$).
-
Thanks for a great hint ;) – Chan Feb 1 '11 at 4:34
HINT $\rm\quad a-b\ +\ (a+b)\ i\ =\ (1+i)\ (a+b\ i)\ \$ provides a slick proof using Gaussian integers. This reveals the arithmetical essence of the matter and, hence, suggests obvious generalizations.
-
See here for a generalization using norms. – Bill Dubuque Jun 13 '11 at 16:48 | {"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": 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": 0.8659975528717041, "perplexity": 323.85628292217484}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1414119649048.35/warc/CC-MAIN-20141024030049-00071-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://proofwiki.org/wiki/Derivative_of_Logarithm_at_One | # Derivative of Logarithm at One
## Theorem
Let $\ln x$ be the natural logarithm of $x$ for real $x$ where $x > 0$.
Then:
$\displaystyle \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$
## Proof 1
L'Hôpital's Rule gives:
$\displaystyle \lim_{x \mathop \to c} \frac {f \left({x}\right)} {g \left({x}\right)} = \lim_{x \mathop \to c} \frac {f' \left({x}\right)} {g' \left({x}\right)}$
(provided the appropriate conditions are fulfilled).
Here we have:
$\ln \left({1 + 0}\right) = 0$
$D_x \left({\ln \left({1 + x}\right)}\right) = \dfrac 1 {1 + x}$ from the Chain Rule
$D_x \left({x}\right) = 1$ from Derivative of Identity Function.
Thus:
$\displaystyle \lim_{x \mathop \to 0} \frac {\ln \left({1 + x}\right)} x = \lim_{x \mathop \to 0} \frac {\left({1 + x}\right)^{-1} } 1 = \frac 1 {1 + 0} = 1$
$\blacksquare$
## Proof 2
$\displaystyle \lim_{x \mathop \to 0} \frac {\ln \left({1 + x}\right)} x$ $=$ $\displaystyle \lim_{x \mathop \to 0} \frac {\ln \left({1 + x}\right) - \ln 1} x$ subtract $\ln 1 = 0$ from the numerator, from Logarithm of 1 is 0 $\displaystyle$ $=$ $\displaystyle \left.{\dfrac {\mathrm d} {\mathrm dx} \ln x}\right \vert_{x \mathop = 1}$ Definition of Derivative at a Point $\displaystyle$ $=$ $\displaystyle \frac 1 1$ Derivative of Natural Logarithm Function $\displaystyle$ $=$ $\displaystyle 1$
$\blacksquare$
## Proof 3
Note that this proof does not presuppose Derivative of Natural Logarithm Function.
$\displaystyle \lim_{x \mathop \to 0} \frac {\ln \left({1 + x}\right)} x$ $=$ $\displaystyle \lim_{n \mathop \to \infty} \frac {\ln \left({1 + \frac 1 n}\right)} {\frac 1 n}$ $\displaystyle$ $=$ $\displaystyle \lim_{n \mathop \to \infty} n \ln \left({1 + \frac 1 n}\right)$ $\displaystyle$ $=$ $\displaystyle \lim_{n \mathop \to \infty} \ln \left({\left({1 + \frac 1 n}\right)^n}\right)$ $\displaystyle$ $=$ $\displaystyle \ln e$ Definition of Euler's Number as Limit of Sequence $\displaystyle$ $=$ $\displaystyle 1$ Natural Logarithm of e is 1
$\blacksquare$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9901279807090759, "perplexity": 302.9612936193279}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573173.68/warc/CC-MAIN-20190918003832-20190918025832-00510.warc.gz"} |
http://umj.imath.kiev.ua/article/?lang=en&article=8002 | 2018
Том 70
№ 6
# Ideals and free Pairs in the semigroup β ℤ
Protasov I. V.
Abstract
We prove that the equations ξ+x=mξ+y, x+ξ=y+mξ have no solutions in the semigroup β ℤ for every free ultrafilter ξ and every integer m∈0, 1. We study semigroups generated by the ultrafilters ξ, mξ. For left maximal idempotents, we prove a reduced hypothesis about elements of finite order in β ℤ.
English version (Springer): Ukrainian Mathematical Journal 49 (1997), no. 4, pp 635-642.
Citation Example: Protasov I. V. Ideals and free Pairs in the semigroup β ℤ // Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 573–580.
Full text | {"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": 0.8974226117134094, "perplexity": 3789.3429693487674}, "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-2018-30/segments/1531676592475.84/warc/CC-MAIN-20180721090529-20180721110529-00103.warc.gz"} |
https://ar.mathigon.org/course/intro-statistics/bootstrapping | ## قائمة المصطلحات
حدد واحدة من الكلمات الرئيسية على اليسار ...
# StatisticsBootstrapping
وقت القراءة: ~10 min
Bootstrapping is the use of simulation to approximate the value of the plug-in estimator of a statistical functional which is expressed in terms of independent observations from the input distribution . The key point is that drawing observations from the empirical distribution is the same as drawing times from the list of observations.
Example
Consider the statistical functional the expected difference between the greatest and least of 10 independent observations from . Suppose that 50 observations from are observed, and that is the associated empirical CDF. Explain how may be estimated with arbitrarily small error.
Solution. The value of is defined to be the expectation of a distribution that we have instructions for how to sample from. So we sample 10 times with replacement from , identify the largest and smallest of the 10 observations, and record the difference. We repeat times for some large integer , and we return the sample mean of these values.
By the law of large numbers, the result can be made arbitrarily close to with arbitrarily high probability by choosing sufficiently large.
Although this example might seem a bit contrived, bootstrapping is useful in practice because of a common source of statistical functionals that fit the bootstrap form: standard errors.
Example
Suppose that we estimate the median of a distribution using the plug-in estimator for 75 observations, and we want to produce a confidence interval for . Show how to use bootstrapping to estimate the standard error of the estimator.
Solution. By definition, the standard error of is the square root of the variance of the median of 75 independent draws from . Therefore, the plug-in estimator of the standard error is the square root of the variance of the median of 75 independent draws from . This can be readily simulated. If the observations are stored in a vector X, then
using Random, Statistics, StatsBase
X = rand(75)
std(median(sample(X, 75)) for _ in 1:10^5)
sd(sapply(1:10^5,function(n) {median(sample(X,75,replace=TRUE))}))
returns a very accurate approximation of .
Perhaps the most important caution regarding bootstrapping is that the bootstrap only approximates . It only approximates (where is the underlying true distribution from which the observations are sampled) insofar as we have enough observations for to approximate well.
Exercise
Suppose that is the uniform distribution on . Generate 75 observations from , store them in a vector , and compute the bootstrap estimate of , where is the standard deviation of 75 independent observations from . Use Monte Carlo simulation to directly estimate . Can the gap between your approximations of and be made arbitrarily small by using more bootstrap samples?
Solution. The gap cannot be made arbitrarily small. We would need to get more than 75 samples from the distribution to get closer to the exact value of .
X = rand(75)
std(median(sample(X, 75)) for _ in 1:10^6) # estimate T(ν̂)
std(median(rand(75)) for _ in 1:10^6) # estimate T(ν)
Bruno | {"extraction_info": {"found_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": 0.9741936922073364, "perplexity": 418.0655737690972}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964360803.6/warc/CC-MAIN-20211201143545-20211201173545-00034.warc.gz"} |
https://physics.stackexchange.com/questions/490758/what-are-connections-in-physics | What are Connections in physics?
This question arises from a personal misunderstanding about a conversation with a friend of mine. He asked me a question about the "truly nature" of spinors, i.e., he asked a question to me about what is an spinor object. After a few lines of dialogue, he asked something quite alien to me:
"So, spinors are Levi-Civita connections?"
The relationship between a mathematical object which models physical entities in field theory (a Dirac spinor for example) and a purely mathematical entity like a Levi-Civita connection, still intrigues me.
Now, today I encountered this question here:
Under what representation do the Christoffel symbols transform?
and in the second answer, the user made another relationship between field theory and connections:
"The "Christoffel symbols" are now just the components of a principal connection on that bundle, where a "connection form" is better known to physicists as a gauge field"
I'm asking this question because, from the point of view of elementary general relativity, we are taught that we need a pseudo-riemanninan manifold and a (Levi-Civita) connection to, roughly speaking, make a well-defined notion of derivative of tensor fields. From this point of view a connection is nothing more than a linear map.
So, what are Connections in physics, INDEED?
• A connection in physics is, indeed, a connection, in the strict mathematical sense. A connection has a precise definition and, in physics, we use that definition (sometimes implicitly or unbeknown to us). I'm not sure what else you want to know. – AccidentalFourierTransform Jul 10 at 1:19
• "The relationship between a mathematical object which models physical entities [...] and a purely mathematical entity like a Levi-Civita connection, still intrigues me." What do you think is the difference between a mathematical object and a "purely" mathematical one? Why are Dirac spinors less "purely" mathematical than connections? – knzhou Jul 10 at 1:23
• I just think of connections as covariant derivatives. In GR you want derivatives that are covariant under general coordinate transformations, and “adding in” Christoffel symbols accomplishes this. In gauge theories you want derivatives that are covariant under an internal gauge transformation, and “adding in” gauge fields accomplishes this. There is a level of abstraction at which these are the same thing, but when first learning GR and QFT I’m not convinced you have to understand it. – G. Smith Jul 10 at 2:06
• It wasn’t obvious to the first people doing GR and QFT. If I understand correctly, it emerged after differential geometers generalized manifolds with a tangent space at each point and manifolds with “a Lie group at each point” into the concept of a fiber bundle. – G. Smith Jul 10 at 2:06
• In any case, spinors are not connections, but there are “spin connections” for covariantly differentiating spinors. – G. Smith Jul 10 at 2:09
Connections in physics "are" the same as they are in mathematics but are usually interpreted as field potentials, with the exception of GR.
The interpretation follows naturally from the concept of a covariant derivative: local transformations of the field being studied must not change the physics involved (i.e. the Lagrangian must be invariant) so one introduces another "gauge" field which has dynamics of its own to cancel changes from the matter field's transformation.
Take the case of quantum electrodynamics: The Lagrangian (density) is $$\mathcal{L}=\overline{\psi}\left(i\gamma^\mu\partial_\mu + m\right)\psi$$
You can check that it is invariant under the transformation $$\psi\to e^{i\lambda}\psi$$ when $$\lambda$$ is a constant. To make the transformation local we "promote" $$\lambda$$ to a function, but now we have an offending term $$\overline{\psi}\gamma^\mu\partial_\mu\lambda e^{i\lambda}\psi$$! All is well if we introduce the covariant derivative $$\mathcal{D}_\mu=\partial_\mu-ie_0A_\mu$$ such that $$A_\mu\to A_\mu +\partial_\mu \lambda$$ is the corresponding transformation.
The full Lagrangian is then $$\mathcal{L}=\overline{\psi}\left(i\gamma^\mu\partial_\mu + m\right)\psi + e_0\overline{\psi}\gamma^\mu A_\mu \psi + \frac{1}{4}F^{\mu\nu}F_{\mu\nu}$$
where $$F_{\mu\nu}$$ is the electromagnetic field strength tensor introduced to account for the dynamics of the potential (photon) field $$A_\mu$$
Connections in the context of relativity are instead the gravitational field strength since in our currently accepted theory gravity is not a "gauge field" like the photon field. The identification of gravitation with spacetime curvature makes particles travel according to the geodesic equation which can recover the usual Gauss's Law for gravity in the Newtonian limit.
• "our currently accepted theory gravity is not a "gauge field" like the photon field." Well, "spin connection" $\omega$ is the "gauge field" of gauge theory of gravity, where the gauge group is spin(1,3) (double cover of Lorentz group). – MadMax Jul 10 at 14:18
• @MadMax I was not aware of that. Is the gauge theory of gravity more generally accepted than GR? What books/articles may I find more information in? – Quantumness Jul 11 at 20:59
• Here is a reference for gauge gravity: journals.aps.org/rmp/abstract/10.1103/RevModPhys.48.393 – MadMax Jul 12 at 13:39 | {"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": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8584126234054565, "perplexity": 567.205860179248}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987773711.75/warc/CC-MAIN-20191021120639-20191021144139-00003.warc.gz"} |
https://readings.owlfolio.org/a/sylvain-ruhault/ | Coming up in this year’s CCS is a paper with the provocative subtitle /dev/random is not Robust, and thanks to widespread availability of online preprints, it’s already scored quite a bit of online attention, e.g. at Bruce Schneier’s blog and on Hacker News.
I’m going to pass over the alleged attacks on the Linux kernel CSPRNG, which are adequately covered in both of the above discussions; suffice it to say that in this paper robust is a precisely defined theoretical security property, and there’s a solid case that it’s stronger than necessary to guarantee practical security. What I’m interested in is the simple and very efficient PRNG construction that is provably robust, advertised in the abstract. CSPRNG-robustness (as defined here) may not be necessary for practical security, but if you can have it essentially for free, why wouldn’t you adopt it? Unfortunately it turns out that there’s a major palmed card in their definition of a CSPRNG with input, and another one in the construction itself, rendering the algorithm useless in practice.
The first problem is that the definition of a CSPRNG with input includes a setup phase that produces a random seed which is not known to the adversary. Despite consistently calling this item a public parameter, the text of the paper makes it clear that it must not be known to the adversary, and they consider this both essential and unproblematic:
Unfortunately, it is well known that no deterministic extractor is capable to simultaneously extract good randomness from all efficiently samplable high-entropy distributions (e.g. consider nearly full entropy distribution $I$ which is random, except the first bit of $\operatorname{Extract}(I)$ is zero).
… [Therefore], we chose … to assume the existence of the setup procedure … this will allow one to consider a seeded extractor … which can now extract entropy from all high-entropy distributions. As a warning, this nice extra generality comes at a price that the [seed] is not passed to the [adversarial] distribution sampler …
But this is impossible to achieve in real life. If the seed is generated on the device that will use it, we have a chicken-and-egg problem, because initial startup is precisely when environmental randomness is least available; the authors should have been aware of actual, catastrophic exploits of systems with little or no access to randomness on first boot, e.g. Mining your Ps and Qs. If the seed is not generated on the device that will use it, then the adversary can probably learn it; in particular, if the seed for all devices of a particular model is embedded in a firmware image that anyone can download off the manufacturer’s website, you have accomplished precisely nothing.
The second problem is in the technical details of their proposed CSPRNG construction, which I will now quote verbatim.
Let $\mathbf{G}: \{0,1\}^m \to \{0,1\}^{n+l}$ be a (deterministic) pseudorandom generator where $m < n$. We use the notation $[y]_1^m$ to denote the first $m$ bits of $y \in \{0,1\}^n$. Our construction of PRNG with input has parameters $n$ (state length), $l$ (output length), and $p = n$ (sample length), and is defined as follows:
• $\operatorname{setup}:$ Output $\text{seed} = (X, X^\prime)\leftarrow\{0,1\}^{2n}$.
• $S^\prime = \operatorname{refresh}(S,I):$ Given $\text{seed} = (X, X^\prime)$, current state $S \in \{0,1\}^n$, and a sample $I \in \{0,1\}^n$, output $S^\prime = S \cdot X + I$, where all operations are over $\mathbb{F}_{2^n}$.
• $(S^\prime, R) = \operatorname{next}(S):$ Given $\text{seed} = (X, X^\prime)$ and a state $S \in \{0,1\}^n$, first compute $U = [X^\prime \cdot S]_1^m$. Then output $(S^\prime, R) = \mathbf{G}(U)$.
It’s a blink-and-you-miss-it sort of thing: by all operations are over $\mathbb{F}_{2^n}$ they are specifying that both the refresh and next steps make use of arithmetic over finite, nonprime fields (more usually referred to as $\operatorname{GF}(2^n)$ in crypto literature). $n$ has to be large—section 6.1 of the paper suggests specific fields with $n$ = 489, 579, and 705. The known techniques for this kind of arithmetic are all very slow, require large look-up tables (and therefore permit side-channel attacks), or cannot be used for a software implementation. [1] [2] This makes them impractical for a cryptographic component of an operating system that cannot rely on special-purpose hardware.
Both problems arise from the same source, namely wanting to build the algorithm around a keyed hash so that the entropy extractor is fully general, and then taking a mathematical construct with convenient theoretical but inconvenient practical properties as that hash. If we back down to the use of an unkeyed, but computationally secure, hash as the entropy extractor, we’d have to redo the analysis, but I think it ought to be possible to arrange that, without any secret unknown to the adversary, it is still computationally infeasible for the adversary to bias the entropy pool; and we could then pick that hash for its practical utility.
… Oddly enough, this hypothetical construction looks an awful lot like the existing Linux CSPRNG. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 27, "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": 0.8515177369117737, "perplexity": 696.5214470599424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647251.74/warc/CC-MAIN-20180320013620-20180320033620-00208.warc.gz"} |
http://mathhelpforum.com/discrete-math/145605-euler-path-euler-circuit-problem.html | Thread: Euler path and Euler circuit problem
1. Euler path and Euler circuit problem
For which n does $K_n$ have an Euler path but not an Euler circuit?
2. There can be but one...
Excluding the trivial edgeless case of $K_1$ we can proceed as follows:
If n is odd
Then $K_n$ has all even degrees, and so by a theorem of Euler's we have that there exist an Eulerian circuit, which by technicality admits an Eulerian path (a circuit is a kind of path).
If n is even
Well clearly $K_2$ contains an Eulerian path but not an Euler circuit. However for all even n>2 we know that they have more than 2 vertices of odd degree (for $K_n$ we have all vertices with degree n-1). For there to be an Eulerian path, we can have at MAX two vertices of odd degree, and for even n>2, this condition fails. And so $K_2$ is the only graph the has an Eulerian path but no circuit.
For more info on Eulerian paths and circuits check out How can we tell if a graph has an Euler path or circuit? | {"extraction_info": {"found_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": 0.8573634028434753, "perplexity": 303.09533405950526}, "config": {"markdown_headings": false, "markdown_code": false, "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-2017-47/segments/1510934806422.29/warc/CC-MAIN-20171121185236-20171121205236-00298.warc.gz"} |
https://www.physicsforums.com/threads/probability-question.131474/ | # Probability question
1. Sep 10, 2006
### quasar987
Show that there are $\binom{r}{k}\binom{n-1}{n-r-k}$ solutions to the equation $x_1+...+x_r=n$ for which exactly k of the r terms of the sum are nul.
There are $\binom{r}{k}$ ways of choosing which k of the r $x_i$'s are zero, and there are
$$\binom{n+(r-k)-1}{n}$$
distinc solutions to the resulting equation. What is wrong with that? If nothing, how are the two binomial coefficients equal?
2. Sep 10, 2006
### 0rthodontist
It says exactly k must be null. So once you've chosen your 0's, the remaining variables must take values >= 1.
3. Sep 10, 2006
### AKG
The whole thing doesn't look right. Treat the xi as distinguishable boxes and n as a bunch of n indistinguishable balls. The question is, how many ways can you put n indistinguishable balls into r distinct boxes if exactly k of the r boxes end up empty?
There are $\binom{r}{k}$ ways to choose which boxes will be empty. Now put your n balls in a row, and observe there are n-1 gaps between adjacent balls. Pick r-k-1 of these gaps (there are $\binom{n-1}{r-k-1} = \binom{n-1}{(n-1)-(r-k-1)} = \binom{n-1}{n-(r-k)}$ ways to do this). The bunch of balls occuring before the first chosen gap goes in the first of the boxes that wasn't chosen to be empty. The next bunch of balls (occuring between the first and second chosen gaps) goes in the next of the boxes that wasn't chosen to be empty.
4. Sep 10, 2006
### quasar987
Well spoted 0rthodontist. And AKG too for noticing that there is an error in the question. (the book really says n-r-k)
Similar Discussions: Probability question | {"extraction_info": {"found_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": 0.9736120700836182, "perplexity": 620.66298678713}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689823.92/warc/CC-MAIN-20170924010628-20170924030628-00173.warc.gz"} |
http://math.stackexchange.com/questions/830776/has-this-property-for-algebraic-structures-got-a-name | # Has this property for algebraic structures got a name?
Given a binary operation $*:M\times M\rightarrow M$. I define the extension of $*$ to the subset of $M$ in the usual way (with $A,B \subseteq M$)
$a*A:=\{a\}*A$and $A*a:=A*\{a\}$,
$$A*B:=\bigcup_{a\in A,b\in B}\{a*b\}$$
$$\forall \, c \in M, \; \forall k \in \mathbb{N} \; : \; M \underbrace{* (M * (M * \cdots *(M *}_{k \text{ } *\text{s}} \{c\}))) \subset M * \{c\}$$
• Has this property got a name?
• Are there known non-commutative (and/or non-associative) algebraic structures with this property?
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Non-commutatuve groups have this property, and I think as long as it's associative, any other structure will as well. – Jacob Bond Aug 9 '14 at 16:17
Not every property in mathematics must carry a name, only the most interesting ones. This one doesn't seem particularly deep. – Alex M. Aug 9 '14 at 17:05 | {"extraction_info": {"found_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": 0.8968991041183472, "perplexity": 702.0251495638528}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276567.28/warc/CC-MAIN-20160524002116-00061-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://crypto.stackexchange.com/questions/971/does-aes-have-any-fixed-points/973 | # Does AES have any fixed-points?
Is there any pair of 128-bit strings M and K such that AES$_K$(M) = M?
If yes, how do I go about (efficiently) finding such a pair?
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Short version: It is quite likely that a large proportion of the keys have fixed points, but I don't have any idea on how to find them.
Long version: A stochastic argument
There are $2^{128}!$ permuations of 128-bit blocks, and of these, $!2^{128}$ (this is the subfactorial) are fixpoint-free. It is known that $\lim_{N\to\infty}\frac{!N}{N!} = \frac 1e \approx 0.3679$ (and this limit is approached quite quickly), i.e. slightly more than one third of all permutations are fixpoint-free.
AES-128 selects $2^{128}$ from these $2^{128}!$ permutations. Assuming this selection behaves like a random one, about 63% of all keys have at least one fixed point.
(This argument is valid for any block cipher with sufficiently large blocks, nothing AES-specific here.)
But ...
Of course, AES (and any block cipher usable in practice) is not a random selection of permutations – for example, all permutations selected by AES are even (AES is composed of operations which each leave some bits unchanged, these are even, and compositions of even permutations are even), thus the available space is halved.
There might be some similar property which makes fixpoint-less permutations less or more likely than a random selection, or even make each key (or no key) have fixed points (both the number of fixed-point-free and non-fixed-point-free permutations are much much larger than the number of available AES keys). I don't know anything here (and as Thomas points out, knowing more might indicate a weakness in AES).
Also, this heuristic gives no way of finding these fixed points (other than brute force).
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But AES is not a random subset of the $2^{128}!$ permutations on 128 bits; for example, it generates only even permutations. How do we know it doesn't select only from the derangements? – Fixee Oct 15 '11 at 2:16
Good point. I don't know an answer here (and I'm not sure there is one). Actually I would not be too surprised if AES is made to have fewer fixed points than expected, or none at all. – Paŭlo Ebermann Oct 15 '11 at 2:23
How could it be "made to have fewer fixed points than expected" without introducing some exploitable structure? You had an idea in mind? – ByteCoin Oct 15 '11 at 2:27
Probability Distributions Related to Random Mappings by Bernard Harris from 1960 might be useful. – ByteCoin Oct 15 '11 at 2:32
@ByteCoin: No, I have no ideas. Just the space of fixed-point-free permutations is large enough that we can choose only from those without introducing some weakness by this alone. I hope we get some answers from people which know more about AES (did some cryptanalysis or read the corresponding papers). My argument here is a generic one which is valid (or not) for any block cipher of this block and key size, nothing AES-specific. (I'll edit my answer later after I've got some sleep.) – Paŭlo Ebermann Oct 15 '11 at 2:37
There is no known efficient algorithm for finding a fixed point; the best currently known algorithm is trying random keys and messages until a fixed point is hit, with average computational cost $2^{128}$ evaluation of AES.
@Fixee: the fact that AES (resp. DES) is an even permutation matters only after $2^{128}-2$ (resp. $2^{64}-2$) plaintext/ciphertext pairs are known; that explains why it is considered a non-issue in practice. – fgrieu Apr 30 '12 at 21:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8380157351493835, "perplexity": 958.5720275841799}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1397609539447.23/warc/CC-MAIN-20140416005219-00489-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/30744/one-more-question-about-decay-of-fourier-coefficients | # One more question about decay of Fourier coefficients
Let $$f=\sum f_{s}\exp(2\pi isx)\in C^{(p-1)}[0,1]$$ and $$f^{(p)}\ in\ L_2[0,1]\ \ ( \sum\left|f_{s}\right|^{2}j^{2p}<\infty )$$ Does it imply that $f_s=O(s^{-(p+\psi)})$ for some $\psi>0.5$?
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You seem to be under the impression that any complex valued sequence $(a_s)$ such that $\displaystyle\sum_s|a_s|^2$ converges must be such that $|a_s|=O(s^{-\psi})$ with $\psi>\frac12$ (and if you are not, please forgive me). But this is far from the truth for at least two reasons.
First, it is enough to ask that $|a_s|=O(s^{-1/2}(\log s)^{-\phi})$ for a large enough $\phi$ (and what large enough means here, I will let you discover), a condition which does not imply the existence of $\psi>\frac12$ such that $|a_s|=O(s^{-\psi})$.
Second, $(a_s)$ could be a lacunary sequence. Assume for instance that $a_s=0$ for every $s$ except the powers of $2$, in which case $a_s=1/\log s$. Then (I will let you check that) $\displaystyle\sum_s|a_s|^2$ converges although the condition $|a_s|=O(s^{-\psi})$ is false for every positive $\psi$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9963929653167725, "perplexity": 86.38010031775134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375094634.87/warc/CC-MAIN-20150627031814-00147-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/692146/how-to-prove-int-0-pi-frac-ln2-cos-phi-sqrt2-cos-phid-phi-frac-ln | # How to prove $\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right)$?
How can I prove the following conjectured identity? $$\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi\stackrel?=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right),\tag1$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind: $$K(x)={_2F_1}\left(\frac12,\frac12;\ 1;\ x^2\right)\cdot\frac\pi2.\tag2$$
• $$I(n)=\int_0^\pi(2+\cos\phi)^nd\phi\quad\iff\quad\int_0^\pi\dfrac{\ln(2+\cos x)}{\sqrt{2+\cos x}}dx=I'\Big(-\tfrac12\Big)$$ – Lucian Feb 27 '14 at 8:06
• With Mathematica I got $\mbox{LHS} - \mbox{RHS} = -0.169648$ – Felix Marin Mar 7 '14 at 23:12
• @FelixMarin You should use EllipticK[x^2] to represent $K(x)$ in Mathematica. See the MathWorld link in my question. – Vladimir Reshetnikov Mar 8 '14 at 0:36
• @VladimirReshetnikov $0$ k. I just checked it: $-9.15712 \times 10^{-13}$. – Felix Marin Mar 8 '14 at 4:41
• @Felix Marin that's small enough of a margin of error to be caused by numerical instability/rounding errors – frogeyedpeas Jul 19 '14 at 4:25
$\int^{\pi}_0\cos^{2k+1}\phi d\phi=0$, and $$\int^{\pi}_0\cos^{2k}\phi d\phi=\sqrt{\pi}\Gamma(k+1/2)/\Gamma(k+1)=2^{-2k}\pi\binom{2k}{k}.$$
Therefore \begin{align*} I(n) &=\int^{\pi}_0\sum_{m=0}^{\infty}2^{n-m}\binom{n}{m}\cos^m\phi~d\phi\\ &=2^n\sum_{m=0}^{\infty}2^{-m}\binom{n}{m}\int^{\pi}_0\cos^m\phi~d\phi\\ &=2^n\pi\sum_{k=0}^{\infty}2^{-2k}\binom{n}{2k}2^{-2k}\binom{2k}{k}\\ &=2^n\pi~{}_2F_1\left(\frac{1-n}2,-\frac{n}{2};1;\frac14 \right)\\ &=2^n\pi\left(\frac23\right)^{-n}~{}_2F_1\left(-n,\frac{1}{2};1;\frac23\right)\\ &=3^n\pi~{}_2F_1\left(-n,\frac{1}{2};1;\frac23\right)\\ \end{align*} Using DLMF 15.8.13 with $a=-n$, $b=1/2$ and $z=2/3$.
We note that $I(-1/2)=\frac{2}{\sqrt{3}}K(\sqrt{2/3})$.
Edit: We have $$I(n)=3^n\pi~{}_2F_1\left(-n,\frac{1}{2};1;\frac23\right)=\frac{\pi}{\sqrt{3}}~{}_2F_1\left(n+1,\frac{1}{2};1;\frac23\right)=3^{n+1/2}I(-n-1).$$
Therefore, If we write $J(n)=3^{-n/2}I(n)$, then $J(n)=J(-n-1)$, and consequently $J'(-1/2)=0$.
Thus \begin{align*} I'(-1/2) &=\left.\frac{d}{dn}3^{n/2}J(n)\right|_{n=-1/2}\\ &=\left.\left(3^{n/2}J'(n)+3^{n/2}\frac{\log 3}{2}J(n)\right)\right|_{n=-1/2}\\ &=3^{-1/4}\frac{\log 3}{2}J(-1/2)\\ &=\frac{\log 3}{2}I(-1/2)\\ &=\frac{\log 3}{\sqrt{3}}K(\sqrt{2/3}). \end{align*}
• Here are some formulas. – Lucian Mar 1 '14 at 14:50
• very nice solution +1 – Shobhit Bhatnagar Mar 2 '14 at 7:30
$$I=\int_0^\pi \frac{\ln(2+\cos x)}{\sqrt{2+\cos x}}dx\overset{2+\cos x=t}=\int_1^3\frac{\color{blue}{\ln t}}{\sqrt{t}}\frac{dt}{\sqrt{1-(2-t)^2}}\overset{t\to \frac{3}{t}}=\int_1^3\frac{\color{red}{\ln\left(\frac{3}{t}\right)}}{\sqrt t}\frac{dt}{\sqrt{1-(2-t)^2}}$$ $$\require{cancel}\Rightarrow 2I= \int_1^3\frac{\color{blue}{\cancel{\ln 3}}+\color{red}{\ln 3-\cancel{\ln t}}}{\sqrt t}\frac{dt}{\sqrt{1-(2-t)^2}}\overset{t=2+\cos x}=\ln 3\int_0^\pi\frac{dx}{\sqrt{2+\cos x}}$$ $$\Rightarrow I\overset{x\to 2x}=\frac{\ln 3}{\sqrt 3}\int_0^\frac{\pi}{2}\frac{dx}{\sqrt {\frac{2+\cos(2x)}{3}}}=\frac{\ln 3}{\sqrt 3}\int_0^\frac{\pi}{2} \frac{dx}{\sqrt{1-\frac{2}{3}\sin^2 x}}=\frac{\ln 3}{\sqrt 3}K\left(\sqrt{\frac23}\right)$$
• Very clever use of integral substitutions. +1 – Paramanand Singh Mar 13 at 13:08 | {"extraction_info": {"found_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": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9158865809440613, "perplexity": 2942.9875039309763}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141188146.22/warc/CC-MAIN-20201126113736-20201126143736-00229.warc.gz"} |
https://proofwiki.org/wiki/Category:Definitions/Symbolic_Logic | # Category:Definitions/Symbolic Logic
This category contains definitions related to Symbolic Logic.
Related results can be found in Category:Symbolic Logic.
Symbolic logic is the study of logic in which the logical form of statements is analyzed by using symbols as tools.
Instead of explicit statements, logical formulas are investigated, which are symbolic representations of statements, and compound statements in particular.
In symbolic logic, the rules of reasoning and logic are investigated by means of formal systems, which form a good foundation for the symbolic manipulations performed in this field.
## Subcategories
This category has the following 16 subcategories, out of 16 total.
## Pages in category "Definitions/Symbolic Logic"
The following 56 pages are in this category, out of 56 total. | {"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": 0.8631834983825684, "perplexity": 1298.27833223799}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526506.44/warc/CC-MAIN-20190720091347-20190720113347-00178.warc.gz"} |
http://programming-r-pro-bro.blogspot.com/2011/09/project-euler-problem-6.html | ## Tuesday, 27 September 2011
### Project Euler: problem 6
The sum of the squares of the first ten natural numbers is,
12 + 22 + ... + 102 = 385
The square of the sum of the first ten natural numbers is,
(1 + 2 + ... + 10)2 = 552 = 3025
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 - 385 = 2640.
Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.
This is one quite simple.
# Create a vector with the first hundred natural numbers
x.1 <- (1:100)
# Square each element of the vector of natural numbers, i.e, square each natural number and store in another vector
y.1 <- x.1^2
# Sum the first hundred natural numbers and the squares of each of the first hundred natural numbers and take the difference of these sums
a.1 <- sum(y.1)
b.1 <- (sum(x.1))^2
b.1 - a.1
b.1
After solving a couple of these problems, and after reading some solutions posted by aatrujillo here and here, I realize that my solutions are not general, i.e., they only cater to the problem at hand and hence their scope is very limited. For example, consider the above problem. Had the question asked to do the same analysis on the first 200 natural numbers, I would have to rewrite the entire loop again. I understand that in this case it does not involve much more than changing the size of the x.1 vector, but for a problem that involves more than one loop, it seems to be very "uncool". As a result, I have decided to orient my results towards general solutions and then solve the problem by specifying the parameters. Let's see how that goes. :)
1. No need to crunch numbers; there are formulae to help you.
1 + ... + n = n * (n + 1) / 2
and
1^2 + ... + n^2 = n * (n + 1) * (2 * n + 1) / 6
Rearrange these for a faster, cleaner solution.
2. Richie makes a good point. As you progress further into PE questions the order of complexity of brute force algorithms to solve the problems will increase quite swiftly. And don't forget that PE expects the code to complete in under a minute. So, it is a good idea to look for ingenious algorithms that reduce problem complexity before you get down to crunching numbers.
3. That's true Akhil... I realized when I went a bit deeper in PE problems. Thanks for the advice. :-)
4. This problem can be solved in a single line:
sum(seq(1:100))^2 - sum(seq(1:100)^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": 0.8469318151473999, "perplexity": 364.177891307868}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257825124.22/warc/CC-MAIN-20160723071025-00299-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://homework.cpm.org/category/CCI_CT/textbook/pc/chapter/2/lesson/2.1.1/problem/2-12 | ### Home > PC > Chapter 2 > Lesson 2.1.1 > Problem2-12
2-12.
Salvador wants to evaluate $81^{3/4}$ in a different way. He knows that $3^4=81$, so he substitutes and uses the power property for exponents: $81^{3/4}=(3^4)^{3/4}=3^3=27$. Use Salvador’s technique to evaluate each of these
1. $81^{-3/4}$
$\frac{1}{81^{3/4}}$
1. $27^{2/3}$
$( 3 ^ { 3 } ) ^ { 2 / 3 } = 3 ^ { 2 } = ?$
1. $\left(\frac{125}{64}\right)^{2/3}$
$( ( \frac { 5 } { 4 } ) ^ { 3 } ) ^ { 2 / 3 }$
1. $\left(\frac{1}{9}\right)^{-1/2}$
Hint: Express $\frac{1}{9}$ as power of $3$.
$9^{1/2}$ | {"extraction_info": {"found_math": true, "script_math_tex": 13, "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": 0.9405338168144226, "perplexity": 3546.37400065248}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662520936.24/warc/CC-MAIN-20220517225809-20220518015809-00506.warc.gz"} |
https://math.stackexchange.com/questions/1503745/can-we-use-vertex-degree-to-detect-hamilton-path-and-hamilton-circuit | # can we use vertex degree to detect hamilton path and hamilton circuit?
Hamilton path: goes through every node/vertex exactly once.
Hamilton circuit: goes through every vertex exactly one and ends at the starting node/vertex.
So i am wondering is possible to use degree to detect if a graph have hamilton paths or circuits. Like we do with eulers circuit , if every vertex in the graph have a even degree then we know that eulers circuit exists.
Check the theorems of Ore and Dirac.
According to the theorem of Ore:
Let $G$ be a (finite and simple) graph with $n ≥ 3$ vertices. We denote by $deg v$ the degree of a vertex $v$ in $G$, i.e. the number of incident edges in $G$ to $v$. Then, Ore's theorem states that if
$deg v + deg w ≥$ $n$ for every pair of non-adjacent vertices $v$ and $w$ of $G$
then $G$ is Hamiltonian.
According to the theorem of Dirac:
A simple graph with $n$ vertices $(n ≥ 3)$ is Hamiltonian if every vertex has degree $n / 2$ or greater.
• What do you mean by Hamiltonian? path or circuit – humble24 Oct 29 '15 at 19:00
• Dirac = circuit. Ore = path – kjanko Oct 29 '15 at 19:14
• alright, for Ore do i just pick a random pair of non-adjacent vertices ? do i have to check for every possible pair of vertices , or isit enough with one random pair of non-adjacent vertices ? – humble24 Oct 29 '15 at 19:36
• Every pair of non-adjacent vertices. – kjanko Oct 29 '15 at 19:39
• alright, thanks – humble24 Oct 29 '15 at 19:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8633233904838562, "perplexity": 415.52147244465135}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540484815.34/warc/CC-MAIN-20191206050236-20191206074236-00246.warc.gz"} |
https://www.physicsforums.com/threads/help-pls.48487/ | # Help Pls
1. Oct 18, 2004
### debbie18
Help Pls!!!!
Can someone tell me how to do this question:
1. a) A car is travelling at 34.5 mi/h on a horizontal highway. The acceleration of gravity is 9.8 m/s^2. If the coefficient of static friction between road and tires on a rainy day is 0.064, what is the minimum distance in which the car will stop?
b) What is the stopping distance when the surface is dry and the coefficient of friction is 0.675.
Thanks.
2. Oct 18, 2004
### pervect
Staff Emeritus
What have you done so far?
The following URL might help you with the definition of static and kinetic friction, if that happens to be your problem (it's hard to say, because you haven't attempted to explain how you tried to solve the problem).
http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html
Similar Discussions: Help Pls | {"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": 0.8158553838729858, "perplexity": 520.2431796460668}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818690203.42/warc/CC-MAIN-20170924190521-20170924210521-00707.warc.gz"} |
http://www.mathwarehouse.com/algebra/radicals/how-to-multiply-square-roots.php | # How to Multiply Square Roots
Examples, formula and the Steps!
Vocabulary Refresher
### Examples
##### Example 1 of Multiplying Square roots
Step 1
Check to see if you can simplify either of the square roots(. )If you can, then simplify!
Both square roots are already simplified so skip this step
Step 2
Step 3
##### Example 2. A slightly more complex example
Step 1
Check to see if you can simplify either of the square roots(. )If you can, then simplify!
can indeed be simplified :
Step 2
Step 3
### Practice Problems
Multiply the square roots below and express each answer in simplest radical form.
This problem is similar to example 1 because you can not simplify either of the square roots.
Step 1
Check to see if you can simplify either of the square roots. If you can, then simplify!
Skip this since both square roots are already simplified.
Step 2
Multiply
Step 3
This problem is similar to example 2 because 1 of the square roots can be simplified
Step 1
Check to see if you can simplify either of the square roots. If you can, then simplify!
can be simplified:
Step 2
Multiply
Step 3
is already in simplest form so you are done.
This problem is similar to example 2 because 1 of the square roots can be simplified Step 1
Check to see if you can simplify either of the square roots. If you can, then simplify!
can indeed be simplified:
Step 2
Multiply
Step 3
This problem is similar to example 2 because the square roots can be simplified. The only difference is that both square roots, in this problem, can be simplified.
Step 1
Check to see if you can simplify either of the square roots. If you can, then simplify!
Step 2
Multiply
Step 3
This problem is similar to example 2 because the square roots can be simplified. The only difference is that both square roots, in this problem, can be simplified.
Step 1
Check to see if you can simplify either of the square roots. If you can, then simplify!
Step 2
Multiply
Step 3
This problem is similar to example 2 because the square roots can be simplified. The only difference is that both square roots, in this problem, can be simplified.
Step 1
Check to see if you can simplify either of the square roots. If you can, then simplify!
Step 2
Multiply
Step 3
You may notice that this is the same as the problem prior problem (#6)...except that we have now added some coefficients.
Step 1
Check to see if you can simplify either of the square roots. If you can, then simplify!
Step 2
Multiply
Step 3
### Ultimate Math Solver (Free)
Free Algebra Solver ... type anything in there! | {"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": 0.987812340259552, "perplexity": 982.4733476756453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719079.39/warc/CC-MAIN-20161020183839-00198-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://chemistry.stackexchange.com/questions/16013/fluoride-color-reaction | # Fluoride Color Reaction
I am trying to find visual ways to demonstrate water contaminants. For example, chlorine reacts to N,N-diethyl-p-phenyldiamine (DPD), forming a magenta color. The chemical doesn't need to be accurate, as this is more of pass-fail test of the water. What reacts with fluoride to provide similar results?
Fluoride has a high affinity for $Fe^{3+}$, and $Fe^{3+}$ forms deeply coloured complexes with phenols. The decomposition of the latter by fluoride ion is the basis for colorimetric determination of fluoride. | {"extraction_info": {"found_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": 0.9008089303970337, "perplexity": 3184.3700387257404}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988850.21/warc/CC-MAIN-20210508061546-20210508091546-00569.warc.gz"} |
http://avalondock.codeplex.com/discussions/155488 | # DockablePane Template Re-styling
Wiki Link: [discussion:155488]
larroushe Feb 23, 2010 at 1:17 PM Edited Feb 23, 2010 at 1:17 PM I have used AvalonDock long time ago and the dll version that I am currently referencing is version: 1.1.1509.0 Since that time, I have never migrated to a newer version. Now I need to do a small modification and want to use the same version I have (1.1.1509.0 as mentioned above) since I don't want to change in the project I have. I have the following block of code: I need to modify the background and foreground colors of the header section which has as title "Business Processes" (shown in green above) and NOT the background of the content itself. How can this be applied on the version I am using? I would appreciate any help with clear explanation and code sample to test it. I really need this asap, thank you in advance! Regards, Lara | {"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": 0.8346952199935913, "perplexity": 1353.3467369848258}, "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-2017-30/segments/1500549424564.72/warc/CC-MAIN-20170723142634-20170723162634-00476.warc.gz"} |
https://www.physicsforums.com/threads/spring-constants-power.265709/ | # Homework Help: Spring constants! power
1. Oct 20, 2008
### Jessicaelleig
Two identical massless springs of constant k = 200 N/m are fixed at opposite ends of a level track, as shown in Figure P5.62. A 5.90 kg block is pressed against the left spring, compressing it by 0.150 m. The block (initially at rest) is then released, as shown in Figure P5.62a. The entire track is frictionless except for the section between A and B.
Figure P5.62.
Given that the coefficient of kinetic friction between block and track along AB is µk = 0.0800, and given that the length AB is 0.238 m,
(a) determine the maximum compression of the spring on the right (see Fig. P5.62b).
(b) Determine where the block eventually comes to rest, as measured from A (see Fig. P5.62c).
If anyone can provide some guidance, I would appreciate it.
Thank you
2. Oct 20, 2008
### LowlyPion
Since you only have the one area that has friction - where energy can be robbed from the block - then what you are interested is how much 1 pass over the patch of friction will take away.
W = F*d
so your work each pass removed is u*m*g*d = .08 * 5.9 * 9.8 * .238
The work for the Spring is kx2/2 so knowing how much work goes to KE then gets subtracted each pass over the patch you can figure the rest.
3. Oct 20, 2008
### Jessicaelleig
thanks! anything else? im so confused
4. Oct 20, 2008
### LowlyPion
Any more and I'd be giving you the answers wouldn't I?
And gosh darn it I can't take your tests for you. So what good what that be?
Just draw a diagram and figure it out. | {"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": 0.9050492644309998, "perplexity": 1341.9325870251846}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1542039742253.21/warc/CC-MAIN-20181114170648-20181114192648-00329.warc.gz"} |
https://math.stackexchange.com/questions/2656406/can-integral-equations-be-paired-with-linear-regression-to-fit-a-double-gaussian | # Can integral equations be paired with linear regression to fit a double Gaussian regression?
In the paper https://www.scribd.com/doc/14674814/Regressions-et-equations-integrales (page 6) a method is presented to fit a single nonlinear Gaussian curve to noisy data. Later on in the paper, the same method is employed to fit a double exponential regression (and even more). I'm curious if it would be possible to employ the same technique to fit a double Gaussian regression with scaling constants? To be specific, I want to perform a regression of the following equation to data.
$$f(x)=\frac{c_1}{\sigma_1} \text{exp}\left(-\frac{1}{2}\left(\frac{x-\mu_1}{\sigma_1}\right)^2\right)+\frac{c_2}{\sigma_2} \text{exp}\left(-\frac{1}{2}\left(\frac{x-\mu_2}{\sigma_2}\right)^2\right)$$
Where you have data $x_i, y_i$ with normally distributed noise $\epsilon_i$.
I know nonlinear least squares works pretty well in this case, but having a non-iterative algorithm to do this (and potentially with more Gaussian kernels) would be useful in many situations.
Take for example this data.
0,0.0953435022119166
0.408163265306122,0.165876041641162
0.816326530612245,0.217055023196137
1.22448979591837,0.336625390515636
1.63265306122449,0.502382096769287
2.04081632653061,0.666515393163172
2.44897959183673,0.883208684189565
2.85714285714286,1.12458773291947
3.26530612244898,1.37726169379448
3.6734693877551,1.61866109023969
4.08163265306122,1.82013750956065
4.48979591836735,1.9779288108796
4.89795918367347,2.06828873076259
5.30612244897959,2.08056217287661
5.71428571428571,2.07533074340141
6.12244897959184,2.0104842044372
6.53061224489796,1.94357288893297
6.93877551020408,1.8682413902483
7.3469387755102,1.82822815415585
7.75510204081633,1.84578299714906
8.16326530612245,1.86852992222851
8.57142857142857,1.95624254355303
8.97959183673469,2.01135490423995
9.38775510204082,2.0796994498718
9.79591836734694,2.09619003334345
10.2040816326531,2.03603444176666
10.6122448979592,1.91638500266306
11.0204081632653,1.77786079492498
11.4285714285714,1.55497697393296
11.8367346938776,1.28953163571889
12.2448979591837,1.06603391254518
12.6530612244898,0.812745486024327
13.0612244897959,0.622710458610182
13.469387755102,0.433501473226128
13.8775510204082,0.333280824649249
14.2857142857143,0.190078385939407
14.6938775510204,0.133877864179819
15.1020408163265,0.0812258047441917
15.5102040816327,0.0427020032495271
15.9183673469388,0.0280925506752726
16.3265306122449,0.0283736350602543
16.734693877551,0.00634269170340754
17.1428571428571,0.00825683417491195
17.5510204081633,-0.0074678105115024
17.9591836734694,-0.000778850177607315
18.3673469387755,-0.00739023269152126
18.7755102040816,-0.00232605185006384
19.1836734693878,0.0194459618205399
19.5918367346939,0.000445784296190274
20,-0.000752024542365978
Which, when plotted looks like this:
• The integral equation, which involves the function Erf, is too complicated for an easy use. Theoretically, it requires four numerical integrations. Considering the deviations in numerical integrations and the complicated equations, the method appears of very little interest in practice, compared to usual non-linear regression. That is why this case of application isn't published in the referenced paper – JJacquelin Feb 19 '18 at 9:10
• Thanks @JJacquelin. I know you emphasize that numerical derivatives are really unstable, but do you think if the data was essentially a perfect reproduction of the curve that a differential equation based approach would work in this case? The derivative of a Gaussian is much easier to work with than it's integral. – wdkrnls Feb 19 '18 at 21:43
• I see that you well understood the general principle of the use of differential equations or/and integral equations to transform some problems of non-linear regression into linear regression. Congratulations. – JJacquelin Feb 20 '18 at 7:51
• In fact, the associated differential equation is much simpler than the integral equation as you rightly pointed out. But another condition to obtain a robust process is that the number of independent terms in the differential equation be not higher than the number of parameters in the original equation (6 parameters in the double Gaussian).. – JJacquelin Feb 20 '18 at 7:52
• Unfortunately, the number of independent parameters of the differential equation is 8. As a consequence, the process will not be robust, even with not scattered data. This means that, sometimes and with luck, the computed values of the parameters will not be too bad. But not in the general case. And even worse with scattered data of course. I am sorry for this discouraging result obtained when I was writing the referenced paper. – JJacquelin Feb 20 '18 at 8:25
Mainly my answer is already in comments. Nevertheless, more details are given below (with different notations in order to made easier the typing) :
$$y(x)=a\,e^{-(bx+c)^2}+A\,e^{-(Bx+C)^2} \tag 1$$ $$\int ydx=\frac{a\sqrt{\pi}}{2b}\text{erf}(bx+c)+\frac{A\sqrt{\pi}}{2B}\text{erf}(Bx+C) \tag 2$$ $$\int xydx=-\frac{ac\sqrt{\pi}}{2b^2}\text{erf}(bx+c)-\frac{AC\sqrt{\pi}}{2B^2}\text{erf}(Bx+C)-\frac{a}{2b^2}\,e^{-(bx+c)^2}-\frac{A}{2B^2}\,e^{-(Bx+C)^2}\tag 3$$ $$\int x^2ydx=\frac{a(2c^2+1)\sqrt{\pi}}{4b^3}\text{erf}(bx+c)+\frac{A(2C^2+1)\sqrt{\pi}}{4B^3}\text{erf}(Bx+C)+\frac{a(c-bx)}{2b^3}\,e^{-(bx+c)^2}+\frac{A(C-Bx)}{2B^3}\,e^{-(Bx+C)^2} \tag 4$$ $$\int x^3ydx=-\frac{ac(2c^2+3)\sqrt{\pi}}{4b^4}\text{erf}(bx+c)-\frac{AC(2C^2+3)\sqrt{\pi}}{4B^4}\text{erf}(Bx+C)-\frac{a(c^2+1-bcx+b^2x^2)}{2b^4}\,e^{-(bx+c)^2}-\frac{A(C^2+1-BCx+B^2x^2)}{2B^4}\,e^{-(Bx+C)^2} \tag 5$$ Considering $\text{erf}(bx+c)$ , $\text{erf}(Bx+C)$ , $e^{-(bx+c)^2}$ and $e^{-(Bx+C)^2}$ as unknowns, Eqs.$(2,3,4,5)$ is a linear system which can be solved. Then the explicit expressions of $e^{-(bx+c)^2}$ and $e^{-(Bx+C)^2}$ obtained can be put into Eq.$(1)$. The result is an integral equation which could be theoretically used for linear regression.
But, in practice, the calculus to get to the integral equation is much too complicated. Moreover, there is another counter-argument (number of parameters) as it will be pointed out below. So, the method of integral equation appears not convenient for practical use in the case of double Gaussian regression.
In comments, wdkrnls raised the pertinent remark that instead of an integral equation, a differential equation would be simpler, regardless the question of instability of numerical differentiation. This is true and was considered when the above referenced paper was written.
Let : $\quad y(x)=af(x)+AF(x)\quad$ where $\quad f(x)=e^{-(bx+c)^2}\quad$ and $\quad F(x)=e^{-(Bx+C)^2}$
$$y'(x)= -2ab(bx+c)f-2AB(Bx+C)F$$ $$y''(x)= 2ab^2(2b^2x^2+4bcx+2c^2-1)f+2AB^2(2B^2x^2+4BCx+2C^2-1)F$$ $$\left(\begin{matrix} f \\ F \end{matrix}\right) = \left(\begin{matrix} -2ab(bx+c) & -2AB(Bx+C) \\ 2ab^2(2b^2x^2+4bcx+2c^2-1) & 2AB^2(2B^2x^2+4BCx+2C^2-1) \end{matrix}\right)^{-1} \left(\begin{matrix} y' \\ y'' \end{matrix}\right)$$ We put $f$ and $F$ into $y(x) = \left(\begin{matrix} a & A \end{matrix}\right) \left(\begin{matrix} f \\ F \end{matrix}\right) \quad$ which leads to the differential equation on matrix form :
$$\left(\begin{matrix} a & A \end{matrix}\right) \left(\begin{matrix} -2ab(bx+c) & -2AB(Bx+C) \\ 2ab^2(2b^2x^2+4bcx+2c^2-1) & 2AB^2(2B^2x^2+4BCx+2C^2-1) \end{matrix}\right)^{-1} \left(\begin{matrix} y' \\ y'' \end{matrix}\right) - y =0$$ After calculus and simplification :
$$(b^2-B^2)xy''+4(b^3c-B^3C)y''+2(b^4-B^4)x^2y'+4(b^3c-B^3C)xy'+(2b^2c^2-b^2-2B^2C^2+B^2)y'+4bB(Bb^3-bB^3)x^3y+4bB(b^3C-B^3c+2Bb^2c-2bB^2C)x^2y+4bB(2b^2cC-2B^2Cc+Bbc^2-bBC^2)xy+2bB(2bc^2C-2BC^2c+Bc-bC)y=0$$
This is the differential equation which could be used for linear regression.
But another condition to obtain a robust process is that the number of independent terms in the differential equation be not higher than the number of parameters in the original equation.
This is far to be the case here.
As a consequence, the process will not be robust, even with not scattered data. This means that, sometimes and with luck, the computed values of the parameters will not be too bad. But not in the general case. And even worse with scattered data, of course. I am sorry for this discouraging result obtained when I was writing the referenced paper. That is why the case of double Gaussian is not treated in the paper.
Note: One can obtain an integral equation from the above differential equation with successive integrations. This is an easier way than the direct integration method as shown above. But the number of independent terms will be still larger than with the differential equation. So, the discouraging conclusion is the same.
• Thanks very much for the detailed exposition. – wdkrnls Feb 23 '18 at 13:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9379222989082336, "perplexity": 279.1523072429595}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986702077.71/warc/CC-MAIN-20191020024805-20191020052305-00055.warc.gz"} |
http://en.wikibooks.org/wiki/Signals_and_Systems/Aperiodic_Signals | # Signals and Systems/Aperiodic Signals
## Aperiodic Signals
The opposite of a periodic signal is an aperiodic signal. An aperiodic function never repeats, although technically an aperiodic function can be considered like a periodic function with an infinite period.
## Background
If we consider aperiodic signals, it turns out that we can generalize the Fourier Series sum into an integral named the Fourier Transform. The Fourier Transform is used similarly to the Fourier Series, in that it converts a time-domain function into a frequency domain representation. However, there are a number of differences:
1. Fourier Transform can work on Aperiodic Signals.
2. Fourier Transform is an infinite sum of infinitesimal sinusoids.
3. Fourier Transform has an inverse transform, that allows for conversion from the frequency domain back to the time domain.
## Fourier Transform
This operation can be performed using this MATLAB command:
fft
The Fourier Transform is the following integral:
$\mathcal{F}(f(t)) = F(j\omega) = \int_{-\infty}^\infty f(t)e^{-j\omega t}dt$
## Inverse Fourier Transform
And the inverse transform is given by a similar integral:
$\mathcal{F}^{-1}\left\{F(j\omega) \right\} = f(t) = \frac{1}{2\pi}\int_{-\infty}^\infty F(j\omega) e^{j\omega t} d\omega$
Using these formulas, time-domain signals can be converted to and from the frequency domain, as needed.
### Partial Fraction Expansion
One of the most important tools when attempting to find the inverse fourier transform is the Theory of Partial Fractions. The theory of partial fractions allows a complicated fractional value to be decomposed into a sum of small, simple fractions. This technique is highly important when dealing with other transforms as well, such as the Laplace transform and the Z-Transform.
## Duality
The Fourier Transform has a number of special properties, but perhaps the most important is the property of duality.
We will use a "double-arrow" signal to denote duality. If we have an even signal f, and it's fourier transform F, we can show duality as such:
$f(t) \Leftrightarrow F(j\omega)$
This means that the following rules hold true:
$\mathcal{F}\{f(t)\} = F(j \omega)$ AND $\mathcal{F}\{F(t)\} = f(j\omega)$
Notice how in the second part we are taking the transform of the transformed equation, except that we are starting in the time domain. We then convert to the original time-domain representation, except using the frequency variable. There are a number of results of the Duality Theorem.
### Convolution Theorem
The Convolution Theorem is an important result of the duality property. The convolution theorem states the following:
Convolution Theorem
Convolution in the time domain is multiplication in the frequency domain. Multiplication in the time domain is convolution in the frequency domain.
Or, another way to write it (using our new notation) is such:
$A \times B \Leftrightarrow A * B$
### Signal Width
Another principle that must be kept in mind is that signal-widths in the time domain, and bandwidth in the frequency domain are related. This can be summed up in a single statement:
Thin signals in the time domain occupy a wide bandwidth. Wide signals in the time domain occupy a thin bandwidth.
This conclusion is important because in modern communication systems, the goal is to have thinner (and therefore more frequent) pulses for increased data rates, however the consequence is that a large amount of bandwidth is required to transmit all these fast, little pulses.
## Power and Energy
### Energy Spectral Density
Unlike the Fourier Series, the Fourier Transform does not provide us with a number of discrete harmonics that we can add and subtract in a discrete manner. If our channel bandwidth is limited, in the Fourier Series representation, we can simply remove some harmonics from our calculations. However, in a continuous spectrum, we do not have individual harmonics to manipulate, but we must instead examine the entire continuous signal.
The Energy Spectral Density (ESD) of a given signal is the square of its Fourier transform. By definition, the ESD of a function f(t) is given by F2(jω). The power over a given range (a limited bandwidth) is the integration under the ESD graph, between the cut-off points. The ESD is often written using the variable Ef(jω).
$E_f(j\omega) = F^2(j\omega)$
### Power Spectral Density
The Power Spectral Density (PSD) is similar to the ESD. It shows the distribution of power in the spectrum of a particular signal.
$P_f(j\omega) = \int_{-\infty}^\infty F(j \omega) d\omega$
Power spectral density and the autocorrelation form a Fourier Transform duality pair. This means that:
$P_f(j\omega) = \mathcal{F}[R_{ff}(t)]$
If we know the autocorrelation of the signal, we can find the PSD by taking the Fourier transform. Similarly, if we know the PSD, we can take the inverse Fourier transform to find the autocorrelation signal. | {"extraction_info": {"found_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": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9738968014717102, "perplexity": 398.5142745847137}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1418802766267.61/warc/CC-MAIN-20141217075246-00117-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/29640/how-to-modify-spacing-around-quotation-environment | # How to modify spacing around quotation environment?
I am almost there in my quest to reproduce the rather compact layout of the book I'm translating... one thing to go, though:
I successfully used the enumitem package to modify the vertical and horizontal spacing of the various list environments.
But how do I modify the vertical and horizontal spacing of the quotation environment? I have found lots of information on how to do it for theorems and formulas etc., but none of that seems to apply for quotation (or quote, for that matter).
\documentclass[twocolumn]{scrbook}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{enumitem}
\setlist{leftmargin=*,parsep=0pt,itemsep=2pt,topsep=2pt,partopsep=0pt}
\setlist[enumerate]{widest=0}
\newcommand{\sometext}{Some random text. Not using lipsum as I don't need that much text.}
\begin{document}
\sometext
\begin{itemize}
\item One item.
\item Another item.
\end{itemize}
\sometext
\begin{quotation}
\textbf{Note:} \emph{This has too much whitespace around it.}
\end{quotation}
\sometext
\end{document}
-
The quoting package provides a quoting environment with customizable font, margins, spacing... Just load it with \usepackage{quoting} and use either \begin{quoting}[vskip=0pt] ... \end{quoting} or set it up globally with \quotingsetup{vskip=0pt}:
\documentclass[twocolumn]{scrbook}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{enumitem}
\setlist{leftmargin=*,parsep=0pt,itemsep=2pt,topsep=2pt,partopsep=0pt}
\setlist[enumerate]{widest=0}
\usepackage{quoting}
\quotingsetup{vskip=0pt}
\newcommand{\sometext}{Some random text. Not using lipsum as I don't need that much text.}
\begin{document}
\sometext
\begin{itemize}
\item One item.
\item Another item.
\end{itemize}
\sometext
\begin{quoting}
\textbf{Note:} \emph{This has no extra white\-space around it.}
\end{quoting}
\sometext
\end{document}
-
Ummm, the word whitespace is a well-known case of a failure of TeX's hyphenation mechanism... You may want to add a discretionary hyphen, \-, to fix this in your MWE. :-) – Mico Sep 27 '11 at 10:25
@Mico Thanks for pointing that out, fixed :-) – diabonas Sep 27 '11 at 10:31
I find setting parameters for a package preferrable over redefining, or writing macros myself. (Assuming that the package authors know more about LaTeX than I do, which isn't hard at all.) Thanks for pointing out this package. – DevSolar Sep 27 '11 at 11:22
Global settings may also be changed using \usepackage[vskip=0pt]{quoting}. – lockstep Sep 27 '11 at 17:10
\documentclass[twocolumn]{scrbook}
\makeatletter
\renewenvironment{quotation}
{\list{}{\listparindent=0pt%whatever you need
\itemindent \listparindent
\leftmargin=0pt% whatever you need
\rightmargin=10pt%whatever you need
\topsep=0pt%%%%% whatever you need
\parsep \z@ \@plus\p@}%
\item\relax}
{\endlist}
\makeatother
\newcommand{\sometext}{Some random text. Not using lipsum as I don't need that much text.}
\begin{document}
\sometext
\begin{quotation}
\textbf{Note:} \emph{This has too much whitespace around it.}
\end{quotation}
\sometext
\end{document}
-
I tried to \setlength{\topsep}{0pt} at the global level, but it did not seem to affect the quotation environment, so I thought it had to be some other parameter... but it seems that the \topsep etc. has to be declared explicitly in the environent... is there a simple explanation as to why? – DevSolar Sep 27 '11 at 10:20
your value is always overwritten by the definition of quotation – Herbert Sep 27 '11 at 10:25
That explains it, thank you. – DevSolar Sep 27 '11 at 10:56
On second thought, isn't this somewhat... I won't call it "buggy", but perhaps "flaky"? If the environment ignores global topsep settings, shouldn't it react on some other setting? Having to re-define the environment to change one parameter strikes me as a bit heavy-handed, especially since LaTeX apparently does not provide a way to either determine the standard definition for reference ("what are all the other things done by this environment, so I don't forget them in my redifinition"), or add a single value to the definition without having to redefine it completely... – DevSolar Sep 27 '11 at 14:36 | {"extraction_info": {"found_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": 0.9367733597755432, "perplexity": 2924.2521837261625}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931007607.9/warc/CC-MAIN-20141125155647-00237-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://arxiver.moonhats.com/2014/03/18/hint-of-relic-gravitational-waves-in-the-planck-and-wmap-data-cl/ | # Hint of relic gravitational waves in the Planck and WMAP data [CL]
Relic gravitational waves (RGWs) leave well-understood imprints on the anisotropies in the temperature and polarization of cosmic microwave background (CMB) radiation. In the TT and TE information channels, which have been well observed by WMAP and Planck missions, RGWs compete with density perturbations mainly at low multipoles. It is dangerous to include high-multipole CMB data in the search for gravitational waves, as the spectral indices may not be constants. In this paper, we repeat our previous work [W.Zhao & L.P.Grishchuk, Phys.Rev.D {\bf 82}, 123008 (2010)] by utilizing the Planck TT and WMAP TE data in the low-multipole range $\ell\le100$. We find that our previous result is confirmed {with higher confidence}. The constraint on the tensor-to-scalar ratio from Planck TT and WMAP TE data is $r\in [0.06,~0.60]$ (68% C.L.) with a peak around $r\sim 0.2$. Correspondingly, the spectral index at the pivot wavenumber $k_*=0.002$Mpc$^{-1}$ is $n_s=1.13^{+0.07}_{-0.08}$, which is larger than 1 at more than $1\sigma$ level. So, we conclude that the new released CMB data indicate a stronger hint for the RGWs with the amplitude $r\sim 0.2$, which is hopeful to be confirmed by the imminent BICEP and Planck polarization data.
W. Zhao, C. Cheng and Q. Huang
Tue, 18 Mar 14
43/62 | {"extraction_info": {"found_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": 0.8589975833892822, "perplexity": 1761.66474171828}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687766.41/warc/CC-MAIN-20170921115822-20170921135822-00499.warc.gz"} |
https://www.emathhelp.net/notes/calculus-2/series-and-sequences/ | # List of Notes - Category: Series and Sequences
## Sequences
In simple words sequence is a list of numbers written in definite order: a_1,a_2,...,a_n. a_1 is first term, a_2 is second term, and, in general, a_n is n-th term. We deal with infinite sequences, so each term a_n will have a successor a_(n+1). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8487716317176819, "perplexity": 2349.436745295917}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218187227.84/warc/CC-MAIN-20170322212947-00462-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://asu.pure.elsevier.com/en/publications/chiral-magnetic-plasmons-in-anomalous-relativistic-matter | # Chiral magnetic plasmons in anomalous relativistic matter
E. V. Gorbar, V. A. Miransky, Igor Shovkovy, P. O. Sukhachov
Research output: Contribution to journalArticlepeer-review
28 Scopus citations
## Abstract
The chiral plasmon modes of relativistic matter in background magnetic and strain-induced pseudomagnetic fields are studied in detail using the consistent chiral kinetic theory. The results reveal a number of anomalous features of these chiral magnetic and pseudomagnetic plasmons that could be used to identify them in experiment. In a system with nonzero electric (chiral) chemical potential, the background magnetic (pseudomagnetic) fields not only modify the values of the plasmon frequencies in the long-wavelength limit, but also affect the qualitative dependence on the wave vector. Similar modifications can be also induced by the chiral shift parameter in Weyl materials. Interestingly, even in the absence of the chiral shift and external fields, the chiral chemical potential alone leads to a splitting of plasmon energies at linear order in the wave vector.
Original language English (US) 115202 Physical Review B 95 11 https://doi.org/10.1103/PhysRevB.95.115202 Published - Mar 7 2017
## ASJC Scopus subject areas
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
## Fingerprint
Dive into the research topics of 'Chiral magnetic plasmons in anomalous relativistic matter'. Together they form a unique fingerprint. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.846492350101471, "perplexity": 4117.319441719313}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323587623.1/warc/CC-MAIN-20211025030510-20211025060510-00670.warc.gz"} |
https://www.thestudentroom.co.uk/showthread.php?t=6760452 | # Divergence of series
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Thread starter 1 month ago
#1
Show that sum from 1 to infinity of [ ln n / (n+4) ] diverges.
I try the integral test but I'm not able to integrate by parts.. How do I show this? Please help
Last edited by golgiapparatus31; 1 month ago
0
1 month ago
#2
(Original post by golgiapparatus31)
Show that sum from 1 to infinity of [ln n / (n+4)] diverges.
I try the integral test but I'm not able to integrate by parts.. How do I show this? Please help
This last sum diverges due to integral test.
Note that
Last edited by RDKGames; 1 month ago
0
Thread starter 1 month ago
#3
(Original post by RDKGames)
This last sum diverges due to integral test.
Note that
Thank you!! (PRSOM)
Can you please check my answer to this too?
Using the comparison test, or otherwise, show that the sum from 1 to infinity of (1/(sqrt(n2+3) diverges.
1/sqrt(n2+3) > 1/sqrt(n2+4)
Integrating 1/sqrt(x2+4) gives sinh-1(x/2) so diverges. Hence the sum diverges
Evaluate whether the sum from 1 to infinity of (3n / (2n + 5n)) diverges or converges.
I start by writing that
(3x / (2x + 5x)) < 3x/5x = (3/5)x
because making the denominator smaller makes the value of the fraction larger
Then I calculate the integral of 0.6^x from 1 to infinity, getting 1.17. So the sum converges.
Does this look correct? Thank you!!
I integrate (3/5)^x from 1 to infinity,
0
1 month ago
#4
(Original post by golgiapparatus31)
Thank you!! (PRSOM)
Can you please check my answer to this too?
Using the comparison test, or otherwise, show that the sum from 1 to infinity of (1/(sqrt(n2+3) diverges.
1/sqrt(n2+3) > 1/sqrt(n2+4)
Integrating 1/sqrt(x2+4) gives sinh-1(x/2) so diverges. Hence the sum diverges
Regarding , your proof looks good.
Though why didn't you just immediately integrate into ?
Personally, I would've just used a comparison test.
The harmonic series is well known to diverge. Part of me wants to say this approach is a bit more natural for a real analysis course because inverse sinh may not be an 'obvious' or a studied function whose properties you can just read off without proof.
0
1 month ago
#5
(Original post by golgiapparatus31)
Thank you!! (PRSOM)
Can you please check my answer to this too?
Using the comparison test, or otherwise, show that the sum from 1 to infinity of (1/(sqrt(n2+3) diverges.
1/sqrt(n2+3) > 1/sqrt(n2+4)
Integrating 1/sqrt(x2+4) gives sinh-1(x/2) so diverges. Hence the sum diverges
Evaluate whether the sum from 1 to infinity of (3n / (2n + 5n)) diverges or converges.
I start by writing that
(3x / (2x + 5x)) < 3x/5x = (3/5)x
because making the denominator smaller makes the value of the fraction larger
Then I calculate the integral of 0.6^x from 1 to infinity, getting 1.17. So the sum converges.
Does this look correct? Thank you!!
I integrate (3/5)^x from 1 to infinity,
So, to some extent this is a "style" complaint, but since an integral is actually a pretty complex mathematical "thing" that's defined in terms of limits of sums, it's better style to use direct comparison where possible.
For these particular sums:
for n > 4, for n > 1 (so both diverge due to comparison with 1/n), while for your last case you get convergence since you're comparing to a GP with |ratio| < 1.
Even if you think I'm nitpicking about "style" (what you've done isn't actually wrong), these manipulations/comparisions are usually less work than the integral test and also help you with the kind of approach you'll use increasingly as you study analysis. (A large part of real analysis is finding "nice" functions that bound "not-so-nice" functions).
Last edited by DFranklin; 1 month ago
1
1 month ago
#6
(Original post by golgiapparatus31)
Evaluate whether the sum from 1 to infinity of (3n / (2n + 5n)) diverges or converges.
I start by writing that
(3x / (2x + 5x)) < 3x/5x = (3/5)x
because making the denominator smaller makes the value of the fraction larger
Then I calculate the integral of 0.6^x from 1 to infinity, getting 1.17. So the sum converges.
Does this look correct? Thank you!!
I integrate (3/5)^x from 1 to infinity,
Regarding you are correct to note an important bound;
But there is no need to apply the integral test here! The RHS is a geometric series ... we can be more simple than the integral test here in order to answer this question.
1
Thread starter 1 month ago
#7
(Original post by RDKGames)
Regarding , your proof looks good.
Though why didn't you just immediately integrate into ?
Personally, I would've just used a comparison test.
The harmonic series is well known to diverge. Part of me wants to say this approach is a bit more natural for a real analysis course because inverse sinh may not be an 'obvious' or a studied function whose properties you can just read off without proof.
Thank you! In my book an example was worked out in that way: https://imgur.com/a/pHjZkpg and https://imgur.com/a/eyDuONc, so I first stated a function less than 1/sqrt(n^2+3), then I integrate it.
My book says: "When an alternative expression is used to evaluate a sum, it is known as a comparison test."
Since question asked for comparison test, I did it that way.. :s The book didn't do manipulation like you did so I didn't know that approach.
(Original post by RDKGames)
Regarding you are correct to note an important bound;
But there is no need to apply the integral test here! The RHS is a geometric series ... we can be more simple than the integral test here in order to answer this question.
Thank you!
(Original post by DFranklin)
So, to some extent this is a "style" complaint, but since an integral is actually a pretty complex mathematical "thing" that's defined in terms of limits of sums, it's better style to use direct comparison where possible.
For these particular sums:
for n > 4, for n > 1 (so both diverge due to comparison with 1/n), while for your last case you get convergence since you're comparing to a GP with |ratio| < 1.
Even if you think I'm nitpicking about "style" (what you've done isn't actually wrong), these manipulations/comparisions are usually less work than the integral test and also help you with the kind of approach you'll use increasingly as you study analysis. (A large part of real analysis is finding "nice" functions that bound "not-so-nice" functions).
Thank you! I am not sure how you reached this inequality:
" for n > 4, for n > 1"
How did you get that ?
0
1 month ago
#8
(Original post by golgiapparatus31)
Thank you! I am not sure how you reached this inequality:
" for n > 4, for n > 1"
How did you get that ?
When , we have:
Multiplying these inequalities gives the inequality he stated.
1
Thread starter 1 month ago
#9
Thanks a lot for the insights you gave me, RDKGames asnd DFranklin!!
I will be reading more on this topic to improve my understanding
Thank you very much !!
0
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4.53% | {"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": 0.9359080195426941, "perplexity": 1487.923995519618}, "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-2020-50/segments/1606141197278.54/warc/CC-MAIN-20201129063812-20201129093812-00360.warc.gz"} |
http://mathhelpforum.com/advanced-applied-math/206452-momentum-force-calculations.html | # Math Help - momentum and force calculations
1. ## momentum and force calculations
Hi,
i have had a incident where a piece of glass has smashed due to someone wedging there foot on a revolving door and i need to find out the forces that have been placed on the glass!
the revolving door leaves weigh 100kg each and there are four on this door, the door was travelling at approximately 8 revolutions per minute.
so total weight moving is 400kg but it the one door leaf has been stopped by the persons foot then the force applied by the other 3 door leaves still moving at the out side edge of the door and at approx 600mm from the centre.
the diameter of the door is 2600mm
i hope someone can help as this is a little to advanced for me
Dave
2. ## Re: momentum and force calculations
Originally Posted by sprk001
Hi,
i have had a incident where a piece of glass has smashed due to someone wedging there foot on a revolving door and i need to find out the forces that have been placed on the glass!
the revolving door leaves weigh 100kg each and there are four on this door, the door was travelling at approximately 8 revolutions per minute.
so total weight moving is 400kg but it the one door leaf has been stopped by the persons foot then the force applied by the other 3 door leaves still moving at the out side edge of the door and at approx 600mm from the centre.
the diameter of the door is 2600mm
i hope someone can help as this is a little to advanced for me
Dave
You need more information. You can calculate the change in momentum of the door, and thus the average force required to stop the door, but you need information about how much stress the glass can take before it breaks.
-Dan | {"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": 0.9256101846694946, "perplexity": 935.9082877933586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1430458969644.56/warc/CC-MAIN-20150501054249-00055-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://wiki.nethome.nu/doku.php?id=Scene | scene
# Scene
This module is a part of the NetHomeServer. The Scene is used to activate a set of commands, for example to set the state of all the lamps in a room. A command may for example be to call the on-method of a lamp or an external shell file. The syntax of the command is described here. The supported commands are: set, call, event, exec.
## Attributes
• Delay [get] [set] Between each command the execution halts for the duration of this delay. The delay is specified in seconds and allows fractions like 0.5.
• Command1 [get] [set] The first Command to be executed at the action of the Scene. Se above for syntax.
• Command2 [get] [set] The second Command to be executed at the action of the Scene. Se above for syntax.
• Command3 [get] [set] The third Command to be executed at the action of the Scene. Se above for syntax.
• Command4 [get] [set] The fourth Command to be executed at the action of the Scene. Se above for syntax.
## Actions
• Action This command activates the Scene. | {"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": 0.9467366933822632, "perplexity": 3236.634722884641}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764499695.59/warc/CC-MAIN-20230128220716-20230129010716-00385.warc.gz"} |
http://zbmath.org/?q=an:0797.68092&format=complete | # zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
##### Operators
a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Zeta functions of formal languages. (English) Zbl 0797.68092
The authors define the zeta function $\zeta \left(L\right)$ of a language $L$ by $\zeta \left(L\right)=exp\left({\sum }_{n\ge 1}{a}_{n}{t}^{n}/n\right)$, where ${a}_{n}$ is the number of words of length $n$ in $L$. They define the generalized zeta function $Z\left(S\right)$ of a formal power series $S$ in noncommuting variables by $Z\left(S\right)=exp\left({\sum }_{n\ge 1}\pi \left({S}_{n}\right)/n\right)$, where $\pi \left({S}_{n}\right)$ is the homogeneous part of degree $n$ of $S$ viewed in a canonical way as a formal series in commuting variables. The generalized zeta function of a language $L$ is $Z\left(\underline{L}\right)$, where $\underline{L}$ is the characteristic series of $L$. By definition, a language $L$ is cyclic if (i) $uv\in L$ if and only if $vu\in L$ and (ii) $w\in L$ if and only if ${w}^{n}\in L$ $\left(n\ge 1\right)$. The authors show that if $L$ is a cyclic language then $\zeta \left(L\right)={\prod }_{n\ge 1}{\left(1-{t}^{n}\right)}^{-{\alpha }_{n}}$, where ${\alpha }_{n}$ is the number of conjugation classes of primitive words of length $n$ contained in $L$. If $𝒜$ is a finite automaton over $A$, the trace $\text{tr}\left(𝒜\right)$ of $𝒜$ is the formal power series $\text{tr}\left(𝒜\right)={\sum }_{w\in {A}^{*}}{\alpha }_{w}w$, where ${\alpha }_{w}$ is the number of couples $\left(q,c\right)$ such that $q$ is a state of $𝒜$ and $c$ is a path $q\to q$ in $𝒜$ labelled $w$. The authors prove that $Z\left(\text{tr}\left(𝒜\right)\right)$ is rational. Using the theory of minimal ideals in finite semigroups, they are able to show that the characteristic series of a cyclic recognizable language is a linear combination over $ℤ$ of traces of finite deterministic automata. Consequently, if $L$ is cyclic and recognizable then $\zeta \left(L\right)$ and $Z\left(L\right)$ are rational. As a corollary the authors obtain the rationality of the zeta function of a sofic system in symbolic dynamics. The authors discuss connections to Dwork’s theorem stating that the zeta function of an algebraic variety over a finite field is rational.
This remarkable paper opens up new and interesting research areas.
##### MSC:
68Q45 Formal languages and automata 20M35 Semigroups in automata theory, linguistics, etc. 05A15 Exact enumeration problems, generating functions 37-99 Dynamic systems and ergodic theory (MSC2000) 14G10 Zeta-functions and related questions 37C25 Fixed points, periodic points, fixed-point index theory 68Q70 Algebraic theory of languages and automata | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 45, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8904945254325867, "perplexity": 1010.3866202040446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1386164960531/warc/CC-MAIN-20131204134920-00033-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://mathhelpforum.com/differential-equations/132073-showing-equilibrium-point-inequality.html | Math Help - Showing an equilibrium point with an inequality
1. Showing an equilibrium point with an inequality
Given $\frac{dy}{dt}=r(1-\frac{y}{k})y-Ey$
if $E show that there are two equilibrium points given by $y_{1}=0$ and $y_{2}=k(1-\frac{E}{r})>0$
The demonstration of $y_{1}$ is plain to see, but how do you go about proving the equilibrium point at $y_{2}$?
2. Originally Posted by FoxyGrandma3000
Given $\frac{dy}{dt}=r(1-\frac{y}{k})y-Ey$
if $E show that there are two equilibrium points given by $y_{1}=0$ and $y_{2}=k(1-\frac{E}{r})>0$
The demonstration of $y_{1}$ is plain to see, but how do you go about proving the equilibrium point at $y_{2}$?
I believe that you'll have two distinct eq. points provided that $E \ne r$. From your ODE
$
\frac{dy}{dx} = y\left(r - \frac{ry}{k} - E \right)
$
$y = 0$ is one, the other is found by setting $r - \frac{ry}{k} - E =$0 and solving for $y$. If you want the second eq. point to be positive then you would need $E < r$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 18, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8654094934463501, "perplexity": 283.2018114256757}, "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-2014-52/segments/1418802775656.66/warc/CC-MAIN-20141217075255-00016-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/moment-of-inertia-problem-kleppner-6-18.196999/ | # Moment of inertia problem- kleppner #6.18
1. Nov 8, 2007
### AmaniKaleo
moment of inertia problem-- kleppner #6.18
Ok here's the problem:
[from An Introduction to Mechanics, Kleppner&Kolenkow, problem 6.18, page 281]
Find the period of a pendulum consisting of a disk of mass M and radius R fixed to the end of a rod of length l and mass m. How does the period change if the disk is mounted to the rod by a frictionless bearing so that it is perfectly free to spin?
For both cases---the unclamped disk and the clamped disk---the torque is the same. Correct? The difference between these two cases is in the calculation of the moment of inertia. In the unclamped case, there is only translational motion of the disk. That is, if we were to draw an arrow on the disc at one point in time, that arrow would still point in the same direction at a later point in time. In the clamped case, however, the motion is both rotational and translational. That is, if we were to draw an arrow on the disk as before, the direction of the arrow would change in time.
In the unclamped case, we were able to treat the disk as a point particle. But in the clamped case, we are not able to treat the disk as a point particle because for different points on the disk, the speeds are different.
My big problem is in the calculation of the moment of inertia. I'm just... really confused as to what is truly going on here. I'd love some help ASAP ttt___ttt
In the unclamped case, there are two pivot points---one at the end of the rod, and one in the middle of the disk, which is what is throwing me off. So how do I calculate the moment of inertia of both cases with the parallel axis theorem?
For the clamped case, you could use
Ip = Ic + ml^2 (general case, where Ip is moment of inertia about pivot, and Ic is moment of inertia about center of mass. But here, what is Ic, since there is the disc at the very end? and "m" here is really the mass of the rod plus the mass of the disk?)
And how would the parallel axis theorem be used with the unclamped case???
Last edited: Nov 8, 2007
2. Nov 8, 2007
### AmaniKaleo
I forgot to put in my beautiful picture of the 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": 0.9462042450904846, "perplexity": 312.5031145133969}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1484560280835.22/warc/CC-MAIN-20170116095120-00482-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://mathprelims.wordpress.com/2008/07/16/linfty-is-not-separable/ | # Mathematics Prelims
## July 16, 2008
### l^\infty is not separable
Filed under: Analysis,Functional Analysis — cjohnson @ 6:21 pm
Recall that a a subset $M$ of a metric space $X$ is called dense if the closure of $M$ is the entire space; $\overline{M} = X$. If $M$ is a dense subset of $X$, then every point of $X$ is a limit point of $M$, and so for every $x \in X$ and $r > 0$, the open ball $B_r(x)$ must contain a point of $M$ that is distinct from $x$. We say that a space is separable if it has a countable dense subset.
The space $\ell^\infty$ is the set of all bounded sequences of real (or complex) numbers where the metric is given by $d(x, y) = \sup_{n \in \mathbb{N}} |x_n - y_n|$ where $x = (x_n)_{n \in \mathbb{N}}$ and similarly for $y$.
Consider the set of all sequences that whose entries are made up of zeroes and ones. Obviously this is a subset of $\ell^\infty$. Furthermore, each of these sequences corresponds to the binary representation of a number in $(0, 1)$, and every number in $(0, 1)$ has a binary representation, so a bijective mapping between $(0, 1)$ and our set exists. This means that our set is uncountable. Note that because of the metric on $\ell^\infty$, any two (distinct) elements in the set are distance one apart. If we place a ball of radius $r < \frac{1}{2}$ around each point, then none of these balls will intersect. This tells us that, since any dense subset of $\ell^\infty$ must have an element in each ball, any dense subset of $\ell^\infty$ must be uncountable, so $\ell^\infty$ is not separable.
1. I’m not sure how similar your prelim is to mine, but this is a nice exercise: Can you use what you just proved to show that $(\ell^\infty)^*\neq \ell^1$?
Comment by hilbertthm90 — July 16, 2008 @ 10:26 pm
2. This is not quite a bijection: the sequence (0,0,1,1,1,1,…) corresponds to the same point in (0,1) as the sequence (0,1,0,0,0,0,…). However, if you remove sequences with an infinite succession of ones, then what is left does biject with (0,1), and so that zero/one set of sequences must be uncountable because a subset of it bijects with (0,1).
Comment by Don — February 6, 2009 @ 12:22 pm
3. Oh, with regard to the previous comment, I think that’s quite simple. Separability is preserved by taking duals, and l^1 is separable, so it can’t be the dual of l^inf.
Comment by Don — February 6, 2009 @ 12:24 pm
4. I understand the proof above that says l^inf is not seperable. But I don’t understand how this is not a contradiction of the theorem: An infinite dimensional normed space is seperable iff it has a countable orthonormal basis. (proof is trivial and found almost anywhere on web).
What I don’t get is that surely l^inf is seperable because it has a countable orthonormal basis which is the set e_i = (0,0,0…1,…0,0,0) where the 1 is in the i-th place. The set e_1, e_2, e_3, …… etc is countable and forms a basis of l^inf so surely l^inf is seperable. Right? (well clearly I am wrong, but I am not sure how?)
Comment by Questionmark — June 6, 2011 @ 8:00 am
• The quick answer is that the set {e_1, e_2, e_3, …} is not a basis. It’s a subtle point, but if a set is a basis then everything in the space can be written as a _finite_ linear combination of things in the basis. The set {e_1, e_2, …} isn’t a basis because all the finite linear combinations of elements of that set will have infinitely many zeros.
Comment by cjohnson — June 6, 2011 @ 10:53 am
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": 27, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9610300064086914, "perplexity": 270.05085706042115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128319943.55/warc/CC-MAIN-20170623012730-20170623032730-00712.warc.gz"} |
https://hyperleap.com/topic/Analysis_of_variance | # Analysis of variance
ANOVAanalysis of variance (ANOVA)corrected the meansANOVAsvarianceanalysisanalysis-of-varianceanalyzing the dataANOVA analysisFactorial ANOVA
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among group means in a sample.wikipedia
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### Ronald Fisher
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From 1919 onward, he worked at the Rothamsted Experimental Station for 14 years; there, he analysed its immense data from crop experiments since the 1840s, and developed the analysis of variance (ANOVA).
### Variance
sample variancepopulation variancevariability
The ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation.
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### Design of experiments
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design of experiments.
### Oscar Kempthorne
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This design-based analysis was discussed and developed by Francis J. Anscombe at Rothamsted Experimental Station and by Oscar Kempthorne at Iowa State University.
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### Effect size
Cohen's deffect sizesmagnitude
a discovery are also high) and effect size (a smaller effect size is more prone to Type II error).
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### Restricted randomization
nested datasplit plotNested factors
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### Statistical hypothesis testing
hypothesis testingstatistical teststatistical tests
In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means.
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### Frank Anscombe
Anscombe, FrancisFrancis AnscombeAnscombe
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### Degrees of freedom (statistics)
degrees of freedomdegree of freedomEffective degrees of freedom
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The term is most often used in the context of linear models (linear regression, analysis of variance), where certain random vectors are constrained to lie in linear subspaces, and the number of degrees of freedom is the dimension of the subspace.
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error sum of squaressum of squared errors
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F distributionF''-distributionF'' distribution
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one-way ANOVA1-way ANOVAavailable
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### Interaction (statistics)
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### Charles Sanders Peirce
PeirceC. S. PeirceCharles S. Peirce
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F''-testF testF-statistic
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trendTrend estimationtrends
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MANOVAMultivariate analysis of variance (MANOVA)multi-way ANOVA
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Two-way ANOVAtwo-way linear-model
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### General linear model
multivariate linear regressionmultivariate regressionGLM
ANOVA is considered to be a special case of linear regression which in turn is a special case of the general linear model.
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### Power (statistics)
statistical powerpowerpowerful
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### Analysis of covariance
ANCOVAcovariance analysis
Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression.
### ANOVA on ranks
In statistics, one purpose for the analysis of variance (ANOVA) is to analyze differences in means between groups. | {"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": 0.9146614074707031, "perplexity": 2001.4021804512777}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20200923050545-20200923080545-00334.warc.gz"} |
https://www.physicsforums.com/threads/where-does-the-three-go.829804/ | # Where does the three go?
1. Aug 28, 2015
### zelly15
1. The problem statement, all variables and given/known data
One liter (1000 cm3) of oil is spilled onto a smooth lake. If the oil spreads out uniformly until it makes an oil slick just three molecules thick, with adjacent molecules just touching, estimate the diameter of the oil slick. Assume the oil molecules have a diameter of 2 ✕ 10^-10m.
2. Relevant equations
V= (h)(pi)(r^2)
3. The attempt at a solution
[sqrt 10m^3/(3)pi(2x10^-10)] *2
2. Aug 28, 2015
### Staff: Mentor
You have a cube of oil 1000 cm^3 and once its dispersed you'll have a cylinder of oil whose height is 3 molecules thick.
Pay attention to units conversion as the cube's volume is in cm^3 whereas the diameter of the oil molecule is in meters. | {"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": 0.9414413571357727, "perplexity": 3741.103547805954}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647883.57/warc/CC-MAIN-20180322112241-20180322132241-00437.warc.gz"} |
https://scholar.harvard.edu/jclee/publications/effects-thermodynamic-stability-wind-properties-different-low-mass-black-hole | # The effects of thermodynamic stability on wind properties in different low mass black hole binary states
### Citation:
Chakravorty S, Lee JC, Neilsen J. The effects of thermodynamic stability on wind properties in different low mass black hole binary states. Monthly Notices of the Royal Astronomical Society [Internet]. 2013;436 :560-569.
### Abstract:
We present a systematic theory-motivated study of the thermodynamic stability condition as an explanation for the observed accretion disk wind signatures in different states of low mass black hole binaries (BHB). The variability in observed ions is conventionally explained either by variations in the driving mechanisms or the changes in the ionizing flux or due to density effects, whilst thermodynamic stability considerations have been largely ignored. It would appear that the observability of particular ions in different BHB states can be accounted for through simple thermodynamic considerations in the static limit. Our calculations predict that in the disk dominated soft thermal and intermediate states, the wind should be thermodynamically stable and hence observable. On the other hand, in the powerlaw dominated spectrally hard state the wind is found to be thermodynamically unstable for a certain range of 3.55 <= log \xi <= 4.20. In the spectrally hard state, a large number of the He-like and H-like ions (including e.g. Fe XXV, Ar XVIII and S XV) have peak ion fractions in the unstable ionization parameter (\xi) range, making these ions undetectable. Our theoretical predictions have clear corroboration in the literature reporting differences in wind ion observability as the BHBs transition through the accretion states Lee et al. 2002; Miller et al. 2008; Neilsen & Lee 2009; Blum et al. 2010; Ponti et al. 2012; Neilsen & Homan 2012). While this effect may not be the only one responsible for the observed gradient in the wind properties as a function of the accretion state in BHBs, it is clear that its inclusion in the calculations is crucial to understanding the link between the environment of the compact object and its accretion
Publisher's Version
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https://www.physicsforums.com/threads/help-e-mc-2-and-heat.67442/ | # Help! E=mc^2 and heat
1. Mar 15, 2005
### pinkie
I posted this on the college board, but think it should go here. It is a college course, but at a high school level.
