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http://mathhelpforum.com/calculus/72697-integration-question.html	# Math Help - Integration Question 1. ## Integration Question The Integral of $1/ (2x+3)$ I am wondering if I could pull out the 2 from the denominator? So, I'd end up with $1/(2(x+3/2))$ Then could I just do 1/2 the integral of $1/(x+(3/2))$ which is ln(x + 3/2)? so the answer would be 1/2 LN(x+ 3/2)? Is there a better way, or a correct way to solve this? Thank you 2. Just put $u=2x+3$ and you'll arrive at $\frac12\int\frac{du}u=\frac12\ln\|u\|+k.$ Just back-substitute.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9854274988174438, "perplexity": 639.1482485854813}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500829210.91/warc/CC-MAIN-20140820021349-00046-ip-10-180-136-8.ec2.internal.warc.gz"}
http://utau.wikia.com/wiki/User_blog:BunnyMMD11	# BunnyMMD11 My favorite wikis • I live in Dominican Republic • I was born on September 11 • My occupation is Digital Artist and Dancer • I am Female • ## UTAU Wakane Zuzu. June 23, 2014 by BunnyMMD11 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Wakane Zuzu is a Japanese voice synthesizer (also comes with Romaji). ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Name: Wakane, Zuzu. Height: 1.57 Weight: 49 K. Favorite Color: Yellow and blue. Favorite Food: Brownies. Age: 15. Likes: She loves to dance, Type it Songs and Playing Guitar. Hates: The black color excess. Object: A Giraffe Plush xD. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Features: Hair Color: Black with touches of brown. Eye Color: brown, blue in some cases. Headphones: Grey with yellow lights. Clothing: White Blouse with two yellow stripes, a short tie (yellow)… • ## (UTAU) Wakane Zuzu (School Uniform) June 21, 2014 by BunnyMMD11 Here's My School Version Of My UTAU :3 She's Soooo Cute!!! • ## Wakane Zuzu June 20, 2014 by BunnyMMD11 So... Here She Is, My UTAU: Wakane Zuzu! \(^.^)/ And, Here or here You can see the videos of Zuzu, Hope You Like 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": 0.9500757455825806, "perplexity": 1041.165238270209}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1492917122720.81/warc/CC-MAIN-20170423031202-00395-ip-10-145-167-34.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/integral-of-xsinx-2.345081/	# Integral of (xsinx)^2? 1. Oct 12, 2009 ### emin hi all, I've been trying to integrate this thing for ages.i tried using integration by parts using u=x^2, dv/dx=sin^2 x but it just doesn't seem to end.any help or pointers much appreciated. thanks 2. Oct 12, 2009 ### VeeEight Switch your u=.. and dv=.. around. Another method would be to try an identity, such as the double angle identity for your (sinx)^2 3. Oct 12, 2009 ### emin hmm i hadn't thought of the identity, i'll give it a go. thanks. 4. Oct 12, 2009 ### n!kofeyn I would use your original u-substitution (u=x2), and then use the half-angle formula for sin2x to integrate the dv. Then after you complete the integration by parts the first time, you'll get a sum of two functions in the integral term. One of them you'll be able to immediately integrate, while for the other one you can use integration by parts again. 5. Oct 13, 2009 ### emin i managed to integrate it but i think i may ave made an error. heres the result: x^3/4-x^2/2sin2x-x/2cos2x-sin2x/4 don't know how to put it in proper formulae. anyway thanks for the help. 6. Oct 13, 2009 ### TheoMcCloskey emin - I come up with a slightly different answer. Can you provide details of your solution. The starting approach I took is as follows: $$I = \int x^2 \cdot \sin^2(x)\,dx = \int x^2 \cdot \frac{(1-\cos(2x))}{2} \, dx$$ then let $$2I = \int x^2 \cdot (1-\cos(2x)) \, dx = \frac{x^3}{3} - I_2$$ where $$I_2 = \int x^2 \cdot \cos(2x) \, dx = \int \left (\frac{2x}{2} \right )^2 \cdot \cos(2x) \, d\left ( \frac{2x}{2} \right ) = (1/8) \int s^2 \cdot \cos(s) \, ds$$ where $s=2x$ 7. Oct 13, 2009 ### emin i did get my fractions mixed didn't i? i was doing it on an a4 page and got all the working jumbled. yes that is the approach i took, i should be able to run it through mathematica when i get the chance, and see what answer it comes up with. Similar Discussions: Integral of (xsinx)^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": 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": 0.9438380002975464, "perplexity": 1311.9636642910596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109525.95/warc/CC-MAIN-20170821191703-20170821211703-00292.warc.gz"}
https://www.physicsforums.com/threads/chain-shape-euler-lagrange-equations.195340/	# Chain shape (Euler-Lagrange equations) 1. Nov 1, 2007 ### neworder1 A chain with uniform linear density d and length L is tied at two ends to the ceiling. How to find its shape using Euler-Lagrange equations? (I know it can be done with other methods, but I want to know how to do it using E-L). 2. Nov 1, 2007 ### siddharth First of all, you need to know what quantity is to be minimized. Next, you'll also have to consider the constraint (in the form of an integral) that the total length of the chain is L. So, use a lagrange undetermined multiplier so that you have the functional $g = f + \lambda f_1[/tex], where f is the integrand which needs to be minimized and [itex]f_1$ is the constraint. If you apply the Euler Lagrange equations to g, you'll be able to get the shape of the chain. To find $\lambda$, you'll need to use the constraint. Can you solve it from here? 3. Nov 1, 2007 ### Dick Actually, I don't think it can be done without E-L. How would you do it? 4. Nov 2, 2007 ### neworder1 Yes, it can be done without E-L "manually", i.e. by writing forces, angles etc., but it's a very tedious way.	{"extraction_info": {"found_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": 0.9209146499633789, "perplexity": 387.6562064677368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560279657.18/warc/CC-MAIN-20170116095119-00082-ip-10-171-10-70.ec2.internal.warc.gz"}
http://mathhelpforum.com/algebra/42782-help-solving-log.html	# Math Help - Help with solving a log 1. ## Help with solving a log I am stuck on this one. Can someone show me the steps tp solve it. Thanx 100(1.02)^x/4 =200 2. Originally Posted by whiteowl I am stuck on this one. Can someone show me the steps tp solve it. Thanx 100(1.02)^x/4 =200 $100(1.02)^{x/4} = 200$ .............divide by 100 $\Rightarrow (1.02)^{x/4} = 2$ ..................log both sides $\Rightarrow \ln (1.02)^{x/4} = \ln 2$ ..............apply the rule $\log_a (x^n) = n \log_a x$ $\Rightarrow \frac x4 \ln 1.02 = \ln 2$ ............now solve for $x$, $\ln 1.02 \mbox{ and } \ln 2$ are just constants, and can be treated as such	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9735217094421387, "perplexity": 1245.9356298557536}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1429246656747.97/warc/CC-MAIN-20150417045736-00243-ip-10-235-10-82.ec2.internal.warc.gz"}
http://danielscully.co.uk/projects/mathml-guide/fractions.php	# Daniel I. Scully ## A Beginner's Guide to MathML ### Fractions To write a fraction, or any thing which should be displayed in a similar way (e.g.: binomial coefficients), we can use the <mfrac> element. 1. <mfrac> 2. <mrow> <mi>x</mi><mo>+</mo><mn>2</mn> </mrow> 3. <mn>3</mn> 4. </mfrac> $\frac{x+2}{3}$ The fraction is our first example of a feature which is strange to XML but common to MathML: the importance of order. Within an <mfrac> the first child is defined as the numerator and the second child the denominator. This also means that it must have exactly two children, no more, no less. When either the numerator or denominator are expressions in their own right, they must be grouped together in some other tag such as the <mrow> in the example above. There are also four fraction-specific attributes which can be set to describe the layout of the fraction: Attribute Description Allowed Values linethickness Sets the thickness of the line which separates the numerator and the denominator • number (a multiplier of the default thickness) • thick • medium (default) • thin numalign Sets the alignment of the numerator • left • center (default) • right denomalign Sets the alignment of the numerator • left • center (default) • right bevelled Determines whether the fraction is displayed with the separating line horizontal (false) or at an angle (true) • true • false (default) Here are some examples of their use: #### Example: Binomial coefficients As we said earlier, <mfrac> is not just for fractions. By setting linethickness="0" we can use <mfrac> to display binomial coefficients: 1. <mfenced> 2. <mfrac linethickness="0"> 3. <mn>2</mn> 4. <mn>3</mn> 5. </mfrac> 6. </mfenced> $\left(\genfrac{}{}{0}{}{2}{3}\right)$ #### Example: Alignment Using the 'denomalign' attribute, we can align the denominator in this expression to the right of the fraction. 1. <mfrac denomalign="right"> 2. <mrow> <mi>x</mi><mo>+</mo><mn>2</mn> </mrow> 3. <mn>3</mn> 4. </mfrac> $\frac{x+2}{\hfill 3}$ The same can be done with the numerator using 'numalign'. #### Example: Bevelled fractions When the 'bevelled' attribute is set to 'true', the fraction is displayed with a diagonal separator, instead of the horizontal one in the other examples above. 1. <mfrac bevelled="true"> 2. <mrow> <mi>x</mi><mo>+</mo><mn>2</mn> </mrow> 3. <mn>3</mn> 4. </mfrac> $x+2}{3}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "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": 0.8943942785263062, "perplexity": 4335.2549433164795}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1490218186530.52/warc/CC-MAIN-20170322212946-00475-ip-10-233-31-227.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/induced-current.117159/	# Homework Help: Induced current 1. Apr 10, 2006 ### Punchlinegirl An aluminum ring of radius 5 cm and resistance 0.003 ohms is placed around the center of a long air-core solenoid with 1000 turns per meter and a smaller radius of 3 cm. If the current in the solenoid is increasing at a constant rate of 270 A/s, what is the induced current in the ring? B= $$\munI$$ change in B/change in time = $$\mu$$ n change in current/change in time = $$4 \pi e-7)(1000)(270)$$ = .339 then change in flux/change in time= Achange in B/change in t A= $$\pir^2$$ So A= (.03)^2 3.14 then multiply that by .339 to get 9.58 e -4. Then I divided this by 3 e -4 to get the current and found that it was 3.19.. Last edited: Apr 10, 2006 2. Apr 10, 2006 ### Hootenanny Staff Emeritus I think the question wants you to use Faraday's law. Incidently, thats how I would go about it. -Hoot 3. Apr 10, 2006 ### Punchlinegirl So if I use Faraday's Law, would I do change in flux= Bcos (theta)A Where B= $$\mu$$ I n and would A be the big area minus the small? $$\pi$$ (.05^2)-(.03^2)?	{"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": 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": 0.8981838822364807, "perplexity": 3774.443397763871}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1544376828018.77/warc/CC-MAIN-20181216234902-20181217020902-00459.warc.gz"}
http://mathhelpforum.com/calculus/61788-convergence-series.html	# Thread: Convergence of a series 1. ## Convergence of a series Consider $f (x) = \sum_{n=1} ^ {\infty} 1/{n(1 + nx^2)}$ (a) For what values of $x$ does the series converge? (b) On what intervals of the form $(a,b)$ does the series converge uniformly? (c) On what intervals of the form $(a,b)$ does the series fail to converge uniformly? (d) Is $f$ continuous at all points where the series converges? I would appreciate any help. Meanwhile I am trying to do the question by myself and would let everyone know of any breakthroughs. Thanks 2. Ok, I have done part (a). Using the limit comparison test with the series $1/n^2$ which we know converges we compute the limit to be $x^2$ which in reals is always positive provided it's non zero. If $x = 0$ well then we get the harmonic series which everyone knows diverges. 3. Originally Posted by davidmccormick Consider $f (x) = \sum_{n=1} ^ {\infty} 1/{n(1 + nx^2)}$ (a) For what values of $x$ does the series converge? (b) On what intervals of the form $(a,b)$ does the series converge uniformly? (c) On what intervals of the form $(a,b)$ does the series fail to converge uniformly? (d) Is $f$ continuous at all points where the series converges? I would appreciate any help. Meanwhile I am trying to do the question by myself and would let everyone know of any breakthroughs. Thanks Is this $\sum_{n=1}^{\infty}\frac{1+nx^2}{n}$ or $\sum_{n=1}^{\infty}\frac{1}{n(1+nx^2)}$. I think it is the latter. a) All except 0 consider that $\lim_{n\to\infty}\frac{\frac{1}{n(1+nx^2)}}{\frac{ 1}{n(n+1)}}=x^2$ b) Use the Weirstrass M-test. For which thing to compare it to consider what interval you may definitely say that $\left\|\frac{1}{n(1+nx^2)}\right\|\leqslant\frac{1}{ n(n+1)}$ and then consider the other interval serperately c) This should be found during b0 d) Use the fact that if $\sum_{n=0}^{\infty}f_n(x)$ is uniformly convergent on $[a,b]$ and $f_n(x)\in\mathcal{C}$ than so is $f(x)=\sum_{n=0}^{\infty}f_n(x)$ 4. It is indeed the latter. Thank you for your help. 5. Parts (a) and (d) are fine. For part (b) I managed to show that the series converges uniformly on the intervals $(-{\infty},-1]$ and $[1, {\infty})$ but have no idea how to show whether or not uniform convergence applies to (-1,1) interval. Can you explain a little more please? thanks. 6. Originally Posted by davidmccormick Parts (a) and (d) are fine. For part (b) I managed to show that the series converges uniformly on the intervals $(-{\infty},-1]$ and $[1, {\infty})$ but have no idea how to show whether or not uniform convergence applies to (-1,1) interval. Can you explain a little more please? thanks. Try considering the intervals $\left(a\ne{0},1\right)$ and $\left(-1,a\ne{0}\right))$ and that $\left\|\frac{1}{n (1+nx^2)}\right\|\leqslant\frac{1}{n(1+\min_{(a\ne{ 0},1)}\left\{x^2\right\}n)}$ and use the Limit comparison test on the right hand series. 7. What series can we compare the series on the right hand side of the inequality to (in the limit comparison test)? Also, if we can show this it means that our original series is uniform convergent everywhere except for $x = 0$, and so the answer to part (c) is any interval which doesn't contain 0, right? 8. Originally Posted by davidmccormick What series can we compare the series on the right hand side of the inequality to (in the limit comparison test)? If $0 then $0\leqslant\frac1{n(1+nx^2)}\leqslant\frac1{n(1+na^ 2)}\leqslant \frac1{n^2a^2}$ and you can use the M-test. Similarly on the interval $-1\leqslant x\leqslant -a$. Originally Posted by davidmccormick so the answer to part (c) is any interval which doesn't contain 0, right? Any closed interval which doesn't contain 0. 9. thanks very much for your help.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 32, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9617400765419006, "perplexity": 201.39131875007507}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1386163049615/warc/CC-MAIN-20131204131729-00021-ip-10-33-133-15.ec2.internal.warc.gz"}
https://www.arxiv-vanity.com/papers/astro-ph/0008041/	# [ [ ###### Abstract A combined sample of 79 high and low redshift supernovae Ia (SNe) is used to set constraints on the degree of anisotropy in the Universe out to . First we derive the global most probable values of matter density , the cosmological constant , and the Hubble constant , and find them to be consistent with the published results from the two data sets of Riess et al. 1998 (R98) and Perlmutter et al. 1999 (P99). We then examine the Hubble diagram (HD, i.e., the luminosity-redshift relation) in different directions on the sky by utilising spherical harmonic expansion. In particular, via the analysis of the dipole anisotropy, we divide the sky into the two hemispheres that yield the most discrepant of the three cosmological parameters, and the scatter in each case. The most discrepant values roughly move along the locus (cf. P99), but by no more than along this line. For a perfect FRW universe, Monte Carlo realizations that mimic the current set of SNe yield values higher than the measured in of the cases. We discuss implications for the validity of the Cosmological Principle, and possible calibration problems in the SNe data sets. c ]Constraints on Cosmological Anisotropy out to from Supernovae Ia Kolatt & Lahav ] Tsafrir S. Kolatt and Ofer Lahav Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel Institute of Astronomy, Madingley Rd., CB3 0HA, Cambridge, UK osmology: miscellaneous – cosmology: observations – cosmology: theory – supernovae:general ## 1 Introduction The validity of the Cosmological Principle and the isotropy it implies gained much credibility in recent years. The small fluctuations in the CMB ( on angular scale ) provide the strongest evidence that the universe can be well approximated by the FRW metric on scales larger than (e.g., Peebles 1993; Wu, Lahav, & Rees 1999) On smaller scales () bulk flows of the order indicate that this isotropy breaks down. This is also manifested by significant correlation functions of galaxies and clusters on large scales, and structures like the Supergalactic Plane and the Great Attractor. The transition scale to isotropy and homogeneity is still poorly known, and so is the convergence of the acceleration vector of the Local Group with respect to the CMB. It is therefore important to quantify the degree of homogeneity and isotropy as function of scale. Traditionally this was done by searching for anisotropy in the distribution of radio sources and background radiations [Nan & Cai (1996), Evans (1992), Webster (Webster)]. Several new methods have been suggested to test isotropy and homogeneity on redshift scales of , such as measurements of in situ CMB temperature [Songaila et al. (1994)], the derivation of an independent rest frame from multiple image lens systems [Kochanek et al. (1996)], and Faraday rotation signature due to anisotropic magnetic field [Kronberg (1976), Vallée (1990), Nodland & Ralston (1997)]. The recent use of SNe as distance indicators [Phillips (1993), Perlmutter et al. (1995), Riess et al. (1996)] opened a new opportunity for accurate measurements of anisotropy on cosmological scales that previously have not been accessible. So far the SNe have been used in order to constrain the Hubble constant from a nearby sample and combinations of the matter density and the cosmological constant utilizing SNe at moderate () and high () redshifts. In the future, SNe samples over a wider redshift range will provide separate estimates for the two parameters. It is important to establish the ‘universality’ of the measurements of cosmological parameters from SN, as they are commonly used in joint analysis with other probes such as the CMB, cluster abundance and peculiar velocities [Efstathiou (1999), Efstathiou et al. (1999), Bridle et al. (1999), Bridle et al. (Bridle et al.), Tegmark (1999)]. Assuming a FRW cosmology, a forth measure can be deduced from the ‘Hubble diagram’ (HD; i.e., the luminosity – redshift relation), the measure for the best fit model. For a perfect distance indicator this measure indicates deviations of the local potential (i.e., at the location of the SN) from a pure FRW geometry. However, in the real universe the deviations can also be due to other sources: • Intrinsic (astrophysical) scatter in the SN luminosity-light curve relation. • Scatter due to the location of the SN within the host galaxy & the galaxy type. • Scatter due to dust absorption in the host galaxy, in the intergalactic medium and in our Galaxy. • Gravitational lensing along the l.o.s. to the SN (e.g., an overdensity along the l.o.s. will enhance the apparent luminosity of a SN). Here we explicitly assume that there is no evolution with redshift in the luminosity-light curve relation. Fortunately, most of the abovementioned effects are on the scale of the host galaxy, so with large enough sample they would be averaged out in the calculation of large scale anisotropies. On the other hand, one should worry about ‘anisotropies’ which are simply due to poor matching of different data sets that sample different portions of the sky, or large angular effects due to Galactic extinction. We also note that some of these effects above might be correlated with other measurements, e.g. if the scatter detected in SN Hubble diagram is affected by fluctuations in the potential, then it would be correlated with Integrated SW (or Rees-Schiama) effect in the CMB fluctuations. The outline of this paper is as follows, in §2 we present the unified data set we will be using for the isotropy analysis. The results for cosmological parameters from the entire sample are presented in §3, the anisotropy measurement is discussed in §4, and put in a probabilistic context in §5. We conclude our results in §6. ## 2 The Unified Data set An ideal data set of SNe for the goals we have put forward in the introduction would be a whole-sky homogeneous coverage at various redshifts of SNe. Since such an optimal set does not exist, the closest data set would be the amalgamation of the two existing, published data sets. We unify the samples of the Supernova Cosmology Project (SCP) [Perlmutter et al. (1999)] and that of the High-z Supernova search team (HZS) [Riess et al. (1998)]. These include also the data from low redshift of the Calán-Tololo survey [Hamuy et al. (1996)]. The two groups have different strategy and different nomenclature for the minimization problem by which the cosmological parameters are derived. We have brought the SCP data to comply with the language of the HZS team For each SNe we list its (i) in the CMB frame, (ii) the distance modulus , (iii) errors for these two quantities, (iv) Galactic and . For the SCP data the fiducial magnitude, (cf. P99), is obtained by comparison of the 18 overlapping low redshift SNe as analysed by the two groups, and equating the distance modulus of R98 (table 10) to of P99 (table 2). This procedure is repeated twice, since Riess et al. provide two ways to calculate the distance moduli, “Multi Light Curve Shapes” (LCS) and “Template”. Errors are taken from the tables and a least square minimization is performed in order to obtain the two best fit values of of P99 (and to recover the Hubble constant dependence they omitted in their calculation). The two values are with of 0.952 and 0.763 for the LCS method and the TEMPLATE method respectively. The value of is degenerated with , so different calibrations in the two samples get “absorbed” in the value for . The unified sample consists of 79 SNe altogether, after the exclusion of 6 SNe from P99 (taking their “model C” version) and including the snap-shot survey from R98 along with 1997ck. Figure 1 shows the SNe distribution in Galactic coordinates. The sky coverage is clearly inhomogeneous: the SNe deficiency near the Galactic plane is evident and the clustering of a few of the observed SNe due to the detection procedure is clear. ## 3 Cosmological parameters from the unified sample We follow the statistical analysis as described in R98 and obtain best values for and probability contours in the () plane after integration (i.e. marginalization) over all values and taking into account only physical regions in that plane. P99 include the error due to redshift measurements and peculiar velocities in their magnitude errors, for R98 we followed their procedure, set km s for SNe of and km s for SNe with , and translated to the distance modulus, , units according to the assumed cosmological model in the likelihood function. Figure 2 show the results of the likelihood analysis. The maxima of the likelihood functions are obtained for () values of () and () for the LCS and TEMPLATE method respectively. The contour lines correspond to the , , and confidence levels. ## 4 Anisotropy measurement The natural expansion for anisotropy detection is in spherical harmonics. The current data are too sparse to allow analysis in redshift shells. We expand the four two-dimensional parameter ‘fields’ (for , , , and ) in spherical harmonics. If the isotropy assumption is valid we expect deviations from the average value to be due to noise, and the angular power spectrum should likewise reflect it. This is unless foreground effects alter the signal significantly. The operational way to calculate the expansion coefficients is as follows. • Build a random distribution of points (“mask”) on the sphere. • Construct the four residual fields about the global mean, i.e., , where is , , , or . • Expand the values as obtained at each grid point in Spherical Harmonics up to , i.e. δF(θ,ϕ)=l=lmax∑l=0m=+l∑m=−lamlYml. (1) In order to include more than SNe in each smoothing bin (at least two-parameter fit) we obtain , however the SNe are not distributed uniformly (cf. Fig. 1) and thus a minimum angular resolution of is imposed. That means that for a whole sky coverage the highest significant multipole, , is . There are, though regions that are more densely covered by SNe data and therefore higher multipoles can be assessed as well but at a lower signal-to-noise level. In order to account for the Poisson noise contribution (and thus to the angular power spectrum in quadrature), we run a set of random “masks” and repeat the calculation each time. For each set of the power spectrum coefficients, are computed. The angular power spectrum of the is an order of magnitude and more smaller than the noise level (), in both methods. Figure 3 shows the angular power spectrum, , for the other three fields as calculated from 50 runs with different random mask points. The straight weaker lines show the noise level in each field. The fields in both methods show signals that exceed the noise level for the dipole () and the quadrupole (). Two factors contribute to the noise level, the discrete number of SNe, and the scatter in the luminosity — redshift relation. The former is common to both methods (LCS and TEMPLATE) and therefore the order of magnitude higher noise level for the fields in the LCS method must be due to the latter. The TEMPLATE method seems to provide smaller errors and a better match between the two data sets, as indicated by the lower level of the fiducial magnitude calibration (cf. §2). The angular power spectrum is similar in shape and magnitude in both methods, and lies an order of magnitude to a factor above its noise level. This may indicate there exists a true dipole (or quadrupole) in this field. From the first multipole of angular power spectrum alone, one cannot deduce what is the dipole direction. We therefore turn to look for the direction by other means. We search for largest dipole in , , , and . This has been done in two ways : an actual search over the sky, dividing the SN population in between two hemispheres, and equivalently, by solving a maximization problem of the dipole term with respect to using the computed coefficients. Both methods yield similar results. We then calculate the confidence regions for each hemisphere separately, and look for statistical consistency (overlapping contours). Each test can be applied to each one of the four parameters. Figure 4 verify the fact that the current SNe data best constrain a linear combination of the cosmological parameters , . In all four panels the likelihood maxima move along the line (P99) , sometimes with a large distance between the two maxima for the two disjoint hemispheres (quoted on the plots). In three cases the contour levels overlap significantly (see next section for quantitative evaluation). In the case of the dipole, using the LCS method, there is no overlap between the confidence levels of the two hemispheres. The discrepancy stems from the very assymetric distribution of SNe between the two hemispheres (59 on one versus 20 on the other) and only one SN (1995at) with in the 20 SNe sample. A small error in the distance measurement of this SN, or a systematic deviation of it from the average LCS relation may cause such a discrepancy as we demonstrate in the next section. Elimination of this SN yields a dipole which points away from the original direction, reduced value of , and almost full inclusion of the confidence contour for the larger sample ( SNe) within the confidence level of the remaining SNe. Note that a different “mixture” of redshift distribution to different directions may cause some directions to become more sensitive to one parameter. E.g, SNe at are mostly sensitive to the combination, as opposed to higher weight on as redshift increases (). Figure 1 includes the dipole directions (positive) of in both methods. We observe no coincidence with any Galactic or CMB direction, moreover not all dipoles point to the same direction. The dipole of the field points in both methods toward with and largest difference of for the LCS (TEMPLATE) method. This dipole direction is suspiciously close to the Galactic plane. One worry is that the detected signal is due to the (mis)match between the two data sets. We therefore repeated the computation for each data set separately and verified that though the noise level increases, the results as drawn from each one of the data sets are consistent with the results from the unified set both in magnitude and direction. ## 5 Degree of anisotropy The results of the last section, regarding the spherical harmonic expansion and the various dipole magnitudes, should now be put in an expected distribution in order to draw conclusions about the degree of anisotropy. The hypothesis we are trying to address is that the SN data do not falsify the FRW geometry as a reliable description of the Universe. This strategy is more efficient than addressing specific anisotropic cosmological models [Célérier (2000a), Célérier (2000b)]. We therefore compute the probability distribution of the dipole magnitudes within a FRW universe and confront it with the values obtained for the real Universe. A simple two dimensional Kolmogorov-Smirnov test to falsify the hypothesis that the two contour maps come from the same underlying distribution of cosmological parameters is inadequate here. Since we have used the maximum discrepant values in order to obtain the dipole, the two sub-samples are not randomly selected and therefore can not be confronted in a KS test. The probability distribution depends on the actual cosmological values and to a lesser extent on the power spectrum (via the scatter due to potential fluctuations). For a self consistency check, the underlying cosmology is taken to be the ”best fit” cosmological model (§3), which we then sample by Monte-Carlo simulations. To mimic accurately the SN sample, we use the same angular locations and redshift values as of the observed sample. Luminosity distances, magnitude scatter and peculiar velocities are drawn from Gaussian distributions with the appropriate observed standard deviation. The dipole analysis is repeated for mock catalogs of the SN and the maximal dipole magnitude is calculated to obtain its distribution for the current sampled SNe. Table 1 shows the rejection levels of the hypothesis that the Universe up to can be described by a FRW metric. E.g., using the LCS method and the current sample of SNIa we expect in of all cases to detect a higher value for dipole, than the observed one. Table 1 Isotropy rejection levels using Cosmological Method parameter LCS TEMPLATE 33% 70% 81% 79% 88% 64% ## 6 Discussion By the exploitation of the current available SNe data we have put constraints on the rejection level of the cosmological principle validity up to . A FRW metric is found to be an adequate description of the Universe. In of all realizations of such universes, the dipole signature for anisotropy in the cosmological parameters , and exceeds the observed one. Even though such dipole magnitudes are reasonable in the framework of the FRW model, they may be indicative of non-cosmological contributions to the angular power spectrum. In §1 we listed possible such contributions. If indeed the Universe up to is well represented by a FRW metric then we can exclude large coherent structures at . Such are the structures that may lead to dipole and quadrupole signatures due to coherent gravitational lensing magnification/de-magnification and therefore the latter can be excluded as anisotropy contributors. That leaves small scale (Galactic) foreground effects to be the most likely power contributors. The Galactic disk geometry makes the quadrupole the most significant multipole to be considered, though the solar system offset from the Galactic center may bring about a dipole contribution as well. In the current sample the quadrupole term is only slightly larger than the noise level and does not allow any conclusive results. None of the dipole directions for coincides with the Galactic plane and thus they are probably not correlated with it. Multipoles due to dust extinction may be affirmed by multiple expansion of the residual colors after extinction correction (i.e., R98 and P99 appendices). The one case where two significantly non-overlapping confidence regions are found for two hemispheres that maximize the dipole (LCS), is probably due to a single SN (1995at) for which the individual errors have been underestimated. This case is an exception since the overall values for the HD fits are statistically acceptable. Nevertheless, this case demonstrates the hazard in the draw of conclusions based on a handful of SNe, for which the error in the error estimate is uncertain. In general, the TEMPLATE method provides a better statistical agreement of the data with an FRW model and the current SNe data. This is seen from the magnitude match (cf. §2), tighter constraints from the combined set, smaller noise levels for all multipoles, and smaller values for the dipoles. We conclude that an isotropic universe cannot be rejected by more than a level based on the current SNe data. ## Acknowledgments: This work was supported by the US-Israel Binational Science Foundation, by the Israel Science Foundation, and by grants from NASA and NSF at UCSC. ## References • Bridle et al. (Bridle et al.) • Bridle et al. (1999) Bridle S. L., Eke V. R., Lahav O., Lasenby A. N., Hobson M. P., Cole S., Frenk C. S., Henry J. P., 1999, MNRAS, 310, 565 • Célérier (2000a) Célérier M., 2000a, A&Ap, 353, 63 • Célérier (2000b) Célérier M., 2000b, in ”Proceedings of the XXXVth Rencontres de Moriond, “Energy Densities in the Universe”, astro-ph/0006273 • Efstathiou (1999) Efstathiou G., 1999, MNRAS, 310, 842 • Efstathiou et al. (1999) Efstathiou G., Bridle S. L., Lasenby A. N., Hobson M. P., Ellis R. S., 1999, MNRAS, 303, L47 • Evans (1992) Evans T., 1992, Thesis Haverford Coll., PA. • Hamuy et al. (1996) Hamuy M. et al., 1996, AJ, 112, 2408 • Kochanek et al. (1996) Kochanek C. S., Kolatt T. S., Bartelmann M., 1996, ApJ, 473, 610 • Kronberg (1976) Kronberg P., 1976, in Int. Astron. Union Symp., Vol. 74, p. 367 • Nan & Cai (1996) Nan R., Cai Z., 1996, in IAU Symp. 168: Examining the Big Bang and Diffuse Background Radiations, Vol. 168, p. 491 • Nodland & Ralston (1997) Nodland B., Ralston J. P., 1997, Phys.Rev.Lett., 78, 3043 • Peebles (1993) Peebles P. J. E., 1993, Principles of Physical Cosmology. Princeton Univ. Press, Princeton, NJ • Perlmutter et al. (1999) Perlmutter S. et al., 1999, ApJ, 517, 565 • Perlmutter et al. (1995) Perlmutter S. et al., 1995, ApJ, 440, L41 • Phillips (1993) Phillips M. M., 1993, ApJ, 413, L105 • Riess et al. (1998) Riess A. G. et al., 1998, AJ, 116, 1009 • Riess et al. (1996) Riess A. G., Press W. H., Kirshner R. P., 1996, ApJ, 473, 88 • Songaila et al. (1994) Songaila A. et al., 1994, nat, 371, 43 • Tegmark (1999) Tegmark M., 1999, ApJ, 514, L69 • Vallée (1990) Vallée J. P., 1990, ApJ, 360, 1 • Webster (Webster) • Wu, Lahav, & Rees (Wu et al.)	{"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": 0.9148502945899963, "perplexity": 1480.0371142196013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610703550617.50/warc/CC-MAIN-20210124173052-20210124203052-00476.warc.gz"}
https://wwrenderer-staging.libretexts.org/render-api?sourceFilePath=Library/Utah/Quantitative_Analysis/set6_Applications_of_Derivatives/pr_9.pg&problemSeed=1234567&courseID=anonymous&userID=anonymous&course_password=anonymous&answersSubmitted=0&showSummary=1&displayMode=MathJax&language=en&outputFormat=nosubmit	Consider the function $f(x) = 3 x + 6 x ^ { -1 }$. For this function there are four important intervals: $(-\infty, A]$, $[A,B)$,$(B,C)$, and $[C,\infty)$ where $A$, and $C$ are the critical numbers and the function is not defined at $B$. Find $A$ and $B$ and $C$ For each of the following intervals, tell whether $f(x)$ is increasing (type in INC) or decreasing (type in DEC). $(-\infty, A]$: $[A,B)$: $(B,C]$: $[C,\infty)$: You can earn partial credit on this problem.	{"extraction_info": {"found_math": true, "script_math_tex": 16, "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": 0.9633012413978577, "perplexity": 305.61224125053235}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20230130133622-20230130163622-00555.warc.gz"}
http://mathhelpforum.com/calculus/60420-evaluating-integral.html	# Math Help - Evaluating integral... 1. ## Evaluating integral... How to use cylindrical or spherical coordinates as appropriate to evaluate the integral: ∫ z^2/(x^2+y^2+z^2) dV E where E is the top half of a sphere of radius a>0 that is centered at the origin. Thankyou for your help!! 2. I came up with this... Is this correct? 3. Hello, Hmmm I'm finding $\cos^2(\phi)$ instead of $\cos(\phi)$ Also, there is that coefficient, 2, disturbing me. We once calculated the dxdydz, and my friend found this coefficient. I wasn't able to find the mistake in either of our two computations. But in the wikipedia, there isn't this coefficient. As for the boundaries of your integral, it depends on how you define $\theta$ and $\phi$ 4. Originally Posted by Moo As for the boundaries of your integral, it depends on how you define $\theta$ and $\phi$ $\theta$ is the polar $\theta$ and $\phi$ is the angle of opening from the z-axis. So $\phi$ from 0 to $\frac{\pi}{2}$ would be like a coffee filter completely closed up along the z-axis and then blossoming outward and to rest on the xy-plane. In regards to the question, I'm confused about where the 2 is coming from as well...I will ponder this a little more. Any insight into how you came up with the 2? I'm just not seeing it. 5. Originally Posted by iwonder I came up with this... Is this correct? Your limits of integration are fine, but remember that from rectangular to polar $z=\rho\cos{\phi}$ and $x^2+y^2+z^2=\rho^2$. So $\frac{z^2}{x^2+y^2+z^2}=\frac{\rho^2\cos^2{\phi}}{ \rho^2}=\cos^2{\phi}$. So, then the integral becomes $\int_0^{2\pi}\int_0^a\int_0^\frac{\pi}{2}\rho^2\co s^2\phi\sin\phi d\phi d\rho d\theta$ Perhaps your two came from a trig identity, I'm too tired to think about it..if it came from an identity then it's fine. Also, I have my integration in a little bit different order than you, but since we are in spherical coordinates and the sphere is centered at the origin we can flip the integration limits arbitrarily. I think this is all correct...good luck 6. So, is the first limit of integration from 0 to 2/pi or 2pi? just wanna make sure, thankyou so much! 7. Originally Posted by iwonder So, is the first limit of integration from 0 to 2/pi or 2pi? just wanna make sure, thankyou so much! Should be zero to (2*pi) bad latex! I will edit to avoid confusion. sorry about that! 8. It's okay, thanks elizsimca so much!!!! i already got the answer (2(pi)a^3)/9, hope this is the right answer.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9797366857528687, "perplexity": 696.1826139207202}, "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-2016-22/segments/1464049273946.43/warc/CC-MAIN-20160524002113-00003-ip-10-185-217-139.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/70981/martingale-representation-theorem-for-levy-processes	# Martingale representation theorem for Levy processes Is there an equivalent of martingale representation theorem for Levy processes in some form? I believe there is no such theorem in generality, but maybe there are some specific cases? - Have you taken a look at www2.imperial.ac.uk/~mdavis/docs/MARTREPBBL.PDF ? There seems to be some kind of result for jump processes. – Paul Tupper Jul 24 '11 at 3:29 Hi, Here is a theorem that might answer your question (it is coming from Chesnay, Jeanblanc-Piqué and Yor's book "Mathematical Methods for Financial Markets"). It is theorem (11.2.8.1 page 621) here it is : (edit note : be carefull as mentioned by G. Lowther there's a typo in the book regarding the domain of integration in the conditions over $\psi$ (defined hereafter) ) Let $X$ be an $R^d$ valued Lévy Process and $F^X$ its natural filtration. Let $M$ be an $F^X$-local Martingale. Then there exist an $R^d$-valued predictable process $\phi$ and an predictable function $\psi : R^+ \times \Omega \times R^d\to R$ such that : -$\int_0^t \phi^i(s)^2ds <\infty$ almost surely -$\int_0^t \int_{\|x\|> 1} \|\psi(s,x)\|ds\nu(dx) <\infty$ almost surely -$\int_0^t \int_{\|x\|\le 1} \psi(s,x)^2ds\nu(dx) <\infty$ almost surely and $M_t=M_0+ \sum_{i=0}^d \int_0^t \phi^i(s)dW^i_s + \int_0^t \int_{R^d} \psi(s,x)\tilde{N}(ds,dx)$ Where $\tilde{N}(ds,dx)$ is the compensated measure of the Lévy process $X$ and $\nu$ the associated Lévy measure. Moreover if $(M_t)$ is square integrable martingale then we have : $E[(\int_0^t \phi^i(s)dW^i_s)^2]=E[\int_0^t \phi^i(s)^2ds]<\infty$ and $E[(\int_0^t \int_{R^d} \psi(s,x)\tilde{N}(ds,dx))^2]=E[ \int_0^t ds \int_{R^d} \psi(s,x)^2\nu(dx)]<\infty$ and $\phi$ and $\psi$ are essentially unique. The theorem is not proved in the book but there is a reference to the following parpers : 1/H. Kunita and S. Watanabe. On square integrable martingales. Nagoya J. Math., 30:209–245, 1967 2/H. Kunita. Representation of martingales with jumps and applications to mathematical finance. In H. Kunita, S. Watanabe, and Y. Takahashi, editors, Stochastic Analysis and Related Topics in Kyoto. In honour of Kiyosi Itô, Advanced studies in Pure mathematics, pages 209–233. Oxford University Press, 2004. Regards - That sounds perfectly reasonable, except I think you might the inequalities $\vert x\vert\le1$ and $\vert x\vert > 1$ the wrong way round in the conditions for $\psi$. – George Lowther Jul 25 '11 at 17:37 Also, this answer does not seem to directly answer the question, although you can use the result you state to give a necessary and sufficient condition for a Levy process to satisfy the martingale representation property (satisfied by Brownian motion and by compensated Poisson processes, but not by general Levy processes). – George Lowther Jul 25 '11 at 17:40 @George Lowther : Hello George unless there's a typo in the book I quote I think I got it right. – The Bridge Jul 26 '11 at 10:05 @The Bridge: If that's what it says, then I think there must be a typo. The sum of squares of the jumps of a martingale is finite, $\sum\_{s\le t}(\Delta M_s)^2 < \infty$, because it has finite quadratic variation. That should correspond to $\int\_0^t\int\_{\vert x\vert\le1}\psi(s,x)^2ds\nu(dx) < \infty$. Also, the integrability of the martingale should correspond to $\int\_0^t\int\_{\vert x\vert > 1}\vert\psi(s,x)\vert ds\nu(dx) < \infty$. This is also required so that $\psi$ is $\tilde N$ integrable (almost-surely). – George Lowther Jul 26 '11 at 22:08 Actually, I think it does answer the question fully (once the possible typo is sorted), as it is the correct form of the martingale representation theorem for Levy processes. I was just thinking he meant representation as a stochastic integral wrt the original process, which is clearly not possible except for special cases. But he does say "in some form", and this seems to be the best form. – George Lowther Jul 26 '11 at 23:12	{"extraction_info": {"found_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": 0.9027352929115295, "perplexity": 228.56573426322868}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398474527.16/warc/CC-MAIN-20151124205434-00102-ip-10-71-132-137.ec2.internal.warc.gz"}
https://waseda.pure.elsevier.com/en/publications/experimental-investigation-of-loading-due-to-debris-dams-on-struc	Gabriella Mauti, Jacob Stolle, Tomoyuki Takabatake, Ioan Nistor, Nils Goseberg, Abdolmajid Mohammadian Corresponding author for this work Research output: Contribution to journalArticlepeer-review 1 Citation (Scopus) ## Abstract The entrainment of debris in tsunami-induced floods and storm surges can result in their accumulation on structures, a phenomenon known as debris damming. Such dams can decrease the stability of the affected structures by increasing the area of the flow obstruction, resulting in increased resistance forces. The formation of debris dams can also result in upstream water level rise. This study investigated the influence of idealized debris dam geometry on induced loads and changes in the free surface surrounding a circular column in steady-state flow conditions. Additionally, it investigated the resistance force coefficient of the debris dams. Results show that the presence of debris dams results in a significant increase of loading on structures. The increase in the resistance force was up to 7.7 times greater than the resistance force acting on the column with no debris present. The resistance force coefficients and the change in water depth were functions of the relative dam height and the Froude number, while the porosity had an insignificant impact on the effective resistance force coefficients. Original language English 04020029 Journal of Hydraulic Engineering 146 5 https://doi.org/10.1061/(ASCE)HY.1943-7900.0001731 Published - 2020 May 1 ## ASJC Scopus subject areas • Civil and Structural Engineering • Water Science and Technology • Mechanical Engineering ## Fingerprint Dive into the research topics of 'Experimental Investigation of Loading due to Debris Dams on Structures'. Together they form a unique fingerprint.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8414083123207092, "perplexity": 4128.486948393763}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662584398.89/warc/CC-MAIN-20220525085552-20220525115552-00247.warc.gz"}
http://mathhelpforum.com/calculus/108084-finding-normal-tangent-curve-print.html	# Finding normal and tangent of a curve • October 14th 2009, 04:38 PM Finding normal and tangent of a curve Ok, i generally know how to find a tangent and it's normal by applying the f(a+h)+f(a)/h formula, but this equation apparently has tan(trig) and I have no idea how to deal with it: y=2tan(pix/4) at x =1 • October 14th 2009, 04:50 PM skeeter Quote: Ok, i generally know how to find a tangent and it's normal by applying the f(a+h)+f(a)/h formula, but this equation apparently has tan(trig) and I have no idea how to deal with it: y=2tan(pix/4) at x =1 are you aware of the general formula for the derivative of the tangent function ? $\frac{d}{dx} \tan{u} = \sec^2{u} \cdot \frac{du}{dx}$ finding the value of the derivative using the limit definition is going to be a bit long and overwhelming algebra-wise.	{"extraction_info": {"found_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": 0.8005995154380798, "perplexity": 526.9834203390883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398468233.50/warc/CC-MAIN-20151124205428-00047-ip-10-71-132-137.ec2.internal.warc.gz"}
http://dml.cz/dmlcz/104159	# Article Full entry \| PDF (0.5 MB) Keywords: system reliability; unit reliability; unit importance; essential unit Summary: The availability of a system with dependent units is obtained in the case where the system fails when one of the essential units fails. Markov model is assumed. The system considered consists of $n$ dependent units of which $r\leq n$ units are essential units. A unit is said to be essential if its failure causes the system to fail. The mean and variance of time to system failure are given. Unit reliability is also discussed. References: [1] John G. Kemeny J. Laurie Snell: Finite Markov Chain. D. Van Nostrand Company Inc., New York (1960). MR 0115196 Partner of	{"extraction_info": {"found_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": 0.9665179252624512, "perplexity": 2089.749902194131}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988719041.14/warc/CC-MAIN-20161020183839-00431-ip-10-171-6-4.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/32752/the-comparison-between-the-square-of-the-functional-value-and-the-sum-of-squares/32811	# The comparison between the square of the functional value and the sum of squares of the L^2 norms of function and its Laplacian I was reading a paper where I came across the following argument : For any x in M and for a geodesic ball B(x; epsilon) in a compact Riemannian manifold M with injectivity radius bigger than or equal to epsilon, and for any smooth eigenfunction f of Laplacian on M, we have : the square of f(x) is <= C times ( the square of L^2 norm of f over B(x;epsilon) + square of L^2 norm of L(f) over B(x:epsilon)), where L(f)= Laplacian of f, where C is independent of the Riemannian metric on M. I was unable to see, with my limited Analysis knowledge, why this is true, but they mentioned that it follows from Sobolev's and Garding's inequality, for which they referred to S. Agmon's "Lectures on Elliptic boundary value problems"... still it is unclear to me. N.B.: ihe injectivity radius of a manifold is the smallest of all numbers r such that I can have a geodesic ball of radius r around each point of M. e.g. injectivity radius of the sphere of radius 1 with standard metric is pi, injectivity radius of R^n is infinity etc. Any help ? Thanks in adavance ! - I would guess that the "eigenfunction" hypothesis is probably redundant,probably it should hold for any smooth function f on compact M, at least that is apparent from the way the steps are written in the paper. I cannot attach the paper here, because it is not available from mathscinet etc. – Analysis Now Jul 21 '10 at 6:45 You should probably make the title more descriptive if you want answers. – Harry Gindi Jul 21 '10 at 6:47 Just now I have done it, thanks ! – Analysis Now Jul 21 '10 at 6:55 The dimension of M is not larger than 3, is it? – Pietro Majer Jul 21 '10 at 7:12 Too add to Pietro's comment: roughly speaking Garding allows you to control the Sobolev norm $\\|f\\|_{H^2}$ by $\\|f\\|_{L^2} + \\|\triangle_g f\\|_{L^2}$. And in dimensions $\leq 3$ Sobolev controls $L^\infty$ by $H^2$. If there's any reason why they specify it only for eigenfucntions, it would most likely be for the fact that they want $C$ to be independent of the metric. – Willie Wong Jul 21 '10 at 10:07 You are working on a Riemann surface. That bit of information is rather important, as Sobolev inequalites depends rather much on the dimension of the space. The basic Sobolev inequality is $$\\| f \\|_{L^q(\Omega)} \leq C (\\| \partial f \\|_{L^p(\Omega)} + \\| f\\|_{L^p(\Omega)})$$ where the condition $\frac1p \geq\frac1q \geq \frac1p - \frac1n$ is satisfied (and $\Omega$ needs to be suitably regular). and $C$ depends on the set $\Omega$ and the coefficients $p,q$. If you want the sup norm on the left hand side, you can morally speaking replace $q$ by $\infty$ (so $1/q = 0$ and ask that the second inequality be strict). In any case, in two dimensions by iterating the derivatives, you can actually show that for smooth $f$ $$\|f\| \leq C( \\|f\\|_{L^2} + \\|\partial^2 f\\|_{L^2})$$ using that $0 > 1/2 - 2/2$. (The 2 in the numerator is the number of derivatives. In the denominator in the first term is the Lebesgue exponent, and in the second term is the dimension.) Now, a consequence of Garding's inequality states that for an uniformly elliptic differential operator $L$ of order $k$, one has that $$\\| \partial^k f\\|_{L^2} \leq C (\\| Lf\\|_{L^2} + \\| f\\|_{L^2})$$ so using that the Laplacian is uniformly elliptic of order 2, you can plug Garding's inequality into Sobolev inequality and square the whole expression to get what the authors claim. As to the actual dependence of the constant $C$ on various parameters: off the top of my head I can't remember the details. So I suggest you look it up either in Agmon's book as the authors suggest, or in Gilbarg & Trudinger Elliptic Partial Differential Equations of Second Order or Adams Sobolev Spaces - Oh, and by the way, the fact that $f$ is an eigenfunction is not used in this step. It is used in the next step where the Laplacian $\triangle_g f$ is replaced by $\lambda^2 f$, the eigenvalue. – Willie Wong Jul 21 '10 at 16:19 Thanks very much Mr. Willie Wong, I really appreciate your answer ! – Analysis Now Jul 21 '10 at 16:26 It seems to me that more needs to be said about the constant $C$. As far as I can tell, the constant $C$ depends on the metric $g$ and its Christoffel symbols with respect to local co-ordinates, and I don't think just knowing that the domain is within the injectivity radius is enough to get uniform ellipticity or uniform bounds on the Christoffel symbol. However, the paper in question appears to assume that the metric is hyperbolic (constant Gauss curvature equal to -1). That would be enough. – Deane Yang Jul 21 '10 at 16:49 Deane: is it not enough that $\epsilon$ is sufficiently small? Suppose we have a compact Riemann surface, the Gauss curvature is everywhere bounded, so on geodesic balls of size $\epsilon$ we can bound the Christoffel symbols (in geodesic normal coordinates, say) by something like $\epsilon R$, where $R$ is the upper bound of the Gauss curvature, and so the variations in the metric is bounded by roughly $\epsilon^2 R$ and so in the patch we have uniform ellipticity. I hope I am not missing something obvious. Of course, I agree with you that just "within injectivity radius" is not enough. – Willie Wong Jul 21 '10 at 17:12 Willie, your comment is essentially on the mark, but I didn't see any mention of curvature in either the question or your answer. If you have upper and lower bounds on the Gauss curvature, then you can definitely find co-ordinates in which you get full control over the constant $C$. Surprisingly, geodesic normal co-ordinates don't seem to work, but Jost and Karcher showed that "almost linear co-ordinates" do. But everything is a lot easier, if you just assume the metric is hyperbolic. – Deane Yang Jul 21 '10 at 18:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9718057513237, "perplexity": 217.7882270678841}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1413507443451.12/warc/CC-MAIN-20141017005723-00216-ip-10-16-133-185.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-geometry/82731-homeomorphism-metrizability.html	# Math Help - Homeomorphism and metrizability 1. ## Homeomorphism and metrizability Prove that if X is a metrizable topological space and Y is homeomorphic to X, then Y is metrizable 2. Originally Posted by Andreamet Prove that if X is a metrizable topological space and Y is homeomorphic to X, then Y is metrizable Since X is a metrizable topological space, we have a metric space (X, d). Let Y be a topological space homeomorphic to X and $f:X \rightarrow Y$ be a homeomorphism. Define d' on $Y \times Y$ such that $d'(y_1, y_2) = d( f^{-1}(y_1), f^{-1}(y_2)), y_1, y_2 \in Y$. I'll leave it to check d' is indeed a metric. Since both $f$ and $f^{-1}$ are continuous bijection, we see that $f$ and $f^{-1}$ are isometries, which implies that an open ball of radius r >0 with respect to a metric d on space X corresponds to an open ball of radius r with respect to a metric d' on space Y, and vice versa. Now, the open balls in Y defined by d' can be given as a basis for a topological space Y. Thus, Y is metrizable.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9234822988510132, "perplexity": 260.2817610138693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1405997901076.42/warc/CC-MAIN-20140722025821-00158-ip-10-33-131-23.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/183601/on-the-definition-of-the-hausdorff-distance	# On the definition of the Hausdorff distance $\newcommand{\dist}{\mathrm{dist}\,}$ Let $M$ be a metric space and $\emptyset\neq A,B\subset M$ bounded closed subsets. The Hausdorff distance is defined as $$h(A,B)=\max\{\dist(A,B),\dist(B,A)\},$$ where $$\dist(A,B)=\sup_{x\in A}\inf_{y\in B}d(x,y).$$ Why do we define $\dist(A,B)$ in this way? Is't it possible to replace the supremum by the infimum in the definition of $\dist\!$, that is, define $$\dist_{\mathrm{new}}(A,B)=\inf_{x\in A}\inf_{y\in B}d(x,y).$$ What is the impact of this 'new' definition on the 'Hausdorff distance' given by $$h_{\mathrm{new}}(A,B)=\max\{\dist_{\mathrm{new}}(A,B),\dist_{\mathrm{new}}(B,A)\}?$$ - One problem that arises if you replace the sup by an inf is that the resulting distance function fails to be a pseudometric, as the triangle property fails to hold. For example, consider the sets $A=\{1\}$, $C = \{-1\}$ and $B=\{z\in\mathbb{C} \| \|z\|=1 \}$ in $\mathbb{C}$ with the usual topology. We then have $d(A,B)=d(B,C)=0$, but $d(A,C)=2$. – Old John Aug 17 '12 at 12:16 Another problem (related to @Old John's observation) is that your suggested distance is zero already if $A$ and $B$ share a point while the Hausdorff distance is a genuine distance function. – t.b. Aug 17 '12 at 12:18 The intuition behind Hausdorff distance is to measure “how similar” two sets are in the metric sense. If two sets are in small Hausdorff distance, they are supposed to “look” almost the same. For example, if $A$ was some arbitrary compact set on the plane and $B$ was its countable dense subset, then the Hausdorff distance between them would be zero, which is to be expected, since they “look” pretty much the same, if you don't look too close. You might want to take a look at the picture in the Wikipedia article, I found that it is quite helpful to intuitively see how the distance works. Furthermore, if we take a locally compact metric space $X$, Hausdorff distance turns the set $\mathcal K(X)$ of non-empty compact subsets of $X$ into a well-behaved metric space (into which $X$ naturally isometrically embeds). Your definition could not do such a thing, because it would fail pretty much all axioms of metric except nonnegativity and symmetry. That's not to say that what you defined does not make sense (though, as suggested by t.b., the symmetrisation is unnecessary, because $\inf_{x\in A}\inf_{y\in B}d(x,y)=\inf_{(x,y)\in A\times B} d(x,y)=\inf_{y\in B}\inf_{x\in A}d(x,y)$). It does measure how “close” sets are to one another. It's just that it's not what Hausdorff distance is about. - More specifically, the suggested distance measures the minimal distance of points $(a,b)$ with $a \in A$ and $b \in B$. The symmetrization is unnecessary, since the suggested "distance" simply is $\operatorname{dist}_{\rm new}(A,B) = \inf_{a \in A, b \in B} d(a,b)$ which is already symmetric in $A$ and $B$. – t.b. Aug 17 '12 at 12:40 @t.b.: good point, I incorporated that comment into the answer. – tomasz Aug 17 '12 at 12:47 Here's my intuition on how you could have invented the Hausdorff distance. Hopefully it helps. You want a metric that tells you how far two compact sets are from being the same. And since these sets happen to be subsets of a metric space, you ought to define your metric in terms of the distances between the points of $A$ and $B$. Suppose $A\ne B$. Then either there is a point in $A$ that is not in $B$, or there is a point in $B$ that is not in $A$ (or both). Let's say there is an $a\in A$ with $a\not\in B$. How far is $a$ from being in $B$? Well, the least you have to move $a$ to get it into $B$ is the distance to the closest point in $B$, which is $\inf_{b\in B} d(a,b)$. So that's the distance from $a$ to $B$, which we might as well call $d(a,B)$. Observe that if $a\in B$ then $d(a,B)=0$, and because $B$ is compact, if $a\not\in B$ then $d(a,B)>0$. Now there are lots of points $a\in A$, some of which may be in $B$, and some may not. As long as there exists any $a\not\in B$, that is, any $a$ such that $d(a,B)>0$, we want to know about it. So we ought to take the supremum: $d_1(A,B)=\sup_{a\in A}d(a,B)$. This also means that every point in $A$ is at most $d_1(A,B)$ away from $B$. Finally, $d_1(A,B)$ only tells us if there is a point in $A$ that is far from $B$. We want the Hausdorff distance $d_H(A,B)$ to be large if either there is a point in $A$ far from $B$, or there is a point in $B$ far from $A$. So we define it to be the maximum of both $d_1(A,B)$ and $d_1(B,A)$. And we're done. -	{"extraction_info": {"found_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": 0.9851895570755005, "perplexity": 139.46232249754465}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246662032.70/warc/CC-MAIN-20150417045742-00186-ip-10-235-10-82.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/1841645/smooth-structure-in-reconstruction-theorem	# Smooth structure in reconstruction theorem Let $M,F$ be smooth manifolds, $\{U_i:i\in I\}$ an open cover of $M$ and a cocycle $\{t_{ij}:U_i\cap U_j\to\mathrm{Diff}(F)\}$. In almost any book which discusses fibre bundles, one can find the theorem that says that you can construct a smooth fibre bundle with fibre $F$ from these data, but no one proves it explicitly. So I thought, let's prove it. We take the disjoint union $\coprod_{i\in I}U_i\times F$ and the equivalence relation which relates $(p,f)\in U_i\times F$ and $(q,g)\in U_j\times F$ iff $p=q$ and $f=t_{ij}(p)g$. Then $E$ is the quotient space equipped with the quotient topology, and the map $\pi:E\to M$ sending $\overline{(p,f)}$ to $p$ is continuous. The restrictions $q\|_{U_i\times F}:U_i\times F\to q(U_i\times F)=\pi^{-1}(U_i)$ are homeomorphisms, and should become the local trivialisations. It remains to show that $E$ is a smooth manifold, that $\pi$ is smooth and that these local trivialisations are smooth (and that the topology on $E$ is Hausdorff/second countable), and this is where I got stuck. Does anyone have any idea how the smooth structure on $E$ is defined? It should be defined by the smooth structure on $M$ and $F$, but I don't see how. Edit: obviously $E$ is Hausdorff and second countable because $M$ is and $\pi$ is continuous. Edit 2: a smooth athlas for each $U_i\times F$ is given by $\mathcal{A}_i\{((U\cap U_i)\times V,\phi\|_{U\cap U_i}\times\psi)\,\|\,(U,\phi)\in\mathcal{A}_M,(V,\psi)\in\mathcal{A}_F\}$, clearly. How is then the atlas for the infinite coproduct defined? Is it just $\{(\coprod_{i\in I}W_i,\prod_{i\in I}\Phi_i)\,\|\,(W_i,\Phi_i)\in\mathcal{A}_i\}$? I still don't see how this descends to $E$. As for the transition functions, these are given by $\phi_i\circ\phi_j^{-1}:U_i\cap U_j\times F\to U_i\cap U_j\times F$, and $(\phi_i\circ\phi_j^{-1})(p,f)=(p,t_{ij}(p)f)$, which are smooth. • You give each of the $U_i \times F$ the smooth structure of a product. Since the bundle is covered by such things this suffices to define the smooth structure. – user98602 Jun 27 '16 at 18:39 • Does it not matter that you have an infinite product of these? And how does this smooth structure descend to E? – B. Pasternak Jun 27 '16 at 18:41 • Infinite coproduct, aka disjoint union. Just do one at a time. You just need to see that the transition maps are smooth. – user98602 Jun 27 '16 at 18:42 • Please see my edit. – B. Pasternak Jun 27 '16 at 19:00	{"extraction_info": {"found_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": 0.9655975103378296, "perplexity": 133.26748192806116}, "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-2019-30/segments/1563195527204.71/warc/CC-MAIN-20190721205413-20190721231413-00103.warc.gz"}
https://www.black-holes.org/explore/glossary/30-e/42-epicycle	## Epicycle A secondary circle centered on another, usually larger, circle. Its center moves along the circumference of the main circle. ## Inspiration No amount of experimentation can ever prove me right; a single experiment can prove me wrong. Albert Einstein	{"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": 0.9073979258537292, "perplexity": 4574.27251803523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1544376830305.92/warc/CC-MAIN-20181219005231-20181219031231-00407.warc.gz"}
https://worldwidescience.org/topicpages/a/atomic+fermi+gases.html	Sample records for atomic fermi gases 1. Vortex line in spin-orbit coupled atomic Fermi gases OpenAIRE 2012-01-01 PHYSICAL REVIEW A 85, 013622 (2012) Vortex line in spin-orbit coupled atomic Fermi gases M. Iskin Department of Physics, Koc¸ University, Rumelifeneri Yolu, TR-34450 Sariyer, Istanbul, Turkey (Received 1 December 2011; published 17 January 2012) It has recently been shown that the spin-orbit coupling gives rise to topologically nontrivial and thermodynamically stable gapless superfluid phases when the pseudospin populations of an atomic Fermi gas are imbalanced, with the ... 2. Physics of our Days: Cooling and thermometry of atomic Fermi gases Science.gov (United States) Onofrio, R. 2017-02-01 We review the status of cooling techniques aimed at achieving the deepest quantum degeneracy for atomic Fermi gases. We first discuss some physics motivations, providing a quantitative assessment of the need for deep quantum degeneracy in relevant physics cases, such as the search for unconventional superfluid states. Attention is then focused on the most widespread technique to reach deep quantum degeneracy for Fermi systems, sympathetic cooling of Bose–Fermi mixtures, organizing the discussion according to the specific species involved. Various proposals to circumvent some of the limitations on achieving the deepest Fermi degeneracy, and their experimental realizations, are then reviewed. Finally, we discuss the extension of these techniques to optical lattices and the implementation of precision thermometry crucial to the understanding of the phase diagram of classical and quantum phase transitions in Fermi gases. 3. Population and mass imbalance in atomic Fermi gases NARCIS (Netherlands) Baarsma, J E; Gubbels, K.B.; Stoof, H.T.C. 2010-01-01 We develop an accurate theory of resonantly interacting Fermi mixtures with both spin and mass imbalance. We consider Fermi mixtures with arbitrary mass imbalances but focus, in particular, on the experimentally available Li6-K40 mixture. We determine the phase diagram of the mixture for different i 4. Polaronic atom-trimer continuity in three-component Fermi gases. Science.gov (United States) Nishida, Yusuke 2015-03-20 Recently it has been proposed that three-component Fermi gases may exhibit a new type of crossover physics in which an unpaired Fermi sea of atoms smoothly evolves into that of trimers in addition to the ordinary BCS-BEC crossover of condensed pairs. Here we study its corresponding polaron problem in which a single impurity atom of one component interacts with condensed pairs of the other two components with equal populations. By developing a variational approach in the vicinity of a narrow Feshbach resonance, we show that the impurity atom smoothly changes its character from atom to trimer with increasing the attraction and eventually there is a sharp transition to dimer. The emergent polaronic atom-trimer continuity can be probed in ultracold atoms experiments by measuring the impurity spectral function. Our novel crossover wave function properly incorporating the polaronic atom-trimer continuity will provide a useful basis to further investigate the phase diagram of three-component Fermi gases in more general situations. 5. Thermoelectricity in a junction between interacting cold atomic Fermi gases Science.gov (United States) Sekera, Tibor; Bruder, Christoph; Belzig, Wolfgang 2016-09-01 A gas of interacting ultracold fermions can be tuned into a strongly interacting regime using a Feshbach resonance. Here, we theoretically study quasiparticle transport in a system of two reservoirs of interacting ultracold fermions on the BCS side of the BCS-BEC crossover coupled weakly via a tunnel junction. Using the generalized BCS theory, we calculate the time evolution of the system that is assumed to be initially prepared in a nonequilibrium state characterized by a particle number imbalance or a temperature imbalance. A number of characteristic features like sharp peaks in quasiparticle currents or transitions between the normal and superconducting states are found. We discuss signatures of the Seebeck and the Peltier effects and the resulting temperature difference of the two reservoirs as a function of the interaction parameter (kFa ) -1. The Peltier effect may lead to an additional cooling mechanism for ultracold fermionic atoms. 6. Superfluidity and BCS-BEC crossover of ultracold atomic Fermi gases in mixed dimensions Science.gov (United States) Zhang, Leifeng; Chen, Qijin Atomic Fermi gases have been under active investigation in the past decade. Here we study the superfluid and pairing phenomena of a two-component ultracold atomic Fermi gas in the presence of mixed dimensionality, in which one component is confined on a 1D optical lattice whereas the other is free in the 3D continuum. We assume a short-range pairing interaction and determine the superfluid transition temperature Tc and the phase diagram for the entire BCS-BEC crossover, using a pairing fluctuation theory which includes self-consistently the contributions of finite momentum pairs. We find that, as the lattice depth increases and the lattice spacing decreases, the behavior of Tc becomes very similar to that of a population imbalance Fermi gas in a simple 3D continuum. There is no superfluidity even at T = 0 below certain threshold of pairing strength in the BCS regime. Nonmonotonic Tc behavior and intermediate temperature superfluidity emerge, and for deep enough lattice, the Tc curve will split into two parts. Implications for experiment will be discussed. References: 1. Q.J. Chen, Ioan Kosztin, B. Janko, and K. Levin, Phys. Rev. B 59, 7083 (1999). 2. Chih-Chun Chien, Qijin Chen, Yan He, and K. Levin, Phys. Rev. Lett. 97, 090402(2006). Work supported by NSF of China and the National Basic Research Program of China. 7. Strongly interacting Fermi gases Directory of Open Access Journals (Sweden) Bakr W. 2013-08-01 Full Text Available Strongly interacting gases of ultracold fermions have become an amazingly rich test-bed for many-body theories of fermionic matter. Here we present our recent experiments on these systems. Firstly, we discuss high-precision measurements on the thermodynamics of a strongly interacting Fermi gas across the superfluid transition. The onset of superfluidity is directly observed in the compressibility, the chemical potential, the entropy, and the heat capacity. Our measurements provide benchmarks for current many-body theories on strongly interacting fermions. Secondly, we have studied the evolution of fermion pairing from three to two dimensions in these gases, relating to the physics of layered superconductors. In the presence of p-wave interactions, Fermi gases are predicted to display toplogical superfluidity carrying Majorana edge states. Two possible avenues in this direction are discussed, our creation and direct observation of spin-orbit coupling in Fermi gases and the creation of fermionic molecules of 23Na 40K that will feature strong dipolar interactions in their absolute ground state. 8. Spin-orbit-coupled two-electron Fermi gases of ytterbium atoms Science.gov (United States) Song, Bo; He, Chengdong; Zhang, Shanchao; Hajiyev, Elnur; Huang, Wei; Liu, Xiong-Jun; Jo, Gyu-Boong 2016-12-01 We demonstrate all-optical implementation of spin-orbit coupling (SOC) in a two-electron Fermi gas of 173Yb atoms by coupling two hyperfine ground states with a narrow optical transition. Due to the SU (N ) symmetry of the S10 ground-state manifold which is insensitive to external magnetic fields, an optical ac Stark effect is applied to split the ground spin states, which exhibits a high stability compared with experiments on alkali-metal and lanthanide atoms, and separate out an effective spin-1/2 subspace from other hyperfine levels for the realization of SOC. The dephasing spin dynamics when a momentum-dependent spin-orbit gap is suddenly opened and the asymmetric momentum distribution of the spin-orbit-coupled Fermi gas are observed as a hallmark of SOC. The realization of all-optical SOC for ytterbium fermions should offer a route to a long-lived spin-orbit-coupled Fermi gas and greatly expand our capability of studying spin-orbit physics with alkaline-earth-metal-like atoms. 9. Spin-orbit coupled two-electron Fermi gases of ytterbium atoms CERN Document Server Song, Bo; Zhang, Shanchao; Zou, Yueyang; Haciyev, Elnur; Huang, Wei; Liu, Xiong-Jun; Jo, Gyu-Boong 2016-01-01 We demonstrate the spin-orbit coupling (SOC) in a two-electron Fermi gas of $^{173}$Yb atoms by coupling two hyperfine ground states via the two-photon Raman transition. Due to the SU($N$) symmetry of the $^1$S$_0$ ground-state manifold which is insensitive to external magnetic field, an optical AC Stark effect is applied to split the ground spin states and separate an effective spin-1/2 subspace out from other hyperfine levels for the realization of SOC. With a momentum-dependent spin-orbit gap being suddenly opened by switching on the Raman transition, the dephasing of spin dynamics is observed, as a consequence of the momentum-dependent Rabi oscillations. Moreover, the momentum asymmetry of the spin-orbit coupled Fermi gas is also examined after projection onto the bare spin state and the corresponding momentum distribution is measured for different two-photon detuning. The realization of SOC for Yb fermions may open a new avenue to the study of novel spin-orbit physics with alkaline-earth-like atoms. 10. Effect of the particle-hole channel on BCS-Bose-Einstein condensation crossover in atomic Fermi gases. Science.gov (United States) Chen, Qijin 2016-01-01 BCS-Bose-Einstein condensation (BEC) crossover is effected by increasing pairing strength between fermions from weak to strong in the particle-particle channel, and has attracted a lot of attention since the experimental realization of quantum degenerate atomic Fermi gases. Here we study the effect of the (often dropped) particle-hole channel on the zero T gap Δ(0), superfluid transition temperature Tc, the pseudogap at Tc, and the mean-field ratio 2Δ(0)/, from BCS through BEC regimes, using a pairing fluctuation theory which includes self-consistently the contributions of finite-momentum pairs and features a pseudogap in single particle excitation spectrum. Summing over the infinite particle-hole ladder diagrams, we find a complex dynamical structure for the particle-hole susceptibility χph, and conclude that neglecting the self-energy feedback causes a serious over-estimate of χph. While our result in the BCS limit agrees with Gor'kov et al., the particle-hole channel effect becomes more complex and pronounced in the crossover regime, where χph is reduced by both a smaller Fermi surface and a big (pseudo)gap. Deep in the BEC regime, the particle-hole channel contributions drop to zero. We predict a density dependence of the magnetic field at the Feshbach resonance, which can be used to quantify χph and test different theories. 11. Bragg spectroscopy of strongly interacting Fermi gases Science.gov (United States) Lingham, M. G.; Fenech, K.; Peppler, T.; Hoinka, S.; Dyke, P.; Hannaford, P.; Vale, C. J. 2016-10-01 This article provides an overview of recent developments and emerging topics in the study of two-component Fermi gases using Bragg spectroscopy. Bragg scattering is achieved by exposing a gas to two intersecting laser beams with a slight frequency difference and measuring the momentum transferred to the atoms. By varying the Bragg laser detuning, it is possible to measure either the density or spin response functions which characterize the basic excitations present in the gas. Specifically, one can measure properties such as the dynamic and static structure factors, Tan's universal contact parameter and observe signatures for the onset of pair condensation locally within a gas. 12. Spin diffusion in Fermi gases DEFF Research Database (Denmark) Bruun, Georg 2011-01-01 We examine spin diffusion in a two-component homogeneous Fermi gas in the normal phase. Using a variational approach, analytical results are presented for the spin diffusion coefficient and the related spin relaxation time as a function of temperature and interaction strength. For low temperatures......, strong correlation effects are included through the Landau parameters which we extract from Monte Carlo results. We show that the spin diffusion coefficient has a minimum for a temperature somewhat below the Fermi temperature with a value that approaches the quantum limit ~/m in the unitarity regime... 13. Dark lump excitations in superfluid Fermi gases Institute of Scientific and Technical Information of China (English) Xu Yan-Xia; Duan Wen-Shan 2012-01-01 We study the linear and nonlinear properties of two-dimensional matter-wave pulses in disk-shaped superfluid Fermi gases.A Kadomtsev Petviashvili I (KPI) solitary wave has been realized for superfluid Fermi gases in the limited cases of Bardeen-Cooper-Schrieffer (BCS) regime,Bose-Einstein condensate (BEC) regime,and unitarity regime.Onelump solution as well as one-line soliton solutions for the KPI equation are obtained,and two-line soliton solutions with the same amplitude are also studied in the limited cases.The dependence of the lump propagating velocity and the sound speed of two-dimensional superfluid Fermi gases on the interaction parameter are investigated for the limited cases of BEC and unitarity. 14. Itinerant Ferromagnetism in Ultracold Fermi Gases DEFF Research Database (Denmark) Heiselberg, Henning 2012-01-01 Itinerant ferromagnetism in cold Fermi gases with repulsive interactions is studied applying the Jastrow-Slater approximation generalized to finite polarization and temperature. For two components at zero temperature a second order transition is found at akF ≃ 0.90 compatible with QMC. Thermodyna......Itinerant ferromagnetism in cold Fermi gases with repulsive interactions is studied applying the Jastrow-Slater approximation generalized to finite polarization and temperature. For two components at zero temperature a second order transition is found at akF ≃ 0.90 compatible with QMC... 15. Creation of ultracold molecules from a Fermi gas of atoms OpenAIRE 2003-01-01 Since the realization of Bose-Einstein condensates (BEC) in atomic gases an experimental challenge has been the production of molecular gases in the quantum regime. A promising approach is to create the molecular gas directly from an ultracold atomic gas; for example, atoms in a BEC have been coupled to electronic ground-state molecules through photoassociation as well as through a magnetic-field Feshbach resonance. The availability of atomic Fermi gases provides the exciting prospect of coup... 16. Physics of ultracold Fermi gases revealed by spectroscopies Science.gov (United States) Törmä, Päivi 2016-04-01 This article provides a brief review of how various spectroscopies have been used to investitage many-body quantum phenomena in the context of ultracold Fermi gases. In particular, work done with RF spectroscopy, Bragg spectroscopy and lattice modulation spectroscopy is considered. The theoretical basis of these spectroscopies, namely linear response theory in the many-body quantum physics context is briefly presented. Experiments related to the BCS-BEC crossover, imbalanced Fermi gases, polarons, possible pseudogap and Fermi liquid behaviour and measuring the contact are discussed. Remaining open problems and goals in the field are sketched from the perspective how spectroscopies could contribute. 17. Strongly Interacting Fermi Gases in Two Dimensions Science.gov (United States) 2012-07-17 Svistunov, M. Ku, A. Sommer, L. W. Cheuk, A. Schirotzek, M. W. Zwierlein Feynman diagrams versus Fermi-gas Feynman emulator Nature Physics 8... Feynman emulator. Nature Physics 8, 366 (2012) 4. Jee Woo Park, Cheng-Hsun Wu, Ibon Santiago, Tobias G. Tiecke, Peyman Ahmadi, Martin W. Zwierlein...chapters 7. M. Randeria, W. Zwerger, and M. Zwierlein. The BEC-BCS Crossover and the Unitary Fermi Gas. Lecture Notes in Physics , Volume 836, edited by 18. String Theory Based Predictions for Novel Collective Modes in Strongly Interacting Fermi Gases CERN Document Server Bantilan, H; Ishii, T; Lewis, W E; Romatschke, P 2016-01-01 Very different strongly interacting quantum systems such as Fermi gases, quark-gluon plasmas formed in high energy ion collisions and black holes studied theoretically in string theory are known to exhibit quantitatively similar damping of hydrodynamic modes. It is not known if such similarities extend beyond the hydrodynamic limit. Do non-hydrodynamic collective modes in Fermi gases with strong interactions also match those from string theory calculations? In order to answer this question, we use calculations based on string theory to make predictions for novel types of modes outside the hydrodynamic regime in trapped Fermi gases. These predictions are amenable to direct testing with current state-of-the-art cold atom experiments. 19. Correlations of the upper branch of 1D harmonically trapped two-component fermi gases. Science.gov (United States) Gharashi, Seyed Ebrahim; Blume, D 2013-07-26 We present highly accurate energy spectra and eigenfunctions of small 1D harmonically trapped two-component Fermi gases with interspecies δ-function interactions, and analyze the correlations of the so-called upper branch (i.e., the branch that describes a repulsive Fermi gas consisting of atoms but no molecules) for positive and negative coupling constants. Changes of the two-body correlations as a function of the interspecies coupling strength reflect the competition of the interspecies interaction and the effective repulsion due to the Pauli exclusion principle, and are interpreted as a few-body analog of a transition from a nonmagnetic to a magnetic phase. Moreover, we show that the eigenstate ψadia of the infinitely strongly interacting system with \|n1+n2\|>2 and \|n1-n2\|Fermi-Fermi mapping function to the eigenfunction of the noninteracting single-component Fermi gas. 20. Superfluid Thomas—Fermi approximation for trapped fermi gases Science.gov (United States) Hernández, E. S.; Capuzzi, P.; Szybisz, L. 2009-02-01 We present a generalization of fermionic fluiddynamics to the case of two trapped fermion species with a contact interaction. Within a mean field approximation, we derive coupled equations of motion for the particle densities, particle currents, and anomalous pair density. For an inhomogeneous system, the equilibrium situation with vanishing currents is described by a generalized Thomas-Fermi relation that includes the superfluid gap, together with a new nonlocal gap equation that replaces the usual BCS one. These equations are numericaly solved resorting to a local density approximation (LDA). Density and gap profiles are analyzed in terms of the scattering length, revealing that the current frame can exhibit microscopic details of quantum origin that are frequently absent in more macroscopic scenarios. 1. Superfluid Thomas-Fermi approximation for trapped fermi gases Energy Technology Data Exchange (ETDEWEB) Hernandez, E S; Capuzzi, P; Szybisz, L [Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires (Argentina)], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] 2009-02-01 We present a generalization of fermionic fluiddynamics to the case of two trapped fermion species with a contact interaction. Within a mean field approximation, we derive coupled equations of motion for the particle densities, particle currents, and anomalous pair density. For an inhomogeneous system, the equilibrium situation with vanishing currents is described by a generalized Thomas-Fermi relation that includes the superfluid gap, together with a new nonlocal gap equation that replaces the usual BCS one. These equations are numericaly solved resorting to a local density approximation (LDA). Density and gap profiles are analyzed in terms of the scattering length, revealing that the current frame can exhibit microscopic details of quantum origin that are frequently absent in more macroscopic scenarios. 2. Metastability in spin polarised Fermi gases and quasiparticle decays DEFF Research Database (Denmark) 2011-01-01 We investigate the metastability associated with the first order transition from normal to superfluid phases in the phase diagram of two-component polarised Fermi gases.We begin by detailing the dominant decay processes of single quasiparticles.Having determined the momentum thresholds of each pr... 3. “Hard probes” of strongly-interacting atomic gases Energy Technology Data Exchange (ETDEWEB) Nishida, Yusuke [Los Alamos National Laboratory 2012-06-18 We investigate properties of an energetic atom propagating through strongly interacting atomic gases. The operator product expansion is used to systematically compute a quasiparticle energy and its scattering rate both in a spin-1/2 Fermi gas and in a spinless Bose gas. Reasonable agreement with recent quantum Monte Carlo simulations even at a relatively small momentum k/kF > 1.5 indicates that our large-momentum expansions are valid in a wide range of momentum. We also study a differential scattering rate when a probe atom is shot into atomic gases. Because the number density and current density of the target atomic gas contribute to the forward scattering only, its contact density (measure of short-range pair correlation) gives the leading contribution to the backward scattering. Therefore, such an experiment can be used to measure the contact density and thus provides a new local probe of strongly interacting atomic gases. 4. Microscopy of 2D Fermi gases. Exploring excitations and thermodynamics Energy Technology Data Exchange (ETDEWEB) Morgener, Kai Henning 2014-12-08 This thesis presents experiments on three-dimensional (3D) and two-dimensional (2D) ultracold fermionic {sup 6}Li gases providing local access to microscopic quantum many-body physics. A broad magnetic Feshbach resonance is used to tune the interparticle interaction strength freely to address the entire crossover between the Bose-Einstein-Condensate (BEC) and Bardeen-Cooper-Schrieffer (BCS) regime. We map out the critical velocity in the crossover from BEC to BCS superfluidity by moving a small attractive potential through the 3D cloud. We compare the results with theoretical predictions and achieve quantitative understanding in the BEC regime by performing numerical simulations. Of particular interest is the regime of strong correlations, where no theoretical predictions exist. In the BEC regime, the critical velocity should be closely related to the speed of sound, according to the Landau criterion and Bogolyubov theory. We measure the sound velocity by exciting a density wave and tracking its propagation. The focus of this thesis is on our first experiments on general properties of quasi-2D Fermi gases. We realize strong vertical confinement by generating a 1D optical lattice by intersecting two blue-detuned laser beams under a steep angle. The large resulting lattice spacing enables us to prepare a single planar quantum gas deeply in the 2D regime. The first measurements of the speed of sound in quasi-2D gases in the BEC-BCS crossover are presented. In addition, we present preliminary results on the pressure equation of state, which is extracted from in-situ density profiles. Since the sound velocity is directly connected to the equation of state, the results provide a crosscheck of the speed of sound. Moreover, we benchmark the derived sound from available equation of state predictions, find very good agreement with recent numerical calculations, and disprove a sophisticated mean field approach. These studies are carried out with a novel apparatus which has 5. Universal properties of Fermi gases in arbitrary dimensions DEFF Research Database (Denmark) Valiente, Manuel; T. Zinner, Nikolaj; Molmer, Klaus 2012-01-01 We consider spin-1/2 Fermi gases in arbitrary, integer or non-integer spatial dimensions, interacting via a Dirac delta potential. We first generalize the method of Tan's distributions and implement short-range boundary conditions to arbitrary dimension and we obtain a set of universal relations...... for the Fermi gas. Three-dimensional scattering under very general conditions of transversal confinement is described by an effectively reduced-dimensional scattering length, which we show depends on the three-dimensional scattering length in a universal way. Our formula for non-integer dimensions interpolates... 6. 具有自旋轨道耦合的冷原子费米气中的拓扑超流和FFLO超流❋%Topological superfluids and FFLO superfluids in spin-orbit coupled atomic Fermi gases Institute of Scientific and Technical Information of China (English) 王俊; 高先龙 2015-01-01 It was investigated the properties of spin-orbit coupled atomic fermi gases under a Zeeman field. By solving the Bogoliubove-de Gennes equation self-consistently, it was found that the system supported the topol-ogical superfluid state and the Fulde-Ferrell-Larkin-Ovchinnikov superfluid state respectively when the system under the different strength of Zeeman field and filling factors. When the system turned into topological super-fluid state, a pair of zero-energy Majorana fermions were found.%研究了具有自旋轨道耦合的冷原子费米气在外磁场作用下的物理性质。通过自洽求解Bogoliubove-de Gennes方程，发现了在不同磁场强度和粒子填充数下，体系分别存在拓扑超流态和 Fulde-Ferrell-Larkin-Ovchinnikov超流态。当体系处于拓扑超流态时，存在零能Majorana费米子。 7. Metastability of Bose and Fermi gases on the upper branch Science.gov (United States) LeClair, André; Roditi, Itzhak; Squires, Joshua 2016-12-01 We study three-dimensional Bose and Fermi gases in the upper branch, a phase defined by the absence of bound states in the repulsive interaction regime, within an approximation that considers only two-body interactions. Employing a formalism based on the S matrix, we derive useful analytic expressions that hold on the upper branch in the weak coupling limit. We determine upper branch phase diagrams for both bosons and fermions with techniques valid for arbitrary positive scattering length. 8. Universal properties of Fermi gases in arbitrary dimensions CERN Document Server Valiente, Manuel; Molmer, Klaus 2012-01-01 We consider spin-1/2 Fermi gases in arbitrary, integer or non-integer spatial dimensions, interacting via a Dirac delta potential. We first generalize the method of Tan's distributions and implement short-range boundary conditions to arbitrary dimension and we obtain a set of universal relations for the Fermi gas, which serve as dimensional interpolation/extrapolation formulae in between integer dimensions. We show that, under very general conditions, effective reduced-dimensional scattering lengths due to transversal confinement depend on the original three-dimensional scattering length in a universal way. As a direct consequence, we find that confinement-induced resonances occur in all dimensions different from D=2, without any need to solve the associated multichannel scattering problem. Finally, we show that reduced-dimensional contacts --- related to the tails of the momentum distributions --- are connected to the actual three-dimensional contact through a correction factor of purely geometric origin. 9. George E. Valley, Jr. Prize Talk: Exact relations for Fermi gases with large scattering length Science.gov (United States) Tan, Shina 2011-05-01 Ultracold two-component atomic Fermi gases near broad Feshbach resonances have both strong interactions and relatively long life times, and the strong attractions between fermions lead to remarkable properties such as superfluidity at large percentages of the Fermi temperature. The interactions can often be described by a single parameter, the two-body s-wave scattering length, which determines how the many-body wave function behaves as two atoms get much closer than the average interparticle spacing. This short-range structure of the wave function leads to a number of exact relations among energy, momentum distribution, pressure, and various high-frequency and short-wave properties. All the relations involve a quantity called contact. The exact relations point to a number of independent determinations of the contact, which have been beautifully demonstrated experimentally as well as numerically. This work was supported, in part, by DOE Grant No. DE-FG02-00ER41132. 10. Finite-size Energy of Non-interacting Fermi Gases Energy Technology Data Exchange (ETDEWEB) Gebert, Martin, E-mail: [email protected] [ETH Zürich , Theoretische Physik (Switzerland) 2015-12-15 We study the asymptotics of the difference of the ground-state energies of two non-interacting N-particle Fermi gases in a finite volume of length L in the thermodynamic limit up to order 1/L. We are particularly interested in subdominant terms proportional to 1/L, called finite-size energy. In the nineties (Affleck, Nuc. Phys. B 58, 35–41 1997; Zagoskin and Affleck, J. Phys. A 30, 5743–5765 1997) claimed that the finite-size energy is related to the decay exponent occurring in Anderson’s orthogonality. We prove that the finite-size energy depends on the details of the thermodynamic limit and is therefore non-universal. Typically, it includes an additional linear term in the scattering phase shift. 11. Finite-size Energy of Non-interacting Fermi Gases Science.gov (United States) Gebert, Martin 2015-12-01 We study the asymptotics of the difference of the ground-state energies of two non-interacting N-particle Fermi gases in a finite volume of length L in the thermodynamic limit up to order 1/ L. We are particularly interested in subdominant terms proportional to 1/ L, called finite-size energy. In the nineties (Affleck, Nuc. Phys. B 58, 35-41 1997; Zagoskin and Affleck, J. Phys. A 30, 5743-5765 1997) claimed that the finite-size energy is related to the decay exponent occurring in Anderson's orthogonality. We prove that the finite-size energy depends on the details of the thermodynamic limit and is therefore non-universal. Typically, it includes an additional linear term in the scattering phase shift. 12. Time-of-flight expansion of trapped dipolar Fermi gases: from collisionless to hydrodynamic regime CERN Document Server 2016-01-01 A recent time-of-flight (TOF) expansion experiment with polarized fermionic erbium atoms measured a Fermi surface deformation from a sphere to an ellipsoid due to dipole-dipole interaction, thus confirming previous theoretical predictions. Here we perform a systematic study of the ground-state properties and TOF dynamics for trapped dipolar Fermi gases from the collisionless to the hydrodynamic regime at zero temperature. To this end we solve analytically the underlying Boltzmann-Vlasov equation within the relaxation-time approximation in the vicinity of equilibrium by using a suitable rescaling of the equilibrium distribution. The resulting ordinary differential equations for the respective scaling parameters are then solved numerically for experimentally realistic parameters and relaxation times that correspond to the collisionless, collisional, and hydrodynamic regime. The equations for the collisional regime are first solved in the approximation of a fixed relaxation time, and then this approach is extend... 13. All-optical cooling of Fermi gases via Pauli inhibition of spontaneous emission CERN Document Server Onofrio, Roberto 2016-01-01 A technique is proposed to cool Fermi gases to the regime of quantum degeneracy based on the expected inhibition of spontaneous emission due to the Pauli principle. The reduction of the linewidth for spontaneous emission originates a corresponding reduction of the Doppler temperature, which under specific conditions may give rise to a runaway process through which fermions are progressively cooled. The approach requires a combination of a magneto-optical trap as a cooling system and an optical dipole trap to enhance quantum degeneracy. This results in expected Fermi degeneracy factors $T/T_F$ comparable to the lowest values recently achieved, with potential for a direct implementation in optical lattices. The experimental demonstration of this technique should also indirectly provide a macroscopic manifestation of the Pauli exclusion principle at the atomic physics level. 14. Dimensional BCS-BEC crossover in ultracold Fermi gases Energy Technology Data Exchange (ETDEWEB) Boettcher, Igor 2014-12-10 We investigate thermodynamics and phase structure of ultracold Fermi gases, which can be realized and measured in the laboratory with modern trapping techniques. We approach the subject from a both theoretical and experimental perspective. Central to the analysis is the systematic comparison of the BCS-BEC crossover of two-component fermions in both three and two dimensions. A dimensional reduction can be achieved in experiments by means of highly anisotropic traps. The Functional Renormalization Group (FRG) allows for a description of both cases in a unified theoretical framework. In three dimensions we discuss with the FRG the influence of high momentum particles onto the density, extend previous approaches to the Unitary Fermi Gas to reach quantitative precision, and study the breakdown of superfluidity due to an asymmetry in the population of the two fermion components. In this context we also investigate the stability of the Sarma phase. For the two-dimensional system scattering theory in reduced dimension plays an important role. We present both the theoretically as well as experimentally relevant aspects thereof. After a qualitative analysis of the phase diagram and the equation of state in two dimensions with the FRG we describe the experimental determination of the phase diagram of the two-dimensional BCS-BEC crossover in collaboration with the group of S. Jochim at PI Heidelberg. 15. Beyond Gaussian pair fluctuation theory for strongly interacting Fermi gases Science.gov (United States) Mulkerin, Brendan C.; Liu, Xia-Ji; Hu, Hui 2016-07-01 Interacting Fermi systems in the strongly correlated regime play a fundamental role in many areas of physics and are of particular interest to the condensed matter community. Though weakly interacting fermions are understood, strongly correlated fermions are difficult to describe theoretically as there is no small interaction parameter to expand about. Existing strong-coupling theories rely heavily on the so-called many-body T -matrix approximation that sums ladder-type Feynman diagrams. Here, by acknowledging the fact that the effective interparticle interaction (i.e., the vertex function) becomes smaller above three dimensions, we propose an alternative way to reorganize Feynman diagrams and develop a theoretical framework for interacting Fermi gases beyond the ladder approximation. As an application, we solve the equation of state for three- and two-dimensional strongly interacting fermions and find excellent agreement with experimental [M. J. H. Ku et al., Science 335, 563 (2012), 10.1126/science.1214987] and other theoretical results above temperatures of 0.5 TF . 16. Thomas-Fermi-von Weizsaecker theory of atoms and molecules Energy Technology Data Exchange (ETDEWEB) Benguria, R.; Brezis, H.; Lieb, E.H. 1981-11-02 We place the Thomas-Fermi-von Weizsaecker model of atoms on a firm mathematical footing. We prove existence and uniqueness of solutions of the Thomas-Fermi-von Weizsaecker equation as well as the fact that they minimize the Thomas-Fermi-von Weizsaecker energy functional. Moreover, we prove the existence of bindings for two very dissimilar atoms in the frame of this model. 17. Electron-Atom Collisions in Gases Science.gov (United States) Kraftmakher, Yaakov 2013-01-01 Electron-atom collisions in gases are an aspect of atomic physics. Three experiments in this field employing a thyratron are described: (i) the Ramsauer-Townsend effect, (ii) the excitation and ionization potentials of xenon and (iii) the ion-electron recombination after interrupting the electric discharge. 18. Nicholas Metropolis Award for Outstanding Doctoral Thesis Work in Computational Physics Talk: Equation of State of the Dilute Fermi Gases Science.gov (United States) Chang, Soon Yong 2008-04-01 In the recent years, dilute Fermi gases have played the center stage role in the many-body physics. The gas of neutral alkali atoms such as Lithium-6 and Potassium-40 can be trapped at temperatures below the Fermi degeneracy. The most relevant feature of these gases is that the interaction is tunable and strongly interacting superfluid can be artificially created. I will discuss the recent progress in understanding the ground state properties of the dilute Fermi gases at different interaction regimes. First, I will present the case of the spin symmetric systems where the Fermi gas can smoothly crossover from the BCS regime to the BEC regime. Then, I will discuss the case of the spin polarized systems, where different quantum phases can occur as a function of the polarization. In the laboratory, the trapped Fermi gas shows spatial dependence of the different quantum phases. This can be understood in the context of the local variation of the chemical potential. I will present the most accurate quantum ab initio results and the relevant experiments. 19. Field theory for trapped atomic gases NARCIS (Netherlands) Stoof, H.T.C. 2001-01-01 In this course we give a selfcontained introduction to the quantum field theory for trapped atomic gases, using functional methods throughout. We consider both equilibrium and nonequilibrium phenomena. In the equilibrium case, we first derive the appropriate Hartree-Fock theory for the properties of 20. Field theory for trapped atomic gases NARCIS (Netherlands) Stoof, H.T.C. 2001-01-01 In this course we give a selfcontained introduction to the quantum field theory for trapped atomic gases, using functional methods throughout. We consider both equilibrium and nonequilibrium phenomena. In the equilibrium case, we first derive the appropriate Hartree—Fock theory for the properties of 1. Dimensionality and Finite Number Effect on BCS Transition of Atomic Fermi Gas Institute of Scientific and Technical Information of China (English) CUI Hai-Tao; WANG Lin-Cheng; YI Xue-Xi 2005-01-01 The effect of finite number and dimensionality has been discussed in this paper. The finite number effect has a negative correction to final temperature for 2D or 3D atomic Fermi gases. The changing of final temperature obtained by scanning from BEC region to BCS region are 10% or so with N ≤ 103 and can be negligible when N ＞ 103.However, in 1D atomic Fermi gas, the effect gives a positive correction which greatly changes the final temperature in Fermi gas. This behavior is completely opposed to the 2D and 3D cases and a proper explanation is still to be found.Dimensionality also has a positive correction, in which the more tightly trapping, the higher final temperature one gets with the same particle number. A discussion is also presented. 2. Density-functional theory of strongly correlated Fermi gases in elongated harmonic traps Science.gov (United States) Xianlong, Gao; Polini, Marco; Asgari, Reza; Tosi, M. P. 2006-03-01 Two-component Fermi gases with tunable repulsive or attractive interactions inside quasi-one-dimensional (Q1D) harmonic wells may soon become the cleanest laboratory realizations of strongly correlated Luttiger and Luther-Emery liquids under confinement. We present a microscopic Kohn-Sham density-functional theory of these systems, with specific attention to a gas on the approach to a confinement-induced Feshbach resonance. The theory employs the one-dimensional Gaudin-Yang model as the reference system and transfers the appropriate Q1D ground-state correlations to the confined inhomogeneous gas via a suitable local-density approximation to the exchange and correlation energy functional. Quantitative understanding of the role of the interactions in the bulk shell structure of the axial density profile is thereby achieved. While repulsive intercomponent interactions depress the amplitude of the shell structure of the noninteracting gas, attractive interactions stabilize atomic-density waves through spin pairing. These should be clearly observable in atomic clouds containing of the order of up to 100 atoms. 3. Photon Bubble Turbulence in Cold Atomic Gases CERN Document Server Rodrigues, João D; Ferreira, António V; Terças, Hugo; Kaiser, Robin; Mendonça, José T 2016-01-01 Turbulent radiation flow is ubiquitous in many physical systems where light-matter interaction becomes relevant. Photon bubbling, in particular, has been identified as the main source of turbulent radiation transport in many astrophysical objects, such as stars and accretion disks. This mechanism takes place when radiation trapping in optically dense media becomes unstable, leading to the energy dissipation from the larger to the smaller bubbles. Here, we report on the observation of photon bubble turbulence in cold atomic gases in the presence of multiple scattering of light. The instability is theoretically explained by a fluid description for the atom density coupled to a diffusive transport equation for the photons, which is known to be accurate in the multiple scattering regime investigated here. We determine the power spectrum of the atom density fluctuations, which displays an unusual $\\sim k^{-4}$ scaling, and entails a complex underlying turbulent dynamics resulting from the formation of dynamical bu... 4. Magnetostriction and exchange effects in trapped dipolar Bose and Fermi gases OpenAIRE Baillie, D; Blakie, P. B. 2012-01-01 We examine the magnetostrictive position and momentum space distortions that occur in harmonically confined dipolar Bose and Fermi gases. Direct interactions give rise to position space magnetostriction and exchange interactions give rise to momentum space magnetostriction. While the position space magnetostriction is similar in Bose and Fermi systems, the momentum space magnetostriction is markedly different: the Bose gas momentum distribution distorts in the opposite sense to that of the Fe... 5. Repulsive polarons and itinerant ferromagnetism in strongly polarized Fermi gases DEFF Research Database (Denmark) Massignan, Pietro; Bruun, Georg 2011-01-01 We analyze the properties of a single impurity immersed in a Fermi sea. At positive energy and scattering lengths, we show that the system possesses a well-defined but metastable excitation, the repulsive polaron, and we calculate its energy, quasiparticle residue and effective mass. From a therm... 6. Nonlinear Ramsey Interferometry of Fermi Superfluid Gases in a Double-Well Potential Institute of Scientific and Technical Information of China (English) 蒙红娟; 苟学强; 王文元; 杨阳; 段文山 2012-01-01 The nonlinear Ramsey interferometry of Fermi superfluid gases in a double-well potential is investigated in this paper. We found that the frequency of the Ramsey fringes exactly reflects the strength of nonlinearity, or the scattering length of the Fermi superfluid gases. The cases of sudden limit, the adiabatic limit and the general case are studied. The analytical result is in good agreement with the numerical ones. The adiabatic condition is proposed. In general situation, the zero-frequency point emerge. Finally the possible applications of the theory axe discussed. 7. Thermodynamic characteristics of Fermi gases in a magnetic field Energy Technology Data Exchange (ETDEWEB) Lipovetskii, S.S.; Olesik, A.A.; Sekerzhitskii, V.S. 1987-11-01 Within the framework of statistical thermodynamics of equilibrium systems, general expressions are obtained for the chemical potential, pressure, and magnetic susceptibility for degenerate ideal nonrelativistic electron, proton, and neutron gases in magnetic fields, which exert no pronounced influence on the anomalous magnetic moments of the fermions. 8. Korteweg de Vries Description of One-Dimensional Superfluid Fermi Gases Institute of Scientific and Technical Information of China (English) 徐艳霞; 段文山 2011-01-01 We study one-dimensional matter-wave pulses in cigar-shaped superfluid Fermi gases, including the linear and nonlinear waves of the system. A Korteweg de Vries (KdV) solitary wave is obtained for the superfluid Fermi gases in the limited case of a BEC regime, a BCS regime and unitarity. The dependences of the propagation velocity, amplitude and the width of the solitary wave on the dimensionless interaction parameter y = 1/{kFasc) are given for the limited cases of BEC and unitarity.%We study one-dimensional matter-wave pulses in cigar-shaped superfluid Fermi gases,including the linear and nonlinear waves of the system.A Korteweg de Vries(KdV)solitary wave is obtained for the superfluid Fermi gases in the limited case of a BEC regime,a BCS regime and unitarity.The dependences of the propagation velocity,amplitude and the width of the solitary wave on the dimensionless interaction parameter y =1 /(kFasc)are given for the limited cases of BEC and unitarity. 9. Transdimensional equivalence of universal constants for Fermi gases at unitarity. Science.gov (United States) Endres, Michael G 2012-12-21 I present lattice Monte Carlo calculations for a universal four-component Fermi gas confined to a finite box and to a harmonic trap in one spatial dimension. I obtain the values ξ(1D) = 0.370(4) and ξ(1D) = 0.372(1), respectively, for the Bertsch parameter, a nonperturbative universal constant defined as the (square of the) energy of the untrapped (trapped) system measured in units of the free gas energy. The Bertsch parameter obtained for the one-dimensional system is consistent to within ~1% uncertainties with the most recent numerical and experimental estimates of the analogous Bertsch parameter for a three-dimensional spin-1/2 Fermi gas at unitarity. The finding suggests the intriguing possibility that there exists a universality between two conformal theories in different dimensions. To lend support to this study, I also compute ground state energies for four and five fermions confined to a harmonic trap and demonstrate the restoration of a virial theorem in the continuum limit. The continuum few-body energies obtained are consistent with exact analytical calculations to within ~1.0% and ~0.3% statistical uncertainties, respectively. 10. Transdimensional equivalence of universal constants from universal Fermi gases CERN Document Server Endres, Michael G 2012-01-01 I present lattice Monte Carlo calculations for a universal four-component Fermi gas confined to a finite box and to a harmonic trap in one spatial dimension. I obtain the continuum and thermodynamic limit extrapolated values xi_1d = 0.370(4) and xi_1d = 0.372(1), respectively, for the Bertsch parameter, a nonperturbative universal constant defined as the (square of the) energy of the untrapped (trapped) system measured in units of the free gas energy. The Bertsch parameter for the one-dimensional system is consistent to within ~1% uncertainties with the most recent numerical and experimental estimates of the analogous Bertsch parameter for a three-dimensional spin-1/2 Fermi gas at unitarity. The finding suggests the intriguing possibility that there exists a universality between two conformal theories in different dimensions. To lend support to this study, I also compute continuum extrapolated ground state energies for four and five fermions confined to a harmonic trap and demonstrate the restoration of a Vir... 11. Superfluidity and collective modes in Rashba spin–orbit coupled Fermi gases Energy Technology Data Exchange (ETDEWEB) He, Lianyi, E-mail: [email protected] [Frankfurt Institute for Advanced Studies and Institute for Theoretical Physics, J. W. Goethe University, 60438 Frankfurt am Main (Germany); Huang, Xu-Guang, E-mail: [email protected] [Center for Exploration of Energy and Matter and Physics Department, Indiana University, Bloomington, IN 47408 (United States) 2013-10-15 We present a theoretical study of the superfluidity and the corresponding collective modes in two-component atomic Fermi gases with s-wave attraction and synthetic Rashba spin–orbit coupling. The general effective action for the collective modes is derived from the functional path integral formalism. By tuning the spin–orbit coupling from weak to strong, the system undergoes a crossover from an ordinary BCS/BEC superfluid to a Bose–Einstein condensate of rashbons. We show that the properties of the superfluid density and the Anderson–Bogoliubov mode manifest this crossover. At large spin–orbit coupling, the superfluid density and the sound velocity become independent of the strength of the s-wave attraction. The two-body interaction among the rashbons is also determined. When a Zeeman field is turned on, the system undergoes quantum phase transitions to some exotic superfluid phases which are topologically nontrivial. For the two-dimensional system, the nonanalyticities of the thermodynamic functions and the sound velocity across the phase transition are related to the bulk gapless fermionic excitation which causes infrared singularities. The superfluid density and the sound velocity behave nonmonotonically: they are suppressed by the Zeeman field in the normal superfluid phase, but get enhanced in the topological superfluid phase. The three-dimensional system is also studied. -- Highlights: •The general effective action for Rashba spin–orbit coupled Fermi superfluids is derived. •The evolution of the collective modes manifests the BCS/BEC-rashbon crossover. •The superfluid properties are universal at large spin–orbit coupling. •The sound velocity behaves nonanalytically across the quantum phase transition. 12. Localization of interacting Fermi gases in quasiperiodic potentials Science.gov (United States) Pilati, Sebastiano; Varma, Vipin Kerala 2017-01-01 We investigate the zero-temperature metal-insulator transition in a one-dimensional two-component Fermi gas in the presence of a quasiperiodic potential resulting from the superposition of two optical lattices of equal intensity but incommensurate periods. A mobility edge separating (low-energy) Anderson localized and (high-energy) extended single-particle states appears in this continuous-space model beyond a critical intensity of the quasiperiodic potential. To discern the metallic phase from the insulating phase in the interacting many-fermion system, we employ unbiased quantum Monte Carlo (QMC) simulations combined with the many-particle localization length familiar from the modern theory of the insulating state. In the noninteracting limit, the critical optical-lattice intensity for the metal-insulator transition predicted by the QMC simulations coincides with the Anderson localization transition of the single-particle eigenstates. We show that weak repulsive interactions induce a shift of this critical point towards larger intensities, meaning that repulsion favors metallic behavior. This shift appears to be linear in the interaction parameter, suggesting that even infinitesimal interactions can affect the position of the critical point. 13. Mixtures of ultracold gases: Fermi sea and Bose-Einstein condensate of lithium isotopes Science.gov (United States) Schreck, F. 2003-03-01 This thesis presents studies of quantum degenerate atomic gases of fermionic ^6Li and bosonic ^7Li. Degeneracy is reached by evaporative cooling of ^7Li in a strongly confining magnetic trap. Since at low temperatures direct evaporative cooling is not possible for a polarized fermionic gas, ^6Li is sympathetically cooled by thermal contact with ^7Li. In a first series of experiments both isotopes are trapped in their low-field seeking higher hyperfine states. A Fermi degeneracy of T/T_F=0.25(5) is achieved for 10^5 fermions. For more than 300 atoms, the ^7Li condensate collapses, due to the attractive interatomic interaction in this state. This limits the degeneracy reached for both species. To overcome this limit, in a second series of experiments ^7Li and ^6Li atoms are transferred to their low field seeking lower hyperfine states, where the boson-boson interaction is repulsive but weak. The inter-isotope collisions are used to thermalize the mixture. A ^7Li Bose-Einstein condensate (BEC) of 10^4 atoms immersed in a Fermi sea is produced. The BEC is quasi-one-dimensional and the thermal fraction can be negligible. The measured degeneracies are T/T_C=T/T_F=0.2(1). The temperature is measured using the bosonic thermal fraction, which vanishes at the lowest temperatures, limiting our measurement sensitivity. In a third series of experiments, the bosons are transferred into an optical trap and their internal state is changed to \|F=1,m_F=1rangle, the lowest energy state. A Feshbach resonance is detected and used to produce a BEC with tunable atomic interactions. When the effective interaction between atoms is tuned to be small and attractive, we observe the formation of a matter-wave bright soliton. Propagation of the soliton without spreading over a macroscopic distance of 1.1 mm is observed. Mélanges de gaz ultrafroids: mer de Fermi et condensat de Bose-Einstein des isotopes du lithium Cette thèse décrit l'étude des gaz de fermions ^6Li et de bosons ^7Li dans le 14. Proximity effects in cold gases of multiply charged atoms (Review) Science.gov (United States) Chikina, I.; Shikin, V. 2016-07-01 Possible proximity effects in gases of cold, multiply charged atoms are discussed. Here we deal with rarefied gases with densities nd of multiply charged (Z ≫ 1) atoms at low temperatures in the well-known Thomas-Fermi (TF) approximation, which can be used to evaluate the statistical properties of single atoms. In order to retain the advantages of the TF formalism, which is successful for symmetric problems, the external boundary conditions accounting for the finiteness of the density of atoms (donors), nd ≠ 0, are also symmetrized (using a spherical Wigner-Seitz cell) and formulated in a standard way that conserves the total charge within the cell. The model shows that at zero temperature in a rarefied gas of multiply charged atoms there is an effective long-range interaction Eproxi(nd), the sign of which depends on the properties of the outer shells of individual atoms. The long-range character of the interaction Eproxi is evaluated by comparing it with the properties of the well-known London dispersive attraction ELond(nd) interaction in gases. For the noble gases argon, krypton, and xenon Eproxi>0 and for the alkali and alkaline-earth elements Eproxi neutral complexes into charged fragments. This phenomenon appears consistently in the TF theory through the temperature dependence of the different versions of Eproxi. The anomaly in the thermal proximity effect shows up in the following way: for T ≠ 0 there is no equilibrium solution of TS statistics for single multiply charged atoms in a vacuum when the effect is present. Instability is suppressed in a Wigner-Seitz model under the assumption that there are no electron fluxes through the outer boundary R3 ∝ n-1d of a Wigner-Seitz cell. Eproxi corresponds to the definition of the correlation energy in a gas of interacting particles. This review is written so as to enable comparison of the results of the TF formalism with the standard assumptions of the correlation theory for classical plasmas. The classic 15. Spin-orbit Coupled Fermi Gases and Heavy Solitons in Fermionic Superfluids Science.gov (United States) Cheuk, Lawrence 2013-05-01 The coupling of the spin of electrons to their motional state lies at the heart of topological phases of matter. We have created and detected spin-orbit coupling in an atomic Fermi gas via spin-injection spectroscopy, which characterizes the energy-momentum dispersion and spin composition of the quantum states. For energies within the spin-orbit gap, the system acts as a spin diode. To fully inhibit transport, we open an additional spin gap with radio-frequency coupling, thereby creating a spin-orbit coupled lattice whose spinful band structure we probe. In the presence of s-wave interactions, spin-orbit coupled fermion systems should display induced p-wave pairing and consequently topological superfluidity. Such systems can be described by a relativistic Dirac theory with a mass term that can be made to vary spatially. Topologically protected edge states are expected to occur whenever the mass term changes sign. A system that similarly supports edges states is the strongly interacting atomic Fermi gas near a Feshbach resonance. Topological excitations, such as vortices - line defects - or solitons - planar defects - have been described theoretically for decades in many different physical contexts. In superconductivity and superfluidity they represent a defect in the order parameter and give rise to localized bound states. We have created and directly observed solitons in a fermionic superfluid by imprinting a phase step into the superfluid wavefunction. These are found to be stable for many seconds, allowing us to track their oscillatory motion in the trapped superfluid. Their trapping period increases dramatically as the interactions are tuned from the BEC to the BCS regime. At the Feshbach resonance, their period is an order of magnitude larger than expectations from mean-field Bogoliubov-de Gennes theory, signaling strong effects of bosonic quantum fluctuations and possible filling of Andreev bound states. Our work opens the study of fermionic edge states in 16. Composite-boson approach to molecular Bose-Einstein condensates in mixtures of ultracold Fermi gases Science.gov (United States) Bouvrie, P. Alexander; Tichy, Malte C.; Roditi, Itzhak 2017-02-01 We show that an ansatz based on independent composite bosons [Phys. Rep. 463, 215 (2008), 10.1016/j.physrep.2007.11.003] accurately describes the condensate fraction of molecular Bose-Einstein condensates in ultracold Fermi gases. The entanglement between the fermionic constituents of a single Feshbach molecule then governs the many-particle statistics of the condensate, from the limit of strong interaction to close to unitarity. This result strengthens the role of entanglement as the indispensable driver of composite-boson behavior. The condensate fraction of fermion pairs at zero temperature that we compute matches excellently previous results obtained by means of fixed-node diffusion Monte Carlo methods and the Bogoliubov depletion approximation. This paves the way towards the exploration of the BEC-BCS crossover physics in mixtures of cold Fermi gases with an arbitrary number of fermion pairs as well as the implementation of Hong-Ou-Mandel-like interference experiments proposed within coboson theory. 