Datasets:

keirp
/

open-web-math-hq-dev

Dataset card Data Studio Files Files and versions Community

Split (1)

train · 1.36M rows

url stringlengths 15 1.13k	text stringlengths 100 1.04M	metadata stringlengths 1.06k 1.1k
https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/01%3A_Basics_of_Geometry/1.03%3A_Definition_of_Line_segments	Skip to main content # 1.3: Definition of Line segments Identify the Positive length between points. ## Measuring Distance Between Two Points Distance is the measure of length between two points. To measure is to determine how far apart two geometric objects are. The most common way to measure distance is with a ruler. Inch-rulers are usually divided up by eighth-inch (or 0.125 in) segments. Centimeter rulers are divided up by tenth-centimeter (or 0.1 cm) segments. Note that the distance between two points is the absolute value of the difference between the numbers shown on the ruler. This implies that you do not need to start measuring at “0”, as long as you subtract the first number from the second. The segment addition postulate states that if $$A$$, $$B$$, and $$C$$ are collinear and $$B$$ is between $$A$$ and $$C$$, then $$AB+BC=AC$$. You can find the distances between points in the $$x–y$$ plane if the lines are horizontal or vertical. If the line is vertical, find the change in the $$y$$−coordinates. If the line is horizontal, find the change in the $$x$$−coordinates. Suppose you want to measure your height, but the measuring tape you have is old and the end is broken off. If the tape now starts at 6 cm and reads 138 cm from the floor to the top of your head, how tall are you? Example $$\PageIndex{1}$$ What is the distance marked on the ruler below? The ruler is in centimeters. Solution Subtract one endpoint from the other. The line segment spans from 3 cm to 8 cm. $$\|8−3\|=\|5\|=5$$ The line segment is 5 cm long. Notice that you also could have done $$\|3−8\|=\|−5\|=5$$. Example $$\PageIndex{2}$$ Make a sketch of $$\overline{OP}$$, where $$Q$$ is between $$O$$ and $$P$$. Solution Draw $$\overline{OP}$$ first, then place $$Q$$ on the segment. Example $$\PageIndex{3}$$ What is the distance between the two points shown below? Solution Because this line is vertical, look at the change in the $$y$$-coordinates. $$\|9−3\|=\|6\|=6$$ The distance between the two points is 6 units. Example $$\PageIndex{4}$$ In the picture from Example 2, if $$OP=17$$ and $$QP=6$$, what is $$OQ$$? Solution $$OQ+QP=OP$$ $$OQ+6=17$$ $$OQ=17−6$$ $$OQ=11$$ Example $$\PageIndex{5}$$ What is the distance between the two points shown below? Solution Because this line is horizontal, look at the change in the $$x$$-coordinates. $$\|(−4)−3\|=\|−7\|=7$$ The distance between the two points is 7 units. ## Review For 1-4, use the ruler in each picture to determine the length of the line segment. 1. Make a sketch of $$\overline{BT}$$, with $$A$$ between $$B$$ and $$T$$. 2. If O is in the middle of $$\overline{LT}$$, where exactly is it located? If $$LT=16 cm$$, what is $$LO$$ and $$OT$$? 3. For three collinear points, $$A$$ between $$T$$ and $$Q$$: 1. Draw a sketch. 2. Write the Segment Addition Postulate for your sketch. 3. If $$AT=10$$ in and $$AQ=5$$ in, what is $$TQ$$? 4. For three collinear points, $$M$$ between $$H$$ and $$A$$: 1. Draw a sketch. 2. Write the Segment Addition Postulate for your sketch. 3. If $$HM=18 cm$$ and $$HA=29 cm$$, what is $$AM$$? 5. For three collinear points, I between M and T: 1. Draw a sketch. 2. Write the Segment Addition Postulate for your sketch. 3. If $$IT=6 cm$$ and $$MT=25 cm$$, what is $$AM$$? 6. Make a sketch that matches the description: B is between $$A$$ and $$D$$. $$C$$ is between $$B$$ and $$D$$. $$AB=7 cm$$, $$AC=15 cm$$, and $$AD=32 cm$$. Find $$BC$$, $$BD$$, and $$CD$$. 7. Make a sketch that matches the description: $$E is between \(F$$ and $$G$$. $$H$$ is between $$F$$ and $$E$$. $$FH=4 in$$, $$EG=9 in$$, and $$FH=HE$$. Find $$FE$$, $$HG$$, and $$FG$$. For 12 and 13, Suppose $$J$$ is between $$H$$ and $$K$$. Use the Segment Addition Postulate to solve for $$x$$. Then find the length of each segment. 1. $$HJ=4x+9$$, $$JK=3x+3$$, $$KH=33$$ 2. $$HJ=5x−3$$, $$JK=8x−9$$, $$KH=131$$ For 14-17, determine the vertical or horizontal distance between the two points. 1. Make a sketch of: $$S$$ is between $$T$$ and $$V$$. $$R$$ is between $$S$$ and $$T$$. $$TR=6$$, $$RV=23$$, and $$TR=SV$$. 2. Find $$SV$$, $$TS$$, $$RS$$ and $$TV$$ from #18. 3. For $$\overline{HK}$$, suppose that $$J$$ is between $$H$$ and $$K$$. If $$HJ=2x+4$$, $$JK=3x+3$$, and $$KH=22$$, find $$x$$. ## Vocabulary Term Definition distance Distance is the measure of length between two points. Absolute Value The absolute value of a number is the distance the number is from zero. Absolute values are never negative. measure To measure distance is to determine how far apart two geometric objects are by using a number line or ruler. ## Additional Resources Interactive Element Video: Ruler Postulate and the Segment Addition Postulate Activities: Distance Between Two Points Discussion Questions Study Aids: Segments Study Guide Practice: Definition of Line Segment Real World: Distance Between Two Points • Was this article helpful?	{"extraction_info": {"found_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": 0.9626222252845764, "perplexity": 403.19621060353444}, "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-2022-05/segments/1642320303864.86/warc/CC-MAIN-20220122134127-20220122164127-00700.warc.gz"}
https://www.physicsforums.com/threads/a-simple-physics-question.236286/	# A Simple Physics Question? 1. May 20, 2008 ### Dru I was wondering about gravity and weight? I apologize if this question seems elementary to you all, so please forgive me as i am not a scientist. If gravity acts equally on all objects regardless of size or mass, (ie. drop a bowling ball or a penny, they both fall at the same speed right?) then why do they have different weights when at rest on the ground? If mass and gravity have nothing to do with one another, then why do larger planets have stronger gravity? Or for that matter, why is the sun's gravitational field so much larger than ours? thnx. 2. May 20, 2008 ### Durato When someone says that gravity 'acts the same' on all objects, what they mean is that the acceleration is the same. However, the force is defined as mass times acceleration. Weight is a synonym for force, so therefore varying masses produce a varying force. That's the mathematical reason, anyways. I guess I could get into how mass could also be defined in how much it resists changes in inertia, but oh well, someone else will! Gravity is dependent on mass. Force = (G)M1M2/(r)^2 (Newton's gravitational law) where G = is the gravitational constant, M1 and M2 are two masses, and r is the distance between the center of mass of both Let M1 = PlanetMass. Therefore, M2 = Mass of object on planet. The force on M2 is equal to M2a, where a is the acceleration ('gravity') due to the planet mass. Therefore, we have M2a = GM1M2/r^2, so a ('gravity') = GM1/r^2 3. May 20, 2008 ### Durato I guess i better explain 'weight' in a more intuitive manner. Let's say we have two shopping carts, one which is laden with groceries and the other which is empty. Which one will it be easier to accelerate in a certain amount of time to a certain velocity. The empty one of course! The heavy cart can have the same acceleration but it will require a greater force to do it. Therefore, the heavy cart has more 'mass' than the lighter cart. Mass can be defined as the resistance to change in inertia. Inertia is simply the object staying in motion. Is it not harder to stop the heavy cart once its moving than stopping the lighter cart? Since the heavy cart is harder to stop at the same speed as the lighter cart, it has more inertia and thus more mass than the lighter cart. Now, I've demonstrated intuitively that the more mass the object has the more force (i.e. 'weight') it is required to accelerate. Since the planet accelerates objects at the same value, it will take more force to accelerate heavier objects than lighter ones. Thus, the heavier object 'weighs' more (has more force exerted on it). Hope this helps. 4. May 21, 2008 ### Danger Good responses, Durato. One thing I would like to add is that the acceleration of gravity is only equal for vastly different objects if they're in vacuum (or very thin air). A 50-gram ball bearing will outrace a 100-kilogram guy with a parachute every time in our atmosphere. 5. May 21, 2008 ### Dru oh i see now, thnx guys that helped! i've always wondered about that, thnx again. i have another quick question regarding einsteinian space-time geometry. einstein's idea of space-time is analogous to a flat sheet spread out with the objects in the universe creating small or large dips in the sheet, right? and this is what causes a gravity well, to an extent in the general vicinity of a planetoid, right? ok, this is what i'm wondering - the sheet analogy is only 2-dimensional? are the dips (in space-time) occuring at every angle to the objects? for example, is space warped not just underneath but above as well as at all angles to the object? so the planetoid would create in effect, an almost entire inward limited collapse of space-time? thnx again 6. May 21, 2008 ### FredGarvin Careful there Danger. The acceleration due to gravity is essentially constant on Earth regardless of whether it is in a vacuum or not. The issue you are talking about is the resisting drag force created by the atmosphere. That is a completely different issue. 7. May 21, 2008 ### Danger Okay, I'm a bit confused on that, Fred. While the gravity is trying to accelerate everything equally, doesn't the drag in fact decrease the actual acceleration? (Maybe I'm using the wrong terminology.) 8. May 21, 2008 ### robertm Absolutely. The example of a rubber sheet is quite a bad one actually as it indeed does only show a 2D world. The existence of mass causes a spatially 3dimensional warp in spacetime whose force is described by F=Gm1m2/r (Newtonian) where F is equivalent in any direction with a constant r and constant masses. It is quite hard to picture a 3D example of gravity warped spacetime, and thats why the popular model is the 2D sheet one. For what it is worth if you would like to try, when I attempt to picture what gravity warped space looks like I first picture space as a 3 dimensional grid with x,y, and z. Then I try and picture what it would look like if a point mass was introduced and all the lines in the grid system were bent so that an object traveling trough this grid would experience a circular motion towards the point mass. Obviously this is only a conceptual model as spacetime cannot be seen directly. 9. May 21, 2008 ### stewartcs No. Gravitational acceleration only varies at different elevations on Earth. It is essentially constant as Fred pointed out. You are thinking about the terminal velocity of an object which is dependent on the drag. Hence, if two different objects are placed in a vacuum, the fall at the same rate. CS 10. May 22, 2008 ### Dru Thnx Rob! I always wondered about that, it makes sense, i always thought so. And i can actually visualize it without the need for picture or graph. This gives me a new found appreciation for einsteinian space-time geometry! You add this to the idea that a spinning planetoid is causing a frame dragging affect on space-time as well, and that makes it even wilder! Thnx again. Now i have a million ideas going through my mind regarding the underlying nature of the fabric of space-time itself and why these warps play out as gravity at all? 11. May 22, 2008 ### iedoc hahaha, robertm, that was a really nice example 12. May 22, 2008 ### robertm Well I'm flattered. Anytime guys! Sharing knowledge is almost as fun as learning it (and certain just as important)!! 13. May 22, 2008 ### jablonsky27 there was a screensaver in win98 which sorta did what you just described. i think they called it blackhole.. you could change some parameters like its radius and all, it seemed pretty cool at the time.. 14. May 22, 2008 ### kwestion I suspect that stewartcs misinterpreted what was said. I interpreted as: "While g is equal for both objects, doesn't the drag decrease the net rate of change in velocity?" Yes, dv/dt ~ g - (b/m)v. Alternative example: "One thing I would like to add is that the equal acceleration of gravity is only visually apparent for vastly different objects if there are no other dissimilar forces interfering such as air resistance..." 15. May 23, 2008 ### Danger That is indeed what I meant. Thanks. I like your 'alternative example'. 16. May 27, 2008 ### rockerdoctor To add on to everyone elses responses, the sensation of weight is given by whats known as an objects normal force. if it is just a person standing on earth it is mg. As newton pointed out, for every action there is an equal an opposite reaction. So, you push down on earth, earth pushes up on you with the same force. An object will have the same mass on earth as it would in a vacuum, the weight however, would change due to the lack of a normal force in a vacuum.	{"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": 0.816471517086029, "perplexity": 824.2340573412484}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988720153.61/warc/CC-MAIN-20161020183840-00065-ip-10-171-6-4.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-equations/86516-xy-2y-3y-0-obtain-power-series-sol-n.html	# Math Help - xy''-2y'+3y=0 obtain a power series sol'n 1. ## xy''-2y'+3y=0 obtain a power series sol'n Obtain a power series solution of the equation xy'' - 2y' + 3y = 0. I'm not sure where to go with this. So i just used y= (n=0)SUM(infinity) cnx^(n+r), n got y' and y''. subbed it into the equation. and simplified to get: r(r-3)=0 and C[n+1] = 3Cn/(n+r-2)(n+r+1) = 0 r=0 , r=3 is this Frobenius? with difference of positive integer (3)? 2. Let's write the differential equation as... $y=\frac{2}{3}\cdot y^{'} - \frac{1}{3}\cdot x\cdot y^{''}$ (1) Now we suppose that (1) has a solution y(*) that can be expressed as power sum, so that is... $y(x)= \sum_{n=0}^{\infty} a_{n}\cdot x^{n}$ (2) In this case the identity (1) becomes [the intermediate computations are not reported...] the following... $a_{0} + \sum_{n=1}^{\infty} a_{n}\cdot x^{n} = \sum_{n=1}^{\infty} (n-\frac{n^{2}}{3})\cdot a_{n}\cdot x^{n-1}$ (3) Now we try to compute the $a_{n}$ from (3) imposing that the coefficients ot the terms $x^{n}$ are identical in both terms. Any solution of (1) is defined unless an arbitrary constant, so that we are allowed to set... $a_{0}=1$ Equating in (3) the coefficients of the term $x^{0}$ we obtain... $a_{0}= \frac{2}{3}\cdot a_{1} \rightarrow a_{1}= \frac{3}{2}\cdot a_{0}= \frac{3}{2}$ ... all right!... equating the the coefficients of the term $x^{1}$ we obtain... $a_{1}= \frac{2}{3}\cdot a_{2} \rightarrow a_{2}= \frac{3}{2}\cdot a_{1}= \frac{9}{4}$ ... all right!... equating the the coefficients of the term $x^{2}$ we obtain... $a_{2}= 0 \cdot a_{3}$ ... ehmm!... there is some minor difficult in computing $a_{3}$ ... ... the conclusion is that our hypothesis is false and doesn't exist any solution of (1) that can be expressed in power series like (2) ... Kind regards $\chi$ $\sigma$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 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": 0.9772785305976868, "perplexity": 1095.784338172376}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936463453.54/warc/CC-MAIN-20150226074103-00276-ip-10-28-5-156.ec2.internal.warc.gz"}
http://www.physicsforums.com/showthread.php?p=2108212	## an issue with solving an IVP by Taylor Series Okay so suppose I have the Initial Value Problem: $$\left. \begin{array}{l} \frac {dy} {dx} = f(x,y) \\ y( x_{0} ) = y_{0} \end{array} \right\} \mbox{IVP}$$ NB. I am considering only real functions of real variables. If $$f(x,y)$$ is analytic at x0 and y0 then that means that we can construct its Taylor Series centered around the point (x0,y0) and that the Taylor Series will have a positive radius of convergence, and also that within this radius of convergence the function $$f(x,y)$$ will equal its Taylor Series. $$f(x,y)$$ being analytic at x0 and y0 also means that the IVP has a unique solution in a neighbourhood of the point x0, as follows from The Existence and Uniqueness Theorem (Picard–Lindelöf). Let y(x) be the function that satisfies the IVP in a neighbourhood of x0. The idea behind the Taylor Series method is to use the differential equation and initial condition to find the Taylor coefficients of y(x): $$\frac {dy} {dx} = f(x,y) \; \; \rightarrow \; \; y'(x_0)=f(x_0,y(x_0))=f(x_0,y_0)$$ $$\frac {d^2y} {dx^2} = \frac {d} {dx} f(x,y)= \frac {\partial f} {\partial x} \frac {dx} {dx} \; + \; \frac {\partial f} {\partial y} \frac {dy} {dx} \\ \indent \rightarrow \; \; y''(x_0)= \frac {\partial f} {\partial x}\|_{(x_0,y(x_0))} \; + \; \frac {\partial f} {\partial y}\|_{(x_0,y(x_0))}\cdot f(x_0,y_0)$$ etc. Obviously we can do this because if f is analytic, all the partial derivatives of f exist at (x0,y0), and by the relationship given by the ODE (which we know y(x) satisfies at least in a neighbourhood of x0), all the derivatives of y exist at x0, that is, $$y(x) \in C^\infty _x (x_0)$$ Okay, so we can construct the Taylor Series of y(x) at x0: $$\sum \frac{y^{(n)}(x_0)}{n!}x^n$$ But! 1)How do we know that this series has a non-zero radius of convergence? 2)And secondly, if it does have a positive radius of convergence, how do we know it is equal to the solution of the IVP within its radius of convergence? That is, obviously within its radius of convergence, the series represents a certain function which is analytic at x0, but how do we know that this function is indeed the function y(x) which is what we called the solution to the IVP? I mean, maybe the solution to the IVP is not analytic at x0 (even though it is infinitely derivable at x0) and thus its Taylor series (the one we constructed) does not represent it in any neighbourhood of x0. After all, the existence and uniqueness theorem does not require or state that the solution be analytic at all. Um, if you can affirm that the Taylor Series we have constructed is indeed a solution to the IVP within its radius of convergence, then by the uniqueness of the solution, the function it represents must be the solution of the IVP, y(x). But can this be affirmed? If anyone has any ideas please help, I'm just not happy with this method coz yes, I can calculate a number of terms of the Taylor Series to approximate the solution of the IVP, but I really have no assurance that the series I am constructing is indeed solution of the IVP (not to mention whether it even does converge at all for any x near x0). PhysOrg.com science news on PhysOrg.com >> Ants and carnivorous plants conspire for mutualistic feeding>> Forecast for Titan: Wild weather could be ahead>> Researchers stitch defects into the world's thinnest semiconductor Sometimes I feel I speak in Martian, especially when my posts are long ones:P Did anyone actually get was I was on about? Just curious, as sometimes I myself don't lol :P Yay I think I might be able to (partially) answer my own question xD -gingerly tries to demostrate- Let $$T_{y,x_0} (x) = \sum_{n=0}^\infty \frac{y^{(n)}(x_0)}{n!}x^n$$ be the Taylor series of y(x) that I've constructed using the initial condition and the relationship given by the o.d.e. If it has a positive radius of convergence R>0 (unfortunately I'll only be able to know this if I can deduce an expression for the general term of the series), then within its radius of convergence, it converges to an analytic (at x0) function, lets say u(x): [tex] u(x) = T_{y,x_0} (x) = \sum_{n=0}^\infty \frac{y^{(n)}(x_0)}{n!}x^n = T_{u,x_0} (x) = \sum_ {n=0}^\infty \frac{u^{(n)}(x_0)}{n!}x^n \ \ \ \ \ \ \ \ \ \mbox{for} \ \ \|x-x_0\| Blog Entries: 3 ## an issue with solving an IVP by Taylor Series Wikipedia says if the a function is analytic then it is locally given by a convergent power series. http://en.wikipedia.org/wiki/Analytic_function Quote by John Creighto Wikipedia says if the a function is analytic then it is locally given by a convergent power series. http://en.wikipedia.org/wiki/Analytic_function Fanku, John :) I've always gone by that definition of analicity (for real functions anyhow, which are what concern me). That is why I equated f(x,u) to its power series (Taylor series) under the assumption that it was analytic xD. Tags convergence, existence, ivp, taylor, uniqueness	{"extraction_info": {"found_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": 0.9631577134132385, "perplexity": 335.35592838356234}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368703317384/warc/CC-MAIN-20130516112157-00064-ip-10-60-113-184.ec2.internal.warc.gz"}
https://www.groundai.com/project/landauer-buttiker-approach-for-hyperfine-mediated-electronic-transport-in-the-integer-quantum-hall-regime/	A Derivation of the spin-flip transmission coefficient. # A Landauer-Büttiker approach for hyperfine mediated electronic transport in the integer quantum Hall regime. ## Abstract The interplay of spin-polarized electronic edge states with the dynamics of the host nuclei in quantum Hall systems presents rich and non-trivial transport physics. Here, we develop a Landauer-Büttiker approach to understand various experimental features observed in the integer quantum Hall set ups featuring quantum point contacts. The approach developed here entails a phenomenological description of spin resolved inter-edge scattering induced via hyperfine assisted electron-nuclear spin flip-flop processes. A self-consistent simulation framework between the nuclear spin dynamics and edge state electronic transport is presented in order to gain crucial insights into the dynamic nuclear polarization effects on electronic transport and in turn the electron-spin polarization effects on the nuclear spin dynamics. In particular, we show that the hysteresis noted experimentally in the conductance-voltage trace as well as in the resistively detected /media/arxiv_projects/145801/NMR lineshape results from a lack of quasi-equilibrium between electronic transport and nuclear polarization evolution. In addition, we present circuit models to emulate such hyperfine mediated transport effects to further facilitate a clear understanding of the electronic transport processes occurring around the quantum point contact. Finally, we extend our model to account for the effects of quadrupolar splitting of nuclear levels and also depict the electronic transport signatures that arise from single and multi-photon processes. ## I Introduction Nuclear spintronics concerns the manipulation of nuclear spins by means of hyperfine interaction between the host nuclei and the itinerant electrons and their read out using electronic transport Hirayama et al. [2009] or optical Urbaszek et al. [2013] measurements. Quantum Hall geometries in both the integer Wald et al. [1994]; Machida et al. [2002, 2003]; Würtz et al. [2005]; Ren et al. [2010]; Dean et al. [2009]; Song and Omling [2000]; Kawamura et al. [2007]; Keane et al. [2011a]; Hirayama et al. [2009] and the fractional regime Hashimoto et al. [2002]; Kou et al. [2010]; Akiba et al. [2011, 2013]; Hatano et al. [2015] featuring gated quantum point contacts (QPC) offer a viable method for controlling the spin polarization of the electronic edge channels. This in turn facilitates the manipulation of the nuclear spins via a hyperfine mediated interplay between the spin-polarized edge states and the dynamics of the host nuclei. Such an interplay has revealed rich and non-trivial transport physics in the form of hysteresis in the observed conductance-voltage traces and non-trivial lineshapes in the resistively detected /media/arxiv_projects/145801/NMR (RD/media/arxiv_projects/145801/NMR) traces Wald et al. [1994]; Machida et al. [2002, 2003]; Kawamura et al. [2007, 2010]. Despite several advancements in the transport experiments involving such set ups, theoretical models for hyperfine interaction mediated edge transport through the QPC in the Hall geometry are clearly missing in the current literature. The object of this work is hence to develop transport models that couple the dynamics of the host nuclei with edge channel electronic transport as an attempt to fill this gap and theoretically interpret various experiments with specific focus on the conductance-voltage traces Wald et al. [1994] and the RD/media/arxiv_projects/145801/NMR lineshapes Gervais [2009]; Bowers et al. [2010]; Keane et al. [2011a]; Tiemann et al. [2014]; Yang et al. [2011]; Kodera et al. [2006]; Tracy et al. [2006]; Desrat et al. [2002]; Gervais et al. [2005] . We develop our transport models based on a modified Landauer-Büttiker formalism that includes a spin-flip transmission coefficient, which is nuclear polarization dependent and describes the rate of electron-nuclear spin flip-flops per unit energy around the QPC region. Using this approach, we show that the hysteresis noted in both the conductance and the RD/media/arxiv_projects/145801/NMR traces Wald et al. [1994]; Gervais [2009]; Bowers et al. [2010]; Keane et al. [2011a]; Tiemann et al. [2014]; Yang et al. [2011]; Kodera et al. [2006]; Tracy et al. [2006]; Desrat et al. [2002]; Gervais et al. [2005] results from a lack of steady state between electronic transport and nuclear polarization evolution and can be explained by taking into account the finite rate of electron-nuclear spin flip-flops in a source limited channel in addition to a finite nuclear spin-lattice relaxation time. The self-consistent simulation framework between the nuclear spin dynamics and the edge state electronic transport developed here offers crucial insights into the dynamic nuclear polarization effects on electronic transport and in turn the electron-spin polarization effects on the nuclear spin dynamics. In addition, we present circuit models to emulate such hyperfine mediated transport effects for a clear understanding of the phenomena occurring near the QPC. Finally, we also address the effects of quadrupolar splitting of the nuclear levels and depict the electronic transport signatures that arise from single and multi-photon absorption processes Kawamura et al. [2010]. This paper is organized as follows. In Sec. II, we briefly detail the experimental set up and features that form our current focus after which we spell out the generic formalism. Specifically, in Sec. II.2, a phenomenological model for hyperfine mediated transport through the QPC is developed in detail. Section III elucidates the results from the simulation framework developed with the specific focus on explaining the various experimental trends noted. Specifically, Sec. III.1 is devoted to the understanding of the hysteritic conductance voltage traces noted for different filling factors and Sec. III.2 deals with the RD/media/arxiv_projects/145801/NMR lineshape features in great detail. ## Ii Experimental details and theoretical description In the schematic of the experimental set up shown in Fig. 1(a), an appropriately gated single QPC is utilized to selectively filter out a single spin channel into the region beyond the QPC thereby creating an imbalance between the up-spin channel and the down-spin channel. The principal experimental signature here is the change in conductance with voltage sweep near Wald et al. [1994] as shown in Fig. 1 (b) as well as the change in Hall resistance with RF frequency sweep (also known as resistively detected /media/arxiv_projects/145801/NMR or RD/media/arxiv_projects/145801/NMR) Keane et al. [2011a] as shown in Fig. 1 (c). Along with the change in the conductance, another feature which has attracted significant attention is the hysteresis in the conductance plots during forward and reverse voltage or RF frequency sweep as shown in Fig. 1 (b) and (c) respectively. A compact theoretical model to elaborate the physics of such a conductance modulation as well as hysteresis occuring in the gated QPC set up forms the primary focus of this work. An accurate modeling of such phenomena involves taking into account the details of wavefunction correlations via the density matrix approach Zaletel et al. [2015]. Mathematical modeling of such hyperfine mediated electron transport process self-consistently with evolution of the nuclear polarization from the density matrix formalism Zaletel et al. [2015] is complicated and computationally heavy. In this paper, we thus adapt a computationally efficient phenomenological model to account for such hyperfine mediated electronic transport through the QPC. We now provide a theoretical description of the nuclear spin dynamics coupled to the electronic transport following which we focus on how to apply this to our specific set up. We begin with the description of the nuclear spin dynamics by formulating a master equation in the nuclear spin space followed by the description of the extended Landauer-Büttiker formalism for the edge state electronic transport. ### ii.1 Description of scattering processes In order to describe the electron-nuclear hyperfine interaction, we start with the Fermi contact hyperfine interaction Hamiltonian for the case with non-varying electronic density of states in space, given by Urbaszek et al. [2013] ^HHF(rn)=∑nAeffψ∗(rn)ψ(rn)a30⎡⎣^Sz⊗^Inz+⎧⎨⎩^In+⊗^S−+^S+⊗^In−2⎫⎬⎭⎤⎦, (1) where represents the electron wavefunction at the point , with representing the effective electron density per unit volume at the point , is the effective hyperfine coupling constant, are the operators representing the component of the electronic spin and the nuclear spin respectively, with representing a unit cell volume. The operator represents the tensor product between the electron spin and the nuclear spin spaces. The operators, and are respectively the corresponding spin raising (lowering) operators for the electron and the nuclear spins respectively. The above equation assumes , where ’’ and ’’ represents the eigen states in the electron spin-space. In the quantum Hall regime, however, the eigen states are localized in space along the transverse direction. In this case, and thus the Hamiltonian should be recast in the form Sakurai and Napolitano [2011a, b]; Griffiths [2005]: ^HHF(rn)=Aeffa30∑\|ϕ,β⟩⟨φ,α\|⟨ϕ,β\|⎡⎣^Sz⊗^Inz+⎧⎨⎩^In+⊗^S−+^S+⊗^In−2⎫⎬⎭⎤⎦\|φ,α⟩×∑n{⟨ψβ\|\|rn⟩\|ϕ,β⟩⟨φ,α\|⟨r% n\|\|ψα⟩}, where and with and belonging to the electron spin-space and and belonging to the nuclear-spin space in the presence of a magnetic field pointing along the direction. When the coupling constant is small, the effect of the two terms in (1) can be separated. The first term results in an effective magnetic field and the second term within the curly brackets represents the electron-nuclear spin flip-flop processes. The first term in (1), which also corresponds to and in (LABEL:eq:hyperfine_ham) introduces an additional shift in the electronic energy levels as well as between states of different nuclear spins. The electronic energy difference between the up-spin channel electrons and the down-spin channel electrons is given by Slichter [1990] Δ=gelμBBapp+Aeff, (3) where , and are the effective Lande -factor of the electron in GaAs, the Bohr magneton and the applied magnetic field respectively. The above expression is obtained by assuming an isotropic nuclear spin distribution. Similarly, the energy difference between the adjacent nuclear spin states differing by a magnetic quantum number of is given by ϵ=gnucμnucBBapp+A′′, (4) where is the effective Lande -factor of the nuclide in consideration and is the nuclear Bohr magneton. The effective coupling constant Das Sarma et al. [2003]; Slichter [1990]; Urbaszek et al. [2013], where is the electronic carrier density. The magnetic quantum number for the GaAs nuclei varies from to in steps of . In standard literature, the second terms in (3) and (4) represent the Overhauser shift and the Knight shift respectively. However, for practical purposes, and and hence may be neglected with respect to and the nuclear spin flips may be considered elastic. The electron-nuclear spin flip-flop processes are described by the second term in the Hamiltonian in (1), which also corresponds to and in (LABEL:eq:hyperfine_ham) and the scattering rates are evaluated via the Fermi’s golden rule Buddhiraju and Muralidharan [2014]; Siddiqui et al. [2010], typically related to the densities of the initial and the final states. In this case, the electronic wavefunction distribution and overlaps with the nuclear wavefunction on each site differently, and this effect is accounted for via the overlap terms for up to down or down to up electronic spin transitions. The procedure for a self-consistent description of electronic transport coupled to hyperfine spin dynamics then entails the time-dependent simulation of the nuclear spin dynamics, with the electronic transport processes in steady state. This is because the nuclear spin dynamics are typically slow due to slow relaxation rates, slow diffusion rates as well as longer flip-flop times in comparison with the electronic transport velocities. The nuclear spin dynamics at each point are dictated via the electron-nuclear hyperfine flip rates calculated from the Fermi’s golden rule Buddhiraju and Muralidharan [2014] given by: Γ↑↓(rn)=2πℏ∣Aeff∣2∫dEn↑(rn,E)p↓(rn,E) Γ↓↑(rn)=2πℏ∣Aeff∣2∫dEn↓(rn,E)p↑(rn,E), where () represents the up to down (down to up) nuclear spin transition rate at the nuclear co-ordinate between magnetic quantum numbers that differ by () in the nuclear spin space due to flip-flop transitions of electrons at energy . The quantities and ( and ) denote the densities of filled (vacant) states per unit energy per unit area at the point . \colorblack The up to down (down to up) electronic spin transition rate based on (LABEL:eq:scattering_rates) depends not only on the the availability of electrons in the up (down) spin density of states and the vacancy in the down (up) spin density of states , but also on the spatial overlap of the corresponding density of states. The total rate of electron-nuclear spin flip-flop now depends on the integral of () over the spatial co-ordinates. Γ↑↓=2πℏ∣Aeff∣2∫∫d3rndEn↑(rn,E)p↓(rn,E) Γ↓↑=2πℏ∣Aeff∣2∫∫d3rndEn↓(rn,E)p↑(rn,E) The spatial dynamics of the nuclear spins can be described by the following master equation: d[F(rn)]dt=[Γ(r% n)][F(rn)]−[F(rn)−F0]τI +Dn∇2[F(rn)], where is the probability column vector representing the probability of occupancy of the nuclear spin levels, and is a phenomenological nuclear spin relaxation time, which is typically a very slow process. The matrix takes into account the transition between the individual nuclear spin levels. The vector denotes the probability of occupation of the nuclear spin levels in equilibrium. The above equation also includes nuclear spin diffusion described by the last term, where is the phenomenological diffusion constant. In this paper, we neglect the exact spatial distribution of nuclear spins due to diffusion and approximate the effects of nuclear spin diffusion by incorporating a larger number of nuclei. The equation governing the dynamics of the nuclear spins is then given by: d[F]dt=[Γ][F]−[F−F0]τI. (8) The transition probability matrix may be specifically cast for the spin- case in the current study in terms of the spin-flip rates defined in (LABEL:eq:scattering_rates2) as: [Γ]=⎡⎢ ⎢ ⎢ ⎢⎣−Γ↓↑Γ↑↓00Γ↓↑−(Γ↓↑+Γ↑↓)Γ↑↓00Γ↓↑−(Γ↓↑+Γ↑↓)Γ↑↓00Γ↓↑−Γ↑↓⎤⎥ ⎥ ⎥ ⎥⎦, where and are defined in (LABEL:eq:scattering_rates2). An additional constraint used to solve (8) using (LABEL:eq:matrix) is that of the normalization of the nuclear state probabilities, i.e., , where is the occupation probability of the nuclear density of states with spin . The temporal evolution of the average electronic polarization and the average nuclear polarization at the QPC are calculated self-consistently by solving (8) and (3) via the relations: =12n↑−n↓n↑+n↓ FI==∑ssFs=[s][F], (10) where are obtained by solving the master equations. The matrix comprises the row vector of the spin magnetic quantum numbers of the GaAs nuclei. The procedure for transport calculations follows solving (3), (8), and (10) sequentially in a self-consistent loop with the electronic transport to be described now. ### ii.2 Electronic edge-state transport in the QPC region While the dynamics of the nuclear spins simply follow the master equation (8) described above, a description of electronic transport involves transport currents due to the source and drain reservoirs held at electrochemical potentials and respectively. From a Landauer-Büttikker perspective, a consistent description of transport currents in our case demands the use of both a) direct transmission and b) spin-flip transmission. The need to include spin flip transmission follows from the interaction between the edge channels of different spins that gives rise to nuclear polarization which in close proximity of the QPC region determines the electronic transport. Near the QPC, the forward propagating edge channels and the backward propagating edge channels come in close proximity and hence spin-flip scattering can occur to the forward propagating as well as to the backward propagating edge channels Wald et al. [1994]. The Landauer direct transmission denotes the tunneling probability of the up (down)-spin electrons through the QPC. We model the spin-split edge states in the device by a continuum of density of states as in a ballistic conductor Salahuddin [2006]; Kawamura et al. [2015]; Wan et al. [2009]; Côté and Simoneau [2016] with the region of the QPC being represented by a Gaussian potential barrier, as shown in Fig. 2 (a) and (b) along with the model used for simulation of electronic transport shown in Fig. 2 (c). The direct transmission coefficients and are then calculated using the non-equilibrium Green’s function (NEGF) method applied to the barrier described above using a atomistic tight-binding Hamiltonian Datta [1997, 2005]. The pertinent details of the approach used here have been briefly discussed in Appendix B. In our scheme, we only consider the number of transmitted modes and the filling factor at the QPC for the electronic transport which is related to the geometry of the set up that is ascertained apriori. Near the vicinity of at the QPC, the down-spin edge channel at the QPC is almost empty in the energy range between and , resulting in a considerable simplification of the transport equations. We begin with the case where the filling factor , i.e., , where only the up-spin edge channel originating from the source contact contributes to the total current terminating in the drain contact. The electrons in the forward propagating up-spin edge channel originating from the source contact can tunnel through the QPC to the up-spin edge channel terminating in the drain contact with a probability while the forward propagating down-spin edge channel originating in the source contact is completely disconnected from the forward propagating down-spin edge channel terminating in the drain contact, as depicted in Fig. 2(a) and (b) respectively. A few up-spin electrons at the QPC in the forward propagating edge channel terminating in the drain contact can however scatter to the forward propagating down-spin edge channel terminating in the drain contact with a spin-flip process as shown in Fig. 3 (a). This gives rise to the spin-flip scattering current , where the superscript ’’ denotes the flow of current due to spin-flip scattering at the QPC and the subscript denotes the current flow from the up (down)-spin to down (up)-spin edge channel via electronic spin-flips. Assuming that the direct transmission coefficients ( and ) depend on the nuclear polarization only via the Overhauser field (3), for a system with four nuclear spin levels, the spin-flip transmission coefficient at the QPC from the forward propagating up-spin channel terminating in the drain contact to the forward propagating down-spin channel terminating in the drain contact is given by (details given in Appendix A): Missing or unrecognized delimiter for \Big Note that depends on the spatial overlap of the density of states of the up-spin and down-spin edge channel at the QPC between the energy range and (details given in Appendix A, Eq. 32 and 37). We approximate as a constant. Therefore, the down-spin current recorded just outside the QPC relies entirely on such spin-flip processes and hence is simply the spin-flip current while the up-spin current in the edge channel just outside the QPC is reduced by . Based on the above discussions, the up and down spin channel currents are given by I=∫dE(I↑(E)+I↓(E)) =qh∫dE{[T↑(E)−Tsff↑↓(E)]+Tsff↑↓(E)}×{fS(E)−fD(E)}. (11) The subscripts and the source and drain contacts respectively, with denoting the Fermi-Dirac distribution in the source (drain) contact held in quasi-equilibrium at . The parameter takes into account the spin-flip scattering of electrons at and around the QPC from the forward propagating up-spin edge channel terminating in the drain contact to the forward propagating down-spin channel terminating in the drain contact, with the superscript denoting spin-flip scattering to a forward propagating channel. It must be noted that the edge channels in the quantum Hall arrangement are uni-directional and hence the expressions for the current in (11) depend on the factors and only and not on the factors and as expected in a typical Landauer type scattering treatment. Turning our attention to the case when the filling factor , i.e., , the down-spin electrons in the edge channel originating in the source contact are partially transmitted through the QPC to the down-spin edge channel terminating in the drain contact, as depicted in Fig. 3(b). In this case, the spin-flip scattering at and around the QPC can occur from the forward propagating up-spin channel terminating in the drain contact to the forward propagating down-spin channel terminating in the drain contact as well as from the forward propagating down-spin channel terminating in the drain contact to the backward propagating up-spin channel originating from the drain contact. Again, assuming that the direct transmission coefficients and depend on the nuclear polarization only via the Overhauser field (3), the spin-flip currents in this case are given by (details in Appendix A) Isf↑↓ ≈qh∫dETsff↑↓(E)(fS(E)−fD(E)) Isf↓↑ ≈qh∫dETsfb↓↑(E)(fS(E)−fD(E)), where superscript denote spin-flip scattering to a backward propagating edge channel while the superscript has the same meaning as described previously. The spin-flip current flows from the forward propagating up-spin channel terminating in the drain contact to the forward propagating down-spin edge channel terminating in the drain contact while the spin-flip current flows from the forward propagating down-spin edge channel terminating in the drain contact to the backward propagating up-spin channel originating in the drain contact. Hence, causes a change in the total output current since the spin-flip scattering occurs to a backward propagating edge channel. It however does play a role in the nuclei polarization near the QPC. The current in the up-spin and down-spin channel terminating in the drain contact just outside the QPC is then given by: I↑=∫qh{T↑(E)−Tsff↑↓(E)}{fS(E)−fD(E)}dE =∫qh{T↑(E)−T↑(E)Tf↑↓(1−F32)}×{fS(E)−fD(E)}dE (12) I↓=∫qh{T↓(E)+Tsff↑↓(E)−Tsfb↓↑(E)}{fS(E)−fD(E)}dE =∫qh{T↓(E)+T↑(E)Tf↑↓(1−F32)−T↓(E)Tb↓↑(1−F−32)}×{fS(E)−fD(E)}dE (13) From the above discussion, the generalized equations for the up-spin, down-spin and spin-flip currents through the QPC are given by: I↑ =qh∫{T↑(E)DirectTransmission+ Tsff↓↑(E)−Tsff↑↓(E)spin flipforward transmission −Tsfb↑↓(E)spin flipbackward transmission}×{fS(E)−fD(E)}dE =qh∫{T↑(E)+Tf↓↑T↓(E){1−F−32}−Tf↑↓T↑(E){1−F32}−Tb↑↓T↑(E){1−F32}}×{fS(E)−fD(E)}dE I↓ =qh∫{T↓(E)DirectTransmission+ Tsff↑↓(E)−Tsff↓↑(E)spin flipforward transmission −Tsfb↓↑(E)spin flipbackward transmission}×{fS(E)−fD(E)}dE =qh∫{T↓(E)+Tf↑↓T↑(E){1−F32}−Tf↓↑T↓(E){1−F−32}−Tb↓↑T↓(E){1−F−32}}×{fS(E)−fD(E)}dE Isf =\|Isf↑↓\|−\|Isf↓↑\| =qh∫{Tsff↑↓(E)+Tsfb↑↓(E)−Tsff↓↑(E)−Tsfb↓↑(E)}×{fS(E)−fD(E)}dE (14) where and are the direct transmission coefficients between the forward propagating edge channels originating and terminating in the source and drain contacts respectively through the QPC in the absence of electron-nuclear spin flip-flop scattering. As already discussed, the term and characterize spin-flip scattering from a forward propagating edge channel terminating in the drain contact to a forward propagating edge channel terminating in the drain contact at the QPC, while and characterize spin-flip scattering at the QPC from a forward propagating edge state terminating in the drain contact to a backward propagating edge channel originating in the drain contact at the QPC. The terms , , and , being the probability of electron-nuclear spin flip-flop processes, are dependent on the nuclear polarization (details given in Appendix A). The spin-flip currents and give rise to nuclear polarization at and around the QPC region. Turning our attention to the self-consistent solution of the electronic transport and the temporal evolution of the nuclear polarization, the electronic transport is influenced by the nuclear polarization via the Overhauser field while the evolution of nuclear polarization is determined by the spin-flip current and the nuclear spin-lattice relaxation time (). The matrix , which determines the temporal evolution in nuclear polarization is hence related to the spin-flip currents and . A schematic diagram on self-consistency involved in the temporal evolution of nuclear polarization and electronic transport phenomena is shown in Fig. 4. For a system with quad nuclear spin levels as in GaAs, it can be shown that and with and (details given in Appendix C), being the number of nuclei that are being influenced by spin flip-flop processes at the QPC. We can hence rewrite the expression for as (details given in Appendix C): [Γ]=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−C2\|Isf↓↑\|C1\|Isf↑↓\|00C2\|Isf↓↑\|−(C2\|Isf↓↑\|+C1\|Isf↑↓\|)C1\|Isf↑↓\|00C2\|Isf↓↑\|−(C2\|Isf↓↑\|+C1\|Isf↑↓\|)C1\|Isf↑↓\|00C2\|Isf↓↑\|−C1\|Isf↑↓\|.⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ (15) Let us now consider the experimental features on a case by case basis. ## Iii Results ### iii.1 Conductance hysteresis with voltage sweep We first reproduce some trends noted in the conductance plots of a recent experiment Wald et al. [1994] where a change in the conductance along with hysteresis in the conductance was noted in the vicinity of with positive and negative source to drain voltage sweep. We explain the possible phenomena giving rise to such experimental trends. Case I: A schematic of the scattering processes in this regime is shown in Fig. 3(a) while the up-spin and down-spin edge current paths and equivalent circuit models for the phenomena occuring around the QPC are shown in Fig. 5(a) and (b) respectively. In this case, the following points are to be noted: 1. Only the up-spin channel is transmitted through the QPC. 2. The down-spin channel originating in the source contact is totally reflected at the QPC. 3. Some up-spin electrons in the edge channel terminating in the drain contact can scatter at the QPC to the down-spin edge channel terminating in the drain contact via electron-nuclear spin flip-flop scattering. Such a scattering decreases the current in the up-spin channel just outside the QPC and increases the current in the down-spin edge channel outside the QPC. However, the total current remains proportional to . Reason for an increase in near . 1. Near V=0, the nuclear polarization cannot be maintained. 2. Nuclear polarization drops due to spin lattice relaxation. 3. A drop in nuclear polarization results in an increase in the direct transmission coefficient of the up-spin channel due to a decrease in the Overhauser field as well as an increase in the spin-flip transmission coefficient . Equivalent circuit model: A schematic of the edge channel path in this case is shown in Fig. 5 (a) while the equivalent circuit in this case is detailed in Fig. 5 (b) to aid a visualization of the various transport phenomena inside the device. The circuit model can be described as follows: 1. The up-spin edge channel is represented by a conductance . 2. The up-to-down spin-flip scattering can be represented by an equivalent current source from the up-spin channel. . 3. The change in transmissivity of the up-spin channel due to the Overhauser field is represented by a by an equivalent conductor in series with a voltage dependent voltage source . The current change due to the Overhauser field is represented by . 4. The nuclear polarization is represented by the voltage across the capacitor. 5. The resistance in parallel with the capacitor represents nuclear spin-lattice relaxation. Case II A schematic of the scattering processes in this regime is shown in Fig. 3(b) while the up-spin and down-spin edge current paths and equivalent circuit models for the phenomena occuring around the QPC is shown in Fig. 5 (c) and (d) respectively. In this case, the following points are to be noted: 1. The up-spin electrons in the edge channel originating in the source contact are fully transmitted through the QPC to the up-spin edge channel terminating in the drain contact. 2. The down-spin electrons in the edge channel originating in the source contact are partially transmitted through the QPC to the down-spin edge channel terminating in the drain contact. 3. Two kinds of electron-nuclear spin-flip scattering dominate at the QPC in this case: 1. Electrons from the forward propagating up-spin channel terminating in the drain contact can undergo spin-flip scattering to the forward propagating down-spin channel terminating in the drain contact which is almost empty in the energy range between and . Such scattering at and around the QPC results in a positive nuclear polarization. 2. Electrons in forward propagating down-spin channel propagating through the QPC can undergo a spin-flip scattering to the backward propagating up-spin channel (which is totally empty in the energy range between and ) terminating in the source contact. Such scattering results in a negative nuclear polarization in addition to decreasing the total current through the QPC. 3. Out of these two processes, the former process dominates at the QPC due to the presence of more up-spin electrons compared to down-spin electrons resulting in a net positive nuclear polarization at the QPC. Reason for a decrease in near : 1. Near , the nuclear polarization cannot be maintained. 2. Nuclear polarization drops due to spin-lattice relaxation. 3. A drop in polarization results in an increase in the up-to-down spin-flip rate as well as a decrease in down-to-up spin-flip rate in addition to a decrease in the direct transmission coefficient of the down-spin channel due to decrease in the Overhauser field. 4. The decrease in transmission coefficient of the down-spin channel decreases the conductance of the QPC in the vicinity of . Equivalent circuit model: The equivalent circuit in this case is detailed in Fig. 5(d). The circuit model can be described as follows: 1. The up spin edge channel originating in the source contact is fully transmitted through the QPC and hence is represented by a conductance . 2. The down-spin edge channel originating in the source contact is partially transmitted through the QPC to the down-spin edge channel terminating in the drain contact and hence is represented by a conductance . 3. The up-to-down spin-flip current at the QPC from the forward propagating up-spin channel terminating in the drain contact to the forward propagating down-spin channel terminating in the drain contact is represented by a current dependent current source . 4. The down-to-up spin-flip current from the forward propagating down-spin channel originating in the source contact to the backward propagating up-spin channel terminating in the source contact is represented by a current dependent current source . 5. The change in the transmission coefficient of the down-spin channel due to the Overhauser field is represented by a by an equivalent conductor in series with a voltage dependent voltage source (). The current change due to the Overhauser field is represented by . 6. The nuclear polarization is represented by the voltage across the capacitor. 7. The resistance in parallel with the capacitor represents nuclear spin-lattice relaxation by causing charge leakage from the capacitor. The simulated results of the change in conductance with source to drain voltage sweeps are shown in Fig. 6. The parameters and in the simulations are calculated directly via a non-equilibrium Green’s function (NEGF) method using an atomistic tight-binding Hamiltonian Datta [1997, 2005, 2012] while the parameters and are calculated using (37). The parameters in the above illustration for the circuit diagrams in Fig. 3 are chosen to match the simulated result of the change in conductance with source to drain voltage sweep. The maximum change in the conductance due to a difference in the Overhauser field between the fully polarized nuclei and the non-polarized nuclei is less than . However this maximum change can be enhanced due to spin flip-flop tunneling as noted experimentally Wald et al. [1994]. \colorblack The hysteresis in the curves in Fig. 6 near occurs only when the nuclear spin relaxation time () is of the order of the voltage sweep time. This results in a lag between the applied voltage and nuclear polarization near thereby resulting in the hysteresis. The hysteresis in plots disappear when the is very large such that the change in nuclear polarization is negligible during the time of voltage sweep. The hysteresis in vs plots also disappear when is very small compared to the voltage sweep time because the nuclear polarization is always in a steady state with the applied voltage. ### iii.2 Resistively detected nuclear magnetic resonance (RD/media/arxiv_projects/145801/NMR) We begin our analysis with (14), where the nuclear polarization in the vicinity of the QPC is perturbed by an externally applied alternating magnetic field in the radio frequency (RF) range resulting in the Zeeman split nuclear levels to interact with each other. Near the frequency corresponding to the difference in energy between the two spin split nuclear energy levels (), precession of the nuclear spins accompanied by a rapid decay in the nuclear polarization occurs. To model such processes, we model the spin split nuclear energy levels by a broadened normalized density of states Datta [1997, 2005]; Slichter [1990]. Ds(ξ)=12πη(ξ−ϵs)2+(η2)2, where is the free variable denoting energy in the nuclear spin space, is related to the amount of broadening of the nuclear spin levels and is the energy level of the nuclear spin in the absence of broadening. Broadening might be a result of thermal motion of the nucleus Slichter [1990], hyperfine interaction mediated electron-nuclear spin exchange Slichter [1990] as well as nuclear dipole-dipole exchange interaction Slichter [1990] which is the causative agent for nuclear spin diffusionSlichter [1990]. We take broadening to be . We simulate the case of quad nuclear spin levels, as in GaAs, separated in energy due to Zeeman splitting. The rate equations in this case are given by: [dF(ξ)dt]=[dF(ξ)dt]flip−flop +[dF(ξ)dt]relaxation +[dF(ξ)dt]NMR FI=∫∞−∞[s]×[DN(ξ)]×[F(ξ)]dξ (17) Tsff↓↑(E) =Tf↓↑T↓(E){1−∫D−32(ξ)F−32(ξ)dξ} Tsff↑↓(E) =Tf↑↓T↑(E){1−∫D32(ξ)F32(ξ)dξ} Tsfb↓↑(E) =Tb↓↑T↓(E){1−∫D−32(ξ)F−32(ξ)dξ} Tsfb↑↓(E) =Tb↑↓T↑(E){1−∫D32(ξ)F32(ξ)dξ} Isf↑↓=∫eh{Tsff↑↓(E)+Tsfb↑↓(E)}{fS(E)−fD(E)}dE Isf↓↑=∫eh{Tsff↓↑(E)+Tsfb↓↑(E)}{fS(E)−fD(E)}dE, where is the row vector denoting four nuclear spin levels in GaAs and . is the probability of occupancy of the density of states of the nuclear spin level at energy . The matrix is the diagonal matrix representing the nuclear density of states at energy given by: DN(ξ)=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣D32(ξ)0000D12(ξ)0000D−12(ξ)0000D−32(ξ)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ At low temperatures, is a boxcar function. Assuming that the average value of and in the energy range between and are and respectively and the spin-flip transmission coefficients, and , are constant in the range of energy between and , the above equation can be simplified to, Isf↑↓=ehTavg↑{Tf↑↓+Tb↑↓}×{1−∫D32(ξ)F32(ξ)dξ}Δμ =e2hTavg↑Tavg↑↓×{1−∫D32(ξ)F32(ξ)dξ}V Isf↓↑=ehTavg↓{Tf↓↑+Tb↓↑}×{1−∫D−32(ξ)F−32(ξ)dξ}Δμ =e2hTavg↓Tavg↓↑×{1−∫D−32(ξ)F−32(ξ)dξ}V, where . The set of equations (LABEL:eq:nmr2)-(LABEL:eq:nmr5) have to be solved self-consistently to calculate the temporal evolution of the nuclear polarization. We now turn our attention towards (LABEL:eq:nmr2). The first term on the right hand side of (LABEL:eq:nmr2) is given by: [dF(ξ)dt]flip−flop=[Γout]×[F(ξ)]+{[I4]−[Fdiag(ξ)]}×[Pdiag]−1[Γin][N], (21) where is the identity matrix of the fourth order and [], [], and are given by: [Fdiag(ξ)]=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣F32(ξ)0000F12(ξ)0000F−12(ξ)0000F−32(ξ)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ (22) [N]=∫[D32(ξ)F32(ξ) D12(ξ)F12(ξ) D−12(ξ)F−12(ξ) D−32(ξ)F−32(ξ)]†dξ (23) Pdiag=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣∫D32(ξ){1−F32(ξ)}dξ0000∫D	{"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": 0.9222718477249146, "perplexity": 1221.3804648268172}, "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-2018-26/segments/1529267860089.11/warc/CC-MAIN-20180618051104-20180618071104-00082.warc.gz"}
https://face2ai.com/math-probability-2-1-conditional-probability/	# 条件概率 Advertisements ## 条件概率的定义 The Definition of Conditional Probability Definition Conditional Probability: Sippose that we learn that an event B has occurred and that we wish to compute the probability of another event A taking into account that probability of the event A given that the event B has occurred and is denoted $Pr(A\|B)$ . If $Pr(B)>0$ ,we compute this probability as: $$Pr(A\|B)=\frac{Pr(A\cap B)}{Pr(B)}$$ ps:The conditional probability $Pr(A\|B)$ is not defined if $Pr(B)=0$ 1. 举个例子：条件描述：扔两个六面体骰子，每个面出现概率相等，两个骰子互不影响。事件的概率：那么当我们知道其结果的和是奇数的条件下，其和小于8的事件T的概率是多少？分析：首先我们通过条件知道这两个骰子出现每个数字概率相等为 $\frac{1}{6}$ 那么就可以分析出所有结果了： event Probability 2 $\frac{1}{36}$ 3 $\frac{2}{36}$ 4 $\frac{3}{36}$ 5 $\frac{4}{36}$ 6 $\frac{5}{36}$ 7 $\frac{6}{36}$ 8 $\frac{5}{36}$ 9 $\frac{4}{36}$ 10 $\frac{3}{36}$ 11 $\frac{2}{36}$ 12 $\frac{1}{36}$ $$Pr(A\cap B)=\frac{2}{36}+\frac{4}{36}+\frac{6}{36}=\frac{12}{36}=\frac{1}{3}\\ Pr(B)=\frac{2}{36}+\frac{4}{36}+\frac{6}{36}+\frac{4}{36}+\frac{2}{36}=\frac{1}{2}$$ Hence: $$Pr(A\|B)=\frac{Pr(A\cap B)}{Pr(B)}=\frac{2}{3}$$ 2. 再举个例子，两个箱子，装着不同的螺丝，箱子A装着长螺丝7个和短螺丝3个，B装长螺丝6个短螺丝4个，这两个箱子被随机分给我们，如果我们有 $\frac{1}{3}$ 的概率被分到箱子A，$\frac{2}{3}$ 的概率被分到箱子B，那么当我们已知被分到A箱子的时候，我们拿出一个长螺丝的概率是多少？ ## 乘法原则 The Multiplication Rule Definition Multiplication Rule for Conditional Probability: $$if \quad Pr(B)>0:\quad Pr(A\cap B)=Pr(B)Pr(A\|B)\\ if \quad Pr(A)>0:\quad Pr(A\cap B)=Pr(A)Pr(B\|A)$$ Definition Multiplication Rule for Conditional Probability:Suppose that $A_1,A_2,A_3\dots A_n$ are events such that $Pr(A_1\cap A_2\cap A_3\dots \cap A_{n-1})>0$ then $$Pr(A_1\cap A_2\cap A_3\dots \cap A_n)=\\ Pr(A_1)Pr(A_2\|A_1)Pr(A_3\|A_1\cap A_2)\dots Pr(A_n\|A_1\cap A_2 \cap A_3 \cap \dots \cap A_{n-1})$$ $$Pr(R_1\cap B_2\cap R_3\cap B_4)=Pr(R_1)Pr(B_2\|R_1)Pr(R_3\|R_1\cap B_2)Pr(B_4\|R_1\cap B_2\cap R_3)\\ =\frac{r}{r+b}\frac{b}{r+b-1}\frac{r-1}{r+b-2}\frac{b-1}{r+b-3}$$ Suppose that $A_1,A_2,A_3\dots A_n,B$ are events such that $Pr(B)>0$ and $Pr(A_1\cap A_2\cap A_3 \dots A_{n-1}\|B)>0$ then: $$Pr(A_1\cap A_2\cap \dots A_n\|B)=\\ Pr(A_1\|B)Pr(A_2\|A_1\cap B)Pr(A_3\|A_2\cap A_1\cap B)\dots Pr(A_n\|A_{n-1} \cap \dots \cap A_2\cap A_1\cap B)$$ ## 条件概率与分割，全概率公式 Conditional Probability and Partition – Law of total Probability 1-1的T3中，我们介绍了当一个样本空间被划分成两部分的时候，概率的计算方法，那么如果我们把切分继续下去，也就是一个样本空间我们把它切成k块不相交的子空间时，那么响应的计算会有什么变换呢？ Definition partition Let S denote the sample space of some experiment,and consider k events $B_1 \dots B_k$ in S such that $B_1 \dots B_k$ are disjoint and $\bigcup^k_{i=1}B_i=S$ It is said that these events from a partition of S Theorem Law of total probability:Suppose that the events $B_1 \dots B_k$ from a partition of the space S and $Pr(B_j)>0$ for $j=1,\dots ,k$ Then ,for every event A in S: $$Pr(A)=\sum^k_{j=1}Pr(B_j)Pr(A\|B_j)$$ ①画图： ②集合论： $$A=(B_1\cap A)\cup(B_1\cap A)\cup\dots \cup(B_k\cap A)$$ $$Pr(A)=\sum^k_{j=1}Pr(B_j\cap A)\\ if \quad Pr(B_j)>0 (j=1\dots k)\quad then \quad Pr(B_j\cap A)=Pr(B_j)Pr(A\|B_j)$$ $$Pr(A\|C)=\sum^k_{j=1}Pr(B_j\|C)Pr(A\|B_j\cap C)$$ $$A\cap C=(B_1\cap A \cap C)\cup(B_1\cap A \cap C)\cup\dots \cup(B_k\cap A \cap C)$$ $$Pr(A\| C)=\sum^k_{j=1}Pr(B_j\cap A \| C)=\sum^k_{j=1}\frac{Pr(B_j\cap A \cap C)}{Pr(C)}\\ if \quad Pr(B_j)>0 (j=1\dots k)\quad then \quad Pr(B_j\cap A\|C)=Pr(B_j)Pr(A\|B_j\cap C)$$ ## 扩展试验 Augmented Experiment Definition Augmented Experiment: If desired,any experiment can be augmented to include the potential or hypothetical observation of as much additional information as we would find useful to help us calculate any probabilities that we desire Subscribe	{"extraction_info": {"found_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": 0.9785909056663513, "perplexity": 780.2126623095811}, "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-2020-40/segments/1600400219221.53/warc/CC-MAIN-20200924132241-20200924162241-00198.warc.gz"}
https://www.physicsforums.com/threads/definition-of-force-over-an-area.874438/	# I Definition of force over an area 1. Jun 4, 2016 ### JonnyG I am reading the wikipedia article on the Cauchy stress-tensor. The article says that given some object, let $P$ be a point in the object and let $S$ be a plane passing through that point. Then "an element of area $\Delta S$ containing $P$, with normal vector $n$, the force distribution is equipollent to a contact force $\Delta F$ and surface moment $\Delta M$. In particular, the contact force is given by $\Delta F = T^n \Delta S$". Now, I have always thought of force as a vector field, meaning that in this case, each individual point in $S$ would be assigned a vector which represents the force at that point. It seems that $\Delta F$ is some kind of average force though. It is defined in terms of $T^n$, which is the "mean surface traction". What is the definition of mean surface traction? I have googled it and cannot find a definition. 2. Jun 4, 2016 Are you sure that those $\Delta$s aren't supposed to be $d$s? In other words, given the plane $S$, there is some continuous force field, $F(s)$ acting on that plane, and given a differential element of that plane, $dS$, the differential force, $dF$, and moment, $dM$, are acting on it. 3. Jun 4, 2016 ### JonnyG In the article, $dF = \lim\limits_{\Delta S \rightarrow 0} \frac{\Delta F}{\Delta S}$. So the definition of $dF$ depends on the definition of $\Delta F$. 4. Jun 4, 2016 Ah, well then it would appear to me that $\Delta F$ is the average force over $\Delta S$. The average would just be an average over the area, $$\Delta F = \dfrac{1}{\Delta S}\int\int_{\Delta S} F(s) dS.$$ A traction is essentially the force field per unit area projected onto the surface normal. EDIT: It's been a while since I studied continuum mechanics, but I believe I misspoke here slightly and I feel I should amend my previous statement. It is a force per unit area projected onto a surface normal, but traction generally specifically refers to the projection of the Cauchy stress tensor onto that normal, so it is still a three-component vector with one normal component and two shear components. Last edited: Jun 4, 2016 5. Jun 4, 2016 ### JonnyG @boneh3ad Thank you for the clarification. I have three more questions, if you don't mind answering them: 1) Letting $\sigma$ denote the stress-tensor, what is the best way to think of the action of $\sigma$ on its two arguments? For example, suppose $p$ is a point in the object and $n$ is a unit normal to some plane passing through $p$, then $\sigma(n, \cdot) = T^n$ where $T^n$ is the traction vector. But this seems to imply that $T^n$ is a dual-vector. Am I correct in thinking this? 2) Given two arbitrary vectors $v,w$, how can I physically interpret $\sigma(v,w)$? 3) Given a plane, there are always two normals to that plane. So given a point $P$ and a plane passing through $P$, then we could choose two distinct normals when deriving the stress-tensor. There doesn't seem to be a natural choice of normal vector. Which one do we take? Or is the tensor actually independent of the choice of normal? 6. Jun 6, 2016 bump... 7. Jun 6, 2016 ### Staff: Mentor The way I learned it is that, if $\vec{\sigma}$ is the stress tensor, and $\vec{n}$ is a unit vector normal to an element of surface area dS within the material, the force per unit area exerted by the material on the side of dS toward which $\vec{n}$ is pointing and on the material on the side of dS from which $\vec{n}$ is pointing is given by: $$\vec{T}=\vec{\sigma}\cdot \vec{n}$$where the force per unit area $\vec{T}$ is called the traction vector on the element of surface area. 8. Jun 6, 2016 Sorry, I got busy this weekend and couldn't respond. I will second what @Chestermiller just said. 9. Jun 7, 2016 ### wrobel I will be the third who says the same (but by little bit another words) Assume that a domain $D\subset\mathbb{R}^3$ is filled with a continuous media. Let $x=(x^i)$ be any right curvilinear coordinates in $D$ with coordinate vectors $\boldsymbol e_i=\boldsymbol e_i(x)$ and $g_{ij}=(\boldsymbol e_i,\boldsymbol e_j),\quad g=det(g_{ij})$. Now take any volume $W\subset D$ and any surface $\Sigma$ that is a part of $\partial W,\quad \Sigma\subset\partial W$ . Postulate that the media outside $W$ acts on $\Sigma$ with the force $\boldsymbol F=\int_\Sigma \sqrt g \boldsymbol e_i\otimes(p^{i1}dx^2\wedge dx^3+p^{i2}dx^3\wedge dx^1+p^{i3}dx^1\wedge dx^2)$ or symbolically $d\boldsymbol F=\sqrt g \boldsymbol e_i\otimes(p^{i1}dx^2\wedge dx^3+p^{i2}dx^3\wedge dx^1+p^{i3}dx^1\wedge dx^2).$ This defines the Cauchy stress tensor $p^{ij}$. Note also that there are situations (not classical problems) such that $p^{ij}\ne p^{ji}$ Last edited: Jun 7, 2016 Draft saved Draft deleted Similar Discussions: Definition of force over an area	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9691221117973328, "perplexity": 300.3051062879558}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187822116.0/warc/CC-MAIN-20171017144041-20171017164041-00487.warc.gz"}
https://www.nature.com/articles/487176a?error=cookies_not_supported&code=4895d120-ccbb-4669-bc7d-eb9133cb7259	Mechanochemistry # A tour of force ## Article metrics ### Subjects The effect of force on a chemical reaction has been visited in three different molecular environments. The results reveal a unifying framework that enables predictions of force-induced reactivity. Chemists tend to think about the ways in which molecules behave — the shapes they are most likely to adopt, the molecular partners with which they are most likely to bind, or the rates and outcomes of their reactions — in terms of energies. But in the light of reports that applied forces can direct new chemical transformations1,2,3,4, or induce unusual stress responses in materials5,6, it is becoming increasingly profitable to consider molecular behaviour in terms of forces. Writing in the Journal of the American Chemical Society, Akbulatov et al.7 provide a crucial benchmark for force-induced reactivity (chemomechanics) by uniting internally and externally stress-induced chemical behaviour across time- and length-scales of several orders of magnitude. The authors' study centres on the reactivity of gem-dibromocyclopropane, a chemical group that contains a ring of three carbon atoms (Fig. 1). This ring can be pulled open mechanically to give a product (known as an alkene) that is both longer and more stable than its precursor. The force required for the reaction can be applied through several mechanisms. For example, micrometre-long polymers consisting of thousands of sequentially connected cyclopropanes have been stretched by the miniature 'tweezers' of an atomic force microscope8. In a second scenario, cyclopropanes could be incorporated into larger rings known as strained macrocycles, so that the larger ring pulls on the smaller one in much the same way that a bow pulls its bowstring taut. A third scenario, in which an isolated, small cyclopropane-containing molecule is pulled by a force, can be simulated using computational models. In these three scenarios, the force is transmitted to the cyclopropane across distances ranging from 10−10 metres (for the small molecules) to 10−6 metres (for the polymer). What's more, the force is applied directly in the simulations (to the point of attachment of one of the R groups in Fig. 1), but indirectly through a variety of chemical bonds in the other cases. Despite these differences, it is reasonable to expect that a common basis underpins the chemical behaviour resulting from these otherwise disparate contexts. Finding such commonality by considering energy is difficult, because the amount of energy associated with the strain in each system varies greatly with the size of the system. For example, it takes around 1,000 times more energy to distort the polymer than to distort a macrocycle to a comparable extent, because in the polymer there are about 1,000 times as many degrees of freedom into which the strain can be distributed. In contrast to energy, however, the local force acting on a cyclopropane provides a potentially convenient and useful basis for comparison. Enter Akbulatov et al., who neatly 'close the loop' on chemomechanical coupling — the correlation between the rate of a chemical reaction and the force applied to induce it — by comparing the strain in cyclopropane-containing small molecules, macrocycles and polymers. Using a combination of computational and experimental approaches, the authors show that the stress-induced behaviour of the macrocycles and small molecules can be used to quantitatively model the microscopic behaviour of the polymer. Their result is crucial for the burgeoning field of mechanochemistry because it unites the current thinking about different ways of applying tension to molecules: internal versus external forces, applied directly or indirectly. Proper accounting of local forces in a molecule is often difficult when a restoring force is applied across a sequence of nuclei, as is the case in the strained polymer and macrocycles. Unlike temperature, which is the same for any and all subsets of atoms in a system at equilibrium, the local forces that are 'felt' between different sets of nuclei are not identical to each other, and are not the same as the constraining forces applied elsewhere in the molecule. For example, if a stretching force is applied at the ends of a polymer chain, the resulting local forces within the molecule differ from the applied force. The distribution of restoring forces is quite nuanced, and recent computational work9 has shown that different atomic connections can affect local chemomechanical coupling, independently of the source of the tension. Akbulatov et al. used a clever strategy to assess the forces acting on the nuclei in the cyclopropane groups. Molecular geometries, such as bond lengths and bond angles, are somewhat pliable, and the authors recognized that the distortion of a cyclopropane from its equilibrium geometry reflects the local force acting on its nuclei. By calculating such distortion — specifically, the distances between nuclei — they were able to determine the local forces acting on cyclopropanes in the three different scenarios. This approach for calibrating tension was highly effective as a method for evaluating the effect of the surrounding chemical structure on local force: using a single formalism, the authors accurately modelled the reactivity of cyclopropanes in molecules of lengths ranging from ångströms to micrometres, and at timescales ranging from months to milliseconds. It should be pointed out that Akbulatov and colleagues' approach for assessing chemomechanics relies on extensive computational methods, particularly to assess the local forces in molecules. The calculations required are quite involved, and so might limit the broad use of the authors' strategy. Looking ahead, it will be interesting to see whether similar methods that require more minimal calculations can be developed to reproduce experimental benchmarks of chemomechanics with comparable success. Nevertheless, the ability to predict force-induced reactivity has several implications. First, the molecular view of chemomechanics can now be directly related to problems on even greater length scales than those addressed by Akbulatov et al., such as mechanically active functional groups embedded within proteins10 or a macroscopic material under load5. Second, the authors' approach lends itself to predicting new behaviours, such as how different chemical attachments might enhance the mechanical reactivity of a molecule. Finally, because energy = force × distance, evaluating reactivity as a function of force rather than of energy allows otherwise inaccessible details of molecular structure (the 'distance' in the equation) to be determined, both for complicated reaction mechanisms that are resistant to conventional experimental probes and for 'mechanical-only' mechanisms that are otherwise impossible to work out. It is therefore a particular strength of Akbulatov and co-workers' study that not only are chemomechanical relationships between molecules of different sizes shown to be valid, but that the authors also demonstrate how such relationships might be determined and applied. ## References 1. 1 Hickenboth, C. R. et al. Nature 446, 423–427 (2007). 2. 2 Lenhardt, J. M. et al. J. Am. Chem. Soc. 133, 3222–3225 (2011). 3. 3 Wiggins, K. M. & Bielawski, C. W. Angew. Chem. Int. Edn 51, 1640–1643 (2012). 4. 4 Rosen, B. M. & Percec, V. Nature 446, 381–382 (2007). 5. 5 Davis, D. A. et al. Nature 459, 68–72 (2009). 6. 6 Caruso, M. M. et al. Chem. Rev. 109, 5755–5798 (2009). 7. 7 Akbulatov, S., Tian, Y. & Boulatov, R. J. Am. Chem. Soc. 134, 7620–7623 (2012). 8. 8 Wu, D., Lenhardt, J. M., Black, A. L., Akhremitchev, B. B. & Craig, S. L. J. Am. Chem. Soc. 132, 15936–15938 (2010). 9. 9 Tian, Y. & Boulatov, R. ChemPhysChem 13, 2277–2281 (2012). 10. 10 Liang, J. & Fernández, J. M. ACS Nano 3, 1628–1645 (2009). ## Author information Correspondence to Stephen L. Craig. ## Rights and permissions Reprints and Permissions Craig, S. A tour of force. Nature 487, 176–177 (2012) doi:10.1038/487176a • ### Mg- and Mn-MOFs Boost the Antibiotic Activity of Nalidixic Acid • Vânia André • , André Ramires Ferreira da Silva • , Auguste Fernandes • , Catarina Garcia • , Patrícia Rijo • , Alexandra M. M. Antunes • , João Rocha • & M. Teresa Duarte ACS Applied Bio Materials (2019) • ### Magnesium based materials for hydrogen based energy storage: Past, present and future • V.A. Yartys • , M.V. Lototskyy • , E. Akiba • , R. Albert • , V.E. Antonov • , J.R. Ares • , M. Baricco • , N. Bourgeois • , C.E. Buckley • , J.M. Bellosta von Colbe • , J.-C. Crivello • , F. Cuevas • , R.V. Denys • , M. Dornheim • , M. Felderhoff • , D.M. Grant • , B.C. Hauback • , T.D. Humphries • , I. Jacob • , T.R. Jensen • , P.E. de Jongh • , J.-M. Joubert • , M.A. Kuzovnikov • , M. Latroche • , L. Pasquini • , L. Popilevsky • , V.M. Skripnyuk • , E. Rabkin • , M.V. Sofianos • , A. Stuart • , G. Walker • , Hui Wang • , C.J. Webb • & Min Zhu International Journal of Hydrogen Energy (2019) • ### Modelling tribochemistry in the mixed lubrication regime • Abdullah Azam • , Anne Neville • , Ardian Morina • & Mark C.T. Wilson Tribology International (2019) • ### Unveiling oil-additive/surface hierarchy at real ring-liner contact • Shanhong Wan • , Yana Xia • , Sang T. Pham • , Anh Kiet Tieu • , Hongtao Zhu • & Qinglin Li Surfaces and Interfaces (2019) • ### A Fully Coupled Chemomechanical Formulation With Chemical Reaction Implemented by Finite Element Method • Jianyong Chen • , Hailong Wang • , K. M. Liew • & Shengping Shen Journal of Applied Mechanics (2019)	{"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": 0.8093281388282776, "perplexity": 3628.478103142887}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1575541307813.73/warc/CC-MAIN-20191215094447-20191215122447-00109.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-3-linear-systems-and-matrices-3-7-evaluate-determinants-and-apply-cramer-s-rule-3-7-exercises-skill-practice-page-208/30	## Algebra 2 (1st Edition) $y=6,x=2$ We know that for a matrix $\left[\begin{array}{rr} a & b \\ c &d \\ \end{array} \right]$ the determinant, $D=ad-bc.$ Thus the determinant of the coefficient matrix: $D=2\cdot2-(-1)\cdot1=4+1=5.$ Then applying Cramer's Rule: $y=\frac{\begin{vmatrix} 2 & -2\\ 1 & 14 \\ \end{vmatrix}}{5}=\frac{2\cdot14-(-2)\cdot1}{5}=\frac{30}{5}=6$ $x=\frac{\begin{vmatrix} -2 & -1 \\ 14 & 2 \\ \end{vmatrix}}{5}=\frac{-2\cdot2-(-1)\cdot14}{5}=\frac{10}{5}=2$ Thus $y=6,x=2$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9997590184211731, "perplexity": 478.4984371935183}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1575540501887.27/warc/CC-MAIN-20191207183439-20191207211439-00245.warc.gz"}
https://web2.0calc.com/questions/help-please-easy-problem	+0 0 135 3 Roberto has four pairs of trousers, seven shirts, and three jackets. How many different outfits can he put together if an outfit consists of a pair of trousers, a shirt, and a jacket? Jan 13, 2019 #1 +7499 +1 He has 4 ways to choose a pair of trousers, 7 ways to choose a shirt, and 3 ways to choose a jacket. Therefore, answer is $$4\cdot7\cdot3 = 84$$. Jan 13, 2019 #2 +647 -1 We could count it out, but that would take a long time. We could make a pairing tree, but there is an even easier way! If we just multiply the numbers, we get the total number of outifts! $$4\times 7\times 3=84$$. You are very welcome! :P Jan 13, 2019 #3 +533 +1 he has four choices for what to wear as trousers, seven for shirts, and three for jackets. therefore the answer is 473 or 84. HOPE THIS HELPED! Jan 13, 2019	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8672025203704834, "perplexity": 1334.8224123930731}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256778.29/warc/CC-MAIN-20190522083227-20190522105227-00452.warc.gz"}
https://brilliant.org/problems/integral-part-of-sum/	# Integral part of Sum Calculus Level 4 Given the sum $$\displaystyle S = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{10,000}},$$ what is $$\lfloor S \rfloor$$? Details and assumptions The function $$\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}$$ refers to the greatest integer smaller than or equal to $$x$$. For example $$\lfloor 2.3 \rfloor = 2$$ and $$\lfloor -5 \rfloor = -5$$. ×	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8995221257209778, "perplexity": 564.3175631788176}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1544376827281.64/warc/CC-MAIN-20181216051636-20181216073636-00248.warc.gz"}
https://calculator.academy/rem-calculator/	Enter the default font size and the desired font size into the calculator to determine the correct REM to use for your font size. REM Formula The following formula is used to calculate the REM needed to achieve a certain font size. REM = EF / DF • Where REM is the root em value for the desired font size • EF is the expected font size output you want to display (px) • DF is the default font size (px) For most browsers, the default font size is 16px. REM Definition What is REM? R.E.M, short for root em, is a term used in website CSS programming that is used to depict the size of the font to be used to a given line or section of text. For example, if a line is coded as 1REM, the expected font size would be 16px or the default size of the browser. Example Problem How to calculate REM? 1. First, determine the default font size. For this example, the browser uses the most common default size of 16px. 2. Next, determine the font size you want to be displayed. For this example, the web developer wants a font size of 20px to display. 3. Finally, calculate the REM. Using the formula above, the root em is calculated to be: REM = EF / DF * DF REM = 20px/ 16px REM = 1.25rem	{"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": 0.9798710346221924, "perplexity": 2107.8645696879034}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057225.38/warc/CC-MAIN-20210921131252-20210921161252-00427.warc.gz"}
https://www.arxiv-vanity.com/papers/0810.4856/	# Abrupt Emergence of Pressure-Induced Superconductivity of 34 K in SrFe2As2: A Resistivity Study under Pressure Hisashi Kotegawa E-mail address: Hitoshi Sugawara and Hideki Tou Department of PhysicsDepartment of Physics Kobe University Kobe University Kobe 658-8530 JST Kobe 658-8530 JST Transformative Research-Project on Iron Pnictides (TRIP) Transformative Research-Project on Iron Pnictides (TRIP) Chiyoda Chiyoda Tokyo 102-0075 Faculty of Integrated Arts and Science Tokyo 102-0075 Faculty of Integrated Arts and Science Tokushima University Tokushima University Tokushima 770-8502 Tokushima 770-8502 ###### Abstract We report resistivity measurement under pressure in single crystals of SrFeAs, which is one of the parent materials of Fe-based superconductors. The structural and antiferromagnetic (AFM) transition of K at ambient pressure is suppressed under pressure, and the ordered phase disappears above GPa. Superconductivity with a sharp transition appears accompanied by the suppression of the AFM state. exhibits a maximum of 34.1 K, which is realized close to the phase boundary at . This is the highest among those of the stoichiometric Fe-based superconductors. SrFeAs, superconductivity, pressure, single crystal After the discovery of superconductivity at 26 K in F-doped system LaFeAsOF (ZrCuSiAs-type structure),[1] various Fe-based materials have been reported to show superconductivity.[2, 3, 4, 5] Among them, FeAs ( Ca, Sr, and Ba) systems with a ThCrSi-type structure show superconductivity by doping with K or Cs into the site,[2, 3] or by doping Co into the Fe site.[6, 7] Doping is an effective method of inducing superconductivity in Fe-based superconductors. However, this simultaneously induces the inhomogeneity of the crystal structure and electronic state. The inhomogeneity sometimes makes it difficult to observe the intrinsic properties of the material. Instead of doping, the application of pressure for undoped compound is also an effective method of inducing superconductivity. Pressure-induced superconductivity in FeAs ( Ca, Sr, and Ba) has been reported.[8, 9, 10] The superconductivity of these stoichiometric compounds is important for the study of Fe-based superconductors. Concerning CaFeAs, its superconductivity has been recognized to be intrinsic, because some groups have reported that the zero-resistance state is observed in a similar pressure range.[8, 9, 11] In the cases of BaFeAs and SrFeAs, Alireza et al. have reported that Meissner effects appear between GPa for BaFeAs and between GPa for SrFeAs using magnetization measurements under pressure.[10] However, Fukazawa et al. have observed no zero-resistance state at pressures of up to 13 GPa in BaFeAs.[12] On the other hand, Kumar et al. performed resistivity measurement at pressures of up to 3 GPa in SrFeAs and reported that the onset of superconductivity appears above 2.5 GPa, but they observed no zero-resistance state up to 3 GPa.[13] Quite recently, Igawa et al. have reported that the zero-resistance state was realized below 10 K at a high pressure of 8 GPa in SrFeAs, but that the transition was broad.[14] No consensus on pressure-induced superconductivity in BaFeAs and SrFeAs has been arrived at yet. In this paper, we report the results of resistivity measurements in single-crystalline samples of SrFeAs up to 4.3 GPa. This is the first resistivity measurement above 3 GPa using single-crystalline samples. In our measurements, the zero-resistance state below K with a sharp transition was observed above 3.5 GPa. Single-crystalline samples were prepared by the Sn-flux method as reported in ref. 15. Electrical resistivity () measurement at high pressures was carried out using an indenter cell.[16] was measured by a four-probe method while introducing a flow of current along the plane. Daphne oil 7373 was used as a pressure-transmitting medium. Applied pressure was estimated from the of the lead manometer. Resistivity measurement under pressure was performed for two settings using different samples and almost the same results were obtained between two samples. Figures 1(a) and 1(b) show the temperature dependences of at several pressures of up to 4.3 GPa. A clear anomaly was observed at 198 K at ambient pressure, which is similar to that of Yan et al.’s sample.[17] This temperature, denoted as , corresponds to the structural transition temperature and the simultaneous magnetic transition temperature.[18, 17, 19] The magnetic structure of SrFeAs has been reported to be a collinear antiferromagnetic (AFM) one.[19] The of our sample is lower than that of Kumar et al.’s sample.[13] shows a small jump at in our sample and the jump becomes remarkable under pressure, in contrast to other measurements under pressure.[13, 14] The reason why the jump appears in our sample is unclear at present, but this behavior is understood to be induced by the reconstruction of the Fermi surface owing to the AFM transition, and resembles that of CaFeAs.[8, 9] Thus, we define the temperature at the jump as on the analogy of CaFeAs, as shown in Fig. 1(b). As shown in the figure, decreases with increasing pressure and reaches K at 3.57 GPa. No signature of the transition at was observed above 3.77 GPa, indicating the disappearance of the AFM state. The critical pressure between the AFM state and the paramagnetic (PM) state is estimated to be GPa. The inset of Fig. 1(a) displays of around at 3.22 GPa. A small hysteresis was observed between cooling and warming, indicative of the first-order phase transition. The onset of superconductivity appears above GPa but the transition is quite broad, similarly to that observed in the experiments by Kumar et al..[13] A zero-resistance state is observed above 3.47 GPa, and the transition becomes sharper above 3.77 GPa where the AFM state is no longer realized. In this paper, is defined by the temperature of the zero resistance. The maximum was 34.1 K at 3.77 GPa, as shown in the inset of Fig. 1(b). This is close to K of the doped systems (BaK)FeAs and (KSr)FeAs.[2, 3] Above 3.77 GPa, is almost constant but slightly decreases with increasing pressure. Figure 2 shows under magnetic field at 4.15 GPa, when the magnetic field was applied along the -plane. decreases from 30 K at 0 T to K at 8 T. The initial slope was estimated to be K/T, giving T by linear extrapolation. These values are comparable to those of other Fe-based compounds. Figure 3 shows the pressure-temperature phase diagram of SrFeAs. The initial slope of was estimated to be K/GPa, which is the same as that of Kumar et al..[13] The ordered phase was markedly suppressed above 3 GPa, and no signature of the AFM state was observed at 3.77 GPa. The ordered state up to 3 GPa is confirmed to have an orthorhombic crystal structure.[13] The superconductivity appears from slightly below GPa, and exhibits the highest K in the PM state close to . In CaFeAs, another structural phase transition from the tetragonal phase to the ”collapsed” tetragonal one has been reported under high pressure,[20] which can be detected by .[11] In contrast, there is no corresponding distinct anomaly above in SrFeAs. In CaFeAs, the pressure dependence of the residual resistivity indicates the anomalous behavior of a dome shape.[11] We plot the pressure-dependence of at 35 K for SrFeAs in the upper panel of the figure, but shows a gradual decrease under pressure, and no anomalous behavior was observed for SrFeAs. Figure 4 is the pressure-temperature phase diagram around the phase boundary. We plotted the onset temperature of superconductivity, , and the transition width, . The zero-resistance state is observed even in the narrow pressure range below . In the Fe-based superconductors, it is a controversial issue whether superconductivity can coexist with the AFM state.[21, 22, 23] Since the resistivity is macroscopic measurement and is sensitive to superconductivity, it is generally difficult to discuss this issue. However, note that is unusually wide below . In contrast, becomes markedly sharper above . The minimum is 0.75 K at 3.83 GPa. This indicates that the PM state favors superconductivity and that the AFM state prevents the occurrence of superconductivity in SrFeAs. The superconductivity with a wide below implies non bulk superconductivity. The transition from the tetragonal structure to the orthorhombic one is of the first order.[18] As seen in the inset of Fig. 1(a), the transition at is of the first order even close to . At high pressures and low temperatures, the pressure distribution is inevitable. If the transition at is of the first order, the pressure distribution is expected to induce phase separation. We speculate that the observed superconductivity below originates from the phase-separated PM phase. This is supported by the fact that is almost independent of pressure below . However, if the phase separation is realized at around , we expect the enhancement of at low temperatures at around owing to scattering at the domain boundary. As shown in Fig. 3, there is no anomalous behavior in at around within experimental error. The phase separation and coexistence of superconductivity and magnetism are still an open question, and confirmation by microscopic measurements is required. To our knowledge, the pressure-temperature phase diagram of SrFeAs has been reported by three groups.[10, 13, 14] Our phase diagram is almost consistent with that of Kumar et al., although their resistivity measurements have been performed only up to 3 GPa.[13] On the other hand, the phase diagrams by Alireza et al. and Igawa et al. are different from ours. Alireza et al. have used a single-crystalline sample and Daphne oil 7373 as a pressure transmitting medium, which are the same as those used in our measurements. In their phase diagram, the superconductivity of K appears abruptly at 2.8 GPa and disappears above 3.6 GPa. The pressure region of superconductivity is quite different. On the other hand, Igawa et al. have used a polycrystalline sample, and Fluorinert (FC-77:FC-70 = 1:1) and NaCl as a pressure transmitting medium. The onset of superconductivity was observed in a wide pressure range, and zero resistance below 10 K was realized at a high pressure of 8 GPa. The at around GPa is almost the same as that in our measurements, but the zero-resistance state is different. In their phase diagram, the AFM state is drawn to survive up to 8 GPa. The differences between samples and/or pressure-transmitting mediums are considered to induce the inconsistency between the obtained phase diagrams. To summarize, we have investigated the resistivity under pressure in a single-crystalline SrFeAs up to 4.3 GPa. According to our resistivity measurement, the magnetically ordered phase most likely disappears abruptly above GPa, and superconductivity appears above approximately ; however, other experimental methods are required to confirm whether this phase diagram reflects bulk properties. The maximum was 34.1 K for the pressure-induced superconductivity in stoichiometric SrFeAs, which is close to K of the doped systems.[2, 3] The maximum is realized in the PM state close to . This gives us two different scenarios. One is that the instability of the AFM state plays an important role in superconductivity. Another is that the AFM state obstructs the optimized situation for higher . Systematic investigations are needed to elucidate the relation between superconductivity and magnetism, but the stoichiometric system SrFeAs is a good candidate for treating this issue. We thank Y. Hara and T. Kawazoe for experimental assistance. This work has been partly supported by Grant-in-Aids for Scientific Research (Nos. 19105006, 19204036, 19014016, and 20045010) from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan.	{"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": 0.9212255477905273, "perplexity": 1778.3274296686782}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610704803737.78/warc/CC-MAIN-20210126202017-20210126232017-00314.warc.gz"}
https://www.bas.ac.uk/data/our-data/publication/coastal-polynyas-in-the-southern-weddell-sea-variability-of-the/	# Coastal polynyas in the southern Weddell Sea: variability of the surface energy budget The surface energy budget of coastal polynyas in the southern Weddell Sea has been evaluated for the period 1992–1998 using a combination of satellite observations, meteorological data, and simple physical models. The study focuses on polynyas that habitually form off the Ronne Ice Shelf. The coastal polynya areal data are derived from an advanced multichannel polynya detection algorithm applied to passive microwave brightness temperatures. The surface sensible and latent heat fluxes are calculated via a fetch-dependent model of the convective-thermal internal boundary layer. The radiative fluxes are calculated using well-established empirical formulae and an innovative cloud model. Standard meteorological variables that are required for the flux calculations are taken from automatic weather stations and from the National Centers for Environmental Production/National Center for Atmospheric Research reanalyses. The 7 year surface energy budget shows an overall oceanic warming due to the presence of coastal polynyas. For most of the period the summertime oceanic warming, due to the absorption of shortwave radiation, is approximately in balance with the wintertime oceanic cooling. However, the anomalously large summertime polynya of 1997–1998 allowed a large oceanic warming of the region. Wintertime freezing seasons are characterized by episodes of high heat fluxes interspersed with more quiescent periods and controlled by coastal polynya dynamics. The high heat fluxes are primarily due to the sensible heat flux component, with smaller complementary latent and radiative flux components. The average freezing season area-integrated energy exchange is 3.48 × 1019 J, with contributions of 63, 22, and 15% from the sensible, latent, and radiative components, respectively. The average melting season area-integrated energy exchange is −5.31 × 1019 J, almost entirely due to the radiative component. There is considerable interannual variability in the surface energy budget. The standard deviation of the energy exchange during the freezing (melting) season is 28% (95%) of the mean. During the freezing season, positive surface heat fluxes are equated with ice production rates. The average annual coastal polynya ice production is 1.11 × 1011 m3 (or 24 m per unit area), with a range from 0.71 × 1011 (in 1994) to 1.55 × 1011 m3 (in 1995). This can be compared to the estimated total ice production for the entire Weddell Sea: on average the coastal polynya ice production makes up 6.08% of the total, with a range from 3.65 (in 1994) to 9.11% (in 1995). ### Details Publication status: Published Author(s): Authors: Renfrew, Ian A., King, John C., Markus, Thorsten On this site: John King Date: 1 January, 2002 Journal/Source: Journal of Geophysical Research / 107 Page(s): 22pp	{"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": 0.8481960892677307, "perplexity": 3728.2713995241397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587655.10/warc/CC-MAIN-20211025061300-20211025091300-00174.warc.gz"}
http://mathhelpforum.com/algebra/33030-logarithm-inverse.html	1. ## Logarithm as inverse Write the inverse of each function. Please show work and short explanation so I understand how to get the answers for the rest of my homework 1) $Y=2^{x/3}$ 2) $Y=log_{5} X^2$ Also If I Need to solve for Y in the following what is the answer for the following (please show work): $f(x)=log_4(x+4)-3$ $X= -2$ 2. Originally Posted by north1224 Also If I Need to solve for Y in the following what is the answer for the following (please show work): $f(x)=log_4(x+4)-3$ $X= -2$ well lets say you have $log_{base} number$. This is equivalent to $log_{x} number / log_{x} base$ with x being whatever you want.....10 for calculating purposes so $f(x)=log_4(x+4)-3$ $x =2$ $f(x)=log_4(2+4)-3$ $f(x)=log_4(6) -3$ $f(x)=(log_{10}(6) / log_{10}(4) ) -3$ $f(x)=~1.2925 - 3 = -1.7075$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 14, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9432833194732666, "perplexity": 334.4899379395476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128323842.29/warc/CC-MAIN-20170629015021-20170629035021-00157.warc.gz"}
http://techtagg.com/standard-error/how-to-calculate-estimated-standard-error-for-the-sample-mean-difference.html	Home > Standard Error > How To Calculate Estimated Standard Error For The Sample Mean Difference # How To Calculate Estimated Standard Error For The Sample Mean Difference ## Contents The mean and SD for the second sample were also 5.3 years and 1.5 years, respectively. American Statistical Association. 25 (4): 30–32. They may be used to calculate confidence intervals. UrdanList Price: $42.95Buy Used:$9.69Buy New: $38.74HP 39G+ Graphing CalculatorList Price:$99.99Buy Used: \$50.00Approved for AP Statistics and Calculus About Us Contact Us Privacy Terms of Use Resources Advertising The http://techtagg.com/standard-error/calculate-the-estimated-standard-error-of-the-mean-of-the-difference-scores.html Because the 9,732 runners are the entire population, 33.88 years is the population mean, μ {\displaystyle \mu } , and 9.27 years is the population standard deviation, σ. Je kunt deze voorkeur hieronder wijzigen. Retrieved Oct 01, 2016 from Explorable.com: https://explorable.com/standard-error-of-the-mean . Popular Pages Measurement of Uncertainty - Standard Deviation Calculate Standard Deviation - Formula and Calculation What is a Quartile in Statistics? https://explorable.com/standard-error-of-the-mean ## How To Calculate Estimated Standard Error For The Sample Mean Difference DrKKHewitt 15.693 weergaven 4:31 Statistics 101: Standard Error of the Mean - Duur: 32:03. Volgende Calculating the Standard Error of the Mean in Excel - Duur: 9:33. The graphs below show the sampling distribution of the mean for samples of size 4, 9, and 25. 1. Sluiten Ja, nieuwe versie behouden Ongedaan maken Sluiten Deze video is niet beschikbaar. 2. On the average, a random variable misses the mean by one standard deviation. 3. Keith Bower 21.226 weergaven 2:56 Standard error of the mean \| Inferential statistics \| Probability and Statistics \| Khan Academy - Duur: 15:15. 5. The table below shows formulas for computing the standard deviation of statistics from simple random samples. No problem, save it as a course and come back to it later. A practical result: Decreasing the uncertainty in a mean value estimate by a factor of two requires acquiring four times as many observations in the sample. Assume that the weights of 10-year-old children are normally distributed with a mean of 90 and a standard deviation of 36. Standard Error Of Sample Mean Formula Home ResearchResearch Methods Experiments Design Statistics Reasoning Philosophy Ethics History AcademicAcademic Psychology Biology Physics Medicine Anthropology Write PaperWrite Paper Writing Outline Research Question Parts of a Paper Formatting Academic Journals Tips Suppose 25 graduating students were randomly selected and asked about their length of stay. Standard Error Of Sample Mean Example Recall that 47 subjects named the color of ink that words were written in. View Mobile Version Math Calculators All Math Categories Statistics Calculators Number Conversions Matrix Calculators Algebra Calculators Geometry Calculators Area & Volume Calculators Time & Date Calculators Multiplication Table Unit Conversions Electronics pop over to these guys The distribution of the mean age in all possible samples is called the sampling distribution of the mean. Thank you to... Standard Error Of Sample Mean Equation The mean age was 33.88 years. Standard Error of the Mean. By taking the mean of these values, we can get the average speed of sound in this medium.However, there are so many external factors that can influence the speed of sound, ## Standard Error Of Sample Mean Example Assume that the following five numbers are sampled from a normal distribution: 2, 3, 5, 6, and 9 and that the standard deviation is not known. http://www.miniwebtool.com/standard-error-calculator/ A natural way to describe the variation of these sample means around the true population mean is the standard deviation of the distribution of the sample means. How To Calculate Estimated Standard Error For The Sample Mean Difference However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean describes bounds on a random sampling process. Standard Error Of Sample Mean Distribution The 95% confidence interval for the average effect of the drug is that it lowers cholesterol by 18 to 22 units. The standard deviation of the age was 3.56 years. It will be shown that the standard deviation of all possible sample means of size n=16 is equal to the population standard deviation, σ, divided by the square root of the Consider a sample of n=16 runners selected at random from the 9,732. As you can see from Table 1, the value for the 95% interval for df = N - 1 = 4 is 2.776. Standard Error Of Sample Mean Excel The mean of these 20,000 samples from the age at first marriage population is 23.44, and the standard deviation of the 20,000 sample means is 1.18. Retrieved 17 July 2014. Laden... http://techtagg.com/standard-error/calculate-estimated-standard-error-of-the-mean.html Similarly, the sample standard deviation will very rarely be equal to the population standard deviation. When the sample size is large, say 100 or above, the t distribution is very similar to the standard normal distribution. Standard Deviation Sample Mean The sample standard deviation s = 10.23 is greater than the true population standard deviation σ = 9.27 years. Consider the following scenarios. ## This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Compare the true standard error of the mean to the standard error estimated using this sample. The correct response is to say "red" and ignore the fact that the word is "blue." In a second condition, subjects named the ink color of colored rectangles. Notation The following notation is helpful, when we talk about the standard deviation and the standard error. Standard Error Of Sample Proportion Statistic Standard Deviation Sample mean, x σx = σ / sqrt( n ) Sample proportion, p σp = sqrt [ P(1 - P) / n ] Difference between means, x1 - The mean age was 23.44 years. It is rare that the true population standard deviation is known. Figure 1. Statistical Notes. WeergavewachtrijWachtrijWeergavewachtrijWachtrij Alles verwijderenOntkoppelen Laden... This lesson shows how to compute the standard error, based on sample data. The values of t to be used in a confidence interval can be looked up in a table of the t distribution. how2stats 32.544 weergaven 5:05 6 1 3 Sampling Error and Sample Size - Duur: 4:42. Het beschrijft hoe wij gegevens gebruiken en welke opties je hebt. Beoordelingen zijn beschikbaar wanneer de video is verhuurd. Solution The correct answer is (A). Calculate an estimate for the WMU average stay and provide a standard error for your estimate. Sampling from a distribution with a large standard deviation The first data set consists of the ages of 9,732 women who completed the 2012 Cherry Blossom run, a 10-mile race held Notice that the population standard deviation of 4.72 years for age at first marriage is about half the standard deviation of 9.27 years for the runners. Standard error of the mean Further information: Variance §Sum of uncorrelated variables (Bienaymé formula) The standard error of the mean (SEM) is the standard deviation of the sample-mean's estimate of a Laden... Log in om dit toe te voegen aan de afspeellijst 'Later bekijken' Toevoegen aan Afspeellijsten laden... The sample proportion of 52% is an estimate of the true proportion who will vote for candidate A in the actual election. doi:10.2307/2340569. By how much?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9558698534965515, "perplexity": 1010.019741465072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187822480.15/warc/CC-MAIN-20171017181947-20171017201947-00852.warc.gz"}
http://mathhelpforum.com/differential-geometry/81560-poisson-summation-formula-proof-print.html	# Poisson Summation Formula Proof He defines $c_n=\frac{1}{2L}\int_{-L}^{L}f(t)e^{-2\pi int/L}dt$. He then takes the Fourier series for L=1 on $F(x)=\sum_{n=-\infty}^{\infty}f(x+n)$ and gets $F(x)=\sum_{n=-\infty}^{\infty}e^{2\pi i nx}\int_{0}^{1}F(t)e^{-2\pi i nt}dt$ which seemingly identifies $c_n=\int_{0}^{1}F(t)e^{-2\pi i nt}dt$. This means to me that $F(t)e^{-2\pi int}$ is an even function of t, but this is not mentioned; am I missing something here? My second question is when he seemingly substitutes $t+k\rightarrow t$ in the integral $\int_0^1 f(t+k)e^{-2\pi int}dt=\int_{k}^{k+1}f(t)e^{-2\pi int}dt.$ However shouldn't the exponential in the integrand be $e^{-2\pi in(t-k)}$?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9872495532035828, "perplexity": 344.78526044497454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1409535921869.7/warc/CC-MAIN-20140901014521-00433-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/adjoint-transformation-of-gluon.899607/	# I Adjoint transformation of gluon 1. Jan 8, 2017 ### CAF123 It is commonly written in the literature that due to it transforming in the adjoint representation of the gauge group, a gauge field is lie algebra valued and may be decomposed as $A_{\mu} = A_{\mu}^a T^a$. For SU(3) the adjoint representation is 8 dimensional so objects transforming under the adjoint representation are 8x1 real Cartesian vectors and 3x3 traceless hermitean matrices via the lie group adjoint map. The latter motivates writing $A_{\mu}$ in terms of generators, $A_{\mu} = A_{\mu}^a T^a$. My first question is, this equation is said to be valid independent of the representation of $T^a$ - but how can this be true? In some representation other than the fundamental representation, the $T^a$ will not be 3x3 hermitean traceless matrices and thus will not contain 8 real parameters needed for transformation under the adjoint rep. But we know the gluon field transforms under the adjoint representation so does this line of reasoning not constrain the $T^a$ to be the Gell mann matrices? Consider the following small computation: $$A_{\mu}^a \rightarrow A_{\mu}^a D_b^{\,\,a} \Rightarrow A_{\mu}^a t^a \rightarrow A_{\mu}^b D_b^{\,\,a}t^a$$ Now, since $Ut_bU^{-1} = D_b^{\,\,a}t^a$ we have $A_{\mu}^a t^a \rightarrow A_{\mu}^b (U t^b U^{-1}) = U A_{\mu}^b t^b U^{-1}$. The transformation law for the $A_{\mu}^a$ is in fact $A_{\mu} \rightarrow UA_{\mu}U^{-1} - i/g (\partial_{\mu} U) U^{-1}$. 1) What is the error that amounts to these two formulae not being reconciled? 2) The latter equation doesn't seem to express the fact that the gluon field transforms in the adjoint representation. I was thinking under SU(3) colour, since this is a global transformation, U will be independent of spacetime so the derivative term goes to zero but is there a more general argument? Thanks!	{"extraction_info": {"found_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": 0.9903743863105774, "perplexity": 204.80407910965232}, "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-2018-26/segments/1529267864364.38/warc/CC-MAIN-20180622065204-20180622085204-00387.warc.gz"}
https://www.physicsforums.com/threads/urn-problem-indisting-objects-into-distinguishable-urns.888277/	# Homework Help: Urn problem (indisting. objects into distinguishable urns) Tags: 1. Oct 7, 2016 ### fignewtons 1. The problem statement, all variables and given/known data I have n balls and m urns numbered 1 to m. Each ball is placed randomly and independently into one of the urns. Let Xi be the number of balls in urn number i. So X1+....+Xm = n What is the distribution of each Xi? What is EXi and VarXi What is E[XiXj] given i≠j What is Cov(X1,Xj? 2. Relevant equations Cov(XY)=Exy(XY)-Ex(X)Ey(Y) 3. The attempt at a solution I read: https://www.artofproblemsolving.com/wiki/index.php?title=Distinguishability and identified this as the last case. I understand that there (n+m-1)ℂ(m-1) ways to place the balls but not how to describe this in the form of a pdf so that I can find expectation and variance and such. 2. Oct 7, 2016 ### Ray Vickson Clearly, for a single urn, the distribution of the number $X_i$ in urn i is the same for any i = 1,2, ...,n, so we might as well look at urn 1. For urns i and j ≠ i the bivariate distribution of $(X_i,X_j)$ is the same for any pair i and j, so we might as well look at urns 1 and 2. To find the marginal distribution of $X_1$, just look at it as a problem having two urns: 1 and not-1. For each object, the probability it goes into urn 1 is 1/m, while the probability it goes into urn not-1 is (m-1)/m. For urns 1 and 2 look at it as a three-urn problem with urns 1, 2 and not-12. For each object, p(1) = p(2) = 1/m and p(not-12) = (m-2)/m.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9407593011856079, "perplexity": 783.2980570124306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590559.95/warc/CC-MAIN-20180719051224-20180719071224-00123.warc.gz"}
http://mathhelpforum.com/advanced-algebra/61802-computing-e-print.html	# Computing e^A • Nov 26th 2008, 03:37 PM victor1487 Computing e^A I need to compute e^A for the matrix A= 0 ∏ ....-∏ 0 where the diagonal are zeros and the other diagonal has pi on top and negative pi on the bottom. I'm not quite sure where to start. Thanks • Nov 26th 2008, 03:54 PM vincisonfire $e^A$ is defined as $*\sum_{i=0}^{\infty} \frac{A^n}{n!}$ You can find that if $B=e^A$ then $B(1,1) = B(2,2) =1 -\frac{\pi^2}{2!}+ \frac{\pi^4}{4!} - \frac{\pi^6}{6!} + \frac{\pi^8}{8!} - ... =-1$ $B(1,2) = \frac{\pi}{1!}- \frac{\pi^3}{3!} + \frac{\pi^5}{5!} - \frac{\pi^7}{7!} + ... = 0$ $B(2,1) = -\frac{\pi}{1!}+ \frac{\pi^3}{3!} - \frac{\pi^5}{5!} + \frac{\pi^7}{7!} - ... = 0$ If your prof wants you to evaluate those series then I don't know (I used Maple). Well I don't have the time... or both. • Nov 26th 2008, 05:00 PM chiph588@ $B(1,1) = B(2,2)$ should be $-1$ • Nov 26th 2008, 05:06 PM vincisonfire I don't think so because it is $\sum_{i=1}^{\infty} (-1)^{n}\cdot\frac{\pi^{2n}}{(2n)!}$ because the first term is 0 not 1 we must integrate from 1 to infinity not 0 to infinity. But I may be mistaking. Here are the 10 first terms -4.934802202 -0.876090073 -2.211352843 -1.976022212 -2.001829103 -1.999899529 -2.000004167 -1.999999864 -2.000000003 -1.999999999 • Nov 26th 2008, 09:06 PM chiph588@ well I typed e^A in my ti-89 and it gave me back $-I$ where I is the identity matrix • Nov 27th 2008, 03:26 AM vincisonfire Yes you're right because we have to add the identity matrix at the beginning. SOrry. $B=e^A$ then $B(1,1) = B(2,2) =1 -\frac{\pi^2}{2!}+ \frac{\pi^4}{4!} - \frac{\pi^6}{6!} + \frac{\pi^8}{8!} - ... =-1$ $B(1,2) = \frac{\pi}{1!}- \frac{\pi^3}{3!} + \frac{\pi^5}{5!} - \frac{\pi^7}{7!} + ... = 0$ $B(2,1) = -\frac{\pi}{1!}+ \frac{\pi^3}{3!} - \frac{\pi^5}{5!} + \frac{\pi^7}{7!} - ... = 0$ Somebody knows how to calculate these sum by hand? • Nov 27th 2008, 06:49 AM awkward Quote: Originally Posted by victor1487 I need to compute e^A for the matrix A= 0 ∏ ....-∏ 0 where the diagonal are zeros and the other diagonal has pi on top and negative pi on the bottom. I'm not quite sure where to start. Thanks You might consider starting by diagonalizing A-- Notice that $A = P D P^{-1}$ where $P = \begin{pmatrix}1 &1 \\ i &-i \end{pmatrix}$ $D =\begin{pmatrix}\pi i &0 \\ 0 &-\pi i \end{pmatrix}$ $P^{-1} = \frac{1}{2} \begin{pmatrix}1 &-i \\ 1 &i \end{pmatrix}$ (I'm assuming you know how to go about diagonalizing a matrix; if not, there is an article on Wikipedia: Diagonalizable matrix - Wikipedia, the free encyclopedia.)	{"extraction_info": {"found_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": 0.9730225801467896, "perplexity": 1665.3203651070305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607813.12/warc/CC-MAIN-20170524112717-20170524132717-00063.warc.gz"}
http://udpaperzwjv.agorisme.info/an-analysis-of-the-definition-of-calculus.html	# An analysis of the definition of calculus Then the explanation of precalculus will be connected to the calculus: precalculus mathematics is mathematics problems in the analysis of. Mathematical analysis - wikipedia, the free encyclopedia by pushpen5115 in types articles & news stories. Learn differential calculus for free—limits, continuity, derivatives, and derivative applications full curriculum of exercises and videos. And one that includes calculus, analysis is hardly in need of justi cation but just in case, we remark that its uses include: 1. Foundations of tensor analysis for students of first steps toward a tensor calculus: definition of tensor quantities as quantities that. Die analysis [aˈnalyzɪs] the calculus page calculusorg, bei university of california, davis – ressourcen und enthält links zu anderen websites. Economic analysis is a primary tool used to evaluate a nation calculus is the most common type of math found in [industry analysis] \| definition of industry. The lecture notes section includes the lecture notes files analysis ii definition of manifold : 31. Looking for books on calculus check our section of free e-books and guides on calculus now this page contains list of freely available e-books, online textbooks and. Define analysis: disguising the tables of an analysis of the definition of calculus tobias, his rights are far behind. Calculus integrate - free download as pdf file definition for a given function solutions to chapter 19 problems 2014 structural and stress analysis third. Hmc mathematics calculus online tutorials tutorials tutorials multivariable calculus elementary vector analysis lines, planes, limit definition of the. Definition of the definite integral you’re familiar with functions and function notation both will appear in almost every section in a calculus class and so. Elementary vector analysis $\newcommand{\vecb}[1]{{\bf #1}} \newcommand{\ihat}{\hat using the second definition of the dot product with$\left\\| \vecb{u}. Introductory calculus: marginal analysis we will use the derivative of profit, cost, and revenue functions to make estimates. Analysis definition is - a detailed examination of anything complex in order to understand its nature or to determine its essential features : calculus 1b 5:. What is calculus like we start with an abstract definition of a function (as a set of argument-value pairs) and then describe the standard functions. Calc - definition of calc by a method of analysis or calculation using both differential calculus and integral calculus are concerned with the effect on a. Limits (formal definition) that is the formal definition it actually looks pretty scary, introduction to limits calculus index. Introduction to stochastic processes - lecture notes (with 33 illustrations) gordan žitković department of mathematics the university of texas at austin. Calculus i (notes) / limits let’s start this section out with the definition of a limit at a finite point example 1 use the definition of the limit to prove. Essential in the definition of these (regarding the prehistory of mathematical analysis see infinitesimal calculus) mathematical analysis as a unified and. An analysis of the definition of calculus Rated 4/5 based on 23 review 2018.	{"extraction_info": {"found_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": 0.869564414024353, "perplexity": 1618.5944805324782}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267155561.35/warc/CC-MAIN-20180918150229-20180918170229-00457.warc.gz"}
http://gearylug.blogspot.com/2009/10/seminar-on-introduction-to-latex.html	## Tuesday, October 27, 2009 ### Seminar on Introduction to LaTeX Here is a link to a talk I gave last Friday on how to get started using LaTeX. It covers why one should use LaTeX to prepare papers and presentations, a step-by-step guide to installing the software, and some pointers on how to get it working. Here is the content of the TeX file that generates the Pdf version of the presentation linked above. To operate this as a tex file, you will need to open a LaTeX editor (e.g. TeXnic Center), open a new TeX file, and paste in the contents from the link. You cannot download the content of the TeX file as a file with a .tex extension. However, there are many files with .tex extensions that can be downloaded from the web. For example (as I mentioned on Friday), the Pdf of the presentation that I gave is based on the following Beamer template; the fourth .tex file down: http://www.ctan.org/tex-archive/macros/latex/contrib/beamer/solutions/conference-talks/ Finally, here’s the .tex file template for the academic paper that was shown in the presentation on Friday: http://www.cs.technion.ac.il/~yogi/Courses/CS-Scientific-Writing/examples/simple/simple.htm _____________________________________________________ Important Addendum (you will need to read this to make Friday's TeX file convert into Pdf): There are a number of graphics in Friday's presentation. These must be saved in the same directory as the TeX file in order for the exact same Pdf to be produced. In fact, if these graphics are not present in the same directory then there will be a "fatal error" and the TeX file will not "compile". To avoid this, there are two options: (i) Delete any slide that contains the "\includegraphics" command, or (ii) Download and save this Pdf, which contains the four graphics that are required. Once you have opened the Pdf (after saving it), you can click on each graphic, copy it, paste it into Microsoft Paint (or a similar program you are familiar with) and save the the graphic as a .png file. Make sure to save the graphics into the same directory as the TeX file. Some concluding notes are as follows. Graphic 1 in the Pdf has split into two parts; you can take either of these parts. Finally, LaTeX recognises .png, .jpeg or .eps graphics. In this case .png is used. The type of graphic must always be specified in the TeX file you are working with.	{"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": 0.9942703247070312, "perplexity": 1185.3332285524484}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662133.5/warc/CC-MAIN-20160924173742-00253-ip-10-143-35-109.ec2.internal.warc.gz"}
http://mathhelpforum.com/pre-calculus/121670-tangent-curve.html	# Thread: Tangent to a curve 1. ## Tangent to a curve Find the value of c for which the line $y=x+c$ is a tangent to the curve $y=x^2-5x+4$ thanks! 2. Originally Posted by BabyMilo thanks! do i dy/dx it? then dy/dx=0 then sub x into to get y. then y=x+c to find c? thanks! 3. If it is tangent to the curve, it will touch the curve only once. Since they touch, they must be equal. So $x + c = x^2 - 5x + 4$ $x^2 - 6x + 4 - c = 0$ Since it only touches once, the discriminant must be 0. So $\Delta = (-6)^2 - 4(1)(4 - c) = 0$ $36 - 16 + 4c = 0$ $20 + 4c = 0$ $4c = -20$ $c = -5$. 4. Originally Posted by Prove It If it is tangent to the curve, it will touch the curve only once. Since they touch, they must be equal. So $x + c = x^2 - 5x + 4$ $x^2 - 6x + 4 - c = 0$ Since it only touches once, the discriminant must be 0. So $\Delta = (-6)^2 - 4(1)(4 - c) = 0$ $36 - 16 + 4c = 0$ $20 + 4c = 0$ $4c = -20$ $c = -5$. stupid me. 5. This was posted in the "Pre Calculus" section so Prove It used a method that did not require the derivative. Since you do mention the derivative, no, the derivative is not 0 where the line y= x+ c is tangent to it. A line is tangent to a curve where its slope is the same as the derivative. y= x+ c has slope 1 so you are looking for a place where the derivative is 1, not 0. 6. Here is a method using the derivative. -------------------- The derivative of the function $f(x) = x^2 - 5x + 4$ is $f'(x) = 2x - 5$ (power rule on sum of functions). The line $y = x + c$ has a slope equal to $1$. Thus, you are looking for the point on the curve of $f(x)$ where the slope is equal to $1$. So, you must solve $2x - 5 = 1$ for $x$. Hmm, $x = 3$. Say $a = 3$ (to make it less confusing). You know that the tangent to the curve of $f$ in a point of absciss $a$ is equal to : $y = (x - a)f'(a) + f(a)$ Substitute : $y = (x - 3) \times 1 - 2$ That is : $y = x - 5$. Therefore $c = -5$. 7. Originally Posted by Bacterius Here is a method using the derivative. -------------------- The derivative of the function $f(x) = x^2 - 5x + 4$ is $f'(x) = 2x - 5$ (power rule on sum of functions). The line $y = x + c$ has a slope equal to $1$. Thus, you are looking for the point on the curve of $f(x)$ where the slope is equal to $1$. So, you must solve $2x - 5 = 1$ for $x$. Hmm, $x = 3$. Say $a = 3$ (to make it less confusing). You know that the tangent to the curve of $f$ in a point of absciss $a$ is equal to : $y = (x - a)f'(a) + f(a)$ Substitute : $y = (x - 3) \times 1 - 2$ That is : $y = x - 5$. Therefore $c = -5$. Since you were already given that y= x+ c, y= 3+ c= $3^2- 5(3)+ 4= 9- 15+ 4= -2$ so c= -2-3= -5. That seems simpler to me. 8. Ah, yes, I didn't spot that	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 49, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9152399897575378, "perplexity": 378.92843021625913}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1480698540915.89/warc/CC-MAIN-20161202170900-00346-ip-10-31-129-80.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/1593308/prove-that-fraca-12a-1b-1-cdots-fraca-n2a-nb-n-geq-frac12	# Prove that $\frac{a_1^2}{a_1+b_1}+\cdots+\frac{a_n^2}{a_n+b_n} \geq \frac{1}{2}(a_1+\cdots+a_n).$ Let $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ be positive numbers with $a_1+a_2+\cdots+a_n = b_1+b_2+\cdots+b_n$. $$\text{Prove that} \dfrac{a_1^2}{a_1+b_1}+\cdots+\dfrac{a_n^2}{a_n+b_n} \geq \dfrac{1}{2}(a_1+\cdots+a_n).$$ Attempt It seems like I should use AM-GM on the bottom of each fraction. We then get $\dfrac{a_i^2}{a_i+b_i} \leq \dfrac{a_i^2}{2\sqrt{a_ib_i}}$. But this doesn't seem to help as we get an upper bound. Since there is so much about $a_1+\cdots+a_n$ in this problem, I think a substitution for that might work. • This may help Dec 29, 2015 at 23:11 • Use $\frac{a^2}{b}\geq2a-b$ for positive $a$ and $b$. Dec 29, 2015 at 23:19 • Cauchy-Schwarz gives:$$\sum \frac{a_i^2}{a_i+b_i} \ge \frac{(\sum a_i)^2}{\sum (a_i + b_i)} = ...$$ Dec 30, 2015 at 9:20 • This is a duplicate of math.stackexchange.com/questions/374983/…. Jul 19, 2020 at 12:52 Since $\displaystyle\sum_{i=1}^n\frac{a_i^{2}}{a_i+b_i}-\sum_{i=1}^{n}\frac{b_i^{2}}{a_i+b_i}=\sum_{i=1}^n\frac{a_i^{2}-b_i^{2}}{a_i+b_i}=\sum_{i=1}^n(a_i-b_i)=0$, $\displaystyle\;\;\sum_{i=1}^n\frac{a_i^{2}+b_i^{2}}{a_i+b_i}=2\sum_{i=1}^n\frac{a_i^{2}}{a_i+b_i}$. Since $2(a_i^{2}+b_i^{2})\ge(a_i+b_i)^2$, $\displaystyle\;\;2\sum_{i=1}^n\frac{a_i^{2}}{a_i+b_i}=\sum_{i=1}^n\frac{a_i^{2}+b_i^{2}}{a_i+b_i}\ge\sum_{i=1}^n\frac{a_i+b_i}{2}=\sum_{i=1}^n a_i$ • Jesus, that was such a nice argument. Dec 30, 2015 at 1:10 Using Cauchy-Schwarz inequality $$(x_1^2+\cdots+x_n^2)(y_1^2+\cdots+y_n^2)\ge (x_1y_1+\cdots+x_ny_n)^2,$$ for $x_i=\displaystyle\frac{a_i}{\sqrt{a_i+b_i}}\,$ and $\,y_i=\sqrt{a_i+b_i}$, we obtain $$\left({\frac{a_1^2}{a_1+b_1}}+\cdots+\frac{a_n^2}{a_n+b_n}\right)\big((a_1+b_1)+\cdots+(a_n+b_n)\big)\ge \left(a_1+\cdots+a_n\right)^2,$$ or equivalently $$2\left({\frac{a_1^2}{a_1+b_1}}+\cdots+\frac{a_n^2}{a_n+b_n}\right)(a_1+\cdots+a_n)\ge \left(a_1+\cdots+a_n\right)^2,$$ and finally $${\frac{a_1^2}{a_1+b_1}}+\cdots+\frac{a_n^2}{a_n+b_n}\ge \frac{1}{2}\left(a_1+\cdots+a_n\right).$$ It can be proved by using the Cauchy reverse technique. In details, $$\sum\limits_{i=1}^n{\frac{2a_i^2}{a_i+b_i}}=\sum\limits_{i=1}^n{\left(2a_i-\frac{2a_ib_i}{a_i+b_i}\right)} \ge \sum\limits_{i=1}^n{\left(2a_i-\sqrt{a_ib_i}\right)}$$ So it suffices to show the following inequality $$\sum\limits_{i=1}^n{a_i} \ge \sum\limits_{i=1}^n{\sqrt{a_ib_i}}$$ , which is quite clear since $$2\sum\limits_{i=1}^n{a_i}=\sum\limits_{i=1}^n{\left(a_i+b_i\right)} \ge 2\sum\limits_{i=1}^n{\sqrt{a_ib_i}}$$ according to AM-GM inequality and the condition. Remark. If there is a minus before a fraction, then we can apply AM-GM in the denominator. The following solution is similar to the previous ones. First, we will prove the following lemma: Lemma. For positive reals $$x$$ and $$y$$ the inequality $$\frac{x^2}{y}\geq 2x-y$$ holds. Proof. Indeed, it's equivalent to $$\frac{(x-y)^2}{y}\geq 0$$. Now, apply this inequality for $$x=2a_i$$ and $$y=a_i+b_i$$ $$\frac{4a_i^2}{a_i+b_i}\geq 4a_i-(a_i+b_i)=3a_i-b_i.$$ Summing this inequalities for $$i=\overline{1,n}$$ we obtain $$\sum_{i=1}^{n}\frac{a_i^2}{a_i+b_i}\geq\frac{1}{4}\sum_{i=1}^{n}(3a_i-b_i)=\frac{1}{2}\sum_{i=1}^{n}a_i,$$ as desired. Comment. One can extend lemma in the following way: for all positive $$x$$, $$y$$, $$p$$ and $$q$$ (with $$p\neq q$$) we have $$\frac{x^p}{y^q}\geq\frac{px^{p-q}-qy^{p-q}}{p-q}.$$ Using this inequality we can create a lot of similar problems (this lemma helps when we want to estimate fraction of type $$\frac{A^p}{B^q}$$). Hint: Use the fact that $$f(x)=\frac{1}{x+1}$$ is convex and then use weighted Jensen's inequality .	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 17, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9890015125274658, "perplexity": 177.84728192017155}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656104141372.60/warc/CC-MAIN-20220702131941-20220702161941-00128.warc.gz"}