I feel kind of silly because this is probably a very basic problem, but I'm having problems with physics in general. Would anyone be able to tell me if I'm on the right track for the following question (we are focusing on Einstein's theories of relativity right now)?
1. Is the mass of a frying pan different when it is hot compared to when it is cold?
Yes it is, as E=mc^2 proves. Because the mass is equivalent to the energy of the object, when the energy increases, so does the mass. As energy increases when an object is heated, the mass of the pan will also increase when it is heated. The mass will decrease when the pan is cool and the energy has decreased.
~~
I hope I have this right. Lately I feel very hopeless at physics. Any help or comment would be very much appreciated
2. Mar 15, 2005
### Andrew Mason
Yes. You seem to grasp the essential point about mass-energy equivalence (as long as you understand that the change in mass is so small as to be unmeasureable).
Physics seems hard, in part, because it builds on concepts and principles. If you are having trouble, go back and redo something you think you really understand and work back up to the thing you are trying to understand. You will invariably find that there was something earlier that you thought you understood but didn't quite. Also work on problems, problems and more problems.
And then again physics seems hard because it is. But that just makes it all the more rewarding when you discover that you understand something new. Like E=mc^2.
AM
3. Mar 15, 2005
### pinkie
Thank you. I really appreciate the help. Physics is very new to me, so I guess you are right to tell me to practice. Thanks again. :) | {"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": 0.8933007121086121, "perplexity": 351.7471564775397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1484560280929.91/warc/CC-MAIN-20170116095120-00062-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.pattayamail.com/business/private-sector-opts-for-nuclear-power-22886 | # Private sector opts for nuclear power
1
798
BANGKOK, 27 February 2013 The Thai National Shippers’ Council (TNSC) proposed nuclear power as an alternative energy solution to protect the country’s economy from future risk of power disruption.
TNSC chief Paiboon Polsuwanna said the construction of nuclear plants is a viable option to cope with energy demand during the peak of the hottest season in the country. He added that there has been resistance to nuclear energy after a nuclear disaster in 2011 at Japan’s Fukushima nuclear power plant. However, following Vietnam’s recently announced plans to build 5 nuclear plants, Thailand’s risk levels for a possible nuclear meltdown in close proximity to the country’s territory has already been heightened.
Mr Paiboon also criticized the government’s delayed decision to announce Myanmar’s anticipated routine maintenance on a gas platform during April 5-14. He remarked that urging industrial estates to cut back on power during the 10-day period is not a sustainable solution as a number of sectors such food-processing and frozen goods cannot halt operations for a long period of time.
Recently, the country’s Energy Minister Pongsak Raktapongpaisarn has targeted the development of more coal-fired power plants as a way to meet surging demand. He had also commented that renewables and nuclear power sources were generally too expensive. | {"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": 0.8713516592979431, "perplexity": 4492.341837237389}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320304572.73/warc/CC-MAIN-20220124155118-20220124185118-00392.warc.gz"} |
http://www.neuraldump.com/glossary/continuous-function/ | A function on the set of real numbers is considered to be continuous if its graph on the Cartesian plane is a smooth curve with no “holes” or “jumps.” In other words, any arbitrarily small change to the function’s input corresponds to an arbitrarily small change in its output.
A function is said to be continuous at a point, if there is no hole or jump at that point. If there is, the point is called a discontinuity. So, for a function to be continuous, it must be continuous at every point in its domain.
This definition is usually sufficient to judge a function’s continuity, but it is not mathematically rigorous. More formally, we can say that if f is function mapping I, a subset of the real numbers, to R, the set of real numbers:
Then is continuous at a point c, if the limit of f(x) as x approaches c exists and is equal to f(c). In mathematical notation:
There are 3 rules implicit in this definition:
1. f is defined at c,
2. the limit as x approaches c from the lefthand side of the domain must exist,
3. and that limit must equal f(c).
« Back to Glossary Index | {"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": 0.9624415636062622, "perplexity": 281.3009338238409}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818693866.86/warc/CC-MAIN-20170925235144-20170926015144-00665.warc.gz"} |
https://math.stackexchange.com/questions/967459/metric-topology-induced-by-the-sum-of-two-metrics | Metric topology induced by the sum of two metrics
I have to show the following:
Let $X$ be a set with metrics $d_1$ and $d_2$ inducing metric topologies $\tau_1$ and $\tau_2$.
Define a new metric on $X$ where $d(x,y) = d_1(x,y) + d_2(x,y)$ for all $x,y$ in $X$.
a) Show that the topology $\tau_d$ induced by $d$ is finer than $\tau_1$ and finer than $\tau_2$.
b) Show that if $\tau_1 = \tau_2$, then $\tau_d = \tau_1$.
Part a:
Since the set of all metric open balls is a basis for the metric topology, I'll show that $\tau_d$ is finer than $\tau_1$ by showing that any $d_1$ metric open ball contains a $d$ metric open ball.
Let $B_x^{d_1}(\delta)$ be a metric open ball of radius $\delta$ centered at an arbitrary point $x$ in $X$ for metric $d_1$.
Let $\epsilon$ = $\delta$ + 0 = $\delta$. Thus, the metric open ball of radius $\epsilon$ centered at x for metric $d = d_1 + d_2$ is equal to the $d_1$ metric open ball, ie $B_x^{d1+d2}(\epsilon)$ = $B_x^{d_1}(\delta)$. Thus, $B_x^{d1+d2}(\epsilon) \subseteq B_x^{d_1}(\delta)$. Thus, the topology induced by metric $d$ is finer than the topology induced by metric $d_1$.
A similar argument shows that the topology induced by metric $d$ is finer than the topology induced by metric $d_2$.
I'm stuck on part b. I know that to show $\tau_1$ = $\tau_d$ we just need to show that $\tau_1$ is finer than $\tau_d$, since in part a we showed that $\tau_d$ is finer than $\tau_1$, but I'm not sure how to do that.
At first I wanted to say that $\tau_1 = \tau_2$ means that metric $d=d_1+d_2 = 2d_1$ has multiple kd(x,y) bounded above by $d_1(x,y)$ for all $x,y$ in $X$ when k $\le$ $\frac 12$, but then I realized that $\tau_1 = \tau_2$ only means that they have the same open sets and does not mean that $d1$ and $d2$ are the same metric. So, I'm not sure that there's an argument by means of bounding a constant multiple of $d(x,y)$ by $d_1(x,y)$.
I'm not sure how to approach an argument that every $d$ metric open ball must contain a $d_1$ metric open ball. It's seems kind of obvious that if you extend a $d_1$ metric open ball of radius $\delta_1$ by a non-negative distance $\delta_2$ given by metric $d_2$, a $d_1$ metric open ball of radius no more than $\delta_1$ must be contained in it. But it doesn't seem that we need $\tau_1 = \tau_2$ to argue that, so I think I'm missing something here.
To show that $\tau_d\supseteq\tau_1$ it’s not enough to show that each $d_1$-open ball contains a $d$-open ball: you must show that it contains a $d$-open ball with the same centre. You seem to have tried to do this, but it needs to be said as well, and your argument isn’t quite right, though you have found a $d$-open ball that works. All you have to show is that $B_x^d(\delta)\subseteq B_x^{d_1}(\delta)$. To do this, suppose that $y\in B_x^d(\delta)$; then $d_1(x,y)=d(x,y)-d_2(x,y)\le d(x,y)<\delta$, since $d_2(x,y)\ge 0$, so $y\in B_x^{d_1}(\delta)$. As you say, a similar argument yields the conclusion that $\tau_d\supseteq\tau_2$.
For (b) suppose that $U\in\tau_d$; we need to show that $U\in\tau_1$. Let $x\in U$; by hypothesis there is an $\epsilon>0$ such that $B_x^d(\epsilon)\subseteq U$, and we want a $\delta>0$ such that $B_x^{d_1}(\delta)\subseteq U$. We know that $B_x^{d_2}(\epsilon/2)\in\tau_1$, so there is a $\delta_1>0$ such that $B_x^{d_1}(\delta_1)\subseteq B_x^{d_2}(\epsilon/2)$. Let $\delta=\min\{\delta_1,\epsilon/2\}$; if $y\in B_x^{d_1}(\delta)$, what can you say about $d(x,y)$?
• @user92638: Okay. I’d do it a little differently, but that works. I’d simply say that if $y\in B_x^{d_1}(\delta)$, then $d_1(x,y)<\delta_1$, so $d_2(x,y)<\epsilon/2$. Moreover, $d_1(x,y)<\epsilon/2$, so $d(x,y)=d_1(x,y)+d_2(x,y)<\epsilon$. (I think that it’s a little easier to read this way.) – Brian M. Scott Oct 11 '14 at 8:27
• How does this sound? $$If y \in B_x^{d_1}(\delta), then$$ $$d(x,y) = d_1(x,y) + d_2(x,y) = min\{\delta_1, \epsilon/2\} + d_2(x,y) \tag*{}$$ $$< min\{\delta_1, \epsilon/2\} + \epsilon/2 (Since B_x^{d_1}(\delta) \subseteq B_x^{d_2}(\epsilon/2)) \le \epsilon,$$ so $$y \in B_x^{d_1}(\delta).$$ – user92638 Oct 11 '14 at 8:46
• @user92638: No, you can’t say that $d_1(x,y)=\min\{\delta_1,\epsilon/2\}$: what you know is that $d_1(x,y)$ is actually less than that minimum. Otherwise it’s okay. – Brian M. Scott Oct 11 '14 at 8:51
It suffices to show that if $$\{x_i\}_{i\in I}$$ is a net converging to $$x$$ in $$\tau_1$$ then $$\{x_i\}_{i\in I}$$ converges to $$x$$ in $$\tau_d$$ (because then any set closed in $$\tau_1$$ will be closed in $$\tau_d$$). But if $$x_i\rightarrow x$$ in $$\tau_1$$ then by assumption $$x_i\rightarrow x$$ in $$\tau_2$$ also, i.e. both $$d_1(x_i,x)$$ and $$d_2(x_i,x)$$ tend to $$0$$. Hence their sum does, so $$x_i\rightarrow x$$ in $$d$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 17, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9848958849906921, "perplexity": 61.13940471440427}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315618.73/warc/CC-MAIN-20190820200701-20190820222701-00422.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=75&t=30348&p=94441 | ## Energy from constant pressure versus constant volume RXN
mendozayael_2H
Posts: 31
Joined: Fri Sep 29, 2017 7:06 am
### Energy from constant pressure versus constant volume RXN
Does more energy come from a reaction at constant pressure or at constant volume? Does it have to do with work?
Alexander Peter 1F
Posts: 20
Joined: Wed Jun 28, 2017 3:00 am
### Re: Energy from constant pressure versus constant volume RXN
I'm not sure what you mean by energy coming from a reaction, but to my understanding when the volume is changing energy would be used in the expansion. So, less energy would be coming out from that.
Gurshaan Nagra 2F
Posts: 49
Joined: Thu Jul 27, 2017 3:01 am
### Re: Energy from constant pressure versus constant volume RXN
Are you asking like if the changes in in pressure and volume or the same in their respective situations. | {"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": 0.9614199995994568, "perplexity": 1929.0640306616226}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1606141181482.18/warc/CC-MAIN-20201125071137-20201125101137-00108.warc.gz"} |
http://math.stackexchange.com/questions/629178/solution-of-trigonometric-equations | # Solution of Trigonometric equations
If we were to solve the trigonometric equation $$\sqrt{13-18\tan x} = 6\tan x-3$$ by squaring both the sides, we would get two roots; $\tan x = \frac{2}{3}$ and $\tan x=-\frac{1}{6}$
2/3 is okay, but when we substitute -1/6 in the equation and simplify it, it becomes $\sqrt{16}$ = $-4$
The solution to the question says that this must be rejected, as $\sqrt{16}$ = $|4|$, which cannot be equal to $-4$.
Why is this done? Why can't the square root of $16$ be equated to $-4$?
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Square root is a function and has only one value which is the positive one. Square root is the function $y=\sqrt {x}$. Don't confuse it with $y=x^{2}$, which is a relation.
Hence $\sqrt {16}=4$ and not $-4$.
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Thanks for that perspective. I looked around a bit more, and found that it's a very satisfying answer. Thanks once again! – Dhananjay Gupta Jan 6 '14 at 16:28
please upvote and accept the answer if you liked it.. – Apurv Jan 6 '14 at 16:29
What you need to know is that $$\sqrt{13-18\tan x}=6\tan x-3$$ is not the same as $$13-18\tan x=(6\tan x-3)^2.$$ These are different equations.
Notice that $$\sqrt{13-18\tan x}=6\tan x-3$$ $$\iff 13-18\tan x=(6\tan x-3)^2\ \ \text{and}\ \ 6\tan x-3\ge0$$ $$\iff 13-18\tan x=(6\tan x-3)^2\ \ \text{and}\ \ \tan x\ge\frac 12.$$
So, we know that $\tan x=-1/6$ is not a solution of the equation at the top.
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I see. But i'm still unable to understand why you had to define RHS as a positive value when you squared both sides. Why does the fact that we're taught in elementary math, that sqrt 16 = 4 or -4, invalid here? Is it because the notation $\sqrt{16}$ can only be used for 4, not -4? – Dhananjay Gupta Jan 6 '14 at 16:15
Note that $\sqrt{16}$ is positive. It is already positive. So, $\sqrt{13-18\tan x}\ge0$. For example, if you have an equation $\sqrt x=-2,$ do you think you can solve it? – mathlove Jan 6 '14 at 16:18
Oh okay. So in the original equation, LHS already is positive, which is why, upon solving, RHS cannot be negative. I understand. Thanks a lot for your help! – Dhananjay Gupta Jan 6 '14 at 16:23
You are welcome! – mathlove Jan 6 '14 at 16:35
The real problem is that squaring both sides of an equation may introduce false solutions to the original equation. For example, if you start with $$x = -1$$ and square both sides, you get $$x^2 = 1$$ which has solutions $x=1$ and $x=-1$. Obviously only one of these is a solution to the original equation. You must check your solutions in the original equation if you square and solve that way.
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Why can't the square root of 16 be equated to -4?
It can be... But in a different context... Later on, when you'll learn about complex numbers, they'll teach you that each radical or order n returns n distinct complex values. E.g., $\sqrt[4]1=\{\pm1,\pm i\}$. In other words, a radical isn't really a function, but a relation, since a function, by definition, returns exactly one value. If, however, you do want to transform it into one, then you have to make sure that it fulfills this demand, and the only way to do this is to restrict its value domain. And since we are only working with real numbers (for now), and since people are more familiar with positive numbers than with negative ones, then, by convention, we choose this domain to be restricted to the positive reals, or $\mathbb{R}^+$.
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Yes, i know about complex numbers, and the complex roots of unity, but what i was confused about was the more fundamental fact that my teachers taught me about when I was younger. I now see that my understanding was correct, but because of a lack of proper knowledge of relations and functions, I was getting confused. Thanks for the help! – Dhananjay Gupta Jan 10 '14 at 17:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9338743090629578, "perplexity": 246.15072248003088}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246641054.14/warc/CC-MAIN-20150417045721-00232-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/159970/chain-rule-and-inverse-in-matrix-calculus?answertab=votes | # Chain rule and inverse in matrix calculus
I am having trouble understanding the derivation of some seemingly simple matrix derivatives and am wondering if there is an intuitive (perhaps geometric) explanation. I am reasonably well-versed in multivariate calculus and linear algebra, but am not comfortable with tensor math.
The function I am interested in is $f(t)=\mathbf{B}^T(\mathbf{X}+t\mathbf{Y})^{-1}\mathbf{A}$, where $t$ is a scalar, and $\mathbf{A},\mathbf{B},\mathbf{X},\mathbf{Y}$ are matrices with conformant dimensions.
On the page 24 of the pdf of the appendix on matrix calculus in the book by Jon Dattorro (page 600 of the book), I find the formula for the first derivative of $f(t)$:
$$\frac{df}{dt}=-\mathbf{B}^T(\mathbf{X}+t\mathbf{Y})^{-1}\mathbf{Y}(\mathbf{X}+t\mathbf{Y})^{-1}\mathbf{A}$$
This sort of makes sense to me from my knowledge of calculus of functions of single variable: if you have $g(t)=a(x+ty)^{-1}b=ab(x+ty)^{-1}$, then $\frac{dg}{dt}=-ab(x+ty)^{-2}y=-a(x+ty)^{-1}y(x+ty)^{-1}b$ (from the chain rule and the power rule). That is, there is a clear similarity in the form.
What I don't understand is why the matrix equation for $\frac{df}{dt}$ looks the way it does. Is it due to non-commutativity of matrix multiplication? But how does that come in to this problem exactly? I've found the chain rule for matrix-valued function in the same pdf on page 8 (eq 1749) but I am not sure how to apply it here. Maybe I don't understand something about the calculus of the single-variable functions.
I guess I am asking if there is a way to derive the equation for $\frac{df}{dt}$ "from first principles" without using tensors.
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I think this follows more quickly from the product rule. The derivative of $t \mapsto X + tY$ is $Y$. You have $$0 = \frac{d}{dt}I = \frac{d}{dt} [(X+tY)^{-1}(X+tY)] = \frac{d}{dt}(X+tY)^{-1} * (X+tY) + (X+tY)^{-1}Y$$ and so $$\frac{d}{dt} (X+tY)^{-1} = -(X+tY)^{-1} Y (X+tY)^{-1}.$$ Multiplying on the left and right by $B^T$ and $A$ won't change much.
Let $C=X+tY$ and $D=C^{-1}$. As $C D = I$, then $C' D +C D'=0$. Therefore $D'=-C^{-1} C' C^{-1}$. As $C'=Y$, then $[(X+tY)^{-1}]'=-(X+tY)^{-1}Y~(X+tY)^{-1}$. Finally, $A$ and $B$ are constant matrices, and then $[B(X+tY)^{-1}A]'=-B(X+tY)^{-1}Y~(X+tY)^{-1}A$.
Hint: The mape : $i: GL_n \to GL_n$ sush that $i(u)=u^{-1}$ for all $u \in GL_n$ (where $GL_n$ is the linear group : invertibles matrices so) is differentiable and $i'(u).h=-u^{-1}hu^{-1}$ forall squarre matrixe $h$ and invertible matrix $u$. By composition , and using $v: t \mapsto X+tY$ differentiable $v'(t)=Y$ you can ask to this question. | {"extraction_info": {"found_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": 0.9617515802383423, "perplexity": 72.68943061227021}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1413558066265.80/warc/CC-MAIN-20141017150106-00027-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://chromium.googlesource.com/chromium/deps/libmtp/+/07c54f6511906d0b04833451c6fe8663db77dafd/src/unicode.h | blob: f612c32cd02f7781d4a609afba98b6d4b4206749 [file] [log] [blame]
/** * \file unicode.h * * This file contains general Unicode string manipulation functions. * It mainly consist of functions for converting between UCS-2 (used on * the devices) and UTF-8 (used by several applications). * * For a deeper understanding of Unicode encoding formats see the * Wikipedia entries for * UTF-16/UCS-2 * and UTF-8. * * Copyright (C) 2005-2007 Linus Walleij * * This library is free software; you can redistribute it and/or * modify it under the terms of the GNU Lesser General Public * License as published by the Free Software Foundation; either * version 2 of the License, or (at your option) any later version. * * This library is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU * Lesser General Public License for more details. * * You should have received a copy of the GNU Lesser General Public * License along with this library; if not, write to the * Free Software Foundation, Inc., 59 Temple Place - Suite 330, * Boston, MA 02111-1307, USA. * */ #ifndef __MTP__UNICODE__H #define __MTP__UNICODE__H int ucs2_strlen(uint16_t const * const); char *utf16_to_utf8(LIBMTP_mtpdevice_t*,const uint16_t*); uint16_t *utf8_to_utf16(LIBMTP_mtpdevice_t*, const char*); void strip_7bit_from_utf8(char *str); #endif /* __MTP__UNICODE__H */ | {"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": 0.9345702528953552, "perplexity": 3569.669743600874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572744.7/warc/CC-MAIN-20190916135948-20190916161948-00317.warc.gz"} |
https://brilliant.org/discussions/thread/time-zones/ | # Time Zones
In my geography class we just reviewed time zones. The time zones and meridians and prime meridian are all "unnatural" and just manmade. Many time zones are influenced politically and geographically, which results in awkward (to say the least) shapes and exceptions. Why don't humans just keep longitudes the same as latitudes, at least change the clearly outdated and dysfunctional system, or better yet have a universal time (literally) so that everybody, astronauts or earthlings, are "on the same page"? Yes, time is relative and confusing, but isn't there a better way? Time zones are so illogical and confusing that they are much harder for me to understand than any math concept I've ever approached...
Note by Justin Wong
6 years, 2 months ago
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Why does the US use the US Customary system, and not the metric system? There's your answer.
- 6 years, 2 months ago
But I was referring to international time zones... US decisions don't dictate all international affairs, especially when England was the most powerful country when time zones were created. That isn't much of a satisfying answer.
- 6 years, 1 month ago
That is not the answer to my question. The answer to my question is this: because of custom. It's too difficult to change it.
- 6 years, 1 month ago
Is it really? Some people say it is easier to adapt something than create. I am questioning customs and their relevance in today's world of change.
- 6 years, 1 month ago | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "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": 0.8788367509841919, "perplexity": 3693.452075738665}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570987795403.76/warc/CC-MAIN-20191022004128-20191022031628-00457.warc.gz"} |
http://mathhelpforum.com/calculus/121853-find-equation-tangent-normal-curve.html | # Thread: Find the equation of tangent and normal to the curve
1. ## Find the equation of tangent and normal to the curve
Question : Find the eq of tangent and normal to the curve
$y(x-2)(x-3)-x+7 = 0$
at the point where the curve meets the x-axis
-----------------------------------------------------------------
Solution:
$y = \frac{x-7}{x^2 - 5x +6 }$.............Is that correct?
$\therefore \frac{dy}{dx} = \frac{-x^2 + 14x -29}{(x^2 - 5x +6)^2}$...................Is that correct?????
2. Originally Posted by zorro
Question : Find the eq of tangent and normal to the curve
$y(x-2)(x-3)-x+7 = 0$
at the point where the curve meets the x-axis
-----------------------------------------------------------------
Solution:
$y = \frac{x-7}{x^2 - 5x +6 }$.............Is that correct?
$\therefore \frac{dy}{dx} = \frac{-x^2 + 14x -29}{(x^2 - 5x +6)^2}$...................Is that correct?????
differentiate (x - 7)/(x^2 - 5x + 6) - Wolfram|Alpha
You can do the simplifying of the Wolfram expression and see if it's equivalent to what you got.
3. ## what should i do nxt
Originally Posted by mr fantastic
differentiate (x - 7)/(x^2 - 5x + 6) - Wolfram|Alpha
You can do the simplifying of the Wolfram expression and see if it's equivalent to what you got.
$\frac{dy}{dx} = \frac{- (x^2 - 14x + 31)}{(x^2 - 5x + 6)^2}$
what should i do??????
Now eg of tangent is at $(x_0,y_0)$ is given by
$(y-y_0) = \frac{dy}{dx} (x-x_0)$
but i dont know $(x_0,y_0)$.................what should i do??????
4. Originally Posted by zorro
$\frac{dy}{dx} = \frac{- (x^2 - 14x + 31)}{(x^2 - 5x + 6)^2}$
what should i do??????
Now eg of tangent is at $(x_0,y_0)$ is given by
$(y-y_0) = \frac{dy}{dx} (x-x_0)$
but i dont know $(x_0,y_0)$.................what should i do??????
When the curve meets the x-axis, y = 0. Therefore $y_0 = 0$. To get $x_0$, solve $-x + 7 = 0$ (I hope you see why). | {"extraction_info": {"found_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": 17, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9236097931861877, "perplexity": 879.5132183276818}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164645800/warc/CC-MAIN-20131204134405-00084-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://flyingcoloursmaths.co.uk/ask-uncle-colin-oh-god-not/ | Dear Uncle Colin,
When evaluating an expression, after you’ve dealt with brackets, powers and roots, do you deal with multiplication and division left to right, or does multiplication take precedence?
- BIDMAS Idiocy Doesn’t Make Any Sense
Hi, BIDMAS, and thanks for your message!
There are three answers to this.
If you’re a computer, you evaluate operators of the same precedence - such as multiplication and divide - from left to right, by convention.
If you’re a human, you try to figure out from the context what the person writing the expression meant, regardless of BIDMAS. Someone on Reddit might write 1 / x-3 to mean $\frac{1}{x-3}$ rather than the $\frac{1}{x} - 3$ a strict interpretation of the rules gives.
And if the question is $6 \div 2(1+2)$, you say “This is fake maths, write the bloody thing properly.”
Hope that helps!
- Uncle Colin | {"extraction_info": {"found_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": 0.8982375264167786, "perplexity": 1574.6180599345153}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652663012542.85/warc/CC-MAIN-20220528031224-20220528061224-00761.warc.gz"} |
http://mathhelpforum.com/trigonometry/179573-verifying-trigonomic-identities-print.html | # Verifying Trigonomic Identities
• May 4th 2011, 07:49 PM
school420
Verifying Trigonomic Identities
this is a trigonomic identity, but im unsure how to verify it
1+1/tan(x)=1/sin(x)
• May 4th 2011, 08:03 PM
pickslides
Are you sure this works out?
Take the case where $\displaystyle x = \frac{\pi}{4}$
$\displaystyle 1+ \frac{1}{\tan \left( \frac{\pi}{4}\right)} = 1+\frac{1}{1} = 2$
$\displaystyle \frac{1}{\sin \left(\frac{\pi}{4}\right)}= \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}$
• May 4th 2011, 08:07 PM
school420
for what i was told by my teacher, it was suppossed to be , but i could be wrong , thanks soo much for the help though ! | {"extraction_info": {"found_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": 0.9653781652450562, "perplexity": 4062.5419517557652}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123318.85/warc/CC-MAIN-20170423031203-00209-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Bilinear_time%E2%80%93frequency_distribution | Bilinear time–frequency distribution
Bilinear time–frequency distributions, or quadratic time–frequency distributions, arise in a sub-field of signal analysis and signal processing called time–frequency signal processing, and, in the statistical analysis of time series data. Such methods are used where one needs to deal with a situation where the frequency composition of a signal may be changing over time;[1] this sub-field used to be called time–frequency signal analysis, and is now more often called time–frequency signal processing due to the progress in using these methods to a wide range of signal-processing problems.
Background
Methods for analysing time series, in both signal analysis and time series analysis, have been developed as essentially separate methodologies applicable to, and based in, either the time or the frequency domain. A mixed approach is required in time–frequency analysis techniques which are especially effective in analyzing non-stationary signals, whose frequency distribution and magnitude vary with time. Examples of these are acoustic signals. Classes of "quadratic time-frequency distributions" (or bilinear time–frequency distributions") are used for time–frequency signal analysis. This class is similar in formulation to Cohen's class distribution function that was used in 1966 in the context of quantum mechanics. This distribution function is mathematically similar to a generalized time–frequency representation which utilizes bilinear transformations. Compared with other time–frequency analysis techniques, such as short-time Fourier transform (STFT), the bilinear-transformation (or quadratic time–frequency distributions) may not have higher clarity for most practical signals, but it provides an alternative framework to investigate new definitions and new methods. While it does suffer from an inherent cross-term contamination when analyzing multi-component signals, by using a carefully chosen window function(s), the interference can be significantly mitigated, at the expense of resolution. All these bilinear distributions are inter-convertible to each other, cf. transformation between distributions in time–frequency analysis.
Wigner–Ville distribution
The Wigner–Ville distribution is a quadratic form that measures a local time-frequency energy given by:
${\displaystyle P_{V}f(u,\xi )=\int _{-\infty }^{\infty }f\left(u+{\tfrac {\tau }{2}}\right)f^{*}\left(u-{\tfrac {\tau }{2}}\right)e^{-i\tau \xi }\,d\tau }$
The Wigner–Ville distribution remains real as it is the fourier transform of f(u + τ/2)·f*(u − τ/2), which has Hermitian symmetry in τ. It can also be written as a frequency integration by applying the Parseval formula:
${\displaystyle P_{V}f(u,\xi )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {f}}\left(\xi +{\tfrac {\gamma }{2}}\right){\hat {f}}^{*}\left(\xi -{\tfrac {\gamma }{2}}\right)e^{i\gamma u}\,d\gamma }$
Proposition 1. for any f in L2(R)
${\displaystyle \int _{-\infty }^{\infty }P_{V}f(u,\xi )\,du=|{\hat {f}}(\xi )|^{2}}$
${\displaystyle \int _{-\infty }^{\infty }P_{V}f(u,\xi )\,d\xi =2\pi |f(u)|^{2}}$
Moyal Theorem. For f and g in L2(R),
${\displaystyle 2\pi \left|\int _{-\infty }^{\infty }f(t)g^{*}(t)\,dt\right|^{2}=\iint {P_{V}f(u,\xi )}P_{V}g(u,\xi )\,du\,d\xi }$
Proposition 2 (time-frequency support). If f has a compact support, then for all ξ the support of ${\displaystyle P_{V}f(u,\xi )}$ along u is equal to the support of f. Similarly, if ${\displaystyle {\hat {f}}}$ has a compact support, then for all u the support of ${\displaystyle P_{V}f(u,\xi )}$ along ξ is equal to the support of ${\displaystyle {\hat {f}}}$.
Proposition 3 (instantaneous frequency). If ${\displaystyle f_{a}(t)=a(t)e^{i\phi (t)}}$ then
${\displaystyle \phi '(u)={\frac {\int _{-\infty }^{\infty }\xi P_{V}f_{a}(u,\xi )d\xi }{\int _{-\infty }^{\infty }P_{V}f_{a}(u,\xi )d\xi }}}$
Interference
Let ${\displaystyle f=f_{1}+f_{2}}$ be a composite signal. We can then write,
${\displaystyle P_{V}f=P_{V}f_{1}+P_{V}f_{2}+P_{V}\left[f_{1},f_{2}\right]+P_{V}\left[f_{2},f_{1}\right]}$
where
${\displaystyle P_{V}[h,g](u,\xi )=\int _{-\infty }^{\infty }h\left(u+{\tfrac {\tau }{2}}\right)g^{*}\left(u-{\tfrac {\tau }{2}}\right)e^{-i\tau \xi }d\tau }$
is the cross Wigner–Ville distribution of two signals. The interference term
${\displaystyle I[f_{1},f_{2}]=P_{V}[f_{1},f_{2}]+P_{V}[f_{2},f_{1}]}$
is a real function that creates non-zero values at unexpected locations(close to the origin) in the ${\displaystyle (u,\xi )}$ plane. Interference terms present in a real signal can be avoided by computing the analytic part ${\displaystyle f_{a}(t)}$.
Positivity and smoothing kernel
The interference terms are oscillatory since the marginal integrals vanish and can be partially removed by smoothing ${\displaystyle P_{V}f}$ with a kernel θ
${\displaystyle P_{\theta }f(u,\xi )=\int _{-\infty }^{\infty }{\int _{-\infty }^{\infty }{P_{V}f(u',\xi ')}}\theta (u,u',\xi ,\xi ')\,u'\,d\xi '}$
The time-frequency resolution of this distribution depends on the spread of kernel θ in the neighborhood of ${\displaystyle (u,\xi )}$. Since the interferences take negative values, one can guarantee that all interferences are removed by imposing that
${\displaystyle P_{\theta }f(u,\xi )\geq 0,\qquad \forall (u,\xi )\in {{\mathbf {R} }^{2}}}$
The spectrogram and scalogram are examples of positive time-frequency energy distributions. Let a linear transform ${\displaystyle Tf(\gamma )=\left\langle f,\phi _{\gamma }\right\rangle }$ be defined over a family of time-frequency atoms ${\displaystyle \left\{\phi _{\gamma }\right\}_{\gamma \in \Gamma }}$. For any ${\displaystyle (u,\xi )}$ there exists a unique atom ${\displaystyle \phi _{\gamma (u,\xi )}}$ centered in time-frequency at ${\displaystyle (u,\xi )}$. The resulting time-frequency energy density is
${\displaystyle P_{T}f(u,\xi )=\left|\left\langle f,\phi _{\gamma (u,\xi )}\right\rangle \right|^{2}}$
From the Moyal formula,
${\displaystyle P_{T}f(u,\xi )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }P_{V}f(u',\xi ')P_{V}\phi _{\gamma (u,\xi )}(u',\xi ')\,du'\,d\xi '}$
which is the time frequency averaging of a Wigner–Ville distribution. The smoothing kernel thus can be written as
${\displaystyle \theta (u,u',\xi ,\xi ')={\frac {1}{2\pi }}P_{V}\phi _{\gamma (u,\xi )}(u',\xi ')}$
The loss of time-frequency resolution depends on the spread of the distribution ${\displaystyle P_{V}\phi _{\gamma (u,\xi )}(u',\xi ')}$ in the neighborhood of ${\displaystyle (u,\xi )}$.
Example 1
A spectrogram computed with windowed fourier atoms,
${\displaystyle \phi _{\gamma (u,\xi )}(t)=g(t-u)e^{i\xi t}}$
${\displaystyle \theta (u,u',\xi ,\xi ')={\frac {1}{2\pi }}P_{V}\phi _{\gamma (u,\xi )}(u',\xi ')={\frac {1}{2\pi }}P_{V}g(u'-u,\xi '-\xi )}$
For a spectrogram, the Wigner–Ville averaging is therefore a 2-dimensional convolution with ${\displaystyle P_{V}g}$. If g is a Gaussian window,${\displaystyle P_{V}g}$ is a 2-dimensional Gaussian. This proves that averaging ${\displaystyle P_{V}f}$ with a sufficiently wide Gaussian defines positive energy density. The general class of time-frequency distributions obtained by convolving ${\displaystyle P_{V}f}$ with an arbitrary kernel θ is called a Cohen's class, discussed below.
Wigner Theorem. There is no positive quadratic energy distribution Pf that satisfies the following time and frequency marginal integrals:
${\displaystyle \int _{-\infty }^{\infty }Pf(u,\xi )\,d\xi =2\pi |f(u)|^{2}}$
${\displaystyle \int _{-\infty }^{\infty }Pf(u,\xi )\,du=|{\hat {f}}(\xi )|^{2}}$
Mathematical definition
The definition of Cohen's class of bilinear (or quadratic) time–frequency distributions is as follows:
${\displaystyle C_{x}(t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }A_{x}(\eta ,\tau )\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau ,}$
where ${\displaystyle A_{x}(\eta ,\tau )}$ is the ambiguity function (AF), which will be discussed later; and ${\displaystyle \Phi (\eta ,\tau )}$ is Cohen's kernel function, which is often a low-pass function, and normally serves to mask out the interference in the original Wigner representation, ${\displaystyle \Phi \equiv 1}$.
An equivalent definition relies on a convolution of the Wigner distribution function (WD) instead of the AF :
${\displaystyle C_{x}(t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }W_{x}(\theta ,\nu )\Pi (t-\theta ,f-\nu )\,d\theta \,d\nu =[W_{x}\ast \Pi ](t,f)}$
where the kernel function ${\displaystyle \Pi (t,f)}$ is defined in the time-frequency domain instead of the ambiguity one. In the original Wigner representation, ${\displaystyle \Pi =\delta _{(0,0)}}$. The relationship between the two kernels is the same as the one between the WD and the AF, namely two successive Fourier transforms (cf. diagram).
${\displaystyle \Phi ={\mathcal {F}}_{t}{\mathcal {F}}_{f}^{-1}\Pi }$
i.e.
${\displaystyle \Phi (\eta ,\tau )=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\Pi (t,f)\exp(-j2\pi (t\eta -f\tau ))\,dt\,df,}$
or equivalently
${\displaystyle \Pi (t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau .}$
Ambiguity function
Main article: Ambiguity function
The class of bilinear (or quadratic) time–frequency distributions can be most easily understood in terms of the ambiguity function, an explanation of which follows.
Consider the well known power spectral density ${\displaystyle P_{x}(f)}$ and the signal auto-correlation function ${\displaystyle R_{x}(\tau )}$ in the case of a stationary process. The relationship between these functions is as follows:
${\displaystyle P_{x}(f)=\int _{-\infty }^{\infty }R_{x}(\tau )e^{-j2\pi f\tau }\,d\tau ,}$
${\displaystyle R_{x}(\tau )=\int _{-\infty }^{\infty }x\left(t+{\tfrac {\tau }{2}}\right)x^{*}\left(t-{\tfrac {\tau }{2}}\right)\,dt.}$
For a non-stationary signal ${\displaystyle x(t)}$, these relations can be generalized using a time-dependent power spectral density or equivalently the famous Wigner distribution function of ${\displaystyle x(t)}$ as follows:
${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }R_{x}(t,\tau )e^{-j2\pi f\tau }\,d\tau ,}$
${\displaystyle R_{x}(t,\tau )=x\left(t+{\tfrac {\tau }{2}}\right)x^{*}\left(t-{\tfrac {\tau }{2}}\right).}$
If the Fourier transform of the auto-correlation function is taken with respect to t instead of τ, we get the ambiguity function as follows:
${\displaystyle A_{x}(\eta ,\tau )=\int _{-\infty }^{\infty }x\left(t+{\tfrac {\tau }{2}}\right)x^{*}\left(t-{\tfrac {\tau }{2}}\right)e^{j2\pi t\eta }\,dt.}$
The relationship between the Wigner distribution function, the auto-correlation function and the ambiguity function can then be illustrated by the following figure.
By comparing the definition of bilinear (or quadratic) time–frequency distributions with that of the Wigner distribution function, it is easily found that the latter is a special case of the former with ${\displaystyle \Phi (\eta ,\tau )=1}$. Alternatively, bilinear (or quadratic) time–frequency distributions can be regarded as a masked version of the Wigner distribution function if a kernel function ${\displaystyle \Phi (\eta ,\tau )\neq 1}$ is chosen. A properly chosen kernel function can significantly reduce the undesirable cross-term of the Wigner distribution function.
What is the benefit of the additional kernel function? The following figure shows the distribution of the auto-term and the cross-term of a multi-component signal in both the ambiguity and the Wigner distribution function.
For multi-component signals in general, the distribution of its auto-term and cross-term within its Wigner distribution function is generally not predictable, and hence the cross-term cannot be removed easily. However, as shown in the figure, for the ambiguity function, the auto-term of the multi-component signal will inherently tend to close the origin in the ητ-plane, and the cross-term will tend to be away from the origin. With this property, the cross-term in can be filtered out effortlessly if a proper low-pass kernel function is applied in ητ-domain. The following is an example that demonstrates how the cross-term is filtered out.
Kernel properties
The Fourier transform of ${\displaystyle \theta (u,\xi )}$ is
${\displaystyle {\hat {\theta }}(\tau ,\gamma )=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\theta (u,\xi )e^{-i(u\gamma +\xi \tau )}\,du\,d\xi }$
The following proposition gives necessary and sufficient conditions to ensure that ${\displaystyle P_{\theta }}$ satisfies marginal energy properties like those of the Wigner–Ville distribution.
Proposition: The marginal energy properties
${\displaystyle \int _{-\infty }^{\infty }P_{\theta }f(u,\xi )\,d\xi =2\pi |f(u)|^{2},}$
${\displaystyle \int _{-\infty }^{\infty }P_{\theta }f(u,\xi )\,du=|{\hat {f}}(\xi )|^{2}}$
are satisfied for all ${\displaystyle f\in L^{2}(\mathbf {R} )}$ if and only if
${\displaystyle \forall (\tau ,\gamma )\in \mathbf {R} ^{2}:\qquad {\hat {\theta }}(\tau ,0)={\hat {\theta }}(0,\gamma )=1}$
Some time-frequency distributions
Wigner distribution function
Aforementioned, the Wigner distribution function is a member of the class of quadratic time-frequency distributions (QTFDs) with the kernel function ${\displaystyle \Phi (\eta ,\tau )=1}$. The definition of Wigner distribution is as follows:
${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }x\left(t+{\tfrac {\tau }{2}}\right)x^{*}\left(t-{\tfrac {\tau }{2}}\right)e^{-j2\pi f\tau }\,d\tau .}$
Modified Wigner distribution functions
Affine invariance
We can design time-frequency energy distributions that satisfy the scaling property
${\displaystyle {\frac {1}{\sqrt {s}}}f\left({\tfrac {t}{s}}\right)\longleftrightarrow P_{V}f\left({\tfrac {u}{s}},s\xi \right)}$
as does the Wigner–Ville distribution. If
${\displaystyle g(t)={\frac {1}{\sqrt {s}}}f\left({\tfrac {t}{s}}\right)}$
then
${\displaystyle P_{\theta }g(u,\xi )=P_{\theta }f\left({\tfrac {u}{s}},s\xi \right).}$
This is equivalent to imposing that
${\displaystyle \forall s\in \mathbf {R} ^{+}:\qquad \theta \left(su,{\tfrac {\xi }{s}}\right)=\theta (u,\xi ),}$
and hence
${\displaystyle \theta (u,\xi )=\theta (u\xi ,1)=\beta (u\xi )}$
The Rihaczek and Choi–Williams distributions are examples of affine invariant Cohen's class distributions.
Choi–Williams distribution function
The kernel of Choi–Williams distribution is defined as follows:
${\displaystyle \Phi (\eta ,\tau )=\exp(-\alpha (\eta \tau )^{2}),}$
where α is an adjustable parameter.
Rihaczek distribution function
The kernel of Rihaczek distribution is defined as follows:
${\displaystyle \Phi (\eta ,\tau )=\exp \left(-i2\pi {\frac {\eta \tau }{2}}\right),}$
With this particular kernel a simple calculation proves that
${\displaystyle C_{x}(t,f)=x(t){\hat {x}}^{*}(f)e^{i2\pi tf}}$
Cone-shape distribution function
The kernel of cone-shape distribution function is defined as follows:
${\displaystyle \Phi (\eta ,\tau )={\frac {\sin(\pi \eta \tau )}{\pi \eta \tau }}\exp \left(-2\pi \alpha \tau ^{2}\right),}$
where α is an adjustable parameter. See Transformation between distributions in time-frequency analysis. More such QTFDs and a full list can be found in, e.g., Cohen's text cited.
Spectrum of non-stationary processes
A time-varying spectrum for non-stationary processes is defined from the expected Wigner–Ville distribution. Locally stationary processes appear in many physical systems where random fluctuations are produced by a mechanism that changes slowly in time. Such processes can be approximated locally by a stationary process. Let ${\displaystyle X(t)}$ be a real valued zero-mean process with covariance
${\displaystyle R(t,s)=E[X(t)X(s)]}$
The covariance operator K is defined for any deterministic signal ${\displaystyle f\in L^{2}(\mathbf {R} )}$ by
${\displaystyle Kf(t)=\int _{-\infty }^{\infty }R(t,s)f(s)\,ds}$
For locally stationary processes, the eigenvectors of K are well approximated by the Wigner–Ville spectrum.
Wigner–Ville spectrum
The properties of the covariance ${\displaystyle R(t,s)}$ are studied as a function of ${\displaystyle \tau =t-s}$ and ${\displaystyle u={\frac {t+s}{2}}}$:
${\displaystyle R(t,s)=R\left(u+{\tfrac {\tau }{2}},u-{\tfrac {\tau }{2}}\right)=C(u,\tau )}$
The process is wide-sense stationary if the covariance depends only on ${\displaystyle \tau =t-s}$:
${\displaystyle Kf(t)=\int _{-\infty }^{\infty }C(t-s)f(s)\,ds=C*f(t)}$
The eigenvectors are the complex exponentials ${\displaystyle e^{i\omega t}}$ and the corresponding eigenvalues are given by the power spectrum
${\displaystyle P_{X}(\omega )=\int _{-\infty }^{\infty }C(\tau )e^{-i\omega \tau }\,d\tau }$
For non-stationary processes, Martin and Flandrin have introduced a time-varying spectrum
${\displaystyle P_{X}(u,\xi )=\int _{-\infty }^{\infty }C(u,\tau )e^{-i\xi \tau }\,d\tau =\int _{-\infty }^{\infty }E\left[X\left(u+{\tfrac {\tau }{2}}\right)X\left(u-{\tfrac {\tau }{2}}\right)\right]e^{-i\xi \tau }\,d\tau }$
To avoid convergence issues we suppose that X has compact support so that ${\displaystyle C(u,\tau )}$ has compact support in ${\displaystyle \tau }$. From above we can write
${\displaystyle P_{X}(u,\xi )=E[P_{V}X(u,\xi )]}$
which proves that the time varying spectrum is the expected value of the Wigner–Ville transform of the process X. Here, the Wigner–Ville stochastic integral is interpreted as a mean-square integral:[2]
${\displaystyle P_{V}(u,\xi )=\int _{-\infty }^{\infty }\left\{X\left(u+{\tfrac {\tau }{2}}\right)X\left(u-{\tfrac {\tau }{2}}\right)\right\}e^{-i\xi \tau }\,d\tau }$
References
1. ^ E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” Digital Signal Processing, vol. 19, no. 1, pp. 153–183, January 2009.
2. ^ a wavelet tour of signal processing, Stephane Mallat
• L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995. ISBN 978-0135945322
• B. Boashash, editor, “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003.
• L. Cohen, “Time-Frequency Distributions—A Review,” Proceedings of the IEEE, vol. 77, no. 7, pp. 941–981, 1989.
• S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.
• H. Choi and W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE. Trans. Acoustics, Speech, Signal Processing, vol. 37, no. 6, pp. 862–871, June 1989.
• Y. Zhao, L. E. Atlas, and R. J. Marks, “The use of cone-shape kernels for generalized time-frequency representations of nonstationary signals,” IEEE Trans. Acoustics, Speech, Signal Processing, vol. 38, no. 7, pp. 1084–1091, July 1990.
• B. Boashash, “Heuristic Formulation of Time-Frequency Distributions”, Chapter 2, pp. 29–58, in B. Boashash, editor, Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003.
• B. Boashash, “Theory of Quadratic TFDs”, Chapter 3, pp. 59–82, in B. Boashash, editor, Time-Frequency Signal Analysis & Processing: A Comprehensive Reference, Elsevier, Oxford, 2003. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 95, "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": 0.9517658948898315, "perplexity": 933.8021441542488}, "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-2016-44/segments/1476988719273.37/warc/CC-MAIN-20161020183839-00241-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/684/3/h/d/ | # Properties
Label 684.3.h.d Level $684$ Weight $3$ Character orbit 684.h Self dual yes Analytic conductor $18.638$ Analytic rank $0$ Dimension $4$ CM discriminant -19 Inner twists $4$
# Related objects
## Newspace parameters
Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.h (of order $$2$$, degree $$1$$, minimal)
## Newform invariants
Self dual: yes Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{19})$$ Defining polynomial: $$x^{4} - 11 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$
## $q$-expansion
Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( -3 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( -3 - \beta_{3} ) q^{7} -\beta_{2} q^{11} + ( 2 \beta_{1} + \beta_{2} ) q^{17} + 19 q^{19} + ( \beta_{1} + 3 \beta_{2} ) q^{23} + ( 42 + 3 \beta_{3} ) q^{25} + ( -8 \beta_{1} - 7 \beta_{2} ) q^{35} + ( 43 + \beta_{3} ) q^{43} + ( -7 \beta_{1} + 4 \beta_{2} ) q^{47} + ( 88 + 5 \beta_{3} ) q^{49} + ( -7 - 7 \beta_{3} ) q^{55} + ( 49 - 5 \beta_{3} ) q^{61} + ( 7 - 11 \beta_{3} ) q^{73} + ( 14 \beta_{1} - 3 \beta_{2} ) q^{77} + ( 4 \beta_{1} + 12 \beta_{2} ) q^{83} + ( 141 + 13 \beta_{3} ) q^{85} + 19 \beta_{1} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{7} + O(q^{10})$$ $$4 q - 10 q^{7} + 76 q^{19} + 162 q^{25} + 170 q^{43} + 342 q^{49} - 14 q^{55} + 206 q^{61} + 50 q^{73} + 538 q^{85} + O(q^{100})$$
Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 11 x^{2} + 16$$:
$$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - 6 \nu$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 12 \nu$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{2} - 17$$
$$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 17$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 2 \beta_{1}$$
## Character values
We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$$.
$$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$
## Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
−3.04547 1.31342 −1.31342 3.04547
0 0 0 −9.97368 0 −13.8248 0 0 0
37.2 0 0 0 −5.61478 0 8.82475 0 0 0
37.3 0 0 0 5.61478 0 8.82475 0 0 0
37.4 0 0 0 9.97368 0 −13.8248 0 0 0
$$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
## Inner twists
Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
3.b odd 2 1 inner
57.d even 2 1 inner
## Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.h.d 4
3.b odd 2 1 inner 684.3.h.d 4
4.b odd 2 1 2736.3.o.k 4
12.b even 2 1 2736.3.o.k 4
19.b odd 2 1 CM 684.3.h.d 4
57.d even 2 1 inner 684.3.h.d 4
76.d even 2 1 2736.3.o.k 4
228.b odd 2 1 2736.3.o.k 4
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.h.d 4 1.a even 1 1 trivial
684.3.h.d 4 3.b odd 2 1 inner
684.3.h.d 4 19.b odd 2 1 CM
684.3.h.d 4 57.d even 2 1 inner
2736.3.o.k 4 4.b odd 2 1
2736.3.o.k 4 12.b even 2 1
2736.3.o.k 4 76.d even 2 1
2736.3.o.k 4 228.b odd 2 1
## Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 131 T_{5}^{2} + 3136$$ acting on $$S_{3}^{\mathrm{new}}(684, [\chi])$$.
## Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$3136 - 131 T^{2} + T^{4}$$
$7$ $$( -122 + 5 T + T^{2} )^{2}$$
$11$ $$12544 - 251 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$4096 - 803 T^{2} + T^{4}$$
$19$ $$( -19 + T )^{4}$$
$23$ $$( -1216 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 1678 - 85 T + T^{2} )^{2}$$
$47$ $$11669056 - 10043 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -554 - 103 T + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -15362 - 25 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( -19456 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9754915833473206, "perplexity": 2966.0476135443237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362999.66/warc/CC-MAIN-20211204154554-20211204184554-00288.warc.gz"} |
http://mathhelpforum.com/algebra/118149-help-radicals.html | # Math Help - Help with radicals :(
1. ## Help with radicals :(
i got this answer.. (after a long process)
2/3 * (3)^(3/2)
i know its suppose to be
2√3
the problem is how do i get that..
i got the decimal form but for some reason i can not remember how to get the radical form..
can someone please show me how to get it step by step..
Many Thanks
2. remember:
$\frac{1}{a} = a^{-1}$
and $a^{m}*a^{n}= a^{m+n}$
3. o.O oh okay yeah that helps a lot..
Thanks..
2/3 * 3^1 * 3^(1/2)
2/3 * 3 = 2
3^(1/2) = √3
= 2√3
Thanks dude that really helped me out can't believe i forgot about that... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8230025172233582, "perplexity": 2964.2083521829472}, "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-2014-52/segments/1419447547188.66/warc/CC-MAIN-20141224185907-00047-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/232058-limit-ln-sin-n-n-n-infinity.html | # Math Help - limit of ln|sin(n)|/n, n->infinity
1. ## limit of ln|sin(n)|/n, n->infinity
prove: $\lim_{n}\cfrac {ln|sin(n)|} {n} = 0 , n\rightarrow \infty$
or show it is not true.
thanks
2. ## Re: limit of ln|sin(n)|/n, n->infinity
I think the limit does not exist but im not sure.
Try with Lhopital rule , you will get limn->inf from cot(n) , and that does not exist as the function is oscilating to infinity without aproaching any specific value.
3. ## Re: limit of ln|sin(n)|/n, n->infinity
Since \displaystyle \begin{align*} \sin{(x)} \end{align*} does not have an infinite limit, then I don't see how \displaystyle \begin{align*} \ln{ \left| \sin{(x)} \right| } \end{align*} could possibly have one. So I would expect that the limit does not exist. Thus L'Hospital's Rule should not be used...
4. ## Re: limit of ln|sin(n)|/n, n->infinity
$0 \ge \lim_{n \to \infty} a_{n} \geq \limsup_{n \to \infty} a_{n} = 0$
5. ## Re: limit of ln|sin(n)|/n, n->infinity
Originally Posted by Cartesius24
$0 \geq \displaystyle{\lim_{n \to \infty}} a_{n} \geq \displaystyle{\limsup_{n \to \infty}}~ a_{n} = 0$
corrected latex for our interpreter
6. ## Re: limit of ln|sin(n)|/n, n->infinity
Originally Posted by Cartesius24
$0 \ge \lim_{n \to \infty} a_{n} \geq \limsup_{n \to \infty} a_{n} = 0$
May I ask what you got for the supremum of this sequence?
7. ## Re: limit of ln|sin(n)|/n, n->infinity
I would try to show that for any fixed $\varepsilon>0$ there are infinitely many n's such that $\sin x>1-\varepsilon$ and infinitely many n's such that $|\sin x|<\varepsilon$.
The basic idea is to use Dirichlet principle.
Similar approach was used here: Sine function dense in [−1,1] at MSE.
8. ## Re: limit of ln|sin(n)|/n, n->infinity
Originally Posted by Prove It
May I ask what you got for the supremum of this sequence?
Doen't matter . It is 1. More important is this is finite. | {"extraction_info": {"found_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": 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": 0.9981217384338379, "perplexity": 2868.638364667042}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1448398445033.85/warc/CC-MAIN-20151124205405-00190-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://hackmd.io/@5pwCvlLhSMm2E1skjPTOTQ/elbo | # How I learned to stop worrying and write ELBO (and its gradients) in a billion ways ## Overview <!-- I had a really hard time learning about VAE at the beginning of my PhD, mainly because it seems like every paper is writing the ELBO objective a slightly different way. However, as I mature, my attitude towards this have changed --- now I have learned to embrace the power of the infinitely many transforms of ELBO. this is gonna be a really boring blog for you if you are not interested in VAE, but if you are --> I had a really hard time learning about VAE at the beginning of my PhD. I felt very betrayed spending time deriving and memorising ELBO (the evidence lower bound objective), then seeing yet another paper that writes it in a different way. However, as I mature, my attitude towards this changed --- now I have learned to embrace the power of the seemingly infinitely many forms of ELBO. Thinking back, this transformation really took place when I was introduced by my supervisor [Sid](https://www.robots.ox.ac.uk/~nsid/) to this great series of literature that covers the evolution of ELBO over the last 5, 6 years. Organising all of them and describing them in non-jibberish took some time, but I hope that this will serve as a frustration-free note-to-self for future revisiting to the topic, and also that it can be helpful to people out there who are feeling equally bamboozled as I was a year ago. I will discuss the following papers (click on links for PDF), one in each section --- and trust me they each serve a purpose and tell a whole story: 1. [ELBO surgery](http://approximateinference.org/accepted/HoffmanJohnson2016.pdf) (warm up) \$$\Rightarrow\$$ A more intuitive (visualisable) way to write ELBO 2. [IWAE](https://arxiv.org/pdf/1509.00519.pdf) \$$\Rightarrow\$$ "K steps away" from basic VAE ELBO 3. [Sticking the landing](https://arxiv.org/pdf/1703.09194.pdf) \$$\Rightarrow\$$ What's wrong with ELBO and IWAE? 4. [Tighter isn't better](https://arxiv.org/pdf/1802.04537.pdf) \$$\Rightarrow\$$ What is wrong with IWAE, in particular? 6. [DReG](https://arxiv.org/pdf/1810.04152.pdf) \$$\Rightarrow\$$ How to fix IWAE? ## 0. Standard ELBO Before we dive in, let's look at the most basic form of ELBO first, here it is in all of its glory: \begin{align} \mathcal{L}(\theta, \phi) = \mathbb{E}_{z\sim q(z|x)} \displaystyle \left[\log \frac{p(x,z)}{q(z|x)} \right] < \log p(x),\\ \end{align} where \$$\theta,\phi\$$ denotes the generative and inference model respectively, \$$x\$$ the observation and \$$z\$$ sample from latent space. The objective serves as a lower bound to the marginal likelihood of observation \$$\log p(x)\$$, and the VAE is trained by maximising the likelihood of reconstruction through maximising ELBO. If you have this memorised or tattooed on your arm, we are ready to go! ## 1. A more intuitive (visualisable) guide to ELBO > Paper discussed: [ELBO surgery: yet another way to carve up the variational evidence lower bound](http://approximateinference.org/accepted/HoffmanJohnson2016.pdf), work by Matthew Hoffman and Matthew Johnson. This work provides a very intuitive perspective of the VAE objective by decomposing and rewriting ELBO. For a batch of N observations \$$X = \{x_n\}_{n=1}^N\$$ and their corresponding latent codes \$$Z = \{z_n\}_{n=1}^N\$$, ELBO can be rewritten as: \begin{align} \mathcal{L}(\theta, \phi) &= \underbrace{\left[ \frac{1}{N} \sum^N_{n=1} \mathbb{E}_{q(z_n|x_n)} [\log p(x_n | z_n)] \right]}_{\color{#4059AD}{\text{(1) Average reconstruction}}} - \underbrace{(\log N - \mathbb{E}_{q(z)}[\mathbb{H}[q(x_n|z)]])}_{\color{#EE6C4D}{\text{(2) Index-code mutual info}}} + \underbrace{\text{KL}(q(z)||p(z))}_{\color{#86CD82}{\text{(3) KL between q and p}}} \end{align} Where \$$q(z)\$$ is the marginal, i.e. \$$q(z)=\sum^{N}_{n=1}q(z,x_n) \$$, and for large N can be approximated by the average aggregated posterior \$$q^{\text{avg}}(z)=\frac{1}{N}\sum^N_{n=1}q(z|x_n)\$$. So what is the point of all this? Well, what's interesting with this decomposition is that <span style="color:#4059AD">(1) average reconstruction </span> and <span style="color:#EE6C4D">(2) index-code mutual information</span> have opposing effects on the latent space: - <span style="color:#4059AD">Term (1)</span> encourages accurate reconstruction of observations, which typically forces separated encoding for each \$$x_n\$$; - <span style="color:#EE6C4D">Term (2)</span> maximises the entropy of \$$q(x_n|z)\$$, and thereby promoting overlapping encoding \$$q(z|x_n)\$$ for disdinct observations. We visualise these effects in the graph below for two observations \$$x_1,x_2\$$ and their corresponding latent \$$z_1,z_2\$$. Plain and simple, <span style="color:#4059AD">(1)</span> encourages separate encodings by "squeeshing" each latent code, and <span style="color:#EE6C4D">(2)</span> "stretches" them, resulting in more overlap between \$$z_1\$$ and \$$z_2\$$. <p align='center'><img src="https://i.imgur.com/CbpEkBH.png" alt="drawing" width="450"/></p> Fig. Visualisation of effect of term <span style="color:#4059AD">(1)</span> and <span style="color:#EE6C4D">(2)</span>. Dotted lines represent inference model \$$\phi\$$ and solid lines generative model \$$\theta\$$. This now leaves us with <span style="color:#86CD82">term (3)</span>, which is the **only term that involves prior**. This term regularises the aggregated posterior by prior through minimising the KL distance between \$$q^{\text{avg}}(z)\$$ and \$$p(z)\$$. Theoretically speaking, \$$q^{\text{avg}}(z)\$$ can be arbitrarily close to \$$p(z)\$$ without losing expressivity of posterior; however in practice, when <span style="color:#86CD82">(3)</span> is too large, it always indicate unwanted regularisation effect from prior. Paper [*Disentangling disentanglement in Variational Autoencoders*](https://arxiv.org/pdf/1812.02833.pdf) also did a great job analysing and utilising the effect of these three terms for disentanglement in VAEs, and I strongly recommend that you go and have a look. # 2. "K steps away" from basic ELBO: IWAE > Paper discussed: [Importance Weighted Autoencoders](https://arxiv.org/pdf/1509.00519.pdf), work by Yuri Burda, Roger Grosse & Ruslan Salakhutdinov Hopefully the previous section served as a good warm-up for this blog, and now you have a better intuition on how ELBO affects the graphical model. Now, we will move just a tat away from the original ELBO, to a more advanced K-sampled lower bound estimator: IWAE. Importance Weighted Autoencoders (IWAE), is probably my favourite machine learning trick (and I know about 4). It is a simple and yet powerful way to improve the performance of VAEs, and you're really missing out if you went through the trouble to implement ELBO but stopped there. Here, I will talk about the formultaion of IWAE and its 3 benefits: ***tighter lower bound estimate***, ***importance-weighted gradients*** and ***complex implicit distribution***. ### Formulation IWAE proposes a tighter estimate to \$$\log p(x)\$$. As a reference, here's the original ELBO again: \begin{align} \mathcal{L}(\theta, \phi) = \mathbb{E}_{z\sim q(z|x)} \left[\log \frac{p(x,z)}{q(z|x)}\right] \leq \log p(x) \end{align} A common practice to acquire a better estimate to \$$\log p(x)\$$ with ELBO is to use its multisample variations, by taking \$$K\$$ samples from \$$q(z|x)\$$: \begin{align} \mathcal{L}_{\text{VAE}}(\theta, \phi) = \mathbb{E}_{z_1, z_2 \cdots z_K \sim q(z|x)} \left[\frac{1}{K} \sum_{k=1}^K \log \frac{p(x,z_k)}{q(z_k|x)}\right]\leq \log p(x) \end{align} IWAE simply switch the position between the sum over \$$K\$$ and the \$$\log\$$ of the above, giving us: \begin{align} \mathcal{L}_{\text{IWAE}}(\theta, \phi) = \mathbb{E}_{z_1, z_2 \cdots z_K \sim q(z|x)} \left[\log \frac{1}{K}\sum_{k=1}^K \frac{p(x,z_k)}{q(z_k|x)}\right]\leq \log p(x) \end{align} ### Benefit 1: Tighter lower bound estimate It is easy to see that by [Jensen's inequality](https://www.probabilitycourse.com/chapter6/6_2_5_jensen's_inequality.php), \$$\mathcal{L}_{\text{VAE}}(\theta, \phi)\leq\mathcal{L}_{\text{IWAE}}(\theta, \phi) \$$. This means that IWAE is a tighter lower bound to the marginal log likelihood. ### Benefit 2: Importance-weighted gradients Things become even more interesting if we look at the gradient of IWAE compared to the original ELBO: \begin{align} \nabla_\Theta \mathcal{L}_{\text{VAE}}(\theta,\phi)&=\mathbb{E}_{z_1, z_2 \cdots z_K \sim q(z|x)} \left[\sum_{k=1}^K \frac{1}{K} \nabla_\Theta \log \frac{p(x,z_k)}{q(z_k|x)}\right]\\ \nabla_\Theta \mathcal{L}_{\text{IWAE}}(\theta,\phi)&=\mathbb{E}_{z_1, z_2 \cdots z_K \sim q(z|x)} \left[\sum_{k=1}^K w_k \nabla_\Theta \log \frac{p(x,z_k)}{q(z_k|x)}\right], \end{align} where \begin{align} w_k = \frac{\frac{p(x,z_k)}{q(z_k|x)}}{\sum^K_{i=1}\frac{p(x,z_i)}{q(z_i|x)}} \end{align} So we can see that in the \$$\mathcal{L}_{\text{VAE}}\$$ the gradients of each samples are **equally weighted** by \$$1/K\$$, but in \$$\mathcal{L}_{\text{VAE}}\$$ gradient weights them by their **relative importance** \$$w_k\$$. ### Benefit 3: Complex implicit distribution However, this is not all of it --- authors in the [original paper](https://arxiv.org/pdf/1509.00519.pdf) also showed that IWAE can be interpreted as standard ELBO, but with a more complex (implicit) posterior distribution \$$q_{IW}\$$, thanks to importance sampling. This is probably the most important take-away of IWAE, and I always like go back to this plot from [reinterpreting IWAE](https://arxiv.org/pdf/1704.02916v2.pdf) as an intuitive demonstration of its power:  Here, K is the number of importance-weighted samples taken, and the left-most plot is the true distribution that we are trying to approximate with the 3 different \$$q_{IW}\$$. We can see that when \$$K=1\$$, the IWAE objective reduces to original VAE ELBO, and the approximation to true distribution is poor; as K grows, the approximation becomes more and more accurate. > Side note: Paper [Reinterpreting IWAE](https://arxiv.org/pdf/1704.02916v2.pdf) helped me a lot to understanding the IWAE objective, highly recommended. In addition, [this blog post](http://akosiorek.github.io/ml/2018/03/14/what_is_wrong_with_vaes.html) by Adam Kosiorek is also a very comprehensive interpretation on the topic. # 3. Big! Gradient! Estimator! Variance! > Paper discussed: [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://arxiv.org/pdf/1703.09194.pdf) by Geoffrey Roeder, Yuhuai Wu & David Duvenaud. So far we discussed two variational lower bounds in details, ELBO and IWAE. Now is high time to take them off their pedestals and talk about what's wrong with them --- and as you can guess from the title of this section, this has something to do with gradient variance. ## Recap: 2 types of gradient estimators Despite my best effort to sound very excited about all this, I had definitely struggled to care about things like "gradient variance" in the past, largely because there seems to be so many different Monte Carlo gradient estimators out there. But not too long ago, I realised that there are only two very common ones that you need to care about: REINFORCE estimator and reparametrisation trick. I'm leaving some details about each of them here as a note-to-self, but here's the key thing you need to remember if you want to skip this part and get to the good stuff: - **REINFORCE estimator (<span style="color:#87BBA2;font-weight:bold">score function</span>)**: very general purpose, large variance; - **Reparametrisation (<span style="color:#55828B;font-weight:bold">path derivative</span>)**: less general purpose, much smaller variance. Portal to [next section]((#4.Tighter-lower-bounds-aren't-necessarily-better)). #### REINFORCE estimator (score function) This is commonly used in Reinforcement Learning. It is named score function because it utilises this "cool little logarithm trick": \begin{align} \nabla_{\theta} \log p(x, \theta) = \frac{\nabla_\theta p(x;\theta)}{p(x;\theta)} \end{align} So now, when we try to estimate the gradient of some function \$$f(x)\$$ under the expectation of distribution \$$p(x;\theta)\$$, we can do the following: \begin{align} \nabla_{\theta}\mathbb{E}_{x\sim p(x,\theta)}[f(x)] = \mathbb{E}_{x\sim p(x,\theta)}[f(x)\nabla_{\theta} \log p(x,\theta)] \end{align} and now we can easily estimate the gradient by performing MC sampling --- taking $N$ samples of $\hat{x}\sim p(x,\theta)$: \begin{align} \nabla_{\theta}\mathbb{E}_{x\sim p(x,\theta)}[f(x)] \approx \frac{1}{N}\sum^N_{n=1}f(\hat{x}^{(n)})\nabla_\theta \log p(\hat{x}^{(n)};\theta); \end{align} Keep in mind that this score function estimator estimator, despite being unbiased, has **very large variance** from multiple sources (see [here](https://arxiv.org/pdf/1906.10652.pdf) in section 4.3.1 for details). It is however very flexible and places no requirement on \$$p(x;\theta)\$$ or \$$f(x)\$$ --- hence its popularity. #### Reparametrisation trick (pathwise derivative) I assume you are faimiliar with the reparametrisation trick if you got all the way here, but I am a completionist, so here's a quick recap: The reparametrisation trick utilises the property that for continuous distribution \$$p(x;\theta)\$$, the following sampling processes are equivalent: \begin{align} \hat{x} \sim p(x;\theta) \quad \equiv \quad \hat{x}=g(\hat{\epsilon},\theta) , \hat{\epsilon} \sim p(\epsilon) \end{align} The most common usage of this is seen in VAE, where instead of directly sampling from the posterior, we typically take random sample from a standard Normal distribution \$$\hat{\epsilon} \sim \mathcal{N}(0,1)\$$ and multiply it by the mean and variance of the posterior computed from our inference model. Here's that familiar illustration again as a reminder (image from [Kingma's NeurIPS2015 workshop slides](http://dpkingma.com/wordpress/wp-content/uploads/2015/12/talk_nips_workshop_2015.pdf)): <p align='center'><img src="https://i.imgur.com/wh2MJO6.png" alt="drawing" width="450"/></p> This method is much less general-purpose compared to the score function estimator since it requires \$$p(x)\$$ to be a continuous distribution, and also access to its underlying sampling path. However, by trading-off the generality, we get an estimator with much **lower variance**. > Side note: For readers who're not afraid of gradients, [here](https://arxiv.org/pdf/1906.10652.pdf) is a great survey paper on MC gradient estimators. ## The lurking score function in reparametrisation trick At this point we should all be familiar with reparametrisation trick used in VAEs for gradient estimation, but here we need to formalise it a bit more for the derivation in this section: > Reparametrisation trick express sample \$$z\$$ from parametric distribution \$$q_\phi(z)\$$ as a deterministic function of a random variable \$$\hat{\epsilon}\$$ with some fixed distribution and the parameters \$$\phi\$$, i.e. \$$z=t(\hat{\epsilon}, \phi)\$$. For example, if \$$q_\phi\$$ is a diagonal Gaussian, then for \$$\epsilon \sim \mathcal{N}(0, \mathbb{I}),\ z=\mu+\sigma\hat{\epsilon}\$$. We already know that reparametrisation trick (<span style="color:#55828B">path derivative</span>) has the benefit of lower variance for gradient estimation compared to <span style="color:#87BBA2">score function</span>. The kicker here is --- the gradient of ELBO actually contains a <span style="color:#87BBA2">score function</span> term, causing the estimator to have large variance! To see this, we can first rewrite ELBO as the following: \begin{align} \mathcal{L}(\theta,\phi) = \mathbb{E}_{z\sim q_\phi(z|x)}\left[ \log p_\theta(x|z) + \log p(z) - \log q_\phi(z|x) \right] \end{align} We can then take the total derivative of the term within expectation w.r.t. \$$\phi\$$: \begin{align} \nabla_{\phi} (\hat{\epsilon},\phi) &= \nabla_{\phi} \left[ \log p_\theta(x|z) + \log p(z) - \log q_\phi(z|x) \right]\\ &= \nabla_{\phi} \left[ \log p_\theta(z|x) + \log p(x) - \log q_\phi(z|x) \right]\\ &= \underbrace{\nabla_{z} \left[ \log p_\theta(z|x) - \log q_\phi(z|x) \right] \nabla_{\phi}t(\hat{\epsilon},\phi)}_{\color{#55828B}{\text{path derivative}}} - \underbrace{\nabla_{\phi}\log q_\phi(z|x)}_{\color{#87BBA2}{\text{score function}}} \end{align} So we see that \$$\nabla_{\phi} (\hat{\epsilon},\phi)\$$ decomposes into 2 terms, one <span style="color:#55828B">path derivative</span> component that measures the dependence on \$$\phi\$$ only through sample \$$z\$$; the <span style="color:#87BBA2">score function</span> the dependence on \$$\log q_\phi\$$ directly, without considering how sample \$$z\$$ changes as a function of \$$\phi\$$. So, it is not surprising to learn that the large variance of the <span style="color:#87BBA2">score function</span> term here causes problems: the authors discovered that even when the variational posterior \$$q_\phi(z|x)\$$ completely matches the true posterior \$$p(z|x)\$$, while the <span style="color:#55828B">path derivative</span> component in \$$\nabla_{\phi} (\hat{\epsilon},\phi)\$$ reduces to zero, <span style="color:#87BBA2">score function</span> will have non-zero variance. So what do we do here? Well, authors propose to simply drop the score function component to get an unbiased gradient estimator: \begin{align} \hat{\nabla}_{\phi} (\hat{\epsilon},\phi) &= \underbrace{\nabla_{z} \left[ \log p_\theta(z|x) - \log q_\phi(z|x) \right] \nabla_{\phi}t(\hat{\epsilon},\phi)}_{\color{#55828B}{\text{path derivative}}} \end{align} It sounds a bit wacky at first, but this approach works miracle, as authors show in this plot: <p align='center'><img src="https://i.imgur.com/oSuYLLA.png" alt="drawing" width="400"/></p> As we see clearly here that by using the path derivative only gradient, the variance of gradient estimation is much lower and \$$\phi\$$ converges to the true variational parameters much faster. Note that this large gradient variance problem applies for any ELBO, including both standard VAE and IWAE. However, we will show in the next section that IWAE has its unique problem caused by the K multiple samples，that is --- # 4.Tighter lower bounds aren't necessarily better > Paper discussed: [Tight Variational Bounds are Not Necessarily Better](https://arxiv.org/pdf/1802.04537.pdf), work by Tom Rainforth, Adam R. Kosiorek, Tuan Anh Le, Chris J. Maddison, Maximilian Igl, Frank Wood & Yee Whye Teh This builds on the previous [Sticking the Landing](https://arxiv.org/pdf/1703.09194.pdf) paper, and discovers that the gradient variance caused by score function becomes a even bigger problem when using a multi-sample estimator like IWAE. In here it's not just a variance problem: estimators with **small expected values** need **proportionally smaller variance** to be estimated accurately. In other words, what we really care about here is the expectation-to-variance, or signal-to-noise (SNR) ratio: <!-- \begin{align} \text{SNR}_{\text{M,K}} (\phi) = \frac{\mathbb{E}[\nabla_{\text{M,K}}(\phi)]}{\sigma[\nabla_{\text{M,K}}(\phi)]} \end{align} --> <!--  --> <p align='center'><img src="https://i.imgur.com/X8ukWX7.png" alt="drawing" width="400"/></p> Here \$$\nabla_{M,K}(\phi)\$$ refers to the gradient with respect to \$$\phi\$$, and here the two quantaties we care about are: - \$$M\$$: the number of samples used for Monte Carlo estimate of ELBO's **gradient**; - \$$K\$$: the number of samples used for **IWAE** to estimate a tighter lower bound to \$$\log p(x) \$$. Ideally we want a large SNR for the gradient estimator fo both \$$\theta\$$ and \$$\phi\$$, since smaller SNR indicates that the estimate is completely random. The main contribution of this paper is dicovering the following, very surprising relationships: \begin{align} \text{SNR}(\theta) &= \mathcal{O}(\sqrt{MK})\\ \text{SNR}(\phi) &= \mathcal{O}(\sqrt{M/K}) \end{align} This tells us that while increasing the number of IWAE samples \$$K\$$ get us a tighter lower bound, it actually worsen SNR\$$(\phi)\$$ --- meaning that **a large K hurts the performance of the gradient estimator for** \$$\phi\$$! Also note that the same effect is not observed for the generative model \$$\theta\$$, but the damage on inference model learning cannot simply be mitigated by increasing \$$M\$$. The authors gave a very comprehensive proof to their finding, so I'm going to leave the mathy heavy lifting to the original paper :) We shall march on to the last section of this blog: an elegant solution to solve the large variance in ELBO gradient estimators --- DReG. # 5. How to fix IWAE? > Paper discussed: [Doubly Reparametrised Gradient Estimators for Monte Carlo Objectives](https://arxiv.org/pdf/1810.04152.pdf), work by George Tucker, Dieterich Lawson, Shixiang Gu & Chris J. Maddison. In [section 3](#3.-Big!-Gradient!-Estimator!-Variance!) we talked about the large gradient variance caused by the score function lurking in the gradient estimation, and [section 4](#4.Tighter-lower-bounds-aren't-necessarily-better) about how this is exacerbated for IWAE. I'll put the total derivative we have seen in [section 3](#3.-Big!-Gradient!-Estimator!-Variance!) here as a reference, but to make it more relevant, this time we rewrite it for IWAE that uses \$$K\$$ importance samples: \begin{align} \nabla_{\phi} (\hat{\epsilon},\phi) = \mathbb{E}_{\hat{\epsilon}_{1:K}} \underbrace{\left[\sum_{k=1}^K w_k \nabla_{z} \left[ \log p_\theta(z|x) - \log q_\phi(z|x) \right] \nabla_{\phi}t(\hat{\epsilon},\phi)\right.}_{\color{#55828B}{\text{path derivative}}} - \underbrace{\left.\sum_{k=1}^K w_k \nabla_{\phi}\log q_\phi(z|x)\right]}_{\color{#87BBA2}{\text{score function}}} , \end{align} where \begin{align} w_k =\frac{\tilde{w}_k}{\sum^K_{i=1} \tilde{w}_i}= \frac{\frac{p(x,z_k)}{q(z_k|x)}}{\sum^K_{i=1}\frac{p(x,z_i)}{q(z_i|x)}}. \end{align} > This is not much of a change from the total derivative of original ELBO, as we have mentioned in [section 2](#2.-"K-steps-away"-from-basic-ELBO:-IWAE) that IWAE simply weights the gradients of VAE ElBO by the relative importance of each sample \$$w_k\$$. We have learned that one way to deal with it is to completely remove the score function term. However, is there a better way than completely discarding a term in gradient estimation? Well obviously I wouldn't be asking this question here if the answer weren't yes --- authors in this paper proposed to reduce the variance by **doing another reparametrisation on the score function term**! Here's how: Taking the score function term in the total derivative of IWAE, we can first take the \$$\sum_k\$$ term out of the expectation: \begin{align} \mathbb{E}_{\hat{\epsilon}_{1:K}} \underbrace{\left[\sum_{k=1}^K w_k \nabla_{\phi}\log q_\phi(z|x)\right]}_{\color{#87BBA2}{\text{score function}}} = \sum_{k=1}^K \mathbb{E}_{\hat{\epsilon}_{1:K}} \left[ w_k \nabla_{\phi}\log q_\phi(z|x)\right] \end{align} Now we can just ignore the sum and focus on what's in the expectation \$$\mathbb{E}_{\hat{\epsilon}_{1:K}}\$$. Since the derivative is taken with respect to \$$\phi\$$, we can treat \$$\epsilon\$$, the pseudo sample we take for reparametrisation trick, as a constant. Therefore, it is possible to substitute \$$\epsilon\$$ by the actual sample from our approximated posterior \$$z\$$ --- also a constant as far as \$$\nabla_\phi\$$ is concerned. This way we have: \begin{align} \mathbb{E}_{\hat{\epsilon}_{1:K}} \left[ w_k \nabla_{\phi}\log q_\phi(z|x)\right] &= \mathbb{E}_{z_{1:K}} \left[ w_k \nabla_{\phi}\log q_\phi(z|x)\right]\\\\ &= \mathbb{E}_{z_{-k}} \underbrace{\mathbb{E}_{z_k} \left[ w_k \nabla_{\phi}\log q_\phi(z|x)\right]}_{\text{A }\color{#EE6C4D}{\text{REINFORCE}}\text{ term appears!}} \end{align} :shocked_face_with_exploding_head: By doing this substitution, a REINFORCE term appears! I'll just let that sink in for a bit. >I should clarify that that previously we just had the score function term, but since the expectation is over \$$\hat{\epsilon}\$$ instead of actual samples from \$$q_\phi(z|x)\$$), it is not actually REINFORCE. This is important because REINFORCE and reparametrisation trick are interchangable, as we see below: \begin{align} \underbrace{\mathbb{E}_{q_\phi (z|x)}\left[ f(z)\frac{\partial}{\partial \phi}\log q_\phi(z|x) \right]}_{\text{REINFORCE}} = \underbrace{\mathbb{E}_{\hat{\epsilon}} \left[ \frac{\partial f(z)}{\partial z} \frac{\partial z(\hat{\epsilon}, \phi)}{\partial \phi} \right]}_{\text{reparametrisation trick}} \end{align} If we substitute the above back into the original total derivative of IWAE, after some math montage, we can simplifying it as the following: \begin{align} \nabla_{\phi} (\hat{\epsilon},\phi) = \mathbb{E}_{\hat{\epsilon}_{1:K}} \left[ \sum^K_{k=1} (w_k)^2 \frac{\partial\log \tilde{w}_k}{\partial z_i} \frac{\partial z_i}{\partial \phi} \right] \end{align} This is actually very easy to implement: cheeky little plug, we used this objective in our paper on multimodal VAE learning, you can find the code [here](https://github.com/iffsid/mmvae) that comes with a handy implementation of DReG in pytorch. # We are done! A heartfelt congratulation if you got all the way here, well done! Leave a comment if you have any question, if you find this helpful please share on twitter/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": 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": 22, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.972932755947113, "perplexity": 3222.7977363468885}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710953.78/warc/CC-MAIN-20221204004054-20221204034054-00252.warc.gz"} |
http://openmx.ssri.psu.edu/thread/1532?q=thread/1532 | # Common Pathways Model - total variance not adding to 100
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Common Pathways Model - total variance not adding to 100
Hello
I have run the common Pathways multivariate model for my data. When I total up the specific genetic variance and genetic variance from the latent variable to each of the phenotypes, I am getting less than 100%. I am also getting less than what the model says is the total genetic variance (from stA in the model attached). Is there anything I am doing wrong?
(output for one of my variables:
stA 0.3060186 (overall heritability)
stE 0.6939814
specific heritability (stas^2) 0.07233587
specific environment (stes^2) 0.3732820
latent heritability contribution (h2c) 0.1749177
Latent unique environment (e2c) 0.2400518)
Also I am getting different total heritability values (again from stA in the model) than what I am getting when I run each of the variables through a univariate model. Would this be correct or does it indicate a problem?
Many thanks
Karen
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Joined: 07/31/2009 - 15:12
Two things: First, Word likes
Two things:
First, Word likes to "adjust" formatting in ways that R and most other programs don't like, so try to use plain text files or .R files in the future.
Second, I didn't see a specific error, but you've apparently constrained stA and sdE to total 1.00 based on what you've posted. Is there a relation between stA and stas^2 & stE and stees^2 that I'm not seeing? | {"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": 0.8030918836593628, "perplexity": 3447.8120346577957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1484560281746.82/warc/CC-MAIN-20170116095121-00305-ip-10-171-10-70.ec2.internal.warc.gz"} |
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1. CJM Online first
Wang, Liping; Zhao, Chunyi
Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in $\mathbb B^N$ We consider the following prescribed boundary mean curvature problem in $\mathbb B^N$ with the Euclidean metric: $\begin{cases} \displaystyle -\Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N, \\[2ex] \displaystyle \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} \widetilde K(x) u^{2^\#-1} \quad & \text{on }\mathbb S^{N-1}, \end{cases}$ where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb S^{N-1}, 2^\#=\frac{2(N-1)}{N-2}$. We show that if $\widetilde K(x)$ has a local maximum point, then the above problem has infinitely many positive solutions that are not rotationally symmetric on $\mathbb S^{N-1}$. Keywords:infinitely many solutions, prescribed boundary mean curvature, variational reductionCategories:35J25, 35J65, 35J67
2. CJM 2012 (vol 65 pp. 702)
Taylor, Michael
Regularity of Standing Waves on Lipschitz Domains We analyze the regularity of standing wave solutions to nonlinear Schrödinger equations of power type on bounded domains, concentrating on Lipschitz domains. We establish optimal regularity results in this setting, in Besov spaces and in Hölder spaces. Keywords:standing waves, elliptic regularity, Lipschitz domainCategories:35J25, 35J65 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9747124910354614, "perplexity": 1157.2668179838363}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368705575935/warc/CC-MAIN-20130516115935-00030-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.research-collection.ethz.ch/handle/20.500.11850/429547?show=full | dc.contributor.author
Ibrogimov, Orif O.
dc.date.accessioned
2020-09-02T11:08:29Z
dc.date.available
2020-08-01T03:29:21Z
dc.date.available
2020-09-02T11:08:29Z
dc.date.issued
2020-07
dc.identifier.issn
0129-055X
dc.identifier.issn
1793-6659
dc.identifier.other
10.1142/S0129055X20500154
en_US
dc.identifier.uri
http://hdl.handle.net/20.500.11850/429547
dc.description.abstract
We study the spectrum of the spin-boson Hamiltonian with two bosons for arbitrary coupling alpha >0 in the case when the dispersion relation is a bounded function. We derive an explicit description of the essential spectrum which consists of the so-called two- and three-particle branches that can be separated by a gap if the coupling is sufficiently large. It turns out, that depending on the location of the coupling constant and the energy level of the atom (w.r.t. certain constants depending on the maximal and the minimal values of the boson energy) as well as the validity or the violation of the infrared regularity type conditions, the essential spectrum is either a single finite interval or a disjoint union of at most six finite intervals. The corresponding critical values of the coupling constant are determined explicitly and the asymptotic lengths of the possible gaps are given when a approaches to the respective critical value. Under minimal smoothness and regularity conditions on the boson dispersion relation and the coupling function, we show that discrete eigenvalues can never accumulate at the edges of the two-particle branch. Moreover, we show the absence of the discrete eigenvalue accumulation at the edges of the three-particle branch in the infrared regular case.
en_US
dc.language.iso
en
en_US
dc.publisher
World Scientific
en_US
dc.subject
Spin-boson Hamiltonian
en_US
dc.subject
Fock space
en_US
dc.subject
discrete spectrum
en_US
dc.subject
essential spectrum
en_US
dc.title
Spectral analysis of the spin-boson Hamiltonian with two bosons for arbitrary coupling and bounded dispersion relation
en_US
dc.type
Journal Article
dc.date.published
2019-11-21
ethz.journal.title
Reviews in Mathematical Physics
ethz.journal.volume
32
en_US
ethz.journal.issue
6
en_US
ethz.journal.abbreviated
Rev. Math. Phys.
ethz.pages.start
2050015
en_US
ethz.size
28 p.
en_US
ethz.identifier.wos
ethz.identifier.scopus
ethz.publication.place
Singapore
en_US
ethz.publication.status
published
en_US
ethz.date.deposited
2020-08-01T03:29:26Z
ethz.source
WOS
ethz.eth
yes
en_US
ethz.availability
en_US
ethz.rosetta.installDate
2020-09-02T11:08:40Z
ethz.rosetta.lastUpdated
2021-02-15T16:50:11Z
ethz.rosetta.versionExported
true
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http://ijoer.blogspot.com/2015/09/analytical-solution-with-two-time.html | ## Wednesday, 2 September 2015
### Analytical Solution with Two Time Scales of Circular Restricted Three-Body Problem
Analytical Solution with Two Time Scales of Circular Restricted Three-Body Problem
F.B. Gao1, H.L. Hu2
1School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China
Abstract Analytical solution performs a vital role in a wide variety of deep space exploration missions, especially the periodic solutions which were believed to be the unique avenue to solve three-body problem by Poincaré. As the absence of a general solution for the problem, an approximate analytical solution of the circular restricted three-body problem is addressed by employing multiple scales method in conjunction with some analytical techniques. It is worthwhile to note that the presented solution in three-dimensional space contains two time scales: and ( is a small, dimensionless parameter), which is significative to improve and perfect the known literature.
Keywords Analytical solution, multiple scales, three-body 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": 0.8893119096755981, "perplexity": 1085.007378801877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583829665.84/warc/CC-MAIN-20190122054634-20190122080634-00560.warc.gz"} |
https://gcc.gnu.org/legacy-ml/gcc-patches/2010-08/msg00606.html | This is the mail archive of the [email protected] mailing list for the GCC project.
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# Re: [Patch, Fortran, Doc] Inverse hyperbolic functions
Hi All,
With my remark to the info page for gfortran-4.5, I didn't want to cause so
much trouble ;-)). However, I'm pretty sure that also in the US the correct
name is area (co)sine/tangent hyperbolicus. I think this happened only quite
recently that the functions got wrong names. E.g., in Mathematica they are
also called ArcTanh etc. :-(. If you really want to change the info pages for
gfortran, I'd also say, take the neutral name, "inverse hyperbolic
<function>".
The geometric meaning is clear. If you look at the unit hyperbola, given by
the parametrization
\vec{r}(t)=(cosh t,sinh t); t \in |R,
then t/2 is the area of a sector of the right branch of this hyperbola.
That's pretty much the same as for the usual trigonometric functions. If you
take the parametrization
\vec{r}(t)=(cos t,sin t), t \in [0,2 pi)
then t/2 is the area of the corresponding sector of the circle.
In the case of the circle the angle, t, at the same time is the length of the
arc of this sector.
This latter is not true in the analoguous parametrization of the hyperbola
given above! That's why the inverse hyperbolic functions are named area
functions and not arc functions.
Cheers,
Hendrik.
http://en.wikipedia.org/wiki/Artanh
On Sunday 08 August 2010 19:55:08 Janus Weil wrote:
> Hi Dominique,
>
> > I agree that "Inverse hyperbolic *" is better than "Hyperbolic arc*"
> > (when I was taught hyperbolic functions and their inverses some 45 years
> > ago, they were denoted ch, sh, and th and the inverses argch, argsh,
> > and argth, I have no idea about the today fashion!-).
> >
> > However the inverse hyperbolic functions appear in many places and
> > not in non-Euclidian geometry only, so I am not very fond
> > (to say the least) of "hyperbolic area". Why not
> > "computes the inverse @var{X} of the hyperbolic * (@code{*(X)})."
> > (or any suitable translation from Frenglish to native English!).
>
> well, I'm starting to think that there may be regional differences in
> the naming of the inverse hyperbolics (though I'm not sure about
> this)?
>
> The common naming convention in Germany, I think, is that the inverse
> trigonometrics are called ARCTAN etc, while in contrast the inverse
> hyperbolics are labeled ATANH or ARTANH, where the 'A' or 'AR' in
> front stands for 'area' (and not for 'arc'). This is confirmed by my
> copy of the Mathematical Handbook by Bronstein et al. (which I think
> is Russian by origin), and which even goes into some depth to explain
> where this naming comes from.
>
> However, I just had a look into Abramowitz & Stegun, which has
> ARCTANH, cf. http://www.math.ucla.edu/~cbm/aands/page_86.htm
>
> So maybe it is indeed common to talk about an 'hyperbolic arctangent"
> in the US (and France?). What is the ultimate mathematical instance
> than we should follow? Maybe it is best to stick to the neutral
> "inverse hyperbolic tangent"? Or should we rather mention both naming
> conventions in the manual?