17. Simple Method Obtaining Analytical Expressions of Particle and Kinetic—Energy Densities for One—Dimensional Confined Fermi Gases Institute of Scientific and Technical Information of China (English) YANGXiao－Xue; WUYing 2002-01-01 We develop a simple approach to obtain explicitly exact analytical expressions of particle and kineticenergy densities for noninteracting Fermi gases in one-dimensional harmonic confinement,and in one-dimensional box confinement as well. 18. Simple Method Obtaining Analytical Expressions of Particle and Kinetic-Energy Densities for One-Dimensional Confined Fermi Gases Institute of Scientific and Technical Information of China (English) YANG XiaoXue; WU Ying 2002-01-01 We develop a simple approach to obtain explicitly exact analytical expressions of particle and kinetic-energy densities for noninteracting Fermi gases in one-dimensional harmonic confinement, and in one-dimensional boxconfinement as well. 19. Phase transitions in definite total spin states of two-component Fermi gases CERN Document Server 2016-01-01 Symmetry under permutations of indistinguishable particles, contained in each medium, is one of the fundamental symmetries. Generally, a change in symmetry affects the medium's thermodynamic properties, leading to phase transitions. Permutation symmetry can be changed since, in addition to the conventional symmetric and anti-symmetric states under permutations of bosons and fermions, mathematical group-representation theory allows for non-Abelian permutation symmetry. Such symmetry can be hidden in states with defined total spins of spinor gases, which can be formed in optical cavities. However, the thermodynamic effects of non-Abelian symmetry are unknown. The present work shows that the symmetry reveals itself in spin-independent or coordinate-independent properties of these gases, namely as non-Abelian entropy in thermodynamic properties. In weakly interacting Fermi gases, saturated and unsaturated phases appear associated with fermionic and non-Abelian symmetry under permutations of particle states, respe... 20. Inhomogeneous atomic Bose-Fermi mixtures in cubic lattices. Science.gov (United States) Cramer, M; Eisert, J; Illuminati, F 2004-11-05 We determine the ground state properties of inhomogeneous mixtures of bosons and fermions in cubic lattices and parabolic confining potentials. For finite hopping we determine the domain boundaries between Mott-insulator plateaux and hopping-dominated regions for lattices of arbitrary dimension within mean-field and perturbation theory. The results are compared with a new numerical method that is based on a Gutzwiller variational approach for the bosons and an exact treatment for the fermions. The findings can be applied as a guideline for future experiments with trapped atomic Bose-Fermi mixtures in optical lattices. 1. Ultracold Fermi and Bose gases and Spinless Bose Charged Sound Particles Directory of Open Access Journals (Sweden) Minasyan V. 2011-10-01 Full Text Available We propose a novel approach for investigation of the motion of Bose or Fermi liquid (or gas which consists of decoupled electrons and ions in the uppermost hyperfine state. Hence, we use such a concept as the fluctuation motion of “charged fluid particles” or “charged fluid points” representing a charged longitudinal elastic wave. In turn, this elastic wave is quantized by spinless longitudinal Bose charged sound particles with the rest mass m and charge e 0 . The existence of spinless Bose charged sound particles allows us to present a new model for description of Bose or Fermi liquid via a non-ideal Bose gas of charged sound particles . In this respect, we introduce a new postulation for the superfluid component of Bose or Fermi liquid determined by means of charged sound particles in the condensate, which may explain the results of experiments connected with ultra-cold Fermi gases of spin-polarized hydrogen, 6 Li and 40 K, and such a Bose gas as 87 Rb in the uppermost hyperfine state, where the Bose- Einstein condensation of charged sound particles is realized by tuning the magnetic field. 2. Statistical Mechanical Approach to the Equation of State of Unitary Fermi Gases CERN Document Server De Silva, Theja N 2016-01-01 We combine a Tan's universal relation with a basic statistical mechanical approach to derive a general equation of state for unitary Fermi gases. The universal equation of state is written as a series solution to a self consistent integral equation where the general solution is a linear combination of Fermi functions. By truncating our series solution to four terms with already known exact theoretical inputs at limiting cases, namely the first three virial coefficients and the Bertsch parameter, we find a good agreement with experimental measurements in the entire temperature region in the normal state. Our analytical equation of state agrees with experimental data up to the fugacity $z = 18$, which is a vast improvement over the other analytical equations of state available where the agreements is \\emph{only} up to $z \\approx 7$. 3. Interacting Bose and Fermi Gases in Low Dimensions and the Riemann Hypothesis Science.gov (United States) Leclair, André We apply the S-matrix based finite temperature formalism to nonrelativistic Bose and Fermi gases in 1+1 and 2+1 dimensions. For the (2+1)-dimensional case in the constant scattering length approximation, the free energy is given in terms of Roger's dilogarithm in a way analagous to the thermodynamic Bethe ansatz for the relativistic (1+1)-dimensional case. The 1d fermionic case with a quasiperiodic two-body potential is closely connected with the Riemann hypothesis. 4. Immersing carbon nano-tubes in cold atomic gases OpenAIRE 2013-01-01 We investigate the sympathetic relaxation of a free-standing, vibrating carbon nano-tube that is mounted on an atom chip and is immersed in a cloud of ultra-cold atoms. Gas atoms colliding with the nano-tube excite phonons via a Casimir-Polder potential. We use Fermi's Golden Rule to estimate the relaxation rates for relevant experimental parameters and develop a fully dynamic theory of relaxation for the multi-mode phononic field embedded in a thermal atomic reservoir. Based on currently ava... 5. Observation of the Efimovian expansion in scale-invariant Fermi gases Science.gov (United States) Deng, Shujin; Shi, Zhe-Yu; Diao, Pengpeng; Yu, Qianli; Zhai, Hui; Qi, Ran; Wu, Haibin 2016-07-01 Scale invariance plays an important role in unitary Fermi gases. Discrete scaling symmetry manifests itself in quantum few-body systems such as the Efimov effect. Here, we report on the theoretical prediction and experimental observation of a distinct type of expansion dynamics for scale-invariant quantum gases. When the frequency of the harmonic trap holding the gas decreases continuously as the inverse of time t, the expansion of the cloud size exhibits a sequence of plateaus. The locations of these plateaus obey a discrete geometric scaling law with a controllable scale factor, and the expansion dynamics is governed by a log-periodic function. This marked expansion shares the same scaling law and mathematical description as the Efimov effect. 6. Enhancement effect of mass imbalance on Fulde-Ferrell-Larkin-Ovchinnikov type of pairing in Fermi-Fermi mixtures of ultracold quantum gases Science.gov (United States) Wang, Jibiao; Che, Yanming; Zhang, Leifeng; Chen, Qijin 2017-01-01 Ultracold two-component Fermi gases with a tunable population imbalance have provided an excellent opportunity for studying the exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, which have been of great interest in condensed matter physics. However, the FFLO states have not been observed experimentally in Fermi gases in three dimensions (3D), possibly due to their small phase space volume and extremely low temperature required for an equal-mass Fermi gas. Here we explore possible effects of mass imbalance, mainly in a 6Li–40K mixture, on the one-plane-wave FFLO phases for a 3D homogeneous case at the mean-field level. We present various phase diagrams related to the FFLO states at both zero and finite temperatures, throughout the BCS-BEC crossover, and show that a large mass ratio may enhance substantially FFLO type of pairing. 7. Enhancement effect of mass imbalance on Fulde-Ferrell-Larkin-Ovchinnikov type of pairing in Fermi-Fermi mixtures of ultracold quantum gases Science.gov (United States) Wang, Jibiao; Che, Yanming; Zhang, Leifeng; Chen, Qijin 2017-01-01 Ultracold two-component Fermi gases with a tunable population imbalance have provided an excellent opportunity for studying the exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, which have been of great interest in condensed matter physics. However, the FFLO states have not been observed experimentally in Fermi gases in three dimensions (3D), possibly due to their small phase space volume and extremely low temperature required for an equal-mass Fermi gas. Here we explore possible effects of mass imbalance, mainly in a 6Li–40K mixture, on the one-plane-wave FFLO phases for a 3D homogeneous case at the mean-field level. We present various phase diagrams related to the FFLO states at both zero and finite temperatures, throughout the BCS-BEC crossover, and show that a large mass ratio may enhance substantially FFLO type of pairing. PMID:28051145 8. Quantum Effects of Uniform Bose Atomic Gases with Weak Attraction Institute of Scientific and Technical Information of China (English) CHENG Ze 2011-01-01 @@ We find that uniform Bose atomic gases with weak attraction can undergo a Bardeen-Cooper-Schrieffer(BCS)condensation below a critical temperature.In the BCS condensation state,bare atoms with opposite wave vectors are bound into pairs,and unpaired bare atoms are transformed into a new kind of quasi-particles,i.e.the dressed atoms.The atom-pair system is a condensate or a superfluid and the dressed-atom system is a normal fluid.The critical temperature and the effective mass of dressed atoms are derived analytically.The transition from the BCS condensation state to the normal state is a first-order phase transition.%We find that uniform Bose atomic gases with weak attraction can undergo a Bardeen-Cooper-Schrieffer (BCS)condensation below a critical temperature. In the BCS condensation state, bare atoms with opposite wave vectors are bound into pairs, and unpaired bare atoms are transformed into a new kind of quasi-particles, i.e. the dressed atoms. The atom-pair system is a condensate or a superfluid and the dressed-atom system is a normal fluid. The critical temperature and the effective mass of dressed atoms are derived analytically. The transition from the BCS condensation state to the normal state is a first-order phase transition. 9. Light Propagation in Ultracold Atomic Gases OpenAIRE Bariani, Francesco 2009-01-01 The propagation of light through an ultracold atomic gas is the main topic of the present work. The thesis consists of two parts. In Part I (Chapters 1,2,3), we give a complete description of the 1D photonic bands of a MI of two-level atoms paying attention to both band diagrams and reflectivity spectra. The role of regular periodicity of the system is addressed within a polariton formalism. The scattering on defects inside lattices of three-level atoms is also studied in view of optica... 10. Quantum information entropies of ultracold atomic gases in a harmonic trap Tutul Biswas; Tarun Kanti Ghosh 2011-10-01 The position and momentum space information entropies of weakly interacting trapped atomic Bose–Einstein condensates and spin-polarized trapped atomic Fermi gases at absolute zero temperature are evaluated. We ﬁnd that sum of the position and momentum space information entropies of these quantum systems containing atoms conﬁned in a $D(≤ 3)$-dimensional harmonic trap has a universal form as $S^{(D)}_t = N(a D − b ln N)$, where ∼ 2.332 and = 2 for interacting bosonic systems and a ∼ 1.982 and = 1 for ideal fermionic systems. These results obey the entropic uncertainty relation given by Beckner, Bialynicki-Birula and Myceilski. 11. Comparing and contrasting nuclei and cold atomic gases CERN Document Server Zinner, N T; 10.1088/0954-3899/40/5/053101 2013-01-01 The experimental revolution in ultracold atomic gas physics over the past decades have brought tremendous amounts of new insight to the world of degenerate quantum systems. Here we compare and constrast the developments of cold atomic gases with the physics of nuclei since many concepts, techniques, and nomenclatures are common to both fields. However, nuclei are finite systems with interactions that are typically much more complicated than those of ultracold atomic gases. The simularities and differences must therefore be carefully addressed for a meaningful comparison and to facilitate fruitful crossdisciplinary activity. Universal results from atomic physics should have impact in certain limits of the nuclear domain. In particular, with advances in the trapping of few-body atomic systems we expect a more direct exchange of ideas and results. 12. Transport phenomena in correlated quantum liquids: Ultracold Fermi gases and F/N junctions Science.gov (United States) Li, Hua Landau Fermi-liquid theory was first introduced by L. D. Landau in the effort of understanding the normal state of Fermi systems, where the application of the concept of elementary excitations to the Fermi systems has proved very fruitful in clarifying the physics of strongly correlated quantum systems at low temperatures. In this thesis, I use Landau Fermi-liquid theory to study the transport phenomena of two different correlated quantum liquids: the strongly interacting ultracold Fermi gases and the ferromagnet/normal-metal (F/N) junctions. The detailed work is presented in chapter II and chapter III of this thesis, respectively. Chapter I holds the introductory part and the background knowledge of this thesis. In chapter II, I study the transport properties of a Fermi gas with strong attractive interactions close to the unitary limit. In particular, I compute the transport lifetimes of the Fermi gas due to superfluid fluctuations above the BCS transition temperature Tc. To calculate the transport lifetimes I need the scattering amplitudes. The scattering amplitudes are dominated by the superfluid fluctuations at temperatures just above Tc. The normal scattering amplitudes are calculated from the Landau parameters. These Landau parameters are obtained from the local version of the induced interaction model for computing Landau parameters. I also calculate the leading order finite temperature corrections to the various transport lifetimes. A calculation of the spin diffusion coefficient is presented in comparison to the experimental findings. Upon choosing a proper value of F0a, I am able to present a good match between the theoretical result and the experimental measurement, which indicates the presence of the superfluid fluctuations near Tc. Calculations of the viscosity, the viscosity/entropy ratio and the thermal conductivity are also shown in support of the appearance of the superfluid fluctuations. In chapter III, I study the spin transport in the low 13. Atomic Gases at Negative Kinetic Temperature NARCIS (Netherlands) Mosk, A.P. 2005-01-01 We show that thermalization of the motion of atoms at negative temperature is possible in an optical lattice, for conditions that are feasible in current experiments. We present a method for reversibly inverting the temperature of a trapped gas. Moreover, a negative-temperature ensemble can be coole 14. Tan's contact and the phase distribution of repulsive Fermi gases: Insights from QCD noise analyses CERN Document Server Porter, William J 2016-01-01 Path-integral analyses originally pioneered in the study of the complex-phase problem afflicting lattice calculations of finite-density quantum chromodynamics are generalized to non-relativistic Fermi gases with repulsive interactions. Using arguments similar to those previously applied to relativistic theories, we show that the analogous problem in nonrelativistic systems manifests itself naturally in Tan's contact as a nontrivial cancellation between terms with varied dependence on extensive thermodynamic quantities. We analyze that case under the assumption of gaussian phase distribution, which is supported by our Monte Carlo calculations and perturbative considerations. We further generalize these results to observables other than the contact, as well as to polarized systems and systems with fixed particle number. Our results are quite general in that they apply to repulsive multi-component fermions, are independent of dimensionality or trapping potential, and hold in the ground state as well as at finite... 15. Theoretical Approach to the Gauge Invariant Linear Response Theories for Ultracold Fermi Gases with Pseudogap Directory of Open Access Journals (Sweden) Hao Guo 2015-01-01 Full Text Available Recent experimental progress allows for exploring some important physical quantities of ultracold Fermi gases, such as the compressibility, spin susceptibility, viscosity, optical conductivity, and spin diffusivity. Theoretically, these quantities can be evaluated from suitable linear response theories. For BCS superfluid, it has been found that the gauge invariant linear response theories can be fully consistent with some stringent consistency constraints. When the theory is generalized to stronger than BCS regime, one may meet serious difficulties to satisfy the gauge invariance conditions. In this paper, we try to construct density and spin linear response theories which are formally gauge invariant for a Fermi gas undergoing BCS-Bose-Einstein Condensation (BEC crossover, especially below the superfluid transition temperature Tc. We adapt a particular t-matrix approach which is close to the G0G formalism to incorporate noncondensed pairing in the normal state. We explicitly show that the fundamental constraints imposed by the Ward identities and Q-limit Ward identity are indeed satisfied. 16. Fermi Data.gov (United States) National Aeronautics and Space Administration — Fermi is a powerful space observatory that will open a wide window on the universe. Gamma rays are the highest-energy form of light, and the gamma-ray sky is... 17. Experimental studies of spin-imbalanced Fermi gases in 2D geometries Science.gov (United States) Thomas, John We study the thermodynamics of a quasi-two-dimensional Fermi gas, which is not quite two-dimensional (2D), but far from three dimensional (3D). This system offers opportunities to test predictions that cross interdisciplinary boundaries, such as enhanced superfluid transition temperatures in spin-imbalanced quasi-2D superconductors, and provides important benchmarks for calculations of the phase diagrams. In the experiments, an ultra-cold Fermi gas is confined in an infrared CO2 laser standing-wave, which produces periodic pancake-shaped potential wells, separated by 5.3 μm. To study the thermodynamics, we load an ultra-cold mixture of N1 = 800 spin 1/2 -up and N2 interaction strength and spin imbalance N2/N1. The measured properties are in disagreement with 2D-BCS theory, but can be fit by a 2D-polaron gas model, where each atom is surrounded by a cloud of particle-hole pairs of the opposite spin. However, this model fails to predict a transition to a spin-balanced central region as N2/N1is increased. Supported by the physics divisions of ARO, AFOSR, and NSF and by the Division of Materials Science and Engineering, the Office of Basic Energy Sciences, DOE. 18. The quantum pressure correction to the excitation spectrum of the trapped superfluid Fermi gases in a BEC-BCS crossover Institute of Scientific and Technical Information of China (English) Dong Hang; Ma Yong-Li 2009-01-01 Using quantum hydrodynamic approaches, we study the quantum pressure correction to the collective excitation spectrum of the interacting trapped superfluid Fermi gases in the BEC-BCS crossover. Based on a phenomenological equation of state, we derive hydrodynamic equations of the system in the whole BEC-BCS crossover regime. Beyond the Thomas-Fermi approximation, expressions of the frequency corrections of collective modes for both spherical and axial symmetric traps excited in the BEC-BCS crossover are given explicitly. The corrections of the eigenfrequencies due to the quantum pressure and their dependence on the inverse interaction strength. Anisotropic parameter and particle numbers of the condensate are discussed in detail. 19. Superfluidity versus Bloch oscillations in confined atomic gases. Science.gov (United States) Büchler, H P; Geshkenbein, V B; Blatter, G 2001-09-01 We study the superfluid properties of (quasi) one-dimensional bosonic atom gases/liquids in traps with finite geometries in the presence of strong quantum fluctuations. Driving the condensate with a moving defect we find the nucleation rate for phase slips using instanton techniques. While phase slips are quenched in a ring resulting in a superfluid response, they proliferate in a tube geometry where we find Bloch oscillations in the chemical potential. These Bloch oscillations describe the individual tunneling of atoms through the defect and thus are a consequence of particle quantization. 20. Thermoelectric transport and Peltier cooling of cold atomic gases Science.gov (United States) Grenier, Charles; Kollath, Corinna; Georges, Antoine 2016-12-01 This brief review presents the emerging field of mesoscopic physics with cold atoms, with an emphasis on thermal and 'thermoelectric' transport, i.e. coupled transport of particles and entropy. We review in particular the comparison between theoretically predicted and experimentally observed thermoelectric effects in such systems. We also show how combining well-designed transport properties and evaporative cooling leads to an equivalent of the Peltier effect with cold atoms, which can be used as a new cooling procedure with improved cooling power and efficiency compared to the evaporative cooling currently used in atomic gases. This could lead to a new generation of experiments probing strong correlation effects of ultracold fermionic atoms at low temperatures. xml:lang="fr" 1. Controlling Rydberg atom excitations in dense background gases CERN Document Server Liebisch, Tara Cubel; Engel, Felix; Nguyen, Huan; Balewski, Jonathan; Lochead, Graham; Böttcher, Fabian; Westphal, Karl M; Kleinbach, Kathrin S; Schmid, Thomas; Gaj, Anita; Löw, Robert; Hofferberth, Sebastian; Pfau, Tilman; Pérez-Ríos, Jesús; Greene, Chris H 2016-01-01 We discuss the density shift and broadening of Rydberg spectra measured in cold, dense atom clouds in the context of Rydberg atom spectroscopy done at room temperature, dating back to the experiments of Amaldi and Segr\\e in 1934. We discuss the theory first developed in 1934 by Fermi to model the mean-field density shift and subsequent developments of the theoretical understanding since then. In particular, we present a model whereby the density shift is calculated using a microscopic model in which the configurations of the perturber atoms within the Rydberg orbit are considered. We present spectroscopic measurements of a Rydberg atom, taken in a Bose-Einstein condensate (BEC) and thermal clouds with densities varying from $5\\times10^{14}\\textrm{cm}^{-3}$ to $9\\times10^{12}\\textrm{cm}^{-3}$. The density shift measured via the spectrum's center of gravity is compared with the mean-field energy shift expected for the effective atom cloud density determined via a time of flight image. Lastly, we present calcul... 2. Quantum Control nd Measurement of Spins in Cold Atomic Gases Science.gov (United States) Deutsch, Ivan 2014-03-01 Spins are natural carriers of quantum information given their long coherence time and our ability to precisely control and measure them with magneto-optical fields. Spins in cold atomic gases provide a pristine environment for such quantum control and measurement, and thus this system can act as a test-bed for the development of quantum simulators. I will discuss the progress my group has made in collaboration with Prof. Jessen, University of Arizona, to develop the toolbox for this test-bed. Through its interactions with rf and microwave magnetic fields, whose waveforms are designed through optimal control techniques, we can implement arbitrary unitary control on the internal hyperfine spins of cesium atoms, a 16 dimensional Hilbert space (isomorphic to 4 qubits). Control of the collective spin of the ensemble of many atoms is performed via the mutual coupling of the atomic ensemble to a mode of the electromagnetic field that acts as a quantum data bus for entangling atoms with one another. Internal spin control can be used to enhance the entangling power of the atom-photon interface. Finally, both projective and weak-continuous measurements can be performed to tomograhically reconstruct quantum states and processes. 3. Atom-molecule equilibration in a degenerate Fermi gas with resonant interactions DEFF Research Database (Denmark) Williams, J. E.; Nikuni, T.; Nygaard, Nicolai; 2004-01-01 We present a nonequilibrium kinetic theory describing atom-molecule population dynamics in a two-component Fermi gas with a Feshbach resonance. Key collision integrals emerge that govern the relaxation of the atom-molecule mixture to chemical and thermal equilibrium. Our focus is on the pseudogap... 4. Generalized BEC and crossover theories of superconductors and ultracold Fermi gases Energy Technology Data Exchange (ETDEWEB) Grether, M. [Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 México DF (Mexico); Llano, M. de, E-mail: [email protected] [Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Apdo. Postal 70-360, 04510 México DF (Mexico) 2013-10-15 Highlights: • A generalized BEC (GBEC) formalism of superconductivity is discussed. • GBEC includes BCS and BEC as special cases, as well as the Friedberg-T.D. Lee model. • It leads to substantial enhancements in critical superconducting temperatures. • In ultracold boson or fermion gases divergent scattering lengths are dealt with. -- Abstract: The generalized Bose–Einstein condensation (GBEC) formalism of superconductivity hinges on three separate new ingredients: (a) treatment of Cooper pairs as real bosons, (b) inclusion of two-hole pairs on an equal footing with two-electron ones, and (c) inclusion in the resulting ternary ideal boson–fermion gas of boson–fermion vertex interactions that drive formation/disin-tegration processes. Besides subsuming both BCS and BEC theories as well as the well-known crossover picture as special cases, GBEC leads to several-order-of-magnitude enhancement in the critical superconducting temperature T{sub c}. The crossover picture is applicable also to ultracold atomic clouds, both bosonic and fermionic. But low-density expansions involving the interatomic scattering length a diverge term-by-term around the so-called unitary zone about the Feshbach resonance. However, expanding a in powers of the attractive part of the interatomic potential renders smooth, divergence-free low-density expansions. 5. Statistics of work and orthogonality catastrophe in discrete level systems: an application to fullerene molecules and ultra-cold trapped Fermi gases Directory of Open Access Journals (Sweden) Antonello Sindona 2015-03-01 Full Text Available The sudden introduction of a local impurity in a Fermi sea leads to an anomalous disturbance of its quantum state that represents a local quench, leaving the system out of equilibrium and giving rise to the Anderson orthogonality catastrophe. The statistics of the work done describe the energy fluctuations produced by the quench, providing an accurate and detailed insight into the fundamental physics of the process. We present here a numerical approach to the non-equilibrium work distribution, supported by applications to phenomena occurring at very diverse energy ranges. One of them is the valence electron shake-up induced by photo-ionization of a core state in a fullerene molecule. The other is the response of an ultra-cold gas of trapped fermions to an embedded two-level atom excited by a fast pulse. Working at low thermal energies, we detect the primary role played by many-particle states of the perturbed system with one or two excited fermions. We validate our approach through the comparison with some photoemission data on fullerene films and previous analytical calculations on harmonically trapped Fermi gases. 6. Statistics of work and orthogonality catastrophe in discrete level systems: an application to fullerene molecules and ultra-cold trapped Fermi gases. Science.gov (United States) Sindona, Antonello; Pisarra, Michele; Gravina, Mario; Vacacela Gomez, Cristian; Riccardi, Pierfrancesco; Falcone, Giovanni; Plastina, Francesco 2015-01-01 The sudden introduction of a local impurity in a Fermi sea leads to an anomalous disturbance of its quantum state that represents a local quench, leaving the system out of equilibrium and giving rise to the Anderson orthogonality catastrophe. The statistics of the work done describe the energy fluctuations produced by the quench, providing an accurate and detailed insight into the fundamental physics of the process. We present here a numerical approach to the non-equilibrium work distribution, supported by applications to phenomena occurring at very diverse energy ranges. One of them is the valence electron shake-up induced by photo-ionization of a core state in a fullerene molecule. The other is the response of an ultra-cold gas of trapped fermions to an embedded two-level atom excited by a fast pulse. Working at low thermal energies, we detect the primary role played by many-particle states of the perturbed system with one or two excited fermions. We validate our approach through the comparison with some photoemission data on fullerene films and previous analytical calculations on harmonically trapped Fermi gases. 7. Modeling Strongly Correlated Fermi Systems Using Ultra-Cold Atoms Science.gov (United States) 2008-06-28 exceeds the optical scattering rate Γsc). For the lattice described above, the Lamb Dicke parameter ER/hν = 0.12 and the festina lente criterion Γsc...zero entropy ). Initialization of the quantum register for quantum computations requires a gas of neutral atoms in a near-zero- entropy state...zero- entropy state is prepared by selectively removing atoms in the second band from the lattice potential. optical lattice experiments have 8. Probing superfluid properties in strongly correlated Fermi gases with high spatial resolution Energy Technology Data Exchange (ETDEWEB) Weimer, Wolf 2014-07-01 In this thesis an apparatus to study ultracold fermionic {sup 6}Li with tunable interaction strength and dimensionality is presented. The apparatus is applied to investigate the speed of sound v{sub s} and the superfluid critical velocity v{sub c} across the transition from Bose-Einstein condensation (BEC) to Bardeen-Cooper-Schrieffer (BCS) superfluidity. The results set benchmarks for theories describing strongly correlated systems. To measure v{sub c}, an obstacle, that is formed by a tightly focused laser beam, is moved through a superfluid sample with a constant velocity along a line of constant density. For velocities larger than v{sub c} heating of the gas is observed. The critical velocity is mapped out for various different interaction strengths covering the BEC-BCS crossover. According to the Landau criterion and Bogolyubov theory, v{sub c} should be closely related to v{sub s} in a Bose-Einstein condensate. The measurement of v{sub s} is conducted by creating a density modulation in the centre of the cloud and tracking the excited modulation. The velocities v{sub s} and v{sub c} are measured in a similar range of interaction strengths and in similar samples to ensure comparability. The apparatus which provides the ultracold samples is a two chamber design with a magneto-optical trap that is loaded via a Zeeman slower. The subsequent cooling steps are all-optical and finally create an ultracold oblate atom cloud inside a flat vacuum cell. This cell provides optimal optical access and is placed between two high numerical aperture microscope objectives. These objectives are used to probe the samples in-situ on length scales which are comparable to the intrinsic length scales of the gases. Similarly, optical dipole potentials are employed to manipulate the clouds on the same small length scales. The oblate samples are sufficiently flat such that there spatial extent along the microscope axes is smaller than the depth of field of the objectives. With an 9. Bose-Einstein condensation in dilute atomic gases Science.gov (United States) Arlt, J.; Bongs, K.; Sengstock, K.; Ertmer, W. 2002-02-01 Bose-Einstein condensation is one of the most curious and fascinating phenomena in physics. It lies at the heart of such intriguing processes as superfluidity and superconductivity. However, in most cases, only a small part of the sample is Bose-condensed and strong interactions are present. A weakly interacting, pure Bose-Einstein condensate (BEC) has therefore been called the "holy grail of atomic physics". In 1995 this grail was found by producing almost pure BECs in dilute atomic gases. We review the experimental development that led to the realization of BEC in these systems and explain how BECs are now routinely produced in about 25 laboratories worldwide. The tremendous experimental progress of the past few years is outlined and a number of recent experiments show the current status of the field. Electronic supplementary material to this paper can be obtained by using the Springer LINK server located at http://dx.doi.org/10.1007/s00114-001-0277-8. 10. In-medium bound-state formation and inhomogeneous condensation in Fermi gases in a hard-wall box CERN Document Server Roscher, Dietrich 2016-01-01 The formation of bosonic bound states underlies the formation of a superfluid ground state in the many-body phase diagram of ultracold Fermi gases. We study bound-state formation in a spin- and mass-imbalanced ultracold Fermi gas confined in a box with hard-wall boundary conditions. Because of the presence of finite Fermi spheres, the center-of-mass momentum of the potentially formed bound states can be finite, depending on the parameters controlling mass and spin imbalance as well as the coupling strength. We exploit this observation to estimate the potential location of inhomogeneous phases in the many-body phase diagram as a function of spin- and mass imbalance as well as the box size. Our results suggest that a hard-wall box does not alter substantially the many-body phase diagram calculated in the thermodynamic limit. Therefore, such a box may serve as an ideal trap potential to bring experiment and theory closely together and facilitate the search for exotic inhomogeneous ground states. 11. Synthetic Lorentz force in classical atomic gases via Doppler effect and radiation pressure CERN Document Server Dubček, T; Jukić, D; Aumiler, D; Ban, T; Buljan, H 2014-01-01 We theoretically predict a novel type of synthetic Lorentz force for classical (cold) atomic gases, which is based on the Doppler effect and radiation pressure. A fairly uniform and strong force can be constructed for gases in macroscopic volumes of several cubic millimeters and more. This opens the possibility to mimic classical charged gases in magnetic fields, such as those in a tokamak, in cold atom experiments. 12. Observation of repulsive Fermi polarons in a resonant mixture of ultracold ${}^6$Li atoms CERN Document Server Scazza, F; Massignan, P; Recati, A; Amico, A; Burchianti, A; Fort, C; Inguscio, M; Zaccanti, M; Roati, G 2016-01-01 We employ radio-frequency spectroscopy to investigate a polarized spin-mixture of ultracold ${}^6$Li atoms close to a broad Feshbach scattering resonance. Focusing on the regime of strong repulsive interactions, we observe well-defined coherent quasiparticles even for unitarity-limited interactions. We characterize the many-body system by extracting the key properties of repulsive Fermi polarons: the energy $E_+$, the effective mass $m^$, the residue $Z$ and the decay rate $\\Gamma$. Above a critical interaction, $E_+$ is found to exceed the Fermi energy of the bath while $m^$ diverges and even turns negative. Such findings reveal that the paramagnetic Fermi liquid state becomes thermodynamically unstable towards an energetically favored ferromagnetic phase. 13. High-temperature atomic superfluidity in lattice Bose-Fermi mixtures. Science.gov (United States) Illuminati, Fabrizio; Albus, Alexander 2004-08-27 We consider atomic Bose-Fermi mixtures in optical lattices and study the superfluidity of fermionic atoms due to s-wave pairing induced by boson-fermion interactions. We prove that the induced fermion-fermion coupling is always attractive if the boson-boson on-site interaction is repulsive, and predict the existence of an enhanced BEC-BCS crossover as the strength of the lattice potential is varied. We show that for direct on-site fermion-fermion repulsion, the induced attraction can give rise to superfluidity via s-wave pairing at striking variance with the case of pure systems of fermionic atoms with direct repulsive interactions. 14. A Proposal for measuring Anisotropic Shear Viscosity in Unitary Fermi Gases CERN Document Server Samanta, Rickmoy; Trivedi, Sandip P 2016-01-01 We present a proposal to measure anisotropic shear viscosity in a strongly interacting, ultra-cold, unitary Fermi gas confined in a harmonic trap. We introduce anisotropy in this setup by strongly confining the gas in one of the directions with relatively weak confinement in the remaining directions. This system has a close resemblance to anisotropic strongly coupled field theories studied recently in the context of gauge-gravity duality. Computations in such theories (which have gravity duals) revealed that some of the viscosity components of the anisotropic shear viscosity tensor can be made much smaller than the entropy density, thus parametrically violating the bound proposed by Kovtun, Son and Starinets (KSS): $\\frac {\\eta} {s} \\geq \\frac{1}{4 \\pi}$. A Boltzmann analysis performed in a system of weakly interacting particles in a linear potential also shows that components of the viscosity tensor can be reduced. Motivated by these exciting results, we propose two hydrodynamic modes in the unitary Fermi ga... 15. Mapping the Two-Component Atomic Fermi Gas to the Nuclear Shell-Model DEFF Research Database (Denmark) Özen, C.; Zinner, Nikolaj Thomas 2014-01-01 of the external potential becomes important. A system of two-species fermionic cold atoms with an attractive zero-range interaction is analogous to a simple model of nucleus in which neutrons and protons interact only through a residual pairing interaction. In this article, we discuss how the problem of a two......-component atomic fermi gas in a tight external trap can be mapped to the nuclear shell model so that readily available many-body techniques in nuclear physics, such as the Shell Model Monte Carlo (SMMC) method, can be directly applied to the study of these systems. We demonstrate an application of the SMMC method... 16. The contact in the BCS–BEC crossover for finite range interacting ultracold Fermi gases Energy Technology Data Exchange (ETDEWEB) Caballero-Benítez, Santiago F., E-mail: [email protected]; Paredes, Rosario; Romero-Rochín, Víctor 2013-10-15 Using mean-field theory for the Bardeen–Cooper–Schriefer (BCS) to the Bose–Einstein condensate (BEC) crossover we investigate the ground state thermodynamic properties of an interacting homogeneous Fermi gas. The interatomic interactions modelled through a finite range potential allows us to calculate the thermodynamic behaviour as a function of the potential parameters in the whole crossover region. We concentrate in studying the Contact variable, the thermodynamic conjugate of the inverse of the s-wave scattering length. Our analysis leads to predict a quantum phase transition – like in the case of large potential range. This finding is a direct consequence of the k-dependent energy gap. 17. Ionization of Atoms and the Thomas-Fermi Model for the Electric Field in Crystal Planar Channels Institute of Scientific and Technical Information of China (English) LIU Ying-Tai; ZHANG Qi-Ren; GAO Chun-Yuan 2002-01-01 The electric field in the crystal planar channels is studied by the Thomas Fermi method. The Thomas-Fermi equation and the corresponding boundary conditions are derived for the crystal planar channels. The numericalsolution for the electric field in the channels between (110) planes of the single crystal silicon and the critical angles ofchannelling protons in them are shown. Reasonable agreements with the experimental data are obtained. The resultsshow that the Thomas-Fermi method for the crystal works well in this study, and a microscopic research of the channelelectric field with the contribution of all atoms and the atomic ionization being taken into account is practical. 18. Bilayer honeycomb lattice with ultracold atoms: Multiple Fermi surfaces and incommensurate spin density wave instability Science.gov (United States) Dey, Santanu; Sensarma, Rajdeep 2016-12-01 We propose an experimental setup using ultracold atoms to implement a bilayer honeycomb lattice with Bernal stacking. In the presence of a potential bias between the layers and at low densities, fermions placed in this lattice form an annular Fermi sea. The presence of two Fermi surfaces leads to interesting patterns in Friedel oscillations and RKKY interactions in the presence of impurities. Furthermore, a repulsive fermion-fermion interaction leads to a Stoner instability towards an incommensurate spin density wave order with a wave vector equal to the thickness of the Fermi sea. The instability occurs at a critical interaction strength which goes down with the density of the fermions. We find that the instability survives interaction renormalization due to vertex corrections and discuss how this can be seen in experiments. We also track the renormalization group flows of the different couplings between the fermionic degrees of freedom, and find that there are no perturbative instabilities, and that Stoner instability is the strongest instability which occurs at a critical threshold value of the interaction. The critical interaction goes to zero as the chemical potential is tuned towards the band bottom. 19. Quantum Degenerate Fermi-Bose Mixtures of 40K and 87Rb Atoms in a Quadrupole-Ioffe Configuration Trap Institute of Scientific and Technical Information of China (English) XIONG De-Zhi; CHEN Hai-Xia; WANG Peng-Jun; YU Xu-Dong; GAO Feng; ZHANG Jing 2008-01-01 @@ We report on the attainment of quantum degeneracy of 40K by means of efficient thermal collisions with the evaporatively cooled 87Rb atoms.In a quadrupole-Ioffe configuration trap,potassium atoms are cooled to 0.5 times the Fermi temperature.We obtain up to 7.59 × 105 degenerate fermions 40K. 20. Quantum measurement-induced antiferromagnetic order and density modulations in ultracold Fermi gases in optical lattices Science.gov (United States) Mazzucchi, Gabriel; Caballero-Benitez, Santiago F.; Mekhov, Igor B. 2016-08-01 Ultracold atomic systems offer a unique tool for understanding behavior of matter in the quantum degenerate regime, promising studies of a vast range of phenomena covering many disciplines from condensed matter to quantum information and particle physics. Coupling these systems to quantized light fields opens further possibilities of observing delicate effects typical of quantum optics in the context of strongly correlated systems. Measurement backaction is one of the most funda- mental manifestations of quantum mechanics and it is at the core of many famous quantum optics experiments. Here we show that quantum backaction of weak measurement can be used for tailoring long-range correlations of ultracold fermions, realizing quantum states with spatial modulations of the density and magnetization, thus overcoming usual requirement for a strong interatomic interactions. We propose detection schemes for implementing antiferromagnetic states and density waves. We demonstrate that such long-range correlations cannot be realized with local addressing, and they are a consequence of the competition between global but spatially structured backaction of weak quantum measurement and unitary dynamics of fermions. 1. Quantum measurement-induced antiferromagnetic order and density modulations in ultracold Fermi gases in optical lattices. Science.gov (United States) Mazzucchi, Gabriel; Caballero-Benitez, Santiago F; Mekhov, Igor B 2016-08-11 Ultracold atomic systems offer a unique tool for understanding behavior of matter in the quantum degenerate regime, promising studies of a vast range of phenomena covering many disciplines from condensed matter to quantum information and particle physics. Coupling these systems to quantized light fields opens further possibilities of observing delicate effects typical of quantum optics in the context of strongly correlated systems. Measurement backaction is one of the most funda- mental manifestations of quantum mechanics and it is at the core of many famous quantum optics experiments. Here we show that quantum backaction of weak measurement can be used for tailoring long-range correlations of ultracold fermions, realizing quantum states with spatial modulations of the density and magnetization, thus overcoming usual requirement for a strong interatomic interactions. We propose detection schemes for implementing antiferromagnetic states and density waves. We demonstrate that such long-range correlations cannot be realized with local addressing, and they are a consequence of the competition between global but spatially structured backaction of weak quantum measurement and unitary dynamics of fermions. 2. Atom chip apparatus for experiments with ultracold rubidium and potassium gases Energy Technology Data Exchange (ETDEWEB) Ivory, M. K.; Ziltz, A. R.; Fancher, C. T.; Pyle, A. J.; Sensharma, A.; Chase, B.; Field, J. P.; Garcia, A.; Aubin, S., E-mail: [email protected] [Department of Physics, College of William and Mary, Williamsburg, Virginia 23187 (United States); Jervis, D. [Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7 (Canada) 2014-04-15 We present a dual chamber atom chip apparatus for generating ultracold {sup 87}Rb and {sup 39}K atomic gases. The apparatus produces quasi-pure Bose-Einstein condensates of 10{sup 4} {sup 87}Rb atoms in an atom chip trap that features a dimple and good optical access. We have also demonstrated production of ultracold {sup 39}K and subsequent loading into the chip trap. We describe the details of the dual chamber vacuum system, the cooling lasers, the magnetic trap, the multicoil magnetic transport system, the atom chip, and two optical dipole traps. Due in part to the use of light-induced atom desorption, the laser cooling chamber features a sufficiently good vacuum to also support optical dipole trap-based experiments. The apparatus is well suited for studies of atom-surface forces, quantum pumping and transport experiments, atom interferometry, novel chip-based traps, and studies of one-dimensional many-body systems. 3. Atom chip apparatus for experiments with ultracold rubidium and potassium gases. Science.gov (United States) Ivory, M K; Ziltz, A R; Fancher, C T; Pyle, A J; Sensharma, A; Chase, B; Field, J P; Garcia, A; Jervis, D; Aubin, S 2014-04-01 We present a dual chamber atom chip apparatus for generating ultracold (87)Rb and (39)K atomic gases. The apparatus produces quasi-pure Bose-Einstein condensates of 10(4) (87)Rb atoms in an atom chip trap that features a dimple and good optical access. We have also demonstrated production of ultracold (39)K and subsequent loading into the chip trap. We describe the details of the dual chamber vacuum system, the cooling lasers, the magnetic trap, the multicoil magnetic transport system, the atom chip, and two optical dipole traps. Due in part to the use of light-induced atom desorption, the laser cooling chamber features a sufficiently good vacuum to also support optical dipole trap-based experiments. The apparatus is well suited for studies of atom-surface forces, quantum pumping and transport experiments, atom interferometry, novel chip-based traps, and studies of one-dimensional many-body systems. 4. Comparing and contrasting nuclei and cold atomic gases DEFF Research Database (Denmark) Zinner, Nikolaj Thomas; Jensen, Aksel Stenholm 2013-01-01 , interactions, and relevant length and energy scales of cold atoms and nuclei. Next we address some attempts in nuclear physics to transfer the concepts of condensates in nuclei that can in principle be built from bosonic alpha-particle constituents. We also consider Efimov physics, a prime example of nuclear... 5. Atom Interferometry with Ultracold Quantum Gases in a Microgravity Environment Science.gov (United States) Williams, Jason; D'Incao, Jose; Chiow, Sheng-Wey; Yu, Nan 2015-05-01 Precision atom interferometers (AI) in space promise exciting technical capabilities for fundamental physics research, with proposals including unprecedented tests of the weak equivalence principle, precision measurements of the fine structure and gravitational constants, and detection of gravity waves and dark energy. Consequently, multiple AI-based missions have been proposed to NASA, including a dual-atomic-species interferometer that is to be integrated into the Cold Atom Laboratory (CAL) onboard the International Space Station. In this talk, I will discuss our plans and preparation at JPL for the proposed flight experiments to use the CAL facility to study the leading-order systematics expected to corrupt future high-precision measurements of fundamental physics with AIs in microgravity. The project centers on the physics of pairwise interactions and molecular dynamics in these quantum systems as a means to overcome uncontrolled shifts associated with the gravity gradient and few-particle collisions. We will further utilize the CAL AI for proof-of-principle tests of systematic mitigation and phase-readout techniques for use in the next-generation of precision metrology experiments based on AIs in microgravity. This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. 6. Evidence for ferromagnetic instability in a repulsive Fermi gas of ultracold atoms CERN Document Server Valtolina, G; Amico, A; Burchianti, A; Recati, A; Enss, T; Inguscio, M; Zaccanti, M; Roati, G 2016-01-01 Ferromagnetism is among the most spectacular manifestations of interactions within many-body fermion systems. In contrast to weak-coupling phenomena, it requires strong repulsion to develop, making a quantitative description of ferromagnetic materials notoriously difficult. This is especially true for itinerant ferromagnets, where magnetic moments are not localized into a crystal lattice. In particular, it is still debated whether the simplest case envisioned by Stoner of a homogeneous Fermi gas with short-range repulsive interactions can exhibit ferromagnetism at all. In this work, we positively answer this question by studying a clean model system consisting of a binary spin-mixture of ultracold 6Li atoms, whose repulsive interaction is tuned via a Feshbach resonance. We drastically limit detrimental pairing effects that affected previous studies by preparing the gas in a magnetic domain-wall configuration. We reveal the ferromagnetic instability by observing the softening of the spin-dipole collective mode... 7. Ionization of Atoms and the Thomas－Fermi Model for the Electric Field in Crystal Planar Channels Institute of Scientific and Technical Information of China (English) LIUYing－Tai; ZHANGQi－Ren; 等 2002-01-01 The electric field in the crystal planar channels is studied by the Thomas-Fermi method.The ThomasFermi equation and the corresponding boundary conditions are derived for the crystal palanar channels,The numerical solution for the elctric field in the channels between(110) Planes of the single crystal silicaon and the critical angles of channelling protons in them are shown.Reasonable agreements with the experimental data are obtained.The results show that the Thomas-Fermi method for the crystal works well in this study,and a microscopic research of the channel electric field with the contribution of all atoms and the atomic ionization being taken into account is practical. 8. An effective field theory analysis of Efimov features in heteronuclear mixture of ultracold atomic gases Science.gov (United States) Acharya, Bijaya; Ji, Chen; Platter, Lucas 2016-05-01 Recent experimental studies have unveiled Efimov physics in ultracold atomic gases of heteronuclear mixtures. The recombination features of such atomic systems display universal correlations including discrete scaling invariance. We use Effective Field Theory (EFT) to study the Efimov features of the heteronuclear three-atom systems consisting of two identical bosons which interact with each other through a natural scattering length and with the third particle through a large scattering length. We compute the corrections to the universal correlations by perturbative insertions of the interspecies effective range and the intraspecies scattering length. Such an analysis is relevant for mixtures of ultracold atomic gases near the interspecies Feshbach resonance. Supported by the US Department of Energy under Contract No. DE-AC05-00OR22725 and the National Science Foundation under Grant No. PHY-1516077. 9. Critical Dynamics in Quenched 2D Atomic Gases Science.gov (United States) Larcher, F.; Dalfovo, F.; Proukakis, N. P. 2016-05-01 Non-equilibrium dynamics across phase transitions is a subject of intense investigations in diverse physical systems. One of the key issues concerns the validity of the Kibble-Zurek (KZ) scaling law for spontaneous defect creation. The KZ mechanism has been recently studied in cold atoms experiments. Interesting open questions arise in the case of 2D systems, due to the distinct nature of the Berezinskii-Kosterlitz-Thouless (BKT) transition. Our studies rely on the stochastic Gross-Pitaevskii equation. We perform systematic numerical simulations of the spontaneous emergence and subsequent dynamics of vortices in a uniform 2D Bose gas, which is quenched across the BKT phase transition in a controlled manner, focusing on dynamical scaling and KZ-type effects. By varying the transverse confinement, we also look at the extent to which such features can be seen in current experiments. Financial support from EPSRC and Provincia Autonoma di Trento. 10. State-specific transport properties of partially ionized flows of electronically excited atomic gases Science.gov (United States) Istomin, V. A.; Kustova, E. V. 2017-03-01 State-to-state approach for theoretical study of transport properties in atomic gases with excited electronic degrees of freedom of both neutral and ionized species is developed. The dependence of atomic radius on the electronic configuration of excited atoms is taken into account in the transport algorithm. Different cutoff criteria for increasing atomic radius are discussed and the limits of applicability for these criteria are evaluated. The validity of a Slater-like model for the calculation of state-resolved transport coefficients in neutral and ionized atomic gases is shown. For ionized flows, a method of evaluation for effective cross-sections of resonant charge-transfer collisions is suggested. Accurate kinetic theory algorithms for modelling the state-specific transport properties are applied for the prediction of transport coefficients in shock heated flows. Based on the numerical observations, different distributions over electronic states behind the shock front are considered. For the Boltzmann-like distributions at temperatures greater than 14,000 K, an important effect of electronic excitation on the partial thermal conductivity and viscosity coefficients is found for both neutral and ionized atomic gases: increasing radius of excited atoms causes a strong decrease in these transport coefficients. Similarly, the presence of electronically excited states with increased atomic radii leads to reduced diffusion coefficients. Nevertheless the overall impact of increasing effective cross-sections on the transport properties just behind the shock front under hypersonic reentry conditions is found to be minor since the populations of high-lying electronic energy levels behind the shock waves are low. 11. Coexistence of photonic and atomic Bose-Einstein condensates in ideal atomic gases Directory of Open Access Journals (Sweden) N. Boichenko 2015-12-01 Full Text Available We have studied conditions of photon Bose-Einstein condensate formation that is in thermodynamic equilibrium with ideal gas of two-level Bose atoms below the degeneracy temperature. Equations describing thermodynamic equilibrium in the system were formulated; critical temperatures and densities of photonic and atomic gas subsystems were obtained analytically. Coexistence conditions of these photonic and atomic Bose-Einstein condensates were found. There was predicted the possibility of an abrupt type of photon condensation in the presence of Bose condensate of ground-state atoms: it was shown that the slightest decrease of the temperature could cause a significant gathering of photons in the condensate. This case could be treated as a simple model of the situation known as "stopped light" in cold atomic gas. We also showed how population inversion of atomic levels can be created by lowering the temperature. The latter situation looks promising for light accumulation in atomic vapor at very low temperatures. 12. Fermi and Coulomb correlation effects upon the interacting quantum atoms energy partition CERN Document Server Ruiz, Isela; Holguín-Gallego, Fernando José; Francisco, Evelio; Pendás, Ángel Martín; Rocha-Rinza, Tomás 2016-01-01 The Interacting Quantum Atoms (IQA) electronic energy partition is an important method in the field of quantum chemical topology which has given important insights of different systems and processes in physical chemistry. There have been several attempts to include Electron Correlation (EC) in the IQA approach, for example, through DFT and Hartree-Fock/Coupled-Cluster (HF/CC) transition densities. This work addresses the separation of EC in Fermi and Coulomb correlation and its effect upon the IQA analysis by taking into account spin-dependent one- and two-electron matrices $D^{\\mathrm{HF/CC}}_{p\\sigma q \\sigma}$ and $d^{\\mathrm{HF/CC}}_{p\\sigma q\\sigma r\\tau s\\tau}$ wherein $\\sigma$ and $\\tau$ represent either of the $\\alpha$ and $\\beta$ spin projections. We illustrate this approach by considering BeH$_2$,BH, CN$^-$, HF, LiF, NO$^+$, LiH, H$_2$O$\\cdots$H$_2$O and C$_2$H$_2$, which comprise non-polar covalent, polar covalent, ionic and hydrogen bonded systems. The same and different spin contributions to ($i$... 13. Nonlinear pressure shifts of alkali-metal atoms in inert gases. Science.gov (United States) Gong, F; Jau, Y-Y; Happer, W 2008-06-13 Precise measurements show that the microwave resonance frequencies of ground-state Rb or Cs atoms have a nonlinear dependence on the pressure of the buffer gases Ar and Kr. No nonlinearities were observed in the gases He or N(2). These observations strongly suggest that the nonlinearities are due to the van der Waals molecules that form in Ar and Kr, but not in He or N(2). The nonlinear part of the shifts is largest in the pressure range of a few tens of torr, similar to the operating pressures of gas-cell atomic clocks. The observed shifts are very well described by a simple function, parametrized by the effective three-body formation rate of molecules and by the effective product of the collisionally limited lifetime times the shift of the hyperfine coupling coefficient in the molecule. 14. Fundamental Interactions for Atom Interferometry with Ultracold Quantum Gases in a Microgravity Environment Science.gov (United States) D'Incao, Jose P.; Willians, Jason R. 2015-05-01 Precision atom interferometers (AI) in space are a key element for several applications of interest to NASA. Our proposal for participating in the Cold Atom Laboratory (CAL) onboard the International Space Station is dedicated to mitigating the leading-order systematics expected to corrupt future high-precision AI-based measurements of fundamental physics in microgravity. One important focus of our proposal is to enhance initial state preparation for dual-species AIs. Our proposed filtering scheme uses Feshbach molecular states to create highly correlated mixtures of heteronuclear atomic gases in both their position and momentum distributions. We will detail our filtering scheme along with the main factors that determine its efficiency. We also show that the atomic and molecular heating and loss rates can be mitigated at the unique temperature and density regimes accessible on CAL. This research is supported by the National Aeronautics and Space Administration. 15. Measuring laser carrier-envelope phase effects in the noble gases with an atomic hydrogen calibration standard CERN Document Server Khurmi, Champak; U, Satya Sainadh; Ivanov, I A; Kheifets, A S; Tong, X M; Litvinyuk, I V; Sang, R T; Kielpinski, D 2016-01-01 We present accurate measurements of carrier-envelope phase effects on ionisation of the noble gases with few-cycle laser pulses. The experimental apparatus is calibrated by using atomic hydrogen data to remove any systematic offsets and thereby obtain accurate CEP data on other generally used noble gases such as Ar, Kr and Xe. Experimental results for H are well supported by exact TDSE theoretical simulations however significant differences are observed in case of noble gases. 16. Critical temperature of Bose-Einstein condensation in trapped atomic Bose-Fermi mixtures Energy Technology Data Exchange (ETDEWEB) Albus, A P [Institut fuer Physik, Universitaet Potsdam, D-14469 Potsdam (Germany); Giorgini, S [Dipartimento di Fisica, Universita di Trento, and Istituto Nazionale per la Fisica della Materia, I-38050 Povo (Italy); Illuminati, F [Dipartimento di Fisica, Universita di Salerno, and Istituto Nazionale per la Fisica della Materia, I-84081 Baronissi (Italy); Viverit, L [Dipartimento di Fisica, Universita di Trento, and Istituto Nazionale per la Fisica della Materia, I-38050 Povo (Italy) 2002-12-14 We calculate the shift in the critical temperature of Bose-Einstein condensation for a dilute Bose-Fermi mixture confined by a harmonic potential, to lowest order in both the Bose-Bose and Bose-Fermi coupling constants. The relative importance of the effect on the critical temperature of the boson-boson and boson-fermion interactions is investigated as a function of the parameters of the mixture. The possible relevance of the shift of the transition temperature in current experiments on trapped Bose-Fermi mixtures is discussed. (letter to the editor) 17. Observation of Multibubble Sonoluminescence from Water Saturated with Various Gases during Ultrasonic Atomization Science.gov (United States) Harada, Hisashi; Iwata, Naohiro; Shiratori, Keisuke 2009-07-01 Multibubble sonoluminescence (MBSL) from water saturated with various atmospheric gases was observed using an ultrasonic atomizer (2.4 MHz). The majority of sonoluminescence (SL) in the system did not originate from capillary waves but from acoustic cavitation. The dependence of MBSL intensity on the type of dissolved gas was confirmed. Atomization occurred similarly in all cases. The intensities for the dissolved gases were in the following order: Ar > Air > O2 > N2 ≫He, H2, CO2. The intensity for water saturated with air is higher than those for the O2- and N2-saturated solutions. To examine the effect of gas mixing, MBSL was measured for various ratios of O2 to N2. The maximum intensity was obtained at 40% O2/60% N2. In the regions above and below this ratio, the intensity decreased gradually. To explain this result, the possibilities of Ar rectification and chemical reactions between O2 and N2 gases were also discussed. After examination, it could not be confirmed that Ar rectification occurred. Chemical reactions of O2 with N2 proceed inside the cavitation bubble. 18. Atomic capture and transfer of negative pions stopped in binary mixtures of hydrogen with polyatomic gases Energy Technology Data Exchange (ETDEWEB) Vasilyev, V.A.; Levay, B.; Minkova, A.; Petrukhin, V.I.; Horvath, D. 1985-12-01 The atomic capture and transfer of stopped negative pions have been studied in binary gas mixtures of H/sub 2/+M, where M is CCl/sub 2/F/sub 2/, CClF/sub 3/, CBrF/sub 3/ or SF/sub 6/. The ..pi../sup 0/ yield, versus relative atomic concentration Csub(A) of M, goes through a maximum at Csub(A)proportional0.1 and levels off at zero at high concentrations. This phenomenon together with other observed characteristics of the atomic capture and transfer of pions in these systems is interpreted in the frame of a phenomenological model. The average transfer coefficients anti ..lambda..sub(Z) exhibit a weak concentration dependence. The estimated average atomic capture ratios anti A(Z/H) are lower than those found for noble gases, probably because of the mutual screening of the constituent atoms in the molecules. The probability of pion capture in an atomic orbit is not proportional to the stopping power of the components of the mixture. (orig.). 19. Self-Channeling of High-Power Long-Wave Infrared Pulses in Atomic Gases Science.gov (United States) Schuh, K.; Kolesik, M.; Wright, E. M.; Moloney, J. V.; Koch, S. W. 2017-02-01 We simulate and elucidate the self-channeling of high-power 10 μ m infrared pulses in atomic gases. The major new result is that the peak intensity can remain remarkably stable over many Rayleigh ranges. This arises from the balance between the self-focusing, diffraction, and defocusing caused by the excitation induced dephasing due to many-body Coulomb effects that enhance the low-intensity plasma densities. This new paradigm removes the Rayleigh range limit for sources in the 8 - 12 μ m atmospheric transmission window and enables transport of individual multi-TW pulses over multiple kilometer ranges. 20. Information and backaction due to phase contrast imaging measurements of cold atomic gases: beyond Gaussian states CERN Document Server Ilo-Okeke, Ebubechukwu O 2016-01-01 We further examine a theory of phase contrast imaging (PCI) of cold atomic gases, first introduced by us in Phys. Rev. Lett. {\\bf 112}, 233602 (2014). We model the PCI measurement by directly calculating the entangled state between the light and the atoms due to the ac Stark shift, which induces a conditional phase shift on the light depending upon the atomic state. By interfering the light that passes through the BEC with the original light, one can obtain information of the atomic state at a single shot level. We derive an exact expression for a measurement operator that embodies the information obtained from PCI, as well as the back-action on the atomic state. By the use of exact expressions for the measurement process, we go beyond the continuous variables approximation such that the non-Gaussian regime can be accessed for both the measured state and the post-measurement state. Features such as the photon probability density, signal, signal variance, Fisher information, error of the measurement, and the b... 1. Quantum-Shell Corrections to the Finite-Temperature Thomas-Fermi-Dirac Statistical Model of the Atom Energy Technology Data Exchange (ETDEWEB) Ritchie, A B 2003-07-22 Quantum-shell corrections are made directly to the finite-temperature Thomas-Fermi-Dirac statistical model of the atom by a partition of the electronic density into bound and free components. The bound component is calculated using analytic basis functions whose parameters are chosen to minimize the energy. Poisson's equation is solved for the modified density, thereby avoiding the need to solve Schroedinger's equation for a self-consistent field. The shock Hugoniot is calculated for aluminum: shell effects characteristic of quantum self-consistent field models are fully captures by the present model. 2. Phase space methods for degenerate quantum gases CERN Document Server Dalton, Bryan J; Barnett, Stephen M 2015-01-01 Recent experimental progress has enabled cold atomic gases to be studied at nano-kelvin temperatures, creating new states of matter where quantum degeneracy occurs - Bose-Einstein condensates and degenerate Fermi gases. Such quantum states are of macroscopic dimensions. This book presents the phase space theory approach for treating the physics of degenerate quantum gases, an approach already widely used in quantum optics. However, degenerate quantum gases involve massive bosonic and fermionic atoms, not massless photons. The book begins with a review of Fock states for systems of identical atoms, where large numbers of atoms occupy the various single particle states or modes. First, separate modes are considered, and here the quantum density operator is represented by a phase space distribution function of phase space variables which replace mode annihilation, creation operators, the dynamical equation for the density operator determines a Fokker-Planck equation for the distribution function, and measurable... 3. Exotic pairing in 1D spin-3/2 atomic gases with SO(4 symmetry Directory of Open Access Journals (Sweden) Yuzhu Jiang 2015-06-01 Full Text Available Tuning interactions in the spin singlet and quintet channels of two colliding atoms could change the symmetry of the one-dimensional spin-3/2 fermionic systems of ultracold atoms while preserving the integrability. Here we find a novel SO(4 symmetry integrable point in the spin-3/2 Fermi gas and derive the exact solution of the model using the Bethe ansatz. In contrast to the model with SU(4 and SO(5 symmetries, the present model with SO(4 symmetry preserves spin singlet and quintet Cooper pairs in two sets of SU(2⊗SU(2 spin subspaces. We obtain full phase diagrams, including the Fulde–Ferrel–Larkin–Ovchinnikov like pair correlations, spin excitations and quantum criticality through the generalized Yang–Yang thermodynamic equations. In particular, various correlation functions are calculated by using finite-size corrections in the frame work of conformal field theory. Moreover, within the local density approximation, we further find that spin singlet and quintet pairs form subtle multiple shell structures in density profiles of the trapped gas. 4. Probing the dynamic structure factor of a neutral Fermi superfluid along the BCS-BEC crossover using atomic impurity qubits Science.gov (United States) Mitchison, Mark T.; Johnson, Tomi H.; Jaksch, Dieter 2016-12-01 We study an impurity atom trapped by an anharmonic potential, immersed within a cold atomic Fermi gas with attractive interactions that realizes the crossover from a Bardeen-Cooper-Schrieffer superfluid to a Bose-Einstein condensate. Considering the qubit comprising the lowest two vibrational energy eigenstates of the impurity, we demonstrate that its dynamics probes the equilibrium density fluctuations encoded in the dynamic structure factor of the superfluid. Observing the impurity's evolution is thus shown to facilitate nondestructive measurements of the superfluid order parameter and the contact between collective and single-particle excitation spectra. Our setup constitutes a model of an open quantum system interacting with a thermal reservoir, the latter supporting both bosonic and fermionic excitations that are also coupled to each other. 5. Fermi level de-pinning of aluminium contacts to n-type germanium using thin atomic layer deposited layers Energy Technology Data Exchange (ETDEWEB) Gajula, D. R., E-mail: [email protected]; Baine, P.; Armstrong, B. M.; McNeill, D. W. [School of Electronics, Electrical Engineering and Computer Science, Queen' s University Belfast, Ashby Building, Stranmillis Road, Belfast BT9 5AH (United Kingdom); Modreanu, M.; Hurley, P. K. [Tyndall National Institute, University College Cork, Lee Maltings, Cork (Ireland) 2014-01-06 Fermi-level pinning of aluminium on n-type germanium (n-Ge) was reduced by insertion of a thin interfacial dielectric by atomic layer deposition. The barrier height for aluminium contacts on n-Ge was reduced from 0.7 eV to a value of 0.28 eV for a thin Al{sub 2}O{sub 3} interfacial layer (∼2.8 nm). For diodes with an Al{sub 2}O{sub 3} interfacial layer, the contact resistance started to increase for layer thicknesses above 2.8 nm. For diodes with a HfO{sub 2} interfacial layer, the barrier height was also reduced but the contact resistance increased dramatically for layer thicknesses above 1.5 nm. 6. Site-selected luminescence of atomic europium in the solid rare gases Energy Technology Data Exchange (ETDEWEB) Byrne, Owen; McCaffrey, John G. [Department of Chemistry, National University of Ireland - Maynooth, Maynooth, County Kildare (Ireland) 2011-07-14 Site-selective excitation has been used to simplify complex emission recorded in the visible spectral region for atomic europium isolated in the solid rare gases. In addition to y{sup 8}P resonance fluorescence, excitation of the y{sup 8}P state produces emission from the z{sup 6}P state and the metastable a{sup 10}D state. Very weak emission at 690 nm is tentatively assigned to the J = 9/2 level of the z{sup 10}P state. Eu atoms isolated in the red and blue sites exhibit very different temperature dependence both spectrally and temporally. For the y{sup 8}P state emission the red site atoms exhibit small Stokes shifts and yield radiative lifetimes while the emission from the blue site loses intensity and the temporal profiles shorten dramatically between 10 and 16 K indicating very efficient non-radiative relaxation in this site. An analysis of the Stokes shifts exhibited for the y{sup 8}P state in each site supports the attributions made in a previous publication [O. Byrne and J.G. McCaffrey, J. Chem. Phys. 134, 124501 (2011)] that the smaller blue tetravacancy site has a greater repulsive interaction with the guest. With the exception of the y{sup 8}P state resonance fluorescence, the recorded decay profiles of all the other emissions exhibit multiple components. This behaviour has been attributed to the existence of multiple crystal field levels arising from the splitting of the distinct spin-orbit levels from which emission occurs. 7. Reduction of Fermi level pinning at Au-MoS2 interfaces by atomic passivation on Au surface Science.gov (United States) Min, Kyung-Ah; Park, Jinwoo; Wallace, Robert M.; Cho, Kyeongjae; Hong, Suklyun 2017-03-01 Monolayer molybdenum disulfide (MoS2), which is a semiconducting material with direct band gap of ˜1.8 eV, has drawn much attention for application in field effect transistors (FETs). In this connection, it is very important to understand the Fermi level pinning (FLP) which occurs at metal-semiconductor interfaces. It is known that MoS2 has an n-type contact with Au, which is a high work function metal, representing the strong FLP at Au-MoS2 interfaces. However, such FLP can obstruct the attainment of high performance of field effect devices. In this study, we investigate the reduction of FLP at Au-MoS2 interfaces by atomic passivation on Au(111) using first-principles calculations. To reduce the FLP at Au-MoS2 interfaces, we consider sulfur, oxygen, nitrogen, fluorine, and hydrogen atoms that can passivate the surface of Au(111). Calculations show that passivating atoms prevent the direct contact between Au(111) and MoS2, and thus FLP at Au-MoS2 interfaces is reduced by weak interaction between atom-passivated Au(111) and MoS2. Especially, FLP is greatly reduced at sulfur-passivated Au-MoS2 interfaces with the smallest binding energy. Furthermore, fluorine-passivated Au(111) can form ohmic contact with MoS2, representing almost zero Schottky barrier height (SBH). We suggest that SBH can be controlled depending on the passivating atoms on Au(111). 8. Bosonic models with Fermi-liquid kinematics: realizations and properties Science.gov (United States) Goldbart, Paul; Gopalakrishnan, Sarang; Lamacraft, Austen 2011-03-01 We consider models of interacting bosons in which the single-particle kinetic energy achieves its minimum on a surface in momentum space. The kinematics of such models resembles that resulting from Pauli blocking in Fermi liquids; therefore, Shankar's renormalization-group treatment of Fermi liquids can be adapted to investigate phase transitions in these bosonic systems. We explore possible experimental realizations of such models in cold atomic gases: e.g., via spin-orbit coupling, multimode-cavity-mediated interactions, and Cooper pairing of Fermi gases in spin-dependent lattices. We address the phase structure and critical behavior of the resulting models within the framework of Ref., focusing in particular on Bose-Einstein condensation and on quantum versions of the Brazovskii transition from a superfluid to a supersolid. 9. Non-Equilbrium Fermi Gases Science.gov (United States) 2016-02-02 frequencies ) are smaller than the spontaneous emission rate ge. Unfortunately, this method is invalid in the bare basis for broad resonances, where the...classification in accordance with security classification regulations , e.g. U, C, S, etc. If this form contains classified information, stamp classification...show that the EIT method creates narrow features in the scattering phase shift, enabling control by optical frequency rather than intensity, providing 10. Quantum Simulations of Condensed Matter Systems Using Ultra-Cold Atomic Gases Science.gov (United States) 2013-03-01 Feynman diagrams versus Fermi-‐gas Feynman emulator”, Nature Physics 8, 366...BEC-‐BCS Crossover and the Unitary Fermi Gas”, Lecture Notes in Physics , Volume 836, edited by Wilhelm... Lecture at 100th Anniversary Solvay Conference on Physics , "The Theory of the 11. An interpolatory ansatz captures the physics of one-dimensional confined Fermi systems DEFF Research Database (Denmark) Andersen, Molte Emil Strange; Salami Dehkharghani, Amin; Volosniev, A. G.; 2016-01-01 beyond the Bethe ansatz and bosonisation allow us to predict the behaviour of one-dimensional confined systems with strong short-range interactions, and new experiments with cold atomic Fermi gases have already confirmed these theories. Here we demonstrate that a simple linear combination of the strongly... 12. 75 FR 63867 - DTE Energy; Enrico Fermi Atomic Power Plant Unit 1, Exemption From Certain Security Requirements Science.gov (United States) 2010-10-18 .... Nuclear Regulatory Commission (NRC or the Commission) now or hereafter in effect. Fermi 1 was a fast breeder reactor power plant cooled by sodium and operated at essentially atmospheric pressure. In November... in Monroe County, Michigan. Fermi 1 is a permanently shutdown nuclear reactor facility. The... 13. Strong enhancement of Penning ionization for asymmetric atom pairs in cold Rydberg gases: the Tom and Jerry effect KAUST Repository Efimov, D K 2016-05-18 We consider Penning ionization of Rydberg atom pairs as an Auger-type process induced by the dipole-dipole interaction and employ semiclassical formulae for dipole transitions to calculate the autoionization width as a function of the principal quantum numbers, n d, n i, of both atoms. While for symmetric atom pairs with the well-known increase of the autoionization width with increasing n 0 is obtained, the result for asymmetric pairs is counterintuitive - for a fixed n i of the ionizing atom of the pair, the autoionization width strongly increases with decreasing n d of the de-excited atom. For H Rydberg atoms this increase reaches two orders of magnitude at the maximum of the n d dependence, and the same type of counterintuitive behavior is exhibited also by Na, Rb and Cs atoms. This is a purely quantum-mechanical effect, which points towards existence of optimal (we call them \\'Tom\\' and \\'Jerry\\' for \\'big\\' and \\'small\\') pairs of Rydberg atoms with respect to autoionization efficiency. Building on the model of population redistribution in cold Rydberg gases proposed in [1], we demonstrate that population evolution following the initial laser excitation of Rydberg atoms in state n 0 would eventually lead to the formation of such Tom-Jerry pairs with which feature autoionization widths that are enhanced by several orders of magnitude compared to that of two atoms in the initial laser-excited state n 0. We also show that in the high-density regime of cold Rydberg gas experiments the ionization rate of Tom-Jerry pairs can be substantially larger than the blackbody radiation-induced photoionization rate. © 2016 IOP Publishing Ltd. 14. Metastability and coherence of repulsive polarons in a strongly interacting Fermi mixture DEFF Research Database (Denmark) Kohstall, Cristoph; Zaccanti, Mattheo; Jag, Matthias; 2012-01-01 Ultracold Fermi gases with tunable interactions provide a test bed for exploring the many-body physics of strongly interacting quantum systems1, 2, 3, 4. Over the past decade, experiments have investigated many intriguing phenomena, and precise measurements of ground-state properties have provided...... benchmarks for the development of theoretical descriptions. Metastable states in Fermi gases with strong repulsive interactions5, 6, 7, 8, 9, 10, 11 represent an exciting area of development. The realization of such systems is challenging, because a strong repulsive interaction in an atomic quantum gas...... implies the existence of a weakly bound molecular state, which makes the system intrinsically unstable against decay. Here we use radio-frequency spectroscopy to measure the complete excitation spectrum of fermionic 40K impurities resonantly interacting with a Fermi sea of 6Li atoms. In particular, we... 15. FINAL–REPORT NO. 2: INDEPENDENT CONFIRMATORY SURVEY SUMMARY AND RESULTS FOR THE ENRICO FERMI ATOMIC POWER PLANT, UNIT 1, NEWPORT, MICHIGAN (DOCKET NO. 50 16; RFTA 10-004) Energy Technology Data Exchange (ETDEWEB) Erika Bailey 2011-07-07 The Enrico Fermi Atomic Power Plant, Unit 1 (Fermi 1) was a fast breeder reactor design that was cooled by sodium and operated at essentially atmospheric pressure. On May 10, 1963, the Atomic Energy Commission (AEC) granted an operating license, DPR-9, to the Power Reactor Development Company (PRDC), a consortium specifically formed to own and operate a nuclear reactor at the Fermi 1 site. The reactor was designed for a maximum capability of 430 megawatts (MW); however, the maximum reactor power with the first core loading (Core A) was 200 MW. The primary system was filled with sodium in December 1960 and criticality was achieved in August 1963. 16. Generation of few-cycle laser pulses:Comparison between atomic and molecular gases in a hollow-core fiber Institute of Scientific and Technical Information of China (English) 黄志远; 戴晔; 赵睿睿; 王丁; 冷雨欣 2016-01-01 We numerically study the pulse compression approaches based on atomic or molecular gases in a hollow-core fiber. From the perspective of self-phase modulation (SPM), we give the extensive study of the SPM infl uence on a probe pulse with molecular phase modulation (MPM) effect. By comparing the two compression methods, we summarize their advan-tages and drawbacks to obtain the few-cycle pulses with micro-or millijoule energies. It is also shown that the double pump-probe approach can be used as a tunable dual-color source by adjusting the time delay between pump and probe pulses to proper values. 17. Towards Quantum Turbulence in Cold Atomic Fermionic Superfluids CERN Document Server Bulgac, Aurel; Wlazłowski, Gabriel 2016-01-01 Fermionic superfluids provide a new realization of quantum turbulence, accessible to both experiment and theory, yet relevant to both cold atoms and nuclear astrophysics. In particular, the strongly interacting Fermi gas realized in cold-atom experiments is closely related to dilute neutron matter in the neutron star crust. Unlike the liquid superfluids 4He (bosons) and 3He (fermions), where quantum turbulence has been studied in laboratory for decades, quantum gases, and in particular superfluid Fermi gases stand apart for a number of reasons. Fermi gases admit a rather reliable microscopic description based on density functional theory which describes both static and dynamical phenomena. Cold atom experiments demonstrate exquisite control over particle number, spin polarization, density, temperature, and interacting strength. Topological defects such as domain walls and quantized vortices, which lie at the heart of quantum turbulence, can be created and manipulated with time-dependent external potentials, a... 18. Computer simulations on resonant fluorescence spectra in atomic gases in two monochromatic laser fields of arbitrary intensity and magnetic field Science.gov (United States) Karagodova, Tamara Y. 1996-03-01 In the intense radiation fields with power density from 104W/cm2 to 109W/cm2 the essential modification of electronic states of atoms occurs displaying, in particular, in modifications of resonant fluorescence (rf) spectra. We use 'Fermi golden rule' for calculations of relative intensities and frequencies for rf multiplet for real multilevel initially unexcited atoms in two monochromatic laser fields of arbitrary intensity resonant to adjacent transitions of (Xi) or (Lambda) types and magnetic field, giving the level splittings of different values from Zeeman to Paschen-Back effect. The dependence of quasienergies on parameters obtained with the help of a sorting program permits us to define the values of parameters for which the states of the system are mixed and so to receive the correct probability amplitudes for instantaneous or adiabatic regimes of switching the perturbation. The analysis of the quasienergies and form of rf spectra permits us to get relations between the form of the spectra and modifications of electronic structure of the atom due to radiation fields and external magnetic field. 19. Measuring the spin polarization of alkali-metal atoms using nuclear magnetic resonance frequency shifts of noble gases Directory of Open Access Journals (Sweden) X. H. Liu 2015-10-01 Full Text Available We report a novel method of measuring the spin polarization of alkali-metal atoms by detecting the NMR frequency shifts of noble gases. We calculated the profile of 87Rb D1 line absorption cross sections. We then measured the absorption profile of the sample cell, from which we calculated the 87Rb number densities at different temperatures. Then we measured the frequency shifts resulted from the spin polarization of the 87Rb atoms and calculated its polarization degrees at different temperatures. The behavior of frequency shifts versus temperature in experiment was consistent with theoretical calculation, which may be used as compensative signal for the NMRG closed-loop control system. 20. Measuring the spin polarization of alkali-metal atoms using nuclear magnetic resonance frequency shifts of noble gases Energy Technology Data Exchange (ETDEWEB) Liu, X. H.; Luo, H.; Qu, T. L., E-mail: [email protected]; Yang, K. Y.; Ding, Z. C. [College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha 410073 (China) 2015-10-15 We report a novel method of measuring the spin polarization of alkali-metal atoms by detecting the NMR frequency shifts of noble gases. We calculated the profile of {sup 87}Rb D1 line absorption cross sections. We then measured the absorption profile of the sample cell, from which we calculated the {sup 87}Rb number densities at different temperatures. Then we measured the frequency shifts resulted from the spin polarization of the {sup 87}Rb atoms and calculated its polarization degrees at different temperatures. The behavior of frequency shifts versus temperature in experiment was consistent with theoretical calculation, which may be used as compensative signal for the NMRG closed-loop control system. 1. Fermi liquid theory CERN Document Server Apostol, M 2001-01-01 sup 3 He liquefies at 3.2 K under normal pressure, where its mean inter-particle separation of a few angstroms, is comparable with the range of the interaction potential (and with the mean inter-particle separation in the corresponding ideal gas); its thermal wavelength is about 8 A, so that, under this conditions, sup 3 He is a quantum liquid of fermions, or a Fermi liquid (sometimes called a normal Fermi liquid too). The motion of the sup 3 He atoms in the (repulsive) self-consistent, meanfield potential is affected by inertial effects, i.e. the particles possess an effective mass, and consequently they obey the Fermi distribution, like an ideal Fermi gas. In this paper the Landau's theory of the Fermi liquid is reviewed. (author) 2. Engineering the Dynamics of Effective Spin-Chain Models for Strongly Interacting Atomic Gases DEFF Research Database (Denmark) Volosniev, A. G.; Petrosyan, D.; Valiente, M. 2015-01-01 We consider a one-dimensional gas of cold atoms with strong contact interactions and construct an effective spin-chain Hamiltonian for a two-component system. The resulting Heisenberg spin model can be engineered by manipulating the shape of the external confining potential of the atomic gas. We...... find that bosonic atoms offer more flexibility for tuning independently the parameters of the spin Hamiltonian through interatomic (intra-species) interaction which is absent for fermions due to the Pauli exclusion principle. Our formalism can have important implications for control and manipulation... 3. Emergence of correlated optics in one-dimensional waveguides for classical and quantum atomic gases Science.gov (United States) Ruostekoski, Janne; Javanainen, Juha 2016-09-01 We analyze the emergence of correlated optical phenomena in the transmission of light through a waveguide that confines classical or ultracold quantum degenerate atomic ensembles. The conditions of the correlated collective response are identified in terms of atom density, thermal broadening, and photon losses by using stochastic Monte Carlo simulations and transfer matrix methods of transport theory. We also calculate the "cooperative Lamb shift" for the waveguide transmission resonance, and discuss line shifts that are specific to effectively one-dimensional waveguide systems. 4. Fermi and Szilard CERN Document Server Byers, N 2002-01-01 This talk is about Enrico Fermi and Leo Szilard, their collaboration and involvement in nuclear energy development and decisions to construct and use the atomic bomb in World War II. Fermi and Szilard worked closely together at Columbia in 1939-40 to explore feasibility of a nuclear chain reaction, and then on the physics for construction of the first pile (nuclear reactor). "On matters scientific or technical there was rarely any disagreement between Fermi and myself" Szilard said. But there were sharp differences on other matters. 5. Quantum gases finite temperature and non-equilibrium dynamics CERN Document Server Szymanska, Marzena; Davis, Matthew; Gardiner, Simon 2013-01-01 The 1995 observation of Bose-Einstein condensation in dilute atomic vapours spawned the field of ultracold, degenerate quantum gases. Unprecedented developments in experimental design and precision control have led to quantum gases becoming the preferred playground for designer quantum many-body systems. This self-contained volume provides a broad overview of the principal theoretical techniques applied to non-equilibrium and finite temperature quantum gases. Covering Bose-Einstein condensates, degenerate Fermi gases, and the more recently realised exciton-polariton condensates, it fills a gap by linking between different methods with origins in condensed matter physics, quantum field theory, quantum optics, atomic physics, and statistical mechanics. Thematically organised chapters on different methodologies, contributed by key researchers using a unified notation, provide the first integrated view of the relative merits of individual approaches, aided by pertinent introductory chapters and the guidance of ed... 6. Debye screening and a Thomas - Fermi model of a dyonic atom in a two potential theory of electromagnetism Energy Technology Data Exchange (ETDEWEB) Wolf, C. [North Adams State College, MA (United States) 1993-02-01 We study the screening of a central Abelian dyon by a surrounding dyon cloud in a two potential theory of electromagnetism. A generalized formula for the Debye screening length is obtained and a Thomas - Fermi Model for a charged cloud surrounding a central Dyonic Core is studied. 20 refs. 7. Optical pumping effect in absorption imaging of F=1 atomic gases CERN Document Server Kim, Sooshin; Noh, Heung-Ryoul; Shin, Y 2016-01-01 We report our study of the optical pumping effect in absorption imaging of $^{23}$Na atoms in the $F=1$ hyperfine spin states. Solving a set of rate equations for the spin populations under a probe beam, we obtain an analytic expression for the optical signal of the $F=1$ absorption imaging. Furthermore, we verify the result by measuring the absorption spectra of $^{23}$Na Bose-Einstein condensates prepared in various spin states with different probe beam pulse durations. The analytic result can be used in quantitative analysis of $F=1$ spinor condensate imaging and readily applied to other alkali atoms with $I=3/2$ nuclear spin such as $^{87}$Rb. 8. Velocity-changing collisional effects in nonlinear atomic spectroscopy and photon echo decay in gases Science.gov (United States) Herman, R. M. 1983-01-01 A general theory of atomic dipole coherence under the influence of collisional phase changes, inelastic effects and optically active atom velocity changes, including those due to anisotropic interactions is presented. Velocity change effects are obtained in closed form. Line shapes appear as convolutions of standard pressure broadening contours with velocity-change contours. Width and shift parameters for the He-broadened Na D lines at 2 m bar pressure, 380 K are calculated, as are He-induced photon echo decay rates for these lines. Overall agreement with xperiment is reasonably good. 9. Microscopic Many-Body Theory of Atomic Bose Gases near a Feshbach Resonance NARCIS (Netherlands) Duine, R.A.; Stoof, H.T.C. 2003-01-01 A Feshbach resonance in the s-wave scattering length occurs if the energy of the two atoms in the incoming open channel is close to the energy of a bound state in a coupled closed channel. Starting from the microscopic Hamiltonian that describes this situation, we derive the effective atom–molecule 10. Even-odd spatial nonequivalence for atomic quantum gases with isotropic spin-orbit couplings Science.gov (United States) Singh, G. S.; Gupta, Reena 2014-05-01 A general expression for the density of states (DOS) of power-law trapped d-dimensional ideal quantum gases with isotropic spin-orbit couplings (SOCs) is derived and is found to bifurcate into even- dand odd- d classes. The expressions for the grand potential and hence for several thermodynamic quantities are then shown to be amenable to exact analytical forms provided d is an odd integer. Also, a condition γ transition temperature and the condensate fraction in a 3D Bose gas under combined presence of the harmonic trapping and the Weyl coupling shows that the condensation is favored by the former but disfavored by the latter. This countering behavior is discussed to be in conformity with the exchange-symmetry-induced statistical interactions resulting from these two entities as enunciated recently [Phys. Rev. A 88, 053607 (2013)]. 11. Effective atomic numbers, electron densities, and tissue equivalence of some gases and mixtures for dosimetry of radiation detectors Directory of Open Access Journals (Sweden) Singh Vishwanath P. 2012-01-01 Full Text Available Total mass attenuation coefficients, µm, effective atomic number, Zeff, and effective electron density, Neff, of different gases - carbon dioxide, methane, acetylene, propane, butane, and pentane used in radiation detectors, have been calculated for the photon energy of 1 keV to 100 GeV. Each gas has constant Zeff values between 0.10 to 10 MeV photon energies; however, these values are way far away from ICRU tissue. Carbon dioxide gas shows the closest tissue equivalence in the entire photon energy spectrum. Relative tissue equivalences of the mixtures of gases with respect to ICRU tissue are in the range of 0.998-1.041 for air, argon (4.5% + methane (95.5%, argon (0.5% + carbon dioxide (99.5%, and nitrogen (5% + methane (7% + carbon dioxide (88%. The gas composition of xenon (0.5% + carbon dioxide (99.5% shows 1.605 times higher tissue equivalence compared to the ICRU tissue. The investigated photon interaction parameters are useful for exposure and energy absorption buildup factors calculation and design, and fabrication of gaseous detectors for ambient radiation measurement by the Geiger-Muller detector, ionization chambers and proportional counters. 12. Generation of few-cycle laser pulses: Comparison between atomic and molecular gases in a hollow-core fiber Science.gov (United States) Zhi-Yuan, Huang; Ye, Dai; Rui-Rui, Zhao; Ding, Wang; Yu-Xin, Leng 2016-07-01 We numerically study the pulse compression approaches based on atomic or molecular gases in a hollow-core fiber. From the perspective of self-phase modulation (SPM), we give the extensive study of the SPM influence on a probe pulse with molecular phase modulation (MPM) effect. By comparing the two compression methods, we summarize their advantages and drawbacks to obtain the few-cycle pulses with micro- or millijoule energies. It is also shown that the double pump-probe approach can be used as a tunable dual-color source by adjusting the time delay between pump and probe pulses to proper values. Project supported by the National Natural Science Foundation of China (Grant Nos. 11204328, 61221064, 61078037, 11127901, 11134010, and 61205208), the National Basic Research Program of China (Grant No. 2011CB808101), and the Natural Science Foundation of Shanghai, China (Grant No. 13ZR1414800). 13. All-optical production of 6Li quantum gases Science.gov (United States) Burchianti, A.; Seman, J. A.; Valtolina, G.; Morales, A.; Inguscio, M.; Zaccanti, M.; Roati, G. 2015-03-01 We report efficient production of quantum gases of 6Li using a sub-Doppler cooling scheme based on the D1 transition. After loading in a standard magneto-optical trap, an atomic sample of 109 atoms is cooled at a temperature of 40 μK by a bichromatic D1 gray-molasses. More than 2×107 atoms are then transferred into a high-intensity optical dipole trap, where a two-spin state mixture is evaporatively cooled down to quantum degeneracy. We observe that D1 cooling remains effective in the deep trapping potential, allowing an effective increase of the atomic phase-space density before starting the evaporation. In a total experimental cycle of 11 s, we produce weakly-interacting degenerate Fermi gases of 7×105 atoms at T/TF molecules. We further describe a simple and compact optical system both for high-resolution imaging and for imprinting a thin optical barrier on the atomic cloud; this represents a first step towards the study of quantum tunneling in strongly interacting superfluid Fermi gases. 14. Dark solitons in a Gross-Pitaevskii equation with a power-law nonlinearity: application to ultracold Fermi gases near the Bose-Einstein condensation regime Energy Technology Data Exchange (ETDEWEB) Yan, D; Kevrekidis, P G [Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515 (United States); Frantzeskakis, D J, E-mail: [email protected] [Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 157 84 (Greece) 2011-10-14 In this work, we consider a model of a defocusing nonlinear Schroedinger equation with a variable nonlinearity exponent. This is motivated by the study of a superfluid Fermi gas in the Bose-Einstein condensation (BEC)-Bardeen-Cooper-Schrieffer crossover. In particular, we focus on the relevant mean-field model in the regime from BEC to unitarity and especially consider the modification of the nearly black soliton oscillation frequency due to the variation in the nonlinearity exponent in a harmonic trapping potential. The analytical expressions given as a function of the relevant nonlinearity exponent are corroborated by numerical computations and also extended past the BEC limit. (paper) 15. Optical pumping effect in absorption imaging of F =1 atomic gases Science.gov (United States) Kim, Sooshin; Seo, Sang Won; Noh, Heung-Ryoul; Shin, Y. 2016-08-01 We report our study of the optical pumping effect in absorption imaging of 23Na atoms in the F =1 hyperfine spin states. Solving a set of rate equations for the spin populations in the presence of a probe beam, we obtain an analytic expression for the optical signal of the F =1 absorption imaging. Furthermore, we verify the result by measuring the absorption spectra of 23Na Bose-Einstein condensates prepared in various spin states with different probe-beam pulse durations. The analytic result can be used in the quantitative analysis of F =1 spinor condensate imaging and readily applied to other alkali-metal atoms with I =3 /2 nuclear spin such as 87Rb. 16. Observability of quantum criticality and a continuous supersolid in atomic gases. Science.gov (United States) Diehl, S; Baranov, M; Daley, A J; Zoller, P 2010-04-23 We analyze the Bose-Hubbard model with a three-body hard-core constraint by mapping the system to a theory of two coupled bosonic degrees of freedom. We find striking features that could be observable in experiments, including a quantum Ising critical point on the transition from atomic to dimer superfluidity at unit filling, and a continuous supersolid phase for strongly bound dimers. 17. Classical stochastic measurement trajectories: Bosonic atomic gases in an optical cavity and quantum measurement backaction Science.gov (United States) Lee, Mark D.; Ruostekoski, Janne 2014-08-01 We formulate computationally efficient classical stochastic measurement trajectories for a multimode quantum system under continuous observation. Specifically, we consider the nonlinear dynamics of an atomic Bose-Einstein condensate contained within an optical cavity subject to continuous monitoring of the light leaking out of the cavity. The classical trajectories encode within a classical phase-space representation a continuous quantum measurement process conditioned on a given detection record. We derive a Fokker-Planck equation for the quasiprobability distribution of the combined condensate-cavity system. We unravel the dynamics into stochastic classical trajectories that are conditioned on the quantum measurement process of the continuously monitored system. Since the dynamics of a continuously measured observable in a many-atom system can be closely approximated by classical dynamics, the method provides a numerically efficient and accurate approach to calculate the measurement record of a large multimode quantum system. Numerical simulations of the continuously monitored dynamics of a large atom cloud reveal considerably fluctuating phase profiles between different measurement trajectories, while ensemble averages exhibit local spatially varying phase decoherence. Individual measurement trajectories lead to spatial pattern formation and optomechanical motion that solely result from the measurement backaction. The backaction of the continuous quantum measurement process, conditioned on the detection record of the photons, spontaneously breaks the symmetry of the spatial profile of the condensate and can be tailored to selectively excite collective modes. 18. Molecular ions in ultracold atomic gases: computed electronic interactions for \\MgHion with Rb CERN Document Server Tacconi, Mario 2007-01-01 The electronic structures of the manifold of potential energy surfaces generated in the lower energy range by the interaction of the MgH$^+$(X$^1\\Sigma^+$) cationic molecule with Rb($^2$S), neutral atom are obtained over a broad range of Jacobi coordinates from strongly correlated \\emph{ab initio} calculations which use a Multireference (MR) wavefunction within a Complete Active Space (CAS) approach. The relative features of the lowest five surfaces are analyzed in terms of possible collisional outcomes when employed to model the ultracold dynamics of ionic molecular partners. 19. Magnetic transport apparatus for the production of ultracold atomic gases in the vicinity of a dielectric surface CERN Document Server Haendel, S; Wiles, T P; Hopkins, S A; Cornish, S L 2011-01-01 We present an apparatus designed for studies of atom-surface interactions using quantum degenerate gases of $^{85}$Rb and $^{87}$Rb in the vicinity of a room temperature dielectric surface. The surface to be investigated is a super-polished face of a glass Dove prism mounted in a glass cell under ultra-high vacuum (UHV). To maintain excellent optical access to the region surrounding the surface magnetic transport is used to deliver ultracold atoms from a separate vacuum chamber housing the magneto-optical trap (MOT). We present a detailed description of the vacuum apparatus highlighting the novel design features; a low profile MOT chamber and the inclusion of an obstacle in the transport path. We report the characterization and optimization of the magnetic transport around the obstacle, achieving transport efficiencies of 70% with negligible heating. Finally we demonstrate the loading of a hybrid optical-magnetic trap with $^{87}$Rb and the creation of Bose-Einstein condensates via forced evaporative cooling ... 20. Effective-field-theory analysis of Efimov physics in heteronuclear mixtures of ultracold atomic gases Science.gov (United States) Acharya, Bijaya; Ji, Chen; Platter, Lucas 2016-09-01 We use an effective-field-theory framework to analyze the Efimov effect in heteronuclear three-body systems consisting of two species of atoms with a large interspecies scattering length. In the leading-order description of this theory, various three-body observables in heteronuclear mixtures can be universally parametrized by one three-body parameter. We present the next-to-leading corrections, which include the effects of the finite interspecies effective range and the finite intraspecies scattering length, to various three-body observables. We show that only one additional three-body parameter is required to render the theory predictive at this order. By including the effective range and intraspecies scattering length corrections, we derive a set of universal relations that connect the different Efimov features near the interspecies Feshbach resonance. Furthermore, we show that these relations can be interpreted in terms of the running of the three-body counterterms that naturally emerge from proper renormalization. Finally, we make predictions for recombination observables of a number of atomic systems that are of experimental interest. 1. MeV femtosecond electron pulses from direct-field acceleration in low density atomic gases CERN Document Server Varin, Charles; Hogan-Lamarre, Pascal; Fennel, Thomas; Piché, Michel; Brabec, Thomas 2015-01-01 Using three-dimensional particle-in-cell simulations, we show that few-MeV electrons can be produced by focusing tightly few-cycle radially-polarized laser pulses in a low-density atomic gas. In particular, it is observed that for the few-TW laser power needed to reach relativistic electron energies, longitudinal attosecond microbunching occurs naturally, resulting in femtosecond structures with high-contrast attosecond density modulations. The three-dimensional particle-in-cell simulations show that in the relativistic regime the leading pulse of these attosecond substructures survives to propagation over extended distances, suggesting that it could be delivered to a distant target, with the help of a properly designed transport beamline. 2. Non-equilibrium universality in the dynamics of dissipative cold atomic gases Science.gov (United States) Marcuzzi, M.; Levi, E.; Li, W.; Garrahan, J. P.; Olmos, B.; Lesanovsky, I. 2015-07-01 The theory of continuous phase transitions predicts the universal collective properties of a physical system near a critical point, which for instance manifest in characteristic power-law behaviours of physical observables. The well-established concept at or near equilibrium, universality, can also characterize the physics of systems out of equilibrium. The most fundamental instance of a genuine non-equilibrium phase transition is the directed percolation (DP) universality class, where a system switches from an absorbing inactive to a fluctuating active phase. Despite being known for several decades it has been challenging to find experimental systems that manifest this transition. Here we show theoretically that signatures of the DP universality class can be observed in an atomic system with long-range interactions. Moreover, we demonstrate that even mesoscopic ensembles—which are currently studied experimentally—are sufficient to observe traces of this non-equilibrium phase transition in one, two and three dimensions. 3. An effective field theory analysis of Efimov physics in heteronuclear mixtures of ultracold atomic gases CERN Document Server Acharya, Bijaya; Platter, Lucas 2016-01-01 We use an effective field theory framework to analyze the Efimov effect in heteronuclear three-body systems consisting of two species of atoms with a large interspecies scattering length. In the leading-order description of this theory, various three-body observables in heteronuclear mixtures can be universally parameterized by one three-body parameter. We present the next-to-leading corrections, which include the effects of the finite interspecies effective range and the finite intraspecies scattering length, to various three-body observables. We show that only one additional three-body parameter is required to render the theory predictive at this order. By including the effective range and intraspecies scattering length corrections, we derive a set of universal relations that connect the different Efimov features near the interspecies Feshbach resonance. Furthermore, we show that these relations can be interpreted in terms of the running of the three-body counterterms that naturally emerge from proper renor... 4. Metastable Phases and Dynamics of Low-Dimensional Strongly-Correlated Atomic Quantum Gases Science.gov (United States) Pielawa, Susanne In this thesis we theoretically study low-dimensional, strongly correlated systems of cold atoms, which are not in an equilibrium situation. This is motivated by recent experimental progress, which has made it possible to study quantum many-body physics in a controllable and clean setting; and parameters can be changed during the experiment. In Chapter 2 and 3 we study phases and quantum phase transitions of 'tilted' Mott insulator of bosons. We analyze a variety of lattices and tilt directions in two dimensions: square, decorated square, triangular, and kagome. We show that there are rich possibilities for correlated phases with non-trivial entanglement of pseudospin degrees of freedom encoded in the boson density. For certain configurations three-body interactions are necessary to ensure that the energy of the effective resonant subspace is bounded from below. We find quantum phases with Ising density wave order, with superfluidity transverse to the tilt direction, a quantum liquid state with no broken symmetry. We also find cases for which the resonant subspace is described by effective quantum dimer models. In Chapter 4 we study spin 1/2 chains with a Heisenberg interaction which are coupled in a way that would arise if they are taken off graphene at a zig-zag edge. In Chapter 5 we theoretically analyze interference patterns of parametrically driven one-dimensional cold atomic systems. The parametric driving leads to spatial oscillations in the interference patter, which can be analyzed to obtain the sound velocity of the 1d system, and to probe spin-charge separation. 5. Exploring the thermodynamics of a universal Fermi gas. Science.gov (United States) Nascimbène, S; Navon, N; Jiang, K J; Chevy, F; Salomon, C 2010-02-25 One of the greatest challenges in modern physics is to understand the behaviour of an ensemble of strongly interacting particles. A class of quantum many-body systems (such as neutron star matter and cold Fermi gases) share the same universal thermodynamic properties when interactions reach the maximum effective value allowed by quantum mechanics, the so-called unitary limit. This makes it possible in principle to simulate some astrophysical phenomena inside the highly controlled environment of an atomic physics laboratory. Previous work on the thermodynamics of a two-component Fermi gas led to thermodynamic quantities averaged over the trap, making comparisons with many-body theories developed for uniform gases difficult. Here we develop a general experimental method that yields the equation of state of a uniform gas, as well as enabling a detailed comparison with existing theories. The precision of our equation of state leads to new physical insights into the unitary gas. For the unpolarized gas, we show that the low-temperature thermodynamics of the strongly interacting normal phase is well described by Fermi liquid theory, and we localize the superfluid transition. For a spin-polarized system, our equation of state at zero temperature has a 2 per cent accuracy and extends work on the phase diagram to a new regime of precision. We show in particular that, despite strong interactions, the normal phase behaves as a mixture of two ideal gases: a Fermi gas of bare majority atoms and a non-interacting gas of dressed quasi-particles, the fermionic polarons. 6. Exceptional Points for Nonlinear Schroedinger Equations Describing Bose-Einstein Condensates of Ultracold Atomic Gases Directory of Open Access Journals (Sweden) G. Wunner 2011-01-01 Full Text Available The coalescence of two eigenfunctions with the same energy eigenvalue is not possible in Hermitian Hamiltonians. It is, however, a phenomenon well known from non-hermitian quantum mechanics. It can appear, e.g., for resonances in open systems, with complex energy eigenvalues. If two eigenvalues of a quantum mechanical system which depends on two or more parameters pass through such a branch point singularity at a critical set of parameters, the point in the parameter space is called an exceptional point. We will demonstrate that exceptional points occur not only for non-hermitean Hamiltonians but also in the nonlinear Schroedinger equations which describe Bose-Einstein condensates, i.e., the Gross-Pitaevskii equation for condensates with a short-range contact interaction, and with additional long-range interactions. Typically, in these condensates the exceptional points are also found to be bifurcation points in parameter space. For condensates with a gravity-like interaction between the atoms, these findings can be confirmed in an analytical way. 7. An effective mean field theory for the coexistence of anti-ferromagnetism and superconductivity: Applications to iron-based superconductors and cold Bose-Fermi atomic mixtures Science.gov (United States) Brackett, Jeremy; Newman, Joseph; De Silva, Theja N. 2016-10-01 We study an effective fermion model on a square lattice to investigate the cooperation and competition of superconductivity and anti-ferromagnetism. In addition to particle tunneling and on-site interaction, a bosonic excitation mediated attractive interaction is also included in the model. We assume that the attractive interaction is mediated by spin fluctuations and excitations of Bose-Einstein condensation (BEC) in electronic systems and Bose-Fermi mixtures on optical lattices, respectively. Using an effective mean-field theory to treat both superconductivity and anti-ferromagnetism at equal footing, we study a single effective model relevant for both systems within the Landau energy functional approach and a linearized theory. Within our approaches, we find possible co-existence of superconductivity and anti-ferromagnetism for both electronic and cold-atomic models. Our linearized theory shows while spin fluctuations favor d-wave superconductivity and BEC excitations favor s-wave superconductivity. 8. Mott-Insulator to Liquid Transition and Population Trapping in Ultracold Fermi Gases by Non-Equilibrium Modulation of the Optical Lattice CERN Document Server Frank, Regine 2011-01-01 An ultracold gas of interacting fermionic atoms in a three dimensional optical lattice is considered, where the lattice potential strength is periodically modulated. This non-equilibrium system is nonperturbatively described by means of a Keldysh-Floquet-Green's function approach employing a generalized dynamical mean field theory (DMFT). Strong repulsive interactions between different atoms lead to a Mott-Insulator state for the equilibrium system, but the additional external driving yields a non-equilibrium density of Floquet-states and a transition to a liquid or conducting state. 9. Propagation and scattering of high-intensity X-ray pulses in dense atomic gases and plasmas Energy Technology Data Exchange (ETDEWEB) Weninger, Clemens 2015-10-15 Nonlinear spectroscopy in the X-ray domain is a promising technique to explore the dynamics of elementary excitations in matter. X-rays provide an element specificity that allows them to target individual chemical elements, making them a great tool to study complex molecules. The recent advancement of X-ray free electron lasers (XFELs) allows to investigate non-linear processes in the X-ray domain for the first time. XFELs provide short femtosecond X-ray pulses with peak powers that exceed previous generation synchrotron X-ray sources by more than nine orders of magnitude. This thesis focuses on the theoretical description of stimulated emission processes in the X-ray regime in atomic gases. These processes form the basis for more complex schemes in molecules and provide a proof of principle for nonlinear X-ray spectroscopy. The thesis also includes results from two experimental campaigns at the Linac Coherent Light Source and presents the first experimental demonstration of stimulated X-ray Raman scattering. Focusing an X-ray free electron laser beam into an elongated neon gas target generates an intense stimulated X-ray emission beam in forward direction. If the incoming X-rays have a photon energy above the neon K edge, they can efficiently photo-ionize 1s electrons and generate short-lived core excited states. The core-excited states decay mostly via Auger decay but have a small probability to emit a spontaneous X-ray photon. The spontaneous emission emitted in forward direction can stimulate X-ray emission along the medium and generate a highly directional and intense X-ray laser pulse. If the photon energy of the incoming X-rays however is below the ionization edge in the region of the pre-edge resonance the incoming X-rays can be inelastically scattered. This spontaneous X-ray Raman scattering process has a very low probability, but the spontaneously scattered photons in the beginning of the medium can stimulate Raman scattering along the medium. The 10. EDITORIAL: The 20th European Sectional Conference on Atomic and Molecular Physics of Ionized Gases The 20th European Sectional Conference on Atomic and Molecular Physics of Ionized Gases Science.gov (United States) Petrović, Zoran Lj; Marić, Dragana; Malović, Gordana 2011-03-01 This special issue consists of papers that are associated with invited lectures, workshop papers and hot topic papers presented at the 20th European Sectional Conference on Atomic and Molecular Physics of Ionized Gases (ESCAMPIG XX). This conference was organized in Novi Sad (Serbia) from 13 to 17 July 2010 by the Institute of Physics of the University of Belgrade. It is important to note that this is not a conference 'proceedings'. Following the initial selection process by the International Scientific Committee, all papers were submitted to the journal by the authors and have been fully peer reviewed to the standard required for publication in Plasma Sources Science and Technology (PSST). The papers are based on presentations given at the conference but are intended to be specialized technical papers covering all or part of the topic presented by the author during the meeting. The ESCAMPIG conference is a regular biennial Europhysics Conference of the European Physical Society focusing on collisional and radiative aspects of atomic and molecular physics in partially ionized gases as well as on plasma-surface interaction. The conference focuses on low-temperature plasma sciences in general and includes the following topics: Atomic and molecular processes in plasmas Transport phenomena, particle velocity distribution function Physical basis of plasma chemistry Plasma surface interaction (boundary layers, sheath, surface processes) Plasma diagnostics Plasma and discharges theory and simulation Self-organization in plasmas, dusty plasmas Upper atmospheric plasmas and space plasmas Low-pressure plasma sources High-pressure plasma sources Plasmas and gas flows Laser-produced plasmas During ESCAMPIG XX special sessions were dedicated to workshops on: Atomic and molecular collision data for plasma modeling, organized by Professors Z Lj Petrovic and N Mason Plasmas in medicine, organized by Dr N Puac and Professor G Fridman. The conference topics were represented in the 11. The long-range non-additive three-body dispersion interactions for the rare gases, alkali, and alkaline-earth atoms. Science.gov (United States) Tang, Li-Yan; Yan, Zong-Chao; Shi, Ting-Yun; Babb, James F; Mitroy, J 2012-03-14 The long-range non-additive three-body dispersion interaction coefficients Z(111), Z(112), Z(113), and Z(122) are computed for many atomic combinations using standard expressions. The atoms considered include hydrogen, the rare gases, the alkali atoms (up to Rb), and the alkaline-earth atoms (up to Sr). The term Z(111) arising from three mutual dipole interactions is known as the Axilrod-Teller-Muto coefficient or the DDD (dipole-dipole-dipole) coefficient. Similarly, the terms Z(112), Z(113), and Z(122) arise from the mutual combinations of dipole (1), quadrupole (2), and octupole (3) interactions between atoms and they are sometimes known, respectively, as dipole-dipole-quadrupole, dipole-dipole-octupole, and dipole-quadrupole-quadrupole coefficients. Results for the four Z coefficients are given for the homonuclear trimers, for the trimers involving two like-rare-gas atoms, and for the trimers with all combinations of the H, He, and Li atoms. An exhaustive compilation of all coefficients between all possible atomic combinations is presented as supplementary data. 12. The long-range non-additive three-body dispersion interactions for the rare gases, alkali and alkaline-earth atoms CERN Document Server Tang, Li-Yan; Shi, Ting-Yun; Babb, James F; Mitroy, J 2012-01-01 The long-range non-additive three-body dispersion interaction coefficients $Z_{111}$, $Z_{112}$, $Z_{113}$, and $Z_{122}$ are computed for many atomic combinations using standard expressions. The atoms considered include hydrogen, the rare gases, the alkali atoms (up to Rb) and the alkaline-earth atoms (up to Sr). The term $Z_{111}$, arising from three mutual dipole interactions is known as the Axilrod-Teller-Muto coefficient or the DDD (dipole-dipole-dipole) coefficient. Similarly, the terms $Z_{112}$, $Z_{113}$, and $Z_{122}$ arise from the mutual combinations of dipole (1), quadrupole (2), and octupole (3) interactions between atoms and they are sometimes known, respectively, as DDQ, DDO, and DQQ coefficients. Results for the four $Z$ coefficients are given for the homonuclear trimers, for the trimers involving two like-rare-gas atoms, and for the trimers with all combinations of the H, He, Li atoms. An exhaustive compilation of all coefficients between all possible atomic combinations is presented as supp... 13. Strongly interacting ultracold quantum gases Institute of Scientific and Technical Information of China (English) Hui ZHAI 2009-01-01 This article reviews recent progresses in ul- tracold quantum gases, and it includes three subjects which are the Fermi gases across a Feshbach resonance, quantum gases in the optical lattices and the fast ro- tating quantum gases. In this article, we discuss many basic physics pictures and concepts in quantum gases, for examples, the resonant interaction, universality and condensation in the lowest Landau level; we introduce fundamental theoretical tools for studying these systems, such as mean-field theory for BEC-BCS crossover and for the boson Hubbard model; also, we emphasize the im- portant unsolved problems in the forefront of this field, for instance, the temperature effect in optical lattices. 