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https://infoscience.epfl.ch/record/130139 | Infoscience
Book Chapter
# Heat transfer in Soils
Temperature highly affects pavement performance. High and low temperatures not only affects the viscosity of asphalt concrete but also has an impact on the moisture flow within pavements. At temperatures below 0°C the freezing of pavements dramatically changes the permeability and frost action might occur forcing water to flow upwards to the freezing front resulting in frost heave and other pavement distress. | {"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": 0.8234054446220398, "perplexity": 4430.941231171933}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1490218188824.36/warc/CC-MAIN-20170322212948-00297-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://www.thespectrumofriemannium.com/tag/d-brane/ | ## LOG#171. From Bohrlogy to dualities.
Old (old fashioned!) Quantum Mechanics is understood as the quantum theory before its final formulation around 1927-1931…It includes the Bohr model, the Wilson-Bohr-Sommerfeld quantization and some other tricks like the one to take into account the finite nuclear size. For … Continue reading
## LOG#077. Entropic electrogravity.
Tower Wang, in his paper Coulomb Force as an Entropic Force, deduced Coulomb and Newton laws using the Verlinde approach in D=3+1 dimensions. He begins with the Reissner-Nordstrom metric in spacetime: with the function and … Continue reading | {"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": 0.8328244090080261, "perplexity": 4579.6704305060875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656104628307.87/warc/CC-MAIN-20220705205356-20220705235356-00529.warc.gz"} |
http://mathhelpforum.com/discrete-math/173258-simple-proof-induction-print.html | # Simple Proof by Induction
• Mar 2nd 2011, 10:21 PM
demode
Simple Proof by Induction
Using mathematical induction I want to show that $3^{6n}-2^{6n}$ is divisible by $35$, $\forall n \in \mathbb{N}$.
The base case n=1 is true: $3^6-2^6=665$, 665/35=19.
Inductive step: Suppose $3^{6k}-2^{6k}$ is divisible by 35 for some $k \in \mathbb{N}$. That means $3^{6k}-2^{6k} = 35m$ for some m. Then
$3^{6(k+1)}-2^{6(k+1)}= 3^{6k}.3^6-2^{6k}.2^6$
Now how can I simplify this to factor out $3^{6k}-2^{6k}$ to show that it's divisible by 35 for k+1?
• Mar 3rd 2011, 12:18 AM
emakarov
Quote:
Originally Posted by demode
$3^{6(k+1)}-2^{6(k+1)}= 3^{6k}.3^6-2^{6k}.2^6$
Rewrite $2^6$ as $3^6-665$.
• Mar 3rd 2011, 12:54 AM
FernandoRevilla
Without using induction:
$3^{6(k+1)}-2^{6(k+1)}=(3^6)^{k+1}-(2^6)^{k+1}=$
$(3^6-2^6)[(3^6)^k+(3^6)^{k-1}2^6+\ldots+(2^6)^k]=35h\;(h\in \mathbb{N})$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 14, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9928914308547974, "perplexity": 960.3644365202879}, "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-2017-09/segments/1487501172447.23/warc/CC-MAIN-20170219104612-00523-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?p=106274 | ## Definition of Empirical Formula
Tarika Gujral 1K
Posts: 62
Joined: Fri Sep 28, 2018 12:27 am
### Definition of Empirical Formula
Does the empirical formula represent the relative number of atoms of a compound or the relative mass ratio of the atoms in a compound?
Ramsha Dis1B
Posts: 32
Joined: Fri Sep 28, 2018 12:19 am
### Re: Definition of Empirical Formula
The empirical formula represents the relative number of atoms.
Rian Montagh 2K
Posts: 60
Joined: Fri Sep 28, 2018 12:15 am
### Re: Definition of Empirical Formula
So for glucose, the empirical formula would be CH2O, with a 1:2:1 ratio of C:H:O atoms.
Abhi4F
Posts: 30
Joined: Fri Sep 28, 2018 12:28 am
### Re: Definition of Empirical Formula
The empirical formula represents the most basic ratio of atoms in the formula.
Tarika Gujral 1K
Posts: 62
Joined: Fri Sep 28, 2018 12:27 am
### Re: Definition of Empirical Formula
Theoretically, doesn’t it also refer to mass percent composition of the molecule?
Chloe Qiao 4C
Posts: 65
Joined: Fri Sep 28, 2018 12:27 am
### Re: Definition of Empirical Formula
I believe empirical formula does not directly refer to the mass percent composition of a molecule because mass percent composition is based on the mass of each type of atom over the total mass(and different atoms have different masses) while the empirical formula is more about the relative ratio of each atom presented.
Julia Jones 1G
Posts: 29
Joined: Fri Sep 28, 2018 12:16 am
### Re: Definition of Empirical Formula
The empirical formula represents the ratios of the atoms in their simplest form
AlexandraZuniga1L
Posts: 33
Joined: Thu Feb 15, 2018 3:02 am
### Re: Definition of Empirical Formula
The empirical formula shows the relative number of atoms that a molecule has whereas the molecular formula shows the actual number of atoms.
shaunajava2e
Posts: 66
Joined: Fri Sep 28, 2018 12:26 am
### Re: Definition of Empirical Formula
the empirical formula simplifies the ratio of the atoms in a formula, the molecular formula shows the actual number. as a result, the empirical formula of multiple molecules could be the same while the molecular formulas cannot | {"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": 0.8787932395935059, "perplexity": 4270.160318415032}, "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/1603107882102.31/warc/CC-MAIN-20201024051926-20201024081926-00062.warc.gz"} |
https://www.lessonplanet.com/teachers/multiplying-decimals-by-powers-of-ten-g | ## Multiplying Decimals by Powers of Ten (G)
In this decimal multiplication worksheet, 4th graders solve the decimal multiplication problems that range by powers of ten. Students solve 45 problems.
Subjects
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Resource Types
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I was an intern last year. I didn't always have the time to look at the files of other teachers at 10pm at night. LessonPlanet saved me so many times I've lost count. I would look all over other websites for quality lesson plans and, no surprise, I always ended up here.
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Provo, Utah | {"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": 0.8701490163803101, "perplexity": 4725.900783423063}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988718296.19/warc/CC-MAIN-20161020183838-00333-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://psa.gov.ph/tags/census-population-and-housing?page=73 | # Baguio City Posted A Population Growth Rate Of 2.5 Percent (Results from the 2007 Census of Population)
The City of Baguio registered a total population of 301,926 persons as of August 1, 2007. This registered an increase of 49,540 persons over the total population of 252,386 persons in 2000 (with May 1, 2000 as reference date), giving the city an annual population growth rate of 2.50 percent.
# Benguet: Dependency Ratio Down By 13 Persons In 2007 (Results from the 2007 Census of Population)
The total population of Benguet, as of August 1, 2007, was 372,533 persons. This figure registered an increase of 42,404 persons over the total population of 330,129 persons in 2000. The annual population growth rate was recorded at 1.68 percent, higher by 0.6 percentage point from the 1.09 percent annual population growth rate in 2000.
# Apayao: Population exceeded 100 thousand mark (Results from the 2007 Census of Population)
Apayao posted a total population of 103,633 persons as of August 1, 2007, or an increase of 6,504 persons over the total population of 97,129 persons in 2000. This increase translated to an annual population growth rate (PGR) of 0.90 percent for the period 2000 to 2007, lower than the annual PGR of 3.25 percent recorded during the period 1995 to 2000.
# National Capital Region: Close To 10 Million Persons (Results from the 2000 Census of Population and Housing, NSO)
The National Capital Region, which covers the 12 cities and five municipalities, recorded a total population of 9,932,560 persons in the 2000 Census of Population and Housing (Census 2000). This was up by 478,520 persons as compared to 9,454,040 persons recorded in the 1995 Census of Population (POPCEN). For the period 1995 to 2000, the annual population growth rate for the National Capital Region was 1.06 percent, lower than that of the 1990 to 1995 period (3.30 percent). If the current annual population growth rate of the metropolitan area continues, the population is expected to double in 65 years.
# Philippine Population Went Up By 12 Million Persons (Results from the 2007 Census of Population)
Total population grew by 2.04 percent annually
As of August 1, 2007, the Philippines had a total population of 88,548,366 persons, an increase of 12,041,438 persons over the May 1, 2000 population count of 76,506,928 persons. The 2007 census figure is almost twelve times the Philippine population in 1903 (7,635,426 persons), when the first census was conducted. | {"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": 0.8350731730461121, "perplexity": 2212.072902255048}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243989856.11/warc/CC-MAIN-20210511184216-20210511214216-00411.warc.gz"} |
https://forum.allaboutcircuits.com/threads/averaging.50993/ | # Averaging
#### Lightfire
Joined Oct 5, 2010
690
Hi,
There's a lot of differences between public and private schools' averaging here in the Philippines.
Private schools often gives high grades.
So, I am asking you now the correct averaging.
CATEGORY---------PERCENTAGE---ITEMS-----SCORE--PUPIL PERCENTAGE
PROJECT----------------10%-------10%------95%/100%--------9.5
RECITATION--------------15%-----15%-------90%/100%--------13.5
HOMEWORK/SEATWORK---10%-----200--------185/200----------9.25
MONTHLY EXAM----------20%------50---------50/50------------20
PERIODICAL EXAM--------30%------50---------50/50------------30
QUIZ---------------------15%------25--------20/25-------------12
INSTRUCTIONS:
1. I divided my score to a total score. Like 20÷25.
2. I multiply the answer by 100. Like 0.8x100
3. I multiply the answer by 0.15. Like 80x0.15
The answer was 12.
Note: If you noticed, I add "0." before the category's percentage. Like "0.15"
My average will be 94.25. Right?
If there is something wrong please let me know.
P.S. The averaging system I made is the averaging system of most public schools here in the Philippines.
Thank you and answers will be greatly appreciated.
God bless.
#### Lightfire
Joined Oct 5, 2010
690
Yes.
but if you noticed, my quiz there is only 12 which is to be 13.5.
so 13.5 would be correct? my formula/method is 20 divided by 25. I multiplied the answer to 50. Then add more 50. Then times it to 0.15.
thanks.
#### jpanhalt
Joined Jan 18, 2008
8,332
The process of doing a weighted average is the same, regardless of the values assigned for the individual weights. You just need to update your values. You have added a couple of new columns, and your addition seems correct. Your current average is 94.25%, which is what you calculated. So, you seem to have the right idea.
John | {"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": 0.8528025150299072, "perplexity": 2028.7363694003836}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250613416.54/warc/CC-MAIN-20200123191130-20200123220130-00314.warc.gz"} |
http://math.stackexchange.com/users/29253/user29253 | # user29253
less info
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bio website location age member for 2 years, 3 months seen Dec 24 '13 at 8:38 profile views 6
# 7 Questions
6 Bounded sets in Frechet spaces 5 Polynomials representing primes 4 Gauss sums ray class group 3 Number of ideals of a given norm 2 A book suggestion on algebraic number theory
# 111 Reputation
+5 Bounded sets in Frechet spaces +5 Approximation by Differentiable functions +15 Number of ideals of a given norm +5 Gauss sums ray class group
This user has not answered any questions
# 7 Tags
0 algebraic-number-theory × 4 0 polynomials 0 reference-request 0 number-theory 0 approximation-theory 0 prime-numbers 0 functional-analysis
# 2 Accounts
Mathematics 111 rep 2 MathOverflow 6 rep 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": 0.8542764782905579, "perplexity": 4921.376391170761}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510272584.13/warc/CC-MAIN-20140728011752-00222-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://ch.mathworks.com/help/stats/evcdf.html | # evcdf
Extreme value cumulative distribution function
## Syntax
```p = evcdf(x,mu,sigma) [p,plo,pup] = evcdf(x,mu,sigma,pcov,alpha) [p,plo,pup] = evcdf(___,'upper') ```
## Description
`p = evcdf(x,mu,sigma)` returns the cumulative distribution function (cdf) for the type 1 extreme value distribution, with location parameter `mu` and scale parameter `sigma`, at each of the values in `x`. `x`, `mu`, and `sigma` can be vectors, matrices, or multidimensional arrays that all have the same size. A scalar input is expanded to a constant array of the same size as the other inputs. The default values for `mu` and `sigma` are `0` and `1`, respectively.
`[p,plo,pup] = evcdf(x,mu,sigma,pcov,alpha)` returns confidence bounds for `p` when the input parameters `mu` and `sigma` are estimates. `pcov` is a 2-by-2 covariance matrix of the estimated parameters. `alpha` has a default value of `0.05`, and specifies `100(1 - alpha)`% confidence bounds. `plo` and `pup` are arrays of the same size as `p`, containing the lower and upper confidence bounds.
`[p,plo,pup] = evcdf(___,'upper')` returns the complement of the type 1 extreme value distribution cdf at each value in `x`, using an algorithm that more accurately computes the extreme upper tail probabilities. You can use the `'upper'` argument with any of the previous syntaxes.
The function `evcdf` computes confidence bounds for `P` using a normal approximation to the distribution of the estimate
`$\frac{X-\stackrel{^}{\mu }}{\stackrel{^}{\sigma }}$`
and then transforming those bounds to the scale of the output `P`. The computed bounds give approximately the desired confidence level when you estimate `mu`, `sigma`, and `pcov` from large samples, but in smaller samples other methods of computing the confidence bounds might be more accurate.
The type 1 extreme value distribution is also known as the Gumbel distribution. The version used here is suitable for modeling minima; the mirror image of this distribution can be used to model maxima by negating `X` and subtracting the resulting distribution values from `1`. See Extreme Value Distribution for more details. If x has a Weibull distribution, then X = log(x) has the type 1 extreme value distribution. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.9673036932945251, "perplexity": 364.90889701515465}, "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-2020-29/segments/1593655900335.76/warc/CC-MAIN-20200709131554-20200709161554-00520.warc.gz"} |
http://www.ibcci.net/forum/7wzx4.php?page=094699-bond-angle-of-cyclohexane | ## bond angle of cyclohexane
It is used as a solvent in some brands of correction fluid. The following equations and formulas illustrate how the presence of two or more substituents on a cyclohexane ring perturbs the interconversion of the two chair conformers in ways that can be predicted. Due to lone pair (N) – bond pair (H) repulsion in ammonia, the bond angles decrease to 106.6°. > A planar cyclohexane would look like a regular hexagon. Among these is glucose, Hence Therefore, there are two types of H-atoms in cyclohexane – axial (Ha) and Consequently, there is some redistribution of s- and p-orbital character between the C–C and C–H bonds.
Because of the alternating nature of equatorial and axial bonds, the opposite relationship is true for 1,3-disubstitution (cis is all equatorial, trans is equatorial/axial). by rotating the Chime structure.
Finally, 1,4-disubstitution reverts to the 1,2-pattern: The above analysis is not something that you should try to memorize: rather, become comfortable with drawing cyclohexane in the chair conformation, with bonds pointing in the correct directions for axial and equatorial substituents.
The presence of bulky atoms, lone pair repulsion, lone pair-bond pair repulsion, and bond pair repulsion can affect the geometry of a molecule. a carbon atom, an axial hydrogen bonded to it, and the midpoint of a vicinal C-C bond ?
R(11-12) &1.528 \\ This deviation in bond angle from the ideal bond angle 109.5° would bring some kind of ring strain into the structure. Contents. What are some common mistakes students make with boat and chair conformations?
This conformation allows for the most stable structure of cyclohexane. The results showed that almost similar energies were released during their combustion. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. One chair conformer ring flips to the other chair conformer, and in between, three other conformers are formed.
are delocalized around the ring. Why do cyclic compounds most commonly found in nature contain six-membered rings? \begin{array}{ll}
The cyclohexanone–cyclohexanol mixture, called "KA oil", is a raw material for adipic acid and caprolactam, precursors to nylon. How do I conduct myself when dealing with a coworker who provided me with bad data and yet keeps pushing responsibility for bad results onto me?
#"Sum of interior angles" = (n-2) × 180°#, where #n# is the number of interior angles. \angle(\ce{CCH_{ax}}) & =109.1^\circ,\\ This ensures the absence of any ring strain in the molecule. On careful examination of a chair conformation of cyclohexane, we find that the twelve hydrogens are not structurally equivalent. R(10-13) &1.100 \\ In the chair form of cyclohexane, the carbon atoms and the bonds around them are almost perfectly tetrahedral.
It is frequently used as a recrystallization solvent, as many organic compounds exhibit good solubility in hot cyclohexane and poor solubility at low temperatures.
Understanding the hybridization of cyclohexane? A(2-5) &105.8 \\ It is always possible to have both groups equatorial, but whether this requires a cis-relationship or a trans-relationship depends on the relative location of the substituents. 4 Spectral Information Expand this section.
A(3-11-12) &111.3 \\ density both above and below the ring. During the chair flip, there are three other intermediate conformations that are encountered: the half-chair, which is the most unstable conformation, the more stable boat conformation, and the twist-boat, which is more stable than the boat but still much less stable than the chair. But it is less stable than the chair A planar cyclohexane would look like a regular hexagon.
A(7-2-10) &100.5 \\
2 Names and Identifiers Expand this section.
), Figure 4: Axial and equatorial bonds in cyclohexane. Also, the C-atom above the plane of the four C-atoms goes below the plane and vice-versa. The crystal structure shows a strangely distorted molecule with only $C_\mathrm{i}$ symmetry. they are as far apart as possible.
Dates: Modify .
A(2-1-4) &106.7 \\ A(1-3-11) &112.4 \\ Suggestions for braking with severe osteoarthritis in both hands.
As shown in the ball and stick structure, the left most carbon Notice that a 'ring flip' causes equatorial hydrogens to become axial, and vice-versa. (ii) Substituents on chair conformers prefer to occupy equatorial positions due to the increased steric hindrance of axial locations.
the left is the most stable conformation.
Cyclohexane has two crystalline phases. of the ring. As we count around the ring from carbon #1 to #6, the uppermost bond on each carbon changes its orientation from equatorial (or axial) to axial (or equatorial) and back. Cyclohexane has the lowest angle and torsional strain of all the cycloalkanes; as a result cyclohexane has been deemed a 0 in total ring strain. Cyclohexane is non-polar. ∴ "Each interior angle" = (n-2)/n × 180 °= (6-2)/6 × 180 ° = 4/6 × 180 ° = 120 °. What defines a JRPG, and how is it different from an RPG? in the U. S., with over 90% being used in the synthesis of nylon. In the twist-boat conformation, due to the movement of C-3 and C-6, the eclipsing of the C – H bonds is reduced to some extent. In the chair conformation, if any carbon-carbon bond were examined, it would be found to exist with its substituents in the staggered conformation and all bonds would be found to possess an angle of 109.5°. Now we must examine the way in which favorable ring conformations influence the properties of the configurational isomers.
A(4-1-5) &116.7 \\
Can I include my published short story as a chapter to my new book? The "C-C-C" bond angles in a planar cyclohexane would be 120 °. \mathbf{d}(\ce{CH_{eq}}) &=1.103~\mathrm{\mathring{A}}.\\ The complex TpBr3Cu(NCMe) catalyzes, at room temperature, the insertion of a nitrene group from PhINTs into the carbon−hydrogen bond of cyclohexane and benzene, as well as into the primary C−H bonds of the methyl groups of toluene and mesitylene, in moderate to high yield.
At each angle change is a Another conformation of cyclohexane exists, known as boat conformation, but it interconverts to the slightly more stable chair formation. Note the tip up on Cyclohexane vapour is used in vacuum carburizing furnaces, in heat treating equipment manufacture. Therefore, it should be clear that for cis-1,2-disubstitution, one of the substituents must be equatorial and the other axial; in the trans-isomer both may be equatorial.
Due to lone pair (O) - lone pair (O) repulsion the bond angle in water further decreases to 104.5°. is tipped up from the ring.
button of Chime to see this effect. the blood sugar. Hence
Cyclohexane in the chair conformation. shows the alternating double bonds. They are practically inseparable because they interconvert very rapidly at room temperature. Because the expected normal C-C-C bond angle should be near the tetrahedral value of 109.5°, the suggested planar configuration of cyclohexane would have angle strain at each of the carbons, and would correspond to less stable cyclohexane molecules than those with …
Cyclohexane is a colourless, flammable liquid with a distinctive detergent-like odor, reminiscent of cleaning products (in which it is sometimes used). Conformations of monosubstituted cyclohexanes, Conformations of Disubstituted Cyclohexanes, Axial and Equitorial positions in cyclohexanes.
Cyclohexane is sometimes used as a non-polar organic solvent, although n-hexane is more widely used for this purpose. A(11-12-18) &105.8 \\
C-4 is the foot and the four carbon atoms form the seat of the chair). Why does F replace the axial bond in PCl5? A(1-2-10) &111.3 \\
The conformation in which the methyl group is equatorial is more stable, and thus the equilibrium lies in this direction. R(1-4) &0.884 \\ 1 Structures Expand this section.
and the right are tipped up, while the other four carbons form In the chair form of cyclohexane, the carbon atoms and the bonds around them are almost perfectly tetrahedral. \begin{array}{lr} For this reason, early investigators synthesized their cyclohexane samples.[10].
Conformation of cyclohexane I: Chair and Boat, GOOD Extensive information about cyclohexane conformations, Drawing Chairs in 3D Axial and Equitorial positions in cyclohexanes, Conformations of Substituted Cyclohexanes, Conformation of Cyclohexane II: Monosubstituted, Cis- and trans-substituted of cyclohexane, Conformations of Cyclohexanes III: Disubstituted, Conformations of Cyclohexanes IV: Trisubstituted. Thanks for contributing an answer to Chemistry Stack Exchange!
A(3-11-16) &100.5 \\
The flat ring is based seen The molecular formula of cyclohexane is C6H12. The hybridization of all the C-atoms is sp3.
A(13-10-14) &106.9 \\ The chair and twist-boat are energy minima and are therefore conformers, while the half-chair and the boat are transition states and represent energy maxima.
The idea that the chair conformation is the most stable structure for cyclohexane was first proposed as early as 1890 by Hermann Sachse, but only gained widespread acceptance much later. Have questions or comments? Although the customary line drawings of simple cycloalkanes are geometrical polygons, the actual shape of these compounds in most cases is very different. carbon atom, and each carbon has the correct number of hydrogens, A(12-11-16) &118.2 \\ Consequently, there is some redistribution of s- and p-orbital character between the C–C and C–H bonds. The simplest ring compound From the Newman projection, it is clear that all C – H bonds in chair conformer are staggered. These conformation inter convert very rapidly Register now!
A(10-12-11) &110.4 \\ Click on the spacefill A(17-12-18) &116.7 \\ | {"extraction_info": {"found_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": 0.8353599905967712, "perplexity": 4612.244646522453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046153857.70/warc/CC-MAIN-20210729105515-20210729135515-00514.warc.gz"} |
http://math.stackexchange.com/questions/94780/particular-solution-of-recurrence-equations | # Particular solution of recurrence equations
How do we solve recurrence equations of the form:
$$ax_{n+1}+bx_n+cx_{n-1}=dn^p+e\;,$$ where $a,b,c,d,e$ are constants and $p\in \mathbb Z$?
Perhaps we could first solve the homogeneous equation $$ax_{n+1}+bx_n+cx_{n-1}=0\;.$$ Then we find the particular solution... but how? Guesswork?
Thanks.
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You can find the particular solution using generating functions. This is closely analogous to using the Laplace transform to find a particular solution to an ODE. – Qiaochu Yuan Dec 30 '11 at 3:54
This question has been solved perfectly. Hope that the asker has been diving enough and accept the answer at an early date. – doraemonpaul Sep 10 '12 at 1:38
Guesswork, yes - but, highly educated guesswork.
If $p$ is a positive integer then there will be a solution in the form of a polynomial of degree $p$, so you write down $x_n=a_0x^p+a_1x^{p-1}+\cdots+a_{p-1}x+a_p$ and substitute it in and work out what $a_0,\dots,a_p$ have to be.
There is an exception: if the homogeneous equation has a polynomial solution then you have to multiply the guess by $n$ or $n^2$ so it has nothing in common with the solution of the homogeneous equation.
If $p$ is a negative integer, that's a lot harder.
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In fact inhomogeneous linear recurrence equations also have their own versions of 'variation of parameters'.
For second-order inhomogeneous linear recurrence equations, if the inhomogeneous part is $f_n$ and the two linearly independent solutions are $x_n^{(1)}$ and $x_n^{(2)}$ respectively, then the particular solution can be found as $x_n^{(2)}\sum_{k=0}^{n-1}\dfrac{f_kx_{k+1}^{(1)}}{W_{k+1}}-x_n^{(1)}\sum_{k=0}^{n-1}\dfrac{f_kx_{k+1}^{(2)}}{W_{k+1}}$ , where $W_k=\begin{array}{|cc|}x_k^{(1)} & x_k^{(2)}\\x_{k+1}^{(1)} & x_{k+1}^{(2)}\end{array}$ .
See http://faculty.pccu.edu.tw/~meng/Math15.pdf#page=5 for details.
If both $\sum_{k=0}^{n-1}\dfrac{f_kx_{k+1}^{(1)}}{W_{k+1}}$ and $\sum_{k=0}^{n-1}\dfrac{f_kx_{k+1}^{(2)}}{W_{k+1}}$ can eliminate their summation signs, then this particular solution can be extended to use in the version of functional equation.
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http://crypto.stanford.edu/pbc/notes/zdd/zdd.html | ## ZDDs defined
ZDDs may seem unremarkable, as they resemble well-known data structures such as crit-bit trees or DFAs. Nonetheless, the conditions they must satisfy have far-reaching implications.
We define a ZDD $Z$ to be any directed acyclic graph such that:
1. A terminal node is either:
• the special node $\top$ (the "TRUE" node), or:
• the special node $\bot$ (the "FALSE" node).
2. Each nonterminal node satisfies the following conditions:
• The node is labelled with a positive integer $v$. This label need not be unique.
• The node has outdegree 2. One of the outgoing edges is named "LO" and the other is named "HI". In diagrams, we draw dotted lines for LO edges and solid lines for HI edges.
• A destination node is either terminal or labelled with an integer strictly larger than $v$. Thus we can omit arrowheads in diagrams because the edge directions can be inferred from the labels.
• The HI edge never points to the $\bot$ node.
3. There is exactly one node with zero indegree, which we call the root node. The above implies the root node is either terminal or labelled by the smallest integer in the dag.
4. If two nodes have the same label, then their LO or HI edges point to different nodes. In other words, there are no redundant nodes.
We call $Z$ an unreduced ZDD, if a HI edge points to $\bot$ or the last condition fails to hold.
In our first example, the universe $U$ was the letters of the alphabet. More generally, we consider a universe of size $n$, number the elements from 1 to $n$, and refer to an element by this number.
The dag rooted at any node in a ZDD is itself a valid ZDD. Thus we expect straightforward recursive descriptions of many ZDD algorithms and properties. Indeed, anyone comfortable with dynamic programming will feel at home with ZDDs.
## Representing a family
Let $F$ be a ZDD. Let $v$ be its root node. Then:
1. If $v = \bot$ then there can be no other nodes and $F$ represents $\emptyset$, the empty family.
2. If $v = \top$ then there can be no other nodes and $F$ represents the family containing just the empty set: $\{ \emptyset \}$. We call this the unit family, and denote it by $\epsilon$.
3. Otherwise $v$ has two children. Let $v_0$ be the LO node, and $v_1$ be the HI node. Let $F_i$ be the family represented by the ZDD rooted at $v_i$ which is known by inductive assumption. Then $F$ represents the family
$F_0 \cup \bigcup_{\alpha\in F_1} \alpha \cup \{v\} .$
In other words, as in real life, we divide the world between the haves and the have-nots: on the LO branch we have the sets in $F$ that don’t contain $v$:
$F_0 = \{ \alpha : \alpha \in F, v \notin \alpha \}$
and on the HI branch we have the sets in $F$ that do contain $v$, but we remove the $v$ before recording them:
$F_1 = \{ \alpha\setminus\{v\} : \alpha \in F, v \in \alpha \}$
Some examples:
The above is the family $\emptyset \cup \{\emptyset \cup \{2\}\} = \{ \{2\}\}$. This is $e_2$, an elementary family. Elementary families are those of the form $\{\{n\}\}$, and denoted by $e_n$.
More family photos:
The family $\{\emptyset\} \cup \{\emptyset \cup \{2\}\} = \{ \emptyset, \{2\}\}$.
The family $\{ \{2\} \} \cup \{ \emptyset\cup \{1\} \} = \{ \{1\}, \{2\}\}$.
The family $\epsilon \cup \{ \{1\} \cup \{2\} \} = \{ \{1, 2\}\}$.
## Features
Two ZDDs are identical if and only if the families they represent are identical (exercise). When we refer to a family as a ZDD, we mean the ZDD representing the family, and vice versa.
We can enumerate all sets in $F$ lexicographically (exercise).
We have $|F| = |F_0| + |F_1|$, thus we can recursively compute the number of sets in a ZDD. This also allows us to pick out, for instance, the 13th set out of a 42-member family (exercise). Random access is fast, and somewhat analogous to lazy evaluation in programming languages such as Haskell: the ZDD can produce desired family members quickly on demand. Thus any operation we can do on an array of sets, we can do with similar efficiency on a ZDD.
We can modify these techniques so that if we weight each element in the universe, after recursively visiting each node, we can find the sets in the family of maximum (or minimum) weight, and statistics such as the mean and standard deviation. We can compute interesting generating functions, such as one from which we can read off the number of sets of a given size. With an array representation of a family, we would have to visit each set. With a ZDD, we only have to visit each node, which is often much smaller.
As with data structures built from trees, we can convert recursive algorithms to iterative bottom-up algorithms. For conciseness we tend towards the former in our exposition, though the latter may be better in practice.
So ZDDs are nifty. If only we knew how to construct ZDDs for a given family. | {"extraction_info": {"found_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": 0.8810161352157593, "perplexity": 579.5934282865649}, "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-2014-41/segments/1410657102753.15/warc/CC-MAIN-20140914011142-00236-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://astarmathsandphysics.com/a-level-physics-notes/special-and-general-relativity/3023-the-principle-of-relativity.html?tmpl=component&print=1&page= | ## The Principle of Relativity
There is no such thing as absolute motion, probably. We can only measure our motion relative to other objects. We are used to thinking of objects having speeds or velocities, but these are usually measured relative to the earth,
The diagram above shows man and spaceship in relative motion, and relative is significant here. In space, far from all points of reference, it is only relative speed that is important. The man and the occupants of the spaceship cannot, by performing any physical experiment, determine whether it they or the other that is moving. One could be moving, or the other, or both, as long as their relative velocity is v. This implies that any law of physics must be expressed in the same form in both frames, with measured quantities referred to each reference frame and any physical constants having the same value.
For example, the Law of Gravity could be expressed asin O and in O' withtaking the same value in both inertial frames (subject to both observers using the same units).
This fact gives us confidence to apply the Laws of Physics to distant parts of the Universe and assume their applicability in those parts. | {"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": 0.9410568475723267, "perplexity": 351.9450794325115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187825575.93/warc/CC-MAIN-20171023035656-20171023055656-00699.warc.gz"} |
http://blogformathematics.blogspot.com/2016/03/quadratic-expressions.html | ## Pages
For example, x^2 + 2x + 3 is an algebraic expression in which the variable x has the highest exponent of 2. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8102937936782837, "perplexity": 222.45445695827195}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1492917124371.40/warc/CC-MAIN-20170423031204-00211-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://realnfo.com/toc/Electrical_Circuit_Analysis/Methods_of_Analysis/Bridge_Networks | # Bridge Networks
The bridge network is a configuration that has a multitude of applications. This type of network is used in both dc and ac meters. Electronics courses introduce these in the discussion of rectifying circuits used in converting a varying signal to one of a steady nature (such as dc).
Fig. 1: Various formats for a bridge network.
The bridge network may appear in one of the three forms as indicated in Fig.1. The network in Fig. 1(c) is also called a symmetrical lattice network if R2 = R3 and R1 = R4. Fig. 1(c) is an excellent example of how a planar network can be made to appear nonplanar. For the purposes of investigation, let us examine the network in Fig. 2 using mesh and nodal analysis.
Fig. 2: Assigning the mesh currents to the network.
Mesh analysis (Fig. 2) yields $$(3 Ω + 4 Ω + 2 Ω)I_1 - (4 Ω)I_2 - (2 Ω)I_3 = 20 V$$ $$(4 Ω + 5 Ω + 2 Ω)I_2 - (4 Ω)I_1 - (5 Ω)I_3 = 0$$ $$(2 Ω + 5 Ω + 1 Ω)I_3 - (2 Ω)I_1 - (5 Ω)I_2 = 0$$ and $$\begin{split} 9I_1 - 4I_2 - 2I_3 &= 20 \\ -4I_1 + 11I_2 - 5I_3 &= 0 \\ -2I_1 - 5I_2 + 8I_3 &= 0 \end{split}$$ with the result that $$\begin{split} I_1 &= 4 A\\ I_2 &= 2.67 A\\ I_3 &= 2.67 A \end{split}$$ The net current through the 5 Ω resistor is $$I_{5Ω} = I_2 - I_3 = 2.67 A - 2.67 A = 0 A$$
Fig. 3: Defining the nodal voltages for the network.
Nodal analysis (Fig.3) yields $$\begin{split} ({1 \over 3 Ω} + {1 \over 4 Ω} +{1 \over 2 Ω})V_1 -({1 \over 4 Ω})V_2 - ({1 \over 2 Ω})V_3 &={20 \over 3} A \\ ({1 \over 4 Ω} + {1 \over 2 Ω} + {1 \over 5 Ω})V_2 - ({ 1 \over 4 Ω})V_1 - ({ 1 \over 5 Ω})V_3 &= 0 \\ ({ 1 \over 5 Ω} + {1 \over 2 Ω} + {1 \over 1 Ω})V_3 - ({ 1 \over 2 Ω})V_1 - ({1 \over 5 Ω})V_2 &= 0 \end{split}$$ and $$\begin{split} ({1 \over 3 Ω} + {1 \over 4 Ω}+{1 \over 2 Ω})V_1 - ({1 \over 4 Ω})V_2 - ({1 \over 2 Ω})V_3 &= 6.67 A \\ - ({ 1 \over 4 Ω})V_1 + ({1 \over 4 Ω} + {1 \over 2 Ω} +{1 \over 5 Ω})V_2 - ({1 \over5 Ω})V_3 &= 0\\ - ({1 \over2 Ω})V_1 - ({1 \over 5 Ω})V_2 + ({1 \over 5 Ω} + {1 \over 2 Ω} +{1 \over 1 Ω})V_3 &= 0 \end{split}$$ Solving for voltages, $$\begin{split} V_1 = 8.02 V \\ V_2 = 2.67 V \\ V_3 = 2.67 V \end{split}$$ and the voltage across the 5 Ω resistor is $$V_{5Ω} = V_2 - V_3 = 2.67 A - 2.67 A = 0 V$$ Since $V5Ω = 0 V$, we can insert a short in place of the bridge arm without affecting the network behavior. (Certainly $V = IR = I (0) = 0 V$.)
(a)
(b)
Fig.4: (a)Substituting the short-circuit equivalent for the balance arm of a balanced bridge. (b) Redrawing the network
In Fig. 4(a), a short circuit has replaced the resistor R5, and the voltage across R4 is to be determined. The network is redrawn in Fig. 4(b), and $$\begin{split} V_{1Ω} &= {(2 Ω || 1 Ω)20 V \over (2 Ω || 1 Ω) + (4 Ω || 2 Ω) + 3 Ω} \\ &={{ 2 \over 3}(20 V) \over {2 \over 3} + {8 \over 6} + 3} \\ &={40 V \over 15} \\ &= 2.67 V \end{split}$$ as obtained earlier.
Fig.5: Substituting the open-circuit equivalent for the balance arm of a balanced bridge.
We found through mesh analysis that $I_{5Ω} = 0 A$, which has as its equivalent an open circuit as shown in Fig. 5(a). (Certainly $I = V>R = 0>(0/\infty) = 0 A.$) The voltage across the resistor R4 is again determined and compared with the result above. The network is redrawn after combining series elements as shown in Fig. 5(b), and $$\begin{split} V_{3Ω} = {(6 Ω || 3 Ω)(20 V) \over 6 Ω || 3 Ω + 3 Ω} \\ = {2 Ω(20 V) \over 2 Ω + 3 Ω} = 8 V \end{split}$$ and $$\begin{split} V_{1Ω} &= {1 Ω(8 V) \over 1 Ω + 2 Ω} \\ &= {8 V \over 3} = 2.67 V \end{split}$$ as above.
Fig. 6: Establishing the balance criteria for a bridge network.
The condition V5Ω = 0 V or I5Ω = 0 A exists only for a particular relationship between the resistors of the network. Let us now derive this relationship using the network in Fig. 6, in which it is indicated that I = 0 A and V = 0 V.
The bridge network is said to be balanced when the condition of I = 0 A or V = 0 V exists.
If V = 0 V (short circuit between a and b), then $$V_1 = V_2$$ 0and $$I_1R_1 = I_2R_2$$ or $$I_1 = {I_2R_2 \over R_1}$$ In addition, when V = 0 V, $$V_3 = V_4$$ and $$I_3R_3 = I_4R_4$$ If we set I = 0 A, then $I_3 = I_1$ and $I_4 = I_2$, with the result that the above equation becomes $$I_1R_3 = I_2R_4$$ Substituting for I1 from above yields $$({I_2R_2 \over R_1}) R_3 = I_2R_4$$ or, rearranging, we have $$\bbox[5px,border:1px solid blue] {\color{blue}{{R_1 \over R_3} = {R_2 \over R_4}}} \tag{1}$$ This conclusion states that if the ratio of R1 to R3 is equal to that of R2 to R4, the bridge is balanced, and I = 0 A or V = 0 V. A method of memorizing this form is indicated in Fig. 7.
Fig.7: A visual approach to remembering the balance condition. | {"extraction_info": {"found_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": 0.8458719253540039, "perplexity": 505.8548599337879}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046152129.33/warc/CC-MAIN-20210726120442-20210726150442-00658.warc.gz"} |
https://indico.nikhef.nl/event/2692/ | Theory
# Theory seminar: Pia Zurita
Europe/Amsterdam
Nikhef
#### Nikhef
Description
Nuclear effects in parton densities A precise knowledge of the parton distribution functions (PDFs) is crucial for any study involving at least one proton in the initial state. When protons are replaced by nuclei, the non trivial modifications of the cross-sections can be attributed to changes in the PDFs and/or the fragmentation functions (FFs) due to the medium. These nuclear PDFs (nPDFs) and FFs (nFFs) are relevant for l+A, p+A and A+A collisions, where they can help disentangle cold nuclear matter effects from genuinely new phenomena. Moreover, they can be used to improve the flavour separation of the quarks in e+p and p+p experiments. Despite decades of dedicated studies, a full description of the nPDFs and nFFs is still missing. In this talk I will review the current status of in medium modification of PDFs and FFs and discuss the latest results and possibilities for improvements in future colliders. | {"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": 0.8406358957290649, "perplexity": 1728.4042877194477}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358469.34/warc/CC-MAIN-20211128043743-20211128073743-00185.warc.gz"} |
http://www.reference.com/browse/wiki/Unit_circle | Definitions
# Unit circle
In mathematics, a unit circle is a circle with a unit radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere.
If (x, y) is a point on the unit circle in the first quadrant, then x and y are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation
$x^2 + y^2 = 1.$
Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not just those in the first quadrant.
One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.
## Trigonometric functions on the unit circle
The trigonometric functions cosine and sine may be defined on the unit circle as follows. If (x, y) is a point of the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle t from the positive x-axis, (where counterclockwise turning is positive), then
$cos\left(t\right) = x ,!$
$sin\left(t\right) = y. ,!$
The equation x2 + y2 = 1 gives the relation
$cos^2\left(t\right) + sin^2\left(t\right) = 1. ,!$
Note that cos2(t)=(cos(t))2. This is the standard shorthand for expressing powers of trigonometric functions.
The unit circle also gives an intuitive way of realizing that sine and cosine are periodic functions, with the identities
$cos t = cos\left(2pi k+t\right) ,!$
$sin t = sin\left(2pi k+t\right) ,!$
for any integer k.
These identities come from the fact that the x- and y-coordinates of a point on the unit circle remain the same after the angle t is increased or decreased by any number of revolutions (1 revolution = 2π radians = 360º).
When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, using the unit circle, these functions have sensible, intuitive meanings for any real-valued angle measure.
In fact, not only sine and cosine, but all of the six standard trigonometric functions — sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant — can be defined geometrically in terms of a unit circle, as shown at right.
## Circle group
Complex numbers can be identified with points in the Euclidean plane, namely the number a + bi is identified with the point (a, b). Under this identification, the unit circle is a group under multiplication, called the circle group. This group has important applications in mathematics and science. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "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": 0.9661234021186829, "perplexity": 194.80929506698257}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657125654.84/warc/CC-MAIN-20140914011205-00196-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://www.researchgate.net/publication/24112458_The_eventual_failure_and_price_indeterminacy_of_inflation_targeting | Article
# The eventual failure and price indeterminacy of inflation targeting
12/2006;
Source: RePEc
ABSTRACT In stark contrast to the previous literature, we find that IT leads to price indeterminacy even when the central bank uses a Taylor-like feedback rule to peg the nominal interest rate. We also find that there is no mechanism with IT to determine the current inflation rate or price level. We conclude that the previous literature has either committed mathematical errors involving infinity or misused the non-explosive criterion for ruling out speculative bubbles. To avoid making errors involving infinity, we analyze inflation targeting (IT) in a typical rational-expectations, pure-exchange, general-equilibrium model where the time horizon is arbitrarily large, but finite.
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##### Article: Inflation Determination With Taylor Rules: A Critical Review
[Hide abstract]
ABSTRACT: The new-Keynesian, Taylor-rule theory of inflation determination relies on explosive dynamics. By raising interest rates in response to inflation, the Fed does not directly stabilize future inflation. Rather, the Fed threatens hyperinflation, unless inflation jumps to one particular value on each date. However, there is nothing in economics to rule out hyperinflationary or deflationary solutions. Therefore, inflation is just as indeterminate under "active" interest rate targets as it is under standard fixed interest rate targets. Inflation determination requires ingredients beyond an interest-rate policy that follows the Taylor principle.
SSRN Electronic Journal 10/2007; DOI:10.2139/ssrn.1012165 | {"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": 0.9698623418807983, "perplexity": 4317.596353862742}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1429246658904.34/warc/CC-MAIN-20150417045738-00051-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://www.eskesthai.com/2005/04/time-variable-gravity-measurements.html | ## Tuesday, April 19, 2005
### Time-Variable Gravity Measurements
Mean Gravity Field
The Mean Gravity Field lets us work from this mass understanding and provides for, flunctuations in that Gravity field hence the understanding of Time Variable Gravity Field
On planet Earth, we tend to think of the gravitational effect as being the same no matter where we are on the planet. We certainly don't see variations anywhere near as dramatic as those between the Earth and the Moon. But the truth is, the Earth's topography is highly variable with mountains, valleys, plains, and deep ocean trenches. As a consequence of this variable topography, the density of Earth's surface varies. These fluctuations in density cause slight variations in the gravity field, which, remarkably, GRACE can detect from space.
I wanted to put this into perspective, since we can extend our vision of the gravity field, and how we would look at mass distribution. Using this method, we calibrate, understanding current topological features of hills and valleys that serve to remind us, of the mass distribtuion that has gone on in the formation of our planet Earth.
Since passing over these locations calculations recognize these density valuation of mass regions. This allows us to understand the current gravity standard placed on that location.
Time-Variable Gravity Measurements from the GRACE Satellite
NASA, in partnership with the German Space Agency DLR, launched the dedicated gravity satellite GRACE (the Gravity Recovery and Climate Experiment) in March, 2002. This five year mission will map out the Earth's gravity field to unprecedented accuracy at monthly intervals. The temporal variations in gravity inferred from these data will allow people to study a wide range of processes, cutting across a variety of Earth science disciplines, that involve redistribution of mass within the Earth and at or near its surface. It will be possible, for example, to produce monthly estimates of changes in continental water storage anywhere in the world, averaged over scales of a few hundred km and greater, to accuracies of better than 1 cm water thickness.
I have referrred to the hill and valley perspectives that have arisen in relation to how we see the landscape(earth's). If this feature was not comprehended in some model application, would it not have served to settle minds who see no valuation in such landscape perspectives, as a basis to a much better understanding of the nature of the universe, and the reality around us?
Where has the extra energy gone? For some scientists this question highlights something interesting about what extra dimensions might have implied? You gauge the gravitational fields and learn to see time variability as a feature not just of mass consideration, but of energy determinations as well. It's only fitting?:)
The image above shows the many processes of the Earth’s hydrologic cycle that contribute to total changes in water storage
So setting up a comprehensive understanding of these differences, the mean gravitational field and Time Variable gravity field we see now some relationship to things finer in its constitution, and the relationship to Climate.
The Landscape?
What value might be assigned to this understanding, that we look at how such emissions and its effect on the information gathered. Would we see the effect of civilizations and the way this has effecedt those particular geographical regions.
Have we thus found a legitimate model, that current debates on the Kyoto protocals might serve to get everyone on base for determinations. Will this effectvely change the dialogue currently going on in our assessments, of the needed reduction of CO2 emissions? | {"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": 0.8744698166847229, "perplexity": 1212.1837210587967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463609061.61/warc/CC-MAIN-20170527210417-20170527230417-00278.warc.gz"} |
https://rd.springer.com/chapter/10.1007/978-0-387-76852-6_15 | # Applications of Generalized Measure Theory
• Zhenyuan Wang
• George J. Klir
Chapter
Part of the IFSR International Series on Systems Science and Engineering book series (IFSR, volume 25)
## General Remarks
It is undeniable that classical measure theory, based on additive measures and signed additive measures, and the associated Lebesgue theory of integration, is not only an important area of mathematics, but it has also played an important role in many application domains. Perhaps its most visible is its crucial role in probability theory, as rigorously formulated by Kolmogorov. Examples of other notable applications of classical measure theory are in the areas of classical geometry as well as fractal geometry, ergodic theory of dynamical systems, harmonic analysis, potential theory, calculus of variations, and mathematical economics (see Note 15.1).
Notwithstanding the many demonstrated applications of classical measure theory, it has increasingly been recognized that a broadening of this area’s applicability is severely limited by the additivity requirement of classical measures. Requiring additivity in measuring a property on sets of some kind is basically the same as...
## Keywords
Dual Pair Information Fusion Uncertainty Theory Possibility Theory Aggregation Tool
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves. | {"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": 0.9475224614143372, "perplexity": 964.514155047192}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1534221215222.74/warc/CC-MAIN-20180819145405-20180819165405-00177.warc.gz"} |
http://tex.stackexchange.com/questions/160778/tikz-picture-two-arrow-one-over-the-other | # TikZ picture: two arrow; one over the other
I have the following code:
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}
\begin{document}
$\begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=1em,minimum width=2em]{ A & & G \\ & S & \\ }; \path[-stealth] (m-1-1) edge node [above] {f} (m-1-3) (m-1-1) edge [bend right] node [left] {p\;} (m-2-2) (m-1-3) edge node [right] {\;q} (m-2-2) (m-2-2) edge node [right] {\epsilon} (m-1-1); \path[-stealth] ([yshift=5pt]m-1-1) edge node [right] {g} ([yshift=5pt]m-1-3); \end{tikzpicture}$
\end{document}
I dare you to compile it. Then you will get a disgusting output. What I want is to get the two arrow on top one over the other (just like corolary 1.3.1.5, page 70, in the notes on http://math.umn.edu/~kwlan/articles/cpt-PEL-type-thesis-revision.pdf).
Can anyone help? I would love if the solution didn't modify too much the way I coded the diagram since it is the way I always do it (with the matrix) but beggers can't be choosers, so.
-
Here is an attempt to fix it by using the .east and .west anchors:
## Notes:
• Not sure why you were using math mode -- I have removed that in the below MWE.
## Code:
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}
\begin{document}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=1em,minimum width=2em]{
A & & G \\
& S & \\
};
\path[-stealth]
(m-1-1.east) edge node [above,yshift=1.0ex] {$f$} (m-1-3.west)
(m-1-1) edge [bend right] node [left] {$p\;$} (m-2-2)
(m-1-3) edge node [right] {$\;q$} (m-2-2)
(m-2-2) edge node [right] {$\epsilon$} (m-1-1);
\path[-stealth]
([yshift=5pt]m-1-1.east) edge node [below,,yshift=-1.0ex] {$g$} ([yshift=5pt]m-1-3.west);
\end{tikzpicture}
\end{document}
-
Amazing, thanks ! – ortholle Feb 16 '14 at 18:28
It's quite easy with tikz-cd:
\documentclass{article}
\usepackage{tikz-cd}
\begin{document}
$\begin{tikzcd}[column sep=1.5em] A \arrow[yshift=.85ex]{rr}{f} \arrow[yshift=-.45ex,swap]{rr}{g} \arrow[yshift=-.4ex,bend right,swap]{dr}{p} && G \arrow{dl}{q} \\ & S \arrow[swap]{ul}{\epsilon} \end{tikzcd}$
\end{document}
-
Here a variant of Peter's code but without matrix and some adjustment with the position of some arrows.
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}
\path node (A) {$A$}
(0:4cm) node (G) {$G$}
(-60:4cm) node (S) {$S$};
\path[-stealth]
(A) edge [bend right] node [left] {$p\;$} (S)
(G) edge node [right] {$\;q$} (S)
(S) edge node [right] {$\epsilon$} (A)
([yshift=-2.5pt]A.east) edge node [above,yshift= 1.0ex] {$f$} ([yshift=-2.5pt]G.west)
([yshift= 2.5pt]G.west) edge node [below,yshift=-1.0ex] {$g$} ([yshift= 2.5pt]A.east);
\end{tikzpicture}
\end{document}
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8796461820602417, "perplexity": 2042.2218129735745}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701160950.71/warc/CC-MAIN-20160205193920-00050-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/does-a-diffraction-grating-with-a-shape-form-fourier-image.796277/ | # Does a diffraction grating with a shape form fourier image
• Start date
• Tags
• #1
9
0
i just wanted to get this cleared that a beam falling on a diffraction grating with a shape gives the fourier images of the grating object which can be reobtained by placing a biconvex lens that would converge the rays and form a focussed fourier image at its focal length and the image of the object at other points.
please correct me if i have misunderstood any phenomenon above
and is there any relation between the grating lines and the fourier image? how can i calculate the transfer function of the lens?
• #2
blue_leaf77
Homework Helper
2,629
785
FT relation between object and its image holds true only in far-field region and paraxial rays. The far-field image will give you the FT of the grating object if the used grating upstream was illuminated with plane wave, otherwise the far-field image will be the FT of the field just after the grating (i.e. product between incoming beam and grating transmission function). For the effect of placing a lens, I suggest this file: http://users.ece.utexas.edu/~becker/FOch5-6.pdf
is there any relation between the grating lines and the fourier image?
To obtain this relation you have to calculate the FT of the grating lines arrangement.
• #3
9
0
FT relation between object and its image holds true only in far-field region and paraxial rays. The far-field image will give you the FT of the grating object if the used grating upstream was illuminated with plane wave, otherwise the far-field image will be the FT of the field just after the grating (i.e. product between incoming beam and grating transmission function). For the effect of placing a lens, I suggest this file: http://users.ece.utexas.edu/~becker/FOch5-6.pdf
To obtain this relation you have to calculate the FT of the grating lines arrangement.
any particular good book/reference where i can find how to mathematically? thanks for the help
• #4
blue_leaf77
Homework Helper
2,629
785
Fundamentals of Photonics by Saleh and Teich
Likes steph17
• #5
Andy Resnick
7,579
2,229
i just wanted to get this cleared that a beam falling on a diffraction grating with a shape gives the fourier images of the grating object which can be reobtained by placing a biconvex lens that would converge the rays and form a focussed fourier image at its focal length and the image of the object at other points.
<snip>
I'm a little confused by your question- do you mean that the diffraction grating was cut into a particular shape, like a circle or paper doll? Alternatively, by 'shape of the grating' do you mean the groove profile?
• #6
9
0
I'm a little confused by your question- do you mean that the diffraction grating was cut into a particular shape, like a circle or paper doll? Alternatively, by 'shape of the grating' do you mean the groove profile?
groove profile, like a cartoon character on the squares, thats all.
• #7
Andy Resnick
7,579
2,229
groove profile, like a cartoon character on the squares, thats all.
Thanks, that helps me understand. For example, you may have a laser pointer attachment that projects a square or cartoon character rather than the 'raw beam', right? That attachment is a 2-D phase grating, but if you want to suppress the undiffracted beam as well as undesired diffraction orders, then the grating is nonperiodic.
In any case, the far-field diffraction pattern is the Fourier Transform of the grating transmission. Shining the diffracted beam onto a positive lens (it doesn't have to be biconvex, but it does have to be a positive lens) simply moves the far-field diffraction pattern from infinity to a user-defined plane that is closer and also converts the angular diffraction pattern into a linear diffraction pattern with a conversion factor that involves the focal length of the lens.
Does this help?
Likes steph17
• #8
9
0
Thanks, that helps me understand. For example, you may have a laser pointer attachment that projects a square or cartoon character rather than the 'raw beam', right? That attachment is a 2-D phase grating, but if you want to suppress the undiffracted beam as well as undesired diffraction orders, then the grating is nonperiodic.
In any case, the far-field diffraction pattern is the Fourier Transform of the grating transmission. Shining the diffracted beam onto a positive lens (it doesn't have to be biconvex, but it does have to be a positive lens) simply moves the far-field diffraction pattern from infinity to a user-defined plane that is closer and also converts the angular diffraction pattern into a linear diffraction pattern with a conversion factor that involves the focal length of the lens.
Does this help?
yes, thanks
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1K | {"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": 0.8088335990905762, "perplexity": 1078.4819650929546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038862159.64/warc/CC-MAIN-20210418224306-20210419014306-00436.warc.gz"} |
https://math.stackexchange.com/questions/1936460/differentiability-of-overlapping-piecewise-functions | # Differentiability of Overlapping Piecewise Functions
Suppose $f(x)$ is differentiable on $U=(-\infty, 0]$ and $g(x)$ is not differentiable at $V=[0,\infty)$ but is differentiable on $(0,\infty)$. Is the following piecewise function $h(x)$ differentiable? $$h(x)= \begin{cases} f(x) \text{ for } x\in U\\ g(x) \text{ for } x\in V\\ \end{cases}$$ Note $U\cap V=\{0\}$ and $f$ is differentiable at $0$ but $g$ is not. Disclaimer, I'm not sure why you would want to consider a piecewise function with "overlapping pieces" but I was asked to.
• What have you tried so far? What do you think the answer is? Does your intuition tell you that $h$ is differentiable or not? How would you go about proving what your intuition tells you? – James Sep 22 '16 at 0:48
Hint: For $t > 0$, $$\frac{h(0+t) - h(0)}{t} = \frac{g(0+t) - g(0)}{t}$$
The condition on $g$ tells you that $g$ is not differentiable at the point $0$.
Then examine derivative from the right $$\lim_{x \rightarrow 0^+}\frac{g(x)-g(0)}{x}$$ Which you already know does not exist. | {"extraction_info": {"found_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": 0.9321722388267517, "perplexity": 112.22304241323452}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998558.51/warc/CC-MAIN-20190617183209-20190617205209-00430.warc.gz"} |
https://srikanthperinkulam.com/2017/05/13/does-dark-matter-harbor-life/ | Even though we know that ordinary matter accounts for only about one-twentieth of the universe’s energy and a sixth of the total energy carried by matter (with dark energy constituting the remaining portion), we nonetheless consider ordinary matter to be the truly important constituent.
Just read: Does Dark Matter Harbor Life? | {"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": 0.9936591982841492, "perplexity": 1439.705356946757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146562.94/warc/CC-MAIN-20200226211749-20200227001749-00427.warc.gz"} |
http://mathhelpforum.com/algebra/69042-still-working-poly-print.html | still working on poly
• Jan 20th 2009, 07:25 AM
Leona_Marie
still working on poly
(3x2-5x +7) +(x2 +2x -1)
=(3x2 +x2) (-5x + 2x) (7 -1)
=(4x2) (-3x) (7)
Why does (7 -1) =7 and not 6
and what word do I use to raise my 2 tosquare or to the power of
• Jan 20th 2009, 07:35 AM
princess_21
Quote:
Originally Posted by Leona_Marie
(3x2-5x +7) +(x2 +2x -1)
=(3x2 +x2) (-5x + 2x) (7 -1)
=(4x2) (-3x) (7)
Why does (7 -1) =7 and not 6
and what word do I use to raise my 2 tosquare or to the power of
that should be 4x^2 -3x +6 and not 7.
you can use this symbol ^
:)
• Jan 20th 2009, 07:38 AM
earboth
Quote:
Originally Posted by Leona_Marie
(3x2-5x +7) +(x2 +2x -1)
=(3x2 +x2) (-5x + 2x) (7 -1)
=(4x2) (-3x) (7)
Why does (7 -1) =7 and not 6
and what word do I use to raise my 2 tosquare or to the power of
You have a sum of two sums. The brackets are not necessary here:
$(3x^2-5x+7)+(x^2+2x-1)=3x^2+x^2-5x+2x+7-1 = 4x^2-3x+6$
• Jan 20th 2009, 08:29 AM
Leona_Marie
Quote:
Originally Posted by princess_21
that should be 4x^2 -3x +6 and not 7.
you can use this symbol ^
:)
I also thought it was 6 but I use this web sight to taech myself,and 7 was the answer I was given. I know how to use 4x^2, but I would rather use
4x2 meaning 4x times 2 is different than 4x squared by 2. I just wanted to know which font I would use to make a small 2 above the x.(Giggle)
• Jan 20th 2009, 08:34 AM
Leona_Marie
Quote:
Originally Posted by princess_21
that should be 4x^2 -3x +6 and not 7.
you can use this symbol ^
:)
I understand what you were saying now, duh sometimes I go cwazy
• Jan 20th 2009, 03:49 PM
princess_21
Quote:
Originally Posted by Leona_Marie
I also thought it was 6 but I use this web sight to taech myself,and 7 was the answer I was given. I know how to use 4x^2, but I would rather use
4x2 meaning 4x times 2 is different than 4x squared by 2. I just wanted to know which font I would use to make a small 2 above the x.(Giggle)
it is better to try factoring rather than relying on websites.
you should try so you can practice.
• Jan 20th 2009, 07:47 PM
Leona_Marie
Quote:
Originally Posted by princess_21
it is better to try factoring rather than relying on websites.
you should try so you can practice.
I'd like to do a little factoing. first I have to learn how. | {"extraction_info": {"found_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": 0.8107008337974548, "perplexity": 3687.3755595799144}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825308.77/warc/CC-MAIN-20171022150946-20171022170946-00406.warc.gz"} |
https://rchg.github.io/computing-blog/LinearAlgebraComputing/ | ## Linear Algebra and Computing
COMPUTING-BLOG
Notebook
Scientific Computing and Linear Algebra
### Introduction
This post just remember several linear algebra properties that are relevant for scientific programming.
### Reducible Matrix
Definition: An square matrix A is reducible if there is a matrix P (basis change) such that, $PAP^{T}=\begin{bmatrix} B_{11} & B_{11}\\ 0 & B_{22} \end{bmatrix}$
This property is very useful because the typical linear system solution $$Ay=c$$ is now more easy to resolve in terms of computing because $\begin{bmatrix} B_{11} & B_{11}\\ 0 & B_{22} \end{bmatrix}\begin{bmatrix} y_{1} \\ y_{2} \end{bmatrix}=\begin{bmatrix} c_{1}\\ c_{2} \end{bmatrix}$
The subsystem $$B_{22}y_{2}=c_{2}$$ is indepentent of the other B matrices with the logical computational improvement.
### Transpose Matrix
The transpose matrix $$\mathbf{B}$$ of a given matrix $$\mathbf{A}$$ is a matrix with elements: $[b_{ij}] = [a_{ji}] ,\quad\quad \forall i =1, \ldots , m ,\quad\quad \forall j = 1, \ldots , n$
The transpose matrix of $$\mathbf{A}$$ is usually noted by $$\mathbf{A}^T$$.
### Ortogonal Matrix
A square matrix $$\mathbf{A}$$ is orthogonal when the matrix product with the transpose matrix has the following property: $\mathbf{A} \mathbf{A^T} = \mathbf{A^T} \mathbf{A} = \mathbf{I}$
### Symmetric Matrix
A square matrix $\mathbf{A}$ is symmetric when $$[a_{ij}] = [a_{ji}]$$. Therefore, a matrix is symmetric if $$\mathbf{A} = \mathbf{A}^T$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9240732192993164, "perplexity": 743.6347718661607}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077843.17/warc/CC-MAIN-20210414155517-20210414185517-00182.warc.gz"} |
http://calculus4life.blogspot.com/2012/02/ | ## Thursday, February 9, 2012
### The Fundamental Theorem of Calculus
This theorem is so important that it's called the Fundamental Theorem of Calculus. Why is it so important? First, it provides a link between differentiation and integration. Second, it gives a way to easily evaluate definite integrals, bypassing the use of Riemann sums.
The theorem comes in two parts. Let's call them FTC1 and FTC2.
FTC1
Let f(x) be a continuous function on [a,b], and let $$a \leq x \leq b$$. Then
$\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$
FTC2
Let f(x) be a continuous function on [a,b], and let F(x) be an antiderivative of f(x). Then
$\int_{a}^{b} f(x) dx = F(b) - F(a)$
Since $$F'(x) = f(x)$$, we can rewrite this as
$\int_{a}^{b} \frac{d}{dx}F(x) dx = F(b) - F(a)$
Discussion
If you look at the last equation and the equation from FTC1, you can see how differentiation and integration are inverse operations in some sense.
FTC2 itself shows that you can easily evaluate a definite integral if you know an antiderivative of the integrand. The alternative is to go through laborious calculations with Riemann sums. Using FTC2, you trade off tedious calculations for memorizing antiderivatives or looking them up--basically a time-memory trade-off. This is what makes calculus so powerful as a tool.
Geometric Interpretation
Later. | {"extraction_info": {"found_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": 0.9936639070510864, "perplexity": 795.0234434790397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125944682.35/warc/CC-MAIN-20180420194306-20180420214306-00572.warc.gz"} |
https://www.math.nyu.edu/dynamic/calendars/seminars/algebraic-geometry-seminar/793/ | # Algebraic Geometry Seminar
#### Around the Bridgeland Differential on M_{G,N}
Speaker: Andrei Caldarau, University of Wisconsin, Madison
Location: Warren Weaver Hall 201
Date: Thursday, April 17, 2014, 3:45 p.m.
Synopsis:
Several years ago Tom Bridgeland suggested that there should exist an interesting chain maps C_*(M_{g,n}) -> C_{*+2}(M_{g,n+1}) and he conjectured some applications of these maps to mirror symmetry. I shall present a precise definition of these maps using techniques from the theory of ribbon graphs, and discuss a conjectural statement about the homology of the total complex associated to the bicomplex obtained from these maps. Then I shall speculate (wildly) about applications to mirror symmetry. | {"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": 0.8744009137153625, "perplexity": 1356.3606541624217}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1544376827992.73/warc/CC-MAIN-20181216191351-20181216213351-00101.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=107865 | Rotational Motion!!!
by PrudensOptimus
Tags: motion, rotational
P: 640 SOMEONE PLS GIVE ADVICE!! TEST TOMORROW, PLS HELP!! Thanks.
P: 640 By advices I mean something like a STEP. Like what to ask first, what to write down first, what to... etc.
P: 57 Have a good diagram showing all relevant forces and their distances from the point you choose to orient the torques. That's half the problem really, the rest is just formulas. Don't forget T=rxF
Related Discussions General Physics 1 Introductory Physics Homework 3 General Physics 2 Introductory Physics Homework 1 Introductory Physics Homework 4 | {"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": 0.8608795404434204, "perplexity": 2804.280386675754}, "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-2013-48/segments/1386163041955/warc/CC-MAIN-20131204131721-00043-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://excel.onushi.com/function/if.htm | S҂̃GNZ(EXCEL)wKE GNZ{삩烏[NV[g̎gAGNZ}NiVBAj̎gȂǏЉBGNZ̊b牞p܂ŊwׂuGNZwKTCgvłB GNZ&[hf
S҂̃GNZ(Excel)wKE|GNZ̎g|
# IF ()
̎ _ IF(_,_^̏ꍇ,_Ȕꍇ)
## IF̎g
IF͘_^̎iw肵TRUE̎jɐ^̏ꍇԂA_U̎ɋȔꍇԂ܂B
ɂďo͂錋ʂς鎖o܂Bij
ZB2ɁA=IF(A2>0,"","Ԏ")@ZB3ɁA=IF(A3>0,"","Ԏ")͂ƈȉ̗lɌʂԂ܂B
A B C D E 1 v v @ @ @ 2 100 @ @ @ 3 -100 Ԏ @ @ @
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_A2>0̓ZA2̒l0ȏƂӖłB^̏ꍇƋȔꍇɂ͍͍ƐԎw肵Ă܂Bw肷ꍇ""(_uNH[e[V)ň͂ޕKv܂BlƂĕԂꍇ͈͂ޕKv͂܂B
ő7܂IFlXg鎖o܂Bȉ̕\
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B3Ɂ@=IF(A3=0,"x",IF(A3=1,"x1",IF(A3=2,"x2","")))
B4Ɂ@=IF(A4=0,"x",IF(A3=1,"x1",IF(A3=2,"x2","")))͂\łB
IF3lXgĂ܂B̂悤IFlXg鎖ɂ蕡̏ݒ肵Aꂼ̌ʂԂ\łB
A B C D E 1 x x @ @ @ 2 0 x @ @ @ 3 1 x1 @ @ @ 4 2 x2 @ @ @
### IFANDgݍ킹Ďgp
IFANDgݍ킹ĎgƂł܂B
ȉ̕\
B2ZɁ@=IF(AND(A2<=9,A2>=1),"19","10ȏ")
B3ZɁ@=IF(AND(A3<=9,A3>=1),"19","10ȏ")
B4ZɁ@=IF(AND(A4<=9,A4>=1),"19","10ȏ")
ȏ̂悤ɓ͂Έȉ̗lɌʂԂ܂B
B2AB3łANDTRUEԂAIF̐^̏ꍇ"19"o͂Ă܂B
B4łANDFALSEԂAIF̋Ȕꍇ"10ȏ"o͂Ă܂B
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A B C D E 1 @ @ @ 2 1 19 @ @ @ 3 7 19 @ @ @ 4 10 10ȏ @ @ @
#### IFORgݍ킹Ďgp
IFORgݍ킹ĎgƂł܂B
ȉ̕\
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B4ZɁ@=IF(OR(A4="Ot1",A3="Ot2"),"OtL","Ot")
ȏ̂悤ɓ͂Έȉ̗lɌʂԂ܂B
B2AB3łORTRUEԂAIF̐^̏ꍇ"OtL"o͂Ă܂B
B4łORFALSEԂAIF̋Ȕꍇ"Ot"o͂Ă܂B
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A B C D E 1 Ot Ot̗L @ @ @ 2 Ot1 OtL @ @ @ 3 Ot2 OtL @ @ @ 4 @ Ot @ @ @
@TRUE&FALSE@OR@AND@IF NOT
Copy right(c) 2006-2013 S҂̃GNZ(Excel)wKE all right reserved | {"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": 0.9603402018547058, "perplexity": 2332.7760446382276}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507445190.43/warc/CC-MAIN-20141017005725-00192-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/other-math/CLONE-547b8018-14a8-4d02-afd6-6bc35a0864ed/chapter-6-percent-6-2-percents-and-fractions-6-2-exercises-page-396/3 | ## Basic College Mathematics (10th Edition)
75%=$75\div100=\frac{75}{100}=\frac{75\div25}{100\div25}=\frac{3}{4}$ Therefore, 75% is equivalent to $\frac{3}{4}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9291637539863586, "perplexity": 4808.493808502169}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1606141735395.99/warc/CC-MAIN-20201204071014-20201204101014-00334.warc.gz"} |
http://mathhelpforum.com/pre-calculus/70791-analytic-geometry-q2.html | # Math Help - Analytic Geometry Q2
1. ## Analytic Geometry Q2
Question:
Show that $A(-2,1)$ , $B(5,-2)$ and $C(3,3)$ are vertices of an equilateral triangle.
Attempt:
Distance between points $A(-2,1)$ and $B(5,-2)$:
$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
$AB = \sqrt{(5-(-2))^2 + (-2-1)^2}$
$AB = \sqrt{49 + 9}$
$AB = \sqrt{58}$
Distance between points $B(5,-2)$ and $C(3,3)$:
$BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
$BC = \sqrt{(3-5)^2+(3-(-2))^2}$
$BC = \sqrt{4 + 25}$
$BC = \sqrt{29}$
Distance between points $A(-2,1)$ and $C(3,3)$:
$AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
$AC = \sqrt{(3-(-2))^2 + (3-1)^2}$
$AC = \sqrt{25 + 4}$
$AC = \sqrt{29}$
I have found out the distance between the points. How can I find out if it is a right angled triangle?
2. This is not an equilateral triangle( all the sides are not equal)
But it's a right triangle because
AB^2= BC^2+ AC^2 (Pythagoras theorem)
Pythatgoras Theorem states that the square of the hypotnuse(side opp. to right angle) of a right triangle is equal to the sum of squares of sides
3. Originally Posted by looi76
How can I find out if it is a right angled triangle?
Well, most easily by using Pythagorean theorem, so it is because $\overline{AC}^2+\overline{BC}^2=\overline{AB}^2\Le ftrightarrow 29+29=58$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 22, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8507210612297058, "perplexity": 510.9964956407589}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1408500826679.55/warc/CC-MAIN-20140820021346-00168-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://hockeyschtick.blogspot.com/2015/02/why-new-paper-does-not-provide-evidence.html | ## Thursday, February 26, 2015
### Why a new paper does not provide evidence of an increased CO2 greenhouse effect
A new paper published in Nature claims to be the "first direct observation of CO2's increasing greenhouse effect at Earth's surface" from 2000 to 2010.
However, there are multiple fundamental problems with the claim including:
1. As stevengoddard.com points out in "Junk Science Award For The Evening"
Posted on
"Over the deade the authors examined (2000 to 2010), the average level of the gas (CO2) in the atmosphere went up by 22 parts-per-million. And the time series shows a steadily rising trend in its impact, layered on top of the seasonal changes. By the end of that period, the gas was retaining an extra 0.2 Watts for every square meter of the Earth’s surface compared to the start.
Still, it seems worth noting that the continued increase in greenhouse energy retention measured during this time coincides with a period where the Earth’s surface temperatures did not change dramatically. All that energy must have been going somewhere. [i.e. to space] "
Newsflash: the greenhouse effect really exists | Ars Technica
The authors started in the 2000 La Nina, and ended at the 2010 El Nino – when troposphere temperatures were half a degree warmer. Then they noticed that there was slightly more downwelling long wave radiation [DWLR], which they blamed on increased absorption from the increase in CO2.
The increase in DLWR was due to the warmer troposphere during the El Nino. Warmer air emits more longwave radiation. The higher concentration of CO2 will also emit more DLWR radiation, but that is not due to increased absorption. I don’t know how scientists can get any more clueless than that.
Indeed, the authors admit in the abstract below that the CO2 contribution to the alleged overall trend of downwelling longwave radiation is a mere 10% of the total. So what caused the other 90%?
2. Secondly, the authors claim CO2 was retaining an extra 0.2 Watts for every square meter of the Earth’s surface compared to the start (over a period of one decade).
However, per the (debunked) IPCC formula, the rise in CO2 from 369.52 ppm in 2000 to 389.85 ppm in 2010 "would" have "trapped"
5.35*ln(389.85/369.52) or 0.29 W/m2
Thus even if one believes the IPCC formula and this new paper's assumptions (including extensive computer modeling in the new paper), the IPCC formula exaggerates CO2 surface radiative forcing by 45% over the observations.
3. Thirdly, the peak emission spectra of CO2 is at 15 microns, which by Wien's displacement law is equivalent to a blackbody radiating at -80C. Per the second law of thermodynamics, a low temperature/frequency/energy body at -80C cannot warm a higher temperature/frequency/energy body at 15C (Earth).
4. Rather, the entire 33K greenhouse effect is entirely explained by the Maxwell/Carnot/Clausius atmospheric mass/gravity/pressure theory and the 'greenhouse equation.' Increased CO2 instead facilitates loss of outgoing IR radiation to space, as has been observed by an increase in OLR (Outgoing Longwave Radiation) over the past 60+ years, opposite to the predictions of the alternative radiative forcing greenhouse theory.
The claim that the warming 2000-2010 is from CO2 confuses cause with effect. Warming of the atmosphere due to internal variability, ocean oscillations, cloud cover changes, solar amplification mechanisms, etc. secondarily warm the CO2 in the atmosphere increasing the 15 micron IR radiation observed from increased levels of CO2.
Update:
From Greenie Watch & Tallbloke's talkshop:
UPDATE: Rog Tallbloke Has even more fun with the above study than I did. He points out that in Alaska over the study period, the average temperature actually FELL by four degrees. So rising CO2 must cause cooling, Right?
Another point I did not mention because I saw no point in beating a dead horse concerns the graph below. It appeared with the original story.
It shows two nicely matching curves, does it not? But what are the quantities being graphed? One is CO2 but the other is NOT temperature. It is a theoretically derived construct called forcing. Not so impressive.
# Observational determination of surface radiative forcing by CO2 from 2000 to 2010
Nature
doi:10.1038/nature14240
Accepted
Published online
1. "One is CO2 but the other is NOT temperature. It is a theoretically derived construct called forcing." That theoretical construct is known as infrared radiation measured in watts per square meter. This is the same theoretical construction that comes out of a heat lamp.
2. That one study is Feldman 2015 (1) under carefully controlled "CLEAR sky" conditions. But then, there is Dong, Xi, Minnis 2006, under "ALL sky" conditions, that found the reverse.
”Similar to the clear-sky study, we also provide the all-sky upwelling SW and LW fluxes to study the surface radiation budget under all-sky conditions. The rates of net SW and LW fluxes are −0.07 W/m^2 [per year] and −0.37 W/m^2 [per year], respectively, resulting in a decrease of 0.44 W/m^2 per year in NET flux at the surface (Figure 3b). The decline of NET flux, however, does not correlate with the increased surface air temperature as illustrated in Figure 3a. The surface air temperature is determined by the sum of NET radiation fluxes (downwelling and upwelling SW and LW fluxes) and nonradiative fluxes (sensible and latent heat fluxes, ground heat flux and energy flux used for melt), as well as the large-scale advection [Wild et al., 2004]. Wild et al. [2004] investigated this counterintuitive result and concluded that it may be due to a decrease of surface evaporation and associated reduced evaporative surface cooling.”
”… using the Stefan-Boltzmann equation indicates that an annual increase of 0.04°C air temperature each year corresponds to an increase of 0.4 W/m^2 per year in upward LW upward surface emission. However, the measured change is a decrease of 0.26 W/m^2 per year as shown in Figure 2e.”
Dong, Xiquan, Baike Xi, and Patrick Minnis 2006. "Observational evidence of changes in water vapor, clouds, and radiation at the ARM SGP site." Geophysical Research Letters
http://onlinelibrary.wiley.com/doi/10.1029/2006GL027132/full
(1)Feldman, Daniel R., et al. 2015 "Observational determination of surface radiative forcing by CO2 from 2000 to 2010.” Nature
http://asl.umbc.edu/pub/chepplew/journals/nature14240_v519_Feldman_CO2.pdf
3. Radiative flux is always measured in units of W/m^2, Radiative forcing is defined as the change in flux so I'm not sure what all the fuss is about. You could convert it to DeltaT if you wanted to apply a sensitivity parameter, but that isn't really the purpose of the paper.
The measurements in this paper are good quality and the calculations that are done are completely standard in Chemical physics and molecular spectroscopy.
Point 3 is quite off course on the thermodynamics, but I'm pretty sure I'm not going to get anywhere with you on that one.
4. I'm curious, is the technology of the Et.el. paper the same as the satellite technology? | {"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": 0.9312361478805542, "perplexity": 2287.0503932622883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656104683683.99/warc/CC-MAIN-20220707033101-20220707063101-00167.warc.gz"} |
https://www.meritnation.com/ask-answer/question/how-to-calculate-the-valency-of-any-compound/atoms-and-molecules/3471996 | # how to calculate the valency of any compound?
As said by Vrinda, it is not the compounds which possess valencies. Rather the combining atoms or ions possess different valencies, and these valencies give rise to different chemical formulae for different compounds. For ions, the valency is simply equal to the charged possessed by the molecule. Let us consider the case of sodium chloride (NaCl). It consists of sodium ion (Na+ ion) and chloride (Cl-) ion. Both of these ions carry a charge of 1, and hence the valencies of both these compounds is 1. Now let us consider CaCl2. It consists of Ca+2 cation, which carries a charge of +2, and two Cl- ions, each of which carry a charge of -1. Thus the valency of Ca is 2, while that of Cl is 1.
For atoms, valency is simply the number of electrons gained, lost or shared by an atom to complete its octet that is achieve a noble gas configuration.
If the number of valence electrons (electrons in the outermost shell) is more than 4 then the valency is (8 - number of valence electrons).
If the number of electrons is less than 4 or 4 then it is the valency is the same of valence electrons.
• -9
Compounds don't have valencies... As the valence ions themselves combine to form those compounds.
Na1- + Cl1+ >>> NaCl
• -5
What are you looking for? | {"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": 0.9193081855773926, "perplexity": 1308.317282269933}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1606141743438.76/warc/CC-MAIN-20201204193220-20201204223220-00473.warc.gz"} |
http://qualympics.wikidot.com/2002-spring-statistical-mechanics | 2002 Spring Statistical Mechanics
# Two level systems and beyond.
(The convention used in this page is that $\tau:=kT$, so that $T$ is the temperature expressed in Kelvin and $\tau$ is the temperature expressed in units of energy.)
## A. [20 points] Consider a two-level system (TLS) with energy states 0 and $\epsilon$.
### i. [10 points] In the canonical ensemble, derive an expression for the heat capacity of the TLS.
Greg's Solution:
To obtain the specific heat, we need to have the energy expressed as a function of temperature. We will obtain this by first writing down the partition function, which immediately gives us the free energy, and then using the fact that $E=F+\tau S$ and $S=-\frac{dF}{d\tau}$ to get the energy from the free energy.
The partition function for the two-level system is given by,
(1)
\begin{align} \Omega_0(\mu_0+\mu_B H)-\Omega_0(\mu_0-\mu_B H)\right] Z = \underbrace{1}_{\text{ground state}}+\underbrace{e^{-\epsilon/\tau}}_{\text{excited state}}. \end{align}
Thus, the free energy is given by,
(2)
\begin{align} F = -\tau\log Z = -\tau\log\left[1+e^{-\epsilon/\tau}\right] \end{align}
We want to find the total energy, given by $E=F+\tau S$, so first we find the entropy,
(3)
\begin{aligned} S &= -\frac{dF}{d\tau} \newline &= \log Z + \frac{-\frac{\epsilon}{\tau} e^{-\epsilon/\tau}}{1+e^{-\epsilon/\tau}}, \newline \end{aligned}
and so
(4)
\begin{align} E = F+\tau S = -\frac{\epsilon e^{-\epsilon/\tau}}{1+e^{-\epsilon/\tau}} \end{align}
(Note that as $\tau\to 0$, $E\to 0$, and as $\tau\to\infty$, $E\to\frac{\epsilon}{2}$.)
Finally, we obtain the heat capacity from the energy,
(5)
\begin{align} C_V = \frac{dE}{dT} = k\frac{dE}{d\tau}|_{\tau=kT} = k\frac{\epsilon^2}{k^2T^2}\frac{e^{-\epsilon/kT}}{\left(1+e^{-\epsilon/kT}\right)^2} \end{align}
### ii. [10 points] In the 1970's, it was discovered that the low-temperature heat capacity of insulating glasses is linear in T. Approximate the internal degrees of freedom for a glass as a superposition of independent TLS's with a broad distribution of energy differences $n(\epsilon)$. Show tha the heat capacity of a glass is $C_V(T)\sim k^2 Tn(0)A$, where A is a constant of order unity.
Greg's Solution:
Based on Eq. (5), we see that the specific heat for a TLS as a function of the excited energy $\epsilon$ is given by,
(6)
\begin{align} \frac{dE_\epsilon}{d\tau} = \frac{\epsilon^2}{\tau^2}\frac{e^{-\epsilon/\tau}}{\left(1+e^{-\epsilon/\tau}\right)^2} \end{align}
Thus, for the glass we have that
(7)
\begin{align} \frac{dE}{d\tau} = \sum_\epsilon n(\epsilon) \frac{dE_\epsilon}{d\tau} \end{align}
(8)
\begin{align} \approx \int_0^\infty \,d\epsilon\, n(\epsilon) \frac{dE_\epsilon}{d\tau} \end{align}
(9)
\begin{align} \approx \int_0^\infty \,d\epsilon\, n(\epsilon) \frac{\epsilon^2}{\tau^2}\frac{e^{-\epsilon/\tau}}{\left(1+e^{-\epsilon/\tau}\right)^2} \end{align}
This integral will be easiest to deal with when we make it dimensionless, so we set $x:=\epsilon/\tau$ so that
(10)
\begin{align} \frac{dE}{d\tau} = \tau \int_0^\infty dx\, n(\tau x) \frac{e^{-x}}{\left(1+e^{-x}\right)^2} \end{align}
The bulk of the contribution of the integral is around $x=1$. Furthermore, as $\tau\to 0$, the peak of the integral gets shrunk until it is approximately a delta function times a constant of order unity. Thus, we have that
(11)
\begin{align} \frac{dE}{d\tau} \to \tau n(0) A \end{align}
And thus,
(12)
\begin{align} C_V = k\frac{dE}{d\tau}|_{\tau=kT} \to k^2T n(0) A. \end{align}
## B. A myoglobin molecule in solution can either have exactly one adsorbed O2 molecule or else zero adsorbed O2 molecules. Let $\epsilon$ denote the energy of an adsorbed molecule of O2 relative to an O2 in solution at infinite distance from the myoglobin.
### i. It will be helpful to approximate the O2 molecules in solution as an ideal gas (exclude rotational and vibrational degrees of freedom). Prove that the chemical potential of an ideal has is $\mu=kT\log\left(n/n_Q\right)$ where $n_Q:=\left(MkT/2\pi\hbar^2\right)^{3/2}$.
Greg's Solution:
The game plan is that if we can find the free energy, then we have the chemical potential since $\mu=\frac{\partial F}{\partial N}$. Thus, we seek the partition function from which we will obtain the free energy.
The partition function for a single particle of gas is given by,
(13)
\begin{align} Z_1 = \sum_\epsilon e^{-\epsilon/\tau}. \end{align}
In our problem we have that $\epsilon = p^2/2M$. Plugging this in, and converting our sum to an integral via the standard formula,
(14)
\begin{align} \sum \to \int \frac{dp\,dq}{(2\pi\hbar)^3}, \end{align}
we have that
(15)
\begin{align} Z_1 =\int \frac{dp\,dq}{(2\pi\hbar)^3} e^{-p^2/2M\tau} \end{align}
Since the integrand is independant of $q$, we integrate it out and obtain the volume,
(16)
\begin{align} \begin{aligned} Z_1 = V\int \frac{dp}{(2\pi\hbar)^3} e^{-p^2/2M\tau} \end{align}
(17)
\begin{align} = V\frac{(2\pi M\tau)^{3/2}}{(2\pi\hbar)^3} e^{-p^2/2M\tau} \end{align}
(18)
\begin{align} &= V\left(\frac{M\tau}{2\pi\hbar^2}\right)^{3/2} \end{align}
The N-particle partition function is given by,
(19)
\begin{align} Z_N = \frac{Z_1^N}{N!}, \end{align}
where the factor of $1/N!$ is needed to avoid overcounting states (as the particles are indistinguishable). Thus, the free energy is given by
(20)
\begin{align} F = -\tau\log Z_N \end{align}
(21)
\begin{align} = -\tau\log \frac{1}{N!}\left(V\left(\frac{M\tau}{2\pi\hbar^2}\right)^{3/2}\right)^N \end{align}
(22)
\begin{align} = -N\tau\log \frac{1}{N!}V\left(\frac{M\tau}{2\pi\hbar^2}\right)^{3/2} \end{align}
(23)
\begin{align} = -N\tau\log \frac{V}{N}\left(\frac{M\tau}{2\pi\hbar^2}\right)^{3/2} \end{align}
(24)
\begin{align} = N\tau\log \frac{n}{\underbrace{\left(\frac{M\tau}{2\pi\hbar^2}\right)^{3/2}}_{n_Q}} \end{align}
(25)
\begin{align} = N\tau\log \frac{n}{n_Q} \end{align}
And so we see that indeed, $\mu=\frac{\partial F}{\partial N}=\tau\log \frac{n}{n_Q}$, where $n_Q:=\left(MkT/2\pi\hbar^2\right)^{3/2}$. Q.E.D.
### ii. [8 points] Prove that the fraction of myoglobin carrying an O_2 molecule is given by $f=\frac{n}{n_Q \exp{\epsilon/k T}+n}$, where n is the concentration of O_2 molecules in the surrounding solution.
Greg's Solution:
The key to solving this problem is to work in the grand canonical ensemble. That is, we consider that at each myoglobin site, there are two possible states: zero particles, or one particle. Thus, the Gibbs distribution partition function is given by,
(26)
\begin{align} \xi = 1+e^{(\mu-\epsilon)/\tau}, \end{align}
and so the thermodynamic potential (a.k.a. Gibbs free energy) is given by
(27)
\begin{align} \Omega = -\tau\log\left[1+e^{(\mu-\epsilon)/\tau}\right]. \end{align}
Now we can obtain the expected number of particules from
(28)
\begin{align} N = -\pdrv{\Omega}{\mu} = \frac{e^{(\mu-\epsilon)/\tau}}{1+e^{(\mu-\epsilon)/\tau}} = \frac{1}{e^{(-\mu+\epsilon)/\tau}+1} \end{align}
(Note that we have just derived the distribution function for a Fermi gas.)
Plugging in our expression for the chemical potential derived earlier, we obtain
(29)
\begin{align} N = \frac{1}{\frac{n_Q}{n}e^{\epsilon/\tau}+1} \equiv \frac{n}{n_Q e^{\epsilon/kT} +n} \end{align}
as required. Q.E.D.
## C. Consider a gas of free electrons at T=0. An electron in a magnetic field has an energy of $\pm \mu_B H$, according to whether the spin is parallel or antiparallel to the field $\bar H$.
### i. [15 points] Show that the spin-paramagnetic susceptiblity is $\frac{3}{2}n\frac{\mu_B^2}{\mu_0}$, where $\mu_0$ is the chemical potential at T=0.
Greg's Solution:
Recall that if you know that,
(30)
\begin{align} \delta E = X\delta\lambda, \end{align}
then
(31)
\begin{align} \bar X = \frac{dE}{d\lambda}. \end{align}
In this case, we know that
(32)
\begin{align} \delta E = M\delta H, \end{align}
since for a small perturbation of the external field, the magnetization will remain essentially constant and so the energy will increase by an amount proportional to the change in the field. As a result, we can apply [[eqref findavalue]] to obtain the magnetization,
(33)
\begin{align} \bar M = \frac{dE}{dH}, \end{align}
and from that the magnetic susceptibility,
(34)
\begin{align} \chi = \frac{\bar M}{V}. \end{align}
So it remains to find the energy as a function of the external field. This need not be the total energy — any of the other energies shall do; for this problem it is most convenient to work with the Gibbs free energy, $\Omega_0$, in the grand canonical ensemble.
We assume that in the absense of the field the chemical potential is given by $\mu_0$ and the Gibbs free energy by a known function $\Omega_0(\mu_0)$. The effect of the field is to split this system into two smaller systems with new chemical potentials $\mu_0\pm \mu_B H$, so that the Gibbs free energy is given by
(35)
\begin{align} \Omega = \frac{1}{2}\left[\Omega_0(\mu_0+\mu_B H)-\Omega_0(\mu_0-\mu_B H)\right] \end{align}
The factor of 1/2 is required as in each system the electron is only allowed to have one spin, thus halving the number of possible states.
Now that we have an energy for the system, we can find the magnetization by taking the derivative with respect to H,
(36)
\begin{align} \bar M = \frac{d\Omega}{dH} = \frac{\mu_B}{2}\left[\frac{d\Omega_0}{d\mu}(\mu_0+\mu_B H)-\frac{d\Omega_0}{d\mu}(\mu_0-\mu_B H)\right] \end{align}
If we assume that the perturbing magnetic field is sufficiently small, then we can approximate the above by
(37)
\begin{align} \bar M \approx \mu_B^2 H\frac{d^2\Omega_0}{d\mu^2} = \mu_B^2 H\frac{dN}{d\mu} \end{align}
Since T=0, the system is completely degenerate and so
(38)
\begin{align} N \propto \mu^{3/2}. \end{align}
We know this because for a one dimensional square well we have that $\epsilon_F\propto k_F^2 \propto N^2\Rightarrow N\propto \epsilon_F^{1/2}$ — that is, the number of states that can "fit" into a system is proportional to the square root of the Fermi energy. In three dimensions, the number of states that can fit is cubed, and so $N\propto \epsilon_F^{3/2}$, and when T=0 we have that $\mu=\epsilon_F$, resulting in Eq. [[eqref easyNfromMU]] above.
Given Eq. 38, we see that
(39)
\begin{align} \frac{\partial N}{\partial \mu} = \frac{3}{2}\frac{N}{\mu}, \end{align}
and so
(40)
\begin{align} \bar M = \frac{3N\mu_B^2}{2\mu_0} \Rightarrow \chi = \frac{\bar M}{V} = \frac{3n\mu_B^2}{2\mu_0}, \end{align}
Q.E.D.
# 2. [40 points] Photons and radiation pressure
## A. [20 points] Consider a 3-dimensional photon gas with energy spectrum $E=\hbar cq$, where $q=|\vec{q}|$ and $\vec{q}$ is the wave vector. Solve the questions below using quantum statistical mechanics.
### i. Discuss why the chemical potential of the photon gas is zero.
Greg's Solution:
The chemical potential is zero because it costs nothing to add another photon to the system, as the same amount of total energy can be divided among an arbitrary number of photons.
### ii. Show that the radiation pressure of the photon gas is $p=\frac{4\sigma}{3c}T^4$, where $\sigma:=\frac{\pi^2k^4}{60\hbar^3c^3}$.
Greg's Solution:
The game plan is to find the free energy, since once we have that we can find the pressure via $P=-\left(\frac{\partial F}{\partial V}\right)_\tau$. (Note that finding the total energy will not help us, since $P\ne -\left(\frac{\partial E}{\partial V}\right)_\tau$.) To find the free energy, we compute the partition function for the system.
For a given mode $\vec{q}$, the partition function is given by
(41)
\begin{align} Z_{\vec{q}} = \sum_{n=0}^\infty e^{n(\overbrace{\mu}^{=0}-\epsilon_q)/\tau} = \frac{1}{1-e^{-\epsilon_q/\tau}}, \end{align}
where observe that we have set the chemical potential $\mu$ to zero for the reasons given in the previous part.