>
> Cheers,
> Janus
a
--
Hendrik van Hees
Justus-Liebig Universität Gießen
D-35392 Gießen
http://theorie.physik.uni-giessen.de/~hees/
Index Nav: Message Nav: [Date Index] [Subject Index] [Author Index] [Thread Index] [Date Prev] [Date Next] [Thread Prev] [Thread Next] | {"extraction_info": {"found_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": 0.9606400728225708, "perplexity": 3641.1606232263366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570921.9/warc/CC-MAIN-20220809094531-20220809124531-00250.warc.gz"} |
https://digitalcommons.lsu.edu/gradschool_disstheses/4516/ | ## LSU Historical Dissertations and Theses
1988
Dissertation
#### Degree Name
Doctor of Philosophy (PhD)
Thomas F. Moslow
#### Abstract
A strike-oriented trend of Wilcox oil and gas production in central Louisiana marks the location of an early Tertiary clastic shelf-margin. The shelf-margin contained a central unstable region flanked by two stable regions. The stable regions occurred where there was no significant progradation beyond the location of the underlying shelf-margins. Conversely, the unstable region occurred where progradation extended basinward of the underlying shelf-margins. Seven depositional sequences can be recognized within the shelf-margin. The vertical arrangement of these sequences shows that migration of the margin was negligible throughout Wilcox deposition, thereby suggesting a balance between subsidence and deposition. Through numerical simulation, it was concluded that published values for global cyclic sea-level fluctuations cannot be used to account for the development of these sequences. Rather, a sea-level which experiences variable rates of fall over a 0.5 $\times$ 10$\sp6$ year interval could account for the origin of such sequences. A shale-filled submarine canyon system occurs within the unstable region of the margin. Morphologically, the canyon cross-sectional profile resembles that of an entrenched fluvial system. Conventional cores through sandstone bodies from stable and unstable regions of the margin exhibit similarities in vertical sequence and interval thickness. Both sandstone bodies represent truncated progradational shoreface sequences which were associated with shelf-margin deltas. A computer program (DEPSIM) was developed in order to account for a significant difference observed between transition zone facies of the two shoreface sequences. The shoreface sequence from the unstable region of the margin contains a well-developed transition zone facies in contrast to that from the stable margin. Results from the program suggest that the difference between these two sequences may be explained by the fact that the sequence from the unstable region of the margin formed during a relative sea-level rise. The shoreface sequence from the stable region formed in response to a falling sea-level which resulted in extensive lateral translation of the shoreline. A vertical profile through the simulated shoreface sequence which formed during a sea-level fall therefore exhibits a poorly developed transition zone facies.
415
COinS | {"extraction_info": {"found_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": 0.8181982040405273, "perplexity": 3452.37002397507}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107885126.36/warc/CC-MAIN-20201025012538-20201025042538-00645.warc.gz"} |
http://physics.stackexchange.com/questions/38749/width-of-gaussian-beam-and-refractive-index/38773 | Width of Gaussian Beam and Refractive Index
I know that in free space, the width of a Gaussian beam can be written as $W=W_0\sqrt{1+(\frac{z}{z_0})^{2}}$. However, I was wondering if it was possible to express this width as a function of refractive index instead (since I don't believe a Gaussian beam originating in say, glass, will diverge in the same manner as one in air). Anyone have any ideas?
- | {"extraction_info": {"found_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": 0.9884173274040222, "perplexity": 219.61755964323578}, "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-2014-42/segments/1414119646944.14/warc/CC-MAIN-20141024030046-00059-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/vector-space.685715/ | # Vector Space
1. Apr 15, 2013
### Workout
1. The problem statement, all variables and given/known data
Is the set of all polynomials with positive coefficients a vector space?
It's not.
But after going through the vector space conditions I don't see how it can't be.
2. Apr 15, 2013
### Staff: Mentor
what are the inverse elements of the space?
3. Apr 15, 2013
### Staff: Mentor
If you show your work, it is easier to find your error.
Oh, and you have to specify the field for the vector space (probably R).
Edit: Or use the direct hint given above ;).
4. Apr 15, 2013
### HallsofIvy
Staff Emeritus
One of the properties that a vector space must have is that it is "closed under scalar multiplication". That is, the product of any real number a, and vector v, av is also a number. v= x+ 1 is a "polynomial with positive coefficients" and a= -1 is a number. What is av?
Draft saved Draft deleted
Similar Discussions: Vector Space | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9682785272598267, "perplexity": 1625.1625647531566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825057.91/warc/CC-MAIN-20171022022540-20171022042540-00513.warc.gz"} |
https://www.cut-the-knot.org/m/Geometry/BottemaIsoscelesTriangles.shtml | # On Bottema's Shoulders with a Ladder
### Problem
Segment $AB$ is fixed. $ACA'$ and $BCB'$ are similar isosceles ($AC=A'C$ and $BC=B'C$) triangles, with a given apex angle. $D$ is the intersection of the bisector of $\angle ACA'$ with the circumcircle $(ACA'),$ $E$ the intersection of the bisector of $\angle BCB'$ with the circumcircle $(BCB').$ $(ACA')$ and $(BCB')$ meet at $J$. $G$ is the midpoint of $DE.$
Prove that
1. $D,$ $J,$ and $E$ are collinear,
2. $AB',$ $A'B$ meet at $J,$
3. $AB'=A'B,$
4. $CJ$ is the bisector of $\angle AJB,$
5. If $O_A,$ $O_B$, $O$ are the centers of $(ACA'),$ $(BCB'),$ and $(AGB),$ respectively, then $OO_A=OO_B$ and triangle $O_{A}OO_B$ is similar to triangles $ACA'$ and $BCB'.$
6. Position of $G$ is independent of $C,$
7. Quadrilateral $AJGB$ is cyclic and $G$ is the midpoint of arc $AB.$
### Solution
Let $\alpha = \angle CAA' = \angle CA'A = \angle CBB' = \angle CB'B\space$ and $\beta = \angle ACA'=\angle BCB'$ such that $2\alpha +\beta =180^{\circ}.$ Then, $\angle CJD$ equals half the arc $CD=CA'+A'D,$ making $\angle CJD=\alpha +\beta/2.$ Similarly, $\angle CJE=\alpha +\beta/2.$ Thus each of these angles is right and together they give $180^{\circ},$ meaning that $D,$ $J,$ and $E$ are collinear (#1). (As a byproduct, $CD$ is a diameter in $(ACA'),$ $CE$ is a diameter in $(BCB').)$
$\angle A'JC=\alpha$ while $\angle BJC=\alpha+\beta$ so again $\angle A'JC+\angle BJC=2\alpha+\beta=180^{\circ},$ implying that $A',$ $J,$ and $B$ are collinear. Similarly, $A,$ $J,$ and $B'$ are collinear (#2).
$\angle A'CB=\angle A'CA+\angle ACB=\angle B'CB+\angle ACB=\angle ACB',\space$ implying that triangles $A'CB$ and $ACB'$ are equal by SAS so that also $AB'=A'B$ (#3).
$\angle AJD = \beta/2 = \angle BJE$ while $CJ\perp DE,$ proving #4. $\angle AJB = 2\alpha,$ making each half equal $\alpha.$
To prove #5, note that $O_AO_B\perp CJ$ while $OO_A\perp AJ$ which says that $\angle OO_AO_B=180^{\circ}-\angle AJC=90^{\circ}-\beta/2=\alpha.$ Similarly, $\angle OO_BO_A=\alpha.$
(Hubert Shutrick came up with a triginometric proof of ##6-7.) Denoting the angles of $\Delta ABC$ $\bar{\alpha},$ $\bar{\beta},$ and $\bar{\gamma},$ and the sides $a,$ $b,$ and $c,$ let $D'$ and $E'$ on $AB$ be projections of $D$ and $E.$ Observe that, since $CD$ is a diameter of $(O_A),$ $\angle CAD=90^{\circ},$ implying $\angle DAD'+\alpha=90^{\circ}$
$AD=a\space\mbox{tan}(\beta/2)$ and $AD'=AD\space\mbox{sin}(\bar{\alpha})$ which together give $AD' = a\space\mbox{tan}(\beta/2)\mbox{sin}(\bar{\alpha}).$ Similarly, $BE'= b\space\mbox{tan}(\beta/2)\mbox{sin}(\bar{\beta}).$ But both $a\space\mbox{sin}(\bar{\alpha})$ and $b\space\mbox{sin}(\bar{\beta})$ express the length of $C$-height in $\Delta ABC.$ Therefore, $AD'=BE'.$
If the midpoint of $AB$ is $H,$ then (from the above) $D'H=E'H$ and also
$2GH = a\space\mbox{tan}(\beta/2)\mbox{cos}(\bar{\alpha}) + b\space\mbox{tan}(\beta/2)\mbox{cos}(\bar{\beta}) = AB\space\mbox{tan}(\beta/2).$
Thus, $\angle GAB = \beta/2 = \angle GJB,$ proving #6. #7 follows from the fact that $\angle AJB = 2\alpha = \angle AGB.$
### Acknowledgment
The problem has been posted by Dao Thanh Oai at the CutTheKnotMath facebook page. | {"extraction_info": {"found_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": 0.9597161412239075, "perplexity": 239.69573515694086}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046154053.17/warc/CC-MAIN-20210731043043-20210731073043-00677.warc.gz"} |
http://math.stackexchange.com/questions/69272/rate-of-increase-in-the-area-of-a-square | # Rate of increase in the area of a square
I really do not understand how to do these problems, so many weird math tricks and rules and I am getting caught up on at least a dozen in this problem. Anyways I am supposed to find:
Each side of a square is increasing at a rate of $6 \text{ cm/s}$. At what rate is the area of the square increasing when the area of the square is $16 \text{ cm}^2$?
I think what I need to do is set it equal to 16 or 4, but I am not sure which so the problem will look like $4=s(36)$ but I am not sure what to do with that.
-
You are familiar with opening sentences like the one you wrote here. These bring nothing to your questions and can only alienate people participating to this site, as was already explained to you. It was also already explained to you that maths is NOT (I repeat, NOT) a collection of weird tricks. Why you persist in this vein is a mystery to me. // Regarding your question: what is the side of the initial square? What will be the side of the square at time $t$? Hence? – Did Oct 2 '11 at 16:10
I don't know what the initial square is, I don't think there is a way to find that out. I am not too sure how to find the side of a square, but I do know that the area at time t would be a=t(36) – user138246 Oct 2 '11 at 16:13
The area of the initial square is 16 cm${}^2$ but you do not know the length of the side? – Did Oct 2 '11 at 16:14
Oh yeah I knew that it is 4. – user138246 Oct 2 '11 at 16:16
Yes it is. Next step: what is the length of the side of the square at time $t$? – Did Oct 2 '11 at 16:16
We may define rate as $\frac{dP}{dt}$ , so let's find first derivative of the area formula $P=a^2$
$\frac{dP}{dt}=2a\frac{da}{dt}$ since $P=a^2 \Rightarrow a=\sqrt{P}$
$\frac{dP}{dt}=2\sqrt{P}\frac{da}{dt} \Rightarrow \frac{dP}{dt}=2*4*6 \frac{cm^2}{s} \Rightarrow \frac{dP}{dt}=48\frac{cm^2}{s}$
-
I don't really know what the purpose of all those symbols are, I know what they mean but there are too many varying and changing letters for me to keep track of. – user138246 Oct 2 '11 at 16:26
@Jordan, keep telling yourself that, and you will certainly never learn it. – Henning Makholm Oct 2 '11 at 16:35
@jordan there really are only three essential symbols in pedja's answer: $P$ , $a$ , and $t$. $P$ is the area of the square, $a$ is the side length, and $t$ is just your time variable. $dP$, for example, is just shorthand to say, "the change in P" However, that's pretty meaningless without knowing what P is changing relative to. So, we write $\frac{dP}{dt}$, meaning the change in P relative to the change in t. $\frac{da}{dt}$ follows a similar pattern Pedja's first step, then, is implicit differentiation of the area formula $P = a^2$. From there, everything is replaced by givens. – Drew Christianson Oct 2 '11 at 16:36
I am getting confused by dp/dt and da/dt was that to use the chain rule? I really don't get this at all and it is incredibly frustrating, I have about 14 hours of homework to do today and I am too frustrated to continue already. I need to take a break and come back in a bit. – user138246 Oct 2 '11 at 16:38
@Henning, indeed. – Did Oct 2 '11 at 16:47
pedja's answer does seem to be expressed in a somewhat complicated way.
Let $A$ be the area in square centimeters. Let $s$ be the length of the side in centimeters. Let $t$ be time in seconds.
Then we are given $\dfrac{ds}{dt} = 6$.
We recall that $A = s^2$.
We want $\dfrac{dA}{dt}$ when $A=16$.
$$\frac{dA}{dt} = \frac{d}{dt} s^2 = 2s \frac{ds}{dt}.$$
When $A=16$ then $s=4$ and $ds/dt = 6$. So $$2s\frac{ds}{dt} = 2\cdot4\cdot 6.$$
-
This is incredibly frustrating but I just don't follow what is happening. Isn't the derivative of s 2s? Where does 6 come in? – user138246 Oct 2 '11 at 17:38
You would be less frustrated if you tried to answer the step-by-step questions I am asking you in the comments. Just my two cents. – Did Oct 2 '11 at 17:52
@jordan, the derivative of $s^2$ with respect to s is $2s$. However, we want the derivative of $s^2$ with respect to t, so we have to multiply it by $\frac{ds}{dt}$. I know it seems like magic, but it follows from the chain rule for derivatives. – Drew Christianson Oct 2 '11 at 18:27
Your step by step questions were just making me look like an idiot, like I can't read english, so I stopped reading them. – user138246 Oct 2 '11 at 18:31
@Jordan, I see. You could have said so earlier (you know, simple politeness, sparing others' time, and everything). – Did Oct 2 '11 at 19:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.934814989566803, "perplexity": 314.603120945649}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1408500829210.91/warc/CC-MAIN-20140820021349-00410-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/basic-ciruits-dependent-sources.670084/ | # Homework Help: Basic Ciruits , dependent sources
1. Feb 7, 2013
### Ammar w
1. The problem statement, all variables and given/known data
https://www.diigo.com/item/image/2sb3i/1svo [Broken]
Why don't we flip the signs of VA like this :
https://www.diigo.com/item/image/2sb3i/g2a4 [Broken]
because the current must flow from + to - of the resistor.
2. Relevant equations
none
3. The attempt at a solution
none
Last edited by a moderator: May 6, 2017
2. Feb 7, 2013
### Staff: Mentor
The "+ -" tags on the 15Ω resistor tell you which way to "measure" the potential VA; It's like specifying how to orient the positive and negative leads of a voltmeter on the component.
You could change the tags as you suggest IF you also reverse the polarity of the controlled voltage source 2VA, since its output depends on how VA is "measured".
3. Feb 7, 2013
### Ammar w
thanks
and the KvL equation changes from :
-120+V30+2VA-VA = 0
to :
-120+V30-2VA+VA = 0 | {"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": 0.8521935343742371, "perplexity": 4182.284097882081}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794866938.68/warc/CC-MAIN-20180525024404-20180525044404-00625.warc.gz"} |
http://math.stackexchange.com/questions/284900/finding-descent-direction-of-quadratic-function | # Finding descent direction of quadratic function
I have a quadratic function: $f(x) = 24x_1+14x_2+x_1x_2$ and point $x_0 = (2,10)^T$ with $f(x_0) = 208$
And the first question is "give descent direction r in $x_0$" The second question "is f convex in direction r?
How can I do that? I've already determine gradient and Hessean. But which step should be next? According to descent ditrection I fount such formula in the Inet: $r=-\partial{^2}f(x)\partial{f(x)}$ Is that formula correct?
About second question: If Hessean is positive for any x, it means, that f is convex But how to determine whether f convex $\mathbf{in}$ $\mathbf{direction }$?
-
(Steepest) descent direction is $-\nabla f(x_0)$. Which is $(-34,-16)^T$ in your case.
To find if $f$ is convex in direction $\vec r$, plug $\vec r$ into the quadratic form associated with the Hessian matrix: $\vec r^T H \vec r$ gives the second directional derivative in direction $\vec r$. Positive=convex, negative=concave... Here $$(-34,-16)\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} (-34,-16)^T = 2\cdot 24\cdot 16>0$$ Notice that the sign of $\vec r$ does not matter: changing the vector to opposite does not affect convexity in that direction. | {"extraction_info": {"found_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": 0.838699996471405, "perplexity": 251.7836681768641}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1448398448227.60/warc/CC-MAIN-20151124205408-00022-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.hackmath.net/en/math-problem/5621 | Clock hands
The second hand has a length of 1.5 cm. How long does the endpoint of this hand travel in one day?
Result
s = 13571.68 cm
Solution:
$r = 1.5 \ cm \ \\ o = 2 \pi \cdot \ r = 2 \cdot \ 3.1416 \cdot \ 1.5 \doteq 9.4248 \ cm \ \\ s = 24 \cdot \ 60 \cdot \ o = 24 \cdot \ 60 \cdot \ 9.4248 \doteq 13571.6803 = 13571.68 \ \text{ cm }$
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https://math.stackexchange.com/questions/2968932/show-that-a-symmetric-and-idempotent-matrix-p-is-the-projection-matrix-onto-so | # Show that a symmetric and idempotent matrix $P$ is the projection matrix onto some subspace.
I am reading "Seminar of Linear Algebra" by Kenichi Kanatani.
In this book, there is the following problem:
Show that a symmetric and idempotent matrix $$P$$ is the orthogonal projection matrix onto some subspace.
As is well known, an $$n \times n$$ symmetric matrix $$P$$ has $$n$$ real eigenvalues $$\lambda_1, \dots, \lambda_n$$, the corresponding eigenvectors $$u_1, \dots, u_n$$ being an orthonormal system. If we multiply $$P u_i = \lambda_i u_i$$ by $$P$$ from left on both sides, we have
$$P^2 u_i = \lambda_i P u_i = \lambda_i^2 u_i.$$ If $$P$$ is idempotent, the left side is $$P u_i = \lambda_i u_i$$. Hence, $$\lambda_i = \lambda_i^2$$, i.e., $$\lambda_i = 0, 1$$. Let $$\lambda_1 = \cdots = \lambda_r = 1$$, $$\lambda_{r+1} = \cdots = \lambda_n = 0$$. Then, $$P u_i=u_i, i = 1, \dots, r,$$ $$P u_i = 0, i = r+1, \dots, n.$$ We see that $$P$$ is the orthogonal projection matrix onto the subspace spanned by $$u_1, \dots, u_r$$.
I cannot understand this statement:
As is well known, an $$n \times n$$ symmetric matrix $$P$$ has $$n$$ real eigenvalues $$\lambda_1, \dots, \lambda_n$$, the corresponding eigenvectors $$u_1, \dots, u_n$$ being an orthonormal system.
• I know that an $$n \times n$$ symmetric matrix $$P$$ has $$n$$ real eigenvalues $$\lambda_1, \dots, \lambda_n$$.
• And I know if $$\lambda_i \ne \lambda_j$$, then $$u_i$$ is orthogonal to $$u_j$$.
• And I know that every $$n \times n$$ symmetric matrix $$P$$ doesn't have distinct $$n$$ real eigenvalues.
Let $$P$$ be an $$n \times n$$ symmetric matrix $$P$$.
Are there eigenvecotrs of $$P$$ that are mutually orthogonal?
Yes, for any symmetric matrix $$P$$ of size $$n$$, it is possible to find a set of eigenvectors $$(u_i)_{i=1}^n$$ such that if $$i\neq j$$, then $$u_i . u_j = 0$$ (scalar product), and $$||u_i||^2 = 1$$ for all $$i$$.
Basically, a symmetric matrix always admits a set of eigenvectors which is an orthonormal basis of the vector space.
In matrix notations, it is possible to find an orthogonal matrix $$O$$ and a diagonal matrix $$D$$ (whose diagonal are the eigenvalues of $$P$$) such that $$P = O D O^T = O D O^{-1}$$.
• Thank you very much, seamp. Oct 24 '18 at 10:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 49, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9926213622093201, "perplexity": 56.76653921426292}, "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-2022-05/segments/1642320301263.50/warc/CC-MAIN-20220119033421-20220119063421-00398.warc.gz"} |
https://www.physicsforums.com/threads/algebraic-geometry.159145/ | # Algebraic Geometry
• Start date
• #1
70
0
I'm having trouble understanding the importance of dominance with regards rational maps.
I have the definition that a rational map phi (from some affine variety V to an affine variety W) is dominant if the image of V is not contained in a proper subvariety of W.
In my lecture notes there is some remark about how if we compose phi with another such map psi going in the reverse direction W to V (to get a map V to V) then it may not make sense unless phi is dominant.
Whilst reading around the subject I have noticed that the notion of dominance crops up in a lot of theorem hypotheses so I really feel I ought to get a grip on it.
My problem is this: Rational maps are not generally functions anyway since they may have undefined points. OK. Now, in the remark when it talks about the composition 'not making sense', one would usually assume by the phrase 'not making sense' that it has undefined points - but why are we suddenly bothered about this now when we weren't before?
Any help would be greatly appreciated.
• #2
Hurkyl
Staff Emeritus
Gold Member
14,950
19
A rational map must be defined on an dense subset of the domain, right? Well, what if the image of f lies in the complement of the set where g is defined?
Explicit examples are fun: let's use R^2.
Let f be the map (x, y) --> (1/x, 0)
Let g be the map (x, y) --> (0, 1/y)
What is their composite? (Either way)
Also, I suspect that dominant maps are epimorphisms (f is epimorphic iff g o f = h o f implies g=h), and so play a similar role that surjective maps would in topology.
Last edited:
• #3
matt grime
Homework Helper
9,420
4
You should think of dominance as being 'essentially surjective'. It is the definition that makes rational maps composable. As you say there might be a set of points on which they are not defined - that is a subvariety of dimension 0. Each composition of dominant maps, adds at most finitely many more points that are bad. And a finite sum of finitely many things is still finite, so a dimension 0 subvariety. If you dropped the dominant assumption you can get a situation as in Hall's post above.
Last edited:
• #4
70
0
A rational map must be defined on an dense subset of the domain, right?
I didn't know this but it certainly makes sense now you say it. Topology wasn't a prerequisite for this course so the lecturer has avoided topological notions. (I have done toplogy though.) I know about the Zariski topology so it seems that dense here means a subset A of our variety for which there is no closed set (i.e. a subvariety) which both contains A and is contained in V. So nicely enough NOT being dominant means that our image of V is not a dense subset of W. It's starting to make sense now.
Let f be the map (x, y) --> (1/x, 0)
Let g be the map (x, y) --> (0, 1/y)
What is their composite? (Either way)
Of course the image of either composition is the empty set.
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3K | {"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": 0.8734636902809143, "perplexity": 1024.5077007528218}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056752.16/warc/CC-MAIN-20210919065755-20210919095755-00641.warc.gz"} |
http://mathhelpforum.com/calculus/83785-limit-question.html | Math Help - limit question
1. limit question
Why is this right
$\frac{1}{2}\frac{\lim}{u\rightarrow0}\frac{\sin[u]}{u}=\frac{1}{2}$
I can make it work with L'Hop's but is it legal to use when u approaches 0 and not infinity
2. Yep. You can use L'Hop for this case too. There are also other ways of proving this identity.
3. Thanks Rangr. Can L'Hops be used for u=any N
4. L'Hops can be used if you have 0/0 or 0/infinity or infinity/infinity or infinity/0. i.e. functions not evaluable at their limits, so we evaluate their slopes when they tend towards these limits. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8541437387466431, "perplexity": 2024.2202637459566}, "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-2015-22/segments/1432207924799.9/warc/CC-MAIN-20150521113204-00104-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/33130/cyclic-co-ordinates-implying-the-constant-velocity-motion-of-center-of-mass-of-a?answertab=oldest | # Cyclic co-ordinates implying the constant velocity motion of center of mass of a system of particles
I'm reading the section on Central Force in my textbook (Goldstein's Classical Mechanics has a similar argument in the chapter titled "The Central Force Problem", first section), where we have the following:
The Lagrangian for the system of two particles is found to be $$L=\frac{1}{2}M \dot R^2+\frac{1}{2}\mu \dot r^2-V(r)$$ where $R$ is the position vector of the center of mass of the particles.
The textbook says that since the three components of $R$ do not appear in the Lagrangian, they are cyclic.
(My first question is : Is it referring to the fact that $L$ is not a function of $(x,y,z)$? What about the $V(r)$ term. This introduces a position-dependence, doesn't it?)
We continue "..(therefore) the center of mass is either at rest or moving at a constant velocity, and we can drop the first term of the Lagrangian in our discussion. The effective Lagrangian is now given by $$L=\frac{1}{2}\mu \dot r^2-V(r)$$
"(end quote)
I don't quite see how we conclude that the center of mass is either at rest or moving at a constant velocity based on the fact that $L$ is not a function of ($x,y,z$).
-
The three components of $R$ indeed do not appear in the Lagrangian; $V(r)$ is a function of only $r$ (i.e. the distance between the particles). If $V$ were a function of $R$ it would imply the presence of some external field and you wouldn't be dealing with the same two-body problem anymore.
That the center of mass is either at rest or moving at a constant velocity can easily be seen from the Euler-Lagrange equations for the original Lagrangian. For $R$ E-L equation reads: $$\ddot{R} = 0 .$$ In fact, in the absence of external forces, the center of mass of a system is always at rest or moving at a constant velocity. | {"extraction_info": {"found_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": 0.9798858165740967, "perplexity": 84.11862618413335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1414119648891.34/warc/CC-MAIN-20141024030048-00000-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://mathonline.wikidot.com/the-osculating-circle-at-a-point-on-a-curve | The Osculating Circle at a Point on a Curve
# The Osculating Circle at a Point on a Curve
Definition: Let $\vec{r}(t) = (x(t), y(t), z(t))$ be a vector-valued function that traces out the smooth curve $C$, and let $P$ be the point on $C$ at $t$. The Osculating Circle at $P$ is the circle that best approximates $C$ at $P$. The osculating circle at $P$: 1) Contains the point $P$. 2) Has radius $\rho (t) = \frac{1}{\kappa (t)}$. 3) Has curvature $\kappa (t)$. 4) Shares the same tangent at $P$.
If $y = f(x)$ generates a curve $C$, then the osculating circle at $P$ is also defined to be the circle that best approximates $C$ at $P$ by having radius $\rho (x) = \frac{1}{\kappa (x)}$, curvature $\kappa (x)$, and shares the same tangent at $P$.
Definition: Let $\vec{r}(t) = (x(t), y(t), z(t))$ be a vector-valued function that traces out the smooth curve $C$, and let $P$ be the point on $C$ at $t$. Then Center of Curvature at $P$ is the center of the osculating circle at $P$.
From the definition of an osculating circle, we can calculate the center of curvature which we will denote by $\vec{r_c}(t)$, by the following formula:
(1)
\begin{align} \mathrm{Center \: of \: Curvature} = \vec{r_c}(t) = \vec{r}(t) + \rho (t) \hat{N}(t) \end{align}
This can easily be seen as $\vec{r}(t)$ takes us to the point $P$, and then we travel $\rho (t)$ (the radius of the osculating circle at $P$) in the direction of the unit normal vector $\hat{N}(t)$.
## Example 1
Let $\vec{r}(t) = \left ( t, \frac{1}{2}t^2 + 1, 0 \right )$. Find the equation of the osculating circle at $t = 1$.
We will first simplify this problem down. We should first notice that the graph of $\vec{r}(t)$ represents a plane curve - namely one that lies on the $xy$-plane. Since $x = t$, then $y = \frac{1}{2}t^2 + 1$ implies that $y = \frac{1}{2}x^2 + 1$. So $\vec{r}(t)$ represents the graph of the single variable function $f(x) = \frac{1}{2}x^2 + 1$.
We will first find the curvature of this function by using the formula $\kappa (x) = \frac{ \mid f''(x) \mid}{(1 + (f'(x))^2)^{3/2}}$ and then take the reciprocal to get the radius of curvature, $\rho (x) = \frac{1}{\kappa (x)}$. To use this formula, we must first calculate $f'(x)$ and $f''(x)$. We have that $f'(x) = x$, and $f''(x) = 1$, and so:
(2)
\begin{align} \kappa (x) = \frac{1}{(1 + x^2)^{3/2}} \\ \rho (x) = (1 + x^2)^{3/2} \end{align}
Since $x = t$, the given value $t = 1$ corresponds to $x = 1$. Plugging this into the formula above and we have that $\rho (1) = 2^{3/2}$.
We now need to find the center of curvature, i.e, the center of the osculating circle at $t = 1$. To do so, we need to find the unit normal vector $\hat{N}(1)$. To calculate the unit normal vector, we must first find the unit tangent vector $\hat{T}(1)$ and the unit binormal vector $\hat{B}(1)$ and then take the cross product $\hat{N}(1) = \hat{B}(1) \times \hat{T}(1)$.
First let's calculate the unit tangent vector with the formula $\hat{T}(t) = \frac{\vec{r'}(t)}{\| \vec{r'}(t) \|}$. We note that $\vec{r'}(t) = (1, t, 0)$, and $\| \vec{r'}(t) \| = \sqrt{1 + t^2}$. Therefore $\hat{T}(t) = \left ( \frac{1}{\sqrt{1 + t^2}}, \frac{t}{\sqrt{1+t^2}}, 0 \right )$. Plugging in $t = 1$ and we have that $\hat{T}(1) = \left ( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \right )$.
Now let's calculate the unit binormal vector with the formula $\hat{B}(t) = \frac{\vec{r'}(t) \times \vec{r''}(t)}{\| \vec{r'}(t) \times \vec{r''}(t) \|}$. We note that $\vec{r'}(t) = (1, t, 0)$ and $\vec{r''}(t) = (0, 1, 0)$. Therefore $\vec{r'}(t) \times \vec{r''}(t) = (0, 0, 1)$. Clearly $\| \vec{r'}(t) \times \vec{r''}(t) \| = 1$, so $\hat{B}(t) = (0, 0, 1)$. Plugging in $t = 1$ and we get that $\hat{B}(1) = (0, 0, 1)$.
Therefore the unit normal vector can be obtained as $\hat{N}(1) = \hat{B}(1) \times \hat{T}(1)$:
(3)
\begin{align} \quad \quad \hat{N}(1) = (0,0,1) \times \left ( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \right ) = \left ( - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \right ) \end{align}
Therefore the center of curvature (the center of the circle) at $t = 1$ is:
(4)
\begin{align} \quad \vec{r_c}(1) = \vec{r}(1) + \rho (1) \hat{N}(1) \\ \quad \vec{r_c}(1) = \left (1, \frac{3}{2}, 0 \right) + 2^{3/2} \left ( - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \right ) \\ \quad \vec{r_c}(1) = \left (1, \frac{3}{2}, 0 \right ) + (-2, 2, 0)\\ \quad \vec{r_c}(1) = \left (-1, \frac{7}{2}, 0 \right ) \end{align}
In $\mathbb{R}^2$, the center of the osculating circle at $t = 1$ is thus $\left ( -1, \frac{7}{2} \right )$. We calculated the radius of the circle above ($\rho (1) = 2^{3/2}$) and so the equation of the osculating circle is:
(5)
\begin{align} (x + 1)^2 + \left ( y - \frac{7}{2} \right )^2 = (2^{3/2})^2 \\ (x + 1)^2 + \left ( y - \frac{7}{2} \right )^2 = 8 \end{align}
In $\mathbb{R}^3$, we can represent this circle with the set of parametric equations $\left\{\begin{matrix}x = 2^{3/2} \cos t - 1\\ y = 2^{3/2} \sin t + \frac{7}{2} \\ z = 0\\ \end{matrix}\right.$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 5, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9995002746582031, "perplexity": 127.48358760667678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886802.13/warc/CC-MAIN-20200704232817-20200705022817-00126.warc.gz"} |
https://forum.allaboutcircuits.com/threads/eqn-for-current.4391/ | # Eqn. for current
#### pratapkollu
Joined Dec 13, 2006
42
Some please tell me the equation for CURRENT OUT in the circuit that is given in attachment. And please tell me how to calculate the transfer function( equation for current)
Thanks and regards
Pratap
#### Attachments
• 26.1 KB Views: 33
#### Ron H
Joined Apr 14, 2005
7,014
If you understand the following three things, you can understand the analysis in the attached schematic.
1. Ohm's law.
2. Operational amplifiers (and feedback theory).
3. MOSFETs.
If there is something you don't understand, post it here.
#### Attachments
• 14.6 KB Views: 25
#### pratapkollu
Joined Dec 13, 2006
42
If you understand the following three things, you can understand the analysis in the attached schematic.
1. Ohm's law.
2. Operational amplifiers (and feedback theory).
3. MOSFETs.
If there is something you don't understand, post it here.
HI, Thanx for your explanation, can u please give the eqn for current as function of resistors and input voltage? for ex: I = [r2/ (r1.r3)]. Vi
#### Ron H
Joined Apr 14, 2005
7,014
HI, Thanx for your explanation, can u please give the eqn for current as function of resistors and input voltage? for ex: I = [r2/ (r1.r3)]. Vi
Why don't you provide a schematic with reference designators (r1, r2, etc.), figure out what you think the transfer function is, and we will tell you if it's correct or not.
#### pratapkollu
Joined Dec 13, 2006
42
Why don't you provide a schematic with reference designators (r1, r2, etc.), figure out what you think the transfer function is, and we will tell you if it's correct or not.
I dont know the exact eqn for current , I just gave u that eqn to let u know that I need the eqn for current as function of resisor newtwork and input voltage
Thanks and regards
Pratap
#### Attachments
• 25.5 KB Views: 18
#### beenthere
Joined Apr 20, 2004
15,819
Hi,
What resistor network? The output depends on the gate voltage of the second FET. That comes from the output of the second op amp, which is being driven by the first op amp. The limit is 100 milliamps into a 50 ohm load.
If you can get a transfer curve for the FET, find the voltage on the gate that corresponds to the D-S resistance necessary to allow 100 milliamps into your chosen load. Hint: the circuit is dynamic, so no calculation will be valid except as it applies current into some load. As there is no load in the schematic, output current is zero. Put some resistance between the terminals and see what happens. One suspects further that the 5 volt control level may correspond to 100 ma into 50 ohms.
#### pratapkollu
Joined Dec 13, 2006
42
Hi,
What resistor network? The output depends on the gate voltage of the second FET. That comes from the output of the second op amp, which is being driven by the first op amp. The limit is 100 milliamps into a 50 ohm load.
If you can get a transfer curve for the FET, find the voltage on the gate that corresponds to the D-S resistance necessary to allow 100 milliamps into your chosen load. Hint: the circuit is dynamic, so no calculation will be valid except as it applies current into some load. As there is no load in the schematic, output current is zero. Put some resistance between the terminals and see what happens. One suspects further that the 5 volt control level may correspond to 100 ma into 50 ohms.
Thank you very much for the replies.
#### Ron H
Joined Apr 14, 2005
7,014
I dont know the exact eqn for current , I just gave u that eqn to let u know that I need the eqn for current as function of resisor newtwork and input voltage
Thanks and regards
Pratap
I didn't even look at your equation. I just suggested that
you provide a schematic with reference designators (r1, r2, etc.), figure out what you think the transfer function is, and we will tell you if it's correct or not.
You should be able to do that from the output current equation that I put on the schematic. Just substitute R1 for 499, etc.
#### hgmjr
Joined Jan 28, 2005
9,029
I think what Ronh is asking of you is that you examine the annotations from his attachment and use your schematic containing the reference designators (R1, R2, R3, etc.) to produce the circuit's transfer function.
Ronh's has done a stellar job of annotating the schematic. He has even left the values of all of the resistors and the power supply voltage in the final expression.
hgmjr
#### Ron H
Joined Apr 14, 2005
7,014
I think what Ronh is asking of you is that you examine the annotations from his attachment and use your schematic containing the reference designators (R1, R2, R3, etc.) to produce the circuit's transfer function.
Ronh's has done a stellar job of annotating the schematic. He has even left the values of all of the resistors and the power supply voltage in the final expression.
hgmjr
Yes, that's what I am asking.
Thanks for the compliment. I should point out that the power supply voltage does not appear in the final expression. Perhaps you meant "control voltage".
#### hgmjr
Joined Jan 28, 2005
9,029
Yes, that's what I am asking.
Thanks for the compliment. I should point out that the power supply voltage does not appear in the final expression. Perhaps you meant "control voltage".
Oooooppppssss!!!!!! You're correct. I did have the control voltage in mind when I typed power supply voltage.....
hgmjr | {"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": 0.9079734683036804, "perplexity": 1716.2463176641859}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243991413.30/warc/CC-MAIN-20210512224016-20210513014016-00325.warc.gz"} |
http://mathhelpforum.com/algebra/57608-how-many-square-inches.html | # Thread: how many square inches
1. ## how many square inches
We have a box (which is not open) and is 8 inches wide, 12 inches long, and 6 inches high. How many square inches of cardboard are used?
2. Originally Posted by Mccoy31
We have a box (which is not open) and is 8 inches wide, 12 inches long, and 6 inches high. How many square inches of cardboard are used?
a box has six sides correct
12*6( because the length is 12 and the box has 4 sides which the length is 12)
=72(4) for the amount of sides
=288
Now do the ends 8*6=48(2) for the two ends of the box
96
288+96=384 inches squares
does that match the answer in the book?
3. Thank you very much!
4. Originally Posted by Mccoy31
Thank you very much!
no 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": 0.864646315574646, "perplexity": 1488.2516982376317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1474738660887.60/warc/CC-MAIN-20160924173740-00027-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://www.dsprelated.com/freebooks/pasp/Coupled_Horizontal_Vertical_Waves.html | ### Coupled Horizontal and Vertical Waves
No vibrating string in musical acoustics is truly rigidly terminated, because such a string would produce no sound through the body of the instrument.7.15Yielding terminations result in coupling of the horizontal and vertical planes of vibration. In typical acoustic stringed instruments, nearly all of this coupling takes place at the bridge of the instrument.
Figure 6.20 illustrates the more realistic case of two planes of vibration which are linearly coupled at one end of the string (the bridge''). Denoting the traveling force waves entering the bridge from the vertical and horizontal vibration components by and , respectively, the outgoing waves in each plane are given by
(7.16)
as shown in the figure. In physically symmetric situations, we expect . That is, the transfer function from horizontal to vertical waves is normally the same as the transfer function from vertical to horizontal waves. If we consider a single frequency , then the coupling matrix with is a constant (generally complex) matrix (where denotes the sampling interval as usual). An eigenanalysis of this matrix gives information about the modes of the coupled system and the damping and tuning of these modes [543]. As a simple example, suppose the coupling matrix at some frequency has the form
where and are any complex numbers. This means both string terminations are identical, and the coupling is symmetric (the simplest case in practice). The eigenvectors are easily calculated to be7.16
and the eigenvalues are and , respectively. The eigenvector corresponds to in phase'' vibration of the two string endpoints, i.e., , while corresponds to opposite phase'' vibration, for which . If it happens to be the case that
then the in-phase vibration component will decay faster than the opposite-phase vibration. This situation applies to coupled piano strings [543], as discussed further below. More generally, the two eigenvectors of the coupling frequency-response matrix
correspond to two decoupled polarization planes. That is, at each frequency there are two eigenpolarizations in which incident vibration reflects in the same plane. In general, the eigenplanes change with frequency. A related analysis is given in [543]. By definition of the eigenvectors of , we have
where denotes the th eigenvalue of the coupling-matrix at frequency , where . Since the eigenvector holds the Fourier transform of the incoming waves for mode of the coupled-string system, we see that the eigenvalues have the following interpretation:
The th eigenvalue of the coupling matrix equals the frequency response seen by the th eigenpolarization.
In particular, the modulus of the eigenvalue gives the reflectance magnitude (affecting mode damping), and the angle of the eigenvalue is the phase shift of the reflection, for that mode (affecting tuning of the mode). Use of coupling matrix eigenanalysis to determine mode damping and tuning is explored further in §C.13.
Next Section:
Asymmetry of Horizontal/Vertical Terminations
Previous Section:
Horizontal and Vertical Transverse Waves | {"extraction_info": {"found_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": 0.9861384630203247, "perplexity": 916.125222752652}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400201826.20/warc/CC-MAIN-20200921143722-20200921173722-00237.warc.gz"} |
https://www.ms.u-tokyo.ac.jp/seminar/past_32.html | ## 過去の記録
### 2019年06月12日(水)
#### 代数学コロキウム
17:00-18:00 数理科学研究科棟(駒場) 056号室
On extension of overconvergent log isocrystals on log smooth varieties (Japanese)
[ 講演概要 ]
Kを混標数の完備な非アルキメデス付値体とし，kをその剰余体とする．
Kedlayaおよび志甫の研究により，k上の滑らかな代数多様体Xとその上の単純正規交叉因子Zについて，X ¥setminus Z上の過収束アイソクリスタルのうちZの周りである種のモノドロミーを持つものは，XにZから定まる対数的構造を入れた対数的代数多様体上の収束対数的アイソクリスタルに延長できることが知られている．
### 2019年06月11日(火)
#### 解析学火曜セミナー
16:50-18:20 数理科学研究科棟(駒場) 128号室
Antonio De Rosa 氏 (クーラン数理科学研究所)
Solutions to two conjectures in branched transport: stability and regularity of optimal paths (English)
[ 講演概要 ]
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $\alpha$ is bigger than a critical threshold, but we prove it for every exponent $\alpha \in (0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely solve a conjecture of the book Optimal transportation networks by Bernot, Caselles and Morel. Moreover the robustness of our proof allows us to get the stability for more general lower semicontinuous functional. Furthermore, we prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the aforementioned book. We use the latter result to show the regularity of the optimal networks. (Joint works with Maria Colombo and Andrea Marchese)
### 2019年06月06日(木)
#### 情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 128号室
[ 講演概要 ]
### 2019年06月05日(水)
#### 代数学コロキウム
17:30-18:30 数理科学研究科棟(駒場) 056号室
Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)
[ 講演概要 ]
Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.
### 2019年06月04日(火)
#### PDE実解析研究会
10:30-11:30 数理科学研究科棟(駒場) 056号室
Giuseppe Mingione 氏 (Università di Parma)
Recent progresses in nonlinear potential theory (English)
[ 講演概要 ]
Nonlinear Potential Theory aims at studying the fine properties of solutions to nonlinear, potentially degenerate nonlinear elliptic and parabolic equations in terms of the regularity of the give data. A major model example is here given by the $p$-Laplacean equation
$$-\operatorname{div}(|Du|^{p-2}Du) = \mu \quad\quad p > 1,$$
where $\mu$ is a Borel measure with finite total mass. When $p = 2$ we find the familiar case of the Poisson equation from which classical Potential Theory stems. Although many basic tools from the classical linear theory are not at hand - most notably: representation formulae via fundamental solutions - many of the classical information can be retrieved for solutions and their pointwise behaviour. In this talk I am going to give a survey of recent results in the field. Especially, I will explain the possibility of getting linear and nonlinear potential estimates for solutions to nonlinear elliptic and parabolic equations which are totally similar to those available in the linear case. I will also draw some parallels with what is nowadays called Nonlinear Calderón-Zygmund theory.
#### トポロジー火曜セミナー
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Gluck twist on branched twist spins (JAPANESE)
[ 講演概要 ]
Branched twist spin とは４次元球面上の円作用の特異点集合として定義される埋め込まれた２次元球面であり，スパン結び目やツイストスパン結び目などの２次元結び目の一般化となっている．Gluck は４次元多様体内の２次元結び目に沿った向きを保つ手術は微分同相類を除いて２種類のみであることを示しており，自明でない手術を Gluck twist と呼ぶ．一般に Gluck twist が全空間の微分同相を保つかどうかは知られていないが，Pao によって branched twist spin に沿った Gluck twist は 再び４次元球面と微分同相になることが知られている．本講演では，Pao の結果の別証明として円作用を用いて４次元球面の分解を与え，各ピースが Gluck twist を通してどのように変化するかを説明する．また，２次元結び目に注目したとき，Gluck twist によって branched twist spin は再び branched twist spin になることを証明する．
### 2019年05月29日(水)
#### 代数幾何学セミナー
15:30-17:00 数理科学研究科棟(駒場) 118号室
Minimal log discrepancies of 3-dimensional non-canonical singularities (English)
[ 講演概要 ]
Canonical and terminal singularities, introduced by Reid, appear naturally in minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension 3, while the property of non-canonical singularities is still mysterious. We investigate the difference between canonical and non-canonical singularities via minimal log discrepancies (MLD). We show that there is a gap between MLD of 3-dimensional non-canonical singularities and that of 3-dimensional canonical singularities, which is predicted by a conjecture of Shokurov.
This result on local singularities has applications to global geometry of Calabi–Yau 3-folds. We show that the set of all non-canonical klt Calabi–Yau 3-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3-folds are bounded from above.
#### 代数学コロキウム
17:00-18:00 数理科学研究科棟(駒場) 056号室
On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)
[ 講演概要 ]
PEL型志村多様体のp進整数環上の整モデルは, Abel多様体と付加構造のモジュライ空間として定義される. その幾何的特殊ファイバーのうち, 超特異Abel多様体に対応する点からなる閉部分スキームを超特異部分という. 超特異部分の構造の明示的な記述は, arithmetic intersectionをはじめとする整数論への応用をもつことが知られている.
### 2019年05月28日(火)
#### 諸分野のための数学研究会
10:30-11:30 数理科学研究科棟(駒場) 056号室
[ 講演概要 ]
これまでに独自に開発した細胞内温度イメージング法を用いて、細胞内に時空間的な不均一な温度変動があることを発見した。この温度変化は同じ体積の水と比較すると著しく大きな値であり、細胞内に特殊な熱移動機構が予想された。本研究では人工熱源を用いた際の細胞内の熱移動に関する定量的計測と熱拡散方程式への近似についてその限界と展望を紹介したい。
#### トポロジー火曜セミナー
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
R. Inanc Baykur 氏 (University of Massachusetts)
Exotic four-manifolds via positive factorizations (ENGLISH)
[ 講演概要 ]
We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.
### 2019年05月27日(月)
#### 複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 128号室
Gluing construction of K3 surfaces (Japanese)
[ 講演概要 ]
Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of an elliptic curve embedded in a surface whose normal bundle satisfies "Diophantine condition" in the Picard variety. By applying this theorem, we construct a K3 surface by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of anti-canonical curves of blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the resulting K3 surface is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction. (Based on joint work with Takato Uehara)
### 2019年05月24日(金)
#### 談話会・数理科学講演会
15:30-16:30 数理科学研究科棟(駒場) 002号室
「ポスト量子」暗号と格子暗号 (日本語)
[ 講演概要 ]
### 2019年05月23日(木)
#### 情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 128号室
ビットコインの問題点とその他の仮想通貨、ブロックチェーン (Japanese)
[ 講演概要 ]
### 2019年05月22日(水)
#### 代数幾何学セミナー
15:30-17:00 数理科学研究科棟(駒場) 122号室
Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic
[ 講演概要 ]
In characteristic zero, cotangent bundle of n(>1)-dimensional smooth projective varieties does not contain a big line bundle. This is a part of Bogomolov vanishing and this vanishing plays an important role in the proof of Miyaoka-Yau inequality. In positive characteristic, it is known that Bogomolov vanishing does not hold. There exists a general type surface whose cotangent bundle contains an ample line bundle. So, it is natural to ask when Bogomolov type vanishing holds in positive characteristic. In this talk, I discuss Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic.
### 2019年05月21日(火)
#### トポロジー火曜セミナー
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Maria de los Angeles Guevara 氏 (大阪市立大学)
On the dealternating number and the alternation number (ENGLISH)
[ 講演概要 ]
Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is “close to” alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.
### 2019年05月20日(月)
#### 複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 128号室
Cohomology and normal reduction numbers of normal surface singularities (Japanese)
[ 講演概要 ]
The normal reduction number of a normal surface singularity relates the maximal degree of the generators of associated graded algebra for certain line bundles on resolution spaces. We show fundamental properties of this invariant and formulas for some special cases. This talk is based on the joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.
### 2019年05月16日(木)
#### 情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 128号室
ビットコイン：電子マネー革命 (Japanese)
[ 講演概要 ]
1980年代から暗号分野では電子マネーの研究が行われてきたが，いずれも権威ある(電子マネー)発行機関の存在を前提としていた．ところが，2008年に誕生したビットコインは，特権的な機関は存在せず，非中央集権的な形でマネー（コイン）を発行する仕組みを作り上げた．今回は基本的な暗号機能を利用してどのように非中央集権的にコインを発行するのかなど，ビットコインの仕組みを説明する．
### 2019年05月15日(水)
#### FMSPレクチャーズ
17:30-18:30 数理科学研究科棟(駒場) 122号室
*The date and room have changed.
Gábor Domokos 氏 (Hungarian Academy of Sciences/Budapest University of Technology and Economics)
'Oumuamua, the Gömböc and the Pebbles of Mars (ENGLISH)
[ 講演概要 ]
In this talk I will concentrate on two examples from planetary science, which made the headlines in recent years to highlight the power and significance of nonlinear geometric partial differential equations (PDEs) explaining puzzles presented by Nature. One key link between PDE theory of shape evolution and natural phenomena is the Gömböc, the first mono-monostatic object whose existence was first conjectured by V.I. Arnold in 1995. I will explain the connection and illustrate the process how mathematical models of Nature may be identified.
[ 参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Domokos.pdf
#### FMSPレクチャーズ
15:00-17:20 数理科学研究科棟(駒場) 122号室
J. Scott Carter 氏 (University of South Alabama / Osaka City University)
Part 1 : Categorical analogues of surface singularities
Part 2 : Prismatic Homology (ENGLISH)
[ 講演概要 ]
Part 1 :
Isotopy classes of surfaces that are embedded in 3-space can be described as a free 4-category that has one object and one weakly invertible arrow. That description coincides with a fundamental higher homotopy group. The surface singularities that correspond to cusps and optimal points on folds can be used to develop categorical analogues of swallow-tails and horizontal cusps. In this talk, the 4-category will be constructed from the ground up, and the general structure will be described.
Part 2 :
A qualgebra is a set that has two binary operations whose relationships to each other are similar to the relations between group multiplication and conjugation. The axioms themselves are described in terms of isotopies of knotted trivalent graphs and the handle-body knots that are represented. The moves naturally live in prisms. By using a generalization of the tensor product of chain complexes, a homology theory is presented that encapsulates these axioms and the higher order relations between them. We show how to use this homology theory to give a solution a system of tensor equations related to the Yang-Baxter relation.
[ 参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Carter.pdf
#### 代数幾何学セミナー
15:30-17:00 数理科学研究科棟(駒場) 118号室
On quasi-log canonical pairs
(Japanese)
[ 講演概要 ]
The notion of quasi-log canonical pairs was introduced by Florin Ambro. It is a kind of generalizations of that of log canonical pairs. Now we know that quasi-log canonical pairs are ubiquitous in the theory of minimal models. In this talk, I will explain some basic properties and examples of quasi-log canonical pairs. I will also discuss some new developments around quasi-log canonical pairs. Some parts are joint works with Haidong Liu.
#### 作用素環セミナー
16:45-18:15 数理科学研究科棟(駒場) 126号室
Combinatorial aspects of Borel functions
### 2019年05月14日(火)
#### トポロジー火曜セミナー
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
J. Scott Carter 氏 (University of South Alabama, 大阪市立大学)
Diagrammatic Algebra (ENGLISH)
[ 講演概要 ]
Three main ideas will be explored. First, a higher dimensional category (a category that has arrows, double arrows, triple arrows, and quadruple arrows) that is based upon the axioms of a Frobenius algebra will be outlined. Then these structures will be promoted into one higher dimension so that braiding can be introduced. Second, relationships between braiding and multiplication will be studied from a homological perspective. Third, the next order relations will be used to formulate a system of abstract tensor equations that are analogous to the Yang-Baxter relation. In this way, a broad outline of the notion of diagrammatic algebra will be presented.
### 2019年05月13日(月)
#### 複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 128号室
Some Bonnet--Myers Type Theorems for Transverse Ricci Solitons on Complete Sasaki Manifolds (Japanese)
[ 講演概要 ]
The aim of this talk is to discuss the compactness of complete Ricci solitons and its generalizations. Ricci solitons were introduced by R. Hamilton in 1982 and are natural generalizations of Einstein manifolds. They correspond to self-similar solutions to the Ricci flow and often arise as singularity models of the flow. The importance of Ricci solitons was demonstrated by G. Perelman, where they played crucial roles in his affirmative resolution of the Poincare conjecture.
In this talk, after we review basic facts on Ricci solitons, I would like to introduce some Bonnet--Myers type theorems for complete Ricci solitons. Our results generalize the previous Bonnet--Myers type theorems due to W. Ambrose (1957), J. Cheeger, M. Gromov, and M. Taylor (1982), M. Fernandez-Lopez and E. Garcia-Rio (2008), M. Limoncu (2010, 2012), Z. Qian (1997), Y. Soylu (2017), and G. Wei and W. Wylie (2009). Moreover, I would also like to extend such Bonnet--Myers type theorems to the case of transverse Ricci solitons on complete Sasaki manifolds. Our results generalize the previous Bonnet--Myers type theorems for complete Sasaki manifolds due to I. Hasegawa and M. Seino (1981) and Y. Nitta (2009).
#### 数値解析セミナー
16:50-18:20 数理科学研究科棟(駒場) 056号室
[ 講演概要 ]
### 2019年05月09日(木)
#### 情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 128号室
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https://xianblog.wordpress.com/tag/abc-in-london/ | ## the Frankenstein chronicles
Posted in Statistics with tags , , , , , , , , , , , , , on March 31, 2019 by xi'an
Over a lazy weekend, I watched the TV series The Frankenstein Chronicles, which I found quite remarkable (if definitely Gothic and possibly too gory for some!). Connections with celebrities of (roughly) the time abound: While Mary Shelley makes an appearance in the first season of the series, not only as the writer of the famous novel (already famous in the novel as well) but also as a participant to a deadly experiment that would succeed in the novel (and eventually in the series), Charles Dickens is a constant witness to the unraveling of scary events as Boz the journalist, somewhat running after the facts, William Blake dies in one of the early episodes after painting a series of tarot like cards that eventually explains it all, Ada Lovelace works on the robotic dual of Frankenstein, Robert Peel creates the first police force (which will be called the Bobbies after him!), John Snow’s uncovering of the cholera source as the pump of Broad Street is reinvented with more nefarious reasons, and possibly others. Besides these historical landmarks (!), the story revolves around the corpse trafficking that fed medical schools and plots for many a novel. The (true) Anatomy Act is about to pass to regulate body supply for anatomical purposes and ensues a debate on the end of God that permeates mostly the first season and just a little bit the second season, which is more about State versus Church… The series is not without shortcomings, in particular a rather disconnected plot (which has the appeal of being unpredictable of jumping from one genre to the next) and a repeated proneness of the main character to being a scapegoat, but the reconstitution of London at the time is definitely impressive (although I cannot vouch for its authenticity!). Only the last episode of Season 2 feels a bit short when delivering, by too conveniently tying up all loose threads.
## a Bayesian criterion for singular models [discussion]
Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , on October 10, 2016 by xi'an
[Here is the discussion Judith Rousseau and I wrote about the paper by Mathias Drton and Martyn Plummer, a Bayesian criterion for singular models, which was discussed last week at the Royal Statistical Society. There is still time to send a written discussion! Note: This post was written using the latex2wp converter.]
It is a well-known fact that the BIC approximation of the marginal likelihood in a given irregular model ${\mathcal M_k}$ fails or may fail. The BIC approximation has the form
$\displaystyle BIC_k = \log p(\mathbf Y_n| \hat \pi_k, \mathcal M_k) - d_k \log n /2$
where ${d_k }$ corresponds on the number of parameters to be estimated in model ${\mathcal M_k}$. In irregular models the dimension ${d_k}$ typically does not provide a good measure of complexity for model ${\mathcal M_k}$, at least in the sense that it does not lead to an approximation of
$\displaystyle \log m(\mathbf Y_n |\mathcal M_k) = \log \left( \int_{\mathcal M_k} p(\mathbf Y_n| \pi_k, \mathcal M_k) dP(\pi_k|k )\right) \,.$
A way to understand the behaviour of ${\log m(\mathbf Y_n |\mathcal M_k) }$ is through the effective dimension
$\displaystyle \tilde d_k = -\lim_n \frac{ \log P( \{ KL(p(\mathbf Y_n| \pi_0, \mathcal M_k) , p(\mathbf Y_n| \pi_k, \mathcal M_k) ) \leq 1/n | k ) }{ \log n}$
when it exists, see for instance the discussions in Chambaz and Rousseau (2008) and Rousseau (2007). Watanabe (2009} provided a more precise formula, which is the starting point of the approach of Drton and Plummer:
$\displaystyle \log m(\mathbf Y_n |\mathcal M_k) = \log p(\mathbf Y_n| \hat \pi_k, \mathcal M_k) - \lambda_k(\pi_0) \log n + [m_k(\pi_0) - 1] \log \log n + O_p(1)$
where ${\pi_0}$ is the true parameter. The authors propose a clever algorithm to approximate of the marginal likelihood. Given the popularity of the BIC criterion for model choice, obtaining a relevant penalized likelihood when the models are singular is an important issue and we congratulate the authors for it. Indeed a major advantage of the BIC formula is that it is an off-the-shelf crierion which is implemented in many softwares, thus can be used easily by non statisticians. In the context of singular models, a more refined approach needs to be considered and although the algorithm proposed by the authors remains quite simple, it requires that the functions ${ \lambda_k(\pi)}$ and ${m_k(\pi)}$ need be known in advance, which so far limitates the number of problems that can be thus processed. In this regard their equation (3.2) is both puzzling and attractive. Attractive because it invokes nonparametric principles to estimate the underlying distribution; puzzling because why should we engage into deriving an approximation like (3.1) and call for Bayesian principles when (3.1) is at best an approximation. In this case why not just use a true marginal likelihood?
1. Why do we want to use a BIC type formula?
The BIC formula can be viewed from a purely frequentist perspective, as an example of penalised likelihood. The difficulty then stands into choosing the penalty and a common view on these approaches is to choose the smallest possible penalty that still leads to consistency of the model choice procedure, since it then enjoys better separation rates. In this case a ${\log \log n}$ penalty is sufficient, as proved in Gassiat et al. (2013). Now whether or not this is a desirable property is entirely debatable, and one might advocate that for a given sample size, if the data fits the smallest model (almost) equally well, then this model should be chosen. But unless one is specifying what equally well means, it does not add much to the debate. This also explains the popularity of the BIC formula (in regular models), since it approximates the marginal likelihood and thus benefits from the Bayesian justification of the measure of fit of a model for a given data set, often qualified of being a Bayesian Ockham’s razor. But then why should we not compute instead the marginal likelihood? Typical answers to this question that are in favour of BIC-type formula include: (1) BIC is supposingly easier to compute and (2) BIC does not call for a specification of the prior on the parameters within each model. Given that the latter is a difficult task and that the prior can be highly influential in non-regular models, this may sound like a good argument. However, it is only apparently so, since the only justification of BIC is purely asymptotic, namely, in such a regime the difficulties linked to the choice of the prior disappear. This is even more the case for the sBIC criterion, since it is only valid if the parameter space is compact. Then the impact of the prior becomes less of an issue as non informative priors can typically be used. With all due respect, the solution proposed by the authors, namely to use the posterior mean or the posterior mode to allow for non compact parameter spaces, does not seem to make sense in this regard since they depend on the prior. The same comments apply to the author’s discussion on Prior’s matter for sBIC. Indeed variations of the sBIC could be obtained by penalizing for bigger models via the prior on the weights, for instance as in Mengersen and Rousseau (2011) or by, considering repulsive priors as in Petralia et al. (20120, but then it becomes more meaningful to (again) directly compute the marginal likelihood. Remains (as an argument in its favour) the relative computational ease of use of sBIC, when compared with the marginal likelihood. This simplification is however achieved at the expense of requiring a deeper knowledge on the behaviour of the models and it therefore looses the off-the-shelf appeal of the BIC formula and the range of applications of the method, at least so far. Although the dependence of the approximation of ${\log m(\mathbf Y_n |\mathcal M_k)}$ on ${\mathcal M_j }$, $latex {j \leq k} is strange, this does not seem crucial, since marginal likelihoods in themselves bring little information and they are only meaningful when compared to other marginal likelihoods. It becomes much more of an issue in the context of a large number of models. 2. Should we care so much about penalized or marginal likelihoods ? Marginal or penalized likelihoods are exploratory tools in a statistical analysis, as one is trying to define a reasonable model to fit the data. An unpleasant feature of these tools is that they provide numbers which in themselves do not have much meaning and can only be used in comparison with others and without any notion of uncertainty attached to them. A somewhat richer approach of exploratory analysis is to interrogate the posterior distributions by either varying the priors or by varying the loss functions. The former has been proposed in van Havre et l. (2016) in mixture models using the prior tempering algorithm. The latter has been used for instance by Yau and Holmes (2013) for segmentation based on Hidden Markov models. Introducing a decision-analytic perspective in the construction of information criteria sounds to us like a reasonable requirement, especially when accounting for the current surge in studies of such aspects. [Posted as arXiv:1610.02503] ## approximate Bayesian inference Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , on March 23, 2016 by xi'an Maybe it is just a coincidence, but both most recent issues of Bayesian Analysis have an article featuring approximate Bayesian inference. One is by Daniel Graham and co-authors on Approximate Bayesian Inference for Doubly Robust Estimation, while the other one is by Chris Drovandi and co-authors from QUT on Exact and Approximate Bayesian Inference for Low Integer-Valued Time Series Models with Intractable Likelihoods. The first paper has little connection with ABC. Even though it (a) uses a lot of three letter acronyms [which does not help with speed reading] and (b) relies on moment based and propensity score models. Instead, it relies on Bayesian bootstrap, which suddenly seems to me to be rather connected with empirical likelihood! Except the weights are estimated via a Dirichlet prior instead of being optimised. The approximation lies in using the bootstrap to derive a posterior predictive. I did not spot any assessment or control of the approximation effect in the paper. “Note that we are always using the full data so avoiding the need to choose a summary statistic” (p.326) The second paper connects pMCMC with ABC. Plus pseudo-marginals on the side! And even simplified reversible jump MCMC!!! I am far from certain I got every point of the paper, though, especially the notion of dimension reduction associated with this version of reversible jump MCMC. It may mean that latent variables are integrated out in approximate (marginalised) likelihoods [as explicated in Andrieu and Roberts (2009)]. “The difference with the common ABC approach is that we match on observations one-at-a-time” (p.328) The model that the authors study is an integer value time-series, like the INAR(p) model. Which integer support allows for a non-zero probability of exact matching between simulated and observed data. One-at-a-time as indicated in the above quote. And integer valued tolerances like ε=1 otherwise. In the case auxiliary variables are necessary, the authors resort to the alive particle filter of Jasra et al. (2013), which main point is to produce an unbiased estimate of the (possibly approximate) likelihood, to be exploited by pseudo-marginal techniques. However, unbiasedness sounds less compelling when moving to approximate methods, as illustrated by the subsequent suggestion to use a more stable estimate of the log-likelihood. In fact, when the tolerance ε is positive, the pMCMC acceptance probability looks quite close to an ABC-MCMC probability when relying on several pseudo-data simulations. Which is unbiased for the “right” approximate target. A fact that may actually holds for all ABC algorithms. One quite interesting aspect of the paper is its reflection about the advantage of pseudo-marginal techniques for RJMCMC algorithms since they allow for trans-dimension moves to be simplified, as they consider marginals on the space of interest. Up to this day, I had not realised Andrieu and Roberts (2009) had a section on this aspect… I am still unclear about the derivation of the posterior probabilities of the models under comparison, unless it is a byproduct of the RJMCMC algorithm. A last point is that, for some of the Markov models used in the paper, the pseudo observations can be produced as a random one-time move away from the current true observation, which makes life much easier for ABC and explain why exact simulations can sometimes be produced. (A side note: the authors mention on p.326 that EP is only applicable when the posterior is from an exponential family, while my understanding is that it uses an exponential family to approximate the true posterior.) ## ABC in Sydney, July 3-4, 2014!!! Posted in pictures, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , , , on February 12, 2014 by xi'an After ABC in Paris in 2009, ABC in London in 2011, and ABC in Roma last year, things are accelerating since there will be—as I just learned— an ABC in Sydney next July (not June as I originally typed, thanks Robin!). The workshop on the current developments of ABC methodology thus leaves Europe to go down-under and to take advantage of the IMS Meeting in Sydney on July 7-10, 2014. Hopefully, “ABC in…” will continue its tour of European capitals in 2015! To keep up with an unbroken sequence of free workshops, Scott Sisson has managed to find support so that attendance is free of charge (free as in “no registration fee at all”!) but you do need to register as space is limited. While I would love to visit UNSW and Sydney once again and attend the workshop, I will not, getting ready for Cancún and our ABC short course there. ## fie on fee frenzy! Posted in Mountains, Statistics, Travel, University life with tags , , , , , , , , , , , , on June 11, 2013 by xi'an In the past years, I noticed a clear inflation on conference fees, inflation that I feel unjustified… I already mentioned the huge$720 fees for the Winter Simulation Conference (WSC 2012), which were certainly not all due to the heating bill! Even conferences held by and in universities or societies seem to face the same doom: to stick to conferences I will attend—and do support, to the point of being directly or indirectly involved—, take for instance Bayes 250 in London (RSS Headquarters), £135, Bayes 250 at Duke, $190, both one day-long, and O-Bayes 2013, also at Duke,$480 (in par with JSM fees)… While those later conferences include side “benefits” like meals and banquet, the amount remains large absolutive. Too large. And prohibitive for participants from less-favoured countries (possibly including speakers themselves in the case of O-Bayes 2013). And also counter-productive in the case of both Bayes 250 conferences since we want to get together to celebrate two and a half centuries of Bayesian statistics. Since most of the talks there will be partly commemorative, rather than on the brink of research, I fear some people may have to make a choice to allocate their meagre research funds to other conferences. And I do not understand why universities now consider organising meetings as a source of income rather than as a natural part of their goals.
Now, you may ask, and what about MCMski on which I have more than a modicum of control..?! Well, the sole cost there is renting the conference centre in Chamonix, which is the only place I knew where a large conference could be held. Apart from that, no frill! The coffee breaks will be few and frugal, there will be no free lunch or breakfast or banquet, and no one will get a free entry or a paid invitation. As a result, the registration fee is only 170€ for three days (plus a free satellite meeting the next day), an amount computed on an expected number of participants of 150 and which could lead me to pay the deficit from my own research grants in case I am wrong. (And may I recall the “ABC in…” series, which has been free of fees so far!)
My point, overall, is that we should aim at more frugal meetings, in order to attract larger and more diverse crowds (even though fees are only part of the equation, lodging and travelling can be managed to some extent as long as the workshop is not in too an exotic location). | {"extraction_info": {"found_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": 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": 0.8015974760055542, "perplexity": 905.9319496608853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585405.74/warc/CC-MAIN-20211021102435-20211021132435-00704.warc.gz"} |
https://worldwidescience.org/topicpages/h/hidden+sector+dirac.html | #### Sample records for hidden sector dirac
1. Coupling of Hidden Sector
OpenAIRE
Królikowski, Wojciech
2016-01-01
A hypothetic Hidden Sector of the Universe, consisting of sterile fer\\-mions (sterinos'') and sterile mediating bosons (sterons'') of mass dimension 1 (not 2!) --- the last described by an antisymmetric tensor field --- requires to exist also a scalar isovector and scalar isoscalar in order to be able to construct electroweak invariant coupling (before spontaneously breaking its symmetry). The introduced scalar isoscalar might be a resonant source for the diphoton excess of 750 GeV, sugge...
2. Higgs Portal into Hidden Sectors
CERN Multimedia
CERN. Geneva
2007-01-01
Several attractive theoretical ideas suggest the existence of one or more 'hidden sectors' consisting of standard model singlet fields, some of which may not be too heavy. There is a profound reason to think that the Higgs sector might provide the first access to these hidden sectors. This scenario could affect Higgs phenomenology in drastic ways.
3. Dissipative hidden sector dark matter
Science.gov (United States)
Foot, R.; Vagnozzi, S.
2015-01-01
A simple way of explaining dark matter without modifying known Standard Model physics is to require the existence of a hidden (dark) sector, which interacts with the visible one predominantly via gravity. We consider a hidden sector containing two stable particles charged under an unbroken U (1 )' gauge symmetry, hence featuring dissipative interactions. The massless gauge field associated with this symmetry, the dark photon, can interact via kinetic mixing with the ordinary photon. In fact, such an interaction of strength ε ˜10-9 appears to be necessary in order to explain galactic structure. We calculate the effect of this new physics on big bang nucleosynthesis and its contribution to the relativistic energy density at hydrogen recombination. We then examine the process of dark recombination, during which neutral dark states are formed, which is important for large-scale structure formation. Galactic structure is considered next, focusing on spiral and irregular galaxies. For these galaxies we modeled the dark matter halo (at the current epoch) as a dissipative plasma of dark matter particles, where the energy lost due to dissipation is compensated by the energy produced from ordinary supernovae (the core-collapse energy is transferred to the hidden sector via kinetic mixing induced processes in the supernova core). We find that such a dynamical halo model can reproduce several observed features of disk galaxies, including the cored density profile and the Tully-Fisher relation. We also discuss how elliptical and dwarf spheroidal galaxies could fit into this picture. Finally, these analyses are combined to set bounds on the parameter space of our model, which can serve as a guideline for future experimental searches.
4. Hidden sector behind the CKM matrix
Science.gov (United States)
Okawa, Shohei; Omura, Yuji
2017-08-01
The small quark mixing, described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix in the standard model, may be a clue to reveal new physics around the TeV scale. We consider a simple scenario that extra particles in a hidden sector radiatively mediate the flavor violation to the quark sector around the TeV scale and effectively realize the observed CKM matrix. The lightest particle in the hidden sector, whose contribution to the CKM matrix is expected to be dominant, is a good dark matter (DM) candidate. There are many possible setups to describe this scenario, so that we investigate some universal predictions of this kind of model, focusing on the contribution of DM to the quark mixing and flavor physics. In this scenario, there is an explicit relation between the CKM matrix and flavor violating couplings, such as four-quark couplings, because both are radiatively induced by the particles in the hidden sector. Then, we can explicitly find the DM mass region and the size of Yukawa couplings between the DM and quarks, based on the study of flavor physics and DM physics. In conclusion, we show that DM mass in our scenario is around the TeV scale, and the Yukawa couplings are between O (0.01 ) and O (1 ). The spin-independent DM scattering cross section is estimated as O (10-9) [pb]. An extra colored particle is also predicted at the O (10 ) TeV scale.
5. Discovering hidden sectors with monophoton Z' searches
International Nuclear Information System (INIS)
Gershtein, Yuri; Petriello, Frank; Quackenbush, Seth; Zurek, Kathryn M.
2008-01-01
In many theories of physics beyond the standard model, from extra dimensions to Hidden Valleys and models of dark matter, Z ' bosons mediate between standard model particles and hidden sector states. We study the feasibility of observing such hidden states through an invisibly decaying Z ' at the LHC. We focus on the process pp→γZ ' →γXX † , where X is any neutral, (quasi-) stable particle, whether a standard model neutrino or a new state. This complements a previous study using pp→ZZ ' →l + l - XX † . Only the Z ' mass and two effective charges are needed to describe this process. If the Z ' decays invisibly only to standard model neutrinos, then these charges are predicted by observation of the Z ' through the Drell-Yan process, allowing discrimination between Z ' decays to standard model ν's and invisible decays to new states. We carefully discuss all backgrounds and systematic errors that affect this search. We find that hidden sector decays of a 1 TeV Z ' can be observed at 5σ significance with 50 fb -1 at the LHC. Observation of a 1.5 TeV state requires super-LHC statistics of 1 ab -1 . Control of the systematic errors, in particular, the parton distribution function uncertainty of the dominant Zγ background, is crucial to maximize the LHC search reach.
6. Laser experiments explore the hidden sector
International Nuclear Information System (INIS)
Ahlers, M.
2007-11-01
Recently, the laser experiments BMV and GammeV, searching for light shining through walls, have published data and calculated new limits on the allowed masses and couplings for axion-like particles. In this note we point out that these experiments can serve to constrain a much wider variety of hidden-sector particles such as, e.g., minicharged particles and hidden-sector photons. The new experiments improve the existing bounds from the older BFRT experiment by a factor of two. Moreover, we use the new PVLAS constraints on a possible rotation and ellipticity of light after it has passed through a strong magnetic field to constrain pure minicharged particle models. For masses -7 times the electron electric charge. This is the best laboratory bound and comparable to bounds inferred from the energy spectrum of the cosmic microwave background. (orig.)
7. Strangest man the hidden life of Paul Dirac, quantum genius
CERN Document Server
Farmelo, Graham
2009-01-01
Paul Dirac was among the great scientific geniuses of the modern age. One of the discoverers of quantum mechanics, the most revolutionary theory of the past century, his contributions had a unique insight, eloquence, clarity, and mathematical power. His prediction of antimatter was one of the greatest triumphs in the history of physics. One of Einstein's most admired colleagues, Dirac was in 1933 the youngest theoretician ever to win the Nobel Prize in physics. Dirac's personality is legendary. He was an extraordinarily reserved loner, relentlessly literal-minded and appeared to have no empath
8. Probing hidden sector photons through the Higgs window
International Nuclear Information System (INIS)
Ahlers, M.
2008-07-01
We investigate the possibility that a (light) hidden sector extra photon receives its mass via spontaneous symmetry breaking of a hidden sector Higgs boson, the so-called hidden-Higgs. The hidden-photon can mix with the ordinary photon via a gauge kinetic mixing term. The hidden-Higgs can couple to the Standard Model Higgs via a renormalizable quartic term - sometimes called the Higgs Portal. We discuss the implications of this light hidden-Higgs in the context of laser polarization and light-shining-through-the-wall experiments as well as cosmological, astrophysical, and non-Newtonian force measurements. For hidden-photons receiving their mass from a hidden-Higgs we find in the small mass regime significantly stronger bounds than the bounds on massive hidden sector photons alone. (orig.)
9. Probing hidden sector photons through the Higgs window
International Nuclear Information System (INIS)
Ahlers, Markus; Jaeckel, Joerg; Redondo, Javier; Ringwald, Andreas
2008-01-01
We investigate the possibility that a (light) hidden sector extra photon receives its mass via spontaneous symmetry breaking of a hidden sector Higgs boson, the so-called hidden-Higgs. The hidden-photon can mix with the ordinary photon via a gauge kinetic mixing term. The hidden-Higgs can couple to the standard model Higgs via a renormalizable quartic term - sometimes called the Higgs portal. We discuss the implications of this light hidden-Higgs in the context of laser polarization and light-shining-through-the-wall experiments as well as cosmological, astrophysical, and non-Newtonian force measurements. For hidden-photons receiving their mass from a hidden-Higgs, we find in the small mass regime significantly stronger bounds than the bounds on massive hidden sector photons alone.
10. Searching for hidden sectors in multiparticle production at the LHC
CERN Document Server
Sanchis-Lozano, Miguel-Angel; Moreno-Picot, Salvador
2016-01-01
We study the impact of a hidden sector beyond the Standard Model, e.g. a Hidden Valley model, on factorial moments and cumulants of multiplicity distributions in multiparticle production with a special emphasis on the prospects for LHC results.
11. Abelian hidden sectors at a GeV
International Nuclear Information System (INIS)
Morrissey, David E.; Poland, David; Zurek, Kathryn M.
2009-01-01
We discuss mechanisms for naturally generating GeV-scale hidden sectors in the context of weak-scale supersymmetry. Such low mass scales can arise when hidden sectors are more weakly coupled to supersymmetry breaking than the visible sector, as happens when supersymmetry breaking is communicated to the visible sector by gauge interactions under which the hidden sector is uncharged, or if the hidden sector is sequestered from gravity-mediated supersymmetry breaking. We study these mechanisms in detail in the context of gauge and gaugino mediation, and present specific models of Abelian GeV-scale hidden sectors. In particular, we discuss kinetic mixing of a U(1) x gauge force with hypercharge, singlets or bi-fundamentals which couple to both sectors, and additional loop effects. Finally, we investigate the possible relevance of such sectors for dark matter phenomenology, as well as for low- and high-energy collider searches.
12. The strangest man. The hidden life of Paul Dirac
International Nuclear Information System (INIS)
Farmelo, Graham
2016-01-01
The Strangest Man is the Costa Biography Award-winning account of Paul Dirac, the famous physicist sometimes called the British Einstein. He was one of the leading pioneers of the greatest revolution in twentieth-century science: quantum mechanics. The youngest theoretician ever to win the Nobel Prize for Physics, he was also pathologically reticent, strangely literal-minded and legendarily unable to communicate or empathize. Through his greatest period of productivity, his postcards home contained only remarks about the weather. Based on a previously undiscovered archive of family papers, Graham Farmelo celebrates Dirac's massive scientific achievement while drawing a compassionate portrait of his life and work. Farmelo shows a man who, while hopelessly socially inept, could manage to love and sustain close friendship. The Strangest Man is an extraordinary and moving human story, as well as a study of one of the most exciting times in scientific history.
13. High Energy Colliders and Hidden Sectors
Science.gov (United States)
Dror, Asaf Jeff
This thesis explores two dominant frontiers of theoretical physics, high energy colliders and hidden sectors. The Large Hadron Collider (LHC) is just starting to reach its maximum operational capabilities. However, already with the current data, large classes of models are being put under significant pressure. It is crucial to understand whether the (thus far) null results are a consequence of a lack of solution to the hierarchy problem around the weak scale or requires expanding the search strategy employed at the LHC. It is the duty of the current generation of physicists to design new searches to ensure that no stone is left unturned. To this end, we study the sensitivity of the LHC to the couplings in the Standard Model top sector. We find it can significantly improve the measurements on ZtRtR coupling by a novel search strategy, making use of an implied unitarity violation in such models. Analogously, we show that other couplings in the top sector can also be measured with the same technique. Furthermore, we critically analyze a set of anomalies in the LHC data and how they may appear from consistent UV completions. We also propose a technique to measure lifetimes of new colored particles with non-trivial spin. While the high energy frontier will continue to take data, it is likely the only collider of its kind for the next couple decades. On the other hand, low-energy experiments have a promising future with many new proposed experiments to probe the existence of particles well below the weak scale but with small couplings to the Standard Model. In this work we survey the different possibilities, focusingon the constraints as well as possible new hidden sector dynamics. In particular, we show that vector portals which couple to an anomalous current, e.g., baryon number, are significantly constrained from flavor changing meson decays and rare Z decays. Furthermore, we present a new mechanism for dark matter freezeout which depletes the dark sector through an
14. Supersymmetric leptogenesis with a light hidden sector
International Nuclear Information System (INIS)
De Simone, Andrea
2010-04-01
Supersymmetric scenarios incorporating thermal leptogenesis as the origin of the observed matter-antimatter asymmetry generically predict abundances of the primordial elements which are in conflict with observations. In this paper we pro- pose a simple way to circumvent this tension and accommodate naturally ther- mal leptogenesis and primordial nucleosynthesis. We postulate the existence of a light hidden sector, coupled very weakly to the Minimal Supersymmetric Standard Model, which opens up new decay channels for the next-to-lightest supersymmetric particle, thus diluting its abundance during nucleosynthesis. We present a general model-independent analysis of this mechanism as well as two concrete realizations, and describe the relevant cosmological and astrophysical bounds and implications for this dark matter scenario. Possible experimental signatures at colliders and in cosmic-ray observations are also discussed. (orig.)
15. Searching for hidden sector in multiparticle production at LHC
Directory of Open Access Journals (Sweden)
Miguel-Angel Sanchis-Lozano
2016-03-01
Full Text Available We study the impact of a hidden sector beyond the Standard Model, e.g. a Hidden Valley model, on factorial moments and cumulants of multiplicity distributions in multiparticle production with a special emphasis on the prospects for LHC results.
16. On the LHC sensitivity for non-thermalised hidden sectors
Science.gov (United States)
Kahlhoefer, Felix
2018-04-01
We show under rather general assumptions that hidden sectors that never reach thermal equilibrium in the early Universe are also inaccessible for the LHC. In other words, any particle that can be produced at the LHC must either have been in thermal equilibrium with the Standard Model at some point or must be produced via the decays of another hidden sector particle that has been in thermal equilibrium. To reach this conclusion, we parametrise the cross section connecting the Standard Model to the hidden sector in a very general way and use methods from linear programming to calculate the largest possible number of LHC events compatible with the requirement of non-thermalisation. We find that even the HL-LHC cannot possibly produce more than a few events with energy above 10 GeV involving states from a non-thermalised hidden sector.
17. Helioscope bounds on hidden sector photons
International Nuclear Information System (INIS)
Redondo, J.
2008-01-01
The flux of hypothetical ''hidden photons'' from the Sun is computed under the assumption that they interact with normal matter only through kinetic mixing with the ordinary standard model photon. Requiring that the exotic luminosity is smaller than the standard photon luminosity provides limits for the mixing parameter down to χ -14 , depending on the hidden photon mass. Furthermore, it is pointed point out that helioscopes looking for solar axions are also sensitive to hidden photons. The recent results of the CAST collaboration are used to further constrain the mixing parameter χ at low masses (m γ' <1 eV) where the luminosity bound is weaker. In this regime the solar hidden photon ux has a sizable contribution of longitudinally polarized hidden photons of low energy which are invisible for current helioscopes. (orig.)
18. Rare Z boson decays to a hidden sector
Science.gov (United States)
Blinov, Nikita; Izaguirre, Eder; Shuve, Brian
2018-01-01
We demonstrate that rare decays of the Standard Model Z boson can be used to discover and characterize the nature of new hidden-sector particles. We propose new searches for these particles in soft, high-multiplicity leptonic final states at the Large Hadron Collider. The proposed searches are sensitive to low-mass particles produced in Z decays, and we argue that these striking signatures can shed light on the hidden-sector couplings and mechanism for mass generation.
19. Hidden-Sector Dynamics and the Supersymmetric Seesaw
CERN Document Server
Campbell, Bruce A; Maybury, David W
2008-01-01
In light of recent analyses that have shown that nontrivial hidden-sector dynamics in models of supersymmetry breaking can lead to a significant impact on the predicted low-energy supersymmetric spectrum, we extend these studies to consider hidden-sector effects in extensions of the MSSM to include a seesaw model for neutrino masses. A dynamical hidden sector in an interval of mass scales below the seesaw scale would yield renormalization-group running involving both the anomalous dimension from the hidden sector and the seesaw-extended MSSM renormalization group equations (RGEs). These effects interfere in general, altering the generational mixing of the sleptons, and allowing for a substantial change to the expected level of charged-lepton flavour violation in seesaw-extended MSSM models. These results provide further support for recent theoretical observations that knowledge of the hidden sector is required in order to make concrete low-energy predictions, if the hidden sector is strongly coupled. In parti...
20. Update on hidden sectors with dark forces and dark matter
Energy Technology Data Exchange (ETDEWEB)
Andreas, Sarah
2012-11-15
Recently there has been much interest in hidden sectors, especially in the context of dark matter and ''dark forces'', since they are a common feature of beyond standard model scenarios like string theory and SUSY and additionally exhibit interesting phenomenological aspects. Various laboratory experiments place limits on the so-called hidden photon and continuously further probe and constrain the parameter space; an updated overview is presented here. Furthermore, for several hidden sector models with light dark matter we study the viability with respect to the relic abundance and direct detection experiments.
1. Light hidden-sector U(1)s in string compactifications
Energy Technology Data Exchange (ETDEWEB)
Goodsell, Mark; Ringwald, Andreas
2010-02-15
We review the case for light U(1) gauge bosons in the hidden-sector of heterotic and type II string compactifications, present estimates of the size of their kinetic mixing with the visible-sector hypercharge U(1), and discuss their possibly very interesting phenomenological consequences in particle physics and cosmology. (orig.)
2. Light hidden-sector U(1)s in string compactifications
International Nuclear Information System (INIS)
Goodsell, Mark; Ringwald, Andreas
2010-02-01
We review the case for light U(1) gauge bosons in the hidden-sector of heterotic and type II string compactifications, present estimates of the size of their kinetic mixing with the visible-sector hypercharge U(1), and discuss their possibly very interesting phenomenological consequences in particle physics and cosmology. (orig.)
3. A two particle hidden sector and the oscillations with photons
Energy Technology Data Exchange (ETDEWEB)
Alvarez, Pedro D. [Universidad de Antofagasta, Departamento de Fisica, Antofagasta (Chile); Arias, Paola; Maldonado, Carlos [Universidad de Santiago de Chile, Departmento de Fisica, Santiago (Chile)
2018-01-15
We present a detailed study of the oscillations and optical properties for vacuum, in a model for the dark sector that contains axion-like particles and hidden photons. We provide bounds for the couplings versus the mass, using current results from ALPS-I and PVLAS. We also discuss the challenges for the detection of models with more than one hidden particle in light shining trough wall-like experiments. (orig.)
4. Gauge mediation scenario with hidden sector renormalization in MSSM
International Nuclear Information System (INIS)
Arai, Masato; Kawai, Shinsuke; Okada, Nobuchika
2010-01-01
We study the hidden sector effects on the mass renormalization of a simplest gauge-mediated supersymmetry breaking scenario. We point out that possible hidden sector contributions render the soft scalar masses smaller, resulting in drastically different sparticle mass spectrum at low energy. In particular, in the 5+5 minimal gauge-mediated supersymmetry breaking with high messenger scale (that is favored by the gravitino cold dark matter scenario), we show that a stau can be the next lightest superparticle for moderate values of hidden sector self-coupling. This provides a very simple theoretical model of long-lived charged next lightest superparticles, which imply distinctive signals in ongoing and upcoming collider experiments.
5. Gauge mediation scenario with hidden sector renormalization in MSSM
Science.gov (United States)
Arai, Masato; Kawai, Shinsuke; Okada, Nobuchika
2010-02-01
We study the hidden sector effects on the mass renormalization of a simplest gauge-mediated supersymmetry breaking scenario. We point out that possible hidden sector contributions render the soft scalar masses smaller, resulting in drastically different sparticle mass spectrum at low energy. In particular, in the 5+5¯ minimal gauge-mediated supersymmetry breaking with high messenger scale (that is favored by the gravitino cold dark matter scenario), we show that a stau can be the next lightest superparticle for moderate values of hidden sector self-coupling. This provides a very simple theoretical model of long-lived charged next lightest superparticles, which imply distinctive signals in ongoing and upcoming collider experiments.
6. Cold dark matter from the hidden sector
International Nuclear Information System (INIS)
Arias, Paola; Pontificia Univ. Catolica de Chile, Santiago
2012-02-01
Weakly interacting slim particles (WISPs) such as hidden photons (HP) and axion-like particles (ALPs) have been proposed as cold dark matter candidates. They might be produced non-thermally via the misalignment mechanism, similarly to cold axions. In this talk we review the main processes of thermalisation of HP and we compute the parameter space that may survive as cold dark matter population until today. Our findings are quite encouraging for experimental searches in the laboratory in the near future.
7. Cold dark matter from the hidden sector
Energy Technology Data Exchange (ETDEWEB)
Arias, Paola [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Pontificia Univ. Catolica de Chile, Santiago (Chile). Facultad de Fisica
2012-02-15
Weakly interacting slim particles (WISPs) such as hidden photons (HP) and axion-like particles (ALPs) have been proposed as cold dark matter candidates. They might be produced non-thermally via the misalignment mechanism, similarly to cold axions. In this talk we review the main processes of thermalisation of HP and we compute the parameter space that may survive as cold dark matter population until today. Our findings are quite encouraging for experimental searches in the laboratory in the near future.
8. Exploring a hidden fermionic dark sector
Indian Academy of Sciences (India)
Debasish Majumdar
2017-10-09
Oct 9, 2017 ... background radiation (CMBR) by Planck [1] satellite experiment suggests ... (SM) of particle physics also cannot explain the physics of dark matter. ... the dark sector also achieve mass from the spontaneous breaking of this ...
9. Heavy superpartners with less tuning from hidden sector renormalisation
International Nuclear Information System (INIS)
Hardy, Edward
2014-01-01
In supersymmetric extensions of the Standard Model, superpartner masses consistent with collider bounds typically introduce significant tuning of the electroweak scale. We show that hidden sector renormalisation can greatly reduce such a tuning if the supersymmetry breaking, or mediating, sector runs through a region of strong coupling not far from the weak scale. In the simplest models, only the tuning due to the gaugino masses is improved, and a weak scale gluino mass in the region of 5 TeV may be obtained with an associated tuning of only one part in ten. In models with more complex couplings between the visible and hidden sectors, the tuning with respect to sfermions can also be reduced. We give an example of a model, with low scale gauge mediation and superpartner masses allowed by current LHC bounds, that has an overall tuning of one part in twenty
10. Dark matter and dark forces from a supersymmetric hidden sector
Energy Technology Data Exchange (ETDEWEB)
Andreas, S.; Goodsell, M.D.; Ringwald, A.
2011-09-15
We show that supersymmetric ''Dark Force'' models with gravity mediation are viable. To this end, we analyse a simple supersymmetric hidden sector model that interacts with the visible sector via kinetic mixing of a light Abelian gauge boson with the hypercharge. We include all induced interactions with the visible sector such as neutralino mass mixing and the Higgs portal term. We perform a detailed parameter space scan comparing the produced dark matter relic abundance and direct detection cross-sections to current experiments. (orig.)
11. Low-scale gravity mediation in warped extra dimension and collider phenomenology on hidden sector
International Nuclear Information System (INIS)
Itoh, H.; Okada, N.; Yamashita, T.
2007-01-01
We propose a new scenario of gravity-mediated supersymmetry breaking (gravity mediation) in a supersymmetric Randall-Sundrum model, where the gravity mediation takes place at a low scale due to the warped metric. We investigate collider phenomenology involving the hidden sector field, and find a possibility that the hidden sector field can be produced at the LHC and the ILC. The hidden sector may no longer be hidden. (author)
12. Invisible axion in the hidden sector of no-scale supergravity
International Nuclear Information System (INIS)
Sato, Hikaru
1987-01-01
We propose a new axion model which incorporates the U(1) PQ symmetry into a hidden sector, as well as an observable sector, of no-scale supergravity models. The axion is a spin-zero field in the hidden sector. The U(1) PQ symmetry is naturally embedded in the family symmetry of the no-scale models. Invisible axions live in the gravity hidden sector without conflict with the cosmological and astrophysical constraints. (orig.)