14. Simultaneous Observations of PKS 2155--304 with H.E.S.S., Fermi, RXTE and ATOM: Spectral Energy Distributions and Variability in a Low State Energy Technology Data Exchange (ETDEWEB) Aharonian, F.; /Heidelberg, Max Planck Inst. /Dublin Inst.; Akhperjanian, A.G.; /Yerevan Phys. Inst.; Anton, G.; /Erlangen - Nuremberg U.; Barres de Almeida, U.; /Durham U.; Bazer-Bachi, A.R.; /Toulouse, CESR; Becherini, Y.; /APC, Paris; Behera, B.; /Heidelberg Observ.; Bernlohr, K.; /Heidelberg, Max Planck Inst. /Humboldt U., Berlin; Boisson, C.; /LUTH, Meudon; Bochow, A.; /Heidelberg, Max Planck Inst.; Borrel, V.; /Toulouse, CESR; Brion, E.; /DAPNIA, Saclay; Brucker, J.; /Erlangen - Nuremberg U.; Brun, P.; /DAPNIA, Saclay; Buhler, R.; /Heidelberg, Max Planck Inst.; Bulik, T.; /Warsaw, Copernicus Astron. Ctr.; Busching, I.; /Western Ontario U.; Boutelier, T.; /Grenoble Observ.; Chadwick, P.M.; /Durham U.; Charbonnier, A.; /Paris U., VI-VII; Chaves, R.C.G.; /Heidelberg, Max Planck Inst. /Durham U. /Ecole Polytechnique /Heidelberg, Max Planck Inst. /Annecy, LAPP /Humboldt U., Berlin /Durham U. /Namibia U. /Western Ontario U. /Ecole Polytechnique /Heidelberg, Max Planck Inst. /Durham U. /APC, Paris /Heidelberg, Max Planck Inst. /Dublin Inst. /Annecy, LAPP /Grenoble Observ. /Warsaw, Copernicus Astron. Ctr. /Cracow, INP /Heidelberg, Max Planck Inst. /Heidelberg Observ. /APC, Paris /Montpellier U. /Montpellier U. /Montpellier U. /Heidelberg, Max Planck Inst. /Ecole Polytechnique /Humboldt U., Berlin /Dublin Inst. /Montpellier U. /APC, Paris /SLAC; /more authors.. 2009-05-07 We report on the first simultaneous observations that cover the optical, X-ray, and high-energy gamma-ray bands of the BL Lac object PKS 2155-304. The gamma-ray bands were observed for 11 days, between 2008 August 25 and 2008 September 6 (MJD 54704-54715), jointly with the Fermi Gamma-ray Space Telescope and the HESS atmospheric Cherenkov array, providing the first simultaneous MeV-TeV spectral energy distribution (SED) with the new generation of {gamma}-ray telescopes. The ATOM telescope and the RXTE and Swift observatories provided optical and X-ray coverage of the low-energy component over the same time period. The object was close to the lowest archival X-ray and very high energy (VHE; >100 GeV) state, whereas the optical flux was much higher. The light curves show relatively little ({approx}30%) variability overall when compared to past flaring episodes, but we find a clear optical/VHE correlation and evidence for a correlation of the X-rays with the high-energy spectral index. Contrary to previous observations in the flaring state, we do not find any correlation between the X-ray and VHE components. Although synchrotron self-Compton models are often invoked to explain the SEDs of BL Lac objects, the most common versions of these models are at odds with the correlated variability we find in the various bands for PKS 2155-304. 15. A new numerical approach to solve Thomas-Fermi model of an atom using bio-inspired heuristics integrated with sequential quadratic programming. Science.gov (United States) Raja, Muhammad Asif Zahoor; Zameer, Aneela; Khan, Aziz Ullah; Wazwaz, Abdul Majid 2016-01-01 In this study, a novel bio-inspired computing approach is developed to analyze the dynamics of nonlinear singular Thomas-Fermi equation (TFE) arising in potential and charge density models of an atom by exploiting the strength of finite difference scheme (FDS) for discretization and optimization through genetic algorithms (GAs) hybrid with sequential quadratic programming. The FDS procedures are used to transform the TFE differential equations into a system of nonlinear equations. A fitness function is constructed based on the residual error of constituent equations in the mean square sense and is formulated as the minimization problem. Optimization of parameters for the system is carried out with GAs, used as a tool for viable global search integrated with SQP algorithm for rapid refinement of the results. The design scheme is applied to solve TFE for five different scenarios by taking various step sizes and different input intervals. Comparison of the proposed results with the state of the art numerical and analytical solutions reveals that the worth of our scheme in terms of accuracy and convergence. The reliability and effectiveness of the proposed scheme are validated through consistently getting optimal values of statistical performance indices calculated for a sufficiently large number of independent runs to establish its significance. 16. Revised FINAL–REPORT NO. 2: INDEPENDENT CONFIRMATORY SURVEY SUMMARY AND RESULTS FOR THE ENRICO FERMI ATOMIC POWER PLANT, UNIT 1, NEWPORT, MICHIGAN (DOCKET NO. 50 16; RFTA 10-004) 2018-SR-02-1 Energy Technology Data Exchange (ETDEWEB) Erika Bailey 2011-10-27 The Enrico Fermi Atomic Power Plant, Unit 1 (Fermi 1) was a fast breeder reactor design that was cooled by sodium and operated at essentially atmospheric pressure. On May 10, 1963, the Atomic Energy Commission (AEC) granted an operating license, DPR-9, to the Power Reactor Development Company (PRDC), a consortium specifically formed to own and operate a nuclear reactor at the Fermi 1 site. The reactor was designed for a maximum capability of 430 megawatts (MW); however, the maximum reactor power with the first core loading (Core A) was 200 MW. The primary system was filled with sodium in December 1960 and criticality was achieved in August 1963. The reactor was tested at low power during the first couple years of operation. Power ascension testing above 1 MW commenced in December 1965 immediately following the receipt of a high-power operating license. In October 1966 during power ascension, zirconium plates at the bottom of the reactor vessel became loose and blocked sodium coolant flow to some fuel subassemblies. Two subassemblies started to melt and the reactor was manually shut down. No abnormal releases to the environment occurred. Forty-two months later after the cause had been determined, cleanup completed, and the fuel replaced, Fermi 1 was restarted. However, in November 1972, PRDC made the decision to decommission Fermi 1 as the core was approaching its burn-up limit. The fuel and blanket subassemblies were shipped off-site in 1973. Following that, the secondary sodium system was drained and sent off-site. The radioactive primary sodium was stored on-site in storage tanks and 55 gallon (gal) drums until it was shipped off-site in 1984. The initial decommissioning of Fermi 1 was completed in 1975. Effective January 23, 1976, DPR-9 was transferred to the Detroit Edison Company (DTE) as a 'possession only' license (DTE 2010a). This report details the confirmatory activities performed during the second Oak Ridge Institute for Science and Education 17. Spectral investigations of photoionized plasmas induced in atomic and molecular gases using nanosecond extreme ultraviolet (EUV) pulses Energy Technology Data Exchange (ETDEWEB) Bartnik, A.; Fiedorowicz, H.; Wachulak, P. [Institute of Optoelectronics, Military University of Technology, Kaliskiego 2, 00-908 Warsaw (Poland) 2014-07-15 In this paper, results of spectral investigations of low temperature photoionized plasmas, created by irradiation of gases with intense pulses of extreme ultraviolet (EUV) radiation from a laser-produced plasma (LPP) source, are presented. The LPP source was based on a double-stream KrXe/He gas-puff target irradiated with 4 ns/0.8 J/10 Hz Nd:YAG laser pulses. The most intense emission from the source spanned a relatively narrow spectral region λ ≈ 10–12 nm; however, spectrally integrated intensity at longer wavelengths was also significant. The EUV beam was focused on a gas stream, injected into a vacuum chamber synchronously with the EUV pulses. Irradiation of gases resulted in formation of photoionized plasmas emitting radiation in the EUV range. Radiation spectra, measured for plasmas produced in various gases, are dominated by emission lines, originating from single charged ions. Significant differences in spectral intensities and distributions between plasmas created in neon and molecular gases were observed. 18. Strongly correlated Bose gases Science.gov (United States) Chevy, F.; Salomon, C. 2016-10-01 The strongly interacting Bose gas is one of the most fundamental paradigms of quantum many-body physics and the subject of many experimental and theoretical investigations. We review recent progress on strongly correlated Bose gases, starting with a description of beyond mean-field corrections. We show that the Efimov effect leads to non universal phenomena and to a metastability of the low temperature Bose gas through three-body recombination to deeply bound molecular states. We outline differences and similarities with ultracold Fermi gases, discuss recent experiments on the unitary Bose gas, and finally present a few perspectives for future research. 19. Laser cooling of dense atomic gases by collisional redistribution of radiation and spectroscopy of molecular dimers in a dense buffer gas environment CERN Document Server Saß, Anne; Christopoulos, Stavros; Knicker, Katharina; Moroshkin, Peter; Weitz, Martin 2014-01-01 We study laser cooling of atomic gases by collisional redistribution of fluorescence. In a high pressure buffer gas regime, frequent collisions perturb the energy levels of alkali atoms, which allows for the absorption of a far red detuned irradiated laser beam. Subsequent spontaneous decay occurs close to the unperturbed resonance frequency, leading to a cooling of the dense gas mixture by redistribution of fluorescence. Thermal deflection spectroscopy indicates large relative temperature changes down to and even below room temperature starting from an initial cell temperature near 700 K. We are currently performing a detailed analysis of the temperature distribution in the cell. As we expect this cooling technique to work also for molecular-noble gas mixtures, we also present initial spectroscopic experiments on alkali-dimers in a dense buffer gas surrounding. 20. Enrico Fermi Institute of Scientific and Technical Information of China (English) 李琳 2006-01-01 Enrico Fermi was born in Rome on 29th September, 1901. He attended a local grammar school, and in 1918, he won a fellowship of the Scuola Normale Superiore of Pisa, where he gained his doctor’s degree in physics in 1922, with Professor Puccianti. In 1923, he was awarded a scholarship from the Italian Government. With a Rockefeller Fellowship, in 1924, he moved to Leyden, and later that same year he returned to Italy to occupy for two 1. Enrico Fermi the obedient genius CERN Document Server Bruzzaniti, Giuseppe 2016-01-01 This biography explores the life and career of the Italian physicist Enrico Fermi, which is also the story of thirty years that transformed physics and forever changed our understanding of matter and the universe: nuclear physics and elementary particle physics were born, nuclear fission was discovered, the Manhattan Project was developed, the atomic bombs were dropped, and the era of “big science” began. It would be impossible to capture the full essence of this revolutionary period without first understanding Fermi, without whom it would not have been possible. Enrico Fermi: The Obedient Genius attempts to shed light on all aspects of Fermi’s life - his work, motivation, influences, achievements, and personal thoughts - beginning with the publication of his first paper in 1921 through his death in 1954. During this time, Fermi demonstrated that he was indeed following in the footsteps of Galileo, excelling in his work both theoretically and experimentally by deepening our understanding of the Pauli e... 2. The first example of commensurate adsorption of atomic gas in a MOF and effective separation of xenon from other noble gases KAUST Repository Wang, Hao 2014-01-01 In industry, cryogenic rectification for separating xenon from other noble gases such as krypton and argon is an energy and capital intensive process. Here we show that a microporous metal-organic framework, namely Co 3(HCOO)6 is capable of effective capture and separation of xenon from other noble gases. Henry\\'s constant, isosteric heat of adsorption (Qst), and IAST selectivity are calculated based on single component sorption isotherms. Having the highest Qst reported to date, Co 3(HCOO)6 demonstrates high adsorption capacity for xenon and its IAST selectivity for Xe-Kr is the largest among all MOFs investigated to date. To mimic real world conditions, breakthrough experiments are conducted on Xe-Kr binary mixtures at room temperature and 1 atmosphere. The results are consistent with the calculated data. These findings show that Co 3(HCOO)6 is a promising candidate for xenon capture and purification. Our gas adsorption measurements and molecular simulation study also reveal that the adsorption of xenon represents the first example of commensurate adsorption of atomic gases near ambient conditions. © 2014 The Royal Society of Chemistry. 3. Trends in structural, electronic properties, Fermi surface topology, and inter-atomic bonding in the series of ternary layered dichalcogenides KNi2S2, KNi2Se2, and KNi2Te2 from first principles calculations Science.gov (United States) Bannikov, V. V.; Ivanovskii, A. L. 2013-06-01 By means of the FLAPW-GGA approach, we have systematically studied the structural and electronic properties of tetragonal dichalcogenides KNi2Ch2 (Ch=S, Se, and Te). Our results show that replacements of chalcogens (S→Se→Te) lead to anisotropic deformations of the crystals structure, which are related to the strong anisotropic character of the inter-atomic bonds, where inside the [Ni2Ch2] blocks, mixed covalent-ionic-metallic bonds occur, whereas between the adjacent [Ni2Ch2] blocks and K atomic sheets, ionic bonds emerge. We found that in the sequence KNi2S2→KNi2Se2→KNi2Te2 (i) the overall band structure (where the near-Fermi valence bands are due mainly to the Ni states) is preserved, but the width of the common valence band and the widths of the separate sub-bands and the gaps decrease; (ii) the total DOSs at the Fermi level also decrease; and (iii) for the Fermi surfaces, the most appreciable changes are demonstrated by the hole-like sheets, when a necklace-like topology is formed for the 2D-like sheets and the volume of the closed pockets decreases. Some trends in structural and electronic parameters for ThCr2Si2-type layered dichalcogenides, KNi2Ch2, KFe2Ch2, KCo2Se2, are discussed. 4. Studies on the atomic capture of stopped negative pions in binary mixtures of /sup 3/He with other gases Energy Technology Data Exchange (ETDEWEB) Bannikov, A.V.; Levay, B.; Petrukhin, V.I.; Vasilyev, V.A. (Joint Inst. for Nuclear Research, Dubna (USSR)); Kochenda, L.M.; Markov, A.A.; Medvedev, V.I.; Sokolov, G.L.; Strakovsky, I.I. (Leningrad Nuclear Physics Inst., Gatchina (USSR)); Horvath, D. (Hungarian Academy of Sciences, Budapest. Central Research Inst. for Physics) 1983-07-25 Systematic experimental study has been carried out on the atomic capture of negative pions by /sup 3/He in binary gas mixtures of /sup 3/He + Z, where Z is Ne, Ar, Kr, Xe, N/sub 2/, O/sub 2/, CO/sub 2/ and SF/sub 6/. The results are analysed in the framework of a phenomenological model. It is shown that there is no pion transfer from the /sup 3/He..pi../sup -/ mesic atoms to the heavier Z-atoms. The probabilities of pion capture in the various atoms of the mixtures are found to be proportional to the atomic concentraions, thereby excluding the possibility of a concentration dependence in the atomic capture ratio A(Z//sup 3/He). In contradiction to previous assumptions the probability of pion capture into an atomic orbit is not proportional to the stopping power of the components of the mixture. The atomic capture ratio of pions in a /sup 3/He + /sup 4/He mixture is A(/sup 4/He//sup 3/He) = 0.75 +- 0.13, which might be the indication of an isotopic effect. The branching ratio for the charge-exchange reaction at rest ..pi../sup -/ + /sup 3/He -> ..pi../sup 0/ + /sup 3/H) is found to be 0.128 +- 0.012. 5. Studies on the atomic capture of stopped negative pions in binary mixtures of 3He with other gases Science.gov (United States) Bannikov, A. V.; Lévay, B.; Petrukhin, V. I.; Vasilyev, V. A.; Kochenda, L. M.; Markov, A. A.; Medvedev, V. I.; Sokolov, G. L.; Strakovsky, I. I.; Horváth, D. 1983-07-01 Systematic experimental study has been carried out on the atomic capture of negative pions by 3He in binary gas mixtures of 3He + Z, where Z is Ne, Ar, Kr, Xe, N 2, O 2, CO 2 and sf 6. The results are analysed in the framework of a phenomenological model. It is shown that there is no pion transfer from the 3Heπ - mesic atoms to the heavier Z-atoms. The probabilities of pion capture in the various atoms of the mixtures are found to be proportional to the atomic concentrations, thereby excluding the possibility of a concentration dependence in the atomic capture ratio A( Z/ 3He). In contradiction to previous assumptions the probability of pion capture into an atomic orbit is not proportional to the stopping power of the components of the mixture. The atomic capture ratio of pions in a 3He + 4He mixture is A( 4He/ 3He) = 0.75 ± 0.13 , which might be the indication of an isotopic effect. The branching ratio for the charge-exchange reaction at rest (π - + 3He → π 0 + 3H) is found to be 0.128 ± 0.012. 6. Studies on the atomic capture of stopped negative pions in binary mixtures of /sup 3/He with other gases Energy Technology Data Exchange (ETDEWEB) Bannikov, A.V.; Levay, B.; Petrukhin, V.I.; Vasilyev, V.A. (Joint Inst. for Nuclear Research, Dubna (USSR)); Kochenda, L.M.; Markov, A.A.; Medvedev, V.I.; Sokolov, G.L.; Strakovsky, I.I. (Leningrad Nuclear Physics Inst., Gatchina (USSR)); Horvath, D. (Hungarian Academy of Sciences, Budapest. Central Research Inst. for Physics) 1983-07-25 Systematic experimental study has been carried out on the atomic capture of negative pions by /sup 3/He in binary gas mixtures of /sup 3/He + Z, where Z is Ne, Ar, Kr, Xe, N/sub 2/, O/sub 2/, CO/sub 2/ and SF/sub 6/. The results are analyzed in the framework of a phenomenological model. It is shown that there is no pion transfer from the /sup 3/He..pi../sup -/ mesic atoms to the heavier Z-atoms. The probabilities of pion capture in the various atoms of the mixtures are found to be proportional to the atomic concentraions, thereby excluding the possibility of a concentration dependence in the atomic capture ratio A(Z//sup 3/He). In contradiction to previous assumptions the probability of pion capture into an atomic orbit is not proportional to the stopping power of the components of the mixture. The atomic capture ratio of pions in a /sup 3/He + /sup 4/He mixture is A(/sup 4/He//sup 3/He) = 0.75 +- 0.13, which might be the indication of an isotopic effect. The branching ratio for the charge-exchange reaction at rest ..pi../sup -/ + /sup 3/He -> ..pi../sup 0/ + /sup 3/H is found to be 0.128 +- 0.012. 7. Lifetime measurement of excited atomic and ionic states of some noble gases using the high-frequency deflection technique M B Das; S Karmakar 2005-12-01 High-frequency deflection (HFD) technique with a delayed coincidence single photon counting arrangement is an efficient technique for radiative lifetime measurement. An apparatus for measurement of the radiative lifetime of atoms and molecules has been developed in our laboratory and measurements have been performed with great success in a large number of atoms and ions. The present version of the apparatus is described in this paper together with a brief description of the basic features and performance. 8. Towards quantum turbulence in cold atomic fermionic superfluids Science.gov (United States) Bulgac, Aurel; McNeil Forbes, Michael; Wlazłowski, Gabriel 2017-01-01 Fermionic superfluids provide a new realization of quantum turbulence, accessible to both experiment and theory, yet relevant to phenomena from both cold atoms to nuclear astrophysics. In particular, the strongly interacting Fermi gas realized in cold-atom experiments is closely related to dilute neutron matter in neutron star crusts. Unlike the liquid superfluids 4He (bosons) and 3He (fermions), where quantum turbulence has been studied in laboratory for decades, superfluid Fermi gases stand apart for a number of reasons. They admit a rather reliable theoretical description based on density functional theory called the time-dependent superfluid local density approximation that describes both static and dynamic phenomena. Cold atom experiments demonstrate exquisite control over particle number, spin polarization, density, temperature, and interaction strength. Topological defects such as domain walls and quantized vortices, which lie at the heart of quantum turbulence, can be created and manipulated with time-dependent external potentials, and agree with the time-dependent theoretical techniques. While similar experimental and theoretical control exists for weakly interacting Bose gases, the unitary Fermi gas is strongly interacting. The resulting vortex line density is extremely high, and quantum turbulence may thus be realized in small systems where classical turbulence is suppressed. Fermi gases also permit the study of exotic superfluid phenomena such as the Larkin-Ovchinnikov-Fulde-Ferrell pairing mechanism for polarized superfluids which may give rise to 3D supersolids, and a pseudo-gap at finite temperatures that might affect the regime of classical turbulence. The dynamics associated with these phenomena has only started to be explored. Finally, superfluid mixtures have recently been realized, providing experimental access to phenomena like Andreev-Bashkin entrainment predicted decades ago. Superfluid Fermi gases thus provide a rich forum for addressing 9. Will Allis Prize for the Study of Ionized Gases Lecture: Electron and Photon Collisions with Atoms and Molecules Science.gov (United States) Burke, Philip G. 2012-06-01 After a brief historical introduction this talk will review the broad range of collision processes involving electron and photon collisions with atoms and molecules that are now being considered. Their application in the analysis of astronomical spectra, atmospheric observations and laboratory plasmas will be considered. The talk will review the R-matrix computational method which has been widely used by international collaborations and by other scientists in the field to obtain accurate scattering amplitudes and cross sections of importance in these applications. Results of some recent calculations of electron and photon collisions with atoms and molecules will be presented. In conclusion some challenges for future research will be briefly discussed. 10. Phase Diagram of a Strongly Interacting Spin-Imbalanced Fermi Gas CERN Document Server Olsen, Ben A; Fry, Jacob A; Sheehy, Daniel E; Hulet, Randall G 2015-01-01 We obtain the phase diagram of spin-imbalanced interacting Fermi gases from measurements of density profiles of $^6$Li atoms in a harmonic trap. These results agree with, and extend, previous experimental measurements. Measurements of the critical polarization at which the balanced superfluid core vanishes generally agree with previous experimental results and with quantum Monte Carlo (QMC) calculations in the BCS and unitary regimes. We disagree with the QMC results in the BEC regime, however, where the measured critical polarizations are greater than theoretically predicted. We also measure the equation of state in the crossover regime for a gas with equal numbers of the two fermion spin states. 11. Noble Gases Science.gov (United States) Podosek, F. A. 2003-12-01 The noble gases are the group of elements - helium, neon, argon, krypton, xenon - in the rightmost column of the periodic table of the elements, those which have "filled" outermost shells of electrons (two for helium, eight for the others). This configuration of electrons results in a neutral atom that has relatively low electron affinity and relatively high ionization energy. In consequence, in most natural circumstances these elements do not form chemical compounds, whence they are called "noble." Similarly, much more so than other elements in most circumstances, they partition strongly into a gas phase (as monatomic gas), so that they are called the "noble gases" (also, "inert gases"). (It should be noted, of course, that there is a sixth noble gas, radon, but all isotopes of radon are radioactive, with maximum half-life a few days, so that radon occurs in nature only because of recent production in the U-Th decay chains. The factors that govern the distribution of radon isotopes are thus quite different from those for the five gases cited. There are interesting stories about radon, but they are very different from those about the first five noble gases, and are thus outside the scope of this chapter.)In the nuclear fires in which the elements are forged, the creation and destruction of a given nuclear species depends on its nuclear properties, not on whether it will have a filled outermost shell when things cool off and nuclei begin to gather electrons. The numerology of nuclear physics is different from that of chemistry, so that in the cosmos at large there is nothing systematically special about the abundances of the noble gases as compared to other elements. We live in a very nonrepresentative part of the cosmos, however. As is discussed elsewhere in this volume, the outstanding generalization about the geo-/cosmochemistry of the terrestrial planets is that at some point thermodynamic conditions dictated phase separation of solids from gases, and that the 12. Signals of Bose Einstein condensation and Fermi quenching in the decay of hot nuclear systems Directory of Open Access Journals (Sweden) P. Marini 2016-05-01 Full Text Available We report on first experimental observations of nuclear fermionic and bosonic components displaying different behaviours in the decay of hot Ca projectile-like sources produced in mid-peripheral collisions at sub-Fermi energies. The experimental setup, constituted by the coupling of the INDRA 4π detector array to the forward angle VAMOS magnetic spectrometer, allowed to reconstruct the mass, charge and excitation energy of the decaying hot projectile-like sources. By means of quantum-fluctuation analysis techniques, temperatures and local partial densities of bosons and fermions could be correlated to the excitation energy of the reconstructed system. The results are consistent with the production of dilute mixed systems of bosons and fermions, where bosons experience higher phase-space and energy density as compared to the surrounding fermionic gas. Our findings recall phenomena observed in the study of Bose condensates and Fermi gases in atomic traps despite the different scales. 13. Precision measurements of cross sections of inelastic processes realized in collisions of alkali metal ions with atoms of rare gases CERN Document Server Lomsadze, R A; Mosulishvili, N O; Kezerashvili, R Ya 2015-01-01 This work presents a multifaceted experimental study of collisions of Na$^{+}$ and K$^{+}$ ions in the energy range 0.5 -- 10 keV with He and Ar atoms. Absolute cross sections for charge-exchange, ionization, stripping and excitation were measured using a refined version of the transfer electric field method, angle- and energy-dependent collection of product ions, energy loss, and optical spectroscopy. The experimental data and the schematic correlation diagrams have been employed to analyze and determine the mechanisms for these processes. 14. Precision measurements of cross-sections for inelastic processes in collisions of alkali metal ions with atoms of rare gases Science.gov (United States) Lomsadze, R. A.; Gochitashvili, M. R.; Kezerashvili, R. Ya. 2017-01-01 A multifaceted experimental study of collisions of Na+ and K+ ions in the energy range of 0.5-10 keV with He and Ar atoms is presented. Absolute cross-sections for charge-exchange, ionization, stripping and excitation processes were measured using a refined version of the transfer electric field method, angle- and energy-dependent collection of product ions, energy loss and optical spectroscopy methods. The experimental data and the schematic correlation diagrams are employed to analyze and determine the mechanisms for these processes. 15. Three-body recombination of two-component cold atomic gases into deep dimers in an optical model DEFF Research Database (Denmark) Mikkelsen, Mathias; Jensen, A. S.; Fedorov, D. V. 2015-01-01 We consider three-body recombination into deep dimers in a mass-imbalanced two-component atomic gas. We use an optical model where a phenomenological imaginary potential is added to the lowest adiabatic hyper-spherical potential. The consequent imaginary part of the energy eigenvalue corresponds...... to the decay rate or recombination probability of the three-body system. The method is formulated in details and the relevant qualitative features are discussed as functions of scattering lengths and masses. We use zero-range model in analyses of recent recombination data. The dominating scattering length... 16. Topological insulators in cold-atom gases with non-Abelian gauge fields: the role of interactions Energy Technology Data Exchange (ETDEWEB) Orth, Peter Philipp [Institut fuer Theorie der Kondensierten Materie, Karlsruher Institut fuer Technologie, 76128 Karlsruhe (Germany); Cocks, Daniel; Buchhold, Michael; Hofstetter, Walter [Institut fuer Theoretische Physik, Goethe Universitaet, 60438 Frankfurt am Main (Germany); Rachel, Stephan [Department of Physics, Yale University, New Haven, Connecticut 06520 (United States); Le Hur, Karyn [Department of Physics, Yale University, New Haven, Connecticut 06520 (United States); Center for Theoretical Physics, Ecole Polytechnique, 91128 Palaiseau Cedex (France) 2012-07-01 With the recent technological advance of creating (non)-Abelian gauge fields for ultracold atoms in optical lattices, it becomes possible to study the interplay of topological phases and interactions in these systems. Specifically, we consider a spinful and time-reversal invariant version of the Hofstadter problem. In addition, we allow for a hopping term which does not preserve S{sub z} spin symmetry and a staggered sublattice potential. Without interactions, the parameters can be tuned such that the system is a topological insulator. Using a combination of analytical techniques and the powerful real-space dynamical mean-field (R-DMFT) method, we discuss the effect of interactions and determine the interacting phase diagram. 17. Thermodynamics of Quantum Gases for the Entire Range of Temperature Science.gov (United States) Biswas, Shyamal; Jana, Debnarayan 2012-01-01 We have analytically explored the thermodynamics of free Bose and Fermi gases for the entire range of temperature, and have extended the same for harmonically trapped cases. We have obtained approximate chemical potentials for the quantum gases in closed forms of temperature so that the thermodynamic properties of the quantum gases become… 18. Atoms Institute of Scientific and Technical Information of China (English) 刘洪毓 2007-01-01 Atoms(原子)are all around us.They are something like the bricks (砖块)of which everything is made. The size of an atom is very,very small.In just one grain of salt are held millions of atoms. Atoms are very important.The way one object acts depends on what 19. Feneric Fermi Size Enhancement of Pairing in Mesoscopic Fermi Systems CERN Document Server Farine, M; Schuck, P; Viñas, X 2002-01-01 The finite size dependent enhancement of pairing in mesoscopic Fermi systems is studied under the assumption that the BCS approach is valid and that the two body force is size independent. Different systems are investigated such as superconducting metallic grains and films as well as atomic nuclei. It is shown that the finite size enhancement of pairing in these systems is a surface effect which, when properly included, accounts for the data. 20. Observation of a pairing pseudogap in a two-dimensional Fermi gas. Science.gov (United States) Feld, Michael; Fröhlich, Bernd; Vogt, Enrico; Koschorreck, Marco; Köhl, Michael 2011-11-30 Pairing of fermions is ubiquitous in nature, underlying many phenomena. Examples include superconductivity, superfluidity of (3)He, the anomalous rotation of neutron stars, and the crossover between Bose-Einstein condensation of dimers and the BCS (Bardeen, Cooper and Schrieffer) regime in strongly interacting Fermi gases. When confined to two dimensions, interacting many-body systems show even more subtle effects, many of which are not understood at a fundamental level. Most striking is the (as yet unexplained) phenomenon of high-temperature superconductivity in copper oxides, which is intimately related to the two-dimensional geometry of the crystal structure. In particular, it is not understood how the many-body pairing is established at high temperature, and whether it precedes superconductivity. Here we report the observation of a many-body pairing gap above the superfluid transition temperature in a harmonically trapped, two-dimensional atomic Fermi gas in the regime of strong coupling. Our measurements of the spectral function of the gas are performed using momentum-resolved photoemission spectroscopy, analogous to angle-resolved photoemission spectroscopy in the solid state. Our observations mark a significant step in the emulation of layered two-dimensional strongly correlated superconductors using ultracold atomic gases. 1. Universal spin transport in a strongly interacting Fermi gas. Science.gov (United States) Sommer, Ariel; Ku, Mark; Roati, Giacomo; Zwierlein, Martin W 2011-04-14 Transport of fermions, particles with half-integer spin, is central to many fields of physics. Electron transport runs modern technology, defining states of matter such as superconductors and insulators, and electron spin is being explored as a new carrier of information. Neutrino transport energizes supernova explosions following the collapse of a dying star, and hydrodynamic transport of the quark-gluon plasma governed the expansion of the early Universe. However, our understanding of non-equilibrium dynamics in such strongly interacting fermionic matter is still limited. Ultracold gases of fermionic atoms realize a pristine model for such systems and can be studied in real time with the precision of atomic physics. Even above the superfluid transition, such gases flow as an almost perfect fluid with very low viscosity when interactions are tuned to a scattering resonance. In this hydrodynamic regime, collective density excitations are weakly damped. Here we experimentally investigate spin excitations in a Fermi gas of (6)Li atoms, finding that, in contrast, they are maximally damped. A spin current is induced by spatially separating two spin components and observing their evolution in an external trapping potential. We demonstrate that interactions can be strong enough to reverse spin currents, with components of opposite spin reflecting off each other. Near equilibrium, we obtain the spin drag coefficient, the spin diffusivity and the spin susceptibility as a function of temperature on resonance and show that they obey universal laws at high temperatures. In the degenerate regime, the spin diffusivity approaches a value set by [planck]/m, the quantum limit of diffusion, where [planck]/m is Planck's constant divided by 2π and m the atomic mass. For repulsive interactions, our measurements seem to exclude a metastable ferromagnetic state. 2. Closed-Form Solutions of the Thomas-Fermi in Heavy Atoms and the Langmuir-Blodgett in Current Flow ODEs in Mathematical Physics Directory of Open Access Journals (Sweden) Efstathios E. Theotokoglou 2015-01-01 Full Text Available Two kinds of second-order nonlinear, ordinary differential equations (ODEs appearing in mathematical physics are analyzed in this paper. The first one concerns the Thomas-Fermi (TF equation, while the second concerns the Langmuir-Blodgett (LB equation in current flow. According to a mathematical methodology recently developed, the exact analytic solutions of both TF and LB ODEs are proposed. Both of these are nonlinear of the second order and by a series of admissible functional transformations are reduced to Abel’s equations of the second kind of the normal form. The closed form solutions of the TF and LB equations in the phase and physical plane are given. Finally a new interesting result has been obtained related to the derivative of the TF function at the limit. 3. Umklapp superradiance with a collisionless quantum degenerate Fermi gas. Science.gov (United States) Piazza, Francesco; Strack, Philipp 2014-04-11 The quantum dynamics of the electromagnetic light mode of an optical cavity filled with a coherently driven Fermi gas of ultracold atoms strongly depends on the geometry of the Fermi surface. Superradiant light generation and self-organization of the atoms can be achieved at low pumping threshold due to resonant atom-photon umklapp processes, where the fermions are scattered from one side of the Fermi surface to the other by exchanging photon momenta. The cavity spectrum exhibits sidebands that, despite strong atom-light coupling and cavity decay, retain narrow linewidth, due to absorptionless transparency windows outside the atomic particle-hole continuum and the suppression of broadening and thermal fluctuations in the collisionless Fermi gas. Institute of Scientific and Technical Information of China (English) 郭飞翔; 周晓凡; 赵华 2015-01-01 采用密度矩阵重整化群 ( density-matrix-renormalization-group, DMRG) 方法, 研究梯状光晶格中排斥相互作用费米气体的基态属性. 研究表明, Zeeman场能够激发系统的相分离 (完全极化相和部分极化相), 而自旋轨道耦合效应能抑制相分离, 使整个晶格处于部分极化相, 在不同的强弱排斥相互作用系统中, 极化率会随自旋轨道耦合改变表现出不同的变化规律.%The density-matrix-renormalization-group ( DMRG ) method is used to numerically calculate the ground state of repulsively interacting Fermi atoms loaded in optical ladder lattices. It is found that the system exhibits the spatial separation of a fully spin-polarized phase from the partially polarized phase for the suitable intensity of Zeeman field without the effect of spin-orbit coupled atoms. The spin-orbit coupling drives the fully spin-polarized phase to the partially spin-polarized phase in the whole system. The spin polarizations of weak and strong repulsively interac-ting systems vary differently with spin-orbit interaction strength. 5. The universal sound velocity formula for the strongly interacting unitary Fermi gas Institute of Scientific and Technical Information of China (English) Liu Ke; Chen Ji-Sheng 2011-01-01 Due to the scale invariance, the thermodynamic laws of strongly interacting limit unitary Fermi gas can be similar to those of non-interacting ideal gas. For example, the virial theorem between pressure and energy density of the ideal gas P = 2E/ZV is still satisfied by the unitary Fermi gas. This paper analyses the sound velocity of unitary Fermi gases with the quasi-linear approximation. For comparison, the sound velocities for the ideal Boltzmann, Bose and Fermi gas are also given. Quite interestingly, the sound velocity formula for the ideal non-interacting gas is found to be satisfied by the unitary Fermi gas in different temperature regions. 6. Quantum phases of Fermi-Fermi mixtures in optical lattices OpenAIRE Iskin, M.; de Melo, C. A. R. Sa 2007-01-01 The ground state phase diagram of Fermi-Fermi mixtures in optical lattices is analyzed as a function of interaction strength, population imbalance, filling fraction and tunneling parameters. It is shown that population imbalanced Fermi-Fermi mixtures reduce to strongly interacting Bose-Fermi mixtures in the molecular limit, in sharp contrast to homogeneous or harmonically trapped systems where the resulting Bose-Fermi mixture is weakly interacting. Furthermore, insulating phases are found in ... 7. Mott criticality and pseudogap in Bose-Fermi mixtures. Science.gov (United States) Altman, Ehud; Demler, Eugene; Rosch, Achim 2012-12-07 We study the Mott transition of a mixed Bose-Fermi system of ultracold atoms in an optical lattice, where the number of (spinless) fermions and bosons adds up to one atom per lattice, n(F)+n(B)=1. For weak interactions, a Fermi surface coexists with a Bose-Einstein condensate while for strong interaction the system is incompressible but still characterized by a Fermi surface of composite fermions. At the critical point, the spectral function of the fermions A(k,ω) exhibits a pseudogapped behavior, rising as \|ω\| at the Fermi momentum, while in the Mott phase it is fully gapped. Taking into account the interaction between the critical modes leads at very low temperatures either to p-wave pairing or the transition is driven weakly first order. The same mechanism should also be important in antiferromagnetic metals with a small Fermi surface. 8. Thermodynamic property of gases in the sonoluminescing bubble Institute of Scientific and Technical Information of China (English) AN Yu; LI Guiqin; ZHOU Tieying 2001-01-01 With the theory of statistical physics dealing with chemical reaction (the law of mass action), the different thermodynamic property of noble gases (mono-atomic gases) in a small bubble and diatomic gases in a small bubble semi-quantitatively are analyzed. As bubbles of the mono-atomic and the diatomic gases are compressed, shock waves are produced in both bubbles. Though shock wave leads to sharp increase of pressure and temperature of gases in the bubble, diatomic gas will excitated vibrations and dissociate themselves to mono-atomic gas,these processes will consume many accumulated heat energy and block the further increase of the temperature. Therefore, compare with the mono-atomic gases in the bubble, there will be no enough charged particles ionized to flash for diatomic gases in the bubble, this may be the reason why a bubble of diatomic gases has no single bubble sonoluminescence while a bubble of noble gases has. 9. Irritant gases NARCIS (Netherlands) Meulenbelt, J 2016-01-01 Acute inhalation injury can result from the use of household cleaning agents (e.g. chlorine, ammonia), industrial or combustion gases (e.g. sulfur dioxide, nitrogen oxides) or bioterrorism. The severity of the injury is to a great extent determined by the circumstances of exposure. If exposure was i 10. Greenhouse Gases Science.gov (United States) ... life. Governments all around the world ban and control production and use of several industrial gases that destroy atmospheric ozone and create a hole in the ozone layer . At lower elevations of the atmosphere (the troposphere), ozone is harmful to ... for Future Emissions FAQs How much carbon dioxide is produced when ... 11. Fermi comes to CERN CERN Multimedia NASA 2009-01-01 1. This view from NASA's Fermi Gamma-ray Space Telescope is the deepest and best-resolved portrait of the gamma-ray sky to date. The image shows how the sky appears at energies more than 150 million times greater than that of visible light. Among the signatures of bright pulsars and active galaxies is something familiar -- a faint path traced by the sun. (Credit: NASA/DOE/Fermi LAT Collaboration) 2. The Large Area Telescope (LAT) on Fermi detects gamma-rays through matter (electrons) and antimatter (positrons) they produce after striking layers of tungsten. (Credit: NASA/Goddard Space Flight Center Conceptual Image Lab) 12. Atomic physics CERN Document Server Born, Max 1989-01-01 The Nobel Laureate's brilliant exposition of the kinetic theory of gases, elementary particles, the nuclear atom, wave-corpuscles, atomic structure and spectral lines, electron spin and Pauli's principle, quantum statistics, molecular structure and nuclear physics. Over 40 appendices, a bibliography, numerous figures and graphs. 13. Fermi LAT GRBs Data.gov (United States) National Aeronautics and Space Administration — All analysis results presented here are preliminary and are not intended as an official catalog of Fermi-LAT detected GRBs. Please consult the table's caveat page... 14. Fermi GBM Trigger Catalog Data.gov (United States) National Aeronautics and Space Administration — Fermi is a powerful space observatory that will open a wide window on the universe. Gamma rays are the highest-energy form of light, and the gamma-ray sky is... 15. Enrico Fermi centenary exhibition seminar CERN Multimedia Maximilien Brice 2002-01-01 Photo 01: Dr. Juan Antonio Rubio, Leader of the Education and Technology Transfer Division and CERN Director General, Prof. Luciano Maiani. Photo 03: Luciano Maiani, Welcome and Introduction Photo 09: Antonino Zichichi, The New 'Centro Enrico Fermi' at Via Panisperna Photos 10, 13: Ugo Amaldi, Fermi at Via Panisperna and the birth of Nuclear Medicine Photo 14: Jack Steinberger, Fermi in Chicago Photo 18: Valentin Telegdi, A close-up of Fermi Photo 21: Arnaldo Stefanini, Celebrating Fermi's Centenary in Documents and Pictures. 16. Energy-pressure relation for low-dimensional gases Science.gov (United States) Mancarella, Francesco; Mussardo, Giuseppe; Trombettoni, Andrea 2014-10-01 A particularly simple relation of proportionality between internal energy and pressure holds for scale-invariant thermodynamic systems (with Hamiltonians homogeneous functions of the coordinates), including classical and quantum - Bose and Fermi - ideal gases. One can quantify the deviation from such a relation by introducing the internal energy shift as the difference between the internal energy of the system and the corresponding value for scale-invariant (including ideal) gases. After discussing some general thermodynamic properties associated with the scale-invariance, we provide criteria for which the internal energy shift density of an imperfect (classical or quantum) gas is a bounded function of temperature. We then study the internal energy shift and deviations from the energy-pressure proportionality in low-dimensional models of gases interpolating between the ideal Bose and the ideal Fermi gases, focusing on the Lieb-Liniger model in 1d and on the anyonic gas in 2d. In 1d the internal energy shift is determined from the thermodynamic Bethe ansatz integral equations and an explicit relation for it is given at high temperature. Our results show that the internal energy shift is positive, it vanishes in the two limits of zero and infinite coupling (respectively the ideal Bose and the Tonks-Girardeau gas) and it has a maximum at a finite, temperature-depending, value of the coupling. Remarkably, at fixed coupling the energy shift density saturates to a finite value for infinite temperature. In 2d we consider systems of Abelian anyons and non-Abelian Chern-Simons particles: as it can be seen also directly from a study of the virial coefficients, in the usually considered hard-core limit the internal energy shift vanishes and the energy is just proportional to the pressure, with the proportionality constant being simply the area of the system. Soft-core boundary conditions at coincident points for the two-body wavefunction introduce a length scale, and induce a 17. Temperature dependence of the universal contact parameter in a unitary Fermi gas. Science.gov (United States) Kuhnle, E D; Hoinka, S; Dyke, P; Hu, H; Hannaford, P; Vale, C J 2011-04-29 The contact I, introduced by Tan, has emerged as a key parameter characterizing universal properties of strongly interacting Fermi gases. For ultracold Fermi gases near a Feshbach resonance, the contact depends upon two quantities: the interaction parameter 1/(k(F)a), where k(F) is the Fermi wave vector and a is the s-wave scattering length, and the temperature T/T(F), where T(F) is the Fermi temperature. We present the first measurements of the temperature dependence of the contact in a unitary Fermi gas using Bragg spectroscopy. The contact is seen to follow the predicted decay with temperature and shows how pair-correlations at high momentum persist well above the superfluid transition temperature. 18. Strongly correlated quantum fluids: ultracold quantum gases, quantum chromodynamic plasmas and holographic duality Science.gov (United States) Adams, Allan; Carr, Lincoln D.; Schäfer, Thomas; Steinberg, Peter; Thomas, John E. 2012-11-01 Strongly correlated quantum fluids are phases of matter that are intrinsically quantum mechanical and that do not have a simple description in terms of weakly interacting quasiparticles. Two systems that have recently attracted a great deal of interest are the quark-gluon plasma, a plasma of strongly interacting quarks and gluons produced in relativistic heavy ion collisions, and ultracold atomic Fermi gases, very dilute clouds of atomic gases confined in optical or magnetic traps. These systems differ by 19 orders of magnitude in temperature, but were shown to exhibit very similar hydrodynamic flows. In particular, both fluids exhibit a robustly low shear viscosity to entropy density ratio, which is characteristic of quantum fluids described by holographic duality, a mapping from strongly correlated quantum field theories to weakly curved higher dimensional classical gravity. This review explores the connection between these fields, and also serves as an introduction to the focus issue of New Journal of Physics on ‘Strongly Correlated Quantum Fluids: From Ultracold Quantum Gases to Quantum Chromodynamic Plasmas’. The presentation is accessible to the general physics reader and includes discussions of the latest research developments in all three areas. 19. Atomic polarizabilities Energy Technology Data Exchange (ETDEWEB) Safronova, M. S. [Department of Physics and Astronomy, University of Delaware, Newark, DE 19716 (United States); Mitroy, J. [School of Engineering, Charles Darwin University, Darwin NT 0909 (Australia); Clark, Charles W. [Joint Quantum Institute, National Institute of Standards and Technology and the University of Maryland, Gaithersburg, Maryland 20899-8410 (United States); Kozlov, M. G. [Petersburg Nuclear Physics Institute, Gatchina 188300 (Russian Federation) 2015-01-22 The atomic dipole polarizability governs the first-order response of an atom to an applied electric field. Atomic polarization phenomena impinge upon a number of areas and processes in physics and have been the subject of considerable interest and heightened importance in recent years. In this paper, we will summarize some of the recent applications of atomic polarizability studies. A summary of results for polarizabilities of noble gases, monovalent, and divalent atoms is given. The development of the CI+all-order method that combines configuration interaction and linearized coupled-cluster approaches is discussed. 20. Enrico Fermi exhibition at CERN CERN Document Server 2002-01-01 A touring exhibition celebrating the centenary of Enrico Fermi's birth in 1901 will be on display at CERN (Main Building, Mezzanine) from 12-27 September. You are cordially invited to the opening celebration on Thursday 12 September at 16:00 (Main Building, Council Chamber), which will include speechs from: Luciano Maiani Welcome and Introduction Arnaldo Stefanini Celebrating Fermi's Centenary in Documents and Pictures Antonino Zichichi The New 'Centro Enrico Fermi' at Via Panisperna Ugo Amaldi Fermi at Via Panisperna and the birth of Nuclear Medicine Jack Steinberger Fermi in Chicago Valentin Telegdi A Close-up of Fermi and the screening of a documentary video about Fermi: Scienziati a Pisa: Enrico Fermi (Scientists at Pisa: Enrico Fermi) created by Francesco Andreotti for La Limonaia from early film, photographs and sound recordings (In Italian, with English subtitles - c. 30 mins). This will be followed by an aperitif on the Mezz... 1. A long-lived spin-orbit-coupled degenerate dipolar Fermi gas CERN Document Server Burdick, Nathaniel Q; Lev, Benjamin L 2016-01-01 We describe the creation of a long-lived spin-orbit-coupled gas of quantum degenerate atoms using the most magnetic fermionic element, dysprosium. Spin-orbit-coupling arises from a synthetic gauge field created by the adiabatic following of degenerate dressed states comprised of optically coupled components of an atomic spin. Because of dysprosium's large electronic orbital angular momentum and large magnetic moment, the lifetime of the gas is limited not by spontaneous emission from the light-matter coupling, as for gases of alkali-metal atoms, but by dipolar relaxation of the spin. This relaxation is suppressed at large magnetic fields due to Fermi statistics. We observe lifetimes up to 400 ms, which exceeds that of spin-orbit-coupled fermionic alkali atoms by a factor of 10-100, and is close to the value obtained from a theoretical model. Elastic dipolar interactions are also observed to influence the Rabi evolution of the spin, revealing an interacting fermionic system. The long lifetime of this weakly in... 2. A new look at Thomas–Fermi theory DEFF Research Database (Denmark) Solovej, Jan Philip 2016-01-01 In this short note, we argue that Thomas–Fermi theory, the simplest of all density functional theories, although failing to explain features such as molecular binding or stability of negative ions, is surprisingly accurate in estimating sizes of atoms. We give both numerical, experimental...... and rigorous mathematical evidence for this claim. Motivated by this, we formulate two new mathematical conjectures on the exactness of Thomas–Fermi theory.... 3. A new look at Thomas-Fermi Theory CERN Document Server Solovej, Jan Philip 2016-01-01 In this short note we argue that Thomas-Fermi Theory the simplest of all density functional theories, although failing to explain features such as binding or stability of negative ions, is surprisingly accurate in estimating sizes of atoms. We give both numerical, experimental and rigorous mathematical evidence for this claim. Motivated by this we formulate two new mathematical conjectures on the exactness of Thomas-Fermi Theory. 4. The Fermi's Bayes Theorem CERN Document Server D'Agostini, G 2005-01-01 It is curious to learn that Enrico Fermi knew how to base probabilistic inference on Bayes theorem, and that some influential notes on statistics for physicists stem from what the author calls elsewhere, but never in these notes, {\\it the Bayes Theorem of Fermi}. The fact is curious because the large majority of living physicists, educated in the second half of last century -- a kind of middle age in the statistical reasoning -- never heard of Bayes theorem during their studies, though they have been constantly using an intuitive reasoning quite Bayesian in spirit. This paper is based on recollections and notes by Jay Orear and on Gauss' Theoria motus corporum coelestium'', being the {\\it Princeps mathematicorum} remembered by Orear as source of Fermi's Bayesian reasoning. 5. Fermi comes to CERN CERN Multimedia 2009-01-01 In only 10 months of scientific activity, the Fermi space observatory has already collected an unprecedented wealth of information on some of the most amazing objects in the sky. In a recent talk at CERN, Luca Latronico, a member of the Fermi collaboration, explained some of their findings and emphasized the strong links between High Energy Physics (HEP) and High Energy Astrophysics (HEA). The Fermi gamma-ray telescope was launched by NASA in June 2008. After about two months of commissioning it started sending significant data back to the Earth. Since then, it has made observations that are changing our view of the sky: from discovering a whole new set of pulsars, the greatest total energy gamma-ray burst ever, to detecting an unexplained abundance of high-energy electrons that could be a signature of dark matter, to producing a uniquely rich and high definition sky map in gamma-rays. The high performance of the instrument comes as ... 6. Quantum Gas Microscope for Fermionic Atoms Science.gov (United States) Okan, Melih; Cheuk, Lawrence; Nichols, Matthew; Lawrence, Katherine; Zhang, Hao; Zwierlein, Martin 2016-05-01 Strongly interacting fermions define the properties of complex matter throughout nature, from atomic nuclei and modern solid state materials to neutron stars. Ultracold atomic Fermi gases have emerged as a pristine platform for the study of many-fermion systems. In this poster we demonstrate the realization of a quantum gas microscope for fermionic 40 K atoms trapped in an optical lattice and the recent experiments which allows one to probe strongly correlated fermions at the single atom level. We combine 3D Raman sideband cooling with high- resolution optics to simultaneously cool and image individual atoms with single lattice site resolution at a detection fidelity above 95%. The imaging process leaves the atoms predominantly in the 3D motional ground state of their respective lattice sites, inviting the implementation of a Maxwell's demon to assemble low-entropy many-body states. Single-site resolved imaging of fermions enables the direct observation of magnetic order, time resolved measurements of the spread of particle correlations, and the detection of many-fermion entanglement. NSF, AFOSR-PECASE, AFOSR-MURI on Exotic Phases of Matter, ARO-MURI on Atomtronics, ONR, a Grant from the Army Research Office with funding from the DARPA OLE program, and the David and Lucile Packard Foundation. 7. Atomic phase diagram Institute of Scientific and Technical Information of China (English) LI Shichun 2004-01-01 Based on the Thomas-Fermi-Dirac-Cheng model, atomic phase diagram or electron density versus atomic radius diagram describing the interaction properties of atoms of different kinds in equilibrium state is developed. Atomic phase diagram is established based on the two-atoms model. Besides atomic radius, electron density and continuity condition for electron density on interfaces between atoms, the lever law of atomic phase diagram involving other physical parameters is taken into account, such as the binding energy, for the sake of simplicity. 8. Formation of noble-gas hydrides and decay of solvated protons revisited: diffusion-controlled reactions and hydrogen atom losses in solid noble gases. Science.gov (United States) Tanskanen, Hanna; Khriachtchev, Leonid; Lignell, Antti; Räsänen, Markku; Johansson, Susanna; Khyzhniy, Ivan; Savchenko, Elena 2008-02-07 UV photolysis and annealing of C2H2/Xe, C2H2/Xe/Kr, and HBr/Xe matrices lead to complicated photochemical processes and reactions. The dominating products in these experiments are noble-gas hydrides with general formula HNgY (Ng = noble-gas atom, Y = electronegative fragment). We concentrate on distinguishing the local and global mobility and losses of H atoms, barriers of the reactions, and the decay of solvated protons. Different deposition temperatures change the amount of lattice imperfections and thus the amount of traps for H atoms. The averaged distance between reacting species influencing the reaction kinetics is controlled by varying the precursor concentration. A number of solid-state processes connected to the formation of noble-gas hydrides and decay of solvated protons are discussed using a simple kinetic model. The most efficient formation of noble-gas hydrides is connected with global (long-range) mobility of H atoms leading to the H + Xe + Y reaction. The highest concentration of noble-gas hydrides was obtained in matrices of highest optical quality, which probably have the lowest concentration of defects and H-atom losses. In matrices with high amount of geometrical imperfections, the product formation is inefficient and dominated by a local (short-range) process. The decay of solvated protons is rather local than a global process, which is different from the formation of noble-gas molecules. However, the present data do not allow distinguishing local proton and electron mobilities. Our previous results indicate that these are electrons which move to positively-charged centers and neutralize them. It is believed that the image obtained here for solid xenon is applicable to solid krypton whereas the case of argon deserves special attention. 9. Physics of Ionized Gases Science.gov (United States) Reiss, Howard R.; Smirnov, Boris M. 2001-03-01 A comprehensive textbook and reference for the study of the physics of ionized gases The intent of this book is to provide deep physical insight into the behavior of gases containing atoms and molecules from which one or more electrons have been ionized. The study of these so-called plasmas begins with an overview of plasmas as they are found in nature and created in the laboratory. This serves as a prelude to a comprehensive study of plasmas, beginning with low temperature and "ideal" plasmas and extending to radiation and particle transport phenomena, the response of plasmas to external fields, and an insightful treatment of plasma waves, plasma instabilities, nonlinear phenomena in plasmas, and the study of plasma interactions with surfaces. In all cases, the emphasis is on a clear and unified understanding of the basic physics that underlies all plasma phenomena. Thus, there are chapters on plasma behavior from the viewpoint of atomic and molecular physics, as well as on the macroscopic phenomena involved in physical kinetics of plasmas and the transport of radiation and of charged particles within plasmas. With this grounding in the fundamental physics of plasmas, the notoriously difficult subjects of nonlinear phenomena and of instabilities in plasmas are then treated with comprehensive clarity. 10. Doping dependence of electronic and mechanical properties of GaSe1-xTex and Ga1-xInxSe from first principles Science.gov (United States) Rak, Zs.; Mahanti, S. D.; Mandal, Krishna C.; Fernelius, N. C. 2010-10-01 The electronic and mechanical properties of the hexagonal, layered GaSe doped with Te and In have been studied using first-principles pseudopotential method within density-functional theory. The calculated elastic constants of the end compounds GaSe and InSe compare well with the available experimental and theoretical values. As we go from GaSe to InSe, the elastic constants C13 , C33 , and C44 increase while C11 and C12 decrease, suggesting that the crystal becomes stiffer in the direction perpendicular to the atomic layers and the softer in the direction parallel to the layers, as more substitutional In is incorporated in GaSe. The electronic structure and the formation energies of several defects and simple defect complexes are discussed and the calculated charge transition levels are compared to available experimental data. We demonstrate that In doping may play an important role in the observed enhancement in the structural properties of GaSe. Depending on the Fermi energy, In can either substitute for Ga (InGa) or occupy an interstitial position as a triply charged defect (Ini3+) . While the substitutional In does not change significantly the electronic and mechanical properties of the host, we find that the shear stiffness of GaSe is considerably increased when In is incorporated as charged interstitial impurity. 11. Schottky barrier engineering via adsorbing gases at the sulfur vacancies in the metal–MoS2 interface Science.gov (United States) Su, Jie; Feng, Liping; Zhang, Yan; Liu, Zhengtang 2017-03-01 Sulfur vacancies (S-vacancies) are common in monolayer MoS2 (mMoS2). Finding an effective way to control rather than abolish the effect of S-vacancies on contact properties is vital for the application of mMoS2. Here, we propose the adsorption of gases to passivate the S-vacancies in Pt–mMoS2 interfaces. Results demonstrate that gases are stably and preferentially adsorbed at S-vacancies. The n-type Schottky barriers of Pt–mMoS2 interfaces are reduced significantly upon the adsorption electron-donor gases, especially Cl2. The n-type transport character of the Pt–mMoS2 interface can be changed to p-type by the adsorption of electron-acceptor gases. As the adsorption concentration increases, both n- and p-type Schottky barriers are further reduced, and the lowest n- and p-type Schottky barriers are 0.36 and 0 eV, respectively. Note that the variations in Schottky barriers are independent of the oxidizing ability of gases but relative to the average number of valence electrons per gas atom. Analysis demonstrates that although gases at S-vacancies cannot cause gap states to vanish, and can even enhance Fermi level pinning, they modulate charge redistribution and the potential step at the interface region. Moreover, with increasing adsorption concentration, the valence band maximum of mMoS2 shows the opposite variation tendency to that of the potential step. Our results suggest that adsorption of gases is an effective way to passivate S-vacancies to modulate the transport properties of Pt–mMoS2 interfaces. 12. From few to many. Ultracold atoms in reduced dimensions Energy Technology Data Exchange (ETDEWEB) Wenz, Andre Niklas 2013-12-19 This thesis reports on experimental studies exploring few and many-body physics of ultracold Bose and Fermi gases with reduced dimensionality. These experiments illustrate the versatility and great amount of control over the particle number, the interaction and other degrees of freedom, like the spin, that these generic quantum systems offer. In the first part of this thesis, we use quasi one-dimensional few-particle systems of one to ten fermionic atoms to investigate the crossover from few to many-body physics. This is achieved by measuring the interaction energy between a single impurity atom in a state vertical stroke ↓ right angle which repulsively interacts with an increasing number of majority atoms in a state vertical stroke ↑ right angle. We find that the system quickly approaches the results from the many-body theory, which describes the behavior of a single impurity immersed in a Fermi sea of an infinite number of majority particles. The second part of this thesis presents studies of the time evolution of a bosonic F=1 spinor BEC of {sup 87}Rb atoms. In this system, we investigate the emergence and coarsening of ferromagnetic spin textures from initially unmagnetized samples. While the ferromagnetic domains grow, we observe the development of a spin space anisotropy which is in agreement with the predicted phase-diagram. The last part of this thesis presents our first steps towards the investigation of phase coherence of quasi two-dimensional quantum gases in the crossover from bosonic molecules to fermionic atoms. 13. Leaky Fermi accelerators CERN Document Server Shah, Kushal; Rom-Kedar, Vered; Turaev, Dmitry 2015-01-01 A Fermi accelerator is a billiard with oscillating walls. A leaky accelerator interacts with an environment of an ideal gas at equilibrium by exchange of particles through a small hole on its boundary. Such interaction may heat the gas: we estimate the net energy flow through the hole under the assumption that the particles inside the billiard do not collide with each other and remain in the accelerator for sufficiently long time. The heat production is found to depend strongly on the type of the Fermi accelerator. An ergodic accelerator, i.e. one which has a single ergodic component, produces a weaker energy flow than a multi-component accelerator. Specifically, in the ergodic case the energy gain is independent of the hole size, whereas in the multi-component case the energy flow may be significantly increased by shrinking the hole size. 14. Virial theorem and universality in a unitary fermi gas. Science.gov (United States) Thomas, J E; Kinast, J; Turlapov, A 2005-09-16 Unitary Fermi gases, where the scattering length is large compared to the interparticle spacing, can have universal properties, which are independent of the details of the interparticle interactions when the range of the scattering potential is negligible. We prepare an optically trapped, unitary Fermi gas of 6Li, tuned just above the center of a broad Feshbach resonance. In agreement with the universal hypothesis, we observe that this strongly interacting many-body system obeys the virial theorem for an ideal gas over a wide range of temperatures. Based on this result, we suggest a simple volume thermometry method for unitary gases. We also show that the observed breathing mode frequency, which is close to the unitary hydrodynamic value over a wide range of temperature, is consistent with a universal hydrodynamic gas with nearly isentropic dynamics. 15. Trends in structural, electronic properties, Fermi surface topology, and inter-atomic bonding in the series of ternary layered dichalcogenides KNi{sub 2}S{sub 2}, KNi{sub 2}Se{sub 2}, and KNi{sub 2}Te{sub 2} from first principles calculations Energy Technology Data Exchange (ETDEWEB) Bannikov, V.V. [Institute of Solid State Chemistry, Ural Branch, Russian Academy of Sciences, Pervomaiskaya Street, 91, Ekaterinburg 620990 (Russian Federation); Ivanovskii, A.L., E-mail: [email protected] [Institute of Solid State Chemistry, Ural Branch, Russian Academy of Sciences, Pervomaiskaya Street, 91, Ekaterinburg 620990 (Russian Federation) 2013-06-01 By means of the FLAPW-GGA approach, we have systematically studied the structural and electronic properties of tetragonal dichalcogenides KNi{sub 2}Ch{sub 2} (Ch=S, Se, and Te). Our results show that replacements of chalcogens (S→Se→Te) lead to anisotropic deformations of the crystals structure, which are related to the strong anisotropic character of the inter-atomic bonds, where inside the [Ni{sub 2}Ch{sub 2}] blocks, mixed covalent–ionic–metallic bonds occur, whereas between the adjacent [Ni{sub 2}Ch{sub 2}] blocks and K atomic sheets, ionic bonds emerge. We found that in the sequence KNi{sub 2}S{sub 2}→KNi{sub 2}Se{sub 2}→KNi{sub 2}Te{sub 2} (i) the overall band structure (where the near-Fermi valence bands are due mainly to the Ni states) is preserved, but the width of the common valence band and the widths of the separate sub-bands and the gaps decrease; (ii) the total DOSs at the Fermi level also decrease; and (iii) for the Fermi surfaces, the most appreciable changes are demonstrated by the hole-like sheets, when a necklace-like topology is formed for the 2D-like sheets and the volume of the closed pockets decreases. Some trends in structural and electronic parameters for ThCr{sub 2}Si{sub 2}-type layered dichalcogenides, KNi{sub 2}Ch{sub 2}, KFe{sub 2}Ch{sub 2}, KCo{sub 2}Se{sub 2}, are discussed. 16. Gradient catastrophe and Fermi-edge resonances in Fermi gas. Science.gov (United States) Bettelheim, E; Kaplan, Y; Wiegmann, P 2011-04-22 Any smooth spatial disturbance of a degenerate Fermi gas inevitably becomes sharp. This phenomenon, called the gradient catastrophe, causes the breakdown of a Fermi sea to multiconnected components characterized by multiple Fermi points. We argue that the gradient catastrophe can be probed through a Fermi-edge singularity measurement. In the regime of the gradient catastrophe the Fermi-edge singularity problem becomes a nonequilibrium and nonstationary phenomenon. We show that the gradient catastrophe transforms the single-peaked Fermi-edge singularity of the tunneling (or absorption) spectrum to a sequence of multiple asymmetric singular resonances. An extension of the bosonic representation of the electronic operator to nonequilibrium states captures the singular behavior of the resonances. 17. Mode analysis of fluctuations in two-species Bose-Einstein condensates of atomic gases with a vortex for a component. Characteristic features of compressive and sliding motions Energy Technology Data Exchange (ETDEWEB) Doi, Kensuke; Natsume, Yuhei [Chiba Univ., Graduate School of Science and Technology, Chiba (Japan) 2003-04-01 Characteristic features of fluctuations of Bose-Einstein condensations for systems of two-components in gas phases of alkali-metal atoms trapped by spherical harmonic potentials are discussed on the basis of numerical calculations. We concentrate our attention on the phases in which the spherical state {psi}{sub 1} without vortex is surrounded by {psi}{sub 2} with a vortex for the unit circulation q=1. These states are expressed by Gross-Pitaevskii equation, where a vortex-core is along the z-axis. We investigate properties of collective excitations by the linear analysis for bosonic excitations described as Bogoliubov equations. The behavior of each mode is discussed in relation with the role of interspecies repulsion in addition to that of intraspecies one. We point out the role of the new compressive mode which has two nodes on z-axis, in addition to that of the core mode without a node which have been previously discussed in the single-component system. Furthermore, we would like to emphasize that sliding modes show the branching features into in- and out-of-phase motions with increasing interspecies interaction. The dependence of those branchings on interspecies repulsion is explained by spatial shapes of relevant modes. (author) 18. A new look at Thomas–Fermi theory DEFF Research Database (Denmark) Solovej, Jan Philip 2016-01-01 In this short note, we argue that Thomas–Fermi theory, the simplest of all density functional theories, although failing to explain features such as molecular binding or stability of negative ions, is surprisingly accurate in estimating sizes of atoms. We give both numerical, experimental and rig...... and rigorous mathematical evidence for this claim. Motivated by this, we formulate two new mathematical conjectures on the exactness of Thomas–Fermi theory.......In this short note, we argue that Thomas–Fermi theory, the simplest of all density functional theories, although failing to explain features such as molecular binding or stability of negative ions, is surprisingly accurate in estimating sizes of atoms. We give both numerical, experimental... 19. Observation of hydrodynamic expansion in a strongly-interacting Fermi gas: Signature of superfluidity? Science.gov (United States) O'Hara, K. M.; Hemmer, S. L.; Gehm, M. E.; Thomas, J. E. 2003-05-01 Atomic Fermi gases with magnetically tunable, strong interactions provide a desktop laboratory for exploring new nonperturbative theories in systems ranging from superconductors to neutron stars. We use all-optical methods to produce a highly degenerate, two-component gas of ^6Li atoms in an applied magnetic field (910 G) near a Feshbach resonance where strong interactions are observed [1]. The s-wave scattering length is estimated to be a_S=-10^4 a_0, which is large compared to the interparticle spacing. Exciting new predictions for this regime include unitarity-limited universal interactions [2] and the onset of resonance superfluidity at a very high transition temperature [3-5]. Forced evaporation is accomplished by lowering the trap laser intensity over a period of 3.5 seconds and then recompressing the trap to full depth. Abrupt release of the cloud at 910 G results in a highly anisotropic expansion, where the gas expands rapidly in the transverse directions while remaining nearly stationary in the axial direction [1]. This anisotropic energy release has been predicted recently to be a signature of superfluidity in a Fermi gas [6]. We will discuss interpretations of the data in terms of superfluidity and unitarity-limited collision dynamics. References 1. K. M. O'Hara et al., Science, 298, 2179 (2002). 2. H. Heiselberg, Phys. Rev. A 63, 043606 (2001). 3. M. Holland, et al., Phys. Rev. Lett. 87, 120406 (2001). 4. E. Timmermans, et al., Phys. Lett. A 285, 228 (2001). 5. Y. Ohashi and A. Griffin, Phys. Rev. Lett. 89, 130402 (2002). 6. C. Menotti, et al., Phys. Rev. Lett. 89, 250402 (2002). 20. Chiral non-Fermi liquids Science.gov (United States) Sur, Shouvik; Lee, Sung-Sik 2014-07-01 A non-Fermi liquid state without time-reversal and parity symmetries arises when a chiral Fermi surface is coupled with a soft collective mode in two space dimensions. The full Fermi surface is described by a direct sum of chiral patch theories, which are decoupled from each other in the low-energy limit. Each patch includes low-energy excitations near a set of points on the Fermi surface with a common tangent vector. General patch theories are classified by the local shape of the Fermi surface, the dispersion of the critical boson, and the symmetry group, which form the data for distinct universality classes. We prove that a large class of chiral non-Fermi liquid states exists as stable critical states of matter. For this, we use a renormalization group scheme where low-energy excitations of the Fermi surface are interpreted as a collection of (1+1)-dimensional chiral fermions with a continuous flavor labeling the momentum along the Fermi surface. Due to chirality, the Wilsonian effective action is strictly UV finite. This allows one to extract the exact scaling exponents although the theories flow to strongly interacting field theories at low energies. In general, the low-energy effective theory of the full Fermi surface includes patch theories of more than one universality classes. As a result, physical responses include multiple universal components at low temperatures. We also point out that, in quantum field theories with extended Fermi surface, a noncommutative structure naturally emerges between a coordinate and a momentum which are orthogonal to each other. We show that the invalidity of patch description for Fermi liquid states is tied with the presence of UV/IR mixing associated with the emergent noncommutativity. On the other hand, UV/IR mixing is suppressed in non-Fermi liquid states due to UV insensitivity, and the patch description is valid. 1. Thermodynamic properties of noninteracting quantum gases with spin-orbit coupling Energy Technology Data Exchange (ETDEWEB) He Li [Jiangsu University of Science and Technology, Zhangjiagang, Jiangsu, 215600 (China); Yu Zengqiang [Institute for Advanced Study, Tsinghua University, Beijing, 100084 (China) 2011-08-15 In this brief report we study thermodynamic properties of noninteracting quantum gases with isotropic spin-orbit coupling. At high temperature, coefficients of virial expansion depend on both temperature T and spin-orbit coupling strength {kappa}. For strong coupling, virial expansion is applicable to the temperature region below the conventional degenerate temperature T{sub F}. At low temperature, specific heat is proportional to {radical}(T) in Bose gases and T in Fermi gases. Temperature dependence of the chemical potential of fermions shows a different behavior when the Fermi surface is above and below the Dirac point. 2. Berry Fermi liquid theory Science.gov (United States) Chen, Jing-Yuan; Son, Dam Thanh 2017-02-01 We develop an extension of the Landau Fermi liquid theory to systems of interacting fermions with non-trivial Berry curvature. We propose a kinetic equation and a constitutive relation for the electromagnetic current that together encode the linear response of such systems to external electromagnetic perturbations, to leading and next-to-leading orders in the expansion over the frequency and wave number of the perturbations. We analyze the Feynman diagrams in a large class of interacting quantum field theories and show that, after summing up all orders in perturbation theory, the current-current correlator exactly matches with the result obtained from the kinetic theory. 3. Spin-Seebeck effect in a strongly interacting Fermi gas NARCIS (Netherlands) Wong, C.H.; Stoof, H.T.C.; Duine, R.A. 2012-01-01 We study the spin-Seebeck effect in a strongly interacting, two-component Fermi gas and propose an experiment to measure this effect by relatively displacing spin-up and spin-down atomic clouds in a trap using spin-dependent temperature gradients. We compute the spin-Seebeck coefficient and related 4. New physics of metals: fermi surfaces without Fermi liquids. OpenAIRE Anderson, P W 1995-01-01 I relate the historic successes, and present difficulties, of the renormalized quasiparticle theory of metals ("AGD" or Fermi liquid theory). I then describe the best-understood example of a non-Fermi liquid, the normal metallic state of the cuprate superconductors. 5. Attachment of Surface "Fermi Arcs" to the Bulk Fermi Surface: "Fermi-Level Plumbing" in Topological Metals OpenAIRE Haldane, F. D. M. 2014-01-01 The role of "Fermi arc" surface-quasiparticle states in "topological metals" (where some Fermi surface sheets have non-zero Chern number) is examined. They act as "Fermi-level plumbing" conduits that transfer quasiparticles among groups of apparently-disconnected Fermi sheets with non-zero Chern numbers to maintain equality of their chemical potentials, which is required by gauge invariance. Fermi arcs have a chiral tangential attachment to the surface projections of sheets of the bulk Fermi ... 6. Statistical mechanics of a Feshbach-coupled Bose-Fermi gas in an optical lattice DEFF Research Database (Denmark) Sørensen, Ole Søe; Nygaard, Nicolai; Blakie, P.B. 2009-01-01 We consider an atomic Fermi gas confined in a uniform optical lattice potential, where the atoms can pair into molecules via a magnetic-field-controlled narrow Feshbach resonance. The phase diagram of the resulting atom-molecule mixture in chemical and thermal equilibria is determined numerically... 7. Peltier cooling of fermionic quantum gases. Science.gov (United States) Grenier, Ch; Georges, A; Kollath, C 2014-11-14 We propose a cooling scheme for fermionic quantum gases, based on the principles of the Peltier thermoelectric effect and energy filtering. The system to be cooled is connected to another harmonically trapped gas acting as a reservoir. The cooling is achieved by two simultaneous processes: (i) the system is evaporatively cooled, and (ii) cold fermions from deep below the Fermi surface of the reservoir are injected below the Fermi level of the system, in order to fill the "holes" in the energy distribution. This is achieved by a suitable energy dependence of the transmission coefficient connecting the system to the reservoir. The two processes can be viewed as simultaneous evaporative cooling of particles and holes. We show that both a significantly lower entropy per particle and faster cooling rate can be achieved in this way than by using only evaporative cooling. 8. Peltier Cooling of Fermionic Quantum Gases Science.gov (United States) Grenier, Ch.; Georges, A.; Kollath, C. 2014-11-01 We propose a cooling scheme for fermionic quantum gases, based on the principles of the Peltier thermoelectric effect and energy filtering. The system to be cooled is connected to another harmonically trapped gas acting as a reservoir. The cooling is achieved by two simultaneous processes: (i) the system is evaporatively cooled, and (ii) cold fermions from deep below the Fermi surface of the reservoir are injected below the Fermi level of the system, in order to fill the "holes" in the energy distribution. This is achieved by a suitable energy dependence of the transmission coefficient connecting the system to the reservoir. The two processes can be viewed as simultaneous evaporative cooling of particles and holes. We show that both a significantly lower entropy per particle and faster cooling rate can be achieved in this way than by using only evaporative cooling. 9. Composite-fermionization of the mixture composed of Tonks gas and Fermi gas Institute of Scientific and Technical Information of China (English) Hao Ya-Jiang 2011-01-01 This paper investigates the ground-state properties of the mixture composed of the strongly interacting TonksGirardeau gas and spin polarized Fermi gas confined in one-dimensional harmonic traps, where the interaction between the Bose atoms and Fermi atoms is tunable. With a generalized Bose-Fermi transformation the mixture is mapped into a two-component Fermi gas. The homogeneous Fermi gas is exactly solvable by the Bethe-ansatz method and the ground state energy density can be obtained. Combining the ground-state energy function of the homogeneous system with local density approximation it obtains the ground-state density distributions of inhomogeneous mixture. It is shown that with the increase in boson-fermion interaction, the system exhibits composite-fermionization crossover. 10. FERMI multi-chip module CERN Multimedia This FERMI multi-chip module contains five million transistors. 25 000 of these modules will handle the flood of information through parts of the ATLAS and CMS detectors at the LHC. To select interesting events for recording, crucial decisions are taken before the data leaves the detector. FERMI modules are being developed at CERN in partnership with European industry. 11. Fermi Communications and Public Outreach CERN Document Server Cominsky, L 2015-01-01 The Sonoma State University (SSU) Education and Public Outreach (E/PO) group participates in the planning and execution of press conferences that feature noteworthy Fermi discoveries, as well as supporting social media and outreach websites. We have also created many scientific illustrations for the media, tools for amateur astronomers for use at star parties, and have given numerous public talks about Fermi discoveries. 12. Superconductivity in an electron band just above the Fermi level: possible route to BCS-BEC superconductivity. Science.gov (United States) Okazaki, K; Ito, Y; Ota, Y; Kotani, Y; Shimojima, T; Kiss, T; Watanabe, S; Chen, C-T; Niitaka, S; Hanaguri, T; Takagi, H; Chainani, A; Shin, S 2014-02-28 Conventional superconductivity follows Bardeen-Cooper-Schrieffer(BCS) theory of electrons-pairing in momentum-space, while superfluidity is the Bose-Einstein condensation(BEC) of atoms paired in real-space. These properties of solid metals and ultra-cold gases, respectively, are connected by the BCS-BEC crossover. Here we investigate the band dispersions in FeTe(0.6)Se(0.4)(Tc = 14.5 K ~ 1.2 meV) in an accessible range below and above the Fermi level(EF) using ultra-high resolution laser angle-resolved photoemission spectroscopy. We uncover an electron band lying just 0.7 meV (~8 K) above EF at the Γ-point, which shows a sharp superconducting coherence peak with gap formation below Tc. The estimated superconducting gap Δ and Fermi energy [Symbol: see text]F indicate composite superconductivity in an iron-based superconductor, consisting of strong-coupling BEC in the electron band and weak-coupling BCS-like superconductivity in the hole band. The study identifies the possible route to BCS-BEC superconductivity. 13. Enrico Fermi Symposium at CERN : opening celebration CERN Document Server CERN. Geneva. Audiovisual Unit 2002-01-01 You are cordially invited to the opening celebration on Thursday 12 September at 16:00 (Main Building, Council Chamber), which will include speechs from: Luciano Maiani - Welcome and Introduction Antonino Zichichi - The New 'Centro Enrico Fermi' at Via Panisperna Ugo Amaldi - Fermi at Via Panisperna and the birth of Nuclear Medicine Jack Steinberger - Fermi in Chicago Valentin Telegdi - A Close-up of Fermi Arnaldo Stefanini - Celebrating Fermi's Centenary in Documents and Pictures and the screening of a documentary video about Fermi: Scienziati a Pisa: Enrico Fermi (Scientists at Pisa: Enrico Fermi) created by Francesco Andreotti for La Limonaia from early film, photographs and sound recordings (English version - c. 30 mins). 14. Fermi, Heisenberg y Lawrence Directory of Open Access Journals (Sweden) Ynduráin, Francisco J. 2002-01-01 Full Text Available Not available Los azares de las onomásticas hacen coincidir en este año el centenario del nacimiento de tres de los más grandes físicos del siglo XX. Dos de ellos, Fermi y Heisenberg, dejaron una marca fundamental en la ciencia (ambos, pero sobre todo el segundo y, el primero, también en la tecnología. Lawrence, indudablemente de un nivel inferior al de los otros dos, estuvo sin embargo en el origen de uno de los desarrollos tecnológicos que han sido básicos para la exploración del universo subnuclear en la segunda mitad del siglo que ha terminado hace poco, el de los aceleradores de partículas. 15. Momentum sharing in imbalanced Fermi systems CERN Document Server Hen, O; Weinstein, L B; Piasetzky, E; Hakobyan, H; Higinbotham, D W; Braverman, M; Brooks, W K; Gilad, S; Adhikari, K P; Arrington, J; Asryan, G; Avakian, H; Ball, J; Baltzell, N A; Battaglieri, M; Beck, A; Beck, S May-Tal; Bedlinskiy, I; Bertozzi, W; Biselli, A; Burkert, V D; Cao, T; Carman, D S; Celentano, A; Chandavar, S; Colaneri, L; Cole, P L; Crede, V; DAngelo, A; De Vita, R; Deur, A; Djalali, C; Doughty, D; Dugger, M; Dupre, R; Egiyan, H; Alaoui, A El; Fassi, L El; Elouadrhiri, L; Fedotov, G; Fegan, S; Forest, T; Garillon, B; Garcon, M; Gevorgyan, N; Ghandilyan, Y; Gilfoyle, G P; Girod, F X; Goetz, J T; Gothe, R W; Griffioen, K A; Guidal, M; Guo, L; Hafidi, K; Hanretty, C; Hattawy, M; Hicks, K; Holtrop, M; Hyde, C E; Ilieva, Y; Ireland, D G; Ishkanov, B I; Isupov, E L; Jiang, H; Jo, H S; Joo, K; Keller, D; Khandaker, M; Kim, A; Kim, W; Klein, F J; Koirala, S; Korover, I; Kuhn, S E; Kubarovsky, V; Lenisa, P; Levine, W I; Livingston, K; Lowry, M; Lu, H Y; MacGregor, I J D; Markov, N; Mayer, M; McKinnon, B; Mineeva, T; Mokeev, V; Movsisyan, A; Camacho, C Munoz; Mustapha, B; Nadel-Turonski, P; Niccolai, S; Niculescu, G; Niculescu, I; Osipenko, M; Pappalardo, L L; Paremuzyan, R; Park, K; Pasyuk, E; Phelps, W; Pisano, S; Pogorelko, O; Price, J W; Procureur, S; Prok, Y; Protopopescu, D; Puckett, A J R; Rimal, D; Ripani, M; Ritchie, B G; Rizzo, A; Rosner, G; Rossi, P; Roy, P; Sabatie, F; Schott, D; Schumacher, R A; Sharabian, Y G; Smith, G D; Shneor, R; Sokhan, D; Stepanyan, S S; Stepanyan, S; Stoler, P; Strauch, S; Sytnik, V; Taiuti, M; Tkachenko, S; Ungaro, M; Vlassov, A V; Voutier, E; Watts, D; Walford, N K; Wei, X; Wood, M H; Wood, S A; Zachariou, N; Zana, L; Zhao, Z W; Zheng, X; Zonta, I 2014-01-01 The atomic nucleus is composed of two different kinds of fermions, protons and neutrons. If the protons and neutrons did not interact, the Pauli exclusion principle would force the majority fermions (usually neutrons) to have a higher average momentum. Our high-energy electron scattering measurements using 12C, 27Al, 56Fe and 208Pb targets show that, even in heavy neutron-rich nuclei, short-range interactions between the fermions form correlated high-momentum neutron-proton pairs. Thus, in neutron-rich nuclei, protons have a greater probability than neutrons to have momentum greater than the Fermi momentum. This finding has implications ranging from nuclear few body systems to neutron stars and may also be observable experimentally in two-spin state, ultra-cold atomic gas systems. 16. Density Functional Theory Studies of Magnetically Confined Fermi Gas Institute of Scientific and Technical Information of China (English) 陈宇俊; 马红孺 2001-01-01 A theory is developed for magnetically confined Fermi gas at a low temperature based on the density functional theory. The theory is illustrated by the numerical calculation of the density distributions of Fermi atoms 40K with parameters according to DeMarco and Jin's experiment [Science, 285(1999)1703]. Our results are in close agreement with the experiment. To check the theory, we also performed calculations using our theory at a high temperature, which compared very well to the results of the classical limit. 17. Nanoclusters and Microparticles in Gases and Vapors CERN Document Server Smirnov, Boris M 2012-01-01 Research of processes involving Nanoclusters and Microparticleshas been developing fastin many fields of rescent research, in particular in materials science. To stay at the cutting edge of this development, a sound understanding of the processes is needed. In this work, several processes involving small particles are described, such as transport processes in gases, charging of small particles in gases, chemical processes, atom attachment and quenching of excited atomic particles on surfaces, nucleation, coagulation, coalescence and growth processes for particles and aggregates. This work pres 18. Anisotropic non-Fermi liquids Science.gov (United States) Sur, Shouvik; Lee, Sung-Sik 2016-11-01 We study non-Fermi-liquid states that arise at the quantum critical points associated with the spin density wave (SDW) and charge density wave (CDW) transitions in metals with twofold rotational symmetry. We use the dimensional regularization scheme, where a one-dimensional Fermi surface is embedded in (3 -ɛ ) -dimensional momentum space. In three dimensions, quasilocal marginal Fermi liquids arise both at the SDW and CDW critical points: the speed of the collective mode along the ordering wave vector is logarithmically renormalized to zero compared to that of Fermi velocity. Below three dimensions, however, the SDW and CDW critical points exhibit drastically different behaviors. At the SDW critical point, a stable anisotropic non-Fermi-liquid state is realized for small ɛ , where not only time but also different spatial coordinates develop distinct anomalous dimensions. The non-Fermi liquid exhibits an emergent algebraic nesting as the patches of Fermi surface are deformed into a universal power-law shape near the hot spots. Due to the anisotropic scaling, the energy of incoherent spin fluctuations disperse with different power laws in different momentum directions. At the CDW critical point, on the other hand, the perturbative expansion breaks down immediately below three dimensions as the interaction renormalizes the speed of charge fluctuations to zero within a finite renormalization group scale through a two-loop effect. The difference originates from the fact that the vertex correction antiscreens the coupling at the SDW critical point whereas it screens at the CDW critical point. Science.gov (United States) Gray, Robert H 2015-03-01 The so-called Fermi paradox claims that if technological life existed anywhere else, we would see evidence of its visits to Earth--and since we do not, such life does not exist, or some special explanation is needed. Enrico Fermi, however, never published anything on this topic. On the one occasion he is known to have mentioned it, he asked "Where is everybody?"--apparently suggesting that we do not see extraterrestrials on Earth because interstellar travel may not be feasible, but not suggesting that intelligent extraterrestrial life does not exist or suggesting its absence is paradoxical. The claim "they are not here; therefore they do not exist" was first published by Michael Hart, claiming that interstellar travel and colonization of the Galaxy would be inevitable if intelligent extraterrestrial life existed, and taking its absence here as proof that it does not exist anywhere. The Fermi paradox appears to originate in Hart's argument, not Fermi's question. Clarifying the origin of these ideas is important, because the Fermi paradox is seen by some as an authoritative objection to searching for evidence of extraterrestrial intelligence--cited in the U.S. Congress as a reason for killing NASA's SETI program on one occasion. But evidence indicates that it misrepresents Fermi's views, misappropriates his authority, deprives the actual authors of credit, and is not a valid paradox. 20. Fermi pulsar revolution CERN Document Server Caraveo, Patrizia A 2010-01-01 2009 has been an extraordinary year for gamma-ray pulsar astronomy and 2010 promises to be equally good. Not only have we registered an extraordinary increase in the number of pulsars detected in gamma rays, but we have also witnessed the birth of new sub-families: first of all, the radio-quiet gamma pulsars and later an ever growing number of millisecond pulsars, a real surprise. We started with a sample of 7 gamma-ray emitting neutron stars (6 radio pulsars and Geminga) and now the Fermi-LAT harvest encompasses 24 "Geminga-like" new gamma-ray pulsars, a dozen millisecond pulsars and about thirty radio pulsars. Moreover, radio searches targeted to LAT unidentified sources yielded 18 new radio millisecond pulsars, several of which have been already detected also in gamma rays. Thus, currently the family of gamma-ray emitting neutron stars seems to be evenly divided between classical radio pulsars, millisecond pulsars and radio quiet neutron stars. 1. Kinetic equation for strongly interacting dense Fermi systems CERN Document Server Lipavsky, P; Spicka, V 2001-01-01 We review the non-relativistic Green's-function approach to the kinetic equations for Fermi liquids far from equilibrium. The emphasis is on the consistent treatment of the off-shell motion between collisions and on the non-instant and non-local picture of binary collisions. The resulting kinetic equation is of the Boltzmann type, and it represents an interpolation between the theory of transport in metals and the theory of moderately dense gases. The free motion of particles is renormalised by various mean field and mass corrections in the spirit of Landau's quasiparticles in metals. The collisions are non-local in the spirit of Enskog's theory of non-ideal gases. The collisions are moreover non-instant, a feature which is absent in the theory of gases, but which is shown to be important for dense Fermi systems. In spite of its formal complexity, the presented theory has a simple implementation within the Monte-Carlo simulation schemes. Applications in nuclear physics are given for heavy-ion reactions and th... 2. Atom Skimmers and Atom Lasers Utilizing Them Science.gov (United States) Hulet, Randall; Tollett, Jeff; Franke, Kurt; Moss, Steve; Sackett, Charles; Gerton, Jordan; Ghaffari, Bita; McAlexander, W.; Strecker, K.; Homan, D. 2005-01-01 Atom skimmers are devices that act as low-pass velocity filters for atoms in thermal atomic beams. An atom skimmer operating in conjunction with a suitable thermal atomic-beam source (e.g., an oven in which cesium is heated) can serve as a source of slow atoms for a magneto-optical trap or other apparatus in an atomic-physics experiment. Phenomena that are studied in such apparatuses include Bose-Einstein condensation of atomic gases, spectra of trapped atoms, and collisions of slowly moving atoms. An atom skimmer includes a curved, low-thermal-conduction tube that leads from the outlet of a thermal atomic-beam source to the inlet of a magneto-optical trap or other device in which the selected low-velocity atoms are to be used. Permanent rare-earth magnets are placed around the tube in a yoke of high-magnetic-permeability material to establish a quadrupole or octupole magnetic field leading from the source to the trap. The atoms are attracted to the locus of minimum magnetic-field intensity in the middle of the tube, and the gradient of the magnetic field provides centripetal force that guides the atoms around the curve along the axis of the tube. The threshold velocity for guiding is dictated by the gradient of the magnetic field and the radius of curvature of the tube. Atoms moving at lesser velocities are successfully guided; faster atoms strike the tube wall and are lost from the beam. 3. Davisson-Germer Prize Talk: Many-Body Physics with Atomic Fermions Science.gov (United States) Hulet, Randall 2016-05-01 Ultracold atomic gases confined to optical lattices have proven to be highly versatile and tunable systems for realizing novel quantum states of matter. We are using Fermi gases of 6 Li atoms in our laboratory to explore several goals related to the strong correlations that arise in these systems. We have realized the Hubbard model, which has long been suspected of containing the essential ingredients of high temperature superconductivity. We measured the compressibility of the Mott insulating phase that occurs near half filling (1 atom/site), thus demonstrating the excitation gap of the Mott insulator. Progress in this field, however, has been hampered by an inability to cool to low enough temperatures to achieve the most ambitious goals. To address this problem, we have developed the compensated optical lattice method to enable evaporative cooling in the lattice. With this method, we have cooled the Mott insulator sufficiently far to observe short-range antiferromagnetic correlations using Bragg scattering of light. We are currently exploring new methods for entropy storage and redistribution to achieve even lower entropy in the antiferromagnetic phase. Motivated by the enhancement of quantum correlations in low dimensions, we are also exploring Fermi gases in quasi-one-dimension (1D). A deep 2D optical lattice produces an array of 1D tubes which can be weakly coupled by reducing the lattice depth, thus increasing the lattice hopping t between them. We observe a crossover from 1D-like to 3D-like behavior in the phase separation of a spin-imbalanced Fermi gas with increasing t. While this crossover occurs at a value of t that depends on interaction, we find that the crossover location is universally dependent upon the scaled hopping t /ɛb , where ɛb is the pair binding energy. Finally, I will also report progress on measuring the speed of sound of the charge and spin modes in a 1D Fermi gas. Work supported by an ARO MURI, NSF, and the Robert A Welch Foundation. 4. Relativistic Scott correction for atoms and molecules DEFF Research Database (Denmark) Solovej, Jan Philip; Sørensen, Thomas Østergaard; Spitzer, Wolfgang Ludwig 2010-01-01 We prove the first correction to the leading Thomas-Fermi energy for the ground state energy of atoms and molecules in a model where the kinetic energy of the electrons is treated relativistically. The leading Thomas-Fermi energy, established in [25], as well as the correction given here, are of ... 5. Gases in molten salts CERN Document Server Tomkins, RPT 1991-01-01 This volume contains tabulated collections and critical evaluations of original data for the solubility of gases in molten salts, gathered from chemical literature through to the end of 1989. Within the volume, material is arranged according to the individual gas. The gases include hydrogen halides, inert gases, oxygen, nitrogen, hydrogen, carbon dioxide, water vapor and halogens. The molten salts consist of single salts, binary mixtures and multicomponent systems. Included also, is a special section on the solubility of gases in molten silicate systems, focussing on slags and fluxes. 6. Handbook of purified gases CERN Document Server Schoen, Helmut 2015-01-01 Technical gases are used in almost every field of industry, science and medicine and also as a means of control by government authorities and institutions and are regarded as indispensable means of assistance. In this complete handbook of purified gases the physical foundations of purified gases and mixtures as well as their manufacturing, purification, analysis, storage, handling and transport are presented in a comprehensive way. This important reference work is accompanied with a large number of Data Sheets dedicated to the most important purified gases. 7. Polarons and molecules in a two-dimensional Fermi gas DEFF Research Database (Denmark) Zöllner, Sascha; Bruun, Georg Morten; Pethick, C. J. 2011-01-01 We study an impurity atom in a two-dimensional Fermi gas using variational wave functions for (i) an impurity dressed by particle-hole excitations (polaron) and (ii) a dimer consisting of the impurity and a majority atom. In contrast to three dimensions, where similar calculations predict a sharp...... transition to a dimer state with increasing interspecies attraction, we show that the polaron Ansatz always gives a lower energy. However, the exact solution for a heavy impurity reveals that both a two-body bound state and distortions of the Fermi sea are crucial. This reflects the importance of particle......-hole pairs in lower dimensions and makes simple variational calculations unreliable. We show that the energy of an impurity gives important information about its dressing cloud, for which both Ansätze give inaccurate results.... 8. Quantum dynamics of impurities coupled to a Fermi sea Science.gov (United States) Parish, Meera M.; Levinsen, Jesper 2016-11-01 We consider the dynamics of an impurity atom immersed in an ideal Fermi gas at zero temperature. We focus on the coherent quantum evolution of the impurity following a quench to strong impurity-fermion interactions, where the interactions are assumed to be short range like in cold-atom experiments. To approximately model the many-body time evolution, we use a truncated basis method, where at most two particle-hole excitations of the Fermi sea are included. When the system is initially noninteracting, we show that our method exactly captures the short-time dynamics following the quench, and we find that the overlap between initial and final states displays a universal nonanalytic dependence on time in this limit. We further demonstrate how our method can be used to compute the impurity spectral function, as well as describe many-body phenomena involving coupled impurity spin states, such as Rabi oscillations in a medium or highly engineered quantum quenches. 9. An interpolatory ansatz captures the physics of one-dimensional confined Fermi systems. Science.gov (United States) Andersen, M E S; Dehkharghani, A S; Volosniev, A G; Lindgren, E J; Zinner, N T 2016-01-01 Interacting one-dimensional quantum systems play a pivotal role in physics. Exact solutions can be obtained for the homogeneous case using the Bethe ansatz and bosonisation techniques. However, these approaches are not applicable when external confinement is present. Recent theoretical advances beyond the Bethe ansatz and bosonisation allow us to predict the behaviour of one-dimensional confined systems with strong short-range interactions, and new experiments with cold atomic Fermi gases have already confirmed these theories. Here we demonstrate that a simple linear combination of the strongly interacting solution with the well-known solution in the limit of vanishing interactions provides a simple and accurate description of the system for all values of the interaction strength. This indicates that one can indeed capture the physics of confined one-dimensional systems by knowledge of the limits using wave functions that are much easier to handle than the output of typical numerical approaches. We demonstrate our scheme for experimentally relevant systems with up to six particles. Moreover, we show that our method works also in the case of mixed systems of particles with different masses. This is an important feature because these systems are known to be non-integrable and thus not solvable by the Bethe ansatz technique. 10. Conjugate Fermi holes and its manifestation in He-like systems Energy Technology Data Exchange (ETDEWEB) Sako, Tokuei, E-mail: [email protected] [Laboratory of Physics, College of Science and Technology, Nihon University, 7-24-1 Narashinodai, Funabashi, 274-8501 Chiba (Japan) 2015-12-31 The structure of genuine and conjugate Fermi holes in two-electron atomic systems, namely He and He-like atomic ions, has been studied relying on accurate full configuration interaction wave functions. The standard Fermi hole exists in the vicinity of region in the two-electron coordinate space satisfying the well-known condition, r{sub 1} = r{sub 2}, while the conjugate Fermi hole exists in the vicinity of region close to this genuine Fermi hole but satisfying r{sub 1} ≠ r{sub 2} instead of r{sub 1} = r{sub 2}. Existence of these holes has shown to give an insightful interpretation of the origin of the first Hund rule and of the anomalously strong angular correlation manifested in the series of the singlet-triplet pair of singly-excited states of the aforementioned systems. 11. Capturing Gases in Carbon Honeycomb Science.gov (United States) Krainyukova, Nina V. 2016-12-01 In our recent paper (Krainyukova and Zubarev in Phys Rev Lett 116:055501, 2016. doi: 10.1103/PhysRevLett.116.055501) we reported the observation of an exceptionally stable honeycomb carbon allotrope obtained by deposition of vacuum-sublimated graphite. A family of structures can be built from absolutely dominant {sp}2 -bonded carbon atoms, and may be considered as three-dimensional graphene. Such structures demonstrate high absorption capacity for gases and liquids. In this work we show that the formation of honeycomb structures is highly sensitive to the carbon evaporation temperature and deposition rates. Both parameters are controlled by the electric current flowing through thin carbon rods. Two distinctly different regimes were found. At lower electric currents almost pure honeycomb structures form owing to sublimation. At higher currents the surface-to-bulk rod melting is observed. In the latter case densification of the carbon structures and a large contribution of glassy graphite emerge. The experimental diffraction patterns from honeycomb structures filled with absorbed gases and analyzed by the advanced method are consistent with the proposed models for composites which are different for Ar, Kr and Xe atoms in carbon matrices. 12. Capturing Gases in Carbon Honeycomb Science.gov (United States) Krainyukova, Nina V. 2017-04-01 In our recent paper (Krainyukova and Zubarev in Phys Rev Lett 116:055501, 2016. doi: 10.1103/PhysRevLett.116.055501) we reported the observation of an exceptionally stable honeycomb carbon allotrope obtained by deposition of vacuum-sublimated graphite. A family of structures can be built from absolutely dominant {sp}2-bonded carbon atoms, and may be considered as three-dimensional graphene. Such structures demonstrate high absorption capacity for gases and liquids. In this work we show that the formation of honeycomb structures is highly sensitive to the carbon evaporation temperature and deposition rates. Both parameters are controlled by the electric current flowing through thin carbon rods. Two distinctly different regimes were found. At lower electric currents almost pure honeycomb structures form owing to sublimation. At higher currents the surface-to-bulk rod melting is observed. In the latter case densification of the carbon structures and a large contribution of glassy graphite emerge. The experimental diffraction patterns from honeycomb structures filled with absorbed gases and analyzed by the advanced method are consistent with the proposed models for composites which are different for Ar, Kr and Xe atoms in carbon matrices. 13. The Fermiac or Fermi's Trolley Science.gov (United States) Coccetti, F. 2016-03-01 The Fermiac, known also as Fermi's trolley or Monte Carlo trolley, is an analog computer used to determine the change in time of the neutron population in a nuclear device, via the Monte Carlo method. It was invented by Enrico Fermi and constructed by Percy King at Los Alamos in 1947, and used for about two years. A replica of the Fermiac was built at INFN mechanical workshops of Bologna in 2015, on behalf of the Museo Storico della Fisica e Centro Studi e Ricerche "Enrico Fermi", thanks to the original drawings made available by Los Alamos National Laboratory (LANL). This reproduction of the Fermiac was put in use, and a simulation was developed. 14. Dynamic structure factors and sum rules in two-component quantum gases with spin-orbit coupling%自旋-轨道耦合作用下双组分量子气体中的动力学结构因子与求和规则 Institute of Scientific and Technical Information of China (English) 贺丽; 余增强 2016-01-01 Sum rules for the dynamic structure factors are powerful tools to explore the collective behaviors in many-body systems at zero temperature as well as at finite temperatures. The recent remarkable realization of synthetic spin-orbit (SO) coupling in quantum gases is opening up new perspective to study the intriguing SO effects with ultracold atoms. So far, a specific type of SO coupling, which is generated by a pair of Raman laser beams, has been experimentally achieved in Bose-Einstein condensates of 87Rb and degenerate Fermi gases of 40K and 6Li. In the presence of SO coupling, the dynamic structure factors for the density fluctuation and spin fluctuation satisfy different sum rules. In particular, in the two-component quantum gases with inter-species Raman coupling, the f-sum rule for the spin fluctuation has an additional term proportional to the transverse spin polarization. Due to the coupling between the momentum and spin, the first moment of the dynamic structure factor does not necessarily possess the inversion symmetry, which is in strong contrast to the conventional system without SO coupling. Such an asymmetric behavior could be observed in both Fermi gases and Bose gases with Raman coupling. As a demonstration, we focus on the uniform case at zero temperature in this work. For the non-interacting Fermi gases, the asymmetric first moment appears only when the Raman detuning is finite. The asymmetric amplitude is quite limited, and it vanishes at both zero detuning and infinite detuning. For the weakly interacting Bose gases, the first moment is asymmetric in momentum space even at zero detuning, when the ground state spontaneously breaks the Z2 symmetry in the plane-wave condensation phase. Using the Bogoliubov method, the dynamic structure factor and its first moment are explicitly calculated for various interaction parameters. We find that the asymmetric behavior in the spin channel could be much more significant than in the density channel, and the CERN Document Server Gray, Robert H 2016-01-01 The so-called Fermi paradox claims that if technological life existed anywhere else, we would see evidence of its visits to Earth-and since we do not, such life does not exist, or some special explanation is needed. Enrico Fermi, however, never published anything on this topic. On the one occasion he is known to have mentioned it, he asked 'where is everybody?'- apparently suggesting that we don't see extraterrestrials on Earth because interstellar travel may not be feasible, but not suggesting that intelligent extraterrestrial life does not exist, or suggesting its absence is paradoxical. The claim 'they are not here; therefore they do not exist' was first published by Michael Hart, claiming that interstellar travel and colonization of the galaxy would be inevitable if intelligent extraterrestrial life existed, and taking its absence here as proof that it does not exist anywhere. The Fermi paradox appears to originate in Hart's argument, not Fermi's question. Clarifying the origin of these ideas is important... 16. Holographic optical traps for atom-based topological Kondo devices Science.gov (United States) Buccheri, F.; Bruce, G. D.; Trombettoni, A.; Cassettari, D.; Babujian, H.; Korepin, V. E.; Sodano, P. 2016-07-01 The topological Kondo (TK) model has been proposed in solid-state quantum devices as a way to realize non-Fermi liquid behaviors in a controllable setting. Another motivation behind the TK model proposal is the demand to demonstrate the quantum dynamical properties of Majorana fermions, which are at the heart of their potential use in topological quantum computation. Here we consider a junction of crossed Tonks-Girardeau gases arranged in a star-geometry (forming a Y-junction), and we perform a theoretical analysis of this system showing that it provides a physical realization of the TK model in the realm of cold atom systems. Using computer-generated holography, we experimentally implement a Y-junction suitable for atom trapping, with controllable and independent parameters. The junction and the transverse size of the atom waveguides are of the order of 5 μm, leading to favorable estimates for the Kondo temperature and for the coupling across the junction. Since our results show that all the required theoretical and experimental ingredients are available, this provides the demonstration of an ultracold atom device that may in principle exhibit the TK effect. 17. On the Dynamics of the Fermi-Bose model DEFF Research Database (Denmark) Ögren, Magnus In this talk we formulate and prove results for the exponential matrix representing the dynamics of the Fermi-Bose model in an undepleted bosonic field approximation. A recent application of this model is molecular dimmers dissociating into its atomic compounds. The problem is solved in D spatial....... In particular the results can be used for studies of threedimensional physical systems of arbitrary geometry. We illustrate the generality of our approach by giving numerical results for the dynamics of Glauber type atomic pair correlation functions for a non-isotropic three-dimensional harmonically trapped... 18. Many-electron tunneling in atoms CERN Document Server Zon, B A 1999-01-01 A theoretical derivation is given for the formula describing N-electron ionization of atom by a dc field and laser radiation in tunneling regime. Numerical examples are presented for noble gases atoms. 19. Relationship between Fermi Resonance and Solvent Effects Institute of Scientific and Technical Information of China (English) JIANG Xiu-Lan; LI Dong-Fei; SUN Cheng-Lin; LI Zhan-Long; YANG Guang; ZHOU Mi; LI Zuo-Wei; GAO Shu-Qin 2011-01-01 We theoretically and experimentally study the relationship between Fermi resonance and solvent effects and investigate the Fermi resonance of p-benzoquinone and cyclopentanone in different solvents and the Fermi resonance of CS2 in C6H6 at different concentrations. Also, we investigate the Fermi resonance of C6H6 and CCl4 in their solution at different pressures. It is found that solvent effects can be utilized to search Fermi resonance parameters such as coupling coefficient and spectral intensity ratio, etc., on the other hand, the mechanism of solvent effects can be revealed according to Fermi resonance at high pressure.%@@ We theoretically and experimentally study the relationship between Fermi resonance and solvent effects and investigate the Fermi resonance of p-benzoquinone and cyclopentanone in different solvents and the Fermi resonance of CS2 in C6H6 at different concentrations.Also,we investigate the Fermi resonance of C6H6 and CCl4 in their solution at different pressures.It is found that solvent effects can be utilized to search Fermi resonance parameters such as coupling coefficient and spectral intensity ratio,etc.,on the other hand,the mechanism of solvent effects can be revealed according to Fermi resonance at high pressure. 20. STEM education and Fermi problems Science.gov (United States) Holubova, Renata 2017-01-01 One of the research areas of Physics education is the study of the educational process. Investigations in this area are aimed for example on the teaching and learning process and its results. The conception of STEM education (Science, Technology, Engineering, and Mathematics) is discussed - it is one possible approach to the preparation of the curriculum and the focus on the educational process at basic and secondary schools. At schools in the Czech Republic STEM is much more realized by the application of interdisciplinary relations between subjects Physics-Nature-Technique. In both conceptions the aim is to support pupils' creativity, critical thinking, cross-curricular links. In this context the possibility of using Fermi problems in teaching Physics was discussed (as an interdisciplinary and constructivist activity). The aim of our research was the analysis of Fermi problems solving strategies, the ability of pupils to solve Fermi problems. The outcome of our analysis was to find out methods and teaching strategies which are important to use in teaching - how to solve qualitative and interdisciplinary tasks in physics. In this paper the theoretical basis of STEM education and Fermi problems will be presented. The outcome of our findings based on the research activities will be discussed so as our experiences from 10 years of Fermi problems competition that takes place at the Science Faculty, Palacky University in Olomouc. Changes in competencies of solving tasks by our students (from the point of view in terms of modern, activating teaching methods recommended by theory of Physics education and other science subjects) will be identified. 