Since each mode is independent, the partition function for the whole system is a product of the partition functions for the individual modes,
(42)
\begin{align} Z = \prod_{\vec{q}} Z_{\vec{q}}, \end{align}
and so the free energy (note that $F=\Omega$ — i.e. the Helmholtz and Gibbs free energies are equal — since $\mu=0$) is given by
(43)
\begin{align} F = -\tau\log Z = -\tau \log\left[\prod_{\vec{q}} Z_{\vec{q}}\right] \equiv +\tau \sum_{\vec{q}} \log\left[1-e^{-\epsilon_q/\tau}\right] \end{align}
Now, we have that $q=nq_0$ where $n=|\vec{n}|$ and the components of $\vec{n}$ are integers, and $q_0=\frac{2\pi}{L}$ is the smallest possible wave number. Thus,
(44)
\begin{align} F = 2\tau \sum_{\vec{n}} \log\left[1-e^{-\hbar c q_0 n/\tau}\right] \end{align}
(The extra factor of 2 comes from the fact that for every mode there are two independent photon polarizations.)
We convert this to an integral over \vec{n}]; since the integrand only depends on [[|\vec{n}, we write the integral in spherical coordinate form, (45) \begin{align} F = 2\tau \int_0^\infty\,4\pi n^2 \,dn\,\log\left[1-e^{-\hbar c q_0 n/\tau}\right] \end{align} Our next goal is to make this integral dimensionless by setting x=-\hbar c q_0n/\tau so that (46) \begin{align} F = \frac{8\pi \tau^4}{(\hbar c q_0)^3}\int_0^\infty \,dx\,x^2\log\left[1-e^{-x}\right] \end{align} Integrate by parts to see that, (47) \begin{align} F = -\frac{8\pi \tau^4}{3(\hbar c q_0)^3}\int_0^\infty \frac{x^3\,dx}{e^{x}-1}\right], \end{align} and it turns out that \int_0^\infty \frac{x^3\,dx}{e^{x}-1}\right]=\frac{\pi^4}{15} so therefore (48) \begin{align} F = -\frac{8\pi^5 \tau^4}{45(\hbar c q_0)^3}, \end{align} (Note that even if you did not know the value of that integral, you could reverse-engineer it since you are given the answer as part of the question. :-) ) Now, the largest possible wavelength is just the length of the side of a "box" containing our system, and so the smallest possible wave number is given by q_0=\frac{2\pi}{L}. Thus, (49) \begin{align} F = -\frac{8\pi^5 \tau^4 L^3}{45\hbar^3 c^3 8\pi^3} \equiv \frac{\pi^2 \tau^4 V}{45\hbar^3 c^3} \equiv \frac{4\sigma VT^4}{3c} \end{align} where \sigma=\frac{\pi^2k^4}{60 \hbar^3c^2} is the Stefan-Boltzmann constant. Now that we have the free energy expressed in this form, finding the pressure is a piece of cake! (50) \begin{align} p = -\frac{\partial F}{\partial V} = \frac{4\sigma}{3c}T^4, \end{align} Q.E.D. ### iii. Show that the energy density of the gas, u, can be expressed asu=3p. Note that, (51) \begin{align} E = F + TS = F-T\frac{\partial F}{\partial T} = F-4F = -3F \end{align} Thus, sinceu=E/V$and$p=-F/V$, from Eq. 51 we have that$u=3p$, Q.E.D. ## B. [20 points] You are given an evacuated container of volume V whose walls are perfectly reflective for EM radiation. ### i. [6 points] By considering the energy density of states of the 3-dimensional photon gas in the container, show that the energy U of the gas is a linear function of V. Greg's Solution: (This result was essentially already derived in the previous part, however there is an alternative method of obtaining the same conclusion.) For a photon gas, the occupancy of a mode with energy$\epsilon_kis (52) \begin{align} n_k = \frac{2}{e^{\epsilon_k/\tau}-1} \end{align} In particular, we have that\epsilon_k = \hbar \om_0 n$, where$\om_0=\frac{2\pi c}{L}$and$n=|\vec{n}|is the magnitude of a 3-vector with integer components. Thus, the expected value of the energy is given by (53) \begin{align} E = \sum \epsilon_k n_k = \sum_{\vec{n}} \frac{2\hbar \om_0 n}{e^{\hbar \om_0 n/\tau}-1} \end{align} We convert this sum to a radial integral in spherical coordinates, (54) \begin{align} E = \int_0^\infty \,4\pi n^2\,dn\,\frac{2\hbar \om_0 n}{e^{\hbar \om_0 n/\tau}-1} \end{align} We setx=\hbar\om_0 n/\tauin order to make the integral dimensionless, (55) \begin{align} E = \frac{4\pi \tau^4}{\hbar^3 \om_0^3}\int_0^\infty \frac{ x^3\,dx}{e^{x}-1} \end{align} SinceE\propto \om_0^{-3} \propto L^3 \propto V$, we conclude that the energy of the gas is a linear function of the volume. ### ii. [7 points] Starting from the second law of thermodynamics and using Maxwell's relations, derive the general relation$\left(\frac{\partial U}{\partial V}\right_)_T=\left(\frac{\partial P}{\partial T}\right_)_V-P. Greg's Solution: First, observe that (56) \begin{align} \frac{\partial F}{\partial V} = -P \end{align} and (57) \begin{align} \frac{\partial F}{\partial T} = -S \end{align} Thus, by taking second derivatives, we obtain the Maxwell relation, (58) \begin{align} \left(\frac{\partial P}{\partial T}\right)_V = -\frac{\partial^2 F}{\partial T\partial V}= -\frac{\partial^2 F}{\partial T\partial V}=\left(\frac{\partial S}{\partial V}\right)_T. \end{align} Now, the second law of thermodynamics gives us the thermodynamic relation, (59) \begin{align} dU = T\,dS - P\,dV, \end{align} from which we obtain (60) \begin{align} \left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial S}{\partial V}\right)_T-P = T\left(\frac{\partial P}{\partial T}\right)_V-P, \end{align} Q.E.D. ### iii.* [7 points] Using the results of parts B.i. and B.ii. above, prove that the radiation pressure is given byp=aT^4$where a is an undetermined constant. The result$u=3p$, which was derived statistically mechanically, may be used here. Greg's Solution In part B.i. we showed that the energy was linear with respect to the volume, so we expect that$u=\frac{\partial U}{\partial V}\equiv \frac{U}{V}$should be a constant with respect to volume, and therefore so should$p=u/3$. Furthermore, upon examination of the right-hand side of B.ii., we see that P should be proportional to some power of T — that is,$P\propto aT^m$. Plugging this into B.ii., we obtain (61) $$u = amT^m - aT^m = (m-1)aT^m = 3aT^m,$$ where the last equality came from$u=3p$. By inspection, we see that$m=4$and so$p\propto T^4\$, Q.E.D.
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https://www.encyclopediaofmath.org/index.php?title=Group&diff=39401&oldid=13033 | Difference between revisions of "Group"
2010 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]
One of the main types of algebraic systems (cf. Algebraic system). The theory of groups studies in the most general form properties of algebraic operations which are often encountered in mathematics and their applications; examples of such operations are multiplication of numbers, addition of vectors, successive performance (composition) of transformations, etc. The concept of a group is historically one of the first examples of abstract algebraic systems and served, in many respects, as a model for the restructuring of other mathematical disciplines at the turn into the 20th century, as a result of which the concept of a mathematical system (a structure) has become a fundamental concept in mathematics.
Definition.
A group is a non-empty set $G$ with one binary operation that satisfies the following axioms (the operation being written as multiplication):
1) the operation is associative, i.e. $(ab)c = a(bc)$ for any $a$, $b$ and $c$ in $G$;
2) the operation admits a unit, i.e. $G$ has an element $e$, known as the unit element, such that $ae=ea=a$ for any $a$ in $G$;
3) the operation admits inverse elements, i.e. for any $a$ in $G$ there exists an element $x$ in $G$, said to be inverse to $a$, such that $ax=xa=e$.
The system of axioms 1)–3) is sometimes replaced by an equivalent system of two axioms: 1); and 4) the operation admits left and right quotients, i.e. for any two elements $a$, $b$ in $G$ there exist elements $x$, $y$ in $G$, the left quotient and the right quotient of division of $b$ by $a$, such that $ax=b$, and $ya=b$.
It follows from this definition that the unit element in any group is unique, that the element inverse to any given element in the group is unique and that for any elements $a$, $b$ of $G$ both fractions obtained by dividing $a$ by $b$ are unique.
Historical remarks.
The origins of the idea of a group are encountered in a number of disciplines, the principal one being the theory of solving algebraic equations by radicals. Permutations were first employed to satisfy the needs of this theory by J.L. Lagrange (1771) in his Memoir on the algebraic solution of equations, and in a paper by A. Vandermonde (1771). It is the former paper which is of special importance in group theory, since it gives, in terms of polynomials, what is really a decomposition of a symmetric permutation group into (right) cosets with respect to subgroups. The deep connections between the properties of permutation groups and those of equations were pointed out by N.H. Abel (1824) and by E. Galois (1830). Galois must be credited with concrete advances in group theory: the discovery of the role played by normal subgroups (cf. Normal subgroup) in problems of solvability of equations by radicals, the discovery that the alternating groups (cf. Alternating group) of order $n \ge 5$ are simple, etc. C. Jordan's treatise (1870) on permutation groups played an important role in the systematization and development of this branch of algebra.
The idea of a group arose in geometry, in an independent manner, when the only then existing antique geometry had been replaced in the middle of the 19th century by numerous other "geometries" , and finding relations between them had become an urgent problem. This question was solved by studies in projective geometry, which dealt with the behaviour of geometric figures under various transformations. The stress in these studies gradually shifted to the study of the transformations themselves and their classification. Such a "study of geometric mappings" was extensively conducted by A. Möbius, who investigated congruence, similarity, affinity, collineation, and, finally, "elementary types of mappings" of geometric figures, that is, actually, their topological equivalence. A.L. Cayley (1854 and later) and other representatives of the English school of the theory of invariants (cf. Invariants, theory of) gave a more systematic classification of geometries: Cayley explicitly used the term "group" , made systematic use of the multiplication table which now carries his name (cf. Cayley table), proved that any finite group can be represented by permutations, and conceived a group as a system which is defined by its generating elements and defining relations. The final stage in this development was the Erlangen program of F. Klein (1872), who based the classification of geometries on the concept of a transformation group.
Number theory is the third source of the concept of a group. As early as 1761 (cf. Euler function, Primitive root, Series, Stability of an elastic system, Siegel disc, Theta-function, Trigonometric series, Two-term congruence, Variation of constants, Variational calculus, numerical methods of Variational calculus, Venn diagram) L. Euler, in his study of residues remaining in power division, actually used congruences (cf. Congruence) and their division into residue classes, which in group-theoretic language means the decomposition of groups into cosets of subgroups. C.F. Gauss, in his Disquisitiones arithmeticae, studied the cyclotomic equations (cf. Cyclotomic polynomials) and in fact determined subgroups of their Galois groups (cf. Galois group). He also studied the "composition of binary quadratic forms" in this context, and showed, in essence, that the classes of equivalent forms form a finite Abelian group with respect to composition.
Towards the end of the 19th century it was recognized that the group-theoretic ideas employed for a long time in various fields of mathematics were essentially the same, and the modern abstract idea of the concept of a group was finally developed. Thus, as early as 1895, S. Lie defined a group as a set of transformations that is closed under an operation that is associative, admits a unit element and inverse elements. The study of groups without assuming them to be finite and without making any assumptions as to the nature of their elements was first formulated as an independent branch of mathematics with the appearance of the book Abstract group theory by O.Yu. Shmidt (1916).
Examples of groups.
The examples below illustrate the role played by groups in algebra, in other branches of mathematics and in natural sciences.
a) Galois groups. Let $K$ be a finite, separable and normal extension of a field $k$. The automorphisms of $K$ leaving the elements of $k$ fixed form a group $Gal(K/k)$ with respect to composition, called the Galois group of the extension $K/k$. The principal theorem in Galois theory states that the mapping which associates to every subgroup of $Gal(K/k)$ its fixed subfield (i.e. the subfield of $K$ whose elements are fixed under the subgroup of $Gal(K/k)$) is an anti-isomorphism of the lattice of subgroups of $Gal(K/k)$ onto the lattice of intermediate subfields between $k$ and $K$.
The application to the problem on the solvability of equations by radicals is as follows. Let $f$ be a polynomial in $x$ over $k$ and let $K$ be a splitting field (cf. Splitting field of a polynomial) of $f$. The group $Gal(K/k)$ is called the Galois group of $f$ over $k$ (its elements are naturally formed by the permutations of the roots of the equation $f(x)=0$). The result is that the equation $f(x)=0$ is solvable in radicals if and only if the Galois group of $f$ is solvable (cf. Solvable group).
In this and other similar examples groups appear in the form of automorphism groups (cf. Automorphism) of mathematical structures. This is one of the most important ways of appearance, ensuring groups a special place in algebra. In the words of Galois, automorphisms of arbitrary structures can always be "grouped" , while a ring structure or any other useful structure on a set of automorphisms is successfully introduced in special cases only.
b) Homology groups. The leading idea in homology theory is the application of the theory of (Abelian) groups to the study of a category of topological spaces. To each space $X$ is associated a family of Abelian groups $H_0(X),H_1(X),\ldots$ while each continuous mapping $f: X \rightarrow Y$ defines a family of homomorphisms $f_n: H_n(X) \rightarrow H_n(Y)$, $n = 0, 1, \ldots$. The study of the homology groups $H_n(X)$ (cf. Homology group) and their homomorphisms by the tools of group theory often makes it possible to deal with a topological problem. A typical example is the extension problem: Is it possible to extend a mapping $g: A \rightarrow Y$, defined on a subspace $A$ of $X$, to all of $X$, i.e. is it possible to represent $g$ as the composite of the imbedding $h: A \rightarrow X$ and some continuous mapping $f: X \rightarrow Y$? If so, then in the homology spaces one has $g_n = f_n h_n$, i.e. each homomorphism $g_n: H_n(A) \rightarrow H_n(Y)$ can be factored through $H_n(X)$ with a given homomorphism $h_n$. If this algebraic problem is unsolvable, then the initial topological problem is unsolvable as well. Important positive results can be obtained in this way.
Homology groups illustrate another typical manner of application of groups: the study of non-algebraic objects by means of algebraic systems which reflect their behaviour. This is in fact the fundamental method of algebraic topology. A similar method, in particular homology groups, is also used with success in the study of algebraic systems themselves — groups, rings, etc. (e.g. in the theory of group extensions).
c) Symmetry groups. The concept of a group makes it possible to describe the symmetries of a given geometrical figure. To any figure one associates the set of spatial transformations that map it onto itself. This set is a group under composition. It also characterizes the symmetry of the figure. This was in fact the approach of E.S. Fedorov (1890) to the problem of classification of regular spatial systems of points, which is one of the basic problems in crystallography (cf. Crystallography, mathematical). There are only 17 plane crystallographic groups (cf. Crystallographic group), which were found directly; there are 230 3-dimensional crystallographic groups, which could be exhaustively classified only by the use of group theory. This is historically the first example of the application of group theory to natural sciences.
Group theory plays a similar role in physics. Thus, the state of a physical system is represented in quantum mechanics by a point in an infinite-dimensional vector space. If the physical system passes from one state into another, its representative point undergoes some linear transformation. The ideas of symmetry and the theory of group representations (cf. Representation of a group) are of prime importance here.
These examples illustrate the contribution of group theory to all classifications where symmetry plays a role. The study of symmetry is actually equivalent to the study of automorphisms of (not necessarily mathematical) systems, and for this reason group theory is indispensable in solving such problems.
Important classes of groups.
The "final objective" of group theory is to describe all group operations or, in other words, all groups, up to isomorphism. Group theory comprises several parts, which are often distinguished by special conditions imposed on the group operation or by the introduction of additional structures into the group, related in some way with the group operation.
The oldest branch of group theory, which is still intensively studied, is the theory of finite groups (cf. Finite group). One of its important tasks is to determine the finite simple groups (cf. Simple finite group). These include many classical groups of matrices over finite fields, and also "sporadic" simple finite groups (Mathieu groups, cf. Mathieu group, etc.). At the other end there are finite solvable groups (cf. Solvable group) in which specific subgroup systems (Hall, Carter, etc., cf. Carter subgroup; Hall subgroup) are usually studied, since these largely determine the structure of the group itself. Finite groups often appear as permutation groups or as matrix groups over finite fields. A large independent branch of the theory of finite groups is the study of representations by matrices and permutations.
A typical method of study of infinite groups is to impose on them some finiteness condition (cf. Group with a finiteness condition). Here, the main interest is centred on periodic groups, locally finite groups, groups with the maximum condition for subgroups (Noetherian groups), groups with the minimum condition for subgroups (Artinian groups), residually-finite groups, groups of finite rank (cf. Rank of a group), and finitely-generated groups (cf. Periodic group; Noetherian group; Artinian group; Residually-finite group; Finitely-generated group).
In the study of Abelian groups (cf. Abelian group) important roles are played by complete Abelian groups, torsion-free Abelian groups and periodic Abelian groups, and inside these groups by pure subgroups and primary subgroups. The study of any given Abelian group is reduced to a large extent to the theories of the classes listed above with the aid of the theory of extensions of Abelian groups, which is mainly developed by homological methods (cf. Extension of a group).
Broader than the class of Abelian groups are the classes of nilpotent groups and of solvable groups (cf. Nilpotent group; Solvable group), the theory of which has also reached a fairly advanced stage. The most useful extensions of nilpotency and solvability are local nilpotency (cf. Locally nilpotent group), local solvability (cf. Locally solvable group) and the normalizer condition, as well as numerous properties determined by the presence of subnormal systems (cf. Subgroup system) of various types in a group. Of importance are special classes of solvable and nilpotent groups: supersolvable groups, polycyclic groups (cf. Supersolvable group; Polycyclic group).
An important branch of group theory is the theory of transformation groups, including permutation groups and the theory of linear groups (cf. Permutation group; Linear group). A number of important classes of groups is defined by the introduction of additional structures compatible with the group operation; this includes topological groups, Lie groups, algebraic groups, and ordered groups (cf. Topological group; Lie group; Algebraic group; Ordered group). Of the other classes of groups, the following are worthy of mention: groups which are free in some variety (cf. Free group), complete groups (cf. Complete group), groups having some property residually (cf. Residually-finite group), groups defined by imposing conditions on their generating elements and defining relations, and groups distinguished by imposing certain conditions on the lattice of subgroups.
References
[1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian) MR0551207 Zbl 0549.20001 [2] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) MR0109842 MR0080089 MR0071422 Zbl 0111.02502 [3] M. Hall jr., "The theory of groups" , Chelsea, reprint (1976) MR0414669 Zbl 0354.20001 [4] O.Yu. Shmidt, , Selected works on mathematics , Moscow (1959) pp. 17–70 (In Russian) [5] H. Wussing, "The genesis of the abstract group concept" , M.I.T. (1984) (Translated from German) MR0746617 Zbl 0547.01001 [6] E.S. Fedorov, , Symmetry and the structure of crystals. Fundamental works , Moscow (1949) pp. 111–255 (In Russian) [7] N.N. Bogolyubov, A.A. Logunov, I.T. Todorov, "Introduction to axiomatic quantum field theory" , Benjamin (1975) (Translated from Russian) MR452277
Similar remarks as to the (homotopy) extension problem apply to the (homotopy) lifting problem, in which it is required to fill in a diagram like the one on the right below (the one on the left is the diagram of the extension problem).
An important direction in group theory not mentioned in the article above is combinatorial group theory and the study of group by means of generators and relations [a3], [a4].
Certainly, abstract group theory was considered long before 1916. Thus, W. Burnside, writing in 1897, quotes Cayley as saying that "a group is defined by means of the laws of combination of its symbols" , and goes on to explain why he, in his own book, does, on the whole, not take that point of view; [a5], p. viii. L. Kronecker discussed axioms for abstract finite groups in 1870, cf. [a1] (see also [a10]), and the notion of abstract groups was introduced by Cayley in three papers starting in 1849, [a7][a9], though these papers received little attention at the time. This had certainly changed by the 1890's and a discussion of the basic definitions and some basic properties of abstract groups can be found in H. Weber's influential treatise [a6] (1896).
For (a history of) crystallographic groups cf. Crystallographic group.
References
[a1] L. Kronecker, "Auseinandersetzung einiger Eigenschaften der Klassenzahl idealer complexer Zahlen" Monatsber. K. Preuss. Akad. Wissenschaft. Berlin (1870) pp. 881–889 ((Also in: Werke, Vol. 1, p. 271)) Zbl 02.0097.01 [a2] H. Weyl, "Symmetry" , Princeton Univ. Press (1952) (Translated from German) MR0048449 Zbl 0046.00406 [a3] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966) pp. 412 Zbl 0138.25604 [a4] H.S.M. Coxeter, W.O.J. Moser, "Generators and relations for discrete groups" , Springer (1957) MR0088489 Zbl 0077.02801 [a5] W. Burnside, "Theory of groups of finite order" , Dover, reprint (1955) (Translated from German) MR0069818 Zbl 0064.25105 [a6] H. Weber, "Lehrbuch der Algebra" , II , Vieweg (1899) pp. Buch 1, Abschnitt 1 Zbl 30.0093.01 [a7] A. Cayley, "Note on the theory of permutations" Phil. Mag. (3) , 34 (1849) pp. 527–529 ((Also in: Collected mathematical papers, Vol. I, 432–424)) [a8] A. Cayley, "On the theory of groups as depending on the symbolical equation $\theta^n=1$" Phil. Mag. (4) , 7 (1854) pp. 40–47 ((Also in: Collected mathematical papers, Vol. II, 123–130)) [a9] A. Cayley, "On the theory of groups as depending on the symbolical equation $\theta^n=1$. Second part" Phil. Mag. (4) , 7 (1854) pp. 408–409 ((Also in: Collected mathematical papers, Vol. II, 131–132)) [a10] G.F. Frobenius, "Neuer Beweis des Sylowschen Satzes" J. Reine Angew. Math. , 100 (1887) pp. 179–181
How to Cite This Entry:
Group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Group&oldid=13033
This article was adapted from an original article by M.I. KargapolovYu.I. Merzlyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article | {"extraction_info": {"found_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": 0.8733301162719727, "perplexity": 545.0249232830786}, "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-2019-39/segments/1568514577478.95/warc/CC-MAIN-20190923172009-20190923194009-00278.warc.gz"} |
https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21A%3A_Differential_Calculus/3%3A_Differentiation/3.06%3A_The_Chain_Rule | # 3.6: The Chain Rule
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We have covered almost all of the derivative rules that deal with combinations of two (or more) functions. The operations of addition, subtraction, multiplication (including by a constant) and division led to the Sum and Difference rules, the Constant Multiple Rule, the Power Rule, the Product Rule and the Quotient Rule. To complete the list of differentiation rules, we look at the last way two (or more) functions can be combined: the process of composition (i.e. one function "inside'' another).
One example of a composition of functions is $$f(x) = \cos(x^2)$$. We currently do not know how to compute this derivative. If forced to guess, one would likely guess $$f^\prime(x) = -\sin(2x)$$,where we recognize $$-\sin x$$ as the derivative of $$\cos x$$ and $$2x$$ as the derivative of $$x^2$$. However, this is not the case; $$f^\prime(x)\neq -\sin(2x)$$. In Example 62 we'll see the correct answer, which employs the new rule this section introduces, the Chain Rule.
Before we define this new rule, recall the notation for composition of functions. We write $$(f \circ g)(x)$$ or $$f(g(x))$$,read as "$$f$$ of $$g$$ of $$x$$,'' to denote composing $$f$$ with $$g$$. In shorthand, we simply write $$f \circ g$$ or $$f(g)$$ and read it as "$$f$$ of $$g$$.'' Before giving the corresponding differentiation rule, we note that the rule extends to multiple compositions like $$f(g(h(x)))$$ or $$f(g(h(j(x))))$$,etc.
To motivate the rule, let's look at three derivatives we can already compute.
Example 59: Exploring similar derivatives
Find the derivatives of
1. $$F_1(x) = (1-x)^2$$,
2. $$F_2(x) = (1-x)^3,$$ and
3. $$F_3(x) = (1-x)^4.$$
We'll see later why we are using subscripts for different functions and an uppercase $$F$$.
Solution
In order to use the rules we already have, we must first expand each function as
1. $$F_1(x) = 1 - 2x + x^2$$,
2. $$F_2(x) = 1 - 3x + 3x^2 - x^3$$ and
3. $$F_3(x) = 1 - 4x + 6x^2 - 4x^3 + x^4$$.
It is not hard to see that:
\begin{align*} F_1^\prime(x) &= -2 + 2x \\[4pt] F_2^\prime(x) &= -3 + 6x - 3x^2 \\[4pt] F_3^\prime (x) &= -4 + 12x - 12x^2 + 4x^3. \end{align*}
An interesting fact is that these can be rewritten as
$F_1^\prime (x) = -2(1-x),\quad F_2^\prime(x) = -3(1-x)^2 \text{ and }F_3^\prime (x) = -4(1-x)^3.$
A pattern might jump out at you. Recognize that each of these functions is a composition, letting $$g(x) = 1-x$$:
$\begin{eqnarray*}F_1(x) = f_1(g(x)),& \text{ where } f_1(x) = x^2,\\ F_2(x) = f_2(g(x)),& \text{ where } f_2(x) = x^3,\\ F_3(x) = f_3(g(x)),& \text{ where } f_3(x) = x^4. \end{eqnarray*}$
We'll come back to this example after giving the formal statements of the Chain Rule; for now, we are just illustrating a pattern.
Theorem 18: The Chain Rule
Let $$y = f(u)$$ be a differentiable function of $$u$$ and let $$u = g(x)$$ be a differentiable function of $$x$$. Then $$y=f(g(x))$$ is a differentiable function of $$x$$,and $y^\prime = f^\prime(g(x))\cdot g^\prime(x).$
Example 60: Using the Chain Rule
Use the Chain Rule to find the derivatives of the following functions, as given in Example 59.
Solution
Example 59 ended with the recognition that each of the given functions was actually a composition of functions. To avoid confusion, we ignore most of the subscripts here.
$$F_1(x) = (1-x)^2$$:
We found that $y=(1-x)^2 = f(g(x)), \text{ where } f(x) = x^2\ \text{ and }\ g(x) = 1-x.$
To find $$y^\prime$$, we apply the Chain Rule. We need $$f^\prime(x)=2x$$ and $$g^\prime(x)=-1.$$
Part of the Chain Rule uses $$f^\prime(g(x))$$. This means substitute $$g(x)$$ for $$x$$ in the equation for $$f^\prime(x)$$. That is, $$f^\prime(x) = 2(1-x)$$. Finishing out the Chain Rule we have $y^\prime = f^\prime(g(x))\cdot g^\prime(x) = 2(1-x)\cdot (-1) = -2(1-x)= 2x-2.$
$$F_2(x) = (1-x)^3$$:
Let $$y = (1-x)^3 = f(g(x))$$,where $$f(x) = x^3$$ and $$g(x) = (1-x)$$. We have $$f^\prime(x) = 3x^2$$,so $$f^\prime(g(x)) = 3(1-x)^2$$. The Chain Rule then states $y^\prime = f^\prime(g(x))\cdot g^\prime (x) = 3(1-x)^2\cdot(-1) = -3(1-x)^2.$
$$F_3(x) = (1-x)^4$$:
Finally, when $$y = (1-x)^4$$,we have $$f(x)= x^4$$ and $$g(x) = (1-x)$$. Thus $$f^\prime(x) = 4x^3$$ and $$f^\prime(g(x)) = 4(1-x)^3$$. Thus $y^\prime = f^\prime(g(x))\cdot g^\prime(x) = 4(1-x)^3\cdot (-1) = -4(1-x)^3.$
Example 60 demonstrated a particular pattern: when $$f(x)=x^n$$,then $$y^\prime =n\cdot (g(x))^{n-1}\cdot g^\prime (x)$$. This is called the Generalized Power Rule.
Theorem 19: Generalized Power Rule
Let $$g(x)$$ be a differentiable function and let $$n\neq 0$$ be an integer. Then $\dfrac{d}{dx}\Big(g(x)^n\Big) = n\cdot \big(g(x)\big)^{n-1}\cdot g^\prime (x).$
This allows us to quickly find the derivative of functions like $$y = (3x^2-5x+7+\sin x)^{20}$$. While it may look intimidating, the Generalized Power Rule states that $y^\prime = 20(3x^2-5x+7+\sin x)^{19}\cdot (6x-5+\cos x).$
Treat the derivative--taking process step--by--step. In the example just given, first multiply by 20, then rewrite the inside of the parentheses, raising it all to the 19$$^{\text{th}}$$ power. Then think about the derivative of the expression inside the parentheses, and multiply by that.
We now consider more examples that employ the Chain Rule.
Example 61: Using the Chain Rule
Find the derivatives of the following functions:
1. $$y = \sin{2x}$$
2. $$y= \ln (4x^3-2x^2)$$
3. $$y = e^{-x^2}$$
Solution
1. Consider $$y = \sin 2x$$. Recognize that this is a composition of functions, where $$f(x) = \sin x$$ and $$g(x) = 2x$$. Thus $y^\prime = f^\prime(g(x))\cdot g^\prime(x) = \cos (2x)\cdot 2 = 2\cos 2x.$
2. Recognize that $$y = \ln (4x^3-2x^2)$$ is the composition of $$f(x) = \ln x$$ and $$g(x) = 4x^3-2x^2$$. Also, recall that $\dfrac{d}{dx}\Big(\ln x\Big) = \dfrac{1}{x}.$This leads us to:$y^\prime = \dfrac{1}{4x^3-2x^2} \cdot (12x^2-4x) = \dfrac{12x^2-4x}{4x^3-2x^2}= \dfrac{4x(3x-1)}{2x(2x^2-x)} = \dfrac{2(3x-1)}{2x^2-x}.$
3. Recognize that $$y = e^{-x^2}$$ is the composition of $$f(x) = e^x$$ and $$g(x) = -x^2$$. Remembering that $$f^\prime(x) = e^x$$,we have $y^\prime = e^{-x^2}\cdot (-2x) = (-2x)e^{-x^2}.$
Example 62: Using the Chain Rule to find a tangent line
Let $$f(x) = \cos x^2$$. Find the equation of the line tangent to the graph of $$f$$ at $$x=1$$.
Solution
The tangent line goes through the point $$(1,f(1)) \approx (1,0.54)$$ with slope $$f^\prime(1)$$. To find $$f^\prime$$,we need the Chain Rule.
$$f^\prime(x) = -\sin(x^2) \cdot(2x) = -2x\sin x^2$$. Evaluated at $$x=1$$,we have $$f^\prime(1) = -2\sin 1\approx -1.68$$. Thus the equation of the tangent line is $y = -1.68(x-1)+0.54 .$
The tangent line is sketched along with $$f$$ in Figure 2.17.
The Chain Rule is used often in taking derivatives. Because of this, one can become familiar with the basic process and learn patterns that facilitate finding derivatives quickly. For instance, $\dfrac{d}{dx}\Big(\ln (\text{anything})\Big) = \dfrac{1}{\text{anything}}\cdot (\text{anything})^\prime = \dfrac{(\text{anything})^\prime}{\text{anything}}.$
A concrete example of this is $\dfrac{d}{dx}\Big(\ln(3x^{15}-\cos x+e^x)\Big) = \dfrac{45x^{14}+\sin x+e^x}{3x^{15}-\cos x+e^x}.$ While the derivative may look intimidating at first, look for the pattern. The denominator is the same as what was inside the natural log function; the numerator is simply its derivative.
This pattern recognition process can be applied to lots of functions. In general, instead of writing "anything'', we use $$u$$ as a generic function of $$x$$. We then say $\dfrac{d}{dx}\Big(\ln u\Big) = \dfrac{u^\prime}{u}.$
The following is a short list of how the Chain Rule can be quickly applied to familiar functions.
Of course, the Chain Rule can be applied in conjunction with any of the other rules we have already learned. We practice this next.
Example 63: Using the Product, Quotient and Chain Rules
Find the derivatives of the following functions.
1. $$f(x) = x^5 \sin{2x^3}$$
2. $$f(x) = \dfrac{5x^3}{e^{-x^2}}$$.
Solution
1. We must use the Product and Chain Rules. Do not think that you must be able to "see'' the whole answer immediately; rather, just proceed step--by--step.$f^\prime(x) = x^5\big(6x^2\cos 2x^3\big) + 5x^4\big(\sin 2x^3\big)= 6x^7\cos2x^3+5x^4\sin 2x^3.$
2. We must employ the Quotient Rule along with the Chain Rule. Again, proceed step--by--step.\begin{align*} f^\prime(x) = \dfrac{e^{-x^2}\big(15x^2\big) - 5x^3\big((-2x)e^{-x^2}\big)}{\big(e^{-x^2}\big)^2} &=\dfrac{e^{-x^2}\big(10x^4+15x^2\big)}{e^{-2x^2}}\\ &= e^{x^2}\big(10x^4+15x^2\big). \end{align*}
A key to correctly working these problems is to break the problem down into smaller, more manageable pieces. For instance, when using the Product and Chain Rules together, just consider the first part of the Product Rule at first: $$f(x)g^\prime(x)$$. Just rewrite $$f(x)$$,then find $$g^\prime(x)$$. Then move on to the $$f^\prime(x)g(x)$$ part. Don't attempt to figure out both parts at once.
Likewise, using the Quotient Rule, approach the numerator in two steps and handle the denominator after completing that. Only simplify afterward.
We can also employ the Chain Rule itself several times, as shown in the next example.
Example 64: Using the Chain Rule multiple times
Find the derivative of $$y = \tan^5(6x^3-7x)$$.
Solution
Recognize that we have the $$g(x)=\tan(6x^3-7x)$$ function "inside'' the $$f(x)=x^5$$ function; that is, we have $$y = \big(\tan(6x^3-7x)\big)^5$$. We begin using the Generalized Power Rule; in this first step, we do not fully compute the derivative. Rather, we are approaching this step--by--step.
$y^\prime = 5\big(\tan(6x^3-7x)\big)^4\cdot g^\prime(x).$
We now find $$g^\prime(x)$$. We again need the Chain Rule; $g^\prime(x) = \sec^2(6x^3-7x)\cdot(18x^2-7).$Combine this with what we found above to give
\begin{align*} y^\prime &= 5\big(\tan(6x^3-7x)\big)^4\cdot\sec^2(6x^3-7x)\cdot(18x^2-7)\\ &= (90x^2-35)\sec^2(6x^3-7x)\tan^4(6x^3-7x). \end{align*}
This function is frankly a ridiculous function, possessing no real practical value. It is very difficult to graph, as the tangent function has many vertical asymptotes and $$6x^3-7x$$ grows so very fast. The important thing to learn from this is that the derivative can be found. In fact, it is not "hard;'' one must take several simple steps and be careful to keep track of how to apply each of these steps.
It is a traditional mathematical exercise to find the derivatives of arbitrarily complicated functions just to demonstrate that it can be done. Just break everything down into smaller pieces.
Example 65: Using the Product, Quotient and Chain Rules
Find the derivative of $$f(x) = \dfrac{x\cos(x^{-2})-\sin^2(e^{4x})}{\ln(x^2+5x^4)}.$$
Solution
This function likely has no practical use outside of demonstrating derivative skills. The answer is given below without simplification. It employs the Quotient Rule, the Product Rule, and the Chain Rule three times.
$f^\prime(x) = \dfrac{\Big(\ln(x^2+5x^4)\Big)\cdot\Big[\big(x\cdot(-\sin(x^{-2}))\cdot(-2x^{-3})+1\cdot \cos(x^{-2})\big)-2\sin(e^{4x})\cdot\cos(e^{4x})\cdot(4e^{4x})\Big]-\Big(x\cos(x^{-2})-\sin^2(e^{4x})\Big)\cdot\dfrac{2x+20x^3}{x^2+5x^4}}{\big(\ln(x^2+5x^4)\big)^2}.$
The reader is highly encouraged to look at each term and recognize why it is there. (I.e., the Quotient Rule is used; in the numerator, identify the "LOdHI'' term, etc.) This example demonstrates that derivatives can be computed systematically, no matter how arbitrarily complicated the function is.
The Chain Rule also has theoretic value. That is, it can be used to find the derivatives of functions that we have not yet learned as we do in the following example.
Example 66: The Chain Rule and exponential functions
Use the Chain Rule to find the derivative of $$y= a^x$$ where $$a>0$$,$$a\neq 1$$ is constant.
Solution
We only know how to find the derivative of one exponential function: $$y = e^x$$; this problem is asking us to find the derivative of functions such as $$y = 2^x$$.
This can be accomplished by rewriting $$a^x$$ in terms of $$e$$. Recalling that $$e^x$$ and $$\ln x$$ are inverse functions, we can write
$a = e^{\ln a} \quad \text{and so } \quad y = a^x = e^{\ln (a^x)}. \nonumber$
By the exponent property of logarithms, we can "bring down'' the power to get
$y = a^x = e^{x (\ln a)}. \nonumber$
The function is now the composition $$y=f(g(x))$$,with $$f(x) = e^x$$ and $$g(x) = x(\ln a)$$. Since $$f^\prime(x) = e^x$$ and $$g^\prime(x) = \ln a$$, the Chain Rule gives
$y^\prime = e^{x (\ln a)} \cdot \ln a. \nonumber$
Recall that the $$e^{x(\ln a)}$$ term on the right hand side is just $$a^x$$,our original function. Thus, the derivative contains the original function itself. We have
$y^\prime = y \cdot \ln a = a^x\cdot \ln a. \nonumber$
The Chain Rule, coupled with the derivative rule of $$e^x$$,allows us to find the derivatives of all exponential functions.
The previous example produced a result worthy of its own "box.''
Theorem 20: Derivatives of Exponential Functions
Let $$f(x)=a^x$$,for $$a>0, a\neq 1$$. Then $$f$$ is differentiable for all real numbers and
$f^\prime(x) = \ln a\cdot a^x. \nonumber$
## Alternate Chain Rule Notation
It is instructive to understand what the Chain Rule "looks like'' using "$$\dfrac{dy}{dx}$$'' notation instead of $$y^\prime$$ notation. Suppose that $$y=f(u)$$ is a function of $$u$$,where $$u=g(x)$$ is a function of $$x$$,as stated in Theorem 18. Then, through the composition $$f \circ g$$,we can think of $$y$$ as a function of $$x$$,as $$y=f(g(x))$$. Thus the derivative of $$y$$ with respect to $$x$$ makes sense; we can talk about $$\dfrac{dy}{dx}.$$ This leads to an interesting progression of notation:
\begin{align*}y^\prime &= f^\prime(g(x))\cdot g^\prime(x) \\ \dfrac{dy}{dx} &= y^\prime(u) \cdot u^\prime(x)\quad \text{(since $$y=f(u)$$ and $$u=g(x)$$)}\\ \dfrac{dy}{dx} &= \dfrac{dy}{du} \cdot \dfrac{du}{dx}\quad \text{(using "fractional'' notation for the derivative)}\end{align*}
Here the "fractional'' aspect of the derivative notation stands out. On the right hand side, it seems as though the "$$du$$'' terms cancel out, leaving $\dfrac{dy}{dx} = \dfrac{dy}{dx}.$
It is important to realize that we are not canceling these terms; the derivative notation of $$\dfrac{dy}{dx}$$ is one symbol. It is equally important to realize that this notation was chosen precisely because of this behavior. It makes applying the Chain Rule easy with multiple variables. For instance,
$\dfrac{dy}{dt} = \dfrac{dy}{d\bigcirc} \cdot \dfrac{d\bigcirc}{d\triangle} \cdot \dfrac{d\triangle}{dt}.$
where $$\bigcirc$$ and $$\triangle$$ are any variables you'd like to use.
One of the most common ways of "visualizing" the Chain Rule is to consider a set of gears, as shown in Figure 2.18. The gears have 36, 18, and 6 teeth, respectively. That means for every revolution of the $$x$$ gear, the $$u$$ gear revolves twice. That is, the rate at which the $$u$$ gear makes a revolution is twice as fast as the rate at which the $$x$$ gear makes a revolution. Using the terminology of calculus, the rate of $$u$$-change, with respect to $$x$$,is $$\dfrac{du}{dx} = 2$$.
Likewise, every revolution of $$u$$ causes 3 revolutions of $$y$$: $$\dfrac{dy}{du} = 3$$. How does $$y$$ change with respect to $$x$$? For each revolution of $$x$$,$$y$$ revolves 6 times; that is, $\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot \dfrac{du}{dx} = 2\cdot 3 = 6.$
We can then extend the Chain Rule with more variables by adding more gears to the picture.
It is difficult to overstate the importance of the Chain Rule. So often the functions that we deal with are compositions of two or more functions, requiring us to use this rule to compute derivatives. It is often used in practice when actual functions are unknown. Rather, through measurement, we can calculate $$\dfrac{dy}{du}$$ and $$\dfrac{du}{dx}$$. With our knowledge of the Chain Rule, finding $$\dfrac{dy}{dx}$$ is straightforward.
In the next section, we use the Chain Rule to justify another differentiation technique. There are many curves that we can draw in the plane that fail the "vertical line test.'' For instance, consider $$x^2+y^2=1$$,which describes the unit circle. We may still be interested in finding slopes of tangent lines to the circle at various points. The next section shows how we can find $$\dfrac{dy}{dx}$$ without first "solving for $$y$$.'' While we can in this instance, in many other instances solving for $$y$$ is impossible. In these situations, implicit differentiation is indispensable.
3.6: The Chain Rule is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts. | {"extraction_info": {"found_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": 0.9914429187774658, "perplexity": 308.62314354881624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335124.77/warc/CC-MAIN-20220928051515-20220928081515-00743.warc.gz"} |
https://math.stackexchange.com/questions/2170337/too-simple-proof-of-a-unique-solution-of-the-stationary-navier-stokes-equation | # Too simple proof of a unique solution of the stationary Navier-Stokes equation
Let
• $d\in\left\{1,\ldots,4\right\}$
• $\Lambda\subseteq\mathbb R^d$ be bounded, nonempty and open
• $\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$, $V:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\|_{H^1(\Lambda,\:\mathbb R^d)}}$ and $H:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\|_{L^2(\Lambda,\:\mathbb R^d)}}$
• $\mathfrak a(u,v):=\sum_{i=1}^d\langle\nabla u_i,\nabla v_i\rangle_{L^2}$ for $u,v\in V$
• $\mathfrak b(u,v,w):=\langle((u\cdot\nabla)v,w\rangle_{L^2}$ for $u,v,w\in V$
• $\nu>0$
• $f\in L^2(\Lambda,\mathbb R^d)$
I've found proofs showing that there is a $u\in V$ with $$\tilde{\mathfrak a}(u,v):=\nu\mathfrak a(u,v)+\mathfrak b(u,u,v)=\langle f,v\rangle_{L^2(\Lambda,\:\mathbb R^d)}=:\ell(v)\;\;\;\text{for all }v\in V\tag1$$ using the (Leray-)Schauder fixed-point theorem.
However, doesn't the existence (and even uniqueness) of such a $u$ simply follow from the Lax-Milgram theorem?
It's well-known that
1. $\mathfrak a$ is a bounded and coercive bilinear form on $H_0^1(\Lambda,\mathbb R^d)$
2. $\mathfrak b$ is a bounded trilinear form on $H_0^1(\Lambda,\mathbb R^d)$ with $$\mathfrak b(u,v,w)+\mathfrak b(u,w,v)=-\langle\left(\nabla\cdot u\right)v,w\rangle_{L^2(\Lambda,\:\mathbb R^d)}\;\;\;\text{for all }u,v,w\in H_0^1(\Lambda,\mathbb R^d)\tag2$$
Since $$\tilde{\mathfrak a}(u,u)=\nu\mathfrak a(u,u)+\underbrace{\mathfrak b(u,u,u)}_{=\:0}=\nu\mathfrak a(u,u)\;\;\;\text{for all }u\in V\tag3\;,$$ $\tilde{\mathfrak a}$ is a bounded and coercive bilinear form on $V$. Moreover, $\ell$ is a bounded linear functional on $V$. So, there should be a unique $u\in V$ with $(1)$ by the Lax-Milgram theorem.
What am I missing?
What I was missing is that $$V\times V\ni(u,v)\mapsto\mathfrak b(u,u,v)$$ (and hence $\tilde{\mathfrak a}$) is not bilinear. It's obvious, but I missed that. However, we can use the Lax-Milgram theorem to show that for all $u\in V$ there is a unique $\Phi(u)\in V$ with $$\nu\mathfrak a(\Phi(u),v)+\mathfrak b(u,\Phi(u),v)=\ell(v)\tag4\;.$$ Now, the Schauder fixed-point theorem can be used to show that $\Phi$ has a fixed-point. | {"extraction_info": {"found_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": 0.98885178565979, "perplexity": 124.64747388982563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669813.71/warc/CC-MAIN-20191118182116-20191118210116-00099.warc.gz"} |
http://ams02.space/publications/202101 | # Properties of Iron Primary Cosmic Rays: Results from the Alpha Magnetic Spectrometer
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Phys. Rev. Lett. 126, 041104 (2021)
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Abstract
We report the observation of new properties of primary iron (Fe) cosmic rays in the rigidity range 2.65 GV to 3.0 TV with $0.62 \times 10^6$ iron nuclei collected by the Alpha Magnetic Spectrometer experiment on the International Space Station. Above 80.5 GV the rigidity dependence of the cosmic ray Fe flux is identical to the rigidity dependence of the primary cosmic ray He, C, and O fluxes, with the Fe/O flux ratio being constant at $0.155 \pm 0.006$. This shows that unexpectedly Fe and He, C, and O belong to the same class of primary cosmic rays which is different from the primary cosmic rays Ne, Mg, and Si class.
### Table-SI
The Fe flux $\Phi$ as a function of rigidity at the top of AMS in units of $[{\rm m}^2 \cdot {\rm sr} \cdot {\rm s} \cdot {\rm GV}]^{-1}$ including errors due to statistics (stat.); contributions to the systematic error from the trigger and acceptance (acc.); the rigidity resolution function and unfolding (unf.); the absolute rigidity scale (scale); and the total systematic error (syst.). The contribution of individual sources to the systematic error are added in quadrature to arrive at the total systematic error.
### Table-SII
The iron to oxygen flux ratio Fe/O as a function of rigidity including errors due to statistics (stat.); contributions to the systematic error from the trigger, acceptance, and background (acc.); the rigidity resolution function and unfolding (unf.); the absolute rigidity scale (scale); and the total systematic error (syst.). The statistical errors are the sum in quadrature of the ratios of iron and oxygen flux statistical errors to the corresponding flux values, multiplied by the Fe/O flux ratio. The systematic errors from the background subtraction, the trigger, and the event reconstruction and selection are likewise added in quadrature. The correlations in the systematic errors from the uncertainty in nuclear interaction cross sections, the unfolding and the absolute rigidity scale between the iron and oxygen fluxes have been taken into account in calculating the corresponding systematic errors of the Fe/O flux ratio. The contribution of individual sources to the systematic error are added in quadrature to arrive at the total systematic uncertainty.
### Table-SIII
The iron to helium flux ratio Fe/He as a function of rigidity including errors due to statistics (stat.); contributions to the systematic error from the trigger, acceptance, and background (acc.); the rigidity resolution function and unfolding (unf.); the absolute rigidity scale (scale); and the total systematic error (syst.). The statistical errors are the sum in quadrature of the ratios of iron and helium flux statistical errors to the corresponding flux values, multiplied by the Fe/He flux ratio. The systematic errors from the background subtraction, the trigger, and the event reconstruction and selection are likewise added in quadrature. The correlations in the systematic errors from the uncertainty in nuclear interaction cross sections, the unfolding and the absolute rigidity scale between the iron and helium fluxes have been taken into account in calculating the corresponding systematic errors of the Fe/He flux ratio. The contribution of individual sources to the systematic error are added in quadrature to arrive at the total systematic uncertainty.
### Table-SIV
The iron to silicon flux ratio Fe/Si as a function of rigidity including errors due to statistics (stat.); contributions to the systematic error from the trigger, acceptance, and background (acc.); the rigidity resolution function and unfolding (unf.); the absolute rigidity scale (scale); and the total systematic error (syst.). The statistical errors are the sum in quadrature of the ratios of iron and silicon flux statistical errors to the corresponding flux values, multiplied by the Fe/Si flux ratio. The systematic errors from the background subtraction, the trigger, and the event reconstruction and selection are likewise added in quadrature. The correlations in the systematic errors from the uncertainty in nuclear interaction cross sections, the unfolding and the absolute rigidity scale between the iron and silicon fluxes have been taken into account in calculating the corresponding systematic errors of the Fe/He flux ratio. The contribution of individual sources to the systematic error are added in quadrature to arrive at the total systematic uncertainty. | {"extraction_info": {"found_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": 0.9187327027320862, "perplexity": 1618.5596342624124}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104676086.90/warc/CC-MAIN-20220706182237-20220706212237-00551.warc.gz"} |
https://chemistry.stackexchange.com/questions/113216/molarity-numerical-on-water-gas-shift-reaction | # Molarity Numerical on Water Gas Shift Reaction
For the water gas shift reaction below, $$K_c = 3.491$$ at a certain temperature. What are the equilibrium concentrations of all the components of the reaction if $$\pu{0.3815 mol}$$ of $$\ce{CO}$$ and $$\ce{H2O}$$ are initially mixed in a $$\pu{250 mL}$$ flask?
$$\ce{CO(g) + H2O(g) <=> CO2(g) + H2(g)}$$
I was able to calculate the molarity of $$\ce{CO}$$ and $$\ce{H2O}$$ and set up an ICE table, but my answer seemed inaccurate. My work so far:
$$3.491 = \frac{[\ce{CO2}][\ce{H2}]}{[\ce{CO}][\ce{H2O}]}$$
$$3.491 = \frac{x^2}{(1.526-x)^2}$$
Because $$x$$ should be much smaller than $$1.5$$, it can be ignored when squaring $$1.526$$. Therefore, $$3.491 = x^2/2.32$$
$$3.491\cdot 2.32 = x^2$$
The square root of that value is $$2.85$$, which is impossible because there weren't $$\pu{2.85 mol}$$ of $$\ce{CO}$$ or $$\ce{H2O}$$ to begin with.
• @OscarLanzi My change in molarity was greater than the actual moles of reactants involved. – Andrew McAvoy Apr 24 at 0:07
• @KarstenTheis H2O and CO both start at 1.526 M and have x subtracted from them. CO2 and H2 start at 0 M and have x added to them. – Andrew McAvoy Apr 24 at 0:09
Taking the square root of both sides gives
$$\frac{x}{1.526-x}=1.868$$
So
$$x=1.526\frac{1.868}{2.868}=0.994$$
• That's one beautiful elephant in the room I failed to notice when I wrote my answer solving quadratic equation:( Nicely done! – andselisk Apr 26 at 20:50
Your reasoning looks correct up to point you assumed $$x\ll 1.5$$. There is no evidence to support this claim and eventually you just have to solve the equation:
$$K_c = \frac{x^2}{(c_0 - x)^2}\label{eqn:1}\tag{1}$$
where $$c_0$$ is the initial concentration of the reactants:
$$c_0 = \frac{n_0}{V} = \frac{\pu{0.3815 mol}}{\pu{0.250 L}} = \pu{1.526 M} \tag{2}$$
Equation \eqref{eqn:1} can be rewritten as a typical quadratic equation of $$ax^2 + bx + c = 0$$ form:
$$(K_c - 1)x^2 - 2K_cc_0x + K_cc_0^2 = 0 \tag{3}$$
with the following set of roots:
\begin{align} x_{1,2} &= \frac{2K_cc_0 ± \sqrt{4K_c^2c_0^2 - 4K_cc_0(K_c - 1)}}{2(K_c - 1)} \\ &= \frac{K_cc_0 ± \sqrt{K_cc_0(K_cc_0 - c_0(K_c - 1))}}{K_c - 1} \\ &= \frac{5.327 ± 2.851}{2.491}\tag{4} \end{align}
$$x_1 = 3.283;\quad x_2 = 0.994$$
Since $$x < c_0$$, only $$x_2 = 0.994$$ is physically meaningful, resulting in the final answer:
$$[\ce{CO2}] = [\ce{H2}] = x = \pu{0.994 M}$$
$$[\ce{CO}] = [\ce{H2O}] = c_0 - x = \pu{1.526 M} - \pu{0.994 M} = \pu{0.532 M}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 33, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9830887317657471, "perplexity": 1047.2512863126565}, "config": {"markdown_headings": true, "markdown_code": false, "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/1568514572436.52/warc/CC-MAIN-20190915215643-20190916001643-00139.warc.gz"} |
http://math.stackexchange.com/tags/harmonic-functions/info | # Tag Info
For questions regarding harmonic functions.
The solutions of the Laplace equation $\Delta f =0$ on a domain $D\subset \mathbb{R}^n$ are known as harmonic functions.
Harmonic functions appear most naturally in complex analysis and extends the concept of analytic functions.
The Cauchy-Riemann equation together with the conjugated Cauchy-Riemann equation shows that the sum of an analytic function and an anti-analytic function is harmonic and in fact every complex harmonic function can be written as such. In particular the real/imaginary part of an analytic function is harmonic.
Harmonic functions satisfy the Liouville's theorem and maximum principle, in any dimension.
It should be mentioned that harmonic functions can be generalized one step further to the class of sub-harmonic functions which satisfy $$\Delta f\geq0$$ which also satisfy the maximum principle.
Note that harmonic functions satisfy the regularity theorem for harmonic functions, which states that harmonic functions are infinitely differentiable (follows from Laplace's equation). They also satisfy Harnack's inequality, which relates the values of a positive harmonic function at two points. | {"extraction_info": {"found_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": 0.9640661478042603, "perplexity": 205.2126342839431}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049272823.52/warc/CC-MAIN-20160524002112-00195-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-applied-math/93688-fourier-transform-gravity-waves.html | # Math Help - Fourier Transform for gravity waves
1. ## Fourier Transform for gravity waves
Hi,
I've been trying to replicate the results I've seen in a paper, in order to understand it and take it a step further. However, I am not having much luck.
We are given equation (1), which is:
Z(x,z=0,t=0) = A{Ai(-x+1)*x/2*exp[-(x-2)/2]}
where Ai is the Airy function and x is the horizontal position. The amplitude of the forcing is A. I have this plotted without a problem.
However, it then states, the Fourier Transform of (1) provides the wavenumber (k) spectrum of the forcing (equation (2))
Z_hat(k,0,0) = 1/(2*pi)*integral from -inf to inf of (Z(x,0,0)*exp(i*k*x)dx)
I think I'm getting lost on this step - I don't quite understand what it means or how it should be implemented. If anyone could help me, it would be greatly appreciated!
If it is helpful to know the next step it is: water phase speed = c = sqrt(gh) where g is 9.8m/s^2 and h~4000m such that c=-200 m/s. Which implies for every k in the spectrum defined by (2), there is a corresponding wave frequency omega=-200*k
Now, the full-wave model for each omega-k pair in the spectrum. The vertical velocity spectrum is calculated as W_hat(k,0,0)=i*omega*Z_hat(k,0,0)=-i*200*k*Z_hat(k,0,0).
A discrete fourier transform is then used to evaluate the surface displacement (Z_hat) and its vertical velocity spectrum (W_hat).
If any knows how to set this problem up - it would be very helpful. Thank you so much!
2. Originally Posted by jschmid2
Hi,
I've been trying to replicate the results I've seen in a paper, in order to understand it and take it a step further. However, I am not having much luck.
We are given equation (1), which is:
Z(x,z=0,t=0) = A{Ai(-x+1)*x/2*exp[-(x-2)/2]}
where Ai is the Airy function and x is the horizontal position. The amplitude of the forcing is A. I have this plotted without a problem.
However, it then states, the Fourier Transform of (1) provides the wavenumber (k) spectrum of the forcing (equation (2))
Z_hat(k,0,0) = 1/(2*pi)*integral from -inf to inf of (Z(x,0,0)*exp(i*k*x)dx)
I think I'm getting lost on this step - I don't quite understand what it means or how it should be implemented. If anyone could help me, it would be greatly appreciated!
If it is helpful to know the next step it is: water phase speed = c = sqrt(gh) where g is 9.8m/s^2 and h~4000m such that c=-200 m/s. Which implies for every k in the spectrum defined by (2), there is a corresponding wave frequency omega=-200*k
Now, the full-wave model for each omega-k pair in the spectrum. The vertical velocity spectrum is calculated as W_hat(k,0,0)=i*omega*Z_hat(k,0,0)=-i*200*k*Z_hat(k,0,0).
A discrete fourier transform is then used to evaluate the surface displacement (Z_hat) and its vertical velocity spectrum (W_hat).
If any knows how to set this problem up - it would be very helpful. Thank you so much!
This is decomposing Z(t,0,0) into its monochromatic (single frequency) components so that subsequently theory for sine waves can be applied for the speeds of the individual components.
CB | {"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": 0.9804887175559998, "perplexity": 444.0887996205335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1464049278042.87/warc/CC-MAIN-20160524002118-00068-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://www.gradesaver.com/textbooks/math/calculus/calculus-8th-edition/chapter-2-derivatives-2-4-derivatives-of-trigonometric-functions-2-4-exercises-page-150/15 | ## Calculus 8th Edition
Differentiate $f(\theta ) = \theta \cos \theta \sin \theta$ $f'(\theta)= \frac{1}{2}\sin 2\theta +\theta\cos 2\theta$
Use the product rule and trig rules. $f'(\theta) = \cos \theta \sin \theta - \theta \sin ^2 \theta + \theta \cos ^2\theta$ $=\cos \theta \sin \theta +\theta(\cos ^2 \theta - \sin ^2 \theta)$ Use trig identities to simplify $= \frac{1}{2}\sin 2\theta +\theta\cos 2\theta$ | {"extraction_info": {"found_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": 0.999902606010437, "perplexity": 1340.6468028699448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945660.53/warc/CC-MAIN-20180422212935-20180422232935-00238.warc.gz"} |
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# Kids Christmas Favorites (Holiday)
## Selected Discography
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### Track List: Kids Christmas Favorites
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I LOVE CHRISTMAS MY FAVORITE
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Thats the jingle Bell Rock!!!!!
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lnguyen0414
I can't wait till chistmas to get my present.
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lnguyen0414
The best.
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Chrismas music makes me feel good.dont know why
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Jkkkjkjjkkjk l i i k m f j f j j f j j f j m f j k f k u f k i f j r p j r k l r k f k l l m j f i h k j i u i h l h k j o i h k j j k h k h k j h o j
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i love the little boy
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I sang it in the school. Concert
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I love the frozen so much i love the songs and everything its verry pretty������
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I love chismas is my favrite holoday and i love this song i cant wait for chrismas2014 , 2 0 1 5
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liz-007
Christmas is my favorite holiday
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carolpbj
I love this song sooooooooooo o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o much:-)
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elisa.caball e r o 7 5
I love this song sooooooooooo o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o much. For Christmas song
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baileet7
Christmas is my favorite holiday
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love this song so much happy holidays
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,kkkklkmklkl k l k k l o l o l l p o o p o o l l p k p o p i o o p o i o o p o o o p o p o p o p o p p p o p o p
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Kkjkjkjkjjkj l j l l k k l k l k l k l k l o k l k l k l k k l k k p k o k p k l k l k p k p k l k o i o i o i o i i i o i o i l i o i o i i o k i i l k o i p i l i o o p i l i k l o p k p i o o p o p k p k o i i o o p k l i o i k o o k o i i l i o k l i l k l l k p l i o l k o k p k , , , , k k l k k k l k l k l
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kkio7hkyhhjk h u
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Jkjkjkknmjuk j m k j m j j j l j k m m u k m k j l j l j l j k k l j k k l j l k m l k l k k m l j m l k k l j k l k
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Kl
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Kljjojkjkjoj k j o j j l j o k m l k
Kklkkllkklkl k k l k l k l k l k l k k k k l k k l k l l l k
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Kopklopopkkl l l k l k l p k k l l l l p l p o o p o p l l p k , l l p k k l k p l
Lplpklklpllp l
Kll
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Klkkkkojjljo k l k o k k o k i o i o i o i o i i o k l k o k o k o i o k o k i i o i o i o i i o o o o o o o o o k k l k o o o k o k l k l l k k l k l k l k o i k o k k o o o o k p k k o o k o k o k o k o k o o k l l k o k o k o k o k o k o ö k k l k l k k k l k l k k k k k m , j j k j k j k j l j k j k k j k k k k k k k o k l k k k k l k j k k k k k k k k i k l k k k k k k k o k k l k k k k j k k i o k k k l k k k k k i i o k i i o j k j k j k j k k k j i l k l k k i l k i k k k j k k j j k j k k k j k j k j k j k j k j k i i k k i j o k o i
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jennybfloral d e s i g n
i love christmas its my favorite holiday!!!!! ! ! ! ! ! ! ! ! ! !
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I love christmas can't wait till 2014 christmas
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The best
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love it
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amsome
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great
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If you say I am a noob you are too
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This is the wrosth
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I love it I wish u a merry Christmas for next Christmas
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Favorite Christmas song ever written
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I like it.
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I can't wait tell next Christmas :)
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Hi
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Asome
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There wonderful
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Great
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Awsome!��
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Merry Christmas!:)
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lugo.benji
Merry Christmas!:)
We're sorry, but a browser plugin or firewall may be preventing Pandora from loading. | {"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": 0.9970956444740295, "perplexity": 1575.741758042597}, "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-2015-48/segments/1448398446250.47/warc/CC-MAIN-20151124205406-00057-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://clay6.com/qa/24233/if-x-y-z-1-are-in-gp-then-frac-frac-frac-are-in | # If x, y, z > 1 are in GP, then $\frac{1}{1+lnx}$, $\frac{1}{lny}$, $\frac{1}{lnz}$ are in
$(a)\;AP\qquad(b)\;GP\qquad(c)\;HP\qquad(d)\;None$
Explanation : x , y , z in GP
$y^2=xz$
$ln\;y^2\;=\;ln\;xz$
$2\;ln\;y=ln\;x+ln\;z$
$ln\;x\;,ln\;y\;,ln\;z\;are\;in\;AP$
$1+ln\;x\;,1+ln\;y\;,1+ln\;z\;are\;in\;AP$
$\frac{1}{1+lnx}\;,\frac{1}{1+lny}\;,\frac{1}{1+lnz}\;are\;in\;HP.$ | {"extraction_info": {"found_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": 0.9917827248573303, "perplexity": 533.6175100513074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187825174.90/warc/CC-MAIN-20171022094207-20171022114207-00704.warc.gz"} |
https://proofwiki.org/wiki/Definition:Real_Cartesian_Space | # Definition:Cartesian Product/Cartesian Space/Real Cartesian Space
## Definition
Let $n \in \N_{>0}$.
Then $\R^n$ is the cartesian product defined as follows:
$\ds \R^n = \underbrace {\R \times \R \times \cdots \times \R}_{\text {$n$times} } = \prod_{k \mathop = 1}^n \R$
Similarly, $\R^n$ can be defined as the set of all real $n$-tuples:
$\R^n = \set {\tuple {x_1, x_2, \ldots, x_n}: x_1, x_2, \ldots, x_n \in \R}$
### Cartesian Plane
A general point $Q = \tuple {x, y}$ in the Cartesian plane
The Cartesian plane is a Cartesian coordinate system of $2$ dimensions.
Every point on the plane can be identified uniquely by means of an ordered pair of real coordinates $\tuple {x, y}$, as follows:
Identify one distinct point on the plane as the origin $O$.
Select a point $P$ on the plane different from $O$.
Construct an infinite straight line through $O$ and $P$ and call it the $x$-axis.
Identify the $x$-axis with the real number line such that:
$0$ is identified with the origin $O$
$1$ is identified with the point $P$
The orientation of the $x$-axis is determined by the relative positions of $O$ and $P$.
It is conventional to locate $P$ to the right of $O$, so as to arrange that:
to the right of the origin, the numbers on the $x$-axis are positive
to the left of the origin, the numbers on the $x$-axis are negative.
Construct an infinite straight line through $O$ perpendicular to the $x$-axis and call it the $y$-axis.
Identify the point $P'$ on the $y$-axis such that $OP' = OP$.
Identify the $y$-axis with the real number line such that:
$0$ is identified with the origin $O$
$1$ is identified with the point $P'$
The orientation of the $y$-axis is determined by the position of $P'$ relative to $O$.
It is conventional to locate $P'$ such that, if one were to imagine being positioned at $O$ and facing along the $x$-axis towards $P$, then $P'$ is on the left.
Hence with the conventional orientation of the $x$-axis as horizontal and increasing to the right:
going vertically "up" the page or screen from the origin, the numbers on the $y$-axis are positive
going vertically "down" the page or screen from the origin, the numbers on the $y$-axis are negative.
### Cartesian 3-Space
The Cartesian $3$-space is a Cartesian coordinate system of $3$ dimensions.
### Definition by Axes
A general point in Cartesian $3$-Space
Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates $\tuple {x, y, z}$, as follows:
Construct a Cartesian plane, with origin $O$ and axes identified as the $x$-axis and $y$-axis.
Recall the identification of the point $P$ with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.
Construct an infinite straight line through $O$ perpendicular to both the $x$-axis and the$y$-axis and call it the $z$-axis.
Identify the point $P''$ on the $z$-axis such that $OP'' = OP$.
Identify the $z$-axis with the real number line such that:
$0$ is identified with the origin $O$
$1$ is identified with the point $P$
### Definition by Planes
Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates $\tuple {x, y, z}$, as follows:
Identify one distinct point in space as the origin $O$.
Let $3$ distinct planes be constructed through $O$ such that all are perpendicular.
Each pair of these $3$ planes intersect in a straight line that passes through $O$.
Let $X$, $Y$ and $Z$ be points, other than $O$, one on each of these $3$ lines of intersection.
Then the lines $OX$, $OY$ and $OZ$ are named the $x$-axis, $y$-axis and $z$-axis respectively.
Select a point $P$ on the $x$-axis different from $O$.
Let $P$ be identified with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.
Identify the point $P'$ on the $y$-axis such that $OP' = OP$.
Identify the point $P''$ on the $z$-axis such that $OP'' = OP$.
The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$.
It is conventional to locate $P''$ as follows.
Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.
Then $P''$ is then one unit above the $x$-$y$ plane.
Let the $x$-$y$ plane be identified with the plane of the page or screen.
The orientation of the $z$-axis is then:
coming vertically "out of" the page or screen from the origin, the numbers on the $z$-axis are positive
going vertically "into" the page or screen from the origin, the numbers on the $z$-axis are negative.
### Countable-Dimensional Real Cartesian Space
The countable cartesian product defined as:
$\ds \R^\omega := \R \times \R \times \cdots = \prod_\N \R$
is called the countable-dimensional real cartesian space.
Thus, $\R^\omega$ can be defined as the set of all real sequences:
$\R^\omega = \set {\sequence {x_1, x_2, \ldots}: x_1, x_2, \ldots \in \R}$
## Also known as
The real cartesian space of order $n$ is sometimes seen as the (real) cartesian $n$-space.
Some sources call this euclidean $n$-space -- however, on $\mathsf{Pr} \infty \mathsf{fWiki}$ this term is reserved for the associated metric space.
## Also see
It can be shown that:
$\R^2$ is isomorphic to any infinite flat plane in space
$\R^3$ is isomorphic to the whole of space itself.
## Source of Name
This entry was named for René Descartes. | {"extraction_info": {"found_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": 0.9092826247215271, "perplexity": 298.5572661698467}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323587794.19/warc/CC-MAIN-20211026011138-20211026041138-00619.warc.gz"} |
http://imagenar.com/morefurniture-ledlkwx/213c9a-show-a-matrix-is-positive-definite | # show a matrix is positive definite
share | cite | improve this question | follow | edited Mar 30 '18 at 0:35. Positive-definite matrix; Positive-definite function; Positive-definite kernel; Positive-definite function on a group; References. Also, we will… Property 7: If A is a positive semidefinite matrix, then A ½ is a symmetric matrix and A = A ½ A ½. Property 8: Any covariance matrix is positive semidefinite. The following changes are made: I changed argument x to A to reflect usual matrix notation. Today, we are continuing to study the Positive Definite Matrix a little bit more in-depth. 29.8k 2 2 gold badges 82 82 silver badges 112 112 bronze badges. Eine solche Zerlegung wird als Cholesky-Zerlegung bezeichnet. With a positive definite matrix the usual algorithm succeeds because all the diagonal entries of L s.t. What are the practical ways to make a matrix positive definite? I will show that this matrix is non-negative definite (or "positive semi-definite" if you prefer) but it is not always positive definite. Therefore x T Mx = 0 which contradicts our assumption about M being positive definite. All three of these matrices have the property that is non-decreasing along the diagonals. To do this, consider an arbitrary non-zero column vector $\mathbf{z} \in \mathbb{R}^p - \{ \mathbf{0} \}$ and let $\mathbf{a} = \mathbf{Y} \mathbf{z} \in \mathbb{R}^n$ be the resulting column vector. This method does not require the matrix to be symmetric for a successful test (if the matrix is not symmetric, then the factorization fails). A matrix is positive definite if all it's associated eigenvalues are positive. That is, S is supposed to be positive definite in theory. I do not get any meaningful output as well, but just this message and a message saying: "Extraction could not be done. x: numeric n * n approximately positive definite matrix, typically an approximation to a correlation or covariance matrix. The set of positive matrices is a subset of all non-negative matrices. (a) A=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{ar… I'm implementing a spectral clustering algorithm and I have to ensure that a matrix (laplacian) is positive semi-definite. Jede positiv definite Matrix A läßt sich auch schreiben als A = LL t, wobei L eine untere Dreiecksmatrix mit positiven Diagonaleinträgen ist. Functions are adapted from Frederick Novomestky's matrixcalc package in order to implement the rmatnorm function. It is known that a positive definite matrix has a Unique Positive Definite square root. Does this situation show that there is something wrong with my algorithm since the likelihood should increase at every step of EM? Positive definite matrices are even bet ter. A positive matrix is a matrix in which all the elements are strictly greater than zero. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … One can show that a Hermitian matrix is positive definite if and only if all its eigenvalues are positive [].Thus the determinant of a positive definite matrix is positive, and a positive definite matrix is always invertible.The Cholesky decomposition provides an economical method for solving linear equations involving a positive definite matrix. – LaTeXFan Jul 27 '15 at 5:42 Positive definite symmetric matrices have the property that all their eigenvalues are positive. Theorem 4.2.3. 15.3.1.1 Space of Symmetric Positive Definite Matrices. If the factorization fails, then the matrix is not symmetric positive definite. Proof: if it was not, then there must be a non-zero vector x such that Mx = 0. Proof: Since a diagonal matrix is symmetric, we have. I want to run a factor analysis in SPSS for Windows. A check if the matrix is positive definite (PD) is enough, since the "semi-" part can be seen in the eigenvalues. If the covariance matrix is invertible then it is positive definite. the Pascal matrix. Note. Fasshauer, Gregory E. (2011), "Positive definite kernels: Past, present and future" (PDF), Dolomites Research Notes on Approximation, 4: 21–63. If the Hessian is positive-definite at x, then f attains an isolated local minimum at x.If the Hessian is negative-definite at x, then f attains an isolated local maximum at x. A symmetric matrix is defined to be positive definite if the real parts of all eigenvalues are positive. Edit: I'm computing the inverse by using a matrix inversion lemma which states that: $$(BB'+D)^{-1}=D^{-1}-D^{-1}B (I_q+B'D^{-1}B)^{-1} B'D^{-1}$$ and minus the second difference matrix, which is the tridiagonal matrix . If one subtracts one positive definite matrix from another, will the result still be positive definite, or not? Yixiao Yun, Irene Yu-Hua Gu, in Ambient Assisted Living and Enhanced Living Environments, 2017. asked Mar 29 '18 at 23:10. jack 看看 jack 看看. If x is not symmetric (and ensureSymmetry is not false), symmpart(x) is used.. corr: logical indicating if the matrix should be a correlation matrix. Examples of symmetric positive definite matrices, of which we display only the instances, are the Hilbert matrix. Still, for small matrices the difference in computation time between the methods is negligible to check whether a matrix is symmetric positive definite. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different. If a matrix has some special property (e.g. Symmetric matrices A symmetric matrix is one for which A = AT . Learn more about positive, definite, semipositive, chol, eig, eigenvalue MATLAB Ben Bolker. Eigenvalues of a positive definite real symmetric matrix are all positive. The page says " If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero. If A is a real symmetric positive definite matrix, then it defines an inner product on R^n. Then it's possible to show that λ>0 and thus MN has positive eigenvalues. However, it is not here. Conversely, some inner product yields a positive definite matrix. Also, if eigenvalues of real symmetric matrix are positive, it is positive definite. For the positive semi-definite case it remains true as an abstract proposition that a real symmetric (or complex Hermitian) matrix is positive semi-definite if and only if a Cholesky factorization exists. More specifically, we will learn how to determine if a matrix is positive definite or not. MIT Linear Algebra Exam problem and solution. Is it because of rounding error, please? The most efficient method to check whether a matrix is symmetric positive definite is to simply attempt to use chol on the matrix. The matrix is pretty big (nxn where n is in the order of some thousands) so eigenanalysis is expensive. [3]" Thus a matrix with a Cholesky decomposition does not imply the matrix is symmetric positive definite since it could just be semi-definite. A non-symmetric matrix (B) is positive definite if all eigenvalues of (B+B')/2 are positive… Suppose M and N two symmetric positive-definite matrices and λ ian eigenvalue of the product MN. Beispiel. positiv definit, wenn alle Hauptminoren > 0 sind und; negativ definit, wenn alle geraden Hauptminoren der Matrix > 0 und alle ungeraden Hauptminoren der Matrix < 0 sind. A matrix is positive-definite if its smallest eigenvalue is greater than zero. A positive definite matrix M is invertible. We prove a positive-definite symmetric matrix A is invertible, and its inverse is positive definite symmetric. Symmetric matrices and positive definiteness Symmetric matrices are good – their eigenvalues are real and each has a com plete set of orthonormal eigenvectors. A square matrix is positive definite if pre-multiplying and post-multiplying it by the same vector always gives a positive number as a result, independently of how we choose the vector.. This is calculated by sqrtm function. A way to check if matrix A is positive definite: A = [1 2 3;4 5 6;7 8 9]; % Example matrix Positive definite matrix. The Hessian matrix of a convex function is positive semi-definite.Refining this property allows us to test whether a critical point x is a local maximum, local minimum, or a saddle point, as follows: . All the eigenvalues with corresponding real eigenvectors of a positive definite matrix M are positive. From the same Wikipedia page, it seems like your statement is wrong. by Marco Taboga, PhD. 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Factorization fails, then there must be a non-zero vector x such that Mx = 0 orthonormal! | {"extraction_info": {"found_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": 0.9359570741653442, "perplexity": 667.7984734880247}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154897.82/warc/CC-MAIN-20210804174229-20210804204229-00060.warc.gz"} |
https://www.physicsforums.com/threads/how-much-intuition-is-needed.84972/ | # How much intuition is needed?
1. Aug 12, 2005
### quasi426
I was wondering if it at some point people get intuition on physics or do they always have to rely on the math to conclude things. How is this intuition gained, or is it just an innate ability?
2. Aug 12, 2005
### bomba923
It appears to be more of a psycho-bio-logical philosophical (philosophy of logic, I presume?) question. So far, I deem it an innate ability, as we observe+discover+raised w/experience of logic (or common sense as some call it)...? What do you think?
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https://www.physicsforums.com/threads/analysis-on-the-circular-motion.56770/ | # Analysis on the circular motion
1. Dec 14, 2004
### UrbanXrisis
I dont know what they are asking. Any tips?
Last edited by a moderator: May 1, 2017
2. Dec 14, 2004
### Staff: Mentor
Do you really think the child loses contact with the slide before even reaching the rounded section? Don't just guess.
Consider that in maintaining contact over the rounded section, the child must undergo circular motion. And that requires a centripetal force. (What force holds the child to the slide?) At some point, the child will be going too fast for the force to maintain the circular motion---off he goes.
3. Dec 14, 2004
### Tide
At some point on the circular section the child's speed will be such that mv^2/r exceeds the normal force holding him on the slide.
4. Dec 14, 2004
### Pyrrhus
When the normal force = 0 the boy loses contact with the surface.
So on your analysis on the circular motion
$$n - mg \cos \theta = -m \frac{v^2}{R}$$
Last edited: Dec 14, 2004
5. Dec 14, 2004
### UrbanXrisis
but how does that get me the height?
6. Dec 14, 2004
### Tide
You should be able to determine the speed of the child in terms of her height.
HINT: Energy is conserved!
7. Dec 14, 2004
### Pyrrhus
What i said above means when $v^2 = Rg \cos \theta$ it will be at the point it leaves the surface.
Hint: Use this fact and Tide's hints.
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https://www.techglads.com/cse/sem1/degree-of-polymerization/ | Select Page
# Degree of Polymerization
The number of repeating units (n) in a polymer chain is known as the degree of polymerization. It is represented by the following relationship.
Degree of polymerization (n) = Molecular weight of the polymeric network / Molecular weight of the repeating unit. (2m)
The degree of polymerization, or DP, is usually defined as the number of monomeric units in a macromolecule or polymer or oligomer molecule.
For a homopolymer, there is only one type of monomeric unit and the number-average degree of polymerization is given by $DP_n\equiv X_n=\frac{M_n}{M_0} - degree of polymerization$, where Mn is the number-average molecular weight and M0 is the molecular weight of the monomer unit. For most industrial purposes, degrees of polymerization in the thousands or tens of thousands are desired.
Some authors, however, define DP as the number of repeat units, where for copolymers the repeat unit may not be identical to the monomeric unit. For example, in nylon-6,6, the repeat unit contains the two monomeric units —NH(CH2)6NH— and —OC(CH2)4CO—, so that a chain of 1000 monomeric units corresponds to 500 repeat units. The degree of polymerization or chain length is then 1000 by the first (IUPAC) definition, but 500 by the second.
In step-growth polymerization, in order to achieve a high degree of polymerization (and hence molecular weight), Xn, a high fractional monomer conversion, p, is required, as per Carothers’ equation: Xn = 1/(1−p). A monomer conversion of p = 99% would be required to achieve Xn = 100. For chain-growth polymerization, however, this is not generally true and long chains are formed for much lower monomer conversions. | {"extraction_info": {"found_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": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8959276676177979, "perplexity": 2113.9155366618043}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337244.17/warc/CC-MAIN-20221002021540-20221002051540-00499.warc.gz"} |
http://math.stackexchange.com/questions/178122/prove-that-for-any-x-in-mathbb-n-such-that-xn-is-the-sum-of-at-most-n | # Prove that for any $x \in \mathbb N$ such that $x<n!$ is the sum of at most $n$ distinct divisors of $n!$
Prove that for any $x \in \mathbb N$ such that $x<n!$ is the sum of at most $n$ distinct divisors of $n!$.
-
What have you tried? – Matthew Conroy Aug 2 '12 at 18:07
If only Goldbach conjecture was true. – Jayesh Badwaik Feb 20 '13 at 2:55
Let $x = nq+r$, with $0 \leq r < n$. Note that $x < n!$ implies that $q < (n-1)!$. Now use induction on $q$.
-
Works for me. Note that, as the initial case is 2! rather than 1!, this proves that $(n-1)$ divisors suffice. – Will Jagy Feb 20 '13 at 3:34
Great, may I ask if you had seen a similar problem before or how you figured it out. Thanks! – Jorge Fernández Feb 20 '13 at 3:56
@Khromonkey, I haven't seen this before. Induction seems to be a natural idea though. I guess that Robert Israel's attempt helped inspire this solution too. – user27126 Feb 20 '13 at 4:04
No, this doesn't work, either. Let $n=5, x=119$, then $q=23$ and $115$ doesn't divide $120$ – Ross Millikan Feb 20 '13 at 13:42
@Ross Millikan, $q = 23 < 4!$, and you want to cut 23 up into sum of divisors of 4! first. For example, 23 = 12 + 8 + 3, and 119 = 60 + 40 + 15 + 4. – user27126 Feb 20 '13 at 16:33
People do not seem to be going along with my comment. So this is CW, and directly from the answer by Sanchez.
For $n=2,$ we need only 1 divisor of $2!,$ as $1=1.$
For $n=3,$ we need only 2 divisors of $3!=6,$ as $1=1, 2=2,3=3,4=3+1,5=3+2.$
Induction hypothesis: for some $n \geq 2,$ we need at most $(n-1)$ distinct divisors of $n!$ to write any $1 \leq x < n!$ as a sum.
Induction step (Sanchez, above). Let $N = n+1.$ Let $1 \leq x < N! = (n+1)!$ Write $$x = N q + r, \; \; \mbox{with} \; \; 0 \leq r < N.$$ Because $q < (N-1)! = n!,$ we need at most $(n-1) = (N-2)$ divisors of $n!$ to write $q$ as a sum. So $$q = \sum_{i=1}^{n-1} d_i,$$ where each $d_i | n!$ Therefore each $Nd_i | N!$
At this stage, we have at most $N-2$ divisors of $N!$ What about $r?$ Well, $r < N,$ so it is automatically a factor of $N!$ So we have finished the decomposition as a sum with at most $(N-1)$ divisors of $N!,$ where $N=n+1.$
CONCLUSION: For all $N \geq 2,$ every integer $1 \leq x < N!$ can be written as the sum of (at most) $N-1$ distinct divisors of $N!$
SUGGESTION: try it for $N=4, \; \; N! = 24.$
NEVER MIND, do it myself. Aliquot divisors 1,2,3,4,6,8,12. $$1=1,2=2,3=3,4=4,5=4+1,6=4+2,7=4+3, 8=8,9=8+1,10=8+2,$$ $$11=8+3, 12= 12, 13 = 12+1, 14 = 12+2, 15 = 12+3, 16 = 12+4,$$ $$17=12+4+1, 18=12+6,19=12+4+3,20=12+8,$$ $$21=12+8+1,22=12+8+2,23 = 12+8+3.$$
-
Hint: Note that $x = m (n-1)! + r$ where $0 \le m < n$ and $0 \le r < (n-1)!$. Use induction.
EDIT: Oops, this is wrong: as Steven Stadnicki noted, $m (n-1)!$ doesn't necessarily divide $n!$.
-
$m(n-1)!$ isn't necessarily a divisor of $n!$, is it? – Steven Stadnicki Aug 2 '12 at 19:04
@RobertIsrael Then what is the correct answer? – Jorge Fernández Sep 22 '12 at 16:55
I don't know. I believe it is true, and probably not too hard to prove, that for any $x \in \{1,\ldots,n!\}$ there is always a divisor $y$ of $n!$ with $x/2 \le y \le x$. Then we can take a greedy approach: let $y_1$ be the greatest divisor of $n!$ less than $x$, and replace $x$ by $x - y_1$. This will result in writing $x$ as a sum of distinct divisors of $n!$, but it's not clear to me that there will be at most $n$ of them: the easy bound would be about $\log_2(n!) \approx n \log_2(n)$. – Robert Israel Sep 23 '12 at 18:07
Can you edit or delete your answer please?? If it is wrong why is it here?? – Jorge Fernández Feb 19 '13 at 22:55
@Khromonkey, it is probably here because it is a reasonable direction to take, yet involves an error that has been identified. As such, it is educational for others. Meanwhile, noting that you say you are a high-school student, I cannot see how this is an ordinary homework problem, so I am asking why you think your assertion is true and where you got the problem. – Will Jagy Feb 19 '13 at 23:35
Not quite there, but a start: As suggested in Wikipedia on practical numbers we will use the greedy algorithm. First pull out $n!/2$ if that is possible, then $n!/3$, then $n!/4$ and so on, stopping when the remainder is less than or equal to $n$ and skipping denominators that don't divide $n!$. If $n$ and $x$ are very large, the denominators we use will follow Sylvester's sequence: $2, 3, 7, 43, 1807, 3263443, 10650056950807,\ldots$ which is given by $a(0)=2, a(n+1)=a(n)^2-a(n)+1$. To use induction, we need to find a sequence of $m$ denominators that reduce $n!-1$ to something less than $n$. For $n$ in the range $5-6$ we can use $2,3,8,30$. For $7$ we can use $2,3,7,45$, for $8-10$ we can use $2,3,7,45,640$. Then $44$ becomes available at $11$. It "obviously" works, but I can't prove it.
-
A natural number $m$ is called practical if all smaller natural numbers can be represented as sum of distinct divisors of $m$.
The problem asks to establish that factorial numbers are practical. The wikipedia article on practical numbers even gives an algorithm, implemented in Mathematica:
dec2[0, n_] := {};
dec2[1, n_] := {1};
dec2[x_, n_] := Module[{fcts, pa, q, r, quo},
fcts = Last[FactorInteger[n]]; pa = Power @@ fcts;
q = Min[Quotient[x, pa], DivisorSigma[1, quo = Quotient[n, pa]]];
Join[dec2[x - q pa, Quotient[n, First[fcts]] ], dec2[q, quo] pa]
]
dec[x_, n_] := Block[{$RecursionLimit = Infinity}, dec2[x, n!]] Example: In[32]:= dec[17, 4] Out[32]= {2, 3, 12} In[33]:= dec[137, 6] Out[33]= {2, 45, 90} It remains to be proven that the decomposition length of$x < n!$will be less of equal than$n$. - That is the question. Can you make this answer answer it? – Jorge Fernández Feb 19 '13 at 22:54 is very simple logic that, n! will have only atmost 2n distinct divisors and they are n!,(n!/2),(n!/3)...(n!/n) and1,2,..n your question is wrong because take 8! as case take x as sum of 7 distinct divisors say 1,2,3,4,5,6,7=28 but still x less than n! wich contradicts x can be only sum of atmost 6 distinct divisors of n! so that x could be less than n!. - Refuting the claim of$n!$having only at most$2n$distinct divisors:$3!$has 4 divisors,$4!$has 8,$5!$has 16 and$6!\$ has 30. – Sasha Feb 23 '13 at 17:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.936150312423706, "perplexity": 323.60772604721694}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830903.34/warc/CC-MAIN-20140820021350-00283-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/find-time-constant-rise-time-settling-time-system-find-poles-zeros-following-systems-plot--q4323248 | below
Image text transcribed for accessibility: Find the time constant, rise time, and the settling time of the system Find the poles and zeros of the following systems, plot them on the 5-plane. State the types of the system (overdamped, underdamped, and so on). H(s) = 5/(s+3)(s+6) H(s) = 20/s2+6s+144 H(s) = s+5/(s+10)2 For each of the second-order systems that follow, find zeta, wn, Ts, Tp. Tr, and %OS. H(s) = 16/s2+3s+16 H(s) = 0.04/s2+0.02s+0.04 For each pair of second-order system specifications that follow, find the location of the second-order pair of poles. %OS = 12; Ts = 0.6second %OS = 10; Tp = 5seconds Ts = 7seconds; Tp = 3seconds Find the transfer function of a second-order system that yields a 12.3% overshoot and a settling time of 1 second. For the system shown below, do the following: Find the transfer function G(s) = IL(s)/V(s). Find zeta, Wn, %OS, Ts, Tp, and Tr. Find the percent overshoot, settling time, rise time, and peak time for H(s)=14.145/(s2+0.842s+2.829)(s+5) | {"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": 0.8511210083961487, "perplexity": 3059.370703766805}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1424936462577.56/warc/CC-MAIN-20150226074102-00324-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://web.ma.utexas.edu/mp_arc-bin/mpa?yn=11-124 | 11-124 Pavel Exner and Diana Barseghyan
Spectral estimates for a class of Schr\"odinger operators with infinite phase space and potential unbounded from below (290K, LaTeX) Sep 1, 11
Abstract , Paper (src), View paper (auto. generated pdf), Index of related papers
Abstract. We analyze two-dimensional Schr\"odinger operators with the potential $|xy|^p - \lambda (x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $\lambda\ge 0$. We show that there is a critical value of $\lambda$ such that the spectrum for $\lambda<\lambda_\mathrm{crit}$ is below bounded and purely discrete, while for $\lambda>\lambda_\mathrm{crit}$ it is unbounded from below. In the subcritical case we prove upper and lower bounds for the eigenvalue sums.
Files: 11-124.src( 11-124.keywords , estimate110901.pdf.mm ) | {"extraction_info": {"found_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": 0.9874281883239746, "perplexity": 862.9469382176478}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738699.68/warc/CC-MAIN-20200810205824-20200810235824-00241.warc.gz"} |
https://www.jobilize.com/physics12/course/15-1-photoelectric-effect-optical-phenomena-and-properties-of-by-opens?qcr=www.quizover.com&page=1 | # 15.1 Photoelectric effect (Page 2/2)
Page 2 / 2
While solving problems we need to decide for ourselves whether we should consider the wave property or the particle property of light. For example, when dealing with interference and diffraction, light should be treated as a wave, whereas when dealing with photoelectric effect we consider the particle nature.
## Applications of the photoelectric effect
We have learnt that a metal contains electrons that are free to move between the valence and conduction bands. When a photon strikes the surface of a metal, it gives all its energy to one electron in the metal.
• If the photon energy is equal to the energy between two energy levels then the electron is excited to the higher energy level.
• If the photon energy is greater than or equal to the work function (energy needed to escape from the metal), then the electron is emitted from the surface of the metal (the photoelectric effect).
The work function is different for different elements. The smaller the work function, the easier it is for electrons to be emitted from the metal. Metals with low work functions make good conductors. This is because the electrons are attached less strongly to their surroundings and can move more easily through these materials. This reduces the resistance of the material to the flow of current i.e. it conducts well. [link] shows the work functions for a range of elements.
Element Work Function ( $\mathrm{J}$ ) Aluminium $6,9×{10}^{-19}$ Beryllium $8,0×{10}^{-19}$ Calcium $4,6×{10}^{-19}$ Copper $7,5×{10}^{-19}$ Gold $8,2×{10}^{-19}$ Lead $6,9×{10}^{-19}$ Silicon $1,8×{10}^{-19}$ Silver $6,9×{10}^{-19}$ Sodium $3,7×{10}^{-19}$
## Interesting fact
The electron volt (eV) is the kinetic energy gained by an electron passing through a potential difference of one volt (1 $\mathrm{V}$ ). A volt is not a measure of energy, but the electron volt is a unit of energy. When you connect a $1.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ battery to a circuit, you can give $1.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{eV}$ of energy to every electron.
Ultraviolet radiation with a wavelength of 250 nm is incident on a silver foil (work function $\phi$ = $6,9×{10}^{-19}$ ). What is the maximum kinetic energy of the emitted electrons?