13. The strangest man. The hidden life of Paul Dirac; Der seltsamste Mensch. Das verborgene Leben des Quantengenies Paul Dirac
Energy Technology Data Exchange (ETDEWEB)
Farmelo, Graham
2016-07-01
The Strangest Man is the Costa Biography Award-winning account of Paul Dirac, the famous physicist sometimes called the British Einstein. He was one of the leading pioneers of the greatest revolution in twentieth-century science: quantum mechanics. The youngest theoretician ever to win the Nobel Prize for Physics, he was also pathologically reticent, strangely literal-minded and legendarily unable to communicate or empathize. Through his greatest period of productivity, his postcards home contained only remarks about the weather. Based on a previously undiscovered archive of family papers, Graham Farmelo celebrates Dirac's massive scientific achievement while drawing a compassionate portrait of his life and work. Farmelo shows a man who, while hopelessly socially inept, could manage to love and sustain close friendship. The Strangest Man is an extraordinary and moving human story, as well as a study of one of the most exciting times in scientific history.
14. Recent Developments in Supersymmetric and Hidden Sector Dark Matter
International Nuclear Information System (INIS)
Feldman, Daniel; Liu Zuowei; Nath, Pran
2008-01-01
New results which correlate SUSY dark matter with LHC signals are presented, and a brief review of recent developments in supersymmetric and hidden sector dark matter is given. It is shown that the direct detection of dark matter is very sensitive to the hierarchical SUSY sparticle spectrum and the spectrum is very useful in distinguishing models. It is shown that the prospects of the discovery of neutralino dark matter are very bright on the 'Chargino Wall' due to a copious number of model points on the Wall, where the NLSP is the Chargino, and the spin independent neutralino-proton cross section is maintained at high values in the 10 -44 cm 2 range for neutralino masses up to ∼850 GeV. It is also shown that the direct detection of dark matter along with lepton plus jet signatures and missing energy provide dual, and often complementary, probes of supersymmetry. Finally, we discuss an out of the box possibility for dark matter, which includes dark matter from the hidden sector, which could either consist of extra weakly interacting dark matter (a Stino XWIMP), or milli-charged dark matter arising from the Stueckelberg extensions of the MSSM or the SM.
15. New experimental limit on photon hidden-sector paraphoton mixing
Energy Technology Data Exchange (ETDEWEB)
Afanasev, A. [Department of Physics, Hampton University, Hampton, VA 23668 (United States); Baker, O.K. [Department of Physics, Yale University, PO Box 208120, New Haven, CT 06520 (United States)], E-mail: [email protected]; Beard, K.B. [Muons, Inc., 552 N. Batavia Avenue, Batavia, IL 60510 (United States); Biallas, G.; Boyce, J. [Free Electron Laser Division, Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606 (United States); Minarni, M. [Department of Physics, Universitas Riau (UNRI), Pekanbaru, Riau 28293 (Indonesia); Ramdon, R. [Department of Physics, Hampton University, Hampton, VA 23668 (United States); Shinn, M. [Free Electron Laser Division, Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606 (United States); Slocum, P. [Department of Physics, Yale University, PO Box 208120, New Haven, CT 06520 (United States)
2009-08-31
We report on the first results of a search for optical-wavelength photons mixing with hypothetical hidden-sector paraphotons in the mass range between 10{sup -5} and 10{sup -2} electron volts for a mixing parameter greater than 10{sup -7}. This was a generation-regeneration experiment using the 'light shining through a wall' technique in which regenerated photons are searched for downstream of an optical barrier that separates it from an upstream generation region. The new limits presented here are the most stringent limits to date on the mixing parameter. The present results indicate no evidence for photon-paraphoton mixing for the range of parameters investigated.
16. LHCb - Search for hidden-sector bosons at LHCb
CERN Multimedia
Mauri, Andrea
2016-01-01
A search is presented for a hidden-sector boson, $\\chi$, produced in the decay $B^0 \\rightarrow K^* (892)^0 \\chi$, with $K^* (892)^0 \\rightarrow K^+ \\pi^-$ and $\\chi \\rightarrow \\mu^+ \\mu^-$ . The search is performed using a $pp$-collision data sample collected at $\\sqrt{s}=7$ and 8 TeV with the LHCb detector, corresponding to integrated luminosities of 1 and 2 fb$^{-1}$ respectively. No significant signal is observed in the mass range $214 \\le m_\\chi \\le 4350$ MeV, and upper limits are placed on the branching fraction product $\\mathcal{B}(B^0 \\rightarrow K^* (892)^0 \\chi) \\times \\mathcal{B}(\\chi \\rightarrow \\mu^+ \\mu^- )$ as a function of the mass and lifetime of the $\\chi$ boson. These limits place the most stringent constraints to date on many theories that predict the existence of additional low-mass dark bosons.
17. Minimal hidden sector models for CoGeNT/DAMA events
International Nuclear Information System (INIS)
Cline, James M.; Frey, Andrew R.
2011-01-01
Motivated by recent attempts to reconcile hints of direct dark matter detection by the CoGeNT and DAMA experiments, we construct simple particle physics models that can accommodate the constraints. We point out challenges for building reasonable models and identify the most promising scenarios for getting isospin violation and inelasticity, as indicated by some phenomenological studies. If inelastic scattering is demanded, we need two new light gauge bosons, one of which kinetically mixes with the standard model hypercharge and has mass <2 GeV, and another which couples to baryon number and has mass 6.8±(0.1/0.2) GeV. Their interference gives the desired amount of isospin violation. The dark matter is nearly Dirac, but with small Majorana masses induced by spontaneous symmetry breaking, so that the gauge boson couplings become exactly off-diagonal in the mass basis, and the small mass splitting needed for inelasticity is simultaneously produced. If only elastic scattering is demanded, then an alternative model, with interference between the kinetically mixed gauge boson and a hidden sector scalar Higgs, is adequate to give the required isospin violation. In both cases, the light kinetically mixed gauge boson is in the range of interest for currently running fixed target experiments.
18. Low scale gravity mediation with warped extra dimension and collider phenomenology on the hidden sector
International Nuclear Information System (INIS)
Itoh, Hideo; Okada, Nobuchika; Yamashita, Toshifumi
2006-01-01
We propose a scenario of gravity mediated supersymmetry breaking (gravity mediation) in a supersymmetric Randall-Sundrum model. In our setup, both the visible sector and the hidden sector coexist on the infrared (IR) brane. We introduce the Polonyi model as a simple hidden sector. Because of the warped metric, the effective cutoff scale on the IR brane is 'warped down', so that the gravity mediation occurs at a low scale. As a result, the gravitino is naturally the lightest superpartner (LSP) and contact interactions between the hidden and the visible sector fields become stronger. We address phenomenologies for various IR cutoff scales. In particular, we investigate collider phenomenology involving a scalar field (Polonyi field) in the hidden sector for the case with the IR cutoff around 10 TeV. We find a possibility that the hidden sector scalar can be produced at the LHC and the international linear collider (ILC). Interestingly, the scalar behaves like the Higgs boson of the standard model in the production process, while its decay process is quite different and, once produced, it will provide us with a very clean signature. The hidden sector may be no longer hidden
19. Hidden from view: coupled dark sector physics and small scales
Science.gov (United States)
Elahi, Pascal J.; Lewis, Geraint F.; Power, Chris; Carlesi, Edoardo; Knebe, Alexander
2015-09-01
We study cluster mass dark matter (DM) haloes, their progenitors and surroundings in a coupled dark matter-dark energy (DE) model and compare it to quintessence and Λ cold dark matter (ΛCDM) models with adiabatic zoom simulations. When comparing cosmologies with different expansions histories, growth functions and power spectra, care must be taken to identify unambiguous signatures of alternative cosmologies. Shared cosmological parameters, such as σ8, need not be the same for optimal fits to observational data. We choose to set our parameters to ΛCDM z = 0 values. We find that in coupled models, where DM decays into DE, haloes appear remarkably similar to ΛCDM haloes despite DM experiencing an additional frictional force. Density profiles are not systematically different and the subhalo populations have similar mass, spin, and spatial distributions, although (sub)haloes are less concentrated on average in coupled cosmologies. However, given the scatter in related observables (V_max,R_{V_max}), this difference is unlikely to distinguish between coupled and uncoupled DM. Observations of satellites of Milky Way and M31 indicate a significant subpopulation reside in a plane. Coupled models do produce planar arrangements of satellites of higher statistical significance than ΛCDM models; however, in all models these planes are dynamically unstable. In general, the non-linear dynamics within and near large haloes masks the effects of a coupled dark sector. The sole environmental signature we find is that small haloes residing in the outskirts are more deficient in baryons than their ΛCDM counterparts. The lack of a pronounced signal for a coupled dark sector strongly suggests that such a phenomena would be effectively hidden from view.
20. Diurnal modulation signal from dissipative hidden sector dark matter
Directory of Open Access Journals (Sweden)
R. Foot
2015-09-01
Full Text Available We consider a simple generic dissipative dark matter model: a hidden sector featuring two dark matter particles charged under an unbroken U(1′ interaction. Previous work has shown that such a model has the potential to explain dark matter phenomena on both large and small scales. In this framework, the dark matter halo in spiral galaxies features nontrivial dynamics, with the halo energy loss due to dissipative interactions balanced by a heat source. Ordinary supernovae can potentially supply this heat provided kinetic mixing interaction exists with strength ϵ∼10−9. This type of kinetically mixed dark matter can be probed in direct detection experiments. Importantly, this self-interacting dark matter can be captured within the Earth and shield a dark matter detector from the halo wind, giving rise to a diurnal modulation effect. We estimate the size of this effect for detectors located in the Southern hemisphere, and find that the modulation is large (≳10% for a wide range of parameters.
1. Hidden-sector Spectroscopy with Gravitational Waves from Binary Neutron Stars
Science.gov (United States)
Croon, Djuna; Nelson, Ann E.; Sun, Chen; Walker, Devin G. E.; Xianyu, Zhong-Zhi
2018-05-01
We show that neutron star (NS) binaries can be ideal laboratories to probe hidden sectors with a long-range force. In particular, it is possible for gravitational wave (GW) detectors such as LIGO and Virgo to resolve the correction of waveforms from ultralight dark gauge bosons coupled to NSs. We observe that the interaction of the hidden sector affects both the GW frequency and amplitude in a way that cannot be fitted by pure gravity.
2. 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.
3. WIMPless dark matter from non-Abelian hidden sectors with anomaly-mediated supersymmetry breaking
International Nuclear Information System (INIS)
Feng, Jonathan L.; Shadmi, Yael
2011-01-01
In anomaly-mediated supersymmetry breaking models, superpartner masses are proportional to couplings squared. Their hidden sectors therefore naturally contain WIMPless dark matter, particles whose thermal relic abundance is guaranteed to be of the correct size, even though they are not weakly interacting massive particles. We study viable dark matter candidates in WIMPless anomaly-mediated supersymmetry breaking models with non-Abelian hidden sectors and highlight unusual possibilities that emerge in even the simplest models. In one example with a pure SU(N) hidden sector, stable hidden gluinos freeze out with the correct relic density, but have an extremely low, but natural, confinement scale, providing a framework for self-interacting dark matter. In another simple scenario, hidden gluinos freeze out and decay to visible Winos with the correct relic density, and hidden glueballs may either be stable, providing a natural framework for mixed cold-hot dark matter, or may decay, yielding astrophysical signals. Last, we present a model with light hidden pions that may be tested with improved constraints on the number of nonrelativistic degrees of freedom. All of these scenarios are defined by a small number of parameters, are consistent with gauge coupling unification, preserve the beautiful connection between the weak scale and the observed dark matter relic density, and are natural, with relatively light visible superpartners. We conclude with comments on interesting future directions.
4. A realistic extension of gauge-mediated SUSY-breaking model with superconformal hidden sector
International Nuclear Information System (INIS)
Asano, Masaki; Hisano, Junji; Okada, Takashi; Sugiyama, Shohei
2009-01-01
The sequestering of supersymmetry (SUSY) breaking parameters, which is induced by superconformal hidden sector, is one of the solutions for the μ/B μ problem in gauge-mediated SUSY-breaking scenario. However, it is found that the minimal messenger model does not derive the correct electroweak symmetry breaking. In this Letter we present a model which has the coupling of the messengers with the SO(10) GUT-symmetry breaking Higgs fields. The model is one of the realistic extensions of the gauge mediation model with superconformal hidden sector. It is shown that the extension is applicable for a broad range of conformality breaking scale
5. New ALPS results on hidden-sector lightweights
Energy Technology Data Exchange (ETDEWEB)
Ehret, Klaus [Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg (Germany); Frede, Maik [Laser Zentrum Hannover e.V., Hollerithallee 8, D-30419 Hannover (Germany); Ghazaryan, Samvel [Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg (Germany); Hildebrandt, Matthias [Laser Zentrum Hannover e.V., Hollerithallee 8, D-30419 Hannover (Germany); Knabbe, Ernst-Axel [Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg (Germany); Kracht, Dietmar [Laser Zentrum Hannover e.V., Hollerithallee 8, D-30419 Hannover (Germany); Lindner, Axel, E-mail: [email protected] [Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg (Germany); List, Jenny [Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg (Germany); Meier, Tobias [Max-Planck-Institute for Gravitational Physics, Albert-Einstein-Institute, Institut fuer Gravitationsphysik, Leibniz Universitaet, Hannover, Callinstrasse 38, D-30167 Hannover (Germany); Meyer, Niels; Notz, Dieter; Redondo, Javier; Ringwald, Andreas [Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg (Germany); Wiedemann, Guenter [Hamburger Sternwarte, Gojenbergsweg 112, D-21029 Hamburg (Germany); Willke, Benno [Max-Planck-Institute for Gravitational Physics, Albert-Einstein-Institute, Institut fuer Gravitationsphysik, Leibniz Universitaet, Hannover, Callinstrasse 38, D-30167 Hannover (Germany)
2010-05-31
The ALPS Collaboration runs a 'Light Shining through a Wall' (LSW) experiment to search for photon oscillations into 'Weakly Interacting Sub-eV Particles' (WISPs) often predicted by extensions of the Standard Model. The experiment is set up around a superconducting HERA dipole magnet at the site of DESY. Due to several upgrades of the experiment we are able to place limits on the probability of photon-WISP-photon conversions of a fewx10{sup -25}. These limits result in today's most stringent laboratory constraints on the existence of low mass axion-like particles, hidden photons and minicharged particles.
6. New ALPS results on hidden-sector lightweights
International Nuclear Information System (INIS)
Ehret, Klaus; Ghazaryan, Samvel; Frede, Maik
2010-01-01
The ALPS collaboration runs a ''Light Shining through a Wall'' (LSW) experiment to search for photon oscillations into ''Weakly Interacting Sub-eV Particles'' (WISPs) often predicted by extensions of the Standard Model. The experiment is set up around a superconducting HERA dipole magnet at the site of DESY. Due to several upgrades of the experiment we are able to place limits on the probability of photon-WISP-photon conversions of a few x 10 -25 . These limits result in today's most stringent laboratory constraints on the existence of low mass axion-like particles, hidden photons and minicharged particles. (orig.)
7. Hidden sector dark matter and the Galactic Center gamma-ray excess: a closer look
Science.gov (United States)
Escudero, Miguel; Witte, Samuel J.; Hooper, Dan
2017-11-01
Stringent constraints from direct detection experiments and the Large Hadron Collider motivate us to consider models in which the dark matter does not directly couple to the Standard Model, but that instead annihilates into hidden sector particles which ultimately decay through small couplings to the Standard Model. We calculate the gamma-ray emission generated within the context of several such hidden sector models, including those in which the hidden sector couples to the Standard Model through the vector portal (kinetic mixing with Standard Model hypercharge), through the Higgs portal (mixing with the Standard Model Higgs boson), or both. In each case, we identify broad regions of parameter space in which the observed spectrum and intensity of the Galactic Center gamma-ray excess can easily be accommodated, while providing an acceptable thermal relic abundance and remaining consistent with all current constraints. We also point out that cosmic-ray antiproton measurements could potentially discriminate some hidden sector models from more conventional dark matter scenarios.
8. Hidden Sector Dark Matter and the Galactic Center Gamma-Ray Excess: A Closer Look
Energy Technology Data Exchange (ETDEWEB)
Escudero, Miguel; Witte, Samuel J.; Hooper, Dan
2017-09-20
Stringent constraints from direct detection experiments and the Large Hadron Collider motivate us to consider models in which the dark matter does not directly couple to the Standard Model, but that instead annihilates into hidden sector particles which ultimately decay through small couplings to the Standard Model. We calculate the gamma-ray emission generated within the context of several such hidden sector models, including those in which the hidden sector couples to the Standard Model through the vector portal (kinetic mixing with Standard Model hypercharge), through the Higgs portal (mixing with the Standard Model Higgs boson), or both. In each case, we identify broad regions of parameter space in which the observed spectrum and intensity of the Galactic Center gamma-ray excess can easily be accommodated, while providing an acceptable thermal relic abundance and remaining consistent with all current constraints. We also point out that cosmic-ray antiproton measurements could potentially discriminate some hidden sector models from more conventional dark matter scenarios.
9. New ALPS results on hidden-sector lightweights
Energy Technology Data Exchange (ETDEWEB)
Ehret, Klaus; Ghazaryan, Samvel [Deutsches Elektronen-Synchrotron DESY, Hamburg (Germany); Frede, Maik [Laser Zentrum Hannover e.V. (DE)] (and others)
2010-04-08
The ALPS collaboration runs a ''Light Shining through a Wall'' (LSW) experiment to search for photon oscillations into ''Weakly Interacting Sub-eV Particles'' (WISPs) often predicted by extensions of the Standard Model. The experiment is set up around a superconducting HERA dipole magnet at the site of DESY. Due to several upgrades of the experiment we are able to place limits on the probability of photon-WISP-photon conversions of a few x 10{sup -25}. These limits result in today's most stringent laboratory constraints on the existence of low mass axion-like particles, hidden photons and minicharged particles. (orig.)
10. Big Bang Nucleosynthesis in Visible and Hidden-Mirror Sectors
Directory of Open Access Journals (Sweden)
Paolo Ciarcelluti
2014-01-01
dark matter. The production of ordinary nuclides shows differences from the standard model for a ratio of the temperatures between mirror and ordinary sectors x=T′/T≳0.3, and they present an interesting decrease of the abundance of Li7. For the mirror nuclides, instead, one observes an enhanced production of He4, which becomes the dominant element for x≲0.5, and much larger abundances of heavier elements.
11. Search for dark matter in the hidden-photon sector with a large spherical mirror
CERN Document Server
Veberic, Darko; Doebrich, Babette; Engel, Ralph; Jaeckel, Joerg; Kowalski, Marek; Lindner, Axel; Mathes, Hermann-Josef; Redondo, Javier; Roth, Markus; Schaefer, Christoph; Ulrich, Ralf
2015-01-01
If dark matter consists of hidden-sector photons which kinetically mix with regular photons, a tiny oscillating electric-field component is present wherever we have dark matter. In the surface of conducting materials this induces a small probability to emit single photons almost perpendicular to the surface, with the corresponding photon frequency matching the mass of the hidden photons. We report on a construction of an experimental setup with a large ~14 m2 spherical metallic mirror that will allow for searches of hidden-photon dark matter in the eV and sub-eV range by application of different electromagnetic radiation detectors. We discuss sensitivity and accessible regions in the dark matter parameter space.
12. Looking for a hidden sector in exotic Higgs boson decays with the ATLAS experiment
Directory of Open Access Journals (Sweden)
Andrea Coccaro
2015-12-01
Full Text Available The nature of dark matter (DM is one of the most intriguing questions in particle physics. DM can be postulated to be part of a hidden sector whose interactions with the visible matter are not completely decoupled. The discovery of a fundamental scalar particle compatible with the Higgs boson predicted by the Standard Model paves the way for looking for DM with novel methods. An overview of the searches looking for a hidden sector in exotic Higgs decays and for invisible decays of the Higgs boson within the ATLAS experiment is presented. Prospects for searches with Large Hadron Collider data at a center-of-mass energy of 13 TeV are summarized.
13. Minimal spontaneously broken hidden sector and its impact on Higgs boson physics at the CERN Large Hadron Collider
International Nuclear Information System (INIS)
Schabinger, Robert; Wells, James D.
2005-01-01
Little experimental data bears on the question of whether there is a spontaneously broken hidden sector that has no Standard Model quantum numbers. Here we discuss the prospects of finding evidence for such a hidden sector through renormalizable interactions of the Standard Model Higgs boson with a Higgs boson of the hidden sector. We find that the lightest Higgs boson in this scenario has smaller rates in standard detection channels, and it can have a sizeable invisible final state branching fraction. Details of the hidden sector determine whether the overall width of the lightest state is smaller or larger than the Standard Model width. We compute observable rates, total widths and invisible decay branching fractions within the general framework. We also introduce the 'A-Higgs Model', which corresponds to the limit of a hidden sector Higgs boson weakly mixing with the Standard Model Higgs boson. This model has only one free parameter in addition to the mass of the light Higgs state and it illustrates most of the generic phenomenology issues, thereby enabling it to be a good benchmark theory for collider searches. We end by presenting an analogous supersymmetry model with similar phenomenology, which involves hidden sector Higgs bosons interacting with MSSM Higgs bosons through D-terms
14. Solution to the hierarchy problem from an almost decoupled hidden sector within a classically scale invariant theory
International Nuclear Information System (INIS)
Foot, Robert; Kobakhidze, Archil; Volkas, Raymond R.; McDonald, Kristian L.
2008-01-01
If scale invariance is a classical symmetry then both the Planck scale and the weak scale should emerge as quantum effects. We show that this can be realized in simple scale invariant theories with a hidden sector. The weak/Planck scale hierarchy emerges in the (technically natural) limit in which the hidden sector decouples from the ordinary sector. In this limit, finite corrections to the weak scale are consequently small, while quadratic divergences are absent by virtue of classical scale invariance, so there is no hierarchy problem
15. The hidden winners of renewable energy promotion: Insights into sector-specific wage differentials
International Nuclear Information System (INIS)
Antoni, Manfred; Janser, Markus; Lehmer, Florian
2015-01-01
In light of Germany's energy system transformation, this paper examines differences in employment structures and wage differentials between renewable energy establishments and their sector peers. To do so, we have developed a novel data set by linking company-level information from the German Renewable Energy Federation with administrative establishment-level data from the Institute for Employment Research. Descriptive evidence shows significant differences in wages and several other characteristics between renewable energy establishments and their sector peers. Our estimates give evidence that human capital and other establishment-level characteristics mostly explain the wage differential among manufacturers and energy providers. However, we find a persistent ‘renewable energy wage premium' of more than ten percent in construction/installation activities and architectural/engineering services. We interpret this premium as a positive indirect effect of the promotion of renewable energies for the benefit of employees in renewable energy establishments within these two sectors. - Highlights: • Renewable energy (RE) firms pay considerably more than their non-RE sector peers. • In manufacturing and energy supply, firm attributes explain mainly the wage gap. • In installation, planning and project management one third remains unexplained. • This unexplained rest represents a ‘RE wage premium’ of around 10 percent. • The employees in both sectors are the ‘hidden winners’ of RE promotion.
16. Weak-scale hidden sector and energy transport in fireball models of gamma-ray bursts
International Nuclear Information System (INIS)
Demir, Durmus A.; Mosquera Cuesta, Herman J.
2000-12-01
The annihilation of pairs of very weakly interacting particles in the neighborhood of gamma-ray sources is introduced here as a plausible mechanism to overcome the baryon load problem. This way we can explain how these very high energy gamma-ray bursts can be powered at the onset of very energetic events like supernovae (collapsars) explosions or coalescences of binary neutron stars. Our approach uses the weak-scale hidden sector models in which the Higgs sector of the standard model is extended to include a gauge singlet that only interacts with the Higgs particle. These particles would be produced either during the implosion of the red supergiant star core or at the aftermath of a neutron star binary merger. The whole energetics and timescales of the relativistic blast wave, the fireball, are reproduced. (author)
17. Gauge symmetry breaking in the hidden sector of the flipped SU(5)xU(1) superstring model
Energy Technology Data Exchange (ETDEWEB)
Antoniadis, I.; Rizos, J. (Centre de Physique Theorique, Ecole Polytechnique, 91 - Palaiseau (France)); Tamvakis, K. (Theoretical Physics Div., Univ. Ioannina (Greece))
1992-03-26
We analyze the SU(5)xU(1)'xU(1){sup 4}xSO(10)xSU(4) superstring model with a spontaneously broken hidden sector down to SO(7)xSO(5) taking into account non-renormalizable superpotential terms up to eight order. As a result of the hidden sector breaking the 'exotic' states get a mass and the 'observable' spectrum is composed of the standard three families. In addition, Cabibbo mixing arises at sixth order and an improved fermion mass hierarchy emerges. (orig.).
18. Fermion zero modes in the vortex background of a Chern-Simons-Higgs theory with a hidden sector
Energy Technology Data Exchange (ETDEWEB)
Lozano, Gustavo [Departamento de Física, FCEYN Universidad de Buenos Aires & IFIBA CONICET,Pabellón 1 Ciudad Universitaria, 1428 Buenos Aires (Argentina); Mohammadi, Azadeh [Departamento de Física, Universidade Federal da Paraíba,58.059-970, Caixa Postal 5.008, João Pessoa, PB (Brazil); Schaposnik, Fidel A. [Departamento de Física, Universidad Nacional de La Plata/IFLP/CICBA,CC 67, 1900 La Plata (Argentina)
2015-11-06
In this paper we study a 2+1 dimensional system in which fermions are coupled to the self-dual topological vortex in U(1)×U(1) Chern-Simons theory, where both U(1) gauge symmetries are spontaneously broken. We consider two Abelian Higgs scalars with visible and hidden sectors coupled to a fermionic field through three interaction Lagrangians, where one of them violates the fermion number. Using a fine tuning procedure, we could obtain the number of the fermionic zero modes which is equal to the absolute value of the sum of the vortex numbers in the visible and hidden sectors.
19. Diurnal modulation due to self-interacting mirror and hidden sector dark matter
International Nuclear Information System (INIS)
Foot, R.
2012-01-01
Mirror and more generic hidden sector dark matter models can simultaneously explain the DAMA, CoGeNT and CRESST-II dark matter signals consistently with the null results of the other experiments. This type of dark matter can be captured by the Earth and shield detectors because it is self-interacting. This effect will lead to a diurnal modulation in dark matter detectors. We estimate the size of this effect for dark matter detectors in various locations. For a detector located in the northern hemisphere, this effect is expected to peak in April and can be detected for optimistic parameter choices. The diurnal variation is expected to be much larger for detectors located in the southern hemisphere. In particular, if the CoGeNT detector were moved to e.g. Sierra Grande, Argentina then a 5σ dark matter discovery would be possible in around 30 days of operation
20. Phenomenological constraints imposed by the hidden sector in the flipped SU(5)xU(1) superstring model
Energy Technology Data Exchange (ETDEWEB)
Leontaris, G.K.; Rizos, J.; Tamvakis, K. (Ioannina Univ. (Greece). Theoretical Physics Div.)
1990-06-28
We calculate the trilinear superpotential of the hidden sector of the three generation flipped SU(5)xU(1)xU(1){sup 4}xSO(10)xSU(4) superstring model. We perform a renormalization group analysis of the model taking into account the hidden sector. We find that, in all relevant cases, fractionally charged tetraplets of the hidden SO(6) gauge group are confined at a high scale. Nevertheless, their contribution to the observable U(1) gauge coupling evolution results in a drastic reduction of the available freedom in the values of a{sub 3}(m{sub w}), sin{sup 2}{theta}{sub w} and M{sub x} that allow superunification. (orig.).
1. Search for hidden-sector bosons in $B^0 \\!\\to K^{*0}\\mu^+\\mu^-$ decays
CERN Document Server
2015-10-16
A search is presented for hidden-sector bosons, $\\chi$, produced in the decay ${B^0\\!\\to K^*(892)^0\\chi}$, with $K^*(892)^0\\!\\to K^{+}\\pi^{-}$ and $\\chi\\!\\to\\mu^+\\mu^-$. The search is performed using $pp$-collision data corresponding to 3.0 fb$^{-1}$ collected with the LHCb detector. No significant signal is observed in the accessible mass range $214 \\leq m({\\chi}) \\leq 4350$ MeV, and upper limits are placed on the branching fraction product $\\mathcal{B}(B^0\\!\\to K^*(892)^0\\chi)\\times\\mathcal{B}(\\chi\\!\\to\\mu^+\\mu^-)$ as a function of the mass and lifetime of the $\\chi$ boson. These limits are of the order of $10^{-9}$ for $\\chi$ lifetimes less than 100 ps over most of the $m(\\chi)$ range, and place the most stringent constraints to date on many theories that predict the existence of additional low-mass bosons.
2. Paul Dirac
Science.gov (United States)
Pais, Abraham; Jacob, Maurice; Olive, David I.; Atiyah, Michael F.
2005-09-01
Preface Peter Goddard; Dirac memorial address Stephen Hawking; 1. Paul Dirac: aspects of his life and work Abraham Pais; 2. Antimatter Maurice Jacob; 3. The monopole David Olive; 4. The Dirac equation and geometry Michael F. Atiyah.
3. Mass spectra of four-quark states in the hidden charm sector
International Nuclear Information System (INIS)
Patel, Smruti; Shah, Manan; Vinodkumar, P.C.
2014-01-01
Masses of the low-lying four-quark states in the hidden charm sector (cq anti c anti q; q element of u,d) are calculated within the framework of a non-relativistic quark model. The four-body system is considered as two two-body systems such as diquark-antidiquark (Qq- anti Q anti q) and quark-antiquark-quark-antiquark (Q anti q- anti Qq) molecular-like four-quark states. Here, the Cornell-type potential has been used for describing the two-body interactions among Q-q, anti Q- anti q, Q- anti q, Qq- anti Q anti q and Q anti q- anti Qq, with appropriate string tensions. Our present analysis suggests the following exotic states: X(3823), Z c (3900), X(3915), Z c (4025), ψ (4040), Z 1 (4050) and X(4160) as Q anti q- anti Qq molecular-like four-quark states, while Z c (3885), X(3940) and Y(4140) as the diquark-antidiquark four-quark states. We have been able to assign the J PC values for many of the recently observed exotic states according to their structure. Apart from this, we have identified the charged state Z(4430) recently confirmed by LHCb as the first radial excitation of Zc(3885) with G = + 1 and Y(4360) state as the first radial excitation of Y(4008) with G = - 1 and the state ψ(4415) as the first radial excitation of the ψ(4040) state. (orig.)
4. Dirac materials
OpenAIRE
Wehling, T. O.; Black-Schaffer, A. M.; Balatsky, A. V.
2014-01-01
A wide range of materials, like d-wave superconductors, graphene, and topological insulators, share a fundamental similarity: their low-energy fermionic excitations behave as massless Dirac particles rather than fermions obeying the usual Schrodinger Hamiltonian. This emergent behavior of Dirac fermions in condensed matter systems defines the unifying framework for a class of materials we call "Dirac materials''. In order to establish this class of materials, we illustrate how Dirac fermions ...
5. Search for Hidden-Sector Bosons in B(0)→K(*0)μ(+)μ(-) Decays.
Science.gov (United States)
2015-10-16
A search is presented for hidden-sector bosons, χ, produced in the decay B(0)→K*(892)(0)χ, with K*(892)(0)→K(+)π(-) and χ→μ(+)μ(-). The search is performed using pp-collision data corresponding to 3.0 fb(-1) collected with the LHCb detector. No significant signal is observed in the accessible mass range 214≤m(χ)≤4350 MeV, and upper limits are placed on the branching fraction product B(B(0)→K*(892)(0)χ)×B(χ→μ(+)μ(-)) as a function of the mass and lifetime of the χ boson. These limits are of the order of 10(-9) for χ lifetimes less than 100 ps over most of the m(χ) range, and place the most stringent constraints to date on many theories that predict the existence of additional low-mass bosons.
6. Strong phase transition, dark matter and vacuum stability from simple hidden sectors
Energy Technology Data Exchange (ETDEWEB)
Alanne, Tommi, E-mail: [email protected] [Department of Physics, University of Jyväskylä, P.O. Box 35 (YFL), FI-40014 University of Jyväskylä (Finland); Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki (Finland); Tuominen, Kimmo, E-mail: [email protected] [Department of Physics, University of Helsinki, P.O. Box 64, FI-00014 University of Helsinki (Finland); Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki (Finland); Vaskonen, Ville, E-mail: [email protected] [Department of Physics, University of Jyväskylä, P.O. Box 35 (YFL), FI-40014 University of Jyväskylä (Finland); Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki (Finland)
2014-12-15
Motivated by the possibility to explain dark matter abundance and strong electroweak phase transition, we consider simple extensions of the Standard Model containing singlet fields coupled with the Standard Model via a scalar portal. Concretely, we consider a basic portal model consisting of a singlet scalar with Z{sub 2} symmetry and a model containing a singlet fermion connected with the Standard Model fields via a singlet scalar portal. We perform a Monte Carlo analysis of the parameter space of each model, and we find that in both cases the dark matter abundance can be produced either via freeze-out or freeze-in mechanisms, but only in the latter model one can obtain also a strong electroweak phase transition required by the successful electroweak baryogenesis. We impose the direct search limits and consider systematically the possibility that the model produces only a subdominant portion of the dark matter abundance. We also study the renormalization group evolution of the couplings of the model to determine if the scalar sector of the model remains stable and perturbative up to high scales. With explicit examples of benchmark values of the couplings at weak scale, we show that this is possible. Models of this type are further motivated by the possibility that the excursions of the Higgs field at the end of inflation are large and could directly probe the instability region of the Standard Model.
7. Dirac matter
CERN Document Server
Rivasseau, Vincent; Fuchs, Jean-Nöel
2017-01-01
This fifteenth volume of the Poincare Seminar Series, Dirac Matter, describes the surprising resurgence, as a low-energy effective theory of conducting electrons in many condensed matter systems, including graphene and topological insulators, of the famous equation originally invented by P.A.M. Dirac for relativistic quantum mechanics. In five highly pedagogical articles, as befits their origin in lectures to a broad scientific audience, this book explains why Dirac matters. Highlights include the detailed "Graphene and Relativistic Quantum Physics", written by the experimental pioneer, Philip Kim, and devoted to graphene, a form of carbon crystallized in a two-dimensional hexagonal lattice, from its discovery in 2004-2005 by the future Nobel prize winners Kostya Novoselov and Andre Geim to the so-called relativistic quantum Hall effect; the review entitled "Dirac Fermions in Condensed Matter and Beyond", written by two prominent theoreticians, Mark Goerbig and Gilles Montambaux, who consider many other mater...
8. DIRAC Security
CERN Document Server
Casajús Ramo, A
2006-01-01
DIRAC is the LHCb Workload and Data Management System. Based on a service-oriented architecture, it enables generic distributed computing with lightweight Agents and Clients for job execution and data transfers. DIRAC implements a client-server architecture exposing server methods through XML Remote Procedure Call (XML-RPC) protocol. DIRAC is mostly coded in python. DIRAC security infrastructure has been designed to be a completely generic XML-RPC transport over a SSL tunnel. This new security layer is able to handle standard X509 certificates as well as grid-proxies to authenticate both sides of the connection. Serve and client authentication relies over OpenSSL and py-Open SSL, but to be able to handle grid proxies some modifications have been added to those libraries. DIRAC security infrastructure handles authorization and authorization as well as provides extended capabilities like secure connection tunneling and file transfer. Using this new security infrastructure all LHCb users can safely make use o...
9. Hidden charged dark matter and chiral dark radiation
Science.gov (United States)
Ko, P.; Nagata, Natsumi; Tang, Yong
2017-10-01
In the light of recent possible tensions in the Hubble constant H0 and the structure growth rate σ8 between the Planck and other measurements, we investigate a hidden-charged dark matter (DM) model where DM interacts with hidden chiral fermions, which are charged under the hidden SU(N) and U(1) gauge interactions. The symmetries in this model assure these fermions to be massless. The DM in this model, which is a Dirac fermion and singlet under the hidden SU(N), is also assumed to be charged under the U(1) gauge symmetry, through which it can interact with the chiral fermions. Below the confinement scale of SU(N), the hidden quark condensate spontaneously breaks the U(1) gauge symmetry such that there remains a discrete symmetry, which accounts for the stability of DM. This condensate also breaks a flavor symmetry in this model and Nambu-Goldstone bosons associated with this flavor symmetry appear below the confinement scale. The hidden U(1) gauge boson and hidden quarks/Nambu-Goldstone bosons are components of dark radiation (DR) above/below the confinement scale. These light fields increase the effective number of neutrinos by δNeff ≃ 0.59 above the confinement scale for N = 2, resolving the tension in the measurements of the Hubble constant by Planck and Hubble Space Telescope if the confinement scale is ≲1 eV. DM and DR continuously scatter with each other via the hidden U(1) gauge interaction, which suppresses the matter power spectrum and results in a smaller structure growth rate. The DM sector couples to the Standard Model sector through the exchange of a real singlet scalar mixing with the Higgs boson, which makes it possible to probe our model in DM direct detection experiments. Variants of this model are also discussed, which may offer alternative ways to investigate this scenario.
10. Dirac experiment
International Nuclear Information System (INIS)
Gomez, F.; Adeva, B.; Afanasev, L.; Benayoun, M.; Brekhovskikh, V.; Caragheorgheopol, G.; Cechak, T.; Chiba, M.; Constantinescu, S.; Doudarev, A.; Dreossi, D.; Drijard, D.; Ferro-Luzzi, M.; Gallas, M.V.; Gerndt, J.; Giacomich, R.; Gianotti, P.; Goldin, D.; Gorin, A.; Gortchakov, O.; Guaraldo, C.; Hansroul, M.; Hosek, R.; Iliescu, M.; Jabitski, M.; Kalinina, N.; Karpoukhine, V.; Kluson, J.; Kobayashi, M.; Kokkas, P.; Komarov, V.; Koulikov, A.; Kouptsov, A.; Krouglov, V.; Krouglova, L.; Kuroda, K.-I.; Lanaro, A.; Lapshine, V.; Lednicky, R.; Leruste, P.; Levisandri, P.; Lopez Aguera, A.; Lucherini, V.; Maki, T.; Manuilov, I.; Montanet, L.; Narjoux, J.-L.; Nemenov, L.; Nikitin, M.; Nunez Pardo, T.; Okada, K.; Olchevskii, V.; Pazos, A.; Pentia, M.; Penzo, A.; Perreau, J.-M.; Petrascu, C.; Plo, M.; Ponta, T.; Pop, D.; Riazantsev, A.; Rodriguez, J.M.; Rodriguez Fernandez, A.; Rykaline, V.; Santamarina, C.; Saborido, J.; Schacher, J.; Sidorov, A.; Smolik, J.; Takeutchi, F.; Tarasov, A.; Tauscher, L.; Tobar, M.J.; Trusov, S.; Vazquez, P.; Vlachos, S.; Yazkov, V.; Yoshimura, Y.; Zrelov, P.
2001-01-01
The main objective of DIRAC experiment is the measurement of the lifetime τ of the exotic hadronic atom consisting of π + and π - mesons. The lifetime of this atom is determined by the decay mode π + π - → π 0 π 0 due to the strong interaction. Through the precise relationship between the lifetime and the S-wave pion-pion scattering length difference |a 0 - a 2 | for isospin 0 and 2 (respectively), a measurement of τ with an accuracy of 10% will allow a determination of |a 0 - a 2 | at a 5% precision level. Pion-pion scattering lengths have been calculated in the framework of chiral perturbation theory with an accuracy below 5%. In this way DIRAC experiment will provide a crucial test of the chiral symmetry breaking scheme in QCD effective theories at low energies
11. Dirac experiment
Energy Technology Data Exchange (ETDEWEB)
Gomez, F.; Adeva, B.; Afanasev, L.; Benayoun, M.; Brekhovskikh, V.; Caragheorgheopol, G.; Cechak, T.; Chiba, M.; Constantinescu, S.; Doudarev, A.; Dreossi, D.; Drijard, D.; Ferro-Luzzi, M.; Gallas, M.V.; Gerndt, J.; Giacomich, R.; Gianotti, P.; Goldin, D.; Gorin, A.; Gortchakov, O.; Guaraldo, C.; Hansroul, M.; Hosek, R.; Iliescu, M.; Jabitski, M.; Kalinina, N.; Karpoukhine, V.; Kluson, J.; Kobayashi, M.; Kokkas, P.; Komarov, V.; Koulikov, A.; Kouptsov, A.; Krouglov, V.; Krouglova, L.; Kuroda, K.-I.; Lanaro, A.; Lapshine, V.; Lednicky, R.; Leruste, P.; Levisandri, P.; Lopez Aguera, A.; Lucherini, V.; Maki, T.; Manuilov, I.; Montanet, L.; Narjoux, J.-L.; Nemenov, L.; Nikitin, M.; Nunez Pardo, T.; Okada, K.; Olchevskii, V.; Pazos, A.; Pentia, M.; Penzo, A.; Perreau, J.-M.; Petrascu, C.; Plo, M.; Ponta, T.; Pop, D.; Riazantsev, A.; Rodriguez, J.M.; Rodriguez Fernandez, A.; Rykaline, V.; Santamarina, C.; Saborido, J.; Schacher, J.; Sidorov, A.; Smolik, J.; Takeutchi, F.; Tarasov, A.; Tauscher, L.; Tobar, M.J.; Trusov, S.; Vazquez, P.; Vlachos, S.; Yazkov, V.; Yoshimura, Y.; Zrelov, P
2001-04-01
The main objective of DIRAC experiment is the measurement of the lifetime {tau} of the exotic hadronic atom consisting of {pi}{sup +} and {pi}{sup -} mesons. The lifetime of this atom is determined by the decay mode {pi}{sup +} {pi}{sup -} {yields} {pi}{sup 0} {pi}{sup 0} due to the strong interaction. Through the precise relationship between the lifetime and the S-wave pion-pion scattering length difference |a{sub 0} - a{sub 2}| for isospin 0 and 2 (respectively), a measurement of {tau} with an accuracy of 10% will allow a determination of |a{sub 0} - a{sub 2}| at a 5% precision level. Pion-pion scattering lengths have been calculated in the framework of chiral perturbation theory with an accuracy below 5%. In this way DIRAC experiment will provide a crucial test of the chiral symmetry breaking scheme in QCD effective theories at low energies.
12. Hidden photons in connection to dark matter
Energy Technology Data Exchange (ETDEWEB)
Andreas, Sarah; Ringwald, Andreas [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Goodsell, Mark D. [CPhT, Ecole Polytechnique, Palaiseau (France)
2013-06-15
Light extra U(1) gauge bosons, so called hidden photons, which reside in a hidden sector have attracted much attention since they are a well motivated feature of many scenarios beyond the Standard Model and furthermore could mediate the interaction with hidden sector dark matter.We review limits on hidden photons from past electron beam dump experiments including two new limits from such experiments at KEK and Orsay. In addition, we study the possibility of having dark matter in the hidden sector. A simple toy model and different supersymmetric realisations are shown to provide viable dark matter candidates in the hidden sector that are in agreement with recent direct detection limits.
13. Hidden photons in connection to dark matter
International Nuclear Information System (INIS)
Andreas, Sarah; Ringwald, Andreas; Goodsell, Mark D.
2013-06-01
Light extra U(1) gauge bosons, so called hidden photons, which reside in a hidden sector have attracted much attention since they are a well motivated feature of many scenarios beyond the Standard Model and furthermore could mediate the interaction with hidden sector dark matter.We review limits on hidden photons from past electron beam dump experiments including two new limits from such experiments at KEK and Orsay. In addition, we study the possibility of having dark matter in the hidden sector. A simple toy model and different supersymmetric realisations are shown to provide viable dark matter candidates in the hidden sector that are in agreement with recent direct detection limits.
14. A sequence of Clifford algebras and three replicas of Dirac particle
International Nuclear Information System (INIS)
Krolikowski, W.; Warsaw Univ.
1990-01-01
The embedding of Dirac algebra into a sequence N=1, 2, 3,... of Clifford algebras is discussed, leading to Dirac equations with N=1 additional, electromagnetically ''hidden'' spins 1/2. It is shown that there are three and only three replicas N=1, 3, 5 of Dirac particle if the theory of relativity together with the probability interpretation of wave function is applied both to the ''visible'' spin and ''hidden'' spins, and a new ''hidden exclusion principle''is imposed on the wave function (then ''hidden'' spins add up to zero). It is appealing to explore this idea in order to explain the puzzle of three generations of lepton and quarks. (author)
15. Tight connection between direct and indirect detection of dark matter through Higgs portal couplings to a hidden sector
International Nuclear Information System (INIS)
Arina, Chiara; Josse-Michaux, Francois-Xavier; Sahu, Narendra
2010-01-01
We present a hidden Abelian extension of the standard model including a complex scalar as a dark matter candidate and a light scalar acting as a long range force carrier between dark matter particles. The Sommerfeld enhanced annihilation cross section of the dark matter explains the observed cosmic ray excesses. The light scalar field also gives rise to potentially large cross sections of dark matter on the nucleon, therefore providing an interesting way to probe this model simultaneously at direct and indirect dark matter search experiments. We constrain the parameter space of the model by taking into account the CDMS-II exclusion limit as well as PAMELA and Fermi LAT data.
16. Hidden charged dark matter
International Nuclear Information System (INIS)
Feng, Jonathan L.; Kaplinghat, Manoj; Tu, Huitzu; Yu, Hai-Bo
2009-01-01
Can dark matter be stabilized by charge conservation, just as the electron is in the standard model? We examine the possibility that dark matter is hidden, that is, neutral under all standard model gauge interactions, but charged under an exact (\\rm U)(1) gauge symmetry of the hidden sector. Such candidates are predicted in WIMPless models, supersymmetric models in which hidden dark matter has the desired thermal relic density for a wide range of masses. Hidden charged dark matter has many novel properties not shared by neutral dark matter: (1) bound state formation and Sommerfeld-enhanced annihilation after chemical freeze out may reduce its relic density, (2) similar effects greatly enhance dark matter annihilation in protohalos at redshifts of z ∼ 30, (3) Compton scattering off hidden photons delays kinetic decoupling, suppressing small scale structure, and (4) Rutherford scattering makes such dark matter self-interacting and collisional, potentially impacting properties of the Bullet Cluster and the observed morphology of galactic halos. We analyze all of these effects in a WIMPless model in which the hidden sector is a simplified version of the minimal supersymmetric standard model and the dark matter is a hidden sector stau. We find that charged hidden dark matter is viable and consistent with the correct relic density for reasonable model parameters and dark matter masses in the range 1 GeV ∼ X ∼< 10 TeV. At the same time, in the preferred range of parameters, this model predicts cores in the dark matter halos of small galaxies and other halo properties that may be within the reach of future observations. These models therefore provide a viable and well-motivated framework for collisional dark matter with Sommerfeld enhancement, with novel implications for astrophysics and dark matter searches
17. Search for Hidden Particles
CERN Multimedia
Solovev, V
The SHiP Experiment is a new general-purpose fixed target facility at the SPS to search for hidden particles as predicted by a very large number of recently elaborated models of Hidden Sectors which are capable of accommodating dark matter, neutrino oscillations, and the origin of the full baryon asymmetry in the Universe. Specifically, the experiment is aimed at searching for very weakly interacting long lived particles including Heavy Neutral Leptons - right-handed partners of the active neutrinos; light supersymmetric particles - sgoldstinos, etc.; scalar, axion and vector portals to the hidden sector. The high intensity of the SPS and in particular the large production of charm mesons with the 400 GeV beam allow accessing a wide variety of light long-lived exotic particles of such models and of SUSY. Moreover, the facility is ideally suited to study the interactions of tau neutrinos.
18. DIRAC distributed secure framework
International Nuclear Information System (INIS)
Casajus, A; Graciani, R
2010-01-01
DIRAC, the LHCb community Grid solution, provides access to a vast amount of computing and storage resources to a large number of users. In DIRAC users are organized in groups with different needs and permissions. In order to ensure that only allowed users can access the resources and to enforce that there are no abuses, security is mandatory. All DIRAC services and clients use secure connections that are authenticated using certificates and grid proxies. Once a client has been authenticated, authorization rules are applied to the requested action based on the presented credentials. These authorization rules and the list of users and groups are centrally managed in the DIRAC Configuration Service. Users submit jobs to DIRAC using their local credentials. From then on, DIRAC has to interact with different Grid services on behalf of this user. DIRAC has a proxy management service where users upload short-lived proxies to be used when DIRAC needs to act on behalf of them. Long duration proxies are uploaded by users to a MyProxy service, and DIRAC retrieves new short delegated proxies when necessary. This contribution discusses the details of the implementation of this security infrastructure in DIRAC.
19. DIRAC RESTful API
International Nuclear Information System (INIS)
Casajus Ramo, A; Graciani Diaz, R; Tsaregorodtsev, A
2012-01-01
The DIRAC framework for distributed computing has been designed as a flexible and modular solution that can be adapted to the requirements of any community. Users interact with DIRAC via command line, using the web portal or accessing resources via the DIRAC python API. The current DIRAC API requires users to use a python version valid for DIRAC. Some communities have developed their own software solutions for handling their specific workload, and would like to use DIRAC as their back-end to access distributed computing resources easily. Many of these solutions are not coded in python or depend on a specific python version. To solve this gap DIRAC provides a new language agnostic API that any software solution can use. This new API has been designed following the RESTful principles. Any language with libraries to issue standard HTTP queries may use it. GSI proxies can still be used to authenticate against the API services. However GSI proxies are not a widely adopted standard. The new DIRAC API also allows clients to use OAuth for delegating the user credentials to a third party solution. These delegated credentials allow the third party software to query to DIRAC on behalf of the users. This new API will further expand the possibilities communities have to integrate DIRAC into their distributed computing models.
20. Photoconductivity in Dirac materials
International Nuclear Information System (INIS)
Shao, J. M.; Yang, G. W.
2015-01-01
Two-dimensional (2D) Dirac materials including graphene and the surface of a three-dimensional (3D) topological insulator, and 3D Dirac materials including 3D Dirac semimetal and Weyl semimetal have attracted great attention due to their linear Dirac nodes and exotic properties. Here, we use the Fermi’s golden rule and Boltzmann equation within the relaxation time approximation to study and compare the photoconductivity of Dirac materials under different far- or mid-infrared irradiation. Theoretical results show that the photoconductivity exhibits the anisotropic property under the polarized irradiation, but the anisotropic strength is different between 2D and 3D Dirac materials. The photoconductivity depends strongly on the relaxation time for different scattering mechanism, just like the dark conductivity
1. Hidden Liquidity
OpenAIRE
Cebiroglu, Gökhan; Horst, Ulrich
2012-01-01
We cross-sectionally analyze the presence of aggregated hidden depth and trade volume in the S&P 500 and identify its key determinants. We find that the spread is the main predictor for a stock’s hidden dimension, both in terms of traded and posted liquidity. Our findings moreover suggest that large hidden orders are associated with larger transaction costs, higher price impact and increased volatility. In particular, as large hidden orders fail to attract (latent) liquidity to the market, hi...
2. Dirac, Weyl, Majorana, a review
International Nuclear Information System (INIS)
Uschersohn, J.
1982-05-01
The Dirac equation and the properties of Dirac matrices are presented and discussed. A large number of representations of the Dirac matrices is identified. Special emphasis is put on aspects rarely treated or neglected in textbooks
3. NA62: Hidden Sector Physics
CERN Document Server
Cesarotti, Carissa Joyce
2016-01-01
Modern experimental physics is often probing for new physics by either finding deviations from predictions on extremely precise measurements, or by looking for a new signal that cannot be explained with existing models. The NA62 experiment at CERN does the former by measuring the ultra-rare decay $K^+ \\rightarrow \\pi^+ \ 4. Fermi–Dirac Statistics Indian Academy of Sciences (India) IAS Admin Pauli exclusion principle, Fermi–. Dirac statistics, identical and in- distinguishable particles, Fermi gas. Fermi–Dirac Statistics. Derivation and Consequences. S Chaturvedi and Shyamal Biswas. (left) Subhash Chaturvedi is at University of. Hyderabad. His current research interests include phase space descriptions. 5. The Dirac Sea OpenAIRE Dimock, J. 2010-01-01 We give an alternate definition of the free Dirac field featuring an explicit construction of the Dirac sea. The treatment employs a semi-infinite wedge product of Hilbert spaces. We also show that the construction is equivalent to the standard Fock space construction. 6. Bohr and Dirac* Indian Academy of Sciences (India) IAS Admin We present an account of the work of Niels Bohr and Paul Dirac, their interactions and personal- ities. 1. Introduction. In this essay I would like to convey to my readers some- thing about the personalities and work of Niels Bohr and Paul Dirac, juxtaposed against one another. Let me hope that the portraits I will paint of these ... 7. On the Dirac oscillator International Nuclear Information System (INIS) Rodrigues, R. de Lima 2007-01-01 In the present work we obtain a new representation for the Dirac oscillator based on the Clifford algebra C 7. The symmetry breaking and the energy eigenvalues for our model of the Dirac oscillator are studied in the non-relativistic limit. (author) 8. P A M Dirac Indian Academy of Sciences (India) Home; Journals; Resonance – Journal of Science Education. P A M Dirac. Articles written in Resonance – Journal of Science Education. Volume 8 Issue 8 August 2003 pp 102-110 Classics. XI. The Relation between Mathematics and Physics · P A M Dirac · More Details Fulltext PDF ... 9. Particles and Dirac-type operators on curved spaces International Nuclear Information System (INIS) Visinescu, Mihai 2003-01-01 We review the geodesic motion of pseudo-classical particles in curved spaces. Investigating the generalized Killing equations for spinning spaces, we express the constants of motion in terms of Killing-Yano tensors. Passing from the spinning spaces to the Dirac equation in curved backgrounds we point out the role of the Killing-Yano tensors in the construction of the Dirac-type operators. The general results are applied to the case of the four-dimensional Euclidean Taub-Newman-Unti-Tamburino space. From the covariantly constant Killing-Yano tensors of this space we construct three new Dirac-type operators which are equivalent with the standard Dirac operator. Finally the Runge-Lenz operator for the Dirac equation in this background is expressed in terms of the fourth Killing-Yano tensor which is not covariantly constant. As a rule the covariantly constant Killing-Yano tensors realize certain square roots of the metric tensor. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. On the other hand, the not covariantly constant Killing-Yano tensors are important in generating hidden symmetries. The presence of not covariantly constant Killing-Yano tensors implies the existence of non-standard supersymmetries in point particle theories on curved background. (author) 10. Three Dimensional Dirac Semimetals Science.gov (United States) Zaheer, Saad 2014-03-01 Dirac points on the Fermi surface of two dimensional graphene are responsible for its unique electronic behavior. One can ask whether any three dimensional materials support similar pseudorelativistic physics in their bulk electronic spectra. This possibility has been investigated theoretically and is now supported by two successful experimental demonstrations reported during the last year. In this talk, I will summarize the various ways in which Dirac semimetals can be realized in three dimensions with primary focus on a specific theory developed on the basis of representations of crystal spacegroups. A three dimensional Dirac (Weyl) semimetal can appear in the presence (absence) of inversion symmetry by tuning parameters to the phase boundary separating a bulk insulating and a topological insulating phase. More generally, we find that specific rules governing crystal symmetry representations of electrons with spin lead to robust Dirac points at high symmetry points in the Brillouin zone. Combining these rules with microscopic considerations identifies six candidate Dirac semimetals. Another method towards engineering Dirac semimetals involves combining crystal symmetry and band inversion. Several candidate materials have been proposed utilizing this mechanism and one of the candidates has been successfully demonstrated as a Dirac semimetal in two independent experiments. Work carried out in collaboration with: Julia A. Steinberg, Steve M. Young, J.C.Y. Teo, C.L. Kane, E.J. Mele and Andrew M. Rappe. 11. Equazione di Dirac CERN Document Server Monti, Dalida 1996-01-01 Relativamente poco noto al gran pubblico, il premio Nobel Paul Adrien Maurice Dirac appartiene a quel gruppo di uomini di ingegno che nei primi decenni del secolo contribuirono a dare alla nostra concezione del mondo fisico la sua impronta attuale. Assolutamente cruciali, per una valutazione dell'opera di Dirac, sono gli anni compresi tra il 1925 e il 1931: un periodo in cui il fisico fornisce la prima spiegazione chiara e coerente delle proprietà di spin dell'elettrone (equazione di Dirac) e perviene, in forza della pura deduzione matematica, alla scoperta dell'esistenza dell'elettrone positivo o positrone. 12. In the Dirac tradition Energy Technology Data Exchange (ETDEWEB) Anon. 1988-04-15 It was Paul Dirac who cast quantum mechanics into the form we now use, and many generations of theoreticians openly acknowledge his influence on their thinking. When Dirac died in 1984, St. John's College, Cambridge, his base for most of his lifetime, instituted an annual lecture in his memory at Cambridge. The first lecture, in 1986, attracted two heavyweights - Richard Feynman and Steven Weinberg. Far from using the lectures as a platform for their own work, in the Dirac tradition they presented stimulating material on deep underlying questions. 13. In the Dirac tradition International Nuclear Information System (INIS) Anon. 1988-01-01 It was Paul Dirac who cast quantum mechanics into the form we now use, and many generations of theoreticians openly acknowledge his influence on their thinking. When Dirac died in 1984, St. John's College, Cambridge, his base for most of his lifetime, instituted an annual lecture in his memory at Cambridge. The first lecture, in 1986, attracted two heavyweights - Richard Feynman and Steven Weinberg. Far from using the lectures as a platform for their own work, in the Dirac tradition they presented stimulating material on deep underlying questions 14. DIRAC distributed computing services International Nuclear Information System (INIS) Tsaregorodtsev, A 2014-01-01 DIRAC Project provides a general-purpose framework for building distributed computing systems. It is used now in several HEP and astrophysics experiments as well as for user communities in other scientific domains. There is a large interest from smaller user communities to have a simple tool like DIRAC for accessing grid and other types of distributed computing resources. However, small experiments cannot afford to install and maintain dedicated services. Therefore, several grid infrastructure projects are providing DIRAC services for their respective user communities. These services are used for user tutorials as well as to help porting the applications to the grid for a practical day-to-day work. The services are giving access typically to several grid infrastructures as well as to standalone computing clusters accessible by the target user communities. In the paper we will present the experience of running DIRAC services provided by the France-Grilles NGI and other national grid infrastructure projects. 15. Dark matter scenarios in a constrained model with Dirac gauginos CERN Document Server Goodsell, Mark D.; Müller, Tobias; Porod, Werner; Staub, Florian 2015-01-01 We perform the first analysis of Dark Matter scenarios in a constrained model with Dirac Gauginos. The model under investigation is the Constrained Minimal Dirac Gaugino Supersymmetric Standard model (CMDGSSM) where the Majorana mass terms of gauginos vanish. However,$R$-symmetry is broken in the Higgs sector by an explicit and/or effective$B_\\mu$-term. This causes a mass splitting between Dirac states in the fermion sector and the neutralinos, which provide the dark matter candidate, become pseudo-Dirac states. We discuss two scenarios: the universal case with all scalar masses unified at the GUT scale, and the case with non-universal Higgs soft-terms. We identify different regions in the parameter space which fullfil all constraints from the dark matter abundance, the limits from SUSY and direct dark matter searches and the Higgs mass. Most of these points can be tested with the next generation of direct dark matter detection experiments. 16. Executor Framework for DIRAC Science.gov (United States) Casajus Ramo, A.; Graciani Diaz, R. 2012-12-01 DIRAC framework for distributed computing has been designed as a group of collaborating components, agents and servers, with persistent database back-end. Components communicate with each other using DISET, an in-house protocol that provides Remote Procedure Call (RPC) and file transfer capabilities. This approach has provided DIRAC with a modular and stable design by enforcing stable interfaces across releases. But it made complicated to scale further with commodity hardware. To further scale DIRAC, components needed to send more queries between them. Using RPC to do so requires a lot of processing power just to handle the secure handshake required to establish the connection. DISET now provides a way to keep stable connections and send and receive queries between components. Only one handshake is required to send and receive any number of queries. Using this new communication mechanism DIRAC now provides a new type of component called Executor. Executors process any task (such as resolving the input data of a job) sent to them by a task dispatcher. This task dispatcher takes care of persisting the state of the tasks to the storage backend and distributing them among all the Executors based on the requirements of each task. In case of a high load, several Executors can be started to process the extra load and stop them once the tasks have been processed. This new approach of handling tasks in DIRAC makes Executors easy to replace and replicate, thus enabling DIRAC to further scale beyond the current approach based on polling agents. 17. Executor Framework for DIRAC International Nuclear Information System (INIS) Casajus Ramo, A; Graciani Diaz, R 2012-01-01 DIRAC framework for distributed computing has been designed as a group of collaborating components, agents and servers, with persistent database back-end. Components communicate with each other using DISET, an in-house protocol that provides Remote Procedure Call (RPC) and file transfer capabilities. This approach has provided DIRAC with a modular and stable design by enforcing stable interfaces across releases. But it made complicated to scale further with commodity hardware. To further scale DIRAC, components needed to send more queries between them. Using RPC to do so requires a lot of processing power just to handle the secure handshake required to establish the connection. DISET now provides a way to keep stable connections and send and receive queries between components. Only one handshake is required to send and receive any number of queries. Using this new communication mechanism DIRAC now provides a new type of component called Executor. Executors process any task (such as resolving the input data of a job) sent to them by a task dispatcher. This task dispatcher takes care of persisting the state of the tasks to the storage backend and distributing them among all the Executors based on the requirements of each task. In case of a high load, several Executors can be started to process the extra load and stop them once the tasks have been processed. This new approach of handling tasks in DIRAC makes Executors easy to replace and replicate, thus enabling DIRAC to further scale beyond the current approach based on polling agents. 18. Hidden particle production at the ILC International Nuclear Information System (INIS) Fujii, Keisuke; Itoh, Hideo; Okada, Nobuchika; Hano, Hitoshi; Yoshioka, Tamaki 2008-01-01 In a class of new physics models, the new physics sector is completely or partly hidden, namely, a singlet under the standard model (SM) gauge group. Hidden fields included in such new physics models communicate with the standard model sector through higher-dimensional operators. If a cutoff lies in the TeV range, such hidden fields can be produced at future colliders. We consider a scalar field as an example of the hidden fields. Collider phenomenology on this hidden scalar is similar to that of the SM Higgs boson, but there are several features quite different from those of the Higgs boson. We investigate productions of the hidden scalar at the International Linear Collider (ILC) and study the feasibility of its measurements, in particular, how well the ILC distinguishes the scalar from the Higgs boson, through realistic Monte Carlo simulations. 19. Detecting hidden particles with MATHUSLA Science.gov (United States) Evans, Jared A. 2018-03-01 A hidden sector containing light long-lived particles provides a well-motivated place to find new physics. The recently proposed MATHUSLA experiment has the potential to be extremely sensitive to light particles originating from rare meson decays in the very long lifetime region. In this work, we illustrate this strength with the specific example of a light scalar mixed with the standard model-like Higgs boson, a model where MATHUSLA can further probe unexplored parameter space from exotic Higgs decays. Design augmentations should be considered in order to maximize the ability of MATHUSLA to discover very light hidden sector particles. 20. The origin of the hidden supersymmetry International Nuclear Information System (INIS) Jakubsky, Vit; Nieto, Luis-Miguel; Plyushchay, Mikhail S. 2010-01-01 The hidden supersymmetry and related tri-supersymmetric structure of the free particle system, the Dirac delta potential problem and the Aharonov-Bohm effect (planar, bound state, and tubule models) are explained by a special nonlocal unitary transformation, which for the usual N=2 supercharges has a nature of Foldy-Wouthuysen transformation. We show that in general case, the bosonized supersymmetry of nonlocal, parity even systems emerges in the same construction, and explain the origin of the unusual N=2 supersymmetry of electron in three-dimensional parity even magnetic field. The observation extends to include the hidden superconformal symmetry. 1. κ-deformed Dirac oscillator in an external magnetic field Science.gov (United States) Chargui, Y.; Dhahbi, A.; Cherif, B. 2018-04-01 We study the solutions of the (2 + 1)-dimensional κ-deformed Dirac oscillator in the presence of a constant transverse magnetic field. We demonstrate how the deformation parameter affects the energy eigenvalues of the system and the corresponding eigenfunctions. Our findings suggest that this system could be used to detect experimentally the effect of the deformation. We also show that the hidden supersymmetry of the non-deformed system reduces to a hidden pseudo-supersymmetry having the same algebraic structure as a result of the κ-deformation. 2. DIRAC optimized workload management CERN Document Server Paterson, S K 2008-01-01 The LHCb DIRAC Workload and Data Management System employs advanced optimization techniques in order to dynamically allocate resources. The paradigms realized by DIRAC, such as late binding through the Pilot Agent approach, have proven to be highly successful. For example, this has allowed the principles of workload management to be applied not only at the time of user job submission to the Grid but also to optimize the use of computing resources once jobs have been acquired. Along with the central application of job priorities, DIRAC minimizes the system response time for high priority tasks. This paper will describe the recent developments to support Monte Carlo simulation, data processing and distributed user analysis in a consistent way across disparate compute resources including individual PCs, local batch systems, and the Worldwide LHC Computing Grid. The Grid environment is inherently unpredictable and whilst short-term studies have proven to deliver high job efficiencies, the system performance over ... 3. easyDiracGauginos Energy Technology Data Exchange (ETDEWEB) Abel, Steven [Durham Univ. (United Kingdom). Inst. for Particle Physics Phenomenology; CERN, Geneva (Switzerland); Goodsell, Mark [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany) 2011-02-15 A simple and natural model is presented that gives Dirac gauginos. The configuration is related to ''deconstructed gaugino mediation''. A high energy completion is provided based on existing ISS-like models of deconstructed gaugino mediation. This provides a complete picture of Dirac gauginos that includes the necessary extra adjoint fermions (generated as magnetic quarks of the ISS theory) and supersymmetry breaking (via the ISS mechanism). Moreover the screening of the scalar masses means that they can similar to or less than the gaugino masses, even though the supersymmetry breaking is driven by F-terms. (orig.) 4. easyDiracGauginos International Nuclear Information System (INIS) Abel, Steven; Goodsell, Mark 2011-02-01 A simple and natural model is presented that gives Dirac gauginos. The configuration is related to ''deconstructed gaugino mediation''. A high energy completion is provided based on existing ISS-like models of deconstructed gaugino mediation. This provides a complete picture of Dirac gauginos that includes the necessary extra adjoint fermions (generated as magnetic quarks of the ISS theory) and supersymmetry breaking (via the ISS mechanism). Moreover the screening of the scalar masses means that they can similar to or less than the gaugino masses, even though the supersymmetry breaking is driven by F-terms. (orig.) 5. The Dirac equation International Nuclear Information System (INIS) Thaller, B. 1992-01-01 This monograph treats most of the usual material to be found in texts on the Dirac equation such as the basic formalism of quantum mechanics, representations of Dirac matrices, covariant realization of the Dirac equation, interpretation of negative energies, Foldy-Wouthuysen transformation, Klein's paradox, spherically symmetric interactions and a treatment of the relativistic hydrogen atom, etc., and also provides excellent additional treatments of a variety of other relevant topics. The monograph contains an extensive treatment of the Lorentz and Poincare groups and their representations. The author discusses in depth Lie algebaic and projective representations, covering groups, and Mackey's theory and Wigner's realization of induced representations. A careful classification of external fields with respect to their behavior under Poincare transformations is supplemented by a basic account of self-adjointness and spectral properties of Dirac operators. A state-of-the-art treatment of relativistic scattering theory based on a time-dependent approach originally due to Enss is presented. An excellent introduction to quantum electrodynamics in external fields is provided. Various appendices containing further details, notes on each chapter commenting on the history involved and referring to original research papers and further developments in the literature, and a bibliography covering all relevant monographs and over 500 articles on the subject, complete this text. This book should satisfy the needs of a wide audience, ranging from graduate students in theoretical physics and mathematics to researchers interested in mathematical physics 6. Dirac neutrino masses from generalized supersymmetry breaking International Nuclear Information System (INIS) Demir, D.A.; Everett, L.L.; Langacker, P. 2007-12-01 We demonstrate that Dirac neutrino masses in the experimentally preferred range are generated within supersymmetric gauge extensions of the Standard Model with a generalized supersymmetry breaking sector. If the usual superpotential Yukawa couplings are forbidden by the additional gauge symmetry (such as a U(1) ' ), effective Dirac mass terms involving the ''wrong Higgs'' field can arise either at tree level due to hard supersymmetry breaking fermion Yukawa couplings, or at one-loop due to nonanalytic or ''nonholomorphic'' soft supersymmetry breaking trilinear scalar couplings. As both of these operators are naturally suppressed in generic models of supersymmetry breaking, the resulting neutrino masses are naturally in the sub-eV range. The neutrino magnetic and electric dipole moments resulting from the radiative mechanism also vanish at one-loop order. (orig.) 7. Nonlinear Dirac Equations Directory of Open Access Journals (Sweden) Wei Khim Ng 2009-02-01 Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations. 8. Three Dirac neutrinos International Nuclear Information System (INIS) Joshipura, A.S.; Rindani, S.D. 1991-01-01 The consequences of imposing an exact L e +L τ -L μ symmetry on a 6x6 matrix describing neutrino masses are discussed. The presence of right-handed neutrinos avoids the need of introducing any SU(2) Higgs triplet. Hence the conflict with the CERN LEP data on the Z width found in earlier models with L e +L τ -L μ symmetry is avoided. The L e +L τ -L μ symmetry provides an interesting realization of a recent proposal of Glashow to accommodate the 17-keV Dirac neutrino in the SU(2)xU(1) theory. All the neutrinos in this model are Dirac particles. The solar-neutrino problem can be solved in an extension of the model which generates a large (∼10 -11 μ B ) magnetic moment for the electron neutrino 9. Dirac Material Graphene OpenAIRE Sheka, Elena F. 2016-01-01 The paper presents the author view on spin-rooted properties of graphene supported by numerous experimental and calculation evidences. Dirac fermions of crystalline graphene and local spins of graphene molecules are suggested to meet a strict demand - different orbitals for different spins- which leads to a large spectrum of effects caused by spin polarization of electronic states. The consequent topological non-triviality, making graphene topological insulator, and local spins, imaging graph... 10. Potential scattering of Dirac particles International Nuclear Information System (INIS) Thaller, B. 1981-01-01 A quantum mechanical interpretation of the Dirac equation for particles in external electromagnetic potentials is discussed. It is shown that a consequent development of the Stueckelberg-Feynman theory into a probabilistic interpretation of the Dirac equation corrects some prejudices concerning negative energy states, Zitterbewegung and bound states in repulsive potentials and yields the connection between propagator theory and scattering theory. Limits of the Dirac equation, considered as a wave mechanical equation, are considered. (U.K.) 11. Hidden loss DEFF Research Database (Denmark) Kieffer-Kristensen, Rikke; Johansen, Karen Lise Gaardsvig 2013-01-01 to participate. RESULTS: All children were affected by their parents' ABI and the altered family situation. The children's expressions led the authors to identify six themes, including fear of losing the parent, distress and estrangement, chores and responsibilities, hidden loss, coping and support. The main......PRIMARY OBJECTIVE: The purpose of this study was to listen to and learn from children showing high levels of post-traumatic stress symptoms after parental acquired brain injury (ABI), in order to achieve an in-depth understanding of the difficulties the children face in their everyday lives...... finding indicates that the children experienced numerous losses, many of which were often suppressed or neglected by the children to protect the ill parents. CONCLUSIONS: The findings indicated that the children seemed to make a special effort to hide their feelings of loss and grief in order to protect... 12. Dirac gauginos, gauge mediation and unification International Nuclear Information System (INIS) Benakli, K. 2010-03-01 We investigate the building of models with Dirac gauginos and perturbative gauge coupling unification. Here, in contrast to the MSSM, additional fields are required for unification, and these can naturally play the role of the messengers of supersymmetry breaking. We present a framework within which such models can be constructed, including the constraints that the messenger sector must satisfy; and the renormalisation group equations for the soft parameters, which differ from those of the MSSM. For illustration, we provide the spectrum at the electroweak scale for explicit models whose gauge couplings unify at the scale predicted by heterotic strings. (orig.) 13. Dirac gauginos, gauge mediation and unification Energy Technology Data Exchange (ETDEWEB) Benakli, K. [UPMC Univ. Paris 06 (France). Laboratoire de Physique Theorique et Hautes Energies, CNRS; Goodsell, M.D. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany) 2010-03-15 We investigate the building of models with Dirac gauginos and perturbative gauge coupling unification. Here, in contrast to the MSSM, additional fields are required for unification, and these can naturally play the role of the messengers of supersymmetry breaking. We present a framework within which such models can be constructed, including the constraints that the messenger sector must satisfy; and the renormalisation group equations for the soft parameters, which differ from those of the MSSM. For illustration, we provide the spectrum at the electroweak scale for explicit models whose gauge couplings unify at the scale predicted by heterotic strings. (orig.) 14. Dirac Gauginos, Gauge Mediation and Unification CERN Document Server Benakli, K 2010-01-01 We investigate the building of models with Dirac gauginos and perturbative gauge coupling unification. Here, in contrast to the MSSM, additional fields are required for unification, and these can naturally play the role of the messengers of supersymmetry breaking. We present a framework within which such models can be constructed, including the constraints that the messenger sector must satisfy; and the renormalisation group equations for the soft parameters, which differ from those of the MSSM. For illustration, we provide the spectrum at the electroweak scale for explicit models whose gauge couplings unify at the scale predicted by heterotic strings. 15. Planar Dirac diffusion International Nuclear Information System (INIS) Leo, Stefano de; Rotelli, Pietro 2009-01-01 We present the results of the planar diffusion of a Dirac particle by step and barrier potentials, when the incoming wave impinges at an arbitrary angle with the potential. Except for right-angle incidence this process is characterized by the appearance of spin flip terms. For the step potential, spin flip occurs for both transmitted and reflected waves. However, we find no spin flip in the transmitted barrier result. This is surprising because the barrier result may be derived directly from a two-step calculation. We demonstrate that the spin flip cancellation indeed occurs for each ''particle'' (wave packet) contribution. (orig.) 16. Dirac particle on S2 International Nuclear Information System (INIS) Ferreira, P.L.; Palladino, B.E. 1985-01-01 The problem of a Dirac particle in stationary motion on S 2 - a two dimensional sphere embedded in Euclidean space E 3 - is discussed. It provides a particularly simple case of an exactly solvable constrained Dirac particle whose properties are here studied, with emphasis on its magnetic moment. (Author) [pt 17. LHCb: DIRAC Secure Distributed Platform CERN Multimedia Casajus, A 2009-01-01 DIRAC, the LHCb community grid solution, provides access to a vast amount of computing and storage resources to a large number of users. In DIRAC users are organized in groups with different needs and permissions. In order to ensure that only allowed users can access the resources and to enforce that there are no abuses, security is mandatory. All DIRAC services and clients use secure connections that are authenticated using certificates and grid proxies. Once a client has been authenticated, authorization rules are applied to the requested action based on the presented credentials. These authorization rules and the list of users and groups are centrally managed in the DIRAC Configuration Service. Users submit jobs to DIRAC using their local credentials. From then on, DIRAC has to interact with different Grid services on behalf of this user. DIRAC has a proxy management service where users upload short-lived proxies to be used when DIRAC needs to act on behalf of them. Long duration proxies are uploaded by us... 18. DIRAC: data production management International Nuclear Information System (INIS) Smith, A C; Tsaregorodtsev, A 2008-01-01 The LHCb Computing Model describes the dataflow for all stages in the processing of real and simulated events, and defines the role of LHCb associated Tier-1 and Tier-2 computing centers. The WLCG 'Dress Rehearsal' exercise aims to allow LHC experiments to deploy the full chain of their Computing Models, making use of all underlying WLCG services and resources, in preparation for real data taking. During this exercise simulated RAW physics data, matching the properties of eventual real data, will be uploaded from the LHCb Online storage system to Grid enabled storage. This data will then be replicated to LHCb Tier-1 centers and subsequently processed (reconstructed and stripped). The product of this processing is user analysis data that are distributed to all LHCb Tier-1 centers. DIRAC, LHCbs Workload and Data Management System, supports the implementation of the Computing Model in a data driven, real time and coordinated fashion. In this paper the LHCb Computing Model will be reviewed and the DIRAC components providing the needed functionality to support the Computing Model will be detailed. An evaluation of the preparedness for real data taking will also be given 19. DIRAC: data production management Energy Technology Data Exchange (ETDEWEB) Smith, A C [CERN, CH-1211, Geneva (Switzerland); Tsaregorodtsev, A [CPPM, Marseille (France)], E-mail: [email protected], E-mail: [email protected] 2008-07-15 The LHCb Computing Model describes the dataflow for all stages in the processing of real and simulated events, and defines the role of LHCb associated Tier-1 and Tier-2 computing centers. The WLCG 'Dress Rehearsal' exercise aims to allow LHC experiments to deploy the full chain of their Computing Models, making use of all underlying WLCG services and resources, in preparation for real data taking. During this exercise simulated RAW physics data, matching the properties of eventual real data, will be uploaded from the LHCb Online storage system to Grid enabled storage. This data will then be replicated to LHCb Tier-1 centers and subsequently processed (reconstructed and stripped). The product of this processing is user analysis data that are distributed to all LHCb Tier-1 centers. DIRAC, LHCbs Workload and Data Management System, supports the implementation of the Computing Model in a data driven, real time and coordinated fashion. In this paper the LHCb Computing Model will be reviewed and the DIRAC components providing the needed functionality to support the Computing Model will be detailed. An evaluation of the preparedness for real data taking will also be given. 20. Status of the DIRAC Project International Nuclear Information System (INIS) Casajus, A; Ciba, K; Fernandez, V; Graciani, R; Hamar, V; Mendez, V; Poss, S; Sapunov, M; Stagni, F; Tsaregorodtsev, A; Ubeda, M 2012-01-01 The DIRAC Project was initiated to provide a data processing system for the LHCb Experiment at CERN. It provides all the necessary functionality and performance to satisfy the current and projected future requirements of the LHCb Computing Model. A considerable restructuring of the DIRAC software was undertaken in order to turn it into a general purpose framework for building distributed computing systems that can be used by various user communities in High Energy Physics and other scientific application domains. The CLIC and ILC-SID detector projects started to use DIRAC for their data production system. The Belle Collaboration at KEK, Japan, has adopted the Computing Model based on the DIRAC system for its second phase starting in 2015. The CTA Collaboration uses DIRAC for the data analysis tasks. A large number of other experiments are starting to use DIRAC or are evaluating this solution for their data processing tasks. DIRAC services are included as part of the production infrastructure of the GISELA Latin America grid. Similar services are provided for the users of the France-Grilles and IBERGrid National Grid Initiatives in France and Spain respectively. The new communities using DIRAC started to provide important contributions to its functionality. Among recent additions can be mentioned the support of the Amazon EC2 computing resources as well as other Cloud management systems; a versatile File Replica Catalog with File Metadata capabilities; support for running MPI jobs in the pilot based Workload Management System. Integration with existing application Web Portals, like WS-PGRADE, is demonstrated. In this paper we will describe the current status of the DIRAC Project, recent developments of its framework and functionality as well as the status of the rapidly evolving community of the DIRAC users. 1. Spectral density and a family of Dirac operators International Nuclear Information System (INIS) Niemi, A.J. 1985-01-01 The spectral density for a class Dirac operators is investigated by relating its even and odd parts to the Riemann zeta-function and to the eta-invariant by Atiyah, Padoti and Singer. Asymptotic expansions are studied and a 'hidden' supersymmetry is revealed and used to relate the Dirac operator to a supersymmetric quantum mechanics. A general method for the computation of the odd spectral density is developed, and various applications are discussed. In particular the connection to the fermion number and a relation between the odd spectral density and some ratios of Jost functions and relative phase shifts are pointed out. Chiral symmetry breaking is investigated using methods analogous to those applied in the investigation of the fermion number, and related to supersymmetry breaking in the corresponding quantum mechanical model. (orig.) 2. Quantum mechanics and hidden superconformal symmetry Science.gov (United States) Bonezzi, R.; Corradini, O.; Latini, E.; Waldron, A. 2017-12-01 Solvability of the ubiquitous quantum harmonic oscillator relies on a spectrum generating osp (1 |2 ) superconformal symmetry. We study the problem of constructing all quantum mechanical models with a hidden osp (1 |2 ) symmetry on a given space of states. This problem stems from interacting higher spin models coupled to gravity. In one dimension, we show that the solution to this problem is the Vasiliev-Plyushchay family of quantum mechanical models with hidden superconformal symmetry obtained by viewing the harmonic oscillator as a one dimensional Dirac system, so that Grassmann parity equals wave function parity. These models—both oscillator and particlelike—realize all possible unitary irreducible representations of osp (1 |2 ). 3. DIRAC data production management CERN Document Server Smith, A C 2008-01-01 The LHCb Computing Model describes the dataflow for all stages in the processing of real and simulated events, and defines the role of LHCb associated Tier-1 and Tier-2 computing centers. The WLCG 'Dress Rehearsal' exercise aims to allow LHC experiments to deploy the full chain of their Computing Models, making use of all underlying WLCG services and resources, in preparation for real data taking. During this exercise simulated RAW physics data, matching the properties of eventual real data, will be uploaded from the LHCb Online storage system to Grid enabled storage. This data will then be replicated to LHCb Tier-1 centers and subsequently processed (reconstructed and stripped). The product of this processing is user analysis data that are distributed to all LHCb Tier-1 centers. DIRAC, LHCbs Workload and Data Management System, supports the implementation of the Computing Model in a data driven, real time and coordinated fashion. 4. DIRAC Data Management System CERN Document Server Smith, A C 2007-01-01 The LHCb experiment being built to utilize CERN’s flagship Large Hadron Collider will generate data to be analysed by a community of over 600 physicists worldwide. DIRAC, LHCb’s Workload and Data Management System, facilitates the use of underlying EGEE Grid resources to generate, process and analyse this data in the distributed environment. The Data Management System, presented here, provides real-time, data-driven distribution in accordance with LHCb’s Computing Model. The data volumes produced by the LHC experiments are unprecedented, rendering individual institutes and even countries, unable to provide the computing and storage resources required to make full use of the produced data. EGEE Grid resources allow the processing of LHCb data possible in a distributed fashion and LHCb’s Computing Model is based on this approach. Data Management in this environment requires reliable and high-throughput transfer of data, homogeneous access to storage resources and the cataloguing of data replicas, all of... 5. DIRAC universal pilots Science.gov (United States) Stagni, F.; McNab, A.; Luzzi, C.; Krzemien, W.; Consortium, DIRAC 2017-10-01 In the last few years, new types of computing models, such as IAAS (Infrastructure as a Service) and IAAC (Infrastructure as a Client), gained popularity. New resources may come as part of pledged resources, while others are in the form of opportunistic ones. Most but not all of these new infrastructures are based on virtualization techniques. In addition, some of them, present opportunities for multi-processor computing slots to the users. Virtual Organizations are therefore facing heterogeneity of the available resources and the use of an Interware software like DIRAC to provide the transparent, uniform interface has become essential. The transparent access to the underlying resources is realized by implementing the pilot model. DIRAC’s newest generation of generic pilots (the so-called Pilots 2.0) are the “pilots for all the skies”, and have been successfully released in production more than a year ago. They use a plugin mechanism that makes them easily adaptable. Pilots 2.0 have been used for fetching and running jobs on every type of resource, being it a Worker Node (WN) behind a CREAM/ARC/HTCondor/DIRAC Computing element, a Virtual Machine running on IaaC infrastructures like Vac or BOINC, on IaaS cloud resources managed by Vcycle, the LHCb High Level Trigger farm nodes, and any type of opportunistic computing resource. Make a machine a “Pilot Machine”, and all diversities between them will disappear. This contribution describes how pilots are made suitable for different resources, and the recent steps taken towards a fully unified framework, including monitoring. Also, the cases of multi-processor computing slots either on real or virtual machines, with the whole node or a partition of it, is discussed. 6. The Dirac equation for accountants International Nuclear Information System (INIS) Ord, G.N. 2006-01-01 In the context of relativistic quantum mechanics, derivations of the Dirac equation usually take the form of plausibility arguments based on experience with the Schroedinger equation. The primary reason for this is that we do not know what wavefunctions physically represent, so derivations have to rely on formal arguments. There is however a context in which the Dirac equation in one dimension is directly related to a classical generating function. In that context, the derivation of the Dirac equation is an exercise in counting. We provide this derivation here and discuss its relationship to quantum mechanics 7. Dirac and non-Dirac conditions in the two-potential theory of magnetic charge Science.gov (United States) Scott, John; Evans, Timothy J.; Singleton, Douglas; Dzhunushaliev, Vladimir; Folomeev, Vladimir 2018-05-01 We investigate the Cabbibo-Ferrari, two-potential approach to magnetic charge coupled to two different complex scalar fields, Φ _1 and Φ _2, each having different electric and magnetic charges. The scalar field, Φ _1, is assumed to have a spontaneous symmetry breaking self-interaction potential which gives a mass to the "magnetic" gauge potential and "magnetic" photon, while the other "electric" gauge potential and "electric" photon remain massless. The magnetic photon is hidden until one reaches energies of the order of the magnetic photon rest mass. The second scalar field, Φ _2, is required in order to make the theory non-trivial. With only one field one can always use a duality rotation to rotate away either the electric or magnetic charge, and thus decouple either the associated electric or magnetic photon. In analyzing this system of two scalar fields in the Cabbibo-Ferrari approach we perform several duality and gauge transformations, which require introducing non-Dirac conditions on the initial electric and magnetic charges. We also find that due to the symmetry breaking the usual Dirac condition is altered to include the mass of the magnetic photon. We discuss the implications of these various conditions on the charges. 8. Kinks and the Dirac equation International Nuclear Information System (INIS) Skyrme, T.H.R. 1994-01-01 In a model quantum theory of interacting mesons, the motion of certain conserved particle-like structures is discussed. It is shown how collective coordinates may be introduced to describe them, leading, in lowest approximation, to a Dirac equation. (author) 9. Alternatives to the Dirac equation International Nuclear Information System (INIS) Girvin, S.M.; Brownstein, K.R. 1975-01-01 Recent work by Biedenharn, Han, and van Dam (BHvD) has questioned the uniqueness of the Dirac equation. BHvD have obtained a two-component equation as an alternate to the Dirac equation. Although they later show their alternative to be unitarily equivalent to the Dirac equation, certain physical differences were claimed. BHvD attribute the existence of this alternate equation to the fact that their factorizing matrices were position-dependent. To investigate this, we factor the Klein-Gordon equation in spherical coordinates allowing the factorizing matrices to depend arbitrarily upon theta and phi. It is shown that despite this additional freedom, and without involving any relativistic covariance, the conventional four-component Dirac equation is the only possibility 10. Paul Dirac lectures at CERN CERN Multimedia CERN Bulletin 2010-01-01 When a group of physicists entered the Main Auditorium, during the evening of 29 June, they felt they had opened a time portal. Paul Dirac in front of a blackboard showing his formula. ©Sandra Hoogeboom An attentive audience, dressed in early 1900 costumes, were watching a lecture by the elusive Paul Dirac, presenting for the first time his famous formula on the blackboard. Paul Adrien Maurice Dirac (1902-1984) was a British mathematical physicist at Cambridge, and one of the "fathers" of quantum mechanics. When he first wrote it, in 1928, Dirac was not sure what his formula really meant. As demonstrated by Andersson four year later, what Dirac had written on the blackboard was the first definition of a positron, hence he is credited with having anticipated the existence of antimatter. The actor John Kohl performs as Paul Dirac. ©Sandra Hoogeboom What the group of puzzled physicists were really observing when they entered the CERN Auditorium was the shoo... 11. Massive hidden photons as lukewarm dark matter International Nuclear Information System (INIS) Redondo, Javier; Postma, Marieke 2008-11-01 We study the possibility that a keV-MeV mass hidden photon (HP), i.e. a hidden sector U(1) gauge boson, accounts for the observed amount of dark matter. We focus on the case where the HP interacts with the standard model sector only through kinetic mixing with the photon. The relic abundance is computed including all relevant plasma effects into the photon's self-energy, which leads to a resonant yield almost independent of the HP mass. The HP can decay into three photons. Moreover, if light enough it can be copiously produced in stars. Including bounds from cosmic photon backgrounds and stellar evolution, we find that the hidden photon can only give a subdominant contribution to the dark matter. This negative conclusion may be avoided if another production mechanism besides kinetic mixing is operative. (orig.) 12. Massive hidden photons as lukewarm dark matter Energy Technology Data Exchange (ETDEWEB) Redondo, Javier [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Postma, Marieke [Nationaal Inst. voor Kernfysica en Hoge-Energiefysica (NIKHEF), Amsterdam (Netherlands) 2008-11-15 We study the possibility that a keV-MeV mass hidden photon (HP), i.e. a hidden sector U(1) gauge boson, accounts for the observed amount of dark matter. We focus on the case where the HP interacts with the standard model sector only through kinetic mixing with the photon. The relic abundance is computed including all relevant plasma effects into the photon's self-energy, which leads to a resonant yield almost independent of the HP mass. The HP can decay into three photons. Moreover, if light enough it can be copiously produced in stars. Including bounds from cosmic photon backgrounds and stellar evolution, we find that the hidden photon can only give a subdominant contribution to the dark matter. This negative conclusion may be avoided if another production mechanism besides kinetic mixing is operative. (orig.) 13. Two-loop Dirac neutrino mass and WIMP dark matter OpenAIRE Bonilla, Cesar; Ma, Ernest; Peinado, Eduardo; Valle, Jose W.F. 2018-01-01 We propose a "scotogenic" mechanism relating small neutrino mass and cosmological dark matter. Neutrinos are Dirac fermions with masses arising only in two--loop order through the sector responsible for dark matter. Two triality symmetries ensure both dark matter stability and strict lepton number conservation at higher orders. A global spontaneously broken U(1) symmetry leads to a physical$Diraconthat induces invisible Higgs decays which add up to the Higgs to dark matter mode. This enhan... 14. LHCb: LHCbDirac is a DIRAC extension to support LHCb specific workflows CERN Multimedia Stagni, Federico 2012-01-01 We present LHCbDIRAC, an extension of the DIRAC community Grid solution to handle the LHCb specificities. The DIRAC software has been developed for many years within LHCb only. Nowadays it is a generic software, used by many scientific communities worldwide. Each community wanting to take advantage of DIRAC has to develop an extension, containing all the necessary code for handling their specific cases. LHCbDIRAC is an actively developed extension, implementing the LHCb computing model and workflows. LHCbDIRAC extends DIRAC to handle all the distributed computing activities of LHCb. Such activities include real data processing (reconstruction, stripping and streaming), Monte-Carlo simulation and data replication. Other activities are groups and user analysis, data management, resources management and monitoring, data provenance, accounting for user and production jobs. LHCbDIRAC also provides extensions of the DIRAC interfaces, including a secure web client, python APIs and CLIs. While DIRAC and LHCbDIRAC f... 15. Sociology of Hidden Curriculum Directory of Open Access Journals (Sweden) Alireza Moradi 2017-06-01 Full Text Available This paper reviews the concept of hidden curriculum in the sociological theories and wants to explain sociological aspects of formation of hidden curriculum. The main question concentrates on the theoretical approaches in which hidden curriculum is explained sociologically.For this purpose it was applied qualitative research methodology. The relevant data include various sociological concepts and theories of hidden curriculum collected by the documentary method. The study showed a set of rules, procedures, relationships and social structure of education have decisive role in the formation of hidden curriculum. A hidden curriculum reinforces by existed inequalities among learners (based on their social classes or statues. There is, in fact, a balance between the learner's "knowledge receptions" with their "inequality proportion".The hidden curriculum studies from different major sociological theories such as Functionalism, Marxism and critical theory, Symbolic internationalism and Feminism. According to the functionalist perspective a hidden curriculum has a social function because it transmits social values. Marxists and critical thinkers correlate between hidden curriculum and the totality of social structure. They depicts that curriculum prepares learners for the exploitation in the work markets. Symbolic internationalism rejects absolute hegemony of hidden curriculum on education and looks to the socialization as a result of interaction between learner and instructor. Feminism theory also considers hidden curriculum as a vehicle which legitimates gender stereotypes. 16. Exploring the Hidden Sector @ Low Energies CERN Multimedia CERN. Geneva 2015-01-01 Over the years we have accumulated a large number of indications for physics beyond the standard model. This new physics is often sought-after at high masses and energies. Here collider experiments can bring decisive insights. However, over recent years it has become increasingly clear that new physics can also appear at low energy, but extremely weak coupling. Experiments and observations at this low energy frontier' therefore provide a powerful tool to gain insight into fundamental physics, which is complementary to accelerators. 17. Quasi-Dirac neutrino oscillations Science.gov (United States) Anamiati, Gaetana; Fonseca, Renato M.; Hirsch, Martin 2018-05-01 Dirac neutrino masses require two distinct neutral Weyl spinors per generation, with a special arrangement of masses and interactions with charged leptons. Once this arrangement is perturbed, lepton number is no longer conserved and neutrinos become Majorana particles. If these lepton number violating perturbations are small compared to the Dirac mass terms, neutrinos are quasi-Dirac particles. Alternatively, this scenario can be characterized by the existence of pairs of neutrinos with almost degenerate masses, and a lepton mixing matrix which has 12 angles and 12 phases. In this work we discuss the phenomenology of quasi-Dirac neutrino oscillations and derive limits on the relevant parameter space from various experiments. In one parameter perturbations of the Dirac limit, very stringent bounds can be derived on the mass splittings between the almost degenerate pairs of neutrinos. However, we also demonstrate that with suitable changes to the lepton mixing matrix, limits on such mass splittings are much weaker, or even completely absent. Finally, we consider the possibility that the mass splittings are too small to be measured and discuss bounds on the new, nonstandard lepton mixing angles from current experiments for this case. 18. Hidden photons in beam dump experiments and in connection with dark matter Energy Technology Data Exchange (ETDEWEB) Andreas, Sarah 2012-12-15 Hidden sectors with light extra U(1) gauge bosons, so-called hidden photons, recently received much interest as natural feature of beyond standard model scenarios like string theory and SUSY and because of their possible connection to dark matter. This paper presents limits on hidden photons from past electron beam dump experiments including two new limits from experiments at KEK and Orsay. Additionally, various hidden sector models containing both a hidden photon and a dark matter candidate are discussed with respect to their viability and potential signatures in direct detection. 19. Hidden photons in beam dump experiments and in connection with dark matter International Nuclear Information System (INIS) Andreas, Sarah 2012-12-01 Hidden sectors with light extra U(1) gauge bosons, so-called hidden photons, recently received much interest as natural feature of beyond standard model scenarios like string theory and SUSY and because of their possible connection to dark matter. This paper presents limits on hidden photons from past electron beam dump experiments including two new limits from experiments at KEK and Orsay. Additionally, various hidden sector models containing both a hidden photon and a dark matter candidate are discussed with respect to their viability and potential signatures in direct detection. 20. The many faces of Maxwell, Dirac and Einstein equations a Clifford bundle approach CERN Document Server Rodrigues, Jr, Waldyr A 2016-01-01 This book is an exposition of the algebra and calculus of differential forms, of the Clifford and Spin-Clifford bundle formalisms, and of vistas to a formulation of important concepts of differential geometry indispensable for an in-depth understanding of space-time physics. The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields are shown to have representatives by objects of the same mathematical nature, namely sections of an appropriate Clifford bundle. This approach reveals unity in diversity and suggests relationships that are hidden in the standard formalisms and opens new paths for research. This thoroughly revised second edition also adds three new chapters: on the Clifford bundle approach to the Riemannian or semi-Riemannian differential geometry of branes; on Komar currents in the context of the General Relativity theory; and an analysis of the similarities and main differences between Dirac, Majorana and ELKO spinor fields. The exercises with solut... 1. Hidden photon dark matter search with large metallic mirror International Nuclear Information System (INIS) Doebrich, Babette; Lindner, Axel; Daumiller, Kai; Engel, Ralph; Roth, Markus; Kowalski, Marek 2014-10-01 If Dark Matter is composed of hidden-sector photons that kinetically mix with photons of the visible sector, then Dark Matter has a tiny oscillating electric field component. Its presence would lead to a small amount of visible radiation being emitted from a conducting surface, with the photon frequency given approximately by the mass of the hidden photon. Here, we report on experimental efforts that have started recently to search for such hidden photon Dark Matter in the (sub-)eV regime with a prototype mirror for the Auger fluorescence detector at the Karlsruhe Institute for Technology. 2. Common origin of neutrino mass, dark matter and Dirac leptogenesis Energy Technology Data Exchange (ETDEWEB) Borah, Debasish [Department of Physics, Indian Institute of Technology Guwahati, Assam 781039 (India); Dasgupta, Arnab, E-mail: [email protected], E-mail: [email protected] [Institute of Physics, HBNI, Sachivalaya Marg, Bhubaneshwar 751005 (India) 2016-12-01 We study the possibility of generating tiny Dirac neutrino masses at one loop level through the scotogenic mechanism such that one of the particles going inside the loop can be a stable cold dark matter (DM) candidate. Majorana mass terms of singlet fermions as well as tree level Dirac neutrino masses are prevented by incorporating the presence of additional discrete symmetries in a minimal fashion, which also guarantee the stability of the dark matter candidate. Due to the absence of total lepton number violation, the observed baryon asymmetry of the Universe is generated through the mechanism of Dirac leptogenesis where an equal and opposite amount of leptonic asymmetry is generated in the left and right handed sectors which are prevented from equilibration due to tiny Dirac Yukawa couplings. Dark matter relic abundance is generated through its usual freeze-out at a temperature much below the scale of leptogenesis. We constrain the relevant parameter space from neutrino mass, baryon asymmetry, Planck bound on dark matter relic abundance, and latest LUX bound on spin independent DM-nucleon scattering cross section. We also discuss the charged lepton flavour violation (μ → e γ) and electric dipole moment of electron in this model in the light of the latest experimental data and constrain the parameter space of the model. 