1. Method and system for continuous atomic layer deposition Energy Technology Data Exchange (ETDEWEB) Elam, Jeffrey W.; Yanguas-Gil, Angel; Libera, Joseph A. 2017-03-21 A system and method for continuous atomic layer deposition. The system and method includes a housing, a moving bed which passes through the housing, a plurality of precursor gases and associated input ports and the amount of precursor gases, position of the input ports, and relative velocity of the moving bed and carrier gases enabling exhaustion of the precursor gases at available reaction sites. 2. Gases, liquids and solids CERN Document Server Tabor, David 1969-01-01 It has been tradional to treat gases, liquids and solids as if they were completely unrelated material. However, this book shows that many of their bulk properties can been explained in terms of intermolecular forces. 3. Interaction of atomic oxygen with a graphite surface Science.gov (United States) Mateljevic, Natasa This project was a part of the Multi University Research Initiative (MURI) Center for Materials Chemistry in the Space Environment which seeks to develop a quantitative and predictive understanding of how materials degrade or become passivated in the space environment. This is a critical research area for the Department of Defense (DoD) and National Aeronautics and Space Administration (NASA) given the large and increasing dependence on satellites and manned spacecrafts that reside in, or pass through, the low-Earth orbit (LEO) space environment. In this work, we completed three separate projects. First, we carried out ab initio electronic structure studies of the interaction of oxygen atoms with graphite surfaces. The (O3 P) ground state of oxygen interacts weakly with the graphite surface while the excited (O1D) state interacts more strongly with a binding energy sufficient for a high coverage of oxygen to be maintained on the surface. Thus, it requires a transition from O(3P) to O(1D) in order for oxygen to strongly bind. Since graphite is a semi-metal, it requires a vanishingly small energy to remove an electron of up spin from just below the Fermi level, and replace it with a down spin electron just above the Fermi level; spin-orbit interaction is not required to switch the state of the oxygen atom. We have examined this complexity for the first time and developed guidelines for properly describing chemical reactivity on graphite surfaces. The second project is a kinetic Monte Carlo study of the erosion of graphite by energetic oxygen atoms in LEO and in the laboratory. These simulations, in conjunction with experiments by our MURI collaborators, reveal new insights about reaction pathways. Finally, we have developed a new model for accommodation of energy and momentum in collisions of gases with highly corrugated surfaces. This model promises to be valuable in simulating frictional heating and drag of objects moving through the atmosphere. 4. Fermi resonance in optical microcavities Science.gov (United States) Yi, Chang-Hwan; Yu, Hyeon-Hye; Lee, Ji-Won; Kim, Chil-Min 2015-04-01 Fermi resonance is a phenomenon of quantum mechanical superposition, which most often occurs between normal and overtone modes in molecular systems that are nearly coincident in energy. We find that scarred resonances in deformed dielectric microcavities are the very phenomenon of Fermi resonance, that is, a pair of quasinormal modes interact with each other due to coupling and a pair of resonances are generated through an avoided resonance crossing. Then the quantum number difference of a pair of quasinormal modes, which is a consequence of quantum mechanical superposition, equals periodic orbits, whereby the resonances are localized on the periodic orbits. We derive the relation between the quantum number difference and the periodic orbits and confirm it in an elliptic, a rectangular, and a stadium-shaped dielectric microcavity. 5. On the Fermi Golden Rule DEFF Research Database (Denmark) Jensen, Arne; Nenciu, Gheorghe 2008-01-01 We review and further develop the framework in [9] of the stationary theory of resonances, arising by perturbation of either threshold, or embedded in the continuum, eigenvalues. While in [9] only non/degenerate eigenvalues were considered, here we add some results for the degenerate case. [9] A........ Jensen and G. Nenciu, The Fermi Golden Rule and its form at thresholds in odd dimensions. Comm. Math. Phys 261 (2006), 693-727... 6. Imprisonment dynamics of resonance radiation in gases Energy Technology Data Exchange (ETDEWEB) Kosarev, N I; Shaparev, N Y, E-mail: [email protected] [Institute of Computational Modeling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, 660036 (Russian Federation) 2011-05-28 Imprisonment of resonant radiation in gases on the basis of the numerical solution of the rate balance equation for the excited atoms and the transfer resonant radiation equation is investigated. Calculations of the escape factor for the slab, cylinder and sphere at Doppler and Lorentz forms of absorption and scattering profiles are executed. Calculation results of the escape factor for the cylinder and slab are compared with Holsten's asymptotical solutions. The numerical data for time dependence of a spectrum, the intensity of resonant radiation and the spatial distribution of the excited atomic concentration in a regime of afterglow are also considered. 7. Enrico Fermi and the Dolomites CERN Document Server Battimelli, Giovanni 2014-01-01 Summer vacations in the Dolomites were a tradition among the professors of the Faculty of Mathematical and Physical Sciences at the University of Roma since the end of the XIX century. Beyond the academic walls, people like Tullio Levi-Civita, Federigo Enriques and Ugo Amaldi sr., together with their families, were meeting friends and colleagues in Cortina, San Vito, Dobbiaco, Vigo di Fassa and Selva, enjoying trekking together with scientific discussions. The tradition was transmitted to the next generations, in particular in the first half of the XX century, and the group of via Panisperna was directly connected: Edoardo Amaldi, the son of the mathematician Ugo sr., rented at least during two summers, in 1925 and in 1949, and in the winter of 1960, a house in San Vito di Cadore, and almost every year in the Dolomites; Enrico Fermi was a frequent guest. Many important steps in modern physics, in particular the development of the Fermi-Dirac statistics and the Fermi theory of beta decay, are related to scient... 8. Enrico Fermi and the Dolomites Energy Technology Data Exchange (ETDEWEB) Battimelli, Giovanni, E-mail: [email protected]; Angelis, Alessandro de, E-mail: [email protected] 2014-11-15 Summer vacations in the Dolomites were a tradition among the professors of the Faculty of Mathematical and Physical Sciences at the University of Roma since the end of the XIX century. Beyond the academic walls, people like Tullio Levi-Civita, Federigo Enriques and Ugo Amaldi sr., together with their families, were meeting friends and colleagues in Cortina, San Vito, Dobbiaco, Vigo di Fassa and Selva, enjoying trekking together with scientific discussions. The tradition was transmitted to the next generations, in particular in the first half of the XX century, and the group of via Panisperna was directly connected: Edoardo Amaldi, the son of the mathematician Ugo sr., rented at least during two summers, in 1925 and in 1949, and in the winter of 1960, a house in San Vito di Cadore, and almost every year in the Dolomites; Enrico Fermi was a frequent guest. Many important steps in modern physics, in particular the development of the Fermi-Dirac statistics and the Fermi theory of beta decay, are related to scientific discussions held in the region of the Dolomites. 9. Enrico Fermi and the Dolomites Science.gov (United States) Battimelli, Giovanni; de Angelis, Alessandro 2014-11-01 Summer vacations in the Dolomites were a tradition among the professors of the Faculty of Mathematical and Physical Sciences at the University of Roma since the end of the XIX century. Beyond the academic walls, people like Tullio Levi-Civita, Federigo Enriques and Ugo Amaldi sr., together with their families, were meeting friends and colleagues in Cortina, San Vito, Dobbiaco, Vigo di Fassa and Selva, enjoying trekking together with scientific discussions. The tradition was transmitted to the next generations, in particular in the first half of the XX century, and the group of via Panisperna was directly connected: Edoardo Amaldi, the son of the mathematician Ugo sr., rented at least during two summers, in 1925 and in 1949, and in the winter of 1960, a house in San Vito di Cadore, and almost every year in the Dolomites; Enrico Fermi was a frequent guest. Many important steps in modern physics, in particular the development of the Fermi-Dirac statistics and the Fermi theory of beta decay, are related to scientific discussions held in the region of the Dolomites. 10. Topological Non-Fermi Liquid CERN Document Server Cai, Rong-Gen; Wu, Yue-Liang; Zhang, Yun-Long 2016-01-01 In this paper we investigate the $(2+1)$-dimensional topological non-Fermi liquid in strongly correlated electron system, which has a holographic dual description by Einstein gravity in $(3+1)$-dimensional anti-de Sitter (AdS) space-time. In a dyonic Reissner-Nordstrom black hole background, we consider a Dirac fermion coupled to the background $U(1)$ gauge theory and an intrinsic chiral gauge field $b_M$ induced by chiral anomaly. UV retarded Green's function of the charged fermion in the UV boundary from AdS$_4$ gravity is calculated, by imposing in-falling wave condition at the horizon. We also obtain IR correlation function of the charged fermion at the IR boundary arising from the near horizon geometry of the topological black hole with index $k=0,\\pm 1$. By using the UV retarded Green's function and IR correlation function, we analyze the low frequency behavior of the topological non-Fermi liquid at zero and finite temperatures, especially the relevant non-Fermi liquid behavior near the quantum critical... 11. Fermi Timing and Synchronization System Energy Technology Data Exchange (ETDEWEB) Wilcox, R.; Staples, J.; Doolittle, L.; Byrd, J.; Ratti, A.; Kaertner, F.X.; Kim, J.; Chen, J.; Ilday, F.O.; Ludwig, F.; Winter, A.; Ferianis, M.; Danailov, M.; D' Auria, G. 2006-07-19 The Fermi FEL will depend critically on precise timing of its RF, laser and diagnostic subsystems. The timing subsystem to coordinate these functions will need to reliably maintain sub-100fs synchronicity between distant points up to 300m apart in the Fermi facility. The technology to do this is not commercially available, and has not been experimentally demonstrated in a working facility. Therefore, new technology must be developed to meet these needs. Two approaches have been researched by different groups working with the Fermi staff. At MIT, a pulse transmission scheme has been developed for synchronization of RF and laser devices. And at LBL, a CW transmission scheme has been developed for RF and laser synchronization. These respective schemes have advantages and disadvantages that will become better understood in coming years. This document presents the work done by both teams, and suggests a possible system design which integrates them both. The integrated system design provides an example of how choices can be made between the different approaches without significantly changing the basic infrastructure of the system. Overall system issues common to any synchronization scheme are also discussed. 12. Fermi/non-Fermi mixing in SU($N$) Kondo effect CERN Document Server Kimura, Taro 2016-01-01 We apply conformal field theory analysis to the $k$-channel SU($N$) Kondo system, and find a peculiar behavior in the cases $N > k > 1$, which we call Fermi/non-Fermi mixing: The low temperature scaling is described as the Fermi liquid, while the zero temperature IR fixed point exhibits the non-Fermi liquid signature. We also show that the Wilson ratio is no longer universal for the cases $N > k > 1$. The deviation from the universal value of the Wilson ratio could be used as an experimental signal of the Fermi/non-Fermi mixing. 13. Recognizing nitrogen dopant atoms in graphene using atomic force microscopy DEFF Research Database (Denmark) van der Heijden, Nadine J.; Smith, Daniel; Calogero, Gaetano 2016-01-01 Doping graphene by heteroatoms such as nitrogen presents an attractive route to control the position of the Fermi level in the material. We prepared N-doped graphene on Cu(111) and Ir(111) surfaces via chemical vapor deposition of two different molecules. Using scanning tunneling microscopy images...... as a benchmark, we show that the position of the dopant atoms can be determined using atomic force microscopy. Specifically, the frequency shift-distance curves Delta f(z) acquired above a N atom are significantly different from the curves measured over a C atom. Similar behavior was found for N-doped graphene... 14. Landau Theory of Helical Fermi Liquids. Science.gov (United States) Lundgren, Rex; Maciejko, Joseph 2015-08-07 We construct a phenomenological Landau theory for the two-dimensional helical Fermi liquid found on the surface of a three-dimensional time-reversal invariant topological insulator. In the presence of rotation symmetry, interactions between quasiparticles are described by ten independent Landau parameters per angular momentum channel, by contrast with the two (symmetric and antisymmetric) Landau parameters for a conventional spin-degenerate Fermi liquid. We project quasiparticle states onto the Fermi surface and obtain an effectively spinless, projected Landau theory with a single projected Landau parameter per angular momentum channel that captures the spin-momentum locking or nontrivial Berry phase of the Fermi surface. As a result of this nontrivial Berry phase, projection to the Fermi surface can increase or lower the angular momentum of the quasiparticle interactions. We derive equilibrium properties, criteria for Fermi surface instabilities, and collective mode dispersions in terms of the projected Landau parameters. We briefly discuss experimental means of measuring projected Landau parameters. 15. Synthetic gases production Energy Technology Data Exchange (ETDEWEB) Mazaud, J.P. 1996-06-01 The natural gas or naphtha are the main constituents used for the production of synthetic gases. Several production ways of synthetic gases are industrially used as for example the natural gas or naphtha catalytic reforming, the selective oxidation of natural gas or heavy fuels and the coal oxy-vapo-gasification. The aim of this work is to study the different steps of production and treatment of the synthetic gases by the way of catalytic reforming. The first step is the desulfurization of the hydrocarbons feedstocks. The process used in industry is described. Then is realized the catalytic hydrocarbons reforming process. After having recalled some historical data on the catalytic reforming, the author gives the reaction kinetics and thermodynamics. The possible reforming catalysts, industrial equipments and furnaces designs are then exposed. The carbon dioxide is a compound easily obtained during the reforming reactions. It is a wasteful and harmful component which has to be extracted of the gaseous stream. The last step is then the gases de-carbonation. Two examples of natural gas or naphtha reforming reactions are at last given: the carbon monoxide conversion by steam and the carbon oxides reactions with hydrogen (methanization). (O.M.). 8 figs., 6 tabs. 16. Fe distribution in GaSe and InSe Energy Technology Data Exchange (ETDEWEB) Kovalyuk, Z.D.; Feichuk, P.I.; Shcherbak, L.P.; Zbykovskaya, N.I. 1985-06-01 In this paper, the authors use tagged atoms to determine the effective coefficients of Fe distribution in GaSe and InSe during crystallization of a doped melt by the Bridgman method. The distribution of Fe in GaSe and InSe was studied with the aid of Fe tagged with the radiosotopes VVFe + VZFe. Doping of the material was combined with the processes of synthesis and crystallization. Equations are presented for the calculation of the real impurity distribution in GaSe and InSe crystals. 17. Bioterrorism and the Fermi Paradox Science.gov (United States) Cooper, Joshua 2013-04-01 We proffer a contemporary solution to the so-called Fermi Paradox, which is concerned with conflict between Copernicanism and the apparent paucity of evidence for intelligent alien civilizations. In particular, we argue that every community of organisms that reaches its space-faring age will (1) almost immediately use its rocket-building computers to reverse-engineer its genetic chemistry and (2) self-destruct when some individual uses said technology to design an omnicidal pathogen. We discuss some of the possible approaches to prevention with regard to Homo sapiens' vulnerability to bioterrorism, particularly on a short-term basis. 18. The Low Density Matter (LDM) beamline at FERMI: optical layout and first commissioning. Science.gov (United States) Svetina, Cristian; Grazioli, Cesare; Mahne, Nicola; Raimondi, Lorenzo; Fava, Claudio; Zangrando, Marco; Gerusina, Simone; Alagia, Michele; Avaldi, Lorenzo; Cautero, Giuseppe; de Simone, Monica; Devetta, Michele; Di Fraia, Michele; Drabbels, Marcel; Feyer, Vitaliy; Finetti, Paola; Katzy, Raphael; Kivimäki, Antti; Lyamayev, Viktor; Mazza, Tommaso; Moise, Angelica; Möller, Thomas; O'Keeffe, Patrick; Ovcharenko, Yevheniy; Piseri, Paolo; Plekan, Oksana; Prince, Kevin C; Sergo, Rudi; Stienkemeier, Frank; Stranges, Stefano; Coreno, Marcello; Callegari, Carlo 2015-05-01 The Low Density Matter (LDM) beamline has been built as part of the FERMI free-electron laser (FEL) facility to serve the atomic, molecular and cluster physics community. After the commissioning phase, it received the first external users at the end of 2012. The design and characterization of the LDM photon transport system is described, detailing the optical components of the beamline. 19. Adsorption of Gases on Carbon Nanotubes Science.gov (United States) 2014-01-01 This research focus in studying the interaction between various classical and quantum gases with novel carbon nanostructures, mainly carbon nanotubes (CNTs). Since their discovery by the Japanese physicist Sumio Iijima [1] carbon nanotubes have, experimentally and theoretically, been subjected to many scientific investigation. Studies of adsorption on CNTs are particularly directed toward their better usage in gas storage, gas separation, catalyst, drug delivery, and water purification. We explore the adsorption of different gases entrapped in a single, double, or multi-bundles of CNTs using computer simulations. The first system we investigate consists of Ar and Kr films adsorbed on zigzag or armchair nanotubes. Our simulations revealed that Kr atoms on intermediate size zigzag NTs undergo two phase transitions: A liquid-vapor (L→V), and liquid-commensurate (L→CS) with a fractional coverage of one Kr atoms adsorbed for every four carbon atoms. For Ar on zigzag and armchair NTs, the only transition observed is a L→V. In the second problem, we explore the adsorption of CO2 molecules in a nanotube bundle and calculate the isosteric heat of adsorption of the entrapped molecules within the groove. We observed that the lower the temperature, the higher the isosteric of adsorption. Last, we investigate the adsorption of hydrogen, Helium, and Neon gases on the groove site of two parallel nanotubes. At low temperature, the transverse motion on the plane perpendicular to the tubes' axis is frozen out and as a consequence, the heat capacity is reduced to 1/2. At high temperature, the atoms gain more degree of freedom and as a consequence the heat capacity is 5/2. 20. Observation of the Leggett-Rice effect in a unitary Fermi gas. Science.gov (United States) Trotzky, S; Beattie, S; Luciuk, C; Smale, S; Bardon, A B; Enss, T; Taylor, E; Zhang, S; Thywissen, J H 2015-01-09 We observe that the diffusive spin current in a strongly interacting degenerate Fermi gas of (40)K precesses about the local magnetization. As predicted by Leggett and Rice, precession is observed both in the Ramsey phase of a spin-echo sequence, and in the nonlinearity of the magnetization decay. At unitarity, we measure a Leggett-Rice parameter γ=1.08(9) and a bare transverse spin diffusivity D(0)(⊥)=2.3(4)ℏ/m for a normal-state gas initialized with full polarization and at one-fifth of the Fermi temperature, where m is the atomic mass. One might expect γ=0 at unitarity, where two-body scattering is purely dissipative. We observe γ→0 as temperature is increased towards the Fermi temperature, consistent with calculations that show the degenerate Fermi sea restores a nonzero γ. Tuning the scattering length a, we find that a sign change in γ occurs in the range 0Fermi momentum. We discuss how γ reveals the effective interaction strength of the gas, such that the sign change in γ indicates a switching of branch between a repulsive and an attractive Fermi gas. 1. Shortcut to adiabaticity for an anisotropic unitary Fermi gas CERN Document Server Deng, Shujin; Yu, Qianli; Wu, Haibin 2016-01-01 Coherent control of complex quantum systems is a fundamental requirement in quantum information processing and engineering. Recently developed notion of shortcut to adiabaticity (STA) has spawned intriguing prospects. So far, the most experimental investigations of STA are implemented in the ideal thermal gas or the weakly interacting ultracold Bose gases. Here we report the first demonstration of a many-body STA in a 3D anisotropically trapped unitary Fermi gas. A new dynamical scaling law is demonstrated on such a strongly interacting quantum gas. By simply engineering the frequency aspect ratio of a harmonic trap, the dynamics of the gas can be manipulated and the many-body state can be transferred adiabatically from one stationary state to another one in short time scale without the excitation. The universal scaling both for non-interacting and unitary Fermi gas is also verified. This could be very important for future many-body quantum engineering and the exploration of the fundamental law of the thermod... 2. The greenhouse gases Energy Technology Data Exchange (ETDEWEB) Clarke, R. 1987-01-01 The main greenhouse gases are carbon dioxide, methane, nitrous oxide, CFCs and ozone. They are greenhouse gases as they absorb radiation from the Earth and thus impede its emission back to space. CO{sub 2} is responsible for about half the enhanced greenhouse effect. A global warming of only a few degrees would have a profound effect on climate. Increased levels of CO{sub 2} promote plant growth, but may not benefit agriculture overall. Sea levels may rise. It is difficult to predict the effects of global warming in society. It would be possible to reduce the scale of the greenhouse effect by energy conservation, using alternative energy sources, and possibly by capturing CO{sub 2} from fossil fuel power stations and disposing of it on the ocean floor. 13 refs., 19 figs., 1 tab. 3. On the theory of polarized Fermi liquid OpenAIRE Mineev, V. P. 2004-01-01 The transport equation for transverse vibrations of magnetization in spin polarized Fermi liquid is derived from integral equation for the vertex function. The dispersion law for the transverse spin waves is established. The existance of zero-temperature spin-waves attenuation is confirmed. The problem of similar derivation in ferromagnetic "Fermi liquid" is discussed. 4. Fermi Surface and Antiferromagnetism in Europium Metal DEFF Research Database (Denmark) Andersen, O. Krogh; Loucks, T. L. 1968-01-01 We have calculated the Fermi surface of europium in order to find those features which determine the wave vector of the helical moment arrangement below the Néel point. We find that there are two pieces of Fermi surface: an electron surface at the symmetry point H, which has the shape of rounded-... 5. Vacuum alignment and radiatively induced Fermi scale CERN Document Server Alanne, Tommi 2016-01-01 We extend the discussion about vacuum misalignment by quantum corrections in models with composite pseudo-Goldstone Higgs boson to renormalisable models with elementary scalars. As a concrete example, we propose a framework, where the hierarchy between the unification and the Fermi scale emerges radiatively. This scenario provides an interesting link between the unification and Fermi scale physics. 6. Biased discrete symmetry breaking and Fermi balls CERN Document Server MacPherson, A L; Macpherson, Alick L; Campbell, Bruce A 1994-01-01 The spontaneous breaking of an approximate discrete symmetry is considered, with the resulting protodomains of true and false vacuum being separated by domain walls. Given a strong, symmetric Yukawa coupling of the real scalar field to a generic fermion, the domain walls accumulate a gas of fermions, which modify the domain wall dynamics. The splitting of the degeneracy of the ground states results in the false vacuum protodomain structures eventually being fragmented into tiny false vacuum bags with a Fermi gas shell (Fermi balls), that may be cosmologically stable due to the Fermi gas pressure and wall curvature forces, acting on the domain walls. As fermions inhabiting the domain walls do not undergo number density freeze out, stable Fermi balls exist only if a fermion anti-fermion asymmetry occurs. Fermi balls formed with a new Dirac fermion that possesses no standard model gauge charges provide a novel cold dark matter candidate. 7. Anomalous excitation facilitation in inhomogeneously broadened Rydberg gases CERN Document Server Letscher, Fabian; Niederprüm, Thomas; Ott, Herwig; Fleischhauer, Michael 2016-01-01 When atomic gases are laser driven to Rydberg states in an off resonant way, a single Rydberg atom may enhance the excitation rate of surrounding atoms. This leads to a facilitated excitation referred to as Rydberg anti-blockade. In the usual facilitation scenario, the detuning of the laser from resonance compensates the interaction shift. Here, we discuss a different excitation mechanism, which we call anomalous facilitation. This occurs on the "wrong side" of the resonance and originates from inhomogeneous broadening. The anomalous facilitation may be seen in experiments of attractively interacting atoms on the blue detuned side, where facilitation is not expected to appear. 8. Merge of high harmonic generation from gases and solids and its implications for attosecond science Science.gov (United States) Vampa, G.; Brabec, T. 2017-04-01 High harmonic generation (HHG) in atomic and molecular gases builds the foundation of attosecond science. In recent experiments HHG has been demonstrated in solids for the first time. A theoretical analysis has revealed that one of the mechanisms driving HHG in semiconductors is similar to the one in atomic and molecular gases. As a result, many of the processes developed for attosecond science in gases can be adapted and applied to the condensed matter phase. In this tutorial, the connection between atomic and solid HHG is summarized with covering both theoretical and experimental work, and some implications for attosecond science in solids are presented. 9. Theory of warm ionized gases: equation of state and kinetic Schottky anomaly CERN Document Server Capolupo, Antonio; Illuminati, Fabrizio 2013-01-01 Based on accurate Lennard-Jones type interaction potentials, we derive a closed set of state equations for the description of warm atomic gases in the presence of ionization processes. The specific heat is predicted to exhibit peaks in correspondence to single and multiple ionizations. Such kinetic analogue in atomic gases of the Schottky anomaly in solids is enhanced at intermediate and low atomic densities. The case of adiabatic compression of noble gases is analyzed in detail and the implications on sonoluminescence are discussed. 10. Theory of warm ionized gases: equation of state and kinetic Schottky anomaly. Science.gov (United States) Capolupo, A; Giampaolo, S M; Illuminati, F 2013-10-01 Based on accurate Lennard-Jones-type interaction potentials, we derive a closed set of state equations for the description of warm atomic gases in the presence of ionization processes. The specific heat is predicted to exhibit peaks in correspondence to single and multiple ionizations. Such kinetic analog in atomic gases of the Schottky anomaly in solids is enhanced at intermediate and low atomic densities. The case of adiabatic compression of noble gases is analyzed in detail and the implications on sonoluminescence are discussed. In particular, the predicted plasma electron density in a sonoluminescent bubble turns out to be in good agreement with the value measured in recent experiments. 11. Low temperatures shear viscosity of a two-component dipolar Fermi gas with unequal population Science.gov (United States) Darsheshdar, E.; Yavari, H.; Zangeneh, Z. 2016-07-01 By using the Green's functions method and linear response theory we calculate the shear viscosity of a two-component dipolar Fermi gas with population imbalance (spin polarized) in the low temperatures limit. In the strong-coupling Bose-Einstein condensation (BEC) region where a Feshbach resonance gives rise to tightly bound dimer molecules, a spin-polarized Fermi superfluid reduces to a simple Bose-Fermi mixture of Bose-condensed dimers and the leftover unpaired fermions (atoms). The interactions between dimer-atom, dimer-dimer, and atom-atom take into account to the viscous relaxation time (τη) . By evaluating the self-energies in the ladder approximation we determine the relaxation times due to dimer-atom (τDA) , dimer-dimer (τcDD ,τdDD) , and atom-atom (τAA) interactions. We will show that relaxation rates due to these interactions τDA-1 ,τcDD-1, τdDD-1, and τAA-1 have T2, T4, e - E /kB T (E is the spectrum of the dimer atoms), and T 3 / 2 behavior respectively in the low temperature limit (T → 0) and consequently, the atom-atom interaction plays the dominant role in the shear viscosity in this rang of temperatures. For small polarization (τDA ,τAA ≫τcDD ,τdDD), the low temperatures shear viscosity is determined by contact interaction between dimers and the shear viscosity varies as T-5 which has the same behavior as the viscosity of other superfluid systems such as superfluid neutron stars, and liquid helium. 12. Focus on strongly correlated quantum fluids: from ultracold quantum gases to QCD plasmas Focus on strongly correlated quantum fluids: from ultracold quantum gases to QCD plasmas Science.gov (United States) Adams, Allan; Carr, Lincoln D.; Schaefer, Thomas; Steinberg, Peter; Thomas, John E. 2013-04-01 interdisciplinary appeal and include new studies of high temperature superfluidity, viscosity, spin-transport, spin-imbalanced mixtures, and three-component gases, this last having a close parallel to color superconductivity. Another system important for the field of strongly-interacting quantum fluids was revealed by analysis of data from the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory. Despite naive expectations based on asymptotic freedom that the deconfinement of quarks and gluons at high temperatures would lead to a weakly-interacting quark gluon plasma (QGP), the system appeared to be quite strongly coupled. Subsequent estimates of the viscosity-to-entropy ratio suggest that the system is tantalizingly close to the postulated bound from AdS/CFT calculations. The field is quite dynamic at the moment; new measurements are expected from upgraded detectors at RHIC, and an entirely new energy regime is being opened up by heavy ion collisions at the Large Hadron Collider (LHC) at CERN. On the theoretical side, much work remains to be done to extract the precise values of the transport coefficients, and to characterize the nature of quasi-particle excitations in the plasma. Finally, holographic dualities such as anti-de Sitter/conformal field theory (AdS/CFT) have opened a new theoretical window on strongly correlated fluids. Holography relates strongly-interacting quantum many-body systems to weakly-coupled semi-classical gravitational systems, replacing quasiparticles with geometry and translating various difficult questions about quantum fluids into simple and calculable geometric exercises. Already, some of the earliest lessons of holography, such as the conjectural bound on the viscosity-to-entropy ratio, have had a considerable impact on the theoretical and experimental study of strongly correlated fluids, from RHIC to ultracold atoms. More recently, the study of holographic superconductors, non-Fermi liquids and unitary quantum gases has touched 13. Dynamical structure factor of one-dimensional Bose gases: Experimental signatures of beyond-Luttinger-liquid physics NARCIS (Netherlands) N. Fabbri; M. Panfil; D. Clément; L. Fallani; M. Inguscio; C. Fort; J.-S. Caux 2015-01-01 Interactions are known to have dramatic effects on bosonic gases in one dimension (1D). Not only does the ground state transform from a condensate like state to an effective Fermi sea, but new fundamental excitations, which do not have any higher-dimensional equivalents, are predicted to appear. In 14. Properties of strongly dipolar Bose gases beyond the Born approximation CERN Document Server Ołdziejewski, Rafał 2016-01-01 Strongly dipolar Bose gases can form liquid droplets stabilized by quantum fluctuations. In theoretical description of this phenomenon, low energy scattering amplitude is utilized as an effective potential. We show that for magnetic atoms corrections with respect to Born approximation arise, and derive modified pseudopotential using realistic interaction model. We discuss the resulting changes in collective mode frequencies and droplet stability diagram. Our results are relevant for recent experiments with erbium and dysprosium atoms. 15. On Classical Ideal Gases Directory of Open Access Journals (Sweden) Laurent Chusseau 2013-02-01 Full Text Available We show that the thermodynamics of ideal gases may be derived solely from the Democritean concept of corpuscles moving in vacuum plus a principle of simplicity, namely that these laws are independent of the laws of motion, aside from the law of energy conservation. Only a single corpuscle in contact with a heat bath submitted to a z and t-invariant force is considered. Most of the end results are known but the method appears to be novel. The mathematics being elementary, the present paper should facilitate the understanding of the ideal gas law and of classical thermodynamics even though not-usually-taught concepts are being introduced. 16. Observation of relativistic antihydrogen atoms Science.gov (United States) Blanford, Glenn Delfosse, Jr. 1997-09-01 An observation of relativistic antihydrogen atoms is reported in this dissertation. Experiment 862 at Fermi National Accelerator Laboratory observed antihydrogen atoms produced by the interaction of a circulating beam of high momentum (3 production is outlined within. The cross section corresponds to the process where a high momentum antiproton causes e+e/sp- pair creation near a nucleus with the e+ being captured by the antiproton. Antihydrogen is the first atom made exclusively of antimatter to be detected. The observation experiment's results are the first step towards an antihydrogen spectroscopy experiment which would measure the n = 2 Lamb shift and fine structure. 17. Positron scattering from noble gases future prospects Energy Technology Data Exchange (ETDEWEB) Jones, A C L; Caradonna, P; Makochekanwa, C; Slaughter, D S; Sullivan, J P; Buckman, S J [Centre for Antimatter-Matter Studies, Research School of Physics and Engineering, Australian National University, Canberra, ACT (Australia); Mitroy, J, E-mail: [email protected] [Faculty of Education Health and Science, Charles Darwin University, NT (Australia) 2009-11-01 Recent results for positron scattering from noble gases over an energy range from 0.5 to 60eV are presented. Measurements include the grand total ({sigma}{sub GT}), Ps formation ({sigma}{sub Ps}) and Grand total - Ps formation (({sigma}{sub GT}-P{sub s}) cross sections. Some preliminary DCS results will also be presented. Work on a formulation of modified effective range theory (MERT) is being undertaken to determine the value of the scattering length which may be useful for identifying a bound state. Plans for experiments on metal atoms will be outlined. 18. Sampling and analysis methods for geothermal fluids and gases Energy Technology Data Exchange (ETDEWEB) Watson, J.C. 1978-07-01 The sampling procedures for geothermal fluids and gases include: sampling hot springs, fumaroles, etc.; sampling condensed brine and entrained gases; sampling steam-lines; low pressure separator systems; high pressure separator systems; two-phase sampling; downhole samplers; and miscellaneous methods. The recommended analytical methods compiled here cover physical properties, dissolved solids, and dissolved and entrained gases. The sequences of methods listed for each parameter are: wet chemical, gravimetric, colorimetric, electrode, atomic absorption, flame emission, x-ray fluorescence, inductively coupled plasma-atomic emission spectroscopy, ion exchange chromatography, spark source mass spectrometry, neutron activation analysis, and emission spectrometry. Material on correction of brine component concentrations for steam loss during flashing is presented. (MHR) 19. Quantum chaos on a critical Fermi surface CERN Document Server Patel, Aavishkar A 2016-01-01 We compute parameters characterizing many-body quantum chaos for a critical Fermi surface without quasiparticle excitations. We examine a theory of $N$ species of fermions at non-zero density coupled to a $U(1)$ gauge field in two spatial dimensions, and determine the Lyapunov rate and the butterfly velocity in an extended RPA approximation. The thermal diffusivity is found to be universally related to these chaos parameters, i.e. the relationship is independent of $N$, the gauge coupling constant, the Fermi velocity, the Fermi surface curvature, and high energy details. 20. New frontiers for quantum gases of polar molecules Science.gov (United States) Moses, Steven A.; Covey, Jacob P.; Miecnikowski, Matthew T.; Jin, Deborah S.; Ye, Jun 2017-01-01 Compared to atoms, molecules possess additional degrees of freedom that can be exploited in fundamental tests, ultracold chemistry, and engineering new quantum phases in many-body systems. Here, we review the recent progress in creating and manipulating ultracold bialkali molecules to study quantum gases of polar molecules. 1. Fermi liquid-to-Bose condensate crossover in a two-dimensional ultracold gas experiment Science.gov (United States) Barmashova, T. V.; Mart'yanov, K. A.; Makhalov, V. B.; Turlapov, A. V. 2016-02-01 By controling interparticle interactions, it is possible to transform a fermionic system into a bosonic system and vice versa, while preserving quantum degeneracy. Evidence of such a transformation may be found by monitoring the pressure and interference. The Fermi pressure is an indication of the fermion?ic character of a system, while the interference implies a nonzero order parameter and Bose condensation. Lowering from three to two spatial dimensions introduces new physics and makes the system more difficult to describe due to the increased fluctuations and the reduced applicability of mean field methods. An experiment with a two-dimensional ultracold atomic gas shows a crossover between the Bose and Fermi limits, as evident from the value of pressure and from the interference pattern, and provides data to test models of 2D Fermi and Bose systems, including the most-difficult-to-model strongly coupled systems. 2. Observation of spatial charge and spin correlations in the 2D Fermi-Hubbard model. Science.gov (United States) Cheuk, Lawrence W; Nichols, Matthew A; Lawrence, Katherine R; Okan, Melih; Zhang, Hao; Khatami, Ehsan; Trivedi, Nandini; Paiva, Thereza; Rigol, Marcos; Zwierlein, Martin W 2016-09-16 Strong electron correlations lie at the origin of high-temperature superconductivity. Its essence is believed to be captured by the Fermi-Hubbard model of repulsively interacting fermions on a lattice. Here we report on the site-resolved observation of charge and spin correlations in the two-dimensional (2D) Fermi-Hubbard model realized with ultracold atoms. Antiferromagnetic spin correlations are maximal at half-filling and weaken monotonically upon doping. At large doping, nearest-neighbor correlations between singly charged sites are negative, revealing the formation of a correlation hole, the suppressed probability of finding two fermions near each other. As the doping is reduced, the correlations become positive, signaling strong bunching of doublons and holes, in agreement with numerical calculations. The dynamics of the doublon-hole correlations should play an important role for transport in the Fermi-Hubbard model. 3. Effective dynamics of strongly dissipative Rydberg gases CERN Document Server Marcuzzi, M; Olmos, B; Lesanovsky, I 2014-01-01 We investigate the evolution of interacting Rydberg gases in the limit of strong noise and dissipation. Starting from a description in terms of a Markovian quantum master equation we derive effective equations of motion that govern the dynamics on a "coarse-grained" timescale where fast dissipative degrees of freedom have been adiabatically eliminated. Specifically, we consider two scenarios which are of relevance for current theoretical and experimental studies --- Rydberg atoms in a two-level (spin) approximation subject to strong dephasing noise as well as Rydberg atoms under so-called electromagnetically induced transparency (EIT) conditions and fast radiative decay. In the former case we find that the effective dynamics is described by classical rate equations up to second order in an appropriate perturbative expansion. This drastically reduces the computational complexity of numerical simulations in comparison to the full quantum master equation. When accounting for the fourth order correction in this e... 4. Sir William Ramsay and the noble gases. Science.gov (United States) Davies, Alwyn G 2012-01-01 Sir William Ramsay was one of the world's leading scientists at the end of the 19th century, and in a spectacular period of research between 1894 and 1898, he discovered five new elements. These were the noble gases, helium, neon, argon, krypton, and xenon; they added a whole new group to the Periodic Table of the elements, and provided the keystone to our understanding of the electronic structure of atoms, and the way those electrons bind the atoms together into molecules. For this work he was awarded the Nobel Prize in Chemistry in 1904, the first such prize to come to a British subject. He was also a man of great charm, a good linguist, and a composer and performer of music, poetry and song. This review will trace his career, describe his character and give and account of the chemistry which led to the award of the Nobel Prize. 5. Weyl spin-orbit-coupling-induced interactions in uniform and trapped atomic quantum fluids Science.gov (United States) Gupta, Reena; Singh, G. S.; Bosse, Jürgen 2013-11-01 We establish through analytical and numerical studies of thermodynamic quantities for noninteracting atomic gases that the isotropic three-dimensional spin-orbit coupling, the Weyl coupling, induces interaction which counters “effective” attraction (repulsion) of the exchange symmetry present in zero-coupling Bose (Fermi) gas. The exact analytical expressions for the grand potential and hence for several thermodynamic quantities have been obtained for this purpose in both uniform and trapped cases. It is enunciated that many interesting features of spin-orbit-coupled systems revealed theoretically can be understood in terms of coupling-induced modifications in statistical interparticle potential. The temperature dependence of the chemical potential, specific heat, and isothermal compressibility for a uniform Bose gas is found to have signature of the incipient Bose-Einstein condensation in the very weak coupling regime although the system does not really go in the Bose-condensed phase. The transition temperature in the harmonically trapped case decreases with an increase of coupling strength consistent with the weakening of the statistical attractive interaction. Anomalous behavior of some thermodynamic quantities, partly akin to that in dimensions less than two, appears for uniform fermions as soon as the Fermi level goes down the Dirac point on increasing the coupling strength. It is suggested that the fluctuation-dissipation theorem can be utilized to verify anomalous behaviors from studies of long-wavelength fluctuations in bunching and antibunching effects. 6. Radiatively induced Fermi scale and unification CERN Document Server Alanne, Tommi 2016-01-01 We propose a framework, where the hierarchy between the unification and the Fermi scale emerges radiatively. This work tackles the long-standing question about the connection between the low Fermi scale and a more fundamental scale of Nature. As a concrete example, we study a Pati-Salam-type unification of Elementary-Goldstone-Higgs scenario, where the Standard Model scalar sector is replaced by an SU(4)-symmetric one, and the observed Higgs particle is an elementary pseudo-Goldstone boson. We construct a concrete model where the unification scale is fixed to a phenomenologically viable value, while the Fermi scale is generated radiatively. This scenario provides an interesting link between the unification and Fermi scale physics, and opens up prospects for exploring a wide variety of open problems in particle physics, ranging from neutrinos to cosmic inflation. 7. Gamma-Ray Astrophysics NSSTC Fermi GBM Data.gov (United States) National Aeronautics and Space Administration — The Fermi Gamma-Ray Burst Monitor (GBM) is not a pointed or imaging instrument. To determine fluxes for known sources, we measure the change in the count rate... 8. Fermi: physicist with a capital F Science.gov (United States) Cobal, Marina 2016-12-01 Enrico Fermi – one of the great physicists of the 21st century – was a beacon for every Italian student of physics. This is wonderfully captured in The Pope of Physics by Gino Segrè and Bettina Hoerlin. 9. Fermi Surface of the Most Dilute Superconductor Science.gov (United States) Lin, Xiao; Zhu, Zengwei; Fauqué, Benoît; Behnia, Kamran 2013-04-01 The origin of superconductivity in bulk SrTiO3 is a mystery since the nonmonotonous variation of the critical transition with carrier concentration defies the expectations of the crudest version of the BCS theory. Here, employing the Nernst effect, an extremely sensitive probe of tiny bulk Fermi surfaces, we show that, down to concentrations as low as 5.5×1017cm-3, the system has both a sharp Fermi surface and a superconducting ground state. The most dilute superconductor currently known therefore has a metallic normal state with a Fermi energy as little as 1.1 meV on top of a band gap as large as 3 eV. The occurrence of a superconducting instability in an extremely small, single-component, and barely anisotropic Fermi surface implies strong constraints for the identification of the pairing mechanism. 10. A fast algorithm for finding point sources in the Fermi data stream: FermiFAST Science.gov (United States) Asvathaman, Asha; Omand, Conor; Barton, Alistair; Heyl, Jeremy S. 2017-04-01 We present a new and efficient algorithm for finding point sources in the photon event data stream from the Fermi Gamma-Ray Space Telescope, FermiFAST. The key advantage of FermiFAST is that it constructs a catalogue of potential sources very fast by arranging the photon data in a hierarchical data structure. Using this structure, FermiFAST rapidly finds the photons that could have originated from a potential gamma-ray source. It calculates a likelihood ratio for the contribution of the potential source using the angular distribution of the photons within the region of interest. It can find within a few minutes the most significant half of the Fermi Third Point Source catalogue (3FGL) with nearly 80 per cent purity from the 4 yr of data used to construct the catalogue. If a higher purity sample is desirable, one can achieve a sample that includes the most significant third of the Fermi 3FGL with only 5 per cent of the sources unassociated with Fermi sources. Outside the Galactic plane, all but eight of the 580 FermiFAST detections are associated with 3FGL sources. And of these eight, six yield significant detections of greater than 5σ when a further binned likelihood analysis is performed. This software allows for rapid exploration of the Fermi data, simulation of the source detection to calculate the selection function of various sources and the errors in the obtained parameters of the sources detected. 11. Coexistence of Fermi arcs and Fermi pockets in a high-T(c) copper oxide superconductor. Science.gov (United States) Meng, Jianqiao; Liu, Guodong; Zhang, Wentao; Zhao, Lin; Liu, Haiyun; Jia, Xiaowen; Mu, Daixiang; Liu, Shanyu; Dong, Xiaoli; Zhang, Jun; Lu, Wei; Wang, Guiling; Zhou, Yong; Zhu, Yong; Wang, Xiaoyang; Xu, Zuyan; Chen, Chuangtian; Zhou, X J 2009-11-19 In the pseudogap state of the high-transition-temperature (high-T(c)) copper oxide superconductors, angle-resolved photoemission (ARPES) measurements have seen Fermi arcs-that is, open-ended gapless sections in the large Fermi surface-rather than a closed loop expected of an ordinary metal. This is all the more puzzling because Fermi pockets (small closed Fermi surface features) have been suggested by recent quantum oscillation measurements. The Fermi arcs cannot be understood in terms of existing theories, although there is a solution in the form of conventional Fermi surface pockets associated with competing order, but with a back side that is for detailed reasons invisible to photoemission probes. Here we report ARPES measurements of Bi(2)Sr(2-x)La(x)CuO(6+delta) (La-Bi2201) that reveal Fermi pockets. The charge carriers in the pockets are holes, and the pockets show an unusual dependence on doping: they exist in underdoped but not overdoped samples. A surprise is that these Fermi pockets appear to coexist with the Fermi arcs. This coexistence has not been expected theoretically. 12. Atomic physics: A strange kind of liquid Science.gov (United States) Laburthe-Tolra, Bruno 2016-11-01 Interactions between the magnetic dipoles of dysprosium atoms in an ultracold gas can produce a 'self-bound' droplet. This provides a useful isolated system for probing the quantum-mechanical properties of ultracold gases. See Letter p.259 13. Superfluidity in ultracold gases Science.gov (United States) Campbell, Gretchen 2016-05-01 The study of superfluidity has a long and rich history. In Bose-Einstein condensate, superfluidity gives rise to a number of interesting effects, including quantized vortices and persistent currents. In this seminar I will give an introduction to superfluidity in ultracold atoms, including a discussion of the critical velocity and the spectrum of elementary excitations in superfluid systems. 14. Fermi-Dirac Statistics of Complex Networks Institute of Scientific and Technical Information of China (English) SHEN Yi; ZHU Di-Ling; LIU Wei-Ming 2005-01-01 @@ We investigate phenomena of decline of complex networks by employing and analysing an illness model. Its intrinsic relation with the Fermi distribution is shown and a mapping to Fermi gas is established. The results of numerical simulations are obtained in two ways. We also compare the model with other models, including the dual relationship with the fitness model, and its difference from the Cayley tree model. 15. Fermi breakup and the statistical multifragmentation model Energy Technology Data Exchange (ETDEWEB) Carlson, B.V., E-mail: [email protected] [Departamento de Fisica, Instituto Tecnologico de Aeronautica - CTA, 12228-900 Sao Jose dos Campos (Brazil); Donangelo, R. [Instituto de Fisica, Universidade Federal do Rio de Janeiro, Cidade Universitaria, CP 68528, 21941-972, Rio de Janeiro (Brazil); Instituto de Fisica, Facultad de Ingenieria, Universidad de la Republica, Julio Herrera y Reissig 565, 11.300 Montevideo (Uruguay); Souza, S.R. [Instituto de Fisica, Universidade Federal do Rio de Janeiro, Cidade Universitaria, CP 68528, 21941-972, Rio de Janeiro (Brazil); Instituto de Fisica, Universidade Federal do Rio Grande do Sul, Av. Bento Goncalves 9500, CP 15051, 91501-970, Porto Alegre (Brazil); Lynch, W.G.; Steiner, A.W.; Tsang, M.B. [Joint Institute for Nuclear Astrophysics, National Superconducting Cyclotron Laboratory and the Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824 (United States) 2012-02-15 We demonstrate the equivalence of a generalized Fermi breakup model, in which densities of excited states are taken into account, to the microcanonical statistical multifragmentation model used to describe the disintegration of highly excited fragments of nuclear reactions. We argue that such a model better fulfills the hypothesis of statistical equilibrium than the Fermi breakup model generally used to describe statistical disintegration of light mass nuclei. 16. Reaching a Fermi-superfluid state near an orbital Feshbach resonance Science.gov (United States) Xu, Junjun; Zhang, Ren; Cheng, Yanting; Zhang, Peng; Qi, Ran; Zhai, Hui 2016-09-01 We propose to realize a strongly interacting Fermi superfluid near a narrow Feshbach resonance using the recently discovered "orbital Feshbach resonance." The orbital Feshbach resonance is a type of magnetic field tunable scattering resonance theoretically predicted and experimentally observed recently in the alkaline-earth-metal-like 173Yb atom. We first show that the orbital Feshbach resonance is a narrow resonance in energy, while it is hundreds Gauss wide in terms of magnetic field strength, taking the advantage that the magnetic moment difference between the open and closed channels is quite small. Therefore, this is an ideal platform for the experimental realization of a strongly interacting Fermi superfluid with narrow resonance. We further show that the transition temperature for the Fermi superfluid in this system, especially at the BCS side of the resonance, is even higher than that in a wide resonance, which is also due to the narrow character of this resonance. Our results will encourage experimental efforts to realize Fermi superfluid in the alikaline-earth-metal-like 173Yb system, the properties of which will be complementary to extensively studied Fermi superfluids nearby a wide resonance in alkali-metal 40K and 6Li systems. 17. Understanding and Using the Fermi Science Tools Science.gov (United States) Asercion, Joseph; Fermi Science Support Center 2017-01-01 The Fermi Science Support Center (FSSC) provides information, documentation, and tools for the analysis of Fermi science data, including both the Large-Area Telescope (LAT) and the Gamma-ray Burst Monitor (GBM). Source and binary versions of the Fermi Science Tools can be downloaded from the FSSC website, and are supported on multiple platforms. An overview document, the Cicerone, provides details of the Fermi mission, the science instruments and their response functions, the science data preparation and analysis process, and interpretation of the results. Analysis Threads and a reference manual available on the FSSC website provide the user with step-by-step instructions for many different types of data analysis: point source analysis - generating maps, spectra, and light curves, pulsar timing analysis, source identification, and the use of python for scripting customized analysis chains. We present an overview of the structure of the Fermi science tools and documentation, and how to acquire them. We also provide examples of standard analyses, including tips and tricks for improving Fermi science analysis. 18. Coherence Properties of Strongly Interacting Atomic Vapors in Waveguides Science.gov (United States) 2011-12-31 Molecular and Optical Physics Knoxville, TN, May 16-20, 2006 books, invited reviews, editorials , etc [10] Maxim Olshanii, Quantum Mechanics in Two...Rev. Lett. 99, 230402 (2007) [subcontracted under N00014-06-1-0455] [27] del Campo , A.Muga, J. G., Girardeau, M. D, Stability of spinor Fermi gases 19. Deep inelastic scattering on ultracold gases CERN Document Server Hofmann, Johannes 2016-01-01 We discuss the dynamic structure factor of both Bose and Fermi gases with strong short-range interactions, focussing on the deep inelastic regime of large wave vector transfer $q$. Here, the dynamic structure factor is dominated by a resonance at the free-particle energy $\\hbar \\omega = \\varepsilon_{\\bf q} = \\hbar^2 q^2/2m$ and is described in terms of scaling functions. We show that the high-momentum structure has a rich scaling behavior characterized by two separate scaling regions: first, for frequencies that differ from the single-particle energy by terms of order ${\\cal O}(q)$ (i.e., small deviations compared to the single-particle energy), the dynamic structure factor is described by the impulse approximation (IA) of Hohenberg and Platzman. Second, deviations of order ${\\cal O}(q^2)$ (i.e., of the same order or larger than the single-particle energy) are described by the operator product expansion (OPE), with a universal cross-over connecting both regimes. We use the full asymptotic form to derive vario... 20. Doping Scheme of Semiconducting Atomic Chains Science.gov (United States) Toshishige, Yamada; Saini, Subhash (Technical Monitor) 1998-01-01 Atomic chains, precise structures of atomic scale created on an atomically regulated substrate surface, are candidates for future electronics. A doping scheme for intrinsic semiconducting Mg chains is considered. In order to suppress the unwanted Anderson localization and minimize the deformation of the original band shape, atomic modulation doping is considered, which is to place dopant atoms beside the chain periodically. Group I atoms are donors, and group VI or VII atoms are acceptors. As long as the lattice constant is long so that the s-p band crossing has not occurred, whether dopant atoms behave as donors or acceptors is closely related to the energy level alignment of isolated atomic levels. Band structures are calculated for Br-doped (p-type) and Cs-doped (n-type) Mg chains using the tight-binding theory with universal parameters, and it is shown that the band deformation is minimized and only the Fermi energy position is modified. 1. Population Dynamics in Cold Gases Resulting from the Long-Range Dipole-Dipole Interaction CERN Document Server Mandilara, A; Pillet, P 2009-01-01 We consider the effect of the long range dipole-dipole interaction on the excitation exchange dynamics of cold two-level atomic gase in the conditions where the size of the atomic cloud is large as compared to the wavelength of the dipole transition. We show that this interaction results in population redistribution across the atomic cloud and in specific spectra of the spontaneous photons emitted at different angles with respect to the direction of atomic polarization. 2. Doping of Semiconducting Atomic Chains Science.gov (United States) Toshishige, Yamada; Kutler, Paul (Technical Monitor) 1997-01-01 Due to the rapid progress in atom manipulation technology, atomic chain electronics would not be a dream, where foreign atoms are placed on a substrate to form a chain, and its electronic properties are designed by controlling the lattice constant d. It has been shown theoretically that a Si atomic chain is metallic regardless of d and that a Mg atomic chain is semiconducting or insulating with a band gap modified with d. For electronic applications, it is essential to establish a method to dope a semiconducting chain, which is to control the Fermi energy position without altering the original band structure. If we replace some of the chain atoms with dopant atoms randomly, the electrons will see random potential along the chain and will be localized strongly in space (Anderson localization). However, if we replace periodically, although the electrons can spread over the chain, there will generally appear new bands and band gaps reflecting the new periodicity of dopant atoms. This will change the original band structure significantly. In order to overcome this dilemma, we may place a dopant atom beside the chain at every N lattice periods (N > 1). Because of the periodic arrangement of dopant atoms, we can avoid the unwanted Anderson localization. Moreover, since the dopant atoms do not constitute the chain, the overlap interaction between them is minimized, and the band structure modification can be made smallest. Some tight-binding results will be discussed to demonstrate the present idea. CERN Document Server Sneddon, J 1998-01-01 This volume continues the series'' cutting-edge reviews on developments in this field. Since its invention in the 1920s, electrostatic precipitation has been extensively used in industrial hygiene to remove dust and particulate matter from gases before entering the atmosphere. This combination of electrostatic precipitation is reported upon in the first chapter. Following this, chapter two reviews recent advances in the area of chemical modification in electrothermal atomization. Chapter three consists of a review which deal with advances and uses of electrothermal atomization atomic absorption spectrometry. Flow injection atomic spectroscopy has developed rapidly in recent years and after a general introduction, various aspects of this technique are looked at in chapter four. Finally, in chapter five the use of various spectrometric techniques for the determination of mercury are described. 4. Low-lying excitations in a strongly interacting Fermi gas Science.gov (United States) Vale, Christopher; Hoinka, Sascha; Dyke, Paul; Lingham, Marcus 2016-05-01 We present measurements of the low-lying excitation spectrum of a strongly interacting Fermi gas across the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) crossover using Bragg spectroscopy. By focussing the Bragg lasers onto the central volume of the cloud we can probe atoms at near-uniform density allowing measurement of the homogeneous density-density response function. The Bragg wavevector is set to be approximately half of the Fermi wavevector to probe the collective response. Below the superfluid transition temperature the Bragg spectra dominated by the Bogoliubov-Anderson phonon mode. Single particle excitations become visible at energies greater than twice the pairing gap. As interactions are tuned from the BCS to BEC regime the phonon and single particle modes separate apart and both the pairing gap and speed of sound can be directly read off in certain regions of the crossover. Single particle pair-breaking excitations become heavily suppressed as interactions are tuned from the BCS to BEC regimes. 5. Energy and contact of the one-dimensional Fermi polaron at zero and finite temperature. Science.gov (United States) Doggen, E V H; Kinnunen, J J 2013-07-12 We use the T-matrix approach for studying highly polarized homogeneous Fermi gases in one dimension with repulsive or attractive contact interactions. Using this approach, we compute ground state energies and values for the contact parameter that show excellent agreement with exact and other numerical methods at zero temperature, even in the strongly interacting regime. Furthermore, we derive an exact expression for the value of the contact parameter in one dimension at zero temperature. The model is then extended and used for studying the temperature dependence of ground state energies and the contact parameter. 6. A Fast Algorithm for Finding Point Sources in the Fermi Data Stream: FermiFAST CERN Document Server Ashathaman, Asha; Heyl, Jeremy S 2016-01-01 This paper presents a new and efficient algorithm for finding point sources in the photon event data stream from the Fermi Gamma-Ray Space Telescope. It can rapidly construct about most significant half of the Fermi Third Point Source catalogue (3FGL) with nearly 80% purity from the four years of data used to construct the catalogue. If a higher purity sample is desirable, one can achieve a sample that includes the most significant third of the Fermi 3FGL with only five percent of the sources unassociated with Fermi sources. Outside the galaxy plane, the contamination is essentially negligible. This software allows for rapid exploration of the Fermi data, simulation of the source detection to calculate the selection function of various sources and the errors in the obtained parameters of the sources detected. 7. Anomalous conductance of a strongly interacting Fermi gas through a quantum point contact Science.gov (United States) Liu, Boyang; Zhai, Hui; Zhang, Shizhong 2017-01-01 In this work we study the particle conductance of a strongly interacting Fermi gas through a quantum point contact. With an atom-molecule two-channel model, we compute the contribution to particle conductance by both the fermionic atoms and the bosonic molecules using the Keldysh formalism. Focusing on the regime above the Fermi superfluid transition temperature, we find that the fermionic contribution to the conductance is reduced by interaction compared with the quantized value for the noninteracting case; while the bosonic contribution to the conductance exhibits a plateau with nonuniversal values that is larger than the quantized conductance. This feature is particularly profound at temperature close to the superfluid transition. We emphasize that the enhanced conductance arises because of the bosonic nature of closed channel molecules and the low dimensionality of the quantum point contact. 8. Upgrading Fermi Without Traveling to Space Science.gov (United States) Kohler, Susanna 2016-02-01 The Large Area Telescope (LAT) on board the Fermi Gamma-ray Space Telescope has received an upgrade that increased its sensitivity by a whopping 40% and nobody had to travel to space to make it happen! The difference instead stems from remarkable improvement to the software used to analyze Fermi-LATs data, and it has resulted in a new high-energy map of our sky.Animation (click to watch!) comparing the Pass 7 to the Pass 8 Fermi-LAT analysis, in a region in the constellation Carina. Pass 8 provides more accurate directions for incoming gamma rays, so more of them fall closer to their sources, creating taller spikes and a sharper image. [NASA/DOE/Fermi LAT Collaboration]Pass 8Fermi-LAT has been surveying the whole sky since August 2008. It detects gamma-ray photons by converting them into electron-positron pairs and tracking the paths of these charged particles. But differentiating this signal from the charged cosmic rays that also pass through the detector with a flux that can be 10,000 times larger! is a challenging process. Making this distinction and rebuilding the path of the original gamma ray relies on complex analysis software.Pass 8 is a complete reprocessing of all data collected by Fermi-LAT. The software has gone through many revisions before now, but this is the first revision that has taken into account all of the experience that the Fermi team has gained operating the LAT in its orbital environment.The improvements made in Pass 8 include better background rejection of misclassified charged particles, improvements to the point spread function and effective area of the detector, and an extension of the effective energy range from below 100 MeV to beyond a few hundred GeV. The changes made in Pass 8 have increased the sensitivity of Fermi-LAT by an astonishing 40%.Map of the High-Energy SkySky map of the sources in the 2FHL catalog, classified by their most likely association. Click for a better look! [Ackermann et al. 2016]The first result from the 9. Fermi Normal Coordinates and Fermion Curvature Couplings in General Relativity CERN Document Server Dey, Anshuman; Sarkar, Tapobrata 2014-01-01 We study gravitational curvature effects in circular and radial geodesics in static, spherically symmetric space-times, using Fermi normal coordinates. We first set up these coordinates in the general case, and then use this to study effective magnetic fields due to gravitational curvature in the exterior and interior Schwarzschild, Janis-Newman-Winicour, and Bertrand space-times. We show that these fields can be large for specific parameter values in the theories, and thus might have observational significance. We discuss the qualitative differences of the magnetic field for vacuum space-times and for those seeded by matter. We estimate the magnitude of these fields in realistic galactic scenarios and discuss their possible experimental relevance. Gravitational curvature corrections to the Hydrogen atom spectrum for these space-times are also discussed briefly. 10. Advances in atomic, molecular, and optical physics CERN Document Server Berman, Paul R; Arimondo, Ennio 2006-01-01 Volume 54 of the Advances Series contains ten contributions, covering a diversity of subject areas in atomic, molecular and optical physics. The article by Regal and Jin reviews the properties of a Fermi degenerate gas of cold potassium atoms in the crossover regime between the Bose-Einstein condensation of molecules and the condensation of fermionic atom pairs. The transition between the two regions can be probed by varying an external magnetic field. Sherson, Julsgaard and Polzik explore the manner in which light and atoms can be entangled, with applications to quantum information processing 11. Pairing in a dry Fermi sea. Science.gov (United States) Maier, T A; Staar, P; Mishra, V; Chatterjee, U; Campuzano, J C; Scalapino, D J 2016-06-17 In the traditional Bardeen-Cooper-Schrieffer theory of superconductivity, the amplitude for the propagation of a pair of electrons with momentum k and -k has a log singularity as the temperature decreases. This so-called Cooper instability arises from the presence of an electron Fermi sea. It means that an attractive interaction, no matter how weak, will eventually lead to a pairing instability. However, in the pseudogap regime of the cuprate superconductors, where parts of the Fermi surface are destroyed, this log singularity is suppressed, raising the question of how pairing occurs in the absence of a Fermi sea. Here we report Hubbard model numerical results and the analysis of angular-resolved photoemission experiments on a cuprate superconductor. In contrast to the traditional theory, we find that in the pseudogap regime the pairing instability arises from an increase in the strength of the spin-fluctuation pairing interaction as the temperature decreases rather than the Cooper log instability. 12. Fermi's Paradox - The Last Challenge for Copernicanism? CERN Document Server Cirkovic, Milan M 2009-01-01 We review Fermi's paradox (or the "Great Silence" problem), not only arguably the oldest and crucial problem for the Search for ExtraTerrestrial Intelligence (SETI), but also a conundrum of profound scientific, philosophical and cultural importance. By a simple analysis of observation selection effects, the correct resolution of Fermi's paradox is certain to tell us something about the future of humanity. Already a more than three quarters of a century old puzzle - and a quarter of century since the last major review paper in the field by G. David Brin - Fermi's paradox has generated many ingenious discussions and hypotheses. We analyze the often tacit methodological assumptions built into various answers to this puzzle and attempt a new classification of the numerous solutions proposed in an already huge literature on the subject. Finally, we consider the ramifications of various classes of hypotheses for the practical SETI projects. Somewhat paradoxically, it seems that the class of (neo)catastrophic hypoth... 13. Beyond the 2nd Fermi Pulsar Catalog CERN Document Server Hou, Xian; Reposeur, Thierry; Rousseau, Romain 2013-01-01 Over thirteen times more gamma-ray pulsars have now been studied with the Large Area Telescope on NASA's Fermi satellite than the ten seen with the Compton Gamma-Ray Observatory in the nineteen-nineties. The large sample is diverse, allowing better understanding both of the pulsars themselves and of their roles in various cosmic processes. Here we explore the prospects for even more gamma-ray pulsars as Fermi enters the 2nd half of its nominal ten-year mission. New pulsars will naturally tend to be fainter than the first ones discovered. Some of them will have unusual characteristics compared to the current population, which may help discriminate between models. We illustrate a vision of the future with a sample of six pulsars discovered after the 2nd Fermi Pulsar Catalog was written. 14. Fermi surface properties of paramagnetic NpCd{sub 11} with a large unit cell Energy Technology Data Exchange (ETDEWEB) Homma, Yoshiya; Aoki, Dai; Shiokawa, Yoshinobu [Institute for Materials Research, Tohoku University, Oarai, Ibaraki 311-1313 (Japan); Haga, Yoshinori; Sakai, Hironori; Ikeda, Shugo; Yamamoto, Etsuji; Nakamura, Akio; Onuki, Yoshichika [Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195 (Japan); Settai, Rikio [Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043 (Japan); Takeuchi, Tetsuya [Cryogenic Center, Osaka University, Toyonaka, Osaka 560-0043 (Japan); Yamagami, Hiroshi, E-mail: [email protected] [Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto 603-8555 (Japan) 2010-03-15 We succeeded in growing a high-quality single crystal of NpCd{sub 11} with the cubic BaHg{sub 11}-type structure by the Cd-self flux method. The lattice parameter of a = 9.2968(2) A and crystallographic positions of the atoms were determined by x-ray single-crystal structure analysis. From the results of the magnetic susceptibility and specific heat experiments, this compound is found to be a 5f-localized paramagnet with the singlet ground state in the crystalline electric field (CEF) scheme. Fermi surface properties were measured using the de Haas-van Alphen (dHvA) technique. Long-period oscillations were observed in the dHvA frequency range of 9.1 x 10{sup 5} to 1.9 x 10{sup 7} Oe, indicating small cross-sectional areas of Fermi surfaces, which is consistent with a small Brillouin zone based on a large unit cell. From the results of dHvA and magnetoresistance experiments, the Fermi surface of NpCd{sub 11} is found to consist of many kinds of closed Fermi surfaces and a multiply-connected-like Fermi surface, although the result of energy band calculations based on the 5f-localized Np{sup 3+}(5f{sup 4}) configuration reveals the existence of only closed Fermi surfaces. The corresponding cyclotron effective mass is small, ranging from 0.1 to 0.7 m{sub 0}, which is consistent with a small electronic specific heat coefficient {gamma} {approx_equal} 10mJ/K{sup 2{center_dot}}mol, revealing no hybridization between the 5f electrons and conduction electrons. 15. Pseudogap-generated a coexistence of Fermi arcs and Fermi pockets in cuprate superconductors Science.gov (United States) Zhao, Huaisong; Gao, Deheng; Feng, Shiping 2017-03-01 One of the most intriguing puzzle is why there is a coexistence of Fermi arcs and Fermi pockets in the pseudogap phase of cuprate superconductors? This puzzle is calling for an explanation. Based on the t - J model in the fermion-spin representation, the coexistence of the Fermi arcs and Fermi pockets in cuprate superconductors is studied by taking into account the pseudogap effect. It is shown that the pseudogap induces an energy band splitting, and then the poles of the electron Green's function at zero energy form two contours in momentum space, however, the electron spectral weight on these two contours around the antinodal region is gapped out by the pseudogap, leaving behind the low-energy electron spectral weight only located at the disconnected segments around the nodal region. In particular, the tips of these disconnected segments converge on the hot spots to form the closed Fermi pockets, generating a coexistence of the Fermi arcs and Fermi pockets. Moreover, the single-particle coherent weight is directly related to the pseudogap, and grows linearly with doping. The calculated result of the overall dispersion of the electron excitations is in qualitative agreement with the experimental data. The theory also predicts that the pseudogap-induced peak-dip-hump structure in the electron spectrum is absent from the hot-spot directions. 16. High order harmonic generation in rare gases Energy Technology Data Exchange (ETDEWEB) Budil, Kimberly Susan [Univ. of California, Davis, CA (United States) 1994-05-01 The process of high order harmonic generation in atomic gases has shown great promise as a method of generating extremely short wavelength radiation, extending far into the extreme ultraviolet (XUV). The process is conceptually simple. A very intense laser pulse (I ~1013-1014 W/cm2) is focused into a dense (~1017 particles/cm3) atomic medium, causing the atoms to become polarized. These atomic dipoles are then coherently driven by the laser field and begin to radiate at odd harmonics of the laser field. This dissertation is a study of both the physical mechanism of harmonic generation as well as its development as a source of coherent XUV radiation. Recently, a semiclassical theory has been proposed which provides a simple, intuitive description of harmonic generation. In this picture the process is treated in two steps. The atom ionizes via tunneling after which its classical motion in the laser field is studied. Electron trajectories which return to the vicinity of the nucleus may recombine and emit a harmonic photon, while those which do not return will ionize. An experiment was performed to test the validity of this model wherein the trajectory of the electron as it orbits the nucleus or ion core is perturbed by driving the process with elliptically, rather than linearly, polarized laser radiation. The semiclassical theory predicts a rapid turn-off of harmonic production as the ellipticity of the driving field is increased. This decrease in harmonic production is observed experimentally and a simple quantum mechanical theory is used to model the data. The second major focus of this work was on development of the harmonic "source". A series of experiments were performed examining the spatial profiles of the harmonics. The quality of the spatial profile is crucial if the harmonics are to be used as the source for experiments, particularly if they must be refocused. 17. The Mirage of the Fermi Scale DEFF Research Database (Denmark) Antipin, Oleg; Sannino, Francesco; Tuominen, Kimmo 2013-01-01 The discovery of a light Higgs boson at LHC may be suggesting that we need to revise our model building paradigms to understand the origin of the weak scale. We explore the possibility that the Fermi scale is not fundamental but rather a derived one, i.e. a low energy mirage. We show that this sc......The discovery of a light Higgs boson at LHC may be suggesting that we need to revise our model building paradigms to understand the origin of the weak scale. We explore the possibility that the Fermi scale is not fundamental but rather a derived one, i.e. a low energy mirage. We show... 18. Scattering resonances in a degenerate Fermi gas DEFF Research Database (Denmark) Challis, Katharine; Nygaard, Nicolai; Mølmer, Klaus 2009-01-01 We consider elastic single-particle scattering from a one-dimensional trapped two-component superfluid Fermi gas when the incoming projectile particle is identical to one of the confined species. Our theoretical treatment is based on the Hartree-Fock ground state of the trapped gas and a configur......We consider elastic single-particle scattering from a one-dimensional trapped two-component superfluid Fermi gas when the incoming projectile particle is identical to one of the confined species. Our theoretical treatment is based on the Hartree-Fock ground state of the trapped gas... 19. Supernova Remnants with Fermi Large Area Telescope Directory of Open Access Journals (Sweden) Caragiulo M. 2017-01-01 Full Text Available The Large Area Telescope (LAT, on-board the Fermi satellite, proved to be, after 8 years of data taking, an excellent instrument to detect and observe Supernova Remnants (SNRs in a range of energies running from few hundred MeV up to few hundred GeV. It provides essential information on physical processes that occur at the source, involving both accelerated leptons and hadrons, in order to understand the mechanisms responsible for the primary Cosmic Ray (CR acceleration. We show the latest results in the observation of Galactic SNRs by Fermi-LAT. 20. Clustering in the nuclear Fermi liquid CERN Document Server Ebran, J -P; Niksic, T; Vretenar, D 2012-01-01 We analyze the emergence of various structures in nucleonic matter, such as crystal, clusters, liquid drops and haloes. The formation of clusters indicates that nuclei behave like a Fermi liquid close to the liquid to solid transition. The relevant parameter is the ratio of the dispersion of the single-nucleon wave functions in the nucleus to the inter-nucleon distance. We also discuss the relationship between cluster states in nuclei and the pasta phase in the crust of neutron stars, as a transitional state between a Fermi liquid and a crystal. Haloes and clusters exhibit opposite features with respect to nucleonic localization. 1. Shock instability in dissipative gases OpenAIRE 2011-01-01 Previous experiments have revealed that shock waves in thermally relaxing gases, such as ionizing, dissociating and vibrationally excited gases, can become unstable. To date, the mechanism controlling this instability has not been resolved. Previous accounts of the D'yakov-Kontorovich instability, and Bethe-Zel'dovich-Thompson behaviour could not predict the experimentally observed instability. To address the mechanism controlling the instability, we study the propagation of shock waves in a ... 2. Deep Inelastic Scattering on Ultracold Gases Science.gov (United States) Hofmann, Johannes; Zwerger, Wilhelm 2017-01-01 We discuss Bragg scattering on both Bose and Fermi gases with strong short-range interactions in the deep inelastic regime of large wave vector transfer q , where the dynamic structure factor is dominated by a resonance near the free-particle energy ℏω =ɛq=ℏ2q2/2 m . Using a systematic short-distance expansion, the structure factor at high momentum is shown to exhibit a nontrivial dependence on frequency characterized by two separate scaling regimes. First, for frequencies that differ from the single-particle energy by terms of order O (q ) (i.e., small deviations compared to the single-particle energy), the dynamic structure factor is described by the impulse approximation of Hohenberg and Platzman. Second, deviations of order O (q2) (i.e., of the same order or larger than the single-particle energy) are described by the operator product expansion, with a universal crossover connecting both regimes. The scaling is consistent with the leading asymptotics for a number of sum rules in the large momentum limit. Furthermore, we derive an exact expression for the shift and width of the single-particle peak at large momentum due to interactions, thus extending a result by Beliaev [J. Exp. Theor. Phys. 7, 299 (1958)] for the low-density Bose gas to arbitrary values of the scattering length a . The shift exhibits a maximum around q a ≃1 , which is connected with a maximum in the static structure factor due to strong short-range correlations. For Bose gases with moderate interaction strengths, the theoretically predicted shift is consistent with the value observed by Papp et al. [Phys. Rev. Lett. 101, 135301 (2008), 10.1103/PhysRevLett.101.135301]. Finally, we develop a diagrammatic theory for the dynamic structure factor which accounts for the correlations beyond Bogoliubov theory. It covers the full range of momenta and frequencies and provides an explicit example for the emergence of asymptotic scaling at large momentum. 3. The Physical and Dynamical Properties of Gas that Molds the Fermi Bubbles Science.gov (United States) Jenkins, Edward 2012-10-01 Two sharply defined lobes of gamma-ray emission emerging from the center of our Galaxy, called the Fermi Bubbles, have been discovered in the Galactic halo. Their emissivity appears to be uniform and extends up to 8 kpc on either side of the plane. Accompanying the Fermi Bubbles are excess emissions seen in X-rays and microwaves. It is generally believed that cosmic ray particles emitted from the central portion of the Galactic disk {or perhaps the nucleus itself} are responsible for these emissions. These particles must have been advected into the halo by a wind or shock. Our goal is to gain a better understanding of the nature of this gaseous transport by viewing the UV spectra of bright, extragalactic sources behind one of the Fermi Bubbles and its surrounding regions. We plan to obtain COS spectra of 5 such objects, with the goal of measuring absorption features from Si III, Si IV, C IV and N V. We expect that our mapping of column densities and kinematics of the gases will help us to distinguish a shock from a wind. Moreover, if a shock is present, we should be able to evaluate the product of its age and the density of the gas by comparing the column densities of different species. 4. Parametric optimum analysis of an irreversible Ericsson cryogenic refrigeration cycle working with an ideal Fermi gas Bihong Lin; Yingru Zhao; Jincan Chen 2008-05-01 An irreversible model of an Ericsson cryogenic refrigeration cycle working with an ideal Fermi gas is established, which is composed of two isothermal and two isobaric processes. The influence of both the quantum degeneracy and the finite-rate heat transfer between the working fluid and the heat reservoirs on the performance of the cycle is investigated, based on the theory of statistical mechanics and thermodynamic properties of an ideal Fermi gas. The inherent regeneration losses of the cycle are analyzed. Expressions for several important performance parameters such as the coefficient of performance, cooling rate and power input are derived. By using numerical solutions, the cooling rate of the cycle is optimized for a given power input. The maximum cooling rate and the corresponding parameters are calculated numerically. The optimal regions of the coefficient of performance and power input are determined. Especially, the optimal performance of the cycle in the strong and weak gas degeneracy cases and the high temperature limit is discussed in detail. The analytic expressions of some optimized parameters are derived. Some optimum criteria are given. The distinctions and connections between the Ericsson refrigeration cycles working with the Fermi and classical gases are revealed. 5. A two-dimensional Fermi gas in the BEC-BCS crossover Energy Technology Data Exchange (ETDEWEB) Ries, Martin Gerhard 2016-01-21 This thesis reports on the preparation of a 2D Fermi gas in the BEC-BCS crossover and the observation of the BKT transition into a quasi long-range ordered superfluid phase. The pair momentum distribution of the gas is probed by means of a matter-wave focusing technique which relies on time-of-flight evolution in a weak harmonic potential. This distribution holds the coherence properties of the gas. The quasi long-range ordered phase manifests itself as a sharp low-momentum peak. The temperature where it forms is identified as the transition temperature. By tuning the temperature and the interaction strength, the phase diagram of the 2D Fermi gas in the BEC-BCS crossover is mapped out. The phase coherence is investigated in a self-interference experiment. Furthermore, algebraic decay of correlations is observed in the trap average of the first order correlation function, which is obtained from the Fourier transform of the pair momentum distribution. This is in qualitative agreement with predictions of homogeneous theory for the superfluid phase in a 2D gas. The presented results provide a foundation for future experimental and theoretical studies of strongly correlated 2D Fermi gases. They might thus help to elucidate complex systems such as the electron gas in high-T{sub c} superconductors. 6. Predicted Abundances of Carbon Compounds in Volcanic Gases on Io CERN Document Server Schaefer, L; Schaefer, Laura 2004-01-01 We use chemical equilibrium calculations to model the speciation of carbon in volcanic gases on Io. The calculations cover wide temperature (500-2000 K), pressure (10^-8 to 10^+2 bars), and composition ranges (bulk O/S atomic ratios \\~0 to 3), which overlap the nominal conditions at Pele (1760 K, 0.01 bar, O/S ~ 1.5). Bulk C/S atomic ratios ranging from 10^-6 to 10^-1 in volcanic gases are used with a nominal value of 10^-3 based upon upper limits from Voyager for carbon in the Loki plume on Io. Carbon monoxide and CO2 are the two major carbon gases under all conditions studied. Carbonyl sulfide and CS2 are orders of magnitude less abundant. Consideration of different loss processes (photolysis, condensation, kinetic reactions in the plume) indicates that photolysis is probably the major loss process for all gases. Both CO and CO2 should be observable in volcanic plumes and in Io's atmosphere at abundances of several hundred parts per million by volume for a bulk C/S ratio of 10^-3. 7. Fermi and the Theory of Weak Interactions CERN Document Server Rajasekaran, G 2014-01-01 The history of weak interactions starting with Fermi's creation of the beta decay theory and culminating in its modern avatar in the form of the electroweak gauge theory is described. Discoveries of parity violation, matter-antimatter asymmetry, W and Z bosons and neutrino mass are highlighted. 8. Fermi Large Area Telescope Second Source Catalog CERN Document Server , 2011-01-01 We present the second catalog of high-energy gamma-ray sources detected by the Large Area Telescope (LAT), the primary science instrument on the Fermi Gamma-ray Space Telescope (Fermi), derived from data taken during the first 24 months of the science phase of the mission, which began on 2008 August 4. Source detection is based on the average flux over the 24-month period. The Second Fermi-LAT catalog (2FGL) includes source location regions, defined in terms of elliptical fits to the 95% confidence regions and spectral fits in terms of power-law, exponentially cutoff power-law, or log-normal forms. Also included are flux measurements in 5 energy bands and light curves on monthly intervals for each source. Twelve sources in the catalog are modeled as spatially extended. We provide a detailed comparison of the results from this catalog with those from the first Fermi-LAT catalog (1FGL). Although the diffuse Galactic and isotropic models used in the 2FGL analysis are improved compared to the 1FGL catalog, we att... 9. FERMI LARGE AREA TELESCOPE SECOND SOURCE CATALOG Energy Technology Data Exchange (ETDEWEB) Nolan, P. L.; Ajello, M.; Allafort, A.; Bechtol, K.; Berenji, B.; Blandford, R. D.; Bloom, E. D. [W. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology, Department of Physics and SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305 (United States); Abdo, A. A. [Center for Earth Observing and Space Research, College of Science, George Mason University, Fairfax, VA 22030 (United States); Ackermann, M. [Deutsches Elektronen Synchrotron DESY, D-15738 Zeuthen (Germany); Antolini, E.; Bonamente, E. [Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, I-06123 Perugia (Italy); Atwood, W. B.; Belfiore, A. [Santa Cruz Institute for Particle Physics, Department of Physics and Department of Astronomy and Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95064 (United States); Axelsson, M. [Department of Astronomy, Stockholm University, SE-106 91 Stockholm (Sweden); Baldini, L.; Bellazzini, R. [Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa (Italy); Ballet, J. [Laboratoire AIM, CEA-IRFU/CNRS/Universite Paris Diderot, Service d' Astrophysique, CEA Saclay, 91191 Gif sur Yvette (France); Barbiellini, G. [Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34127 Trieste (Italy); Bastieri, D. [Istituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131 Padova (Italy); Bignami, G. F., E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Istituto Universitario di Studi Superiori (IUSS), I-27100 Pavia (Italy); and others 2012-04-01 We present the second catalog of high-energy {gamma}-ray sources detected by the Large Area Telescope (LAT), the primary science instrument on the Fermi Gamma-ray Space Telescope (Fermi), derived from data taken during the first 24 months of the science phase of the mission, which began on 2008 August 4. Source detection is based on the average flux over the 24 month period. The second Fermi-LAT catalog (2FGL) includes source location regions, defined in terms of elliptical fits to the 95% confidence regions and spectral fits in terms of power-law, exponentially cutoff power-law, or log-normal forms. Also included are flux measurements in five energy bands and light curves on monthly intervals for each source. Twelve sources in the catalog are modeled as spatially extended. We provide a detailed comparison of the results from this catalog with those from the first Fermi-LAT catalog (1FGL). Although the diffuse Galactic and isotropic models used in the 2FGL analysis are improved compared to the 1FGL catalog, we attach caution flags to 162 of the sources to indicate possible confusion with residual imperfections in the diffuse model. The 2FGL catalog contains 1873 sources detected and characterized in the 100 MeV to 100 GeV range of which we consider 127 as being firmly identified and 1171 as being reliably associated with counterparts of known or likely {gamma}-ray-producing source classes. 10. Radiatively Induced Fermi Scale in Grand Unification DEFF Research Database (Denmark) Alanne, Tommi; Meroni, Aurora; Sannino, Francesco; 2016-01-01 We consider Grand Unified Theories in which the hierarchy between the unification and the Fermi scale emerges radiatively. Within the Pati-Salam framework, we show that it is possible to construct a viable model where the Higgs is an elementary pseudo-Goldstone boson, and the correct hierarchy... 11. Switchable Fermi surface sheets in greigite NARCIS (Netherlands) Zhang, B.; de Wijs, G. A.; de Groot, R. A. 2012-01-01 Greigite (Fe3S4) and magnetite (Fe3O4) are isostructural and isoelectronic ferrimagnets with quite distinct properties. Electronic structure calculations reveal greigite is a normalmetal in contrast to half-metallic magnetite. Greigite shows a complex Fermi surface with a unique influence of relativ 12. Fermi detected blazars seen by INTEGRAL CERN Document Server Beckmann, V; Soldi, S 2009-01-01 Multiwavelength observations are essential to constrain physical parameters of the blazars observed by Fermi/LAT. Among the 187 AGN significantly detected in public INTEGRAL data above 20 keV by the imager IBIS/ISGRI, 20 blazars were detected. 15 of these sources allowed significant spectral extraction. They show hard X-ray spectra with an average photon index of 2.1+-0.1 and a hard X-ray luminosity of L(20-100 keV) = 1.3e46 erg/s. 15 of the INTEGRAL blazars are also visible in the first 16 months of the Fermi/LAT data, thus allowing to constrain the inverse Compton branch in these cases. Among others, we analyse the LAT data of four blazars which were not included in the Fermi LAT Bright AGN Sample based on the first 3 months of the mission: QSO B0836+710, H 1426+428, RX J1924.8-2914, and PKS 2149-306. Especially for blazars during bright outbursts, as already observed simultaneously by INTEGRAL and Fermi (e.g. 3C 454.3 and Mrk 421), INTEGRAL provides unique spectral coverage up to several hundred keV. We pr... 13. Fermi Large Area Telescope Second Source Catalog Science.gov (United States) Nolan, P. L.; Abdo, A. A.; Ackermann, M.; Ajello, M; Allafort, A.; Antolini, E; Bonnell, J.; Cannon, A.; Celik O.; Corbet, R.; Davis, D. S.; DeCesar, M. E.; Ferrara, E. C.; Gehrels, N.; Harding, A. K.; Hays, E.; Johnson, T. E.; McConville, W.; McEnery, J. E; Perkins, J. S.; Racusin, J. L; Scargle, J. D.; Stephens, T. E.; Thompson, D. J.; Troja, E. 2012-01-01 We present the second catalog of high-energy gamma-ray sources detected by the Large Area Telescope (LAT), the primary science instrument on the Fermi Gamma-ray Space Telescope (Fermi), derived from data taken during the first 24 months of the science phase of the mission, which began on 2008 August 4. Source detection is based on the average flux over the 24-month period. The Second Fermi-LAT catalog (2FGL) includes source location regions, defined in terms of elliptical fits to the 95% confidence regions and spectral fits in terms of power-law, exponentially cutoff power-law, or log-normal forms. Also included are flux measurements in 5 energy bands and light curves on monthly intervals for each source. Twelve sources in the catalog are modeled as spatially extended. We provide a detailed comparison of the results from this catalog with those from the first Fermi-LAT catalog (1FGL). Although the diffuse Galactic and isotropic models used in the 2FGL analysis are improved compared to the 1FGL catalog, we attach caution flags to 162 of the sources to indicate possible confusion with residual imperfections in the diffuse model. The 2FGL catalog contains 1873 sources detected and characterized in the 100 11eV to 100 GeV range of which we consider 127 as being firmly identified and 1171 as being reliably associated with counterparts of known or likely gamma-ray-producing source classes. 14. Automatic Cloud Bursting under FermiCloud Energy Technology Data Exchange (ETDEWEB) Wu, Hao [Fermilab; Shangping, Ren [IIT; Garzoglio, Gabriele [Fermilab; Timm, Steven [Fermilab; Bernabeu, Gerard [Fermilab; Kim, Hyun Woo; Chadwick, Keith; Jang, Haengjin [KISTI, Daejeon; Noh, Seo-Young [KISTI, Daejeon 1900-01-01 Cloud computing is changing the infrastructure upon which scientific computing depends from supercomputers and distributed computing clusters to a more elastic cloud-based structure. The service-oriented focus and elasticity of clouds can not only facilitate technology needs of emerging business but also shorten response time and reduce operational costs of traditional scientific applications. Fermi National Accelerator Laboratory (Fermilab) is currently in the process of building its own private cloud, FermiCloud, which allows the existing grid infrastructure to use dynamically provisioned resources on FermiCloud to accommodate increased but dynamic computation demand from scientists in the domains of High Energy Physics (HEP) and other research areas. Cloud infrastructure also allows to increase a private cloud’s resource capacity through “bursting” by borrowing or renting resources from other community or commercial clouds when needed. This paper introduces a joint project on building a cloud federation to support HEP applications between Fermi National Accelerator Laboratory and Korea Institution of Science and Technology Information, with technical contributions from the Illinois Institute of Technology. In particular, this paper presents two recent accomplishments of the joint project: (a) cloud bursting automation and (b) load balancer. Automatic cloud bursting allows computer resources to be dynamically reconfigured to meet users’ demands. The load balance algorithm which the cloud bursting depends on decides when and where new resources need to be allocated. Our preliminary prototyping and experiments have shown promising success, yet, they also have opened new challenges to be studied 15. Thermodynamics and statistical mechanics. [thermodynamic properties of gases Science.gov (United States) 1976-01-01 The basic thermodynamic properties of gases are reviewed and the relations between them are derived from the first and second laws. The elements of statistical mechanics are then formulated and the partition function is derived. The classical form of the partition function is used to obtain the Maxwell-Boltzmann distribution of kinetic energies in the gas phase and the equipartition of energy theorem is given in its most general form. The thermodynamic properties are all derived as functions of the partition function. Quantum statistics are reviewed briefly and the differences between the Boltzmann distribution function for classical particles and the Fermi-Dirac and Bose-Einstein distributions for quantum particles are discussed. 16. Thermodynamics of ultracold Bose gases at a dimensional crossover Science.gov (United States) Labouvie, Ralf; Vogler, Andreas; Guarrera, Vera; Ott, Herwig 2013-05-01 We have studied the thermodynamics of ultracold Bose gases in the crossover from a three-dimensional to a one-dimensional regime. In our experiment, we use a focused electron-beam to probe in situ atomic density distributions with high temporal and spatial resolution. Starting with a Bose-Einstein-Condensate in a single beam optical dipole trap we can create one-dimensional systems by loading the atoms in a two-dimensional blue-detuned optical lattice. With increasing strength of the lattices we go from a three-dimensional into a one-dimensional system. Furthermore we tune the interaction strengths of the one-dimensional quantum-gases from weak (quasi-condensate) to strong (Tonks-Girardeau). By measuring the density profiles and applying an inverse Abel-Transformation we extract the equation of states of these systems and characterize the crossover from the three-dimensional to the one-dimensional regime. 17. 76 FR 1197 - Detroit Edison Company, FERMI 2; Exemption Science.gov (United States) 2011-01-07 ... COMMISSION Detroit Edison Company, FERMI 2; Exemption 1.0 Background Detroit Edison Company (DECo) (the licensee) is the holder of Facility Operating License No. NFP-43 which authorizes operation of the Fermi 2... exemption stated that a tornado swept across the Fermi 2 property on June 6, 2010, and that the... 18. 75 FR 15748 - Detroit Edison Company; Fermi 2; Exemption Science.gov (United States) 2010-03-30 ... COMMISSION Detroit Edison Company; Fermi 2; Exemption 1.0 Background Detroit Edison Company (the licensee) is the holder of Facility Operating License No. NPF-43, which authorizes operation of Fermi 2. The...- September 11, 2001, security orders. It is from five of these new requirements that Fermi 2 now seeks... 19. Nonequilibrium statistical mechanics in one-dimensional bose gases Science.gov (United States) Baldovin, F.; Cappellaro, A.; Orlandini, E.; Salasnich, L. 2016-06-01 We study cold dilute gases made of bosonic atoms, showing that in the mean-field one-dimensional regime they support stable out-of-equilibrium states. Starting from the 3D Boltzmann-Vlasov equation with contact interaction, we derive an effective 1D Landau-Vlasov equation under the condition of a strong transverse harmonic confinement. We investigate the existence of out-of-equilibrium states, obtaining stability criteria similar to those of classical plasmas. 20. 40 CFR 1065.750 - Analytical gases. Science.gov (United States) 2010-07-01 ... 40 Protection of Environment 32 2010-07-01 2010-07-01 false Analytical gases. 1065.750 Section... ENGINE-TESTING PROCEDURES Engine Fluids, Test Fuels, Analytical Gases and Other Calibration Standards § 1065.750 Analytical gases. Analytical gases must meet the accuracy and purity specifications of... 1. Spectral anisotropy of a photoresponse from heterojunctions based on GaSe and InSe layered crystals Science.gov (United States) Katerinchuk, V. N.; Kudrynskyi, Z. R.; Kovalyuk, Z. D. 2014-03-01 The object of investigation is photoresponse spectra taken from the cleaved end face of heterojunctions formed by GaSe and InSe anisotropic crystals. Spectra taken from the as-prepared and chemically processed faces of the heterojunctions are compared. A modified method of growing GaSe crystals with a virgin end face is suggested, and the surface of GaSe crystals thus grown is examined by atomic force microscopy. 2. Kinetic theory the nature of gases and of heat CERN Document Server Brush, Stephen G 1965-01-01 Kinetic Theory, Volume I: The Nature of Gases and of Heat covers the developments in area of kinetic theory, statistical mechanics, and thermodynamics. This book is organized into two parts encompassing 11 chapters. The book starts with an overview of the history of atomism, the caloric theory, the conservation of energy, the virial theorem, and atomic magnitudes. The second part deals first with the delineation of observed phenomena of motions through the repulsion theory. This part also considers other forces of nature, including fire and heat, with emphasis on the nature of motion of these 3. Three-Body Recombination in Cold Atomic Gases CERN Document Server Sørensen, Peder K 2013-01-01 Systems of three particles show a surprising feature in their bound state spectrum: a series of geometrically scaled states, known as Efimov states. These states have not yet been observed directly, but many recent experiments show indirect evidence of their existence via the so-called recombination process. The theories that predict the Efimov states also predicts either resonant enhancement of the recombination process or suppression by destructive interference, depending on the sign of the interaction between the particles. The theories predict universal features for the Efimov states, for instance that the geometric scaling factor is 22.7, meaning that one state is 22.7 times larger than its lower lying neighbour state. This thesis seeks to investigate non-universal effects by incorporating additional information about the physical interactions into the universal theories. 4. The Chemistry of the Noble Gases, Understanding the Atom Series. Science.gov (United States) Chernick, Cedric L. The history of the discovery, isolation, characterization, production and use of argon, krypton, xenon, helium, and radon is followed by an account of early attempts to react them with other elements. The use of the electron shell theory of valence to explain their inertness and the reactions of chemists to the production of xenon compounds is… 5. [Effect on Fermi Resonance by Some External Fields: Investigation of Fermi Resonance According to Raman Spectra]. Science.gov (United States) Jiang, Xiu-lan; Sun, Cheng-lin; Zhou, Mi; Li, Dong-fei; Men, Zhi-wei; Li, Zuo-wei; Gao, Shu-qin 2015-03-01 Fermi resonance is a phenomenon of molecular vibrational coupling and energy transfer occurred between different groups of a single molecule or neighboring molecules. Many properties of Fermi resonance under different external fields, the investigation method of Raman spectroscopy as well as the application of Fermi resonance, etc need to be developed and extended further. In this article the research results and development about Fermi resonance obtained by Raman spectral technique were introduced systematically according to our work and the results by other researchers. Especially, the results of the behaviors of intramolecular and intermolecular Fermi resonance of some molecules under some external fields such as molecular field, pressure field and temperature field, etc were investigated and demonstrated in detail according to the Raman spectra obtained by high pressure DAC technique, temperature variation technique as well as the methods we planed originally in our group such as solution concentration variation method and LCOF resonance Raman spectroscopic technique, and some novel properties of Fermi resonance were found firstly. Concretely, (1) Under molecular field. a. The Raman spectra of C5H5 N in CH3 OH and H2O indicates that solvent effect can influence Fermi resonance distinctly; b. The phenomena of the asymmetric movement of the Fermi resonance doublets as well as the fundamental involved is tuned by the Fermi resonance which had not been found by other methods were found firstly by our variation solution concentration method; c. The Fermi resonance properties can be influenced distinctly by the molecular group reorganization induced by the hydrogen bond and anti-hydrogen bond in solution; d. Fermi resonance can occurred between C7 H8 and m-C8H10, and the Fermi resonance properties behave quite differently with the solution concentration; (2) Under pressure field. a. The spectral lines shift towards high wavenumber with increasing pressure, and 6. Conduction of molecular electronic devices: Qualitative insights through atom-atom polarizabilities Energy Technology Data Exchange (ETDEWEB) Stuyver, T.; Fias, S., E-mail: [email protected]; De Proft, F.; Geerlings, P. [ALGC, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel (Belgium); Fowler, P. W. [Department of Chemistry, University of Sheffield, Sheffield S3 7HF (United Kingdom) 2015-03-07 The atom-atom polarizability and the transmission probability at the Fermi level, as obtained through the source-and-sink-potential method for every possible configuration of contacts simultaneously, are compared for polycyclic aromatic compounds. This comparison leads to the conjecture that a positive atom-atom polarizability is a necessary condition for transmission to take place in alternant hydrocarbons without non-bonding orbitals and that the relative transmission probability for different configurations of the contacts can be predicted by analyzing the corresponding atom-atom polarizability. A theoretical link between the two considered properties is derived, leading to a mathematical explanation for the observed trends for transmission based on the atom-atom polarizability. 7. Conduction of molecular electronic devices: qualitative insights through atom-atom polarizabilities. Science.gov (United States) Stuyver, T; Fias, S; De Proft, F; Fowler, P W; Geerlings, P 2015-03-07 The atom-atom polarizability and the transmission probability at the Fermi level, as obtained through the source-and-sink-potential method for every possible configuration of contacts simultaneously, are compared for polycyclic aromatic compounds. This comparison leads to the conjecture that a positive atom-atom polarizability is a necessary condition for transmission to take place in alternant hydrocarbons without non-bonding orbitals and that the relative transmission probability for different configurations of the contacts can be predicted by analyzing the corresponding atom-atom polarizability. A theoretical link between the two considered properties is derived, leading to a mathematical explanation for the observed trends for transmission based on the atom-atom polarizability. 8. Desulphurization of exhaust gases in chemical processes Energy Technology Data Exchange (ETDEWEB) Asperger, K.; Wischnewski, W. 1981-01-01 The sulfur content of exhaust gases can be reduced by: desulphurization of fuels; modification of processes; or treatment of resultant gases. In this paper a few selected examples from the chemical industry in the German Democratic Republic are presented. Using modified processes and treating the resultant gases, the sulphuric content of exhaust gases is effectively reduced. Methods to reduce the sulfur content of exhaust gases are described in the field of production of: sulphuric acid; viscose; fertilizers; and paraffin. 9. Fermi Arc Evolution and Doping Mechanism in High-Temperature Superconductors Science.gov (United States) Sunko, Denis K.; Pelc, Damjan; Požek, Miroslav; Despoja, Vito; Lazic, Predrag 2015-03-01 We calculate realistic Fermi surface (FS) evolution of La2-xSrxCuO4 (LSCO) with Sr doping within an extensive ab-initio framework including advanced band-unfolding techniques. We show that ordinary Kohn-Sham DFT+U can reproduce the observed metal-insulator transition and arc growth, when not restricted to the paramagnetic solution space. We elucidate both arc protection and the inadequacy of the rigid-band picture as consequences of a rapid change in orbital symmetry at the Fermi energy: the material undergoes a dimensional crossover along the Fermi surface, between the nodal (2D) and antinodal (3D) regions. In LSCO, this crossover accounts for FS arcs and the antinodal pseudogap, otherwise ubiquitous phenomena in high-Tc cuprates. The same calculation shows that the Sr hole stays localized in the vicinity of the dopand atom, indicating that metallization of the Cu-O plane is due to an orbital transition between Cu and O planar sites, originally proposed by Mazumdar in 1989. We can directly observe effects of the transition in charge transfers among in-plane atoms, which are different than predicted by non-interacting coherent models. This ionic doping'' mechanism has close parallels to modern views on the metallization of interfaces. 10. Landau damping in a dipolar Bose-Fermi mixture in the Bose-Einstein condensation (BEC) limit Science.gov (United States) Moniri, S. M.; Yavari, H.; Darsheshdar, E. 2016-12-01 By using a mean-field approximation which describes the coupled oscillations of condensate and noncondensate atoms in the collisionless regime, Landau damping in a dilute dipolar Bose-Fermi mixture in the BEC limit where Fermi superfluid is treated as tightly bounded molecules, is investigated. In the case of a uniform quasi-two-dimensional (2D) case, the results for the Landau damping due to the Bose-Fermi interaction are obtained at low and high temperatures. It is shown that at low temperatures, the Landau damping rate is exponentially suppressed. By increasing the strength of dipolar interaction, and the energy of boson quasiparticles, Landau damping is suppressed over a broader temperature range. 11. Superconductivity. Fermi arcs in a doped pseudospin-1/2 Heisenberg antiferromagnet. Science.gov (United States) Kim, Y K; Krupin, O; Denlinger, J D; Bostwick, A; Rotenberg, E; Zhao, Q; Mitchell, J F; Allen, J W; Kim, B J 2014-07-11 High-temperature superconductivity in cuprates arises from an electronic state that remains poorly understood. We report the observation of a related electronic state in a noncuprate material, strontium iridate (Sr2IrO4), in which the distinct cuprate fermiology is largely reproduced. Upon surface electron doping through in situ deposition of alkali-metal atoms, angle-resolved photoemission spectra of Sr2IrO4 display disconnected segments of zero-energy states, known as Fermi arcs, and a gap as large as 80 millielectron volts. Its evolution toward a normal metal phase with a closed Fermi surface as a function of doping and temperature parallels that in the cuprates. Our result suggests that Sr2IrO4 is a useful model system for comparison to the cuprates. 12. The 1st Fermi LAT SNR Catalog: the Impact of Interstellar Emission Modeling CERN Document Server Brandt, T J; de Palma, F; Johannesson, G; Tibaldo, L 2013-01-01 Galactic interstellar emission contributes substantially to Fermi LAT observations in the Galactic plane, the location of the majority of supernova remnants (SNRs). To explore some systematic effects on SNRs' properties caused by interstellar emission modeling, we have developed a method comparing the official LAT interstellar emission model results to eight alternative models. We created the eight alternative Galactic interstellar models by varying a few input parameters to GALPROP, namely the height of the cosmic ray propagation halo, cosmic ray source distribution in the Galaxy, and atomic hydrogen spin temperature. We have analyzed eight representative SNRs chosen to encompass a range of Galactic locations, extensions, and spectral properties using the eight different interstellar emission models. We will present the results and method in detail and discuss the implications for studies such as the 1st Fermi LAT SNR Catalog. 13. Evidence for an excited-state Efimov trimer in a three-component Fermi gas. Science.gov (United States) Williams, J R; Hazlett, E L; Huckans, J H; Stites, R W; Zhang, Y; O'Hara, K M 2009-09-25 We observe enhanced three-body recombination in a three-component ;{6}Li Fermi gas attributable to an excited Efimov trimer state intersecting the three-atom scattering threshold near 895 G. From measurements of the recombination rate we determine the Efimov parameters kappa_{} and eta_{} for the universal region above 600 G which includes three overlapping Feshbach resonances. The value of kappa_{*} also predicts the locations of loss features previously observed near 130 and 500 G [T. B. Ottenstein, Phys. Rev. Lett. 101, 203202 (2008)10.1103/PhysRevLett.101.203202; J. H. Huckans, Phys. Rev. Lett. 102, 165302 (2009)10.1103/PhysRevLett.102.165302] suggesting they are associated with a ground-state Efimov trimer near threshold. We also report on the realization of a degenerate three-component Fermi gas with approximate SU(3) symmetry. 14. Bandstructure and Fermi Surfaces of CeRh3B2 Science.gov (United States) Yamauchi, Kunihiko; Yanase, Akira; Harima, Hisatomo 2010-04-01 The electronic bandstructure and the Fermi surfaces of ferromagnetic CeRh3B2 are calculated by using FLAPW and LSDA+U method. As assuming several kinds of the ground state to describe the 4f electronic state, we propose a fully orbital- and spin-polarized state \| lz=0, sx=1/2 > as the ground state, instead of the conventional \\mathit{LS}-coupled CEF ground state, generally expected in typical 4f compounds. This is supported by the fact that both the observed magnetic moment and the observed dHvA frequencies are well explained by the calculated electronic structure and the Fermi surfaces. The unconventional ground state is stabilized by the strong 4f-4f direct mixing between the neighbored Ce atoms along the extremely small distance along the c-axis in the hexagonal crystal cell. 15. Study of Optical Band Gap of CuO Using Fermi's Golden Rule Science.gov (United States) Nemade, K. R.; Waghuley, S. A. 2012-05-01 Quantum size effect where the electronic and optical properties of solids are altered due to changes in the band structures, enhanced the surface/volume ratio in nano dimensions forces more than 33% of the atoms to be on the surface (for 10nm dot 35), which drastically altering the physical properties such as having lower melting temperature and lower sintering temperature, and higher diffusion force at elevated temperatures. Consequently, its Fermi's golden rule analysis becomes crucial. Cupric oxide (CuO) is an important transition metal oxide with the basis of several high temperature superconductors and giant magnetoresistance materials. In present investigation, optical Band Gap from UV data using Fermi's golden rule for single step chemically synthesized CuO was computed. 16. Magnetar Observations in the Fermi Era Science.gov (United States) Kouveliotou, Chryssa 2009-01-01 NASA s Fermi Observatory was launched June 11, 2009; the Fermi Gamma Ray Burst Monitor (GBM) began normal operations on July 14, about a month after launch, when the trigger algorithms were enabled. In the first 8 months of operations we recorded emission of three magnetar sources; of these, only one was an old magnetar: SGR 1806+20. The other two detections were: SGR J0501+4516, newly discovered with Swift and extensively monitored with both Swift and GBM, and SGR J1550-5418, a source originally classified as an Anomalous X-ray Pulsar (AXP). I report below on the current status of the analyses efforts of all these GBM data sets, combined with data from other satellites (Spitzer, RXTE, Chandra, Swift). 17. Magnetar Observations with Fermi/GBM Science.gov (United States) Kouveliotou, Chryssa 2009-01-01 NASA's Fermi Observatory was launched June 11, 2009; the Fermi Gamma Ray Burst Monitor (GBM) began normal operations on July 14, about a month after launch, when the trigger algorithms were enabled. In the first year of operations we recorded emission from four magnetar sources; of these, only one was an old magnetar: SGR 1806+20. The other three detections were: SGR J0501+4516, newly discovered with Swift and extensively monitored with both Swift and GBM, SGR J1550-5418, a source originally classified as an Anomalous X-ray Pulsar (AXP) and a very recently discovered new source, SGR 0418+5729. I report below on the current status of the analyses efforts of the GBM data. 18. Relativistic Beaming Effect in Fermi Blazars J. H. Fan; D. Bastieri; J. H. Yang; Y. Liu; D. X. Wu; S. H. Li 2014-09-01 The most identified sources observed by Fermi/LAT are blazars, based on which we can investigate the emission mechanisms and beaming effect in the -ray bands for blazars. Here, we used the compiled around 450 Fermi blazars with the available X-ray observations to estimate their Doppler factors and compared them with the integral -ray luminosity in the range of 1–100 GeV. It is interesting that the integral -ray luminosity is closely correlated with the estimated Doppler factor, log = (2.95 ± 0.09) log + 43.59 ± 0.08 for the whole sample. When the dependence of the correlation between them and the X-ray luminosity is removed, the correlation is still strong, which suggests that the -ray emissions are strongly beamed. 19. Quantum gravity as a Fermi liquid CERN Document Server Alexander, Stephon H S 2008-01-01 We present a reformulation of loop quantum gravity with a cosmological constant and no matter as a Fermi-liquid theory. When the topological sector is deformed and large gauge symmetry is broken, we show that the Chern-Simons state reduces to Jacobson's degenerate sector describing 1+1 dimensional propagating fermions with nonlocal interactions. The Hamiltonian admits a dual description which we realize in the simple BCS model of superconductivity. On one hand, Cooper pairs are interpreted as wormhole correlations at the de Sitter horizon; their number yields the de Sitter entropy. On the other hand, BCS is mapped into a deformed conformal field theory reproducing the structure of quantum spin networks. When area measurements are performed, Cooper-pair insertions are activated on those edges of the spin network intersecting the given area, thus providing a description of quantum measurements in terms of excitations of a Fermi sea to superconducting levels. The cosmological constant problem is naturally addres... 