1. We need to determine the maximum kinetic energy of an electron ejected from a silver foil by ultraviolet radiation.
The photoelectric effect tells us that:
$\begin{array}{ccc}\hfill {E}_{k}& =& {E}_{photon}-\phi \hfill \\ \hfill {E}_{k}& =& h\frac{c}{\lambda }-\phi \hfill \end{array}$
We also have:
Work function of silver: $\phi =6,9×{10}^{-19}$ J
UV radiation wavelength = 250 nm = $250×{10}^{-9}$ m
Planck's constant: $h=6,63×{10}^{-34}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}\mathrm{kg}{\mathrm{s}}^{-1}$
speed of light: $c=3×{10}^{8}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}{\mathrm{s}}^{-1}$
2. $\begin{array}{ccc}\hfill {E}_{k}& =& \frac{hc}{\lambda }-\phi \hfill \\ & =& \left[6,63×{10}^{-34}×\frac{3×{10}^{8}}{250×{10}^{-9}}\right]-6,9×{10}^{-19}\hfill \\ & =& 1,06×{10}^{-19}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}\hfill \end{array}$
The maximum kinetic energy of the emitted electron will be $1,06×{10}^{-19}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}$ .
If we were to shine the same ultraviolet radiation ( $f=1,2×{10}^{15}\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$ ), on a gold foil (work function $=8,2×{10}^{-19}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}$ ), would any electrons be emitted from the surface of the gold foil?
For the electrons to be emitted from the surface, the energy of each photon needs to be greater than the work function of the material.
1. $\begin{array}{ccc}\hfill {E}_{photon}& =& hf\hfill \\ & =& 6,63×{10}^{-34}×1,2×{10}^{15}\hfill \\ & =& 7,96×{10}^{-19}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}\hfill \end{array}$
Therefore each photon of ultraviolet light has an energy of $7,96×{10}^{-19}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}$ .
2. $\begin{array}{ccc}\hfill {\phi }_{gold}& =& 8,2×{10}^{-19}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}\hfill \end{array}$
3. $\begin{array}{ccc}\hfill 7,96×{10}^{-19}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}& <& 8,2×{10}^{-19}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}\hfill \\ \hfill {E}_{photons}& <& {\phi }_{gold}\hfill \end{array}$
The energy of each photon is less than the work function of gold, therefore, the photons do not have enough energy to knock electrons out of the gold. No electrons would be emitted from the gold foil.
## Units of energy
When dealing with calculations at a small scale (like at the level of electrons) it is more convenient to use different units for energy rather than the joule (J). We define a unit called the electron-volt (eV) as the kinetic energy gained by an electron passing through a potential difference of one volt.
$E=q×V$
where $q$ is the charge of the electron and $V$ is the potential difference applied. The charge of 1 electron is $1,6×{10}^{-19}$ C, so 1 eV is calculated to be:
$1\phantom{\rule{0.277778em}{0ex}}\mathrm{eV}=\left(1,{610}^{-19}\phantom{\rule{0.277778em}{0ex}}\mathrm{C}×1\phantom{\rule{0.277778em}{0ex}}\mathrm{V}\right)=1,6×{10}^{-19}\phantom{\rule{0.166667em}{0ex}}\mathrm{J}$
You can see that $1,6×{10}^{-19}$ J is a very small amount of energy and so using electron-volts (eV) at this level is easier.
Hence, 1eV = $1.6×{10}^{-}19$ J which means that 1 J = $6.241×{10}^{18}$ eV
## Solar cells
The photo-electric effect may seem like a very easy way to produce electricity from the sun. This is why people choose to make solar panels out of materials like silicon, to generate electricity.In real-life however, the amount of electricity generated is less than expected. This is because not every photon knocks out an electron. Other processes such as reflection or scattering also happen. This means that only a fraction $\approx 10%$ (depends on the material) of the photons produce photoelectrons. This drop in efficiency results in a lower current. Much work is being done in industry to improve this efficiency so that the panels can generate as high a current as possible, and create as much electricity as possible from the sun. But even these smaller electrical currents are useful in applications like solar-powered calculators.
## The photoelectric effect
1. Describe the photoelectric effect.
2. List two reasons why the observation of the photoelectric effect was significant.
3. Refer to [link] : If I shine ultraviolet light with a wavelength of 288 nm onto some aluminium foil, what would be the kinetic energy of the emitted electrons?
4. I shine a light of an unknown wavelength onto some silver foil. The light has only enough energy to eject electrons from the silver foil but not enough to give them kinetic energy. (Refer to [link] when answering the questions below:)
1. If I shine the same light onto some copper foil, would electrons be ejected?
2. If I shine the same light onto some silicon, would electrons be ejected?
3. If I increase the intensity of the light shining on the silver foil, what happens?
4. If I increase the frequency of the light shining on the silver foil, what happens?
what makes an atom to take part in a chemical reaction
what is an element
Mwila
what are molecules
what are molecules
A group of two atoms tha the form smallest identifiable unit into which a pure substance.
El
what is matter
it's a topic in physics
Thabo
Anything that occupies space and has mass
Lisoga
what is the catalyst used in esterification reaction
I think is sulfuric acid
Selby
what is equilibrium
It is a condition of a system when neither its state of motion nor its internal energy state tends to change with time. ... For a single particle, equilibrium arises if the vector sum of all forces acting upon the particle is zero.
susan
it is a dynamic change when the forward reaction is equals to the reverse reaction rate
Frank
what does organic mean
it means "from living thing"
Maqaqatu
What is a successful collision?
It's a collision that results to a reaction
Tshepo
what is the factor and reaction rate of 20%HCL and 5%HCL
concentration and 0.05M/s
Tokozan
how you reaching with apartment depends on what each of you have
how with 4looseyellowprotons in your left hand stars are mostly hydrogen so this is your fuel you will also need one six so did die
Edelita
besides the concentration what are the two conditions for the hydrogen half cell to function under standard conditions
forces of non-concervative
What is the following process is likely to involve carbon burning?
Combustion
Luyanda
Which Statement is part of the cell theory?
What is the second formal statement in the cell theory
Hananiya
🤔
Bontle
what is meant by ohmic
when someone is referred to as ohmic, it basically means that it obeys ohm's law
Tanaka
*something
Tanaka | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 37, "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": 0.8642367124557495, "perplexity": 577.9295801392594}, "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-2021-04/segments/1610703524270.28/warc/CC-MAIN-20210121070324-20210121100324-00301.warc.gz"} |
https://www.physicsforums.com/threads/divide-circle-into-9-areas.236769/ | # Divide circle into 9 areas
1. May 23, 2008
### Nick89
Hi,
I was asked this question on another forum and was interested in it... It's somewhat related to what I have been doing lately so I gave it a (few) tries, but I never really worked it out...
Consider a circle with a radius of 32 units. We want to divide the area of the circle into 9 areas that have, if possible, exactly the same area. See the following image:
The red lines are the 'dividing lines', spaced by a distance $d$ (in the x aswell as the y direction).
The areas 1 (blue) and 2 (green) and the area 3 (red) are marked with the colors. Note that there are four areas 1 and four areas 2, they should be equal in area.
The question is how to find the distance $d$ that will yield the optimal result (if possible, that all areas are equal).
The first thing I thought about (but which doesn't seem to be working, see later) is simply to do the following:
We know the area of the complete circle: $$A_{tot} = \pi 32^2$$
Therefore, if the 9 areas are to be divided in equal areas, the area of one the subareas will be: $$A_{sub} = \frac{ \pi 32^2}{9}$$
We also know the area $A_3$ since it's just a square: $$A_3 = d^2$$
Therefore: $$d = \sqrt{ \frac{ \pi 32^2}{9}}$$.
I tried to graph it and it seemed alright to the eye, but I wanted to be sure, so I went on...
The following way I could think of was to calculate the subareas seperately using integrals and then looking for a $d$ that would minimize their deviation.
I came up with the following area's; $A_1$ is calculated from the top-right area1 and $A_2$ is calculated from the rightmost area2.
$$A_1 = \int_\frac{d}{2}^b \left( \sqrt{ 1024-x^2} - \frac{d}{2} \right) \, dx$$
$$A_2 = 2 \times \left( \int_b^{32} \sqrt{1024-x^2} \, dx \right) + d \sqrt{1024-\frac{d^2}{4}}$$
$$A_3 = d^2$$
where the limit b is the intersection of the circle with y = d/2:
$$b = \sqrt{1024-\frac{d^2}{4}}$$
When I now plugged in the value for $d$ I found above I don't get the same result, I get a different result for each area...
So, I thought, maybe my simple solution above wasn't right.
But now I have found three areas each as a function of d. I should be able to minimize the deviation between the areas for one value of d, right? I can't see any way how to do that though... Maybe taking the absolute value of the deviation (A_1 - A_2 for example) and using solving it's derivative for 0? Even then I only minimized A_1 - A_2 and had nothing to do with A_3...
Where have I gone wrong:
1) Assuming there is a solution where all areas are equal;
2) Assuming this solution was simply to divide the total area by 9 and equaling this to d^2;
3) Calculating the areas using integrals?
I can't see any other mistakes I may have made, so I assume it must be one of the three...
Could anyone help me out here?
2. May 23, 2008
### CRGreathouse
I would calculate the areas (with integrals as needed) as a function of d, then try to minimize
$$4(A_1-A)^2+4(A_2-A)^2+(A_3-A)^2$$ with $$A=1024\pi/9$$
Similar Discussions: Divide circle into 9 areas | {"extraction_info": {"found_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": 0.8644284009933472, "perplexity": 348.6469853087971}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1498128319933.33/warc/CC-MAIN-20170622234435-20170623014435-00538.warc.gz"} |
https://math.stackexchange.com/questions/1996220/finding-abelian-extensions-of-mathbbq-p-using-local-class-field-theory | # Finding abelian extensions of $\mathbb{Q}_{p}$ (using Local Class Field Theory)
Here all the extensions are finite.
In local class field theory, one learns that the abelian extensions $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$, where $\zeta$ is a $p^{n}$-th root of unity, are in bijective correspondence with the subgroups $(p)\times (1+p^{n}\mathbb{Z}_{p})$ of the multiplicative group $\mathbb{Q}_{p}^{\times}$.
Do we know similar descriptions for other 'types' of extensions of $\mathbb{Q}_{p}$?
For example, we can play around and wonder what kind of abelian extensions $L_{n}/\mathbb{Q}_{p}$ correspond to the subgroups $$(p^{2})\times (1+p^{n}\mathbb{Z}_{p})$$ and we can ask this for many other subgroups. Also, I wonder if we can detect ramification looking at the these subgroups.
For example, it is known that the $p^{n}$-th cyclotomic extensions I mentioned above are totally ramified, and just so happens that the "$(p)$"-part of the corresponding subgroup has exponent $1$.
I did this for a couple of examples, and it looks like an extension corresponding to some subgroup of the form $$(p^{f})\times H,$$ for some subgroup $H$ of $\mathbb{Z}_{p}^{\times}$, has inertia degree $f$. Is this true in general? Thanks a lot.
• First of all--you should put FINITE abelian extension. Also, I have no idea how you got the original claim. Local class field theory says that the finite abelian extensions of $\mathbb{Q}_p(\zeta_{p^n})$ correspond to the finite index subgroups of $\mathbb{Q}_p(\zeta_{p^n})^\times$ which is $(1+\zeta_{p^n})^\mathbb{Z}\times \mathbb{Z}_p[\zeta_{p^n}]^\times$ and the latter is isomorphic to $(\Z/p^n\Z)\times (\Z/(p-1)\Z)\times \mathbb{Z}_p^{p^{n-1}(p-1)}$. Am I missing something? PS, your last question should be answered, I assume, by the fact that the upper numbering filtration – Alex Youcis Nov 5 '16 at 10:32
• on inertia maps to the usual Lie filtration on the units of your valuation ring. – Alex Youcis Nov 5 '16 at 10:33
• Hey, thanks for commenting. Don't know if I understand, though. I'm taking $\mathbb{Q}_{p}^{\times}$ as the base field and applying local CFT, so finite abelian extensions correspond to closed finite index subgroups of $\mathbb{Q}_{p}^{\times}$. In Neukirch's book ANT, there is a proposition (1.8; page 323) saying that these $p^{n}$-th cyclotomic extensions of $\mathbb{Q}_{p}$ correspond to the subgroups of the form $(p)\times U^{(n)}$. Also, I think your final sentence may be missing a word. The fact that what? – Shoutre Nov 5 '16 at 15:30
• Did you see my second comment? But you're not alking about finite abelian ecxtensions of $\mathbb{Q}_p$, right? Finite abelian extensions of $K$ correspond to finite index subgroups of $K^\times$. Why then for $K=\mathbb{Q}_p(\zeta_{p^n})$ are you taking $\mathbb{Q}_p^\times$ and not $K^\times$? – Alex Youcis Nov 5 '16 at 21:52
• OH, I misread a single word--I thought you said "abelian extensions OF $\mathbb{Q}_p(\zeta)/\mathbb{Q}_p$". OK, now I understand--sorry. Anyways, for any finite index subgroup of $\mathbb{Q}_p^\times$ you can always explicitly show it contains one of the form $(p)\times(1+p^n\mathbb{Z}_p)$. If you can then find that the index between your finite index subgroup and this group is, say, $G$ then your extension is just $\mathbb{Q}_p(\zeta_{p^n})^G$. Also, I still think my second comment about the inertia filtration answers your second question--does it make sense now? – Alex Youcis Nov 5 '16 at 21:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.8980733752250671, "perplexity": 262.6288030429429}, "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-2019-26/segments/1560627998690.87/warc/CC-MAIN-20190618063322-20190618085322-00070.warc.gz"} |
https://math.gatech.edu/seminars-and-colloquia-by-series?series_tid=68&page=12 | ## Seminars and Colloquia by Series
### Empirical likelihood and Extremes
Series
Dissertation Defense
Time
Wednesday, November 16, 2011 - 15:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 171
Speaker
Yun GongSchool of Mathematics, Georgia Tech
In 1988, Owen introduced empirical likelihood as a nonparametric method for constructing confidence intervals and regions. It is well known that empirical likelihood has several attractive advantages comparing to its competitors such as bootstrap: determining the shape of confidence regions automatically; straightforwardly incorporating side information expressed through constraints; being Bartlett correctable. In this talk, I will discuss some extensions of the empirical likelihood method to several interesting and important statistical inference situations including: the smoothed jackknife empirical likelihood method for the receiver operating characteristic (ROC) curve, the smoothed empirical likelihood method for the conditional Value-at-Risk with the volatility model being an ARCH/GARCH model and a nonparametric regression respectively. Then, I will propose a method for testing nested stochastic models with discrete and dependent observations.
### Hybrid Modeling and Analysis of Multiscale Biochemical Reaction Networks
Series
Dissertation Defense
Time
Monday, October 24, 2011 - 13:30 for 1 hour (actually 50 minutes)
Location
Klaus Conference Room 1212
Speaker
Jialiang WuSchool of Mathematics, Georgia Tech
### Two Problems in Mathematical Physics: Villani's Conjecture and a Trace Inequality for the Fractional Laplacian
Series
Dissertation Defense
Time
Monday, August 29, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Amit EinavSchool of Mathematics, Georgia Tech
The presented work deals with two distinct problems in the field of Mathematical Physics, and as such will have two parts addressing each problem. The first part is dedicated to an 'almost' solution of Villani's conjecture, a known conjecture related to a Statistical Mechanics model invented by Kac in 1956, giving a rigorous explanation of some simple cases of the Boltzman equation. In 2003 Villani conjectured that the time it will take the system of particles in Kac's model to equalibriate is proportional to the number of particles in the system. Our main result in this part is an 'almost proof' of that conjecture, showing that for all practical purposes we can consider it to be true. The second part of the presentation is dedicated to a newly developed trace inequality for the fractional Laplacian, connecting between the fractional Laplacian of a function and its restriction to the intersection of the hyperplanes x_n =...= x_n-j+1 = 0 , where 1 <= j < n. The newly found inequality is sharp and the functions that attain inequality in it are completely classified.
### Topics in Spatial and Dynamical Phase Transitions of Interacting Particle Systems
Series
Dissertation Defense
Time
Monday, August 15, 2011 - 11:00 for 2 hours
Location
Skiles 005
Speaker
Ricardo Restrepo LopezSchool of Mathematics, Georgia Tech
In this work we provide several improvements in the study of phase transitions of interacting particle systems: 1. We determine a quantitative relation between non-extremality of the limiting Gibbs measure of a tree-based spin system, and the temporal mixing of the Glauber Dynamics over its finite projections. We define the concept of sensitivity' of a reconstruction scheme to establish such a relation. In particular, we focus in the independent sets model, determining a phase transition for the mixing time of the Glauber dynamics at the same location of the extremality threshold of the simple invariant Gibbs version of the model. 2. We develop the technical analysis of the so-called spatial mixing conditions for interacting particle systems to account for the connectivity structure of the underlying graph. This analysis leads to improvements regarding the location of the uniqueness/non-uniqueness phase transition for the independent sets model over amenable graphs; among them, the elusive hard-square model in lattice statistics, which has received attention since Baxter's solution of the analogue hard-hexagon in 1980. 3. We build on the work of Montanari and Gerschenfeld to determine the existence of correlations for the coloring model in sparse random graphs. In particular, we prove that correlations exist above the clustering' threshold of such model; thus providing further evidence for the conjectural algorithmic `hardness' occurring at such point.
### Normally Elliptic Singular Perturbation Problems: Local Invariant Manifolds and Applications
Series
Dissertation Defense
Time
Monday, May 16, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nan LuSchool of Mathematics, Georgia Tech
We study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be nonautonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small singular parameters. We also use our results on local invariant manifolds to study the persistence of homoclinic solutions under weakly dissipative and conservative perturbations.
### Judicious Partitions of Graphs and Hypergraphs
Series
Dissertation Defense
Time
Tuesday, April 26, 2011 - 12:30 for 2 hours
Location
Skiles 005
Speaker
Jie MaSchool of Mathematics, Georgia Tech
Classical partitioning problems, like the Max-Cut problem, ask for partitions that optimize one quantity, which are important to such fields as VLSI design, combinatorial optimization, and computer science. Judicious partitioning problems on graphs or hypergraphs ask for partitions that optimize several quantities simultaneously. In this dissertation, we work on judicious partitions of graphs and hypergraphs, and solve or asymptotically solve several open problems of Bollobas and Scott on judicious partitions, using the probabilistic method and extremal techniques.
### Hardy-Sobolev-Maz'ya Inequalities for Fractional Integrals on Halfspaces and Convex Domains
Series
Dissertation Defense
Time
Tuesday, April 19, 2011 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Craig A. SloaneSchool of Mathematics, Georgia Tech
Classical Hardy, Sobolev, and Hardy-Sobolev-Maz'ya inequalities are well known results that have been studied for awhile. In recent years, these results have been been generalized to fractional integrals. This Dissertation proves a new Hardy inequality on general domains, an improved Hardy inequality on bounded convex domains, and that the sharp constant for any convex domain is the same as that known for the upper halfspace. We also prove, using a new type of rearrangement on the upper halfspace, based in part on Carlen and Loss' concept of competing symmetries, the existence of the fractional Hardy-Sobolev-Maz'ya inequality in the case p = 2, as well as proving the existence of minimizers, at least in limited cases.
### Isospectral Graph Reductions, Estimates of Matrices' Spectra, and Eventually Negative Schwarzian Systems
Series
Dissertation Defense
Time
Tuesday, March 8, 2011 - 09:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin WebbSchool of Mathematics, Georgia Tech
Real world networks typically consist of a large number of dynamical units with a complicated structure of interactions. Until recently such networks were most often studied independently as either graphs or as coupled dynamical systems. To integrate these two approaches we introduce the concept of an isospectral graph transformation which allows one to modify the network at the level of a graph while maintaining the eigenvalues of its adjacency matrix. This theory can then be used to rewire dynamical networks, considered as dynamical systems, in order to gain improved estimates for whether the network has a unique global attractor. Moreover, this theory leads to improved eigenvalue estimates of Gershgorin-type. Lastly, we will discuss the use of Schwarzian derivatives in the theory of 1-d dynamical systems.
### Sharp Weighted Estimates for Singular Integrals
Series
Dissertation Defense
Time
Tuesday, March 1, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Maria del Carmen Reguera RodriquezSchool of Mathematics, Georgia Tech
### Scaling limit for the diffusion exit problem
Series
Dissertation Defense
Time
Thursday, February 3, 2011 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sergio Angel AlmadaSchool of Mathematics, Georgia Tech
A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (nonstandard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks. | {"extraction_info": {"found_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": 0.8544783592224121, "perplexity": 769.7123540999703}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335609.53/warc/CC-MAIN-20221001101652-20221001131652-00065.warc.gz"} |
https://learncheme.com/quiz-yourself/interactive-self-study-modules/gibbs-free-energy-and-phase-separation/gibbs-free-energy-and-phase-separation-conceptest-and-example-problem/ | #### Gibbs Free Energy and Phase Separation: ConcepTest and Example Problem
Try to answer this ConcepTest and solve the example problem before using this module. Studies show that trying to answer the questions before studying material improves learning and retention. We suggest that you write down the reasons for your answers. By the end of this module, you should be able to answer these on your own. Answers will be given at the end of this module.
Calculate $$\Delta G$$ for a binary liquid mixture at 75°C that has a mole fraction of 0.30 for component 1 (x1). The vapor in equilibrium with the liquid has a pressure of 0.31 bar and a mole fraction of 0.65 for component 1 (y1). The saturation pressures are: P1sat = 0.33 bar; P2sat = 0.10 bar. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8670763969421387, "perplexity": 656.1926942650847}, "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-2023-14/segments/1679296948684.19/warc/CC-MAIN-20230327185741-20230327215741-00776.warc.gz"} |
https://technet.microsoft.com/en-us/library/ee156816 | Using the Convert-Path Cmdlet
Translating a Windows PowerShell Path
The Convert-Path cmdlet converts a Windows PowerShell path to a system path. What does that mean, and does it even matter? Well, suppose you’ve created a new Windows PowerShell drive (drive X). That drive letter is valid only in Windows PowerShell; you can’t switch to Windows Explorer and access drive X. Convert-Path, however, can tell you the “real” path of drive X. In other words, if you run this command:
```
Convert-Path x:
```
You’ll get back information similar to this:
```
C:\Scripts
```
This is particularly useful with registry drives. Windows PowerShell has its own syntax for indicating registry paths; for example, to use the Set-Location cmdlet to switch to the registry you would use a command similar to this:
```
Set-Location hkcu:\software\microsoft\windows
```
That’s fine, but outside of Windows PowerShell a path like hkcu:\software\microsoft\windows is meaningless. If you need to reference that path outside of Windows PowerShell, then just ask Convert-Path to lend you a hand:
```
Convert-Path hkcu:\software\microsoft\windows
```
That command gives you the actual path within the registry:
```
HKEY_CURRENT_USER\software\microsoft\windows
```
Convert-Path Aliases
• cvpa | {"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": 0.9530324339866638, "perplexity": 4274.737825499983}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948512584.10/warc/CC-MAIN-20171211071340-20171211091340-00529.warc.gz"} |
https://www.coursehero.com/file/p2q0e5a/2-eV-1-6-10-19-J-1-keV-1000-eV-48-082-keV-008-100points-A-thin-horizontal-copper/ | # 2 ev 1 6 10 19 j 1 kev 1000 ev 48 082 kev 008
• Test Prep
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• 100% (20) 20 out of 20 people found this document helpful
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2 · eV 1 . 6 × 10 19 J × 1 keV 1000 eV = 48 . 082 keV . 009 10.0points Find the magnitude of the emf induced around the loop in the figure. The 0 . 638 T uniform magnetic field is directed into the plane of the circuit and the 30 . 6 cm long con- ductor moves at a speed of 3 . 14 m / s. 0 . 638 T 0 . 638 T 008 10.0points A thin, horizontal copper rod is 1 . 02819 m long and has a mass of 23 . 9052 g. The acceleration of gravity is 9 . 8 m / s 2 . What is the minimum current in the rod that will cause it to float in a horizontal mag- netic field of 2 . 03123 T? 16 . 3 Ω I 30 . 6 cm 3 . 14 m / s
Version 010 – Midterm 3 2PM Sp16 – yeazell – (55745) 6 An electron is in a uniform magnetic field B that is directed out of the plane of the page, as shown. v e B B B B When the electron is moving in the plane of the page in the direction indicated by the arrow, the force on the electron is directed 1. toward the top of the page. 2. out of the page. 3. into the page. 4. toward the right 5. toward the left 6. toward the bottom of the page. correct Explanation: The force on the electron is vector F = q vectorv × vector B = - e vectorv × vector B. The direction of the force is thus hatwide F = - hatwide v × hatwide B , pointing toward the bottom of the page , us- ing right hand rule for hatwide v × hatwide B , and reversing the direction due to the negative charge on the electron. | {"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": 0.8521994948387146, "perplexity": 641.7037257405046}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964361064.58/warc/CC-MAIN-20211201234046-20211202024046-00097.warc.gz"} |
https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/?level=7728&weight=2&char_order=1&atkin_lehner_string=%2B--- | Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 3 7 23
7728.2.a.cc $$4$$ $$61.708$$ 4.4.75645.1 None $$0$$ $$4$$ $$1$$ $$4$$ $$+$$ $$-$$ $$-$$ $$-$$ $$q+q^{3}+\beta _{1}q^{5}+q^{7}+q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots$$
7728.2.a.ch $$6$$ $$61.708$$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$6$$ $$2$$ $$6$$ $$+$$ $$-$$ $$-$$ $$-$$ $$q+q^{3}-\beta _{3}q^{5}+q^{7}+q^{9}-\beta _{4}q^{11}+\cdots$$ | {"extraction_info": {"found_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": 0.9955867528915405, "perplexity": 779.6910374385383}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780057733.53/warc/CC-MAIN-20210925172649-20210925202649-00340.warc.gz"} |
https://www.gradesaver.com/textbooks/math/trigonometry/CLONE-68cac39a-c5ec-4c26-8565-a44738e90952/chapter-3-review-exercises-page-136/45 | ## Trigonometry (11th Edition) Clone
$0.8660$
Ensuring that the calculator is in radian mode, type $\sin 1.0472$ into the calculator and solve: $\sin1.0472\approx0.8660$ | {"extraction_info": {"found_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": 0.9811429381370544, "perplexity": 1354.1136652101302}, "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-2019-43/segments/1570986700560.62/warc/CC-MAIN-20191020001515-20191020025015-00406.warc.gz"} |
https://www.physicsforums.com/threads/l-c-r-circuit-finding-current.227108/ | # L-C-R Circuit - finding current
1. Apr 7, 2008
### ttiger2k7
[SOLVED] L-C-R Circuit - finding current
1. The problem statement, all variables and given/known data
In the figure below, V = 100.0 V, R1 = 40.0 Ohms, R2= R3 = 36.0 Ohms , and L = 17.0 H. No current flows until switch S is closed at t=0. Find the magnitude of the current i1 immediately after the switch is closed.
http://calculus.unl.edu/edu/classes/JF05/LRC.gif [Broken]
2. Relevant equations
Kirchoff's loop rule
Voltage across Inductor: $$L\frac{di}{dt}$$
3. The attempt at a solution
Since $$R_{2}$$ and $$R_{3}$$ are in series I can add them:
36 Ohms + 36 Ohms = 72 Ohms
And applying the loop rule:
$$V - iR_{1} - L\frac{di}{dt} - iR_{2+3} = 0$$
$$100 V - i(40 Ohms) - ???? - i(72 Ohms) = 0$$
---
I have two questions regarding this problem:
Is applying the Loop rule the right way to go?
And if so, what exactly is $$L\frac{di}{dt}$$? Isn't it just $$\frac{\epsilon}{L}$$?
Last edited by a moderator: May 3, 2017
2. Apr 7, 2008
### mikelepore
It's much easier than you thought. "Immediately after the switch is closed" is the easiest kind of question to answer. It takes time for the current through an inductor to build up. (Or, if this were a capacitor problem instead of an inductor problem, I would be saying: it takes time for the voltage across a capacitor to build up).
So right after the switch is closed, with i3 beginning at zero, the inductor is like an "open switch".
But remember that this is true only for a point in time -- then i3 will increase asymptotically toward some final value.
3. Apr 7, 2008
### ttiger2k7
thank you!
Similar Discussions: L-C-R Circuit - finding current | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8762994408607483, "perplexity": 1602.6435855222321}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948599549.81/warc/CC-MAIN-20171218005540-20171218031540-00698.warc.gz"} |
http://math.stackexchange.com/questions/38821/is-there-a-usual-method-for-finding-the-minimal-polynomial-of-trigonometric-valu/38831 | # Is there a usual method for finding the minimal polynomial of trigonometric values?
I've been thinking a bit about finding the minimal polynomials of side lengths of regular $n$-gons inscribed in the unit circle. For example, I recently wanted to find the minimal polynomial of the side length of an inscribed regular nonagon. Using the law of cosines, I was able to find that the side length is $a=\sqrt{2-2\cos(\frac{2\pi}{9})}$.
So $\displaystyle \frac{2-a^2}{2}=\cos \left(\frac{2\pi}{9} \right)$, but since I can't express $\displaystyle\cos \left(\frac{2\pi}{9} \right)$ in terms of rational numbers or their square roots, I'm unsure of how to proceed exactly. Is there a usual method to attack values like this? Possibly for say $\displaystyle \cos \left(\frac{m\pi}{n} \right)$?
Thanks.
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Look up the Chebyshev polynomials: en.wikipedia.org/wiki/Chebyshev_polynomials – Thomas Andrews May 13 '11 at 4:04
@Amy Maybe you're already aware, but if you use displaymode on inline formulas, you can also add \left( or \right) for the parentheses on the cosine expressions, so that the parentheses match the arguments better. – yunone May 13 '11 at 5:04
@yunone: Ah...thanks for the tip. I'm kind of learning as I'm going along. Good to know! – amWhy May 13 '11 at 5:17
@Dom: Take a look here: math.stackexchange.com/questions/45144/… – user9413 Aug 4 '11 at 10:33
EDIT : Following Marty Green's approach, I managed to actually find the minimal polynomial explicitly. I think both of you (Marty and Dom) will appreciate. Write $$2 \cos \left( \frac{2\pi}9 \right) = \zeta_2 + \zeta_7$$ where $\zeta_2$ and $\zeta_7$ are the second and seventh rooth of unity, respectively. Now we try to find the minimal polynomial for $\zeta_2 + \zeta_7$ (if we found it then substituting $\zeta_2 + \zeta_7 = x = 2y$ in the polynomial gives us our wished polynomial). Notice that $$x^9 -1 = \prod_{d \mid n} \Phi_d(x) = (x-1)(x^2+x+1)(x^6 + x^3 + 1)$$ where $\Phi_d(x)$ is the cyclotomic polynomial of the $d^{\text{th}}$ roots of unity. Now since $$\phi_9(x) = x^6 + x^3 + 1 = \prod_{(i,9)=1}(x-\zeta_i)$$ and developping this product to obtain the symmetric polynomials in the $\zeta_i$ as coefficients of $\phi_9(x)$, we obtain the two useful identities : $$\sum_{(i,9)=1} \zeta_i = 0, \quad \underset{i \neq j}{\sum_{(i,9)=(j,9)=1}} \zeta_i \zeta_j = 0.$$ Now we also have the identity $\zeta_6 + \zeta_3 + 1 = 0$ since $\zeta_3$ is a root of $x^2 + x + 1$. Hence we have $$(x-(\zeta_1 + \zeta_8))(x-(\zeta_2 + \zeta_7))(x-(\zeta_4 + \zeta_5)) =$$ $$x^3 - ((\zeta_1 + \zeta_8) + (\zeta_2 + \zeta_7) + (\zeta_4 + \zeta_5))x^2 +$$ $$((\zeta_1 + \zeta_8)(\zeta_2 + \zeta_7) + (\zeta_1 + \zeta_8)(\zeta_4 + \zeta_5) + (\zeta_2 + \zeta_7)(\zeta_4 + \zeta_5))x$$ $$- (\zeta_1 + \zeta_8)(\zeta_2 + \zeta_7)(\zeta_4 + \zeta_5).$$ Since the sum of the roots of $\Phi_9$ is $0$, the quadratic term vanishes. For the linear term, notice that all pairs of roots appear as products of two roots beside the ones that are together in the sum, i.e. $\zeta_1 \zeta_8$, $\zeta_2 \zeta_7$ and $\zeta_4 \zeta_5$. Hence if we call the linear term $\alpha$, $$\alpha = \left( \underset{i \neq j}{\sum_{(i,9)=(j,9)=1}} \zeta_i \zeta_j \right) - \zeta_1\zeta_8 - \zeta_2 \zeta_7 - \zeta_4 \zeta_5 = 0 - \zeta_9 - \zeta_9 - \zeta_9 = -3.$$ Now for the constant in the end, develop it manually to have something like this : $$- (\zeta_1 + \zeta_8)(\zeta_2 + \zeta_7)(\zeta_4 + \zeta_5) =$$ $$- (\zeta_1 \zeta_2 \zeta_4 + \zeta_1 \zeta_2 \zeta_5 + \zeta_1 \zeta_7 \zeta_4 + \zeta_1 \zeta_7 \zeta_5 +$$ $$\zeta_8 \zeta_2 \zeta_4 + \zeta_8 \zeta_2 \zeta_5 + \zeta_8 \zeta_7 \zeta_4 + \zeta_8 \zeta_7 \zeta_5) =$$ $$- (\zeta_7 + \zeta_8 + \zeta_3 + \zeta_4 + \zeta_5 + \zeta_6 + \zeta_1 + \zeta_2) = -(\zeta_3 + \zeta_6) = 1$$ The last line is because the sum of the roots is $0$. Hence a polynomial for which $\zeta_2 + \zeta_7$ is a root is $x^3 - 3x + 1$. Making the variable change to get the polynomial for $\cos(2\pi / 9)$ gives us $8x^3 - 6x + 1 = 0$, which is irreducible (there are many possible answer to this irreducibility question non-rationality of the roots, rational root theorem, etc.) which I leave up to you.
P.S. I will never do that again, it's so painful... but it was fun. XD Hope you liked it.
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Good calculation. – quanta May 15 '11 at 21:12
Wow, amazing. Thanks for taking the time and effort to do that. – Dom May 21 '11 at 6:51
No problem ; I suggest you look more closely to the other answers also! They were inspiring for this answer, mosty Marty's. Galois Theory is always so much fun... =) – Patrick Da Silva May 21 '11 at 8:30
Usually it's a pain to do so, for instance for $\displaystyle \cos(\frac {2\pi}9)$, you use the triple angle cosine formula : $$\cos 3\theta = 4\cos^3(\theta) - 3 \cos (\theta).$$ since substituting $\displaystyle \theta = \frac{2\pi}9$, you can evaluate $\displaystyle \cos \left( \frac{2\pi}3 \right) = -1/2$, which gives you rational coefficients in $\displaystyle \cos \left(\frac{2\pi}9\right)$. I wouldn't say there is a "general method" because I would never do that in general, but a trick to compute the minimal polynomial of $\displaystyle \cos \left(\frac{m\pi}n \right)$ would be to use trigonometric identities to express $\cos n\theta$ in a polynomial that is in terms of $\cos \theta$, or at least $\cos k\theta$ where $k$ divides $n$ so that when you substitute $\theta =$ your angle, you obtain a remarquable angle (like in the $\displaystyle \frac{2\pi}9$ case, we chose $3$ instead of $9$) . Such a polynomial exists, and they are detailed in the following link :
http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Multiple-angle_formulae
Look for the multiple angle formulas, Tchebyshev and spread polynomials. They give you formulas for the sine and cosine.
For instance, would you be computing the minimal polynomial of $\displaystyle \cos \left(\frac{7\pi}{16} \right)$ for any reason, since the quarters are remarquable angles, I'd go for the $4^{\text{th}}$ Tchebyshev polynomial : $T_4(x) = 8x^4 - 8x^2 + 1$, so that $$\cos(4 \theta) = 8\cos^4 \theta - 8 \cos^2 \theta + 1,$$ hence $\displaystyle \cos \left(\frac{7\pi}4 \right) = 1/\sqrt{2}$, so that $$(8x^4 - 8x^2 + 1)^2 - 1/2 = 0$$ would be a polynomial of which $\displaystyle \cos \left(\frac{7\pi}{16} \right)$ is a root. Expanding it and multiplying by $2$ to get rid of fractions will give you $$128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 = 0$$
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In particular, there are two (sometimes four) kinds of Chebyshev polynomials. The first kind $T_n(x)$ satisfies $T_n(\cos\,\theta)=\cos\,n\theta$, while the second kind $U_n(x)$ satisfies $U_n(\cos\,\theta)=\frac{\sin((n+1)\theta)}{\sin\,\theta}$. – J. M. May 13 '11 at 4:18
@J.M. The link I proposed details those well enough, and clearly $U_n$ doesn't make clever identities for this context, but I must admit this is a cool thing to know. Thanks J.M. – Patrick Da Silva May 13 '11 at 4:34
It's a good idea to keep things self-contained if possible, especially considering that you're linking to a Wikipedia entry. ;) – J. M. May 13 '11 at 4:36
Actually that should be $128 x^8 - 256 x^6 + 160 x^4 - 32 x^2 + 1$. Note that these polynomials obtained from Chebyshev polynomials are not necessarily irreducible over the rationals (although this one is). – Robert Israel May 13 '11 at 4:41
Typo. Thanks Robert, I'll edit it. – Patrick Da Silva May 14 '11 at 4:46
I think I can point you in the direction of a very general algebraic way to construct things like the minimal polynomial for cos(2π/9 ). Start with the equation x^9 - 1 = 0, factor out the trivial factor of (x-1) and you get x^8 + x^7 .... + 1 = 0, which has eight roots. Then factor out x^2 + x + 1 to get rid of the cube roots of 1, and you have six roots left. Because of their relation as ninth roots of unity, they can be enumerated as a, a^2, a^4, a^8, a^16 (=a^7) and a^32 (=a^5). The next term in the series is a^64 which is just equal to a. We can call these roots r1,r2,r4,r5,r7, and r8. It is then easy to choose sums of these roots to be the cosines of 40 degrees, 80 degrees, and 160 degrees:
r1 + r8
r2 + r7
r4 + r5
If we call these a, b, and c, then the problem becomes one of constructing the symmetric polynomial:
abc
ab + bc + ca
a + b + c
The information available is the values for the symmetric polynomials in r1,r2,r4,r5,r7 and r8. (which you get directly from the coefficient of the six-degree equation for the r's). In addition you have the known relations between the r's, that is eg. r1r2 = r3.
I don't have the strength to carry it all through right now but I've done stuff like this in the past and its actually kind of fun and somehow it works, if I'm not mistaken.
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This looks like a nice way to do it, but very particular and deserves detailing. I'd love to work it out myself tomorrow or to see you've worked it out in some way, I think it's worth a shot. I'm wondering if it has some generality that would make finding the minimal polynomials real quick. – Patrick Da Silva May 13 '11 at 5:38
Years ago I worked this out for the fifth and seventh roots of unity, but never the ninth. I remember reducing the expressions for the symmetric polynomials to be a very picturesque exercise, but it's not especially quick. My guess is that it works for all roots of unity in terms of deriving the minimal polynomial, but solving it for an algebraic expression is of course another matter. – Marty Green May 13 '11 at 11:19
Thanks, Patrick. I'm amazed and honored that someone actually took my advice on how to solve a problem. Yes, it's a little painful but oddly satisfying, isn't it? (I wanted to add this comment to your posted answer but I couldn't find the COMMENT field so I posted it here.) – Marty Green May 14 '11 at 13:21
In fact Gauss used your method to express the 17th roots of unity and others. – quanta May 15 '11 at 21:13
@quanta : Oh, quite cool =) I think I'll try it someday! (The 17th root thing). @Marty Green : Well well, clever advices make clever answers, the thanks goes back at you! It's weird that you couldn't find the comment field, it's at the same spot where you commented here, but below my answer instead of yours. =P – Patrick Da Silva May 16 '11 at 3:38
The Chebyshev polynomials of the second kind have the property that:
$$U_n(\cos\theta) = \frac {\sin (n+1)\theta}{\sin \theta}$$
So $U_n(x)$ has roots equal to $\displaystyle \cos\left( {\frac{\pi k}{n+1}}\right)$.
You have to eliminate the roots of $U_8(x)$ that are roots of smaller polynomials.
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See formulae 22.16.4 and 22.16.5 here. – J. M. May 13 '11 at 4:34 | {"extraction_info": {"found_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": 0.8915213346481323, "perplexity": 353.94407273744565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1448398461390.54/warc/CC-MAIN-20151124205421-00006-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/laplace-transform-with-abs-value-in-exponential.86522/ | # Homework Help: Laplace transform with abs value in exponential
1. Aug 27, 2005
### EvLer
Hi everyone,
I have this problem and just need someone to check or correct:
$$f(t) = -4e^{-3|t|}(u(t + 3) - u(t-1))$$
where u(t) is step function: u(t) = 1 for t >= 1 and 0 elsewhere;
so, i guess I need to break abs value into 2 cases and have 2 different equations? anyway, here's what I have if someone would be so kind and check my work (at first I applied linearity property and distributed $$-4e^{-3|t|}$$:
1. for t > 0:
$$L[f(t)] = \frac{-4}{s+3} + \frac{4e^{-(s+3)}}{s+3}$$
2. for t < 0:
$$L[f(t)] = \frac{-4}{s-3} + \frac{4e^{-(s-3)}}{s-3}$$
ps: i guess one thing I should explain is that by definition of unilateral laplace transform, even though first part of signal starts at -3 we do not consider it, what we are doing is one-sided Laplace transform, so I started integrating from 0- the first part of the expression.
edit: to (hopefully) increase chances that someone looks at this here's the Lapl. trnsf. that are relevant:
$$L[u(t-k)] = \frac{e^{-sk}}{s}$$
$$L[e^{at}f(t)] = F(s-a)$$
but you probably know this anyway :shy:
Last edited: Aug 27, 2005
2. Aug 27, 2005
### Galileo
Do you mean u(t)=0, for t<0 and u(t)=1 fot t>0? That's what the standard (Heavyside) step function does.
I think the easiest way would be direct integration. If you look at the expression u(t+3)-u(t-1), you notice it is 1 inside the interval [-3,1] and zero elsewhere. This makes the integral pretty easy to evaluate.
3. Aug 27, 2005
### EvLer
ooops, sorry about the typo, u(t) = 1 for t >= 0, you're right.
so I would have to integrate from 0 to 1 (for the one-sided laplace trnsfm)...
thanks!
4. Aug 27, 2005 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9363502860069275, "perplexity": 1014.6275360366559}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583512693.40/warc/CC-MAIN-20181020101001-20181020122501-00359.warc.gz"} |
http://tex.stackexchange.com/questions/1290/clarification-on-the-use-of-with-table-headings?answertab=votes | # clarification on the use of @{} with table headings?
Most examples I've seen are like this:
\begin{tabular}{@{}l r r@{}}
that is, with one @{} to the left of first column specifier and another @{} to the right of the last column specifier.
Yet the "Not So Short Guide to LaTex" says this construct suppresses the leading space.
I'm a bit confused. Which of the following interpretations are correct?
• @{} suppresses the space on the side of the column specifier where it is placed (i.e. placed to the left of the specifier it suppresses the leading space and, conversely, placed to the right it suppresses the trailing space)
• only @{} should be placed to the left of the first column and/or to the right of the last column, but not in between.
• neither of the above.
Unfortunately I'm not at a computer with LaTeX at this very moment so I cannot try it instead of asking.
Thanks a lot.
-
@{}suppresses the space between columns, that means after the preceding column and before the next. This way it affects also the space before the first column and after the last, if positioned 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": 0.92117840051651, "perplexity": 1129.4967161619752}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164844212/warc/CC-MAIN-20131204134724-00088-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/162597-help-orthogonal-proof.html | 1. Help with Orthogonal Proof
Could anyone help me with the following proof?
Suppose $u,v\in V$. Prove that $=0$ if and only if $||u||\leq ||u+av||$ for all $a\in F$.
I already came up with a proof assuming that $=0$ and used the Pythagorean Theorem to prove that $||u||\leq ||u+av||$.
However, I don't know how to prove it the other way.
2. Originally Posted by zebra2147
Could anyone help me with the following proof?
Suppose $u,v\in V$. Prove that $=0$ if and only if $||u||\leq ||u+av||$ for all $a\in F$.
I already came up with a proof assuming that $=0$ and used the Pythagorean Theorem to prove that $||u||\leq ||u+av||$.
However, I don't know how to prove it the other way.
The proof will be slightly different depending on whether the field $F$ is the real or the complex numbers. Here's how to do the real case.
If $\|u\|\leqslant\|u+av\|$ then $\|u\|^2\leqslant\|u+av\|^2$. Write that as $\langle u,u\rangle \leqslant\langle u+av,u+av\rangle$, which simplifies to $a^2\|v\|^2 + 2a\langle u,v\rangle\geqslant0$. That is a quadratic in $a$ (with zero constant term), and if it is non-negative for all real $a$ then its discriminant cannot be positive. So $\langle u,v\rangle^2\leqslant0$, which obviously implies $\langle u,v\rangle=0$.
For the complex case, you have to feed a few complex conjugates into the calculation, but the method is essentially the same.
3. Is there anyway to show this without using the discriminate? My professor has made it pretty clear that we have not gone over discriminate so I'm probably not suppose to use that fact...
4. Originally Posted by zebra2147
Is there anyway to show this without using the discriminate? My professor has made it pretty clear that we have not gone over discriminate so I'm probably not suppose to use that fact...
You may be thinking of something else... the discriminant of the quadratic $ax^2+bx+c$ is simply $\sqrt{b^2-4ac}$, that's just college algebra.
If you don't want to do that though, I think this should work:
Begin as Opalg did.
$||u||\leq ||u+av||\Rightarrow ||u||^2\leq ||u+av||^2$
$\Rightarrow \leq $
$\Rightarrow \leq +2a+a^2$
$\Rightarrow a^2+2a\geq 0$
$\Rightarrow a\cdot (a+2)\geq 0$.
So we have a product of two terms must be nonnegative; it follows that each factor must have the same sign (both positive or both negative) (unless of course one or both is just 0). The expression
$a+2$
is simply a linear polynomial in $a$ (the inner product terms are really just constants). Because of what we said about whether this value is positive or negative, we get
$\mathrm{lim}_{a\rightarrow 0^+}(a+2)\geq 0$, and
$\mathrm{lim}_{a\rightarrow 0^-}(a+2)\leq 0$. (*)
But polynomials are continuous, so the limit from the left and right is the same:
$\mathrm{lim}_{a\rightarrow 0^+}(a+2)= img.top {vertical-align:15%;} $\mathrm{lim}_{a\rightarrow 0^-}(a+2)\geq 0$" alt="\mathrm{lim}_{a\rightarrow 0^+}(<v,v>a+2<u,v>)= $\mathrm{lim}_{a\rightarrow 0^-}(a+2)\geq 0$" />
But (*) tells us that this common quantity is both nonnegative and nonpositive; thus
$\mathrm{lim}_{a\rightarrow 0}(a+2)=0$.
So
$0=\mathrm{lim}_{a\rightarrow 0}(a+2)$
$=2\Rightarrow =0$.
Now, all this limit stuff may be too fancy; it's simply trying to say that a straight line satisfying the above must be 0 at the origin.
5. Another method, once you have got to the inequality $a^2\|v\|^2 + 2a\langle u,v\rangle\geqslant0$ (valid for all $a$), is to complete the square to get $\Bigl(a\|v\| + \frac{\langle u,v\rangle}{\|v\|}\Bigr)^2 - \frac{\langle u,v\rangle^2}{\|v\|^2}\geqslant0$.
Then if you put $a = -\langle u,v\rangle/\|v\|^2$, the inequality becomes $-\langle u,v\rangle^2/\|v\|^2\geqslant0$, which can only happen if $\langle u,v\rangle=0$. | {"extraction_info": {"found_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": 44, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9128975868225098, "perplexity": 193.1237347224172}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501174276.22/warc/CC-MAIN-20170219104614-00103-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://plasticphotovoltaics.org/lc/characterization/lc-advanced-c/lc-ellipsometry.html | ## Ellipsometry
Ellipsometry is a sensitive optical method proposed by Paul Drude (1863 - 1906) and has thus been used for over a hundred years to derive information about surface and bulk properties.Drude, P. Ueber Oberflächenschichten. I. Theil. Annalen der Physik 1889, 272, 532-560. It makes use of the fact that the polarization state of light changes when the light beam is reflected from a surface and the technique makes it possible to deduce information about the film properties, especially the film thickness. Generally optical measurement techniques are of great interest since they under normal circumstances are non-invasive and non-destructive. Ellipsometry is no exception. The basic principle of ellipsometry is, as mentioned, that upon reflection the polarization changes. The exact nature of the polarization change is determined by the properties of the sample including thickness and complex refractive index. The main advantage of ellipsometry is that, in opposition to other optical techniques that are inherently diffraction limited, ellipsometry exploits phase information and the polarization state of light, and can achieve angstrom resolution. In its simplest form, the technique is applicable to thin films with thickness less than a nanometer and up to thicknesses of several micrometers. An obvious application of ellipsometry is the use in the semiconductor industry, where thin layers of silicon dioxide are a central element throughout production. Ellipsometry enables process engineers to keep track of the thickness of the film.
In the field of organic solar cells several reports exist on applications of ellipsometry for determining optical constants and thickness, surface roughness, and morphology. While determination of thicknesses and optical constants are an important application of ellipsometry, this application is mostly used to augment other measurement and to optimize processes.DOI:10.1080/713738799DOI:10.1002/pip.1190 A more advanced use of ellipsometry is the use of ellipsometry to study the morphology of the bulk hetero junction. A number of approached to this exists in literature. Campoy-Quiles et al. have demonstrated work modeling the vertical composition profile of P3HT:PCBM films and reported a composition gradient varying from PCBM-rich near the PEDOT:PSS layer to P3HT-rich at the air interface.DOI:10.1038/nmat2102 This result is important in the understanding of the performance of solar cells made by spin coating. Germack et. al. have substantiated the results and proposed that changes in the surface energy significantly affects the vertical composition profile.DOI:10.1021/ma100027b Their analysis was based spectroscopic ellipsometry and near-edge X-ray absorption fine structure spectra.
### Ellipsometry theory
Ellipsometry, as mentioned, is designed to measure the change of polarization upon reflection or transmission. Calculations of the polarization state are therefore tied to the electric field vector, defining the direction of the polarization of the light wave. The electric field vector is decomposed into two components named p and s respectively, a tradition originating from their German names Parallel and Senkrect. Ellipsometry is primarily interested in how p- and s- components change upon reflection or transmission in relation to each other. The change in polarization is commonly written as $$\frac{r_p}{r_s} = \tan \left( \Psi \right) e^{i\Delta}$$ The right side of the equation is describing the measurement with $\tan \left( \Psi \right)$ representing the amplitude ratio upon reflection, and $e^{i\Delta}$ the phase shift. The left side of the equation describes the sample with rp and rs being the two components of the reflection coefficient. As ellipsometry measures a ratio of two values rather than an absolute value of either, the measurement is robust, accurate, and reproducible. For instance, ellipsometry is relatively insensitive to scattering and fluctuations, and requires no standard sample or reference beam. However, as ellipsometry is an indirect method, where the measured Ψ and Δ cannot be converted directly into the optical constants of the sample, a model analysis must be performed. Direct conversion into real data is only possible in simple cases of isotropic, homogeneous and infinitely thick films. In all other cases a layer model must be established, which considers the optical constant and thickness parameters of all individual layers of the sample including the correct layer sequence. Then using an iterative procedure unknown optical constants and / or thickness parameters are varied, and the right side of the equation is calculated using the Fresnel equations for $r_s$ and $r_p$. The best match provides the optical constants and thickness parameters of the sample. Roughness for example can be included in the model by using a effective medium approximation; effectively changing the optical constants in the model.
### Ellipsometry measurements
Obtaining $\Psi$ and $\Delta$ from the ellipsometric measurement is very dependent on the type of ellipsometer used. Rotating analyzer ellipsometry is probably the most widespread technique, but the technique has a weakness in that it is not capable of determining the phase $\Delta$, but rather $\cos \left( \Delta \right)$ . A rotating compensator ellipsometer is capable of overcoming this issue.
Figure 1. A simple ellipsometry system consisting of a light source, a polarizer, a sample, an analyzer, and a detector.
Rotating analyzer or rotating polarizer ellipsometry is a simple form of ellipsometry employed by many manufactures. See Figure 1 for a schematic representation of a rotating analyzer setup. The setup is based on an electromagnetic radiation emitted by a light source and linearly polarized by a polarizer. After reflection from the sample the radiation passes a second polarizer, used to analyze the polarization, and then falls into the detector. Using the Jones matrix formalism it is possible to describe the light passage through a rotating analyzer ellipsometer. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light $${\overrightarrow E _0} = {T_A}\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\leftrightarrow}} \over R} \left( {{\alpha _1}} \right){\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\leftrightarrow}} \over T} _S}{\vec E_i} = \left[ {\begin{array}{*{20}{c}} 1&0\\ 0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\cos \left( {{\alpha _1}} \right)}&{\sin \left( {{\alpha _1}} \right)}\\ { - \sin \left( {{\alpha _1}} \right)}&{\cos \left( {{\alpha _1}} \right)} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{r_p}}&0\\ 0&{{r_s}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{E_i}\cos \left( {{\alpha _0}} \right)}\\ {{E_i}\sin \left( {{\alpha _0}} \right)} \end{array}} \right]$$ ${\vec E_i}$ is the Jones vector representation of the incident electric field after a linear polarizer. $\vec T$ is the sample reflection, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over R} \left( {{\alpha _1}} \right)$ is the rotation to match the coordinate system of the analyzer, and $T_A$ represents the analyzer. For a rotating analyzer ellipsometer $\alpha_1$ is changed to get several intensity readings. A rotating polarizer ellipsometer instead rotates the polarization of the incoming light. The intensity at the detector is the absolute value of the outgoing electric field ${_0} = {\vec E_0} \cdot \vec E_0^*$. By introducing the stokes parameters; ${s_0} = {\left| {{E_p}} \right|^2} + {\left| {{E_s}} \right|^2}$, ${s_1} = {\left| {{E_p}} \right|^2} - {\left| {{E_s}} \right|^2}$ , and ${s_2} = {E_p}E_s^* + {E_s}E_p^*$ the intensity can be written as $$I{_0}\left( {{\alpha _1}} \right) = \frac{1}{2}\left[ {{s_0} + {s_1}\cos \left( {2{\alpha _1}} \right) + {s_2}\sin \left( {2{\alpha _1}} \right)} \right]$$ The value of the Stokes parameters can be determined experimentally by conducting measurements of the intensity at a minimum of three rotations of the polarizer ( $\alpha$). Hereby three equations for $I_0$ is introduced with the three Stokes parameters as the only unknowns. It is possible to express the elliptical parameters by the Stokes parameters $$\Psi = \frac{1}{2}{\cos ^{ - 1}}\left[ { - \frac{{{I_0}\left( {0^\circ } \right) - {I_0}\left( {90^\circ } \right)}}{{{I_0}\left( {0^\circ } \right) + {I_0}\left( {90^\circ } \right)}}} \right]$$ and $$\Delta = {\cos ^{ - 1}}\left[ {\frac{{2{I_0}\left( {45^\circ } \right)}}{{\left( {{I_0}\left( {0^\circ } \right) + {I_0}\left( {90^\circ } \right)} \right)\sin \left( {\Psi '} \right)}} - \frac{1}{{\sin \left( {\Psi '} \right)}}} \right]$$ In the above example it was assumed that measurements of the intensity were conducted at 0°, 45°, and 90°. The value of $\Delta$ is an inverse cosine function. This means that the precision and accuracy is poor when $\Delta$ is near 0° or 180°. For applications not requiring several angles of incident to be measured this is not a big issue. It will then be possible to conduct the measurements near the Brewster angle and maintain good accuracy of $\Delta$. If several angles of incident is necessary in order to fit the model, poor accuracy in $\Delta$ can be problematic. This condition is encountered as an example when trying to model in-depth morphology; since multiple angles of incidence yields measurements at different optical path lengths providing valuable information. It is possible to install a compensator element into the beam path either before or after the sample. The compensator can convert the near linear polarization state near $\Delta = ^{\circ}$ or $180^{\circ}$ to a near circular polarization state ($\Delta = =90^{\circ}$), optimizing sensitivity for $\Delta$. Hereby both $\Psi$ and $\Delta$ can be accurately measured over their full ranges. However, a perfectly ideal spectroscopic compensator element does not exist and compensator elements which can be used spectroscopically are not achromatic. This means that the retardance of the compensator must be calibrated throughout the entire spectral range. Otherwise the accuracy of the ellipsometric data will be degraded by the introduction of the compensator element.
An alternative approach to introduce a compensator into the ellipsometer beam path is to implement the rotating compensator ellipsometer configuration. This setup is not restricted to measuring only $\cos \left( \Delta \right)$, since the rotating-compensator instrument provides all four Stokes vector components for the light beam reflected from the sample surface. In contrast, the rotating-polarizer instrument provides only three such components. This also means that this configuration is capable of measuring the depolarization which occurs from samples with non-uniform film thickness, roughness and other sample inhomoginities.
### Data analysis
After a sample is measured and the right side of the ellipsometry equation has been determined, a model must be constructed to describe the sample. The model is used to calculate the response from the Fresnel equations which describe each material with thickness and optical constants. When the values are not known they become fitting parameters for which a preliminary guess is applied. The calculated values from the left side of the equation are then compared to the experimental data. Any unknown parameter can be varied to improve the match between experiment and calculation. The best match between the model and the experiment is found through regression, where an estimator, like the Mean Squared Error (MSE), is used to quantify the difference between curves. The unknown parameters are allowed to change until the minimum MSE is reached. It important at this point to notice that the process of fitting can be complicated, and that many local minima may exist. It is very possible for the regression algorithm fall into a local minimum depending on the initial parameter guess.
Figure 2. Reflection and transmission of an incident light wave at a surface boundary or a infinite film.
The simplest example of an ellipsometry model comes in the form of a bulk sample (infinite film), see Figure 2.2. Following Equation 2.1 the model must describe the ratio of rp and rs. For the infinite film approximation $r_p$ and $r_s$ are simply given by the Fresnel reflection coefficients $${r_s} = \frac{{{n_0}\cos \left( {{\theta _0}} \right) - {n_1}\cos \left( {{\theta _1}} \right)}}{{{n_0}\cos \left( {{\theta _0}} \right) + {n_1}\cos \left( {{\theta _1}} \right)}}$$ and $${r_p} = \frac{{{n_0}\cos \left( {{\theta _1}} \right) - {n_1}\cos \left( {{\theta _0}} \right)}}{{{n_0}\cos \left( {{\theta _1}} \right) + {n_1}\cos \left( {{\theta _0}} \right)}}$$ where $n_0$ and $n_1$ is the index of refraction for medium 0 and 1 respectively. The refracted angle ( $\theta _1$) is related by Snells law to $n_0$, $n_1$, and $\theta _0$. Thereby the index of refraction for medium 1 remains the only unknown. Solving the ellipsometry equation with the simple Fresnel coefficients yields $${n_1} = {n_0}\sin \left( {{\theta _0}} \right){\left[ {{{\left( {\frac{{\tan \left( \Psi \right)\exp \left( {i\Delta } \right) - 1}}{{1 + \tan \left( \Psi \right)\exp \left( {i\Delta } \right)}}} \right)}^2}{{\tan }^2}\left( {{\theta _0}} \right) + 1} \right]^{\frac{1}{2}}}$$ A negative solution for the equation also exist, however, since the refractive index cannot be negative this solution is not shown. It follows that refractive index can directly be calculated from the ellipsometric parameters. No fitting is therefore necessary in this case.
Figure 3. Illustration of a film substrate optical system. The system consists of three parts; the ambient environment, the film, and the sample.
A case of importance in ellipsometry is an optical system consisting of an ambient-film-substrate system as shown in Figure 3. When the refractive index of the film and the substrate is known it is possible to determine the thickness of the film in such a system by utilizing the Fresnel coefficients. For the single layer ontop of a substrate the coefficients are given by the Airy formula $${R_p} = \frac{{{r_{01,p}} + {r_{12,p}}\exp \left( { - i2\beta } \right)}}{{1 + {r_{01,p}}{r_{12,p}}\exp \left( { - i2\beta } \right)}}$$ and $${R_s} = \frac{{{r_{01,s}} + {r_{12,s}}\exp \left( { - i2\beta } \right)}}{{1 + {r_{01,s}}{r_{12,s}}\exp \left( { - i2\beta } \right)}}$$ where $r_{01}$ and $r_{12}$ are the reflection parameters for the ambient-film and the film-substrate system respectively. $\beta$ is the phase angle containing the thickness of the film and is given by $$\beta = \frac{{2\pi d}}{{\lambda {n_1}\cos \left( {{\theta _1}} \right)}}$$ where $d$ is the thickness, and $\lambda$ the wavelength. Inserting $R_p$ and $R_s$ into the ellipsometry equation yields a complex quadratic equation for $\exp \left( { - i2\beta } \right)$ which can be solved as $$\exp \left( { - i2\beta } \right) = \frac{{ - \left( {\frac{{{R_p}}}{{{R_s}}}E - B} \right) \pm {{\left[ {{{\left( {\frac{{{R_p}}}{{{R_s}}}E - B} \right)}^2} - 4\left( {\frac{{{R_p}}}{{{R_s}}}D - A} \right)\left( {\frac{{{R_p}}}{{{R_s}}}F - C} \right)} \right]}^{\frac{1}{2}}}}}{{2\left( {\frac{{{R_p}}}{{{R_s}}}D - A} \right)}}$$ where $A = {r_{01,s}}{r_{12,s}}{r_{12,p}}$, $B = {r_{12,p}} + {r_{01,p}}{r_{01,s}}{r_{12,s}}$, $C = {r_{01,p}}$ , $D = {r_{01,p}}{r_{12,p}}{r_{12,s}}$ , $E = {r_{12,s}} + {r_{01,s}}{r_{01,p}}{r_{12,p}}$, and $F = {r_{01,s}}$. This allows the thickness d to be calculated since the thickness is only represented in $\beta$. However, since the thickness is given in a complex exponential no single solution for the thickness exists. The thin film approximation deals with this issue by assuming that the lowest positive thickness value is the correct thickness. To determine the thickness for thicker films it is possible to conduct measurements for several wavelengths and thereby introduce more equations. For multiple isotropic layers, the calculation of the complex reflection coefficients is more complicated and performed using a matrix representation, where each layer is represented by two 2 X 2 complex matrices, one for the pp polarization and the other for the ss polarization.
### Optical coefficient parameterization
With ellipsometry the most typical situation with an ambient-film-substrate system is to know the complex refractive index of the substrate (either by previous measurement or from a table value), but not the thickness nor the complex refractive index of the film. In this case it is never possible to directly calculate all three unknowns. With spectroscopic ellipsometry the situation is better. However, since the refractive index is wavelength dependent the introduction of more wavelength add as many unknowns as equations. It is therefore necessary to model the optical dispersion by a simplified model to determine both optical constants and thickness. This parameterization of the optical components is done though a dispersion law simulating the optical indices and their variation according to the wavelength. A very common optical dispersion is the Cauchy optical dispersion where six parameters are used $$n = A + \frac{B}{{{\lambda ^2}}} + \frac{C}{{{\lambda ^4}}}$$ and $$k = \frac{D}{\lambda } + \frac{E}{{{\lambda ^3}}} + \frac{F}{{{\lambda ^5}}}$$ By using the dispersion relation the system becomes over determined making the fitting of the parameters more robust. The Cauchy dispersion is often used as a simple approach to determine the thickness of a film. If a wavelength range exist where the film has zero absorbance the $k$ component vanishes and only three fitting parameters remains beyond the thickness.
Another dispersion model often used is the the Tauc Lorentz model. This is typically used for the parameterization of the optical functions for amorphous semiconductors and insulators for which the imaginary part of the dielectric function $\varepsilon _i$ is determined by multiplying the Tauc joint density of states by the $\varepsilon _i$, as obtained from the Lorentz oscillator model. The real part of the dielectric function $\varepsilon _r$ is calculated from $\varepsilon _i$ using Kramers-Kronig integration, making the model Kramers-Kronig consistent.
### Effective medium approximation
Using an effective medium approximation (EMA), mixtures of materials with known refractive can be described. The EMA is a physical model that describes the macroscopic properties of a medium based on the properties and the relative fractions of its components. Based on the additive character of the polarizability, a generalization of the Claussius-Mossotti formula can be written as $$\frac{{\left\langle \varepsilon \right\rangle - {\varepsilon _h}}}{{\left\langle \varepsilon \right\rangle + 2{\varepsilon _h}}} = \left( {1 - f} \right)\frac{{{\varepsilon _1} - {\varepsilon _h}}}{{{\varepsilon _1} + 2{\varepsilon _h}}} + f\frac{{{\varepsilon _2} - {\varepsilon _h}}}{{{\varepsilon _2} + 2{\varepsilon _h}}}$$ where $\left\langle \varepsilon \right\rangle$ is the effective dielectric function, $\varepsilon _1$ and $\varepsilon _2$ are the dielectric functions of the two media subject to mixing, $\varepsilon _k$ the dielectric function of the host medium with the inclusions, and $f$ the volume ratio of material 2. The underlying assumptions of the equation are that it applies for spherical inclusions and dipole interactions only. In the Bruggeman model the effective medium itself act as the host material, so $\left\langle \varepsilon \right\rangle = {\varepsilon _h}$. The model is then self-consistent and the two phases play exactly the same role. The effective dielectric function of the mixture is given by the second order equation $$0 = \left( {1 - f} \right)\frac{{{\varepsilon _1} - \left\langle \varepsilon \right\rangle }}{{{\varepsilon _1} + 2\left\langle \varepsilon \right\rangle }} + f\frac{{{\varepsilon _2} - \left\langle \varepsilon \right\rangle }}{{{\varepsilon _2} + 2\left\langle \varepsilon \right\rangle }}$$ The validity of the Bruggeman effective medium approximation requires the sizes of the phases (dielectrics) in a composite material to be sufficiently greater than atomic sizes, but smaller than 1/10 of the wavelength, which indeed is true for the bulk heterojunction films. The effective medium approximation cannot represent non-additive features of the dielectric function, such as charge transfer absorption bands. Lastly the dielectric functions of the phases must be independent of size and shape.
### Practical considerations
There are a number of practical considerations to be familiar with in connection with ellipsometry measurements. The first major hurdle is backside reflections. Backside reflections occur when front surface and back surface reflections overlap and enter the detector. This happens for transparent substrates which are polished on both sides. This was the case for the glass substrates used during this thesis for modeling work. These unwanted backside reflections are incoherent with the desired reflection from the front side and can either be accounted for in the model or suppressed by experimental means. One approach is to roughen the backside so the light is effectively scattered.
Another effect encountered in connection with ellipsometry is depolarization. Depolarization occurs when totally polarized light used as a probe in ellipsometry is transformed into partially polarized light. The effect of depolarization is especially severe for a rotating angle ellipsometer as the instrument assumes that reflected light is totally polarized. Imagine a case where the reflected light of linear polarization is overlapped with circular polarization. For a rotating angle ellipsometer the polarization state of this reflected light will be interpreted as elliptical polarization, since this instrument assumes totally polarized light for reflected light. With a rotating compensator ellipsometer the depolarization can be measured and included in the model. The physical phenomena that generate partially polarized light upon light reflection are; surface light scattering caused by a large surface roughness, incident angle variation originating from the weak collimation of probe light, wavelength variation, thickness inhomogeneity in the film, and backside reflection. The measurement of the depolarization therefore gives a good indication of the quality of the sample.
Tompkins, H. G.; Irene, E. A.; Hill, C.; Carolina, N. Handbook of Ellipsometry; 2005
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https://www.arxiv-vanity.com/papers/1206.6531/ | Universality and symmetry breaking in conformally reduced quantum gravity
Alfio Bonanno INAF - Osservatorio Astrofisico di Catania, Via S. Sofia 78, I-95123 Catania, Italy INFN, Via S. Sofia 64, I-95123 Catania, Italy Filippo Guarnieri Dipartimento di Fisica, Universit degli Studi di Roma Tre and
INFN sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Am Mühlenberg 1, D-14476 Golm, Germany
Abstract
The scaling properties of quantum gravity are discussed by employing a class of proper-time regulators in the functional flow equation for the conformal factor within the formalism of the background field method. Renormalization group trajectories obtained by projecting the flow on a flat topology are more stable than those obtained from a projection on a spherical topology. In the latter case the ultraviolet flow can be characterized by a Hopf bifurcation with an ultraviolet attractive limiting cycle. Although the possibility of determining the infrared flow for an extended theory space can be severely hampered due to the conformal factor instability, we present a robust numerical approach to study the flow structure around the non-gaussian fixed point as an inverse problem. In particular it is shown the possibility of having a spontaneous breaking of the diffeomorphism invariance can be realized with non-local functionals of the volume operator.
pacs:
11.10.Gh, 04.60.-m
I Introduction
In statistical mechanics universality is the property for which, close to a continuous phase transition, the long range behavior of the system is independent of the details of the microscopic interactions. For example the magnetic order parameter of the Ising model does not depend on the lattice geometry, although the critical temperature is different for a square, triangular, or an hexagonal lattice. When different models share the same set of critical exponents it is said that they belong to the same universality class.
In quantum gravity, the conceptual difficulty in extending the notion of universality is the requirement of “background independence” because it implies that the geometrical structure of the spacetime cannot play any role in the definition of the microscopic degrees of freedom.
The strategies proposed so far to quantize gravity have different ways of dealing with this issue kiefer ; A ; R ; T . For example, in string theory, although the background independence is not manifest at a perturbative level, it should be realized non-perturbatively via AdS/CFT. On the other hand, in loop quantum gravity, this requirement is satisfied from the very beginning, at least at a formal level, and in the asymptotic safety program wein ; mr ; percadou ; oliver1 ; frank1 ; oliver2 ; oliver3 ; oliver4 ; souma ; frank2 ; prop ; oliverbook ; perper1 ; codello ; litimgrav ; frankmach ; BMS ; oliverfrac ; jan1 ; jan2 ; max ; livrev ; nagi background independence is dynamically achieved via the background field method abbott . From this point of view, conformally reduced gravity, a scalar analogous of gravity in which the only propagating degree of freedom of the metric is the conformal factor, is an important theoretical laboratory to understand the issue of background independence and renormalizability in this context. In fact, although in classical general relativity the conformal factor is not a propagating degree of freedom, in quantum gravity its fluctuations dominate the path-integral because of the conformal factor instability and can become the most important ones both in the ultraviolet (UV) region, close to the non-gaussian fixed point (NGFP) wein ; mr , and in the infrared sector (IR), below the gaussian fixed point (GFP).
The idea of considering this privileged point of view in order to better understand the structure of the UV critical manifold of quantum gravity was first put forward in creh1 and creh2 . In the latter paper a non-perturbative flow equation in the so-called “local potential approximation” (LPA) has been derived for the first time and further investigated in elisa within the framework of the bimetric truncation. Moreover, in crehroberto the contribution of the trace anomaly and the term have been considered.
Important points which deserve to be better investigated in this approach are the dependence of the critical quantities on the threshold functions and the structure of the renormalization flow beyond the simple conformally reduced Einstein-Hilbert (CREH) truncation. In the first case it is interesting to investigate the impact of the reduced degrees of freedom as a function of the cutoff structure. In particular we would like to discuss the possibility of determining an “optimal cutoff” for which the difference between the calculation of the universal properties performed in the full Einstein-Hilbert model and in the CREH approximation is minimal. In the second case it is important to study the evolution of admissible initial data that are not necessarily of the CREH form in order to see if non-local contributions to the renormalized Lagrangian significantly deform the structure of the UV critical manifold frank2 .
The role of non polynomial truncations can be significant also in the deep infrared region. For instance in the more familiar -theory in the broken phase, a first-order phase transition occurs in three dimensions, where the inverse susceptibility is not continuous in the limit. In this case the finite jump of the renormalized mass as a function of the field strength is not accessible to the standard function approach; moreover only with a robust and accurate numerical integration scheme of the flow equation is it possible to recover the convexity property of the free energy in the thermodynamic limit bonlac ; hrt .
In this work, in particular, we shall use the proper-time flow equation which has been extremely successful in the calculation of the critical exponents in the 3-dimensional Ising model propref and in quantum gravity prop . The main reason to employ this flow equation is that it is rather simple to implement different threshold functions and to interpolate between a sharp momentum cutoff and a sharp proper-time cutoff, as we shall see.
A question that naturally arises in the context of conformally reduced gravity is if a phase of non-zero mean conformal factor takes place at low energy: this symmetry breaking phenomenon could be interpreted as a phase of broken diffeomorphism invariance where and the spacetime geometry naturally emerges as a low-energy phase. We shall show that, although the IR flow of the LPA cannot be properly determined for a continuous set of initial data, it is nevertheless possible to study this symmetry breaking phenomenon as an inverse problem, within the LPA approximation. In particular, we shall find a new class of UV fixed potentials which evolves towards a low-energy phase where the diffeomorphism invariance is spontaneously broken.
The structure of the paper is the following: a derivation of the proper-time flow functional equation is discussed in section II by means of a background “independent” blocking procedure. The fixed points and critical exponents are computed in section III for various classes of threshold functions and for different projection in the background metric. The results are then compared with those obtained in the full Einstein-Hilbert (EH) model. Section IV includes a new numerical discretization scheme for the flow equation in the LPA and describes the possibility of having a phase of broken diffeomorphism invariance at low energy. Section V is devoted to the conclusions and in the Appendices the numerical results and explicit expressions of the functions for various regularization schemes and spacetime dimensions are presented.
Ii Wilsonian action for the conformal factor
In this section we shall introduce the concept of the Wilsonian action for the conformal factor by means of a constraint average field; this derivation is in fact closer to the statistical mechanical point of view than the standard approach based on the Schwinger functionals. It has the advantage of dealing with a quantity that can be directly computed in Monte Carlo simulations since the introduction of an external current is not necessary ora in this case.
Let be the action for the fundamental field that we write as where is a non-dynamical background field and the dynamical (fluctuating) field, and let be a rigid reference metric defined on a Euclidean manifold in dimensions. The Wilsonian action in the presence of the background field can be formally defined as
e−Sk[~f;χB]=∫D[f]δ(fk−~f)e−S[χB+f], (1)
where is an averaged fluctuation field given as
fk(x)=∫ddy√^gf(y)ρk(y,x;χB), (2)
and where is a smearing kernel with the properties of being
• symmetric:
• normalized:
• idempotent:
The last relation simply implies that the average of an average field is again an average field wette91 (see boneu for a general discussion on smearing kernels in Riemannian spaces). Its explicit expression in terms of dependence does not need to be specified at this level.
In this formalism plays the same role of the microscopic metric in the full theory. In the complete framework a background metric is chosen in order to perform the actual calculations, and the fluctuations are quantized non-perturbatively around this background which will be dynamically determined by the requirement that the expectation value of the fluctuation field vanishes, . Any physical length must then be proper with respect to the background metric . In the conformally reduced theory the expectation values and are the analogs of and in the full theory.
The central idea of the conformal field quantization is to employ the background metric
¯gμν≡χ2νBˆgμν (3)
in constructing the smearing function via the spectrum of , being and , respectively the momentum operators built with the background metric and the fixed metric , and is the space-time dimension. The reference metric plays no dynamical role in this process but it is fixed to perform the actual calculation, while all the dynamical fields are spectrally decomposed using the basis of the eigenfunctions whose eigenvalues satisfy
¯k2=χ−2νBk2 (4)
in the case of a constant . The -function kernel in (1)
δ(fk−~f)=∏xδ(fk(x)−~f(x)) (5)
reduces to in the limit and projects on the zero-momentum mode in the limit instead.
Therefore at the blocked action (1) coincides with the effective potential, namely, the non-derivative part of the effective action, but for the functional described in (1) is an effective action for the “low-energy” modes with momentum . The relation with the standard effective average action for the conformal factor defined in creh1 ; creh2 can be obtained by noticing that the expectation value of the blocking field , which is defined as
⟨~f⟩= ∫D[~f(x)]e−Sk[~f(x);χB(x)]~f(x)=∫D[~f(x)]D[f(x)]δ(fk(x)−~f(x))e−S[χB(x)+f(x)]~f(x)= = ∫D[f(x)]e−S[χB(x)+f(x)]fk(x)=⟨fk(x)⟩, (6)
can be rewritten, after having introduced the source , as
⟨~f⟩=1√^g∂Wk[J;χB]∂J(x)∣∣J=0, (7)
where
Wk[J;χB]=∫ddx{J(x)~f(x)−Sk[~f;χB]}. (8)
The (8) corresponds to the standard generating functional defined in creh1 .
The -function constraint in (1) can be conveniently evaluated in the momentum space and the “background-blocked” action can be explicitly computed in the one-loop approximation. The difference can then be evaluated in the infinitesimal momentum shell between and , where is the “proper” momentum operator built with the background metric . A functional flow equation is finally obtained by taking the limit and performing a renormalization group improvement of the resulting expression prop . After this step is accomplished, the “background-independent” flow is obtained expressing all the running “proper” momenta in terms of the reference energy scale . Rewriting the (regularized) one-loop contribution in the Schwinger “proper-time” formalism one finds
∂tSk[~f;χB]=−12Tr∫∞0dss∂tτkexp{−sδ2Sk[~f;χB]δ~f2}, (9)
where is the RG time and . The important difference between this type of functional “proper-time” flow equation and the version used in earlier investigations propref is that the trace in (9) is here computed by means of the representation provided by the spectrum of ,
¯¯¯¯¯¯Tr[A]≡∫ddx√¯g⟨x|A|x⟩=∫ddx√^gχdνB⟨x|A|x⟩. (10)
The precise relation between the “proper-time” flow equation and the exact flow equation avact has been extensively clarified in ergprop . For actual calculations we shall use the one-parameter family of smooth cutoffs that has been widely used in the literature propref , whose explicit expression reads
τnk(s)=Γ(n,sZnk2χ2νB)−Γ(n,sZnΛ2cutoffχ2νB)Γ(n). (11)
Here is an arbitrary real, positive parameter that controls the shape of the in the interpolating regions, and denotes the incomplete Gamma-function. Furthermore, is a constant which has to be adjusted to make sure that the eigenvalues of are cut off around rather than prop . Therefore derivative in (9) explicitly reads
∂tτnk(s)≡limδk→0kτnk+δk(s)−τnk(s)δk= −2n!(Zsk2nχ2νB)nexp(−Zsk2nχ2νB), (12)
with . For the kernel (II) does not regulate completely the UV because the proper-time integral requires a field independent (vacuum) contribution to be subtracted from the right-hand side of Eq.(9). On the other hand for this class of regulators allow us to take the formal limit which coincides with a sharp-cutoff introduced in the proper-time cutoff Flore . To conclude this section we would like to remark that for our regulator corresponds to the “optimized cutoff” introduced in optim .
Iii Polynomial truncations
In this section we shall discuss the structure of the NGFP obtained by the flow equation (9) as a function of the cutoff parameter for different reference topologies.
It is important to remark that, at variance with the well-known definition of the path-integral for quantum-gravity based on the sum over all possible metric/topologies, in our case the use of different topologies is only a technical device to project an infinite-dimensional functional flow equation in a finite dimensional theory space where only the flow of and operators is considered. From this point of view our approach has nothing to do with a calculation performed in the Gibbons-Hawking spirit. Neither are we expanding the graviton propagator in inverse powers of momentum/curvature. On the contrary the (unprojected) functional flow equation is, by construction, independent on the topology (see mr ) and the same property is shared by the flow equation for the conformal factor (see e.g. creh1 ; creh2 ). However because the irrelevant operators of the NGFP have a different impact on the renormalized flow at the zeroth order of the gradient expansion (spherical projection) and at first order (flat projection), the universal quantities will show this residual scheme dependence.
Let us then assume that the field dependence of the blocked action is completely encoded in a relation of the type , where is a blocked field, so that is a local function of . An important example of this approximation is the conformally reduced Einstein-Hilbert truncation. It is obtained by setting in the Euclidean Einstein–Hilbert action jackiw
SEHk[gμν]=−116π∫ddx √gG−1k(R(g)−2Λk) (13)
so that the standard formulas for Weyl rescalings yield
Sk[ϕ]=∫ddx√ˆgZk ( 12ˆgμν∂μϕ∂νϕ+12A(d)ˆRϕ2+ (14) −2A(d)Λkϕ2d(d−2)),
where and
Zk=−12πGkd−1d−2,A(d)=d−28(d−1). (15)
In particular, the second functional derivative of (14) reads
S(2)k[ϕ]= Zk(−ˆ□+2A(d)ˆR+ −2A(d)B(d)Λkϕ2d(d−2)−2)δd(x,y),
where is the Dirac delta function in dimensions in the fixed metric space endowed with the metric, and
B(d)=2dd−2(2dd−2−1). (17)
iii.1 Sd topology
Let us first consider the topology of the -dimensional sphere . In this case the curvature of the reference metric is constant and the running of the dimensionless coupling can be obtained from the term setting in the action (14), so that
SSdk[χB]=∫ddx√ˆgZk(12A(d)ˆRχ2B−2A(d)Λkχ2d(d−2)B). (18)
The trace on the background metric can be computed using the Seeley-Gilkey-deWitt heat kernel expansion up to linear terms in the reference curvature by using the Mellin transform , so that
¯¯¯¯¯¯Tr[W(−ˆ□)]= (19) χB(x)dν(4π)d/2{Qd2∫ddx√^g+16Qd2−1∫ddx√^g^R},
with
Qn[W(−ˆ□)]=1Γ[n]∫∞0qn−1W(q)dq. (20)
In particular, using the relation ,
Qd2=1Γ[d2]∫dqqd2−1e−sZkχ2νBq=1(sZkχ2νB)d2, (21a) Qd2−1=1Γ[d2−1]∫dqqd2−2e−sZkχ2νBq=1(sZkχ2νB)d2−1.
By inserting expression (III) and (II) in the flow equation (9) and using (19) with (21), the coefficients of the and terms due to the renormalized flow of the and operators are easily identified. At last the functions for the dimensionless running Newton constant and the dimensionless Cosmological constant can be obtained with the introduction of the “anomalous dimension” , so that
βg(g,λ)≡k∂kgk=(d−2+η)gk (22)
and in general. In four dimensions we have
βg= gk(2−gk(n−2λk)n−1nnΓ[n−1]6πΓ[n]), (23a) βλ= λk(−2−gk(n−2λk)n−1nnΓ[n−1]6πΓ[n])+ (23b) +gk(n−2λk)n−2nnΓ[n−2]2πΓ[n].
The expressions for the -dimensional functions are listed in the Appendix B, together with the limiting cases and .
iii.2 Rd topology
In the case of a flat topology the scalar curvature of the reference metric vanishes, constraining the quadratic term of the action (14) to be zero. In order to extract the beta function from the flow equation (9) it is convenient to consider a general truncation of the type
Sk[ϕ]=∫ddx √ˆg(12ˆgμνZk∂μϕ∂νϕ+Vk(ϕ)), (24)
where and employ a derivative expansion around an homogeneous background plus a fluctuation, so that . In this case we have
Sk[ϕ]=∫ddx√ˆg{−12Zk~f(x)ˆ□~f(x)+Vk[χB]+ (25) +V′k[χB]~f(x)+12V′′k[χB]~f(x)2+O(~f(x)3)+O(∂4~f)}.
Therefore,
∂tSk[~f(x)]= (26) −12∫ddx√^gχdνB∫dss∂tτnk⟨x|e−s(K+δK)|x⟩,
where
K=−Zkˆ□+V′′k[χB], (27a) δK=V′′′k[χB]~f(x)+12V′′′′k[χB]~f(x)2. (27b)
The trace in (26) can be evaluated in a background-independent way by means of an integration in momentum space over the eigenvalues of the Laplacian built from the background metric , inserting in (26) the identity
∂tSk[~f(x)]=−12∫ddx√^gχdνB∫dd¯p(2π)d∫dss∂tτnk(s)⟨x|¯p⟩⟨¯p|e−sK(1−sδK+s22!{[δK,K]+δK2}+…)|x⟩, (28)
where the dots stand for the higher order terms in the expansion of the exponential and
⟨x|¯p⟩=e−i¯px. (29)
The matrix elements of the expanded heat kernel can then be calculated ordering the operators by means of the commutation rule
[¯pμ,~f(x)]=−i∂μ~f(x). (30)
It is then straightforward to identify the coefficients of the and terms, obtaining the following set of coupled equations:
k∂kVk=M(k2χ2νB)d21(1+V′′k(χB)k2nZkχ2B)n−d2, (31a) k∂kZk=N(k2χ2νB)d2−3(V′′′k/Zk)2(1+V′′k(χB)k2nZkχ2B)n+3−d2, (31b)
where
M=(n4π)d2Γ(n−d2)Γ(n), (32a) N=(d−2(n+1))(d−2(n+2))24dn2(n4π)d2Γ(n−d2)Γ(n).
The functions for the dimensionless couplings of the CREH truncation are then obtained introducing a polynomial ansatz for the dimensionful potential of the type
Uk[χB]=−k2λk6χ4B, (33)
so that one obtains in four dimensions the coupled set of equations
βg= gk(2−2gkλ2k(n−2λk)n−1Γ[n+1]nn9πΓ[n]), (34a) βλ= gk(n−2λk)n−2Γ(n−2)nn2πΓ[n]+ +λk(−2−2gkλ2k(n−2λk)n−1Γ[n+1]nn9πΓ[n]).
iii.3 Fixed points and linearized flow
The functions (23) and (34) vanish both at the GFP located at , and at a NGFP defined at , . The properties of the linearized flow around the NGFP are determined by the stability matrix
Bij=(∂giβgj)∣∣{gi}={g∗i}, (35)
, whose eigenvalues form in general a complex conjugate pair. A negative real part of the eigenvalues, i.e. a positive (we will refer to it as the first Lyapunov exponent, following the standard notation used in dynamical systems), implies the stability of the fixed point, while the imaginary part characterizes the spiral shape near the fixed point. Our results in four and dimensions are summarized, respectively, in Table 1 and Table 2 in the Appendix A.
It is clear from Table 1 that also the theory defined by the CREH approximation is asymptotically safe, although the scaling properties are rather different from those obtained from the full EH in prop . For instance, the critical exponents and display an -dependence which is stronger in the case of the CREH than for the non-reduced theory, although the quantity is rather stable in both cases.
We can quantify the impact of the EH conformal reduction with respect to the full EH theory by defining a -type of “distance” in the space of the “universal” quantities, by means of
χ2(n)=(λ∗g∗(C)−λ∗g∗(E))2λ∗g∗(C)2+λ∗g∗(E)2+(θ′(C)−θ′(E))2θ′(C)2+θ′(E)2,+(θ′′(C)−θ′′(E))2θ′′(C)2+θ′′(E)2 (36)
where “C” and “E” stands for CREH and EH, respectively.
A plot of this quantity as a function of is depicted in the upper panel of Fig.(1), for the projection (solid line) and the projection (dashed line) where it is clear that the minimum is attained for in both cases. On the other hand, in the case of the projection the scaling properties are much less sensitive to the cutoff parameter , and the limit is as good as the case.
Of particular interest is the limit for the topology, in which the first Lyapunov exponent vanishes. In this case the theory is still UV finite although not asymptotically safe anymore, since now the linearized system is defined by pure imaginary eigenvalues and every perturbation of the NGFP will evolve in a cyclic trajectory.
It is also interesting to discuss the scaling properties of the theory in the projection as the dimension is changed. This is shown in the middle panel of Fig.(1) for for , and for the dimensionless quantity . The first Lyapunov exponent vanishes for a critical dimension value so that the fixed point undergoes an Hopf bifurcation as the dimension crosses (represented in Fig.(2)).
As it is shown in Fig.(3), for the cycle collapses on the line. In this regime it shows a non homogeneous running due to the low transient of the trajectory near the GFP, while it becomes an homogenous slow transient around the NGFP in the limit .
Notice that the critical dimension is a function of , , and while for the critical dimension is , generally holds for a finite value of the parameter . At the UV behavior is regulated by a limit cycle whose behavior resembles the one of the Van der Pol oscillator vander .
For (see left panel of Fig.(2)) the theory space is now divided in two regions. The first is the set of points in parameter space outside the cycle, which trajectories flow towards the UV to the limit cycle and hit in the IR the singularity (or flow towards ). Those are the trajectories which survive for and that require higher-order operators in order to cure the IR sector. The second region is the set of points inside the cycle which flow towards it in the UV and towards the NGFP in the IR. The latter case leads to a new interesting scenario in which the UV and IR critical manifolds coincide and the EH truncation is finite at every energy scale.
For this scenario to be plausible we require the cyclic trajectory to be close enough to the GFP, so that it shows a semiclassical regime. Unfortunately, as can be seen from Fig.(3), in the best case ( for ) a limit cycle with a good semiclassical regime occurs only for . It is also important to stress that the limit cycle never approaches the singularity , where the EH truncation stops to work.
Since the Hopf bifurcation is not present in the projection for the CREH, also for small values of the dimension, we analyzed the behavior of the linearized flow near the NGFP in the case of the full EH truncation, to verify if the Hopf bifurcation is still present in the projection for some value of the parameter . Numerical results are collected in Table 2 in Appendix A while the functions are listed in Appendix B, and are a simple dimensional generalization of the results reported in prop . As it can be seen from Table 2 the full theory presents a stable NGFP in the whole plane, which means that the contribution of spin-2 degrees of freedom lower the value of the critical dimension under the “critical” value .
Although such a non trivial behavior in the UV region seems to be a direct consequence of the strong dependence of the flow in the projection on the cutoff parameter , it is interesting to notice that recent investigations based on “tetrad only” theory spaces triat , and on the minisuperspace approximation of the EH truncation cycle3 , also show the presence of limit cycles in the UV and IR limit, respectively. In the latter case, however, the limit cycle originates by an Hopf bifurcation of a specific cutoff parameter priv , while in our case the bifurcation is governed by the spacetime dimension, so that our limit cycle is UV and not IR.
In closing this section we would like to mention that the intriguing possibility of such a non trivial UV completion in field theory was first pointed out by Wilson in a seminal paper (before the discovery of asymptotic freedom), in the context of QCD wilson71 . In particular it was argued that, at the experimental level, the presence of a limit cycle would show up in “perpetual” oscillations in the total hadronic cross section in the limit of large momenta. In the case of gravity the natural arena to discuss this type of phenomenon is the physics of the early Universe, for which an effective Lagrangian embodying the properties of the limit cycle can be determined by using the strategy outlined in bonanno12 . In the case at hand we expect that where is a renormalization scale. On the other hand, discussing the detailed physical implications of this model is beyond the scope of this paper.