3. The Dirac medals of the ICTP. 1993 Energy Technology Data Exchange (ETDEWEB) NONE 1996-12-31 The Dirac Medals of the International Centre for Theoretical Physics (ICTP) were instituted in 1985. These are awarded yearly to outstanding physicists, on Diracs birthday - 8th August- for contributions to theoretical physics. The document includes the lectures of the three Dirac Medalists for 1993: Professor Sergio Ferrara, Professor Daniel Z. Freedman, and Professor Peter van Nieuwenhuizen. A separate abstract was prepared for each lecture 4. A Dirac algebraic approach to supersymmetry International Nuclear Information System (INIS) Guersey, F. 1984-01-01 The power of the Dirac algebra is illustrated through the Kaehler correspondence between a pair of Dirac spinors and a 16-component bosonic field. The SO(5,1) group acts on both the fermion and boson fields, leading to a supersymmetric equation of the Dirac type involving all these fields. (author) 5. Interlayer magnetoresistance in multilayer Dirac electron systems: motion and merging of Dirac cones OpenAIRE Assili, Mohamed; Haddad, Sonia 2013-01-01 We theoretically study the effect of the motion and the merging of Dirac cone on the interlayer magnetoresistance in multilayer graphene like systems. This merging, which could be induced by a uniaxial strain, gives rise in monolayer Dirac electron system to a topological transition from a semi-metallic phase to an insulating phase where Dirac points disappear. Based on a universal Hamiltonian proposed to describe the motion and the merging of Dirac points in two dimensional Dirac electron cr... 6. Signatures of a hidden cosmic microwave background. Science.gov (United States) Jaeckel, Joerg; Redondo, Javier; Ringwald, Andreas 2008-09-26 If there is a light Abelian gauge boson gamma' in the hidden sector its kinetic mixing with the photon can produce a hidden cosmic microwave background (HCMB). For meV masses, resonant oscillations gammagamma' happen after big bang nucleosynthesis (BBN) but before CMB decoupling, increasing the effective number of neutrinos Nnu(eff) and the baryon to photon ratio, and distorting the CMB blackbody spectrum. The agreement between BBN and CMB data provides new constraints. However, including Lyman-alpha data, Nnu(eff) > 3 is preferred. It is tempting to attribute this effect to the HCMB. The interesting parameter range will be tested in upcoming laboratory experiments. 7. Dirac, Prof. Paul Adrien Maurice Indian Academy of Sciences (India) Home; Fellowship. Fellow Profile. Elected: 1935 Honorary. Dirac, Prof. Paul Adrien Maurice Nobel Laureate (Physics) - 1933. Date of birth: 8 August 1902. Date of death: 20 October 1984. YouTube; Twitter; Facebook; Blog. Academy News. IAS Logo. 29th Mid-year meeting. Posted on 19 January 2018. The 29th Mid-year ... 8. Dirac, Jordan and quantum fields International Nuclear Information System (INIS) Darrigol, O. 1985-01-01 The case of two principal physicists of quantum mechanics is specially chose: Paul Dirac and Pascual Jordan. They gave a signification and an importance very different to the notion of quantum field, and in particular to the quantized matter wave one. Through their formation and motivation differences, such as they are expressed in their writings, this deep difference is tentatively understood [fr 9. about the Dirac Delta Function(?) Indian Academy of Sciences (India) V Balakrishnan is in the. Department of ... and sweet as befits this impatient age. It said (in its en- ... to get down to real work by shutting down the system and reverting to ... the Dirac delta function" - but do note the all-important question mark in ... 10. Superconductivity in doped Dirac semimetals Science.gov (United States) Hashimoto, Tatsuki; Kobayashi, Shingo; Tanaka, Yukio; Sato, Masatoshi 2016-07-01 We theoretically study intrinsic superconductivity in doped Dirac semimetals. Dirac semimetals host bulk Dirac points, which are formed by doubly degenerate bands, so the Hamiltonian is described by a 4 ×4 matrix and six types of k -independent pair potentials are allowed by the Fermi-Dirac statistics. We show that the unique spin-orbit coupling leads to characteristic superconducting gap structures and d vectors on the Fermi surface and the electron-electron interaction between intra and interorbitals gives a novel phase diagram of superconductivity. It is found that when the interorbital attraction is dominant, an unconventional superconducting state with point nodes appears. To verify the experimental signature of possible superconducting states, we calculate the temperature dependence of bulk physical properties such as electronic specific heat and spin susceptibility and surface state. In the unconventional superconducting phase, either dispersive or flat Andreev bound states appear between point nodes, which leads to double peaks or a single peak in the surface density of states, respectively. As a result, possible superconducting states can be distinguished by combining bulk and surface measurements. 11. Dirac Magnons in Honeycomb Ferromagnets Directory of Open Access Journals (Sweden) Sergey S. Pershoguba 2018-01-01 Full Text Available The discovery of the Dirac electron dispersion in graphene [A. H. Castro Neto, et al., The Electronic Properties of Graphene, Rev. Mod. Phys. 81, 109 (2009RMPHAT0034-686110.1103/RevModPhys.81.109] led to the question of the Dirac cone stability with respect to interactions. Coulomb interactions between electrons were shown to induce a logarithmic renormalization of the Dirac dispersion. With a rapid expansion of the list of compounds and quasiparticle bands with linear band touching [T. O. Wehling, et al., Dirac Materials, Adv. Phys. 63, 1 (2014ADPHAH0001-873210.1080/00018732.2014.927109], the concept of bosonic Dirac materials has emerged. We consider a specific case of ferromagnets consisting of van der Waals-bonded stacks of honeycomb layers, e.g., chromium trihalides CrX_{3} (X=F, Cl, Br and I, that display two spin wave modes with energy dispersion similar to that for the electrons in graphene. At the single-particle level, these materials resemble their fermionic counterparts. However, how different particle statistics and interactions affect the stability of Dirac cones has yet to be determined. To address the role of interacting Dirac magnons, we expand the theory of ferromagnets beyond the standard Dyson theory [F. J. Dyson, General Theory of Spin-Wave Interactions, Phys. Rev. 102, 1217 (1956PHRVAO0031-899X10.1103/PhysRev.102.1217, F. J. Dyson, Thermodynamic Behavior of an Ideal Ferromagnet, Phys. Rev. 102, 1230 (1956PHRVAO0031-899X10.1103/PhysRev.102.1230] to the case of non-Bravais honeycomb layers. We demonstrate that magnon-magnon interactions lead to a significant momentum-dependent renormalization of the bare band structure in addition to strongly momentum-dependent magnon lifetimes. We show that our theory qualitatively accounts for hitherto unexplained anomalies in nearly half-century-old magnetic neutron-scattering data for CrBr_{3} [W. B. Yelon and R. Silberglitt, Renormalization of Large-Wave-Vector Magnons in 12. Dirac Magnons in Honeycomb Ferromagnets Science.gov (United States) Pershoguba, Sergey S.; Banerjee, Saikat; Lashley, J. C.; Park, Jihwey; Ågren, Hans; Aeppli, Gabriel; Balatsky, Alexander V. 2018-01-01 The discovery of the Dirac electron dispersion in graphene [A. H. Castro Neto, et al., The Electronic Properties of Graphene, Rev. Mod. Phys. 81, 109 (2009), 10.1103/RevModPhys.81.109] led to the question of the Dirac cone stability with respect to interactions. Coulomb interactions between electrons were shown to induce a logarithmic renormalization of the Dirac dispersion. With a rapid expansion of the list of compounds and quasiparticle bands with linear band touching [T. O. Wehling, et al., Dirac Materials, Adv. Phys. 63, 1 (2014), 10.1080/00018732.2014.927109], the concept of bosonic Dirac materials has emerged. We consider a specific case of ferromagnets consisting of van der Waals-bonded stacks of honeycomb layers, e.g., chromium trihalides CrX3 (X =F , Cl, Br and I), that display two spin wave modes with energy dispersion similar to that for the electrons in graphene. At the single-particle level, these materials resemble their fermionic counterparts. However, how different particle statistics and interactions affect the stability of Dirac cones has yet to be determined. To address the role of interacting Dirac magnons, we expand the theory of ferromagnets beyond the standard Dyson theory [F. J. Dyson, General Theory of Spin-Wave Interactions, Phys. Rev. 102, 1217 (1956), 10.1103/PhysRev.102.1217, F. J. Dyson, Thermodynamic Behavior of an Ideal Ferromagnet, Phys. Rev. 102, 1230 (1956), 10.1103/PhysRev.102.1230] to the case of non-Bravais honeycomb layers. We demonstrate that magnon-magnon interactions lead to a significant momentum-dependent renormalization of the bare band structure in addition to strongly momentum-dependent magnon lifetimes. We show that our theory qualitatively accounts for hitherto unexplained anomalies in nearly half-century-old magnetic neutron-scattering data for CrBr3 [W. B. Yelon and R. Silberglitt, Renormalization of Large-Wave-Vector Magnons in Ferromagnetic CrBr3 Studied by Inelastic Neutron Scattering: Spin-Wave Correlation 13. Graphene based d-character Dirac Systems Science.gov (United States) Li, Yuanchang; Zhang, S. B.; Duan, Wenhui From graphene to topological insulators, Dirac material continues to be the hot topics in condensed matter physics. So far, almost all of the theoretically predicted or experimentally observed Dirac materials are composed of sp -electrons. By using first-principles calculations, we find the new Dirac system of transition-metal intercalated epitaxial graphene on SiC(0001). Intrinsically different from the conventional sp Dirac system, here the Dirac-fermions are dominantly contributed by the transition-metal d-electrons, which paves the way to incorporate correlation effect with Dirac-cone physics. Many intriguing quantum phenomena are proposed based on this system, including quantum spin Hall effect with large spin-orbital gap, quantum anomalous Hall effect, 100% spin-polarized Dirac fermions and ferromagnet-to-topological insulator transition. 14. POVMs and hidden variables International Nuclear Information System (INIS) Stairs, Allen 2007-01-01 Recent results by Paul Busch and Adan Cabello claim to show that by appealing to POVMs, non-contextual hidden variables can be ruled out in two dimensions. While the results of Busch and Cabello are mathematically correct, interpretive problems render them problematic as no hidden variable proofs 15. Partially Hidden Markov Models DEFF Research Database (Denmark) Forchhammer, Søren Otto; Rissanen, Jorma 1996-01-01 Partially Hidden Markov Models (PHMM) are introduced. They differ from the ordinary HMM's in that both the transition probabilities of the hidden states and the output probabilities are conditioned on past observations. As an illustration they are applied to black and white image compression where... 16. 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. 17. DIRAC pilot framework and the DIRAC Workload Management System International Nuclear Information System (INIS) Casajus, Adrian; Graciani, Ricardo; Paterson, Stuart; Tsaregorodtsev, Andrei 2010-01-01 DIRAC, the LHCb community Grid solution, has pioneered the use of pilot jobs in the Grid. Pilot Jobs provide a homogeneous interface to an heterogeneous set of computing resources. At the same time, Pilot Jobs allow to delay the scheduling decision to the last moment, thus taking into account the precise running conditions at the resource and last moment requests to the system. The DIRAC Workload Management System provides one single scheduling mechanism for jobs with very different profiles. To achieve an overall optimisation, it organizes pending jobs in task queues, both for individual users and production activities. Task queues are created with jobs having similar requirements. Following the VO policy a priority is assigned to each task queue. Pilot submission and subsequent job matching are based on these priorities following a statistical approach. 18. DIRAC pilot framework and the DIRAC Workload Management System Energy Technology Data Exchange (ETDEWEB) Casajus, Adrian; Graciani, Ricardo [Universitat de Barcelona (Spain); Paterson, Stuart [CERN (Switzerland); Tsaregorodtsev, Andrei, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [CPPM Marseille (France) 2010-04-01 DIRAC, the LHCb community Grid solution, has pioneered the use of pilot jobs in the Grid. Pilot Jobs provide a homogeneous interface to an heterogeneous set of computing resources. At the same time, Pilot Jobs allow to delay the scheduling decision to the last moment, thus taking into account the precise running conditions at the resource and last moment requests to the system. The DIRAC Workload Management System provides one single scheduling mechanism for jobs with very different profiles. To achieve an overall optimisation, it organizes pending jobs in task queues, both for individual users and production activities. Task queues are created with jobs having similar requirements. Following the VO policy a priority is assigned to each task queue. Pilot submission and subsequent job matching are based on these priorities following a statistical approach. 19. DIRAC in Large Particle Physics Experiments Science.gov (United States) Stagni, F.; Tsaregorodtsev, A.; Arrabito, L.; Sailer, A.; Hara, T.; Zhang, X.; Consortium, DIRAC 2017-10-01 The DIRAC project is developing interware to build and operate distributed computing systems. It provides a development framework and a rich set of services for both Workload and Data Management tasks of large scientific communities. A number of High Energy Physics and Astrophysics collaborations have adopted DIRAC as the base for their computing models. DIRAC was initially developed for the LHCb experiment at LHC, CERN. Later, the Belle II, BES III and CTA experiments as well as the linear collider detector collaborations started using DIRAC for their computing systems. Some of the experiments built their DIRAC-based systems from scratch, others migrated from previous solutions, ad-hoc or based on different middlewares. Adaptation of DIRAC for a particular experiment was enabled through the creation of extensions to meet their specific requirements. Each experiment has a heterogeneous set of computing and storage resources at their disposal that were aggregated through DIRAC into a coherent pool. Users from different experiments can interact with the system in different ways depending on their specific tasks, expertise level and previous experience using command line tools, python APIs or Web Portals. In this contribution we will summarize the experience of using DIRAC in particle physics collaborations. The problems of migration to DIRAC from previous systems and their solutions will be presented. An overview of specific DIRAC extensions will be given. We hope that this review will be useful for experiments considering an update, or for those designing their computing models. 20. Observable lepton number violation with predominantly Dirac nature of active neutrinos Energy Technology Data Exchange (ETDEWEB) Borah, Debasish [Department of Physics, Indian Institute of Technology Guwahati,Assam-781039 (India); Dasgupta, Arnab [Institute of Physics, HBNI,Sachivalaya Marg, Bhubaneshwar-751005 (India) 2017-01-17 We study a specific version of SU(2){sub R}×SU(2){sub L}×U(1){sub B−L} models extended by discrete symmetries where the new physics sector responsible for tiny neutrino masses at leading order remains decoupled from the new physics sector that can give rise to observable signatures of lepton number violation such as neutrinoless double beta decay. More specifically, the dominant contribution to light neutrino masses comes from a one-loop Dirac mass. At higher loop level, a tiny Majorana mass also appears which remains suppressed by many order of magnitudes in comparison to the Dirac mass. Such a model where the active neutrinos are predominantly of Dirac type, also predicts observable charged lepton flavour violation like μ→3e,μ→eγ and multi-component dark matter. 1. Halogenated arsenenes as Dirac materials International Nuclear Information System (INIS) Tang, Wencheng; Sun, Minglei; Ren, Qingqiang; Wang, Sake; Yu, Jin 2016-01-01 Highlights: • We have revealed the presence of Dirac cone in fully-halogenated arsenene compounds. • All fully-halogenated arsenene except As_2I_2 would spontaneously form and stable in defending the thermal fluctuation in room temperature. - Abstract: Arsenene is the graphene-like arsenic nanosheet, which has been predicted very recently [S. Zhang, Z. Yan, Y. Li, Z. Chen, and H. Zeng, Angewandte Chemie, 127 (2015) 3155–3158]. Using first-principles calculations, we systematically investigate the structures and electronic properties of fully-halogenated arsenenes. Formation energy analysis reveals that all the fully-halogenated arsenenes except iodinated arsenene are energetically favorable and could be synthesized. We have revealed the presence of Dirac cone in fully-halogenated arsenene compounds. They may have great potential applications in next generation of high-performance devices. 2. DIRAC: Secure web user interface International Nuclear Information System (INIS) Casajus Ramo, A; Sapunov, M 2010-01-01 Traditionally the interaction between users and the Grid is done with command line tools. However, these tools are difficult to use by non-expert users providing minimal help and generating outputs not always easy to understand especially in case of errors. Graphical User Interfaces are typically limited to providing access to the monitoring or accounting information and concentrate on some particular aspects failing to cover the full spectrum of grid control tasks. To make the Grid more user friendly more complete graphical interfaces are needed. Within the DIRAC project we have attempted to construct a Web based User Interface that provides means not only for monitoring the system behavior but also allows to steer the main user activities on the grid. Using DIRAC's web interface a user can easily track jobs and data. It provides access to job information and allows performing actions on jobs such as killing or deleting. Data managers can define and monitor file transfer activity as well as check requests set by jobs. Production managers can define and follow large data productions and react if necessary by stopping or starting them. The Web Portal is build following all the grid security standards and using modern Web 2.0 technologies which allow to achieve the user experience similar to the desktop applications. Details of the DIRAC Web Portal architecture and User Interface will be presented and discussed. 3. Hidden gauge symmetry International Nuclear Information System (INIS) O'Raifeartaigh, L. 1979-01-01 This review describes the principles of hidden gauge symmetry and of its application to the fundamental interactions. The emphasis is on the structure of the theory rather than on the technical details and, in order to emphasise the structure, gauge symmetry and hidden symmetry are first treated as independent phenomena before being combined into a single (hidden gauge symmetric) theory. The main application of the theory is to the weak and electromagnetic interactions of the elementary particles, and although models are used for comparison with experiment and for illustration, emphasis is placed on those features of the application which are model-independent. (author) 4. Search for hidden particles with the SHiP experiment Energy Technology Data Exchange (ETDEWEB) Hagner, Caren; Bick, Daniel; Bieschke, Stefan; Ebert, Joachim; Schmidt-Parzefall, Walter [Universitaet Hamburg, Institut fuer Experimentalphysik, Luruper Chaussee 149, 22761 Hamburg (Germany) 2016-07-01 Many theories beyond the standard model predict long lived neutral (hidden) particles. There might be a whole Hidden Sector (HS) of weakly interacting particles, which cannot be detected in existing high energy experiments. The SHiP experiment (Search for Hidden Particles) requires a high intensity beam dump, which could be realized by a new facility at the CERN SPS accelerator. New superweakly interacting particles with masses below O(10) GeV could be produced in the beam dump and detected in a general purpose Hidden Sector (HS) detector. In addition there will be a dedicated tau neutrino subdetector. I present the major requirements and technical challenges for the HS detector and discuss how the HS can be accessed through several portals: neutrino portal, scalar portal, vector portal and many more. 5. Double Dirac cones in phononic crystals KAUST Repository Li, Yan 2014-07-07 A double Dirac cone is realized at the center of the Brillouin zone of a two-dimensional phononic crystal (PC) consisting of a triangular array of core-shell-structure cylinders in water. The double Dirac cone is induced by the accidental degeneracy of two double-degenerate Bloch states. Using a perturbation method, we demonstrate that the double Dirac cone is composed of two identical and overlapping Dirac cones whose linear slopes can also be accurately predicted from the method. Because the double Dirac cone occurs at a relatively low frequency, a slab of the PC can be mapped onto a slab of zero refractive index material by using a standard retrieval method. Total transmission without phase change and energy tunneling at the double Dirac point frequency are unambiguously demonstrated by two examples. Potential applications can be expected in diverse fields such as acoustic wave manipulations and energy flow control. 6. Data Management System of the DIRAC Project CERN Multimedia Haen, Christophe; Tsaregorodtsev, Andrei 2015-01-01 The DIRAC Interware provides a development framework and a complete set of components for building distributed computing systems. The DIRAC Data Management System (DMS) offers all the necessary tools to ensure data handling operations for small and large user communities. It supports transparent access to storage resources based on multiple technologies, and is easily expandable. The information on data files and replicas is kept in a File Catalog of which DIRAC offers a powerful and versatile implementation (DFC). Data movement can be performed using third party services including FTS3. Bulk data operations are resilient with respect to failures due to the use of the Request Management System (RMS) that keeps track of ongoing tasks. In this contribution we will present an overview of the DIRAC DMS capabilities and its connection with other DIRAC subsystems such as the Transformation System. The DIRAC DMS is in use by several user communities now. The contribution will present the experience of the LHCb exper... 7. The DIRAC Data Management System (poster) CERN Document Server Haen, Christophe 2015-01-01 The DIRAC Interware provides a development framework and a complete set of components for building distributed computing systems. The DIRAC Data Management System (DMS) offers all the necessary tools to ensure data handling operations for small and large user communities. It supports transparent access to storage resources based on multiple technologies, and is easily expandable. The information on data files and replicas is kept in a File Catalog of which DIRAC offers a powerful and versatile implementation (DFC). Data movement can be performed using third party services including FTS3. Bulk data operations are resilient with respect to failures due to the use of the Request Management System (RMS) that keeps track of ongoing tasks. In this contribution we will present an overview of the DIRAC DMS capabilities and its connection with other DIRAC subsystems such as the Transformation System. The DIRAC DMS is in use by several user communities now. The contribution will present the experience of the LHCb exper... 8. Double Dirac cones in phononic crystals KAUST Repository Li, Yan; Wu, Ying; Mei, Jun 2014-01-01 A double Dirac cone is realized at the center of the Brillouin zone of a two-dimensional phononic crystal (PC) consisting of a triangular array of core-shell-structure cylinders in water. The double Dirac cone is induced by the accidental degeneracy of two double-degenerate Bloch states. Using a perturbation method, we demonstrate that the double Dirac cone is composed of two identical and overlapping Dirac cones whose linear slopes can also be accurately predicted from the method. Because the double Dirac cone occurs at a relatively low frequency, a slab of the PC can be mapped onto a slab of zero refractive index material by using a standard retrieval method. Total transmission without phase change and energy tunneling at the double Dirac point frequency are unambiguously demonstrated by two examples. Potential applications can be expected in diverse fields such as acoustic wave manipulations and energy flow control. 9. Hidden twelve-dimensional super Poincare symmetry in eleven dimensions International Nuclear Information System (INIS) Bars, Itzhak; Deliduman, Cemsinan; Pasqua, Andrea; Zumino, Bruno 2004-01-01 First, we review a result in our previous paper, of how a ten-dimensional superparticle, taken off-shell, has a hidden eleven-dimensional super Poincare symmetry. Then, we show that the physical sector is defined by three first-class constraints which preserve the full eleven-dimensional symmetry. Applying the same concepts to the eleven-dimensional superparticle, taken off-shell, we discover a hidden twelve-dimensional super Poincare symmetry that governs the theory 10. The Dirac medals of the ICTP. 1993 International Nuclear Information System (INIS) 1995-01-01 The Dirac Medals of the International Centre for Theoretical Physics (ICTP) were instituted in 1985. These are awarded yearly to outstanding physicists, on Dirac's birthday - 8th August- for contributions to theoretical physics. The document includes the lectures of the three Dirac Medalists for 1993: Professor Sergio Ferrara, Professor Daniel Z. Freedman, and Professor Peter van Nieuwenhuizen. A separate abstract was prepared for each lecture 11. LHCbDIRAC as Apache Mesos microservices OpenAIRE Haen, Christophe; Couturier, Benjamin 2017-01-01 The LHCb experiment relies on LHCbDIRAC, an extension of DIRAC, to drive its offline computing. This middleware provides a development framework and a complete set of components for building distributed computing systems. These components are currently installed and run on virtual machines (VM) or bare metal hardware. Due to the increased workload, high availability is becoming more and more important for the LHCbDIRAC services, and the current installation model is showing its limitations. A... 12. The hidden universe International Nuclear Information System (INIS) Disney, M. 1985-01-01 Astronomer Disney has followed a somewhat different tack than that of most popular books on cosmology by concentrating on the notion of hidden (as in not directly observable by its own radiation) matter in the universe 13. Locating Hidden Servers National Research Council Canada - National Science Library Oeverlier, Lasse; Syverson, Paul F 2006-01-01 .... Announced properties include server resistance to distributed DoS. Both the EFF and Reporters Without Borders have issued guides that describe using hidden services via Tor to protect the safety of dissidents as well as to resist censorship... 14. LHCbDIRAC as Apache Mesos microservices Science.gov (United States) Haen, Christophe; Couturier, Benjamin 2017-10-01 The LHCb experiment relies on LHCbDIRAC, an extension of DIRAC, to drive its offline computing. This middleware provides a development framework and a complete set of components for building distributed computing systems. These components are currently installed and run on virtual machines (VM) or bare metal hardware. Due to the increased workload, high availability is becoming more and more important for the LHCbDIRAC services, and the current installation model is showing its limitations. Apache Mesos is a cluster manager which aims at abstracting heterogeneous physical resources on which various tasks can be distributed thanks to so called “frameworks” The Marathon framework is suitable for long running tasks such as the DIRAC services, while the Chronos framework meets the needs of cron-like tasks like the DIRAC agents. A combination of the service discovery tool Consul together with HAProxy allows to expose the running containers to the outside world while hiding their dynamic placements. Such an architecture brings a greater flexibility in the deployment of LHCbDirac services, allowing for easier deployment maintenance and scaling of services on demand (e..g LHCbDirac relies on 138 services and 116 agents). Higher reliability is also easier, as clustering is part of the toolset, which allows constraints on the location of the services. This paper describes the investigations carried out to package the LHCbDIRAC and DIRAC components into Docker containers and orchestrate them using the previously described set of tools. 15. Black holes, hidden symmetries, and complete integrability. Science.gov (United States) Frolov, Valeri P; Krtouš, Pavel; Kubizňák, David 2017-01-01 The study of higher-dimensional black holes is a subject which has recently attracted vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higher-dimensional black holes with the spherical horizon topology and described by the Kerr-NUT-(A)dS metrics are very similar to the properties of the well known four-dimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higher-dimensional Kerr-NUT-(A)dS black hole spacetimes. We start with discussion of the Killing and Killing-Yano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a "seed object" which generates all these symmetries. It determines the form of the geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic motion, and separability of the Hamilton-Jacobi, Klein-Gordon, and Dirac equations. The review also contains a discussion of different applications of the developed formalism and its possible generalizations. 16. Integrating out the Dirac sea International Nuclear Information System (INIS) Karbstein, Felix 2009-01-01 We introduce a new method for dealing with fermionic quantum field theories amenable to a mean-field-type approximation. In this work we focus on the relativistic Hartree approximation. Our aim is to integrate out the Dirac sea and derive a no-sea effective theory'' with positive energy single particle states only. As the derivation of the no-sea effective theory involves only standard Feynman diagrams, our approach is quite general and not restricted to particular space-time dimensions. We develop and illustrate the approach in the ''large N'' limit of the Gross-Neveu model family in 1+1 dimensions. As the Gross-Neveu model has been intensely studied and several analytical solutions are known for this model, it is an ideal testing ground for our no-sea effective theory approach. The chiral Gross-Neveu model, also referred to as 1+1 dimensional Nambu-Jona-Lasinio model, turns out to be of particular interest. In this case, we explicitly derive a consistent effective theory featuring both elementary ''π meson'' fields and (positive energy) ''quark'' fields, starting from a purely fermionic quantum field theory. In the second part of this work, we apply our approach to the Walecka model in 1+1 and 3+1 dimensions. As the Dirac sea caused considerable difficulties in attempts to base nuclear physics on field theoretic models like the Walecka model, mean-field calculations were typically done without the sea. We confront several of these mean-field theory results with our no-sea effective theory approach. The potential of our approach is twofold. While the no-sea effective theory can be utilized to provide new analytical insights in particular parameter regimes, it also sheds new light on more fundamental issues as the explicit emergence of effective, Dirac-sea induced multi-fermion interactions in an effective theory with positive energy states only. (orig.) 17. Dirac tensor with heavy photon Energy Technology Data Exchange (ETDEWEB) Bytev, V.V.; Kuraev, E.A. [Joint Institute of Nuclear Research, Moscow (Russian Federation). Bogoliubov Lab. of Theoretical Physics; Scherbakova, E.S. [Hamburg Univ. (Germany). 1. Inst. fuer Theoretische Physik 2012-01-15 For the large-angles hard photon emission by initial leptons in process of high energy annihilation of e{sup +}e{sup -} {yields} to hadrons the Dirac tensor is obtained, taking into account the lowest order radiative corrections. The case of large-angles emission of two hard photons by initial leptons is considered. This result is being completed by the kinematics case of collinear hard photons emission as well as soft virtual and real photons and can be used for construction of Monte-Carlo generators. (orig.) 18. Higher-dimensional black holes: hidden symmetries and separation of variables International Nuclear Information System (INIS) Frolov, Valeri P; Kubiznak, David 2008-01-01 In this paper, we discuss hidden symmetries in rotating black hole spacetimes. We start with an extended introduction which mainly summarizes results on hidden symmetries in four dimensions and introduces Killing and Killing-Yano tensors, objects responsible for hidden symmetries. We also demonstrate how starting with a principal CKY tensor (that is a closed non-degenerate conformal Killing-Yano 2-form) in 4D flat spacetime one can 'generate' the 4D Kerr-NUT-(A)dS solution and its hidden symmetries. After this we consider higher-dimensional Kerr-NUT-(A)dS metrics and demonstrate that they possess a principal CKY tensor which allows one to generate the whole tower of Killing-Yano and Killing tensors. These symmetries imply complete integrability of geodesic equations and complete separation of variables for the Hamilton-Jacobi, Klein-Gordon and Dirac equations in the general Kerr-NUT-(A)dS metrics 19. The Dirac equation in classical statistical mechanics International Nuclear Information System (INIS) Ord, G.N. 2002-01-01 The Dirac equation, usually obtained by 'quantizing' a classical stochastic model is here obtained directly within classical statistical mechanics. The special underlying space-time geometry of the random walk replaces the missing analytic continuation, making the model 'self-quantizing'. This provides a new context for the Dirac equation, distinct from its usual context in relativistic quantum mechanics 20. Dirac and Weyl semimetals with holographic interactions NARCIS (Netherlands) Jacobs, V.P.J. 2015-01-01 Dirac and Weyl semimetals are states of matter exhibiting the relativistic physics of, respectively, the Dirac and Weyl equation in a three-dimensional bulk material. These three-dimensional semimetals have recently been realized experimentally in various crystals. Theoretically, especially the 1. A fractional Dirac equation and its solution International Nuclear Information System (INIS) Muslih, Sami I; Agrawal, Om P; Baleanu, Dumitru 2010-01-01 This paper presents a fractional Dirac equation and its solution. The fractional Dirac equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange equations of motion. We present a Lagrangian and a Hamiltonian for the fractional Dirac equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Dirac equation which is the same as that obtained using the fractional variational principle. Eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard Dirac equation. We also provide expressions for the path integral quantization for the fractional Dirac field which, in the limit α → 1, approaches to the path integral for the regular Dirac field. It is hoped that the fractional Dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics. 2. New solitons connected to the Dirac equation International Nuclear Information System (INIS) Grosse, H. 1984-01-01 Imposing isospectral invariance for the one dimensional Dirac operator leads to systems of nonlinear partial differential equations. By constructing reflectionless potentials of the Dirac equation we obtain a new type of solitons for a system of modified Korteweg-de Vries equations. (Author) 3. Effects of acceleration through the Dirac sea International Nuclear Information System (INIS) Hacyan, S. 1986-01-01 The effects of acceleration through massive scalar and spin-1/2 fields are investigated. It is shown that the density-of-states factor in a uniformly accelerated frame takes a complicated form, but the energy spectrum exhibits a Bose-Einstein or Fermi-Dirac distribution function. In particular, the Dirac sea shows thermal-like effects 4. Semi-Dirac points in phononic crystals KAUST Repository Zhang, Xiujuan; Wu, Ying 2014-01-01 of rubber, in which the acoustic wave velocity is lower than that in water, the semi-Dirac dispersion can be characterized by an effective medium theory. The effective medium parameters link the semi-Dirac point to a topological transition in the iso 5. LHCbDIRAC as Apache Mesos microservices CERN Multimedia Couturier, Ben 2016-01-01 The LHCb experiment relies on LHCbDIRAC, an extension of DIRAC, to drive its offline computing. This middleware provides a development framework and a complete set of components for building distributed computing systems. These components are currently installed and ran on virtual machines (VM) or bare metal hardware. Due to the increased load of work, high availability is becoming more and more important for the LHCbDIRAC services, and the current installation model is showing its limitations. Apache Mesos is a cluster manager which aims at abstracting heterogeneous physical resources on which various tasks can be distributed thanks to so called "framework". The Marathon framework is suitable for long running tasks such as the DIRAC services, while the Chronos framework meets the needs of cron-like tasks like the DIRAC agents. A combination of the service discovery tool Consul together with HAProxy allows to expose the running containers to the outside world while hiding their dynamic placements. Such an arc... 6. Dirac operators on coset spaces International Nuclear Information System (INIS) Balachandran, A.P.; Immirzi, Giorgio; Lee, Joohan; Presnajder, Peter 2003-01-01 The Dirac operator for a manifold Q, and its chirality operator when Q is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when Q=G/H, where G and H are compact connected Lie groups and G is simple. An elementary discussion of the differential geometric and bundle theoretic aspects of G/H, including its projective modules and complex, Kaehler and Riemannian structures, is presented for this purpose. An attractive feature of our approach is that it transparently shows obstructions to spin- and spin c -structures. When a manifold is spin c and not spin, U(1) gauge fields have to be introduced in a particular way to define spinors, as shown by Avis, Isham, Cahen, and Gutt. Likewise, for manifolds like SU(3)/SO(3), which are not even spin c , we show that SU(2) and higher rank gauge fields have to be introduced to define spinors. This result has potential consequences for string theories if such manifolds occur as D-branes. The spectra and eigenstates of the Dirac operator on spheres S n =SO(n+1)/SO(n), invariant under SO(n+1), are explicitly found. Aspects of our work overlap with the earlier research of Cahen et al 7. Thermal dark matter through the Dirac neutrino portal Science.gov (United States) Batell, Brian; Han, Tao; McKeen, David; Haghi, Barmak Shams Es 2018-04-01 We study a simple model of thermal dark matter annihilating to standard model neutrinos via the neutrino portal. A (pseudo-)Dirac sterile neutrino serves as a mediator between the visible and the dark sectors, while an approximate lepton number symmetry allows for a large neutrino Yukawa coupling and, in turn, efficient dark matter annihilation. The dark sector consists of two particles, a Dirac fermion and complex scalar, charged under a symmetry that ensures the stability of the dark matter. A generic prediction of the model is a sterile neutrino with a large active-sterile mixing angle that decays primarily invisibly. We derive existing constraints and future projections from direct detection experiments, colliders, rare meson and tau decays, electroweak precision tests, and small scale structure observations. Along with these phenomenological tests, we investigate the consequences of perturbativity and scalar mass fine tuning on the model parameter space. A simple, conservative scheme to confront the various tests with the thermal relic target is outlined, and we demonstrate that much of the cosmologically-motivated parameter space is already constrained. We also identify new probes of this scenario such as multibody kaon decays and Drell-Yan production of W bosons at the LHC. 8. 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. 9. Integrating out the Dirac sea Energy Technology Data Exchange (ETDEWEB) Karbstein, Felix 2009-07-08 We introduce a new method for dealing with fermionic quantum field theories amenable to a mean-field-type approximation. In this work we focus on the relativistic Hartree approximation. Our aim is to integrate out the Dirac sea and derive a no-sea effective theory'' with positive energy single particle states only. As the derivation of the no-sea effective theory involves only standard Feynman diagrams, our approach is quite general and not restricted to particular space-time dimensions. We develop and illustrate the approach in the ''large N'' limit of the Gross-Neveu model family in 1+1 dimensions. As the Gross-Neveu model has been intensely studied and several analytical solutions are known for this model, it is an ideal testing ground for our no-sea effective theory approach. The chiral Gross-Neveu model, also referred to as 1+1 dimensional Nambu-Jona-Lasinio model, turns out to be of particular interest. In this case, we explicitly derive a consistent effective theory featuring both elementary ''{pi} meson'' fields and (positive energy) ''quark'' fields, starting from a purely fermionic quantum field theory. In the second part of this work, we apply our approach to the Walecka model in 1+1 and 3+1 dimensions. As the Dirac sea caused considerable difficulties in attempts to base nuclear physics on field theoretic models like the Walecka model, mean-field calculations were typically done without the sea. We confront several of these mean-field theory results with our no-sea effective theory approach. The potential of our approach is twofold. While the no-sea effective theory can be utilized to provide new analytical insights in particular parameter regimes, it also sheds new light on more fundamental issues as the explicit emergence of effective, Dirac-sea induced multi-fermion interactions in an effective theory with positive energy states only. (orig.) 10. Paul Dirac: the purest soul in physics International Nuclear Information System (INIS) Berry, M. 1998-01-01 Paul Dirac published the first of his papers on ''The Quantum Theory of the Electron'' seventy years ago this month. Published in the Proceedings of the Royal Society (London) in February and March 1928, the papers contained one of the greatest leaps of imagination in 20th century physics. The Dirac equation, derived in those papers, is one of the most important equations in physics. Dirac showed that the simplest wave satisfying the requirements of quantum mechanics and relativity was not a simple number but had four components. He found that the logic that led to the theory was, although deeply sophisticated, in a sense beautifully simple. Much later, when someone asked him ''How did you find the Dirac equation?'' he is said to have replied: ''I found it beautiful''. In addition to explaining the magnetic and spin properties of the electron, the equation also predicts the existence of antimatter. Because Dirac was a quiet man - famously quiet, indeed - he is not well known outside physics, although this is slowly changing. In 1995 a plaque to Dirac was unveiled at Westminster Abbey in London and last year Institute of Physics Publishing, which is based in Bristol, named its new building Dirac House. In this article the author recalls the achievements of the greatest physicists of the 20th century. (UK) 11. Dirac fermions in blue-phosphorus International Nuclear Information System (INIS) Li, Yuanchang; Chen, Xiaobin 2014-01-01 We propose that Dirac cones can be engineered in phosphorene with fourfold-coordinated phosphorus atoms. The key is to separate the energy levels of the in-plane (s, p x , and p y ) and out-of-plane (p z ) oribtals through the sp 2 configuration, yielding respective σ- and π-character Dirac cones, and then quench the latter. As a proof-of-principle study, we create σ-character Dirac cones in hydrogenated and fluorinated phosphorene with a honeycomb lattice. The obtained Dirac cones are at K-points, slightly anisotropic, with Fermi velocities of 0.91 and 1.23 times that of graphene along the ΓK and KM direction, and maintain good linearity up to ∼2 eV for holes. A substantive advantage of a σ-character Dirac cone is its convenience in tuning the Dirac gap via in-plane strain. Our findings pave the way for development of high-performance electronic devices based on Dirac materials. (letter) 12. Dirac cones in isogonal hexagonal metallic structures Science.gov (United States) Wang, Kang 2018-03-01 A honeycomb hexagonal metallic lattice is equivalent to a triangular atomic one and cannot create Dirac cones in its electromagnetic wave spectrum. We study in this work the low-frequency electromagnetic band structures in isogonal hexagonal metallic lattices that are directly related to the honeycomb one and show that such structures can create Dirac cones. The band formation can be described by a tight-binding model that allows investigating, in terms of correlations between local resonance modes, the condition for the Dirac cones and the consequence of the third structure tile sustaining an extra resonance mode in the unit cell that induces band shifts and thus nonlinear deformation of the Dirac cones following the wave vectors departing from the Dirac points. We show further that, under structure deformation, the deformations of the Dirac cones result from two different correlation mechanisms, both reinforced by the lattice's metallic nature, which directly affects the resonance mode correlations. The isogonal structures provide new degrees of freedom for tuning the Dirac cones, allowing adjustment of the cone shape by modulating the structure tiles at the local scale without modifying the lattice periodicity and symmetry. 13. New limits on hidden photons from past electron beam dumps International Nuclear Information System (INIS) Andreas, Sarah; Niebuhr, Carsten; Ringwald, Andreas 2012-09-01 Hidden sectors with light extra U(1) gauge bosons, so called hidden photons, have recently attracted some attention because they are a common feature of physics beyond the Standard Model like string theory and SUSY and additionally are phenomenologically of great interest regarding recent astrophysical observations. The hidden photon is already constrained by various laboratory experiments and presently searched for in running as well as upcoming experiments. We summarize the current status of limits on hidden photons from past electron beam dump experiments including two new limits from such experiments at KEK and Orsay that have so far not been considered. All our limits take into account the experimental acceptances obtained from Monte Carlo simulations. 14. New limits on hidden photons from past electron beam dumps Energy Technology Data Exchange (ETDEWEB) Andreas, Sarah; Niebuhr, Carsten; Ringwald, Andreas 2012-09-15 Hidden sectors with light extra U(1) gauge bosons, so called hidden photons, have recently attracted some attention because they are a common feature of physics beyond the Standard Model like string theory and SUSY and additionally are phenomenologically of great interest regarding recent astrophysical observations. The hidden photon is already constrained by various laboratory experiments and presently searched for in running as well as upcoming experiments. We summarize the current status of limits on hidden photons from past electron beam dump experiments including two new limits from such experiments at KEK and Orsay that have so far not been considered. All our limits take into account the experimental acceptances obtained from Monte Carlo simulations. 15. The Dirac-Milne cosmology Science.gov (United States) Benoit-Lévy, Aurélien; Chardin, Gabriel 2014-05-01 We study an unconventional cosmology, in which we investigate the consequences that antigravity would pose to cosmology. We present the main characteristics of the Dirac-Milne Universe, a cosmological model where antimatter has a negative active gravitational mass. In this non-standard Universe, separate domains of matter and antimatter coexist at our epoch without annihilation, separated by a gravitationally induced depletion zone. We show that this cosmology does not require a priori the Dark Matter and Dark Energy components of the standard model of cosmology. Additionally, inflation becomes an unnecessary ingredient. Investigating this model, we show that the classical cosmological tests such as primordial nucleosynthesis, Type Ia supernovæ and Cosmic Microwave Background are surprisingly concordant. 16. The hidden values DEFF Research Database (Denmark) Rasmussen, Birgitte; Jensen, Karsten Klint “The Hidden Values - Transparency in Decision-Making Processes Dealing with Hazardous Activities”. The report seeks to shed light on what is needed to create a transparent framework for political and administrative decisions on the use of GMOs and chemical products. It is our hope that the report... 17. The Dirac equation and its solutions Energy Technology Data Exchange (ETDEWEB) Bagrov, Vladislav G. [Tomsk State Univ., Tomsk (Russian Federation). Dept. of Quantum Field Theroy; Gitman, Dmitry [Sao Paulo Univ. (Brazil). Inst. de Fisica; P.N. Lebedev Physical Institute, Moscow (Russian Federation); Tomsk State Univ., Tomsk (Russian Federation). Faculty of Physics 2013-07-01 The Dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly. In particular, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation. 18. The Dirac equation and its solutions CERN Document Server Bagrov, Vladislav G 2014-01-01 Dirac equations are of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly.In particular, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation. 19. Counter-diabatic driving for Dirac dynamics Science.gov (United States) Fan, Qi-Zhen; Cheng, Xiao-Hang; Chen, Xi 2018-03-01 In this paper, we investigate the fast quantum control of Dirac equation dynamics by counter-diabatic driving, sharing the concept of shortcut to adiabaticity. We systematically calculate the counter-diabatic terms in different Dirac systems, like graphene and trapped ions. Specially, the fast and robust population inversion processes are achieved in Dirac system, taking into account the quantum simulation with trapped ions. In addition, the population transfer between two bands can be suppressed by counter-diabatic driving in graphene system, which might have potential applications in opt-electric devices. 20. Quantum geometry of the Dirac fermions International Nuclear Information System (INIS) Korchemskij, G.P. 1989-01-01 The bosonic path integral formalism is developed for Dirac fermions interacting with a nonabelian gauge field in the D-dimensional Euclidean space-time. The representation for the effective action and correlation functions of interacting fermions as sums over all bosonic paths on the complex projective space CP 2d-1 , (2d=2 [ D 2] is derived where all the spinor structure is absorbed by the one-dimensional Wess-Zumino term. It is the Wess-Zumino term that ensures all necessary properties of Dirac fermions under quantization. i.e., quantized values of the spin, Dirac equation, Fermi statistics. 19 refs 1. The Dirac equation and its solutions International Nuclear Information System (INIS) Bagrov, Vladislav G.; Gitman, Dmitry; P.N. Lebedev Physical Institute, Moscow; Tomsk State Univ., Tomsk 2013-01-01 The Dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly. In particular, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation. 2. Scalar potentials and the Dirac equation International Nuclear Information System (INIS) Bergerhoff, B.; Soff, G. 1994-01-01 The Dirac equation is solved for various types of scalar potentials. Energy eigenvalues and normalized bound-state wave functions are calculated analytically for a scalar 1/r-potential as well as for a mixed scalar and Coulomb 1/r-potential. Also continuum wave functions for positive and negative energies are derived. Similarly, we investigate the solutions of the Dirac equation for a scalar square-well potential. Relativistic wave functions for scalar Yukawa and exponential potentials are determined numerically. Finally, we also discuss solutions of the Dirac equation for scalar linear and quadratic potentials which are frequently used to simulate quark confinement. (orig.) 3. Wigner function for the Dirac oscillator in spinor space International Nuclear Information System (INIS) Ma Kai; Wang Jianhua; Yuan Yi 2011-01-01 The Wigner function for the Dirac oscillator in spinor space is studied in this paper. Firstly, since the Dirac equation is described as a matrix equation in phase space, it is necessary to define the Wigner function as a matrix function in spinor space. Secondly, the matrix form of the Wigner function is proven to support the Dirac equation. Thirdly, by solving the Dirac equation, energy levels and the Wigner function for the Dirac oscillator in spinor space are obtained. (authors) 4. Dirac equation in magnetic-solenoid field Energy Technology Data Exchange (ETDEWEB) Gavrilov, S.P. [Dept. Fisica e Quimica, UNESP, Campus de Guaratingueta (Brazil); Gitman, D.M.; Smirnov, A.A. [Instituto de Fisica, Universidade de Sao Paulo (Brazil) 2004-07-01 We consider the Dirac equation in the magnetic-solenoid field (the field of a solenoid and a collinear uniform magnetic field). For the case of Aharonov-Bohm solenoid, we construct self-adjoint extensions of the Dirac Hamiltonian using von Neumann's theory of deficiency indices. We find self-adjoint extensions of the Dirac Hamiltonian and boundary conditions at the AB solenoid. Besides, for the first time, solutions of the Dirac equation in the magnetic-solenoid field with a finite radius solenoid were found. We study the structure of these solutions and their dependence on the behavior of the magnetic field inside the solenoid. Then we exploit the latter solutions to specify boundary conditions for the magnetic-solenoid field with Aharonov-Bohm solenoid. (orig.) 5. SU(4) proprerties of the Dirac equation International Nuclear Information System (INIS) Linhares, C.A.; Mignaco, J.A. 1985-09-01 The Dirac equation in four dimensions has an intimate connection with the representations of the group SU(4). This connection is shown in detail and subsequent properties are displayed in the continuum as well as in the lattice description [pt 6. New symmetries for the Dirac equation International Nuclear Information System (INIS) Linhares, C.A.; Mignaco, J.A. 1990-01-01 The Dirac equation in four dimension is studied describing fermions, both as 4 x 4 matrices and differential forms. It is discussed in both formalisms its properties under transformations of the group SU(4). (A.C.A.S.) [pt 7. On the level order for Dirac operators International Nuclear Information System (INIS) Grosse, H. 1987-01-01 We start from the Dirac operator for the Coulomb potential and prove within first order perturbation theory that degenerate levels split in a definite way depending on the sign of the Laplacian of the perturbing potential. 9 refs. (Author) 8. Data acquisition software for DIRAC experiment International Nuclear Information System (INIS) Ol'shevskij, V.G.; Trusov, S.V. 2000-01-01 The structure and basic processes of data acquisition software of DIRAC experiment for the measurement of π + π - atom life-time are described. The experiment is running on PS accelerator of CERN. The developed software allows one to accept, record and distribute to consumers up to 3 Mbytes of data in one accelerator supercycle of 14.4 s duration. The described system is used successfully in the DIRAC experiment starting from 1998 year 9. New exact solutions of the Dirac equation International Nuclear Information System (INIS) Bagrov, V.G.; Gitman, D.M.; Zadorozhnyj, V.N.; Lavrov, P.M.; Shapovalov, V.N. 1980-01-01 Search for new exact solutions of the Dirac and Klein-Gordon equations are in progress. Considered are general properties of the Dirac equation solutions for an electron in a purely magnetic field, in combination with a longitudinal magnetic and transverse electric fields. New solutions for the equations of charge motion in an electromagnetic field of axial symmetry and in a nonstationary field of a special form have been found for potentials selected concretely 10. Deuteron stripping reactions using dirac phenomenology Science.gov (United States) Hawk, E. A.; McNeil, J. A. 2001-04-01 In this work deuteron stripping reactions are studied using the distorted wave born approximation employing dirac phenomenological potentials. In 1982 Shepard and Rost performed zero-range dirac phenomenological stripping calculations and found a dramatic reduction in the predicted cross sections when compared with similar nonrelativistic calculations. We extend the earlier work by including full finite range effects as well as the deuteron's internal D-state. Results will be compared with traditional nonrelativistic approaches and experimental data at low energy. 11. Solvable linear potentials in the Dirac equation International Nuclear Information System (INIS) Dominguez-Adame, F.; Gonzalez, M.A. 1990-01-01 The Dirac equation for some linear potentials leading to Schroedinger-like oscillator equations for the upper and lower components of the Dirac spinor have been solved. Energy levels for the bound states appear in pairs, so that both particles and antiparticles may be bound with the same energy. For weak coupling, the spacing between levels is proportional to the coupling constant while in the strong limit those levels are depressed compared to the nonrelativistic ones 12. Leptons as systems of Dirac particles International Nuclear Information System (INIS) Borstnik, N.M.; Kaluza, M. 1988-03-01 Charged leptons are treated as systems of three equal independent Dirac particles in an external static effective potential which has a vector and a scalar term. The potential is constructed to reproduce the experimental mass spectrum of the charged leptons. The Dirac covariant equation for three interacting particles is discussed in order to comment on the magnetic moment of leptons. (author). 9 refs, 2 figs, 4 tabs 13. Dirac equation on a curved surface Energy Technology Data Exchange (ETDEWEB) Brandt, F.T., E-mail: [email protected]; Sánchez-Monroy, J.A., E-mail: [email protected] 2016-09-07 The dynamics of Dirac particles confined to a curved surface is examined employing the thin-layer method. We perform a perturbative expansion to first-order and split the Dirac field into normal and tangential components to the surface. In contrast to the known behavior of second order equations like Schrödinger, Maxwell and Klein–Gordon, we find that there is no geometric potential for the Dirac equation on a surface. This implies that the non-relativistic limit does not commute with the thin-layer method. Although this problem can be overcome when second-order terms are retained in the perturbative expansion, this would preclude the decoupling of the normal and tangential degrees of freedom. Therefore, we propose to introduce a first-order term which rescues the non-relativistic limit and also clarifies the effect of the intrinsic and extrinsic curvatures on the dynamics of the Dirac particles. - Highlights: • The thin-layer method is employed to derive the Dirac equation on a curved surface. • A geometric potential is absent at least to first-order in the perturbative expansion. • The effects of the extrinsic curvature are included to rescue the non-relativistic limit. • The resulting Dirac equation is consistent with the Heisenberg uncertainty principle. 14. The DIRAC Web Portal 2.0 Science.gov (United States) Mathe, Z.; Casajus Ramo, A.; Lazovsky, N.; Stagni, F. 2015-12-01 For many years the DIRAC interware (Distributed Infrastructure with Remote Agent Control) has had a web interface, allowing the users to monitor DIRAC activities and also interact with the system. Since then many new web technologies have emerged, therefore a redesign and a new implementation of the DIRAC Web portal were necessary, taking into account the lessons learnt using the old portal. These new technologies allowed to build a more compact, robust and responsive web interface that enables users to have better control over the whole system while keeping a simple interface. The web framework provides a large set of “applications”, each of which can be used for interacting with various parts of the system. Communities can also create their own set of personalised web applications, and can easily extend already existing ones with a minimal effort. Each user can configure and personalise the view for each application and save it using the DIRAC User Profile service as RESTful state provider, instead of using cookies. The owner of a view can share it with other users or within a user community. Compatibility between different browsers is assured, as well as with mobile versions. In this paper, we present the new DIRAC Web framework as well as the LHCb extension of the DIRAC Web portal. 15. Dirac Fermions in an Antiferromagnetic Semimetal Science.gov (United States) Tang, Peizhe; Zhou, Quan; Xu, Gang; Zhang, Shou-Cheng; Shou-Cheng Zhang's Group Team, Prof. Analogues of the elementary particles have been extensively searched for in condensed matter systems for both scientific interest and technological applications. Recently, massless Dirac fermions were found to emerge as low energy excitations in materials now known as Dirac semimetals. All the currently known Dirac semimetals are nonmagnetic with both time-reversal symmetry and inversion symmetry "". Here we show that Dirac fermions can exist in one type of antiferromagnetic systems, where both and "" are broken but their combination "" is respected. We propose orthorhombic antiferromagnet CuMnAs as a candidate, analyze the robustness of the Dirac points under symmetry protections, and demonstrate its distinctive bulk dispersions as well as the corresponding surface states by ab initio calculations. Our results provide a possible platform to study the interplay of Dirac fermion physics and magnetism. We acknowledge the DOE, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DE-AC02-76SF00515, NSF under Grant No.DMR-1305677 and FAME, one of six centers of STARnet. 16. Hidden neural networks DEFF Research Database (Denmark) Krogh, Anders Stærmose; Riis, Søren Kamaric 1999-01-01 A general framework for hybrids of hidden Markov models (HMMs) and neural networks (NNs) called hidden neural networks (HNNs) is described. The article begins by reviewing standard HMMs and estimation by conditional maximum likelihood, which is used by the HNN. In the HNN, the usual HMM probability...... parameters are replaced by the outputs of state-specific neural networks. As opposed to many other hybrids, the HNN is normalized globally and therefore has a valid probabilistic interpretation. All parameters in the HNN are estimated simultaneously according to the discriminative conditional maximum...... likelihood criterion. The HNN can be viewed as an undirected probabilistic independence network (a graphical model), where the neural networks provide a compact representation of the clique functions. An evaluation of the HNN on the task of recognizing broad phoneme classes in the TIMIT database shows clear... 17. A hidden history OpenAIRE Peppers, Emily 2008-01-01 The Cultural Collections Audit project began at the University of Edinburgh in 2004, searching for hidden treasures in its 'distributed heritage collections' across the university. The objects and collections recorded in the Audit ranged widely from fine art and furniture to historical scientific and teaching equipment and personalia relating to key figures in the university's long tradition of academic excellence. This information was gathered in order to create a central database of informa... 18. Distinguishing Hidden Markov Chains OpenAIRE Kiefer, Stefan; Sistla, A. Prasad 2015-01-01 Hidden Markov Chains (HMCs) are commonly used mathematical models of probabilistic systems. They are employed in various fields such as speech recognition, signal processing, and biological sequence analysis. We consider the problem of distinguishing two given HMCs based on an observation sequence that one of the HMCs generates. More precisely, given two HMCs and an observation sequence, a distinguishing algorithm is expected to identify the HMC that generates the observation sequence. Two HM... 19. Rare-Region-Induced Avoided Quantum Criticality in Disordered Three-Dimensional Dirac and Weyl Semimetals Directory of Open Access Journals (Sweden) J. H. Pixley 2016-06-01 Full Text Available We numerically study the effect of short-ranged potential disorder on massless noninteracting three-dimensional Dirac and Weyl fermions, with a focus on the question of the proposed (and extensively theoretically studied quantum critical point separating semimetal and diffusive-metal phases. We determine the properties of the eigenstates of the disordered Dirac Hamiltonian (H and exactly calculate the density of states (DOS near zero energy, using a combination of Lanczos on H^{2} and the kernel polynomial method on H. We establish the existence of two distinct types of low-energy eigenstates contributing to the disordered density of states in the weak-disorder semimetal regime. These are (i typical eigenstates that are well described by linearly dispersing perturbatively dressed Dirac states and (ii nonperturbative rare eigenstates that are weakly dispersive and quasilocalized in the real-space regions with the largest (and rarest local random potential. Using twisted boundary conditions, we are able to systematically find and study these two (essentially independent types of eigenstates. We find that the Dirac states contribute low-energy peaks in the finite-size DOS that arise from the clean eigenstates which shift and broaden in the presence of disorder. On the other hand, we establish that the rare quasilocalized eigenstates contribute a nonzero background DOS which is only weakly energy dependent near zero energy and is exponentially small at weak disorder. We also find that the expected semimetal to diffusive-metal quantum critical point is converted to an avoided quantum criticality that is “rounded out” by nonperturbative effects, with no signs of any singular behavior in the DOS at the energy of the clean Dirac point. However, the crossover effects of the avoided (or hidden criticality manifest themselves in a so-called quantum critical fan region away from the Dirac energy. We discuss the implications of our results for 20. Viability of Dirac phase leptogenesis International Nuclear Information System (INIS) Anisimov, Alexey; Blanchet, Steve; Di Bari, Pasquale 2008-01-01 We discuss the conditions for a non-vanishing Dirac phase δ and mixing angle θ 13 , sources of CP violation in neutrino oscillations, to be uniquely responsible for the observed matter–antimatter asymmetry of the Universe through leptogenesis. We show that this scenario, that we call δ-leptogenesis, is viable when the degenerate limit for the heavy right-handed (RH) neutrino spectrum is considered. We derive an interesting joint condition on sinθ 13 and the absolute neutrino mass scale that can be tested in future neutrino oscillation experiments. In the limit of the hierarchical heavy RH neutrino spectrum, we strengthen the previous result that δ-leptogenesis is only very marginally allowed, even when the production from the two heavier RH neutrinos is taken into account. An improved experimental upper bound on sinθ 13 and/or an account of quantum kinetic effects could completely rule out this option in the future. Therefore, δ-leptogenesis can be also regarded as motivation for models with degenerate heavy neutrino spectrum 1. Naturally light Dirac neutrino in Left-Right Symmetric Model Energy Technology Data Exchange (ETDEWEB) Borah, Debasish [Department of Physics, Indian Institute of Technology Guwahati, Assam-781039 (India); Dasgupta, Arnab, E-mail: [email protected], E-mail: [email protected] [Institute of Physics, HBNI, Sachivalaya Marg, Bhubaneshwar-751005 (India) 2017-06-01 We study the possibility of generating tiny Dirac masses of neutrinos in Left-Right Symmetric Model (LRSM) without requiring the existence of any additional symmetries. The charged fermions acquire masses through a universal seesaw mechanism due to the presence of additional vector like fermions. The neutrinos acquire a one-loop Dirac mass from the same additional vector like charged leptons without requiring any additional discrete symmetries. The model can also be extended by an additional Z {sub 2} symmetry in order to have a scotogenic version of this scenario predicting a stable dark matter candidate. We show that the latest Planck upper bound on the effective number of relativistic degrees of freedom N {sub eff}=3.15 ± 0.23 tightly constrains the right sector gauge boson masses to be heavier than 3.548 TeV . This bound on gauge boson mass also affects the allowed values of right scalar doublet dark matter mass from the requirement of satisfying the Planck bound on dark matter relic abundance. We also discuss the possible implications of such a scenario in charged lepton flavour violation and generating observable electric dipole moment of leptons. 2. First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals KAUST Repository Mei, Jun; Wu, Ying; Chan, C. T.; Zhang, Zhao-Qing 2012-01-01 By using the k•p method, we propose a first-principles theory to study the linear dispersions in phononic and photonic crystals. The theory reveals that only those linear dispersions created by doubly degenerate states can be described by a reduced Hamiltonian that can be mapped into the Dirac Hamiltonian and possess a Berry phase of -π. Linear dispersions created by triply degenerate states cannot be mapped into the Dirac Hamiltonian and carry no Berry phase, and, therefore should be called Dirac-like cones. Our theory is capable of predicting accurately the linear slopes of Dirac and Dirac-like cones at various symmetry points in a Brillouin zone, independent of frequency and lattice structure. © 2012 American Physical Society. 3. First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals KAUST Repository Mei, Jun 2012-07-24 By using the k•p method, we propose a first-principles theory to study the linear dispersions in phononic and photonic crystals. The theory reveals that only those linear dispersions created by doubly degenerate states can be described by a reduced Hamiltonian that can be mapped into the Dirac Hamiltonian and possess a Berry phase of -π. Linear dispersions created by triply degenerate states cannot be mapped into the Dirac Hamiltonian and carry no Berry phase, and, therefore should be called Dirac-like cones. Our theory is capable of predicting accurately the linear slopes of Dirac and Dirac-like cones at various symmetry points in a Brillouin zone, independent of frequency and lattice structure. © 2012 American Physical Society. 4. Localization of hidden Chua's attractors International Nuclear Information System (INIS) Leonov, G.A.; Kuznetsov, N.V.; Vagaitsev, V.I. 2011-01-01 The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. In the present Letter for localization of hidden attractors of Chua's circuit it is suggested to use a special analytical-numerical algorithm. -- Highlights: → There are hidden attractors: basin doesn't contain neighborhoods of equilibria. → Hidden attractors cannot be reached by trajectory from neighborhoods of equilibria. → We suggested special procedure for localization of hidden attractors. → We discovered hidden attractor in Chua's system, L. Chua in his work didn't expect this. 5. Dirac Sea and its Evolution Science.gov (United States) Volfson, Boris 2013-09-01 The hypothesis of transition from a chaotic Dirac Sea, via highly unstable positronium, into a Simhony Model of stable face-centered cubic lattice structure of electrons and positrons securely bound in vacuum space, is considered. 13.75 Billion years ago, the new lattice, which, unlike a Dirac Sea, is permeable by photons and phonons, made the Universe detectable. Many electrons and positrons ended up annihilating each other producing energy quanta and neutrino-antineutrino pairs. The weak force of the electron-positron crystal lattice, bombarded by the chirality-changing neutrinos, may have started capturing these neutrinos thus transforming from cubic crystals into a quasicrystal lattice. Unlike cubic crystal lattice, clusters of quasicrystals are "slippery" allowing the formation of centers of local torsion, where gravity condenses matter into galaxies, stars and planets. In the presence of quanta, in a quasicrystal lattice, the Majorana neutrinos' rotation flips to the opposite direction causing natural transformations in a category comprised of three components; two others being positron and electron. In other words, each particle-antiparticle pair "e-" and "e+", in an individual crystal unit, could become either a quasi- component "e- ve e+", or a quasi- component "e+ - ve e-". Five-to-six six billion years ago, a continuous stimulation of the quasicrystal aetherial lattice by the same, similar, or different, astronomical events, could have triggered Hebbian and anti-Hebbian learning processes. The Universe may have started writing script into its own aether in a code most appropriate for the quasicrystal aether "hardware": Eight three-dimensional "alphabet" characters, each corresponding to the individual quasi-crystal unit shape. They could be expressed as quantum Turing machine qubits, or, alternatively, in a binary code. The code numerals could contain terminal and nonterminal symbols of the Chomsky's hierarchy, wherein, the showers of quanta, forming the 6. Hidden Liquidity: Determinants and Impact OpenAIRE Gökhan Cebiroglu; Ulrich Horst 2012-01-01 We cross-sectionally analyze the presence of aggregated hidden depth and trade volume in the S&P 500 and identify its key determinants. We find that the spread is the main predictor for a stock’s hidden dimension, both in terms of traded and posted liquidity. Our findings moreover suggest that large hidden orders are associated with larger transaction costs, higher price impact and increased volatility. In particular, as large hidden orders fail to attract (latent) liquidity to the market, ... 7. Semi-Dirac points in phononic crystals KAUST Repository Zhang, Xiujuan 2014-01-01 A semi-Dirac cone refers to a peculiar type of dispersion relation that is linear along the symmetry line but quadratic in the perpendicular direction. It was originally discovered in electron systems, in which the associated quasi-particles are massless along one direction, like those in graphene, but effective-mass-like along the other. It was reported that a semi-Dirac point is associated with the topological phase transition between a semi-metallic phase and a band insulator. Very recently, the classical analogy of a semi-Dirac cone has been reported in an electromagnetic system. Here, we demonstrate that, by accidental degeneracy, two-dimensional phononic crystals consisting of square arrays of elliptical cylinders embedded in water are also able to produce the particular dispersion relation of a semi-Dirac cone in the center of the Brillouin zone. A perturbation method is used to evaluate the linear slope and to affirm that the dispersion relation is a semi-Dirac type. If the scatterers are made of rubber, in which the acoustic wave velocity is lower than that in water, the semi-Dirac dispersion can be characterized by an effective medium theory. The effective medium parameters link the semi-Dirac point to a topological transition in the iso-frequency surface of the phononic crystal, in which an open hyperbola is changed into a closed ellipse. This topological transition results in drastic change in wave manipulation. On the other hand, the theory also reveals that the phononic crystal is a double-zero-index material along the x-direction and photonic-band-edge material along the perpendicular direction (y-direction). If the scatterers are made of steel, in which the acoustic wave velocity is higher than that in water, the effective medium description fails, even though the semi-Dirac dispersion relation looks similar to that in the previous case. Therefore different wave transport behavior is expected. The semi-Dirac points in phononic crystals described in 8. Strain engineering of Dirac cones in graphyne Energy Technology Data Exchange (ETDEWEB) Wang, Gaoxue; Kumar, Ashok; Pandey, Ravindra, E-mail: [email protected] [Department of Physics, Michigan Technological University, Houghton, Michigan 49931 (United States); Si, Mingsu [Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000 (China) 2014-05-26 6,6,12-graphyne, one of the two-dimensional carbon allotropes with the rectangular lattice structure, has two kinds of non-equivalent anisotropic Dirac cones in the first Brillouin zone. We show that Dirac cones can be tuned independently by the uniaxial compressive strain applied to graphyne, which induces n-type and p-type self-doping effect, by shifting the energy of the Dirac cones in the opposite directions. On the other hand, application of the tensile strain results into a transition from gapless to finite gap system for the monolayer. For the AB-stacked bilayer, the results predict tunability of Dirac-cones by in-plane strains as well as the strain applied perpendicular to the plane. The group velocities of the Dirac cones show enhancement in the resistance anisotropy for bilayer relative to the case of monolayer. Such tunable and direction-dependent electronic properties predicted for 6,6,12-graphyne make it to be competitive for the next-generation electronic devices at nanoscale. 9. Interlayer magnetoresistance in multilayer Dirac electron systems: motion and merging of Dirac cones International Nuclear Information System (INIS) Assili, M; Haddad, S 2013-01-01 We theoretically study the effect of the motion and the merging of Dirac cones on the interlayer magnetoresistance in multilayer graphene-like systems. This merging, which can be induced by a uniaxial strain, gives rise in a monolayer Dirac electron system to a topological transition from a semi-metallic phase to an insulating phase whereby Dirac points disappear. Based on a universal Hamiltonian, proposed to describe the motion and the merging of Dirac points in two-dimensional Dirac electron crystals, we calculate the interlayer conductivity of a stack of deformed graphene-like layers using the Kubo formula in the quantum limit where only the contribution of the n = 0 Landau level is relevant. A crossover from a negative to a positive interlayer magnetoresistance is found to take place as the merging is approached. This sign change of the magnetoresistance can also result from a coupling between the Dirac valleys, which is enhanced as the magnetic field amplitude increases. Our results describe the behavior of the magnetotransport in the organic conductor α-(BEDT) 2 I 3 and in a stack of deformed graphene-like systems. The latter can be simulated by optical lattices or microwave experiments in which the merging of Dirac cones can be observed. (paper) 10. Interlayer magnetoresistance in multilayer Dirac electron systems: motion and merging of Dirac cones Science.gov (United States) Assili, M.; Haddad, S. 2013-09-01 We theoretically study the effect of the motion and the merging of Dirac cones on the interlayer magnetoresistance in multilayer graphene-like systems. This merging, which can be induced by a uniaxial strain, gives rise in a monolayer Dirac electron system to a topological transition from a semi-metallic phase to an insulating phase whereby Dirac points disappear. Based on a universal Hamiltonian, proposed to describe the motion and the merging of Dirac points in two-dimensional Dirac electron crystals, we calculate the interlayer conductivity of a stack of deformed graphene-like layers using the Kubo formula in the quantum limit where only the contribution of the n = 0 Landau level is relevant. A crossover from a negative to a positive interlayer magnetoresistance is found to take place as the merging is approached. This sign change of the magnetoresistance can also result from a coupling between the Dirac valleys, which is enhanced as the magnetic field amplitude increases. Our results describe the behavior of the magnetotransport in the organic conductor α-(BEDT)2I3 and in a stack of deformed graphene-like systems. The latter can be simulated by optical lattices or microwave experiments in which the merging of Dirac cones can be observed. 11. Interlayer magnetoresistance in multilayer Dirac electron systems: motion and merging of Dirac cones. Science.gov (United States) Assili, M; Haddad, S 2013-09-11 We theoretically study the effect of the motion and the merging of Dirac cones on the interlayer magnetoresistance in multilayer graphene-like systems. This merging, which can be induced by a uniaxial strain, gives rise in a monolayer Dirac electron system to a topological transition from a semi-metallic phase to an insulating phase whereby Dirac points disappear. Based on a universal Hamiltonian, proposed to describe the motion and the merging of Dirac points in two-dimensional Dirac electron crystals, we calculate the interlayer conductivity of a stack of deformed graphene-like layers using the Kubo formula in the quantum limit where only the contribution of the n = 0 Landau level is relevant. A crossover from a negative to a positive interlayer magnetoresistance is found to take place as the merging is approached. This sign change of the magnetoresistance can also result from a coupling between the Dirac valleys, which is enhanced as the magnetic field amplitude increases. Our results describe the behavior of the magnetotransport in the organic conductor α-(BEDT)2I3 and in a stack of deformed graphene-like systems. The latter can be simulated by optical lattices or microwave experiments in which the merging of Dirac cones can be observed. 12. Dirac operator on spaces with conical singularities International Nuclear Information System (INIS) Chou, A.W. 1982-01-01 The Dirac operator on compact spaces with conical singularities is studied via the separation of variables formula and the functional calculus of the Dirac Laplacian on the cone. A Bochner type vanishing theorem which gives topological obstructions to the existence of non-negative scalar curvature k greater than or equal to 0 in the singular case is proved. An index formula relating the index of the Dirac operator to the A-genus and Eta-invariant similar to that of Atiyah-Patodi-Singer is obtained. In an appendix, manifolds with boundary with non-negative scalar curvature k greater than or equal to 0 are studied, and several new results on constructing complete metrics with k greater than or equal to on them are obtained 13. LHCb: Monitoring the DIRAC Distribution System CERN Multimedia Nandakumar, R; Santinelli, R 2009-01-01 DIRAC is the LHCb gateway to any computing grid infrastructure (currently supporting WLCG) and is intended to reliably run large data mining activities. The DIRAC system consists of various services (which wait to be contacted to perform actions) and agents (which carry out periodic activities) to direct jobs as required. An important part of ensuring the reliability of the infrastructure is the monitoring and logging of these DIRAC distributed systems. The monitoring is done collecting information from two sources - one is from pinging the services or by keeping track of the regular heartbeats of the agents, and the other from the analysis of the error messages generated by both agents and services and collected by the logging system. This allows us to ensure that he components are running properly and to collect useful information regarding their operations. The process status monitoring is displayed using the SLS sensor mechanism which also automatically allows one to plot various quantities and also keep ... 14. DIRAC - Distributed Infrastructure with Remote Agent Control CERN Document Server Tsaregorodtsev, A; Closier, J; Frank, M; Gaspar, C; van Herwijnen, E; Loverre, F; Ponce, S; Graciani Diaz, R.; Galli, D; Marconi, U; Vagnoni, V; Brook, N; Buckley, A; Harrison, K; Schmelling, M; Egede, U; Bogdanchikov, A; Korolko, I; Washbrook, A; Palacios, J P; Klous, S; Saborido, J J; Khan, A; Pickford, A; Soroko, A; Romanovski, V; Patrick, G N; Kuznetsov, G; Gandelman, M 2003-01-01 This paper describes DIRAC, the LHCb Monte Carlo production system. DIRAC has a client/server architecture based on: Compute elements distributed among the collaborating institutes; Databases for production management, bookkeeping (the metadata catalogue) and software configuration; Monitoring and cataloguing services for updating and accessing the databases. Locally installed software agents implemented in Python monitor the local batch queue, interrogate the production database for any outstanding production requests using the XML-RPC protocol and initiate the job submission. The agent checks and, if necessary, installs any required software automatically. After the job has processed the events, the agent transfers the output data and updates the metadata catalogue. DIRAC has been successfully installed at 18 collaborating institutes, including the DataGRID, and has been used in recent Physics Data Challenges. In the near to medium term future we must use a mixed environment with different types of grid mid... 15. DIRAC - The Distributed MC Production and Analysis for LHCb CERN Document Server Tsaregorodtsev, A 2004-01-01 DIRAC is the LHCb distributed computing grid infrastructure for MC production and analysis. Its architecture is based on a set of distributed collaborating services. The service decomposition broadly follows the ARDA project proposal, allowing for the possibility of interchanging the EGEE/ARDA and DIRAC components in the future. Some components developed outside the DIRAC project are already in use as services, for example the File Catalog developed by the AliEn project. An overview of the DIRAC architecture will be given, in particular the recent developments to support user analysis. The main design choices will be presented. One of the main design goals of DIRAC is the simplicity of installation, configuring and operation of various services. This allows all the DIRAC resources to be easily managed by a single Production Manager. The modular design of the DIRAC components allows its functionality to be easily extended to include new computing and storage elements or to handle new tasks. The DIRAC system al... 16. Cloud flexibility using DIRAC interware International Nuclear Information System (INIS) Albor, Víctor Fernandez; Miguelez, Marcos Seco; Silva, Juan Jose Saborido; Pena, Tomas Fernandez; Muñoz, Victor Mendez; Diaz, Ricardo Graciani 2014-01-01 Communities of different locations are running their computing jobs on dedicated infrastructures without the need to worry about software, hardware or even the site where their programs are going to be executed. Nevertheless, this usually implies that they are restricted to use certain types or versions of an Operating System because either their software needs an definite version of a system library or a specific platform is required by the collaboration to which they belong. On this scenario, if a data center wants to service software to incompatible communities, it has to split its physical resources among those communities. This splitting will inevitably lead to an underuse of resources because the data centers are bound to have periods where one or more of its subclusters are idle. It is, in this situation, where Cloud Computing provides the flexibility and reduction in computational cost that data centers are searching for. This paper describes a set of realistic tests that we ran on one of such implementations. The test comprise software from three different HEP communities (Auger, LHCb and QCD phenomelogists) and the Parsec Benchmark Suite running on one or more of three Linux flavors (SL5, Ubuntu 10.04 and Fedora 13). The implemented infrastructure has, at the cloud level, CloudStack that manages the virtual machines (VM) and the hosts on which they run, and, at the user level, the DIRAC framework along with a VM extension that will submit, monitorize and keep track of the user jobs and also requests CloudStack to start or stop the necessary VM's. In this infrastructure, the community software is distributed via the CernVM-FS, which has been proven to be a reliable and scalable software distribution system. With the resulting infrastructure, users are allowed to send their jobs transparently to the Data Center. The main purpose of this system is the creation of flexible cluster, multiplatform with an scalable method for software distribution for 17. Cloud flexibility using DIRAC interware Science.gov (United States) Fernandez Albor, Víctor; Seco Miguelez, Marcos; Fernandez Pena, Tomas; Mendez Muñoz, Victor; Saborido Silva, Juan Jose; Graciani Diaz, Ricardo 2014-06-01 Communities of different locations are running their computing jobs on dedicated infrastructures without the need to worry about software, hardware or even the site where their programs are going to be executed. Nevertheless, this usually implies that they are restricted to use certain types or versions of an Operating System because either their software needs an definite version of a system library or a specific platform is required by the collaboration to which they belong. On this scenario, if a data center wants to service software to incompatible communities, it has to split its physical resources among those communities. This splitting will inevitably lead to an underuse of resources because the data centers are bound to have periods where one or more of its subclusters are idle. It is, in this situation, where Cloud Computing provides the flexibility and reduction in computational cost that data centers are searching for. This paper describes a set of realistic tests that we ran on one of such implementations. The test comprise software from three different HEP communities (Auger, LHCb and QCD phenomelogists) and the Parsec Benchmark Suite running on one or more of three Linux flavors (SL5, Ubuntu 10.04 and Fedora 13). The implemented infrastructure has, at the cloud level, CloudStack that manages the virtual machines (VM) and the hosts on which they run, and, at the user level, the DIRAC framework along with a VM extension that will submit, monitorize and keep track of the user jobs and also requests CloudStack to start or stop the necessary VM's. In this infrastructure, the community software is distributed via the CernVM-FS, which has been proven to be a reliable and scalable software distribution system. With the resulting infrastructure, users are allowed to send their jobs transparently to the Data Center. The main purpose of this system is the creation of flexible cluster, multiplatform with an scalable method for software distribution for several 18. Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations International Nuclear Information System (INIS) Esteban, M.J.; Georgiev, V.; Sere, E. 1995-01-01 The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic field. We prove the existence of soliton-like solutions of Maxwell-Dirac in (3+1)-Minkowski space-time. The solutions obtained are regular, stationary in time, and localized in space. They are found by a variational method, as critical points of an energy functional. This functional is strongly indefinite and presents a lack of compactness. We also find soliton-like solutions for the Klein-Gordon-Dirac system, arising in the Yukawa model. (author). 32 refs 19. All-Metallic Vertical Transistors Based on Stacked Dirac Materials OpenAIRE Wang, Yangyang; Ni, Zeyuan; Liu, Qihang; Quhe, Ruge; Zheng, Jiaxin; Ye, Meng; Yu, Dapeng; Shi, Junjie; Yang, Jinbo; Lu, Jing 2014-01-01 It is an ongoing pursuit to use metal as a channel material in a field effect transistor. All metallic transistor can be fabricated from pristine semimetallic Dirac materials (such as graphene, silicene, and germanene), but the on/off current ratio is very low. In a vertical heterostructure composed by two Dirac materials, the Dirac cones of the two materials survive the weak interlayer van der Waals interaction based on density functional theory method, and electron transport from the Dirac ... 20. On Huygens' principle for Dirac operators associated to electromagnetic fields Directory of Open Access Journals (Sweden) CHALUB FABIO A.C.C. 2001-01-01 Full Text Available We study the behavior of massless Dirac particles, i.e., solutions of the Dirac equation with m = 0 in the presence of an electromagnetic field. Our main result (Theorem 1 is that for purely real or imaginary fields any Huygens type (in Hadamard's sense Dirac operators is equivalent to the free Dirac operator, equivalence given by changes of variables and multiplication (right and left by nonzero functions. 1. Dirac equations for generalised Yang-Mills systems International Nuclear Information System (INIS) Lechtenfeld, O.; Nahm, W.; Tchrakian, D.H. 1985-06-01 We present Dirac equations in 4p dimensions for the generalised Yang-Mills (GYM) theories introduced earlier. These Dirac equations are related to the self-duality equations of the GYM and are checked to be elliptic in a 'BPST' background. In this background these Dirac equations are integrated exactly. The possibility of imposing supersymmetry in the GYM-Dirac system is investigated, with negative results. (orig.) 2. A framework for unified Dirac gauginos Directory of Open Access Journals (Sweden) Benakli Karim 2017-01-01 Full Text Available We identify the Minimal Dirac Gaugino Supersymmetric Standard Model (MDGSSM as the minimal field content with Dirac gauginos allowing unification of gauge coupling. We stress that its parameter space describes also other most popular models as the MSSM, NMSSM and MRSSM. We discuss the generation of trilinear couplings in models of gauge mediation that has been overlooked in the past. We study the different source of Higgs mixings and constraints from the ƿ parameter. Finally, we provide new experimental limits on the masses of the scalar octets. 3. Dirac particle tunneling from black rings International Nuclear Information System (INIS) Jiang Qingquan 2008-01-01 Recent research shows that Hawking radiation can be treated as a quantum tunneling process, and Hawking temperatures of Dirac particles across the horizon of a black hole can be correctly recovered via the fermion tunneling method. In this paper, motivated by the fermion tunneling method, we attempt to apply the analysis to derive Hawking radiation of Dirac particles via tunneling from black ring solutions of 5-dimensional Einstein-Maxwell-dilaton gravity theory. Finally, it is interesting to find that, as in the black hole case, fermion tunneling can also result in correct Hawking temperatures for the rotating neutral, dipole, and charged black rings. 4. Kapitza–Dirac effect with traveling waves International Nuclear Information System (INIS) Hayrapetyan, Armen G; Götte, Jörg B; Grigoryan, Karen K; Petrosyan, Rubik G 2015-01-01 We report on the possibility of diffracting electrons from light waves traveling inside a dielectric medium. We show that, in the frame of reference which moves with the group velocity of light, the traveling wave acts as a stationary diffraction grating from which electrons can diffract, similar to the conventional Kapitza–Dirac effect. To characterize the Kapitza–Dirac effect with traveling light waves, we make use of the Hamiltonian Analogy between electron optics and quantum mechanics and apply the Helmholtz–Kirchhoff theory of diffraction. (fast track communication) 5. Are Dirac electrons faster than light? International Nuclear Information System (INIS) De Angelis, G.F. 1986-01-01 This paper addresses the problem of path integral solutions of the Dirac equation. The path integral construction of the Dirac propagator which extends Fynman's checkerboard rule in more than one space dimension is discussed. A distinguished feature of such extension is the fact that the speed of a relativistic electron is actually greater than the speed of light when the space has more than one dimension. A technique employed in obtaining an extension to higher space dimension is described which consists in comparing continuity equations of quantum mechanical origin with forward Kolmogorov equations for suitable chosen classes of random processes 6. Hidden Risk Factors for Women Science.gov (United States) ... A.S.T. Quiz Hidden Stroke Risk Factors for Women Updated:Nov 22,2016 Excerpted from "What Women Need To Know About The Hidden Risk Factors ... 2012) This year, more than 100,000 U.S. women under 65 will have a stroke. Stroke is ... 7. New exact solutions of the Dirac equation. 8 International Nuclear Information System (INIS) Bagrov, V.G.; Gitman, D.M.; Zadorozhnyj, V.N.; Sukhomlin, N.B.; Shapovalov, V.N. 1978-01-01 The paper continues the investigation into the exact solutions of the Dirac, Klein-Gordon, and Lorentz equations for a charge in an external electromagnetic field. The fields studied do not allow for separation of variables in the Dirac equation, but solutions to the Dirac equation are obtained 8. Kondo effect in three-dimensional Dirac and Weyl systems NARCIS (Netherlands) Mitchell, Andrew K.; Fritz, Lars 2015-01-01 Magnetic impurities in three-dimensional Dirac and Weyl systems are shown to exhibit a fascinatingly diverse range of Kondo physics, with distinctive experimental spectroscopic signatures. When the Fermi level is precisely at the Dirac point, Dirac semimetals are in fact unlikely candidates for a 9. Dirac cones beyond the honeycomb lattice : a symmetry based approach NARCIS (Netherlands) Miert, G. van; de Morais Smith, Cristiane 2016-01-01 Recently, several new materials exhibiting massless Dirac fermions have been proposed. However, many of these do not have the typical graphene honeycomb lattice, which is often associated with Dirac cones. Here, we present a classification of these different two-dimensional Dirac systems based on 10. SARAH 3.2: Dirac gauginos, UFO output, and more Science.gov (United States) Staub, Florian 2013-07-01 . Nature of problem: To use Madgraph for new models it is necessary to provide the corresponding model files which include all information about the interactions of the model. However, the derivation of the vertices for a given model and putting those into model files which can be used with Madgraph is usually very time consuming. Dirac gauginos are not present in the minimal supersymmetric standard model (MSSM) or many extensions of it. Dirac mass terms for vector superfields lead to new structures in the supersymmetric (SUSY) Lagrangian (bilinear mass term between gaugino and matter fermion as well as new D-terms) and modify also the SUSY renormalization group equations (RGEs). The Dirac character of gauginos can change the collider phenomenology. In addition, they come with an extended Higgs sector for which a precise calculation of the 1-loop masses has not happened so far. Solution method: SARAH calculates the complete Lagrangian for a given model whose gauge sector can be any direct product of SU(N) gauge groups. The chiral superfields can transform as any, irreducible representation with respect to these gauge groups and it is possible to handle an arbitrary number of symmetry breakings or particle rotations. Also the gauge fixing is automatically added. Using this information, SARAH derives all vertices for a model. These vertices can be exported to model files in the UFO which is supported by Madgraph and other codes like GoSam, MadAnalysis or ALOHA. The user can also study models with Dirac gauginos. In that case SARAH includes all possible terms in the Lagrangian stemming from the new structures and can also calculate the RGEs. The entire impact of these terms is then taken into account in the output of SARAH to UFO, CalcHep, WHIZARD, FeynArts and SPheno. Reasons for new version: SARAH provides, with this version, the possibility of creating model files in the UFO format. The UFO format is supposed to become a standard format for model files which should be 11. Sterile neutrino, hidden dark matter and their cosmological signatures International Nuclear Information System (INIS) Das, Subinoy 2012-01-01 Though thermal dark matter has been the central idea behind the dark matter candidates, it is highly possible that dark matter of the universe is non-thermal in origin or it might be in thermal contact with some hidden or dark sector but not with standard model. Here we explore the cosmological bounds as well as the signatures on two types of non-thermal dark matter candidates. First we discuss a hidden dark matter with almost no interaction (or very feeble) with standard model particles so that it is not in thermal contact with visible sector but we assume it is thermalized with in a hidden sector due to some interaction. While encompassing the standard cold WIMP scenario, we do not require the freeze-out process to be non-relativistic. Rather, freeze-out may also occur when dark matter particles are semi-relativistic or relativistic. Especially we focus on the warm dark matter scenario in this set up and find the constraints on the warm dark matter mass, cross-section and hidden to visible sector temperature ratio which accounts for the observed dark-matter density, satisfies the Tremaine-Gunn bound on dark-matter phase space density and has a free-streaming length consistent with cosmological constraints on the matter power spectrum. Our method can also be applied to keV sterile neutrino dark matter which is not thermalized with standard model but is thermalized with in a dark sector. The second part of this proceeding focuses on an exotic dark matter candidate which arises from the existence of eV mass sterile neutrino through a late phase transition. Due to existence of a strong scalar force the light sterile states get trapped into stable degenerate micro nuggets. We find that its signature in matter power spectra is close to a warm dark matter candidate. 12. Hidden solution to the μ/Bμ problem in gauge mediation International Nuclear Information System (INIS) Roy, Tuhin S.; Schmaltz, Martin 2008-01-01 We propose a solution to the μ/B μ problem in gauge mediation. The novel feature of our solution is that it uses dynamics of the hidden sector, which is often present in models with dynamical supersymmetry breaking. We give an explicit example model of gauge mediation where a very simple messenger sector generates both μ and B μ at one loop. The usual problem, that B μ is then too large, is solved by strong renormalization effects from the hidden sector which suppress B μ relative to μ. Our mechanism relies on an assumption about the signs of certain incalculable anomalous dimensions in the hidden sector. Making these assumptions not only allows us to solve the μ/B μ problem but also leads to a characteristic superpartner spectrum which would be a smoking gun signal for our mechanism. 13. Dirac's aether in relativistic quantum mechanics International Nuclear Information System (INIS) Petroni, N.C.; Bari Univ.; Vigier, J.P. 1984-01-01 The paper concerns Dirac's aether model, based on a stochastic covariant distribution of subquantum motions. Stochastic derivation of the relativistic quantum equations; deterministic nonlocal interpretation of the Aspect-Rapisarda experiments on the EPR paradox; and photon interference with itself; are all discussed. (U.K.) 14. Dirac's minimum degree condition restricted to claws NARCIS (Netherlands) Broersma, Haitze J.; Ryjacek, Z.; Schiermeyer, I. 1997-01-01 Let G be a graph on n 3 vertices. Dirac's minimum degree condition is the condition that all vertices of G have degree at least . This is a well-known sufficient condition for the existence of a Hamilton cycle in G. We give related sufficiency conditions for the existence of a Hamilton cycle or a 15. On the Dirac groups of rank n International Nuclear Information System (INIS) Ferreira, P.L.; Alcaras, J.A.C. 1980-01-01 The group theoretical properties of the Dirac groups of rank n are discussed together with the properties and construction of their IR's. The cases n even and n odd show distinct features. Furthermore, for n odd, the cases n=4K+1 and n=4K+3 exhibit some different properties too. (Author) [pt 16. Higher dimensional supersymmetric quantum mechanics and Dirac ... Indian Academy of Sciences (India) We exhibit the supersymmetric quantum mechanical structure of the full 3+1 dimensional Dirac equation considering mass' as a function of coordinates. Its usefulness in solving potential problems is discussed with specific examples. We also discuss the physical' significance of the supersymmetric states in this formalism. 17. Applications of Dirac's Delta Function in Statistics Science.gov (United States) Khuri, Andre 2004-01-01 The Dirac delta function has been used successfully in mathematical physics for many years. The purpose of this article is to bring attention to several useful applications of this function in mathematical statistics. Some of these applications include a unified representation of the distribution of a function (or functions) of one or several… 18. On Kaehler's geometric description of dirac fields International Nuclear Information System (INIS) Goeckeler, M.; Joos, H. 1983-12-01 A differential geometric generalization of the Dirac equation due to E. Kaehler seems to be an appropriate starting point for the lattice approximation of matter fields. It is the purpose of this lecture to illustrate several aspects of this approach. (orig./HSI) 19. SU(4) properties of the Dirac equation International Nuclear Information System (INIS) Linhares, C.A.; Mignaco, J.A. 1988-01-01 The Dirac equation in four dimensions has an intimate connection with the representations of the group SU(4). This connection is shown in detail and subsequente properties are displayed in the continuum as well as in the lattice description. (author) [pt 20. The Dirac operator on the Fuzzy sphere International Nuclear Information System (INIS) Grosse, H. 1994-01-01 We introduce the Fuzzy analog of spinor bundles over the sphere on which the non-commutative analog of the Dirac operator acts. We construct the complete set of eigenstates including zero modes. In the commutative limit we recover known results. (authors) 1. Mass and oscillations of Dirac neutrinos International Nuclear Information System (INIS) Collot, J. 1989-01-01 In the most economical extension of the standard model, we have presented the theory of massive Dirac neutrinos. We have particularly emphasized that, in this model, a complete analogy between quarks and leptons can be erected and predicts neutrino flavor oscillations. We have reviewed the last experimental results concerning kinetic neutrino mass experiments and neutrino oscillation investigations 2. First level trigger of the DIRAC experiment International Nuclear Information System (INIS) Afanas'ev, L.G.; Karpukhin, V.V.; Kulikov, A.V.; Gallas, M. 2001-01-01 The logic of the first level trigger of the DIRAC experiment at CERN is described. A parallel running of different trigger modes with tagging of events and optional independent prescaling is realized. A CAMAC-based trigger system is completely computer controlled 3. Evolution kernel for the Dirac field International Nuclear Information System (INIS) Baaquie, B.E. 1982-06-01 The evolution kernel for the free Dirac field is calculated using the Wilson lattice fermions. We discuss the difficulties due to which this calculation has not been previously performed in the continuum theory. The continuum limit is taken, and the complete energy eigenfunctions as well as the propagator are then evaluated in a new manner using the kernel. (author) 4. Probabilistic solution of the Dirac equation International Nuclear Information System (INIS) Blanchard, P.; Combe, P. 1985-01-01 Various probabilistic representations of the 2, 3 and 4 dimensional Dirac equation are given in terms of expectation with respect to stochastic jump processes and are used to derive the nonrelativistic limit even in the presence of an external electromagnetic field. (orig.) 5. Poisson geometry from a Dirac perspective Science.gov (United States) Meinrenken, Eckhard 2018-03-01 We present proofs of classical results in Poisson geometry using techniques from Dirac geometry. This article is based on mini-courses at the Poisson summer school in Geneva, June 2016, and at the workshop Quantum Groups and Gravity at the University of Waterloo, April 2016. 6. Analysis of changing hidden energy flow in Vietnam International Nuclear Information System (INIS) Nguyen Thi Anh Tuyet; Ishihara, Keiichi N. 2006-01-01 The energy consumption in production process is changing especially in developing countries by substituting technology. Input-output analysis for energy flows has been developing and is one of the best solutions for investigating macroscopic exchanges of both economy and energy. Since each element in the Leontief inverse contains both direct and indirect effects of any change in final demand, to separate those direct and indirect effects, the power series expansion is available. In this work, the changes of embodied energy intensity in Vietnam from 1996 to 2000 were analyzed using the structural decomposition and its power series expansion. By illustrating the change of causal relationship between direct energy consumption and embodied energy consumption, the change of hidden energy flow, which indicates how the changing embodied energy builds up the change of direct energy consumption in every sector, can be seen. In the case study, the rice processing sector, which is one of the important food processing sectors in Vietnam, is focused. By drawing a diagrammatic map for the change of hidden energy flow, it is clarified that in the case of raising embodied energy intensity, cultivation sector and trade and repaired service sector are the main contributors, and, on the contrary, in the case of reducing embodied energy intensity, paper pulp sector is the main contributor 7. A Route to Dirac Liquid Theory: A Fermi Liquid Description for Dirac Materials Science.gov (United States) Gochan, Matthew; Bedell, Kevin Since the pioneering work developed by L.V. Landau sixty years ago, Fermi Liquid Theory has seen great success in describing interacting Fermi systems. While much interest has been generated over the study of non-Fermi Liquid systems, Fermi Liquid theory serves as a formidable model for many systems and offers a rich amount of of results and insight. The recent classification of Dirac Materials, and the lack of a unifying theoretical framework for them, has motivated our study. Dirac materials are a versatile class of materials in which an abundance of unique physical phenomena can be observed. Such materials are found in all dimensions, with the shared property that their low-energy fermionic excitations behave as massless Dirac fermions and are therefore governed by the Dirac equation. The most popular Dirac material, graphene, is the focus of this work. We present our Fermi Liquid description of Graphene. We find many interesting results, specifically in the transport and dynamics of the system. Additionally, we expand on previous work regarding the Virial Theorem and its impact on the Fermi Liquid parameters in graphene. Finally, we remark on viscoelasticity of Dirac Materials and other unusual results that are consequences of AdS-CFT. 8. The GridPP DIRAC project - DIRAC for non-LHC communities CERN Document Server Bauer, D; Currie, R; Fayer, S; Huffman, A; Martyniak, J; Rand, D; Richards, A 2015-01-01 The GridPP consortium in the UK is currently testing a multi-VO DIRAC service aimed at non-LHC VOs. These VOs (Virtual Organisations) are typically small and generally do not have a dedicated computing support post. The majority of these represent particle physics experiments (e.g. NA62 and COMET), although the scope of the DIRAC service is not limited to this field. A few VOs have designed bespoke tools around the EMI-WMS & LFC, while others have so far eschewed distributed resources as they perceive the overhead for accessing them to be too high. The aim of the GridPP DIRAC project is to provide an easily adaptable toolkit for such VOs in order to lower the threshold for access to distributed resources such as Grid and cloud computing. As well as hosting a centrally run DIRAC service, we will also publish our changes and additions to the upstream DIRAC codebase under an open-source license. We report on the current status of this project and show increasing adoption of DIRAC within the non-LHC communiti... 9. The GridPP DIRAC project - DIRAC for non-LHC communities Science.gov (United States) Bauer, D.; Colling, D.; Currie, R.; Fayer, S.; Huffman, A.; Martyniak, J.; Rand, D.; Richards, A. 2015-12-01 The GridPP consortium in the UK is currently testing a multi-VO DIRAC service aimed at non-LHC VOs. These VOs (Virtual Organisations) are typically small and generally do not have a dedicated computing support post. The majority of these represent particle physics experiments (e.g. NA62 and COMET), although the scope of the DIRAC service is not limited to this field. A few VOs have designed bespoke tools around the EMI-WMS & LFC, while others have so far eschewed distributed resources as they perceive the overhead for accessing them to be too high. The aim of the GridPP DIRAC project is to provide an easily adaptable toolkit for such VOs in order to lower the threshold for access to distributed resources such as Grid and cloud computing. As well as hosting a centrally run DIRAC service, we will also publish our changes and additions to the upstream DIRAC codebase under an open-source license. We report on the current status of this project and show increasing adoption of DIRAC within the non-LHC communities. 