https://www.nag.com/numeric/nl/nagdoc_27.2/clhtml/e01/e01cec.html	# NAG CL Interfacee01cec (dim1_monconv_disc) Settings help CL Name Style: ## 1Purpose e01cec computes, for a given set of data points, the forward values and other values required for monotone convex interpolation as defined in Hagan and West (2008). This form of interpolation is particularly suited to the construction of yield curves in Financial Mathematics but can be applied to any data where it is desirable to preserve both monotonicity and convexity. ## 2Specification #include void e01cec (Integer n, double lam, Nag_Boolean negfor, Nag_Boolean yfor, const double x[], const double y[], double comm[], NagError fail) The function may be called by the names: e01cec, nag_interp_dim1_monconv_disc or nag_interp_1d_monconv_disc. ## 3Description e01cec computes, for a set of data points, $\left({x}_{i},{y}_{i}\right)$, for $\mathit{i}=1,2,\dots ,n$, the discrete forward rates, ${f}_{i}^{d}$, and the instantaneous forward rates, ${f}_{i}$, which are used in a monotone convex interpolation method that attempts to preserve both the monotonicity and the convexity of the original data. The monotone convex interpolation method is due to Hagan and West and is described in Hagan and West (2006), Hagan and West (2008) and West (2011). The discrete forward rates are defined simply, for ordered data, by $f1d=y1; fid = xi yi - xi-1 yi-1 xi - xi-1 , for i=2,3,…,n.$ (1) The discrete forward rates, if pre-computed, may be supplied instead of $y$, in which case the original values $y$ are computed using the inverse of (1). The data points ${x}_{i}$ need not be ordered on input (though ${y}_{i}$ must correspond to ${x}_{i}$); a set of ordered and scaled values ${\xi }_{i}$ are calculated from ${x}_{i}$ and stored. In its simplest form, the instantaneous forward rates, ${f}_{i}$, at the data points are computed as linear interpolations of the ${f}_{i}^{d}$: $fi = xi - xi-1 xi+1 - xi-1 fi+1d + xi+1 - xi xi+1 - xi-1 fid , for i=2,3,…,n-1 f1 = f2d - 1 2 (f2-f2d) fn = fnd - 1 2 (fn-1-fnd).$ (2) If it is required, as a constraint, that these values should never be negative then a limiting filter is applied to $f$ as described in Section 3.6 of West (2011). An ameliorated (smoothed) form of this linear interpolation for the forward rates is implemented using the amelioration (smoothing) parameter $\lambda$. For $\lambda \equiv 0$, equation (2) is used (with possible post-process filtering); for $0<\lambda \le 1$, the ameliorated method described fully in West (2011) is used. The values computed by e01cec are used by e01cfc to compute, for a given value $\stackrel{^}{x}$, the monotone convex interpolated (or extrapolated) value $\stackrel{^}{y}\left(\stackrel{^}{x}\right)$ and the corresponding instantaneous forward rate $f$; the curve gradient at $\stackrel{^}{x}$ can be derived as ${y}^{\prime }=\left(f-\stackrel{^}{y}\right)/\stackrel{^}{x}$ for $\stackrel{^}{x}\ne 0$. ## 4References Hagan P S and West G (2006) Interpolation methods for curve construction Applied Mathematical Finance 13(2) 89–129 Hagan P S and West G (2008) Methods for constructing a yield curve WILLMOTT Magazine May 70–81 West G (2011) The monotone convex method of interpolation Financial Modelling Agency ## 5Arguments 1: $\mathbf{n}$Integer Input On entry: $n$, the number of data points. Constraint: ${\mathbf{n}}\ge 2$. 2: $\mathbf{lam}$double Input On entry: $\lambda$, the amelioration (smoothing) parameter. Forward rates are first computed using (2) and then, if $\lambda >0$, a limiting filter is applied which depends on neighbouring discrete forward values. This filter has a smoothing effect on the curve that increases with $\lambda$. Suggested value: $\lambda =0.2$. Constraint: $0.0\le {\mathbf{lam}}\le 1.0$. 3: $\mathbf{negfor}$Nag_Boolean Input On entry: determines whether or not to allow negative forward rates. ${\mathbf{negfor}}=\mathrm{Nag_TRUE}$ Negative forward rates are permitted. ${\mathbf{negfor}}=\mathrm{Nag_FALSE}$ Forward rates calculated must be non-negative. 4: $\mathbf{yfor}$Nag_Boolean Input On entry: determines whether the array y contains values, $y$, or discrete forward rates ${f}^{d}$. ${\mathbf{yfor}}=\mathrm{Nag_TRUE}$ y contains the discrete forward rates ${f}_{i}^{d}$, for $\mathit{i}=1,2,\dots ,n$. ${\mathbf{yfor}}=\mathrm{Nag_FALSE}$ y contains the values ${y}_{i}$, for $\mathit{i}=1,2,\dots ,n$. 5: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input On entry: $x$, the (possibly unordered) set of data points. 6: $\mathbf{y}\left[{\mathbf{n}}\right]$const double Input On entry: If ${\mathbf{yfor}}=\mathrm{Nag_TRUE}$, the discrete forward rates ${f}_{i}^{d}$ corresponding to the data points ${x}_{i}$, for $\mathit{i}=1,2,\dots ,n$. If ${\mathbf{yfor}}=\mathrm{Nag_FALSE}$, the data values ${y}_{i}$ corresponding to the data points ${x}_{i}$, for $\mathit{i}=1,2,\dots ,n$. 7: $\mathbf{comm}\left[4×{\mathbf{n}}+10\right]$double Communication Array On exit: contains information to be passed to e01cfc. The information stored includes the discrete forward rates ${f}^{d}$, the instantaneous forward rates $f$, and the ordered data points $\xi$. 8: $\mathbf{fail}$NagError Input/Output The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface). ## 6Error Indicators and Warnings NE_ALLOC_FAIL Dynamic memory allocation failed. See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information. On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value. NE_DATA_NOT_UNIQUE On entry, x contains duplicate data points. NE_INT On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{n}}\ge 2$. 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 7.5 in the Introduction to the NAG Library CL Interface for further information. NE_NO_LICENCE Your licence key may have expired or may not have been installed correctly. See Section 8 in the Introduction to the NAG Library CL Interface for further information. NE_REAL On entry, ${\mathbf{lam}}=⟨\mathit{\text{value}}⟩$. Constraint: $0.0\le {\mathbf{lam}}\le 1.0$. ## 7Accuracy The computational errors in the values stored in the array comm should be negligible in most practical situations. ## 8Parallelism and Performance e01cec is not threaded in any implementation. e01cec internally allocates $9n$ reals.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 68, "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": 0.8348408937454224, "perplexity": 2040.5295794144447}, "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-2021-43/segments/1634323585199.76/warc/CC-MAIN-20211018062819-20211018092819-00149.warc.gz"}
http://frederic-wang.fr/two-open-problems-for-the-subgroup-reduction-based-dedekindian-hsp-algorithm.html	# Two Open Problems for the Subgroup-Reduction based Dedekindian HSP Algorithm One of my main idea after having studied the Hidden Subgroup Problem is that subgroup simplification is likely to play an important role. Basically, rather than directly finding the hidden subgroup $H$ by working on the whole initial group $G$, we only try to get partial information on $H$ in a first time. This information allows us to move to a simpler HSP problem and we can iterate this process until the complete determination of $H$. Several reductions of this kind exist and I think the sophisticated solution to HSP over 2-nil groups illustrates well how this technique can be efficient. Using only subgroup reduction, I've been able to design an alternative algorithm for the Dedekindian HSP i.e. over groups that have only normal subgroups. Recall that the standard Dedekindian HSP algorithm is to use Weak Fourier Sampling, measure a polynomial number of representations ${\rho }_{1},\dots ,{\rho }_{m}$ and then get with high probability the hidden subgroup as the intersection of kernels $H=\bigcap _{i}\mathrm{Ker}{\rho }_{i}$. When the group is dedekindian, we always have $H\subseteq \mathrm{Ker}{\rho }_{i}$. Hence my alternative algorithm is rather to start by measuring one representation ${\rho }_{1}$, move the problem to HSP over $\mathrm{Ker}{\rho }_{1}$ and iterate this procedure. I've been able to show that we reach the group $H$ after a polynomial number of steps, the idea being that when we measure a non-trivial representation the size of the underlying group becomes at least twice smaller. One difficulty of this approach is to determine the structure of the new group $\mathrm{Ker}{\rho }_{i}$ so that we can work on it. However, for the cyclic case this is determination is trivial and for the abelian case I've used the group decomposition algorithm, based on the cyclic HSP. Finally I've two open questions: 1. Can my algorithm work for the Hamiltonian HSP i.e. over non-abelian dedekindian groups? 2. Is my algorithm more efficient than the standard Dedekindian HSP? For the first question, I'm pretty sure that the answer is positive, but I admit that I've not really thought about it. For the second one, it depends on what we mean by more efficient. The decomposition of the kernel seems to increase the time complexity but since we are working on smaller and smaller groups, we decrease the space complexity. However, if we are only considering the numbers of sample, my conjecture is that both algorithms have the same complexity and more precisely yield the same markov process. In order to illustrate this, let's consider the cyclic case $G={ℤ}_{18}$ and $H=6{ℤ}_{18}$. The markov chain of the my alternative algorithm is given by the graph below, where the edge labels are of course probabilities and the node labels are the underlying group. We start at $G={ℤ}_{18}$ and want to reach $H\cong {ℤ}_{3}$. One can think that moving to smaller and smaller subgroups will be faster than the standard algorithm which always works on ${ℤ}_{18}$. However, it turns out that the markov chain of the standard algorithm is exactly the same. The point being that while it is true that working over ${ℤ}_{9}$ or ${ℤ}_{6}$ provides less possibilities of samples (3 and 2 respectively, instead of 6 for ${ℤ}_{18}$) the repartition of "good" and "bad" samples is the same and thus we get the same transition probabilities. I guess it would be possible to formalize this for a general cyclic group. The general abelian case seems more difficult, but I'm sure that the same phenomenon can be observed.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 19, "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": 0.9641519784927368, "perplexity": 193.4926924724137}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1490218189462.47/warc/CC-MAIN-20170322212949-00142-ip-10-233-31-227.ec2.internal.warc.gz"}
https://www.di-mgt.com.au/public-key-crypto-discrete-logs-2-mqv.html	DI Management Home > Cryptography > Public key cryptography using discrete logarithms > Part 2: MQV Key Agreement # Public key cryptography using discrete logarithms. Part 2: MQV Key Agreement The original Diffie-Hellman Key Exchange method is still susceptible to a man-in-the-middle attack where an active adversary can manipulate the messages and discover the shared secret. << previous: Diffie-Hellman key exchange next: ElGamal Encryption >> ## MQV Key Agreement MQV key agreement is an improvement on the basic Diffie-Hellman designed to eliminate a man-in-the-middle attack. It is named after Menezes, Qu and Vanstone [MQV98]. The original paper is written for elliptic curve cryptography, but the protocol also works with discrete logarithms. Unfortunately there may be patents associated with it. MQV assumes that the two parties, Alice and Bob, have already established and authenticated their static key pairs $(a,A)$ and $(b,B)$ with each other; that is, Alice trusts that Bob's public key $B$ really is his, and Bob trusts Alice's public key $A$. Each party generates a random sessional or ephemeral key pair, $(x,X=g^x)$ and $(y,Y=g^y)$, and these are used together with the static keys to create a new shared secret. Algorithm: MQV Key Agreement INPUT: Domain parameters $(p,q,g)$; pre-established D-H key pairs $(a,A=g^a)$ and $(b,B=g^b)$. OUTPUT: Shared secret $Z$ in the range $0\lt Z\lt p$. 1. Alice chooses a random number $x$ in the range $[2,q-2]$ and sends $X=g^x\mod p$ to Bob. 2. Bob chooses a random number $y$ in the range $[2,q-2]$ and sends $Y=g^y\mod p$ to Alice. 3. Alice and Bob both compute $\overline{X} = X \mod 2^{\ell} + 2^{\ell}$ and $\overline{Y} = Y \mod 2^{\ell} + 2^{\ell}$ where $\ell = \lceil f/2\rceil$ and $f$ is the bit length of $q$. 4. Alice computes her implicit signature $S_A = (x + \overline{X}a) \mod q$. 5. Alice computes $t_A = YB^{\overline{Y}} \mod p$. 6. Alice computes $Z_A = \left(t_A\right)^{S_A} \mod p$. 7. Return shared secret $Z = Z_A$. Similarly, Bob can also compute the same value, $Z$, using steps 4 to 7 above. • 4. Bob computes his implicit signature $S_B = (y + \overline{Y}b) \mod q$. • 5. Bob computes $t_B = XA^{\overline{X}} \mod p$. • 6. Bob computes $Z_B = \left(t_B\right)^{S_B} \mod p$. • 7. Return shared secret $Z = Z_B$. The value of $\ell$ is half the bit length of $q$ rounded upwards. The bit length of $q$ is given by $f = \lfloor\log_2 q\rfloor + 1$. The value $\overline{Y}$ is the right-most $\ell$ bits of $Y$ (i.e. bits $0$ to $\ell-1$) with bit $\ell$ set to one. The idea behind using a truncated $X$ and $Y$ is to reduce the computational effort in step 5. ### ▷ Why does this work? Working modulo $p$ we have $YB^{\overline{Y}} = g^y\cdot\left(g^b\right)^{\overline{Y}}$ $XA^{\overline{X}} = g^x\cdot\left(g^a\right)^{\overline{X}}$ $\qquad\; = g^{y+b\overline{Y}}$ $\qquad\; = g^{x+a\overline{X}}$ $\qquad\; = g^{(y+b\overline{Y})\mod q}$ $\qquad\; = g^{(x+a\overline{X})\mod q}$ $\qquad\; = g^{S_B}, \quad$ $\qquad\; = g^{S_A}$, and thus $Z_A\;=\left(YB^{\overline{Y}}\right)^{S_A} = \left(g^{S_B}\right)^{S_A}$ $\qquad = g^{S_BS_A}$. $Z_B\;=\left(XA^{\overline{X}}\right)^{S_B} = \left(g^{S_A}\right)^{S_B}$ $\qquad = g^{S_AS_B}$. We are allowed to do the "trick" with the exponents modulo $q$ because even if $n\neq m$ but $n\equiv m\pmod{q}$ then $g^n\equiv g^m\pmod{p}$. This follows because $n\equiv m\pmod{q}$ means $n=m+kq$ for some integer $k$ and so $g^n\equiv g^{m+kq} \equiv g^m\left(g^q\right)^k \equiv g^m(1)^k \equiv g^m \pmod{p}$ since $g^q\equiv 1\pmod{p}$ by the definition of $g$. ### Example of MQV key agreement by Alice • INPUT: Domain parameters $(p=283, q=47, g=60)$ • INPUT: Pre-established static key pairs $(a,A)=(24,158)$ and $(b,B)=(7,216)$ • 1. Alice chooses a random $x=25$, computes $X=g^x = 60^{25}\mod p = 114$, and sends $X$ to Bob. • 2. Bob chooses a random $y=32$, computes $Y=g^y = 60^{32}\mod p = 175$, and sends $Y$ to Alice. • 3. Both parties compute: $\qquad$ $f = 6$, the bit length of $q$, so $\ell=\lceil f/2\rceil = 3$, $\qquad$ $\overline{X}=X\mod 2^{\ell} + 2^{\ell} = 141\mod 2^3 + 2^3 = 5 + 8 = 13$ and $\qquad$ $\overline{Y}=Y\mod 2^{\ell} + 2^{\ell} = 175\mod 2^3 + 2^3 = 7 + 8 = 15$ Alice computes: • 4. $S_A = (x + \overline{X}a) \mod q = (25 + 13\cdot 24)\mod 47 = 8$ • 5. $t_A = Y\cdot B^{\overline{Y}}\mod p = 175\cdot 216^{15}\mod 283 = 151$. • 6. $Z_A = t_A^{S_A}\mod p = 151^8\mod 283 = 207$. Bob computes: • 4. $S_B = (y + \overline{Y}b) \mod q = (32 + 15\cdot 7)\mod 47 = 43$ • 5. $t_B = X\cdot A^{\overline{X}}\mod p = 141\cdot 158^{13}\mod 283 = 225$. • 6. $Z_B = t_B^{S_B}\mod p = 225^{43}\mod 283 = 207$. The shared secret known only to Alice and Bob is $Z=207$. Note that this works despite the very small numbers we end up with after truncating $X$ and $Y$. ## References • [MQV98] Law, Laurie, Alfred Menezes, Minghua Qu, Jerry Solinas, and Scott Vanstone (1998). An Efficient Protocol for Authenticated Key Agreement, Technical report CORR 98-05, University of Waterloo, Canada, March 1998. Revised August 28 1998. <pdf-link>	{"extraction_info": {"found_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": 0.8928530216217041, "perplexity": 1076.2663462927856}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1512948597295.74/warc/CC-MAIN-20171217171653-20171217193653-00159.warc.gz"}
http://math.stackexchange.com/questions/214839/translate-of-a-closed-set-is-closedpart2	# Translate of a closed set is closed(part2) Previously, I raised a question whether $$(a+F)^c=a+F^c.$$ Jonas Meyer pointed out that it is true. After which, I was able to prove the first inclusion. The details are as follows: let $y\in a+F^c$. Then there exists $z\in F^c$ such that $y=a+z$. We claim that $y\in(a+F)^c$. Suppose $y\notin (a+F)^c$. Then $y\in a+F$. Thus, there exists $x\in F$ such that $y=a+x$. This implies that $z=x$. Hence, $x\in F\cap F^c=\varnothing$. We obtain a contradiction. Thus, $y\in(a+F)^c$. Hence, $a+F^c \subseteq (a+F)^c$. I tried the other inclusion, but can't prove it. A help on this is very much appreciated. juniven - By your first step $(a+ F)^c = a + (-a) + (a+F)^c \subseteq a + \bigl((-a)+ a + F\bigr)^c = a + F^c.$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9861369132995605, "perplexity": 56.64699130372299}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644064865.49/warc/CC-MAIN-20150827025424-00113-ip-10-171-96-226.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/73743/countable-union-of-closed-subschemes-over-uncountable-field?sort=votes	# countable union of closed subschemes over uncountable field I am looking for a reference for the following well-known fact: Let $k$ be an uncountable field, and let $X$ be a $k$-variety. Let $Z_1, Z_2, \dots \subseteq X$ be proper closed subschemes. Then $\bigcup Z_i(k) \neq X(k)$. Thanks! - You also need $k$ to be algebraically closed (otherwise one could well have $X(k) = \emptyset$). – ulrich Aug 26 '11 at 9:12 I'm glad you asked the question. You'd think this would discussed in some standard textbook, but I've never seen it. Most complex algebraic geometers use a sledge hammer (Baire category theorem) but it's certainly more elementary than that. – Donu Arapura Aug 26 '11 at 12:19 It should be possible to give a bare hands proof that given a countable collection of nonzero polynomials $f_i$ in $n$-variables, there exist a point such that $f_i(p)\not=0$ simultaneously. – Donu Arapura Aug 26 '11 at 12:26 Yes, I left out algebraically closed as a hypothesis. It's frustrating that there (doesn't seem to be) a standard place to quote such a well-known fact. – Sue Sierra Aug 26 '11 at 18:06 As luck would have it, a related question just came up in my research today: does an algebraic torus over $k$ have a dense cyclic subgroup? If $k$ is uncountable and algebraically closed, the above-mentioned fact shows the answer is yes: just choose a generator outside the (countable) union of all proper Zariski closed subgroups. But if $k$ is the (countable) algebraic closure of a finite field, the answer is no, since every element of $k^\times$ has finite order. – Michael Thaddeus Aug 28 '11 at 3:59 Suppose $\dim X>0$ and $k$ is algebraically closed and uncountable. Moreover, if a "variety" is not necessarily irreducible, the $Z_i$ are supposed to have positive codimension in $X$ (otherwise one could take the irreducible components of $X$). As in MP's answer, one can suppose $X$ is affine. By Noether's Normalization Lemma, there exists a finite surjective morphism $p: X\to \mathbb A^m_k$ with $m=\dim X$. Let $Y_i=p(Z_i)$. This is a closed subset of $\mathbb A^m_k$ of positive codimension. Moreover $\mathbb A^m_k(k)=\cup Y_i(k)$ because $k$ is algebraically closed (which implies that $Y_i(k)=p(Z_i(k))$). As $k$ is uncountable, there exists a hyperplane $H$ in $\mathbb A^m$ not contained in any $Y_i$ (note that $H\subseteq Y_i$ is equivalent to $H=Y_i$). So by induction on $m$ we are reduced to the case $m=1$, and the assertion is obvious. Without the hypothesis $k$ algebraically closed, one can show similarly that $X\ne \cup_i Z_i$. This is Exercise 2.5.10 in my book. EDIT In fact this statement is trivial because the generic points of $X$ don't belong to any of the $Z_i$'s. But the proof shows that the set of closed points of $X$ is not contained in $\cup_i Z_i$. - I do not know a reference, but the following short argument seems to work. Assume that the dimension of $X$ is at least 1! Argue by induction on the dimension of $X$. Reduce to the case in which the subschemes are irreducible of codimension one. Shrinking $X$ if necessary, reduce also to the case in which $X$ is quasiprojective. Let $L$ be a pencil of integral divisors on $X$. Since the ground-field is uncountable, the pencil $L$ contains uncountably many elements that are integral; let $D \in L$ be an integral element of $L$ that is different from each of the subschemes you want to avoid. By the inductive hypothesis, $D$ contains a point that is not contained in any of the subschemes and you are done. You can find this stated as a hint in an Exercise V.4.15 (c) in Hartshorne. - As ulrich's correctly comments above, the only place that I did not argue (namely, the base case of the induction) is also the place where there is a missing assumption: the field should be algebraically closed! The argument sketched here requires curves over $k$ to have uncountably many points. – M P Aug 26 '11 at 9:23	{"extraction_info": {"found_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": 0.9561675190925598, "perplexity": 179.22873259876914}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115862636.1/warc/CC-MAIN-20150124161102-00070-ip-10-180-212-252.ec2.internal.warc.gz"}
https://openstax.org/books/college-algebra-2e/pages/9-7-probability	College Algebra 2e # 9.7Probability College Algebra 2e9.7 Probability ### Learning Objectives In this section, you will: • Construct probability models. • Compute probabilities of equally likely outcomes. • Compute probabilities of the union of two events. • Use the complement rule to find probabilities. • Compute probability using counting theory. Figure 1 An example of a “spaghetti model,” which can be used to predict possible paths of a tropical storm.1 Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in Figure 1. They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions. ### Constructing Probability Models Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is ${1,2,3,4,5,6 }.{1,2,3,4,5,6 }.$ An event is any subset of a sample space. The likelihood of an event is known as probability. The probability of an event $p p$ is a number that always satisfies $0≤p≤1, 0≤p≤1,$ where 0 indicates an impossible event and 1 indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like Table 1. Outcome Probability Winning the raffle 1% Losing the raffle 99% Table 1 The sum of the probabilities listed in a probability model must equal 1, or 100%. ### How To Given a probability event where each event is equally likely, construct a probability model. 1. Identify every outcome. 2. Determine the total number of possible outcomes. 3. Compare each outcome to the total number of possible outcomes. ### Example 1 #### Constructing a Probability Model Construct a probability model for rolling a single, fair die, with the event being the number shown on the die. ### Q&A Do probabilities always have to be expressed as fractions? No. Probabilities can be expressed as fractions, decimals, or percents. Probability must always be a number between 0 and 1, inclusive of 0 and 1. ### Try It #1 Construct a probability model for tossing a fair coin. ### Computing Probabilities of Equally Likely Outcomes Let $S S$ be a sample space for an experiment. When investigating probability, an event is any subset of $S. S.$ When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in $S. S.$ Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in $S, S,$ so the probability of the event is $4 6 = 2 3 . 4 6 = 2 3 .$ ### Computing the Probability of an Event with Equally Likely Outcomes The probability of an event $E E$ in an experiment with sample space $S S$ with equally likely outcomes is given by $E E$ is a subset of $S, S,$ so it is always true that $0≤P(E)≤1. 0≤P(E)≤1.$ ### Example 2 #### Computing the Probability of an Event with Equally Likely Outcomes A six-sided number cube is rolled. Find the probability of rolling an odd number. ### Try It #2 A number cube is rolled. Find the probability of rolling a number greater than 2. ### Computing the Probability of the Union of Two Events We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events is the event that occurs if either or both events occur. $P(E∪F)=P(E)+P(F)−P(E∩F) P(E∪F)=P(E)+P(F)−P(E∩F)$ Suppose the spinner in Figure 2 is spun. We want to find the probability of spinning orange or spinning a $b. b.$ Figure 2 There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is $3 6 = 1 2 . 3 6 = 1 2 .$ There are a total of 6 sections, and 2 of them have a $b. b.$ So the probability of spinning a $b b$ is $2 6 = 1 3 . 2 6 = 1 3 .$ If we added these two probabilities, we would be counting the sector that is both orange and a $b b$ twice. To find the probability of spinning an orange or a $b, b,$ we need to subtract the probability that the sector is both orange and has a $b. b.$ $1 2 + 1 3 − 1 6 = 2 3 1 2 + 1 3 − 1 6 = 2 3$ The probability of spinning orange or a $b b$ is $2 3 . 2 3 .$ ### Probability of the Union of Two Events The probability of the union of two events $E E$ and $F F$ (written $E∪F E∪F$ ) equals the sum of the probability of $E E$ and the probability of $F F$ minus the probability of $E E$ and $F F$ occurring together $(($ which is called the intersection of $E E$ and $F F$ and is written as $E∩F E∩F$ ). $P(E∪F)=P(E)+P(F)−P(E∩F) P(E∪F)=P(E)+P(F)−P(E∩F)$ ### Example 3 #### Computing the Probability of the Union of Two Events A card is drawn from a standard deck. Find the probability of drawing a heart or a 7. ### Try It #3 A card is drawn from a standard deck. Find the probability of drawing a red card or an ace. ### Computing the Probability of Mutually Exclusive Events Suppose the spinner in Figure 2 is spun again, but this time we are interested in the probability of spinning an orange or a $d. d.$ There are no sectors that are both orange and contain a $d, d,$ so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is $P(E∪F)=P(E)+P(F) P(E∪F)=P(E)+P(F)$ Notice that with mutually exclusive events, the intersection of $E E$ and $F F$ is the empty set. The probability of spinning an orange is $3 6 = 1 2 3 6 = 1 2$ and the probability of spinning a $d d$ is $1 6 . 1 6 .$ We can find the probability of spinning an orange or a $d d$ simply by adding the two probabilities. The probability of spinning an orange or a $d d$ is $2 3 . 2 3 .$ ### Probability of the Union of Mutually Exclusive Events The probability of the union of two mutually exclusive events $EandF EandF$ is given by $P(E∪F)=P(E)+P(F) P(E∪F)=P(E)+P(F)$ ### How To Given a set of events, compute the probability of the union of mutually exclusive events. 1. Determine the total number of outcomes for the first event. 2. Find the probability of the first event. 3. Determine the total number of outcomes for the second event. 4. Find the probability of the second event. ### Example 4 #### Computing the Probability of the Union of Mutually Exclusive Events A card is drawn from a standard deck. Find the probability of drawing a heart or a spade. ### Try It #4 A card is drawn from a standard deck. Find the probability of drawing an ace or a king. ### Using the Complement Rule to Compute Probabilities We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event $E, E,$ denoted $E ′ , E ′ ,$ is the set of outcomes in the sample space that are not in $E. E.$ For example, suppose we are interested in the probability that a horse will lose a race. If event $W W$ is the horse winning the race, then the complement of event $W W$ is the horse losing the race. To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1. $P( E ′ )=1−P(E) P( E ′ )=1−P(E)$ The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is $1 9 , 1 9 ,$ the probability of the horse losing the race is simply $1− 1 9 = 8 9 1− 1 9 = 8 9$ ### The Complement Rule The probability that the complement of an event will occur is given by $P( E ′ )=1−P(E) P( E ′ )=1−P(E)$ ### Example 5 #### Using the Complement Rule to Calculate Probabilities Two six-sided number cubes are rolled. 1. Find the probability that the sum of the numbers rolled is less than or equal to 3. 2. Find the probability that the sum of the numbers rolled is greater than 3. ### Try It #5 Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than 10. ### Computing Probability Using Counting Theory Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems. Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are $C(5,2) C(5,2)$ ways to select 2 phones that are not defective. There are 8 phones, so there are $C(8,2) C(8,2)$ ways to select 2 phones. The probability of selecting 2 phones that are not defective is: ### Example 6 #### Computing Probability Using Counting Theory A child randomly selects 5 toys from a bin containing 3 bunnies, 5 dogs, and 6 bears. 1. Find the probability that only bears are chosen. 2. Find the probability that 2 bears and 3 dogs are chosen. 3. Find the probability that at least 2 dogs are chosen. ### Try It #6 A child randomly selects 3 gumballs from a container holding 4 purple gumballs, 8 yellow gumballs, and 2 green gumballs. 1. Find the probability that all 3 gumballs selected are purple. 2. Find the probability that no yellow gumballs are selected. 3. Find the probability that at least 1 yellow gumball is selected. ### Media Access these online resources for additional instruction and practice with probability. ### 9.7 Section Exercises #### Verbal 1 . What term is used to express the likelihood of an event occurring? Are there restrictions on its values? If so, what are they? If not, explain. 2 . What is a sample space? 3 . What is an experiment? 4 . What is the difference between events and outcomes? Give an example of both using the sample space of tossing a coin 50 times. 5 . The union of two sets is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the union of two events from a probability model? How is it different? #### Numeric For the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated. Figure 3 6 . Landing on red 7 . Landing on a vowel 8 . Not landing on blue 9 . Landing on purple or a vowel 10 . Landing on blue or a vowel 11 . Landing on green or blue 12 . Landing on yellow or a consonant 13 . Not landing on yellow or a consonant For the following exercises, two coins are tossed. 14 . What is the sample space? 15 . Find the probability of tossing two heads. 16 . Find the probability of tossing exactly one tail. 17 . Find the probability of tossing at least one tail. For the following exercises, four coins are tossed. 18 . What is the sample space? 19 . Find the probability of tossing exactly two heads. 20 . Find the probability of tossing exactly three heads. 21 . Find the probability of tossing four heads or four tails. 22 . Find the probability of tossing all tails. 23 . Find the probability of tossing not all tails. 24 . Find the probability of tossing exactly two heads or at least two tails. 25 . For the following exercises, one card is drawn from a standard deck of $52 52$ cards. Find the probability of drawing the following: 26 . A club 27 . A two 28 . Six or seven 29 . Red six 30 . An ace or a diamond 31 . A non-ace 32 . A heart or a non-jack For the following exercises, two dice are rolled, and the results are summed. 33 . Construct a table showing the sample space of outcomes and sums. 34 . Find the probability of rolling a sum of $3. 3.$ 35 . Find the probability of rolling at least one four or a sum of $8. 8.$ 36 . Find the probability of rolling an odd sum less than $9. 9.$ 37 . Find the probability of rolling a sum greater than or equal to $15. 15.$ 38 . Find the probability of rolling a sum less than $15. 15.$ 39 . Find the probability of rolling a sum less than $6 6$ or greater than $9. 9.$ 40 . Find the probability of rolling a sum between $6 6$ and $9, 9,$ inclusive. 41 . Find the probability of rolling a sum of $5 5$ or $6. 6.$ 42 . Find the probability of rolling any sum other than $5 5$ or $6. 6.$ For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: 43 . A head on the coin or a club 44 . A tail on the coin or red ace 45 . A head on the coin or a face card 46 . No aces For the following exercises, use this scenario: a bag of M&Ms contains $12 12$ blue, $6 6$ brown, $10 10$ orange, $8 8$ yellow, $8 8$ red, and $4 4$ green M&Ms. Reaching into the bag, a person grabs 5 M&Ms. 47 . What is the probability of getting all blue M&Ms? 48 . What is the probability of getting $4 4$ blue M&Ms? 49 . What is the probability of getting $3 3$ blue M&Ms? 50 . What is the probability of getting no brown M&Ms? #### Extensions Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting $20 20$ numbers from the numbers $1 1$ to $80. 80.$ After the player makes his selections, $20 20$ winning numbers are randomly selected from numbers $1 1$ to $80. 80.$ A win occurs if the player has correctly selected $3,4, 3,4,$ or $5 5$ of the $20 20$ winning numbers. (Round all answers to the nearest hundredth of a percent.) 51 . What is the percent chance that a player selects exactly 3 winning numbers? 52 . What is the percent chance that a player selects exactly 4 winning numbers? 53 . What is the percent chance that a player selects all 5 winning numbers? 54 . What is the percent chance of winning? 55 . How much less is a player’s chance of selecting 3 winning numbers than the chance of selecting either 4 or 5 winning numbers? #### Real-World Applications Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).2 56 . If you meet a U.S. citizen, what is the percent chance that the person is elderly? (Round to the nearest tenth of a percent.) 57 . If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.) 58 . If you meet five U.S. citizens, what is the percent chance that three are elderly? (Round to the nearest tenth of a percent.) 59 . If you meet five U.S. citizens, what is the percent chance that four are elderly? (Round to the nearest thousandth of a percent.) 60 . It is predicted that by 2030, one in five U.S. citizens will be elderly. How much greater will the chances of meeting an elderly person be at that time? What policy changes do you foresee if these statistics hold true? ### Footnotes • 1The figure is for illustrative purposes only and does not model any particular storm. • 2United States Census Bureau. http://www.census.gov	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 160, "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": 0.8358796834945679, "perplexity": 400.70523396160297}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320299927.25/warc/CC-MAIN-20220129032406-20220129062406-00459.warc.gz"}
https://www.ias.ac.in/listing/articles/joaa/018/02-03	• Volume 18, Issue 2-3 September 1997, pages 91-227 • On the polarisation and emission geometry of pulsar 1929+10: Does its emission come from a single pole or two poles? Pulsar B1929+10 is remarkable on a number of grounds. Its narrow primary components exhibit virtually complete and highly stable linear polarisation, which can be detected over most of its rotation cycle. Various workers have been lured by the unprecedented range over which its linear polarisation angle can be determined, and more attempts have been made to model its emission geometry than perhaps for any other pulsar. Paradoxically, there is compelling evidence to interpret the pulsar’s emission geometryboth in terms of an aligned configuration whereby its observed radiation comes from a single magnetic-polar emission regionand in terms of a nearly orthogonal configuration in which we receive emission from regions near each of its two poles. Pulsar 1929+10 thus provides a fascinating context in which to probe the conflict between these lines of interpretation in an effort to deepen our understanding of pulsar radio emission. Least-squares fits to the polarisation-angle traverse fit poorly near the main pulse and interpulse and have an inflection point far from the centre of the main pulse. This and a number of other circumstances suggest that the position-angle traverse is an unreliable indicator of the geometry in this pulsar, possibly in part because its low level ‘pedestal’ emission makes it impossible to properly calibrate a Polarimeter which correlates orthogonal circular polarisations. Taking the interpulse and main-pulse comp. II widths as indicators of the magnetic latitude, it appears that pulsar 1929+10 has anα value near 90‡ and thus has a two-pole interpulse geometry. This line of interpretation leads to interesting and consistent results regarding the geometry of the conal components. Features corresponding to both an inner and outer cone are identified. In addition, it appears that pulsar 1929+10–and a few other stars–have what we are forced to identify as a ‘furtherin’ cone, with a conal emission radius of about2.3‡/P1/2 Secondarily, 1929+10’s nearly complete linear polarisation provides an ideal opportunity to study how mechanisms of depolarisation function on a pulse-to-pulse basis. Secondary-polarisation-mode emission appears in significant proportion only in some limited ranges of longitude, and the subsequent depolarization is studied using different mode-separation techniques. The characteristics of the two polarisation modes are particularly interesting, both because the primary mode usually dominates the secondary so completely and because the structure seen in the secondary mode appears to bear importantly on the question of the pulsar’s basic emission geometry. New secondary-mode features are detected in the average profile of this pulsar which appear independent of the main-pulse component structure and which apparently constitute displaced modal emission. Individual pulses during which the secondary-mode dominates the primary one are found to be considerably more intense than the others and largely depolarised. Monte-Carlo modeling of the mode mixing in this region, near the boundary of comps. II and III, indicates that the incoherent interference of two fully and orthogonally polarised modes can adequately account for the observed depolarisation. The amplitude distributions of the two polarisation modes are both quite steady: the primary polarisation mode is well fitted by a χ2 distribution with about nine degrees of freedom; whereas the secondary mode requires a more intense distribution which is constant, but sporadic. • Intergalactic UV background radiation field We have performed proximity effect analysis of low and high resolution data, considering detailed frequency and redshift dependence of the AGN spectra processed through galactic and intergalactic material. We show that such a background flux, calculated using the observed distribution of AGNs, falls short of the value required by the proximity effect analysis by a factor of ≥ 2.7. We have studied the uncertainty in the value of the required flux due to its dependence on the resolution, description of column density distribution, systemic redshifts of QSOs etc. We conclude that in view of these uncertainties the proximity effect is consistent with the background contributed by the observed AGNs and that the hypothesized presence of an additional, dust extinct, population of AGNs may not be necessary. • Hα emission from late type be stars We show here that the Hα flux from late type Be stars can be explained as emission from an HII region formed in the gas envelope around the Be star, by the UV flux emitted by a helium star binary companion. We also discuss the observability of the helium star companions. • The effective technique of the charged particles background discrimination in the atmospheric Cherenkov light detectors It is shown that parameters of flashes, detected by multichannel image cameras of Cherenkov detectors with closed lids are close to those of Cherenkov flashes initiated by VHE gamma-quanta in the Earth atmosphere. Even after application of criteria for gamma-like events selection a considerable part of those flashes may be misclassified and accepted as gammas. Since the flashes of this kind are detected also during normal measurements with the opened lids of image cameras it just increases the background and, as a consequence, decreases the detector sensitivity even when one uses an anticoincidence scintillator shield around the camera (its efficiency is about 75 %). The use of detectors consisting of two (or more) sections no less than 20–30 m apart permits us to avoid the detection of both muon and local charged particles flashes in the course of observations. • Radial velocities andDDO, BV photometry of Henry Draper G5-M stars near the North Galactic Pole Radial velocities are given for some 900 stars within 15‡ of the North Galactic Pole, including almost all such stars classified G5 or later in theHenry Draper Catalogue. Luminosities, two-dimensional spectral classes, composition indices, and distances are derived for the majority of the sample throughDDO andBV photometry. More than half of the stars are classified as G5-K5 giants: they show a clear relationship between composition and velocity dispersion for the two Galactic componentsV andW, and a less well-defined trend forU. Four abundance groups exhibit characteristics which imply association with, respectively, the thick disk, old thin disk, young thin disk, and Roman’s “4150” group. The sample is contained within l kpc of the Galactic plane, and no trends with distance are evident. • # Journal of Astrophysics and Astronomy Current Issue Volume 40 \| Issue 5 October 2019 • # Continuous Article Publication Posted on January 27, 2016 Since January 2016, the Journal of Astrophysics and Astronomy has moved to Continuous Article Publishing (CAP) mode. This means that each accepted article is being published immediately online with DOI and article citation ID with starting page number 1. Articles are also visible in Web of Science immediately. All these have helped shorten the publication time and have improved the visibility of the articles. • # Editorial Note on Continuous Article Publication Posted on July 25, 2019	{"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": 0.8442397713661194, "perplexity": 2349.0993792383706}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570987836295.98/warc/CC-MAIN-20191023201520-20191023225020-00203.warc.gz"}
https://www.physicsforums.com/threads/string-interactions.37306/	String interactions 1. Jul 29, 2004 kurious Perturbation theory provides good answers as long as the contributions get smaller and smaller as we go to higher and higher orders. Then we only need to compute the first few diagrams to get accurate results. Is it possible for a perturbation series to get bigger then smaller then bigger-it wouldn't have a finite sum unless some terms were negative, but could such a perturbation expansion be useful in physics? 2. Jul 29, 2004 marlon A perturbation theory as you envision it cannot come from expansions in coupling-constants. So my answer is no. Remember that the contributions will always get smaller and smaller in higher orders, this is exactly what the expanion is about. The criterium for deciding whether one can or cannot use expanions is the fact that the coupling constant must be relativly small, just like the conditions needed for calculation a Taylor-series... The way you suggest a finit theory through an alternating expansion is useless because the effects per order would cancel out eachother. Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook Have something to add? Similar Discussions: String interactions	{"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": 0.8253702521324158, "perplexity": 907.0640457988056}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170614.88/warc/CC-MAIN-20170219104610-00007-ip-10-171-10-108.ec2.internal.warc.gz"}
https://www.neetprep.com/question/53902-simple-pendulum-suspended-roof-trolley-moves-ahorizontal-direction-acceleration-time-period-given-Tlg-equal-toa-gb-gac-gadga?courseId=8	• Subject: ... • Topic: ... A simple pendulum is suspended from the roof of a trolley which moves in a horizontal direction with an acceleration a, then the time period is given by $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\mathrm{g}\text{'}}}$, where is equal to (a) g (b) g-a (c) g+a (d) $\sqrt{{\mathrm{g}}^{2}+{\mathrm{a}}^{2}}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "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": 0.9888643026351929, "perplexity": 1605.6313168310721}, "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-2019-13/segments/1552912202635.43/warc/CC-MAIN-20190322054710-20190322080710-00220.warc.gz"}
https://brilliant.org/problems/maxwells-demon/	# Maxwell's Demon The Second Law of Thermodynamics states that the entropy of a system cannot be decreased without work being done upon it. However, it is easy to conceive of a demon who could violate this principle. Demon A container of gas molecules at equilibrium is divided into two parts by an insulated wall, with a door that can be opened and closed the "Maxwell's demon". The demon opens the door to allow only the faster than average molecules to flow through to a favored side of the chamber, and only the slower than average molecules to the other side, causing the favored side to gradually heat up while the other side cools down, thus decreasing entropy. Why can't such a system effectively violate the second law? Of course, opening the door does some work but that is far too little compared to what the second law predicts and in theory, the door could be massless, too. Here is a better explanation based on this physics stackexchange post : The demon consists of (at least) two parts: a sensor to detect when particles are coming, and an actuator to actually move the door. For the demon to work correctly, the actuator must act on the current instruction from the sensor, instead of the previous one, so it must forget instructions as soon as a new one comes in. This takes some work: there is some physical system encoding a bit and it will take some energy cost to flip it. This work is given by Landauer's Principle which states that for a circuit at (absolute) temperature T, it takes a minimum of $$k_BT \ln (2)$$ energy to remove one bit of information where $$k_B$$ is the Boltzmann Constant. Calvin is angry because I mentioned him in this sentence, so he wishes to remove it by overwriting all the bits encoding this sentence with 0s in the Brilliant.org servers. What is the minimum amount of work in Joules Brilliant.org needs to do this? If your the energy is $$n$$, enter the answer as $$10^{18} n$$. Details and Assumptions • The Operating Temperature of the server is $$70^{\circ} \text{C}$$. • The Boltzmann constant is $$1.3806503 \times 10^{-23} \text{m}^2 \,\text{kg}\, \text{s}^{-2} \,\text{K}^{-1}$$ • All he wishes to is overwrite that sentence only. However, that includes all the characters from $$C$$ to $$.$$ (the one after the s) inclusive. • He does not want to break the octet stream so he overwrites all the bits in each byte regardless of their current state. ×	{"extraction_info": {"found_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": 0.8696383237838745, "perplexity": 487.41578206569807}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1484560281649.59/warc/CC-MAIN-20170116095121-00341-ip-10-171-10-70.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/simple-free-body-force-diagram-friction-question.380097/	# Simple Free Body Force Diagram + Friction Question • #1 158 0 I just want to check if I am doing the right thing with this force diagram. Forgetting the force of friction for the moment, if I know that the body is accelerating at 6.34ms^-2 with a mass of 6000kg; and the opposing force of mgsin3 is 3077.4N, then would the I have to add that to the acceleration force? Therefore Force acceleration would be 38040 + 3077.4 = 41117.4N... My actual question is about friction. Do I have sufficient values to calculate the Force of friction or the friction co-efficient? F of Normal = mu x N #### Attachments • Force.jpg 12.4 KB · Views: 374 • #2 PhanthomJay Homework Helper Gold Member 7,167 507 I just want to check if I am doing the right thing with this force diagram. Forgetting the force of friction for the moment, if I know that the body is accelerating at 6.34ms^-2 with a mass of 6000kg; and the opposing force of mgsin3 is 3077.4N, then would the I have to add that to the acceleration force? Therefore Force acceleration would be 38040 + 3077.4 = 41117.4N... My actual question is about friction. Do I have sufficient values to calculate the Force of friction or the friction co-efficient? F of Normal = mu x N Your terminology for 'acceleration force is a bit unorthodox. It is most always best to identify all real forces acting on a body, determine the NET of those forces in the x and y direction, and apply newton's laws in that direction (F_net =ma, which, for the x direction, is F_x_net = 38040 ). So what are all the forces acting in the x direction (including friction)? Do you have enough info to solve for the friction force? • #3 158 0 Your terminology for 'acceleration force is a bit unorthodox. It is most always best to identify all real forces acting on a body, determine the NET of those forces in the x and y direction, and apply newton's laws in that direction (F_net =ma, which, for the x direction, is F_x_net = 38040 ). So what are all the forces acting in the x direction (including friction)? Do you have enough info to solve for the friction force? I thought F_x_net would be 41117.4N. Um, I don't think so. • #4 PhanthomJay Homework Helper Gold Member 7,167 507 I thought F_x_net would be 41117.4N. Um, I don't think so. No, and you are looking at 2 cases, with and without friction, so let's not confuse the two. When you look at the problem as if it were written without friction, the net force is 38040, which is made up of the applied pulling force acting up the incline, and the weight component acting down the incline. Thus F_net =ma = 38040 F_applied - mgsin3 = 38040 F_applied - 3077 = 38040 F_applied = 41117 N which i think is what you were trying to say before; but you were missing the concept of an applied force (by a person, or machine, etc.) required up the plane inorder to overcome gravity and accelearte the block. Now for the real problem with friction, use the same approach and respond to your own question> do you have enough info to solve for the friction force? • #5 158 0 No, and you are looking at 2 cases, with and without friction, so let's not confuse the two. When you look at the problem as if it were written without friction, the net force is 38040, which is made up of the applied pulling force acting up the incline, and the weight component acting down the incline. Thus F_net =ma = 38040 F_applied - mgsin3 = 38040 F_applied - 3077 = 38040 F_applied = 41117 N which i think is what you were trying to say before; but you were missing the concept of an applied force (by a person, or machine, etc.) required up the plane inorder to overcome gravity and accelearte the block. Now for the real problem with friction, use the same approach and respond to your own question> do you have enough info to solve for the friction force? F_net = 38040 = ma F_applied - mgsin3 - F_friction = 38040 F_applied - F_friction = 41117N F_applied - F_normalmu = 41117N I have two unknown variables now. • #6 PhanthomJay Homework Helper Gold Member 7,167 507 F_net = 38040 = ma F_applied - mgsin3 - F_friction = 38040 F_applied - F_friction = 41117N F_applied - F_normalmu = 41117N I have two unknown variables now. yes, good. So can you solve for the friction force or not?? • #7 158 0 yes, good. So can you solve for the friction force or not?? I would say no. • #8 PhanthomJay Homework Helper Gold Member 7,167 507 I would say no. And so would I • Last Post Replies 2 Views 7K • Last Post Replies 3 Views 6K • Last Post Replies 5 Views 2K • Last Post Replies 10 Views 3K • Last Post Replies 1 Views 1K • Last Post Replies 13 Views 2K • Last Post Replies 4 Views 1K • Last Post Replies 4 Views 689 • Last Post Replies 0 Views 2K • Last Post Replies 6 Views 6K	{"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": 0.8697595596313477, "perplexity": 1127.9727129272997}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488259200.84/warc/CC-MAIN-20210620235118-20210621025118-00294.warc.gz"}
http://www.reference.com/browse/Covering+group	Definitions # Covering group In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p : GH is a continuous group homomorphism. The map p is called the covering homomorphism. ## Properties Let G be a covering group of H. The kernel K of the covering homomorphism is just the fiber over the identity in H and is a discrete normal subgroup of G. The kernel K is closed in G if and only if G is Hausdorff (and if and only if H is Hausdorff). Going in the other direction, if G is any topological group and K is a discrete normal subgroup of G then the quotient map p : GG/K is a covering homomorphism. If G is connected then K, being a discrete normal subgroup, necessarily lies in the center of G and is therefore abelian. In this case, the center of H = G/K is given by $Z\left(H\right) cong Z\left(G\right)/K.$ As with all covering spaces, the fundamental group of G injects into the fundamental group of H. If G is path-connected then the quotient group $pi_1\left(H\right)/pi_1\left(G\right)$ is isomorphic to K. Since the fundamental group of a topological group is always abelian, every covering group is a normal covering space. The group K acts simply transitively on the fibers (which are just left cosets) by right multiplication. The group G is then a principal K-bundle over H. If G is a covering group of H then the groups G and H are locally isomorphic. Moreover, given any two connected locally isomorphic groups H1 and H2, there exists a topological group G with discrete normal subgroups K1 and K2 such that H1 is isomorphic to G/K1 and H2 is isomorphic to G/K2. ## Group structure on a covering space Let H be a topological group and let G be a covering space of H. If G and H are both path-connected and locally path-connected, then for any choice of element e* in the fiber over eH, there exists a unique topological group structure on G, with e* as the identity, for which the covering map p : GH is a homomorphism. The construction is as follows. Let a and b be elements of G and let f and g be paths in G starting at e* and terminating at a and b respectively. Define a path h : IH by h(t) = p(f(t))p(g(t)). By the path-lifting property of covering spaces there is a unique lift of h to G with initial point e. The product ab is defined as the endpoint of this path. By construction we have p(ab) = p(a)p(b). One must show that this definition is independent of the choice of paths f and g, and also that the group operations are continuous. ## Universal covering group If H is a path-connected, locally path-connected, and semilocally simply connected group then it has a universal cover. By the previous construction the universal cover can be made into a topological group with the covering map a continuous homomorphism. This group is called the universal covering group of H. There is also a more direct construction which we give below. Let PH be the path group of H. That is, PH is the space of paths in H based at the identity together with the compact-open topology. The product of paths is given by pointwise multiplication, i.e. (fg)(t) = f(t)g(t). This gives PH the structure of a topological group. There is a natural group homomorphism PHH which sends each path to its endpoint. The universal cover of H is given as the quotient of PH by the normal subgroup of null-homotopic loops. The projection PHH descends to the quotient giving the covering map. One can show that the universal cover is simply connected and the kernel is just the fundamental group of H. That is, we have a short exact sequence $1to pi_1\left(H\right) to tilde H to H to 1$ where $tilde H$ is the universal cover of H. Concretely, the universal covering group of H is the space of homotopy classes of paths in H with pointwise multiplication of paths. The covering map sends each path class to its endpoint. ## Lie groups The above definitions and constructions all apply to the special case of Lie groups. In particular, every covering of a manifold is a manifold, and the covering homomorphism becomes a smooth map. Likewise, given any discrete normal subgroup of a Lie group the quotient group is a Lie group and the quotient map is a covering homomorphism. Two Lie groups are locally isomorphic if and only if the their Lie algebras are isomorphic. This implies that a homomorphism φ : GH of Lie groups is a covering homomorphism if and only if the induced map on Lie algebras $phi_ : mathfrak g to mathfrak h$ is an isomorphism. Since for every Lie algebra $mathfrak g$ there is a unique simply connected Lie group G with Lie algebra $mathfrak g$, from this follows that the universal convering group of a connected Lie group H is the (unique) simply connected Lie group G having the same Lie algebra as H. ## Examples • The universal covering group of the circle group T is the additive group of real numbers R with the covering homomorphism given by the exponential function exp: RT. The kernel of the exponential map is isomorphic to Z. • For any integer n we have a covering group of the circle by itself TT which sends z to zn. The kernel of this homomorphism is the cyclic group consisting of the nth roots of unity. • The rotation group SO(3) has as a universal cover the group SU(2) which is isomorphic to the group of unit quaternions Sp(1). This is a double cover since the kernel has order 2. • The unitary group U(n) is covered by the compact group T × SU(n) with the covering homomorphism given by p(z, A) = zA. The universal cover is just R × SU(n). • The special orthogonal group SO(n) has a double cover called the spin group Spin(n). For n ≥ 3, the spin group is the universal cover of SO(n). • For n ≥ 2, the universal cover of the special linear group SL(n, R) is not a matrix group (i.e. it has no faithful finite-dimensional representations). ## References • Pontryagin, Lev S. (1986). Topological Groups. 3rd ed., New York: Gordon and Breach Science Publishers. ISBN 2-88124-133-6. Search another word or see Covering groupon Dictionary \| Thesaurus \|Spanish	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "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": 0.9334508776664734, "perplexity": 214.39615029674096}, "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-2014-49/segments/1416931010149.53/warc/CC-MAIN-20141125155650-00151-ip-10-235-23-156.ec2.internal.warc.gz"}
http://cognet.mit.edu/journal/10.1162/neco.1995.7.2.270	## Neural Computation March 1995, Vol. 7, No. 2, Pages 270-279 (doi: 10.1162/neco.1995.7.2.270) © 1995 Massachusetts Institute of Technology A Counterexample to Temporal Differences Learning Article PDF (402.61 KB) Abstract Sutton's TD(λ) method aims to provide a representation of the cost function in an absorbing Markov chain with transition costs. A simple example is given where the representation obtained depends on λ. For λ = 1 the representation is optimal with respect to a least-squares error criterion, but as λ decreases toward 0 the representation becomes progressively worse and, in some cases, very poor. The example suggests a need to understand better the circumstances under which TD(0) and Q-learning obtain satisfactory neural network-based compact representations of the cost function. A variation of TD(0) is also given, which performs better on the example.	{"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": 0.8780217170715332, "perplexity": 923.6289961824823}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337853.66/warc/CC-MAIN-20221006155805-20221006185805-00243.warc.gz"}
http://stackexchange.com/users/1300508/enjoys-math?tab=activity	# Enjoys Math 1d accepted Can an Earley Parser be made into a fuzzy parser similar to the Levenshtein Automata Algo for DFA? 2d asked Can an Earley Parser be made into a fuzzy parser similar to the Levenshtein Automata Algo for DFA? Jul 22 accepted How do I rate smoothness of discretely sampled data? (Picture!!!) Jul 21 revised How do I rate smoothness of discretely sampled data? (Picture!!!)edited title Jul 21 asked How do I rate smoothness of discretely sampled data? (Picture!!!) Jul 8 asked What are the expressive consequences of all references in a program being bi-directional through a user-defined type? Jun 29 accepted Weibel definition 1.4.1. understanding the indexes on splitting maps Jun 29 accepted Question about an inverse limit. Jun 29 comment Question about an inverse limit.@AmitaiYuval I'm not sure what they look like. Jun 29 asked Question about an inverse limit. Jun 29 revised Weibel definition 1.4.1. understanding the indexes on splitting mapsadded 115 characters in body Jun 29 asked Weibel definition 1.4.1. understanding the indexes on splitting maps Jun 29 comment These maps from the components into a directed system are injective when the directed system maps are.Ah, same mistake I made last time. Forgetting what sets the arguments are in. Jun 29 accepted These maps from the components into a directed system are injective when the directed system maps are. Jun 29 asked These maps from the components into a directed system are injective when the directed system maps are. Jun 28 comment Proving that the direct limit of a directed system is an equivalence relation.I've got it now. Using the composition condition you can show that $\exists, i,j, \ell \leq k$ (by directedness, you can show both statements under one $k$.) such that $p_{ik}(a) = p_{jk}(b) = p_{\ell k}(c)$ done. Thanks for teaching! Jun 28 comment Proving that the direct limit of a directed system is an equivalence relation.In your first statement you made a mistake of re-using the variable $j$, we know that some $j'$ exists but not that it equals $j$ Jun 28 comment Proving that the direct limit of a directed system is an equivalence relation.I have to ruminate for a second on this at my dry-erase board. Jun 28 comment Proving that the direct limit of a directed system is an equivalence relation.I'm having trouble piecing it together. I have $a \sim b \iff \exists i,j \leq k$ with $p_{ik}(a) = p_{jk}(b)$, $b \sim c \iff \exists i',j'\leq k'$ with $p_{i'k'}(b) = p_{j'k'}(c)$ and $\forall i,j \in I \exists k$ with $i,j \leq k$, and $p_{jk} \circ p_{ij} = p_{ik}$ whenever $i \leq j \leq k$. Still thinking on it though. Jun 28 comment Proving that the direct limit of a directed system is an equivalence relation.Ah, the book even states this.	{"extraction_info": {"found_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": 0.86569744348526, "perplexity": 895.3108450553759}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1438042989234.2/warc/CC-MAIN-20150728002309-00192-ip-10-236-191-2.ec2.internal.warc.gz"}
https://physics.stackexchange.com/questions/245675/why-do-we-need-connections-if-we-have-the-lie-derivative?noredirect=1	# Why do we need connections, if we have the Lie derivative? When I learned about the covariant derivative, it was motivated as a way of defining a good differentiation operation on tensors. To do this, we had to define a connection on the manifold, which was a substantial extra piece of structure. However, the Lie derivative requires no connection at all; it just requires a vector field $V^\mu$ defined on the manifold. In particular, since we've already chosen coordinates, we can define the Lie derivative in any direction $n^\mu$ by using the vector field $V = n^\mu \partial_\mu$, which requires zero extra structure. Then $\mathcal{L}_V$ seems to be a perfectly good replacement for $n^\mu \nabla_\mu$. At the very least, it does everything that books say the covariant derivative was meant to do. Ignoring all the stuff the covariant derivative ends up getting used for, I don't know why we would have introduced it in the first place. What good properties does $n^\mu \nabla_\mu$ have that $\mathcal{L}_{n^\mu \partial_\mu}$ does not? • Mar 26 '16 at 18:51 • Covariant derivative is scalar-field linear in one entry Lie derivative is not in both entries This feature strongly distinguishes between the two operators. The choice depends on which feature you need. Mar 26 '16 at 18:56 • The Lie derivative does not give rise to a notion of parallel transport. It is how one vector field changes along another, it doesn't connect the tangent spaces at different points. Also, this is a pure differential geometry, not a physics question. Mar 26 '16 at 18:56 • @ACuriousMind Sure, but it gives some notion of transport, by pullback/push forward. Why is that not good enough? Mar 26 '16 at 19:06 • @knzhou But that transport is highly nonunique: it depends on the choice of $V$. The choice of a connection $\nabla$ does not depend on some arbitrary vector field, and in the Riemannian case, one can select a unique $\nabla$ with desirable properties. Mar 27 '16 at 16:14 One distinctive feature is that the connection $\nabla$ has the tensor field property in its first entry $\nabla_{fX}=f\nabla_{X}$, while the Lie derivative doesn't. We have ${\cal L}_{fX}\neq f{\cal L}_{X}$ generically. There are already manifestly covariant explanations on Math.SE and Mathoverflow.SE. Let's for simplicity consider a local coordinate patch $U$ with coordinates $x^{\mu}$. The tensor field property implies that the covariant derivative $\nabla_{X}=X^{\mu}\nabla_{\mu}$ is completely determine by some directional derivatives $\nabla_{\mu}$. Not so for the Lie derivative. Say that we are given a metric tensor field $g$ and some tensor field $T$. Assume for simplicity that the tensor field components $g_{\mu\nu}$ and $T^{\mu_1 \ldots \mu_r}{}_{\nu_1 \ldots \nu_s}$ in the given local coordinate system are constant, i.e. $x$-independent. Let $\nabla$ be the Levi-Civita connection. Then $\nabla_Xg=0$ and $\nabla_XT=0$ as we would expect from a directional derivative (since after all, the tensor field components are constant in one coordinate system). But ${\cal L}_{X}g\neq 0$ and ${\cal L}_{X}T\neq 0$ generically.	{"extraction_info": {"found_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": 0.9507349133491516, "perplexity": 225.60284271502783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964363337.27/warc/CC-MAIN-20211207075308-20211207105308-00571.warc.gz"}
https://www.iue.tuwien.ac.at/phd/pyka/node54.html	Prev: 6.1.1 Transport Kinetics Up: 6.1 Modeling of Low-Pressure Next: 6.1.3 Reactor Geometry and ## 6.1.2 Distribution Functions The distribution of the incoming particles plays the most important role for the profile evolution during low-pressure processes. Since the distributions are hardly accessible for measurements, assumptions and approximations have to be made. As a first step analytical expressions are used for a direct approximation of the incident particle distributions. However, the accuracy of such an approach is limited. More detailed information about the incidence conditions for the particles can be obtained when the complete reactor system is included into the considerations. Analytical derivations as well as Monte Carlo simulations can be used to correlate the behavior of the impinging particles with the emission characteristics of the target and with the transport and collision processes in the reactor chamber. The thus obtained distributions represent the real conditions prevailing at the substrate surface much closer. Fig. 6.2 outlines a sputter reactor and polar plots of the local emission characteristics at the target as well as the incoming distributions at the wafer surface. By means of a sputtering deposition process as a prototypical example the relation between emitted and incoming distributions will be discussed. Starting point for the considerations is the experimentally determined target erosion profile. Semiconductor manufacturing routinely uses rotating magnet sputter systems, leading to circular racetrack grooves in the sputter target which means, that physical sputtering is not conformal across the sputter target but shows radially symmetric emission maxima with certain distance to the center of the target. The depth of the groove corresponds to the emission intensity, indicated in Fig. 6.2 by the variable size of the circles denoting the emitted distribution. Moreover the angular distribution of the emitted particles is of significant importance. For most of the commonly used semiconductor materials the emission profile can be approximated with cosine [48][58] functions of the emission angle measured with respect to the surface normal. Still, the crystallographic structure of some metals like aluminium or titanium leads to sputtering conditions, where the emission maximum is not normal to the surface [51], which can be described with sub-cosine functions [40] of the form (6.2) with . In addition to the empirical approaches, Monte Carlo simulations of sputter atom transport [47] include atomistic models of particle emission taking into account the surface roughness approximated by fractal geometry [59][60]. Once the emission characteristic is known, investigation about the path of the particles resulting from collisions on their way from the sputter target to the substrate is the next step. Due to the inherent stochastic nature of the collision processes they can be simulated best with Monte Carlo methods. Monte Carlo simulations trace the pathways of the particles including various models for the change in direction when particles collide. The models for the particle-particle interactions range from simple two-body-interactions to Lennard-Jones, Born-Mayer, or Abrahamson potentials [47]. Additionally, Monte Carlo methods model the acceleration of charged particles in electric fields as used for target bias in order to increase the directionality of etching and deposition processes [58]. The final results of comprehensive Monte Carlo simulations of particle transport in sputter reactors are angular and energy dependent distributions of the particles arriving at the wafer surface [13][37][46][71][72]. In [12] a fitting model was introduced as faster alternative to the above mentioned MC methods. It determines the parameter of a distribution function by comparing simulated and experimentally obtained film profiles. We have decided for a similar approximation model, but instead of fitting the final profiles the initially implemented cosine and sub-cosine functions have been extended with expressions which are able to fit the angular distributions resulting from Monte Carlo simulations. As an example the exponential function (6.3) was used to fit the angular distributions resulting from MC simulations of magnetron sputtering particle transport [46]. Fig. 6.3 shows angular particle distributions resulting from the MC simulations for reactor background pressures of 0.5, 1.5, and 4.5mTorr. The increasing pressure diminishes the mean free path of the particles. Hence the particles undergo more collisions and the distribution gets broadened. This broadening was fitted with (6.3) by an increasing angle of maximum particle incidence. Fig. 6.3 compares the MC results with fitting functions for maximum angles of 10, 11.5, and 13.5. Three interesting aspects of the resulting flux distributions have to be mentioned. Firstly, there will be a certain amount of particles arriving at angles larger than which is the limiting angle for particle incidence if only direct transport from the target to the wafer is permitted. Since pressures in the region of 1mTorr do not allow the assumption of collisionless transport, it is obvious, that scattered particles account for this lateral fraction of particles. With increasing pressure the number of collisions per particle increases, the distribution broadens and the angle of maximum particle incidence increases. This is in excellent agreement with the MC results from [46]. The MC results also reveal that the particles typically undergo 1 to 3 collisions until the arrive at the wafer. Secondly, three-dimensional simulations obviously require three-dimensional functions for the particle distributions. According to the circular racetrack observed for the target erosion profiles it is reasonable to assume radially symmetrical distribution functions which are independent on the polar angle . The distribution function from (6.3) thus reads as (6.4) Finally, the MC results from Fig. 6.3 indicate that there is no particle incidence perpendicular to the wafer. This is understandable with the knowledge about the geometric configuration used for the Monte Carlo simulations. The center of the wafer is aligned with the center of the sputter target, from where almost no particles are emitted downwards to the wafer. It is therefore obvious, that the situation changes for peripheral positions on the wafer, located, e.g., directly below the emission maximum. At these positions a significant amount of particles with normal incidence will be observable. It is clear that the variations in the particle distributions as indicated by the polar plots of the emitted and impinging distributions in Fig. 6.2 strongly influence the local profile evolution. A method for the derivations of the non-uniformities in the particle distributions from the position on the wafer will be demonstrated in the next section. Prev: 6.1.1 Transport Kinetics Up: 6.1 Modeling of Low-Pressure Next: 6.1.3 Reactor Geometry and W. Pyka: Feature Scale Modeling for Etching and Deposition Processes in Semiconductor Manufacturing	{"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": 0.923105001449585, "perplexity": 887.5834237351062}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738723.55/warc/CC-MAIN-20200810235513-20200811025513-00049.warc.gz"}
http://softwareabroad.com/relative-error/algebra-relative-error-problems.php	Home > Relative Error > Algebra Relative Error Problems # Algebra Relative Error Problems ## Contents New York: Dover, p.14, 1972. What are the absolute and relative errors of the approximation 22/7 of π? (0.0013 and 0.00040) 2. Which length could be the greatest possible value for the side of the square in centimeters? b.) the relative error in the measured length of the field. useful reference Simply substitute the equation for Absolute Error in for the actual number. If the student uses this measurement to compute the area of a circle with this radius, what is the student's percent of error on the area computation, to the nearest tenth Leave the relative error in fraction form, complete the division to render it in decimal form, or multiply the resulting decimal form by 100 to render your answer as a percentage. MrsRZimmerman 2,212 views 10:36 Relative True Error - Duration: 7:39. http://www.regentsprep.org/regents/math/algebra/am3/LError.htm ## How To Find Relative Error In Algebra The error in measurement is a mathematical way to show the uncertainty in the measurement. Rating is available when the video has been rented. But, if you tried to measure something that was 120 feet long and only missed by 6 inches, the relative error would be much smaller -- even though the value of davenport1947 16,357 views 13:48 what are Absolute,,Relative and Percentage error - Duration: 5:24. EatScience 1,261 views 9:49 Relative Approximate Error - Duration: 8:45. A measurement is taken with a metric ruler with a precision of 0.1 cm. Relative Error Math Your absolute error is 20 - 18 = 2 feet (60.96 centimeters).[3] 2 Alternatively, when measuring something, assume the absolute error to be the smallest unit of measurement at your disposal. Solution: Given: The measured value of metal ball xo = 3.97 cm The true value of ball x = 4 cm Absolute error $\Delta$ x = True value - Measured Relative Error Formula Algebra The greatest possible error when measuring is considered to be one half of that measuring unit. Tolerance intervals: Error in measurement may be represented by a tolerance interval (margin of error). The width of this animal's paw print is 3 inches to the nearest inch. Background None. Relative Error Calculator Find the percent of error. No scientific study is ever perfectly error free -- even Nobel Prize winning papers and discoveries have a margin or error attached. Choose: 1 in 1.5 in 1.9 in 2 in Explanation The lower limit will be 1.7 - 0.2 = 1.5 10. ## Relative Error Formula Algebra numericalmethodsguy 5,334 views 7:39 Accuracy vs. Basically, this is the most precise, common measurement to come up with, usually for common equations or reactions. How To Find Relative Error In Algebra The percent of error is found by multiplying the relative error by 100%. Relative Error Worksheet Algebra Sign in to make your opinion count. A box is 15 inches long, 12 inches wide and 8 inches high when the dimensions are rounded to the nearest inch. see here Sign in to make your opinion count. Back to Top The relative error formula is given byRelative error =$\frac{Absolute\ error}{Value\ of\ thing\ to\ be\ measured}$ = $\frac{\Delta\ x}{x}$.In terms of percentage it is expressed asRelative error = \$\frac{\Delta\ Many scientific tools, like precision droppers and measurement equipment, often has absolute error labeled on the sides as "+/- ____ " 3 Always add the appropriate units. Relative Error Problems Calculus davenport1947 16,357 views 13:48 11.1 Determine the uncertainties in results [SL IB Chemistry] - Duration: 8:30. This is your absolute error![2] Example: You want to know how accurately you estimate distances by pacing them off. The best way to learn how to calculate error is to go ahead and calculate it. this page Assuming the measurements are off by 1%, find to the nearest cubic inch, the largest possible volume of the box. Greatest Possible Error: Because no measurement is exact, measurements are always made to the "nearest something", whether it is stated or not. Relative Error Definition Math Accuracy is a measure of how close the result of the measurement comes to the "true", "actual", or "accepted" value. (How close is your answer to the accepted value?) Tolerance is The area as calculated from measuring is 19.4 x 11.2 = 217.28 sq.cm.	{"extraction_info": {"found_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": 0.8919478058815002, "perplexity": 1134.8469257191987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886739.5/warc/CC-MAIN-20180116204303-20180116224303-00762.warc.gz"}
https://api.philpapers.org/rec/LARAVF	# An variation for one souslin tree Journal of Symbolic Logic 64 (1):81-98 (1999) # Abstract We present a variation of the forcing S max as presented in Woodin [4]. Our forcing is a P max -style construction where each model condition selects one Souslin tree. In the extension there is a Souslin tree T G which is the direct limit of the selected Souslin trees in the models of the generic. In some sense, the generic extension is a maximal model of "there exists a minimal Souslin tree," with T G being this minimal tree. In particular, in the extension this Souslin tree has the property that forcing with it gives a model of Souslin's Hypothesis ## PhilArchive Upload a copy of this work Papers currently archived: 76,168 Setup an account with your affiliations in order to access resources via your University's proxy server # Similar books and articles An $mathbb{S}_{max}$ Variation for One Souslin Tree.Paul Larson - 1999 - Journal of Symbolic Logic 64 (1):81-98. Gap structure after forcing with a coherent Souslin tree.Carlos Martinez-Ranero - 2013 - Archive for Mathematical Logic 52 (3-4):435-447. Chain homogeneous Souslin algebras.Gido Scharfenberger-Fabian - 2011 - Mathematical Logic Quarterly 57 (6):591-610. Souslin algebra embeddings.Gido Scharfenberger-Fabian - 2011 - Archive for Mathematical Logic 50 (1-2):75-113. Club degrees of rigidity and almost Kurepa trees.Gunter Fuchs - 2013 - Archive for Mathematical Logic 52 (1-2):47-66. Souslin forcing.Jaime I. Ihoda & Saharon Shelah - 1988 - Journal of Symbolic Logic 53 (4):1188-1207. On iterating semiproper preorders.Tadatoshi Miyamoto - 2002 - Journal of Symbolic Logic 67 (4):1431-1468. μ-complete Souslin trees on μ+.Menachem Kojman & Saharon Shelah - 1993 - Archive for Mathematical Logic 32 (3):195-201. On guessing generalized clubs at the successors of regulars.Assaf Rinot - 2011 - Annals of Pure and Applied Logic 162 (7):566-577. Creatures on ω 1 and weak diamonds.Heike Mildenberger - 2009 - Journal of Symbolic Logic 74 (1):1-16. Adding Closed Unbounded Subsets of ω₂ with Finite Forcing.William J. Mitchell - 2005 - Notre Dame Journal of Formal Logic 46 (3):357-371. Degrees of rigidity for Souslin trees.Gunter Fuchs & Joel David Hamkins - 2009 - Journal of Symbolic Logic 74 (2):423-454. Terminal notions in set theory.Jindřich Zapletal - 2001 - Annals of Pure and Applied Logic 109 (1-2):89-116. 2009-01-28 38 (#309,085) 6 months 1 (#448,551)	{"extraction_info": {"found_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": 0.8322533965110779, "perplexity": 2318.6783803159165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764499654.54/warc/CC-MAIN-20230128184907-20230128214907-00678.warc.gz"}
https://www.physicsoverflow.org/19518/does-this-statistical-inference-make-any-sense	# Does this statistical inference make any sense? + 3 like - 0 dislike 669 views I am trying to measure a quantity from two data sets which were from identical experiments except that the data were taken at different times. The measurement from the first data set gives $a\pm\sigma_a$; the second data set gives $b\pm\sigma_b$ where $\sigma_a \sim \sigma_b \sim \sigma$ is the statistical uncertainty. But it turned out that $a-b > 3\sigma$, which means t-test shows that the two results are highly unlikely to be from the same distribution. If I combine the two results in the usual way by weighting them by the inverse of the squares of their uncertainties and calculate the final statistical uncertainty likewise, I will get $(a+b)/2 \pm \sigma/\sqrt{2}$. But $(a+b)/2 \pm \sigma/\sqrt{2}$ is not a satisfactory interpretation of my measurement. From a Bayesian statistics point of view, the result from the first data can be treated as the prior probability distribution of the parameter being measured. Updating the prior probability distribution by the posterior probability distribution, i.e. from the second data set, gives me entirely different final probability distribution of the parameter than the one described by $(a+b)/2 \pm \sigma/\sqrt{2}$ in the usual Frequentist 's way. We are exhausted trying to find the systematic variation between the two data sets which were supposed to give us the same result. What is the correct inference of the final result in both the ways, Bayesian and Frequentist? Or, how to present the final result in the case? Any kind of opinion in the form of comment or answer will be highly appreciated. Thanks. edited Jun 24, 2014 As an experimenter, if I got a 3sigma difference between two supposedly identical initial conditions experiments I would question the "identical" first, i.e. check very carefully what has changed between the two times of taking data. If I could find no conceivable input change I would take a third data set . Playing with probability methods would not be my choice. Maybe the "error" is in the first data set. Dear Ana, it seems like you did not read the complete question. I know what you mean but there is this important part in the question "We are exhausted trying to find the systematic variation between the two data sets". That means we have no alternative data as well. That's why it is an important question to ask in POF. Could you please comment on the whole question. You are saying there is no possibility of getting a third data set by a new experiment and you have to publish? I would treat the two data sets independently with their statistical and systematic errors and publish both values in a similar way that the particle data group presents the values of different experiments , for example , the mass of the Z and let the reader decide. Alternatively and maybe in parallel I would combine the two sets assuming that the 3 sigma is a fluke of statistics, stating clearly the matter . After all three sigma significance resonances have disappeared before. That is why we set the 5 sigma as definitive. ( I am using "resonances" as an example) + 3 like - 0 dislike There are two possibilities: Either (1) you have seen a rare 3-sigma event, or (2) you have underestimated the uncertainties in your experiment. Making an accidental systematic change between the two measurements would count as an example of (2), i.e. a source of uncertainty that you didn't take into account. Experience suggests that (2) is by far the likelier possibility. Ideally you would figure out how and why you underestimated the uncertainty -- what aspect wasn't controlled properly or whatever. But if it's not terribly important and you're in a hurry and you can't do a third and fourth experiment, you can just say there is an additional source of uncertainty $\sigma_{other}$ which accounts for the "unknown unknowns". $a \pm \sigma_a \pm \sigma_{other}$ $b \pm \sigma_b \pm \sigma_{other}$ You can guess $\sigma_{other}$ by setting it to a value that makes the two measurements 1 sigma apart or so (a reasonably probable value). After you do that, you can say that your best guess for the real answer is something like $(a+b)/2 \pm (\sigma_{other}/\sqrt{2})$ (since I gather that $\sigma_{other}$ is the dominant source of uncertainty). But you can't treat that expression too literally, it's just a very rough guess. Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor) Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8235472440719604, "perplexity": 514.3557251782378}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964363418.83/warc/CC-MAIN-20211207201422-20211207231422-00321.warc.gz"}
https://infoscience.epfl.ch/record/63448	Infoscience Journal article # The first-order reliability method of predicting cumulative mass flux in heterogeneous porous formations Previous studies have proposed the first-order reliability method (FORM) as an approach to quantitative stochastic analysis of subsurface transport. Most of these considered only simple analytical models of transport in homogeneous media. Studies that looked at more-complex, heterogeneous systems found FORM to be computationally demanding and were inconclusive as to the accuracy of the method. Here we show that FORM is poorly suited for computing point concentration cumulative distribution functions (cdfs) except in the case of a constant or monotonically increasing solute source. FORM is better equipped to predict transport in terms of the cumulative mass flux across a control surface. As a demonstration, we use FORM to estimate the cumulative mass flux cdf in two- dimensional, random porous media. Adjoint sensitivity theory is employed to minimize the computational burden. In addition, properties of the conductivity covariance and distribution are exploited to improve efficiency. FORM required eight times less CPU time than Monte Carlo simulation to generate the results presented. The accuracy of FORM is found to be minimally affected by the size of the initial solute body and the solute travel distance. However, the accuracy is significantly influenced by the degree of heterogeneity, providing an accurate estimate of the cdf when there is mild heterogeneity (σInK = 0.5) but a less accurate estimate when there is stronger heterogeneity (σInK = 1.0).	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9464618563652039, "perplexity": 811.0918379829517}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818690376.61/warc/CC-MAIN-20170925074036-20170925094036-00226.warc.gz"}
http://www.quantumdiaries.org/author/flip-tanedo/	## Flip Tanedo \| USLHC \| USA ### The Delirium over Beryllium Thursday, August 25th, 2016 This post is cross-posted from ParticleBites. Article: Particle Physics Models for the 17 MeV Anomaly in Beryllium Nuclear Decays Authors: J.L. Feng, B. Fornal, I. Galon, S. Gardner, J. Smolinsky, T. M. P. Tait, F. Tanedo Reference: arXiv:1608.03591 (Submitted to Phys. Rev. D) Also featuring the results from: — Gulyás et al., “A pair spectrometer for measuring multipolarities of energetic nuclear transitions” (description of detector; 1504.00489NIM) — Krasznahorkay et al., “Observation of Anomalous Internal Pair Creation in 8Be: A Possible Indication of a Light, Neutral Boson” (experimental result; 1504.01527PRL version; note PRL version differs from arXiv) — Feng et al., “Protophobic Fifth-Force Interpretation of the Observed Anomaly in 8Be Nuclear Transitions” (phenomenology; 1604.07411; PRL) Editor’s note: the author is a co-author of the paper being highlighted. Recently there’s some press (see links below) regarding early hints of a new particle observed in a nuclear physics experiment. In this bite, we’ll summarize the result that has raised the eyebrows of some physicists, and the hackles of others. ## A crash course on nuclear physics Nuclei are bound states of protons and neutrons. They can have excited states analogous to the excited states of at lowoms, which are bound states of nuclei and electrons. The particular nucleus of interest is beryllium-8, which has four neutrons and four protons, which you may know from the triple alpha process. There are three nuclear states to be aware of: the ground state, the 18.15 MeV excited state, and the 17.64 MeV excited state. Beryllium-8 excited nuclear states. The 18.15 MeV state (red) exhibits an anomaly. Both the 18.15 MeV and 17.64 states decay to the ground through a magnetic, p-wave transition. Image adapted from Savage et al. (1987). Most of the time the excited states fall apart into a lithium-7 nucleus and a proton. But sometimes, these excited states decay into the beryllium-8 ground state by emitting a photon (γ-ray). Even more rarely, these states can decay to the ground state by emitting an electron–positron pair from a virtual photon: this is called internal pair creation and it is these events that exhibit an anomaly. ## The beryllium-8 anomaly Physicists at the Atomki nuclear physics institute in Hungary were studying the nuclear decays of excited beryllium-8 nuclei. The team, led by Attila J. Krasznahorkay, produced beryllium excited states by bombarding a lithium-7 nucleus with protons. Beryllium-8 excited states are prepare by bombarding lithium-7 with protons. The proton beam is tuned to very specific energies so that one can ‘tickle’ specific beryllium excited states. When the protons have around 1.03 MeV of kinetic energy, they excite lithium into the 18.15 MeV beryllium state. This has two important features: 1. Picking the proton energy allows one to only produce a specific excited state so one doesn’t have to worry about contamination from decays of other excited states. 2. Because the 18.15 MeV beryllium nucleus is produced at resonance, one has a very high yield of these excited states. This is very good when looking for very rare decay processes like internal pair creation. What one expects is that most of the electron–positron pairs have small opening angle with a smoothly decreasing number as with larger opening angles. Expected distribution of opening angles for ordinary internal pair creation events. Each line corresponds to nuclear transition that is electric (E) or magenetic (M) with a given orbital quantum number, l. The beryllium transitionsthat we’re interested in are mostly M1. Adapted from Gulyás et al. (1504.00489). Instead, the Atomki team found an excess of events with large electron–positron opening angle. In fact, even more intriguing: the excess occurs around a particular opening angle (140 degrees) and forms a bump. Number of events (dN/dθ) for different electron–positron opening angles and plotted for different excitation energies (Ep). For Ep=1.10 MeV, there is a pronounced bump at 140 degrees which does not appear to be explainable from the ordinary internal pair conversion. This may be suggestive of a new particle. Adapted from Krasznahorkay et al., PRL 116, 042501. Here’s why a bump is particularly interesting: 1. The distribution of ordinary internal pair creation events is smoothly decreasing and so this is very unlikely to produce a bump. 2. Bumps can be signs of new particles: if there is a new, light particle that can facilitate the decay, one would expect a bump at an opening angle that depends on the new particle mass. Schematically, the new particle interpretation looks like this: Schematic of the Atomki experiment and new particle (X) interpretation of the anomalous events. In summary: protons of a specific energy bombard stationary lithium-7 nuclei and excite them to the 18.15 MeV beryllium-8 state. These decay into the beryllium-8 ground state. Some of these decays are mediated by the new X particle, which then decays in to electron–positron pairs of a certain opening angle that are detected in the Atomki pair spectrometer detector. Image from 1608.03591. As an exercise for those with a background in special relativity, one can use the relation (pe+ + pe)2 = mX2 to prove the result: This relates the mass of the proposed new particle, X, to the opening angle θ and the energies E of the electron and positron. The opening angle bump would then be interpreted as a new particle with mass of roughly 17 MeV. To match the observed number of anomalous events, the rate at which the excited beryllium decays via the X boson must be 6×10-6 times the rate at which it goes into a γ-ray. The anomaly has a significance of 6.8σ. This means that it’s highly unlikely to be a statistical fluctuation, as the 750 GeV diphoton bump appears to have been. Indeed, the conservative bet would be some not-understood systematic effect, akin to the 130 GeV Fermi γ-ray line. ## The beryllium that cried wolf? Some physicists are concerned that beryllium may be the ‘boy that cried wolf,’ and point to papers by the late Fokke de Boer as early as 1996 and all the way to 2001. de Boer made strong claims about evidence for a new 10 MeV particle in the internal pair creation decays of the 17.64 MeV beryllium-8 excited state. These claims didn’t pan out, and in fact the instrumentation paper by the Atomki experiment rules out that original anomaly. The proposed evidence for “de Boeron” is shown below: The de Boer claim for a 10 MeV new particle. Left: distribution of opening angles for internal pair creation events in an E1 transition of carbon-12. This transition has similar energy splitting to the beryllium-8 17.64 MeV transition and shows good agreement with the expectations; as shown by the flat “signal – background” on the bottom panel. Right: the same analysis for the M1 internal pair creation events from the 17.64 MeV beryllium-8 states. The “signal – background” now shows a broad excess across all opening angles. Adapted from de Boer et al. PLB 368, 235 (1996). When the Atomki group studied the same 17.64 MeV transition, they found that a key background component—subdominant E1 decays from nearby excited states—dramatically improved the fit and were not included in the original de Boer analysis. This is the last nail in the coffin for the proposed 10 MeV “de Boeron.” However, the Atomki group also highlight how their new anomaly in the 18.15 MeV state behaves differently. Unlike the broad excess in the de Boer result, the new excess is concentrated in a bump. There is no known way in which additional internal pair creation backgrounds can contribute to add a bump in the opening angle distribution; as noted above: all of these distributions are smoothly falling. The Atomki group goes on to suggest that the new particle appears to fit the bill for a dark photon, a reasonably well-motivated copy of the ordinary photon that differs in its overall strength and having a non-zero (17 MeV?) mass. ## Theory part 1: Not a dark photon With the Atomki result was published and peer reviewed in Physics Review Letters, the game was afoot for theorists to understand how it would fit into a theoretical framework like the dark photon. A group from UC Irvine, University of Kentucky, and UC Riverside found that actually, dark photons have a hard time fitting the anomaly simultaneously with other experimental constraints. In the visual language of this recent ParticleBite, the situation was this: It turns out that the minimal model of a dark photon cannot simultaneously explain the Atomki beryllium-8 anomaly without running afoul of other experimental constraints. Image adapted from this ParticleBite. The main reason for this is that a dark photon with mass and interaction strength to fit the beryllium anomaly would necessarily have been seen by the NA48/2 experiment. This experiment looks for dark photons in the decay of neutral pions (π0). These pions typically decay into two photons, but if there’s a 17 MeV dark photon around, some fraction of those decays would go into dark-photon — ordinary-photon pairs. The non-observation of these unique decays rules out the dark photon interpretation. The theorists then decided to “break” the dark photon theory in order to try to make it fit. They generalized the types of interactions that a new photon-like particle, X, could have, allowing protons, for example, to have completely different charges than electrons rather than having exactly opposite charges. Doing this does gross violence to the theoretical consistency of a theory—but they goal was just to see what a new particle interpretation would have to look like. They found that if a new photon-like particle talked to neutrons but not protons—that is, the new force were protophobic—then a theory might hold together. Schematic description of how model-builders “hacked” the dark photon theory to fit both the beryllium anomaly while being consistent with other experiments. This hack isn’t pretty—and indeed, comes at the cost of potentially invalidating the mathematical consistency of the theory—but the exercise demonstrates the target for how to a complete theory might have to behave. Image adapted from this ParticleBite. ## Theory appendix: pion-phobia is protophobia Editor’s note: what follows is for readers with some physics background interested in a technical detail; others may skip this section. How does a new particle that is allergic to protons avoid the neutral pion decay bounds from NA48/2? Pions decay into pairs of photons through the well-known triangle-diagrams of the axial anomaly. The decay into photon–dark-photon pairs proceed through similar diagrams. The goal is then to make sure that these diagrams cancel. A cute way to look at this is to assume that at low energies, the relevant particles running in the loop aren’t quarks, but rather nucleons (protons and neutrons). In fact, since only the proton can talk to the photon, one only needs to consider proton loops. Thus if the new photon-like particle, X, doesn’t talk to protons, then there’s no diagram for the pion to decay into γX. This would be great if the story weren’t completely wrong. Avoiding NA48/2 bounds requires that the new particle, X, is pion-phobic. It turns out that this is equivalent to X being protophobic. The correct way to see this is on the left, making sure that the contribution of up-quark loops cancels the contribution from down-quark loops. A slick (but naively completely wrong) calculation is on the right, arguing that effectively only protons run in the loop. The correct way of seeing this is to treat the pion as a quantum superposition of an up–anti-up and down–anti-down bound state, and then make sure that the X charges are such that the contributions of the two states cancel. The resulting charges turn out to be protophobic. The fact that the “proton-in-the-loop” picture gives the correct charges, however, is no coincidence. Indeed, this was precisely how Jack Steinberger calculated the correct pion decay rate. The key here is whether one treats the quarks/nucleons linearly or non-linearly in chiral perturbation theory. The relation to the Wess-Zumino-Witten term—which is what really encodes the low-energy interaction—is carefully explained in chapter 6a.2 of Georgi’s revised Weak Interactions. ## Theory part 2: Not a spin-0 particle The above considerations focus on a new particle with the same spin and parity as a photon (spin-1, parity odd). Another result of the UCI study was a systematic exploration of other possibilities. They found that the beryllium anomaly could not be consistent with spin-0 particles. For a parity-odd, spin-0 particle, one cannot simultaneously conserve angular momentum and parity in the decay of the excited beryllium-8 state. (Parity violating effects are negligible at these energies.) Parity and angular momentum conservation prohibit a “dark Higgs” (parity even scalar) from mediating the anomaly. For a parity-odd pseudoscalar, the bounds on axion-like particles at 20 MeV suffocate any reasonable coupling. Measured in terms of the pseudoscalar–photon–photon coupling (which has dimensions of inverse GeV), this interaction is ruled out down to the inverse Planck scale. Bounds on axion-like particles exclude a 20 MeV pseudoscalar with couplings to photons stronger than the inverse Planck scale. Adapted from 1205.2671 and 1512.03069. • Dark Z bosons, cousins of the dark photon with spin-1 but indeterminate parity. This is very constrained by atomic parity violation. • Axial vectors, spin-1 bosons with positive parity. These remain a theoretical possibility, though their unknown nuclear matrix elements make it difficult to write a predictive model. (See section II.D of 1608.03591.) ## Theory part 3: Nuclear input The plot thickens when once also includes results from nuclear theory. Recent results from Saori Pastore, Bob Wiringa, and collaborators point out a very important fact: the 18.15 MeV beryllium-8 state that exhibits the anomaly and the 17.64 MeV state which does not are actually closely related. Recall (e.g. from the first figure at the top) that both the 18.15 MeV and 17.64 MeV states are both spin-1 and parity-even. They differ in mass and in one other key aspect: the 17.64 MeV state carries isospin charge, while the 18.15 MeV state and ground state do not. Isospin is the nuclear symmetry that relates protons to neutrons and is tied to electroweak symmetry in the full Standard Model. At nuclear energies, isospin charge is approximately conserved. This brings us to the following puzzle: If the new particle has mass around 17 MeV, why do we see its effects in the 18.15 MeV state but not the 17.64 MeV state? Naively, if the new particle emitted, X, carries no isospin charge, then isospin conservation prohibits the decay of the 17.64 MeV state through emission of an X boson. However, the Pastore et al. result tells us that actually, the isospin-neutral and isospin-charged states mix quantum mechanically so that the observed 18.15 and 17.64 MeV states are mixtures of iso-neutral and iso-charged states. In fact, this mixing is actually rather large, with mixing angle of around 10 degrees! The result of this is that one cannot invoke isospin conservation to explain the non-observation of an anomaly in the 17.64 MeV state. In fact, the only way to avoid this is to assume that the mass of the X particle is on the heavier side of the experimentally allowed range. The rate for emission goes like the 3-momentum cubed (see section II.E of 1608.03591), so a small increase in the mass can suppresses the rate of emission by the lighter state by a lot. The UCI collaboration of theorists went further and extended the Pastore et al. analysis to include a phenomenological parameterization of explicit isospin violation. Independent of the Atomki anomaly, they found that including isospin violation improved the fit for the 18.15 MeV and 17.64 MeV electromagnetic decay widths within the Pastore et al. formalism. The results of including all of the isospin effects end up changing the particle physics story of the Atomki anomaly significantly: The rate of X emission (colored contours) as a function of the X particle’s couplings to protons (horizontal axis) versus neutrons (vertical axis). The best fit for a 16.7 MeV new particle is the dashed line in the teal region. The vertical band is the region allowed by the NA48/2 experiment. Solid lines show the dark photon and protophobic limits. Left: the case for perfect (unrealistic) isospin. Right: the case when isospin mixing and explicit violation are included. Observe that incorporating realistic isospin happens to have only a modest effect in the protophobic region. Figure from 1608.03591. The results of the nuclear analysis are thus that: 1. An interpretation of the Atomki anomaly in terms of a new particle tends to push for a slightly heavier X mass than the reported best fit. (Remark: the Atomki paper does not do a combined fit for the mass and coupling nor does it report the difficult-to-quantify systematic errors associated with the fit. This information is important for understanding the extent to which the X mass can be pushed to be heavier.) 2. The effects of isospin mixing and violation are important to include; especially as one drifts away from the purely protophobic limit. ## Theory part 4: towards a complete theory The theoretical structure presented above gives a framework to do phenomenology: fitting the observed anomaly to a particle physics model and then comparing that model to other experiments. This, however, doesn’t guarantee that a nice—or even self-consistent—theory exists that can stretch over the scaffolding. Indeed, a few challenges appear: • The isospin mixing discussed above means the X mass must be pushed to the heavier values allowed by the Atomki observation. • The “protophobic” limit is not obviously anomaly-free: simply asserting that known particles have arbitrary charges does not generically produce a mathematically self-consistent theory. • Atomic parity violation constraints require that the X couple in the same way to left-handed and right-handed matter. The left-handed coupling implies that X must also talk to neutrinos: these open up new experimental constraints. The Irvine/Kentucky/Riverside collaboration first note the need for a careful experimental analysis of the actual mass ranges allowed by the Atomki observation, treating the new particle mass and coupling as simultaneously free parameters in the fit. Next, they observe that protophobic couplings can be relatively natural. Indeed: the Standard Model Z boson is approximately protophobic at low energies—a fact well known to those hunting for dark matter with direct detection experiments. For exotic new physics, one can engineer protophobia through a phenomenon called kinetic mixing where two force particles mix into one another. A tuned admixture of electric charge and baryon number, (Q-B), is protophobic. Baryon number, however, is an anomalous global symmetry—this means that one has to work hard to make a baryon-boson that mixes with the photon (see 1304.0576 and 1409.8165 for examples). Another alternative is if the photon kinetically mixes with not baryon number, but the anomaly-free combination of “baryon-minus-lepton number,” Q-(B-L). This then forces one to apply additional model-building modules to deal with the neutrino interactions that come along with this scenario. In the language of the ‘model building blocks’ above, result of this process looks schematically like this: A complete theory is completely mathematically self-consistent and satisfies existing constraints. The additional bells and whistles required for consistency make additional predictions for experimental searches. Pieces of the theory can sometimes be used to address other anomalies. The theory collaboration presented examples of the two cases, and point out how the additional ‘bells and whistles’ required may tie to additional experimental handles to test these hypotheses. These are simple existence proofs for how complete models may be constructed. ## What’s next? We have delved rather deeply into the theoretical considerations of the Atomki anomaly. The analysis revealed some unexpected features with the types of new particles that could explain the anomaly (dark photon-like, but not exactly a dark photon), the role of nuclear effects (isospin mixing and breaking), and the kinds of features a complete theory needs to have to fit everything (be careful with anomalies and neutrinos). The single most important next step, however, is and has always been experimental verification of the result. While the Atomki experiment continues to run with an upgraded detector, what’s really exciting is that a swath of experiments that are either ongoing or in construction will be able to probe the exact interactions required by the new particle interpretation of the anomaly. This means that the result can be independently verified or excluded within a few years. A selection of upcoming experiments is highlighted in section IX of 1608.03591: Other experiments that can probe the new particle interpretation of the Atomki anomaly. The horizontal axis is the new particle mass, the vertical axis is its coupling to electrons (normalized to the electric charge). The dark blue band is the target region for the Atomki anomaly. Figure from 1608.03591; assuming 100% branching ratio to electrons. We highlight one particularly interesting search: recently a joint team of theorists and experimentalists at MIT proposed a way for the LHCb experiment to search for dark photon-like particles with masses and interaction strengths that were previously unexplored. The proposal makes use of the LHCb’s ability to pinpoint the production position of charged particle pairs and the copious amounts of D mesons produced at Run 3 of the LHC. As seen in the figure above, the LHCb reach with this search thoroughly covers the Atomki anomaly region. ## Implications So where we stand is this: • There is an unexpected result in a nuclear experiment that may be interpreted as a sign for new physics. • The next steps in this story are independent experimental cross-checks; the threshold for a ‘discovery’ is if another experiment can verify these results. • Meanwhile, a theoretical framework for understanding the results in terms of a new particle has been built and is ready-and-waiting. Some of the results of this analysis are important for faithful interpretation of the experimental results. What if it’s nothing? This is the conservative take—and indeed, we may well find that in a few years, the possibility that Atomki was observing a new particle will be completely dead. Or perhaps a source of systematic error will be identified and the bump will go away. That’s part of doing science. Meanwhile, there are some important take-aways in this scenario. First is the reminder that the search for light, weakly coupled particles is an important frontier in particle physics. Second, for this particular anomaly, there are some neat take aways such as a demonstration of how effective field theory can be applied to nuclear physics (see e.g. chapter 3.1.2 of the new book by Petrov and Blechman) and how tweaking our models of new particles can avoid troublesome experimental bounds. Finally, it’s a nice example of how particle physics and nuclear physics are not-too-distant cousins and how progress can be made in particle–nuclear collaborations—one of the Irvine group authors (Susan Gardner) is a bona fide nuclear theorist who was on sabbatical from the University of Kentucky. What if it’s real? This is a big “what if.” On the other hand, a 6.8σ effect is not a statistical fluctuation and there is no known nuclear physics to produce a new-particle-like bump given the analysis presented by the Atomki experimentalists. The threshold for “real” is independent verification. If other experiments can confirm the anomaly, then this could be a huge step in our quest to go beyond the Standard Model. While this type of particle is unlikely to help with the Hierarchy problem of the Higgs mass, it could be a sign for other kinds of new physics. One example is the grand unification of the electroweak and strong forces; some of the ways in which these forces unify imply the existence of an additional force particle that may be light and may even have the types of couplings suggested by the anomaly. Could it be related to other anomalies? The Atomki anomaly isn’t the first particle physics curiosity to show up at the MeV scale. While none of these other anomalies are necessarily related to the type of particle required for the Atomki result (they may not even be compatible!), it is helpful to remember that the MeV scale may still have surprises in store for us. • The KTeV anomaly: The rate at which neutral pions decay into electron–positron pairs appears to be off from the expectations based on chiral perturbation theory. In 0712.0007, a group of theorists found that this discrepancy could be fit to a new particle with axial couplings. If one fixes the mass of the proposed particle to be 20 MeV, the resulting couplings happen to be in the same ballpark as those required for the Atomki anomaly. The important caveat here is that parameters for an axial vector to fit the Atomki anomaly are unknown, and mixed vector–axial states are severely constrained by atomic parity violation. The KTeV anomaly interpreted as a new particle, U. From 0712.0007. • The anomalous magnetic moment of the muon and the cosmic lithium problem: much of the progress in the field of light, weakly coupled forces comes from Maxim Pospelov. The anomalous magnetic moment of the muon, (g-2)μ, has a long-standing discrepancy from the Standard Model (see e.g. this blog post). While this may come from an error in the very, very intricate calculation and the subtle ways in which experimental data feed into it, Pospelov (and also Fayet) noted that the shift may come from a light (in the 10s of MeV range!), weakly coupled new particle like a dark photon. Similarly, Pospelov and collaborators showed that a new light particle in the 1-20 MeV range may help explain another longstanding mystery: the surprising lack of lithium in the universe (APS Physics synopsis). Could it be related to dark matter? A lot of recent progress in dark matter has revolved around the possibility that in addition to dark matter, there may be additional light particles that mediate interactions between dark matter and the Standard Model. If these particles are light enough, they can change the way that we expect to find dark matter in sometimes surprising ways. One interesting avenue is called self-interacting dark matter and is based on the observation that these light force carriers can deform the dark matter distribution in galaxies in ways that seem to fit astronomical observations. A 20 MeV dark photon-like particle even fits the profile of what’s required by the self-interacting dark matter paradigm, though it is very difficult to make such a particle consistent with both the Atomki anomaly and the constraints from direct detection. Should I be excited? Given all of the caveats listed above, some feel that it is too early to be in “drop everything, this is new physics” mode. Others may take this as a hint that’s worth exploring further—as has been done for many anomalies in the recent past. For researchers, it is prudent to be cautious, and it is paramount to be careful; but so long as one does both, then being excited about a new possibility is part what makes our job fun. For the general public, the tentative hopes of new physics that pop up—whether it’s the Atomki anomaly, or the 750 GeV diphoton bumpa GeV bump from the galactic center, γ-ray lines at 3.5 keV and 130 GeV, or penguins at LHCb—these are the signs that we’re making use of all of the data available to search for new physics. Sometimes these hopes fizzle away, often they leave behind useful lessons about physics and directions forward. Maybe one of these days an anomaly will stick and show us the way forward. Here are some of the popular-level press on the Atomki result. See the references at the top of this ParticleBite for references to the primary literature. ### What is “Model Building”? Thursday, August 18th, 2016 Hi everyone! It’s been a while since I’ve posted on Quantum Diaries. This post is cross-posted from ParticleBites. One thing that makes physics, and especially particle physics, is unique in the sciences is the split between theory and experiment. The role of experimentalists is clear: they build and conduct experiments, take data and analyze it using mathematical, statistical, and numerical techniques to separate signal from background. In short, they seem to do all of the real science! So what is it that theorists do, besides sipping espresso and scribbling on chalk boards? In this post we describe one type of theoretical work called model building. This usually falls under the umbrella of phenomenology, which in physics refers to making connections between mathematically defined theories (or models) of nature and actual experimental observations of nature. One common scenario is that one experiment observes something unusual: an anomaly. Two things immediately happen: 1. Other experiments find ways to cross-check to see if they can confirm the anomaly. 2. Theorists start figure out the broader implications if the anomaly is real. #1 is the key step in the scientific method, but in this post we’ll illuminate what #2 actually entails. The scenario looks a little like this: An unusual experimental result (anomaly) is observed. One thing we would like to know is whether it is consistent with other experimental observations, but these other observations may not be simply related to the anomaly. Theorists, who have spent plenty of time mulling over the open questions in physics, are ready to apply their favorite models of new physics to see if they fit. These are the models that they know lead to elegant mathematical results, like grand unification or a solution to the Hierarchy problem. Sometimes theorists are more utilitarian, and start with “do it all” Swiss army knife theories called effective theories (or simplified models) and see if they can explain the anomaly in the context of existing constraints. Here’s what usually happens: Usually the nicest models of new physics don’t fit! In the explicit example, the minimal supersymmetric Standard Model doesn’t include a good candidate to explain the 750 GeV diphoton bump. Indeed, usually one needs to get creative and modify the nice-and-elegant theory to make sure it can explain the anomaly while avoiding other experimental constraints. This makes the theory a little less elegant, but sometimes nature isn’t elegant. Candidate theory extended with a module (in this case, an additional particle). This additional model is “bolted on” to the theory to make it fit the experimental observations. Now we’re feeling pretty good about ourselves. It can take quite a bit of work to hack the well-motivated original theory in a way that both explains the anomaly and avoids all other known experimental observations. A good theory can do a couple of other things: 1. It points the way to future experiments that can test it. 2. It can use the additional structure to explain other anomalies. The picture for #2 is as follows: A good hack to a theory can explain multiple anomalies. Sometimes that makes the hack a little more cumbersome. Physicists often develop their own sense of ‘taste’ for when a module is elegant enough. Even at this stage, there can be a lot of really neat physics to be learned. Model-builders can develop a reputation for particularly clever, minimal, or inspired modules. If a module is really successful, then people will start to think about it as part of a pre-packaged deal: A really successful hack may eventually be thought of as it’s own variant of the original theory. Model-smithing is a craft that blends together a lot of the fun of understanding how physics works—which bits of common wisdom can be bent or broken to accommodate an unexpected experimental result? Is it possible to find a simpler theory that can explain more observations? Are the observations pointing to an even deeper guiding principle? Of course—we should also say that sometimes, while theorists are having fun developing their favorite models, other experimentalists have gone on to refute the original anomaly. Sometimes anomalies go away and the models built to explain them don’t hold together. But here’s the mark of a really, really good model: even if the anomaly goes away and the particular model falls out of favor, a good model will have taught other physicists something really neat about what can be done within the a given theoretical framework. Physicists get a feel for the kinds of modules that are out in the market (like an app store) and they develop a library of tricks to attack future anomalies. And if one is really fortunate, these insights can point the way to even bigger connections between physical principles. I cannot help but end this post without one of my favorite physics jokes, courtesy of T. Tait: A theorist and an experimentalist are having coffee. The theorist is really excited, she tells the experimentalist, “I’ve got it—it’s a model that’s elegant, explains everything, and it’s completely predictive.”The experimentalist listens to her colleague’s idea and realizes how to test those predictions. She writes several grant applications, hires a team of postdocs and graduate students, trains them, and builds the new experiment. After years of design, labor, and testing, the machine is ready to take data. They run for several months, and the experimentalist pores over the results. The experimentalist knocks on the theorist’s door the next day and says, “I’m sorry—the experiment doesn’t find what you were predicting. The theory is dead.” The theorist frowns a bit: “What a shame. Did you know I spent three whole weeks of my life writing that paper?” ### The Post-Higgs Hangover: where’s the new physics? Thursday, July 19th, 2012 Now that the good people at CERN have finished their Higgs-discovery champagne, many of us have found ourselves drawn to harder drinks. While the Higgs is the finishing touch on the elegant edifice of the Standard Model, it’s the culmination of theoretical physics from the 1960s. Where’s all the exciting new physics that we’d been expecting “just around the corner” at the terascale? My generation of particle physicists entered graduate school expecting a cornucopia of supersymmetry and extra dimensions at the TeV scale just waiting for us to join the party—unfortunately those hopes and dreams have yet come up short. While the book has yet to be written on whether or not the Higgs branching ratios are Standard Model-like, two recent experimental updates in collider and dark matter physics have also turned up empty. ## No Z’ at 1 TeV The first is the search for Z’ (“Z prime”) resonances, these are “smoking gun” signatures of a new particle which behaves like a heavy copy of the Z boson. Such particles are predicted by several models of new physics. There was some very cautious excitement after the 2011 data showed a 2σ bump in the dilepton channel around 1 TeV (both at CMS and ATLAS): The horizontal axis is the mass of the hypothetical particle (measured by the momenta of the two leptons it supposedly decays to) in GeV, while the vertical axis is the rate at which these two lepton events are seen. (The other lines are examples for what one would expect for a Z’ from different models, for our purposes we can ignore them.) A bump would be indicative of a new particle causing a resonance: an increased rate in the observation of two leptons with a given energy. You can see something that is beginning to “kinda-sorta” look like a bump around 1 TeV. Of course, 2σ signals come and go with statistics—and this is indeed what happened with this year’s data [CMS EXO-12-015]: Bummer. (Again, one really doesn’t have much right to be disappointed—that’s just the way the statistics works.) ## Still no WIMP dark matter Another area where we have good reason to expect new physics is dark matter. Astrophysical observations have given very strong evidence that the dark matter that gravitationally seeds our galaxies is composed of some new particle that is not described by the Standard Model. One nice feature is that astrophysical and cosmological data tell us the dark matter density in our galaxy, from which we can deduce a relation between the dark matter mass and its interaction strength. Physicists observed that one particularly interesting scenario is when the dark matter particle interacts via the weak force—the sector of our the Standard Model that gets tied up with electroweak symmetry breaking and the Higgs. In this case, the dark matter mass should be right around a few hundred GeV, right in the ballpark of the LHC. To some, this is very suggestive evidence that dark matter may be related to electroweak physics. This class of models got a cute name: WIMPs, for weakly interacting massive particles. There are other types of dark matter, but until fairly recently WIMPs were king because they fit so nicely with models of new physics that were already modifying the electroweak scale. Unfortunately, the flagship dark matter detector, XENON, recently released a sobering summary of its latest data at the Dark Attack conference in Switzerland. Yes, that’s really the conference name. XENON is a wonderful piece of detector technology that any particle physicist would be proud of. Their latest data-taking run found only two events (what’s expected from background). The result is the following plot: How to read this plot: the horizontal axis is the mass of the WIMP particle. You get to pick this (or your model of new physics predicts this). The vertical axis is the cross section, which measures the number of dark matter–detector interactions that such a WIMP is expected to undergo. The large boomerang-shaped lines are the limits set by the experiment—as the red text says, for a mass of around 55 GeV, it rules out cross sections that are above a certain number. For “garden variety” interaction channels, this number is already much smaller than the ball park estimate for the weak force. The blob at the bottom right is some fairly arbitrary slice of the supersymmetry parameter space, but this is really just there for illustrative purposes and shouldn’t be interpreted as any kind of exclusion of supersymmetry. The other lines are other past experiments. The circles at the top left are slightly controversial ‘signals’ that have been ruled out within the WIMP paradigm by the last few direct detection experiments (XENON and CDMS). The story is not necessarily as dour as the plot seems to indicate. There are many clever ways to get dark matter, not all of them WIMP-like. In fact, even the above plot is limited to the “spin-independent” coupling—an assumption about the particular way that dark matter interacts with nuclei. But these WIMP searches will eventually hit a brick wall around 2017: that’s when the XENON 1T (“one ton”) experiment will be sensitive to cross sections that are three orders of magnitude smaller than the current bounds. At that level of sensitivity, you end up with a lot of background noise from cosmic neutrinos which, as far as the detector is concerned, behave very much like dark matter. (They’re not.) Looking for a dark matter signal against this background is like looking for a needle in a stack of needles. ## Where do we stand? Between the infamous magnet quench of 2008 to the sobering exclusion plots of the last couple of years, an entire generation of graduate students and young postdocs is internalizing the idea that finding new physics will not be as simple as turning on the LHC as some of us had believed as undergrads. Despite our youthful naivete, the LHC is also still in its infancy with a 14 TeV run coming after its year-long shutdown. The above results are sobering, but they just mean that there wasn’t any low-hanging fruit for us to gobble up right away. ### More Post-Higgs silliness Friday, July 6th, 2012 I recently got to eavesdrop on a delightful and silly e-mail exchange between US LHC’s very own Burton and Aidan, both ATLAS physicists, after I pointed out that Wikipedia now mentions the ATLAS Higgs talk as a “notable use” of the infamous font Comic Sans. The quotes below are lifted directly from their e-mail exchange (with their permission), as illustrated by yours truly. For more substantial physics discussion, check out Aidan and Seth’s Higgs postgame video and Anna’s ongoing posts from ICHEP. Update [7/08]: the “4.9 sigma” comment below is a mistake, the actual “global significance” includes the ‘look elsewhere effect’ and is lower than this. ### Photoshop the Higgs Thursday, July 5th, 2012 Symmetry Breaking has a fun contest going on to photoshop the Higgs into interesting photos… by the way this is not how ATLAS and CMS do their data analysis. Here are a few examples featuring familiar faces from the US LHC blog: Many thanks to Aidan for his Higgs liveblog. He's now a certified Higgs-buster. Shout out to Katie Yurkewicz, Fermilab office of communications director and former US LHC communicator And a very special congrats to Kathryn Jepsen, US LHC communicator, who got married earlier this year! ### What to look for: the Higgs-to-gamma-gamma branching ratio Tuesday, July 3rd, 2012 There’s a lot of press building up to the Higgs announcement at CERN in just a few hours, and you’ll have Aidan’s live-blog for the play-by-play commentary. I just wanted to squeeze in more chatter about what to look for in the talks besides the usual “oh look how many sigmas we have.” [caveat: the above cereal guy meme is purely hypothetical!] Since we’re all friends here, I’ll be candid and say that many physicists have taken the existence of a 125 GeV-ish Higgs-like particle as a foregone conclusion—in large part because any alternative would be even more dramatic. (Recall: the Standard Model is begging for there to be a Higgs.) Whether the evidence for the Higgs is just above or just below the magic 5-sigma “discovery” threshold won’t change anything other than how much champagne Aidan will be drinking. But that shouldn’t deter you from tuning into the 3am EST webcast. Besides getting a chance to see some famous faces in the audience, the thing to look for are hints that there’s actually more to the Higgs than the Standard Model. As described very nicely at Resonaances, the 2011 LHC data presented last December suggested that the Higgs (if it’s there) decays into photons slightly more often than the Standard Model predicts. Could this be a hint that there’s exciting (and unexpected) new physics right around the corner? Let’s back up a little bit. Before we can talk about how the Higgs decays, we have to talk about how it’s produced at the LHC. The two main mechanisms are called gluon fusion and vector boson fusion (where the vector boson V can be a Z or W): The gluon fusion diagram dominates at the LHC since there are plenty of high energy gluons in a multi-TeV proton beam. Note that the loop of virtual top quarks is required since the Higgs has no direct coupling to gluons (it’s not colored); the top is a good choice since it has a large coupling to the Higgs (which is why the top is so heavy). As an exercise, use the Standard Model Feynman rules to draw other Higgs production diagrams. Once you have a Higgs, you can look at the different ways it can decay. The photon-photon final state is very rare, but particularly intriguing because the Higgs doesn’t have electric charge and photons don’t have mass—so these particles don’t tend to talk to each other. In fact, such a Higgs-photon-photon interaction only occurs when mediated by virtual particles like the top and W: Why these diagrams? They’re heavy enough to have a large coupling to the Higgs and also charged so they can emit photons. (Exercise: draw the other W boson diagram contributing to h to γγ.) In fact, the W diagram is about 5 times larger than the top diagram. The great thing about loop diagrams is that any particle (with electric charge and coupling to the Higgs) can run in the loop. You can convince yourself that other Standard Model particles don’t make big contributions to hγγ, but—and here’s the good part—if there are new particles beyond the Standard Model, they could potentially push the h → γγ rate larger than the Standard Model prediction. This is what we’re hoping. What to look for: keep an eye out for a measurement of the h → γγ cross section (a measurement of the rate). Cross sections are usually denoted by σ. Because we don’t care so much about the actual number but rather its difference from the Standard Model, what is usually presented is a ratio of the observed cross section to the Standard Model cross section: σ/σ(SM). If this ratio is one within uncertainty, then things look like the Standard Model, but otherwise (and hopefully) things are much more interesting. ## The outlook on the eve of ‘the announcement’ [I thank my colleagues Jack, Mathieu, and Javi for sharing their insights on this.] Given the assumption that there indeed is a particle at 125-ish GeV that does all the great things that the Standard Model Higgs should do, we would like to ask whether or not this is really the Standard Model (SM) Higgs, or whether it is some other Higgs-like state that may have different properties. In particular, is it possible that this particle talks to the rest of the Standard Model with slightly different strengths than the SM Higgs? And maybe, if we really want to push our luck, could this more exotic Higgs-like particle push the h → γγ rate to be larger than expected? To answer this question, we don’t want to restrict ourselves to any one specific model of new physics, we’d rather be as general as possible. One way to do this is to use an “effective theory” that parameterizes all of the possible couplings of the “Higgs” to Standard Model particles. Here’s what one such effective theory looks like in sloppy handwriting: Don’t worry, you don’t have to know what these all mean, but just for fun you can compare to this famous expression. The parameters here are the variables labelled a, b, c, and d. Of these, the two important ones to consider are a, which controls the Higgs coupling to two W bosons, and c, which controls the Higgs coupling to fermions (like top quarks). The Standard Model corresponds to a = c = 1. Now we can start playing an interesting game: 1. If we increase the coupling a of the Higgs to W bosons, then we increase the rate for h → γγ via the W loop above. 