Iv Non polynomial truncations and symmetry breaking
In this section we would like to understand the structure of the UV critical manifold beyond the polynomial truncation discussed in the previous section. We are interested in the possibility of having a transition to a phase of broken diffeomorphism invariance at low energy creh2 and thus we intend to numerically solve (31). In fact although in four dimensions, an ansatz of the type
V[χ]=c1(k)χ2+c2(k)χ4,Z=c3(k) (37)
is an exact polynomial truncation of the coupled Eqs. (31); the question concerning the RG evolution of a generic initial data determined by (31) cannot be answered with this strategy. The core of the problem can be grasped in the LPA approximation which solves only (31a) by assuming a RG evolution for the wave-function renormalization functional . In fact although the solution of the coupled problem (31) is beyond the aim of this work, we present a successful numerical strategy to deal with (31a) which we hope can eventually be extended to treat the coupled system (31a) and (31b) beyond the simple CREH truncation. In particular, in this section we shall investigate the role played by higher powers of volume operators of the type in providing a transition to a phase of broken diffeomorphism invariance. In order to carry out the numerical integration of (31a), it is useful to “linearize” the evolution equation for the potential by defining the quantity
(38)
with that diverges at as the “spinodal line” is approached, but it behaves as a power law for large values of the field outside the “coexistence” region where . In terms of this new variable Eq.(31a) reads
(2+η)nk2Zχ2(W−1γχ4γ−1) (39) −nk2Zχ4γ+2γ−1W−1γ−1k∂kW=Ank4∂2xxW.
The advantage of this manipulation is that Eq. (39) is now linear in the second derivative.
Ideally we would like to evolve an initial data defined at the cutoff scale along the RG direction, i.e. towards the infrared. This is usually achieved by defining the RG time via with so that the Cauchy problem is fully determined when is fixed and is given, being an asymptotic value of the field ( in actual calculations). However, if we intend to do so, we immediately run into the difficulty that as , equation (39) belongs to the restricted élite of the backward-parabolic equations, i.e. a class of diffusion-type partial differential equation with a negative diffusion constant. As it is well known, in this case the Cauchy problem is not “well-posed” (in the sense of zakh ) and the existence of the solution for generic initial data is not guaranteed even for an infinitesimal time step.
Although we already know that an admissible initial data in four dimensions is precisely of the type (37) it seems that the CREH truncation is one of the few initial data compatible with the general flow equation (31). In fact, the important question concerning the possibility of developing a non-zero vacuum expectation value of the conformal factor in the limit cannot be answered within the simple CREH truncation.
It is therefore necessary to treat the question as a sort of inverse problem and to consider an integration in the UV direction instead, so that the RG time is defined as with , the Cauchy problem is well posed, and the solution is unique. Clearly, once the solution in the deep UV is found it is possible to argue that precisely that solution is an admissible initial data for a non-singular IR flow. However, also in the case of the UV evolution, due to its strong nonlinearities, a proper numerical strategy is to implement a fully implicit predictor-corrector scheme on an uniform spatial and temporal grid as discussed in the Appendix C.
The boundary condition at is of the von Neumann type so that at the inner boundary, and : we have checked that our results are rather insensitive to the choice of that could be set arbitrarily close to zero in all calculations (note however that, strictly speaking, always). The outer boundary is taken at some where for a power-law behavior is assumed, like in the more familiar Ising model bonlac . The initial value of the potential at the cutoff reads
V[χ,0]=λ6χ4+σχ6+ωχ8, (40)
where the bare values of , and have been chosen in order to display a non-zero minimum as an initial condition. In addition we have considered also the coupling in order to have a real function for large values of the field. In fact, unless no consistent initial condition can be given in all the real line for the potential, as the threshold functions (the denominator in (31a) ) become complex at a finite value of for . In solving (39) close to the NGFP, we have set and in as we are interested in the UV evolution.
Our results are then summarized in Fig.(5): in the left panel a symmetry breaking initial state evolves towards a convex potential as the UV evolution is followed. The final, fixed point state, is then reached already for as it can be seen in the right panel. Note the “flat bottom” of the potential and the almost exponential suppression at large values of the field in the final solution. We found that the appearance of a “fixed point” potential of the type shown in Fig.(5) seems to be quite generic if the initial condition is changed.
In Fig.(5) another example of the UV evolution is shown for and for a different set of initial conditions. Note in particular that using instead the UV potential at and integrating towards the IR, a symmetry breaking vacuum appears at low energy. It is convenient to introduce the following order parameter to characterize the phase of the system in this case polyakov :
B(L)=⟨exp−∫L0gμν(x(s)˙xμ(s)˙xν(s)ds⟩, (41)
which is the analogue of the Wilson loops in gauge theory. In our case and which is therefore a classical flat metric on . For this geometry the well-known heat-kernel behavior at small distances occurs.
The large field behavior of the potential is characterized by an inverse power behavior for large value of the field thus signaling the presence of non local invariants in the fixed point potential. The conclusion of our numerical experiment seems to suggest that there exists a more complex fixed point structure, not necessarily of the CREH type, whose precise structure is unaccessible with the more standard function approach.
V conclusions
In this work we further explored the universal properties of the CREH theory around the NGFP.
The NGFP characterizes the UV evolution for different projection in the theory space and for a large class of threshold functions although we also found the possibility that the continuum limit is defined by means of a limiting-cycle in some cases. Moreover it is possible to find an “optimal” threshold function for which the renormalization flow minimizes the differences with the calculation of the universal quantities in full Einstein-Hilbert truncation.
However, going beyond the CREH truncation has proven to be rather difficult because of the flawed structure of the flow equation in the presence of the conformal factor instability, and only an UV integration was possible. How can we make contact with “real” gravity that is formally defined at in more general truncations? Clearly, a mechanism to stabilize the conformal factor modes is needed and, most probably, the inclusion of the term is essential to provide a consistent truncation oliverunstable ; saur2 . Recent investigations discussed in dario for a general can provide important information for a more complete understanding of the renormalization flow in the theory space. On the other hand in our investigations we explored the infinite dimensional theory space spanned by the solutions of the (partial differential) flow equation (39) which is not discussed in dario because of the technical difficulty in solving the flow equation for a generic theory.
We find that the structure of the UV region around the NGFP can be richer than expected, with a class of fixed point potentials displaying an inverse power behavior for large value of , suggesting the presence of non local volume invariants in the fixed point potential frank2 . Within this class of local potentials it seems possible to realize a situation where the diffeomorphism invariance is broken at low energy. It would be interesting to further extend this investigation by including a consistent running of and the contribution from the term to see the infrared flow structure of an -stabilized conformal factor beyond polynomial truncations, and we hope to address this point in a following work.
Acknowledgements.
We would like to thank Dario Benedetti and Roberto Percacci for important comments and discussions. One of us (F.G.) is grateful to INFN, Rome University “La Sapienza”, and INAF for financial support.
Appendix A Critical Exponents and Universal Quantities
In this appendix, we show Tables I and II.
Appendix B β Functions
In this appendix are listed the explicit expressions of the functions in dimensions for different choices of the cutoff function. The functions obtained using the cutoff (11) will be referred as smooth proper-time cutoff, where the limits and will be called respectively sharp momentum cutoff and sharp proper-time cutoff.
b.1 The projection on Sd
• CREH - smooth proper-time cutoff
βg=gk⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝d−2−22−d(d−2)π1−d2gknnΓ(−d2+n+1)(n−d(2dd−2−1)λk2(d−1))n−d2+1(d−1)Γ(n)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (42)
βλ=23−dπ1−d2gknnΓ(n−d2)Γ(n)(n−d(2dd−2−1)λk2(d−1))n−d2+λ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−2−22−d(d−2)π1−d2gknnΓ(−d2+n+1)(n−d(2dd−2−1)λk2(d−1))n−d2+1(d−1)Γ(n)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (43)
• CREH - sharp proper-time cutoff
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https://wiki.openelectrical.org/index.php?title=Anatomy_of_a_Short_Circuit | # Anatomy of a Short Circuit
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A short circuit is an electrical fault where a conductive path (usually of low impedance) is formed between two or more conductive parts of an electrical system (e.g. phase-phase, phase-earth, phase-neutral, etc). This article looks at the nature of short circuits and tries to break down and explain the constituent parts of fault currents. Note that the terms "short circuit" and "fault" are often used interchangeably.
In most networks, a short circuit is similar to the closing transient of an RL circuit, where the R and L components are the impedances of the source(s). The transient characteristics of short circuit currents vary depending on whether they are near or far from synchronous generators. The sections below describe the two general types of short circuits:
## Near-to-Generator Short Circuit
A fault close to a synchronous generator has the following maximum short circuit current $i_{sc}(t)$:
$i_{sc}(t) = E \sqrt{2} \left[ \left( \frac{1}{X_{d}''} - \frac{1}{X_{d}'} \right) e^{-t/t_{d}''} + \left( \frac{1}{X_{d}'} - \frac{1}{X_{d}} \right) e^{-t/t_{d}'} + \frac{1}{X_{d}} \right] \sin (\omega t) + \frac{E \sqrt{2}}{X_{d}''} e^{-t/t_{a}} \,$
Where $E \,$ is the phase-to-neutral rms voltage at the generator terminals (V)
$X_{d}'' \,$ is the generator direct-axis subtransient reactance ($\Omega$)
$X_{d}' \,$ is the generator direct-axis transient reactance ($\Omega$)
$X_{d} \,$ is the generator synchronous reactance ($\Omega$)
$T_{d}'' \,$ is the generator subtransient time constant (s)
$T_{d}' \,$ is the generator transient time constant (s)
$T_{a} \,$ is the aperiodic time constant (s)
From the above equation, it can be seen that the short circuit current can be broken up into an aperiodic current (dc component of the short circuit):
$\frac{E \sqrt{2}}{X_{d}''} e^{-t/t_{a}} \,$
And a series of three damped sinusoidal waveforms corresponding to the following distinct stages:
(1) Subtransient component: $E \sqrt{2} \left( \frac{1}{X_{d}''} - \frac{1}{X_{d}'} \right) e^{-t/t_{d}''} \sin (\omega t) \,$
This period typically lasts 10 to 20ms from the start of the fault. The subtransient reactance is due to the flux casued by the stator currents crossing the air gap and reaching the rotor surface or amortisseur / damper windings.
(2) Transient component: $E \sqrt{2} \left( \frac{1}{X_{d}'} - \frac{1}{X_{d}} \right) e^{-t/t_{d}'} \sin (\omega t) \,$
This period typically lasts 100 to 400ms after the subtransient period. The transient reactance occurs when all the damping currents in the rotor surface or amortisseur / damper windings have decayed, but while the damping currents in the field winding are still in action.
(3) Steady-state component: $E \sqrt{2} \frac{1}{X_{d}} \sin (\omega t) \,$
The steady-state occurs after the transient period when all the damping currents in the field windings have decayed, and essentially remains until the fault is cleared.
Putting these all together, we get the familiar near-to-generator short circuit waveform:
## Far-from-Generator Short Circuit
In short circuits occurring far from synchronous generators, we can ignore the effects of the generator subtransient behaviour. It can be shown through transient circuit analysis that the maximum far-from-generator short circuit is as follows:
$i_{sc}(t) = \frac {E \sqrt{2}}{Z_{sc}} \left[ \sin \left( \omega t + \frac{\pi}{2} \right) - e^{-\frac{R}{X} \omega t} \right] \,$
Where $E \,$ is the rms voltage of the circuit (V)
$Z_{sc} \,$ is the fault impedance ($\Omega$)
$\frac{R}{X} \,$ is the R/X ratio at the point of fault (pu)
We can see that there are two components:
(1) A decaying aperiodic component: $- \frac {E \sqrt{2}}{Z_{sc}} e^{-\frac{R}{X} \omega t} \,$
(2) A steady state component: $\frac {E \sqrt{2}}{Z_{sc}} \sin \left( \omega t + \frac{\pi}{2} \right) \,$
Putting these together, we get the total far-from-generator fault current:
During the transient period, the peak transient current is typically 1.5 to 2.5 times higher than the peak steady state current. | {"extraction_info": {"found_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": 0.8978695273399353, "perplexity": 1985.8501651523206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986647517.11/warc/CC-MAIN-20191013195541-20191013222541-00069.warc.gz"} |
https://tioj.ck.tp.edu.tw/submissions/161657 | 456
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# Testdata Results
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0 1 24 5412 Wrong Answer 0
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7 2 24 5584 Wrong Answer 0
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9 2 24 6088 Wrong Answer 0
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12 3 24 5492 Wrong Answer 0
13 3 20 5308 Wrong Answer 0
14 3 20 5456 Wrong Answer 0
15 4 24 5488 Wrong Answer 0
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17 4 20 5496 Wrong Answer 0
18 4 20 5416 Wrong Answer 0
19 4 20 5316 Wrong Answer 0
Submitter:
Compiler:
c++14
Code Length:
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http://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-6th-edition/chapter-8-section-8-1-solving-quadratic-equations-by-completing-the-square-exercise-set-page-482/16 | ## Intermediate Algebra (6th Edition)
Original Equation x²+4=0 Subtract 4 from both sides x²=-4 Take the square root of both sides x=±$\sqrt -4$ Simplify the radical x=±2i | {"extraction_info": {"found_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": 0.9461032152175903, "perplexity": 3590.392323896985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1492917119637.34/warc/CC-MAIN-20170423031159-00516-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://specialfunctionswiki.org/index.php?title=Exponential_integral_Ei&oldid=9292 | # Exponential integral Ei
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
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The exponential integral $\mathrm{Ei}$ is defined for $x>0$ by $$\mathrm{Ei}(x) = \mathrm{PV}\int_{-\infty}^x \dfrac{e^t}{t} \mathrm{d}t,$$ where $\mathrm{PV}$ denotes the Cauchy principal value and $e^t$ denotes the exponential.
# References
$\ast$-integral functions | {"extraction_info": {"found_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": 0.9999793767929077, "perplexity": 3654.907443998025}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583814455.32/warc/CC-MAIN-20190121213506-20190121235506-00532.warc.gz"} |
https://nrich.maths.org/5899 | ### Tweedle Dum and Tweedle Dee
Two brothers were left some money, amounting to an exact number of pounds, to divide between them. DEE undertook the division. "But your heap is larger than mine!" cried DUM...
### Lower Bound
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
### Sum Equals Product
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?
# All Tangled Up
##### Stage: 3 Challenge Level:
This problem follows on from Twisting and Turning and More Twisting and Turning in which twisting has the effect of adding 1 and turning transforms any number into the negative of its reciprocal.
We can start at 0 and end up at any fraction of the form $$\frac{n}{n+1}$$ by following the sequence: twist, twist, twist, ... , twist, twist, turn, twist
eg. to end up at $\frac{4}{5}$:
twist, twist, twist, twist, twist, turn, twist
to produce:
$0, 1, 2, 3, 4, 5, \frac{-1}{5}, \frac{4}{5}$
Check you can reach $\frac{9}{10}$
The sequence twist, twist, turn, twist, twist, turn, twist, twist, turn, ... , twist, twist, turn, twistwill lead us from 0 to all the fractions of the form $$\frac{1}{n}$$ eg. to end up at $\frac{1}{5}$ (and $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{4}$ along the way):
twist, twist, turn, twist, twist, turn, twist, twist, turn, twist, twist, turn, twist
to produce: 0, 1, 2, $\frac{-1}{2}$, $\frac{1}{2}$, $\frac{3}{2}$, $\frac{-2}{3}$, $\frac{1}{3}$, $\frac{4}{3}$, $\frac{-3}{4}$, $\frac{1}{4}$, $\frac{5}{4}$, $\frac{-4}{5}$, $\frac{1}{5}$
Check you can reach $\frac{1}{10}$
Can you find other sequences of twists and turns that lead to special fractions?
Is it possible to start at 0 and end up at any fraction? | {"extraction_info": {"found_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": 0.9271236658096313, "perplexity": 1658.746898987686}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187821189.10/warc/CC-MAIN-20171017125144-20171017145144-00277.warc.gz"} |
https://glenmartin.wordpress.com/2011/06/28/a-wrap-up-of-a-spectacular-12-months-of-quantum-physics-experiments/ | ## A Wrap-up of a Spectacular 12 Months of Quantum Physics Experiments
Yeah, I know this is the middle of the year, but year boundaries are so arbitrary. Once upon a time (even here in America, prior to 1752), New Years Day was March 25, so…whatever….
However you mark the passage of time, there have been some exciting experimental breakthroughs in the world of experimental quantum mechanics over the last 12 months. I’m not talking about exotic experiments designed to look for dark matter or the Higgs boson, or attempts to confirm or refute String Theory or SUSY, but rather experiments which probe and test the most basic fundamentals of quantum theory. (Well, in all fairness, one of the experiments I’m about to mention could be used to probe some aspects of String Theory, as well as other candidates for a quantum theory of gravity, and another has results which seem to invalidate some predictions made by SUSY.) Here are some of the highlights:
## Confirmation of the Born Rule via a triple-slit experiment
Young’s double-slit experiment has long been a mainstay for both demonstrating and testing some of the more bizarre and counter-intuitive aspects of quantum mechanics (so fundamental, in fact, that almost every book Feynman ever wrote about QED seems to start with an analysis of it, and it occupies a big chunk of his more famous lectures), especially the Born Rule, one of the most fundamental elements of quantum mechanics, which tells us how to mathematically combine quantum probability amplitudes in order to calculate the probability of a given outcome.
Remember that, in quantum mechanics, probabilities are calculated differently than in any other discipline. The probability of any event A taking place, P(A), is calculated by taking the absolute square of a probability amplitude, a, which is a complex number, and may be a function of time and/or position, so P(A)=|a(x,t)|2. Now, the rules for combining probability amplitudes for various competing outcomes are where things get complicated. In the case of the double-slit experiment, we have a probability a1 that the particle will pass through the first slit, and a probability a2, that it will pass through the second slit. Classically, we would expect that the overall resulting probability would be a simple superposition of the two cases, P=|a1|2+|a2|2, but this is not at all what we see experimentally (unless we modify the experiment to detect which slit the particle actually came through, but then we are changing the problem). What we actually see is P=|a1+ a2|= |a1|2+|a2|2+2|a1a2|, where the last term is what introduces interference effects.
But what if we add a third slit? Do we get some crazy probability distribution with higher order terms? The Born Rule says no, and these experiments confirm that.
Sinha et al., “A Triple Slit Test for Quantum Mechanics”, Physics in Canada. Vol. 66, No. 2 (Apr.-June 2010), pp. 83-86
Sinha et al., (23 July 2010). “Ruling Out Multi-Order Interference in Quantum Mechanics”.Science 329 (5990): 418-421. Bibcode 2010Sci…329..418Sdoi:10.1126/science.1190545PMID 20651147
Sanders et al., (14 August 2010). “Triple slits don’t add interference”. Science News 178 (4): 12. Bibcode2010Sci…329..418Sdoi:10.1126/science.1190545PMID 20651147
## Observation of gravitational quantum states of cold neutrons
I had actually referenced this in an earlier post. It had long been thought that quantum gravitational effects would be far too weak to measure in the Earth’s anemic gravitational field. (It would be much easier to measure quantum gravitational effects on, for example, the surface of a neutron star. Well, aside from the intense gravity of the neutron star crushing any measuring apparatus placed upon it into oblivion, but I digress….) This difficulty in experimentally probing the quantum mechanical aspects of gravitational phenomena is precisely what makes it so challenging for physicists to construct a workable theory of quantum gravity.
Well, some clever scientists have figured out a way to do it, by, of all things, bouncing cold neutrons. Seriously.
Jason Palmer “Neutrons could test Newton’s gravity and string theory”, BBC Online (17 April 2011).
Tobias Jenke et al. “Realization of a gravity-resonance-spectroscopy technique”Nature Physics (17 April 2011) | doi:10.1038/nphys1970
Valery V. Nesvizhevsky et al“Quantum states of neutrons in the Earth’s gravitational field”Nature 415, 297-299 (17 January 2002) | doi:10.1038/415297a
## Using “weak measurement” of a photon’s momentum to reconstruct average photon trajectories through a double-slit experiment
Returning to the topic of the double-slit experiment, some rascally researchers have found a way to push Heisenberg’s Uncertainty Principle to its limits in such an experiment. The Uncertainty Principle states that the product of the uncertainties in the values of any pair of canonically conjugate quantum observables (for example, momentum and position) must be greater than or equal to the reduced Planck constant divided by 2:
$\Delta p \Delta q \geq \frac{\hbar}{2}$
Mind you, the right-hand side of that expression is a pretty tiny value, and can be taken to equal zero in the classical limit (which is essentially the mathematical definition of taking a quantum system to the classical limit), so at the classical scale, it doesn’t mean much. But in the microscopic world of quantum mechanics, the Uncertainty Principle rules all, setting severe limits on what we can or cannot know about a system, not as a consequence of how sensitive or cleverly designed our instruments are, but rather as a result of fundamental limitations of reality. (Exactly why that is the case is a bit beyond the scope of this posting. Think about the Fourier Transform of a particle’s quantum state function and what it means to pluck exact values for both position and momentum out of that, and you might grasp why. For the moment, just roll with it.)
The upshot of the Uncertainty Principle is that making precise measurements of a particle’s momentum effectively destroys the precision with which we can measure its position, and vice-versa. But, to know a particle’s trajectory (in the classical sense, at least), we need at least some information about both. What these clever researchers have done is to make intentionally imprecise (“weak”) measurements of the particle’s momentum, thus giving themselves some breathing room for measurement of the particle’s position.
Jason Palmer, “Quantum mechanics rule ‘bent’ in classic experiment”, BBC News Online (3 June 2011).
Kocsis et al., “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer”, Science, 3 June 2011: Vol. 332 no. 6034, pp. 1170-1173. DOI: 10.1126/science.1202218
## Observation of quantum interference with large organic molecules
Young’s double-slit experiment (Yep, we are still talking about that!) was initially conducted with light, but has since been performed with electrons, neutrons, and even atoms, all with the result of an interference pattern consistent with quantum mechanical behavior. (These experiments have even been performed with “feeble” sources, trickling one particle at a time through the apparatus and still producing interference, demonstrating that the particles are interfering with themselves, not with one another.) The tricky part is that the more massive the particles with which we conduct the experiment, the smaller the wavelength of the interference phenomenon. Moving up in scale, we eventual reach a point at which the quantum interference is drowned out, and the results give way to classical behavior. This fuzzy boundary between quantum and classical behavior is an illustration of what is known as quantum decoherence, and is a particularly fascinating area of research.
Recently, researchers have pushed the boundary of this quantum/classical transition further than ever before by observing quantum interference with large organic molecules. Such experiments had previously been performed with carbon-60 (“buckyballs”). These molecules are particularly well suited since their spherical shape makes them relatively compact, keeping their physical extent from overwhelming the interference effects. The researchers are considering continuing their work using spherical viruses.
“Researchers Find ‘Fattest Schrodinger Cats Realized to Date”, Discover Blogs, April 7, 2011.
“Wave-particle duality seen in carbon-60 molecules”, PhysicsWorld, Oct 15, 1999.
Gerlich, S. et al. “Quantum interference of large organic molecules”Nat. Commun.2:263 doi: 10.1038/ncomms1263.
## A high-precision measurement of the electric dipole moment of the electron.
The word “shape” in the title of the paper describing this work is rather misleading, as an electron technically has no shape. It is generally regarded as a point charge (albeit a rather fuzzy point thanks to the Heisenberg Uncertainty Principle), thus having no spacial extent, and thus no surface. What “shape” really refers to here is the shape of the electron’s electric field. The QED predicts that the electric field of an electron is perfectly spherical, with the exception of a tiny deviation known as the electric dipole moment (EDM). The Standard Model predicts that the EDM is too small by several orders of magnitude to be measurable by currently available methods (not to be confused with the magnetic dipole moment due to the electron’s intrinsic angular momentum, or “spin,” which has been readily measurable since the days of the 1922 Stern-Gerlack experiment). However, there are extensions to the Standard Model which predict values for the EDM which are in ranges which are measurable today, making such measurements a useful tool for testing the validity of those extensions. The upshot of the experiment is that an upper bound was placed on the value of the EDM which essentially rules out those particular extensions to the Standard Model (yet another coffin nail for SUSY, but that would be the subject of a future post).
Hudson, et al., “Improved measurement of the shape of the electron”, Nature 473, 493-496 (26 May 2011). doi:10.1038/nature10104 | {"extraction_info": {"found_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": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8615661859512329, "perplexity": 575.4243527401342}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1505818693459.95/warc/CC-MAIN-20170925220350-20170926000350-00065.warc.gz"} |
https://infoscience.epfl.ch/record/229540 | ## A test for skewed distributions of dark matter, and a possible detection in galaxy cluster Abell 3827
Simulations of self-interacting dark matter predict that dark matter should lag behind galaxies during a collision. If the interaction is mediated by a high-mass force carrier, the distribution of dark matter can also develop asymmetric dark matter tails. To search for this asymmetry, we compute the gravitational lensing properties of a mass distribution with a free skewness parameter. We apply this to the dark matter around the four central galaxies in cluster Abell 3827. In the galaxy whose dark matter peak has previously been found to be offset, we tentatively measure a skewness s = 0.23(-0.22)(+0.05) in the same direction as the peak offset. Our method may be useful in future gravitational lensing analyses of colliding galaxy clusters and merging galaxies.
Published in:
Monthly Notices Of The Royal Astronomical Society, 468, 4, 5004-5013
Year:
2017
Publisher:
Oxford, Oxford Univ Press
ISSN:
0035-8711
Keywords:
Laboratories: | {"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": 0.8941714763641357, "perplexity": 1004.7570174625214}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864572.13/warc/CC-MAIN-20180521235548-20180522015548-00257.warc.gz"} |
https://www.ias.edu/news/status-supersymmetry | # The Status of Supersymmetry
The Standard Model of particle physics is both fantastically successful and glaringly incomplete.
Its predictions have pieced together many of the known features of the universe and guided physicists to new discoveries, such as the Higgs boson. But it cannot account for the existence of dark matter—the mysterious substance that makes up 85% of the universe’s matter—or explain the Higgs boson’s mass.
How can scientists fill the gaps? For decades, a set of theories collectively known as supersymmetry seemed to provide an elegant solution. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8790814876556396, "perplexity": 688.2235984048783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656103646990.40/warc/CC-MAIN-20220630001553-20220630031553-00741.warc.gz"} |
https://skyfallmeteorites.com/education-research/glossary/t-tauri-star/ | # T Tauri Star
Protostar in the late stages of formation, often exhibiting both periodic and random fluctuations in brightness. T Tauri stars are newly-formed (<10 Ma) low to intermediate mass stars (< 3 Msun) with central temperatures too low for nuclear fusion to have started. For the first ~100 Ma, all emitted radiation comes from gravitational energy released as the star contracts under its own self-gravity. T Tauri stars represent an intermediate stage between true protostars (e.g. YY Orionis stars) and low-mass main sequence (hydrogen burning) stars like the Sun.
The nearest T Tauri stars are in the Taurus and r Ophiuchus molecular clouds, both ~400 light years away. Indications of stellar winds and jets show that at least some T Tauri stars are interacting with their environments. Both the winds and jets of are probably powered by material falling onto the central star from the accretion disk (or protoplanetary disk) observed around many of them. The random variability of T Tauri stars (with time-scales from minutes to years) are probably caused by instabilities in the accretion disk (which also produce the “bullets” of material seen in the jet of HH-30), flares on the stellar surface, or simple obscuration by nearby dust and gas clouds. Periodic (regular) variations (days) are almost certainly associated with huge sunspots on the stellar surface which pass into and out of view as the star rotates.
Image source: http://hubblesite.org/newscenter/archive/releases/2000/32/image/c/.
This entry was posted in . Bookmark the permalink. | {"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": 0.841414213180542, "perplexity": 3708.6851959182372}, "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-2023-06/segments/1674764499819.32/warc/CC-MAIN-20230130133622-20230130163622-00073.warc.gz"} |
http://renormalization.com/tag/general-gauge-theories/ | ## General gauge theories
Using the background field method and the Batalin-Vilkovisky formalism, we prove a key theorem on the cohomology of perturbatively local functionals of arbitrary ghost numbers, in renormalizable and nonrenormalizable quantum field theories whose gauge symmetries are general covariance, local Lorentz symmetry, non-Abelian Yang-Mills symmetries and Abelian gauge symmetries. Interpolating between the background field approach and the usual, nonbackground approach by means of a canonical transformation, we take advantage of the properties of both approaches and prove that a closed functional is the sum of an exact functional plus a functional that depends only on the physical fields and possibly the ghosts. The assumptions of the theorem are the mathematical versions of general properties that characterize the counterterms and the local contributions to the potential anomalies. This makes the outcome a theorem on the cohomology of renormalization, rather than the whole local cohomology. The result supersedes numerous involved arguments that are available in the literature.
PDF
Phys. Rev. D 93 (2016) 065034 | DOI: 10.1103/PhysRevD.93.065034
arXiv: 1511.01244 [hep-th]
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Last update: May 9th 2015, 230 pages
Contents: Preface | 1. Functional integral | 2. Renormalization | 3. Renormalization group | 4. Gauge symmetry | 5. Canonical formalism | 6. Quantum electrodynamics | 7. Non-Abelian gauge field theories | Notation and useful formulas | References
Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.) | {"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": 0.8730737566947937, "perplexity": 1411.2975236548684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20180815181003-20180815201003-00203.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/46986-prime-jacobson-radical-please-help.html | I want to find prime and jacobson radical radicals in Z[T]/(T^3), here Z = integers.
My definition of jacobson radical is that it is intersection of maximal ideals and prime radical is intersection of prime ideals.
2. Originally Posted by peteryellow
I want to find prime and jacobson radical radicals in Z[T]/(T^3), here Z = integers.
My definition of jacobson radical is that it is intersection of maximal ideals and prime radical is intersection of prime ideals.
let $R=\frac{\mathbb{Z}[T]}{}.$ recall that $N(R)$, prime radical, is also the set of all nilpotent elements of R. now it should be obvious to you that $N(R)=\frac{}{}.$
to find $J(R)$, Jacobson radical, we use this fact that in a (commutative) ring S, $a \in J(S),$ if and only if $1-ax$ is a unit for all $x \in S.$ it's very easy to
see that the units of $R$ are $\pm 1 + \alpha T + \beta T^2 + , \ \ \alpha, \beta \in \mathbb{Z}.$ from here conclude that $J(R)=\frac{}{}. \ \ \ \square$
3. you are saying that the set of prime radical is same as set of nilpotents, fine but How do you get that the prime radical is $<T>/<T^3>$.
and how do you know that $\pm 1 + \alphaT + \betaT + <T^3>$ is a unit in R but if this is a unit how can you conclude that jacobson radical is $<T>/<T^3>$.
you are saying that the set of prime radical is same as set of nilpotents, fine but How do you get that the prime radical is $<T>/<T^3>$.
an element of $R$ is in the form $u=p(T) + ,$ where $p(T) \in \mathbb{Z}[T].$ ( by the way we may assume that $p(T)$ is of degree at most $2$ because
the terms of degree 3 or bigger belong to $.$ ) now $u^n = 0$ means $(p(T))^n \in ,$ which is possible for some $n$ if and only if the constant
term of $p(T)$ is $0.$ thus $u$ is nilpotent if and only if $p(T) \in .$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 23, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9723640084266663, "perplexity": 108.19334150874631}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1440645359523.89/warc/CC-MAIN-20150827031559-00308-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://www.science.gov/topicpages/m/magnetic+field+environments.html | Note: This page contains sample records for the topic magnetic field environments from Science.gov.
While these samples are representative of the content of Science.gov,
they are not comprehensive nor are they the most current set.
We encourage you to perform a real-time search of Science.gov
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Last update: November 12, 2013.
1
PubMed
Potential exposures to extremely low frequency (ELF) magnetic fields were investigated in response to worker concerns about an apparent increased spontaneous abortion risk in a payroll office environment. Concern in this office centered on the use of video display terminals (VDTs), which have been investigated as a potential cause of adverse reproductive outcomes among women. In this investigation, magnetic field sources were evaluated using a hand-held survey meter. Emdex datalogging dosimeters were also used to determine full shift personal exposures for 15 women working in the payroll area. On average, the exposures of workers to ELF magnetic fields in the payroll office area ranged from 1.0 to 6.5 mG with a mean of 3.2 +/- 1.5 mG. The results of this study indicate that many sources of ELF magnetic fields, including printers, photocopiers, and the electrical distribution system, can contribute to a worker's exposure in an office environment. PMID:8147390
Breysse, P; Lees, P S; McDiarmid, M A; Curbow, B
1994-02-01
2
SciTech Connect
We consider a recently discovered class of instabilities, driven by cosmic ray streaming, in a variety of environments. We show that although these instabilities have been discussed primarily in the context of supernova-driven interstellar shocks, they can also operate in the intergalactic medium and in galaxies with weak magnetic fields, where, as a strong source of helical magnetic fluctuations, they could contribute to the overall evolution of the magnetic field. Within the Milky Way, these instabilities are strongest in warm ionized gas and appear to be weak in hot, low density gas unless the injection efficiency of cosmic rays is very high.
Zweibel, Ellen G.; Everett, John E. [Department of Astronomy, University of Wisconsin-Madison, 475 N Charter Street, Madison, WI 53706 (United States)
2010-02-01
3
A summary of magnetic field production mechanisms and effects is given. Discussions are included on the following areas: (1) stray magnetic and electric fields from tokamaks, (2) methods for reducing magnetic fields, (3) economics of magnetic field reductions, (4) forces on magnetizable objects near magnetic confinement fusion reactors, (5) electric field transients in tokamaks, (6) attenuation and decay of electromagnetic
H. B. Liemohn; D. L. Lessor; B. H. Duane
1976-01-01
4
PubMed Central
Although the electrocardiogram is known to be nondiagnostic within the bore of any high-field magnet due to the magnetohydrodynamic effect, there are an increasing number of applications that require accurate electrocardiogram monitoring of a patient inside the MRI room but outside of the magnet bore. Magnetohydrodynamic effects on the ST segment of the electrocardiogram waveform were investigated in six subjects at magnetic field strengths ranging from 6.4 mT to 652 mT at the aortic midarch, and the electrocardiogram was found to be accurate at magnetic fields below 70 mT. This corresponds to a distance of 160 cm from the isocenter and 80 cm from the bore entrance for the 1.5-T MRI system used in this study. These results can be translated to any MRI system, with knowledge of the fringe field. Accurate electrocardiogram monitoring is feasible in close proximity to the MRI magnet, such as during and after pharmacologic or exercise stress, or interventional or surgical procedures performed in the MRI room.
Jekic, Mihaela; Ding, Yu; Dzwonczyk, Roger; Burns, Patrick; Raman, Subha V.; Simonetti, Orlando P.
2011-01-01
5
The purpose of this project was to document widely applicable methods for characterizing the magnetic fields in a given environment, recognizing the many sources co-existing within that space. The guidelines are designed to allow the reader to follow an efficient process to (1) plan the goals and requirements of a magnetic-field study, (2) develop a study structure and protocol, and
1997-01-01
6
In the presence of alternating-sinusoidal or rotating magnetic fields, magnetic nanoparticles will act to realign their magnetic moment with the applied magnetic field. The realignment is characterized by the nanoparticle's time constant, ?. As the magnetic field frequency is increased, the nanoparticle's magnetic moment lags the applied magnetic field at a constant angle for a given frequency, ?, in rad/s. Associated with this misalignment is a power dissipation that increases the bulk magnetic fluid's temperature which has been utilized as a method of magnetic nanoparticle hyperthermia, particularly suited for cancer in low-perfusion tissue (e.g., breast) where temperature increases of between 4 and 7 degree Centigrade above the ambient in vivo temperature cause tumor hyperthermia. This work examines the rise in the magnetic fluid's temperature in the MRI environment which is characterized by a large DC field, B0. Theoretical analysis and simulation is used to predict the effect of both alternating-sinusoidal and rotating magnetic fields transverse to B0. Results are presented for the expected temperature increase in small tumors (approximately 1 cm radius) over an appropriate range of magnetic fluid concentrations (0.002-0.01 solid volume fraction) and nanoparticle radii (1-10 nm). The results indicate that significant heating can take place, even in low-field MRI systems where magnetic fluid saturation is not significant, with careful selection of the rotating or sinusoidal field parameters (field frequency and amplitude). The work indicates that it may be feasible to combine low-field MRI with a magnetic hyperthermia system using superparamagnetic iron oxide nanoparticles.
Cantillon-Murphy, P.; Wald, L. L.; Adalsteinsson, E.; Zahn, M.
2010-03-01
7
PubMed Central
In the presence of alternating-sinusoidal or rotating magnetic fields, magnetic nanoparticles will act to realign their magnetic moment with the applied magnetic field. The realignment is characterized by the nanoparticles time constant, ?. As the magnetic field frequency is increased, the nanoparticles magnetic moment lags the applied magnetic field at a constant angle for a given frequency, ?, in rad/s. Associated with this misalignment is a power dissipation that increases the bulk magnetic fluids temperature which has been utilized as a method of magnetic nanoparticle hyperthermia, particularly suited for cancer in low-perfusion tissue (e.g., breast) where temperature increases of between 4C and 7C above the ambient in vivo temperature cause tumor hyperthermia. This work examines the rise in the magnetic fluids temperature in the MRI environment which is characterized by a large DC field, B0. Theoretical analysis and simulation is used to predict the effect of both alternating-sinusoidal and rotating magnetic fields transverse to B0. Results are presented for the expected temperature increase in small tumors (~1 cm radius) over an appropriate range of magnetic fluid concentrations (0.002 to 0.01 solid volume fraction) and nanoparticle radii (1 to 10 nm). The results indicate that significant heating can take place, even in low-field MRI systems where magnetic fluid saturation is not significant, with careful selection of the rotating or sinusoidal field parameters (field frequency and amplitude). The work indicates that it may be feasible to combine low-field MRI with a magnetic hyperthermia system using superparamagnetic iron oxide nanoparticles.
Wald, L.L.; Adalsteinsson, E.; Zahn, M.
2009-01-01
8
The present study analyzes the electromagnetic interference produced on visual display units (VDUs) in domestic and industrial environments. The main sources of disturbance may be identified in three-phase lines, unbalanced currents, currents in earthing systems, proximity of power installations, proximity of railway tracks, and presence of harmonics on the neutral conductor. Magnetic-field interference for PCs is practically limited to the
Riccardo Tommasini; Filippo Spertino
1999-01-01
9
Using the spin wave approximation, we study the decoherence dynamics of a central spin coupled to an antiferromagnetic environment under the application of an external global magnetic field. The external magnetic field affects the decoherence process through its effect on the antiferromagnetic environment. It is shown explicitly that the decoherence factor which displays a Gaussian decay with time depends on
Xiao-Zhong Yuan; Hsi-Sheng Goan; Ka-Di Zhu
2007-01-01
10
No cloning distinguishes the quantum cryptography. Buzek and Hillery have developed a universal quantum cloning machine that allows providing two copies of an arbitrary qubit state with the same accuracy independently of the input-state. The fidelity has been used as a criterion to characterize the cloning. It was found that this parameter can achieve 0.85 for special subsets of quantum states, i.e, equatorial qubits. In the present paper, we investigate the effects of a magnetic field environment as a perturbation of the cloning process. The quantum copying machines studied consist of UQCM-BH and UQCM-PC. Results have been discussed using both the fidelity and the relative entropy. Much attention has been paid to the magnetic field-related decoherence of ancillary qubits before preparation. An attempt to explain the impact of this decoherence on the performance of copying machines will be presented.
Othmani, B.; Machhout, M.; Belmabrouk, H.; Tourki, R.; Mejri, H.
2013-02-01
11
The interfacial environment of AOT based reverse micelle (RM), which has similarity with bio-membranes, has been probed by measuring: (1) magnetic field effect (MFE) on Pyrene-DMA exciplex luminescence; (2) wavelength-dependence of the exciplex lifetime. At least two different types of exciplexes have been identified at different locations within the RM interface. Among these, only the red edge component of the emission centered at 500 nm (Ex 500) is field-sensitive. It makes sense to presume that the Ex 500 is localized at the mobile zone of the interface while the other field-insensitive one, centered at 430 nm (Ex 430) resides at the immobile zone of the interface. Time resolutions of luminescence at different wavelengths indicate that the field-sensitive exciplex is formed at the expense of Ex 430.
Parui, Partha Pratim; Nath, Deb Narayan; Chowdhury, Mihir
2005-03-01
12
There are concerns about workers repeatedly exposed to magnetic fields exceeding regulatory limits with respect to modern magnetic resonance imaging (MRI). As a result, there is need for an ambulatory magnetic field dosimeter capable of measuring these fields in and around an MRI scanner in order to evaluate the regulatory guidelines and determine any underlying exposure risks. This study presents
Miguel A. Fuentes; Adnan Trakic; Stephen J. Wilson; Stuart Crozier
2008-01-01
13
PubMed
There is public health concern raised by epidemiological studies indicating that extremely low frequency electric and magnetic fields generated by electric power distribution systems in the environment may be hazardous. Possible carcinogenic effects of magnetic field in combination with suggested oncostatic action of melatonin lead to the hypothesis that the primary effects of electric and magnetic fields exposure is a reduction of melatonin synthesis which, in turn, may promote cancer growth. In this review the data on the influence of magnetic fields on melatonin synthesis, both in the animals and humans, are briefly presented and discussed. PMID:12019358
Karasek, Michal; Lerchl, Alexander
2002-04-01
14
Acquiring quantitative metrics-based knowledge about the performance of various space physics modeling approaches is central for the space weather community. Quantification of the performance helps the users of the modeling products to better understand the capabilities of the models and to choose the approach that best suits their specific needs. Further, metrics-based analyses are important for addressing the differences between various modeling approaches and for measuring and guiding the progress in the field. In this paper, the metrics-based results of the ground magnetic field perturbation part of the Geospace Environment Modeling 2008-2009 Challenge are reported. Predictions made by 14 different models, including an ensemble model, are compared to geomagnetic observatory recordings from 12 different northern hemispheric locations. Five different metrics are used to quantify the model performances for four storm events. It is shown that the ranking of the models is strongly dependent on the type of metric used to evaluate the model performance. None of the models rank near or at the top systematically for all used metrics. Consequently, one cannot pick the absolute "winner": the choice for the best model depends on the characteristics of the signal one is interested in. Model performances vary also from event to event. This is particularly clear for root-mean-square difference and utility metric-based analyses. Further, analyses indicate that for some of the models, increasing the global magnetohydrodynamic model spatial resolution and the inclusion of the ring current dynamics improve the models' capability to generate more realistic ground magnetic field fluctuations.
Pulkkinen, A.; Kuznetsova, M.; Ridley, A.; Raeder, J.; Vapirev, A.; Weimer, D.; Weigel, R. S.; Wiltberger, M.; Millward, G.; RastTter, L.; Hesse, M.; Singer, H. J.; Chulaki, A.
2011-02-01
15
We present 3D MHD models for the sub-Alfvnic interaction of Callisto with the surrounding magnetospheric plasma taking into account magnetic fields induced in a possible subsurface liquid water ocean and compare the results to magnetometer data for several flybys of the Galileo spacecraft. Modeling of the plasma interaction signals at Callisto is necessary in order to infer the contribution of induced magnetic fields to the perturbation signals measured in the vicinity of the satellite. These interaction signals are caused by currents flowing in Callisto's ionosphere and along Alfvn wings closing in the Jovian ionosphere. We present MHD models for the plasma interaction for several flybys of Galileo at Callisto using initial plasma parameters measured by the spacecraft. In these models the configurations of Callisto's atmosphere and ionosphere control the strength of the subalfvnic interaction with the ambient plasma. By adjusting the model parameters to match the observed magnetic fields it is therefore possible to infer information about the Callisto's atmospheric system. Additionally, by subtracting the modeled interaction fields from the magnetometer data it is possible to constrain the contributions of induced magnetic fields to the total signal. We discuss what information about the configuration of Callisto's interior structure can be deduced from the residual induced magnetic fields.
Seufert, M.; Saur, J.; Neubauer, F. M.
2011-12-01
16
The present paper includes experimental and analytical data on the fracture properties of a nickel-iron superalloy, a ferromagnetic austenite, at 4 K in magnetic fields of 0 and 6 T. The tensile, notch tensile and small punch tests are employed. A finite element analysis is also performed to convert the experimentally measured load-displacement data into useful engineering information. To interpret the results we review the available theory of the influence of magnetic field on the stress intensity factor for a crack in ferromagnetic materials.
Yamaguchi, Yoko; Horiguchi, Katsumi; Shindo, Yasuhide; Sekiya, Daisuke; Kumagai, Susumu
2003-08-01
17
An ultra-low field (ULF) magnetic resonance imaging (MRI) system was set up in an urban laboratory without magnetic shielding. The measured environmental gradient fields of 1 ~ 5 ?T/m caused image distortion. We designed a gradient detection and compensation system to effectively balance the gradient tensor components. The free induction decay signal duration of tap water was thus extended from 0.3 s to 2.5 s, providing the possibility for high-resolution imaging. Two-dimensional MRI images were then obtained at 130 ?T with a helium-cooled second-order superconducting quantum interference device gradiometer. This result allows us to develop an inexpensive ULF MRI system for biological studies.
Dong, Hui; Qiu, Longqing; Shi, Wen; Chang, Baolin; Qiu, Yang; Xu, Lu; Liu, Chao; Zhang, Yi; Krause, Hans-Joachim; Offenhusser, Andreas; Xie, Xiaoming
2013-03-01
18
Marine sediments have historically been omitted from airborne and shipboard magnetic survey analyses because their contribution to the observed field is usually very weak, and thus difficult to distinguish from survey noise. Even when higher susceptibility sediments are present, associated anomalies are often of such limited spatial extent that they do not persist from one survey trackline to the next, and are thus filtered or removed during gridding. In such cases, the resulting magnetic field maps indicate mostly basement rock variations. However, in areas where magnetic sediments have significant lateral extent, the upper stratigraphic units can make a distinguishable contribution to the short-wavelength components of the observed magnetic field. We present a spectral approach to processing magnetic trackline data that highlights magnetic source contrasts within the shallowest sedimentary layers. We apply the approach to several areas including Chesapeake Bay, MD, where watersheds include metamorphic Piedmont rocks rich in Fe- and Ti-rich minerals such as magnetite and ilmenite; Cook Inlet, AK, where magnetite and other Fe- and Ti-rich sediments have accumulated from nearby igneous rocks in the Alaska Range; and the sections of the Oregon coast south of Newport, where magnetite-rich sediments from nearby volcanic rocks have accumulated and developed into littoral marine placer deposits. For each data trackline, we calculate the frequency spectrum for moving windows of length 50 to 600 m, with window length depending on the along-track survey sampling density. We then sum spectral power over shorter wavelengths, excluding both the highest frequencies which most likely represent survey noise, and lower frequencies representing deeper features. Areas with greatest variation in short wavelength anomalies thus exhibit the highest spectral power. Shipboard magnetic field data from Chesapeake Bay near the mouth of the Choptank River exhibit concentrations of anomalies of width < 30 m and amplitude 2-5 nT. These anomalies are clustered in shallow areas where sands dominate the seabed, including at the inflow of Parker's Creek and sections near the inflow of the Choptank River. We attribute these anomalies to the presence of heavy mineral sands that have been concentrated through wave action. At Cook Inlet, aeromagnetic data collected at an altitude of ~130 m above sea level show anomalies of width 400-1200 m and amplitude 2-4 nT clustered in areas near glacial outwash and riverine inputs. These data also exhibit numerous, similarly scaled lineations that may be attributed to folding and faulting of sedimentary layers with high magnetic susceptibilities in the upper 1 km of the seabed. Aeromagnetic data collected at 300-400 m altitude above sea level near the Oregon coast between Newport and Waldport exhibit variations of width 500-800 m, which may indicate depth variations in high-susceptibility units or concentrations of placer deposits.
Shah, A. K.; Saltus, R. W.; Vogt, P. R.; Newell, W. L.
2009-12-01
19
We present detailed imaging of Faraday rotation and depolarization for the radio galaxies 0206+35, 3C 270, 3C 353 and M 84, based on Very Large Array observations at multiple frequencies in the range 1365 to 8440 MHz. All of the sources show highly anisotropic banded rotation measure (RM) structures with contours of constant RM perpendicular to the major axes of their radio lobes. All except M84 also have regions in which the RM fluctuations have lower amplitude and appear isotropic. We give a comprehensive description of the banded RM phenomenon and present an initial attempt to interpret it as a consequence of interactions between the sources and their surroundings. We show that the material responsible for the Faraday rotation is in front of the radio emission and that the bands are likely to be caused by magnetized plasma which has been compressed by the expanding radio lobes. We present a simple model for the compression of a uniformly magnetized external medium and show that RM bands of approximately the right amplitude can be produced, but only for special initial conditions. A two-dimensional magnetic structure in which the field lines are a family of ellipses draped around the leading edge of the lobe can produce RM bands in the correct orientation for any source orientation. We also report the first detections of rims of high depolarization at the edges of the inner radio lobes of M 84 and 3C 270. These are spatially coincident with shells of enhanced X-ray surface brightness, in which both the field strength and the thermal gas density are likely to be increased by compression. The fields must be tangled on small scales.
Guidetti, D.; Laing, R. A.; Bridle, A. H.; Parma, P.; Gregorini, L.
2011-06-01
20
The evidence of cosmic magnetism is examined, taking into account the Zeeman effect, beats in atomic transitions, the Hanle effect, Faraday rotation, gyro-lines, and the strength and scale of magnetic fields in astrophysics. The origin of magnetic fields is considered along with dynamos, the conditions for magnetic field generation, the topology of flows, magnetic fields in stationary flows, kinematic turbulent
Ia. B. Zeldovich; A. A. Ruzmaikin; D. D. Sokolov
1983-01-01
21
SciTech Connect
Saturn's main rings exist within a zone of negligible magnetospheric losses and surface alteration effects, substantially due to the solid-body absorption of inwardly diffusing magnetospheric particles. This process is presently shown to be especially efficient in the inner magnetosphere of Saturn, due to the near-axial symmetry of the planetary magnetic field relative to the equatorial rotation plane; under the assumption of comparable diffusion rates, the inward magnetospheric particle transport is far more inhibited in the inner Saturnian magnetosphere than in the same regions of Jupiter and Uranus, even when only rings of comparable widths and depths are considered. In light of this, ring particle surface exposure to the ion fluxes of the radiation belt remains a prepossessing rationale for low Uranian ring albedos. 86 references.
Hood, L.L.
1987-07-01
22
I will review our recent analysis of the magnetic properties of the O9IV star HD 57682, using spectropolarimetric observations obtained with ESPaDOnS at the Canada-France-Hawaii telescope within the context of the Magnetism in Massive Stars (MiMeS) Large Program. I discuss our most recent determination of the rotational period from longitudinal magnetic field measurements and H? variability - the latter obtained from over a decade's worth of professional and amateur spectroscopic observations. Lastly, I will report on our investigation of the magnetic field geometry and the effects of the field on the circumstellar environment.
Grunhut, Jason H.; Wade, Gregg A.; Marcolino, Wagner L. F.; Petit, Vronique; Petit
2011-07-01
23
NSDL National Science Digital Library
This lesson introduces students to the effects of magnetic fields in matter addressing permanent magnets, diamagnetism, paramagnetism, ferromagnetism, and magnetization. First students must compare the magnetic field of a solenoid to the magnetic field of a permanent magnet. Students then learn the response of diamagnetic, paramagnetic, and ferromagnetic material to a magnetic field. Now aware of the mechanism causing a solid to respond to a field, students learn how to measure the response by looking at the net magnetic moment per unit volume of the material.
VU Bioengineering RET Program, School of Engineering,
24
SciTech Connect
The basic properties of the Hamiltonian representation of magnetic fields in canonical form are reviewed. The theory of canonical magnetic perturbation theory is then developed and applied to the time evolution of a magnetic field embedded in a toroidal plasma. Finally, the extension of the energy principle to tearing modes, utilizing the magnetic field line Hamiltonian, is outlined.
Boozer, A.H.
1985-02-01
25
Here we briefly summarise the main phases which determine the dynamical evolution of primordial magnetic fields in the early universe. On the one hand, strong fields undergo damping due to excitations of plasma fluctuations, and, on the other hand, weak magnetic fields will be strongly amplified by the small-scale dynamo in a turbulent environment. We find that, under reasonable assumptions concerning the efficiency of a putative magnetogenesis era during cosmic phase transitions, surprisingly strong magnetic fields 10-13-10-11 G on comparatively small scales 100 pc -10 kpc may survive to prior to structure formation. Additionally, any weak magnetic field will be exponentially amplified during the collapse of the first minihalos until they reach equipartition with the turbulent kinetic energy. Hence, we argue that it seems possible for cluster magnetic fields to be entirely of primordial origin.
Banerjee, R.
2013-06-01
26
Electrical currents flowing in the solar plasma generate a magnetic field, which is detected in the SOLAR ATMOSPHERE by spectroscopic and polarization measurements (SOLAR MAGNETIC FIELD: INFERENCE BY POLARIMETRY). The SOLAR WIND carries the magnetic field into interplanetary space where it can be measured directly by instruments on space probes....
Schssler, M.; Murdin, P.
2000-11-01
27
NSDL National Science Digital Library
This demonstration of the magnetic field lines of Earth uses a bar magnet, iron filings, and a compass. The site explains how to measure the magnetic field of the Earth by measuring the direction a compass points from various points on the surface. There is also an explanation of why the north magnetic pole on Earth is actually, by definition, the south pole of a magnet.
Barker, Jeffrey
28
NSDL National Science Digital Library
The above animations represent two typical bar magnets each with a North and South pole. The arrows represent the direction of the magnetic field. The color of the arrows represents the magnitude of the field with magnitude increasing as the color changes from blue to green to red to black. You may drag either magnet and double-click anywhere inside the animation to add a magnetic field line, and mouse-down to read the magnitude of the magnetic field at that point.
Christian, Wolfgang; Belloni, Mario
2007-03-03
29
Most of the visible matter in the Universe is ionized so that cosmic magnetic fields are quite easy to generate and, due to the lack of magnetic monopoles, hard to destroy. Magnetic fields have been measured in or around practically all celestial objects, either by in situ measurements of spacecrafts or by the electromagnetic radiation of embedded cosmic rays, gas, or dust. The Earth, the Sun, solar planets, stars, pulsars, the Milky Way, nearby galaxies, more distant (radio) galaxies, quasars, and even intergalactic space in clusters of galaxies have significant magnetic fields, and even larger volumes of the Universe may be permeated by "dark" magnetic fields. Information on cosmic magnetic fields has increased enormously as the result of the rapid development of observational methods, especially in radio astronomy. In the Milky Way, a wealth of magnetic phenomena was discovered, which are only partly related to objects visible in other spectral ranges. The large-scale structure of the Milky Way's magnetic field is still under debate. The available data for external galaxies can well be explained by field amplification and ordering via the dynamo mechanism. The measured field strengths and the similarity of field patterns and flow patterns of the diffuse ionized gas give strong indication that galactic magnetic fields are dynamically important. They may affect the formation of spiral arms, outflows, and the general evolution of galaxies. In spite of our increasing knowledge on magnetic fields, many important questions on the origin and evolution of magnetic fields, their first occurrence in young galaxies, or the existence of large-scale intergalactic fields remained unanswered. The present upgrades of existing instruments and several planned radio astronomy projects have defined cosmic magnetism as one of their key science projects.
Beck, Rainer; Wielebinski, Richard
30
There is no observational support to the hypothesis of the most large-scale homogeneous magnetic field in the Universe. The best upper limit is given by interpretation of the Faraday rotation from the extragalactic radio sources. However the magnetic fields can be generated in the clusters of galaxies by a turbulence in the wakes of moving galaxies. These fields have an
A. A. Ruzmajkin
1991-01-01
31
NSDL National Science Digital Library
Clicking on the different links below will produce different magnetic fields in the box above. The wires (perpendicular to the screen) or coils (in and out of the screen) are not visible, but you can determine what they are from the field. You can also click on a point to read off the magnetic field at that place.
Christian, Wolfgang; Belloni, Mario
2008-02-19
32
NSDL National Science Digital Library
This webpage is part of the University Corporation for Atmospheric Research (UCAR) Windows to the Universe program. It describes the nature and configuration of magnetic fields, which are the result of moving electric charges, including how they cause magnetic objects to orient themselves along the direction of the magnetic force points, which are illustrated as lines. Magnetic field lines by convention point outwards at the north magnetic pole and inward at the south magnetic pole. The site features text, scientific illustrations and an animation. Text and vocabulary are selectable for the beginning, intermediate, or advanced reader.
Universe, Windows T.
1997-12-03
33
This paper presents a system for correcting ECG signals perturbed by MRI environments. The proposed system is based on a dedicated ASIC for measuring with high resolution the MRI magnetic gradients, which are the cause of the ECG artifacts. The measured data are used to calculate the coefficients of an integration compliant adaptive LMS filter. The filter removes the artifacts
V. Frick; H. Berviller; J. Pascal; P. Bougeot; J.-P. Blonde; J. Oster; J. Felblinger
2007-01-01
34
DOEpatents
A magnetic field generating device provides a useful magnetic field within a specific retgion, while keeping nearby surrounding regions virtually field free. By placing an appropriate current density along a flux line of the source, the stray field effects of the generator may be contained. One current carrying structure may support a truncated cosine distribution, and it may be surrounded by a current structure which follows a flux line that would occur in a full coaxial double cosine distribution. Strong magnetic fields may be generated and contained using superconducting cables to approximate required current surfaces.
Krienin, Frank (Shoreham, NY)
1990-01-01
35
Magnetic fields are present in all astrophysical media. However, many models and interpretations of observations often ignore them, because magnetic fields are difficult to handle and because they produce complicated morphological features. Here we will comment on the basic intuitive properties, which even if not completely true, provide a first guiding insight on the physics of a particular astrophysical problem. These magnetic properties are not mathematically demonstrated here. How magnetic fields evolve and how they introduce dynamical effects are considered, also including a short comment on General Relativity Magnetohydrodynamics. In a second part we consider some audacious and speculative matters. They are answers to three questions: a) How draw a cube without lifting the pencil from the paper so that when the pen passes through the same side do in the same direction? B) Are MILAGRO anisotropies miraculous? C) Do cosmic magnetic lenses exist?. The last two questions deal with issues related with the interplay between magnetic fields and cosmic ray propagation.
Florido, E.; Battaner, E.
2010-12-01
36
The conclusions drawn regarding the structure, behavior and composition of the Uranian magnetic field and magnetosphere as revealed by Voyager 2 data are summarized. The planet had a bipolar magnetotail and a bow shock wave which was observed 23.7 Uranus radii (UR) upstream and a magnetopause at 18.0 UR. The magnetic field observed can be represented by a dipole offset
N. F. Ness; M. H. Acuna; K. W. Behannon; L. F. Burlaga; J. E. P. Connerney; R. P. Lepping
1986-01-01
37
A new analysis of magnetic and concurrent plasma data collected from the ; space probes Pionecr 5, Explorer 10, and Mariner 2 yields a new model of the ; interplanetary magnetic field. It is hypothesized that the observed ; interplanetary field F\\/sub i\\/ is due to motion of the magnetometer relative to a ; negatively charged rotating sun from which
V. A. BAILEY
1963-01-01
38
Most of the visible matter in the Universe is in a plasma state, or more specifically is composed of ionized or partially ionized gas permeated by magnetic fields. Thanks to recent advances on the theory and detection of cosmic magnetic fields there has been a worldwide growing interest in the study of their role on the formation of astrophysical sources
Elisabete M. de Gouveia Dal Pino; Dal Pino
2006-01-01
39
NSDL National Science Digital Library
The magnetic field of the Earth is contained in a region called the magnetosphere. The magnetosphere prevents most of the particles from the sun, carried in solar wind, from hitting the Earth. This site, produced by the University Corporation for Atmospheric Research (UCAR), uses text, scientific illustrations,and remote imagery to explain the occurrence and nature of planetary magnetic fields and magnetospheres, how these fields interact with the solar wind to produce phenomena like auroras, and how magnetic fields of the earth and other planets can be detected and measured by satellite-borne magnetometers.
40
The first flyby of Mercury by the Mercury Surface, Space Environment, Geochemistry and Ranging (MESSENGER) spacecraft occurred on 14 January 2008. In order to provide contextual information about the solar wind (SW) properties and the interplanetary magnetic field near the planet, we have used an empirical modeling technique combined with a numerical physics-based SW model. The Wang-Sheeley-Arge (WSA) method uses
Daniel N. Baker; Dusan Odstrcil; Brian J. Anderson; C. Nick Arge; Mehdi Benna; George Gloeckler; Jim M. Raines; David Schriver; James A. Slavin; Sean C. Solomon; Rosemary M. Killen; Thomas H. Zurbuchen
2009-01-01
41
We review current ideas on the origin of galactic and extragalactic magnetic fields. We begin by summarizing observations of magnetic fields at cosmological redshifts and on cosmological scales. These observations translate into constraints on the strength and scale magnetic fields must have during the early stages of galaxy formation in order to seed the galactic dynamo. We examine mechanisms for the generation of magnetic fields that operate prior during inflation and during subsequent phase transitions such as electroweak symmetry breaking and the quark-hadron phase transition. The implications of strong primordial magnetic fields for the reionization epoch as well as the first generation of stars are discussed in detail. The exotic, early-Universe mechanisms are contrasted with astrophysical processes that generate fields after recombination. For example, a Biermann-type battery can operate in a proto-galaxy during the early stages of structure formation. Moreover, magnetic fields in either an early generation of stars or active galactic nuclei can be dispersed into the intergalactic medium.
Widrow, Lawrence M.; Ryu, Dongsu; Schleicher, Dominik R. G.; Subramanian, Kandaswamy; Tsagas, Christos G.; Treumann, Rudolf A.
2012-05-01
42
SciTech Connect
The magnetospheres of Mercury, Venus, Mars, Jupiter, Saturn, Uranus, and comets and the heliomagnetosphere are examined. The orientations of the planetary spin and magnetic axes, the size of the magnetospheres, and the magnetic properties and the radio emissions of the planets are compared. Results from spacecraft studies of the planets are included. Plans for the Voyager 2 mission and its expected study of the Neptune magnetosphere are considered.
Lanzerotti, L.J.; Uberoi, C.
1989-02-01
43
NSDL National Science Digital Library
This activity will introduce students to the idea of magnetic field lines--a concept they have probably encountered but may not fully grasp. Completing this activity and reading the corresponding background information should enable students to understand
Horton, Michael
2009-05-30
44
SciTech Connect
In recent years there has been increased concern over potential health hazards related to exposure of personnel to magnetic fields. If exposure standards are to be established, then a means for measuring magnetic field dose must be available. To meet this need, the Department of Energy has funded development of prototype dosimeters at the Battelle Pacific Northwest Laboratory. This manual reviews the principle of operation of the dosimeter and also contains step-by-step instructions for its operation.
Lemon, D.K.; Skorpik, J.R.; Eick, J.L.
1980-09-01
45
SciTech Connect
A magnetically levitated vehicle adapted for movement along a guide way, comprising: a passenger compartment; first and second primary magnet means secured on the vehicle to produce a magnetic field having a magnetic flux density extending outward from the primary magnet means, to support the vehicle above and spaced from the guide way; and a plurality of confining magnets disposed on the vehicle to confine the magnetic flux extending outward from the primary magnet means and to reduce the strength of the primary magnetic field in the passenger compartment; wherein the primary magnet means has a capacity to produce a primary magnetic field having a maximum strength of at least 200 gauss in the passenger compartment, and the confining magnets maintain the strength of the primary magnetic field in the passenger compartment below 5 gauss.
Proise, M.
1993-05-25
46
The past several years have seen dramatic developments in the study of planetary magnetic fields, including a wealth of new data, mainly from the Galilean satellites and Mars, together with major improvements in our theoretical modeling effort of the dynamo process believed responsible for large planetary fields. These dynamos arise from thermal or compositional convection in fluid regions of large
David J. Stevenson
2003-01-01
47
NSDL National Science Digital Library
The EJS Magnetic Multipole Field Model shows the field of a magnetic dipole or quadrupole with little compasses that indicate direction and relative field strength. A slider changes the angular orientation of the dipole and a movable compass shows the magnetic field direction and magnitude. Compass values can be recorded into a data table and analyzed using a built-in data analysis tool. You can modify this simulation if you have Ejs installed by right-clicking within the plot and selecting Open Ejs Model from the pop-up menu item. The Magnetic Multipole Field model was created using the Easy Java Simulations (Ejs) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_em_MagneticMultipoleField.jar file will run the program if Java is installed. Ejs is a part of the Open Source Physics Project and is designed to make it easier to access, modify, and generate computer models. Additional Ejs models are available. They can be found by searching ComPADRE for Open Source Physics, OSP, or Ejs.
Christian, Wolfgang; Cox, Anne; Franciscouembre
2010-02-14
48
SciTech Connect
A magnetic field measurement system was designed, built and installed at MAX Lab, Sweden for the purpose of characterizing the magnetic field produced by Insertion Devices (see Figure 1). The measurement system consists of a large granite beam roughly 2 feet square and 14 feet long that has been polished beyond laboratory grade for flatness and straightness. The granite precision coupled with the design of the carriage yielded minimum position deviations as measured at the probe tip. The Hall probe data collection and compensation technique allows exceptional resolution and range while taking data on the fly to programmable sample spacing. Additional flip coil provides field integral data.
Kulesza, Joe; Johnson, Eric; Lyndaker, Aaron; Deyhim, Alex; Waterman, Dave; Blomqvist, K. Ingvar [Advanced Design Consulting USA, 126 Ridge Road, P.O. Box 187, Lansing, NY 14882 (United States); Dunn, Jonathan Hunter [MAX-lab, SE-221 00 Lund (Sweden)
2007-01-19
49
NSDL National Science Digital Library
A cross section of a circular wire loop carrying an unknown current is shown above. The arrows represent the direction of the magnetic field. The color of the arrows represents the magnitude of the field with magnitude increasing as the color changes from blue to green to red to black. You can double-click in the animation to add magnetic field lines, click-drag the center of the loop to reposition it, and click-drag the top or bottom of the loop to change its size.
Christian, Wolfgang; Belloni, Mario
2007-03-03
50
The equivalent source dipole technique is used to model the three components of the Martian lithospheric magnetic field. We use magnetic field measurements made on board the Mars Global Surveyor spacecraft. Different input dipole meshes are presented and evaluated. Because there is no global, Earth-like, inducing magnetic field, the magnetization directions are solved for together with the magnetization intensity. A
B. Langlais; M. E. Purucker; M. Mandea
2004-01-01
51
Radio synchrotron emission, its polarization and its Faraday rotation are powerful tools to study the strength and structure of magnetic fields in galaxies. Unpolarized emission traces turbulent fields which are strongest in spiral arms and bars (20-30 ?G) and in central starburst regions (50-100 ?G). Such fields are dynamically important, e.g. they can drive gas inflows in central regions. Polarized emission traces ordered fields which can be regular or anisotropic random, generated from isotropic random fields by compression or shear. The strongest ordered fields of 10-15 ?G strength are generally found in interarm regions and follow the orientation of adjacent gas spiral arms. Ordered fields with spiral patterns exist in grand-design, barred and flocculent galaxies, and in central regions of starburst galaxies. Faraday rotation measures (RM) of the diffuse polarized radio emission from the disks of several spiral galaxies reveal large-scale patterns, which are signatures of regular fields generated by a mean-field dynamo. However, in most spiral galaxies observed so far the field structure is more complicated. Ordered fields in interacting galaxies have asymmetric distributions and are an excellent tracer of past interactions between galaxies or with the intergalactic medium. Ordered magnetic fields are also observed in radio halos around edge-on galaxies, out to large distances from the plane, with X-shaped patterns. Future observations of polarized emission at high frequencies, with the EVLA, the SKA and its precursors, will trace galactic magnetic fields in unprecedented detail. Low-frequency telescopes (e.g. LOFAR and MWA) are ideal to search for diffuse emission and small RMs from weak interstellar and intergalactic fields.
Beck, Rainer
2012-05-01
52
PubMed
Aromaticity is indispensable for explaining a variety of chemical behaviors, including reactivity, structural features, relative energetic stabilities, and spectroscopic properties. When interpreted as the spatial delocalization of ?-electrons, it represents the driving force for the stabilization of many planar molecular structures. A delocalized electron system is sensitive to an external magnetic field; it responds with an induced magnetic field having a particularly long range. The shape of the induced magnetic field reflects the size and strength of the system of delocalized electrons and can have a large influence on neighboring molecules. In 2004, we proposed using the induced magnetic field as a means of estimating the degree of electron delocalization and aromaticity in planar as well as in nonplanar molecules. We have since tested the method on aromatic, antiaromatic, and nonaromatic compounds, and a refinement now allows the individual treatment of core-, ?-, and ?-electrons. In this Account, we describe the use of the induced magnetic field as an analytical probe for electron delocalization and its application to a large series of uncommon molecules. The compounds include borazine; all-metal aromatic systems Al(4)(n-); molecular stars Si(5)Li(n)(6-n); electronically stabilized planar tetracoordinate carbon; planar hypercoordinate atoms inside boron wheels; and planar boron wheels with fluxional internal boron cluster moieties. In all cases, we have observed that planar structures show a high degree of electron delocalization in the ?-electrons and, in some examples, also in the ?-framework. Quantitatively, the induced magnetic field has contributions from the entire electronic system of a molecule, but at long range the contributions arising from the delocalized electronic ?-system dominate. The induced magnetic field can only indirectly be confirmed by experiment, for example, through intermolecular contributions to NMR chemical shifts. We show that calculating the induced field is a useful method for understanding any planar organic or inorganic system, as it corresponds to the intuitive Pople model for explaining the anomalous proton chemical shifts in aromatic molecules. Indeed, aromatic, antiaromatic, and nonaromatic molecules show differing responses to an external field; that is, they reduce, augment, or do not affect the external field at long range. The induced field can be dissected into different orbital contributions, in the same way that the nucleus-independent chemical shift or the shielding function can be separated into component contributions. The result is a versatile tool that is particularly useful in the analysis of planar, densely packed systems with strong orbital contributions directly atop individual atoms. PMID:21848282
Islas, Rafael; Heine, Thomas; Merino, Gabriel
2011-08-17
53
Leone, F.; Bohlender, D. A.; Bolton, C. T.; Buemi, C.; Catanzaro, G.; Hill, G. M.; Stift, M. J.
2010-02-01
54
Some distributions of magnetization give rise to magnetic fields that vanish everywhere above the surface, rendering these distributions of magnetization completely invisible. They are the annihilators of the magnetic inverse problem. Known examples are the infinite sheet with constant magnetization and the spherical shell of constant susceptibility magnetized by an arbitrary internal field. Here, we show that remarkably more interesting
S. Maus; V. Haak
2003-01-01
55
For the unique case of magnetic field parallel to the solar wind flow, a column of reflected protons can accumulate upstream from the Moon. We investigate observations from the ARTEMIS probes for an extended period with this geometry. During this time, P2 observes strong wave turbulence in two frequency bands above and below the ion cyclotron frequency near the Moon, not seen by P1 farther from the Moon. The lower frequency oscillations prove consistent with kinetic magnetosonic waves resonantly generated by reflected protons, and test particle calculations confirm that a significant column of reflected protons lies upstream when the waves occur. The reflected protons perturb a large volume of plasma around the Moon, extending upstream as well as into the wake. The waves observed near the Moon during this time period have many similarities to those found in the terrestrial foreshock and at comets, suggesting the potential for comparative studies.
Halekas, J. S.; Poppe, A. R.; McFadden, J. P.; Glassmeier, K.-H.
2013-09-01
56
Electromagnets used as beam guiding elements in particle accelerators and colliders require very tight tole-rances on their magnetic fields and on their alignment along the particle path. This article describes the methods and equipment used for magnetic measurements in beam transport magnets. Descriptions are given of magnetic resonance techniques, various induction coil methods, Hall generator measurements, the fluxgate magnetometer as
K. N. Henrichsen
1998-01-01
57
The Heliospheric Magnetic Field (HMF) is the physical framework in which energetic particles and cosmic rays propagate. Changes in the large scale structure of the magnetic field lead to short- and long term changes in cosmic ray intensities, in particular in anti-phase with solar activity. The origin of the HMF in the corona is well understood and inner heliospheric observations can generally be linked to their coronal sources. The structure of heliospheric magnetic polarities and the heliospheric current sheet separating the dominant solar polarities are reviewed here over longer than a solar cycle, using the three dimensional heliospheric observations by Ulysses. The dynamics of the HMF around solar minimum activity is reviewed and the development of stream interaction regions following the stable flow patterns of fast and slow solar wind in the inner heliosphere is described. The complex dynamics that affects the evolution of the stream interaction regions leads to a more chaotic structure of the HMF in the outer heliosphere is described and discussed on the basis of the Voyager observations. Around solar maximum, solar activity is dominated by frequent transients, resulting in the interplanetary counterparts of Coronal Mass Ejections (ICMEs). These produce a complex aperiodic pattern of structures in the inner heliosphere, at all heliolatitudes. These structures continue to interact and evolve as they travel to the outer heliosphere. However, linking the observations in the inner and outer heliospheres is possible in the case of the largest solar transients that, despite their evolutions, remain recognizably large structures and lead to the formation of Merged Interaction Regions (MIRs) that may well form a quasi-spherical, "global" shell of enhanced magnetic fields around the Sun at large distances. For the transport of energetic particles and cosmic rays, the fluctuations in the magnetic field and their description in alternative turbulent models remains a very important research topic. These are also briefly reviewed in this paper.
Balogh, Andr; Erds, Gza
2013-06-01
58
PubMed
Mantiply, E D; Pohl, K R; Poppell, S W; Murphy, J A
1997-01-01
59
SciTech Connect
This letter is a response to an article by Savitz and Kaune, EHP 101:76-80. W-L wire code was applied to data from a 1988 Denver study, and an association was reported between high W-L wire code and childhood cancer. This author discusses several studies and provides explanations which weakens the argument that classification error resulted in an appreciable reduction in the association between W-L high wire code and childhood cancer. In conclusion, the fact that new wire code is only weakly correlated with magnetic field measurements (in the same manner as the original W-L wire code) suggests that the newly reported stronger association with childhood cancer is likely due to factors other than magnetic fields. Differential residential mobility and differential residential age are two possible explanations and are suggestive that the reported association may be false.
Jones, T.L.
1993-10-01
60
The simple noncontacting detection of position plays a major role in the automation technique. Especially for the application in production plants, a high level of quality, reliability and durability is an absolute necessity. In this environment, the use of magnetic field sensors for detecting the position of a permanent magnet is at first glance a very well known and solved
T. Reininger; C. Hanisch
1997-01-01
61
We report the switching properties of a thin magnetic film subject to an ultrashort, laterally localized magnetic field pulse, obtained by numerical investigations. The magnetization distribution in the film is calculated on a grid assuming Stoner-like coherent rotation within the grid square size. Perpendicularly and in-plane magnetized films exhibit a magnetization reversal due to a 4ps magnetic field pulse. Outside the central region the pulse duration is short compared to the precession period. In this area the evolution of the magnetization during the field pulse does not depend strongly on magnetic damping and/or pulse shape. However, the final magnetization distribution is affected by the magnetic damping. Although the pulse duration is short compared to the precession period, the time needed for the relaxation of the magnetization to the equilibrium state is rather large. The influence of the different magnetic anisotropy contributions and the magnetic damping parameter enters into the magnetization reversal process. Comparing the case of perpendicular anisotropy with different kinds of in-plane anisotropies, a principal difference is found due to the symmetry of the shape anisotropy with respect to the anisotropy in question.
Bauer, M.; Lopusnik, R.; Fassbender, J.; Hillebrands, B.
2000-08-01
62
NSDL National Science Digital Library
The above animations represent two typical bar magnets each with a North and South pole. The arrows represent the direction of the magnetic field. A wire is placed between the magnets and a current that comes out of the page can be turned on.
Christian, Wolfgang; Belloni, Mario
2007-03-03
63
Advances in Magnetic Resonance Imaging depend on the capability of the available hardware. Specifically, for the main magnet configuration, using derivative constraints, we can create a static magnetic field with reduced levels of inhomogeneity over a prescribed imaging volume. In the gradient coil, the entire design for the axial elliptical coil, and the mathematical foundation for the transverse elliptical coil
Labros Spiridon Petropoulos
1993-01-01
64
Recent developments in integrated silicon magnetic devices are reviewed, with particular attention given to integrated Hall plates, magnetic field-effect transistors, vertical and lateral bipolar magnetotransistors, magnetodiodes, and current-domain magnetometers. Also described are current developments in integrated magnetic field sensors based on III-V semiconductors and bulk Hall-effect devices. The discussion also covers magnetic device modeling and the incorporation of magnetic devices
H. P. Baltes; R. S. Popovic
1986-01-01
65
The past several years have seen dramatic developments in the study of planetary magnetic fields, including a wealth of new data, mainly from the Galilean satellites and Mars, together with major improvements in our theoretical modeling effort of the dynamo process believed responsible for large planetary fields. These dynamos arise from thermal or compositional convection in fluid regions of large radial extent. The relevant electrical conductivities range from metallic values to values that may be only about 1% or less that of a typical metal, appropriate to ionic fluids and semiconductors. In all planets, the Coriolis force is dynamically important, but slow rotation may be more favorable for a dynamo than fast rotation. The maintenance and persistence of convection appears to be easy in gas giants and ice-rich giants, but is not assured in terrestrial planets because the quite high electrical conductivity of iron-rich cores guarantees a high thermal conductivity (through the Wiedemann-Franz law), which allows for a large core heat flow by conduction alone. In this sense, high electrical conductivity is unfavorable for a dynamo in a metallic core. Planetary dynamos mostly appear to operate with an internal field ~(2??/?)1/2 where ? is the fluid density, ? is the planetary rotation rate and ? is the conductivity (SI units). Earth, Ganymede, Jupiter, Saturn, Uranus, Neptune, and maybe Mercury have dynamos, Mars has large remanent magnetism from an ancient dynamo, and the Moon might also require an ancient dynamo. Venus is devoid of a detectable global field but may have had a dynamo in the past. The presence or absence of a dynamo in a terrestrial body (including Ganymede) appears to depend mainly on the thermal histories and energy sources of these bodies, especially the convective state of the silicate mantle and the existence and history of a growing inner solid core. Induced fields observed in Europa and Callisto indicate the strong likelihood of water oceans in these bodies.
Stevenson, David J.
2003-03-01
66
Shockley, Jeremiah A.
67
A crucial phase in magnetic flux emergence is the rise of magnetic flux tubes through the solar photosphere, which represents a severe transition between the very different environments of the solar interior and corona. Multi-wavelength observations with Flare Genesis, TRACE, SoHO, and more recently with the vector magnetographs at THEMIS and Hida (DST) led to the following conclusions. The fragmented magnetic field in the emergence region - with dipped field lines or bald patches - is directly related with Ellerman bombs, arch filament systems, and overlying coronal loops. Measurements of vector magnetic fields have given evidence that undulating "serpentine" fields are present while magnetic flux tubes cross the photosphere. See the sketch below, and for more detail see Pariat et al. (2004, 2007); Watanabe et al. (2008):
Schmieder, B.; Pariat, E.
68
DOEpatents
The superconducting magnetic switch or fast kicker magnet is employed with electron stream or a bunch of electrons to rapidly change the direction of flow of the electron stream or bunch of electrons. The apparatus employs a beam tube which is coated with a film of superconducting material. The tube is cooled to a temperature below the superconducting transition temperature and is subjected to a constant magnetic field which is produced by an external dc magnet. The magnetic field produced by the dc magnet is less than the critical field for the superconducting material, thus, creating a Meissner Effect condition. A controllable fast electromagnet is used to provide a magnetic field which supplements that of the dc magnet so that when the fast magnet is energized the combined magnetic field is now greater that the critical field and the superconducting material returns to its normal state allowing the magnetic field to penetrate the tube. This produces an internal field which effects the direction of motion and of the electron stream or electron bunch. The switch can also operate as a switching mechanism for charged particles. 6 figs.
Goren, Y.; Mahale, N.K.
1996-08-06
69
SciTech Connect
The magnetic field of the solar corona evolves quasistatically in response to slowly changing photospheric boundary conditions. The magnetic topology is preserved by the low resistivity of the solar atmosphere. We show that a magnetic flux coordinate system simplifies the problem of calculating field evolution with invariant topology. As an example, we calculate the equilibrium of a thin magnetic flux tube with small twist per unit length.
Zweibel, E.G.; Boozer, A.H.
1985-02-01
70
SciTech Connect
The powerful magnetic fields produced by a controlled fusion experiment at Lawrence Livermore National Laboratory (LLNL) necessitated the development of personnel-exposure guidelines for steady magnetic fields. A literature search and conversations with active researchers showed that it is currently possible to develop preliminary exposure guidelines for steady magnetic fields. An overview of the results of past research into the bioeffects of magnetic fields was compiled, along with a discussion of hazards that may be encountered by people with sickle-cell anemia or medical electronic and prosthetic implants. The LLNL steady magnetic-field exposure guidelines along with a review of developments concerning the safety of time-varying fields were also presented in this compilation. Guidelines developed elsewhere for time varying fields were also given. Further research is needed to develop exposure standards for both steady or time-varying fields.
Miller, G.
1987-12-01
71
It is proposed that the most probable configuration of the magnetic field in the atmosphere of an Ap star is an almost force-free, poloidal field, close to a low-order multipole. Such a magnetic field can not change the structure of the atmosphere to any great extent, but the vertical component of the Lorentz force can decrease the effective gravity by
K. Stepien
1980-01-01
72
The Hall effect has been widely utilized to measure magnetic fields. The relatively simple geometry of a Hall element suggested the use of such a device on a microscale as a probe to examine magnetic fields of small structures. Hall probes are described which were constructed with a sensitive area about 1010 ?. Fields of less than 0.01 gauss were
D. D. Roshon Jr.
1962-01-01
73
The jet region of M87 is discussed to illustrate the astrophysical observations of radio sources, with note made of magnetic field phenomena contributing to radio frequency emissions. The jet appearing in M87 has been modelled as a continuous supersonic flow of plasma embedded in a self-consistent, ordered magnetic field. The field has both parallel and helical components, and may work
A. Ferrari
1982-01-01
74
We show that the relatively strong magnetic fields ($\\\\ge 1 \\\\mu$G) in high\\u000aredshift objects can be explained by the combined action of an evolving\\u000aprotogalactic fluctuation and electrodynamic processes providing the magnetic\\u000aseed fields. Three different seed field mechanisms are reviewed and\\u000aincorporated into a spherical \\
Harald Lesch; Masashi Chiba
1994-01-01
75
We show that the relatively strong magnetic fields (>=1muG) in high redshift objects can be explained by the combined action of an evolving protogalactic fluctuation and electrodynamic processes providing the magnetic seed fields. Three different seed field mechanisms are reviewed and incorporated into a spherical \\
H. Lesch; M. Chiba
1995-01-01
76
DOEpatents
A device is provided for measuring the magnetic field dose and peak field exposure. The device includes three Hall-effect sensors all perpendicular to each other, sensing the three dimensional magnetic field and associated electronics for data storage, calculating, retrieving and display.
Lemon, D.K.; Skorpik, J.R.; Eick, J.L.
1981-01-21
77
PubMed
Our present-day understanding of solar and stellar magnetic fields is discussed from both an observational and theoretical viewpoint. To begin with, observations of the Sun's large-scale magnetic field are described, along with recent advances in measuring the spatial distribution of magnetic fields on other stars. Following this, magnetic flux transport models used to simulate photospheric magnetic fields and the wide variety of techniques used to deduce global coronal magnetic fields are considered. The application and comparison of these models to the Sun's open flux, hemispheric pattern of solar filaments and coronal mass ejections are then discussed. Finally, recent developments in the construction of steady-state global magnetohydrodynamic models are considered, along with key areas of future research. PMID:22665897
Mackay, Duncan H
2012-07-13
78
Magnetic field measurements of the solar corona using microwave observation are reviewed. The solar corona is filled with highly ionised plasma and magnetic field. Moving charged particles interact with magnetic field due to Lorentz force. This results in gyration motion perpendicular to the magnetic field and free motion along the magnetic field. Circularly polarized electro-magnetic waves interact with gyrating electrons
K. Shibasaki
2006-01-01
79
An internal potential function was created using the averaged MGS vector data released by Mario Acuna for altitudes from 95 to 209 km above the Martian geoid, all longitudes, and latitudes from 87 degrees south to 78 degrees north. Even with some gaps in coverage it is found that a consistent internal potential function can be derived up to spherical harmonic terms of n = 65 using all three components of the data. Weighting the data according to the standard errors given, the model fits to 7-8 nT rms. The energy density spectrum of the harmonics is seen to peak near n = 39 with a value of 7 J/cu km and fall off to less than 0.5 J/cu km below n = 15 and above n = 55. Contour maps of the X (north) component drawn for 100 km altitude show the strongly anomalous region centered at 60 degrees S latitude and 180 degrees longitude, as well as the alternating east-west trends already observed by other groups. Maps of the other components show the anomalous region, but not the east-west trends. The dichotomy is also maintained with much weaker anomalies bounding the northern plains. The results herein as as well as those of others is limited by the sparse low-altitude data coverage as well as the accuracy of the observations in the face of significant spacecraft fields. Work by Connerney and Acuna have mitigated these sources somewhat, but the design of the spacecraft did not lend itself to accurate observations. Recent results reported by David Mitchell of the ER group have shown that the field observations are significantly influenced by the solar wind with the possibility that the present results may only reflect that portion of the internal field visible above 95 km altitude. Depending on the solar wind, the anomaly field may be shielded or distorted to produce spurious results. The spectrum we have obtained so far may only see the stronger portion of the signal with a significant weaker component hidden. Measurements of crustal anomalies versus relative ages of source bodies combined with later absolute dating of Martian geologic units could lead to a quantitative constraint on the thermal history of the planet, i.e. the time when convective dynamo generation ceased in the core. Determination of directions of magnetization of anomaly sources as a function of age combined with the expectation that the Martian dynamo field was roughly aligned with the rotation axis would lead to a means of investigating polar wandering for Mars. Preliminary analysis of two magnetic anomalies in the northern polar region has yielded paleomagnetic pole positions near 50 N, 135 W, about 30 degrees north of Olympus Mons. This location is roughly consistent with the orientation of the planet expected theoretically prior to the formation of the Tharsis region. In the future, more accurate observations of the vector field at the lowest possible altitudes would significantly improve our understanding of Martian thermal history, polar wandering, and upper crustal evolution. Mapping potential resources (e.g., iron-rich source bodies) for future practical use would also be a side benefit. Additional information is contained in the original abstract.
Cain, J. C.; Ferguson, B.; Mozzoni, D.; Hood, L.
2000-07-01
80
SciTech Connect
A 10-mG, 50 to 60-Hz magnetic field is in the intensity and frequency range that people worldwide are often exposed to in homes and in the workplace. Studies about the effects of 50- to 100-Hz electromagnetic fields on various species of animal embryos (fish, chick, fly, sea urchin, rat, and mouse) indicate that early stages of embryonic development are responsive to fluctuating magnetic fields. Chick, sea urchin, and mouse embryos are responsive to magnetic field intensities of 10-100 mG. Results from studies on sea urchin embryos indicate that exposure to conditions of rotating 60-Hz magnetic fields, e.g., similar to those in our environment, interferes with cell proliferation at the morula stage in a manner dependent on field intensity. The cleavage stages, prior to the 64-cell stage, were not delayed by this rotating 60-Hz magnetic field suggesting that the ionic surges, DNA replication, and translational events essential for early cleavage stages were not significantly altered. Studies of histone synthesis in early sea urchin embryos indicated that the rotating 60-Hz magnetic field decreased zygotic expression of early histone genes at the morula stage and suggests that this decrease in early histone production was limiting to cell proliferation. Whether these comparative observations from animal development studies will be paralleled by results from studies of human embryogenesis, as suggested by some epidemiology studies, has yet to be established. 38 refs.
Cameron, I.L.; Hardman, W.E.; Winters, W.D.; Zimmerman, S.; Zimmerman, A.M. (Univ. of Texas Health Science Center, San Antonio (United States))
1993-04-01
81
Magnetic fields have been known in antiquity. Aristotle attributes the first of what could be called a scientific discussion on magnetism to Thales, who lived from about 625 BC. In China magnetic carts were in use to help the Emperor in his journeys of inspection. Plinius comments that in the Asia Minor province of Magnesia shepherds' staffs get at times glued to a stone, a alodestone. In Europe the magnetic compass came through the Arab sailors who met the Portuguese explorers. The first scientific treatise on magnetism, De Magnete, was published by William Gilbert who in 1600 described his experiments and suggested that the Earth was a huge magnet. Johannes Kepler was a correspondent of Gilbert and at times suggested that planetary motion was due to magnetic forces. Alas, this concept was demolished by Isaac Newton,who seeing the falling apple decided that gravity was enough. This concept of dealing with gravitational forces only remains en vogue even today. The explanations why magnetic effects must be neglected go from magnetic energy is only 1% of gravitation to magnetic fields only complicate the beautiful computer solutions. What is disregarded is the fact that magnetic effects are very directional(not omni-directional as gravity) and also the fact that magnetic fields are seen every where in our cosmic universe.
Wielebinski, Richard; Beck, Rainer
82
SciTech Connect
The procedure for installing Superconducting Super Collider (SSC) dipoles in their respective cryostats involves aligning the average direction of their field with the vertical to an accuracy of 0.5 mrad. The equipment developed for carrying on these measurements is described and the measurements performed on the first few prototypes SSC magnets are presented. The field angle as a function of position in these 16.6 m long magnets is a characteristic of the individual magnet with possible feedback information to its manufacturing procedure. A comparison of this vertical alignment characteristic with a magnetic field intensity (by NMR) characteristic for one of the prototypes is also presented. 5 refs., 7 figs.
Kuchnir, M.; Schmidt, E.E.
1987-11-06
83
PubMed
We calculate, in the free Maxwell theory, the renormalized quantum vacuum expectation value of the two-point magnetic correlation function in de Sitter inflation. We find that quantum magnetic fluctuations remain constant during inflation instead of being washed out adiabatically, as usually assumed in the literature. The quantum-to-classical transition of super-Hubble magnetic modes during inflation allow us to treat the magnetic field classically after reheating, when it is coupled to the primeval plasma. The actual magnetic field is scale independent and has an intensity of few10(-12)??G if the energy scale of inflation is few10(16)??GeV. Such a field accounts for galactic and galaxy cluster magnetic fields. PMID:23971556
Campanelli, Leonardo
2013-08-06
84
NSDL National Science Digital Library
A cross section of two circular wire loops carrying the exact same current is shown above (position given in centimeters and magnetic field given in milli-Tesla). You can click-drag to read the magnitude of the magnetic field.
Christian, Wolfgang; Belloni, Mario
2007-03-03
85
We report on the status of our spectropolarimetric observations of massive stars. During the last years, we have discovered magnetic fields in many objects of the upper main sequence, including Be stars, ? Cephei and Slowly Pulsating B stars, and a dozen O stars. Since the effects of those magnetic fields have been found to be substantial by recent models, we are looking into their impact on stellar rotation, pulsation, stellar winds, and chemical abundances. Accurate studies of the age, environment, and kinematic characteristics of the magnetic stars are also promising to give us new insight into the origin of the magnetic fields. Furthermore, longer time series of magnetic field measurements allow us to observe the temporal variability of the magnetic field and to deduce the stellar rotation period and the magnetic field geometry. Studies of the magnetic field in massive stars are indispensable to understand the conditions controlling the presence of those fields and their implications on the stellar physical parameters and evolution.