10. Identifying Dirac cones in carbon allotropes with square symmetry Energy Technology Data Exchange (ETDEWEB) Wang, Jinying [College of Chemistry and Molecular Engineering, Peking University, Beijing 100871 (China); Huang, Huaqing; Duan, Wenhui [Department of Physics, Tsinghua University, Beijing 100084 (China); Liu, Zhirong, E-mail: [email protected] [College of Chemistry and Molecular Engineering, Peking University, Beijing 100871 (China); State Key Laboratory for Structural Chemistry of Unstable and Stable Species and Beijing National Laboratory for Molecular Sciences (BNLMS), Peking University, Beijing 100871 (China) 2013-11-14 A theoretical study is conducted to search for Dirac cones in two-dimensional carbon allotropes with square symmetry. By enumerating the carbon atoms in a unit cell up to 12, an allotrope with octatomic rings is recognized to possess Dirac cones under a simple tight-binding approach. The obtained Dirac cones are accompanied by flat bands at the Fermi level, and the resulting massless Dirac-Weyl fermions are chiral particles with a pseudospin of S = 1, rather than the conventional S = 1/2 of graphene. The spin-1 Dirac cones are also predicted to exist in hexagonal graphene antidot lattices. 11. Hidden attractors in dynamical systems Science.gov (United States) Dudkowski, Dawid; Jafari, Sajad; Kapitaniak, Tomasz; Kuznetsov, Nikolay V.; Leonov, Gennady A.; Prasad, Awadhesh 2016-06-01 Complex dynamical systems, ranging from the climate, ecosystems to financial markets and engineering applications typically have many coexisting attractors. This property of the system is called multistability. The final state, i.e., the attractor on which the multistable system evolves strongly depends on the initial conditions. Additionally, such systems are very sensitive towards noise and system parameters so a sudden shift to a contrasting regime may occur. To understand the dynamics of these systems one has to identify all possible attractors and their basins of attraction. Recently, it has been shown that multistability is connected with the occurrence of unpredictable attractors which have been called hidden attractors. The basins of attraction of the hidden attractors do not touch unstable fixed points (if exists) and are located far away from such points. Numerical localization of the hidden attractors is not straightforward since there are no transient processes leading to them from the neighborhoods of unstable fixed points and one has to use the special analytical-numerical procedures. From the viewpoint of applications, the identification of hidden attractors is the major issue. The knowledge about the emergence and properties of hidden attractors can increase the likelihood that the system will remain on the most desirable attractor and reduce the risk of the sudden jump to undesired behavior. We review the most representative examples of hidden attractors, discuss their theoretical properties and experimental observations. We also describe numerical methods which allow identification of the hidden attractors. 12. Predictions for the Dirac C P -violating phase from sum rules Science.gov (United States) Delgadillo, Luis A.; Everett, Lisa L.; Ramos, Raymundo; Stuart, Alexander J. 2018-05-01 We explore the implications of recent results relating the Dirac C P -violating phase to predicted and measured leptonic mixing angles within a standard set of theoretical scenarios in which charged lepton corrections are responsible for generating a nonzero value of the reactor mixing angle. We employ a full set of leptonic sum rules as required by the unitarity of the lepton mixing matrix, which can be reduced to predictions for the observable mixing angles and the Dirac C P -violating phase in terms of model parameters. These sum rules are investigated within a given set of theoretical scenarios for the neutrino sector diagonalization matrix for several known classes of charged lepton corrections. The results provide explicit maps of the allowed model parameter space within each given scenario and assumed form of charged lepton perturbations. 13. Squark production in R-symmetric SUSY with Dirac gluinos. NLO corrections Energy Technology Data Exchange (ETDEWEB) Diessner, Philip [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Kotlarski, Wojciech [Technische Univ. Dresden (Germany). Inst. fuer Kern- und Teilchenphysik; Warsaw Univ. (Poland). Faculty of Physics; Liebschner, Sebastian; Stoeckinger, Dominik [Technische Univ. Dresden (Germany). Inst. fuer Kern- und Teilchenphysik 2017-11-15 R-symmetry leads to a distinct realisation of SUSY with a significantly modified coloured sector featuring a Dirac gluino and a scalar colour octet (sgluon). We present the impact of R-symmetry on squark production at the 13 TeV LHC. We study the total cross sections and their NLO corrections from all strongly interacting states, their dependence on the Dirac gluino mass and sgluon mass as well as their systematics for selected benchmark points. We find that tree-level cross sections in the R-symmetric model are reduced compared to the MSSM but the NLO K-factors are generally larger in the order of ten to twenty per cent. In the course of this work we derive the required DREG → DRED transition counterterms and necessary on-shell renormalisation constants. The real corrections are treated using FKS subtraction, with results cross checked against an independent calculation employing the two cut phase space slicing method. 14. Squark production in R-symmetric SUSY with Dirac gluinos. NLO corrections International Nuclear Information System (INIS) Diessner, Philip; Kotlarski, Wojciech; Warsaw Univ.; Liebschner, Sebastian; Stoeckinger, Dominik 2017-11-01 R-symmetry leads to a distinct realisation of SUSY with a significantly modified coloured sector featuring a Dirac gluino and a scalar colour octet (sgluon). We present the impact of R-symmetry on squark production at the 13 TeV LHC. We study the total cross sections and their NLO corrections from all strongly interacting states, their dependence on the Dirac gluino mass and sgluon mass as well as their systematics for selected benchmark points. We find that tree-level cross sections in the R-symmetric model are reduced compared to the MSSM but the NLO K-factors are generally larger in the order of ten to twenty per cent. In the course of this work we derive the required DREG → DRED transition counterterms and necessary on-shell renormalisation constants. The real corrections are treated using FKS subtraction, with results cross checked against an independent calculation employing the two cut phase space slicing method. 15. Variational Infinite Hidden Conditional Random Fields NARCIS (Netherlands) Bousmalis, Konstantinos; Zafeiriou, Stefanos; Morency, Louis-Philippe; Pantic, Maja; Ghahramani, Zoubin 2015-01-01 Hidden conditional random fields (HCRFs) are discriminative latent variable models which have been shown to successfully learn the hidden structure of a given classification problem. An Infinite hidden conditional random field is a hidden conditional random field with a countably infinite number of 16. A Hidden Twelve-Dimensional SuperPoincare Symmetry In Eleven Dimensions Energy Technology Data Exchange (ETDEWEB) Bars, Itzhak; Deliduman, Cemsinan; Pasqua, Andrea; Zumino, Bruno 2003-12-13 First, we review a result in our previous paper, of how a ten-dimensional superparticle, taken off-shell, has a hidden eleven-dimensional superPoincare symmetry. Then, we show that the physical sector is defined by three first-class constraints which preserve the full eleven-dimensional symmetry. Applying the same concepts to the eleven dimensional superparticle, taken off-shell, we discover a hidden twelve dimensional superPoincare symmetry that governs the theory. 17. Prediction of super-heavy N⁎ and Λ⁎ resonances with hidden beauty International Nuclear Information System (INIS) Wu Jiajun; Zhao Lu; Zou, B.S. 2012-01-01 The meson-baryon coupled channel unitary approach with the local hidden gauge formalism is extended to the hidden beauty sector. A few narrow N ⁎ and Λ ⁎ resonances around 11 GeV are predicted as dynamically generated states from the interactions of heavy beauty mesons and baryons. Production cross sections of these predicted resonances in pp and ep collisions are estimated as a guide for the possible experimental search at relevant facilities. 18. Managing Hidden Costs of Offshoring DEFF Research Database (Denmark) Larsen, Marcus M.; Pedersen, Torben 2014-01-01 This chapter investigates the concept of the ‘hidden costs’ of offshoring, i.e. unexpected offshoring costs exceeding the initially expected costs. Due to the highly undefined nature of these costs, we position our analysis towards the strategic responses of firms’ realisation of hidden costs....... In this regard, we argue that a major response to the hidden costs of offshoring is the identification and utilisation of strategic mechanisms in the organisational design to eventually achieving system integration in a globally dispersed and disaggregated organisation. This is heavily moderated by a learning......-by-doing process, where hidden costs motivate firms and their employees to search for new and better knowledge on how to successfully manage the organisation. We illustrate this thesis based on the case of the LEGO Group.... 19. The Hidden Costs of Offshoring DEFF Research Database (Denmark) Møller Larsen, Marcus; Manning, Stephan; Pedersen, Torben 2011-01-01 of offshoring. Specifically, we propose that hidden costs can be explained by the combination of increasing structural, operational and social complexity of offshoring activities. In addition, we suggest that firm orientation towards organizational design as part of an offshoring strategy and offshoring......This study seeks to explain hidden costs of offshoring, i.e. unexpected costs resulting from the relocation of business tasks and activities outside the home country. We develop a model that highlights the role of complexity, design orientation and experience in explaining hidden costs...... experience moderate the relationship between complexity and hidden costs negatively i.e. reduces the cost generating impact of complexity. We develop three hypotheses and test them on comprehensive data from the Offshoring Research Network (ORN). In general, we find support for our hypotheses. A key result... 20. Child Abuse: The Hidden Bruises Science.gov (United States) ... for Families - Vietnamese Spanish Facts for Families Guide Child Abuse - The Hidden Bruises No. 5; Updated November 2014 The statistics on physical child abuse are alarming. It is estimated hundreds of thousands ... 1. Atlas of solar hidden photon emission Energy Technology Data Exchange (ETDEWEB) Redondo, Javier [Departamento de Física Teórica, Universidad de Zaragoza,Pedro Cerbuna 12, E-50009, Zaragoza (Spain); Max-Planck-Institut für Physik, Werner-Heisenberg-Institut,Föhringer Ring 6, 80805 München (Germany) 2015-07-20 Hidden photons, gauge bosons of a U(1) symmetry of a hidden sector, can constitute the dark matter of the universe and a smoking gun for large volume compactifications of string theory. In the sub-eV mass range, a possible discovery experiment consists on searching the copious flux of these particles emitted from the Sun in a helioscope setup à la Sikivie. In this paper, we compute in great detail the flux of HPs from the Sun, a necessary ingredient for interpreting such experiments. We provide a detailed exposition of transverse photon-HP oscillations in inhomogenous media, with special focus on resonance oscillations, which play a leading role in many cases. The region of the Sun emitting HPs resonantly is a thin spherical shell for which we justify an averaged-emission formula and which implies a distinctive morphology of the angular distribution of HPs on Earth in many cases. Low mass HPs with energies in the visible and IR have resonances very close to the photosphere where the solar plasma is not fully ionised and requires building a detailed model of solar refraction and absorption. We present results for a broad range of HP masses (from 0–1 keV) and energies (from the IR to the X-ray range), the most complete atlas of solar HP emission to date. 2. Atlas of solar hidden photon emission Energy Technology Data Exchange (ETDEWEB) Redondo, Javier, E-mail: [email protected] [Departamento de Física Teórica, Universidad de Zaragoza, Pedro Cerbuna 12, E-50009, Zaragoza, España (Spain) 2015-07-01 Hidden photons, gauge bosons of a U(1) symmetry of a hidden sector, can constitute the dark matter of the universe and a smoking gun for large volume compactifications of string theory. In the sub-eV mass range, a possible discovery experiment consists on searching the copious flux of these particles emitted from the Sun in a helioscope setup à la Sikivie. In this paper, we compute in great detail the flux of HPs from the Sun, a necessary ingredient for interpreting such experiments. We provide a detailed exposition of transverse photon-HP oscillations in inhomogenous media, with special focus on resonance oscillations, which play a leading role in many cases. The region of the Sun emitting HPs resonantly is a thin spherical shell for which we justify an averaged-emission formula and which implies a distinctive morphology of the angular distribution of HPs on Earth in many cases. Low mass HPs with energies in the visible and IR have resonances very close to the photosphere where the solar plasma is not fully ionised and requires building a detailed model of solar refraction and absorption. We present results for a broad range of HP masses (from 0–1 keV) and energies (from the IR to the X-ray range), the most complete atlas of solar HP emission to date. 3. Extracting hidden-photon dark matter from an LC-circuit International Nuclear Information System (INIS) Arias, Paola; Arza, Ariel; Gamboa, Jorge; Mendez, Fernando; Doebrich, Babette 2015-01-01 We point out that a cold dark matter condensate made of gauge bosons from an extra hidden U(1) sector - dubbed hidden photons - can create a small, oscillating electric density current. Thus, they could also be searched for in the recently proposed LC-circuit setup conceived for axion cold dark matter search by Sikivie, Sullivan and Tanner. We estimate the sensitivity of this setup for hidden-photon cold dark matter and we find it could cover a sizable, so far unexplored parameter space. (orig.) 4. Extracting hidden-photon dark matter from an LC-circuit International Nuclear Information System (INIS) Arias, Paola; Arza, Ariel; Gamboa, Jorge; Mendez, Fernando 2014-11-01 We point out that a cold dark matter condensate made of gauge bosons from an extra hidden U(1) sector - dubbed hidden-photons - can create a small, oscillating electric density current. Thus, they could also be searched for in the recently proposed LC-circuit setup conceived for axion cold dark matter search by Sikivie, Sullivan and Tanner. We estimate the sensitivity of this setup for hidden-photon cold dark matter and we find it could cover a sizable, so far unexplored parameter space. 5. Extracting Hidden-Photon Dark Matter From an LC-Circuit CERN Document Server Arias, Paola; Döbrich, Babette; Gamboa, Jorge; Méndez, Fernando 2015-01-01 We point out that a cold dark matter condensate made of gauge bosons from an extra hidden U(1) sector - dubbed hidden- photons - can create a small, oscillating electric density current. Thus, they could also be searched for in the recently proposed LC-circuit setup conceived for axion cold dark matter search by Sikivie, Sullivan and Tanner. We estimate the sensitivity of this setup for hidden-photon cold dark matter and we find it could cover a sizable, so far unexplored parameter space. 6. Hidden Statistics of Schroedinger Equation Science.gov (United States) Zak, Michail 2011-01-01 Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods. 7. Hidden Curriculum: An Analytical Definition Directory of Open Access Journals (Sweden) Mohammad Reza Andarvazh 2018-03-01 Full Text Available Background: The concept of hidden curriculum was first used by Philip Jackson in 1968, and Hafferty brought this concept to the medical education. Many of the subjects that medical students learn are attributed to this curriculum. So far several definitions have been presented for the hidden curriculum, which on the one hand made this concept richer, and on the other hand, led to confusion and ambiguity.This paper tries to provide a clear and comprehensive definition of it.Methods: In this study, concept analysis of McKenna method was used. Using keywords and searching in the databases, 561 English and 26 Persian references related to the concept was found, then by limitingthe research scope, 125 abstracts and by finding more relevant references, 55 articles were fully studied.Results: After analyzing the definitions by McKenna method, the hidden curriculum is defined as follows: The hidden curriculum is a hidden, powerful, intrinsic in organizational structure and culture and sometimes contradictory message, conveyed implicitly and tacitly in the learning environment by structural and human factors and its contents includes cultural habits and customs, norms, values, belief systems, attitudes, skills, desires and behavioral and social expectations can have a positive or negative effect, unplanned, neither planners nor teachers, nor learners are aware of it. The ultimate consequence of the hidden curriculum includes reproducing the existing class structure, socialization, and familiarizing learners for transmission and joining the professional world.Conclusion: Based on the concept analysis, we arrived at an analytical definition of the hidden curriculum that could be useful for further studies in this area.Keywords: CONCEPT ANALYSIS, HIDDEN CURRICULUM, MCKENNA’S METHOD 8. Hidden conformal symmetry of extremal black holes International Nuclear Information System (INIS) Chen Bin; Long Jiang; Zhang Jiaju 2010-01-01 We study the hidden conformal symmetry of extremal black holes. We introduce a new set of conformal coordinates to write the SL(2,R) generators. We find that the Laplacian of the scalar field in many extremal black holes, including Kerr(-Newman), Reissner-Nordstrom, warped AdS 3 , and null warped black holes, could be written in terms of the SL(2,R) quadratic Casimir. This suggests that there exist dual conformal field theory (CFT) descriptions of these black holes. From the conformal coordinates, the temperatures of the dual CFTs could be read directly. For the extremal black hole, the Hawking temperature is vanishing. Correspondingly, only the left (right) temperature of the dual CFT is nonvanishing, and the excitations of the other sector are suppressed. In the probe limit, we compute the scattering amplitudes of the scalar off the extremal black holes and find perfect agreement with the CFT prediction. 9. Dirac phenomenology and hyperon-nucleus interactions Energy Technology Data Exchange (ETDEWEB) Mares, J; Jennings, B K [TRIUMF, Vancouver, British Columbia (Canada); Cooper, E D [Fraser Valley Univ. College, Chilliwack, British Columbia (Canada). Dept. of Physics 1993-05-01 We discuss various aspects of hyperon-nucleus interactions in the relativistic mean field theory. First, characteristics of {Lambda}, {Sigma} and {identical_to} hypernuclei, as well as multi strange baryonic objects, are investigated. The spin-orbit splittings and magnetic moments are shown to be very sensitive to the value of the tensor coupling f{omega}y. Second, optical potentials for {Lambda} and {Sigma} scattering off nuclei are developed based on a global nucleon-nucleon Dirac optical potential and SU(3) symmetry. The tensor coupling has a large effect on the predictions for the analyzing power. Third, the Dirac approach is used in the calculations of the non-mesonic decay of {Lambda} hypernuclei. The large discrepancy between the decay rates and data suggests the need for additional meson exchanges. (authors). 62 refs.,7 figs., 6 tabs. 10. Classical electromagnetic radiation of the Dirac electron Science.gov (United States) Lanyi, G. 1973-01-01 A wave-function-dependent four-vector potential is added to the Dirac equation in order to achieve conservation of energy and momentum for a Dirac electron and its emitted electromagnetic field. The resultant equation contains solutions which describe transitions between different energy states of the electron. As a consequence it is possible to follow the space-time evolution of such a process. This evolution is shown in the case of the spontaneous emission of an electromagnetic field by an electron bound in a hydrogen-like atom. The intensity of the radiation and the spectral distribution are calculated for transitions between two eigenstates. The theory gives a self-consistent deterministic description of some simple radiation processes without using quantum electrodynamics or the correspondence principle. 11. Floquet-Engineered Valleytronics in Dirac Systems. Science.gov (United States) Kundu, Arijit; Fertig, H A; Seradjeh, Babak 2016-01-08 Valley degrees of freedom offer a potential resource for quantum information processing if they can be effectively controlled. We discuss an optical approach to this problem in which intense light breaks electronic symmetries of a two-dimensional Dirac material. The resulting quasienergy structures may then differ for different valleys, so that the Floquet physics of the system can be exploited to produce highly polarized valley currents. This physics can be utilized to realize a valley valve whose behavior is determined optically. We propose a concrete way to achieve such valleytronics in graphene as well as in a simple model of an inversion-symmetry broken Dirac material. We study the effect numerically and demonstrate its robustness against moderate disorder and small deviations in optical parameters. 12. Crucial test of the Dirac cosmologies International Nuclear Information System (INIS) Steigman, G. 1978-01-01 In a cosmology consistent with the Cosmological Principle (large scale, statistical isotropy and homogeneity of the universe), a Planck spectrum is not preserved as the universe evolves unless the number of photons in a comoving volume is conserved. It is shown that a large class of cosmological models based on Dirac's Large Numbers Hypothesis (LNH) violate this constraint. The observed isotropy and spectral distribution of the microwave background radiation thus provide a crucial test of such cosmologies. After reviewing the LNH, the general evolution of radiation spectra in cosmologies consistent with the cosmological principle is outlined. It is shown that the predicted deviations from a Planck spectrum for Dirac cosmologies (as well as for ''tired-light'' cosmologies) are enormous. The Planckian (or near-Planckian) spectral form for the microwave radiation provides a crucial test, failed by such cosmologies 13. Excitation spectrum of correlated Dirac fermions Science.gov (United States) Jalali, Z.; Jafari, S. A. 2015-04-01 Motivated by the puzzling optical conductivity measurements in graphene, we speculate on the possible role of strong electronic correlations on the two-dimensional Dirac fermions. In this work we employ the slave-particle method to study the excitations of the Hubbard model on honeycomb lattice, away from half-filling. Since the ratio U/t ≈ 3.3 in graphene is not infinite, double occupancy is not entirely prohibited and hence a finite density of doublonscan be generated. We therefore extend the Ioff-Larkin composition rule to include a finite density of doublons. We then investigate the role played by each of these auxiliary particles in the optical absorption of strongly correlated Dirac fermions. 14. Dirac gauginos in low scale supersymmetry breaking International Nuclear Information System (INIS) Goodsell, Mark D.; Tziveloglou, Pantelis 2014-01-01 It has been claimed that Dirac gaugino masses are necessary for realistic models of low-scale supersymmetry breaking, and yet very little attention has been paid to the phenomenology of a light gravitino when gauginos have Dirac masses. We begin to address this deficit by investigating the couplings and phenomenology of the gravitino in the effective Lagrangian approach. We pay particular attention to the phenomenology of the scalar octets, where new decay channels open up. This leads us to propose a new simplified effective scenario including only light gluinos, sgluons and gravitinos, allowing the squarks to be heavy – with the possible exception of the third generation. Finally, we comment on the application of our results to Fake Split Supersymmetry 15. Dirac operator, chirality and random matrix theory International Nuclear Information System (INIS) Pullirsch, R. 2001-05-01 Quantum Chromodynamics (QCD) is considered to be the correct theory which describes quarks and gluons and, thus, all strong interaction phenomena from the fundamental forces of nature. However, important properties of QCD such as the physical mechanism of color confinement and the spontaneous breaking of chiral symmetry are still not completely understood and under extensive discussion. Analytical calculations are limited, because in the low-energy regime where quarks are confined, application of perturbation theory is restricted due to the large gluon coupling. A powerful tool to investigate numerically and analytically the non-perturbative region is provided by the lattice formulation of QCD. From Monte Carlo simulations of lattice QCD we know that chiral symmetry is restored above a critical temperature. As the chiral condensate is connected to the spectral density of the Dirac operator via the Banks-Casher relation, the QCD Dirac spectrum is an interesting object for detailed studies. In search for an analytical expression of the infrared limit of the Dirac spectrum it has been realized that chiral random-matrix theory (chRMT) is a suitable tool to compare with the distribution and the correlations of the small Dirac eigenvalues. Further, it has been shown that the correlations of eigenvalues on the scale of mean level spacings are universal for complex physical systems and are given by random-matrix theory (Rm). This has been formulated as the Baghouse-Giannoni-Schmit conjecture which states that spectral correlations of a classically chaotic system are given by RMT on the quantum level. The aim of this work is to analyze the relationship between chiral phase transitions and chaos to order transitions in quantum field theories. We study the eigenvalues of the Dirac operator for Quantum Electrodynamics (QED) with compact gauge group U(1) on the lattice. This theory shows chiral symmetry breaking and confinement in the strong coupling region. Although being 16. Renormalization group evolution of Dirac neutrino masses International Nuclear Information System (INIS) Lindner, Manfred; Ratz, Michael; Schmidt, Michael Andreas 2005-01-01 There are good reasons why neutrinos could be Majorana particles, but there exist also a number of very good reasons why neutrinos could have Dirac masses. The latter option deserves more attention and we derive therefore analytic expressions describing the renormalization group evolution of mixing angles and of the CP phase for Dirac neutrinos. Radiative corrections to leptonic mixings are in this case enhanced compared to the quark mixings because the hierarchy of neutrino masses is milder and because the mixing angles are larger. The renormalization group effects are compared to the precision of current and future neutrino experiments. We find that, in the MSSM framework, radiative corrections of the mixing angles are for large tan β comparable to the precision of future experiments 17. Dirac equation in Kerr space-time Energy Technology Data Exchange (ETDEWEB) Iyer, B R; Kumar, Arvind [Bombay Univ. (India). Dept. of Physics 1976-06-01 The weak-field low-velocity approximation of Dirac equation in Kerr space-time is investigated. The interaction terms admit of an interpretation in terms of a 'dipole-dipole' interaction in addition to coupling of spin with the angular momentum of the rotating source. The gravitational gyro-factor for spin is identified. The charged case (Kerr-Newman) is studied using minimal prescription for electromagnetic coupling in the locally intertial frame and to the leading order the standard electromagnetic gyro-factor is retrieved. A first order perturbation calculation of the shift of the Schwarzchild energy level yields the main interesting result of this work: the anomalous Zeeman splitting of the energy level of a Dirac particle in Kerr metric. 18. On an uninterpretated tensor in Dirac's theory International Nuclear Information System (INIS) Costa de Beauregard, O. 1989-01-01 Franz, in 1935, deduced systematically from the Dirac equation 10 tensorial equations, 5 with a mechanical interpretation, 5 with an electromagnetic interpretation, which are also consequences of Kemmer's formalism for spins 1 and 0; Durand, in 1944, operating similarly with the second order Dirac equation, obtained, 10 equations, 5 of which expressing the divergences of the Gordon type tensors. Of these equations, together with the tensors they imply, some are easily interpreted by reference to the classical theories, some other remain uniterpreted. Recently (1988) we proposed a theory of the coupling between Einstein's gravity field and the 5 Franz mechanical equations, yielding as a bonus the complete interpretation of the 5 Franz mechanical equations. This is an incitation to reexamine the 5 electromagnetic equations. We show here that two of these, together with one of the Durand equations, implying the same tensor, remain uninterpreted. This is proposed as a challenge to the reader's sagacity [fr 19. LHCb: Pilot Framework and the DIRAC WMS CERN Multimedia Graciani, R; Casajus, A 2009-01-01 DIRAC, the LHCb community Grid solution, has pioneered the use of pilot jobs in the Grid. Pilot jobs provide a homogeneous interface to an heterogeneous set of computing resources. At the same time, pilot jobs allow to delay the scheduling decision to the last moment, thus taking into account the precise running conditions at the resource and last moment requests to the system. The DIRAC Workload Management System provides one single scheduling mechanism for jobs with very different profiles. To achieve an overall optimisation, it organizes pending jobs in task queues, both for individual users and production activities. Task queues are created with jobs having similar requirements. Following the VO policy a priority is assigned to each task queue. Pilot submission and subsequent job matching are based on these priorities following a statistical approach. Details of the implementation and the security aspects of this framework will be discussed. 20. Transversal Dirac families in Riemannian foliations International Nuclear Information System (INIS) Glazebrook, J.F.; Kamber, F.W. 1991-01-01 We describe a family of differential operators parametrized by the transversal vector potentials of a Riemannian foliation relative to the Clifford algebra of the foliation. This family is non-elliptic but in certain ways behaves like a standard Dirac family in the absolute case as a result of its elliptic-like regularity properties. The analytic and topological indices of this family are defined as elements of K-theory in the parameter space. We indicate how the cohomology of the parameter space is described via suitable maps to Fredholm operators. We outline the proof of a theorem of Vafa-Witten type on uniform bounds for the eigenvalues of this family using a spectral flow argument. A determinant operator is also defined with the appropriate zeta function regularization dependent on the codimension of the foliation. With respect to a generalized coupled Dirac-Yang-Mills system, we indicate how chiral anomalies are located relative to the foliation. (orig.) 1. Hidden U (1 ) gauge symmetry realizing a neutrinophilic two-Higgs-doublet model with dark matter Science.gov (United States) Nomura, Takaaki; Okada, Hiroshi 2018-04-01 We propose a neutrinophilic two-Higgs-doublet model with hidden local U (1 ) symmetry, where active neutrinos are Dirac type, and a fermionic dark matter (DM) candidate is naturally induced as a result of remnant symmetry even after the spontaneous symmetry breaking. In addition, a physical Goldstone boson arises as a consequence of two types of gauge singlet bosons and contributes to the DM phenomenologies as well as an additional neutral gauge boson. Then, we analyze the relic density of DM within the safe range of direct detection searches and show the allowed region of dark matter mass. 2. Permanent Magnet Dipole for DIRAC Design Report CERN Document Server Vorozhtsov, Alexey 2012-01-01 Two dipole magnets including one spare unit are needed for the for the DIRAC experiment. The proposed design is a permanent magnet dipole. The design based on Sm2Co17 blocks assembled together with soft ferromagnetic pole tips. The magnet provides integrated field strength of 24.6 10-3 T×m inside the aperture of 60 mm. This Design Report summarizes the main magnetic and mechanic design parameters of the permanent dipole magnets. 3. Dirac monopole without strings: monopole harmonics International Nuclear Information System (INIS) Wu, T.T.; Yang, C.N. 1983-01-01 Using the ideas developed in a previous paper which are borrowed from the mathematics of fiber bundles, it is shown that the wave function psi of a particle of charge Ze around a Dirac monopole of strength g should be regarded as a section. The section is without discontinuities. Thus the monopole does not possess strings of singularities in the field around it. The eigensections of the angular momentum operators are monopole harmonics which are explicitly exhibited. 7 references, 2 figures, 1 table 4. Dispersionless wave packets in Dirac materials International Nuclear Information System (INIS) Jakubský, Vít; Tušek, Matěj 2017-01-01 We show that a wide class of quantum systems with translational invariance can host dispersionless, soliton-like, wave packets. We focus on the setting where the effective, two-dimensional Hamiltonian acquires the form of the Dirac operator. The proposed framework for construction of the dispersionless wave packets is illustrated on silicene-like systems with topologically nontrivial effective mass. Our analytical predictions are accompanied by a numerical analysis and possible experimental realizations are discussed. 5. Dirac operators and Killing spinors with torsion International Nuclear Information System (INIS) Becker-Bender, Julia 2012-01-01 On a Riemannian spin manifold with parallel skew torsion, we use the twistor operator to obtain an eigenvalue estimate for the Dirac operator with torsion. We consider the equality case in dimensions four and six. In odd dimensions we describe Sasaki manifolds on which equality in the estimate is realized by Killing spinors with torsion. In dimension five we characterize all Killing spinors with torsion and obtain certain naturally reductive spaces as exceptional cases. 6. Data acquisition software for DIRAC experiment CERN Document Server Olshevsky, V G 2001-01-01 The structure and basic processes of data acquisition software of the DIRAC experiment for the measurement of pi /sup +/ pi /sup -/ atom lifetime are described. The experiment is running on the PS accelerator of CERN. The developed software allows one to accept, record and distribute up to 3 Mbytes of data to consumers in one accelerator supercycle of 14.4 s duration. The described system is successfully in use in the experiment since its startup in 1998. (13 refs). 7. Data acquisition software for DIRAC experiment Science.gov (United States) Olshevsky, V.; Trusov, S. 2001-08-01 The structure and basic processes of data acquisition software of the DIRAC experiment for the measurement of π +π - atom lifetime are described. The experiment is running on the PS accelerator of CERN. The developed software allows one to accept, record and distribute up to 3 Mbytes of data to consumers in one accelerator supercycle of 14.4 s duration. The described system is successfully in use in the experiment since its startup in 1998. 8. Data acquisition software for DIRAC experiment International Nuclear Information System (INIS) Olshevsky, V.; Trusov, S. 2001-01-01 The structure and basic processes of data acquisition software of the DIRAC experiment for the measurement of π + π - atom lifetime are described. The experiment is running on the PS accelerator of CERN. The developed software allows one to accept, record and distribute up to 3 Mbytes of data to consumers in one accelerator supercycle of 14.4 s duration. The described system is successfully in use in the experiment since its startup in 1998 9. Hidden ion population: Revisited International Nuclear Information System (INIS) Olsen, R.C.; Chappell, C.R.; Gallagher, D.L.; Green, J.L.; Gurnett, D.A. 1985-01-01 Satellite potentials in the outer plasmasphere range from near zero to +5 to +10 V. Under such conditions ion measurements may not include the low energy core of the plasma population. In eclipse, the photoelectron current drops to zero, and the spacecraft potential can drop to near zero volts. In regions where the ambient plasma density is below 100 cm -3 , previously unobserved portions of the ambient plasma distribution function can become visible in eclipse. A survey of the data obtained from the retarding ion mass spectrometer (RIMS) on Dynamics Explorer 1 shows that the RIMS detector generally measured the isotropic background in both sunlight and eclipse in the plasma-sphere. Absolute density measurements for the ''hidden'' ion population are obtained for the first time using the plasma wave instrument observations of the upper hybrid resonance. Agreement in total density is found in sunlight and eclipse measurements at densities above 80 cm -3 . In eclipse, agreement is found at densities as low as 20 cm -3 . The isotropic plasma composition is primarily H + , with approx.10% He + , and 0.1 to 1.0% O + . A low energy field-aligned ion population appears in eclipse measurements outside the plasmasphere, which is obscured in sunlight. These field-aligned ions can be interpreted as field-aligned flows with densities of a few particles per cubic centimeter, flowing at 5-20 km/s. The problem in measuring these field-aligned flows in sunlight is the masking of the high energy tail of the field-aligned distribution by the isotropic background. Effective measurement of the core of the magnetospheric plasma distribution awaits satellites with active means of controlling the satellite potential 10. Quantum transport through 3D Dirac materials Energy Technology Data Exchange (ETDEWEB) Salehi, M. [Department of Physics, Sharif University of Technology, Tehran 11155-9161 (Iran, Islamic Republic of); Jafari, S.A., E-mail: [email protected] [Department of Physics, Sharif University of Technology, Tehran 11155-9161 (Iran, Islamic Republic of); Center of Excellence for Complex Systems and Condensed Matter (CSCM), Sharif University of Technology, Tehran 1458889694 (Iran, Islamic Republic of) 2015-08-15 Bismuth and its alloys provide a paradigm to realize three dimensional materials whose low-energy effective theory is given by Dirac equation in 3+1 dimensions. We study the quantum transport properties of three dimensional Dirac materials within the framework of Landauer–Büttiker formalism. Charge carriers in normal metal satisfying the Schrödinger equation, can be split into four-component with appropriate matching conditions at the boundary with the three dimensional Dirac material (3DDM). We calculate the conductance and the Fano factor of an interface separating 3DDM from a normal metal, as well as the conductance through a slab of 3DDM. Under certain circumstances the 3DDM appears transparent to electrons hitting the 3DDM. We find that electrons hitting the metal-3DDM interface from metallic side can enter 3DDM in a reversed spin state as soon as their angle of incidence deviates from the direction perpendicular to interface. However the presence of a second interface completely cancels this effect. 11. Quantum transport through 3D Dirac materials International Nuclear Information System (INIS) Salehi, M.; Jafari, S.A. 2015-01-01 Bismuth and its alloys provide a paradigm to realize three dimensional materials whose low-energy effective theory is given by Dirac equation in 3+1 dimensions. We study the quantum transport properties of three dimensional Dirac materials within the framework of Landauer–Büttiker formalism. Charge carriers in normal metal satisfying the Schrödinger equation, can be split into four-component with appropriate matching conditions at the boundary with the three dimensional Dirac material (3DDM). We calculate the conductance and the Fano factor of an interface separating 3DDM from a normal metal, as well as the conductance through a slab of 3DDM. Under certain circumstances the 3DDM appears transparent to electrons hitting the 3DDM. We find that electrons hitting the metal-3DDM interface from metallic side can enter 3DDM in a reversed spin state as soon as their angle of incidence deviates from the direction perpendicular to interface. However the presence of a second interface completely cancels this effect 12. Spectrum of the Wilson Dirac operator at finite lattice spacings DEFF Research Database (Denmark) Akemann, G.; Damgaard, Poul Henrik; Splittorff, Kim 2011-01-01 We consider the effect of discretization errors on the microscopic spectrum of the Wilson Dirac operator using both chiral Perturbation Theory and chiral Random Matrix Theory. A graded chiral Lagrangian is used to evaluate the microscopic spectral density of the Hermitian Wilson Dirac operator...... as well as the distribution of the chirality over the real eigenvalues of the Wilson Dirac operator. It is shown that a chiral Random Matrix Theory for the Wilson Dirac operator reproduces the leading zero-momentum terms of Wilson chiral Perturbation Theory. All results are obtained for fixed index...... of the Wilson Dirac operator. The low-energy constants of Wilson chiral Perturbation theory are shown to be constrained by the Hermiticity properties of the Wilson Dirac operator.... 13. Wave Functions for Time-Dependent Dirac Equation under GUP Science.gov (United States) Zhang, Meng-Yao; Long, Chao-Yun; Long, Zheng-Wen 2018-04-01 In this work, the time-dependent Dirac equation is investigated under generalized uncertainty principle (GUP) framework. It is possible to construct the exact solutions of Dirac equation when the time-dependent potentials satisfied the proper conditions. In (1+1) dimensions, the analytical wave functions of the Dirac equation under GUP have been obtained for the two kinds time-dependent potentials. Supported by the National Natural Science Foundation of China under Grant No. 11565009 14. Stargate of the Hidden Multiverse Directory of Open Access Journals (Sweden) Alexander Antonov 2016-02-01 Full Text Available Concept of Monoverse, which corresponds to the existing broad interpretation of the second postulate of the special theory of relativity, is not consistent with the modern astrophysical reality — existence of the dark matter and the dark energy, the total mass-energy of which is ten times greater than the mass-energy of the visible universe (which has been considered as the entire universe until very recent . This concept does not allow to explain their rather unusual properties — invisibility and lack of baryon content — which would seem to even destroy the very modern understanding of the term ‘matter’. However, all numerous alternative concepts of Multiverses, which have been proposed until today, are unable to explain these properties of the dark matter and dark energy. This article describes a new concept: the concept of the hidden Multiverse and hidden Supermultiverse, which mutual invisibility of parallel universes is explained by the physical reality of imaginary numbers. This concept completely explains the phenomenon of the dark matter and the dark energy. Moreover, it is shown that the dark matter and the dark energy are the experimental evidence for the existence of the hidden Multiverse. Described structure of the hidden Multiverse is fully consistent with the data obtained by the space stations WMAP and Planck. An extremely important property of the hidden Multiverse is an actual possibility of its permeation through stargate located on the Earth. 15. Preliminary Results of the CASCADE Hidden Sector Photon Search CERN Document Server Woollett, Nathan; Burt, Graeme; Chattopadhyay, Swapan; Dainton, John; Dexter, Amos; Goudket, Phillipe; Jenkins, Michael; Kalliokoski, Matti; Moss, Andrew; Pattalwar, Shrikant; Thakker, Trina; Williams, Peter 2015-01-01 Light shining through a wall experiments can be used to make measurements of photon-WISP couplings. The first stage of the CASCADE experiment at the Cockcroft Institute of Accelerator Science and Technology is intended to be a proof-of-principle experiment utilising standard microwave technologies to make a modular, cryogenic HSP detector to take advantage of future high-power superconducting cavity tests. In these proceedings we will be presenting the preliminary results of the CASCADE LSW experiment showing a peak expected exclusion of1.10 \\times 10^{-8}$in the mass range from 1.96$\\mu$eV to 5.38$\\mu$eV, exceeding current limits. 16. Full utilization of semi-Dirac cones in photonics Science.gov (United States) Yasa, Utku G.; Turduev, Mirbek; Giden, Ibrahim H.; Kurt, Hamza 2018-05-01 In this study, realization and applications of anisotropic zero-refractive-index materials are proposed by exposing the unit cells of photonic crystals that exhibit Dirac-like cone dispersion to rotational symmetry reduction. Accidental degeneracy of two Bloch modes in the Brillouin zone center of two-dimensional C2-symmetric photonic crystals gives rise to the semi-Dirac cone dispersion. The proposed C2-symmetric photonic crystals behave as epsilon-and-mu-near-zero materials (ɛeff≈ 0 , μeff≈ 0 ) along one propagation direction, but behave as epsilon-near-zero material (ɛeff≈ 0 , μeff≠ 0 ) for the perpendicular direction at semi-Dirac frequency. By extracting the effective medium parameters of the proposed C4- and C2-symmetric periodic media that exhibit Dirac-like and semi-Dirac cone dispersions, intrinsic differences between isotropic and anisotropic materials are investigated. Furthermore, advantages of utilizing semi-Dirac cone materials instead of Dirac-like cone materials in photonic applications are demonstrated in both frequency and time domains. By using anisotropic transmission behavior of the semi-Dirac materials, photonic application concepts such as beam deflectors, beam splitters, and light focusing are proposed. Furthermore, to the best of our knowledge, semi-Dirac cone dispersion is also experimentally demonstrated for the first time by including negative, zero, and positive refraction states of the given material. 17. Quasiparticle dynamics in reshaped helical Dirac cone of topological insulators. Science.gov (United States) Miao, Lin; Wang, Z F; Ming, Wenmei; Yao, Meng-Yu; Wang, Meixiao; Yang, Fang; Song, Y R; Zhu, Fengfeng; Fedorov, Alexei V; Sun, Z; Gao, C L; Liu, Canhua; Xue, Qi-Kun; Liu, Chao-Xing; Liu, Feng; Qian, Dong; Jia, Jin-Feng 2013-02-19 Topological insulators and graphene present two unique classes of materials, which are characterized by spin-polarized (helical) and nonpolarized Dirac cone band structures, respectively. The importance of many-body interactions that renormalize the linear bands near Dirac point in graphene has been well recognized and attracted much recent attention. However, renormalization of the helical Dirac point has not been observed in topological insulators. Here, we report the experimental observation of the renormalized quasiparticle spectrum with a skewed Dirac cone in a single Bi bilayer grown on Bi(2)Te(3) substrate from angle-resolved photoemission spectroscopy. First-principles band calculations indicate that the quasiparticle spectra are likely associated with the hybridization between the extrinsic substrate-induced Dirac states of Bi bilayer and the intrinsic surface Dirac states of Bi(2)Te(3) film at close energy proximity. Without such hybridization, only single-particle Dirac spectra are observed in a single Bi bilayer grown on Bi(2)Se(3), where the extrinsic Dirac states Bi bilayer and the intrinsic Dirac states of Bi(2)Se(3) are well separated in energy. The possible origins of many-body interactions are discussed. Our findings provide a means to manipulate topological surface states. 18. Hidden worlds in quantum physics CERN Document Server Gouesbet, Gérard 2014-01-01 The past decade has witnessed a resurgence in research and interest in the areas of quantum computation and entanglement. This new book addresses the hidden worlds or variables of quantum physics. Author Gérard Gouesbet studied and worked with a former student of Louis de Broglie, a pioneer of quantum physics. His presentation emphasizes the history and philosophical foundations of physics, areas that will interest lay readers as well as professionals and advanced undergraduate and graduate students of quantum physics. The introduction is succeeded by chapters offering background on relevant concepts in classical and quantum mechanics, a brief history of causal theories, and examinations of the double solution, pilot wave, and other hidden-variables theories. Additional topics include proofs of possibility and impossibility, contextuality, non-locality, classification of hidden-variables theories, and stochastic quantum mechanics. The final section discusses how to gain a genuine understanding of quantum mec... 19. Hidden Crises and Communication: An Interactional Analysis of Hidden Crises NARCIS (Netherlands) dr. Annette Klarenbeek 2011-01-01 In this paper I describe the ways in which the communication discipline can make a hidden crisis transparent. For this purpose I examine the concept of crisis entrepreneurship from a communication point of view. Using discourse analysis, I analyse the discursive practices of crisis entrepreneurs in 20. Hidden Crises and Communication : An Interactional Analysis of Hidden Crises NARCIS (Netherlands) dr. Annette Klarenbeek 2011-01-01 In this paper I describe the ways in which the communication discipline can make a hidden crisis transparent. For this purpose I examine the concept of crisis entrepreneurship from a communication point of view. Using discourse analysis, I analyse the discursive practices of crisis entrepreneurs in 1. Local moment formation in Dirac electrons International Nuclear Information System (INIS) Mashkoori, M; Mahyaeh, I; Jafari, S A 2015-01-01 Elemental bismuth and its compounds host strong spin-orbit interaction which is at the heart of topologically non-trivial alloys based on bismuth. These class of materials are described in terms of 4x4 matrices at each v point where spin and orbital labels of the underlying electrons are mixed. In this work we investigate the single impurity Anderson model (SIAM) within a mean field approximation to address the nature of local magnetic moment formation in a generic Dirac Hamiltonian. Despite the spin-mixing in the Hamiltonian, within the Hartree approximation it turns out that the impuritys Green function is diagonal in spin label. In the three dimensional Dirac materials defined over a bandwidth D and spin-orbit parameter γ, that hybridizes with impurity through V, a natural dimensionless parameter V 2 D/2πγ 3 emerges. So neither the hybridization strength, V, nor the spin-orbit coupling γ, but a combination thereof governs the phase diagram. By tuning chemical potential and the impurity level, we present phase diagram for various values of Hubbard U. Numerical results suggest that strong spin-orbit coupling enhances the local moment formation both in terms of its strength and the area of the local moment region. In the case that we tune the chemical potential in a similar way as normal metal we find that magnetic region is confined to μ ≥ ε 0 , in sharp contrast to 2D Dirac fermions. If one fixes the chemical potential and tunes the impurity level, phase diagram has two magnetic regions which corresponds to hybridization of impurity level with lower and upper bands. (paper) 2. The Dirac distorted wave Born approximation International Nuclear Information System (INIS) Cooper, T.; Sherif, H.S.; Johansson, J.; Sawafta, R.I. 1985-02-01 The purpose of this investigation is to illuminate the assumptions which are made when one writes down a Dirac DWBA matrix element. Due to the strong nature of the nucleon-nucleon potentials it is difficult to justify some of the steps involved in the general case; however by limiting ourselves to situations where only one (interacting) nucleon is present we can side-step this difficulty. We conclude the excellent agreement with the experiment justifies, a posteriori, the procedure, however we would like to remind the reader that, at least for proton inelastic scattering to collective states, the same quality of agreement can be obtained purely within a Schrodinger formalism 3. Total angular momentum from Dirac eigenspinors International Nuclear Information System (INIS) Szabados, Laszlo B 2008-01-01 The eigenvalue problem for Dirac operators, constructed from two connections on the spinor bundle over closed spacelike 2-surfaces, is investigated. A class of divergence-free vector fields, built from the eigenspinors, are found, which, for the lowest eigenvalue, reproduce the rotation Killing vectors of metric spheres, and provide rotation BMS vector fields at future null infinity. This makes it possible to introduce a well-defined, gauge invariant spatial angular momentum at null infinity, which reduces to the standard expression in stationary spacetimes. The general formula for the angular momentum flux carried away by the gravitational radiation is also derived 4. Incomplete Dirac reduction of constrained Hamiltonian systems Energy Technology Data Exchange (ETDEWEB) Chandre, C., E-mail: [email protected] 2015-10-15 First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac’s theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac–Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed. The relevance of this procedure for infinite dimensional Hamiltonian systems is exemplified. 5. Dispersionless wave packets in Dirac materials Czech Academy of Sciences Publication Activity Database Jakubský, Vít; Tušek, M. 2017-01-01 Roč. 378, MAR (2017), s. 171-182 ISSN 0003-4916 R&D Projects: GA ČR(CZ) GJ15-07674Y; GA ČR GA17-01706S Institutional support: RVO:61389005 Keywords : quantum systems * wave packets * dispersion * dirac materials Subject RIV: BE - Theoretical Physics OBOR OECD: Atomic, molecular and chemical physics ( physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect) Impact factor: 2.465, year: 2016 6. Agriculture Sectors Science.gov (United States) The Agriculture sectors comprise establishments primarily engaged in growing crops, raising animals, and harvesting fish and other animals. Find information on compliance, enforcement and guidance on EPA laws and regulations on the NAICS 111 & 112 sectors. 7. Electroweak-charged bound states as LHC probes of hidden forces Science.gov (United States) Li, Lingfeng; Salvioni, Ennio; Tsai, Yuhsin; Zheng, Rui 2018-01-01 We explore the LHC reach on beyond-the-standard model (BSM) particles X associated with a new strong force in a hidden sector. We focus on the motivated scenario where the SM and hidden sectors are connected by fermionic mediators ψ+,0 that carry SM electroweak charges. The most promising signal is the Drell-Yan production of a ψ±ψ¯ 0 pair, which forms an electrically charged vector bound state ϒ± due to the hidden force and later undergoes resonant annihilation into W±X . We analyze this final state in detail in the cases where X is a real scalar ϕ that decays to b b ¯, or a dark photon γd that decays to dileptons. For prompt X decays, we show that the corresponding signatures can be efficiently probed by extending the existing ATLAS and CMS diboson searches to include heavy resonance decays into BSM particles. For long-lived X , we propose new searches where the requirement of a prompt hard lepton originating from the W boson ensures triggering and essentially removes any SM backgrounds. To illustrate the potential of our results, we interpret them within two explicit models that contain strong hidden forces and electroweak-charged mediators, namely λ -supersymmetry (SUSY) and non-SUSY ultraviolet extensions of the twin Higgs model. The resonant nature of the signals allows for the reconstruction of the mass of both ϒ± and X , thus providing a wealth of information about the hidden sector. 8. Entry deterrence and hidden competition NARCIS (Netherlands) Lavrutich, Maria; Huisman, Kuno; Kort, Peter This paper studies strategic investment behavior of firms facing an uncertain demand in a duopoly setting. Firms choose both investment timing and the capacity level while facing additional uncertainty about market participants, which is introduced via the concept of hidden competition. We focus on 9. Adaptive Partially Hidden Markov Models DEFF Research Database (Denmark) Forchhammer, Søren Otto; Rasmussen, Tage 1996-01-01 Partially Hidden Markov Models (PHMM) have recently been introduced. The transition and emission probabilities are conditioned on the past. In this report, the PHMM is extended with a multiple token version. The different versions of the PHMM are applied to bi-level image coding.... 10. The Hidden Dimensions of Databases. Science.gov (United States) Jacso, Peter 1994-01-01 Discusses methods of evaluating commercial online databases and provides examples that illustrate their hidden dimensions. Topics addressed include size, including the number of records or the number of titles; the number of years covered; and the frequency of updates. Comparisons of Readers' Guide Abstracts and Magazine Article Summaries are… 11. Dirac's equation and the nature of quantum field theory International Nuclear Information System (INIS) Plotnitsky, Arkady 2012-01-01 This paper re-examines the key aspects of Dirac's derivation of his relativistic equation for the electron in order advance our understanding of the nature of quantum field theory. Dirac's derivation, the paper argues, follows the key principles behind Heisenberg's discovery of quantum mechanics, which, the paper also argues, transformed the nature of both theoretical and experimental physics vis-à-vis classical physics and relativity. However, the limit theory (a crucial consideration for both Dirac and Heisenberg) in the case of Dirac's theory was quantum mechanics, specifically, Schrödinger's equation, while in the case of quantum mechanics, in Heisenberg's version, the limit theory was classical mechanics. Dirac had to find a new equation, Dirac's equation, along with a new type of quantum variables, while Heisenberg, to find new theory, was able to use the equations of classical physics, applied to different, quantum-mechanical variables. In this respect, Dirac's task was more similar to that of Schrödinger in his work on his version of quantum mechanics. Dirac's equation reflects a more complex character of quantum electrodynamics or quantum field theory in general and of the corresponding (high-energy) experimental quantum physics vis-à-vis that of quantum mechanics and the (low-energy) experimental quantum physics. The final section examines this greater complexity and its implications for fundamental physics. 12. Special function solutions of the free particle Dirac equation International Nuclear Information System (INIS) Strange, P 2012-01-01 The Dirac equation is one of the fundamental equations in physics. Here we present and discuss two novel solutions of the free particle Dirac equation. These solutions have an exact analytical form in terms of Airy or Mathieu functions and exhibit unexpected properties including an enhanced Doppler effect, accelerating wavefronts and solutions with a degree of localization. (paper) 13. On oscillations of neutrinos with Dirac and Majorana masses International Nuclear Information System (INIS) Bilenky, S.M.; Hosek, J.; Petcov, S.T.; Bylgarska Akademiya na Naukite, Sofia) 1980-01-01 Pontecorvo neutrino beam oscillations are discussed assuming both Dirac and Majorana neutrino mass terms. It is proved that none of possible experiments on neutrino oscillations, including those on effects of CP violation, can distinguish between these two possibilities. Neutrino oscillations with concomitant Dirac and Majorana mass terms are also considered 14. Equivalence of Dirac quantization and Schwinger's action principle quantization International Nuclear Information System (INIS) Das, A.; Scherer, W. 1987-01-01 We show that the method of Dirac quantization is equivalent to Schwinger's action principle quantization. The relation between the Lagrange undetermined multipliers in Schwinger's method and Dirac's constraint bracket matrix is established and it is explicitly shown that the two methods yield identical (anti)commutators. This is demonstrated in the non-trivial example of supersymmetric quantum mechanics in superspace. (orig.) 15. Upper-Division Student Difficulties with the Dirac Delta Function Science.gov (United States) Wilcox, Bethany R.; Pollock, Steven J. 2015-01-01 The Dirac delta function is a standard mathematical tool that appears repeatedly in the undergraduate physics curriculum in multiple topical areas including electrostatics, and quantum mechanics. While Dirac delta functions are often introduced in order to simplify a problem mathematically, students still struggle to manipulate and interpret them.… 16. Multi-component bi-Hamiltonian Dirac integrable equations Energy Technology Data Exchange (ETDEWEB) Ma Wenxiu [Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700 (United States)], E-mail: [email protected] 2009-01-15 A specific matrix iso-spectral problem of arbitrary order is introduced and an associated hierarchy of multi-component Dirac integrable equations is constructed within the framework of zero curvature equations. The bi-Hamiltonian structure of the obtained Dirac hierarchy is presented be means of the variational trace identity. Two examples in the cases of lower order are computed. 17. Relativistic Spinning Particle without Grassmann Variables and the Dirac Equation Directory of Open Access Journals (Sweden) A. A. Deriglazov 2011-01-01 Full Text Available We present the relativistic particle model without Grassmann variables which, being canonically quantized, leads to the Dirac equation. Classical dynamics of the model is in correspondence with the dynamics of mean values of the corresponding operators in the Dirac theory. Classical equations for the spin tensor are the same as those of the Barut-Zanghi model of spinning particle. 18. The algebraic manipulation program DIRAC on IBM personal computers International Nuclear Information System (INIS) Grozin, A.G.; Perlt, H. 1989-01-01 The version DIRAC (2.2) for IBM compatible personal computers is described. It is designed to manipulate algebraically with polynomials and tensors. After a short introduction concerning implementation and usage on personal computers an example program is given. It contains a detailed user's guide to DIRAC (2.2) and, additionally some useful applications. 4 refs 19. A matricial approach for the Dirac-Kahler formalism International Nuclear Information System (INIS) Goto, M. 1987-01-01 A matricial approach for the Dirac-Kahler formalism is considered. It is shown that the matrical approach i) brings a great computational simplification compared to the common use of differential forms and that ii) by an appropriate choice of notation, it can be extended to the lattice, including a matrix Dirac-Kahler equation. (author) [pt 20. Dirac in 20th century physics: a centenary assessment International Nuclear Information System (INIS) Sanyuk, Valerii I; Sukhanov, Alexander D 2003-01-01 Current views on Dirac's creative heritage and on his role in the formation and development of quantum physics and in shaping the physical picture of the world are discussed. Dirac's fundamental ideas in later life (1948 - 1984) and their current development are given considerable attention. (from the history of physics) 1. Remarks about singular solutions to the Dirac equation International Nuclear Information System (INIS) Uhlir, M. 1975-01-01 In the paper singular solutions of the Dirac equation are investigated. They are derived in the Lorentz-covariant way of functions proportional to static multipole fields of scalar and (or) electromagnetic fields and of regular solutions of the Dirac equations. The regularization procedure excluding divergences of total energy, momentum and angular momentum of the spinor field considered is proposed 2. The new physician as unwitting quantum mechanic: is adapting Dirac's inference system best practice for personalized medicine, genomics, and proteomics? Science.gov (United States) Robson, Barry 2007-08-01 What is the Best Practice for automated inference in Medical Decision Support for personalized medicine? A known system already exists as Dirac's inference system from quantum mechanics (QM) using bra-kets and bras where A and B are states, events, or measurements representing, say, clinical and biomedical rules. Dirac's system should theoretically be the universal best practice for all inference, though QM is notorious as sometimes leading to bizarre conclusions that appear not to be applicable to the macroscopic world of everyday world human experience and medical practice. It is here argued that this apparent difficulty vanishes if QM is assigned one new multiplication function @, which conserves conditionality appropriately, making QM applicable to classical inference including a quantitative form of the predicate calculus. An alternative interpretation with the same consequences is if every i = radical-1 in Dirac's QM is replaced by h, an entity distinct from 1 and i and arguably a hidden root of 1 such that h2 = 1. With that exception, this paper is thus primarily a review of the application of Dirac's system, by application of linear algebra in the complex domain to help manipulate information about associations and ontology in complicated data. Any combined bra-ket can be shown to be composed only of the sum of QM-like bra and ket weights c(), times an exponential function of Fano's mutual information measure I(A; B) about the association between A and B, that is, an association rule from data mining. With the weights and Fano measure re-expressed as expectations on finite data using Riemann's Incomplete (i.e., Generalized) Zeta Functions, actual counts of observations for real world sparse data can be readily utilized. Finally, the paper compares identical character, distinguishability of states events or measurements, correlation, mutual information, and orthogonal character, important issues in data mining and biomedical analytics, as in QM. 3. The hidden dragons. Science.gov (United States) Zeng, Ming; Williamson, Peter J 2003-10-01 Most multinational corporations are fascinated with China. Carried away by the number of potential customers and the relatively cheap labor, firms seeking a presence in China have traditionally focused on selling products, setting up manufacturing facilities, or both. But they've ignored an important development: the emergence of Chinese firms as powerful rivals--in China and also in the global market. In this article, Ming Zeng and Peter Williamson describe how Chinese companies like Haier, Legend, and Pearl River Piano have quietly managed to grab market share from older, bigger, and financially stronger rivals in Asia, Europe, and the United States. Global managers tend to offer the usual explanations for why Chinese companies don't pose a threat: They aren't big enough or profitable enough to compete overseas, the managers say, and these primarily state-owned companies are ill-financed and ill-equipped for global competition. As the government's policies about the private ownership of companies changed from forbidding the practice to encouraging it, a new breed of Chinese companies evolved. The authors outline the four types of hybrid Chinese companies that are simultaneously tackling the global market. China's national champions are using their advantages as domestic leaders to build global brands. The dedicated exporters are entering foreign markets on the strength of their economies of scale. The competitive networks have taken on world markets by bringing together small, specialized companies that operate in close proximity. And the technology upstarts are using innovations developed by China's government-owned research institutes to enter emerging sectors such as biotechnology. Zeng and Williamson identify these budding multinationals, analyze their strategies, and evaluate their weaknesses. 4. Graphene Dirac point tuned by ferroelectric polarization field Science.gov (United States) Wang, Xudong; Chen, Yan; Wu, Guangjian; Wang, Jianlu; Tian, Bobo; Sun, Shuo; Shen, Hong; Lin, Tie; Hu, Weida; Kang, Tingting; Tang, Minghua; Xiao, Yongguang; Sun, Jinglan; Meng, Xiangjian; Chu, Junhao 2018-04-01 Graphene has received numerous attention for future nanoelectronics and optoelectronics. The Dirac point is a key parameter of graphene that provides information about its carrier properties. There are lots of methods to tune the Dirac point of graphene, such as chemical doping, impurities, defects, and disorder. In this study, we report a different approach to tune the Dirac point of graphene using a ferroelectric polarization field. The Dirac point can be adjusted to near the ferroelectric coercive voltage regardless its original position. We have ensured this phenomenon by temperature-dependent experiments, and analyzed its mechanism with the theory of impurity correlation in graphene. Additionally, with the modulation of ferroelectric polymer, the current on/off ratio and mobility of graphene transistor both have been improved. This work provides an effective method to tune the Dirac point of graphene, which can be readily used to configure functional devices such as p-n junctions and inverters. 5. Optical analogue of relativistic Dirac solitons in binary waveguide arrays Energy Technology Data Exchange (ETDEWEB) Tran, Truong X., E-mail: [email protected] [Department of Physics, Le Quy Don University, 236 Hoang Quoc Viet str., 10000 Hanoi (Viet Nam); Max Planck Institute for the Science of Light, Günther-Scharowsky str. 1, 91058 Erlangen (Germany); Longhi, Stefano [Department of Physics, Politecnico di Milano and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche, Piazza L. da Vinci 32, I-20133 Milano (Italy); Biancalana, Fabio [Max Planck Institute for the Science of Light, Günther-Scharowsky str. 1, 91058 Erlangen (Germany); School of Engineering and Physical Sciences, Heriot-Watt University, EH14 4AS Edinburgh (United Kingdom) 2014-01-15 We study analytically and numerically an optical analogue of Dirac solitons in binary waveguide arrays in the presence of Kerr nonlinearity. Pseudo-relativistic soliton solutions of the coupled-mode equations describing dynamics in the array are analytically derived. We demonstrate that with the found soliton solutions, the coupled mode equations can be converted into the nonlinear relativistic 1D Dirac equation. This paves the way for using binary waveguide arrays as a classical simulator of quantum nonlinear effects arising from the Dirac equation, something that is thought to be impossible to achieve in conventional (i.e. linear) quantum field theory. -- Highlights: •An optical analogue of Dirac solitons in nonlinear binary waveguide arrays is suggested. •Analytical solutions to pseudo-relativistic solitons are presented. •A correspondence of optical coupled-mode equations with the nonlinear relativistic Dirac equation is established. 6. Bosonic Analogue of Dirac Composite Fermi Liquid Science.gov (United States) Mross, David; Alicea, Jason; Motrunich, Olexei The status of particle-hole symmetry has long posed a challenge to the theory of the quantum Hall effect. It is expected to be present in the half-filled Landau level, but is absent in the conventional field theory, i.e., the composite Fermi liquid. Recently, Son proposed an alternative, explicitly particle-hole symmetric theory which features composite fermions that exhibit a Dirac dispersion. In my talk, I will introduce an analogous particle-hole-symmetric metallic state of bosons at odd-integer filling. This state hosts composite fermions whose energy dispersion features a quadratic band touching and corresponding 2 Ï Berry flux, protected by particle-hole and discrete rotation symmetries. As in the Dirac composite Fermi liquid introduced by Son, breaking particle-hole symmetry recovers the familiar Chern-Simons theory. I will discuss realizations of this phase both in 2D and on bosonic topological insulator surfaces, as well as its signatures in experiments and simulations. 7. Veselago focusing of anisotropic massless Dirac fermions Science.gov (United States) Zhang, Shu-Hui; Yang, Wen; Peeters, F. M. 2018-05-01 Massless Dirac fermions (MDFs) emerge as quasiparticles in various novel materials such as graphene and topological insulators, and they exhibit several intriguing properties, of which Veselago focusing is an outstanding example with a lot of possible applications. However, up to now Veselago focusing merely occurred in p-n junction devices based on the isotropic MDF, which lacks the tunability needed for realistic applications. Here, motivated by the emergence of novel Dirac materials, we investigate the propagation behaviors of anisotropic MDFs in such a p-n junction structure. By projecting the Hamiltonian of the anisotropic MDF to that of the isotropic MDF and deriving an exact analytical expression for the propagator, precise Veselago focusing is demonstrated without the need for mirror symmetry of the electron source and its focusing image. We show a tunable focusing position that can be used in a device to probe masked atom-scale defects. This study provides an innovative concept to realize Veselago focusing relevant for potential applications, and it paves the way for the design of novel electron optics devices by exploiting the anisotropic MDF. 8. White dwarfs, the galaxy and Dirac's cosmology International Nuclear Information System (INIS) Stothers, R. 1976-01-01 Reference is made to the apparent absence, or deficiency, of white dwarfs fainter than about 10 -4 L solar mass. An explanation is here proposed on the basis of Dirac's cosmological hypothesis that the gravitational constant, G, has varied with the time elapsed since the beginning of the expansion of the Universe as t -1 and the number of particles in the Universe has increases as t 2 , if the measurements are made in atomic units. For a white dwarf the Chandrasekhar mass limit is a collection of fundamental constants proportional to Gsup(-3/2) and therefore increases with time as tsup(3/2). In the 'additive' version of Dirac's theory the actual mass, M, of a relatively small object like a star remains essentially unchanged by the creation of new matter in the Universe and hence a white dwarf will become more stable with the course of time; but in the 'multiplicative' version of the theory, M increases as t 2 and may eventually exceed the Chandrasekhar limit, and if this happens, gravitational collapse of the white dwarf into an invisible black hole or neutron star will quickly occur. It is considered interesting to find whether the 'multiplicative' theory may have a bearing on the apparent deficiency of faint white dwarfs, and to consider whether there are any possible consequences for galactic evolution. This is here discussed. (U.K.) 9. White dwarfs, the galaxy and Dirac's cosmology Energy Technology Data Exchange (ETDEWEB) Stothers, R [National Aeronautics and Space Administration, Greenbelt, Md. (USA). Goddard Space Flight Center 1976-08-05 Reference is made to the apparent absence, or deficiency, of white dwarfs fainter than about 10/sup -4/L solar mass. An explanation is here proposed on the basis of Dirac's cosmological hypothesis that the gravitational constant, G, has varied with the time elapsed since the beginning of the expansion of the Universe as t/sup -1/ and the number of particles in the Universe has increases as t/sup 2/, if the measurements are made in atomic units. For a white dwarf the Chandrasekhar mass limit is a collection of fundamental constants proportional to Gsup(-3/2) and therefore increases with time as tsup(3/2). In the 'additive' version of Dirac's theory the actual mass, M, of a relatively small object like a star remains essentially unchanged by the creation of new matter in the Universe and hence a white dwarf will become more stable with the course of time; but in the 'multiplicative' version of the theory, M increases as t/sup 2/ and may eventually exceed the Chandrasekhar limit, and if this happens, gravitational collapse of the white dwarf into an invisible black hole or neutron star will quickly occur. It is considered interesting to find whether the 'multiplicative' theory may have a bearing on the apparent deficiency of faint white dwarfs, and to consider whether there are any possible consequences for galactic evolution. This is here discussed. 10. Dirac operators and Killing spinors with torsion; Dirac-Operatoren und Killing-Spinoren mit Torsion Energy Technology Data Exchange (ETDEWEB) Becker-Bender, Julia 2012-12-17 On a Riemannian spin manifold with parallel skew torsion, we use the twistor operator to obtain an eigenvalue estimate for the Dirac operator with torsion. We consider the equality case in dimensions four and six. In odd dimensions we describe Sasaki manifolds on which equality in the estimate is realized by Killing spinors with torsion. In dimension five we characterize all Killing spinors with torsion and obtain certain naturally reductive spaces as exceptional cases. 11. Axial anomaly and index theorem for Dirac-Kaehler fermions International Nuclear Information System (INIS) Fonseca Junior, C.A.L. da. 1985-02-01 Some aspects of topological influence on gauge field theory are analysed, considering the geometry and differential topology methods. A review of concepts of differential forms, fibered spaces, connection and curvature, showing an interpretation of gauge theory in this context, is presented. The question of fermions, analysing in details the Dirac-Kaehler which fermionic particle is considered a general differential form, is studied. It is shown how the explicit expressions in function of the Dirac spinor components vary with the Dirac matrix representation. The Dirac-Kahler equation contains 4 times (in 4 dimensions) the Dirac equation, each particle being associated an ideal at left of the algebra of general differential forms. These ideals and the SU(4) symmetry among them are also studied on the point of view of spinors and, the group of reduction to one of the ideals is identified as the Cartan subalgebra of this SU(4). Finally, the axial anomaly is calculated through the functional determinant given by the Dirac-Kaehler operator. The regularization method is the Seeley's coefficients. From that results a comparison of the index theorems for the twisted complexes of signature and spin, which proportionality is given by the number of the algebra ideals contained in the Dirac-Kaehler equation and which also manifests in the respective axial anomaly equations. (L.C.) [pt 12. Hidden supersymmetry and large N International Nuclear Information System (INIS) Alfaro, J. 1988-01-01 In this paper we present a new method to deal with the leading order in the large-N expansion of a quantum field theory. The method uses explicitly the hidden supersymmetry that is present in the path-integral formulation of a stochastic process. In addition to this we derive a new relation that is valid in the leading order of the large-N expansion of the hermitian-matrix model for any spacetime dimension. (orig.) 13. Topological insulators Dirac equation in condensed matter CERN Document Server Shen, Shun-Qing 2017-01-01 This new edition presents a unified description of these insulators from one to three dimensions based on the modified Dirac equation. It derives a series of solutions of the bound states near the boundary, and describes the current status of these solutions. Readers are introduced to topological invariants and their applications to a variety of systems from one-dimensional polyacetylene, to two-dimensional quantum spin Hall effect and p-wave superconductors, three-dimensional topological insulators and superconductors or superfluids, and topological Weyl semimetals, helping them to better understand this fascinating field. To reflect research advances in topological insulators, several parts of the book have been updated for the second edition, including: Spin-Triplet Superconductors, Superconductivity in Doped Topological Insulators, Detection of Majorana Fermions and so on. In particular, the book features a new chapter on Weyl semimetals, a topic that has attracted considerable attention and has already b... 14. New experimental proposals for testing Dirac equation International Nuclear Information System (INIS) Camacho, Abel; Macias, Alfredo 2004-01-01 The advent of phenomenological quantum gravity has ushered us in the search for experimental tests of the deviations from general relativity predicted by quantum gravity or by string theories, and as a by-product of this quest the possible modifications that some field equations, for instance, the motion equation of spin-1/2-particles, have already been considered. In the present Letter a modified Dirac equation, whose extra term embraces a second-order time derivative, is taken as mainstay, and three different experimental proposals to detect it are put forward. The novelty in these ideas is that two of them do not fall within the extant approaches in this context, to wit, red-shift, atomic interferometry, or Hughes-Drever type-like experiments 15. Topological Insulators Dirac Equation in Condensed Matters CERN Document Server Shen, Shun-Qing 2012-01-01 Topological insulators are insulating in the bulk, but process metallic states around its boundary owing to the topological origin of the band structure. The metallic edge or surface states are immune to weak disorder or impurities, and robust against the deformation of the system geometry. This book, Topological insulators, presents a unified description of topological insulators from one to three dimensions based on the modified Dirac equation. A series of solutions of the bound states near the boundary are derived, and the existing conditions of these solutions are described. Topological invariants and their applications to a variety of systems from one-dimensional polyacetalene, to two-dimensional quantum spin Hall effect and p-wave superconductors, and three-dimensional topological insulators and superconductors or superfluids are introduced, helping readers to better understand this fascinating new field. This book is intended for researchers and graduate students working in the field of topological in... 16. DIRAC reliable data management for LHCb CERN Document Server Smith, A C 2008-01-01 DIRAC, LHCb's Grid Workload and Data Management System, utilizes WLCG resources and middleware components to perform distributed computing tasks satisfying LHCb's Computing Model. The Data Management System (DMS) handles data transfer and data access within LHCb. Its scope ranges from the output of the LHCb Online system to Grid-enabled storage for all data types. It supports metadata for these files in replica and bookkeeping catalogues, allowing dataset selection and localization. The DMS controls the movement of files in a redundant fashion whilst providing utilities for accessing all metadata. To do these tasks effectively the DMS requires complete self integrity between its components and external physical storage. The DMS provides highly redundant management of all LHCb data to leverage available storage resources and to manage transient errors in underlying services. It provides data driven and reliable distribution of files as well as reliable job output upload, utilizing VO Boxes at LHCb Tier1 sites ... 17. Dark matter asymmetry in supersymmetric Dirac leptogenesis International Nuclear Information System (INIS) Choi, Ki-Young; Chun, Eung Jin; Shin, Chang Sub 2013-01-01 We discuss asymmetric or symmetric dark matter candidate in the supersymmetric Dirac leptogenesis scenario. By introducing a singlet superfield coupling to right-handed neutrinos, the overabundance problem of dark matter can be evaded and various possibilities for dark matter candidate arise. If the singlino is the lightest supersymmetric particle (LSP), it becomes naturally asymmetric dark matter. On the other hand, the right-handed sneutrino is a symmetric dark matter candidate whose relic density can be determined by the usual thermal freeze-out process. The conventional neutralino or gravitino LSP can be also a dark matter candidate as its non-thermal production from the right-handed sneutrino can be controlled appropriately. In our scenario, the late-decay of heavy supersymmetric particles mainly produces the right-handed sneutrino and neutrino which is harmless to the standard prediction of the Big-Bang Nucleosynthesis 18. Topology-optimized dual-polarization Dirac cones Science.gov (United States) Lin, Zin; Christakis, Lysander; Li, Yang; Mazur, Eric; Rodriguez, Alejandro W.; Lončar, Marko 2018-02-01 We apply a large-scale computational technique, known as topology optimization, to the inverse design of photonic Dirac cones. In particular, we report on a variety of photonic crystal geometries, realizable in simple isotropic dielectric materials, which exhibit dual-polarization Dirac cones. We present photonic crystals of different symmetry types, such as fourfold and sixfold rotational symmetries, with Dirac cones at different points within the Brillouin zone. The demonstrated and related optimization techniques open avenues to band-structure engineering and manipulating the propagation of light in periodic media, with possible applications to exotic optical phenomena such as effective zero-index media and topological photonics. 19. Majorana mass term, Dirac neutrinos and selective neutrino oscillations International Nuclear Information System (INIS) Leung, C.N. 1987-01-01 In a theory of neutrino mixing via a Majorana mass term involving only the left-handed neutrinos there exist selection rules for neutrino oscillations if true Dirac and/or exactly zero mass eigenstates are present. In the case of three neutrino flavours no oscillation is allowed if the mass spectrum contains one Dirac and one nondegenerate Majorana massive neutrino. The origin of these selection rules and their implications are discussed and the number of possible CP-violating phases in the lepton mixing matrix when Dirac and Majorana mass eigenstates coexist is given. (orig.) 20. [p,q] {ne} i{Dirac_h} Energy Technology Data Exchange (ETDEWEB) Costella, J P 1995-05-22 In this short note, it is argued that [p, q] {ne} i{Dirac_h}, contrary to the oiginal claims of Born and Jordan, and Dirac. Rather, [p, q] is equal to something that is infinitesimally different from i{Dirac_h}. While this difference is usually harmless, it does provide the solution of the Born-Jordan trace paradox of [p, q]. More recently, subtleties of a very similar form have been found to be of fundamental importance in quantum field theory. 3 refs. 1. A strong-topological-metal material with multiple Dirac cones OpenAIRE Ji, Huiwen; Pletikosić, I; Gibson, Q. D.; Sahasrabudhe, Girija; Valla, T.; Cava, R. J. 2015-01-01 We report a new, cleavable, strong-topological-metal, Zr2Te2P, which has the same tetradymite-type crystal structure as the topological insulator Bi2Te2Se. Instead of being a semiconductor, however, Zr2Te2P is metallic with a pseudogap between 0.2 and 0.7 eV above the fermi energy (EF). Inside this pseudogap, two Dirac dispersions are predicted: one is a surface-originated Dirac cone protected by time-reversal symmetry (TRS), while the other is a bulk-originated and slightly gapped Dirac cone... 2. Dirac particle in a box, and relativistic quantum Zeno dynamics International Nuclear Information System (INIS) Menon, Govind; Belyi, Sergey 2004-01-01 After developing a complete set of eigenfunctions for a Dirac particle restricted to a box, the quantum Zeno dynamics of a relativistic system is considered. The evolution of a continuously observed quantum mechanical system is governed by the theorem put forth by Misra and Sudarshan. One of the conditions for quantum Zeno dynamics to be manifest is that the Hamiltonian is semi-bounded. This Letter analyzes the effects of continuous observation of a particle whose time evolution is generated by the Dirac Hamiltonian. The theorem by Misra and Sudarshan is not applicable here since the Dirac operator is not semi-bounded 3. Gauging hidden symmetries in two dimensions International Nuclear Information System (INIS) Samtleben, Henning; Weidner, Martin 2007-01-01 We initiate the systematic construction of gauged matter-coupled supergravity theories in two dimensions. Subgroups of the affine global symmetry group of toroidally compactified supergravity can be gauged by coupling vector fields with minimal couplings and a particular topological term. The gauge groups typically include hidden symmetries that are not among the target-space isometries of the ungauged theory. The gaugings constructed in this paper are described group-theoretically in terms of a constant embedding tensor subject to a number of constraints which parametrizes the different theories and entirely encodes the gauged Lagrangian. The prime example is the bosonic sector of the maximally supersymmetric theory whose ungauged version admits an affine e 9 global symmetry algebra. The various parameters (related to higher-dimensional p-form fluxes, geometric and non-geometric fluxes, etc.) which characterize the possible gaugings, combine into an embedding tensor transforming in the basic representation of e 9 . This yields an infinite-dimensional class of maximally supersymmetric theories in two dimensions. We work out and discuss several examples of higher-dimensional origin which can be systematically analyzed using the different gradings of e 9 4. Pseudoclassical supergauge model for a (2 + 1) Dirac particle International Nuclear Information System (INIS) Gitman, D.M.; Gonsalves, A.E.; Tyhtin, I.V. 1997-01-01 A new pseudo-classical supergauge model of the Dirac particle in 2 + 1 dimensions is proposed. Two ways of its quantization are discussed. Both reproduce the minimal quantum theory of the particle. 24 refs 5. Magnetotransport in Layered Dirac Fermion System Coupled with Magnetic Moments Science.gov (United States) Iwasaki, Yoshiki; Morinari, Takao 2018-03-01 We theoretically investigate the magnetotransport of Dirac fermions coupled with localized moments to understand the physical properties of the Dirac material EuMnBi2. Using an interlayer hopping form, which simplifies the complicated interaction between the layers of Dirac fermions and the layers of magnetic moments in EuMnBi2, the theory reproduces most of the features observed in this system. The hysteresis observed in EuMnBi2 can be caused by the valley splitting that is induced by the spin-orbit coupling and the external magnetic field with the molecular field created by localized moments. Our theory suggests that the magnetotransport in EuMnBi2 is due to the interplay among Dirac fermions, localized moments, and spin-orbit coupling. 6. Using OSG Computing Resources with (iLC)Dirac CERN Document Server AUTHOR|(SzGeCERN)683529; Petric, Marko 2017-01-01 CPU cycles for small experiments and projects can be scarce, thus making use of all available resources, whether dedicated or opportunistic, is mandatory. While enabling uniform access to the LCG computing elements (ARC, CREAM), the DIRAC grid interware was not able to use OSG computing elements (GlobusCE, HTCondor-CE) without dedicated support at the grid site through so called 'SiteDirectors', which directly submit to the local batch system. This in turn requires additional dedicated effort for small experiments on the grid site. Adding interfaces to the OSG CEs through the respective grid middleware is therefore allowing accessing them within the DIRAC software without additional sitespecific infrastructure. This enables greater use of opportunistic resources for experiments and projects without dedicated clusters or an established computing infrastructure with the DIRAC software. To allow sending jobs to HTCondor-CE and legacy Globus computing elements inside DIRAC the required wrapper classes were develo... 7. Invariance properties of the Dirac equation with external electro ... Indian Academy of Sciences (India) . Introduction. The objective of this short paper is to investigate the invariance properties of the Dirac equation with external electro-magnetic field. There exists a large number of literatures on the problem beginning almost from the formulation ... 8. LHCb: Analysing DIRAC's Behavior using Model Checking with Process Algebra CERN Multimedia Remenska, Daniela 2012-01-01 DIRAC is the Grid solution designed to support LHCb production activities as well as user data analysis. Based on a service-oriented architecture, DIRAC consists of many cooperating distributed services and agents delivering the workload to the Grid resources. Services accept requests from agents and running jobs, while agents run as light-weight components, fulfilling specific goals. Services maintain database back-ends to store dynamic state information of entities such as jobs, queues, staging requests, etc. Agents use polling to check for changes in the service states, and react to these accordingly. A characteristic of DIRAC's architecture is the relatively low complexity in the logic of each agent; the main source of complexity lies in their cooperation. These agents run concurrently, and communicate using the services' databases as a shared memory for synchronizing the state transitions. Although much effort is invested in making DIRAC reliable, entities occasionally get into inconsistent states, leadi... 9. Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations KAUST Repository Lorz, Alexander 2011-01-17 Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses coexist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution. © Taylor & Francis Group, LLC. 10. Dirac directional emission in anisotropic zero refractive index photonic crystals. Science.gov (United States) He, Xin-Tao; Zhong, Yao-Nan; Zhou, You; Zhong, Zhi-Chao; Dong, Jian-Wen 2015-08-14 A certain class of photonic crystals with conical dispersion is known to behave as isotropic zero-refractive-index medium. However, the discrete building blocks in such photonic crystals are limited to construct multidirectional devices, even for high-symmetric photonic crystals. Here, we show multidirectional emission from low-symmetric photonic crystals with semi-Dirac dispersion at the zone center. We demonstrate that such low-symmetric photonic crystal can be considered as an effective anisotropic zero-refractive-index medium, as long as there is only one propagation mode near Dirac frequency. Four kinds of Dirac multidirectional emitters are achieved with the channel numbers of five, seven, eleven, and thirteen, respectively. Spatial power combination for such kind of Dirac directional emitter is also verified even when multiple sources are randomly placed in the anisotropic zero-refractive-index photonic crystal. 11. Elastic gauge fields and Hall viscosity of Dirac magnons Science.gov (United States) Ferreiros, Yago; Vozmediano, María A. H. 2018-02-01 We analyze the coupling of elastic lattice deformations to the magnon degrees of freedom of magnon Dirac materials. For a honeycomb ferromagnet we find that, as happens in the case of graphene, elastic gauge fields appear coupled to the magnon pseudospinors. For deformations that induce constant pseudomagnetic fields, the spectrum around the Dirac nodes splits into pseudo-Landau levels. We show that when a Dzyaloshinskii-Moriya interaction is considered, a topological gap opens in the system and a Chern-Simons effective action for the elastic degrees of freedom is generated. Such a term encodes a phonon Hall viscosity response, entirely generated by quantum fluctuations of magnons living in the vicinity of the Dirac points. The magnon Hall viscosity vanishes at zero temperature, and grows as temperature is raised and the states around the Dirac points are increasingly populated. 12. Science in culture the life of Paul Dirac CERN Multimedia Abbott, A 2000-01-01 The life of Paul Dirac has been used as the theme of a show held underground at the Delphi experiment at CERN. The 'Oracle of Delphi' was created as an outreach project and has been extremely successful (1 p). 13. Large optical conductivity of Dirac semimetal Fermi arc surface states Science.gov (United States) Shi, Li-kun; Song, Justin C. W. 2017-08-01 Fermi arc surface states, a hallmark of topological Dirac semimetals, can host carriers that exhibit unusual dynamics distinct from that of their parent bulk. Here we find that Fermi arc carriers in intrinsic Dirac semimetals possess a strong and anisotropic light-matter interaction. This is characterized by a large Fermi arc optical conductivity when light is polarized transverse to the Fermi arc; when light is polarized along the Fermi arc, Fermi arc optical conductivity is significantly muted. The large surface spectral weight is locked to the wide separation between Dirac nodes and persists as a large Drude weight of Fermi arc carriers when the system is doped. As a result, large and anisotropic Fermi arc conductivity provides a novel means of optically interrogating the topological surfaces states of Dirac semimetals. 14. Asymmetric dark matter and the hadronic spectra of hidden QCD Science.gov (United States) Lonsdale, Stephen J.; Schroor, Martine; Volkas, Raymond R. 2017-09-01 The idea that dark matter may be a composite state of a hidden non-Abelian gauge sector has received great attention in recent years. Frameworks such as asymmetric dark matter motivate the idea that dark matter may have similar mass to the proton, while mirror matter and G ×G grand unified theories provide rationales for additional gauge sectors which may have minimal interactions with standard model particles. In this work we explore the hadronic spectra that these dark QCD models can allow. The effects of the number of light colored particles and the value of the confinement scale on the lightest stable state, the dark matter candidate, are examined in the hyperspherical constituent quark model for baryonic and mesonic states. 15. From a world-sheet supersymmetry to the Dirac equation International Nuclear Information System (INIS) Mankoc Borstnik, N. 1991-10-01 Starting from a classical action for a point particle with a local world-sheet supersymmetry, the Dirac equation follows with operators α-vector, β-vector γ-vector being defined in the Grassmann space as differential operators and having all the properties of the corresponding Dirac matrices except that α-vector and β-vector are anti-Hermitian rather than Hermitian. Such a particle interacts with an external field as expected. (author). 7 refs 16. The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials International Nuclear Information System (INIS) Contreras-Astorga, Alonso; Schulze-Halberg, Axel 2014-01-01 We introduce the confluent version of the quantum-mechanical supersymmetry formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results [L. M. Nieto, A. A. Pecheritsin, and B. F. Samsonov, “Intertwining technique for the one-dimensional stationary Dirac equation,” Ann. Phys. 305, 151–189 (2003) 17. Topological Crystalline Insulators and Dirac Octets in Anti-perovskites OpenAIRE Hsieh, Timothy H.; Liu, Junwei; Fu, Liang 2014-01-01 We predict a new class of topological crystalline insulators (TCI) in the anti-perovskite material family with the chemical formula A$_3$BX. Here the nontrivial topology arises from band inversion between two$J=3/2$quartets, which is described by a generalized Dirac equation for a "Dirac octet". Our work suggests that anti-perovskites are a promising new venue for exploring the cooperative interplay between band topology, crystal symmetry and electron correlation. 18. Relativistic Photoionization Computations with the Time Dependent Dirac Equation Science.gov (United States) 2016-10-12 Naval Research Laboratory Washington, DC 20375-5320 NRL/MR/6795--16-9698 Relativistic Photoionization Computations with the Time Dependent Dirac... Photoionization Computations with the Time Dependent Dirac Equation Daniel F. Gordon and Bahman Hafizi Naval Research Laboratory 4555 Overlook Avenue, SW...Unclassified Unlimited Unclassified Unlimited 22 Daniel Gordon (202) 767-5036 Tunneling Photoionization Ionization of inner shell electrons by laser 19. Dirac and Weyl fermion dynamics on two-dimensional surface International Nuclear Information System (INIS) Kavalov, A.R.; Sedrakyan, A.G.; Kostov, I.K. 1986-01-01 Fermions on 2-dimensional surface, embedded into a 3-dimensional space are investigated. The determinant of induced Dirac operator for the Dirac and Weyl fermions is calculated. The reparametrization-invariant effective action is determined by conformal anomaly (giving Liouville action) and also by Lorentz anomaly leading to Wess-Zumino term, the structure of which at d=3 is determined by the Hopf topological invariant of the S 3 → S 2 map 20. Einstein-Cartan-Dirac theory in (1+2)-dimensions Energy Technology Data Exchange (ETDEWEB) Dereli, Tekin [Koc University, Department of Physics, Istanbul (Turkey); Oezdemir, Nese [Istanbul Technical University, Department of Physics Engineering, Istanbul (Turkey); Sert, Oezcan [Pamukkale University, Department of Physics, Denizli (Turkey) 2013-01-15 Einstein-Cartan theory is formulated in (1+2) dimensions using the algebra of exterior differential forms. A Dirac spinor is coupled to gravity and the field equations are obtained by a variational principle. The space-time torsion is found to be given algebraically in terms of a quadratic spinor condensate field. Circularly symmetric, exact solutions that collapse to AdS{sub 3} geometry in the absence of the Dirac condensate are found. (orig.) 1. Lie algebras for the Dirac-Clifford ring International Nuclear Information System (INIS) Mignaco, J.A.; Linhares, C.A. 1992-01-01 It is shown in a general way that the Dirac-Clifford ring formed by the Dirac matrices and all their products, for all even and odd spacetime dimensions D, span the cumulation algebras SU(2 D/2 ) for even D and SU(2 (D- 1 )/2 ) + SU(2 (D-1)/2 ) for odd D. Some physical consequences of these results are discussed. (author) 2. Neural network real time event selection for the DIRAC experiment CERN Document Server Kokkas, P; Tauscher, Ludwig; Vlachos, S 2001-01-01 The neural network first level trigger for the DIRAC experiment at CERN is presented. Both the neural network algorithm used and its actual hardware implementation are described. The system uses the fast plastic scintillator information of the DIRAC spectrometer. In 210 ns it selects events with two particles having low relative momentum. Such events are selected with an efficiency of more than 0.94. The corresponding rate reduction for background events is a factor of 2.5. (10 refs). 3. The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials Energy Technology Data Exchange (ETDEWEB) Contreras-Astorga, Alonso, E-mail: [email protected]; Schulze-Halberg, Axel, E-mail: [email protected], E-mail: [email protected] [Department of Mathematics and Actuarial Science and Department of Physics, Indiana University Northwest, 3400 Broadway, Gary, Indiana 46408 (United States) 2014-10-15 We introduce the confluent version of the quantum-mechanical supersymmetry formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results [L. M. Nieto, A. A. Pecheritsin, and B. F. Samsonov, “Intertwining technique for the one-dimensional stationary Dirac equation,” Ann. Phys. 305, 151–189 (2003)]. 4. The Dirac equation in the Lobachevsky space-time International Nuclear Information System (INIS) Paramonov, D.V.; Paramonova, N.N.; Shavokhina, N.S. 2000-01-01 The product of the Lobachevsky space and the time axis is termed the Lobachevsky space-time. The Lobachevsky space is considered as a hyperboloid's sheet in the four-dimensional pseudo-Euclidean space. The Dirac-Fock-Ivanenko equation is reduced to the Dirac equation in two special forms by passing from Lame basis in the Lobachevsky space to the Cartesian basis in the enveloping pseudo-Euclidean space 5. Simulation of Zitterbewegung by modelling the Dirac equation in Metamaterials OpenAIRE Ahrens, Sven; Jiang, Jun; Sun, Yong; Zhu, Shi-Yao 2015-01-01 We develop a dynamic description of an effective Dirac theory in metamaterials, in which the wavefunction is modeled by the corresponding electric and magnetic field in the metamaterial. This electro-magnetic field can be probed in the experimental setup, which means that the wavefunction of the effective theory is directly accessible by measurement. Our model is based on a plane wave expansion, which ravels the identification of Dirac spinors with single-frequency excitations of the electro-... 6. Nonlinear Excitations in Strongly-Coupled Fermi-Dirac Plasmas OpenAIRE Akbari-Moghanjoughi, M. 2012-01-01 In this paper we use the conventional quantum hydrodynamics (QHD) model in combination with the Sagdeev pseudopotential method to explore the effects of Thomas-Fermi nonuniform electron distribution, Coulomb interactions, electron exchange and ion correlation on the large-amplitude nonlinear soliton dynamics in Fermi-Dirac plasmas. It is found that in the presence of strong interactions significant differences in nonlinear wave dynamics of Fermi-Dirac plasmas in the two distinct regimes of no... 7. Revivals of zitterbewegung of a bound localized Dirac particle International Nuclear Information System (INIS) Romera, Elvira 2011-01-01 In this paper a bound localized Dirac particle is shown to exhibit a revival of the zitterbewegung (ZB) oscillation amplitude. These revivals go beyond the known quasiclassical regenerations in which the ZB oscillation amplitude is decreasing from period to period. This phenomenon is studied in a Dirac oscillator and it is shown that it is possible to set up wave packets in which there is a regeneration of the initial ZB amplitude. 8. Bound-state Dirac eigenvalues for scalar potentials International Nuclear Information System (INIS) Ram, B.; Arafah, M. 1981-01-01 The Dirac equation is solved with a linear and a quadratic scalar potential using an approach in which the Dirac equation is first transformed to a one-dimensional Schroedinger equation with an effective potential. The WKB method is used to obtain the energy eigenvalues. The eigenvalues for the quadratic scalar potential are real just as they are for the linear potential. The results with the linear potential agree well with those obtained by Critchfield. (author) 9. [P.A.M. Dirac and antimatter applied to medicine]. Science.gov (United States) Kulenović, Fahrudin; Vobornik, Slavenka; Dalagija, Faruk 2003-01-01 Regarding to the hundredth anniversary of P. Dirac birth, it was made review on life and work of this genius in the history of physics and science generally. His ingenious scientific work, that significantly marked contemporary time, was presented in the simplest way with aim to approach more number of readers. Special accent was put on application of Dirac's ideas about antiparticles in medical practice. 10. Hidden measurements, hidden variables and the volume representation of transition probabilities OpenAIRE Oliynyk, Todd A. 2005-01-01 We construct, for any finite dimension$n$, a new hidden measurement model for quantum mechanics based on representing quantum transition probabilities by the volume of regions in projective Hilbert space. For$n=2$our model is equivalent to the Aerts sphere model and serves as a generalization of it for dimensions$n \\geq 3$. We also show how to construct a hidden variables scheme based on hidden measurements and we discuss how joint distributions arise in our hidden variables scheme and th... 11. Particle creation and Dirac's large number hypothesis; and Reply International Nuclear Information System (INIS) Canuto, V.; Adams, P.J.; Hsieh, S.H.; Tsiang, E.; Steigman, G. 1976-01-01 The claim made by Steigman (Nature; 261:479 (1976)), that the creation of matter as postulated by Dirac (Proc. R. Soc.; A338:439 (1974)) is unnecessary, is here shown to be incorrect. It is stated that Steigman's claim that Dirac's large Number Hypothesis (LNH) does not require particle creation is wrong because he has assumed that which he was seeking to prove, that is that rho does not contain matter creation. Steigman's claim that Dirac's LNH leads to nonsensical results in the very early Universe is superficially correct, but this only supports Dirac's contention that the LNH may not be valid in the very early Universe. In a reply Steigman points out that in Dirac's original cosmology R approximately tsup(1/3) and using this model the results and conclusions of the present author's paper do apply but using a variation chosen by Canuto et al (T approximately t) Dirac's LNH cannot apply. Additionally it is observed that a cosmological theory which only predicts the present epoch is of questionable value. (U.K.) 12. The Lorentz-Dirac equation in light of quantum theory International Nuclear Information System (INIS) Nikishov, A.I. 1996-01-01 To high accuracy, an electron in ultrarelativistic motion 'sees' an external field in its rest frame as a crossed field (E=H, E·H=0). In this case, quantum expressions allow the introduction of a local intensity of the radiation, which determines the radiative term of the force of radiative reaction. For γ=(1-v2)-1/2>> 1 this term is much larger than the mass term, i.e., the term with xd3do. Under these conditions, the reduced Lorentz-Dirac equation, which is obtained from the full Lorentz-Dirac equation by eliminating the terms xd3do and xe on the right side using the equation of motion without taking into account the force of radiative reaction, is equivalent to good accuracy to the original Lorentz-Dirac equation. Exact solutions to the reduced Lorentz-Dirac equation are obtained for a constant field and the field of a plane wave. For γ∼1 a local expression for the radiative term cannot be obtained quantitatively from the quantum expressions. In this case the mass (Lorentz-Dirac) terms in the original and reduced Lorentz-Dirac equations are not small compared to the radiative term. The predictions of these equations, which depend appreciably on the mass terms, are therefore less reliable 13. The DIRAC Data Management System and the Gaudi dataset federation CERN Document Server Haen, Christophe; Frank, Markus; Tsaregorodtsev, Andrei 2015-01-01 The DIRAC Interware provides a development framework and a complete set of components for building distributed computing systems. The DIRAC Data Management System (DMS) offers all the necessary tools to ensure data handling operations for small and large user communities. It supports transparent access to storage resources based on multiple technologies, and is easily expandable. The information on data files and replicas is kept in a File Catalog of which DIRAC offers a powerful and versatile implementation (DFC). Data movement can be performed using third party services including FTS3. Bulk data operations are resilient with respect to failures due to the use of the Request Management System (RMS) that keeps track of ongoing tasks.In this contribution we will present an overview of the DIRAC DMS capabilities and its connection with other DIRAC subsystems such as the Transformation System. This paper also focuses on the DIRAC File Catalog, for which a lot of new developments have been carried out, so that LH... 14. Split Dirac Supersymmetry: An Ultraviolet Completion of Higgsino Dark Matter Energy Technology Data Exchange (ETDEWEB) Fox, Patrick J. [Fermilab; Kribs, Graham D. [Oregon U.; Martin, Adam [Notre Dame U. 2014-10-07 Motivated by the observation that the Higgs quartic coupling runs to zero at an intermediate scale, we propose a new framework for models of split supersymmetry, in which gauginos acquire intermediate scale Dirac masses of$\\sim 10^{8-11}$GeV. Scalar masses arise from one-loop finite contributions as well as direct gravity-mediated contributions. Like split supersymmetry, one Higgs doublet is fine-tuned to be light. The scale at which the Dirac gauginos are introduced to make the Higgs quartic zero is the same as is necessary for gauge coupling unification. Thus, gauge coupling unification persists (nontrivially, due to adjoint multiplets), though with a somewhat higher unification scale$\\gtrsim 10^{17}$GeV. The$\\mu\$-term is naturally at the weak scale, and provides an opportunity for experimental verification. We present two manifestations of Split Dirac Supersymmetry. In the "Pure Dirac" model, the lightest Higgsino must decay through R-parity violating couplings, leading to an array of interesting signals in colliders. In the "Hypercharge Impure" model, the bino acquires a Majorana mass that is one-loop suppressed compared with the Dirac gluino and wino. This leads to weak scale Higgsino dark matter whose overall mass scale, as well as the mass splitting between the neutral components, is naturally generated from the same UV dynamics. We outline the challenges to discovering pseudo-Dirac Higgsino dark matter in collider and dark matter detection experiments.