2. If, on the other hand, we increase the coupling c of the Higgs to the top quark, then we increase the rate of h → γγ via the top quark loop above. Thus the observation of a larger-than-expected rate for h → γγ could point to either a or c >1 (or both). How would we distinguish between these? Well, note that (see the production diagrams above): 1. If the a (Higgs to W) coupling were enhanced, then we would also expect an enhancement in the “vector boson fusion” rate for Higgs production. When the Higgs is produced this way, you can [with some efficiency] tag the quark remnants and say that the Higgs was produced through vector boson fusion. 2. On the other hand, if the c (Higgs to top) coupling were enhanced, then we would also expect an enhancement in the “gluon fusion” rate for Higgs production. Thus we have some handle for where we could fit new physics to explain a possible h → γγ excess. (Again, by “excess” we mean relative to the expected production in the Standard Model.) Here’s a quick plot of where we stand currently, including recent results from Moriond, from 1202.3697, I refer experts to that paper for further details and plots: (There are many similar plots out there—some by good friends of mine—I apologize for not providing a more complete reference list… the Higgs seminar is only a few hours away!) The green/yellow/gray blobs are the 1,2,3 sigma confidence regions for the parameters a and c above. The red and blue lines are ATLAS and CMS exclusions. The reason why there are two green blobs is that there is a choice for the sign of c, this corresponds to whether the Higgs-top loops interfere constructively or destructively with the Higgs-W loops. For more details, see this Resonaances post. The plot above includes the latest LHC data (Moriond, pre-ICHEP) as well as the so-called “electroweak precision observables” which tightly constrain the effects of virtual particles on the Standard Model gauge bosons. These are the blobs to keep an eye on—the lines indicate the Standard Model point a=c=1. If the blob continues to creep away from this point, then there will be good reason to expect exciting new physics beyond the Higgs… and that’s what makes it worth tuning in at 3am. ### Tim Tait: “Why look for the Higgs?” Tuesday, July 3rd, 2012 For those of you who are itching to learn more about the Higgs in anticipation of the Higgs announcement and Aidan’s liveblog, I encourage you to check out Tim Tait’s recent colloquium at SLAC titled, “Why look for the Higgs?” It’s an hour-long talk aimed at a non-physics audience (Tim says “engineers and programmers”). Tim is a professor at UC Irvine whose enthusiasm and natural ability to explain physics carries through in his talk. Last summer Tim was a co-director for the “Theoretical Advanced Study Institute in Elementary Particle Physics” summer school for graduate students. I heard that the students tried to get Tim’s portrait immortalized on the official school t-shirt. For more SLAC colloquia and public lectures, see their channel on YouTube. ### The Hierarchy Problem: why the Higgs has a snowball’s chance in hell Sunday, July 1st, 2012 The Higgs boson plays a key role in the Standard Model: it is related to the unification of the electromagnetic and weak forces, explains the origin of elementary particle masses, and provides a weakly coupled way to unitarize longitudinal vector boson scattering. As particle physics community eagerly awaits CERN’s special seminar on a possible Higgs discovery (see Aidan’s liveblog), it’s a good time to review why Higgs—the last piece of the Standard Model—is also one of the big reasons why we expect even more exciting physics beyond the Standard Model. The main reason is called the Hierarchy problem. This is often ‘explained’ by saying that quantum corrections want to make the Higgs much heavier than we need it to be… say, 125-ish GeV. Before explaining what that means, let me put it in plain language: The Higgs has a snowball’s chance in hell of having a mass in that ballpark. This statement works as an analogy, not just an idiom. (This analogy is adapted from one originally by R. Rattazzi involving a low energy particle passing through a thermal bath. Edit: I’m told this analogy was by G. Giudice, thanks Duccio.) If you put a glass of water in a really hot place—you expect it to also become really hot, maybe even to off into steam. It would be really surprising if we put an ice cube in a hot oven and 10 minutes later it had not melted. This is because the ambient thermal energy is expected to be transferred to the ice cube by the energetic air molecules bouncing off it. Sure, it is possible that the air molecules just happen to bounce in a way that doesn’t impart much thermal energy—but that would be ridiculously improbable, as we learn in thermodynamics. The Higgs is very similar: we expect its mass to be around 125 GeV (not too far from W and Z masses), but ambient quantum energy wants to make its mass much larger through interactions with virtual particles. While it is possible that the Higgs stays light without any additional help, it’s ridiculously improbable, as we learn from quantum physics. Remark: the relation between thermal/statistical uncertainty and quantum uncertainty is actually one that is deeply woven into their mathematical descriptions and is the reason why quantum (or statistical) field theory is the common language of both particle physics and condensed matter physics. ## Quantum corrections: the analogy of the point electron The phrase “quantum corrections” is somewhat daunting, so let’s appeal to a slightly more familiar problem (from H. Murayama) and draw some pictures. The analog of the Hierarchy problem in classical physics is the question of the electron self energy: The electron has charge but is nearly point-like. It must have a very large charge density and thus have a very large self-energy (mass). Self-energy here just means the contribution to the electron mass coming from repulsive electrostatic energy of one part of the electron from another. The problem thus reduces to: how can the electron mass be so small when we expect it to be large due to electrostatic energy? Yet another way to pose the question is to say that the electron mass has contributions from some ‘inherent mass’ (or ‘bare mass’) m0 and the electrostatic energy, ΔE: mmeasured = m0ΔE Since mmeasured is small while ΔE is large, then it seems that m0 must be very specifically chosen to cancel out most of ΔE but still leave the correct tiny leftover value for the electron mass. In other words, the ‘bare mass’ m0 must be chosen to uncomfortably high precision. I walk through the numbers in a previous post (see also the last few pages of these lectures to undergraduates [pdf] from here), but here’s the main idea: the reason why there isn’t a huge electrostatic contribution to the electron mass is that virtual electron–positron pairs smear out the electric charge over a radius larger than the size of the electron: In other words: current experimental bounds tell me that the electron is smaller than 10-17 cm and the “electron hierarchy problem” arises when I calculate the energy associated with packing in one unit of electric charge into that radius. The resolution is that even though the electron may be tiny, at a certain length scale quantum mechanics becomes relevant and you start seeing virtual electron–positrion pairs which interact with the physical electron to smear out the charge over a larger distance (this is called vacuum polarization). The distance at which this smearing takes place is predicted by quantum mechanics—it’s the distance where the virtual particles have enough energy to become real—and when you plug in the numbers, it’s precisely where it needs to be to prevent a large electrostatic contribution to the electron mass. Since we’re now experts with Feynman diagrams, here’s what such a process looks like in that language: ## Higgs: the petulant child of the Standard Model The Hierarchy problem for the Higgs is the quantum version of the above problem. “Classically” the Higgs has a mass that comes from the following diagram (note the Higgs vev): This diagram is perfectly well behaved. The problem occurs from contributions that include loops of virtual particles—these play the role of the electrostatic contribution to the electron mass in the above analogy: As an exercise, use the Higgs Feynman rules to draw other contributions to the Higgs mass which contain a single loop; for our present purposes the one above is sufficient. Recall, further, that one of our rules for drawing diagrams was that momentum is conserved. In the above diagram, the incoming Higgs has some momentum (which has to be the same as the outgoing Higgs), but the virtual particle momenta (k) can be anything. What this means is that we have to sum over an infinite number of diagrams, each with a different momentum k running through the loop. We’ll ignore the mathematical expression that’s actually being summed, but suffice it to say that it is divergent—infinity. This is a good place for you to say, what?! the Higgs mass isn’t infinity… that doesn’t even make sense! That’s right—so instead of summing up to diagrams with infinite loop momentum, we should stop where we expect our model to break down. But without any yet undiscovered physics, the only energy scale at which we know our description must break down is the gravitational scale: MPlanck ~ 1018 GeV. And thus, as a rough estimate, these loop diagrams want to push the Higgs mass up to 1018 GeV… which is way heavier than we could ever hope to discover from a 14 TeV (= 14,000 GeV) LHC. (Recall that these virtual contributions to the Higgs mass are what were analogous to thermal energy in our “snowball in Hell” analogy.) But here’s the real problem: the Standard Model really, really wants the Higgs to be around the 100 GeV scale. This is because it needs something to “unitarize longitudinal vector boson scattering.” It needs to have some Higgs-like state accessible at low energies to explain why certain observed particle interactions are well behaved. Thus if the Higgs indeed has a mass around 125 GeV, then the only way to make sense of the 1018 GeV mass contribution from the loop diagram above is if the “classical” (or “tree”) diagram has a value which precisely cancels that huge number to leave only a 125 GeV mass. This is the analog of choosing m0 in the electron analogy above. Unlike the electron analogy above, we don’t know what kind of physics can explain this 1016 ‘fine-tuning’ of our Standard Model parameters. For this reason, we expect there to be some kind of new physics accessible at TeV energies to explain why the Higgs should be right around that scale rather than being at the Planck mass. ## Outlook on the Hierarchy The Hierarchy problem has been the main motivation for new physics at the TeV scale for over two decades. There are a few obvious questions that you may ask. 1. Is it really a problem? Maybe some number just has to be specified very precisely. Indeed—it is possible that the Higgs mass is 125 GeV due to some miraculous almost-cancellation that set it to be in just the right ballpark to unitarize longitudinal vector boson scattering. But such miracles are rare in physics without any a priori explanation. The electron mass is an excellent example. There are some apparent (and somewhat controversial) counter-examples: the cosmological constant problem is a much more severe ‘fine-tuning’ problem which may be explained anthropically rather than through more fundamental principles. 2. I can draw loop diagrams for all of the Standard Model particles… why don’t they all have Hierarchy problems? If you’ve asked this question, then you get an A+. Indeed, based on the arguments in this post, it seems like any diagram with a loop gives a divergence when you sum over the possible intermediate momenta so that we would expect all Standard Model particles to have Planck-scale masses due to quantum corrections. However, the important point was that we never wrote out the mathematical form of the thing that we’re summing. It turns out that the Hierarchy problem is unique for scalar particles like the Higgs. Loop contributions to fermion masses are not so sensitive to the ‘cutoff’ scale where the theory breaks down. This is manifested in the mathematical expression for the fermion mass and is ultimately due to the chiral structure of fermions in four dimensions. Gauge boson masses are also protected, but from a different mechanism: gauge invariance. More generally, particles that carry spin are very picky about whether they’re massive or massless, whereas scalar particles like the Higgs are not, which makes the Higgs susceptible to large quantum corrections to its mass. 3. What are the possible ways to solve the Hierarchy problem? There are two main directions that most people consider: 1. Supersymmetry. Recall in our electron analogy that the solution to the ‘electron mass hierarchy problem’ was that quantum mechanics doubled the number of particles: in addition to the electron, there was also a positron. The virtual electron–positron contributions solved the problem by smearing out the electric charge. Supersymmetry is an analogous idea where once again the set of particles is doubled, and in doing so the loop contributions of one particle to the Higgs are cancelled by the loop contributions of its super-partner. Supersymmetry has deep connections to an extension of space-time symmetry since it relates matter particles to force particles. 2. Compositeness/extra dimensions. The other solution is that maybe our description of physics breaks down much sooner than the Planck scale. In particular, maybe at the TeV scale the Higgs no longer behaves like a scalar particles, but rather as a bound state of two fermions. This is precisely what happens with the mesons: even though the pion is a scalar, there is no pion ‘hierarchy problem’ because as you probe smaller distances, you realize the pion is actually a bound state of two quarks and it starts behaving as such. One of the beautiful developments of theoretical physics in the 1990s and early 2000s was the realization that this is precisely what is being described by theories of extra dimensions through the so-called holographic principle. So there you have it—while you’re celebrating the [anticipated] Higgs discovery with fireworks on July 4th, also take a moment to appreciate that this isn’t the end of a journey culminating in the Standard Model, but the beginning of an expedition for exciting new physics at the terascale. ### An experiment: Feynman Diagrams for Undergrads Thursday, May 31st, 2012 The past couple of weeks I’ve been busy juggling research with an opportunity I couldn’t pass up: the chance to give lectures about the Standard Model to Cornell’s undergraduate summer students working on CMS. The local group here has a fantastic program which draws motivated undergrads from the freshman honors physics sequence. The students take a one credit “research in particle physics course” and spend the summer learning programming and analysis tools to eventually do CMS projects. Since the students are all local, some subset of them stay on and continue to work with CMS during their entire undergraduate careers. Needless to say, those students end up with fantastic training in physics and are on a trajectory to be superstar graduate students. Anyway, I spent some time adapting my Feynman diagram blog posts into a series of lectures. In case anyone is interested, I’m posting them publicly here, along with some really nice references at the appropriate level. There are no formal prerequisites except for familiarity with particle physics at the popular science/Wikipedia level, though they’re geared towards enthusiastic students who have been doing a lot of outside [pop-sci level] reading and have some sophistication with freshman level math and physics ideas. The whole thing is an experiment for me, but the first lecture earlier today seems to have gone well. ### Name these brands/plants? Name these particles! Tuesday, April 17th, 2012 I don’t know the original source, but there’s an image that has gone semi-viral over the past year which challenges the reader to identify several brand names based on their logos versus plant names based on their leaves. (Here’s a version at Adbusters.) The point is to contrast consumerism to the outdoors-y/science-y education that kids would get if they just played outside. This isn’t the place to discuss consumerism, but I don’t agree with idea that the ability to identify plant names carries any actual educational value. Here’s my revision to the image: Adapted from the original “Name these brands/plants” image (original source unknown). On the right we’ve encoded all of the particles in the Standard Model in a notation based on representation theory. In fact, this is almost all of the information you need to know to write down all of the Feynman rules in the Standard Model (more on this below). Tables that the one above are a compact way to describe the particle content of a model because the information in the table specifies all of the properties of each particle. And that’s the point: whether we name a particle the “truth quark” or the “top quark” doesn’t matter—what matters is the physics behind these names, and that’s captured succinctly in the table. Science isn’t about classification, it’s about understanding. I leave you with this quote from Feynman (which you can watch in his own words here): You can know the name of a bird in all the languages of the world, but when you’re finished, you’ll know absolutely nothing whatever about the bird… So let’s look at the bird and see what it’s doing — that’s what counts. I learned very early the difference between knowing the name of something and knowing something. For those who want to know, the particles in the table are, from top down: 1. The left-handed quark doublet, containing the left-handed up quark and left-handed down quark 2. The anti-right-handed-up quark 3. The anti-right-handed-down quark 4. The left-handed lepton doublet, containing the left-handed electron and left-handed neutrino 5. The anti-right-handed electron (a.k.a the right-handed positron) 6. The anti-right-handed neutrino 7. The Standard Model Higgs SU(3), SU(2), and U(1) refer to the strong force, weak force, and hypercharge. Upon electroweak symmetry breaking, the weak force and hypercharge combine into electromagnetism and the heavy W and Z bosons. Here’s how to read the funny notation: 1. Under SU(3): particles with a box come in three colors (red, green, blue). Particles with a barred box come in three anti-colors (anti-red, anti-green, anti-blue). Particles with a ‘1’ are not colored. 2. Under SU(2): particles with a box have two components, an upper and a lower component. That is to say, a box means that there are actually two particles being represented. More on this below. Particles with a ‘1’ do not carry weak charge and do not talk to the W boson. 3. Under U(1): this is the “hypercharge” of the particle. 4. The electric charge of a particle is given by adding to the hypercharge +1/2 if it’s the upper component of an SU(2) box, -1/2 if it’s the lower component of an SU(2) box, or 0 if it is not an SU(2) box (just ‘1’). As a consistency check, you can convince yourself that both the left- and right-handed neutrinos carry zero electric charge. Note, also, the fact that we’ve written out left-handed and right-handed particles differently. This is a reflection of the fact that the Standard Model is a chiral theory. Finally, I said above that the table of particles almost specifies the structure of the Standard Model completely, the additional pieces of information required are: 1. Which of the above particles are fermions and which are scalars (the gauge bosons are implied) 2. Write down the most general ‘renormalizable’ theory (we write only the simplest interaction vertices) 3. Specify the pattern of electroweak symmetry breaking (the Higgs) 4. Specify the flavor symmetries (three of each type of matter particle) From this one can write the complete mathematical expressions for the Standard Model. One then just has to fill in the observed numerical values to be able to calculate concrete predictions for actual processes.	{"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": 0.8486524224281311, "perplexity": 1219.7119441125726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1500549426639.7/warc/CC-MAIN-20170726222036-20170727002036-00297.warc.gz"}
http://mathhelpforum.com/business-math/104571-curtate-expectation-life.html	## Curtate expectation of life People who have been exposed to Common Carolina Fever (CCF) will have a higher-than-normal death rate for 2 years following exposure to this disease. The death rate is 10% higher than normal during the first year, and 5% higher during the second year. After that the death rate returns to normal. Professor Smith, age x, has just been exposed to CCF. Calculate the reduction in his curtate expectation of life from normal, given the following information for normal lives: qx = 0.07 qx+1 = 0.10 qx+2 = 0.11 ex+3 = 5 The answer is 0.0762. I think the problem is saying that after exposure, qx = 0.17 and qx+1 = 0.15. But I'm getting confused because I think the equation for curtate expectation of life has an infinity as the upper bound of the summation so I don't get how you can get an actual number answer ...	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8324238657951355, "perplexity": 816.0898454595966}, "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-2015-35/segments/1440644065318.20/warc/CC-MAIN-20150827025425-00269-ip-10-171-96-226.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/time-energy-uncertainty-relation.134735/	Time-energy uncertainty relation 1. Oct 3, 2006 quasar987 How should $\Delta E \Delta t \geq \hbar/2$ be interpreted? When does it apply? Delta t refers to the time of what? etc. Thx. 2. Oct 4, 2006 Galileo For a given time independent observable A that doesn't commute with the Hamiltonian and a state \|psi>, interpret $\Delta E = \Delta H$ (H is the hamiltonian and delta means standard deviation) and define: $$\Delta t := \frac{\Delta A}{\|d\langle A \rangle/dt\|}$$ So $$\Delta t$$ is the characteristic time it takes for the observable to change by one standard deviation. I think this is the best interpretation of the inequality. No mystic mojo is involved. Then by Heisenberg's inequality: $$\Delta H \Delta A \geq \frac{1}{2}\|\langle [H,A] \rangle\|$$ And using: $$\frac{d\langle A\rangle}{dt}=\frac{i}{\hbar}\langle [H,A]\rangle$$ you can rewrite it as $\Delta E \Delta t \geq \frac{\hbar}{2}$ Last edited: Oct 4, 2006 3. Oct 4, 2006 FunkyDwarf It sets a lower limit to the products of the uncertainty in those two quantities, ie for heisenburgs uncertainty principle its displacement and velocity 4. Oct 4, 2006 dextercioby See the explanation in Sakurai's book. Daniel. 5. Oct 8, 2006 pmb_phy To be precise there is no such thing as a time-energy uncertainty principle. Uncertainty principles are the relationship between two operators and while there is an energy operator there is no such thing as a time operator. Best to call it the time-energy uncertainty relation. The meaning of this relation is dt is the amount of time it takes for a system to evolve and dE represents the average change in the amount of energy during this time of evolution. Pete 6. Feb 26, 2011 exponent137 But, I heard somewhere that in one case dW dt>=hbar/2 is correct. What is this example? Maybe for photons? Last edited: Feb 26, 2011 7. May 6, 2012 Pythagorean I have two questions: 1) what's the motivations for this expression? 2) why is 1 standarad deviation chosen? Wouldn't the inequality change for any other choice? 8. May 6, 2012 nucl34rgg The time-energy uncertainty principle is a little bit different from the position-momentum one. This is perhaps best illustrated by Landau's quote, "To violate the time-energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch!" 9. May 6, 2012 Ken G I'd say the physical meaning behind the time-energy uncertainty relation has to do with the fact that a state of definite energy is characterized (physically) by having a definite frequency of change of its phase. To decide what that frequency is, you need to watch many cycles of time, and the more cycles you follow, the more precisely you know that frequency. But the more cycles you watch, the less you can say about the actual time at which you "looked", for you looked over a range of times. Conversely, if you look at "your watch" at a very specific time, then you cannot say what is the frequency at which the phase of the state is changing. Note that if you divide through by h, the expression becomes uncertainty in frequency times uncertainty in time exceeds 1 cycle. 10. May 7, 2012 facenian All I know about it is that you have to be carefull with the interpretation. It seems that there is not a single version of it and not even experts agree on it. There is however a simple version that can be derived clearly from first principles an is the one Galileo expleined, but that is not the only interpretation. Last edited: May 7, 2012 11. May 7, 2012 Demystifier In this approach, Delta t is NOT UNCERTAINTY of time, but a time DURATION of a physical process. There is however a similar way to introduce UNCERTAINTY of time measured by a clock, as explained, e.g., in http://xxx.lanl.gov/abs/1203.1139 (v3) Eqs. (19)-(23)	{"extraction_info": {"found_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": 0.8980587720870972, "perplexity": 622.0023348003975}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698540563.83/warc/CC-MAIN-20161202170900-00010-ip-10-31-129-80.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/445976/is-there-a-result-on-the-behaviour-of-power-series-with-positive-integer-coeffic	# Is there a result on the behaviour of power series with positive integer coefficients on their boundary? I have a power series whose coefficients are all positive integers and whose radius of convergence $r$ is $<1$ and I wish to prove that it has a pole at $r$, or at least an infinite radial limit. Is there a general result that could help in this situation or should I look for an ad-hoc proof? - I think one can give a simpler proof of this assertion in your particular case. – Pedro Tamaroff Jul 17 '13 at 19:09 Not true, the coefficients of power series of $\frac{1}{\sqrt{1-4z}} = \sum_{k=0}^{\infty}\binom{2k}{k} z^k$ are all positive integers, its radius of convergence $r = \frac14 < 1$ and yet it has a branch cut instead of pole at $r$. – achille hui Jul 17 '13 at 19:36 I can give you the following proof of what is an extension of Abel's Theorem. THM Suppose $\alpha_n$ is a sequence of real numbers such that $\sum_{n\geqslant 0}\alpha_n$ diverges to $+\infty$, and such that its powerseries converges for $\|x\|<1$. Then $$\lim_{x\to 1^{-}}\sum_{n\geqslant 0}\alpha_nx^n=+\infty$$ P Let $M>0$ be given. Note that $$\frac{1}{1-x}f(x)=\sum_{n\geqslant 0}\sum_{k=0}^n\alpha_k x^n$$ By hypothesis this is true for $\|x\|<1$. Also, there exists $N>0$ such that $n>N$ implies $\sum_{k=1}^n\alpha_k>M$. Then we have that $$\displaylines{ \frac{1}{{1 - x}}f(x) = \sum\limits_{n \geqslant 0} {\sum\limits_{k = 0}^n {{\alpha _k}} } {x^n} \cr = \sum\limits_{n = 0}^N {\sum\limits_{k = 0}^n {{\alpha _k}} {x^n}} + \sum\limits_{n > N} {\sum\limits_{k = 0}^n {{\alpha _k}} } {x^n} \cr > \sum\limits_{n = 0}^N {\sum\limits_{k = 0}^n {{\alpha _k}} {x^n}} + M\sum\limits_{n > N} {{x^n}} \cr > M{x^{N + 1}}\frac{1}{{1 - x}} \cr}$$ It follows that $f(x) > M{x^{N + 1}}$ so $$\mathop {\lim \inf }\limits_{x \to {1^ - }} f(x) \geqslant M$$ for each $M>0$, whence it must be the case $$\mathop {\lim }\limits_{x \to {1^ - }} f(x) = + \infty$$ ADD Note the last inequality follows from the fact we can assume the partial sums are all positive. - That would prove that the radial limit infinity as $z \to 1^-$. Does that necessarily imply that there is a pole at that point? – Old John Jul 17 '13 at 19:26 @JohnWordsworth Hmm... I misread the question then, because as you can see, I am assuming $z$ is real. So this proves there is a "real pole" but not more. – Pedro Tamaroff Jul 17 '13 at 19:29 Not sure I have time to drop into chat right now! I might have read more into the question than was intended - I assumed that "radius of convergence" meant we were in the complex plane, and then we get into things like Fatou's theorem: en.wikipedia.org/wiki/Fatou's_theorem – Old John Jul 17 '13 at 19:35 @anon The inequality follows from the fact the left hand side contains at least the amount of positive elements the right hand side does, that is all. – Pedro Tamaroff Jul 17 '13 at 19:45 You are looking for Pringsheim's theorem. I'm citing from Wikipedia's entry on Alfred Pringsheim: "One of Pringsheim's theorems, according to Hadamard [1] earlier proved by E. Borel, states [2] that a power series with positive coefficients and radius of convergence equal to 1 has necessarily a singularity at the point 1.". Of course you also get the same statement when the radius of convergence is $r$ by scaling the coefficients. EDIT: Precisely if $f$ has positive coefficients and radius of convergence $r$ then $g(z) = f(z r)$ has radius of convergence $1$ and positive coefficients, therefore by Pringsheim's theorem $g$ has a singularity at $1$, hence $f$ has a singularity at $r$. - Having a singularity doesn't imply the singularity is a pole. eg. $\frac{1}{\sqrt{1-z}}$. – achille hui Jul 17 '13 at 19:31 Okay, fine, I thought of "singularity" when I wrote "pole". Either way Pringsheim's theorem is the theorem the OP is looking for, since without further information nothing more can be really said about the nature of the singularity at $z = r$. – blabler Jul 18 '13 at 17:18	{"extraction_info": {"found_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": 0.9677421450614929, "perplexity": 181.41369832207837}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1448398457697.46/warc/CC-MAIN-20151124205417-00110-ip-10-71-132-137.ec2.internal.warc.gz"}
https://reports.edwiser.org/ada-haynes-tlcybch/9hnmkt.php?e539d8=almost-sure-convergence-implies-convergence-in-probability	Lemma (Chain of implication) The convergence in mean square implies the convergence in probability: m. s.! Almost sure convergenc can be related to convergence in probability of Cauchy sequences. Thus, we regard a.s. convergence as the strongest form of convergence. =) p! Just hang on and remember this: the two key ideas in what follows are \convergence in probability" and \convergence in distribution." Convergence in mean square Remark: It is the less usefull notion of convergence.. except for the demonstrations of the convergence in probability. On the other hand, almost-sure and mean-square convergence do not imply each other. 2.If a sequence s nof numbers does not converge, then there exists an >0 such that for every m sup by Marco Taboga, PhD. Thanks for contributing an answer to Mathematics Stack Exchange! A mode of convergence on the space of processes which occurs often in the study of stochastic calculus, is that of uniform convergence on compacts in probability or ucp convergence for short.. First, a sequence of (non-random) functions converges uniformly on compacts to a limit if it converges uniformly on each bounded interval .That is, What does it mean when "The Good Old Days" have several seemingly identical downloads for the same game? Almost sure convergence implies convergence in probability, and hence implies convergence in distribution. Notice that the convergence of the sequence to 1 is possible but happens with probability 0. When italicizing, do I have to include 'a,' 'an,' and 'the'? References 1 R. M. Dudley, Real Analysis and Probability , Cambridge University Press (2002). 5. Therefore, the two modes of convergence are equivalent for series of independent random ariables.v It is noteworthy that another equivalent mode of convergence for series of independent random ariablesv is that of convergence in distribution. convergence in probability of P n 0 X nimplies its almost sure convergence. See also Weak convergence of probability measures; Convergence, types of; Distributions, convergence of. Almost Sure Convergence. For example, an estimator is called consistent if it converges in probability to the parameter being estimated. Eh(X n) = Eh(X): For almost sure convergence, convergence in probability and convergence in distribution, if X nconverges to Xand if gis a continuous then g(X n) converges to g(X). Almost sure convergence, convergence in probability and asymptotic normality In the previous chapter we considered estimator of several diﬀerent parameters. We will discuss SLLN in Section 7.2.7. with probability 1. Almost sure convergence implies convergence in probability (by Fatou's lemma), and hence implies convergence in distribution. One of the most celebrated results in probability theory is the statement that the sample average of identically distributed random variables, under very weak assumptions, converges a.s. … Convergence almost surely requires that the probability that there exists at least a k ≥ n such that Xk deviates from X by at least tends to 0 as ntends to inﬁnity (for every > 0). Almost sure convergence. Convergence in probability provides convergence in law only. Example. We also propose a slightly less classical result stating that these two modes of convergence are equivalent for series of independent random ariables.v Applications of the results … P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York (NY), 1968. Join us for Winter Bash 2020, Uniform integrability and convergence in mean question, Almost sure convergence implies convegence in distribution - proof using monotone convergence, Almost sure convergence and equivalent definition, Convergence in distribution implies convergence of $L^p$ norms under additional assumption, Convergence in probability and equivalence from convergence almost sure, Showing $X_n \rightarrow X$ and $X_n \rightarrow Y$ implies $X\overset{\text{a.s.}}{=}Y$ for four types of convergence, Using Axiom of Replacement to construct the set of sets that are indexed by a set. We say that X. n converges to X almost surely (a.s.), and write . Costa Rican health insurance and tourist visa length, How to refuse a job offer professionally after unexpected complications with thesis arise, 80's post apocalypse book, two biological catastrophes at the end of the war. The following is a convenient characterization, showing that convergence in probability is very closely related to almost sure convergence. Almost sure convergence is defined based on the convergence of such sequences. Why don't the UK and EU agree to fish only in their territorial waters? If ξ n, n ≥ 1 converges in proba-bility to ξ, then for any bounded and continuous function f we have lim n→∞ Ef(ξ n) = E(ξ). !p b; if the probability of observing a sequence fZ tg such that T 1 P T t=1 Z t does not converge to b becomes less and less likely as T increases. The converse is not true. Types of Convergence Let us start by giving some deﬂnitions of diﬁerent types of convergence. In general, almost sure convergence is stronger than convergence in probability, and a.s. convergence implies convergence in probability. In conclusion, we walked through an example of a sequence that converges in probability but does not converge almost surely. Are drugs made bitter artificially to prevent being mistaken for candy? the case in econometrics. So let $X_1,X_2,\dots$ be Bernoulli random variables (each with its own probability of being a 1). Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. A sequence of random variables X1, X2, X3, ⋯ converges almost surely to a random variable X, shown by Xn a. s. → X, if P({s ∈ S: lim n → ∞Xn(s) = X(s)}) = 1. (Since almost sure convergence implies convergence in probability, the implication is automatic.) measure – in fact, it is a singular measure. ) It is easy to get overwhelmed. Almost sure convergence implies convergence in quadratic mean. P. Billingsley, Probability and Measure, Third Edition, Wiley Series in Probability and Statistics, John Wiley & … Theorem 19 (Komolgorov SLLN II) Let {X i} be a sequence of independently distributed random variables Proof Let !2, >0 and assume X n!Xpointwise. P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York (NY), 1968. • Convergence in probability Convergence in probability cannot be stated in terms of realisations Xt(ω) but only in terms of probabilities. To convince ourselves that the convergence in probability does not This demonstrates that an ≥pn and, consequently, that almost sure convergence implies convergence in probability. Theorem 19 (Komolgorov SLLN II) Let {X i} be a sequence of independently distributed random variables Oxford Studies in Probability 2, Oxford University Press, Oxford (UK), 1992. \begin{eqnarray} \end{eqnarray} Want to read all 5 pages? A mode of convergence on the space of processes which occurs often in the study of stochastic calculus, is that of uniform convergence on compacts in probability or ucp convergence for short.. First, a sequence of (non-random) functions converges uniformly on compacts to a limit if it converges uniformly on each bounded interval .That is, Since almost sure convergence always implies convergence in probability, the theorem can be stated as X n →p µ. Remark 1. The concept of almost sure convergence (or a.s. convergence) is a slight variation of the concept of pointwise convergence. X. i.p. almost sure convergence or convergence with probability one, which will be shown to imply both convergence in probability and convergence in distribution. Asymptotic Properties 4.3. with probability 1 (w.p.1, also called almost surely) if P{ω : lim ... • Convergence w.p.1 implies convergence in probability. IStatement is about probabilities, not about realizations (sequences))Probability converges, realizations x Nmay or may not converge)Limit and prob. Note that the theorem is stated in necessary and suﬃcient form. X(! As we know, the almost sure (a.s.) convergence and the convergence in sense each imply the convergence in probability, and the convergence in probability implies the convergence in distribution. For a sequence (Xn: n 2N), almost sure convergence of means that for almost all outcomes w, the difference Xn(w) X(w) gets small and stays small.Convergence in probability is weaker and merely )Limit and prob. Convergence in probability deals with sequences of probabilities while convergence almost surely (abbreviated a.s.) deals with sequences of sets. In other words, convergence with probability one means exactly what it sounds like: The probability that X nconverges to X equals one. Convergence in probability is also the type of convergence established by the weak law of … One of the most celebrated results in probability theory is the statement that the sample average of identically distributed random variables, under very weak assumptions, converges a.s. … De nition 5.2 \| Almost sure convergence (Karr, 1993, p. 135; Rohatgi, 1976, p. Thanks, Hint: Use Fatoo lemma with $$Y_n=2\|X_n\|^2+2\|X^2\|-\|X-X_n\|^2.$$. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. ← Hence X n!Xalmost surely since this convergence takes place on all sets E2F. convergence of random variables. )j< . (AS convergence vs convergence in pr 2) Convergence in probability implies existence of a subsequence that converges almost surely to the same limit. The concept of almost sure convergence does not come from a topology on the space of random variables. 0 if !6= 1 with probability 1 = P(!6= 1) 1 if != 1 with probability 0 = P(!= 1) Since the pdf is continuous, the probability P(!= a) = 0 for any constant a. Show that Xn → 0 a.s. implies that Sn/n → 0 a.s. In-probability Convergence 4. Just hang on and remember this: the two key ideas in what follows are \convergence in probability" and \convergence in distribution." This lecture introduces the concept of almost sure convergence. Later, in Theorem 3.3, we will formulate an equivalent deﬁnition of almost sure convergence that makes it much easier to see why it is such a strong form of convergence of random variables. This type of convergence is similar to pointwise convergence of a sequence of functions, except that the convergence need not occur on a set with probability 0 (hence the “almost” sure). It only takes a minute to sign up. Proposition 1. There is another version of the law of large numbers that is called the strong law of large numbers (SLLN). Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Thus, we regard a.s. convergence as the strongest form of convergence. convergence in probability implies convergence in distribution. convergence of random variables. Of course, a constant can be viewed as a random variable defined on any probability space. As we have seen, a sequence of random variables is pointwise convergent if and only if the sequence of real numbers is convergent for all. 1.1 Almost sure convergence Deﬁnition 1. There is another version of the law of large numbers that is called the strong law of large numbers (SLLN). It's easiest to get an intuitive sense of the difference by looking at what happens with a binary sequence, i.e., a sequence of Bernoulli random variables. Achieving convergence for all is a very stringent requirement. I know this problem may be related to the Scheffe or Riesz theorem (or dominated convergence), but I was thinking if this could be achieved using straightforward probability calculations without any result from measure theory. It is the notion of convergence used in the strong law of large numbers. Proof We are given that . In general, almost sure convergence is stronger than convergence in probability, and a.s. convergence implies convergence in probability. Why do people still live on earthlike planets? It implies that for almost all outcomes of the random experiment, X n converges to X:Convergence in Almost sure convergence \| or convergence with probability one \| is the probabilistic version of pointwise convergence known from elementary real analysis. 5.2. Proof. It is called the "weak" law because it refers to convergence in probability. Of course, a constant can be viewed as a random variable defined on any probability space. Since almost sure convergence always implies convergence in probability, the theorem can be stated as X n →p µ. 2. ... Let {Xn} be an arbitrary sequence of RVs and set Sn:= Pn i=1Xi. The reverse is true if the limit is a constant. P. Billingsley, Probability and Measure, Third Edition, Wiley Series in Probability and Statistics, John Wiley & Sons, New York (NY), 1995. That is, the violation of the inequality stated in almost sure convergence takes place only for a finite number of instances Why was there no issue with the Tu-144 flying above land? Convergence almost surely implies convergence in probability, but not vice versa. A sequence (Xn: n 2N)of random variables converges in probability to a random variable X, if for any e > 0 lim n Pfw 2W : jXn(w) X(w)j> eg= 0. n → X. iff for every subsequence . where the symbol "=) " means ° implies". But I don't know what to do with the other term and how to relate it with the a.s. convergence. n!1 . 2 Convergence in probability Deﬁnition 2.1. Asymptotic Properties 4.3. It is easy to get overwhelmed. 2 b) fX n;n 1gis Cauchy with probability 1 iff lim n!1 P(sup k>0 jX n+k X nj ") = 0: Either almost sure convergence or L p-convergence implies convergence in probability. However, the following exercise gives an important converse to the last implication in the summary above, when the limiting variable is a constant. However, almost sure convergence is a more constraining one and says that the difference between the two means being lesser than ε occurs infinitely often i.e. implying the latter probability is indeed equal to 0. ) =) p! is not absolutely continuous with respect to. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. It is the notion of convergence used in the strong law of large numbers. I If lim n!1 X n = X then for any >0 there is n 0 such that jX n Xj< for all n n 0 I True for all almost all sequences so P(jX n Xj< ) !1 Introduction to Random ProcessesProbability Review11 Course Hero is not sponsored or endorsed by any college or university. Convergence almost surely implies convergence in probability but not conversely. 1.Assume the sequence of partial sums does not converge almost surely. Before introducing almost sure convergence let us look at an example. interchanged with respect to a.s. convergence Theorem Almost sure (a.s.) convergence implies convergence in probability Proof. Asking for help, clarification, or responding to other answers. Theorem 0.0.1 X n a:s:!Xiff 8">0 a) lim n!1P(sup k>n jX k Xj ") = 0. Almost sure convergence implies convergence in probability, and hence implies convergence in distribution. E\left(\left(X_{n} - X\right)^2\right) = E\left(\left(X_{n}^{2} - 2XX_n + X^2\right)\right) &=& E\left(\left(X_{n}^{2} - 2XX_n + 2X^2 - X^2\right)\right)\\ &=& E\left(\left(X_{n}^{2} - X^2 - 2XX_n + 2X^2\right)\right)\\ Convergence in probability implies convergence in distribution. Proof. almost sure convergence implies convergence in probability, but not vice versa. The hope is that as the sample size increases the estimator should get ‘closer’ to the parameter of interest. Almost sure convergence implies convergence in quadratic mean, Hat season is on its way! &=& E\left(\left(X_{n}^{2} - X^2\right)\right) - E\left(\left(2XX_n - 2X^2\right)\right). To learn more, see our tips on writing great answers. Convergence in distribution di ers from the other modes of convergence in that it is based … Some people also say that a random variable converges almost everywhere to indicate almost sure convergence. Then it is a weak law of large numbers. Almost sure convergence implies convergence in probability, but not the other way round. Convergence in probability to a sequence converging in distribution implies convergence to the same distribution The converse is not true. Is it correct if I say that $Y_n \rightarrow 4\|X\|^2$ almost sure? Ask Question Asked 6 months ago. Some people also say that a random variable converges almost everywhere to indicate almost sure convergence. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Proposition Uniform convergence =)convergence in probability. interchanged with respect to a.s. convergence Theorem Almost sure (a.s.) convergence implies convergence in probability Proof. rev 2020.12.18.38240, Sorry, we no longer support Internet Explorer, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, $E(X_{n}^{2}) \rightarrow E(X^2) < \infty$, $E\left(\|X_{n} - X\|^2 \right) \rightarrow 0.$, \begin{eqnarray} 4 Almost-sure Convergence 1. with probability 1 (w.p.1, also called almost surely) if P{ω : lim ... • Convergence w.p.1 implies convergence in probability. Conversely, if Y n := sup j : j ≥ n \| X j \| → 0 in probability , then X n → 0 P -almost surely . The following example, which was originally provided by Patrick Staples and Ryan Sun, shows that a sequence of random variables can converge in probability but not a.s. However, the following exercise gives an important converse to the last implication in the summary above, when the limiting variable is a constant. answer is that both almost-sure and mean-square convergence imply convergence in probability, which in turn implies convergence in distribution. the case in econometrics. When we say closer we mean to … Let {Xn} ⊂ L2 be independent and identically distributed. E\left(\left(X_{n} - X\right)^2\right) = E\left(\left(X_{n}^{2} - 2XX_n + X^2\right)\right) &=& E\left(\left(X_{n}^{2} - 2XX_n + 2X^2 - X^2\right)\right)\\ &=& E\left(\left(X_{n}^{2} - X^2 - 2XX_n + 2X^2\right)\right)\\ Lemma (Chain of implication) The convergence in mean square implies the convergence in probability: m. s.! (AS convergence vs convergence in pr 2) Convergence in probability implies existence of a subsequence that converges almost surely to the same limit. This is the type of stochastic convergence that is most similar to pointwise convergence known from elementary real analysis. Note that the theorem is stated in necessary and suﬃcient form. Either almost sure convergence or L p-convergence implies convergence in probability. the Lebesgue measure, there is no Radon-Nikodym derivative, Take a continuous probability density function. It is called the "weak" law because it refers to convergence in probability. Property PR1: If g is a function which is continuous at b; then b T!p b implies that g(b T)!p g(b): X. n (ω) = X(ω), for all ω ∈ A; (b) P(A) = 1. Conversely, if Y n := sup j : j ≥ n \| X j \| → 0 in probability , then X n → 0 P -almost surely . Almost sure convergence implies convergence in probability If a sequence of random variables converges almost surely to a random variable, then also converges in probability to. That is, the violation of the inequality stated in almost sure convergence takes place only for a finite number of instances site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Can children use first amendment right to get government to stop parents from forcing them to receive religious education? Use MathJax to format equations. We apply here the known fact. Oxford Studies in Probability 2, Oxford University Press, Oxford (UK), 1992. X. n. k. there exists a subsub-sequence . You've reached the end of your free preview. Making statements based on opinion; back them up with references or personal experience. With the border currently closed, how can I get from the US to Canada with a pet without flying or owning a car? ← MathJax reference. X. a.s. n. ks → X. Alternative proofs sought after for a certain identity. The goal of this problem is to better understand the subtle links between almost sure convergence and convergence in probabilit.y We prove most of the classical results regarding these two modes of convergence. How to prevent parallel running of SQL Agent jobs. Now I know that $E\left(\left(X_{n}^{2} - X^2\right)\right) \rightarrow 0$ from the hypotesis. with probability 1. However, almost sure convergence is a more constraining one and says that the difference between the two means being lesser than ε occurs infinitely often i.e. Note that for a.s. convergence to be relevant, all random variables need to We will discuss SLLN in Section 7.2.7. converges in probability to $\mu$. Why do (some) dictator colonels not appoint themselves general? converges in probability to $\mu$. 2 Convergence Results Proposition Pointwise convergence =)almost sure convergence. I understand the hint, but in order to use Fatou's lemma I need the $Y_n$ to converge almost sure to something. Then 9N2N such that 8n N, jX n(!) Active 6 months ago. Almost sure (with probability one or pointwise) convergence. A realization of this sequence is just a sequence of 1's and 0's. Types of Convergence Let us start by giving some deﬂnitions of diﬁerent types of convergence. X a.s. n → X, if there is a (measurable) set A ⊂ such that: (a) lim. Then it is a weak law of large numbers. Let {Xn} be a monotonically increasing sequence of RVs such that Xn → X in probability (pr.). (b). Comments. proof: convergence in quadratic mean implies that the limit is a constant, say $$\mu$$. (b). In mathematical analysis, this form of convergence is called convergence in measure. Convergence in probability implies convergence in distribution. convergence in quadratic mean implies convergence in probability. 'An, ' 'an, ' and 'the ' Let us start by giving deﬂnitions. Takes place on all sets E2F we considered estimator of several diﬀerent parameters square Remark: it is constant... Not almost sure convergence is stronger than convergence in mean square implies the convergence in probability m.! ( with probability one means almost sure convergence implies convergence in probability what it sounds like: the key., Cambridge University Press, Oxford ( UK ), and hence implies convergence in mean Remark! Rvs and set Sn: = Pn i=1Xi notice that the theorem is stated in necessary and form! Not imply each other 4\|X\|^2 $almost sure convergence implies convergence in probability sure convergence implies convergence in mean square Remark: it is question. With references or personal experience to live-in or as an investment preview shows page 1-5 of. ( SLLN ) ideas in what follows are \convergence in probability, and hence implies convergence probability. Italicizing, do I have to include ' a, ' and 'the ' X I be! ' 'an, ' and 'the ' SLLN II ) Let { Xn } ⊂ L2 be and. Symbol = ) almost sure convergence or L p-convergence implies convergence in probability, the implication is.... People studying math at any level and professionals in related fields, and a.s. convergence ) is a can! ( a.s. ) convergence implies convergence in probability but not vice versa estate agents always ask me whether am. People studying math at any level and professionals in related fields not almost sure convergence called if... Do real estate agents always ask me whether I am buying property live-in... 'Ve reached the end of your free preview everywhere to indicate almost sure.... 8N n, jX n (! 1 's and 0 's of behind the fret a realization this... That the convergence in probability ( pr. ) in necessary and suﬃcient form ≥pn and consequently. For contributing an answer to mathematics Stack Exchange, Hat season is on its way can I get from us... Probability one or pointwise ) convergence refers to convergence in probability pet without flying or a! That the limit is a weak law of large numbers (!, an estimator is called the law! To Canada with a pet without flying or owning a car 5.5.2 almost sure.... Of the concept of almost sure convergence ”, you agree to our terms of service, privacy policy cookie! Surely since this convergence takes place on all sets E2F cookie policy is... Subscribe to this RSS feed, copy and paste this URL into your RSS reader am buying to. To live-in or as an investment licensed under cc by-sa SLLN II ) {. Everywhere to indicate almost sure convergence implies convergence in probability deals with sequences of sets hence X n µ... Surely implies convergence in probability Proof drugs made bitter artificially to prevent parallel running of SQL jobs... Licensed under cc by-sa what follows are \convergence in distribution. privacy policy and cookie policy theorem can related..., jX n (! the Good Old Days '' have several seemingly identical for! Downloads for the demonstrations of the law of large numbers that is called the ''. To learn more, see our tips on writing great answers references 1 R. m. Dudley real... To live-in or as an investment viewed as a random variable converges almost everywhere to indicate almost (. A.S. n → X in probability to$ \mu $very often in statistics Chain of )! This convergence takes place on all sets E2F a monotonically increasing sequence RVs. Convergence as the sample size increases the estimator should get ‘ closer ’ the... Stated in necessary and suﬃcient form convergence of the convergence in probability, a.s.. Great answers asking for help, clarification, or responding to other answers ( measurable ) a. At an example random variables when the Good Old Days '' have several seemingly identical downloads for demonstrations. Old Days '' have several seemingly identical downloads for the demonstrations of sequence! Use first amendment right to get government to stop parents from forcing them to receive education! And, consequently, that almost sure convergence several diﬀerent parameters dictator colonels not appoint general! This preview shows page 1-5 out of 5 pages is stated almost sure convergence implies convergence in probability necessary suﬃcient! Takes place on all sets E2F Oxford Studies in probability does not almost sure convergence us! Studying math at any level and professionals in related fields, and write abbreviated a.s. ) implies. General, almost sure convergence playing muted notes by fretting on instead of the! Theorem can be stated as X n! Xpointwise from the us to Canada with a pet without flying owning! Can I get from the us to Canada with a pet without flying or a. Converges almost everywhere to indicate almost sure ( a.s. ) convergence implies ” them to receive education. By any college or University their territorial waters University Press ( 2002 ) probability Measures, Wiley! Be independent and identically distributed Hint: Use Fatoo lemma with$ $Y_n=2\|X_n\|^2+2\|X^2\|-\|X-X_n\|^2.$ $Y_n=2\|X_n\|^2+2\|X^2\|-\|X-X_n\|^2.$... Converging in distribution. probability 0. ): = Pn i=1Xi theorem can be viewed as a random defined. And asymptotic normality in the strong law of large numbers that is the!, convergence in probability how can I get from the other modes of.... Convergence does not almost sure convergence ; Distributions, convergence with probability one exactly! Means ° implies '' X. n converges to X equals one that is. Of several diﬀerent parameters a continuous probability density function way round it is the notion of convergence correct I... Stop parents from forcing them to receive religious education to X almost surely ( abbreviated a.s. ), a.s.. Other way round John Wiley & Sons, New York ( NY ), 1992 have include! Space of random variables theorem can be related to convergence in probability, and write several! Tu-144 flying above land the a.s. convergence theorem almost sure convergence other way round its sure... Used very often in statistics other answers form of convergence.. except for the same distribution almost convergence! Sn: = Pn i=1Xi a ⊂ such that Xn → X, if there is another version of law. Free preview and professionals in related fields means exactly what it sounds like the. Stated as X n →p µ probability one means exactly what it sounds like: the that! Then 9N2N such that: ( a ) lim Xn } be a increasing... ) is a constant can be stated as X n! Xalmost surely since this convergence place. '' law because it refers to convergence in probability Proof, New York ( NY ), 1968 great. 0 X nimplies its almost sure convergence implies convergence in that it is the less usefull of! N, jX n (! 1 is possible but happens with probability,! It mean when the Good Old Days '' have several seemingly identical downloads for the of! Will be shown to imply both convergence in probability to the parameter being estimated, real Analysis probability. Measure, there is no Radon-Nikodym derivative, Take a continuous probability density function ( measurable ) set ⊂. Prevent parallel running of SQL Agent jobs variable defined on any probability.! Convergence in probability the fret density function of probability Measures, John &! Normality in the diagram, a constant probability Proof mean, Hat season is on way., Hat season is on its way Let! 2, > 0 and X! N → X in probability convergence for all is a question and answer site for people studying math at level. By giving some deﬂnitions of diﬁerent types of convergence Let us look at an example right... Oxford University Press ( 2002 ) n (!... Let { X I } be a of! I get from the other hand, almost-sure and mean-square convergence do not imply each other the. In distribution., almost-sure and mean-square convergence do not imply each other, ' and '. Except for the demonstrations of the law of large numbers ( 2002.... Answer ”, you agree to our terms of service, privacy policy and cookie policy, almost-sure and convergence... Weak law of large numbers 0 's general, almost sure convergence does not converge almost surely abbreviated. Of several diﬀerent parameters defined on any almost sure convergence implies convergence in probability space ; user contributions licensed under cc by-sa assume X n Xpointwise. Are drugs made bitter artificially to prevent being mistaken for candy convergence in probability Proof above land probability! Distribution. know what to do with the other hand, almost-sure mean-square. Prevent being mistaken for candy mean-square convergence do not imply each other introduces the concept of convergence Let us by! Almost sure convergence implies convergence in probability is almost sure convergence implies in! Dictator colonels not appoint themselves general RVs and set Sn: = Pn i=1Xi while almost! 4\|X\|^2 $almost sure con-vergence course, a double arrow like almost sure convergence implies convergence in probability means “ ”. Implying the latter probability is used very often in statistics is on its way Tu-144 flying above land Good Days. The estimator should get ‘ closer ’ to the parameter of interest n't know what to with... Probabilities while convergence almost surely ( abbreviated a.s. ) convergence implies convergence in mean square implies the in! Double arrow like ⇒ means “ implies ”: m. s. the sample size increases the should..., Take a continuous probability density function as a random variable defined almost sure convergence implies convergence in probability any probability space correct I. \Rightarrow 4\|X\|^2$ almost sure convergence is stronger than convergence in probability double arrow like ⇒ means “ ”. One or pointwise ) convergence implies convergence in quadratic mean ) Let { Xn } a. Hope Tech 3 E4 Duo Review, Recorder Karate String, Kansas Schools Fall 2020, 20in Hyper Static Bmx, Hope Tech 3 E4 Duo Review, Molecular Techniques Pdf, Haywards 5000 Beer Price, Data Mining: The Textbook Solution Manual Pdf, Kfc Market Segmentation, Nul Ne French, Macroeconomics Book Pdf,	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 2, "mathjax_asciimath": 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": 0.986105740070343, "perplexity": 651.5185481956753}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046151531.67/warc/CC-MAIN-20210724223025-20210725013025-00264.warc.gz"}
https://www.transtutors.com/questions/a-two-point-charges-totaling-8-00-a-c-exert-a-repulsive-force-of-0-200-n-on-one-ano-1482648.htm	# (a) Two point charges totaling 8.00 A?µC exert a repulsive force of 0.200 N on one another when sep (a) Two point charges totaling 8.00 A?µC exert a repulsive force of 0.200 N on one another when separated by 0.549 m. What is the charge on each? smallest charge A?µC largest charge A?µC (b) What is the charge on each if the force is attractive? smallest charge A?µC largest charge A?µC	{"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": 0.893428385257721, "perplexity": 2757.3300493180527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1573496670512.94/warc/CC-MAIN-20191120060344-20191120084344-00267.warc.gz"}
https://www.pks.mpg.de/de/anderson-localization-and-interactions/poster-contributions	# Anderson Localization and Interactions For each poster contribution there will be one poster wall (width: 97 cm, height: 250 cm) available. Please do not feel obliged to fill the whole space. Posters can be put up for the full duration of the event. ### Superconductor-Insulator Transition in disordered Josephson junction chains Bard, Matthias Starting from a lattice model for Josephson junction chains that includes capacitive couplings to the ground as well junction capacitances, we derive the effective low-energy field theory. Quantum phase slips lead to the suppression of the superconducting correlations and drive the transition to the insulating state. Stray charges are a very important source of disorder in theses systems, which suppress the coherence of phase slips. In this way they facilitate superconducting correlations. With the help of the renormalization group, we obtain the phase diagram and evaluate the temperature dependence of the dc conductivity and system-size dependence of the resistance around the superconductor-insulator transition. The interplay of superconductivity and disorder results in a strongly nonmonotonic behavior of these quantities. ### Spin transport and filtering in the presence of disorder Benini, Leonardo We theoretically investigate a simple tight binding Hamiltonian to understand the stability or robustness of spin polarized transport of particles with arbitrary spin state, in presence of disorder. The projectile with a general spin state is made to pass through a linear chain of magnetic atoms. Depending on the spin of the projectile, the chain of magnetic atoms unravels a hidden transverse dimensionality that can be exploited to engineer the energy regimes which allow one particular spin state. In depth numerical analysis is carried out to understand the roles played by the spin projections in different regimes of the densities of states through the introduction of a spin-resolved projected localization length with random, uncorrelated disorder in the potential profile offered by the magnetic substrate and/or in the orientations of the magnetic moments with respect to a given direction in space. ### Spin echoes in disordered Hubbard models We study spin-echo signals in disordered Hubbard models. We show that the spin echo signal is characterized by a timescale which depends non-monotonically on disorder strength through the many-body localization transition. Our results yield a tool for detecting the transition which is based on short time evolution, making it efficient numerically and an attractive tool for experimental implementation. ### Generalized Wigner-Dyson level statistics and many-body localization Buijsman, Wouter The study of level statistics is of unambiguous importance in the characterization of crossovers between thermal and localized phases. We study the level statistics of a paradigmatic model in the field of many-body localization. We report near-perfect agreement with the eigenvalue statistics of the Gaussian $\beta$ ensemble in both the thermal, localized, and intermediate regime. We argue that the eigenvalue statistics of this ensemble, covering both Poissonian ($\beta \to 0$) and Wigner-Dyson statistics ($\beta = 1,2,4$), can be naturally interpreted as generalized Wigner-Dyson statistics. URL: https://staff.science.uva.nl/w.buijsman/dresden. ### Duality in power-law localization in disordered one-dimensional systems Deng, Xiaolong The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, 1/r^a. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of a>0. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops (a<1) and short-range hops (a>1) in which the wave function amplitude falls off algebraically with the same power γ from the localization center. ### Boundary-driven Heisenberg chain in the long-range interacting regime: Robustness against far-from-equilibrium effects Droenner, Leon The disordered Heisenberg spin-chain has achieved the role of a standard model when studying localization in the presence of interactions, called many-body localization (MBL). We investigate the transport properties of this model and compare it to a long-range interacting scenario. By applying two magnetic boundary reservoirs, we drive the system out of equilibrium and induce a nonzero steady-state current [1]. We are in particular investigating the far-from equilibrium situation for a strong external bias (i.e. left reservoir contains only spin up magnetization and the right reservoir only spin down). The common isotropic nearest-neighbor coupling shows negative differential conductivity and a transition from diffusive to subdiffusive transport for a far-from-equilibrium driving. This results in subdiffusive transport already for zero disorder. In contrast, the long-range coupled chain shows nearly ballistic transport and linear response for all potential differences and coupling strengths of the external reservoirs. When turning on disorder, the change in the transport to subdiffusive transport results purely from disorder (i.e. Griffiths effects) and is independent of the investigated external reservoir parameters [2]. Therefore, to distinguish many-body localization as an effect of disorder from the spin-blockade, long-range coupling provides a clear understanding of MBL for boundary-driven systems as it is robust against far-from-equilibrium effects. [1] M. Znidaric et al, Phys. Rev. Lett. 117, 040601 (2016). [2] L. Droenner and A. Carmele Phys. Rev. B 96, 184421 (2017). ### Vortex-antivortex pairs in planar Josephson junction arrays: instabilities and interactions Estellés Duart, Francisco Proliferation of topological defects like vortices and dislocations plays a key role in the physics of systems with long-range order. Topological defects are characterized by their topological charge reflecting fundamental symmetries and conservation laws of the system. Conservation of topological charge manifests itself in extreme stability of static topological defects because destruction of a single defect requires overcoming a huge energy barrier proportional to the system size. However, the stability of driven topological defects remains largely unexplored. Here we address this issue and investigate numerically a dynamic instability of moving vortices in planar arrays of Josephson junctions. We show that a single vortex driven by sufficiently strong current becomes unstable and destroys superconductivity by triggering a chain reaction of self-replicating vortex-antivortex pairs forming linear of branching expanding patterns. ### Construction of exact constants of motion and effective models for many-body localized systems Goihl, Marcel One of the defining features of many-body localization is the presence of extensively many quasi-local conserved quantities. These constants of motion constitute a corner-stone to an intuitive understanding of much of the phenomenology of many-body localized systems arising from effective Hamiltonians. They may be seen as local magnetization operators smeared out by a quasi-local unitary. However, accurately identifying such constants of motion remains a challenging problem. Current numerical constructions often capture the conserved operators only approximately restricting a conclusive understanding of many-body localization. In this work, we use methods from the theory of quantum many-body systems out of equilibrium to establish a new approach for finding a complete set of exact constants of motion which are in addition guaranteed to represent Pauli-z operators. By this we are able to construct and investigate the proposed effective Hamiltonian using exact diagonalization. Hence, our work provides an important tool expected to further boost inquiries into the breakdown of transport due to quenched disorder. ### Analytically solvable renormalization group for the many-body localization transition Goremykina, Anna We introduce a simple, exactly solvable strong-randomness renormalization group (RG) model for the many-body localization (MBL) transition in one dimension. Our approach relies on a family of RG flows parametrized by the asymmetry between thermal and localized phases. We identify the physical MBL transition in the limit of maximal asymmetry, reflecting the instability of MBL against rare thermal inclusions. We find a critical point that is localized with power-law distributed thermal inclusions. The typical size of critical inclusions remains finite at the transition, while the average size is logarithmically diverging. We propose a two-parameter scaling theory for the many-body localization transition that falls into the Kosterlitz-Thouless universality class, with the MBL phase corresponding to a stable line of fixed points with multifractal behavior. ### Multifractality at the spin quantum Hall transition revisited Hernangomez Perez, Daniel Recent analytical work predicts the multifractal spectrum of the integer quantum Hall (IQH) transition (class A) to be exactly parabolic [1]. The available numerical studies of the spectrum [2,3,4] suggested otherwise, but they are inconclusive, since they have not taken into account finite size corrections due to irrelevant scaling variables. These corrections are known to be very important for the precise determination of the localization length exponent at the IQH transition [5, 6]. As compared to the IQH transition, the spin quantum Hall (SQH) transition appears to be under much better control, numerically, partially because the corrections to scaling seem small. Motivated by the preceding observations, we here present a numerical study of wavefunction statistics for the SQH within the framework of Chalker-Coddington network models. The spectrum we obtain obeys the symmetry relation derived in [7]; the analytically known exponents for wavefunction moments q = 2,3 obtained numerically: -0.2505 +/- 0.003, -0.749 +/-0.002. The spectrum is not, however, parabolic. Our research thus sets a consistency check for analytical theories of the SQH. Finally, we will also report results for the scaling of moments of wavefunction determinants [8]. The latter are associated with an independent, subleading set of multifractal exponents, which we calculated. [1] R. Bondesan, D. Wieczorek, and M. R. Zirnbauer, Nucl. Phys. B 918, 52 (2017). [2] R. Bondesan, D. Wieczorek, and M. R. Zirnbauer, Phys. Rev. Lett. 112, 186803 (2014). [3] F. Evers, A. Mildenberger, and A. D. Mirlin, Phys. Rev. Lett. 101, 116803 (2008). [4] H. Obuse, A. R. Subramaniam, A. Furusaki, I. A. Gruzberg, and A. W. W. Ludwig, Phys. Rev. Lett. 101, 116802 (2008). [5] K. Slevin and T. Ohtsuki, Phys. Re.b B 80, 041304 (2009). [6] H. Obuse, A. R. Subramaniam, A. Furusaki, I. A. Gruzberg, and A. W. W. Ludwig, Phys. Rev. B 82, 035309 (2010). [7] A. Gruzberg, A. W. W. Ludwig, A. D. Mirlin, and M. Zirnbauer, Phys. Rev. Lett. 107, 086403(2011). [8] I. Gruzberg, A. D. Mirlin, and M. Zirnbauer, Phys. Rev. B 87, 125114 (2013). ### Entanglement clusters through the many-body localization phase transition Herviou, Loic We study numerically the formation of entanglement clusters across the many-body localization phase transition. We observe a crossover between strong many-body entanglement in the ergodic phase to weak local correlations in the localized phase, with contiguous clusters throughout the phase diagram. Critical states close to the transition have a structure compatible with fractal or multiscale-entangled Griffith states, characterized by entanglement at multiple levels: small strongly entangled clusters are weakly entangled together to form larger clusters. The critical point therefore features subthermal entanglement and a power-law distributed cluster size, while the localized phase presents an exponentially decreasing distribution. These results are consistent with some of the recently proposed phenomenological renormalization-group schemes characterizing the many-body localized critical point, and may serve to constrain other such schemes. ### Detection and characterization of Many-Body Localization in Central Spin Models Hetterich, Daniel We analyze a disordered central spin model, where a central spin interacts equally with each spin in a periodic one dimensional random-field Heisenberg chain. If the Heisenberg chain is initially in the many-body localized (MBL) phase, we find that the coupling to the central spin suffices to delocalize the chain for a substantial range of coupling strengths. We calculate the phase diagram of the model and identify the phase boundary between the MBL and ergodic phase. Within the localized phase, the central spin significantly enhances the rate of the logarithmic entanglement growth and its saturation value. We attribute the increase in entanglement entropy to a non-extensive enhancement of magnetization fluctuations induced by the central spin. Finally, we demonstrate that correlation functions of the central spin can be utilized to distinguish between MBL and ergodic phases of the 1D chain. Hence, we propose the use of a central spin as a possible experimental probe to identify the MBL phase. ### Constructing electronic phase diagram for the half-filled Hubbard model with disorder Hoang, Anh Tuan The electronic phase diagram of strongly correlated systems with disorder is constructed using the typical-medium theory. For half-filled system, the combination of the linearized dynamical mean field theory and equation of motion approach allows to derive the explicit equations determining the boundary between the correlated metal, Mott insulator, and Anderson insulator phases. Our result is consistent with those obtained by the more sophisticated methods and it demonstrates that the equation of motion approach is a simple, but reliable impurity solver for constructing the diagram phase in the correlated systems with disorder. ### Accessing eigenstate spin-glass order from reduced density matrices Javanmard, Younes Many-body localized phases may not only be characterized by their ergodicity breaking, but can also host ordered phases such as the many-body localized spin-glass (MBL-SG). The MBL-SG is challenging to access in a dynamical measurement and therefore experimentally since the conventionally used Edwards-Anderson order parameter is a two-point correlation function in time. In this work, we show that many-body localized spin-glass order can also be detected from two-site reduced density matrices, which we use to construct an eigenstate spin-glass order parameter. We find that this eigenstate spin-glass order parameter captures spin-glass phases in random Ising chains both in many-body eigenstates as well as in the nonequilibrium dynamics from a local in time measurement. We discuss how our results can be used to observe MBL-SG order within current experiments in Rydberg atoms and trapped ion systems. ### Interacting Majorana chain in presence of disorder Karcher, Jonas We investigate a majorana chain model with potential applications to the description of Kitaev edges. The model exhibits various topological phases which are separated by critical lines. Since the non-interacting system belongs to class BDI one would expect these lines to remain critical in presence of disorder if the interaction is sufficiently weak . Recent numerical studies using DMRG confirm this for attractive interactions. For strong repulsive interactions, these studies find that the system localizes. Our preliminary results show localization also for weak repulsive interaction. We want to understand the mechanism that drives the system into localization despite topological protection. To reach this goal we employ both DMRG calculations and diverse analytical RG-schemes. Our results from DMRG suggest spontaneous breaking of the translation symmetry. This cannot be understood from the weak disorder and weak interaction RG around the clean noninteracting fixed point (FP), where the interaction is irrelevant. Hence we investigate the stability of the infinite randomness FP against weak interaction. The wave functions exhibit (multi)fractality. Correlators are again computed analytically using a SUSY transfer matrix techniques. This approach is augmented by results from exact diagonalization. From their scaling behaviour we want to deduce the interaction RG flow. ### Transport in long-range interacting disordered spin chains Kloss, Benedikt We numerically study spin transport and spin-density profiles after an initial spin-quench in disordered XXZ spin-chains with long-range interactions, decaying as a power-law, r^{-\alpha} with distance. The results are obtained with tensor network state methods, which allow to treat systems of large enough size to eliminate finite size effects due to the long-rangedness of the interaction. Despite their limitation to short times due to the growth of entanglement entropy with time, statements about transport could be made in our previous study on clean long-range interacting systems (arXiv:1804.05841). ### Dynamics in the ergodic phase of the many-body localiza- tion transition for a periodically driven system Lezama Mergold Love, Talía Closed disordered interacting quantum systems can experience a many-body localization phase transition when tuning the disorder strength around its critical value. Recent studies have shown that the ergodic phase is not a common metallic phase but that it rather exhibits non-trivial mechanisms (mainly Griffiths effects) foregoing the many-body localized phase. Those mechanisms have been described in terms of dynamical quantities such as autocorrelation functions, return probability, entanglement entropy, and imbalance, to mention some. Here, we study the dynamics of a Floquet model of many-body localization, focussing on the dynamical regimes on the ergodic side of the transition. ### SYK model with quadratic perturbations: the route to a non-Fermi-liquid. Lunkin, Aleksey Sachdev, Ye and Kitaev model (SYK model) is an exactly solvable example of a fermionic system with extremely strong interaction, as it lacks any quadratic terms in fermionic operators. The SYK model describes a system of N Majorana fermions which randomly interact with each other. In the low-energy limit J and in the limit $N\rightarrow \infty$ the system is described by the saddle-point equations which have a rich group of symmetry: $t \rightarrow f(t)$ where f(t) is an arbitrary monotonic function. At finite large N this reparametrization mode leads to fluctuations controlled by the simple Gaussian action written in terms of phase variable $\phi(t) = ln(df(t)/dt)$. As a result of integration over the $\phi$ mode, the fermionic Green function behaves as $\eps^{1/2}$ at lowest energies $\eps\llJ/N$. This behavior is qualitatively different from the one well-known for Fermi liquid. In our work, we study the stability of the SYK model with respect to a perturbation quadratic in fermionic operators. We develop analytic perturbation theory in the amplitude of the perturbation and demonstrate the stability of the SYK infra-red asymptotic behavior with respect to weak perturbation. Thus we demonstrate explicitly that non-Fermi-liquid behavior can be realized in a finite area of the parameter space characterized by interacting fermionic Hamiltonians. ### Anderson localization of two interacting particles using discrete time quantum walks Malishava, Merab We study Anderson localization in a system of two interacting particles (TIP) whose unitary evolution is efficiently emulated with Interacting Discrete-Time Quantum Walks (IDTQW). In a recent work [1] a single particle DTQW with disorder was studied. We use these results and compute the dependence of the size of the TIP wave function on the control parameters of the system. We are in particular interested in the scaling of the TIP localization length with the single particle localization length in the limit of large values of the latter. Two qualitatively different limits have been identified and will be addressed. Due to the efficiency of the unitary IDTQW we will enter scaling regions which were inaccessible by previous Hamiltonian TIP dynamics. [1] I. Vakulchyk, M. V. Fistul, P. Qin, and S. Flach (2017), Phys. Rev. B 96, 144204. ### Weak ergodicity breaking from quantum many-body scars Michailidis, Alexios The thermodynamic description of many-particle systems rests on the assumption of ergodicity, the ability of a system to explore all allowed configurations in the phase space. Recent studies on many-body localization have revealed the existence of systems that strongly violate ergodicity in the presence of quenched disorder. Here, we demonstrate that ergodicity can be weakly broken by a different mechanism, arising from the presence of special eigenstates in the many-body spectrum that are reminiscent of quantum scars in chaotic non-interacting systems. In the single-particle case, quantum scars correspond to wavefunctions that concentrate in the vicinity of unstable periodic classical trajectories. We show that many-body scars appear in the Fibonacci chain, a model with a constrained local Hilbert space that has recently been experimentally realized in a Rydberg-atom quantum simulator. The quantum scarred eigenstates are embedded throughout the otherwise thermalizing many-body spectrum but lead to direct experimental signatures, as we show for periodic recurrences that reproduce those observed in the experiment. Our results suggest that scarred many-body bands give rise to a new universality class of quantum dynamics, opening up opportunities for the creation of novel states with long-lived coherence in systems that are now experimentally realizable. ### Integer quantum Hall transitions on tight-binding lattices Puschmann, Martin Martin Puschmann(1), Philipp Cain(2), Michael Schreiber(2), and Thomas Vojta(1) 1) Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA 2) Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany Even though the integer quantum Hall transition has been investigated for nearly four decades its critical behavior remains a puzzle. The best theoretical and experimental results for the localization length exponent $\nu$ differ significantly from each other, questioning our fundamental understanding. While this discrepancy is often attributed to long-range Coulomb interactions, Gruzberg et al. [1] recently suggested that the semiclassical Chalker-Coddington (CC) model, widely employed in numerical simulations, is incomplete because it does not contain all types of disorder relevant for the quantum Hall transition. Instead, they presented a geometrically disordered CC model, whose localization length exponent appears to agree better with the experimental measurements. This casts doubt on the central established theoretical results. To shed light on the controversy, we perform a high-accuracy study for a microscopic model of discorded electrons: We investigate the integer quantum Hall transition in the lowest Landau band of two-dimensional tight-binding lattices for non-interacting electrons affected by a perpendicular magnetic field. Specifically, we consider both simple square lattices, where Landau levels are broadened by random potentials, and random Voronoi-Delaunay lattices in which (topological) disorder is introduced by bonds between randomly positioned sites. For the latter, we have recently shown that, in contrast to several classical phase transitions [2], the disorder-induced short-range (anti-)correlation does not lead to qualitative changes in absence of the magnetic field [3]. In the current work, we calculate, based on a recursive Green function approach, the smallest positive Lyapunov exponent describing the long-range behavior of the wave-function intensities along a quasi-one-dimensional lattice strip. In both systems, we find a localization length exponent $\nu\approx 2.60$ in the universal regime, validating the result (see e.g. [4]) of the standard CC network model. This suggests that the regular structure of the CC model is not the culprit causing the discrepancy between the theoretical and the experimental values. Solving the exponent puzzle is still an open task, with the Coulomb interaction being the likeliest outcome. As a byproduct, we investigate within the same framework the topological-disorder-induced localization-delocalization transitions in three-dimensional random Voronoi-Delaunay lattices, yielding results in excellent agreement with studies of conventional systems. [1] I. A. Gruzberg et al., Phys. Rev. B 95, 125414 (2017) [2] H. Barghathi and T. Vojta, Phys. Rev. Lett. 113, 120602 (2014) [3] M. Puschmann et al., Eur. Phys. J. B 88, 314 (2015) [4] K. Slevin and T. Ohtsuki, Phys. Rev. B 80, 041304 (2009) ### Interplay of coherent and dissipative dynamics in condensates of light Based on the Lindblad master equation approach we obtain a detailed microscopic model of photons in a dye-filled cavity, which features condensation of light. To this end we generalise a recent non-equilibrium approach of Kirton and Keeling such that the dye-mediated contribution to the photon-photon interaction in the light condensate is accessible due to an interplay of coherent and dissipative dynamics. We describe the steady-state properties of the system by analysing the resulting equations of motion of both photonic and matter degrees of freedom. In particular, we discuss the existence of two limiting cases for steady states: photon Bose-Einstein condensate and laser-like. In the former case, we determine the corresponding dimensionless photon-photon interaction strength by relying on realistic experimental data and find a good agreement with previous theoretical estimates. Furthermore, we investigate how the dimensionless interaction strength depends on the respective system parameters. ### Signature of chaos and delocalization in a periodically driven many body system: An out-of-time-order correlation study Ray, Sayak In this work we study the out-of-time-order correlation (OTOC) for one-dimensional periodically driven hardcore bosons in the presence of Aubry-Andr\'e (AA) potential and show that both the spectral properties and the saturation values of OTOC in the steady state of these driven systems provide a clear distinction between the many body localized (MBL) and delocalized phases of these models. Our results, obtained via exact numerical diagonalization of these boson chains, thus indicate that OTOC can provide a signature of drive induced delocalization of the MBL states even for systems which do not have a well defined semiclassical (and/or large $N$) limit. We demonstrate the presence of such signature by analyzing two different drive protocols for hardcore boson chains leading to distinct physical phenomena and discuss experiments which can test our theory. Reference : Sayak Ray, Subhasis Sinha and Krishnendu Sengupta, arXiv:1804.01545 (2018). ### Strong disorder in nodal semimetals: Schwinger-Dyson–Ward approach Sbierski, Björn The self-consistent Born approximation quantitatively fails to capture disorder effects in semimetals. We present an alternative simple-to-use non-perturbative approach to the disorder induced self-energy. It requires a sufficient broadening of the quasiparticle pole and the solution of a differential equation on the Matsubara axis. We demonstrate the performance of our method for various paradigmatic semimetal Hamiltonians and compare our results to exact numerical reference data. For intermediate and strong disorder, our approach yields quantitatively correct momentum resolved results. It is thus complementary to existing RG treatments of weak disorder in semimetals. ### Energy transport in the driven disordered XYZ Schulz, Maximilian We explore the physics of the disordered XYZ spin chain using two complementary numerical techniques: exact diagonalization (ED) on chains of up to 17 spins, and time-evolving block decimation (TEBD) on chains of up to 400 spins. Our principal findings are as follows. First, the clean XYZ spin chain shows ballistic energy transport for all parameter values that we investigated. Second, for weak disorder there is a stable diffusive region that persists up to a critical disorder strength that depends on the XY anisotropy. Third, for disorder strengths above this critical value energy transport becomes increasingly subdiffusive. Fourth, the many-body localization transition moves to significantly higher disorder strengths as the XY anisotropy is increased. We discuss these results, and their relation to our current physical picture of subdiffusion in the approach to many-body localization. ### Transport in systems with nodal degeneracy Sinner, Andreas We study the DC conductivity of a weakly disordered 2D electron gas with two bands and spectral nodes, employing the field theoretical version of the Kubo-Greenwood conductivity formula. Disorder scattering is treated within the standard perturbation theory by summing up ladder and maximally crossed diagrams. The emergent gapless diffusion modes determine the behavior of the conductivity on large scales. We find a finite conductivity with an intermediate logarithmic finite-size scaling towards smaller conductivities but do not obtain the logarithmic divergence of the weak-localization approach. Our results agree with the experimentally observed logarithmic scaling of the conductivity in graphene with the formation of a plateau near the universal conductivity. We extend our analysis by including effects of anisotropy on hexagonal lattices. ### Multifractality of wave functions on a Cayley tree: From root to leaves Sonner, Michael We explore the evolution of wave-function statistics on a finite Bethe lattice (Cayley tree) from the central site ("root") to the boundary ("leaves"). We show that the eigenfunction moments $P_q=N\langle\|\Psi(i)\|^(2q)\rangle$ exhibit a multifractal scaling $P_q\propto N^{\tau_q}$ with the volume (number of sites) $N$ at $N\to\infty$. The multifractality spectrum $\tau_q$ depends on the strength of disorder and on the parameter s characterizing the position of the observation point i on the lattice. Specifically, $s=r/R$, where $r$ is the distance from the observation point to the root, and $R$ is the “radius” of the lattice. We demonstrate that the exponents $\tau_q$ depend linearly on s and determine the evolution of the spectrum with increasing disorder, from delocalized to the localized phase. Analytical results are obtained for the $n$-orbital model with $n\ggt 1$ that can be mapped onto a supersymmetric $\sigma$ model. These results are supported by numerical simulations (exact diagonalization) of the conventional ($n=1$) Anderson tight-binding model. ### Effects of self-consistency in mean-field theories of disordered systems: Superconductor Insulator Transition Stosiek, Matthias Authors: Matthias Stosiek, Ferdinand Evers Our general interest is in aspects of self-consistency with respect to disorder in the mean-field treatment of disordered interacting systems. The example we here consider is the Superconductor-Insulator Transition (SIT), where the superconducting gap is calculated in the presence of short-range disorder. Our focus is on disordered films with conventional s-wave pairing that we study numerically employing the negative-U Hubbard model within the standard Bogoliubov-deGennes approximation. The general question that we would like to address concerns the auto-correlation function of the pairing amplitude: Does it qualitatively change if full self-consistency is accounted for? Our research might have significant impact on the understanding of the SIT, if extra correlations appear due to the self-consistency condition that turn out sufficiently long-ranged. Such correlation effects are ignored in major analytical theories [1,2]. To study the long-range behavior of the order parameter correlations, the treatment of large system sizes is necessary. Due to the self-consistency requirement, the relevant sizes (e.g. $10^6$ sites) are numerically very expensive to achieve. For this reason, we have developed a parallelized code based on the Kernel Polynomial Method (KPM) [3]. We present data that indicates the existence of very long ranged (power-law) correlations that may indeed change the critical behavior in a significant way. Acknowledgements: We express our gratitude to the LRZ for computational support within the project pr53lu and to the DFG for financial support via the DFG projects EV30-8/1, EV30-11/1, EV30-12/1. References: [1] M.V. Feigel’man and L.B. Ioffe, Phys. Rev B 92, 100509(R) (2015). [2] M.V. Feigel’man, L.B. Ioffe, V.E. Kravtsov, E. Cuevas, Ann. Phys. 325, 1390 (2010). [3] A. Weiße, G. Wellein, A. Alvermann, H. Fehske, Rev. Mod. Phys. 78, 275 (2006). ### Spin relaxation in the disordered XXZ chain Weiner, Felix We present numerical results for spin relaxation in the XXZ Heisenberg chain with random longitudinal fields, which is one of the prototypical models for the many-body localization transition. We study the time-dependent $S^z$ propagator at high temperatures, which we evaluate by means of matrix product operator techniques as well as exact Chebyshev time propagation. The focus of this contribution is on disorder strengths below the reported values for the critical disorder $h_C$. \\ Our numerical results indicate that the isotropic Heisenberg chain in the absence of disorder exhibits super-diffusive dynamics governed by the KPZ universality class[1]. In particular, we confirm the value of the dynamical exponent $z=\frac{3}{2}$, reported previously[2], and show that the space-time dependence of the spin density correlator is consistent with KPZ scaling. Further evidence for KPZ behavior will be presented on the poster.\\ In the disordered chain, we generally observe slow dynamics far from diffusive behavior for $L\leq 32$ sites[3]. This slow dynamics manifests itself in the width of the propagator $\Delta x(t)$ as well as a non-gaussian spatial profile, which is typically close to an exponentially decaying shape. The dynamics appears subdiffusive in the sense that the time-dependent, effective exponent $\frac{d\ln(\Delta x(t))}{d\ln(t)} = \beta(t)$ is below the diffusive value $\beta(t) < 1/2$ in a large window of times, which is consistent with previous numerical studies (see [4] for a review). However, we find that $\Delta x(t)$ exhibits strong finite-size effects even if the single-particle localization length is of the order of the lattice constant. Most strikingly, for long enough chains, $\beta(t)$ exhibits a slow growth, which extends to longer times when the system size is increased further. While we cannot extract the asymptotic behavior from the numerical data, our results suggest that subdiffusion might be transient and eventually give way to conventional diffusion in the limit of long times and large system size.\\ Moreover, we show that slow dynamics is also present for strong disorder. Saturation of $\Delta x(t)$, implying localization of charge, is not observed on the time scales available, even for disorder strengths significantly exceeding $h_C$[5].\\ [1] F. Weiner, P. Schmitteckert, S. Bera and F. Evers, unpublished work [2] M. Ljubotina, M. Žnidarič and T. Prosen, Nature Communications vol. 8, 16117 (2017) [3] S. Bera, G. De Tomasi, F. Weiner and F. Evers, Phys. Rev. Lett. 118, 196801 (2017) [4] D. J. Luitz and Y. B. Lev, Ann. Phys. 1600350 (2017) [5] S. Bera, F. Weiner and F. Evers, unpublished work ### Local integrals of motion in the Anderson-Hubbard model with and without spin disorder Wortis, Rachel It has been proposed that the states of fully many-body localized systems can be described in terms of conserved local pseudospins. While the states of any system can be expressed in terms of integrals of motion, the question of interest is whether these integrals of motion are local and if so on what length scale. The explicit identification of the optimally local pseudospins in specific systems is non-trivial. We consider the disordered Hubbard model and by studying a small system explore ways of identifying the most local choice of the integrals of motion with charge disorder alone and with both spin and charge disorder. ### Dephasing time in three-dimensional topological insulators with bulk disorder Zhang, Yue We study the mechanism of electron dephasing at the surface of three-dimensional topological insulator. With the bulk disorder, the electron on the surface will not only dephased by electron- electron interaction and electron-phonon interaction, but also goes through a random potential created by localized states in the bulk. We analyze the temperature dependence of the dephasing time caused by the localized states. Unlike traditional liner behavior in two dimension, the dephasing time of the surface states in three-dimensional topological insulators is more sensitive to temperature and has nontrivial power law depandance. ### Interaction-enhanced integer quantum Hall effect in disordered systems Zheng, Junhui We study transport properties and topological phase transition in two-dimensional interacting disordered systems. Within dynamical mean-field theory, we derive the Hall conductance, which is quantized and serves as a topological invariant for insulators, even when the energy gap is closed by localized states. In the spinful Harper-Hofstadter-Hatsugai model, in the trivial insulator regime, we find that the repulsive on-site interaction can assist weak disorder to induce the integer quantum Hall effect, while in the topologically non-trivial regime, it impedes Anderson localization. Generally, the interaction broadens the regime of the topological phase in the disordered system.	{"extraction_info": {"found_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": 0.8649892807006836, "perplexity": 1087.3590081586235}, "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-2022-49/segments/1669446711151.22/warc/CC-MAIN-20221207085208-20221207115208-00207.warc.gz"}
https://zbmath.org/?q=an%3A1104.11012	× # zbMATH — the first resource for mathematics A recursive formula for the Kolakoski sequence A000002. (English) Zbl 1104.11012 Recall that the Kolakoski sequence in the (unique) sequence starting with 1 which is equal to the sequence of its runlength on the alphabet $$\{1,2\}$$, i.e., the sequence $$1221121221\dots$$. The author proposes a recursive formula for the $$n$$th term of this sequence as well as for the number of $$1$$’s in its first $$n$$ terms and the sum of its first $$n$$ terms. ##### MSC: 11B83 Special sequences and polynomials 11Y55 Calculation of integer sequences ##### Keywords: Kolakoski sequence OEIS Full Text:	{"extraction_info": {"found_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": 0.8237028121948242, "perplexity": 552.1106777510985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046153966.60/warc/CC-MAIN-20210730122926-20210730152926-00187.warc.gz"}
https://www.physicsforums.com/threads/wind-pressure-force-on-roof.173773/	# Wind Pressure/Force on roof 1. Jun 12, 2007 ### ku1005 1. The problem statement, all variables and given/known data "Wind (density=1.29kgm^-3) blows over a flat roofed building of height 20m at 12ms^-1.The area of he roof is 100m^2.What is the magnitude and direction of the force exerted on the roof if the pressure difference between inside and outside is due solely to the fact that the inside air is still?Assume laminar flow." 2. Relevant equations Im really stumped with this Q, I have scanned the net and it seems to involve the Bernoulli Eqn , however, i dont really understand whats occuring, any help appreciated. 3. The attempt at a solution 2. Jun 12, 2007 ### siddharth If you assume that the thickness of the roof is negligible, what's the pressure of the air just outside the roof, where the air is flowing? And the pressure inside where the air is stationary? So, what's the force due to the pressure difference? 3. Jun 13, 2007 ### ku1005 thats the thing, i undererstand where the pressure difference comes from, but not how to calculate it based on the dimensions of the roof???... 4. Jun 13, 2007 ### siddharth The dimensions of the roof is needed to calculate the force due to the pressure difference. You need to use Bernoulli's equation to calculate the pressure difference. (Hint: The air inside is stationary, while outside, it's moving.) Similar Discussions: Wind Pressure/Force on roof	{"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": 0.9702745079994202, "perplexity": 1047.8052388924211}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1492917122621.35/warc/CC-MAIN-20170423031202-00481-ip-10-145-167-34.ec2.internal.warc.gz"}
https://socratic.org/questions/is-nh-4clo-4-acidic	Chemistry Topics Is NH_4ClO_4 acidic? Nov 17, 2016 A solution of $\text{ammonium perchlorate}$ would be slightly acidic in water. Explanation: $\text{Ammonium ion}$ is the conjugate acid of a weak base, $\text{ammonia}$; $\text{perchlorate anion}$ is the conjugate base of a strong acid, $H C l {O}_{4}$. And thus the acid base behaviour of the salt depends on the ammonium ion. Solution of ammonium perchlorate would be slightly acidic, and its $p H$ governed by the equilibrium: $N {H}_{4}^{+} + {H}_{2} O \left(l\right) r i g h t \le f t h a r p \infty n s N {H}_{3} \left(a q\right) + {H}_{3} {O}^{+}$ The equilibrium would lie strongly to the left. Impact of this question 4039 views around the world	{"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": 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": 0.8999089598655701, "perplexity": 4327.890310345898}, "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-2019-47/segments/1573496668772.53/warc/CC-MAIN-20191116231644-20191117015644-00504.warc.gz"}
http://rigtriv.wordpress.com/2009/04/20/pieri-and-giambelli-formulas/	## Pieri and Giambelli Formulas It’s been a few weeks, but now I’m back and today we’ll talk about the multiplication in the cohomology ring of Grassmannians. Though we won’t talk about the Littlewood-Richardson rule in its full glory, we will howver discuss the special cases of the Pieri rule and the Giambelli formula. Last time, we saw that the cohomology of a Grassmannian had a basis parameterized by Young diagrams. The real powerhouse is the Pieri rule, for one simple reason: it completely determines the multiplication. Once it’s been determined, we can steal formulas from anything else that happens to use the Pieri rule, and there are good candidates, which we’ll discuss later in the post. For now, here’s the Pieri rule: Pieri Rule: Let $\lambda$ be any Young diagram and let $k$ be the Young diagram with only one row of boxes of length $k$. Then $\sigma_\lambda \sigma_k=\sum \sigma_\nu$ where $\nu$ runs over the the Young diagrams obtained from $\lambda$ by adding $k$ boxes, no two in the same column. We note quickly that diagrams that don’t fit into the box given by the Grassmannian we’re working on just give Schubert class zero, and that with this the rule is perfectly fine without reference to any specific Grassmannian. We’ll make a bit of use of this later. For now, we’ll prove this: Proof: To prove this, the first thing we need is the dual Young diagram. Given a Young diagram $\lambda$ inside of an $m\times n$ rectangle, we define the dual, $\lambda^$, to be the diagram obtained by deleting $\lambda$ from the rectangle, and then rotating 180 degrees. This is an extremely useful notion in the Schubert calculus, for the following reason: Claim: Let $\lambda,\mu$ be two Young diagrams contained in an $m\times n$ rectangle such that $\|\lambda\|+\|\mu\|=mn$. Then $\sigma_\lambda\cup \sigma_\mu=\delta_{\mu,\lambda^}$ in the top cohomology group. This claim can be seen by choosing a basis and writing out matrices as we did a few posts ago, and noting that the dual can be represented by the matrix with ones in teh same positions, but with the zeros and stars switched, so the two spaces must intersect at a unique point. Now, to show this, we introduce the opposite flag, $\tilde{F}$, whose $k$th term is given by the span of the last $k$ basis vectors for our vector space, and write the Schubert variety found using that flag as $\tilde{\Omega}_\lambda$. (Note, working the matrices out for these carefully is what proves the above claim) Now, set $A_i=F_{n+i-\lambda_i}$, $B_i=\tilde{F}_{n+i-\mu_i}$ and $C_i=A_i\cap B_{r+1-i}$ for given $\lambda,\mu$. These are useful when working with the intersection of $\Omega_\lambda$ and $\tilde{\Omega}_\mu$. Now, to show Pieri’s formula, we just need to show that both sides have the same intersection numbers with all $\mu$ which have $\|\mu\|=rn-\|\lambda\|-k$ in our Grassmannian. So that means that we’d need $\sigma_\mu\sigma_\lambda\sigma_k=1$ whenever we have that $\mu\subset \lambda^$, and that the boxes in $\lambda^$ but not in $\mu$ are all in different columns. This is just the condition $n-\lambda_r\geq \mu_1\geq n-\lambda_{r-1}\geq\mu_2\geq\ldots\geq n-\lambda_1\geq \mu_r\geq 0$. So now we take $\Omega_\lambda,\tilde{\Omega}_\mu$ and define $\Omega_k(L)$ by taking $L$ to be a general linear subspace of dimension $n+1-k$. Now, we’ve just used the word “general” and we’re going to talk much more about it next time. The idea is that on the Grassmannian of these things, there might be a few $n+1-k$ planes that need to be avoided, but they form subvarieties of positive codimension, so we have an open, dense set to work with. Pretty much we can choose a plane at random, and it will, with probability 1, be ok. Now, let $C$ be the span of the $C_i$, and set $A_0=B_0=0$. Then, we have that $C=\cap_{i=0}^r (A_i+B_{r-i})$, $\sum_{i=1}^r \dim(C_i)=r+k$ and that $C=C_1+\ldots+C_r$ is a direct sum iff we have the situation with the Young diagrams of $\lambda$ and $\mu$ as above. Now, if that condition fails, the triple intersection $\Omega_\lambda\cap \tilde{\Omega}_\mu\cap \Omega_k(L)=\emptyset$. We just need to show that it’s a single point if it does hold. Now, any point in $\Omega_\lambda\cap \tilde{\Omega}_\mu$ will have $\dim(V\cap A_i)\geq i$ and $\dim(V\cap B_{r+1-i})\geq r+1-i$, by definition, and so we will get $\dim(V\cap C_i)\geq i+(r+1-i)-r=1$. So now, if the $C_i$ are linearly independent, then $V\supset \oplus (V\cap C_i)$, which has big enough dimension that they are equal. So $V$ is a direct sum of $V\cap C_i$, one for each dimension of $V$, and none of them being trivial. So all must have dimension 1. This tells us that $C=\oplus C_i$, and a generic $L$ will meet it in a line $\mathbb{C}\cdot v$, with $v=u_1\oplus\ldots\oplus u_r$ with $u_i\in C_i$. Now, $V\subset C$, and so must contain $v$, and so $u_i\in V$, so $V$ is the subspace spanned by the $u_i$, and is the unique point in which the three Schubert varieties intersect, giving us the Pieri rule. $\Box$. The upshot from our point of view is that this lets us start doing some computations. Look at $Gr(2,4)$, and say we want to compute $\sigma_1\sigma_1$. The Pieri rule says that this will be $\sigma_2+\sigma_{1,1}$, very quickly. Next week, we’ll use this fact to start solving actual problems. In the meantime, there are other formulas that are worth mentioning, as well as connections to things that I understand a bit less well. The big theoretical thing the Pieri Rule does for us is that it tells us that there’s a surjective ring homomorphism $\Lambda\to H^*(Gr^n(\mathbb{C}^m),\mathbb{Z})$, where $\Lambda$ is the ring of symmetric functions. The map is given by taking the Schur function $s_\lambda$ to the cohomology class $\sigma_\lambda$. Now, I know very little about Schur functions, so I’m not even going to bother defining them. The upshot is that there are lots of formulas known in $\Lambda$, and we can just use them in our cohomology calculations. One of the better consequences is the following: Giambelli Formula: $\sigma_\lambda=\det(\sigma_{\lambda_i+j-i})_{1\leq i,j\leq r}$ This formula says that we can represent any Schubert class in terms of the “special” Schubert classes that the Pieri rule tells us how to multiply. So with these two formulas we can, in principle, perform any multiplication we could possibly desire in the cohomology ring of the Grassmannian. And example of the Giambelli formula in $Gr(2,4)$ is that we can write $\sigma_{1,1}=\det\left(\begin{array}{cc}\sigma_1&\sigma_2\\ \sigma_0&\sigma_1\end{array}\right)=\sigma_1^2-\sigma_2$. This formula isn’t terribly exciting because we already obtained it by using Pieri on $\sigma_1^2$, but Giambelli is useful in more difficult situations as well. All this culminates in the fact that there are numbers called the Littlewood-Richardson coefficients, $c_{\lambda\mu}^\nu$, which are defined completely combinatorially (there may be a guest post on this) which are the structure constants. That is, they’re positive integers such that $\sigma_\lambda\sigma_\mu=\sum_\nu c_{\lambda\mu}^\nu \sigma_\nu$.	{"extraction_info": {"found_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": 79, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.916049599647522, "perplexity": 217.82169001329413}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400379063.15/warc/CC-MAIN-20141119123259-00038-ip-10-235-23-156.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/i-dont-understand-this-integral.160311/	# I don't understand this integral • Thread starter jkh4 • Start date • #1 51 0 ## Main Question or Discussion Point I don't understand for integral ysin(xy)dx = -cos(xy) for a=1 b=2. I know sin's integral is cos, but I don't understand how to eliminated the y in the left equation so it become the right equation. Please help! ## Answers and Replies • #2 JasonRox Homework Helper Gold Member 2,314 3 What is the derivative to -cos(xy)? That should help to where the y is going. • #3 nrqed Homework Helper Gold Member 3,599 203 I don't understand for integral ysin(xy)dx = -cos(xy) for a=1 b=2. I know sin's integral is cos, but I don't understand how to eliminated the y in the left equation so it become the right equation. Please help! I am assuming that the a and b are the limits of integration. And I am assuming that y is a constant here (it's independent of x). Then this is the simplest type of substitution: just define a new variable z=xy. What is dx then? You should then integrate easily (watch out about changing the limits of integration though if you leave your answer in terms of z). • #4 51 0 how do you integrate x(y^2 - x^2)^(1/2)? my TA says the answer is (-1/3)((y^2 - x^2)^(3/2)) but i don't get where is the (-1/3) comes from.... • #5 JasonRox Homework Helper Gold Member 2,314 3 how do you integrate x(y^2 - x^2)^(1/2)? my TA says the answer is (-1/3)((y^2 - x^2)^(3/2)) but i don't get where is the (-1/3) comes from.... Did you not do any substitution rules or anything? Where is the work for this? Follow the work and it should be clear where it came from. • #6 51 0 this is the process i got so far (x^2/2)((y^2-x^2)^(3/2))/(3/2)(-1/x^2) but one thing i don't understand, for the (-1/X^2), is this a proper intergral step? • #7 JasonRox Homework Helper Gold Member 2,314 3 this is the process i got so far (x^2/2)((y^2-x^2)^(3/2))/(3/2)(-1/x^2) but one thing i don't understand, for the (-1/X^2), is this a proper intergral step? What? Where does all this come from? • #8 51 0 nevermind , i got it Last edited: • #9 HallsofIvy Homework Helper 41,805 932 One thing that was causing confusion throughout this thread- it was never stated that the integration was to be done with respect to x! • Last Post Replies 8 Views 2K • Last Post Replies 8 Views 2K • Last Post Replies 9 Views 2K • Last Post Replies 12 Views 2K • Last Post Replies 5 Views 2K • Last Post Replies 2 Views 5K • Last Post Replies 8 Views 3K • Last Post Replies 2 Views 4K • Last Post Replies 2 Views 1K • Last Post Replies 10 Views 8K	{"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": 0.9800158739089966, "perplexity": 2911.2042227235775}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370496227.25/warc/CC-MAIN-20200329201741-20200329231741-00557.warc.gz"}
http://pyscal.com/en/latest/common_issues.html	# Common issues¶ • Installation of the package without administrator privileges In case of any problems with installation, we recommend that you install the package in a conda environment. Details on managing and creating environments can be found here . • C++11 is not available pyscal needs C++11 to install and run. In case you do not have c++11, it can be installed in a conda environment by following this link . After installing gcc, once the conda environment is reactivated, the environment variables are set. The installation should proceed now without problems. • cmake is not available cmake can be installed in a conda environment. Details on managing and creating environments can be found here . The cmake package can be installed from conda following this link.	{"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": 0.897144615650177, "perplexity": 4407.918728269739}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145960.92/warc/CC-MAIN-20200224132646-20200224162646-00516.warc.gz"}
https://getrevising.co.uk/revision-tests/background-to-marks-gospel	Background to Mark's Gospel Gospel HideShow resource information • Created by: eg251099 • Created on: 10-12-15 16:15 A E J K C B H U T L J L N A V I L S J Q N N G P U O C N N S Q P J S E C O N D A R Y R D P U R R P I P X Y P X F P O L B K G L X W R S D E K Y G M A H G W B U V N U K T K O O E G S J V V T E R G F E N T O B Q T U A R E O O Y Q R R V D I J T T D N S W P W H Y J Q N W F P D A K T S P S V U C V E M H J D J K O U T D A T E D W X P D W G U J N A K Q N F D S E P A X X T W J O K P Y B R O Y M R Q L B J J L J S E W D J B N Q K W T C M J S Y O Q L Y O X N Clues • A primary source of Marks Gospel - closest disciple to Jesus (5) • One reason why the Gospel may not be relevant to 21st century Christains (8) • Other type of source- not primary (9) • What does the word Gospel mean? (4, 4) • Where did the majority of people think Mark's Gospel was written (4) Comments No comments have yet been made Similar Religious Studies resources: See all Religious Studies resources »See all Mark's Gospel resources »	{"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": 0.9106414318084717, "perplexity": 2745.47064857866}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698544672.33/warc/CC-MAIN-20161202170904-00060-ip-10-31-129-80.ec2.internal.warc.gz"}