20. Unconventional Fermi surface in an insulating state Energy Technology Data Exchange (ETDEWEB) Harrison, Neil [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Tan, B. S. [Cambridge Univ., Cambridge (United Kingdom); Hsu, Y. -T. [Cambridge Univ., Cambridge (United Kingdom); Zeng, B. [National High Magnetic Field Lab., Tallahassee, FL (United States); Hatnean, M. Ciomaga [Univ. of Warwick, Coventry (United Kingdom); Zhu, Z. [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Hartstein, M. [Cambridge Univ., Cambridge (United Kingdom); Kiourlappou, M. [Cambridge Univ., Cambridge (United Kingdom); Srivastava, A. [Cambridge Univ., Cambridge (United Kingdom); Johannes, M. D. [Center for Computational Materials Science, Washington, DC (United States); Murphy, T. P. [National High Magnetic Field Lab., Tallahassee, FL (United States); Park, J. -H. [National High Magnetic Field Lab., Tallahassee, FL (United States); Balicas, L. [National High Magnetic Field Lab., Tallahassee, FL (United States); Lonzarich, G. G. [Cambridge Univ., Cambridge (United Kingdom); Balakrishnan, G. [Univ. of Warwick, Coventry (United Kingdom); Sebastian, Suchitra E. [Cambridge Univ., Cambridge (United Kingdom) 2015-07-17 Insulators occur in more than one guise; a recent finding was a class of topological insulators, which host a conducting surface juxtaposed with an insulating bulk. Here, we report the observation of an unusual insulating state with an electrically insulating bulk that simultaneously yields bulk quantum oscillations with characteristics of an unconventional Fermi liquid. We present quantum oscillation measurements of magnetic torque in high-purity single crystals of the Kondo insulator SmB6, which reveal quantum oscillation frequencies characteristic of a large three-dimensional conduction electron Fermi surface similar to the metallic rare earth hexaborides such as PrB6 and LaB6. As a result, the quantum oscillation amplitude strongly increases at low temperatures, appearing strikingly at variance with conventional metallic behavior. 1. The basis of the Fermi liquid theory CERN Document Server Apostol, M 2001-01-01 Interaction may affect drastically the many-particle ensembles; for instance an attraction, even weak, between electrons, binds them up in pairs, leading to superconductivity; interacting fermions in one dimension get bosonized; anisotropic fermions with 'nested' Fermi surfaces become non-homogeneous, when interacting, and develop charge- or spin- density waves. All these are different phases, and appear as symmetry breakings, spontaneous or induced; they are also termed as instabilities of the many-body systems, under interaction. Hints toward their nature are often obtained through studying the interacting two-particle problem, scattering included. In this paper the basis of the Fermi liquid theory is shown, and electronic liquid is briefly discussed. (author) 2. A Probabilistic Analysis of the Fermi Paradox CERN Document Server Solomonides, Evan; Terzian, Yervant 2016-01-01 The fermi paradox uses an appeal to the mediocrity principle to make it seem counter-intuitive that humanity has not been contacted by extraterrestrial intelligence. A numerical, statistical analysis was conducted to determine whether this apparent loneliness is, in fact, unexpected. An inequality was derived to relate the frequency of life arising and developing technology on a suitable planet in the galaxy, the average length of time since the first broadcast of such a civilization, and a constant term. An analysis of the sphere reached thus far by human communication was also conducted, considering our local neighborhood and planets of particular interest. We clearly show that human communication has not reached a number of stars and planets adequate to expect an answer. These analyses both conclude that the Fermi paradox is not, in fact, unexpected. By the mediocrity principle and numerical modeling, it is actually unlikely that the Earth would have been reached by extraterrestrial communication at this p... 3. Thickness dependence on the optoelectronic properties of multilayered GaSe based photodetector Science.gov (United States) 2016-08-01 Two-dimensional (2D) layered materials exhibit unique optoelectronic properties at atomic thicknesses. In this paper, we fabricated metal-semiconductor-metal based photodetectors using layered gallium selenide (GaSe) with different thicknesses. The electrical and optoelectronic properties of the photodetectors were studied, and these devices showed good electrical characteristics down to GaSe flake thicknesses of 30 nm. A photograting effect was observed in the absence of a gate voltage, thereby implying a relatively high photoresponsivity. Higher values of the photoresponsivity occurred for thicker layers of GaSe with a maximum value 0.57 AW-1 and external quantum efficiency of of 132.8%, and decreased with decreasing GaSe flake thickness. The detectivity was 4.05 × 1010 cm Hz1/2 W-1 at 532 nm laser wavelength, underscoring that GaSe is a promising p-type 2D material for photodetection applications in the visible spectrum. 4. Fermi Large Area Telescope Second Source Catalog Science.gov (United States) Nolan, P. L.; Abdo, A. A.; Ackermann, M.; Ajello, M.; Allafort, A.; Antolini, E.; Atwood, W. B.; Axelsson, M.; Baldini, L.; Ballet, J.; Barbiellini, G.; Bastieri, D.; Bechtol, K.; Belfiore, A.; Bellazzini, R.; Berenji, B.; Bignami, G. F.; Blandford, R. D.; Bloom, E. D.; Bonamente, E.; Bonnell, J.; Borgland, A. W.; Bottacini, E.; Bouvier, A.; Brandt, T. J.; Bregeon, J.; Brigida, M.; Bruel, P.; Buehler, R.; Burnett, T. H.; Buson, S.; Caliandro, G. A.; Cameron, R. A.; Campana, R.; Cañadas, B.; Cannon, A.; Caraveo, P. A.; Casandjian, J. M.; Cavazzuti, E.; Ceccanti, M.; Cecchi, C.; Çelik, Ö.; Charles, E.; Chekhtman, A.; Cheung, C. C.; Chiang, J.; Chipaux, R.; Ciprini, S.; Claus, R.; Cohen-Tanugi, J.; Cominsky, L. R.; Conrad, J.; Corbet, R.; Cutini, S.; D'Ammando, F.; Davis, D. S.; de Angelis, A.; DeCesar, M. E.; DeKlotz, M.; De Luca, A.; den Hartog, P. R.; de Palma, F.; Dermer, C. D.; Digel, S. W.; Silva, E. do Couto e.; Drell, P. S.; Drlica-Wagner, A.; Dubois, R.; Dumora, D.; Enoto, T.; Escande, L.; Fabiani, D.; Falletti, L.; Favuzzi, C.; Fegan, S. J.; Ferrara, E. C.; Focke, W. B.; Fortin, P.; Frailis, M.; Fukazawa, Y.; Funk, S.; Fusco, P.; Gargano, F.; Gasparrini, D.; Gehrels, N.; Germani, S.; Giebels, B.; Giglietto, N.; Giommi, P.; Giordano, F.; Giroletti, M.; Glanzman, T.; Godfrey, G.; Grenier, I. A.; Grondin, M.-H.; Grove, J. E.; Guillemot, L.; Guiriec, S.; Gustafsson, M.; Hadasch, D.; Hanabata, Y.; Harding, A. K.; Hayashida, M.; Hays, E.; Hill, A. B.; Horan, D.; Hou, X.; Hughes, R. E.; Iafrate, G.; Itoh, R.; Jóhannesson, G.; Johnson, R. P.; Johnson, T. E.; Johnson, A. S.; Johnson, T. J.; Kamae, T.; Katagiri, H.; Kataoka, J.; Katsuta, J.; Kawai, N.; Kerr, M.; Knödlseder, J.; Kocevski, D.; Kuss, M.; Lande, J.; Landriu, D.; Latronico, L.; Lemoine-Goumard, M.; Lionetto, A. M.; Llena Garde, M.; Longo, F.; Loparco, F.; Lott, B.; Lovellette, M. N.; Lubrano, P.; Madejski, G. M.; Marelli, M.; Massaro, E.; Mazziotta, M. N.; McConville, W.; McEnery, J. E.; Mehault, J.; Michelson, P. F.; Minuti, M.; Mitthumsiri, W.; Mizuno, T.; Moiseev, A. A.; Mongelli, M.; Monte, C.; Monzani, M. E.; Morselli, A.; Moskalenko, I. V.; Murgia, S.; Nakamori, T.; Naumann-Godo, M.; Norris, J. P.; Nuss, E.; Nymark, T.; Ohno, M.; Ohsugi, T.; Okumura, A.; Omodei, N.; Orlando, E.; Ormes, J. F.; Ozaki, M.; Paneque, D.; Panetta, J. H.; Parent, D.; Perkins, J. S.; Pesce-Rollins, M.; Pierbattista, M.; Pinchera, M.; Piron, F.; Pivato, G.; Porter, T. A.; Racusin, J. L.; Rainò, S.; Rando, R.; Razzano, M.; Razzaque, S.; Reimer, A.; Reimer, O.; Reposeur, T.; Ritz, S.; Rochester, L. S.; Romani, R. W.; Roth, M.; Rousseau, R.; Ryde, F.; Sadrozinski, H. F.-W.; Salvetti, D.; Sanchez, D. A.; Saz Parkinson, P. M.; Sbarra, C.; Scargle, J. D.; Schalk, T. L.; Sgrò, C.; Shaw, M. S.; Shrader, C.; Siskind, E. J.; Smith, D. A.; Spandre, G.; Spinelli, P.; Stephens, T. E.; Strickman, M. S.; Suson, D. J.; Tajima, H.; Takahashi, H.; Takahashi, T.; Tanaka, T.; Thayer, J. G.; Thayer, J. B.; Thompson, D. J.; Tibaldo, L.; Tibolla, O.; Tinebra, F.; Tinivella, M.; Torres, D. F.; Tosti, G.; Troja, E.; Uchiyama, Y.; Vandenbroucke, J.; Van Etten, A.; Van Klaveren, B.; Vasileiou, V.; Vianello, G.; Vitale, V.; Waite, A. P.; Wallace, E.; Wang, P.; Werner, M.; Winer, B. L.; Wood, D. L.; Wood, K. S.; Wood, M.; Yang, Z.; Zimmer, S. 2012-04-01 We present the second catalog of high-energy γ-ray sources detected by the Large Area Telescope (LAT), the primary science instrument on the Fermi Gamma-ray Space Telescope (Fermi), derived from data taken during the first 24 months of the science phase of the mission, which began on 2008 August 4. Source detection is based on the average flux over the 24 month period. The second Fermi-LAT catalog (2FGL) includes source location regions, defined in terms of elliptical fits to the 95% confidence regions and spectral fits in terms of power-law, exponentially cutoff power-law, or log-normal forms. Also included are flux measurements in five energy bands and light curves on monthly intervals for each source. Twelve sources in the catalog are modeled as spatially extended. We provide a detailed comparison of the results from this catalog with those from the first Fermi-LAT catalog (1FGL). Although the diffuse Galactic and isotropic models used in the 2FGL analysis are improved compared to the 1FGL catalog, we attach caution flags to 162 of the sources to indicate possible confusion with residual imperfections in the diffuse model. The 2FGL catalog contains 1873 sources detected and characterized in the 100 MeV to 100 GeV range of which we consider 127 as being firmly identified and 1171 as being reliably associated with counterparts of known or likely γ-ray-producing source classes. We dedicate this paper to the memory of our colleague Patrick Nolan, who died on 2011 November 6. His career spanned much of the history of high-energy astronomy from space and his work on the Large Area Telescope (LAT) began nearly 20 years ago when it was just a concept. Pat was a central member in the operation of the LAT collaboration and he is greatly missed. 5. Pulsar Timing with the Fermi LAT CERN Document Server Ray, Paul S; Parent, Damien; PSC, the Fermi 2010-01-01 We present an overview of precise pulsar timing using data from the Large Area Telescope (LAT) on Fermi. We describe the analysis techniques including a maximum likelihood method for determining pulse times of arrival from unbinned photon data. In addition to determining the spindown behavior of the pulsars and detecting glitches and timing noise, such timing analyses allow the precise determination of the pulsar position, thus enabling detailed multiwavelength follow up. 6. Higher time derivatives, stability and Fermi Statistics CERN Document Server Lopez-Sarrion, Justo 2011-01-01 We show that statistics is crucial for the instability problem derived from higher time derivatives. In fact, and contrary to previous statements, we check that when dealing with Fermi systems, the Hamiltonian is well bounded and the quantum states are normalizable. Although, ghost states are still present, they do not affect unitarity under certain conditions. We first analyze a quantum oscillator involving Grassman variables and then we generalize it to a Dirac field. Finally, we discuss some physical implications 7. EIS: the scattering beamline at FERMI. Science.gov (United States) Masciovecchio, Claudio; Battistoni, Andrea; Giangrisostomi, Erika; Bencivenga, Filippo; Principi, Emiliano; Mincigrucci, Riccardo; Cucini, Riccardo; Gessini, Alessandro; D'Amico, Francesco; Borghes, Roberto; Prica, Milan; Chenda, Valentina; Scarcia, Martin; Gaio, Giulio; Kurdi, Gabor; Demidovich, Alexander; Danailov, Miltcho B; Di Cicco, Andrea; Filipponi, Adriano; Gunnella, Roberto; Hatada, Keisuke; Mahne, Nicola; Raimondi, Lorenzo; Svetina, Cristian; Godnig, Roberto; Abrami, Alessandro; Zangrando, Marco 2015-05-01 The Elastic and Inelastic Scattering (EIS) beamline at the free-electron laser FERMI is presented. It consists of two separate end-stations: EIS-TIMEX, dedicated to ultrafast time-resolved studies of matter under extreme and metastable conditions, and EIS-TIMER, dedicated to time-resolved spectroscopy of mesoscopic dynamics in condensed matter. The scientific objectives are discussed and the instrument layout illustrated, together with the results from first exemplifying experiments. 8. Nonequilibrium steady states of ideal bosonic and fermionic quantum gases Science.gov (United States) Vorberg, Daniel; Wustmann, Waltraut; Schomerus, Henning; Ketzmerick, Roland; Eckardt, André 2015-12-01 We investigate nonequilibrium steady states of driven-dissipative ideal quantum gases of both bosons and fermions. We focus on systems of sharp particle number that are driven out of equilibrium either by the coupling to several heat baths of different temperature or by time-periodic driving in combination with the coupling to a heat bath. Within the framework of (Floquet-)Born-Markov theory, several analytical and numerical methods are described in detail. This includes a mean-field theory in terms of occupation numbers, an augmented mean-field theory taking into account also nontrivial two-particle correlations, and quantum-jump-type Monte Carlo simulations. For the case of the ideal Fermi gas, these methods are applied to simple lattice models and the possibility of achieving exotic states via bath engineering is pointed out. The largest part of this work is devoted to bosonic quantum gases and the phenomenon of Bose selection, a nonequilibrium generalization of Bose condensation, where multiple single-particle states are selected to acquire a large occupation [Phys. Rev. Lett. 111, 240405 (2013), 10.1103/PhysRevLett.111.240405]. In this context, among others, we provide a theory for transitions where the set of selected states changes, describe an efficient algorithm for finding the set of selected states, investigate beyond-mean-field effects, and identify the dominant mechanisms for heat transport in the Bose-selected state. 9. Signatures of an annular Fermi sea Science.gov (United States) Jo, Insun; Liu, Yang; Pfeiffer, L. N.; West, K. W.; Baldwin, K. W.; Shayegan, M.; Winkler, R. 2017-01-01 The concept of a Fermi surface, the constant-energy surface containing all the occupied electron states in momentum, or wave-vector (k ) , space plays a key role in determining electronic properties of conductors. In two-dimensional (2D) carrier systems, the Fermi surface becomes a contour which, in the simplest case, encircles the occupied states. In this case, the area enclosed by the contour, which we refer to as the Fermi sea (FS), is a simple disk. Here we report the observation of an FS with a new topology, namely, an FS in the shape of an annulus. Such an FS is expected in a variety of 2D systems where the energy band dispersion supports a ring of extrema at finite k , but its experimental observation has been elusive. Our study provides (1) theoretical evidence for the presence of an annular FS in 2D hole systems confined to wide GaAs quantum wells and (2) experimental signatures of the onset of its occupation as an abrupt rise in the sample resistance, accompanied by a sudden appearance of Shubnikov-de Haas oscillations at an unexpectedly high frequency whose value does not simply correspond to the (negligible) density of holes contained within the annular FS. 10. A Probabilistic Analysis of the Fermi Paradox Science.gov (United States) Solomonides, Evan; Terzian, Yervant 2016-06-01 The Fermi paradox uses an appeal to the mediocrity principle to make it seem counterintuitive that humanity has not been contacted by extraterrestrial intelligence. A numerical, statistical analysis was conducted to determine whether this apparent loneliness is, in fact, unexpected. An inequality was derived to relate the frequency of life arising and developing technology on a suitable planet in the galaxy; the average length of time since the first broadcast of such a civilization; and a constant term. An analysis of the sphere reached thus far by human communication was also conducted, considering our local neighborhood and planets of particular interest. These analyses both conclude that the Fermi paradox is not, in fact, unexpected. By the mediocrity principle and numerical modeling, it is actually unlikely that the Earth would have been reached by extraterrestrial communication at this point. We predict that under 1% of the galaxy has been reached at all thus far, and we do not anticipate to be reached until approximately 50% of stars/planets have been reached. We offer a prediction that we should not expect this until at least 1,500 years in the future. Thus the Fermi paradox is not a shocking observation- or lack thereof- and humanity may very well be contacted within our species’ lifespan (we can begin to expect to be contacted 1,500 years in the future). 11. A hybrid Fermi-Ulam-bouncer model Energy Technology Data Exchange (ETDEWEB) Leonel, Edson D; McClintock, P V E [Department of Physics, Lancaster University, Lancaster LA1 4YB (United Kingdom) 2005-01-28 Some dynamical and chaotic properties are studied for a classical particle bouncing between two rigid walls, one of which is fixed and the other moves in time, in the presence of an external field. The system is a hybrid, behaving not as a purely Fermi-Ulam model, nor as a bouncer, but as a combination of the two. We consider two different kinds of motion of the moving wall: (i) periodic and (ii) random. The dynamics of the model is studied via a two-dimensional nonlinear area-preserving map. We confirm that, for periodic oscillations, our model recovers the well-known results of the Fermi-Ulam model in the limit of zero external field. For intense external fields, we establish the range of control parameters values within which invariant spanning curves are observed below the chaotic sea in the low energy domain. We characterize this chaotic low energy region in terms of Lyapunov exponents. We also show that the velocity of the particle, and hence also its kinetic energy, grow according to a power law when the wall moves randomly, yielding clear evidence of Fermi acceleration. 12. Superconducting instability in non-Fermi liquids CERN Document Server Mandal, Ipsita 2016-01-01 We use renormalization group (RG) analysis and dimensional regularization techniques to study potential superconductivity-inducing four-fermion interactions in systems with critical Fermi surfaces of general dimensions ($m$) and co-dimensions ($d-m$), arising as a result of quasiparticle interaction with a gapless Ising-nematic order parameter. These are examples of non-Fermi liquid states in $d$ spatial dimensions. Our formalism allows us to treat the corresponding zero-temperature low-energy effective theory in a controlled approximation close to the upper critical dimension $d=d_c(m)$. The fixed points are identified from the RG flow equations, as functions of $d$ and $m$. We find that the flow towards the non-Fermi liquid fixed point is preempted by Cooper pair formation for both the physical cases of $(d=3, m=2)$ and $(d=2, m=1)$. In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point. 13. Women in Physics in Fermi's Time CERN Document Server Byers, N 2003-01-01 Enrico Fermi lived from 1901 to 1955, a period of great progress in physics and a period in which opportunities for women to study and work in institutions of higher learning increased significantly in Europe and North America. Though there are a few examples of women who made important contributions to physics in the 18th century such as Emilie du Chatelet and Laura Bassi, it was only in Fermi's time that the number began to increase significantly. It is remarkable that almost immediately after they gained entrance to laboratories and universities, among them appeared women of great creative ability who made lasting contributions to physics. This talk is mainly about some of these whose scientific lives are not as well known as their contributions deserve - Emmy Noether, Marietta Blau, Irene Joliot-Curie, Lise Meitner. Additionally, some outstanding women whose work played a role in Enrico Fermi's life in physics are noted - Ida Tacke Noddack, Tatiana Ehrenfest-Afanaseva, Leona Woods Marshall Libby, and Mari... 14. Hydrophobic encapsulation of hydrocarbon gases. Science.gov (United States) Leontiev, Alexander V; Saleh, Anas W; Rudkevich, Dmitry M 2007-04-26 [reaction: see text] Encapsulation data for hydrophobic hydrocarbon gases within a water-soluble hemicarcerand in aqueous solution are reported. It is concluded that hydrophobic interactions serve as the primary driving force for the encapsulation, which can be used for the design of gas-separating polymers with intrinsic inner cavities. 15. Explosion limits for combustible gases Institute of Scientific and Technical Information of China (English) TONG Min-ming; WU Guo-qing; HAO Ji-fei; DAI Xin-lian 2009-01-01 Combustible gases in coal mines are composed of methane, hydrogen, some multi-carbon alkane gases and other gases. Based on a numerical calculation, the explosion limits of combustible gases were studied, showing that these limits are related to the concentrations of different components in the mixture. With an increase of C4H10 and C6H14, the Lower ExplosionLimit (LEL) and Upper Explosion-Limit (UEL) of a combustible gas mixture will decrease clearly. For every 0.1% increase in C4H10 and C6H14, the LEL decreases by about 0.19% and the UEL by about 0.3%. The results also prove that, by increasing the amount of H2, the UEL of a combustible gas mixture will increase considerably. If the level of H2 increases by 0.1%, the UEL will increase by about 0.3%. However, H2 has only a small effect on the LEL of the combustible gas mixture. Our study provides a theoretical foundation for judging the explosion risk of an explosive gas mixture in mines. 16. Spectrum-splitting approach for Fermi-operator expansion in all-electron Kohn-Sham DFT calculations Science.gov (United States) Motamarri, Phani; Gavini, Vikram; Bhattacharya, Kaushik; Ortiz, Michael 2017-01-01 We present a spectrum-splitting approach to conduct all-electron Kohn-Sham density functional theory (DFT) calculations by employing Fermi-operator expansion of the Kohn-Sham Hamiltonian. The proposed approach splits the subspace containing the occupied eigenspace into a core subspace, spanned by the core eigenfunctions, and its complement, the valence subspace, and thereby enables an efficient computation of the Fermi-operator expansion by reducing the expansion to the valence-subspace projected Kohn-Sham Hamiltonian. The key ideas used in our approach are as follows: (i) employ Chebyshev filtering to compute a subspace containing the occupied states followed by a localization procedure to generate nonorthogonal localized functions spanning the Chebyshev-filtered subspace; (ii) compute the Kohn-Sham Hamiltonian projected onto the valence subspace; (iii) employ Fermi-operator expansion in terms of the valence-subspace projected Hamiltonian to compute the density matrix, electron density, and band energy. We demonstrate the accuracy and performance of the method on benchmark materials systems involving silicon nanoclusters up to 1330 electrons, a single gold atom, and a six-atom gold nanocluster. The benchmark studies on silicon nanoclusters revealed a staggering fivefold reduction in the Fermi-operator expansion polynomial degree by using the spectrum-splitting approach for accuracies in the ground-state energies of ˜10-4Ha/atom with respect to reference calculations. Further, numerical investigations on gold suggest that spectrum splitting is indispensable to achieve meaningful accuracies, while employing Fermi-operator expansion. 17. Core-dominance parameter, black hole mass and jet-disc connection in Fermi blazars OpenAIRE Chen, Y. Y.; Zhang, X.; Zhang, H. J.; X. L. Yu 2015-01-01 We study the relationship between jet power and accretion for Fermi and non-Fermi blazars, respectively. We also compare the relevant parameter between them. Our main results are as follows. (i) Fermi and non-Fermi blazars have significant difference in redshift, black hole mass, and broad line luminosity. (ii) Fermi blazars have higher average core-dominance parameter than non-Fermi blazars, which suggests that Fermi blazars have strong beaming effect. (iii) We find significant correlation b... 18. Spectral backbone of excitation transport in ultracold Rydberg gases Science.gov (United States) Scholak, Torsten; Wellens, Thomas; Buchleitner, Andreas 2014-12-01 The spectral structure underlying excitonic energy transfer in ultracold Rydberg gases is studied numerically, in the framework of random matrix theory, and via self-consistent diagrammatic techniques. Rydberg gases are made up of randomly distributed, highly polarizable atoms that interact via strong dipolar forces. Dynamics in such a system is fundamentally different from cases in which the interactions are of short range, and is ultimately determined by the spectral and eigenvector structure. In the energy levels' spacing statistics, we find evidence for a critical energy that separates delocalized eigenstates from states that are localized at pairs or clusters of atoms separated by less than the typical nearest-neighbor distance. We argue that the dipole blockade effect in Rydberg gases can be leveraged to manipulate this transition across a wide range: As the blockade radius increases, the relative weight of localized states is reduced. At the same time, the spectral statistics, in particular, the density of states and the nearest-neighbor level-spacing statistics, exhibits a transition from approximately a 1-stable Lévy to a Gaussian orthogonal ensemble. Deviations from random matrix statistics are shown to stem from correlations between interatomic interaction strengths that lead to an asymmetry of the spectral density and profoundly affect localization properties. We discuss approximations to the self-consistent Matsubara-Toyozawa locator expansion that incorporate these effects. 19. Biosignature Gases in H2-Dominated Atmospheres on Rocky Exoplanets CERN Document Server Seager, S; Hu, R 2013-01-01 (Abridged) Super Earth exoplanets are being discovered with increasing frequency and some will be able to retain stable H2-dominated atmospheres. We study biosignature gases on exoplanets with thin H2 atmospheres and habitable surface temperatures, by using a model atmosphere with photochemistry, and biomass estimate framework for evaluating the plausibilty of a range of biosignature gas candidates. We find that photochemically produced H atoms are the most abundant reactive species in H2 atmospheres. In atmospheres with high CO2 levels, atomic O is the major destructive species for some molecules. In sun-Earth-like UV radiation environments, H (and in some cases O) will rapidly destroy nearly all biosignature gases of interest. The lower UV fluxes from UV quiet M stars would produce a lower concentration of H (or O) for the same scenario, enabling some biosignature gases to accumulate. The favorability of low-UV radiation environments to in an H2 atmosphere is closely analogous to the case of oxidized atmosp... 20. Quasiparticle lifetime in a mixture of Bose and Fermi superfluids. Science.gov (United States) Zheng, Wei; Zhai, Hui 2014-12-31 In this Letter, we study the effect of quasiparticle interactions in a Bose-Fermi superfluid mixture. We consider the lifetime of a quasiparticle of the Bose superfluid due to its interaction with quasiparticles in the Fermi superfluid. We find that this damping rate, i.e., the inverse of the lifetime, has quite a different threshold behavior at the BCS and the BEC side of the Fermi superfluid. The damping rate is a constant near the threshold momentum in the BCS side, while it increases rapidly in the BEC side. This is because, in the BCS side, the decay process is restricted by the constraint that the fermion quasiparticle is located near the Fermi surface, while such a restriction does not exist in the BEC side where the damping process is dominated by bosonic quasiparticles of the Fermi superfluid. Our results are related to the collective mode experiment in the recently realized Bose-Fermi superfluid mixture. 1. Solution of the Problem of the Couette Flow for a Fermi Gas with Almost Specular Boundary Conditions Science.gov (United States) Bedrikova, E. A.; Latyshev, A. V. 2016-06-01 A solution of the Couette problem for a Fermi gas is constructed. The kinetic Bhatnagar-Gross-Krook (BGK) equation is used. Almost specular boundary conditions are considered. Formulas for the mass flux and the heat flux of the gas are obtained. These fluxes are proportional to the difference of the tangential momentum accommodation coefficients of the molecules. An expression for the viscous drag force acting on the walls of the channel is also found. An analysis of the macroparameters of the gas is performed. The limit to classical gases is taken. The obtained results are found to go over to the known results in this limit. 2. Dipolar Excitations of a Trapped Bose-Fermi Mixture Institute of Scientific and Technical Information of China (English) FUXiao-Wei; LIUXia-Ji; HUHui; LIShi-Qun 2004-01-01 We study the dipolar excitation of a trapped Bose-Fermi mixture at zero temperature, by using a scalingansatz formalism and Thomas-Fermi approximation at mean-field level. We show that both frequencies of the low-lying and high-lying modes are strongly affected by the Bose-Fermi interaction. Possible implication of our results to the recent experiment has been commented. 3. Dipolar Excitations of a Trapped Bose-Fermi Mixture Institute of Scientific and Technical Information of China (English) FU Xiao-Wei; LIU Xia-Ji; HU Hui; LI Shi-Qun 2004-01-01 We study the dipolar excitation of a trapped Bose-Fermi mixture at zero temperature, by using a scaling ansatz formalism and Thomas-Fermi approximation at mean-field level. We show that both frequencies of the low-lying and high-lying modes are strongly affected by the Bose-Fermi interaction. Possible implication of our results to the recent experiment has been commented. 4. Theoretical Insight into Shocked Gases Energy Technology Data Exchange (ETDEWEB) Leiding, Jeffery Allen [Los Alamos National Lab. (LANL), Los Alamos, NM (United States) 2016-09-29 I present the results of statistical mechanical calculations on shocked molecular gases. This work provides insight into the general behavior of shock Hugoniots of gas phase molecular targets with varying initial pressures. The dissociation behavior of the molecules is emphasized. Impedance matching calculations are performed to determine the maximum degree of dissociation accessible for a given flyer velocity as a function of initial gas pressure. 5. Atmospheric Chemistry and Greenhouse Gases Energy Technology Data Exchange (ETDEWEB) Ehhalt, D.; Prather, M.; Dentener, F.; Derwent, R.; Dlugokencky, Edward J.; Holland, E.; Isaksen, I.; Katima, J.; Kirchhoff, V.; Matson, P.; Midgley, P.; Wang, M.; Berntsen, T.; Bey, I.; Brasseur, G.; Buja, L.; Collins, W. J.; Daniel, J. S.; DeMore, W. B.; Derek, N.; Dickerson, R.; Etheridge, D.; Feichter, J.; Fraser, P.; Friedl, R.; Fuglestvedt, J.; Gauss, M.; Grenfell, L.; Grubler, Arnulf; Harris, N.; Hauglustaine, D.; Horowitz, L.; Jackman, C.; Jacob, D.; Jaegle, L.; Jain, Atul K.; Kanakidou, M.; Karlsdottir, S.; Ko, M.; Kurylo, M.; Lawrence, M.; Logan, J. A.; Manning, M.; Mauzerall, D.; McConnell, J.; Mickley, L. J.; Montzka, S.; Muller, J. F.; Olivier, J.; Pickering, K.; Pitari, G.; Roelofs, G.-J.; Rogers, H.; Rognerud, B.; Smith, Steven J.; Solomon, S.; Staehelin, J.; Steele, P.; Stevenson, D. S.; Sundet, J.; Thompson, A.; van Weele, M.; von Kuhlmann, R.; Wang, Y.; Weisenstein, D. K.; Wigley, T. M.; Wild, O.; Wuebbles, D.J.; Yantosca, R.; Joos, Fortunat; McFarland, M. 2001-10-01 Chapter 4 of the IPCC Third Assessment Report Climate Change 2001: The Scientific Basis. Sections include: Executive Summary 2414.1 Introduction 2434.2 Trace Gases: Current Observations, Trends and Budgets 2484.3 Projections of Future Emissions 2664.4 Projections of Atmospheric Composition for the 21st Century 2674.5 Open Questions 2774.6 Overall Impact of Global Atmospheric Chemistry Change 279 6. Global warming and greenhouse gases OpenAIRE Belić Dragoljub S. 2006-01-01 Global warming or Climate change refers to long-term fluctuations in temperature, precipitation, wind, and other elements of the Earth's climate system. Natural processes such as solar-irradiance variations, variations in the Earth's orbital parameters, and volcanic activity can produce variations in climate. The climate system can also be influenced by changes in the concentration of various gases in the atmosphere, which affect the Earth's absorption of radiation. 7. 40 CFR 90.312 - Analytical gases. Science.gov (United States) 2010-07-01 ... expiration date stated by the gas supplier must be recorded. (b) Pure gases. The required purity of the gases... a concentration of propane higher than what a gas supplier considers to be safe may be substituted... choice of diluent (zero air or purified nitrogen) between the calibration and span gases. If... 8. Charge transfer effects on the Fermi surface of Ba0.5K 0.5Fe2As2 KAUST Repository Nazir, Safdar 2011-01-31 Ab-initio calculations within density functional theory are performed to obtain a more systematic understanding of the electronic structure of iron pnictides. As a prototypical compound we study Ba0.5K 0.5Fe2As2 and analyze the changes of its electronic structure when the interaction between the Fe2As 2 layers and their surrounding is modified. We find strong effects on the density of states near the Fermi energy as well as the Fermi surface. The role of the electron donor atoms in iron pnictides thus cannot be understood in a rigid band picture. Instead, the bonding within the Fe2As 2 layers reacts to a modified charge transfer from the donor atoms by adapting the intra-layer Fe-As hybridization and charge transfer in order to maintain an As3- valence state. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. 9. Evolution of electron Fermi surface with doping in cobaltates. Science.gov (United States) Ma, Xixiao; Lan, Yu; Qin, Ling; Kuang, Lülin; Feng, Shiping 2016-08-24 The notion of the electron Fermi surface is one of the characteristic concepts in the field of condensed matter physics, and it plays a crucial role in the understanding of the physical properties of doped Mott insulators. Based on the t-J model, we study the nature of the electron Fermi surface in the cobaltates, and qualitatively reproduce the essential feature of the evolution of the electron Fermi surface with doping. It is shown that the underlying hexagonal electron Fermi surface obeys Luttinger's theorem. The theory also predicts a Fermi-arc phenomenon at the low-doped regime, where the region of the hexagonal electron Fermi surface along the [Formula: see text]-K direction is suppressed by the electron self-energy, and then six disconnected Fermi arcs located at the region of the hexagonal electron Fermi surface along the [Formula: see text]-M direction emerge. However, this Fermi-arc phenomenon at the low-doped regime weakens with the increase of doping. 10. A Nonlocal Poisson-Fermi Model for Ionic Solvent CERN Document Server Xie, Dexuan; Eisenberg, Bob; Scott, L Ridgway 2016-01-01 We propose a nonlocal Poisson-Fermi model for ionic solvent that includes ion size effects and polarization correlations among water molecules in the calculation of electrostatic potential. It includes the previous Poisson-Fermi models as special cases, and its solution is the convolution of a solution of the corresponding nonlocal Poisson dielectric model with a Yukawa-type kernel function. Moreover, the Fermi distribution is shown to be a set of optimal ionic concentration functions in the sense of minimizing an electrostatic potential free energy. Finally, numerical results are reported to show the difference between a Poisson-Fermi solution and a corresponding Poisson solution. 11. Nonlocal Poisson-Fermi model for ionic solvent. Science.gov (United States) Xie, Dexuan; Liu, Jinn-Liang; Eisenberg, Bob 2016-07-01 We propose a nonlocal Poisson-Fermi model for ionic solvent that includes ion size effects and polarization correlations among water molecules in the calculation of electrostatic potential. It includes the previous Poisson-Fermi models as special cases, and its solution is the convolution of a solution of the corresponding nonlocal Poisson dielectric model with a Yukawa-like kernel function. The Fermi distribution is shown to be a set of optimal ionic concentration functions in the sense of minimizing an electrostatic potential free energy. Numerical results are reported to show the difference between a Poisson-Fermi solution and a corresponding Poisson solution. 12. Induced interactions in a superfluid Bose-Fermi mixture DEFF Research Database (Denmark) Kinnunen, Jami; Bruun, Georg 2015-01-01 -particle and collective excitations of the Fermi gas give rise to an induced interaction between the bosons, which varies strongly with momentum and frequency. It diverges at the sound mode of the Fermi superfluid, resulting in a sharp avoided crossing feature and a corresponding sign change of the interaction energy...... shift in the excitation spectrum of the BEC. In addition, the excitation of quasiparticles in the Fermi superfluid leads to damping of the excitations in the BEC. Besides studying induced interactions themselves, we can use these prominent effects to systematically probe the strongly interacting Fermi... 13. Theory of the Fermi-level energy in semiconductor superlattices Energy Technology Data Exchange (ETDEWEB) Luscombe, J.H. (Central Research Laboratories, Texas Instruments Incorporated, Dallas, Texas (USA)); Aggarwal, R. (Central Research Laboratories, Texas Instruments Incorporated, Dallas, Texas (USA) Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts (USA)); Reed, M.A. (Central Research Laboratories, Texas Instruments Incorporated, Dallas, Texas (USA) Department of Electrical Engineering, Yale University, New Haven, Connecticut (USA)); Frensley, W.R. (Central Research Laboratories, Texas Instruments Incorporated, Dallas, Texas (USA) Department of Electrical Engineering, University of Texas at Dallas, Richardson, Texas (USA)); Luban, M. (Iowa Univ., Iowa City, IA (USA). Dept. of Physics and Astronomy Ames Lab., IA (USA)) 1991-09-15 A theoretical study of the properties of the Fermi level in semiconductor superlattices (SL's) is made which is based upon the carrier occupation of the minibands in thermal equilibrium. We find, for a fixed carrier density and temperature, that the SL Fermi level can differ significantly from that obtained using commonly employed three-dimensional approximations, depending upon the relative spacings and widths of the minibands, with the SL Fermi level being higher than the corresponding bulk value. We find that the SL Fermi level is a sensitive function of the relative widths of the quantum wells and barriers. 14. Atom chip based generation of entanglement for quantum metrology CERN Document Server Riedel, Max F; Li, Yun; Hänsch, Theodor W; Sinatra, Alice; Treutlein, Philipp 2010-01-01 Atom chips provide a versatile quantum laboratory on a microchip' for experiments with ultracold atomic gases. They have been used in experiments on diverse topics such as low-dimensional quantum gases, cavity quantum electrodynamics, atom-surface interactions, and chip-based atomic clocks and interferometers. A severe limitation of atom chips, however, is that techniques to control atomic interactions and to generate entanglement have not been experimentally available so far. Such techniques enable chip-based studies of entangled many-body systems and are a key prerequisite for atom chip applications in quantum simulations, quantum information processing, and quantum metrology. Here we report experiments where we generate multi-particle entanglement on an atom chip by controlling elastic collisional interactions with a state-dependent potential. We employ this technique to generate spin-squeezed states of a two-component Bose-Einstein condensate and show that they are useful for quantum metrology. The obser... 15. Emergent physics: Fermi-point scenario. Science.gov (United States) Volovik, Grigory 2008-08-28 The Fermi-point scenario of emergent gravity has the following consequences: gravity emerges together with fermionic and bosonic matter; emergent fermionic matter consists of massless Weyl fermions; emergent bosonic matter consists of gauge fields; Lorentz symmetry persists well above the Planck energy; space-time is naturally four dimensional; the Universe is naturally flat; the cosmological constant is naturally small or zero; the underlying physics is based on discrete symmetries; 'quantum gravity' cannot be obtained by quantization of Einstein equations; and there is no contradiction between quantum mechanics and gravity, etc. 16. Optical Observations Of Fermi LAT Monitored Blazars Science.gov (United States) Cook, Kyle; Carini, M. T. 2009-01-01 For the past 8 years the Bell Observatory at Western Kentucky University has been conducting R band monitoring of the variability of approximately 50 Blazars. A subset of these objects are being routinely observed with the LAT instrument on-board the Fermi Space Telescope. Adding the Robotically Controlled Telescope (RCT) at Kitt Peak National Observatory and observations with the AZT-11 telescope at the Crimean Astrophysical Observatory (CRAO), we are intensively monitoring the Blazars on the Lat monitoring list. We present the results of our long term monitoring of the LAT monitored Blazars, as well as the recent contemporaneous optical R band observations we have obtained of the LAT Blazars. 17. The FERMI free-electron lasers. Science.gov (United States) Allaria, E; Badano, L; Bassanese, S; Capotondi, F; Castronovo, D; Cinquegrana, P; Danailov, M B; D'Auria, G; Demidovich, A; De Monte, R; De Ninno, G; Di Mitri, S; Diviacco, B; Fawley, W M; Ferianis, M; Ferrari, E; Gaio, G; Gauthier, D; Giannessi, L; Iazzourene, F; Kurdi, G; Mahne, N; Nikolov, I; Parmigiani, F; Penco, G; Raimondi, L; Rebernik, P; Rossi, F; Roussel, E; Scafuri, C; Serpico, C; Sigalotti, P; Spezzani, C; Svandrlik, M; Svetina, C; Trovó, M; Veronese, M; Zangrando, D; Zangrando, M 2015-05-01 FERMI is a seeded free-electron laser (FEL) facility located at the Elettra laboratory in Trieste, Italy, and is now in user operation with its first FEL line, FEL-1, covering the wavelength range between 100 and 20 nm. The second FEL line, FEL-2, a high-gain harmonic generation double-stage cascade covering the wavelength range 20-4 nm, has also completed commissioning and the first user call has been recently opened. An overview of the typical operating modes of the facility is presented. 18. Shear Viscosity of a Unitary Fermi Gas OpenAIRE Wlazłowski, Gabriel; Magierski, Piotr; Drut, Joaquín E. 2012-01-01 We present the first ab initio determination of the shear viscosity eta of the Unitary Fermi Gas, based on finite temperature quantum Monte Carlo calculations and the Kubo linear-response formalism. We determine the temperature dependence of the shear viscosity to entropy density ratio eta/s. The minimum of eta/s appears to be located above the critical temperature for the superfluid-to-normal phase transition with the most probable value being eta/s approx 0.2 hbar/kB, which almost saturates... 19. -Rays Radiation of High Redshift Fermi Blazars W. G. Liu; S. H. Fu; X. Zhang; L. Ma; Y. B. Li; D. R. Xiong 2014-09-01 Based on the 31 high redshift ( > 2) Flat Spectral Radio Quasars (FSRQs), which is from the second Fermi-LAT AGNs catalogue (2LAC), we studied the correlation between flux densities (R, K, ) in the radio, infrared and -ray wave bands. We found that there is a significant positive correlation between and R, and a weak anticorrelation between and K in the average state. For high redshift blazars, we argue that the seed photon of -ray emission mainly comes from the jet itself and partially from the dusty torus. 20. Holographic non-Fermi-liquid fixed points. Science.gov (United States) Faulkner, Tom; Iqbal, Nabil; Liu, Hong; McGreevy, John; Vegh, David 2011-04-28 Techniques arising from string theory can be used to study assemblies of strongly interacting fermions. Via this 'holographic duality', various strongly coupled many-body systems are solved using an auxiliary theory of gravity. Simple holographic realizations of finite density exhibit single-particle spectral functions with sharp Fermi surfaces, of a form distinct from those of the Landau theory. The self-energy is given by a correlation function in an infrared (IR) fixed-point theory that is represented by a two-dimensional anti de Sitter space (AdS(2)) region in the dual gravitational description. Here, we describe in detail the gravity calculation of this IR correlation function. 1. Turbomolecular Pumps for Holding Gases in Open Containers Science.gov (United States) Keller, John W.; Lorenz, John E. 2010-01-01 Proposed special-purpose turbomolecular pumps denoted turbotraps would be designed, along with mating open containers, to prevent the escape of relatively slowly (thermal) moving gas molecules from the containers while allowing atoms moving at much greater speeds to pass through. In the original intended applications, the containers would be electron-attachment cells, and the contained gases would be vapors of alkali metal atoms moving at thermal speeds that would be of the order of a fraction of 300 meters per second. These cells would be parts of apparatuses used to measure fluxes of neutral atoms incident at kinetic energies in the approximate range of 10 eV to 10 keV (corresponding to typical speeds of the order of 40,000 m/s and higher). The incident energetic neutral atoms would pass through the cells, wherein charge-exchange reactions with the alkali metal atoms would convert the neutral atoms to negative ions, which, in turn, could then be analyzed by use of conventional charged-particle optics. 2. Ultralong-range order in the Fermi-Hubbard model with long-range interactions Science.gov (United States) van Loon, Erik G. C. P.; Katsnelson, Mikhail I.; Lemeshko, Mikhail 2015-08-01 We use the dual boson approach to reveal the phase diagram of the Fermi-Hubbard model with long-range dipole-dipole interactions. By using a large-scale finite-temperature calculation on a 64 ×64 square lattice we demonstrate the existence of a novel phase, possessing an "ultralong-range" order. The fingerprint of this phase—the density correlation function—features a nontrivial behavior on a scale of tens of lattice sites. We study the properties and the stability of the ultralong-range-ordered phase, and show that it is accessible in modern experiments with ultracold polar molecules and magnetic atoms. 3. On directly measuring relative Fermi energies of noble metals and their alloys Science.gov (United States) Kleiman, G. G.; Sundaram, V. S.; Rogers, J. D. 1981-09-01 We present the first evidence of direct measurement of relative Fermi energies in alloys and between pure metals. From applying the "atomic" concept of core hole final state screening, the Auger energy shift of noble metal A equals EFA- EF( x). High resolution Auger shifts in P1- xtCux, AuxCu1- x and AuxAg1- x demonstrate experimental verification of this simple relation. We find E FCuE FAu ≅ - 0.2 eV, and E FPt ≅ E FCu and E FAg ≅ E FAu. 4. Fermi Large Area Telescope Observations of the Dark Accelerator HESS J1745-303 Science.gov (United States) Yeung, Paul 2016-12-01 Reviewing the two MeV-GeV investigations in the field of the HESS J1745-303 performed using Fermi Large Area Telescope data, we confirmed that the emission peak comfortably coincides with ‘Region A’ in the TeV regime, which is the brightest part of this feature. The MeV–TeV spectrum can be precisely described by a single power-law. Also, recent investigation has shown that the MeV-GeV feature is elongated from ‘Region A’ toward the north-west, which is similar to the case of large- scale atomic/molecular gas distribution. 5. Search of the Earth Limb Fermi Data and Non-Galactic Center Region Fermi Data for Signs of Narrow Lines CERN Document Server Bloom, E; Izaguirre, E; Snyder, A; Albert, A; Winer, B; Yang, Z; Essig, R 2013-01-01 Since the spring of 2012 there have been many papers published using Fermi LAT public data that claim evidence for narrow spectral lines coming from the region of the Galactic center. This study uses non-Galactic center Fermi-LAT data from survey mode observations, and Earth limb Fermi data to test the dark matter interpretation of this feature and better understand its origins. 6. Effects of traces of molecular gases (hydrogen, nitrogen) in glow discharges in noble gases Science.gov (United States) Steers, E. B. M.; Smid, P.; Hoffmann, V. 2008-07-01 molecular gases can change the discharge impedance, alter the sputtering rate and crater profile and cause changes in the absolute and relative intensities of lines in both the atomic and ionic spectra of the sample element and the plasma gas. The authors wish to acknowledge financial support from EC funded Analytical Glow Discharge Research Training Network GLADNET, contract no. MRTN-CT-2006-035459. P. Smid thanks the Deutsche Forschungsgemeinschaft (Ref 436 TSE 17/7/06) for support while carrying out experiments at IFW Dresden. 7. Bending Two-Dimensional Materials To Control Charge Localization and Fermi-Level Shift. Science.gov (United States) Yu, Liping; Ruzsinszky, Adrienn; Perdew, John P 2016-04-13 High-performance electronics requires the fine control of semiconductor conductivity. In atomically thin two-dimensional (2D) materials, traditional doping technique for controlling carrier concentration and carrier type may cause crystal damage and significant mobility reduction. Contact engineering for tuning carrier injection and extraction and carrier type may suffer from strong Fermi-level pinning. Here, using first-principles calculations, we predict that mechanical bending, as a unique attribute of thin 2D materials, can be used to control conductivity and Fermi-level shift. We find that bending can control the charge localization of top valence bands in both MoS2 and phosphorene nanoribbons. The donor-like in-gap edge-states of armchair MoS2 ribbon and their associated Fermi-level pinning can be removed by bending. A bending-controllable new in-gap state and accompanying direct-indirect gap transition are predicted in armchair phosphorene nanoribbon. We demonstrate that such emergent bending effects are realizable. The bending stiffness as well as the effective thickness of 2D materials are also derived from first principles. Our results are of fundamental and technological relevance and open new routes for designing functional 2D materials for applications in which flexuosity is essential. 8. Antiprotonic atom formation and spectroscopy-ASACUSA experiment at CERN-AD CERN Document Server Widmann, E 1999-01-01 This talk describes the experiments on atomic spectroscopy and atomic collisions as proposed by the ASACUSA collaboration for the forthcoming AD facility at CERN. They consist of high-precision spectroscopy of antiprotonic atoms, the study of anti-protonic atom formation processes, and stopping power and ionization measurements in low-pressure gases. (18 refs). 9. Atom chips CERN Document Server Reichel, Jakob 2010-01-01 This book provides a stimulating and multifaceted picture of a rapidly developing field. The first part reviews fundamentals of atom chip research in tutorial style, while subsequent parts focus on the topics of atom-surface interaction, coherence on atom chips, and possible future directions of atom chip research. The articles are written by leading researchers in the field in their characteristic and individual styles. 10. Atomic energy CERN Multimedia 1996-01-01 Interviews following the 1991 co-operation Agreement between the Department of Atomic Energy (DAE) of the Government of India and the European Organization for Nuclear Research (CERN) concerning the participation in the Large Hadron Collider Project (LHC) . With Chidambaram, R, Chairman, Atomic Energy Commission and Secretary, Department of Atomic Energy, Department of Atomic Energy (DAE) of the Government of India and Professor Llewellyn-Smith, Christopher H, Director-General, CERN. 11. Pulsar Candidates Toward Fermi Unassociated Sources CERN Document Server Frail, D A; Jagannathan, P; Intema, H T 2016-01-01 We report on a search for steep spectrum radio sources within the 95% confidence error ellipses of the Fermi unassociated sources from the Large Array Telescope (LAT). Using existing catalogs and the newly released GMRT all-sky survey at 150 MHz we identify compact radio sources that are bright at MHz frequencies but faint or absent at GHz frequencies. Such steep spectrum radio sources are rare and constitute a sample of pulsar candidates, selected independently of period, dispersion measure, interstellar scattering and orbital parameters. We find point-like, steep spectrum candidates toward 11 Fermi sources. Based on the gamma-ray/radio positional coincidence, the rarity of such radio sources, and the properties of the 3FGL sources themselves, we argue that many of these sources could be pulsars. They may have been missed by previous radio periodicity searches due to interstellar propagation effects or because they lie in an unusually tight binary. If this hypothesis is correct, then renewed gamma-ray and ra... 12. Cores in Dwarf Galaxies from Fermi Repulsion CERN Document Server Randall, Lisa; Unwin, James 2016-01-01 We show that Fermi repulsion can lead to cored density profiles in dwarf galaxies for sub-keV fermionic dark matter. We treat the dark matter as a quasi-degenerate self-gravitating Fermi gas and calculate its density profile assuming hydrostatic equilibrium. We find that suitable dwarf galaxy cores of larger than 130 pc can be achieved for fermion dark matter with mass in the range 70 eV - 400 eV. While in conventional dark matter scenarios, such sub-keV thermal dark matter would be excluded by free streaming bounds, the constraints are ameliorated in models with dark matter at lower temperature than conventional thermal scenarios, such as the Flooded Dark Matter model that we have previously considered. Modifying the arguments of Tremaine and Gunn we derive a conservative lower bound on the mass of fermionic dark matter of 70 eV and a stronger lower bound from Lyman-$\\alpha$ clouds of about 470 eV, leading to slightly smaller cores than have been observed. We comment on this result and how the tension is rel... 13. Spiraling Fermi arcs in Weyl materials Science.gov (United States) Li, Songci; Andreev, Anton In Weyl materials the valence and conduction electron bands touch at an even number of isolated points in the Brillouin zone. In the vicinity of these points the electron dispersion is linear and may be described by the massless Dirac equation. This results in nontrivial topology of Berry connection curvature. One of its consequences is the existence of peculiar surface electron states whose Fermi surfaces form arcs connecting projections of the Weyl points onto the surface plane. Band bending near the boundary of the crystal also produces surface states. We show that in Weyl materials band bending near the crystal surface gives rise to spiral structure of energy surfaces of arc states. The corresponding Fermi surface has the shape of a spiral that winds about the projection of the Weyl point onto the surface plane. The direction of the winding is determined by the helicity of the Weyl point and the sign of the band bending potential. For close valleys arc state morphology may be understood in terms of avoided crossing of oppositely winding spirals. This work is supported by the U.S. Department of Energy Office of Science, Basic Energy Sciences under Award Number DE-FG02-07ER46452. 14. Fermi level stabilization energy in cadmium oxide Energy Technology Data Exchange (ETDEWEB) Speaks, D. T.; Mayer, M. A.; Yu, K. M.; Mao, S. S.; Haller, E. E.; Walukiewicz, W. 2010-04-08 We have studied the effects of high concentrations of native point defects on the electrical and optical properties of CdO. The defects were introduced by irradiation with high energy He+, Ne+, Ar+ and C+ ions. Increasing the irradiation damage with particles heavier than He+ increases the electron concentration until a saturation level of 5x1020 cm-3 is reached. In contrast, due to the ionic character and hence strong dynamic annealing of CdO, irradiation with much lighter He+ stabilizes the electron concentration at a much lower level of 1.7x1020 cm-3. A large shift of the optical absorption edge with increasing electron concentration in irradiated samples is explained by the Burstein-Moss shift corrected for electron-electron and electron-ion interactions. The saturation of the electron concentration and the optical absorption edge energy are consistent with a defect induced stabilization of the Fermi energy at 1 eV above the conduction band edge. The result is in a good agreement with previously determined Fermi level pinning energies on CdO surfaces. The results indicate that CdO shares many similarities with InN, as both materials exhibit extremely large electron affinities and an unprecedented propensity for n-type conductivity. 15. Fermi LAT Observations of LS 5039 Energy Technology Data Exchange (ETDEWEB) Abdo, A.A.; /Naval Research Lab, Wash., D.C. /Federal City Coll.; Ackermann, M.; /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept.; Ajello, M.; /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept.; Atwood, W.B.; /UC, Santa Cruz; Axelsson, M.; /Stockholm U. /Stockholm U., OKC; Baldini, L.; /INFN, Pisa; Ballet, J.; /DAPNIA, Saclay; Barbiellini, G.; /INFN, Trieste /Trieste U.; Bastieri, D.; /INFN, Padua /Padua U.; Baughman, B.M.; /Ohio State U.; Bechtol, K.; /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept.; Bellazzini, R.; /INFN, Pisa; Berenji, B.; /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept.; Blandford, R.D.; /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept.; Bloom, E.D.; /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept.; Bonamente, E.; /INFN, Perugia /Perugia U.; Borgland, A.W.; /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept.; Bregeon, J.; /INFN, Pisa; Brez, A.; /INFN, Pisa; Brigida, M.; /Bari U. /INFN, Bari; Bruel, P.; /Ecole Polytechnique /Washington U., Seattle /Padua U. /Bari U. /INFN, Bari /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept. /IASF, Milan /Milan Polytechnic /DAPNIA, Saclay /ASDC, Frascati /INFN, Perugia /Perugia U. /NASA, Goddard /NASA, Goddard /CSST, Baltimore /DAPNIA, Saclay /Naval Research Lab, Wash., D.C. /George Mason U. /NASA, Goddard /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept. /INFN, Perugia /Perugia U. /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept. /Montpellier U. /Sonoma State U. /Stockholm U. /Stockholm U., OKC /DAPNIA, Saclay /NASA, Goddard /CSST, Baltimore /SLAC /ASDC, Frascati /Naval Research Lab, Wash., D.C. /INFN, Trieste /Bari U. /INFN, Bari /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept. /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept. /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept. /Stanford U., HEPL /KIPAC, Menlo Park /Stanford U., Phys. Dept. /SLAC /Grenoble Observ. /CENBG, Gradignan /CENBG, Gradignan /Montpellier U.; /more authors.. 2012-03-29 The first results from observations of the high-mass X-ray binary LS 5039 using the Fermi Gamma-ray Space Telescope data between 2008 August and 2009 June are presented. Our results indicate variability that is consistent with the binary period, with the emission being modulated with a period of 3.903 {+-} 0.005 days; the first detection of this modulation at GeV energies. The light curve is characterized by a broad peak around superior conjunction in agreement with inverse Compton scattering models. The spectrum is represented by a power law with an exponential cutoff, yielding an overall flux (100 MeV-300 GeV) of 4.9 {+-} 0.5(stat) {+-} 1.8(syst) x 10{sup -7} photon cm{sup -2} s{sup -1}, with a cutoff at 2.1 {+-} 0.3(stat) {+-} 1.1(syst) GeV and photon index {Gamma} = 1.9 {+-} 0.1(stat) {+-} 0.3(syst). The spectrum is observed to vary with orbital phase, specifically between inferior and superior conjunction. We suggest that the presence of a cutoff in the spectrum may be indicative of magnetospheric emission similar to the emission seen in many pulsars by Fermi. 16. Orientifolding of the ABJ Fermi gas CERN Document Server Okuyama, Kazumi 2016-01-01 The grand partition functions of ABJ theory can be factorized into even and odd parts under the reflection of fermion coordinate in the Fermi gas approach. In some cases, the even/odd part of ABJ grand partition function is equal to that of $\\mathcal{N}=5$ $O(n)\\times USp(n')$ theory, hence it is natural to think of the even/odd projection of grand partition function as an orientifolding of ABJ Fermi gas system. By a systematic WKB analysis, we determine the coefficients in the perturbative part of grand potential of such orientifold ABJ theory. We also find the exact form of the first few "half-instanton" corrections coming from the twisted sector of the reflection of fermion coordinate. For the Chern-Simons level $k=2,4,8$ we find closed form expressions of the grand partition functions of orientifold ABJ theory, and for $k=2,4$ we prove the functional relations among the grand partition functions conjectured in arXiv:1410.7658. 17. Fermi Liquid Instabilities in the Spin Channel Energy Technology Data Exchange (ETDEWEB) Wu, Congjun; /Santa Barbara, KITP; Sun, Kai; Fradkin, Eduardo; /Illinois U., Urbana; Zhang, Shou-Cheng; /Stanford U., Phys. Dept. 2010-03-16 We study the Fermi surface instabilities of the Pomeranchuk type in the spin triplet channel with high orbital partial waves (F{sub l}{sup a} (l > 0)). The ordered phases are classified into two classes, dubbed the {alpha} and {beta}-phases by analogy to the superfluid {sup 3}He-A and B-phases. The Fermi surfaces in the {alpha}-phases exhibit spontaneous anisotropic distortions, while those in the {beta}-phases remain circular or spherical with topologically non-trivial spin configurations in momentum space. In the {alpha}-phase, the Goldstone modes in the density channel exhibit anisotropic overdamping. The Goldstone modes in the spin channel have nearly isotropic underdamped dispersion relation at small propagating wavevectors. Due to the coupling to the Goldstone modes, the spin wave spectrum develops resonance peaks in both the {alpha} and {beta}-phases, which can be detected in inelastic neutron scattering experiments. In the p-wave channel {beta}-phase, a chiral ground state inhomogeneity is spontaneously generated due to a Lifshitz-like instability in the originally nonchiral systems. Possible experiments to detect these phases are discussed. 18. Massive Fermi gas in the expanding universe Science.gov (United States) Trautner, Andreas 2017-03-01 The behavior of a decoupled ideal Fermi gas in a homogeneously expanding three-dimensional volume is investigated, starting from an equilibrium spectrum. In case the gas is massless and/or completely degenerate, the spectrum of the gas can be described by an effective temperature and/or an effective chemical potential, both of which scale down with the volume expansion. In contrast, the spectrum of a decoupled massive and non-degenerate gas can only be described by an effective temperature if there are strong enough self-interactions such as to maintain an equilibrium distribution. Assuming perpetual equilibration, we study a decoupled gas which is relativistic at decoupling and then is red-shifted until it becomes non-relativistic. We find expressions for the effective temperature and effective chemical potential which allow us to calculate the final spectrum for arbitrary initial conditions. This calculation is enabled by a new expansion of the Fermi-Dirac integral, which is for our purpose superior to the well-known Sommerfeld expansion. We also compute the behavior of the phase space density under expansion and compare it to the case of real temperature and real chemical potential. Using our results for the degenerate case, we also obtain the mean relic velocity of the recently proposed non-thermal cosmic neutrino background. 19. Fermi's paradox: The last challenge for copernicanism? Directory of Open Access Journals (Sweden) Ćirković M.M. 2009-01-01 Full Text Available We review Fermi's paradox (or the 'Great Silence' problem, not only arguably the oldest and crucial problem for the Search for ExtraTerrestrial Intelligence (SETI, but also a conundrum of profound scientific, philosophical and cultural importance. By a simple analysis of observation selection effects, the correct resolution of Fermi's paradox is certain to tell us something about the future of humanity. Already more than three quarters of century old puzzle and a quarter of century since the last major review paper in the field by G. David Brin has generated many ingenious discussions and hypotheses. We analyze the often tacit methodological assumptions built in various answers to this puzzle and attempt a new classification of the numerous solutions proposed in an already huge literature on the subject. Finally, we consider the ramifications of various classes of hypotheses for the practical SETI projects. Somewhat paradoxically, it seems that the class of (neocatastrophic hypotheses gives, on the balance, the strongest justification to optimism regarding our current and near-future SETI efforts. 20. Fermi's Paradox - The Last Challenge For Copernicanism? Directory of Open Access Journals (Sweden) Ćirković, M. M. 2009-06-01 Full Text Available We review Fermi's paradox (or the "Great Silence" problem, not only arguably the oldest and crucial problem for the Search for ExtraTerrestrial Intelligence (SETI, but also a conundrum of profound scientific, philosophical and cultural importance. By a simple analysis of observation selection effects, the correct resolution of Fermi's paradox is certain to tell us something about the future of humanity. Already more than three quarters of century old puzzle -- and a quarter of century since the last major review paper in the field by G. David Brin -- has generated many ingenious discussions and hypotheses. We analyze the often tacit methodological assumptions built in various answers to this puzzle and attempt a new classification of the numerous solutions proposed in an already huge literatureon the subject. Finally, we consider the ramifications of variousclasses of hypotheses for the practical SETI projects. Somewhatparadoxically, it seems that the class of (neocatastrophichypotheses gives, on the balance, the strongest justification tooptimism regarding our current and near-future SETI efforts. 1. Universal Nonequilibrium Properties of Dissipative Rydberg Gases Science.gov (United States) Marcuzzi, Matteo; Levi, Emanuele; Diehl, Sebastian; Garrahan, Juan P.; Lesanovsky, Igor 2014-11-01 We investigate the out-of-equilibrium behavior of a dissipative gas of Rydberg atoms that features a dynamical transition between two stationary states characterized by different excitation densities. We determine the structure and properties of the phase diagram and identify the universality class of the transition, both for the statics and the dynamics. We show that the proper dynamical order parameter is in fact not the excitation density and find evidence that the dynamical transition is in the "model A " universality class; i.e., it features a nontrivial Z2 symmetry and a dynamics with nonconserved order parameter. This sheds light on some relevant and observable aspects of dynamical transitions in Rydberg gases. In particular it permits a quantitative understanding of a recent experiment [C. Carr, Phys. Rev. Lett. 111, 113901 (2013)] which observed bistable behavior as well as power-law scaling of the relaxation time. The latter emerges not due to critical slowing down in the vicinity of a second order transition, but from the nonequilibrium dynamics near a so-called spinodal line. 2. Kinetic approach to granular gases. Science.gov (United States) Puglisi, A; Loreto, V; Marini Bettolo Marconi, U; Vulpiani, A 1999-05-01 We address the problem of the so-called "granular gases," i.e., gases of massive particles in rapid movement undergoing inelastic collisions. We introduce a class of models of driven granular gases for which the stationary state is the result of the balance between the dissipation and the random forces which inject energies. These models exhibit a genuine thermodynamic limit, i.e., at fixed density the mean values of kinetic energy and dissipated energy per particle are independent of the number N of particles, for large values of N. One has two regimes: when the typical relaxation time tau of the driving Brownian process is small compared with the mean collision time tau(c) the spatial density is nearly homogeneous and the velocity probability distribution is Gaussian. In the opposite limit tau>tau(c) one has strong spatial clustering, with a fractal distribution of particles, and the velocity probability distribution strongly deviates from the Gaussian one. Simulations performed in one and two dimensions under the Stosszahlansatz Boltzmann approximation confirm the scenario. Furthermore, we analyze the instabilities bringing to the spatial and the velocity clusterization. Firstly, in the framework of a mean-field model, we explain how the existence of the inelasticity can lead to a spatial clusterization; on the other hand, we discuss, in the framework of a Langevin dynamics treating the collisions in a mean-field way, how a non-Gaussian distribution of velocity can arise. The comparison between the numerical and the analytical results exhibits an excellent agreement. 3. Mechanics of liquids and gases CERN Document Server Loitsyanskii, L G; Jones, W P 1966-01-01 Mechanics of Liquids and Gases, Second Edition is a 10-chapter text that covers significant revisions concerning the dynamics of an ideal gas, a viscous liquid and a viscous gas.After an expanded introduction to the fundamental properties and methods of the mechanics of fluids, this edition goes on dealing with the kinetics and general questions of dynamics. The next chapters describe the one-dimensional pipe flow of a gas with friction, the elementary theory of the shock tube; Riemann's theory of the wave propagation of finite intensity, and the theory of plane subsonic and supersonic flows. 4. Pole-Based Approximation of the Fermi-Dirac Function Institute of Scientific and Technical Information of China (English) Lin LIN; Jianfeng LU; Lexing YING; Weinan E 2009-01-01 Two approaches for the efficient rational approximation of the Fermi-Dirac function are discussed: one uses the contour integral representation and conformal map-ping, and the other is based on a version of the multipole representation of the Fermi-Dirac function that uses only simple poles. Both representations have logarithmic computational complexity. They are of great interest for electronic structure calculations. 5. Tan's distributions and Fermi-Huang pseudopotential in momentum space DEFF Research Database (Denmark) Valiente, Manuel 2012-01-01 form of the Fourier-transformed pseudopotential remains very simple. Operator forms for the so-called Tan's selectors, which, together with Fermi-Huang pseudopotential, largely simplify the derivation of Tan's universal relations for the Fermi gas, are here derived and are also very simple. A momentum... 6. Don't Just Stand There--Teach Fermi Problems! Science.gov (United States) Robinson, A. W. 2008-01-01 Fermi problems, or order of magnitude estimates, are often used in introductory physics courses. In this paper I will show that first year students studying physics at university do not arrive with the skill set to solve these problems, and they have to be actively taught how to solve them. Once they have been shown how to solve Fermi problems,… 7. Some Aspects of Statistical Thermodynamics of a Magnetized Fermi Gas CERN Document Server 2015-01-01 We show that at the Landau ground state a Fermi gas remains precisely a three-dimensional for an arbitrary magnetic field in radical contrast to the previous claims that the perpendicular component of the pressure of a Fermi gas vanishes at the Landau ground state and therefore, it becomes strictly a one-dimensional gas. 8. Shubnikov-de Haas oscillation of KTaO3 based electron gases Science.gov (United States) Miao, Ludi; Du, Renzhong; Li, Qi; Qi Li's Lab Team Two-dimensional electron gases (2DEGs) at transition metal oxide (TMO) surfaces and interfaces have attracted much attention due to their exotic properties such as superconductivity, and ferromagnetism. Recently, 5 d TMOs are hotly investigated due to their strong spin-orbit coupling (SOC), an indispensable element for topolotical insulating states. Among them, KTaO3 not only hosts 2DEGs but also involves strong SOC. We have created KTaO3 based electron gases, with low temperature mobility as large as 8000cm2V-1s-1. Shubnikov de Haas oscillations in magnetoresistance have been observed at 1.8 K for field applied along various directions. Contributions from dxy and dxz / yz bands are both seen. These oscillation curves exhibit a field direction dependence with 4-fold symmetry, revealing the cubic symmetry of Fermi surface of KTaO3 based electon gases. Moreover, the intercept of oscillation indices is close to 0.5, a typical value for systems that involve strong SOC. Our results provide unique insights into the electronic structures of KTaO3 based electron gases via magnetotransport measurements. 9. Changing Horses in Midstream: Fermi LAT Computing and SCons Science.gov (United States) Bogart, J. R.; Golpayegani, N. 2011-07-01 (For the Fermi LAT Collaboration) Several years into GLAST (now Fermi) offline software development it became evident we would need a replacement for our original build system, the Configuration Management Tool (CMT) developed at CERN, in order to support Mac users and to keep pace with newer compilers and operating system versions on our traditional platforms, Linux and Windows. The open source product SCons emerged as the only viable alternative and development began in earnest several months before Fermi's successful launch in June of 2008. Over two years later the conversion is nearing completion. This paper describes the conversion to and our use of SCons, concentrating on the resulting environment for users and developers and how it was achieved. Topics discussed include SCons and its interaction with Fermi code, GoGui, a cross-platform gui for Fermi developers, and issues specific to Windows developer support. 10. The Spectral Backbone of Excitation Transport in Ultra-Cold Rydberg Gases CERN Document Server Scholak, Torsten; Buchleitner, Andreas 2014-01-01 The spectral structure underlying excitonic energy transfer in ultra-cold Rydberg gases is studied numerically, in the framework of random matrix theory, and via self-consistent diagrammatic techniques. Rydberg gases are made up of randomly distributed, highly polarizable atoms that interact via strong dipolar forces. Dynamics in such a system is fundamentally different from cases in which the interactions are of short range, and is ultimately determined by the spectral and eigenvector structure. In the energy levels' spacing statistics, we find evidence for a critical energy that separates delocalized eigenstates from states that are localized at pairs or clusters of atoms separated by less than the typical nearest-neighbor distance. We argue that the dipole blockade effect in Rydberg gases can be leveraged to manipulate this transition across a wide range: As the blockade radius increases, the relative weight of localized states is reduced. At the same time, the spectral statistics -- in particular, the den... 11. Wigner’s phase-space function and atomic structure: II. Ground states for closed-shell atoms DEFF Research Database (Denmark) Springborg, Michael; Dahl, Jens Peder 1987-01-01 display and analyze the function for the closed-shell atoms helium, beryllium, neon, argon, and zinc in the Hartree-Fock approximation. The quantum-mechanical exact results are compared with those obtained with the approximate Thomas-Fermi description of electron densities in phase space.... 12. Cloud processing of soluble gases Science.gov (United States) Laj, P.; Fuzzi, S.; Facchini, M. C.; Lind, J. A.; Orsi, G.; Preiss, M.; Maser, R.; Jaeschke, W.; Seyffer, E.; Helas, G.; Acker, K.; Wieprecht, W.; Möller, D.; Arends, B. G.; Mols, J. J.; Colvile, R. N.; Gallagher, M. W.; Beswick, K. M.; Hargreaves, K. J.; Storeton-West, R. L.; Sutton, M. A. Experimental data from the Great Dun Fell Cloud Experiment 1993 were used to investigate interactions between soluble gases and cloud droplets. Concentrations of H 2O 2, SO 2, CH 3COOOH, HCOOH, and HCHO were monitored at different sites within and downwind of a hill cap cloud and their temporal and spatial evolution during several cloud events was investigated. Significant differences were found between in-cloud and out-of-cloud concentrations, most of which could not be explained by simple dissolution into cloud droplets. Concentration patterns were analysed in relation to the chemistry of cloud droplets and the gas/liquid equilibrium. Soluble gases do not undergo similar behaviour: CH 3COOH simply dissolves in the aqueous phase and is outgassed upon cloud dissipation; instead, SO 2 is consumed by its reaction with H 2O 2. The behaviour of HCOOH is more complex because there is evidence for in-cloud chemical production. The formation of HCOOH interferes with the odd hydrogen cycle by enhancing the liquid-phase production of H 2O 2. The H 2O 2 concentration in cloud therefore results from the balance of consumption by oxidation of SO 2 in-cloud production, and the rate by which it is supplied to the system by entrainment of new air into the clouds. 13. Precision measurement of the sound velocity in an ultracold fermi gas through the BEC-BCS crossover Science.gov (United States) A trapped Fermi gas near a collisional resonance provides a unique laboratory for testing many-body theories in a variety of fields. The ultracold Fermi gas produced in our lab is comprised of the lowest two spin states of 6Li. At 834 G there is a collisional or Feshbach resonance between the two spin states. The scattering length between trapped atoms of opposing spins far exceeds the interparticle spacing of the gas. On resonance, a strongly interacting, unitary, Fermi gas is created which exhibits universal behavior. The unitary Fermi gas is a prototype for other exotic systems in nature from nuclear matter to neutron stars and high temperature superconductors. For magnetic fields less than 834 G the scattering length is positive, and pairs Fermi atoms can form molecular dimers. These dimers, comprised of two fermions, are bosons. At ultracold temperatures the molecular bosons populate the lowest energy level and form a Bose Einstein Condensate (BEC). For magnetic fields greater than 834G the scattering length between fermions in opposing spin states is negative, like Cooper pairs formed between electrons in a superconductor. The Bardeen, Cooper, and Shriefer (BCS) theory was developed to describe the pairing effect in the context of superconductors. In our experiment we produce an ultracold unitary gas. By tuning the magnetic field to either side of the Feshbach resonance we can transform the gas into a weakly interacting BEC or BCS superfluid. Therefore, the region near a Feshbach resonance is called the BEC-BCS crossover. This dissertation presents a precision measurement of the hydrodynamic sound velocity in an ultracold Fermi gas near a Feshbach resonance. The sound velocity is measured at various magnetic fields both above and below resonance. Moreover, we are able compare our measurements to theoretical descriptions of hydrodynamic sound propagation. Further, our measurement of sound velocity exactly reproduces the non-perturbative case, eliminating the 14. FermiLib v0.1 Energy Technology Data Exchange (ETDEWEB) 2017-02-27 FermiLib is an open source software package designed to facilitate the development and testing of algorithms for simulations of fermionic systems on quantum computers. Fermionic simulations represent an important application of early quantum devices with a lot of potential high value targets, such as quantum chemistry for the development of new catalysts. This software strives to provide a link between the required domain expertise in specific fermionic applications and quantum computing to enable more users to directly interface with, and develop for, these applications. It is an extensible Python library designed to interface with the high performance quantum simulator, ProjectQ, as well as application specific software such as PSI4 from the domain of quantum chemistry. Such software is key to enabling effective user facilities in quantum computation research. 15. The Gamma-ray Sky with Fermi Science.gov (United States) Thompson, David 2012-01-01 Gamma rays reveal extreme, nonthermal conditions in the Universe. The Fermi Gamma-ray Space Telescope has been exploring the gamma-ray sky for more than four years, enabling a search for powerful transients like gamma-ray bursts, novae, solar flares, and flaring active galactic nuclei, as well as long-term studies including pulsars, binary systems, supernova remnants, and searches for predicted sources of gamma rays such as dark matter annihilation. Some results include a stringent limit on Lorentz invariance derived from a gamma-ray burst, unexpected gamma-ray variability from the Crab Nebula, a huge gamma-ray structure associated with the center of our galaxy, surprising behavior from some gamma-ray binary systems, and a possible constraint on some WIMP models for dark matter. 16. Diffusive Shock Acceleration the Fermi Mechanism CERN Document Server Baring, M G 1997-01-01 The mechanism of diffusive Fermi acceleration at collisionless plasma shock waves is widely invoked in astrophysics to explain the appearance of non-thermal particle populations in a variety of environments, including sites of cosmic ray production, and is observed to operate at several sites in the heliosphere. This review outlines the principal results from the theory of diffusive shock acceleration, focusing first on how it produces power-law distributions in test-particle regimes, where the shock dynamics are dominated by the thermal populations that provide the seed particles for the acceleration process. Then the importance of non-linear modifications to the shock hydrodynamics by the accelerated particles is addressed, emphasizing how these subsequently influence non-thermal spectral formation. 17. The Gamma-ray Sky with Fermi Energy Technology Data Exchange (ETDEWEB) Thompson, D.J. [NASA Goddard Space Flight Center, Greenbelt, Maryland, 20771 (United States) 2013-10-15 Gamma rays reveal extreme, nonthermal conditions in the Universe. The Fermi Gamma-ray Space Telescope has been exploring the gamma-ray sky for more than four years, enabling a search for powerful transients like gamma-ray bursts, solar flares, and flaring active galactic nuclei, as well as long-term studies including pulsars, binary systems, supernova remnants, and searches for predicted sources of gamma rays such as clusters of galaxies. Some results include a stringent limit on Lorentz invariance violation derived from a gamma-ray burst, unexpected gamma-ray variability from the Crab Nebula, a huge gamma-ray structure in the direction of the center of our Galaxy, and strong constraints on some Weakly Interacting Massive Particle (WIMP) models for dark matter. 18. Entanglement rules for holographic Fermi surfaces Science.gov (United States) Roychowdhury, Dibakar 2016-08-01 In this paper, based on the notion of Gauge/Gravity duality, we explore the laws of entanglement thermodynamics for most generic classes of Quantum Field Theories with hyperscaling violation. In our analysis, we note that for Quantum Field Theories with compressible quark like excitation, the first law of entanglement thermodynamics gets modified due to the presence of an additional term that could be identified as the entanglement chemical potential associated with hidden Fermi surfaces of the boundary theory. Most notably, we find that the so called entanglement chemical potential does not depend on the size of the entangling region and is purely determined by the quark d.o.f. encoded within the entangling region. 19. Distinguishing short and long Fermi GRBs CERN Document Server Tarnopolski, Mariusz 2015-01-01 Two classes of GRBs, short and long, have been determined without any doubts, and are usually ascribed to different progenitors, yet these classes overlap for a variety of descriptive parameters. A subsample of 46 long and 22 short $Fermi$ GRBs with estimated Hurst Exponents (HEs), complemented by minimum variability time-scales (MVTS) and durations ($T_{90}$) is used to perform a supervised Machine Learning (ML) and Monte Carlo (MC) simulation using a Support Vector Machine (SVM) algorithm. It is found that while $T_{90}$ itself performs very well in distinguishing short and long GRBs, the overall success ratio is higher when the training set is complemented by MVTS and HE. These results may allow to introduce a new (non-linear) parameter that might provide less ambiguous classification of GRBs. 20. Entanglement rules for holographic Fermi surfaces Directory of Open Access Journals (Sweden) Dibakar Roychowdhury 2016-08-01 Full Text Available In this paper, based on the notion of Gauge/Gravity duality, we explore the laws of entanglement thermodynamics for most generic classes of Quantum Field Theories with hyperscaling violation. In our analysis, we note that for Quantum Field Theories with compressible quark like excitation, the first law of entanglement thermodynamics gets modified due to the presence of an additional term that could be identified as the entanglement chemical potential associated with hidden Fermi surfaces of the boundary theory. Most notably, we find that the so called entanglement chemical potential does not depend on the size of the entangling region and is purely determined by the quark d.o.f. encoded within the entangling region. 1. The Gamma-ray Universe through Fermi Science.gov (United States) Thompson, David J. 2012-01-01 Gamma rays, the most powerful form of light, reveal extreme conditions in the Universe. The Fermi Gamma-ray Space Telescope and its smaller cousin AGILE have been exploring the gamma-ray sky for several years, enabling a search for powerful transients like gamma-ray bursts, novae, solar flares, and flaring active galactic nuclei, as well as long-term studies including pulsars, binary systems, supernova remnants, and searches for predicted sources of gamma rays such as dark matter annihilation. Some results include a stringent limit on Lorentz invariance derived from a gamma-ray burst, unexpected gamma-ray variability from the Crab Nebula, a huge ga.nuna-ray structure associated with the center of our galaxy, surprising behavior from some gamma-ray binary systems, and a possible constraint on some WIMP models for dark matter. 2. Depletion of the nuclear Fermi sea CERN Document Server Rios, A; Dickhoff, W H 2009-01-01 The short-range and tensor components of the bare nucleon-nucleon interaction induce a sizeable depletion of low momenta in the ground state of a nuclear many-body system. The self-consistent Green's function method within the ladder approximation provides an \\textit{ab-initio} description of correlated nuclear systems that accounts properly for these effects. The momentum distribution predicted by this approach is analyzed in detail, with emphasis on the depletion of the lowest momentum state. The temperature, density, and nucleon asymmetry (isospin) dependence of the depletion of the Fermi sea is clarified. A connection is established between the momentum distribution and the time-ordered components of the self-energy, which allows for an improved interpretation of the results. The dependence on the underlying nucleon-nucleon interaction provides quantitative estimates of the importance of short-range and tensor correlations in nuclear systems. Directory of Open Access Journals (Sweden) Ćirković Milan M. 2005-01-01 Full Text Available One of the most interesting problems in the nascent discipline of astrobiology is more than half-century old Fermi's paradox: why, considering extraordinary young age of Earth and the Solar System in the Galactic context, don't we perceive much older intelligent communities or signposts of their activity? In spite of a vigorous research activity in recent years, especially bolstered by successes of astrobiology in finding extrasolar planets and extremophiles, this problem (also known as the "Great Silence" or "astrosociological" paradox remains as open as ever. In a previous paper, we have discussed a particular evolutionary solution suggested by Karl Schroeder based on the currently dominant evolutionary doctrine of adaptationism. Here, we extend that discussion with emphasis on the problems such a solution is bound to face, and conclude that it is ultimately quite unlikely. . 4. Structure of Inert Gases Adsorbed in MCM-41 Science.gov (United States) Evans, Dylan; Sokol, Paul One-dimensional quantum liquids of 3He or 4He have generated recent interest for investigation in the Luttinger liquid model. Unfortunately, current studies lack a clear demonstration of definitively one-dimensional behavior. We propose using the templated, porous material, MCM-41, as a host for an atomic Luttinger liquid. In general, the pores of MCM-41 are too wide to provide a strictly one-dimensional environment, so we investigate preplating these pores with inert gases to effectively reduce their diameter. We present the results of studies of the structure of inert gases in MCM-41. Nitrogen sorption isotherms were used to characterize the sample. Then, using inert gases as adsorbates, we determined the minimum effective pore diameter that can be achieved in our sample before capillary condensation takes over. X-ray powder diffraction (XRD) was performed on the ideally preplated sample to investigate the structure of the adsorbates in the nanopores. The XRD measurements are compared to simulations of core-shell cylinder model scattering, and the validity of the model is assessed. The prospects for creating a definitively one-dimensional channel for the application of studying the structure and dynamics of helium confined in one dimension are discussed. This work was supported by the National Science Foundation under Grant DGE-1069091. 5. Doping Scheme in Atomic Chain Electronics Science.gov (United States) 1997-01-01 Due to the dramatic reduction in MOS size, there appear many unwanted effects. In these small devices, the number of dopant atoms in the channel is not macroscopic and electrons may suffer significantly different scattering from device to device since the spatial distribution of dopant atoms is no longer regarded as continuous. This prohibits integration, while it is impossible to control such dopant positions within atomic scale. A fundamental solution is to create electronics with simple but atomically precise structures, which could be fabricated with recent atom manipulation technology. All the constituent atoms are placed as planned, and then the device characteristics are deviation-free, which is mandatory for integration. Atomic chain electronics belongs to this category. Foreign atom chains or arrays form devices, and they are placed on the atomically flat substrate surface. We can design the band structure and the resultant Fermi energy of these structures by manipulating the lattice constant. Using the tight-binding theory with universal parameters, it has been predicted that isolated Si chains and arrays are metallic, Mg chains are insulating, and Mg arrays have metallic and insulating phases [1]. The transport properties along a metallic chain have been studied, emphasizing the role of the contact to electrodes [2]. For electronic applications, it is essential to establish a method to dope a semiconducting chain, which is to control the Fermi energy position without altering the original band structure. If we replace some of the chain atoms with dopant atoms randomly, the electrons will see random potential along die chain and will be localized strongly in space (Anderson localization). However, if we replace periodically, although the electrons can spread over the chain, there will generally appear new bands and band gaps reflecting the new periodicity of dopant atoms. This will change the original band structure significantly. In order to overcome 6. New perspectives for noble gases in oceanography Science.gov (United States) Aeschbach, Werner 2016-08-01 Conditions prevailing in regions of deep water formation imprint their signature in the concentrations of dissolved noble gases, which are conserved in the deep ocean. Such "recharge conditions" including temperature, salinity, and interactions with sea ice are important in view of ocean-atmosphere CO2 partitioning. Noble gases, especially the temperature sensitive Kr and Xe, are well-established tracers to reconstruct groundwater recharge conditions. In contrast, tracer oceanography has traditionally focused on He isotopes and the light noble gases Ne and Ar, which could be analyzed at the required high precision. Recent developments of analytical and data interpretation methods now provide fresh perspectives for noble gases in oceanography. 7. [Study of Fermi resonance by means of solution concentration variation]. Science.gov (United States) Jiang, Xiu-lan; Li, Dong-fei; Chen, Yuan-zheng; Zhou, Mi; Sun, Cheng-lin; Yang, Guang; Li, Zuo-wei; Gao, Shu-qin 2011-05-01 The values of Raman scattering coefficients of some molecules in which Fermi resonance occurs vary with solution concentration variation. We measured the Raman spectra of some solvents such as CCl4, CS2, C6H6, etc by changing the concentration of the solutions ranging from 10% to 100% in volume. As a result, the authors obtained the general law of Fermi resonance. We found some weak Fermi resonance phenomena as well that the two bands of Raman spectrum shift asymmetrically and that the fundamental of overtone is tuned by Fermi resonance and moves towards the same direction with the overtone simultaneously, which is same as the results Bier K. D. obtained by means of high-pressure technique. By means of this method, the authors demonstrated the conclusion that only the fundamental in combinations which has the same symmetry as the fundamental involved in Fermi resonance directly can its intensity variation influence the Fermi resonance. In this article, the authors present a new method to study Fermi resonance. This method is valuable in the identification and the assignment of spectral lines of solutions, the determination of molecular configuration of enzyme, the discrimination of isomer, as well as the influences on the molecular structures and properties caused by hydrogen bond. 8. Instability in Shocked Granular Gases CERN Document Server 2013-01-01 Shocks in granular media, such as vertically oscillated beds, have been shown to develop instabilities. Similar jet formation has been observed in explosively dispersed granular media. Our previous work addressed this instability by performing discrete-particle simulations of inelastic media undergoing shock compression. By allowing finite dissipation within the shock wave, instability manifests itself as distinctive high density non-uniformities and convective rolls within the shock structure. In the present study we have extended this work to investigate this instability at the continuum level. We modeled the Euler equations for granular gases with a modified cooling rate to include an impact velocity threshold necessary for inelastic collisions. Our results showed a fair agreement between the continuum and discrete-particle models. Discrepancies, such as higher frequency instabilities in our continuum results may be attributed to the absence of higher order effects. 9. Instability in shocked granular gases Science.gov (United States) Sirmas, Nick; Falle, Sam; Radulescu, Matei 2014-05-01 Shocks in granular media, such as vertically oscillated beds, have been shown to develop instabilities. Similar jet formation has been observed in explosively dispersed granular media. Our previous work addressed this instability by performing discrete-particle simulations of inelastic media undergoing shock compression. By allowing finite dissipation within the shock wave, instability manifests itself as distinctive high density non-uniformities and convective rolls within the shock structure. In the present study we have extended this work to investigate this instability at the continuum level. We modeled the Euler equations for granular gases with a modified cooling rate to include an impact velocity threshold necessary for inelastic collisions. Our results showed a fair agreement between the continuum and discrete-particle models. Discrepancies, such as higher frequency instabilities in our continuum results may be attributed to the absence of higher order effects. 10. Strong photoassociation in a degenerate fermi gas Science.gov (United States) Rvachov, Timur; Jamison, Alan; Jing, Li; Son, Hyungmok; Ebadi, Sepehr; Jiang, Yijun; Zwierlein, Martin; Ketterle, Wolfgang 2016-05-01 Despite many studies there remain open questions about strong photoassociation in ultracold gases. We study the effects of strong photoassociation in ultracold fermions. Photoassociation occurs only at short range and thus can be used as a tool to probe and control the two-body correlation function in an interacting many-body system. We study the effects of strong photoassociation in 6 Li, the onset of saturation, and its effects on spin polarized and interacting spin-mixtures. This work was funded by the NSF, ARO-MURI, SAMSUNG, and NSERC. 11. Neutron physics for nuclear reactors unpublished writings by Enrico Fermi CERN Document Server Fermi, Enrico; Pisanti, O 2010-01-01 This unique volume gives an accurate and very detailed description of the functioning and operation of basic nuclear reactors, as emerging from yet unpublished papers by Nobel Laureate Enrico Fermi. In the first part, the entire course of lectures on Neutron Physics delivered by Fermi at Los Alamos is reported, according to the version made by Anthony P French. Here, the fundamental physical phenomena are described very clearly and comprehensively, giving the appropriate physics grounds for the functioning of nuclear piles. In the second part, all the patents issued by Fermi (and coworkers) on 12. X.509 Authentication/Authorization in FermiCloud Energy Technology Data Exchange (ETDEWEB) Kim, Hyunwoo [Fermilab; Timm, Steven [Fermilab 2014-11-11 We present a summary of how X.509 authentication and authorization are used with OpenNebula in FermiCloud. We also describe a history of why the X.509 authentication was needed in FermiCloud, and review X.509 authorization options, both internal and external to OpenNebula. We show how these options can be and have been used to successfully run scientific workflows on federated clouds, which include OpenNebula on FermiCloud and Amazon Web Services as well as other community clouds. We also outline federation options being used by other commercial and open-source clouds and cloud research projects. 13. Fermi Large Area Telescope Bright Gamma-ray Source List Energy Technology Data Exchange (ETDEWEB) Abdo, Aous A.; /Naval Research Lab, Wash., D.C.; Ackermann, M.; /KIPAC, Menlo Park /SLAC; Ajello, M.; /KIPAC, Menlo Park /SLAC; Atwood, W.B.; /UC, Santa Cruz; Axelsson, M.; /Stockholm U., OKC /Stockholm U.; Baldini, L.; /INFN, Pisa; Ballet, J.; /DAPNIA, Saclay; Band, D.L.; /NASA, Goddard /NASA, Goddard; Barbiellini, Guido; /INFN, Trieste /Trieste U.; Bastieri, Denis; /INFN, Padua /Padua U.; Bechtol, K.; /KIPAC, Menlo Park /SLAC; Bellazzini, R.; /INFN, Pisa; Berenji, B.; /KIPAC, Menlo Park /SLAC; Bignami, G.F.; /Pavia U.; Bloom, Elliott D.; /KIPAC, Menlo Park /SLAC; Bonamente, E.; /INFN, Perugia /Perugia U.; Borgland, A.W.; /KIPAC, Menlo Park /SLAC; Bregeon, J.; /INFN, Pisa; Brigida, M.; /Bari U. /INFN, Bari; Bruel, P.; /Ecole Polytechnique; Burnett, Thompson H.; /Washington U., Seattle /Bari U. /INFN, Bari /KIPAC, Menlo Park /SLAC /IASF, Milan /IASF, Milan /DAPNIA, Saclay /ASDC, Frascati /INFN, Perugia /Perugia U. /KIPAC, Menlo Park /SLAC /George Mason U. /Naval Research Lab, Wash., D.C. /NASA, Goddard /KIPAC, Menlo Park /SLAC /INFN, Perugia /Perugia U. /KIPAC, Menlo Park /SLAC /Montpellier U. /Sonoma State U. /Stockholm U., OKC /Royal Inst. Tech., Stockholm /Stockholm U. /KIPAC, Menlo Park /SLAC /ASDC, Frascati /NASA, Goddard /Maryland U. /Naval Research Lab, Wash., D.C. /INFN, Trieste /Pavia U. /Bari U. /INFN, Bari /KIPAC, Menlo Park /SLAC /UC, Santa Cruz /KIPAC, Menlo Park /SLAC /KIPAC, Menlo Park /SLAC /KIPAC, Menlo Park /SLAC /Montpellier U. /Bari U. /INFN, Bari /Ecole Polytechnique /NASA, Goddard; /more authors.. 2009-05-15 Following its launch in 2008 June, the Fermi Gamma-ray Space Telescope (Fermi) began a sky survey in August. The Large Area Telescope (LAT) on Fermi in three months produced a deeper and better resolved map of the {gamma}-ray sky than any previous space mission. We present here initial results for energies above 100 MeV for the 205 most significant (statistical significance greater than {approx}10{sigma}) {gamma}-ray sources in these data. These are the best characterized and best localized point-like (i.e., spatially unresolved) {gamma}-ray sources in the early mission data. 14. Fermi LAT View of a Sample of Flaring -Ray AGNs S. Buson; D. Bastieri; F. D’Ammando; G. Tosti 2014-09-01 In the first 3.5 years of operations, Fermi detected several sources whose flaring activity brought them to exceed daily fluxes brighter than ( > 100MeV) > 10-6 ph cm-2 s-1. These episodes were promptly reported to the scientific community by the Fermi collaboration by means of astronomer telegrams (ATels). We focus our attention on the sample composed by these flaring sources, most of which are blazars, known to be extremely variable over the whole electromagnetic spectrum, from radio to -ray energies. We study properties of the selected sample and compare them to general characteristics of the Fermi source catalogue. 15. Temporal dynamics of Bose-condensed gases Energy Technology Data Exchange (ETDEWEB) Trujillo Martinez, Mauricio 2014-03-19 We perform a detailed quantum dynamical study of non-equilibrium trapped, interacting Bose-condensed gases. We investigate Josephson oscillations between interacting Bose-Einstein condensates confined in a finite size double-well trap and the non-trivial time evolution of a coherent state placed at the center of a two dimensional optical lattice. For the Josephson oscillations three time scales appear. We find that Josephson junction can sustain multiple undamped oscillations up to a characteristic time scale τ{sub c} without exciting atoms out of the condensates. Beyond the characteristic time scale τ{sub c} the dynamics of the junction are governed by fast, non-condensed particles assisted Josephson tunnelling as well as the collisions between non-condensed particles. In the non-condensed particles dominated regime we observe strong damping of the oscillations due to inelastic collisions, equilibrating the system leading to an effective loss of details of the initial conditions. In addition, we predict that an initially self-trapped BEC state will be destroyed by these fast dynamics. The time evolution of a coherent state released at the center of a two dimensional optical lattice shows a ballistic expansion with a decreasing expansion velocity for increasing two-body interactions strength and particle number. Additionally, we predict that if the two-body interactions strength exceeds a certain value, a forerunner splits up from the expanding coherent state. We also observe that this system, which is prepared far from equilibrium, can evolve to a quasistationary non-equilibrium state. 16. Path-Integral Monte Carlo Determination of the Fourth-Order Virial Coefficient for a Unitary Two-Component Fermi Gas with Zero-Range Interactions. Science.gov (United States) Yan, Yangqian; Blume, D 2016-06-10 The unitary equal-mass Fermi gas with zero-range interactions constitutes a paradigmatic model system that is relevant to atomic, condensed matter, nuclear, particle, and astrophysics. This work determines the fourth-order virial coefficient b_{4} of such a strongly interacting Fermi gas using a customized ab initio path-integral Monte Carlo (PIMC) algorithm. In contrast to earlier theoretical results, which disagreed on the sign and magnitude of b_{4}, our b_{4} agrees within error bars with the experimentally determined value, thereby resolving an ongoing literature debate. Utilizing a trap regulator, our PIMC approach determines the fourth-order virial coefficient by directly sampling the partition function. An on-the-fly antisymmetrization avoids the Thomas collapse and, combined with the use of the exact two-body zero-range propagator, establishes an efficient general means to treat small Fermi systems with zero-range interactions. 17. Nanoscale atomic waveguides with suspended carbon nanotubes CERN Document Server Peano, V; Kasper, A; Egger, R 2005-01-01 We propose an experimentally viable setup for the realization of one-dimensional ultracold atom gases in a nanoscale magnetic waveguide formed by single doubly-clamped suspended carbon nanotubes. We show that all common decoherence and atom loss mechanisms are small guaranteeing a stable operation of the trap. Since the extremely large current densities in carbon nanotubes are spatially homogeneous, our proposed architecture allows to overcome the problem of fragmentation of the atom cloud. Adding a second nanowire allows to create a double-well potential with a moderate tunneling barrier which is desired for tunneling and interference experiments with the advantage of tunneling distances being in the nanometer regime. 18. 40 CFR 86.1514 - Analytical gases. Science.gov (United States) 2010-07-01 ... carbon monoxide on a dry basis. (b) If the raw CO sampling system specified in 40 CFR part 1065 is used, the analytical gases specified in 40 CFR part 1065, subpart H, shall be used. (c) If a CVS sampling system is used, the analytical gases specified in 40 CFR part 1065, subpart H, shall be used.... 19. 40 CFR 91.312 - Analytical gases. Science.gov (United States) 2010-07-01 ... stated by the gas supplier for each calibration gas. (b) Pure gases. The required purity of the gases is... purified synthetic air which contains a concentration of propane higher than what a gas supplier considers... manufacturer must be consistent in the choice of diluent (zero air or purified nitrogen) between... 20. Controlling interactions between highly-magnetic atoms with Feshbach resonances CERN Document Server Kotochigova, Svetlana 2014-01-01 This paper reviews current experimental and theoretical progress in the study of dipolar quantum gases of ground and meta-stable atoms with a large magnetic moment. We emphasize the anisotropic nature of Feshbach resonances due to coupling to fast-rotating resonant molecular states in ultracold s-wave collisions between magnetic atoms in external magnetic fields. The dramatic differences in the distribution of resonances of magnetic $^7$S$_3$ chromium and magnetic lanthanide atoms with a submerged 4f shell and non-zero electron angular momentum is analyzed. We focus on Dysprosium and Erbium as important experimental advances have been recently made to cool and create quantum-degenerate gases for these atoms. Finally, we describe progress in locating resonances in collisions of meta-stable magnetic atoms in electronic P states with ground-state atoms, where an interplay between collisional anisotropies and spin-orbit coupling exists.	{"extraction_info": {"found_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": 0.8650729656219482, "perplexity": 2774.3150044891213}, "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/1508187823067.51/warc/CC-MAIN-20171018180631-20171018200631-00376.warc.gz"}
https://www.physicsforums.com/threads/what-simple-concept-am-i-missing-about-moments.679399/	# What simple concept am I missing about moments? 1. Mar 19, 2013 ### mindheavy In a two-dimensional statics problem involving finding a moment about a point, I don't understand how the result is either in the positive/negative z direction. I realize moments are found by the cross product, and the cross product requires the answer to be perpendicular to the plane that the two vectors form. If I look down at the surface of my desk and take that to be an x-y plane. a pencil laying on the plane is held stationary at it's left end, and the right end is made to rotate clockwise. This tells me that I can expect my moment to be in the negative z direction (say, going down through the surface of the desk). This makes no sense to me, what concept am I missing here? 2. Mar 19, 2013 ### tiny-tim hi mindheavy! it's a convention we could define it other way round (ie moment up for clockwise) (like we could say electrons have positive charge, protons have negative charge) but we don't, we say moment down for clockwise, moment up for anti-clockwise what don't you like about that? 3. Mar 19, 2013 ### mindheavy What I don't like is thinking of the pen rotating clockwise and knowing that the 'answer' for the moment is going down through the desk. Is it that I don't really get what a moment is? It's easy for me to understand that a force applied to the pen causes it to rotate, but why is the moment saying the result is perpendicular? Why is the result not just the direction of rotation? Another way I'm looking at it: I'm imagining standing at a slot machine, the lever is directly in front of me. When I pull this lever down, there's a force acting on it going straight out either left or right? This makes no sense to me. 4. Mar 19, 2013 ### tiny-tim hi mindheavy! it's not a force (it's not even a vector, it's a pseudovector) it's a moment (or torque or couple) imagine that lever is twice as long, with the pivot in the middle you'd produce the same effect by pulling one end down and the other end up the total force is zero only the total moment is non-zero but the direction of rotation is perpendicular to the desk! here's a question for you: how would you describe the direction of rotation of the earth? (and if you wanted to make the earth rotate faster, what would you call the direction of the moment you should apply?) 5. Mar 19, 2013 ### mindheavy Ok, this is starting to become clearer. Getting a vector when calculating a moment made me think of a force, but if it isn't actually a force, it will be easier to grasp. I would describe the earth as rotating about it's vertical axis, and the direction of this rotation is perpendicular to that axis. Is that a main idea here, being perpendicular to the axis? 6. Mar 19, 2013 ### tiny-tim nooo, the direction of rotation is the axis that's the only way we can unambiguously define rotation! the earth's axis is N-S perpendicular would be any diameter through the equator, but which one? through ecuador? through kenya? isn't the only sensible direction of rotation the line through the poles? 7. Mar 19, 2013 ### mindheavy I thought it would be the line perpendicular to the axis of rotation, at any point along that axis, maybe I have some more reading to do :) 8. Mar 26, 2013 ### timthereaper A vector is really just a magnitude and a direction. When vector analysis was developed, someone thought, "Hey, we can apply this to physics!". Forces have a magnitude and a direction, positions can have a magnitude and a direction from an origin, so using vector-based math seemed like a perfect fit. Some scientists figured out that if you do the vector cross product of a (relative) position and a force, you get a vector whose magnitude corresponded exactly to the magnitude of a moment created around a point. The direction of that vector is a result of the cross product operation (following the right-hand rule). In vector analysis, that cross product vector is perpendicular to the plane that the two original vectors lie in. Translating that into the physical problem, they determined that the direction pointed along the axis of rotation and perpendicular to the plane of the created moment. They also figured out that which way the direction vector pointed identified which way the moment "spins" on that axis. The moment vector is still just a magnitude and direction, but it tells you in what plane the actual "rotation" occurs, which way it's "spinning", and what the magnitude of the moment is. Similar Discussions: What simple concept am I missing about moments?	{"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": 0.829991340637207, "perplexity": 573.581442888763}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806316.80/warc/CC-MAIN-20171121040104-20171121060104-00468.warc.gz"}
https://fgiesen.wordpress.com/2013/05/13/trig-identities-from-complex/	There’s tons of useful trig identities. You could spend the time to learn them by heart, or just look them up on Wikipedia when necessary. But I’ve always had problems remembering where the signs and such go when trying to memorize this directly. At least for me, what worked way better is this: spend a few hours familiarizing yourself with complex numbers if you haven’t done so already; after that, most identities that you need in practice are easy to derive from Euler’s formula: $e^{ix} = \exp(ix) = \cos(x) + i \sin(x)$ Let’s do the basic addition formulas first. Euler’s formula gives: $\cos(x+y) + i \sin(x+y) = \exp(i(x+y)) = \exp(ix) \exp(iy)$ and once we apply the identity again we get: $(\cos(x) + i \sin(x)) (\cos(y) + i \sin(y))$ multiplying out: $(\cos(x) \cos(y) - \sin(x) \sin(y)) + i (\sin(x) \cos(y) + \cos(x) \sin(y))$ The terms in parentheses are all real numbers; equating them with our original expression yields the result $\cos(x+y) = \cos(x) \cos(y) - \sin(x) \sin(y)$ $\sin(x+y) = \sin(x) \cos(y) + \cos(x) \sin(y)$ Both addition formulas for the price of one. (In fact, this exploits that the addition formulas for trigonometric functions and the addition formula for exponents are really the same thing). The main point being that if you know complex multiplication, you never have to remember what the grouping of factors and the signs are, something I used to have trouble remembering. Plugging in x=y into the above also immediately gives the double-angle formulas: $\cos(2x) = \cos(x)^2 - \sin(x)^2$ $\sin(2x) = 2 \sin(x) \cos(x)$ so if you know the addition formulas there’s really no reason to learn these separately. Then there’s the well-known $\cos(x)^2 + \sin(x)^2 = 1$ but it’s really just the Pythagorean theorem in disguise (since cos(x) and sin(x) are the side lengths of a right-angled triangle). So not really a new formula either! Moving either the cosine or sine terms to the right-hand side gives the two immensely useful equations: $\cos(x)^2 = 1 - \sin(x)^2$ $\sin(x)^2 = 1 - \cos(x)^2$ In particular, that second one is perfect if you need the sine squared of an angle that you only have the cosine of (usually because you’ve determined it using a dot product). Judicious application of these two tends to be a great way to simplify superfluous math in shaders (and elsewhere), one of my pet peeves. For practice, let’s apply these two identities to the cosine double-angle formula: $\cos(2x) = \cos(x)^2 - \sin(x)^2 = 2 \cos(x)^2 - 1 \Leftrightarrow cos(x)^2 = (cos(2x) + 1) / 2$ $\cos(2x) = \cos(x)^2 - \sin(x)^2 = 1 - 2 \sin(x)^2 \Leftrightarrow sin(x)^2 = (1 - cos(2x)) / 2$ why, it’s the half-angle formulas! Fancy meeting you here! Can we do something with the sine double-angle formula too? Well, it’s not too fancy, but we can get this: $\sin(2x) = 2 \sin(x) \cos(x) \Leftrightarrow \sin(x) \cos(x) = \sin(2x) / 2$ Now, let’s go back to the original addition formulas and let’s see what happens when we plug in negative values for y. Using $\sin(-x) = -\sin(x)$ and $\cos(-x) = \cos(x)$, we get: $\cos(x-y) = \cos(x) \cos(y) + \sin(x) \sin(y)$ $\sin(x-y) = \sin(x) \cos(y) - \cos(x) \sin(y)$ Hey look, flipped signs! This means that we can now add these to (or subtract them from) the original formulas to get even more identities! $\cos(x+y) + \cos(x-y) = 2 \cos(x) \cos(y)$ $\cos(x-y) - \cos(x+y) = 2 \sin(x) \sin(y)$ $\sin(x+y) + \sin(x-y) = 2 \sin(x) \cos(y)$ $\sin(x+y) - \sin(x-y) = 2 \cos(x) \sin(y)$ It’s the product-to-sum identities this time. I got one more! We’ve deliberately flipped signs and then added/subtracted the addition formulas to get the above set. What if we do the same trick in reverse to get rid of those x+y and x-y terms? Let’s set $x = (a + b)/2$ and $y = (b - a)/2$ and plug that into the identities above and we get: $\cos(b) + \cos(a) = 2 \cos((a+b)/2) \cos((b-a)/2)$ $\cos(a) - \cos(b) = 2 \sin((a + b)/2) \sin((b - a)/2)$ $\sin(b) + \sin(a) = 2 \sin((a + b)/2) \cos((b - a)/2)$ Ta-dah, it’s the sum-to-product identities. Now, admittedly, we’ve taken quite a few steps to get here, and looking these up when you need them is going to be faster than walking through the derivation (if you ever need them in the first place – I don’t think I’ve ever used the product/sum identities in practice). But still, working these out is a good exercise, and a lot less likely to go wrong (at least for me) than memorizing lots of similar formulas. (I never can get the signs right that way) Bonus exercise: work out general expressions for $\cos(x)^n$ and $\sin(x)^n$. Hint: $\cos(x) = (\exp(ix) + \exp(-ix))/2$ $\sin(x) = (\exp(ix) - \exp(-ix))/2i$. And I think that’s enough for now. (At some later point, I might do an extra post about one of the sneakier trig techniques: the Weierstrass substitution). From → Maths	{"extraction_info": {"found_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": 31, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.947822630405426, "perplexity": 500.12251333704245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187823284.50/warc/CC-MAIN-20171019122155-20171019142155-00046.warc.gz"}