15. Dirac equation in low dimensions: The factorization method
Energy Technology Data Exchange (ETDEWEB)
Sánchez-Monroy, J.A., E-mail: [email protected] [Instituto de Física, Universidade de São Paulo, 05508-090, São Paulo, SP (Brazil); Quimbay, C.J., E-mail: [email protected] [Departamento de Física, Universidad Nacional de Colombia, Bogotá, D. C. (Colombia); CIF, Bogotá (Colombia)
2014-11-15
We present a general approach to solve the (1+1) and (2+1)-dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein–Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials. - Highlights: • The low-dimensional Dirac equation in the presence of static potentials is solved. • The factorization method is generalized for energy-dependent Hamiltonians. • The shape invariance is generalized for energy-dependent Hamiltonians. • The stability of the Dirac sea is related to the existence of supersymmetric partner Hamiltonians.
16. P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation
International Nuclear Information System (INIS)
Duran, Antonio J; Gruenbaum, F Alberto
2006-01-01
The solution of several instances of the Schroedinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585-600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Gruenbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461-84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein
17. Understanding quaternions and the Dirac belt trick
International Nuclear Information System (INIS)
Staley, Mark
2010-01-01
The Dirac belt trick is often employed in physics classrooms to show that a 2π rotation is not topologically equivalent to the absence of rotation whereas a 4π rotation is, mirroring a key property of quaternions and their isomorphic cousins, spinors. The belt trick can leave the student wondering if a real understanding of quaternions and spinors has been achieved, or if the trick is just an amusing analogy. The goal of this paper is to demystify the belt trick and to show that it suggests an underlying four-dimensional parameter space for rotations that is simply connected. An investigation into the geometry of this four-dimensional space leads directly to the system of quaternions, and to an interpretation of three-dimensional vectors as the generators of rotations in this larger four-dimensional world. The paper also shows why quaternions are the natural extension of complex numbers to four dimensions. The level of the paper is suitable for undergraduate students of physics.
18. DIRAC: reliable data management for LHCb
International Nuclear Information System (INIS)
Smith, A C; Tsaregorodtsev, A
2008-01-01
DIRAC, LHCb's Grid Workload and Data Management System, utilizes WLCG resources and middleware components to perform distributed computing tasks satisfying LHCb's Computing Model. The Data Management System (DMS) handles data transfer and data access within LHCb. Its scope ranges from the output of the LHCb Online system to Grid-enabled storage for all data types. It supports metadata for these files in replica and bookkeeping catalogues, allowing dataset selection and localization. The DMS controls the movement of files in a redundant fashion whilst providing utilities for accessing all metadata. To do these tasks effectively the DMS requires complete self integrity between its components and external physical storage. The DMS provides highly redundant management of all LHCb data to leverage available storage resources and to manage transient errors in underlying services. It provides data driven and reliable distribution of files as well as reliable job output upload, utilizing VO Boxes at LHCb Tier1 sites to prevent data loss. This paper presents several examples of mechanisms implemented in the DMS to increase reliability, availability and integrity, highlighting successful design choices and limitations discovered
19. Hidden inventory and safety considerations
International Nuclear Information System (INIS)
Anderson, A.R.; James, R.H.; Morgan, F.
1976-01-01
Preliminary results are described of the evaluation of residual plutonium in a process line used for the production of experimental fast reactor fuel. Initial attention has been focussed on a selection of work boxes used for processing powders and solutions. Amounts of material measured as ''hidden inventory'' are generally less than 0.1 percent of throughput but in one box containing very complex equipment the amount was exceptionally about 0.5 percent. The total surface area of the box and the installed equipment appears to be the most significant factor in determining the amount of plutonium held-up as ''hidden inventory,'' representing an average of about 4 x 10 -4 g cm -2 . Present results are based on gamma spectrometer measurements but neutron techniques are being developed to overcome some of the inherent uncertainties in the gamma method. It is suggested that the routine use of sample plates of known surface area would be valuable in monitoring the deposition of plutonium in work boxes
20. Hidden costs of nuclear power
International Nuclear Information System (INIS)
England, R.W.
1979-01-01
Mr. England contends that these hidden costs add up to a figure much higher than those that appear in the electric utilities' profit and loss account - costs that are borne by Federal taxpayers, by nuclear industry workers, and by all those people who must share their environment with nuclear facilities. Costs he details are additional deaths and illnesses resulting from exposure to radiation, and the use of tax dollars to clean up the lethal garbage produced by those activities. He asserts that careless handling of uranium ore and mill tailings in past years has apparently resulted in serious public health problems in those mining communities. In another example, Mr. England states that the failure to isolate uranium tailings physically from their environment has probably contributed to an acute leukemia rate in Mesa County, Colorado. He mentions much of the technology development for power reactors being done by the Federal government, not by private reactor manufacturers - thus, again, hidden costs that do not show up in electric bills of customers. The back end of the nuclear fuel cycle as a place for Federally subsidized research and development is discussed briefly. 1 figure, 2 tables
1. The Hidden Reason Behind Children's Misbehavior.
Science.gov (United States)
Nystul, Michael S.
1986-01-01
Discusses hidden reason theory based on the assumptions that: (1) the nature of people is positive; (2) a child's most basic psychological need is involvement; and (3) a child has four possible choices in life (good somebody, good nobody, bad somebody, or severely mentally ill.) A three step approach for implementing hidden reason theory is…
2. Hidden neural networks: application to speech recognition
DEFF Research Database (Denmark)
Riis, Søren Kamaric
1998-01-01
We evaluate the hidden neural network HMM/NN hybrid on two speech recognition benchmark tasks; (1) task independent isolated word recognition on the Phonebook database, and (2) recognition of broad phoneme classes in continuous speech from the TIMIT database. It is shown how hidden neural networks...
3. Insight: Exploring Hidden Roles in Collaborative Play
Directory of Open Access Journals (Sweden)
Tricia Shi
2015-06-01
Full Text Available This paper looks into interaction modes between players in co-located, collaborative games. In particular, hidden traitor games, in which one or more players is secretly working against the group mission, has the effect of increasing paranoia and distrust between players, so this paper looks into the opposite of a hidden traitor – a hidden benefactor. Rather than sabotaging the group mission, the hidden benefactor would help the group achieve the end goal while still having a reason to stay hidden. The paper explores what games with such a role can look like and how the role changes player interactions. Finally, the paper addresses the divide between video game and board game interaction modes; hidden roles are not common within video games, but they are of growing prevalence in board games. This fact, combined with the exploration of hidden benefactors, reveals that hidden roles is a mechanic that video games should develop into in order to match board games’ complexity of player interaction modes.
4. Hidden variables and locality in quantum theory
International Nuclear Information System (INIS)
Shiva, Vandana.
1978-12-01
The status of hidden variables in quantum theory has been debated since the 1920s. The author examines the no-hidden-variable theories of von Neumann, Kochen, Specker and Bell, and finds that they all share one basic assumption: averaging over the hidden variables should reproduce the quantum mechanical probabilities. Von Neumann also makes a linearity assumption, Kochen and Specker require the preservation of certain functional relations between magnitudes, and Bell proposes a locality condition. It has been assumed that the extrastatistical requirements are needed to serve as criteria of success for the introduction of hidden variables because the statistical condition is trivially satisfied, and that Bell's result is based on a locality condition that is physically motivated. The author shows that the requirement of weak locality, which is not physically motivated, is enough to give Bell's result. The proof of Bell's inequality works equally well for any pair of commuting magnitudes satisfying a condition called the degeneracy principle. None of the no-hidden-variable proofs apply to a class of hidden variable theories that are not phase-space reconstructions of quantum mechanics. The author discusses one of these theories, the Bohm-Bub theory, and finds that hidden variable theories that re all the quantum statistics, for single and sequential measurements, must introduce a randomization process for the hidden variables after each measurement. The philosophical significance of this theory lies in the role it can play in solving the conceptual puzzles posed by quantum theory
5. Hidden supersymmetry and Fermion number fractionalization
International Nuclear Information System (INIS)
Akhoury, R.
1985-01-01
This paper discusses how a hidden supersymmetry of the underlying field theories can be used to interpret and to calculate fermion number fractionalization in different dimensions. This is made possible by relating it to a corresponding Witten index of the hidden supersymmetry. The closely related anomalies in odd dimensions are also discussed
6. Constraints on hidden photons from current and future observations of CMB spectral distortions
International Nuclear Information System (INIS)
Kunze, Kerstin E.; Vázquez-Mozo, Miguel Á.
2015-01-01
A variety of beyond the standard model scenarios contain very light hidden sector U(1) gauge bosons undergoing kinetic mixing with the photon. The resulting oscillation between ordinary and hidden photons leads to spectral distortions of the cosmic microwave background. We update the bounds on the mixing parameter χ 0 and the mass of the hidden photon m γ' for future experiments measuring CMB spectral distortions, such as PIXIE and PRISM/COrE. For 10 −14 eV∼< m γ' ∼< 10 −13 eV, we find the kinetic mixing angle χ 0 has to be less than 10 −8 at 95% CL. These bounds are more than an order of magnitude stronger than those derived from the COBE/FIRAS data
7. Double Dirac Point Semimetal in Two-Dimensional Material: Ta2Se3
OpenAIRE
Ma, Yandong; Jing, Yu; Heine, Thomas
2017-01-01
Here, we report by first-principles calculations one new stable 2D Dirac material, Ta2Se3 monolayer. For this system, stable layered bulk phase exists, and exfoliation should be possible. Ta2Se3 monolayer is demonstrated to support two Dirac points close to the Fermi level, achieving the exotic 2D double Dirac semimetal. And like 2D single Dirac and 2D node-line semimetals, spin-orbit coupling could introduce an insulating state in this new class of 2D Dirac semimetals. Moreover, the Dirac fe...
8. 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)....
9. Hidden symmetry in the presence of fluxes
International Nuclear Information System (INIS)
Kubiznak, David; Warnick, Claude M.; Krtous, Pavel
2011-01-01
We derive the most general first-order symmetry operator for the Dirac equation coupled to arbitrary fluxes. Such an operator is given in terms of an inhomogeneous form ω which is a solution to a coupled system of first-order partial differential equations which we call the generalized conformal Killing-Yano system. Except trivial fluxes, solutions of this system are subject to additional constraints. We discuss various special cases of physical interest. In particular, we demonstrate that in the case of a Dirac operator coupled to the skew symmetric torsion and U(1) field, the system of generalized conformal Killing-Yano equations decouples into the homogeneous conformal Killing-Yano equations with torsion introduced in D. Kubiznak et al. (2009) and the symmetry operator is essentially the one derived in T. Houri et al. (2010) . We also discuss the Dirac field coupled to a scalar potential and in the presence of 5-form and 7-form fluxes.
10. Dirac matrices for Chern-Simons gravity
Energy Technology Data Exchange (ETDEWEB)
Izaurieta, Fernando; Ramirez, Ricardo; Rodriguez, Eduardo [Departamento de Matematica y Fisica Aplicadas, Universidad Catolica de la Santisima Concepcion, Alonso de Ribera 2850, 4090541 Concepcion (Chile)
2012-10-06
A genuine gauge theory for the Poincare, de Sitter or anti-de Sitter algebras can be constructed in (2n- 1)-dimensional spacetime by means of the Chern-Simons form, yielding a gravitational theory that differs from General Relativity but shares many of its properties, such as second order field equations for the metric. The particular form of the Lagrangian is determined by a rank n, symmetric tensor invariant under the relevant algebra. In practice, the calculation of this invariant tensor can be reduced to the computation of the trace of the symmetrized product of n Dirac Gamma matrices {Gamma}{sub ab} in 2n-dimensional spacetime. While straightforward in principle, this calculation can become extremely cumbersome in practice. For large enough n, existing computer algebra packages take an inordinate long time to produce the answer or plainly fail having used up all available memory. In this talk we show that the general formula for the trace of the symmetrized product of 2n Gamma matrices {Gamma}{sub ab} can be written as a certain sum over the integer partitions s of n, with every term being multiplied by a numerical cofficient {alpha}{sub s}. We then give a general algorithm that computes the {alpha}-coefficients as the solution of a linear system of equations generated by evaluating the general formula for different sets of tensors B{sup ab} with random numerical entries. A recurrence relation between different coefficients is shown to hold and is used in a second, 'minimal' algorithm to greatly speed up the computations. Runtime of the minimal algorithm stays below 1 min on a typical desktop computer for up to n = 25, which easily covers all foreseeable applications of the trace formula.
11. Hidden scale invariance of metals
DEFF Research Database (Denmark)
Hummel, Felix; Kresse, Georg; Dyre, Jeppe C.
2015-01-01
Density functional theory (DFT) calculations of 58 liquid elements at their triple point show that most metals exhibit near proportionality between the thermal fluctuations of the virial and the potential energy in the isochoric ensemble. This demonstrates a general “hidden” scale invariance...... of metals making the condensed part of the thermodynamic phase diagram effectively one dimensional with respect to structure and dynamics. DFT computed density scaling exponents, related to the Grüneisen parameter, are in good agreement with experimental values for the 16 elements where reliable data were...... available. Hidden scale invariance is demonstrated in detail for magnesium by showing invariance of structure and dynamics. Computed melting curves of period three metals follow curves with invariance (isomorphs). The experimental structure factor of magnesium is predicted by assuming scale invariant...
12. Hidden Valley Search at ATLAS
CERN Document Server
Verducci, M
2011-01-01
A number of extensions of the Standard Model result in neutral and weakly-coupled particles that decay to multi hadrons or multi leptons with macroscopic decay lengths. These particles with decay paths that can be comparable with ATLAS detector dimensions represent, from an experimental point of view, a challenge both for the trigger and for the reconstruction capabilities of the ATLAS detector. We will present a set of signature driven triggers for the ATLAS detector that target such displaced decays and evaluate their performances for some benchmark models and describe analysis strategies and limits on the production of such long-lived particles. A first estimation of the Hidden Valley trigger rates has been evaluated with 6 pb-1 of data collected at ATLAS during the data taking of 2010.
13. Dirac dark matter and b →s ℓ+ℓ- with U(1) gauge symmetry
Science.gov (United States)
Celis, Alejandro; Feng, Wan-Zhe; Vollmann, Martin
2017-02-01
We revisit the possibility of a Dirac fermion dark matter candidate in the light of current b →s ℓ+ℓ- anomalies by investigating a minimal extension of the Standard Model with a horizontal U(1 ) ' local symmetry. Dark matter stability is protected by a remnant Z2 symmetry arising after spontaneous symmetry breaking of U(1 ) '. The associated Z' gauge boson can accommodate current hints of new physics in b →s ℓ+ℓ- decays, and acts as a vector portal between dark matter and the visible sector. We find that the model is severely constrained by a combination of precision measurements at flavor factories, LHC searches for dilepton resonances, as well as direct and indirect dark matter searches. Despite this, viable regions of the parameter space accommodating the observed dark matter relic abundance and the b →s ℓ+ℓ-anomalies still persist for dark matter and Z ' masses in the TeV range.
14. Relativistic space-charge-limited current for massive Dirac fermions
Science.gov (United States)
Ang, Y. S.; Zubair, M.; Ang, L. K.
2017-04-01
A theory of relativistic space-charge-limited current (SCLC) is formulated to determine the SCLC scaling, J ∝Vα/Lβ , for a finite band-gap Dirac material of length L biased under a voltage V . In one-dimensional (1D) bulk geometry, our model allows (α ,β ) to vary from (2,3) for the nonrelativistic model in traditional solids to (3/2,2) for the ultrarelativistic model of massless Dirac fermions. For 2D thin-film geometry we obtain α =β , which varies between 2 and 3/2, respectively, at the nonrelativistic and ultrarelativistic limits. We further provide rigorous proof based on a Green's-function approach that for a uniform SCLC model described by carrier-density-dependent mobility, the scaling relations of the 1D bulk model can be directly mapped into the case of 2D thin film for any contact geometries. Our simplified approach provides a convenient tool to obtain the 2D thin-film SCLC scaling relations without the need of explicitly solving the complicated 2D problems. Finally, this work clarifies the inconsistency in using the traditional SCLC models to explain the experimental measurement of a 2D Dirac semiconductor. We conclude that the voltage scaling 3 /2 <α <2 is a distinct signature of massive Dirac fermions in a Dirac semiconductor and is in agreement with experimental SCLC measurements in MoS2.
15. Industrial sector
International Nuclear Information System (INIS)
Ainul Hayati Daud; Hazmimi Kasim
2010-01-01
The industrial sector is categorized as related to among others, the provision of technical and engineering services, supply of products, testing and troubleshooting of parts, systems and industrial plants, quality control and assurance as well as manufacturing and processing. A total of 161 entities comprising 47 public agencies and 114 private companies were selected for the study in this sector. The majority of the public agencies, 87 %, operate in Peninsular Malaysia. The remainders were located in Sabah and Sarawak. The findings of the study on both public agencies and private companies are presented in subsequent sections of this chapter. (author)
16. Agricultural sector
International Nuclear Information System (INIS)
Ainul Hayati Daud; Hazmimi Kasim
2010-01-01
The applications of nuclear technology in agriculture sector cover the use of the technology at every aspects of agricultural activity, starting from the seed to harvesting as well as the management of plantations itself. In this sector, a total of 55 entities comprising 17 public agencies and 38 private companies were selected for the study. Almost all, 91 % of them are located in Peninsular Malaysia; the rest operates in Sabah and Sarawak. The findings of the study in the public agencies and private companies are presented in the next sections. (author)
17. Baryon states with hidden charm in the extended local hidden gauge approach
International Nuclear Information System (INIS)
Uchino, T.; Oset, E.; Liang, Wei-Hong
2016-01-01
The s-wave interaction of anti DΛ c , anti DΣ c , anti D * Λ c , anti D * Σ c and anti DΣ c * , anti D * Σ c * , is studied within a unitary coupled channels scheme with the extended local hidden gauge approach. In addition to the Weinberg-Tomozawa term, several additional diagrams via the pion exchange are also taken into account as box potentials. Furthermore, in order to implement the full coupled channels calculation, some of the box potentials which mix the vector-baryon and pseudoscalar-baryon sectors are extended to construct the effective transition potentials. As a result, we have observed six possible states in several angular momenta. Four of them correspond to two pairs of admixture states, two of anti DΣ c - anti D * Σ c with J = 1/2, and two of anti DΣ c * - anti D * Σ c * with J = 3/2. Moreover, we find a anti D * Σ c resonance which couples to the anti DΛ c channel and one spin degenerated bound state of anti D * Σ c * with J = 1/2,5/2. (orig.)
18. Baryon states with hidden charm in the extended local hidden gauge approach
Energy Technology Data Exchange (ETDEWEB)
Uchino, T.; Oset, E. [Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigacion de Paterna, Departamento de Fisica Teorica y IFIC, Valencia (Spain); Liang, Wei-Hong [Guangxi Normal University, Department of Physics, Guilin (China)
2016-03-15
The s-wave interaction of anti DΛ{sub c}, anti DΣ{sub c}, anti D{sup *}Λ{sub c}, anti D{sup *}Σ{sub c} and anti DΣ{sub c}{sup *}, anti D{sup *}Σ{sub c}{sup *}, is studied within a unitary coupled channels scheme with the extended local hidden gauge approach. In addition to the Weinberg-Tomozawa term, several additional diagrams via the pion exchange are also taken into account as box potentials. Furthermore, in order to implement the full coupled channels calculation, some of the box potentials which mix the vector-baryon and pseudoscalar-baryon sectors are extended to construct the effective transition potentials. As a result, we have observed six possible states in several angular momenta. Four of them correspond to two pairs of admixture states, two of anti DΣ{sub c} - anti D{sup *}Σ{sub c} with J = 1/2, and two of anti DΣ{sub c}{sup *} - anti D{sup *}Σ{sub c}{sup *} with J = 3/2. Moreover, we find a anti D{sup *}Σ{sub c} resonance which couples to the anti DΛ{sub c} channel and one spin degenerated bound state of anti D{sup *}Σ{sub c}{sup *} with J = 1/2,5/2. (orig.)
19. Dirac equation and optical wave propagation in one dimension
Energy Technology Data Exchange (ETDEWEB)
Gonzalez, Gabriel [Catedras CONACYT, Universidad Autonoma de San Luis Potosi (Mexico); Coordinacion para la Innovacion y la Aplicacion de la Ciencia y la Tecnologia, Universidad Autonoma de San Luis Potosi (Mexico)
2018-02-15
We show that the propagation of transverse electric (TE) polarized waves in one-dimensional inhomogeneous settings can be written in the form of the Dirac equation in one space dimension with a Lorentz scalar potential, and consequently perform photonic simulations of the Dirac equation in optical structures. In particular, we propose how the zero energy state of the Jackiw-Rebbi model can be generated in an optical set-up by controlling the refractive index landscape, where TE-polarized waves mimic the Dirac particles and the soliton field can be tuned by adjusting the refractive index. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
20. Shot noise in systems with semi-Dirac points
International Nuclear Information System (INIS)
Zhai, Feng; Wang, Juan
2014-01-01
We calculate the ballistic conductance and shot noise of electrons through a two-dimensional stripe system (width W ≫ length L) with semi-Dirac band-touching points. We find that the ratio between zero-temperature noise power and mean current (the Fano factor) is highly anisotropic. When the transport is along the linear-dispersion direction and the Fermi energy is fixed at the semi-Dirac point, the Fano factor has a universal value F = 0.179 while a minimum conductivity exists and scales with L 1∕2 . Along the parabolic dispersion direction, the Fano factor at the semi-Dirac point has a contact-independent limit exceeding 0.9, which varies weakly with L due to the common-path interference of evanescent waves. Our findings suggest a way to discern the type of band-touching points
1. Spin-1 Dirac-Weyl fermions protected by bipartite symmetry
Energy Technology Data Exchange (ETDEWEB)
Lin, Zeren [College of Chemistry and Molecular Engineering, Peking University, Beijing 100871 (China); School of Physics, Peking University, Beijing 100871 (China); Liu, Zhirong, E-mail: [email protected] [College of Chemistry and Molecular Engineering, Peking University, Beijing 100871 (China); Center for Nanochemistry, Beijing National Laboratory for Molecular Sciences (BNLMS), Peking University, Beijing 100871 (China)
2015-12-07
We propose that bipartite symmetry allows spin-1 Dirac-Weyl points, a generalization of the spin-1/2 Dirac points in graphene, to appear as topologically protected at the Fermi level. In this spirit, we provide methodology to construct spin-1 Dirac-Weyl points of this kind in a given 2D space group and get the classification of the known spin-1 systems in the literature. We also apply the workflow to predict two new systems, P3m1-9 and P31m-15, to possess spin-1 at K/K′ in the Brillouin zone of hexagonal lattice. Their stability under various strains is investigated and compared with that of T{sub 3}, an extensively studied model of ultracold atoms trapped in optical lattice with spin-1 also at K/K′.
2. New approaches for searching for the Dirac magnetic monopole
International Nuclear Information System (INIS)
Kukhtin, V.V.; Krivokhizhin, V.G.; Stetsenko, S.G.; Cheplakov, A.P.
2012-01-01
Three new approaches, not applied earlier, are proposed to search for the Dirac monopole - an object whose existence was proposed by P.Dirac more than 80 years ago to explain the electrical charge quantization. The first approach assumes that the monopole must be accelerated by a magnetic field, and such acceleration is constant in the magnetic field which is homogeneous and constant. The conclusion about the object movement nature can be drawn by measuring the time marks for equidistant registering planes. The second approach is supposed to reconstruct the movement trajectory in the homogeneous and permanent electrical field, which is the circle or its part for the magnetic monopole. The third approach is based on the constancy of energy losses by Dirac monopole due to medium ionization in the multilayer passive dielectric tracking detectors placed in the homogeneous and permanent electrical field
3. Time-dependent massless Dirac fermions in graphene
Energy Technology Data Exchange (ETDEWEB)
Khantoul, Boubakeur, E-mail: [email protected] [Department of Mathematics, City University London, Northampton Square, London EC1V 0HB (United Kingdom); Department of Physics, University of Jijel, BP 98, Ouled Aissa, 18000 Jijel (Algeria); Fring, Andreas, E-mail: [email protected] [Department of Mathematics, City University London, Northampton Square, London EC1V 0HB (United Kingdom)
2015-10-30
Using the Lewis–Riesenfeld method of invariants we construct explicit analytical solutions for the massless Dirac equation in 2+1 dimensions describing quasi-particles in graphene. The Hamiltonian of the system considered contains some explicit time-dependence in addition to one resulting from being minimally coupled to a time-dependent vector potential. The eigenvalue equations for the two spinor components of the Lewis–Riesenfeld invariant are found to decouple into a pair of supersymmetric invariants in a similar fashion as the known decoupling for the time-independent Dirac Hamiltonians. - Highlights: • An explicit analytical solution for a massless 2+1 dimensional time-dependent Dirac equation is found. • All steps of the Lewis–Riesenfeld method have been carried out.
4. Maxwell-Like Equations for Free Dirac Electrons
Science.gov (United States)
Bruce, S. A.
2018-03-01
In this article, we show that the wave equation for a free Dirac electron can be represented in a form that is analogous to Maxwell's electrodynamics. The electron bispinor wavefunction is explicitly expressed in terms of its real and imaginary components. This leads us to incorporate into it appropriate scalar and pseudo-scalar fields in advance, so that a full symmetry may be accomplished. The Dirac equation then takes on a form similar to that of a set of inhomogeneous Maxwell's equations involving a particular self-source. We relate plane wave solutions of these equations to waves corresponding to free Dirac electrons, identifying the longitudinal component of the electron motion, together with the corresponding Zitterbewegung ("trembling motion").
5. Dirac vacuum: Acceleration and external-field effects
International Nuclear Information System (INIS)
Jauregui, R.; Torres, M.; Hacyan, S.
1991-01-01
The quantization of the massive spin-1/2 field in Rindler coordinates is considered, including the effects of a background magnetic field. We calculate the expectation values of conserved quantities such as the stress-energy tensor, current density, and spin distribution, as detected by an accelerated observer. The ratio of the energy and particle densities is given by a Fermi-Dirac distribution, but the spectrum of these quantities takes in general a complicated form that cannot be simply interpreted as a thermal spectrum. For the free-particle case the spectrum of the energy-stress tensor has a Fermi-Dirac form only in the massless limit. In the presence of the magnetic field the Dirac vacuum is magnetized and exhibits plasmalike properties
6. String effects on Fermi-Dirac correlation measurements
International Nuclear Information System (INIS)
Duran Delgado, R.M.; Gustafson, G.; Loennblad, L.
2007-01-01
We investigate some recent measurements of Fermi-Dirac correlations by the LEP collaborations indicating surprisingly small source radii for the production of baryons in e + e - annihilation at the Z 0 peak. In hadronization models there is besides the Fermi-Dirac correlation effect also a strong dynamical (anti-) correlation. We demonstrate that the extraction of the pure FD effect is highly dependent on a realistic Monte Carlo event generator, both for separation of those dynamical correlations that are not related to Fermi-Dirac statistics, and for corrections of the data and background subtractions. Although the model can be tuned to well reproduce single particle distributions, there are large model uncertainties when it comes to correlations between identical baryons. We therefore, unfortunately, have to conclude that it is at present not possible to draw any firm conclusion about the source radii relevant for baryon production at LEP. (orig.)
7. Inverse scattering scheme for the Dirac equation at fixed energy
International Nuclear Information System (INIS)
Leeb, H.; Lehninger, H.; Schilder, C.
2001-01-01
Full text: Based on the concept of generalized transformation operators a new hierarchy of Dirac equations with spherical symmetric scalar and fourth component vector potentials is presented. Within this hierarchy closed form expressions for the solutions, the potentials and the S-matrix can be given in terms of solutions of the original Dirac equation. Using these transformations an inverse scattering scheme has been constructed for the Dirac equation which is the analog to the rational scheme in the non-relativistic case. The given method provides for the first time an inversion scheme with closed form expressions for the S-matrix for non-relativistic scattering problems with central and spin-orbit potentials. (author)
8. Luciano Maiani and Jean Iliopoulos awarded the Dirac Medal
CERN Multimedia
2007-01-01
Luciano Maiani, when he was Director-General of CERN. Jean Iliopoulos in 1999. (©CNRS Photothèque - Julien Quideau)On 8 August, the 2007 Dirac Medal, one of the most prestigious prizes in the fields of theoretical physics and mathematics, was awarded to Luciano Maiani, professor at Rome’s La Sapienza University and former Director-General of CERN, and to Jean Iliopoulos, emeritus Director of Research at the CNRS Laboratory of Theoretical Physics. The medal was awarded to both physicists for their joint "work on the physics of the charm quark, a major contribution to the birth of the Standard Model, the modern theory of Elementary Particles." Founded by the Abdus Salam International Centre for Theoretical Physics (ICTP) in 1985, the Dirac Medal is awarded annually on 8 August, the birthday of the famous physicist Paul Dirac, winner of the 1933 Nobel Prize for Physics. It is awarded to ...
9. Dirac charge dynamics in graphene by infrared spectroscopy
International Nuclear Information System (INIS)
Martin, Michael C; Li, Z.Q.; Henriksen, E.A.; Jiang, Z.; Hao, Z.; Martin, Michael C; Kim, P.; Stormer, H.L.; Basov, Dimitri N.
2008-01-01
A remarkable manifestation of the quantum character of electrons in matter is offered by graphene, a single atomic layer of graphite. Unlike conventional solids where electrons are described with the Schroedinger equation, electronic excitations in graphene are governed by the Dirac hamiltonian. Some of the intriguing electronic properties of graphene, such as massless Dirac quasiparticles with linear energy-momentum dispersion, have been confirmed by recent observations. Here, we report an infrared spectromicroscopy study of charge dynamics in graphene integrated in gated devices. Our measurements verify the expected characteristics of graphene and, owing to the previously unattainable accuracy of infrared experiments, also uncover significant departures of the quasiparticle dynamics from predictions made for Dirac fermions in idealized, free-standing graphene. Several observations reported here indicate the relevance of many-body interactions to the electromagnetic response of graphene
10. Accidental degeneracy of double Dirac cones in a phononic crystal
KAUST Repository
Chen, Ze-Guo; Ni, Xu; Wu, Ying; He, Cheng; Sun, Xiao-Chen; Zheng, Li-Yang; Lu, Ming-Hui; Chen, Yan-Feng
2014-01-01
Artificial honeycomb lattices with Dirac cone dispersion provide a macroscopic platform to study the massless Dirac quasiparticles and their novel geometric phases. In this paper, a quadruple-degenerate state is achieved at the center of the Brillouin zone in a two-dimensional honeycomb lattice phononic crystal, which is a result of accidental degeneracy of two double-degenerate states. In the vicinity of the quadruple-degenerate state, the dispersion relation is linear. Such quadruple degeneracy is analyzed by rigorous representation theory of groups. Using method, a reduced Hamiltonian is obtained to describe the linear Dirac dispersion relations of this quadruple-degenerate state, which is well consistent with the simulation results. Near such accidental degeneracy, we observe some unique properties in wave propagating, such as defect-insensitive propagating character and the Talbot effect.
11. Accidental degeneracy of double Dirac cones in a phononic crystal
KAUST Repository
Chen, Ze-Guo
2014-04-09
Artificial honeycomb lattices with Dirac cone dispersion provide a macroscopic platform to study the massless Dirac quasiparticles and their novel geometric phases. In this paper, a quadruple-degenerate state is achieved at the center of the Brillouin zone in a two-dimensional honeycomb lattice phononic crystal, which is a result of accidental degeneracy of two double-degenerate states. In the vicinity of the quadruple-degenerate state, the dispersion relation is linear. Such quadruple degeneracy is analyzed by rigorous representation theory of groups. Using method, a reduced Hamiltonian is obtained to describe the linear Dirac dispersion relations of this quadruple-degenerate state, which is well consistent with the simulation results. Near such accidental degeneracy, we observe some unique properties in wave propagating, such as defect-insensitive propagating character and the Talbot effect.
12. Electronic structure of a graphene superlattice with massive Dirac fermions
International Nuclear Information System (INIS)
Lima, Jonas R. F.
2015-01-01
We study the electronic and transport properties of a graphene-based superlattice theoretically by using an effective Dirac equation. The superlattice consists of a periodic potential applied on a single-layer graphene deposited on a substrate that opens an energy gap of 2Δ in its electronic structure. We find that extra Dirac points appear in the electronic band structure under certain conditions, so it is possible to close the gap between the conduction and valence minibands. We show that the energy gap E g can be tuned in the range 0 ≤ E g ≤ 2Δ by changing the periodic potential. We analyze the low energy electronic structure around the contact points and find that the effective Fermi velocity in very anisotropic and depends on the energy gap. We show that the extra Dirac points obtained here behave differently compared to previously studied systems
13. A survey of hidden-variables theories
CERN Document Server
Belinfante, F J
1973-01-01
A Survey of Hidden-Variables Theories is a three-part book on the hidden-variable theories, referred in this book as """"theories of the first kind"""". Part I reviews the motives in developing different types of hidden-variables theories. The quest for determinism led to theories of the first kind; the quest for theories that look like causal theories when applied to spatially separated systems that interacted in the past led to theories of the second kind. Parts II and III further describe the theories of the first kind and second kind, respectively. This book is written to make the literat
14. A classification of hidden-variable properties
International Nuclear Information System (INIS)
Brandenburger, Adam; Yanofsky, Noson
2008-01-01
Hidden variables are extra components added to try to banish counterintuitive features of quantum mechanics. We start with a quantum-mechanical model and describe various properties that can be asked of a hidden-variable model. We present six such properties and a Venn diagram of how they are related. With two existence theorems and three no-go theorems (EPR, Bell and Kochen-Specker), we show which properties of empirically equivalent hidden-variable models are possible and which are not. Formally, our treatment relies only on classical probability models, and physical phenomena are used only to motivate which models to choose
15. PREFACE: International Workshop on Dirac Electrons in Solids 2015
Science.gov (United States)
Ogata, M.; Suzumura, Y.; Fuseya, Y.; Matsuura, H.
2015-04-01
It is our pleasure to publish the Proceedings of the International Workshop on Dirac Electrons in Solids held in University of Tokyo, Japan, for January 14-15, 2015. The workshop was organized by the entitled project which lasted from April 2012 to March 2015 with 10 theorists. It has been supported by a Grand-in-Aid for Scientific Research (A) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan. The subjects discussed in the workshop include bismuth, organic conductors, graphene, topological insulators, new materials including Ca3PbO, and new directions in theory (superconductivity, orbital susceptibility, etc). The number of participants was about 70 and the papers presented in the workshop include four invited talks, 16 oral presentations, and 23 poster presentations. Dirac electron systems appear in various systems, such as graphene, quasi-two-dimensional organic conductors, bismuth, surface states in topological insulators, new materials like Ca3PbO. In these systems, characteristic transport properties caused by the linear dispersion of Dirac electrons and topological properties, have been extensively discussed. In addition to these, there are many interesting research fields such as Spin-Hall effect, orbital diamagnetism due to interband effects, Landau levels characteristic to Dirac dispersion, anomalous interlayer transport phenomena and magnetoresistance, the effects of spin-orbit interaction, and electron correlation. The workshop focused on recent developments of theory and experiment of Dirac electron systems in the above materials. We note that all papers published in this volume of Journal of Physics: Conference Series were peer reviewed. Reviews were performed by expert referees with professional knowledge and high scientific standards in this field. Editors made efforts so that the papers may satisfy the criterion of a proceedings journal published by IOP Publishing. We hope that all the participants of the workshop
16. Chemistry at the dirac point of graphene
Science.gov (United States)
Sarkar, Santanu
device mobility. To this end, we find that the organometallic hexahapto metal complexation chemistry of graphene, in which the graphene pi-band constructively hybridizes with the vacant d-orbitals of transition metals, allows the fabrication of field effect devices which retain a high degree of the mobility with enhanced on-off ratio. In summary, we find that the singular electronic structure of graphene at the Dirac point governs the chemical reactivity of graphene and this chemistry will play a vital role in propelling graphene to assume its role as the next generation electronic material beyond silicon.
17. Out-of-Bounds Hydrodynamics in Anisotropic Dirac Fluids
Science.gov (United States)
Link, Julia M.; Narozhny, Boris N.; Kiselev, Egor I.; Schmalian, Jörg
2018-05-01
We study hydrodynamic transport in two-dimensional, interacting electronic systems with merging Dirac points at charge neutrality. The dispersion along one crystallographic direction is Dirac-like, while it is Newtonian-like in the orthogonal direction. As a result, the electrical conductivity is metallic in one and insulating in the other direction. The shear viscosity tensor contains six independent components, which can be probed by measuring an anisotropic thermal flow. One of the viscosity components vanishes at zero temperature leading to a generalization of the previously conjectured lower bound for the shear viscosity to entropy density ratio.
18. Geometric interpretation for the Dirac field in curved space
International Nuclear Information System (INIS)
Ranganathan, D.
1987-01-01
The imposition of the condition of length invariance on a Weyl manifold that does not lead uniquely to general relativity is shown. Rather, in this limit, the Weyl vector field can be interpreted as a Dirac current. The action is also the same as the Einstein Dirac one, if and only if, the spinor field is anticommuting. The allowed interactions are greatly restricted. They are only minimal gauge couplings and Yukawa interactions with a scalar field transforming according to the rules of Utiyama [Prog. Theor. Phys. 53, 565 (1975)
19. Levinson theorem for Dirac particles in n dimensions
International Nuclear Information System (INIS)
Jiang Yu
2005-01-01
We study the Levinson theorem for a Dirac particle in an n-dimensional central field by use of the Green function approach, based on an analysis of the n-dimensional radial Dirac equation obtained through a simple algebraic derivation. We show that the zero-momentum phase shifts are related to the number of bound states with |E|< m plus the number of half-bound states of zero momenta--i.e., |E|=m--which are denoted by finite, but not square-integrable, wave functions
20. Dirac potentials in a coupled channel approach to inelastic scattering
International Nuclear Information System (INIS)
Mishra, V.K.; Clark, B.C.; Cooper, E.D.; Mercer, R.L.
1990-01-01
It has been shown that there exist transformations that can be used to change the Lorentz transformation character of potentials, which appear in the Dirac equation for elastic scattering. We consider the situation for inelastic scattering described by coupled channel Dirac equations. We examine a two-level problem where both the ground and excited states are assumed to have zero spin. Even in this simple case we have not found an appropriate transformation. However, if the excited state has zero excitation energy it is possible to find a transformation
1. Adaptive Multigrid Algorithm for the Lattice Wilson-Dirac Operator
International Nuclear Information System (INIS)
Babich, R.; Brower, R. C.; Rebbi, C.; Brannick, J.; Clark, M. A.; Manteuffel, T. A.; McCormick, S. F.; Osborn, J. C.
2010-01-01
We present an adaptive multigrid solver for application to the non-Hermitian Wilson-Dirac system of QCD. The key components leading to the success of our proposed algorithm are the use of an adaptive projection onto coarse grids that preserves the near null space of the system matrix together with a simplified form of the correction based on the so-called γ 5 -Hermitian symmetry of the Dirac operator. We demonstrate that the algorithm nearly eliminates critical slowing down in the chiral limit and that it has weak dependence on the lattice volume.
2. Radiative heat transfer in 2D Dirac materials
International Nuclear Information System (INIS)
Rodriguez-López, Pablo; Tse, Wang-Kong; Dalvit, Diego A R
2015-01-01
We compute the radiative heat transfer between two sheets of 2D Dirac materials, including topological Chern insulators and graphene, within the framework of the local approximation for the optical response of these materials. In this approximation, which neglects spatial dispersion, we derive both numerically and analytically the short-distance asymptotic of the near-field heat transfer in these systems, and show that it scales as the inverse of the distance between the two sheets. Finally, we discuss the limitations to the validity of this scaling law imposed by spatial dispersion in 2D Dirac materials. (paper)
3. Rigid particle revisited: Extrinsic curvature yields the Dirac equation
Energy Technology Data Exchange (ETDEWEB)
Deriglazov, Alexei, E-mail: [email protected] [Depto. de Matemática, ICE, Universidade Federal de Juiz de Fora, MG (Brazil); Laboratory of Mathematical Physics, Tomsk Polytechnic University, 634050 Tomsk, Lenin Ave. 30 (Russian Federation); Nersessian, Armen, E-mail: [email protected] [Yerevan State University, 1 Alex Manoogian St., Yerevan 0025 (Armenia); Laboratory of Mathematical Physics, Tomsk Polytechnic University, 634050 Tomsk, Lenin Ave. 30 (Russian Federation)
2014-03-01
We reexamine the model of relativistic particle with higher-derivative linear term on the first extrinsic curvature (rigidity). The passage from classical to quantum theory requires a number of rather unexpected steps which we report here. We found that, contrary to common opinion, quantization of the model in terms of so(3.2)-algebra yields massive Dirac equation. -- Highlights: •New way of canonical quantization of relativistic rigid particle is proposed. •Quantization made in terms of so(3.2) angular momentum algebra. •Quantization yields massive Dirac equation.
4. Job monitoring on DIRAC for Belle II distributed computing
Science.gov (United States)
Kato, Yuji; Hayasaka, Kiyoshi; Hara, Takanori; Miyake, Hideki; Ueda, Ikuo
2015-12-01
We developed a monitoring system for Belle II distributed computing, which consists of active and passive methods. In this paper we describe the passive monitoring system, where information stored in the DIRAC database is processed and visualized. We divide the DIRAC workload management flow into steps and store characteristic variables which indicate issues. These variables are chosen carefully based on our experiences, then visualized. As a result, we are able to effectively detect issues. Finally, we discuss the future development for automating log analysis, notification of issues, and disabling problematic sites.
5. Zero-energy eigenstates for the Dirac boundary problem
International Nuclear Information System (INIS)
Hortacsu, M.; Rothe, K.D.; Schroer, B.
1980-01-01
As an alternative to the method of spherical compactification for the Dirac operator in instanton background fields we study the correct method of 'box-quantization': the Atiyah-Patodi-Singer spectral boundary condition. This is the only self-adjoint boundary condition which respects the charge conjugation property and the γ 5 symmetry, apart form the usual breaking due to zero modes. We point out the relevance of this approach to the computation of instanton determinants and other problems involving Dirac spinors. (orig.)
6. Dispersive estimates for massive Dirac operators in dimension two
Science.gov (United States)
Erdoğan, M. Burak; Green, William R.; Toprak, Ebru
2018-05-01
We study the massive two dimensional Dirac operator with an electric potential. In particular, we show that the t-1 decay rate holds in the L1 →L∞ setting if the threshold energies are regular. We also show these bounds hold in the presence of s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of t-1(log t) - 2 is attained for large t, at the cost of logarithmic spatial weights. The free Dirac equation does not satisfy this bound due to the s-wave resonances at the threshold energies.
7. Hole doped Dirac states in silicene by biaxial tensile strain
KAUST Repository
Kaloni, Thaneshwor P.; Cheng, Yingchun; Schwingenschlö gl, Udo
2013-01-01
The effects of biaxial tensile strain on the structure, electronic states, and mechanical properties of silicene are studied by ab-initio calculations. Our results show that up to 5% strain the Dirac cone remains essentially at the Fermi level, while higher strain induces hole doped Dirac states because of weakened Si–Si bonds. We demonstrate that the silicene lattice is stable up to 17% strain. It is noted that the buckling first decreases with the strain (up to 10%) and then increases again, which is accompanied by a band gap variation. We also calculate the Grüneisen parameter and demonstrate a strain dependence similar to that of graphene.
8. Hole doped Dirac states in silicene by biaxial tensile strain
KAUST Repository
Kaloni, Thaneshwor P.
2013-03-11
The effects of biaxial tensile strain on the structure, electronic states, and mechanical properties of silicene are studied by ab-initio calculations. Our results show that up to 5% strain the Dirac cone remains essentially at the Fermi level, while higher strain induces hole doped Dirac states because of weakened Si–Si bonds. We demonstrate that the silicene lattice is stable up to 17% strain. It is noted that the buckling first decreases with the strain (up to 10%) and then increases again, which is accompanied by a band gap variation. We also calculate the Grüneisen parameter and demonstrate a strain dependence similar to that of graphene.
9. Dirac neutrinos and hybrid inflation from string theory
International Nuclear Information System (INIS)
Antusch, Stefan; Eyton-Williams, Oliver J.; King, Steve F.
2005-01-01
We consider a possible scenario for the generation of Dirac neutrino masses motivated by type-I string theory. The smallness of the neutrino Yukawa couplings is explained by an anisotropic compactification with one compactification radius larger than the others. In addition to this we utilise small Yukawa couplings to develop strong links between the origin of neutrino masses and the physics driving inflation. We construct a minimal model which simultaneously accommodates small Dirac neutrino masses leading to bi-large lepton mixing as well as an inflationary solution to the strong CP and to the μ problem
10. Twisting dirac fermions: circular dichroism in bilayer graphene
Science.gov (United States)
Suárez Morell, E.; Chico, Leonor; Brey, Luis
2017-09-01
Twisted bilayer graphene is a chiral system which has been recently shown to present circular dichroism. In this work we show that the origin of this optical activity is the rotation of the Dirac fermions’ helicities in the top and bottom layer. Starting from the Kubo formula, we obtain a compact expression for the Hall conductivity that takes into account the dephasing of the electromagnetic field between the top and bottom layers and gathers all the symmetries of the system. Our results are based in both a continuum and a tight-binding model, and they can be generalized to any two-dimensional Dirac material with a chiral stacking between layers.
11. Kinetic mixing of the photon with hidden U(1)s in string phenomenology
International Nuclear Information System (INIS)
Abel, S.A.; Khoze, V.V.; Jaeckel, J.
2008-03-01
Embeddings of the standard model in type II string theory typically contain a variety of U(1) gauge factors arising from D-branes in the bulk. In general, there is no reason why only one of these - the one corresponding to weak hypercharge - should be massless. Observations require that standard model particles must be neutral (or have an extremely small charge) under additional massless U(1)s, i.e. the latter have to belong to a so called hidden sector. The exchange of heavy messengers, however, can lead to a kinetic mixing between the hypercharge and the hidden-sector U(1)s, that is testable with near future experiments. This provides a powerful probe of the hidden sectors and, as a consequence, of the string theory realisation itself. In the present paper, we show, using a variety of methods, how the kinetic mixing can be derived from the underlying type II string compactification, involving supersymmetric and nonsupersymmetric configurations of D-branes, both in large volumes and in warped backgrounds with fluxes. We first demonstrate by explicit example that kinetic mixing occurs in a completely supersymmetric set-up where we can use conformal field theory techniques. We then develop a supergravity approach which allows us to examine the phenomenon in more general backgrounds, where we find that kinetic mixing is natural in the context of flux compactifications. We discuss the phenomenological consequences for experiments at the low-energy frontier, searching for signatures of light, sub-electronvolt or even massless hidden-sector U(1) gauge bosons and minicharged particles. (orig.)
12. Kinetic mixing of the photon with hidden U(1)s in string phenomenology
Energy Technology Data Exchange (ETDEWEB)
Abel, S.A.; Khoze, V.V. [Durham Univ. (United Kingdom). Inst. for Particle Physics Phenomenology; Goodsell, M.D. [Laboratoire de Physique Theorique et Hautes Energies, Paris (France); Jaeckel, J. [Durham Univ. (United Kingdom). Inst. for Particle Physics Phenomenology]|[Heidelberg Univ. (Germany). Inst. fuer Theoretische Physik; Ringwald, A. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2008-03-15
Embeddings of the standard model in type II string theory typically contain a variety of U(1) gauge factors arising from D-branes in the bulk. In general, there is no reason why only one of these - the one corresponding to weak hypercharge - should be massless. Observations require that standard model particles must be neutral (or have an extremely small charge) under additional massless U(1)s, i.e. the latter have to belong to a so called hidden sector. The exchange of heavy messengers, however, can lead to a kinetic mixing between the hypercharge and the hidden-sector U(1)s, that is testable with near future experiments. This provides a powerful probe of the hidden sectors and, as a consequence, of the string theory realisation itself. In the present paper, we show, using a variety of methods, how the kinetic mixing can be derived from the underlying type II string compactification, involving supersymmetric and nonsupersymmetric configurations of D-branes, both in large volumes and in warped backgrounds with fluxes. We first demonstrate by explicit example that kinetic mixing occurs in a completely supersymmetric set-up where we can use conformal field theory techniques. We then develop a supergravity approach which allows us to examine the phenomenon in more general backgrounds, where we find that kinetic mixing is natural in the context of flux compactifications. We discuss the phenomenological consequences for experiments at the low-energy frontier, searching for signatures of light, sub-electronvolt or even massless hidden-sector U(1) gauge bosons and minicharged particles. (orig.)
13. Tilted Dirac Cone Effect on Interlayer Magnetoresistance in α-(BEDT-TTF)2I3
Science.gov (United States)
Tajima, Naoya; Morinari, Takao
2018-04-01
We report the effect of Dirac cone tilting on interlayer magnetoresistance in α-(BEDT-TTF)2I3, which is a Dirac semimetal under pressure. Fitting of the experimental data by the theoretical formula suggests that the system is close to a type-II Dirac semimetal.
14. UV Photography Shows Hidden Sun Damage
Science.gov (United States)
... mcat1=de12", ]; for (var c = 0; c UV photography shows hidden sun damage A UV photograph gives ... developing skin cancer and prematurely aged skin. Normal photography UV photography 18 months of age: This boy's ...
15. Coding with partially hidden Markov models
DEFF Research Database (Denmark)
Forchhammer, Søren; Rissanen, J.
1995-01-01
Partially hidden Markov models (PHMM) are introduced. They are a variation of the hidden Markov models (HMM) combining the power of explicit conditioning on past observations and the power of using hidden states. (P)HMM may be combined with arithmetic coding for lossless data compression. A general...... 2-part coding scheme for given model order but unknown parameters based on PHMM is presented. A forward-backward reestimation of parameters with a redefined backward variable is given for these models and used for estimating the unknown parameters. Proof of convergence of this reestimation is given....... The PHMM structure and the conditions of the convergence proof allows for application of the PHMM to image coding. Relations between the PHMM and hidden Markov models (HMM) are treated. Results of coding bi-level images with the PHMM coding scheme is given. The results indicate that the PHMM can adapt...
16. Hidden costs, value lost: uninsurance in America
National Research Council Canada - National Science Library
Committee on the Consequences of Uninsurance
2003-01-01
Hidden Cost, Value Lost , the fifth of a series of six books on the consequences of uninsurance in the United States, illustrates some of the economic and social losses to the country of maintaining...
17. The hidden epidemic: confronting sexually transmitted diseases
National Research Council Canada - National Science Library
Eng, Thomas R; Butler, William T
.... In addition, STDs increase the risk of HIV transmission. The Hidden Epidemic examines the scope of sexually transmitted infections in the United States and provides a critical assessment of the nation's response to this public health crisis...
18. Perspective: Disclosing hidden sources of funding.
Science.gov (United States)
Resnik, David B
2009-09-01
In this article, the author discusses ethical and policy issues related to the disclosure of hidden sources of funding in research. The author argues that authors have an ethical obligation to disclose hidden sources of funding and that journals should adopt policies to enforce this obligation. Journal policies should require disclosure of hidden sources of funding that authors know about and that have a direct relation to their research. To stimulate this discussion, the author describes a recent case: investigators who conducted a lung cancer screening study had received funding from a private foundation that was supported by a tobacco company, but they did not disclose this relationship to the journal. Investigators and journal editors must be prepared to deal with these issues in a manner that promotes honesty, transparency, fairness, and accountability in research. The development of well-defined, reasonable policies pertaining to hidden sources of funding can be a step in this direction.
19. Results from the solar hidden photon search (SHIPS)
International Nuclear Information System (INIS)
Schwarz, Matthias; Schneide, Magnus; Susol, Jaroslaw; Wiedemann, Guenter; Redondo, Javier
2015-02-01
We present the results of a search for transversely polarised hidden photons (HPs) with ∝3 eV energies emitted from the Sun. These hypothetical particles, known also as paraphotons or dark sector photons, are theoretically well motivated for example by string theory inspired extensions of the Standard Model. Solar HPs of sub-eV mass can convert into photons of the same energy (photon<->HP oscillations are similar to neutrino flavour oscillations). At SHIPS this would take place inside a long light-tight high-vacuum tube, which tracks the Sun. The generated photons would then be focused into a low-noise photomultiplier at the far end of the tube. Our analysis of 330 h of data (and 330 h of background characterisation) reveals no signal of photons from solar hidden photon conversion. We estimate the rate of newly generated photons due to this conversion to be smaller than 25 mHz/m 2 at the 95%C.L. Using this and a recent model of solar HP emission, we set stringent constraints on χ, the coupling constant between HPs and photons, as a function of the HP mass.
20. Results from the Solar Hidden Photon Search (SHIPS)
Energy Technology Data Exchange (ETDEWEB)
Schwarz, Matthias [Hamburger Sternwarte, Gojenbergsweg 112, D-21029 Hamburg (Germany); Knabbe, Ernst-Axel; Lindner, Axel [Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg (Germany); Redondo, Javier [Departamento de Física Teórica, Universidad de Zaragoza, Pedro Cerbuna 12, E-50009, Zaragoza (Spain); Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 München (Germany); Ringwald, Andreas [Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg (Germany); Schneide, Magnus; Susol, Jaroslaw; Wiedemann, Günter [Hamburger Sternwarte, Gojenbergsweg 112, D-21029 Hamburg (Germany)
2015-08-07
We present the results of a search for transversely polarised hidden photons (HPs) with ∼3 eV energies emitted from the Sun. These hypothetical particles, known also as paraphotons or dark sector photons, are theoretically well motivated for example by string theory inspired extensions of the Standard Model. Solar HPs of sub-eV mass can convert into photons of the same energy (photon ↔ HP oscillations are similar to neutrino flavour oscillations). At SHIPS this would take place inside a long light-tight high-vacuum tube, which tracks the Sun. The generated photons would then be focused into a low-noise photomultiplier at the far end of the tube. Our analysis of 330 h of data (and 330 h of background characterisation) reveals no signal of photons from solar hidden photon conversion. We estimate the rate of newly generated photons due to this conversion to be smaller than 25 mHz/m{sup 2} at the 95% C.L. Using this and a recent model of solar HP emission, we set stringent constraints on χ, the coupling constant between HPs and photons, as a function of the HP mass.
1. Results from the Solar Hidden Photon Search (SHIPS)
Energy Technology Data Exchange (ETDEWEB)
Schwarz, Matthias; Schneide, Magnus; Susol, Jaroslaw; Wiedemann, Günter [Hamburger Sternwarte, Gojenbergsweg 112, D-21029 Hamburg (Germany); Knabbe, Ernst-Axel; Lindner, Axel; Ringwald, Andreas [Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg (Germany); Redondo, Javier, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Departamento de Física Teórica, Universidad de Zaragoza, Pedro Cerbuna 12, E-50009, Zaragoza (Spain)
2015-08-01
We present the results of a search for transversely polarised hidden photons (HPs) with ∼ 3 eV energies emitted from the Sun. These hypothetical particles, known also as paraphotons or dark sector photons, are theoretically well motivated for example by string theory inspired extensions of the Standard Model. Solar HPs of sub-eV mass can convert into photons of the same energy (photon ↔ HP oscillations are similar to neutrino flavour oscillations). At SHIPS this would take place inside a long light-tight high-vacuum tube, which tracks the Sun. The generated photons would then be focused into a low-noise photomultiplier at the far end of the tube. Our analysis of 330 h of data (and 330 h of background characterisation) reveals no signal of photons from solar hidden photon conversion. We estimate the rate of newly generated photons due to this conversion to be smaller than 25 mHz/m{sup 2} at the 95% C.L . Using this and a recent model of solar HP emission, we set stringent constraints on χ, the coupling constant between HPs and photons, as a function of the HP mass.
2. Results from the solar hidden photon search (SHIPS)
Energy Technology Data Exchange (ETDEWEB)
Schwarz, Matthias; Schneide, Magnus; Susol, Jaroslaw; Wiedemann, Guenter [Hamburg Univ. (Germany). Sternwarte; Knabbe, Ernst-Axel; Lindner, Axel; Ringwald, Andreas [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Redondo, Javier [Zaragoza Univ. (Spain). Dept. de Fisica Teorica; Max-Planck-Institut fuer Physik, Muenchen (Germany)
2015-02-15
We present the results of a search for transversely polarised hidden photons (HPs) with ∝3 eV energies emitted from the Sun. These hypothetical particles, known also as paraphotons or dark sector photons, are theoretically well motivated for example by string theory inspired extensions of the Standard Model. Solar HPs of sub-eV mass can convert into photons of the same energy (photon<->HP oscillations are similar to neutrino flavour oscillations). At SHIPS this would take place inside a long light-tight high-vacuum tube, which tracks the Sun. The generated photons would then be focused into a low-noise photomultiplier at the far end of the tube. Our analysis of 330 h of data (and 330 h of background characterisation) reveals no signal of photons from solar hidden photon conversion. We estimate the rate of newly generated photons due to this conversion to be smaller than 25 mHz/m{sup 2} at the 95%C.L. Using this and a recent model of solar HP emission, we set stringent constraints on χ, the coupling constant between HPs and photons, as a function of the HP mass.
3. Energy sector
International Nuclear Information System (INIS)
1995-01-01
Within the framework of assessing the state of the environment in Lebanon, this chapter describes primary energy demand, the electricity generating sector and environmental impacts arising from the energy sector.Apart from hydropower and traditional energy sources, which together represent 1.7% of energy consumption, all energy in Lebanon derives from imported petroleum products and some coal.Tables present the imports of different petroleum products (Gasoil, Kerosene, fuel oil, coal etc...), their use, the energy balance and demand.Energy pricing and pricing policies, formal and informal electricity generations in Lebanon are described emphasized by tables. The main environmental impacts are briefly summarized. Thermal power stations give rise to emissions of Sulphur dioxide (SO 2 ), particulates, oxides of nitrogen (NO x ) and CO/CO 2 from combustion of primary fuel informally generated power from both industry and domestic consumption produce particulate materials and emissions of NO x and SO 2 projected emissions of SO 2 from the power sector with the present generating capacity and with the new combined cycle power plants in operation are shown. Other environmental impacts are described. Recommendations for supply and environment policy are presented
4. Petro Rents, Political Institutions, and Hidden Wealth
DEFF Research Database (Denmark)
Andersen, Jørgen Juel; Johannesen, Niels; Lassen, David Dreyer
2017-01-01
Do political institutions limit rent seeking by politicians? We study the transformation of petroleum rents, almost universally under direct government control, into hidden wealth using unique data on bank deposits in offshore financial centers that specialize in secrecy and asset protection. Our...... rulers is diverted to secret accounts. We find very limited evidence that shocks to other types of income not directly controlled by governments affect hidden wealth....
5. Hidden charm molecules in a finite volume
International Nuclear Information System (INIS)
Albaladejo, M.; Hidalgo-Duque, C.; Nieves, J.; Oset, E.
2014-01-01
In the present paper we address the interaction of charmed mesons in hidden charm channels in a finite box. We use the interaction from a recent model based on heavy quark spin symmetry that predicts molecules of hidden charm in the infinite volume. The energy levels in the box are generated within this model, and several methods for the analysis of these levels ("inverse problem") are investigated. (author)
6. Workplace ageism: discovering hidden bias.
Science.gov (United States)
Malinen, Sanna; Johnston, Lucy
2013-01-01
BACKGROUND/STUDY CONTEXT: Research largely shows no performance differences between older and younger employees, or that older workers even outperform younger employees, yet negative attitudes towards older workers can underpin discrimination. Unfortunately, traditional "explicit" techniques for assessing attitudes (i.e., self-report measures) have serious drawbacks. Therefore, using an approach that is novel to organizational contexts, the authors supplemented explicit with implicit (indirect) measures of attitudes towards older workers, and examined the malleability of both. This research consists of two studies. The authors measured self-report (explicit) attitudes towards older and younger workers with a survey, and implicit attitudes with a reaction-time-based measure of implicit associations. In addition, to test whether attitudes were malleable, the authors measured attitudes before and after a mental imagery intervention, where the authors asked participants in the experimental group to imagine respected and valued older workers from their surroundings. Negative, stable implicit attitudes towards older workers emerged in two studies. Conversely, explicit attitudes showed no age bias and were more susceptible to change intervention, such that attitudes became more positive towards older workers following the experimental manipulation. This research demonstrates the unconscious nature of bias against older workers, and highlights the utility of implicit attitude measures in the context of the workplace. In the current era of aging workforce and skill shortages, implicit measures may be necessary to illuminate hidden workplace ageism.
7. Hidden slow pulsars in binaries
Science.gov (United States)
Tavani, Marco; Brookshaw, Leigh
1993-01-01
The recent discovery of the binary containing the slow pulsar PSR 1718-19 orbiting around a low-mass companion star adds new light on the characteristics of binary pulsars. The properties of the radio eclipses of PSR 1718-19 are the most striking observational characteristics of this system. The surface of the companion star produces a mass outflow which leaves only a small 'window' in orbital phase for the detection of PSR 1718-19 around 400 MHz. At this observing frequency, PSR 1718-19 is clearly observable only for about 1 hr out of the total 6.2 hr orbital period. The aim of this Letter is twofold: (1) to model the hydrodynamical behavior of the eclipsing material from the companion star of PSR 1718-19 and (2) to argue that a population of binary slow pulsars might have escaped detection in pulsar surveys carried out at 400 MHz. The possible existence of a population of partially or totally hidden slow pulsars in binaries will have a strong impact on current theories of binary evolution of neutron stars.
8. Dirac's Conception of the Magnetic Monopole, and its Modern Avatars
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 10; Issue 12. Dirac's Conception of the Magnetic Monopole, and its Modern Avatars. Sunil Mukhi. Volume 10 Issue 12 December 2005 pp 193-202. Fulltext. Click here to view fulltext PDF. Permanent link:
9. Spinors, tensors and the covariant form of Dirac's equation
International Nuclear Information System (INIS)
Chen, W.Q.; Cook, A.H.
1986-01-01
The relations between tensors and spinors are used to establish the form of the covariant derivative of a spinor, making use of the fact that certain bilinear combinations of spinors are vectors. The covariant forms of Dirac's equation are thus obtained and examples in specific coordinate systems are displayed. (author)
10. U matrix construction for Quantum Chromodynamics through Dirac brackets
International Nuclear Information System (INIS)
Santos, M.A. dos.
1987-09-01
A procedure for obtaining the U matrix using Dirac brackets, recently developed by Kiefer and Rothe, is applied for Quantum Chromodynamics. The correspondent interaction Lagrangian is the same obtained by Schwinger, Christ and Lee, using independent methods. (L.C.J.A.)
11. Dirac particle in a constant magnetic field: path integral treatment
Energy Technology Data Exchange (ETDEWEB)
Merdaci, A.; Boudiaf, N.; Chetouani, L. [Univ. Mentouri, Constantine (Algeria). Dept. de Physique
2008-05-15
The Green functions related to a Dirac particle in a constant magnetic field are calculated via two methods, global and local, by using the supersymmetric formalism of Fradkin and Gitman. The energy spectrum as well as the corresponding wave functions are extracted following these two approaches. (orig.)
12. Dirac particle in a constant magnetic field: path integral treatment
International Nuclear Information System (INIS)
Merdaci, A.; Boudiaf, N.; Chetouani, L.
2008-01-01
The Green functions related to a Dirac particle in a constant magnetic field are calculated via two methods, global and local, by using the supersymmetric formalism of Fradkin and Gitman. The energy spectrum as well as the corresponding wave functions are extracted following these two approaches. (orig.)
13. A rational interpretation of the Dirac equation for the electron
International Nuclear Information System (INIS)
Koga, T.
1975-01-01
Rationalization of the interpretation of the Dirac equation for the electron lies beyond the conventional scope of quantum mechanics. This difficulty motivates a revision of the system of quantum mechanics through which the indeterministic trait is eliminated from the system. (author)
14. Holographic interaction effects on transport in Dirac semimetals
NARCIS (Netherlands)
Jacobs, V.P.J.; Vandoren, S.; Stoof, H.T.C.
2014-01-01
Strongly interacting Dirac semimetals are investigated using a holographic model especially geared to compute the single-particle correlation function for this case, including both interaction effects and non-zero temperature. We calculate the (homogeneous) optical conductivity at zero chemical
15. Survey on Dirac equation in general relativity theory
International Nuclear Information System (INIS)
Paillere, P.
1984-10-01
Starting from an infinitesimal transformation expressed with a Killing vector and using systematically the formalism of the local tetrades, we show that, in the area of the general relativity, the Dirac equation may be formulated only versus the four local vectors which determine the gravitational potentials, their gradients and the 4-vector potential of the electromagnetic field [fr
16. Dirac electronics states in graphene systems: optical spectroscopy studies
Czech Academy of Sciences Publication Activity Database
Orlita, Milan; Potemski, M.
2010-01-01
Roč. 25, č. 6 (2010), 063001/1-063001/22 ISSN 0268-1242 R&D Projects: GA AV ČR KAN400100652 Institutional research plan: CEZ:AV0Z10100521 Keywords : graphene * Dirac fermions Subject RIV: BM - Solid Matter Physics ; Magnetism Impact factor: 1.323, year: 2010
17. A new derivation of Dirac's magnetic monopole strength
International Nuclear Information System (INIS)
Panat, P V
2003-01-01
A new derivation of the strength of Dirac's magnetic monopole is presented which does not require an explicit form of the magnetic induction in terms of g, the magnetic pole strength. The derivation essentially uses a modification of Faraday's law of induction and quantization of angular momentum
18. LHCb : The DIRAC Web Portal 2.0
CERN Multimedia
Mathe, Zoltan; Lazovsky, N; Stagni, Federico
2015-01-01
For many years the DIRAC interware (Distributed Infrastructure with Remote Agent Control) has had a web interface, allowing the users to monitor DIRAC activities and also interact with the system. Since then many new web technologies have emerged, therefore a redesign and a new implementation of the DIRAC Web portal were necessary, taking into account the lessons learnt using the old portal. These new technologies allowed to build a more compact and more responsive web interface that is robust and that enables users to have more control over the whole system while keeping a simple interface. The framework provides a large set of "applications", each of which can be used for interacting with various parts of the system. Communities can also create their own set of personalised web applications, and can easily extend already existing web applications with a minimal effort. Each user can configure and personalise the view for each application and save it using the DIRAC User Profile service as RESTful state prov...
19. On the representation of generalized Dirac (Clifford) algebras
International Nuclear Information System (INIS)
Srivastava, T.
1981-10-01
Some results of Brauer and Weyl and of Jordan and Wigner on irreducible representations of generalized Dirac (Clifford) algebras have been proved, adopting a new and simple approach which (i) makes the whole subject straightforward for physicists and (ii) simplifies the demonstration of the fundamental theorem of Pauli. (author)
20. Chiral Tricritical Point: A New Universality Class in Dirac Systems
Science.gov (United States)
Yin, Shuai; Jian, Shao-Kai; Yao, Hong
2018-05-01
Tricriticality, as a sister of criticality, is a fundamental and absorbing issue in condensed-matter physics. It has been verified that the bosonic Wilson-Fisher universality class can be changed by gapless fermionic modes at criticality. However, the counterpart phenomena at tricriticality have rarely been explored. In this Letter, we study a model in which a tricritical Ising model is coupled to massless Dirac fermions. We find that the massless Dirac fermions result in the emergence of a new tricritical point, which we refer to as the chiral tricritical point (CTP), at the phase boundary between the Dirac semimetal and the charge-density wave insulator. From functional renormalization group analysis of the effective action, we obtain the critical behaviors of the CTP, which are qualitatively distinct from both the tricritical Ising universality and the chiral Ising universality. We further extend the calculations of the chiral tricritical behaviors of Ising spins to the case of Heisenberg spins. The experimental relevance of the CTP in two-dimensional Dirac semimetals is also discussed.
1. Hydrogenated arsenenes as planar magnet and Dirac material
Energy Technology Data Exchange (ETDEWEB)
Zhang, Shengli; Cai, Bo; Zeng, Haibo, E-mail: [email protected], E-mail: [email protected] [Institute of Optoelectronics and Nanomaterials, Herbert Gleiter Institute of Nanoscience, College of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094 (China); Hu, Yonghong [Institute of Optoelectronics and Nanomaterials, Herbert Gleiter Institute of Nanoscience, College of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094 (China); School of Nuclear Technology and Chemistry and Biology, Hubei University of Science and Technology, Xianning 437100 (China); Hu, Ziyu, E-mail: [email protected], E-mail: [email protected] [Beijing Computational Science Research Center, Beijing 100084 (China)
2015-07-13
Arsenene and antimonene are predicted to have 2.49 and 2.28 eV band gaps, which have aroused intense interest in the two-dimensional (2D) semiconductors for nanoelectronic and optoelectronic devices. Here, the hydrogenated arsenenes are reported to be planar magnet and 2D Dirac materials based on comprehensive first-principles calculations. The semi-hydrogenated (SH) arsenene is found to be a quasi-planar magnet, while the fully hydrogenated (FH) arsenene is a planar Dirac material. The buckling height of pristine arsenene is greatly decreased by the hydrogenation, resulting in a planar and relatively low-mass-density sheet. The electronic structures of arsenene are also evidently altered after hydrogenating from wide-band-gap semiconductor to metallic material for SH arsenene, and then to Dirac material for FH arsenene. The SH arsenene has an obvious magnetism, mainly contributed by the p orbital of the unsaturated As atom. Such magnetic and Dirac materials modified by hydrogenation of arsenene may have potential applications in future optoelectronic and spintronic devices.