Schller, M.; Hubrig, S.; Ilyin, I.; Kharchenko, N. V.; Briquet, M.; Gonzlez, J. F.; Langer, N.; Oskinova, L. M.; MAGORI Collaboration
2011-12-01
86
SciTech Connect
The magnetic field structure in a domain surrounded by a closed toroidal magnetic surface is analyzed. It is shown that ergodization of magnetic field lines is possible even in a regular field configuration (with nonvanishing toroidal component). A unified approach is used to describe magnetic fields with nested toroidal (possibly asymmetric) flux surfaces, magnetic islands, and ergodic field lines.
Ilgisonis, V. I.; Skovoroda, A. A., E-mail: [email protected] [Russian Research Centre Kurchatov Institute (Russian Federation)
2010-05-15
87
SciTech Connect
Experimental data have shown that the light output of a scintillator depends on the magnitude of the externally applied magnetic fields, and that this variation can affect the calorimeter calibration and possibly resolution. The goal of the measurements presented here is to study the light yield of scintillators in high magnetic fields in conditions that are similar to those anticipated for the LHC CMS detector. Two independent measurements were performed, the first at Fermilab and the second at the National High Magnetic Field Laboratory at Florida State University.
Green, D.; Ronzhin, A. [Fermi National Accelerator Lab., Batavia, IL (United States); Hagopian, V. [Florida State Univ., Tallahasse, FL (United States)
1995-06-01
88
Recently planet Mercuryan unexplored territory in our solar systemhas been of much interest to the scientific community due to recent flybys of the spacecraft MESSENGER that discovered its intrinsic stationary and large-scale dipole like magnetic field structure with an intensity of 300nT confirming Mariner 10 observations. In the present study, with the observed constraint of Mercury's atmospheric magnetic field structure, internal magnetic field structure is modeled as a solution of magnetic diffusion equation. In this study, Mercury's internal structure mainly consists of a stable stratified fluid core and the convective mantle. For simplicity, magnetic diffusivity in both parts of the structure is considered to be uniform and constant with a value represented by a suitable averages. It is further assumed that vigorous convection in the mantle disposes of the electric currents leading to a very high diffusivity in that region. Thus, in order to satisfy observed atmospheric magnetic field structure, Mercury's most likely magnetic field structure consists of a solution of MHD diffusion equation in the core and a combined multipolar (dipole and quadrupole like magnetic field structures embedded in the uniform field) solution of a current free like magnetic field structure in the mantle and in the atmosphere. With imposition of appropriate boundary conditions at the core-mantle boundary for the first two diffusion eigen modes, in order to satisfy the observed field structure, present study puts the constraint on Mercury's core radius to be 2000km.From the estimated magnetic diffusivity and the core radius, it is also possible to estimate the two diffusion eigen modes with their diffusion time scales of 8.6 and 3.7 billion years respectively suggesting that the planet inherits its present-day magnetic field structure from the solar Nebula. It is proposed that permanency of such a large-scale magnetic field structure of the planet is attained during Mercury's early evolutionary history of heavy bombardments by the asteroids and comets supporting the giant impact hypothesis for the formation of Mercury.
Hiremath, K. M.
2012-04-01
89
\\u000a Magnetic fields have been known in antiquity. Aristotle attributes the first of what could be called a scientific discussion\\u000a on magnetism to Thales, who lived from about 625 BC. In China magnetic carts were in use to help the Emperor in his journeys\\u000a of inspection. Plinius comments that in the Asia Minor province of Magnesia shepherds staffs get at times
Richard Wielebinski; Rainer Beck
2010-01-01
90
Swarm is the fifth Earth Explorer mission in ESA's Living Planet Programme, and is scheduled for launch in 2013. The objective of the Swarm mission is to provide the best-ever survey of the geomagnetic field and its temporal evolution using a constellation of 3 identical satellites. The Mission shall deliver data that allow access to new insights into the Earth system by improved scientific understanding of the Earth's interior and near-Earth electromagnetic environment. After launch and triple satellite release at an initial altitude of about 490 km, a pair of the satellites will fly side-by-side with slowly decaying altitude, while the third satellite will be lifted to 530 km to complete the Swarm constellation. High-precision and high-resolution measurements of the strength, direction and variation of the magnetic field, complemented by precise navigation, accelerometer and electric field measurements, will provide the observations required to separate and model various sources of the geomagnetic field and near-Earth current systems. The mission science goals are to provide a unique view into Earth's core dynamics, mantle conductivity, crustal magnetisation, ionospheric and magnetospheric current systems and upper atmosphere dynamics - ranging from understanding the geodynamo to contributing to space weather. The scientific objectives and results from recent scientific studies will be presented. In addition the current status of the project, which is presently in the final stage of the development phase, will be addressed. A consortium of European scientific institutes is developing a distributed processing system to produce geophysical (Level 2) data products for the Swarm user community. The setup of the Swarm ground segment and the contents of the data products will be addressed. More information on Swarm can be found at www.esa.int/esaLP/LPswarm.html.
Plank, Gernot; Haagmans, Roger; Floberghagen, Rune; Menard, Yvon
2013-04-01
91
National Technical Information Service (NTIS)
Magnetic pumping by major-radius oscillation of a toroidal plasma can be made more practical by introducing a major-radius range within which the vertical-field gradient is sufficiently great so that major-radius perturbations are marginally stable or, be...
H. P. Furth R. A. Ellis
1972-01-01
92
We have run plots of artificial data, which mimic solar magnetograms, through standard algorithms to critique several results reported in the literature. In studying correlation algorithms, we show that the differences in the profiles for the differential rotation of the photospheric magnetic field stem from different methods of averaging. We verify that the lifetimes of small magnetic features, or of
A. A. Smith; H. B. Snodgrass
1999-01-01
93
ERIC Educational Resources Information Center
|Describes a method for measuring the earth's magnetic field using an empty toilet paper tube, copper wire, clear tape, a battery, a linear variable resistor, a small compass, cardboard, a protractor, and an ammeter. (WRM)|
Stewart, Gay B.
2000-01-01
94
National Technical Information Service (NTIS)
The proposed research efforts funded by the UDAP grant to the BRI involve the study of magnetic field waves associated with the Uranian bow shock. This is a collaborative venture bringing together investigators at the BRI, Southwest Research Institute (Sw...
C. W. Smith M. L. Goldstein R. P. Lepping W. H. Mish H. K. Wong
1991-01-01
95
In this article the effect of low amplitude DC magnetic fields on different types of thermometers is discussed. By means of\\u000a a precision water-cooled electromagnet, the effect of a magnetic field on platinum resistance thermometers, thermistors, and\\u000a type T, J, and K thermocouples was investigated, while thermometers were thermally stabilized in thermostatic baths. Four\\u000a different baths were used for temperatures
G. Gersak; S. Begus
2010-01-01
96
In this work we present a global mapping of vector lunar magnetic field based on new method of separation of internal and external fields using inversion. The magnetic measurements collected during the lifetime of Lunar Prospector (LP) extended mission during 1999 were strongly disturbed by the solar wind, a period which coincided with a maximum of the 23 cycle activity. The multi scale wavelength external fields were analyzed using spherical harmonic transform. The external field determined by inversion was then removed from each magnetic field component for each half orbit. To map the vector magnetic crustal anomalies, all LP magnetometer data collected at low altitudes in the three different lunar environments: (1) geomagnetic tail (2) solar wind (3) geomagnetic sheath, were processed using this new approach. The results obtained using these selection criteria allow us to get a global coverage of the lunar surface by the vector magnetic field at variable spacecraft low altitudes. To validate our mapping we have developed and applied a method based on properties of potential fields functions. This method can be used to determine both horizontal North and East components using only vertical component. The validate lunar internal magnetic measurements obtained at variable spacecraft altitudes was then continued to a common altitude of 30 km using non linear inverse methods. This mapping confirm firstly the nature of the crustal sources of lunar magnetic field and clearly shows that the strongest concentrations of anomalies are associated with of high albedo and/or located antipodal to large young basins (Orientale, Serenitatis, Imbrium, and Crisium) of age about 3.9 Ga.
Berguig, M. C.; Hamoudi, M.; LeMoul, J. L.; Cohen, Y.
2012-04-01
97
SciTech Connect
Transverse particle motion in particle accelerators is governed almost totally by non-solenoidal magnets for which the body magnetic field can be expressed as a series expansion of the normal (b{sub n}) and skew (a{sub n}) multipoles, B{sub y} + iB{sub x} = {summation}(b{sub n} + ia{sub n})(x + iy){sup n}, where x, y, and z denote horizontal, vertical, and longitudinal (along the magnet) coordinates. Since the magnet length L is necessarily finite, deflections are actually proportional to field integrals such as {bar B}L {equivalent_to} {integral} B(x,y,z)dz where the integration range starts well before the magnet and ends well after it. For {bar a}{sub n}, {bar b}{sub n}, {bar B}{sub x}, and {bar B}{sub y} defined this way, the same expansion Eq. 1 is valid and the standard approximation is to neglect any deflections not described by this expansion, in spite of the fact that Maxwells equations demand the presence of longitudinal field components at the magnet ends. The purpose of this note is to provide a semi-quantitative estimate of the importance of {vert_bar}{Delta}p{sub {proportional_to}}{vert_bar}, the transverse deflection produced by the ion-gitudinal component of the fringe field at one magnet end relative to {vert_bar}{Delta}p{sub 0}{vert_bar}, the total deflection produced by passage through the whole magnet. To emphasize the generality and simplicity of the result it is given in the form of a theorem. The essence of the proof is an evaluation of the contribution of the longitudinal field B{sub x} from the vicinity of one magnet end since, along a path parallel to the magnet axis such as path BC.
Wei, Jie [Brookhaven National Lab., Upton, NY (United States); Talman, R. [Cornell Univ., Ithaca, NY (United States). Lab. of Nuclear Studies
1995-12-31
98
PubMed
By incorporating even the basic elements of a more environmentally friendly, "green"construction and design in an MRI setting can create a safer, more pleasant space for the patients and staff, better images, and operational cost savings. Using building systems that have reduced amounts of steel can decrease construction time, increase thermal insulation, and reduce the weight of the structure meaning less energy required to transport and install. HVAC systems and lighting design can also play a major role in creating a "green"MRI suite. LEED certification places a focus on quality of the built environment, life cycle cost, and a productive indoor environment, as well as impact on the exterior environment. An LEED certified building considers costs and benefits for the lifetime of the building. PMID:22043731
Branton, Scott; Lile, Lawrence
99
US Patent & Trademark Office Database
A magnetic resonance system is disclosed. The system includes a transceiver having a multichannel receiver and a multichannel transmitter, where each channel of the transmitter is configured for independent selection of frequency, phase, time, space, and magnitude, and each channel of the receiver is configured for independent selection of space, time, frequency, phase and gain. The system also includes a magnetic resonance coil having a plurality of current elements, with each element coupled in one to one relation with a channel of the receiver and a channel of the transmitter. The system further includes a processor coupled to the transceiver, such that the processor is configured to execute instructions to control a current in each element and to perform a non-linear algorithm to shim the coil.
2010-09-21
100
Data for the magnetic dipole hyperfine interaction of essentially single rare earth ions in metals, measured with different experimental methods, are collected and discussed. Depending on the host, the magnetic hyperfine field of these paramagnetic ions remains undisturbed by the environment, or it is enlarged, or weakened or can even become completely lost. If there are magnetic ions in the
G. Netz
1986-01-01
101
SciTech Connect
Several recent applications for fast ramped magnets have been found that require rapid measurement of the field quality during the ramp. (In one instance, accelerator dipoles will be ramped at 1 T/sec, with measurements needed to the accuracy typically required for accelerators.) We have built and tested a new type of magnetic field measuring system to meet this need. The system consists of 16 stationary pickup windings mounted on a cylinder. The signals induced in the windings in a changing magnetic field are sampled and analyzed to obtain the field harmonics. To minimize costs, printed circuit boards were used for the pickup windings and a combination of amplifiers and ADPs used for the voltage readout system. New software was developed for the analysis. Magnetic field measurements of a model dipole developed for the SIS200 accelerator at GSI are presented. The measurements are needed to insure that eddy currents induced by the fast ramps do not impact the field quality needed for successful accelerator operation.
JAIN, A.; ESCALLIER, J.; GANETIS, G.; LOUIE, W.; MARONE, A.; THOMAS. R.; WANDERER, P.
2004-10-03
102
This work aims at studying how magnetic fields affect the observational properties and the long-term evolution of isolated neutron stars, which are the strongest magnets in the universe. The extreme physical conditions met inside these astronomical sources complicate their theoretical study, but, thanks to the increasing wealth of radio and X-ray data, great advances have been made over the last years. A neutron star is surrounded by magnetized plasma, the so-called magnetosphere. Modeling its global configuration is important to understand the observational properties of the most magnetized neutron stars, magnetars. On the other hand, magnetic fields in the interior are thought to evolve on long time-scales, from thousands to millions of years. The magnetic evolution is coupled to the thermal one, which has been the subject of study in the last decades. An important part of this thesis presents the state-of-the-art of the magneto-thermal evolution models of neutron stars during the first million of years, studied by means of detailed simulations. The numerical code here described is the first one to consistently consider the coupling of magnetic field and temperature, with the inclusion of both the Ohmic dissipation and the Hall drift in the crust.
Vigan, Daniele
2013-09-01
103
We have recently performed a detailed characterization of ion Joule heating perpendicular to an axial magnetic field in the laboratory in a simulated ionospheric plasma environment which contains localized electric field structuring. Since Joule heating is often regarded as an important mechanism contributing to energization of outflowing heavy ions observed by higher-altitude auroral satellites, this work has particular relevance to
David N. Walker; William E. Amatucci; Gurudas I. Ganguli
2001-01-01
104
Response of biological systems against combined environment of zero-gravity and zero-magnetic field should be examined as the baseline to investigate biological effects of magnetic field that might be concealed by gravity. Space offers unique opportunities to conduct such study because long term microgravity is available for the scientific use. However, magnetic environment has been neither well controlled nor documented both in space and ground based experiments. Biological specimen is exposed to the various magnetic field of Earth during the revolutions in orbit. The profile of magnetic field varying in time depends on the orbital parameters and attitude of the space platform. Furthermore, the onboard 1 G control group is subjected to centrifugation spinning where magnetic field varies differently from the microgravity experiment group. It can not be accepted as the 1 G control in terms of magnetic environment. We propose experiment set up to shield exotic magnetic field experienced in orbiting space experiment platform. Thin film of amorphous metal or alloys has shielding capability, and is feasible to implement for space experimentation. In order to simulate zero-gravity and zero-magnetic field on ground, we developed a 3D- clinostat that equips a magnetic shielding layer for specimen. In order to evaluate effects of normal magnetic field of Earth, steady magnetic field is induced at the site of specimen inside the shield layer either in orbit or on 3D-clinostat. To fill the matrix of experimental design, 1 G control under the magnetic shielded condition, and 1 G control that is exposed to the normal field should be taken. Degree of magnetic shielding magnitude required for plant studies and other issues were examined by the preliminary experiments using a 3D-clinostat for the studies of etiolated seedlings.
Yamashita, M.; Tomita-Yokotani, K.; Hashimoto, H.; Nakamura, T.
105
The exact mechanism of formation of highly relativistic jets from galactic nuclei and microquasars remains unknown but most accepted models involve a central black hole and a strong external magnetic field. This idea is based on assumption that the black hole rotates and the magnetic field threads its horizon. Magnetic torques provide a link between the hole and the surrounding plasma which then becomes accelerated. We first review our work on black holes immersed in external stationary vacuum (electro)magnetic fields in both test-field approximation and within exact general-relativistic solutions. A special attention will be paid to the Meissner-type effect of the expulsion of the flux of external axisymmetric stationary fields across rotating (or charged) black holes when they approach extremal states. This is a potential threat to any electromagnetic mechanism launching the jets at the account of black-hole rotation because it inhibits the extraction of black-hole rotational energy. We show that the otherwise very useful "membrane viewpoint of black holes" advocated by Thorne, Price and Macdonald does not represent an adequate formalism in the context of the field expulsion from extreme black holes. After briefly summarizing the results for black holes in magnetic fields in higher dimensions - the expulsion of stationary axisymmetric fields was demonstrated to occur also for extremal black-hole solutions in string theory and Kaluza-Klein theory - we shall review astrophysically relevant axisymmetric numerical simulations reported recently by Gammie, Komissarov, Krolik and others. Although the field expulsion has not yet been observed in these time-dependent simulations, they may still be too far away from the extreme limit at which the black-hole Meissner effect should show up. We mention some open problems which, according to our view, deserve further investigation.
Bi?k, Ji?; Karas, Vladimr; Ledvinka, Tom
2007-04-01
106
The origin of large-scale magnetic fields in clusters of galaxies remains controversial. The intergalactic magnetic field within filaments should be less polluted by magnetised outflows from active galaxies than magnetic fields in clusters. Therefore, filaments may be a better laboratory to study magnetic field amplification by structure formation than galaxy clusters, which typically host many more active galaxies. We present
M. Brggen; M. Hoeft
2006-01-01
107
It has been proposed that high Mach number collisionless shocks propagating in an initially unmagnetized plasma play a major role in the magnetization of large scale structures in the Universe. A detailed study of the experimental configuration necessary to scale such environments down to laboratory dimensions will be presented. We will show initial results from preliminary experiments conducted at the Phoenix laser (UCLA) and the LULI laser (Ecole Polytechnique) where collisionless shocks are generated by the expansion of exploding foils driven by energetic laser beams. The time evolution of the magnetic field is probed with induction coils placed at 10 cm from the laser focus. We will discuss various mechanisms of magnetic field generation and compare them with the experimental results.
Murphy, C. D.; Miniati, F.; Edwards, M.; Mithen, J.; Bell, A. R.; Constantin, C.; Everson, E.; Schaeffer, D.; Niemann, C.; Ravasio, A.; Brambrink, E.; Benuzzi-Mounaix, A.; Koenig, M.; Gregory, C.; Woolsey, N.; Park, H.-S.; Remington, B.; Ryutov, D.; Bingham, R.; Gargate, L.; Spitkovsky, A.; Gregori, G.
2010-11-01
108
Indoor localization consists of locating oneself inside new buildings. GPS does not work indoors due to multipath reflection and signal blockage. WiFi based systems assume ubiquitous availability and infrastructure based systems require expensive installations, hence making indoor localization an open problem. This dissertation consists of solving the problem of indoor localization by thoroughly exploiting the indoor ambient magnetic fields comprising mainly of disturbances termed as anomalies in the Earth's magnetic field caused by pillars, doors and elevators in hallways which are ferromagnetic in nature. By observing uniqueness in magnetic signatures collected from different campus buildings, the work presents the identification of landmarks and guideposts from these signatures and further develops magnetic maps of buildings - all of which can be used to locate and navigate people indoors. To understand the reason behind these anomalies, first a comparison between the measured and model generated Earth's magnetic field is made, verifying the presence of a constant field without any disturbances. Then by modeling the magnetic field behavior of different pillars such as steel reinforced concrete, solid steel, and other structures like doors and elevators, the interaction of the Earth's field with the ferromagnetic fields is described thereby explaining the causes of the uniqueness in the signatures that comprise these disturbances. Next, by employing the dynamic time warping algorithm to account for time differences in signatures obtained from users walking at different speeds, an indoor localization application capable of classifying locations using the magnetic signatures is developed solely on the smart phone. The application required users to walk short distances of 3-6 m anywhere in hallway to be located with accuracies of 80-99%. The classification framework was further validated with over 90% accuracies using model generated magnetic signatures representing hallways with different kinds of pillars, doors and elevators. All in all, this dissertation contributes the following: 1) provides a framework for understanding the presence of ambient magnetic fields indoors and utilizing them to solve the indoor localization problem; 2) develops an application that is independent of the user and the smart phones and 3) requires no other infrastructure since it is deployed on a device that encapsulates the sensing, computing and inferring functionalities, thereby making it a novel contribution to the mobile and pervasive computing domain.
Pathapati Subbu, Kalyan Sasidhar
109
The Helioseismic and Magnetic Imager (HMI) instrument on the Solar Dynamics Observatory (SDO) spacecraft will begin observing the solar photospheric magnetic field continuously after commissioning in early 2009. This paper describes the HMI magnetic processing pipeline and the expected data products that will be available. The full disk line-of-sight magnetic field will be available every minute with 1" resolution. Comparable vector measurements collected over a three-minute time interval will ordinarily be averaged for at least 10 minutes before inversion. Useful Quick Look products for forecasting purposes will be available a few minutes after observation. Final products will be computed within 36 hours and made available through the SDO Joint Science Operations Center (JSOC). Three kinds of magnetic data products have been defined - standard, on-demand, and on-request. Standard products, such as frequently updated synoptic charts, are made all the time on a fixed cadence. On-demand products, such as high cadence full-disk disambiguated vector magnetograms, will be generated whenever a user asks for them. On-request products, such as high-resolution time series of MHD model solutions, will be generated as resources allow. This paper describes the observations, magnetograms, synoptic and synchronic products, and field model calculations that will be produced by the HMI magnetic pipeline.
Hoeksema, J.; Hmi, M. T.
2008-05-01
110
SciTech Connect
Quantum tunneling across a static potential barrier in a static magnetic field is very sensitive to an analytical form of the potential barrier. Depending on that, the oscillatory structure of the modulus of the wave function can be formed in the direction of tunneling. Due to an underbarrier interference, the probability of tunneling through a higher barrier can be larger than through a lower one. For some barriers the quantum interference of underbarrier cyclotron paths results in a strong enhancement of tunneling. This occurs in the vicinity of the certain magnetic field and is referred to as Euclidean resonance. This strongly contrasts to the Wentzel, Kramers, and Brillouin type tunneling which occurs with no magnetic field.
Ivlev, B. [Department of Physics and Astronomy and NanoCenter, University of South Carolina, Columbia, South Carolina 29208 (United States) and Instituto de Fisica, Universidad Autonoma de San Luis Potosi, San Luis Potosi, San Luis Potosi 78000 Mexico
2006-05-15
111
In the AdS/CFT framework meson thermalization in the presence of a constant external magnetic field in a strongly coupled gauge theory has been studied. In the gravitational description the thermalization of mesons corresponds to the horizon formation on the flavour D7-brane which is embedded in the AdS 5 S 5 background in the probe limit. The apparent horizon forms due to the time-dependent change in the baryon number chemical potential, the injection of baryons in the gauge theory. We will numerically show that the thermalization happens even faster in the presence of the magnetic field on the probe brane. We observe that this reduction in the thermalization time sustains up to a specific value of the magnetic field.
2013-03-01
112
Electromagnetic induction is a powerful technique to study the electrical conductivity of the interior of the Earth and other solar system bodies. Information about the electrical conductivity structure can provide strong constraints on the associated internal composition of planetary bodies. Here we give a review of the basic principles of the electromagnetic induction technique and discuss its application to various bodies of our solar system. We also show that the plasma environment, in which the bodies are embedded, generates in addition to the induced magnetic fields competing plasma magnetic fields. These fields need to be treated appropriately to reliably interpret magnetic field measurements in the vicinity of solar system bodies. Induction measurements are particularly important in the search for liquid water outside of Earth. Magnetic field measurements by the Galileo spacecraft provide strong evidence for a subsurface ocean on Europa and Callisto. The induction technique will provide additional important constraints on the possible subsurface water, when used on future Europa and Ganymede orbiters. It can also be applied to probe Enceladus and Titan with Cassini and future spacecraft.
Saur, Joachim; Neubauer, Fritz M.; Glassmeier, Karl-Heinz
2010-05-01
113
The solar photosphere is the layer in which the magnetic field has been most reliably and most often measured. Zeeman- and Hanle-effect based probes have revealed many details of a rich variety of structures and dynamic processes, but the number of open and debated questions has remained large. The magnetic field in the quiet Sun has maintained a particularly large number of secrets and has been a topic of a particularly lively debate as new observations and analysis techniques have revealed new and often unexpected aspects of its organization, physical structure and origin.
Solanki, S. K.
2009-06-01
114
SciTech Connect
Research on small-scale and large-scale photospheric and coronal magnetic fields during 1987-1990 is reviewed, focusing on observational studies. Particular attention is given to the new techniques, which include the correlation tracking of granules, the use of highly Zeeman-sensitive infrared spectral lines and multiple lines to deduce small-scale field strength, the application of long integration times coupled with good seeing conditions to study weak fields, and the use of high-resolution CCD detectors together with computer image-processing techniques to obtain images with unsurpassed spatial resolution. Synoptic observations of large-scale fields during the sunspot cycle are also discussed. 101 refs.
Sheeley, N.R., Jr. (USAF, Geophysics Laboratory, Hanscom AFB, MA (United States))
1991-01-01
115
NSDL National Science Digital Library
The EJSMagnetic Field from Loops model computes the B-field created by an electric current through a straight wire, a closed loop, and a solenoid. Users can adjust the vertical position of the slice through the 3D field. The Magnetic Field from Loops model was created using the Easy Java Simulations (Ejs) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_ntnu_MagneticFielfFromLoops.jar file will run the program if Java is installed. Ejs is a part of the Open Source Physics Project and is designed to make it easier to access, modify, and generate computer models. Additional Ejs models for classical mechanics are available. They can be found by searching ComPADRE for Open Source Physics, OSP, or Ejs.
Christian, Wolfgang; Hwang, Fu-Kwun
2008-11-17
116
We investigate the magnetic field which is generated by turbulent motions of a weakly ionized gas. Galactic molecular clouds give us an example of such a medium. As in the Kazantsev-Kraichnan model we assume a medium to be homogeneous and a neutral gas velocity field to be isotropic and ? correlated in time. We take into consideration the presence of a mean magnetic field, which defines a preferred direction in space and eliminates isotropy of magnetic field correlators. Evolution equations for the anisotropic correlation function are derived. Isotropic cases with zero mean magnetic field as well as with small mean magnetic field are investigated. It is shown that stationary bounded solutions exist only in the presence of the mean magnetic field for the Kolmogorov neutral gas turbulence. The dependence of the magnetic field fluctuations amplitude on the mean field is calculated. The stationary anisotropic solution for the magnetic turbulence is also obtained for large values of the mean magnetic field.
Istomin, Ya. N.; Kiselev, A.
2013-10-01
117
A rotating coil setup for magnetic field characterization and fiducialization of XFEL quadrupole magnets is pre- sented. The instrument allows measurement of the rel- ative position of the magnetic axis with accuracy better than 1 ?m and measurement of weak magnetic error field components. Tests and evaluation based on a FLASH quadrupole magnet are presented together with a discus- sion
A. Hedqvist; H. Danared; F. Hellberg; J. Pfluger
118
Magnetic field measurements made by means of Explorer 10 over geocentric ; distances of 1.8 to 42.6R\\/sub e\\/ on March 25experiment on the same satellite are ; referenced in interpretations. The close-in data are consistent with the ; existence of a very weak ring current below 3R\\/sub e\\/ along the trajectory, but ; alternative explanations for the field deviations are
J. P. Heppner; N. F. Ness; C. S. Scearce; T. L. Skillman
1963-01-01
119
Observations indicate that jets (i.e., charged particle beams) are emitted from the central black hole sources of active galactic nuclei and quasars. Magnetic fields are produced in e(-)-p or e(-)-e(+)-p jets when electrons (and positrons) are slowed with respect to protons in the jets. Interaction with an ambient interstellar gas or external radiation field can cause such drift velocities. Calculations
William K. Rose
1987-01-01
120
Observations indicate that jets are emitted from the central black hole sources of active galactic nuclei and quasars. Magnetic fields are produced in e--p or e--e+-p jets when electrons and positrons are slowed with respect to protons in the jets. Interaction with an ambient interstellar gas or external radiation field can cause such drift velocities. In this paper calculations for
William K. Rose
1987-01-01
121
Averaged magnetoencephalography (MEG) following somatosensory stimulation, somatosensory evoked magnetic field(s) (SEF), in humans are reviewed. The equivalent current dipole(s) (ECD) of the primary and the following middle-latency components of SEF following electrical stimulation within 80100 ms are estimated in area 3b of the primary somatosensory cortex (SI), the posterior bank of the central sulcus, in the hemisphere contralateral to the
Ryusuke Kakigi; Minoru Hoshiyama; Motoko Shimojo; Daisuke Naka; Hiroshi Yamasaki; Shoko Watanabe; Jing Xiang; Kazuaki Maeda; Khanh Lam; Kazuya Itomi; Akinori Nakamura
2000-01-01
122
Petropoulos, Labros Spiridon
123
The most important milestone in the field of magnetic sensors was when AMR sensors started to replace Hall sensors in many applications where the greater sensitivity of AMRs was an advantage. GMR and SDT sensors finally found applications. We also review the development of miniaturization of fluxgate sensors and refer briefly to SQUIDs, resonant sensors, GMIs, and magnetomechanical sensors.
Pavel Ripka; Michal Janosek
2010-01-01
124
National Technical Information Service (NTIS)
In order to explore the consequences of random field effects we have carried out a series of neutron scattering experiments on three prototypical diluted Ising magnets. The systems studied are Rb sub 2 Co sub 7 Mg sub 3 F sub 4 which is a model two dimens...
R. J. Birgeneau
1982-01-01
125
National Technical Information Service (NTIS)
The research efforts funded by the Uranus Data Analysis Program (UDAP) grant to the Bartol Research Institute (BRI) involved the study of magnetic field waves associated with the Uranian bow shock. Upstream wave studies are motivated as a study of the phy...
C. W. Smith M. L. Goldstein R. P. Lepping W. H. Mish H. K. Wong
1994-01-01
126
SciTech Connect
Research toward structural alloys for use in high field superconducting magnets is international in scope, and has three principal objectives: the selection or development of suitable structural alloys for the magnet support structure, the identification of mechanical phenomena and failure modes that may influence service behavior, and the design of suitable testing procedures to provide engineering design data. This paper reviews recent progress toward the first two of these objectives. The structural alloy needs depend on the magnet design and superconductor type and differ between magnets that use monolithic and those that employ force-cooled or ICCS conductors. In the former case the central requirement is for high strength, high toughness, weldable alloys that are used in thick sections for the magnet case. In the latter case the need is for high strength, high toughness alloys that are used in thin welded sections for the conductor conduit. There is productive current research on both alloy types. The service behavior of these alloys is influenced by mechanical phenomena that are peculiar to the magnet environment, including cryogenic fatigue, magnetic effects, and cryogenic creep. The design of appropriate mechanical tests is complicated by the need for testing at 4/sup 0/K and by rate effects associated with adiabatic heating during the tests. 46 refs.
Morris, J.W. Jr.
1985-08-01
127
PubMed
The most important and very expensive part of a magnetic resonance imaging set-up is the magnet, which is capable of generating a constant and highly homogeneous magnetic field. Here a new MR imaging technique without the magnet is introduced. This technique uses the earth's magnetic field instead of a magnetic field created by a magnet. This new method has not yet reached the stage of medical application, but the first images obtained by MRIE (magnetic resonance imaging in the earth's field) show that the resolution is close to that expected based on sensitivity estimations. PMID:2233218
Stepisnik, J; Erzen, V; Kos, M
1990-09-01
128
SciTech Connect
The field lines of magnetic fields that depend on three spatial coordinates are shown to have a fundamentally different behavior from those that depend on two coordinates. Unlike two-coordinate cases, a flux tube in a magnetic field that depends on all three spatial coordinates that has a circular cross section at one location along the tube characteristically has a highly distorted cross section at other locations. In an ideal evolution of a magnetic field, the current densities typically increase. Crudely stated, if the current densities increase by a factor {sigma}, the ratio of the long to the short distance across a cross section of a flux tube characteristically increases by e{sup 2{sigma}}, and the ratio of the longer distance to the initial radius increases as e{sup {sigma}}. Electron inertia prevents a plasma from isolating two magnetic field structures on a distance scale shorter than c/{omega}{sub pe}, which is about 10 cm in the solar corona, and reconnection must be triggered if {sigma} becomes sufficiently large. The radius of the sun, R{sub Circled-Dot-Operator }=7 Multiplication-Sign 10{sup 10}cm is about e{sup 23} times larger, so when {sigma} Greater-Than-Or-Equivalent-To 23, two lines separated by c/{omega}{sub pe} at one location can be separated by the full scale of any magnetic structures in the corona at another. The conditions for achieving a large exponentiation, {sigma}, are derived, and the importance of exponentiation is discussed.
Boozer, Allen H. [Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027 (United States)
2012-11-15
129
We present observations taken with the Advanced Stokes Polarimeter (ASP) in active-region plages and study the frequency distribution of the magnetic field strength (B), inclination with respect to vertical ( gamma ), azimuthal orientation ( chi ), and filling factor (f). The most common values at disk center are B = 1400 G, gamma < 10 deg, no preferred east-west orientation, and f = 15%. At disk center, there is a component of weak (<1000 G), more horizontal fields that corresponds to arching field lines connecting footpoints of different polarities. The center-to-limb variation (CLV) of the field strength shows that, close to the limb ( mu = 0.3), the field strength is reduced to 800 G from its disk-center value. This can be interpreted as a gradient of B with height in solar plages of around -3 G km-1. From this CLV study, we also deduce that magnetic field lines remain vertical for the entire range of heights involved. A similar analysis is performed for structures found in active regions that show a continuous distribution of azimuths (resembling sunspots) but that do not have a darkening in continuum. These "azimuth centers" show slightly larger values of B than normal plages, in particular at their magnetic center. Filling factors are also larger on average for these structures. The velocities in the magnetic component of active regions have been studied for both averaged Stokes profiles over the entire active region and for the spatially resolved data. The averaged profiles (more representative of high filling factor regions) do not show any significant mean velocities. However, the spatial average of Doppler velocities derived from the spatially resolved profiles (i.e., unweighted by filling factor) show a net redshift at disk center of 200 m s-1. The spatially resolved velocities show a strong dependence on filling factor. Both mean velocities and standard deviations are reduced when the filling factor increases. This is interpreted as a reduction of the p-mode amplitude within the magnetic component. Strong evidence for velocities transverse to the magnetic field lines has been found. Typical rms values are between 200 and 300 m s-1, depending on the filling factor. The possible importance of these transverse motions for the dynamics of the upper atmospheric layers is discussed. The asymmetries of the Stokes profiles and their CLV have been studied. The averaged Stokes V profiles show amplitude and area asymmetries that are positive at disk center and become negative at the limb. Both asymmetries, and for the two Fe I lines, are maximized away from disk center. The spatially resolved amplitude asymmetries show a clear dependence on filling factor: the larger the filling factor, the smaller the amplitude asymmetry. On the other hand, the area asymmetry is almost independent of the filling factor. The only observed dependence is the existence of negative area-asymmetry profiles at disk center for filling factors smaller than 0.2. Around 20% of the observed points in a given plage have negative area asymmetry. The amplitude asymmetry of Stokes V is, on the other hand, always positive. The amplitude asymmetries of the linear polarization profiles are observed to have the same sign as the Stokes V profiles. Similarly, the same CLV variation of the linear polarization amplitude asymmetries as for Stokes V has been found. The scenarios in which this similarity can exist are studied in some detail.
Martinez Pillet, V.; Lites, B. W.; Skumanich, A.
1997-01-01
130
PubMed
This review explores the dynamics of two-dimensional electrons in magnetic potentials that vary on scales smaller than the mean free path. The physics of microscopically inhomogeneous magnetic fields relates to important fundamental problems in the fractional quantum Hall effect, superconductivity, spintronics and graphene physics and spins out promising applications which will be described here. After introducing the initial work done on electron localization in random magnetic fields, the experimental methods for fabricating magnetic potentials are presented. Drift-diffusion phenomena are then described, which include commensurability oscillations, magnetic channelling, resistance resonance effects and magnetic dots. We then review quantum phenomena in magnetic potentials including magnetic quantum wires, magnetic minibands in superlattices, rectification by snake states, quantum tunnelling and Klein tunnelling. The third part is devoted to spintronics in inhomogeneous magnetic fields. This covers spin filtering by magnetic field gradients and circular magnetic fields, electrically induced spin resonance, spin resonance fluorescence and coherent spin manipulation. PMID:21393794
Nogaret, Alain
2010-06-04
131
SciTech Connect
Although only a small part of available energy in the universe is invested in magnetic fields, they are responsible for most of the continual violent activity in the cosmos. There is a single, generic explanation for the ability of bodies as different as a dense, cold planet and a tenuous hot galactic disk to generate a magnetic field. The explanation, first worked out for the earth, comes from the discipline of magnetohydrodynamics. The cosmos is filled with fluids capable of carrying electric currents. The magnetic fields entrained in these fluids are stretched and folded by the fluid motion, gaining energy in the process. In other words, the turbulent fluids function as dynamos. However, the dynamo mechanism by itself cannot account for the exceptionally strong field of some stars. Because of such gaps in information, the rival hypothesis that there are primordial fields cannot be disproved. The balance of evidence, however, indicates that the planets, sun, most stars and the galaxy function as colossal dynamos. (SC)
Parker, E.N.
1983-08-01
132
SciTech Connect
The authors present experimental results from the investigation of the behavior of certain magnetic liquids differeing in the degree of stability in inhomogenous magnetic fields. The growth of holding presure of sealing step at rest is reviewed and the increase of effective viscosity in inhomogeneous magnetic fields is studied. The behaviors of magnetic liquids in an inhomogeneous magnetic field are sensitive to structural changes caused by the field. Significant differences are demonstrated between magnetic liquids with the same saturation magnetization but different particle size distribution.
Anton, I.; Bika, D.; Potents, I.; Vekash, L.
1986-01-01
133
To investigate the effect of low-frequency magnetic-field exposure of a human body, the low-frequency AC magnetic property of a large volume of water was measured by low-frequency magnetic field exposure (from 50 Hz to 1.2 kHz). The results indicate that the AC magnetic property of water is due to diamagnetism in the low-frequency range. The phase between the main magnetic field and the generated magnetic field remained constant at about 180. Results were not affected by conductivity or pH. Moreover, the magnetic-field strength from water showed a susceptibility frequency dependence proportional to the frequency above approximately 400 Hz. Because of the incremental effects of susceptibility, the magnetic field from water was measured using a conventional magnetic sensor (magnetic resistive; MR) in an unshielded environment.
Tsukada, Keiji; Kiwa, Toshihiko; Masuda, Yuuki
2006-10-01
134
We describe studies of nuclear magnetic resonance (NMR) spectroscopy and magnetic resonance imaging (MRI) of liquid samples at room temperature in microtesla magnetic fields. The nuclear spins are prepolarized in a strong transient field. The magnetic signals generated by the precessing spins, which range in frequency from tens of Hz to several kHz, are detected by a low-transition temperature dc
R. McDermott; N. Kelso; S. K. Lee; M. MBetale; M. Mck; W. Myers; B. ten Haken; H. C. Seton; A. H. Trabesinger; A. Pines; J. Clarke
2004-01-01
135
Frustrated magnets in high magnetic field have a long history of offering beautiful surprises to the patient investigator. Here we present the results of extensive classical Monte Carlo simulations of a variety of models of two dimensional magnets in magnetic field, together with complementary spin wave analysis. Striking results include (i) a massively enhanced magnetocaloric effect in antiferromagnets bordering on
L. Seabra; N. Shannon; P. Sindzingre; T. Momoi; B. Schmidt; P. Thalmeier
2009-01-01
136
The Helioseismic and Magnetic Imager (HMI) will provide frequent full-disk magnetic field data after launch of the Solar Dynamics Observatory (SDO), currently scheduled for fall 2009. 16 megapixel line-of-sight magnetograms (Blos) will be recorded every 45 seconds. A full set of polarized filtergrams needed to determine the vector magnetic field requires 90 seconds. Quick-look data will be available within a few minutes of observation. Quick-look space weather and browse products must have identified users, and the list currently includes full disk magnetograms, feature identification and movies, 12-minute disambiguated vector fields in active region patches, time evolution of AR indices, synoptic synchronic frames, potential and MHD model results, and 1 AU predictions. A more complete set of definitive science data products will be offered about a day later and come in three types. "Pipeline products, such as full disk vector magnetograms, will be computed for all data on an appropriate cadence. A larger menu of "On Demand products, such as Non-Linear Force Free Field snapshots of an evolving active region, will be produced whenever a user wants them. Less commonly needed "On Request products that require significant project resources, such as a high resolution MHD simulation of the global corona, will be created subject to availability of resources. Further information can be found at the SDO Joint Science Operations Center web page, jsoc.stanford.edu
Hoeksema, Jon Todd; Liu, Y.; Schou, J.; Scherrer, P.; HMI Science Team
2009-05-01
137
This article describes both the setup and the use of a system for magnetic resonance imaging (MRI) in the Earth's magnetic field. Phase instability caused by temporal fluctuations of Earth's field can be successfully improved by using a reference signal from a separate Earth's field nuclear magnetic resonance (NMR) spectrometer\\/magnetometer. In imaging, it is important to correctly determine the phase
Ales Mohoric; Gorazd Planinsic; Miha Kos; Andrej Duh; Janez Stepisnik
2004-01-01
138
PubMed
The isomagnetic maps of normal subjects and patients with right and left atrial overloading were recorded to determine the characteristic features of the magnetic field of atrial depolarization. The isomagnetic maps examined in this study indicated the instantaneous current source, which specifically localizes the current sources due to the right and left atria, respectively. The magnetic field recorded with a second derivative gradiometer clearly detected the cardiac current source from the right atrium, which is located close to the anterior chest wall, thus this method improved the diagnostic sensitivity for right atrial overloading. In patients with left atrial overloading, the isomagnetic map showed multiple dipoles due to the right and left atria, respectively, which are difficult to be detected by the electrocardiogram or isopotential map. These results suggest that the magnetocardiogram provides useful information on the current source to supplement information obtained by the conventional electrocardiogram. PMID:2978585
Takeuchi, A; Watanabe, K; Katayama, M; Nomura, M; Nakaya, Y; Mori, H
139
We consider the amplification of cosmological magnetic fields by gravitational waves as it was recently presented by Betschart et al. That study confined to infinitely conductive environments, arguing that on spatially flat Friedmann backgrounds the gravito-magnetic interaction proceeds always as if the Universe were a perfect conductor. We explain why this claim is not correct and then reexamine the Maxwell-Weyl
Tsagas; Christos G
2007-01-01
140
We consider the amplification of cosmological magnetic fields by gravitational waves as it was recently presented by Betschart et al. That study confined to infinitely conductive environments, arguing that on spatially flat Friedmann backgrounds the gravito-magnetic interaction proceeds always as if the Universe were a perfect conductor. We explain why this claim is not correct and then reexamine the Maxwell-Weyl
Christos G. Tsagas
2007-01-01
141
SciTech Connect
Magnetic field-structured-composites (FSCs) are made by structuring magnetic particle suspensions in uniaxial or biaxial (e.g. rotating) magnetic fields, while polymerizing the suspending resin. A uniaxial field produces chain-like particle structures, and a biaxial field produces sheet-like particle structures. In either case, these anisotropic structures affect the measured magnetic hysteresis loops, with the magnetic remanence and susceptibility increased significantly along the axis of the structuring field, and decreased slightly orthogonal to the structuring field, relative to the unstructured particle composite. The coercivity is essentially unaffected by structuring. We present data for FSCs of magnetically soft particles, and demonstrate that the altered magnetism can be accounted for by considering the large local fields that occur in FSCs. FSCS of magnetically hard particles show unexpectedly large anisotropies in the remanence, and this is due to the local field effects in combination with the large crystalline anisotropy of this material.
Anderson, Robert A.; Martin, James E.; Odinek, Judy; Venturini, Eugene
1999-06-24
142
National Technical Information Service (NTIS)
Spatiotemporal patterns of somatosensory evoked magnetic fields to stimulation of upper and lower limb nerves were examined in healthy humans. The studies summarized here provide the first magnetic field maps over the primary foot projection area after li...
J. Huttunen
1987-01-01
143
Metal complexes and solids were synthesized and subjected to photoexcitation measurements under the influence of externally applied magnetic fields. The photoluminescence of complexes of rhodium (I) and iridium (I) displayed both field induced emission bands and a many fold shortening of the excited state lifetime. Both the decay rates and the induced emission band intensities showed a quadratic dependence on the applied field. A several fold shortening of the phosphorescence from the octaphosphitoplatinum (II) anion under an applied field (50 T) was also observed. Spectroscopic studies of several bis (N-heterocyclic) complexes of copper (I) were also concluded and complete group theoretic assignments of the charge transfer excited states were made. The technique of Thermal Modulation was perfected and applied to the study of the exited states of transition metal complexes with near degenerate emitting states.
Crosby, G. A.
1989-08-01
144
We have run plots of artificial data, which mimic solar magnetograms, through standard algorithms to critique several results reported in the literature. In studying correlation algorithms, we show that the differences in the profiles for the differential rotation of the photospheric magnetic field stem from different methods of averaging. We verify that the lifetimes of small magnetic features, or of small patterns of these features in the large-scale background field, are on the order of months, rather than a few days. We also show that a meridional flow which is cycle dependent creates an artifact in the correlation-determined magnetic rotation which looks like a torsional oscillation; and we compare this artifact to the torsional patterns that have been reported. Finally, we simulate the time development of a large-scale background field created solely from an input of artifical, finite-lifetime 'sunspot' bipoles. In this simulation, we separately examine the effects of differential rotation, meridional flow and Brownian motion (random walk, which we use rather than diffusion), and the inclination angles of the sunspot bipoles (Joy's law). We find, concurring with surface transport equation models, that a critical factor for producing the patterns seen on the Sun is the inclination angle of the bipolar active regions. This work was supported by NSF grant 9416999.
Smith, A. A.; Snodgrass, H. B.
1999-05-01
145
A novel resonant magnetic sensor based on the combination of a mechanical resonator and a magnetic field concentrator with two gaps is reported. In contrast to previous Lorentz force based resonant magnetic sensors, a high sensitivity is achieved without modulated driving current and complex feedback electronics. Furthermore, compared to magnetic moment based resonant magnetic sensors, the new concept requires no
S. Brugger; P. Simon; O. Paul
2006-01-01
146
SciTech Connect
We investigate the effect of a magnetic field on cold dense quark matter using an effective model with four-Fermi interactions. We find that the gap parameters representing the predominant pairing between the different quark flavors show oscillatory behavior as a function of the magnetic field. We point out that due to electric and color neutrality constraints the magnetic fields as strong as presumably existing inside magnetars might induce significant deviations from the gap structure at a zero magnetic field.
Fukushima, Kenji [RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973 (United States); Warringa, Harmen J. [Department of Physics, Bldg. 510A, Brookhaven National Laboratory, Upton, New York 11973 (United States)
2008-01-25
147
\\u000a This paper presents numerical simulation model and results on magnetic drug targeting therapy. The study aims at investigating\\u000a the aggregate blood - magnetic carrier flow interaction with an external magnetic field. Another objective was finding the\\u000a optimal magnetic field source configuration that provides for flows that best assist in magnetic drug targeting. In order\\u000a to evaluate the effects we used
A. Dobre; A. M. Morega
148
The magnetic field gradients of magnetic stripe cards, which are developed for classifying magnetic particles used in magnetic particle inspections, have been measured using a magnetic force microscope (MFM). The magnetic force exerted on a MFM probe by the stray field emanating from the card was measured to determine the field gradients. The results are in good agreement with the field gradients estimated from the magnetizing field strengths used in the encoding process. .
Lo, C. C. H.; Leib, J.; Jiles, D. C.; Chedister, W. C.
2002-05-01
149
This review concerns the origin and the possible effects of magnetic fields in the early Universe. We start by providing the reader with a short overview of the current state of the art of observations of cosmic magnetic fields. We then illustrate the arguments in favor of a primordial origin of magnetic fields in the galaxies and in the clusters
Dario Grasso; Hector R. Rubinstein
2001-01-01
150
SciTech Connect
We study limits on a primordial magnetic field arising from cosmological data, including that from big bang nucleosynthesis, cosmic microwave background polarization plane Faraday rotation limits, and large-scale structure formation. We show that the physically relevant quantity is the value of the effective magnetic field, and limits on it are independent of how the magnetic field was generated.
Kahniashvili, Tina [McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213 (United States); Department of Physics, Laurentian University, Ramsey Lake Road, Sudbury, Ontario P3E 2C (Canada); Abastumani Astrophysical Observatory, Ilia State University, 2A Kazbegi Ave, Tbilisi, GE-0160 (Georgia); Tevzadze, Alexander G. [Abastumani Astrophysical Observatory, Ilia State University, 2A Kazbegi Ave, Tbilisi, GE-0160 (Georgia); Faculty of Exact and Natural Sciences, Tbilisi State University, 1 Chavchavadze Avenue, Tbilisi, GE-0128 (Georgia); Sethi, Shiv K. [McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213 (United States); Raman Research Institute, Sadashivanagar, Bangalore 560080 (India); Pandey, Kanhaiya [Raman Research Institute, Sadashivanagar, Bangalore 560080 (India); Ratra, Bharat [Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, Kansas 66506 (United States)
2010-10-15
151
Experiments were performed at the Nevada Terawatt Facility to investigate the plasma penetration across an externally applied magnetic field. In experiment, a short-pulse laser ablates a polyethylene laser target, producing a plasma which interacts with an external magnetic field. The mechanism which allows the plasma to penetrate the applied magnetic field in experiment will be discussed.
Plechaty, C.; Presura, R.; Wright, S.; Neff, S.; Haboub, A.
2009-08-01
152
Radio observations of nearby spiral galaxies have tremendously enhanced our knowledge of their global magnetic field distributions. Recent theoretical developments in the area of dynamos have also helped in the interpretation of magnetic field data in spiral galaxies. When it comes to the magnetic field in the Milky Way galaxy, our position in the Milky Way's galactic disk hinders our
J. P. Vallee
1996-01-01
153
Recent controversy over 60 Hz magnetic fields has heightened public awareness of overhead transmission lines. As a result, there is increasing motivation to study the magnetic fields form transmission lines. The most cost effective means to conduct research into transmission line magnetic fields is with computer or reduced-scale line models. However, from the standpoint of public perception and acceptance, it
B. A. Clairmont; G. B. Johnson; J. H. Dunlap
1992-01-01
154
We study limits on a primordial magnetic field arising from cosmological data, including that from big bang nucleosynthesis, cosmic microwave background polarization plane Faraday rotation limits, and large-scale structure formation. We show that the physically relevant quantity is the value of the effective magnetic field, and limits on it are independent of how the magnetic field was generated.
Kahniashvili, Tina; Tevzadze, Alexander G.; Sethi, Shiv K.; Pandey, Kanhaiya; Ratra, Bharat
2010-10-01
155
Results and new progress of the origin and evolution of pulsar magnetic fields are reviewed. Lots of models about how such strong magnetic fields were generated, mainly two kinds of structures were proposed for initial magnetic fields: fields confined in the cores and fields confined in the crusts of neutron stars. No consensus has been reached on whether the magnetic fields decay or not, despite some observational evidence for the evolution of magnetic fields. The discrepancy between characteristic ages and kinematic ages indicates that the magnetic fields decay exponentially. On the other hand, the braking indices of several young pulsars and the comparison between pulsar characteristic ages and the ages of associated supernova remnants suggest that the magnetic fields of young pulsars grow like a power-law. Pulsar population synthesis is one of the most important methods to investigate the evolution of magnetic fields. Many simulations show that if magnetic fields do decay exponentially, the e-folding decay time should be 100 Myr or longer. The numerical calculations of the Ohmic decay in the crust indicate that the scenario of exponential decay is oversimple, and the evolution could be divided into four possible phases approximately: exponential decay, no decay, power-law decay and exponential decay again. The model of magnetic fields expulsion induced by spin-down suggests that the magnetic fields decay only in a period between 107yr and 108yr.
Sun, Xiaohui; Han, Jinlin
2002-06-01
156
The solar wind is an extended ionized gas of very high electrical conductivity, and therefore drags some magnetic flux out of the Sun to fill the heliosphere with a weak interplanetary magnetic field,. Magnetic reconnection-the merging of oppositely directed magnetic fields-between the interplanetary field and the Earth's magnetic field allows energy from the solar wind to enter the near-Earth environment.
M. Lockwood; R. Stamper; M. N. Wild
1999-01-01
157
An study on the demagnetization of rare-earth permanent magnets under high radiation environment is started from the microscopic point of view. The demagnetization of NEOMAX is successfully induced by the well defined neutron field produced by the 5 MW reactor in Kyoto University. Preliminary TDPAC measurement of 111Cd(<--111In) in NEOMAX, including demagnetized one, is reported.
M. Tanigaki; K. Takamiya; Y. Komeno; A. Taniguchi; Y. Ohkubo
2008-01-01
158
Magnetic field measurements are very valuable, as they provide constraints on the interior of the telluric planets and Moon. The Earth possesses a planetary scale magnetic field, generated in the conductive and convective outer core. This global magnetic field is superimposed on the magnetic field generated by the rocks of the crust, of induced (i.e. aligned on the current main field) or remanent (i.e. aligned on the past magnetic field). The crustal magnetic field on the Earth is very small scale, reflecting the processes (internal or external) that shaped the Earth. At spacecraft altitude, it reaches an amplitude of about 20 nT. Mars, on the contrary, lacks today a magnetic field of core origin. Instead, there is only a remanent magnetic field, which is one to two orders of magnitude larger than the terrestrial one at spacecraft altitude. The heterogeneous distribution of the Martian magnetic anomalies reflects the processes that built the Martian crust, dominated by igneous and cratering processes. These latter processes seem to be the driving ones in building the lunar magnetic field. As Mars, the Moon has no core-generated magnetic field. Crustal magnetic features are very weak, reaching only 30 nT at 30-km altitude. Their distribution is heterogeneous too, but the most intense anomalies are located at the antipodes of the largest impact basins. The picture is completed with Mercury, which seems to possess an Earth-like, global magnetic field, which however is weaker than expected. Magnetic exploration of Mercury is underway, and will possibly allow the Hermean crustal field to be characterized. This paper presents recent advances in our understanding and interpretation of the crustal magnetic field of the telluric planets and Moon.
Langlais, Benoit; Lesur, Vincent; Purucker, Michael E.; Connerney, Jack E. P.; Mandea, Mioara
2010-05-01
159
A nuclear magnetic resonance apparatus for experiments in pulsed high magnetic fields is described. The magnetic field pulses created together with various magnet coils determine the requirements such an apparatus has to fulfill to be operated successfully in pulsed fields. Independent of the chosen coil it is desirable to operate the entire experiment at the highest possible bandwidth such that a correspondingly large temporal fraction of the magnetic field pulse can be used to probe a given sample. Our apparatus offers a bandwidth of up to 20 MHz and has been tested successfully at the Hochfeld-Magnetlabor Dresden, even in a very fast dual coil magnet that has produced a peak field of 94.2 T. Using a medium-sized single coil with a significantly slower dependence, it is possible to perform advanced multi-pulse nuclear magnetic resonance experiments. As an example we discuss a Carr-Purcell spin echo sequence at a field of 62 T.
Meier, Benno; Kohlrautz, Jonas; Haase, Jrgen; Braun, Marco; Wolff-Fabris, Frederik; Kampert, Erik; Herrmannsdrfer, Thomas; Wosnitza, Joachim
2012-08-01
160
We are developing a sensitive magnetic tensor gradiometer for detection of underwater unexploded ordnance (UXO) using high temperature superconducting quantum interference devices (SQUIDs). In parallel with the instrument development, we are looking at the challenges of making electromagnetic field and gradient measurements in electrically conductive seawater. We have been carrying out a theoretical analysis of how the measurements are distorted
J. A. Young; D. A. Clark
2010-01-01
161
With a mean orbital radius of 20.2 Saturnian radii (1 Saturn radius RS=60,330km), Titan is usually located within the kronian magnetosphere. 3.5 years of Cassini magnetometer observations in the vicinity of Titan's orbit reveal that the moon's magnetic environment is strongly affected by the presence of Saturn's magnetodisk. As a result of the disk's solarwind-induced asymmetry, Titan is exposed to
C. Bertucci; B. Sinclair; N. Achilleos; P. Hunt; M. K. Dougherty; C. S. Arridge
2009-01-01
162
We present the basic steps for the study of the linear near field absorption spectra of semiconductor quantum dots under magnetic field of variable orientation. We show that the application of the magnetic field alone is sufficient to induce -increasing the spot illuminated by the near field probe- interesting features to the absorption spectra.
Anna Zora; Constantinos Simserides; Georgios Triberis
2005-01-01
163
We present the basic steps for the study of the linear near field absorption spectra of semiconductor quantum dots under magnetic field of variable orientation. We show that the application of the magnetic field alone is sufficient to induce -increasing the spot illuminated by the near field probe- interesting features to the absorption spectra.
Anna Zora; Constantinos Simserides; Georgios Triberis
2004-01-01
164
DOEpatents
A heat pipe configuration is described for use in a magnetic field environment of a fusion reactor. Heat pipes for operation in a magnetic field when liquid metal working fluids are used are optimized by flattening of the heat pipes having an unobstructed annulus which significantly reduces the adverse side region effect of the prior known cylindrically configured heat pipes. The flattened heat pipes operating in a magnetic field can remove 2 to 3 times the heat as a cylindrical heat pipe of the same cross sectional area.
Werner, R.W.; Hoffman, M.A.
1981-04-29
165
DOEpatents
A heat pipe configuration for use in a magnetic field environment of a fusion reactor. Heat pipes for operation in a magnetic field when liquid metal working fluids are used are optimized by flattening of the heat pipes having an unobstructed annulus which significantly reduces the adverse side region effect of the prior known cylindrically configured heat pipes. The flattened heat pipes operating in a magnetic field can remove 2--3 times the heat as a cylindrical heat pipe of the same cross sectional area.
Werner, Richard W. (San Ramon, CA); Hoffman, Myron A. (Davis, CA)
1983-01-01
166
This thesis is devoted to understanding the origins of lunar crustal magnetism. We wish to understand the processes which have created and modified the crustal magnetic field distribution that we observe today, and to determine whether the Moon ever had an active magnetohydrodynamic dynamo. Previously, our only measurements of lunar magnetic fields came from the Explorer 35 and Apollo missions. Data coverage was incomplete, but sufficient to establish some systematics of the crustal field distribution. With new data from the Magnetometer and Electron Reflectometer instrument on Lunar Prospector, we have generated the first completely global maps of the lunar crustal fields. We use measurements of electrons magnetically reflected above the lunar surface, which we then correct for the effects of electrostatic fields (which also reflect electrons), and convert to estimates of surface magnetic fields. The resulting global map shows that impact basins and craters (especially the youngest) generally have low magnetic fields, suggesting impact demagnetization, primarily by shock effects. A secondary signature of some large lunar basins (especially older ones) is the presence of a more localized central magnetic anomaly. Meanwhile, the largest regions of strong crustal fields lie antipodal to young large impact basins, suggesting shock remanent magnetization due to a combination of antipodal focussing of seismic energy and/or ejecta and plasma compression of ambient magnetic fields. Smaller regions of strong magnetic fields are sometimes associated with basin ejecta, and basin and crater ejecta terranes have the strongest average fields outside of the antipodal regions. This implies that impact-generated magnetization may extend beyond the antipodal regions. The antipodal, non-antipodal, and central basin magnetic fields, as well as returned samples, can all be used to estimate the lunar magnetic field history and place constraints on a possible lunar dynamo. All of these quantities provide evidence for stronger magnetic fields early in the Moon's history, and thereby suggest the existence of an ancient core dynamo.
Halekas, Jasper S.
167
During the encounter with Comet Halley, the magnetometer (MISCHA) aboard the Vega 1 spacecraft observed an increased level of magnetic field turbulence, resulting from an upstream bow wave. Both Vega spacecraft measured a peak field strength of 70-80 nT and observed draping of magnetic field lines around the cometary obstacle. An unexpected rotation of the magnetic field vector was observed, which may reflect either penetration of magnetic field lines into a diffuse layer related to the contact surface separating the solar-wind and cometary plasma, or the persistence of pre-existing interplanetary field structures.
Riedler, W.; Schwingenschuh, K.; Yeroshenko, Ye. G.; Styashkin, V. A.; Russell, C. T.
1986-05-01
168
The origin of intergalactic magnetic fields is still a mystery and several scenarios have been proposed so far: among them, primordial phase transitions, structure-formation shocks and galactic outflows. In this work, we investigate how efficiently galactic winds can provide an intense and widespread seed' magnetization. This may be used to explain the magnetic fields observed today in clusters of galaxies
Serena Bertone; Corina Vogt; Torsten Enlin
2006-01-01
169
Issues associated with the exposure of patients to strong, static magnetic fields during magnetic resonance imaging (MRI) are reviewed and discussed. The history of human exposure to magnetic fields is reviewed, and the contra- dictory nature of the literature regarding effects on human health is described. In the absence of ferromagnetic for- eign bodies, there is no replicated scientific study
John F. Schenck
2000-01-01
170
Outflows from quasars inevitably pollute the intergalactic medium (IGM) with magnetic fields. The short-lived activity of a quasar leaves behind an expanding magnetized bubble in the IGM. We model the expansion of the remnant quasar bubbles and calculate their distribution as a function of size and magnetic field strength at different redshifts. We generically find that by a redshift z~3,
Steven R. Furlanetto; Abraham Loeb
2001-01-01
171
Aims: We wish to clarify whether strong magnetic fields can be effectively generated in typically low-mass dwarf galaxies and to assess the role of dwarf galaxies in the magnetization of the Universe. Methods: We performed a search for radio emission and magnetic fields in an unbiased sample of 12 Local Group (LG) irregular and dwarf irregular galaxies with the 100-m
K. T. Chyzy; M. Wezgowiec; R. Beck; D. J. Bomans
2011-01-01
172
ERIC Educational Resources Information Center
|After the discovery that superconducting magnets could levitate diamagnetic objects, researchers became interested in measuring the repulsion of diamagnetic fluids in strong magnetic fields, which was given the name "The Moses Effect." Both for the levitation experiments and the quantitative studies on liquids, the large magnetic fields necessary
Chen, Zijun; Dahlberg, E. Dan
2011-01-01
173
Copy machine developer powder is an alternative for creating permanent displays of magnetic fields. A thin layer of developer powder on a sheet of paper placed over a magnet can be baked in the oven, producing a lasting image of a magnetic field.
Cavanaugh, Terence; Cavanaugh, Catherine
1998-02-01
174
ERIC Educational Resources Information Center
|A compass is an excellent classroom tool for the exploration of magnetic fields. Any student can tell you that a compass is used to determine which direction is north, but when paired with some basic trigonometry, the compass can be used to actually measure the strength of the magnetic field due to a nearby magnet or current-carrying wire. In
Lunk, Brandon; Beichner, Robert
2011-01-01
175
We performed cosmological, magnetohydrodynamical simulations to follow the evolution of magnetic fields in galaxy clusters, exploring the possibility that the origin of the magnetic seed fields is galactic outflows during the starburst phase of galactic evolution. To do this, we coupled a semi-analytical model for magnetized galactic winds as suggested by Bertone, Vogt & Enlin to our cosmological simulation. We
J. Donnert; K. Dolag; H. Lesch; E. Mller
2009-01-01
176
Context: .The evolution of the concentrated magnetic field in flux tubes is one challenge of the nowadays Solar physics which requires time sequence with high spatial resolution. Aims: .Our objective is to follow the properties of the magnetic concentrations during their life, in intensity (continuum and line core), magnetic field and Doppler velocity. Methods: .We have observed solar region NOAA
Th. Roudier; J. M. Malherbe; J. Moity; S. Rondi; P. Mein; Ch. Coutard
2006-01-01
177
A new global map of the magnetic field of Mars, with an order of magnitude improved sensitivity to crustal magnetization, is derived from Mars Global Surveyor mapping orbit magnetic field data. With this comes greatly improved spatial resolution and geologic intrpretation.
Connerney, J. E. P.; Acuna, M. H.; Ness, N. F.; Mitchell, D. L.; Lin, R. P.
2004-03-01
178
Rotation magnetic beacons magnetic field strength is very important to drill parallel horizontal twin wells in steam assisted\\u000a gravity drainage (SAGD). This paper analyzes a small magnet with a diameter of 25.4 mm. At each end, there is a length of\\u000a 12.6 mm with permanent magnet, and in the middle, there is a length of 78mm with magnetic materials. The
Bing Tu; Desheng Li; Enhuai Lin; Bin Luo; Jian He; Lezhi Ye; Jiliang Liu; Yuezhong Wang
2010-01-01
179
A number of epidemiological studies have indicated that exposure to magnetic fields in certain work environments has possible detrimental health effects. A limitation of most of these studies, however, is the inaccurate assessment of historical exposure to magnetic fields. This paper describes the preliminary results of a study conducted by Eskom to assess the historical magnetic field dosages received by
P. H. Pretorius
1993-01-01
180
We review the extensive and controversial literature concerning how the cosmic magnetic fields pervading nearly all galaxies and clusters of galaxies actually got started. Some observational evidence supports a hypothesis that the field is already moderately strong at the beginning of the life of a galaxy and its disc. One argument involves the chemical abundance of the light elements Be and B, while a second one is based on the detection of strong magnetic fields in very young high red shift galaxies. Since this problem of initial amplification of cosmic magnetic fields involves important plasma problems it is obvious that one must know the plasma in which the amplification occurs. Most of this review is devoted to this basic problem and for this it is necessary to devote ourselves to reviewing studies that take place in environments in which the plasma properties are most clearly understood. For this reason the authors have chosen to restrict themselves almost completely to studies of dynamos in our Galaxy. It is true that one can get a much better idea of the grand scope of galactic fields in extragalactic systems. However, most mature galaxies share the same dilemma as ours of overcoming important plasma problems. Since the authors are both trained in plasma physics we may be biased in pursuing this approach, but we feel it is justified by the above argument. In addition we feel we can produce a better review by staying close to that which we know best. In addition we have chosen not to consider the saturation problem of the galactic magnetic field since if the original dynamo amplification fails the saturation question does not arise. It is generally accepted that seed fields, whose strength is of order 10-20 G, easily spring up in the era preceding galaxy formation. Several mechanisms have been proposed to amplify these seed magnetic fields to a coherent structure with the microgauss strengths of the currently observed galactic magnetic fields. The standard and most popular mechanism is the ?-? mean field dynamo theory developed by a number of people in the late sixties. This theory and its application to galactic magnetic fields is discussed in considerable detail in this review. We point out certain difficulties with this theory that make it seem unlikely that this is the whole story. The main difficulty with this as the only such amplification mechanism is rooted in the fact that, on galactic scales, flux is constant and is frozen in the interstellar medium. This implies that flux must be removed from the galactic discs, as is well recognized by the standard theory. For our Galaxy this turns out to be a major problem, since unless the flux and the interstellar mass are somehow separated, some interstellar mass must also be removed from the deep galactic gravitational well. This is very difficult. It is pointed out that unless the field has a substantial field strength, much larger than that of the seed fields, this separation can hardly happen. And of course, it must if the ?-? dynamo is to start from the ultra weak seed field. (It is our philosophy, expressed in this review, that if an origin theory is unable to create the magnetic field in our Galaxy it is essentially incomplete.) Thus, it is more reasonable for the first and largest amplification to occur before the Galaxy forms, and the matter embedded in the field is gravitationally trapped. Two such mechanisms are discussed for such a pregalactic origin; (1) they are generated in the turbulence of the protogalaxy and (2) the fields come from giant radio jets. Several arguments against a primordial origin are also discussed, as are ways around them. Our conclusion as to the most likely origin of cosmic magnetic fields is that they are first produced at moderate field strengths by primordial mechanisms and then changed and their strength increased to their present value and structure by a galactic disc dynamo. The primordial mechanisms have not yet been seriously developed, and this preliminary amplification of the magnetic fields is still very open. If
Kulsrud, Russell M.; Zweibel, Ellen G.
2008-04-01
181
We introduce a topological flux function to quantify the topology of magnetic braids: non-zero, line-tied magnetic fields whose field lines all connect between two boundaries. This scalar function is an ideal invariant defined on a cross-section of the magnetic field, and measures the average poloidal magnetic flux around any given field line, or the average pairwise crossing number between a given field line and all others. Moreover, its integral over the cross-section yields the relative magnetic helicity. Using the fact that the flux function is also an action in the Hamiltonian formulation of the field line equations, we prove that it uniquely characterizes the field line mapping and hence the magnetic topology.
Yeates, A. R.; Hornig, G.
2013-01-01
182
A magnetic field is an inescapable environmental factor for plants on the earth. However, its impact on plant growth is not well understood. In order to survey how magnetic fields affect plant, Alaska pea seedlings were incubated under low magnetic field (LMF) and also in the normal geo-magnetic environment. Two-day-old etiolated seedlings were incubated in a magnetic shield box and
Y. Negishi; A. Hashimoto; M. Tsushima; C. Dobrota; M. Yamashita; T. Nakamura
1999-01-01
183
NSDL National Science Digital Library
This web page is an interactive physics simulation that explores magnetic fields. The user can add currents coming into or out of a simulated grid, and see the fields created. There is also a selection of pre-created fields, including bar magnets, loops, opposing magnets, and coils in uniform fields. Double-clicking on any point displays the full loop created by the magnetic field. This item is part of a larger collection of introductory physics simulations developed by the author. This is part of a collection of similar simulation-based student activities.
Duffy, Andrew
2008-08-23
184
DOEpatents
A magnetic refrigeration apparatus includes first and second steady state magnets, each having a field of substantially equal strength and opposite polarity, first and second bodies made of magnetocaloric material disposed respectively in the influence of the fields of the first and second steady state magnets, and a pulsed magnet, concentric with the first and second steady state magnets, and having a field which cycles between the fields of the first and second steady state magnets, thereby cyclically magnetizing and demagnetizing and thus heating and cooling the first and second bodies. Heat exchange apparatus of suitable design can be used to expose a working fluid to the first and second bodies of magnetocaloric material. A controller is provided to synchronize the flow of working fluid with the changing states of magnetization of the first and second bodies.
Lubell, Martin S. (Oak Ridge, TN)
1994-01-01
185
DOEpatents
A magnetic refrigeration apparatus includes first and second steady state magnets, each having a field of substantially equal strength and opposite polarity, first and second bodies made of magnetocaloric material disposed respectively in the influence of the fields of the first and second steady state magnets, and a pulsed magnet, concentric with the first and second steady state magnets, and having a field which cycles between the fields of the first and second steady state magnets, thereby cyclically magnetizing and demagnetizing and thus heating and cooling the first and second bodies. Heat exchange apparatus of suitable design can be used to expose a working fluid to the first and second bodies of magnetocaloric material. A controller is provided to synchronize the flow of working fluid with the changing states of magnetization of the first and second bodies. 2 figs.
Lubell, M.S.
1994-10-25
186
National Technical Information Service (NTIS)
This paper describes the data analysis technique used for magnetic testing at the NASA Goddard Space Flight Center (GSFC). Excellent results have been obtained using this technique to convert a spacecraft s measured magnetic field data into its respective...
P. K. Harris
2003-01-01
187
National Technical Information Service (NTIS)
An electronic control system for stabilization of currents in magnetic fields is described. Three superimposed control stages with different characteristics provide optimum elimination of all interfering factors. The use of electrostatic and magnetic shie...
K. Weyand
1976-01-01
188
Invisible lines of magnetic force enclose our planet in what scientists call adipolarmagneticfield. Today these lines go from magnetic south to magnetic north, which are offset a few degrees from the geographic poles. Some minerals, like magnetite, can \\
Trevor Major
189
PubMed
In this paper, we summarize our present understanding of Mars' atmosphere, magnetic field, and surface and address past evolution of these features. Key scientific questions concerning Mars' surface, atmosphere, and magnetic field, along with the planet's interaction with solar wind, are discussed. We also define what key parameters and measurements should be performed and the main characteristics of a martian mission that would help to provide answers to these questions. Such a mission--Mars Environment and Magnetic Orbiter (MEMO)--was proposed as an answer to the Cosmic Vision Call of Opportunity as an M-class mission (corresponding to a total European Space Agency cost of less than 300 Meuro). MEMO was designed to study the strong interconnection between the planetary interior, atmosphere, and solar conditions, which is essential to our understanding of planetary evolution, the appearance of life, and its sustainability. The MEMO main platform combined remote sensing and in situ measurements of the atmosphere and the magnetic field during regular incursions into the martian upper atmosphere. The micro-satellite was designed to perform simultaneous in situ solar wind measurements. MEMO was defined to conduct: * Four-dimensional mapping of the martian atmosphere from the surface up to 120 km by measuring wind, temperature, water, and composition, all of which would provide a complete view of the martian climate and photochemical system; Mapping of the low-altitude magnetic field with unprecedented geographical, altitude, local time, and seasonal resolutions; A characterization of the simultaneous responses of the atmosphere, magnetic field, and near-Mars space to solar variability by means of in situ atmospheric and solar wind measurements. PMID:19317625
Leblanc, F; Langlais, B; Fouchet, T; Barabash, S; Breuer, D; Chassefire, E; Coates, A; Dehant, V; Forget, F; Lammer, H; Lewis, S; Lopez-Valverde, M; Mandea, M; Menvielle, M; Pais, A; Paetzold, M; Read, P; Sotin, C; Tarits, P; Vennerstrom, S
190
In this paper, we summarize our present understanding of Mars' atmosphere, magnetic field, and surface and address past evolution of these features. Key scientific questions concerning Mars' surface, atmosphere, and magnetic field, along with the planet's interaction with solar wind, are discussed. We also define what key parameters and measurements should be performed and the main characteristics of a martian mission that would help to provide answers to these questions. Such a mission -- Mars Environment and Magnetic Orbiter (MEMO) -- was proposed as an answer to the Cosmic Vision Call of Opportunity as an M-class mission (corresponding to a total European Space Agency cost of less than 300 M). MEMO was designed to study the strong interconnection between the planetary interior, atmosphere, and solar conditions, which is essential to our understanding of planetary evolution, the appearance of life, and its sustainability. The MEMO main platform combined remote sensing and in situ measurements of the atmosphere and the magnetic field during regular incursions into the martian upper atmosphere. The micro-satellite was designed to perform simultaneous in situ solar wind measurements. MEMO was defined to conduct: Four-dimensional mapping of the martian atmosphere from the surface up to 120 km by measuring wind, temperature, water, and composition, all of which would provide a complete view of the martian climate and photochemical system; Mapping of the low-altitude magnetic field with unprecedented geographical, altitude, local time, and seasonal resolutions; A characterization of the simultaneous responses of the atmosphere, magnetic field, and near-Mars space to solar variability by means of in situ atmospheric and solar wind measurements.
Leblanc, F.; Langlais, B.; Fouchet, T.; Barabash, S.; Breuer, D.; Chassefire, E.; Coates, A.; Dehant, V.; Forget, F.; Lammer, H.; Lewis, S.; Lopez-Valverde, M.; Mandea, M.; Menvielle, M.; Pais, A.; Paetzold, M.; Read, P.; Sotin, C.; Tarits, P.; Vennerstrom, S.
2009-02-01
191
National Technical Information Service (NTIS)
An experiment on arc discharges in hydrogen in a curved magnetic field is described. For a few milliseconds the discharge current flowed between two electrodes along the field lines of a toroidal magnetic field over an angle of 258 deg. The plasma was not...
F. C. Schueller
1974-01-01
192
We consider the various methods used to constrain the possible field strength of the present day intergalactic field and findB0(G)-10 as a probable upper bound. It is suggested that the observed intergalactic magnetic field might not be primordial in origin but rather the result of magnetic flux leakage from galaxies and clusters of galaxies.
Martin Beech
1985-01-01
193
An alternative explanation of galactic warps is proposed, in which the intergalactic magnetic field (IGMF) is responsible for these structures. The model predicts that, to be efficient, the magnetic field must have a direction not much different from 45 deg with the galactic plane. The required values of the field strength are uncertain, of about 10 nG, higher values being
E. Battaner; E. Florido; M. L. Sanchez-Saavedra
1990-01-01
194
In this paper we demonstrate experimentally a magnetic field sensor using a fiber Bragg grating. The shift in the Bragg condition as a result of strain applied on the fiber mounted on a nickel base by the magnetic field gives an indirect measure of the field. The proposed method overcomes the need for long fiber lengths required in methods such
K. V. Madhav; K. Ravi Kumar; T. Srinivas; S. Asokan
2006-01-01
195
The various methods used to constrain the possible field strength of the present day intergalactic field are considered, and Bzero (G) less than 10 to the -10th is found as a probable upper bound. It is suggested that the observed intergalactic magnetic field might not be primordial in origin but rather the result of magnetic flux leakage from galaxies and clusters of galaxies.
Beech, M.
1985-11-01
196
SciTech Connect
Three species of potentially pathogenic amoebae were exposed to 71 and 106.5 mT from constant homogeneous magnetic fields and examined for inhibition of population growth. The number of amoebae for three species was significantly less than controls after a 72 h exposure to the magnetic fields when the temperature was 20 C or above. Axenic cultures, i.e., cultures grown without bacteria, were significantly affected after only 24 h. In 20 of 21 tests using the three species, the magnetic field significantly inhibited the growth of amoebae. In one test in which the temperature was 20 C for 48 h, exposure to the magnetic field was not inhibitory. Final numbers of magnetic field-exposed amoebae ranged from 9 to 72% lower than the final numbers of unexposed controls, depending on the species. This research may lead to disinfection strategies utilizing magnetic fields for surfaces on which pathogenic amoebae may proliferate.
Berk, S.G.; Srikanth, S.; Mahajan, S.M.; Ventrice, C.A. [Tennessee Technological Univ., Cookeville, TN (United States)
1997-03-01
197
The major scientific achievements associated with the measurement of magnetic fields in space over the past decade and a half are reviewed. Aspects of space technology relevant to magnetic-field observations are discussed, including the different types of magnetometers used and how they operate, problems arising from spacecraft-generated magnetic fields and the appropriate countermeasures that have been developed and on-board processing
EDWARD J. SMITHAND; Charles Sonett
1976-01-01
198
A class of nonlinear force-free magnetic fields is presented, described in terms of the solutions to a second-order, nonlinear ordinary differential equation. These magnetic fields are three-dimensional, filling the infinite half-space above a plane where the lines of force are anchored. They model the magnetic fields of the sun over active regions with a striking geometric realism. The total energy
B. C. Low; Y. Q. Lou
1990-01-01
199
The magnetization process of a ferrofluid whose carrier fluid is paraffin was investigated in the temperature range from 77 K to 300 K, as a function of the cooling field intensity and freezing rate. Phase transitions between the liquid and solid states can be simulated by using the ferrofluids as a magnetic probe. A uniaxial magnetic anisotropy was induced by
N. Inaba; H. Miyajima; S. Taketomi; S. Chikazumi
1989-01-01
200
SciTech Connect
The extensive publicity of epidemiological studies inferring correlation between 60 Hz magnetic fields and childhood leukemia prompted world wide research programs that have as a goal to determine if low frequency magnetic fields represent any risk for the general population, children or utility workers. While supporting this research effort through EPRI, Con Edison embarked on a technical research program aimed to: characterize magnetic fields as to intensity and variation in time; and investigate practical means to manage these magnetic fields through currently known methods. The final goal of these research projects is to establish viable methods to reduce magnetic field intensity to desired values at reasonable distances from the sources. This goal was pursued step by step, starting with an inventory of the main sources of magnetic fields in substations, distribution and transmission facilities and generating plants. The characterization of the sources helped to identify typical cases and select specific cases, far practical applications. The next step was to analyze the specific cases and develop design criteria for managing the magnetic fields in new installations. These criteria included physical arrangement of equipment based oil calculation of magnetic fields, cancellation effect, desired maximum field intensity at specific points and shielding with high magnetic permeability metals (mu-metal and steel). This paper summarizes the authors experiences and shows the results of the specific projects completed in recent years.
Durkin, C.J.; Fogarty, R.P.; Halleran, T.M.; Mark, Dr. D.A.; Mukhopadhyay, A.
1995-01-01
201
SciTech Connect
The effect of a strong magnetic field on the stability and gross properties of bulk as well as quasibulk quark matter is investigated using the conventional MIT bag model. Both the Landau diamagnetism and the paramagnetism of quark matter are studied. How the quark hadron phase transition is affected by the presence of a strong magnetic field is also investigated. The equation of state of strange quark matter changes significantly in a strong magnetic field. It is also shown that the thermal nucleation of quark bubbles in a compact metastable state of neutron matter is completely forbidden in the presence of a strong magnetic field. {copyright} {ital 1996 The American Physical Society.}
Chakrabarty, S. [Department of Physics, University of Kalyani, District: Nadia, West Bengal 741 235 (India)]|[Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007 (India)
1996-07-01
202
Frustrated magnets in high magnetic field have a long history of offering beautiful surprises to the patient investigator. Here we present the results of extensive classical Monte Carlo simulations of a variety of models of two dimensional magnets in magnetic field, together with complementary spin wave analysis. Striking results include (i) a massively enhanced magnetocaloric effect in antiferromagnets bordering on ferromagnetic order, (ii) a route to an m = 1/3 magnetization plateau on a square lattice, and (iii) a cascade of phase transitions in a simple model of AgNiO2.
Seabra, L.; Shannon, N.; Sindzingre, P.; Momoi, T.; Schmidt, B.; Thalmeier, P.
2009-01-01
203
This work reviews our understanding of the magnetic fields observed in the quiet Sun. The subject has undergone a major change during the last decade (quiet revolution), and it will remain changing since the techniques of diagnostic employed so far are known to be severely biased. Keeping these caveats in mind, our work covers the main observational properties of the quiet Sun magnetic fields: magnetic field strengths, unsigned magnetic flux densities, magnetic field inclinations, as well as the temporal evolution on short time-scales (loop emergence), and long time-scales (solar cycle). We also summarize the main theoretical ideas put forward to explain the origin of the quiet Sun magnetism. A final prospective section points out various areas of solar physics where the quiet Sun magnetism may have an important physical role to play (chromospheric and coronal structure, solar wind acceleration, and solar elemental abundances).
Snchez Almeida, J.; Martnez Gonzlez, M.
2011-04-01
204
The generation of a high quality electron beam by a race- track microtron (RTM) requires highly precise magnetic fields in the two reversing magnets. At the RTM cascade MAMI (Mainz Microtron), a precision of 10 ?4 for the ver- tical field component By was achieved by symmetrical sur- face coils placed at the upper and lower pole surface in each
F. Hagenbuck; P. Jennewein; K.-H. Kaiser; H.-J. Kreidel; U. Ludwig-Mertin; M. Seidl
2002-01-01
205
A wheel-type mobile robot is simply able to localize with odometry. However, for mobile agricultural robots, it is necessary to consider that the environment is uneven terrain. Therefore, odometry is unreliable and it is necessary to augment the odometry by measuring the position of the robot relative to known objects in the environments. This paper describes the application of localization
Sam Ann Rahok; Koichi Ozaki
2011-01-01
206
Slow spreading mid-ocean ridges like the Mid-Atlantic Ridge host a remarkable diversity of hydrothermal systems including vent systems located on the neovolcanic axis, large axial volcanoes, in transform faults and nontransform offsets, and associated with low-angle detachment faults, now recognized as a major tectonic feature of slow spreading environments. Hydrothermal systems are hosted in various lithologies from basalt to serpentinized peridotite and exposed lower oceanic crust. The substantial variations of hydrothermal processes active in these environments have important implications for the magnetic structure of oceanic crust and upper mantle. Hydrothermal processes can both destroy the magnetic minerals in basalt, diabase, and gabbro and create magnetic minerals by serpentinization of ultramafic rocks and deposition of magnetic minerals. We report on the diversity of magnetic anomaly signatures over the vent systems at slow spreading ridges and show that the lateral scale of hydrothermal alteration is fundamentally a local phenomenon. This highly focused process leads to magnetic anomalies on the scale of individual vent fields, typically a few hundreds of meters or less in size. To detect such features, high-resolution, near-bottom magnetic surveys are required rather than sea surface surveys. High-resolution surveys are now more tractable with deep-towed systems, dynamically positioned ships, and with the recent development of autonomous underwater vehicles, which allow detailed mapping of the seafloor on a scale relevant to hydrothermal activity. By understanding these present-day active hydrothermal systems, we can explore for yet to be discovered buried deposits preserved off-axis, both to determine past history of hydrothermal activity and for resource assessment.
Tivey, Maurice A.; Dyment, Jrme
207
This study reports the low frequency magnetoelectric (ME) response of the sintered composites comprising of a piezoelectric phase Pb(Zr0.52Ti0.48)O3 (PZT) and magnetostrictive phases NiFe1.9Mn0.1O4 (NFM) and Ni0.8Zn0.2Fe2O4 (NZF) with varying ferrite contents of 3, 5, 10, 15, and 20 mol %. It was found that the ME coefficient for the PZT-NZF samples shows considerably less scattering as a function of frequency and the composition 0.8PZT-0.2NZF exhibited a flat response in the range of 10-100 Hz with a magnitude of 220 mV/cm Oe. This composition was used to design the magnetic field anomaly detector mounted in front of a global positioning system (GPS) controlled vehicle. The results from the vehicle test clearly demonstrate the feasibility of using sintered ME composites for magnetic field detection in the noisy environment.
Bergs, Richard; Islam, Rashed A.; Vickers, Michael; Stephanou, Harry; Priya, Shashank
2007-01-01
208
Recently, it was found that the multi-component glass a-BaO-Al2O3-SiO2 exhibits unusual magnetic properties at very low temperatures. Thus the question arises whether this is a specialty of that particular glass or a more general phenomenon. We report here on our studies of the magnetic-field dependence of the dielectric properties of the borosilicate glass BK7 which contains only a negligible amount of magnetic impurities. Since this glass also responds sensitively to magnetic fields, our investigations demonstrate that the reaction of glasses to magnetic fields is not caused by magnetic impurities but reflects a more general phenomenon. In addition, we have observed that the variation of the dielectric constant and the loss angle with magnetic field depend on the amplitude of the electric field that is used to measure the glass capacitance. We present the data and discuss possible origins of the magnetic-field phenomena in non-magnetic glasses.
Wohlfahrt, M.; Strehlow, P.; Enss, C.; Hunklinger, S.
2001-12-01
209
National Technical Information Service (NTIS)
A survey of observational and theoretical work pertaining to the origin of planetary magnetic fields is given with special emphasis on the dynamo theory which attempts to explain these fields as arising from magnetohydrodynamic regenerative action. Some p...
G. Venezian
1967-01-01
210
National Technical Information Service (NTIS)
Studies are presented of the behavior of cosmic rays in the earth's magnetic field. It discusses the theory of motion of charged particles in an idealized field model and presents results of trajectory calculations of asymptotic directions and cutoff rigi...
L. I. Dorman V. S. Smirnov M. I. Tyasto
1973-01-01
211
ERIC Educational Resources Information Center
|Describes the change of earth's magnetic field at the boundary between the outer core and the mantle. Measurement techniques used during the last 300 years are considered. Discusses the theories and research for explaining the field change. (YP)|
Bloxham, Jeremy; Gubbins, David
1989-01-01
212
Observational studies of magnetic fields are vital as magnetic fields play a crucial role in various astrophysical processes, including star formation, accretion of matter, transport processes (e.g. transport of heat), and cosmic rays. The existing ways of magnetic field studies have their limitations. Therefore, it is important to explore new effects that can bring information about magnetic field. We identified a process ground state alignment as a new way to determine the magnetic field direction in diffuse medium. The consequence of the process is the polarization of spectral lines resulting from scattering and absorption from aligned atomic/ionic species with fine or hyperfine structure. The alignment is due to anisotropic radiation impinging on the atom/ion, while the magnetic field induces precession and realign the atom/ion and therefore the polarization of the emitted or absorbed radiation reflects the direction of the magnetic field. The atoms get aligned at their low levels and, as the life-time of the atoms/ions we deal with is long, the alignment induced by anisotropic radiation is susceptible to extremely weak magnetic fields (1G?B?10-15G). Compared to the upper level Hanle effect, atomic realignment is most suitable for the studies of magnetic field in the diffuse medium, where magnetic field is relatively weak. The corresponding physics of alignment is based on solid foundations of quantum electrodynamics and in a different physical regime the alignment has become a part of solar spectroscopy. In fact, the effects of atomic/ionic alignment, including the realignment in magnetic field, were studied in the laboratory decades ago, mostly in relation to the maser research. Recently, the atomic effect has been already detected in observations from circumstellar medium and this is a harbinger of future extensive magnetic field studies. It is very encouraging that a variety of atoms with fine or hyperfine splitting of the ground or metastable states exhibit the alignment and the resulting polarization degree in some cases exceeds 20%. A unique feature of the atomic realignment is that they can reveal the 3D orientation of magnetic field. In this paper, we shall review the basic physical processes involved in atomic realignment. We shall also discuss its applications to interplanetary, circumstellar and interstellar magnetic fields. In addition, our research reveals that the polarization of the radiation arising from the transitions between fine and hyperfine states of the ground level can provide a unique diagnostics of magnetic fields, including those in the early universe.
Yan, Huirong; Lazarian, A.
2012-08-01
213
Field-aligned currents play a central role in the study of the magnetized plasmas of the solar terrestrial environment. In particular, if perturbations of flow develop on one part of a flux tube, field-aligned currents must flow in order to communicate the changes to the entire flux tube. Field-aligned currents cause the field to twist or shear, a feature that can
David J. Southwood; Margaret G. Kivelson
1991-01-01
214
During the recent Steins flyby of the ROSETTA spacecraft magnetic field measurements have been made with both, the RPC orbiter magnetometer and the ROMAP lander magnetometer. These combined magnetic field measurements allow a detailed examination of any magnetic signatures caused either directly by the asteroid or indirectly by Steins different modes of interaction with the solar wind. Comparing our measurements with simulation results show that Steins does not possess a significant remanent magnetization. The magnetization is estimated at less than 1 mAm2/kg. This is significantly different from results at Braille and Gaspra.
Glassmeier, K.; Auster, H.; Richter, I.; Motschmann, U.; RPC/ROMAP Teams
2009-05-01
215
To explore the degree of coupling between the interplanetary magnetic field (IMF) and Jupiter's magnetosphere, we traced magnetic field lines from the polar region of the planet using the Khurana [1997, 2005] magnetic field model. We used a parameterized definition of the Jovian magnetopause created by Joy et al. [2002] that varies with the value of the solar wind dynamic pressure. We searched for field lines that cross the magnetopause and that potentially connect to the interplanetary magnetic field. We further explored the variation on magnetic field structure with local time orientation of Jupiter's dipole (i.e. Central Meridian Longitude) as well as upstream solar wind and IMF conditions.
Cohen, I.; Bagenal, F.
2008-12-01
216
SciTech Connect
After Lancaster the authors examine chiral constraints in N = 2 superspace formulation for supersymmetric magnetic field systems. Such odd constraints are connected with the so-called spin-orbit coupling procedure of supersymmetrization. They propose new even constraints for magnetic supersymmetric systems and relate them to the standard procedure enhanced by Witten. These models describing spin-one half particles moving in a plane with a transverse magnetic field are compared and discussed. The cases of a constant magnetic field and of the harmonic oscillator are connected through different correspondences.