https://www.physicsforums.com/threads/magnetic-field-of-wire-on-rectangular-loop.160744/	# Magnetic Field of wire on rectangular loop 1. Mar 14, 2007 ### BrettB 1. The problem statement, all variables and given/known data I have attached a diagram. In case you can't view it, it shows an infintely long wire $I_i = 5.00$ A on the positive y direction. 0.100 m to the right, there is a rectangular loop of dimensions 0.150 m x 0.450 m, the long side is parallel to the infinitely long wire to its left. The current is $I_2 = 10.0$ A and also flowing up (in the position closest to the infinitely long wire). a) Find the magnitude and direction of the net force exerted on the loop by the magnetic field created by the wire. b) Find the force on the top, horizontal segment of the loop. Calculate this three ways: net force, average force, and the force on the midpoint. 2. Relevant equations Bio-Savaart equation. 3. The attempt at a solution I didn't have any trouble with (a). The force on the top and bottom horizontal segments just cancel, so I can disregard them. The rest is pretty simple. Part (b) is giving me lots of trouble. I am not 100% sure I am calculating the magnetic force of the infinitely long wire on the horizontal part of the loop properly, and I really am not certain how to handle the vertical segments of the loop. These are not infinite, so there must be some error introduced in this case, but I have no idea how to approximate it. Would the errors just cancel, and the net force of these also be zero? Here is what I have done so far, I hope this is ok: For the net force: $$B = \int_{0.100}^{0.250} \frac{(5.00)(10.0)(\mu_0)}{2\pi x}\,dx = 1.00268(\ln(x))_{0.100}^{0.250} = 9.19 \times 10^{-6}$$ For the average force, I just used the mean value theorem, and multiplied the above by 1/(0.250-0.100) to get 6.12 x 10^{-15}. To calculate the force on the midpoint, I first calculate the force from the vertical wire: $$\frac{\mu_0(5.00)}{2\pi(0.100+\frac{0.150}{2})}\hat{i} = 5.73\times 10^{-6}\hat{i}$$ Then the other force: $$\frac{\mu_0(10.0)}{2\pi(0.100 + \frac{0.150}{2})}\hat{j} = 1.15 \times 10^{-5}\hat{j}$$ Since these are perpendicular, I got the rest as $$\sqrt{(5.73\times 10^{-6})^2 + (1.15 \times 10^{-5})^2} = 1.28 \times 10^{-5}$$ Intuitively, I would have expected closer values with all these results, so I am surprised they are so far apart, and that makes me question my work. Thanks, Brett #### Attached Files: • ###### magnetism.bmp File size: 14.6 KB Views: 357 Last edited: Mar 14, 2007 2. Mar 14, 2007 ### BrettB Wow, nobody? Well, I fixed a couple of typos, but I am no closer to certainty. If I've violated some rule, I'm sorry. I'm new here :-) But I would really appreciate some help. Thanks, Brett 3. Mar 15, 2007 ### hage567 You haven't done anything wrong, I think you've done a good job of presenting your problem. I just honestly am not sure I can help! 4. Mar 15, 2007 ### Staff: Mentor I'm a bit puzzled by this question. I would interpret it as: Calculate the force exerted on the top horizontal segment by the magnetic field created by the wire in three different ways and compare. For the "net force" you integrated to find the actual force from the wire on the segment. OK. (Don't really know what "net force" means in this context.) By "average force", I would have calculated the force per unit length at each end of the segment, averaged them, then multiplied by the length of the segment. For "force at midpoint", I would have calculated the force per unit length at the midpoint, then multiplied by the length of the segment. Grasping at straws here, since the question isn't clear to me. 5. Mar 15, 2007 ### hage567 I can point out a couple of things for you to double check. I think you are getting B and F mixed up when you are using the equations. Not sure, just a thought. 6. Mar 15, 2007 ### BrettB Thank you both, VERY much. Hage567, yes, I was calculating the completely wrong thing, thank you. I think I still must be doing something wrong, since I get answers that are very different, or I don't understand why they should be this different. I'm also wondering why using the mean-value theorem wouldn't be the right way to calculate the "average" value in this case. Using Doc Al's suggestion to compute the average, I get for the left end of the horiztonal wire: $$\frac{(1.26 \times 10^{-6})(5.00)(10.0)}{2\pi (0.100)} = 1.00268 \times 10^{-4}$$ For the right end: $$\frac{(1.26 \times 10^{-6})(5.00)(10.0)}{2\pi (0.250)} = 4.01070 \times 10^{-5}$$ These values would be the force per meter, I believe. Taking the average, and multiplying by the length of the wire, I get 1.05 x 10^-5. For the force/m at the midpoint: $$\frac{(1.26\times 10^{-6})(5.00)(10.0)}{2\pi(0.100 + \frac{0.150}{2})} = 5.72958 \times 10^{-5}$$ and multiplying by 0.150 (the length of the wire), I get 8.59 x 10^-6, which is closer to the total force. But not very. Is this the right way to go about it? The only conclusion I can come to from this is that, since the force doesn't vary linearly (if you look at the integral, it is a logarithmic function), using these other ways is not the right way to estimate the force on a perpendicular wire. Thanks! 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https://repository.lboro.ac.uk/articles/journal_contribution/Temporal_analysis_of_a_microkernel/9402617/1	Hussak_Temporal_analysis_of_a_microkernel.pdf (454.78 kB) # Temporal analysis of a microkernel journal contribution posted on 04.02.2009, 12:40 Temporal logic techniques have been proposed as a way of achieving a very natural transition from informal requirements to a formal specification of the requirements. The paper presents a case study of a real-life system developed using such techniques. Both a top-level specification and implementation semantics are given in temporal logic. In particular, the progression from statements in English to temporal logic is highlighted. A correctness proof that the implemented system satisfies the specification has been produced. • Science ## Department • Computer Science ## Citation HUSSAK, W., 1995. Temporal analysis of a microkernel. Software Engineering Journal, 10 (1), pp. 21-26 ## Version VoR (Version of Record) 1995 ## Notes This article was published online in the journal, Software Engineering Journal [© IEEE]. It is also available at: http://ieeexplore.ieee.org/ Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. 0268-6961 en	{"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": 0.8189252614974976, "perplexity": 1340.3596351358135}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153966.52/warc/CC-MAIN-20210730091645-20210730121645-00529.warc.gz"}
http://www.researchgate.net/researcher/6252912_A_Burrows	# A. Burrows Princeton University, Princeton, New Jersey, United States Are you A. Burrows? ## Publications (123)454.82 Total impact • Source ##### Article: A Spitzer Search for Transits of Radial Velocity Detected Super-Earths [Hide abstract] ABSTRACT: Unlike hot Jupiters or other gas giants, super-Earths are expected to have a wide variety of compositions, ranging from terrestrial bodies like our own to more gaseous planets like Neptune. Observations of transiting systems, which allow us to directly measure planet masses and radii and constrain atmospheric properties, are key to understanding the compositional diversity of the planets in this mass range. Although Kepler has discovered hundreds of transiting super-Earth candidates over the past four years, the majority of these planets orbit stars that are too far away and too faint to allow for detailed atmospheric characterization and reliable mass estimates. Ground-based transit surveys focus on much brighter stars, but most lack the sensitivity to detect planets in this size range. One way to get around the difficulty of finding these smaller planets in transit is to start by choosing targets that are already known to contain super-Earth sized bodies detected using the radial velocity technique. Here we present results from a Spitzer program to observe six of the most favorable RV detected super-Earth systems, including HD 1461, HD 7924, HD 156668, HIP 57274, and GJ 876. We find no evidence for transits in any of their 4.5 micron flux light curves, and place limits on the allowed transit depths and corresponding planet radii that rule out even the most dense and iron-rich compositions for these objects. We also observed HD 97658, but the observation window was based on a possible ground-based transit detection (Henry et al. 2011) that was later ruled out; thus the window did not include the predicted time for the transit detection recently made by MOST (Dragomir et al. 2013). The Astrophysical Journal 10/2013; 781(2). DOI:10.1088/0004-637X/781/2/103 · 6.28 Impact Factor • ##### Article: 3.6 and 4.5 micron Full-orbit Phase Curves of the Hot-Saturn HD 149026b [Hide abstract] ABSTRACT: The extrasolar planet HD 149026b, discovered in 2005, was among the first to be observed to transit its host star as seen from earth. Since its discovery, several observational campaigns have targeted HD 149026b in wavelengths from the visible to the infrared to obtain both transmission and emission broadband measurements. These measurements have revealed that HD 149026b has a radius similar to that of Saturn, but a density more akin to that of Neptune. This suggest that HD 149026b is enriched in heavy elements much like the ice giants of our solar system. Half-orbit phase curve observations of HD 149026b at 8 microns (Knutson et al., 2009) suggest that the day-to-night transport of heat is fairly efficient for this planet. However, further phase curve observations at other infrared wavelengths are needed to better constrain the planet’s energy budget, location of hot and cold regions, as well as possible chemical gradients in the planet’s atmosphere. Here we present an analysis of the full-orbit phase-curves of HD 149026b at 3.6 and 4.5 microns. We discuss the implications of the combined phase-curve information at 3.6, 4.5, and 8 microns and compare the observations to theoretical phase curves derived from three-dimensional atmospheric models that consider a range of possible heavy element enrichments in HD 149026b’s atmosphere. • Source ##### Article: Direct Imaging of a Cold Jovian Exoplanet in Orbit around the Sun-like Star GJ 504 [Hide abstract] ABSTRACT: Several exoplanets have recently been imaged at wide separations of >10 AU from their parent stars. These span a limited range of ages (<50 Myr) and atmospheric properties, with temperatures of 800--1800 K and very red colors (J - H > 0.5 mag), implying thick cloud covers. Furthermore, substantial model uncertainties exist at these young ages due to the unknown initial conditions at formation, which can lead to an order of magnitude of uncertainty in the modeled planet mass. Here, we report the direct imaging discovery of a Jovian exoplanet around the Sun-like star GJ 504, detected as part of the SEEDS survey. The system is older than all other known directly-imaged planets; as a result, its estimated mass remains in the planetary regime independent of uncertainties related to choices of initial conditions in the exoplanet modeling. Using the most common exoplanet cooling model, and given the system age of 160 [+350, -60] Myr, GJ 504 b has an estimated mass of 4 [+4.5, -1.0] Jupiter masses, among the lowest of directly imaged planets. Its projected separation of 43.5 AU exceeds the typical outer boundary of ~30 AU predicted for the core accretion mechanism. GJ 504 b is also significantly cooler (510 [+30, -20] K) and has a bluer color (J-H = -0.23 mag) than previously imaged exoplanets, suggesting a largely cloud-free atmosphere accessible to spectroscopic characterization. Thus, it has the potential of providing novel insights into the origins of giant planets, as well as their atmospheric properties. The Astrophysical Journal 07/2013; 774(1). DOI:10.1088/0004-637X/774/1/11 · 6.28 Impact Factor • Source ##### Article: CASTRO: A New Compressible Astrophysical Solver. III. Multigroup Radiation Hydrodynamics [Hide abstract] ABSTRACT: We present a formulation for multigroup radiation hydrodynamics that is correct to order \$O(v/c)\$ using the comoving-frame approach and the flux-limited diffusion approximation. We describe a numerical algorithm for solving the system, implemented in the compressible astrophysics code, CASTRO. CASTRO uses an Eulerian grid with block-structured adaptive mesh refinement based on a nested hierarchy of logically-rectangular variable-sized grids with simultaneous refinement in both space and time. In our multigroup radiation solver, the system is split into three parts, one part that couples the radiation and fluid in a hyperbolic subsystem, another part that advects the radiation in frequency space, and a parabolic part that evolves radiation diffusion and source-sink terms. The hyperbolic subsystem and the frequency space advection are solved explicitly with high-order Godunov schemes, whereas the parabolic part is solved implicitly with a first-order backward Euler method. Our multigroup radiation solver works for both neutrino and photon radiation. The Astrophysical Journal Supplement Series 07/2012; 204(1). DOI:10.1088/0067-0049/204/1/7 · 14.14 Impact Factor • Source ##### Article: Correlated Gravitational Wave and Neutrino Signals from General-Relativistic Rapidly Rotating Iron Core Collapse [Hide abstract] ABSTRACT: We present results from a new set of 3D general-relativistic hydrodynamic simulations of rotating iron core collapse. We assume octant symmetry and focus on axisymmetric collapse, bounce, the early postbounce evolution, and the associated gravitational wave (GW) and neutrino signals. We employ a finite-temperature nuclear equation of state, parameterized electron capture in the collapse phase, and a multi-species neutrino leakage scheme after bounce. The latter captures the important effects of deleptonization, neutrino cooling and heating and enables approximate predictions for the neutrino luminosities in the early evolution after core bounce. We consider 12-solar-mass and 40-solar-mass presupernova models and systematically study the effects of (i) rotation, (ii) progenitor structure, and (iii) postbounce neutrino leakage on dynamics, GW, and, neutrino signals. We demonstrate, that the GW signal of rapidly rotating core collapse is practically independent of progenitor mass and precollapse structure. Moreover, we show that the effects of neutrino leakage on the GW signal are strong only in nonrotating or slowly rotating models in which GW emission is not dominated by inner core dynamics. In rapidly rotating cores, core bounce of the centrifugally-deformed inner core excites the fundamental quadrupole pulsation mode of the nascent protoneutron star. The ensuing global oscillations (f~700-800 Hz) lead to pronounced oscillations in the GW signal and correlated strong variations in the rising luminosities of antineutrino and heavy-lepton neutrinos. We find these features in cores that collapse to protoneutron stars with spin periods <~ 2.5 ms and rotational energies sufficient to drive hyper-energetic core-collapse supernova explosions. Hence, joint GW + neutrino observations of a core collapse event could deliver strong evidence for or against rapid core rotation. [abridged] Physical review D: Particles and fields 04/2012; 86(2). DOI:10.1103/PhysRevD.86.024026 · 4.86 Impact Factor • ##### Article: New Imaging of the beta Pictoris Planet and Debris Disk [Hide abstract] ABSTRACT: We present new direct imaging results on the planet and debris disk surrounding 12 Myr-old beta Pictoris. Using an updated version of our reduction pipeline, we extract a new detection of beta Pic b from 2008 VLT/NaCo data at a sub-Jupiter projected separation ( 4 AU), about 1.5 lambda/D. We also obtain a high signal-to-noise rereduction of L' data taken in 2009 December. Intriguingly, the planet's orbit is aligned with the major axis of the outer disk (Omega 31 deg) but is probably misaligned with the warp/inclined disk at 80 AU, often cited as a signpost for the planet's existence. We also present new images of the beta Pic debris disk and discuss any evidence for a 2nd massive gas giant planet in the system. • ##### Article: Grism Spectroscopy Of The Eclipse Of Corot-2b At 1.1-1.7μM [Hide abstract] ABSTRACT: Here we present HST eclipse spectroscopy spanning 1.1 to 1.7 μm of the CoRoT-2 system using the G141 grism on WFC3. These near-infrared data serve to complement the pre-existing and already-published optical CoRoT data and warm Spitzer infrared observations. CoRoT-2b, the sole planet known in the system, is a member of the Very Hot Jupiter (VHJ) class of exoplanets, and the secondary eclipse was measured with three separate HST visits. We find the albedo of CoRoT-2b upon comparison of optical data, which comprises both thermal and reflected spectral components, and the thermal spectrum as constrained by the Spitzer and WFC3 infrared data. Analysis of the data required characterization of the persistence on the WFC3 detector as it manifests for these observations; it is not insignificant. After compiling results of flux patterns for the majority of the seventeen exoplanets studied in this HST program, we find the extent of the persistence, a linear, additive effect that is strongly dependent upon stimulating flux and time of/since exposure, and subtract it to leave only the true flux received from the CoRoT-2 system in and out of secondary eclipse. • ##### Article: Infrared Spectroscopy of the Transiting Exoplanets HD189733b and XO-1 Using Hubble WFC3 in Spatial Scan Mode [Hide abstract] ABSTRACT: Infrared transmission spectroscopy of the exoplanets HD189733b and XO-1 has been previously reported by Swain et al. and Tinetti et al. based on observations using the NICMOS instrument on the Hubble Space Telescope. The robustness of those results has been questioned, because derivation of the exoplanetary spectrum required decorrelating strong instrumental systematic effects in the NICMOS data. We here discuss results from HST/WFC3 grism 1.1-1.7 micron spectroscopy of these planets during transit. WFC3 instrumental signatures are smaller in both amplitude and complexity as compared to NICMOS. Moreover, we use a new spatial scan mode to trail the stars perpendicular to the dispersion direction during WFC3 exposures, and this increases the efficiency of the observations and reduces persistence effects in the detector. We derive the 1.4-micron water absorption spectrum of these planets during transit, discuss implications for these exoplanetary atmospheres, and compare our results to the NICMOS spectroscopy. • Source ##### Article: The Hydrodynamic Origin of Neutron Star Kicks [Hide abstract] ABSTRACT: We present results from a suite of axisymmetric, core-collapse supernova simulations in which hydrodynamic recoil from an asymmetric explosion produces large proto-neutron star (PNS) velocities. We use the adaptive-mesh refinement code CASTRO to self-consistently follow core-collapse, the formation of the PNS and its subsequent acceleration. We obtain recoil velocities of up to 620 km/s at ~1 s after bounce. These velocities are consistent with the observed distribution of pulsar kicks and with PNS velocities obtained in other theoretical calculations. Our PNSs are still accelerating at several hundred km/s at the end of our calculations, suggesting that even the highest velocity pulsars may be explained by hydrodynamic recoil in generic, core-collapse supernovae. Monthly Notices of the Royal Astronomical Society 12/2011; 423(2). DOI:10.1111/j.1365-2966.2012.21002.x · 5.23 Impact Factor • ##### Article: A Warm Spitzer Survey of Atmospheric Circulation Patterns [Hide abstract] ABSTRACT: The atmospheres of close-in extrasolar planets experience strong, asymmetrically distributed radiative forcing that can potentially lead to dramatic variations in both temperature and composition between the day- and night-side hemispheres. However, secondary eclipse observations only tell us about the properties of the dayside atmosphere, while transmission spectroscopy probes the region around the day-night terminator. By measuring changes in the infrared emission spectra of these planets as a function of orbital phase, we can resolve thermal and compositional gradients in these atmospheres, allowing us to obtain a complete picture of their local properties. The most extensively studied planet to date, HD 189733b, appears to have a relatively modest day-night temperature gradient as seen in the 8 and 24 micron Spitzer bands, suggesting that compositional gradients in this atmosphere are likely to be minimal. We present new, full-orbit phase curves at 3.6 and 4.5 um obtained with warm Spitzer, which we use to construct improved multi-color maps and to constrain variations in the pressure-temperature profile and atmospheric composition as a function of longitude. We also present preliminary results for complementary full-orbit observations of HAT-P-7b in the same bands, and discuss an emerging pattern in which the most highly irradiated (>2000 K) planets appear to undergo a shift towards large day-night temperature gradients, perhaps due to Lorentz braking or other MHD processes. • ##### Conference Paper: Thermal Phase Variations of WASP-12b [Hide abstract] ABSTRACT: The short-period planet WASP-12b is among the hottest known transiting planets. Space- and ground-based secondary eclipse depths imply that this planet has a C/O ratio greater than 1 (Madhusudhan et al. 2011), in stark contrast to the chemistry in the Solar System and the assumed chemistry of other planets. These same eclipse data put the planet's day-side effective temperature at 3000 K. This indicates a low albedo and poor recirculation of heat to the night-side, as has been found for all of the hottest transiting giant planets (Cowan & Agol 2011b). But these trends were based solely on day-side observations (eclipse depths) rather than full phase variations, which directly probe night-side temperature. The short period (1.1 day) and inflated radius (1.8 RJ) of WASP-12b has led to speculation that it may be undergoing Roche-lobe overflow (Li et al. 2010, Lai et al. 2010), and UV observations by Fossati et al. (2010) seem to support this idea. We have recently obtained thermal phase curves of this planet with Warm Spitzer (PI:Machalek; PID 70060). Our data include two eclipses, a transit, and full phase coverage at each of 3.6 and 4.5 micron. Because of the planet's high temperature and large size, this is one of the highest S/N phase curves yet obtained with Spitzer. These data (currently being analyzed) will allow us to directly measure the planet's night-side temperature and the longitudinal offset of its day-side hot-spot. Since the 3.6 and 4.5 micron bands probe different depths in the atmosphere, we will strongly constrain climate models for the hottest gas giants. The high precision transit and eclipse photometry offered by Spitzer will also allow us to search for signs of accretion in this system. AAS/Division for Extreme Solar Systems Abstracts; 09/2011 • ##### Article: The Atmospheric Circulation of Eccentric Hot Jupiter HAT-P-2b [Hide abstract] ABSTRACT: The Spitzer warm mission has already greatly expanded the field of exoplanet characterization with over 3000 hours of time dedicated to exoplanet observations. Observations of eclipsing systems with Spitzer are at the heart of these advances, as they allow us to move beyond simple mass and period estimates to determine planetary radius, dayside emission, and emission variations as a function of orbital phase. The eclipsing system HAT-P-2 is of special interest because the massive Jovian sized planet in this system is on a highly eccentric orbit (e=0.5171). Because HAT-P-2b's orbit is eccentric, the planet is subject to time variable heating and probable non-synchronous rotation. Circulation patterns that we expect to develop in HAT-P-2b's atmosphere will likely vary with both planetary local time and orbital phase. Here we present an analysis of two full-orbit light curves for the HAT-P-2 system obtained at 3.6 and 4.5 microns during the first two years of the Spitzer warm mission and discuss the observational constraints imposed on the atmospheric circulation of HAT-P-2b. Additionally, three-dimensional atmospheric models that incorporate realistic radiative transfer will be presented to further elucidate possible global scale circulations patterns present in the atmosphere of HAT-P-2b. Support for this work was provided by NASA. • ##### Article: Characterizing Hot Jupiter Atmospheres with Hubble WFC3 [Hide abstract] ABSTRACT: We present transit and eclipse spectroscopy of Very Hot Jupiter atmospheres using the newly installed Wide Field Camera 3 (WFC3) spectrograph on the Hubble Space Telescope (HST). We include results from several hot Jupiter planets, but we focus particularly on spectra of WASP-4b in both transmission and emission, the first time this has been achieved using this instrument. These data are already in hand. Our preliminary analysis indicates they do not require substantial decorrelation against external measurements to correct for systematic errors, resulting in robust results that avoid the ambiguities in interpretation faced by earlier work with NICMOS. Recent work by Madhusudhan and Seager highlighted degeneracies in the interpretation of low-resolution Spitzer spectra of hot Jupiters. The G141 grism on WFC3 (1.1-1.7 microns) spans a 1.4 micron water feature, allowing us to constrain water vapor abundance and break this degeneracy. We measure the 1.5 micron continuum flux, which in conjunction with the Spitzer data allows us to constrain the thermal spectrum of the planet and obtain a more precise energy budget relevant to understanding WASP-4b’s abnormally large radius. This work is part of the Cycle-18 Large Program 12181, and we acknowledge support by NASA, NSF GRFP, and the HST Project. • ##### Conference Paper: A Comparative Study of the Atmospheres of Transiting Exoplanets [Hide abstract] ABSTRACT: Spitzer's extended warm mission gives us the opportunity to perform comparative studies of exoplanets' atmospheres. We describe the techniques and methods involved in our recently accepted Spitzer Exploration Science Proposal (Proposal ID 80016; PI: J. Krick; 619 hours) to obtain high-precision 4.5 micron phase curves for 22 transiting hot Jupiter systems, along with observations of secondary eclipses of 7 of these systems. The principal goal of this study, which will quadruple the number of phase curve observations to date, is to map longitudinal temperature distribution of the planetary atmospheres and to assess the following questions: (1) What is the contrast between exoplanetary day- and nightside temperatures, i. e., how efficiently is the incident energy redistributed? (2) Are the weather phenomena in the exoplanetary atmospheres stable over long periods of time? (3) How do the temperature distributions on the planetary surfaces correlate with astrophysical properties of the star-planet systems? To answer these questions with our proposed Spitzer observations, we will employ a novel observing technique involving snapshot observing to reduce telescope time requirements, and PCRS peak-up to IRAC to increase the pointing accuracy and thus minimize the photometric error due to intrapixel sensitivity variation. AAS/Division for Extreme Solar Systems Abstracts; 09/2011 • Source ##### Article: CASTRO: A New Compressible Astrophysical Solver. II. Gray Radiation Hydrodynamics [Hide abstract] ABSTRACT: We describe the development of a flux-limited gray radiation solver for the compressible astrophysics code, CASTRO. CASTRO uses an Eulerian grid with block-structured adaptive mesh refinement based on a nested hierarchy of logically-rectangular variable-sized grids with simultaneous refinement in both space and time. The gray radiation solver is based on a mixed-frame formulation of radiation hydrodynamics. In our approach, the system is split into two parts, one part that couples the radiation and fluid in a hyperbolic subsystem, and another parabolic part that evolves radiation diffusion and source-sink terms. The hyperbolic subsystem is solved explicitly with a high-order Godunov scheme, whereas the parabolic part is solved implicitly with a first-order backward Euler method. The Astrophysical Journal Supplement Series 05/2011; 196(2). DOI:10.1088/0067-0049/196/2/20 · 14.14 Impact Factor • ##### Article: Comparative Atmospheric Study of Exoplanets [Hide abstract] ABSTRACT: Spitzer's extended warm mission gives us the opportunity to contribute to its legacy by performing comparative science on atmospheres of extrasolar planets. The goal of this proposal is to obtain high quality 4.5 micron phase curves for 22 transiting hot Jupiter systems, which represent a complete sample of systems that can be studied with Spitzer in a reasonable amount of time. The resulting dataset will not only quadruple the number of phase curve observations to date, but also populate gaps in parameter space explored by current phase curve studies. The combination of our phase curves with well-known literature ephemerides and observations of secondary eclipses will produce maps of the longitudinal brightness/temperature distributions in the planetary atmospheres. These maps can be used to calculate energy redistribution efficiencies between the hot dayside and cooler nightside -- exoplanetary weather. Our observations focus on the following three questions: (1) What is the contrast between exoplanetary day- and nightside temperatures, i.e., how efficiently is the incident energy redistributed? (2) Are the weather phenomena in the exoplanetary atmospheres stable over long periods of time? (3) How do the temperature distributions on the planetary surfaces correlate with astrophysical properties of the star-planet systems? To answer these questions with our proposed Spitzer observations, we will employ a novel observing technique using snapshot observing to reduce telescope time requirements, and PCRS peak-up to IRAC to increase the pointing accuracy and thus minimize the photometric error due to intrapixel sensitivity variation. The analysis of this comprehensive data set will elevate the study of phase curves to the level of comparative atmospheric studies outside the solar system. • ##### Article: Analysis of HAT-P-2b Warm Spitzer Full Orbit Light Curve [Hide abstract] ABSTRACT: The Spitzer warm mission has already greatly expanded the field of exoplanet characterization with over 3000 hours of time dedicated to exoplanet observations. Observations of eclipsing systems with Spitzer are at the heart of these advances, as they allow us to move beyond simple mass and period estimates to determine planetary radius, dayside emission, and emission variations as a function of orbital phase. The eclipsing system HAT-P-2 is of special interest because the massive Jovian sized planet in this system is on a highly eccentric orbit (e=0.5171). Because HAT-P-2b's orbit is eccentric, the planet is subject to time variable heating and probable non-synchronous rotation. Circulation patterns that we expect to develop in HAT-P-2b's atmosphere will likely vary with both planetary local time and orbital phase. Here we present an analysis of a full orbit light curve from the HAT-P-2 system obtained during the most recent cycle of the Spitzer warm mission and discuss the constraints it imposes on the atmospheric circulation of HAT-P-2b. Support for this work was provided by NASA. • Source ##### Article: Dynamics and gravitational wave signature of collapsar formation. [Hide abstract] ABSTRACT: We perform 3+1 general relativistic simulations of rotating core collapse in the context of the collapsar model for long gamma-ray bursts. We employ a realistic progenitor, rotation based on results of stellar evolution calculations, and a simplified equation of state. Our simulations track self-consistently collapse, bounce, the postbounce phase, black hole formation, and the subsequent early hyperaccretion phase. We extract gravitational waves from the spacetime curvature and identify a unique gravitational wave signature associated with the early phase of collapsar formation. Physical Review Letters 04/2011; 106(16):161103. DOI:10.1103/PhysRevLett.106.161103 · 7.73 Impact Factor • ##### Article: Advances in the Theory of Giant Exoplanets David S. Spiegel, A. Burrows [Hide abstract] ABSTRACT: We review recent results on the spectra of giant planets, including calculations specific to objects in the Kepler field (HAT-P-7b and TrES-2), and calculations of emergent radiation and transit spectra associated with general circulation models. We also present recent results on the potential habitability of terrestrial planets on eccentric orbits. • ##### Article: A Subaru/MMT Study of the Atmospheres of the Planets Orbiting HR 8799 [Hide abstract] ABSTRACT: We present new Subaru data and a rereduction of archival MMT data for the HR 8799 planetary system from 1 to 5 microns. We compare photometry from these data to that for field L/T dwarfs and synthetic SEDs for substellar objects with wide range of temperatures, gravities, and cloud prescriptions. Our analysis strongly suggests that HR 8799bcd have atmospheres dissimilar from almost any brown dwarf over a wide wavelength range. Best-fit models yield temperatures of 900-1100K and masses between 7 and 9 Mjupiter, roughly consistent with values derived from cooling models, and suggest that extensive cloud coverage is a likely source of the SED differences between these planets and brown dwarfs with similar temperatures. #### Publication Stats 4k Citations 454.82 Total Impact Points #### Institutions • ###### Princeton University • Department of Astrophysical Sciences Princeton, New Jersey, United States • ###### TRI/Princeton Princeton, New Jersey, United States • ###### The University of Arizona • • Department of Astronomy • • Department of Planetary Sciences Tucson, Arizona, United States • ###### Franklin and Marshall College Marshall, Texas, United States • ###### 1997 • Department of Meteorology • ###### New Mexico State University • Department of Astronomy Las Cruces, NM, United States • ###### Vanderbilt University • Department of Physics and Astronomy Nashville, Michigan, United States • ###### French National Centre for Scientific Research Lutetia Parisorum, Île-de-France, France	{"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": 0.8726680874824524, "perplexity": 3570.773397057608}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246658904.34/warc/CC-MAIN-20150417045738-00058-ip-10-235-10-82.ec2.internal.warc.gz"}
http://www.mitpress.mit.edu/authors/stephen-i-gallant	# Stephen I. Gallant ## Neural Network Learning and Expert Systems Neural Network Learning and Expert Systems is the first book to present a unified and in-depth development of neural network learning algorithms and neural network expert systems. Especially suitable for students and researchers in computer science, engineering, and psychology, this text and reference provides a systematic development of neural network learning algorithms from a computational perspective, coupled with an extensive exploration of neural network expert systems which shows how the power of neural network learning can be harnessed to generate expert systems automatically. Features include a comprehensive treatment of the standard learning algorithms (with many proofs), along with much original research on algorithms and expert systems. Additional chapters explore constructive algorithms, introduce computational learning theory, and focus on expert system applications to noisy and redundant problems. For students there is a large collection of exercises, as well as a series of programming projects that lead to an extensive neural network software package. All of the neural network models examined can be implemented using standard programming languages on a microcomputer. Stephen l. Gallant taught courses in neural network learning and expert systems as Associate Professor of Computer Science at Northeastern University. He is currently a Senior Scientist at HNC, Inc.	{"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": 0.8057955503463745, "perplexity": 835.9583532120561}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131317570.85/warc/CC-MAIN-20150323172157-00116-ip-10-168-14-71.ec2.internal.warc.gz"}
https://forum.allaboutcircuits.com/threads/i-am-a-math-rock-simple-transposition-question.34914/	# I am a math rock. Simple transposition question Discussion in 'Math' started by SgtWookie, Feb 27, 2010. 1. ### SgtWookie Thread Starter Expert Jul 17, 2007 22,202 1,791 Terrible to admit, but I'm a math rock. I've been struggling with this for a day, and I simply don't "get it" - this is high-school stuff, and that was a long time ago. Where am I going wrong? The idea is to get L over on the left side of the equation. Somehow I've managed to be involved in complex electronics and software for many years, but frankly I suck at math. I don't understand where I'm going wrong here. Fo=1/2pisqrt(LxCs) (this works) L =4pi()^2((1/(D73E73/1E6))^2/5e-8) (wrong!) Various things I've tried: To=2pisqrt(LxCs) To2pi^2=4pi^2LxCs =((d73e73/1e6)/pi()2)^2/5e-8 To=2PI()SQRT(F690.05) <=that works; F69=L(uH) 0=2PI()SQRT(F690.05)/To 0/2PI()^2=(F690.05)^2/To 0/2PI()^2/F69^2=0.05^2/To To=2PI()SQRT(F690.05) <=that works; F69=L(uH) To=2PI()SQRT(F690.05/1E12) <=that works F=1/(2pi()sqrt(F660.05/1E12)) <=that works F=1/(2pi()sqrt(F65/1E60.05/1E6)) <=that works L=4pi()^2(L67^2/(.05/1e6)) <- garbage L=4pi()^2((1/d68e68)^2/.05) <- garbage Where am I going wrong? 2. ### hgmjr Moderator Jan 28, 2005 9,030 218 $f_o=\frac{1}{2\pi\sqrt{(LC)}}$ Square both sides: ${f_o}^2=\frac{1^2}{{4\pi^2{\sqrt{(LC)}}^2}$ Simplify ${f_o}^2=\frac{1}{{{(4\pi^2LC)}}}$ Multiply both sides by L: ${f_o}^2L=L\frac{1}{{{(4\pi^2LC)}}}$ L cancels on the right: ${f_o}^2L=\frac{1}{{{(4\pi^2C)}}}$ Divide both sides by ${f_o}^2$ $L=\frac{1}{{{(4\pi^2C){f_o}^2}}}$ Substitution allows us to exhange omega for 2pif $L=\frac{1}{{{(\omega^2C)}}}$ We are all rocks in the math quarry. It is just a matter of size. hgmjr Last edited: Feb 27, 2010 3. ### SgtWookie Thread Starter Expert Jul 17, 2007 22,202 1,791 Thanks for your reply hgmjr, which I'm sure took a lot of work to get all of the tex graphcs right. But I still don't know how to get L over to the left side, but more importantly, how to do so. Like I said, I'm a math rock. You've tried to provide instructions that would be adequate for most, but for some reason I have a really hard time with it. Thanks, Wook 4. ### hgmjr Moderator Jan 28, 2005 9,030 218 Which of the steps is not clear? Maybe I can break it down a bit more. hgmjr Jul 7, 2009 1,585 142 The key idea in manipulating equations is that you have to do the same thing to both sides -- otherwise it doesn't remain an equality. Thus, suppose you have an equation $a = \frac {b} {L}$ and you want the L on the other side. The strategy used is to perform the same operation on both sides so that the L disappears from the right side and goes to the left side. Here, we can multiply both sides by L: $L a = \frac {b} {L} L$ which reveals our strategy, because L divided by L is just 1. To see this, rewrite things slightly (using the associative law of multiplication) to get $L a = b \frac {L} {L}$ Thus, we "cancel out" the L's on the right and we are left with the equation $L a = b$ Analogously, to move the a to the right hand side, we just divide both sides by a to get $L = \frac {b} {a}$ 6. ### hgmjr Moderator Jan 28, 2005 9,030 218 Someonesdad has captured the essence of the technique needed to manupulate the equation indeed any equation. This particular equation has an added factor in that L is contained in a square root term. This is what required the squaring of both sides of the equation to liberate L from the square root term. This squaring of both sides is consistent with someonesdad's statement that what is done to one side of the equation must be done to the other side of the equation if the equality is to be maintained. hgmjr 7. ### Louise New Member Jan 17, 2010 13 1 If I may be so bold as to make a tiny suggestion - something that helps me sometimes (and my maths ability might be accurately described as 'vanishingly small') is to try to re-arrange the same equation using numbers instead of symbols. The same rules apply. 8. ### Wendy Moderator Mar 24, 2008 21,306 2,912 I figure we are as close as two people can be technically with such different back grounds. I find I'm having trouble with a lot of the math at this site. I do what can, and ask for help with the rest, same as you. There was a time I would claim I knew some calculus. However, it rusts like the rest of my skills, but I really don't have a use for it very often. It is pretty much gone now. 9. ### hgmjr Moderator Jan 28, 2005 9,030 218 If it helps. The above attempt came off the tracks when both sides of the equation were divided by To in STEP 2. The left side of the equation should have been 1 rather than 0. This is because To/To = 1. I think it is safe to assume that in these equations references such as F69 and F66 are artifacts of the cut and paste operation from the spreadsheet you are using. Everything you did above was kosher. In the above box, I can't easily interpret what the cell references equate to so I am unable to determine where things went south. Hope this helps some. hgmjr 10. ### SgtWookie Thread Starter Expert Jul 17, 2007 22,202 1,791 Thanks folks, I figured it out. It's just a simple Excel spreadsheet that I've been using for testing an assortment of toroids I've had kicking around here for awhile. I'm using a 74HC14 hex Schmitt-trigger inverter IC with three 100nF caps and a couple of resistors as a buffered Pierce oscillator; from an article published in EDN's Design Ideas, and measured the time of the output square wave with my O-scope (haven't built a freq. counter yet; still in the works...). EDN article: http://www.edn.com/article/CA6462564.html I record the divisions and time/div, and out pops the inductance in uH and approximate AL value. It's pretty decent down to maybe 10uH, but then the propagation delay in the 74HC14 starts kicking in along with parasitics from the X7R caps I used. A known 977nH inductor measures 2uH, so it's not all that bad. One of the caps is used as a supply bypass cap for the 74HC14. The other two caps are on either side of the inductor under test, so they're essentially in series for a total of 50nF. I was forgetting to shift my decimal places around where they belonged. Adding in a few /1e6 and 1e-6 fixed things up. L(uH)=1/(4PI()^2(1/(scope_divisionsuSecs_per_division0.000001))^2(0.050.000001))*1000000 11. ### Harrington New Member Dec 19, 2009 86 3 Here try this I'm not brilliant at maths myself but this the way we solve problems like this see attached hope this helps you in the future with your calculations I use this from time to time you can also solve matrices maths with this and calculus Its a dos version but it will run on windows XP Hope this helps you Mark • ###### Derive.zip File size: 221.3 KB Views: 23 Related Forum Posts: 1. Replies: 3 Views: 1,307 2. Replies: 58 Views: 8,588 3. Replies: 8 Views: 3,110 4. Replies: 8 Views: 2,982 5. Replies: 3 Views: 4,279	{"extraction_info": {"found_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": 13, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8625181317329407, "perplexity": 1386.0363044628302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1537267157216.50/warc/CC-MAIN-20180921151328-20180921171728-00466.warc.gz"}
http://nylogic.org/talks/grounded-martins-axiom	# Grounded Martin’s axiom Set theory seminarFriday, November 8, 201310:00 am # Grounded Martin’s axiom ### The CUNY Graduate Center I will present a weakening of Martin’s axiom which asserts the existence of partial generics only for ccc posets contained in a ccc ground model. This principle, named the grounded Martin’s axiom, emerges naturally in the analysis of the Solovay-Tennenbaum proof of the consistency of MA. While the grounded MA has some of the combinatorial consequences of MA, it will be shown to be more flexible (being consistent with a singular continuum, for example) and more robust under forcing (being preserved in a strong way under both adding a Cohen or a random real). Miha Habič is a graduate student at the CUNY GC. He got his Masters Degree in Mathematics at the University of Ljubljana, Slovenia. His interests lie in the area of infinitary computability, forcing and large cardinals. Posted by on November 2nd, 2013	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8125031590461731, "perplexity": 1446.230201936959}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038067400.24/warc/CC-MAIN-20210412113508-20210412143508-00435.warc.gz"}
http://crypto.stackexchange.com/questions/797/is-diffie-hellman-mathematically-the-same-as-rsa/803	Is Diffie-Hellman mathematically the same as RSA? Is the Diffie-Hellman key exchange the same as RSA? Diffie Hellman allows key exchange on a observed wire – but so can RSA. Alice and Bob want to exchange a key – Big brother is watching everything. 1. Bob makes an fresh RSA key pair and sends his public key to Alice. 2. Alice makes a random session key and sends it to Bob encrypted with Bob's public key. 3. Bob decrypts the session key with his private key. Alice and Bob have exchanged a key despite the fact that anybody can observe all the traffic. The maths of RSA and Hiffie Hellman are remarkably similar, both involving modular exponentiation. They both work because $(AB)^C \bmod N$ can be done in two steps, i.e. by calculating $X = A^C \bmod N$ on one side of the transaction, and $X^B \bmod N$ on the other – this trick is the basis of both Hiffie Hellman and RSA. Which make me wonder: Are they really the same thing? Can we algebraically prove that the correctness of RSA implies the correctness of Diffie-Hellman? - migrated from security.stackexchange.comSep 27 '11 at 15:53 This question came from our site for Information security professionals. Both RSA and Diffie-Hellman work with modular exponentiation. But they work in a different way: In RSA, there are two exponentiations which invert each other, i.e. we have $e$ and $d$ such that $(x^e)^d \equiv x$ for all $x$. E.g. if $\square^e$ is the encryption, $\square^d$ is the corresponding decryption. To create this pair of $e$ and $d$ (or derive one from the other), we need the prime factorization of the modulus $m$ (which thus should be private). In Diffie-Hellman, both the basis $g$ and the modulus $m$ of the exponentiation is fixed. The exponents $x$ and $y$ are randomly chosen private keys, and we use the fact that $(g^x)^y \equiv g^{x\cdot y} \equiv (g^y)^x$, where $g^x$ and $g^y$ are public, while $x$ and $y$ are (and stay) private, and $g^{x·y}$ is the shared secret. To break RSA, we would have to get $x$ from $x^d$, $m$ and $d$ - this could be called the discrete $d$-th root problem. (The best known way to do this is to factor $m$ to get $e$ ... and if you have $e$, this also can be used to factor $m$). To break Diffie-Hellman, we have to get $g^{x·y}$ from $g^x$, $g^y$, $m$ and $g$. The best known way to do this would be to get $x$ from $g^x$ or $y$ from $g^y$ (and $m$ and $g$), the so-called discrete logarithm problem. (Incidentially, it is not proven that this is really the best way, i.e. that there is not a faster way to do this.) - Diffie-Hellman and RSA are distinct and do not use the same "trick". In Diffie-Hellman, commutativity is used: $(g^a)^b = (g^b)^a$. Both Alice and Bob do two modular exponentiations each (Alice chooses $a$, computes $g^a$ and sends it to Bob, receives $g^b$ from Bob, and finally computes $(g^b)^a$). Security relies on the difficulty of discrete logarithm: given a prime $p$, an integer $g$, and $g^x \mod p$, it is utterly difficult to find $x$. In RSA, there is no commutativity involved; Alice and Bob do only one modular exponentiation each; computations are not done modulo a prime $p$, but modulo a non-prime $n$. Alice chooses a random $m$, computes $m^e \mod n$ and sends it to Bob; Bob computes $(m^e)^d \mod n$ which is equal to $m$ because $d$ and $e$ have been chosen for this to work. RSA relies on the difficulty of extracting $e$-th roots: given $n$, $e$ and $m^e \mod n$, it is utterly difficult to find $m$ -- unless you know the "magic trap", i.e. $d$ (or the factorization of $n$)(if $n$ was prime, finding $m$ would be easy). Although both algorithms involve modular exponentiations, they are quite different in how they work, what they provide, and what hard problems they rely on. Note the difference: in discrete logarithm, you have $g$ and $g^x$, and seek $x$; in $e$-th roots, you have $m^e$ and $e$, and seek $m$. Any asymmetric encryption algorithm (such as RSA) can be used as a key exchange algorithm, in the way you describe (to "exchange" a key, Alice selects a random blob and encrypts it with Bob's public key). SSL/TLS does that. The converse is not true: you cannot generically transform a key exchange algorithm "alone" into an asymmetric encryption algorithm (but you can use the exchanged key with a symmetric encryption algorithm like AES; there again, SSL does that when using Diffie-Hellman). - The mathematical problems behind DH and RSA are similar but not known to be directly related. It is still an open question if an oracle breaking DH can be used to construct another oracle that breaks RSA (or vice versa). It is mostly believed that the two problems are not reducible to each other in poly time. However, the complexity of the fastest known DH and RSA breaking algorithms are very close. Therefore, both DH and RSA recommended key sizes are the same. I should add that in the case of Elliptic Curve-DH, there is no better way to solve the DH problem than by generic algorithms and so the key size is much shorter. - DH and RSA are definitely not the same. DH is a key exchange algorithm, RSA an encryption/signing algorithm. Now as to whether are operate in the the same way in the scenario you describe. They can indeed be used for a similar purpose. However, the operate on a fundamentally different mathematical problem. DH uses DLP (discrete logarithm), trusting on the hardness of finding x when g^x=H and g and H are known. RSA trusts on the prime factorization problem: the hardness of finding p(prime) and q(prime) when n=(p-1)(q-1) is known. If the DL problem would be solved, making it feasible to compute x in a reasonable time, DH and ElGamal would both be broken. RSA wouldn't, if I'm not mistaken; RSA would suffer when the prime factorization problem would be solved. For this reason, you cannot use DH to prove RSA or the other way around. - Correction: in RSA, $n=(p-1)(q-1)$ (or $lcm(p-1,q-1)$) is not known (it's easy to factor given that value). Instead, the hard problem is determining x given e and $x^e mod pq$; in constrast, one way of attacking DH is to attack the hard problem of determining e given x and $x^e mod p$ – poncho Sep 27 '11 at 16:02 I think I might be wrong, but I thought I've heard the claim that breaking discrete log breaks/threatens RSA as well. Maybe this should be a question. – Ethan Heilman Sep 27 '11 at 16:29 Well, there are two ways that claim may be correct: 1) if you can solve the DLOG problem in a composite modulus, then you can factor that modulus. If your Oracle to solve DLOG works only in a prime modulus, there's no obvious way to use that to factor; 2) a lot of approaches for attacking the DLOG problem can be used (with some changes) to attack the factorization problem (and vica versa). Now, that's not for all known approaches. – poncho Sep 27 '11 at 17:14 No, they are not the same. The short answer is that Diffie-Hellman is for negotiating a secret between parties who don't already share one, while RSA uses existing key material to protect data. It's actually quite a major difference. Both are "security" related, of course, but DH is starting from scratch while RSA is using a system that's already in place. - RSA security relies on the assumption that the "RSA problem" is hard. The RSA problem is to find the plaintext given only $n$, $e$, and the ciphertext. It is easy to see that an efficient factoring algorithm breaks RSA, but the converse is unknown: ie, if we had an oracle that could efficiently break RSA, could we efficiently factor $n$?. Interestingly, quantum computers break BOTH algorithms via similar techniques (assuming DH is in $Z^_p$) Actually, DH relies on the "Diffie-Hellman problem" (given $g^x$ and $g^y$, find $g^{x·y}$, which might be easier than the discrete log problem (and is easier in some groups, I think). – Paŭlo Ebermann Jan 18 '12 at 21:35	{"extraction_info": {"found_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": 0.8887925744056702, "perplexity": 616.5607732597848}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999638008/warc/CC-MAIN-20140305060718-00046-ip-10-183-142-35.ec2.internal.warc.gz"}
https://www.varsitytutors.com/hotmath/hotmath_help/topics/operations-on-sets.html	# Operations on Sets Recall that a set is a collection of elements. Given sets $A$ and $B$ , we can define the following operations: Operation Notation Meaning Intersection $A\cap B$ all elements which are in both $A$ and $B$ Union $A\cup B$ all elements which are in either $A$ or $B$ (or both) Difference $A-B$ all elements which are in $A$ but not in $B$ Complement $\stackrel{¯}{A}$ (or ${A}^{C}$ ) all elements which are not in $A$ Example 1: Let $A=\left\{1,2,3,4\right\}$ and let $B=\left\{3,4,5,6\right\}$ . Then: $A\cap B=\left\{3,4\right\}$ $A\cup B=\left\{1,2,3,4,5,6\right\}$ $A-B=\left\{1,2\right\}$ Example 2: Let $A=\left\{y,z\right\}$ and let $B=\left\{x,y,z\right\}$ . Then:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 21, "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": 0.9612295627593994, "perplexity": 207.4643603236722}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1492917125719.13/warc/CC-MAIN-20170423031205-00376-ip-10-145-167-34.ec2.internal.warc.gz"}
http://questionpaper.org/progressions/	Progressions Formulas, Tricks and Shortcuts Arithmetic Progressions Page - 2 Geometric Progressions Page - 3 Progressions Important Questions Page - 6 Sequences, following specific patterns are called progressions. Arithmetic progression (A.P). In this Chapter, besides discussing more about A.P.; arithmetic mean, geometric mean, relationship between A.M. and G.M., special series in forms of sum to n terms of consecutive natural numbers, sum to n terms of squares of natural numbers and sum to n terms of cubes of natural numbers will also be studied. Sequences Let the number of person’s ancestors for the first, second, third, ..., tenth generations are 2, 4, 8, 16, 32, ..., 1024. These numbers form what we call a sequence. Consider the successive quotients that we obtain in the division of 10 by 3 at different steps of division. In this process we get 3,3.3,3.33,3.333, ... and so on. These quotients also form a sequence. The various numbers occurring in a sequence are called its term . We denote the terms of a sequence by $a_{1},a_{2},a_{3},.....,a_{n}$ etc, the subscripts denote the position of the term. The $n^{th}$ term is the number at the position of the sequence and is denoted by $a_{n}$ The $n^{th}$ term is also called the general term of the sequence. A sequence is called infinite, if it is not a finite sequence. For example, the sequence of successive quotients mentioned above is an infinite sequence, infinite in the sense that it never ends .Often, it is possible to express the rule, which yields the various terms of a sequencein terms of algebraic formula. Consider for instance, the sequence of even natural number 2, 4, 6,... Here, $a_{1}=2=2\times 1,a_{2}=4=2\times 2,a_{3}=6=2\times 3,........a_{23}=46=2\times 23,a_{24}=48=2\times 24$, and so on. In some cases, an arrangement of numbers such as 1, 1, 2, 3, 5, 8,.. has no visible pattern, but the sequence is generated by the recurrence relation given by. $a_{1}=a_{2}=1,a_{3}=a_{1}+a_{2},a_{n}=a_{n+2}+a_{n-1},n> 2$ Above sequence is called Fibonacci sequence Series $a_{1},a_{2},a_{3},.....,a_{n}$ be a given sequence. Then, the expression$a_{1}+a_{2}+a_{3}+.....+a_{n}$ s called the series associated with the given sequence. The series is finite or infinite according as the given sequence is finite or infinite. Series are often represented in compact form called sigma notation as means indicating the summation involved .Thus , the series $a_{1}+a_{2}+a_{3}+.....+a_{n}$ is abbreviated . as $\sum_{k-1}^{n}$ Note: When the series is used, it refers to the indicated sum not to the sum itself. For example, 1 + 3 + 5 + 7 is a finite series with four terms. When we use the phrase “ sum of a series,” we will mean the number that results from adding the terms, the sum of the series is 16.	{"extraction_info": {"found_math": true, "script_math_tex": 10, "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": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.970672070980072, "perplexity": 309.6462629604256}, "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-2019-09/segments/1550247479885.8/warc/CC-MAIN-20190216045013-20190216071013-00403.warc.gz"}
https://rd.springer.com/chapter/10.1007/978-3-319-63133-2_1	Advertisement # Chapter 1 Basics of Electrostatics • Andrea Macchi • Giovanni Moruzzi • Francesco Pegoraro Chapter ## Abstract The electric charge. The electric field. The superposition principle. Gauss’s law. Symmetry considerations. The electric field of simple charge distributions (plane layer, straight wire, sphere). Point charges and Coulomb’s law. The equations of electrostatics. Potential energy and electric potential. The equations of Poisson and Laplace. Electrostatic energy. Multipole expansions. The field of an electric dipole. Topics. The electric charge. The electric field. The superposition principle. Gauss’s law. Symmetry considerations. The electric field of simple charge distributions (plane layer, straight wire, sphere). Point charges and Coulomb’s law. The equations of electrostatics. Potential energy and electric potential. The equations of Poisson and Laplace. Electrostatic energy. Multipole expansions. The field of an electric dipole. Units. An aim of this book is to provide formulas compatible with both SI (French: Système International d’Unités) units and Gaussian units in Chapters , while only Gaussian units will be used in Chapters . This is achieved by introducing some system-of-units-dependent constants. The first constant we need is Coulomb’s constant, $$k_\mathrm{e}$$, which for instance appears in the expression for the force between two electric point charges $$q_1$$ and $$q_2$$ in vacuum, with position vectors $$\mathbf {r}_1$$ and $$\mathbf {r}_2$$, respectively. The Coulomb force acting, for instance, on $$q_1$$ is \begin{aligned} \mathbf {f}_1=k_\mathrm{e}\frac{q_1q_2}{\|\mathbf {r}_1-\mathbf {r}_2\|^2}\,\hat{\mathbf {r}}_{12}\,, \end{aligned} (1.1) where $$k_\mathrm{e}$$ is Coulomb’s constant, dependent on the units used for force, electric charge, and length. The vector $$\mathbf {r}_{12}=\mathbf {r}_1-\mathbf {r}_2$$ is the distance from $$q_2$$ to $$q_1$$, pointing towards $$q_1$$, and $$\hat{\mathbf {r}}_{12}$$ the corresponding unit vector. Coulomb’s constant is \begin{aligned} k_\mathrm{e}=\left\{ \begin{array}{ll} \displaystyle \frac{1}{4\pi \varepsilon _0} 8.987\,\cdots \times 10^9\ \mathrm {N\cdot m^2\cdot C}^{-2} \simeq 9\times 10^9 \text{ m/F }&{}\quad \text{ SI }\\ 1&{}\quad \text{ Gaussian. } \end{array} \right. \end{aligned} (1.2) Constant $$\varepsilon _0\simeq 8.854\, 187\, 817\, 620\,\cdots \times 10^{-12}$$ F/m is the so-called “dielectric permittivity of free space”, and is defined by the formula \begin{aligned} \varepsilon _0=\frac{1}{\mu _0 c^2}\,, \end{aligned} (1.3) where $$\mu _0=4\pi \times 10^{-7}$$ H/m (by definition) is the vacuum magnetic permeability, and c is the speed of light in vacuum, $$c=299\, 792\, 458$$ m/s (this is a precise value, since the length of the meter is defined from this constant and the international standard for time). Basic equations The two basic equations of this Chapter are, in differential and integral form, \begin{aligned} \varvec{\nabla }\cdot \mathbf {E}&=4\pi k_\mathrm{e}\, \varrho \,,&\oint _S\mathbf {E}\cdot \mathrm {d}\mathbf {S}&=4\pi k_\mathrm{e}\int _V\varrho \,\mathrm {d}^3 r \; \end{aligned} (1.4) \begin{aligned} \varvec{\nabla }\times \mathbf {E}&=0 \,,&\oint _C\mathbf {E}\cdot \mathrm {d}\mathbf {\ell }&=0 \; . \end{aligned} (1.5) where $$\mathbf {E}(\mathbf {r}, t)$$ is the electric field, and $$\varrho (\mathbf {r}, t)$$ is the volume charge density, at a point of location vector $$\mathbf {r}$$ at time t. The infinitesimal volume element is $$\mathrm{d}^3r=\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$. In (1.4) the functions to be integrated are evaluated over an arbitrary volume V, or over the surface S enclosing the volume V. The function to be integrated in (1.5) is evaluated over an arbitrary closed path C. Since $$\varvec{\nabla }\times \mathbf {E}=0$$, it is possible to define an electric potential $$\varphi =\varphi (\mathbf {r})$$ such that \begin{aligned} \mathbf {E}=-\varvec{\nabla }\varphi \; . \end{aligned} (1.6) The general expression of the potential generated by a given charge distribution $$\varrho (\mathbf {r})$$ is \begin{aligned} \varphi (\mathbf {r}) =k_\mathrm{e}\int _V\frac{\varrho (\mathbf {r}')}{\|\mathbf {r} -\mathbf {r}'\|}\,\mathrm{d}^3 r'\;. \end{aligned} (1.7) The force acting on a volume charge distribution $$\varrho (\mathbf {r})$$ is \begin{aligned} \mathbf {f}=\int _V\varrho (\mathbf {r}')\,\mathbf {E}(\mathbf {r}')\,\mathrm{d}^3 r'\;. \end{aligned} (1.8) As a consequence, the force acting on a point charge q located at $$\mathbf {r}$$ (which corresponds to a charge distribution $$\varrho (\mathbf {r}')=q\delta (\mathbf {r}-\mathbf {r}')$$, with $$\delta (\mathbf {r})$$ the Dirac-delta function) is \begin{aligned} \mathbf {f}=q\,\mathbf {E}(\mathbf {r})\; . \end{aligned} (1.9) The electrostatic energy $$U_{\mathrm{es}}$$ associated with a given distribution of electric charges and fields is given by the following expressions \begin{aligned} U_\mathrm{es}= & {} \int _V \frac{\mathbf {E}^2}{8\pi k_\mathrm{e}}\,\mathrm {d}^3 r \; . \end{aligned} (1.10) \begin{aligned} U_\mathrm{es}= & {} \frac{1}{2}\int _V \varrho \,\varphi \,\mathrm {d}^3r \; , \end{aligned} (1.11) Equations (1.101.11) are valid provided that the volume integrals are finite and that all involved quantities are well defined. The multipole expansion allows us to obtain simple expressions for the leading terms of the potential and field generated by a charge distribution at a distance much larger than its extension. In the following we will need only the expansion up to the dipole term, \begin{aligned} \varphi (\mathbf {r}) \simeq k_\mathrm{e}\left( \frac{Q}{r}+\frac{\mathbf {p}\cdot \mathbf {r}}{r^3} +\ldots \right) \; , \end{aligned} (1.12) where Q is the total charge of the distribution and the electric dipole moment is \begin{aligned} \mathbf {p}\equiv \int _V\mathbf {r}'\rho (\mathbf {r}')\mathrm{d}^3\mathbf {r}' \; . \end{aligned} (1.13) If $$Q=0$$, then $$\mathbf {p}$$ is independent on the choice of the origin of the reference frame. The field generated by a dipolar distribution centered at $$\mathbf {r}=0$$ is \begin{aligned} \mathbf {E}=k_\mathrm{e}\frac{3\hat{\mathbf {r}}(\mathbf {p}\cdot \hat{\mathbf {r}})-\mathbf {p}}{r^3} \; . \end{aligned} (1.14) We will briefly refer to a localized charge distribution having a dipole moment as “an electric dipole” (the simplest case being two opposite point charges $$\pm q$$ with a spatial separation $$\mathbf {\delta }$$, so that $$\mathbf {p}=q\mathbf {\delta }$$). A dipole placed in an external field $$\mathbf {E}_\mathrm{ext}$$ has a potential energy \begin{aligned} U_\mathrm{p}=-\mathbf {p}\cdot \mathbf {E}_\mathrm{ext} \; . \end{aligned} (1.15) ## 1.1 Overlapping Charged Spheres We assume that a neutral sphere of radius R can be regarded as the superposition of two “rigid” spheres: one of uniform positive charge density $$+\varrho _0$$, comprising the nuclei of the atoms, and a second sphere of the same radius, but of negative uniform charge density $$-\varrho _0$$, comprising the electrons. We further assume that its is possible to shift the two spheres relative to each other by a quantity $$\mathbf {\delta }$$, as shown in Fig. 1.1, without perturbing the internal structure of either sphere. Find the electrostatic field generated by the global charge distribution a) in the “inner” region, where the two spheres overlap, b) in the “outer” region, i.e., outside both spheres, discussing the limit of small displacements $$\delta \ll R$$. ## 1.2 Charged Sphere with Internal Spherical Cavity A sphere of radius a has uniform charge density $$\varrho$$ over all its volume, excluding a spherical cavity of radius $$b<a$$, where $$\varrho =0$$. The center of the cavity, $$O_b$$ is located at a distance $$\mathbf {d}$$, with $$\|\mathbf {d}\|<(a-b)$$, from the center of the sphere, $$O_a$$. The mass distribution of the sphere is proportional to its charge distribution. a) Find the electric field inside the cavity. Now we apply an external, uniform electric field $$\mathbf {E}_0$$. Find b) the force on the sphere, c) the torque with respect to the center of the sphere, and the torque with respect to the center of mass. ## 1.3 Energy of a Charged Sphere A total charge Q is distributed uniformly over the volume of a sphere of radius R . Evaluate the electrostatic energy of this charge configuration in the following three alternative ways: a) Evaluate the work needed to assemble the charged sphere by moving successive infinitesimals shells of charge from infinity to their final location. b) Evaluate the volume integral of $$u_{\mathrm{E}}=\|\mathbf {E}\|^2/(8\pi k_\mathrm{e})$$ where $$\mathbf {E}$$ is the electric field [Eq. (1.10)]. c) Evaluate the volume integral of $$\varrho \,\phi /2$$ where $$\varrho$$ is the charge density and $$\phi$$ is the electrostatic potential [Eq. (1.11)]. Discuss the differences with the calculation made in b). ## 1.4 Plasma Oscillations A square metal slab of side L has thickness h, with $$h\ll L$$ . The conduction-electron and ion densities in the slab are $$n_{\mathrm{e}}$$ and $$n_{i}=n_{\mathrm{e}}/Z$$, respectively, Z being the ion charge. An external electric field shifts all conduction electrons by the same amount $$\delta$$, such that $$\|\delta \|\ll h$$, perpendicularly to the base of the slab. We assume that both $$n_{\mathrm{e}}$$ and $$n_{i}$$ are constant, that the ion lattice is unperturbed by the external field, and that boundary effects are negligible. a) Evaluate the electrostatic field generated by the displacement of the electrons. b) Evaluate the electrostatic energy of the system. Now the external field is removed, and the “electron slab” starts oscillating around its equilibrium position. c) Find the oscillation frequency, at the small displacement limit ($$\delta \ll h$$). ## 1.5 Mie Oscillations Now, instead of a the metal slab of Problem 1.4, consider a metal sphere of radius R. Initially, all the conduction electrons ($$n_{\mathrm{e}}$$ per unit volume) are displaced by $$-\mathbf {\delta }$$ (with $$\delta \ll R$$) by an external electric field, analogously to Problem 1.1. a) At time $$t=0$$ the external field is suddenly removed. Describe the subsequent motion of the conduction electrons under the action of the self-consistent electrostatic field, neglecting the boundary effects on the electrons close to the surface of the sphere. b) At the limit $$\delta \rightarrow 0$$ (but assuming $$en_{\mathrm{e}}\delta =\sigma _0$$ to remain finite, i.e., the charge distribution is a surface density), find the electrostatic energy of the sphere as a function of $$\delta$$ and use the result to discuss the electron motion as in point a). ## 1.6 Coulomb explosions At $$t=0$$ we have a spherical cloud of radius R and total charge Q , comprising N point-like particles. Each particle has charge $$q=Q/N$$ and mass m. The particle density is uniform, and all particles are at rest. a) Evaluate the electrostatic potential energy of a charge located at a distance $$r<R$$ from the center at $$t=0$$. b) Due to the Coulomb repulsion, the cloud begins to expand radially, keeping its spherical symmetry. Assume that the particles do not overtake one another, i.e., that if two particles were initially located at $$r_{\scriptscriptstyle 1}(0)$$ and $$r_{\scriptscriptstyle 2}(0)$$, with $$r_{\scriptscriptstyle 2}(0)>r_{\scriptscriptstyle 1}(0)$$, then $$r_{\scriptscriptstyle 2}(t)>r_{\scriptscriptstyle 1}(t)$$ at any subsequent time $$t>0$$. Consider the particles located in the infinitesimal spherical shell $$r_0<r_{\mathrm{s}}<r_0+\mathrm {d}r$$, with $$r_0+\mathrm{d}r<R$$, at $$t=0$$. Show that the equation of motion of the layer is \begin{aligned} m\,\frac{\mathrm {d}^2r_{\mathrm{s}}}{\mathrm {d}t^2}=k_\mathrm{e}\frac{qQ}{r_{\mathrm{s}}^2 } \left( \frac{r_0}{R}\right) ^3 \end{aligned} (1.16) c) Find the initial position of the particles that acquire the maximum kinetic energy during the cloud expansion, and determinate the value of such maximum energy. d) Find the energy spectrum, i.e., the distribution of the particles as a function of their final kinetic energy. Compare the total kinetic energy with the potential energy initially stored in the electrostatic field. e) Show that the particle density remains spatially uniform during the expansion. ## 1.7 Plane and Cylindrical Coulomb Explosions Particles of identical mass m and charge q are distributed with zero initial velocity and uniform density $$n_0$$ in the infinite slab $$\|x\|<a/2$$ at $$t=0$$. For $$t>0$$ the slab expands because of the electrostatic repulsion between the pairs of particles. a) Find the equation of motion for the particles, its solution, and the kinetic energy acquired by the particles. b) Consider the analogous problem of the explosion of a uniform distribution having cylindrical symmetry. ## 1.8 Collision of two Charged Spheres Two rigid spheres have the same radius R and the same mass M, and opposite charges $$\pm Q$$. Both charges are uniformly and rigidly distributed over the volumes of the two spheres. The two spheres are initially at rest, at a distance $$x_0\gg R$$ between their centers, such that their interaction energy is negligible compared to the sum of their “internal" (construction) energies. a) Evaluate the initial energy of the system. The two spheres, having opposite charges, attract each other, and start moving at $$t=0$$. b) Evaluate the velocity of the spheres when they touch each other (i.e. when the distance between their centers is $$x=2R$$). c) Assume that, after touching, the two spheres penetrate each other without friction. Evaluate the velocity of the spheres when the two centers overlap ($$x=0$$). ## 1.9 Oscillations in a Positively Charged Conducting Sphere An electrically neutral metal sphere of radius a contains N conduction electrons. A fraction f of the conduction electrons ($$0<f<1$$) is removed from the sphere, and the remaining $$(1-f)N$$ conduction electrons redistribute themselves to an equilibrium configurations, while the N lattice ions remain fixed. a) Evaluate the conduction-electron density and the radius of their distribution in the sphere. Now the conduction-electron sphere is rigidly displaced by $$\varvec{\delta }$$ relatively to the ion lattice, with $$\|\varvec{\delta }\|$$ small enough for the conduction-electron sphere to remain inside the ion sphere. b) Evaluate the electric field inside the conduction-electron sphere. c) Evaluate the oscillation frequency of the conduction-electron sphere when it is released. ## 1.10 Interaction between a Point Charge and an Electric Dipole An electric dipole $$\mathbf {p}$$ is located at a distance $$\mathbf {r}$$ from a point charge q , as in Fig. 1.5. The angle between $$\mathbf {p}$$ and $$\mathbf {r}$$ is $$\theta$$. a) Evaluate the electrostatic force on the dipole. b) Evaluate the torque acting on the dipole. ## 1.11 Electric Field of a Charged Hemispherical Surface A hemispherical surface of radius R is uniformly charged with surface charge density $$\sigma$$. Evaluate the electric field and potential at the center of curvature (hint: start from the electric field of a uniformly charged ring along its axis). ## Copyright information © Springer International Publishing AG 2017 ## Authors and Affiliations • Andrea Macchi • 1 Email author • Giovanni Moruzzi • 1 • Francesco Pegoraro • 1 1. 1.Department of Physics “Enrico Fermi”University of PisaPisaItaly	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9898863434791565, "perplexity": 732.3336405263501}, "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-2019-22/segments/1558232257845.26/warc/CC-MAIN-20190525004721-20190525030721-00026.warc.gz"}
http://mathhelpforum.com/statistics/206377-fibonacci-sequence-question-print.html	# Fibonacci Sequence question Show 40 post(s) from this thread on one page Page 1 of 2 12 Last • Oct 30th 2012, 05:42 AM Arkious Fibonacci Sequence question Hey all, Im hoping you guys could point me in the right direction after getting myself rather lost with a question on my assignment. I am not putting the actual question as i would like to work it out myself. here is a question i have made up that is similar with what i think to be correct. Could anyone verify this or point me in the right direction? Thanks! Let Fn be the Fibonacci sequence. Use the Fibonacci recurrence relation to express Fn+3 in terms of Fn+1 and Fn. Hence show that Fn+1 = 1/2(Fn+5 - Fn) for n=0,1,2,... the next question i have is the same but with Fn+1 = 1/2(Fn+5 + Fn) then i have to decide wether the formulas would remain true if the sequence Fn were replaced by a sequence with the same recurrence relation as the Fibonacci sequence but with different initial terms, then justify. So if i look just at the 1st one at the moment, i think that i would get something like this; Fn+1 = 1/2(Fn+5 - Fn) for n=0,1,2,... Fn = 1/2(Fn+1 - Fn+1) Fn = 1/2(Fn+2 + Fn+1) if that is not correct, how do i work this out properly as im rather stumped :-S • Oct 30th 2012, 06:27 AM a tutor Re: Fibonacci Sequence question Quote: Originally Posted by Arkious Hence show that Fn+1 = 1/2(Fn+5 - Fn) for n=0,1,2,... but this is not true. Presumably you meant $F_{n+1}=\frac{1}{2}(F_{n+3}-F_n)$. • Oct 30th 2012, 06:57 AM Arkious Re: Fibonacci Sequence question Yeah that's the question I actually have, I just didn't want to ask the question I have been given. I'm not 100% with Fibonacci yet. But if you could explain how I would complete the question you have mentioned, I would much much appreciate it • Oct 30th 2012, 10:23 PM Salahuddin559 Re: Fibonacci Sequence question Dont know what you have asked above, the notation is confusing, It gives 2 different expressions for Fn+1 alone. I will try to give you the proof for Fn+1 = 1/2(Fn+3 - Fn). Fn+3 = Fn+2 + Fn+1 = (Fn+1 + Fn) + Fn+1 = 2Fn+1 + Fn. Just rearrange terms here. Salahuddin Maths online • Oct 31st 2012, 12:17 AM Arkious Re: Fibonacci Sequence question Yeah that's the problem, it's very confusing! If i didn't have to show that Fn+1 = 1/2(Fn+3 - Fn) for n=0,1,2,... I would have been able to do it. Thanks for your input :-) anyone got any ideas though? I'm no further forward. • Oct 31st 2012, 03:53 AM a tutor Re: Fibonacci Sequence question Did you try to express $F_{n+3}$ in terms of $F_{n+1}$ and $F_n$ ? You could start by writing $F_{n+3}$ in terms of $F_{n+2}$ and $F_{n+1}$. • Oct 31st 2012, 05:16 AM Arkious Re: Fibonacci Sequence question right, this is what i have done so far... in Fibonacci terms; Fn+3 = Fn+2 + Fn+1 Fn+2 = (Fn+2 + Fn) Fn+1 = (Fn+2 - Fn) Fn = (Fn+2 -Fn+1) (if thats correct) But then i need to express Fn+1 = 1/2(Fn+3 - Fn) • Oct 31st 2012, 06:02 AM Salahuddin559 Re: Fibonacci Sequence question in Fibonacci terms; Fn+3 = Fn+2 + Fn+1 Fn+2 = (Fn+1 + Fn) Correct for these two, but now, get rid of the Fn+2 in the first equation, since we do not have any Fn+2 in our formula. Substitute your Fn+2 = Fn + Fn+1 there, and rearrange terms. Salahuddin Maths online • Nov 1st 2012, 04:06 AM Arkious Re: Fibonacci Sequence question but this still doesnt make sense... how do i then show that Fn+1 = 1/2(Fn+3 - Fn). surely above is all wrong? • Nov 1st 2012, 04:23 AM Arkious Re: Fibonacci Sequence question Fn+3 = Fn+2 + Fn+1 (2) = (1) + (1) Fn+2 = Fn+1 + Fn (1) = (1) + (0) Fn+1 = Fn - Fn-1 (1) = (0) - (-1) Fn = Fn-1 - Fn-2 (0) = (-1) - (-1) Thats proved that those fibonacci terms are correct, but then how do i then show Fn+1 = 1/2(Fn+3 - Fn)? • Nov 1st 2012, 04:44 AM Salahuddin559 Re: Fibonacci Sequence question Guys, here is the steps. Fn+3 = Fn+2 + Fn+1 But since Fn+2 = Fn+1 + Fn Fn+3 = Fn+2 + Fn+1 = (Fn+1 + Fn) + Fn+1 = 2Fn+1 + Fn Remove Fn, but subtracting it both sides, Fn+3 - Fn = 2Fn+1. In other words, Fn+1 = 1/2(Fn+3 - Fn). This is the same style for many other problems, get it? Salahuddin Maths online • Nov 1st 2012, 04:46 AM Arkious Re: Fibonacci Sequence question Fn+1 = 1/2(Fn+3 - Fn) (5) = 1/2 ((13) - (3)) (5) = 1/2 (10) am i supposed to find the solution for Fn then use that to find Fn+3? I have tried multiple ways and cant find a solution that works for more than one number :-( • Nov 1st 2012, 04:58 AM Salahuddin559 Re: Fibonacci Sequence question Hi Arkious, could you be more specific, I am not able to understand what you are saying. (Basically if you have Fn and Fn+1, you can find Fn+3. But why are you finding the solutions, what is the actual problem?). Salahuddin Maths online • Nov 1st 2012, 05:54 AM Arkious Re: Fibonacci Sequence question Let Fn be the Fibonacci sequence. Use the Fibonacci recurrence relation to express Fn+3 in terms of Fn+1 and Fn. Hence show that Fn+1 = 1/2(Fn+3 - Fn) for n=0,1,2,... That is the question i have to answer. My issue is how to show that Fn+1 = 1/2(Fn+3 - Fn). I dont think that these are of any use for the question i have to answer. Fn+3 = Fn+2 + Fn+1 Fn+2 = Fn+1 + Fn Fn+1 = Fn - Fn-1 Fn = Fn-1 - Fn-2 All i need to know is how to answer that question and i think that what i have done so far has confused me more. I just don't understand how im supposed to show that Fn+1 = 1/2(Fn+3 - Fn) for Fn+3 & Fn+1 • Nov 1st 2012, 06:09 AM a tutor Re: Fibonacci Sequence question Quote: Originally Posted by Arkious Fn+3 = Fn+2 + Fn+1 Fn+2 = Fn+1 + Fn These are what you need. Use the second one to substitute for Fn+2 in the first. Fn+3 = Fn+2 + Fn+1 becomes Fn+3 = _?_+_?_ + Fn+1. Show 40 post(s) from this thread on one page Page 1 of 2 12 Last	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8195459842681885, "perplexity": 2912.9732338695803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886126017.3/warc/CC-MAIN-20170824004740-20170824024740-00217.warc.gz"}
https://www.physicsforums.com/threads/skin-effect.613289/	# Skin Effect 1. Jun 11, 2012 ### CheyenneXia Hey I know the conclusion of skin effect since secondary year of school, but I still didnt get it. I know the reason is because of eddy current. I couldnt include the link here as I havent posted 10 yet. For the link, just search "Skin Effect" at Wikipedia. First paragraph of the cause in Wikipedia says "The counter EMF is strongest at the center of the conductor, and forces the conducting electrons to the outside of the conductor, as shown in the diagram on the right." I didnt really get it. Why the counter EMF is strongest? Even if it is strongest, why electrons are going to be driven outside? Shouldnt we have to consider about the positive or negetive then decide whether being driven outside or inside? Also, the explaination of the third picture on the right "Skin depth is due to the circulating eddy currents (arising from a changing H field) cancelling the current flow in the center of a conductor and reinforcing it in the skin." Well, it is true when I in read increases. But what if I decreases, aren't eddy currents in the picture should circulate the opposite direction. By that case, eddy current actually cancel the current flow in the skin and enforce the current in the center! Also, I believe eddy current is larger in the skin as H is larger compared to it in the center. I know skin effect is correct. But I do not know why it works that way. Thanks 2. Jun 13, 2012 ### Staff: Mentor Thread moved to the EE forum for better views. Here is the link you are referring to: http://en.wikipedia.org/wiki/Skin_effect . 3. Jun 13, 2012 ### the_emi_guy You are right, the Wiki page is a little misleading on this point. This is not an electrostatic situation where electrons are being forced to move from the center to the edges. "I" is an AC current, the electrons in the conductor are wiggling back and forth (a relatively small distance). The counter EMF is simply impeding the electrons in the center from wiggling, electrons on the skin are free to wiggle. Because fewer electrons in the cross section of the conductor are participating in this dance, the resistance of the conductor is lowered. Draw out the main current "I", the H field, and the eddy current "Iw" as sinusoids, and bear in mind that they are phase shifted due to the derivatives in Maxwell's equations. For example, when "I" decreases but is still directed upward, the H field will be in the opposite direction, but it will be decreasing in magnitude. This leads to eddy current that opposes "I". (Lenz's Law). 4. Jun 13, 2012 ### Antiphon Eddy currents is a bad description of this phenomenon. Eddies are circular vortex-like motions. This happens to describe the currents that flow when the magnetic field points into the surface. But it doesn't accurately describe the much more common situation of a plane wave striking a metallic surface or the penetration of AC current into a wire- again fields are parallel to, not normal to a surface. Having established that, the skin depth is controlled by a dissipative phenomenon depending (for finite conductors) on the resistivity of the metal. The penetration is zero for the ideal conductor and as the resistivity increases the penetration increases. As with any dissipative wave propagation the amplitude decays exponentially. 5. Jun 13, 2012 ### CheyenneXia Hey, EMI Guy, I can understand "The counter EMF is simply impeding the electrons in the center from wiggling", but why "electrons on the skin are free to wiggle"? No counter EMF on the skin? I think there should be. Is it because electrons cannot wiggle as free as in the centre at all directions? "Because fewer electrons in the cross section of the conductor are participating in this dance, the resistance of the conductor is lowered. " I thought the resistance is increased as the effective cross section is smaller. Am I correct? "when "I" decreases but is still directed upward, the H field will be in the opposite direction, but it will be decreasing in magnitude." If "I" dereases but still directed upward, I thought the original H field is still in the same direction but decreasing in magnitude, which causes eddy current inducing H filed at the same direction of original H. Eddy current here is opposite of the one shown in the graph at wiki page. That's what confused me. If I am wrong, tell me where. Thanks for your help. Last edited: Jun 13, 2012 6. Jun 13, 2012 ### CheyenneXia Hey Antiphon, thanks for your help. But I do not really understand your explanation. Do I have to look into "dissipative phenomenon" to understand skin effect? I know the conclusion of the skin depth and the factors affecting it. But I need to understand the root cause of skin effect. 7. Jun 13, 2012 ### Antiphon That search will probably not help you. The root cause of the skin effect is that the penetration of fields into conductors takes time. The lower the resistivity of the material, the longer it will take. For example, work out what the resistivity of a wire would be that had a skin depth of 1 mm at a frequency of 1 Hz. It will be an almost perfect conductor. Now if you had a long 1 meter thick wire made of it then when you turn on a DC current, at first it will flow along the outside. Slowly the DC current will penetrate into the conductor in exactly the same manner as if heat were penetrating into the side of a cold glass cylinder. After around twenty minutes the current inside would begin to approach the level at the surface. Now clearly if you apply AC to this wire you can see how the field penetration won't get very far; it would be as if you were alternately heating and cooling the outside of a glass cylinder. If you do it fast enough there will be very little temperature variation going on in the core. Does that make sense so far? Ok, the thermal analogy is good but not exact. In the wire there is in fact induction going on just like the Wikipedia article says but there is also resistive dissipation. The correct picture looks like an LR circuit where you have a DC voltage source at the left, an inductor at the top, the resistor on the right. If the resistor is zero, the current in the inductor will steadily increase but you will measure no output voltage on the resistor. If there is a non-zero resistance the voltage will ramp up at a rate governed by the RL time constant (almost the skin effect equation!). Now the full physically correct circuit analog is an infinite repeating ladder with series inductors and shunt resistors. When you turn on the source, a voltage "front" (measured across the shunt resistors) will "diffuse" through this network. -That is unless the resistors are zero ohms and then the current will only flow in the first inductor- the "skin" of the LR ladder! Does this physics make sense to you? Last edited: Jun 13, 2012 8. Jun 17, 2012 ### CheyenneXia Thanks. I kinda understand it now. Similar Discussions: Skin Effect	{"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": 0.8130226731300354, "perplexity": 728.7671021870761}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823114.39/warc/CC-MAIN-20171018195607-20171018215607-00779.warc.gz"}
https://socratic.org/questions/5a7e2ec07c0149612990a763	Physics Topics # What is the form of energy acquired from food? Feb 10, 2018 It's known as a Calorie, which is then stored as potential energy in the form of chemical bonds or chemical energy #### Explanation: The food that you eat is eventually broken down into three main typical of chemicals: carbohydrates, proteins(amino acids), and fats. The energy is stored in the chemical bonds of the these molecules. These molecules are eventually processed to make the energy molecules called ATP that provides energy that the body needs. So food energy is also called chemical energy. Fundamentally, chemical bonds are electrical in origin, as molecules are made of nuclei with positive charge and electrons with negative charges. When chemical reactions happened, nuclei and electrons are reshuffled, giving up the electric potential energy stored in these molecules. The pathways food into energy is best illustrated here. ##### Impact of this question 386 views around the world	{"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": 0.8466586470603943, "perplexity": 1391.562924910316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348493151.92/warc/CC-MAIN-20200605045722-20200605075722-00363.warc.gz"}
http://www.computer.org/csdl/trans/tm/2009/09/ttm2009091167-abs.html	Subscribe Issue No.09 - September (2009 vol.8) pp: 1167-1179 Giovanna Carofiglio , Alcatel-Lucent Bell Labs, France Carla-Fabiana Chiasserini , Politecnico di Torino, Torino Michele Garetto , Università degli Studi di Torino, Torino Emilio Leonardi , Politecnico di Torino, Torino ABSTRACT A fundamental issue arising in mobile ad hoc networks (MANETs) is the selection of the optimal path between any two nodes. A method that has been advocated to improve routing efficiency is to select the most stable path so as to reduce the latency and the overhead due to route reconstruction. In this work, we study both the availability and the duration probability of a routing path that is subject to link failures caused by node mobility. In particular, we focus on the case where the network nodes move according to the Random Direction model, and we derive both exact and approximate (but simple) expressions of these probabilities. Through our results, we study the problem of selecting an optimal route in terms of path availability. Finally, we propose an approach to improve the efficiency of reactive routing protocols. INDEX TERMS Mobile ad hoc networks, routing, modeling and analysis. CITATION Giovanna Carofiglio, Carla-Fabiana Chiasserini, Michele Garetto, Emilio Leonardi, "Route Stability in MANETs under the Random Direction Mobility Model", IEEE Transactions on Mobile Computing, vol.8, no. 9, pp. 1167-1179, September 2009, doi:10.1109/TMC.2009.20 REFERENCES	{"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": 0.8908514976501465, "perplexity": 1111.3486176120873}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042990112.92/warc/CC-MAIN-20150728002310-00201-ip-10-236-191-2.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/nuclear-magneton.124413/	# Nuclear Magneton 1. Jun 22, 2006 ### Lambda Thanks 2. Jun 22, 2006 ### Meir Achuz If you know what a Bohr magneton is, a nuclear magneton is the same thing, except using the mass of a paroton instead of the mass of an electron. Specifically it is defined by \mu_N=e hbar/M_p, with the numerical value \mu_N=3.15X10^-14 MeV. It arises from the connection betrween angular momentum and magnetic moment. Using the Dirac equation, the magnetic moment of a spin 1/2 particle proton would be given by \mu_p=\mu_N. But the proton has an anomaloous magnetic moment so that its actual value is \mu_p=g\mu_N, where g is called the "g value" or more commonly nowadays g is called "the magnetic moment" given in units olf the nuclear magneton, \mu_N. This is what is meant by the statement "The magnetic moment of the proton is 2.79", because that is the value of g for the proton. The nuclear magneton is just a convenient unit in which to express magnetic mooments of hadrons.	{"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": 0.9947580695152283, "perplexity": 1087.1619757972637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1487501170864.16/warc/CC-MAIN-20170219104610-00434-ip-10-171-10-108.ec2.internal.warc.gz"}
https://xianblog.wordpress.com/2012/03/06/seminar-at-crest-on-predictive-estimation/	## seminar at CREST on predictive estimation On Thursday, March 08, Éric Marchand (from Université de Sherbrooke, Québec, where I first heard of MCMC!, and currently visiting Université de Montpellier 2) will give a seminar at CREST. It is scheduled at 2pm in ENSAE (ask the front desk for the room!) and is related to a recent EJS paper with Dominique Fourdrinier, Ali Righi, and Bill Strawderman: here is the abstract from the paper (sorry, the pictures from Roma are completely unrelated, but I could not resist!): We consider the problem of predictive density estimation for normal models under Kullback-Leibler loss (KL loss) when the parameter space is constrained to a convex set. More particularly, we assume that $X \sim \mathcal{N}_p(\mu,v_x\mathbf{I})$ is observed and that we wish to estimate the density of $Y \sim \mathcal{N}_p(\mu,v_y\mathbf{I})$ under KL loss when μ is restricted to the convex set C⊂ℝp. We show that the best unrestricted invariant predictive density estimator p̂U is dominated by the Bayes estimator p̂πC associated to the uniform prior πC on C. We also study so called plug-in estimators, giving conditions under which domination of one estimator of the mean vector μ over another under the usual quadratic loss, translates into a domination result for certain corresponding plug-in density estimators under KL loss. Risk comparisons and domination results are also made for comparisons of plug-in estimators and Bayes predictive density estimators. Additionally, minimaxity and domination results are given for the cases where: (i) C is a cone, and (ii) C is a ball. ### One Response to “seminar at CREST on predictive estimation” 1. […] Éric Marchand came to give a talk last week, we discussed about minimality and Bayesian estimation for […] This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"extraction_info": {"found_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": 2, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9316325783729553, "perplexity": 2110.4042980715203}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046150307.84/warc/CC-MAIN-20210724160723-20210724190723-00476.warc.gz"}
http://lenrexplained.com/tag/lenr-theory/	## Q&A on the NAE Peter Gluck of Ego-out engages Edmund Storms on the NAE http://egooutpeters.blogspot.ro/2017/01/jan-23-2017-lenr-info-questions.html Question If NAE are nanocracks – why is there a limit for their number/density? What is the limiting factor? Answer The cracks are generated by stress generated by the change in volume when D reacts with Pd. The cracks form at weak regions in the structure. A limit to the number of weak regions exists in a structure. Once crack formation has relieved the stress, no further cracks can form. This is basic material behavior having nothing unusual about the process until the Hydroton forms. For reasons yet unknown, once the critical size crack forms, it can then support the LENR process. Question Are those active cracks special in some way or is it only a problem of size? Answer The gap size is the critical condition. A size too large can not support LENR. Question If temperature is a factor, how? Answer Temperature determines how fast D can get to the NAE by diffusion from its site in the surrounding lattice. Question Will the processes at 70, 400, 800, 12000 C be qualitatively the same, or will be some changes in the mechanism? Answer The mechanism is not changed by temperature. Temperature ONLY changes how fast the fuel (D or H) can get to where it can fuse. Question How and why do the NAE resist and survive the nuclear process? Answer The gap is filled with a chemical structure consisting of chains of D. These chains (Hydrotons) fuse by an unknown process and are destroyed. The gap remains in which more Hydroton can form. The gap can remain because the energy is released slowly without causing destruction of the local lattice structure. As I have been saying, one unique and required feature of LENR is the slow rate at which energy is released. Of course, this process is only slow when compared to the hot fusion process. Cold fusion is actually better described as slow fusion. Question Piantelli said he had excess heat for months. The Rossi heat effect seems to be OK for 6 months. Why is the duration of the PdD excess heat a problem? Answer Many people have seen the process last for a long time. In my case, it stops only when I cause it to stop because want to go on to other studies. Question What do you think and which factors play a role for the claimed greater density of NAE in NiH then in PdD – metallurgy, morphology? Perhaps we have to consider that Pd D works with deuterium and NiH with protium. Answer Ni does not take up as much hydrogen isotope as Pd, hence the stress is less compared to Pd. Also, Ni is stronger than Pd, thereby preventing the stress from producing much cracking. Rossi found a way to produce the active cracks in Ni powder where each grain could contain a number of active cracks. Arata was able to activate Pd powder with impressive power production. Clearly, powder allows more NAE to form within the same weight of material. Work in Japan is taking advantage of this conclusion using Pd. by ## JCMNS Vol. 20 publishes two Storms papers The Journal of Condensed Matter Nuclear Science JCMNS Vol. 20 [.pdf] has published Anomalous Energy Produced by PdD and How Basic Behavior Can Guide A Search for an Explanation both by Edmund Storms, LENERGY, LLC. Anomalous Energy Produced by PdD on pages 81-99 reports on the “production of anomalous energy using two different samples and the behavior of this energy when temperature, deuterium content of the material, and applied current are changed. Storms has determined that high-loading is not a necessary condition to initiate the reaction, and that the single greatest factor is temperature. The second paper How Basic Behavior Can Guide A Search for an Explanation pages 100-138 is a systematic narrowing of LENR models as examined by their assumptions and logical consequences. Jettisoning all ideas that rely on imagined events, the theoretical field is pared down with only the most robust theoretical elements surviving. An attempt to provide a reasoned approach to an explanation continues the further evolution of Nanocrack Theory, where experimental evidence is supreme. http://lenr-canr.org/wordpress/?page_id=1495 http://lenr-canr.org/acrobat/BiberianJPjcondenseds.pdf by ## LENR theory paper revised with more detail Edmund StormsHow basic behavior of LENR can guide a search for an explanation [.pdf] has been revised with more details. How basic behavior of LENR can guide a search for an explanation [.pdf] by Edmund Storms ABSTRACT The LENR effect was identified 27 years ago by Profs. Fleischmann and Pons as production of extra energy in a normal chemical structure, in this case PdD. Over a thousand published papers now support the discovery and the energy is shown to result from fusion of hydrogen isotopes without the need to apply energy and without energetic radiation being produced. By conventional standards, the claims are impossible. Nevertheless, a new phenomenon has been discovered requiring acceptance and understanding. The major behaviors and their present understanding are described in this paper and are used to suggest how an effective explanation might be constructed. Once again, science has been forced to either reject the obvious or accept the impossible. In this case, the normal skepticism needs to be ignored in order to determine if this promised energy source is real and can provide the ideal energy so critically needed. INTRODUCTION Low Energy Nuclear Reaction (LENR) or Cold Fusion was introduced to the world 27 years ago by Fleischmann and Pons(1), Univ. Utah, with expectation of great benefit to mankind. Instead, their claim for a new kind of fusion was quickly rejected (2), an attitude that continues even today. Over the years, several thousand papers addressed the subject with a large fraction supporting the claim(3). Mastery of about 1000 papers is now required to understand the effect. A description of all the known behaviors and all proposed explanations would require much more than a single review paper. Here, only the tip of the large iceberg will be examined along with some original results not published elsewhere. The selection of behaviors is designed to focus attention on only the essential conditions required to cause the LENR effect. Limits will be set using observed behavior in order to evaluate proposed explanations. The new kind of nuclear interaction needed to explain LENR is expected to fall within these limits. In other words, boundaries need to be identified to keep the imagination from running wild. The LENR effect is assumed consistent with all rules normally applied to conventional chemical and nuclear behavior. Nevertheless, a novel mechanism is clearly operating and needs to be acknowledged. Many conditions needing consideration are not quantitative or lend themselves to mathematical analysis. While frustrating to conventional scientists, these unique behaviors must be made part of a successful explanation. Quantitative behaviors can be used to expand understanding once the basic process is understood. An effective explanation needs to solve several difficult problems. The Coulomb barrier needs to be overcome without using more energy than is normally available in a chemical structure at room temperature. Neutron formation, which has been suggested by several theoreticians (4, 5), is prohibited because the required energy of 0.78 MeV and the required neutrino can not be expected to be available at the same site at the same time. Once fusion has occurred, the mechanism must then dissipate the huge nuclear energy released by the process without producing local destruction of the chemical structure or energetic radiation. The mechanism must also account for various transmutation reactions known to occur. Failure to combine these events in a way that is consistent with known chemical and nuclear behavior dooms most efforts to explain the process. In contrast, a single mechanism is proposed in this paper to cause all observed behavior while being consistent with known chemical and nuclear behavior. This paper has two parts, with the first describing the important observations on which an explanation must be based. The second part uses a few assumptions combined with these chosen behaviors to provide an explanation about how LENR can be initiated using a proposed mechanism. This mechanism is clearly much different from that causingn the conventional hot fusion process. Ironically, this conflict is used to reject the claims for LENR rather than guiding a search for the cause of the difference. Consequently, this difference must be clearly understood before the novel features of LENR can be explored. Unlike hot fusion, LENR takes place in and requires a chemical structure to operate. The role of this structure must be understood before physics is applied to understanding subsequent nuclear process. Clearly, a unique and rare condition must form in the structure in which a nuclear process can function. The nature of this condition is discussed following the discussion of hot fusion. Continue reading How basic behavior of LENR can guide a search for an explanation – Revised here. by ## How basic behavior of LENR can guide a search for an explanation How basic behavior of LENR can guide a search for an explanation – Revised by Edmund Storms LENRGY LLC Santa Fe, NM, 87501 (4/2/16) ABSTRACT The LENR effect was identified 27 years ago by Profs. Fleischmann and Pons as production of extra energy by a normal chemical structure, in this case PdD. Over a thousand published papers now support the discovery and the energy is shown to result from fusion of hydrogen isotopes without the need to apply energy and without energetic radiation being produced. By conventional standards, the claims are impossible. Nevertheless, a new phenomenon has been discovered requiring acceptance and understanding. The major behaviors and their present understanding are described in this paper and are used to suggest how an effective explanation might be constructed. Once again, science has been forced to either reject the obvious or accept the impossible. In this case, the normal skepticism needs to be ignored in order to determine if this promised energy source is real and can provide the ideal energy so critically needed. INTRODUCTION Low Energy Nuclear Reaction (LENR) or Cold Fusion was introduced to the world 27 years ago by Fleischmann and Pons(1), Univ. Utah, with expectation of great benefit to mankind. Instead, their claim for a new kind of fusion was quickly rejected (2), an attitude that continues even today. Over the years, several thousand papers addressed the subject with a large fraction supporting the claim(3). Mastery of about 1000 papers is now required to understand the effect. A description of all the known behaviors and all proposed explanations would require much more than a single review paper. Here, only the tip of the large iceberg will be examined along with some original results not published elsewhere. The selection of behaviors is designed to focus attention on only the essential conditions required to cause the LENR effect. Limits will be set using observed behavior in order to evaluate proposed explanations. The new kind of nuclear interaction needed to explain LENR is expected to fall within these limits. In other words, boundaries need to be identified to keep the imagination from running wild. The LENR effect is assumed consistent with all rules normally applied to conventional chemical and nuclear behavior. Nevertheless, a novel mechanism is clearly operating and needs to be acknowledged. Many conditions needing consideration are not quanitative or lend themselves to mathematical analysis. While frustrating to conventional scientists, these unique behaviors must be made part of a successful explanation. Quantitative behaviors can be used to expand understanding once the basic process is understood. The present paper has two parts, with the first describing the important observations on which an explanation must be based. The second part uses a few assumptions combined with these chosen behaviors to provide an explanation about how LENR can be initiated and the resulting mechanism. The LENR mechanism is clearly much different from that causing the conventional hot fusion process. Ironically, this conflict is used to reject the claims for LENR rather than guiding a search for the cause of the difference. This difference must be clearly understood before the novel features of LENR can be explored. Consequently, the hot fusion process is discussed first. Unlike hot fusion, LENR takes place in and requires a chemical structure to operate. The role of this structure must be understood before physics is applied to understanding subsequent nuclear process. Clearly, a unique and rare condition must form in the structure in which a nuclear process can function. The nature of this condition is discussed following the discussion of hot fusion. The nature of the hot fusion mechanism Because LENR involves fusion of hydrogen, the conventional fusion process, called hot fusion, needs to be understood in relation to LENR. For the last 75 years, the hot fusion method has been applied in various ways, including in the ITER(4) facility now being constructed in France using magnetic confinement and in the National Ignition Facility(5) in Livermore, CA with lasers being used to create the required energetic plasma. These methods use high energy to overcome the Coulomb barrier by brute force. This large applied energy changes the fusion rate in plasma as shown by the log-log plot in Fig. 1. The energy applied to LENR is no more than 1 eV. FIGURE 1. Effect of energy on the fusion rate in plasma for different combinations of hydrogen isotopes as result of the hot fusion process. (Wikipedia) Hot fusion can also be initiated by bombarding a material by energetic deuterons. In this case, the fusion rate is slightly greater at low applied energy compared to when the same energy is applied to plasma, as can be seen in Fig. 2. Even so, the overall fusion rate FIGURE 2. Comparison between the fusion rate in plasma (Bare Cross-Section) and when fusion occurs in a solid material as the result of applying energy to the bombarding D+ ions, as shown by the X-axis. A value of unity occurs when the rate in plasma is equal to the rate using a target material.(6) decreases as applied energy is reduced. In other words, the environment in a material can slightly increase the fusion rate but it does not significantly offset the reduction in the rate as applied energy is lowered. While the electrons clearly help lower the barrier to achieve hot fusion, this effect alone would seem too small to explain the LENR process, although it might make a small contribution. In any case, the measured shielding effect applies only to the hot fusion mechanism. Perhaps more effective shielding during LENR might be expected if the shielding electrons were contained in the unique nuclear-environment rather than having a random and lower concentration in the general where hot fusion interaction takes place. An evaluation of just how the electrons function during LENR compared to hot fusion requires LENR not be viewed as extension of hot fusion. Once the nuclei of deuterium have fused by hot fusion, the assembly breaks into fragments, which dissipate the excess mass-energy as kinetic energy. Easily detected energetic neutrons, tritium, protons, and He3 are produced in equal amounts. This process is understood and is consistent with conventional expectations. A similar result occurs when muons are used to bring the nuclei close enough to cause fusion. In other words, no matter whether energy is used to overcome the Coulomb barrier by brute force or the separation is reduced by using the heavy muon(7-10), the same energy dissipation process results. No other method for energy dissipation as result of a fusion reaction was known to occur in nature until “cold fusion” was discovered. Clearly, the mechanisms causing hot fusion and cold fusion are significantly different because LENR does not lead to fragmentation of the nuclear products. Cold fusion is novel because it does not require significant applied energy to overcome the Coulomb barrier and it does not result in fragmentation of the fusion product as occurs during hot fusion. This difference has caused much skepticism about the reality of LENR. After all, experience and teaching deny any possibility of spontaneous fusion taking place in an ordinary chemical structure without the need to apply significant energy. This apparent contradiction is resolved by proposing the cold fusion process takes place in a unique structure, called the nuclear-active-environment (NAE) where a novel mechanism can operate. Questions about how this structure forms, where in the chemical structure this formation takes place, the nature of the unique conditions at the NAE, and the nuclear mechanism operating therein are explored later in this paper. Role of chemical structure Because the LENR process takes place within a chemical structure, it must play by the rules such a structure imposes. This conclusion is critical to understanding the LENR process. These rules include the Laws of Thermodynamics and the Phase Rule. Local energy cannot spontaneously increase without violating the Second Law of Thermodynamics and the local concentration of ambient energy is limited by how much energy the chemical bonds can tolerate before melting or decomposition results. Simply stated, energy cannot go up hill and its density cannot exceed the strength of the container. If a novel mechanism is proposed to concentrate energy in order to cause nuclear fusion, why it is not found to affect chemical reactions? After all, if such a process were possible, it would be expected to operate in normal chemicals and cause chemical effects before the local energy had increased enough to cause a nuclear reaction. For example, the mechanism of energy transfer to electrons proposed by Widom and Larsen(11, 12) would be expected to make many normal chemical compounds unstable. Furthermore, how such a proposed violation of the Second Law of Thermodynamics can function in PdD needs to be justified. Similar conflicts with the laws of thermodynamics and normal chemical behavior create a similar weakness in many explanations now being proposed. Normally, nuclear reactions of any kind are not affected by the chemical environment because the energy states are too different and local energy density cannot be increased according to the Second Law of Thermodynamics. Amazingly, the normal level of local ambient energy is sufficient to initiate the LENR process at high rate on rare occasions. Explaining how this “magic” takes place is the first of two basic challenges. The second challenge involves how the resulting energy is dissipated as heat. Once fusion occurs, the structure must convert the excess mass-energy to heat without causing local melting. After all, local destruction of the active site would stop further heat production and severely limit the amount of energy produced by LENR, which is not experienced. Although local melting is occasionally seen, it is not sufficient to create a limit to the amount of power or its stability over time. Thus, both the presence of a little local melting and the absence of extensive melting have to be explained. Several different chemical structures have been found to support LENR, with PdD given the most attention. Consequently, PdD is the focus of further discussion. Palladium deuteride has attracted interest for about the last 100 years(13) during which time it has been studied extensively. Although the palladium can acquire hydrogen up to about PdD0.98±0.02, nothing about the overall behavior would suggest an ability to host a fusion reaction. The structure is face-centered-cubic (fcc) and exists in two slightly different forms having the same crystal structure based on the Pd sublattice. The alpha phase occurs between pure Pd and about PdD0.05, and the beta phase forms near PdD0.6 when 1 atm of D2 pressure is applied at 20° C. A two-phase region exists between these two compositions. The beta phase continues to acquire D atoms at random sites in the fcc sublattice as pressure is increased, finally reaching the upper limit of the fcc phase. Fig 3 shows the structure when all lattice sites are fully filled by deuterium. Another phase is expected to form and grow in amount as the overall D/Pd ratio increases beyond the upper limit to the fcc phase, similar to the behavior of other metallic hydrides.(14, 15) In other words, any composition in excess of PdD0.98 would be expected to be a two-phase mixture of the fcc and another phase having a different structure and increased stoichiometry. In the absence of the rare double occupancy(16, 17) of normal lattice sites, the deuterium nuclei are too far apart to fuse. Achieving close approach without violating the rules of chemistry and without producing fragmentation typical of hot fusion remains a serious challenge discussed in a later section. Identifying where the NAE is located and what form it takes in the material has created a problem for many proposed explanations. Many explanations assume the fusion process takes place in a modification of the fcc structure when the D/Pd ratio is large. Formation of such a structure would be apparent because its formation would cause changes in various properties. A search for the expected change can be made by examining several known properties, such as resistivity and lattice parameter as a function of D/Pd. The lattice parameter can be seen to have a linear(18-21) relationship to composition with no indication of a two-phase region forming within the limits of the beta phase. Both the pressure and resistivity(22) also show no sign of a change in crystal structure(23) over the composition range of interest. In every way, all properties are consistent with a normal fcc structure being present within the composition range in which LENR is found to occur. FIGURE 3. Crystal structure of the face-centered-cubic PdD when all deuterium sites (small purple) are filled. (Wikipedia) On the other hand, Fukai(24) reported formation of a phase change when high pressure is applied at high temperature to PdH. This structure is proposed to also form under normal conditions during electrodeposition.(25) Superabundant vacancies are proposed to form in the metal sublattice. A similar structure change is proposed to be caused by deformation induced vacancies.(26) This behavior might also occur when repeated loading and deloading of PdD causes the structure to expand, producing what Storms(27) calls excess volume. Nevertheless, this condition does not explain LENR because the presence of excess volume over about 2% is found to inhibit LENR(28) rather than aid the reaction as would be expected if formation of metal atom vacancies were required to support LENR. Even though the proposed vacancies are not associated with the LENR process, a unique condition is expected to form in the PdD in order for LENR to take place. This conclusion is consistent with common experience. When a piece of Pd is found to be nuclear active, most of the entire batch is also found to be nuclear active. In addition, once the sample is made nuclear active, the LENR process using that piece becomes reproducible and robust. Obviously, treatment of the entire batch of Pd creates stable conditions in which the LENR process can be initiated and supported for extended times. Unfortunately, these conditions are hard to produce because their unique characteristic is unknown and rarely formed. Even when certain important initial conditions are present, an additional special treatment is required before the nuclear process can be produced by PdD. These observations are important because they show a treatment is possible to make large amounts of palladium nuclear active. A suggested combination of conditions is described later in this paper. Initially, the LENR reaction was thought to take place anywhere in the PdD structure. Later studies reveal both helium(29, 30) and tritium(31) form only very near the surface and not within the bulk material or on the surface where nanoparticles might be present when electrolysis is used. Transmutation products are also detected mainly in the surface region. Based on the known behavior of helium in PdH(32, 33), the nuclear reactions apparently take place within a region perhaps no more than 10 μm wide, extending from the surface. We now need to discover the nature of the unique condition forming within this narrow band. The condition does not appear to involve a phase change, creation of vacancies in the hydride structure, creation of nanoparticles on the surface, nor does it require a high concentration of deuterium. Formation of NAE would appear to require conditions formed by a unique process, which apparently only forms near the surface. IMPORTANT OBSERVED BEHAVIOR Formation of the NAE In order for fusion to take place, the reacting nuclei must obviously be in the same place at the same time. This condition is not normally present. Normally, the D atoms are located too distant to fuse. For atoms to assemble in a chemical structure, Gibbs energy must be released while the material achieves a different stable state. Generally, the atoms in a chemical structure are close to their equilibrium condition and do not contain excess energy or have the ability to form another crystal structure unless the conditions are significantly changed. Simply increasing the D/Pd ratio does not create sufficient energy to change the structure in order to initiate the LENR process. Furthermore, for the process to be as rare and as difficult to initiate as is observed, the conditions for releasing this energy must be equally rare and difficult to create. To make the problem even more challenging, once the NAE is formed, LENR must operate at a significant rate without further change in conditions. These conditions immediately place a limit on any proposed condition in which LENR can take place. Most samples of PdD do not host the LENR process regardless of the deuterium content presumably because the unique NAE is not initially present in the material. This conclusion suggests the NAE is not related to any of the features normally found in a chemical structure, such as vacancies, dislocations, and occupancy of unusual lattice sites. After all, if the NAE were related to these common features, the effect would be initiated more easily and more often. Multiple occupancy of the normal deuterium-atom vacancy must also be rejected based on this conclusion because, if such occupancy were possible, it would be present in all material under normal conditions and cause LENR with greater frequency. Nevertheless, a rare condition must form as result of some kind of treatment in order to account for occasional success. Failure to initiate LENR simply means this treatment was not successful in producing the required NAE. Once produced, the NAE appears to be stable and relatively constant in amount as indicated by production of relatively constant power. Experience reveals another important behavior. When part of a batch of palladium can be made nuclear active, the remainder of the batch is found to be active. This activation treatment does not simply involve reaction with D but also requires extended electrolysis and/or repeated deloading and loading with D. This behavior is important because it revels a condition can be created throughout an entire batch of Pd as result of a common treatment that can eventually host the LENR process. In other words, the physical treatment before reacting with deuterium affects later initiation of LENR. Once the nuclei are assembled in the NAE, a unique process must reduce the Coulomb barrier perhaps by a tunneling mechanism without using energy beyond that which is normally available at room temperature. Immediately, we are confronted by a problem. Normal chemical structures are known not to support nuclear reactions without significant energy being applied to bombarding ions. After all, the Coulomb barrier keeps nuclei separated and allows chemical structures to form in the first place by interaction between the electrons. The energy required to force the nuclei close enough to fuse is well in excess of the energy holding the atoms in the structure and in excess of the electron energy. This well-known and accepted behavior suggests a need to form a novel arrangement between the nuclei in the NAE designed to avoid this limitation. In summary, two separate processes have to be considered. The first is creation of the NAE. The second is formation of a structure of H and/or D within the NAE having the ability to fuse. This nuclear process is separate from the structure of the NAE, but needs to be consistent with it. A description of the fusion process is a job for physics while identification of the NAE is a job for chemistry. Thus, we are forced to acknowledge an uncomfortable marriage between two normally independent branches of science, with chemistry being applied first to identify the NAE. Nature of the NAE Two different kinds of NAE have been suggested. Many researchers place the LENR process in the normal crystal structure where vacancies or dislocations might be present. Different variations of the crystal lattice are proposed, including formation of nanoparticles and active sites on the surface of the structure. In contrast, Storms(34) places the NAE in cracks having a critically small gap, which are separate from and chemically independent of the crystal structure. Such an environment can have properties much different from a crystal structure, including a high negative charge. Resolving this fundamental difference in proposed location of the NAE is critical to understanding the LENR process because the chosen location sets the logic on the correct path to discovering the mechanism. A choice of the wrong path will result in arriving at the wrong understanding. In order to contrast these two proposed conditions, the well documented suggestion by Hagelstein et al.(35) is explored. The Hagelstein idea is based on formation of a new phase in the normal fcc structure, such as suggested by Fukai and Okuma(36). This phase is proposed to form on occasion after deuterium content has exceeded D/Pd=0.85, thereby causing formation of palladium atom vacancies. Deuterium atoms fill the vacant sites and form a structure in which fusion is proposed to occur. The resulting mass-energy is dissipated by phonons. Evidence for this proposed phase change could be obtained by searching for a discontinuity in the various properties. As noted above, such a search reveals no evidence for a phase change within the composition range of the beta phase. In addition, X-ray and neutron diffraction studies of the face-centered-cubic structure reveal no phase change in this composition range. Using a similar argument, all the other explanations of LENR involving changes in gross structure can be rejected. The NAE is apparently a feature outside of the thermodynamic behavior and its presence does not affect the measured physical properties. While arguments based on the absence of behavior are usually ignored, in this case the failure of the physical properties to respond to the change required to form a NAE is an important characteristic of the LENR process. The author, in several previous papers (37-39), proposes the NAE resides in nanocracks resulting from stress relief. These gaps exist outside of the chemical properties and are not influenced by the limitations imposed by the chemical structure. As long as a gap having a critically small width is created, deuterons are proposed to enter the gap and form a structure that can be described in many different ways. This structure then experiences fusion by a novel mechanism. The required gap width is rarely created because most cracks would quickly become too wide to host the required hydrogen structure. Success in creating the NAE involves creating modest stress and applying it to a structure containing many weak regions having similar ability to form small cracks. This condition might be created by accident as result of various intended and accidental treatments applied during a study, thus accounting for occasional success without apparent reason. Although large cracks are often seen when LENR occurs, the cracks having the ability to act as the NAE are too small to be easily detected and can be overlooked. In fact, unless these structures are sought using high magnification, they would be impossible to detect. Experience shows the critical initial condition can also be created in a batch of material by a yet to be identified pretreatment. This realization encourages the search for such a treatment from which production of large amounts of nuclear active material can be expected to result. Deciding which explanation should be explored is important because they each propose entirely different treatments to cause the LENR process. The wrong choice of explanation can lead a researcher down the wrong path with much wasted effort. Power production The LENR effect was first identified by its ability to produce energy in amounts greater than would be possible by any chemical reaction. This energy has been produced when Pd is used as the cathode in an electrolytic cell using an electrolyte consisting of D2O+LiOD. When a Pd cathode is initially subjected to this treatment, the deuterium concentration in the Pd increases while energy is absorbed by the reaction, as shown in Fig. 4. Energy is absorbed because the energy used to decompose the D2O into D2 and O2 is greater than is recovered when the resulting D2 reacts with Pd, thereby causing an overall endothermic reaction. FIGURE 4. The D/Pd ratio and resulting power when Pd is reacted with D2O using the electrolytic method. All D made available by the applied current initially reacts with the Pd. The amount reacted is reduced only gradually as the upper limit is reached. No excess energy is produced even after the average D/Pd ratio becomes very large. The total amount of energy/mole Pd absorbed by the process is noted. (Storms, www.LENRexplained.com) The enthalpy of formation for deuterium can be calculated using the data in Fig. 4. For this purpose, the total amount of D reacted every six minutes is divided by the amount of energy absorbed during this time, from which the amount of energy used to decomposed the D2O is subtracted. As can be seen in Fig. 5, the electrolytic method applied to a solid piece of Pd gives values for the partial enthalpy of formation similar to the values obtained when D2 is reacted directly with Pd nanopowder. Both reactions show that chemical energy is released when Pd reacts with D2 and the amount decreases as the D/Pd ratio increases. FIGURE 5. Enthalpy of formation calculated using the data shown in Fig. 4 based on the amount of D reacted every 6 minutes, the amount of power measured during this time, and the amount of energy used to decompose the D2O from which the D results. The reaction of D2 with Pd is exothermic. The Sakamoto et al. (40) line is obtained using their reported linear equation, which is then extrapolated from D/Pd= 0.85 to 0.98, and their reported D2 pressure. The pressure of D2 is also obtained from the review by Santandrea and Behrens(41). (Storms, www.LENRexplained.com) The equilibrium deuterium activity, as pressure, is also plotted to show the large range in values being applied to the material. This quantity can be described as pressure only when the gas phase is present and is in equilibrium with the solid. The deviation from ideal behavior, called fugacity, is not taken into account. No excess energy was produced even though a very high D/Pd ratio was reached. Additional treatment was later required to start the LENR process. No additional phase forms in this composition range, such as proposed by Fukai, as indicated by the smooth unbroken variation of ΔH and pressure. Also, the smooth unbroken change in resistivity observed by McKubre et al.(22) while LENR took place is consistent with this conclusion. The effect of temperature on power production for various D/Pd ratios is compared in Fig. 6. Samples having D/Pd = 0.80 and 0.48 produce the same amount of power at the same temperature. Removal of all deuterium stops power production. Clearly, power is not as sensitive to the deuterium content as previous studies suggest(42). Nevertheless, some D is required for LENR to function. The Arrhenius plot (Fig. 7), using the data in Fig. 6 (D/Pd=0.8), shows the activation energy for the LENR process to be nearly equal to the value for diffusion of D in PdD. In other words, the rate of the fusion process is sensitive to the rate at which D can get to the site where fusion takes place and it is not sensitive to the concentration of D in the surrounding lattice. The fusion process can be proposed to rapidly convert deuterium in the NAE to fusion products, after which new D has to move relatively slowly from the surrounding lattice in order to supply additional fuel to the active sites. The rate of energy production is determined by the rate at which D can get to the NAE. By analogy, this is similar to the speed of a car being determined by how fast gas is delivered to the engine and not related to the amount of gas in the tank or the reaction rate within each cylinder. The resulting equation allows the resulting power to be predicted when temperature is increased. FIGURE 6. Effect of temperature on power production when three different amounts of deuterium are present in the sample. (Storms, www.LENRexplained.com) FIGURE 7. Comparison between the rate of diffusion of D in PdD and production of LENR power as a function of 1/T. The similar slopes created by the data suggest both processes are affected by the same mechanism, i.e. diffusion of D though PdD. (43) Probability of forming the NAE Figure 8 compares power produced by 157 studies reported before 2007. Notice that most studies produce power at relative low levels. On a few occasions, a large amount of power is observed with the number of reports rapidly decreasing as the reported power increases. The number of reports, shown in Fig. 8 can be compared to FIGURE 8. Histogram of power production vs the number of reported values. A probability function, shown as the dashed line, is used to fit the data to bins at 10 watt intervals. predicted behavior based on an assumed probability of causing increased power once power production is possible. In other words, the probability of forming additional NAE once the conditions allow some NAE to form can be estimated and compared to the behavior to see if the assumption fits. If 300 attempts are made to initiate LENR and the probability of producing 10 watts is 0.3, the probability of producing 20 watts would be 0.3×0.3, and the probability of producing 30 watts would be 0.3×0.3×0.3 etc. The number of predicted successful observations at each power level is shown by the dashed line. The relatively good fit to the observed behavior suggests the power is caused by an increasing number of active sites whose production is caused by a random process, with more power resulting as the number of NAE sites is increased by a process having low probability. The probability of producing any power at all would be expected to be much less than producing additional power once conditions allow some NAE to form. Continue reading How basic behavior of LENR can guide a search for an explanation How basic behavior of LENR can guide a search for an explanation pdf TOC INTRODUCTION The nature of the hot fusion mechanism Role of chemical structure IMPORTANT OBSERVED BEHAVIOR Formation of the NAE Nature of the NAE Power Production Probability of forming the NAE Helium Production Tritium Production Transmutation Production SUMMARY OF BEHAVIOR CREATING A THEORY ASSUMPTIONS LENR Initiation as a Chemical Reaction Nuclear Process Applied to LENR Role of h4 formation Consequence of LENR using a mixture of d and p Transmutation How does the fusion process work? Reduction of Coulomb barrier Dissipation of excess mass energy Photons as the energy dissipation method Electrons as the energy dissipation method Phonons as the energy dissipation method Storage of energy after fusion Effect of different variables TESTABLE PREDICTIONS Creation of the NAE DISCUSSION SUMMARY by ## Shift theoretical focus from nuclear consequences to chemical beginnings I would like to emphasize one other aspect of LENR that is frequently overlooked: Fusion can be caused by two different mechanisms. The common one, called hot fusion, involves applying high energy to the reacting nuclei. This approach takes advantage of the increased reaction rate applied energy provides. Science has ignored what happens when fusion takes place at low energy because the rates are too low to study the process. Discovery of LENR, called cold fusion, has revealed how the fusion rate can be increased without using applied energy. However this process requires a unique condition I identify as the NAE. As many people have noted, the NAE acts like a catalyst so that applied energy is no longer required. This being the case, the essential understanding of the LENR process resides in the nature of the NAE. Creation of the NAE makes LENR possible and the unique fusion mechanism operates only within the confines of the NAE. A condition is created within the NAE in which the Coulomb barrier can be overcome without applied energy, and mass-energy can be converted to heat energy without producing the high-energy radiation normally associated with nuclear interaction. The magic of LENR takes place in and only in the NAE. This concept does not conflict with or violate any physical law because this unique condition has yet to be explored by science. This is virgin territory having no relationship to hot fusion or to the concepts obtained from studies of the hot fusion mechanism. Therefore, identifying and describing the NAE is essential to creating a useful theory about LENR. Because the NAE is part of a chemical structure, the chemical conditions must be part of this understanding. Unlike hot fusion, which occurs in plasma, cold fusion is strongly influenced by the chemical properties of the material in which NAE forms. Most theories mistakenly ignore the chemical requirements. Without a chemical structure and its chemical behavior, the NAE cannot form and LENR cannot occur. Consequently, LENR is first and foremost a chemical process with nuclear consequences. Thus, the focus should be shifted from the nuclear consequences to those conditions required to form the NAE and to its role in hosting the nuclear reactions. This idea might be a bridge too far for many theoreticians, nevertheless, I strongly suggest an effort be made to cross the bridge rather than keep trying to swim the river. by	{"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": 0.8177908658981323, "perplexity": 970.0862046853388}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573104.24/warc/CC-MAIN-20220817183340-20220817213340-00579.warc.gz"}