2. Hydrogenated arsenenes as planar magnet and Dirac material
International Nuclear Information System (INIS)
Zhang, Shengli; Cai, Bo; Zeng, Haibo; Hu, Yonghong; Hu, Ziyu
2015-01-01
Arsenene and antimonene are predicted to have 2.49 and 2.28 eV band gaps, which have aroused intense interest in the two-dimensional (2D) semiconductors for nanoelectronic and optoelectronic devices. Here, the hydrogenated arsenenes are reported to be planar magnet and 2D Dirac materials based on comprehensive first-principles calculations. The semi-hydrogenated (SH) arsenene is found to be a quasi-planar magnet, while the fully hydrogenated (FH) arsenene is a planar Dirac material. The buckling height of pristine arsenene is greatly decreased by the hydrogenation, resulting in a planar and relatively low-mass-density sheet. The electronic structures of arsenene are also evidently altered after hydrogenating from wide-band-gap semiconductor to metallic material for SH arsenene, and then to Dirac material for FH arsenene. The SH arsenene has an obvious magnetism, mainly contributed by the p orbital of the unsaturated As atom. Such magnetic and Dirac materials modified by hydrogenation of arsenene may have potential applications in future optoelectronic and spintronic devices
3. Hydrogenated arsenenes as planar magnet and Dirac material
Science.gov (United States)
Zhang, Shengli; Hu, Yonghong; Hu, Ziyu; Cai, Bo; Zeng, Haibo
2015-07-01
Arsenene and antimonene are predicted to have 2.49 and 2.28 eV band gaps, which have aroused intense interest in the two-dimensional (2D) semiconductors for nanoelectronic and optoelectronic devices. Here, the hydrogenated arsenenes are reported to be planar magnet and 2D Dirac materials based on comprehensive first-principles calculations. The semi-hydrogenated (SH) arsenene is found to be a quasi-planar magnet, while the fully hydrogenated (FH) arsenene is a planar Dirac material. The buckling height of pristine arsenene is greatly decreased by the hydrogenation, resulting in a planar and relatively low-mass-density sheet. The electronic structures of arsenene are also evidently altered after hydrogenating from wide-band-gap semiconductor to metallic material for SH arsenene, and then to Dirac material for FH arsenene. The SH arsenene has an obvious magnetism, mainly contributed by the p orbital of the unsaturated As atom. Such magnetic and Dirac materials modified by hydrogenation of arsenene may have potential applications in future optoelectronic and spintronic devices.
4. Fermi-Dirac statistics and the number theory
OpenAIRE
Kubasiak, A.; Korbicz, J.; Zakrzewski, J.; Lewenstein, M.
2005-01-01
We relate the Fermi-Dirac statistics of an ideal Fermi gas in a harmonic trap to partitions of given integers into distinct parts, studied in number theory. Using methods of quantum statistical physics we derive analytic expressions for cumulants of the probability distribution of the number of different partitions.
5. Axial anomaly and index theorem for Dirac-Kaehler fermions
International Nuclear Information System (INIS)
Linhares, C.A.; Mignaco, J.A.; Monteiro, M.A.R.
1985-01-01
We present the calculation of the axial anomaly for Dirac-Kaehler fermions in two and four dimensions applying the procedure developed by Seeley to the signature operator in the twisted complex. The result is equal to the one for the twisted spin complex times 2 n/2 (n:number of dimensions) and agrees with the expressions from the index theorem. (author) [pt
6. Axial anomaly and index theorem for Dirac-Kaehler fermions
International Nuclear Information System (INIS)
Linhares, C.A.; Mignaco, J.A.; Rego Monteiro, M.A.
1985-01-01
We present a calculation of the axial anomaly for Dirac-Kaehler fermions in two and four dimensions applying the procedure developed by Seeley to the signature operator in the twisted complex. The result is equal to the one for the twisted spin complex times 2sup(π/2) (n: number of dimensions) and agrees with the expressions from the index theorem. (orig.)
7. New exact solutions of the Dirac equation. 11
International Nuclear Information System (INIS)
Bagrov, V.G.; Noskov, M.D.
1984-01-01
Investigations into determining new exact solutions of relativistic wave equations started in another paper were continued. Exact solutions of the Dirac, Klein-Gordon equations and classical relativistic equations of motion in four new types of external electromagnetic fields were found
8. Majorana zero modes in Dirac semimetal Josephson junctions
Science.gov (United States)
Li, Chuan; de Boer, Jorrit; de Ronde, Bob; Huang, Yingkai; Golden, Mark; Brinkman, Alexander
We have realized proximity-induced superconductivity in a Dirac semimetal and revealed the topological nature of the superconductivity by the observation of Majorana zero modes. As a Dirac semimetal, Bi0.97Sb0.03 is used, where a three-dimensional Dirac cone exists in the bulk due to an accidental touching between conduction and valence bands. Electronic transport measurements on Hall-bars fabricated out of Bi0.97Sb0.03 flakes consistently show negative magnetoresistance for magnetic fields parallel to the current, which is associated with the chiral anomaly. In perpendicular magnetic fields, we see Shubnikov-de Haas oscillations that indicate very low carrier densities. The low Fermi energy and protection against backscattering in our Dirac semimetal Josephson junctions provide favorable conditions for a large contribution of Majorana zero modes to the supercurrent. In radiofrequency irradiation experiments, we indeed observe these Majorana zero modes in Nb-Bi0.97Sb0.03-Nb Josephson junctions as a 4 π periodic contribution to the current-phase relation.
9. Dirac Magnon Nodal Loops in Quasi-2D Quantum Magnets.
Science.gov (United States)
Owerre, S A
2017-07-31
In this report, we propose a new concept of one-dimensional (1D) closed lines of Dirac magnon nodes in two-dimensional (2D) momentum space of quasi-2D quantum magnetic systems. They are termed "2D Dirac magnon nodal-line loops". We utilize the bilayer honeycomb ferromagnets with intralayer coupling J and interlayer coupling J L , which is realizable in the honeycomb chromium compounds CrX 3 (X ≡ Br, Cl, and I). However, our results can also exist in other layered quasi-2D quantum magnetic systems. Here, we show that the magnon bands of the bilayer honeycomb ferromagnets overlap for J L ≠ 0 and form 1D closed lines of Dirac magnon nodes in 2D momentum space. The 2D Dirac magnon nodal-line loops are topologically protected by inversion and time-reversal symmetry. Furthermore, we show that they are robust against weak Dzyaloshinskii-Moriya interaction Δ DM magnon edge modes.
10. Radiationless Zitterbewegung of Dirac particles and mass formula
International Nuclear Information System (INIS)
Noboru Hokkyo.
1987-06-01
The Zitterbewegung of the Dirac particle is given a visual representation by solving the two-component difference form of the Dirac equation. It is seen that the space-time trajectory of a Dirac particle can be pictured as a correlated whole of a network of zigzags of left- and right-handed chiral neutrino-like line elements. These zigzags can feel the curl of the external electromagnetic vector potential and give rise to the spin magnetic interaction, confirming Schroedinger's earlier intuitive picture of the spin as the orbital angular momentum of the Zitterbewegung. The network of zigzags associated with an electron splits and reunites in passing through the slits in the electron beam interference experiment. It is proposed to interpret Nambu's empirical mass formula m n =(n/2)137m e =(n/2)((h/2π)/cL), n=integer, as a radiationless condition for the Zitterbewegung of the hadronic Dirac particle of the linear spatial extension of the order of the classical electron radius L=e 2 /m e c 2 . (author). 20 refs, 4 figs
11. Qualitative analysis of trapped Dirac fermions in graphene
Czech Academy of Sciences Publication Activity Database
Jakubský, Vít; Krejčiřík, David
2014-01-01
Roč. 349, OCT (2014), s. 268-287 ISSN 0003-4916 R&D Projects: GA ČR(CZ) GA14-06818S Institutional support: RVO:61389005 Keywords : graphene * Dirac fermion * confinement * Varitional principle Subject RIV: BE - Theoretical Physics Impact factor: 2.103, year: 2014
12. Dirac fermions in nontrivial topology black hole backgrounds
International Nuclear Information System (INIS)
Gozdz, Marek; Nakonieczny, Lukasz; Rogatko, Marek
2010-01-01
We discuss the behavior of the Dirac fermions in a general spherically symmetric black hole background with a nontrivial topology of the event horizon. Both massive and massless cases are taken into account. We will conduct an analytical study of intermediate and late-time behavior of massive Dirac hair in the background of a black hole with a global monopole and dilaton black hole pierced by a cosmic string. In the case of a global monopole swallowed by a static black hole, the intermediate late-time behavior depends on the mass of the Dirac field, the multiple number of the wave mode, and the global monopole parameter. The late-time behavior is quite independent of these factors and has a decay rate proportional to t -5/6 . As far as the black hole pierced by a cosmic string is concerned, the intermediate late-time behavior depends only on the hair mass and the multipole number of the wave mode, while the late-time behavior dependence is the same as in the previous case. The main modification stems from the topology of the S 2 sphere pierced by a cosmic string. This factor modifies the eigenvalues of the Dirac operator acting on the transverse manifold.
13. Hawking radiation of Dirac particles in the hot NUT-Kerr-Newman spacetime
International Nuclear Information System (INIS)
Ahmed, M.
1991-01-01
The Hawking radiation of charged Dirac particles on the horizons of the hot NUT-Kerr-Newman spacetime is studied in this paper. To this end, we obtain the radial decoupled Dirac equation for the electron in the hot NUT-Kerr-Newman spacetime. Next we solve the Dirac equation near the horizons. Finally, by analytic continuation, the Hawking thermal spectrum formula of Dirac particles is obtained. The problem of the Hawking evaporation of Dirac particles in the hot NUT-Kerr-Newman background is thus solved. (orig.)
14. Analysis of DIRAC's behavior using model checking with process algebra
International Nuclear Information System (INIS)
Remenska, Daniela; Templon, Jeff; Willemse, Tim; Bal, Henri; Verstoep, Kees; Fokkink, Wan; Charpentier, Philippe; Lanciotti, Elisa; Roiser, Stefan; Ciba, Krzysztof; Diaz, Ricardo Graciani
2012-01-01
DIRAC is the grid solution developed to support LHCb production activities as well as user data analysis. It consists of distributed services and agents delivering the workload to the grid resources. Services maintain database back-ends to store dynamic state information of entities such as jobs, queues, staging requests, etc. Agents use polling to check and possibly react to changes in the system state. Each agent's logic is relatively simple; the main complexity lies in their cooperation. Agents run concurrently, and collaborate using the databases as shared memory. The databases can be accessed directly by the agents if running locally or through a DIRAC service interface if necessary. This shared-memory model causes entities to occasionally get into inconsistent states. Tracing and fixing such problems becomes formidable due to the inherent parallelism present. We propose more rigorous methods to cope with this. Model checking is one such technique for analysis of an abstract model of a system. Unlike conventional testing, it allows full control over the parallel processes execution, and supports exhaustive state-space exploration. We used the mCRL2 language and toolset to model the behavior of two related DIRAC subsystems: the workload and storage management system. Based on process algebra, mCRL2 allows defining custom data types as well as functions over these. This makes it suitable for modeling the data manipulations made by DIRAC's agents. By visualizing the state space and replaying scenarios with the toolkit's simulator, we have detected race-conditions and deadlocks in these systems, which, in several cases, were confirmed to occur in the reality. Several properties of interest were formulated and verified with the tool. Our future direction is automating the translation from DIRAC to a formal model.
15. Analysis of DIRAC's behavior using model checking with process algebra
Science.gov (United States)
Remenska, Daniela; Templon, Jeff; Willemse, Tim; Bal, Henri; Verstoep, Kees; Fokkink, Wan; Charpentier, Philippe; Graciani Diaz, Ricardo; Lanciotti, Elisa; Roiser, Stefan; Ciba, Krzysztof
2012-12-01
DIRAC is the grid solution developed to support LHCb production activities as well as user data analysis. It consists of distributed services and agents delivering the workload to the grid resources. Services maintain database back-ends to store dynamic state information of entities such as jobs, queues, staging requests, etc. Agents use polling to check and possibly react to changes in the system state. Each agent's logic is relatively simple; the main complexity lies in their cooperation. Agents run concurrently, and collaborate using the databases as shared memory. The databases can be accessed directly by the agents if running locally or through a DIRAC service interface if necessary. This shared-memory model causes entities to occasionally get into inconsistent states. Tracing and fixing such problems becomes formidable due to the inherent parallelism present. We propose more rigorous methods to cope with this. Model checking is one such technique for analysis of an abstract model of a system. Unlike conventional testing, it allows full control over the parallel processes execution, and supports exhaustive state-space exploration. We used the mCRL2 language and toolset to model the behavior of two related DIRAC subsystems: the workload and storage management system. Based on process algebra, mCRL2 allows defining custom data types as well as functions over these. This makes it suitable for modeling the data manipulations made by DIRAC's agents. By visualizing the state space and replaying scenarios with the toolkit's simulator, we have detected race-conditions and deadlocks in these systems, which, in several cases, were confirmed to occur in the reality. Several properties of interest were formulated and verified with the tool. Our future direction is automating the translation from DIRAC to a formal model.
16. More on the hypercharge portal into the dark sector
International Nuclear Information System (INIS)
Domingo, Florian; Lebedev, Oleg; Ringwald, Andreas; Mambrini, Yann; Quevillon, Jeremie
2013-05-01
If the hidden sector contains more than one U(1) groups, additional dim-4 couplings (beyond the kinetic mixing) between the massive U(1) fields and the hypercharge generally appear. These are of the form similar to the Chern-Simons interactions. We study the phenomenology of such couplings including constraints from laboratory experiments and implications for dark matter. The hidden vector fields can play the role of dark matter whose characteristic signature would be monochromatic gamma ray emission from the galactic center. We show that this possibility is consistent with the LHC and other laboratory constraints, as well as astrophysical bounds.
17. More on the hypercharge portal into the dark sector
Energy Technology Data Exchange (ETDEWEB)
Domingo, Florian; Lebedev, Oleg; Ringwald, Andreas [DESY Hamburg (Germany). Theory Group; Mambrini, Yann; Quevillon, Jeremie [Paris-Sud Univ., Orsay (France). Laboratoire de Physique Theorique
2013-05-15
If the hidden sector contains more than one U(1) groups, additional dim-4 couplings (beyond the kinetic mixing) between the massive U(1) fields and the hypercharge generally appear. These are of the form similar to the Chern-Simons interactions. We study the phenomenology of such couplings including constraints from laboratory experiments and implications for dark matter. The hidden vector fields can play the role of dark matter whose characteristic signature would be monochromatic gamma ray emission from the galactic center. We show that this possibility is consistent with the LHC and other laboratory constraints, as well as astrophysical bounds.
18. Magnetic moments of confined quarks and baryons in an independent-quark model based on Dirac equation with power-law potential
International Nuclear Information System (INIS)
Barik, N.; Das, M.
1983-01-01
The effect of confinement on the magnetic moment of a quark has been studied in a simple independent-quark model based on the Dirac equation with a power-law potential. The magnetic moments so obtained for the constituent quarks, which are found to be significantly different from their corresponding Dirac moments, are used in predicting the magnetic moments of baryons in the nucleon octet as well as those in the charmed and b-flavored sectors. We not only get an improved result for the proton magnetic moment, but the calculation for the rest of the nucleon octet also turns out to be in reasonable agreement with experiment. The overall predictions for the charmed and b-flavored baryons are also comparable with other model predictions
19. A CPT-even and Lorentz-Violating nonminimal coupling in the Dirac equation
International Nuclear Information System (INIS)
Ferreira Junior, Manoel; Casana, M.R.; Santos, Frederico E.P. dos; Silva, E.O.; Passos, E.
2013-01-01
Full text: The Standard Model Extension (SME) has been the usual framework for investigating signals of Lorentz violation in physical systems. It is the natural framework for studying properties of physical systems with Lorentz-violation since it includes Lorentz-violating terms in all sectors of the minimal standard model. The Lorentz-violating (LV) terms are generated as vacuum expectation values of tensors defined in a high energy scale. This framework has inspired a great deal of investigation in recent years. Such works encompass several distinct aspects involving fermion systems and radiative corrections, CPT- probing experiments, the electromagnetic CPT- and Lorentz-odd term, the 19 electromagnetic CPT-even coefficients. Recently, some studies involving higher dimensional operators have also been reported with great interest, including nonminimal interactions. These many contributions have elucidated the effects induced by Lorentz violation and served to set up stringent upper bounds on the LV coefficients. In the present work, we propose a new CPT-even, dimension-five, nonminimal coupling linking the fermionic and gauge fields in the context of the Dirac equation, involving the CPT-even tensor of the gauge term of the SME. By considering the nonrelativistic limit of the modified Dirac equation, we explicitly evaluate the new contributions to the nonrelativistic Hamiltonian. These new terms imply a direct correction on the anomalous magnetic moment, a kind of electrical Zeeman-like effect on the atomic spectrum, and a Rashba-like coupling term. These effects are then used to impose upper bounds on the magnitude of the non minimally coupled LV coefficients at the level of 1 part in 10 16 . (author)
20. A CPT-even and Lorentz-Violating nonminimal coupling in the Dirac equation
Energy Technology Data Exchange (ETDEWEB)
Ferreira Junior, Manoel; Casana, M.R.; Santos, Frederico E.P. dos; Silva, E.O. [UFMA, Sao Luis (Brazil); Passos, E. [UFCG, Campina Grande, PB (Brazil)
2013-07-01
Full text: The Standard Model Extension (SME) has been the usual framework for investigating signals of Lorentz violation in physical systems. It is the natural framework for studying properties of physical systems with Lorentz-violation since it includes Lorentz-violating terms in all sectors of the minimal standard model. The Lorentz-violating (LV) terms are generated as vacuum expectation values of tensors defined in a high energy scale. This framework has inspired a great deal of investigation in recent years. Such works encompass several distinct aspects involving fermion systems and radiative corrections, CPT- probing experiments, the electromagnetic CPT- and Lorentz-odd term, the 19 electromagnetic CPT-even coefficients. Recently, some studies involving higher dimensional operators have also been reported with great interest, including nonminimal interactions. These many contributions have elucidated the effects induced by Lorentz violation and served to set up stringent upper bounds on the LV coefficients. In the present work, we propose a new CPT-even, dimension-five, nonminimal coupling linking the fermionic and gauge fields in the context of the Dirac equation, involving the CPT-even tensor of the gauge term of the SME. By considering the nonrelativistic limit of the modified Dirac equation, we explicitly evaluate the new contributions to the nonrelativistic Hamiltonian. These new terms imply a direct correction on the anomalous magnetic moment, a kind of electrical Zeeman-like effect on the atomic spectrum, and a Rashba-like coupling term. These effects are then used to impose upper bounds on the magnitude of the non minimally coupled LV coefficients at the level of 1 part in 10{sub 16}. (author) | {"extraction_info": {"found_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": 0.8541883826255798, "perplexity": 2267.5807980332957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589237.16/warc/CC-MAIN-20180716080356-20180716100356-00293.warc.gz"} |
https://arxiv.org/abs/1408.6892 | math.PR
(what is this?)
# Title:Rare-event Analysis for Extremal Eigenvalues of white Wishart matrices
Abstract: In this paper we consider the extreme behavior of the extremal eigenvalues of white Wishart matrices, which plays an important role in multivariate analysis. In particular, we focus on the case when the dimension of the feature p is much larger than or comparable to the number of observations n, a common situation in modern data analysis. We provide asymptotic approximations and bounds for the tail probabilities of the extremal eigenvalues. Moreover, we construct efficient Monte Carlo simulation algorithms to compute the tail probabilities. Simulation results show that our method has the best performance amongst known approximation approaches, and furthermore provides an efficient and accurate way for evaluating the tail probabilities in practice.
Subjects: Probability (math.PR) Cite as: arXiv:1408.6892 [math.PR] (or arXiv:1408.6892v2 [math.PR] for this version)
## Submission history
From: Gongjun Xu [view email]
[v1] Fri, 29 Aug 2014 00:20:15 UTC (36 KB)
[v2] Tue, 26 Jul 2016 00:37:39 UTC (387 KB) | {"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": 0.8706954717636108, "perplexity": 856.9791143066237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1552912201885.28/warc/CC-MAIN-20190319032352-20190319054352-00236.warc.gz"} |
http://mathhelpforum.com/trigonometry/47876-solving-trigo.html | # Math Help - Solving trigo
1. ## Solving trigo
cos3X=sinX
Find X, 0 degree < x < 360 degree
2. Originally Posted by ose90
cos3X=sinX
Find X, 0 degree < x < 360 degree
There's the hard way and the easy way. Personally I like the easy way best:
From the complementary angle formulae: $\cos (3x) = \sin x \Rightarrow \cos (3x) = \cos (90 - x)$.
Case 1: $3x = 90 - x + 360 n$ where n is an integer.
Solve for x and substitute appropriate values of n to get all values of x satisfying the given domain.
Case 2: $3x = -(90 - x) + 360 n$ where n is an integer.
Solve for x and substitute appropriate values of n to get all values of x satisfying the given domain.
3. Thank you very much! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9313631057739258, "perplexity": 902.0135479775904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1438042992201.62/warc/CC-MAIN-20150728002312-00096-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/77391/reverse-mathematics-strength-of-lipschitz-functions-are-somewhere-differentiabl/78726 | reverse mathematics strength of “Lipschitz functions are somewhere differentiable”
What is the reverse mathematics strength of
"For all Lipschitz functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ?
(defined using epsilon-delta, so not requiring that there exist a function witnessing the convergence)
Since Lipschitz functions are differentiable almost everywhere,
I would guess the answer is "is equivalent to WWKL$_0$ (over RCA$_0$)".
-
Are you just looking for a lower bound, or an upper bound too? Did you check whether the "standard" proof works in $\mathsf{WWKL}_0$? – François G. Dorais Oct 6 '11 at 22:19
I'm looking for both a lower bound and an upper bound. $\;$ I couldn't find the "standard" proof anywhere. – Ricky Demer Oct 6 '11 at 22:30
You should probably add the "logic" tag, right? – Jason Rute Oct 7 '11 at 15:32
Your theorem should be true in every $\omega$-model of $\mathsf{RCA}_0$ as follows. The following paper
• Brattka, Miller and Nies. Randomness and Differentiability. Submitted. (preprint)
proves that every computable function on $[0,1]$ that is Lipschitz is differentiable on all computable randoms. Further, from this proof, I believe you can extract a single computable martingale $M$ such that if $x$ is a point of differentiability of $f$, then $M$ does not succeed on $x$. Unlike ML randomness---which corresponds to $\mathsf{WWKL}_0$---with a test for computable randomness, i.e. the computable martingale $M$, you can compute a real $y$ such that $M$ doesn't succeed on $y$.
Hence there is (I believe) a real $y$ computable from the function $f$ for which $f$ is differentiable at $x$.
Hence your theorem holds in every $\omega$-model of $\mathsf{RCA}_0$.
This implies that your theorem is true in $\mathsf{RCA}_0$ or $\mathsf{RCA}_0$ plus some first order principle.
I should mention, I learned of this trick from Steve Simpson (in the case of Schnorr randomness).
Of course to see if it is provable in ${RCA}_0$, you should check that level of induction/collection used. A recent paper with Jeremy Avigad and Ed Dean,
• Avigad, Dean and Rute. Algorithmic Randomness, reverse mathematics, and the dominated convergence theorem. Submitted (preprint)
gives an example where the dominated convergence thereom, in a sense, corresponds to $2$-randomness, but the reverse math strength of the dominated convergence thereom was a bit higher, since $\Sigma^0_2$ collection was needed.
(You do not not seem to need this, but...) even though you know a point $x$ such that $f$ is differentiable, that doesn't mean you know what the derviative is at that point. Actually, you can't even compute the derivative $f'$ in the $L^1$ norm.
There are more published and unpublished works related to the diffentiability of computable functions. One good summary of the recent work in this area is the Logic Blog on Andre Nies's website. Another is this talk by Nies.
There are also three results in preparation that may be of value. I proved, as did Pathak, Rojas and Simpson independently, that the Lebesgue differentiation theorem holds on Schnorr randoms. In particular this implies that a more restrictive notion of computable Lipschitz function is differentiable on Schnorr randoms (see Nies' talk mentioned above). Freer, Kjos-Hanssen and Nies, have a paper in preparation about Lipschitz functions and differentiability, which in a sense, "reverses" the above results. (Again, this is mentioned in the talk by Nies.)
-
Is "that test" [$\hspace{.01 in} f$ being differentiable at $x$] or [the martingale]? – Ricky Demer Oct 6 '11 at 23:29
By test I meant the martingale. It is computable from the function. If f is differentiable at x, then it cannot succeed on the martingale. There may be a null set of differentiable points that succeed on the martingale as well. – Jason Rute Oct 6 '11 at 23:38
"If $f$ is differentiable at $x$, then it cannot succeed on the martingale." $\hspace{2 in}$ ... except for a null set of points $f$ is differentiable at? – Ricky Demer Oct 6 '11 at 23:50
Sorry, I said the converse! Let me try to say it correct this time. As you know, the set of non-differentiable points of $f$ is null. From $f$ we can compute a martingale $M$ such that if $f$ is not differentiable at $x$, then $M$ succeeds on $x$. (I said "$x$ succeeds on the martingale" before but that was incorrect terminology---the martingale succeeds on $x$.) Now, since we can compute from $M$ some $y$ on which $M$ does not succeed, we can also computed a real for which $f$ is differentiable. – Jason Rute Oct 7 '11 at 2:33
Do you know a link to "Brattka, Miller and Nies, Randomness and Differentiability"? $\;$ I can't find it online. – Ricky Demer Oct 7 '11 at 7:37
Here it is, the newest version
http://dl.dropbox.com/u/370127/papers/randomnessanalysisjuly_2011.pdf
What jason says is right, even for nondecreasing functions. It is in our paper, Subection 5.1
Hence there is (I believe) a real $y$ computable from the function $f$ for which $f$ is differentiable at $x$.
Hence your theorem holds in every $\omega$-model of RCA0.
This implies that your theorem is true in RCA0 or RCA0 plus some first order principle.
Not sure what these fo principles do, I guess it's standard stuff?
-
Dear Andre, Welcome to MO! You can use tex-style math as you normally would. The site also uses markdown, which is very useful has some weird side effects from time to time. – François G. Dorais Oct 23 '11 at 1:15
Andre, I suspect that the theorem goes through in $\mathsf{RCA}_0$, but I would need to walk through the proof carefully to be sure. In case you or another reader is unfamiliar with Reverse Math, the axioms of $\mathsf{RCA}_0$ are basically the Peano axioms but with only induction on $\Sigma_1$ formulas (with second order parameters). Also there is a comprehension axiom for $\Delta_1$ formulas (with parameters). This comprehension axiom ensures the existence of computable sets. (.../...) – Jason Rute Oct 24 '11 at 23:01
(.../...) It is a common mistake (that I have made in the past) to assume that if a set existence principle (like the one asked about) is computable, then it is provable in $\mathsf{RCA}_0$. A common example is the infinite pigeon-hole principle. Given a computable coloring, an infinite homogeneous set is computable: pick the correct color (by guessing correctly) and enumerate the set. But to prove it, one needs $\mathsf{B}\Sigma_2$, a collection principle between $\Sigma_1$ induction and $\Sigma_2$ induction. – Jason Rute Oct 24 '11 at 23:09
And yes, this is standard reverse math stuff. The standard resource is Steve Simpson's book Subsystems of Second Order Arithmetic. The first chapter is online at Steve's website: math.psu.edu/simpson/sosoa/chapter1.pdf – Jason Rute Oct 24 '11 at 23:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9398270845413208, "perplexity": 653.506811586126}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936463660.11/warc/CC-MAIN-20150226074103-00080-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/factorize-in-q-x-r-x-and-c-x.126788/ | # Homework Help: Factorize in Q[x] , R[x] and C[X]
1. Jul 21, 2006
### kezman
Factorize in Q[x] , R[x] and C[X]
$$4x^6 + 8x^5 - 3x^4 - 19x^3 - 26x^2 - 15x - 3$$
I couldnt finish it but this is up to where I did:
$$4(x+1/2)^2 (x^4+x^3-2x^2-3x-3)$$
2. Jul 21, 2006
### arildno
In the original one, 1 is not a root, whereas 1 is a root in your second one.
3. Jul 21, 2006
### kezman
Im sorry I cant see that.
In the second gives -6
4. Jul 21, 2006
### arildno
Sorry, my mistake.
5. Jul 21, 2006
### 0rthodontist
Well, my calculator says that the remaining real solutions of your fourth-degree factor are $$\pm \sqrt{3}$$ but I don't know how you could find those by hand.
6. Jul 23, 2006
### kezman
Yes thats right thanks
7. Jul 23, 2006
### kezman
That was a problem in my exam
8. Jul 23, 2006
### 0rthodontist
Well--how do you find them by hand then, seeing how they are irrational?
9. Jul 23, 2006
### StatusX
It's easy to check if a polynomial has any rational roots. To see if it has irrational roots of the form $\sqrt{r}$ for r rational, you can break up the polynomial into even and odd terms and substitute in $\sqrt{r}$, which will leave something like $e(r)+o(r) \sqrt{r}$. For r rational, this will only be zero if e(r)=o(r)=0. In this case,
$$x^4+x^3-2x^2-3x-3 = (x^4+2x^2-3) + (x^3-3x)=0$$
plugging in x= $\sqrt{r}$:
$$(r^2+2r-3) + \sqrt{r} (r-3)$$
We see that r=3 is a solution to both terms, and so $x=\pm \sqrt{3}$ is a root of the original polynomial. This process can be generalized to cube roots, etc, but I doubt if it's very useful unless you know in advance that you'll have a root of this form.
Last edited: Jul 23, 2006 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9019418954849243, "perplexity": 1297.1151016294366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1544376823895.25/warc/CC-MAIN-20181212134123-20181212155623-00550.warc.gz"} |
http://tex.stackexchange.com/users/18/jos%c3%a9-figueroa-ofarrill?tab=activity&sort=comments | # José Figueroa-O'Farrill
less info
reputation
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bio website maths.ed.ac.uk/~jmf location Edinburgh, Scotland age 51 member for 4 years, 5 months seen 1 hour ago profile views 287
I am a mathematical physicist at the Maxwell Institute for Mathematical Sciences and the School of Mathematics of The University of Edinburgh, in sunny Scotland. I am a founding member of the Edinburgh Mathematical Physics Group and regular contributor to its blog.
Nov3 comment Square root sign has dotted line (PGF/TikZ + XeLaTeX). Is this a bug? Is there a workaround? Indeed! That works! (But this is surely not the intended behaviour, right?) Mar15 comment “Undefined control sequence” error with fontspec Excellent! Many thanks! This solves my problem. Mar11 comment \pounds misbehaving with the eulervm package even in T1 encoding Ah, I didn't know -- thanks! Mar11 comment \pounds misbehaving with the eulervm package even in T1 encoding One question: why do you recommend utf8 instead of utf8x? I must admit I forgot why I made that choice when I thought about that problem. Mar11 comment \pounds misbehaving with the eulervm package even in T1 encoding Thanks. I am undecided about leaving the £ outside of math mode. I tend to agree that they detract from the maths, yet at the same the currency is a system of units and just as in physics formulae, the units are often part of the equation. Mar11 comment \pounds misbehaving with the eulervm package even in T1 encoding In the meantime I'm putting the £ outside math-mode, so it's not as if this is critical, but I would like to know the answer. Feb28 comment Changing the font for “alert” in beamer Many thanks! I didn't think to look in the code :( Aug28 comment In TexLive 2009, 'tlmgr' tells me there are 'no updates available' for months now. Really? If I may answer my own question... I just came across a thread in the tex-live mailing list which seems to suggest that TeXLive 2010 pretest is also frozen prior to release. Aug28 comment In TexLive 2009, 'tlmgr' tells me there are 'no updates available' for months now. Really? I'm running a pretest version of TeXLive 2010, but I also find with tlmgr (and what I believe to be a correct repository) that no updates are available. Is this also the expected behaviour? Aug25 comment \colon for maps in opposite direction Yes, I understand and agree that there is a semantic distinction. Alas, I find $f:A\to B$ much easier to parse (in the source) than $f\colon A \to B$, not to mention easier to type. So will not be using \colon any time soon :( Aug25 comment \colon for maps in opposite direction Why is \colon preferred over :? I just checked a random sample of books in my office and the overwhelming majority uses the symmetrical spacing of :. The exceptions (among the 15 or so books I checked) are Atiyah-MacDonald, Besse and Kobayashi-Nomizu. I have to admit that I never sat down to think about this before. Aug20 comment New language - hyphenation @Arthur: Yes! That's the one. Many thanks. Aug19 comment New language - hyphenation @Konrad: Hyphenation in English is a bit like English Common Law. It's by precedent: so you have to see how it was done in the past. Nowadays it's recorded in the dictionaries, of course. Hyphenation in TeX has a curious history, though: it's the result of a couple of PhD theses supervised by Knuth in Stanford. I'll try to fish out the reference later. Aug16 comment New language - hyphenation I understand -- I was simply suggesting using the Castillian hyphenation patterns which already exist. Aug16 comment Aquamac AucTeX should open PDF in external viewer, not render it in Emacs buffer I think that Aquamacs (version 2.0 (or 2.1?) by the way: 23.1 is the GNU Emacs version on which it is based) should read the system's default application to open PDF files out of the box without any need for customisation. Aug14 comment What fonts are available for LaTeX? I realise that your question mentions LaTeX explicitly, so you may already be aware of Xe(La)TeX and do not wish to consider it. But just in case, Xe(La)TeX allows you to use pretty much any font present in your system. Aug11 comment New language - hyphenation Hyphenation in English is a pain (and worth a PhD thesis or two) but I remember learning the Castillian hyphenation rules in school as a child, so it would seem to be much simpler (unless I'm missing something, of course). Is hyphenation in Asturian all that different from Castillian? Aug10 comment What is the correct way to do delimiters? In English, "goniometric" functions are more commonly known as trigonometric functions. I don't think that this is a TeXnical term :) Aug10 comment How to determine the true size of a font? msw: thanks -- can you point to some literature on (attempts at) a precise definition of the size of a fontface? I'm curious. Thanks in advance. Aug10 comment Showcase of beautiful typography done in TeX & friends Lev, I'm still finding typos in my own thesis, written over 20 years ago. (Let me just point out, though, that Feynman's name is inexplicably also misspelled in the references. Don't tell me you wrote the bibliography by hand...) | {"extraction_info": {"found_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": 0.8333070278167725, "perplexity": 2137.8190764446254}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1419447550512.5/warc/CC-MAIN-20141224185910-00058-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://mathhelpforum.com/pre-calculus/97460-slope-curve-print.html | # slope of curve
• August 9th 2009, 08:54 AM
live_laugh_luv27
slope of curve
For $f(x)=5x^2$ at the point (2,20), find
a) slope of curve
b) an equation of the tangent line
c) an equation of the normal line
Here is what I have so far:
a) $m=20$
b) $y-20=20(x-2)$
c) $y-20=-\frac{1}{20}(x-2)$
Is my work correct? Thanks!
• August 9th 2009, 09:04 AM
Plato
Quote:
Originally Posted by live_laugh_luv27
For $f(x)=5x^2$ at the point (2,20), find
a) slope of curve
b) an equation of the tangent line
c) an equation of the normal line
Here is what I have so far:
a) $m=20$
b) $y-20=20(x-2)$
c) $y-20=-\frac{1}{20}(x-2)$
Is my work correct?
YES | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9142326712608337, "perplexity": 2616.758499568879}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1418802768205.146/warc/CC-MAIN-20141217075248-00047-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://arxiv.org/abs/1605.08174 | cs.LG
# Title:Adiabatic Persistent Contrastive Divergence Learning
Abstract: This paper studies the problem of parameter learning in probabilistic graphical models having latent variables, where the standard approach is the expectation maximization algorithm alternating expectation (E) and maximization (M) steps. However, both E and M steps are computationally intractable for high dimensional data, while the substitution of one step to a faster surrogate for combating against intractability can often cause failure in convergence. We propose a new learning algorithm which is computationally efficient and provably ensures convergence to a correct optimum. Its key idea is to run only a few cycles of Markov Chains (MC) in both E and M steps. Such an idea of running incomplete MC has been well studied only for M step in the literature, called Contrastive Divergence (CD) learning. While such known CD-based schemes find approximated gradients of the log-likelihood via the mean-field approach in E step, our proposed algorithm does exact ones via MC algorithms in both steps due to the multi-time-scale stochastic approximation theory. Despite its theoretical guarantee in convergence, the proposed scheme might suffer from the slow mixing of MC in E step. To tackle it, we also propose a hybrid approach applying both mean-field and MC approximation in E step, where the hybrid approach outperforms the bare mean-field CD scheme in our experiments on real-world datasets.
Comments: 22 pages, 2 figures Subjects: Machine Learning (cs.LG); Machine Learning (stat.ML) Cite as: arXiv:1605.08174 [cs.LG] (or arXiv:1605.08174v2 [cs.LG] for this version)
## Submission history
From: Hyeryung Jang [view email]
[v1] Thu, 26 May 2016 07:26:25 UTC (48 KB)
[v2] Tue, 14 Feb 2017 10:52:07 UTC (48 KB) | {"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": 0.8525808453559875, "perplexity": 1858.8778625140649}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540514893.41/warc/CC-MAIN-20191208202454-20191208230454-00308.warc.gz"} |
http://www.catalystframework.org/calendar/2006/12 | # Day 12 - Producing PDFs with Template::Latex
Using Template::Latex to produce PDFs in Catalyst
## LaTeX
LaTeX (pronounced "lay-tek" by those in the know) is a typesetting language that can be used to produce DVI ("Device Independent"), Postscript, or PDF files. However the main problem is that LaTeX (and TeX, upon which LaTeX is based) is one of those "easy to learn, hard to master languages". If you don't know LaTeX and are interested, there are a numerous tutorials around the web - hit Google. For the sake of completeness I will show a minimal example of a LaTeX file:
\documentclass[a4paper]{article}
\begin{document}
Look at me! I was typeset using \LaTeX
And I'm a second paragraph that
is split over two source lines.
\end{document}
If you run pdflatex (assuming you have it installed) and type those five lines in, you will end up with a small, beautifully presented PDF file. Easy wasn't it?
## Template::Latex
So now that you know LaTeX (*cough*) we can go on and produce PDF documents from inside Catalyst. You could simply do a system('pdflatex') call, except for the fact that TeX and LaTeX really don't like taking data from STDIN - I guess STDIN wasn't so common back in the 1970's when TeX was first written.
So rather than going through the hassle of creating a temporary directory, saving the .tex file, parsing the output log and getting the output file ourselves, we will use the Template::Latex module by Andy Wardley which handles all of this for us. Great!
Once you've installed Template::Latex (and the LaTeX system itself if you don't already have it installed) it's time to create a Catalyst application:
$catalyst.pl PDFTest$ cd PDFTest
$perl ./script/pdftest_create.pl view TT TT Now we need to create our action that will generate the PDF file. In lib/PDFTest/Controller/Root.pm add the following code: sub pdf : Local { my ($self, $c) = @_; if ($c->forward( 'PDFTest::View::TT' ) ) {
# Only set the content type if we sucessfully processed the template
$c->response->content_type('application/pdf');$c->response->header('Content-Disposition', "attachment; filename=PDFTest.pdf")
}
}
Now let's create the template. Create a new file in your editor of choice, and save it as root/pdf with the following contents:
[%- USE Latex;
FILTER latex("pdf") %]
\documentclass{article}
\begin{document}
I'm a doument created from [% name %]
\end{document}
[% END -%]
(If you are using this in an existing application and you are using a wrapper, make sure that this template isn't wrapped, or you will get the binary PDF data nicely placed inside your wrapper.)
## Testing It Out
We've created a template that uses the LaTeX filter, and an action that will cause it to be rendered. Let's test it!
$perl ./script/pdftest_server.pl <output snipped> You can connect to your server at http://localhost:3000 Since we called our action pdf, let's visit http://localhost:3000/pdf. Now with any luck your browser should display the PDF inline or prompt you to save what to do with a file called "PDFTest.pdf". However if you are using Catalyst::Runtime version 5.7006 (which is the latest at the time of writing) then you will be seeing an error like the following: Couldn't render template "latex error - pdflatex exited with errors: " This is caused by a bug in the standalone server only (version 5.7006 - this will be fixed in newer versions) where all system calls return -1. Hmm, so we could use one of the other server modes, but they make developing harder. So instead let's use a dirty hack to get round the problem. WARNING: This is very dirty and prone to break for different versions, but it worked for me with pdfeTeX 3.141592-1.21a-2.2 (Web2C 7.5.4) and Template::Latex 2.17. Don't use this in production code. Use it for development only. You have been warned. The way this hack works is be defining a system sub in the Template::Latex package that will get called instead of Perl's built-in system. What this 'replacement' system call does is first call the built-in system, and it then examines the last line of the log file to see if it says "Output written on filename.pdf", in which case it returns 0, else it returns the return value of the built-in system call. Rather than explaining how to go about this hack, I will just include what the pdf sub should look like: sub pdf : Local { my ($self, $c) = @_; { no warnings 'redefine'; # right here, you can tell bad things will happen local *Template::Latex::system = sub { my$ret = system(@_);
my ($filename) =$_[0] =~ m[\\input{(.*?)}] ;
my $fh = new IO::File "${filename}.log"
or die "Unable to open pdflatex logfile ${filename}.log:$!";
my $line; while ( defined($_ = $fh->getline) ) {$line = $_; } return 0 if$line =~ /^Output written on ${filename}.pdf $$\d+ pages?, \d+ bytes?$$.$/;
return $ret; } if$c->engine =~ /^Catalyst::Engine::HTTP/;
if ($c->forward( 'PDFTest::View::TT')) {$c->response->content_type('application/pdf');
}
}
}
## Caveat
Hacks like the one detailed above are generally a really bad idea, because you are messing with the internals of another package. This is asking for trouble. It's included so that you can make a working example with the development server under Catalyst 5.7006 - the bug should be fixed in subsequent versions.
### AUTHOR
Ash Berlin <[email protected]> | {"extraction_info": {"found_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": 0.9173454642295837, "perplexity": 4286.5586461747225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1440645335509.77/warc/CC-MAIN-20150827031535-00239-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/before-the-04-tsunami-an-earthquake-so-violent-it-even-shook-gravity.128586/ | # Before the ’04 Tsunami, an Earthquake So Violent It Even Shook Gravity
1. Aug 10, 2006
### scott1
How is it possible to have a drop in gravity when a earth quake happens?
Last edited by a moderator: Apr 22, 2017
2. Aug 10, 2006
### Danger
Not my field, but I can't see it happening on the whole. If the quake was big enough, maybe it caused a temporary shift in the centre of gravity. If the satellite was aimed at a small, specific area, maybe there was a local drop.
3. Aug 12, 2006
### cesiumfrog
Aerial measurements of gravity are useful for various purposes, eg. identifying economically feasable ore deposits. Try googling "Gravity Recovery and Climate Experiment"?
4. Aug 12, 2006
### Rach3
A large mass was suddenly displaced, thus the earth's gravitational field was slightly altered, very suddenly. A planet is not a uniform sphere; there are inhomogenities in mass distribution, which can be mapped by measuring the gravitational field. Look at the picture - the scale is in parts per 10^-5, these are very tiny differnces.
Note carefully - "drop in gravity" is not a change of physics, it's simply the result of a change of topography.
Similar Discussions: Before the ’04 Tsunami, an Earthquake So Violent It Even Shook Gravity | {"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": 0.828403651714325, "perplexity": 2882.221624696934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823731.36/warc/CC-MAIN-20171020044747-20171020064747-00739.warc.gz"} |
http://www.cs.cornell.edu/courses/cs2800/2017sp/lectures/lec12-probability.html | Lecture 12: probability
• Reading: Peter Cameron's Notes on Probability 1.1–1.4
• Definitions
• Probability space, sample space, probability measure, event, outcome
• examples of modeling experiments as sample spaces
• one die, sum of two dice, height of people
• basic properties of probability spaces
• probability of complement, probabilities ≤ 1
Occasionally I assert properties of sets. For example, in the previous lecture I asserted that $$A \cup B = (A \setminus B) \cup (A \cap B) \cup (B \setminus A)$$, while in today's lecture I asserted that if $$A \subseteq S$$, then $$A \cup (S \setminus A) = S$$.
On your homework, you may assert these kinds of properties without proof as long as:
1. They are clearly stated.
2. They are true
3. They do not trivialize the problem
For example, if asked to prove that $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$, it is not enough to say "this is obvious", but it is fine to say that this is obvious in the context of another proof.
To show you how such a proof would go, I gave the following example:
Claim: If $$E \subseteq S$$ then $$E \cup (S \setminus E) = S$$.
Proof: We must show that if $$x$$ is in the left hand side, then it is in the right hand side, and vice-versa; this is what it means for two sets to be equal.
First, choose an arbitrary $$x \in E \cup (S \setminus E)$$. By definition of $$\cup$$, either $$x \in E$$ or $$x \in S \setminus E$$. In the former case, since $$E \subseteq S$$, we see that $$x \in S$$, while in the latter case, $$x \in S$$ because $$S \setminus E$$ is the set of elements of $$S$$ that don't appear in $$E$$. In either case, $$x \in S$$, completing the proof in this direction.
For the other direction, assume $$x \in S$$. Then either $$x \in E$$ or $$x \notin E$$. In the former case, $$x \in E \cup (S \setminus E)$$ by definition of $$\cup$$. In the latter case, by definition of $$\setminus$$, we see that $$x \in S \setminus E$$, so that $$x \in E \cup (S \setminus E)$$, again by definition of $$\cup$$. In either case, $$x$$ is in $$E \cup (S \setminus E)$$, completing the proof in this direction.
Here is another example:
Claim: $$E \cap (S \setminus E) = \emptyset$$.
Proof: by contradiction. Suppose $$E \cap (S \setminus E) \neq \emptyset$$. Then there exists some $$x \in E \cap (S \setminus E)$$. By definition of $$\cap$$, we see $$x \in E$$ and $$x \in S \setminus E$$. By definition of $$\setminus$$, we see that $$x \notin E$$, but this contradicts the fact that $$x \in E$$.
Definitions
I will use $$2^S$$ and $$Pow(S)$$ interchangably to refer to the power set of $$S$$; I prefer $$2^S$$ (it's shorter) but did not want to use it before we proved that $$|2^S| = 2^{|S|}$$. Recall that the power set of $$S$$ is the set of all subsets of $$S$$.
A probability space is a set $$S$$ (called the sample space) paired with a function $$Pr : 2^S → \mathbb{R}$$, satisfying:
1. for all $$E \subseteq S$$, $$Pr(E) \geq 0$$.
2. $$Pr(S) = 1$$.
3. If $$E_1$$ and $$E_2$$ are disjoint, then $$Pr(E_1 \cup E_2) = Pr(E_1) + Pr(E_2)$$.
$$Pr$$ is called the probability function or probability measure.
The elements of $$S$$ are called outcomes; the subsets of $$S$$ are called events. Thus the probability measure assigns a (non-negative) real number to every event.
Important: The probability of $$E$$ is not $$|E|/|S|$$. This is true for some probability spaces, but not all. Assuming that $$Pr(E) = |E|/|S|$$ will lead to incorrect answers for most problems.
Examples
To model the throw of a single six-sided die, we could choose the sample space $$S = \{1, 2, \dots, 6\}$$. If we wanted to assume that all outcomes were equally likely, we could define $$Pr(E) = |E|/6$$, but this is only one possible definition; we could certainly model a die with different likelihoods for different sides, which would give a different function.
There are many ways to model a throw of two dice. On possible sample space is
$S_1 = \{1, 2, 3, \dots, 12\}$
Another possible sample space is $$S_2 = N \times N$$ where $$N = \{1,2,\dots,6\}$$.
There are a few things that determine a good choice:
• there should be enough outcomes to describe the experiments we are interested. For example, the first sample space wouldn't allow us to ask "what is the probability that both dice are even?")
• it's nice to model the problem in a way that makes the probability function easy to write down. With $$S_1$$, writing down the probabilities is non-trivial, whereas it is easier to write down the probability function for $$S_2$$ if both dice are fair.
Another example: suppose we wanted to perform an experiment by selecting a student from the room uniformly at random and sampling their height. Possible sample spaces include:
• $$\mathbb{R}$$
• some subset of $$\mathbb{R}$$, such as the positive reals, or the set of numbers having somebody in the room with that height
• the set of people in the room.
Again, these are all perfectly reasonable ways to model the experiment (they will of course have different probability functions). However, some of them make it easier to write down the probability function.
Properties of probability spaces
Everything else that we know about probability is derived from the definition. Here are some examples:
Notation: if there is a sample space that is clear from context, I will write $$\bar{E}$$ (read "$$E$$ complement") for $$S \setminus E$$.
Claims about probability all assume that $$S$$ and $$Pr$$ form a probability space; I will not explicitly write this down.
Claim: $$Pr(E) + Pr(\bar{E}) = 1$$ (alternatively, $$Pr(\bar{E}) = 1 - Pr(E)$$).
Proof: By above, $$E$$ and $$\bar{E}$$ are disjoint, so \begin{aligned} Pr(E) + Pr(\bar{E}) &= Pr(E \cup \bar{E}) && \text{by rule 3} \\ &= Pr(S) && \text{since E \cup \bar{E} = S} \\ &= 1 && \text{by rule 2} \\ \end{aligned}
Claim: For all $$E$$, $$Pr(E) \leq 1$$.
Proof: For the sake of contradiction, suppose there were some $$E$$ with $$Pr(E) > 1$$. By rule 2, we know $$Pr(\bar{E}) \geq 0$$. Adding these inequalities together, we see that $$Pr(E) + Pr(\bar{E}) > 1 + 0 = 1$$. But by the previous claim, we know that $$Pr(E) + Pr(\bar{E}) = 1$$; this is a contradiction. | {"extraction_info": {"found_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": 0.9969528317451477, "perplexity": 172.40343820656878}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592309.94/warc/CC-MAIN-20180721032019-20180721052019-00628.warc.gz"} |
https://www.intmath.com/blog/mathematics/roller-door-problem-7665 | # Roller Door problem
By Murray Bourne, 28 Nov 2012
In an earlier article, I discussed the Length of an Archimedean Spiral. We saw we can find the length of such a spiral using integration, once we know the equation of the spiral and the beginning and end points. The equation for an Archimedean Spiral is in polar form, and is determined by an angle θ, the amount of turn of the spiral.
Roller door.
[Image source]
Brent, an engineer with a roller door company, asked an interesting question in the comments for that article. He posed a "real-life" issue he faced.
His problem is the reverse. He knows length of the spiral already (it's the height of the roller door, plus the bit that's wrapped around the drum wheel), but he needs to know the number of turns of the drum for the given height of the door and radius of the drum.
This is not such a straightforward problem.
I suggested he could produce a table for already known door heights and number of turns. and then use interpolation to find the required number of turns (at least approximately).
Brent also turned to PhysicsForums to ask for help. He wrote:
Hi,
I'm an engineer designing a spring system for a garage roller door. I need to know the number of turns of the door for all the size combinations.
I've found this page which gives a good equation for finding the length if you know the number of turns, starting radius and gap between spirals:
The equation of the spiral is r = x + yθ, with x = starting radius, y = gap/(), and to find L we're taking the integral from a = 0 to b = 2πn (where n = turns).
When you know n, this is straightfoward, and even I could work that out. But it's been a decade since I've done anything like this, so I was wondering if anyone could help me solve for n in this integral:
My way of solving this is to find L for n=1,2,3,4,5 etc, graph it in Excel and use "find trendline" to get an equation. Any help appreciated, thanks.
The response there was not very helpful.
Brent provided his Excel spreadsheet so I could see what he was doing.
## Specific Example
Brent's roller door has a barrel radius of 127.5 cm and the thickness of the door slats (called the "curtain") is 4.0 cm. This means the radius will increase by 4.0 cm for each turn of the barrel.
So x = 127.5 and y = 4/(2π)=0.6366 in this case.
The following daigram (not to scale) indicates our situation:
Applying the integration formula above, we have the length of the curtain wrapped around the barrel for 1 turn:
Similarly, for 2 turns (that is, 0 to 4π), we have:
Continuing on for 3 up to 6 turns, we obtain the following values:
Turns Length
0
0
1
814
2
1654
3
2516
4
3406
5
4320
6
5259
Next, we use Excel to plot these values and to draw a trendline through them. We choose "polynomial of degree 2" because a consideration of the integral and an inspection of the Excel graph (when extrapolating beyond 6 turns) shows it is parabolic.
Excel can tell us the equation for the graph produced, and it is included on the chart.
We need to translate the "y" and "x" variables on Excel's chart for our own situation. The vertical scale is the length of the curtain, and the horizontal scale is the number of turns. Also, the "0.1667" value is a rounding error, which we can discard.
So in our case, we would write the Length of the curtain (L) for the number of turns, n as:
Solving for n (using the quadratic formula) gives us 2 solutions. The first one is:
The second solution gives a negative result for n, so we discard it.
We now check our work (by comparing a known value).
If the length is 3406 cm, then the number of turns given by our new formula is:
This is certainly close to the 4 turns we expected.
So Brent could now use this formula to find the number of turns for any roller door's curtain length given a barrel radius of 127.5 cm and a curtain thickness of 4 cm.
## Linear Solution
The Excel graph for the range of turns 0 to 6 is very nearly linear. Using the "Linear" trendline option in Excel gives the model for the Length given the number of turns as:
L = 876.5n - 62.4
Solving for n gives:
Applying this simpler formula to some of our known values gives:
For length L = 814, number of turns, n = 0.99989. (expected 1)
For length L = 1654, number of turns, n = 1.9582. (expected 2)
For length L = 5529, number of turns, n = 6.0712. (expected 6)
So for lengths of the roller door curtain less than about 10 m, this simpler linear model gives acceptable results.
## Other solutions
Neil provided another solution in the comments.
He proposed:
where
n = number of turns
OL = overall length
ID = inner diameter of the spiral
OD = outer diameter of the spiral
However, Brent pointed out this can't be used because he doesn't know the outer diameter until he knows the number of turns.
Jacob then weighed in with another proposal, which involved taking an average of the inner and outer radius and then applying the circumference of a circle to that average.
Jacob suggests if we don't know the number of turns, we can use this formula to find it:
Applying that formula to our example above, with y = 4 and ID = 127.5×2 = 255 and our known case where L = 3406 gives:
This is very close to the 4 turns we expected.
Applying the formula for another known case, L = 5259, we obtain 6.08 turns, nicely close to the expected value of 6.
## Conclusion
Brent has a choice of formulas to use, all of which give acceptable results. The linear model is the easiest.
Parabolic model:
Linear model:
The model suggested by Neil and refined by Jacob:
Here is a table that shows the values from the given formulas.
Length
(via calculus)
Turns
(original)
Turns
(parabolic model)
Turns
(linear model)
Turns
(Neil/Jacob model)
0 0 0 0 0
814 1 1.0001 0.9999 1.0158
1654 2 2.0014 1.9582 2.0318
2516 3 2.9992 2.9417 3.0431
3406 4 4.0004 3.9571 4.0571
4320 5 5.0005 4.9999 5.069
5259 6 6.0006 6.0712 6.0802
### 9 Comments on “Roller Door problem”
1. Thomas A Buckley says:
Two different simple non calculus methods yielded identical results, which differ greatly from the table used to derive the concluding solution above. So calculus people, recheck!
For example you say starting radius x = 127.5 cm. Taking center of door thickness for length, allowing inner shortening to be balanced by outer lengthening when rolled,
effective starting radius = 127.5 + 4/2 = 129.5 cm!
.....................................Length cm....Length cm
1.........................2Pi(127.5+2)=814........880
2............2Pi(127.5+6)+814=1652.......1917
3.....2Pi(127.5+10)+1652=2516.......3111
4.....2Pi(127.5+14)+2516=3405.......4462
5.....2Pi(127.5+18)+3405=4320.......5970
6.....2Pi(127.5+22)+4320=5259.......7636
After the 1st turn each diameter increases by 4 cm.
OK, so my method is NOT exact, but it should not be so different from the results obtained by calculus.
2nd Method.
Number of Turns n
= [SqRt( Lt/pi + r^2)-r]/t
L = Door length, t = door thickness, r = barrel radius.
CSA of door = Lt
This area must fit in the space between concentric circles.
Taking R as radius when door is wound on barrel, then
Lt = piR^2 - pir^2 . . ie both areas are the same.
R = SqRt( Lt/pi + r^2) Eq1
Thickness of door on barrel = R-r
Number of turns n required
= (R-r) /t . . now substitute RHS of Eq1 for R
= [SqRt( Lt/pi + r^2)-r]/t
Note R disappears, and providing the desired variables,
door Length and thickness, and barrel radius, yields the required number of Turns. Also fractional turns are valid, eg 4.5 turns means 4.5 turns. This equation gives the same results as my 1st method table above.
Still cannot get html codes in my equations, sorry Murray.
2. Thomas A Buckley says:
Jacob's formula returns 5.2 turns. The 12.8 turns result is in error because barrel DIAMETER is 127.5 x2=255, not 127.5/2!
My equation returns 5.15 turns. That is good agreement
for different approximate methods, casting further doubt that the variables for door length of 4462cm requires only 4 turns as the earlier table shows.
3. Alan says:
Brent's best bet is to get Mathcad! A numerical solution is then obtained almost trivially in a couple of lines.
4. Murray says:
@Thomas - Thank you! The errors were due to my not dividing the curtain width 4 by 2pi.
I have re-written and fixed that whole part, plus added a simpler linear model.
Your approximations using the average radius certainly make sense for this problem, and align very closely with the (corrected) values via calculus.
I must have been half asleep when looking at Jacob's formula. That part has been corrected in the article, too!
@Alan - Good suggestion, but Mathcad is probably overkill for this problem...
5. brent says:
Alan - I needed to obtain a formula because I needed to do this calculation about 1000 times: for every size and width of door.
Thanks.
6. Calla says:
Hello! I am using this problem in a mathematical exploration assignment for school. I understand most of the solution, except for the line, 'x = 127.5 and y = 4/(2?)=0.6366 in this case'. Why do we divide the width of the curtain (4 cm) by 2pi? What do x and y refer to in this line?
Any help would be greatly appreciated.
7. Calla says:
I am confused by one aspect of this article: the author divides 4 (the distance between each arm) by 2pi in order to get b, 0.6366. This follows Brent's assertion that in the equation r=x + y pheta, y=gap/2pi.
But in a previous article ("Length of an Archimedean Spiral") it was stated that,
Total angle turned by spiral in radians = 2?b
Distance between each arm (or 'gap') = 2?b divided by the number of turns
Thus, how can b be derived from dividing 2pi by the gap? I would think that:
b = (gap x number of turns) divided by 2?
based on the previous article. Could anyone help in explaining this to me? It would be greatly appreciated.
8. Eric Blaise says:
I am actually glad I do not have to worry about this. I am pretty sure I did a similar problem in geometry but cannot use it today for the life of me. Most garage doors continue rolling until they encounter resistance and then stop. either way, I am pretty sure there is an application for this somewhere.
Eric
9. brent says:
Eric, the number of turns is needed to calculate how many turns the springs need to sized for. You can't do any spring calculations without having a good way to calculate turns of the door.
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https://www.mecharithm.com/screws-a-geometric-description-of-twists-in-robotics/ | Blog Post
# Screws: a Geometric Description of Twists in Robotics
In the previous lesson, we learned about velocities in robotics. We became familiar with angular and linear velocities and saw that stacking them together gives us the twist. We also saw how we could change the frame of reference for angular velocities and twists.
This lesson is about screws as a geometric interpretation for twists and how they can be used to express configurations in robotics.
This lesson is part of the series of lessons on foundations necessary to express robot motions. For the complete comprehension of the Fundamentals of Robot Motions and the tools required to represent the configurations, velocities, and forces causing the motion, please read the following lessons (note that more lessons will be added in the future):
https://www.mecharithm.com/category/learning-robotics-mechatronics/fundamentals-of-robotics-course/fundamentals-of-robot-motions/
Also, reading some lessons from the base lessons of the Fundamentals of Robotics course are deemed invaluable.
Ok, now let’s see how twists and screws are related.
In a couple of lessons before, we saw that every rigid body displacement could be obtained by a finite rotation about and translation along a fixed screw axis 𝘚 and we became familiar with the exponential coordinates of robot motions. We saw that in order for us to define the screw axis, we needed to learn about velocities, and now that we have the necessary foundation for it, we can go back and look at the screw motion again. These tools will equip us for the robot kinematics lesson in the near future.
In the Velocities in Robotics lesson, we learned that the angular velocity ω could be represented by a unit axis ω̂ and the rate of rotation θ̇ about this axis. Similarly, we can express a twist (that has an angular component and a linear component) as a screw axis 𝘚 and the rate θ̇ about this axis:
$\mathcal{V} = \mathcal{S} \dot{\theta}$
Any robot configuration can be achieved by starting from the home (fixed) reference frame at time t = 0 (T(0) = I) and then integrating this twist over a specified time to reach the final configuration, as we also saw before.
On the other hand, any rigid body velocity has a linear component and an angular component equivalent to the instantaneous velocity about some screw axis.
Suppose that the configuration of the body frame relative to the space frame at any time is represented by a rotation about and translation along the screw axis 𝘚 with the rate θ̇ (a scalar that shows how fast the body moves along the screw). The figure below shows this representation:
The screw axis 𝘚 can be represented in two ways.
## {q,ŝ,h} Interpretation of the Screw Axis
First, the screw axis can be represented by any point on the axis q ∈ ℝ3, the unit vector in the direction of the screw axis ŝ, and the screw pitch h, which is the linear speed along the screw axis divided by the angular speed θ̇ about the screw axis: {q,ŝ,h}.
The twist about the screw axis 𝘚 represented by {q,ŝ,h} can be defined as:
$\mathcal{V} = \begin{pmatrix} \omega\\ v \end{pmatrix} = \begin{pmatrix} \hat{s}\dot{\theta}\\ -\hat{s}\dot{\theta} \times q+h\hat{s}\dot{\theta} \end{pmatrix}$
ω is the angular velocity, and it is in the direction of the unit vector ŝ with the magnitude θ̇, and the linear velocity v has two parts. hŝθ̇ is due to translation along the screw axis (in the ŝ direction) that exists when the screw has non-zero pitch and ω×r = ŝθ̇×(-q) = -ŝθ̇×q is due to the linear motion at the origin which is the result of the rotation about the screw axis. We saw before that the linear velocity during the circular motion (that has zero pitch = no translational motion) is tangential to the circular path, and it can be calculated by the cross product of the angular velocity and the radius of the path (this is in the plane orthogonal to ŝ):
Now think in the reverse fashion. If we have the twist 𝒱 = (ω,ν) and we want to find the screw axis {q,ŝ,h} and θ̇ that can generate the same twist, we will have two cases.
Case one is where we have rotational motion (ω ≠ 0), and the screw pitch is finite. If the angular velocity is not zero (we also have rotational motion), then the pitch h is finite, and θ̇ is the norm of the angular velocity vector (θ̇ = ||ω||). Then because ω = ŝθ̇ = ŝ||ω||, we can find ŝ = ω/||ω|| = ω̂. The pitch can be calculated using the equation h = ω̂Tν/||ω|| and q is chosen so that the term -ŝθ̇×q provides the portion of ν orthogonal to the screw axis.
Case two is where there is no rotational motion (ω = 0), and thus the pitch h is infinite and ŝ = ν/||ν|| and θ̇ is the linear speed ||ν|| along ŝ.
In this representation of the screw axis 𝘚, the pitch can be infinite, q is not unique (any point along the screw axis can be used), and it is a cumbersome collection. Hence, we opt for an alternative representation for the screw axis 𝘚.
## Screw Interpretation of a Twist in Robotics
As an alternative representation, the screw axis 𝘚 is defined as follows. First, we choose a reference frame and then define the screw axis 𝘚 as the 6-vector in that frame’s coordinates as:
$\mathcal{S} = \begin{pmatrix} \mathcal{S}_{\omega}\\ \mathcal{S}_v \end{pmatrix} = \begin{pmatrix} \text{angular velocity when }\dot{\theta} = 1\\ \text{linear velocity of the origin when } \dot{\theta} = 1 \end{pmatrix}$
𝘚ω is the 3D unit angular velocity when the rate θ̇ = 1 and 𝘚ν is the 3D linear velocity of the origin of the frame when the rate is 1. We can conclude that the screw axis is a normalized twist and thus, the twist 𝒱 = (ω,ν) can be represented by the multiplication of the screw axis 𝘚 by the scalar rate θ̇ (𝘚θ̇). In this case, as we discussed, the screw axis 𝘚 can be defined using a normalized version of the twist 𝒱 = (ω,ν) corresponding to motion along the screw.
As before, we will have two cases:
Case one is when there is a rotational component and ω≠0, and therefore the pitch h is finite. The angular component 𝘚ω of the screw axis is nonzero and the twist, 𝒱 is normalized by the norm of the angular velocity vector:
$\mathcal{S} = \frac{\mathcal{V}}{||\omega||} = (\frac{\omega}{||\omega||},\frac{v}{||\omega||} )$
θ̇ = ||ω|| is the angular speed about the screw axis such that 𝘚 θ̇ = 𝒱. ||𝘚ω|| = 1 because we normalized it with the angular speed. 𝘚ν is arbitrary, with no constraints on it.
Case two is when there is no rotational motion (ω=0), and thus the pitch h is infinite. In this case, the motion is a purely linear motion with no rotation. The angular component is zero, and the linear part is a unit vector. In this case, the twist, 𝒱 is normalized by the length of the linear velocity vector:
$\mathcal{S} = \frac{\mathcal{V}}{||v||} = (0,\frac{v}{||v||} )$
θ̇ = ||ν|| is the linear speed along the screw axis such that 𝘚 θ̇ = 𝒱. For this case, 𝘚ω = 0, ||𝘚ν|| = 1 since we normalized it.
Note here that six numbers are needed to represent the screw axis 𝘚 = (𝘚ω,𝘚ν), but the space of all screws is five-dimensional (5D), and this is because either 𝘚ω or 𝘚v has a unit length. As discussed, if both the angular and linear components of the screw are non-zero meaning that we have a rotational motion as well, then the screw is defined so that ||𝘚ω|| = 1.
Now let’s see how we can define the matrix representation of the screw axis 𝘚.
## Matrix Representation of the Screw Axis 𝘚
Since the screw axis is the normalized version of the twist, then the matrix representation of the screw axis 𝘚 = (𝘚ω,𝘚v) can be defined as (for more information on why it has this matrix form, please refer to the lesson on the velocities in robotics):
$[\mathcal{S}] = \begin{pmatrix} [\mathcal{S}_{\omega}] & \mathcal{S}_v\\ o & 0 \end{pmatrix} \in se(3)$
Now let’s see how we can change the frame of reference in which a screw axis 𝘚 is defined.
## Change of Frame of Reference of a Screw Axis 𝘚
As we saw in the lesson on the velocities in robotics, the adjoint transformation could be used to change the frame of reference of the twist, and since the screw axis is a normalized twist, we can use the adjoint transformation to change the frame of reference of the screw axis as well:
$\mathcal{S}_a = [Ad_{T_{ab}}] \mathcal{S}_b\\ \mathcal{S}_b = [Ad_{T_{ba}}] \mathcal{S}_a$
𝘚a is the screw axis representation in the frame {a}, and 𝘚b is the screw axis representation in the frame {b}.
If the screw axis 𝘚 is expressed in coordinates of the body frame {b}, 𝘚b, then 𝒱b = (ωbb) = 𝘚bθ̇ is called the body twist which is not affected by choice of the space frame, and if the screw axis 𝘚 is expressed in coordinates of the space frame {s}, 𝘚s, then 𝒱s= (ωss) = 𝘚sθ̇ is called the spatial twist which is not affected by choice of the body frame. Thus, we only need to define the frame in which the twist (or screw) is represented. No other frames matter. A spatial twist depends on the {s} frame, and a body twist depends on the {b} frame.
Now, based on our knowledge about the screw axis, let’s go back to the exponential coordinate representation of rigid body motions.
## Exponential Coordinate Representation of Rigid Body Motions
In the exponential coordinate representation, 𝘚θ ∈ ℝ6:
• If pitch of the screw axis 𝘚 = (𝘚ω,𝘚v) is finite then we have rotational motion and θ is the angle of rotation about the screw axis.
• If the pitch is infinite then the motion is pure translation with no rotation and θ is the linear distance travelled along the screw axis.
As we saw in the lesson about the exponential coordinates of rotation, the matrix exponential e[ω̂]θ is equal to the rotation matrix that can act on a vector or a frame and can rotate it from the initial orientation to the final orientation. Similarly, the matrix representation of the screw axis can be used in the matrix exponential e[𝘚]θ for rigid body motions. Thus, the matrix exponential for rigid body motions can map the elements of the Lie algebra se(3) to the elements of the Lie group SE(3):
exp: [𝘚]θ ∈ se(3) → T ∈ SE(3)
And this means that exponentiation takes the initial configuration of the frame to the final configuration of the frame by following along and about a screw axis 𝘚 by θ.
And the matrix logarithm is the invert of the matrix exponential and finds the matrix representation of the exponential coordinates 𝘚θ:
log: T ∈ SE(3) → [𝘚]θ ∈ se(3)
And this means that if we have a given configuration, we want to find the screw axis and θ such that if followed along and about this screw axis by that amount gives the same configuration.
The normalized (unit) screw axis for full spatial motions is similar to the normalized angular velocity axis for pure rotations.
As we found a closed-form solution for the matrix exponential e[ω̂]θ for orientations before, let’s examine if we can do the same for the matrix exponential e[𝘚]θ for rigid body motions. Let the screw axis be 𝘚 = (𝘚ω,𝘚ν) then we will have two cases:
• If we have rotational motion, then for any distance θ ∈ ℝtravelled along the axis, the matrix exponential for rigid body motions can be written as:
$e^{[\mathcal{S}]\theta} = \begin{pmatrix} e^{[\mathcal{S}_\omega]\theta} & \underbrace{(I\theta + (1-cos\theta)[\mathcal{S}_\omega] + (\theta-sin\theta)[\mathcal{S}_\omega]^2)}_{G(\theta)}\mathcal{S}_v\\ o & 1 \end{pmatrix}$
• And if the rotational part is zero and the screw axis is pure translation with no rotation, then:
$e^{[\mathcal{S}]\theta} = \begin{pmatrix} I & \mathcal{S}_v\theta\\ o & 1 \end{pmatrix}$
Here, θ is the linear distance traveled. The matrix I in the upper left of the matrix shows that the orientation does not change, and the motion is pure translation.
The proof is similar to the approach that we used in the lesson about the exponential coordinates of rotation for the matrix exponential.
The inverse problem says that given an arbitrary configuration (R,p) ∈ SE(3), we can always find a screw axis 𝘚 = (𝘚ω,𝘚ν) and a scalar θ such that
$e^{[\mathcal{S}]\theta} = \begin{pmatrix} R & p\\ o & 1 \end{pmatrix}$
And as we saw before, the matrix
$[\mathcal{S}]\theta = \begin{pmatrix} [\mathcal{S}_{\omega}]\theta & \mathcal{S}_v\theta\\ o & 0 \end{pmatrix} \in se(3)$
is called the matrix logarithm of T = (R,p).
To solve the inverse problem, we follow the following algorithm:
Given (R,p) written as T ∈ SE(3), find a θ ∈ [0,π] and a screw axis 𝘚 = (𝘚ω,𝘚ν) ∈ ℝsuch that e[𝘚]θ = T. The vector 𝘚θ ∈ ℝ6 comprises the exponential coordinates for T, and the matrix [𝘚]θ ∈ se(3) is the matrix logarithm of T.
(a) If R = I then set 𝘚ω = 0 and 𝘚ν = p/||p|| and θ = ||p||.
(b) Otherwise, use the matrix logarithm on SO(3) that we learned in the exponential coordinates for rotations lesson to determine 𝘚ω (= ω̂ ) and θ for R. The 𝘚ν is calculated as 𝘚ν = G-1(θ)p where G-1(θ) = 1/θ I – 1/2[𝘚ω] + (1/θ – 1/2 cotθ/2)[𝘚ω]2.
Note that every single-degree-of-freedom joint (revolute joint, a prismatic joint, and a helical joint) of a robot that we talked about in the degrees of freedom lesson has a joint axis defined by a screw axis, and thus, we can conclude that the matrix exponential and the logarithm can be used to study the robot kinematics as we will see in the coming lessons:
Now let’s see some examples that use all the knowledge that we have learned thus far to find solutions.
### Example: Homogenous Transformation Matrix to Exponential Coordinates of Motion
Suppose that the configuration of the body frame relative to the space frame is as the following figure:
In which the origin of the {b} frame is at (3,0,0) in terms of the space frame coordinates. The configuration of the {b} frame relative to the {s} frame, as we learned in the lesson about the homogenous transformation matrices, can be found using the transformation matrix Tsb as:
$T_{sb} = \begin{pmatrix} 0 & -1 & 0 & 3\\ 0 & 0 & -1 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$
We want to find the screw motion (the screw axis 𝘚 and the amount of traveled distance θ about the screw axis) that can generate the same configuration.
Since the orientation of the body frame is not the same as the orientation of the space frame, then we have a rotational motion. Using the approach we learned in the lesson about the exponential coordinates of orientation for the matrix logarithm of rotations, we can easily find the unit axis and the amount of rotation about this axis that can produce the given orientation as:
$1+2c_{\theta} = r_{11} + r_{22} + r_{33} = 0 \rightarrow \theta = 120^{o} = 2.093 rad$
$\begin{split} [\hat{\omega}] & = \frac{1}{2sin\theta}(R-R^T)\\ & = \frac{1}{\sqrt 3} \begin{pmatrix} 0 & -1 & -1\\ 1 & 0 & -1\\ 1 & 1 & 0 \end{pmatrix} \end{split} \rightarrow \hat{\omega} = \frac{1}{\sqrt 3} \begin{pmatrix} 1\\ -1\\ 1 \end{pmatrix}$
So, a rotation of 120o about the unit axis calculated above will create the same orientation. Now, using the second approach to calculate the screw axis, we can say that the
$\mathcal{S}_{\omega} = \hat{\omega} = \frac{1}{\sqrt 3}\begin{pmatrix} 1\\ -1\\ 1 \end{pmatrix}$
And
$\begin{split} \mathcal{S}_v & = G^{-1}(\theta)p \\ & = (\frac{1}{\theta}I – \frac{1}{2} [\mathcal{S}_{\omega}] + (\frac{1}{\theta} -\frac{1}{2}cot\frac{\theta}{2} )[\mathcal{S}_{\omega}]^2)p\\ & = \begin{pmatrix} 1.055\\ -1.055\\ -0.677 \end{pmatrix} \end{split}$
And thus the screw axis can be defined as:
$\mathcal{S} = \begin{pmatrix} \mathcal{S}_{\omega}\\ \mathcal{S}_v \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt 3}\\ -\frac{1}{\sqrt 3}\\ \frac{1}{\sqrt 3}\\ 1.055\\ -1.055\\ -0.6769 \end{pmatrix}$
Therefore, a screw motion about the screw axis 𝘚 calculated above with the amount of θ calculated earlier produces the same configuration defined by the homogenous transformation matrix Tsb.
For more practice, let’s find the {q,ŝ,h} representation of the screw axis and draw it. Since we have rotational motion, then we use the equations for the first case and can calculate the screw axis parameters as:
$\begin{matrix} \hat{s} = \mathcal{S}_{\omega} = 0.5774\begin{pmatrix} 1\\ -1\\ 1 \end{pmatrix}\\ h = \frac{{\mathcal{S}_{\omega}^T} v}{||\omega||} = 0.8275\\ q = \begin{pmatrix} 1\\ 1\\ 0 \end{pmatrix} \end{matrix}$
Note that q is not unique and the calculated q is only one feasible answer that can be calculated solving the equation v = -ŝθ̇×q + hŝθ̇. Therefore, the screw axis can be visualized as:
Now let’s see another example.
### Example: Screw Motion Corresponding to a Given Twist
We want to find and visualize the screw axis which if followed at a specific rate will correspond to the given twist 𝒱 = (0,2,2,4,0,0).
From the screw representation of a twist, we can write:
$\mathcal{V} = \begin{pmatrix} 0\\ 2\\ 2\\ 4\\ 0\\ 0 \end{pmatrix} = \mathcal{S}\dot{\theta}$
Since the rotational part is not zero, we also have rotational motion and we should normalize the twist using the norm of the angular velocity:
$\mathcal{V} = 2\sqrt{2}\begin{pmatrix} 0\\ 1/\sqrt{2}\\ 1/\sqrt{2}\\ 2/\sqrt{2}\\ 0\\ 0 \end{pmatrix}$
Therefore, it is easy to see that the rate θ̇ = 2.828 and the screw axis is equal to:
$\mathcal{S} = \begin{pmatrix} 0\\ 0.7071\\ 0.7071\\ 1.4142\\ 0\\ 0 \end{pmatrix}$
Therefore the {q,ŝ,h} representation of the screw axis can easily be calculated as:
$\begin{matrix} \hat{s} = \mathcal{S}_{\omega} = \begin{pmatrix} 0\\ 0.7071\\ 0.7071\\ \end{pmatrix}\\ h = 0\\ q = \begin{pmatrix} 0\\ 1\\ -1\\ \end{pmatrix} \end{matrix}$
And therefore, the screw axis can be visualized as:
Since the screw pitch is zero, the motion is pure rotation with no translational motion about the screw axis. Therefore, a screw motion about the above screw axis by the rate of θ̇ calculated above can produce the given twist (𝘚θ̇).
### Example: Exponential Coordinates of Motion to Homogenous Transformation Matrix
In this example, we want to go backward and find the homogenous transformation matrix corresponding to the given exponential coordinates of the motion.
Suppose that the exponential coordinates of the motion are given by the following matrix:
$\mathcal{S} \theta = \begin{pmatrix} 0\\ 1\\ 2\\ 3\\ 0\\ 0 \end{pmatrix}$
In order to find the homogenous transformation matrix representing the same configuration, we should find the matrix exponential corresponding to the exponential coordinates. Since the upper matrix part is not zero, we have rotational motion and thus the rotational part of the screw axis should be normalized. Thus we can write:
$\mathcal{S} \theta = \sqrt 5 \begin{pmatrix} 0\\ \frac{1}{\sqrt 5}\\ \frac{2}{\sqrt 5}\\ \frac{3}{\sqrt 5}\\ 0\\ 0 \end{pmatrix} \rightarrow \theta = 2.23 rad = 128^{o}, \mathcal{S} = \begin{pmatrix} 0\\ \frac{1}{\sqrt 5}\\ \frac{2}{\sqrt 5}\\ \frac{3}{\sqrt 5}\\ 0\\ 0 \end{pmatrix}$
From the screw axis that we calculated, it will be easy to calculate the matrix exponential of the motion through the following process:
$\mathcal{S}_{\omega} = \begin{pmatrix} 0\\ 0.447\\ 0.8944 \end{pmatrix} \rightarrow [\mathcal{S}_{\omega}] = \begin{pmatrix} 0 & -0.8944 & 0.447\\ 0.8944 & 0 & 0\\ -0.447 & 0 & 0 \end{pmatrix}$
And using the Rodrigues’ formula that we learned in the lesson about the exponential coordinates of orientation, we can find the rotational part of the transformation matrix as:
$\begin{split} e^{[\mathcal{S}_{\omega}]\theta } & = I + sin\theta [\mathcal{S}_{\omega}] + (1-cos\theta) [\mathcal{S}_{\omega}]^2 \\ & = \begin{pmatrix} -0.6121 & -0.7070 & 0.3533\\ 0.7070 & -0.2899 & 0.6447\\ -0.3533 & 0.6447 & 0.6778 \end{pmatrix} \end{split}$
And the linear part can be calculated as
$\begin{split} G(\theta){\mathcal{S}_{v}} & = (I\theta + (1-cos\theta)[\mathcal{S}_{\omega}]+(\theta – sin\theta)[\mathcal{S}_{\omega}]^2)\mathcal{S}_{v}\\ & = \begin{pmatrix} 0.7908 & -1.4422 & 0.7208\\ 1.4422 & 1.0785 & 0.5755\\ -0.7208 & 0.5755 & 1.9424 \end{pmatrix}\begin{pmatrix} 1.345\\ 0\\ 0 \end{pmatrix}\\ & = \begin{pmatrix} 1.0637\\ 1.9398\\ -0.9695 \end{pmatrix} \end{split}$
Therefore the homogenous transformation matrix representing the same configuration can be calculated as:
$T_{sb} = e^{[\mathcal{S}]\theta} \underbrace{T_{sb}(0)}_{=I} = \begin{pmatrix} -0.6121 & -0.7070 & 0.3533 & 1.0637\\ 0.7070 & -0.2899 & 0.6447 & 1.9398\\ -0.3533 & 0.6447 & 0.6778 & -0.9695\\ 0 & 0 & 0 & 1 \end{pmatrix}$
This homogenous transformation matrix represents the same configuration as the configuration of the frame after going through a screw motion about the defined screw axis. The initial configuration is the identity matrix since the {b} frame is initially coincident with the frame {s}.
Now let’s see how we can find the body frame’s final configuration after traveling a distance θ along the screw axis 𝘚 if the screw axis is defined in the space or the body frame.
## Body Frame’s Final Configuration After Travelling along the Screw Axis Defined in the Space or the Body Frame
Suppose that the space frame {s} and the body frame {b} are configured in space as the following figure:
The configuration of the body frame relative to the space frame can be found using the matrix Tsb. We would like to know the body frame’s final configuration, Tsb’ if it travels a distance θ along the screw axis 𝘚. We would have two cases since 𝘚 can be represented in either {b} or {s} frame.
• If the screw axis 𝘚 is expressed in the {b} frame then the final configuration of the body frame can be calculated using the equation Tsb’ = Tsb e[𝘚b. In this case, the transformation matrix representation of the {b} frame relative to the {s} frame, Tsb, is post-multiplied by the matrix exponential.
• If the screw axis 𝘚 is expressed in the {s} frame then the final configuration of the body frame can be calculated using the equation Tsb’ = e[𝘚s Tsb. In this case, the transformation matrix representation of the {b} frame relative to the {s} frame, Tsb, is pre-multiplied by the matrix exponential.
That’s going to wrap up today’s lesson. We hope that it gave you a good understanding of screws in robotics. In the next lesson, we will talk about forces in robotics. Stay Tuned! See you in the next lesson!
The video version of the current lesson can be watched at the link below:
Thanks for reading this post. You can also find the other posts on the Fundamentals of Robotics Course in the link below:
https://www.mecharithm.com/category/learning-robotics-mechatronics/fundamentals-of-robotics-course/
References (some links are affiliate):
📘 Textbooks:
Modern Robotics: Mechanics, Planning, and Control by Frank Park and Kevin Lynch
https://amzn.to/3tvQs4G
A Mathematical Introduction to Robotic Manipulation by Murray, Lee, and Sastry
https://amzn.to/36fcyQr
A good book about Screw Theory in Robotics:
https://amzn.to/3Ncq5Zo
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• Samuele
I am student in robotics. This lesson provides the best explanation of screw theory that I could find online so far! It shed some light on this difficult topic.
• Thanks Samuele,
Great to hear that. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8675172924995422, "perplexity": 613.755674746353}, "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-2022-27/segments/1656104585887.84/warc/CC-MAIN-20220705144321-20220705174321-00419.warc.gz"} |
http://andrewkay.name/maths/fmp/problem.php?type=linalg/MatrixArithmetic&level=3&seed=831078 | Formulaic Maths Problems - get another problem: level 1, level 2, level 3 - link to this problem
Find $\begin{pmatrix} 19 \\ -13 \\ 0 \end{pmatrix} + 9\begin{pmatrix} -14 \\ 16 \\ -30 \end{pmatrix}-6\begin{pmatrix} 0 \\ 14 \\ -6 \end{pmatrix}-6\begin{pmatrix} -16 \\ -7 \\ -6 \end{pmatrix}$.
$\begin{pmatrix} \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar0c0m0}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar1c0m0}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar2c0m0}}\hspace{35px}}~} \end{pmatrix} + \begin{pmatrix} \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar0c0m1}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar1c0m1}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar2c0m1}}\hspace{35px}}~} \end{pmatrix} + \begin{pmatrix} \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar0c0m2}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar1c0m2}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar2c0m2}}\hspace{35px}}~} \end{pmatrix} + \begin{pmatrix} \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar0c0m3}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar1c0m3}}\hspace{35px}}~} \\ \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Ar2c0m3}}\hspace{35px}}~} \end{pmatrix}$ $\begin{pmatrix} 19 \\ -13 \\ 0 \end{pmatrix} + \begin{pmatrix} -126 \\ 144 \\ -270 \end{pmatrix} + \begin{pmatrix} 0 \\ -84 \\ 36 \end{pmatrix} + \begin{pmatrix} 96 \\ 42 \\ 36 \end{pmatrix}$
$= \begin{pmatrix} \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Br0c0}}\hspace{35px}}~} \\ \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Br1c0}}\hspace{35px}}~} \\ \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{Br2c0}}\hspace{35px}}~} \end{pmatrix}$ $= \begin{pmatrix} -11 \\ 89 \\ -198 \end{pmatrix}$ | {"extraction_info": {"found_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": 0.8294718265533447, "perplexity": 2687.6982448090475}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1516084891539.71/warc/CC-MAIN-20180122193259-20180122213259-00220.warc.gz"} |
http://mathhelpforum.com/pre-calculus/122306-equation-perpendicular-bisector.html | # Math Help - Equation of perpendicular bisector
1. ## Equation of perpendicular bisector
Next Question .... Sorry :P
The line with equation x+3y=12 meets the x and y axes at the points A and B respectively. Find the equation of the perpendicular bisector of AB
2. This requires a couple of steps. The words look all googly and weird but its pretty basic:
You have the equation for a line. You are told to find the bisector (the midpoint) of the line formed by the points A and B, which are your X and Y intercepts. Therefore:
$x+3y=12 \Longrightarrow x-12=3y \Longrightarrow \frac{12-x}{3}=y \Longrightarrow 4-\frac{1}{3}x=y$
To find the X-int, we set Y=0 and solve:
$0=4-\frac{1}{3}x \Longrightarrow \frac{1}{3}x=4 \Longrightarrow x=12$
Alternatively, we could have used out original equation.
To find the Y=int, we set X=0 and solve. This one is straight forward. Y=4.
Now we have the co-ordinates of point A and B: (12,0) and (0,4) respectively. Our question asks us to find the equation of the perp-bisector. Now for a line to be perpendicular to another line, it must have the a slope equal to the negative reciprocal of the line it is to be perpendicular to. The slope of our original line is simple to see: $-\frac{1}{3}$. Therefore the negative reciprocal is 3. Thus the slope of our perpendicular line is 3.
But we aren't done. We have a very specific point we need to use in order to make sure this line bisects the X and Y intercepts of our original equation. Thus, we need to use the midpoint formula $\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}$, to find the midpoint between (12,0) and (0,4):
$\frac{12+0}{2},\frac{0+4}{2} \Longrightarrow (6,2)$
Now, we have a point (6,2), a slope of 3 and a form called the point-slope form of an equation:
$y-2=3(x-12) \Longrightarrow y=3x-36+2 \Longrightarrow y=3x-34$
3. Originally Posted by Math-DumbassX
Next Question .... Sorry :P
The line with equation x+3y=12 meets the x and y axes at the points A and B respectively. Find the equation of the perpendicular bisector of AB | {"extraction_info": {"found_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": 0.854242742061615, "perplexity": 310.6030308831163}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1394011155638/warc/CC-MAIN-20140305091915-00099-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/108330/when-is-the-graph-of-a-function-a-dense-set?sort=votes | # When is the graph of a function a dense set ?
Let f: R -> R be any function. When is the graph of f dense in R^2 ? The only examples I know for this are for non-measurable functions, but is that a necessary condition ?
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of course not, e.g. the graph of a function which is non-zero on the rationals may have a dense graph. – Pietro Majer Sep 28 '12 at 11:24
The Conway base 13 function is probably a standard example: the graph is dense because any restriction of the function to an open interval is surjective. But meanwhile, since the graph of the base 13 function is easily defined by an arithmetic property of the digits, the function is Borel and hence measurable. More information is available at Is Conway's base-13 function measurable?
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For each $q\in\mathbb Q$ choose a sequence $x_{q,n}\to q$ $(n\to\infty)$ such that all $x_{q,n}$ are pairwise distinct. For $\mathbb Q=\lbrace r_n:n\in\mathbb N\rbrace$ define $f(x_{q,n})=r_n$ and $f(x)=0$ for $x\notin \lbrace x_{q,n}: q\in\mathbb Q, n\in\mathbb N\rbrace.$ Then $f$ is measurable and its graph is dense.
However this class of function suggests a solution to your problem, for example $f(a+ \sqrt{2} b ) = a - \sqrt{2} b$ whenever $a,b \in \mathbb{Q}$ and $f \equiv 0$ out of $\mathbb{Q} ( \sqrt{2} )$. In this way you have a measurable function (because it is $0$ out of a countable set) and it has the property of the dense graph. | {"extraction_info": {"found_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": 0.8766199946403503, "perplexity": 165.15687393278924}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1397609535095.9/warc/CC-MAIN-20140416005215-00456-ip-10-147-4-33.ec2.internal.warc.gz"} |
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