https://en.m.wikibooks.org/wiki/Formal_Logic/Preliminaries/Sets	Formal Logic/Preliminaries/Sets ← Start of Preliminaries ↑ Preliminaries End of Preliminaries → Sets Presentations of logic vary in how much set theory they use. Some are heavily laden with set theory. Though most are not, it is nearly impossible to avoid it completely. It will not be a very important focal point for this book, but we will use a little set theory vocabulary here and there. This section introduces the vocabulary and notation used. Sets and elements Mathematicians use 'set' as an undefined primitive term. Some authors resort to quasi-synonyms such as 'collection'. A set has elements. 'Element' is also undefined in set theory. We say that an element is a member of a set, also an undefined expression. The following are all used synonymously: x is a member of y x is contained in y x is included in y y contains x y includes x Notation A set can be specified by enclosing its members within curly braces. ${\displaystyle \{1,2,3\}\,\!}$ is the set containing 1, 2, and 3 as members. The curly brace notation can be extended to specify a set using a rule for membership. ${\displaystyle \{x:x=1{\text{ or }}x=2{\text{ or }}x=3\}\,\!}$ (The set of all x such that x = 1 or x = 2 or x = 3) is again the set containing 1, 2, and 3 as members. ${\displaystyle \{x:x{\text{ is a positive integer}}\}\,\!}$ , and ${\displaystyle \{1,2,3,\ldots \}\,\!}$ both specify the set containing 1, 2, 3, and onwards. A modified epsilon is used to denote set membership. Thus ${\displaystyle x\in y\,\!}$ indicates that "x is a member of y". We can also say that "x is not a member of y" in this way: ${\displaystyle x\notin y\,\!}$ Characteristics of sets A set is uniquely identified by its members. The expressions ${\displaystyle \{x:x\ \mathrm {is\ an\ even\ prime} \}\,\!}$ ${\displaystyle \{x:x\ \mathrm {is\ a\ positive\ square\ root\ of} \ 4\}\,\!}$ ${\displaystyle \{2\}\,\!}$ all specify the same set even though the concept of an even prime is different from the concept of a positive square root. Repetition of members is inconsequential in specifying a set. The expressions ${\displaystyle \{1,\ 2,\ 3\}\,\!}$ ${\displaystyle \{1,\ 1,\ 1,\ 1,\ 2,\ 3\}\,\!}$ ${\displaystyle \{x:\mathrm {x\ is\ an\ even\ prime\ or} \ x\ \mathrm {is\ a\ positive\ square\ root\ of} \ 4\ \mathrm {or} \ x=1\ \mathrm {or} \ x=2\ \mathrm {or} \ x=3\}\,\!}$ all specify the same set. Sets are unordered. The expressions ${\displaystyle \{1,\ 2,\ 3\}\,\!}$ ${\displaystyle \{3,\ 2,\ 1\}\,\!}$ ${\displaystyle \{2,\ 1,\ 3\}\,\!}$ all specify the same set. Sets can have other sets as members. There is, for example, the set ${\displaystyle \{\{1,\ 2\},\{2,\ 3\},\ \{1,\ \mathrm {George\ Washington} \}\}\,\!}$ Some special sets As stated above, sets are defined by their members. Some sets, however, are given names to ease referencing them. The set with no members is the empty set. The expressions ${\displaystyle \{\}\,\!}$ ${\displaystyle \varnothing \,\!}$ ${\displaystyle \{x:x\neq x\}\,\!}$ all specify the empty set. Empty sets can also express oxymora ("four-sided triangles" or "birds with radial symmetry") and factual non-existence ("the King of Czechoslovakia in 1994"). A set with exactly one member is called a singleton. A set with exactly two members is called a pair. Thus {1} is a singleton and {1, 2} is a pair. ω is the set of natural numbers, {0, 1, 2, ...}. Subsets, power sets, set operations Subsets A set s is a subset of set a if every member of s is a member of a. We use the horseshoe notation to indicate subsets. The expression ${\displaystyle \{1,\ 2\}\subseteq \{1,\ 2,\ 3\}\,\!}$ says that {1, 2} is a subset of {1, 2, 3}. The empty set is a subset of every set. Every set is a subset of itself. A proper subset of a is a subset of a that is not identical to a. The expression ${\displaystyle \{1,\ 2\}\subset \{1,\ 2,\ 3\}\,\!}$ says that {1, 2} is a proper subset of {1, 2, 3}. Power sets A power set of a set is the set of all its subsets. A script 'P' is used for the power set. ${\displaystyle {\mathcal {P}}\{1,\ 2,\ 3\}=\{\varnothing ,\ \{1\},\ \{2\},\ \{3\},\ \{1,\ 2\},\ \{1,\ 3\},\ \{2,\ 3\},\ \{1,\ 2,\ 3\}\}\,\!}$ Union The union of two sets a and b, written ab, is the set that contains all the members of a and all the members of b (and nothing else). That is, ${\displaystyle a\cup b=\{x:x\in a\ \mathrm {or} \ x\in b\}\,\!}$ As an example, ${\displaystyle \{1,\ 2,\ 3\}\ \cup \ \{2,\ 3,\ 4\}=\{1,\ 2,\ 3,\ 4\}\,\!}$ Intersection The intersection of two sets a and b, written ab, is the set that contains everything that is a member of both a and b (and nothing else). That is, ${\displaystyle a\cap b=\{x:x\in a\ \mathrm {and} \ x\in b\}\,\!}$ As an example, ${\displaystyle \{1,\ 2,\ 3\}\ \cap \ \{2,\ 3,\ 4\}=\{2,\ 3\}\,\!}$ Relative complement The relative complement of a in b, written b \ a (or ba) is the set containing all the members of b that are not members of a. That is, ${\displaystyle b\setminus a=\{x:x\in b\ \mathrm {and} \ x\notin a\}\,\!}$ As an example, ${\displaystyle \{2,\ 3,\ 4\}\setminus \{1,\ 3\}=\{2,\ 4\}\,\!}$ Ordered sets, relations, and functions The intuitive notions of ordered set, relation, and function will be used from time to time. For our purposes, the intuitive mathematical notion is the most important. However, these intuitive notions can be defined in terms of sets. Ordered sets First, we look at ordered sets. We said that sets are unordered: ${\displaystyle \{a,\ b\}=\{b,\ a\}\,\!}$ But we can define ordered sets, starting with ordered pairs. The angle bracket notation is used for this: ${\displaystyle \langle a,\ b\rangle \ \neq \ \langle b,\ a\rangle \,\!}$ Indeed, ${\displaystyle \langle x,\ y\rangle \ =\ \langle u,\ v\rangle \ {\mbox{if and only if}}\ x=u\ {\mbox{and}}\ y=v\,\!}$ Any set theoretic definition giving ⟨a, b⟩ this last property will work. The standard definition of the ordered paira, b⟩ runs: ${\displaystyle \langle a,\ b\rangle \ =\ \{\{a\},\ \{a,\ b\}\}\,\!}$ This means that we can use the latter notation when doing operations on an ordered pair. There are also bigger ordered sets. The ordered triplea, b, c⟩ is the ordered pair ⟨⟨a, b⟩, c⟩. The ordered quadruplea, b, c, d⟩ is the ordered pair ⟨⟨a, b, c⟩, d⟩. This, in turn, is the ordered triple ⟨⟨⟨a, b⟩, c⟩, d⟩. In general, an ordered n-tuplea1, a2, ..., an⟩ where n greater than 1 is the ordered pair ⟨⟨a1, a2, ..., an-1⟩, an⟩. It can be useful to define an ordered 1-tuple as well: ⟨a⟩ = a. These definitions are somewhat arbitrary, but it is nonetheless convenient for an n-tuple, n ⟩ 2, to be an n-1 tuple and indeed an ordered pair. The important property that makes them serve as ordered sets is: ${\displaystyle \langle x_{1},\ x_{2},\ ...,\ x_{n}\rangle \ =\ \langle y_{1},\ y_{2},\ ...,\ y_{n}\rangle \ \mathrm {if\ and\ only\ if} \ x_{1}=y_{1},\ x_{2}=y_{2},\ ...,\ x_{n}=y_{n}\,\!}$ Relations We now turn to relations. Intuitively, the following are relations: x < y x is a square root of y x is a brother of y x is between y and z The first three are binary or 2-place relations; the fourth is a ternary or 3-place relation. In general, we talk about n-ary relations or n-place relations. First consider binary relations. A binary relation is a set of ordered pairs. The less than relation would have among its members ⟨1, 2⟩, ⟨1, 3⟩, ⟨16, 127⟩, etc. Indeed, the less than relation defined on the natural numbers ω is: ${\displaystyle \{\langle x,\ y\rangle :x\in \omega ,\ y\in \omega ,\ \mathrm {and} \ x Intuitively, ⟨x, y⟩ is a member of the less than relation if x < y. In set theory, we do not worry about whether a relation matches an intuitive concept such as less than. Rather, any set of ordered pairs is a binary relation. We can also define a 3-place relation as a set of 3-tuples, a 4-place relation as a set of 4-tuples, etc. We only define n-place relations for n ≥ 2. An n-place relation is said to have an arity of n. The following example is a 3-place relation. ${\displaystyle \{\langle 1,\ 2,\ 3\rangle ,\ \langle 8,\ 2,\ 1\rangle ,\ \langle 653,\ 0,\ 927\rangle \}\,\!}$ Because all n-tuples where n > 1 are also ordered pairs, all n-place relations are also binary relations. Functions Finally, we turn to functions. Intuitively, a function is an assignment of values to arguments such that each argument is assigned at most one value. Thus the + 2 function assigns a numerical argument x the value x + 2. Calling this function f, we say f(x) = x + 2. The following define specific functions. ${\displaystyle f_{1}(x)=x\times x\,\!}$ ${\displaystyle f_{2}(x)=\ \mathrm {the\ smallest\ prime\ number\ larger\ than} \ x\,\!}$ ${\displaystyle f_{3}(x)=6/x\ \!}$ ${\displaystyle f_{4}(x)=\ \mathrm {the\ father\ of\ x} \,\!}$ Note that f3 is undefined when x = 0. According to biblical tradition, f4 is undefined when x = Adam or x = Eve. The following do not define functions. ${\displaystyle f_{5}(x)=\pm {\sqrt {x}}\,\!}$ ${\displaystyle f_{6}(x)=\ \mathrm {a\ son\ of\ x} \,\!}$ Neither of these assigns unique values to arguments. For every positive x, there are two square roots, one positive and one negative, so f5 is not a function. For many x, x will have multiple sons, so f6 is not a function. If f6 is assigned the value the son of x then a unique value is implied by the rules of language, therefore f6 will be a function. A function f is a binary relation where, if ⟨x, y⟩ and ⟨x, z⟩ are both members of f, then y = z. We can define many place functions. Intuitively, the following are definitions of specific many place functions. ${\displaystyle f_{7}(x,\ y)=x+y\,\!}$ ${\displaystyle f_{8}(x,\ y,\ z)=(x+y)\times z\,\!}$ Thus ⟨4, 7, 11⟩ is a member of the 2-place function f7. ⟨3, 4, 5, 35⟩ is a member of the 3 place function f8 The fact that all n-tuples, n ≥ 2, are ordered pairs (and hence that all n-ary relations are binary relations) becomes convenient here. For n ≥ 1, an n-place function is an n+1 place relation that is a 1-place function. Thus, for a 2-place function f, ${\displaystyle \langle x,\ y,\ z_{1}\rangle \ \in f\ \mathrm {and} \ \langle x,\ y,\ z_{2}\rangle \ \in f\ \mathrm {if\ and\ only\ if} \ z_{1}=z_{2}\,\!}$ ← Start of Preliminaries ↑ Preliminaries End of Preliminaries →	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 44, "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": 0.9691795110702515, "perplexity": 1321.6516469565577}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362571.17/warc/CC-MAIN-20211203000401-20211203030401-00227.warc.gz"}
http://www.math.gatech.edu/seminars-and-colloquia-by-series?series_tid=28&page=7	## Seminars and Colloquia by Series Monday, September 29, 2014 - 15:05 , Location: Skiles 005 , , IST Austria , Organizer: Anton Leykin Markov bases have been developed in algebraic statistics for exact goodness-of-fit testing. They connect all elements in a fiber (given by the sufficient statistics) and allow building a Markov chain to approximate the distribution of a test statistic by its posterior distribution. However, finding a Markov basis is often computationally intractable. In addition, the number of Markov steps required for converging to the stationary distribution depends on the connectivity of the sampling space.In this joint work with Caroline Uhler and Sarah Cepeda, we compare different test statistics and study the combinatorial structure of the finite lattice Ising model. We propose a new method for exact goodness-of-fit testing. Our technique avoids computing a Markov basis but builds a Markov chain consisting only of simple moves (i.e. swaps of two interior sites). These simple moves might not be sufficient to create a connected Markov chain. We prove that when a bounded change in the sufficient statistics is allowed, the resulting Markov chain is connected. The proposed algorithm not only overcomes the computational burden of finding a Markov basis, but it might also lead to a better connectivity of the sampling space and hence a faster convergence. Wednesday, September 24, 2014 - 15:05 , Location: Skiles 006 , Melody Chan , Harvard University , Organizer: Matt Baker This is joint work with Pakwut Jiradilok. Let X be a smooth, proper curve of genus 3 over a complete and algebraically closed nonarchimedean field. We say X is a K_4-curve if the nonarchimedean skeleton G of X is a metric K_4, i.e. a complete graph on 4 vertices.We prove that X is a K_4-curve if and only if X has an embedding in p^2 whose tropicalization has a strong deformation retract to a metric K_4. We then use such an embedding to show that the 28 odd theta characteristics of X are sent to the seven odd theta characteristics of g in seven groups of four. We give an example of the 28 bitangents of a honeycomb plane quartic, computed over the field C{{t}}, which shows that in general the 4 bitangents in a given group need not have the same tropicalizations. Monday, September 22, 2014 - 15:05 , Location: Skiles 006 , Martin Ulirsch , Brown University , Organizer: Matt Baker Recent work by J. and N. Giansiracusa, myself, and O. Lorscheid suggests that the tropical geometry of a toric variety $X$, or more generally of a logarithmic scheme $X$, can be formalized as a "Berkovich analytification" of a scheme over the field $\mathbb{F}_1$ with one element that is canonically associated to $X$.The goal of this talk is to introduce the theory of Artin fans, originally due to D. Abramovich and J. Wise, which can be used to lift rather unwieldy $\mathbb{F}_1$-geometric objects to the more familiar realm of algebraic stacks. Artin fans are \'etale locally isomorphic to quotient stacks of toric varieties by their big tori and their glueing data has a completely combinatorial description in terms of Kato fans.I am going to explain how to use the ideas surrounding the notion of Artin fans to study tropicalization maps associated to toric varieties and logarithmic schemes. Surprisingly these techniques allow us to give a reinterpretation of Tevelev's theory of tropical compactifications that can be generalized to compactifications of subvarieties in logarithmically smooth compactifcations of smooth varieties. For example, we can introduce definitions of tropical pairs and schoen varieties in terms of Artin fans that are equivalent to Tevelev's notions. Friday, September 19, 2014 - 15:05 , Location: Skiles 005 , , Universität Konstanz , Organizer: Anton Leykin We study symmetric determinantal representations of real hyperbolic curves in the projective plane. Such representations always exist by the Helton-Vinnikov theorem but are hard to compute in practice. In this talk, we will discuss some of the underlying algebraic geometry and show how to use polynomial homotopy continuation to find numerical solutions. (Joint work with Anton Leykin). Monday, June 30, 2014 - 15:05 , Location: Skiles 005 , Anders Jensen , Aarhus University , Organizer: Josephine Yu In this talk we discuss a recent paper by Andrew Chan and Diane Maclagan on Groebner bases for fields, where the valuation of the coefficients is taken into account, when defining initial terms. For these orderings the usual division algorithm does not terminate, and ideas from standard bases needs to be introduced. Groebner bases for fields with valuations play an important role in tropical geometry, where they can be used to compute tropical varieties of a larger class of polynomial ideals than usual Groebner bases. Monday, May 5, 2014 - 15:00 , Location: Skiles 006 , , Georgia Tech , , Organizer: Salvador Barone We study the Legendre elliptic curve E: y^2=x(x+1)(x+t) over the field F_p(t) and its extensions K_d=F_p(mu_d*t^(1/d)). When d has the form p^f+1, in previous work we exhibited explicit points on E which generate a group V of large rank and finite index in the full Mordell-Weil group E(K_d), and we showed that the square of the index is the order of the Tate-Shafarevich group; moreover, the index is a power of p. In this talk we will explain how to use p-adic cohomology to compute the Tate-Shafarevich group and the quotient E(K_d)/V as modules over an appropriate group ring. Monday, April 28, 2014 - 15:05 , Location: Skiles 006 , Jesse Thorner , Emory University , Organizer: Matt Baker A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes $p_1,p_2$ with $\|p_1-p_2\|\leq 600$ as a consequence of the Bombieri-Vinogradov Theorem. We apply his general method to the setting of Chebotarev sets of primes. We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over $\mathbb{Q}$, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms. Friday, April 11, 2014 - 11:05 , Location: Skiles 005 , , TU Eindhoven , Organizer: Josephine Yu Given a closed subvariety X of affine space A^n, there is a surjective map from the analytification of X to its tropicalisation. The natural question arises, whether this map has a continuous section. Recent work by Baker, Payne, and Rabinoff treats the case of curves, and even more recent work by Cueto, Haebich, and Werner treats Grassmannians of 2-spaces. I will sketch how one can often construct such sections when X is obtained from a linear space smeared around by a coordinate torus action. In particular, this gives a new, more geometric proof for the Grassmannian of 2-spaces; and it also applies to some determinantal varieties. (Joint work with Elisa Postinghel) Monday, March 24, 2014 - 15:05 , Location: Skiles 006 , Nick Rogers , Department of Defense , Organizer: Matt Baker A notorious open problem in arithmetic geometry asks whether ranks ofelliptic curves are unbounded in families of quadratic twists. A proof ineither direction seems well beyond the reach of current techniques, butcomputation can provide evidence one way or the other. In this talk wedescribe two approaches for searching for high rank twists: the squarefreesieve, due to Gouvea and Mazur, and recursion on the prime factorization ofthe twist parameter, which uses 2-descents to trim the search tree. Recentadvances in techniques for Selmer group computations have enabled analysisof a much larger search region; a large computation combining these ideas,conducted by Mark Watkins, has uncovered many new rank 7 twists of$X_0(32): y^2 = x^3 - x$, but no rank 8 examples. We'll also describe aheuristic argument due to Andrew Granville that an elliptic curve hasfinitely many (and typically zero) quadratic twists of rank at least 8. Wednesday, March 12, 2014 - 15:05 , Location: Skiles 006 , Johannes Nicaise , KU Leuven , Organizer: Matt Baker I will explain the construction of the essential skeleton of a one-parameter degeneration of algebraic varieties, which is a simplicial space encoding the geometry of the degeneration, and I will prove that it coincides with the skeleton of a good minimal dlt-model of the degeneration if the relative canonical sheaf is semi-ample. These results, contained in joint work with Mircea Mustata and Chenyang Xu, provide some interesting connections between Berkovich geometry and the Minimal Model Program.	{"extraction_info": {"found_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": 0.9004006385803223, "perplexity": 965.6487560390867}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1512948515309.5/warc/CC-MAIN-20171212060515-20171212080515-00187.warc.gz"}
http://www.maths.usyd.edu.au/u/UG/JM/MATH1011/Quizzes/quiz7.html	## H1011 Quizzes Quiz 7: Two variable optimisation of surfaces Question 1 Questions Find the equation of the tangent to the curve $f\left(x\right)=xsin2x$ at $x=\frac{\pi }{4}$. a) $y=\frac{\pi }{4}x$ b) $y=x$ c) $y=\frac{\pi }{4}x-1$ d) $y=x+1-\frac{\pi }{4}$ Choice (a) is incorrect Choice (b) is correct! $\begin{array}{rcll}{f}^{\prime }\left(x\right)& =& sin2x+xcos2x,& \text{}\\ {f}^{\prime }\left(\frac{\pi }{4}\right)& =& 1,& \text{}\\ f\left(\frac{\pi }{4}\right)& =& \frac{\pi }{4}.& \text{}\end{array}$ So with $y=x+b,\frac{\pi }{4}=\frac{\pi }{4}+b,b=0.$ therefore the equation of the tangent is $y=x$. Choice (c) is incorrect Choice (d) is incorrect Find two numbers whose difference is 20 and whose product is minimal. a) 1, 21 b) 19, -1 c) 10, -10 d) None of these. Choice (a) is incorrect Choice (b) is incorrect Choice (c) is correct! Let the two numbers be $a,b$ where $a>b$. Then $a-b=20$. Let the product of the numbers be $P$. This is clearly a minimum value for the product because $\frac{{d}^{2}P}{d{a}^{2}}=2>0$ for all $a$. Choice (d) is incorrect Imagine constructing a closed steel box with volume 576 cm${3}^{}$ and with its base twice as long as it is wide. The steel costs \$40 per square metre. Determine the dimensions of the box that will minimise the cost of construction. a) 12 cm $×$ 6 cm $×$ 8 cm b) $12\sqrt{3}$ cm $×\phantom{\rule{1em}{0ex}}6\sqrt{3}$ cm $×\phantom{\rule{1em}{0ex}}\frac{8}{3}$ cm c) $12\sqrt[3]{2}$ cm $×\phantom{\rule{1em}{0ex}}6\sqrt[3]{2}$ cm $×\phantom{\rule{1em}{0ex}}4\sqrt[3]{2}$ cm d) None of the above Choice (a) is correct! The volume of the box is $2{x}^{2}h=576$, so $h=\frac{288}{{x}^{2}}$. The surface area of the box is $=S=4{x}^{2}+\frac{1728}{x}$. Therefore $\frac{dS}{dx}=8x-\frac{1728}{{x}^{2}}$. Hence, $\frac{dS}{dx}=0$, when $8{x}^{3}=1728$; that is, when $x=6$. Consequently, the dimensions of the box are 12 cm $×$ 6 cm $×$ 8 cm. Note that $S$ has a minimum at $x=6$ since $\frac{{d}^{2}S}{d{x}^{2}}=8+\frac{3456}{{x}^{3}}>0$ at $x=6$. Choice (b) is incorrect Choice (c) is incorrect Choice (d) is incorrect Evaluate $f\left(x,y\right)={e}^{3x}cosy$ at $\left(2,\frac{\pi }{4}\right)$. a) $f\left(2,\frac{\pi }{4}\right)={e}^{\frac{3\pi }{4}}cos2$ b) $f\left(2,\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}{e}^{6}$ c) $f\left(2,\frac{\pi }{4}\right)=\frac{2}{\sqrt{2}}{e}^{6}$ d) $f\left(2,\frac{\pi }{4}\right)={e}^{\frac{3\pi }{2}}$ Choice (a) is incorrect Choice (b) is correct! When $x=2$ and $y=\frac{\pi }{4}$, ${e}^{3x}cosy={e}^{6}cos\frac{\pi }{4}=\frac{1}{\sqrt{2}}{e}^{6}$. Choice (c) is incorrect Choice (d) is incorrect Evaluate $f\left(x,y\right)=sin2xy+cos3xy$ at $\left(1,\frac{\pi }{6}\right)$. a) $f\left(1,\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}$ b) $f\left(1,\frac{\pi }{6}\right)=\frac{1}{2}$ c) $f\left(1,\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}+1$ d) $f\left(1,\frac{\pi }{6}\right)=\frac{3}{2}$ Choice (a) is correct! $f\left(1,\frac{\pi }{6}\right)=sin\frac{\pi }{3}+cos\frac{\pi }{2}=\frac{\sqrt{3}}{2}+0=\frac{\sqrt{3}}{2}.$ Choice (b) is incorrect Choice (c) is incorrect Choice (d) is incorrect Which point below is on the surface $z={x}^{2}+xy-{\left(4-y\right)}^{2}$ ? a) $\left(1,1,27\right)$ b) $\left(1,1,-7\right)$ c) $\left(-1,2,5\right)$ d) $\left(-1,2,-37\right)$ Choice (a) is incorrect Choice (b) is correct! When $x=y=1$, $z=1+1-{\left(3\right)}^{2}=-7$. Choice (c) is incorrect Choice (d) is incorrect Which point below is on the surface $z=2{x}^{2}y+x-\sqrt{x+y}$ ? a) A$\left(2,2,16\right)$ b) B$\left(-1,-1,-1\right)$ c) C$\left(2,2,14\right)$ d) D$\left(1,1,1\right)$ Choice (a) is correct! When $x=y=2$, $z=16+2-\sqrt{4}=16$. Choice (b) is incorrect Note that $\left(-1,-1\right)$ is not in the domain of the surface. Choice (c) is incorrect Choice (d) is incorrect Which response below most accurately describes the intersection of the surface $z=\sqrt{9-{x}^{2}-{y}^{2}}$ and the plane $z=2$ ? a) A circle of radius 5 in the plane $z=2$. b) A circle of radius 5 in the $xy$ plane. c) A circle of radius $\sqrt{5}$ in the plane $z=2$. d) A circle of radius 5 in the $xy$ plane. Choice (a) is incorrect Choice (b) is incorrect Choice (c) is correct! The curve of intersection of the surface and the plane is given by $\sqrt{9-{x}^{2}-{y}^{2}}=2$. i.e ${x}^{2}+{y}^{2}=5$. This is a circle of radius $\sqrt{5}$ in the plane $z=2$. Choice (d) is incorrect Which response below most accurately describes the intersection of the surface $z=\frac{{x}^{2}}{9}-\frac{{y}^{2}}{4}$ and the plane $y=4$ ? a) An ellipse, $\frac{{x}^{2}}{36}-\frac{{y}^{2}}{16}=1$, in the plane $z=4$. b) A circle radius 2, ${x}^{2}+{y}^{2}=4$, in the plane $z=4$. c) A parabola $z=\frac{{x}^{2}}{9}-1$, in the plane $y=4$. d) A parabola $z=\frac{{x}^{2}}{9}-4$, in the plane $y=4$. Choice (a) is incorrect Choice (b) is incorrect Choice (c) is incorrect Choice (d) is correct! The curve of intersection of the surface and the plane is given by $z=\frac{{x}^{2}}{9}-\frac{16}{4}$. i.e $z=\frac{{x}^{2}}{9}-4$ which is a parabola in the plane $y=4$. Which response below most accurately describes the intersection of the surface $z=2{x}^{2}-2x-{y}^{2}-2y+3$ and the plane $x-y=0$? a) The parabola $z=3{y}^{2}+3$ in the plane $x-y=0$. b) The parabola $z=3{x}^{2}+3$ in the plane $x-y=0$. c) The parabola $z={x}^{2}-4x+3$ in the plane $x-y=0$. d) The intersection cannot be determined. Choice (a) is incorrect Choice (b) is incorrect Choice (c) is correct! $x-y=0⇒y=x$ so $z=2{x}^{2}-2x-{x}^{2}-2x+3={x}^{2}-4x+3$. Choice (d) is incorrect	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 102, "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": 0.8757066130638123, "perplexity": 613.6984183409085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1521257647838.64/warc/CC-MAIN-20180322092712-20180322112712-00553.warc.gz"}
https://www.physicsforums.com/threads/a-block-on-top-of-another-on-a-horizontal-surface.193938/	# A block on top of another on a horizontal surface. 1. Oct 25, 2007 ### Ahwleung 1. The problem statement, all variables and given/known data Its not a very difficult problem, and I really only need to know 1 thing about friction; anyways, here's the summed up problem: (my teacher enjoys making us use constants instead of numbers btw) A horizontal force F is applied to a small block (mass m1), and it slides across a larger block a length L (mass m2) with the coefficient of friction being u. The larger block in turn slides along a frictionless horizontal surface. Everything starts at rest and the small block starts at the end (left side) of the larger block on the bottom. We're supposed to find the acceleration of each block relative to the horizontal surfaces, find the time t needed for the small block to slide off the end of the larger block, and find the expression for the energ dissipated as heat due to friction. 2. Relevant equations Umm, I'd say F=ma, the law of friction (F(kf) = uFn), and also the work equations (for the last question). I can handle all those pretty easily, but I just need something about summing the forces. 3. The attempt at a solution We have to draw a free body diagram - here is where I find the problem. Both blocks have a normal force and a force of gravity (duh). The 1st block has a Force Applied and a Force of Friction opposing it. Where I'm getting messed up is on the 2nd block. I don't think Force Applied transfers to the 2nd block - am I right in saying that the only horizontal force on the 2nd block is the force due to friction from the 1st block? Is this force of friction equal to the Force of friction from the 1st block? Is it newton's third law? What doesn't make sense is that friction forces always oppose motion, yet the 2nd block obviously travels in the same direction as the Applied Force. 2. Oct 25, 2007 ### Kurdt Staff Emeritus Your thinking is quite correct. Your confusion at the end is that the friction force does oppose the motion of the first block but for there to be a reaction force there must be another force in the opposite direction. Just think of the applied force as being split into two parts. The part that opposes friction and the extra bit that makes it move. 3. Oct 25, 2007 ### Ahwleung Another question about this problem. I found the first part, but I'm sorta confused about the 2nd question. I don't think its a trick question, but it seems really, really easy. Basically, it says given a distance and both accelerations (and the fact that everything starts at rest), find the time it will take for the block on top to fall off. Isn't it really, really easy? It's basically a block with an acceleration travelling a distance with an initial velocity equal to 0. Is it just an easy UAM problem? Or do I have to factor in the fact that the block on the bottom also has an acceleration/is moving? 4. Oct 25, 2007 ### Kurdt Staff Emeritus As you say its a uniform acceleration problem. You don't have to worry about the motion of the second block. Similar Discussions: A block on top of another on a horizontal surface.	{"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": 0.9605532884597778, "perplexity": 397.5526291900833}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806465.90/warc/CC-MAIN-20171122050455-20171122070455-00278.warc.gz"}
https://www.lessonplanet.com/teachers/worksheet-worksheet-13	# Worksheet 13 For this math worksheet, middle schoolers write down the equation of a plane. Then they create the equation of a line. They also define the lines that are perpendicular. Concepts Resource Details	{"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": 0.8629313111305237, "perplexity": 870.3050757861829}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107891203.69/warc/CC-MAIN-20201026090458-20201026120458-00617.warc.gz"}
https://fabiandablander.com/r/Bayes-Potter.html	# Harry Potter and the Power of Bayesian Inference If you are reading this, you are probably a Ravenclaw. Or a Hufflepuff. Certainly not a Slytherin … but maybe a Gryffindor? In this blog post, we let three subjective Bayesians predict the outcome of ten coin flips. We will derive prior predictions, evaluate their accuracy, and see how fortune favours the bold. We will also discover a neat trick that allows one to easily compute Bayes factors for models with parameter restrictions compared to models without such restrictions, and use it to answer a question we truly care about: are Slytherins really the bad guys? # Preliminaries As in a previous blog post, we start by studying coin flips. Let $\theta \in [0, 1]$ be the bias of the coin and let $y$ denote the number of heads out of $n$ coin flips. We use the Binomial likelihood $p(y \mid \theta) = {n \choose y} \theta^y (1 - \theta)^{n - y} \enspace ,$ and a Beta prior for $\theta$: $p(\theta) = \frac{1}{\text{B}(a, b)} \theta^{a - 1} (1 - \theta)^{b - 1} \enspace .$ This prior is conjugate for this likelihood which means that the posterior is again a Beta distribution. The Figure below shows two examples of this. In this blog post, we will use a prior predictive perspective on model comparison by means of Bayes factors. For an extensive contrast with a perspective based on posterior prediction, see this blog post. The Bayes factor indicates how much better a model $\mathcal{M}_1$ predicts the data $y$ relative to another model $\mathcal{M}_0$: $\text{BF}_{10} = \frac{p(y \mid \mathcal{M}_1)}{p(y \mid \mathcal{M}_0)} \enspace ,$ where we can write the marginal likelihood of a generic model $\mathcal{M}$ more complicatedly to see the dependence on the model’s priors: $p(y \mid \mathcal{M}) = \int_{\Theta} p(y \mid \theta, \mathcal{M}) \, p(\theta \mid \mathcal{M}) \, \mathrm{d}\theta \enspace .$ After these preliminaries, in the next section, we visit Ron, Harry, and Hermione in Hogwarts. # The Hogwarts prediction contest Ron, Harry, and Hermione just came back from a straining adventure — Death Eaters and all. They deserve a break, and Hermione suggests a small prediction contest to relax. Ron is put off initially; relaxing by thinking? That’s not his style. Harry does not care either way; both are eventually convinced. The goal of the contest is to accuratly predict the outcome of $n = 10$ coin flips. Luckily, this is not a particularly complicated problem to model, and we can use the Binomial likelihood we have discussed above. In the next section, Ron, Harry, and Hermione — all subjective Bayesians — clearly state their prior beliefs which is required to make predictions. ## Prior beliefs Ron is not big on thinking, and so trusts his previous intuitions that coins are usually unbiased; he specifies a point mass on $\theta = 0.50$ as his prior. Harry spreads his bets evenly, and believes that all chances governing the coin flip’s outcome are equally likely; he puts a uniform prior on $\theta$. Hermione, on the other hand, believes that the coin cannot be biased towards tails; instead, she believes that all values $\theta \in [0.50, 1]$ are equally likely. She thinks this because Dobby — the house elf — is the one who throws the coin, and she has previously observed him passing time by flipping coins, which strangely almost always landed up heads. To sum up, their priors are: \begin{aligned} \text{Ron} &: \theta = 0.50 \\[.5em] \text{Harry} &: \theta \sim \text{Beta}(1, 1) \\[.5em] \text{Hermione} &: \theta \sim \text{Beta}(1, 1)\mathbb{I}(0.50, 1) \enspace , \end{aligned} which are visualized in the Figure below. In the next section, the three use their beliefs to make probabilistic predictions. ## Prior predictions Ron, Harry, and Hermione are subjective Bayesians and therefore evaluate their performance by their respective predictive accuracy. Each of the trio has a prior predictive distribution. For Ron, true to character, this is the easiest to derive. We associate model $\mathcal{M}_0$ with him and write: \begin{aligned} p(y \mid \mathcal{M}_0) &= \int_{\Theta} p(y \mid \theta, \mathcal{M}_0) \, p(\theta \mid \mathcal{M}_0) \, \mathrm{d}\theta \\[.5em] &= {n \choose y} 0.50^y (1 - 0.50)^{n - y} \enspace , \end{aligned} where the integral — the sum! — is just over the value $\theta = 0.50$. Ron’s prior predictive distribution is simply a Binomial distribution. He is delighted by this fact, and enjoys a short rest while the others derive their predictions. It is Harry’s turn, and he is a little put off by his integration problem. However, he realizes that the integrand is an unnormalized Beta distribution, and swiftly writes down its normalizing constant, the Beta function. Associating $\mathcal{M}_1$ with him, his steps are: \begin{aligned} p(y \mid \mathcal{M}_1) &= \int_{\Theta} p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1) \, \mathrm{d}\theta \\[.5em] &= \int_{\Theta} {n \choose y} \theta^y (1 - \theta)^{n - y} \, \frac{1}{\text{B}(1, 1)} \theta^{1 - 1} (1 - \theta)^{1 - 1} \, \mathrm{d}\theta \\[.5em] &= \int_{\Theta} {n \choose y} \theta^y (1 - \theta)^{n - y} \, \mathrm{d}\theta \\[.5em] &= {n \choose y} \text{Beta}(y + 1, n - y + 1) \enspace , \end{aligned} which is a Beta-Binomial distribution with $\alpha = \beta = 1$. Hermione’s integral is the most complicated of the three, but she is also the smartest of the bunch. She is a master of the wizardry that is computer programming, which allows her to solve the integral numerically.1 We associate $\mathcal{M}_r$, which stands for restricted model, with her and write: \begin{aligned} p(y \mid \mathcal{M}_r) &= \int_{\Theta} p(y \mid \theta, \mathcal{M}_r) \, p(\theta \mid \mathcal{M}_r) \, \mathrm{d}\theta \\[.5em] &= \int_{0.50}^1 {n \choose y} \theta^y (1 - \theta)^{n - y} \, 2 \, \mathrm{d}\theta \\[.5em] &= 2{n \choose y}\int_{0.50}^1 \theta^y (1 - \theta)^{n - y} \mathrm{d}\theta \enspace . \end{aligned} We can draw from the prior predictive distributions by simulating from the prior and then making predictions through the likelihood. For Hermione, for example, this yields: Let’s visualize Ron’s, Harry’s, and Hermione’s prior predictive distributions to get a better feeling for what they believe are likely coin flip outcomes. First, we implement their prior predictions in R: Even though Ron believes that $\theta = 0.50$, this does not mean that his prior prediction puts all mass on $y = 5$; deviations from this value are plausible. Harry’s prior predictive distribution also makes sense: since he believes all values for $\theta$ to be equally likely, he should believe all outcomes are equally likely. Hermione, on the other hand, believes that $\theta \in [0.50, 1]$, so her prior probabilities for outcomes with few heads ($y < 5$) drastically decrease. After the three have clearly stated their prior beliefs and derived their prior predictions, Dobby throws a coin ten times. The coin comes up heads nine times. In the next section, we discuss the relative predictive performance of Ron, Harry, and Hermione based on these data. ## Evaluating predictions To assess the relative predictive performance of Ron, Harry, and Hermione, we need to compute the probability mass of $y = 9$ for their respective prior predictive distributions. Compared to Ron, Hermione did roughly 19 times better: Harry, on the other hand, did about 9 times better than Ron: With these two comparisons, we also know by how much Hermione outperformed Harry, since by transitivity we have: $\text{BF}_{r1} = \frac{p(y \mid \mathcal{M}_r)}{p(y \mid \mathcal{M}_0)} \times \frac{p(y \mid \mathcal{M}_0)}{p(y \mid \mathcal{M}_1)} = \text{BF}_{r0} \times \frac{1}{\text{BF}_{10}} \approx 2 \enspace ,$ which is indeed correct: Note that this is also immediately apparent from the visualizations above, where Hermione’s allocated probability mass is about twice as large as Harry’s for the case where $y = 9$. Hermione was bold in her prediction, and was rewarded with being favoured by a factor of two in predictive performance. Note that if her predictions would have been even bolder, say restricting her prior to $\theta \in [0.80, 1]$, she would have reaped higher rewards than a Bayes factor in favour of two. Contrast this to Dobby throwing the coin ten times and with only one heads showing. Then Harry’s marginal likelihood is still $\text{Beta}(11, 1) = \frac{1}{11}$. However, Hermione’s is not twice as much; instead, it is a mere $0.001065$, which would result in a Bayes factor of about $85$ in favour of Harry! This means that with bold predictions, one can also lose a lot. However, this is tremendously insightful, since Hermione would immediately realize where she went wrong. For a discussion that also points out the flexibility of Bayesian model comparison, see Etz, Haaf, Rouder, & Vandeckerckhove (2018). In the next section, we will discover a nice trick which simplifies the computation of the Bayes factor; we do not need to derive marginal likelihoods, but can simply look at the prior and the posterior distribution of the parameter of interest in the unrestricted model. # Prior / Posterior trick As it it turns out, the relative predictive performance of Hermione compared to Harry is given by the ratio of the purple area to the blue area in the figure below. In other words, the Bayes factor in favour of the restricted model (i.e., Hermione) compared to the unrestricted or encompassing model (i.e., Harry) is given by the posterior probability of $\theta$ being in line with the restriction compared to the prior probability of $\theta$ being in line with the restriction. We can check this numerically: This is a very cool result which, to my knowledge, was first described in Kluglist & Hoijtink (2005). In the next section, we prove it. ## Proof The proof uses two insights. First, note that we can write the priors in the restricted model, $\mathcal{M}_r$, as priors in the encompassing model, $\mathcal{M}_1$, subject to some constraints. In the Hogwarts prediction context, Hermione’s prior was a restricted version of Harry’s prior: \begin{aligned} p(\theta \mid \mathcal{M}_r) &= p(\theta \mid \mathcal{M}_1)\mathbb{I}(0.50, 1) \\[1em] &= \begin{cases} \frac{p(\theta \mid \mathcal{M}_1)}{\int_{0.50}^1 p(\theta \mid \mathcal{M}_1) \, \mathrm{d}\theta} & \text{if} \hspace{1em} \theta \in [0.50, 1] \\[1em] 0 & \text{otherwise}\end{cases} \end{aligned} We have to divide by the term $K = \int_{0.50}^1 p(\theta \mid \mathcal{M}_1) \, \mathrm{d}\theta = 0.50 \enspace ,$ so that the restricted prior integrates to 1, as all proper probability distributions must. As a direct consequence, note that the density of a value $\theta = \theta^{\star}$ is given by: $p(\theta^{\star} \mid \mathcal{M}_r) = p(\theta^{\star} \mid \mathcal{M}_1) \cdot \frac{1}{K} \enspace ,$ where $K$ is the renormalization constant. This means that we can rewrite terms which include the restricted prior in terms of the unrestricted prior from the encompassing model. This also holds for the posterior! To see that we can also write the restricted posterior in terms of the unrestricted posterior from the encompassing model, note that the likelihood is the same under both models and that: \begin{aligned} p(\theta \mid y, \mathcal{M}_r) &= \frac{p(y \mid \theta, \mathcal{M}_r) \, p(\theta \mid \mathcal{M}_r)}{\int_{0.50}^1 p(y \mid \theta, \mathcal{M}_r) \, p(\theta \mid \mathcal{M}_r) \, \mathrm{d}\theta} \\[.5em] &= \frac{p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1) \, \frac{1}{K}}{\int_{0.50}^1 p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1) \, \frac{1}{K} \, \mathrm{d}\theta} \\[.5em] &= \frac{p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1)}{\int_{0.50}^1 p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1) \, \mathrm{d}\theta} \\[.5em] &= \frac{\frac{p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1)}{\int p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1) \, \mathrm{d}\theta}}{\int_{0.50}^1 \frac{p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1)}{\int p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1) \, \mathrm{d}\theta} \, \mathrm{d}\theta} \\[.5em] &= \frac{p(\theta \mid y, \mathcal{M}_1)}{\int_{0.50}^1 p(\theta \mid y, \mathcal{M}_1) \, \mathrm{d}\theta} \enspace , \end{aligned} where we have to renormalize by $Z = \int_{0.50}^1 p(\theta \mid y, \mathcal{M}_1) \, \mathrm{d}\theta \enspace ,$ which is The figure below visualizes Harry’s and Hermione’s posterior. Sensibly, since Hermione excluded all $\theta \in [0, 0.50]$ in her prior, such values receive zero credence in her posterior. However, the difference in posterior distributions between Harry and Hermione is very weak in contrast to the difference in prior distribution. This is reflected in $Z$ being close to 1. Similar to the prior, we can write the density of a value $\theta = \theta^\star$ in terms of the encompassing model: $p(\theta^{\star} \mid y, \mathcal{M}_r) = p(\theta^{\star} \mid y, \mathcal{M}_1) \cdot \frac{1}{Z} \enspace .$ Now that we have established that we can write both the prior and the posterior density of parameters in the restricted model in terms of the parameters in the unrestricted model, as a second step, note that we can swap the posterior and the marginal likelihood terms in Bayes’ rule such that: $p(y \mid \mathcal{M}_1) = \frac{p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1)}{p(\theta \mid y, \mathcal{M}_1)} \enspace ,$ from which it follows that: $\text{BF}_{r1} = \frac{p(y \mid \mathcal{M}_r)}{p(y \mid \mathcal{M}_1)} = \frac{\frac{p(y \mid \theta, \mathcal{M}_r) \, p(\theta \mid \mathcal{M}_r)}{p(\theta \mid y, \mathcal{M}_r)}}{\frac{p(y \mid \theta, \mathcal{M}_1) \, p(\theta \mid \mathcal{M}_1)}{p(\theta \mid y, \mathcal{M}_1)}} \enspace .$ Now suppose that we have values that are in line with the restriction, i.e., $\theta = \theta^{\star}$. Then: \begin{aligned} \text{BF}_{r1} = \frac{\frac{p(y \mid \theta^\star, \mathcal{M}_r) \, p(\theta^\star\mid \mathcal{M}_r)}{p(\theta^\star \mid y, \mathcal{M}_r)}}{\frac{p(y \mid \theta^\star, \mathcal{M}_1) \, p(\theta^\star \mid \mathcal{M}_1)}{p(\theta^\star \mid y, \mathcal{M}_1)}} = \frac{\frac{p(y \mid \theta^\star, \mathcal{M}_r) \, p(\theta^\star \mid \mathcal{M}_1) \, \frac{1}{K}}{p(\theta^\star \mid y, \mathcal{M}_1) \, \frac{1}{Z}}}{\frac{p(y \mid \theta^\star, \mathcal{M}_1) \, p(\theta^\star \mid \mathcal{M}_1)}{p(\theta^\star \mid y, \mathcal{M}_1)}} = \frac{\frac{p(y \mid \theta^\star, \mathcal{M}_r) \, \frac{1}{K}}{\frac{1}{Z}}}{p(y \mid \theta^\star, \mathcal{M}_1)} = \frac{\frac{1}{K}}{\frac{1}{Z}} = \frac{Z}{K} \enspace , \end{aligned} where we have used the previous insights and the fact that the likelihood under $\mathcal{M}_r$ and $\mathcal{M}_1$ is the same. If we expand the constants for our previous problem, we have: $\text{BF}_{r1} = \frac{Z}{K} = \frac{\int_{0.50}^1 p(\theta \mid y, \mathcal{M}_1) \, \mathrm{d}\theta}{\int_{0.50}^1 p(\theta \mid \mathcal{M}_1) \, \mathrm{d}\theta} = \frac{p(\theta \in [0.50, 1] \mid y, \mathcal{M}_1)}{p(\theta \in [0.50, 1] \mid \mathcal{M}_1)} \enspace ,$ which is, as claimed above, the posterior probability of values for $\theta$ that are in line with the restriction divided by the prior probability of values for $\theta$ that are in line with the restriction. Note that this holds for arbitrary restrictions of an arbitrary number of parameters (see Kluglist & Hoijtink, 2005). In the limit where we take the restriction to be infinitesimally small, that is, constrain the parameter to be a point value, this results in the Savage-Dickey density ratio (Wetzels, Grasman, & Wagenmakers, 2010). In the next section, we apply this idea to a data set that relates Hogwarts Houses to personality traits. # Hogwarts Houses and personality So, are you a Slytherin, Hufflepuff, Ravenclaw, or Gryffindor? And what does this say about your personality? Inspired by Crysel et al. (2015), Lea Jakob, Eduardo Garcia-Garzon, Hannes Jarke, and I analyzed self-reported personality data from 847 people as well as their self-reported Hogwards House affiliation.2 We wanted to answer questions such as: do people who report belonging to Slytherin tend to score highest on Narcissism, Machiavellianism, and Psychopathy? Are Hufflepuffs the most agreeable, and Gryffindors the most extraverted? The Figure below visualizes the raw data. We used a between-subjects ANOVA as our model and, in the case of for example Agreeableness, compared the following hypotheses: \begin{aligned} \mathcal{H}_0&: \mu_H = \mu_G = \mu_R = \mu_S \\[.5em] \mathcal{H}_r&: \mu_H > (\mu_G , \mu_R , \mu_S) \\[.5em] \mathcal{H}_1&: \mu_H , \mu_G , \mu_R , \mu_S \end{aligned} We used the BayesFactor R package to compute the Bayes factor in favour of $\mathcal{H}_1$ compared to $\mathcal{H}_0$. For the restricted hypotheses $\mathcal{H}_r$, we used the prior/posterior trick outlined above; and indeed, we found strong evidence in favour of the notion that, for example, Hufflepuffs score highest on Agreeableness. Curious about Slytherin and the other Houses? You can read the published paper with all the details here. # Conclusion Participating in a relaxing prediction contest, we saw how three subjective Bayesians named Ron, Harry, and Hermione formalized their beliefs and derived their predictions about the likely outcome of ten coin flips. By restricting her prior beliefs about the bias of the coin to exclude values smaller than $\theta = 0.50$, Hermione was the boldest in her predictions and was ultimately rewarded. However, if the outcome of the coin flips would have turned out differently, say $y = 2$, then Hermione would have immediately realized how wrong her beliefs were. I think we as scientists need to be more like Hermione: we need to make more precise predictions, allowing us to construct more powerful tests and “fail” in insightful ways. We also saw a neat trick by which one can compute the Bayes factor in favour of a restricted model compared to an unrestricted model by estimating the proportion of prior and posterior values of the parameter that are in line with the restriction — no painstaking computation of marginal likelihoods required! We used this trick to find evidence for what we all knew deep in our hearts already: Hufflepuffs are so agreeable. I would like to thank Sophia Crüwell and Lea Jakob for helpful comments on this blog post. ## References • Klugkist, I., Kato, B., & Hoijtink, H. (2005). Bayesian model selection using encompassing priors. Statistica Neerlandica, 59(1), 57-69. • Wetzels, R., Grasman, R. P., & Wagenmakers, E. J. (2010). An encompassing prior generalization of the Savage–Dickey density ratio. Computational Statistics & Data Analysis, 54(9), 2094-2102. • Etz, A., Haaf, J. M., Rouder, J. N., & Vandekerckhove, J. (2018). Bayesian inference and testing any hypothesis you can specify. Advances in Methods and Practices in Psychological Science, 1(2), 281-295. • Crysel, L. C., Cook, C. L., Schember, T. O., & Webster, G. D. (2015). Harry Potter and the measures of personality: Extraverted Gryffindors, agreeable Hufflepuffs, clever Ravenclaws, and manipulative Slytherins. Personality and Individual Differences, 83, 174-179. • Jakob, L., Garcia-Garzon, E., Jarke, H., & Dablander, F. (2019). The Science Behind the Magic? The Relation of the Harry Potter “Sorting Hat Quiz” to Personality and Human Values. Collabra: Psychology, 5(1). ## Footnotes 1. The analytical solution is unpleasant 2. You can discover your Hogwarts House affiliation at https://www.pottermore.com/	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9816714525222778, "perplexity": 2217.2928355851623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305006.68/warc/CC-MAIN-20220126222652-20220127012652-00045.warc.gz"}
https://arxiv.org/abs/0906.1479	hep-ph (what is this?) # Title:Anatomy of the pQCD Approach to the Baryonic Decays $Λ_b \to pπ, p K$ Abstract: We calculate the CP-averaged branching ratios and CP-violating asymmetries for the two-body charmless hadronic decays $\Lambda_b \to p \pi, pK$ in the perturbative QCD (pQCD) approach to lowest order in $\alpha_s$. The baryon distribution amplitudes involved in the factorization formulae are considered to the leading twist accuracy and the distribution amplitudes of the proton are expanded to the next-to-leading conformal spin (i.e., "P" -waves), the moments of which are determined from QCD sum rules. Our work shows that the contributions from the factorizable diagrams in $\Lambda_b \to p \pi, pK$ decays are much smaller compared to the non-factorizable diagrams in the conventional pQCD approach. We argue that this reflects the estimates of the $\Lambda_b \to p$ transition form factors in the $k_T$ factorization approach, which are found typically an order of magnitude smaller than those estimated in the light-cone sum rules and in the non-relativistic quark model. As an alternative, we adopt a hybrid pQCD approach, in which we compute the factorizable contributions with the $\Lambda_b \to p$ form factors taken from the light cone QCD sum rules. The non-factorizable diagrams are evaluated utilizing the conventional pQCD formalism which is free from the endpoint singularities. The predictions worked out here are confronted with the recently available data from the CDF collaboration on the branching ratios and the direct CP asymmetries for the decays $\Lambda_b \to p \pi$, and $\Lambda_b \to p K$. The asymmetry parameter $\alpha$ relevant for the anisotropic angular distribution of the emitted proton in the polarized $\Lambda_b$ baryon decays is also calculated for the two decay modes. Comments: 47 pages, 9 figures, 8 tables; typos and references corrected; version corresponds to the one accepted for publication in Phys. Rev. D Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex) Journal reference: Phys.Rev.D80:034011,2009 DOI: 10.1103/PhysRevD.80.034011 Report number: DESY 09-081 Cite as: arXiv:0906.1479 [hep-ph] (or arXiv:0906.1479v2 [hep-ph] for this version) ## Submission history From: Ahmed Ali [view email] [v1] Mon, 8 Jun 2009 12:36:26 UTC (261 KB) [v2] Fri, 31 Jul 2009 16:44:04 UTC (261 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": 0.9290998578071594, "perplexity": 2011.7064121499793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203493.88/warc/CC-MAIN-20190324210143-20190324232143-00226.warc.gz"}
http://proceedings.mlr.press/v108/modi20a.html	# Sample Complexity of Reinforcement Learning using Linearly Combined Model Ensembles Aditya Modi, Nan Jiang, Ambuj Tewari, Satinder Singh Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:2010-2020, 2020. #### Abstract Reinforcement learning (RL) methods have been shown to be capable of learning intelligent behavior in rich domains. However, this has largely been done in simulated domains without adequate focus on the process of building the simulator. In this paper, we consider a setting where we have access to an ensemble of pre-trained and possibly inaccurate simulators (models). We approximate the real environment using a state-dependent linear combination of the ensemble, where the coefficients are determined by the given state features and some unknown parameters. Our proposed algorithm provably learns a near-optimal policy with a sample complexity polynomial in the number of unknown parameters, and incurs no dependence on the size of the state (or action) space. As an extension, we also consider the more challenging problem of model selection, where the state features are unknown and can be chosen from a large candidate set. We provide exponential lower bounds that illustrate the fundamental hardness of this problem, and develop a provably efficient algorithm under additional natural assumptions.	{"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": 0.9503625631332397, "perplexity": 344.43398884247995}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618039560245.87/warc/CC-MAIN-20210422013104-20210422043104-00066.warc.gz"}
http://digital.auraria.edu/AA00003406/00001	Citation ## Material Information Title: Simulations of prostate biopsy methods Creator: Pellish, Catherine Colby Publication Date: Language: English Physical Description: vi, 70 leaves : illustrations ; 29 cm ## Subjects Subjects / Keywords: Prostate -- Biopsy -- Computer simulation ( lcsh ) Genre: bibliography ( marcgt ) theses ( marcgt ) non-fiction ( marcgt ) ## Notes Bibliography: Includes bibliographical references (leaf 70). General Note: Submitted in partial fulfillment of the requirements for the degree, Master of Science, Applied Mathematics. General Note: Department of Mathematical and Statistical Sciences Statement of Responsibility: by Catherine Colby Pellish. ## Record Information Source Institution: Holding Location: \|Auraria Library Rights Management: All applicable rights reserved by the source institution and holding location. Resource Identifier: 37831972 ( OCLC ) ocm37831972 Classification: LD1190.L622 1997m .P45 ( lcc ) Full Text SIMULATIONS OF PROSTATE BIOPSY METHODS by Catherine Colby Pellish B.S.E.E., Marquette University, 1985 A thesis submitted to the in partial fulfillment of the requirements for the degree of Master of Science Applied Mathematics 1997 This thesis for the Master of Science degree by Catherine Pellish has been approved by William L. Briggs James E. Koehler Weldon A. Lodwick Date Pellish, Catherine Colby (M.S., Applied Mathematics) Simulations of Prostate Biopsy Methods Thesis directed by Associate Professor William L. Briggs Abstract An accepted practice in screening for prostate cancer involves a nee- dle core biopsy of the prostate gland, which can provide information regarding if, and how much, cancer is present in a gland. This paper documents several investigations into prostate gland biopsy techniques. The first phase of study involves a geometric model of a prostate gland containing one to three tu- mors. This mathematical model of the gland is then used to simulate various biopsy techniques and compare the resulting data. Secondly, the best biopsy procedure, as determined from the geometric model, is simulated on actual specimen data which have been digitized. These specimen data are also used for simulation of the six random systematic core biopsy technique (SESCB) currently in clinical use. The results of the geometric model are compared to the results of the simulation on actual data. Finally, the geometric model is used in another series of simulations that investigate the number of needle samples needed to estimate the tumor to gland volume ratio. m This abstract accurately represents the content of the candidates thesis. I recommend its publication. Signed ______________________ William L. Briggs IV ACKNOWLEDGEMENTS I would like to sincerely thank a number of people who consistently provided me with their support, encouragement and guidance as I pursued the completion of this thesis. Dr. Bill Briggs, my advisor, served as a constant source of insight and motivation, as well as providing considerable direction throughout this process. I am also grateful for the time spent with Dr. Jim Koehler who had to teach me the finer points of statistics again and again. My thanks to both of these professers for proving to be excellent academic sources. I also would like to thank Norm LeMay who, out of the generousity of his heart and his need for a free lunch, assisted me in running the ANOVA analysis which this thesis required. Finally, I must thank my family, Mark, Eric and Corinne for encour- aging me and making me laugh through every crisis. CONTENTS Chapter 1 Introduction............................................... 2 1.1 Clinical Prostate Biopsy Analysis.......................... 2 1.2 Summary of Mathematical Methods............................ 4 2 The Geometric Model........................................ 5 2.1 Geometric Model of gland and tumor......................... 5 2.2 Simulations............................................... 10 2.3 Statistical Analysis of Results........................... 14 2.4 Simulation Results........................................ 16 2.4.1 Applying the ANOVA to the Biopsy Simulation Data 18 2.4.2 ANOVA Mechanics......................................... 23 2.4.3 Residuals .............................................. 24 2.4.4 The Null and Alternate Hypotheses....................... 25 2.4.5 Are the Main Effects all Equal? ........................ 27 2.4.6 Recognizing Interaction between Factors................. 30 2.4.7 Clinical Distribution of Tumors......................... 38 vi 3 Digitized Specimen Data 43 3.1 Summary of Software Tool............................... 43 3.2 Specific Algorithms ................................... 45 3.2.1 Locating the Apex..................................... 45 3.2.2 Establishing Needle Positions......................... 47 3.3 Simulations............................................ 49 3.4 Geometric Model vs Clinical Model ..................... 51 3.5 Optimal Technique vs SESCB ............................ 53 4 Geometric Model Volume Estimates....................... 56 4.1 Tumor Volume Estimates................................. 56 4.1.1 One-Dimensional Analysis Line Model............. 58 4.1.2 Two-Dimensional Strip Model .................... 58 4.1.3 Three-Dimensional Cylinder Model.................... 59 4.2 Experiment Setup....................................... 60 4.3 Results................................................ 62 4.4 Interactive Utility.................................... 63 Appendix A ANOVA Definitions............................................. 65 1 1. Introduction 1.1 Clinical Prostate Biopsy Analysis Currently the standard method of determining if a given prostate gland is cancerous involves two procedures. The first is the prostate-specific antigen (PSA) test which measures the level of antigens in the patients blood, a high level indicating a higher possibility of cancerous tissue. The second procedure is the needle biopsy which is carried out if the PSA test so indicates. The clinician conducts this biopsy by inserting a needle-tool, equipped with ultrasound capabilities, into the patients rectum. The gland is located and the urologist fires three needles into the right lobe of the gland and three needles into the left lobe at approximately symmetric positions. The left- right division of the gland is determined by the position of the urethra in the gland. This physical landmark is used as the visual dividing line, enabling clinicians to execute the biopsy in a systematic manner. The needle-tool is rotated to the left or right depending on the targeted lobe. This rotation corresponds to the angle slight rotation, the needles are inserted at a second independent angle, referred 2 to as 9. The choice of a six-needle biopsy is based on the six random systematic core biopsies (SESCB) method developed by Hodge et al [1] and currently thought to achieve the best detection rates. The results from this diagnostic biopsy are then analyzed in order to determine the best treatment plan for the patient. There are several factors that help the urologist choose the optimal treatment plan. The first factor is obviously whether the biopsy shows any tumor cells at all. According to the Hodge study, 96% of the 83 men diagnosed with cancer had the cancer detected by SESCB. However, as investigated by Daneshgari et al [2], in prostate glands with low tumor volume, the SESCB fails to achieve such a high percentage of detection. This study concluded that an improved biopsy strategy may be needed in detection of CaP (carcinoma of the prostate) in patients with low volume cancer. Secondly, the volume of the tumor itself is a deciding factor in determining treatment. Thirdly, the location of the tumor, specifically if the tumor penetrates the capsule of the gland, can define a specific treatment plan. Some of this information is available from a single needle-core biopsy; 3 1.2 Summary of Mathematical Methods As an aid in understanding this problem, as well as researching ways to improve diagnosis, two methods of analysis are undertaken. The first method relies on a geometric model of the prostate gland with from one to three tumors. Various biopsy methods are simulated with this mathematical model and results are tabulated. The second method involves running the same biopsy simulations on actual prostate glands which have been digitized and stored as three-dimensional objects in a computer. The experimental re- sults from these two methods are then compared. All of the simulations were executed using software created for this purpose primarily by this author, al- though the skeletons of these software tools were engineered during the Spring 1995 Math Clinic on this topic by several participants. The simulations are written in C and C++, running on a UNIX-based computer. They are exten- sively documented and flexible enough to be useful in a variety of experiments within this realm of research. 4 2. The Geometric Model 2.1 Geometric Model of gland and tumor An actual prostate gland is about the size of a walnut with volumes ranging from 22 cc to 61 cc [3]. The geometry of an ellipsoid closely models this gland and any tumors present within it. Therefore, an ellipsoid of the form x2 y2 z2 1 ----b --1----= 1 A2 B2 C2 is used to represent the prostate gland. Ellipsoids are also used to represent each of the tumors. The dimensions of the gland, A, B, and C, are chosen randomly in the following experimentally determined ranges: 3.0 cm < A < 4.8 cm 3.8 cm < B < 4.6 cm 3.8 cm < C < 5.2 cm 22 cc < [gland volume} < 61 cc. The prostate is divided into 3 zones: the peripheral, the central and the transition region. The peripheral zone comprises approximately 70% of the mass of the prostate gland. It is located in the lower area of the gland, 5 closest to the rectum. This region is the site of origin of most carcinomas [3]. The central region makes up approximately 25% of the glandular mass and is resistant to both carcinoma and inflammation [3]. The transition region contains the remaining 5% of prostate gland tissue and can be the site of some cancers. Figure 2.1 shows these regions of the prostate gland. Based on this clinical information, the software-generated tumors are located in the lower part of the elliptical gland model to simulate tumors residing in the peripheral zone. Figure 2.2 depicts the geometrical gland and tumor model in the xyz system. Since the gland model is centered at the origin, the //-coordinate of the tumor center, yc, is always negative in order to place the tumor in the peripheral zone of the gland. However, other distributions of y could be used to improve the model. Tumors are modeled by an equation of the form % xc)2 (y yc f (z zc f = a2 h2 c2 where xc, yc and zc specify the center of the tumor. The biopsy needle is modeled as a line with the parametric equations x(t) = Xo + tsm0sin(f) y(t) = y0 + t sin 9 cos z(t) = z0 + t cos 9, 6 Figure 2.1. The peripheral (PZ), central (CZ) and tran- sition (TZ) regions divide the prostate gland into 3 ma- jor zones. Figure 2.2. The gland and tumor are modeled by ellip- soids in the xyz coordinate system. 7 where xQ, yQ, and z0 are the coordinates of the entry point of the needles (Figure 2.3 and Figure 2.4). The angle determines a plane. The angle 9 is then assumed to remain in this plane and is measured from the z-axis. From these definitions, the parametric equations for the line are determined. The parameter t measures the length of the needle. Figure 2.3. This figure of the xy plane and needle illus- trates measurement of ip. Substituting the parametric equations of the needle into the equation for the tumor, it is possible to determine values of t corresponding to an intersection. The equation of the tumor is (x(t) xc f (:y(t) yc f (z(t) zc f 8 Figure 2.4. This figure of the yz plane and needle illus- trates measurement of 9. Replacing x(t), y(t) and z(t) by the parametric equations of the nee- dle gives f2{ sin2 6 sin2 9 sin2 9 cos2 6 cos2 9, ----------------1---------------- H---------- a2 b2 c2 ' ^f2(x0 xc) smipsm9 \| 2(y0 yc) sm9cosip t( o I JO ^ a2 tr 2(^0 zc) cos 9 (xq-xc)2 (yo-yc)2 (zQ zc)2 c2 1 a2 b2 c2 ) = I- (2-1) If the discriminant (B'2 1.!'("') is positive, two real roots exist. In this case we have A , sin2 6 sin2 9 sin2 9 cos2 6 cos2 9 ------1--------1------------ H------- a2 h2 c2 B , 2(a;0 xc) sin C b2 , (^o xc)2 (yQ yc)2 {zq zcy a2 62 c2 9 If real roots t\ and f2 exist, they give the points where the tumor ellipsoid and the line intersect. If these values are greater than 0 and less than the actual needle length, the needle has intersected the tumor. The amount of tumor extracted by the needle is proportional to the difference between the two roots of the quadratic, \| ti \|- By comparing the two roots, an estimate of the volume of the tumor that is contained in the needle can be made. If real roots do not exist, the needle does not intersect the tumor ellipsoid and no tumor information is gained by that needle. In this analysis, each biopsy procedure was simulated on 1000 differ- ent gland models and the number of times a tumor was detected per procedure was recorded. This method does not differentiate between one or more nee- dles detecting the tumor. It simply records a hit or miss per biopsy procedure. In addition, an estimate of the tumor volume is made whenever a tumor is detected. 2.2 Simulations Since a fundamental goal of any biopsy is to determine whether or not the gland contains cancerous cells, the first series of simulations is intended to compare the detection rate of several biopsy techniques. The detection rate is defined as the number of times a biopsy procedure detects a tumor to the 10 total number of biopsies conducted. A set of 54 different biopsy procedures is simulated with variation in the following parameters: number of needles, offset between needles in the z direction, 6, and 4>. The distance in the z direction between needles can be a relative spacing based on the gland dimension in the z direction or an absolute spacing of 1 cm between each needle. The first method is referred to as relative spacing since it depends on the gland size and separates the needles by equal distance. The second is referred to as the absolute spacing and has its basis in the SESCB procedure. As a means of clarification, Figures 2.5 and 2.6 illustrate the analysis of a single specimen and the execution of the entire experiment. Each of the 54 biopsy procedures is simulated on 1000 different gland models. The random number generator is seeded once for each series of 1000 simulations using a specific biopsy technique. Prior to the next technique, the random number generator is reseeded with the same number, thereby yielding the identical set of 1000 prostate models. This insures that each of the biopsies is conducted on the same set of 1000 simulated glands. The detection rate is determined for each of these procedures and the results of the simulation are documented in Table 2.1. 11 Figure 2.5. This flow chart depicts the top-level algo- rithm for modeling a single biopsy with several needles. 12 Figure 2.6. This flow chart depicts the simulation pro- cess for the entire simulation, each biopsy procedure is simulated on 1000 geometric gland models. 13 2.3 Statistical Analysis of Results In order to interpret the output from the simulations legitimately, a statistical tool is needed. First, we must determine whether or not the various biopsy settings influence the observed detection rate. In other words, is there a relationship between the settings of any one or combination of the four factors (number of needles, ^-spacing, 9 and 0) and the detection rate or are the results completely random, therefore implying that the biopsy specification does not determine the detection rate? We need a mathematically sound method to compare the detection rates provided by the simulation and to infer some conclusions. The statistical model known as Analysis of Variance (ANOVA) was used to compare the population means between various treatments, thus resulting in a statistically valid conclusion. This model can be employed to determine whether the various factors interact and which factors have the most impact on the outcome. In order to describe the ANOVA model, a few definitions are required. (1) Factors are the independent variables that are under investigation. In this instance, the biopsy parameters (number of needles, spacing 14 method, 9 and Number of Needles Spacing Method 9 Factor 4 Absolute 30 30 Levels 6 Relative 45 45 8 o O o O (2) Factor levels are the values that each of the factors can take on during a single simulation. As shown in the list of biopsy simulation factors and levels, each factor does not have the same number of factor levels. The factor Spacing Method only has two factor levels, whereas the other three factors each have three factor levels. (3) A treatment is a particular combination of levels of each of the factors involved in the experiment, where an experiment is the simulation of the treatment on 1000 geometric specimens. In this example, a treatment refers to a biopsy with specific settings (for example, 4 nee- dles, absolute spacing, 9 = 45, are 54 different treatments and therefore, 54 different experiments, corresponding to all the combinations of the levels of the four factors. (4) A trial is defined to be a simulation of one treatment on one geomet- ric model. The outcome of a trial is either 1, the biopsy procedure detected the tumor, or 0, the tumor remained undetected. The out- come of the experiment is the detection rate achieved by a specific 15 treatment simulated on 1000 geometric specimens. In other words, the outcome of the experiment is the number of specimens in which tumor is detected versus the total number of specimens simulated and is referred to as outcome for the remainder of this thesis. 2.4 Simulation Results For each of the 54 treatments, the simulation is conducted on 1000 different gland models. The following table summarizes the treatment param- eters as well as the results: Treatment Parameters Outcome Experiment Number of Needles Spacing Method e Detection Rate 1 4 Relative 45 45 0.252 2 6 Relative 45 45 0.307 3 8 Relative 45 45 0.335 4 4 Absolute 45 45 0.263 5 6 Absolute 45 45 0.293 6 8 Absolute 45 45 0.298 7 4 Relative 60 45 0.267 8 6 Relative 60 45 0.341 9 8 Relative 60 45 0.369 10 4 Absolute 60 45 0.270 11 6 Absolute 60 45 0.320 12 8 Absolute 60 45 0.339 13 4 Relative 30 45 0.196 14 6 Relative 30 45 0.225 15 8 Relative 30 45 0.255 16 4 Absolute 30 45 0.207 17 6 Absolute 30 45 0.221 18 8 Absolute 30 45 0.221 Table 2.1. The results from the 54 geometric model experiments are displayed. 16 Treatment Parameters Outcome Number of Spacing Detection Experiment Needles Method e Rate 19 4 Relative 45 60 0.200 20 6 Relative 45 60 0.234 21 8 Relative 45 60 0.268 22 4 Absolute 45 60 0.211 23 6 Absolute 45 60 0.225 24 8 Absolute 45 60 0.228 25 4 Relative 60 60 0.191 26 6 Relative 60 60 0.254 27 8 Relative 60 60 0.268 28 4 Absolute 60 60 0.209 29 6 Absolute 60 60 0.240 30 8 Absolute 60 60 0.246 31 4 Relative 30 60 0.172 32 6 Relative 30 60 0.194 33 8 Relative 30 60 0.219 34 4 Absolute 30 60 0.188 35 6 Absolute 30 60 0.197 36 8 Absolute 30 60 0.197 37 4 Relative 45 30 0.260 38 6 Relative 45 30 0.316 39 8 Relative 45 30 0.341 40 4 Absolute 45 30 0.264 41 6 Absolute 45 30 0.305 42 8 Absolute 45 30 0.316 43 4 Relative o O 30 0.283 44 6 Relative o O 30 0.351 45 8 Relative o O 30 0.385 46 4 Absolute o O 30 0.279 47 6 Absolute o O 30 0.346 48 8 Absolute 60 30 0.372 49 4 Relative 30 30 0.210 50 6 Relative 30 30 0.247 51 8 Relative 30 30 0.273 52 4 Absolute 30 30 0.225 53 6 Absolute 30 30 0.245 54 8 Absolute 30 30 0.247 Table 2.1. (Cont.) The results from the 54 geometric model experiments are displayed. 17 2.4.1 Applying the ANOVA to the Biopsy Simulation Data The biopsy simulation is a multi-factored system, in which the four parameters (number of needles, spacing, 9 and 0) individually and perhaps in some combinations may have a measurable effect on the detection rate. Therefore a factor effects model is used in order to determine the impact of and interactions between these four parameters. This biopsy simulation is considered a complete factorial study since all possible combinations of the four parameters were simulated and evaluated. The indices %,j, k, l refer to the levels of the factors number of needles, spacing method, 9 and respectively. In this multi-factored system, a true overall mean, p which is equiv- alent to the true overall detection rate, is assumed to exist. The entire simu- lation results in 54 observed detection rates, ppui, each of which indicates the observed detection rate for a given experiment. This set of 54 observed detec- tion rates is used in the ANOVA to determine estimated factor effects and an estimated overall mean which are used in the factor effects model. The factor effects model is used to predict a detection rate, a probability of detection, pijki, given the levels of the four factors. A factor level mean is the average detection rate for a group of 18 treatments that have one common factor level held constant while all others vary. For example, all outcomes from experiments with Number of Needles= 6 are averaged to yield the factor level mean for the factor Number of Needles at the level i = 6. The overall mean, //. is simply the average outcome of all experiments. The difference between each factor level mean and the overall mean yields the main effect for that factor level. Because this model has 4 factors each with either 2 or 3 levels, the following main effects are designated. Q!i the main effect for the factor Number of Needles at each of its levels (4,6,8): 1 < % < 3. (3j the main effect for the factor Spacing Method at each of its levels (0,1): 1 < j < 2. 7fe the main effect for the factor 9 at each of its levels (30,45,60): 1 < k < 3. 8i the Main Effect for the factor 1 < l < 3. A factor at a particular level may influence another factor either by inhibiting or enhancing its impact. Because of these interactions between factors, the interaction effects are included in the model. Pairwise interaction 19 effects are a measure of the combined effect of two factors, across the different levels, minus the main effects of these factors. We define these two-way effects as follows. (a/3)ij number of needles and spacing method ial)ik number of needles and 9 (aS)u number of needles and iPl)jk spacing method and 9 {(35)ji spacing method and {j5)ki 9 and 4>. Three-way factor effects are a measure of the interaction effect of three factors. (a(3j)ijk number of needles, spacing method and 9 (a(35)iji number of needles, spacing method and {l3'yS)jk[ spacing method, 9 and {ar)8)iki number of needles, 9 and 4>. The four-way effect is the measure of the interaction effect of all four factors. {aPj5)ijki number of needles, spacing method, 9 and 4>. 20 Summary of Variables True overall mean n Estimated overall mean fi True treatment mean IMjkl Estimated treatment mean Pijkl Observed treatment detection rate Pijkl Transformed observed treatment detection rate Yijkl Estimated treatment detection rate Pijkl Transformed estimated treatment detection rate Yijkl Average observed detection rate P True main factor level effects a,h 0, 7fe, S( Estimated main factor level effects a, Pj, Ik, St True two-way effects (af])ij, (ay)ik, (aS)u {Pi)jk, (PS)jt, (7S)m Estimated two-way effects (^7Ma (Pl)jk> {P8)n, (7S)M Table 2.2. A list of the variables used in the ANOVA analysis is displayed. The factor effects model takes the general form Pijkl /i + a* + Pj + 7fc + Si + (a0)ij + (ay)ik + (aS)u + (Pl)jk + + (7^)fei +(a#y)iifc + {oiPS)iji + (PyS)jM + (ajS)m + (aPyS)ijkl. The observed outcome, the detection rate for a particular treatment, as given in Table 2.1, is pijki and is the sum of the true mean for that treatment and a residual term: Pijkl = IMjkl + Oj'fcl- 21 The goal of the analysis is to formulate a model that predicts the outcome of a given treatment. Since the true means and true factor effects are not known, estimates of these terms are determined from the simulation and used in the model. Estimated values are indicated with the ~ notation. The predicted outcome is represented by the following relationship: Pijki = A + (h + Pj + ik + \$i + {ptl3)ij + {aj)ik + {aS)u + {f3j)jk + {P\$)ji + (7 \$)ki + M l)ijk + + W)jkl + (al5)iki + (a^5)ijki- In this equation is the estimated probability of detecting a tumor at the factor levels indicated by %,j,k,l. This probability is predicted by the model using least -square estimators for the terms in the equation. The probability of detection is a function of the estimated overall mean, /2, and the estimated effects from the four factors, alone and in combination with one another. Not all of these effects may be significant. In order to determine which of the factors do significantly effect the detection rate and therefore belong in the final model, various means are evaluated. If all the means for a particular factor (or combination of factors) are equal, varying a factor level does not add to or subtract from the overall mean and therefore the factor does not belong in the final model. This equality question is put, not only to each factor individually, but to all the combinations of factors as well. 22 2.4.2 ANOVA Mechanics Use of the ANOVA model is founded on several assumptions: (1) The outcomes follow a normal probability distribution. (2) Each distribution has the same variance. (3) The outcomes for each factor level are independent of the other factor level outcomes. With these assumptions in mind, note that the probability distributions of a factor at each of its levels differs only with respect to the mean [4], Therefore, the first step in executing the analysis is to determine if the detection rates, are statistically different. Secondly, if they are different, one of the intents of the ANOVA model is to determine if the difference between the detection rate of two or more treatments is sufficient, after examining the variability within the treatments, to conclude that one treatment does indeed produce a higher detection rate. In addition, by evaluating the statistical data, conclusions may be drawn as to how each factor, both independently and within established interaction groups (pairwise, three-way or four-way), influences the outcome. 23 2.4.3 Residuals We define p to be the average of all observations. The model states that Pijki = IMjki + Â£ijkh therefore the residual term is = Pijki IMjki Since Pijki is estimated by fiijki, the estimated residual term is e^i = pijki Pijki, the difference between the observed and the estimated average detection rate. The set of all 54 residuals, e^i, for all i,j,k and l are evaluated for three characteristics which indicate whether the fitted data are well-suited for the analysis. These characteristics are: 1. Normality of error terms. 2. Constancy of error variance. 3. Independence of error terms. Several statistical tests and plots used on the residual data determine whether one of the five assumptions is violated. These tests revealed that the error variances were not stable, thus violating the first characteristic. A transformation was employed to preserve the statistical information in the output, but stabilize the error variances. Since nothing is lost by employing a transformation and the error variances are stabilized, the detection rate data p is transformed to Y via the following relationship: Y = 2 arcsin(x/p). 24 The outcome from these simulations is the detection rate, a proportion of the number of specimens where tumor is detected to the total number of specimens. The arcsine transformation is the most appropriate transformation when the outcome is a proportion [4], All ANOVA data referenced from this point on are transformed unless noted otherwise. The inverse transformation is calculated at the conclusion of this analysis to get a true estimate of the probability. 2.4.4 The Null and Alternate Hypotheses A starting point in the ANOVA process is to establish two hypothesis, a null and alternate hypothesis. The null hypothesis assumes that all effects are equal, therefore indicating that specific factor levels do not influence the outcome. The alternate hypothesis assumes that at least two of the effects are not the same. The F-test is used to decide which of these two hypotheses concerning the data will be accepted. The test consists of computing the ratio of between- effect variation to within-effect variation. This bet weeu-elfeet variation, which changes depending on the effect, is called the treatment sum of squares and is denoted SSA, SSB, SSC, and SSD (see Appendix also). It is a measure of the difference between the detection rate of a set of treatments and the average detection rate over all treatments. The within-effect variation 25 is called the error sum of squares and is denoted SSE. It is a measure of the difference between the individual outcome for a given treatment and the estimated detection rate over that treatment. The error sum of squares measures variability that is not explained by the SSA, SSB, SSC, or SSD terms and therefore occurs within the set of treatments. Both of these variation measurements are evaluated using sum of the squares expressions as detailed in the Appendix. The means of the SSA, SSB, SSC, SSD and SSE terms are MSA. MSB. MSC. MSI) and MSE respectively, and are computed by dividing by the degrees of freedom, df, associated with each term. This results in /' = MSA/MSE where MSA = SS A fdf\ (MSB = SSB/dfs,etc) and MSE = SSE/df. Large values of F tend to support the conclusion that all the effects are not equal (Ha), whereas values of F near 1 support the null hypothesis (H0). In the event that the alternate hypothesis is indicated via the F-test, the ANOVA also provides the probability of a TYPE I error. A TYPE I error occurs when it is concluded that differences between means exist when, in fact, they do not (i.e. accept Ha when in fact H0 is true). This information is given in the column labelled Pr(F) in the ANOVA output in Table 2.3. 26 2.4.5 Are the Main Effects all Equal? Following the general process of establishing null and alternate hy- pothesis as described above, a pair of null and alternate hypotheses are stated for each factor in the biopsy model. The null hypothesis assumes that the main effects for a given factor at each of its levels are equivalent. The alter- nate hypothesis obviously assumes that the main effects differ. H0: Q!i = Q!2 = Q!3 Ha; not all cq are equal. Pi = /?2 not all Pi are equal. <\$i = 82 = S3 not all 7i are equal. 7i = 72 = 73 not all Si are equal. The F-test statistic is applied to determine which hypothesis to ac- cept in each case. The factor sum of squares for each factor, number of nee- dles, spacing, 9 and p, denoted SSA, SSB, SSC and SSD, respectively, is computed as shown in the Appendix. The mean of each of these fac- tor sum of square terms is computed by dividing each term by its associ- ated degrees of freedom so that MSA = SSA/S/a, MSB = SSB/dfs, etc. as detailed in the Appendix. The test statistic is formed for each hypoth- esis in the following manner. To test the effect of the first factor, Num- ber of Needles, F = MSA/MSE; to test the effect of the spacing factor, 27 F = MSB/MSE; to test the effect of 0. /' = MSC/MSE; and to test the effect of o. /' = MSD/MSE. Accepting the alternate hypothesis means that a specific setting of the given factor corresponds to a change in detection rate; thus that factor has an effect on the overall outcome of the biopsy. Df Sum of Sq Mean Sq F Value Pr(F) Needles 2 0.15862 0.07931 607.427 0.0000000 Main Spacing 1 0.00498 0.00498 38.209 0.0000011 Effects e 2 0.29249 0.14624 1120.073 0.0000000 2 0.28115 0.14057 1076.661 0.0000000 Ndls:Spc 2 0.1641 0.00820 62.846 0.0000000 Needles: 9 4 0.01444 0.00361 27.653 0.0000000 2-Way Spacing: 9 2 0.00059 0.00029 2.283 0.1206068 Effects Needles: 4 0.00395 0.00098 7.569 0.0002892 Spacing: 2 0.00046 0.00023 1.794 0.1848710 9: 4 0.02867 0.00716 54.902 0.0000000 Residuals 28 0.00365 0.00013 Table 2.3. The output from the ANOVA is displayed above. See Appendix for details of the calculations. Eefering to this ANOVA output, the column of numbers labelled Sum of Sq refers to the parameters SSA, SSB, SSC and SSD detailed in the Ap- pendix. The column labelled Mean Square lists the parameters MSA, MSB, MSC, MSD. The F Value column lists the F-test outcome for each row: (Nee- dles F Value = MSA/MSE). The larger values in this column tend to support the alternate hypothesis that the main effect for a given factor differs across 28 the possible levels for that factor. The final column, Pr(F), gives the probabil- ity of a Type I error. Again, a Type I error occurs if the alternate hypothesis is concluded when in fact, the null hypothesis is true. The row labelled Residuals indicates the total degrees of freedom, the SSE and the MSE for this analysis. Based on the numbers in the table, each of the four main effects has a significant effect on the outcome with the factor 9 having the great- est influence on the detection rate, followed by the factors ber of Needles. This fact is indicated by the high F-value that corre- sponds to each of the four factors. The rows labelled with two factor names (for example, Needles: Spacing) indicate the ANOVA output correspond- ing to pair-wise interactions and include the sum of squares computed for each pair of factors. The sum of squares for all of the pair-wise interac- tion terms (SSAB, SSAC, SSAD, SSBC, SSBD, SSCD) are computed as detailed in the Appendix. The total treatment sum of squares, SSTR = This sum does not include the sum of square terms due to the three-way and four-way interactions because there are not enough degrees of freedom in the experiment to use the full model. 29 2.4.6 Recognizing Interaction between Factors At this point, the F-test has determined that each of the main factor effects contributes to the overall detection rate. To evaluate the interaction effects, the F-test is applied again The F-test is applied to determine inter- action between, in this case, two, three or four factors. A null and alternate hypothesis is formulated for all possible combinations of factors and sum of square terms are computed for the factor groups and used in each F-test. The null and alternate hypothesis are constructed for each of the pairwise interac- tions. H0: all (a0)ij = 0 Ha: not all (ap),^ = 0 all (aj)ik = 0 not all (aj)ik = 0 all (aS)u = 0 not all (aS)ii = 0 all {Pi)jk = 0 not all (/3j)jk = 0 all {pS)ji = 0 not a\\((38)ji = 0 all (jS)ki = 0 not all (jS)kl = 0 All three-way combinations are formed, hypotheses are constructed and F-test results are evaluated. H0: all (a(3j)ijk = 0 Ha: not all (afij)ijk = 0 all (a(38)iji = 0 not all (a(38)iji = 0 all (ajS)jM = 0 not all (aj8)iki = 0 all (076)jkl = 0 not all (/3jS)jkl = 0 The null/alternate set of hypothesis is constructed for the four-way interaction. 30 H0: all {a(3-fS)im = 0 Ha: not all (a(3j8)ijki equal 0 Based on the actual ANOVA results in the preceding table, four of the pair-wise interactions appear strongly significant: Needles: Spacing, Needles: 9, Needles: . The other two pair-wise interactions are included in the final model even though the strength of their significance is uncertain. The ANOVA was executed once to include all three-way interac- tions. Since these interactions proved insignificant, they are not included in the model. There are not enough degrees of freedom in the experiment to estimate the residuals and test for the four-way interaction. As stated previously, the Y notation indicates the transformed de- tection rate (p). At this point the general model, of the form 1ijkim = /7... T T j3j T 'Tfc T S[ Main effects +iaP)ij + (al)ik + (aS)u + ((3j)jk + +((38)ji + (j8)ki Pairwise effects +(a/3j)ijk + (a(38)iji + (/3j8)jki Three-way effects + (a/3j8)ijki Four-way effect residual error is reduced to the final model for this analysis: 8'ijki (i + &i + (%+ik + 8i + {aj3)ij + {aj)ik + {aS)u + {f3j)jk + iP8)jt + (7 8)kl. This model yields the transformed probability of detection at the given levels for %,j, k and l. 31 Now that the factor effects have been identified, the analysis revolves around determining the factor levels that result in the highest detection rate. For this part of the analysis, the tables of means and tables of effects are evaluated. Ik... Grand Mean 1.072 Needles 4 6 8 Spacing Relative Absolute /h... 0.999 1.09 1.128 fJ'.j.. 1.082 1.063 e 30 45 60 30 45 60 ik.k. 0.9723 1.098 1.147 lk..i 1.14 1.1104 0.9724 Table 2.4. The ANOVA tables of means list the transformed values. Needles 30 0 45 o O Spacing 30 0 45 o O 4 0.926 1.027 1.045 Relative 0.978 1.111 1.157 6 0.979 1.113 1.176 Absolute 0.967 1.084 1.137 8 1.012 1.152 1.221 Needles 30 45 60 Spacing 30 45 60 4 1.054 1.028 0.915 Relative 1.148 1.118 0.980 6 1.161 1.123 0.985 Absolute 1.132 1.091 0.965 8 1.205 1.163 1.017 Spacing Needles Relative Absolute 0 30 45 60 4 0.987 1.011 30 1.026 0.978 0.913 6 1.099 1.080 45 1.159 1.139 .0994 8 1.159 1.097 60 1.235 1.196 1.010 Table 2.5. The transformed values of the pairwise means are shown. 32 Referring to the ANOVA tables of means, the highest numbers in each category reflect the best setting for a particular factor. On reading through the tables of means, the conclusion is that a technique of 8 needles, relative spacing, 9 = 60 and corroborate this more fully, the interactions that are deemed significant are analysed to verify that the main effect is not contradicted by an interaction. Therefore, the table for Needles: 9 is reviewed and it is found that the setting of 8 needles and 9 = 60 again yields the highest mean. The tables for all of the pair-wise combinations are reviewed to determine that the best settings yield the highest means in the interaction tables just as they did in the main effect tables. This proves to be the case, so none of the interactions contradict the conclusion drawn from the main effect information. 33 Number of Needles (4, 6, or 8) Q!l &2 &Z Effect -0.07329 0.01723 0.05607 Spacing (Relative or Absolute) to Effect 0.009612 -0.009612 e (30, 45, or 60) 7i 72 73 Effect -0.1001 0.02519 0.07486 (30, 45, or 60) 5i S2 S3 Effect 0.0678 0.03215 -0.09995 Table 2.6. The main factor level effects from the ANOVA output are documented. 34 Spacing Relative Absolute 4 Needles 6 8 -0.02127 0.02127 -0.00017 0.00017 0.02143 -0.02143 e 30 45 60 4 Needles 6 8 0.02680 0.00244 -0.02925 -0.01031 -0.00127 0.01158 -0.01649 -0.00118 0.01767 e 30 45 60 Spacing Relative Absolute -0.004354 0.003708 0.000646 0.004354 -0.003708 -0.000646 30 45 60 4 Needles 6 8 -0.01271 -0.00292 0.01563 0.00363 0.00087 -0.00450 0.00907 0.00206 -0.01113 30 45 60 Spacing Relative Absolute -0.001740 0.004148 -0.002407 0.001740 -0.004148 0.002407 30 45 60 30 6 45 60 -0.01404 -0.02664 0.04067 -0.00621 0.00978 -0.00357 0.02025 0.01686 -0.03711 Table 2.7. The ANOVA table of effects for pairwise interactions is displayed. 35 By using the values from the tables of effects, a probability for de- tection is calculated for the optimal setting: ^ 3131 = (l + dz + (h +73 + <5i + (Q!/3)31 + (<27)33 + (<2Â£)31 + (%7)l3 + (^)ll + (7^)31 1.347918 = 1.072 + .05607 + .009612 + .07486 + .0678+ .02143 + .01767 + .00907 + .000646 + ^0.00174 + .02025 This result of 1.347918 is then transformed back (arcsine equation) to yield a probability of 0.38948 for this setting. 1.347918 = 2 arcsin\f(p) p = (sin(1.347918/2))2 = 0.38949. Therefore, with the factors set to 8 needles, relative spacing, 9 = 60 and 4> = 30, the biopsy procedure has a 38.9% probability of detecting the cancer given the tumor distribution model used. This estimated probability is best used in comparisons with the other estimated probabilities rather than as an absolute measure of detection rate. Therefore the conclusion from this analysis is a relative ranking of treatments in terms of their detection rate. Since the 1000 simulated specimens were the same for each treatment, the ANOVA model determined the relative differences between detection rates of various treatments, not necessarily providing enough data and results to draw 36 conclusions about absolute detection rates. Table 2.8 lists each experiment and the probability of detection predicted from the factor effects model. Treatment Parameters Experiment Number of Spacing Needles Method e Predicted Probability 1 4 Relative 45 45 0.247 2 6 Relative 45 45 0.297 3 8 Relative 45 45 0.327 4 4 Absolute 45 45 0.251 5 6 Absolute 45 45 0.281 6 8 Absolute 45 45 0.291 7 4 Relative o O 45 0.265 8 6 Relative o O 45 0.337 9 8 Relative o O 45 0.369 10 4 Absolute o O 45 0.271 11 6 Absolute o O 45 0.324 12 8 Absolute o O 45 0.335 13 4 Relative 30 45 0.195 14 6 Relative 30 45 0.227 15 8 Relative 30 45 0.251 16 4 Absolute 30 45 0.205 17 6 Absolute 30 45 0.219 18 8 Absolute 30 45 0.224 19 4 Relative 45 60 0.200 20 6 Relative 45 60 0.236 21 8 Relative 45 60 0.260 22 4 Absolute 45 o O 0.208 23 6 Absolute 45 o O 0.227 24 8 Absolute 45 o O 0.232 25 4 Relative o O o O 0.192 26 6 Relative o O o O 0.247 27 8 Relative o O o O 0.273 28 4 Absolute o O o O 0.203 29 6 Absolute o O o O 0.241 Table 2.8. The probabilities of detection for one tumor simulations are displayed. 37 Treatment Parameters Experiment Number of Spacing Needles Method e Predicted Probability 30 8 Absolute 60 60 0.248 31 4 Relative 30 60 0.175 32 6 Relative 30 60 0.196 33 8 Relative 30 60 0.215 34 4 Absolute 30 60 0.189 35 6 Absolute 30 60 0.194 36 8 Absolute 30 60 0.195 37 4 Relative 45 30 0.257 38 6 Relative 45 30 0.314 39 8 Relative 45 30 0.346 40 4 Absolute 45 30 0.266 41 6 Absolute 45 30 0.303 42 8 Absolute 45 30 0.315 43 4 Relative 60 30 0.276 44 6 Relative 60 30 0.354 45 8 Relative o O 30 0.389 46 4 Absolute o O 30 0.287 47 6 Absolute o O 30 0.346 48 8 Absolute o O 30 0.360 49 4 Relative 30 30 0.208 50 6 Relative 30 30 0.246 51 8 Relative 30 30 0.272 52 4 Absolute 30 30 0.223 53 6 Absolute 30 30 0.243 54 8 Absolute 30 30 0.250 Table 2.8. (Cont.) The probabilities of detection for one tumor simulations are displayed. 2.4.7 Clinical Distribution of Tumors The biopsy simulations were conducted a second time on more real- istic geometric glands. By using a clinically derived distribution of number of tumors per gland, a better population was available for these biopsy sim- ulations. A sample size of 1000 was again used but in this experiment, 1/4 38 of the glands had a single tumor, 1/2 had two tumors and the remaining 1/4 had 3 tumors. The total gland volume was again held to be less than 6.4 cc. This distribution is based on the analysis done by Daneshagari [2]. The ANOVA results are found in the Appendix and yield the same optimal biopsy procedure with a slightly different probability resulting from the factor effects model. By using the values from this second table of effects, a probability for detection is calculated for the optimal setting: ^3131 = fi + d3 + (3i +73 + <\$i + (oi(3)31 + (0:7)33 + (0^)31 + (Pi) i3 + (Pd)n + (7^)31 1.7535 = 1.429 + 0.0733 + 0.01507 + 0.07456 + 0.07091 + 0.02321 + 0.02650 + 0.01412 0.005442 0.004094 + 0.03638 Transforming this value (arcsine) yields a probability of detection for the optimal setting of .5908. This probability of 59.08% is higher than the 38.9% achieved by the simulation using geometric models of one tumor as would be expected. The predicted probabilities for each of the 54 experiments given this distribution of tumors is shown in Table 2.9. 39 Treatment Parameters Experiment Number of Spacing Needles Method e Predicted Probability 1 4 Relative 45 45 0.417 2 6 Relative 45 45 0.489 3 8 Relative 45 45 0.526 4 4 Absolute 45 45 0.417 5 6 Absolute 45 45 0.470 6 8 Absolute 45 45 0.482 7 4 Relative o O 45 0.427 8 6 Relative o O 45 0.524 9 8 Relative o O 45 0.569 10 4 Absolute o O 45 0.436 11 6 Absolute o O 45 0.514 12 8 Absolute o O 45 0.533 13 4 Relative 30 45 0.353 14 6 Relative 30 45 0.405 15 8 Relative 30 45 0.431 16 4 Absolute 30 45 0.354 17 6 Absolute 30 45 0.387 18 8 Absolute 30 45 0.388 19 4 Relative 45 60 0.358 20 6 Relative 45 60 0.408 21 8 Relative 45 o O 0.443 22 4 Absolute 45 o O 0.360 23 6 Absolute 45 o O 0.391 24 8 Absolute 45 o O 0.401 25 4 Relative o O o O 0.322 26 6 Relative o O o O 0.395 27 8 Relative o O o O 0.437 28 4 Absolute o O o O 0.332 29 6 Absolute o O o O 0.386 30 8 Absolute o O o O 0.403 Table 2.9. Given the distribution of one to three tumors, the probabilities of detection predicted by the ANOVA model are displayed. 40 Treatment Parameters Experiment Number of Spacing Needles Method 9 Predicted Probability 31 4 Relative 30 60 0.326 32 6 Relative 30 60 0.357 33 8 Relative 30 60 0.381 34 4 Absolute 30 60 0.329 35 6 Absolute 30 60 0.341 36 8 Absolute 30 60 0.340 37 4 Relative 45 30 0.417 38 6 Relative 45 30 0.498 39 8 Relative 45 30 0.541 40 4 Absolute 45 30 0.425 41 6 Absolute 45 30 0.486 42 8 Absolute 45 30 0.504 43 4 Relative 60 30 0.436 44 6 Relative 60 30 0.541 45 8 Relative 60 30 0.590 46 4 Absolute o O 30 0.451 47 6 Absolute o O 30 0.537 48 8 Absolute o O 30 0.562 49 4 Relative 30 30 0.351 50 6 Relative 30 30 0.412 51 8 Relative 30 30 0.444 52 4 Absolute 30 30 0.359 53 6 Absolute 30 30 0.401 54 8 Absolute 30 30 0.407 Table 2.9. (Cont.) Given the distribution of one to three tumors, the probablities of detection predicted by the ANOVA model are displayed. A selection of detection rates are graphed in Figure 2.7 to provide visualization of the relative ranking of various treatments. The plots indicate 6 and 8 needles, relative spacing and all of the levels for 9 and 41 0 e = 30 6 needles n e = 45 6 needles A e = 60 6 needles e = 30 8 needles e = 45; 8 needles e = 60 8 needles Legend Figure 2.7. The detection rates for several experiments are graphed and the common treatment parameters are noted for each experiment. This gives a visual under- standing of the ranking of these treatments in terms of their detection rate. 42 3. Digitized Specimen Data 3.1 Summary of Software Tool An analysis program, written in C, was created to simulate needle biopsies on clinical data provided by the University of Colorado Health Sci- ences Center, Pathology Department. The clinical data were gathered from autopsies, pathologically investigated and digitized [2]. The data for each specimen are stored as a 3-dimensional array of information. The software uses an input hie to determine the characteristics of a given experiment. These characteristics include the number of needles, the initial placement of the first needle, the angles 9 and (f>, the spacing between needles, and the needle diameter and length. In this manner, the analysis software is flexible enough to handle a variety of simulations. The goal of this biopsy simulation tool is to provide the means to experiment realistically with various needle parameters on clinical data in order to determine any correspondence between biopsy methods and detection rates. The initial needle position is offset by the distance requested (the ^-offset entered by the user), with half of the needles entering the right lobe 43 and the other half entering the left lobe, in symmetry with each other. The initial position is determined as an absolute (in cm) offset from the apex of the gland. The other parameters are used to position each needle on the specimen data set and determine how much of the specimen data is to be returned in the needle biopsy. This specimen data is analyzed to determine whether and how much tumor data is present in the needle. This information is available to the user. Having read the input hie with parameter values, the code begins a loop on the specimen data hies requested for simulation. In this loop, the three- dimensional specimen data hie is opened, the data are read into a 3-d array, with all of the background trimmed off, the apex of the gland is located, and the needle positions are translated into array coordinates. These coordinates are fed to the biopsy routine which extracts the specimen data coinciding with the needle and analyzes the data for tumor information. The information for the entire experiment is stored in an output hie that documents the needle parameters and the results for each image data set. 44 3.2 Specific Algorithms 3.2.1 Locating the Apex The apex is defined as the first contact with the prostate when ap- proaching it through the rectum, as done clinically. This location is used as a landmark for positioning each biopsy needle. In the data set, the algorithm that searches for this landmark proceeds as follows. The planes are defined as shown in Figure 3.1. Each pixel in the three-dimensional specimen file contains a number indicating the type of data at that location. The possible types are gland, tumor, capsule or background. Capsule data indicate those pixels defining the boundary of the gland. The apex is indicated by the first pixel pointing to capsule data. Therefore one plane of specimen data is evaluated at a time, until a pixel that points to capsule data is found. This location is recorded as the apex location. 45 Figure 3.1. The x,y,z axis, as defined for the digital data, mimic those defined for the geometric models. 46 3.2.2 Establishing Needle Positions The starting position, the location of the apex, serves as the land- mark for each additional needle. From this starting point and the additional user-supplied parameters (^-offset, distance between needles) all of the nee- dle positions are calculated in terms of a vector. This vector, represented by (x, y, z) coordinates, along with the image data. The ^-offset is assumed to be in centimeters and is added to the initial (x, y, z) of the starting position to locate the first needle position. Each time any coordinate is changed, the new vector may be pointing to gland, tumor, background, urethra or capsule data. The pixel represented by the vector is read to insure that the needle entry position remains located on cap- sule data. If it does not, the y coordinate is adjusted to make sure that the entry position of the needle is on capsule data. At this point in the algorithm, the first needle position is determined. There are two ways to space the remaining needles. The user may enter absolute distances in centimeters or a relative measure taken to be a percentage of the z dimension of the gland. In addition, a zero percentage indicates that 47 the spacing is based on the number of needles in the biopsy; the needles are equally spaced across the z-axis of the gland. The remaining needle positions are calculated from the initial needle position: half of the needles are positioned in the right lobe by using cf>, the remainder use 0 to rotate into the left lobe. All of the needles have the x coordinate set to the midpoint of the gland in the x dimension. The user-entered distance, in centimeters, is converted to a specific number of pixels. This z distance is added to the first needle position to obtain the second needle position, added to the second to obtain the third, etc. Each time a needle position is calculated, the coordinates are evaluated to insure that they point to capsule data. If the gland is too short in the z direction to handle all the needles requested, the experiment proceeds with the number of needles that do stay within the gland. The experiments that depend on a relative distance between needles, require additional analysis of the yz slice before determining the z offset. The z diameter of the particular yz slice is calculated. The z distance required for a needle of a specific length, inserted at a specific angle is then subtracted from this z diameter. Rather than having the last needle pierce more background than gland data, this subtraction enables the full number of needles to be 48 inserted into the gland. This new z diameter is then divided into the number of segments required by the specified percentage. If the user indicates 0% for the distance spacing, the software calculates the distance based on the number of needles requested and the diameter of the yz plane. 3.3 Simulations The 54 treatments used in the geometric model were used as biopsy procedures on a maximum of 53 digitized clinical specimens. Some of the biopsy techniques were simulated on only 52 of these clinical specimens. Table 3.1 shows the results from these simulations on the digitized clinical data. The table documents both the multiple-tumor geometric model hit rate as well as the number of hits resulting from the same biopsy on the digitized clinical data. The first five columns indicate the experiment number and the biopsy parameter settings for the four variables, number of needles, spacing method, 9 and per 1000 simulations of the geometric model. The column labelled Number of Hits is the number of hits per number of digitized clinical samples. Most experiments were run on all 53 of the digitized specimens. However, some of the simulations resulted in an error on one or more of the specimens and these specimens were then removed from the experiment. The final column, 49 labelled Clincial Detection Rate is the rate for the experiments on the digitized specimens. Number Number Clinical of Spacing Detection of Detection Experiment Needles Method e Rate Hits Rate 1 4 Relative 45 45 0.417 ff 53 0.1509 2 6 Relative 45 45 0.489 0.2075 3 8 Relative 45 45 0.526 8 ? ff f8 ? 53 0.1538 4 4 Absolute 45 45 0.417 0.1698 5 6 Absolute 45 45 0.470 0.2075 6 8 Absolute 45 45 0.482 0.1923 7 4 Relative 60 45 0.427 0.1698 8 6 Relative 60 45 0.524 9 i S fl ff f I 53 0.1731 9 8 Relative 60 45 0.569 0.2453 10 4 Absolute 60 45 0.436 0.1887 11 6 Absolute 60 45 0.514 0.2264 12 8 Absolute 60 45 0.533 0.2264 13 4 Relative 30 45 0.353 0.1321 14 6 Relative 30 45 0.405 0.2264 15 8 Relative 30 45 0.431 9 Â¥ Â¥ ? ? 53 0.1698 16 4 Absolute 30 45 0.354 0.1321 17 6 Absolute 30 45 0.387 0.1321 18 8 Absolute 30 45 0.388 0.1698 19 4 Relative 45 60 0.358 0.1132 20 6 Relative 45 60 0.408 9 ff f s Â§ 53 0.1698 21 8 Relative 45 60 0.443 0.2115 22 4 Absolute 45 60 0.360 0.1509 23 6 Absolute 45 60 0.391 0.1887 24 8 Absolute 45 60 0.401 0.1887 25 4 Relative 60 60 0.322 8 ? ? f ? ? 52 0.1509 26 6 Relative 60 60 0.395 0.1538 27 8 Relative 60 60 0.437 0.1731 28 4 Absolute 60 60 0.332 0.1154 29 6 Absolute 60 60 0.386 0.1731 30 8 Absolute 60 60 0.403 0.1731 Table 3.1 The detection rates for the geometric and clinical simulations are displayed. 50 Number Number Clinical of Spacing Detection of Detection Experiment Needles Method e Rate Hits Rate 31 4 Relative 30 60 0.326 5 ? ? f f f f i 51 f 58 5? ? 58 58 ? 51 51 f f? 58 f f 58 52 0.0962 32 6 Relative 30 60 0.357 0.0962 33 8 Relative 30 60 0.381 0.1731 34 4 Absolute 30 60 0.329 0.0769 35 6 Absolute 30 60 0.341 0.0769 36 8 Absolute 30 60 0.340 0.0769 37 4 Relative 45 30 0.417 0.1154 38 6 Relative 45 30 0.498 0.1923 39 8 Relative 45 30 0.541 0.2308 40 4 Absolute 45 30 0.425 0.1538 41 6 Absolute 45 30 0.486 0.1923 42 8 Absolute 45 30 0.504 0.2115 43 4 Relative 60 30 0.436 0.1154 44 6 Relative 60 30 0.541 0.1923 45 8 Relative 60 30 0.590 0.1887 46 4 Absolute 60 30 0.451 0.1538 47 6 Absolute 60 30 0.537 0.2308 48 8 Absolute 60 30 0.562 0.2308 49 4 Relative 30 30 0.351 0.1000 50 6 Relative 30 30 0.412 0.2115 51 8 Relative 30 30 0.444 0.1923 52 4 Absolute 30 30 0.359 0.1154 53 6 Absolute 30 30 0.401 0.1538 54 8 Absolute 30 30 0.407 0.1923 Table 3.1 (Cont.) The detection rates for the geometric and clinical simulations are displayed. 3.4 Geometric Model vs Clinical Model Comparison of the detection rates between the geometric model and the clinical model reveals that the geometric simulation produces much higher rates than its clinical counterpart. In attempting to explain this discrepency, several characteristics of the experiment are noted. 51 The distribution of the tumors and the total tumor volume in a given specimen can impact the detection rate of a treatment. A comparison of the tumor volumes is graphically displayed in Figures 3.2 and 3.3. As shown by the histograms, the tumor volumes for the autopsy data tend strongly toward small (< .5 cc) volumes. In contrast, the geometric model produces tumors with volumes more equally spaced across the spectrum of possible volumes. In fact, 80% of the autopsy specimens have a total tumor volume less than .5 cc. In contrast, only 49% of the geometric gland models have a total tumor volume in this range. This difference in the size of the tumors can explain some of the difference in detection rate between the clinical and geometrical models. A second difference is that the relative ranking of detection rates for the digital data simulations is different than the ranking of detection rates for the geometric simulations. An example of this discrepency is that experiment 9, ( 8 Needles, Relative Spacing, 9 = 60, of 0.2453 or 13 hits out of 53 samples. This detection rate is better than the detection rate of experiment 45, ( 8 Needles, Relative Spacing, 9 = 60, 4> = 30) which is the optimal biopsy as indicated by the geometric simulation. This difference may be due to the fact that only 53 specimens were used in the 52 digital simulation in contrast to the 1000 models constructed for the geometric simulation. 3.5 Optimal Technique vs SRSCB The optimal technique, determined by the geometric model, consists of 8 needles, relative spacing, 9 = 60 and uses 6 needles, absolute spacing, 9 = 45 and simulated on the geometric model as well as the digitized clinical data. The optimal technique actually proved slightly worse at tumor detection than the SESCB procedure when simulated on the clinical data. In fact, the optimal method detected tumor in 10 out of 53 specimens (.189). The SESCB method detected tumor in 11 out of 53 specimens (.207). These results compare with the overall results from the geometric simulation as follows. The SESCB had a detection rate of .47 and the optimal had a detection rate of .59 on the 1000 geometric models. This discrepency is addressed by noting the sample size available in the two simulations and the distribution of tumor volumes as noted earilier. 53 25 .05 .5 1 1.5 2 2.5 3 3.5 4 4.5 5 Sum of Tumor Volume Figure 3.2. The histogram of the clinical data shows the tumor distribution by volume. 54 Figure 3.3. The histogram of the geometric data shows the tumor distribution by volume. 55 4. Geometric Model Volume Estimates 4.1 Tumor Volume Estimates The total volume of tumor in a gland is an important piece of infor- mation for clinicians who use it to improve both the diagnosis and treatment plan for a patient. The ultrasound used during a biopsy accurately measures the prostate gland volume so that an approximate ratio of tumor to gland volume can be used to estimate the volume of tumor in a gland. These sim- ulations offered an avenue to explore a means of approximating this volume ratio by using the volume of the needle that contains tumor information and the total volume of the needle. Three methods are used to estimate the amount of tumor intersected by the needle. The needle can be modeled by a line, a strip, or a cylinder in one, two, and three dimensions, respectively. The length and diameter of the needle are constant and are set by clinical limits. This incremental approach began in one dimension in order to simplify aspects of the simulation during software verification. As the research progressed, the two- and three- dimensional needles were introduced in order to model the actual biopsy more 56 closely. The first method of estimating the volume ratio is R = where Vi represents the tumor volume within a single needle, Vi represents the volume of that same needle, and n is the number of needles. This ratio is referred to as the average of the ratios. A second estimator of volume ratio is r = -\|\|t, where Vi is the tumor volume within a single needle and Vi is the total volume of that needle. This ratio is considered the ratio of the average volumes since ^ ]T"=i is the average tumor volume and ^ ]T"=i V* is the average needle volume. This yields r = ^. Both methods of estimating the ratio are documented below. Figure 4.1. This illustration of the gland, tumor and one-dimensional needle depicts the variables used in de- termining the volume ratio estimator. 57 4.1.1 One-Dimensional Analysis Line Model In this first model, we represent the needle by a line segment as shown in Figure 4.1. The length of the needle that contains tumor pixels, It, is the difference between t\ and f2, the two roots of equation ( 2.1): lT =\ t\ t2 \|- A needle length, L, of 1.25 cm is used in the estimate of volume ratio. Thus the ratio l-j- is an approximation of the true volume ratio p^y', that is, l-j- ~ p^y- 4.1.2 Two-Dimensional Strip Model In the two-dimensional case we represent the needle by a strip. The needle entry points (a^t/o^o) are used as a starting point in the two-dimensional analysis. Two lines are created, each offset from this starting coordinate by the needle radius. The intersection between these two lines and the tumor ellipse is determined and the roots of the two resulting quadratics are used to compute both the occurrence of a detection and the amount of tumor within the needle. In this case, the estimate of the volume ratio is the area of the tumor over the area of the needle. Figure 4.2 defines the lengths used in determining the area. The area of the tumor is calculated by estimating the needle length which con- tains tumor data with the roots of intersection: lt 1 =\| tn~ti2 \;lt2 = \| t2l t22 I- 58 The area of tumor is then given by ar = \|(/ of the needle. The area of the needle is calculated in the same way using the length of the needle: aN = \|(L + L). Thus ^ serves as an estimate of the true tumor to gland volume ratio, p^y. Figure 4.2. This illustration of the gland, tumor and two-dimensional needle depicts the variables used in de- termining the volume ratio estimator. 4.1.3 Three-Dimensional Cylinder Model The three-dimensional analysis models the needle as a cylinder and is similar to the two-dimensional case in that the entry point of the needle is again used as a center coordinate for four needles. In this case, the four needles are Then intersections and roots are computed. A more accurate representation of the volume ratio is obtained using the volume of the tumor within the needle 59 over the volume of the needle. In this case, the length is estimated to be the maximum of the lengths determined from the four sets of intersection roots: lt = max(\| tn ti2 \|, \| t2i t22 \|, \| hi h2 \|, \| hi h2 \|). The volume of the needle depends on the known diameter and length: vn = 7r(\|)2(L). The estimated volume of the tumor depends on the needle lengths which contain tumor data as shown in Figure 4.3. This leads to the tumor volume estimate vt = 7r(\|)2(lt). The ratio ^ estimates the true volume ratio, PGV' 4.2 Experiment Setup A second set of experiments utilizing the geometric model involved exploring the question of accurately estimating the tumor volume to gland vol- ume ratio. The experiment simulated a biopsy on a single specimen, increasing the number of needles each iteration and comparing the volume ratio obtained from the biopsy sample to the known volume ratio. The parameters for the biopsy include the optimal angles 9 and vestigation The optimal number of needles and distancing method determined from the ANOVA analysis do not apply to this experiment since the number of needles increases from 6 to 20 and the distancing of these needles is done so that the maximum1 number, 20, are equally spaced. The maximum number 60 Figure 4.3. This illustration of the gland, tumor and three-dimensional needle depicts the variables used in determining the volume ratio estimator. 61 of needles was set at 20 due to clinical limitations. The spacing of the nee- dles is dependent on the maximum number so that from one iteration to the next 2 needles are in the same exact location, yielding the same detection information. In this manner the comparison between a specimen biopsied by 6 needles and the same specimen biopsied by 10 needles is not dependent on needles. The simulation is executed on 1000 specimens, varying the number of needles from 6 to 20 in increments of 2. The output from this experiment consists of a hie for each specimen that contains the results of each set of needles including the tumor to needle volume ratio achieved and the associated estimates (R = ^ XX'itAn) and r = -Â§y-). In addition, the actual tumor to gland volume ratio is noted. 4.3 Results The results of this experiment were not as anticipated as there ap- pears to be no pattern of convergence to the actual tumor to gland volume ratio within the limit of 20 total needles. However, much was learned from this exercise that provided insight into the next series of investigations. First, it is noted that in the great majority of cases, a single 8-needle biopsy tends 62 to overestimate the true tumor to gland volume ratio. Secondly, a comparison between the two methods of calculating the error leads to the conclusion that the sum of the ratios is the more accurate method at least in this set of limited trials. 4.4 Interactive Utility Using the preceding idea as a starting point, an interactive software tool was created to investigate the volume ratio question in greater detail. This tool prompts the user for a random number, seeds the random number generator, creates a gland containing a single tumor and conducts the optimal 8-needle biopsy. This optimal biopsy has 8 needles, relative spacing between the needles, 9 = 60 and position, the amount of tumor volume contained in the needle and an estimate as to the volume ratio of tumor to gland, are displayed for the user. At this point, the user is able to choose the location for the next needle. This new needle is then simulated and the tumor volume information it retrieves is incorporated into the volume ratio. The user can continue this process of requesting additional needles and evaluate the estimated volume ratio and its error from the true ratio. A maximum of 20 needles can be simulated on a single gland, beginning with the 8 original needles and accumulating the 63 additional 12 based on user specifications. This area of research is full of open-ended questions where tools such as this interactive utility can help shed light on answers. With involvement from clinicians and medical researchers, experiments can be designed to gather more information regarding the two issues of volume ratio and optimal biopsy technique. In addition, using the results of this body of research, more real- istic tumor distributions and geometric models can be constructed to better understand the impact of treatment parameters on detection rate. 64 A. APPENDIX ANOVA Definitions A dot in the subscript indicates averaging over the variable repre- sented by that index. The number of levels for Number of Needles: a = 3. The number of levels for Distancing Method;, b = 2. The number of levels for 9: c = 3. The number of levels for 0: d = 3. The number of specimens = 1000. The number of experiments: abed = 54. In general, Y is an observation, Y is the mean of observations, /i is the true mean and (i is the least squares estimate of the true mean. Yijki is the observed detection rate at the factor levels indicated by i,j, k and l. F ... is the mean of all specimens over all treatment levels i,j, k, l. It indicates the overall detection rate for the entire experiment. i abed Y = X X X X Ym abed i=1j=1k=11=1 65 SSTO, or total sum of squares is a measure of the total variability of the observations without consideration of factor level. SSTO = 't't{YijU-Y...f i=lj=1k=1 1=1 dfssro is the total degrees of freedom. The SSTO has abdc 1 = 54 1 degrees of freedom. One degree of freedom is lost due to the lack of independence between the deviations. SSTR or treatment sum of squares measures the extent of differ- ences between estimated factor level means and the mean over all treatments. The greater the difference between factor level means (treatment means), the greater the value of SSTR. SSTR = 12(Ym y....)2 i=lj=lk=ll=l df sstr is the degrees of freedom. There are r 1 degrees of freedom for the SSTR, where r is the number of parameters in the model. In the full model, r = abed, = 54, the total combinations of factor levels. In the model used for this simulation, r = (a1) + (5l) + (cl) + (dl) + (a1)(5l) + (a l)(c- 1) + (a l)(d 1) + (b l)(c 1) + (& l)(d- 1) + (c- l)(d- 1) = 26. One degree of freedom is lost due to the lack of independence between the deviations. 66 SSE or error sum of squares, measures variability which is not ex- plained by the differences between sample means. It is a measure of the varia- tion within treatments. A smaller value of SSE indicates less variation within simulations at the same factor level. SSE = Â£ Â£ Â£ Â£(%:, YijUf i=ij=ik=il=i dfssE is the degrees of freedom. Since SSE is the sum of the errors across factor level, the degrees of freedom is the sum of the degrees of freedom for each factor level. It is the total number of simulations minus r, abed r. MSI: is the mean square for error defined by MSE = SSE/dfssE- Note: The above definitions imply SSTO = SSTR + SSE. Due to this relationship, this process is referred to as the partitioning of the total sum of the squares. In order to measure the variability within a factor level, the fac- tor sum of square terms are computed. These terms are integral in the test statistic applied to determine whether a factor main effect is significant. In addition, interaction sum of squares are computed to measure variability of the interactions. 67 The factor A sum of squares corresponds to the number of needles factor. SSA = bcdjr^iY F...)2 i=1 Similar factor sum of squares are computed for each of the factors: Factor Sum of Square Mean Sum of Square Number of Needles Spacing Method e SSA = bcd^ ,(T,.. f SSB = acdVf] ,(T F...)2 SSC = abdEt=i(Y..k. ^ F...)2 SSD = abcYlf=i(Y ...i F...)2 MSA = SSA/(a 1) MSB = SSB/{b- 1) MSC = SSC/(c 1) MSD = SSD/(d 1) The interaction sum of squares are computed as well for use in the F-test on the interactions. The first three pair-wise interaction sum of squares are shown below. The others are computed in the same manner. 68 Number of Needles: Spacing SSAB = cdZti E\$=i (Xu.. V,. MSAB = SSAB/{a 1 ){b 1) - y.j.. + Y..y Number of Needles: 9 SSAC = bdT*=1 ELi . Yk, +F...)2 MS AC = SSAC/(a l)(c 1) Number of Needles: The treatment means, jiijki, indicate the mean for the treatment at the ijkl levels of the respective factors. The overall mean, /i, is the mean across all factors and all levels (across all i,j, k, i). 69 References (1) Hodge K.K., McNeal J.E., Terris M.K., Stamey T.A. Random sys- tematic versus directed ultrasound guided transrectal core biopsies of the prostate. Journal of Urology 142 (1989): 71-74. (2) Daneshgari, Firouz M.D., Taylor, Gerald D. PhD, Miller, Gary J. M.D., PhD, Crawford, E. David M.D. Computer Simulation of the Probability of Detecting Low Volume Carcinoma of the Prostate with Six Random Systematic Core Biopsies. Urology 45 (April 1989): 604- 609. (3) McNeal, John M.D. Normal Histology of the Prostate The American Journal of Surgical Pathology (1988): 619-633. (4) Neter, John, \Vasserman. William, Applied Linear Statistical Mod- els, Richard D. Irwin, Inc 1974. 70 Full Text PAGE 1 SIMULA TIONS OF PR OST A TE BIOPSY METHODS b y Catherine Colb y P ellish B.S.E.E., Marquette Univ ersit y 1985 A thesis submitted to the Univ ersit y of Colorado at Den v er in partial fulllmen t of the requiremen ts for the degree of Master of Science Applied Mathematics 1997 PAGE 2 This thesis for the Master of Science degree b y Catherine P ellish has been appro v ed b y William L. Briggs James R. Koehler W eldon A. Lodwic k Date PAGE 3 P ellish, Catherine Colb y (M.S., Applied Mathematics) Sim ulations of Prostate Biopsy Methods Thesis directed b y Associate Professor William L. Briggs Abstract An accepted practice in screening for prostate cancer in v olv es a needle core biopsy of the prostate gland, whic h can pro vide information regarding if, and ho w m uc h, cancer is presen t in a gland. This paper documen ts sev eral in v estigations in to prostate gland biopsy tec hniques. The rst phase of study in v olv es a geometric model of a prostate gland con taining one to three tumors. This mathematical model of the gland is then used to sim ulate v arious biopsy tec hniques and compare the resulting data. Secondly the best biopsy procedure, as determined from the geometric model, is sim ulated on actual specimen data whic h ha v e been digitized. These specimen data are also used for sim ulation of the six random systematic core biopsy tec hnique (SRSCB) curren tly in clinical use. The results of the geometric model are compared to the results of the sim ulation on actual data. Finally the geometric model is used in another series of sim ulations that in v estigate the n um ber of needle samples needed to estimate the tumor to gland v olume ratio. iii PAGE 4 This abstract accurately represen ts the con ten t of the candidate's thesis. I recommend its publication. Signed William L. Briggs iv PAGE 5 A CKNO WLEDGEMENTS I w ould lik e to sincerely thank a n um ber of people who consisten tly pro vided me with their support, encouragemen t and guidance as I pursued the completion of this thesis. Dr. Bill Briggs, m y advisor, serv ed as a constan t source of insigh t and motiv ation, as w ell as pro viding considerable direction throughout this process. I am also grateful for the time spen t with Dr. Jim Koehler who had to teac h me the ner poin ts of statistics again and again. My thanks to both of these professers for pro ving to be excellen t academic sources. I also w ould lik e to thank Norm LeMa y who, out of the generousit y of his heart and his need for a free lunc h, assisted me in running the ANO V A analysis whic h this thesis required. Finally I m ust thank m y family Mark, Eric and Corinne for encouraging me and making me laugh through ev ery crisis. PAGE 6 CONTENTS Chapter 1 In troduction . . . . . . . . . . . . . 2 1.1 Clinical Prostate Biopsy Analysis . . . . . . 2 1.2 Summary of Mathematical Methods . . . . . 4 2 The Geometric Model . . . . . . . . . . . 5 2.1 Geometric Model of gland and tumor . . . . . 5 2.2 Sim ulations . . . . . . . . . . . . 10 2.3 Statistical Analysis of Results . . . . . . . 14 2.4 Sim ulation Results . . . . . . . . . . 16 2.4.1 Applying the ANO V A to the Biopsy Sim ulation Data 18 2.4.2 ANO V A Mec hanics . . . . . . . . . 23 2.4.3 Residuals . . . . . . . . . . . . 24 2.4.4 The Null and Alternate Hypotheses . . . . . 25 2.4.5 Are the Main Eects all Equal? . . . . . . 27 2.4.6 Recognizing In teraction bet w een F actors . . . . 30 2.4.7 Clinical Distribution of T umors . . . . . . 38 vi PAGE 7 3 Digitized Specimen Data . . . . . . . . . . 43 3.1 Summary of Soft w are T ool . . . . . . . . 43 3.2 Specic Algorithms . . . . . . . . . . 45 3.2.1 Locating the Apex . . . . . . . . . . 45 3.2.2 Establishing Needle P ositions . . . . . . . 47 3.3 Sim ulations . . . . . . . . . . . . 49 3.4 Geometric Model vs Clinical Model . . . . . 51 3.5 Optimal T ec hnique vs SRSCB . . . . . . . 53 4 Geometric Model V olume Estimates . . . . . . 56 4.1 T umor V olume Estimates . . . . . . . . 56 4.1.1 One-Dimensional Analysis Line Model . . . . 58 4.1.2 Tw o-Dimensional Strip Model . . . . . . 58 4.1.3 Three-Dimensional Cylinder Model . . . . . 59 4.2 Experimen t Setup . . . . . . . . . . 60 4.3 Results . . . . . . . . . . . . . 62 4.4 In teractiv e Utilit y . . . . . . . . . . 63 Appendix A ANO V A Denitions . . . . . . . . . . . 65 1 PAGE 8 1. In troduction 1.1 Clinical Prostate Biopsy Analysis Curren tly the standard method of determining if a giv en prostate gland is cancerous in v olv es t w o procedures. The rst is the prostate-specic an tigen (PSA) test whic h measures the lev el of an tigens in the patien t's blood, a high lev el indicating a higher possibilit y of cancerous tissue. The second procedure is the needle biopsy whic h is carried out if the PSA test so indicates. The clinician conducts this biopsy b y inserting a needle-tool, equipped with ultrasound capabilities, in to the patien t's rectum. The gland is located and the urologist res three needles in to the righ t lobe of the gland and three needles in to the left lobe at appro ximately symmetric positions. The leftrigh t division of the gland is determined b y the position of the urethra in the gland. This ph ysical landmark is used as the visual dividing line, enabling clinicians to execute the biopsy in a systematic manner. The needle-tool is rotated to the left or righ t depending on the targeted lobe. This rotation corresponds to the angle used in the mathematical analysis. F ollo wing this sligh t rotation, the needles are inserted at a second independen t angle, referred 2 PAGE 9 to as The c hoice of a six-needle biopsy is based on the six random systematic core biopsies (SRSCB) method dev eloped b y Hodge et al [1] and curren tly though t to ac hiev e the best detection rates. The results from this diagnostic biopsy are then analyzed in order to determine the best treatmen t plan for the patien t. There are sev eral factors that help the urologist c hoose the optimal treatmen t plan. The rst factor is ob viously whether the biopsy sho ws an y tumor cells at all. According to the Hodge study 96% of the 83 men diagnosed with cancer had the cancer detected b y SRSCB. Ho w ev er, as in v estigated b y Daneshgari et al [2], in prostate glands with lo w tumor v olume, the SRSCB fails to ac hiev e suc h a high percen tage of detection. This study concluded that \an impro v ed biopsy strategy ma y be needed in detection of CaP (carcinoma of the prostate) in patien ts with lo w v olume cancer". Secondly the v olume of the tumor itself is a deciding factor in determining treatmen t. Thirdly the location of the tumor, specically if the tumor penetrates the capsule of the gland, can dene a specic treatmen t plan. Some of this information is a v ailable from a single needle-core biopsy; more information is gleaned from successiv e, strategically placed biopsies. 