https://uwaterloo.ca/library/geospatial/population-growth-waterloo-through-1867-present	# Population Growth of Waterloo through 1867 to the present In celebration of Canada's 150th anniversary, the Geospatial Centre has created a graphic visualizing the population growth of Waterloo from 1855 to 2016. Using ArcGIS Online's storybook application, you can see the increase in Waterloo's population over the years. The historical maps that were used to create the population density maps can be seen in the storymap. View story map	{"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": 0.9005299806594849, "perplexity": 2520.941796723178}, "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-2020-45/segments/1603107907213.64/warc/CC-MAIN-20201030033658-20201030063658-00565.warc.gz"}
https://researchportal.port.ac.uk/en/publications/a-search-for-gravitational-waves-from-binary-mergers-with-a-singl	# A search for gravitational waves from binary mergers with a single observatory Alexander H. Nitz, Thomas Dent, Gareth S. Davies, Ian Harry Research output: Contribution to journalArticlepeer-review 16 Downloads (Pure) ## Abstract We present a search for merging compact binary gravitational-wave sources that produce a signal appearing solely or primarily in a single detector. Past analyses have heavily relied on coincidence between multiple detectors to reduce non-astrophysical background. However, for $\sim40\%$ of the total time of the 2015-2017 LIGO-Virgo observing runs only a single detector was operating. We discuss the difficulties in assigning significance and calculating the probability of astrophysical origin for candidates observed primarily by a single detector, and suggest a straightforward resolution using a noise model designed to provide a conservative assessment given the observed data. We also describe a procedure to assess candidates observed in a single detector when multiple detectors are observing. We apply these methods to search for binary black hole (BBH) and binary neutron star (BNS) mergers in the open LIGO data spanning 2015-2017. The most promising candidate from our search is 170817+03:02:46UTC (probability of astrophysical origin $p_{\rm astro} \sim 0.4$): if astrophysical, this is consistent with a BBH merger with primary mass $67_{-15}^{+21}\,M_{\odot}$, suggestive of a hierarchical merger origin. We also apply our method to the analysis of GW190425 and find $p_{\rm astro} \sim 0.5$, though this value is highly dependent on assumptions about the noise and signal models. Original language English 169 9 The Astrophysical Journal 897 2 https://doi.org/10.3847/1538-4357/ab96c7 Published - 15 Jul 2020 • astro-ph.HE • gr-qc • GW_HIGHLIGHT ## Fingerprint Dive into the research topics of 'A search for gravitational waves from binary mergers with a single observatory'. Together they form a unique fingerprint. • ### Data availability statement for 'A search for gravitational waves from binary mergers with a single observatory'. Nitz, A. H. (Creator), Dent, T. (Creator), Davies, G. S. (Creator) & Harry, I. (Creator), IOP Publishing, 15 Jul 2020 Dataset	{"extraction_info": {"found_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": 0.8433706760406494, "perplexity": 3055.4310469346583}, "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-2022-21/segments/1652662526009.35/warc/CC-MAIN-20220519074217-20220519104217-00010.warc.gz"}
http://math.stackexchange.com/questions/294660/trying-to-prove-let-e-be-a-hilbert-a-module-then-e-langle-e-e-rangle-i	# Trying to prove: Let $E$ be a Hilbert $A$-module. Then, $E\langle E,E\rangle$ is norm dense in $E$. Let $E$ be a Hilbert $A$-module. Then, $E\langle E,E\rangle$ is norm dense in $E$. I am having trouble proving this. I believe $\langle E,E\rangle$ is a $C^$-algebra. If I can show this, then the proof is easy since all $C^$-algebras have an approximate identity. It seems like it shouldn't be too hard, but I am having trouble showing it. Thank you. - What precisely do the notations $E\langle E,E\rangle$ and $\langle E,E\rangle$ refer to? In any case, a stronger statement can be found here: math.stackexchange.com/questions/163485/… – Jonas Meyer Feb 4 '13 at 17:14 They are in fact equal See lemma 2.2.3. of book Hilbert C-Modules by M. Manuilov page20 But I only point out that $\langle E,E\rangle$ is a C-subalgebra of A and so $E\langle E,E\rangle\subset E$; conversely since any x in E can be written as $x=y\langle y,y\rangle$ we have $E\subset E\langle E,E\rangle.$ - For those who don't have that book, could you describe a bit about what that citation says? – robjohn Feb 5 '13 at 7:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9729763269424438, "perplexity": 191.19993694913248}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010607072/warc/CC-MAIN-20140305091007-00076-ip-10-183-142-35.ec2.internal.warc.gz"}
https://stacks.math.columbia.edu/tag/0FQU	## 24.2 Conventions In this chapter we hold on to the convention that ring means commutative ring with $1$. If $R$ is a ring, then an $R$-algebra $A$ will be an $R$-module $A$ endowed with an $R$-bilinear map $A \times A \to A$ (multiplication) such that multiplication is associative and has an identity. In other words, these are unital associative $R$-algebras such that the structure map $R \to A$ maps into the center of $A$. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9559610486030579, "perplexity": 229.63483383530047}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764499831.97/warc/CC-MAIN-20230130232547-20230131022547-00369.warc.gz"}
https://www.physicsforums.com/threads/calculating-voltage-which-voltmeter-is-showing.900427/	Calculating voltage which voltmeter is showing Tags: 1. Jan 15, 2017 akaliuseheal 1. The problem statement, all variables and given/known data Schematic of circuit is not given, only the text which I translated into English. Using a voltmeter with internal resistance of 6k ohm, voltage between two points, 1 and 2 of a circuit of constant current, is measured to be 8v. Then, using a voltmeter with internal resistance of 10k ohm, between the same points, voltage is measured to be 12v. What voltage will the voltmeter with internal resistance of 15k ohm measure? 2. Relevant equations I=U/R 3. The attempt at a solution I tried to solve it by calculating currents getting 1.3mA and 1.2mA in first and second case, but wasn't sure what to do next. These voltmeters should be represented as resistors and the answer is 16V. 2. Jan 15, 2017 Staff: Mentor You're told that the current in the circuit is constant, which I take to mean that the current leaving node 2 is the same as the current entering node 1: So $I$ is an unknown constant and $R$ is some unknown resistance lying between nodes 1 and 2. That's two unknowns. Fortunately you were given two cases where the meter resistance and the measured voltage are given, so you can construct two equations in two unknowns. 3. Jan 15, 2017 akaliuseheal I would like to thank you for looking into this but I am unsure on how to do that. I mean, how would system of equations look like? 4. Jan 15, 2017 Staff: Mentor Start symbolically: don't plug in any numbers, just use variables. See if you can write an expression for the voltage $V$ in terms of $I$, $R$ and $R_m$. Or, write an expression for $I$ in terms of $V$, $R$ and $R_m$. Either way is fine (although the latter may be more straight forward). 5. Jan 15, 2017 akaliuseheal So like voltage/current divider? V = (Rm \|\| R) * I 6. Jan 15, 2017 Staff: Mentor Sure, that would work. Or since the voltage is the same across both resistors it's easy to write the sum of the currents: $I = \frac{V}{R} + \frac{V}{R_m}$ Whatever you are more comfortable with. 7. Jan 15, 2017 akaliuseheal So it's like this. R=30k ohm I = 0,0016A Replacing those values gets me the voltage of 16V. Thanks, was struggling with this trivial problem. 8. Jan 15, 2017 Staff: Mentor Draft saved Draft deleted Similar Discussions: Calculating voltage which voltmeter is showing	{"extraction_info": {"found_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": 0.9061651825904846, "perplexity": 855.142365621211}, "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/1512948515165.6/warc/CC-MAIN-20171212041010-20171212061010-00577.warc.gz"}
https://scholarship.rice.edu/handle/1911/102242?show=full	dc.contributor.author Huang, Yin 2018-06-19T17:50:45Z 2018-06-19T17:50:45Z 2016-05 Huang, Yin. "Born Waveform Inversion in Shot Coordinate Domain." (2016) https://hdl.handle.net/1911/102242. https://hdl.handle.net/1911/102242 This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/96240 The goal of this thesis is to integrate Born waveform inversion, variable projection algorithm and model extension concept to get a method that can improve the long scale background model updates reliably and efficiently from seismic data. Born waveform inversion is a partially linearized version of full waveform inversion based on Born (linearized) modeling, in which the earth model is separated into a smooth long scale background model and an oscillatory short scale reflectivity and both are updated to fit observed trace data. Because kinematic variables (background model) are updated, Born waveform inversion has the same feature as full waveform inversion: very sensitive to initial model when solved by gradient based optimization method (almost the only possible method because of the problem scale). Extended Born waveform inversion allows reflectivity to depend on additional parameters, potentially enlarging the convexity domain by enlarging the searching model space and permitting data fit throughout the inversion process and in turn reducing the sensitivity to initial model. Extended or not, the Born waveform inversion objective function is quadratic in the reflectivity, so that a nested optimization approach is available: minimize over reflectivity in an inner stage, then minimize the background-dependent result in a second, outer stage, which results in a reduced objective function of the background model only (VPE method). This thesis integrates the nested optimization approach into an inversion scheme and analyzes that the accuracy of the solution to the inner optimization is crucial for a robust outer optimization and both model extension and the nested optimization are necessary for a successful Born waveform inversion. And then we propose a flexibly preconditioned least squares migration scheme (FPCG) that significantly improves the convergence of iterative least squares migration and produces high resolution images with balanced amplitude. The proposed scheme also improves the efficiency of the solution to the inner stage of the nested optimization scheme and the accuracy of the gradient, and thus potentially improves the efficiency of the VPE method. However, a theoretical error estimate in the gradient computation of the VPE method is still hard to obtain, and we explain the reason and illustrate with numerical examples. 133 pp Born Waveform Inversion in Shot Coordinate Domain Technical report May 2016 TR16-02 Text	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8484936356544495, "perplexity": 1407.6366333855738}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316021.66/warc/CC-MAIN-20190821131745-20190821153745-00319.warc.gz"}
https://www.physicsforums.com/threads/work-constant-pressure-changing-volume.145847/	# Work- constant pressure, changing volume 1. Nov 28, 2006 ### physics1234 Find the work done by a gas when it is expanded from a volume of 1.0 L to 3.3 L at a constant pressure of 2.8 atm. I converted the atm to Pascals by multiplying 2.8 x 101325 to get 283710. Then I used the formula W=p(deltaV) so 283710 x (3.3-1.0) = 652533 I don't understand why this is the wrong answer. 2. Nov 28, 2006 ### physics1234 Nevermind, I got it :-) Thanks anyway! 3. Jul 25, 2010 ### ahmed-shaqlaw hi brother you must convert the volume to cubic meter m3 , and to convert the pressure to kpa not pascal , by multiplying by 101.3 , try the new answer.... sorry my english is not so good	{"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": 0.9210379123687744, "perplexity": 3131.320628585476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560282937.55/warc/CC-MAIN-20170116095122-00235-ip-10-171-10-70.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/323795/the-number-of-non-trivial-polynomial-solutions-of-the-differential-equation-x3	# The number of non-trivial polynomial solutions of the differential equation $x^3y'(x)=y(x^2)$ I came across the following problem that says: The number of non-trivial polynomial solutions of the differential equation $x^3y'(x)=y(x^2)$ is which of the following? $(1)0\space (2)1 \space (3)3 (4)\infty.$ Can someone point me in the right direction? Thanks in advance for your time. - Does the $y(x^2)$ on the right side mean $y \cdot x^2$, or on the other hand that the function $y$ is evaluated at input $x^2$? – coffeemath Mar 7 at 17:44 It is the function $y$ which is evaluated at input $x^2.$ – user52976 Mar 7 at 17:46 The only polynomial that could fit such a description is quadratic. So try $y=ax^2+bx+c$ and see the resulting system of equations by setting the coefficients of both sides equal. – Maesumi Mar 7 at 17:51 For a polynomial of order $n$, this reads: $$3+(n-1)=2n \to n=2$$ Write this out: $$2ax^4+bx^3=ax^4+bx^2+c \to ax^4+bx^2(x-1)-c=0$$ So, the answer is zero, since no such non-trivial polynomials exist (non-trivial forth degree polynomials have at most 4 real roots). Well, if you proceed using the ansatz (thanks @user33640 for the "correction") $$y(x) = \sum_{n=0}^N a_n x^n$$ and substitute it directly in the equation, you end up with $$x^3 \sum_{n=1}^N n a_n x^{n-1} = \sum_{n=0}^N a_n x^{2n}.$$ This means that $$\sum_{n=1}^N n a_n x^{n+2} - \sum_{n=0}^N a_n x^{2n} = 0$$ which tells you that all $a_n = 0$ for $n=2m+1$. This in turn help us to propose an improved ansatz $$y(x) = \sum_{n=0}^N b_{2n} x^{2n}.$$ Again, substituting in the ode, we have $$\sum_{n = 1}^N 2n b_{2n} x^{2n+2} = \sum_{n=0}^N b_{2n} x^{4n}$$ Now, the only way to this equation to be satisfied, is if $$2N + 2 = 4N \quad \Longrightarrow \quad N=1$$ Then $y(x) = b_0 + b_2 x^2$ implies that $$2 b_2 x^4 = b_0 + b_2 x^4$$ which means $b_0 = b_2 = 0$ hence no non-trivial solution 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": 0.9422094821929932, "perplexity": 176.69106995707605}, "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-2013-48/segments/1387345758566/warc/CC-MAIN-20131218054918-00022-ip-10-33-133-15.ec2.internal.warc.gz"}
https://worldofinterest.wordpress.com/2013/03/06/is-equity-cheap/	# Is Equity Cheap? This is the question which is dominating financial media at the moment. Case-Schiller point out that their cyclically adjusted PE ration (CAPE10), is above historically adjusted norms. My thesis in this post is that the CAPE10 actually indicates that the stock market is cheaply-fairly valued depending on the expected future path of interest rates. In fact, I am going to use Shiller’s own data to explain why I think he is wrong. Equity Valuations The most theoretically sound way to value equities is the present discounted cash flow. That is to say, the current intrinsic value of a stock is the sum of all future profits discounted by the risk free rate. Of course, we can never be sure about the future, so is born the concept of a risk premium, that equities should be valued slightly lower than their fair value because we have a high degree of uncertainty about the future. My preferred way of saying this in maths is: $\frac{\text{Price}}{\text{current earnings}} = \sum^{\infty}_{t=0} \exp((-R+G-\rho)t) \approx \frac{1}{R-G+\rho}$ where we have defined that R is the risk free interest rate, G is the growth of the company, and $\latex \rho$ is the equity risk premium, which is at heart, the uncertainty about growth and interest rates. The approximation follows from the formula for a geometric sum if one assumes that $R-G+\rho$ is small. Obviously, such a valuation builds in some fairly crucial assumptions. Firstly, that growth is smaller than the interest rate and the risk premium. A share whose earnings growth continually out earns interest rate and uncertainty premiums is worth infinity. Of course, this occurs because we assumed growth was a fixed parameter, which it is not, but such simplifications are sufficient for our current purpose. Using the Case Shiller data, and using the above analysis, I have plotted the PE ratio for ten year US treasuries along side the CAPE10 ratio for the S&P500, since 1950. Notice the incredible correlation between bond returns and equity returns over the proceeding decades as tail risks of WW2, the cold war, and thermonuclear war abated. So the question, “are stocks cheap”, is now seen in its proper context. Stocks are cheap if you expect interest rates to stay at their current levels for an extended period. If you expect that interest rates will normalise around 5% in the not to distant future, then stocks are fairly valued. I see think that interest rates are still some years away from 5%, and see rather evidence of continued tail risk from the Eurozone’s self inflicted misery. Notice quite what an aberration the tech bubble was. Even at the end of the dot com crash stocks were not especially cheap compared to bonds. Notice that stocks were incontrovertibly cheap post financial crisis in 2009. Anyway, I am cautiously optimistic in the long run/medium run for stocks, but still worried about the Eurozone. The ongoing crises in Europe seems temporarily in the background of investor’s minds, either because they believe that the ECB will bow to pressure and follow the other central banks, or because they believe that liquidity from other parts of the world will alleviate Europe’s problems despite ECB intransigence. I would not bet on either. Disclosure, I am 30% in cash, 70% long equities, with 0% European exposure.	{"extraction_info": {"found_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": 2, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8059871792793274, "perplexity": 1216.226766817228}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187826049.46/warc/CC-MAIN-20171023130351-20171023150351-00179.warc.gz"}
https://www.physicsforums.com/threads/find-electric-field-in-these-regions-of-a-spherical-shell.906110/	Find ELectric field in these regions of a spherical shell 1. Mar 2, 2017 grandpa2390 1. The problem statement, all variables and given/known data a thick spherical shell carries charge density k/r^2 a<r<b find E in the three regions r<a a<r<b b<r 2. Relevant equations E dot da = Q/ε 3. The attempt at a solution I can't understand why, when integrating, they choose for ii to integrate between a and r, iii and the between a and b for iii 2. Mar 2, 2017 haruspex What do you know about the field inside a uniformly charged spherical shell? 3. Mar 2, 2017 grandpa2390 that it's uniform at the surface? I don't know what you are asking. what I have been able to gather is that they are integrating the volume of the shell. integral of 4 pi r^2 dr for a<r<b the volume is from a - r for b<r the volume is from a-b I don't know why. 4. Mar 2, 2017 haruspex No, that there is no field produced inside a uniformly charged spherical shell. This is a fundamental result of enormous importance in these problems. The equally important result for outside the shell is that the field there is the same as if all of the charge were concentrated at the sphere's centre. The same pair of results applies (of course) to gravitational fields from uniform spherical mass distributions. Can you see how this explains the integration range? 5. Mar 2, 2017 grandpa2390 We are trying to capture all of the "mass" below our boundary. Between A and B we want to capture all the mass from a to wherever r is. If R is greater then B then we want all of the "mass" less then r which is from a to b ??? 6. Mar 2, 2017 haruspex Hence the integration range from a to r. Hence the integration range from a to b. Please try to explain more clearly what it is that you do not understand. 7. Mar 2, 2017 grandpa2390 No you answered it. Or at the very least, you slapped some sense into my brain, pointed... pushed my brain into the right direction. I don't know. When you compared it to mass, it just made sense to me suddenly. I don't know. I was thinking it should have been integrated between the boundaries stated. integrated from a to b, and then from b to infinity. I didn't get it until your last reply :) Then I just restated what I got from you in my own words for verification to make sure whether I had it : ) 8. Mar 2, 2017 haruspex Verified. Draft saved Draft deleted Similar Discussions: Find ELectric field in these regions of a spherical shell	{"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": 0.8575940132141113, "perplexity": 869.5839187643285}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891811794.67/warc/CC-MAIN-20180218062032-20180218082032-00335.warc.gz"}
https://www.physicsforums.com/threads/definitions-and-properties-of-limits-handwriting-attached.674672/	# Definitions and properties of limits (handwriting attached) 1. Feb 26, 2013 ### tolove I'm not entirely sure on the properties of limits, but this seems to work. Could someone look over this for me? http://imgur.com/6zCHYo5 2. Feb 26, 2013 ### SammyS Staff Emeritus What is it that you're trying to do? #### Attached Files: • ###### 6zCHYo5.jpg File size: 80.1 KB Views: 102 3. Feb 26, 2013 ### tolove Not really a problem here, just wanting to make sure I'm doing this correctly. I'm trying to show that ∫ y' dx = ∫ dy through definitions. 4. Feb 26, 2013 ### SammyS Staff Emeritus What are you using for the definition of the indefinite integral? The Riemann sum is generally used for the definite integral. Draft saved Draft deleted Similar Discussions: Definitions and properties of limits (handwriting attached)	{"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": 0.980728268623352, "perplexity": 3599.5802276792374}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105291.88/warc/CC-MAIN-20170819012514-20170819032514-00253.warc.gz"}
https://cs.stackexchange.com/questions/103502/how-to-prove-that-x-is-a-regular-language-if-x-is-derived-from-l	# How to prove that $\{\$x\$\}$ is a regular language if $x$ is derived from $L=\{w\}$ by substituting substrings? Prove that if $$L$$ is regular over $$\Sigma=\{0,1,2\}$$ then the following language over $$\{0,1,2,\\}$$ is also regular: $$G=\{\x\\|\exists w\in L: x\text{ is derived from }w\text{ by substituting } 01 \text{ with }\\ \}$$ For example, if $$10112\in L$$ then $$\1\\12\\in G$$. I think this can be solved using closure properties of regular languages. 1) Let $$H=\L\$$. $$H$$ is also regular because it's concatenation. 2) Let $$h:\Sigma\to \Sigma^$$ be defined as follows: $$h(\)=h(0)=h(1)=\$$ Then: $$G=h^{-1}(H)\cap $$\Sigma^\\\Sigma^)^+\$$ which is regular because $$h^{-1}$$, intersection and regular expressions are closed in regular languages. I wonder if my proof using closure is correct or automaton should be built in this case? In addition if I managed to think of a regular expression to describe $$G$$ then this alone would've proved that $$G$$ is regular? ## 1 Answer Unfortunately, your argument breaks down at step 2). For example, let $$L=\{01\}$$. Then $$G =\{\\\\\}.$$ $$H=\{\01\\}$$. $$h^{-1}(H)=\emptyset$$ since every word in the range of $$h$$ contains neither 0 or 1. $$h^{-1}(H)\cap $\Sigma^\\\Sigma^*)^+\=\emptyset.$$ However, $$G$$ is not empty. If I managed to think of a regular expression to describe G then would this alone have proved that G is regular? Of course. However, it looks like it is not immediate to figure out a regular expression for $$G$$ even if we have been given the regular expression for $$L$$ and DFA for $$L$$. Here is a way to show $$G$$ is regular by DFA. Let the DFA for $$L$$ be $$(\Sigma,Q, q_0,\delta_L, F)$$. Define (an incomplete) DFA $$D$$ with alphabet $$\{0,1,2,\\}$$, states $$Q\times \{s_0, s_1, o\}$$, initial state $$(q_0, s_0)$$, accepting state $$F\times\{s_0, o\}$$, transition function $$\delta_D$$ such that $$\delta_D((q, s_0), 0)= (\delta_L(q, 0), o)$$ $$\delta_D((q, s_0), 1)= (\delta_L(q, 1), s_0)$$ $$\delta_D((q, s_0), 2)= (\delta_L(q, 2), s_0)$$ $$\delta_D((q, s_0),$$= (\delta_L(q, 0), s_1)$$ $$\delta_D((q, s_1), $= (\delta_L(q, 1), s_0)$$ $$\delta_d((q, o), 0)= \delta_L(q, 0), o)$$ $$\delta_d((q, o), 2)= \delta_L(q, 2), s_0)$$ $$\delta_d((q, o), \)= \delta_L(q, 1), s_1)$$ Here is how we can understand the states • state $$(q,o)$$ corresponds to the states that are reached by words that end with $$0$$. • state $$(q,s_1)$$ corresponds to the states that are reached by words that end with odd number of $$\$$'s. • state $$(q,s_0)$$ corresponds to the states that are reached by words that end with $$1$$ or $$2$$ or even number of $$\$$'s. We can check that $$G=\L(D)\$$. Exercise. Prove that if $$L$$ is regular over $$\Sigma=\{0,1\}$$ then the following language over $$\{0,1,2\}$$ is also regular. $$G=\{x\mid \exists w\in L: x\text{ is derived from }w\text{ by substituting } 00 \text{ and } 11 \text{ with } 2 \text{ from left to right} \}$$ For example, if $$w=1000111110$$, then $$x=1202210$$. • Can you please explain why do you have states based on what letter reached the end of the words? For example, I'd define the transition functions as follows: $$\delta_D((q,2),0)=(\delta(q,2),0)\\\delta_D((q,\),0)=(\delta(q,0),1)\\\delta_D((q,\),1)=(\delta(q,1),0)$$.In my example we start out with state $0$. When we read the first $\$$we flip the state to 1. When we read the second \$$ we flip the state back to$0$. When we read$2$nothing changes so we stay in state$0$as well. In the end we can add$\$$'s from the beginning and end of each word. – Yos Jan 28 '19 at 20:38 • If we know the last letters of the words that reaches a state s, then we can specified what will be the next state if the word is extended with another letter. Note that we must ensure words containing 01 and words with isolated \$$ should not be accepted. Jan 28 '19 at 20:47 • Exactly. That is why incomplete DFA is popular since there is no need to write any transition that goes to a dead state. In fact, we do not have to specify that dead state. Jan 28 '19 at 20:58 • I understand the transitions now, thanks! I still don't why you concatenate dollar sign here: $\delta_D((q, s_1), \$)= (\delta_L(q, 1), s_0)\? – Yos Jan 28 '19 at 21:05 • Updated with a few corrections of typos. 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https://www.physicsforums.com/threads/what-type-of-of-force-is-applied-force-conservative-nonconservative.768294/	# What type of of force is applied force? Conservative ,Nonconservative 1. Aug 30, 2014 ### Miraj Kayastha In a closed loop when we apply an applied force on an object the object starts at point A and stops at point A. Since the displacement is 0, Work done by the applied force on the object is = F x s x cosθ = 0 So the net work done by the applied force is 0 but why is applied force a non-conservative force? 2. Aug 30, 2014 ### ehild When the force is not constant during the motion the work is calculated by integration. Have you studied calculus? ehild 3. Aug 30, 2014 ### Miraj Kayastha Yes I have. 4. Aug 30, 2014 ### ehild The work done along a curved path is $W=\int F_s d_s$ where Fs is the component of the force tangent to the segment of the curve and ds is the length of the line segment. If you apply a force of constant magnitude, always tangent to the curve, $W=F \int d_s =F L$ where L is the length of the curve. The work is not zero along a closed loop. If you push a crate along a rough horizontal surface with force F you do FD work during D distance. When you push it back with force of the same magnitude, you do FD work again. The displacement is zero, but the net work is not. In general, the work depends on the way it is done. The conservative forces are exceptions, their work does not depend on the path taken between the initial and final points. ehild Last edited: Aug 30, 2014 5. Aug 31, 2014 ### Miraj Kayastha Here when you said "when we push it back with the same magnitude, you do F x D work again" I think we should also account the direction of the force because the force is constant both in magnitude and direction throughout the motion. So shouldn't the work done by the force on the crate when crate is coming back be - F x D ? And then the net work is zero. 6. Aug 31, 2014 ### jbriggs444 The D in the work formula is not a "distance". It is a "displacement". The distinction is that a distance is always positive and has no direction. A displacement has a direction. When you push the crate back toward the starting point, both force and direction are reversed. So it's -F x -D. 7. Aug 31, 2014 ### ehild The elementary work can be written either as dot product of the force vector with the displacement vector, $dW=\vec F \cdot \vec{dr}$ or as the product of two scalars, Fs the component of force along the path taken and the elementary path length, ds dW=Fsds. ehild	{"extraction_info": {"found_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": 0.9582477807998657, "perplexity": 471.80020650550864}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860127878.90/warc/CC-MAIN-20160428161527-00140-ip-10-239-7-51.ec2.internal.warc.gz"}
https://reciprocalsystem.org/paper/permittivity-permeability-and-the-speed-of-light-in-the-reciprocal-system	# PERMITTIVITY, PERMEABILITY AND THE SPEED OF LIGHT IN THE RECIPROCAL SYSTEM Introduction Physics textbooks do not provide a theoretical derivation of Newton‘s law of gravitation or Coulomb’s laws of electrostatics and magnetostatics; they are simply stated as empirical truths. By contrast, books on the Reciprocal System, such as Ref. [1], [2], [3], do provide a theoretical derivation of these laws. This paper will take a closer look at the terms of the electrostatic and magnetostatic equations--the terms of the gravitational equation have already been discussed in detail, most recently in Ref. [3]. Unlike the force of gravitation the forces of electrostatics and magnetostatics can be reduced by the intervening media between the charges. The question to be answered is: what are the dimensions of all the terms of Coulomb’s laws? Permittivity Coulomb’s law of electrostatic attraction is expressed as Q1 Q² Fe = ke ——— (1) r² where Q1 and Q² are the electric charges, r is the distance between them, Fe is the force, and ke is the proportionality constant. In the Reciprocal System all physical quantities are expressed in terms of space and time only; there are no separate dimensions for mass or charge. So, electric charge is not taken as a fundamental entity and given an independent unit. Larson has deduced that the dimensions of force and charge are Fe = [t/s²] Q = [t/s] In the gravitational expression, the second mass and the distance are considered to be dimensionless ratios. Such a procedure could be used in analyzing the electrostatic expression; however I find it more fruitful to treat the second charge and the distance as having dimensions. In this case Coulomb‘s law expressed in dimensions is [t/s] [t/s] [t/s²] = ke ———– (2) [s²] For this equation to be dimensionally correct, the dimensions of ke must be ke = [s²/t] (3) These are the dimensions of permittivity (in the Reciprocal System). However, the conventional expression for the coefficient of the law is ke = 1/(4 pÎ) (4) where Î is the permittivity, expressed in farads/meter. Thus the derivation gives a result that is the inverse of the usual coefficient. But in the Reciprocal System the farad is reducible to a length. So in the conventional units that are used, permittivity turns out to be dimensionless whereas physically it is not. It should not then be surprising that the numerical values of permittivity are inverted. Instead of saying that the permittivity of air is 1.0006 times that of free space, I would say that it is 1/1.0006 = .9994 times that of free space. This actually sounds better! (The other part of the coefficient (1/4p) of the conventional expression was put in for practical reasons having nothing to do with basic physics; there is no point to keeping it in the Reciprocal System). Of course, the end result-the calculated force--must be the same in both systems. Permeability Coulomb‘s law of magnetostatics is M1 M² Fm = km ——— (5) r² where M1 and M² are the magnetic charges, Fm is the force, r the separation distance, and km the proportionaiity constant. Larson has deduced that the dimensions of magnetic charge are M = [t²/s²] (6) Then, expressed dimensionally, Coulomb‘s law for magnetostatics is [ t²/s² ] [ t² /s² ] [t/s²] km ——————— (7) [s²] For this equation to be dimensionally correct, the dimensions of km must be km = [s4/t³] (8) But in the Reciprocal System magnetic permeability (symbol Ã) has the dimensions t /s . Thus km = 1/Ã . This time the derivation from the Reciprocal System is in exact accord with the conventional Kennelly system, with magnetic charges or poles expressed in webers (volt-sec). The only thing awkward here is the name “permeability” . On the basis of the equation, a better name for this quantity would be “impermeability” . Besides, as Larson has pointed out “permeability” is the magnetic analog of electric resistance (t² /s³ * t/s). Perhaps for parallelism with the revised electrostatic expression we should put the reciprocal of what is now called permeability into the numerator of the law but call it by the same name. Thus the higher the electric force between charges, the higher the (reciprocal) “permittivity” ; and the higher the magnetic force between magnetic charges, the higher the (reciprocal) “permeability” . Permittivity, Permeability, and the Speed of Light One way to confirm the identification of the dimensions of permittivity and permeability is to use them in the same expression. One such expression is Maxwell‘s famous result from electromagnetic theory: c = 1/(ÎoÃo )½ (9) where c is the speed of light, Îo is the permittivity of free space, and Ão is the permeability of free space. In the dimensional terms of the Reciprocal System, the equation is [s/t] = 1/[(s²/t)(t³/s4)]½ (10) As expected, the dimensions check out fine. These new results should help clarify electrostatics and magnetostatics for both students and working scientists and engineers. References 1. Dewey B. Larson, The Structure of the Physical Unieerse (Portland, Oregon: North Pacific Publishers, 1959). Note: in this, the first presentation of the Reciprocal System, the permittivity and permeability were treated as dimensionless (p. 82). 2. Dewey B. Larson, Nothing But Motion (Portland, Oregon: North Pacific Publishers, 1979). 3. Dewey B. Larson, Basic Properties of Matter (Salt Lake City, Utah: International Society of Unified Science2 1988). Note: the dimensions of permittivity are stated as [s²/t] on p. 172; the dimensions of permeability are stated as [t³ /s4] on p. 222. APPENDIX: SAMPLE CALCULATIONS 1. Electrostatics What is the force exerted by a charge of one coulomb on another charge of one coulomb one km away, in air? From eq. 3, the value of the permittivity in free space is snat ²/tnat = (4.55881610-6)²/1.52065510-16 = 136670.11 cm²/sec. In air, the value is .9994 times this, or 136588.11 cm²/sec. Now a coulomb is defined as the electrostatic charge which when placed at a distance of 1 meter from an equal charge of the same sign produces a repulsive force of 8.9875510-9 N. In space-time terms, this force is 8.98755109 N * 105 dynes/N * (7.31688910-6 sec/cm²) / (3.2722310² dynes) = 20096664 sec/cm² . Then from Coulomb‘s law (for a vacuum) we have 20096664 sec/cm² = 136670.11 cm²/sec * Q² sec²/cm² / 10000 cm². Solving for Q gives 1212.6213 see/em per coulomb. So for the problem at hand we have Fe - 136588.111212.6213²/100000² = 20.084604 sec/cm². Converting back to conventional units we have 20.084604 sec/cm² 3.2722310² dynes/7.31688910-6 sec/cm² * 10-5 N/dynes = 8982.1567 N. This agrees with the value from experiment. 2. Magnetostatics What is the force exerted by a magnetic pole with a strenath of one weber against another magnetic pole of equal strength one km away, in vacuum? The value of the permeability of free space is t³/s4 = (1.52065510-16)³/(4.55881610-6)4 = 8.141107310-27 sec /cm4 . Now a weber may be defined as the strength of a magnetic pole which exerts in a vacuum a force of 63325.74 N upon another magnetic pole of the same strength one meter away. In space-time terms this force is 63325.74 N 105 dynes/N 2 (7.31688910-6 sec/cm²)/(3.2722310² dynes) = 141.59989 sec/cm . From Coulomb’s magnestatic law we get 141.59989 sec/cm² = (1/8.141107310 -27 sec3/cm4) M²sec4/cm4/10000 cm². Solving for M gives 1.073675910-10 sec²/cm² per weber. So for the problem at hand we have Fm = (1/8�141107310-27 sec³/cm4) * (1.073675910-10 sec /cm ) / (100000 cm) = 1.41599890 sec/cm. Converting back to conventional units we have 1.415998910-4 sec/cm² * 3.2722310² dynes/(7.31688910-6 sec/cm ) * 10-5 N dyne = 0633257 N. This agrees with the value from experiment. SUMMARY: 1. permittivity of free space = 136670.11 cm²/sec 2. permeabilit of free s ace = 8.1411073 10 sec /cm 3. one coulomb = 1212.6213 sec/cm 4. one weber = 1.0736759-10 10 sec²/cm² International Society of Unified Science Reciprocal System Research Society Salt Lake City, UT 84106 USA Theme by Danetsoft and Danang Probo Sayekti inspired by Maksimer	{"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": 0.9533984065055847, "perplexity": 2143.345816441754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107865665.7/warc/CC-MAIN-20201023204939-20201023234939-00381.warc.gz"}
https://www.vedantu.com/physics/superconductivity	# Superconductivity ## Superconductivity Meaning Superconductivity meaning simply states that there is no resistance or almost zero resistance in the material or any object. A material or an object that shows such properties is known as a superconductor. The conductivity referred to here is the electrical conductivity of a material. When the electrical conductivity is to the full potential facing almost to completely zero resistance in a material any of the magnetic flux fields are expelled from the material. The zero resistance is achieved by lowering the temperature of the material which leads to a decrease in the resistance of the material and leading to an increase in the conductivity. In a superconductor as well the same method is applied to achieve superior conductivity. ## Who Discovered Superconductivity and What is a Superconductor? A brief introduction of superconductivity and superconductors is given before. In order to explain superconductivity, it is necessary to note that materials possess certain physical properties that cause resistance to electrical conductivity through the material. This characteristic of the material varies with temperature changes. If the temperature of the material is increased the resistance increases whereas if the temperature of the material is decreased the resistance decreases. This phenomenon is exploited for achieving the highest conductivity of superconductor. In 1911, Heike Kamerlingh Onnes, a Dutch physicist, discovered the superconductivity phenomenon. Currently, the research for the explanation of the phenomenon is done using quantum mechanics as it cannot be completely explained by the concept of perfect conductivity in classical physics. One of the important physical properties exhibited by a conducting material exhibiting superconductivity meaning is that there is no magnetic flux field present in the material as the presence of magnetic flux fields leads to a loss in energy and an indication of the presence of resistance in the material. Superconductor definition can be given as a material that incorporates the superconductivity meaning as a part of its physical properties. Normally when the temperature of a conductor is decreased there is an increase in conductivity as one move to absolute zero temperatures. But superconductors are those special materials in which after a certain critical temperature the resistance drops to zero value and the conductivity thus reaches the maximum. This is a critical point that has to be noted while defining what is a superconductor and explaining the superconductor definition. At this point, while decreasing the temperature below the critical temperature, the conductivity of superconductor is maximum and there is the complete ejection of any magnetic field flux from the material as well. In superconductivity the conductivity of a material becomes such that when an electric current is passed through a loop of such a superconductor the electric current will keep flowing through it indefinitely without any need of a power supply. This can lead to the creation of self-sustaining energy sources solving innumerable problems such as power surges and costly electricity. And because there is no loss of energy due to the resistance of the material the electricity available will be much cheaper when such superconducting material sources are used as power sources. ### Properties of Superconductors In the superconductor definition, the electrical properties arising due to unique and specialised physical properties play an important role, as what is a superconductor without any such interesting electrical properties. One of such properties is the zero electrical DC resistance present in the material. This is a common property of all superconductors irrespective of physical properties of the material such as the heat capacity, critical temperatures (as they can be different for different materials), etc. Also as defined above in the superconductivity phenomenon key role is played by a decrease in the temperature. Although different materials have different critical temperatures once the temperature drops down from the critical temperatures the resistance falls to absolute zero. Thus, it indicates that superconductivity meaning in a superconductor is a thermal property and hence after having reached a superconducting state the phenomenon is independent of the physical properties of the material. All the superconducting materials behave in the same manner. When material changes from a non-superconducting state to a superconducting state there are significant changes in the physical properties of the material which are the characteristics of phase transitions. When the temperature drops below a thermal superconductor there is an ejection of the magnetic field. But when there is an external magnetic field applied to the superconductor and which is more than the critical magnetic field, the superconductor leaves the superconducting state and starts to behave as a normal conductor. This change in the phase of the superconducting material occurs due to the changes in the Gibbs free energy. In the superconducting phase, the Gibbs free energy of the conductor is lower than the normal non-superconducting phase of free energy. When a finite amount of free energy is applied externally to the superconductor through the external magnetic field, the free energy increases quadratically in the superconductor and it reaches the normal free energy value and thus a phase transition takes place in the conductor from the superconducting phase to non-superconducting phase. Thus, these properties of the superconductor make it possible to be used for a variety of purposes. Superconductor examples and their applications are mentioned below. ### Applications of Superconductors Superconductors are noted for their zero DC electrical resistance. Hence, most of the applications of the superconductor examples are because of their properties which provide advantages such as low power loss because of less dissipation of energy, high-speed operations because of zero resistance and continuous flowing electrical current, and high sensitivity. The usual and well-known superconductor examples are mercury superconductors, niobium-tin superconductors, lanthanum-barium-copper oxide superconductors, and yttrium-barium copper oxide superconductors. The examples of applications of superconductors include medical MRI/NMR devices, magnetic-energy storage systems, motors, generators, transformers, computer parts and sensitive devices for the measurement of magnetic fields, electrical currents, etc. Future possible applications involve high-performance smart grids, electric power transmission, transformers, electric motors (in vehicles like maglev trains), magnetic levitation devices, superconducting magnetic refrigerators, etc. FAQs (Frequently Asked Questions) 1. What is a Superconductor? Ans: A superconductor is an object that provides zero resistance to electrical currents at very low temperatures. It incorporates the definition of superconductivity phenomenon which is a physical phenomenon dependent on the physical properties of the material that gives rise to the electrical resistance in a conductor. Thus, a superconductor is conducting material that shows zero resistance to electric current when the temperature of the material is taken below a critical temperature. 2. What is Superconductivity? Ans: The complete disappearance of the electrical resistance from a conducting material when the temperature of the material is taken below a characteristic temperature and the ejection of any magnetic flux field from the material is known as superconductivity. The temperature at which such a change occurs is called critical temperature and is different for different materials.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9318909049034119, "perplexity": 374.38948271428336}, "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-2021-49/segments/1637964362918.89/warc/CC-MAIN-20211203182358-20211203212358-00432.warc.gz"}
https://www.spp2026.de/members-guests/19-member-pages/prof-dr-daniel-lenz	# Members & Guests ## Prof. Dr. Daniel Lenz Friedrich-Schiller-Universität Jena E-mail: daniel.lenz(at)uni-jena.de Telephone: +49 3641 9 46 131 Homepage: http://www.analysis-lenz.uni-jena.de/ ## Project 19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces 59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces ## Publications within SPP2026 We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods. Journal Journal de Mathématiques Pures et Appliquées. (9) Volume 126 Pages 109--143 Link to preprint version Link to published version We study pairs of Dirichlet forms related by an intertwining order isomorphisms between the associated $$L^2$$-spaces. We consider the measurable, the topological and the geometric setting respectively. In the measurable setting, we deal with arbitrary (irreducible) Dirichlet forms and show that any intertwining order isomorphism is necessarily unitary (up to a constant). In the topological setting we deal with quasi-regular forms and show that any intertwining order isomorphism induces a quasi-homeomorphism between the underlying spaces. In the geometric setting we deal with both regular Dirichlet forms as well as resistance forms and essentially show that the geometry defined by these forms is preserved by intertwining order isomorphisms. In particular, we prove in the strongly local regular case that intertwining order isomorphisms induce isometries with respect to the intrinsic metrics between the underlying spaces under fairly mild assumptions. This applies to a wide variety of metric measure spaces including $$\mathrm{RCD}(K,N)$$-spaces, complete weighted Riemannian manifolds and complete quantum graphs. In the non-local regular case our results cover in particular graphs as well as fractional Laplacians as arising in the treatment of $$\alpha$$-stable Lévy processes. For resistance forms we show that intertwining order isomorphisms are isometries with respect to the resistance metrics. Our results can can be understood as saying that diffusion always determines the Hilbert space, and -- under natural compatibility assumptions -- the topology and the geometry respectively. As special instances they cover earlier results for manifolds and graphs.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9292271733283997, "perplexity": 2534.143648380495}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20220707033101-20220707063101-00125.warc.gz"}
https://homework.cpm.org/category/CC/textbook/cca2/chapter/4/lesson/4.1.2/problem/4-25	### Home > CCA2 > Chapter 4 > Lesson 4.1.2 > Problem4-25 4-25. Ted needs to find the point of intersection for the lines $y=18x−30$ and $y=−22x+50$. He takes out a piece of graph paper and then realizes that he can solve this problem without graphing. Explain how Ted is going to accomplish this, and then find the point of intersection. Homework Help ✎ Set the two expressions for $y$ equal to each other and solve algebraically.	{"extraction_info": {"found_math": true, "script_math_tex": 3, "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": 0.9026674032211304, "perplexity": 350.5063416290561}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986682037.37/warc/CC-MAIN-20191018104351-20191018131851-00208.warc.gz"}
http://mathoverflow.net/questions/138596/inequality-for-complex-hankel-function	# Inequality for complex Hankel function Let $x>1$ and $0<\varphi<\frac{\pi}{2}$ be fixed. I would like to show that for any $s>0$, the following inequality holds: $$\left\| H_{\frac{is}{e^{i\varphi } \cos \varphi}}^{\left( 1 \right)} \left( {\frac{is}{e^{i\varphi } \cos \varphi }x} \right) \right\| \le \left\| H_{is}^{\left( 1 \right)} \left( isx \right) \right\| = iH_{is}^{\left( 1 \right)} \left( isx \right),$$ where $H_\nu^{(1)}(z)$ is the Hankel function. It would be enough to show that for any fixed $v>0$, $x>1$, $$[0,+\infty) \ni u \mapsto \left\| {H_{u + iv}^{\left( 1 \right)} \left( {\left( {u + iv} \right)x} \right)} \right\|$$ is a decreasing function. From Debye's asymptotic formula, we have $$\mathop {\lim }\limits_{s \to + \infty } \left\| \frac{H_{\frac{is}{e^{i\varphi } \cos \varphi }}^{\left( 1 \right)} \left( {\frac{is}{e^{i\varphi } \cos \varphi}x} \right)}{H_{is}^{\left( 1 \right)} \left( isx \right)} \right\| = \sqrt {\cos \varphi } < 1.$$ So for large $s$ the inequality is valid. However, I could not find any suitable representation of the (absolute value of the) Hankel function to show the inequality for every $s>0$. - Does this fail for $x < 1$? – Suvrit Aug 6 at 4:52 Numerical computations suggest that it does. – Gary Aug 6 at 10:37	{"extraction_info": {"found_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": 0.9530256986618042, "perplexity": 124.13090268686656}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1386163051684/warc/CC-MAIN-20131204131731-00069-ip-10-33-133-15.ec2.internal.warc.gz"}
http://www-old.newton.ac.uk/programmes/MOS/seminars/2011031014001.html	# MOS ## Seminar ### Introduction to a motivic point of view on the cohomology of moduli spaces of bundles on curves Heinloth, J (Amsterdam) Thursday 10 March 2011, 14:00-15:00 Seminar Room 1, Newton Institute #### Abstract For moduli spaces of vector bundles on curves and some moduli spaces of Higgs bundles, it is possible to compute their cohomology groups in a geometric way, i.e., one can describe the space by a cut-and-paste procedure in terms of cells and symmetric products of the base curve. This gives a rather explicit description of the "motive" of the space. For moduli space of vector bundles this is due to Behrend and Dhillon, relying on an argument of Bifet, Ghione, and Letizia. We will try to give an introduction to this point of view on cohomology calcuations for moduli spaces. #### Video The video for this talk should appear here if JavaScript is enabled. If it doesn't, something may have gone wrong with our embedded player. We'll get it fixed as soon as possible.	{"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": 0.8313778638839722, "perplexity": 613.9700174855025}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1448398468971.92/warc/CC-MAIN-20151124205428-00014-ip-10-71-132-137.ec2.internal.warc.gz"}
https://machinelearningmastery.com/a-gentle-introduction-to-multivariate-calculus	# A Gentle Introduction to Multivariate Calculus It is often desirable to study functions that depend on many variables. Multivariate calculus provides us with the tools to do so by extending the concepts that we find in calculus, such as the computation of the rate of change, to multiple variables. It plays an essential role in the process of training a neural network, where the gradient is used extensively to update the model parameters. In this tutorial, you will discover a gentle introduction to multivariate calculus. After completing this tutorial, you will know: • A multivariate function depends on several input variables to produce an output. • The gradient of a multivariate function is computed by finding the derivative of the function in different directions. • Multivariate calculus is used extensively in neural networks to update the model parameters. Let’s get started. A Gentle Introduction to Multivariate Calculus Photo by Luca Bravo, some rights reserved. ## Tutorial Overview This tutorial is divided into three parts; they are: • Re-Visiting the Concept of a Function • Derivatives of Multi-Variate Functions • Application of Multivariate Calculus in Machine Learning ## Re-Visiting the Concept of a Function We have already familiarised ourselves with the concept of a function, as a rule that defines the relationship between a dependent variable and an independent variable. We have seen that a function is often represented by y = f(x), where both the input (or the independent variable), x, and the output (or the dependent variable), y, are single real numbers. Such a function that takes a single, independent variable and defines a one-to-one mapping between the input and output, is called a univariate function. For example, let’s say that we are attempting to forecast the weather based on the temperature alone. In this case, the weather is the dependent variable that we are trying to forecast, which is a function of the temperature as the input variable. Such a problem can, therefore, be easily framed into a univariate function. However, let’s say that we now want to base our weather forecast on the humidity level and the wind speed too, in addition to the temperature. We cannot do so by means of a univariate function, where the output depends solely on a single input. Hence, we turn our attention to multivariate functions, so called because these functions can take several variables as input. Formally, we can express a multivariate function as a mapping between several real input variables, n, to a real output: For example, consider the following parabolic surface: f(x, y) = x2 + 2y2 This is a multivariate function that takes two variables, x and y, as input, hence n = 2, to produce an output. We can visualise it by graphing its values for x and y between -1 and 1. Three-Dimensional Plot of a Parabolic Surface Similarly, we can have multivariate functions that take more variables as input. Visualising them, however, may be difficult due to the number of dimensions involved. We can even generalize the concept of a function further by considering functions that map multiple inputs, n, to multiple outputs, m: These functions are more often referred to as vector-valued functions. ## Derivatives of Multi-Variate Functions Recall that calculus is concerned with the study of the rate of change. For some univariate function, g(x), this can be achieved by computing its derivative: The generalization of the derivative to functions of several variables is the gradient. – Page 146, Mathematics of Machine Learning, 2020. The technique to finding the gradient of a function of several variables involves varying each one of the variables at a time, while keeping the others constant. In this manner, we would be taking the partial derivative of our multivariate function with respect to each variable, each time. The gradient is then the collection of these partial derivatives. – Page 146, Mathematics of Machine Learning, 2020. In order to visualize this technique better, let’s start off by considering a simple univariate quadratic function of the form: g(x) = x2 Line Plot of a Univariate Quadratic Function Finding the derivative of this function at some point, x, requires the application of the equation for g’(x) that we have defined earlier. We can, alternatively, take a shortcut by using the power rule to find that: g’(x) = 2x Furthermore, if we had to imagine slicing open the parabolic surface considered earlier, with a plane passing through y = 0, we realise that the resulting cross-section of f(x, y) is the quadratic curve, g(x) = x2. Hence, we can calculate the derivative (or the steepness, or slope) of the parabolic surface in the direction of x, by taking the derivative of f(x, y) but keeping y constant. We refer to this as the partial derivative of f(x, y) with respect to x, and denote it by to signify that there are more variables in addition to x but these are not being considered for the time being. Therefore, the partial derivative with respect to x of f(x, y) is: We can similarly hold x constant (or, in other words, find the cross-section of the parabolic surface by slicing it with a plane passing through a constant value of x) to find the partial derivative of f(x, y) with respect to y, as follows: What we have essentially done is that we have found the univariate derivative of f(x, y) in each of the x and y directions. Combining the two univariate derivatives as the final step, gives us the multivariate derivative (or the gradient): The same technique remains valid for functions of higher dimensions. ## Application of Multivariate Calculus in Machine Learning Partial derivatives are used extensively in neural networks to update the model parameters (or weights). We had seen that, in minimizing some error function, an optimization algorithm will seek to follow its gradient downhill. If this error function was univariate, and hence a function of a single independent weight, then optimizing it would simply involve computing its univariate derivative. However, a neural network comprises many weights (each attributed to a different neuron) of which the error is a function. Hence, updating the weight values requires that the gradient of the error curve is calculated with respect to all of these weights. This is where the application of multivariate calculus comes into play. The gradient of the error curve is calculated by finding the partial derivative of the error with respect to each weight; or in other terms, finding the derivative of the error function by keeping all weights constant except the one under consideration. This allows each weight to be updated independently of the others, to reach the goal of finding an optimal set of weights. This section provides more resources on the topic if you are looking to go deeper. ## Summary In this tutorial, you discovered a gentle introduction to multivariate calculus. Specifically, you learned: • A multivariate function depends on several input variables to produce an output. • The gradient of a multivariate function is computed by finding the derivative of the function in different directions. • Multivariate calculus is used extensively in neural networks to update the model parameters. Do you have any questions? ## Get a Handle on Calculus for Machine Learning! #### Feel Smarter with Calculus Concepts ...by getting a better sense on the calculus symbols and terms Discover how in my new Ebook: Calculus for Machine Learning It provides self-study tutorials with full working code on: differntiation, gradient, Lagrangian mutiplier approach, Jacobian matrix, and much more... ### 8 Responses to A Gentle Introduction to Multivariate Calculus 1. Jayaganthan R July 24, 2021 at 10:39 am # Dear Dr Stefania Thank you for your excellent presentation of multivariate calculus. It’s beautifully explained to follow. With best regards Jayaganthan • Stefania Cristina July 24, 2021 at 5:58 pm # Thank you, Jayaganthan. Glad to hear that you have found it useful. 2. Meshack Owira Amimo July 25, 2021 at 4:26 am # A wonderful introduction to how calculus is relevant to help one have a command of ML and artificial intelligence. Thanks for the simple free-flowing explanation n narrative on a subject regarded by many as abstract • Stefania Cristina July 26, 2021 at 1:12 am # Thank you for the kind words, Meshack. 3. Ashwin Norbert July 26, 2021 at 9:38 pm # A detailed presentation and explanation of multivariant calculus. Thanks a lot Stefania. • Stefania Cristina July 27, 2021 at 6:08 am #	{"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": 0.8482038974761963, "perplexity": 499.610254842306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296950383.8/warc/CC-MAIN-20230402043600-20230402073600-00215.warc.gz"}
http://math.stackexchange.com/questions/30595/prove-the-reduction-formula?answertab=active	# Prove the reduction formula The question is to "prove the reduction formula" $$\int{ \frac{ x^2 }{ \left(a^2 + x^2\right)^n } dx } = \frac{ 1 }{ 2n-2 } \left( -\frac{x}{ \left( a^2+x^2 \right)^{n-1} } + \int{ \frac{dx}{ \left( a^2 + x^2 \right)^{n-1} } } \right)$$ What I got is Set $u = x$ $du = dx$ $\displaystyle{ dv = \frac{ x }{ \left( a^2 + x^2 \right)^{n} } dx }$ $\displaystyle{ v = \frac{ 1 }{ 2(n+1) \left( a^2 + x^2 \right)^{n+1} } }$ So I got $$\frac{ 1 }{ 2n+2 } \left( \frac{x}{ \left( a^2 + x^2 \right)^{n+1}} - \int{ \frac{dx}{ \left( a^2+x^2 \right)^{n+1} } } \right)$$ Which I believe is correct. They are subtracting from n in the integration step and I'm not sure why - +1 for, as usual, showing your work. – Arturo Magidin Apr 3 '11 at 1:19 You went wrong when you integrated $dv$. You have $dv = x(a^2+x^2)^{-n}\,dx$. When you integrate, you add one to the exponent. But adding one to $-n$ gives $-n+1 = -(n-1)$. So $$v = \frac{1}{2(-n+1)}(a^2+x^2)^{-n+1} = \frac{1}{2(1-n)(a^2+x^2)^{n-1}}.$$ The minus sign from integration by parts can be cancelled out by switching the sign of $2(1-n)$ to get $2(n-1) = 2n-2$. If you use the correct value of $v$, I think you will have no trouble establishing the formula.	{"extraction_info": {"found_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": 0.9229817986488342, "perplexity": 306.3531875093351}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1387345777160/warc/CC-MAIN-20131218054937-00071-ip-10-33-133-15.ec2.internal.warc.gz"}
http://mathhelpforum.com/math-software/230566-simplying-linear-equation-get-quartic-q-using-maple-using-descarte-s.html	# Math Help - Simplying linear equation to get quartic in q with using Maple and using Descarte’s 1. ## Simplying linear equation to get quartic in q with using Maple and using Descarte’s Using the maple I am trying to get quardic in q from this big linear equation. Then use Descarte’s rule of signs to determine the number of positive roots. \frac{\gammaqP_Q}{k_p(1-q)P_C} = \frac{I\alpha}{k_f+k_d+\frac{k_n\lambda_b\gamma qP_Q}{\lambda_b\gammaqP_Q+k_p\lambda_r(1-q)^2}+k_p(1-q)} Values of parameters are given below: $I=1200$ $k_f = 6.710.^7$ $k_d = 6.0310.^8$ $k_n = 2.9210.^9$ $k_p = 4.9410.^9$ $\alpha = 1.1443710.^(-3)$ $\lambda_b = 0.87e-2$ $\lambda_r = 835$ $\gamma = 2.74$ $P_C = 310.^(11)$ $P_Q = 2.8710.^(10)$ => I tried the code in maple to get quartic in q but DOES NOT WORKS. Code: II := 1200: k_f := 6.710.^7: k_d := 6.0310.^8: k_n := 2.9210.^9: k_p := 4.9410.^9: alpha := 1.1443710.^(-3): lambda_b := 0.87e-2: lambda_r := 835: ggamma := 2.74: P_C := 310.^11: P_Q := 2.8710.^10: eq := ggammaqP_Q/(k_p(1-q)P_C) = IIalpha/(k_f+k_d+k_nlambda_bggammaqP_Q/(lambda_bggammaqP_Q+k_plambda_r(1-q)^2)+k_p(1-q)): simply(eq, q); My lecturer want me to manipulate the equation and get a quartic in q before substituting the values of parameters into the equation. After that,use Descarte’s rule of signs to determine the number of positive roots. Then write Q=1-q to get second quartic in Q and repeat rule of signs to determine number of steady states of q less than 1. And do the substition of parameters if necessary. Now, its kind of hard for me what he wants because to get quartic in q first from the equation is hard to do by hand , so i have to use in maple which is not working then use Descarte’s rule of signs. 2. ## Re: Simplying linear equation to get quartic in q with using Maple and using Descarte Posting a second time will not get you an answer any faster. Perhaps try different variable names. Try using semicolons at the end of every statement instead of colons? Also, try solve(eqn,q) instead of simplify(eqn,q)? 3. ## Re: Simplying linear equation to get quartic in q with using Maple and using Descarte Originally Posted by SlipEternal Posting a second time will not get you an answer any faster. Perhaps try different variable names. Try using semicolons at the end of every statement instead of colons? Also, try solve(eqn,q) instead of simplify(eqn,q)? Maple understand the colons not semicolons sir. And i did tried the solve(eqn,q) to get the result for q which i got it. but this time i am trying to manipulate the equation to get quartic in q, so that i can use Descarte’s rule of signs to determine the number of positive roots. And after that do substitute parameters in and get the results for q. 4. ## Re: Simplying linear equation to get quartic in q with using Maple and using Descarte Cross multiplying by hand should not be a problem. \begin{align}& \left[\left(k_f + k_d + k_p(1-q) \right)\left(\lambda_b\cdot \gamma \cdot q \cdot P_Q + k_p\cdot \lambda_r\cdot (1-q)^2\right) + k_n\cdot \lambda_b\cdot \gamma \cdot q \cdot P_Q \right]\left(\gamma \cdot q \cdot P_Q\right) \\ = & I\cdot \alpha\left( k_p(1-q)P_C \right)\left( \lambda_b\cdot \gamma \cdot q \cdot P_Q + k_p\cdot \lambda_r\cdot (1-q)^2 \right)\end{align} Ask maple to simplify that expression.	{"extraction_info": {"found_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": 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": 0.9958752393722534, "perplexity": 1751.4367719363968}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1419447548655.55/warc/CC-MAIN-20141224185908-00082-ip-10-231-17-201.ec2.internal.warc.gz"}
https://socratic.org/questions/56e81b1311ef6b3478778796#243766	Physics Topics # What is the relationship between the radius of circular motion and the centripetal force, if the mass undergoing the circular motion is kept constant? Yes. The short answer is that it's right there in the formula: $F = \frac{m {v}^{2}}{r}$ or $F = m {\omega}^{2} r$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8839406967163086, "perplexity": 169.09327290367384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662558030.43/warc/CC-MAIN-20220523132100-20220523162100-00388.warc.gz"}
https://www.hepdata.net/record/42378	Measurement of the $Z Z \gamma$ and $Z \gamma \gamma$ couplings in $p\bar{p}$ collisions at $\sqrt{s} = 1.8$ TeV Phys.Rev.Lett. 75 (1995) 1028 Abstract (data abstract) FNAL-COLLIDER. Measurement of the Z Z GAMMA and Z GAMMA GAMMA couplings in PBAR P collisions at sqrt(s) = 1.8 TeV.	{"extraction_info": {"found_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": 0.9999616146087646, "perplexity": 4614.658815764835}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1544376829997.74/warc/CC-MAIN-20181218225003-20181219011003-00610.warc.gz"}
https://geo.libretexts.org/Courses/University_of_California_Davis/GEL_056%3A_Introduction_to_Geophysics/Geophysics_is_everywhere_in_geology.../04%3A_Plate_Tectonics/4.01%3A_The_Forces_Driving_Plate_Motions	# 4.1: The Forces Driving Plate Motions $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ The motion of tectonic plates is driven by convection in the mantle. In simple terms, convection is the idea that dense, cold things sink, and buoyant, warm things rise. In the earth the cold sinking things are slabs (subducting plates) and the warm things are plumes, or just rising material from deeper in the mantle. There are three main forces that determine the rate at which tectonic plates move as part of the mantle convection system: • slab pull: the force due to the weight of the cold, dense sinking tectonic plate • ridge push: the force due to the buoyancy of the hot mantle rising to the surface beneath the ridge. • viscous drag: the force opposing motion of the plate and slab past the viscous mantle underneath or on the side. This force balance is given by: $F_{ridge-push}+F_{slab-pull}-F_{viscous-drag}=0$ Because the viscous drag depends explicitly on the plate velocity, it is possible to predict the speed plate motion when the forces are balanced (under certain simplifying assumptions). Recall from Chapter 1 that stress is related to force as $$\sigma=\frac{F}{A}$$ where $$A = L W$$. Rearranging, $$F=\sigma A$$. When considering the force balance on the a plate it is useful to using a simplified geometry as shown below and consider the force per unit length (or ridge or subduction zone) $F= \sigma A/W = \sigma L \label{force}$ As seen in the figure below, $$W$$ is the length of the plate along the ridge and subduction zone. By considering the force per unit length (in the W direction), we can then consider the forces working along a profiles from the ridge to the subduction zone and ignore any variation in the ridge-parallel direction. ## Viscous Drag One of the mechanisms for plate motion that is a result of convection is viscous drag. This concept relates to ideas we discussed earlier, including the Couette Flow and viscous stress. The drag on the base of the oceanic lithosphere can both drive or resist plate motion, depending on the relative motion between the plate and the underlying mantle. For this discussion, we will consider that the plate is being pulled or pushed by slab-pull and ridge push and drags the asthenosphere with it. Therefore viscsous drag oppose the motion of the plate (or slab) and acts to slow it down. From the figure above, we can derive an equation to describe plate motion due to viscous drag. $V=V_o(1-\frac{y}{h})$ $\dot{\varepsilon}=\frac{1}{2}\frac{dV}{dy}=-\frac{V_o}{2h}$ \begin{align}\sigma &=2\eta\dot{\varepsilon} \\[4pt] &=2\eta(\frac{-V_o}{2h}) \\[4pt] &=\frac{-V_o\eta}{h} \end{align} The negative sign indicates an opposition to plate motion, i.e., the viscous drag. We now have an equation that describes the magnitude of drag: $\|\sigma\|=\frac{V_o\eta}{h}$ placing this into our previous force equation (Equation \ref{force}): $F=\sigma L = \frac{V_o\eta}{h} L$ Let's do an example calculating viscous drag under a plate. Example Viscous Drag Under the Plate Assume that: • the plate drags the mantle underneath in a layer that is 100 km thick (h~100 km). • the mantle viscosity is $$\eta ~ 1\times10^{18}$$ Pa(s). • the plate velocity is Vo~5 cm/yr. (Note that this is equivalent to 0.05 m/yr = 50 mm/yr = 50 km/my • the plate length from the ridge to the subduction zone is about 5000 km From the strain-rate, we can get the stress as shown above, and then multiply by the length of the plate Using these numbers, we can calculate the force on the descending plate, $$F=1.59 \times 10^{11}$$ N/m. The above example is for the viscous drag just under the plate, but now we will also consider a case where there are stresses acting on both sides of the subducting slab. As the slab sinks into the mantle, the viscous mantle on both sides resists the sinking motion of the slab. For this simple example, $\sigma=\frac{V_o\eta_{av}}{h}$ $F=\sigma L_{slab}$ $F_{vis.}=2F=2\sigma L_{slab}=\frac{2V_o\eta_{av}}{h}L_{slab}$ Example Viscous Drag on Two Sides of a Plate As we did in the previous example, consider viscous drag on a slab, but now on both sides of the plate. We are given that • The slab sinks at the same speed that the plate is moving: Vo=Vplate~5 $$\frac{cm}{yr}$$, • Again, the shearing layer on both sides is 100 km wide (h~100 km), • The viscosity of the mantle increases with depth, for this example we'll assume its constant $$\eta_{av} =10^{21}$$ Pa s , • The slab length (from the base of the lithosphere to the upper/lower mantle boundary is Lslab = 560 m. In this case, $$F = 78 \times 10^{13}$$ N/m. If we change $$\eta$$ to $$10^{21}$$, then equals $$F=1.78x10^{10}$$ N/m. Notice that the stress is proportional to the velocity $$\sigma \alpha V$$ so as the velocity increases so does the stress opposing the sinking. However, the biggest changes come from the viscosity because it can changes by x10-100 in the upper mantle, thus this change has the biggest effect on the viscous drag. The viscous drag is also proportional to the length of the slab, so the resistance also increases as the slab gets longer (but so does the slab pull force). ## Slab-Pull Force The main force on a subducting plate is the slab-pull force. This is the force due to the density contrast between a slab and its surroundings, ie., the mass anomaly of the slab in the mantle. We can approximate this force by thinking of the sinking slab a a vertical rectangle (like the viscous drag example) with a width, $$w$$, hanging down a distance, $$d$$ from the Earth's surface with a temperature $$\Delta T$$ different than its surroundings. The density as a function of temperature is given by $\rho (T)=\rho_m(1+\alpha (T_m - T))$ where $$\rho_m$$ is the reference mantle density at $$T = T_m$$ and cold material has a higher density, and $$\alpha$$ is the thermal expansion coefficient and is ~$$2 \times 10^{-5}$$ 1/K in the sinking slab. For the slab, we can rewrite this as: $\rho (T)=\rho_m(1+\alpha \Delta T)$ where $$\Delta T=T_m-T_{slab}$$ The density anomaly is then $\Delta \rho= \rho_m\alpha\Delta T$ This is the extra density of the slab relative to the mantle. To calculate the force per unit area, we need to determine the mass anomaly per unit area, $$\Delta M = \Delta \rho A_{slab}$$, where A_{slab} is the cross sectional area of the slab along the profile (width across times length into the mantle). The temperature in the slab is related to the temperature of the subducting plate. Recall from the chapter on thermal diffusion that the temperature of a tectonic plate is given by the half-space cooling model. Once the plate starts sinking, the cold slab warms up, and the surrounding mantle cools down. However, the total temperature anomaly (temperature difference times volumes) stays the same. (Remember that there is conservation of energy, so the total energy of the cold slab is conserved). So, if we integrate the temperature profile solution from the half-space cooling model over the thickness of this plate, this will also be the thermal anomaly of the the slab. Finally, because the density anomaly is just $\Delta \rho=\rho_m\alpha\Delta T$, we can get the total density anomaly by just multiplying by $$\rho_m \alpha$$. So, the trick to figuring out the slab pull force is the first determine the density anomaly of the subducting plate just before it enters the trench. Then imagine taking a length of plate equal to the length of the slab and rotating it en masse in to the mantle. The density anomaly of the sinking slab will be the same. To get the density anomaly, we integrate the temperature equation for the plate from the half-space cooling model. $T(z, t)=(T_s-T_m)erfc(\frac{z}{2\sqrt{\kappa t}})+T_m$ Here $$z$$ is the depth through the subducting plate, before it sinks, or the width through the slab while it is sinking. $\Delta T=T_m - T(z, t)$ $\Delta T=-(T_s-T_m)erfc(\frac{z}{2\sqrt{\kappa t}})$ $\Delta T=(T_s-T_m)erfc(\frac{z}{2\sqrt{\kappa t}})$ Then, multiplying the $$\rho_m \alpha$$, $\Delta \rho=\rho_m\alpha\Delta T$ $\Delta \rho=\rho_m\alpha\Delta T(z)$ Next we integrate over the width (depth) to the get the mass per unit length (length parallel to the trench) in a single profile, (we will need to sum this over the length of the slab): $\Delta m=\int_{0}^{z\rightarrow\infty}{\Delta \rho(z)dz}$ $\Delta m=\rho_m\alpha(T_m-T_s)\int_{0}^{\infty}erfc(\frac{z}{2\sqrt{\kappa t}})dz$ Now we use variable substitution so we can solve the integral Let $$q=\frac{z}{2\sqrt{\kappa t}}$$ where $$z\rightarrow\infty$$ and $$q\rightarrow\infty$$. $\frac{dq}{dz}=\frac{1}{2\sqrt{\kappa t}}$ $dz=2\sqrt{\kappa t}dq$ Now substitute into the integral, $\Delta m=\rho_m\alpha(T_m-T_s)2\sqrt{\kappa t}\int_{0}^{\infty}erfc(q)dq$ The complementary error function integral solution can be found in any integral table (or search online) and equals $$\sqrt{\frac{1}{\pi}}$$ $\Delta m=\rho_m\alpha(T_m-T_s)2\sqrt{\frac{\kappa t}{\pi}}$ The above equation gives the mass anomaly (relative to the surrounding mantle) per unit length. Finally, we can determine the total mass anomaly by multiplying by the length of the slab: $\Delta M=\Delta m L_{slab}$ The slab pull force is then given by, $F_{slab-pull}=\Delta m L_{slab} g$ and subsituting for $$\Delta m$$, $F_{sp}=2 \rho_m\alpha (T_m-T_s) g L_{slab}\sqrt{\frac{\kappa t}{\pi} }$ Example Slab-Pull Force We are given that • $$\rho_o$$=3300 kg/m$$^3$$, • $$\alpha =2e-5$$ 1/K, g=9.81 m/s$$^2$$, • Tm-Ts=1400 K, • $$\kappa =1e-6$$ m$$^2$$/s, • t=age of slab at time of subduction, • Lslab=560 km. We know that $$F_{sp}=\Delta m L_{slab} g$$ Thus, the force per unit length of the slab-pull is Fsp=2.88x1013 N/m You might notice that this force is 100 - 1000x more than the viscous drag force. ## Ridge-Push Force Let's cover a final force a subducting plate would experience, the ridge-push force. This force results from the elevation of oceanic ridges above the seafloor. This difference in height leads to pressure that 'pushes' the plate away from the ridge. Ridge push is the simplest force in some ways, as all its components can be easily examined. Simply put, to determine ridge-push forces, we need to do two things. First, look at the isostatic balance and the depth of oceans w(t). Second, we need to look at the force balance on the plate. $T(t)=(T_s-T_m)erfc(\frac{z}{2\sqrt{\kappa t}})+T_m$ $\rho(T)=\rho_m(1+\alpha\Delta T)$ $\Delta T=T_m-T(t)$ T(t) is <Tm+$$\Delta T$$ because of more density. Balance pressures: P1=P2 $drp_wg+\int_{dr}^{dr+w}\rho (T)gdz+\int_{dr+w}^{z_c}\rho (T)gdz=(dr+w)\rho_wg + \int_{dr+w}^{z_c}\rho (T)gdz$ $\int_{dr}^{dr+w}\rho_mdz+\int_{dr+w}^{z_c}\rho_mdz=wp_w+\int_{dr+w}^{z_c}\rho (z)dz$ $w\rho_m-wp_w=\int_{dr+w}^{z_c}\rho (z)-\rho_mdz$ $\rho (z)=\rho (T)\rightarrow T(z)$ Like we did above, we will use the complementary error function and find its solution in a integral table. $w(\rho_m-\rho_w)=\rho_m\alpha(T_m-T_s)\int_{0}^{\infty}erfc(\frac{z'}{2\sqrt{\kappa t}})dz'$ The complementary error function integral solution is $$2\sqrt{\frac{\kappa t}{\pi}}$$ $w=\frac{2\rho_m\alpha(T_m-T_o)}{\rho_m-\rho_w}\sqrt{\frac{\kappa t}{\pi}}$ An important thing to note is that the depth of the ocean basins deepen at ~$$\sqrt{t}$$ (the square root of age) ⇒balanced pressures (isostasy) Ridge-push is very similar to the balanced pressures idea, but uses balanced forces instead. ⇒$$\Delta P$$⇒drives flow across A. No net force⇒no $$\Delta P$$. $F\backsim\Delta P A$ This is the force per unit width. Example Ridge-Push The forces in this example should balance. After we write the integrals for the various forces, we should have an expression where FRP=F1-F2-F3. $F_1=\int_{0}^{z_c}\rho_mgzdz=F_s$ Here, $$\rho_mgz$$ represents the pressure, dz is XA⇒Area⇒wXL$$\frac{area}{w}$$=L We can do a simple integral in depth, this is the same pressure from the mantle below a plate. Rewriting F1 0→zc, 0→w + w-zL $F_1=\int_{0}^{w}\rho_mgzdz+\int_{w}^{w+z_c}\rho_mgzdz+\int_{0}^{2L}\rho_mg\bar{z}d\bar{z}$ Where we let $$\bar{z}=z-w$$ $F_2=\int_o^w\rho_wgz=F_4$ $$\rho_wgz$$ is P2. Here, we do not need to integrate over shape of w. $F_3=\int_0^{z_L}P_Ld\bar{z}$ $P_L=\rho_w g w+\int_0^{\bar{z}} \rho_L (T)g d\bar{z}$ Here, $$\rho_w g w$$ is the pressure due to the water and $$\rho_L$$ is the temperature-dependent density of the lithosphere. Remember our force balance equation: $F_{RP}=F_1-F_2-F_3$ Plugging everything in and skipping some of the basic algebra we get: $F_{RP}=g[\frac{1}{2}(\rho_m-\rho_w)w^2+\kappa t\rho_m\alpha(T_m+T_s)]$ We get the term w2 because PL~w and we integrated PL. From isostasy, we can plug in $w(t)=\frac{2\rho_m\alpha(T_m-T_s)}{(\rho_m-\rho_w)}\sqrt{\frac{\kappa t}{\pi}}$ $F_{RP}=g \rho_m\alpha(T_m-T_s) \kappa t (1+\frac{2p_m\alpha(T_m-T_s)}{\pi(\rho_m-\rho_w)})$ Notes on Ridge-Push A few things to note: isostasy→controlling w(t) The pressure across the lithosphere→ridge-push force FRP$$\alpha$$t age of the oldest part of plate Now that we have our general solution for the ridge-push force, if we are given actual values, we can solve it. If g=9.81, k=1e-6$$\frac{m^2}{s}$$, $$\rho_m$$=3300$$\frac{ky}{m^3}$$, Tm-Ts=1400 K, $$\alpha$$=2x10-5$$\frac{1}{K}$$, $$\rho_m$$=1000$$\frac{kg}{m^3}$$, we can finally find the force $F_{RP}=3.24x10^{12}\frac{N}{m}$ 4.1: The Forces Driving Plate Motions is shared under a CC BY-SA license and was authored, remixed, and/or curated by Magali Billen.	{"extraction_info": {"found_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": 0.9707909226417542, "perplexity": 1079.4887664399778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662577757.82/warc/CC-MAIN-20220524233716-20220525023716-00457.warc.gz"}
http://math.stackexchange.com/users/20570/eladidan?tab=activity	Reputation Next privilege 250 Rep. Sep 24 awarded Autobiographer Jul 21 revised Show solution to ODE's fourier series is a series of sines only edited body Jul 21 comment Show solution to ODE's fourier series is a series of sines only Then I don't get it: The ODE itself has nothing to do with this result? It all comes strictly from the boundary values? Jul 21 comment Show solution to ODE's fourier series is a series of sines only @AndrewD I guess this is theorem: en.wikipedia.org/wiki/… I guess we can't use the theorem since $u$ doesn't necessarily has a bounded variation. Jul 21 comment Show solution to ODE's fourier series is a series of sines only @AndrewD There was actually another segment in the question asking us to explain why we cannot apply that theorem to conclude 2 but since I didn't know the theorem or its terms to quote, I didn't bring it in. Could you give a link to the theorem in question? We are also hinted that we should derive 2 by developing the coefficients of the Fourier series. Jul 21 asked Show solution to ODE's fourier series is a series of sines only Jul 19 accepted Find roots of $3z^{100} - e^z$ in the unit disc. Jul 15 awarded Teacher Jul 14 revised Find roots of $3z^{100} - e^z$ in the unit disc. circle->disc Jul 14 answered Find roots of $3z^{100} - e^z$ in the unit disc. Jul 14 revised Find roots of $3z^{100} - e^z$ in the unit disc. correction from the comments Jul 14 comment Find roots of $3z^{100} - e^z$ in the unit disc. @AntonioVargas oh boy, what a blunder. Thanks for setting me straight Jul 14 comment Find roots of $3z^{100} - e^z$ in the unit disc. @AntonioVargas, on $0$, $e^z=1$ is greater than $3z^{100}=0$ and on $1$ for example, $3z^{100}=3$ and $e^z=e<3$ Jul 14 asked Find roots of $3z^{100} - e^z$ in the unit disc. Jul 11 accepted Show that $\sum_{k=1}^{n}a_ke^{2 \pi ikx}$ has a root in $\left[ 0,1 \right]$ Jul 11 awarded Commentator Jul 11 comment Show that $\sum_{k=1}^{n}a_ke^{2 \pi ikx}$ has a root in $\left[ 0,1 \right]$ D'oh! I poisoned the internet Jul 11 asked Show that $\sum_{k=1}^{n}a_ke^{2 \pi ikx}$ has a root in $\left[ 0,1 \right]$ Jul 11 comment Show that the complex closed line integral $\oint\frac{\mathrm{d}z}{p(z)}$ is $0$ ($p$ is polynomial) Yeah, the fact that the roots are distinct is relevant to the rest of the question which I didn't present here. I just didn't want to leave it out in case it somehow is needed Jul 10 accepted Show that the complex closed line integral $\oint\frac{\mathrm{d}z}{p(z)}$ is $0$ ($p$ is polynomial)	{"extraction_info": {"found_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": 0.9128331542015076, "perplexity": 395.8343192201669}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1454701164289.84/warc/CC-MAIN-20160205193924-00106-ip-10-236-182-209.ec2.internal.warc.gz"}
https://risk.asmedigitalcollection.asme.org/FEDSM/proceedings-abstract/FEDSM2021/85307/V003T08A005/1120962?searchresult=1	## Abstract This paper numerically investigates unsteady behavior of cloud cavitation, in particular, to elucidate the induced shock wave emission. To do this, we consider a submerged water-jet injection into still water through a nozzle and make some numerical analysis of two-dimensional multiphase flows by Navier-Stokes equations. In our previous study [7], we have shown that twin vortices symmetrically appear in the injected water, which plays an essential role in performing the unsteady behavior of a cloud of bubbles. In this paper, we further illustrate the elementary process of the emission of the shock waves. First, we set up the mixture model of liquid and gas in Lagrangian description by the SPH method, together with the details on the treatment of boundary conditions. Second, we show the velocity fields of the multiphase flow to illustrate the inception, growth as well as the collapse of the cloud. In particular, we explain the mechanism of the collapse of the cloud in view of the motion of the twin vortices. Further, we investigate the pressure fields of the multiphase flow in order to demonstrate how the shock wave is emitted associated with the collapse of the cloud. Finally, we show that a small shock wave may be released prior to the main shock wave emission. This content is only available via PDF.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8423721790313721, "perplexity": 380.3674011451146}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711396.19/warc/CC-MAIN-20221209112528-20221209142528-00019.warc.gz"}
https://physics.stackexchange.com/questions/208778/interpretation-of-the-electromagnetic-field-strength-tensor-as-a-spin-1-field	# Interpretation of the electromagnetic field strength tensor as a spin-1 field If I understand correctly the electromagnetic field strength tensor $F_{\mu\nu}$ could be considered as a spin-1 field. In that case, what can one say about the total spin and the $z$-component of the spin for this field? Also, how is $F_{\mu\nu}$'s spin related to that of the photon field ($A_{\mu}$)? The spin of the electromagnetic field tensor $F_{\mu\nu}$ is best understood by writing it as a spinor. A spin 1 field is a represented by a symmetric spinor $\xi^{AB}$ or by a dotted symmetric spinor $\eta_{\dot{A}\dot{B}}$. In order to get the field transforming correctly under parity, the electromagnetic field has to be a direct sum using the symmetric spinor and it's complex conjugate dotted spinor. $$F_{\mu\nu} \sim \xi^{AB}\oplus [\xi^{}]_{\dot{A}\dot{B}}$$ The symmetric spinor $\xi^{AB}$ has three independent complex components $\xi^{11},\xi^{12}=\xi^{21},\xi^{22}$. Linear combinations of these components correspond to the three complex components of the electromagnetic field $B^{r}+iE^{r}$. The source free Maxwell equations are obtained by acting on the symmetric spinor with the Hermitian momentum operator $\hat{p}^{\dot{A}}_{\ B}$ $$\hat{p}^{\dot{A}}_{\ B}\xi^{BC}=0$$ This equation is similar to the Dirac equation for a massive spin 1/2 field. The photon is massless, so it has helicity = $\pm 1$ instead of spin (essentially spin 1 along or against the direction of flight). The photon has two helicity degrees of freedom, but the spin 1 field $F_{\mu\nu} \sim \xi^{AB}\oplus [\xi^{}]_{\dot{A}\dot{B}}$ has six real components. The Maxwell equations $\hat{p}^{\dot{A}}_{\ B}\xi^{BC}=0$ project the spinor onto a two-dimensional subspace. This is as far as I know how to answer the question at present. The gauge field $A^{\mu}$ is a four vector so it ought to be a Hermitian spinor field of type $X^{\dot{A}}_{\ B}$ which is the tensor product of two spin 1/2 fields. It has four components, so the gauge fixing must come in to reduce four to the two helicity degrees of freedom.	{"extraction_info": {"found_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": 0.9611865878105164, "perplexity": 158.22574813481518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1593655879738.16/warc/CC-MAIN-20200702174127-20200702204127-00510.warc.gz"}
https://socratic.org/questions/for-a-field-trip-11-students-rode-in-cars-the-rest-filled-eight-buses-how-many-s	Algebra Topics # For a field trip 11 students rode in cars the rest filled eight buses. How many students were in each bus if 315 students were on the trip? Mar 4, 2018 $38$ students on each bus. #### Explanation: If $11$ of $315$ students were not on the buses then $315 - 11 = 304$ students were on the buses. If these $304$ students were divided evenly among $8$ buses there would be $\frac{304}{8} = 38$ students on each bus. ##### Impact of this question 229 views around the world You can reuse this answer	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 7, "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": 0.9924147725105286, "perplexity": 3562.1914028212527}, "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-2018-51/segments/1544376823228.36/warc/CC-MAIN-20181209232026-20181210013526-00552.warc.gz"}
http://math.stackexchange.com/questions/134024/compact-hausdorff-space-and-continuity	# compact Hausdorff space and continuity Let $X$ be a Hausdorff space. Suppose $f:X\rightarrow \mathbb{R}$ is such that $\{(x,f(x)):x\in X\}$ is a compact subset of $X\times \mathbb{R}$. How would I show $f$ is continuous? - Show that the pre-image of a closed set under $f$ is closed (lift a closed set first to $X \times \mathbb{R}$ via the projection $X \times \mathbb{R} \to \mathbb{R}$, intersect the result with the graph and project down to $X$). Alternatively, note that $p_X(x,f(x)) = x$ is a continuous bijection from the graph to $X$. The hypotheses imply that this is a homeomorphism. Then observe that $f = p_{\mathbb R} \circ (p_{X})^{-1}$ is a composition of continuous 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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9699909090995789, "perplexity": 60.77216030640278}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049274119.75/warc/CC-MAIN-20160524002114-00211-ip-10-185-217-139.ec2.internal.warc.gz"}
https://inordinatum.wordpress.com/2012/07/29/directed-polymers/	# inordinatum Physics and Mathematics of Disordered Systems ## A very brief introduction to directed polymers A classical problem in the field of disordered systems is an elastic string (or elastic polymer) stretched between two points in a random environment. One is interested in knowing how the polymer geometry (roughness), ground state energy, and excitations change due to the disordered environment. A fundamental question is if the polymer becomes rough for arbitrarily weak disorder, or if there is a phase transition between a weakly disordered (smooth) and a strongly disordered (rough) phase. In the following I will try explain some basic results on the simple case of a two-dimensional directed polymer (i.e. a “stretched” polymer which does not contain loops or overhangs). I will explain why for large polymer lengths $L$, moments of the partition sum $Z$ grow as $\ln \overline{Z^n} \propto n^3 L$. I will show why this indicates a rough polymer with roughness exponent $\zeta=\frac{2}{3}$. This analysis holds for arbitrarily weak disorder, meaning that in two dimensions disorder is always relevant, and the polymer is always in the rough phase. Although none of these observations are new, the existing results are often buried in technical literature (there is not even a Wikipedia page describing the directed polymer problem). I hope this self-contained summary will be useful for people interested in a quick introduction to the subject. Some original references where more details can be found are cited at the end of the post. ## The directed polymer partition sum, and its moments We can consider a polymer configuration as a trajectory for a particle moving between two points. We call the polymer directed if the particle always moves from the initial towards the final point, i.e. if the trajectory does not contain loops or overhangs. Formally, this means that the polymer configuration in $1+d$ dimensions can be parametrized by giving $d$ transversal coordinates $(x_1...x_d)=\vec{x}$ as a function of the longitudinal coordinate $t$ (distance along the path). The partition sum is then given by a path integral $Z = \int \mathcal{D}[\vec{x}]e^{-\int_0^L \mathrm{d}t \, \left[(\dot{\vec{x}})^2 + \eta(\vec{x},t)\right]}$ $(\dot{\vec{x}})^2$ is the elastic energy term, which is minimized for straight paths. $\eta(\vec{x},t)$ is the energy it costs the polymer to pass through the site $(\vec{x},t)$ (and which is minimized for rough paths picking out the minimal energy sites). We model a disordered environment by taking $\eta$ to be uncorrelated Gaussian white noise: $\displaystyle \overline{\eta(\vec{x}_1,t_1)\eta(\vec{x}_2,t_2)} = 2c\delta^{(d)}(\vec{x}_1-\vec{x}_2)\delta(t_1-t_2)$ Applying the Feynman-Kac formula, the partition sum $Z(\vec{x},t)$ satisfies the following stochastic differential equation: $\displaystyle \partial_t Z(\vec{x},t) = \nabla^2 Z(\vec{x},t) +\eta(\vec{x},t) Z(\vec{x},t)$ $Z$ is a random quantity, because $\eta$ is random. Since computing the distribution of $Z$ is difficult, we will consider its moments, defined as $\displaystyle \Psi_n(\vec{x}_1...\vec{x}_n,t) := \overline{Z(\vec{x}_1,t)\cdots Z(\vec{x}_n,t)}$ The time evolution of $\Psi_n$ is determined by applying the Itô formula to the equation for $Z$: $\displaystyle \partial_t\Psi_n(\vec{x}_1...\vec{x}_n,t) = \left[\sum_{j=1}^n \nabla_{\vec{x}_j}^2 + 2c\sum_{j This is, up to an overall sign, the Schrödinger equation for $n$ bosons interacting through pairwise pointlike attraction of strength $c$. In one dimension, this is called the Lieb-Liniger model. It is easy to determine the ground state of this system exactly, which will allow us to obtain the leading behaviour of $Z^n$ for large $t$. ## Solution in one spatial dimension: Bethe ansatz ground state In $d=1$, the ground state of the Hamiltonian $H_n$ defined above has energy $E_n = \frac{c^2}{12}n(n^2-1)$ and is given by $\displaystyle \Psi_n(\vec{x}_1...\vec{x}_n) \propto \exp\left(-\frac{c}{2}\sum_{j This is a special case of the general Bethe ansatz ubiquitious in modern theoretical physics. Its application to the directed polymer problem, as discussed here, was recognized by Kardar (see references below). To check that this ansatz for $\Psi$ gives, indeed, an eigenstate, let us take two derivatives with respect to $x_j$: $\displaystyle \begin{array}{lcl} \partial_{x_j}^2 \Psi_n & = & \partial_{x_j} \left[-\frac{c}{2}\sum_{j \neq k} \text{sgn}(x_j-x_k) \Psi_n\right] \\ & = & \left[-c\sum_{j \neq k} \delta(x_j-x_k)\right] \Psi_n + \left[-\frac{c}{2}\sum_{k \neq j} \text{sgn}(x_j-x_k)\right]^2 \Psi_n \end{array}$ Taking the sum over all $j$ we get $\displaystyle \sum_{j=1}^n \partial_{x_j}^2 \Psi_n = -2c\sum_{j < k} \delta(x_j-x_k) \Psi_n + \frac{c^2}{4}\sum_{j=1}^n \left[\sum_{j \neq k} \text{sgn}(x_j-x_k)\right]^2 \Psi_n$ To compute the second term, note that it is symmetric under permutations of the $x_j$. Let us assume (without loss of generality) that $x_1 < x_2 < ... < x_n$. Then $\displaystyle \begin{array}{lcl} \sum_{j=1}^n \left[\sum_{j \neq k} \text{sgn}(x_j-x_k)\right]^2 & = & \sum_{j=1}^n \left[\sum_{k>j} (-1)+\sum_{k Thus, we see that as claimed, $\displaystyle H_n \Psi_n = E_n \Psi_n$ with the energy $E_n = \frac{c^2}{12}n(n^2-1)$. To see that this is, indeed, the ground state, is beyond the scope of this post (essentially it would require one to take a finite system volume $V$, construct all excited states using the Bethe ansatz, and check that they all have higher energy. Since one knows the number of eigenstates in a finite volume, this would exclude any lower-energy state). Believing that this is, indeed, the ground state our argument shows that for large polymer length $L$, the moments of the partition sum $Z$ grow as: $\displaystyle \overline{Z^n (L)} \propto e^{\frac{c^2}{12}n(n^2-1) L}$ ## Implications for the string geometry The asymptotics of $\overline{Z^n}$ derived above is consistent with a free energy that scales with polymer length $L$ as $\ln Z = F =: f L^\frac{1}{3}$. Furthermore, the distribution $P(f)$ of the rescaled free energy should have a tail decaying as $\displaystyle P(f) \propto e^{-f^{\frac{3}{2}}} \quad \text{for } f \gg 1$ To see this, let us compute $\overline{Z^n}$ assuming this form for $P(f)$: $\displaystyle \overline{Z^n} = \int \mathrm{d}f e^{n f L^{\frac{1}{3}}-f^{\frac{3}{2}}}$ For large $n$, we expect $f$ to be large; thus the integral can be approximated by the maximum of the exponent. It occurs at $f \propto \left(n L^{\frac{1}{3}}\right)^2$ and thus we get $\displaystyle \ln \overline{Z^n} \propto L n^3$, consistent with the large-$n$ behaviour of the explicit formula for $\overline{Z^n}$ derived above. Thus, the asymptotics of $\overline{Z^n}$ allowed us to determine the tail of the free energy distribution $P(f)$ and the scaling $F = f L^\frac{1}{3}$. The scaling of the typical displacement $x$ with polymer length $L$ follows: $\displaystyle L^\frac{1}{3} \sim F \sim \int_0^L \mathrm{d}t (\dot{x})^2 \sim L \left(\frac{x}{L}\right)^2 \Rightarrow x \sim L^\frac{2}{3}$ Thus, the polymer is rough in the sense that the typical fluctuations don’t remain bounded, but increase with polymer length as $\displaystyle \overline{\left[x(L)-x(0)\right]^2}\sim L^{2\zeta}$ with the so-called roughness exponent $\zeta= \frac{2}{3}$. This whole analysis remains valid at an arbitrarily weak coupling strength $c$. Thus, the polymer is rough for arbitrarily weak disorder (or, equivalently, arbitrarily high temperatures). In order to observe a true phase transition between weak-disorder and strong-disorder behaviour one needs to go to higher dimensions. ## Higher dimensions and phase transitions As we have now seen, in two dimensions (one longitudinal + one transversal) the directed polymer is always in a rough phase. It turns out (but it would take too long to explain here) that the same holds in three (1+2) dimensions. However, starting from one longitudinal + three transversal dimensions, weak disorder is insufficient to make the polymer rough. In other words, there is a high-temperature phase where the moments of $Z$ scale as $\ln \overline{Z^n} \propto n^2$, indicating Gaussian fluctuations, i.e. purely thermal roughness. ## References The original argument for obtaining the moments of $Z$ from the Bethe ansatz solution of the Lieb-Liniger model is due to Kardar (see e.g. Nucl. Phys. B 290, 1989 and the references therein). The saddle-point argument for the tail of the free energy distribution is due to Zhang (see e.g. PRB 42, 1990). If you feel some other reference should be included please let me know! Written by inordinatum July 29, 2012 at 9:42 pm ### 10 Responses 1. […] few posts ago I briefly talked about the problem of a directed polymer in a random medium. In that post, I discussed the scaling behaviour of the polymer fluctuations due to the disordered […] 2. A time derivative, $\partial_t \Psi_n$, seems to be missing in front of the evolution equation for $\Psi$, the Schrodinger equation. Great web notes, congratulations! Juan M Lopez September 16, 2016 at 3:45 pm • Thanks, I just corrected that! Glad you enjoyed it 🙂 inordinatum September 17, 2016 at 9:34 am 3. […] a cartesian lattice in 1+1 dimensions, the roughness exponent is known exactly (see e.g. my earlier introduction to directed polymers) and, in some cases, even the distribution of the free energy can be computed (see my earlier post […] 4. Great Introduction again! I missed exactly how one gets the ground energy value. Is their a simple counting argument? Thanks a lot GUILLAUME A January 31, 2018 at 4:27 pm • Thanks! The ground energy value is calculated in section “Solution in one spatial dimension: Bethe ansatz ground state” – the calculations there show that when you apply the Hamiltonian (including the Laplacian and the delta-functions sum) to the wave function $\Psi_n$ you obtain again $\Psi_n$, multiplied by $E_n$… Let me know if something is unclear there! inordinatum February 3, 2018 at 4:03 pm • Oh yes you are right! I was wondering how does one finds the eigen vectors/values together by some general method… but of course it is all enclosed in the Bethe ansatz itself ! My bad, I never learned that properly, now I know at least that bit 😉 Thanks again! GUILLAUME A February 3, 2018 at 7:00 pm • Let me try to ask a more ‘clever’ question this time: So as it looks, Bethe ansatz works pretty well for delta correlated potentials, I guess we can work out generalizations ? And as a corollary, could you indicate maybe a few cases where the roughness exponent is modified in specific ways ? I am interested to delve deeper into it. Again : Great blog! GUILLAUME A February 3, 2018 at 7:18 pm • Thanks 🙂 The roughness exponent is actually pretty robust, on large scales it appears independently of the small-scale details of the system. I.e., even if you smoothen the delta-correlated potential slightly, or add small thermal fluctuations, it will remain the same. A different roughness exponent is obtained if you modify the universality class of your model, e.g. by changing the spatial dimension or by introducing long-range correlations (e.g. long-range elastic coupling or long-range correlated disorder). See e.g. the following papers and the references therein: – Peng, Havlin, Schwartz, Eugene Stanley 1991, “Directed-polymer and ballistic-deposition growth with correlated noise”, available on ResearchGate – Rosso, Krauth 2002, “Roughness at the depinning threshold for a long-range elastic string”, arXiv: cond-mat/0107527 – Fedorenko, Le Doussal, Wiese 2006, “Statics and dynamics of elastic manifolds in media with long-range correlated disorder”, arxiv:cond-mat/0609234 inordinatum February 5, 2018 at 9:16 pm • Fascinating! GUILLAUME A February 6, 2018 at 10:04 am	{"extraction_info": {"found_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": 76, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9383302330970764, "perplexity": 1132.218997098604}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647406.46/warc/CC-MAIN-20180320111412-20180320131412-00768.warc.gz"}
https://mathoverflow.net/questions/380008/a-p-adic-logarithm-as-a-limit-of-discrete-logs	# A p-adic logarithm as a limit of discrete logs I've been searching for something similar to the argument below for about a week now and I just must be missing out on the right key words. Can someone point me in the right direction and/or let me know where I'm going wrong? I haven't worked out the details yet; spending most of my time trying to find the article where someone must've worked through all this before... Let $$p$$ be an odd prime. Let $$g$$ be a primitive root mod $$p^n$$ for all $$n \ge 2$$. Then for any $$a \in \mathbb{Z}$$ relatively prime to $$p$$, we can define the index $$\text{ind}_{(g,n)}(a) = k_n$$ where $$g^{k_n} \equiv a \mod p^n$$. Such $$k_n$$ are only defined modulo $$\phi(p^n) = (p-1)p^{n-1}$$. But it also must be true that $$k_n \equiv k_{n-1} \mod (p-1)p^{n-2}$$. Thus for some $$0 \le c_i \le p$$ and $$0\le c_0 \le p-1$$, $$k_n \equiv c_0 + c_1(p-1)+c_2(p-1)p+c_3(p-1)p^2+\cdots+c_{n-1}(p-1)p^{n-2} \mod (p-1)p^{n-1}.$$ Hence $$\{k_n\}$$ converges $$p$$-adically to some $$k \in \mathbb{Q}_p$$. Define $$\text{ind}_g(a)=k$$. Seems reasonable to believe then that $$g^k = a$$ in $$\mathbb{Q}_p$$. Questions: • Initially the domain is integers relatively prime to $$p$$. It seems like these methods can be extended to $$z \in \mathbb{Q}$$ with $$\|z\|_p = 1$$ (for $$z=\frac{a}{b}$$, just use $$ab^{-1}$$ computed mod $$p^n$$). So perhaps it can be defined on any unit of $$\mathbb{Q}_p$$? • How does this relate (if at all) to the $$p$$-adic logarithm that shows up in all my internet searches? It seems that $$\text{ind}_g$$ is defined on the boundary of the domain of $$\log_p$$. For $$p\ge 3$$ if $$g$$ is a generator of $$(\Bbb{Z}/(p^2))^\times$$ then it is a generator of all $$(\Bbb{Z}/(p^k))^\times$$. For $$a\in \Bbb{Z}_p^\times$$ there is a unique $$l_{g,k}(a)\in \Bbb{Z}/(p-1)p^{k-1}\Bbb{Z}$$ such that $$a= g^{l_{g,k}(a)}\bmod p^k$$. For $$m, $$l_{g,k}(a)= l_{g,m}(a) \bmod (p-1)p^{m-1}$$ thus $$l_{g,k}(a)= l_{g,m}(a) \bmod p^{m-1}$$ and hence $$l_g(a)=\lim_{k\to \infty}l_{g,k}(a)$$ converges in $$\Bbb{Z}_p$$ and $$a=\lim_{k\to \infty} g^{l_{g,k}(a)}$$. Let $$c_k\in 0\ldots p^{k-1}-1,c_k = l_{g,k}(a)\bmod p^{k-1}$$. Then $$a=\lim_{k\to \infty} g^{c_k}$$ iff $$a=1\bmod p$$. In other words the full discrete logarithm of a $$p$$-adic number is an element of $$\Bbb{Z}/(p-1) \times \Bbb{Z}_p$$ and when keeping only the $$\Bbb{Z}_p$$ part we lose the $$a\bmod p$$ information. If $$c\in 1+p\Bbb{Z}_p$$ then $$l_g(c) = \frac{\log_p(c)}{\log_p(g^{p-1})}\in \Bbb{Z}_p$$ where $$\log_p$$ is the $$p$$-adic logarithm $$\log_p(1+pb)=\sum_{n\ge 1}\frac{p^n(-1)^{n-1} b^n}{n}, b\in \Bbb{Z}_p$$ For $$a,a'\in \Bbb{Z}_p^\times$$, $$l_g(a)=l_g(a')\in \Bbb{Z}_p$$ iff $$a/a'$$ has finite order in $$\Bbb{Z}_p^\times$$ iff $$a^{p-1}=(a')^{p-1}$$. $$\log_p(g^{p-1})\ne 1$$ when $$g$$ is an integer. $$\log_p(1+pb)=1$$ iff $$1+pb=\sum_{n\ge 0}\frac{p^n b^n}{n!}=\exp_p(1)$$ which is not in $$\Bbb{Q}\cap \Bbb{Z}_p$$. $$\log_p$$ is the discrete logarithm in base $$\exp_p(1)$$, ie. $$\log_p(1+pb)= \lim_{k\to \infty} l_{\exp_p(1)\bmod p^k,k}(1+pb)$$. • This is great stuff. Is there a resource (book, article, etc.) that gives more detail? Or is this just, as my advisor used to say, known by those who know things? Jan 2, 2021 at 19:28 • Also, this doesn't answer the question of if the $k_n$ as constructed work. The $k_n$ explicitly contain the $\mathbb{Z}/(p-1)$ information. So we should be able to retain $a \mod p$. Mar 20 at 17:30 For whatever it's worth, below is a paper of mine that discusses lifting the elliptic curve discrete log in $$E(\mathbb F_p$$) to $$E(\mathbb Z/p^2\mathbb Z)$$ and eventually to $$E(\mathbb Z_p)$$. It gives two methods, neither of which allows one to solve the ECDLP, but they fail for somewhat different reasons, which I always found kind of interesting. BTW, you might want to add a cryptography tag to your question, since clearly it's very relevant there. Again talking about elliptic curves, in the case the $$\#E(\mathbb F_p)=p$$, then in fact this sort of lifting procedure does work, because you can (more-or-less) multiply the point by $$p$$ to get into the formal group while maintaining the order coming from the mod $$p$$ point. That's the only situation that I know of where such a lifting works. I mention it because in some sense it shows what goes wrong in the classical $$\mathbb F_p$$ case, namely the order of the group $$\mathbb F_p^*$$ is $$p-1$$, which is not a multiple of $$p$$. Lifting and Elliptic Curve Discrete Logarithms, Conference: Selected Areas in Cryptography, 15th International Workshop, SAC 2008, Sackville, New Brunswick, Canada, August 14-15, DOI: 10.1007/978-3-642-04159-4_6	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 78, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9858222007751465, "perplexity": 572.360055208107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949701.56/warc/CC-MAIN-20230401063607-20230401093607-00497.warc.gz"}
http://gdaymath.com/lessons/fractions/3-5-percentages/	## Fractions are Hard! ### 3.5 Percentages In the first century B.C.E., Roman Emperor Augustus levied for the first time a tax of one part per hundred on the proceeds of all goods sold at markets and auctions in ancient Rome. From the Latin phrase per centum meaning “by the hundred” came the term percent. A percent is just a fraction with denominator one hundred (the number of pies per 100 boys). For example, $$\dfrac{1}{2}$$ written as a fraction with denominator one hundred is $$\dfrac{1}{2}=\dfrac{1\times 50}{2\times 50}=\dfrac{50}{100}$$. We say that $$\dfrac{1}{2}$$ equals 50 percent, and write $$\dfrac{1}{2}=50\%$$. Levying a tax of 1 dollar for every 100 dollars results in $$\dfrac{1}{100}$$, one one-hundredth, of all cash exchanged being given to the government. This is tax rate of $$1\%$$. The symbol $$\%$$ developed in Italy during the 1500s. Clerks started shortening per cento to P 00, which then eventually became $$%$$. Working backwards is actually easier. For example, 80% is a fraction with denominator 100. We have $$80\%=\dfrac{80}{100}$$ which is equivalent to the fraction $$\dfrac{4}{5}$$. Some more examples: $$\dfrac{3}{10}=\dfrac{30}{100}=100\%$$ $$2 = \dfrac{200}{100}=200\%$$ $$0.632=\dfrac{0.632}{1}=\dfrac{0.632\times 100}{1 \times 100}=\dfrac{63.2}{100}=63.2\%$$ Question: Nervous Nelly wants a rule for understanding percentages. Someone once told her that to write a number $$x$$ as a percentage, just multiply that number by 100 and write a $$\%$$ sign after the result. For example: $$\dfrac{1}{2}\times 100 = 50$$ and indeed $$\dfrac{1}{2}$$ is $$50\%$$. $$0.632\times 100 = 63.2$$ and indeed $$0.632$$ is $$63.2\%$$. $$\dfrac{3}{7}\times 100 = \dfrac{300}{7}=42\dfrac{6}{7}$$ and indeed $$\dfrac{3}{27}$$ is $$42\dfrac{6}{7}\%$$. One day, maybe in the distant future, Nervous Nelly will ask why this rule works. What would you say to her? How would you explain why this rule is true? (And did she even need to memorize this as rule in the first place?) And backwards $$53\% = \dfrac{53}{100}$$ $$25\% = \dfrac{25}{100}=\dfrac{1}{4}$$ $$150\% = \dfrac{150}{100}=1\dfrac{1}{2}$$ $$1000\% = \dfrac{1000}{100}=10$$ Question: The price of a Nice-and-Spicy Soup Cup yesterday was $1.50. It’s price went up to$1.80 overnight. What percentage increase is that? Question: The ancient Romans also spoke of parts per thousand. Today this is called, if it is ever used, per millage with the symbol o/oo. For example, 467 o/oo equals $$\dfrac{467}{1000}$$, and one tenth, which is $$\dfrac{1}{10} = \dfrac{100}{1000}$$, equals $$100$$ o/oo. Convert each of these per millages into fractions a) 40 o/oo b) 500 o/oo c) 6000 o/oo d) 2 o/oo Convert each of the following fractions into per millages: a) $$\dfrac{1}{100}$$ b) $$\dfrac{2}{5}$$ c) $$30\%$$ d) $$1$$ e) $$0$$ ## Books Take your understanding to the next level with easy to understand books by James Tanton. BROWSE BOOKS ## Guides & Solutions Dive deeper into key topics through detailed, easy to follow guides and solution sets. BROWSE GUIDES ## Donations Consider supporting G'Day Math! with a donation, of any amount. Your support is so much appreciated and enables the continued creation of great course content. Thanks! Donations can be made via PayPal and major credit cards. A PayPal account is not required. Many thanks! DONATE	{"extraction_info": {"found_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": 0.8040818572044373, "perplexity": 3701.760488930401}, "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-2017-30/segments/1500549424088.27/warc/CC-MAIN-20170722162708-20170722182708-00120.warc.gz"}