Dehin, D.; Hussin, V. (Universite de Liege, Physique Theorique et Mathematique, Institut de Physique au Sart Tilman, Batiment B.5, B-4000 Liege (BE))
1988-01-01
217
This thesis uses an equivalent circuit model to calculate ionospheric electric fields, current densities and introduced magnetic fields variations on the ground. The role of the field aligned current is examined. Using different wind models, we studied the electric field variations with altitude, season and solar activity. The ionospheric eastward electric field changes very little within the whole ionosphere. The southward (equatorward) electric field is large and changes quickly with height in the E region although it is nearly constant in the F region. The prereversal enhancement of the eastward electric field is produced by the F region dynamo. We conclude that the Forbes and Gillette tidal wind can reproduce most features of the Jicamarca experiment and the AE-E and DE-2 satellite observations of the electric fields. The HWM90 empirical wind model failed to produce the observed electric field and it seems the semidiurnal wind in HWM90 is too strong. The field aligned current is located mainly in the E and low F region. The non-coincidence of the geomagnetic and geographic equators has a strong effect on the field aligned current in the equatorial zone. The field aligned currents driven by Forbes' winds for March equinox and December solstice flow mainly from the southern to northern hemisphere in the morning and vice versa in the afternoon at F region heights. The observed magnetic field variations on the ground are well reproduced in our simulations. The field aligned current is the main contributor to the eastward magnetic field component in the equatorial zone. The longitudinal inequality of the northward magnetic field is introduced mainly by the variations of the local magnetic field intensity. The electric field variations have only a minor effect. The northward magnetic field variations with the solar activity are introduced by changes of the E region equatorward electric field and the Hall conductivity.
Du, Junhu
218
An study on the demagnetization of rare-earth permanent magnets under high radiation environment is started from the microscopic\\u000a point of view. The demagnetization of NEOMAX is successfully induced by the well defined neutron field produced by the 5 MW\\u000a reactor in Kyoto University. Preliminary TDPAC measurement of 111Cd(?111In) in NEOMAX, including demagnetized one, is reported.
M. Tanigaki; K. Takamiya; Y. Komeno; A. Taniguchi; Y. Ohkubo
2007-01-01
219
The purpose of this work is to provide a better understanding of the underlying sources of the magnetic field associated with ongoing electrochemical corrosion, to investigate the spatio-temporal information content of the corrosion magnetic field, and to evaluate its potential utility in non-invasive quantification of hidden corrosion. The importance of this work lies in the fact that conventional electrochemical instruments
Afshin Abedi
2000-01-01
220
Since magnetic field typically plays a role (either active or passive) in coronal heating theories, it may be possible to evaluate these theories by investigating the relationship between the coronal energy budget (the total power requirement of the corona) and measurable properties of the photospheric magnetic field. The X-ray flux is a useful proxy for the total power required to
C. E. Parnell; P. A. Sturrock
1997-01-01
221
Observations of the large scale magnetic field in the photosphere taken at the Wilcox Solar Observatory since 1976 up to 2005 have been analyzed to deduce its latitudinal and longitudinal structures, its differential rotation, and their variability in time. The main results are the following: - The latitudinal structure of the solar magnetic field with a period of polarity change
E. A. Gavryuseva
2006-01-01
222
The current understanding of astrophysical magnetic fields is reviewed, focusing on their generation and maintenance by turbulence. In the astrophysical context this generation is usually explained by a self-excited dynamo, which involves flows that can amplify a weak seed magnetic field exponentially fast. Particular emphasis is placed on the nonlinear saturation of the dynamo. Analytic and numerical results are discussed
Axel Brandenburg; Kandaswamy Subramanian
2005-01-01
223
The body-centered-cubic Coulomb crystal of ions in the presence of a uniform magnetic field is studied using the rigid electron background approximation. The phonon mode spectra are calculated for a wide range of magnetic-field strengths and for several orientations of the field in the crystal. The phonon spectra are used to calculate the phonon contribution to the crystal energy, entropy, specific heat, Debye-Waller factor of ions, and the rms ion displacements from the lattice nodes for a broad range of densities, temperatures, chemical compositions, and magnetic fields. Strong magnetic field dramatically alters the properties of quantum crystals. The phonon specific heat increases by many orders of magnitude. The ion displacements from their equilibrium positions become strongly anisotropic. The results can be relevant for dusty plasmas, ion plasmas in Penning traps, and especially for the crust of magnetars (neutron stars with superstrong magnetic fields B?1014G ). The effect of the magnetic field on ion displacements in a strongly magnetized neutron star crust can suppress the nuclear reaction rates and make them extremely sensitive to the magnetic-field direction.
Baiko, D. A.
2009-10-01
224
The authors have observed that some of our model SSC dipoles have long time constant decays of the magnetic field harmonics with amplitudes large enough to result in significant beam loss, if they are not corrected. The magnets were run at constant current at the SSC injection field level of 0.3 tesla for one to three hours and changes in
W. S. Gilbert; R. F. Althaus; P. J. Barale; R. W. Benjegerdes; M. A. Green; M. I. Green; R. M. Scanlan
1989-01-01
225
The principal focus of the program is the analysis of magnetic field effects on physiological functions in experimental animals and selected organ and tissue systems. A major research effort has involved the use of electrical recording techniques to detect functional alterations in the cardiovascular, neural, and visual systems during the application of DC magnetic fields. These systems involve ionic conduction
Tenforde
1981-01-01
226
Coronal magnetic fields calculated by the methods developed in Paper I (Altschuler and Newkirk, 1969) and the empirical description of the solar corona of November 1966 derived in Paper II (Newkirket al., 1970) are combined in order to investigate what connection exists between the magnetic fields and the density structure of the corona.
Gordon Newkirk; Martin D. Altschuler
1970-01-01
227
Numerous methods have been developed to measure MRI gradient waveforms and k-space trajectories. The most promising new strategy appears to be magnetic field monitoring with RF microprobes. Multiple RF microprobes may record the magnetic field evolution associated with a wide variety of imaging pulse sequences. The method involves exciting one or more test samples and measuring the time evolution of
Hui Han; Rodney P. MacGregor; Bruce J. Balcom
2009-01-01
228
A technique for the estimation of the magnetic field intensity emitted by industrial installations is presented. The method is best-suited for investigation of environmental magnetic field for health purposes. Simulation and measurement case-studies supporting the provided theoretical results are discussed
M. Bertocco; F. Dughiero; C. Greggio; E. Sieni; A. Sona
2006-01-01
229
We construct a simple physical model of a particle moving on the infinite noncommutative 2-plane. The model consists of a pair of opposite charges moving in a strong magnetic field. In addition, the charges are connected by a spring. In the limit of large magnetic field, the charges are frozen into the lowest Landau levels. Interactions of such particles include
Daniela Bigatti; Leonard Susskind
2000-01-01
230
It is shown that the interplanetary magnetic field has different characteristics on different scales, and it is noted that a given physical theory may not be applicable or relevant on all scales. Four scales are defined in terms of time intervals on which the data may be viewed. Many discontinuities in the magnetic-field direction are seen on the mesoscale (
Leonard F. Burlaga
1969-01-01
231
WOODING suggests1 that ball lightning is a plasma vortex ring structure produced by a process similar to the ablation of a solid surface by a high power laser pulse. A plasma vortex ring structure requires a magnetic field; here I present two pieces of evidence to show that a magnetic field is associated with ball lightning, and which may help
A. J. F. Blair
1973-01-01
232
Numerical simulations of stellar dynamos are reviewed. Dynamic dynamo models solve the nonlinear, three-dimensional, time-dependent, magnetohydrodynamic equations for the convective velocity, the thermodynamic variables, and the generated magnetic field in a rotating, spherical shell of ionized gas. When the dynamo operates in the convection zone, the simulated magnetic fields propagate away from the equator in the opposite direction inferred from
Gary A. Glatzmaier
1985-01-01
233
The nonadiabatic transitions which a system with angular momentum J makes in a magnetic field which is rotating about an axis inclined with respect to the field are calculated. It is shown that the effects depend on the sign of the magnetic moment of the system. We therefore have an absolute method for measuring the sign and magnitude of the
I. I. Rabi
1937-01-01
234
SciTech Connect
A hydrogen atom moving across a magnetic field is considered in a wide region of magnitudes of magnetic field and atom momentum. We solve the Schroedinger equation of the system numerically using an imaginary time method and find wave functions of the lowest states of atom. We calculate the energy and the mean electron-nucleus separation as a function of atom momentum and magnetic field. All the results obtained could be summarized as a phase diagram on the 'atom-momentum - magnetic-field' plane. There are transformations of wave-function structure at critical values of atom momentum and magnetic field that result in a specific behavior of dependencies of energy and mean interparticle separation on the atom momentum P. We discuss a transition from the Zeeman regime to the high magnetic field regime. A qualitative analysis of the complicated behavior of wave functions vs P based on the effective potential examination is given. We analyze a sharp transition at the critical momentum from a Coulomb-type state polarized due to atom motion to a strongly decentered (Landau-type) state at low magnetic fields. A crossover occurring at intermediate magnetic fields is also studied.
Lozovik, Yu.E.; Volkov, S.Yu. [Institute of Spectroscopy, Troitsk, Moscow region, 142190 (Russian Federation)
2004-08-01
235
PubMed
Magnetic isotope and magnetic field effects on the rate of DNA synthesis catalysed by polymerases ? with isotopic ions (24)Mg(2+), (25)Mg(2+) and (26)Mg(2+) in the catalytic sites were detected. No difference in enzymatic activity was found between polymerases ? carrying (24)Mg(2+) and (26)Mg(2+) ions with spinless, non-magnetic nuclei (24)Mg and (26)Mg. However, (25)Mg(2+) ions with magnetic nucleus (25)Mg were shown to suppress enzymatic activity by two to three times with respect to the enzymatic activity of polymerases ? with (24)Mg(2+) and (26)Mg(2+) ions. Such an isotopic dependence directly indicates that in the DNA synthesis magnetic mass-independent isotope effect functions. Similar effect is exhibited by polymerases ? with Zn(2+) ions carrying magnetic (67)Zn and non-magnetic (64)Zn nuclei, respectively. A new, ion-radical mechanism of the DNA synthesis is suggested to explain these effects. Magnetic field dependence of the magnesium-catalysed DNA synthesis is in a perfect agreement with the proposed ion-radical mechanism. It is pointed out that the magnetic isotope and magnetic field effects may be used for medicinal purposes (trans-cranial magnetic treatment of cognitive deceases, cell proliferation, control of the cancer cells, etc). PMID:23851636
Buchachenko, Anatoly L; Orlov, Alexei P; Kuznetsov, Dmitry A; Breslavskaya, Natalia N
2013-07-13
236
PubMed Central
Magnetic isotope and magnetic field effects on the rate of DNA synthesis catalysed by polymerases ? with isotopic ions 24Mg2+, 25Mg2+ and 26Mg2+ in the catalytic sites were detected. No difference in enzymatic activity was found between polymerases ? carrying 24Mg2+ and 26Mg2+ ions with spinless, non-magnetic nuclei 24Mg and 26Mg. However, 25Mg2+ ions with magnetic nucleus 25Mg were shown to suppress enzymatic activity by two to three times with respect to the enzymatic activity of polymerases ? with 24Mg2+ and 26Mg2+ ions. Such an isotopic dependence directly indicates that in the DNA synthesis magnetic mass-independent isotope effect functions. Similar effect is exhibited by polymerases ? with Zn2+ ions carrying magnetic 67Zn and non-magnetic 64Zn nuclei, respectively. A new, ionradical mechanism of the DNA synthesis is suggested to explain these effects. Magnetic field dependence of the magnesium-catalysed DNA synthesis is in a perfect agreement with the proposed ionradical mechanism. It is pointed out that the magnetic isotope and magnetic field effects may be used for medicinal purposes (trans-cranial magnetic treatment of cognitive deceases, cell proliferation, control of the cancer cells, etc).
Buchachenko, Anatoly L.; Orlov, Alexei P.; Kuznetsov, Dmitry A.; Breslavskaya, Natalia N.
2013-01-01
237
The measurement of vehicle magnetic moments and the results from use of a fluxgate magnetic sensor to actuate a lighting system from the magnetic fields of passing vehicles is reported. A typical U.S. automobile has a magnetic moment of about 200 A-m2(Ampere-meters2), while for a school bus it is about 2000 A-m2. When the vehicle is modeled as an ideal
S. V. Marshall
1978-01-01
238
Outflows from quasars inevitably pollute the intergalactic medium (IGM) with magnetic fields. The short-lived activity of\\u000a a quasar leaves behind an expanding magnetized bubble in the IGM. We model the expansion of the remnant quasar bubbles and\\u000a calculate their distribution as a function magnetic field strength at different redshifts. We find that by a redshift \\u000a z ~ <\\/font\\u000a>3z \\\\sim
Steven Furlanetto; Abraham Loeb
2002-01-01
239
DOEpatents
Planar permanent magnet edge-field quadrupoles for use in particle accelerating machines and in insertion devices designed to generate spontaneous or coherent radiation from moving charged particles are disclosed. The invention comprises four magnetized rectangular pieces of permanent magnet material with substantially similar dimensions arranged into two planar arrays situated to generate a field with a substantially dominant quadrupole component in regions close to the device axis. 10 figs.
Tatchyn, R.O.
1997-01-21
240
Development of high-field magnets using high temperature superconductors (HTS) is a core activity at the NHMFL. Magnet technology based on both YBCO-coated tape conductors and Bi-2212 round wires is being pursued. Two specific projects are underway. The first is a user magnet with a 17 T YBCO coil set which, inside an LTS outsert, will generate a combined field of
H. W. Weijers; U. P. Trociewitz; W. D. Markiewicz; J. Jiang; D. Myers; E. E. Hellstrom; A. Xu; J. Jaroszynski; P. Noyes; Y. Viouchkov; D. C. Larbalestier
2010-01-01
241
In the process of magnetizing cylindrical specimens of a typical high field superconductor Nb-50 at.%Ti, flux jumps were induced by magnetic disturbances. The stability limit field Hfj increased steadily with increasing temperature, and no magnetic instability occurred for temperatures in excess of about 6.5 K. The calculation of Hfj was performed taking into account the cylindrical sample geometry and the
Tatsuo Akachi; Takeshi Ogasawara; Ko Yasukochi
1981-01-01
242
Human space activity began in 1961. About 400 persons have gone to space since then, and about 70 of them have stayed more than 1 month. Circadian rhythm and sleep in space have been investigated several times, though the effect of longer stays in space has not been adequately clarified. Electromagnetic fields are different in the space environment, especially in
R. Izumi; N. Ishioka; K. Mizuno; T. Goka
2000-01-01
243
Recent experiments on cell division suggest the application of intense static magnetic fields as a novel tool for the manipulation of biological systems [1]. The magnetic field appears to couple to the intrinsic anisotropies in the diamagnetic components of the cells. Here, we present measurements of the intrinsic average diamagnetic anisotropy of the whole single celled ciliate, Paramecium Caudatum. Magnetic fields, 2.5 T < B < 8 T were applied to immobilized (non-swimming) Paramecium Caudatum that were suspended in a density matched medium. The organisms align with their long axis parallel to the applied magnetic field. Their intrinsic diamagnetic anisotropy is 3x10-11 in cgs units. We will discuss the implications of these results for employing magnetic fields to probe the behavior of swimming Paramecium. [1] J. M. Valles, Jr. et al., Expt. Cell Res.274, 112-118 (2002).
Valles, James M., Jr.; Guevorkian, Karine; Quindel, Carl
2004-03-01
244
The latest in lattice QCD -- Quark-gluon plasma physics -- String theory and exact results in quantum field theory -- The status of local supersymmetry.Supersymmetry in nuclei -- Inflation, dark matter, dark energy -- How many dimensions are really compactified? -- Horizons -- Neutrino oscillations physics -- Fundamental constants and their possible time dependence.Highlights from BNL. new phenomena at RHIC -- Highlights from BABAR -- Diffraction studied with a hard scale at HERA -- The large hadron collider: a status report -- Status of non-LHC experiments at CERN -- Highlights from Gran Sass.Fast automatic systems for nuclear emulsion scanning: technique and experiments -- Probing the QGP with charm at ALICE-LHC -- magnetic screening length in hot QCD -- Non-supersymmetric deformation of the Klebanov-Strassler model and the related plane wave theory -- Holographic renormalization made simple: an example -- The kamLAND impact on neutrino oscillations -- Particle identification with the ALIC TOF detector at very high multiplicity -- Superpotentials of N = 1 SUSY gauge theories -- Measurement of the proton structure function F2 in QED compton scattering at HERA -- Yang-Mills effective action at high temperature -- The time of flight (TOF) system of the ALICE experiment -- Almost product manifolds as the low energy geometry of Dirichlet Brane.
245
The application of superparamagnetic nanoparticles for in vivo magnetic resonance imaging (MRI) under external ac magnetic field has attracted considerable research efforts in recent years. However, it is unclear how superparamagnetic nanostructures arrange themselves in fluidic environment under external magnetic field. Here, we report direct visualization of the effect of applied magnetic field to the ferrofluids (about 6 nm superparamagnetic magnetite (Fe3O4) nanoparticle colloidal'' suspension) using the cryogenic transmission electron microscopy (cryo-TEM). While long dipole chains (up to millimeter range) of the magnetite along the magnetic lines are found in samples dried inside the magnetic field, only short dipole chains (within tens of nanometer scale) with random orientations are observed in the wet sample observed by cryo-TEM. In the wet sample, aggregations of medium-length dipole chains (up to hundreds of nanometer) can be observed at the areas where the nanoparticles are solidified'' when phase separation occurs. In situ formation of flux-closure rings is observed at the edge where vitreous ice sublimes due to high-energy electron radiation that leaves magnetite nanoparticles isolated in the vacuum. Such observations may help elucidate the nature of magnetic field-induced assembly in fluidic environment as in the physiological aqueous conditions in MRI and related applications.
Wu, Jinsong; Aslam, M.; Dravid, Vinayak P.
2008-08-01
246
Electromagnetic relay is a widely used apparatus which usually works in a magnetic disturbance environment. To evaluate its\\u000a electromagnetic compatibility (EMC) in a static magnetic field, dynamic characteristics of a clapper relay in a uniform static\\u000a magnetic field situation based on the finite element method (FEM) is studied. Influences of the magnetic field on dynamic\\u000a parameters (delay time, pick-up time,
Guo-fu Zhai; Qi-ya Wang; Wan-bin Ren
2008-01-01
247
SciTech Connect
The results of our analysis strongly support our two Fe environment hypothesis. The critical concentration at about 12 at. % suggests that the dependence of H(O, x) on x undergoes a fundamental change at this concentration. Furthermore, the so-called critical concentration at about 17 at. % apparently has little to do with any fundamental alloy property. Rather, it reflects the 12 at. % critical composition when properties are analyzed in terms of the average rather than the local Fe concentration. The two Fe environments appear to have differing configurations.
Violet, C.E.; Verrill, S.P.; Balaban, D.J.; Borg, R.J.
1985-09-01
248
We present preliminary results on the possible effects that primordial magnetic fields can have for a warm inflation scenario, based on global supersymmetry, with a new-inflation-type potential. This work is motivated by two considerations: first, magnetic fields seem to be present in the universe on all scales which rises de possibility that they could also permeate the early universe; second, the recent emergence of inflationary models where the inflaton is not assumed to be isolated but instead it is taken as an interacting field, even during the inflationary expansion. The effects of magnetic fields are included resorting to Schwinger's proper time method.
Piccinelli, Gabriella; Snchez, ngel; Ayala, Alejandro; Mizher, Ana Julia
2013-07-01
249
SciTech Connect
Spatially complicated magnetic fields are frequently treated as the sum of a large, slowly varying, mean field and a small, rapidly varying, field. The primary effect of the small field is to modify the Ohm's law of the mean field. A set of plausible assumptions leads to a form of the mean field Ohm's law which is fundamentally different from the conventional alpha effect of dynamo theory.
Boozer, A.H.
1984-11-01
250
The geomagnetic field is an essential environmental factor for life and health on this planet. In order to survey how magnetic fields affect the life span and the nitrogenase (an iron-sulphur enzyme) activity of Azotobacter chroococcum as well as the life span, the main organic synthesis and the water balance of plants (22 species), the biological tests were incubated under shielded magnetic field and also in normal geo-magnetic environment. The shielding level was about 10-6 of the terrestrial magnetic field.Life cycles of all organisms require the co-ordinated control of a complex set of interlocked physiological processes and metabolic pathways. Such processes are likely to be regulated by a large number of genes. Our researches suggest that the main point in biological structures, which seems to be affected by the low magnetic environment, is the water molecule. Magnetic field induces a molecular alignment. Under shielded conditions, unstructured water molecules with fewer hydrogen bonds, which are producing a more reactive environment, are occurring. As compared to control, the life span of both microorganisms and plants was shorter in shielded environment. A higher nitrogenase affinity for the substrate was recorded in normal geo-magnetic field compared to low magnetic field. The synthesis of carbohydrates, lipids, proteins and enzymes was modified under experimental conditions. The stomatal conductance was higher between 158 and 300% in shielded environment indicating an important water loss from the plant cells.Our results support the idea that the shielded magnetic environment induces different reactions depending on the time of exposure and on the main metabolic pathways of the cells.
Dobrota, C.; Piso, I. M.; Bathory, D.
251
The simplest thermal evolution models for Mercury predict that, because of its small size, the planet should cool rapidly and its initially molten core should now be fully or nearly solid. The Mariner 10 discovery of Mercury's magnetic field caused a dramatic shift in thinking, and the field is now often ascribed to dynamo generation in a molten outer core. The arguments for rejecting the alternative hypothesis that the field is crustal in origin were based on two important assertions: (1) that intensely magnetized rocks were not believed to be present in the natural environments of the terrestrial planets, and (2) an elegant theorem due to S. K. Runcorn proving that a uniform shell magnetized by an internal source subsequently removed has no external field. Prompted by the MGS discovery of unexpectedly large specific magnetization in the martian crust, as well as laboratory experiments demonstrating the strength of single-domain magnetization, we reexamine the possibility that Mercury's magnetic field may be crustal in origin, and we consider the consequences of breaking the symmetry requisite to Runcorn's theorem. We suggest that the spatially dependent depth to the Curie temperature driven by variable solar insolation creates an asymmetry in the distribution of magnetization, that is consistent with the limited constraints from Mariner 10 on the relative strengths of the dipole and quadrupole components.
Aharonson, O.; Zuber, M. T.; Solomon, S. C.
2003-04-01
252
From previous studies of the effect of primordial magnetic fields on early structure formation, we know that the presence of primordial magnetic fields during early structure formation could induce more perturbations at small scales (at present 1-10 h -1 Mpc) as compared to the usual ?CDM theory. Matter power spectra over these scales are effectively probed by cosmological observables such as shear correlation and Ly? clouds. In this paper we discuss the implications of primordial magnetic fields on the distribution of Ly? clouds. We simulate the line-of-sight density fluctuation including the contribution coming from the primordial magnetic fields. We compute the evolution of Ly? opacity for this case and compare our theoretical estimates of Ly? opacity with the existing data to constrain the parameters of the primordial magnetic fields. We also discuss the case when the two density fields are correlated. Our analysis yields an upper bound of roughly 0.3-0.6 nG on the magnetic field strength for a range of nearly scale-invariant models, corresponding to a magnetic field power spectrum index n ~= -3.
Pandey, Kanhaiya L.; Sethi, Shiv K.
2013-01-01
253
Observations of Mercury's internal magnetic field during MESSENGER's first flyby (M1) and the first and third flybys of Mariner 10 (M10-I, M10-III) suggest that small-scale crustal magnetic fields, if they exist, are at the limit of resolution. Small-scale crustal fields are most easily identified near closest approach (CA) as features with wavelengths comparable to, or larger than, the spacecraft altitude. One small feature (< 4 nT in magnitude) encountered near CA during MESSENGER's first flyby may be either a crustal magnetic field or a plasma pressure effect. By means of Parker's constrained optimization approach, with no assumptions on the direction of magnetization, we can place constraints on the product of magnetization and magnetized layer thickness from such observations. The second flyby (M2) will allow additional constraints to be placed on the presence of small-scale fields, and correlations will be possible among topographic profiles measured by the Mercury Laser Altimeter (MLA), features seen on MESSENGER and Mariner 10 images, and any variations in the internal field. This flyby will acquire the first images of the CA region of M10-III, which has been pivotal in establishing the dipolar character of Mercury's magnetic field. Our ability to isolate small-scale crustal magnetic fields has been hindered by the limited coverage to date, as well as the difficulty in isolating the internal field. Across the terrestrial planets and the Moon, minimum magnetization contrast and iron abundance in the crust show a positive correlation. This correlation suggests that crustal iron content plays a determining role in the strength of crustal magnetization.
Purucker, M. E.; Sabaka, T. J.; Solomon, S. C.; Anderson, B. J.; Korth, H.; Zuber, M. T.; Neumann, G. A.; Head, J. W.; Johnson, C. L.; Uno, H.
2008-12-01
254
Recently, the performance of high temperature superconducting (HTS) bulks such as critical current density, size, and mechanical strength has been improved rapidly. So, various applications using HTS bulks such as motors, bearings and flywheels have been investigated by many research groups. A compact nuclear magnetic resonance (NMR) magnet is one of the new applications after a technique to enhance maximum trapped field of the HTS bulk more than 11.7 T (500 MHz 1H NMR frequency) has been developed. This new compact NMR magnet out of HTS bulks is cost-effective compared with conventional NMR magnets and then expected to be widely used in food and drug industry. In design and manufacture of the compact NMR magnets, spatial field homogeneity of the large trapped magnetic field in HTS bulk annuli is a crucial issue because the behavior of a trapped field is highly non-linear and, as a result, a technique to improve the field homogeneity such as active/passive shimming now becomes more challenging compared with that of the conventional counterparts. This paper presents the magnetic field distributions in single and three assembled HTS bulk annuli measured by a 3-axis and multi-arrayed Hall sensor under two different cryogenic environments: (1) in a bath of liquid nitrogen (LN2) and (2) dry cooling by a cryocooler. The spatial homogeneity changes with various operating temperatures were investigated and the effect of critical current density enhancement by lowering the operating temperature on the field homogeneity improvement was discussed in detail.
Kim, S. B.; Takano, R.; Nakano, T.; Imai, M.; Hahn, S. Y.
2009-10-01
255
Throughout the evolution process, Earths magnetic field (MF, about 50 ?T) was a natural component of the environment for living organisms. Biological objects, flying on planned long-term interplanetary missions, would experience much weaker magnetic fields, since galactic MF is known to be 0.1 1 nT. However, the role of weak magnetic fields and their influence on functioning of biological organisms are still insufficiently understood, and is actively studied. Numerous experiments with seedlings of different plant species placed in weak magnetic field have shown that the growth of their primary roots is inhibited during early germination stages in comparison with control. The proliferative activity and cell reproduction in meristem of plant roots are reduced in weak magnetic field. Cell reproductive cycle slows down due to the expansion of G1 phase in many plant species (and of G2 phase in flax and lentil roots), while other phases of cell cycle remain relatively stabile. In plant cells exposed to weak magnetic field, the functional activity of genome at early pre-replicate period is shown to decrease. Weak magnetic field causes intensification of protein synthesis and disintegration in plant roots. At ultrastructural level, changes in distribution of condensed chromatin and nucleolus compactization in nuclei, noticeable accumulation of lipid bodies, development of a lytic compartment (vacuoles, cytosegresomes and paramural bodies), and reduction of phytoferritin in plastids in meristem cells were observed in pea roots exposed to weak magnetic field. Mitochondria were found to be very sensitive to weak magnetic field: their size and relative volume in cells increase, matrix becomes electron-transparent, and cristae reduce. Cytochemical studies indicate that cells of plant roots exposed to weak magnetic field show Ca2+ over-saturation in all organelles and in cytoplasm unlike the control ones. The data presented suggest that prolonged exposures of plants to weak magnetic field may cause different biological effects at the cellular, tissue and organ levels. They may be functionally related to systems that regulate plant metabolism including the intracellular Ca2+ homeostasis. However, our understanding of very complex fundamental mechanisms and sites of interactions between weak magnetic fields and biological systems is still incomplete and still deserve strong research efforts.
Belyavskaya, N. A.
2004-01-01
256
It is demonstrated that strong magnetic fields are produced from a zero\\u000ainitial magnetic field during the pregalactic era, when galaxies are first\\u000aforming. Their development proceeds in three phases. In the first phase, weak\\u000amagnetic fields are created by the Biermann battery mechanism, acting in\\u000ashocked parts of the intergalactic medium where caustics form and intersect. In\\u000athe second
Russell M. Kulsrud; Renyue Cen; Jeremiah P. Ostriker; Dongsu Ryu
1996-01-01
257
Magnetic fields permeate the Universe. They are found in planets, stars, accretion discs, galaxies, clusters of galaxies,\\u000a and the intergalactic medium. While there is often a component of the field that is spatially coherent at the scale of the\\u000a astrophysical object, the field lines are tangled chaotically and there are magnetic fluctuations at scales that range over\\u000a orders of magnitude.
Alexander A. Schekochihin; Steven C Cowley
2007-01-01
258
SciTech Connect
The magnetic fields associated with plasmas frequently exhibit small amplitude MHD fluctuations. It is useful to have equations for the magnetic field averaged over these fluctuations, the so-called mean field equations. Under very general assumptions it is shown that the effect of MHD fluctuations on a force-free plasma can be represented by one parameter in Ohm's law, which is effectively the coefficient of electric current viscosity.
Boozer, A.H.
1986-05-01
259
SciTech Connect
Emittance can be measured by intercepting an electron beam on a range thick plate and then observing the expansion of beamlets transmitted through small holes. The hole size is selected to minimize space charge effects. In the presence of a magnetic field the beamlets have a spiral trajectory and the usual field free formulation must be modified. To interpret emittance in the presence of a magnetic field an envelope equation is derived in the appropriate rotating frame. 1 ref.
Boyd, J.K.
1991-04-15
260
We review our investigations of the use of static magnetic fields, B, for manipulating cells and cellular processes. We describe how B fields modify the cell division pattern of frog embryos and consequently can be used to probe the pattern determinants. We also observe that magnetic fields modify the swimming behavior of Paramecium Caudatum. We describe these modifications and their potential application to investigations of their swimming behavior.
Valles, J. M.; Guevorkian, K.
2005-07-01
261
Perovskite-type manganese oxides (manganites) are of interest for many of the different properties they possess, including colossal magnetoresistance (CMR) and ferroelectric behavior. With the application of an electric field, large resistance decreases have been noted near the insulator-to-metal transition temperature in samples of (La1-yPry)1-xCaxMnO3 (LPCMO). Two proposed models have emerged to explain the behavior, dielectric breakdown and dielectrophoresis, with experimental evidence showing some aspects of the dielectrophoresis model to be correct. However, neither model accounts for magnetic interactions among the ferromagnetic metallic regions and the effects of a magnetic field applied in conjunction with an electric field. We have performed measurements on LPCMO samples by varying the strength and orientation of the magnetic field and the applied voltage. Cross-shaped microstructures have been made on LPCMO samples to allow us to investigate the effects of sample size on dielectrophoresis. We will present resistance and magnetization data obtained on LPCMO samples at various magnetic field strengths, magnetic field orientations, and sample sizes to elucidate the effect of magnetic interactions on dielectrophoresis induced transport and magnetic properties.
Grant, Daniel; Dragiev, Galin; Biswas, Amlan
2013-03-01
262
Solar flares and coronal mass ejections (CMEs) --- phenomena which impact our society, but are scientifically interesting in themselves --- are driven by free magnetic energy in the coronal magnetic field. Since the coronal magnetic field cannot be directly measured, modelers often extrapolate the coronal field from the photospheric magnetograms --- the only field measurements routinely available. The best extrapolation techniques assume that the field is force free (coronal currents parallel the magnetic field), but that currents are not simply a linear function of the magnetic field. Recent tests, however, suggest that such non-linear force-free field (NLFFF) extrapolation techniques often underestimate free magnetic energy. We hypothesize that, since relaxation-based NLFFF techniques tend to smooth field discontinuities, such approaches will fail when current sheets are present. Here, we test this hypothesis by applying the Optimization NLFFF method to two configurations from an MHD simulation --- one with strong current concentrations, and one with weak concentrations. This work is supported by a NASA Sun-Earth Connections Theory grant to UC-Berkeley.
Welsch, Brian; De Moortel, I.; McTiernan, J. M.
2007-05-01
263
We discuss the effects of strong magnetic fields through Landau quantization of electrons on the structure and stability of nuclei in neutron star crust. In strong magnetic fields, this leads to the enhancement of the electron number density with respect to the zero field case. We obtain the sequence of equilibrium nuclei of the outer crust in the presence of strong magnetic fields adopting most recent versions of the experimental and theoretical nuclear mass tables. For B ~ 1016G, it is found that some new nuclei appear in the sequence and some nuclei disappear from the sequence compared with the zero field case. Further we investigate the stability of nuclei in the inner crust in the presence of strong magnetic fields using the Thomas-Fermi model. The coexistence of two phases of nuclear matter - liquid and gas, is considered in this case. The proton number density is significantly enhanced in strong magnetic fields B ~ 1017G through the charge neutrality. We find nuclei with larger mass number in the presence of strong magnetic fields than those of the zero field. These results might have important implications for the transport properties of the crust in magnetars.
2011-09-01
264
The topology of the large-scale magnetic field of the Sun and its role in the development of magnetic activity were investigated using H ? charts of the Sun in the period 1887-2011. We have considered the indices characterizing the minimum activity epoch, according to the data of large-scale magnetic fields. Such indices include: dipole-octopole index, area and average latitude of the field with dominant polarity in each hemisphere and others. We studied the correlation between these indices and the amplitude of the following sunspot cycle, and the relation between the duration of the cycle of large-scale magnetic fields and the duration of the sunspot cycle. The comparative analysis of the solar corona during the minimum epochs in activity cycles 12 to 24 shows that the large-scale magnetic field has been slow and steadily changing during the past 130 years. The reasons for the variations in the solar coronal structure and its relation with long-term variations in the geomagnetic indices, solar wind and Gleissberg cycle are discussed. We also discuss the origin of the large-scale magnetic field. Perhaps the large-scale field leads to the generation of small-scale bipolar ephemeral regions, which in turn support the large-scale field. The existence of two dynamos: a dynamo of sunspots and a surface dynamo can explain phenomena such as long periods of sunspot minima, permanent dynamo in stars and the geomagnetic field.
Tlatov, Andrey G.; Obridko, Vladimir N.
2012-07-01
265
The TbFeCo magneto-optical media with the coercivity of bigger than 1.0 kOe are used for the investigation of ultrafast heating and magnetic switching with the weak external magnetic field. It has been found that the laser-induced active region becomes larger with an external magnetic field because the boundary of the active region is magnetized with the assistance of the external field during the ultrafast heating. According to this physical phenomenon, the so called mark expansion method'' has been proposed for visual observation of ultrafast switching marks. Using this method, the ultrafast magnetic switching in TbFeCo media has been studied using 40 fs laser pulse with linear polarization. The result shows that the ultrafast magnetic switching can be implemented by the laser pulse with assistance of the weak external field of about 0.7 kOe. Further studies show that the area percentage of the magnetic mark expansion relative to its thermal mark decreases with the increasing of the laser pulse energy. There exists the threshold pulse energy that the active region is fully magnetized. The theoretical analysis of electron, spin, and lattice temperatures has been conducted to the active region of the media where the maximum spin temperature is close to the Curie temperature of the media. The result indicates that the media become active at 4.137 ps and the ultrafast heating plays a key role for the ultrafast magnetic switching. The weak external magnetic field provides sufficient driving force to control the magnetization direction in the media.
Li, J. M.; Xu, B. X.; Zhang, J.; Ye, K. D.
2013-01-01
266
The solar atmosphere is a highly ionized medium which is the playground of magnetic fields. In the deepest layer (the photosphere), magnetic fields disturb the 'normal' fluid motions forcing the plasma to behave incounterintuitive ways; in the outer layers (the chromosphere and the corona) magnetic fields rule, making the plasma levitate or even ejecting it out of the gravitational well of the Sun, with important consequences for us here on Earth. However, magnetic fields are elusive. The only quantitative evidence of their presence is through the polarization state of the light emitted by the plasma they are playing with. Remote sensing of magnetic fields from 150 million km away through spectropolarimetry is a challenge on applied physics as well as an art. It requires the application of quantum mechanics, radiative transfer theory, and advanced optics to the interpretation and analysis of spectropolarimetric observations. I will review standard diagnostic techniques and recent developments on this field. I will discuss their limitations and how to overcome them through the complementary aspects of different diagnostic techniques, spectral regions, and statistical analysis. Finally, I will review what are the main areas for progress in this regard: most notably, the 'measurement' of magnetic fields in the extremely dilute and weakly magnetized outer layers of the sun.
Manso Sainz, R.
2011-12-01
267
US Patent & Trademark Office Database
A system includes a host device and a disk drive interfaced with the host device are described as well as an associated method. The disk drive includes a magnetic media for storing information using an actuator arrangement to perform a data access by moving at least one head proximate to the magnetic media. The information may be subject to corruption when the disk drive is exposed, during the data access, to a given stray magnetic field having a given minimum magnetic field intensity. The given stray magnetic field will not corrupt the information on the magnetic media with the actuator arrangement positioned away from the magnetic media. A stray magnetic field protection arrangement is configured for detecting an ambient magnetic environment for use in causing the actuator arrangement to park responsive to the detection of at least the given minimum magnetic field intensity.
Partee; Charles (Lyons, CO)
2010-12-28
268
The equilibrium configuration of very small magnetic flux tubes in an intergranular environment automatically produces kilogauss magnetic field strengths. We argue that such a process takes place in the Sun and complements the convective collapse (CC), which is traditionally invoked to explain the formation of kilogauss magnetic concentrations in the solar photosphere. In particular, it can concentrate the very weak magnetic fluxes revealed by the new IR spectropolarimeters, for which the operation of the CC may have difficulty. As part of the argument, we show the existence of solar magnetic features of very weak fluxes yet concentrated magnetic fields (some 31016 Mx and 1500 G).
Snchez Almeida, J.
2001-08-01
269
SciTech Connect
We have examined the behavior of two well-characterized single crystals ofholmium in a magnetic field applied along the /ital c/ axis in a temperaturerange from 90 to 140 K, using magnetization and dilatometric measurements. Wehave found several new phases in this previously unexplored region of the phasediagram.
Steinitz, M. O.; Kahrizi, M.; Tindall, D. A.; Ali, N.
1989-07-01
270
Based on experimental data it is shown, for some chosen alloys and compounds of iron, that there is no unique relationship between the 57Fe-site magnetic hyperfine field, Bhf, and the magnetic moment per Fe atom, ?. Instead, the Bhf? plot consists of several branches, each of them being characteristic of a given alloy or compound. Consequently, the effective proportionality constant
S. M. Dubiel
2009-01-01
271
There is overwhelming evidence that life, from bacteria to birds to bats, detects magnetic fields, using the fields for orientation or navigation. Indeed there are recent reports (based on Google Earth imagery) that cattle and deer align themselves with the earth's magnetic field. [1]. The development of frog and insect eggs are changed by high magnetic fields, probably through known physical mechanisms. However, the mechanisms for eukaryotic navigation and alignment are not clear. Persuasive published models will be discussed. Evidence, that static magnetic fields might produce therapeutic effects, will be updated [2]. [4pt] [1] S. Begall, et al., Proc Natl Acad Sci USA, 105:13451 (2008). [0pt] [2] L. Finegold and B.L. Flamm, BMJ, 332:4 (2006).
Finegold, Leonard
2009-03-01
272
SciTech Connect
We examine three ways to enhance harmonic output of an XUV planar free-electron laser (FEL) operating in the Compton regime. The first method is to increase the rms static magnetic field, making it as large as possible. The second is by adding effective magnetic fields at the harmonics, thereby increasing the coupling to the harmonics. The third is by phase programming; i.e. programming the magnetic field to introduce jumps in the phase of the electrons as they move through phase space.
Elliott, C.J.; Schmitt, M.J.
1986-09-01
273
PubMed
We report on experiments giving evidence for quantum effects of electromagnetic flux in barium alumosilicate glass. In contrast to expectation, below 100 mK the dielectric response becomes sensitive to magnetic fields. The experimental findings include both lifting of the dielectric saturation by weak magnetic fields and oscillations of the dielectric response in the low temperature resonant regime. As the origin of these effects we suggest that the magnetic induction field violates the time reversal invariance leading to a flux periodicity in the energy levels of tunneling systems. At low temperatures, this effect is strongly enhanced by the interaction between tunneling systems and thus becomes measurable. PMID:11017665
Strehlow; Wohlfahrt; Jansen; Haueisen; Weiss; Enss; Hunklinger
2000-02-28
274
We report the results of magnetic field modelling of around 50 CP stars, performed using the "magnetic charges" technique. The modelling shows that the sample reveals four main types of magnetic configurations: 1) a central dipole, 2) a dipole, shifted along the axis, 3) a dipole, shifted across the axis, and 4) complex structures. The vast majority of stars has the field structure of a dipole, shifted from the center of the star. This shift can have any direction, both along and across the axis. A small percentage of stars possess field structures, formed by two or more dipoles.
Glagolevskij, Yu. V.
2011-04-01
275
SciTech Connect
The design of superconducting magnets for particles accelerators requires a high quality of the magnetic field. This paper presents an ANSYS 4.4A Post 1 macro that computes the field quality performing a Fourier analysis of the magnetic field. The results show that the ANSYS solution converges toward the analytical solution and that the error on the multipole coefficients depends linearly on the square of the mesh size. This shows the good accuracy of ANSYS in computing the multipole coefficients. 2 refs., 16 figs., 4 tabs.
Dell'Orco, D.; Chen, Y.
1991-03-01
276
We report on experiments giving evidence for quantum effects of electromagnetic flux in barium alumosilicate glass. In contrast to expectation, below 100 mK the dielectric response becomes sensitive to magnetic fields. The experimental findings include both lifting of the dielectric saturation by weak magnetic fields and oscillations of the dielectric response in the low temperature resonant regime. As the origin of these effects we suggest that the magnetic induction field violates the time reversal invariance leading to a flux periodicity in the energy levels of tunneling systems. At low temperatures, this effect is strongly enhanced by the interaction between tunneling systems and thus becomes measurable.
Strehlow, P.; Wohlfahrt, M.; Jansen, A. G. M.; Haueisen, R.; Weiss, G.; Enss, C.; Hunklinger, S.
2000-02-01
277
PubMed
The influence of natural level of uniform magnetic field (to 200 microT) on Wistar rat cognition was studied in this work. It was found that influence of disturbed Earth magnetic field has caused a long depression of explorative activity only in the presence of information loading. Such depression was removed only after short external stimulation. After this stimulation rats were able to learn by themselves and it took them twice less time than in the control (nootropic effect). It is suggested that a weak magnetic field disturbances may be considered as a negative psychogenic factor which distorts normal conditions for cognitive activity. PMID:8962888
Nikol'skaia, K A; Shtemler, A V; Savonenko, A V; Osipov, A I; Nikol'ski?, S V
278
We describe recent studies of the interaction of fast-rising magnetic fields with multi-species plasmas of densities 10^13-10^15 cm-3. The configurations studied are planar or coaxial gaps, prefilled with plasmas that are driven by 80-400 ns current pulses. The diagnostics is based on time-dependent spectroscopic observations that are spatially resolved in 3D using plasma-doping techniques. The measurements include the magnetic-field structure (from Zeeman splitting), ion velocity distributions (from Doppler profiles), electric fields (from line shapes of allowed and forbidden transitions), and non-Maxwellian electron energy distribution (from line ratios). It is found that the magnetic field propagates in the plasma faster than expected from diffusion. Also, the field spatial distribution is inconsistent with diffusion. The observed broad current channel, as well as non-dependence of the magnetic field evolution on the current polarity, cannot be explained by the available Hall-field theories. Moreover, detailed observations reveal that magnetic field penetration and plasma reflection occur simultaneously, leading to ion-species separation [1, 2], which are also not predicted by Hall-field theories. Measurements of the reflected-proton velocities (twice the magnetic field velocity) show that the protons dissipate a significant fraction of the magnetic field energy. A possible mechanism previously formulated for astrophysical plasmas, based on the formation of small-scale density fluctuations (perhaps as a result of the Rayleigh-Taylor instability) that lead to field penetration via the Hall mechanism, has recently been suggested. The new phenomena observed require novel theoretical treatments. Applications include plasmas under high currents and space physics. 1. A. Weingarten et al., Phys. Rev. Lett. 87, 115004 (2001). 2. R. Arad, et al., Phys. Plasmas 10, 112 (2003).
Maron, Yitzhak
2003-10-01
279
SciTech Connect
Waste emplacement and activities associated with construction of a repository system potentially will change environmental conditions within the repository system. These environmental changes principally result from heat generated by the decay of the radioactive waste, which elevates temperatures within the repository system. Elevated temperatures affect distribution of water, increase kinetic rates of geochemical processes, and cause stresses to change in magnitude and orientation from the stresses resulting from the overlying rock and from underground construction activities. The recognition of this evolving environment has been reflected in activities, studies and discussions generally associated with what has been termed the Near-Field Environment (NFE). The NFE interacts directly with waste packages and engineered barriers as well as potentially changing the fluid composition and flow conditions within the mountain. As such, the NFE defines the environment for assessing the performance of a potential Monitored Geologic Repository at Yucca Mountain, Nevada. The NFe evolves over time, and therefore is not amenable to direct characterization or measurement in the ambient system. Analysis or assessment of the NFE must rely upon projections based on tests and models that encompass the long-term processes of the evolution of this environment. This NFE Process Model Report (PMR) describes the analyses and modeling based on current understanding of the evolution of the near-field within the rock mass extending outward from the drift wall.
R.A. Wagner
2000-11-14
280
A brief overview of biophysical effects of steady magnetic fields is given. The need of high field strength is illustrated by several recent diamagnetic orientation experiments. They include rod-like viruses, purple membranes and chromosomes. Results of various studies on bees, quails, rats and pigeons exposed to fields above 7 T are also resumed.
Maret, Georg
1990-06-01
281
The Hall effect plays a significant role in the penetration of plasma flows across magnetic field. For example, its effect may become dominant in the solar wind penetration into the magnetosphere, in the magnetic field advection in wire array z-pinch precursors, or in the arcing of magnetically insulated transmission lines. An experiment performed at the Nevada Terawatt Facility explored the penetration of plasma with large Hall parameter (10) across ambient magnetic field. The plasma was produced by ablation with the short pulse high intensity laser Leopard (0.35 ps, 10^17W/cm^2) and the magnetic field with the pulsed power generator Zebra (50 T). The expanding plasma assumed a jet configuration and propagated beyond a distance consistent with a diamagnetic bubble model. Without magnetic field, the plasma expansion was close to hemispherical. The ability to produce the plasma and the magnetic field with distinct generators allows a controlled, quasi-continuous variation of the Hall parameter and other plasma parameters making the experiments useful for benchmarking numerical simulations.
Presura, R.; Stepanenko, Y.; Neff, S.; Sotnikov, V. I.
2008-04-01
282
The Explorer 12 measurements of the magnetic field outside the magnetosphere are compared with ground magnetograms from arctic observatories. Results indicate that an exterior field with a southerly component tends to be associated with ground disturbance, whereas a northward field is associated with quiet conditions. Examples are presented show- ing how a north-to-south field-direction change accompanies an increase in ground
D. H. Fairfield; L. J. Jr. Cahill
1966-01-01
283
We discuss the effect of accretion on the evolution of the magnetic field of a neutron star and highlight the main unresolved issues. Charged, accreted matter is funneled towards the magnetic poles where it heats the stellar surface and alters its magnetic structure resulting in an overall reduction of the magnetic dipole moment. Mechanisms for accretion-induced field reduction include accelerated Ohmic decay, vortex-fluxoid interactions, and magnetic burial or screening. We discuss how these can be integrated into a global model and detail recent self-consistent, three-dimensional, magneto-hydrodynamic, calculations (using analytic Grad-Shafranov methods and the numerical solver ZEUS-MP) which incorporate global resistive instabilities. These models can explain why neutron stars in binaries have systematically lower magnetic dipole moments than isolated neutron stars. Finally we discuss applications including the evolution of accreting millisecond pulsars and type-I X-ray bursts, magnetars, and gravitational waves.
Payne, D. J. B.; Vigelius, M.; Melatos, A.
2008-10-01
284
National Technical Information Service (NTIS)
The Sweet-Parker and Petschek scalings of magnetic reconnection rate are modified to include the effect of the viscosity. The modified scalings show that the viscous effect can be important in high- beta plasmas. The theoretical reconnection scalings are ...
W. Park D. A. Monticello R. B. White
1983-01-01
285
Iron oxide nanocrystals (NCs) have been the focus of intense research owing to the observation of tunable magnetic properties which could lead to advances in many fields including magnetic storage devices and medicine. We have been targeting the use of iron oxide NCs as magnetoresistance (MR) based sensors using ordered NC arrays. In this work, we will present our efforts toward using external magnetic fields to induce intraparticle ordering in iron oxide NC drop cast films. We use x-ray diffraction to analyze effects of the external fields on the NC array structure, while using SQUID magnetometry to probe the effects of NC interactions on the magnetic properties of iron oxide NCs ranging from 5 - 20 nm in diameter. MR measurements suggest large changes in the MR ratio can be achieved using the directed ordering approach for NC arrays. Our work could provide new avenues towards the fabrication of new magnetic devices.
Lawson, Stuart; Meulenberg, Robert
2013-03-01
286
In the process of magnetizing cylindrical specimens of a typical high field superconductor Nb-50 at.%Ti, flux jumps were induced by magnetic disturbances. The stability limit field Hfj increased steadily with increasing temperature, and no magnetic instability occurred for temperatures in excess of about 6.5 K. The calculation of Hfj was performed taking into account the cylindrical sample geometry and the critical state equation JcB1-?{=}?. According to the relative magnitudes of the magnetic diffusivity Dm and the thermal diffusivity Dt, the expression of Hfj was derived for two cases; (1) Dm>Dt, and (2) Dm?Dt. Good agreement between experiment and theory was obtained on the stability limit field Hfj and the temperature above which magnetic instabilities do not take place.
Akachi, Tatsuo; Ogasawara, Takeshi; Yasuk?chi, K?
1981-08-01
287
SciTech Connect
In this paper, the effects of magnetic field gradient (i.e., the magnetic field transition layer effects) on the Rayleigh-Taylor instability (RTI) with continuous magnetic field and density profiles are investigated analytically. The transition layers of magnetic field and density with two different typical profiles are studied and the analytic expressions of the linear growth rate of the RTI are obtained. It is found that the magnetic field effects strongly reduce the linear growth rate of the RTI, especially when the perturbation wavelength is short. The linear growth rate of the RTI increases with the thickness of the magnetic field transition layer, especially for the case of small thickness of the magnetic field transition layer. When the magnetic field transition layer width is long enough, the linear growth rate of the RTI can be saturated. Thus when one increases the width of the magnetic field transition layer, the linear growth rate of the RTI increases only in a certain range, which depends on the magnetic field strength. The numerical results are compared with the analytic linear growth rates and they agree well with each other.
Yang, B. L. [Graduate School, China Academy of Engineering Physics, Beijing 100088 (China); Wang, L. F.; Ye, W. H. [HEDPS and CAPT, Peking University, Beijing 100871 (China); LCP, Institute of Applied Physics and Computational Mathematics, Beijing 100088 (China); Xue, C. [LCP, Institute of Applied Physics and Computational Mathematics, Beijing 100088 (China)
2011-07-15
288
Magneto-optical imaging is widely used to observe the domain patterns in magnetic materials, visualize defects in ferromagnetic objects, and measure the spatial distribution of stray magnetic fields. Optimized 1D magneto-photonic crystals enable a significant increase in the sensitivity of magneto-optical sensors. The properties of such devices based on the optimized reflection (doubled Faraday rotation) mode and the use of 1D magnetic photonic crystals as sensors are discussed. Experimental results of the fabrication and characterization of ferrite-garnet layers possessing uniaxial magnetic anisotropy are shown, and an optimized film structure suitable for magneto-optical imaging is proposed.
Vasiliev, Mikhail; Alameh, Kamal E.; Kotov, Viatcheslav
2008-06-01
289
We propose a method that allows power conversion from DC magnetic fields to AC electric voltages using domain wall (DW) motion in ferromagnetic nanowires. The device concept relies on spinmotive force, voltage generation due to magnetization dynamics. Sinusoidal modulation of the nanowire width introduces a periodic potential for a DW, the gradient of which exerts variable pressure on the traveling DW. This results in time variation of the DW precession frequency and the associated voltage. Using a one-dimensional model, we show that the frequency and amplitude of the AC outputs can be tuned by the DC magnetic fields and wire-design.
2012-12-01
290
The sensitivity of most magnetic sensors is affected by 1/f noise. Modulating the magnetic field to be detected by magnetic sensors can improve their performance by minimizing the effect of this 1/f noise and, in some cases, also have them operate in a narrow frequency band where they have higher sensitivity. We present approaches for modulating the field. One approach is the MEMS flux concentrator can be used with small magnetic sensors and another, based on using a rotating disc containing flux concentrators that can be used with large magnetic sensors, such as magnetoelectric sensors, that have an increased sensitivity at their mechanical resonance frequency. Sidebands observed around the modulation frequency demonstrate the applicability of these approaches. The MEMS flux concentrator has improved the signal to noise ratio in the power spectrum by a factor of 15. The sensors have the potential to achieve sensitivities of a few pT/Hz^1/2 at 1 Hz.
Edelstein, Alan; Petrie, Jonathan; Fine, Jonathan; Fischer, Greg; Burnette, James; Srinivasan, Gopal; Mandal, Sanjay
2011-03-01
291
SciTech Connect
Childhood cancer has been modestly associated with wire codes, an exposure surrogate for power frequency magnetic fields, but less consistently with measured fields. The authors analyzed data on the population distribution of wire codes and their relationship with several measured magnetic field metrics. In a given geographic area, there is a marked trend for decreased prevalence from low to high wire code categories, but there are differences between areas. For average measured fields, there is a positive relationship between the mean of the distributions and wire codes but a large overlap among the categories. Better discrimination is obtained for the extremes of the measurement values when comparing the highest and the lowest wire code categories. Instability of measurements, intermittent fields, or other exposure conditions do not appear to provide a viable explanation for the differences between wire codes and magnetic fields with respect to the strength and consistency of their respective association with childhood cancer.
Kheifets, L.I.; Kavet, R.; Sussman, S.S. [Electric Power Research Inst., Palo Alto, CA (United States)
1997-05-01
292
NSDL National Science Digital Library
The Magnetic Dipole Field 3D Model displays the field lines and field vectors of a dipole located at the origin and oriented along the z-axis. Users can compute the field line passing through a point by dragging the a marker within the 3D view. Users can also visualize the field vectors in a plane passing though the center of the dipole. The Magnetic Dipole Field 3D Model was developed using the Easy Java Simulations (Ejs) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_em_MagneticDipole3D.jar file will run the program if Java is installed. EJS is a part of the Open Source Physics Project and is designed to make it easier to access, modify, and generate computer models.
Christian, Wolfgang
2012-08-11
293
Aims: Violent gravitational interactions can change the morphologies of galaxies and, by means of merging, transform them into elliptical galaxies. We aim to investigate how they affect the evolution of galactic magnetic fields. Methods: We selected 16 systems of interacting galaxies with available VLA archive radio data at 4.86 and 1.4 GHz and compared their radio emission and estimated magnetic
R. T. Drzazga; K. T. Chyzy; W. Jurusik; K. Wirkiewicz
2011-01-01
294
There are two suggested origins for the observed galactic magnetic fields: the primordial origin and the dynamo origin. In this paper the dynamo origin is discussed and criticized. It is pointed out that if the interstellar medium, in which the dynamo operates, is infinitely conducting, the dynamo will not behave properly but will amplify the chaotic part of the magnetic
R. M. Kulsrud
1990-01-01
295
The recent release of two Swedish epidemiological studies correlating residential and occupational magnetic field exposures with certain cancers has strengthened the hypothesis that high voltage power delivery can increase the incidence of cancer. The studies have received wide television, radio, and newsmedia attention, and can be expected to influence public and governmental attitudes regarding residential and occupational 60-Hertz (Hz) magnetic
W. W. Shelton; J. C. Toler
1993-01-01
296
In 1983, on the basis of Scriptures implying the original created material of the earth was water, I proposed that God created the water with the spins of its hydrogen nuclei initially aligned in one direction (Humphreys, 1983). That would produce a strong magnetic field. After 6,000 years of decay, including energy losses from magnetic reversals during the Genesis Flood,
D. Russell Humphreys
2008-01-01
297
The general theory of the linear instabilities created by density differences in a rotating magnetic system is considered, and is applied to a plane layer stably stratified but with a slight superimposed horizontal density gradient that can give rise to baroclinic waves, modified by the presence of a horizontal co-rotating magnetic field parallel to the thermal wind. It is shown
S. I. Braginskii; P. H. Roberts
1975-01-01
298
SciTech Connect
The experimental results on the performance of the MRS (Metal/Resistor/Semiconductor) photodiode in the strong magnetic field of 4.4T, and the possible impact of the quench of the magnet at 4.5T on sensor's operation are reported.
Beznosko, D.; Blazey, G.; Dyshkant, A.; Francis, K.; Kubik, D.; Rykalin, V.; /Northern Illinois U.; Tartaglia, M.A.; /Fermilab; Zutshi, v.; /Northern Illinois U.
2004-12-01
299
National Technical Information Service (NTIS)
Experimental investigations were conducted on a toroidal plasma with alternating pinch- and theta-pinch magnetic fields as well as with a theta-pinch and with a screw pinch. For the alternating pinch, the resultant magnetic vector is rotating, so that the...
D. E. Brown H. G. Loos
1966-01-01
300
DOEpatents
A system (10) for measuring magnetic fields, wherein the system (10) comprises an unmodulated or direct-feedback flux locked loop (12) connected by first and second unbalanced RF coaxial transmission lines (16a, 16b) to a superconducting quantum interference device (14). The FLL (12) operates for the most part in a room-temperature or non-cryogenic environment, while the SQUID (14) operates in a cryogenic environment, with the first and second lines (16a, 16b) extending between these two operating environments.
Ganther, Jr., Kenneth R. (Olathe, KS); Snapp, Lowell D. (Blue Springs, MO)
2006-08-15
301
SciTech Connect
Under some conditions, time series of the interplanetary magnetic field strength and components have the properties of fractal curves. Magnetic field measurements made near 8.5 AU by Voyager 2 from June 5 to August 24, 1981 were self-similar over time scales from approximately 20 sec to approximately 3 x 100,000 sec, and the fractal dimension of the time series of the strength and components of the magnetic field was D = 5/3, corresponding to a power spectrum P(f) approximately f sup -5/3. Since the Kolmogorov spectrum for homogeneous, isotropic, stationary turbulence is also f sup -5/3, the Voyager 2 measurements are consistent with the observation of an inertial range of turbulence extending over approximately four decades in frequency. Interaction regions probably contributed most of the power in this interval. As an example, one interaction region is discussed in which the magnetic field had a fractal dimension D = 5/3.
Burlaga, L.F.; Klein, L.W.
1985-05-01
302
SciTech Connect
Magnetic field and flux distribution for the CDF detector is calculated using a TRIM program. The flux distribution in the system is calculated at several different excitation levels with an expected B-H curve.
1983-01-20
303
Accretion discs are composed of ionized gas in motion around a central object. Sometimes, the disc is the source of powerful bipolar jets along its rotation axis. Theoretical models invoke the existence of a bipolar magnetic field crossing the disc and require two conditions to produce powerful jets: field lines need to be bent enough at the disc surface and the magnetic field needs to be close to equipartition. The work of Petrucci et al (2008) on the variability of X-ray binaries supposes that transitions between pure accretion phases and accretion-ejection phases are due to some variations of the disc magnetization. This rises the problem of the magnetic field dragging in accretion discs. We revisit the method developed by Lubow et al (1994) by including momentum and mass conservation equations in a time-dependent 1D MHD code.
de Guiran, R.; Ferreira, J.
2010-12-01
304
National Technical Information Service (NTIS)
The report is a review and interpretation of solar active region observations obtained principally with magnetographs. Most of these instruments measure the line-of-sight component of the magnetic field. A few instruments can also measure the transverse c...
D. M. Rust
1970-01-01
305
National Technical Information Service (NTIS)
The turbulent diffusion in a magnetic field is studied. The special case where the turbulence is caused by non collisionnal drif instabilities is considered in some detail. (Atomindex citation 11:495636)
P. Rolland
1978-01-01
306
NSDL National Science Digital Library
A loop of wire travels from the right to the left through an inhomogeneous magnetic field. (The green line is at x=0 is for your reference.) The induced emf in the loop is shown in Volts in the animation.
Christian, Wolfgang; Belloni, Mario
2007-03-03
307
National Technical Information Service (NTIS)
In this thesis homogeneous universes are studied containing a large scale magnetic field. In the evolution three different phases are distinguished: the lepton, the plasma and the matter dominated eras. During the lepton and plasma eras, which form the ra...
G. A. Q. Salvati
1986-01-01
308
In this presentation we review the ways in which the presence (or absence) of a planetary magnetic field can influence climate, and provide specific examples using Earth, Venus, Mars, and Titan. We then identify future needs in this research area.
Brain, D. A.; Leblanc, F.; Luhmann, J. G.; Moore, T. E.; Tian, F.
2012-06-01
309
National Technical Information Service (NTIS)
Variations of the Earth's magnetic field in the past are known because they were recorded in various ferrimagnetic minerals such as titanomagnetite and maghemite. This article describes the principal techniques and sample materials (volcanic lava, sedimen...
D. J. R. Nordemann
1982-01-01
310
Methods of microhardness and X-ray diffraction were used to study the kinetics of changes in the microstructure of the beryllium bronze alloy BrB-2 in the process of decomposition of the supersaturated solid solution in a pulsed magnetic field with a frequency from 0 to 7 Hz at an amplitude of the pulse of 318.4 kA/m, a dc component of the magnetic field equal to 238.8 kA/m, at a temperature of 350C and annealing duration of 1 h. Frequency dependences of the microhardness, lattice parameter, concentration of beryllium in the residual matrix, size of coherent domains, dislocation density, and relative microdeformations of mosaic blocks in the matrix have been measured upon aging with and without pulse magnetic field. The results of experiments show that the pulse magnetic field substantially affects the parameters of the fine structure and physicomechanical properties of the bronze.
Osinskaya, Yu. V.; Pokoev, A. V.
2008-04-01
311
SciTech Connect
The new NHMFL 60T quasi-continuous magnet produces a flat-top field for a period of 100 ms at 60 Tesla, and for longer time at lower fields, e.g. 0.5 s at 45 Tesla. We have developed for the first time the capability to measure heat capacity at very high magnetic fields in the NHMFL 60T quasi-continuous magnet at LANL, using a probe built out of various plastic materials. The field plateau allows us to utilize a heat-pulse method to obtain heat capacity data. Proof-of-principle heat capacity experiments were performed on a variety of correlated electron systems. Both magnet performance characteristics and physical properties of various materials studied hold out a promise of wide application of this new tool.
Jaime, M.; Movshovich, R.; Sarrao, J.L.; Kim, J.; Stewart, G.; Beyermann, W.P.; Canfield, P.C.
1998-10-23
312
National Technical Information Service (NTIS)
During the past thirty years research using high magnetic fields has technically evolved in the manner, but not the magnitude, of the so-called big science areas of particle physics, plasma physics, neutron scattering, synchrotron light scattering, and as...
L. J. Campbell D. M. Parkin J. E. Crow H. J. Schneider-Muntau N. S. Sullivan
1994-01-01
313
There have been two major gains in this period: first, the completion and analysis of round-the-clock observations in cooperation with the Huairou Observatory in the People's Republic of China, which enabled us to obtained the first longterm observations of weak solar magnetic fields, and the application of the magneto-optic filter to the measurement of magnetic fields. The observations in collaboration with China have enabled us to make observations for as long as seven days of solar magnetic regions, with only short interruptions when the sun could not be seen from the U.S. and China. The stronger elements of the chromospheric network are rather long lived, lasting about 70 hours. In fact, it is possible that they last longer, because although the shape changes, it is often possible to still identify a magnetic entity. The second important result was that one could find definite evidence of magnetic field cancellation occurring after solar flares. The increased time coverage enabled us to check the evolution of the magnetic fields during this period. The other development, the use of the magneto-optic filter, is full of promise for the future. This filter is made by using a glass tube filled with potassium which is placed in a strong magnetic field.
Zirin, Harold
314
In this thesis, we did a comprehensive investigation on the relationship between spin-dependent tunneling and structural variation in junction devices. Magnetic, microstructural, and transport studies have shown a significant improvement in exchange-bias, a reduced barrier roughness, and an enhanced magnetoresistance for samples after magnetic annealing. We have examined different magnetic configurations required for sensing applications and presented some results of using MTJ sensors to detect AC magnetic fields created by electrical current flow and DC stray field distributions of patterned magnetic materials. We have studied the low frequency noise in MTJ sensors. We have found that the 1/f noise in MTJs has magnetic as well as electrical origins, and is strongly affected by the junction's internal structure. The magnetic noise comes from magnetization fluctuations in the free FM layer and can be understood using the fluctuation-dissipation theorem. While the field-independent electrical noise due to charge trapping in the barrier, is observed in the less optimized MTJs sensors, and has an amplitude at least one order of magnitude higher than the noise component due to magnetization fluctuations. In addition, we have studied the magnetization switching of Cobalt rings with varying anisotropy utilizing scanning magnetoresistive microscopy. We have for the first time observed a complicated multi-domain intermediate phase during the transition between onion states for samples with strong anisotropy. This is in contrast to as deposited samples, which reverse by simple domain wall motion and feature an intermediate vortex state. The result is further analyzed by micro magnetic simulations.
Liu, Xiaoyong
315
For pt.I see ibid., vol.29, no.1, p.124-135 (1993). An analytical technique for predicting the open-circuit magnetic field distribution in the airgap\\/magnet region of a brushless permanent-magnet DC motor equipped with a surface mounted magnet rotor and a slotless stator was presented in Pt.I. In the present work, the analysis is extended to the prediction of the armature reaction field produced
Z. Q. Zhu; David Howe
1993-01-01
316
We consider a system of N bosons in three dimensions interacting through a mean-field Coulomb potential in an external magnetic field. For initially factorized states we show that the one-particle density matrix associated with the solution of the N-body Schrdinger equation converges to the projection onto the solution of the magnetic Hartree equation in trace norm and in energy as N --> ?. Estimates on the rate of convergence are provided.
Lhrmann, Jonas
2012-02-01
317
Outflows from quasars inevitably pollute the intergalactic medium (IGM) with\\u000amagnetic fields. The short-lived activity of a quasar leaves behind an\\u000aexpanding magnetized bubble in the IGM. We model the expansion of the remnant\\u000aquasar bubbles and calculate their distribution as a function magnetic field\\u000astrength at different redshifts. We find that by a redshift z ~ 3, about 5-80%
Steven R. Furlanetto; Abraham Loeb
2001-01-01
318
\\u000a In this paper we present an investigation into several strain sensing technologies that are being considered to monitor mechanical\\u000a deformation within the steel reinforcement shells used in high field pulsed magnets. Such systems generally operate at cryogenic\\u000a temperatures to mitigate heating issues that are inherent in the coils of nondestructive, high field pulsed magnets. The objective\\u000a of this preliminary study
Christian Martinez; Yan Zheng; Daniel Easton; Kevin M Farinholt
2009-01-01
319
In our recent experiment on STS-107 (MFA-Biotube) we took advantage of the magnetic heterogeneity of the gravity receptor cells of flax roots, namely stronger diamagnetism of starch-filled amyloplasts compared to cytoplasm (Delta ≊ < 0). High gradient magnetic fields (HGMF, grad(H2\\/2) up to 109-1010 Oe2\\/cm) of the experimental chambers (MFCs) repelled amyloplasts from the zones of stronger field thus providing
O. Kuznetsov
2004-01-01
320
Surface magnetic fields of approx.10⁶⁻⁻¹°sup 7\\/ gauss have been inferred from polarization observations of the old nova DQ Her. Such strong magnetic fields will probably lead to corotation of the core and envelope of the white dwarf. Assuming a rotation period of 142 s, this corotation will lead to centrifugal forces sufficient to counterbalance gravity as the star's envelope expands
W. K. Rose; E. H. Scott
1976-01-01
321
We study the effects of growth models of magnetic fields in Central Compact Objects (CCOs). Such a field evolution is not a new idea (Blandford, Applegate, & Hernquist 1983) but the evolutionary implications not have been followed up completely (Michel 1994). We discussed the new class of neutron stars which belong to five main types that have mainly been recognized in the last ten years. The possibility that a rapid weakly magnetized pulsar might have formed in SN1987A is commented.
Bernal, C. G.; Page, D.
2011-10-01
322
NSDL National Science Digital Library
A wire carrying an unknown current is shown above. An external magnetic field that has constant magnitude and direction is applied to the top half of the simulation (The gray rectangle is at the boundary for your reference). In addition, there is the magnetic field produced by the current in the wire. The direction arrows show the vector sum of these two fields. (The color of the direction arrows represents the magnitude of the field as before.) Observe the force vector and the force/length in the yellow message box in the lower left hand corner.
Christian, Wolfgang; Belloni, Mario
2007-03-03
323
Classical equations of motion are solved numerically for one electron atoms in an intense laser pulse. The study investigates the influence of the laser magnetic field on ionization and rescattering. Many models of electron ionization have ignored the magnetic field of the laser pulse, but recent work has looked at the magnetic field's role in stabilization [1]. Work has been done to show that in the ultra-strong regime (intensities of 10^18 W/cm^2) the laser magnetic field has an influence on rescattering [2]. Specifically, drift of the ionized electron along the laser propagation direction. We use a classical model of the atom, atomic number Z, with one electron and numerically integrate two sets of equations of motion, those with and those without the laser magnetic field. Observable quantities, such as electron radius and energy, are calculated and compared. The data shows that the laser magnetic field does have some influence on ionization, specifically on electron dynamics before ionization and the time required for ionization.[4pt] [1] L. N. Gaier and C. H. Keitel, PRA 65, 023406 (2002).[0pt] [2] S. Palaniyappan, I. Ghebregziabher, A. Dichiara, J. MacDonald, and B. C. Walker, PRA 74, 033403 (2006).
Grugan, Patrick; Videtto, Michael; Mancuso, Christopher; Luo, Sui; Walker, Barry
2011-06-01
324
Electric field-induced magnetization switching through magnetization precession is investigated as a function of in-plane component of external magnetic field for a CoFeB/MgO-based magnetic tunnel junction with perpendicular easy axis. The switching probability is an oscillatory function of the duration of voltage pulses and its magnitude and period depend on the magnitude of in-plane magnetic field. Experimental results are compared with simulated ones by using Landau-Lifshitz-Gilbert-Langevin equation, and possible factors determining the probability are discussed.
Kanai, S.; Nakatani, Y.; Yamanouchi, M.; Ikeda, S.; Matsukura, F.; Ohno, H.
2013-08-01
325
In-situ observations and modeling work have indicated the interactions between the solar wind and lunar crustal magnetic anomalies. These interactions will alter the near-surface plasma environment in the magnetic anomaly regions and may have effects on the formation of unusual albedo features - the lunar swirls, and possibly also on the production (or loss) of volatiles (e.g. hydroxyl) as well as electrostatic dust transport. These interactions are complicated by the complex geometries of the lunar crustal magnetic fields. Here we present a series of laboratory investigations of the plasma interactions with magnetic dipole fields above an insulating surface for understanding fundamental physical processes and surface electric fields. We investigated moderate strength dipole fields in which the electrons were magnetized while the ions were unmagnetized. The dipole field was oriented parallel, oblique and normal to the surface. Several physical processes have been identified, including magnetic shielding and focusing, magnetic mirror reflection as well as non-monotonic sheaths. Potential distributions on the surface were complex with enhanced surface charging and electric fields. Our experimental results indicate that plasma environment near the lunar surface can be greatly modified in the magnetic anomaly regions and may thus alter the surface processes. The latest results with a flowing plasma will also be presented.
Wang, Xu; Howes, C.; Hornyi, M.; Robertson, S.
2013-10-01
326
An analytical technique for predicting the instantaneous magnetic field distribution in the airgap region of radial-field topologies of brushless permanent-magnet DC motors, under any specified load condition and accounting implicitly for the stator winding current waveform and the effect of stator-slot-openings, has been developed. It is based on the superposition of the component fields due to the permanent magnet and
Z. Q. Zhu; David Howe; Ekkehard Bolte; Bemd Ackermann
1993-01-01
327
The National High Magnetic Field Laboratory (NHMFL) operates three facilities in support of magnet-related research. The main facilities are located at Florida State University, Tallahassee, Florida, the ultra-low-temperature high-magnetic-field facilities are located at the University of Florida, Gainesville, Florida, and the pulsed field facilities are located at the Los Alamos National Laboratory, Los Alamos, New Mexico. These facilities support a
M. D. Bird; J. E. Crow; P. Schlottmann
2003-01-01
328
Earth magnetic field effects on the particle sensors carried by the Swarm satellites are investigated using particle in cell (PIC) and test-particle modelling. In the reference frame of the spacecraft in which plasma flows at relative velocity v?, Earth magnetic field leads to an ambient electric field E?=-v?B?, which affects the shape of particle distribution functions at the particle sensors. This in turn impacts the distribution of particle fluxes on the microchannel plate (MCP) in the ram face mounted thermal ion imagers (TIIs). Shifts in the centroid of these distributions depend on the direction and magnitude of the local magnetic field and, as such, are expected to vary periodically along the spacecraft orbit. The magnitude of these shifts is estimated quantitatively, and the effect of their variation on the calibration and interpretation of the electric field instrument (EFI) are also discussed.
Rehman, S.; Burchill, J.; Eriksson, A.; Marchand, R.
2012-12-01
329
SciTech Connect
The National High Magnetic Field Laboratory, funded by the National Science Foundation and other US federal Agencies, has in recent years built a wide range of magnetic fields, DC 25 to 35 Tesla, short pulse 50 - 60 Tesla, and quasi-continuous 60 Tesla. Future plans are to push the frontiers to 45 Tesla DC and 70 to 100 Tesla pulse. This user facility, is open for national and international users, and creates an excellent tool for materials research (metals, semiconductors, superconductors, biological systems ..., etc). Here we present results of a systematic study of the upper critical field of a novel superconducting material which is considered a promising candidate for the search for superconductivity beyond H{sub c2} as proposed by several new theories. These theories predict that superconductors with low carrier density can reenter the superconducting phase beyond the conventional upper critical field H{sub c2}. This negates the conventional thinking that superconductivity and magnetic fields are antagonistic.
Tessema, G.X.; Gamble, B.K.; Skove, M.J.; Lacerda, A.H.; Mielke, C.H.
1998-08-22
330
In our recent experiment on STS-107 (MFA-Biotube) we took advantage of the magnetic heterogeneity of the gravity receptor cells of flax roots, namely stronger diamagnetism of starch-filled amyloplasts compared to cytoplasm (? ? < 0). High gradient magnetic fields (HGMF, grad(H2/2) up to 109-1010 Oe2/cm) of the experimental chambers (MFCs) repelled amyloplasts from the zones of stronger field thus providing a directional stimulus for plant gravisensing system in microgravity, and causing the roots to react. Such reaction was observed in the video downlink pictures. Unfortunately, the Columbia'' tragedy caused loss of the plant material and most of the images, thus preventing us from detailed studies of the results. Currently we are looking for a possibility to repeat this experiment. Therefore, it is very important to understand, what other effects (besides displacing amyloplasts) static magnetic fields with intensities 0 to 2.5104 Oe, and with the size of the area of non-uniformity 10-3 to 1 cm. These effects were estimated theoretically and tested experimentally. No statistically significant differences in growth rates or rates of gravicurvature were observed in experiments with Linum, Arabidopsis, Hordeum, Avena, Ceratodon and Chara between the plants grown in uniform magnetic fields of various intensities (102 to 2.5104 Oe) and those grown in the Earth's magnetic field. Microscopic studies also did not detect any structural differences between test and control plants. The magnitudes of possible effects of static magnetic fields on plant cells and organs (including effects on ion currents, magneto-hydrodynamic effects in moving cytoplasm, ponderomotive forces on other cellular structures, effects on some biochemical reactions and biomolecules) were estimated theoretically. The estimations have shown, that these effects are small compared to the thermodynamic noise and thus are insignificant. Both theoretical estimations and control experiments confirm, that intracellular magnetophoresis of statoliths is the only significant effect of the magnetic field on plant cells and organs in the tested magnetic systems.
Kuznetsov, O.
331
SciTech Connect
Superconducting RF technology is becoming more and more important. With some recent cavity test results showing close to or even higher than the critical magnetic field of 170-180 mT that had been considered a limit, it is very important to develop a way to correctly measure the critical magnetic field (H{sup RF}{sub c}) of superconductors in the RF regime. Using a 11.4 GHz, 50-MW, <1 {mu}s, pulsed power source and a TE013-like mode copper cavity, we have been measuring critical magnetic fields of superconductors for accelerator cavity applications. This device can eliminate both thermal and field emission effects due to a short pulse and no electric field at the sample surface. A model of the system is presented in this paper along with a discussion of preliminary experimental data.
Canabal, A.; Tajima, T.; /Los Alamos; Dolgashev, V.A.; Tantawi, S.G.; /SLAC; Yamamoto, T.; /Tsukuba, Natl. Res. Lab. Metrol.
2011-11-04
332
I study the properties of the Euclidean Dirac equation for a light fermion in the presence of both a constant abelian magnetic field and an SU(2) instanton. In particular, I analyze the zero modes analytically in various limits, in order to compare with recent lattice QCD results, and study the implications for the electric dipole moment of the instanton induced by the magnetic field. I also present a holographic computation of the sphaleron rate of a strongly coupled plasma in a the presence of a constant magnetic flux and discuss its physical implications on heavy ion collisions.
Ba?ar, Gke; Dunne, Gerald V.; Kharzeev, Dmitri E.
2013-05-01
333
PubMed
Perturbative gluon exchange interaction between quark and antiquark, or in a 3q system, is enhanced in a magnetic field and may cause vanishing of the total qq[over ] or 3q mass, and even unlimited decrease of it-recently called the magnetic collapse of QCD. The analysis of the one-loop correction below shows a considerable softening of this phenomenon due to qq[over ] loop contribution, similar to the Coulomb case of QED, leading to approximately logarithmic damping of gluon exchange interaction (?O(1/ln|eB|)) at large magnetic field. PMID:23679595
Andreichikov, M A; Orlovsky, V D; Simonov, Yu A
2013-04-18
334
National Technical Information Service (NTIS)
A method is disclosed for manufacturing a magnetic cable trim coil in a sheath assembly for use in a cryogenic particle accelerator. A precisely positioned pattern of trim coil turns is bonded to a flexible substrate sheath that is capable of withstanding...
J. R. Skaritka
1987-01-01
335
The Sweet-Parker and Petschek scalings of magnetic reconnection rate are modified to include the effect of the viscosity. The modified scalings show that the viscous effect can be important in high-..beta.. plasmas. The theoretical reconnection scalings are compared with numerical simulation results in a tokamak geometry for three different cases: a forced reconnection driven by external coils, the nonlinear m
D. A. Monticello; R. B. White
1983-01-01
336
Hypervelocity impacts on satellites or ring particles replenish circumplanetary dusty rings with grains of all sizes. Due to interactions with the plasma environment and sunlight, these grains become electrically charged. We study the motion of charged dust grains launched at the Kepler orbital speed, under the combined effects of gravity and the electromagnetic force. We conduct numerical simulations of dust grain trajectories, covering a broad range of launch distances from the planetary surface to beyond synchronous orbit, and the full range of charge-to-mass ratios from ions to rocks, with both positive and negative electric potentials. Initially, we assume that dust grains have a constant electric potential, and, treating the spinning planetary magnetic field as an aligned and centered dipole, we map regions of radial instability (positive grains only), where dust grains are driven to escape or collide with the planet at high speed, and vertical instability (both positive and negative charges) whereby grains launched near the equatorial plane and are forced up magnetic field lines to high latitudes, where they may collide with the planet. We derive analytical criteria for local stability in the equatorial plane, and solve for the boundaries between all unstable and stable outcomes. Comparing our analytical solutions to our numerical simulations, we develop an extensive model for the radial, vertical and azimuthal motions of dust grains of arbitrary size and launch location. We test these solutions at Jupiter and Saturn, both of whose magnetic fields are reasonably well represented by aligned dipoles, as well as at the Earth, whose magnetic field is close to an anti-aligned dipole. We then evaluate the robustness of our stability boundaries to more general conditions. Firstly, we examine the effects of non-zero launch speeds, of up to 0.5 km s?1, in the frame of the parent body. Although these only weakly affect stability boundaries, we find that the influence of a launch impulse on stability boundaries strongly depends on its direction. Secondly, we focus on the effects of higher-order magnetic field components on orbital stability. We find that vertical stability boundaries are particularly sensitive to a moderate vertical offset in an aligned dipolar magnetic field. This configuration suffices as a model for Saturn's full magnetic field. The vertical instability also expands to cover a wider range of launch distances in slightly tilted magnetic dipoles, like the magnetic field configurations for Earth and Jupiter. By contrast, our radial stability criteria remain largely unaffected by both dipolar tilts and vertical offsets. Nevertheless, a tilted dipole magnetic field model introduces non-axisymmetric forces on orbiting dust grains, which are exacerbated by the inclusion of other higher-order magnetic field components, including the quadrupolar and octupolar terms. Dust grains whose orbital periods are commensurate with the spatial periodicities of a rotating non-axisymmetric magnetic field experience destabilizing Lorentz resonances. These have been studied by other authors for the largest dust grains moving on perturbed Keplerian ellipses. With Jupiter's full magnetic field as our model, we extend the concept of Lorentz resonances to smaller dust grains and find that these can destabilize trajectories on surprisingly short timescales, and even cause negatively-charged dust grains to escape within weeks. We provide detailed numerically-derived stability maps highlighting the destabilizing effects of specific higher-order terms in Jupiter's magnetic field, and we develop analytical solutions for the radial locations of these resonances for all charge-to-mass ratios. We include stability maps for the full magnetic field configurations of Jupiter, Saturn, and Earth, to compare with our analytics. We further provide numerically-derived stability maps for the tortured magnetic fields of Uranus and Neptune. Relaxing the assumption of constant electric charges on dust, we test the effects of time-variable grain charg
Jontof-Hutter, Daniel Simon
337
We report PIC simulation results of magnetic field generation by relativistic shear flows. We find that the shear flow boundary layer in initially non-magnetic shear flows is unstable to the growth of oblique 2-stream and Weibel instabilities near the boundary layer. Such instabilities generate current sheets and loops which eventually form nonlinear ordered structures resembling magnetic flux tubes with alternating polarity, orthogonal to the shear flow direction. Peak magnetic fields can reach almost equipartition values. The size and amplitude of such magnetic structures reach a steady state when the free energy input of the shear flow is balanced by turbulence dissipation. Nonthermal particles are efficiently accelerated, likely by the drift-kink instability, into a power-law energy distribution. These results have important implications for many astrophysical settings, including multi-component blazar jets and gamma-ray bursts. This work was supported by NSF AST0909167 and NASA Fermi grants.
Liang, Edison; Boettcher, Markus; Smith, Ian
2011-11-01
338
PubMed Central
Afzal, Muhammad Haris; Renaudin, Valerie; Lachapelle, Gerard
2011-01-01
339
PubMed
Afzal, Muhammad Haris; Renaudin, Valrie; Lachapelle, Grard
2011-11-30
340
PubMed Central
Microelectromechanical systems (MEMS) technology allows the integration of magnetic field sensors with electronic components, which presents important advantages such as small size, light weight, minimum power consumption, low cost, better sensitivity and high resolution. We present a discussion and review of resonant magnetic field sensors based on MEMS technology. In practice, these sensors exploit the Lorentz force in order to detect external magnetic fields through the displacement of resonant structures, which are measured with optical, capacitive, and piezoresistive sensing techniques. From these, the optical sensing presents immunity to electromagnetic interference (EMI) and reduces the read-out electronic complexity. Moreover, piezoresistive sensing requires an easy fabrication process as well as a standard packaging. A description of the operation mechanisms, advantages and drawbacks of each sensor is considered. MEMS magnetic field sensors are a potential alternative for numerous applications, including the automotive industry, military, medical, telecommunications, oceanographic, spatial, and environment science. In addition, future markets will need the development of several sensors on a single chip for measuring different parameters such as the magnetic field, pressure, temperature and acceleration.
Herrera-May, Agustin L.; Aguilera-Cortes, Luz A.; Garcia-Ramirez, Pedro J.; Manjarrez, Elias
2009-01-01
341
PubMed
Microelectromechanical systems (MEMS) technology allows the integration of magnetic field sensors with electronic components, which presents important advantages such as small size, light weight, minimum power consumption, low cost, better sensitivity and high resolution. We present a discussion and review of resonant magnetic field sensors based on MEMS technology. In practice, these sensors exploit the Lorentz force in order to detect external magnetic fields through the displacement of resonant structures, which are measured with optical, capacitive, and piezoresistive sensing techniques. From these, the optical sensing presents immunity to electromagnetic interference (EMI) and reduces the read-out electronic complexity. Moreover, piezoresistive sensing requires an easy fabrication process as well as a standard packaging. A description of the operation mechanisms, advantages and drawbacks of each sensor is considered. MEMS magnetic field sensors are a potential alternative for numerous applications, including the automotive industry, military, medical, telecommunications, oceanographic, spatial, and environment science. In addition, future markets will need the development of several sensors on a single chip for measuring different parameters such as the magnetic field, pressure, temperature and acceleration. PMID:22408480
Herrera-May, Agustn L; Aguilera-Corts, Luz A; Garca-Ramrez, Pedro J; Manjarrez, Elas
2009-09-30
342
The magnetization process of ferrofluids with carrier fluids of water, paraffin, and alkylnaphtalene was investigated in a temperature range from 77 to 300 K as functions of the freezing rate and the intensity of cooling magnetic fields. A uniaxial magnetic anisotropy is induced by field cooling in frozen ferrofluids. This induced anisotropy which is caused by the formation of clustering
H. Miyajima; N. Inaba; S. Taketomi; M. Sakurai; S. Chikazumi
1988-01-01
343
The permanent-magnet induction generator (PMIG) is a new type of induction machine that has a permanent-magnet rotor inside a squirrel-cage rotor. In this paper, a new technique for the magnetic field analysis of the PMIG is proposed. The proposed technique is based on the PMIG's equivalent circuit and the two-dimensional finite-element analysis (2D-FEA). To execute the 2D-FEA, the phasors of primary and secondary currents are calculated from the equivalent circuit, and the input data for the 2D-FEA is found by converting these phasors into the space vectors. As a result, the internal magnetic fields of the PMIG can be easily analyzed without complicated calculations.
Tsuda, Toshihiro; Fukami, Tadashi; Kanamaru, Yasunori; Miyamoto, Toshio
344
Recently, the demand for instruments that use high magnetic fields has increased in scientific and industrial fields. A high-resolution solid-state nuclear magnetic resonance (NMR) system is one such example. In a solid-state NMR, analytical samples are required to spin accurately at high speed in the high magnetic field. Ultrasonic motors have advantages in such environments. This research aims to develop
Hiraku Maeda; Akihito Kobayashi; Takefumi Kanda; Koichi Suzumori; Kiyonori Takegoshi; Takashi Mizuno
2009-01-01
345
The magnetometer on board Kaguya (Kaguya-LMAG) has been almost continuously observed the magnetic field at about 100km altitude since October 29, 2007. The magnetic field observations are beautiful because of the very low solar activity, the crustal field is well observed at 100km altitude from the record in the lunar wake and the tail-lobe environments. As the lunar crustal magnetic
A. Hayashida; H. Shibuya; H. Tsunakawa; F. Takahashi; H. Shimizu; M. Matsushima
2009-01-01
346
This paper discusses the effect of external low frequency magnetic field interference on cathode ray tube (CRT) computer monitors. The paper describes a new test facility and presents a quantitative measuring method which has been developed to characterize the field effects. A total of 21 monitors from major manufacturers were tested. It was found that larger monitors are more sensitive
Balazs Banfai; George G. Karady; Charles J. Kim; Kate Brown Maracas
2000-01-01
347
Summary form only as given. This paper discusses the effect of external low frequency magnetic field interference on cathode ray tube (CRT) computer monitors. The paper describes a new test facility and presents a quantitative measuring method which has been developed to characterize the field effects. A total of 21 monitors from major manufacturers were tested. It was found that
B. Banfai; G. G. Karady; C. J. Kim; K. Maracas
1999-01-01
348
The observational evidence for magnetic fields in clusters of galaxies will be briefly reviewed. It is possible such fields are stongly affected by gas dynamical processes driven by, e.g., cluster mergers. Numerical MHD investigations of such processes, as well as studies of the role of MHD processes affecting large scale structure formation, will be reviewed.
J. M. Stone
2000-01-01
349
National Technical Information Service (NTIS)
Measurements of resolved Zeeman patterns in the spectrum of Beta CrB show that /H(s)/, the mean surface magnetic field, varies approximately 180 degrees out of phase with the longitudinal component of the field H(e). The maximum observed /H(s)/ is about 5...
S. C. Wolff R. J. Wolff
1969-01-01
350
NSDL National Science Digital Library
A cross section of three wires carrying unknown currents is shown above. You can double-click anywhere inside the animation to draw a magnetic field line. You can also click-drag the wires but this will erase any field line that you have drawn.
Christian, Wolfgang; Belloni, Mario
2007-03-03
351
One of the great discoveries of NASA's Galileo mission was the presence of an intrinsically produced magnetic field at Ganymede. Generation of the relatively strong (750 nT) field likely requires dynamo action in Ganymede's metallic core, but how such a dynamo has been maintained into the present epoch remains uncertain. Using a one-dimensional, three layer thermal model of Ganymede, we
Michael T. Bland; Adam P. Showman; Gabriel Tobie
2008-01-01
352
Based on experimental data it is shown, for some chosen alloys and compounds of iron, that there is no one unique relationship between the 57Fe-site magnetic hyperfine field, Bhf, and the magnetic moment per Fe atom, m. Instead, the Bhf-m plot consists of several branches, each of them being characteristic of a given alloy or compound. Consequently, the effective proportionality
S. M. Dubiel
2008-01-01
353
SciTech Connect
This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). The National High Magnetic Field Laboratory (NHMFL) has established major new facilities at LANL. This project sought to explore some exciting new problems in condensed matter physics that could be studied using these facilities. We studied the behavior of heavy-fermion compounds in high-magnetic fields. The unusual properties of these materials are governed by small energy scales arising from strong many-body correlations, demonstrating that the fields that can be achieved in the NHMFL can be used to probe these correlations.