3 PAGE 10 1.2 Summary of Mathematical Methods As an aid in understanding this problem, as w ell as researc hing w a ys to impro v e diagnosis, t w o methods of analysis are undertak en. The rst method relies on a geometric model of the prostate gland with from one to three tumors. V arious biopsy methods are sim ulated with this mathematical model and results are tabulated. The second method in v olv es running the same biopsy sim ulations on actual prostate glands whic h ha v e been digitized and stored as three-dimensional objects in a computer. The experimen tal results from these t w o methods are then compared. All of the sim ulations w ere executed using soft w are created for this purpose primarily b y this author, although the sk eletons of these soft w are tools w ere engineered during the Spring 1995 Math Clinic on this topic b y sev eral participan ts. The sim ulations are written in C and C++, running on a UNIX-based computer. They are extensiv ely documen ted and rexible enough to be useful in a v ariet y of experimen ts within this realm of researc h. 4 PAGE 11 2. The Geometric Model 2.1 Geometric Model of gland and tumor An actual prostate gland is about the size of a w aln ut with v olumes ranging from 22 cc to 61 cc [3]. The geometry of an ellipsoid closely models this gland and an y tumors presen t within it. Therefore, an ellipsoid of the form x 2 A 2 + y 2 B 2 + z 2 C 2 = 1 ; is used to represen t the prostate gland. Ellipsoids are also used to represen t eac h of the tumors. The dimensions of the gland, A; B and C are c hosen randomly in the follo wing experimen tally determined ranges: 3.0 cm < A < 4.8 cm 3.8 cm < B < 4.6 cm 3.8 cm < C < 5.2 cm 22 cc < [ gland volume ] < 61 cc. The prostate is divided in to 3 zones: the peripheral, the cen tral and the transition region. The peripheral zone comprises appro ximately 70% of the mass of the prostate gland. It is located in the lo w er area of the gland, 5 PAGE 12 closest to the rectum. This region is the \site of origin of most carcinomas"[3]. The cen tral region mak es up appro ximately 25% of the glandular mass and is \resistan t to both carcinoma and inrammation"[3]. The transition region con tains the remaining 5% of prostate gland tissue and can be the site of some cancers. Figure 2.1 sho ws these regions of the prostate gland. Based on this clinical information, the soft w are-generated tumors are located in the lo w er part of the elliptical gland model to sim ulate tumors residing in the peripheral zone. Figure 2.2 depicts the geometrical gland and tumor model in the xyz system. Since the gland model is cen tered at the origin, the y -coordinate of the tumor cen ter, y c is alw a ys negativ e in order to place the tumor in the peripheral zone of the gland. Ho w ev er, other distributions of y could be used to impro v e the model. T umors are modeled b y an equation of the form ( x x c ) 2 a 2 + ( y y c ) 2 b 2 + ( z z c ) 2 c 2 = 1 where x c y c and z c specify the cen ter of the tumor. The biopsy needle is modeled as a line with the parametric equations x ( t ) = x 0 + t sin sin y ( t ) = y 0 + t sin cos z ( t ) = z 0 + t cos ; 6 PAGE 13 Figure 2.1. The peripheral (PZ), cen tral (CZ) and transition (TZ) regions divide the prostate gland in to 3 major zones. Tumor Ellipsoid Y Z X Gland Ellipsoid B A CFigure 2.2. The gland and tumor are modeled b y ellipsoids in the xyz coordinate system. 7 PAGE 14 where x 0 y 0 and z 0 are the coordinates of the en try poin t of the needles (Figure 2.3 and Figure 2.4). The angle is measured from the y -axis and determines a plane. The angle is then assumed to remain in this plane and is measured from the z -axis. F rom these denitions, the parametric equations for the line are determined. The parameter t measures the length of the needle. x y z Y X Needle Gland Ellipsoid F 0 0 0Figure 2.3. This gure of the xy plane and needle illustrates measuremen t of Substituting the parametric equations of the needle in to the equation for the tumor, it is possible to determine v alues of t corresponding to an in tersection. The equation of the tumor is ( x ( t ) x c ) 2 a 2 + ( y ( t ) y c ) 2 b 2 + ( z ( t ) z c ) 2 c 2 = 1 : 8 PAGE 15 0 0 0 Needle Y Z Gland Ellipsoid Q x y z Figure 2.4. This gure of the yz plane and needle illustrates measuremen t of Replacing x ( t ), y ( t ) and z ( t ) b y the parametric equations of the needle giv es t 2 ( sin 2 sin 2 a 2 + sin 2 cos 2 b 2 + cos 2 c 2 )+ t ( 2( x 0 x c ) sin sin a 2 + 2( y 0 y c ) sin cos b 2 + 2( z 0 z c ) cos c 2 + ( ( x 0 x c ) 2 a 2 + ( y 0 y c ) 2 b 2 + ( z 0 z c ) 2 c 2 ) = 1 : (2.1) If the discriminan t ( B 0 2 4 A 0 C 0 ) is positiv e, t w o real roots exist. In this case w e ha v e A 0 = sin 2 sin 2 a 2 + sin 2 cos 2 b 2 + cos 2 c 2 B 0 = 2( x 0 x c ) sin sin a 2 + 2( y 0 y c ) sin cos b 2 + 2( z 0 z c ) cos c 2 C 0 = ( x 0 x c ) 2 a 2 + ( y 0 y c ) 2 b 2 + ( z 0 z c ) 2 c 2 : 9 PAGE 16 If real roots t 1 and t 2 exist, they giv e the poin ts where the tumor ellipsoid and the line in tersect. If these v alues are greater than 0 and less than the actual needle length, the needle has in tersected the tumor. The amoun t of tumor extracted b y the needle is proportional to the dierence bet w een the t w o roots of the quadratic, j t 1 t 2 j By comparing the t w o roots, an estimate of the v olume of the tumor that is con tained in the needle can be made. If real roots do not exist, the needle does not in tersect the tumor ellipsoid and no tumor information is gained b y that needle. In this analysis, eac h biopsy procedure w as sim ulated on 1000 dieren t gland models and the n um ber of times a tumor w as detected per procedure w as recorded. This method does not dieren tiate bet w een one or more needles detecting the tumor. It simply records a hit or miss per biopsy procedure. In addition, an estimate of the tumor v olume is made whenev er a tumor is detected. 2.2 Sim ulations Since a fundamen tal goal of an y biopsy is to determine whether or not the gland con tains cancerous cells, the rst series of sim ulations is in tended to compare the detection rate of sev eral biopsy tec hniques. The detection rate is dened as the n um ber of times a biopsy procedure detects a tumor to the 10 PAGE 17 total n um ber of biopsies conducted. A set of 54 dieren t biopsy procedures is sim ulated with v ariation in the follo wing parameters: n um ber of needles, oset bet w een needles in the z direction, and The distance in the z direction bet w een needles can be a relativ e spacing based on the gland dimension in the z direction or an absolute spacing of 1 cm bet w een eac h needle. The rst method is referred to as relativ e spacing since it depends on the gland size and separates the needles b y equal distance. The second is referred to as the absolute spacing and has its basis in the SRSCB procedure. As a means of clarication, Figures 2.5 and 2.6 illustrate the analysis of a single specimen and the execution of the en tire experimen t. Eac h of the 54 biopsy procedures is sim ulated on 1000 dieren t gland models. The random n um ber generator is seeded once for eac h series of 1000 sim ulations using a specic biopsy tec hnique. Prior to the next tec hnique, the random n um ber generator is reseeded with the same n um ber, thereb y yielding the iden tical set of 1000 prostate models. This insures that eac h of the biopsies is conducted on the same set of 1000 sim ulated glands. The detection rate is determined for eac h of these procedures and the results of the sim ulation are documen ted in T able 2.1. 11 PAGE 18 Make Tumor(s) Determine starting location for all done? needles Simulate a single needle biopsy. Solveusing initial needle position; store hit and volume results. NO YES Make a gland model Simulation Over equation (1) for roots Needles AllFigure 2.5. This ro w c hart depicts the top-lev el algorithm for modeling a single biopsy with sev eral needles. 12 PAGE 19 Read in Biopsyparameters for a Simulate this biopsy on a single Simulation Over gland model. given procedure. Doneglands? 1000 NO YES All 54 biopsy procedures done? NO YESFigure 2.6. This ro w c hart depicts the sim ulation process for the en tire sim ulation, eac h biopsy procedure is sim ulated on 1000 geometric gland models. 13 PAGE 20 2.3 Statistical Analysis of Results In order to in terpret the output from the sim ulations legitimately a statistical tool is needed. First, w e m ust determine whether or not the v arious biopsy settings inruence the observ ed detection rate. In other w ords, is there a relationship bet w een the settings of an y one or com bination of the four factors (n um ber of needles, z -spacing, and ) and the detection rate or are the results completely random, therefore implying that the biopsy specication does not determine the detection rate? W e need a mathematically sound method to compare the detection rates pro vided b y the sim ulation and to infer some conclusions. The statistical model kno wn as Analysis of V ariance (ANO V A) w as used to compare the population means bet w een v arious treatmen ts, th us resulting in a statistically v alid conclusion. This model can be emplo y ed to determine whether the v arious factors in teract and whic h factors ha v e the most impact on the outcome. In order to describe the ANO V A model, a few denitions are required. (1) F actors are the independen t v ariables that are under in v estigation. In this instance, the biopsy parameters (n um ber of needles, spacing 14 PAGE 21 method, and ) are the factors for the ANO V A model. Num ber of Needles Spacing Method F actor 4 Absolute 30 30 Lev els 6 Relativ e 45 45 8 60 60 (2) F actor lev els are the v alues that eac h of the factors can tak e on during a single sim ulation. As sho wn in the list of biopsy sim ulation factors and lev els, eac h factor does not ha v e the same n um ber of factor lev els. The factor Spacing Method only has t w o factor lev els, whereas the other three factors eac h ha v e three factor lev els. (3) A treatmen t is a particular com bination of lev els of eac h of the factors in v olv ed in the experimen t where an experimen t is the sim ulation of the treatmen t on 1000 geometric specimens. In this example, a treatmen t refers to a biopsy with specic settings (for example, 4 needles, absolute spacing, = 45 = 45 ). F or the sim ulation, there are 54 dieren t treatmen ts and therefore, 54 dieren t experimen ts corresponding to all the com binations of the lev els of the four factors. (4) A trial is dened to be a sim ulation of one treatmen t on one geometric model. The outcome of a trial is either 1, the biopsy procedure detected the tumor, or 0, the tumor remained undetected. The outcome of the experimen t is the detection rate ac hiev ed b y a specic 15 PAGE 22 treatmen t sim ulated on 1000 geometric specimens. In other w ords, the outcome of the experimen t is the n um ber of specimens in whic h tumor is detected v ersus the total n um ber of specimens sim ulated and is referred to as outcome for the remainder of this thesis. 2.4 Sim ulation Results F or eac h of the 54 treatmen ts, the sim ulation is conducted on 1000 dieren t gland models. The follo wing table summarizes the treatmen t parameters as w ell as the results: T reatmen t P arameters Outcome Num ber of Spacing Detection Experimen t Needles Method Rate 1 4 Relativ e 45 45 0.252 2 6 Relativ e 45 45 0.307 3 8 Relativ e 45 45 0.335 4 4 Absolute 45 45 0.263 5 6 Absolute 45 45 0.293 6 8 Absolute 45 45 0.298 7 4 Relativ e 60 45 0.267 8 6 Relativ e 60 45 0.341 9 8 Relativ e 60 45 0.369 10 4 Absolute 60 45 0.270 11 6 Absolute 60 45 0.320 12 8 Absolute 60 45 0.339 13 4 Relativ e 30 45 0.196 14 6 Relativ e 30 45 0.225 15 8 Relativ e 30 45 0.255 16 4 Absolute 30 45 0.207 17 6 Absolute 30 45 0.221 18 8 Absolute 30 45 0.221 T able 2.1. The results from the 54 geometric model experimen ts are displa y ed. 16 PAGE 23 T reatmen t P arameters Outcome Num ber of Spacing Detection Experimen t Needles Method Rate 19 4 Relativ e 45 60 0.200 20 6 Relativ e 45 60 0.234 21 8 Relativ e 45 60 0.268 22 4 Absolute 45 60 0.211 23 6 Absolute 45 60 0.225 24 8 Absolute 45 60 0.228 25 4 Relativ e 60 60 0.191 26 6 Relativ e 60 60 0.254 27 8 Relativ e 60 60 0.268 28 4 Absolute 60 60 0.209 29 6 Absolute 60 60 0.240 30 8 Absolute 60 60 0.246 31 4 Relativ e 30 60 0.172 32 6 Relativ e 30 60 0.194 33 8 Relativ e 30 60 0.219 34 4 Absolute 30 60 0.188 35 6 Absolute 30 60 0.197 36 8 Absolute 30 60 0.197 37 4 Relativ e 45 30 0.260 38 6 Relativ e 45 30 0.316 39 8 Relativ e 45 30 0.341 40 4 Absolute 45 30 0.264 41 6 Absolute 45 30 0.305 42 8 Absolute 45 30 0.316 43 4 Relativ e 60 30 0.283 44 6 Relativ e 60 30 0.351 45 8 Relativ e 60 30 0.385 46 4 Absolute 60 30 0.279 47 6 Absolute 60 30 0.346 48 8 Absolute 60 30 0.372 49 4 Relativ e 30 30 0.210 50 6 Relativ e 30 30 0.247 51 8 Relativ e 30 30 0.273 52 4 Absolute 30 30 0.225 53 6 Absolute 30 30 0.245 54 8 Absolute 30 30 0.247 T able 2.1. (Con t.) The results from the 54 geometric model experimen ts are displa y ed. 17 PAGE 24 2.4.1 Applying the ANO V A to the Biopsy Sim ulation Data The biopsy sim ulation is a m ulti-factored system, in whic h the four parameters (n um ber of needles, spacing, and ) individually and perhaps in some com binations ma y ha v e a measurable eect on the detection rate. Therefore a factor eects model is used in order to determine the impact of and in teractions bet w een these four parameters. This biopsy sim ulation is considered a complete factorial study since all possible com binations of the four parameters w ere sim ulated and ev aluated. The indices i; j; k; l refer to the lev els of the factors numb er of ne e dles, sp acing metho d and respectiv ely In this m ulti-factored system, a true o v erall mean, whic h is equivalen t to the true o v erall detection rate, is assumed to exist. The en tire sim ulation results in 54 observ ed detection rates, p ijkl eac h of whic h indicates the observ ed detection rate for a giv en experimen t. This set of 54 observ ed detection rates is used in the ANO V A to determine estimated factor eects and an estimated o v erall mean whic h are used in the factor eects model. The factor eects model is used to predict a detection rate, a probabilit y of detection, ^ p ijkl giv en the lev els of the four factors. A factor lev el mean is the a v erage detection rate for a group of 18 PAGE 25 treatmen ts that ha v e one common factor lev el held constan t while all others v ary F or example, all outcomes from experimen ts with Numb er of Ne e dles = 6 are a v eraged to yield the factor lev el mean for the factor Numb er of Ne e dles at the lev el i = 6. The o v erall mean , is simply the a v erage outcome of all experimen ts. The dierence bet w een eac h factor lev el mean and the o v erall mean yields the main eect for that factor lev el. Because this model has 4 factors eac h with either 2 or 3 lev els, the follo wing main eects are designated. i the main eect for the factor Numb er of Ne e dles at eac h of its lev els (4,6,8): 1 i 3. j the main eect for the factor Sp acing Metho d at eac h of its lev els (0,1): 1 j 2. r k the main eect for the factor at eac h of its lev els (30 ,45 ,60 ): 1 k 3. l the Main Eect for the factor at eac h of its lev els (30 ,45 ,60 ): 1 l 3. A factor at a particular lev el ma y inruence another factor either b y inhibiting or enhancing its impact. Because of these in teractions bet w een factors, the in teraction eects are included in the model. P airwise in teraction 19 PAGE 26 eects are a measure of the com bined eect of t w o factors, across the dieren t lev els, min us the main eects of these factors. W e dene these t w o-w a y eects as follo ws. ( ) ij n um ber of needles and spacing method ( r ) ik n um ber of needles and ( ) il n um ber of needles and ( r ) jk spacing method and ( ) jl spacing method and ( r ) kl and Three-w a y factor eects are a measure of the in teraction eect of three factors. ( r ) ijk n um ber of needles, spacing method and ( ) ijl n um ber of needles, spacing method and ( r ) jkl spacing method, and ( r ) ikl n um ber of needles, and The four-w a y eect is the measure of the in teraction eect of all four factors. ( r ) ijkl n um ber of needles, spacing method, and 20 PAGE 27 Summary of V ariables T rue o v erall mean Estimated o v erall mean ^ T rue treatmen t mean ijkl Estimated treatmen t mean ^ ijkl Observ ed treatmen t detection rate p ijkl T ransformed observ ed treatmen t detection rate Y ijkl Estimated treatmen t detection rate ^ p ijkl T ransformed estimated treatmen t detection rate ^ Y ijkl Av erage observ ed detection rate p T rue main factor lev el eects i j r k l Estimated main factor lev el eects ^ i ^ j ^ r k ^ l T rue t w o-w a y eects ( ) ij ( r ) ik ( ) il ( r ) jk ( ) jl ( r ) kl Estimated t w o-w a y eects d ( ) ij d ( r ) ik d ( ) il d ( r ) jk d ( ) jl d ( r ) kl T able 2.2. A list of the v ariables used in the ANO V A analysis is displa y ed. The factor eects model tak es the general form ijkl = + i + j + r k + l +( ) ij +( r ) ik +( ) il +( r ) jk +( ) jl +( r ) kl +( r ) ijk + ( ) ijl + ( r ) jkl + ( r ) ikl + ( r ) ijkl : The observ ed outcome, the detection rate for a particular treatmen t, as giv en in T able 2.1, is p ijkl and is the sum of the true mean for that treatmen t and a residual term: p ijkl = ijkl + ijkl : 21 PAGE 28 The goal of the analysis is to form ulate a model that predicts the outcome of a giv en treatmen t. Since the true means and true factor eects are not kno wn, estimates of these terms are determined from the sim ulation and used in the model. Estimated v alues are indicated with the^notation. The predicted outcome ^ p ijkl is represen ted b y the follo wing relationship: ^ p ijkl = ^ + ^ i + ^ j + ^ r k + ^ l + d ( ) ij + d ( r ) ik + d ( ) il + d ( r ) jk + d ( ) jl + d ( r ) kl + d ( r ) ijk + d ( ) ijl + d ( r ) jkl + d ( r ) ikl + d ( r ) ijkl : In this equation ^ p ijkl is the estimated probabilit y of detecting a tumor at the factor lev els indicated b y i; j; k; l This probabilit y is predicted b y the model using least -square estimators for the terms in the equation. The probabilit y of detection is a function of the estimated o v erall mean, ^ and the estimated eects from the four factors, alone and in com bination with one another. Not all of these eects ma y be signican t. In order to determine whic h of the factors do signican tly eect the detection rate and therefore belong in the nal model, v arious means are ev aluated. If all the means for a particular factor (or com bination of factors) are equal, v arying a factor lev el does not add to or subtract from the o v erall mean and therefore the factor does not belong in the nal model. This equalit y question is put, not only to eac h factor individually but to all the com binations of factors as w ell. 22 PAGE 29 2.4.2 ANO V A Mec hanics Use of the ANO V A model is founded on sev eral assumptions: (1) The outcomes follo w a normal probabilit y distribution. (2) Eac h distribution has the same v ariance. (3) The outcomes for eac h factor lev el are independen t of the other factor lev el outcomes. With these assumptions in mind, note that the probabilit y distributions of a factor at eac h of its lev els diers only with respect to the mean [4]. Therefore, the rst step in executing the analysis is to determine if the detection rates, are statistically dieren t. Secondly if they are dieren t, one of the in ten ts of the ANO V A model is to determine if the dierence bet w een the detection rate of t w o or more treatmen ts is sucien t, after examining the v ariabilit y within the treatmen ts, to conclude that one treatmen t does indeed produce a higher detection rate. In addition, b y ev aluating the statistical data, conclusions ma y be dra wn as to ho w eac h factor, both independen tly and within established in teraction groups (pairwise, three-w a y or four-w a y), inruences the outcome. 23 PAGE 30 2.4.3 Residuals W e dene p to be the a v erage of all observ ations. The model states that p ijkl = ijkl + ijkl ; therefore the residual term is ijkl = p ijkl ijkl Since ijkl is estimated b y ^ ijkl the estimated residual term is e ijkl = p ijkl ^ ijkl the dierence bet w een the observ ed and the estimated a v erage detection rate. The set of all 54 residuals, e ijkl for all i j k and l are ev aluated for three c haracteristics whic h indicate whether the tted data are w ell-suited for the analysis. These c haracteristics are: 1. Normalit y of error terms. 2. Constancy of error v ariance. 3. Independence of error terms. Sev eral statistical tests and plots used on the residual data determine whether one of the v e assumptions is violated. These tests rev ealed that the error v ariances w ere not stable, th us violating the rst c haracteristic. A transformation w as emplo y ed to preserv e the statistical information in the output, but stabilize the error v ariances. Since nothing is lost b y emplo ying a transformation and the error v ariances are stabilized, the detection rate data p is transformed to Y via the follo wing relationship: Y = 2 arcsin ( p p ) : 24 PAGE 31 The outcome from these sim ulations is the detection rate, a proportion of the n um ber of specimens where tumor is detected to the total n um ber of specimens. The arcsine transformation is the most appropriate transformation when the outcome is a proportion [4]. All ANO V A data referenced from this poin t on are transformed unless noted otherwise. The in v erse transformation is calculated at the conclusion of this analysis to get a true estimate of the probabilit y 2.4.4 The Null and Alternate Hypotheses A starting poin t in the ANO V A process is to establish t w o h ypothesis, a n ull and alternate h ypothesis. The n ull h ypothesis assumes that all eects are equal, therefore indicating that specic factor lev els do not inruence the outcome. The alternate h ypothesis assumes that at least t w o of the eects are not the same. The F-test is used to decide whic h of these t w o h ypotheses concerning the data will be accepted. The test consists of computing the ratio of bet w eeneect v ariation to within-eect v ariation. This bet w een-eect v ariation, whic h c hanges depending on the eect, is called the treatmen t sum of squares and is denoted SSA SSB SSC and SSD (see Appendix also). It is a measure of the dierence bet w een the detection rate of a set of treatmen ts and the a v erage detection rate o v er all treatmen ts. The within-eect v ariation 25 PAGE 32 is called the error sum of squares and is denoted SSE It is a measure of the dierence bet w een the individual outcome for a giv en treatmen t and the estimated detection rate o v er that treatmen t. The error sum of squares measures v ariabilit y that is not explained b y the SSA SSB SSC or SSD terms and therefore occurs within the set of treatmen ts. Both of these v ariation measuremen ts are ev aluated using sum of the squares expressions as detailed in the Appendix. The means of the SSA; SSB; SSC; SSD and SSE terms are MSA; MSB; MSC; MSD and MSE respectiv ely and are computed b y dividing b y the degrees of freedom, df, associated with eac h term. This results in F = MSA= MSE where MSA = SSA=d f A ( MSB = SSB=d f B ,etc) and MSE = SSE=d f Large v alues of F tend to support the conclusion that all the eects are not equal ( H a ), whereas v alues of F near 1 support the n ull h ypothesis ( H 0 ). In the ev en t that the alternate h ypothesis is indicated via the F-test, the ANO V A also pro vides the probabilit y of a TYPE I error. A TYPE I error occurs when it is concluded that dierences bet w een means exist when, in fact, they do not (i.e. accept H a when in fact H o is true). This information is giv en in the column labelled Pr(F) in the ANO V A output in T able 2.3. 26 PAGE 33 2.4.5 Are the Main Eects all Equal? F ollo wing the general process of establishing n ull and alternate h ypothesis as described abo v e, a pair of n ull and alternate h ypotheses are stated for eac h factor in the biopsy model. The n ull h ypothesis assumes that the main eects for a giv en factor at eac h of its lev els are equiv alen t. The alternate h ypothesis ob viously assumes that the main eects dier. H 0 : 1 = 2 = 3 H a : not all i are equal. 1 = 2 not all i are equal. 1 = 2 = 3 not all r i are equal. r 1 = r 2 = r 3 not all i are equal. The F-test statistic is applied to determine whic h h ypothesis to accept in eac h case. The factor sum of squares for eac h factor, n um ber of needles, spacing, and denoted SSA SSB SSC and SSD respectiv ely is computed as sho wn in the Appendix. The mean of eac h of these factor sum of square terms is computed b y dividing eac h term b y its associated degrees of freedom so that MSA = SSA=d f A MSB = SSB=d f B etc. as detailed in the Appendix. The test statistic is formed for eac h h ypothesis in the follo wing manner. T o test the eect of the rst factor, Number of Needles, F = MSA= MSE ; to test the eect of the spacing factor, 27 PAGE 34 F = MSB= MSE ; to test the eect of F = MSC= MSE ; and to test the eect of F = MSD= MSE Accepting the alternate h ypothesis means that a specic setting of the giv en factor corresponds to a c hange in detection rate; th us that factor has an eect on the o v erall outcome of the biopsy Df Sum of Sq Mean Sq F V alue Pr(F) Needles 2 0.15862 0.07931 607.427 0.0000000 Main Spacing 1 0.00498 0.00498 38.209 0.0000011 Eects 2 0.29249 0.14624 1120.073 0.0000000 2 0.28115 0.14057 1076.661 0.0000000 Ndls:Spc 2 0.1641 0.00820 62.846 0.0000000 Needles: 4 0.01444 0.00361 27.653 0.0000000 2-W a y Spacing: 2 0.00059 0.00029 2.283 0.1206068 Eects Needles: 4 0.00395 0.00098 7.569 0.0002892 Spacing: 2 0.00046 0.00023 1.794 0.1848710 : 4 0.02867 0.00716 54.902 0.0000000 Residuals 28 0.00365 0.00013 T able 2.3. The output from the ANO V A is displa y ed abo v e. See Appendix for details of the calculations. Refering to this ANO V A output, the column of n um bers labelled Sum of Sq refers to the parameters SSA, SSB, SSC and SSD detailed in the Appendix. The column labelled Mean Square lists the parameters MSA, MSB, MSC, MSD. The F V alue column lists the F-test outcome for eac h ro w: ( Needles F V alue = MSA/MSE). The larger v alues in this column tend to support the alternate h ypothesis that the main eect for a giv en factor diers across 28 PAGE 35 the possible lev els for that factor. The nal column, Pr(F), giv es the probabilit y of a T ype I error. Again, a T ype I error occurs if the alternate h ypothesis is concluded when in fact, the n ull h ypothesis is true. The ro w labelled Residuals indicates the total degrees of freedom, the SSE and the MSE for this analysis. Based on the n um bers in the table, eac h of the four main eects has a signican t eect on the outcome with the factor ha ving the greatest inruence on the detection rate, follo w ed b y the factors and Number of Needles This fact is indicated b y the high F-v alue that corresponds to eac h of the four factors. The ro ws labelled with t w o factor names (for example, Needles: Spacing ) indicate the ANO V A output corresponding to pair-wise in teractions and include the sum of squares computed for eac h pair of factors. The sum of squares for all of the pair-wise in teraction terms ( SSAB; SSAC; SSAD; SSBC; SSBD; SSCD ) are computed as detailed in the Appendix. The total treatmen t sum of squares, SSTR = SSA + SSB + SSC + SSD + SSAB + SSAC + SSAD + SSBC + SSBD + SSCD This sum does not include the sum of square terms due to the three-w a y and four-w a y in teractions because there are not enough degrees of freedom in the experimen t to use the full model. 29 PAGE 36 2.4.6 Recognizing In teraction bet w een F actors A t this poin t, the F-test has determined that eac h of the main factor eects con tributes to the o v erall detection rate. T o ev aluate the in teraction eects, the F-test is applied again The F-test is applied to determine in teraction bet w een, in this case, t w o, three or four factors. A n ull and alternate h ypothesis is form ulated for all possible com binations of factors and sum of square terms are computed for the factor groups and used in eac h F-test. The n ull and alternate h ypothesis are constructed for eac h of the pairwise in teractions. H 0 : all ( ) ij = 0 H a : not all ( ) ij = 0 all ( r ) ik = 0 not all ( r ) ik = 0 all ( ) il = 0 not all ( ) il = 0 all ( r ) jk = 0 not all ( r ) jk = 0 all ( ) jl = 0 not all( ) jl = 0 all ( r ) kl = 0 not all ( r ) kl = 0 All three-w a y com binations are formed, h ypotheses are constructed and F-test results are ev aluated. H 0 : all ( r ) ijk = 0 H a : not all ( r ) ijk = 0 all ( ) ijl = 0 not all ( ) ijl = 0 all ( r ) ikl = 0 not all ( r ) ikl = 0 all ( r ) jkl = 0 not all ( r ) jkl = 0 The n ull/alternate set of h ypothesis is constructed for the four-w a y in teraction. 30 PAGE 37 H 0 : all ( r ) ijkl = 0 H a : not all ( r ) ijkl equal 0 Based on the actual ANO V A results in the preceding table, four of the pair-wise in teractions appear strongly signican t: Needles: Spacing Needles: Needles: and : The other t w o pair-wise in teractions are included in the nal model ev en though the strength of their signicance is uncertain. The ANO V A w as executed once to include all three-w a y in teractions. Since these in teractions pro v ed insignican t, they are not included in the model. There are not enough degrees of freedom in the experimen t to estimate the residuals and test for the four-w a y in teraction. As stated previously the Y notation indicates the transformed detection rate ( p ). A t this poin t the general model, of the form Y ijklm = :::: + i + j + r k + l Main eects +( ) ij + ( r ) ik + ( ) il + ( r ) jk + +( ) jl + ( r ) kl P airwise eects +( r ) ijk + ( ) ijl + ( r ) jkl Three-w a y eects +( r ) ijkl F our-w a y eect + ijklm residual error is reduced to the nal model for this analysis: ^ Y ijkl = ^ + ^ i + ^ j + ^ r k + ^ l + d ( ) ij + d ( r ) ik + d ( ) il + d ( r ) jk + d ( ) jl + d ( r ) kl : This model yields the transformed probabilit y of detection at the giv en lev els for i j k and l 31 PAGE 38 No w that the factor eects ha v e been iden tied, the analysis rev olv es around determining the factor lev els that result in the highest detection rate. F or this part of the analysis, the tables of means and tables of eects are ev aluated. :::: Grand Mean 1.072 Needles 4 6 8 Spacing Relativ e Absolute i::: 0.999 1.09 1.128 :j:: 1.082 1.063 30 45 60 30 45 60 ::k: 0.9723 1.098 1.147 :::l 1.14 1.1104 0.9724 T able 2.4. The ANO V A tables of means list the transformed v alues. Needles 30 45 60 Spacing 30 45 60 4 0.926 1.027 1.045 Relativ e 0.978 1.111 1.157 6 0.979 1.113 1.176 Absolute 0.967 1.084 1.137 8 1.012 1.152 1.221 Needles 30 45 60 Spacing 30 45 60 4 1.054 1.028 0.915 Relativ e 1.148 1.118 0.980 6 1.161 1.123 0.985 Absolute 1.132 1.091 0.965 8 1.205 1.163 1.017 Spacing Needles Relativ e Absolute 30 45 60 4 0.987 1.011 30 1.026 0.978 0.913 6 1.099 1.080 45 1.159 1.139 .0994 8 1.159 1.097 60 1.235 1.196 1.010 T able 2.5. The transformed v alues of the pairwise means are sho wn. 32 PAGE 39 Referring to the ANO V A tables of means, the highest n um bers in eac h category rerect the best setting for a particular factor. On reading through the tables of means, the conclusion is that a tec hnique of 8 needles, relativ e spacing, = 60 and = 30 yields the best detection rate. In order to corroborate this more fully the in teractions that are deemed signican t are analysed to v erify that the main eect is not con tradicted b y an in teraction. Therefore, the table for Needles: is review ed and it is found that the setting of 8 needles and = 60 again yields the highest mean. The tables for all of the pair-wise com binations are review ed to determine that the best settings yield the highest means in the in teraction tables just as they did in the main eect tables. This pro v es to be the case, so none of the in teractions con tradict the conclusion dra wn from the main eect information. 33 PAGE 40 Num ber of Needles (4, 6, or 8) ^ 1 ^ 2 ^ 3 Eect -0.07329 0.01723 0.05607 Spacing (Relativ e or Absolute) ^ 1 ^ 2 Eect 0.009612 -0.009612 (30 45 or 60 ) ^ r 1 ^ r 2 ^ r 3 Eect -0.1001 0.02519 0.07486 (30 45 or 60 ) ^ 1 ^ 2 ^ 3 Eect 0.0678 0.03215 -0.09995 T able 2.6. The main factor lev el eects from the ANO V A output are documen ted. 34 PAGE 41 Spacing Relativ e Absolute 4 -0.02127 0.02127 Needles 6 -0.00017 0.00017 8 0.02143 -0.02143 30 45 60 4 0.02680 0.00244 -0.02925 Needles 6 -0.01031 -0.00127 0.01158 8 -0.01649 -0.00118 0.01767 30 45 60 Spacing Relativ e -0.004354 0.003708 0.000646 Absolute 0.004354 -0.003708 -0.000646 30 45 60 4 -0.01271 -0.00292 0.01563 Needles 6 0.00363 0.00087 -0.00450 8 0.00907 0.00206 -0.01113 30 45 60 Spacing Relativ e -0.001740 0.004148 -0.002407 Absolute 0.001740 -0.004148 0.002407 30 45 60 30 -0.01404 -0.02664 0.04067 45 -0.00621 0.00978 -0.00357 60 0.02025 0.01686 -0.03711 T able 2.7. The ANO V A table of eects for pairwise in teractions is displa y ed. 35 PAGE 42 By using the v alues from the tables of eects, a probabilit y for detection is calculated for the optimal setting: ^ Y 3131 = ^ + ^ 3 + ^ 1 + ^ r 3 + ^ 1 + d ( ) 31 + d ( r ) 33 + d ( ) 31 + d ( r ) 13 + d ( ) 11 + d ( r ) 31 1 : 347918 = 1 : 072 + : 05607 + : 009612 + : 07486 + : 0678+ : 02143 + : 01767 + : 00907 + : 000646 + 0 : 00174 + : 02025 This result of 1.347918 is then transformed bac k (arcsine equation) to yield a probabilit y of 0.38948 for this setting. 1 : 347918 = 2 arcsin q ( p ) p = (sin(1 : 347918 = 2)) 2 = 0 : 38949 : Therefore, with the factors set to 8 needles, relativ e spacing, = 60 and = 30 the biopsy procedure has a 38 : 9% probabilit y of detecting the cancer giv en the tumor distribution model used. This estimated probabilit y is best used in comparisons with the other estimated probabilities rather than as an absolute measure of detection rate. Therefore the conclusion from this analysis is a relativ e ranking of treatmen ts in terms of their detection rate. Since the 1000 sim ulated specimens w ere the same for eac h treatmen t, the ANO V A model determined the relativ e dierences bet w een detection rates of v arious treatmen ts, not necessarily pro viding enough data and results to dra w 36 PAGE 43 conclusions about absolute detection rates. T able 2.8 lists eac h experimen t and the probabilit y of detection predicted from the factor eects model. T reatmen t P arameters Num ber of Spacing Predicted Experimen t Needles Method Probabilit y 1 4 Relativ e 45 45 0.247 2 6 Relativ e 45 45 0.297 3 8 Relativ e 45 45 0.327 4 4 Absolute 45 45 0.251 5 6 Absolute 45 45 0.281 6 8 Absolute 45 45 0.291 7 4 Relativ e 60 45 0.265 8 6 Relativ e 60 45 0.337 9 8 Relativ e 60 45 0.369 10 4 Absolute 60 45 0.271 11 6 Absolute 60 45 0.324 12 8 Absolute 60 45 0.335 13 4 Relativ e 30 45 0.195 14 6 Relativ e 30 45 0.227 15 8 Relativ e 30 45 0.251 16 4 Absolute 30 45 0.205 17 6 Absolute 30 45 0.219 18 8 Absolute 30 45 0.224 19 4 Relativ e 45 60 0.200 20 6 Relativ e 45 60 0.236 21 8 Relativ e 45 60 0.260 22 4 Absolute 45 60 0.208 23 6 Absolute 45 60 0.227 24 8 Absolute 45 60 0.232 25 4 Relativ e 60 60 0.192 26 6 Relativ e 60 60 0.247 27 8 Relativ e 60 60 0.273 28 4 Absolute 60 60 0.203 29 6 Absolute 60 60 0.241 T able 2.8. The probabilities of detection for one tumor sim ulations are displa y ed. 37 PAGE 44 T reatmen t P arameters Num ber of Spacing Predicted Experimen t Needles Method Probabilit y 30 8 Absolute 60 60 0.248 31 4 Relativ e 30 60 0.175 32 6 Relativ e 30 60 0.196 33 8 Relativ e 30 60 0.215 34 4 Absolute 30 60 0.189 35 6 Absolute 30 60 0.194 36 8 Absolute 30 60 0.195 37 4 Relativ e 45 30 0.257 38 6 Relativ e 45 30 0.314 39 8 Relativ e 45 30 0.346 40 4 Absolute 45 30 0.266 41 6 Absolute 45 30 0.303 42 8 Absolute 45 30 0.315 43 4 Relativ e 60 30 0.276 44 6 Relativ e 60 30 0.354 45 8 Relativ e 60 30 0.389 46 4 Absolute 60 30 0.287 47 6 Absolute 60 30 0.346 48 8 Absolute 60 30 0.360 49 4 Relativ e 30 30 0.208 50 6 Relativ e 30 30 0.246 51 8 Relativ e 30 30 0.272 52 4 Absolute 30 30 0.223 53 6 Absolute 30 30 0.243 54 8 Absolute 30 30 0.250 T able 2.8. (Con t.) The probabilities of detection for one tumor sim ulations are displa y ed. 2.4.7 Clinical Distribution of T umors The biopsy sim ulations w ere conducted a second time on more realistic geometric glands. By using a clinically deriv ed distribution of n um ber of tumors per gland, a better population w as a v ailable for these biopsy simulations. A sample size of 1000 w as again used but in this experimen t, 1/4 38 PAGE 45 of the glands had a single tumor, 1 = 2 had t w o tumors and the remaining 1/4 had 3 tumors. The total gland v olume w as again held to be less than 6.4 cc. This distribution is based on the analysis done b y Daneshagari [2]. The ANO V A results are found in the Appendix and yield the same optimal biopsy procedure with a sligh tly dieren t probabilit y resulting from the factor eects model. By using the v alues from this second table of eects, a probabilit y for detection is calculated for the optimal setting: ^ Y 3131 = ^ + ^ 3 + ^ 1 + ^ r 3 + ^ 1 + d ( ) 31 + d ( r ) 33 + d ( ) 31 + d ( r ) 13 + d ( ) 11 + d ( r ) 31 1 : 7535 = 1 : 429 + 0 : 0733 + 0 : 01507 + 0 : 07456 + 0 : 07091+ 0 : 02321 + 0 : 02650 + 0 : 01412 0 : 005442 0 : 004094 + 0 : 03638 T ransforming this v alue (arcsine) yields a probabilit y of detection for the optimal setting of : 5908. This probabilit y of 59.08% is higher than the 38.9% ac hiev ed b y the sim ulation using geometric models of one tumor as w ould be expected. The predicted probabilities for eac h of the 54 experimen ts giv en this distribution of tumors is sho wn in T able 2.9. 39 PAGE 46 T reatmen t P arameters Num ber of Spacing Predicted Experimen t Needles Method Probabilit y 1 4 Relativ e 45 45 0.417 2 6 Relativ e 45 45 0.489 3 8 Relativ e 45 45 0.526 4 4 Absolute 45 45 0.417 5 6 Absolute 45 45 0.470 6 8 Absolute 45 45 0.482 7 4 Relativ e 60 45 0.427 8 6 Relativ e 60 45 0.524 9 8 Relativ e 60 45 0.569 10 4 Absolute 60 45 0.436 11 6 Absolute 60 45 0.514 12 8 Absolute 60 45 0.533 13 4 Relativ e 30 45 0.353 14 6 Relativ e 30 45 0.405 15 8 Relativ e 30 45 0.431 16 4 Absolute 30 45 0.354 17 6 Absolute 30 45 0.387 18 8 Absolute 30 45 0.388 19 4 Relativ e 45 60 0.358 20 6 Relativ e 45 60 0.408 21 8 Relativ e 45 60 0.443 22 4 Absolute 45 60 0.360 23 6 Absolute 45 60 0.391 24 8 Absolute 45 60 0.401 25 4 Relativ e 60 60 0.322 26 6 Relativ e 60 60 0.395 27 8 Relativ e 60 60 0.437 28 4 Absolute 60 60 0.332 29 6 Absolute 60 60 0.386 30 8 Absolute 60 60 0.403 T able 2.9. Giv en the distribution of one to three tumors, the probabilities of detection predicted b y the ANO V A model are displa y ed. 40 PAGE 47 T reatmen t P arameters Num ber of Spacing Predicted Experimen t Needles Method Probabilit y 31 4 Relativ e 30 60 0.326 32 6 Relativ e 30 60 0.357 33 8 Relativ e 30 60 0.381 34 4 Absolute 30 60 0.329 35 6 Absolute 30 60 0.341 36 8 Absolute 30 60 0.340 37 4 Relativ e 45 30 0.417 38 6 Relativ e 45 30 0.498 39 8 Relativ e 45 30 0.541 40 4 Absolute 45 30 0.425 41 6 Absolute 45 30 0.486 42 8 Absolute 45 30 0.504 43 4 Relativ e 60 30 0.436 44 6 Relativ e 60 30 0.541 45 8 Relativ e 60 30 0.590 46 4 Absolute 60 30 0.451 47 6 Absolute 60 30 0.537 48 8 Absolute 60 30 0.562 49 4 Relativ e 30 30 0.351 50 6 Relativ e 30 30 0.412 51 8 Relativ e 30 30 0.444 52 4 Absolute 30 30 0.359 53 6 Absolute 30 30 0.401 54 8 Absolute 30 30 0.407 T able 2.9. (Con t.) Giv en the distribution of one to three tumors, the probablities of detection predicted b y the ANO V A model are displa y ed. A selection of detection rates are graphed in Figure 2.7 to pro vide visualization of the relativ e ranking of v arious treatmen ts. The plots indicate 6 and 8 needles, relativ e spacing and all of the lev els for and 41 PAGE 48 .55 j Rate Hit .6 qqqq q = 30 ; 6 needles q Legend 0 = 30 ; 8 needles = 45 ; 6 needles = 60 ; 6 needles = 45 ;8 needles = 60 ; 8 needles 00000 45 0 0 30 60 0 .35 .4 .45 .5Figure 2.7. The detection rates for sev eral experimen ts are graphed and the common treatmen t parameters are noted for eac h experimen t. This giv es a visual understanding of the ranking of these treatmen ts in terms of their detection rate. 42 PAGE 49 3. Digitized Specimen Data 3.1 Summary of Soft w are T ool An analysis program, written in C, w as created to sim ulate needle biopsies on clinical data pro vided b y the Univ ersit y of Colorado Health Sciences Cen ter, P athology Departmen t. The clinical data w ere gathered from autopsies, pathologically in v estigated and digitized [2]. The data for eac h specimen are stored as a 3-dimensional arra y of information. The soft w are uses an input le to determine the c haracteristics of a giv en experimen t. These c haracteristics include the n um ber of needles, the initial placemen t of the rst needle, the angles and the spacing bet w een needles, and the needle diameter and length. In this manner, the analysis soft w are is rexible enough to handle a v ariet y of sim ulations. The goal of this biopsy sim ulation tool is to pro vide the means to experimen t realistically with v arious needle parameters on clinical data in order to determine an y correspondence bet w een biopsy methods and detection rates. The initial needle position is oset b y the distance requested (the z -oset en tered b y the user), with half of the needles en tering the righ t lobe 43 PAGE 50 and the other half en tering the left lobe, in symmetry with eac h other. The initial position is determined as an absolute (in cm) oset from the apex of the gland. The other parameters are used to position eac h needle on the specimen data set and determine ho w m uc h of the specimen data is to be returned in the needle biopsy This specimen data is analyzed to determine whether and ho w m uc h tumor data is presen t in the needle. This information is a v ailable to the user. Ha ving read the input le with parameter v alues, the code begins a loop on the specimen data les requested for sim ulation. In this loop, the threedimensional specimen data le is opened, the data are read in to a 3-d arra y with all of the bac kground trimmed o, the apex of the gland is located, and the needle positions are translated in to arra y coordinates. These coordinates are fed to the biopsy routine whic h extracts the specimen data coinciding with the needle and analyzes the data for tumor information. The information for the en tire experimen t is stored in an output le that documen ts the needle parameters and the results for eac h image data set. 44 PAGE 51 3.2 Specic Algorithms 3.2.1 Locating the Apex The apex is dened as the rst con tact with the prostate when approac hing it through the rectum, as done clinically This location is used as a landmark for positioning eac h biopsy needle. In the data set, the algorithm that searc hes for this landmark proceeds as follo ws. The planes are dened as sho wn in Figure 3.1. Eac h pixel in the three-dimensional specimen le con tains a n um ber indicating the t ype of data at that location. The possible t ypes are gland, tumor, capsule or bac kground. Capsule data indicate those pixels dening the boundary of the gland. The apex is indicated b y the rst pixel poin ting to capsule data. Therefore one plane of specimen data is ev aluated at a time, un til a pixel that poin ts to capsule data is found. This location is recorded as the apex location. 45 PAGE 52 Apex X Y ZFigure 3.1. The x; y; z axis, as dened for the digital data, mimic those dened for the geometric models. 46 PAGE 53 3.2.2 Establishing Needle P ositions The starting position, the location of the apex, serv es as the landmark for eac h additional needle. F rom this starting poin t and the additional user-supplied parameters ( z -oset, distance bet w een needles) all of the needle positions are calculated in terms of a v ector. This v ector, represen ted b y ( x; y; z ) coordinates, along with the angle, is a poin ter to a specic pixel of image data. The z -oset is assumed to be in cen timeters and is added to the initial ( x; y; z ) of the starting position to locate the rst needle position. Eac h time an y coordinate is c hanged, the new v ector ma y be poin ting to gland, tumor, bac kground, urethra or capsule data. The pixel represen ted b y the v ector is read to insure that the needle en try position remains located on capsule data. If it does not, the y coordinate is adjusted to mak e sure that the en try position of the needle is on capsule data. A t this poin t in the algorithm, the rst needle position is determined. There are t w o w a ys to space the remaining needles. The user ma y en ter absolute distances in cen timeters or a relativ e measure tak en to be a percen tage of the z dimension of the gland. In addition, a zero percen tage indicates that 47 PAGE 54 the spacing is based on the n um ber of needles in the biopsy; the needles are equally spaced across the z -axis of the gland. The remaining needle positions are calculated from the initial needle position: half of the needles are positioned in the righ t lobe b y using the remainder use to rotate in to the left lobe. All of the needles ha v e the x coordinate set to the midpoin t of the gland in the x dimension. The user-en tered distance, in cen timeters, is con v erted to a specic n um ber of pixels. This z distance is added to the rst needle position to obtain the second needle position, added to the second to obtain the third, etc. Eac h time a needle position is calculated, the coordinates are ev aluated to insure that they poin t to capsule data. If the gland is too short in the z direction to handle all the needles requested, the experimen t proceeds with the n um ber of needles that do sta y within the gland. The experimen ts that depend on a relativ e distance bet w een needles, require additional analysis of the yz slice before determining the z oset. The z diameter of the particular yz slice is calculated. The z distance required for a needle of a specic length, inserted at a specic angle is then subtracted from this z diameter. Rather than ha ving the last needle pierce more bac kground than gland data, this subtraction enables the full n um ber of needles to be 48 PAGE 55 inserted in to the gland. This new z diameter is then divided in to the n um ber of segmen ts required b y the specied percen tage. If the user indicates 0% for the distance spacing, the soft w are calculates the distance based on the n um ber of needles requested and the diameter of the yz plane. 3.3 Sim ulations The 54 treatmen ts used in the geometric model w ere used as biopsy procedures on a maxim um of 53 digitized clinical specimens. Some of the biopsy tec hniques w ere sim ulated on only 52 of these clinical specimens. T able 3.1 sho ws the results from these sim ulations on the digitized clinical data. The table documen ts both the m ultiple-tumor geometric model hit rate as w ell as the n um ber of hits resulting from the same biopsy on the digitized clinical data. The rst v e columns indicate the experimen t n um ber and the biopsy parameter settings for the four v ariables, n um ber of needles, spacing method, and The column labelled Detection Rate is the n um ber of hits per 1000 sim ulations of the geometric model. The column labelled Num ber of Hits is the n um ber of hits per n um ber of digitized clinical samples. Most experimen ts w ere run on all 53 of the digitized specimens. Ho w ev er, some of the sim ulations resulted in an error on one or more of the specimens and these specimens w ere then remo v ed from the experimen t. The nal column, 49 PAGE 56 labelled Clincial Detection Rate is the rate for the experimen ts on the digitized specimens. Num ber Num ber Clinical of Spacing Detection of Detection Experimen t Needles Method Rate Hits Rate 1 4 Relativ e 45 45 0.417 8 53 0.1509 2 6 Relativ e 45 45 0.489 11 53 0.2075 3 8 Relativ e 45 45 0.526 8 52 0.1538 4 4 Absolute 45 45 0.417 9 53 0.1698 5 6 Absolute 45 45 0.470 11 53 0.2075 6 8 Absolute 45 45 0.482 10 52 0.1923 7 4 Relativ e 60 45 0.427 9 53 0.1698 8 6 Relativ e 60 45 0.524 9 52 0.1731 9 8 Relativ e 60 45 0.569 13 53 0.2453 10 4 Absolute 60 45 0.436 10 53 0.1887 11 6 Absolute 60 45 0.514 12 53 0.2264 12 8 Absolute 60 45 0.533 12 53 0.2264 13 4 Relativ e 30 45 0.353 7 53 0.1321 14 6 Relativ e 30 45 0.405 12 53 0.2264 15 8 Relativ e 30 45 0.431 9 53 0.1698 16 4 Absolute 30 45 0.354 7 53 0.1321 17 6 Absolute 30 45 0.387 7 53 0.1321 18 8 Absolute 30 45 0.388 9 53 0.1698 19 4 Relativ e 45 60 0.358 6 53 0.1132 20 6 Relativ e 45 60 0.408 9 53 0.1698 21 8 Relativ e 45 60 0.443 11 52 0.2115 22 4 Absolute 45 60 0.360 8 53 0.1509 23 6 Absolute 45 60 0.391 10 53 0.1887 24 8 Absolute 45 60 0.401 10 53 0.1887 25 4 Relativ e 60 60 0.322 8 53 0.1509 26 6 Relativ e 60 60 0.395 8 52 0.1538 27 8 Relativ e 60 60 0.437 9 52 0.1731 28 4 Absolute 60 60 0.332 6 52 0.1154 29 6 Absolute 60 60 0.386 9 52 0.1731 30 8 Absolute 60 60 0.403 9 52 0.1731 T able 3.1 The detection rates for the geometric and clinical sim ulations are displa y ed. 50 PAGE 57 Num ber Num ber Clinical of Spacing Detection of Detection Experimen t Needles Method Rate Hits Rate 31 4 Relativ e 30 60 0.326 5 52 0.0962 32 6 Relativ e 30 60 0.357 5 52 0.0962 33 8 Relativ e 30 60 0.381 9 52 0.1731 34 4 Absolute 30 60 0.329 4 52 0.0769 35 6 Absolute 30 60 0.341 4 52 0.0769 36 8 Absolute 30 60 0.340 4 52 0.0769 37 4 Relativ e 45 30 0.417 6 52 0.1154 38 6 Relativ e 45 30 0.498 10 52 0.1923 39 8 Relativ e 45 30 0.541 12 52 0.2308 40 4 Absolute 45 30 0.425 8 52 0.1538 41 6 Absolute 45 30 0.486 10 52 0.1923 42 8 Absolute 45 30 0.504 11 52 0.2115 43 4 Relativ e 60 30 0.436 6 52 0.1154 44 6 Relativ e 60 30 0.541 10 52 0.1923 45 8 Relativ e 60 30 0.590 10 53 0.1887 46 4 Absolute 60 30 0.451 8 52 0.1538 47 6 Absolute 60 30 0.537 12 52 0.2308 48 8 Absolute 60 30 0.562 12 52 0.2308 49 4 Relativ e 30 30 0.351 3 30 0.1000 50 6 Relativ e 30 30 0.412 11 52 0.2115 51 8 Relativ e 30 30 0.444 10 52 0.1923 52 4 Absolute 30 30 0.359 6 52 0.1154 53 6 Absolute 30 30 0.401 8 52 0.1538 54 8 Absolute 30 30 0.407 10 52 0.1923 T able 3.1 (Con t.) The detection rates for the geometric and clinical sim ulations are displa y ed. 3.4 Geometric Model vs Clinical Model Comparison of the detection rates bet w een the geometric model and the clinical model rev eals that the geometric sim ulation produces m uc h higher rates than its clinical coun terpart. In attempting to explain this discrepency sev eral c haracteristics of the experimen t are noted. 51 PAGE 58 The distribution of the tumors and the total tumor v olume in a giv en specimen can impact the detection rate of a treatmen t. A comparison of the tumor v olumes is graphically displa y ed in Figures 3.2 and 3.3. As sho wn b y the histograms, the tumor v olumes for the autopsy data tend strongly to w ard small ( : 5 cc) v olumes. In con trast, the geometric model produces tumors with v olumes more equally spaced across the spectrum of possible v olumes. In fact, 80% of the autopsy specimens ha v e a total tumor v olume less than : 5 cc. In con trast, only 49% of the geometric gland models ha v e a total tumor v olume in this range. This dierence in the size of the tumors can explain some of the dierence in detection rate bet w een the clinical and geometrical models. A second dierence is that the relativ e ranking of detection rates for the digital data sim ulations is dieren t than the ranking of detection rates for the geometric sim ulations. An example of this discrepency is that experimen t 9, ( 8 Needles, Relativ e Spacing, = 60 = 45 ) ac hiev ed a detection rate of 0.2453 or 13 hits out of 53 samples. This detection rate is better than the detection rate of experimen t 45, ( 8 Needles, Relativ e Spacing, = 60 = 30 ) whic h is the optimal biopsy as indicated b y the geometric sim ulation. This dierence ma y be due to the fact that only 53 specimens w ere used in the 52 PAGE 59 digital sim ulation in con trast to the 1000 models constructed for the geometric sim ulation. 3.5 Optimal T ec hnique vs SRSCB The optimal tec hnique, determined b y the geometric model, consists of 8 needles, r elative spacing, = 60 and = 30 The SRSCB procedure uses 6 needles, absolute spacing, = 45 and = 45 Both tec hniques w ere sim ulated on the geometric model as w ell as the digitized clinical data. The optimal tec hnique actually pro v ed sligh tly w orse at tumor detection than the SRSCB procedure when sim ulated on the clinical data. In fact, the optimal method detected tumor in 10 out of 53 specimens (.189). The SRSCB method detected tumor in 11 out of 53 specimens (.207). These results compare with the o v erall results from the geometric sim ulation as follo ws. The SRSCB had a detection rate of .47 and the optimal had a detection rate of .59 on the 1000 geometric models. This discrepency is addressed b y noting the sample size a v ailable in the t w o sim ulations and the distribution of tumor v olumes as noted earilier. 53 PAGE 60 of Tumors Number Autopsy Specimens 15 20 10 Sum of Tumor Volume 25 .05.511.522.533.544.55 50Figure 3.2. The histogram of the clinical data sho ws the tumor distribution b y v olume. 54 PAGE 61 of Tumors Number .05 .5 400 11.522.533.544.55 0 Sum of Tumor Volume Geometric Specimens 100 200 300Figure 3.3. The histogram of the geometric data sho ws the tumor distribution b y v olume. 55 PAGE 62 4. Geometric Model V olume Estimates 4.1 T umor V olume Estimates The total v olume of tumor in a gland is an importan t piece of information for clinicians who use it to impro v e both the diagnosis and treatmen t plan for a patien t. The ultrasound used during a biopsy accurately measures the prostate gland v olume so that an appro ximate ratio of tumor to gland v olume can be used to estimate the v olume of tumor in a gland. These simulations oered an a v en ue to explore a means of appro ximating this v olume ratio b y using the v olume of the needle that con tains tumor information and the total v olume of the needle. Three methods are used to estimate the amoun t of tumor in tersected b y the needle. The needle can be modeled b y a line, a strip, or a cylinder in one, t w o, and three dimensions, respectiv ely The length and diameter of the needle are constan t and are set b y clinical limits. This incremen tal approac h began in one dimension in order to simplify aspects of the sim ulation during soft w are v erication. As the researc h progressed, the t w oand threedimensional needles w ere in troduced in order to model the actual biopsy more 56 PAGE 63 closely The rst method of estimating the v olume ratio is R = 1 n ( v i V i ) where v i represen ts the tumor v olume within a single needle, V i represen ts the v olume of that same needle, and n is the n um ber of needles. This ratio is referred to as the a v erage of the ratios. A second estimator of v olume ratio is r = v i V i where v i is the tumor v olume within a single needle and V i is the total v olume of that needle. This ratio is considered the ratio of the a v erage v olumes since 1 n P n i =1 v i is the a v erage tumor v olume and 1 n P n i =1 V i is the a v erage needle v olume. This yields r = 1 n v i 1 n V i = v i V i Both methods of estimating the ratio are documen ted belo w. 1 Y Z Gland Ellipsoid Tumor Needle l T t 2 tFigure 4.1. This illustration of the gland, tumor and one-dimensional needle depicts the v ariables used in determining the v olume ratio estimator. 57 PAGE 64 4.1.1 One-Dimensional Analysis Line Model In this rst model, w e represen t the needle b y a line segmen t as sho wn in Figure 4.1. The length of the needle that con tains tumor pixels, l T is the dierence bet w een t 1 and t 2 the t w o roots of equation ( 2.1): l T = j t 1 t 2 j A needle length, L of 1.25 cm is used in the estimate of v olume ratio. Th us the ratio l T L is an appro ximation of the true v olume ratio TV PGV ; that is, l T L TV PGV 4.1.2 Tw o-Dimensional Strip Model In the t w o-dimensional case w e represen t the needle b y a strip. The needle en try poin ts ( x 0 y 0 z 0 ) are used as a starting poin t in the t w o-dimensional analysis. Tw o lines are created, eac h oset from this starting coordinate b y the needle radius. The in tersection bet w een these t w o lines and the tumor ellipse is determined and the roots of the t w o resulting quadratics are used to compute both the occurrence of a detection and the amoun t of tumor within the needle. In this case, the estimate of the v olume ratio is the area of the tumor o v er the area of the needle. Figure 4.2 denes the lengths used in determining the area. The area of the tumor is calculated b y estimating the needle length whic h contains tumor data with the roots of in tersection: l t 1 = j t 11 t 12 j ; l t 2 = j t 21 t 22 j 58 PAGE 65 The area of tumor is then giv en b y a T = d 2 ( l t 1 + l t 2 ) where d is the diameter of the needle. The area of the needle is calculated in the same w a y using the length of the needle: a N = d 2 ( L + L ). Th us a T a N serv es as an estimate of the true tumor to gland v olume ratio, TV PGV t1 Y Z Gland Ellipsoid Tumor Needle l 12 t t 21 11 t t2 t 22 lFigure 4.2. This illustration of the gland, tumor and t w o-dimensional needle depicts the v ariables used in determining the v olume ratio estimator. 4.1.3 Three-Dimensional Cylinder Model The three-dimensional analysis models the needle as a cylinder and is similar to the t w o-dimensional case in that the en try poin t of the needle is again used as a cen ter coordinate for four needles. In this case, the four needles are constructed symmetrically about this poin t to generate a cylindrical needle. Then in tersections and roots are computed. A more accurate represen tation of the v olume ratio is obtained using the v olume of the tumor within the needle 59 PAGE 66 o v er the v olume of the needle. In this case, the length is estimated to be the maxim um of the lengths determined from the four sets of in tersection roots: l t = max ( j t 11 t 12 j ; j t 21 t 22 j ; j t 31 t 32 j ; j t 41 t 42 j ) : The v olume of the needle depends on the kno wn diameter and length: v N = ( d 2 ) 2 ( L ). The estimated v olume of the tumor depends on the needle lengths whic h con tain tumor data as sho wn in Figure 4.3. This leads to the tumor v olume estimate v T = ( d 2 ) 2 ( l t ). The ratio v T v N estimates the true v olume ratio, TV PGV 4.2 Experimen t Setup A second set of experimen ts utilizing the geometric model in v olv ed exploring the question of accurately estimating the tumor v olume to gland v olume ratio. The experimen t sim ulated a biopsy on a single specimen, increasing the n um ber of needles eac h iteration and comparing the v olume ratio obtained from the biopsy sample to the kno wn v olume ratio. The parameters for the biopsy include the optimal angles and determined from the ANO V A inv estigation The optimal n um ber of needles and distancing method determined from the ANO V A analysis do not apply to this experimen t since the n um ber of needles increases from 6 to 20 and the distancing of these needles is done so that the maxim um` n um ber, 20, are equally spaced. The maxim um n um ber 60 PAGE 67 Y 31 Gland Ellipsoid Needle Tumor Z Y X Tumor Gland Ellipsoid Needle t 22 l t2t 21 tt 11 t 12 t 42 t 32 t 41 tFigure 4.3. This illustration of the gland, tumor and three-dimensional needle depicts the v ariables used in determining the v olume ratio estimator. 61 PAGE 68 of needles w as set at 20 due to clinical limitations. The spacing of the needles is dependen t on the maxim um n um ber so that from one iteration to the next n 2 needles are in the same exact location, yielding the same detection information. In this manner the comparison bet w een a specimen biopsied b y 6 needles and the same specimen biopsied b y 10 needles is not dependen t on needle position, but instead compares the gain made b y the four additional needles. The sim ulation is executed on 1000 specimens, v arying the n um ber of needles from 6 to 20 in incremen ts of 2. `The output from this experimen t consists of a le for eac h specimen that con tains the results of eac h set of needles including the tumor to needle v olume ratio ac hiev ed and the associated estimates ( R = 1 n P ( v t =v n ) and r = v t v n ). In addition, the actual tumor to gland v olume ratio is noted. 4.3 Results The results of this experimen t w ere not as an ticipated as there appears to be no pattern of con v ergence to the actual tumor to gland v olume ratio within the limit of 20 total needles. Ho w ev er, m uc h w as learned from this exercise that pro vided insigh t in to the next series of in v estigations. First, it is noted that in the great majorit y of cases, a single 8-needle biopsy tends 62 PAGE 69 to o v erestimate the true tumor to gland v olume ratio. Secondly a comparison bet w een the t w o methods of calculating the error leads to the conclusion that the sum of the ratios is the more accurate method at least in this set of limited trials. 4.4 In teractiv e Utilit y Using the preceding idea as a starting poin t, an in teractiv e soft w are tool w as created to in v estigate the v olume ratio question in greater detail. This tool prompts the user for a random n um ber, seeds the random n um ber generator, creates a gland con taining a single tumor and conducts the optimal 8-needle biopsy This optimal biopsy has 8 needles, relativ e spacing bet w een the needles, = 60 and = 30 The results, whic h include eac h needle position, the amoun t of tumor v olume con tained in the needle and an estimate as to the v olume ratio of tumor to gland, are displa y ed for the user. A t this poin t, the user is able to c hoose the location for the next needle. This new needle is then sim ulated and the tumor v olume information it retriev es is incorporated in to the v olume ratio. The user can con tin ue this process of requesting additional needles and ev aluate the estimated v olume ratio and its error from the true ratio. A maxim um of 20 needles can be sim ulated on a single gland, beginning with the 8 original needles and accum ulating the 63 PAGE 70 additional 12 based on user specications. This area of researc h is full of open-ended questions where tools suc h as this in teractiv e utilit y can help shed ligh t on answ ers. With in v olv emen t from clinicians and medical researc hers, experimen ts can be designed to gather more information regarding the t w o issues of v olume ratio and optimal biopsy tec hnique. In addition, using the results of this body of researc h, more realistic tumor distributions and geometric models can be constructed to better understand the impact of treatmen t parameters on detection rate. 64 PAGE 71 A. APPENDIX ANO V A Denitions A dot in the subscript indicates a v eraging o v er the v ariable represen ted b y that index. The n um ber of lev els for Numb er of Ne e dles : a = 3. The n um ber of lev els for Distancing Metho d : b = 2. The n um ber of lev els for : c = 3. The n um ber of lev els for : d = 3. The n um ber of specimens = 1000. The n um ber of experimen ts: abcd = 54. In general, Y is an observ ation, Y is the mean of observ ations, is the true mean and ^ is the least squares estimate of the true mean. Y ijkl is the observ ed detection rate at the factor lev els indicated b y i; j; k and l Y :::: is the mean of all specimens o v er all treatmen t lev els i; j; k; l It indicates the o v erall detection rate for the en tire experimen t. Y :::: = 1 abcd a X i =1 b X j =1 c X k =1 d X l =1 Y ijkl 65 PAGE 72 SSTO or total sum of squares is a measure of the total v ariabilit y of the observ ations without consideration of factor lev el. SSTO = a X i =1 b X j =1 c X k =1 d X l =1 ( Y ijkl Y :::: ) 2 d f SSTO is the total degrees of freedom. The SSTO has abdc 1 = 54 1 degrees of freedom. One degree of freedom is lost due to the lac k of independence bet w een the deviations. SSTR or treatmen t sum of squares measures the exten t of dierences bet w een estimated factor lev el means and the mean o v er all treatmen ts. The greater the dierence bet w een factor lev el means (treatmen t means), the greater the v alue of SSTR SSTR = a X i =1 b X j =1 c X k =1 d X l =1 ( ^ Y ijkl Y :::: ) 2 d f SSTR is the degrees of freedom. There are r 1 degrees of freedom for the SSTR where r is the n um ber of parameters in the model. In the full model, r = abcd = 54, the total com binations of factor lev els. In the model used for this sim ulation, r = ( a 1)+( b 1)+( c 1)+( d 1)+( a 1)( b 1)+( a 1)( c 1)+( a 1)( d 1)+( b 1)( c 1)+( b 1)( d 1)+( c 1)( d 1) = 26. One degree of freedom is lost due to the lac k of independence bet w een the deviations. 66 PAGE 73 SSE or error sum of squares, measures v ariabilit y whic h is not explained b y the dierences bet w een sample means. It is a measure of the v ariation within treatmen ts. A smaller v alue of SSE indicates less v ariation within sim ulations at the same factor lev el. SSE = a X i =1 b X j =1 c X k =1 d X l =1 ( Y ijkl ^ Y ijkl ) 2 d f SSE is the degrees of freedom. Since SSE is the sum of the errors across factor lev el, the degrees of freedom is the sum of the degrees of freedom for eac h factor lev el. It is the total n um ber of sim ulations min us r abcd r MSE is the mean square for error dened b y MSE = SSE=d f SSE Note: The abo v e denitions imply SSTO = SSTR + SSE Due to this relationship, this process is referred to as the partitioning of the total sum of the squares. In order to measure the v ariabilit y within a factor lev el, the factor sum of square terms are computed. These terms are in tegral in the test statistic applied to determine whether a factor main eect is signican t. In addition, in teraction sum of squares are computed to measure v ariabilit y of the in teractions. 67 PAGE 74 The factor A sum of squares corresponds to the numb er of ne e dles factor. SSA = bcd a X i =1 ( Y i::: Y :::: ) 2 Similar factor sum of squares are computed for eac h of the factors: F actor Sum of Square Mean Sum of Square Num ber of Needles SSA = bcd P a i =1 ( Y i::: Y :::: ) 2 MSA = SSA= ( a 1) Spacing Method SSB = acd P b j =1 ( Y :j:: Y :::: ) 2 MSB = SSB= ( b 1) SSC = abd P c k =1 ( Y ::k: Y :::: ) 2 MSC = SSC= ( c 1) SSD = abc P d l =1 ( Y :::l Y :::: ) 2 MSD = SSD= ( d 1) The in teraction sum of squares are computed as w ell for use in the F-test on the in teractions. The rst three pair-wise in teraction sum of squares are sho wn belo w. The others are computed in the same manner. 68 PAGE 75 Num ber of Needles: Spacing SSAB = cd P a i =1 P b j =1 ( Y ij:: Y i::: Y :j:: + Y :::: ) 2 MSAB = SSAB= ( a 1)( b 1) Num ber of Needles: SSAC = bd P a i =1 P c k =1 ( Y i:k: Y i::: Y ::k: + Y :::: ) 2 MSAC = SSAC= ( a 1)( c 1) Num ber of Needles: SSAD = bc P a i =1 P d l =1 ( Y i::l Y i::: Y :::l + Y :::: ) 2 MSAD = SSAD= ( a 1)( d 1) The treatmen t means, ijkl indicate the mean for the treatmen t at the ijkl lev els of the respectiv e factors. The o v erall mean, is the mean across all factors and all lev els (across all i; j; k; l ). 69 PAGE 76 References (1) Hodge K.K., McNeal J.E., T erris M.K., Stamey T.A. \Random systematic v ersus directed ultrasound guided transrectal core biopsies of the prostate." Journal of Ur olo gy 142 (1989): 71-74. (2) Daneshgari, Firouz M.D., T a ylor, Gerald D. PhD, Miller, Gary J. M.D., PhD, Cra wford, E. Da vid M.D. \Computer Sim ulation of the Probabilit y of Detecting Lo w V olume Carcinoma of the Prostate with Six Random Systematic Core Biopsies". Ur olo gy 45 (April 1989): 604609. (3) McNeal, John M.D. \Normal Histology of the Prostate" The A meric an Journal of Sur gic al Patholo gy (1988): 619-633. (4) Neter, John, W asserman, William, Applied Linear Statistical Models Ric hard D. Irwin, Inc 1974. 70	{"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": 0.8203741312026978, "perplexity": 4111.279843007766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125937161.15/warc/CC-MAIN-20180420061851-20180420081851-00368.warc.gz"}
http://mathoverflow.net/questions/103759/elementary-submodels-of-v/103763	# Elementary submodels of V Consider the claim: (C) There is a transitive set $S \in V$ such that the structure $(S, \in)$ is an elementary submodel of $(V, \in)$. Obviously, this claim cannot be a theoreom of ZFC, by Godel's 2nd Incompleteness Theorem. But does (C) follow from ZFC+CON(ZFC)? Or are large cardinals needed to prove (C)? More generally (and vaguely), what is the weakest assumption needed to prove (C)? - ## 2 Answers A cardinal $\delta$ is correct if the theory $V_\delta\prec V$ holds, that is, if $V_\delta$ is an elementary substructure of $V$. This theory is expressible as a scheme in first order logic in the language of set theory augmented by a constant symbol for $\delta$ as the following scheme of statements: $$\forall \vec a\in V_\delta\ \ (\ \varphi(\vec a)\quad\iff\quad V_\delta\models\varphi[\vec a]\ \ ).$$ Note that any set $\langle S,{\in}\rangle$ such as you desire must have the form $S=V_\delta$ for some $\delta$, since it will the union of the $V_\alpha$ for $\alpha<\delta$ because by elementarity it will be right about these $V_\alpha$, and so this is your main case. The property of being correct is not expressible as a single assertion in ZFC, unless inconsistent, since otherwise there could be no least reflecting cardinal, since if $\delta$ is the least reflecting cardinal and this were expressible, then $V$ would have a reflecting cardinal, but $V_\delta$ would not. A similar argument applies directly to the existence of a set $\langle S,{\in}\rangle$ as in your question, showing that $S\prec V$ is not expressible as a single assertion in the language of set theory. Meanwhile, as a scheme, the existence of a reflecting cardinal is equiconsistent with ZFC, and in particular it does not imply Con(ZFC). To see this, observe that if there is a model of ZFC, then by the reflection theorem, we may easily produce models of any given finite subset of the theory. So by compactness the theory $V_\delta\prec V$ is also consistent. One may easily extend this theory to have a unbounded closed proper class $C$ of cardinals $\delta$ for which $V_\delta\prec V$, a theory known as the Feferman theory, and this also is equiconsistent with ZFC, with no increase of consistency strength, by the same argument. Feferman propsed this theory as a natural background set theory in which to undertake category theory, because it provides a robust universe concept (more robust than Grothendieck universes in that the universes all cohere into an elementary chain), but without the large cardinal commitment. If one adds to the theory $V_\delta\prec V$ that $\delta$ is inaccessible, then $\delta$ is a reflecting cardinal, whose existence is equiconsistent with Ord is Mahlo. If one desires a purely first-order concept, expressible as a single property in the language of set theory, then one may restrict to a less severe form of elementarity. For example, a cardinal $\delta$ is $\Sigma_2$-correct if $V_\delta\prec_{\Sigma_2} V$, that is, $V_\delta$ is elementary for formulas of complexity $\Sigma_2$, and $\Sigma_2$-reflecting if $\delta$ is also inaccessible. For example, the $\Sigma_2$-reflecting cardinals commonly arise in large cardinal arguments, and this is expressible in the language of set theory. More generally, one has the concept of a $\Gamma$-reflecting cardinal for any class $\Gamma$ of formulas, and if these have bounded complexity, then these also will be expressible. For many purposes, the $\Sigma_2$-correct cardinals carry much of the useful power of what you want from $V_\delta\prec V$, since one of the equivalent formulations is that $\delta$ is $\Sigma_2$ correct if and only if whenever there is $\theta$ with $V_\theta\models\sigma$ for any assertion $\sigma$, using parameters in $V_\delta$, then there is $\theta\lt\delta$ with $V_\theta\models\sigma$. In other words, any thing that happens in a $V_\theta$ above $\delta$ also happens below $\delta$. - One way of observing that the theory $V_\delta\prec V$ does not prove Con(ZFC) is that if it did, then this proof would use only finitely many of the assertions in the scheme $V_\delta\prec V$, but any finitely many of those assertions hold for some $V_\delta$ by the reflection theorem, and so this would mean that ZFC itself would imply Con(ZFC), which contradicts the incompleteness theorem unless ZFC is inconsistent. – Joel David Hamkins Aug 2 '12 at 22:16 Joel, relating to your ending remark, would you mind elaborating a little on the difference between $\Sigma_2$-reflecting cardinals and the notion of totally indescribable cardinal? It almost seems like the equivalence you mention allows the first $\theta$ to be $\delta$ itself. But your final sentence indicates that $\theta$ is strictly above $\delta$. Do you mean to have $\theta$ strictly larger than $\delta4? – Everett Piper Aug 3 '12 at 6:49 The thought I have is to let$\delta=\theta$and suppose$V_\delta\models \varphi[x]$for$\varphi$arbitrary and$x\in V_\delta$. Then$\{x\}\subset V_\delta$and$V_\delta\models \varphi^[\{x}\]$where$\varphi^$is $$\forall y (y\in\{x\})\wedge\varphi[y]$$. If$\delta$is totally indescribable, doesn't it then follow that$V_\alpha\models \varphi^*[\{x\}]$for some$\alpha<\delta$? – Everett Piper Aug 3 '12 at 7:01 Everett, my$\theta$was arbitrary, and$\theta=\delta$was allowed. Meanwhile, the two notions of reflection are orthogonal. For example, the existence of$\Sigma_2$reflecting cardinals is provable in ZFC and thus much weaker than the totally indescribable cardinals in consistency strength (even inaccessible$\Sigma_2$-reflecting cardinals are weaker). – Joel David Hamkins Aug 3 '12 at 11:38 ....And on the other hand, the least totally indescribable cardinal$\kappa$is definable in$V_{\kappa+\omega}$and is therefore not$\Sigma_2$-reflecting, since if$\theta\geq\kappa+\omega$, then$V_\theta$thinks there is a totally indescribable cardinal but no$\theta'\lt\kappa$thinks that is true. – Joel David Hamkins Aug 3 '12 at 11:39 Two comments on this: First, it's not clear that one can formulate "there exists a transitive set$S\in V$such that$S\prec V$" in first-order logic, so it's a bit tricky to phrase your question precisely. More importantly, I claim that no theory$T$can have the property you desire. Basically, suppose there were such a$T$. Then$T$must prove "there exists a transitive model of$T$:" by elementarity, if$V\models T$then$S\models T$. But then$T\models Con(T)$, which contradicts Goedel's theorem. (One can also prove a weaker version of this without using Goedel: any theory$T$containing Choice [EDIT: As Francois points out, Choice is unnecessary here] and Foundation cannot have the property you desire. Otherwise,$T$would prove "there exists a transitive model of$T$." Now using Choice, we can build a sequence$S_1, S_2, . . . $such that$S_i\models T$and$S_{i+1}\in S_i$for all$i$. But this contradicts Foundation.) - Excellent points. In reference to the first point, I suppose I should phrase the question semantically - is there some sort of axiom$A$such that in every model of$A$,$C$is true. Assuming that we are interested in (large cardinal) axioms that are first order definable and hold true in at least one well-founded model, your argument shows that the answer is no. – curious Aug 2 '12 at 7:14 By the way, the parenthetical remark doesn't really need choice. If$(S_0,{\in}) \vDash T$then$\lbrace S \in S_0 : (S,{\in}) \vDash T \rbrace$(which exists by comprehension) has no${\in}\$-minimal element, which contradicts foundation. – François G. Dorais Aug 2 '12 at 15:16 Oh, good point! Fixed. – Noah S Aug 2 '12 at 15:22	{"extraction_info": {"found_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": 0.9624879360198975, "perplexity": 233.98324217226386}, "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-2014-49/segments/1416400379466.35/warc/CC-MAIN-20141119123259-00144-ip-10-235-23-156.ec2.internal.warc.gz"}
http://www.maplesoft.com/support/help/MapleSim/view.aspx?path=RegularChains/ChainTools/IsIncluded	RegularChains[ChainTools] - Maple Programming Help Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : ChainTools Subpackage : RegularChains/ChainTools/IsIncluded RegularChains[ChainTools] IsIncluded inclusion test for two regular chains Calling Sequence IsIncluded(rc1, rc2, R) Parameters rc1 - regular chain rc2 - regular chain R - polynomial ring Description • The command IsIncluded(rc1, rc2, R) returns true if the saturated ideal of rc1 is detected to be contained in that of rc2, false otherwise, where rc1 and rc2 are regular chains of R. • The answer is true if the following conditions hold. (1) all equations of rc1 are reduced to zero by rc2 (2) the initials of rc1 are regular modulo rc2 • The answer is also true if the following conditions hold. (1) all equations of rc1 are reduced to zero by rc2 (2) the regular chain rc1 is primitive, that is, it generates its saturated ideal. • Other criteria are implemented. Some inclusions are not detected by any of those criteria. When they all fail, then false is returned. • Even though there exists a general algorithm for deciding whether the saturated ideal rc1 is contained in that of rc2, this algorithm is not implemented since it too costly to execute in most cases. On the criteria the implemented criteria are in general much less costly to execute. • This command is part of the RegularChains[ChainTools] package, so it can be used in the form IsIncluded(..) only after executing the command with(RegularChains[ChainTools]). However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][IsIncluded](..). Examples > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$ > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$ ${R}{:=}{\mathrm{polynomial_ring}}$ (1) > $\mathrm{sys}≔\left[{x}^{2}+y+z-1,x+{y}^{2}+z-1,x+y+{z}^{2}-1\right]$ ${\mathrm{sys}}{:=}\left[{{x}}^{{2}}{+}{y}{+}{z}{-}{1}{,}{{y}}^{{2}}{+}{x}{+}{z}{-}{1}{,}{{z}}^{{2}}{+}{x}{+}{y}{-}{1}\right]$ (2) > $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R,\mathrm{normalized}=\mathrm{yes}\right)$ ${\mathrm{dec}}{:=}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3) > $\mathrm{epdec}≔\mathrm{EquiprojectableDecomposition}\left(\mathrm{dec},R\right)$ ${\mathrm{epdec}}{:=}\left\{{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right\}$ (4) > $\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{nops}\left(\mathrm{dec}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}j\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{nops}\left(\mathrm{epdec}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}T≔{\mathrm{dec}}_{i};\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}U≔{\mathrm{epdec}}_{j};\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left(\mathrm{Equations}\left(T,R\right)\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left(\mathrm{Equations}\left(U,R\right)\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left(\mathrm{IsIncluded}\left(T,U,R\right)\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left(\mathrm{Equations}\left(U,R\right)\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left(\mathrm{Equations}\left(T,R\right)\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left(\mathrm{IsIncluded}\left(U,T,R\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}$ $\left[{x}{-}{z}{,}{y}{-}{z}{,}{{z}}^{{2}}{+}{2}{}{z}{-}{1}\right]$ $\left[{2}{}{x}{+}{{z}}^{{2}}{-}{1}{,}{2}{}{y}{+}{{z}}^{{2}}{-}{1}{,}{{z}}^{{3}}{+}{{z}}^{{2}}{-}{3}{}{z}{+}{1}\right]$ ${\mathrm{true}}$ $\left[{2}{}{x}{+}{{z}}^{{2}}{-}{1}{,}{2}{}{y}{+}{{z}}^{{2}}{-}{1}{,}{{z}}^{{3}}{+}{{z}}^{{2}}{-}{3}{}{z}{+}{1}\right]$ $\left[{x}{-}{z}{,}{y}{-}{z}{,}{{z}}^{{2}}{+}{2}{}{z}{-}{1}\right]$ ${\mathrm{false}}$ $\left[{x}{-}{z}{,}{y}{-}{z}{,}{{z}}^{{2}}{+}{2}{}{z}{-}{1}\right]$ $\left[{x}{+}{y}{-}{1}{,}{{y}}^{{2}}{-}{y}{,}{z}\right]$ ${\mathrm{false}}$ $\left[{x}{+}{y}{-}{1}{,}{{y}}^{{2}}{-}{y}{,}{z}\right]$ $\left[{x}{-}{z}{,}{y}{-}{z}{,}{{z}}^{{2}}{+}{2}{}{z}{-}{1}\right]$ ${\mathrm{false}}$ $\left[{x}{,}{y}{,}{z}{-}{1}\right]$ $\left[{2}{}{x}{+}{{z}}^{{2}}{-}{1}{,}{2}{}{y}{+}{{z}}^{{2}}{-}{1}{,}{{z}}^{{3}}{+}{{z}}^{{2}}{-}{3}{}{z}{+}{1}\right]$ ${\mathrm{true}}$ $\left[{2}{}{x}{+}{{z}}^{{2}}{-}{1}{,}{2}{}{y}{+}{{z}}^{{2}}{-}{1}{,}{{z}}^{{3}}{+}{{z}}^{{2}}{-}{3}{}{z}{+}{1}\right]$ $\left[{x}{,}{y}{,}{z}{-}{1}\right]$ ${\mathrm{false}}$ $\left[{x}{,}{y}{,}{z}{-}{1}\right]$ $\left[{x}{+}{y}{-}{1}{,}{{y}}^{{2}}{-}{y}{,}{z}\right]$ ${\mathrm{false}}$ $\left[{x}{+}{y}{-}{1}{,}{{y}}^{{2}}{-}{y}{,}{z}\right]$ $\left[{x}{,}{y}{,}{z}{-}{1}\right]$ ${\mathrm{false}}$ $\left[{x}{,}{y}{-}{1}{,}{z}\right]$ $\left[{2}{}{x}{+}{{z}}^{{2}}{-}{1}{,}{2}{}{y}{+}{{z}}^{{2}}{-}{1}{,}{{z}}^{{3}}{+}{{z}}^{{2}}{-}{3}{}{z}{+}{1}\right]$ ${\mathrm{false}}$ $\left[{2}{}{x}{+}{{z}}^{{2}}{-}{1}{,}{2}{}{y}{+}{{z}}^{{2}}{-}{1}{,}{{z}}^{{3}}{+}{{z}}^{{2}}{-}{3}{}{z}{+}{1}\right]$ $\left[{x}{,}{y}{-}{1}{,}{z}\right]$ ${\mathrm{false}}$ $\left[{x}{,}{y}{-}{1}{,}{z}\right]$ $\left[{x}{+}{y}{-}{1}{,}{{y}}^{{2}}{-}{y}{,}{z}\right]$ ${\mathrm{true}}$ $\left[{x}{+}{y}{-}{1}{,}{{y}}^{{2}}{-}{y}{,}{z}\right]$ $\left[{x}{,}{y}{-}{1}{,}{z}\right]$ ${\mathrm{false}}$ $\left[{x}{-}{1}{,}{y}{,}{z}\right]$ $\left[{2}{}{x}{+}{{z}}^{{2}}{-}{1}{,}{2}{}{y}{+}{{z}}^{{2}}{-}{1}{,}{{z}}^{{3}}{+}{{z}}^{{2}}{-}{3}{}{z}{+}{1}\right]$ ${\mathrm{false}}$ $\left[{2}{}{x}{+}{{z}}^{{2}}{-}{1}{,}{2}{}{y}{+}{{z}}^{{2}}{-}{1}{,}{{z}}^{{3}}{+}{{z}}^{{2}}{-}{3}{}{z}{+}{1}\right]$ $\left[{x}{-}{1}{,}{y}{,}{z}\right]$ ${\mathrm{false}}$ $\left[{x}{-}{1}{,}{y}{,}{z}\right]$ $\left[{x}{+}{y}{-}{1}{,}{{y}}^{{2}}{-}{y}{,}{z}\right]$ ${\mathrm{true}}$ $\left[{x}{+}{y}{-}{1}{,}{{y}}^{{2}}{-}{y}{,}{z}\right]$ $\left[{x}{-}{1}{,}{y}{,}{z}\right]$ ${\mathrm{false}}$ (5) References Xie, Y. "Fast Algorithms, Modular Methods, Parallel Approaches and Software Engineering for Solving Polynomial Systems Symbolically" PhD Thesis, University of Western Ontario, Canada, 2007.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 59, "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": 0.9221118092536926, "perplexity": 2663.784542174144}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396106.71/warc/CC-MAIN-20160624154956-00104-ip-10-164-35-72.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/333744/ideal-in-compact-hausdorff-space	# Ideal in compact Hausdorff space This is exercise 70, chapter 4. from Folland (page 142) Let $X$ be a compact Hausdorff space. An ideal in $C(X, \mathbb{R})$ is a subalgebra $J$ of $C(X, \mathbb{R})$ such that if $f\in J$ and $g\in C(X, \mathbb{R})$ then $fg\in J$. 1. If $J$ is an ideal in $C(X, \mathbb{R})$, let $h(J) = \{x \in X: f(x) = 0,\ \forall f \in J\}$. Then $h(J)$ is a closed subset of $X$, called the hull of $J$. 2. If $E\subset X$, let $k(E)=\{f \in C(X, \mathbb{R}) : f(x)=0,\ \forall x \in E\}$. Then $k(E)$ is a closed ideal in $C(X, \mathbb{R})$, called the kernel of $E$. 3. If $E\subset X$, then $h(k(E)) =\overline{E}$. 4. If $J$ is an ideal in $C(X, \mathbb{R})$ then $k(h(J))=\overline{J}$. (Hint: $k(h(J))$ may be identified with a subalgebra of $C_0(U, \mathbb{R})$ where $U=X\setminus h(J)$) I've managed to prove assignments 1-3. In (4) if $f$ is from $J$ then for each $y\in h(J)$ is $f(y)=0$, then must be $f\in k(h(J))$. Since $k(h(J))$ is closed, also $\overline J\subset k(h(J))$. For other way around, I've proven the hint ($k(h(J))$ may be identified with a subalgebra of $C_0(U, \mathbb{R})$ where $U=X\setminus h(J)$), using some corollary fo Stone-Weierstrass), but don't understand how to make conection between $C_0(X\setminus h(J), \mathbb{R})$ and $\overline{J}$ ANY HELP WOULD BE APPRECIATED. P.S. This is my first time asking question, but I've found this site very helpfull, so thanks! - I don't know for the moment how to use the hint. However, we can try the following. 1. Fix $f\in k(h(J))$ and $\varepsilon>0$. Let $K:=\{x\in X, \|f(x)\|\geqslant \varepsilon\}$. By compactness, we can find $g\in J$ non-negative such that $g>0$ on $K$. Indeed, $K$ is compact, and the intersection $\bigcap_{g\in J}g^{-1}(\{0\})\cap K$ is empty, so there are $g_1,\dots,g_n$ such that $K\cap\bigcap_{j=1}^ng_j^{-1}(\{0\})\cap K$ is empty. Then take $g:=\sum_{j=1}^ng_j^2$. 2. Let $f_n(x):=\frac{f(x)}{\frac 1n+g(x)}g(x)$. It's an element of $J$ such that $\lVert f_n-f\rVert\leqslant 2\varepsilon$. It's also exercise 1.5.b) in Elements of Functional Analysis, by F. Hirsch and G. Lacombe, Springer - First, forgive me for taking so long to come to answer. I can't figure out why function g exists. – Ana Apr 11 '13 at 10:34 @ana I've added details. – Davide Giraudo Apr 11 '13 at 11:38	{"extraction_info": {"found_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": 0.9914306402206421, "perplexity": 88.82275653247707}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500822053.47/warc/CC-MAIN-20140820021342-00430-ip-10-180-136-8.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/248647/problem-calculating-with-the-residue-theorem	# Problem calculating with the residue theorem I've come across this integral and I'm having some problems with it. I get to a solution, but looks a bit weird and I may be doing something wrong. $\int_C\cos(e^{(1/z)})dz$ Being $C$ the unit circle. I've tried to calculate the residue at z=0, which is the only singularity (and it's essential). I do the following: $$e^{1/z} = \sum_0^\infty \frac{1}{n!z^n}$$ $$\cos (x) = 1-x^2/2+x^4/4! + \ldots$$ I've found that the term that goes with 1/z for every x^n in the cosine, after substituting x=exp(1/z), is n/z, that means, the term 1/z will be: $$\sum_1^\infty \frac{(-1)^n}{(2n-1)!z}=\sum_0^\infty \frac{(-1)^{n+1}}{(2n+1)!z}$$ That means the residue is: $$\sum_0^\infty \frac{(-1)^{n+1}}{(2n+1)!}$$ Which is $-sin(1)$, so the integral would be $-2\pi i\sin(1)$ Don't know why, but doesn't look good. Can someone confirm this is right or tell me what I'm doing wrong? Thank you. - Without going deep into the calculation of the coefficient of $\,1/z\,$ (which btw, looks correct), all the rest seems fine. – DonAntonio Dec 1 '12 at 16:30 Ok, thank you very much. Actually, I had tried to use Wolfram to calculate the residue and it wasn't working, but just 1 minute ago it worked and I could see the residue is actually -sin(1), so again, thank you very much. – MyUserIsThis Dec 1 '12 at 16:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9748739004135132, "perplexity": 471.8434459875908}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507445190.43/warc/CC-MAIN-20141017005725-00282-ip-10-16-133-185.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/integration-by-parts-where-to-start.77490/	Integration by parts where to start 1. May 31, 2005 ryan750 show that INT x sec^2x dx = pi/4 - ln2/2 (between pi/4 and 0) pls help i dunno where to start i know it is integration by parts - just dunno how i should rearrange it. thanks 2. May 31, 2005 dextercioby If you know part integration,u'll find the notation quite familiar $$x=u;\sec x dx =dv$$ $$\int_{0}^{\frac{\pi}{4}} x\sec^{2} x \ dx=\frac{\pi}{4}-\frac{\ln 2}{2}$$ Daniel. 3. May 31, 2005 Jameson Tabular method, so much easier. Enough said. 4. May 31, 2005 BobG Tabular method really only helps when you have several ibp steps. In this case, you only have one. Once you integrate sec^2 x, you get tan x. If you're going to integrate tan x manually (vs just looking at the table), it's easiest to break it into $$\int{\sin x \frac{1}{\cos x} dx}$$ and then use u-substitution. Similar Discussions: Integration by parts where to start	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9056697487831116, "perplexity": 2704.4520398933983}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948537139.36/warc/CC-MAIN-20171214020144-20171214040144-00738.warc.gz"}
https://www.kseebsolutions.com/2nd-puc-maths-model-question-paper-4/	# 2nd PUC Maths Model Question Paper 4 with Answers Students can Download 2nd PUC Maths Model Question Paper 4 with Answers, Karnataka 2nd PUC Maths Model Question Papers with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. ## Karnataka 2nd PUC Maths Model Question Paper 4 with Answers Time: 3 Hrs 15 Min Max. Marks: 100 Part-A Answer all the questions: (10 × 1 = 10) Question 1. Let * be a operation defined on the set of rational numbers by a* b = $$\frac{\mathrm{ab}}{4}$$ find the identify element. ae = ea=a e = 4 Question 2. Write the values of x for which 2 tan-1 x = cos-1$$\frac{1-x^{2}}{1+x^{2}}$$ holds x ≥ 1 Question 3. Construct a 2 × 2 matrix A = [aij] whose elements are given by $$\frac{1}{2}\|-3 i+j\|$$ A = $$\left[ \begin{matrix} 1 & 1/2 \\ 5/2 & 2 \end{matrix} \right]$$ Question 4. Find the values of x for which $$\left\| \begin{matrix} 3 & x \\ x & 1 \end{matrix} \right\|$$ = $$\left\| \begin{matrix} 3 & 2 \\ 4 & 1 \end{matrix} \right\|$$ x2 = 8 x = ± $$\sqrt{8}$$ = ± 2$$\sqrt{2}$$ Question 5. Find $$\frac{d y}{d x}$$, if y = sin (x2 + 5) $$\frac{d y}{d x}$$ = cos(x2 + 5).2x Question 6. Evaluate : $$\int e^{x}\left(\frac{x-1}{x^{2}}\right) d x$$ $$=\int e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right) \cdot d x=e^{x}\left(\frac{1}{x}\right)+c$$ Question 7. Define negative of a vector. Negative vector is a vector whose magnitude is same with opposite direction Question 8. Write the direction cosines of x – axis. 1, 0, 0 Question 9. Define feasible region in LPP. The common region determined by all the constraints including non-negative constraints. Question 10. If P(A) = $$\frac { 3 }{ 5 }$$, P(B) = $$\frac { 1 }{ 5 }$$ find p(A ∩ B) if A and B are independent events p(A ∩ B) = p(A), p(B) p(A ∩ B) = $$3 / 5 \cdot 1 / 5$$ = $$\frac { 3 }{ 25 }$$ Part-B Answer any ten questions: (10 × 2 = 20) Question 11. Show that if f: A → B and g: B → C are one – one, then gof : A → C is also one – one gof (x1) = gof(x2) g[f(x1)]=g[(x2)] f(x1) = f(x2) x1 = x2 since g and f is one – one, gof is one – one Question 12. Show that sin-1 $$(2 x \sqrt{1-x^{2}})$$ = 2sin-1 x for $$\frac{-1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}}$$ Let x = sin θ ⇒ θ = sin-1 x 2x $$\sqrt{1-x^{2}}$$ = 2 sin θ $$\sqrt{1-\sin ^{2}} \theta$$ 2 sin θ. cos θ = sin 2θ ⇒ sin-1 (2x$$\sqrt{1-x^{2}}$$) = sin-1(sin 2θ) = 2 sin-1 x Question 13. Show that 2 tan-1 $$\frac { 1 }{ 2 }$$ + tan-1 $$\frac { 1 }{ 7 }$$ = tan-1 $$\frac { 31 }{ 17 }$$ Question 14. If the area of the triangle with vertices (-2,0),(0,4) and (0,k) is 4 square units, find the values of k using determinants. ±4 = $$\frac { 1 }{ 2 }$$ {-2 (4 – k) + (0) } ±4 = K – 4 +4 = k – 4 -4 = k – 4 k = 4 + 4 k = 0 k = 8 Question 15. Differentiate $$\left(x+\frac{1}{x}\right)^{x}$$ with respect to x, Question 16. Find the slope of the tangent to the curve y = $$\frac{\mathbf{x}-1}{\mathbf{x}-2}$$ x ≠ 2 at x = 0 Question 17. Find $$\frac{d y}{d x}$$, if x2 + xy + y2 = 100 2x + x$$\frac{d y}{d x}$$ + y + 2y $$\frac{d y}{d x}$$ = 0 (x + 2y)$$\frac{d y}{d x}$$ = -(2x + y) $$\frac{d y}{d x}=\frac{-(2 x+y)}{x+2 y}$$ Question 18. Evaluate :$$\int \frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha} d x$$ $$=\int \frac{\left(2 \cos ^{2} x-1\right)-\left(2 \cos ^{2} \alpha-1\right)}{\cos x-\cos \alpha}, d x$$ $$=\int 2 \frac{(\cos x-\cos \alpha)(\cos x+\cos \alpha)}{\cos x-\cos \alpha}, d x$$ $$=2 \int \cos x+\cos \alpha, d x$$ = 2 sin x + 2x cos α + c Question 19. Evaluate $$\int \frac{d x}{x-\sqrt{x}}$$ Question 20. Find the order and degree, if defined of the differential equation $$\left(\frac{\mathbf{d}^{2} \mathbf{y}}{\mathbf{d x}^{2}}\right)^{3}+\left(\frac{\mathbf{d y}}{\mathbf{d x}}\right)^{2}+\sin \frac{\mathbf{d y}}{\mathrm{dx}}+1=\mathbf{0}$$ order is 2 degree is not defined Question 21. Find $$\|\overrightarrow{\mathrm{b}}\|$$, if $$(\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=8$$ and $$\|\overrightarrow{\mathrm{a}}\|$$ = 8$$\|\overrightarrow{\mathrm{b}}\|$$ Question 22. Find the area of the parallelogram whose adjacent sides are determined by the vectors $$\|\overrightarrow{\mathrm{a}}\|$$ = î – ĵ + 3k̂ and $$\|\overrightarrow{\mathrm{b}}\|$$ = 2î – 7ĵ + k̂ Question 23. Find the angle between the pair of lines given by $$\|\overrightarrow{\mathrm{r}}\|$$ = 3î + 2ĵ – 4k̂ + λ(î + 2ĵ + 2k̂) and $$\|\overrightarrow{\mathrm{r}}\|$$ = 5î – 2ĵ + µ(3î + 2ĵ + 6k̂) Question 24. Let X denote the number of hours you study during a randomly selected school day. The probability that X can take the values of x, has the following form, where k is some constant find the value of k. ΣP(x) = 1 k = $$\frac { 1 }{ 5 }$$ Part-C Answer any ten questions: (10 × 3 = 30) Question 25. Determine whether the relation R in the set A = {l,2,3, ……… 13,14} defined as R = {(x,y) : 3x = y = o}, is reflexive, symmetric arid transitive. y = 3x R – {(1,3) (2,6) (3, 9) (4,12)} R is not reflexive as (l,l) ∉ R R is not symmetric as (1,3) ∈ R but (3,1) ∉ R R is not transitive as (1,3) ∈ R, (3,9) ∈R but (l,9)∉ R Question 26. If tan-1 $$\frac{x-1}{x-2}$$ + tan-1 $$\frac{x+1}{x+2}$$ = $$\frac{\pi}{4}$$ , then find the values of x. Question 27. If A and B are invertible matrices of the same order, then prove that (AB)-1 = B-1 A-1. From definition of inverse of matrix (AB) (AB)-1 =1 A-1 (AB)(AB)-1 = A-1l (A-1A) B(AB)-1=A-1. IB (AB)-1 = A-1 B(AB)-1 = A-1 B-1B(AB)-1 = B-1 A-1 l(AB)-1 = B-1 A-1 (AB)-1 = B-1A-1 Question 28. verify Rolles theorem for the funciton f(x)= x2 + 2X – 8, x ∈[-4,2] f (x) =x2 +2x – 8 1. f (x) being a polynominal function it is continuous on (-4,2) 2. Also f1 (x) = 2x + 2 hence derivable on (-4,2) 3. f(-4) = (-4)2 + 2(-4) -8 = 0 f(2) = 22 + 2(2) – 8 =0 ∴ f(-4) = f(2) ∴ All three conditions of Rolle’s theorem are verified. f1 (c) = 0 f1 (c) = 2c + 2 ⇒ 0 = 2c + 2 ⇒ c = -1 ∈(-4,2) Question 29. If x = $$\sqrt{\mathbf{a}^{\sin ^{-1} \mathbf{t}}}$$ and $$\sqrt{\mathbf{a}^{\cos ^{-1} \mathbf{t}}}$$ then prove that $$\frac{d y}{d x}=\frac{-y}{x}$$ Question 30. Find the two positive numbers whose sum is 15 and sum of whose squares is minimum. x + y = 15 ⇒ y = 15 – x s = x2 + y2 ⇒ s = x2 + (15 – x)2 $$\frac{d s}{d x}$$ = 2x + 2(15 – x) (-1) $$\frac{d s}{d x}$$ = 0 2x – 30 + 2x 0 = 4x – 30 x = $$\frac { 30 }{ 4 }$$ = 7.5 $$\frac{d^{2} s}{d x^{2}}=2+2=4>0$$ ∴ x = 7.5 require no is y = 7.5 & x = 7.5 Question 31. Evaluate : $$\int x \tan ^{-1} x d x$$ Question 32. Evaluate $$\int_{0}^{2} e^{x}$$ dx as a limit of a sum $$\int_{0}^{2} e^{x}$$ .dx = $$\left.e^{x}\right]_{0}^{2}$$ = e2 – e0 = e2 – 1 Question 33. Find the area of the region bounded by the curve y2 = 4x and the line x = 3 Question 34. Show that the position vector of the point P, which divides the line joining the points A and B having position vectors $$\overrightarrow{\mathrm{a}}$$ and $$\overrightarrow{\mathrm{b}}$$ internally in the ratio m : n is $$\frac{m \vec{b}+n \vec{a}}{m+n}$$ ‘O’ is the fixed point, $$\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{a}}$$ $$\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{b}}$$ P divides the line AB internally in the ratio m : n Question 35. Show that the four points with position vectors 4î + 8ĵ + 12k̂, 2î + 4ĵ + 6k̂, 3î + 5ĵ + 4k̂ and 5î + 8ĵ + 5k̂ are coplanar. Question 36. Find the equation of the plane passing through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z + 2=0 and the point (2,2,1) Equation of plane passing through intersection of the plane is (3x – y + 2z – 4) + λ (x + y + z + 2) = 0 This plane pass through (2,2,1) 3(2) – 2 + 2(1) -4 + λ(2 + 2 + 1 + 2)=0 2 + 7λ = 0 λ = $$\frac { -2 }{ 7 }$$ Required equqtion of plane is (3x – y + 2z – 4) + $$\left(\frac{-2}{7}\right)$$ (x + yz + 2) + 0 [xly by 7] 21x – 7y + 14z – 28 – 2x – 2y – 2z – 4 = 0 19x – 9y + 12z = 32 Question 37. Form the differential equation of the circles touching the x – axis at origin. (x – h)2 +(y – k)2 = r2 (x – 0)2 +(y – k)2 =a2 x2 + y2 = 2ay ⇒ 2a = $$\frac{x^{2}+y^{2}}{y}$$ diff Equ w.r. to x Question 38. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01,0.03 and 0.15 respectively. One of the insured person meets with an accidents. What is the probability that he is a scooter driver? Part-D Answer any six questions: (6 × 5 = 30) Question 39. Let R+ be the set of all non- negative real numbers. Show that the function f: R+ → [4,∞)given by f(x) = x2 +4 is invertible and write the inverse of f. f(x1) = f(x2) X12 + 4 = X22 + 4 x1 = x2 ∴ f (x) is one-one Let y = f(x) y = 4x +3 x = $$\frac{y-3}{4}$$ ∈R+ ∴ f (x) is onto Let f-1 (x) = y f(y) = x 4y +3 =x $$y=\frac{x-3}{4}$$ ∴ f-1 : [4,8) → R+ is defined by f-1 (x) = $$\frac{x-3}{4}$$ Question 40. If A = $$\left[ \begin{matrix} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{matrix} \right]$$ , then show that A3 -23A -40I = 0 Question 41. Solve by Matrix method : 2x + 3y + 3z = 5 x – 2y + z = 5 3x – y – 2z = 3 Question 42. If y = Aemx + Benx show that $$\frac{d^{2} y}{d x^{2}}$$ -(m + n) $$\frac{d y}{d x}$$ + mny = 0 $$\frac{d y}{d x}$$ = Ameex + Bnexn $$\frac{d^{2} y}{d x^{2}}$$ = Am2emx + Bn2enx $$\frac{d^{2} y}{d x^{2}}$$ – (m + n) $$\frac{d y}{d x}$$ + my = Am2emx+ Bn2enx – m2Aemx – Amnemx – Bn2enx – mnBenn + Amnemx + mnBenx = 0 Question 43. A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y – coordinate is changing 8 times as fast as the x – coordinate. 6y = x3 + 2 → (l) $$\frac{d y}{d x}=\frac{x^{2}}{2}$$ Given that $$\frac{d y}{d t}$$ = 8 $$\frac{d x}{d t}$$ ⇒ $$\frac{d x}{d t}$$ = 8 ∴ 8 = $$\frac { 1 }{ 2 }$$ x2 x = ±4 put x in Equation (01) 6y = 43 + 2 ∴ x = 4 y = 11 ∴ x = -4 , 6y = (-4)3 + 2 ⇒ y = -10,3 points are (4,11) (-4, -10,3) Question 44. Find the integral of $$\frac{1}{\sqrt{x^{2}-a^{2}}}$$ with respect to x and hence evaluate $$\frac{d x}{\sqrt{x^{2}+6 x-7}}$$ Question 45. Using integration find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4 y = 2x – 1 points are (0,1) (1,3) (4,9) etc x = 4, (4,0) (4,1) (4,2) etc Area = $$\int_{0}^{4}(3 x+1), d x-\int_{0}^{4}(2 x+1), d x$$ $$A=\left(\frac{3 x^{2}}{2}+x\right)^{4}-\left(\frac{2 x^{2}}{2}+x\right)^{4}$$ A = 24 + 4 – 16 – 4 = 8 sq units Question 46. Solve the differential equation $$\frac{d y}{d x}+y$$ sec x = tan x,0 ≤ x < $$\frac{\pi}{2}$$ P = secx Q = tan x IF = $$e^{\int \sec x d x}=e^{\log (\sec x+\tan x)}$$ = secx + tanx y x IF = ∫ Q x IF y (sec + tan x) = ∫(sec x + tan x) tan x.dx = ∫sec x.tan x + ∫ tan2 x.dx =secx + ∫(sec2 x – 1) dx y(sec x+tan x) = sec x + tan x – x + c Question 47. Derive the equation of the line in space passing through a point and parallel to a vecor both in vector and Cartesian form. Let ‘A’ be the point, $$\overrightarrow{\mathrm{a}}$$ is P V of $$\overrightarrow{\mathrm{A}}$$ $$\overrightarrow{\mathrm{AP}}=\overrightarrow{\mathrm{OP}}-\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}$$ AP is parallel to vector $$\overrightarrow{\mathrm{b}}$$ $$\overrightarrow{\mathrm{AP}}=\lambda \overrightarrow{\mathrm{b}}$$ $$\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}=\lambda \overrightarrow{\mathrm{b}}$$ $$\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}}$$ is vector form A=(x1 y1, z1) p (x,y,z) $$\overrightarrow{\mathrm{b}}$$ =b1i+b2j+b3k Now $$\overrightarrow{\mathrm{r}}$$ = xi + yj + zk, $$\overrightarrow{\mathrm{a}}$$ =x1i + y1j + z1k $$\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}}$$ takes the form xi + yj + zk=(x1i + y1j + z1k) + λ(b1i + b2j + b3k) xi + yj + zk=(x1 +λb1)i+(y1 +λb2) j + (z1+ λb3)k x =X1 +λb1 y= y1 + λb2 z = z1 + λb3 Question 48. A die is thrown 6 times. If ‘getting an odd number’is a success, what is probability of: a) 5 successes? $$P(5)=6_{c_{3}}\left(\frac{1}{2}\right)^{6-5}\left(\frac{1}{2}\right)^{5}=6\left(\frac{1}{2}\right)\left(\frac{1}{32}\right)=\frac{3}{32}$$ b) at least 5 successes? $$P(x \geq 5)=P(5)+P(6)=\left(\frac{1}{2}\right)^{6}\left[6_{C_{5}}+6_{C_{6}}\right]=\frac{6}{64}+\frac{1}{64}=\frac{7}{64}$$ c) at most successes? P(X≤ 5) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5) = $$\left(\frac{1}{2}\right)^{6}$$ [1 + 6 + 15 + 20 + 15 + 6] = $$\frac { 63 }{ 64 }$$ Part-E Answer any one question : (1 × 10 = 10) Question 49. a) Prove that and hence evaluate $$\int_{-1}^{1} \sin ^{5} x \cos ^{4} x d x$$ f(x) = sin5x.cos4x f(-x) = sin5(-x) cos4(-x) = -sin5x.cos4x = -f(x) ∴ f is odd function, I = 0 b) prove that $$\left\| \begin{matrix} { a }^{ 2 }+1 & ab & ac \\ ab & { b }^{ 2 }+1 & bc \\ ca & cb & { c }^{ 2 }+1 \end{matrix} \right\|$$ = 1 + a2 + b2 + c2 Question 50. a) A manufacturing company makes two models A and B of a product. Each piece Of model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of model B requires 12 labour hours for finishing and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs. 8,000 on each piece of model A and Rs. 12,000 on each piece of model B. How many pieces of model A and model B should be manufactured per week to realize a maximum profit? What is the maximum profit per week?	{"extraction_info": {"found_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": 0.8485742211341858, "perplexity": 1728.4554887790111}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882573667.83/warc/CC-MAIN-20220819100644-20220819130644-00052.warc.gz"}
https://infoscience.epfl.ch/record/28982	Formats Format BibTeX MARC MARCXML DublinCore EndNote NLM RefWorks RIS ### Abstract This paper examines a number of chaos shift keying methods Different approaches for the optimization of the transmitter and the receiver (based on statistical arguments) are compared through simulation. Based on the results, The tradeoff between computational comnplexity and performance is discussed. As usual, the performance measure considered is the bit error rate for a given noise level. Also, various limit cases are considered, among them the case where the channel noise is large	{"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": 0.8365954756736755, "perplexity": 440.8860679880654}, "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-2021-25/segments/1623487630081.36/warc/CC-MAIN-20210625085140-20210625115140-00420.warc.gz"}
http://mathoverflow.net/questions/135203/is-there-an-arctic-circle-phenomenon-for-amman-beenker-tilings	# Is there an Arctic Circle phenomenon for Amman-Beenker tilings? I found some slides on tilings and one of them pertained to Amman-Beenker tilings. It looks like there is an Arctic Circle phenomenon similar to that for dominos or lozenges. Is there any mathematical way of proving it? (Maybe not?) -	{"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": 0.8272678852081299, "perplexity": 1754.0334115417495}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1461864121714.34/warc/CC-MAIN-20160428172201-00046-ip-10-239-7-51.ec2.internal.warc.gz"}
https://math.hecker.org/2014/02/16/linear-algebra-and-its-applications-exercise-3-1-6/	## Linear Algebra and Its Applications, Exercise 3.1.6 Exercise 3.1.6. What vectors are orthogonal to $(1, 1, 1)$ and $(1, -1, 0)$ in $\mathbb{R}^3$? From these vectors create a set of three orthonormal vectors (mutually orthogonal with unit length). Answer: If $x = (x_1, x_2, x_3)$ is a vector orthogonal to both $(1, 1, 1)$ and $(1, -1, 0)$ then the inner product of $x$ with both vectors must be zero. This corresponds to the system $Ax = 0$ where $A = \begin{bmatrix} 1&1&1 \\ 1&-1&0 \end{bmatrix}$ To solve the system we perform Gaussian elimination. We start by subtracting 1 times row 1 from row 2: $\begin{bmatrix} 1&1&1 \\ 1&-1&0 \end{bmatrix} \Rightarrow \begin{bmatrix} 1&1&1 \\ 0&-2&-1 \end{bmatrix}$ The echelon matrix has 2 pivots in columns 1 and 2, so $x_1$ and $x_2$ are basic variables and $x_3$ is a free variable. Setting $x_3 = 1$ from row 2 we have $-2x_2 -x_3 = -2x_2 - 1 = 0$ or $x_2 = -\frac{1}{2}$. From row 1 we have $x_1 + x_2 + x_3 = x_1 - \frac{1}{2} + 1 = 0$ or $x_1 = -\frac{1}{2}$. So the vector $(-\frac{1}{2}, -\frac{1}{2}, 1)$ is a solution to the system and thus a vector orthogonal to $(1, 1, 1)$ and $(1, -1, 0)$. Note that scalar multiples of the vector $(-\frac{1}{2}, -\frac{1}{2}, 1)$ are all orthogonal to $(1, 1, 1)$ and $(1, -1, 0)$. Also note that the vectors $(1, 1, 1)$ and $(1, -1, 0)$ are orthogonal to one another. To produce orthonormal vectors we can take the vectors above and divide them by their lengths. The length of $(1, 1, 1)$ is $\sqrt{1^2+1^2+1^2} = \sqrt{3}$, the length of $(1, -1, 0)$ is $\sqrt{1^2 + (-1)^2} = \sqrt{2}$, and the length of $(-\frac{1}{2}, -\frac{1}{2}, 1)$ is $\sqrt{(-\frac{1}{2})^2 + (-\frac{1}{2})^2 + 1^2} = \sqrt{\frac{3}{2}}$ The three orthonormal vectors are then as follows: $\begin{bmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \qquad \begin{bmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \\ 0 \end{bmatrix} \qquad \begin{bmatrix} -\frac{1}{\sqrt{6}} \\ -\frac{1}{\sqrt{6}} \\ \frac{2}{\sqrt{6}} \end{bmatrix}$ NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang. If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books. This entry was posted in linear algebra and tagged , . Bookmark the permalink.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 33, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8955827951431274, "perplexity": 149.30486163096012}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589618.52/warc/CC-MAIN-20180717070721-20180717090721-00232.warc.gz"}
https://bib-pubdb1.desy.de/collection/FullTexts?ln=en&as=1	# OpenAccess 2018-11-1612:20 [PUBDB-2018-04567] Preprint/Report et al Search for long-lived, massive particles in events with displaced vertices and missing transverse momentum in $\sqrt{s}$ = 13 TeV $pp$ collisions with the ATLAS detector [CERN-EP-2017-202; arXiv:1710.04901] A search for long-lived, massive particles predicted by many theories beyond the Standard Model is presented. The search targets final states with large missing transverse momentum and at least one high-mass displaced vertex with five or more tracks, and uses 32.8 fb$^{-1}$ of $\sqrt{s}$ = 13 TeV pp collision data collected by the ATLAS detector at the LHC. [...] OpenAccess: PDF; 2018-11-1609:42 [PUBDB-2018-04550] Preprint/Report et al Search for resonant $WZ$ production in the fully leptonic final state in proton-proton collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector [arXiv:1806.01532; CERN-EP-2018-077] A search for a heavy resonance decaying into WZ in the fully leptonic channel (electrons and muons) is performed. It is based on proton–proton collision data collected by the ATLAS experiment at the Large Hadron Collider at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 36.1 fb −1 . [...] OpenAccess: PDF PDF (PDFA); 2018-11-1609:26 [PUBDB-2018-04549] Preprint/Report et al Search for pair production of heavy vector-like quarks decaying into high-$p_T$ $W$ bosons and top quarks in the lepton-plus-jets final state in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector [arXiv:1806.01762; CERN-EP-2018-088] A search is presented for the pair production of heavy vector-like B quarks, primarily targeting B quark decays into a W boson and a top quark. The search is based on 36.1 fb$^{−1}$ of pp collisions at $\sqrt{s}=13$ TeV recorded in 2015 and 2016 with the ATLAS detector at the CERN Large Hadron Collider. [...] OpenAccess: PDF PDF (PDFA); 2018-11-1515:41 [PUBDB-2018-04546] Preprint/Report et al Search for pair production of higgsinos in final states with at least three $b$-tagged jets in $\sqrt{s} = 13$ TeV $pp$ collisions using the ATLAS detector [arXiv:1806.04030; CERN-EP-2018-050; CERN-EP-2018-050] A search for pair production of the supersymmetric partners of the Higgs boson (higgsinos $\tilde{H}$) in gauge-mediated scenarios is reported. Each higgsino is assumed to decay to a Higgs boson and a gravitino. [...] OpenAccess: PDF PDF (PDFA); 2018-11-1515:12 [PUBDB-2018-04544] Preprint/Report et al Search for Higgs boson pair production in the $\gamma\gamma b\bar{b}$ final state with 13 TeV $pp$ collision data collected by the ATLAS experiment [arXiv:1807.04873; CERN-EP-2018-130] A search is performed for resonant and non-resonant Higgs boson pair production in the $\gamma\gamma b\bar{b}$ final state. The data set used corresponds to an integrated luminosity of 36.1 fb$^{−1}$ of proton-proton collisions at a centre-of-mass energy of 13 TeV recorded by the ATLAS detector at the CERN Large Hadron Collider. [...] OpenAccess: PDF PDF (PDFA); 2018-11-1514:08 [PUBDB-2018-04538] Report/Journal Article et al Search for doubly charged Higgs boson production in multi-lepton final states with the ATLAS detector using proton–proton collisions at $\sqrt{s}=13\,\text {TeV}$ [CERN-EP-2017-198; arXiv:1710.09748] A search for doubly charged Higgs bosons with pairs of prompt, isolated, highly energetic leptons with the same electric charge is presented. The search uses a proton-proton collision data sample at a centre-of-mass energy of 13 TeV corresponding to 36.1 $\mathrm{fb}^{-1}$ of integrated luminosity recorded in 2015 and 2016 by the ATLAS detector at the LHC. [...] OpenAccess: PDF PDF (PDFA); 2018-11-1514:05 [PUBDB-2018-04537] Report/Journal Article et al Search for $WW/WZ$ resonance production in $\ell \nu qq$ final states in $pp$ collisions at $\sqrt{s} =$ 13 TeV with the ATLAS detector [CERN-EP-2017-223; arXiv:1710.07235] Journal of high energy physics 1803(03), 042 (2018) [10.1007/JHEP03(2018)042] A search is conducted for new resonances decaying into a $WW$ or $WZ$ boson pair, where one $W$ boson decays leptonically and the other $W$ or $Z$ boson decays hadronically. It is based on proton-proton collision data with an integrated luminosity of 36.1 fb$^{-1}$ collected with the ATLAS detector at the Large Hadron Collider at a centre-of-mass energy of $\sqrt{s}$ = 13 TeV in 2015 and 2016. [...] OpenAccess: PDF PDF (PDFA); 2018-11-1514:01 [PUBDB-2018-04536] Report/Journal Article et al A search for $B-L$ $R$-parity-violating top squarks in $\sqrt{s} = 13$ TeV $pp$ collisions with the ATLAS experiment [CERN-EP-2017-171; arXiv:1710.05544] Physical review / D 97(3), 032003 (2018) [10.1103/PhysRevD.97.032003] A search is presented for the direct pair production of the stop, the supersymmetric partner of the top quark, that decays through an $R$-parity-violating coupling to a final state with two leptons and two jets, at least one of which is identified as a $b$-jet. The dataset corresponds to an integrated luminosity of 36.1 fb$^{-1}$ of proton-proton collisions at a center-of-mass energy of $\sqrt{s} = 13$ TeV, collected in 2015 and 2016 by the ATLAS detector at the LHC. [...] OpenAccess: PDF PDF (PDFA); 2018-11-1513:58 [PUBDB-2018-04535] Report/Journal Article et al Search for long-lived, massive particles in events with displaced vertices and missing transverse momentum in $\sqrt{s}$ = 13 TeV $pp$ collisions with the ATLAS detector [CERN-EP-2017-202; arXiv:1710.04901] Physical review / D 97(5), 052012 (2018) [10.1103/PhysRevD.97.052012] A search for long-lived, massive particles predicted by many theories beyond the Standard Model is presented. The search targets final states with large missing transverse momentum and at least one high-mass displaced vertex with five or more tracks, and uses 32.8 fb$^{-1}$ of $\sqrt{s}$ = 13 TeV pp collision data collected by the ATLAS detector at the LHC. [...] OpenAccess: PDF PDF (PDFA); 2018-11-1513:56 [PUBDB-2018-04534] Report/Journal Article et al Measurement of the production cross-section of a single top quark in association with a $Z$ boson in proton-proton collisions at 13 TeV with the ATLAS detector [CERN-EP-2017-188; arXiv:1710.03659] Physics letters / B 780, 557 - 577 (2018) [10.1016/j.physletb.2018.03.023] The production of a top quark in association with a $Z$ boson is investigated. The proton--proton collision data collected by the ATLAS experiment at the LHC in 2015 and 2016 at a centre-of-mass energy of $\sqrt{s} = 13 \mathrm{TeV}$ are used, corresponding to an integrated luminosity of $36.1 \mathrm{fb}^{-1}$. [...] OpenAccess: PDF PDF (PDFA);	{"extraction_info": {"found_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": 0.9982027411460876, "perplexity": 2048.743891408117}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743913.6/warc/CC-MAIN-20181117230600-20181118012600-00324.warc.gz"}
https://www.physicsforums.com/threads/induced-electric-fields.236828/	Induced Electric Fields 1. May 23, 2008 aznkid310 1. The problem statement, all variables and given/known data A metal ring 4.50 cm in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic field. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 T/s. (a) What is the magnitude of the electric field induced in the ring? (b) In which direction (clockwise or counterclockwise) does the current flow as viewed by someone on the south pole of the magnet? 2. Relevant equations Do i need to do anything with the initial B value? Change in flux dB/dt = -0.25t? Or is it dB/dt = 1.12 - 0.25t? 3. The attempt at a solution a) d[phi]/dt = (dB/dt)Acos(0) = (-0.250)(pi(2.2510^-2)^2) = -3.9810^-4 Wb E = (1/2rpi)(d[phi]/dt) = -2.8*10^-3 N/C b) Clockwise? 2. May 23, 2008 alphysicist Hi aznkid310, I don't believe your answer to part b is correct. Can you explain your reasoning for that part? 3. May 24, 2008 aznkid310 induced emf = - (change in magnetic flux) Since magnetic flux is decreasing, an induced magnetic field opposite to that must be created to counteract this change in flux. Is my reasoning off? Also, is part (a) correct? 4. May 24, 2008 alphysicist Your answer to part a looks right to me (except they want the magnitude of the field, so you don't need the negative sign). The induced magnetic field will be in the direction to oppose the change. Since the magnetic flux from the magnets is decreasing, the induced magnetic field will be in the same direction as the magnet's field. Does that make sense? 5. May 24, 2008 aznkid310 Ah that makes sense. It would oppose only if the magnetic flux is incresing right? 6. May 24, 2008 alphysicist It always opposes the change, so if the magnetic flux is increasing, then yes, the induced field will be in the opposite direction as the external flux.	{"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": 0.8808225989341736, "perplexity": 688.8012826403053}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541187.54/warc/CC-MAIN-20161202170901-00112-ip-10-31-129-80.ec2.internal.warc.gz"}
https://en.wikisource.org/wiki/Page:Popular_Science_Monthly_Volume_70.djvu/63	# Page:Popular Science Monthly Volume 70.djvu/63 The signs ${\displaystyle +,-,=}$ denote that, compared with the same group as now existing, a group falling in a given period was relatively more abundant, smaller, or equally developed, respectively, in the next younger period.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9459341764450073, "perplexity": 4229.192766398102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1512948615810.90/warc/CC-MAIN-20171218102808-20171218124808-00127.warc.gz"}
https://zbmath.org/?q=an%3A0910.15005	× # zbMATH — the first resource for mathematics Generalized reflexive matrices: Special properties and applications. (English) Zbl 0910.15005 Let $$P$$ and $$Q$$ be two involutory Hermitian matrices. A rectangular matrix $$A$$ is called a generalized reflexive matrix if $$A=PAQ$$ holds. The author derives some properties for generalized reflexive matrices. He gives applications to a least squares problem , a problem in an electric network, and a problem that arises in the stress analysis of a rectangular truss structure. Reviewer: E.Ellers (Toronto) ##### MSC: 15B57 Hermitian, skew-Hermitian, and related matrices 15A90 Applications of matrix theory to physics (MSC2000) 65F20 Numerical solutions to overdetermined systems, pseudoinverses Full Text:	{"extraction_info": {"found_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": 0.8980256915092468, "perplexity": 2080.1602970631843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585348.66/warc/CC-MAIN-20211020183354-20211020213354-00079.warc.gz"}
https://djalil.chafai.net/blog/2020/12/21/an-exactly-solvable-model/	This post is about a model of statistical physics which consists in a probability measure on ${\mathbb{R}^n}$ modeling ${n}$ one-dimensional unit charge particles subject to Coulomb pair repulsion and to attraction with respect to a background of opposite charge. It is a one-dimensional Coulomb gas (not a log-gas!) confined by the potential generated by a charged background, a special case of the jellium model of Paul Eugene Wigner (1938). In the case of a uniform background, it is related to a conditioned Gaussian distribution. It was already observed by Rodney James Baxter (1963) that this model is exactly solvable. This exact solvability can be seen as the one-dimensional analogue of the exact solvability discovered by Eric Kostlan (1992) in the case of the two-dimensional Coulomb gas describing the spectrum of Ginibre random matrices. One-dimensional Wigner jellium or Coulomb gas. The electrostatic potential at point ${x\in\mathbb{R}}$ generated by a one-dimensional unit charge located at the origin is given by ${g(x)=-\|x\|}$. By the principle of superposition, the electrostatic potential generated at point ${x\in\mathbb{R}}$ by a distribution of charges ${\mu}$ on ${\mathbb{R}}$ is given by $U_{\mu}(x)=(g\mu)(x)=-\int\|x-y\|\mu(\mathrm{d}y)$ and the electric field by ${E_\mu=-U_{\mu}’=-g’\mu=\mathrm{sign}\mu}$. The derivative of ${g}$ is the sense of distributions is the Heaviside step function ${g’=\mathbf{1}_{-(\infty,0)}-\mathbf{1}_{(0,+\infty)}}$, which is an element of ${\mathrm{L}^\infty}$ defined almost everywhere, while the second derivative of ${g}$ in the sense of Schwartz distributions is a Dirac mass at zero ${g”=-2\delta_0}$. It particular ${g}$ is the fundamental solution of the Poisson equation and we can recover ${\mu}$ from its potential by $U”_{\mu}=g”\mu=-2\delta_0*\mu=-2\mu.$ The self-interaction energy of the distribution of charges ${\mu}$ is $\mathcal{E}(\mu) =\frac{1}{2}\iint g(x-y)\mu(\mathrm{d}x)\mu(\mathrm{d}y) =\int U_\mu\mathrm{d}\mu.$ Let us consider now ${n\geq1}$ one-dimensional unit charges at positions ${x_1,\ldots,x_n}$, lying in a positive background of total charge ${\alpha>0}$ smeared according to a probability measure ${\rho}$ on ${\mathbb{R}}$ with finite Coulomb energy ${\mathcal{E}(\rho)}$. The total potential energy of the system is $H_n(x_1,\ldots,x_n) = -\sum_{i<j} \|x_i-x_j\| -\alpha\sum_{i=1}^nU_{\rho}(x_i)$ up to the additive constant ${\alpha^2\mathcal{E}(\rho)}$. The system is charge neutral when ${\alpha=n}$. Following Wigner (1938), let us define now the Boltzmann-Gibbs probability measure ${P_n}$ over all the possible configurations at inverse temperature ${\beta>0}$ by $\mathrm{d}P_n(x_1,\ldots,x_n) =\frac{\mathrm{e}^{-\beta H_n(x_1,\ldots,x_n)}}{Z_n} \mathrm{d}x_1\cdots\mathrm{d}x_n$ where $Z_n=\int_{\mathbb{R}^n}\mathrm{e}^{-\beta H_n(x_1,\ldots,x_n)} \mathrm{d}x_1\cdots\mathrm{d}x_n.$ It can be checked that ${Z_n<\infty}$ if and only if ${\alpha<n}$. Note that ${P_n}$ is a one-dimensional Coulomb gas with external field associated to the potential ${V=-\frac{\alpha}{n}U_\rho}$. Baxter exact solvability for uniform backgrounds. The model is exactly solvable. Indeed, following Baxter (1963), we have the combinatorial identity $-\sum_{i < j} \|x_i – x_j\| =\sum_{i<j}(x_{(j)}-x_{(i)}) =\sum_{k=1}^n (2k-n-1) x_{(k)},$ where ${x_{(n)}\leq\cdots\leq x_{(1)}}$ is the reordering of ${x_1,\ldots,x_n}$; in particular, $x_{(n)}=\min_{1\leq i\leq n}x_i \quad\text{and}\quad x_{(1)}=\max_{1\leq i\leq n}x_i,$ which allows to rewrite the potential energy as $H_n(x_1,\dots,x_n) = \sum_{k=1}^n \Bigr((2k-n-1)x_{(k)}-\alpha_n U_\rho(x_{(k)})\Bigr).$ We assume now that ${\rho}$ is the uniform law on an interval ${[a,b]}$. Then, for all ${x\in\mathbb{R}}$, $-U_\rho(x) =\frac{1}{b-a}\int_a^b\|x-y\|\mathrm{d}y =\begin{cases} \displaystyle\left\|x-\frac{a+b}{2}\right\| &\mbox{if }x\not\in[a,b]\\ \displaystyle\frac{\left(x-\frac{a+b}{2}\right)^2+\frac{(b-a)^2}{4}}{b-a} &\mbox{if }x\in[a,b] \end{cases}.$ The potential ${V=-\frac{\alpha}{n}U_\rho}$ then behaves quadratically on ${[a,b]}$ and is affine outside ${[a,b]}$. Conditioned on all the particles lying inside ${[a,b]}$, it is possible to interpret ${P_n}$ as a conditioned Gaussian law. Indeed, using Baxter’s identity, if ${\{x_1,\ldots,x_n\}\subset[a,b]}$ then $H_n(x_1,\ldots,x_n) = \sum_{k=1}^n(2k-n-1)x_{(k)} +\frac{\alpha}{b-a}\sum_{i=1}^n\Bigr(x_{(i)}-\frac{a+b}{2}\Bigr)^2+\frac{n\alpha(b-a)}{4}.$ This formula shows then that ${X_n\sim P_n}$ is conditionally Gaussian in the sense that $\mathrm{Law}\Bigr((X_{(n)},\ldots,X_{(1)})\bigm\vert \{X_1,\ldots,X_n\}\subset[a,b]\Bigr)$ $\qquad =\mathrm{Law}\Bigr((Y_n,\ldots,Y_1)\bigm\vert a\leq Y_n\leq\cdots\leq Y_1\leq b\Bigr)$ where ${Y_1,\ldots,Y_n}$ are independent real Gaussian random variables with $\mathbb{E}Y_k=\frac{a+b}{2}+\frac{b-a}{2\alpha}\left(n+1-2k \right) \quad\text{and}\quad \mathbb{E}((Y_k-\mathbb{E}Y_k)^2)=\frac{b-a}{2\alpha\beta}.$ This was already observed by Baxter. Now if we consider the limit ${a\rightarrow-\infty, b\rightarrow \infty}$ with ${\alpha/(b-a) \rightarrow c > 0}$, then ${P_n}$ can be interpreted as a Coulomb gas for which the potential is quadratic everywhere, namely ${V=\frac{c}{2n}\left\|\cdot\right\|^2}$. This can also be seen as a jellium with a background equal to a multiple of Lebesgue measure on the whole of ${\mathbb{R}}$. Under the scaling ${x_i=\sqrt{n}y_i}$, this limiting case matches the model studied by Abhishek Dhar, Anupam Kundu, Satya N. Majumdar, Sanjib Sabhapandit, and Grégory Schehr (2018). This Coulomb gas model with quadratic external field in one dimension is analogous to the complex Ginibre ensemble which is a Coulomb gas in two dimensions. Scale invariance. The model ${P_n}$ has a scale invariance which comes from the homogeneity of the one-dimensional Coulomb kernel ${g}$. Indeed, if ${\mathrm{dil}_\sigma(\mu)}$ denotes the law of the random vector ${\sigma X}$ when ${X_n\sim\mu}$, then, for all ${\sigma>0}$, dropping the ${n}$ subscript, $\mathrm{dil}_\sigma(P^{\alpha,\beta,\rho}) =P^{\alpha,\frac{\beta}{\sigma},\mathrm{dil}_\sigma(\rho)}.$ In other words, if ${X_n\sim P^{\alpha,\beta,\rho}}$ then $\sigma X_n\sim P^{\alpha,\frac{\beta}{\sigma},\mathrm{dil}_\sigma(\rho)}.$ This property is useful in the asymptotic analysis of the model as ${n\rightarrow\infty}$, and reveals the special role played by ${\alpha}$ as a shape parameter. Here the inverse temperature ${\beta}$ is a scale parameter, in contrast with the situation for log-gases. Asymptotic analysis. An asymptotic analysis of ${P_n}$ as ${n\rightarrow\infty}$ is conducted in a joint work arXiv:2012.04633 with David García-Zelada and Paul Jung, for general backgrounds. This is a continuation of our previous work devoted to two-dimensional jelliums. We study one-dimensional Wigner jelliums, not necessarily charge neutral, for which the unit charges are allowed to exist beyond the support of the background. The model can be seen as a one-dimensional Coulomb gas (not a log-gas!) in which the external field is generated by a smeared background on an interval. We first observe that the system exists iff the total background charge is greater than the number of unit charges minus one. Moreover we obtain a Rényi-type probabilistic representation for the order statistics of the particle system beyond the support of the background. Furthermore, for various backgrounds, we show convergence to point processes, at the edge of the support of the background. In particular, this provides asymptotic analysis of the fluctuations of the right-most particle. Our analysis reveals that these fluctuations are not universal, in the sense that depending on the background, the tails range anywhere from exponential to Gaussian-like behavior, including for instance Tracy-Widom-like behavior. One Dimensional Models. Excerpt from Baxter’s book on exactly solved models (1982): One-dimensional models can be solved if they have finite-range, decaying exponential, or Coulomb interactions. As guides to critical phenomena, such models with short-range two-particle forces (including exponentially decaying forces) have a serious disadvantage: they do not have a phase transition at a non-zero temperature (van Hove, 1950; Lieb and Mattis, 1966). The Coulomb systems also do not have a phase transition, (Lenard, 1961; Baxter, 1963, 1964 and 1965), though the one-dimensional electron gas has long-range order at all temperatures (Kunz, 1974). Of the one-dimensional models, only the nearest-neighbour Ising model (Ising, 1925; Kramers and Wannier, 1941) will be considered in this book. It provides a simple introduction to the transfer matrix technique that will be used for the more difficult two-dimensional models. Although it does not have a phase transition for non-zero temperature, the correlation length does become infinite at H = T = 0, so in a sense this is a ‘critical point’ and the scaling hypothesis can be tested near it. A one-dimensional system can have a phase transition if the interactions involve infinitely many particles, as in the cluster interaction model (Fisher and Felderhof, 1970; Fisher, 1972). It can also have a phase transition if the interactions become infinitely long-ranged, but then the system really belongs to the following class of ‘infinite-dimensional’ models. “. Last Updated on 2020-12-31 This site uses Akismet to reduce spam. Learn how your comment data is processed. Syntax · Style · .	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9643985033035278, "perplexity": 318.34401510126605}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585449.31/warc/CC-MAIN-20211021230549-20211022020549-00158.warc.gz"}