https://paperswithcode.com/paper/on-the-power-of-adaptivity-in-matrix	# On the Power of Adaptivity in Matrix Completion and Approximation 14 Jul 2014 We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence assumptions on the matrix row space that are necessary for passive sampling procedures... (read more) PDF Abstract # Code Edit Add Remove Mark official No code implementations yet. Submit your code now	{"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": 0.8926270604133606, "perplexity": 2267.615284010358}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1606141733120.84/warc/CC-MAIN-20201204010410-20201204040410-00260.warc.gz"}
https://www.physicsforums.com/threads/concept-of-measurable-functions.110596/	# Concept of measurable functions 1. Feb 14, 2006 ### Castilla First I had learned this definition of a "measurable function" (Apostol): "Let I be an interval. A function f: I -> R is said to be measurable on I if there exists a sequence of step functions (s_n(x) ) such that s_n(x) -> f(x) as n -> infinity for almost all x in I." But now in other sources (for example, the web notes of proffesor Dung Le) I have found this definition: "A function f: E -> R is measurable if E is a measurable set and for each real number r the set { x in E / f(x) > r } is measurable." First I thought Apostol and Le were talking about different things, but afterwards I found that both use their definitions to prove similar theorems, as this one: (In Apostol version): "Suppose that I is an interval and that f, g are measurable functions on I. Then so are f + g, f - g, fg, \|f\|, max {f,g}, min {f,g}". (In Dung Le version): "Let f: E-> R and g: E -> R be measurable functions. Then the functions (k is a real) kf, f+ g, \|f\| and fg are measurable." I was trying to learn the Fundamental Theorem of Calculus with Lebesgue integrals. Dung Le has it in his notes, but he uses the second definition of a measurable function and I have learned the basics of Lebesgue integration in Apostol, which only uses the first definition. Furthermore, to proof the FTC I see that Dung Le use a lemma by which "the function equivalent to the infimun of a set of measurable functions is also measurable". But I dont know if this lemma has an equivalent in the Apostol approach and with Apostol's definition of a measurable function. Is there is some way of reunite these two definitions in one? 2. Feb 14, 2006 ### matt grime I'm pretty sure that these are equivalent definitions. Measurable is such a universal term I do not believe there would be two inequivalent definitions for it. Certainly Apostols' is equivalent to the second (step functions are measurable in the second sense, and hence since measurable functions are closed under limits, anything in Apostol's definition satisfies Le's), and the converse is to prove that step functions are dense in Le's definition which seems clear if messy: split the range up into intervals, inverse image of each is measurable, approximate with step functions which we can therefore do, yeah, ought to work. Last edited: Feb 14, 2006 3. Feb 20, 2006 ### Castilla My "problem" is that I have learn some things about Lebesgue integration in Apostol's book but now it seems that to understand the Fundamental Theorem of Calculus for Lebesgue Integrals what I have learned helps little or nothing and I have to begin from zero with arid measure theory. 1. In the Apostol approach, the Lebesgue integral on an interval I is defined first for step functions (in that case is the same that the Riemann integral). 2. Then we define the "upper functions". f is an upper function iff: - there exists a sequence of increasing step functions (s_n(x)) which converges to f(x) almost everwhere on I. - there exists lim (n->oo) integral s_n(x)dx, and we define integral f(x)dx = said limit. 3. Then we define Lebesgue functions this way: v is a Lebesgue function if there exists two upper functions f, g such that v = f - g. We learn here the monotone convergente theorem, the beppo levi's theorem and the Lebesgue dominated theorem. 4. Then we define measurable functions (see Apostol's definition on my first post) and we learned, among others, the "diferentiation under the integral sign" theorem for lebesgue integrals. All of this with little or nothing of Measure Theory. But now I want to learn the FTC for Lebesgue integrals and in all sources I have found this theorem is embebbed on an approach completely based on dull, arid measure theory. There must exist some book which shares Apostol approach and includes the FTC!! Or not? Similar Discussions: Concept of measurable functions	{"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": 0.9868186712265015, "perplexity": 843.5255217139963}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109670.98/warc/CC-MAIN-20170821211752-20170821231752-00511.warc.gz"}
http://mathhelpforum.com/pre-calculus/126647-i-need-explanation.html	# Thread: I need an explanation 1. ## I need an explanation Can anyone explain to me the Fibonacci rabbits problem in a simple way? 2. Just think of a line, and think of three points. You go down to the first point, then the second, and at the third the line diverges. Once the line has diverged the old line diverges at each point, but the new line must wait two points. so we start with 1 line, it goes down to a point. Diverges and so on. 1. 1. (diverges) 2. (old diverges) 3. (old & 1new diverges) 5. (old and 1new diverges & 2new diverges) 8. annnd there is gets too complex for me to think in my head...	{"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": 0.8352718949317932, "perplexity": 1284.3870413278944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541066.79/warc/CC-MAIN-20161202170901-00080-ip-10-31-129-80.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/64407/is-the-extra-condition-in-this-definition-superfluous	# Is the extra condition in this definition superfluous? I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me if he is right. Definition: Suppose two curves $C_1$ and $C_2$ have a regular point P in common. Take a point A on $C_1$ near P and let $D_A$ be the point on $C_2$ closest to A (i.e orthogonal projection of A on $C_2$) then $C_2$ has contact of order $n$ with $C_1$ at P if $$\lim_{A\to P} \dfrac{ \mbox{dist}(A,D_A)}{[\mbox{dist}(A,P)]^k}= \begin{cases} c\neq 0, & \text{if } \; k=n+1 \\ 0, & \text{if } \; k\leq n \end{cases}$$ Argument: $$\lim_{A\to P} \frac{AD_A}{(AP)^{n+1}} =c \neq 0$$ then $$\lim_{A\to P} \frac{AD_A}{(AP)^{n}} = \lim_{A\to P} AP \cdot \frac{AD_A}{(AP)^{n+1}}$$ $$= \lim_{A\to P} AP \cdot \lim_{A\to P} \frac{AD_A}{(AP)^{n+1}} = 0 \cdot c =0$$ The same process can be followed to show that the limit is zero for all $k\leq n$ Note: The above definition is claimed by the authors to be extracted from this original source - I am suspicious: the argument does not use the fact that $c \neq 0$. Instead you need that $\|c\| < \infty$. [Added: I think that the definition is an emphatic way of saying "the smallest $k$ such that $\lim_{A \to P} \frac{AD_A}{AP^{k+1}}$ is nonzero".] – Srivatsan Sep 15 '11 at 18:22 @Srivatsan why not? It uses the fact that the limit for $k=n+1$ is $c\neq 0$ to show that the limit would be zero for every $k\leq n$. Re: additional comment, I agree, specially after seeing this argument. I need a confirmation that nothing fishy is going on as this refutes an argument presented in two books. – kuch nahi Sep 16 '11 at 6:38 @Srivatsan Yes. So I guess this person's argument is right. The statement should be that order is the minimum $n$ for which the stated limit is non zero and finite. – kuch nahi Sep 16 '11 at 6:43 Nonzero, yes. But I am not sure about finite. (Also note that the limit is nonzero for $k=n+1$, not $k=n$. So it's more like the the maximum $n$ for which the limit is zero.) – Srivatsan Sep 16 '11 at 6:46 @Srivatsan if it is not finite then we cannot conclude that $0\cdot c =0$ and the argument won't stand. – kuch nahi Sep 16 '11 at 6:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9663727283477783, "perplexity": 234.39982872546622}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1448398445208.17/warc/CC-MAIN-20151124205405-00131-ip-10-71-132-137.ec2.internal.warc.gz"}
http://www.planetmath.org/integrationunderintegralsign	# integration under integral sign Let $I(\alpha)\;=\;\int_{a}^{b}\!f(x,\,\alpha)\,dx.$ where $f(x,\,\alpha)$ is continuous in the rectangle $a\leqq x\leqq b,\,\quad\alpha_{1}\leqq\alpha\leqq\alpha_{2}.$ Then $\alpha\mapsto I(\alpha)$ is continuous and hence integrable (http://planetmath.org/RiemannIntegrable) on the interval$\alpha_{1}\leqq\alpha\leqq\alpha_{2}$; we have $\int_{\alpha_{1}}^{\alpha_{2}}I(\alpha)\,d\alpha\;=\;\int_{\alpha_{1}}^{\alpha% _{2}}\left(\int_{a}^{b}\!f(x,\,\alpha)\,dx\right)d\alpha.$ This is a double integral over a in the $x\alpha$-plane, whence one can change the order of integration (http://planetmath.org/FubinisTheorem) and accordingly write $\int_{\alpha_{1}}^{\alpha_{2}}\left(\int_{a}^{b}\!f(x,\,\alpha)\,dx\right)d% \alpha\;=\;\int_{a}^{b}\left(\int_{\alpha_{1}}^{\alpha_{2}}\!f(x,\,\alpha)\,d% \alpha\right)dx.$ Thus, a definite integral depending on a parametre may be integrated with respect to this parametre by performing the integration under the integral sign. Example. For being able to evaluate the improper integral $I\;=\;\int_{0}^{\infty}\frac{e^{-ax}-e^{-bx}}{x}\,dx\qquad(a>0,\;b>0),$ we may interprete the integrand as a definite integral: $\frac{e^{-ax}-e^{-bx}}{x}\;=\;\operatornamewithlimits{\Big{/}}_{\!\!\!\alpha=b% }^{\,\quad a}\!\frac{e^{-\alpha x}}{x}\;=\;\int_{a}^{b}\!e^{-\alpha x}\,d\alpha.$ Accordingly, we can calculate as follows: $\displaystyle I$ $\displaystyle\;=\;\int_{0}^{\infty}\left(\int_{a}^{b}\!e^{-\alpha x}\,d\alpha% \right)dx$ $\displaystyle\;=\;\int_{a}^{b}\left(\int_{0}^{\infty}\!e^{-\alpha x}\,dx\right% )d\alpha$ $\displaystyle\;=\;\int_{a}^{b}\left(\operatornamewithlimits{\Big{/}}_{\!\!\!x=% 0}^{\,\quad\infty}\!-\frac{e^{-\alpha x}}{\alpha}\right)d\alpha$ $\displaystyle\;=\;\int_{a}^{b}\!\frac{1}{\alpha}\,d\alpha\;=\;% \operatornamewithlimits{\Big{/}}_{\!\!\!a}^{\,\quad b}\!\ln{\alpha}$ $\displaystyle\;=\;\ln\frac{b}{a}$ Title integration under integral sign IntegrationUnderIntegralSign 2013-03-22 18:46:27 2013-03-22 18:46:27 pahio (2872) pahio (2872) 5 pahio (2872) Theorem msc 26A42 FubinisTheorem DifferentiationUnderIntegralSign RelativeOfExponentialIntegral	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 16, "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": 0.9994356632232666, "perplexity": 1665.6927239080617}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867977.85/warc/CC-MAIN-20180527004958-20180527024958-00094.warc.gz"}
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http://myafricanschool.com/grade12workenergyandpower	Thursday, February 2, 2023 ## Grade 12 Work, Energy and Power #### 1.Introduction We use the term ‘work’ in everyday conversation to mean many different things. We talk about going to work, doing homework, working in class. Physicists mean something very speciﬁc when they talk about work. In Physics we use the term work to describe the process of transferring energy from object or system to another or converting energy from one form to another. You will learn that work and energy are closely related to Newton’s laws of motion. You shall see that the energy of an object is its capacity to do work and doing work is the process of transferring energy from one object or form to another by means of a force. In other words, • an object with lots of energy can do lots of work. • when object A transfers energy to object B, the energy of object A decreases by the same amount as the energy of object B increases, we say that object A does work on object B. Lifting objects or throwing them requires that you do work on them. Even making an electrical current ﬂow requires that something do work. Objects or systems must have energy to be able to do work on other objects or systems by transferring some of their energy. • Units and unit conversions — Physical Sciences, Grade 10, Science skills • Equations — Mathematics, Grade 10, Equations and inequalities • Techniques of vector addition —Physical Sciences, Grade 10, Vectors and scalars • Newton’s laws — Physical Sciences, Grade 11, Forces • Force diagrams — Physical Sciences, Grade 11, Forces #### 2. Work We cover different topics in different chapters in different grades but that doesn’t mean that they are not related. In fact, it is very important to note that all of the different topics related to mechanics (forces, mechanical energy, momentum, rectilinear motion) actually form a consistent picture of the same physical system. There have been examples where we’ve shown the same results using two methods, for example determining speed or velocity using equations of motion or conservation of mechanical energy. Learning about work will help us tie everything we’ve learnt about previously together. Work will allow us to connect energy transfer to forces, which we have already linked to momentum and the equations of motion. When a force tends to act in or against the direction of motion of an object we say that the force is doing work on the object. Speciﬁcally, work is deﬁned mathematically in terms of the force and the displacement of the object. DEFINITION: Work When a force acts in or against the direction of motion of an object, work is done on the object.W = FΔx cos θ Important: cos θ tells you the relative direciton of the force and the displacment which is important. If the component of the force along the direction of the displacement is opposite in direction to the displacement then the sign of the displacement vector and force vector will be different. This is regardless of which direction was chosen as a positive direction. Let us look at some examples to understand this properly. In the images below the grey dot represents an object. A force, $$\vec{F}$$, acts on the object. The object moves through a displacement, Δ$$\vec{x}$$. What is the sign of the work done in each case? It is only the direction of the force on the object that matters and not the direction from the source of the force to the object. In Figure 5.3 both powerlifters are exerting an upwards force on the weights. On the left the weight is being pulled upwards and on the right it is being pushed upwards. Weight lifting is a good context in which to think about work because it helps to identify misconceptions introduced by everyday use of the term ‘work’. In the two cases in Figure 5.3 everyone would describe moving the weights upwards as very hardwork. From a physics perspective, as the powerlifters lift the weight they are exerting a force in the direction of the displacement so positive work is done on the weights. Consider the strongman walking in Figure 5.4. He carries two very heavy sleds as far as he can in a competition. What work is the man doing on the sleds and why? Most people would say he is working very hard because the sleds are heavy to carry but from a physics perspective he is doing no work on the sleds. The reason that he does no work is because the force he exerts is directly upwards to balance the force of gravity and the displacement is in the horizontal direction. Therefore there is no component of the force in the direction of displacement (θ = 90°) and no work done. His muscles do need to use their energy reserves to maintain the force to balance gravity. That does not result in energy transfer to the sleds. Investigation: Is work done? Decide whether or not work is done in the following situations. Remember that for work to be done a force must be applied in the direction of motion and there must be a displacement. Identify which two objects are interacting, what the action-reaction pairs of forces are and why the force described is or isn’t doing work. 1. Max pushes against a wall and becomes tired. 2. A book falls off a table and free falls to the ground. 3. A rocket accelerates through space. 4. A waiter holds a tray full of plates above his head with one arm and carries it straight across the room at constant speed. For each of the above pictures, the force vector is acting in the same direction as the displacement vector. As a result, the angle θ = 0°because there is no difference in angle between the direction of applied force and the direction of displacement. The work done by a force can then be positive or negative. This sign tells us about the direction of the energy transfer. Work is a scalar so the sign should not be misinterpreted to mean that work is a vector. Work is deﬁned as energy transfer, energy is a scalar quantity and the sign indicates whether energy was increased or decreased. • If $$\vec{F}$$applied acts or has a component acting in the same direction as the motion, then positive work is being done. In this case the object on which the force is applied gains energy. • If the direction of motion and $$\vec{F}$$applied are opposite, then negative work is being done. This means that energy is lost and the object exerting the force gains energy. For example, if you try to push a car uphill by applying a force up the slope and instead the car rolls down the hill you are doing negative work on the car. Alternatively, the car is doing positive work on you! applied Worked example 1: Calculating work on a car when speeding up. QUESTION A car is travelling along a straight horizontal road. A force of 500 N is applied to the car in the direction that it is travelling, speeding it up. While it is speeding up is covers a distance of 20 m. Calculate the work done on the car. SOLUTION Step 1: Analyse the question to determine what information is provided • The magnitude of the force applied is F = 500 N. • The distance moved is Δx = 20 m. • The applied force and distance moved are in the same direction. Therefore, the angle between the force and displacement is θ = 0° These quantities are all in SI units, so no unit conversions are required. Step 2: Analyse the question to determine what is being asked • We are asked to ﬁnd the work done on the car. We know from the deﬁnition that work done is W = FΔx cosθ. Step 3: Next we substitute the values and calculate the work done W = FΔx cosθ. =(500) (20) (cos 0) =(500) (20) (1) = 10 000 J Remember that the answer must be positive, as the applied force and the displacement are in the same direction. In this case, the car gains kinetic energy. Worked example 2: Calculating work on the car while braking What happens when the applied force and the motion are not parallel? By using the formula W = FΔx cosθ , we are actually calculating the component of the applied force in the direction of motion. Note that the component of the force perpendicular to the direction of motion does no work. Worked example 3: Calculating work done on a box pulled at an angle QUESTION Calculate the work done on a box, if it is pulled 5 m along the ground by applying a force of F = 20 N at an angle of 60° to the horizontal. SOLUTION Step 1: Analyse the question to determine what information is provided • The force applied is F=20 N • The distance moved is Δx = 5 m along the ground • The angle between the applied force and the motion is θ=60 These quantities are in the correct units so we do not need to perform any unit conversions. Step 2: Analyse the question to determine what is being asked We are asked to ﬁnd the work done on the box. Step 3: Substitute and calculate the work done Now we can calculate the work done on the box: W = FΔx cos θ =(20) (5) (cos 60) = 50 J Note that the answer is positive as the component of the force parallel to the direction of motion is in the same direction as the motion. The work done on the box is 50J. Exercise 5 – 1: Work 1.A 10 N force is applied to push a block across a frictionless surface for a displacement of 5,0 m to the right. The block has a weight of 20 N. Determine the work done by the following forces: normal force, weight , applied force. 2. A 10 N frictional force slows a moving block to a stop after a displacement of 5,0 m to the right. The block has a weight of 20 N Determine the work done by the following forces: normal force, weight, frictional force. 3. A 10 N force is applied to push a block across a frictional surface at constant speed for a displacement of 5,0 m to the right. The block has a weight of 20 N and the frictional force is 10 N. Determine the work done by the following forces: normal force, weight, applied force and frictional force. 4.An object with a weight of 20 N is sliding at constant speed across a frictionless surface for a displacement of 5 m to the right. Determine if there is any work done. 5.An object with a weight of 20 N is pulled upward at constant speed by a 20 N force for a vertical displacement of 5m . Determine if there is any work done. 6. Before beginning its descent, a roller coaster is always pulled up the ﬁrst hill to high initial height. Work is done on the roller coaster to achieve this initial height. A coaster designer is considering three different incline angles of the hill at which to drag the 2000 kg car train to the top of the 60 m high hill. In each case, the force applied to the car will be applied parallel to the hill. Her critical question is: which angle would require the least work? Analyse the data, determine the work done in each case, and answer this critical question. 7. A traveller carries a 150 N suitcase up four ﬂights of stairs (a total height of 12 m) and then pushes it with a horizontal force of 60 N at a constant speed of 0,25 m.s-1 for a horizontal distance of 50 m on a frictionless surface. How much work does the traveller do on the suitcase during this entire trip? 8. A parent pushes down on a pram with a force of 50 N at an angle of 30 to the horizontal. The pram is moving on a frictionless surface. If the parent pushes the pram for a horizontal distance of 30 m, how work is done on the pram? 9. How much work is done by the force required to raise a 2000 N lift 5 ﬂoors vertically at a constant speed? The vertical distance between ﬂoors is 5 m high. 10. A student with a mass of 60 kg runs up three ﬂights of stairs in 15 s, covering a vertical distance of 10 m. Determine the amount of work done by the student to elevate her body to this height. Net work We have only looked at a single force acting on an object. Sometimes more than one force acts at the same time (we dealt with this in Grade 11). We call the work done after taking all the forces into account the net work done. If there is only one force acting then the work it does, if any, is the net work done. In this case there are two equivalent approaches we can adopt to ﬁnding the net work done on the object. We can: • Approach 1: calculate the work done by each force individually and then sum them taking the signs into account. If one force does positive work and another does the same amount of work but it is negative then they cancel out. • Approach 2: calculate the resultant force from all the forces acting and calculate the work done using the resultant force. This will be equivalent to Approach 1. If the resultant force parallel to the direction of motion is zero, no net work will be done. Remember that work done tells you about the energy transfer to or from an object by means of a force. That is why we can have zero net work done even if multiple large forces are acting on an object. Forces that result in positive work increase the energy of the object, forces that result in negative work reduce the energy of an object. If as much energy is transferred to an object as is transferred away then the ﬁnal result is that the object gains no energy overall. Worked example 4: Approach 1, calculating the net work on a car QUESTION The same car is now accelerating forward, but friction is working against the motion of the car. A force of 300 N is applied forward on the car while it is travelling 20 m forward. A frictional force of 100 N acts to oppose the motion. Calculate the net work done on the car. Only forces with a component in the plane of motion are shown on the diagram. No work is done by Fg or FNormal as they act perpendicular to the direction of motion. SOLUTION Step 1: Analyse the question to determine what information is provided • The force applied is Fapplied=300 N forwards. • The force of friction is Ffriction=100 N opposite to the direction of motion. • The distance moved is Δx = 20 m. The applied force and distance moved are in the same plane so we can calculate friction the work done by the applied forward force and the work done by the force of friction backwards. Step 2: Analyse the question to determine what is being asked We are asked to ﬁnd the net work done on the car. We know from the deﬁnition that work done is W = FΔx cosθ As mentioned before, there is an alternative method to solving the same problem, which is to determine the net force acting on the car and to use this to calculate the work. This means that the vector forces acting in the plane of motion must be added to get the net force $$\vec{F}$$net. The net force is then applied over the displacement to get the net work Wnet. Worked example 5: Approach 2, calculating the net force QUESTION The same car is now accelerating forward, but friction is working against the motion of the car. A force of 300 N is applied forward on the car while it is travelling 20m forward. A frictional force of 100 N acts to oppose the motion. Calculate the net work done on the car. SOLUTION Step 1: Analyse the question to determine what information is provided • The force applied is $$\vec{F}$$applied=300 N forwards. • The force of friction is $$\vec{F}$$friction=100 N backwards. • The distance moved is Δx = 20 m. • The applied forces $$\vec{F}$$applied = 300 N and the force of friction $$\vec{F}$$friction= 100 N are in the same plane as the distance moved. Therefore, we can add the vectors. As vectors require direction, we will say that forward is positive and therefore backward is negative. Note, the force of friction is acting at 180° friction i.e. backwards and so is acting in the opposite vector direction i.e. negative. These quantities are all in the correct units, so no unit conversions are required. Step 2: Analyse the question to determine what is being asked • We are asked to ﬁnd the net work done on the car. We know from the deﬁnition that work done is Wnet=FnetΔxcosθ Step 3: We calculate the net force acting on the car, and we convert this into net work. First we draw the force diagram: Let forwards (to the left in the picture) be positive. We know that the motion of the car is in the horizontal direction so we can neglect the force due to gravity, $$\vec{F}$$g , and the normal force, $$\vec{N}$$. Note: if the car were on a slope we would need to calculate the component of gravity parallel to the slope. $$\vec{F}$$net=$$\vec{F}$$applied+$$\vec{F}$$friction =(+300)+(-100) $$\vec{F}$$ = 200 N forwards $$\vec{F}$$net is pointing in the same direction as the displacement, therefore the angle between the force and displacement is θ = 0. IMPORTANT! The two different approaches give the same result but it is very important to treat the signs correctly. The forces are vectors but work is a scalar so they shouldn’t be interpreted in the same way. Wnet= FnetΔxcosθ =(200) (20)cos(0) = 4000 J #### 3. Work Theorem Conservative and non-conservative forces In Grade 10, you saw that mechanical energy was conserved in the absence of non-conservative forces. It is important to know whether a force is an conservative force or an non-conservative force in the system, because this is related to whether the force can change an object’s total mechanical energy when it does work on an object. When the only forces doing work are conservative forces (for example, gravitational and spring forces), energy changes forms – from kinetic to potential (or vice versa); yet the total amount of mechanical energy (EK+ Ep) is conserved. For example, as an object falls in a gravitational ﬁeld from a high elevation to a lower elevation, some of the object’s potential energy is changed into kinetic energy. However, the sum of the kinetic and potential energies remain constant. Investigation: Non-conservative forces We can investigate the effect of non-conservative forces on an object’s total mechanical energy by rolling a ball along the ﬂoor from point A to point B. In the absence of friction and other non-conservative forces, the ball should slide along the ﬂoor and its speed should be the same at positions A and B. Since there are no non-conservative forces acting on the ball, its total mechanical energy at points A and B are equal. Now, let’s investigate what happens when there is friction (an non-conservative force) acting on the ball. Roll the ball along a rough surface or a carpeted ﬂoor. What happens to the speed of the ball at point A compared to point B? If the surface you are rolling the ball along is very rough and provides a large non-conservative frictional force, then the ball should be moving much slower at point B than at point A. Let’s compare the total mechanical energy of the ball at points A and B: However, in this case, VA ≠ VB and therefore ETotal,A ≠ ETotal,B. Since VA>VB ETotal,A> ETotal,B Therefore, the ball has lost mechanical energy as it moves across the carpet. However,although the ball has lost mechanical energy, energy in the larger system has still been conserved. In this case, the missing energy is the work done by the carpet through applying a frictional force on the ball. In this case the carpet is doing negative work on the ball. When an non-conservative force (for example friction, air resistance, applied force) does work on an object, the total mechanical energy (EK+ Ep) of that object changes. If positive work is done, then the object will gain energy. If negative work is done, then the object will lose energy. When a net force does work on an object, then there is always a change in the kinetic energy of the object. This is because the object experiences an acceleration and therefore a change in velocity. This leads us to the work-energy theorem. DEFINITION: Work-energy theorem The work-energy theorem states that the work done on an object by the net force is equal to the change in its kinetic energy: WNet=ΔEk=Ek,f-Ek,i Worked example 6: Work-energy theorem QUESTION A 1 kg brick is dropped from a height of 10 m. Calculate the work that has been done on the brick between the moment it is released and the moment when it hits the ground. Assume that air resistance can be neglected. SOLUTION Step 1: Determine what is given and what is required • Mass of the brick: m = 1 kg. • Initial height of the brick: hi= 10 m. • Final height of the brick: hfi= 0 m. • We are required to determine the work done on the brick as it hits the ground. Step 2: Determine how to approach the problem The brick is falling freely, so energy is conserved. We know that the work done is equal to the difference in kinetic energy. The brick has no kinetic energy at the moment it is dropped, because it is stationary. When the brick hits the ground, all the brick’s potential energy is converted to kinetic energy. Step 3: Determine the brick’s potential energy at hi Ep= m.g.hi =(1) (9,8) (10) = 98 J Step 4: Determine the work done on the brick The brick had 98 J of potential energy when it was released and 0 J of kinetic energy. When the brick hit the ground, it had 0 J of potential energy and 98 J of kinetic energy. Therefore Ek,i= 0 J and Ek,f= 98 J. From the work-energy theorem: Wnet = ΔEk = Ek,fEk,i = 98 – 0 = 98 J Hence, 98 J of work was done on the brick. Worked example 7: Work-energy theorem 2 QUESTION The driver of a 1000 kg car travelling at a speed of 16,7 m.s-1 applies the car’s brakes when he sees a red light. The car’s brakes provide a frictional force of 8000 N. Determine the stopping distance of the car. SOLUTION Step 1: Determine what is given and what is required We are given: • mass of the car: m = 1000 kg • speed of the car: v = 16,7 m.s-1 • frictional force of brakes: $$\tilde{F}$$= 8000 N We are required to determine the stopping distance of the car. Step 2: Determine how to approach the problem We apply the work-energy theorem. We know that all the car’s kinetic energy is lost to friction. Therefore, the change in the car’s kinetic energy is equal to the work done by the frictional force of the car’s brakes.Therefore, we ﬁrst need to determine the car’s kinetic energy at the moment of braking using: Ek=½mv2 This energy is equal to the work done by the brakes. We have the force applied by the brakes, and we can use: W = FΔx cosθ to determine the stopping distance. Worked example 8: Block on an inclined plane [credit: OpenStax College] QUESTION A block of 2 kg is pulled up along a smooth incline of length 10 m and height 5 m by applying an non-conservative force. At the end of incline, the block is released from rest to slide down to the bottom. Find the 1. work done by the non-conservative force, 2. the kinetic energy of the block at the end of round trip, and 3. the speed at the end of the round trip. We have represented the non-conservative force on the force diagram with an arbitrary vector$$\tilde{F}$$ acts only during upward journey. Note that the block is simply released at the end of the upward journey. We need to ﬁnd the work done by the non-conservative force only during the upward journey. WF=WF(up)+WF(down)=WF(up)+0=WF(up) The kinetic energies in the beginning and at the end of the motion up the slope are zero. We can conclude that sum of the work done by all three forces is equal to zero during the upward motion. The change in kinetic energy is zero which implies that the net work done is zero. WNet = WF(up)+Wg(up)+WN(up) 0 = WF(up)+Wg(up)+WN(up) If we know the work done by the other two forces (normal force and gravity), then we can calculate the work done by the non-conservative force, F, as required. Step 3: Work done by normal force during upward motion The block moves up the slope, the normal force is perpendicular to the slope and, therefore, perpendicular to the direction of motion. Forces that are perpendicular to the direction of motion do no work. WNet = WF(up)+Wg(up)+WN(up) 0 = WF(up)+Wg(up)+WN(up) WF(up) = -Wg(up) IMPORTANT! Be careful not to be confused by which angle has been labelled α and which θ. α is not the angle between the force and the direction of motion but the incline of the plane in this particular problem. It is important to understand which symbol represents which physical quantity in the equations you have learnt. Hence, the work done by the non-conservative force during the round trip is WF(up) = WF(up)= – Wg(up) = – (-98) = 98 J Step 5: Kinetic energy at the end of round trip The kinetic energy at the end of the upward motion was zero but it is not zero at the end of the entire downward motion. We can use the work-energy theorem to analyse the whole motion: W(round trip) = Ek,f– Ek,i = Ek,f– 0 = Ek,f Exercise 5 – 2: Energy 1. Fill in the table with the missing information using the positions of the 1 kg ball in the diagram below combined with the work-energy theorem. 2. A falling ball hits the ground at 10 m.s-1 in a vacuum. Would the speed of the ball be increased or decreased if air resistance were taken into account. Discuss using the work-energy theorem. 3. A pendulum with mass 300 g is attached to the ceiling. It is pulled up to point A which is a height h = 30 cm from the equilibrium position. Calculate the speed of the pendulum when it reaches point B (the equilibrium point). Assume that there are no non-conservative forces acting on the pendulum. #### 4. Conservation of energy There are two categories of forces we will consider, conservative and non-conservative. DEFINITION: Conservative force A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. A conservative force results in stored or potential energy and we can define a potential energy (Ep ) for any conservative force. Gravity is a conservative force and we studied gravitational potential energy in Grade 10. We now have all the concepts we need to actually deduce this ourselves. Let us consider pushing a ball up a number of different slopes. Figure 5.5: Three different slopes are shown, all rising to a height of h . The imaginary right- angled triangle is shown for each slope. d is the length of the slope. α is the angle the slope makes with the horizontal. The slope, of length d is the hypotenuse of an imaginary right-angled triangle. The work done by gravity while pushing a ball of mass, m , up each of the slopes can be This ﬁnal result is independent of the angle of the slope. This is because sinα opposite/hypotenuse=h/d and so the distance cancels out. If the ball moves down the slope the only change is the sign, the work done by gravity still only depends on the change in height. This is why mechanical energy includes gravitational potential energy and is conserved. If an object goes up a distance h gravity does negative work, if it moves back down h gravity does positive work, but the absolute amount of work is the same so you ‘get it back’, no matter what path you take! This means that the work done by gravity will be same for the ball moving up any of the slopes because the end position is at the same height. The different slopes do not end in exactly the same position in the picture. If we break each slope into two sections as show in Figure 5.6 then we have 3 different paths to precisely the same end-point. In this case the total work done by gravity along each path is the sum of the work done on each piece which is just related to the height. The total work done is related to the total height. Figure 5.6: Three different paths that lead from the same start-point to the same-end point. Each path leads to the same overall change in height, h, and, therefore, the same work done by gravity. There are other examples, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. The total work done by a conservative force results in a change in potential energy, ΔEp. If the conservative force does positive work then the change in potential energy is negative. Therefore: DEFINITION: Non-conservative force A non-conservative force is one for which work done on the object depends on the path taken by the object. IMPORTANT! Non-conservative forces do not imply that total energy is not conserved. Total energy is always conserved. Non-conservative forces mean that mechanical energy isn’t conserved in a particular system which implies that the energy has been transferred in a process that isn’t reversible. Friction is a good example of a non-conservative force because if removes energy from the system so the amount of mechanical energy is not conserved. Non-conservative forces can also do positive work thereby increasing the total mechanical energy of the system. The energy transferred to overcome friction depends on the distance covered and is converted to thermal energy which can’t be recovered by the system. Non-conservative forces and work-energy theorem We know that the net work done will be the sum of the work done by all of the individual forces: When the non-conservative forces oppose the motion, the work done by the non-conservative forces is negative, causing a decrease in the mechanical energy of the system. When the non-conservative forces do positive work, energy is added to the system. If the sum of the non-conservative forces is zero then mechanical energy is conserved. Worked example 9: Sliding footballer [credit: OpenStax College Physics] QUESTION Consider the situation shown where a football player slides to a stop on level ground. Using energy considerations, calculate the distance the 65,0 kg football player slides, given that his initial speed is 6,00 m.s-1 and the force of friction against him is a constant 450 N. SOLUTION Step 1: Analyse the problem and determine what is given Friction stops the player by converting his kinetic energy into other forms, including thermal energy. In terms of the work-energy theorem, the work done by friction, which is negative, is added to the initial kinetic energy to reduce it to zero. The work done by friction is negative, because F is in the opposite direction of the motion (that is,θ = 180 of, and so cos θ = 1). Thus Wnon-conservative = –FfΔx There is no change in potential energy. Step 3: Quote the ﬁnal answer The footballer comes to a stop after sliding for 2,60 m. Discussion The most important point of this example is that the amount of non conservative work equals the change in mechanical energy. For example, you must work harder to stop a truck, with its large mechanical energy, than to stop a mosquito. Worked example 10: Sliding up a slope [credit: OpenStax College Physics] QUESTION The same 65,0 kg footballer running at the same speed of 6,00 m.s-1 dives up the inclined embankment at the side of the ﬁeld. The force of friction is still 450 N as it is the same surface, but the surface is inclined at 5o . How far does he slide now? SOLUTION Step 1: Analyse the question Friction stops the player by converting his kinetic energy into other forms, including thermal energy, just in the previous worked example. The difference in this case is that the height of the player will change which means a non-zero change to gravitational potential energy. The work done by friction is negative, because Ff is in the opposite direction of the motion (that is, θ = 180). We sketch the situation showing that the footballer slides a distance d up the slope. As might have been expected, the footballer slides a shorter distance by sliding uphill.Note that the problem could also have been solved in terms of the forces directly and the work energy theorem, instead of using the potential energy. This method would have required combining the normal force and force of gravity vectors, which no longer cancel each other because they point in different directions, and friction, to ﬁnd the net force. You could then use the net force and the net work to ﬁnd the distance d that reduces the kinetic energy to zero. By applying conservation of energy and using the potential energy instead, we need only consider the gravitational potential energy, without combining and resolving force vectors. This simpliﬁes the solution considerably. Exercise 5 – 3: Energy conservation 1. A 60,0 kg skier with an initial speed of 12,0 m.s-1 coasts up a 2,50 m-high rise as shown in the ﬁgure. Find her ﬁnal speed at the top, given that the coefﬁcient of friction between her skis and the snow is 0,0800. (Hint: Find the distance traveled up the incline assuming a straight-line path as shown in the ﬁgure.) 2. a) How high a hill can a car coast up (engine disengaged) if work done by friction is negligible and its initial speed is 110 km.h-1 ? b) If, in actuality, a 750 kg car with an initial speed of 110 km.h-1 is observed to coast up a hill to a height 22,0 m above its starting point, how much thermal energy was generated by friction? c) What is the average force of friction if the hill has a slope 2,5 above the horizontal? 3. A bullet traveling at 100 m/s just pierces a wooden plank of 5 m. What should be the speed (in m/s) of the bullet to pierce a wooden plank of same material, but having a thickness of 10m? #### 5. Power Now that we understand the relationship between work and energy, we are ready to look at a quantity related the rate of energy transfer. For example, a mother pushing a trolley full of groceries can take 30 s or 60 s to push the trolley down an aisle. She does the same amount of work, but takes a different length of time. We use the idea of power to describe the rate at which work is done Power is defined as the rate at which work is done or the rate at which energy is transfered to or from a system. The mathematical definition for power is: P = W/t IMPORTANT! In the case where the force and the velocity are in opposite directions the power will be negative. The unit of power is watt (symbol W). Worked example 11: Power calculation 1 QUESTION Calculate the power required for a force of 10 N applied to move a 10 kg box at a speed of 1 m·s−1 over a frictionless surface. SOLUTION Step 1: Determine what is given and what is required. • We are given the force, F = 10 N. • We are given the speed, v = 1 m · s−1 . • We are required to calculate the power required Step 2: Draw a force diagram Step 3: Determine how to approach the problem from the force diagram, we see that the weight of the box is acting at right angles to the direction of motion. The weight does not contribute to the work done and does not contribute to the power calculation. We can therefore calculate power from: P = F · v . Step 4: Calculate the power required P = F · v = (10 N) 1 m · s−1 = 10 W Step 5: Write the final answer 10 W of power are required for a force of 10 N to move a 10 kg box at a speed of 1 m·s−1 Machines are designed and built to do work on objects. All machines usually have a power rating. The power rating indicates the rate at which that machine can do work upon other objects. A car engine is an example of a machine which is given a power rating. The power rating relates to how rapidly the car can accelerate. Suppose that a 50 kW engine could accelerate the car from 0 km·hr−1 to 60 km·hr−1 in 16 s. Then a car with four times the power rating (i.e. 200 kW) could do the same amount of work in a quarter of the time. That is, a 200 kW engine could accelerate the same car from 0 km·hr−1 to 60 km·hr−1 in 4s. Worked example 12: Power calculation 2 QUESTION A forklift lifts a crate of mass 100 kg at a constant velocity to a height of 8 m over a time of 4 s. The forklift then holds the crate in place for 20 s. Calculate how much power the forklift exerts in lifting the crate? How much power does the forklift exert in holding the crate in place? SOLUTION Step 1: Determine what is given and what is required We are given: • mass of crate: m=100 kg • height that crate is raised: h=8 m • time taken to raise crate: tr = 4 s • time that crate is held in place: ts = 20 sWe are required to calculate the power exerted. Step 2: Determine how to approach the problem We can use: to calculate power. The force required to raise the crate is equal to the weight of the crate. Step 3: Calculate the power required to raise the crate Step 4: Calculate the power required to hold the crate in place While the crate is being held in place, there is no displacement. This means there is no work done on the crate and therefore there is no power exerted. Step 5: Write the final answer 1960 W of power is exerted to raise the crate and no power is exerted to hold the crate in place. Worked example 13: Stair climb QUESTION Step 3: Quote the final answer The power generated is 538,0 W. The woman does 1764 J of work to move up the stairs compared with only 120 J to increase her kinetic energy; thus, most of her power output is required for climbing rather than accelerating. Worked example 14: A borehole QUESTION What is the power required to pump water from a borehole that has a depth h = 15,0 m at a rate of 20,0 l·s−1 ? SOLUTION Step 1: Analyse the question We know that we will have to do work on the water to overcome gravity to raise it a certain height. If we ignore any inefficiencies we can calculate the work, and power, required to raise the mass of water the appropriate height. We know how much water is required in a single second. We can first determine the mass of water: 20,0 l × 1 kg/1l = 20,0 kg. The water will also have non-zero kinetic energy when it gets to the surface because it needs to be flowing. The pump needs to move 20,0 kg from the depth of the borehole every second, we know the depth so we know the speed that the water needs to be moving is v = h/t = 15,0/1 = 15,0 m·s−1 . Experiment: Simple measurements of human power You can perform various physical activities, for example lifting measured weights or climbing a flight of stairs to estimate your output power, using a stop watch. Note: the human body is not very efficient in these activities, so your actual power will be much greater than estimated here. Exercise 5 – 4: Power 1. [IEB 2005/11 HG] Which of the following is equivalent to the SI unit of power: a) V·A b) V·A−1 c) kg·m·s−1 d) kg·m·s−2 2. Two students, Bill and Bob, are in the weight lifting room of their local gym. Bill lifts the 50 kg barbell over his head 10 times in one minute while Bob lifts the 50 kg barbell over his head 10 times in 10 seconds. Who does the most work? Who delivers the most power? Explain your answers. 3. Jack and Jill ran up the hill. Jack is twice as massive as Jill; yet Jill ascended the same distance in half the time. Who did the most work? Who delivered the most power? Explain your answers. 4. When doing a chin-up, a physics student lifts her 40 kg body a distance of 0,25 m in 2 s. What is the power delivered by the student’s biceps? 5. The unit of power that is used on a monthly electricity account is kilowatt-hours (symbol kWh). This is a unit of energy delivered by the flow of 1 kW of electricity for 1 hour. Show how many joules of energy you get when you buy 1 kWh of electricity. 6. An escalator is used to move 20 passengers every minute from the first floor of a shopping mall to the second. The second floor is located 5-meters above the first floor. The average passenger’s mass is 70 kg. Determine the power requirement of the escalator in order to move this number of passengers in this amount of time. #### 6. Chapter Summary Exercise 5 – 5: 1. How much work does a person do in pushing a shopping trolley with a force of 200 N over a distance of 80 m in the direction of the force? 2. How much work does the force of gravity do in pulling a 20 kg box down a 45◦ frictionless inclined plane of length 18 m? 3. [IEB 2001/11 HG1] Of which one of the following quantities is kg·m2 ·s−3 the base S.I. unit? 4. [IEB 2003/11 HG1] A motor is used to raise a mass m through a vertical height h in time t. What is the power of the motor while doing this? 5. [IEB 2002/11 HG1] An electric motor lifts a load of mass M vertically through a height h at a constant speed v. Which of the following expressions can be used to calculate the power transferred by the motor to the load while it is lifted at constant speed? 6. A set 193 kg containers need to be lifted onto higher floors during a building operation. The distance that they need to be raised is 7.5 m, at constant speed. The project manager has determined that in order to keep to budget and time this has to happen in as close to 5,0 s as possible. The power ratings for three motors are listed as 1,0 kW, 3,5 kW, and 5,5 kW. 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https://math.stackexchange.com/questions/2373253/how-is-moving-the-last-digit-of-a-number-to-the-front-and-multiplying-related-to	# How is moving the last digit of a number to the front and multiplying related to multiplicative orders? There seems to be a relationship between multiplicative orders modulo $n$ and a puzzle Presh Talwalkar gave a few days ago at https://www.youtube.com/watch?v=1lHDCAIsyb8 I'm hoping someone can give a more rigorous explanation of the pattern I see. $\bullet\textbf{ The Puzzle }\bullet$ He states the puzzle as: "What Positive Number Doubles When Its Last Digit Becomes Its First Digit?" For example, the smallest solution - assuming base-10 representation - is $105263157894736842$. Which means that $$(2)\cdot105263157894736842 \quad\ \ \$$ $$\|\|$$ $$210526315789473684$$ Naturally, one can proceede to find solutions in other bases. The base-5 solution is $$(2)\cdot102342_5 \quad\ \ \$$ $$\|\|$$ $$210234_5$$ $\bullet\textbf{ Multiplicative Order Connection }\bullet$ The base-10 solutions is $18$ digits and the base-5 solution is $6$ digits. I wrote a program to generate the smallest solution in all bases and then counted the number of digits in each such solution. Here's the sequence, starting with base-2 $$2,4,3,6,10,12,4,8,18,6,11,20,...$$ $$\uparrow \quad\quad\quad\quad\ \uparrow \quad\quad$$ $$\text{Base-} 5 \quad\quad\quad \text{Base-} 10 \quad\quad$$ A quick search on http://oeis.org/ shows that these numbers are the multiplicative order of $$2\ (\text{mod } 2B-1)$$ where $B$ is the representation base we are working in. See http://oeis.org/A002326 This puzzle can be further generalized by finding numbers that triple instead of double upon moving the last digit to the front. Let's introduce an arbitrary multiplying factor: $k$ So the solutions mentioned above and Talwalkar's puzzle are a particular case when $k=2$ For a different example, let's look at $k=5$ in base-7. The smallest solution is $$(5) \cdot 1013042156536245_7 \quad\ \ \$$ $$\|\|$$ $$5101304215653624_7$$ This solution is $16$ digits long. As we did before for base-10, we can write out a sequence of the number of digits in each base's solution starting with base-2 $$6,6,9,2,14,16,4,5,42,18,...$$ $$\uparrow\ \$$ $$\text{Base-}7\ \$$ Another search on http://oeis.org/ quickly shows that these numbers are the multiplicative order of $$5\ (\text{mod } 5\cdot7-1)\quad\text{or rather}\quad 5 \ (\text{mod }34)$$ $\bullet\textbf{ A Conjecture }\bullet$ The pattern might be clear now. After checking other cases, it seems that the smallest positive number in any base $B$ - which is $k$ times the value gotten by moving it's last digit to the front - has a number of digits equal to the multiplicative order of $$k\ (\text{mod } Bk-1)$$ I can see how multiplicative order would be related to this problem. But I can't find an exact reason why this relationship should be so. Is there an intuitive reason? Given some solution in base $B$ - let's start by assigning the variable $x$ to the digit that get's moved to the front and assigning to the variable $y$ to the rest of the number (Talwalkar did this in his video). So if we have a multiplying factor of $k$ we are looking for a solution to $$k(By+x)=B^nx+y$$ $$\text{where}\quad 0<x<B \quad\text{and}\quad n > \text{log}_B(y) \quad\text{and}\quad B\geq2 \quad\text{and}\quad 2\leq k < B$$ rearranging we get $$y = \frac{x(B^n-k)}{Bk-1}$$ Since $y$ is an integer we conclude that either $x$ or $B^n-k$ is divisible by $Bk-1$. But $x$ can't be divisible by $Bk-1$ because we have $$x<B<2B-1\leq Bk-1\quad\Rightarrow\quad x<Bk-1$$ Therefore $B^n-k$ must be divisible by $Bk-1$. Accordingly we write $$B^n-k\equiv 0\ (\text{mod }Bk-1)$$ $$\Updownarrow$$ $$\quad\quad\quad\quad\quad\quad k\equiv B^n\ (\text{mod }Bk-1) \quad\quad\quad\quad\quad(1)$$ We are almost there. Next we need to note a simple property of our modular arithmetic, namely that $Bk$ has a remainder of $1$ when divided by $Bk-1$ (duh). So we say that $$Bk\equiv 1\ (\text{mod }Bk-1)$$ $$\Downarrow$$ $$\quad\quad\quad\quad\quad\quad (Bk)^n\equiv 1\ (\text{mod }Bk-1) \quad\quad\quad\quad\quad(2)$$ Now we multiply the left and right sides of equations $(1)$ and $(2)$ respectively and observe $$k\cdot(Bk)^n\equiv B^n\cdot1\ (\text{mod }Bk-1)$$ $$\Downarrow$$ $$k^{n+1}B^n\equiv B^n\ (\text{mod }Bk-1)$$ $$\Downarrow$$ $$k^{n+1}\equiv 1\ (\text{mod }Bk-1)$$ Which means that $n+1$ is the multiplicative order of $k\ (\text{mod }Bk-1)$. But before we conclude that the solution also has $n+1$ digits, more needs to be said. The smallest solution has $1$ as it's first digit. Since $x$ is the first digit of the solution times $k$, we can conclude that $x=k$. Now to put it all together, our solution is $$By+x=\frac{1}{k}(B^nx+y)=\frac{1}{k}(B^nk+y)=B^n+\frac{y}{k}$$ which has $n+1$ digits. $\bullet\textbf{ Conclusion }\bullet$ We saw that the multiplicative order of $k\ (\text{mod }Bk-1)$ is the number of digits in the smallest solution to Talwalkar's puzzle for the base $B$ and multiplying factor $k$. It should be noted that we can also conclude that $k$ and $B$ have the same multiplicative order here. As follows $$k\cdot1\equiv B^n\cdot(Bk)\ (\text{mod }Bk-1)$$ $$\Downarrow$$ $$k\equiv B^{n+1}k\ (\text{mod }Bk-1)$$ $$\Downarrow$$ $$1\equiv B^{n+1}\ (\text{mod }Bk-1)$$ The smallest solution has the same number of digits as the multiplicative order of $B\ (\text{mod }Bk-1)$ and of $k\ (\text{mod }Bk-1)$ • "Either $x$ or $B^n-k$ is divisible by $Bk-1$" is only obvious if $Bk-1$ is prime. May 3, 2020 at 13:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8699411153793335, "perplexity": 102.51905426017143}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882573145.32/warc/CC-MAIN-20220818003501-20220818033501-00789.warc.gz"}
https://www.cranewarningsystemsatlanta.com/post/how-does-wind-affect-tower-cranes	Search # How Does Wind Affect Tower Cranes? We often take the raw power of wind for granted, considering it a mere nuisance blowing leaves and litter around like some weakling—except, of course, when there is a severe storm that we have to deal with. However, with tower cranes, the case is entirely different, where even the seemingly humble air currents can cause significant disaster due to “amplified” wind loads. # Effects of Wind on Tower Cranes Air is a mixture of various gases, where each gas in the mixture has a certain density. When the wind blows, the molecules of the gases in the air gain energy and are moved at different velocities depending on their size and mass. When these molecules encounter a surface in their path, they give some of their energy to the surface and consequently exert pressure onto it. This pressure is defined as: VP = KVS2, where • VP is the wind pressure • K is the density of a gas, which for design purposes is considered constant at 0.613 • VS is the wind speed As can be seen from the above equation, there exists a squared relationship between win pressure and wind speed. So if you double the wind speed, the wind pressure increases by a factor of four times. This is a crucial point to note because it underscores how a small increase in wind speed can significantly increase its impact. Wind speed itself is a function of height, i.e., wind speed increases with height. So the higher you go above the ground, the harder the wind tends to blow. Since tower cranes are erected at great heights, and sometimes even on top of buildings, the wind can easily reach speed levels where its raw power can become unforgiving and brutal for crane operations. It can destabilize a tower crane or the load on the hook and create potential accident situations. With the center of impact also at a significant distance from the ground, the overturning moment can get further amplified, making the operations even more difficult and dangerous for crane operators. That’s why it’s recommended that tower cranes should always be operated below maximum permissible speed and crane operators should continuously monitor crane wind speed when performing a lift. Here, at Crane Warning Systems Atlanta, we sell a variety of crane safety instrumentation systems, including crane Wind Speed Indicator system. Visit our online store to learn more about our crane-exclusive wind speed indicators. 3 views See All ### Follow us on Social Media Crane Warning Systems Atlanta 1-877-672-2951 Toll Free 1-770-888-8083 Direct 1-678-261-1438 Fax [email protected] Email Be The First To Know 6175 Hickory Flat Hwy Suite 110-376 Canton, GA 30115 Crane Warning Systems Warehouse crane anti two block system crane wireless anti two block system 1 ton gantry crane Crane warning systems Atlanta anti two block system crane Atlanta warehouse crane Atlanta wireless anti two block system Atlanta crane hoist warehouse crane system crane trolley Rayco wylei systems crane warning systems Rayco electronics two block systems	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8231353163719177, "perplexity": 3696.303828588453}, "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-2020-40/segments/1600400245109.69/warc/CC-MAIN-20200926200523-20200926230523-00668.warc.gz"}
https://readsblog.com/midpoint-method-with-matlab/	# Midpoint Method – Numerical Differentiation with MATLAB Midpoint Method is numerical method to solve the first order ordinary differential equation with given initial condition. The midpoint method is also known as 2nd order Runge-Kutta method, which improve the Euler method by adding midpoint in step which is given better accuracy by order one. The Midpoint method is converge faster than Euler method. The local error of Midpoint method is O(h3) At here, we write the code of Midpoint Method in MATLAB step by step. MATLAB is easy way to solve complicated problems that are not solve by hand or impossible to solve at page. MATLAB is develop for mathematics, therefore MATLAB is the abbreviation of MATrix LABoratory. The formula of Modified Euler method is $y_{n+1}=y_{n}+hf(t_{n}+\frac{h}{2},y_{n}+\frac{h}{2}f(t_{n},y_{n}))$ At here, we solve the differential equation $\frac{dy}{dt}=y-t^{2}+1$ by using Euler method with the help of MATLAB. % Midpoint method to solve the ordinary differential equation % Midpoint method fall in the category of Runge-Kutta method of order 2 clear all; close all; clc; f=inline('y-t^2+1'); x0=input('Enter x0='); y0=input('Enter y0='); xn=input('Enter upper limit of interval xn='); h=input('Enter width (equal space) h='); n=(xn-x0)/h; fprintf('--------------------------------------------\n') fprintf(' x y ynew\n'); fprintf('--------------------------------------------\n') for i=1:n y1=y0+hf(x0+h/2,y0+hf(x0,y0)/2); fprintf('%f %f %f \n',x0,y0,y1) y0=y1; x0=x0+h; end	{"extraction_info": {"found_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": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8545920252799988, "perplexity": 785.532406381261}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710962.65/warc/CC-MAIN-20221204040114-20221204070114-00399.warc.gz"}
https://www.whoi.edu/what-we-do/understand/departments-centers-labs/po/po-events/po-chapman-lecture-series/2010-chapman-lecture/	2010 Chapman Lecture Wave-driven Circulation in the Nearshore and Coastal Ocean Tuba Ozkan-Haller College of Oceanic and Atmospheric Sciences Fall, 2010 Clark Building, Room 507 Reception to follow Surface gravity waves propagating towards beaches shoal as they encounter shallower water, and they break in the surf zone. The surf zone region is therefore characterized by decreasing wave momentum with distance towards shore. Surf zone setup and alongshore currents (often referred to as "longshore" currents) are forced by the resulting forces. On alongshore-uniform beaches the resulting setup of the water surface and longshore current are also alongshore-uniform. On alongshore-variable bathymetry the setup displays alongshore non-uniformity, and associated alongshore pressure gradients also contribute to the alongshore momentum balance. Using simulations for several field sites, we find that cases associated with strong longshore currents often indicate a primary balance between the wave forcing and the friction term, with a secondary balance between the advective acceleration and the pressure gradient terms. In the remainder of the talk, this secondary balance is analyzed with the objective of highlighting the effect of the advective terms for conditions of strong longshore currents and weak alongshore variability in the wave forcing. A perturbation analysis is carried out and reveals that a nondimensional parameter (comprised of the length scales of the alongshore variability, the local water depth, longshore current strength and frictional parameter) controls the response of the longshore current to alongshore variability. The resulting simplified model suggests that the nonlinear response is a filtered version of the solution to the linear problem (when nonlinear advective terms are neglected), and the effect of the nonlinearity can be described as an attenuation and spatial shifting of the response to alongshore variability.	{"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": 0.8284950256347656, "perplexity": 2701.107968182414}, "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-2019-51/segments/1575541308604.91/warc/CC-MAIN-20191215145836-20191215173836-00034.warc.gz"}
http://nanoscale.blogspot.com/2012/12/quantum-spin-liquids-neat-stuff.html	Thursday, December 20, 2012 Quantum spin liquids - neat stuff! There is a new result in this week's issue of Nature that is very neat (and my college classmate Young Lee is the PI - small world!). The experiment is an inelastic neutron scattering measurement that looks at a material with the unusual name herbertsmithite, and reports evidence that this material is a "quantum spin liquid". I'll try to break down the physics here into reasonably accessible bite-sized chunks. First, what is a spin liquid? Imagine having a bunch of localized spins on a lattice. You can picture these like little bar magnets. In this case, the spins are the unpaired d electrons of the copper atoms in the herbertsmithite structure. In general, the spins in a solid (this particular one is an insulator) "talk" to each other via the exchange interaction. What this really means is that there are interactions between the spins so that the spins prefer a particular relative orientation to each other. In this case, the interaction between the electron spins is antiferromagnetic, meaning that for spins on two neighboring Cu atoms, having the spins be oppositely directed saves some energy (17 meV) compared to having the spins aligned. As the temperature is lowered, an ensemble of spins will tend to find whatever configuration minimizes the total energy (the ground state). In a ferromagnet, that will be a state with the spins all aligned with their neighbors. In a perfect antiferromagnet, that would be a state where the spins are all antialigned with their neighbors. Both of these are ordered ground states, in that there is some global arrangement of the spins (with a particular symmetry) that wins at T = 0. The problem in herbertsmithite is, because of the spatial arrangement of the Cu atoms (in a Kagome lattice), it's impossible to have every spin antialigned with all of its neighbors. This is an example of geometric frustration. As a result, even as T gets very low, it would appear that the spins in herbertsmithite never order, even though they interact with their neighbors very strongly. This is an analog to the liquid state, where the molecules of a liquid clearly interact very strongly with their neighbors (they bump right into each other!), but they do not form a spatially ordered arrangement (that would be a solid). Why a quantum spin liquid? Two reasons. First, I cheated in my description above. While we can talk classically about antialigned spins, we really should say that pairs of spins want to form singlets, meaning quantum mechanically entangled antialigned states with net spin zero. So, you can think of this spin liquid state as involving a big entangled mess of spins, where each spin is somehow trying to be entangled in a singlet state with each of its nearest neighbors. This is very complicated to treat theoretically. Second, the fluctuations that dominate in this situation are quantum fluctuations, rather than thermally driven fluctuations. Quantum fluctuations will persist all the way down to T = 0. What's special about a quantum spin liquid? Well, the low energy excitations of a quantum spin liquid can be very weird. If you imagine reaching into the material and flipping one spin so that it's now energetically "unhappy" in terms of its neighbors, what you find is that you can start flipping spins and end up with "spinon" excitations that travel through the material, having spin-1/2 but no charge, and other exotic properties. This is described reasonably well here. Importantly, these excitations have effects that are seen in measurable properties, like heat capacity and how the system can take up and lose energy. So what did the experimenters do? They grew large, very pure single crystals of herbertsmithite, and fired neutrons at them. Knowing the energies and momenta of the incident neutrons, and measuring the energies and momenta of the scattered neutrons, they were able to map out the properties of the excitations, showing that they really do look like what one expects for a quantum spin liquid. Why should you care? This is a great example of seeing exotic properties (like these weird spin excitations) that emerge because of the collective response of a large number of particles. A single Cu ion or unit cell of the crystal doesn't do this stuff - you need lots of spins. Moreover, this is now a system where we can study what this weird, highly quantum-entangled does - I think it's very very far from practical applications, but you never know. Looks like a very nice piece of work. paris cox said... this is fantastic! i can't wait until there is a killer app for this. sure it is at low T, but big things start out small. give it time and something will become great of it! Anonymous said... My impression is that QSLs aren't the real goal of people who study QSLs; the next trick is to dope it and see high-temperature superconductivity, isn't it? (of course, doping is the hard problem -- ever thus for science) Anonymous said... Is this the first example of an experimental quantum spin liquid? rob said... i wonder how long it will take some homepath to say their concoctrions are quantum spin liquids? Douglas Natelson said... Anon12:19, certainly some people are interested in doping some spin liquids to look for superconductivity. I recall that Anderson and others put about the idea that some of the physics of the cuprates may be spin liquid stuff (resonating valence bonds, etc.). In the case of herbersmithite, though, I don't think that's credible, because the stuff has a big gap (a couple of eV). Anon2:20, it's the first experimental system that's been probed like this and really seems to have the characteristic excitation spectrum. joelhelton said... I'd like to add a comment to Doug's response to Anonymous 2:20. A good point made very well by Leon Balents (Nature 464, 199) is that experimental signatures of a QSL are difficult to come by since the state is often defined primarily by what it ISN'T (not an ordered magnet, not a spin glass, not a valence bond solid). Therefore there are plenty of reports in the literature of new frustrated magnets with antiferromagnetic interactions that are measured to low temperatures without any evidence of magnetic Bragg peaks, spin glassiness, or muon oscillations. These may well be described as a spin liquid, but in those cases the data really only show a lack of any evidence that it isn't a QSL. This paper is a step forward in the search for experimental signatures of QSLs because the relatively large single crystal samples and new advances in neutron spectroscopy instruments allow for positive evidence in the form of the measured excitation spectrum. (typo in original post corrected) Anonymous said... thanks for posting.. furniture jati said... 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https://www.physicsforums.com/threads/final-angular-velocity-of-a-top.220272/	# Final Angular Velocity of a Top 1. Mar 6, 2008 ### damedesfeuers 1. A top is a toy that is made to spin on its pointed end by pulling on the string wrapped around the body of the top. The string has a length of 64cm and is wound around the top at a spot where its radius is 2.0 cm. The thickness of the string is negligible. The top is initially at rest. Someone pulls the free end of thes stirng, theryb undwinding it and givig the top and an angular acceleration of +12 rad/s^2. What is the final angular velocity of the top when the string is completely unwound? 2. Relevant equations $$\alpha$$ = $$\omega$$/t $$\omega$$ = $$\theta$$ /t Circumfrence: $$\pi$$r^2 3. The attempt at a solution First I found that the circle's circumference is 4$$\pi$$. Then I divided 64cm/4$$\pi$$ to find that the rope wraps around 5.09 times. I know that 360 degrees is equal to 2$$\pi$$radians, so 5.09 x 360 degrees = 1833.5 degrees. Then 1833.5 degrees x ($$\pi$$ radian / 180 degrees ) = 10.2 radians Therefore I know the angular displacement, which is from 0 to 10.2 radians Then I am stumped on how to find the time from that. I know that I can find the final angular velocity by using the time in the angular acceleration formula. Because Angular acceleration is equal to the final velocity / time. I know this because the initial velocity of the top was zero. In short I can't figure out how to find the time. 2. Mar 6, 2008 ### dynamicsolo There are two related ways to do this, because one is a method where the time remains concealed. In the approach you took, you know that the cord is 64 cm long and the circumference it is wrapped around is $$4\pi$$ cm. Therefore the top will turn 5.09 cycles (as you said) or, maybe a bit more helpfully, 64 cm/2 cm = 32 radians (since arclength around a circle is radius times angle). You can make a kinematic equation for constant angular acceleration analogous to the one for linear motion with constant linear acceleration. Instead of x_f = (1/2)·a·(t^2) , starting from rest at x=0, you have theta_f = (1/2)·(alpha)·(t^2) , starting from rest at theta = 0 . The top turns through 32 radians by the time the string unwinds and alpha = 12 rad/sec^2 , so (t^2) = 2·(32)/12 sec^2 . You can get the time and the final angular velocity from there. The other way combines the angle and angular velocity equations for constant acceleration to produce an "angular velocity-squared" equation analogous to the "velocity-squared" equation for linear kinematics. Instead of (v_f)^2 = (v_i)^2 + 2 · (a) · (delta_x) , we have (omega_f)^2 = (omega_i)^2 + 2 · (alpha) · (delta_theta). You know that omega_i = 0, you know alpha, and you know the total angular displacement is delta_theta = 32 radians. Solve for omega_f . BTW, I think you dropped a factor here. The 5.09 cycles is correct, but since a cycle is $$2\pi$$ radians, which is slightly more than six, the total number of radians would have to be somewhat over 30 (in fact, exactly 32, as calculated above). Last edited: Mar 6, 2008 3. Mar 7, 2008 ### damedesfeuers oh thank you, I actually figured it out this morning, but i didn't think I could delete the thread. And I used the method where the time remained concealed! Thank you again!	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9591148495674133, "perplexity": 962.1448019590168}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281162.88/warc/CC-MAIN-20170116095121-00083-ip-10-171-10-70.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/6317/is-the-identity-functor-the-terminal-object-of-the-category-of-endofunctors-on	# Is the identity functor the terminal object of the category of endofunctors on $C$? It seems to me not, since this would seem to imply that for all functors $F$ and all objects $A$ in $C$ there exists a morphism $F(A) \to A$ (making all functors co-pointed?). However, intuitively it seems like the identity functor acts like a terminal object; a monad $M$ on $C$ is a monoid on $[C, C]$ where the "unit" is a natural transformation $η : I \to M$, while for a monoidal set $S$ in Set the unit is a function $e : 1 \to S$. So am I misunderstanding something, or are my intuitions leading me astray? - The difference between those two examples is that in $\textbf{Set}$ the monoidal operation is the categorical product (so the identity object is the terminal object), whereas this is not true in the category of endofunctors on $\mathbf{C}$. (I believe the latter has a product if and only if $\mathbf{C}$ does, and then it is the pointwise product. It follows that the terminal object, if it exists, is the functor which sends all objects to $\mathbf{1}$ and all morphisms to the unique morphism $\mathbf{1} \to \mathbf{1}$. In particular, it's not the identity functor.)	{"extraction_info": {"found_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": 0.9648256897926331, "perplexity": 64.10274348404897}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860123023.37/warc/CC-MAIN-20160428161523-00034-ip-10-239-7-51.ec2.internal.warc.gz"}
http://mathhelpforum.com/math-challenge-problems/83640-differential-equation.html	1. ## A differential equation I won't be posting new problems for a couple of weeks. This one is nice: Suppose $n \geq 1$ and $f: (a,b) \longrightarrow \mathbb{R}$ is a $C^n$ function (see here for the definition) and $f(x)f'(x)f''(x) \cdots f^{(n)}(x) = 0,$ for all $a < x < b.$ Show that $f$ is a polynomial of degree at most $n-1.$ 2. Originally Posted by NonCommAlg I won't be posting new problems for a couple of weeks. This one is nice: Suppose $n \geq 1$ and $f: (a,b) \longrightarrow \mathbb{R}$ is a $C^n$ function (see here for the definition) and $f(x)f'(x)f''(x) \cdots f^{(n)}(x) = 0,$ for all $a < x < b.$ Show that $f$ is a polynomial of degree at most $n-1.$ if we assume that f(x) is a polynomial of the degree n-1, we get, $\large f^{n}(x^{n-1})=0$ If f(x) has the degree n-2, then the (n-1)th derivative of f becomes zero. The same holds for all degrees <(or equal to) (n-1) hence, the given product always becomes vanishes. However, if we now start making the degrees > (n-1) Lets assume that the function is a polynomial of degree n. I'll take the simplest case here. assuming $\large f(x)=x^{n}$ Then, the nth derivative of the function: $\large f^{n}(x^{n})=n!$ And all the other derivatives would be of the form: $\large f^{r}(x^{n})=n(n-1)(n-2)...(n-r+1)x^{n-r}$ it follows that that none of the derivatives will now be zero hence, the product cannot be zero.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 18, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9773522019386292, "perplexity": 284.0680460849286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20170322212948-00459-ip-10-233-31-227.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/119882/hahn-banach-extend-the-functional-by-continuity	# Hahn-Banach. Extend the functional by continuity Let $E$ be a dense linear subspace of a normed vector space $X$, and let $Y$ be a Banach space. Suppose $T_{0}\in\mathcal{L}(E,Y)$ is a bounded linear operator from $E$ to $Y$. Show that $T_{0}$ can be extended to $T\in\mathcal{L}(X,Y)$ (by continuity) without increasing its norm. I have a dumb question: Given the Hahn-Banach theorem, what's to prove here? It seems to be the immediate consequence of that theorem. If I am wrong, please show me how to prove this. Thank you! - Hahn-Banach has nothing to do with the problem at hand (and one only speaks of functionALs if $Y$ is the ground field). The key words here are uniform continuity and completeness of $Y$. – t.b. Mar 14 '12 at 1:59 @t.b. Thanks. I still need to think about this. I agree that I can not apply that theorem directly. – user16859 Mar 14 '12 at 4:04 Yes, a bounded linear operator is Lipschitz continuous by definition: $\\|Tx_1 - Tx_2\\|_Y \leq \\|T\\|\,\\|x_1 - x_2\\|_X$. Lipschitz continuity implies uniform continuity. The reason I phrased it the way I did is that it is a general fact that if $f_0: D \to Y$ is uniformly continuous where $D \subset X$ is dense in a metric space $X$ and $Y$ is a complete metric space then $f_0$ admits a unique extension to a (uniformly) continuous $f: X \to Y$. Applying this in the present situation you get the extension $T$ from this general fact and linearity of $T$ follows from uniqueness of the extension – t.b. Mar 15 '12 at 2:36 Oh, no, no Tietze at all. It's exactly the same argument as the one azarel outlines. You'll see that you won't use that $T_0$ is linear when you define $T$ (or $f$ as azarel write), you'll only need that when verifying that it is linear... (and since nobody gave you a vote so far, here we go :)) – t.b. Mar 15 '12 at 2:50 That's a consequence of the reverse triangle inequality and the definition of $x_n \to x$: $\|\\|x_n\\| - \\|x\\|\| \leq \\|x_n - x\\| \to 0$, so $\\|x_n\\| \to \\|x\\|$. – t.b. Mar 15 '12 at 6:17 Hahn-Banach only apply if $Y=\mathbb R$. For this particular problem you want to show that if $(x_n)$ converges to $x$ then $T_0(x_n)$ is a Cauchy sequence and then define $f(x)$ as the limit of the sequence. Finally you need to show that the map is a well-defined bounded linear function.	{"extraction_info": {"found_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": 0.9881006479263306, "perplexity": 106.01476904260238}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1448398466260.18/warc/CC-MAIN-20151124205426-00344-ip-10-71-132-137.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/21990/proof-square-matrix-has-maximal-rank-if-and-only-if-it-is-invertible/21994	# Proof - Square Matrix has maximal rank if and only if it is invertible Could someone help me with the proof that a square matrix has maximal rank if and only if it is invertible? Thanks to everybody - What is a quadratic matrix? – Qiaochu Yuan Feb 14 '11 at 12:43 @Qiaochu Yuan he obviously means square matrix – Listing Feb 14 '11 at 12:46 Yeah sorry for my english :) – markzzz Feb 14 '11 at 13:27 @user3123: I asked because it sounded like the OP could have been referring to a quadratic form rather than a matrix. – Qiaochu Yuan Feb 14 '11 at 14:01 please make your posts self-contained. Don't rely on the subject: put the entire information on the body. – Arturo Magidin Feb 14 '11 at 14:18 Suppose $A\in F^{n \times n}$. If A is invertible then there is a matrix B such that $AB=I$ so the standard basis $e_i$ (the columns of I) is in the image of A (these vectors are just the image Av where v are the columns of B) - this shows that $\dim (Im(A)) = n$. On the other hand, if $\dim (Im (A))=n$ then for every i there is $v_i$ such that $A v_i = e_i$. Let B be the matrix with columns $v_i$ then $AB=I$ and A is invertible.	{"extraction_info": {"found_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": 0.9085732698440552, "perplexity": 310.27001639082204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1448398460263.61/warc/CC-MAIN-20151124205420-00129-ip-10-71-132-137.ec2.internal.warc.gz"}
https://socratic.org/questions/how-do-you-evaluate-6times8div4	Algebra Topics # How do you evaluate 6times8div4? Dec 20, 2016 $\left(6 \times 8\right) \div 4 = 48 \div 4 = 12$. #### Explanation: The standard convention is that multiplication and division have the same importance, so when deciding which to do first in a situation like this, we just evaluate left to right. Going left to right, we'll first do the multiplication in this case. So we have $6 \times 8 \div 4 = \left(6 \times 8\right) \div 4 = 48 \div 4 = 12$. ##### Impact of this question 122 views around the world You can reuse this answer	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9786780476570129, "perplexity": 1139.0030655466412}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575513.97/warc/CC-MAIN-20190922114839-20190922140839-00119.warc.gz"}
http://www.cs.columbia.edu/~rocco/papers/icalp09malicious.html	Learning Halfspaces with Malicious Noise. A. Klivans and P. Long and R. Servedio. Journal of Machine Learning Research 10(Dec), 2009, pp. 2715--2740. Preliminary version in 36th International Conference on Automata, Languages and Programming (ICALP), 2009, pp. 609-621. Abstract: We give new algorithms for learning halfspaces in the challenging {\it malicious noise} model, where an adversary may corrupt both the labels and the underlying distribution of examples. Our algorithms can tolerate malicious noise rates exponentially larger than previous work in terms of the dependence on the dimension $n$, and succeed for the fairly broad class of all isotropic log-concave distributions. We give poly$(n, 1/\eps)$-time algorithms for solving the following problems to accuracy $\epsilon$: • Learning origin-centered halfspaces in $\R^n$ with respect to the uniform distribution on the unit ball with malicious noise rate $\eta = \Omega(\eps^2/\log(n/\eps)).$ (The best previous result was $\Omega(\eps/(n \log (n/\eps))^{1/4})$.) • Learning origin-centered halfspaces with respect to any isotropic log-concave distribution on $\R^n$ with malicious noise rate $\eta = \Omega(\eps^{3}/\log(n/\epsilon)).$ This is the first efficient algorithm for learning under isotropic log-concave distributions in the presence of malicious noise. • We also give a poly$(n,1/\eps)$-time algorithm for learning origin-centered halfspaces under any isotropic log-concave distribution on $\R^n$ in the presence of \emph{adversarial label noise} at rate $\eta = \Omega(\eps^{2}/\log(1/\eps))$. In the adversarial label noise setting (or agnostic model), labels can be noisy, but not example points themselves. Previous results could handle $\eta = \Omega(\eps)$ but had running time exponential in an unspecified function of $1/\eps$. Our analysis crucially exploits both concentration and anti-concentration properties of isotropic log-concave distributions. Our algorithms combine an iterative outlier removal procedure using Principal Component Analysis together with smooth'' boosting. pdf of conference 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": 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": 0.8988268375396729, "perplexity": 1093.0159532688085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281353.56/warc/CC-MAIN-20170116095121-00149-ip-10-171-10-70.ec2.internal.warc.gz"}
https://www.nag.com/numeric/nl/nagdoc_26.2/nagdoc_cl26.2/html/g22/g22ybc.html	# NAG C Library Function Document ## nag_blgm_lm_describe_data (g22ybc) Note: please be advised that this function is classed as ‘experimental’ and its interface may be developed further in the future. Please see Section 3.1.1 in How to Use the NAG Library and its Documentation for further information. ## 1Purpose nag_blgm_lm_describe_data (g22ybc) describes a data matrix. ## 2Specification #include #include void nag_blgm_lm_describe_data (void *hddesc, Integer nobs, Integer nvar, const Integer levels[], Integer lvnames, const char vnames[], NagError fail) ## 3Description Let $D$ denote a data matrix with $n$ observations on ${m}_{d}$ independent variables, denoted ${V}_{1},{V}_{2},\dots ,{V}_{{m}_{d}}$. The $j$th independent variable, ${V}_{j}$ can be classified as either binary, categorical, ordinal or continuous, where: Binary ${V}_{j}$ can take the value $1$ or $0$. Categorical ${V}_{j}$ can take one of ${L}_{j}$ distinct values or levels. Each level represents a discrete category but does not necessarily imply an ordering. The value used to represent each level is therefore arbitrary and, by convention and for convenience, is taken to be the integers from $1$ to ${L}_{j}$. Ordinal As with a categorical variable ${V}_{j}$ can take one of ${L}_{j}$ distinct values or levels. However, unlike a categorical variable, the levels of an ordinal variable imply an ordering and hence the value used to represent each level is not arbitrary. For example, ${V}_{j}=4$ implies a value that is twice as large as ${V}_{j}=2$. Continuous ${V}_{j}$ can take any real value. nag_blgm_lm_describe_data (g22ybc) returns a G22 handle containing a description of a data matrix, $D$. The data matrix makes no distinction between binary, ordinal or continuous variables. A name can also be assigned to each variable. If names are not supplied then the default vector of names, $\left\{\text{'V1'},\text{'V2'},\dots \right\}$ is used. None. ## 5Arguments 1: $\mathbf{hddesc}$void Input/Output On entry: must be set to NULL. As an alternative an existing G22 handle may be supplied in which case this function will destroy the supplied G22 handle as if nag_blgm_handle_free (g22zac) had been called. On exit: holds a G22 handle to the internal data structure containing a description of the data matrix, $D$. You must not change the G22 handle other than through the functions in Chapter g22. 2: $\mathbf{nobs}$IntegerInput On entry: $n$, the number of observations in the data matrix, $D$. Constraint: ${\mathbf{nobs}}\ge 0$. 3: $\mathbf{nvar}$IntegerInput On entry: ${m}_{d}$, the number of variables in the data matrix, $D$. Constraint: ${\mathbf{nvar}}\ge 0$. 4: $\mathbf{levels}\left[{\mathbf{nvar}}\right]$const IntegerInput On entry: ${\mathbf{levels}}\left[\mathit{j}-1\right]$ contains the number of levels associated with the $\mathit{j}$th variable of the data matrix, for $\mathit{j}=1,2,\dots ,{\mathbf{nvar}}$. If the $j$th variable is binary, ordinal or continuous, ${\mathbf{levels}}\left[j-1\right]$ should be set to $1$; otherwise ${\mathbf{levels}}\left[j-1\right]$ should be set to the number of levels associated with the $j$th variable and the corresponding column of the data matrix is assumed to take the value $1$ to ${\mathbf{levels}}\left[j-1\right]$. Constraint: ${\mathbf{levels}}\left[\mathit{i}-1\right]\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{nvar}}$. 5: $\mathbf{lvnames}$IntegerInput On entry: the number of variable names supplied in vnames. Constraint: ${\mathbf{lvnames}}=0$, or ${\mathbf{nvar}}$. 6: $\mathbf{vnames}\left[{\mathbf{lvnames}}\right]$const char Input On entry: if ${\mathbf{lvnames}}\ne 0$, ${\mathbf{vnames}}\left[\mathit{j}-1\right]$ must contain the name of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,{\mathbf{nvar}}$. If ${\mathbf{lvnames}}=0$, vnames is not referenced and may be NULL. The names supplied in vnames should be at most $50$ characters long and be unique. If a name longer than $50$ characters is supplied it will be truncated. Variable names must not contain any of the characters +.-:^()@. 7: $\mathbf{fail}$NagError Input/Output The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation). ## 6Error Indicators and Warnings NE_ALLOC_FAIL Dynamic memory allocation failed. See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information. NE_ARRAY_SIZE On entry, ${\mathbf{lvnames}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nvar}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{lvnames}}=0$, or ${\mathbf{nvar}}$. On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value. NE_HANDLE On entry, hddesc is not NULL or a recognised G22 handle. NE_INT On entry, ${\mathbf{nobs}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{nobs}}\ge 0$. On entry, ${\mathbf{nvar}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{nvar}}\ge 0$. NE_INT_ARRAY On entry, $j=〈\mathit{\text{value}}〉$ and ${\mathbf{levels}}\left[j-1\right]=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{levels}}\left[\mathit{i}-1\right]\ge 1$. NE_INTERNAL_ERROR An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance. See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information. NE_INVALID_FORMAT On entry, variable name $i$ contains one more invalid characters, $i=〈\mathit{\text{value}}〉$. NE_NO_LICENCE Your licence key may have expired or may not have been installed correctly. See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information. NE_NON_UNIQUE On entry, variable names $i$ and $j$ are not unique (possibly due to truncation), $i=〈\mathit{\text{value}}〉$ and $j=〈\mathit{\text{value}}〉$. Maximum variable name length is $50$. On entry, variable names $i$ and $j$ are not unique, $i=〈\mathit{\text{value}}〉$ and $j=〈\mathit{\text{value}}〉$. NW_TRUNCATED At least one variable name was truncated to $50$ characters. Each truncated name is unique and will be used in all output. Not applicable. ## 8Parallelism and Performance nag_blgm_lm_describe_data (g22ybc) is not threaded in any implementation. None. ## 10Example This example performs a linear regression using nag_regsn_mult_linear (g02dac). The linear regression model is defined via a text string which is parsed using nag_blgm_lm_formula (g22yac). The corresponding design matrix associated with the model and the dataset described via a call to nag_blgm_lm_describe_data (g22ybc) is generated using nag_blgm_lm_design_matrix (g22ycc). Verbose labels for the parameters of the model are constructed using information returned in vinfo by nag_blgm_lm_submodel (g22ydc). See also the examples in nag_blgm_lm_formula (g22yac), nag_blgm_lm_design_matrix (g22ycc) and nag_blgm_lm_submodel (g22ydc). ### 10.1Program Text Program Text (g22ybce.c) ### 10.2Program Data Program Data (g22ybce.d) ### 10.3Program Results Program Results (g22ybce.r) ## 11Optional Parameters As well as the optional parameters common to all G22 handles described in nag_blgm_optset (g22zmc) and nag_blgm_optget (g22znc), a number of additional optional parameters can be specified for a G22 handle holding the description of a data matrix as returned by nag_blgm_lm_describe_data (g22ybc) in hddesc. Each writeable optional parameter has an associated default value; to set any of them to a non-default value, use nag_blgm_optset (g22zmc). The value of an optional parameter can be queried using nag_blgm_optget (g22znc). The remainder of this section can be skipped if you wish to use the default values for all optional parameters. The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 11.1. ### 11.1Description of the Optional Parameters For each option, we give a summary line, a description of the optional parameter and details of constraints. The summary line contains: • a parameter value, where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively; • the default value. Keywords and character values are case and white space insensitive. Number of Observations $i$ If queried, this optional parameter will return $n$, the number of observations in the data matrix. Number of Variables $i$ If queried, this optional parameter will return ${m}_{d}$, the number of variables in the data matrix. Storage Order $a$ Default $\text{}=\mathrm{OBSVAR}$ This optional parameter states how the data matrix, $D$, will be stored in its input array. If ${\mathbf{Storage Order}}=\mathrm{OBSVAR}$, ${D}_{ij}$, the value for the $j$th variable of the $i$th observation of the data matrix is stored in ${\mathbf{dat}}\left[\left(j-1\right)×{\mathbf{pddat}}+i-1\right]$. If ${\mathbf{Storage Order}}=\mathrm{VAROBS}$, ${D}_{ij}$, the value for the $j$th variable of the $i$th observation of the data matrix is stored in ${\mathbf{dat}}\left[\left(i-1\right)×{\mathbf{pddat}}+j-1\right]$. Where dat is the input parameter of the same name in nag_blgm_lm_design_matrix (g22ycc). Constraint: ${\mathbf{Storage Order}}=\mathrm{OBSVAR}$ or $\mathrm{VAROBS}$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 103, "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": 0.8123012781143188, "perplexity": 1410.6113889322316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046153966.52/warc/CC-MAIN-20210730091645-20210730121645-00185.warc.gz"}
https://arxiv.org/abs/1911.02795	astro-ph.GA # Title:Estimating the molecular gas mass of low-redshift galaxies from a combination of mid-infrared luminosity and optical properties Abstract: We present CO(J=1-0) and/or CO(J=2-1) spectroscopy for 31 galaxies selected from the ongoing MaNGA survey, obtained with multiple telescopes. This sample is combined with CO observations from the literature to study the correlation of the CO luminosities ($L_{\rm CO(1-0)}$) with the mid-infrared luminosities at 12 ($L_{12 \mu m}$) and 22 $\mu$m ($L_{\rm 22 \mu m}$), as well as the dependence of the residuals on a variety of galaxy properties. The correlation with $L_{\rm 12 \mu m}$ is tighter and more linear, but galaxies with relatively low stellar masses and blue colors fall significantly below the mean $L_{\rm CO(1-0)}-L_{\rm 12\mu m}$ relation. We propose a new estimator of the CO(1-0) luminosity (and thus the total molecular gas mass) that is a linear combination of three parameters: $L_{\rm 12 \mu m}$, $M_\ast$ and $g-r$. We show that, with a scatter of only 0.18 dex in log $(L_{\rm CO(1-0)})$, this estimator provides unbiased estimates for galaxies of different properties and types. An immediate application of this estimator to a compiled sample of galaxies with only CO(J=2-1) observations yields a distribution of the CO(J=2-1) to CO(J=1-0) luminosity ratios ($R21$) that agrees well with the distribution of real observations, in terms of both the median and the shape. Application of our estimator to the current MaNGA sample reveals a gas-poor population of galaxies that are predominantly early-type and show no correlation between molecular gas-to-stellar mass ratio and star formation rate, in contrast to gas-rich galaxies. We also provide alternative estimators with similar scatters, based on $r$ and/or $z$ band luminosities instead of $M_\ast$. These estimators serve as cheap and convenient $M_{\rm mol}$ proxies to be potentially applied to large samples of galaxies, thus allowing statistical studies of gas-related processes of galaxies. Comments: 24 pages, 12 figures, accepted for publication in ApJ Subjects: Astrophysics of Galaxies (astro-ph.GA) Cite as: arXiv:1911.02795 [astro-ph.GA] (or arXiv:1911.02795v2 [astro-ph.GA] for this version) ## Submission history From: Yang Gao [view email] [v1] Thu, 7 Nov 2019 08:17:55 UTC (2,114 KB) [v2] Mon, 11 Nov 2019 02:39:49 UTC (2,114 KB)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.884438157081604, "perplexity": 2419.1815078914665}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1573496670006.89/warc/CC-MAIN-20191119042928-20191119070928-00557.warc.gz"}
https://math.stackexchange.com/questions/3145476/how-to-calculate-textcov-haty-ij-haty-kj-if-y-ij-mu-a-i	# How to calculate $\text{cov}(\hat{Y}_{ij}, \hat{Y}_{kj})$ if $Y_{ij} = \mu + a_i + b_j + e_{ij}$? Let's assume we have the model following Two-Factor model without replications : $$Y_{ij} = \mu + a_i + b_j + e_{ij}, \; i=1,\dots,p \; \text{and} \; j=1,\dots, q$$ I am interested in calculating the covariance : $$\text{cov}(\hat{Y}_{ij}, \hat{Y}_{kj}), \quad i \neq k$$ I know that the estimators can be written as : $$\hat{Y}_{ij} = \overline{Y}_{i\cdot} + \overline{Y}_{\cdot j} - \overline{Y}_{\cdot \cdot} \quad\text{and}\quad \hat{Y}_{ik} = \overline{Y}_{i\cdot} + \overline{Y}_{\cdot k} - \overline{Y}_{\cdot \cdot}$$ That would make the covariance expression : $$\text{cov}(\hat{Y}_{ij}, \hat{Y}_{kj}) = \text{cov}(\overline{Y}_{i\cdot} + \overline{Y}_{\cdot j} - \overline{Y}_{\cdot \cdot}, \overline{Y}_{i\cdot} + \overline{Y}_{\cdot k} - \overline{Y}_{\cdot \cdot})$$ I know that I can break this covariance up in combinations, but I am having trouble calculating each one. Except from the term $$\text{cov}(\overline{Y}_{\cdot \cdot}, \overline{Y}_{\cdot \cdot}) = \sigma^2/pq$$, I seem to struggle to find the rest of them. Any tips on the calculations ?	{"extraction_info": {"found_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": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8931541442871094, "perplexity": 152.27388591769082}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578529606.64/warc/CC-MAIN-20190420100901-20190420122901-00116.warc.gz"}
https://gaugehow.com/sine-bar/	# Sine Bar ### Introduction Sine bar is a precision angle measuring instrument along with slip gauges. The name suggests that Sine bar work on sine principle. Slip gauge used to build up the height of sine bar. The required angle is obtained when the difference in height between the two rollers is equal to the Sine of the angle multiplied by the distance between the centers of the rollers. ### How to measure with Sine bar ? Sine Bar is an indirect method of measurement. Sine Bar is working on Sine principle that is the ratio of length between two roller and height differences. In the above picture, we are taking a very simple case. Height differences angle H1-H2 is zero, so, after calculation, our result is 180 degree. ### Pros & Corns of sine bar Some Advantages of sine bar are 1. It is precise and accurate 2. Its design is quite simple 3. Availability of Sine bar is high	{"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": 0.9813063740730286, "perplexity": 1564.2538726832252}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1593655891654.18/warc/CC-MAIN-20200707044954-20200707074954-00559.warc.gz"}
https://www.on-time.hu/cn2tvd2/what-is-the-second-fundamental-theorem-of-calculus-bd3689	# what is the second fundamental theorem of calculus in Egyéb - 2020-12-30 In the upcoming lessons, we’ll work through a few famous calculus rules and applications. d x dt Example: Evaluate . A slight change in perspective allows us to gain even more insight into the meaning of the definite integral. Using the Second Fundamental Theorem of Calculus, we have . As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. In this article, let us discuss the first, and the second fundamental theorem of calculus, and evaluating the definite integral using the theorems in detail. This video provides an example of how to apply the second fundamental theorem of calculus to determine the derivative of an integral. Consider the function f(t) = t. For any value of x > 0, I can calculate the de nite integral Z x 0 f(t)dt = Z x 0 tdt: by nding the area under the curve: 18 16 14 12 10 8 6 4 2 Ð 2 Ð 4 Ð 6 Ð 8 Ð 10 Ð 12 Ð 14 Ð 16 Ð 18 It states that if f (x) is continuous over an interval [a, b] and the function F (x) is defined by F (x) = ∫ a x f (t)dt, then F’ (x) = f (x) over [a, b]. However, this, in my view is different from the proof given in Thomas'-calculus (or just any standard textbook), since it does not make use of the Mean value theorem anywhere. Example. identify, and interpret, ∫10v(t)dt. The second fundamental theorem of calculus is basically a restatement of the first fundamental theorem. All that is needed to be able to use this theorem is any antiderivative of the integrand. Define . Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives. The first fundamental theorem of calculus states that, if f is continuous on the closed interval [a,b] and F is the indefinite integral of f on [a,b], then int_a^bf(x)dx=F(b)-F(a). The second part of the theorem (FTC 2) gives us an easy way to compute definite integrals. Fundamental Theorem Of Calculus: The original function lets us skip adding up a gajillion small pieces. dx 1 t2 This question challenges your ability to understand what the question means. Applying the fundamental theorem of calculus tells us $\int_{F(a)}^{F(b)} \mathrm{d}u = F(b) - F(a)$ Your argument has the further complication of working in terms of differentials — which, while a great thing, at this point in your education you probably don't really know what those are even though you've seen them used enough to be able to mimic the arguments people make with them. We use the chain rule so that we can apply the second fundamental theorem of calculus. As we learned in indefinite integrals , a primitive of a a function f(x) is another function whose derivative is f(x). When we do this, F(x) is the anti-derivative of f(x), and f(x) is the derivative of F(x). Let Fbe an antiderivative of f, as in the statement of the theorem. The Second Fundamental Theorem of Calculus shows that integration can be reversed by differentiation. Solution. Pick a function f which is continuous on the interval [0, 1], and use the Second Fundamental Theorem of Calculus to evaluate f(x) dx two times, by using two different antiderivatives. The second fundamental theorem of calculus states that if f(x) is continuous in the interval [a, b] and F is the indefinite integral of f(x) on [a, b], then F'(x) = f(x). It has two main branches – differential calculus and integral calculus. It has gone up to its peak and is falling down, but the difference between its height at and is ft. The second part states that the indefinite integral of a function can be used to calculate any definite integral, \int_a^b f(x)\,dx = F(b) - F(a). The Second Part of the Fundamental Theorem of Calculus The second part tells us how we can calculate a definite integral. Moreover, with careful observation, we can even see that is concave up when is positive and that is concave down when is negative. Find (a) F(π) (b) (c) To find the value F(x), we integrate the sine function from 0 to x. The Fundamental Theorem of Calculus is a theorem that connects the two branches of calculus, differential and integral, into a single framework. F0(x) = f(x) on I. Introduction. That's fundamental theorem of calculus. Area Function It can be used to find definite integrals without using limits of sums . (1) This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. Using The Second Fundamental Theorem of Calculus This is the quiz question which everybody gets wrong until they practice it. This sketch investigates the integral definition of a function that is used in the 2nd Fundamental Theorem of Calculus as a form of an anti-derivativ… Finding derivative with fundamental theorem of calculus: chain rule Our mission is to provide a free, world-class education to anyone, anywhere. The second figure shows that in a different way: at any x-value, the C f line is 30 units below the A f line. (a) To find F(π), we integrate sine from 0 to π:. It looks very complicated, but … The Second Fundamental Theorem of Calculus is our shortcut formula for calculating definite integrals. Fix a point a in I and de ne a function F on I by F(x) = Z x a f(t)dt: Then F is an antiderivative of f on the interval I, i.e. Let f be a continuous function de ned on an interval I. The second fundamental theorem can be proved using Riemann sums. To assist with the determination of antiderivatives, the Antiderivative [ Maplet Viewer ][ Maplenet ] and Integration [ Maplet Viewer ][ Maplenet ] maplets are still available. - The variable is an upper limit (not a lower limit) and the lower limit is still a constant. PROOF OF FTC - PART II This is much easier than Part I! Find the derivative of . The fundamental theorem of calculus has two separate parts. The Fundamental theorem of calculus links these two branches. This means we're accumulating the weighted area between sin t and the t-axis from 0 to π:. Second Fundamental Theorem of Calculus We have seen the Fundamental Theorem of Calculus , which states: If f is continuous on the interval [ a , b ], then In other words, the definite integral of a derivative gets us back to the original function. The First Fundamental Theorem of Calculus shows that integration can be undone by differentiation. Specifically, for a function f that is continuous over an interval I containing the x-value a, the theorem allows us to create a new function, F(x), by integrating f from a to x. The Fundamental Theorem of Calculus relates three very different concepts: The definite integral $\int_a^b f(x)\, dx$ is the limit of a sum. So we evaluate that at 0 to get big F double prime at 0. Second Fundamental Theorem of Calculus. Let f(x) = sin x and a = 0. Thus if a ball is thrown straight up into the air with velocity the height of the ball, second later, will be feet above the initial height. As antiderivatives and derivatives are opposites are each other, if you derive the antiderivative of the function, you get the original function. Calculus is the mathematical study of continuous change. There are several key things to notice in this integral. Khan Academy is a 501(c)(3) nonprofit organization. The real goal will be to figure out, for ourselves, how to make this happen: Using First Fundamental Theorem of Calculus Part 1 Example. While the two might seem to be unrelated to each other, as one arose from the tangent problem and the other arose from the area problem, we will see that the fundamental theorem of calculus does indeed create a link between the two. The Second Fundamental Theorem of Calculus establishes a relationship between a function and its anti-derivative. Now deﬁne a new function gas follows: g(x) = Z x a f(t)dt By FTC Part I, gis continuous on [a;b] and differentiable on (a;b) and g0(x) = f(x) for every xin (a;b). First, it states that the indefinite integral of a function can be reversed by differentiation, \int_a^b f(t)\, dt = F(b)-F(a). MATH 1A - PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS 3 3. Also, this proof seems to be significantly shorter. }\) What is the statement of the Second Fundamental Theorem of Calculus? A ball is thrown straight up from the 5 th floor of the building with a velocity v(t)=−32t+20ft/s, where t is calculated in seconds. We saw the computation of antiderivatives previously is the same process as integration; thus we know that differentiation and integration are inverse processes. It states that if a function F(x) is equal to the integral of f(t) and f(t) is continuous over the interval [a,x], then the derivative of F(x) is equal to the function f(x): . And then we evaluate that at x equals 0. According to me, This completes the proof of both parts: part 1 and the evaluation theorem also. The fundamental theorem of calculus connects differentiation and integration , and usually consists of two related parts . The Fundamental Theorem of Calculus formalizes this connection. Note that the ball has traveled much farther. Section 5.2 The Second Fundamental Theorem of Calculus Motivating Questions. - The integral has a variable as an upper limit rather than a constant. And then we know that if we want to take a second derivative of this function, we need to take a derivative of the little f. And so we get big F double prime is actually little f prime. Let be a continuous function on the real numbers and consider From our previous work we know that is increasing when is positive and is decreasing when is negative. How does the integral function $$A(x) = \int_1^x f(t) \, dt$$ define an antiderivative of $$f\text{? The Second Fundamental Theorem of Calculus provides an efficient method for evaluating definite integrals. The First Fundamental Theorem of Calculus. The first part of the theorem says that if we first integrate \(f$$ and then differentiate the result, we get back to the original function $$f.$$ Part $$2$$ (FTC2) The second part of the fundamental theorem tells us how we can calculate a definite integral. This is not in the form where second fundamental theorem of calculus can be applied because of the x 2. The Fundamental Theorem of Calculus shows that di erentiation and Integration are inverse processes. Second Fundamental Theorem of Calculus. Problem. Here, the "x" appears on both limits. In this wiki, we will see how the two main branches of calculus, differential and integral calculus, are related to each other. A few observations. The fundamental theorem of calculus justifies the procedure by computing the difference between the antiderivative at the upper and lower limits of the integration process. The Second Fundamental Theorem is one of the most important concepts in calculus. 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Single framework process as integration ; thus we know that differentiation and integration, and interpret, ∫10v ( )! • ### Legutóbbi hozzászólások © 2014 OnTime, Minden jog fenntartva!	{"extraction_info": {"found_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": 0.9804986715316772, "perplexity": 374.87885861653194}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662534693.28/warc/CC-MAIN-20220520223029-20220521013029-00111.warc.gz"}
http://math.sns.it/paper/3249/	# On the monotonicity of perimeter of convex bodies created by stefani on 01 Dec 2016 modified on 04 Jan 2018 [BibTeX] Accepted Paper Inserted: 1 dec 2016 Last Updated: 4 jan 2018 Journal: Journal of Convex Analysis Volume: 25 Number: 1 Year: 2018 ArXiv: 1612.00295 PDF Abstract: Let $n\ge2$ and let $\Phi\colon\mathbb{R}^n\to[0,\infty)$ be a positively $1$-homogeneous and convex function. Given two convex bodies $A\subset B$ in $\mathbb{R}^n$, the monotonicity of anisotropic $\Phi$-perimeters holds, i.e. $P_\Phi(A)\le P_\Phi(B)$. In this note, we prove a quantitative lower bound on the difference of the $\Phi$-perimeters of $A$ and $B$ in terms of their Hausdorff distance.	{"extraction_info": {"found_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": 0.820051372051239, "perplexity": 1391.3098114625745}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1534221215284.54/warc/CC-MAIN-20180819184710-20180819204710-00175.warc.gz"}
https://tex.stackexchange.com/questions/394666/unicode-characters-with-xelatex-without-changing-the-font	# Unicode characters with XeLaTeX without changing the font I need to write a document using a given template (with specific fonts and font sizes for title, headers, body etc.). The document is in English, but I need to type some Unicode characters (for instance to type some names). I would like to compile with xelatex, but all the font settings of the template are lost if I load the fontspec package (before) loading the package: \usepackage{fontspec} It is a bit better if I load fontspec after loading the .sty template file, but still quite different from what I would get with basic latex. I know I can use some tricks (e.g. \'e for é, which works so I guess that the fonts that are used support correcly the needed utf8 subset), but I was wondering if there was a generic solution for such cases, i.e. support utf8 file without changing the current font settings ? I am not sure to understand exactly what happens when fontspec is loaded. Many comments suggested that I do not need to use xelatex to get utf8 working. I have to say that I got used to it, to avoid problems of that kind: ! Package inputenc Error: Unicode char − (U+2212) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.77 This is a test with long hyphen "− " and unbreakable space " ". ? For this example, I just have in the preamble: \documentclass{article} \usepackage[utf8]{inputenc} […] How should I avoid such problems if compiling with pdflatex? Should I manually define each unsupported character with \DeclareUnicodeCharacter as suggested sometimes? It seems to me that relying on xelatex is a better practice, but I might be wrong. I have to admit that most of the time I use xelatex more for the extended utf8 support than for the ability of changing the font with fontspec. • you haven't given enough detail to answer. No font has the entire unicode range so you need to pick a font (or fonts) that have the characters you need. It may be that you can find a single font to cover all the needed characters or you may have to switch fonts when switching between (say) European or Japanese characters. there is no general answer. It depends on what font style you want and what character ranges you need. – David Carlisle Oct 5 '17 at 11:14 • Let's say for now that I only need some french accents (é, è). I think that many fonts support accents. And if i can produce an "é" by typing "\'e" (without fontsec), I guess it means that the font supports it. So it should work by typing directly "é" in the source, no ? – Zooky Oct 5 '17 at 11:24 • the inputs of \'e and é are the same (latex converts one to the other before looking at the font) so that really doesn't tell you anything much but if you just want to cover French then more or less any font is going to work (but also there are not so many advantages to using xetex. There is no general answer you need to say what fonts you are using. Also why are you using xetex and fontspec if the requirement is to get the same fonts as an existing pdftex setup? why not use pdftex? – David Carlisle Oct 5 '17 at 12:02 • You don't need xelatex to use UTF8 in your input. You could add \usepackage[utf8]{inputenc} and use pdflatex... – Thruston Oct 5 '17 at 12:54 • I do not think that the input of \'e and é are the same, because the first one is processed correctly, but not the second one (with xelatex without loading fontspec). I use xelatex because I usually do use more utf8 characters (starting with short and long non-breaking spaces, em dash etc., and sometimes other alphabets). I was referring to french accents just to get a simple example; I am sure that I can find some workaround for these specific characters, but it would just be more convenient to use directly xelatex. – Zooky Oct 5 '17 at 13:14	{"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": 0.861868143081665, "perplexity": 1138.025482481322}, "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-2019-18/segments/1555578527720.37/warc/CC-MAIN-20190419121234-20190419143234-00429.warc.gz"}
http://www.reference.com/browse/d'Alembert's%20principle	Related Searches Definitions # D'Alembert's principle D'Alembert's principle, also known as the Lagrange-D'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. The principle states that the sum of the differences between the forces acting on a system and the time derivatives of the momenta of the system itself along a virtual displacement consistent with the constraints of the system, is zero. Thus, in symbols d'Alembert's principle is, $sum_\left\{i\right\} \left(mathbf \left\{F\right\}_\left\{i\right\} - m_i mathbf\left\{a\right\}_i \right)cdot delta mathbf r_i = 0$. $mathbf \left\{F\right\}_i$ are the applied forces $delta mathbf r_i$ is the virtual displacement of the system, consistent with the constraints $mathbf m_i$ are the masses of the particles in the system $mathbf a_i$ are the accelerations of the particles in the system $m_i mathbf a_i$ together as products represent the time derivatives of the system momenta $i$ is an integer used to indicate (via subscript) a variable corresponding to a particular particle It is the dynamic analogue to the principle of virtual work for applied forces in a static system and in fact is more general than Hamilton's principle, avoiding restriction to holonomic systems. If the negative terms in accelerations are recognized as inertial forces, the statement of d'Alembert's principle becomes The total virtual work of the impressed forces plus the inertial forces vanishes for reversible displacements. This above equation is often called d'Alembert's principle, but it was first written in this variational form by Joseph Louis Lagrange. D'Alembert's contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish. That is to say that the generalized forces $\left\{mathbf Q\right\}_\left\{j\right\}$ need not include constraint forces. ## Derivation Consider Newton's law for a system of particles, i. The total force on each particle is $mathbf \left\{F\right\}_\left\{i\right\}^\left\{\left(T\right)\right\} = m_i mathbf \left\{a\right\}_i$. $mathbf \left\{F\right\}_\left\{i\right\}^\left\{\left(T\right)\right\}$ are the total forces acting on the system's particles $m_i mathbf \left\{a\right\}_i$ are the inertial forces that result from the total forces Moving the inertial forces to the left gives an expression that can be considered to represent quasi-static equilibrium, but which is really just a small algebraic manipulation of Newton's law: $mathbf \left\{F\right\}_\left\{i\right\}^\left\{\left(T\right)\right\} - m_i mathbf \left\{a\right\}_i = mathbf 0$. Considering the virtual work, $delta W$, done by the total and inertial forces together through an arbitrary virtual displacement, $delta mathbf r_i$, of the system leads to a zero identity, since the forces involved sum to zero for each particle. $delta W = sum_\left\{i\right\} mathbf \left\{F\right\}_\left\{i\right\}^\left\{\left(T\right)\right\} cdot delta mathbf r_i - sum_\left\{i\right\} m_i mathbf\left\{a\right\}_i cdot delta mathbf r_i = 0$ At this point it should be noted that the original vector equation could be recovered by recognizing that the work expression must hold for arbitrary displacements. Separating the total forces into applied forces, $mathbf F_i$, and constraint forces, $mathbf C_i$, yields $delta W = sum_\left\{i\right\} mathbf \left\{F\right\}_\left\{i\right\} cdot delta mathbf r_i + sum_\left\{i\right\} mathbf \left\{C\right\}_\left\{i\right\} cdot delta mathbf r_i - sum_\left\{i\right\} m_i mathbf\left\{a\right\}_i cdot delta mathbf r_i = 0$ If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces, the constraint forces do no work. Such displacements are said to be consistent with the constraints. This leads to the formulation of d'Alembert's principle, which states that the difference of applied forces and inertial forces for a dynamic system does no virtual work: $delta W = sum_\left\{i\right\} \left(mathbf \left\{F\right\}_\left\{i\right\} - m_i mathbf\left\{a\right\}_i \right)cdot delta mathbf r_i = 0$. There is also a corresponding principle for static systems called the principle of virtual work for applied forces. ## D'Alembert's principle of inertial forces D'Alembert showed that one can transform an accelerating rigid body into an equivalent static system by adding the so-called "inertial force" and "inertial torque" or moment. The inertial force must act through the center of mass and the inertial torque can act anywhere. The system can then be analyzed exactly as a static system subjected to this "inertial force and moment" and the external forces. The advantage is that, in the equivalent static system' one can take moments about any point (not just the center of mass). This often leads to simpler calculations because any force (in turn) can be eliminated from the moment equations by choosing the appropriate point about which to apply the moment equation (sum of moments = zero). In textbooks of engineering dynamics this is sometimes referred to as d'Alembert's principle. ### Example for plane 2D motion of a rigid body For a planar rigid body, moving in the plane of the body (the x-y plane), and subjected to forces and torques causing rotation only in this plane, the inertial force is $mathbf\left\{F\right\}_i = - mddot\left\{mathbf\left\{r\right\}_c\right\}$ where $mathbf\left\{r\right\}_c$ is the position vector of the centre of mass of the body, and $m$ is the mass of the body. The inertial torque (or moment) is $T_i = -Iddot\left\{theta\right\}$ where $I$ is the moment of inertia of the body. If, in addition to the external forces and torques acting on the body, the inertia force acting through the center of mass is added and the inertial torque is added (acting around the centre of mass is as good as anywhere) the system is equivalent to one in static equilibrium. Thus the equations of static equilibrium $sum F_x = 0$ $sum F_y = 0$ $sum T = 0$ hold. The important thing is that $sum T$ is the sum of torques (or moments, including the inertial moment and the moment of the inertial force) taken about any point. The direct application of Newton's laws requires that the angular acceleration equation be applied only about the center of mass. ## References Search another word or see d'Alembert's principleon Dictionary \| Thesaurus \|Spanish	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 28, "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": 0.9319310784339905, "perplexity": 299.2585727577483}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929411.0/warc/CC-MAIN-20150521113209-00286-ip-10-180-206-219.ec2.internal.warc.gz"}

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