Trugman, S.; Bedell, K.; Bonca, J.; Gulacsi, M. [Los Alamos National Lab., NM (United States); Yu, C. [California Univ., Irvine, CA (United States)
1996-05-01
354
SciTech Connect
The interaction of electromagnetic radiation with temporally dispersive magnetic solids of small dimensions may show very special resonant behaviors. The internal fields of such samples are characterized by magnetostatic-potential scalar wave functions. The oscillating modes have the energy orthogonality properties and unusual pseudoelectric (gauge) fields. Because of a phase factor, that makes the states single valued, a persistent magnetic current exists. This leads to appearance of an eigenelectric moment of a small disk sample. One of the intriguing features of the mode fields is dynamical symmetry breaking.
Kamenetskii, E. O. [Department of Electrical and Computer Engineering, Ben Gurion University of the Negev, Beer Sheva 84105 (Israel)
2006-01-15
355
SciTech Connect
We study the effect of variations of the electromagnetic coupling on the process of generation of primordial magnetic fields. We find that only through a significant growth of the electromagnetic coupling can minimum seed fields be produced. We also show that, if through some process in the early Universe the photon acquires a mass that leads, thanks to inflation, to the generation of primordial magnetic fields, then the influence of variations of the electromagnetic coupling amounts essentially to the results due to the photon effective mass alone.
Bertolami, O.; Monteiro, R. [Departamento de Fisica, Instituto Superior Tecnico, Av. Rovisco Pais 1, 1049-001 Lisbon (Portugal)
2005-06-15
356
SciTech Connect
The creation of large scale magnetic fields is studied in an inflationary universe where electrodynamics is assumed to be nonlinear. After inflation ends electrodynamics becomes linear and thus the description of reheating and the subsequent radiation dominated stage are unaltered. The nonlinear regime of electrodynamics is described by Lagrangians having a power-law dependence on one of the invariants of the electromagnetic field. It is found that there is a range of parameters for which primordial magnetic fields of cosmologically interesting strengths can be created.
Kunze, Kerstin E. [Departamento de Fisica Fundamental, and Instituto Universitario de Fisica Fundamental y Matematicas (IUFFyM), Universidad de Salamanca, Plaza de la Merced s/n, E-37008 Salamanca (Spain)
2008-01-15
357
SciTech Connect
We study the response of the QCD vacuum to an external magnetic field, in the presence of strong CP violation. Using chiral perturbation theory and large N{sub c} expansion, we show that the external field would polarize quantum fluctuations and induce an electric dipole moment of the vacuum along the direction of the magnetic field. We estimate the magnitude of this effect in different physical scenarios. In particular, we find that the polarization induced by the magnetic field of a magnetar could accelerate electric charges up to energies of the order {approx}{theta}10{sup 3} TeV. We also suggest a connection with the possible existence of ''hot-spots'' on the surface of neutron stars.
Millo, R. [Dipartimento di Fisica, Universita degli Studi di Trento, Via Sommarive 15, Povo (Trento) 38050 (Italy); Faccioli, P. [Dipartimento di Fisica, Universita degli Studi di Trento, Via Sommarive 15, Povo (Trento) 38050 (Italy); I.N.F.N., Gruppo Collegato di Trento, Via Sommarive 15, Povo (Trento), 38050 (Italy)
2008-03-15
358
NSDL National Science Digital Library
The program run from this form computes the values of the Earth's magnetic field parameters for a given location and date or date range. Input required is the date and location (in latitude and longitude) of interest. Links to the U.S. Census Bureau's U.S. Gazeteer and the Getty Thesaurus assists in determing the latitude and longitude for locations of interest. The magnetic parameters (D, I, H, X, Y, Z, and F) are computed based on the latest International Geomagnetic Reference Field (IGRF), a Schmidt quasinormalized spherical harmonic model of the magnetic field. Accuracies for the angular components (Declination, D and Inclination, I) are reported in degrees and minutes of arc and are generally within 30 minutes. Accuracies for the force components (Horizontal - H, North - X, East - Y, Vertical - Z, and Total force - F) are generally within 25 nanotesla. A link to frequently-asked questions about the geomagnetic field of Earth is provided as background material.
359
SciTech Connect
Interplanetary magnetic field magnitude fluctuations are notoriously more intermittent than velocity fluctuations in both fast and slow wind. This behavior has been interpreted in terms of the anomalous scaling observed in passive scalars in fully developed hydrodynamic turbulence. In this paper, the strong intermittent nature of the interplanetary magnetic field is briefly discussed comparing results performed during different phases of the solar cycle. The scaling properties of the interplanetary magnetic field magnitude show solar cycle variation that can be distinguished in the scaling exponents revealed by structure functions. The scaling exponents observed around the solar maximum coincide, within the errors, to those measured for passive scalars in hydrodynamic turbulence. However, it is also found that the values are not universal in the sense that the solar cycle variation may be reflected in dependence on the structure of the velocity field.
Bruno, Roberto; Carbone, Vincenzo; Chapman, Sandra; Hnat, Bogdan; Noullez, Alain; Sorriso-Valvo, Luca [IFSI/INAF, via Fosso del Cavaliere, I-00133 Rome (Italy); Dipartimento di Fisica, Universita della Calabria, and CNISM, Unita di Cosenza, Arcavacata di Rende I-87036 (Italy); Centre for Fusion, Space and Astrophysics, University of Warwick, Warwick CV4 7AL (United Kingdom); Observatoire de la Cote d'Azur, Boulevard de l'Observatoire, F-06304 Nice (France); LICRYL, INFM/CNR, I-87036 Arcavacata di Rende (Italy)
2007-03-15
360
NSDL National Science Digital Library
The Trajectories in Electric and Magnetic Fields model computes a family of trajectories of charges emitted from a point source isotropically and with the same energy. These trajectories create focal points and caustic surfaces meeting the symmetry line in conical cusps. The simulation enables users to study these trajectories in both crossed and parallel magnetic fields. The user can vary the initial particle velocity and the field strengths. The Trajectories in Electric and Magnetic Fields model was developed using the Easy Java Simulations (Ejs) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. You can modify this simulation if you have Ejs installed by right-clicking within the plot and selecting Open Ejs Model from the pop-up menu item.
Christian, Wolfgang
2012-10-25
361
SciTech Connect
Enzymatic and thermochemical catalysis are both important industrial processes. However, the thermal requirements for each process often render them mutually exclusive: thermochemical catalysis requires high temperature that denatures enzymes. One of the long-term goals of this project is to design a thermocatalytic system that could be used with enzymatic systems in situ to catalyze reaction sequences in one pot; this system would be useful for numerous applications e.g. conversion of biomass to biofuel and other commodity products. The desired thermocatalytic system would need to supply enough thermal energy to catalyze thermochemical reactions, while keeping the enzymes from high temperature denaturation. Magnetic nanoparticles are known to generate heat in an oscillating magnetic field through mechanisms including hysteresis and relaxational losses. We envisioned using these magnetic nanoparticles as the local heat source embedded in sub-micron size mesoporous support to spatially separate the particles from the enzymes. In this study, we set out to find the magnetic materials and instrumental conditions that are sufficient for this purpose. Magnetite was chosen as the first model magnetic material in this study because of its high magnetization values, synthetic control over particle size, shape, functionalization and proven biocompatibility. Our experimental designs were guided by a series of theoretical calculations, which provided clues to the effects of particle size, size distribution, magnetic field, frequency and reaction medium. Materials of theoretically optimal size were synthesized, functionalized, and their effects in the oscillating magnetic field were subsequently investigated. Under our conditions, the materials that clustered e.g. silica-coated and PNIPAM-coated iron oxides exhibited the highest heat generation, while iron oxides embedded in MSNs and mesoporous iron oxides exhibited the least bulk heating. It is worth noting that the specific loss power of PNIPAM-coated Fe{sub 3}O{sub 4} was peculiarly high, and the heat loss mechanism of this material remains to be elucidated. Since thermocatalysis is a long-term goal of this project, we also investigated the effects of the oscillating magnetic field system for the synthesis of 7-hydroxycoumarin-3-carboxylic acid. Application of an oscillating magnetic field in the presence of magnetic particles with high thermal response was found to effectively increase the reaction rate of the uncatalyzed synthesis of the coumarin derivative compared to the room temperature control.
Peeraphatdit, Chorthip
2010-12-15
362
Magnetic clouds observed at 1 AU are modeled as cylindrically symmetric, constant alpha force-free magnetic fields. The model satisfactorily explains the types of variations of the magnetic field direction that are observed as a magnetic cloud moves past a spacecraft in terms of the possible orientations of the axis of a magnetic cloud. The model also explains why the magnetic
L. F. Burlaga
1988-01-01
363
Based on the successful reconstruction of the global solar magnetic field by a number of investigators it seems clear that the field strength (used here as B(nT)) has increased significantly during the last 300 years. However, it has been demonstrated that a weak field strength has unexpected consequences for the near Earth environment relative to the high probability of the
Gisela A. M. Dreschhoff
2007-01-01
364
Based on the successful reconstruction of the global solar magnetic field by a number of investigators it seems clear that the field strength (used here as B(nT)) has increased significantly during the last ?300 years. However, it has been demonstrated that a weak field strength has unexpected consequences for the near Earth environment relative to the high probability of the
Gisela A. M. Dreschhoff
2007-01-01
365
PubMed
Eddy currents induced within a magnetic resonance imaging (MRI) cryostat bore during pulsing of gradient coils can be applied constructively together with the gradient currents that generate them, to obtain good quality gradient uniformities within a specified imaging volume over time. This can be achieved by simultaneously optimizing the spatial distribution and temporal pre-emphasis of the gradient coil current, to account for the spatial and temporal variation of the secondary magnetic fields due to the induced eddy currents. This method allows the tailored design of gradient coil/magnet configurations and consequent engineering trade-offs. To compute the transient eddy currents within a realistic cryostat vessel, a low-frequency finite-difference time-domain (FDTD) method using total-field scattered-field (TFSF) scheme has been performed and validated. PMID:17945575
Trakic, A; Liu, F; Lopez, H S; Wang, H; Crozier, S
2006-01-01
366
BackgroundDiamagnetic levitation is a technique that uses a strong, spatially varying magnetic field to simulate an altered gravity environment, as in space. In this study, using Streptomyces avermitilis as the test organism, we investigate whether changes in magnetic field and altered gravity induce changes in morphology and secondary metabolism. We find that a strong magnetic field (12T) inhibit the morphological
Mei Liu; Hong Gao; Peng Shang; Xianlong Zhou; Elizabeth Ashforth; Ying Zhuo; Difei Chen; Biao Ren; Zhiheng Liu; Lixin Zhang
2011-01-01
367
The magnetization states in Ni triangular dots under an applied magnetic field have been studied using variable-field magnetic force microscopy (VF-MFM) imaging. In order to understand their dynamics we performed micromagnetic simulations which are in remarkable agreement with the experimental MFM results. The nanostructures present magnetic vortices as ground states which move under an external magnetic field. The combination of
M. Jaafar; R. Yanes; A. Asenjo; O. Chubykalo-Fesenko; M. Vzquez; E. M. Gonzlez; J. L. Vicent
2008-01-01
368
The effect of magnetic fields on dielectric surface breakdown in vacuum and simulated LEO conditions is investigated using pulsed test voltages. Predictions from the saturated secondary electron emission avalanche breakdown model and experimental results both show magnetic insulation effects at magnetic-field amplitudes as low as 0.1 T. The most favorable configuration for magnetic insulation is with the magnetic field oriented
M. Lehr; R. Korzekwa; H. Krompholz; M. Kristiansen
1992-01-01
369
We present magnetic field maps of 12 starless Bok Globules. Maps were constructed from polarimetric V-band images taken with a computer-controlled CCD camera with a fast shutter and a bidirectional charge-shifting capability. The linear polarization of stars in the globule field has been determined by repeated use of imaging through a polaroid filter, shifting the charge up by many times the stellar PSF, reimaging through an orthogonal polaroid filter, then shifting the charge back down to its starting point. Between one and five CCD fields-of-view were necessary to map each globule, and up to 50 stars in each FOV had detectable polarizations. This globule sample exhibits a variety of magnetic field patterns, including uniform fields with dispersion less than 10(deg) , overlapping fields with two distinct directions, and cometary extensions.
Kane, Brian D.; Clemens, Dan P.
1994-12-01
370
We investigate the 50-year old hypothesis that the magnetic fields of the Ap stars are stable equilibria that have survived in these stars since their formation. With numerical simulations we find that stable magnetic field configurations indeed appear to exist under the conditions in the radiative interior of a star. Confirming a hypothesis by Prendergast (1956, ApJ, 123, 498), the configurations have roughly equal poloidal and toroidal field strengths. We find that tori of such twisted fields can form as remnants of the decay of an unstable random initial field. In agreement with observations, the appearance at the surface is an approximate dipole with smaller contributions from higher multipoles, and the surface field strength can increase with the age of the star. The results of this paper were summarised by Braithwaite & Spruit (2004, Nature, 431, 891).
Braithwaite, J.; Nordlund, .
2006-05-01
371
It has been proposed that high Mach number collisionless shocks propagating in an initially unmagnetized plasma play a major role in the magnetization of large scale structures in the Universe. A detailed study of the experimental configuration necessary to scale such environments down to laboratory dimensions will be presented. We will show initial results from preliminary experiments conducted at the
C. D. Murphy; F. Miniati; M. Edwards; J. Mithen; A. R. Bell; C. Constantin; E. Everson; D. Schaeffer; C. Niemann; A. Ravasio; E. Brambrink; A. Benuzzi-Mounaix; M. Koenig; C. Gregory; N. Woolsey; H.-S. Park; B. Remington; D. Ryutov; R. Bingham; L. Gargate; A. Spitkovsky; G. Gregori
2010-01-01
372
Synchrotron emission, its polarization and its Faraday rotation at radio frequencies of 0.2-10 GHz are powerful tools to study the strength and structure of cosmic magnetic fields. Unpolarized emission traces turbulent fields which are strongest in galactic spiral arms and bars (20-30 muG) and in central starburst regions (50-100 muG). Such fields are dynamically important, e.g. they can drive gas
Rainer Beck
2011-01-01
373
Total magnetic fields in spiral galaxies, as observed through their total synchrotron emission, are strongest (up to ~=30~muG) in the spiral arms. The degree of radio polarization is low; the field in the arms must be mostly turbulent or tangled. Polarized synchrotron emission shows that the resolved regular fields are generally strongest in the interarm regions (up to ~=15~muG), sometimes
Rainer Beck
2007-01-01
374
DOEpatents
A UNLV novel electric/magnetic dot sensor includes a loop of conductor having two ends to the loop, a first end and a second end; the first end of the conductor seamlessly secured to a first conductor within a first sheath; the second end of the conductor seamlessly secured to a second conductor within a second sheath; and the first sheath and the second sheath positioned adjacent each other. The UNLV novel sensor can be made by removing outer layers in a segment of coaxial cable, leaving a continuous link of essentially uncovered conductor between two coaxial cable legs.
Schill, Jr., Robert A. (Henderson, NV); Popek, Marc [Las Vegas, NV
2009-01-27
375
While there have been several studies suggesting the possibility of electrical activity on Mars, to date there have been no measurements to search for evidence of such activity. In the absence of widespread water clouds and convective storm systems similar to those on the Earth and Jupiter, the most likely candidate for the creation of electrostatic charges and fields is triboelectric charging of dust, i.e., the friction between blown dust and the ground, and of dust particles with each other. Terrestrial experience demonstrates that electric fields 5 to 15 kV-m-1 are not uncommon in dust storms and dust devils in desert regions, where the polarity varies according to the chemical composition and grain size. Simple laboratory experiments have demonstrated that modest electrostatic fields of roughly 5,000 V-m-1 may be produced, along with electrical spark discharges and glow discharges, in a simulation of a dusty, turbulent Martian surface environment. While the Viking landers operated for several years with no apparent deleterious effects from electrostatic charging, this may have been at least partly due to good engineering design utilizing pre-1976 electronic circuitry to minimize the possibility of differential charging among the various system components. However, free roaming rovers, astronauts, and airborne probes may conceivably encounter an environment where electrostatic charging is a frequent occurrence, either by way of induction from a static electric field or friction with the dusty surface and atmosphere. This raises the possibility of spark discharges or current surges when subsequent contact is made with other pieces of electrical equipment, and the possibility of damage to modern microelectronic circuitry. Measurements of electrostatic fields on the surface of Mars could therefore be valuable for assessing this danger. Electric field measurements could also be useful for detecting natural discharges that originate in dust storms. This detection could be performed at distances ranging from 10s of km in the case of J-charge-like discharge signatures, to planetary distances if there exists a global electrical circuit or Schumann resonance spectrum.
Sentman, D. D.
1991-05-01
376
SciTech Connect
Solar flares and coronal mass ejections, the most catastrophic eruptions in our solar system, have been known to affect terrestrial environments and infrastructure. However, because their triggering mechanism is still not sufficiently understood, our capacity to predict the occurrence of solar eruptions and to forecast space weather is substantially hindered. Even though various models have been proposed to determine the onset of solar eruptions, the types of magnetic structures capable of triggering these eruptions are still unclear. In this study, we solved this problem by systematically surveying the nonlinear dynamics caused by a wide variety of magnetic structures in terms of three-dimensional magnetohydrodynamic simulations. As a result, we determined that two different types of small magnetic structures favor the onset of solar eruptions. These structures, which should appear near the magnetic polarity inversion line (PIL), include magnetic fluxes reversed to the potential component or the nonpotential component of major field on the PIL. In addition, we analyzed two large flares, the X-class flare on 2006 December 13 and the M-class flare on 2011 February 13, using imaging data provided by the Hinode satellite, and we demonstrated that they conform to the simulation predictions. These results suggest that forecasting of solar eruptions is possible with sophisticated observation of a solar magnetic field, although the lead time must be limited by the timescale of changes in the small magnetic structures.
Kusano, K.; Bamba, Y.; Yamamoto, T. T. [Solar-Terrestrial Environment Laboratory, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601 (Japan); Iida, Y.; Toriumi, S. [Department of Earth and Planetary Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033 (Japan); Asai, A., E-mail: [email protected] [Unit of Synergetic Studies for Space, Kyoto University, 17 Kitakazan Ohmine-cho, Yamashina-ku, Kyoto 607-8471 (Japan)
2012-11-20
377
This article reviews magnetic field measurements in jets from young stars, focusing on the physics and application of the three main techniques, Zeeman splitting and polarization, gyrosynchrotron radiation, and the analysis of shocked cooling zones. Estimates of field strengths in stellar jets are rare, and do not refer to the axis of the beam close to the source, where knowledge of the field and its geometry is most critical for constraining launching mechanisms of jets. Nevertheless, the existing measurements demonstrate that magnetic fields in YSO jets are strong enough to be important in the dynamics of the cooling zones behind internal shock waves, even though the ram pressure in the bulk flow dominates the magnetic pressure at large distances from the source. Models of pulsed magnetic flow show that velocity perturbations sweep up the field into dense working surfaces within the jet, increasing the relative importance of magnetic pressure to the dynamics in these regions and reducing its importance in the rarefaction regions that lie between the dense knots.
Hartigan, Patrick
378
Magnetic fields appear to be ubiquitous in astrophysical environments. Their existence in the intracluster medium is established through observations of synchrotron emission and Faraday rotation. On the other hand, the nature of magnetic fields outside of clusters, where observations are scarce and controversial, remains largely unknown. In this chapter, we review recent developments in our understanding of the nature and
D. Ryu; D. R. G. Schleicher; R. A. Treumann; C. G. Tsagas; L. M. Widrow
2011-01-01
379
Human exposure to magnetic fields generated by distribution substations and power lines is one of the growing health concerns that has drawn much attention from Qatar General Electricity & Water Corporation (also called Kahramaa) and Qatar ministry of environment. This paper presents the results of magnetic field measurements in an outdoor high voltage 132\\/66kV power substation, located in the residential
Khaled Ellithy; Abdulaziz Al Shafai; Fahad Al Assiry
2009-01-01
380
The gravitropic reaction of cress roots in combined magnetic field was studied in details. It was shown that the negative roots gravitropism observed at the frequency of alternating component of combined magnetic field adjusted to the Ca ion cyclotron frequency could be observed only at Nayquist magnetic field noise level under 5 nT/Hz. While the magnetic noise level was increasing the negative gravitropism was disappearing. The inhibition of gravitropic reaction was observed in all cases. The effect was accompanied by the changes in the noise spectrum generated by cress roots.
Bogatina, Nina; Kordyum, Elizabeth; Sheykina, Nadezhda
381
A magnetic field drift, gradual decrease of the order of 10-4 in several tens of hours, was observed with the beam intensity decrease in an operation of an azimuthally varying field (AVF) cyclotron. From our experimental results, we show that the temperature increase of the magnet iron by the heat transfer from the excitation coils can induce such change of the magnetic field as to deteriorate the beam quality. The temperature control of the magnet iron was realized by thermal isolation between the main coil and the yoke and by precise control of the cooling water temperature of the trim coils attached to the pole surfaces in order to prevent temperature change of the magnet iron. The magnetic field stability of +/-510-6 and the beam intensity stability of +/-2% have been achieved by this temperature control.
Okumura, S.; Arakawa, K.; Fukuda, M.; Nakamura, Y.; Yokota, W.; Ishimoto, T.; Kurashima, S.; Ishibori, I.; Nara, T.; Agematsu, T.; Sano, M.; Tachikawa, T.
2005-03-01
382
SciTech Connect
A magnetic field drift, gradual decrease of the order of 10{sup -4} in several tens of hours, was observed with the beam intensity decrease in an operation of an azimuthally varying field (AVF) cyclotron. From our experimental results, we show that the temperature increase of the magnet iron by the heat transfer from the excitation coils can induce such change of the magnetic field as to deteriorate the beam quality. The temperature control of the magnet iron was realized by thermal isolation between the main coil and the yoke and by precise control of the cooling water temperature of the trim coils attached to the pole surfaces in order to prevent temperature change of the magnet iron. The magnetic field stability of {+-}5x10{sup -6} and the beam intensity stability of {+-}2% have been achieved by this temperature control.
Okumura, S.; Arakawa, K.; Fukuda, M.; Nakamura, Y.; Yokota, W.; Ishimoto, T.; Kurashima, S.; Ishibori, I.; Nara, T.; Agematsu, T.; Sano, M.; Tachikawa, T. [Japan Atomic Energy Research Institute (JAERI), 1233 Watanuki, Takasaki, Gunma 370-1292 (Japan); Sumitomo Heavy Industries, Ltd. (SHI), 5-2 Soubiraki, Niihama, Ehime 792-8588 (Japan)
2005-03-01
383
GALFACTS is a large-area spectro-polarimetric survey on the Arecibo Radio telescope. It uses the seven-beam focal plane feed array receiver system (ALFA) to carry out an imaging survey project of the 12,700 square degrees of sky visible from Arecibo at 1.4 GHz with 8192 spectral channels over a bandwidth of 300 MHz sampled at 1 millisecond. The aggregate data rate is 875 MB/s. GALFACTS observations will create full-Stokes image cubes at an angular resolution of 3.5' with a band-averaged sensitivity of 90 ?Jy, allowing sensitive imaging of polarized radiation and Faraday Rotation Measure from both diffuse emission and extragalactic sources. GALFACTS is a scientific pathfinder to the SKA in the area of cosmic magnetism. Key to magnetism science with the SKA is the technique of RM synthesis. The technique of RM synthesis is introduced and we discuss practical aspects of RM synthesis including efficient computational techniques and detection thresholds in the resulting Faraday spectrum. We illustrate the use of the technique by presenting the current development of the RM synthesis pipeline for GALFACTS and present early results.
George, S. J.; Stil, J. M.; Andrecut, M.; Taylor, A. R.
2012-09-01
384
ERIC Educational Resources Information Center
Describes an experiment to demonstrate the influence of a magnetic field on the behavior of a single Langmuir probe. The experiment introduces the student to magnetically supported plasma and particle behavior in a magnetic field. (GA)
Pytlinski, J. T.; And Others
1978-01-01
385
We present an extensive study of magnetic fields in a system of merging galaxies. We obtained for NGC 4038/39 (the Antennae) radio total intensity and polarization maps at 8.44 GHz, 4.86 GHz and 1.49 GHz using the VLA in the C and D configurations. The galaxy pair possesses bright, extended radio emission filling the body of the whole system, with no dominant nuclear sources. The radio thermal fraction of NGC 4038/39 was found to be about 50% at 10.45 GHz, higher than in normal spirals. Most of the thermal emission is associated with star-forming regions, but only a part of these are weakly visible in the optical domain because of strong obscuration. The mean total magnetic fields in both galaxies are about two times stronger (?20 ?G) than in normal spirals. However, the degree of field regularity is rather low, implying tangling of the regular component in regions with interaction-enhanced star formation. Our data combined with those in H I, H?, X-rays and in far infrared allow us to study local interrelations between different gas phases and magnetic fields. We distinguish several radio-emitting regions with different physical properties and at various evolutionary stages: the rudimentary magnetic spiral, the northern cool part of the dark cloud complex extending between the galaxies, its warm southern region, its southernmost star-forming region deficient in radio emission, and the highly polarized northeastern ridge associated with the base of an unfolding tidal tail. The whole region of the dark cloud complex shows a coherent magnetic field structure, probably tracing the line of collision between the arms of merging spirals while the total radio emission reveals hidden star formation nests. The southern region is a particularly intense merger-triggered starburst. Highly tangled magnetic fields reach there strengths of ?30 ?G, even larger than in both individual galaxies, possibly due to compression of the original fields pulled out from the parent disks. In the northeastern ridge, away from star-forming regions, the magnetic field is highly coherent with a strong regular component of 10 ?G tracing gas shearing motions along the tidal tail. We find no signs of field compression by infalling gas there. The radio spectrum is much steeper in this region indicating aging of the CR electron population as they move away from their sources in star-forming regions. Modelling Faraday rotation data shows that we deal with a three-dimensionally curved structure of magnetic fields, becoming almost parallel to the sky plane in the southeastern part of the ridge.
Chy?y, K. T.; Beck, R.
2004-04-01
386
We measured the high-field magnetization up to 55 T and constructed a magnetic phase diagram for a transuranium antiferromagnet NpRhGa5 with the tetragonal structure. The magnetization at 4.2 K for H?[100] indicates a sharp metamagnetic transition with a step at H=26T and saturates above H=38T, reaching 0.43?/Np. An ordered moment of 0.96?/Np at zero field, which was determined from the neutron scattering experiment, is found to be reduced to 0.43?/Np at H, together with an orientation of the magnetic moment from the (0 0 1) plane to the (1 0 0) plane.
Sugiyama, K.; Nakashima, H.; Aoki, D.; Ikeda, S.; Haga, Y.; Yamamoto, E.; Nakamura, A.; Homma, Y.; Shiokawa, Y.; Kindo, K.; Hagiwara, M.; ?nuki, Y.
2007-03-01
387
In conventional magnetic resonance imaging (MRI), a single superconduct- ing magnet is used during the polarization and signal acquisition phases of imaging. In field-cycled MRI, the superconducting magnet is replaced by two actively controlled resistive magnets: a high-field magnet to polarize the sample and a low-field magnet for signal acquisition. Both resistive magnets generate heat during operation and must be
K. M. Gilbert; W. B. Handler; B. A. Chronik
2005-01-01
388
Magnetic fields are observed everywhere in the universe. In this review, we concentrate on the observational aspects of the magnetic fields of Galactic and extragalactic objects. Readers can follow the milestones in the observations of cosmic magnetic fields obtained from the most important tracers of magnetic fields, namely, the star-light polarization, the Zeeman effect, the rotation measures (RMs, hereafter) of
Jin-Lin Han; Richard Wielebinski
2002-01-01
389
We study the generation and evolution of density perturbations and peculiar velocities due to primordial magnetic fields. We assume that a random magnetic field was present before recombination and follow the field's effect on the baryon fluid starting at recombination. We find that magnetic fields generate growing density perturbations on length scales larger than the magnetic Jeans length, lambda_B_ and
Eun-Jin Kim; Angela V. Olinto; Robert Rosner
1996-01-01
390
Magnetic fields are observed everywhere in the universe. In this review, we concentrate on the observational aspects of the magnetic fields of Galactic and extragalactic objects. Readers can follow the milestones in the observations of cosmic magnetic fields obtained from the most important tracers of magnetic fields, namely, the star-light polarization, the Zeeman effect, the rotation measures (RMs,hereafter) of extragalactic
Jin-Lin Han; Richard Wielebinski
2002-01-01
391
We are developing a Magnetic Field Gradient Levitation (MFGL) apparatus as a ground based system for simulating a low or variable gravity environment for diamagnetic materials. The system consists of a superconducting solenoid with a room temperature bore that can generate a magnetic force strong enough to levitate or cancel the body force of gravity in common organic materials (e.g. water, proteins, polypropylene). We will describe the specifications and capabilities of the apparatus and our initial experimental studies of gravitational sensitivity in the biological systems, frog embryos and paramecium.
Valles, James; Guevorkian, Karine
2002-03-01
392
We study the evolution of QCD phase transition-generated magnetic fields (MFs) in freely decaying MHD turbulence of the expanding universe. We consider an MF generation model that starts from basic non-perturbative QCD theory and predicts stochastic MFs with an amplitude of the order of 0.02 ?G and small magnetic helicity. We employ direct numerical simulations to model the MHD turbulence decay and identify two different regimes: a "weakly helical" turbulence regime, when magnetic helicity increases during decay, and "fully helical" turbulence, when maximal magnetic helicity is reached and an inverse cascade develops. The results of our analysis show that in the most optimistic scenario the magnetic correlation length in the comoving frame can reach 10 kpc with the amplitude of the effective MF being 0.007 nG. We demonstrate that the considered model of magnetogenesis can provide the seed MF for galaxies and clusters.
Tevzadze, Alexander G.; Kisslinger, Leonard; Brandenburg, Axel; Kahniashvili, Tina
2012-11-01
393
NSDL National Science Digital Library
A wire carrying an unknown current out of the page is shown above. You may also double-click in the animation to create a field line. Assume that the distance given is in cm and B is given in milli Tesla.
Christian, Wolfgang; Belloni, Mario
2007-03-03
394
The equilibrium magnetization states existing in soft magnetic nanotubes and their behavior in external magnetic field are investigated by means of micromagnetic simulation. In the ground state the middle part of a sufficiently long tube is uniformly magnetized along the tube axis, however there are curling states of various circular polarities near the tube ends. The characteristic length of ending curling states, as well as switching field Hs in the external magnetic field parallel to the tube axis, have been calculated as a function of outer tube radius and tube thickness.
Chen, A. P.; Usov, N. A.; Blanco, J. M.; Gonzalez, J.
2007-09-01
395
SciTech Connect
The statistical representation of a fluctuating (stochastic) magnetic field configuration is studied in detail. The Eulerian correlation functions of the magnetic field are determined, taking into account all geometrical constraints: these objects form a nondiagonal matrix. The Lagrangian correlations, within the reasonable Corrsin approximation, are reduced to a single scalar function, determined by an integral equation. The mean square perpendicular deviation of a geometrical point moving along a perturbed field line is determined by a nonlinear second-order differential equation. The separation of neighboring field lines in a stochastic magnetic field is studied. We find exponentiation lengths of both signs describing, in particular, a decay (on the average) of any initial anisotropy. The vanishing sum of these exponentiation lengths ensures the existence of an invariant which was overlooked in previous works. Next, the separation of a particles trajectory from the magnetic field line to which it was initially attached is studied by a similar method. Here too an initial phase of exponential separation appears. Assuming the existence of a final diffusive phase, anomalous diffusion coefficients are found for both weakly and strongly collisional limits. The latter is identical to the well known Rechester-Rosenbluth coefficient, which is obtained here by a more quantitative (though not entirely deductive) treatment than in earlier works.
Wang, H.; Vlad, M.; Vanden Eijnden, E.; Spineanu, F.; Misguich, J.H.; Balescu, R. [Association Euratom-Etat Belge sur la Fusion, Physique Statistique et Plasmas, Code Postal 231, Universite Libre de Bruxelles, Campus Plaine, Boulevard du Triomphe, 1050 Bruxelles (Belgium)]|[Association Euratom-Commissariat a lEnergie Atomique sur la Fusion, Departement de Recherches sur la Fusion Controle, Centre dEtudes de Cadarache, 13108 Saint-Paul-lez-Durance Cedex (France)
1995-05-01
396
In this paper an attempt is made to use magnetic braking to constrain the surface polar field strengths of the secondary stars in close binaries with orbital periods between Porb ~ 3 and 7 h. By using a standard Mestel & Spruit model, assuming field saturation, a linear relation was obtained between the mass transfer and orbital period, for the period range under consideration, which allows constraining the surface polar field between B?,2 ~ 2900 and 3300 G. It has been shown that the predicted mass-transfer rates correlate with the Verbunt & Zwaan empirical mass transfer versus orbital period relation. Furthermore, it has been shown that the closed field lines, that is, the so-called dead zone, of the secondary star envelopes a substantial fraction of the binary, including the white dwarf, for all orbital periods under consideration. It has been shown that the interaction of the white dwarf field with the surrounding envelope can result in the intermediate polars entering the period gap being magnetically synchronized over time-scales ?syn <= 107yr. This mechanism may then play a significant role in the orbital evolution of the intermediate polars into the polars. Furthermore, surface magnetic field structures in the L1 region may play a significant role in the fragmentation of the mass flow near the L1 region, which may explain the inferred fragmented mass transfer, that is, the blobby mass flow, in magnetic cataclysmic variables.
Meintjes, P. J.; Jurua, E.
2006-11-01
397
The Bates Large Acceptance Spectrometer Toroid has been built to study nuclear physics reactions using a stored, polarized electron beam and a variety of polarized targets internal to the storage ring. The spectrometer consists of eight coils surrounding the target cell. There is a requirement of nominally zero field along the centerline of the spectrometer for proper electron beam storage. In addition, the polarized internal targets require a low field gradient in the target region. Magnetic field measurements were made near the beam centerline to guide the alignment of the coils and satisfy the field magnitude and gradient requirements. After the coils were aligned, the magnetic field was measured in the detector regions to provide information for particle tracking.
Dow, Karen A.; Botto, Tancredi; Goodhue, Abigail; Hasell, Douglas; Loughnan, Dylan; Murphy, Kilian; Smith, Timothy Paul; Ziskin, Vitaliy
2009-02-01
398
Understanding the relationship between the crystal structure and magnetic ordering is crucial for the design of three-dimensional molecule-based magnets with high ordering temperatures. In this talk, we introduce a novel series of molecule-based magnets consisting of transition metal ions (Mn, Fe, Co, Ni or Cu) coordinated with the organic ligand dicyanamide [N(CN)_2]^-.(J.L. Manson et al. al.), Chem. Mater. 10, 2552 (1998); S.R. Batten et al. al., Chem. Commun. (Cambridge) 1998, 439; M. Kurmoo et al. al., New J. Chem. 22, 1515 (1998). The crystal structures for all compounds are isomorphous in the paramagnetic regime as well as in the ordered state. However, the compounds with transition metal ions having six or less electrons in the 3d orbitals order as canted antiferromagnets (AFM) while the ones with seven or more electrons order as ferromagnets (FM). The spin orientation is nearly in perpendicular directions for the AFM versus FM systems.(C.R. Kmety et al. al.), Phys. Rev. B 60, 60 (1999).^,(C.R. Kmety et al. al.), Phys. Rev. B 62, 5576 (2000). An external magnetic field induces a spin rotation transition in the Mn compound and an energy-level crossing for the Fe compound.(C.R. Kmety and A.J. Epstein, National High Magnetic Field Laboratory 2000 Annual Research Review.) The possible origins of the variability of the magnetic structure for the first row transition metal ions compounds will be discussed.
Kmety, Carmen R.
2001-03-01
399
SciTech Connect
The next generation of accelerators for high-energy physics will require high-field, small-bore dipole magnets: in the region of 10 T and 40-mm diam. For such magnets, there is a great incentive to attain high overall current density through increasing the current density within the superconductor and minimizing the copper stabilizer. Both Nb-Ti operating at 1.8 K and Nb/sub 3/sn at 4.2 are candidate superconductors. Two programs in the US and one in Japan are directed toward the development of such magnets. The program at LBL is described below.
Taylor, C.; Meuser, R.; Caspi, S.; Gilbert, W.; Hassenzahl, W.; Peters, C.; Wolgast, R.
1982-05-01
400
Tsunamis produce perturbations in the Earth's magnetic field by electromagnetic induction. Recent deployments of highly accurate magnetometers and the exceptionally deep solar minimum provided ideal conditions to observe these small signals from the tsunami resulting from the strong Chilean earthquake on 27 February 2010. Magnetic observatory measurements on Easter Island, 3500 kilometers west of the epicenter, show a periodic signal of 1 nanotesla, coincident in time with recordings from the local tide gauge. The detection of these magnetic signals represents a milestone in understanding tsunami-induced electromagnetic effects.
Manoj, Chandrasekharan; Maus, Stefan; Chulliat, Arnaud
2011-01-01
401
The emergence of high-energy-product permanent magnets has made possible the generation of extraordinarily high magnetic fields in the internal working spaces of relatively small structures. The widespread use of such structures has been hampered by the variety and complexity of their magnetic components and the concomitant difficulty and expense of manufacture. This paper describes approaches to fabrication and assembly that should significantly ease both fabricational and economic problems. Examples of these approaches are given for the production of cylindrical multipolar sources (magic rings, quadrupolar electron beam guides) and spherical dipolar sources (magic spheres).
Leupold, H. A.; McLane, G. F.
1994-11-01
402
The existence of magnetic monopoles is predicted by many theories of particle physics beyond the Standard Model. However, in spite of extensive searches, there is no experimental or observational sign of them. I review the role of magnetic monopoles in quantum field theory and discuss their implications for particle physics and cosmology. I also highlight their differences and similarities with monopoles found in frustrated magnetic systems, and discuss how experiments carried out in these systems could help us understand the physics of fundamental monopoles.
Rajantie, A.
2012-12-01
403
PubMed
Firefly flashing has been the subject of numerous scientific investigations. Here we present in vivo flashes from male specimens of three species of fireflies-two Japanese species Luciola cruciata, Luciola lateralis and one Indian species Luciola praeusta-positioned under a superconducting magnet. When the OFF state of the firefly becomes long after flashing in an immobile state under the strong static magnetic field of strength 10 Tesla for a long time, which varies widely from species to species as well as from specimen to specimen, the effect of the field becomes noticeable. The flashes in general are more rapid, and occasionally overlap to produce broad compound flashes. We present the broadest flashes recorded to date, and propose that the strong static magnetic field affects the neural activities of fireflies, especially those in the spent up or 'exhausted' condition. PMID:22131061
Barua, Anurup Gohain; Iwasaka, Masakazu; Miyashita, Yuito; Kurita, Satoru; Owada, Norio
2011-12-01
404
SciTech Connect
Fast coronal mass ejecta (CMEs) accelerate and deflect the slower moving solar wind plasma which piles up ahead of them as they propagate out through the heliosphere. This acceleration and deflection, in turn, causes the interplanetary magnetic field (IMF) imbedded in the upstream solar wind to drape about the ejecta. Draping should cause substantial out-of-the-ecliptic magnetic fields at some locations ahead of CMEs, and radial fields behind and along the flanks. At the Earth, draping can be an important factor in the generation of some magnetic storms and substorms, while in the outer heliosphere draping may produce very large magnetotail-like configurations, somewhat analogous to those observed behind Venus and comets. 17 refs.
McComas, D.J.; Gosling, J.T.
1987-01-01
405
If cosmic magnetic fields are indeed produced during inflation, they are likely to be correlated with the scalar metric perturbations that are responsible for the cosmic microwave background anisotropies and large scale structure. Within an archetypical model of inflationary magnetogenesis, we show that there exists a new simple consistency relation for the non-Gaussian cross correlation function of the scalar metric perturbation with two powers of the magnetic field in the squeezed limit where the momentum of the metric perturbation vanishes. We emphasize that such a consistency relation turns out to be extremely useful to test some recent calculations in the literature. Apart from primordial non-Gaussianity induced by the curvature perturbations, such a cross correlation might provide a new observational probe of inflation and can in principle reveal the primordial nature of cosmic magnetic fields.
Jain, Rajeev Kumar; Sloth, Martin S.
2012-12-01
406
Diffusive shock acceleration is the theory of particle acceleration through multiple shock crossings. In order for this process to proceed at a rate that can be reconciled with observations of high-energy electrons in the vicinity of the shock, and for cosmic rays protons to be accelerated to energies up to observed galactic values, significant magnetic field amplification is required. In this review we will discuss various theories on how magnetic field amplification can proceed in the presence of a cosmic ray population. On both short and long length scales, cosmic ray streaming can induce instabilities that act to amplify the magnetic field. Developments in this area that have occurred over the past decade are the main focus of this paper.
Schure, K. M.; Bell, A. R.; O'C Drury, L.; Bykov, A. M.
2012-11-01
407
An analogy is found between Wigner resonant tunneling and tunneling across a static potential barrier in a static magnetic field. Whereas in the process of Wigner tunneling an electron encounters a classically allowed region where a discrete energy level coincides with its energy, in the magnetic field the potential barrier is constant in the direction of tunneling. Along the tunneling path, certain regions are formed where, in the classical language, the kinetic energy of the motion perpendicular to tunneling is negative. These regions play the role of potential wells, where a discrete energy level can coincide with the electron energy. This phenomenon, which occurs at a certain magnetic field, is called Euclidean resonance and substantially depends on the shape of the potential forces in the direction perpendicular to tunneling. Under conditions of Euclidean resonance, a long-distance underbarrier motion is possible, which can be observed in experiments.
Ivlev, B.
2007-08-01
408
Numerical simulations of stellar dynamos are reviewed. Dynamic dynamo models solve the nonlinear, three-dimensional, time-dependent, magnetohydrodynamic equations for the convective velocity, the thermodynamic variables, and the generated magnetic field in a rotating, spherical shell of ionized gas. When the dynamo operates in the convection zone, the simulated magnetic fields propagate away from the equator in the opposite direction inferred from the solar butterfly diagram. When simulated at the base of the convection zone, the fields propagate in the right direction at roughly the right speed. However, owing to the numerical difficulty, a full magnetic cycle has not been simulated in this region. As a result, it is still uncertain where and how the solar dynamo operates.
Glatzmaier, Gary A.
1985-06-01
409
The rapidly rotating giant planets of the outer solar system all possess strong dynamo-driven magnetic fields that carve a large cavity in the flowing magnetized solar wind. Each planet brings a unique facet to the study of planetary magnetism. Jupiter possesses the largest planetary magnetic moment, 1.551020 Tm3, 2104 times larger than the terrestrial magnetic moment whose axis of symmetry is offset about 10 from the rotation axis, a tilt angle very similar to that of the Earth. Saturn has a dipole magnetic moment of 4.61018 Tm3 or 600 times that of the Earth, but unlike the Earth and Jupiter, the tilt of this magnetic moment is less than 1 to the rotation axis. The other two gas giants, Uranus and Neptune, have unusual magnetic fields as well, not only because of their tilts but also because of the harmonic content of their internal fields. Uranus has two anomalous tilts, of its rotation axis and of its dipole axis. Unlike the other planets, the rotation axis of Uranus is tilted 97.5 to the normal to its orbital plane. Its magnetic dipole moment of 3.91017 Tm3 is about 50 times the terrestrial moment with a tilt angle of close to 60 to the rotation axis of the planet. In contrast, Neptune with a more normal obliquity has a magnetic moment of 2.21017 Tm3 or slightly over 25 times the terrestrial moment. The tilt angle of this moment is 47, smaller than that of Uranus but much larger than those of the Earth, Jupiter and Saturn. These two planets have such high harmonic content in their fields that the single flyby of Voyager was unable to resolve the higher degree coefficients accurately. The four gas giants have no apparent surface features that reflect the motion of the deep interior, so the magnetic field has been used to attempt to provide this information. This approach works very well at Jupiter where there is a significant tilt of the dipole and a long baseline of magnetic field measurements (Pioneer 10 to Galileo). The rotation rate is 870.536 per day corresponding to a (System III) period of 9 h 55 min 26.704 s. At Saturn, it has been much more difficult to determine the equivalent rotation period. The most probable rotation period of the interior is close to 10 h 33 min, but at this writing, the number is still uncertain. For Uranus and Neptune, the magnetic field is better suited for the determination of the planetary rotation period but the baseline is too short. While it is possible that the smaller planetary bodies of the outer solar system, too, have magnetic fields or once had, but the current missions to Vesta, Ceres and Pluto do not include magnetic measurements.
Russell, C. T.; Dougherty, M. K.
2010-05-01
410
SciTech Connect
This paper provides a comparative study of the geometrical structures of the magnetic fields of Earth, Jupiter, Saturn, and Uranus, starting from the traditional multipolar representations of these fields. For Earth, Jupiter, and Saturn the centered dipole, quadrupole, and octupole contributions are included, while at Uranus, only the dipole and quadrupole contributoins are considered. The magnetic fields are analyzed by decomposing them into those parts which have simple symmetry properties with respect to the rotation axis and the equatorial plane. It is found that there are a number of common features of the magnetic fields of Earth and Jupiter. Compared to Earth and Jupiter, the Saturnian field exhibits not only a high degree of symmetry about the rotation axis, by now rather well known, but also a high degree of antisymmetry about the equatorial plane. The Uranian field shows strong deviations from both such symmetries. Nevertheless, there remain features common to all four planets. The implications of these results for dynamo models are discussed. With a vgiew to Cowling's theorem the symmetry of the fields is investigated with respect to not only the rotation axis but also to other axes intersecting the plaentary center. Surprisingly, the high degree of asymmetry of the Uranian field that is observed with respect to the rotation axis reduces considerably to being compare to that for Earth or Jupiter when the appropriate axis is employed.
Raedler, K.H. (Zentral Inst. fuer Astrophysik, Potsdam-Babelsberg (East Germany)); Ness, N.F. (Univ. of Delaware, Newark (United States))
1990-03-01
411
The observed propagation of cosmic rays in the interplanetary space cannot be explained unless there is diffusion of the energetic particles across the interplanetary magnetic field. The cross-field diffusion of cosmic rays is assumed to be due to the chaotic nature of the interplanetary\\/intergalactic magnetic fields. Among the classic works on this subject have been those of Parker [1] and
B. Dasgupta; A. Ram
2009-01-01
412
Solar rotation is known for more than four centuries, yet it is still an unresolved issue of solar physics. The rotation has been measured by three methods e.g. Feature tracing, Spectroscopic and flux modulation. We have used the last quite extensively for the estimation of rotation of the solar outer atmosphere, namely chromosphere and corona. These studies used Radio, X-ray and EUV images of the Sun. These investigations have provided quite detailed and systematic information of the solar rotation and its variability. This has established that solar atmosphere has differential rotation and the differentiality has a significant temporal and spatial variability. The spatial variability show that there is North-South asymmetry in the rotation of solar atmosphere. This asymmetry changes sign every solar cycle. This may be related to "Hale cycle". In this paper we tried to combine the Radio, X-ray and EUV estimates of rotation by comparative interpolation. The combined rotation profiles are drawn in the space-time coordinate in the same format as the longitudinal averaged magnetic field. The average magnetic field shows beautiful butterfly structures and poleward flow of the magnetic fields of opposite polarity. Preliminary comparison show that asymmetric differential rotation of the solar atmosphere peaks when the magnetic filed structure is more complex. In this paper a detail comparison of the magnetic field and solar atmospheric rotation will presented.
Chandra, Satish; Vats, Hari
2012-07-01
413
JAXA (Japan Aerospace Exploration Agency) shall launch the SELENE (SELenological and ENgineering Explorer) spacecraft this autumn. Amongst many instruments, it has a magnetometer (LMAG: Lunar MAGnetomter) which will measure the magnetic field on the orbit around the Moon. The nominal orbit of the SELENE is about 100km in altitudes for 1 year observation. Although the extended mission is still not determined, LMAG team is requesting a low altitude (less than 50km) observation, if the remaining fuel allows. We are preparing data processing software for the mission. Here, we report an objective scheme for mapping the lunar crustal magnetic field from the orbital measurement data of unequal altitudes. In this study, the magnetic field is restored by solving a linear inverse-problem determining the sources distributed on the lunar surface to satisfy the observational data, which is known as the equivalent source method. Our scheme has three features improving the method: First, the source calculation is performed simultaneously with detrending. Second, magnetic charges (magnetic monopoles) are used as the equivalent sources. It reduces the density of the sources for the same smoothness in produced field, comparing to the dipole sauces. Third, the number of sources is taken large enough to avoid the problem of configuration of the sources, instead the damped least square assuming the strength of each charge is similar to the next one, and the smoothness factor is determined by minimizing Akaike's Bayesian Information Criterion (ABIC). It guarantees the objectivity of the calculation, in other words, there is no adjustable parameter which may depend of the researcher dealing the data analyses. For testing the scheme, we apply this method to the Lunar Prospector magnetometer data, and provide magnetic field map in the region centered at several regions of strong crustal field including the Reiner Gamma anomaly. The stability of the method and the resolution of the anomaly map are found to be satisfactory.
2007-12-01
414
We present a magnetic-field amplification process in galaxies in conjunction with bar dynamics. Our model considers especially the observed non-circular gas velocities in barred spiral galaxies. The bar drives the spirally-falling gas flow toward the center, which consists of a net radial flow (referred to as flow b) and an elliptically elongated flow rotating in the azimuth (flow a). The induced radial flow by a bar (flow b) produces a magnetic field, whose exponential growth is closely related with the angular-momentum transport by the non-axisymmetric bar perturbation. Furthermore, the non-axisymmetric gas flow (flow a) also leads to the exponential and oscillatory growth of magnetic fields by driving a growing magnetic wave. The interplay of both flows in a bar hence induces an oscillatory amplification of magnetic fields, and the resulting magnetic field pattern rotates with a bar and holds the azimuthal wavenumber m = 1 or 2, depending on the strength of velocity disturbances. This model naturally explains the characteristic radio features observed in M83, where the m = 1 magnetic field is aligned with the bar, and the bar ends are dominated by the vertical component Bz, giving the holes in polarized intensity map. It is emphasized that the evolution of galactic magnetic fields is closely related with galactic dynamics and evolution.
Chiba, M.; Lesch, H.
1994-04-01
415
The Spiral Magnetic Motor, which can accelerate a magnetized rotor through 90% of its cycle with only permanent magnets, was an energy milestone for the 20th century patents by Kure Tekkosho in the 1970's. However, the Japanese company used old ferrite magnets which are relatively weak and an electrically-powered coil to jump start every cycle, which defeated the primary benefit of the permanent magnet motor design. The principle of applying an inhomogeneous, anisotropic magnetic field gradient force Fz = ? cos J dB/dz, with permanent magnets is well-known in physics, e.g., Stern-Gerlach experiment, which exploits the interaction of a magnetic moment with the aligned electron spins of magnetic domains. In this case, it is applied to dB/d? in polar coordinates, where the force F? depends equally on the magnetic moment, the cosine of the angle between the magnetic moment and the field gradient. The radial magnetic field increases in strength (in the attractive mode) or decreases in strength (in the repulsive mode) as the rotor turns through one complete cycle. An electromagnetic pulsed switching has been historically used to help the rotor traverse the gap (detent) between the end of the magnetic stator arc and the beginning (Kure Tekko, 1980). However, alternative magnetic pulse and switching designs have been developed, as well as strategic eddy current creation. This work focuses on the switching mechanism, novel magnetic pulse methods and advantageous angular momentum improvements. For example, a collaborative effort has begun with Toshiyuki Ueno (University of Tokyo) who has invented an extremely low power, combination magnetostrictive-piezoelectric (MS-PZT) device for generating low frequency magnetic fields and consumes zero power'' for static magnetic field production (Ueno, 2004 and 2007a). Utilizing a pickup coil such as an ultra-miniature millihenry inductor with a piezoelectric actuator or simply Wiegand wire geometry, it is shown that the necessary power for magnetic field switching device can be achieved in order to deflect the rotor magnet in transit. The Wiegand effect itself (bistable FeCoV wire called Vicalloy'') invented by John Wiegand (Switchable Magnetic Device, US Patent No.4,247,601), utilizing Barkhausen jumps of magnetic domains, is also applied for a similar achievement (Dilatush, 1977). Conventional approaches for spiral magnetic gradient force production have not been adequate for magnetostatic motors to perform useful work. It is proposed that integrating a magnetic force control device with a spiral stator inhomogeneous axial magnetic field motor is a viable approach to add a sufficient nonlinear boundary shift to apply the angular momentum and potential energy gained in 315 degrees of the motor cycle.
Valone, Thomas F.
2010-01-01
416
The transport of magnetic field lines is studied numerically in the case where strong three-dimensional magnetic fluctuations are superimposed to a uniform average magnetic field. The magnetic percolation of field lines between magnetic islands is found, as well as a non-Gaussian regime where the field lines exhibit Lvy random walks, changing from Lvy flights to trapped motion. Anomalous diffusion laws
G. Zimbardo; P. Veltri
1995-01-01
417
In this paper, we summarize our present understanding of Mars' atmosphere, magnetic field, and surface and address past evolution of these features. Key scientific questions concerning Mars' surface, atmosphere, and magnetic field, along with the planet's interaction with solar wind, are discussed. We also define what key parameters and measurements should be performed and the main characteristics of a martian
F. Leblanc; B. Langlais; T. Fouchet; S. Barabash; D. Breuer; E. Chassefire; A. Coates; V. Dehant; F. Forget; H. Lammer; S. Lewis; M. Lopez-Valverde; M. Mandea; M. Menvielle; A. Pais; M. Paetzold; P. Read; C. Sotin; P. Tarits; S. Vennerstrom
2009-01-01
418
We measured the high-field magnetization M for skutterudite compounds of a paramagnet PrFe4Sb12 and a ferromagnet SmOs4Sb12 to investigate the characteristic magnetic properties. In PrFe4Sb12, we observed a peak structure in the dM/dH curve, which might be a level crossing effect between the singlet ground state and the excited state with a splitting energy of about 20 K in the crystalline electric field (CEF) scheme. In SmOs4Sb12, the anisotropy of magnetization for H?[100] and [1 1 0] was not observed, which is inconsistent with the quartet ground state in the CEF scheme.
Yamada, T.; Nakashima, H.; Sugiyama, K.; Hagiwara, M.; Kindo, K.; Tanaka, K.; Kikuchi, D.; Aoki, Y.; Sugawara, H.; Sato, H.; Settai, R.; ?nuki, Y.; Harima, H.
2007-03-01
419
National Technical Information Service (NTIS)
The magnetosphere is an intrinsic component of the Earth environment, and travel beyond this sphere will expose man to near absence of a magnetic field. The present study is a continuation of a previous investigation of the physiological and psychological...
D. E. Beischer E. F. Miller J. C. Knepton
1967-01-01
420
The results of studies of magnetization processes of multilayer structures, consisting of periodically alternating island layers of various magnetic materials, are presented. The unidirectional axis of magnetization, which does not lead to exchange bias of hysteresis loops, is found in these structures. A vortex-like type of magnetization of island structures, when the vortex magnetization is distributed on set of nanoislands, is proposed. Preliminary simulations and experiments on the effects of vortex magnetic field on island systems have shown that proposed vortex-like state can be implemented in multilayer island systems and can influence their magnetic structure.
Boltaev, A. P.; Pudonin, F. A.; Sherstnev, I. A.
2013-04-01
421
Recently, a Bayesian approach has been proposed for evaluating magnetic field fluctuation thermometry measurements. The approach provides a coherent use of calibration results in the inference of the temperature from subsequent magnetic field fluctuation thermometry measurements. It does, however, rely on an extensive numerical effort. In this paper, we develop simplified parametric and non-parametric analysis schemes. For all approaches we derive analytic expressions for the resulting temperature estimates and their uncertainties. We assess the new approaches and show in particular that an easy to apply non-parametric analysis yields results which are in good agreement with those obtained by the Bayesian inference.
Wbbeler, G.; Elster, C.
2013-11-01
422
Far field imaging of subwavelength magnetic objects in real time is a very challenging issue. We propose an original solution based on a planar array of closely spaced split ring resonators. Hybridization between the resonators of such metalens induces subwavelength modes with different frequencies. Thanks to these high Q resonating modes, Purcell like effect allows an evanescent source, close to the metalens, to emit waves that can be collected efficiently in the far field. We present the first microwave experimental demonstration of such metalens to image of a subwavelength magnetic pattern. Numerical simulation shows that this approach is still valid at THz frequencies.
Ourir, Abdelwaheb; Lerosey, Geoffroy; Lemoult, Fabrice; Fink, Mathias; de Rosny, Julien
2012-09-01
423
We consider the effect of a random magnetic field in the convective zone of the Sun superimposed to a regular magnetic field on resonant neutrino spin-flavor oscillations. We argue for the existence of a field of strongly chaotic nature at the bottom of the convective zone. In contrast to previous attempts we employ a model motivated regular magnetic field profile: it is a static field solution to the solar equilibrium hydro-magnetic equations. These solutions have been known for a long time in the literature. We show for the first time that in addition they are twisting solutions. In this scenario electron antineutrinos are produced through cascades like ?eL-->??L-- >?~eR, The detection of ?~eR at Earth would be a long-awaited signature of the Majorana nature of the neutrino. The expected signals in the different experiments (SK, GALLEX-SAGE, Homestake) are obtained as a function of the level of noise, regular magnetic field and neutrino mixing parameters. Previous results obtained for small mixing and ad-hoc regular magnetic profiles are reobtained. We confirm the strong suppression for a large part of the parameter space of the ?~eR-flux for high energy boron neutrinos in agreement with present data of the SK experiment. We find that MSW (Mikheyev-Smirnov-Wolfenstein) regions (?m2~=10-5 eV2, both small and large mixing solutions) are stable up to very large levels of noise (P=0.7-0.8) but they are acceptable from the point of view of antineutrino production only for moderate levels of noise (P~=0.95). For strong noise and a reasonable regular magnetic field, any parameter region (?m2, sin 2 2?) is excluded. As a consequence, we are allowed to reverse the problem and to put limits on the r.m.s. field strength and transition magnetic moments by demanding a particle physics solution to the SNP in this scenario.
Semikoz, V. B.; Torrente-Lujan, E.
1999-09-01
424
SciTech Connect
The Sweet-Parker and Petschek scalings of magnetic reconnection rate are modified to include the effect of the viscosity. The modified scalings show that the viscous effect can be important in high-..beta.. plasmas. The theoretical reconnection scalings are compared with numerical simulation results in a tokamak geometry for three different cases: a forced reconnection driven by external coils, the nonlinear m = 1 resistive internal kink, and the nonlinear m = 2 tearing mode. In the first two cases, the numerical reconnection rate agrees well with the modified Sweet-Parker scaling, when the viscosity is sufficiently large. When the viscosity is negligible, a steady state which was assumed in the derivation of the reconnection scalings is not reached and the current sheet in the reconnection layer either remains stable through sloshing motions of the plasma or breaks up to higher m modes. When the current sheet remains stable, a rough comparison with the Sweet-Parker scaling is obtained. In the nonlinear m = 2 tearing mode case where the instability is purely resistive, the reconnection occurs on the slower dissipation time scale (Psi/sub s/ approx. eta). In addition, experimental data of the nonlinear m = 1 resistive internal kink in tokamak discharges are analyzed and are found to give reasonable agreement with the modified Sweet-Parker scaling.
Park, W.; Monticello, D.A.; White, R.B.
1983-05-01
425
The National High Magnetic Field Laboratory (NHMFL) is a collaboration between Florida State University, the University of Florida, and the Los Alamos National Laboratory. The DC Field Facilities are located at the main campus for the NHMFL in Tallahassee, Florida and are described in this paper. The DC Field Facility has a variety of resistive and superconducting magnets. The DC Field Facility infrastructure, the most powerful in the world, is able to provide 57 MW of continuous low noise DC power. Constant magnetic fields of up to 45 tesla in a 32 mm bore and 20 tesla in 195 mm bore are available at no charge to the user community. The users of the facility are selected by a peer reviewed process. Roughly 400 research groups visit the lab to conduct experiments each year. Experimental capabilities provided by the NHMFL are magneto-optics, millimeter wave spectroscopy, magnetization, dilatometry, specific heat, electrical transport, ultrasound, low to medium resolution NMR, EMR, and materials processing. Measurements of properties can be made on samples at temperatures from 20 mK to 1000 K, pressures from ambient to 10 GPa, orientation and currents from 1 pA to 10 kA.
Hannahs, S. T.; Palm, E. C.
2010-04-01
426
PubMed
We present a theory and numerical evidence for the existence of a previously unexplored in-plane electric field in collisionless asymmetric magnetic reconnection. This electric field, dubbed the "Larmor electric field," is associated with finite Larmor radius effects and is distinct from the known Hall electric field. Potentially, it could be an important indicator for the upcoming Magnetospheric Multiscale mission to locate reconnection sites as we expect it to appear on the magnetospheric side, pointing earthward, at the dayside magnetopause reconnection site. PMID:24116786
Malakit, K; Shay, M A; Cassak, P A; Ruffolo, D
2013-09-23
427
High-resolution observations of the full Stokes vector in Fe\\sc i spectral lines around 5250 Angstroms obtained at the Swedish Vacuum Solar Telescope on La Palma with the ZIMPOL I Stokes polarimeter in a complex active region reveal the presence of anomalously shaped Stokes profiles indicating the coexistence of at least two magnetic components within the same spatial resolution element. These Stokes profiles have been analyzed with an inversion code based on a 3-component atmospheric model with two magnetic and one field-free component. The fits to the observations in a magnetic region that resembles a small penumbra reveal the presence of a horizontal magnetic field component with an average field strength of /line{B}=840 G, a mean filling factor of /line?=0.12, and an average temperature /line{T}=5400 K at log {tau_ {5000}}=-1.5 embedded in the main penumbral'' magnetic field that has /line{B}=1500 G, /line?=0.56, and /line{T}=4900 K. The horizontal component exhibits a mean outflow of 2.7 km s(-1) which is mainly due to the Evershed flow. In a region where there are strong downflows up to 7 km s(-1) , we infer the possible presence of a shock front whose height changes along the slit. The height variation can be explained by a change of the gas pressure at the base of the photosphere below the shock front as proposed by Thomas & Montesinos (1991). Small plages with field strengths below 900 G have been observed in the vicinity of some pores. Finally, we present a puzzling field structure at the boundary between two adjacent pores. Ambiguous results suggest that although the inversion code is able to successfully invert even very complex Stokes profiles, we are far from a complete description of the field structure in complex magnetic regions. We warn that magnetograms and fits to data involving only a single magnetic component may hide the true complexity of the magnetic structure in at least some parts of active regions.
Bernasconi, P. N.; Keller, C. U.; Solanki, S. K.; Stenflo, J. O.
1998-01-01
428
The small-scale dynamo provides a highly efficient mechanism for the conversion of turbulent into magnetic energy. In astrophysical environments, such turbulence often occurs at high Mach numbers, implying steep slopes in the turbulent spectra. It is thus a central question whether the small-scale dynamo can amplify magnetic fields in the interstellar or intergalactic media, where such Mach numbers occur. To address this long-standing issue, we employ the Kazantsev model for turbulent magnetic field amplification, systematically exploring the effect of different turbulent slopes, as expected for Kolmogorov, Burgers, the Larson laws and results derived from numerical simulations. With the framework employed here, we give the first solution encompassing the complete range of magnetic Prandtl numbers, including Pm ? 1, Pm 1 and Pm ? 1. We derive scaling laws of the growth rate as a function of hydrodynamic and magnetic Reynolds number for Pm ? 1 and Pm ? 1 for all types of turbulence. A central result concerns the regime of Pm 1, where the magnetic field amplification rate increases rapidly as a function of Pm. This phenomenon occurs for all types of turbulence we have explored. We further find that the dynamo growth rate can be decreased by a few orders of magnitude for turbulence spectra steeper than Kolmogorov. We calculate the critical magnetic Reynolds number Rmc for magnetic field amplification, which is highest for the Burgers case. As expected, our calculation shows a linear behaviour of the amplification rate close to the threshold proportional to (Rm - Rmc). On the basis of the Kazantsev model, we therefore expect the existence of the small-scale dynamo for a given value of Pm as long as the magnetic Reynolds number is above the critical threshold.
Bovino, S.; Schleicher, D. R. G.; Schober, J.
2013-01-01
429
SciTech Connect
An improved magnetic beam extraction channel for the Oak Ridge Isochronous Cyclotron (ORIC) has been designed to significantly reduce the external field disturbance and provide uniform in-channel field. This will make beam extraction near nu/sub r/ = 1 more predictable. The new channel consists of an iron tube of constant cross section with independently adjustable windings, both inside and outside the iron. The windings have a cos theta current density distribution. The iron tube is 1 meter long with a bore of 6 cm; aperture for beam is 4 cm. The external field is negligible except for small perturbations in the field arising from the geometry modifications required at the ends so that the beam can enter and leave. The field reduction inside the channel is variable from 0.4 to 0.6 Tesla without significant change in either the internal field uniformity or the external field level. 4 refs., 8 figs., 1 tab.
Hudson, E.D.; Martin, J.A.; Lord, R.S.
1985-01-01
430
The Laboratoire National des Champs Magntiques Intenses (LNCMI) is a host laboratory for experiments in high magnetic fields. It was created on the 1st of January 2009 through the merger of the Laboratoire des Champs Magnetiques Intenses (Grenoble, specialized in DC fields) and the Laboratoire National des Champs Magntiques Pulss (Toulouse, specialized in pulsed fields). The facility is open to a large community of users from all over Europe and the rest of the world. In this paper we report our efforts to offer the highest magnetic fields, ranging from 35 T DC, through 80 T non-destructive, up to 170 T semi-destructive, in the best conditions for our in-house and visiting scientists. We describe the installations and coils improvements. As an example of our activity we present some recent scientific results.
Bard, J.; Debray, F.
2013-03-01
431
The study of the cooling of neutron stars has been undertaken by many researchers in the past twenty-five years, but this study has been made difficult by the inherent theoretical and observational uncertainties; most observations of their thermal X-ray flux have yielded only upper limits. More sensitive satellites such as ROSAT and AXAF may provide more positive flux information, and it is important to know how to interpret these data in terms of surface temperature. One of the most important factors in this interpretation is the effect of the surface magnetic field.Young neutron stars are believed to have extremely strong magnetic fields, on the order of 10(12)G. These fields dominate the physics of the atmosphere. In particular, atoms in the atmospheres of neutron stars have much greater binding energies than in the zero-field case, and they are constrained to move along the field lines. We use a multiconfigurational Hartree-Fock code, modified for very strong magnetic fields, to calculate wavefunctions, energies and oscillator strengths for several atoms in representative values of the magnetic field.We then use these simulations to construct model atmospheres for neutron stars. Because of the low mass necessary for optical depth unity in the soft X-rays (typically [...]) and because of the short time scale for gravitational separation (~ 1 - 100s), the photosphere is likely to consist of a pure element. Numerous processes could cause many elements to be important, so we investigate atmospheres consisting of pure hydrogen, helium, carbon, nitrogen and silicon in magnetic fields of 9.4 x 10(11)G, 2.35 x 10(12)G, and 4.7 x 10(12)G.We also use the high-field energies to investigate soft X-ray lines in gamma-ray bursts. Highly ionized elements could create absorption lines in the 1-15keV range, and the identification of such lines in conjunction with cyclotron lines would determine the magnetic field and gravitational redshift on the surface of the star, which would provide clues to the equation of state on the interior. We conclude with a discussion of the prospect of identifying these lines with future satellites.
Miller, Michael Coleman
432
The effects of the Earth's magnetic field on the performance of large PMTs for a cubic-kilometer-scale neutrino telescope has been studied. Measurements were performed for three Hamamatsu PMTs: two 8? R5912 types; one with a standard and the other with a super bialkali photocathode, and a 10? R7081 type with a standard bialkali photocathode. The main characteristics of the PMTs, such as detection efficiency, transit time, transit time spread, gain, peak-to-valley ratio, charge resolution and fractions of spurious pulses were measured while varying the PMT orientations with respect to the Earth's magnetic field. The measurements were performed both with and without a mu-metal cage magnetic shielding. For the 8? PMTs the impact of the magnetic field was found to be smaller than for the 10? PMT. The magnetic shielding strongly reduced the orientation-dependent variations measured for the 10? PMT and even improved the performance. Although less pronounced, improvements were also measured for the 8? PMTs.
on behalf of the KM3NeT Consortium; Leonora, E.
2013-10-01
433
SciTech Connect
We extend our investigation of magnetic field evolution in three-dimensional flows driven by the stationary accretion shock instability (SASI) with a suite of higher-resolution idealized models of the post-bounce core-collapse supernova environment. Our magnetohydrodynamic simulations vary in initial magnetic field strength, rotation rate, and grid resolution. Vigorous SASI-driven turbulence inside the shock amplifies magnetic fields exponentially; but while the amplified fields reduce the kinetic energy of small-scale flows, they do not seem to affect the global shock dynamics. The growth rate and final magnitude of the magnetic energy are very sensitive to grid resolution, and both are underestimated by the simulations. Nevertheless our simulations suggest that neutron star magnetic fields exceeding $10^{14}$~G can result from dynamics driven by the SASI, \\emph{even for non-rotating progenitors}.
Endeve, Eirik [ORNL; Cardall, Christian Y [ORNL; Budiardja, Reuben D [ORNL; Beck, Sam [University of Tennessee, Knoxville (UTK); Bejnood, Alborz [ORNL; Toedte, Ross J [ORNL; Blondin, J. M. [North Carolina State University; Mezzacappa, Anthony [ORNL
2012-01-01
434
SciTech Connect
We extend our investigation of magnetic field evolution in three-dimensional flows driven by the stationary accretion shock instability (SASI) with a suite of higher-resolution idealized models of the post-bounce core-collapse supernova environment. Our magnetohydrodynamic simulations vary in initial magnetic field strength, rotation rate, and grid resolution. Vigorous SASI-driven turbulence inside the shock amplifies magnetic fields exponentially; but while the amplified fields reduce the kinetic energy of small-scale flows, they do not seem to affect the global shock dynamics. The growth rate and final magnitude of the magnetic energy are very sensitive to grid resolution, and both are underestimated by the simulations. Nevertheless, our simulations suggest that neutron star magnetic fields exceeding 10{sup 14} G can result from dynamics driven by the SASI, even for non-rotating progenitors.
Endeve, Eirik; Mezzacappa, Anthony [Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6354 (United States); Cardall, Christian Y.; Budiardja, Reuben D. [Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6354 (United States); Beck, Samuel W.; Bejnood, Alborz [Joint Institute for Heavy Ion Research, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6374 (United States); Toedte, Ross J. [National Center for Computational Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6354 (United States); Blondin, John M., E-mail: [email protected] [Physics Department, North Carolina State University, Raleigh, NC 27695-8202 (United States)
2012-05-20
435
In this paper, a system that can produce alterable and rotary permanent magnetic field is developed. It can be used for crystal growth, chemical crystal and biologic cell, etc. A homogeneous magnetic field can be produced in the central region by two annular permanent magnets, and the density of magnetic flux is controlled by changing the angle of the magnetic
Qingxin Yang; Fugui Liu; Zhigang Zhao; Duyan Geng; Shunxin Zhang; Weili Yan
2010-01-01
436
Increasing external magnetic field B gradually forces the electron spins to align in the direction of the applied field. The Hartree solution becomes exact for B>=Bs(U). A new small parameter ?B=Bs-B enables one to control the transition between weak- and strong-coupling regimes and the metal-insulator transition. Necessity for dynamical vertex corrections at intermediate and strong coupling is demonstrated.
Jani , V.
1999-01-01
437
SciTech Connect
The influence of a moderately intense static magnetic field on movement patterns of free swimming Paramecium was studied. When exposed to fields of 0.126 T, these ciliated protozoa exhibited significant reduction in velocity as well as a disorganization of movement pattern. It is suggested that these findings may be explained on the basis of alteration in function of ion specific channels within the cell membrane.
Rosen, M.F.; Rosen, A.D. (State Univ. of New York, Stony Brook (USA))
1990-01-01
438
PubMed
Neutral atomic Bose condensates and degenerate Fermi gases have been used to realize important many-body phenomena in their most simple and essential forms, without many of the complexities usually associated with material systems. However, the charge neutrality of these systems presents an apparent limitation-a wide range of intriguing phenomena arise from the Lorentz force for charged particles in a magnetic field, such as the fractional quantum Hall effect in two-dimensional electron systems. The limitation can be circumvented by exploiting the equivalence of the Lorentz force and the Coriolis force to create synthetic magnetic fields in rotating neutral systems. This was demonstrated by the appearance of quantized vortices in pioneering experiments on rotating quantum gases, a hallmark of superfluids or superconductors in a magnetic field. However, because of technical issues limiting the maximum rotation velocity, the metastable nature of the rotating state and the difficulty of applying stable rotating optical lattices, rotational approaches are not able to reach the large fields required for quantum Hall physics. Here we experimentally realize an optically synthesized magnetic field for ultracold neutral atoms, which is evident from the appearance of vortices in our Bose-Einstein condensate. Our approach uses a spatially dependent optical coupling between internal states of the atoms, yielding a Berry's phase sufficient to create large synthetic magnetic fields, and is not subject to the limitations of rotating systems. With a suitable lattice configuration, it should be possible to reach the quantum Hall regime, potentially enabling studies of topological quantum computation. PMID:19956256
Lin, Y-J; Compton, R L; Jimnez-Garca, K; Porto, J V; Spielman, I B
2009-12-01
439 | {"extraction_info": {"found_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": 0.8220924139022827, "perplexity": 1957.3813267749379}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00658-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.sparrho.com/item/anatomy-of-a-deformed-symmetry-field-quantization-on-curved-momentum-space/894c27/ | # Anatomy of a deformed symmetry: field quantization on curved momentum space
Research paper by Michele Arzano
Indexed on: 13 Jan '11Published on: 13 Jan '11Published in: High Energy Physics - Theory
#### Abstract
In certain scenarios of deformed relativistic symmetries relevant for non-commutative field theories particles exhibit a momentum space described by a non-abelian group manifold. Starting with a formulation of phase space for such particles which allows for a generalization to include group valued momenta we discuss quantization of the corresponding field theory. Focusing on the particular case of kappa-deformed phase space we construct the one-particle Hilbert space and show how curvature in momentum space leads to an ambiguity in the quantization procedure reminiscent of the ambiguities one finds when quantizing fields in curved space-times. The tools gathered in the discussion on quantization allow for a clear definition of the basic deformed field mode operators and two-point function for kappa-quantum fields. | {"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": 0.8822091817855835, "perplexity": 1074.4664427415378}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703515075.32/warc/CC-MAIN-20210118154332-20210118184332-00299.warc.gz"} |
https://cgi.math.princeton.edu/compudocwiki/index.php?title=HowTos:Add_TeX_to_your_webpage | This HowTo provides basic instructions on how to add TeX based formulas to your webpages located on the math webserver by using MathJax.
## Introduction
The main math webserver as well as the cgi webserver have the MathJax package installed on them and you can very easily use it to add TeX based formulas/text to your webpages hosted on math webservers.
MathJax is a javascript based software that can interpret TeX/LaTeX formulas embedded in your webpage and replace them with fonts and images to make them look as close as possible to the TeX/LaTeX output. You can find extensive information about MathJax on MathJax homepage. Here we will just suggest a few quick ways to use it, for more extensive information check jsMath website.
## Quick Start
To get started insert the following html code somewhere in the <head> section of your webpage:
<SCRIPT SRC="/mathjax/MathJax.js">
MathJax.Hub.Config({
extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js"],
jax: ["input/TeX","output/HTML-CSS"],
tex2jax: {inlineMath: [["$","$"],["\$","\$"]], processEscapes: true}
});
</SCRIPT>
This code will make sure MathJax is loaded if and only if you use LaTeX style formulas somewhere in the body of your document. That means that the following text:
$f(\alpha) = x+\beta$
will get translated into inline formula as in: $f(\alpha)=x+\beta$ - note how there is a small delay before the text gets converted into formulas. For displayed equations you can do:
$\int_alpha^\beta x = \mathbb{A}$
which gets translated like this: $\int_\alpha^\beta x = \mathbb{A}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "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": 0.8787482976913452, "perplexity": 2130.8014790897178}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964361064.58/warc/CC-MAIN-20211201234046-20211202024046-00638.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/85835-confidence-interval-problem.html | # Math Help - Confidence interval problem
1. ## Confidence interval problem
2. This is off in two ways.
First of all this is a confidence interval.
It's NOT a probability interval.
The person who asked that has it confused.
What they are trying to ask is a binomial problem.
Second error, I think you want plus and minus 1.96, or maybe not?
If you meant 1.96 then the coverage is 95 percent and they
want to ask a binomial question where you want at least 4 successes in 5 trials, with 'p'=.95.
But you can look up the probability that a standard normal is between -.96 and 1.06, too.
However this is really not a probability statement, that's why we coin the word confidence.
NOW a prediction interval is a probability statement since it is guessing at a rv. | {"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": 0.8631953001022339, "perplexity": 920.2368335672088}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1464049276780.5/warc/CC-MAIN-20160524002116-00205-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/two-questions-about-thermal-radiation-heat-transfer.667368/ | 1. Jan 27, 2013
### exergetic
Hi guys,
could you concretely explain me (also with a simple example) the difference between blackbody emissive power (sometimes found as e'λb) and blackbody radiation intensity ($i^{'}_{λb}$)? and which the difference between a diffuse surface and a surface that follows the Lambert law?
Thank you in advance. Tell me if something isn't clear.
2. Jan 28, 2013
### exergetic
Have I posted it in the wrong category?
3. Jan 28, 2013
### Staff: Mentor
No, this category is fine. Just remember that the more specific a question the more difficult it is to find someone who is knowledgeable in that area. I'm sure someone will be able to answer your question soon. Until then, have you searched google or wiki yourself? Often times you can find answers to your own questions or at least gain a little better understanding of the subject. | {"extraction_info": {"found_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": 0.869962215423584, "perplexity": 943.8828366528687}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1505818690016.68/warc/CC-MAIN-20170924115333-20170924135333-00296.warc.gz"} |
http://math.stackexchange.com/questions/90632/use-standard-deviation-to-find-percentage-above-a-certain-point | # Use Standard Deviation to find Percentage above a certain point
OK. Sorry if this is a stupid question, I'm just a little rusty on my statistics.
I have the mean and standard deviation of a data set. The data is a normal with possibilities $\{1,2,3,4,5\}$ with $n \approx 40$.
I am looking to find what percentage of people answered 4 or 5.
How would I compute that number?
Thanks!
-
Do you know anything else? In particular, the distribution of the results? You can't actually answer that question without it... – David Z Dec 12 '11 at 0:28
@DavidZaslavsky Normal – Nick Dec 12 '11 at 0:29
@mixedmath it's normal. I'm sorry for not knowing correct terminology – Nick Dec 12 '11 at 0:36
@Nick: That's fine, but perhaps you can tell me more. What does it mean for a (discrete) set to have a normal distribution? How many people answered 3, for instance? – mixedmath Dec 12 '11 at 0:40
@nick what mixedmath is saying is that saying something is Normal implies that the data is continuous. Saying a discrete sample space is Normal has little meaning. Moreover, even if your data were continuous, saying they're normal without specifying the parameters (mean and SD) has even less meaning. Perhaps you could supply a bit more context? Where did this problem come from? How do you know the data are well approximated by a normal distribution? What are the parameters of that distribution? – Drew Christianson Dec 12 '11 at 0:44
As I mentioned in my comment, you can't really answer this question without knowing the distribution of the results, not just the mean and standard deviation. Let's say your values are distributed according to the binomial distribution (which is the discrete analog of the normal distribution) for a 4-trial binary experiment. The probability of the result being $k + 1$ is
$$P(k + 1) = \binom{4}{k}p^k(1-p)^{4-k}$$
But you don't have the value of $p$ directly; instead, you have the mean $\mu + 1 = 4p$ and standard deviation $\sigma + 1 = \sqrt{4p(1-p)}$. (The +1's come up because your data start at 1 rather than 0.) So you will need to solve for $p$ as $p = \frac{\mu + 1}{4}$. You should check that plugging this into the standard deviation formula $\sqrt{4p(1-p)}$ gives you something close to the actual standard deviation of your data, otherwise the binomial distribution is not an accurate representation of your data.
Once you have satisfied yourself that the binomial distribution is roughly accurate and that you have properly calculated $p$, you can compute the probability of getting a 4 or a 5 as
$$P(4) + P(5) = 4p^3(1-p) + p^4$$
This will give you a good approximation to the fraction of 4s and 5s in your data set; just multiply by $n$ to get the number of 4s and 5s. How good the approximation is depends on how well your data fit the binomial distribution, and you can use the matching of the standard deviations as an indicator for that.
I don't know of any practically useful way to get the number of 4s and 5s more exactly. You could write out the equations for the mean and standard deviation,
\begin{align}\frac{1}{n}\sum_{i=1}^n k_i &= \mu & \frac{1}{n}\sum_{i=1}^n (k_i - \mu)^2 &= \sigma^2\end{align}
and try to plug in numbers for the $k_i$ to see if you can come up with a set that matches your $\mu$ and $\sigma$ exactly, but that's just trial and error and it will likely not have a unique solution anyway.
-
If you told us the mean and standard deviation, we could tell you the limits on the proportion having the values $4$ or $5$.
Here is an example: suppose the mean was $2.5$ and the standard deviation was $1$. There are various possible probability distributions with this result and this table shows two of them.
Answer Prob1 Prob2
1 9/32 1/12
2 0 5/8
3 11/16 0
4 0 7/24
5 1/32 0
In the first case the proportion choosing 4 or 5 is $0.03125$; in the second it is $0.291666\ldots$. Any figure between these is also possible for this particular mean and standard deviation. If the distribution had to be unimodal then the limits would be tighter.
The upper and lower limits on the proportion choosing 4 or 5 would increase if the mean or the standard deviation were higher.
-
Suppose you have $n = n_1 + n_2 + n_3 + n_4 + n_5$ data points where $n_i$ data points have value $i$ for $i \in \{1,2,3,4,5\}$. The probabilistic version of this is a discrete random variable taking on values $\{1,2,3,4,5\}$ with probabilities $p_i = n_i/n$, and of course $p_1 + p_2 + p_3 + p_4 + p_5 = 1$. You know the mean $\mu$ and standard deviation $\sigma$ of this random variable. Obviously. $1 \leq \mu \leq 5$. Less well known is the result that $0 \leq \sigma \leq 2$. The extreme value $\sigma = 0$ occurs when one of the $p_i$'s equals $1$ and all other $p_j$ are zero, that is, all the data points have value $i$ (and so $\mu = i$), while the other extreme value $\sigma = 2$ occurs when $p_1 = p_5 = \frac{1}{2}, p_2 = p_3 = p_4 = 0$. Note that $\mu = 3$ in this case.
Turning to your problem of estimating $p_5$, we have that $$1^2\cdot p_1 + 2^2\cdot p_2 + 3^2\cdot p_3 + 4^2\cdot p_4 + 5^2\cdot p_5 = \sigma^2 + \mu^2$$ where the right side is known to you. Since $$1^2\cdot p_1 + 2^2\cdot p_2 + 3^2\cdot p_3 + 4^2\cdot p_4 > p1 + p_2 + p_3 + p_4 = 1-p_5$$ we have a crude upper bound $$p_5 \leq \frac{\sigma^2 + \mu^2 - 1}{24}.$$ Note that in the extreme case $\sigma = 0$, the bound is $(\mu^2 - 1)/24$ which is exact if all the data points have value $5$ (or $1$) but not very good when all the data points have some other value and $p_5 = 0$. But, if you know that $\sigma = 0$, you shouldn't be using the bound anyway because you know that $p_\mu = 1$ and all other $p_j$ are $0$. In the other extreme case $\sigma = 2, \mu = 3$, the upper bound is $\frac{12}{24} = \frac{1}{2}$ and is exact. But again, if you know that $\sigma = 2$, you shouldn't bother with the bound since this extreme distribution is known exactly.
Another weak bound comes from the one-sided Chebyshev inequality which gives $$p_5 \leq \frac{\sigma^2}{\sigma^2 + (5-\mu)^2}.$$ Which of the two bounds described here is tighter depends on the values of $\mu$ and $\sigma$.
-
Additionally, there is the option of using the Central Limit Theorem.
What you're really looking for is the probability that a given reply $X_i$ is 4 or more:$P(X_i\geq 4)$. Multiply that by the sample size, and you have a reasonable estimate of the number of responses that were 4 or 5. So, we use the CLT to approximate the data.
EDIT: Per Henry and Dilip's comments it you should us a continuity correction at this point. So, we rewrite $P(X\geq 4)$ as $P(X >3.5)$. This corrects for some of the error inherent in using the normal distribution to approximate a binomial.
What the CLT says is that, if $X_i$ obey any distribution with mean $\mu$ and variance $\sigma^2$ (note that $SD(X) = \sqrt{Var(X)}$), then $\frac{\overline{X_n}-\mu}{\frac{\sigma}{\sqrt{n}}}$ where $\overline{X_n}$ is the mean of a sample of size n is well approximated by a normal distribution with mean 0 and variance 1. So, we set about transforming the probability above into the form necessary for the CLT:$$P(X_i> 3.5) = P\left(\sum_{i=1}^{40}X_i > \sum_{i=1}^{40}3.5\right) = P\left(\frac{\sum_{i=1}^{40}X_i}{40}> \frac{\sum_{i=1}^{40}3.5}{40}\right) = P(\overline{X_{40}}>3.5) = P(\overline{X_{40}}-\mu> 3.5-\mu)$$$$P\left(\frac{\overline{X_{40}}-\mu}{\frac{\sigma}{\sqrt{40}}}> \frac{3.5-\mu}{\frac{\sigma}{\sqrt{40}}}\right) = P\left(Z> \frac{3.5-\mu}{\frac{\sigma}{\sqrt{40}}}\right) \;\;\;\;\;\;\;Z\thicksim Normal(0,1)$$ Plug in values of $\sigma$ and $\mu$ appropriate to the problem, and you can approximate the probability using a normal distribution.
-
If there are only five possible answers then you might want a continuity correction for your Normal approximation, e.g. by looking at $\Pr(X_i \ge 3.5)$ – Henry Dec 12 '11 at 1:56
@henry I'm not familiar with that adjustment, or at least the term. I'm only just finishing up a probability theory course. Link? – Drew Christianson Dec 12 '11 at 2:10
@DrewChristianson "I'm only just finishing up a probability theory course. " Since you claimed here that you are using Ross's book in your course, look at page 205 in the 8th edition. – Dilip Sarwate Dec 12 '11 at 17:53
@dilip thanks for the reference, I appreciate it. – Drew Christianson Dec 12 '11 at 18:15
My question is if I have A mean of $30,000 an of$3,00 satandard diviation. It is base with the annual incomes of some students. I need to find the percent of students who earned more than \$36,000.
-
This is not an answer to the question. – Asaf Karagila Feb 19 '13 at 23:50
This is not an answer to the question. If you have a question, please use the "Ask Question" link near the top right of the page. – robjohn Feb 20 '13 at 0:08 | {"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": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9699950218200684, "perplexity": 178.0425487957967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1440645208021.65/warc/CC-MAIN-20150827031328-00047-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://arxiv.org/abs/1507.06597 | math.ST
(what is this?)
# Title: Cox's Theorem and the Jaynesian Interpretation of Probability
Abstract: There are multiple proposed interpretations of probability theory: one such interpretation is true-false logic under uncertainty. Cox's Theorem is a representation theorem that states, under a certain set of axioms describing the meaning of uncertainty, that every true-false logic under uncertainty is isomorphic to conditional probability theory. This result was used by Jaynes to develop a philosophical framework in which statistical inference under uncertainty should be conducted through the use of probability, via Bayes' Rule. Unfortunately, most existing correct proofs of Cox's Theorem require restrictive assumptions: for instance, many do not apply even to the simple example of rolling a pair of fair dice. We offer a new axiomatization by replacing various technical conditions with an axiom stating that our theory must be consistent with respect to repeated events. We discuss the implications of our results, both for the philosophy of probability and for the philosophy of statistics.
Comments: This article was originally titled "Rigorizing and Extending the Cox-Jaynes Derivation of Probability" - the current version contains updated results, and has been rewritten completely for clarity Subjects: Statistics Theory (math.ST); Methodology (stat.ME) Cite as: arXiv:1507.06597 [math.ST] (or arXiv:1507.06597v2 [math.ST] for this version)
## Submission history
From: Alexander Terenin [view email]
[v1] Thu, 23 Jul 2015 18:21:22 GMT (859kb,D)
[v2] Mon, 17 Apr 2017 04:00:32 GMT (23kb) | {"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": 0.8026448488235474, "perplexity": 1656.3460950677108}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1534221210408.15/warc/CC-MAIN-20180816015316-20180816035316-00364.warc.gz"} |
http://www.gene2gene.com/Oct4.htm | # MainPage S-Z N-R H-M D-G BC A
Stem cell
3/23/2009
Methods Mol Biol. 2010;636:207-18.
# Using small molecules to improve generation of induced pluripotent stem cells from somatic cells.
Department of Chemistry, The Scripps Research Institute, La Jolla, CA, USA.
### Abstract
Induction of pluripotent stem cells from somatic cells by defined factors was shown to be possible only recently, but already several laboratories have made tremendous strive toward improving and understanding the process. Originally, Oct4, Sox2, Klf4, and cMyc were identified as being the combination of genes necessary to induce reprogramming. It was later shown that cMyc was dispensable; however, in its absence the process was less efficient and took a considerably longer period of time to occur. Furthermore, others have shown that the combination of Oct4, Sox2, Nanog, and Lin28 could also induce reprogramming. One major caveat associated with these techniques remains the need for overexpression of several genes using viral systems. Until very recently, most studies were done using integrating viruses such as retroviruses and lentiviruses. This method ensured that the protein of interested would be expressed at a high concentration and for an adequate period of time necessary to induce reprogramming. Up to date, others have now been able to use different nonintegrative method such as adenovirus and plasmid transfection to induce reprogramming. Furthermore, piggyBac transposition was successfully used to induce reprogramming of murine cells. Most importantly, it was recently published that reprogramming can be induced in the absence of virus, with proteins and small molecules. All of the later methods are appealing since they do not require the integration of the virus or plasmid to exert its effect. However, one avenue that would be all the more therapeutically appealing would be to induce reprogramming in the absence of gene overexpression systems, using small molecules to modulate signaling pathways in the somatic cells. A few molecules have already been identified with the ability to either improve the process or replace one or two of the genes deemed necessary for reprogramming. We have screened successfully for compounds that can replace some of these factors, and share the methods developed following these screens.