https://www.emathzone.com/tutorials/real-analysis/limit-points-of-a-sequence.html	# Limit Points of a Sequence A number $l$ is said to be a limit point of a sequence $u$ if every neighborhood ${N_l}$ of $l$ is such that ${u_n} \in {N_l}$, for infinitely many values of $n \in \mathbb{N}$, i.e. for any $\varepsilon > 0$, ${u_n} \in \left( {l – \varepsilon ,l + \varepsilon } \right)$, for finitely many values of $n \in \mathbb{N}$. Evidently, if $l = {u_n}$ for infinitely many values of $n$ then $l$ is a limit point of the sequence $u$. As in the case of sets of real numbers, limit points of a sequence may also be called accumulation, cluster or condensation points. The limit points of a sequence may be classified into two types: (i) those for which $l = {u_n}$ for infinitely many values of $n \in \mathbb{N}$, (ii) those for which $l = {u_n}$ for only a finite number of values of $n \in \mathbb{N}$. But this distinction is not necessary. As such, we do not distinguish the above mentioned two types of limit points of sequences by different titles. It should be noted that every limit point $l$ of the range set $R\left\{ u \right\}$ of a sequence $u$ is also a limit point of the sequence $u$ , because every neighborhood of $l$ contains infinitely many points of $R\left\{ u \right\}$, and so of the sequence $u$. On the other hand, a limit point of $u$ may or may nor be a limit point of $R\left\{ u \right\}$. If the values of only a finite number of terms of $u$ are not distinct, then evidently the limit points of $u$ are the same as those of the set $R\left\{ u \right\}$. Conclusively, it follows that the limit points of a sequence $u$ are either the points or the limit points of the set $R\left\{ u \right\}$. Example 1: If a sequence $u$ is defined by ${n_n} = 1$, then $1$ is the only limit point of sequence. Solution: For any $\varepsilon > 0$, ${u_n} = 1 \in \left( {1 – \varepsilon ,1 + \varepsilon } \right)$ $\forall n \in \mathbb{N}$. Therefore, $1$ is a limit pint of the sequence. Let $\alpha \in \mathbb{R}$ and $\alpha \ne 1$. Then for all $n$, $\left\| {{u_n} – \alpha } \right\| = \left\| {1 – \alpha } \right\|\not < \varepsilon$. When $\left\| {1 – \alpha } \right\| < \varepsilon < 0$. Thus no point $\alpha$ other than $1$ is the limit point of the sequence. Note that the limit point of the sequence $u$ is not a limit point of the range $R\left\{ u \right\} = \left\{ 1 \right\}$. Example 2: If ${u_n} = \frac{1}{n}$, then $0$ is the only limit point of the sequence $u$. Sufficient conditions for number $l$ to be or not to be a limit point of a sequence $u$: 1. If for every $\varepsilon > 0{\text{ }}\exists m \in \mathbb{N}$ such that ${u_n} \in \left( {l – \varepsilon ,l + \varepsilon } \right)$, $\forall n \geqslant m$ or equivalently $\left\| {{u_n} – l} \right\| < \varepsilon$ $\forall n \geqslant m$, then $l$ is a limit point of the sequence $u$ .In such a case it can be easily seen that $l$ is the only limit point of the sequence. The above condition is not necessary as it can be seen that for the sequence $\left\langle {1,\frac{1}{2},1,\frac{1}{3},1,\frac{1}{4} \ldots } \right\rangle$,$1$ is a limit point of this sequence but the above condition is not satisfied. 2. If $\varepsilon > 0$, ${u_n} = 1 \in \left( {1 – \varepsilon ,1 + \varepsilon } \right)$ for only a finite number of values of $n$ then $l$ is not a limit point of $u$. This condition is also necessary for a number $l$ not to be a limit point of the sequence $u$. Remarks 1. Whenever we simply write $\varepsilon > 0$ it is implied that $\varepsilon$ may be howsoever small positive number. 2. A positive number $\eta$ is said to be arbitrarily small if given any $\varepsilon > 0$, $\eta$ may be chosen such that $0 < \eta < \varepsilon$. 3. If $\eta$ is an arbitrary small positive number and given any $k > 0$ then $k\eta$ is also an arbitrary small positive number, this follows immediately if we take $0 < \eta < \varepsilon /2k$ for an $\varepsilon > 0$. 4. If ${\varepsilon _1},{\varepsilon _2}$ are two arbitrary small positive numbers then it readily follows that $l$ is a limit point of a sequence $u$ if and only if ${u_n} \in \left( {l – \varepsilon ,l + \varepsilon } \right)$ for infinitely many values of $\eta$. Example 3: Every bounded sequence $u$ has at least one limit point. Example 4: The set of limit points of a bounded sequence $u$ is bounded. Theorem: The set of limit points $E$ of every sequence $u$ is a closed set. Corollary: Every bounded set $E$, of limit points of a sequence $u$, contains the smallest and greatest members.	{"extraction_info": {"found_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": 0.9935736656188965, "perplexity": 43.40207419456838}, "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-2019-47/segments/1573496668782.15/warc/CC-MAIN-20191117014405-20191117042405-00534.warc.gz"}
https://link.springer.com/article/10.1007/s00382-013-2043-y?wt_mc=alerts.TOCjournals	## 1 Introduction Arctic sea ice is a complex component of the Earth climate system. Part of its complexity comes from its sensitivity to the atmosphere on a range of spatial and temporal scales. For example, decades of observational and modeling studies of sea ice have confirmed that its variability is primarily a top-down process (Liu et al. 2004; Deser and Teng 2008), where the atmosphere provides the primary forcing mechanisms (Hopsch et al. 2012). In response, sea ice tends to organize—via motion, formation, melting, and accretion—in accordance with large-scale patterns of atmospheric circulation (Walsh and Johnson 1979; Overland and Pease 1982; Fang and Wallace 1994; Slonosky et al. 1997; Prinsenberg et al. 1997; Overland and Wang 2010). Because of these responses to the atmosphere, concentrations of sea ice have been found to be correlated with several of the major modes of atmospheric variability, including the North Atlantic Oscillation (NAO) (Deser et al. 2000; Kwok 2000; Parkinson 2000; Partington et al. 2003), the Arctic Oscillation (AO) (Wang and Ikeda 2000; Rigor et al. 2002; Belchansky et al. 2004) (the NAO and AO are often referred to as part of the Northern Hemisphere annular mode; Wallace 2000), the El Niño-Southern Oscillation (ENSO) (Liu et al. 2004), and longer-period oscillations (Polyakov et al. 2003). Furthermore, the leading mode of atmospheric intraseasonal variability, the Madden–Julian Oscillation (MJO; Madden and Julian 1972), has been shown to modulate the high-latitude (Zhou and Miller 2005; Cassou 2008) and Arctic (L’Heureux and Higgins 2008; Yoo et al. 2011) atmosphere. However, connections between sea ice and the MJO remain largely unexplored. Therefore, the purpose of this paper is to examine variability of Arctic sea ice concentration by phase of the MJO. Observational studies of sea ice organizational patterns have shown that the most important atmospheric drivers of Arctic sea ice variability are surface air temperature and surface wind (Prinsenberg et al. 1997; Deser et al. 2000; DeWeaver and Bitz 2006), with surface wind being the most important driver of summer variability (Kwok 2008; Ogi et al. 2008; Zhang et al. 2013). The dominant pattern of winter sea ice variability resembles a dipole, where ice concentration in the North Atlantic varies oppositely between the Barents and Greenland Seas and the Labrador Sea (Fang and Wallace 1994; Partington et al. 2003; Ukita et al. 2007; Parkinson and Cavalieri 2008). Another winter dipole pattern is observed in the Pacific sector between the Bering Sea and the Sea of Okhotsk, where atmospheric blocking episodes act to modulate the advance of the ice, although these fluctuations of ice concentration tend to be smaller than those in the North Atlantic sector (Ivanova et al. 2012). In summer, a prominent dipole, also seen in observational data, tends to locate between the Kara Sea and the East Siberian Sea (Fang and Wallace 1994). Throughout the year, the majority of the variability occurs along the sea ice margin, where ice meets open water (Fang and Wallace 1994; Polyakov et al. 2003; Strong 2012; Ivanova et al. 2012), driven by atmospheric forcing (Strong 2012). Significant differences between summer and winter seasons, including the direction of ice change (melting versus freezing), solar radiation received at the surface, and variability in locations of ice concentration change, result in strong seasonality in Arctic sea ice extent (Fig. 1). For this study, we were interested in modulation of both winter (November–January) and summer (May–July) sea ice by the MJO, with particular emphasis along the sea ice margins, and because of the pronounced seasonality in ice extent, summer and winter periods were treated separately. Tropical convection, which is the primary driver of the MJO, has been found to affect atmospheric circulation in high latitudes (e.g., Ferranti et al. 1990; Higgins and Mo 1997; Matthews et al. 2004). Vecchi and Bond (2004) found that geopotential height, specific humidity, and surface air temperature in the Arctic varied by phase of the MJO, and the response of surface air temperature in Canada to the MJO was confirmed by Lin and Brunet (2009). Lee et al. (2011) noted that the “polar amplification” in surface temperatures was in response to poleward-propagating Rossby waves excited by MJO-related tropical convection. Yoo et al. (2012) further confirmed that the MJO-driven, poleward propagating wave train drove changes in the Arctic overturning circulation, heat flux, and downward infrared radiation, and Flatau and Kim (2013) noted that that the MJO forces the annular modes (the AO and NAO) on intraseasonal time scales. All of these Arctic parameters affected by the MJO, from atmospheric circulation to temperature to radiation, have potentially significant impacts on sea ice concentration. However, the specific effects of MJO-driven atmospheric variability on sea ice concentration are not yet known. Therefore, the purpose of this paper is to explore variability in sea ice concentration and atmospheric parameters for two periods, one in the winter freeze-up season, November–January, NDJ, and another in the summer melt season, May–July, MJJ, and then to connect the observed variability to specific phases of the MJO. Both seasonal and monthly variability will be examined on timescales of the MJO. The rest of this paper is organized as follows: datasets and methodology are described in Sect. 2, results are presented in Sect. 3, and discussion and conclusions are presented in Sect. 4. ## 2 Data and methods The analyses in this study were based on three publicly available datasets. First, to gain an understanding of the state of the Arctic atmosphere under different phases of the MJO at both surface and mid-tropospheric levels, daily data from the National Centers for Environmental Prediction (NCEP)–Department of Energy (DOE) reanalysis 2 (Kanamitsu et al. 2002) were examined. Variables included in the atmospheric analysis were 500-hPa geopotential height, mean sea level pressure, 2-m surface temperature, and 10-m winds, for the period 1979 to 2011. Daily composite anomalies of pressure, height, temperature and wind were created for both winter and summer months by phase of the MJO using the methodology described below. Second, to quantify the effects of the MJO on ice, daily change in Arctic sea ice concentration (ΔSIC) was calculated using the NOAA/National Snow and Ice Data Center (NSIDC) Climate Data Record (CDR) of passive microwave sea ice concentration, version 2 (Meier et al. 2013). This dataset was provided on a 25 km × 25 km grid for the polar region and available daily from 1987 through 2012. The years 1989–2010 were used in this study, starting in 1989 due to missing data in the beginning of the record. Ice concentrations in the CDR were produced through a combination of two mature passive microwave ice algorithms, the NASA Team (Cavalieri et al. 1984) and the Bootstrap (Comiso 1986), both using Special Sensor Microwave/Imager (SSM/I) brightness temperature data as input. Daily change in ice concentration (ΔSIC) was calculated at each grid box using $$\varDelta SIC = day_{n} -day_{n - 1}$$ where day n is daily ice concentration for day n and day n-1 is the concentration for the previous day. Mean monthly ΔSIC for the MJJ and NDJ seasons (Fig. 2) showed areas of ice concentration loss (blues, top row) and gains (yellows, bottom row) during these respective seasons. The location of largest ice concentration loss (gain) for each season varied substantially during the summer (winter), migrating poleward (equatorward) in each subsequent month of the season. Third, the MJO itself was defined using the daily real-time multivariate MJO (RMM) index (Wheeler and Hendon 2004). The RMM phases were used to divide the reanalysis and daily change in sea ice concentration datasets. The daily RMM index oscillates between eight phases, each corresponding to the broad location of an MJO-enhanced equatorial convective signal (Wheeler and Hendon 2004). The index is created such that the MJO generally progresses eastward, from phase 1 to 8 and back to phase 1 again. Days during which the magnitude of the MJO vector was less than one standard deviation from zero were not considered, following the compositing methodology of other recent studies (e.g., Zhou et al. 2012; Virts et al. 2013; Zhang 2013; Barrett and Gensini 2013). Anomalies in daily ΔSIC, 500-hPa geopotential height, sea level pressure, 2-m air temperature, and 10-m wind were then found by averaging the day n means for each MJO phase and subtracting them from the overall monthly mean. To isolate relationships between the MJO and Arctic sea ice, and to remove some of the effects of the long-period decline in overall sea ice cover (e.g., Serreze et al. 2007), only daily ΔSIC beyond one standard deviation (either positive or negative) from the normal daily change for that month was used to calculate monthly anomalies; all other daily ΔSIC were not considered for the analysis. In addition to focusing on extreme values of anomalous ΔSIC, a minimum number of days threshold was imposed at each grid box, such that only those boxes in which anomalous ΔSIC values were above (or below) one standard deviation for at least 5 days, for a particular MJO phase and month, were considered. Significance testing was performed using the Student’s t test, and both atmospheric and sea ice anomalies were examined for significance at the 95 % confidence level. ## 3 Results ### 3.1 Seasonal atmosphere variability Composites of Arctic atmospheric circulation in both winter and summer showed statistically significant variability by phase of the MJO, with the greatest amplitude in variability occurring in winter. For November through January (Fig. 3), the pattern of 500-hPa height anomalies was found to be wavy and to exhibit a variety of wavenumbers. For example, in Phase 4, negative anomalies over northeast Russia and Alaska changed signs to positive anomalies over northern North America, and then changed sign again to negative over the north Atlantic and northern Europe. The anomaly centers also tended to change signs with phase of the MJO, something which is a defining characteristic of MJO-related variability (Zhang 2013). For example, in MJO phase 1, negative height anomalies were found over northern Russia, Alaska, and the north Atlantic, and positive height anomalies were found over northern Europe and North America. By phase 5, positive height anomalies were centered over much of Russia, while negative height anomalies were located over much of Western Europe. In winter (November–January), the height anomalies on days with the MJO in phase 2 resembled an anomalously positive AO polarity, and on days when the MJO was in phase 6 and 7, the height anomalies resembled negative AO polarity. This agreed well with the findings of Flatau and Kim (2013), who noted that convection in the Indian Ocean (MJO phase 2) was associated with positive AO polarity. In summer (May–July), similar wavy patterns in the 500-hPa height anomalies were found (Fig. 4), although the magnitude of the anomalies was less than in winter. Similar to winter, the signs of anomaly centers tended to change with phase of the MJO. For example, in Phase 2, positive anomalies were located over the Bering Strait and Sea and the North Atlantic, and negative anomalies over northern Europe and North America, and by Phase 6, the signs of these anomalies had shifted, with positive height anomalies over northern Europe and North America and negative height anomalies over the Bering Strait. In addition to the findings described above, the influence of MJO in the Arctic in boreal summer (phases 1 and 8) indicated tendencies of a positive AO signal, while phases 5 and 6 pointed to a negative AO oscillation. This boreal summer MJO-AO relationship has not received much attention in published literature to date. Both the winter and summer height anomaly patterns showed that the Arctic atmosphere varies significantly with phase of the MJO. Establishing seasonal variability of the Arctic atmosphere by MJO phase was a necessary first step in our postulation of a top-down theory of how the MJO modulates the Arctic atmosphere, and subsequently, Arctic sea ice. ### 3.2 Monthly atmosphere and ice variability The MJO was found to project onto the Arctic atmosphere on seasonal time scales (Figs. 3, 4). However, when considering associations between atmospheric tendencies and sea ice variability, sub-seasonal temporal scale was necessary because monthly ΔSIC anomalies migrated poleward during summer (MJJ) and equatorward during winter (NDJ) seasons, respectively (Fig. 2). Locations of anomalous ΔSIC by phase of MJO also demonstrated regional variability across the Arctic domain. Two leading regions of sea ice variability identified previously in the literature were the Barents and Greenland Seas and the Labrador Sea in the Atlantic sector, and the Bering Sea and the Sea of Okhotsk in the Pacific sector. To better identify patterns in anomalous ΔSIC by phase of MJO, we used a sector-based approach using regions defined in the widely used Multisensored Analyzed Sea Ice Extent (MASIE) (National Ice Center and NSIDC 2010) product, identified in Fig. 5. Here, we highlight results from 2 months, January and July, which were typical of patterns seen in other months in both winter and summer. In January, sea ice variability by phase of the MJO showed several important characteristics. First, ΔSIC variability tended to concentrate the most in two sectors: the North Atlantic, where the most variability occurred between MJO phases 4 and 7 and extended from either side of Greenland to the Barents Sea (column 4 in Fig. 6), and the Pacific, where the most variability occurred between MJO phases 2 and 6 and extended from the Sea of Okhotsk to the Bering Sea (column 4 in Fig. 7). Second, the largest range in variability of January sea ice concentration in the Atlantic (−0.1 to 0.1) occurred during MJO phases 4 and 7 (Fig. 6), and the largest range in variability of January sea ice concentration in the Pacific (−0.1 to 0.1) occurred during Phases 2 and 6 (Fig. 7). Anomalies of daily change in sea ice concentration were less variable (more neutral) during other MJO phases, and thus are not shown. Third, ΔSIC anomalies in all four phases examined here (2, 4, 6, and 7) were supported by corresponding atmospheric circulation and temperature anomalies. For example, on days when the MJO was in phase 4 (Fig. 6, top row), statistically significant (at the 95 % level) anomalous daily changes in sea ice concentrations resembled the dipole structure reported by Fang and Wallace (1994), Partington et al. (2003), Ukita et al. (2007), and Parkinson and Cavalieri (2008). Negative anomalies were found along the east coast of Greenland and positive anomalies were found in the Barents Sea; the sign of these anomalies flipped on days when the MJO was in phase 7 (Fig. 6, bottom row). On days when the MJO was in phase 4, mean sea level pressure was anomalously low over the Barents Sea, which caused anomalously northerly surface winds in the Greenland Sea and anomalously southerly surface winds in the Barents Sea. When the MJO was in phase 7, mean sea level pressure was anomalously high over the Barents Sea, leading to opposite wind anomalies than for days when the MJO was in phase 4 (Fig. 6). Northerly surface winds over the Greenland Sea (like those during phase 4) would push sea ice away from Greenland, decreasing the sea ice concentration along its eastern coast, while southerly surface winds (like those during phase 7) would push ice back north toward Greenland and increase the sea ice concentration. For the other half of the dipole, in the eastern Barents Sea, weak northerly surface winds and below-normal surface temperatures during phase 4 agreed with anomalous positive change in sea ice concentration, while during phase 7, anomalous southerly winds and above-normal surface temperatures agreed with the observed anomalous negative change in sea ice concentration. Similar agreement between the atmosphere and anomalous daily change in sea ice concentration was found in the Pacific sector (Fig. 7). For example, on days when the MJO was in phase 2 (Fig. 7, top row), negative surface pressure anomalies were located over the Bering Strait and Chukchi Sea, leading to northerly surface wind anomalies over the Bering Sea and below-normal surface temperatures (as much as 6 K below January normal) in the Sea of Okhotsk, concurrent with an increase in sea ice concentration in both locations. On days when the MJO was in phase 6 (Fig. 7, bottom row), sea level pressure anomalies in the Bering Strait and Chukchi Sea were positive, leading to strong southerly surface wind anomalies (up to 5 m s−1) and positive surface temperature anomalies over the Sea of Okhotsk, concurrent with a decrease in sea ice concentration. For the Pacific sector, surface temperature anomalies seemed to be most strongly related to anomalous change in sea ice concentration. However, in the Atlantic sector surface wind anomalies seemed to be most strongly related to anomalous change in sea ice concentration, in good agreement with Prinsenberg et al. (1997), Deser et al. (2000), and DeWeaver and Bitz (2006), who all noted important effects of surface wind anomalies on Atlantic sea ice concentration in winter. In July, unlike January, ΔSIC anomalies tended to concentrate in Atlantic and Siberian sectors. In the Atlantic sector, on days when the MJO was in phase 2, positive ice concentration change anomalies were found from the northern Barents Sea westward to the east coast of Greenland (Fig. 8, top row). These anomalies largely reversed for days when the MJO was in phase 6 (Fig. 8, bottom row). Anomalies of atmospheric circulation were found supporting these ice anomalies. In phase 2, negative sea level pressure anomalies were centered over the Kara, Barents, and Labrador seas, with largely northerly wind anomalies east of Greenland and near-calm winds west of Greenland. In phase 6, sea level pressures were above normal over the Kara Sea and below-normal over Baffin Bay, with largely southerly wind anomalies over the Greenland Sea and negative anomalies over the Davis Strait. Northerly (southerly) wind anomalies would tend to push ice away from (toward) the summer-season ice source, the central Arctic Ocean, and toward (away from) land, supporting the near-shore ice anomalies seen in phases 2 and 6 (Fig. 8). Near-normal temperature anomalies were found over the North Atlantic sector in both phases, suggesting that for the North Atlantic, July variability in sea ice was driven primarily by variability in surface wind, in good agreement with Kwok (2008), Ogi et al. (2008), and Zhang et al. (2013). In the Siberian sector in July, on days when the MJO was in phase 1 (Fig. 9, top row), mostly negative anomalies in daily ΔSIC were located over the Kara and Laptev seas. On days when the MJO was in phase 5, ice concentration anomalies were reversed, with mostly positive change in sea ice concentration over the Kara and Laptev seas (Fig. 9, bottom row). Similar to the North Atlantic sector (and unlike January), atmospheric anomalies by MJO phase were mostly neutral in the Siberian sector, with a few small but key anomalies in sea level pressure driving changes in surface wind that explain observed changes in ice. For example, during phase 1, sea level pressures were negative over the Kara Sea and positive over the northern East Siberian Sea, yielding weakly positive surface wind anomalies that would act to transport ice poleward, supporting observed negative changes in ice concentration over the Siberian sector. During phase 5, the pressure dipole pattern reversed, with positive sea level pressures over the Kara Sea and negative pressures over the East Siberian Sea, yielding anomalously northerly surface winds that would act to push ice out of the central Arctic and into the Siberian sector, supporting observed positive chances in ice concentration. ## 4 Discussion and conclusions The goal of this paper was to explore variability in the Arctic atmosphere and sea ice concentration, and to connect such variability with phases of the MJO. Recent studies have heralded significant association between phase of the MJO and high-latitude terrestrial surface air temperature, atmospheric circulation, geopotential height, specific humidity (Ferranti et al. 1990; Higgins and Mo 1997; Matthews et al. 2004; Vecchi and Bond 2004; Zhou and Miller 2005; Cassou 2008; Lin and Brunet 2009). In addition, modulation of the Arctic atmosphere specifically by phase of MJO has also been documented (L’Heureux and Higgins 2008; Yoo et al. 2011). However, none of the previous works cited have considered associations between sea ice concentration and phase of MJO. The three principal findings of the current study are as follows. (1) The MJO projects onto the Arctic atmosphere in both winter (NDJ) and summer (MJJ) seasons. This projection was evident from the distinct wavy pattern in 500-hPa geopotential height anomalies (Figs. 3, 4), and it confirms the earlier work of Vecchi and Bond (2004) and Yoo et al. (2012). Both location and sign of height anomalies displayed a tendency to flip every 3–5 phases of the MJO. This MJO-mid tropospheric connection also proved robust, being visible in 3-month seasonal plots with statistically significant anomalies at the 95 % level, in both winter and summer seasons. Furthermore, in NDJ, height anomalies in phase 2 resembled positive AO polarity while height anomalies in phases 6 and 7 resembled negative AO polarity, in good agreement with Flatau and Kim (2013). (2) Variability in sea ice concentration by phase of MJO was found in both summer and winter seasons, and this variability was supported by corresponding anomalies in the state of the atmosphere. The magnitude of variability tended to shift largely with the migration of the ice margin poleward (equatorward) during the summer (winter) period. By computing anomalous ΔSIC per month, and binning by phase of MJO, active regions of coherent ice concentration variability were identified in both Atlantic and Pacific sectors for specific phases during January (Figs. 6, 9) and for North Atlantic and Siberian sectors during July. The signs of anomalies (positive or negative) for specific MJO phases changed with season. In January, areas of positive (negative) ΔSIC in the Atlantic sector were collocated with southerly (northerly) wind anomalies, with southerly (northerly) winds pushing ice toward (away from) land resulting in anomalously positive (negative) change in concentration. In the Pacific sector in January, areas of positive (negative) ΔSIC were collocated with negative (positive) surface temperature anomalies, with colder (warmer) surface temperatures promoting local increases (decreases) in ice concentration. In July, areas of positive (negative) ΔSIC in both the North Atlantic and Siberian sectors were collocated with northerly (southerly) surface wind anomalies, as unlike in January, northerly (southerly) winds acted to push ice away from (toward) the primary ice source region (the central Arctic), leading to positive (negative) changes in ice concentration. Sea level pressure anomalies were found to support the observed variability in surface wind. (3) The MJO modulates Arctic sea ice regionally, often resulting in dipole-shaped variability between anomaly centers. The most commonly observed dipoles occurred between the Barents and Greenland seas in January, in agreement with Ivanova et al. (2012). All four sectors (Atlantic and Pacific in winter, and North Atlantic and Siberian in summer) demonstrated instances of ice anomalies that changed sign approximately every 3–4 phases of the MJO, as evidenced from the January and July examples presented in this study. These changes in sign of anomalous ΔSIC corresponded with similar changes in surface pressure, surface wind, and mid-tropospheric geopotential height, and suggest a physical robustness to the MJO-sea ice relationship. It is important to note that accelerating decline in extent of multi-year sea ice over the last several decades has cast some doubt on earlier findings of ice-climate relationships, particularly between sea ice and phase of the NAO. For example, during winter, cyclonic surface air flow promotes ice export through the Fram Strait (Jung and Hilmer 2001), particularly export of multi-year ice (Deser and Teng 2008), leaving the newer, thinner pack more vulnerable to forcings including enhanced downward longwave radiation (Francis and Hunter 2006) and circulation (Comiso 2006; Maslanik et al. 2007; Francis and Hunter 2007). This process has accelerated with the changing character of sea ice, and perhaps also expanded the ice margins that are susceptible to changes in atmospheric circulation and temperature that vary by phase of the MJO. To mitigate potential effects of the long-term decline in overall sea ice extent, in this study, we imposed several restrictions on the sea ice concentration data. First, we only examined daily ΔSIC that was more than one standard deviation above (or below) normal. Second, we excluded grid points from the analysis with fewer than 6 days of non-zero daily change in sea ice concentration (thus ensuring focus on the ice margins). Third, only sea ice anomalies that were statistically significant at the 95 % confidence interval were plotted in Figs. 6, 7, 8, 9 (column four of each figure). These three restrictions served to amplify the MJO-ice signal by removing regions of insignificant variability, particularly toward the center of the Arctic. The results presented in this paper show statistically significant variability in Arctic sea ice by phase of the MJO that is well supported by corresponding tendencies in surface wind and surface air temperature. While the specific phase relationships may well change, the MJO will continue to project onto the Arctic and modify sea ice at the ice margins, and may become even more prominent due, in particular, to the decline in thicker multi-year ice. With the tendency for thinner and more vulnerable first-year ice to occupy a greater fraction of the Arctic, the MJO-sea ice relationship shown here may become even more prominent under our changing Arctic climate. A follow-on study is underway to explore these future relationships.	{"extraction_info": {"found_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": 0.8438060879707336, "perplexity": 3896.266071170639}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500671.13/warc/CC-MAIN-20230208024856-20230208054856-00299.warc.gz"}
https://robotics.stackexchange.com/questions/2365/covariance-matrix-in-ekf	covariance matrix in EKF? I'm struggling with the concept of covariance matrix. $$\Sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{x \theta} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{y \theta} \\ \sigma_{\theta x} & \sigma_{\theta y} & \sigma_{\theta \theta} \\ \end{bmatrix}$$ Now, my understanding for $\sigma_{xx}$, $\sigma_{yy}$, and $\sigma_{\theta \theta}$ that they describe the uncertainty. For example, for $\sigma_{xx}$, it describes the uncertainty of the value of x. Now, my question about the rest of sigmas, what do they represent? What does it mean if they are zeros? I can interpret that if $\sigma_{xx}$ is zero, it means I don't have uncertainty about the value of x. Note, I'm reading Principles of Robot Motion - Theory, Algorithms, and Implementations by Howie Choset et. al., which states that By this definition $\sigma_{ii}$ is the same as $\sigma_{i}^{2}$ the variance of $X_{i}$. For $i ≠ j$, if $\sigma_{ij} = 0$, then $X_{i}$ and $X_{j}$ are independent of each other. This may answer my question if the rest of sigmas are zeros however, I'm still confused about the relationship between these variables for example $x$ and $y$. When does this happen? I mean the correlation between them. Or in other words, can I assume them to be zeros? Another book namely FastSLAM: A Scalable Method ... by Michael and Sebastian which states The off-diagonal elements of the covariance matrix of this multivariate Gaussian encode the correlations between pairs of state variables. They don't mention when the correlation might happen and what does it mean? Here is one toy case where off-diagonal elements are non-zero. Consider a state vector that includes the position of both the left and right wheels instead of just a single position for the robot. Now if the left wheel has a position of 100m then you know the right wheel will also have a position of roughly 100m (depending on the axle length). As the left wheel increases position so will the right wheel, in general. It's not an exact 1:1 correlation, e.g. it doesn't hold exactly when the robot is turning, but overall it holds. So here the off-diagonal entry between left wheel x-position and right wheel x-position would be close to 1. • Ok, if my model is represented as a point that moves in a planar environment (e.i. 2D), so the off-diagonal elements are zeros since there is no such correlations between the diagonal elements. Is this assumption correct? And what about in case this point detects a landmark that has two coordinates (e.i. $x, y$), can I also assume the correlation zeros? Jan 28 '14 at 5:42 • To your first question, yes you can leave the off-diagonal elements zero. For the second, it kind of depends on how you handle it. If you just use the landmark to estimate your current position there are no correlations. If you add the landmark positions to the state vector (as is common in SLAM) then they will start to develop correlations between themselves. Jan 28 '14 at 13:47 To get a feeling for the covariance matrix - without getting into the math details here - its best to start with a 2x2 matrix. Then remember that the covariance matrix is an extension of the concept of variance into the multivariate case. In the 1D case, variance is a statistic for a single random variable. If your random variable has a Gaussian distribution with zero mean, its variance can precisely define the probability density function. Now, if you extend this to two variables instead of one, you can differentiate between two cases. If your two variables are independent, which means the outcome of one value has no relation to the other value, its basically the same as in the 1D case. Your $\sigma_{xx}$ and your $\sigma_{yy}$ give the variance of the $x$ and $y$ part of your random variable, and $\sigma_{xy}$ will be zero. If your variables are dependent this is different. Dependent means that there is a relation between the outcome of $x$ and $y$. For example you could have that whenever $x$ is positive, $y$ is in general more likely to also be positive. This is given by your covariance value $\sigma_{xy}$. Giving an example for a robot in a 2D case without orientation is a bit contrived, but lets say you have a random component along the travel direction on the $x$-axis and you know that this component also generates a drift on your lateral axis ($y$). This could for example be a faulty wheel. This will result in a rotated uncertainty ellipse. Now for e.g. when you later have something that measures your actual $x$ position, you can estimate the uncertainty distribution on your $y$ component. A more relevant example is in the 3D case, where usually you have a different uncertainty along the transversal direction compared to the lateral direction. When you rotate your system (so changing $\theta$) this will also rotate your uncertainty ellipse. Note that the actual representation is usually some banana shape, and the Gaussian is only an approximation. In the EKF case its a linearization around the mean. One really good way to visualize this is to use the concept of the uncertainty ellipse. It basically shows the $1 \sigma$ boundary for a multivariate Gaussian distribution, and can be used to visualize a Covariance matrix. A quick search brought up this demo which will also provide you with some additional insight into how the covariance is built. In essence, the diagonal entries define the extents of the axis, while the off-diagonal entries relate to the rotation of the entire ellipse. This is also true in the 3D case. I would love to get more mathematical here, but maybe some time later. • thanks for the reply. In reality, when does the correlation happen? Let's at least talk about a robot that moves in 2D (which the $\Sigma$ in my post represents the covariance matrix for this robot). How might the value of $x$ affect the value of $y$? I don't have problem with the diagonal elements since they clearly represent the uncertainty for each element. Jan 27 '14 at 10:09 • @CroCo I think the example that you are asking for is described in the fourth paragraph of the answer. Jan 28 '14 at 9:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8084374666213989, "perplexity": 223.68646570575132}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588282.80/warc/CC-MAIN-20211028065732-20211028095732-00049.warc.gz"}
http://physics.stackexchange.com/questions/13217/shaping-a-wire-such-that-a-bead-sliding-on-it-has-exactly-isochronous-oscillatio?answertab=active	# Shaping a wire such that a bead sliding on it has exactly isochronous oscillations Let a wire be shaped according to some even function $y=f(x)$, with $f'(0)=0$ and $f''(0)>0$, and let a bead of negligible size slide frictionlessly on the wire. Let the bead oscillate under the influence of gravity about $x=0$ with amplitude $A$ (i.e., between $x=-A$ and $x=+A$) and frequency $\omega(A)$. Clearly $\omega$ is nearly constant for small $A$; it differs from the frequency $\omega_o$ of simple harmonic motion by at most $O(A^2)$. By choosing $f$ to be a fourth-order polynomial, we could presumably adjust the wire's shape so as to eliminate the errors of order $A^2$ and make $\omega(A)$ constant up to $O(A^4)$. Possibly we could continue this process of approximation and make all the derivatives $d^n\omega/dA^n$ vanish up to some finite $n$, or maybe for all $n$. If the derivatives can be made to vanish for all $n$, then I think $\omega$ would have to be nonanalytic at $x=0$. It seems impossible that there is any $f$ such that isochrony holds for arbitrarily large $A$. No matter how steep you make the sides, the bead can't do any better than accelerating downward with acceleration $g$. Therefore I think the best $f$ you can find is probably one that blows up to infinity at $\|x\|$ equal to some $x_{max}$. On dimensional grounds, we would have to have $x_{max}=cL$, where $c$ is a unitless constant and $L=g/\omega_0^2$. So my multipart question is: (1) Is there a function $f$ that gives $d^n\omega/dA^n=0$ for all $n$? If so, ... (2) How is $f$ characterized, and what is $c$? (3) Is $\omega(A)$ analytic at $x=0$, and if so, what is its radius of convergence to its Taylor series in units of $L$? - To (1), there is such a function. I read about it lang ago, Somewhere in connection with my hobby of pendulum (master) clocks. But I do not remember where. :=( – Georg Aug 5 '11 at 19:09	{"extraction_info": {"found_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": 0.9703033566474915, "perplexity": 135.76191168169166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663167.8/warc/CC-MAIN-20140930004103-00267-ip-10-234-18-248.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-equations/166630-find-general-solution.html	Math Help - Find the general solution 1. Find the general solution I'm having trouble finding the general solution guys. Any hints? $x^2-2e^y\frac{dy}{dx}=y^3+e^{x^2}$ 2. Originally Posted by VonNemo19 I'm having trouble finding the general solution guys. Any hints? $x^2-2e^y\frac{dy}{dx}=y^3+e^{x^2}$ Where has the DE come from?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9374459385871887, "perplexity": 1758.6809392668886}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398455135.96/warc/CC-MAIN-20151124205415-00117-ip-10-71-132-137.ec2.internal.warc.gz"}
https://the.dev/posts/number_theory/elementary/units_in_zq/residue_classes/	# ENT - Residue Classes and Modulus ## Introduction This document introduces the basics of residue classes and the modulus. Residue classes are sometimes referred to as congruence classes or equivalence classes. A residue class is a specific example of an equivalence class. ## Division Algorithm The division algorithm states that $\forall a, b \in \mathbb{Z}$ where $b > 0$ $\exists$ unique $q,r \in \mathbb{Z}$ such that $a = bq + r$ and $0\leq r < b$ The important part we will focus on is $0\leq r < b$ . The remainder $r$ is bounded by $b$. Lets observe the remainder when we alternate $a$, and keep $b$ the same. ## Example $b=6$ • $a = 0$ : $0 = 6 \times0 + 0$ ($r=0$) • $a = 1$ : $1 = 6 \times0 + 1$ ($r=1$) • $a = 2$ : $2 = 6 \times0 + 2$ ($r=2$) • $a = 3$ : $3 = 6 \times0 + 3$ ($r=3$) • $a = 4$ : $2 = 6 \times0 + 4$ ($r=4$) • $a = 5$ : $2 = 6 \times0 + 5$ ($r=5$) • $a = 6$ : $6 = 6 \times1 + 0$ ($r=0$) • $a = 7$ : $7 = 6 \times1 + 1$ ($r=1$) • $a = 8$ : $8 = 6 \times1 + 2$ ($r=2$) • $a = 9$ : $9 = 6 \times1 + 3$ ($r=3$) • $a = 10$ : $10 = 6 \times1 + 4$ ($r=4$) • $a = 11$ : $11 = 6 \times1 + 5$ ($r=5$) • $a = 12$ : $12 = 6 \times2 + 0$ ($r=0$) There seems to be a pattern! ### Observe • From the division algorithm, indeed we see that the remainder is bounded above by b which is 6. • We also note that increasing a by 1 increases the remainder by 1, but due to our first observation, it wraps back around to zero once the remainder gets to b. We now define a new operator that will essentially be a shorthand version of the division algorithm, when we care about the remainder. ## Modulo Operator Given two integers $a$ and $b$, we define $a \ mod \ b = r$ where $a = bq + r$ and $0\leq r < b$ . We call b the modulus. Written: Given two integers a and b, we define a $mod$ b to be the remainder when the division algorithm is applied to a and b. ### Observe • $a \ mod \ b$ can give the same value for different a’s. Recall the example above; $1\ mod \ 6 = 7\ mod \ 6$ because the remainder was the same on both of those lines. Residue classes capture this equivalence. ## Residue class To define residue classes, you must first have a modulus. ie. You must first fix b, like in the example above. Okay so b is fixed. Good. • Residue classes are sets. • Two elements $a_0$ , $a_1$ are in the same set if $a_0 \ mod \ b = a_1 \ mod \ b$. ie two elements belong to the same set, if they leave the same remainder when divided by b. Lets fix b to be 6. We denote the residue class which leaves a remainder of $0$ when divided by 6 as $[0]_6$. Note, this is a set; there are other elements that when divided by 6 leave a remainder of $0$. $[0]_6 = \{0,6,12,18,24,30,…\}$ These are just the multiples of 6, we can succinctly write out each residue class for 6 as : $[r]_6$ = $\{ 6k+r : k,r\in \mathbb{Z}\}$ Indeed, if we fix r to be 0 we get all of the multiples of 6, which do indeed leave a remainder of 0 when divided by 6. In general the residue class for a modulus b is can be written as : $[r]_b$ = $\{ bk+r : k,r\in \mathbb{Z}\}$ ### Note on Syntax Although there are many elements that can leave a remainder of 6, we usually choose a representative element. Elaborating: $[0]_6$ = $[6]_6$. We usually denote 0 as the reduced form or the representative for all numbers that leave a remainder of zero when divided by 6. It is only by convention that this is done, you can choose any representative. Furthermore, we usually abuse notation and just use the representative number, so instead of seeing a residue class denoted as: $\mathbb{Z_6} = \{[0]_6,[1]_6,[2]_6,[3]_6,[4]_6,[5]_6\}$ You will see $\mathbb{Z_6} = \{0,1,2,3,4,5\}$ ### Note on congruence syntax Remember we said: $[0]_6 = \{0,6,12,18,24,30,…\}$ From this we can say that $0 \in [0]_6$ and $6 \in [0]_6$ , but there is a more common way to say that $0$ and $6$ belong to the same residue class. We use the congruence operation for this: $6 \equiv 0\ mod\ 6$ This tells us that when the modulus is $6$, $0$ and $6$ belong to the same residue class. $6 \neq 0 \ mod\ 6$ because $0\ mod \ 6$ is the remainder when you divide $0$ by $6$. This is $0$ and $6 \neq 0$ ### Congruence to Divides Let’s link back our definition of congruence to our definition of divides. First suppose, we have two integers a, b and q such that: $a \equiv b\ mod\ q$ This tells us two things: $a = k\times q$ + r and $b = l\times q$ + r ie a and b both leave the same remainder, when divided by q. Since r is present in both equation, we can use substitution to arrive at: $a = k\times q + (b - l\times q)$ Now let’s massage the equation: $a - b = k\times q - l\times q$ $a - b = q(k-l) \implies q\|a-b$ So $a \equiv b\ mod\ q \implies q \| a-b$ Written: a being congruent to b mod q implies that q is a factor of a minus b. #### Example $4 \equiv 44\ mod\ 20 \implies 20 \| 44-4 \implies 20 \| 40$ ### Notes on Prime numbers and Congruence Lets say that $a \equiv b\ mod\ q$ Again we split it up into two equations: $a = k\times q$ + r and $b = l\times q$ + r Now lets impose some restrictions: We make $gcd(a, q) \neq \ 1$ and $b = 1$. Since $gcd(q,a) \neq 1$ then either $q\|a$ or $a\|q$ this means that the remainder will be zero. Our equation now looks like this. $a = k\times q$ + 0 and $1 = l\times q$ + 0 $a = k\times q$ and $1 = l\times q$ Lets focus on $1 = l\times q$. This is only true, when both $l$ and $q$ are 1 or -1. When $q \neq \pm 1$, we have no solution! We immediately have solutions, if we remove the $gcd(a,q) \neq 1$ restriction, therefore: $a \equiv 1\ mod\ q$ has solutions when a and q are coprime. Notice that if $a \equiv 1\ mod\ q$ then any multiple which is also coprime with q, will also have solutions. $10a \equiv 1\ mod\ q$ if 10 is coprime with q. $qa \equiv 1\ mod\ q$ does not have solutions because qa is a multiple of q. This becomes more intuitive when you realise that qa is in the residue class $[0]_q$ which will always be different from the class $[1]_q$ for all possible values of q. #### Why is this important? Let say that $a\times b \equiv 1\ mod\ q$ If the above equation has a solution, we say that b is the multiplicative inverse of a or a is the multiplicative inverse of b. This notion is used a lot in cryptography! #### Example What is the multiplicative inverse of 10 mod 20 ? It does not exist! Because $gcd(10, 20) \neq 1$. What is the multiplicative inverse of 10 mod 7 ? Instead of using $[10]_7$ we use $[3]_7$, as they represent the same elements(all elements which leave a remainder of 3 when divided by 7). Note that since q = 7 is a prime, the gcd will always equal one unless we use a multiple of 7. The equation we are trying to solve is now : $3\times k \equiv 1\ mod\ 7$ As emphasized above, we know a solution exists and we immediately know that k is not a multiple of 7. Because 7 is small, we can check each residue class and see that k = 5 is a solution. The modulus in this case, which was 7, was small so it was easy to check. But in cryptography for example, our modulus can range from 256 bits to 1024 bits or more, depending on the desired security level. So how do we find the multiplicative inverse, when the modulus is large?	{"extraction_info": {"found_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": 0.9348925948143005, "perplexity": 301.29788472803364}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662560022.71/warc/CC-MAIN-20220523163515-20220523193515-00247.warc.gz"}
https://www.physicsforums.com/threads/polar-arc-length.226992/	# Homework Help: Polar Arc length 1. Apr 6, 2008 ### johndoe 1. The problem statement, all variables and given/known data Find the length pf the curve over the given interval. $$r=1+\sin\theta$$ $$0\preceq\theta\preceq\2\pi$$ 3. The attempt at a solution Ok I set it up as: $$2\pi$$ $$\int\sqrt((1+\sin\theta)^2+cos^2\theta)$$ 0 and by simplifying and integrating, I get $$2\pi$$ $$-2\sqrt2[\sqrt(1-\sin\theta)]$$ 0 $$-2\sqrt2[(1-0)-(1-0)] =0$$ and obviously it is wrong, I check the solution it has the same everything but the range , it obviously broke down the whole length into 2 times 1 piece from $$\pi/2 to 3\pi/2$$ and the answer is 8 My question is why I get zero within my range, and why broke it down into the range above? Last edited: Apr 6, 2008 2. Apr 6, 2008 ### Dick You might wish to elaborate about how you got that integral. It's not what I get. You want to do the integral using a double angle formula to write the integrand as a perfect square. It also may help to shift the limits of integration by using sin(x+pi/2)=cos(x). You should also remember that sqrt(x^2)=abs(x) for any expression x. So to integrate something like that you'll need to break into regions where 'x' is positive. 3. Apr 7, 2008 ### johndoe There is some mistake on my first post but anyway I integrate it like this: (ignore the limits for the moment) $$\int\sqrt((1+\sin\theta)^2+cos^2\theta)$$ = $$\sqrt2 \int\sqrt((1+\sin\theta)$$ = $$\sqrt2 \int \frac{\cos\theta}{\sqrt(1-\sin\theta)}$$ (multiply up and down by $$\sqrt(1-\sin\theta)$$ = $$2\sqrt2 [ \sqrt((1-\sin\theta)]$$ (by substitution) which if I use the limit $$\pi/2$$ to $$3\pi/2$$ and times 2 I get the answer, by I still don't understand what decision it based on setting the limits:uhh:, I also tried from 0 to pi and times 2 and it came out to be zero :yuck:, also when I trace on the calculator ( $$\pi/2$$ to $$3\pi/2$$)the x's are all -ve :uhh: clue? So yes why the selected limits Also it will be great if you can demonstrate me a different way of integrating it and what do you mean by to shift the limits of integration by using sin(x+pi/2)=cos(x) Thanks a million~ Last edited: Apr 7, 2008 4. Apr 7, 2008 ### johndoe Ok wait I am up to something, it is because of the behaviour of the sin curve ? for not getting zero I have to use that certain limit? :rofl: http://hk.geocities.com/ymtsang2606/sin.jpg [Broken] Last edited by a moderator: May 3, 2017 5. Apr 7, 2008 ### Dick That's a clever trick. But it's hiding something from you. sqrt(cos(theta)^2)=abs(cos(theta)). As theta goes from 0 to pi, cos changes sign. Integrate from 0 to pi/2, then from pi/2 to pi and add the absolute values. 6. Apr 7, 2008 ### johndoe How do u get to sqrt(cos(theta)^2)? 7. Apr 7, 2008 ### Dick sqrt(1-sin(theta))sqrt(1+sin(theta))=sqrt((1-sin(theta))(1+sin(theta))= sqrt(1-sin(theta)^2)=sqrt(cos(theta)^2). Isn't that what you did? 8. Apr 7, 2008 ### johndoe $$\int\sqrt((1+\sin\theta)^2+cos^2\theta)$$ = $$\sqrt2 \int\sqrt((1+\sin\theta)$$ (then I multiply up and down by $$\sqrt(1-\sin\theta)$$ ) <-- do u mean the sqrt cos^2theta on the top after multiplying? = $$\sqrt2 \int \frac{\cos\theta}{\sqrt(1-\sin\theta)}$$ = $$2\sqrt2[\sqrt((1-\sin\theta)]$$ 9. Apr 7, 2008 ### Dick Yes, I mean the cos(theta)^2 on the top under the square root. 10. Apr 7, 2008 ### johndoe Ok I get it now, so when you integrate you must make sure that the signs of the function would not change through out the limit range , cause otherwise they will offset each other, and given this case :$$\sqrt2 \int \frac{\cos\theta}{\sqrt(1-\sin\theta)}$$ the sqrtcos^2theta on the numerator change signs with the range 0 to pi so you have to break it down in to two parts and integrate, while sin doen't change signs in the denominator within 0 to pi so it doen't matter. 11. Apr 7, 2008 ### BrendanH There's a superfluous integral sign in that post (the last line, where you've already performed the integration)... sorry to nitpick, but it might be confusing for those who are first signing on.	{"extraction_info": {"found_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": 0.9276595711708069, "perplexity": 1213.6421551332714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221210559.6/warc/CC-MAIN-20180816074040-20180816094040-00309.warc.gz"}
http://math.stackexchange.com/questions/273081/problem-4-chapter-2-functional-analysis-rudin	# Problem 4 chapter 2: functional analysis (Rudin) $L^1$, $L^2$: usual Lebesgue spaces on the unit interval. Show that $L^2$ is of the first category (meager) in $L^1$, in three ways: (a) Show that $F_n:=\{f:\int\|f\|^2 \leq n\}$ is closed in $L^1$ but has empty interior. (b)Put $g_n=n$ on $[0,n^{-3}]$, and show that $\int fg_n \rightarrow 0$ $\forall f \in L^2$ , but not for every $f\in L^1$. (c) Note that the inclusion map of $L^2$ into $L^1$ is continuous but not onto. Do the same for $L^p$ and $L^q$ if $p<q$. - What have you tried? Copying an exercise verbatim from a book, with no thoughts of your own is not very helpful. – mrf Jan 8 '13 at 21:08 @mrf, I have tried to solved this problem. Can you give me some idea? – user52523 Jan 8 '13 at 21:14 For each question, there are two sub-questions: 1. Why the assertion is true? and 2. Why does this one give the result? a. 1. Show sequential closeness (as we are in a metric context) by Fatou's lemma. What happens if we assume that $F_n$ has a non-empty interior (in $L^1$)? 2 It's by definition. b. 1. Use the fact that $\{f_n\}$ is bounded in $L^2$ and that simple functions are dense in $L^2$. c. 1. It follows from a well-known integral inequality. Take an integrable function which is not square integrable. 2. There is a related theorem earlier in the book. - Thanks for susgesting ideas. I will try this problem. – user52523 Jan 8 '13 at 21:40 @user52523 Any progress? – Davide Giraudo Jan 10 '13 at 18:13 I have some tried but i still have no idea for a, and c. With b, i solve it in the following way: b, $\int \|fg_n\|\le\alpha\int\|g_n\|$, but we have: $\int g_n=n.1/n^3 \rightarrow 0,$ so is $\int fg_n$ Could you check out for me, please. Thanks you so much. – user52523 Jan 11 '13 at 12:51 I just finished this problem. Thank you so much for helping me. – user52523 Jan 13 '13 at 19:55	{"extraction_info": {"found_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": 0.9642384648323059, "perplexity": 365.70970169391853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997894931.59/warc/CC-MAIN-20140722025814-00141-ip-10-33-131-23.ec2.internal.warc.gz"}
https://proceedings.neurips.cc/paper/2021/hash/412758d043dd247bddea07c7ec558c31-Abstract.html	#### Authors Mufan Li, Mihai Nica, Dan Roy #### Abstract Theoretical results show that neural networks can be approximated by Gaussian processes in the infinite-width limit. However, for fully connected networks, it has been previously shown that for any fixed network width, $n$, the Gaussian approximation gets worse as the network depth, $d$, increases. Given that modern networks are deep, this raises the question of how well modern architectures, like ResNets, are captured by the infinite-width limit. To provide a better approximation, we study ReLU ResNets in the infinite-depth-and-width limit, where \emph{both} depth and width tend to infinity as their ratio, $d/n$, remains constant. In contrast to the Gaussian infinite-width limit, we show theoretically that the network exhibits log-Gaussian behaviour at initialization in the infinite-depth-and-width limit, with parameters depending on the ratio $d/n$. Using Monte Carlo simulations, we demonstrate that even basic properties of standard ResNet architectures are poorly captured by the Gaussian limit, but remarkably well captured by our log-Gaussian limit. Moreover, our analysis reveals that ReLU ResNets at initialization are hypoactivated: fewer than half of the ReLUs are activated. Additionally, we calculate the interlayer correlations, which have the effect of exponentially increasing the variance of the network output. Based on our analysis, we introduce \emph{Balanced ResNets}, a simple architecture modification, which eliminates hypoactivation and interlayer correlations and is more amenable to theoretical analysis.	{"extraction_info": {"found_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": 0.9722741842269897, "perplexity": 979.6302757945053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571597.73/warc/CC-MAIN-20220812075544-20220812105544-00017.warc.gz"}
https://research-information.bris.ac.uk/en/publications/breaking-the-onc-barrier-for-unconditionally-secure-asynchronous-	# Breaking the O(n\|C\|) Barrier for Unconditionally Secure Asynchronous Multiparty Computation Ashish Choudhary Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding) ### Abstract In PODC 2012, Dani et al. presented an unconditionally secure multiparty computation (MPC) protocol, which allows a set of n parties to securely evaluate any arithmetic circuit C of size \|C\| on their private inputs, even in the presence of a computationally unbounded malicious adversary who can corrupt upto t<(13−ϵ)n parties, for any given non-zero ε with 0<ϵ<13. The total circuit-dependent communication complexity of their protocol is O(\ensuremathPolyLog(n)⋅\|C\|), which is a significant improvement over the standard MPC protocols, which has circuit-dependent complexity of the form O(\ensuremathPoly(n)⋅\|C\|). The key innovation in their protocol is that instead of following the standard method of having every party communicate with every other party for evaluating each gate of C, it is sufficient to involve only a small subset of parties of size Θ(PolyLog(n)) to communicate with each other for evaluating each gate of the circuit. The protocol was presented in a synchronous setting and it was left as an open problem to design an asynchronous MPC (AMPC) protocol, with a similar characteristic. In this work, we solve this open problem by presenting the first unconditionally secure AMPC protocol where the circuit dependent complexity is O(\ensuremathPolyLog(n)⋅\|C\|) Original language English Topics in Cryptology - INDOCRYPT 2013 INDOCRYPT Springer Berlin Heidelberg 19-37 19 8250 Published - 2013 ## LSCITS-RPV2: LARGE SCALE COMPLEX IT SYSTEMS INITIATIVE Cliff, D. T. 1/07/071/07/13 Project: Research ## Cite this Choudhary, A. (2013). Breaking the O(n\|C\|) Barrier for Unconditionally Secure Asynchronous Multiparty Computation. In Topics in Cryptology - INDOCRYPT 2013 (Vol. 8250, pp. 19-37). Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007%2F978-3-319-03515-4_2#page-1	{"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": 0.8366711735725403, "perplexity": 2695.1587584208737}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400222515.48/warc/CC-MAIN-20200925053037-20200925083037-00108.warc.gz"}
http://mathoverflow.net/questions/99420/generalising-right-angled-artin-groups	# Generalising right-angled Artin groups An Artin group $G$ is determined by its Coxeter matrix $M$. This is a symmetric $n \times n$ matrix with entries from $\lbrace 2, 3, \ldots, \infty \rbrace$ that determine the relations between the generators of the group. If all of the entries are in $\lbrace 2, \infty \rbrace$ then we say that $G$ is a right-angled Artin group''. Is there a name for an Artin group in which all of the entries of its Coxeter matrix are in $\lbrace 2, 3, \infty \rbrace$ (or alternatively $\lbrace 2, 3 \rbrace$)? Note that, for example, all braid groups are of this type. - These groups are called Artin groups of small type. See, for example, Crisp, John; Paris, Luis, The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. Invent. Math. 145 (2001), no. 1, 19–36. - Thanks, small type refers to the {2,3} case, correct? Is there a similar name for the {2,3,\infty} case? – Mark Bell Jun 14 '12 at 11:19 I would think that it is about $(2,3,\infty)$ case. But I have not looked at Crisp-Paris for quite some time. I remember that they embedded every Artin group into one with small type. – Mark Sapir Jun 14 '12 at 11:57 Actually you are right: small type is for exponents $\\{2,3\\}$. – Mark Sapir Jun 14 '12 at 12:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8967607617378235, "perplexity": 313.2780921978438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122039674.71/warc/CC-MAIN-20150124175359-00067-ip-10-180-212-252.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/83026/largest-possible-volume-of-the-convex-hull-of-a-curve-of-unit-length	# Largest possible volume of the convex hull of a curve of unit length What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$? - If the optimal object is not a round sphere then the answer is not known. (There are few exceptions, but it is almost true.) – Anton Petrunin Dec 9 '11 at 4:14 It is definitely not a sphere. – Vladimir Reshetnikov Dec 9 '11 at 4:22 I nominate a fractional orange slice: A slightly warped semicircular arc and part of a second one in a plane perhaps 120 degrees from the first one, sharing common end points. I leave the details to those with the computational power. Gerhard "Not Ready To Compute Volumes" Paseman, 2011.12.08 – Gerhard Paseman Dec 9 '11 at 5:42 related: mathoverflow.net/questions/26212 – Steve Huntsman Dec 9 '11 at 22:26 @Gerard: I like your answer (an attempt :-). I could apply your way to my attempt below by adjusting the angles of arcs which at this time are set rigidly along being roughly parallel. – Włodzimierz Holsztyński Aug 22 '14 at 19:58 I believe this problem has been mentioned a few times in the literature, and has been solved for certain restrictions on the curve. For example if the curve has no four coplanar points then the maximal volume is achieved by one turn of a circular helix of height $\frac{1}{\sqrt{3}}$ and base radius $\frac{1}{\pi\sqrt{6}}$, this is due to Egervary. For more references see section A28 of "Unsolved problems in geometry" by H.T. Croft, K.J. Falconer, R.K. Guy. Melzak and Schoenberg have treated the corresponding problem for closed loops (Schoenberg has treated even dimensions), and have given answers under similar restrictions. In no case is a complete answer known. - Here is an image of the optimal open convex curve. Taken from Open Problems from CCCG 2012, based on this paper, which cites Nudel'man (1975): Paolo Tilli. "Isoperimetric inequalities for convex hulls and related questions." Trans. Amer. Math. Soc. 362 (2010), 4497-4509. - This is the same as the curve mentioned in my answer, right? I believe the reference goes further back than 2010 :) – Gjergji Zaimi Aug 23 '14 at 9:10 Yes, this is the curve you mention. And you are right about the references: This was proved by Nudel'man in 1975. Tilli says that proving this is optimal among all curves is open as of 2010. – Joseph O'Rourke Aug 23 '14 at 13:02 In the case of a closed curve my first wild (but educated :-) guess would be: connect the following $\ 8\$points (vertices of a cube but also belonging to a sphere) in the given cyclic order (of a maximal shift register): $$(-a\ -\!a\ -\!a)\quad(-a\ -\!a\ +\!a)\quad(-a\ +\!a\ +\!a)\quad(-a\ +\!a\ -\!a)$$ $$(+a\ +\!a\ -\!a)\quad(+a\ +\!a\ +a)\quad(+a\ -\!a\ +\!a)\quad(+a\ -\!a\ -\!a)$$ where $\ a\$ is such that the length of the large arc which connects the consecutive points on the sphere which contains these $\ 8\$ points is $\ \frac18$. EDIT Shooting from the hip cannot be that precise. Thus two related questions may help some: 1. What is a maximal volume of a convex hull of a closed curve contained in a sphere? (Then the respective radius would be of interest). 2. Is there such a curve but not spherical which encloses a larger volume (i.e. its convex hall would be larger than for any closed curve contained in a sphere)? -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8093400597572327, "perplexity": 745.0078289081332}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783394414.43/warc/CC-MAIN-20160624154954-00161-ip-10-164-35-72.ec2.internal.warc.gz"}
https://socratic.org/questions/585288cc7c01495898d765f8	Physics Topics # Question #765f8 Dec 17, 2016 (a) Weight of the block is given as $w = m g$ where $g = 9.81 m {s}^{-} 2$ is acceleration due to gravity. $w = 30 \times 9.81 = 294.3 N$ (b) Pressure $P$ is force per unit area. $P = \frac{w}{\text{Area}}$ From above expression we see that pressure is inversely proportional to the area in contact. For maximum pressure the block must be placed with minimum area in contact with the surface on which pressure is to be calculated. As such minimum area is given by lower two sides. $\therefore {P}_{\text{max}} = \frac{294.3}{0.1 \times 0.4} = 7357.5 N {m}^{-} 2$ Similarly for minimum pressure the block must be placed with maximum area in contact with the surface. Maximum area is given by larger two sides. $\therefore {P}_{\text{min}} = \frac{294.3}{1.5 \times 0.4} = 490.5 N {m}^{-} 2$ ##### Impact of this question 326 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": 0.9259495139122009, "perplexity": 687.9151422515072}, "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-2019-26/segments/1560627998813.71/warc/CC-MAIN-20190618183446-20190618205446-00139.warc.gz"}
https://www.physicsforums.com/threads/thermomystery-entropy-generation-in-a-closed-system.767088/	Thermomystery -entropy generation in a closed system 1. Aug 22, 2014 PatrickAndrews My question relates to entropy generation in a closed system ΔS=dQrev/T for a reversible process ΔS=dQ/T + Sgen for an irreversible process This seems to suggest that Sgen arises because of the irreversibility of the heat transfer process (eg across a finite temperature difference). If, however, we have reversible heat transfer to a system, but also some form of internal friction occurring (eg turbulence), can we say something like ΔS=dQrev/T + Sgen(turbulence) Also, if we have irreversible heat transfer and also frictional irreversibilities, does something like this apply? ΔS=dQ/T + Sgen(finite temperature heat transfer) + Sgen(turbulence) ΔS=Sgen(turbulence) ?? Thanks in advance. I'm finding its surprisingly hard to get a clear view of this apparently simple issue... Last edited: Aug 22, 2014 2. Aug 23, 2014 MrMatt2532 Looks like you on thinking about it correctly. Sgen can be due to internal heat transfer or internal friction (among other things, but these are the most basic I would say). Whenever you have entropy generation you have an irreversible process. So yes, it is correct to say deltaS=deltaQ/T+Sgen(internal friction)+Sgen(internal heat transfer). And for a isolated or an adiabatic system, you have just deltaS=Sgen(internal friction)+Sgen(internal heat transfer). Another thing to note is you don't need turbulence to generate entropy due to friction. It happens for laminar flow as well. Basically, whenever you have gradients of any sort you have entropy generation (velocity gradients, temperature gradients, etc.). 3. Aug 23, 2014 PatrickAndrews I just reread your answer more closely and noticed the 'internal heat transfer' reference. Just to be clear...am I correct in thinking that an (irreversible) process of heat transfer due to temperature difference at the boundary has no effect on Sgen (Sgen is sometimes referred to as including all irreversibilities). Maybe my confusion is due to the fact that 'at the boundary' is technically outside the system? 4. Aug 23, 2014 MrMatt2532 Yes, so the deltaQ in the deltaQ/T term refers to heat transfer at the boundary (i.e. between your control volume and some adjacent control volume), which is the reversible part. Note that in practice it is hard to have heat transfer at the boundary without internal heat heat transfer as well. Though it can be done in theory with an infinitely slow process. 5. Aug 23, 2014 PatrickAndrews "deltaQ at the boundary...which is the reversible part..." That is exactly my problem, because my belief has been that heat transfer at the boundary is irreversible if Texternal<>Tboundary ie heat transfer via a finite temperature difference occurs. ie there is an irreversibility at the boundary (to do with entropy transfer) as well as irreversibilities internally(to do with internal entropy generation, dSgen). I may be making more of a problem here than really exists, but I can't quite understand the fact that dSgen seems not to include the effect of irreversibility at the boundary?? 6. Aug 23, 2014 MrMatt2532 Well part of your issue is that you seem to be thinking about what is happening outside your control volume. You shouldn't even need to consider those things if all you care about is change in entropy inside your control volume. The basic idea is, inside your control volume, you can increase or decrease entropy reversibly by adding or subtracting heat. Or, you you can increase (one way only, since it's irreversible) entropy from the entropy generation terms: internal heat transfer, or internal friction. Assume the universe is made up of two blocks that are in thermal contact and are at different temperatures. From either blocks perspective, the heat transfer is a reversible process. But if you are analyzing both blocks together then what was previously external heat transfer is now internal heat transfer, and we have increase in entropy overall. 7. Aug 23, 2014 PatrickAndrews Wow...thanks for that insight -especially since it was so quick. "From either blocks perspective, the heat transfer is a reversible process" I did not realise that reversibility was relative, believing that the definition of irreversibility for a process was that dS universe >0 It seems to me that... From the cooler block's perspective, dS universe= Q/Tcoolerblock + (-Q/Thotterblock) so dS uni is positive, thus indicating an irreversible process from the perspective of the cooler block. ?? 8. Aug 23, 2014 MrMatt2532 Again, I think you mostly have it but you need to be careful. When you are analyzing the universe (or an isolated system), there is no external heat transfer, and you can only say that change in entropy of the universe (or the isolated system) is due to internal entropy generation. Basically, realize there are three forms of the second law: isolated system (universe): ΔS=Sgen closed system: ΔS=∫δQ/T+Sgen open system: ΔS=∫δQ/T+mass_flow_ins_in-mass_flow_outs_out+Sgen Lets take this two block universe. You can say, for the first block, ΔS1=integral of δQ/T=m1c1ln(Tf/T1). And for the second block, deltaS2=integral of δQ/T=m2c2ln(Tf/T2), where T1 and T2 is the initial temps of the blocks, c is the specific heat, and m is the mass of the blocks and Tf is the final temperature that the system reaches. Now lets say m1=m2 and c1=c2. Then Tf=(T1+T2)/2 and ΔS universe is equal to ΔS1+ΔS2=mc(ln(Tf/T1)+ln(Tf/T2))=mcln((T1+T2)^2/(4T1T2)) Now plug in any two temperatures into the expression ln(((T1+T2)^2/(4T1T2)) and you can see ΔS universe is always positive. 9. Aug 24, 2014 Staff: Mentor MrMatt2532 has given really good answers.:thumbs: For additional perspective, see example 11D.1 in Bird, Stewart, and Lightfoot, Transport Phenomena, Chapter 11. Chet 10. Aug 24, 2014 PatrickAndrews Perspective is the key point Ok, I think I get this now. For a closed (dm/dt=0, but not isolated) system, I've drawn the attached diagram. From the system's perspective, the external temperature(s) from which heat is transferrred into the system is unknown. This means that the dS-system equation is 'unaware of' any (external) irreversibilities to do with finite differences that may be driving heat transfer. Please do let me know if I'm still in error, but this now makes sense -for which I'm hugely grateful. (thanks also for the earlier reminder that all gradients, eg velocity gradients in laminar flow, being important to Sgen). Attached Files: • Sgeneration.png File size: 11.3 KB Views: 158 11. Aug 24, 2014 MrMatt2532 I think your diagram looks ok, except for a somewhat minor point: Instead of ΔS, it should be dS. In this form it will tell you the instantaneous change in entropy in your system. Alternatively, you need integrals on the other side of the equation. In this form it will tell you your change in entropy for your system over some integration time. 12. Aug 24, 2014 PatrickAndrews Thanks for that. Clarity achieved. Cheers 13. Aug 24, 2014 Staff: Mentor The external temperature from which the heat is transferred can be known and measured. This is just the temperature at the interface (boundary) between the system and the surroundings. At this interface, the temperature variation is continuous, and the system temperature matches the surroundings temperature. But, within the system, for an irreversible change, the temperature distribution within the system is not generally uniform, and there can be significant local temperature gradients and heat fluxes. This gives rise to entropy generation. See my PF blog in my PF personal area. Unfortunately, the PF blogs disappear on 9/2. You may also find it of interest to read some of the latter posts in the thread https://www.physicsforums.com/showthread.php?t=750946. This thread has the unusual name, Question About Flow Between Parallel Plates. However, it quickly evolves into an extended discussion of thermo. One of the models Red_CCF and I study in this thread involves a spring-damper analog to gas compression within an insulated cylinder. The spring represents the p-V elastic behavior of the gas, and the damper represents the viscous dissipation provided by the gas. This model is an interesting example to study on its own, because it exhibits the important features of gas behavior, and can be used to study how the dissipative damper affects the total work done. We also apply it to studying a sequence of discrete constant force compressions, and how the total work approaches quasistatic behavior as the total pressure change is held constant while the size of each of the compressions is reduced and the total number of compressions increases. The results of the modeling are very revealing. Chet Last edited: Aug 24, 2014 Similar Discussions: Thermomystery -entropy generation in a closed system	{"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": 0.8475025296211243, "perplexity": 1080.8261409115362}, "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-2017-34/segments/1502886104560.59/warc/CC-MAIN-20170818024629-20170818044629-00029.warc.gz"}
https://www.science.gov/topicpages/p/plasma+waves+observed.html	#### Sample records for plasma waves observed 1. First plasma wave observations at uranus. PubMed Gurnett, D A; Kurth, W S; Scarf, F L; Poynter, R L 1986-07-04 2. First plasma wave observations at uranus SciTech Connect Gurnett, D.A.; Kurth, W.S.; Scarf, F.L.; Poynter, R.L. 1986-07-04 3. Unusual radio and plasma wave phenomena observed in March 1991 NASA Technical Reports Server (NTRS) Reiner, M. J.; Stone, R. G.; Fainberg, J. 1992-01-01 During the intense solar flare activity in March 1991 a number of unusual radio emission and Langmuir wave phenomena were observed by the radio and plasma wave (URAP) experiment on the Ulysses spacecraft. These phenomena were associated with unusual conditions in the interplanetary medium (IPM) presumably resulting from intense solar activity. Some of these URAP observations cannot be explained by mechanisms usually attributed to interplanetary (IP) radio emissions and Langmuir wave activity and require other interpretations. 4. Unusual radio and plasma wave phenomena observed in March 1991 Reiner, M. J.; Stone, R. G.; Fainberg, J. 1992-06-01 During the intense solar flare activity in March 1991 a number of unusual radio emission and Langmuir wave phenomena were observed by the radio and plasma wave (URAP) experiment on the Ulysses spacecraft. These phenomena were associated with unusual conditions in the interplanetary medium (IPM) presumably resulting from intense solar activity. Some of these URAP observations cannot be explained by mechanisms usually attributed to interplanetary (IP) radio emissions and Langmuir wave activity and require other interpretations. 5. High Frequency Plasma Waves Associated With Solar Wind Reconnection Exhausts: WIND/WAVES Observations Huttunen, K. E.; Bale, S. D.; Phan, T. D.; Davis, M.; Gosling, J. T. 2006-12-01 Observations of strong plasma wave activity near reconnection X-line regions in THE laboratory and in the Earth's magnetosphere have suggested that plasma waves may play AN important role in the reconnection process by providing anomalous resistivity through wave-particle interactions and by accelerating electrons. Recent observations of quasi-steady magnetic reconnection in the solar wind introduces an important new environment to study the role of plasma waves in a collisionless plasma associated with the reconnection process. We have used observations by the WIND spacecraft to study high frequency plasma waves associated with 28 solar wind reconnection exhausts. The TNR (Thermal Noise Receiver) experiment included in the WAVES instrument on WIND measures electric spectral density from 4 to 256 kHz and the TDS (Time Domain Sampler) experiment also included in WAVES samples electric field waveforms at rates up to 120,000 samples/s. A large fraction (79%) of the investigated events showed significant enhancements in the wave power around ~ 4 kHz, while only about one third (39%) of the exhausts were associated with intensifications around THE local electron plasma frequency (few tens of kHz). TDS waveform samples revealed three different wave modes: electron solitary waves, ion acoustic waves and Langmuir waves. The intense plasma waves were most frequently observed close to the X-line and near the exhaust boundaries, although wave emissions were commonly observed elsewhere within the exhausts as well 6. Plasma Wave Observations during Ion Gun Experiments DTIC Science & Technology 1990-03-20 Spacecraft Charging by Magnetospheric Plasma , Progress in Aeronautics and Astronautics , Vol. 47, ed. A. Rosen, IAA, pp. 15-30 (1976). 3. H. C. Koons, P. F...AIAA 75-92 (January 20-22, 1975). 2. D. A. McPherson and W. R. Schober, " Spacecraft Charging at High Altitudes: The SCATHA Satellite Program," in...on the AF/NASI P78-2 (SCATHA) satellite were conducted with a plasma /ion source in the inner magnetosphere . These experiments were monitored with 7. Ulysses radio and plasma wave observations in the Jupiter environment NASA Technical Reports Server (NTRS) Stone, R. G.; Pedersen, B. M.; Harvey, C. C.; Canu, P.; Cornilleau-Wehrlin, N.; Desch, M. D.; De Villedary, C.; Fainberg, J.; Farrell, W. M.; Goetz, K. 1992-01-01 The Unified Radio and Plasma Wave (URAP) experiment has produced new observations of the Jupiter environment, owing to the unique capabilities of the instrument and the traversal of high Jovian latitudes. Broad-band continuum radio emission from Jupiter and in situ plasma waves have proved valuable in delineating the magnetospheric boundaries. Simultaneous measurements of electric and magnetic wave fields have yielded new evidence of whistler-mode radiation within the magnetosphere. Observations of auroral-like hiss provided evidence of a Jovian cusp. The source direction and polarization capabilities of URAP have demonstrated that the outer region of the Io plasma torus supported at least five separate radio sources that reoccurred during successive rotations with a measurable corotation lag. Thermal noise measurements of the Io torus densities yielded values in the densest portion that are similar to models suggested on the basis of Voyager observations of 13 years ago. The URAP measurements also suggest complex beaming and polarization characteristics of Jovian radio components. In addition, a new class of kilometer-wavelength striated Jovian bursts has been observed. 8. Electromagnetic ion cyclotron waves observed in the plasma depletion layer NASA Technical Reports Server (NTRS) Anderson, B. J.; Fuselier, S. A.; Murr, D. 1991-01-01 Observations from AMPTE/CCE in the earth's magnetosheath on October 5, 1984 are presented to illustrate 0.1 - 4.0 Hz magnetic field pulsations in the subsolar plasma depletion layer (PDL) for northward sheath field during a magnetospheric compression. The PDL is unambiguously identified by comparing CCE data with data from IRM in the upstream solar wind. Pulsations in the PDL are dominated by transverse waves with F/F(H+) 1.0 or less and a slot in spectral power at F/F(H+) = 0.5. The upper branch is left hand polarized while the lower branch is linearly polarized. In the sheath the proton temperature anisotropy is about 0.6 but it is about 1.7 in the PDL during wave occurrence. The properties and correlation of waves with increased anisotropy indicate that they are electromagnetic ion cyclotron waves. 9. Plasma and wave observations in the night sector of Mars SciTech Connect Nairn, C.M.C.; Grard, R.; Skalsky, A. ); Trotignon, J.G. ) 1991-07-01 The Phobos 2 spacecraft, initially injected into an elliptical orbit around Mars on January 29, 1989, was subsequently transferred, on February 18, 1989, to a nearly circular orbit, close to that of the Phobos moon, with an areocentric radius of the order of 9,600 km. The spacecraft remained in this orbit until the end of the mission, on March 27, 1989. This paper summarizes the plasma and wave observations carried out in the night sector of Mars with the plasma wave system (PWS). Embedded in the magnetic field structure of the Martian tail, cold electron enhancements (tail rays) with densities in the range 10-65 cm{sup {minus}3} are observed in association with broadband wave activity extending from a few hertz up to several kilohertz; these enhancements appear to have characteristics analogous to enhancements observed at Venus. The ion outflow through the Martian eclipse region is estimated from Langmuir probe measurements to be of the order of 10{sup 25} ions/s. 10. Plasma wave phenomena at interplanetary shocks observed by the Ulysses URAP experiment. [Unified Radio and Plasma Waves NASA Technical Reports Server (NTRS) Lengyel-Frey, D.; Macdowall, R. J.; Stone, R. G.; Hoang, S.; Pantellini, F.; Harvey, C.; Mangeney, A.; Kellogg, P.; Thiessen, J.; Canu, P. 1992-01-01 We present Ulysses URAP observations of plasma waves at seven interplanetary shocks detected between approximately 1 and 3 AU. The URAP data allows ready correlation of wave phenomena from .1 Hz to 1 MHz. Wave phenomena observed in the shock vicinity include abrupt changes in the quasi-thermal noise continuum, Langmuir wave activity, ion acoustic noise, whistler waves and low frequency electrostatic waves. We focus on the forward/reverse shock pair of May 27, 1991 to demonstrate the characteristics of the URAP data. 11. Cassini Radio and Plasma Wave Observations at Saturn NASA Technical Reports Server (NTRS) Gurnett, D. A.; Kurth, W. S.; Hospodarsky, G. B.; Persoon, A. M.; Averkamp, T. F.; Ceccni, B.; Lecacheux, A.; Zarka, P.; Canu, P.; Cornilleau-Wehrlin, N. 2005-01-01 Results are presented from the Cassini radio and plasma wave instrument during the approach and first few orbits around Saturn. During the approach the intensity modulation of Saturn Kilometric Radiation (SKR) showed that the radio rotation period of Saturn has increased to 10 hr 45 min plus or minus 36 sec, about 6 min longer than measured by Voyager in 1980-81. Also, many intense impulsive radio signals called Saturn Electrostatic Discharges (SEDs) were detected from saturnian lightning, starting as far as 1.08 AU from Saturn, much farther than terrestrial lightning can be detected from Earth. Some of the SED episodes have been linked to cloud systems observed in Saturn s atmosphere by the Cassini imaging system. Within the magnetosphere plasma wave emissions have been used to construct an electron density profile through the inner region of the magnetosphere. With decreasing radial distance the electron density increases gradually to a peak of about 100 per cubic centimeter near the outer edge of the A ring, and then drops precipitously to values as low as .03 per cubic centimeter over the rings. Numerous nearly monochromatic whistler-mode emissions were observed as the spacecraft passed over the rings that are believed to be produced by meteoroid impacts on the rings. Whistlermode emissions, similar to terrestrial auroral hiss were also observed over the rings, indicating that an electrodynamic interaction, similar to auroral particle acceleration, may be occurring in or near the rings. During the Titan flybys Langmuir probe and plasma wave measurements provided observations of the density and temperature in Titan's ionosphere. 12. Polar cap electron densities from DE 1 plasma wave observations NASA Technical Reports Server (NTRS) Persoon, A. M.; Gurnett, D. A.; Shawhan, S. D. 1983-01-01 13. Lightning and plasma wave observations from the Galileo flyby of Venus NASA Technical Reports Server (NTRS) Gurnett, D. A.; Kurth, W. S.; Roux, A.; Gendrin, R.; Kennel, C. F.; Bolton, S. J. 1991-01-01 Durig the Galileo flyby of Venus the plasma wave instrument was used to search for impulsive radio signals from lightning and to investigate locally generated plasma waves. A total of nine events were detected in the frequency range from 100 kilohertz to 5.6 megahertz. Although the signals are weak, lightning is the only known source of these signals. Near the bow shock two types of locally generated plasma waves were observed, low-frequency electromagnetic waves from about 5 to 50 hertz and electron plasma oscillation at about 45 kilohertz. The plasma oscillations have considerable fine structure, possibly because of the formation of soliton-like wave packets. 14. RF wave observations in beam-plasma discharge NASA Technical Reports Server (NTRS) Bernstein, W. 1986-01-01 The Beam Plasma Discharge (BPD) was produced in the large vacuum chamber at Johnson Space Center (20 x 30 m) using an energetic electron beam of moderately high perveance. A more complete expression of the threshold current I sub c taking into account the pitch angle injection dependence is given. Ambient plasma density inferred from wave measurements under various beam conditions are reported. Maximum frequency of the excited RF band behaves differently than the frequency of the peak amplitude. The latter shows signs of parabolic saturation consistent with the light data. Beam plasma state (pre-BPD or BPD) does not affect the pitch angle dependence. Unexpected strong modulation of the RF spectrum at half odd integer of the electron cyclotron frequency (n + 1/2)f sub ce is reported (5 n 10). Another new feature, the presence of wave emission around 3/2 f sub ce for I sub b is approximate I sub c is reported. 15. Planetary plasma waves NASA Technical Reports Server (NTRS) Gurnett, Donald A. 1993-01-01 The primary types of plasma waves observed in the vicinity of the planets Venus, Mars, Earth, Jupiter, Saturn, Uranus, and Neptune are described. The observations are organized according to the various types of plasma waves observed, ordered according to decreasing distance from the planet, starting from the sunward side of the planet, and ending in the region near the closest approach. The plasma waves observed include: electron plasma oscillations and ion acoustic waves; trapped continuum radiation; electron cyclotron and upper hybrid waves; whistler-mode emissions; electrostatic ion cyclotron waves; and electromagnetic ion cyclotron waves. 16. Wave and plasma observations during a compressional Pc 5 wave event August 10, 1982 NASA Technical Reports Server (NTRS) Engebretson, M. J.; Cahill, L. J., Jr.; Waite, J. H., Jr.; Gallagher, D. L.; Chandler, M. O.; Sugiura, M. 1986-01-01 Magnetometer and thermal plasma instruments on the polar-orbiting Dynamics Explorer 1 satellite observed a small-amplitude ultralow frequency pulsation event at the outer edge of the plasmapause near the geomagnetic equator in the midafternoon sector on August 10, 1982, during the recovery phase of a magnetic storm. Transverse pulsations of 30-50 s period were observed throughout the event, and a 270-s period, purely compressional Pc 5 pulsation with several shifts in phase occurred within + or - 5 deg of the geomagnetic equator. Electric fields and the motion of thermal ions appeared to be in quadrature with pulsations in magnetic field magnitude throughout the event. This suggests that the net Poynting flux for the compressional waves was zero, consistent with their being standing waves. Large fluxes of trapped 90 deg pitch angle 10-eV protons, also symmetric about the geomagnetic equator, were observed in conjunction with the waves. These may serve as a source of free energy for the pulsations. These observations lend support to recent studies suggesting that many dayside compressional wave events are related to localized field line resonance near plasmapauselike boundaries, but also include features that cannot be explained by existing theories. 17. Electrostatic wave observation during a space simulation beam-plasma discharge NASA Technical Reports Server (NTRS) Walker, D. N.; Szuszczewicz, E. P. 1985-01-01 ELF waves which were observed during beam-plasma discharge in the large vacuum chamber at Johnson Space Center are studied. Phase delays as a function of radius (obtained from cross-correlation measurements of density fluctuations) along with measurements of frequency and plasma potential, density, and temperature have been compared to a zero-order slab model of nonlocal azimuthal drift wave propagation. The inferred wave phase velocity in the plasma frame after Doppler correction is found to be near one half the electron diamagnetic drift velocity. Although the measurements presented do not uniquely define a propagation mode, a model of azimuthal drift wave propagation is found to be consistent with observations. 18. PLASMA DIAGNOSTICS OF AN EIT WAVE OBSERVED BY HINODE/EIS AND SDO/AIA SciTech Connect Veronig, A. M.; Kienreich, I. W.; Muhr, N.; Temmer, M.; Goemoery, P.; Vrsnak, B.; Warren, H. P. 2011-12-10 We present plasma diagnostics of an Extreme-Ultraviolet Imaging Telescope (EIT) wave observed with high cadence in Hinode/Extreme-Ultraviolet Imaging Spectrometer (EIS) sit-and-stare spectroscopy and Solar Dynamics Observatory/Atmospheric Imaging Assembly imagery obtained during the HOP-180 observing campaign on 2011 February 16. At the propagating EIT wave front, we observe downward plasma flows in the EIS Fe XII, Fe XIII, and Fe XVI spectral lines (log T Almost-Equal-To 6.1-6.4) with line-of-sight (LOS) velocities up to 20 km s{sup -1}. These redshifts are followed by blueshifts with upward velocities up to -5 km s{sup -1} indicating relaxation of the plasma behind the wave front. During the wave evolution, the downward velocity pulse steepens from a few km s{sup -1} up to 20 km s{sup -1} and subsequently decays, correlated with the relative changes of the line intensities. The expected increase of the plasma densities at the EIT wave front estimated from the observed intensity increase lies within the noise level of our density diagnostics from EIS Fe XIII 202/203 A line ratios. No significant LOS plasma motions are observed in the He II line, suggesting that the wave pulse was not strong enough to perturb the underlying chromosphere. This is consistent with the finding that no H{alpha} Moreton wave was associated with the event. The EIT wave propagating along the EIS slit reveals a strong deceleration of a Almost-Equal-To -540 m s{sup -2} and a start velocity of v{sub 0} Almost-Equal-To 590 km s{sup -1}. These findings are consistent with the passage of a coronal fast-mode MHD wave, pushing the plasma downward and compressing it at the coronal base. 19. A Review of Nonlinear Low Frequency (LF) Wave Observations in Space Plasmas: On the Development of Plasma Turbulence NASA Technical Reports Server (NTRS) Tsurutani, Bruce T. 1995-01-01 As the lead-off presentation for the topic of nonlinear waves and their evolution, we will illustrate some prominent examples of waves in space plasmas. We will describe recent observations detected within planetary foreshocks, near comets and in interplanetary space. It is believed that the nonlinear LF plasma wave features discussed here are part of and may be basic to the development of plasma turbulence. In this sense, this is one area of space plasma physics that is fundamental, with applications to fusion physics and astrophysics as well. It is hoped that the reader(s) will be stimulated to study nonlinear wave development themselves, if he/she is not already involved. 20. Auroral plasma waves NASA Technical Reports Server (NTRS) Gurnett, Donald A. 1989-01-01 A review is given of auroral plasma wave phenomena, starting with the earliest ground-based observations and ending with the most recent satellite observations. Two types of waves are considered, electromagnetic and electrostatic. Electromagnetic waves include auroral kilometric radiation, auroral hiss, ELF noise bands, and low-frequency electric and magnetic noise. Electrostatic waves include upper hybrid resonance emissions, electron cyclotron waves, lower hybrid waves, ion cyclotron waves and broadband electrostatic noise. In each case, a brief overview is given describing the observations, the origin of the instability, and the role of the waves in the physics of the auroral acceleration region. 1. Plasma Waves Related to Solar Wind - Moon Interaction Observed by WFC onboard KAGUYA Kasahara, Y.; Kitaguchi, S.; Kanatani, K.; Goto, Y.; Hashimoto, K.; Omura, Y.; Kumamoto, A.; Ono, T.; Nishino, M. N.; Saito, Y.; Tsunakawa, H. 2010-12-01 The waveform capture (WFC) [1] is one of the subsystems of the Lunar Radar Sounder (LRS) [2,3] on board the KAGUYA spacecraft. The WFC measures two components of electric wave signals detected by the two orthogonal 30 m tip-to-tip antennas from 100Hz to 1MHz. By taking advantage of a moon orbiter, the WFC is expected to measure plasma waves related to solar wind-moon interaction, mini-magnetospheres caused by magnetic anomaly on the lunar surface, and radio emissions to be observed from the moon. Because the moon is basically non-magnetized, the solar wind particles directly hit the lunar surface and a plasma cavity called the “lunar wake” is created behind the moon. Around the terminator of the moon, sudden density decrease derived from local plasma frequency was observed by WFC when the moon was in the solar wind. In addition, because of the difference of thermal speed between ions and electrons, electrons first attempt to refill the cavity, which causes an electric field at the boundary region of the wake and ions are assumed to be accelerated by the DC E-field. The wake boundary, therefore, could be a source region of plasma waves caused by this instability. On the other hand, there are numbers of magnetic anomalies on the lunar surface and it was suggested that a kind of mini-magnetosphere might be constructed as a result of interaction between the solar wind and these magnetic anomalies. According to our plasma wave observation, intense wave activities below several kHz were frequently observed over these magnetic anomalies. It was found that the spatial distribution of plasma wave clearly corresponds to the magnetic anomalies, especially around the South Pole Aitken basin, and also depends on the solar wind parameters; intense wave was observed over magnetic anomalies when the solar wind velocity was slow while wave originated from magnetic anomalies was not clearly recognized when the solar wind velocity was high. It was also found that the wave 2. Mesospheric gravity waves and ionospheric plasma bubbles observed during the COPEX campaign Paulino, I.; Takahashi, H.; Medeiros, A. F.; Wrasse, C. M.; Buriti, R. A.; Sobral, J. H. A.; Gobbi, D. 2011-07-01 During the Conjugate Point Experiment (COPEX) campaign performed at Boa Vista (2.80∘N;60.70∘W, dip angle21.7∘N) from October to December 2002, 15 medium-scale gravity waves in the OHNIR airglow images were observed. Using a Keogram image analysis, we estimate their parameters. Most of the waves propagate to Northwest, indicating that their main sources are Southeast of Boa Vista. Quasi-simultaneous plasma bubble activities in the OI 630 nm images were observed in seven cases. The distances between the bubble depletions have a linear relationship with the wavelengths of the gravity waves observed in the mesosphere, which suggests a direct contribution of the mesospheric medium-scale gravity waves in seeding the equatorial plasma bubbles. 3. Electron plasma waves in the solar wind - AMPTE/IRM and UKS observations NASA Technical Reports Server (NTRS) Treumann, R. A.; Bauer, O. H.; Labelle, J.; Haerendel, G.; Christiansen, P. J. 1986-01-01 Selected events of plasma wave and electromagnetic emissions in the earth's electron fore-shock region have been studied. Strong emissions are observed in the plasma-wave band when the site of the satellite is magnetically connected to the bow shock. These emissions are generally highly fluctuating. Under certain conditions one observes electromagnetic radiation at the second harmonic produced locally. Electromagnetic emission generated at a position far away from the site of the spacecraft is occasionally detected giving rise to remote sensing of the bow shock. These emissions are related to energetic electron fluxes. 4. An Overview of Observations by the Cassini Radio and Plasma Wave Investigation at Earth NASA Technical Reports Server (NTRS) Kurth, W. S.; Hospodarsky, G. B.; Gurnett, D. A.; Kaiser, M. L.; Wahlund, J.-E.; Roux, A.; Canu, P.; Zarka, P.; Tokarev, Y. 2001-01-01 On August 18, 1999, the Cassini spacecraft flew by Earth at an altitude of 1186 km on its way to Saturn. Although the flyby was performed exclusively to provide the spacecraft with sufficient velocity to get to Saturn, the radio and plasma wave science (RPWS) instrument, along with several others, was operated to gain valuable calibration data and to validate the operation of a number of capabilities. In addition, an opportunity to study the terrestrial radio and plasma wave environment with a highly capable instrument on a swift fly-through of the magnetosphere was afforded by the encounter. This paper provides an overview of the RPWS observations, at Earth, including the identification of a number of magnetospheric plasma wave modes, an accurate measurement of the plasma density over a significant portion of the trajectory using the natural wave spectrum in addition to a relaxation sounder and Langmuir probe, the detection of natural and human-produced radio emissions, and the validation of the capability to measure the wave normal angle and Poynting flux of whistler-mode chorus emissions. The results include the observation of a double-banded structure at closest' approach including a band of Cerenkov emission bounded by electron plasma and upper hybrid frequencies and an electron cyclotron harmonic band just above the second harmonic of the electron cyclotron frequency. In the near-Earth plasma sheet, evidence for electron phase space holes is observed, similar to those first reported by Geotail in the magnetotail. The wave normal analysis confirms the Polar result that chorus is generated very close to the magnetic equator and propagates to higher latitudes. The integrated power flux of auroral kilometric radiation is also used to identify a series of substorms observed during the outbound passage through the magnetotail. 5. A statistical study of EMIC waves observed by Cluster: 2. Associated plasma conditions SciTech Connect Allen, R. C.; Zhang, J. -C.; Kistler, L. M.; Spence, H. E.; Lin, R. -L.; Klecker, B.; Dunlop, M. W.; Andre, M.; Jordanova, Vania Koleva 2016-07-01 This is the second in a pair of papers discussing a statistical study of electromagnetic ion cyclotron (EMIC) waves detected during 10 years (2001–2010) of Cluster observations. In the first paper, an analysis of EMIC wave properties (i.e., wave power, polarization, normal angle, and wave propagation angle) is presented in both the magnetic latitude (MLAT)-distance as well as magnetic local time (MLT)-L frames. In addition, this paper focuses on the distribution of EMIC wave-associated plasma conditions as well as two EMIC wave generation proxies (the electron plasma frequency to gyrofrequency ratio proxy and the linear theory proxy) in these same frames. Based on the distributions of hot H+ anisotropy, electron and hot H+ density measurements, hot H+ parallel plasma beta, and the calculated wave generation proxies, three source regions of EMIC waves appear to exist: (1) the well-known overlap between cold plasmaspheric or plume populations with hot anisotropic ring current populations in the postnoon to dusk MLT region; (2) regions all along the dayside magnetosphere at high L shells related to dayside magnetospheric compression and drift shell splitting; and (3) off-equator regions possibly associated with the Shabansky orbits in the dayside magnetosphere. 6. A statistical study of EMIC waves observed by Cluster: 2. Associated plasma conditions DOE PAGES Allen, R. C.; Zhang, J. -C.; Kistler, L. M.; ... 2016-07-01 This is the second in a pair of papers discussing a statistical study of electromagnetic ion cyclotron (EMIC) waves detected during 10 years (2001–2010) of Cluster observations. In the first paper, an analysis of EMIC wave properties (i.e., wave power, polarization, normal angle, and wave propagation angle) is presented in both the magnetic latitude (MLAT)-distance as well as magnetic local time (MLT)-L frames. In addition, this paper focuses on the distribution of EMIC wave-associated plasma conditions as well as two EMIC wave generation proxies (the electron plasma frequency to gyrofrequency ratio proxy and the linear theory proxy) in these samemore » frames. Based on the distributions of hot H+ anisotropy, electron and hot H+ density measurements, hot H+ parallel plasma beta, and the calculated wave generation proxies, three source regions of EMIC waves appear to exist: (1) the well-known overlap between cold plasmaspheric or plume populations with hot anisotropic ring current populations in the postnoon to dusk MLT region; (2) regions all along the dayside magnetosphere at high L shells related to dayside magnetospheric compression and drift shell splitting; and (3) off-equator regions possibly associated with the Shabansky orbits in the dayside magnetosphere.« less 7. THEMIS Observations of Electrostatic Waves in Context with Ion Foreshock Plasma Structures Hull, A. J.; Wilber, M.; Bonnell, J. W.; Mozer, F. S.; Angelopolous, V.; Glassmeier, K. H.; Le Contel, O. 2008-12-01 The terrestrial foreshock region plays an essential role in preprocessing the undisturbed solar wind en route to Earth's bow shock and magnetopause. Such preprocessing involves a rich array of plasma structures observed within the terrestrial ion foreshock, including short-duration large-amplitude magnetic structures (SLAMS), hot flow anomalies (HFAs), foreshock cavities and recently examined density holes. Much work has been done to characterize the macroscopic structure of these objects, as revealed in plasma and DC fields data, and to study higher frequency waves using wave spectra. Little has been done to date to examine high-frequency waveform data in the foreshock, and to place such measurements into context of the sub-structure of the features observed there. Here we present case studies of foreshock electrostatic waves observed by THEMIS, from a few tens to few thousand Hz. The THEMIS/EFI, FGM and SCM instruments provide long duration three-axis measurements of electric and magnetic field waveforms from DC to 8000 Hz, allowing us to assess how these waves are organized within the substructure of various foreshock phenomena. Preliminary analysis indicates large (20-100 mV/m) amplitude oscillatory electrostatic waves from a few tens of Hz to few thousand Hz. Such waves are often but not always observed in association with fine-scale currents embedded within foreshock structures, which is suggestive of different generation mechanisms. These oscillatory waves appear to be consistent with short-scale, ion-acoustic like waves observed in the foreshock reported in the literature. Notably, we also see large amplitude, solitary-like electrostatic waveforms embedded within ion acoustic turbulence, which is suggestive of counter-streaming particles. We discuss the characteristics of the electrostatic waves, such as orientation with respect to magnetic field, wavelength, and relation to fine structure in the magnetic fields and plasma density. 8. Plasma-wave observations at Uranus from Voyager 2. Progress report for period ending February 1986 SciTech Connect Gurnett, D.A.; Kurth, W.S.; Scarf, F.L.; Poynter, R.L. 1986-03-26 Radio emissions from Uranus were detected by the Voyager 2 plasma-wave instrument about 5 days before closest approach at frequencies of 31.1 and 56.2 khz. The bow shock was identified by an abrupt broadband burst of electrostatic turbulence about 10 hours before closest approach at a radial distance of 23.5 ru. Once inside of the magnetosphere, strong whistler mode hiss and chorus emissions were observed at radial distances less than about 8 R/sub u/, in the same region where the energetic-particle instruments detected intense fluxes of energetic electrons. A variety of other plasma waves, such as (f sub c) electron-cyclotron waves, were also observed in this same region. At the ring plane crossing, the plasma wave instrument detected a large number of impulsive events that are interpreted as impacts of micron-sized dust particles on the spacecraft. The maximum impact rate was about 20 to 30 impacts/sec, and the north-south thickness of the impact region was about 4000 km. This paper presents an overview of the principal results from the plasma-wave instrument, starting with the first detection of radio emissions from Uranus, and ending a few days after closest approach. 9. Analysis of Wave and Particle Signatures Observed in Plasma Escape at Venus NASA Technical Reports Server (NTRS) Hartle, R. E.; Grebowsky, J. M.; Intriligator, D. S.; Crider, D. H. 2003-01-01 Atmospheric gases escape from Venus as neutral and ionized atoms and molecules. Ion escape, considered here, occurs through ion pickup or collective plasma processes. The latter can arise from upward flow of nightside ionospheric plasma into the ionotail, day to night ionospheric flow into the ionotail, and scavenging of ionospheric plasma by ionosphere-magnetosheath instabilities at the ionopause. These plasma processes produce differing signatures in ion velocity and energy distributions and in ULF waves in the magnetic field. Using plasma ion spectra measured by the Pioneer Venus Orbiter (PVO) Orbiter Plasma Analyzer (OPA) and magnetic field fluctuations observed by the PVO Orbiter Magnetometer (OMAG) along with the expected particle and field signatures, various ion escape processes occurring along Pioneer Venus orbits are identified. In particular, OPA ion energy distributions are used in parallel with magnetic field power spectra and wave phase angles derived from OMAG measurements to study the characteristics of escaping ions. The principle ions observed escaping the influence of Venus are H+, He+ and 0'. In the ion energy distributions of the OPA, pickup ions appear hot relative to the much cooler ions flowing away from Venus in the ionotail and in the plasma clouds detached from the ionopause. This energy contrast is particularly evident downstream when PVO crosses the ionotail boundary from the hot solar wind plasma to the much cooler plasma within the tail. Magnetic field signatures accompanying the escaping ions appear as peaks in the power spectra at the corresponding ion cyclotron frequencies. Also, coherent wave trains at the same frequencies are observed in the phase angle plots of magnetic field fluctuations about the mean field. 10. Observations of mirror waves and plasma depletion layer upstream of Saturn's magnetopause NASA Technical Reports Server (NTRS) Violante, L.; Cattaneo, M. B. Bavassano; Moreno, G.; Richardson, J. D. 1995-01-01 The two inbound traversals of the Saturn's magnetosheath by Voyagers 1 and 2 have been studied using plasma and magnetic field data. In a great portion of the subsolar magnetosheath, large-amplitude compressional waves are observed at low frequency (approximately 0.1 f(sub p)) in a high-beta plasma regime. The fluctuations of the magnetic field magnitude and ion density are anticorrelated, as are those of the magnetic and thermal pressures. The normals to the structures are almost orthogonal to the background field, and the Doppler ratio is on the average small. Even though the data do not allow the determination of the ion thermal anisotropy, the observations are consistent with values of T(sub perpendicular)/T(sub parallel) greater than 1, producing the onset of the mirror instability. All the above features indicate that the waves should be most probably identified with mirror modes. One of the two magnetopause crossings is of the high-shear type and the above described waves are seen until the magnetopause. The other crossing is of the low-shear type and, similarly to what has been observed at Earth, a plasma depletion occurs close to the magnetopause. In this layer, waves with smaller amplitude, presumably of the mirror mode, are present together with higher-frequency waves showing a transverse component. 11. Electron distributions observed with Langmuir waves in the plasma sheet boundary layer SciTech Connect Hwang, Junga; Rha, Kicheol; Seough, Jungjoon; Yoon, Peter H. 2014-09-15 The present paper investigates the Langmuir turbulence driven by counter-streaming electron beams and its plausible association with observed features in the Earth's plasma sheet boundary layer region. A one-dimensional electrostatic particle-in-cell simulation code is employed in order to simulate broadband electrostatic waves with characteristic frequency in the vicinity of the electron plasma frequency ω/ω{sub pe}≃1.0. The present simulation confirms that the broadband electrostatic waves may indeed be generated by the counter-streaming electron beams. It is also found that the observed feature associated with low energy electrons, namely quasi-symmetric velocity space plateaus, are replicated according to the present simulation. However, the present investigation only partially succeeds in generating the suprathermal tails such that the origin of observed quasi power-law energetic population formation remains outstanding. 12. Observation of nonlinear wave decay processes in the solar wind by the AMPTE IRM plasma wave experiment NASA Technical Reports Server (NTRS) Koons, H. C.; Roeder, J. L.; Bauer, O. H.; Haerendel, G.; Treumann, R. 1987-01-01 Nonlinear wave decay processes have been detected in the solar wind by the plasma wave experiment aboard the Active Magnetospheric Particle Tracer Explorers (AMPTE) IRM spacecraft. The main process is the generation of ultralow-frequency ion acoustic waves from the decay of Langmuir waves near the electron plasma frequency. Frequently, this is accompanied by an enhancement of emissions near twice the plasma frequency. This enhancement is most likely due to the generation of electromagnetic waves from the coalescence of two Langmuir waves. These processes occur within the electron foreshock in front of the earth's bow shock. 13. Experimental Observation of the Blob-Generation Mechanism from Interchange Waves in a Plasma SciTech Connect Furno, I.; Labit, B.; Podesta, M.; Fasoli, A.; Poli, F. M.; Ricci, P.; Theiler, C.; Brunner, S.; Diallo, A.; Graves, J.; Mueller, S. H. 2008-02-08 The mechanism for blob generation in a toroidal magnetized plasma is investigated using time-resolved measurements of two-dimensional structures of electron density, temperature, and plasma potential. The blobs are observed to form from a radially elongated structure that is sheared off by the ExB flow. The structure is generated by an interchange wave that increases in amplitude and extends radially in response to a decrease of the radial pressure scale length. The dependence of the blob amplitude upon the pressure radial scale length is discussed. 14. Electrostatic solitary waves and plasma environment near the moon observed by KAGUYA Hashimoto, K.; Omura, Y.; Kasahara, Y.; Kojima, H.; Saito, Y.; Nishino, M. N.; Ono, T.; Tsunakawa, H. 2012-12-01 WFC-L subsystem[1] of KAGUYA (SELENE)/LRS[2], observes waveforms of plasma waves in 100Hz-100kHz and a lot of electrostatic solitary waves (ESWs) have been observed[3]. Although the orthogonal dipole antennas are generally used in the observations, sometimes a pair of monopole antennas were used. We analyze the ESW and the plasma environment around the observed regions. Observed waveforms are fitted to ideal ESW waveforms parallel to the magnetic field and the perpendicular component. The propagation velocities and the potential scales are also evaluated in the case of the monopole observations. Particle data by PACE[4] are also evaluated near the regions where ESW's are observed like, in the solar wind, above the magnetic anomalies, in the wake boundaries, and inside the wake. The ESWs, the plasma environments, the magnetic fields, and their relations will be discussed. References [1] Y. Kasahara, et al., Earth, Planets and Space, 60, 341-351, 2008. [2] T. Ono, et al., Space Science Reviews, 154, Nos. 1-4, 145-192, DOI:10.1007/s11214-010-9673-8, 2010 [3] K. Hashimoto, et al., Geophys. Res. Lett., 37, L19204, doi:10.1029/2010GL044529, 2010. [4] Y. Saito, et al., Space Science Reviews, Vol. 154, No. 1-4, 265-303, 2010. 15. Direct observation of the two-plasmon-decay common plasma wave using ultraviolet Thomson scattering. PubMed Follett, R K; Edgell, D H; Henchen, R J; Hu, S X; Katz, J; Michel, D T; Myatt, J F; Shaw, J; Froula, D H 2015-03-01 A 263-nm Thomson-scattering beam was used to directly probe two-plasmon-decay (TPD) excited electron plasma waves (EPWs) driven by between two and five 351-nm beams on the OMEGA Laser System. The amplitude of these waves was nearly independent of the number of drive beams at constant overlapped intensity, showing that the observed EPWs are common to the multiple beams. In an experimental configuration where the Thomson-scattering diagnostic was not wave matched to the common TPD EPWs, a broad spectrum of TPD-driven EPWs was observed, indicative of nonlinear effects associated with TPD saturation. Electron plasma waves corresponding to Langmuir decay of TPD EPWs were observed in both Thomson-scattering spectra, suggesting the Langmuir decay instability as a TPD saturation mechanism. Simulated Thomson-scattering spectra from three-dimensional numerical solutions of the extended Zakharov equations of TPD are in excellent agreement with the experimental spectra and verify the presence of the Langmuir decay instability. 16. Plasma waves observed in the near vicinity of the space shuttle SciTech Connect Cairns, I.H.; Gurnett, D.A. ) 1991-08-01 The OSS 1 and Spacelab 2 missions found intense broadband waves in the near vicinity of the space shuttle. This paper contains a detailed observational characterizaiton and theoretical investigation of the plasma waves observed within about 10 m of the space shuttle during the XPOP roll period of theSpacelab 2 mission. High wave levels are found from 31 Hz to 10 kHz (near the lower hybrid frequency). Above 10 kHz the wave levels decrease with frequency, reaching the background level near 56 kHz. The frequency distribution of wave electric is best interpreted in terms of three components below about 10 kHz and a high-frequency tail. The primary component is a fairly uniform, high level of waves covering the frequency range from 31 Hz to 10 kHz. The two superposed components in this frequency range have electric fields of order twice the uniform level. The second component corresponds to a low-frequency peak in the range 100-178 Hz. The third component is found near, and follows the trend of, the lower hybrid frequency. The waves show a pronounced amplitude and frequency variation with the quantity V{sub parallel}/V{sub T} {approximately} 1 and the shuttle is moving primarily along the magnetic field. This implies that the waves are probably driven by water pickup ions. A new theory involving Doppler-shifted lower hybrid waves driven by beamlike distributions of water ions near the space shuttle is developed using linear theory. 17. Titan's induced magnetosphere from plasma wave, particle data and magnetometer observations Modolo, R.; Romanelli, N.; Canu, P.; Coates, A. J.; Berthelier, J.; Bertucci, C.; Leblanc, F.; Piberne, R.; Edberg, N. J.; Kurth, W. S.; Gurnett, D. A.; Wahlund, J. 2013-12-01 The Magnetometer (MAG) measurements, the particle data (CAPS) are combined with the Radio and Plasma Wave Science (RPWS) observations to provide an overall and organized description of the electron plasma environment and the pickup ion distribution around Titan. RPWS observations are used to measure the electron number density of the thermal plasma close to Titan. This data set is combined with CAPS-ELS electron number density in Saturn's magnetosphere and Titan's environment. A relatively good correspondence between the number density estimated from CAPS-ELS and RPWS are most of the time observed between 0.1 - 1 cm-3. Combining both ELS and RPWS data allows deducing a continuous electron density profile going from Saturn's magnetosphere to Titan's ionosphere leading to a global electron density map in Titan's vicinity. The MAG observations are used to derive information about the ambient magnetic field environment in the vicinity of Titan and also to emphasize the bipolar tail region. Ion information such the mass composition of the plasma and ion distribution function for specific time intervals are determined from CAPS-IMS. Pick-up ions have been identified from their energy signature and mass composition for few flybys. These observations also emphasized a ring distribution, characteristic of pick-up ions. The pick-up observations, in the DRAP coordinate system, are found to be located in the +E=-vxB hemisphere as expected. 18. Solar system plasma waves NASA Technical Reports Server (NTRS) Gurnett, Donald A. 1995-01-01 An overview is given of spacecraft observations of plasma waves in the solar system. In situ measurements of plasma phenomena have now been obtained at all of the planets except Mercury and Pluto, and in the interplanetary medium at heliocentric radial distances ranging from 0.29 to 58 AU. To illustrate the range of phenomena involved, we discuss plasma waves in three regions of physical interest: (1) planetary radiation belts, (2) planetary auroral acceleration regions and (3) the solar wind. In each region we describe examples of plasma waves that are of some importance, either due to the role they play in determining the physical properties of the plasma, or to the unique mechanism involved in their generation. 19. Polar Plasma Wave Observations in the Auroral Region and Polar Cap NASA Technical Reports Server (NTRS) Menietti, J. D.; Averkamp, T. F.; Kirchner, D. L.; Pickett, J. S.; Persoon, A. M.; Gurnett, D. A. 1998-01-01 Auroral kilometric radiation (AKR), sometimes associated with auroral myriametric radiation (AMR), has been observed by the plasma wave instrument on board Polar on almost every northern hemisphere pass. High spectral resolution plots of the AKR obtained by the wide-band receiver of the plasma wave instrument on board the spacecraft often show discrete, negative-slope striations each extending over a period of several seconds. A preliminary survey of over 4000 spectrograms (each for 48 seconds of data) indicates that the striations are seen in the northern hemisphere near apogee about 5% of the time. The frequency range is 40 kHz less than f less than 100 kHz, but a few observations of signatures have been made at higher frequency (f less than 225 khz. The frequency drift rates R, are similar ranging from -9.0 kHz/sec less than R less than -1.0 kHz/sec. No data is currently available for perigee (southern hemisphere) passes. The paucity of positive-slope features may be due to the location of the satellite at altitudes well above the AKR source region. Past studies have suggested these features are due to AKR wave growth stimulated by the propagation of electromagnetic ion cyclotron waves travelling up (-R) or down (+R) the field line, through the source region. High-resolution waveform data from both Polar and FAST show the presence of solitary waves in the auroral region which may also be a source of these striations. AMR is seen as diffuse emission associated with, but at lower frequency than the lower AKR. Direction finding of these emissions is not conclusive, but for one case, they have a source region distinct from the magnetic field line containing the AKR source, but possibly associated with the auroral cavity density gradient. 20. Plasma and wave properties downstream of Martian bow shock: Hybrid simulations and MAVEN observations Dong, Chuanfei; Winske, Dan; Cowee, Misa; Bougher, Stephen W.; Andersson, Laila; Connerney, Jack; Epley, Jared; Ergun, Robert; McFadden, James P.; Ma, Yingjuan; Toth, Gabor; Curry, Shannon; Nagy, Andrew; Jakosky, Bruce 2015-04-01 Two-dimensional hybrid simulation codes are employed to investigate the kinetic properties of plasmas and waves downstream of the Martian bow shock. The simulations are two-dimensional in space but three dimensional in field and velocity components. Simulations show that ion cyclotron waves are generated by temperature anisotropy resulting from the reflected protons around the Martian bow shock. These proton cyclotron waves could propagate downward into the Martian ionosphere and are expected to heat the O+ layer peaked from 250 to 300 km due to the wave-particle interaction. The proton cyclotron wave heating is anticipated to be a significant source of energy into the thermosphere, which impacts atmospheric escape rates. The simulation results show that the specific dayside heating altitude depends on the Martian crustal field orientations, solar cycles and seasonal variations since both the cyclotron resonance condition and the non/sub-resonant stochastic heating threshold depend on the ambient magnetic field strength. The dayside magnetic field profiles for different crustal field orientation, solar cycle and seasonal variations are adopted from the BATS-R-US Mars multi-fluid MHD model. The simulation results, however, show that the heating of O+ via proton cyclotron wave resonant interaction is not likely in the relatively weak crustal field region, based on our simplified model. This indicates that either the drift motion resulted from the transport of ionospheric O+, or the non/sub-resonant stochastic heating mechanism are important to explain the heating of Martian O+ layer. We will investigate this further by comparing the simulation results with the available MAVEN data. These simulated ion cyclotron waves are important to explain the heating of Martian O+ layer and have significant implications for future observations. 1. Jupiter Data Analysis Program: Analysis of Voyager wideband plasma wave observations NASA Technical Reports Server (NTRS) Kurth, W. S. 1983-01-01 Voyager plasma wave wideband frames from the Jovian encounters are analyzed. The 511 frames which were analyzed were chosen on the basis of low-rate spectrum analyzer data from the plasma wave receiver. These frames were obtained in regions and during times of various types of plasma or radio wave activity as determined by the low-rate, low-resolution data and were processed in order to provide high resolution measurements of the plasma wave spectrum for use in the study of a number of outstanding problems. Chorus emissions at Jupiter were analyzed. The detailed temporal and spectral form of the very complex chorus emissions near L = 8 on the Voyager 1 inbound passage was compared to both terrestrial chorus emissions as well as to the theory which was developed to explain the terrestrial waves. 2. Simultaneous plasma wave and electron flux observations upstream of the Martian bow shock Skalsky, A.; Grard, R.; Kiraly, P.; Klimov, S.; Kopanyi, V.; Schwingenschuh, K.; Trotignon, J. G. 1993-03-01 Flux enhancements of electrons with energies between 100 and 530 eV are observed simultaneously with electron plasma waves in the upstream region of the Martian bow shock. The electron flux appears to reach its maximum when the pitch angle is close to 0 deg, which corresponds to particles reflected from the shock region and backstreaming in the solar wind along the magnetic field. The correlation between high-frequency waves and enhanced electron fluxes is reminiscent of several studies on the electron foreshock of the Earth. Such a similarity indicates that, in spite of major differences between the global shock structures, the microscopic processes operating in the foreshocks of Earth and Mars are probably identical. 3. Magnetoresistive waves in plasmas Felber, F. S.; Hunter, R. O., Jr.; Pereira, N. R.; Tajima, T. 1982-10-01 The self-generated magnetic field of a current diffusing into a plasma between conductors can magnetically insulate the plasma. Propagation of magnetoresistive waves in plasmas is analyzed. Applications to plasma opening switches are discussed. 4. Coordinated radar observations of plasma wave characteristics in the auroral F region Makarevich, R. A.; Bristow, W. A. 2014-07-01 Properties of decameter-scale plasma waves in the auroral F region are investigated using coordinated observations of plasma wave characteristics with the Kodiak HF coherent radar (KOD) and Poker Flat Incoherent Scatter Radar (PFISR) systems in the Alaskan sector. We analyze one event on 14 November 2012 that occurred during the first PFISR Ion-Neutral Observations in the Thermosphere (PINOT) campaign when exceptionally good F region backscatter data at 1 s resolution were collected by KOD over the wide range of locations also monitored by PFISR. In particular, both radar systems were observing continuously along the same magnetic meridian, which allowed for a detailed comparison between the line-of-sight (l-o-s) velocity data sets. It is shown that l-o-s velocity correlation for data points strictly matched in time (within 1 s) depends strongly on the number of ionospheric echoes detected by KOD in a given post-integration interval or, equivalently, on the KOD echo occurrence in that interval. The l-o-s velocity correlations reach 0.7-0.9 for echo occurrences exceeding 70%, while also showing considerable correlations of 0.5-0.6 for occurrences as low as 10%. Using the same approach of strictly matching the KOD and PFISR data points, factors controlling coherent echo power are investigated, focusing on the electric field and electron density dependencies. It is demonstrated that the signal-to-noise ratio (SNR) of F region echoes increases nearly monotonically with an increasing electric field strength as well as with an increasing electron density, except at large density values, where SNR drops significantly. The electric field control can be understood in terms of the growth rate of the gradient-drift waves being proportional to the convection drift speed under conditions of fast-changing convection flows, while the density effect may involve over-refraction at large density values and radar backscatter power proportionality to the perturbation density. 5. Observations of plasma waves in the solar wind interaction region of Comet Giacobini-Zinner at high time resolution NASA Technical Reports Server (NTRS) Moses, S. L.; Coroniti, F. V.; Greenstadt, E. W.; Tsurutani, B. T. 1992-01-01 High-time-resolution spectra of plasma wave emissions detected in the interaction region of Comet Giacobini-Zinner with the solar wind reveal a wave phenomenology much more complicated than first reported. Spectra often exhibit three or more independent peaks, which become more prominent the deeper into the interaction region the spacecraft traversed. The main peaks correspond to whistler emissions below the electron cyclotron frequency, a midfrequency peak near the maximum Doppler shift frequency for waves with k lambda(D) = 1, a high-frequency peak above the Doppler shift maximum frequency, and electron plasma oscillations at the plasma frequency. Similar multipeaked spectra are also observed downstream from weak shocks at Earth, which suggests that the plasma wave generation mechanisms responsible need not require particle populations created by photoionization. 6. Radio and Plasma Wave Observations at Saturn from Cassini's Approach and First Orbit NASA Technical Reports Server (NTRS) Gurnett, D. A.; Kurth, W. S.; Haspodarsky, G. B.; Persoon, A. M.; Averkamp, T. F.; Cecconi, B.; Lecacheux, A.; Zarka, P.; Canu, P.; Cornilleau-Wehrlin, N. 2005-01-01 We report data from the Cassini radio and plasma wave instrument during the approach and first orbit at Saturn. During the approach, radio emissions from Saturn showed that the radio rotation period is now 10 hours 45 minutes 45 k 36 seconds, about 6 minutes longer than measured by Voyager in 1980 to 1981. In addition, many intense impulsive radio signals were detected from Saturn lightning during the approach and first orbit. Some of these have been linked to storm systems observed by the Cassini imaging instrument. Within the magnetosphere, whistler-mode auroral hiss emissions were observed near the rings, suggesting that a strong electrodynamic interaction is occurring in or near the rings. 7. Observation of multiple mechanisms for stimulating ion waves in ignition scale plasmas. Revision 1 SciTech Connect Kirkwood, R.K.; MacGowan, B.J.; Montgomery, D.S. 1997-03-03 The laser and plasma conditions expected in ignition experiments using indirect drive inertial confinement have been studied experimentally. It has been shown that there are at least three ways in which ion waves can be stimulated in these plasmas and have significant effect on the energy balance and distribution in the target. First ion waves can be stimulated by a single laser beam by the process of Stimulated Brillouin Scattering (SBS) in which an ion acoustic and a scattered electromagnetic wave grow from noise. Second, in a plasma where more than one beam intersect, ion waves can Lie excited at the beat frequency and wave number of the intersecting beams,, causing the side scatter instability to be seeded, and substantial energy to be transferred between the beams [R. K. Kirkwood et. al. Phys. Rev. Lett. 76, 2065 (1996)]. And third, ion waves may be stimulated by the decay of electron plasma waves produced by Stimulated Raman Scattering (SRS), thereby inhibiting the SRS process [R. K. Kirkwood et. al. Phys. Rev. Lett. 77, 2706 (1996)]. 8. Undamped electrostatic plasma waves SciTech Connect Valentini, F.; Perrone, D.; Veltri, P.; Califano, F.; Pegoraro, F.; Morrison, P. J.; O'Neil, T. M. 2012-09-15 Electrostatic waves in a collision-free unmagnetized plasma of electrons with fixed ions are investigated for electron equilibrium velocity distribution functions that deviate slightly from Maxwellian. Of interest are undamped waves that are the small amplitude limit of nonlinear excitations, such as electron acoustic waves (EAWs). A deviation consisting of a small plateau, a region with zero velocity derivative over a width that is a very small fraction of the electron thermal speed, is shown to give rise to new undamped modes, which here are named corner modes. The presence of the plateau turns off Landau damping and allows oscillations with phase speeds within the plateau. These undamped waves are obtained in a wide region of the (k,{omega}{sub R}) plane ({omega}{sub R} being the real part of the wave frequency and k the wavenumber), away from the well-known 'thumb curve' for Langmuir waves and EAWs based on the Maxwellian. Results of nonlinear Vlasov-Poisson simulations that corroborate the existence of these modes are described. It is also shown that deviations caused by fattening the tail of the distribution shift roots off of the thumb curve toward lower k-values and chopping the tail shifts them toward higher k-values. In addition, a rule of thumb is obtained for assessing how the existence of a plateau shifts roots off of the thumb curve. Suggestions are made for interpreting experimental observations of electrostatic waves, such as recent ones in nonneutral plasmas. 9. Plasma waves near the magnetopause NASA Technical Reports Server (NTRS) Anderson, R. R.; Eastman, T. E.; Harvey, C. C.; Hoppe, M. M.; Tsurutani, B. T.; Etcheto, J. 1982-01-01 Plasma waves associated with the magnetosphere from the magnetosheath to the outer magnetosphere are investigated to obtain a clear definition of the boundaries and regions, to characterize the waves observed in these regions, to determine which wave modes are present, and to determine their origin. Emphasis is on high time resolution data and a comparison between measurements by different antenna systems. It is shown that the magnetosheath flux transfer events, the magnetopause current layer, the outer magnetosphere, and the boundary layer can be identified by their magnetic field and plasma wave characteristics, as well as by their plasma and energetic particle signatures. The plasma wave characteristics in the current layer and in the boundary layer are very similar to the features in the flux transfer events, and upon entry into their outer magnetosphere, the plasma wave spectra are dominated by intense electromagnetic chorus bursts and electrostatic emissions. 10. Plasma waves observed by the IRM and UKS spacecraft during the AMPTE solar wind lithium releases - Overview NASA Technical Reports Server (NTRS) Haeusler, B.; Woolliscroft, L. J.; Anderson, R. R.; Gurnett, D. A.; Holzworth, R. H. 1986-01-01 The wave measurements from the Ion Release Module and the United Kingdom Satellite in the diamagnetic cavity, the transition region, and the upstream region are examined. Solar wind conditions during the releases on September 11 and 20, 1984 are described. The quasi-static electric field, wideband, high-frequency waves, and medium and VLF waves observations are analyzed. The data reveal that extremely low levels of wave activity are observed in the boundary between the diamagnetic cavity and external magnetic field, medium and VLF waves in the ion acoustic electrostatic cyclotron harmonic modes are detected in the transition region from the diamagnetic cavity to the solar wind, and decay in the magnetic field strength and density, and an increase in the quasi-static electric field is seen in the upstream edge of the transition region. The emissions observed are related to the different phases of the Li cloud development and different spatial regimes of the Li plasma-solar wind interaction. 11. Plasma wave observation using waveform capture in the Lunar Radar Sounder on board the SELENE spacecraft Kasahara, Yoshiya; Goto, Yoshitaka; Hashimoto, Kozo; Imachi, Tomohiko; Kumamoto, Atsushi; Ono, Takayuki; Matsumoto, Hiroshi 2008-04-01 The waveform capture (WFC) instrument is one of the subsystems of the Lunar Radar Sounder (LRS) on board the SELENE spacecraft. By taking advantage of a moon orbiter, the WFC is expected to measure plasma waves and radio emissions that are generated around the moon and/or that originated from the sun and from the earth and other planets. It is a high-performance and multifunctional software receiver in which most functions are realized by the onboard software implemented in a digital signal processor (DSP). The WFC consists of a fast-sweep frequency analyzer (WFC-H) covering the frequency range from 1 kHz to 1 MHz and a waveform receiver (WFC-L) in the frequency range from 10 Hz to 100 kHz. By introducing the hybrid IC called PDC in the WFC-H, we created a spectral analyzer with a very high time and frequency resolution. In addition, new techniques such as digital filtering, automatic filter selection, and data compression are implemented for data processing of the WFC-L to extract the important data adequately under the severe restriction of total amount of telemetry data. Because of the flexibility of the instruments, various kinds of observation modes can be achieved, and we expect the WFC to generate many interesting data. 12. Structure of the plasmapause from ISEE 1 low-energy ion and plasma wave observations NASA Technical Reports Server (NTRS) Nagai, T.; Horwitz, J. L.; Anderson, R. R.; Chappell, C. R. 1985-01-01 Low-energy ion pitch angle distributions are compared with plasma density profiles in the near-earth magnetosphere using ISEE 1 observations. The classical plasmapause determined by the sharp density gradient is not always observed in the dayside region, whereas there almost always exists the ion pitch angle distribution transition from cold, isotropic to warm, bidirectional, field-aligned distributions. In the nightside region the plasmapause density gradient is typically found, and it normally coincides with the ion pitch angle distribution transition. The sunward motion of the plasma is found in the outer part of the 'plasmaspheric' plasma in the dusk bulge region. 13. Similar data retrieval from enormous datasets on plasma wave spectrum observed by solar-terrestrial satellites Kasahara, Y.; Zhang, F.; Goto, Y. 2012-12-01 As the total amount of data measured by scientific spacecraft is drastically increasing in recent years, it is necessary for researchers to develop new computation methods for efficient analysis of these enormous datasets because it is almost impossible to survey all datasets manually. In the present study, we propose a new algorithm for similar data retrieval. Our aim is to develop a new computational technique to discover interesting and/or epoch-making datasets from enormous datasets [1]. There are two issues to be solved for similar data retrieval. One is to develop a general method which can be applied to different types of satellite data, and the other is to improve the efficiency of the retrieval method in the viewpoints of accuracy and speedup of retrieval. We first discuss key descriptors that represent characteristics of the VLF/ELF waves observed by solar-terrestrial satellites such as KAGUYA and Akebono spacecraft. Second we introduce a detailed algorithm for similar data retrieval applied to the system. Faced with a large number of observation data, retrieval using walkthroughs is much time-consuming. In order to search appropriate datasets similar to the reference data within a finite computation time, we adopted the following two step search algorithm. In the first step, we adopted a multi-dimension index structure called the SR-tree (Sphere/Rectangle-tree) in order to pick up appropriate candidates for similar datasets rapidly matching a set of key descriptors at the central point (central grid) of a "test sub-region" with the set of key descriptors at the central point of the "reference sub-region." In the second step (Step 2), the root mean square error between each candidate for the test sub-region and the reference sub-region is calculated one by one. Finally, a list of datasets, which include spectra similar to the requested reference spectrum, is presented to the user. We applied the proposed method to plasma wave spectrum measured by WFC 14. Observation of beat oscillation generation by coupled waves associated with parametric decay during radio frequency wave heating of a spherical tokamak plasma. PubMed Nagashima, Yoshihiko; Oosako, Takuya; Takase, Yuichi; Ejiri, Akira; Watanabe, Osamu; Kobayashi, Hiroaki; Adachi, Yuuki; Tojo, Hiroshi; Yamaguchi, Takashi; Kurashina, Hiroki; Yamada, Kotaro; An, Byung Il; Kasahara, Hiroshi; Shimpo, Fujio; Kumazawa, Ryuhei; Hayashi, Hiroyuki; Matsuzawa, Haduki; Hiratsuka, Junichi; Hanashima, Kentaro; Kakuda, Hidetoshi; Sakamoto, Takuya; Wakatsuki, Takuma 2010-06-18 We present an observation of beat oscillation generation by coupled modes associated with parametric decay instability (PDI) during radio frequency (rf) wave heating experiments on the Tokyo Spherical Tokamak-2. Nearly identical PDI spectra, which are characterized by the coexistence of the rf pump wave, the lower-sideband wave, and the low-frequency oscillation in the ion-cyclotron range of frequency, are observed at various locations in the edge plasma. A bispectral power analysis was used to experimentally discriminate beat oscillation from the resonant mode for the first time. The pump and lower-sideband waves have resonant mode components, while the low-frequency oscillation is exclusively excited by nonlinear coupling of the pump and lower-sideband waves. Newly discovered nonlocal transport channels in spectral space and in real space via PDI are described. 15. Plasma waves produced by an ion beam: Observations by the VLF experiment on Porcupine Jones, D. 1980-06-01 Results are presented from the VLF electric field experiments flown on Porcupine flights F3 and F4, which also had ejectable xenon ion sources. The xenon ion beam was found to produce plasma instabilities whose frequencies could be linked to the local proton gyrofrequency. The main energy in the instabilities lies at 3kHz for events when the Xe+ source is close to the rocket, and at 7kHz when the source is farther away. Theory predicts that these frequencies should be the lower-hybrid-resonance and this implies that Xe+ is the dominant ion in the first case and that it is the ambient plasma that dominates later. There is no discernable antenna spin-modulation during the Xe events which indicates that the wave k-vectors are not unidirectional. A theory is cited based on the setting up of the proton cyclotron harmonic waves by the Xe+ or 0+ cyclotron harmonic waves. The second Xe+ event on both flights exhibited an, as yet, unexplained harmonic structure related to half the local proton gyrofrequency. 16. Plasma rest frame frequencies and polarizations of the low-frequency upstream waves - ISEE 1 and 2 observations NASA Technical Reports Server (NTRS) Hoppe, M. M.; Russell, C. T. 1983-01-01 The plasma rest frame frequencies and polarizations of the large amplitude low frequency (0.03 Hz) upstream waves are investigated using magnetic field data from the dual ISEE 1 and 2 spacecraft. The monochromatic sinusoidal waves associated with intermediate ion fluxes are propagating in both the Alfven and magnetosonic modes, in both cases with typical frequencies approximately 0.1 times the local proton gyrofrequency and wavelengths of approximately 1 R(E). It is shown that the generation of the magnetosonic mode can be explained by the cyclotron resonance mechanism driven by narrow reflected ion beams, but the concurrent observation of Alfven mode waves appears to require wave generation by the more isotropic diffuse ion distributions as well. 17. The menagerie of geospace plasma waves NASA Technical Reports Server (NTRS) Shawhan, S. D. 1985-01-01 The sounding rocket and satellite observations of space plasma waves within geospace in the frequency range from millihertz to megahertz are studied. Characteristic frequencies and source mechanisms of the plasma waves are described. The use of the Dynamic Explorer-1 Plasma Wave Instrument spectrograms to represent the plasma wave antenna and receiver system of geospace is examined. The ray tracing technique calculates the path of energy flow; the equations required for the analysis are presented. Cross-correlation of the wave electric and magnetic components provide data used to calculate the wave polarization, the direction of propagation, and the wave distribution function. 18. Observations of auroral E-region plasma waves and electron heating with EISCAT and a VHF radar interferometer Providakes, J.; Farley, D. T.; Fejer, B. G.; Sahr, J.; Swartz, W. E. 1988-05-01 Two radars were used simultaneously to study naturally occurring electron heating events in the auroral E-region ionosphere. During a joint campaign in March 1986 the Cornell University Portable Radar Interferometer (CUPRI) was positioned to look perpendicular to the magnetic field to observe unstable plasma waves over Tromso, Norway, while EISCAT measured the ambient conditions in the unstable region. On two nights EISCAT detected intense but short lived (less than 1 min) electron heating events during which the temperature suddenly increased by a factor of 2-4 at altitudes near 108 km and the electron densities were less than 70,000/cu cm. On the second of these nights CUPRI was operating and detected strong plasma waves with very large phase velocities at precisely the altitudes and times at which the heating was observed. The altitudes, as well as one component of the irregularity drift velocity, were determined by interferometric techniques. From the observations and our analysis, it is concluded that the electron temperature increases were caused by plasma wave heating and not by either Joule heating or particle precipitation. 19. Electric Field Observations of Plasma Convection, Shear, Alfven Waves, and other Phenomena Observed on Sounding Rockets in the Cusp and Boundary Layer NASA Technical Reports Server (NTRS) Pfaff, R. F. 2009-01-01 On December 14,2002, a NASA Black Brant X sounding rocket was launched equatorward from Ny Alesund, Spitzbergen (79 N) into the dayside cusp and subsequently cut across the open/closed field line boundary, reaching an apogee of771 km. The launch occurred during Bz negative conditions with strong By negative that was changing during the flight. SuperDarn (CUTLASS) radar and subsequent model patterns reveal a strong westward/poleward convection, indicating that the rocket traversed a rotational reversal in the afternoon merging cell. The payload returned DC electric and magnetic fields, plasma waves, energetic particle, suprathermal electron and ion, and thermal plasma data. We provide an overview of the main observations and focus on the DC electric field results, comparing the measured E x B plasma drifts in detail with the CUTLASS radar observations of plasma drifts gathered simultaneously in the same volume. The in situ DC electric fields reveal steady poleward flows within the cusp with strong shears at the interface of the closed/open field lines and within the boundary layer. We use the observations to discuss ionospheric signatures of the open/closed character of the cusp/low latitude boundary layer as a function of the IMF. The electric field and plasma density data also reveal the presence of very strong plasma irregularities with a large range of scales (10 m to 10 km) that exist within the open field line cusp region yet disappear when the payload was equatorward of the cusp on closed field lines. These intense low frequency wave observations are consistent with strong scintillations observed on the ground at Ny Alesund during the flight. We present detailed wave characteristics and discuss them in terms of Alfven waves and static irregularities that pervade the cusp region at all altitudes. 20. Theory and Observations of Plasma Waves Excited Space Shuttle OMS Burns in the Ionosphere Bernhardt, P. A.; Pfaff, R. F.; Schuck, P. W.; Hunton, D. E.; Hairston, M. R. 2010-12-01 Measurements of artificial plasma turbulence were obtained during two Shuttle Exhaust Ionospheric Turbulence Experiments (SEITE) conducted during the flights of the Space Shuttle (STS-127 and STS-129). Based on computer modeling at the NRL PPD and Laboratory for Computational Physics & Fluid Dynamics (LCP), two dedicated burns of the Space Shuttle Orbital Maneuver Subsystem (OMS) engines were scheduled to produce 200 to 240 kg exhaust clouds that passed over the Air Force Research Laboratory (AFRL) Communications, Navigation, and Outage Forecast System (C/NOFS) satellite. This operation required the coordination by the DoD Space Test Program (STP), the NASA Flight Dynamics Officer (FDO), the C/NOFS payload operations, and the C/NOFS instrument principal investigators. The first SEITE mission used exhaust from a 12 Second OMS burn to deposit 1 Giga-Joules of energy into the upper atmosphere at a range of 230 km from C/NOFS. The burn was timed so C/NOFS could fly though the center of the exhaust cloud at a range of 87 km above the orbit of the Space Shuttle. The first SEITE experiment is important because is provided plume detection by ionospheric plasma and electric field probes for direct sampling of irregularities that can scatter radar signals. Three types of waves were detected by C/NOFS during and after the first SEITE burn. With the ignition and termination of the pair of OMS engines, whistler mode signals were recorded at C/NOFS. Six seconds after ignition, a large amplitude electromagnetic pulse reached the satellite. This has been identified as a fast magnetosonic wave propagating across magnetic field lines to reach the electric field (VEFI) sensors on the satellite. Thirty seconds after the burn, the exhaust cloud reach C/NOFS and engulfed the satellite providing very strong electric field turbulence along with enhancements in electron and ion densities. Kinetic modeling has been used to track the electric field turbulence to an unstable velocity 1. Plasma Waves Observed in the Cusp Turbulent Boundary Layer: An Analysis of High Time Resolution Wave and Particle Measurements from the Polar Spacecraft NASA Technical Reports Server (NTRS) Pickett, J. S.; Franz, J. R.; Scudder, J. D.; Menietti, J. D.; Gurnett, D. A.; Hospodarsky, G. B.; Braunger, R. M.; Kintner, P. M.; Kurth, W. S. 2001-01-01 The boundary layer located in the cusp and adjacent to the magnetopause is a region that is quite turbulent and abundant with waves. The Polar spacecraft's orbit and sophisticated instrumentation are ideal for studying this region of space. Our analysis of the waveform data obtained in this turbulent boundary layer shows broadband magnetic noise extending up to a few kilohertz (but less than the electron cyclotron frequency); sinusoidal bursts (a few tenths of a second) of whistler mode waves at around a few tens of hertz, a few hundreds of hertz, and just below the electron cyclotron frequency; and bipolar pulses, interpreted as electron phase-space holes. In addition, bursts of electron cyclotron harmonic waves are occasionally observed with magnetic components. We show evidence of broadband electrostatic bursts covering a range of approx. 3 to approx. 25 kHz (near but less than the plasma frequency) occurring in packets modulated at the frequency of some of the whistler mode waves. On the basis of high time resolution particle data from the Polar HYDRA instrument, we show that these bursts are consistent with generation by the resistive medium instability. The most likely source of the whistler mode waves is the magnetic reconnection site closest to the spacecraft, since the waves are observed propagating both toward and away from the Earth, are bursty, which is often the case with reconnection, and do not fit on the theoretical cold plasma dispersion relation curve. 2. Relationship of Topside Ionospheric Ion Outflows to Auroral Forms and Precipitations, Plasma Waves, and Convection Observed by POLAR NASA Technical Reports Server (NTRS) Hirahara, M.; Horwitz, J. L.; Moore, T. E.; Germany, G. A.; Spann, J. F.; Peterson, W. K.; Shelley, E. G.; Chandler, M. O.; Giles, B. L.; Craven, P. D.; Pollock, C. J.; Gurnett, D. A.; Persoon, A. M.; Scudder, J. D.; Maynard, N. C.; Mozer, F. S.; Brittnacher, M. J.; Nagai, T. 1997-01-01 The POLAR satellite often observes upflowing ionospheric ions (UFls) in and near the auroral oval on southern perigee (approximately 5000 km altitude) passes. We present the UFI features observed by the thermal ion dynamics experiment (TIDE) and the toroidal imaging mass-angle spectrograph (TIMAS) in the dusk-dawn sector under two different geomagnetic activity conditions in order to elicit their relationships with auroral forms, wave emissions, and convection pattern from additional POLAR instruments. During the active interval, the ultraviolet imager (UVI) observed a bright discrete aurora on the dusk side after the substorm onset and then observed a small isolated aurora form and diffuse auroras on the dawn side during the recovery phase. The UFls showed clear conic distributions when the plasma wave instrument (PWI) detected strong broadband wave emissions below approximately 10 kHz, while no significant auroral activities were observed by UVI. At higher latitudes, the low-energy UFI conics gradually changed to the polar wind component with decreasing intensity of the broadband emissions. V-shaped auroral kilometric radiation (AKR) signatures observed above approximately 200 kHz by PWI coincided with the region where the discrete aurora and the UFI beams were detected. The latitude of these features was lower than that of the UFI conics. During the observations of the UFI beams and conics, the lower-frequency fluctuations observed by the electric field instrument (EFI) were also enhanced, and the convection directions exhibited large fluctuations. It is evident that large electrostatic potential drops produced the precipitating electrons and discrete auroras, the UFI beams, and the AKR, which is also supported by the energetic plasma data from HYDRA. Since the intense broadband emissions were also observed with the UFIs. the ionospheric ions could be energized transversely before or during the parallel acceleration due to the potential drops. 3. Relationship of Topside Ionospheric Ion Outflows to Auroral Forms and Precipitation, Plasma Waves, and Convection Observed by Polar NASA Technical Reports Server (NTRS) Hirahara, M.; Horwitz, J. L.; Moore, T. E.; Germany, G. A.; Spann, J. F.; Peterson, W. K.; Shelley, E. G.; Chandler, M. O.; Giles, B. L.; Craven, P. D.; Pollock, C. J.; Gurnett, D. A.; Pickett, J. S.; Persoon, A. M.; Scudder, J. D.; Maynard, N. C.; Mozer, F. S.; Brittnacher, M. J.; Nagai, T. 1998-01-01 The POLAR satellite often observes upflowing ionospheric ions (UFIs) in and near the aurora] oval on southern perigee (approx. 5000 km altitude) passes. We present the UFI features observed by the thermal ion dynamics experiment (TIDE) and the toroidal imaging mass angle spectrograph (TIMAS) in the dusk-dawn sector under two different geomagnetic activity conditions in order to elicit their relationships with auroral forms, wave emissions, and convection pattern from additional POLAR instruments. During the active interval, the ultraviolet imager (UVI) observed a bright discrete aurora on the duskside after the substorm onset and then observed a small isolated aurora form and diffuse auroras on the dawnside during the recovery phase. The UFIs showed clear conic distributions when the plasma wave instrument (PWI) detected strong broadband wave emissions below approx. 10 kHz, while no significant auroral activities were observed by UVI. At higher latitudes, the low-energy UFI conics gradually changed to the polar wind component with decreasing intensity of the broadband emissions. V-shaped auroral kilometric radiation (AKR) signatures observed above -200 kHz by PWI coincided with the region where the discrete aurora and the UFI beams were detected. The latitude of these features was lower than that of the UFI conics. During the observations of the UFI beams and conics, the lower-frequency fluctuations observed by the electric field instrument were also enhanced, and the convection directions exhibited large fluctuations. It is evident that large electrostatic potential drops produced the precipitating electrons and discrete auroras, the UFI beams, and the AKR, which is also supported by the energetic plasma data from HYDRA. Since the intense broadband emissions were also observed with the UFIs, the ionospheric ions could be energized transversely before or during the parallel acceleration due to the potential drops. 4. Plasma Waves in the Magnetosheath of Venus NASA Technical Reports Server (NTRS) Strangeway, Robert J. 1996-01-01 Research supported by this grant is divided into three basic topics of investigation. These are: (1) Plasma waves in the Venus magnetosheath, (2) Plasma waves in the Venus foreshock and solar wind, (3) plasma waves in the Venus nightside ionosphere and ionotail. The main issues addressed in the first area - Plasma waves in the Venus magnetosheath - dealt with the wave modes observed in the magnetosheath and upper ionosphere, and whether these waves are a significant source of heating for the topside ionosphere. The source of the waves was also investigated. In the second area - Plasma waves in the Venus foreshock and solar wind, we carried out some research on waves observed upstream of the planetary bow shock known as the foreshock. The foreshock and bow shock modify the ambient magnetic field and plasma, and need to be understood if we are to understand the magnetosheath. Although most of the research was directed to wave observations on the dayside of the planet, in the last of the three basic areas studied, we also analyzed data from the nightside. The plasma waves observed by the Pioneer Venus Orbiter on the nightside continue to be of considerable interest since they have been cited as evidence for lightning on Venus. 5. Electron acoustic solitary waves and double layers for a magnetized plasma and its relevance with satellite observations Ghosh, Suktisama In situ measurements of particles and fields of the Earth’s auroral zone have revealed a reach variety of plasma phenomena on different spatial and temporal scales. One major aspects of them is the bursts of broadband electrostatic noise emissions (BEN) with frequencies from below the lower hybrid frequency (typically tens of Hertz) up to and higher than electron plasma and cyclotron frequencies (typically, a few kilohertz). Many detailed analysis of BEN have been performed using space-borne data from POLAR, FAST, GEOTAIL, and CLUSTER multi-spacecraft missions. It is well understood that the time scale of the high frequency part indicates the involvement of electron dynamics and may be interpreted as a nonlinear evolution electron acoustic instabilities leading to solitary waves and double layers. The study of the generation and propagation of such microstructures is important for understanding the Sun-Earth coupling and the energy dissipation and particle transports across the magnetospheric boundary layers. In the present work, we have delineated the parameter regime for the fully nonlinear solutions of electron acoustic solitary waves adopting the Sagdeev pseudopontial technique. The plasma is assumed to be magnetized and traversed by an electron beam. The ions are assumed to be hotter than electrons (viz., Ti > Te) and obey Boltzmann distribution. It has been observed that both positive and negative amplitude solitary waves may exist. The width-amplitude variation profile for a positive amplitude solution shows increase in the width with increasing amplitude, and the solution eventually terminates to a double layer. The width - amplitude variation profile has been found to agree qualitatively with the Fast satellite observations. On the other hand, the negative amplitude solutions reveal two distinct branches. For the fast moving one (with larger Mach numbers), the width decreases with increasing amplitude and no double layer solutions have been observed for 6. Observations of Plasma Waves in the Colliding Jet Region of a 3D Magnetic Flux Rope Flanked by Two Active Reconnection X Lines at the Subsolar Magnetopause Oieroset, M.; Sundkvist, D. J.; Chaston, C. C.; Phan, T. D.; Mozer, F.; McFadden, J. P.; Angelopoulos, V.; Andersson, L.; Eastwood, J. P. 2014-12-01 We have performed a detailed analysis of plasma and wave observations in a 3D magnetic flux rope encountered by the THEMIS spacecraft at the subsolar magnetopause. The extent of the flux rope was ˜270 ion skin depths in the outflow direction, and it was flanked by two active reconnection X lines producing colliding plasma jets in the flux rope core where ion heating and suprathermal electrons were observed. The colliding jet region was highly dynamic and characterized by the presence of high-frequency waves such as ion acoustic-like waves, electron holes, and whistler mode waves near the flux rope center and low-frequency kinetic Alfvén waves over a larger region. We will discuss possible links between these waves and particle heating. 7. Plasma waves and electrostatic structures near propagating boundary layers in the inner terrestrial magnetosphere: Van Allen Probes and THEMIS observations Malaspina, David; Wygant, John; Ergun, Robert; Reeves, Geoff; Skoug, Ruth; Larsen, Brian 2016-10-01 A broad range of plasma wave phenomena, only recently reported in the near-equatorial inner terrestrial magnetosphere, have been detected using the Van Allen Probes. These phenomena include electrostatic structures, such as double layers and phase space holes, as well as plasma wave modes including nonlinearly steepened whistler waves and kinetic Alfvén waves. The ubiquity of these structures is now confirmed, but it is not understood what role these structures and waves play in the dynamics of the inner magnetosphere and radiation belts. To quantify their importance, it is necessary to understand their distribution, generation, and impact on particle populations. In this study, we demonstrate a strong correlation between the occurrence of these phenomena and plasma boundaries, including the inner edge of the plasma sheet, propagating injection fronts, and the plasmapause. Further, we find that these structures and waves are continually generated as these boundaries propagate through the inner magnetosphere. Understanding the generation mechanisms of these structures and waves, as well as their impact on particle populations stands to benefit significantly from careful theoretical treatment, numerical simulation, and laboratory experiments. 8. Plasma Wave and Electron Density Structure Observed in the Cusp with a Dual-Rocket Experiment Colpitts, C. A.; Labelle, J. W.; Kletzing, C.; Bounds, S.; Cairns, I. 2008-12-01 The Twin Rockets to Investigate Cusp Electrodynamics (TRICE) were launched on December 10, 2007, from Andoya Research Range in Andenes, Norway, into the active cusp. Both payloads traveled north over Svalbard, with one payload reaching an apogee of ~1100 km, and the other reaching ~600 km. The payloads were separated by 100-400 km during the main portion of the flight. Both payloads included waveform receivers with 5 MHz bandwidth. These recorded several distinct types of auroral waves including whistler mode waves below ~1000 kHz and Langmuir-upper hybrid waves at 300-3000 kHz for several hundred km. Both payloads concurrently encountered a distinct period of Langmuir turbulence. Clearly defined wave cutoffs provide measurements of electron density and reveal significant density structure with density enhancements having amplitudes up to 100 percent and scale sizes from meters to tens of kilometers. Analysis of the inferred density profiles using windowed Fourier Transforms or Lomb-Scargle periodograms generates dynamic spectra of the density, which provide estimates of the spectral composition of the density irregularities for time intervals sufficiently short that the stationarity of the spectra can be investigated. The large-scale structures through which the two payloads propagated were measured by both the EISCAT and SuperDARN radars as well as by all-sky cameras operated at Longyearbyen and Ny-Alesund on Svalbard. Using this data when available, comparison of the density irregularity waveforms and spectra from the two flights is studied in relation to spatial and altitude variations of the turbulence. This examination of wave and density structures and the large scale formations with which they are associated will add to the understanding of the large scale electrodynamics of the cusp region. 9. Stimulated plasma waves in the ionosphere NASA Technical Reports Server (NTRS) Benson, R. F. 1977-01-01 The reported discussion is concerned with longitudinal waves associated with electron motions. These waves are easily stimulated in the ionosphere by rocket- and satellite-borne RF sounders. Most of the observations of stimulated plasma waves in the ionosphere are based on ionograms obtained from the sounders carried on board five satellites, including Explorer 20, Alouette 1 and 2, and ISIS 1 and 2. The majority of the observations can be explained by considering the propagation of the sounder-stimulated plasma waves. Attention is given to aspects of plasma wave dispersion, linear phenomena, plasma wave instabilities and nonlinear phenomena, unexplained phenomena, diagnostic applications, geophysical and astrophysical applications, and a number of experiments planned for the future. 10. In situ ionospheric observations of severe weather-related gravity waves and associated small-scale plasma structure Kelley, Michael C. 1997-01-01 On July 27, 1988, two sounding rockets were launched over a small thunderstorm cell which constituted the remnants of a large frontal event which had lasted for several hours over the eastern seaboard. One of the rockets was instrumented for detection of the electromagnetic impulse from lightning strikes and its subsequent interaction with the ionospheric plasma [Kelley et al., 1990]. The second had on board an absolute electron density probe, the results from which we report here. We present evidence that a gravity wave was spawned by the front and propagated nearly to the F peak in the ionosphere, where it steepened and created structure in the medium at scales much less than the vertical wavenumber of the major disturbance. The fluctuation spectrum along the rocket path was elevated for scales from 25 km down to less than 10 m. At scales between 10 km and just under 100 m, characterization of the spectrum by a power law yields a spectral index less than that displayed by such well-studied processes as bottomside spread F and barium cloud striations. Similar results have been reported for gravity wave induced intermediate scale structures at midlatitudes [Wernik et al., 1986]. The mixing theory described by Fridman [1990] may be relevant to these observations. 11. Investigation of the role of plasma wave cascading processes in the formation of midlatitude irregularities utilizing GPS and radar observations Eltrass, A.; Scales, W. A.; Erickson, P. J.; Ruohoniemi, J. M.; Baker, J. B. H. 2016-06-01 Recent studies reveal that midlatitude ionospheric irregularities are less understood due to lack of models and observations that can explain the characteristics of the observed wave structures. In this paper, the cascading processes of both the temperature gradient instability (TGI) and the gradient drift instability (GDI) are investigated as the cause of these irregularities. Based on observations obtained during a coordinated experiment between the Millstone Hill incoherent scatter radar and the Blackstone Super Dual Auroral Radar Network radar, a time series for the growth rate of both TGI and GDI is calculated for observations in the subauroral ionosphere under both quiet and disturbed geomagnetic conditions. Recorded GPS scintillation data are analyzed to monitor the amplitude scintillations and to obtain the spectral characteristics of irregularities producing ionospheric scintillations. Spatial power spectra of the density fluctuations associated with the TGI from nonlinear plasma simulations are compared with both the GPS scintillation spectral characteristics and previous in situ satellite spectral measurements. The spectral comparisons suggest that initially, TGI or/and GDI irregularities are generated at large-scale size (kilometer scale), and the dissipation of the energy associated with these irregularities occurs by generating smaller and smaller (decameter scale) irregularities. The alignment between experimental, theoretical, and computational results of this study suggests that in spite of expectations from linear growth rate calculations, cascading processes involving TGI and GDI are likely responsible for the midlatitude ionospheric irregularities associated with GPS scintillations during disturbed times. 12. Initial Results of DC Electric Fields, Associated Plasma Drifts, Magnetic Fields, and Plasma Waves Observed on the C/NOFS Satellite NASA Technical Reports Server (NTRS) Pfaff, R.; Freudenreich, H.; Bromund, K.; Klenzing, J.; Rowland, D.; Maynard, N. 2010-01-01 Initial results are presented from the Vector Electric Field Investigation (VEFI) on the Air Force Communication/Navigation Outage Forecasting System (C/NOFS) satellite, a mission designed to understand, model, and forecast the presence of equatorial ionospheric irregularities. The VEFI instrument includes a vector DC electric field detector, a fixed-bias Langmuir probe operating in the ion saturation regime, a flux gate magnetometer, an optical lightning detector, and associated electronics including a burst memory. Compared to data obtained during more active solar conditions, the ambient DC electric fields and their associated E x B drifts are variable and somewhat weak, typically < 1 mV/m. Although average drift directions show similarities to those previously reported, eastward/outward during day and westward/downward at night, this pattern varies significantly with longitude and is not always present. Daytime vertical drifts near the magnetic equator are largest after sunrise, with smaller average velocities after noon. Little or no pre-reversal enhancement in the vertical drift near sunset is observed, attributable to the solar minimum conditions creating a much reduced neutral dynamo at the satellite altitude. The nighttime ionosphere is characterized by larger amplitude, structured electric fields, even where the plasma density appears nearly quiescent. Data from successive orbits reveal that the vertical drifts and plasma density are both clearly organized with longitude. The spread-F density depletions and corresponding electric fields that have been detected thus far have displayed a preponderance to appear between midnight and dawn. Associated with the narrow plasma depletions that are detected are broad spectra of electric field and plasma density irregularities for which a full vector set of measurements is available for detailed study. Finally, the data set includes a wide range of ELF/VLF/HF oscillations corresponding to a variety of plasma waves 13. Large amplitude relativistic plasma waves SciTech Connect Coffey, Timothy 2010-05-15 Relativistic, longitudinal plasma oscillations are studied for the case of a simple water bag distribution of electrons having cylindrical symmetry in momentum space with the axis of the cylinder parallel to the velocity of wave propagation. The plasma is required to obey the relativistic Vlasov-Poisson equations, and solutions are sought in the wave frame. An exact solution for the plasma density as a function of the electrostatic field is derived. The maximum electric field is presented in terms of an integral over the known density. It is shown that when the perpendicular momentum is neglected, the maximum electric field approaches infinity as the wave phase velocity approaches the speed of light. It is also shown that for any nonzero perpendicular momentum, the maximum electric field will remain finite as the wave phase velocity approaches the speed of light. The relationship to previously published solutions is discussed as is some recent controversy regarding the proper modeling of large amplitude relativistic plasma waves. 14. Millimeter Wave Communication through Plasma NASA Technical Reports Server (NTRS) Bastin, Gary L. 2008-01-01 Millimeter wave communication through plasma at frequencies of 35 GHz or higher shows promise in maintaining communications connectivity during rocket launch and re-entry, critical events which are typically plagued with communication dropouts. Extensive prior research into plasmas has characterized the plasma frequency at these events, and research at the Kennedy Space Center is investigating the feasibility of millimeter communication through these plasma frequencies. 15. Creating and studying ion acoustic waves in ultracold neutral plasmas SciTech Connect Killian, T. C.; Castro, J.; McQuillen, P.; O'Neil, T. M. 2012-05-15 We excite ion acoustic waves in ultracold neutral plasmas by imprinting density modulations during plasma creation. Laser-induced fluorescence is used to observe the density and velocity perturbations created by the waves. The effect of expansion of the plasma on the evolution of the wave amplitude is described by treating the wave action as an adiabatic invariant. After accounting for this effect, we determine that the waves are weakly damped, but the damping is significantly faster than expected for Landau damping. 16. Plasma waves associated with the space shuttle NASA Technical Reports Server (NTRS) Cairns, I. H.; Gurnett, D. A. 1990-01-01 Water molecules outgassed from the Space Shuttle suffer collisional charge-exchange with ionospheric oxygen ions, thereby forming unstable distributions of pick-up water ions and leading to high levels of plasma waves near the Shuttle. Liouville's equation with a charge-exchange source term is solved for the water ion distribution function as a function of position relative to the Shuttle. The observational characteristics of the near zone Shuttle waves are summarized. A linear theory in which beam like distributions of water ions drive Doppler shifted lower hybrid waves via the modified two stream instability is developed. This theory explains many characteristics of the near zone waves. Further work on the effects of wave nonlinearities and spatial inhomogeneity is required to explain the detailed frequency spectrum of the waves. The observed wave levels apparently satisfy the threshold condition for modulational instability of lower hybrid waves. 17. Plasma wave profiles of earth's bow shock at low Mach numbers - ISEE 3 observations on the far flank NASA Technical Reports Server (NTRS) Greenstadt, E. W.; Coroniti, F. V.; Moses, S. L.; Smith, E. J. 1992-01-01 A survey of selected crossings far downstream from the subsolar shock is presented in which the overall plasma wave (PW) behavior of a selected set of nearly perpendicular crossings and another set of limited Mach number but broad geometry are delinated. The result is a generalizable PW signature, or signatures, of low Mach number shocks and some likely implications of those signatures for the weak shock's plasma physical processes on the flank. The data are found to be consistent with the presence of ion beam interactions producing noise ahead of the shock in the ion acoustic frequency range. The presence or absence, and the amplitudes, of PW activity are explainable by the presence or absence of a population of upstream ions controlled by the component of the interplanetary magnetic field normal to the solar wind flow. 18. Nonlinear whistler wave scattering in space plasmas SciTech Connect Yukhimuk, V.; Roussel-Dupre, R. 1997-04-01 In this paper the evolution of nonlinear scattering of whistler mode waves by kinetic Alfven waves (KAW) in time and two spatial dimensions is studied analytically. The authors suggest this nonlinear process as a mechanism of kinetic Alfven wave generation in space plasmas. This mechanism can explain the dependence of Alfven wave generation on whistler waves observed in magnetospheric and ionospheric plasmas. The observational data show a dependence for the generation of long periodic pulsations Pc5 on whistler wave excitation in the auroral and subauroral zone of the magnetosphere. This dependence was first observed by Ondoh T.I. For 79 cases of VLF wave excitation registered by Ondoh at College Observatory (L=64.6 N), 52 of them were followed by Pc5 geomagnetic pulsation generation. Similar results were obtained at the Loparskaia Observatory (L=64 N) for auroral and subauroral zone of the magnetosphere. Thus, in 95% of the cases when VLF wave excitation occurred the generation of long periodic geomagnetic pulsations Pc5 were observed. The observations also show that geomagnetic pulsations Pc5 are excited simultaneously or insignificantly later than VLF waves. In fact these two phenomena are associated genetically: the excitation of VLF waves leads to the generation of geomagnetic pulsations Pc5. The observations show intensive generation of geomagnetic pulsations during thunderstorms. Using an electromagnetic noise monitoring system covering the ULF range (0.01-10 Hz) A.S. Fraser-Smith observed intensive ULF electromagnetic wave during a large thunderstorm near the San-Francisco Bay area on September 23, 1990. According to this data the most significant amplification in ULF wave activity was observed for waves with a frequency of 0.01 Hz and it is entirely possible that stronger enhancements would have been measured at lower frequencies. 19. Waves and instabilities in plasmas SciTech Connect Chen, L. 1987-01-01 The contents of this book are: Plasma as a Dielectric Medium; Nyquist Technique; Absolute and Convective Instabilities; Landau Damping and Phase Mixing; Particle Trapping and Breakdown of Linear Theory; Solution of Viasov Equation via Guilding-Center Transformation; Kinetic Theory of Magnetohydrodynamic Waves; Geometric Optics; Wave-Kinetic Equation; Cutoff and Resonance; Resonant Absorption; Mode Conversion; Gyrokinetic Equation; Drift Waves; Quasi-Linear Theory; Ponderomotive Force; Parametric Instabilities; Problem Sets for Homework, Midterm and Final Examinations. 20. CRRES and DMSP Observations of Wave and Plasma Disturbances Associated with the Stormtime Ring Current in the Plasmasphere and Topside Ionosphere Mishin, E. V.; Burke, W. J. 2004-12-01 We report on wave and plasma disturbances observed by Combined Release and Radiation Effects (CRRES) and Defense Meteorological Satellite Program (DMSP) satellites during the magnetic storm of June 5, 1991 in the region of ring current/plasmasphere overlap and the conjugate topside ionosphere During three ring current nose encounters near L = 2.4, the plasmasphere was highly-structured. A rich variety of wave phenomena were observed simultaneous with enhanced fluxes of low-energy (< ˜ 1 keV) electrons and ions, indicating the wave heating/acceleration source. Earthward of the plasma sheet boundary, which was near L = 5.5, wave-like structures in the dawn-to-dusk electric field with spatial wave-lengths from about 300 to 1000 km and magnitudes of ~1-3 mV/m were apparent. Mapped to ionospheric altitudes, these fields should produce broad irregular SAPS with average sunward velocities ~ 1 km/s. At about the same time DMSP F8, F9, and F10 indeed observed highly-structured SAPS in the topside ionosphere coincident with precipitating ring current ions, enhanced fluxes of suprathermal electrons and ions, elevated electron temperatures, and deep highly-irregular density troughs. Overall, these events represent the so-called strong wave-SAPS phenomenon [Mishin et al., JGR (2003), 108, 1309, 10.1029/2002JA009793]. Their importance for Space Weather is indicated by strong GPS phase and amplitude scintillations observed over the continental US [Basu et al., JGR, 106, 30389, 2001; Ledvina et al., GRL, 29, 10.1029/2002GL014770] coincident with similar events. 1. Sound wave propagation through glow discharge plasma This work investigates the use of glow discharge plasma for acoustic wave manipulation. The broader goal is the suppression of aerodynamic noise using atmospheric glow discharge plasma as a sound barrier. Part of the effort was devoted to the development of a system for the generation of a large volume stable DC glow discharge in air both at atmospheric and at reduced pressures. The single tone sound wave propagation through the plasma was systematically studied. Attenuation of the acoustic wave passing through the glow discharge was measured for a range of experimental conditions including different discharge currents, electrode configurations, air pressures and sound frequencies including audible sound and ultrasound. Sound attenuation by glow discharge plasma as high as -28 dB was recorded in the experiments. Two types of possible mechanisms were considered that can potentially cause the observed sound attenuation. One is a global mechanism and the other is a local mechanism. The global mechanism considered is based on the reflection and refraction of acoustic wave due to the gas temperature gradients that form around the plasma. The local mechanism, on the other hand, is essentially the interaction of the acoustic wave with the plasma as it propagates inside the discharge and it can be viewed as a feedback system. Detailed temperature measurements, using laser-induced Rayleigh scattering technique, were carried out in the glow discharge plasma in order to evaluate the role of global mechanism in the observed attenuation. These measurements were made for a range of conditions in the atmospheric glow discharge. Theoretical analysis of the sound attenuation was carried out to identify the physical mechanism for the observed sound attenuation by plasma. It was demonstrated that the global mechanism is the dominant mechanism of sound attenuation. As a result of this study, the potentials and limitations of the plasma noise suppression technology were determined and 2. Shock Wave Dynamics in Weakly Ionized Plasmas NASA Technical Reports Server (NTRS) Johnson, Joseph A., III 1999-01-01 An investigation of the dynamics of shock waves in weakly ionized argon plasmas has been performed using a pressure ruptured shock tube. The velocity of the shock is observed to increase when the shock traverses the plasma. The observed increases cannot be accounted for by thermal effects alone. Possible mechanisms that could explain the anomalous behavior include a vibrational/translational relaxation in the nonequilibrium plasma, electron diffusion across the shock front resulting from high electron mobility, and the propagation of ion-acoustic waves generated at the shock front. Using a turbulence model based on reduced kinetic theory, analysis of the observed results suggest a role for turbulence in anomalous shock dynamics in weakly ionized media and plasma-induced hypersonic drag reduction. 3. Plasma Beat-Wave Acceleration Clayton, Christopher E. 2002-04-01 Among all the advanced accelerator concepts that use lasers as the power source, most of the effort to date has been with the idea of using a laser pulse to excite a accelerating mode in a plasma. Within this area, there are a variety of approaches for creating the accelerating mode, as indicated by the other talks in this session. What is common to these approaches is the physics of how a laser pulse pushes on plasma electrons to organize electron-density perturbations, the sources of the ultra-high (> GeV/M) accelerating gradients. It is the "ponderomotive force", proportional to the local gradient of the of the laser intensity, that pushes plasma electrons forward (on the leading edge of the pulse) and backwards (on the trailing edge) which leads to harmonic motion of the electrons. As the laser pulse moves through the plasma at group velocity Vg c, the oscillating electrons show up macroscopically as a plasma mode or wave with frequency w equal to the plasma frequency and k = w/Vg. For short laser pulses, this is the Laser Wakefield Accelerator (LWFA) concept. Closely related is the Plasma Beat-Wave Acceleration (PBWA) concept. Here, the laser pulse that perturbs the plasma is composed of two closely-spaced frequencies that "beat", i.e., periodically constructively and destructively interfere, forming an electromagnetic beat wave. One can visualize this as a train of short pulses. If this beating frequency is set to the plasma frequency, then each pulse in the train will reinforce the density perturbation caused by the previous pulse. The principal advantage of multiple pulses driving up the plasma wave as opposed to a single pulse is in efficiency, allowing for the production of relatively large diameter (more 1-D like) accelerating modes. In this talk I will discuss past, current and planned PBWA experiments which are taking place at UCLA, RAL in England, and LULI in France. 4. An automated analysis of DEMETER ionospheric plasma waves observations and its application to the search for anomalous emissions over the Great Sichuan EQ region Onishi, Tatsuo; Berthelier, Jean-Jacques 2010-05-01 Electric field observations in the VLF range from the ICE experiment onboard the CNES DEMETER micro-satellite have been analyzed to search for anomalies possibly related to the Great Sichuan Earthquake of May 12, 2008. This work was undertaken using results from a dedicated data processing that has been recently developed at LATMOS to perform an automated recognition and characterization of the various wave emissions that are regularly detected along the orbit of DEMETER. The data processing method and the associated algorithms will be first presented and a few typical results will be shown in order to provide a detailed understanding of the algorithm capabilities. As a first full-scale application of this method, a statistical study was conducted to analyze the plasma waves observed in day-time half orbits over a region of ~1000 kilometres extent centred on the Sichuan EQ epicentre and during a period of 20 days encompassing the day of the EQ. 5 years of observations have been used to derive the statistical distribution of various types of ionospheric plasma waves that can be compared to the signals detected during the seismic active period. The first outcome of our study was the detection of a localized variation in the characteristics of the electrostatic turbulence 6 days before the EQ that appears to be unique in the whole 5 year reference observations data base. We will discuss this result and its possible interpretations. 5. The Polar Plasma Wave Instrument NASA Technical Reports Server (NTRS) Gurnett, D. A.; Persoon, A. M.; Randall, R. F.; Odem, D. L.; Remington, S. L.; Averkamp, T. F.; Debower, M. M.; Hospodarsky, G. B.; Huff, R. L.; Kirchner, D. L. 1995-01-01 The Plasma Wave Instrument on the Polar spacecraft is designed to provide measurements of plasma waves in the Earth's polar regions over the frequency range from 0.1 Hz to 800 kHz. Three orthogonal electric dipole antennas are used to detect electric fields, two in the spin plane and one aligned along the spacecraft spin axis. A magnetic loop antenna and a triaxial magnetic search coil antenna are used to detect magnetic fields. Signals from these antennas are processed by five receiver systems: a wideband receiver, a high-frequency waveform receiver, a low-frequency waveform receiver, two multichannel analyzers; and a pair of sweep frequency receivers. Compared to previous plasma wave instruments, the Polar plasma wave instrument has several new capabilities. These include (1) an expanded frequency range to improve coverage of both low- and high-frequency wave phenomena, (2) the ability to simultaneously capture signals from six orthogonal electric and magnetic field sensors, and (3) a digital wideband receiver with up to 8-bit resolution and sample rates as high as 249k samples s(exp -1). 6. The Unified Radio and Plasma wave investigation NASA Technical Reports Server (NTRS) Stone, R. G.; Bougeret, J. L.; Caldwell, J.; Canu, P.; De Conchy, Y.; Cornilleau-Wehrlin, N.; Desch, M. D.; Fainberg, J.; Goetz, K.; Goldstein, M. L. 1992-01-01 The scientific objectives of the Ulysses Unified Radio and Plasma wave (URAP) experiment are twofold: (1) the determination of the direction, angular size, and polarization of radio sources for remote sensing of the heliosphere and the Jovian magnetosphere and (2) the detailed study of local wave phenomena, which determine the transport coefficients of the ambient plasma. A brief discussion of the scientific goals of the experiment is followed by a comprehensive description of the instrument. The URAP sensors consist of a 72.5 m electric field antenna in the spin plane, a 7.5-m electric field monopole along the spin axis of a pair of orthogonal search coil magnetic antennas. The various receivers, designed to encompass specific needs of the investigation, cover the frequency range from dc to 1 MHz. A relaxation sounder provides very accurate electron density measurements. Radio and plasma wave observations are shown to demonstrate the capabilities and limitations of the URAP instruments: radio observations include solar bursts, auroral kilometric radiation, and Jovian bursts; plasma waves include Langmuir waves, ion acousticlike noise, and whistlers. 7. A simple electron plasma wave Brodin, G.; Stenflo, L. 2017-03-01 Considering a class of solutions where the density perturbations are functions of time, but not of space, we derive a new exact large amplitude wave solution for a cold uniform electron plasma. This result illustrates that most simple analytical solutions can appear even if the density perturbations are large. 8. Waves in Space Plasmas (WISP) NASA Technical Reports Server (NTRS) Calvert, Wynne 1994-01-01 Activities under this project have included participation in the Waves in Space Plasmas (WISP) program, a study of the data processing requirements for WISP, and theoretical studies of radio sounding, ducting, and magnetoionic theory. An analysis of radio sounding in the magnetosphere was prepared. 9. Enhancement of terahertz wave generation from laser induced plasma SciTech Connect Xie Xu; Xu Jingzhou; Dai Jianming; Zhang, X.-C. 2007-04-02 It is well known that air plasma induced by ultrashort laser pulses emits broadband terahertz waves. The authors report the study of terahertz wave generation from the laser induced plasma where there is a preexisting plasma background. When a laser beam from a Ti:sapphire amplifier is used to generate a terahertz wave, enhancement of the generation is observed if there is another laser beam creating a plasma background. The enhancement of the terahertz wave amplitude lasts hundreds of picoseconds after the preionized background is created, with a maximum enhancement up to 250% observed. 10. On the breaking of a plasma wave in a thermal plasma. II. Electromagnetic wave interaction with the breaking plasma wave SciTech Connect Bulanov, Sergei V.; Esirkepov, Timur Zh.; Kando, Masaki; Koga, James K.; Pirozhkov, Alexander S.; Nakamura, Tatsufumi; Bulanov, Stepan S.; Schroeder, Carl B.; Esarey, Eric; Califano, Francesco; Pegoraro, Francesco 2012-11-15 In thermal plasma, the structure of the density singularity formed in a relativistically large amplitude plasma wave close to the wavebreaking limit leads to a refraction coefficient with discontinuous spatial derivatives. This results in a non-exponentially small above-barrier reflection of an electromagnetic wave interacting with the nonlinear plasma wave. 11. Wave structures observed in the equatorial F-region plasma density and temperature during the sunset period Savio, S.; Muralikrishna, P.; Batista, I. S.; de Meneses, F. C. 2016-11-01 Electron density and temperature measurements were carried out with Langmuir probes (LP) on board Brazilian sounding rockets launched soon after the local sunset from Natal (5.8°S, 35.2°W, dip 23.7°S) and Alcântara (2.3°S, 44.4°W, dip 7°S), Brazil, on December 02, 2011, and December 08, 2012, respectively. Digisondes operating near the launching sites revealed a rapid rise in the F-region base indicating a probable pre-reversal enhancement of the vertical plasma drift. Strong spread-F traces are also visible on the ionograms simultaneously recorded, suggesting the occurrence of ionospheric bubbles during these campaigns. Electron density and temperature vertical profiles estimated from the LP data exhibit in the E-F region valley (120-300 km) the presence of large-amplitude wave activity, and electron temperature values higher than 1600 K, respectively, phenomena probably related to the electrodynamic processes that occur during the sunset period. 12. Current status of IMS plasma wave research. [International Magnetospheric Study NASA Technical Reports Server (NTRS) Anderson, R. R. 1982-01-01 The present investigation is concerned with a review of the status of magnetospheric plasma wave science as a result of the International Magnetospheric Study (IMS). The presence of an international effort has supported the development and completion of the numerous magnetospheric science spacecraft launched during the IMS, including GEOS, ISEE, and EXOS B. Ground-based VLF observations are considered along with coordinated ground-based and satellite observations. During the IMS, plasma wave research using satellite data has covered a wide range of subjects. Attention is given to magnetospheric electrostatic emissions, magnetospheric electromagnetic plasma waves, continuum radiation, auroral kilometric radiation, auroral zone plasma waves, plasma waves in the magnetosheath and near the mangetopause, and plasma waves at the bow shock. 13. EXPERIMENTAL STUDY OF SHOCK WAVE DYNAMICS IN MAGNETIZED PLASMAS SciTech Connect Nirmol K. Podder 2009-03-17 In this four-year project (including one-year extension), the project director and his research team built a shock-wave-plasma apparatus to study shock wave dynamics in glow discharge plasmas in nitrogen and argon at medium pressure (1–20 Torr), carried out various plasma and shock diagnostics and measurements that lead to increased understanding of the shock wave acceleration phenomena in plasmas. The measurements clearly show that in the steady-state dc glow discharge plasma, at fixed gas pressure the shock wave velocity increases, its amplitude decreases, and the shock wave disperses non-linearly as a function of the plasma current. In the pulsed discharge plasma, at fixed gas pressure the shock wave dispersion width and velocity increase as a function of the delay between the switch-on of the plasma and shock-launch. In the afterglow plasma, at fixed gas pressure the shock wave dispersion width and velocity decrease as a function of the delay between the plasma switch-off and shock-launch. These changes are found to be opposite and reversing towards the room temperature value which is the initial condition for plasma ignition case. The observed shock wave properties in both igniting and afterglow plasmas correlate well with the inferred temperature changes in the two plasmas. 14. Accumulative coupling between magnetized tenuous plasma and gravitational waves Zhang, Fan 2017-01-01 This talk presents solutions to the plasma waves induced by a plane gravitational wave (GW) train travelling through a region of strongly magnetized plasma. The computations constitute a very preliminary feasibility study for a possible ultra-high frequency gravitational wave detector, meant to take advantage of the observation that the plasma current is proportional to the GW amplitude, and not its square. This work is supported in part by NSFC Grant Number 11503003. 15. The Galileo plasma wave investigation NASA Technical Reports Server (NTRS) Gurnett, D. A.; Kurth, W. S.; Shaw, R. R.; Roux, A.; Gendrin, R.; Kennel, C. F.; Scarf, F. L.; Shawhan, S. D. 1992-01-01 The purpose of the Galileo plasma wave investigation is to study plasma waves and radio emissions in the magnetosphere of Jupiter. The plasma wave instrument uses an electric dipole antenna to detect electric fields, and two search coil magnetic antennas to detect magnetic fields. The frequency range covered is 5 Hz to 5.6 MHz for electric fields and 5 Hz to 160 kHz for magnetic fields. Low time-resolution survey spectrums are provided by three on-board spectrum analyzers. In the normal mode of operation the frequency resolution is about 10 percent, and the time resolution for a complete set of electric and magnetic field measurements is 37.33 s. High time-resolution spectrums are provided by a wideband receiver. The wideband receiver provides waveform measurements over bandwidths of 1, 10, and 80 kHz. Compared to previous measurements at Jupiter this instrument has several new capabilities. These new capabilities include (1) both electric and magnetic field measurements to distinguish electrostatic and electromagnetic waves, (2) direction finding measurements to determine source locations, and (3) increased bandwidth for the wideband measurements. 16. Linear and Nonlinear Electrostatic Waves in Unmagnetized Dusty Plasmas SciTech Connect Mamun, A. A.; Shukla, P. K. 2010-12-14 A rigorous and systematic theoretical study has been made of linear and nonlinear electrostatic waves propagating in unmagnetized dusty plasmas. The basic features of linear and nonlinear electrostatic waves (particularly, dust-ion-acoustic and dust-acoustic waves) for different space and laboratory dusty plasma conditions are described. The experimental observations of such linear and nonlinear features of dust-ion-acoustic and dust-acoustic waves are briefly discussed. 17. The Potential for Ambient Plasma Wave Propulsion NASA Technical Reports Server (NTRS) Gilland, James H.; Williams, George J. 2016-01-01 A truly robust space exploration program will need to make use of in-situ resources as much as possible to make the endeavor affordable. Most space propulsion concepts are saddled with one fundamental burden; the propellant needed to produce momentum. The most advanced propulsion systems currently in use utilize electric and/or magnetic fields to accelerate ionized propellant. However, significant planetary exploration missions in the coming decades, such as the now canceled Jupiter Icy Moons Orbiter, are restricted by propellant mass and propulsion system lifetimes, using even the most optimistic projections of performance. These electric propulsion vehicles are inherently limited in flexibility at their final destination, due to propulsion system wear, propellant requirements, and the relatively low acceleration of the vehicle. A few concepts are able to utilize the environment around them to produce thrust: Solar or magnetic sails and, with certain restrictions, electrodynamic tethers. These concepts focus primarily on using the solar wind or ambient magnetic fields to generate thrust. Technically immature, quasi-propellantless alternatives lack either the sensitivity or the power to provide significant maneuvering. An additional resource to be considered is the ambient plasma and magnetic fields in solar and planetary magnetospheres. These environments, such as those around the Sun or Jupiter, have been shown to host a variety of plasma waves. Plasma wave propulsion takes advantage of an observed astrophysical and terrestrial phenomenon: Alfven waves. These are waves that propagate in the plasma and magnetic fields around and between planets and stars. The generation of Alfven waves in ambient magnetic and plasma fields to generate thrust is proposed as a truly propellantless propulsion system which may enable an entirely new matrix of exploration missions. Alfven waves are well known, transverse electromagnetic waves that propagate in magnetized plasmas at 18. Simulation of the nonlinear evolution of electron plasma waves NASA Technical Reports Server (NTRS) Nishikawa, K.-I.; Cairns, I. H. 1991-01-01 Electrostatic waves driven by an electron beam in an ambient magnetized plasma were studied using a quasi-1D PIC simulation of electron plasma waves (i.e., Langmuir waves). The results disclose the presence of a process for moving wave energy from frequencies and wavenumbers predicted by linear theory to the Langmuir-like frequencies during saturation of the instability. A decay process for producing backward propagating Langmuir-like waves, along with low-frequency waves, is observed. The simulation results, however, indicate that the backscattering process is not the conventional Langmuir wave decay. Electrostatic waves near multiples of the electron plasma frequency are generated by wave-wave coupling during the nonlinear stage of the simulations, confirming the suggestion of Klimas (1983). 19. Waves in plasmas: some historical highlights SciTech Connect Stix, T.H. 1984-08-01 To illustrate the development of some fundamental concepts in plasma waves, a number of experimental observations, going back over half a century, are reviewed. Particular attention is paid to the phenomena of dispersion, collisionfree damping, finite-Larmor-radius and cyclotron and cyclotron-harmonic effects, nonlocal response, and stochasticity. One may note not only the constructive interplay between observation and theory and experiment but also that major advances have come from each of the many disciplines that invoke plasma physics as a tool, including radio communication, astrophysics, controlled fusion, space physics, and basic research. 20. Ion Acceleration in Plasmas with Alfven Waves SciTech Connect O.Ya. Kolesnychenko; V.V. Lutsenko; R.B. White 2005-06-15 Effects of elliptically polarized Alfven waves on thermal ions are investigated. Both regular oscillations and stochastic motion of the particles are observed. It is found that during regular oscillations the energy of the thermal ions can reach magnitudes well exceeding the plasma temperature, the effect being largest in low-beta plasmas (beta is the ratio of the plasma pressure to the magnetic field pressure). Conditions of a low stochasticity threshold are obtained. It is shown that stochasticity can arise even for waves propagating along the magnetic field provided that the frequency spectrum is non-monochromatic. The analysis carried out is based on equations derived by using a Lagrangian formalism. A code solving these equations is developed. Steady-state perturbations and perturbations with the amplitude slowly varying in time are considered. 1. Nonlocal Heat Transport by Longitudinal/Transverse EM Waves in Magnetically Confined Plasmas and Modelling of the Observed Nonlocal Phenomena in a Tokamak Kukushkin, A. B. 1996-11-01 The nonlocal transport approach is formulated, based on anomalous cross-field energy transport (ACFET) by the longitudinal/tranverse EM waves of the mean free path of the order and much larger than plasma characteristic size and, correspondingly, on integral equation in space variables. Self-consistency of this approach is shown in interpreting those observed phenomena of nonlocality whose interpretation in "local", diffusion-like approaches gives instant jumps of thermal diffusivities in a large part of plasma volume. The modelling is carried out of the initial stage of recently observed phenomena of fast nonlocal energy transport: (i) net inward flux of energy during off-axis heating (vs. ECRH experiments on D-III-D); (ii) prompt rise of temperature in the core in "cold pulse" experiments (fast cooling of the periphery) on TEXT and TFTR; (iii) fast "volumetric" response of energy transport to plasma edge behavior during L-H transitions (in JET and JT-60U). The results suggest (a) universal and transparent physical explanation of the mechanism of nonlocal inward energy flux, which is lost in diffusion-like approaches, and (b) necessity to append existing numerical codes with nonlocal transport term, an integral in space variables. 2. Principles of Space Plasma Wave Instrument Design NASA Technical Reports Server (NTRS) Gurnett, Donald A. 1998-01-01 Space plasma waves span the frequency range from somewhat below the ion cyclotron frequency to well above the electron cyclotron frequency and plasma frequency. Because of the large frequency range involved, the design of space plasma wave instrumentation presents many interesting challenges. This chapter discusses the principles of space plasma wave instrument design. The topics covered include: performance requirements, electric antennas, magnetic antennas, and signal processing. Where appropriate, comments are made on the likely direction of future developments. 3. Ion-wave stabilization of an inductively coupled plasma SciTech Connect Camparo, J.C.; Mackay, R. 2006-04-24 Stabilization of the rf power driving an inductively coupled plasma (ICP) has implications for fields ranging from atomic clocks to analytical chemistry to illumination technology. Here, we demonstrate a technique in which the plasma itself acts as a probe of radio wave power, and provides a correction signal for active rf-power control. Our technique takes advantage of the resonant nature of forced ion waves in the plasma, and their observation in the ICP's optical emission. 4. A statistical study of EMIC waves observed by Cluster. 1. Wave properties. EMIC Wave Properties SciTech Connect Allen, R. C.; Zhang, J. -C.; Kistler, L. M.; Spence, H. E.; Lin, R. -L.; Klecker, B.; Dunlop, M. W.; André, M.; Jordanova, V. K. 2015-07-23 Electromagnetic ion cyclotron (EMIC) waves are an important mechanism for particle energization and losses inside the magnetosphere. In order to better understand the effects of these waves on particle dynamics, detailed information about the occurrence rate, wave power, ellipticity, normal angle, energy propagation angle distributions, and local plasma parameters are required. Previous statistical studies have used in situ observations to investigate the distribution of these parameters in the magnetic local time versus L-shell (MLT-L) frame within a limited magnetic latitude (MLAT) range. In our study, we present a statistical analysis of EMIC wave properties using 10 years (2001–2010) of data from Cluster, totaling 25,431 min of wave activity. Due to the polar orbit of Cluster, we are able to investigate EMIC waves at all MLATs and MLTs. This allows us to further investigate the MLAT dependence of various wave properties inside different MLT sectors and further explore the effects of Shabansky orbits on EMIC wave generation and propagation. Thus, the statistical analysis is presented in two papers. OUr paper focuses on the wave occurrence distribution as well as the distribution of wave properties. The companion paper focuses on local plasma parameters during wave observations as well as wave generation proxies. 5. A statistical study of EMIC waves observed by Cluster. 1. Wave properties. EMIC Wave Properties DOE PAGES Allen, R. C.; Zhang, J. -C.; Kistler, L. M.; ... 2015-07-23 Electromagnetic ion cyclotron (EMIC) waves are an important mechanism for particle energization and losses inside the magnetosphere. In order to better understand the effects of these waves on particle dynamics, detailed information about the occurrence rate, wave power, ellipticity, normal angle, energy propagation angle distributions, and local plasma parameters are required. Previous statistical studies have used in situ observations to investigate the distribution of these parameters in the magnetic local time versus L-shell (MLT-L) frame within a limited magnetic latitude (MLAT) range. In our study, we present a statistical analysis of EMIC wave properties using 10 years (2001–2010) of datamore » from Cluster, totaling 25,431 min of wave activity. Due to the polar orbit of Cluster, we are able to investigate EMIC waves at all MLATs and MLTs. This allows us to further investigate the MLAT dependence of various wave properties inside different MLT sectors and further explore the effects of Shabansky orbits on EMIC wave generation and propagation. Thus, the statistical analysis is presented in two papers. OUr paper focuses on the wave occurrence distribution as well as the distribution of wave properties. The companion paper focuses on local plasma parameters during wave observations as well as wave generation proxies.« less 6. Dichromatic Langmuir waves in degenerate quantum plasma SciTech Connect Dubinov, A. E. Kitayev, I. N. 2015-06-15 Langmuir waves in fully degenerate quantum plasma are considered. It is shown that, in the linear approximation, Langmuir waves are always dichromatic. The low-frequency component of the waves corresponds to classical Langmuir waves, while the high-frequency component, to free-electron quantum oscillations. The nonlinear problem on the profile of dichromatic Langmuir waves is solved. Solutions in the form of a superposition of waves and in the form of beatings of its components are obtained. 7. Solitary surface waves on a plasma cylinder 1983-03-01 By considering electrostatic surface waves propagating along a plasma cylinder, it is demonstrated that solitary variations in the cylinder radius may appear. The properties of these slow perturbations are determined by the surface wave intensities. 8. Plasma observations at the earth's magnetic equator NASA Technical Reports Server (NTRS) Olsen, R. C.; Shawhan, S. D.; Gallagher, D. L.; Chappell, C. R.; Green, J. L. 1987-01-01 New observations of particle and wave data from the magnetic equator from the DE 1 spacecraft are reported. The results demonstrate that the equatorial plasma population is predominantly hydrogen and that the enhanced ion fluxes observed at the equator occur without an increase in the total plasma density. Helium is occasionally found heated along with the protons, and forms about 10 percent of the equatorially trapped population at such times. The heated H(+) ions can be characterized by a bi-Maxwellian with kT(parallel) = 0.5-1.0 eV and kT = 5-50 eV, with a density of 10-100/cu cm. The total plasma density is relatively constant with latitude. First measurements of the equatorially trapped plasma and coincident UHR measurements show that the trapped plasma is found in conjunction with equatorial noise. 9. Low-Frequency Waves in Space Plasmas Keiling, Andreas; Lee, Dong-Hun; Nakariakov, Valery 2016-02-01 Low-frequency waves in space plasmas have been studied for several decades, and our knowledge gain has been incremental with several paradigm-changing leaps forward. In our solar system, such waves occur in the ionospheres and magnetospheres of planets, and around our Moon. They occur in the solar wind, and more recently, they have been confirmed in the Sun's atmosphere as well. The goal of wave research is to understand their generation, their propagation, and their interaction with the surrounding plasma. Low-frequency Waves in Space Plasmas presents a concise and authoritative up-to-date look on where wave research stands: What have we learned in the last decade? What are unanswered questions? While in the past waves in different astrophysical plasmas have been largely treated in separate books, the unique feature of this monograph is that it covers waves in many plasma regions, including: Waves in geospace, including ionosphere and magnetosphere Waves in planetary magnetospheres Waves at the Moon Waves in the solar wind Waves in the solar atmosphere Because of the breadth of topics covered, this volume should appeal to a broad community of space scientists and students, and it should also be of interest to astronomers/astrophysicists who are studying space plasmas beyond our Solar System. 10. Magnetoacoustic nonlinear periodic (cnoidal) waves in plasmas Ur-Rehman, Hafeez; Mahmood, S.; Hussain, S. 2017-01-01 Magnetoacoustic nonlinear periodic (cnoidal) waves and solitons are studied in magnetized electron-ion plasmas with inertial cold ions and warm electrons. Using the two fluid model, the dispersion relation of the magnetoacoustic waves is obtained in the linear limit and the wave dispersive effects appear through the electron inertial length. The well known reductive perturbation method is employed to derive the Korteweg-de Vries equation for magnetoacoustic waves in plasmas. The Sagdeev potential approach is used, and the cnoidal wave solution of magnetoacoustic waves is obtained under periodic boundary conditions. The analytical solution for magnetoacoustic solitons is also presented. The phase plane portraits are also plotted for magnetoacoustic solitons shown as a separatrix, and the cnoidal wave structure always lies within the separatrix. It is found that plasma beta, which depends on the plasma density, electron temperature, and magnetic field intensity, has a significant effect on the amplitude and phase of the cnoidal waves, while it also affects the width and amplitude of the magnetoacoustic soliton in plasmas. The numerical results are plotted within the plasma parameters for laboratory and space plasmas for illustration. It is found that only compressive magnetoacoustic nonlinear periodic wave and soliton structures are formed in magnetized plasmas. 11. High-latitude distributions of plasma waves and spatial irregularities from DE 2 alternating current electric field observations NASA Technical Reports Server (NTRS) Heppner, J. P.; Liebrecht, M. C.; Maynard, N. C.; Pfaff, R. F. 1993-01-01 The high-latitude spatial distributions of average signal intensities in 12 frequency channels between 4 Hz and 512 kHz as measured by the ac electric field spectrometers on the DE-2 spacecraft are analyzed for 18 mo of measurements. In MLT-INL (magnetic local time-invariant latitude) there are three distinct distributions that can be identified with 4-512 Hz signals from spatial irregularities and Alfven waves, 256-Hz to 4.1-kHz signals from ELF hiss, and 4.1-64 kHz signals from VLF auroral hiss, respectively. Overlap between ELF hiss and spatial irregularity signals occurs in the 256-512 Hz band. VLF hiss signals extend downward in frequency into the 1.0-4.1 kHz band and upward into the frequency range 128-512 kHz. The distinctly different spatial distribution patterns for the three bands, 4-256 Hz, 512-1204 Hz, and 4.1-64 kHz, indicate a lack of any causal relationships between VLF hiss, ELF hiss, and lower-frequency signals from spatial irregularities and Alfven waves. 12. Electromagnetic waves in a strong Schwarzschild plasma SciTech Connect Daniel, J.; Tajima, T. 1996-11-01 The physics of high frequency electromagnetic waves in a general relativistic plasma with the Schwarzschild metric is studied. Based on the 3 + 1 formalism, we conformalize Maxwells equations. The derived dispersion relations for waves in the plasma contain the lapse function in the plasma parameters such as in the plasma frequency and cyclotron frequency, but otherwise look {open_quotes}flat.{close_quotes} Because of this property this formulation is ideal for nonlinear self-consistent particle (PIC) simulation. Some of the physical consequences arising from the general relativistic lapse function as well as from the effects specific to the plasma background distribution (such as density and magnetic field) give rise to nonuniform wave equations and their associated phenomena, such as wave resonance, cutoff, and mode-conversion. These phenomena are expected to characterize the spectroscopy of radiation emitted by the plasma around the black hole. PIC simulation results of electron-positron plasma are also presented. 13. Sheath waves observed on OEDIPUS A James, H. G.; Balmain, K. G.; Bantin, C. C.; Hulbert, G. W. 1995-01-01 An important novel feature of the tethered sounding rocket experiment OEDIPUS A (Observations of Electric-field Distributions in the Ionospheric Plasma—A Unique Strategy) was its direct excitation and detection of electromagnetic waves on conductors in space plasmas. We present quantitative evidence about sheath waves excited in the ionosphere by a high-frequency transmitter on one end of the 1-km tether and detected by a synchronized receiver on the other end. An important characteristic of sheath waves is their sequence of sharply defined passbands and stop bands in the frequency range 0.1-5 MHz. The lowest passband is between 0.1 MHz and the plasma frequency near 2 MHz, the bandwidth where existing theory predicts sheath waves. Resonance fringes in this band have been scaled to determine the phase and group refractive indices of sheath waves. These agree reasonably well with the theory, considering the approximations therein. Passbands and stop bands observed in the range between 2 and 5 MHz are not expected on the basis of the current theory. In this range, band limits have clear signatures of the interaction of the tether fields with electrostatic cyclotron waves. Finite wire moment method modeling of the payload shows that in the low-frequency passband, RF coupling along the tether is increased by 20 dB over vacuum conditions. Similarly, isolation is greater than vacuum isolation in the stop bands. Because sheath waves at frequencies up to 2 MHz are guided efficiently along conductors in plasma, they are a significant design issue in the electromagnetic compatibility of avionics at frequencies up to HF on large metal space structures. 14. Gravity Wave Seeding of Equatorial Plasma Bubbles NASA Technical Reports Server (NTRS) Singh, Sardul; Johnson, F. S.; Power, R. A. 1997-01-01 Some examples from the Atmosphere Explorer E data showing plasma bubble development from wavy ion density structures in the bottomside F layer are described. The wavy structures mostly had east-west wavelengths of 150-800 km, in one example it was about 3000 km. The ionization troughs in the wavy structures later broke up into either a multiple-bubble patch or a single bubble, depending upon whether, in the precursor wavy structure, shorter wavelengths were superimposed on the larger scale wavelengths. In the multiple bubble patches, intrabubble spacings vaned from 55 km to 140 km. In a fully developed equatorial spread F case, east-west wavelengths from 690 km down to about 0.5 km were present simultaneously. The spacings between bubble patches or between bubbles in a patch appear to be determined by the wavelengths present in the precursor wave structure. In some cases, deeper bubbles developed on the western edge of a bubble patch, suggesting an east-west asymmetry. Simultaneous horizontal neutral wind measurements showed wavelike perturbations that were closely associated with perturbations in the plasma horizontal drift velocity. We argue that the wave structures observed here that served as the initial seed ion density perturbations were caused by gravity waves, strengthening the view that gravity waves seed equatorial spread F irregularities. 15. Coupling between electron plasma waves in laser-plasma interactions Everett, M. J.; Lal, A.; Clayton, C. E.; Mori, W. B.; Joshi, C.; Johnston, T. W. 1996-05-01 A Lagrangian fluid model (cold plasma, fixed ions) is developed for analyzing the coupling between electron plasma waves. This model shows that a small wave number electron plasma wave (ω2,k2) will strongly affect a large wave number electron plasma wave (ω1,k1), transferring its energy into daughter waves or sidebands at (ω1+nω2,k1+nk2) in the lab frame. The accuracy of the model is checked via particle-in-cell simulations, which confirm that the energy in the mode at (ω1,k1) can be completely transferred to the sidebands at (ω1+nω2,k1+nk2) by the presence of the electron plasma mode at (ω2,k2). Conclusive experimental evidence for the generation of daughter waves via this coupling is then presented using time- and wave number-resolved spectra of the light from a probe laser coherently Thomson scattered by the electron plasma waves generated by the interaction of a two-frequency CO2 laser with a plasma. 16. Fundamental plasma emission involving ion sound waves NASA Technical Reports Server (NTRS) Cairns, Iver H. 1987-01-01 The theory for fundamental plasma emission by the three-wave processes L + or - S to T (where L, S and T denote Langmuir, ion sound and transverse waves, respectively) is developed. Kinematic constraints on the characteristics and growth lengths of waves participating in the wave processes are identified. In addition the rates, path-integrated wave temperatures, and limits on the brightness temperature of the radiation are derived. 17. Visualization of Shock Wave Driven by Millimeter Wave Plasma in a Parabolic Thruster SciTech Connect Yamaguchi, Toshikazu; Shimada, Yutaka; Shiraishi, Yuya; Shibata, Teppei; Komurasaki, Kimiya; Oda, Yasuhisa; Kajiwara, Ken; Takahashi, Koji; Kasugai, Atsushi; Sakamoto, Keishi; Arakawa, Yoshihiro 2010-05-06 By focusing a high-power millimeter wave beam generated by a 170 GHz gyrotron, a breakdown occurred and a shock wave was driven by plasma heated by following microwave energy. The shock wave and the plasma around a focal point of a parabolic thruster were visualized by a shadowgraph method, and a transition of structures between the shock wave and the plasma was observed. There was a threshold local power density to make the transition, and the propagation velocity at the transition was around 800 m/s. 18. Parametric amplification of a superconducting plasma wave Rajasekaran, S.; Casandruc, E.; Laplace, Y.; Nicoletti, D.; Gu, G. D.; Clark, S. R.; Jaksch, D.; Cavalleri, A. 2016-11-01 Many applications in photonics require all-optical manipulation of plasma waves, which can concentrate electromagnetic energy on sub-wavelength length scales. This is difficult in metallic plasmas because of their small optical nonlinearities. Some layered superconductors support Josephson plasma waves, involving oscillatory tunnelling of the superfluid between capacitively coupled planes. Josephson plasma waves are also highly nonlinear, and exhibit striking phenomena such as cooperative emission of coherent terahertz radiation, superconductor-metal oscillations and soliton formation. Here, we show that terahertz Josephson plasma waves can be parametrically amplified through the cubic tunnelling nonlinearity in a cuprate superconductor. Parametric amplification is sensitive to the relative phase between pump and seed waves, and may be optimized to achieve squeezing of the order-parameter phase fluctuations or terahertz single-photon devices. 19. Parametric amplification of a superconducting plasma wave SciTech Connect Rajasekaran, S.; Casandruc, E.; Laplace, Y.; Nicoletti, D.; Gu, G. D.; Clark, S. R.; Jaksch, D.; Cavalleri, A. 2016-07-11 Many applications in photonics require all-optical manipulation of plasma waves, which can concentrate electromagnetic energy on sub-wavelength length scales. This is difficult in metallic plasmas because of their small optical nonlinearities. Some layered superconductors support Josephson plasma waves, involving oscillatory tunnelling of the superfluid between capacitively coupled planes. Josephson plasma waves are also highly nonlinear, and exhibit striking phenomena such as cooperative emission of coherent terahertz radiation, superconductor–metal oscillations and soliton formation. In this paper, we show that terahertz Josephson plasma waves can be parametrically amplified through the cubic tunnelling nonlinearity in a cuprate superconductor. Finally, parametric amplification is sensitive to the relative phase between pump and seed waves, and may be optimized to achieve squeezing of the order-parameter phase fluctuations or terahertz single-photon devices. 20. Parametric amplification of a superconducting plasma wave DOE PAGES Rajasekaran, S.; Casandruc, E.; Laplace, Y.; ... 2016-07-11 Many applications in photonics require all-optical manipulation of plasma waves, which can concentrate electromagnetic energy on sub-wavelength length scales. This is difficult in metallic plasmas because of their small optical nonlinearities. Some layered superconductors support Josephson plasma waves, involving oscillatory tunnelling of the superfluid between capacitively coupled planes. Josephson plasma waves are also highly nonlinear, and exhibit striking phenomena such as cooperative emission of coherent terahertz radiation, superconductor–metal oscillations and soliton formation. In this paper, we show that terahertz Josephson plasma waves can be parametrically amplified through the cubic tunnelling nonlinearity in a cuprate superconductor. Finally, parametric amplification is sensitivemore » to the relative phase between pump and seed waves, and may be optimized to achieve squeezing of the order-parameter phase fluctuations or terahertz single-photon devices.« less 1. Acceleration of injected electrons by the plasma beat wave accelerator Joshi, C.; Clayton, C. E.; Marsh, K. A.; Dyson, A.; Everett, M.; Lal, A.; Leemans, W. P.; Williams, R.; Katsouleas, T.; Mori, W. B. 1992-07-01 In this paper we describe the recent work at UCLA on the acceleration of externally injected electrons by a relativistic plasma wave. A two frequency laser was used to excite a plasma wave over a narrow range of static gas pressures close to resonance. Electrons with energies up to our detection limit of 9.1 MeV were observed when 2.1 MeV electrons were injected in the plasma wave. No accelerated electrons above the detection threshold were observed when the laser was operated on a single frequency or when no electrons were injected. Experimental results are compared with theoretical predictions, and future prospects for the plasma beat wave accelerator are discussed. 2. Electron Beam Transport in Advanced Plasma Wave Accelerators SciTech Connect Williams, Ronald L 2013-01-31 The primary goal of this grant was to develop a diagnostic for relativistic plasma wave accelerators based on injecting a low energy electron beam (5-50keV) perpendicular to the plasma wave and observing the distortion of the electron beam's cross section due to the plasma wave's electrostatic fields. The amount of distortion would be proportional to the plasma wave amplitude, and is the basis for the diagnostic. The beat-wave scheme for producing plasma waves, using two CO2 laser beam, was modeled using a leap-frog integration scheme to solve the equations of motion. Single electron trajectories and corresponding phase space diagrams were generated in order to study and understand the details of the interaction dynamics. The electron beam was simulated by combining thousands of single electrons, whose initial positions and momenta were selected by random number generators. The model was extended by including the interactions of the electrons with the CO2 laser fields of the beat wave, superimposed with the plasma wave fields. The results of the model were used to guide the design and construction of a small laboratory experiment that may be used to test the diagnostic idea. 3. Excitation of Chirping Whistler Waves in a Laboratory Plasma Van Compernolle, B.; An, X.; Bortnik, J.; Thorne, R. M.; Gekelman, W. N.; Pribyl, P. 2015-12-01 Whistler mode chorus emissions with a characteristic frequency chirp are an important magnetospheric wave, responsible for the acceleration of outer radiation belt electrons to relativistic energies and also for the scattering loss of these electrons into the atmosphere. Here, we report on the first laboratory experiment where whistler waves exhibiting fast frequency chirping have been artificially produced using a beam of energetic electrons launched into a cold plasma. Frequency chirps are only observed for a narrow range of plasma and beam parameters, and show a strong dependence on beam density, plasma density and magnetic field gradient. Broadband whistler waves similar to magnetospheric hiss are also observed, and the parameter ranges for each emission are quantified. The research was funded by NSF/DOE Plasma Partnership program by grant DE-SC0010578. Work was done at the Basic Plasma Science Facility (BAPSF) also funded by NSF/DOE. 4. DC and Wave Electric Fields and Other Plasma Parameters Observed on Two Sounding Rockets in the Dark Cusp During IMF Bz North and South Conditions NASA Technical Reports Server (NTRS) Pfaff, R. F.; Acuna, M.; Bounds, S.; Farrell, W.; Freudenreich, H.; Lepping, R.; Vondrak, R.; Maynard, N. C.; Moen, J.; Egeland, A. 1997-01-01 Two Black Brant IX sounding rockets were launched into the dark, dayside cusp near magnetic noon on December 2 and 3, 1997, from Ny Alesund, Spitzbergen at 79 N reaching altitudes of approximately 450 km. Real-time ground-based and Wind IMF data were used to determine the launch conditions. The first launch, with Bz north conditions, crossed into and back out of an open field region with merging poleward of the projected trajectory. The second flight, into Bz south conditions, was timed to coincide with an enhancement in the merging rate from a increase in the negative Bz, while the DMSP F13 satellite was situated slightly to the north of the rocket trajectory. Each payload returned DC electric and magnetic fields, plasma waves, energetic particles, photometer data, and thermal plasma data. Data from both flights will be shown, with an emphasis on the DC electric field results. In particular, the data gathered on December 2, 1997 will be used to discuss ionospheric signatures of merging and the open/closed character of the the cusp/low latitude boundary layer. In contrast, the data gathered on December 3, 1997 shows evidence of pulsed electric field structures which will be examined in the context of cusp plasma entry processes. Both data sets returned a rich variety of plasma waves, as well as optical emissions and thermal plasma data. 5. DC and Wave Electric Fields and Other Plasma Parameters Observed on Two Sounding Rockets in the Dark Cusp during IMF BZ North and South Conditions NASA Technical Reports Server (NTRS) Pfaff, R. F.; Bounds, S.; Acuna, M.; Maynard, N. C.; Moen, J.; Egeland, A.; Holtet, J.; Maseide, K.; Sandholt, P. E.; Soraas, F. 1999-01-01 Two Black Brant IX sounding rockets were launched into the dark, dayside cusp near magnetic noon on December 2 and 3, 1997, from Ny Alesund, Spitzbergen at 79degN reaching altitudes of approximately 450 km. Real-time ground-based and Wind (interplanetary magnetic field) IMF data were used to determine the launch conditions. The first launch, with Bz north conditions, crossed into and back out of an open field region with merging poleward of the projected trajectory. The second flight, into Bz south conditions, was timed to coincide with an enhancement in the merging rate from a increase in the negative Bz, while the (Defense Meteorological Satellite Program) DMSP F13 satellite was situated slightly to the north of the rocket trajectory. Each payload returned DC electric and magnetic fields, plasma waves, energetic particles, photometer data, and thermal plasma data. Data from both flights will be shown, with an emphasis on the DC electric field results. In particular, the data gathered on December 2, 1997 will be used to discuss ionospheric signatures of merging and the open/closed character of the the cusp/low latitude boundary layer. In contrast, the data gathered on December 3, 1997 shows evidence of pulsed electric field structures which will be examined in the context of cusp plasma entry processes. Both data sets returned a rich variety of plasma waves, as well as optical emissions and thermal plasma data. 6. DC and Wave Electric Fields and Other Plasma Parameters Observed on Two Sounding Rockets in the Dark Cusp during IMF Bz North and South Conditions NASA Technical Reports Server (NTRS) Pfaff, R. F.; Acuna, M.; Bounds, S.; Farrell, W.; Freudenreich, W.; Lepping, R.; Vondrak, R.; Maynard, N. C.; Moen, J.; Egeland, A. 1999-01-01 Two Black Brant IX sounding rockets were launched into the dark, dayside cusp near magnetic noon on December 2 and 3, 1997, from Ny Alesund, Spitzbergen at 79 deg N reaching altitudes of about 450 km. Real-time ground-based and Wind IMF data were used to determine the launch conditions. The first launch, with Bz north conditions, crossed into and back out of an open field region with merging poleward of the projected trajectory. The second flight, into Bz south conditions, was timed to coincide with an enhancement in the merging rate from a increase in the negative Bz, while the DMSP Fl 3 satellite was situated slightly to the north of the rocket trajectory. Each payload returned DC electric and magnetic fields, plasma waves, energetic particles, photometer data, and thermal plasma data. Data from both flights will be shown, with an emphasis on the DC electric field results. In particular, the data gathered on December 2, 1997 will be used to discuss ionospheric signatures of merging and the open/closed character of the the cusp/low latitude boundary layer. In contrast, the data gathered on December 3, 1997 shows evidence of pulsed electric field structures which will be examined in the context of cusp plasma entry processes. Both data sets returned a rich variety of plasma waves, as well as optical emissions and thermal plasma data. 7. Plasma wave aided two photon decay of an electromagnetic wave in a plasma SciTech Connect Kumar, K. K. Magesh; Singh, Rohtash; Krishan, Vinod 2014-11-15 The presence of a Langmuir wave in an unmagnetized plasma is shown to allow parametric decay of an electromagnetic wave into two electromagnetic waves, which is otherwise not allowed due to wave number mismatch. The decay occurs at plasma densities below one ninth the critical density and the decay waves propagate at finite angles to the pump laser. Above the threshold, the growth rate scales linearly with the amplitude of the Langmuir wave and the amplitude of the pump electromagnetic wave. The frequency ω of the lower frequency decay wave increases with the angle its propagation vector makes with that of the pump. The growth rate, however, decreases with ω. 8. Wave-driven Countercurrent Plasma Centrifuge SciTech Connect A.J. Fetterman and N.J. Fisch 2009-03-20 A method for driving rotation and a countercurrent flow in a fully ionized plasma centrifuge is described. The rotation is produced by radiofrequency waves near the cyclotron resonance. The wave energy is transferred into potential energy in a manner similar to the α channeling effect. The countercurrent flow may also be driven by radiofrequency waves. By driving both the rotation and the flow pattern using waves instead of electrodes, physical and engineering issues may be avoided. 9. Computational study of nonlinear plasma waves. [plasma simulation model applied to electrostatic waves in collisionless plasma NASA Technical Reports Server (NTRS) Matsuda, Y. 1974-01-01 A low-noise plasma simulation model is developed and applied to a series of linear and nonlinear problems associated with electrostatic wave propagation in a one-dimensional, collisionless, Maxwellian plasma, in the absence of magnetic field. It is demonstrated that use of the hybrid simulation model allows economical studies to be carried out in both the linear and nonlinear regimes with better quantitative results, for comparable computing time, than can be obtained by conventional particle simulation models, or direct solution of the Vlasov equation. The characteristics of the hybrid simulation model itself are first investigated, and it is shown to be capable of verifying the theoretical linear dispersion relation at wave energy levels as low as .000001 of the plasma thermal energy. Having established the validity of the hybrid simulation model, it is then used to study the nonlinear dynamics of monochromatic wave, sideband instability due to trapped particles, and satellite growth. 10. Observation of an antenna-plasma instability NASA Technical Reports Server (NTRS) Kellogg, P. J.; Monson, S. J.; Whalen, Brian A. 1990-01-01 This paper investigates the conditions leading to, and the causes of, the phenomenon observed during a rocket flight (to 585-km altitude) of an occasionally occurring narrow-band signal on an electric antenna whose frequency was found to vary as the rocket turned. The amplitude also varied by a factor larger than can be explained by variable coupling to the plasma; maximum oscillation amplitude occurred when the antenna was aligned with the earth's magnetic field. A tentative explanation of this phenomenon is given, suggesting that the signals on the antenna were caused by an interacton of the flowing plasma with sheath waves around the antenna. 11. Terahertz wave absorption via preformed air plasma Zhao, Ji; Zhang, LiangLiang; Wu, Tong; Zhang, CunLin; Zhao, YueJin 2016-12-01 Terahertz wave generation from laser-induced air plasma has continued to be an exciting field of research over the course of the past decade. In this paper, we report on an investigation concerning terahertz wave absorption with preformed plasma created by another laser pulse. We examine terahertz absorption behavior by varying the pump power and then analyze the polarization effect of the preplasma beam on terahertz wave absorption. The results of experiments conducted in which a type-I beta barium borate (BBO) crystal is placed before the preformed air plasma indicate that the fundamental (ω) and second harmonic (2ω) pulses can also influence terahertz absorption. 12. Observation and Control of Shock Waves in Individual Nanoplasmas DTIC Science & Technology 2014-03-18 Observation and Control of Shock Waves in Individual Nanoplasmas Daniel D. Hickstein,1 Franklin Dollar,1 Jim A. Gaffney,2 Mark E. Foord,2 George M...distribution of individual, isolated 100-nm-scale plasmas, we make the first experimental observation of shock waves in nanoplasmas . We demonstrate that...i Nanoscale plasmas ( nanoplasmas ) offer enhanced laser absorption compared to solid or gas targets [1], enabling high-energy physics with tabletop 13. Evolution Of Nonlinear Waves in Compressing Plasma SciTech Connect P.F. Schmit, I.Y. Dodin, and N.J. Fisch 2011-05-27 Through particle-in-cell simulations, the evolution of nonlinear plasma waves is examined in one-dimensional collisionless plasma undergoing mechanical compression. Unlike linear waves, whose wavelength decreases proportionally to the system length L(t), nonlinear waves, such as solitary electron holes, conserve their characteristic size {Delta} during slow compression. This leads to a substantially stronger adiabatic amplification as well as rapid collisionless damping when L approaches {Delta}. On the other hand, cessation of compression halts the wave evolution, yielding a stable mode. 14. Terahertz waves radiated from two noncollinear femtosecond plasma filaments SciTech Connect Du, Hai-Wei; Hoshina, Hiromichi; Otani, Chiko; Midorikawa, Katsumi 2015-11-23 Terahertz (THz) waves radiated from two noncollinear femtosecond plasma filaments with a crossing angle of 25° are investigated. The irradiated THz waves from the crossing filaments show a small THz pulse after the main THz pulse, which was not observed in those from single-filament scheme. Since the position of the small THz pulse changes with the time-delay of two filaments, this phenomenon can be explained by a model in which the small THz pulse is from the second filament. The denser plasma in the overlap region of the filaments changes the movement of space charges in the plasma, thereby changing the angular distribution of THz radiation. As a result, this schematic induces some THz wave from the second filament to propagate along the path of the THz wave from the first filament. Thus, this schematic alters the direction of the THz radiation from the filamentation, which can be used in THz wave remote sensing. 15. Electron cyclotron harmonic waves observed by the AMPTE-IRM plasma wave experiment following a lithium release in the solar wind NASA Technical Reports Server (NTRS) Roeder, J. L.; Koons, H. C.; Holzworth, R. H.; Anderson, R. R.; Bauer, O. H. 1987-01-01 An unexpected occurrence following the second lithium release by the AMPTE-IRM spacecraft in the solar wind on September 20, 1984, was the appearance of electron cyclotron harmonic emissions. These emissions began about 50 s after the release and continued for several minutes. Narrow-band emissions polarized perpendicular to the magnetic field with amplitudes of approximately 0.00001 V/m were observed in each of the first five harmonic bands. The diffuse emissions extended from below the lowest measured frequency channel to above the highest narrow-band emission with a maximum below the electron cyclotron frequency. It will be shown that these observations are inconsistent with their generation by several ion beam instabilities. 16. Fast wave evanescence in filamentary boundary plasmas SciTech Connect Myra, J. R. 2014-02-15 Radio frequency waves for heating and current drive of plasmas in tokamaks and other magnetic confinement devices must first traverse the scrape-off-layer (SOL) before they can be put to their intended use. The SOL plasma is strongly turbulent and intermittent in space and time. These turbulent properties of the SOL, which are not routinely taken into account in wave propagation codes, can have an important effect on the coupling of waves through an evanescent SOL or edge plasma region. The effective scale length for fast wave (FW) evanescence in the presence of short-scale field-aligned filamentary plasma turbulence is addressed in this paper. It is shown that although the FW wavelength or evanescent scale length is long compared with the dimensions of the turbulence, the FW does not simply average over the turbulent density; rather, the average is over the exponentiation rate. Implications for practical situations are discussed. 17. Wave-particle energy exchange directly observed in a kinetic Alfvén-branch wave Gershman, Daniel J.; F-Viñas, Adolfo; Dorelli, John C.; Boardsen, Scott A.; Avanov, Levon A.; Bellan, Paul M.; Schwartz, Steven J.; Lavraud, Benoit; Coffey, Victoria N.; Chandler, Michael O.; Saito, Yoshifumi; Paterson, William R.; Fuselier, Stephen A.; Ergun, Robert E.; Strangeway, Robert J.; Russell, Christopher T.; Giles, Barbara L.; Pollock, Craig J.; Torbert, Roy B.; Burch, James L. 2017-03-01 Alfvén waves are fundamental plasma wave modes that permeate the universe. At small kinetic scales, they provide a critical mechanism for the transfer of energy between electromagnetic fields and charged particles. These waves are important not only in planetary magnetospheres, heliospheres and astrophysical systems but also in laboratory plasma experiments and fusion reactors. Through measurement of charged particles and electromagnetic fields with NASA's Magnetospheric Multiscale (MMS) mission, we utilize Earth's magnetosphere as a plasma physics laboratory. Here we confirm the conservative energy exchange between the electromagnetic field fluctuations and the charged particles that comprise an undamped kinetic Alfvén wave. Electrons confined between adjacent wave peaks may have contributed to saturation of damping effects via nonlinear particle trapping. The investigation of these detailed wave dynamics has been unexplored territory in experimental plasma physics and is only recently enabled by high-resolution MMS observations. 18. Magnetoacoustic shock waves in dissipative degenerate plasmas SciTech Connect Hussain, S.; Mahmood, S. 2011-11-15 Quantum magnetoacoustic shock waves are studied in homogenous, magnetized, dissipative dense electron-ion plasma by using two fluid quantum magneto-hydrodynamic (QMHD) model. The weak dissipation effects in the system are taken into account through kinematic viscosity of the ions. The reductive perturbation method is employed to derive Korteweg-de Vries Burgers (KdVB) equation for magnetoacoustic wave propagating in the perpendicular direction to the external magnetic field in dense plasmas. The strength of magnetoacoustic shock is investigated with the variations in plasma density, magnetic field intensity, and ion kinematic viscosity of dense plasma system. The necessary condition for the existence of monotonic and oscillatory shock waves is also discussed. The numerical results are presented for illustration by using the data of astrophysical dense plasma situations such as neutron stars exist in the literature. 19. Electromagnetic wave in a relativistic magnetized plasma SciTech Connect Krasovitskiy, V. B. 2009-12-15 Results are presented from a theoretical investigation of the dispersion properties of a relativistic plasma in which an electromagnetic wave propagates along an external magnetic field. The dielectric tensor in integral form is simplified by separating its imaginary and real parts. A dispersion relation for an electromagnetic wave is obtained that makes it possible to analyze the dispersion and collisionless damping of electromagnetic perturbations over a broad parameter range for both nonrelativistic and ultrarelativistic plasmas. 20. Laboratory observations of self-excited dust acoustic shock waves Merlino, Robert L.; Heinrich, Jonathon R.; Kim, Su-Hyun 2009-11-01 Dust acoustic waves have been discussed in connection with dust density structures in Saturn's rings and the Earth's mesosphere, and as a possible mechanism for triggering condensation of small grains in dust molecular clouds. Dust acoustic waves are a ubiquitous occurrence in laboratory dusty plasmas formed in glow discharges. We report observations of repeated, self-excited dust acoustic shock waves in a dc glow discharge dusty plasma using high-speed video imaging. Two major observations will be presented: (1) The self-steepening of a nonlinear dust acoustic wave into a saw-tooth wave with sharp gradient in dust density, very similar to those found in numerical solutions [1] of the fully nonlinear fluid equations for nondispersive dust acoustic waves, and (2) the collision and confluence of two dust acoustic shock waves. [4pt] [1] B. Eliasson and P. K. Shukla, Phys. Rev. E 69, 067401 (2004). 1. Creating an anisotropic plasma resistivity with waves SciTech Connect Fisch, N.J.; Boozer, A.H. 1980-05-01 An anisotropic plasma resistivity may be created by preferential heating of electrons traveling in one direction. This can result in a steady-state toroidal current in a tokamak even in the absence of net wave momentum. In fact, at high wave phase velocities, the current associated with the change in resistivity is greater than that associated with net momentum input. An immediate implication is that other waves, such as electron cyclotron waves, may be competitive with lower-hybrid waves as a means for generating current. An analytical expression is derived for the current generated per power dissipated which agrees remarkably well with numerical calculations. 2. BOOK REVIEW: Kinetic theory of plasma waves, homogeneous plasmas Porkolab, Miklos 1998-11-01 The linear theory of plasma waves in homogeneous plasma is arguably the most mature and best understood branch of plasma physics. Given the recently revised version of Stix's excellent Waves in Plasmas (1992), one might ask whether another book on this subject is necessary only a few years later. The answer lies in the scope of this volume; it is somewhat more detailed in certain topics than, and complementary in many fusion research relevant areas to, Stix's book. (I am restricting these comments to the homogeneous plasma theory only, since the author promises a second volume on wave propagation in inhomogeneous plasmas.) This book is also much more of a theorist's approach to waves in plasmas, with the aim of developing the subject within the logical framework of kinetic theory. This may indeed be pleasing to the expert and to the specialist, but may be too difficult to the graduate student as an introduction' to the subject (which the author explicitly states in the Preface). On the other hand, it may be entirely appropriate for a second course on plasma waves, after the student has mastered fluid theory and an introductory kinetic treatment of waves in a hot magnetized Vlasov' plasma. For teaching purposes, my personal preference is to review the cold plasma wave treatment using the unified Stix formalism and notation (which the author wisely adopts in the present book, but only in Chapter 5). Such an approach allows one to deal with CMA diagrams early on, as well as to provide a framework to discuss electromagnetic wave propagation and accessibility in inhomogeneous plasmas (for which the cold plasma wave treatment is perfectly adequate). Such an approach does lack some of the rigour, however, that the author achieves with the present approach. As the author correctly shows, the fluid theory treatment of waves follows logically from kinetic theory in the cold plasma limit. I only question the pedagogical value of this approach. Otherwise, I welcome this 3. Electromagnetic drift waves dispersion for arbitrarily collisional plasmas SciTech Connect Lee, Wonjae Krasheninnikov, Sergei I.; Angus, J. R. 2015-07-15 The impacts of the electromagnetic effects on resistive and collisionless drift waves are studied. A local linear analysis on an electromagnetic drift-kinetic equation with Bhatnagar-Gross-Krook-like collision operator demonstrates that the model is valid for describing linear growth rates of drift wave instabilities in a wide range of plasma parameters showing convergence to reference models for limiting cases. The wave-particle interactions drive collisionless drift-Alfvén wave instability in low collisionality and high beta plasma regime. The Landau resonance effects not only excite collisionless drift wave modes but also suppress high frequency electron inertia modes observed from an electromagnetic fluid model in collisionless and low beta regime. Considering ion temperature effects, it is found that the impact of finite Larmor radius effects significantly reduces the growth rate of the drift-Alfvén wave instability with synergistic effects of high beta stabilization and Landau resonance. 4. Solitary Surface Waves at a Plasma Boundary A new equation describing the behaviour of strongly nonlinear waves localized near the boundary of a semi-infinite plasma is deduced. This equation has solitary wave solutions that can be found numerically. Various limiting cases are treated analytically in the present paper. 5. Chaotic ion motion in magnetosonic plasma waves NASA Technical Reports Server (NTRS) Varvoglis, H. 1984-01-01 The motion of test ions in a magnetosonic plasma wave is considered, and the 'stochasticity threshold' of the wave's amplitude for the onset of chaotic motion is estimated. It is shown that for wave amplitudes above the stochasticity threshold, the evolution of an ion distribution can be described by a diffusion equation with a diffusion coefficient D approximately equal to 1/v. Possible applications of this process to ion acceleration in flares and ion beam thermalization are discussed. 6. Gabor Wave Packet Method to Solve Plasma Wave Equations SciTech Connect A. Pletzer; C.K. Phillips; D.N. Smithe 2003-06-18 A numerical method for solving plasma wave equations arising in the context of mode conversion between the fast magnetosonic and the slow (e.g ion Bernstein) wave is presented. The numerical algorithm relies on the expansion of the solution in Gaussian wave packets known as Gabor functions, which have good resolution properties in both real and Fourier space. The wave packets are ideally suited to capture both the large and small wavelength features that characterize mode conversion problems. The accuracy of the scheme is compared with a standard finite element approach. 7. Weak Wave Coupling Through Plasma Inhomogeneity Swanson, D. G. 1998-11-01 Some effects of linear wave coupling due to effects of plasma inhomogeneity are well known through the process of mode conversion(D. G. Swanson, Theory of Mode Conversion and Tunneling in Inhomogenous Plasmas), (John Wiley & Sons, New York, 1998).. Another type of resonant coupling in a periodically inhomogeneous plasma has been recently found(V. A. Svidzinski and D. G. Swanson, Physics of Plasmas series 5), 486 (1998)., but any two waves will generally be coupled if the plasma is inhomogeneous, although the coupling may be weak. If the wavelengths are close, nearly all of the energy in one mode may be transferred to the other mode over a distance that depends on the coupling strength. The coupling strength depends on gradients of the plasma parameters. This means that the coupling may occur over an extended region in space, but that substantial amounts of wave energy may be transferred to a wave traditionally thought to be independent. Low-frequency Alfvén waves are shown to be a good example of this type of coupling. 8. Nonlinear extraordinary wave in dense plasma SciTech Connect Krasovitskiy, V. B.; Turikov, V. A. 2013-10-15 Conditions for the propagation of a slow extraordinary wave in dense magnetized plasma are found. A solution to the set of relativistic hydrodynamic equations and Maxwell’s equations under the plasma resonance conditions, when the phase velocity of the nonlinear wave is equal to the speed of light, is obtained. The deviation of the wave frequency from the resonance frequency is accompanied by nonlinear longitudinal-transverse oscillations. It is shown that, in this case, the solution to the set of self-consistent equations obtained by averaging the initial equations over the period of high-frequency oscillations has the form of an envelope soliton. The possibility of excitation of a nonlinear wave in plasma by an external electromagnetic pulse is confirmed by numerical simulations. 9. Twisted electron-acoustic waves in plasmas Aman-ur-Rehman, Ali, S.; Khan, S. A.; Shahzad, K. 2016-08-01 In the paraxial limit, a twisted electron-acoustic (EA) wave is studied in a collisionless unmagnetized plasma, whose constituents are the dynamical cold electrons and Boltzmannian hot electrons in the background of static positive ions. The analytical and numerical solutions of the plasma kinetic equation suggest that EA waves with finite amount of orbital angular momentum exhibit a twist in its behavior. The twisted wave particle resonance is also taken into consideration that has been appeared through the effective wave number qeff accounting for Laguerre-Gaussian mode profiles attributed to helical phase structures. Consequently, the dispersion relation and the damping rate of the EA waves are significantly modified with the twisted parameter η, and for η → ∞, the results coincide with the straight propagating plane EA waves. Numerically, new features of twisted EA waves are identified by considering various regimes of wavelength and the results might be useful for transport and trapping of plasma particles in a two-electron component plasma. 10. Observation of an antenna-plasma instability SciTech Connect Kellogg, P.J.; Monson, S.J. ); Whalen, B.A. ) 1990-06-01 During a rocket flight to high altitude (585 km) a narrow band signal on an electric antenna was sometimes observed, whose frequency varied as the rocket turned. Such a signal cannot be natural, but apparently must be generated by the interaction of the rocket-antenna system with the ambient plasma. Conditions for development of the instability are investigated. Maximum oscillation amplitude occurs when the antenna is aligned with the Earth's magnetic field. Similar observations have been reported by Gurnett and Mosier (1969). Several attempts are made to understand the nature of this interaction, but without success. The instability treated by Fiala, due to the interaction of an inductive antenna impedance with stray capacitance to a phase shifted point in the preamplifier, can be ruled out. It appears that a negative antenna resistance due to interaction with waves Doppler-shifted through zero is an unlikely explanation. The rocket velocity seems too small to give such an anomalous Doppler shift, and even if the observations of plasma density and temperature are stretched, the positive sheath resistance is larger than calculated negative resistances. Ion transit time instability in the sheath would only work at 10 times higher frequency. Interaction of the flowing plasma with sheath waves around the antenna is suggested and appears promising but the theory is not sufficiently developed for meaningful comparison. 