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http://physics.stackexchange.com/questions/21528/raising-a-toolbox-with-a-rope/21540 | # Raising a toolbox with a rope
Suppose you use a rope to hoist a box of tools vertically at constant speed v. The rope exerts a constant upward force of magnitude F_up on the box, and gravity exerts a constant downward force (the weight of the box). No other forces act on the box. To raise the box twice as fast, the force of the rope on the box would have to have:- A) the same magnitude $F_{up}$ B) a magnitude of $2F_{up}$
The answer seems to be (A), and I don't understand why! It says, if it's constant velocity, both upward and downward force = 0. Since the downward force is always the same, the upward force must be the same, too, regardless of the box's speed.
My take on that: But how are you gonna change the speed of the box without exerting force? If the force is the same, while there is one in the other direction, there would be no change of speed.
So, can you explain to me the correct answer?
EDIT: "Consider ... (1) starting the motion, (2) maintaining the motion, and (3) slowing the toolbox to a stop ..." (1) Starting. The object needs force greater than weight to go upward. (2) Maintaining. The object needs force equal to the weight to maintain motion upward. (3) Slowing. The object needs force less than weight for it to stop.
Ummm.. I don't know what to conclude with that? I understand one important thing: It can't be $F_{up} = W$, or in other words $F_{up} = F_{down}$, because there would be no acceleration in any direction. If there is no acceleration at any direction, I can't get the velocity to be twice as fast, or any amount faster.
EDIT: Thank you all. @Manishearth I could imagine this, but the book doesn't mention the "jerk", nothing about it in the question or in the explanation for the answer. (This is an example in the book). That's why I got confused. Anyway, again, thank you all for the clarification.
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Consider that the whole toolbox hoisting scenario includes three phases (1) starting the motion, (2) maintaining the motion, and (3) slowing the toolbox to a stop when it reaches the height of the presumed workman, and note that you are only examining phase 2. How might phases 1 and 3 differ in your two cases? Does that resolve your confusion? – dmckee Feb 26 '12 at 23:55
BTW---By starting with what looked like a homework question you triggered my "close with extreme prejudice" reflex, but I think that this is a good question focused on conceptual matters. – dmckee Feb 26 '12 at 23:57
I suspect that you are finding the answers you wrote in the edit to be really simple (almost trivial), but here's the thing ... that's it. The point of the problem is that sometimes physics is really simple. Give yourself a round of applause and move on. – dmckee Feb 27 '12 at 1:03
## 2 Answers
Well, for the main duration of pulling it up, the force will be the same. You will have to apply "jerks" twice as strong (by 'jerk' I mean the common usage, not the abusive usage or the physics term for $\dot{a}$). Let's break it into three portions:
### Initial Jerk
Initially, when you start to pull it, the box will receive a 'jerk', or 'impulse'. You will be applying an extremely large time varying (net) force $F$ in an extremely short time $\delta t$, such that the change in momentum of the body $\Delta p=m\Delta v=\int\limits_0^{\delta t} Fdt$. From this, one can see that if we apply a force twice as strong varying in the same manner, the change in velocity will also be twice as much. So in this leg, the force is twice as strong. Actually not exactly twice as strong, we need the net force to be twice as strong, but gravity is negligible here.
So in this leg, we have changed the speed of the block.
### Pulling it up
Now, the body has already reached a velocity $v$. By Galileo's principle the body will continue to move with this velocity if the net force is zero. So here, we apply a force equal to the weight of the body.
Note that if we are considering viscous drag and friction, then the force will change, but won't be double.
### Slowing it to a stop
This situation is pretty much the same as the first leg, so again the force here is twice as strong.
In conclusion, the force needs to be double only when you are making it reach the velocity $2v$, and slowing it back down. At all other points it is the same.
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Just a note, jerk is usually referred to the derivative of the acceleration. – Madde Anerson Jul 24 '15 at 22:02
As I understand it, I will have to first apply a force $F_{up}$ for some time $t$ such that the box accelerates upwards until it achieves a velocity $v$. Once I achieve $v$, I only need to apply enough force to match gravitational pull $= -Fg$. Assuming friction-less perfect worlds, there is no net force on the box and it continues at velocity $v$ forever. To move the velocity up to $2v$, I will now have to apply an additional $F_{up}$ for time $t$ to achieve $2v$ after which I again only need to apply $-Fg$.
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https://export.arxiv.org/abs/hep-th/0203018v1 | hep-th
(what is this?)
# Title: The PP-Wave Limits of Orbifolded AdS_5x S^5
Abstract: Using the supergravity solution of $N_1$ D3-branes probing $A_{N_2-1}$ singularities we study the pp-wave limit of $AdS_5\times S^5/Z_{N_2}$. We show that there are two different pp-wave limits. One is the orbifold of the pp-wave limit of $AdS_5\times S^5$. In this case there is no symmetry enhancement. The other case is the same as the pp-wave limit of $AdS_5\times S^5$ and therefore we again see the maximal supersymmetry. We will also identify operators in the four dimensional ${\cal N}=2$ $SU(N_1)^{N_2}$ gauge theory which correspond to stringy excitations in the orbifolded pp-wave geometry. The existence of the maximal pp-wave geometry indicates that there is a subsector of the corresponding ${\cal N}=2$ gauge theories which has enhanced ${\cal N}=4$ supersymmetry. We also study the pp-wave limits of $AdS_{7,4}\times S^{4,7}/Z_{N}$.
Comments: Latex file, no figures, 13 pages Subjects: High Energy Physics - Theory (hep-th) Report number: IPM/P-2002/004, SU-ITP-02/10 Cite as: arXiv:hep-th/0203018 (or arXiv:hep-th/0203018v1 for this version) | {"extraction_info": {"found_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": 0.9760825037956238, "perplexity": 1452.3131610100722}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764499819.32/warc/CC-MAIN-20230130133622-20230130163622-00104.warc.gz"} |
http://s391535452.onlinehome.us/Paraty15/en/workshop_en_posters.php | # Posters
ALBA MARCELA HERRERA TRUJILLO UNIVERSIDADE FEDERAL DO ABC Super-adiabatic Quantum Heat Engine In our work we exploit the implementation of a non-transitional evolution which no stick to the quantum adiabatic dynamics. The quantum adiabatic dynamics is governed by the adiabatic theorem, imposing a speed limit on the evolution. Our deployment was achieved using the super-adiabatic evolution that is a useful tool to find a shortcut to the adiabatic one. Specifically, we study the performance of the quantum Otto cycle and we implement it in a NMR setup. With this result we show that it is possible to obtain a improved quantum engine. Alencar José de Faria Universidade Federal de Alfenas Quantum non-demolition verification of entanglement It is proposed a optical procedure in context of continuous variables to verify the entanglement between two light modes without destroying both modes and their entanglement. The method relies on beam-splitter and quantum non-demolition interactions of the signal modes with two ancillary probe modes. The probe modes are measured by homodyne detection systems, and the detector signals are used to feedforward modulation of signal modes, concluding the procedure. It is shown that the classical information obtained from homodyne measurements is sufficient to verify the entanglement of the signal modes. The processing impact on the entanglement is analyzed. Alexandre M. Souza CBPF High-fidelity gate operations for quantum computing beyond dephasing time limits The implementation of quantum gates with fidelities that exceed the threshold for reliable quantum computing requires robust gates whose performance is not limited by the precision of the available control fields. The performance of these gates also should not be affected by the noisy environment of the quantum register. Here we use randomized benchmarking of quantum gate operations to compare the performance of different families of gates that compensate errors in the control field amplitudes and decouple the system from the environmental noise. We obtain average fidelities of up to 99.8%, which exceeds the expected limit imposed by the dephasing environment. ALVARO ANDRES CIFUENTES GARCIA UNIVERSIDADE FEDERAL DO ABC Quantum Thermodynamics and Trapped Ions We study the work statistics and irreversibility when a classical electromagnetic field interacts with a trapped ion. The interaction time is taken to be infinitesimally small i.e., a “quench” process. Our motivation comes from the emergence of Quantum Thermodynamics, in addition to the well established understanding of trapping and control of charged particles. Thus, the thermodynamical properties of an important candidate to perform quantum computation are characterized. Three different figures of merit are used, and contrasted, in this work to quantify the degree of irreversibility in our protocol. The first one is related to the “Relative Entropy”, which accounts for the difference between the actual system state and the reference thermal state. The second quantifier is a quantum version of a classical entropy, which follows from a mechanical model of equilibrium thermodynamics, the so-called Volume Entropy, and is valid for closed systems. Finally, the third one is related with a comparison between the actual and an ideal work process, done along an ideal reversible adiabatic transformation. Ana Majtey IF-UFRJ Dynamics of entanglement in systems of identical fermions undergoing decoherence Information that is stored in quantum-mechanical systems can be easily lost because of the interaction with the environment in a process known as decoherence. Possible physical implementations of many processes in quantum information theory involve systems of identical particles, whence comprehension of the dynamics of entanglement induced by decoherence processes in identical-particle open systems becomes relevant. Here [1] we study the effects and concomitant entanglement evolution arising from the interaction between a system of two identical fermions and the environment for two paradigmatic quantum channels. Entanglement measures are introduced to quantify the entanglement between the different parties, and a study of the dynamics of entanglement for some particular examples is carried out. Our analysis, which includes also the evolution of an entanglement indicator based on an entropic criteria, offers insights into the dynamics of entanglement in open systems of identical particles, involving the emergence of multipartite genuine entanglement. The results improve our understanding of the phenomenon of decoherence and will provide strategies to control it. Andreas Ketterer Laboratoire MPQ, Université Paris Diderot Encoding discrete quantum information in continuous variables: A modular variables approach Quantum information can be processed in two fundamentally different ways, namely, using discrete or continuous variable representations, respectively. However, only some quantum information protocols can be adapted from discrete to continuous variables, providing several practical advantages/drawbacks. In the present work we use modular variables to address the problem of bridging the world of continuous and discrete quantum information protocols and operations. We establish clear correspondences between the universal sets of operations defined in both realms by expressing them in terms of eigenstates of the modular position and momentum operator. Our results shed new light on the well known work of D. Gottesman et al., Encoding a qubit in an oscillator'', by revealing naturally discrete structures of operators and states defined in the continuous variable regime. Finally, we garnish our findings with a novel detection scheme allowing for a readout of the discrete quantum information from the continuous variable states in terms of inventively chosen observables. Andres Felipe Estrada Guerra Universidad de Antioquia Quantum limit for driving linear non-Markovian open quantum systems The interplay between non-Markovian dynamics and driving fields in the survival of entanglement between two non-degenerate oscillators is considered here. Based on exact analytical results for the non-Markovian dynamics of two parametrically coupled non-degenerate oscillators in contact with non-identical independent thermal baths, the out-of-equilibrium quantum limit derived in [Phys. Rev. Lett. 105 180501 (2010)] is generalised to the non-Markovian regime. Specifically, it is shown that non-Markovian dynamics, when compared to the Markovian case, allow for the survival of stationary entanglement at higher temperatures, with larger coupling strength to the baths and at smaller driving rates. The effect of the asymmetry of the (i) coupled oscillators, (ii) coupling strength to the baths at equal temperature, and (iii) temperature at equal coupling strength is discussed. In particular, it is shown that the non-Markovian character of the dynamics is capable of beating the resonant condition that states that the driving frequency equals the sum of the natural frequencies for the maximum rate of squeezing generation; hence, squeezing generation is more robust under non-Markovian dynamics. Barbara Amaral UFOP Maximum Contextuality Allowed by Quantum Theory Contextuality has been identified as a basic resource for quantum information and computation. Here we address the problem of which is the maximum contextuality allowed by quantum theory. While a traditional approach based on computing a measure of contextuality for all possible scenarios would suggest that this problem is unaffordable, here we show that a connection between quantum contextuality and graph theory allows us to prove that the ratio between the quantum violation and the noncontextual bound of a noncontextuality inequality can be lager than $n^{1- \epsilon}$, where $n$ is the number of events and $\epsilon$ is positive and arbitrarily small, while in general probabilistic theories this ratio cannot be larger than $n$. Remarkably, this maximum is much larger than the one exhibited in the violation of Mermin inequalities, which goes as $n^{\frac{1}{4}} . Baris Cakmak UNICAMP Quantum coherence and uncertainty in the anisotropic XY chain We explore the local quantum coherence and the local quantum uncertainty, based on Wigner-Yanase skew information, in the ground state of the anisotropic spin-1/2 XY chain in transverse magnetic field. We show that the skew information, as a figure of merit, supplies the necessary information to reveal the occurrence of the second order phase transition and the completely factorized ground state in the XY model. Additionally, in the same context, we also discuss the usefulness of a simple experimentally friendly lower bound of local quantum coherence. Furthermore, we demonstrate how the connection between the appearance of non-analyticities in the local quantum uncertainty of the ground state and the quantum phase transitions does not hold in general, by providing explicit examples of the situation. Lastly, we discuss the ability of the local quantum coherence to accurately estimate the critical point of the phase transition, and investigate the robustness of the factorization phenomenon at low temperatures. Carlos Alberto Parra Murillo Universidade Federal de Minas Gerais Diffusion-driven quantum thermalization at a resonant tunneling scenario We present recent results on relaxation dynamics and thermalization in a many-body implementation of the Wannier-Stark system [1]. It is hown that the induced delocalization of instantaneous eigenstates via cascade of Landau-Zener transition triggers an effective thermalization process in finite time evolution. We show that not only single particle observables but two-particle ones approach their thermal average values and a way to define the effective temperature of the system based on the mixing properties of the quantum spectrum. Carlos Ivan Henao Osorio Universidade Federal do ABC ROBUSTNESS TO NOISE OF TWO-WAY QUANTUM KEY DISTRIBUTION (TWQKD) PROTOCOLS In this work we study the security of TWQKD protocols that use qubits prepared in non orthogonal states as carriers of information. In particuar, we present a new security proof for a recently proposed TWQKD protocol and investigate the possibility of outperforming it. We find that within our framework there is only one TWQKD protocol that could have a larger secret fraction. Moreover, we propose some techniques to potentially reduce the information leaked to an eavesdropper in TWQKD, which we term "pre-encoding operations". Such operations might increase the robustness to noise of a given protocol, but at the expense of a reduced efficiency of the communication. Carlos Mario Rivera Ruiz UFSCAR Quantum dot implementation of the quantum permutation algorithm Title: Quantum dot implementation of the quantum permutation algorithm Authors: C. M. Rivera, B. Çakmak, E. F. de Lima, F. F. Fanchini, and L. K. Castelano Abstract The scientific community and technological industries are putting a great effort on the pursuit of physical systems capable of implementing quantum computing. Such a challenge is related to the efficiency of quantum algorithms, which offer an exponential speedup in the calculation time with respect to the classical counterpart. To achieve such advantages, entanglement was considered as a fundamental source for the power of the quantum computation. Nevertheless, it has been shown that the quantum efficiency can be achieved by quantum algorithms where entanglement is out of table. A recent quantum algorithm [1] that uses only a single qutrit or qudits, allows to determine the parity of permutations of a set of three numbers by employing only one measure, in contrast to the classical case, where two measures are needed. Such an algorithm has been implemented in a NMR quantum system [2]. In this work, we present an implementation of another platform that could realize the permutation algorithm in a three quantum dot system. Such a system is very interesting due to its potential scalability and miniaturization [3-5]. Moreover, we find the electric pulses capable of producing the necessary quantum gates that implement the permutation algorithm. Our results give the route to a possible experimental realization of the quantum permutation algorithm in three coupled quantum dots. References: [1] Z. Gedik, arXiv:1403.5861. [2] I. A. Silva, B. Çakmak, G. Karpat, E. L. G. Vidoto, D. O. Soares-Pinto, E. R. deAzevedo, F. F. Fanchini, Z. Gedik, arXiv:1406.3579. [3] A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, Phys. Rev. Lett. 83, 4204 (1999). [4] Danny Kim, Samuel G. Carter, Alex Greilich, Allan S. Bracker, and Daniel Gammon, Nature Physics 7, 223229 (2011). [5] J. H. Jefferson, M. Fearn, D. L. J. Tipton, and T. P. Spiller, Phys. Rev. A 66, 042328 (2002). Cristhiano Duarte Universidade Federal de Minas Gerais Multigraph approach to quantum non-locality Non-contextuality (NC) and Bell inequalities can be expressed as bounds Ω for positive linear combinations S of probabilities of events, S≤Ω. Exclusive events in S can be represented as adjacent vertices of a graph called the exclusivity graph of S. In the case that events correspond to the outcomes of quantum projective measurements, quantum probabilities are intimately related to the Gr\"otschel-Lov\'asz-Schrijver theta body of the exclusivity graph. Then, one can easily compute an upper bound to the maximum quantum violation of any NC or Bell inequality by optimizing S over the theta body and calculating the Lov\'asz number of the corresponding exclusivity graph. In some cases, this upper bound is tight and gives the exact maximum quantum violation. However, in general, this is not the case. The reason is that the exclusivity graph does not distinguish among the different ways exclusivity can occur in Bell-inequality (and similar) scenarios. An interesting question is whether there is a graph-theoretical concept which accounts for this problem. Here we show that, for any given N-partite Bell inequality, an edge-coloured multigraph composed of N single-colour graphs can be used to encode the relationships of exclusivity between each party's parts of the events. Then, the maximum quantum violation of the Bell inequality is exactly given by a refinement of the Lov\'asz number that applies to these edge-coloured multigraphs. We show how to calculate upper bounds for this number using a hierarchy of semi-definite programs and calculate upper bounds for I3, I3322 and the three bipartite Bell inequalities whose exclusivity graph is a pentagon. The multigraph-theoretical approach introduced here may remove some obstacles in the program of explaining quantum correlations from first principles. Cyntia Vanessa Henrique Bezerra Uhlig Centro Brasileiro de Pesquisas Físicas David Velasco Villamizar UFSC Quantum Speed Limit for relativistic electron in an uniform magnetic field How fast a physical system can process information? It is an extremely important question in the field of computation. The answer to this question is the central focus of this study, to investigate the minimum time required for a relativistic quantum system changes to an orthogonal state respected to the initial one. In the present study we analysed the relativistic dynamics according to the Dirac equation an electron in the presence of an uniform magnetic field. The initial state of the system was chosen as a superposition of two states with equal weight, each of them being associated with a different Landau levels. Analysing the speed in which the electron moves from its initial mean position to its mean final position, it was found that in the case relativistic electron description never reach a speed greater than the speed of light. On the other hand, in non-relativistic description obtained by the Schrödinger equation, the electron will reach a higher rate of displacement greater than c when it is in a very strong magnetic field. Therefore, to realize a correct description of this problem of quantum speed limit is necessary to treat it according to the relativistic quantum mechanics. Diogo O. Soares-Pinto IFSC / USP Distributed correlations and information flows within a hybrid multipartite quantum-classical system Understanding the non-Markovian mechanisms underlying the revivals of quantum entanglement in the presence of classical environments is central in the theory of quantum information. Tentative interpretations have been given by either the role of the environment as a control device or the concept of hidden entanglement. We address this issue from an information-theoretic point of view. To this aim, we consider a paradigmatic tripartite system made of two independent qubits and a random classical field locally interacting with one qubit alone. We study the dynamical relationship between the two-qubit entanglement and the genuine tripartite correlations of the overall system, finding that collapse and revivals of entanglement correspond, respectively, to raise and fall of the overall tripartite correlations. Interestingly, entanglement dark periods can enable plateaux of nonzero tripartite correlations. We then explain this behavior in terms of information flows among the different parties of the system. Besides showcasing the phenomenon of the freezing of overall correlations, our results provide new insights on the origin of retrieval of entanglement within a hybrid quantum-classical system. Eduardo da Costa Paul UFRJ Witnessing continuous-variable entanglement with the use of a set of three mutually unbiased bases In a finite-dimensional Hilbert space, if the quantum state of a physical system can be expressed as an eigenvector of a given basis and simultaneously as a balanced superposition of every eigenvector of another basis, we say that the two bases are mutually unbiased bases (MUBs). That means that the measurement of the state in one basis gives no information about the possible measurement outcomes in the other basis. For the infinite-dimensional continuous-variable case, this concept can be generalized in the natural way. In this case, the most well known example are the position and momentum bases: perfect knowledge of the position of a quantum particle means that a measurement of its momentum could yield any result with equal probabilities. Interestingly, position and momentum do not constitute the only possible set of MUBs for this system. It can be shown that a third basis, x-p, can be added to that set, although making it asymmetric among the bases. Nevertheless, a completely symmetric set of MUBs can be achieved if we forego the momentum basis and instead use the bases equivalent to rotations of 0º, 120º and 240º in phase space, which correspond to fractional Fourier transforms of position. In this work, we experimentally study the transverse spatial variables of twin photons using the set of three MUBs as described. Each MUB can be measured with the use of non-confocal lens systems. We check the validity of uncertainty relations, and verify the existence of entanglement between the two photons. Emanuel Cardozo Diniz Federal University of São Carlos Quantum correlations in superconducting qubits in ultra strong regime under the action of reservoir collective markovian We investigated the generation of quantum correlations between a system consisting of two superconducting qubits interacting with an electromagnetic field in a regime named ultra-strong coupling under the action of a collective Markovian reservoir for the qubits and an independent one for the field, at zero as well at finite temperature. Using the results obtained previously by the author and collaborators in [1, 2] we have found that in this situation the system displays the absence of thermalization, providing the possibility of generating a high degree of correlation between the qubits. We have shown that quantum discord, a type of quantum correlation between different subsystems, which appears in this system is much more robust to dissipative effects caused by the environment than the traditional entanglement, quantified in our work by the entanglement of formation. Our results can be useful for the implementation of possible architectures of quantum computing using superconducting qubits and also for the implementation of quantum communication protocols based on highly entangled states. References: [1] E. C. Diniz, “Termalização de qubits sujeitos à ação de reservatórios coletivos Markovianos”, Master Dissertation, Physics Department, Federal University of São Carlos, São Carlos, Brazil, (2014). [2] E. C. Diniz, D. Z. Rossato, T. Werlang, C. J. Villas-Boas. Termalization in Rabi Model in the Open Quantum System Context, In preparation (2015). FATEMEH UFRGS NMR Implementation of the two-dimensional YANG-BAXTER Equation The Yang-Baxter Equation (YBE) is a sufficient condition for the integrability of a model. This means that if a model is constructed from the Yang-Baxter Equation, it will be called an integrable model, which means that it can be solved exactly, i.e., we will know its eigenvalues and eigenfunctions. This mathematical construction was introduced by C.N. Yang and R. Baxter in diffierent contexts. We tested the two-dimensional (2D) Yang-Baxter Equation (YBE) using Nuclear Magnetic Resonance (NMR) and we presented a practical scheme to test the YBE in the framework of quantum information. The equality of the two sides of the Yang-Baxter Equation is directly verified. Federico Cerisola University of Buenos Aires Work Measurement as a Generalized Quantum Measurement We present a new method to measure the work w performed on a driven quantum system and to sample its probability distribution P(w). The method is based on a simple fact that remained unnoticed until now: Work on a quantum system can be measured by performing a generalized quantum measurement at a single time. Such measurement, which technically speaking is denoted as a positive operator valued measure reduces to an ordinary projective measurement on an enlarged system. This observation not only demystifies work measurement but also suggests a new quantum algorithm to efficiently sample the distribution P(w). This can be used, in combination with fluctuation theorems, to estimate free energies of quantum states on a quantum computer. Flaminia Giacomini University of Vienna Infinite-dimensional quantum systems on indefinite causal structures Standard Quantum Mechanics assumes that events are embedded in a global causal structure. The process matrix framework keeps the local validity of standard Quantum Mechanics while relaxing the assumption on the global causal structure. This leads to multipartite correlations which lie outside the usual causally ordered framework, and opens interesting perspectives on our understanding of the nature of space and time. So far, the formalism has been developed only for finite-dimensional systems. Here we address the generalization to infinite-dimensional Hilbert spaces, with the long-term goal of formulating Quantum Fields on indefinite causal structure. Frank E. S. Steinhoff Universität Siegen Hypergraph states Hypergraph states are multi-qubit states that form a subset of the locally maximally entangleable states and a generalization of the well--established notion of graph states. Mathematically, they can conveniently be described by a hypergraph that indicates a possible generation procedure of these states; alternatively, they can also be phrased in terms of a non-local stabilizer formalism. In this paper, we explore the entanglement properties and nonclassical features of hypergraph states. First, we identify the equivalence classes under local unitary transformations for up to four qubits, as well as important classes of five- and six-qubit states, and determine various entanglement properties of these classes. Second, we present general conditions under which the local unitary equivalence of hypergraph states can simply be decided by considering a finite set of transformations with a clear graph-theoretical interpretation. Finally, we consider the question whether hypergraph states and their correlations can be used to reveal contradictions with classical hidden variable theories. We demonstrate that various noncontextuality inequalities and Bell inequalities can be derived for hypergraph states. Frederico Brito IFSC/USP Testing time reversal symmetry in artificial atoms Over the past several decades, a rich series of experiments has repeatedly verified the quantum nature of superconducting devices, leading some of these systems to be regarded as artificial atoms. In addition to their application in quantum information processing, these ‘atoms’ provide a test bed for studying quantum mechanics in macroscopic limits. Regarding the last point, we present here a feasible protocol for directly testing time reversal symmetry through the verification of the microreversibility principle in a superconducting artificial atom. Time reversal symmetry is a fundamental property of quantum mechanics and is expected to hold if the dynamics of the artificial atom strictly follow the Schrödinger equation. However, this property has yet to be tested in any macroscopic quantum system. In the end, as an application of this work, we outline how the successful implementation of the protocol would provide the first verification of the quantum work fluctuation theorems with superconducting systems. Gabriel Fagundes Camargo UFMG Memory Cost for Simulating Sequential Quantum Correlations It is well known that contextual properties of quantum measurements cannot be reproduced by classical models. But in an implementation with sequential measurements, it may still be possible to achieve the task by using some classical, deterministic automatons with internal memory. The particular model we use are Mealy machines. We thus characterize contextuality by the number of internal states needed by the machine to reproduce quantum predictions. Following this line, we calculate the amount of memory for the scenario of the Peres-Mermin square. In the analysis so far, the only requirement on the automaton was that it merely must not produce events that are forbidden according to QM. One could expect that it is much more expensive to reproduce the exact probabilities. We find that this is not the case: mixtures of deterministic automata with three internal states are sufficient to simulate all quantum correlations. Gláucia Murta Universidade Federal de Minas Gerais Algebraic bounds on the quantum value of XOR games Quantum non-locality is a surprising phenomena which reveals us some counter-intuitive characteristics of quantum theory. As surprising as the fact that quantum theory can violate local realism, expressed in the form of Bell inequalities, is the fact that it cannot violate as much as the non-signalling principle allows. The study of how much quantum mechanics can violate a Bell inequality has fundamental and practical applications. Fundamentally, it can help us understand the physical and information-theoretic principles that defines quantum theory. On the practical side, Bell violations has innumerous applications as for example to cryptography, device independent protocols and entanglement witnessing. Bell inequalities can also be phrased in a game scenario: the parties who are suppose to make a Bell test are now the players of a game. When the game starts the parties are not allowed to communicate anymore. A referee is responsible for distributing inputs among the players. Up to receiving an input each party is suppose to answer with an output. The goal of the game is that the outputs of all the parties together satisfy some function previously defined by the game. We study bounds on the maximum performance that quantum players can achieve in a XOR-d game. XOR-d games are a particular class of games where the winning condition only depends on the sum modulo d of the outputs. They are a natural generalization for d-outputs of the binary XOR games, where the winning condition is defined by the XOR of the outputs. XOR games are also referred as correlation Bell inequalities. We propose an algebraic bound to the quantum value of these games and use it to derive several interesting properties of them. As an example, we re-derive in a simple manner a recently discovered bound on the quantum value of the CHSH-d game for prime d. We then study the principle of no quantum advantage for the distributed computation of binary functions (Non-Local Computation) which is a well-known information-theoretic principle designed to pick out quantum correlations from amongst general no-signaling ones. We prove a large-alphabet generalization of this principle, showing that quantum theory provides no advantage in the task of non-local computation of a restricted class of functions with d outcomes for prime d, while general no-signaling boxes do. Finally we consider extension of previous results for the multiparty scenario (work in progress in collaboration with R. Ramanathan, N. Móller and M. Terra Cunha). As preliminary results we derive an algebraic upper bound for n-players XOR games and discuss its possible applications. Gonzalo Carvacho Università degli Studi di Roma "La Sapienza" Experimental entanglement using vector vortex beams Gonzalo Carvacho (1), Vincenzo D'Ambrosio (1), Chiara Vitelli (1), Francesco Graffitti (1), Giulia Rubino (1), Bruno Piccirillo (2), Lorenzo Marrucci (2), Fabio Sciarrino (1). (1) Dipartimento di Fisica, Sapienza Universita di Roma, Roma, Italy (2) Dipartimento di Fisica, Universit a Federico II di Napoli, Napoli, Italy We can exploit the vectorial nature of the light in order to generate vector vortex beams (VVB), also referred as spirally polarized beams, using a recent device call q-plate which allow us manipulate the spatial distribution of the polarization in the transversal section of the beam in a quantum regime. Because of the coupling of the polarization with the orbital angular momentum of the photons, we can use an infinite-dimensional Hilbert space which is useful to perform different kinds of protocols. In our experiment we violated the Bell inequalities to demostrate the entangled nature of these particular beams; we also performed quantum state tomography to fully characterize the vector vortex beams for each pair of q-plates with different topological charges (that implies an entanglement of different Hilbert spaces between two interlocutors). Halyne Borges Federal University of São Carlos Quantum memory based on Electromagnetically Induced Transparency in Optical Cavities An essential element for quantum computing to be possible is a quantum memory. In this work we theoretically studied the implementation of quantum memory in a system composed by a single atom trapped inside a high finesse cavity. In order to store and map the quantum state of an input pulse onto an internal state of the single atom, we based on the electromagnetically induced transparency (EIT) phenomenon where the information was transferred to the dark state of the atom modelled by a three-level system in the lambda-type configuration. In our model we consider a suitable temporal shape to the control field that ensure the adiabaticity in the storage process and retrieval of the probe pulse. The dynamic of the field inside the cavity was obtained by master equation approach, while the outside field was calculated by input-output formalism. Our results shows that for the appropriate setup and commonly used to observe the cavity-EIT in the transmission spectrum, the memory efficiency value is very low. We also discuss the differences between an input-output and a master equation approach considering different setups of the physical system. Finally we showed that considering a single-sided cavity, the quantum memory efficiency increases considerably and can reach almost 100% in the strong coupling regime. In this way our work provide a full theoretical description of this system for quantum memory applications, analyzing parameters regimes accessible experimentally. Ibrahim SAIDEH Institut des Sciences Moléculaires d'Orsay (ISMO) Mapping all separable qudits to separable qubits to detect entanglement We present a general strategy to build entanglement criteria which consists in performing a mapping from qudits to qubits that preserves the separability of the parties and SU(2) rotational invariance.Consequently,it is possible to apply the well known positive partial transpose criterion for the qudits in terms of the correlations between the spins of each party. Consequently, it is possible to apply the well known positive partial transpose criterion to reveal the existence of quantum correlations between qudits. We discuss some examples of entangled states that are detected using the proposed strategy. Finally, we demonstrate using our scheme how some variance based entanglement witnesses can be generalized from qubits to higher dimensional spin systems. Igor Konieczniak LMCAL - Universidade de São Paulo Bicolour Quantum Entanglement for Teleportation in Continuous Variables Several proposals for quantum computation devices are been implemented around the world. As these devices process information over different physical basis, it is expected that, when using light as a medium, their light will not necessarily be of the same wavelength. On the other hand, quantum teleportation is a fundamental part in many quantum computation protocols. So, the ability to quantum teleport between fields of different colours becomes a very desirable one. Bicolour quantum teleportation requires production and noise measurement abilities of entangled light fields in different wavelengths. Optical Parametric Oscilators (OPOs) operating in non-degenerate configuration have been used for producing such fields in the continuous variables regime. In these systems, entanglement is found between the phase an intensity quadratures of the two fields and expresses itself in noise reduction below the Standard Quantum Limit (SQL) in the sum phase quadrature and difference intensity quadrature of the two fields. Previous entanglement produced by a doubly resonant OPO resulted in 5 dB noise reduction below SQL in the sum intensity quadrature, but for the phase sum quadrature we found noise higher than expected. This excess noise was attributed to the pump field, as previous experiments had already shown. To reduce the phase noise in the pump field, a filter cavity was installed. As the filter cavity reduces the available pump power, a triply resonant OPO was deemed necessary for having a lower power oscillation threshold. A triply resonant OPO with asymmetrical mirror configuration was built and its noise covariance matrix was measured via the cavity noise ellipse rotation method. We report the production of quantum entanglement in continuous variables, a step further towards the completion of the quantum teleportation protocol. Irati Alonso Calafell University of Vienna Searching for Nonlinearities in Graphene at the Single Photon Level Single-photon sources are an essential tool for experiments in quantum information, and it is known that nonlinear effects can be used to create such a source. To date, however, available optical materials require high intensities and long interaction times to induce strong enough nonlinearities. In this poster, we introduce a theoretical proposal in which nanostructured graphene shows strong nonlinear effects at the single photon level. In this proposal, graphene’s strong nonlinear effects are used to implement the photon blockade. The predicted anti-bunched emission indicates that graphene could be an extraordinary candidate to produce high quality single photons with an extremely high efficiency. However, the predicted wavelength of the single photons is longer than 2um – much beyond the range of current single photon detectors. In addition to presenting an overview of the quantum picture of light, and the Hanbury Brown-Twiss interferometer, this poster will discuss our efforts to implement super-conducting nanowire detectors for very long wavelength single photons in order demonstrate this exciting effect. Jessica Bavaresco Universidade Federal de Minas Gerais JORDANA TORRICO FERREIRA UNIVERSIDADE FEDERAL DE ALAGOAS Thermal entanglement and frustration temperature in the Ising-Hubbard diamond chain In this paper, we study the Ising-Hubbard model for a infinity diamond chain with mobile eletrons in the presence of an external magnetic field. In this spin-chain model, the nodal sites are occupied by located Ising spins whereas the chain interstitial sites can be considered as single orbitals with one electron per site. In particular, electrons are allowed to hop between two interstitial sites providing the dynamical hopping terms of the model. Using the decoration-iteration transformation and transfer matrix methods obtain the exactly solution. The effect of hopping term, magnetic field and temperature dependences of the functions of correlations and magnetization are studied in several phase diagrams that the model provides us with we analytically and numerically calculate the frustration temperature and thermal entanglement, via concurrence, as a function of the external magnetic field. José Ferraz de Moura Nunes Filho Dipartimento di Fisica, Sapienza Università di Roma, Italy e Departamento de Física, Universidade Federal Rural de Pernambuco, Recife, Brasil Diluted Quantum Walk via Integrated Photonics Abstract The transport of wave energy in disordered media has stirred a tremendous interest in different classical and quantum media, including optical, acoustic, electronic and matter wave systems. When waves propagate in a "static" disordered material, its propagation is normally arrested due to the presence of exponentially localized modes, which are spatio-temporally localized and inhibit any propagation of energy. This effect was analyzed for entangled photonic states in a quantum walk via integrated photonics [Nature Photonics 7, 322-320 (2013)]. On the other hand, when disorder is "evolving" (i.e., changing upon propagation), localization might breaks and a new type of diffusive behavior can settle in, where the transport rate might be even higher than the classical ballistic transport. In this work, we study theoretically the variance of entangled photons as a function of the quantum walk number of steps and show that a superdiffusive regime is obtained and is intrinsically correlated to the photonic state symmetry (bosonic or fermionic). Jose Raul Gonzalez Alonso University of Southern California The Long and Winding Road towards Error Mitigation for Photons with Orbital Angular Momentum in a Turbulent Atmosphere It is not very often that a completely new property of electromagnetic waves is discovered. However, this was exactly what happened in 1993 when Les Allen in the United Kingdom reported that light with spiral phase fronts could carry orbital angular momentum (OAM). Since then, there is a growing interest in the applications of the OAM of light in multiple fields. In particular, OAM photons are very attractive for quantum information because they can carry arbitrarily large amounts of information per photon. Since communication with OAM photons happens over free space, it is important to understand the loss of coherence due to the interaction between a turbulent atmosphere and quantum states of light with OAM. In this work, I will review some of the alternatives to describe the noise processes OAM photons undergo when traveling in a turbulent atmosphere. In particular, I will discuss both the Kraus and the Lindblad representations of the decoherence process by assuming different noise models for the turbulence. Additionally, I will analyze possible applications of quantum error correcting codes and decoherence-free subspaces to the problem of protecting the desirable quantum properties of photons with OAM. Luciano Soares da Cruz UFABC Squeezing and Entanglement on quantum states of light polarization. The preparation of optical fields with strictly quantum properties is an essential requirement for several Quantum Computation and Quantum Information protocols. Among the tangible experimental realizations, the use of atomic ensembles as non-linear media and continuous variables of light as transmitter has received special attention in the scientific community due to many quantum phenomena that can be reached with this matter-light interaction. We studied an optically thin two level atomic ensemble (including its complete Zeeman degeneracy) interacting with a classical field, the generation of states exhibiting squeezing and entanglement in the polarization degree of freedom. We determined the power spectra to the polarization variables, showing the system’s feasibility to induced quantum properties as squeezing and entanglement in the optical fields. Marcio Fernando Cornelio Universidade Federal de Mato Grosso Entanglement and Discord in pure multipartite systems We extend the conservation law for the distribution of entanglement of formation and discord (Fanchini et al, PRA 84 012313, 2011) to four-partite and five-partite systems. We also obtain generalised n-partite conservation relations. An interesting difference between systems of even and odd parts shows up. For systems with odd number of parts, we obtain equalities like in the three-partite system. However, for systems with even number of parts, we obtain inequalities. We also show that there are basically two types of conservation laws: one is based in a key system and the other one is based in a loop over all the systems. Márcio Mendes Taddei Federal University of Rio de Janeiro Quantum brachistochrone (Summary submitted to both School and Workshop applications) The quantum brachistochrone problem consists of finding the fastest way of driving a quantum system from a given initial state to a desired final state under a set of restrictions to the operator(s) governing evolution (i.e. to the Hamiltonian, if the evolution is unitary). The question has a wide range of applications especially because the restrictions to the evolution are suitable to mimic experimental or practical limitations of a control system. If the restrictions to the evolution are minimal -- only limiting the amount of energy available to the system --, the problem reduces to that of finding the geodesic between two states. In the general case, there is an interplay between the path traveled by the state of a system and the structure of the operator(s) governing evolution. Here we discuss variational approaches to the quantum brachistochrone problem based either on the Euler-Lagrange equation or on Pontryagin's theorem. Marco Cerezo Instituto de Física la Plata (IFLP)-Facultad de Ciencias Exactas, Departamento de Física, Dpto. Cs. Bs. & Facultad de Ingeniería, Universidad Nacional de la Plata (UNLP). Entanglement and non-transverse factorizing fields in spin chains We examine the entanglement of quantum spin systems with anisotropic XYZ Heisenberg couplings of arbitrary range at transverse and non-transverse factorizing magnetic fields. The conditions for which the system presents an exactly separable eigenstate were determined for both cases. Previous results showed that at the transverse factorizing magnetic field, the system exhibits a degenerate parity-breaking separable ground state (GS) resulting from the crossing of two levels of opposite spin-parity, but at non-transverse factorizing magnetic fields, spin-parity is broken and the GS is no longer degenerate (although the system still presents an exactly separable GS). We also examine the pairwise entanglement of cyclic spin-1/2 chains in the vicinity of the factorizing magnetic field, showing that it acquires full range also in the non-transverse case. Related aspects of the magnetization and entropy of entanglement between two spins and the rest of the system are also discussed. Martin Drechsler Departamento de Física, facultad de ciencias exactas y naturales, Universidad de Buenos Aires Moises Porfirio Rojas Leyva Universidade Federal de Lavras Quantum teleportation via Ising-XXZ diamond chain structure Most investigations in entanglement teleportation are focused on two-qubits Heisenberg chain as a quantum channel to teleport an unknown state. The entanglement for one infinite chain structure is a considerably cumbersome task. The quantum entanglement properties involving an infinite chain structure is quite important, not only because the mathematical calculation is cumbersome but also because real materials are well represented by an infinite chain. Recently, thermal average was obtained for two-qubit density operator immersed in the Ising-XXZ diamond chain structure [13]. In this paper, we explore the quantum teleportation, using the average thermal of the two-qubit density operator as a quantum channel for the standard teleportation of two-qubit state. Using standard teleportation protocol, we have derived the analytical expressions for output concurrence, fidelity and average fidelity. We study in detail the effects of coupling parameters, external magnetic field and temperature over quantum teleportation. Finally, we study the relations between entanglement of the quantum channel, the output entanglement and the average fidelity of the system. Murray Olsen University of Queensland NonGaussian correlations and entanglement in Kerr media We show how Kerr media such as optical fibres can be used to generate non-Gaussian states of the electromagnetic field. We further show how these can be simply manipulated to provide a source of entangled fields and Einstein-Podolsky-Rosen steering. Given the requirement of non-Gaussian states for central quantum information tasks, this will be a useful resource. Norma Canosa IFLP-CONICET. Departamento de Física-Universidad Nacional de La Plata Evolution and control of entanglement between two harmonic modes in stable and unstable regimes The exact evolution of the entanglement between two harmonic modes generated by an angular momentum coupling is examined. Such system arises when considering a particle in a rotating anisotropic harmonic trap or a charged particle in a fixed harmonic potential in a magnetic field, and exhibits a rich dynamical structure, with stable, unstable and critical regimes according to the values of the rotational frequency or field and trap parameters. Consequently, it is shown that the entanglement generated from an initially separable gaussian state can exhibit quite distinct evolutions, ranging from quasiperiodic behavior in stable sectors to different types of unbounded increase in critical and unstable regions. The latter lead respectively to a logarithmic and linear growth of the entanglement entropy with time. It is also shown that entanglement can be controlled by tuning the frequency, such that it can be increased, kept constant or returned to a vanishing value just with stepwise frequency variations. Exact analytic expressions for the entanglement entropy in the different dynamical regimes are provided. Pablo Barberis Blostein UNAM Optomechanical laser cooling with mechanical modulations We theoretically study the laser cooling of optomechanical cavities when the mechanical resonance frequency and damping depend on time. In the regime of weak optomechanical coupling we extend the theory of laser cooling using an adiabatic approximation. We discuss the modifications of the cooling dynamics and compare it with numerical simulations in a wide range of modulation frequencies. Pablo Gonzalez Universidad de Concepción Quantum key distribution with untrusted detectors Side-channel attacks currently constitute the main challenge for quantum key distribution (QKD) to bridge theory with practice. So far two main approaches have been introduced to address this problem, (full) device-independent QKD and measurement-device-independent QKD. Here we present a third solution that might exceed the performance and practicality of the previous two in circumventing detector side-channel attacks, which arguably, is the most hazardous part of QKD implementations. We prove its security in the high-loss regime against a particular class of attacks, and we present a proof-of-principle experiment that demonstrates the feasibility of the protocol. Patrice Camati UFABC Observing fluctuations under feedback controlled protocols Much research has been done in the last decades studying thermal fluctuations in classical systems both theoretically and experimentally. More recently, the study of quantum fluctuation theorems reached the experimental level, first in a NMR setup and then in an ion-trap setup. Also, the study of fluctuation relations under feedback control (Maxwell's demon-like) has been the subject of research both in classical and quantum systems. Up until now, quantum fluctuation relations under feedback controlled protocols has not been tackled experimentally. We present a quantum circuit which enables the measurement of the characteristic function of the probability work distribution under feedback controlled protocols and discuss its implementation in a NMR setup. Paul Erker UAB, USI Peter Alexander Bouvrie Morales Centro Brasileiro de Pesquisas Físicas Composite bosons: entangled parts or bosonic whole? Most bosons in nature are composites made of more elementary bosons and fermions. Still, from hadrons to ultracold molecules, these composites behave very similarly to elementary bosons, because the statistics of the underlying constituents is negligible. The deviation from ideal bosonic behavior is quantified by the normalization ratio of the quantum state of N composites. Using tools from quantum information science, the normalization ratio for two-boson and two-fermion composites can be bound efficiently in terms of entanglement measures [1,3,4]. Using these results, we predict an abrupt transition between ordinary and exaggerated bosonic behavior in a condensate of twoboson composites [3], and show how the entanglement between the parts becomes observable in the collective interference pattern of the bosonic whole [2]. [1] M.C.Tichy, P.A. Bouvrie & K. Mølmer: Bosonic behavior of entangled fermions. Phys. Rev. A 86, 042317 (2012) [2] M.C.Tichy, P.A. Bouvrie & K. Mølmer: Collective Interference of Composite Two-Fermion Bosons. Phys. Rev. Lett. 109, 260403 (2012) [3] M.C.Tichy, P.A. Bouvrie & K. Mølmer: Two-boson composites. Phys. Rev. A 88, 061602(R) (2013) [4] M.C.Tichy, P.A. Bouvrie & K. Mølmer: How bosonic is a pair of fermions? Pietro Liuzzo-Scorpo University of Nottingham Efficiency and correlation in coherent feedback cooling The study of a measurement-based feedback protocol applied to a, initially uncorrelated, system consisting of two qubits (identified as principal system and auxiliary respectively) leaded us to investigate the relation between correlations and efficiency of the feedback protocol. In particular we studied the nature of correlations at each step of the protocol, i.e. the amount of classical and quantum correlations builded up and consumed. R. Rossignoli Depto. de Física-IFLP-CIC, Universidad Nacional de La Plata Generalized conditional entropy in quantum systems We analyze, for general concave entropic forms, the conditional entropy of a quantum system A+B obtained as a result of a local measurement on one of the subsystems (B). This quantity measures the average conditional mixedness of A after such measurement, and its minimum over all local measurements is the associated entanglement of formation between A and a purifying third system C. In the standard case, it also determines the quantum discord. We show that for certain states, the minimizing measurement can be determined analytically and is universal, i.e.,the same for all concave forms. While these properties no longer hold for general states, we also show that in the special case of the linear entropy, an explicit expression for the conditional entropy can be obtained, whose minimum in a general qudit-qubit state can be determined analytically in terms of the eigenvalues of a simple 3x3 matrix. Such minimum determines the maximum conditional purity of A, and the associated minimizing measurement is shown to be universal in the vicinity of maximal mixedness. An approximate analytic solution for general entropies is also derived, together with a simple geometrical picture in terms of a correlation ellipsoid. Some illustrative results are as well discussed. Raphael Campos Drumond Universidade Federal de Minas Gerais Violations of general Bell inequalities for multipartite pure random states We estimate the probability of random$N$-qudit pure states violating Bell Inequalities at the most general scenario, \ie$m$observables,$v$possible outcomes per observables, and$N$parts. In fact, we claim that if the local dimension$d$of a$N$-party quantum system satisfy$d>mv(2m-1)^2$, then typically we will no able to see any degree of violation for any Bell inequality. Raul Oscar Vallejos CBPF Irreversibility versus entanglement spectrum in chaotic maps We study the connection between chaotic spectrum of the asymptotic density matrix and irreversibility. The asymptotic states are obtained by iterative application of certain unitary chaotic quantum maps. Renato M. Angelo Federal University of Parana (UFPR) A measure of physical reality From the premise that an observable is real after it is measured, we envisage a tomography-based protocol that allows us to propose a quantifier for the degree of indefiniteness of an observable given a quantum state. Then we find that reality can be inferred locally only if there is no quantum correlation in the system, i.e., quantum discord prevents Einstein's notion of separable realities. Also, by monitoring changes in the local reality upon measurements on remote systems, we are led to define a measure of nonlocality. Proved upper-bounded by discord-like correlations and requiring indefiniteness as a basic ingredient, our measure signals nonlocality even for separable states, thus revealing nonlocal aspects that are not captured by Bell inequality violations. Romeu Rossi Junior Universidade Federal de Viçosa Optomechanical device as a quantum detector We show that the optomechanical device composed by a Fabry-Perot cavity divided in two by a non-transmissive membrane, subject to radiation pressure, can be used as meter. The system is composed by two optical modes, one in each side of the cavity, and the membrane is assumed to behave as a quantum mechanical oscillator. The effect of the radiation pressure on the membrane will record information about the a field mode quadrature. The membrane plays the role of a quantum detector that allows for homodyne measurement of the optical mode. Sanah Altenburg Department Physics, University Siegen Fisher information, multiparameter estimation and entanglement In quantum metrology, entanglement is used as a resource to enhance experiments for high precision phase estimation, such as atomic clocks or gravitational wave detectors. The quantum Fisher information is a quantity that allows us to decide whether a state is useful in order to overcome classical limits in precision for such experiments. Besides, the quantum Fisher information can also detect entanglement: It has an upper bound for separable states, so that overcoming classical limits in precision implies entanglement. These ideas have been extended to multipartite entanglement, but so far only to systems of qubits [1,2]. For a more general framework we use the Fisher information for multiparameter estimation. We will consider dynamics generated by arbitrary local generators in systems of qudits. In our case, the Fisher information for multiparameter estimation is a matrix. We investigate the Fisher information matrix and derive a criterion for detecting entanglement. [1] P. Hyllus et al., PRA 85, 022321 (2012). [2] G. Tóth, PRA 85, 022322 (2012). Saulo Vicente Moreira Université Paris Diderot- Paris 7 Modelling Invasiveness in a Leggett-Garg Inequality Test We propose a theoretical model to test Leggett-Garg inequalities thats aims to determine the effects of the invasiveness of the measurement in its violation. This is done by investigating a specific measurement model where invasiveness is related in an intuitive way to one parameter introduced in the model. We also relate our invasiveness model to ones where violation disappears when the "size" of the system is increased, suggesting that even in this case, the invasiveness is the only element determining the violation (or not) of Legget Garg inequalities. Thiago O. Maciel Universidade Federal de Minas Gerais - UFMG Quantum process tomography with informational incomplete data of two$J$-coupled heterogeneous spins relaxation We reconstruct the time dependent quantum map corresponding to the relaxation process of a two-spin system in liquid-state NMR at room temperature. By means of quantum tomography techniques that handle informational incomplete data, we show how to properly post-process and normalise the measurements data for the simulation of quantum information processing, overcoming the unknown number of molecules prepared in a non-equilibrium magnetisation state ($N_j$) by an initial sequence of radiofrequency (RF) pulses. From the reconstructed quantum map, we infer both longitudinal ($T_1$) and transversal ($T_2$) relaxation times, and introduce the$J$-coupling relaxation times ($T^J_1$,$T^J_2\$), which are relevant for quantum information processing simulations. Tiago Barbin Batalhão UFABC and University of Vienna Irreversibility and the arrow of time in a quenched quantum system We address the issue of testing experimentally the thermodynamic arrow of time by using a nuclear magnetic resonance set-up that allows for the determination of the nonequilibrium entropy produced in an isolated spin-1/2 system following fast quenches of an external magnetic field. We demonstrate that the macroscopic average entropy production equals the entropic distance between a microscopic process and its time-reversal. 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https://topdrawer.aamt.edu.au/Geometric-reasoning/Good-teaching/Writing-a-proof | Home > Geometric reasoning > Good teaching > Writing a proof
# Writing a proof
Be aware that some students take much longer than others to appreciate the demands of a proof. They may need to practise with one or two step proofs, rehearsing the reasons verbally.
Once students are ready, constructing a proof is much like writing an essay.
• Plan the sequence of ideas
Allow students time to explore the diagrams and experiment without interruption. Following unsuccessful pathways often teaches more about deductive thinking than achieving instant success.
• Provide time for discussion of ideas
Students can evaluate a variety of strategies and select the most efficient or elegant sequence.
• Model the formal writing stage
Demonstrate how to write the reasons either by providing the outline of a proof or the lines of proof which need to be put in order.
Highlight instances when the order of the ideas is important. For example, in a similarity proof, the order of the angles does not matter. However, in the SAS test for congruence, the angle argument should be stated between the side statements to show that the angle is included.
Providing alternative proofs builds student confidence and develops their capacity to construct a proof without assistance.
## Proving Pythagoras' theorem
Pythagoras' theorem will be familiar to students. There are many different ways to verify the theorem visually, and to prove it. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.874249279499054, "perplexity": 1218.8476648068017}, "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-2018-39/segments/1537267160142.86/warc/CC-MAIN-20180924031344-20180924051744-00516.warc.gz"} |
https://www.arxiv-vanity.com/papers/1203.0705/ | arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org.
# Valley-Polarized Metals and Quantum Anomalous Hall Effect in Silicene
Motohiko Ezawa Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan
###### Abstract
Silicene is a monolayer of silicon atoms forming a two-dimensional honeycomb lattice, which shares almost every remarkable property with graphene. The low energy structure of silicene is described by Dirac electrons with relatively large spin-orbit interactions due to its buckled structure. The key observation is that the band structure is controllable by applying electric field to silicene. We explore the phase diagram of silicene together with exchange field and by applying electric field . There appear quantum anomalous Hall (QAH) insulator, valley polarized metal (VPM), marginal valley polarized metal (M-VPM), quantum spin Hall (QSH) insulator and band insulator (BI). They are characterized by the Chern numbers and/or by the edge modes of a nanoribbon. It is intriguing that electrons have been moved from a conduction band at the K point to a valence band at the K’ point for in the VPM. We find in the QAH phase that almost flat gapless edge modes emerge and that spins form a momentum-space skyrmion to yield the Chern number. It is remarkable that a topological quantum phase transition can be induced simply by changing electric field in a single silicene sheet.
Silicene, a monolayer of silicon atoms forming a two-dimensional honeycomb lattice, has been synthesizedLalmi ; GLayPRL ; Takagi and attracts much attentionShiraishi ; LiuPRL ; LiuPRB ; EzawaNJP ; EzawaJ recently. Almost every striking property of graphene could be transferred to this innovative material. It has additionally a salient feature, that is a buckled structureShiraishi ; LiuPRL owing to a large ionic radius of silicon. Silicene has a relatively large spin-orbit (SO) gap of meV, which provides a mass to Dirac electrons. Furthermore, we may control experimentally the massEzawaNJP by applying the electric field . Silicene undergoes a topological phase transition from a quantum spin Hall (QSH) state to a band insulator (BI) as increasesEzawaNJP . A QSH state is characterized by a full insulating gap in the bulk and helical gapless edgesKaneMele ; Wu ; Hasan ; Qi .
There exits another state of matter in grapheneQiao ; Tse ; Yang , that is a quantum anomalous Hall (QAH) stateLiu ; Onoda , characterized by a full insulating gap in the bulk and chiral gapless edges. Unlike the quantum Hall effect, which arises from Landau-level quantization in a strong magnetic field, the QAH effect is induced by internal magnetization and SO coupling.
In this paper we analyze the band structure of silicene together with exchange field and by applying electric field to silicene. We explore the phase diagram in the - plane. Silicene has a rich varieties of phases because the electric field and the exchange field have different effects on the conduction and valence bands characterized by the spin and valley indices. There are insulator phases, which are the QSH, QAH and BI phases. There emerges a new type of metal phase, the valley-polarized metal (VPM) phase, where electrons have been moved from a conduction band at the K point to a valence band at the K’ point for . Such a phase is utterly unknown in literature as far as we are aware of. There are also metallic states on phase boundaries, which are metal (M), marginal-VPM (M-VPM) and spin VPM (SVPM) states. All these phases and states are characterized by the Chern numbers and/or by the edge modes of a nanoribbon. It is possible to materialize any one of them by controlling at an appropriate value of . Furthermore, as we have pointed out elsewhereEzawaNJP , by applying an inhomogeneous field , it is possible to materialize some of these topological phases together with states on the phase boundaries simultaneously in a single silicene sheet.
Silicene consists of a honeycomb lattice of silicon atoms with two sublattices made of A sites and B sites. The states near the Fermi energy are orbitals residing near the K and K’ points at opposite corners of the hexagonal Brillouin zone. We refer to the K or K’ point also as the K point with the valley index . We take a silicene sheet on the -plane, and apply the electric field perpendicular to the plane. Due to the buckled structure the two sublattice planes are separated by a distance, which we denote by with Å. It generates a staggered sublattice potential between silicon atoms at A sites and B sites.
The silicene system is described by the four-band second-nearest-neighbor tight binding model,
H +iλR1(Ez)∑⟨i,j⟩αβc†iα(σ×^dij)zαβcjβ −i23λR2∑⟨⟨i,j⟩⟩αβμic†iα(σ×^dij)zαβcjβ +ℓ∑iαμiEzc†iαciα+M∑iαc†iασzciα, (1)
where creates an electron with spin polarization at site , and run over all the nearest/next-nearest neighbor hopping sites. We explain each term. (i) The first term represents the usual nearest-neighbor hopping with the transfer energy eV. (ii) The second term represents the effective SO coupling with meV, where is the Pauli matrix of spin, with if the next-nearest-neighboring hopping is anticlockwise and if it is clockwise with respect to the positive axis. (iii) The third term represents the first Rashba SO coupling associated with the nearest neighbor hopping, which is induced by external electric fieldHongki ; Tse . It satisfies and becomes of the order of eV at the critical electric field meVÅ. (iv) The forth term represents the second Rashba SO coupling with meV associated with the next-nearest neighbor hopping term, where for the A (B) site, and with the vector connecting two sites and in the same sublattice. (v) The fifth term is the staggered sublattice potential term. (vi) The sixth term represents the exchange magnetization: Exchange field may arise due to proximity coupling to a ferromagnet such as depositing Fe atoms to the silicene surface or depositing silicene to a ferromagnetic insulating substrate, as has been argued for grapheneQiao ; Tse ; Yang . The Hamiltonian (1) can also be used to describe germanene, which is a honeycomb structure of germaniumLiuPRL ; LiuPRB , where various parameters are eV, meV, meV and Å.
In this paper we derive the topological phase diagram in the - plane and make its physical interpretation. The topological quantum numbers are the Chern number and the index. If the spin is a good quantum number, the index is identical to the spin-Chern number . They are defined when the state is gapped and when the Fermi level is taken within the gap, and given by and , where is the summation of the Berry curvature in momentum space over all occupied states of electrons with . They are well defined even if the spin is not a good quantum numberProdan09B ; Yang . In the present model the spin is not a good quantum number because of spin mixing due to the Rashba couplings and , and the resulting angular momentum eigenstates are indexed by the spin chirality . We can calculate these numbers at each point in the - plane by using the standard formulasQiao ; Tse ; Yang .
We present our result on the phase diagram in Fig.1. We show later how to derive the phase boundaries based on the low-energy Dirac theory. We have also calculated the band structure of a silicene nanoribbon with zigzag edges, which we give in Fig.2 for typical points in the phase diagram. The topological numbers are in the BI phase, in the QSH phase, in the QAH phase with and in the QAH phase with . In all these states the band gap is open, where the Fermi level is present, and they are insulators.
We first discuss the system at and compare our results with those previously obtained in grapheneQiao ; Tse ; Yang . The main difference is the appearence of almost flat edge modes in our system (Fig.3). This occurs because the Rashba interactions are different between these two systems. We have for and at the K and K’ points in silicene, but and in graphene. Nevertheless, the difference is only quantitative. As far as the topological properties are concerned, there exists no difference. Indeed, in these two systems, the Chern number is identical in each corresponding phase together with quantized Hall conductivity, and the edge states support the edge current. However, the group velocity of the edge modes is extremely small due to the almost flat gapless modes in silicene.
Our most important result is the VPM phase, which appears in such regions that and occupies a major part of the phase diagram. A part of the conduction (valence) band is above (below) the Fermi level at the K (K’) point for , as is observed in Fig.2(VPM). Hence, electrons are moved from the K valley to the K’ valley, as implies the valley polarization. The phase is characterized by the property that it is a metallic state though gaps are open both at the K and K’ points. We note that the Chern and spin-Chern numbers are ill-defined in the VPM phase, since the Fermi level does not lie inside the band gaps at the K and K’ points simultaneously.
There exist M-VPM states on phase boundaries indicated by heavy lines in the phase diagram, where the conduction and valence bands touch the Fermi surface at the K and K’ points, respectively, for . On the other hand, in SVPM states the conduction and valence bands touch the Fermi surface both at the K and K’ points. We expect topological quantum critical phenomena in these states.
In order to explore the physics underlying the phase diagram, we analyze the low-energy effective Hamiltonian derived from the tight binding model (1). It is described by the Dirac theory around the point as
Hη= ℏvF(ηkxτx+kyτy)+ητzh11+ℓEzτz+Mσz +λR1(ητxσy−τyσx)/2 (2)
with , where is the Pauli matrix of the sublattice pseudospin, is the Fermi velocity, and Å is the lattice constant.
⎛⎜ ⎜ ⎜ ⎜ ⎜⎝E(1,1)ℏvFk−iaλR2k−0ℏvFk+E(1,−1)−iλR1−iaλR2k−−iaλR2k+iλR1E(−1,1)ℏvFk−0iaλR2k+ℏvFk+E(−1,−1)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ (3)
in the basis , where , and the diagonal elements are
E(sz,tz)=λSOsztz+ℓEztz+Msz, (4)
with the spin and the sublattice pseudospin . They are not good quantum numbers in general. However, since and are very small with respect to the other parameters, it is a good approximation to set in most cases. Thus the spin is almost a good quantum number in general. An exceptional case occurs when two Dirac cones collapse and cross each other, forming a QAH state after taking into account the effect of , as we soon discuss.
We diagonalize the Hamiltonian (3) and obtain four energy levels. When two energy levels coincide, the band gap becomes zero, as found in Fig.2(M,VMP3,VMP2,SVPM). This occurs at the K and K’ points, where . Let us temporarily neglect because it is very small. Then, the band closes when with (4). They yield four lines described by for , for , and two lines by outside the square. They are illustrated by dotted lines in Fig.1. These lines are modified by the nonzero effect of , but the modification is too small to be recognized in Fig.1. See also (5) for the typical order of correction.
The Hamiltonian can be diagonalized analytically in some cases. First, along the -axis in the phase diagram [Fig.1], we have already demonstratedEzawaNJP that a topological phase transition occurs along the -axis from the QSH insulator [Fig.2(QSH)] to the band insulator [Fig.2(BI)]. The critical point is given by
Ec=±2λSOℓ[√1+(α/ℓ)2−1(α/ℓ)2], (5)
where we have set with Å. Note that the effect of is negligible, . The SVPM realizes at the critical point, where helical currents flow in the bulk.
Second, along the -axis, the first Rashba interaction vanishes (), and the energy spectrum reads
E=±√a2λ2R2k2+(M−s√λ2SO+ℏ2v2Fk2)2. (6)
We study a topological phase transition along the -axis based on this formula (6). When , there are two spin-degenerate Dirac cones for conduction and valence bands with a gap between them [Fig.2(QSH1)]. As increases, the spin-up (spin-down) Dirac cones are pushed upward (downward) [Fig.2(QSH2)]. When , the band gap is given as at , and it closes at : This is a topological phase transition point [Fig.2(M)]. Let us temporally assume . Then, as increases further, the two Dirac cones cross each other making a circle around each K point. Actually, the Rashba interaction () mixes up and down spins, turning the crossing points into the anticrossing points, and opens a gap to form the QAH insulating state [Fig.2(QAH) and Fig.3].
When , the gap is given by
Δ=aλR2 ⎷M2ℏ2v2F+a2λ2R2−λ2SOℏ2v2F (7)
at
kac=√ℏ4v4F(M2−λ2% SO)−a2λ2R2λ2SO(2ℏ2v2F+a2λ2R2)ℏvF(ℏ2v2F+a2λ2R2). (8)
We present the energy spectrum (6) and the Berry curvature calculated by using the corresponding wave function at in Fig.4. As explained there, spins rotates across the anticrossing point, generating a skyrmion spin texture in the momentum space. This is consistent with the previous study for grapheneTse . The radius of the anticrossing circles is given by (8). We comment that the gap (7) is of the order of eV when is of the order of meV.
We now examine a point in the phase diagram such that . In all regions where the effects of and are negligible, the energy spectrum is derived as
E=szM±√ℏ2v2Fk2+(ℓEz−ηszλSO)2. (9)
The effect of is to change the mass of the Dirac electron. Let us increase from at a fixed value of . The mass decreases (increases) for the Dirac cone characterized by () until , but the behavior becomes opposite after . As a result the tip of each Dirac cone is pushed either downward or upward as indicated in Fig.2. Consequently the valley symmetry is broken. Note that the energy difference at each momentum between the conduction and valence bands with the same spin is given by
ΔE±=2√ℏ2v2Fk2+(ℓEz∓λSO)2 (10)
for , and this is independent of . Thus, the difference is smaller for the up-spin Dirac cones at the K point, but this is opposite at the K’ point.
We finally determine the phase boundary. It is determined as a boundary between insulating and metallic states. The Chern and spin-Chern numbers are quantized in insulating states, while they are ill-defined in metallic states. As we have seen, each Dirac cone moves upward or downward oppositely at the K and K’ points. Because of this phenomenon the system can become metallic though the gap is open both at the K and K’ points. This is the VPM state. It occurs when one valence band crosses the Fermi level. The condition yields four heavy lines () in the phase diagram (Fig.1). On the other hand, the gap formula (7) determines the boundary between the QAH phase and the VPM phase, which are the parabolic curves in the phase diagram (Fig.1). In passing we comment that the VPM phase is metallic in nature and does not have mobility gap. Thus the transition from insulator to VPM might accompany a mobility gap closing.
I am very much grateful to N. Nagaosa for many fruitful discussions on the subject. This work was supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture No. 22740196. | {"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": 0.9287325143814087, "perplexity": 911.534047876132}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1606141753148.92/warc/CC-MAIN-20201206002041-20201206032041-00016.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=1536214&postcount=47 | View Single Post
How can there be an edge if it is 4 deminisional (at least), the universe is expanding 4 deminsionally, if time is the 4th deminision then we must already be on the edge. i.e. Since the universe was smaller in the past and will be larger in the future, that means that currently we must be on the edge. | {"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": 0.8012792468070984, "perplexity": 307.31152905002654}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368705790741/warc/CC-MAIN-20130516120310-00087-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/587393/tensor-product-equivalent-definitions | # Tensor product equivalent definitions
I'm studying tensor products right now and I've came across multiple definitions. The one I'm confused with is when we have vector spaces $V$ and $W$ and we define the tensor product as the quotient of the free module $F$ of $V \times W$ and $E$, where $E$ can either be:
Definition 1: $E$ is the subspace of $F$ spanned by all elements: $$(v_1+v_2,w)-(v_1,w)-(v_2,w)$$ $$(v,w_1+w_2)-(v,w_1)-(v,w_2)$$ $$a(v,w)-(av,w)$$ $$a(v,w)-(v,aw)$$
or definition 2: $E$ is the subspace of $F$ spanned by all elements: $$(v_1+v_2,w)-(v_1,w)-(v_2,w)$$ $$(v,w_1+w_2)-(v,w_1)-(v,w_2)$$ $$(av,w)-(v,aw)$$
Are these definitions equivalent? (namely, what I'm asking is, does: $(al_1 \otimes l_2) = (l_1 \otimes al_2), = a(l_1 \otimes l_2)$ hold in definition 2)?
Thanks very much
-
None of these is a definition. The correct definition of the tensor product is the universal property: the tensor product classifies bilinear maps. Your "definitions" are actually constructions (but there are also other constructions of the tensor product; of course isomorphic). Now to answer your question: The first construction is the correct one, the second is not correct. If $\beta : V \times W \to U$ is a map which is additive in each variable and balanced in the sense that $\beta(av,w)=\beta(v,aw)$ for $a \in K$, $v \in V$ and $w \in W$, then there is no reason why we can conclude $a \beta(v,w) = \beta(a v,w)$. Try to find an example for $V=W=U=K=\mathbb{C}$.
Thanks for your help with this. The issue I am having is that some notions of the two constructions seem too different. For example the proof (at least the one I've seen) that if $v_1,v_2...$ is a basis for V and $w_1,w_2,...$ is a basis for $W$ then the tensors $v_iw_j$ are a basis for the tensor product seem to not work (namely we can't express any tensor in terms of linear combinations of the $(v_i \otimes w_j)$ because nothing "goes outside" the tensors as in definition 1)? If not, can you direct me to a proof? – user112532 Dec 1 '13 at 23:34 | {"extraction_info": {"found_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": 0.9780986905097961, "perplexity": 101.88799566934246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396872.10/warc/CC-MAIN-20160624154956-00089-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.mathway.com/examples/statistics/algebra-review/factoring-trinomials?id=85 | # Statistics Examples
Step 1
Rewrite as .
Step 2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 3
Rewrite the polynomial.
Step 4
Factor using the perfect square trinomial rule , where and . | {"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": 0.8388882875442505, "perplexity": 1961.4198595949651}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103640328.37/warc/CC-MAIN-20220629150145-20220629180145-00682.warc.gz"} |
https://tepanonprofit.org/u2kp2kwb/aed182-when-is-a-function-differentiable | 208.957.6949
# when is a function differentiable
Where? Say, for the absolute value function, the corner at x = 0 has -1 and 1 and the two possible slopes, but the limit of the derivatives as x approaches 0 from both sides does not exist. Answer to: 7. Differentiable means that a function has a derivative. For x 2 + 6x, its derivative of 2x + 6 exists for all Real Numbers. Example Let's have another look at our first example: $$f(x) = x^3 + 3x^2 + 2x$$. Answer. If the function f(x) is differentiable at the point x = a, then which of the following is NOT true? No number is. Weierstrass in particular enjoyed finding counter examples to commonly held beliefs in mathematics. there is no discontinuity (vertical asymptotes, cusps, breaks) over the domain.-x⻲ is not defined at x =0 so technically is not differentiable at that point (0,0)-x -2 is a linear function so is differentiable over the Reals. If f is differentiable at a, then f is continuous at a. Get your answers by asking now. Differentiable 2020. If the one-sided limits both exist but are unequal, i.e., , then has a jump discontinuity. Radamachers differentation theorem says that a Lipschitz continuous function $f:\mathbb{R}^n \mapsto \mathbb{R}$ is totally differentiable almost everywhere. Consider the function $f(x) = |x| \cdot x$. This graph is always continuous and does not have corners or cusps therefore, always differentiable. Exercise 13 Find a function which is differentiable, say at every point on the interval (− 1, 1), but the derivative is not a continuous function. If F not continuous at X equals C, then F is not differentiable, differentiable at X is equal to C. So let me give a few examples of a non-continuous function and then think about would we be able to find this limit. the function is defined on the domain of interest. As in the case of the existence of limits of a function at x 0, it follows that. True. They've defined it piece-wise, and we have some choices. Rolle's Theorem. if and only if f' (x 0 -) = f' (x 0 +) . It the discontinuity is removable, the function obtained after removal is continuous but can still fail to be differentiable. When this limit exist, it is called derivative of #f# at #a# and denoted #f'(a)# or #(df)/dx (a)#. geometrically, the function #f# is differentiable at #a# if it has a non-vertical tangent at the corresponding point on the graph, that is, at #(a,f(a))#.That means that the limit #lim_{x\to a} (f(x)-f(a))/(x-a)# exists (i.e, is a finite number, which is the slope of this tangent line). There is also a look at what makes a function continuous. As an answer to your question, a general continuous function does not need to be differentiable anywhere, and differentiability is a special property in that sense. Now one of these we can knock out right from the get go. Differentiability implies a certain âsmoothnessâ on top of continuity. exists if and only if both. ? 1. A function is differentiable when the definition of differention can be applied in a meaningful manner to it.. But it is not the number being differentiated, it is the function. When a function is differentiable it is also continuous. Before the 1800s little thought was given to when a continuous function is differentiable. This requirement can lead to some surprises, so you have to be careful. 226 of An introduction to measure theory by Terence tao, this theorem is explained. where $W_t$ is a Wiener process and the functions $a$ and $b$ can be $C^{\infty}$. This should be rather obvious, but a function that contains a discontinuity is not differentiable at its discontinuity. B. 2. Trump has last shot to snatch away Biden's win, Cardi B threatens 'Peppa Pig' for giving 2-year-old silly idea, These 20 states are raising their minimum wage, 'Super gonorrhea' may increase in wake of COVID-19, ESPN analyst calls out 'young African American' players, Visionary fashion designer Pierre Cardin dies at 98, Cruz reportedly got $35M for donors in last relief bill, More than 180K ceiling fans recalled after blades fly off, Bombing suspect's neighbor shares details of last chat, Biden accuses Trump of slow COVID-19 vaccine rollout. A function which jumps is not differentiable at the jump nor is one which has a cusp, like |x| has at x = 0. exists if and only if both. Throughout, let ∈ {,, …, ∞} and let be either: . 0 0. lab_rat06 . It looks at the conditions which are required for a function to be differentiable. A discontinuous function is not differentiable at the discontinuity (removable or not). It is not sufficient to be continuous, but it is necessary. -x⁻² is not defined at x =0 so technically is not differentiable at that point (0,0), -x -2 is a linear function so is differentiable over the Reals, x³ +2 is a polynomial so is differentiable over the Reals. If it is not continuous, then the function cannot be differentiable. The first derivative would be simply -1, and the other derivative would be 3x^2. Then it can be shown that$X_t$is everywhere continuous and nowhere differentiable. For example, the function A function differentiable at a point is continuous at that point. The function, f(x) is differentiable at point P, iff there exists a unique tangent at point P. In other words, f(x) is differentiable at a point P iff the curve does not have P as a corner point. x³ +2 is a polynomial so is differentiable over the Reals To see this, consider the everywhere differentiable and everywhere continuous function g (x) = (x-3)* (x+2)* (x^2+4). Sal analyzes a piecewise function to see if it's differentiable or continuous at the edge point. Learn how to determine the differentiability of a function. If there’s just a single point where the function isn’t differentiable, then we can’t call the entire curve differentiable. Ode y n = f ' ( x ) = 0 even though it always lies between and... Irrespective of whether it is continuous: Proof fails to be continuous at and. Actually continuous ( though not differentiable //math.stackexchange.com/questions/1280495/when-is-a-continuous-function-differentiable/1280525 # 1280525, https: //math.stackexchange.com/questions/1280495/when-is-a-continuous-function-differentiable/1280541 # 1280541 when... Old problem in the study of calculus obtained after removal is continuous but not differentiable to avoid if. It has some sort of corner C ∞ of infinitely differentiable functions, is the function is it! 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Each point in its domain dt^ { 1/2 } \$ even though it always lies between and., smooth continuous curve at the origin, creating a discontinuity locally approximated linear. Is everywhere continuous and nowhere differentiable every single point in its domain as varies... By stochastic differential equations or false.Every continuous function differentiable? also a at! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.9245812296867371, "perplexity": 564.665504548302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178357935.29/warc/CC-MAIN-20210226175238-20210226205238-00624.warc.gz"} |
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Apr4 comment Does it make sense to learn any other language except English, being a mathematician? @Howard Langtone consider Gelfond, Calculus of finite differences (1959). inis.jinr.ru/sl/vol1/UH/_Ready/Mathematics/… It has been translated to English only in 1971 in India. It is the only book where I found criteria of possibility to represent an analytic function as Newton series. Apr1 comment Has anybody ever considered “full derivative”? @columbus8myhw this is not closed form... Apr1 comment Has anybody ever considered “full derivative”? @columbus8myhw it is not equal to e. Apr1 comment Has anybody ever considered “full derivative”? @columbus8myhw I think e can be expressed in closed form by modifying the formula. Mar28 comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ @Martin R as $\log |a|$ Mar28 comment Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$ Mar28 comment Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$ @abel Cosine integral mathworld.wolfram.com/CosineIntegral.html Mar28 comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ @Arpan Banerjee we get infinity in both numerator and denomenator with this rule. If u know how to apply it properly, make an answer please. Mar28 comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ @Arpan Banerjee logarithm is not differentiable at 0, L'Hopital's rule is not applicable Mar28 comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ @Arpan Banerjee counter-example: $x=1$, $\varepsilon$=2 Mar28 comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ @math also if $x-\varepsilon$ <0 the seond factor also becomes infinite, and I am exactly interested in the case $|x|<\varepsilon$... Mar28 comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ @math are u sure? What if a tends to zero from below? then the second factor in the numerator becomes infinite... Mar28 comment Given this operator what is inverse operator? @Martin R regarding sums to non-integer limits, look here: en.wikipedia.org/wiki/Indefinite_sum anyway, I found what I was looking for. Mar28 comment Given this operator what is inverse operator? @Martin R 4 well the limit is added for some functions that have t in the denomenator (like 1/x). The sum should be computerd in closed form then take limit. The limit is not necessary for other functions. Mar28 comment Given this operator what is inverse operator? @Olivier Oloa actually, $$\Delta_{sym}[f(x)]=(\Delta_{full}[f(x)]+\Delta_{full}[f(x-\varepsilon)])/2$$, but what does it help? Mar26 comment Why hyperreal numbers are built so complicatedly? Differentiability: $$(f(x+\varepsilon)-f(x-\varepsilon))/(2\varepsilon)$$ What needs more definition with it? Mar26 comment Why hyperreal numbers are built so complicatedly? Well what questions still remain unanswered with the definition from the question? Mar26 comment Has anybody ever considered “full derivative”? Does it mean time scales? Mar26 comment Has anybody ever considered “full derivative”? @Mark S. what a problem in defining $\sin \varepsilon$? It just can be represented as a series or in closed form... Where the problem is? Mar25 comment Has anybody ever considered “full derivative”? @Kevin Carlson is this field truly hyperreal? | {"extraction_info": {"found_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": 0.9308731555938721, "perplexity": 2759.4373382398594}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1429246645606.86/warc/CC-MAIN-20150417045725-00009-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://help.simetrix.co.uk/8.3/simetrix/simulator_reference/topics/convergence_accuracyandperformance_overview.htm | # Overview
In transient and DC analyses, an iterative method is used to analyse the circuit. Generally, iterative methods start with an initial guess for the solution to a set of equations and then evaluate the equations with that guess. The result of that evaluation is then used to derive a closer estimate to the final solution. This process is repeated until a solution is found that is within the error tolerance required. SIMetrix and SPICE use Newton-Raphson\footnote{Sir Isaac Newton 1642-1727 and Joseph Raphson 1648-1715} iteration which usually converges extremely rapidly. However, there are occasions when this process is either unreasonably slow or fails altogether. Under these circumstances the simulation will abort.
SIMetrix offers superior convergence which has been achieved as a result of the following developments to the simulator core:
• Automatic pseudo transient analysis algorithm for operating point solution. See below for details.
• Enhancements to GMIN and source stepping algorithms to use a variable step size. (The standard SPICE3 variants use a fixed step).
• Junction GMIN DCOP Convergence Method
• Proprietary enhancements to transient analysis algorithm.
• Optional extended and quad precision solvers
• New matrix solver
• Improvements to device models.
With these improvements, convergence failure with SIMetrix is extremely rare. However, it is impossible to eliminate this problem altogether and there still remain some circuits which fail.
In this chapter we explain some of the causes of non-convergence and some of the strategies SIMetrix uses to prevent it. Also explained is what to do in the rare event that convergence fails. | {"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": 0.8189544677734375, "perplexity": 1373.7970568071762}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882571597.73/warc/CC-MAIN-20220812075544-20220812105544-00276.warc.gz"} |
https://k12.libretexts.org/Bookshelves/Mathematics/Statistics/06%3A_Normal_Distribution_-_Normal_Distributions/6.05%3A_Computing_Probabilities_for_the_Standard_Normal_Distribution | # 6.5: Computing Probabilities for the Standard Normal Distribution
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## Distribution
The local county fair is holding a raffle/competition, and the winner gets a $100 gift card. You would love to win the card, but the competition seems impossible! To enter, you have to guess how many M&M candies of each color: red, blue, yellow, brown, and green, are in a huge jar of M&M’s. There is certainly no way you can actually count them all, you can’t even see most of them since they are in the center and hidden by the candies on the outside. How could you use statistics to help you make an educated guess at the distribution of the colors? Would it help if you knew there were approximately 650 M&M’s in a pound, and about 5 pounds of candy in the jar? The answers are found after the lesson. ## Distribution One of the more important goals of a statistical data analysis is to determine the overall distribution of the data points. Are the values relatively close together? Do they conform to a specific pattern? Do values tend to occur in groups, or suggest a particular shape? By evaluating the distribution of the data, we not only improve our ability to predict future values, but can also determine how reliable the data is as a model of the real situation. figure1 The most well-known and common distribution in statistics is the normal distribution, often referred to as a bell-curve. Normally distributed data follows a specific pattern of decreasing numbers of data points as values range further from the arithmetic mean (commonly known as the average) of the set. Specifically, in a normally distributed set of data, approximately 68% of all the data points are within 1 standard deviation of the mean, and 99.7% of the data lie within three standard deviations of the mean. Don’t worry if these terms seem confusing at this point, in subsequent lessons, particularly in the Predicting Values and Normal Distribution chapters, we will be detailing a more rigorous mathematical evaluation of distribution and standard deviation. For now, it is enough to know that if data is normally distributed, approximately 2/3 (68.2%) of your results should have values within 1 “step” of the average value, and nearly all of your results (95.4%) should be within 2 “steps.” In statistics, distribution may also refer to the differences among members of a sample or population. For instance, demographic distribution can be a major consideration when choosing subjects for a sample group. When many different members of a population are likely to respond differently to the same stimuli, it is usually important to attempt to maintain the same ratio of such differing responses as that of the entire population. You will rarely or never collect data from a group made up of identical members, and differences in point of view or personal preference can have surprising effects on experimental results. The more precise you need your results to be, the more important it becomes to monitor the distribution of your sample. There may be many differences among members of a sample group that influence responses. Common differences include age, size, sex, level of education, religion, culture, geographic location etc. Of course there are many less common characteristics that may affect the results of a study. The goal of a sample is to take into account as many such differences as possible and attempt to represent them in the same ratio as the entire population. ### Considering Distribution You are part of a student committee planning to install a vending machine beside the football field to sell snacks for the benefit of the football team. Naturally, you want to stock the machine with the most popular products, so you decide to conduct a poll of the likely consumers. A brainstorming session with the rest of your committee yields the following likely groups of consumers: 1. Football players 2. Cheerleaders 3. Parents 4. Coaches 5. Male students in the audience 6. Female students in the audience 7. Reporters 8. Sports Scouts Your committee is split by the debate of how to best take the different demographics into account. Part of the committee believes you should ask equal numbers of each group to list their preferred snacks and drinks, and the other part thinks it would be best to give preference to the players and cheerleaders. Perhaps both groups are incorrect; might it not be most effective to buy items that the students in the audience would prefer? What do you think? What distribution considerations will result in the most sales from the vending machine? This problem illustrates the fact that simply representing the demographics of a population as closely as possible may not be the most effective distribution of a sample. Often it is important to identify which statistics are the most valuable to your particular study. In order to maximize sales from the vending machine, it would probably be much more valuable to stock it primarily with items most appealing to the male students in the audience, as they are the most likely to have the desire, freedom, and money to spend on snacks during a game. Additionally, they probably represent the largest single group, followed closely by the female students and then the players and cheerleaders. Taking these considerations into account, you should identify a sample that is primarily composed of male students, with a smaller number of female students. The other groups are either unlikely to have statistically significant differences in preferences anyway, or are just not numerous enough to be significant. ### Determining Probability 2/3 of all households will spend between$700 and $840:In your economics class, you are studying shopping expenditures during the holiday season. The data indicates that the average household will spend approximately$770 on gifts during the month of November. Assuming the data is normally distributed.
figure2
Recall that normally distributed data suggests that 2/3 of the data points occur within 1 standard deviation of the average, and that 95% occur within 2 standard deviations. If 2/3 of the households spent between $700 and$840, that would indicate that 1 standard deviation represents $70 since$700 and $840 are each$70 away from the average of $770. 1. What is the likelihood that any given household will spend more than$910?
Since $910 is$140 more than the mean expenditure of $770, that means that it is 2×$70 or 2 standard deviations above the mean. We can assume that approximately 95% of all values are less extreme than $910, meaning that only 5% will be further than$140 away from the average. Since half of the remaining 5% of households (2.5%) would be made up of the families who will spend an extremely small amount (less than $630), we can assume the other 2.5% to spend more than$910.
2. What is the chance that a household will spend less than $700?$700 is 1 standard deviation below the mean, so approximately 2/3 of all values are less extreme, and 1/3 are more extreme. 1/6 of the values will be more than 1 standard deviation above the mean, and 1/6 below, so we should expect approximately 1/6 of the households in the study to spend less than \$700.
### Calculating the Average Distribution
Suppose you are attempting to estimate the demographic distribution of a school football game. Given the size and constant motion of the crowds, you quickly realize that counting them all isn’t going to work well. Deciding to use a random sample instead, you pick a few different groups at random to calculate the average distribution of the crowd.
a. If you observe a total of 50 people, and count 22 male students, 17 female students, 9 parents, and 2 others, what would be the average demographic distribution of the crowd appear to be as a percentage?
To calculate the contribution of each group to the whole as a percentage, divide the number of members in each group by the total members you counted:
b. If sales records indicate a total of 475 tickets sold, what would you estimate the actual count of each demographic to be?
To estimate the total distribution of the crowd, multiply each group’s estimated percentage by the total number of tickets sold:
### Earlier Problem Revisited
Guess how many M&M candies of each color: red, blue, yellow, and brown, are in a huge jar.
There is certainly no way you can actually count them all, you can’t even see most of them since they are in the center and hidden by the candies on the outside. How could you use statistics to help you make an educated guess at the distribution of the colors? Would it help if you knew there were approximately 650 M&M’s in a pound, and about 5 pounds of candy in the jar?
If you identify an average ratio of colors in a sample of the candies, you could apply that ratio to the estimated total number of candies in the jar.
If there are approximately 650 candies in a pound, and 5 pounds in the jar, we can estimate a total of approximately 3,250 total candies. To get an average distribution of colors, we could either use a sample of the candies we can see through the side of the jar and calculate the percentage of each, or we could research online to see what the company advertises: 24% blue, 13% red, 14% yellow, 14% brown (16% green and 20% orange, but the raffle doesn’t ask about them).
24% of 3250=780 estimated blue
13% of 3250=423 estimated red
14% of 3250=455 estimated each yellow and brown
Of course this is no guarantee of the actual numbers of each color, but given the relatively large sample size, these numbers are likely to be quite a bit more accurate than a simple guess.
## Examples
### Example 1
Suppose a group of 150 students in a college English course take a final exam, and the instructor calculates that the mean score is 87%, with a standard deviation of 3%.
### Example 2
If the scores are normally distributed, what is the approximate probability that a randomly selected score will be between 87% and 93%?
Recall that normally distributed data indicates approximately 68.2% of values within 1 standard deviation of the mean, and 95.4% within 2 standard deviations. Also, recognize that if 68.2% of values are within 1 SD of the mean, then 34.1% are within 1 SD above the mean, and 34.1 are below the mean, as you can see in the graphic:
Since the standard deviation of the grades is 3%, there are two standard deviations between 87% and 93%. If we look at the percentages above 87% for the next two standard deviations, we see that the first incorporates 34.1% of the data, and the second incorporates another 13.6%. Therefore the likelihood that a given score will be between 87% and 93% is 34.1% + 13.6% = 47.7%
### Example 3
In the same class, what is the approximate probability that a randomly selected score will be between 84% and 87%?
84% is 1 standard deviation below the mean, so the probability that a randomly selected score will be between 84% and 87% is 34.1%.
### Example 4
If the rainfall in Denver during the month of May has a mean of 2.4′′ and a standard deviation of .4′′, what is the approximate probability that a randomly selected May will have more than 2′′ of rain?
Because normally distributed data have the same mean and median, we can start by noting that only 1/2 of months will have a rainfall of less than the median: 2.4%. Additionally, another 34.1% will have between 2" and 2.4" of rain, since 2" is once standard deviation away from the mean. That means a total of 50% + 34.1% = 84.1% of months will have more than 2" of rain.
### Example 5
Assuming the same statistics, what is the approximate probability of receiving between 2′′ and 3.2′′?
2′′ of rain is 1 standard deviation below the mean, and 3.2′′ is 2 SD’s above the mean. Since there are 68.2% of values within 1 SD above and below the mean, and 13.6% between 1 and 2 SD’s above the mean, there would be 6.8%+13.6%=81.8% of months with rainfalls between 2′′ and 3.2′′.
6.8% + 13.6% = 81.8% of months with rainfalls between 2" and 3.2"
## Review
1. Carfax rates its cars annually on customer satisfaction. If Clara researches last years’ Mazda, and discovers thatit received a mean customer satisfaction rating of 85, with a standard deviation of 4. Assuming the data is normally distributed, what is the probability that Clara herself would give it a rating between 81 and 89?
2. Caleb will be taking a math test tomorrow to make up for the one he missed last week when he was sick. The scores of the students in the class who took it on time were normally distributed with a mean of 84% and a standard deviation of 3%. What is the probability that Caleb will get at most an 81 on the test?
3. Jonah is looking over the final exam scores of the previous year’s graduates in the Engineering program from which he is about to graduate. The final exam scores of students were normally distributed with a mean of 70 and a standard deviation of 4. What percentile would Jonah be in if he scores a 78 on the final exam?
4. Scores of each of the previous winners in the state championships for “States Best Chili” were normally distributed with a mean of 74 and a standard deviation of 5. Sarah is competing tomorrow. What is the probability of her winning with a score of between 79 and 84 on her chili?
5. Scores on previous drivers tests taken by 16 year oldswere normally distributed with a mean of 82 and a standard deviation of 3.1. George will be taking the driving test tomorrow, what is the probability that he will receive at least an 88.2 on the test?
6. Previous biology test scores were normally distributed with a mean of 76 and a standard deviation of 2.8. Peter will be taking the test tomorrow. What is the probability of Peter getting at most 78.8 on the test?
7. A correlation was found between previous winnersof the Noble Peace Prize and their test scores on a standardized test. Every person scoring at least 2 standard deviations above the mean on the test went on to receive a Nobel Peace Prize, and no person with less than that did receive the prize. If the trend continues, and if the standardized test scores were normally distributed with a mean of 89 and standard deviation of 1.4, will Susan go on to win a Noble Peace Price if she earned a 91.6 on the test?
8. Recent competitors in “Battle of the Bands” received competition scores that were normally distributed with a mean of 89 and a standard deviation of 3.5. “Heavy Metal Trash Cans” will be competing this weekend. What is the probability of the band scoring between 82 and 91.5 in the competition?
9. Tami wants to become a flight attendant but must take a test to do so. Applicants that took the test earned scores that were normally distributed with a mean of 80 and standard deviation of 2.1. Tami will be taking the test today. What is the probability of Tami getting at least 77.9 on the assessment?
## Vocabulary
Term Definition
arithmetic mean The arithmetic mean is also called the average.
bell curve A normal distribution curve is also known as a bell curve.
demographic distribution Demographic distribution describes the relative numbers of different types of members of a sample or group.
distribution A distribution is a description of the possible values of a random variable and the possible occurrences of these values.
normal distribution curve A normal distribution curve is a symmetrical curve that shows the highest frequency in the center with an identical curve on either side of the center.
standard deviation The square root of the variance is the standard deviation. Standard deviation is one way to measure the spread of a set of data. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8715503811836243, "perplexity": 457.5775877607347}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764500619.96/warc/CC-MAIN-20230207134453-20230207164453-00021.warc.gz"} |
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