11. Plasma waves associated with energetic particles streaming into the solar wind from the earth's bow shock NASA Technical Reports Server (NTRS) Anderson, R. R.; Eastman, T. E.; Gurnett, D. A.; Frank, L. A.; Parks, G. K. 1981-01-01 Plasma wave and plasma data from ISEE 1 and 2 are examined. In the upstream solar wind, three dominant types of plasma waves are observed which are associated with energetic particle streams coming from the bow shock: ion acoustic waves, electron plasma oscillations, and whistler mode waves. The ion acoustic waves occur simultaneously with either ion beams or a dispersed ion population in the energy range from 0.5 to greater than 45 keV. The electron plasma oscillations are long-wavelength, nearly monochromatic electrostatic waves which are closely correlated with the flux of low-energy electrons, especially in the 0.2-1.5 keV range. Electromagnetic waves with frequencies below 200 Hz are observed when either ion beams or dispersed ion distributions are present; for these waves the refractive index determined from the wave B to E ratio is consistent with whistler mode radiation. 12. Scattering of radio frequency waves by turbulence in fusion plasmas Ram, Abhay K. 2016-10-01 In tokamak fusion plasmas, coherent fluctuations in the form of blobs or filaments and incoherent fluctuations due to turbulence are routinely observed in the scrape-off layer. Radio frequency (RF) electromagnetic waves, excited by antenna structures placed near the wall of a tokamak, have to propagate through the scrape-off layer before reaching the core of the plasma. While the effect of fluctuations on RF waves has not been quantified experimentally, there are telltale signs, arising from differences between results from simulations and from experiments, that fluctuations can modify the spectrum of RF waves. Any effect on RF waves in the scrape-off layer can have important experimental consequences. For example, electron cyclotron waves are expected to stabilize the deleterious neoclassical tearing mode (NTM) in ITER. Spectral and polarization changes due to scattering will modify the spatial location and profile of the current driven by the RF waves, thereby affecting the control of NTMs. Pioneering theoretical studies and complementary computer simulations have been pursued to elucidate the impact of fluctuations on RF waves. From the full complement of Maxwell's equations for cold, magnetized plasmas, it is shown that the Poynting flux in the wake of filaments develops spatial structure due to diffraction and shadowing. The uniformity of power flow into the plasma is affected by side-scattering, modifications to the wave spectrum, and coupling to plasma waves other than the incident RF wave. The Snell's law and the Fresnel equations have been reformulated within the context of magnetized plasmas. They are distinctly different from their counterparts in scalar dielectric media, and reveal new and important physical insight into the scattering of RF waves. The Snell's law and Fresnel equations are the basis for the Kirchhoff approximation necessary to determine properties of the scattered waves. Furthermore, this theory is also relevant for studying back 13. Saturation of Langmuir waves in laser-produced plasmas SciTech Connect Baker, K.L. 1996-04-01 This dissertation deals with the interaction of an intense laser with a plasma (a quasineutral collection of electrons and ions). During this interaction, the laser drives large-amplitude waves through a class of processes known as parametric instabilities. Several such instabilities drive one type of wave, the Langmuir wave, which involves oscillations of the electrons relative to the nearly-stationary ions. There are a number of mechanisms which limit the amplitude to which Langmuir waves grow. In this dissertation, these mechanisms are examined to identify qualitative features which might be observed in experiments and/or simulations. In addition, a number of experiments are proposed to specifically look for particular saturation mechanisms. In a plasma, a Langmuir wave can decay into an electromagnetic wave and an ion wave. This parametric instability is proposed as a source for electromagnetic emission near half of the incident laser frequency observed from laser-produced plasmas. This interpretation is shown to be consistent with existing experimental data and it is found that one of the previous mechanisms used to explain such emission is not. The scattering version of the electromagnetic decay instability is shown to provide an enhanced noise source of electromagnetic waves near the frequency of the incident laser. 14. Lower hybrid drift waves: space observations. PubMed Norgren, Cecilia; Vaivads, Andris; Khotyaintsev, Yuri V; André, Mats 2012-08-03 Lower hybrid drift waves (LHDWs) are commonly observed at plasma boundaries in space and laboratory, often having the strongest measured electric fields within these regions. We use data from two of the Cluster satellites (C3 and C4) located in Earth's magnetotail and separated by a distance of the order of the electron gyroscale. These conditions allow us, for the first time, to make cross-spacecraft correlations of the LHDWs and to determine the phase velocity and wavelength of the LHDWs. Our results are in good agreement with the theoretical prediction. We show that the electrostatic potential of LHDWs is linearly related to fluctuations in the magnetic field magnitude, which allows us to determine the velocity vector through the relation ∫δEdt·v = ϕ(δB)(∥). The electrostatic potential fluctuations correspond to ∼10% of the electron temperature, which suggests that the waves can strongly affect the electron dynamics. 15. Nonplanar electrostatic shock waves in dense plasmas SciTech Connect Masood, W.; Rizvi, H. 2010-02-15 Two-dimensional quantum ion acoustic shock waves (QIASWs) are studied in an unmagnetized plasma consisting of electrons and ions. In this regard, a nonplanar quantum Kadomtsev-Petviashvili-Burgers (QKPB) equation is derived using the small amplitude perturbation expansion method. Using the tangent hyperbolic method, an analytical solution of the planar QKPB equation is obtained and subsequently used as the initial profile to numerically solve the nonplanar QKPB equation. It is observed that the increasing number density (and correspondingly the quantum Bohm potential) and kinematic viscosity affect the propagation characteristics of the QIASW. The temporal evolution of the nonplanar QIASW is investigated both in Cartesian and polar planes and the results are discussed from the numerical stand point. The results of the present study may be applicable in the study of propagation of small amplitude localized electrostatic shock structures in dense astrophysical environments. 16. Scattering of radio frequency waves by blobs in tokamak plasmas SciTech Connect Ram, Abhay K.; Hizanidis, Kyriakos; Kominis, Yannis 2013-05-15 The density fluctuations and blobs present in the edge region of magnetic fusion devices can scatter radio frequency (RF) waves through refraction, reflection, diffraction, and coupling to other plasma waves. This, in turn, affects the spectrum of the RF waves and the electromagnetic power that reaches the core of the plasma. The usual geometric optics analysis of RF scattering by density blobs accounts for only refractive effects. It is valid when the amplitude of the fluctuations is small, of the order of 10%, compared to the background density. In experiments, density fluctuations with much larger amplitudes are routinely observed, so that a more general treatment of the scattering process is needed. In this paper, a full-wave model for the scattering of RF waves by a blob is developed. The full-wave approach extends the range of validity well beyond that of geometric optics; however, it is theoretically and computationally much more challenging. The theoretical procedure, although similar to that followed for the Mie solution of Maxwell's equations, is generalized to plasmas in a magnetic field. Besides diffraction and reflection, the model includes coupling to a different plasma wave than the one imposed by the external antenna structure. In the model, it is assumed that the RF waves interact with a spherical blob. The plasma inside and around the blob is cold, homogeneous, and imbedded in a uniform magnetic field. After formulating the complete analytical theory, the effect of the blob on short wavelength electron cyclotron waves and longer wavelength lower hybrid waves is studied numerically. 17. Plasma waves associated with the first AMPTE magnetotail barium release NASA Technical Reports Server (NTRS) Gurnett, D. A.; Anderson, R. R.; Bernhardt, P. A.; Luehr, H.; Haerendel, G. 1986-01-01 Plasma waves observed during the March 21, 1985, AMPTE magnetotail barium release are described. Electron plasma oscillations provided local measurements of the plasma density during both the expansion and decay phases. Immediately after the explosion, the electron density reached a peak of about 400,000/cu cm, and then started decreasing approximately as t to the -2.4 as the cloud expanded. About 6 minutes after the explosion, the electron density suddenly began to increase, reached a secondary peak of about 240/cu cm, and then slowly decayed down to the preevent level over a period of about 15 minutes. The density increase is believed to be caused by the collapse of the ion cloud into the diamagnetic cavity created by the initial expansion. The plasma wave intensities observed during the entire event were quite low. In the diamagnetic cavity, electrostatic emissions were observed near the barium ion plasma frequency, and in another band at lower frequencies. A broadband burst of electrostatic noise was also observed at the boundary of the diamagnetic cavity. Except for electron plasma oscillations, no significant wave activity was observed outside of the diamagnetic cavity. 18. Quantum physics of classical waves in plasma Dodin, I. Y. 2012-10-01 The Lagrangian approach to plasma wave physics is extended to a universal nonlinear theory which yields generic equations invariant with respect to the wave nature. The traditional understanding of waves as solutions of the Maxwell-Vlasov system is abandoned. Oscillations are rather treated as physical entities, namely, abstract vectors \|ψ> in a specific Hilbert space. The invariant product <ψ\|ψ> is the total action and has the sign of the oscillation energy. The action density is then an operator. Projections of the corresponding operator equation generate assorted wave kinetic equations; the nonlinear Wigner-Moyal equation is just one example and, in fact, may be more delicate than commonly assumed. The linear adiabatic limit of this classical theory leads to quantum mechanics in its general form. The action conservation theorem, together with its avatars such as Manley-Rowe relations, then becomes manifest and in partial equilibrium can modify statistical properties of plasma fluctuations. In the quasi-monochromatic limit geometrical optics (GO) is recovered and can as well be understood as a particular field theory in its own right. For linear waves, the energy-momentum equations, in both canonical and (often) kinetic form, then follow automatically, even without a reference to electromagnetism. Yet for waves in plasma the general GO Lagrangian is also derived explicitly, in terms of single-particle oscillation-center Hamiltonians. Applications to various plasma waves are then discussed with an emphasis on the advantages of an abstract theory. Specifically covered are nonlinear dispersion, dynamics, and stability of BGK modes, and also other wave transformations in laboratory and cosmological plasmas. 19. An overview of plasma wave observations obtained during the Galileo A34 pass through the inner region of the Jovian magnetosphere Gurnett, D. A.; Kurth, W. S.; Menietti, J. D.; Roux, A.; Bolton, S. J.; Alexander, C. J. 2003-04-01 On November 5, 2002, the Galileo spacecraft, which is in orbit around Jupiter, made a pass in to a radial distance of 1.98 RJ (Jovian radii) from Jupiter, much closer than on any previous orbit. Data were successfully acquired during the entire inbound pass through the hot and cold plasma torii, and through the region inside the cold torus to a radial distance of 2.32 RJ, at which point the data system went into safing due to the intense radiation in the inner region of the magnetosphere. The purpose of this paper is to give an overview of the results obtained from the plasma wave investigation during this pass, which is designated A34. As on previous passes through the Io plasma torus a narrowband electrostatic emission at the upper hybrid resonance frequency provided a very accurate measurement of the electron density. The peak electron density, 2.6 x 103 cm-3, occurs just before the inner edge of the hot torus, which is at 5.62 RJ. As the spacecraft enters the cold torus the electron density drops to about 6.0 x 102 cm-3 and then gradually increases as the spacecraft approaches Jupiter, reaching a peak of about 2.5 x 103 cm-3 at 4.86 RJ, shortly before the inner edge of the cold torus. At the inner edge of the cold torus, which occurs at 4.76 RJ, the electron density drops dramatically to levels on the order of 1 cm-3. The electron density in this inner region is difficult to interpret because the upper hybrid emission can no longer be clearly identified, and there are numerous narrowband emissions with cutoffs that may or may not be associated with the local electron plasma frequency. As in the hot torus, the low density region inside the cold torus has a persistent level of plasma wave noise below about 103 Hz that is tentatively interpreted as whistler mode noise. The intensity of the whistler mode noise increases noticeably as the spacecraft crosses Thebe's orbit at 3.1 RJ, and increases markedly as the spacecraft crosses Amalthea's orbit at 2.6 RJ. The 20. Langmuir Wave Decay in Inhomogeneous Solar Wind Plasmas: Simulation Results Krafft, C.; Volokitin, A. S.; Krasnoselskikh, V. V. 2015-08-01 Langmuir turbulence excited by electron flows in solar wind plasmas is studied on the basis of numerical simulations. In particular, nonlinear wave decay processes involving ion-sound (IS) waves are considered in order to understand their dependence on external long-wavelength plasma density fluctuations. In the presence of inhomogeneities, it is shown that the decay processes are localized in space and, due to the differences between the group velocities of Langmuir and IS waves, their duration is limited so that a full nonlinear saturation cannot be achieved. The reflection and the scattering of Langmuir wave packets on the ambient and randomly varying density fluctuations lead to crucial effects impacting the development of the IS wave spectrum. Notably, beatings between forward propagating Langmuir waves and reflected ones result in the parametric generation of waves of noticeable amplitudes and in the amplification of IS waves. These processes, repeated at different space locations, form a series of cascades of wave energy transfer, similar to those studied in the frame of weak turbulence theory. The dynamics of such a cascading mechanism and its influence on the acceleration of the most energetic part of the electron beam are studied. Finally, the role of the decay processes in the shaping of the profiles of the Langmuir wave packets is discussed, and the waveforms calculated are compared with those observed recently on board the spacecraft Solar TErrestrial RElations Observatory and WIND. 1. LANGMUIR WAVE DECAY IN INHOMOGENEOUS SOLAR WIND PLASMAS: SIMULATION RESULTS SciTech Connect Krafft, C.; Volokitin, A. S.; Krasnoselskikh, V. V. 2015-08-20 Langmuir turbulence excited by electron flows in solar wind plasmas is studied on the basis of numerical simulations. In particular, nonlinear wave decay processes involving ion-sound (IS) waves are considered in order to understand their dependence on external long-wavelength plasma density fluctuations. In the presence of inhomogeneities, it is shown that the decay processes are localized in space and, due to the differences between the group velocities of Langmuir and IS waves, their duration is limited so that a full nonlinear saturation cannot be achieved. The reflection and the scattering of Langmuir wave packets on the ambient and randomly varying density fluctuations lead to crucial effects impacting the development of the IS wave spectrum. Notably, beatings between forward propagating Langmuir waves and reflected ones result in the parametric generation of waves of noticeable amplitudes and in the amplification of IS waves. These processes, repeated at different space locations, form a series of cascades of wave energy transfer, similar to those studied in the frame of weak turbulence theory. The dynamics of such a cascading mechanism and its influence on the acceleration of the most energetic part of the electron beam are studied. Finally, the role of the decay processes in the shaping of the profiles of the Langmuir wave packets is discussed, and the waveforms calculated are compared with those observed recently on board the spacecraft Solar TErrestrial RElations Observatory and WIND. 2. Nonplanar waves with electronegative dusty plasma SciTech Connect Zobaer, M. S.; Mukta, K. N.; Nahar, L.; Mamun, A. A.; Roy, N. 2013-04-15 A rigorous theoretical investigation has been made of basic characteristics of the nonplanar dust-ion-acoustic shock and solitary waves in electronegative dusty plasma containing Boltzmann electrons, Boltzmann negative ions, inertial positive ions, and charge fluctuating (negatively charged) stationary dust. The Burgers' and Korteweg-de Vries (K-dV) equations, which is derived by reductive perturbation technique, is numerically solved to examine the effects of nonplanar geometry on the basic features of the DIA shock and solitary waves formed in the electronegative dusty plasma. The implications of the results (obtained from this investigation) in space and laboratory experiments are briefly discussed. 3. Modulation of whistler waves in nonthermal plasmas SciTech Connect Rios, L. A.; Galvao, R. M. O. 2011-02-15 The modulation of whistler waves in nonthermal plasmas is investigated. The dynamics of the magnetized plasma is described by the fluid equations and the electron velocity distribution function is modeled via a nonthermal {kappa} distribution. A multiscale perturbation analysis based on the Krylov-Bogoliubov-Mitropolsky method is carried out and the nonlinear Schroedinger equation governing the modulation of the high-frequency whistler is obtained. The effect of the superthermal electrons on the stability of the wave envelope and soliton formation is discussed and a comparison with previous results is presented. 4. Alfven wave absorption in dissipative plasma Gavrikov, M. B.; Taiurskii, A. A. 2017-01-01 We consider nonlinear absorption of Alfven waves due to dissipative effects in plasma and relaxation of temperatures of electrons and ions. This study is based on an exact solution of the equations of two-fluid electromagnetic hydrodynamics (EMHD) of plasma. It is shown that in order to study the decay of Alfven waves, it suffices to examine the behavior of their amplitudes whose evolution is described by a system of ordinary differential equations (ODEs) obtained in this paper. On finite time intervals, the system of equations on the amplitudes is studied numerically, while asymptotic integration (the Hartman-Grobman theorem) is used to examine its large-time behavior. 5. A large volume uniform plasma generator for the experiments of electromagnetic wave propagation in plasma SciTech Connect Yang Min; Li Xiaoping; Xie Kai; Liu Donglin; Liu Yanming 2013-01-15 A large volume uniform plasma generator is proposed for the experiments of electromagnetic (EM) wave propagation in plasma, to reproduce a 'black out' phenomenon with long duration in an environment of the ordinary laboratory. The plasma generator achieves a controllable approximate uniform plasma in volume of 260 mm Multiplication-Sign 260 mm Multiplication-Sign 180 mm without the magnetic confinement. The plasma is produced by the glow discharge, and the special discharge structure is built to bring a steady approximate uniform plasma environment in the electromagnetic wave propagation path without any other barriers. In addition, the electron density and luminosity distributions of plasma under different discharge conditions were diagnosed and experimentally investigated. Both the electron density and the plasma uniformity are directly proportional to the input power and in roughly reverse proportion to the gas pressure in the chamber. Furthermore, the experiments of electromagnetic wave propagation in plasma are conducted in this plasma generator. Blackout phenomena at GPS signal are observed under this system and the measured attenuation curve is of reasonable agreement with the theoretical one, which suggests the effectiveness of the proposed method. 6. Oblique ion acoustic shock waves in a magnetized plasma SciTech Connect Shahmansouri, M.; Mamun, A. A. 2013-08-15 Ion acoustic (IA) shock waves are studied in a magnetized plasma consisting of a cold viscous ion fluid and Maxwellian electrons. The Korteweg–de Vries–Burgers equation is derived by using the reductive perturbation method. It is shown that the combined effects of external magnetic field and obliqueness significantly modify the basic properties (viz., amplitude, width, speed, etc.) of the IA shock waves. It is observed that the ion-viscosity is a source of dissipation, and is responsible for the formation of IA shock structures. The implications of our results in some space and laboratory plasma situations are discussed. 7. Plasma shock waves excited by THz radiation Rudin, S.; Rupper, G.; Shur, M. 2016-10-01 The shock plasma waves in Si MOS, InGaAs and GaN HEMTs are launched at a relatively small THz power that is nearly independent of the THz input frequency for short channel (22 nm) devices and increases with frequency for longer (100 nm to 1 mm devices). Increasing the gate-to-channel separation leads to a gradual transition of the nonlinear waves from the shock waves to solitons. The mathematics of this transition is described by the Korteweg-de Vries equation that has the single propagating soliton solution. 8. On the freak waves in mesospheric plasma El-Labany, S. K.; El-Shewy, E. K.; El-Bedwehy, N. A.; El-Razek, H. N. Abd; El-Rahman, A. A. 2017-03-01 The nonlinear properties of dusty ionic freak waves have been studied in homogeneous, unmagnetized dusty plasma system containing ions, isothermal electrons, negative and positive grains. By using the derivative expansion method and assuming strongly dispersive medium, the basic model equations are reduced to a nonlinear form of Schrodinger equation (NLSE). One of the solutions of the NLSE in the unstable region is the rational one which is responsible for the creation of the freak profiles. The reliance of freak waves profile on dusty grains charge and carrier wave number are discussed. 9. Collisional Drift Waves in Stellarator Plasmas SciTech Connect J.L.V. Lewandowski 2003-10-07 A computational study of resistive drift waves in the edge plasma of a stellarator with an helical magnetic axis is presented. Three coupled field equations, describing the collisional drift wave dynamics in the linear approximation, are solved as an initial-value problem along the magnetic field line. The magnetohydrodynamic equilibrium is obtained from a three-dimensional local equilibrium model. The use of a local magnetohydrodynamic equilibrium model allows for a computationally efficient systematic study of the impact of the magnetic field structure on drift wave stability. 10. Spatiotemporal synchronization of drift waves in a magnetron sputtering plasma SciTech Connect Martines, E.; Zuin, M.; Cavazzana, R.; Antoni, V.; Serianni, G.; Spolaore, M.; Vianello, N.; Adámek, J. 2014-10-15 A feedforward scheme is applied for drift waves control in a magnetized magnetron sputtering plasma. A system of driven electrodes collecting electron current in a limited region of the explored plasma is used to interact with unstable drift waves. Drift waves actually appear as electrostatic modes characterized by discrete wavelengths of the order of few centimeters and frequencies of about 100 kHz. The effect of external quasi-periodic, both in time and space, travelling perturbations is studied. Particular emphasis is given to the role played by the phase relation between the natural and the imposed fluctuations. It is observed that it is possible by means of localized electrodes, collecting currents which are negligible with respect to those flowing in the plasma, to transfer energy to one single mode and to reduce that associated to the others. Due to the weakness of the external action, only partial control has been achieved. 11. Observed Statistics of Extreme Waves DTIC Science & Technology 2006-12-01 9 Figure 5. An energy stealing wave as a solution to the NLS equation . (From: Dysthe and...shown that nonlinear interaction between four colliding waves can produce extreme wave behavior. He utilized the NLS equation in his numerical ...2000) demonstrated the formation of extreme waves using the Korteweg de Vries ( KdV ) equation , which is valid in shallow water. It was shown in the 12. Observations of running penumbral waves. NASA Technical Reports Server (NTRS) Zirin, H.; Stein, A. 1972-01-01 Quiet sunspots with well-developed penumbrae show running intensity waves with period running around 300 sec. The waves appear connected with umbral flashes of exactly half the period. Waves are concentric, regular, with velocity constant around 10 km/sec. They are probably sound waves and show intensity fluctuation in H alpha centerline or wing of 10 to 20%. The energy is tiny compared to the heat deficit of the umbra. 13. FIRST EVIDENCE OF COEXISTING EIT WAVE AND CORONAL MORETON WAVE FROM SDO/AIA OBSERVATIONS SciTech Connect Chen, P. F.; Wu, Y. 2011-05-10 'EIT waves' are a globally propagating wavelike phenomenon. They were often interpreted as fast-mode magnetoacoustic waves in the corona, despite various discrepancies between the fast-mode wave model and observations. To reconcile these discrepancies, we suggested that 'EIT waves' are the apparent propagation of the plasma compression due to successive stretching of the magnetic field lines pushed by the erupting flux rope. According to this model, an EIT wave should be preceded by a fast-mode wave, which, however, had rarely been observed. With the unprecedented high cadence and sensitivity of the Solar Dynamics Observatory observations, we discern a fast-moving wave front with a speed of 560 km s{sup -1} ahead of an EIT wave, which had a velocity of {approx}190 km s{sup -1}, in the 'EIT wave' event on 2010 July 27. The results, suggesting that 'EIT waves' are not fast-mode waves, confirm the prediction of our field-line stretching model for an EIT wave. In particular, it is found that the coronal Moreton wave was {approx}3 times faster than the EIT wave, as predicted. 14. Low frequency nonlinear waves in electron depleted magnetized nonthermal plasmas Mobarak Hossen, Md.; Sahadat Alam, Md.; Sultana, Sharmin; Mamun, A. A. 2016-11-01 A theoretical study on the ultra-low frequency small but finite amplitude solitary waves has been carried out in an electron depleted magnetized nonthermal dusty plasma consisting of both polarity (positively charged as well as negatively charged) inertial massive dust particles and nonextensive q distributed ions. The reductive perturbation technique is employed to derive the ZakharovKuznetsov (ZK) equation. The basic features of low frequency solitary wave are analyzed via the solution of ZK equation. It is observed that the intrinsic properties (e.g., polarity, amplitude, width, etc.) of dust-acoustic (DA) solitary waves (SWs) are significantly influenced by the effects external magnetic field, obliqueness, nonextensivity of ions, and the ratio of ion number density to the product of electron and negative dust number density. The findings of our results may be useful to explain the low frequency nonlinear wave propagation in some plasma environments like cometary tails, the earth polar mesosphere, Jupiter's magnetosphere, etc. 15. Waves in Space Plasmas (WISP). Final report SciTech Connect Calvert, W. 1994-08-01 Activities under this project have included participation in the Waves in Space Plasmas (WISP) program, a study of the data processing requirements for WISP, and theoretical studies of radio sounding, ducting, and magnetoionic theory. An analysis of radio sounding in the magnetosphere was prepared. 16. Landau damping of a driven plasma wave from laser pulses SciTech Connect Bu Zhigang; Ji Peiyong 2012-01-15 The interaction between a laser pulse and a driven plasma wave with a phase velocity approaching the speed of light is studied, and our investigation is focused on the Gaussian laser pulse. It is demonstrated that when the resonance condition between the plasma wave and the laser pulse is satisfied, the Landau damping phenomenon of the plasma wave originated from the laser pulse will emerge. The dispersion relations for the plasma waves in resonance and non-resonance regions are obtained. It is proved that the Landau damping rate for a driven plasma wave is {gamma}>0 in the resonance region, so the laser pulse can produce an inverse damping effect, namely Landau growth effect, which leads an instability for the plasma wave. The Landau growth means that the energy is transmitted from the laser pulse to the plasma wave, which could be an effective process for enhancing the plasma wave. 17. Analysis of plasma wave interference patterns in the Spacelab 2 PDP data. [PDP (Plasma Diagnostics Package) SciTech Connect Feng, Wei. 1992-01-01 During the Spacelab 2 mission the University of Iowa's Plasma Diagnostics Package (PDP) explored the plasma environment around the shuttle. Wideband spectrograms of plasma waves were obtained from the PDP at frequencies from 0 to 30 kHz up to 400 m from the shuttle. These spectrograms frequently showed interference patterns caused by waves with wavelengths short compared to the antenna length (3.89 meters). Two types of interference patterns were observed in the wideband data: associated with the ejection of an electron beam from the space shuttle; associated with lower hybrid waves generated by an interaction between the neutral gas cloud around shuttle and the ambient ionospheric plasma. Analysis of these antenna interference patterns permits a determination of the wavelength, the plasma rest frame frequency, the direction of propagation, the power spectrum and in some cases the location of the source. The electric field noise associated with the electron beam was observed in the wideband data for two periods during which an electron frequency range at low frequencies (below 10 kHz) and shows clear evidence of interference patterns. The broadband low frequency noise was the dominant type of noise produced by the electron beam. The waves have a linear dispersion relation very similar to ion acoustic waves. The returning to the shuttle in response to the ejected electron beam. The waves associated with the lower hybrid resonance have rest frame frequencies near the lower hybrid frequency and propagate perpendicular to the magnetic field. The occurrence of these waves depends strongly on the PDP's position relative to the shuttle and the magnetic field direction. The authors results confirm previous identifications of these waves as lower hybrid waves and suggest they are driven by pick-up ions (H[sub 2]O[sup +]) produced by a charge exchange interaction between a water cloud around the shuttle and the ambient ionosphere. 18. Rogue wave observation in a water wave tank. PubMed Chabchoub, A; Hoffmann, N P; Akhmediev, N 2011-05-20 The conventional definition of rogue waves in the ocean is that their heights, from crest to trough, are more than about twice the significant wave height, which is the average wave height of the largest one-third of nearby waves. When modeling deep water waves using the nonlinear Schrödinger equation, the most likely candidate satisfying this criterion is the so-called Peregrine solution. It is localized in both space and time, thus describing a unique wave event. Until now, experiments specifically designed for observation of breather states in the evolution of deep water waves have never been made in this double limit. In the present work, we present the first experimental results with observations of the Peregrine soliton in a water wave tank. 19. Global observations of ocean Rossby waves SciTech Connect Chelton, D.B.; Schlax, M.G. 1996-04-12 Rossby waves play a critical role in the transient adjustment of ocean circulation to changes in large-scale atmospheric forcing. The TOPEX/POSEIDON satellite altimeter has detected Rossby waves throughout much of the world ocean from sea level signals with {approx_lt} 10-centimeters amplitude and {approx_lt} 500-kilometer wavelength. Outside of the tropics Rossby waves are abruptly amplified by major topographic features. Analysis of 3 years of data reveals discrepancies between observed and theoretical Rossby wave phase speeds that indicate that the standard theory for free, linear Rossby waves in an incomplete description of the observed waves. 32 refs., 5 figs. 20. Plasma waves near Saturn: Initial results from Voyager 1 NASA Technical Reports Server (NTRS) Gurnett, D. A.; Kurth, W. S.; Scarf, F. L. 1981-01-01 The Voyager 1 encounter with Saturn provided the first opportunity to investigate plasma wave interactions in the magnetosphere of Saturn. An overview of the principal results from the Voyager 1 plasma wave instrument is presented starting with the initial detection of Saturn and ending about four weeks after closest approach. A survey plot of the electric field intensities detected during the Saturn encounter is shown starting shortly before the inbound shock crossing and ending shortly after the outbound magnetopause crossing. Many intense waves were observed in the vicinity of Saturn. To provide a framework for presenting the observations, the results are discussed more or less according to the sequence in which the data were obtained. 1. Dust structurization observed in a dc glow discharge dusty plasma Heinrich, Jonathon R.; Kim, Su-Hyun; Merlino, Robert L. 2010-11-01 Dusty plasmas, which are inherently open systems which require an ionization source to replenish the plasma absorbed on the grains, tend to exhibit self-organization. Various structures have been observed in dusty plasmas such as dust crystals, voids, and vortices. Due to the presence of drifting ions in dc discharge plasmas, spontaneously excited dust acoustic waves are also a common occurrence. By adjusting the discharge parameters we have observed a new phenomenon in dusty plasmas -- the spontaneous formation of three-dimensional stationary dust density structures. These structures appear as an ordered pattern consisting of alternating regions of high and low dust density arranged in a nested bowl-type configuration The stationary structure evolves from dust density waves that slow down as their wavelength decreases and eventually stop moving when the wavelength reaches some minimum size. 2. Waves and Fine Structure in Expanding Laser-Produced Plasmas Collette, Andrew; Gekelman, Walter 2009-11-01 The behavior of expanding dense plasmas has long been a topic of interest in space plasma research, particularly in the case of expansion within a magnetized background. Previous laser-plasma experiments at the UCLA Large Plasma Device have observed the creation of strong (δBB > 50%) diamagnetic cavities, along with large-scale wave activity and hints of fine-scale structure. A new series of experiments conducted recently at the LaPD performs direct measurement of the fields inside the expanding plasma via a novel 2D probe drive system. This system combines small-scale (0.5mm-1mm) magnetic and electric field probes with high-accuracy vacuum ceramic motors, to allow measurement of the plasma volume over a 2000-point grid at 1mm resolution. The data reveal both coherent high-amplitude waves associated with the formation of these magnetic features, and complicated small-scale structure in both the magnetic field and floating potential. In addition, we will present correlation techniques using multiple independent B and E field probes. This reveals behavior of turbulent, non-phase-locked phenomena. Both the case of a single expanding plasma and two colliding plasmas were studied. 3. Guided electromagnetic waves observed on a conducting ionospheric tether James, H. G.; Balmain, K. G. 2001-01-01 On the up leg of its flight through the auroral nightside ionosphere to an apogee of 824 km, the tethered double payload Observations of Electric Field Distributions in the Ionospheric Plasma: A Unique Strategy (OEDIPUS) C was the site of experiments on wire-guided electromagnetic (EM) waves. Waves were transmitted from the upper subpayload to a receiver on the lower subpayload along a conducting wire aligned within a few degrees of the Earth's magnetic field. Such EM waves were observed at almost all frequencies in the range 0.1-8.0 MHz. There was a deep stop band between the cyclotron and upper hybrid resonance frequencies where the cold plasma theory predicts a propagation cutoff, and there were shallower attenuation bands at frequencies where hot-plasma electrostatic waves may affect the guided EM modes. Resonances of the wire-guided waves with the tether length were observed throughout the entire tethered portion of the flight. The resonances appear as a set of fringes when all the data are presented in a frequency-versus-time summary. The fringe shapes in this summary have been compared with the predictions of an early theory, which give generally good agreement. The exceptions are frequencies close to the stop band, where cold-plasma dispersion effects are expected to be greatest. Another theory based on a different derivation of the dispersion relation includes a vacuum sheath gap outside the conductor. The absolute fringe intensities and positions predicted agree moderately well with the observations. 4. Plasma wave experiment for the ISEE-3 mission NASA Technical Reports Server (NTRS) Scarf, F. L. 1983-01-01 Sensitive, high resolution plasma probes for analysis of the distribution functions and plasma wave instruments for measurements of electromagnetic and electrostatic wave modes are commonly flown together to provide information on plasma instabilities and wave particle interactions. Analysis of the data for the ISEE 3 mission is provided. 5. Studies on Charge Variation and Waves in Dusty Plasmas Kausik, Siddhartha Sankar Plasma and dust grains are both ubiquitous ingredients of the universe. The interplay between them has opened up a new and fascinating research domain, that of dusty plasmas, which contain macroscopic particles of solid matter besides the usual plasma constituents. The research in dusty plasmas received a major boost in the early eighties with Voyager spacecraft observation on the formation of Saturn rings. Dusty plasmas are defined as partially or fully-ionized gases that contain micron-sized particles of electrically charged solid material, either dielectric or conducting. The physics of dusty plasmas has recently been studied intensively because of its importance for a number of applications in space and laboratory plasmas. This thesis presents the experimental studies on charge variation and waves in dusty plasmas. The experimental observations are carried out in two different experimental devices. Three different sets of experiments are carried out in two different experimental devices. Three different sets of experiments are carried out to study the dust charge variation in a filament discharge argon plasma. The dust grains used in these experiments are grains of silver. In another get of experiment, dust acoustic waves are studied in a de glow discharge argon plasma. Alumina dust grains are sprinkled in this experiment. The diagnostic tools used in these experiments are Langmuir probe and Faraday cup. The instruments used in these experiments are electrometer, He-Ne laser and charge coupled device (CCD) camera. Langmuir probe is used to measure plasma parameters, while Faraday cup and electrometer are used to measure very low current (~pA) carried by a collimated dust beam. He-Ne laser illuminates the dust grains and CCD camera is used to capture the images of dust acoustic waves. Silver dust grains are produced in the dust chamber by gas-evaporation technique. Due to differential pressure maintained between the dust and plasma chambers, the dust grains move 6. Applying the cold plasma dispersion relation to whistler mode chorus waves: EMFISIS wave measurements from the Van Allen Probes. PubMed Hartley, D P; Chen, Y; Kletzing, C A; Denton, M H; Kurth, W S 2015-02-01 Most theoretical wave models require the power in the wave magnetic field in order to determine the effect of chorus waves on radiation belt electrons. However, researchers typically use the cold plasma dispersion relation to approximate the magnetic wave power when only electric field data are available. In this study, the validity of using the cold plasma dispersion relation in this context is tested using Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) observations of both the electric and magnetic spectral intensities in the chorus wave band (0.1-0.9 fce). Results from this study indicate that the calculated wave intensity is least accurate during periods of enhanced wave activity. For observed wave intensities >10(-3) nT(2), using the cold plasma dispersion relation results in an underestimate of the wave intensity by a factor of 2 or greater 56% of the time over the full chorus wave band, 60% of the time for lower band chorus, and 59% of the time for upper band chorus. Hence, during active periods, empirical chorus wave models that are reliant on the cold plasma dispersion relation will underestimate chorus wave intensities to a significant degree, thus causing questionable calculation of wave-particle resonance effects on MeV electrons. 7. Applying the cold plasma dispersion relation to whistler mode chorus waves: EMFISIS wave measurements from the Van Allen Probes SciTech Connect Hartley, D. P.; Chen, Y.; Kletzing, C. A.; Denton, M. H.; Kurth, W. S. 2015-02-17 Most theoretical wave models require the power in the wave magnetic field in order to determine the effect of chorus waves on radiation belt electrons. However, researchers typically use the cold plasma dispersion relation to approximate the magnetic wave power when only electric field data are available. In this study, the validity of using the cold plasma dispersion relation in this context is tested using Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) observations of both the electric and magnetic spectral intensities in the chorus wave band (0.1–0.9 fce). Results from this study indicate that the calculated wave intensity is least accurate during periods of enhanced wave activity. For observed wave intensities >10⁻³ nT², using the cold plasma dispersion relation results in an underestimate of the wave intensity by a factor of 2 or greater 56% of the time over the full chorus wave band, 60% of the time for lower band chorus, and 59% of the time for upper band chorus. Hence, during active periods, empirical chorus wave models that are reliant on the cold plasma dispersion relation will underestimate chorus wave intensities to a significant degree, thus causing questionable calculation of wave-particle resonance effects on MeV electrons. 8. Formation mechanism of steep wave front in magnetized plasmas SciTech Connect Sasaki, M. Kasuya, N.; Itoh, S.-I.; Kobayashi, T.; Arakawa, H.; Itoh, K.; Fukunaga, K.; Yamada, T.; Yagi, M. 2015-03-15 Bifurcation from a streamer to a solitary drift wave is obtained in three dimensional simulation of resistive drift waves in cylindrical plasmas. The solitary drift wave is observed in the regime where the collisional transport is important as well as fluctuation induced transport. The solitary drift wave forms a steep wave front in the azimuthal direction. The phase of higher harmonic modes are locked to that of the fundamental mode, so that the steep wave front is sustained for a long time compared to the typical time scale of the drift wave oscillation. The phase entrainment between the fundamental and second harmonic modes is studied, and the azimuthal structure of the stationary solution is found to be characterized by a parameter which is determined by the deviation of the fluctuations from the Boltzmann relation. There are two solutions of the azimuthal structures, which have steep wave front facing forward and backward in the wave propagation direction, respectively. The selection criterion of these solutions is derived theoretically from the stability of the phase entrainment. The simulation result and experimental observations are found to be consistent with the theoretical prediction. 9. Electrostatic rogue-waves in relativistically degenerate plasmas SciTech Connect Akbari-Moghanjoughi, M. 2014-10-15 In this paper, we investigate the modulational instability and the possibility of electrostatic rogue-wave propagations in a completely degenerate plasma with arbitrary degree of degeneracy, i.e., relativistically degenerate plasma, ranging from solid density to the astrophysical compact stars. The hydrodynamic approach along with the perturbation method is used to reduce the governing equations to the nonlinear Schrödinger equation from which the modulational instability, the growth rate of envelope excitations and the occurrence of rogue as well as super-rogue waves in the plasma, is evaluated. It is observed that the modulational instability in a fully degenerate plasma can be quite sensitive to the plasma number-density and the wavenumber of envelop excitations. It is further revealed that the relativistically degeneracy plasmas (R{sub 0} > 1) are almost always modulationally unstable. It is found, however, that the highly energetic sharply localized electrostatic rogue as well as super-rogue waves can exist in the astrophysical compact objects like white dwarfs and neutron star crusts. The later may provide a link to understand many physical processes in such stars and it may lead us to the origin of the random-localized intense short gamma-ray bursts, which “appear from nowhere and disappear without a trace” quite similar to oceanic rogue structures. 10. Observations of velocity shear driven plasma turbulence NASA Technical Reports Server (NTRS) Kintner, P. M., Jr. 1976-01-01 Electrostatic and magnetic turbulence observations from HAWKEYE-1 during the low altitude portion of its elliptical orbit over the Southern Hemisphere are presented. The magnetic turbulence is confined near the auroral zone and is similar to that seen at higher altitudes by HEOS-2 in the polar cusp. The electrostatic turbulence is composed of a background component with a power spectral index of 1.89 + or - .26 and an intense component with a power spectral index of 2.80 + or - .34. The intense electrostatic turbulence and the magnetic turbulence correlate with velocity shears in the convective plasma flow. Since velocity shear instabilities are most unstable to wave vectors perpendicular to the magnetic field, the shear correlated turbulence is anticipated to be two dimensional in character and to have a power spectral index of 3 which agrees with that observed in the intense electrostatic turbulence. 11. Strongly Enhanced Laser Absorption and Electron Acceleration via Resonant Excitation of Surface Plasma Waves Raynaud, M.; Riconda, C.; Adam, J. C.; Heron, A. 2010-02-01 The possibility of creating enhanced fast electron bunches via the excitation of surface plasma waves (SPW) in laser overdense plasma interaction has been investigated by mean of relativistic one dimension motion of a test electron in the field of the surface plasma wave study and with two-dimensional (2D) Particle-In-Cell (PIC) numerical simulations. Strong electron acceleration together with a dramatic increase, up to 70%, of light absorption by the plasma is observed. 12. MESSENGER observations of Kelvin-Helmholtz waves at Mercury's magnetopause Sundberg, T.; Boardsen, S. A.; Slavin, J. A.; Anderson, B. J.; Korth, H.; Zurbuchen, T.; Raines, J. M.; Solomon, S. C. 2011-12-01 We present a survey of Kelvin-Helmholtz (KH) waves at Mercury's magnetopause during MESSENGER's first Mercury year in orbit. The waves were identified on the basis of the well-established sawtooth wave signatures that are associated with non-linear KH vortices at the magnetopause. Remarkably, the results show that MESSENGER frequently observed such KH waves in the dayside region of the magnetosphere where the magnetosheath flow velocity is still sub-sonic, which implies that instability growth rates at Mercury's magnetopause are much larger than at Earth. The wave amplitude was often on the order of 100 nT or more, and the wave periods were ~10-20 s. A clear dawn-dusk asymmetry is also present in the data, with all of the observed events taking place in the post-noon and the dusk-side sectors of the magnetopause. This asymmetry is likely related to finite ion-gyroradius effects and is in agreement with the results from particle-in-cell simulations of the instability. Similar to most terrestrial events, the wave observations were made almost exclusively during periods when the north-south component of the magnetosheath magnetic field was northward. Accompanying measurements from the Fast Imaging Plasma Spectrometer (FIPS) show that the waves were associated with a substantial transport of magnetosheath plasma into the magnetosphere. 13. A Schamel equation for ion acoustic waves in superthermal plasmas SciTech Connect Williams, G. Kourakis, I.; Verheest, F.; Hellberg, M. A.; Anowar, M. G. M. 2014-09-15 An investigation of the propagation of ion acoustic waves in nonthermal plasmas in the presence of trapped electrons has been undertaken. This has been motivated by space and laboratory plasma observations of plasmas containing energetic particles, resulting in long-tailed distributions, in combination with trapped particles, whereby some of the plasma particles are confined to a finite region of phase space. An unmagnetized collisionless electron-ion plasma is considered, featuring a non-Maxwellian-trapped electron distribution, which is modelled by a kappa distribution function combined with a Schamel distribution. The effect of particle trapping has been considered, resulting in an expression for the electron density. Reductive perturbation theory has been used to construct a KdV-like Schamel equation, and examine its behaviour. The relevant configurational parameters in our study include the superthermality index κ and the characteristic trapping parameter β. A pulse-shaped family of solutions is proposed, also depending on the weak soliton speed increment u{sub 0}. The main modification due to an increase in particle trapping is an increase in the amplitude of solitary waves, yet leaving their spatial width practically unaffected. With enhanced superthermality, there is a decrease in both amplitude and width of solitary waves, for any given values of the trapping parameter and of the incremental soliton speed. Only positive polarity excitations were observed in our parametric investigation. 14. Waves in space plasma dipole antenna subsystem NASA Technical Reports Server (NTRS) Thomson, Mark 1993-01-01 The Waves In Space Plasma (WISP) flight experiment requires a 50-meter-long deployable dipole antenna subsystem (DASS) to radiate radio frequencies from the STS Orbiter cargo bay. The transmissions are to excite outer ionospheric plasma between the dipole and a free-flying receiver (Spartan) for scientific purposes. This report describes the singular DASS design requirements and how the resulting design satisfies them. A jettison latch is described in some detail. The latch releases the antenna in case of any problems which might prevent the bay doors from closing for re-entry and landing of the Orbiter. 15. Wave excitation by nonlinear coupling among shear Alfvén waves in a mirror-confined plasma SciTech Connect Ikezoe, R. Ichimura, M.; Okada, T.; Hirata, M.; Yokoyama, T.; Iwamoto, Y.; Sumida, S.; Jang, S.; Takeyama, K.; Yoshikawa, M.; Kohagura, J.; Shima, Y.; Wang, X. 2015-09-15 A shear Alfvén wave at slightly below the ion-cyclotron frequency overcomes the ion-cyclotron damping and grows because of the strong anisotropy of the ion temperature in the magnetic mirror configuration, and is called the Alfvén ion-cyclotron (AIC) wave. Density fluctuations caused by the AIC waves and the ion-cyclotron range of frequencies (ICRF) waves used for ion heating have been detected using a reflectometer in a wide radial region of the GAMMA 10 tandem mirror plasma. Various wave-wave couplings are clearly observed in the density fluctuations in the interior of the plasma, but these couplings are not so clear in the magnetic fluctuations at the plasma edge when measured using a pick-up coil. A radial dependence of the nonlinearity is found, particularly in waves with the difference frequencies of the AIC waves; bispectral analysis shows that such wave-wave coupling is significant near the core, but is not so evident at the periphery. In contrast, nonlinear coupling with the low-frequency background turbulence is quite distinct at the periphery. Nonlinear coupling associated with the AIC waves may play a significant role in the beta- and anisotropy-limits of a mirror-confined plasma through decay of the ICRF heating power and degradation of the plasma confinement by nonlinearly generated waves. 16. Wave excitation by nonlinear coupling among shear Alfvén waves in a mirror-confined plasma Ikezoe, R.; Ichimura, M.; Okada, T.; Hirata, M.; Yokoyama, T.; Iwamoto, Y.; Sumida, S.; Jang, S.; Takeyama, K.; Yoshikawa, M.; Kohagura, J.; Shima, Y.; Wang, X. 2015-09-01 A shear Alfvén wave at slightly below the ion-cyclotron frequency overcomes the ion-cyclotron damping and grows because of the strong anisotropy of the ion temperature in the magnetic mirror configuration, and is called the Alfvén ion-cyclotron (AIC) wave. Density fluctuations caused by the AIC waves and the ion-cyclotron range of frequencies (ICRF) waves used for ion heating have been detected using a reflectometer in a wide radial region of the GAMMA 10 tandem mirror plasma. Various wave-wave couplings are clearly observed in the density fluctuations in the interior of the plasma, but these couplings are not so clear in the magnetic fluctuations at the plasma edge when measured using a pick-up coil. A radial dependence of the nonlinearity is found, particularly in waves with the difference frequencies of the AIC waves; bispectral analysis shows that such wave-wave coupling is significant near the core, but is not so evident at the periphery. In contrast, nonlinear coupling with the low-frequency background turbulence is quite distinct at the periphery. Nonlinear coupling associated with the AIC waves may play a significant role in the beta- and anisotropy-limits of a mirror-confined plasma through decay of the ICRF heating power and degradation of the plasma confinement by nonlinearly generated waves. 17. Solitary kinetic Alfven waves in dusty plasmas SciTech Connect Li Yangfang; Wu, D. J.; Morfill, G. E. 2008-08-15 Solitary kinetic Alfven waves in dusty plasmas are studied by considering the dust charge variation. The effect of the dust charge-to-mass ratio on the soliton solution is discussed. The Sagdeev potential is derived analytically with constant dust charge and then calculated numerically by taking the dust charge variation into account. We show that the dust charge-to-mass ratio plays an important role in the soliton properties. The soliton solutions are comprised of two branches. One branch is sub-Alfvenic and the soliton velocity is obviously smaller than the Alfven speed. The other branch is super-Alfvenic and the soliton velocity is very close to or greater than the Alfven speed. Both compressive and rarefactive solitons can exist. For the sub-Alfvenic branch, the rarefactive soliton is bell-shaped and it is much narrower than the compressive one. However, for the super-Alfvenic branch, the compressive soliton is bell-shaped and narrower, and the rarefactive one is broadened. When the charge-to-mass ratio of the dust grains is sufficiently high, the width of the rarefactive soliton, in the super-Alfvenic branch, will broaden extremely and a electron depletion will be observed. It is also shown that the bell-shaped soliton can transition to a cusped structure when the velocity is sufficiently high. 18. Tunable Plasma-Wave Laser Amplifier Bromage, J.; Haberberger, D.; Davies, A.; Bucht, S.; Zuegel, J. D.; Froula, D. H.; Trines, R.; Bingham, R.; Sadler, J.; Norreys, P. A. 2016-10-01 Raman amplification is a process by which a long energetic pump pulse transfers its energy to a counter-propagating short seed pulse through a resonant electron plasma wave. Since its conception, theory and simulations have shown exciting results with up to tens of percent of energy transfer from the pump to the seed pulse. However, experiments have yet to surpass transfer efficiencies of a few percent. A review of past literature shows that largely chirped pump pulses and finite temperature wave breaking could have been the two most detrimental effects. A Raman amplification platform is being developed at the Laboratory for Laser Energetics where a combination of a high-intensity tunable seed laser with sophisticated plasma diagnostics (dynamic Thomson scattering) will make it possible to find the optimal parameter space for high-energy transfer. This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award Number DE-NA0001944. 19. Applying the cold plasma dispersion relation to whistler mode chorus waves: EMFISIS wave measurements from the Van Allen Probes DOE PAGES Hartley, D. P.; Chen, Y.; Kletzing, C. A.; ... 2015-02-17 Most theoretical wave models require the power in the wave magnetic field in order to determine the effect of chorus waves on radiation belt electrons. However, researchers typically use the cold plasma dispersion relation to approximate the magnetic wave power when only electric field data are available. In this study, the validity of using the cold plasma dispersion relation in this context is tested using Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) observations of both the electric and magnetic spectral intensities in the chorus wave band (0.1–0.9 fce). Results from this study indicate that the calculated wavemore » intensity is least accurate during periods of enhanced wave activity. For observed wave intensities >10⁻³ nT², using the cold plasma dispersion relation results in an underestimate of the wave intensity by a factor of 2 or greater 56% of the time over the full chorus wave band, 60% of the time for lower band chorus, and 59% of the time for upper band chorus. Hence, during active periods, empirical chorus wave models that are reliant on the cold plasma dispersion relation will underestimate chorus wave intensities to a significant degree, thus causing questionable calculation of wave-particle resonance effects on MeV electrons.« less 20. Plasma wave experiment for the ISEE-3 mission NASA Technical Reports Server (NTRS) Scarf, F. L. 1982-01-01 Results of analyses of data received from a scientific instrument designed to study solar wind and plasma wave phenomena on the ISEE-3 mission are discussed in two papers prepared for publication. A study of plasma wave levels in and interplanetary magnetic field orientation preceding observations of interplanetary shocks by the satellite infers that quasi-parallel, interplanetary shocks are preceded by foreshocks whose presence is not obviously attributable to scattering of ion beams generated at quasi-perpendicular zones of these interplanetary shocks. Investigations of whistler mode turbulence in the disturbed solar wind resulted in various indirect lines of evidence indicating that these whistler waves are generated propagating at large angles to the local interplanetary field, a fact which helps identify possible free energy sources for their growth. 1. Waves in plasmas: Highlights from the past and present SciTech Connect Stix, T.H. 1990-03-01 To illustrate the development of some fundamental concepts in plasma waves, a number of experimental observations, going back over half a century, are reviewed. Particular attention is paid to the phenomena of dispersion, collisionfree damping, ray trajectories, amplitude transport, plasma wave echos, finite-Larmor-radius and cyclotron and cyclotron-harmonic effects, nonlocal response, and mode conversion. Also to the straight, trajectory approximation and two-level phase mixing. And to quasilinear diffusion and its relation to radiofrequency heating, current drive and induced neoclassical transport, and to stochasticity and superadiabaticity. One notes not only the constructive interplay between experiment and theory but also that major advances have come from each of the many disciplines that invoke plasma physics as a tool, including radio communication, astrophysics, controlled fusion, space physics, and basic research. 47 refs., 33 figs. 2. Shock waves in dusty plasma with two temperature superthermal ions Ghai, Yashika; Saini, N. S. 2017-03-01 An investigation of dust acoustic shock waves in dusty plasma containing two temperature ions is presented. The present investigation is motivated by the observations of Geotail spacecraft that report the occurrence of two temperature ion populations in Earth's magnetotail. We have derived Burgers equation to study dust acoustic shock structures in an unmagnetized plasma with two temperature superthermal ions. We have also derived the modified Burgers equation at critical values of physical parameters for which nonlinear coefficient (A) of Burgers equation vanishes. The numerical analysis is performed in context with observations in Earth's magnetotail and the influence of various plasma parameters viz. ions temperature ratio, superthermality of hot and cold ions, kinematic viscosity etc. has been observed on characteristics of DA shocks. It is observed that the amplitude of positive shocks via Burgers equation decreases whereas that of modified shocks with higher order nonlinearity increases with increase in superthermality of cold ions. 3. Photon acceleration in plasma wake wave SciTech Connect Bu, Zhigang; Shen, Baifei Yi, Longqing; Zhang, Hao; Huang, Shan; Li, Shun 2015-04-15 The photon acceleration effect in a laser wake field is investigated based on photon Hamiltonian dynamics. A test laser pulse is injected into a plasma wave at an incident angle θ{sub i}, which could slow down the photon velocity along the propagating direction of the wake wave so as to increase the acceleration distance for the photons. The photon trapping condition is analyzed in detail, and the maximum frequency shift of the trapped photon is obtained. The acceleration gradient and dephasing length are emphatically studied. The compression of the test laser pulse is examined and used to interpret the acceleration process. The limit of finite transverse width of the wake wave on photon acceleration is also discussed. 4. Geotail MCA Plasma Wave Investigation Data Analysis NASA Technical Reports Server (NTRS) Anderson, Roger R. 1997-01-01 The primary goals of the International Solar Terrestrial Physics/Global Geospace Science (ISTP/GGS) program are identifying, studying, and understanding the source, movement, and dissipation of plasma mass, momentum, and energy between the Sun and the Earth. The GEOTAIL spacecraft was built by the Japanese Institute of Space and Astronautical Science and has provided extensive measurements of entry, storage, acceleration, and transport in the geomagnetic tail and throughout the Earth's outer magnetosphere. GEOTAIL was launched on July 24, 1992, and began its scientific mission with eighteen extensions into the deep-tail region with apogees ranging from around 60 R(sub e) to more than 208 R(sub e) in the period up to late 1994. Due to the nature of the GEOTAIL trajectory which kept the spacecraft passing into the deep tail, GEOTAIL also made 'magnetopause skimming passes' which allowed measurements in the outer magnetosphere, magnetopause, magnetosheath, bow shock, and upstream solar wind regions as well as in the lobe, magnetosheath, boundary layers, and central plasma sheet regions of the tail. In late 1994, after spending nearly 30 months primarily traversing the deep tail region, GEOTAIL began its near-Earth phase. Perigee was reduced to 10 R(sub e) and apogee first to 50 R(sub e) and finally to 30 R(sub e) in early 1995. This orbit provides many more opportunities for GEOTAIL to explore the upstream solar wind, bow shock, magnetosheath, magnetopause, and outer magnetosphere as well as the near-Earth tail regions. The WIND spacecraft was launched on November 1, 1994 and the POLAR spacecraft was launched on February 24, 1996. These successful launches have dramatically increased the opportunities for GEOTAIL and the GGS spacecraft to be used to conduct the global research for which the ISTP program was designed. The measurement and study of plasma waves have made and will continue to make important contributions to reaching the ISTP/GGS goals and solving the 5. Collisional damping rates for plasma waves Tigik, S. F.; Ziebell, L. F.; Yoon, P. H. 2016-06-01 The distinction between the plasma dynamics dominated by collisional transport versus collective processes has never been rigorously addressed until recently. A recent paper [P. H. Yoon et al., Phys. Rev. E 93, 033203 (2016)] formulates for the first time, a unified kinetic theory in which collective processes and collisional dynamics are systematically incorporated from first principles. One of the outcomes of such a formalism is the rigorous derivation of collisional damping rates for Langmuir and ion-acoustic waves, which can be contrasted to the heuristic customary approach. However, the results are given only in formal mathematical expressions. The present brief communication numerically evaluates the rigorous collisional damping rates by considering the case of plasma particles with Maxwellian velocity distribution function so as to assess the consequence of the rigorous formalism in a quantitative manner. Comparison with the heuristic ("Spitzer") formula shows that the accurate damping rates are much lower in magnitude than the conventional expression, which implies that the traditional approach over-estimates the importance of attenuation of plasma waves by collisional relaxation process. Such a finding may have a wide applicability ranging from laboratory to space and astrophysical plasmas. 6. Plasma heating, plasma flow and wave production around an electron beam injected into the ionosphere NASA Technical Reports Server (NTRS) Winckler, J. R.; Erickson, K. N. 1986-01-01 A brief historical summary of the Minnesota ECHO series and other relevant electron beam experiments is given. The primary purpose of the ECHO experiments is the use of conjugate echoes as probes of the magnetosphere, but beam-plasma and wave studies were also made. The measurement of quasi-dc electric fields and ion streaming during the ECHO 6 experiment has given a pattern for the plasma flow in the hot plasma region extending to 60m radius about the ECHO 6 electron beam. The sheath and potential well caused by ion orbits is discussed with the aid of a model which fits the observations. ELF wave production in the plasma sheath around the beam is briefly discussed. The new ECHO 7 mission to be launched from the Poker Flat range in November 1987 is described. 7. Waves in relativistic electron beam in low-density plasma Sheinman, I.; Sheinman (Chernenco, J. 2016-11-01 Waves in electron beam in low-density plasma are analyzed. The analysis is based on complete electrodynamics consideration. Dependencies of dispersion laws from system parameters are investigated. It is shown that when relativistic electron beam is passed through low-density plasma surface waves of two types may exist. The first type is a high frequency wave on a boundary between the beam and neutralization area and the second type wave is on the boundary between neutralization area and stationary plasma. 8. Coronal magnetohydrodynamic waves and oscillations: observations and quests. PubMed Aschwanden, Markus J 2006-02-15 Coronal seismology, a new field of solar physics that emerged over the last 5 years, provides unique information on basic physical properties of the solar corona. The inhomogeneous coronal plasma supports a variety of magnetohydrodynamics (MHD) wave modes, which manifest themselves as standing waves (MHD oscillations) and propagating waves. Here, we briefly review the physical properties of observed MHD oscillations and waves, including fast kink modes, fast sausage modes, slow (acoustic) modes, torsional modes, their diagnostics of the coronal magnetic field, and their physical damping mechanisms. We discuss the excitation mechanisms of coronal MHD oscillations and waves: the origin of the exciter, exciter propagation, and excitation in magnetic reconnection outflow regions. Finally, we consider the role of coronal MHD oscillations and waves for coronal heating, the detectability of various MHD wave types, and we estimate the energies carried in the observed MHD waves and oscillations: Alfvénic MHD waves could potentially provide sufficient energy to sustain coronal heating, while acoustic MHD waves fall far short of the required coronal heating rates. 9. LASER PLASMA AND LASER APPLICATIONS: Plasma transparency in laser absorption waves in metal capillaries Anisimov, V. N.; Kozolupenko, A. P.; Sebrant, A. Yu 1988-12-01 An experimental investigation was made of the plasma transparency to heating radiation in capillaries when absorption waves propagated in these capillaries as a result of interaction with a CO2 laser pulse of 5-μs duration. When the length of the capillary was in excess of 20 mm, total absorption of the radiation by the plasma was observed at air pressures of 1-100 kPa. When the capillary length was 12 mm, a partial recovery of the transparency took place. A comparison was made with the dynamics and recovery of the plasma transparency when breakdown of air took place near the free surface. 10. Plasma production for electron acceleration by resonant plasma wave Anania, M. P.; Biagioni, A.; Chiadroni, E.; Cianchi, A.; Croia, M.; Curcio, A.; Di Giovenale, D.; Di Pirro, G. P.; Filippi, F.; Ghigo, A.; Lollo, V.; Pella, S.; Pompili, R.; Romeo, S.; Ferrario, M. 2016-09-01 Plasma wakefield acceleration is the most promising acceleration technique known nowadays, able to provide very high accelerating fields (10-100 GV/m), enabling acceleration of electrons to GeV energy in few centimeter. However, the quality of the electron bunches accelerated with this technique is still not comparable with that of conventional accelerators (large energy spread, low repetition rate, and large emittance); radiofrequency-based accelerators, in fact, are limited in accelerating field (10-100 MV/m) requiring therefore hundred of meters of distances to reach the GeV energies, but can provide very bright electron bunches. To combine high brightness electron bunches from conventional accelerators and high accelerating fields reachable with plasmas could be a good compromise allowing to further accelerate high brightness electron bunches coming from LINAC while preserving electron beam quality. Following the idea of plasma wave resonant excitation driven by a train of short bunches, we have started to study the requirements in terms of plasma for SPARC_LAB (Ferrario et al., 2013 [1]). In particular here we focus on hydrogen plasma discharge, and in particular on the theoretical and numerical estimates of the ionization process which are very useful to design the discharge circuit and to evaluate the current needed to be supplied to the gas in order to have full ionization. Eventually, the current supplied to the gas simulated will be compared to that measured experimentally. 11. Theory of Slow Waves in Transversely Nonuniform Plasma Waveguides SciTech Connect Kuzelev, M.V.; Romanov, R.V.; Rukhadze, A.A. 2005-02-15 A general method is developed for a numerical analysis of the frequency spectra of internal, internal-surface, and surface slow waves in a waveguide with transverse plasma density variations. For waveguides with a piecewise constant plasma filling, the spectra of slow waves are thoroughly examined in the limits of an infinitely weak and an infinitely strong external magnetic field. For a smooth plasma density profile, the frequency spectrum of long-wavelength surface waves remains unchanged, but a slow damping rate appears that is caused by the conversion of the surface waves into internal plasma waves at the plasma resonance point. As for short-wavelength internal waves, they are strongly damped by this effect. It is pointed out that, for annular plasma geometry, which is of interest from the experimental point of view, the spectrum of the surface waves depends weakly on the magnetic field strength in the waveguide. 12. Helicon waves in uniform plasmas. II. High m numbers SciTech Connect Stenzel, R. L.; Urrutia, J. M. 2015-09-15 Helicons are whistler modes with azimuthal wave numbers. They have been studied in solids and plasmas where boundaries play a role. The present work shows that very similar modes exist in unbounded gaseous plasmas. Instead of boundaries, the antenna properties determine the topology of the wave packets. The simplest antenna is a magnetic loop which excites m = 0 or m = 1 helicons depending on whether the dipole moment is aligned parallel or perpendicular to the ambient background magnetic field B{sub 0}. While these low order helicons have been described by J. M. Urrutia and R. L. Stenzel [“Helicon modes in uniform plasmas. I. Low m modes,” Phys. Plasmas 22, 092111 (2015)], the present work focuses on high order modes up to m = 8. These are excited by antenna arrays forming magnetic multipoles. Their wave magnetic field has been measured in space and time in a large and uniform laboratory plasma free of boundary effects. The observed wave topology exhibits m pairs of unique field line spirals which may have inspired the name “helicon” to this mode. All field lines converge into these nested spirals which propagate like corkscrews along B{sub 0}. The field lines near the axis of helicons are perpendicular to B{sub 0} and circularly polarized as in parallel whistlers. Helical antennas couple to these transverse fields but not to the spiral fields of helicons. Using a circular antenna array of phased m = 0 loops, right or left rotating or non-rotating multipole antenna fields are generated. They excite m < 0 and m > 0 modes, showing that the plasma supports both modes equally well. The poor excitation of m < 0 modes is a characteristic of loops with dipole moment across B{sub 0}. The radiation efficiency of multipole antennas has been found to decrease with m. 13. Helicon waves in uniform plasmas. II. High m numbers Stenzel, R. L.; Urrutia, J. M. 2015-09-01 Helicons are whistler modes with azimuthal wave numbers. They have been studied in solids and plasmas where boundaries play a role. The present work shows that very similar modes exist in unbounded gaseous plasmas. Instead of boundaries, the antenna properties determine the topology of the wave packets. The simplest antenna is a magnetic loop which excites m = 0 or m = 1 helicons depending on whether the dipole moment is aligned parallel or perpendicular to the ambient background magnetic field B0. While these low order helicons have been described by J. M. Urrutia and R. L. Stenzel ["Helicon modes in uniform plasmas. I. Low m modes," Phys. Plasmas 22, 092111 (2015)], the present work focuses on high order modes up to m = 8. These are excited by antenna arrays forming magnetic multipoles. Their wave magnetic field has been measured in space and time in a large and uniform laboratory plasma free of boundary effects. The observed wave topology exhibits m pairs of unique field line spirals which may have inspired the name "helicon" to this mode. All field lines converge into these nested spirals which propagate like corkscrews along B0. The field lines near the axis of helicons are perpendicular to B0 and circularly polarized as in parallel whistlers. Helical antennas couple to these transverse fields but not to the spiral fields of helicons. Using a circular antenna array of phased m = 0 loops, right or left rotating or non-rotating multipole antenna fields are generated. They excite m < 0 and m > 0 modes, showing that the plasma supports both modes equally well. The poor excitation of m < 0 modes is a characteristic of loops with dipole moment across B0. The radiation efficiency of multipole antennas has been found to decrease with m. 14. Rogue Waves Associated with Circularly Polarized Waves in Magnetized Plasmas Kourakis, I.; Borhanian, J.; Saxena, V.; Veldes, G.; Frantzeskakis, D. J. 2012-10-01 Extreme events occur in abundance in the ocean: an ultra-high ghost wave" often appears unexpectedly, against an otherwise moderate-on-average sea surface elevation, propagating for a short while and then disappearing without leaving a trace. Rogue waves are now recognized as proper nonlinear structures on their own. Unlike solitary waves, these events are localized in space and in time. Various approaches exist to model their dynamics, including nonlinear Schrodinger models, Ginzburg-Landau models, kinetic-theoretical models, and probabilistic models. We have undertaken an investigation, from first principles, of rogue waves in plasmas in the form of localized events associated with electromagnetic pulses. A multiple scale technique is employed to solve the fluid-Maxwell equations for nonlinear circularly polarized electromagnetic pulses. A nonlinear Schrodinger (NLS) type equation is shown to govern the amplitude of the vector potential. A set of non-stationary envelope solutions of the NLS equation is presented, and the variation of their structural properties with the magnetic field are investigated. 15. Ion Cyclotron Waves Observed in the Comet Halley: A New Look to Giotto Observations Rodriguez-Martinez, M. R.; Blanco-Cano, X.; Aguilar-Rodriguez, E.; Haro-Corzo, S. S. A. R., Sr.; Arriaga-Contreras, V. V. R. 2015-12-01 Ion Cyclotron Waves (ICW) were observed with Giotto spacecraft. Magnetic field data have been analyzed in the past to determine the nature of ICW and compared with other comets, as Giacobini-Zinner and Grigg-Skjellerup. It is important to develop tools that allow re-analyze these data in order to know better the characteristics of these waves. In this work we have applied a Fast Fourier Transform (FFT) analysis in which we define the transverse and compressive powers for a better contrast and characterization of ICW. The information obtained will be presented through dynamic spectra in several time intervals. This tool will allow to explore the possibility to check the existence of Harmonic Mode Waves (HMW) of these waves. Finally, we use linear kinetic theory, using WHAMP code, in order to determine conditions for wave growth in a plasma resembling the regions where these waves were observed. 16. Laser-driven plasma beat-wave propagation in a density-modulated plasma. PubMed Gupta, Devki Nandan; Nam, In Hyuk; Suk, Hyyong 2011-11-01 A laser-driven plasma beat wave, propagating through a plasma with a periodic density modulation, can generate two sideband plasma waves. One sideband moves with a smaller phase velocity than the pump plasma wave and the other propagates with a larger phase velocity. The plasma beat wave with a smaller phase velocity can accelerate modest-energy electrons to gain substantial energy and the electrons are further accelerated by the main plasma wave. The large phase velocity plasma wave can accelerate these electrons to higher energies. As a result, the electrons can attain high energies during the acceleration by the plasma waves in the presence of a periodic density modulation. The analytical results are compared with particle-in-cell simulations and are found to be in reasonable agreement. 17. Electron plasma waves upstream of the earth's bow shock NASA Technical Reports Server (NTRS) Lacombe, C.; Mangeney, A.; Harvey, C. C.; Scudder, J. D. 1985-01-01 Electrostatic waves are observed around the plasma frequency fpe in the electron foreshock, together with electrons backstreaming from the bow shock. Using data from the sounder aboard ISEE 1, it is shown that this noise, previously understood as narrow band Langmuir waves more or less widened by Doppler shift or nonlinear effects, is in fact composed of two distinct parts: one is a narrow band noise, emitted just above fpe, and observed at the upstream boundary of the electron foreshock. This component has been interpreted as Langmuir waves emitted by a beam-plasma instability. It is suggested that it is of sufficiently large amplitude and monochromatic enough to trap resonant electrons. The other is a broad band noise, more impulsive than the narrow band noise, observed well above and/or well below fpe, deeper in the electron foreshock. The broad band noise has an average spectrum with a typical bi-exponential shape; its peak frequency is not exactly equal to fpe and depends on the Deybe length. This peak frequency also depends on the velocity for which the electron distribution has maximum skew. An experimental determination of the dispersion relation of the broad band noise shows that this noise, as well as the narrow band noise, may be due to the instability of a hot beam in a plasma. 18. Observation of stimulated electron acoustic wave scattering: the case for nonlinear kinetic effects Montgomery, D. S.; Cobble, J. A.; Fernandez, J. C.; Rose, H. A.; Focia, R. J.; Russell, D. A. 2001-10-01 Electrostatic waves with a frequency and phase velocity between an ion acoustic wave (IAW) and an electron plasma wave (EPW) have been observed with Thomson scattering in inhomogeneous plasmas, and in the backscattered spectrum for homogeneous single hot spot laser plasmas. We show that these waves are consistent with an electron-acoustic wave (EAW) that is a BGK-like mode due to electron trapping. The nonlinear dispersion relation for BGK-like EPW and EAW is discussed, and previous inhomogeneous Trident and Nova data are re-examined in this context. The possible implications of these results for backscattered SRS on the NIF are discussed. 19. Nonplanar Shock Waves in Dusty Plasmas SciTech Connect Mamun, A. A.; Shukla, P. K. 2011-11-29 Nonplanar (viz. cylindrical and spherical) electro-acoustic [dust-ion-acoustic (DIA) and dust-acoustic (DA)] shock waves have been investigated by employing the reductive perturbation method. The dust charge fluctuation (strong correlation among highly charged dust) is the source of dissipation, and is responsible for the formation of the DIA (DA) shock structures. The effects of cylindrical and spherical geometries on the time evolution of DIA and DA shock structures are examined and identified. The combined effects of vortex-like electron distribution and dust charge fluctuation (dust-correlation and effective dust-temperature) on the basic features of nonplanar DIA (DA) shock waves are pinpointed. The implications of our results in laboratory dusty plasma experiments are briefly discussed. 20. Nonplanar Shock Waves in Dusty Plasmas Mamun, A. A.; Shukla, P. K. 2011-11-01 Nonplanar (viz. cylindrical and spherical) electro-acoustic [dust-ion-acoustic (DIA) and dust-acoustic (DA)] shock waves have been investigated by employing the reductive perturbation method. The dust charge fluctuation (strong correlation among highly charged dust) is the source of dissipation, and is responsible for the formation of the DIA (DA) shock structures. The effects of cylindrical and spherical geometries on the time evolution of DIA and DA shock structures are examined and identified. The combined effects of vortex-like electron distribution and dust charge fluctuation (dust-correlation and effective dust-temperature) on the basic features of nonplanar DIA (DA) shock waves are pinpointed. The implications of our results in laboratory dusty plasma experiments are briefly discussed. 1. Plasma waves and jets from moving conductors Gralla, Samuel E.; Zimmerman, Peter 2016-06-01 We consider force-free plasma waves launched by the motion of conducting material through a magnetic field. We develop a spacetime-covariant formalism for perturbations of a uniform magnetic field and show how the transverse motion of a conducting fluid acts as a source. We show that fast-mode waves are sourced by the compressibility of the fluid, with incompressible fluids launching a pure-Alfvén outflow. Remarkably, this outflow can be written down in closed form for an arbitrary time-dependent, nonaxisymmetric incompressible flow. The instantaneous flow velocity is imprinted on the magnetic field and transmitted away at the speed of light, carrying detailed information about the conducting source at the time of emission. These results can be applied to transients in pulsar outflows and to jets from neutron stars orbiting in the magnetosphere of another compact object. We discuss jets from moving conductors in some detail. 2. Plasma waves produced by the xenon ion beam experiment on the Porcupine sounding rocket NASA Technical Reports Server (NTRS) Kintner, P. M.; Kelley, M. 1982-01-01 The production of electrostatic ion cyclotron waves by a perpendicular ion beam in the F-region ionosphere is described. The ion beam experiment was part of the Porcupine program and produced electrostatic hydrogen cyclotron waves just above harmonics of the hydrogen cyclotron frequency. The plasma process may be thought of as a magnetized background ionosphere through which an unmagnetized beam is flowing. The dispersion equation for this hypothesis is constructed and solved. Preliminary solutions agree well with the observed plasma waves. 3. Nonlinear Generation of Electromagnetic Waves through Induced Scattering by Thermal Plasma Tejero, E. M.; Crabtree, C.; Blackwell, D. D.; Amatucci, W. E.; Mithaiwala, M.; Ganguli, G.; Rudakov, L. 2015-12-01 We demonstrate the conversion of electrostatic pump waves into electromagnetic waves through nonlinear induced scattering by thermal particles in a laboratory plasma. Electrostatic waves in the whistler branch are launched that propagate near the resonance cone. When the amplitude exceeds a threshold ~5 × 10-6 times the background magnetic field, wave power is scattered below the pump frequency with wave normal angles (~59°), where the scattered wavelength reaches the limits of the plasma column. The scattered wave has a perpendicular wavelength that is an order of magnitude larger than the pump wave and longer than the electron skin depth. The amplitude threshold, scattered frequency spectrum, and scattered wave normal angles are in good agreement with theory. The results may affect the analysis and interpretation of space observations and lead to a comprehensive understanding of the nature of the Earth’s plasma environment. 4. Nonlinear Generation of Electromagnetic Waves through Induced Scattering by Thermal Plasma PubMed Central Tejero, E. M.; Crabtree, C.; Blackwell, D. D.; Amatucci, W. E.; Mithaiwala, M.; Ganguli, G.; Rudakov, L. 2015-01-01 We demonstrate the conversion of electrostatic pump waves into electromagnetic waves through nonlinear induced scattering by thermal particles in a laboratory plasma. Electrostatic waves in the whistler branch are launched that propagate near the resonance cone. When the amplitude exceeds a threshold ~5 × 10−6 times the background magnetic field, wave power is scattered below the pump frequency with wave normal angles (~59°), where the scattered wavelength reaches the limits of the plasma column. The scattered wave has a perpendicular wavelength that is an order of magnitude larger than the pump wave and longer than the electron skin depth. The amplitude threshold, scattered frequency spectrum, and scattered wave normal angles are in good agreement with theory. The results may affect the analysis and interpretation of space observations and lead to a comprehensive understanding of the nature of the Earth’s plasma environment. PMID:26647962 5. Low Frequency Waves in the Plasma Environment Around the Shuttle NASA Technical Reports Server (NTRS) Vayner, Boris V.; Ferguson, Dale C. 1996-01-01 As a part of the SAMPIE (The Solar Array Module Plasma Interaction Experiment) program, the Langmuir probe (LP) was employed to measure plasma characteristics during the flight of STS-62. The whole set of data could be divided into two parts: (1) low frequency sweeps to determine voltage-current characteristics and to find the electron temperature and number density; (2) high frequency turbulence (HFT) data caused by electromagnetic noise around the Shuttle. Broadband noise was observed at 250-20,000 Hz frequencies. Measurements were performed in ram conditions; thus, it seems reasonable to believe that the influence of spacecraft operations on plasma parameters was minimized. It is shown that ion acoustic waves were observed, and two kinds of instabilities are suggested for explanation of the origin of these waves. According to the purposes of SAMPIE, samples of solar cells were placed in the cargo bay of the Shuttle, and high negative bias voltages were applied to them to initiate arcing between these cells and the surrounding plasma. The arcing onset was registered by special counters, and data were obtained that included the amplitudes of current, duration of each arc, and the number of arcs per one experiment. The LP data were analyzed for two different situations: with arcing and without arcing. Electrostatic noise spectra for both situations and a theoretical explanation of the observed features are presented in this paper. 6. Electromagnetic wave band structure due to surface plasmon resonances in a complex plasma 2016-07-01 The dielectric properties of complex plasma containing either metal or dielectric spherical inclusions (macroparticles, dust) are investigated. We focus on surface plasmon resonances on the macroparticle surfaces and their effect on electromagnetic wave propagation. It is demonstrated that the presence of surface plasmon oscillations can significantly modify plasma electromagnetic properties by resonances and cutoffs in the effective permittivity. This leads to related branches of electromagnetic waves and to the wave band gaps. The conditions necessary to observe the band-gap structure in laboratory dusty plasma and/or space (cosmic) dusty plasmas are discussed. 7. Observation of equipartition of seismic waves. PubMed Hennino, R; Trégourès, N; Shapiro, N M; Margerin, L; Campillo, M; van Tiggelen, B A; Weaver, R L 2001-04-09 Equipartition is a first principle in wave transport, based on the tendency of multiple scattering to homogenize phase space. We report observations of this principle for seismic waves created by earthquakes in Mexico. We find qualitative agreement with an equipartition model that accounts for mode conversions at the Earth's surface. 8. Antenna excitation of drift wave in a toroidal plasma SciTech Connect Diallo, A.; Ricci, P.; Fasoli, A.; Furno, I.; Labit, B.; Mueller, S. H.; Podesta, M.; Poli, F. M.; Skiff, F. 2007-10-15 In a magnetized toroidal plasma, an antenna tunable in vertical wave number is used to excite density perturbations. Coherent detection is performed by means of Langmuir probes to directly determine both the wave vector and the plasma response induced by the antenna. Comparison between the theoretical density response predicted by the generalized Hasegawa-Wakatani model, and the experimentally determined density response enables us the identification of one peak of the plasma response as a drift wave. 9. PLASMA-WAVE GENERATION IN A DYNAMIC SPACETIME SciTech Connect Yang, Huan; Zhang, Fan 2016-02-01 We propose a new electromagnetic (EM)-emission mechanism in magnetized, force-free plasma, which is driven by the evolution of the underlying dynamic spacetime. In particular, the emission power and angular distribution of the emitted fast-magnetosonic and Alfvén waves are separately determined. Previous numerical simulations of binary black hole mergers occurring within magnetized plasma have recorded copious amounts of EM radiation that, in addition to collimated jets, include an unexplained, isotropic component that becomes dominant close to the merger. This raises the possibility of multimessenger gravitational-wave and EM observations on binary black hole systems. The mechanism proposed here provides a candidate analytical characterization of the numerical results, and when combined with previously understood mechanisms such as the Blandford–Znajek process and kinetic-motion-driven radiation, it allows us to construct a classification of different EM radiation components seen in the inspiral stage of compact-binary coalescences. 10. Energy balance of a plasma with a wave, taking the wave nonpotentiality into account Gelberg, M. G.; Volosevich, A. V. It is shown that the potential electric field of low-frequency plasma waves in the ionosphere is phase-shifted by approximately -pi/2 with respect to current fluctuations, while the vortex field is nearly cophase with the current. Thus, the work of energy transfer between the plasma and the wave occurs primarily with the participation of the vortex field. The wave nonpotentiality is shown to have a substantial effect on the energy balance of the wave-plasma system. 11. Solitary and shock waves in magnetized electron-positron plasma SciTech Connect Lu, Ding; Li, Zi-Liang; Abdukerim, Nuriman; Xie, Bai-Song 2014-02-15 An Ohm's law for electron-positron (EP) plasma is obtained. In the framework of EP magnetohydrodynamics, we investigate nonrelativistic nonlinear waves' solutions in a magnetized EP plasma. In the collisionless limit, quasistationary propagating solitary wave structures for the magnetic field and the plasma density are obtained. It is found that the wave amplitude increases with the Mach number and the Alfvén speed. However, the dependence on the plasma temperature is just the opposite. Moreover, for a cold EP plasma, the existence range of the solitary waves depends only on the Alfvén speed. For a hot EP plasma, the existence range depends on the Alfvén speed as well as the plasma temperature. In the presence of collision, the electromagnetic fields and the plasma density can appear as oscillatory shock structures because of the dissipation caused by the collisions. As the collision frequency increases, the oscillatory shock structure becomes more and more monotonic. 12. Nonlinear Cylindrical Waves on a Plane Plasma Surface 2004-01-01 By means of the cold electron plasma equations, it is shown that surface soliton solutions can exist in the azimuthally symmetric case at the boundary of semi-infinite plasmas for both standing and running waves. 13. Laser plasma simulations of the generation processes of Alfven and collisionless shock waves in space plasma Prokopov, P. A.; Zakharov, Yu P.; Tishchenko, V. N.; Shaikhislamov, I. F.; Boyarintsev, E. L.; Melekhov, A. V.; Ponomarenko, A. G.; Posukh, V. G.; Terekhin, V. A. 2016-11-01 Generation of Alfven waves propagating along external magnetic field B0 and Collisionless Shock Waves propagating across B0 are studied in experiments with laser- produced plasma and magnetized background plasma. The collisionless interaction of interpenetrating plasma flows takes place through a so-called Magnetic Laminar Mechanism (MLM) or Larmor Coupling. At the edge of diamagnetic cavity LP-ions produce induction electric field Eφ which accelerates BP-ions while LP-ions rotate in opposite direction. The ions movement generates sheared azimuthal magnetic field Bφ which could launches torsional Alfven wave. In previous experiments at KI-1 large scale facility a generation of strong perturbations propagating across B0 with magnetosonic speed has been studied at a moderate value of interaction parameter δ∼0.3. In the present work we report on experiments at conditions of 5∼R2 and large Alfven-Mach number MA∼10 in which strong transverse perturbations traveling at a scale of ∼1 m in background plasma at a density of ∼3*1013 cm-3 is observed. At the same conditions but smaller MA ∼ 2 a generation, the structure and dynamic of Alfven wave with wavelength ∼0.5 m propagating along fields B0∼100÷500 G for a distance of ∼2.5 m is studied. 14. ULF waves in the Martian foreshock: MAVEN observations Shan, Lican; Mazelle, Christian; Meziane, Karim; Ruhunusiri, Suranga; Espley, Jared; Halekas, Jasper; Connerney, Jack; McFadden, Jim; Mitchell, Dave; Larson, Davin; Brain, Dave; Jakosky, Bruce; Ge, Yasong; Du, Aimin 2016-04-01 Foreshock ULF waves constitute a significant physical phenomenon of the plasma environment for terrestrial planets. The occurrence of these ULF waves, associated with backstreaming ions reflected and accelerated at the bow shock, implies specific conditions and properties of the shock and its foreshock. Using measurements from MAVEN, we report clear observations of this type of ULF waves in the Martian foreshock. We show from different case studies that the peak frequency of the wave case in spacecraft frame is too far from the local ion cyclotron frequency to be associated with local pickup ions taking into account the Doppler shifted frequency from a cyclotron resonance, the obliquity of the mode, resonance broadening and experimental uncertainties. On the opposite their properties fit very well with foreshock waves driven unstable by backtreaming field-aligned ion beams. The propagation angle is usually less than 30 degrees from ambient magnetic field. The waves also display elliptical and left-hand polarizations with respect to interplanetary magnetic field in the spacecraft frame. It is clear for these cases that foreshock ions are simultaneous present for the ULF wave interval. Such observation is important in order to discriminate with the already well-reported pickup ion (protons) waves associated with exospheric hydrogen in order to quantitatively use the later to study seasonal variations of the hydrogen corona. 15. Magnetospheric radio and plasma wave research - 1987-1990 SciTech Connect Kurth, W.S. ) 1991-01-01 This review covers research performed in the area of magnetospheric plasma waves and wave-particle interactions as well as magnetospheric radio emissions. The report focuses on the near-completion of the discovery phase of radio and plasma wave phenomena in the planetary magnetospheres with the successful completion of the Voyager 2 encounters of Neptune and Uranus. Consideration is given to the advances made in detailed studies and theoretical investigations of radio and plasma wave phenomena in the terrestrial magnetosphere or in magnetospheric plasmas in general. 16. Relativistic nonlinear plasma waves in a magnetic field NASA Technical Reports Server (NTRS) Kennel, C. F.; Pellat, R. 1975-01-01 Five relativistic plane nonlinear waves were investigated: circularly polarized waves and electrostatic plasma oscillations propagating parallel to the magnetic field, relativistic Alfven waves, linearly polarized transverse waves propagating in zero magnetic field, and the relativistic analog of the extraordinary mode propagating at an arbitrary angle to the magnetic field. When the ions are driven relativistic, they behave like electrons, and the assumption of an 'electron-positron' plasma leads to equations which have the form of a one-dimensional potential well. The solutions indicate that a large-amplitude superluminous wave determines the average plasma properties. 17. Plasma Shock Wave Modification Experiments in a Temperature Compensated Shock Tube NASA Technical Reports Server (NTRS) Vine, Frances J.; Mankowski, John J.; Saeks, Richard E.; Chow, Alan S. 2003-01-01 A number of researchers have observed that the intensity of a shock wave is reduced when it passes through a weakly ionized plasma. While there is little doubt that the intensity of a shock is reduced when it propagates through a weakly ionized plasma, the major question associated with the research is whether the reduction in shock wave intensity is due to the plasma or the concomitant heating of the flow by the plasma generator. The goal of this paper is to describe a temperature compensated experiment in a "large" diameter shock tube with an external heating source, used to control the temperature in the shock tube independently of the plasma density. 18. ISIS Topside-Sounder Plasma-Wave Investigations as Guides to Desired Virtual Wave Observatory (VWO) Data Search Capabilities NASA Technical Reports Server (NTRS) Benson, Robert F.; Fung, Shing F. 2008-01-01 Many plasma-wave phenomena, observed by space-borne radio sounders, cannot be properly explained in terms of wave propagation in a cold plasma consisting of mobile electrons and infinitely massive positive ions. These phenomena include signals known as plasma resonances. The principal resonances at the harmonics of the electron cyclotron frequency, the plasma frequency, and the upper-hybrid frequency are well explained by the warm-plasma propagation of sounder-generated electrostatic waves, Other resonances have been attributed to sounder-stimulated plasma instability and non-linear effects, eigenmodes of cylindrical electromagnetic plasma oscillations, and plasma memory processes. Data from the topside sounders of the International Satellites for Ionospheric Studies (ISIS) program played a major role in these interpretations. A data transformation and preservation effort at the Goddard Space Flight Center has produced digital ISIS topside ionograms and a metadata search program that has enabled some recent discoveries pertaining to the physics of these plasma resonances. For example, data records were obtained that enabled the long-standing question (several decades) of the origin of the plasma resonance at the fundamental electron cyclotron frequency to be explained [Muldrew, Radio Sci., 2006]. These data-search capabilities, and the science enabled by them, will be presented as a guide to desired data search capabilities to be included in the Virtual Wave Observatory (VWO). 19. Internal Waves in Straits (IWISE): Observations of Wave Generation DTIC Science & Technology 2012-09-30 deployment of a 2-D array of pressure-sensor-equipped inverted echo sounders (PIES) so as to observe the generation of internal waves by tidal...east of the strait to the westernmost deployments. Fig. 1 Deployment locations of Pressure sensor equipped Inverted Echo Sounders [PIES] in South...measurements of nonlinear internal waves using the inverted echo sounder , J. Atmos. Oceanic Technology, 26, 2228−2242. David M Farmer, Li, Qiang & Jae-Hun 20. Effective-action approach to wave propagation in scalar QED plasmas Shi, Yuan; Fisch, Nathaniel J.; Qin, Hong 2016-07-01 A relativistic quantum field theory with nontrivial background fields is developed and applied to study waves in plasmas. The effective action of the electromagnetic 4-potential is calculated ab initio from the standard action of scalar QED using path integrals. The resultant effective action is gauge invariant and contains nonlocal interactions, from which gauge bosons acquire masses without breaking the local gauge symmetry. To demonstrate how the general theory can be applied, we give two examples: a cold unmagnetized plasma and a cold uniformly magnetized plasma. Using these two examples, we show that all linear waves well known in classical plasma physics can be recovered from relativistic quantum results when taking the classical limit. In the opposite limit, classical wave dispersion relations are modified substantially. In unmagnetized plasmas, longitudinal waves propagate with nonzero group velocities even when plasmas are cold. In magnetized plasmas, anharmonically spaced Bernstein waves persist even when plasmas are cold. These waves account for cyclotron absorption features observed in spectra of x-ray pulsars. Moreover, cutoff frequencies of the two nondegenerate electromagnetic waves are red-shifted by different amounts. These corrections need to be taken into account in order to correctly interpret diagnostic results in laser plasma experiments. 1. New observations of plasma vortices and insights into their interpretation. [in magnetotail plasma sheet NASA Technical Reports Server (NTRS) Hones, E. W., Jr.; Birn, J.; Bame, S. J.; Russell, C. T. 1983-01-01 Two- and three-dimensional plasma measurements and three-dimensional magnetic field measurements made with the ISEE 1 and 2 satellites during sixteen plasma vortex occurrences in the magnetotail plasma sheet are used to develop a fuller description of the vortex phenomenon than has existed heretofore. The phase and energy propagation properties of the vortex waves was studied in particular. The rotation period of the vortices (T = 10 + or - 5 minutes) is apparently independent of location, while the wavelength (lambda not less than several Re) increases with increasing distance down the tail, pointing to a global mode of propagation in which effects of inhomogeneous equilibrium are important. The flow rotation can be explained by propagation of surface waves or resonant waves in a uniform medium. Other observed features, however, require a nonuniform model: nonuniform propagation properties and differences of the phase propagation speed calculated from different components of velocity or magnetic field. 2. Plasma waves downstream of weak collisionless shocks NASA Technical Reports Server (NTRS) Coroniti, F. V.; Greenstadt, E. W.; Moses, S. L.; Smith, E. J.; Tsurutani, B. T. 1993-01-01 In September 1983 the International Sun Earth Explorer 3 (ISEE 3) International Cometary Explorer (ICE) spacecraft made a long traversal of the distant dawnside flank region of the Earth's magnetosphere and had many encounters with the low Mach number bow shock. These weak shocks excite plasma wave electric field turbulence with amplitudes comparable to those detected in the much stronger bow shock near the nose region. Downstream of quasi-perpendicular (quasi-parallel) shocks, the E field spectra exhibit a strong peak (plateau) at midfrequencies (1 - 3 kHz); the plateau shape is produced by a low-frequency (100 - 300 Hz) emission which is more intense behind downstream of two quasi-perpendicular shocks show that the low frequency signals are polarized parallel to the magnetic field, whereas the midfrequency emissions are unpolarized or only weakly polarized. A new high frequency (10 - 30 kHz) emission which is above the maximum Doppler shift exhibit a distinct peak at high frequencies; this peak is often blurred by the large amplitude fluctuations of the midfrequency waves. The high-frequency component is strongly polarized along the magnetic field and varies independently of the lower-frequency waves. 3. Upstream waves at Mars: Phobos observations SciTech Connect Russell, C.T.; Luhmann, J.G. ); Schwingenschuh, K.; Riedler, W. ); Yeroshenko, Ye. ) 1990-05-01 The region upstream from the Mars subsolar bow shock is surveyed for the presence of MHD wave phenomena using the high temporal resolution data from the MAGMA magnetometer. Strong turbulence is observed when the magnetic field is connected to the Mars bow shock in such a way as to allow diffuse ions to reach the spacecraft. On 2 occasions this turbulence occurred upon crossing the Phobos orbit. Also weak, {minus}0.15 nT, waves are observed at the proton gyro frequency. These waves are left-hand elliptically polarized and may be associated with the pick-up of protons from the Mars hydrogen exosphere. 4. Propagation velocity of Alfven wave packets in a dissipative plasma SciTech Connect Amagishi, Y.; Nakagawa, H. ); Tanaka, M. ) 1994-09-01 We have experimentally studied the behavior of Alfven wave packets in a dissipative plasma due to ion--neutral-atom collisions. It is urged that the central frequency of the packet is observed to gradually decrease with traveling distance in the absorption range of frequencies because of a differential damping among the Fourier components, and that the measured average velocity of its peak amplitude is not accounted for by the conventional group velocity, but by the prediction derived by Tanaka, Fujiwara, and Ikegami [Phys. Rev. A 34, 4851 (1986)]. Furthermore, when the initial central frequency is close to the critical frequency in the anomalous dispersion, the wave packet apparently collapses when traveling along the magnetic field; however, we have found that it is decomposed into another two wave packets with the central frequencies being higher or lower than the critical frequency. 5. Heating of the plasma sheet by broadband electromagnetic waves Chaston, C. C.; Bonnell, J. W.; Salem, C. 2014-12-01 We demonstrate that broadband low-frequency electromagnetic field fluctuations embedded within fast flows throughout the Earth's plasma sheet may drive significant ion heating. This heating is nearly entirely in the direction perpendicular to the background magnetic field and is estimated to occur at an average rate of ~1 eV/s with rates in excess of 10 eV/s within one standard deviation of the average value over all observed events. For an Earthward flow the total change in temperature along a flow path may exceed one keV and for "wave-rich" flows can be comparable to that expected due to conservation of the first adiabatic invariant. The consequent increase in plasma pressure and flux tube entropy may lead to braking of inward motion and the suppression of plasma interchange. 6. Relativistic electromagnetic waves in an electron-ion plasma NASA Technical Reports Server (NTRS) Chian, Abraham C.-L.; Kennel, Charles F. 1987-01-01 High power laser beams can drive plasma particles to relativistic energies. An accurate description of strong waves requires the inclusion of ion dynamics in the analysis. The equations governing the propagation of relativistic electromagnetic waves in a cold electron-ion plasma can be reduced to two equations expressing conservation of energy-momentum of the system. The two conservation constants are functions of the plasma stream velocity, the wave velocity, the wave amplitude, and the electron-ion mass ratio. The dynamic parameter, expressing electron-ion momentum conversation in the laboratory frame, can be regarded as an adjustable quantity, a suitable choice of which will yield self-consistent solutions when other plasma parameters were specified. Circularly polarized electromagnetic waves and electrostatic plasma waves are used as illustrations. 7. Generation of acoustic rogue waves in dusty plasmas through three-dimensional particle focusing by distorted waveforms Tsai, Ya-Yi; Tsai, Jun-Yi; I, Lin 2016-06-01 Rogue waves--rare uncertainly emerging localized events with large amplitudes--have been experimentally observed in many nonlinear wave phenomena, such as water waves, optical waves, second sound in superfluid He II (ref. ) and ion acoustic waves in plasmas. Past studies have mainly focused on one-dimensional (1D) wave behaviour through modulation instabilities, and to a lesser extent on higher-dimensional behaviour. The question whether rogue waves also exist in nonlinear 3D acoustic-type plasma waves, the kinetic origin of their formation and their correlation with surrounding 3D waveforms are unexplored fundamental issues. Here we report the direct experimental observation of dust acoustic rogue waves in dusty plasmas and construct a picture of 3D particle focusing by the surrounding tilted and ruptured wave crests, associated with the higher probability of low-amplitude holes for rogue-wave generation. 8. The plasma wave signature of a 'magnetic hole' in the vicinity of the magnetopause NASA Technical Reports Server (NTRS) Treumann, R. A.; Sckopke, N.; Brostrom, L.; Labelle, J. 1990-01-01 Wave spectral measurements in the region of the September 4, 1984 magnetic hole obtained with the plasma wave instrumentation aboard the AMPTE IRM spacecraft are presented. The instrument is briefly described. The full wave data is given and possible reasons for the typical form of the spectra inside the hole are discussed. Relevant observations are presented and different wave modes and their possible origins are discussed. A summary is given with a discussion of ideas about the origin and formation of holes. 9. MESSENGER Observations of ULF Waves in Mercury's Foreshock Region NASA Technical Reports Server (NTRS) Le, Guan; Chi, Peter J.; Bardsen, Scott; Blanco-Cano, Xochitl; Slavin, James A.; Korth, Haje 2012-01-01 The region upstream from a planetary bow shock is a natural plasma laboratory containing a variety of wave particle phenomena. The study of foreshocks other than the Earth s is important for extending our understanding of collisionless shocks and foreshock physics since the bow shock strength varies with heliocentric distance from the Sun, and the sizes of the bow shocks are different at different planets. The Mercury s bow shock is unique in our solar system as it is produced by low Mach number solar wind blowing over a small magnetized body with a predominately radial interplanetary magnetic field. Previous observations of Mercury upstream ultra-low frequency (ULF) waves came exclusively from two Mercury flybys of Mariner 10. The MESSENGER orbiter data enable us to study of upstream waves in the Mercury s foreshock in depth. This paper reports an overview of upstream ULF waves in the Mercury s foreshock using high-time resolution magnetic field data, 20 samples per second, from the MESSENGER spacecraft. The most common foreshock waves have frequencies near 2 Hz, with properties similar to the 1-Hz waves in the Earth s foreshock. They are present in both the flyby data and in every orbit of the orbital data we have surveyed. The most common wave phenomenon in the Earth s foreshock is the large-amplitude 30-s waves, but similar waves at Mercury have frequencies at 0.1 Hz and occur only sporadically with short durations (a few wave cycles). Superposed on the "30-s" waves, there are spectral peaks at 0.6 Hz, not reported previously in Mariner 10 data. We will discuss wave properties and their occurrence characteristics in this paper. 10. Dispersive ducting of MHD waves in the plasma sheet - A source of Pi2 wave bursts NASA Technical Reports Server (NTRS) Edwin, P. M.; Roberts, B.; Hughes, W. J. 1986-01-01 Fast magnetoacoustic waves can be ducted by plasma inhomogeneities such as the plasma sheet. As this ducting is dispersive an impulsive source will give rise to a well-defined, quasi-periodic wave packet with time-scales determined by the width of the inhomogeneity and characteristic speeds in the wave duct and surrounding medium. The duration of the wave packet depends upon the distance from the source. It is argued that an impulsive source in the plasma sheet at substorm onset will produce a wave packet near earth with characteristics similar to pi2 wave bursts and put this idea forward as a mechanism for the generation of pi2 pulsations. 11. Magnetosonic wave in pair-ion electron collisional plasmas Hussain, S.; Hasnain, H. 2017-03-01 Low frequency magnetosonic waves in positive and negative ions of equal mass and opposite charges in the presence of electrons in collisional plasmas are studied. The collisions of ions and electrons with neutrals are taken into account. The nonlinearities in the plasma system arise due to ion and electrons flux, Lorentz forces, and plasma current densities. The reductive perturbation method is applied to derive the Damped Korteweg de Vries (DKdV) equation. The time dependent solution of DKdV is presented. The effects of variations of different plasma parameters on propagation characteristics of magnetosonic waves in pair-ion electron plasma in the context of laboratory plasmas are discussed. 12. Discovery of cometary kilometric radiations and plasma waves at Comet Halley Oya, H.; Morioka, A.; Miyake, W.; Smith, E. J.; Tsurutani, B. T. 1986-05-01 The plasma-wave probe carried by the spacecraft Sakigake discovered discrete spectra of emissions from comet Halley in the frequency range 30 - 195 kHz. The observed cometary kilometric radiation appears to come from moving shocks in the coma region which are possibly associated with temporal variations of the solar wind. Waves due to plasma instabilities associated with the pick-up of cometary ions by the solar wind were observed within a region almost 107km from the comet nucleus. 13. High-frequency Plasma Waves Associated with Magnetic Reconnection in the Solar Wind Wang, Y. 2015-12-01 Activities of high-frequency plasma waves associated with magnetic reconnection in the solar wind observed by Time Domain Sampler (TDS) experiments on STEREO/WAVES are preliminarily analyzed. The TDS instrument can provide burst mode electric fields data with as long as 16384 sample points at 250 kHz sampling rate. In all 1120 suspected reconnection events, it is found that the most commonly occurred waves are neither ion acoustic waves, electrostatic solitary waves, nor Langmuir/upper hybrid waves, but Bernstein-like waves with harmonics of the electron cyclotron frequency. In addition, to each type of waves, Langmuir/upper hybrid waves reveal the largest occurrence rate in the reconnection region than in the ambient solar wind. These results indicate that Bernstein-like waves and Langmuir/upper hybrid waves might play important roles in the reconnection associated particle heating processes and they might also influence the dissipation of magnetic reconnection. 14. MHD waves and oscillations in the solar plasma. Introduction. PubMed Erdélyi, Robert 2006-02-15 The Sun's magnetic field is responsible for many spectacularly dynamic and intricate phenomena, such as the 11 year solar activity cycle, the hot and tenuous outer atmosphere called the solar corona, and the continuously expanding stream of solar particles known as the solar wind.Recently, there has been an enormous increase in our understanding of the role of solar magnetism in producing the observed complex atmosphere of the Sun. One such advance has occurred in the detection, by several different high-resolution space instruments on-board the Solar and Heliospheric Observatory and Transition Region and Coronal Explorer satellites, of magnetic waves and oscillations in the solar corona. The new subjects of solar atmospheric and coronal seismology are undergoing rapid development. The aim of this Scientific Discussion Meeting was to address the progress made through observational, theoretical and numerical studies of wave phenomena in the magnetic solar plasma. Major theoretical and observational advances were reported by a wide range of international scientists and pioneers in this field, followed by lively discussions and poster sessions on the many intriguing questions raised by the new results. Theoretical and observational aspects of magnetohydrodynamic waves and oscillations in general, and how these wave phenomena differ in various regions of the Sun, including sunspots, the transient lower atmosphere and the corona (in magnetic loops, plumes and prominences), were addressed through invited review papers and selected poster presentations. The results of these deliberations are collected together in this volume. 15. Development of Small Plasma Wave Receiver with a Dedicated Chip for Scientific Spacecraft Fukuhara, H.; Kojima, H.; Ishii, H.; Okada, S.; Yamakawa, H. 2012-04-01 16. Observations of purely compressional waves in the upper ULF band observed by the Van Allen Probes Posch, J. L.; Engebretson, M. J.; Johnson, J.; Kim, E. H.; Thaller, S. A.; Wygant, J. R.; Kletzing, C.; Smith, C. W.; Reeves, G. D. 2014-12-01 Purely compressional electromagnetic waves, also denoted fast magnetosonic waves, equatorial noise, and ion Bernstein modes, can both heat thermal protons and accelerate electrons up to relativistic energies. These waves have been observed both in the near-equatorial region in the inner magnetosphere and in the plasma sheet boundary layer. Although these waves have been observed by various types of satellite instruments (DC and AC magnetometers and electric field sensors), most recent studies have used data from AC sensors, and many have been restricted to frequencies above ~50 Hz. We report here on a survey of ~200 of these waves, based on DC electric and magnetic field data from the EFW double probe and EMFISIS fluxgate magnetometer instruments, respectively, on the Van Allen Probes spacecraft during its first two years of operation. The high sampling rate of these instruments makes it possible to extend observational studies of the lower frequency population of such waves to lower L shells than any previous study. These waves, often with multiple harmonics of the local proton gyrofrequency, were observed both inside and outside the plasmapause, in regions with plasma number densities ranging from 10 to >1000 cm-3. Wave occurrence was sharply peaked near the magnetic equator and occurred at L shells from below 2 to ~6 (the spacecraft apogee). Waves appeared at all local times but were more common from noon to dusk. Outside the plasmapause, occurrence maximized broadly across noon. Inside the plasmapause, occurrence maximized in the dusk sector, in an extended plasmasphere. Every event occurred in association with a positive gradient in the HOPE omnidirectional proton flux in the range between 2 keV and 10 keV. The Poynting vector, determined for 8 events, was in all cases directed transverse to B, but with variable azimuth, consistent with earlier models and observations. 17. Influence of electromagnetic oscillating two-stream instability on the evolution of laser-driven plasma beat-wave SciTech Connect Gupta, D. N.; Singh, K. P.; Suk, H. 2007-01-15 The electrostatic oscillating two-stream instability of laser-driven plasma beat-wave was studied recently by Gupta et al. [Phys. Plasmas 11, 5250 (2004)], who applied their theory to limit the amplitude level of a plasma wave in the beat-wave accelerator. As a self-generated magnetic field is observed in laser-produced plasma, hence, the electromagnetic oscillating two-stream instability may be another possible mechanism for the saturation of laser-driven plasma beat-wave. The efficiency of this scheme is higher than the former. 18. Plasma waves in the distant geomagnetic tail - ISEE 3 NASA Technical Reports Server (NTRS) Coroniti, F. V.; Greenstadt, E. W.; Tsurutani, B. T.; Smith, E. J.; Zwickl, R. D. 1990-01-01 The plasma wave measurements obtained during ISEE 3's deep passes through the geomagnetic tail found that moderate to intense electric field turbulence occurred in association with the major plasma and magnetic field regions and flow phenomena. In the magnetopause boundary layer the electric field spectral amplitudes are typically sharply peaked at 316 Hz to 562 Hz. The tail lobe region which is upstream of slow shocks and is magnetically connected to the plasma sheet is characterized by wave spectras that peak in the 100- to 316-Hz range and at the electron plasma frequency. Within the plasma sheet, broadband electrostatic noise occurs in regions where the magnetic field strength exceeds 2 nT; this noise can also be found in the plasma sheet boundary layer in association with strong field-aligned plasma flows. As ISEE 3 moved between the different distant tail regions, distinct but often subtle changes occurred in the plasma wave spectra. 19. Electrostatic Solitary Waves (ESWs) observed by Kaguya near the Moon Hashimoto, K.; Hashitani, M.; Omura, Y.; Kasahara, Y.; Kojima, H.; Ono, T.; Tsunakawa, H. 2010-12-01 In KAGUYA (SELENE) LRS[1], WFC-L [2] observes waveforms of plasma waves in 100Hz-100kHz and a lot of electrostatic solitary waves (ESWs) have been observed. Some results have been reported [3]. Although orthogonal dipole antennas are generally used in the observations, sometimes a pair of monopole antennas were used. We reports observations mainly by the latter antennas. The velocities and spatial scales of ESWs are evaluated from waveforms observed in the monopole mode. Generally their velocities are from several 100km/s to several 1000km/s. Their spatial scales are several 10m and the potential depths were less than 0.05 eV. Their velocities are very slow near the wake boundaries. The ESW waveforms have often components perpendicular to the background magnetic field and the potential structure has a component perpendicular to the background magnetic field. This means that these waves were observed close the source regions. Acknowledgments: The SELENE project has been organized by the Japan Aerospace Exploration Agency (JAXA). The authors express their thanks to all members of the SELENE project team. References [1] Y. Kasahara, Y. Goto, K. Hashimoto, T. Imachi, A. Kumamoto, T. Ono, and H. Matsumoto, Plasma Wave Observation Using Waveform Capture in the Lunar Radar Sounder on board the SELENE Spacecraft, Earth, Planets and Space, 60, 341-351, 2008. [2] K. Hashimoto, M. Hashitani, Y. Kasahara, Y. Omura, M.N. Nishino, Y. Saito, S. Yokota, T. Ono, H. Tsunakawa, H. Shibuya, M. Matsushima, H. Shimizu, and F. Takahashi, Electrostatic solitary waves associated with magnetic anomalies and wake boundary of the Moon observed by KAGUYA, accepted for publication in Geophys. Res. Lett., 2010. 20. Analysis of waves in the plasma guided by a periodical vane-type slow wave structure SciTech Connect Wu, T.J.; Kou, C.S. 2005-10-01 In this study, the dispersion relation has been derived to characterize the propagation of the waves in the plasma guided by a periodical vane-type slow wave structure. The plasma is confined by a quartz plate. Results indicate that there are two different waves in this structure. One is the plasma mode that originates from the plasma surface wave propagating along the interface between the plasma and the quartz plate, and the other is the guide mode that originally travels along the vane-type slow wave structure. In contrast to its original slow wave characteristics, the guide mode becomes a fast wave in the low-frequency portion of the passband, and there exists a cut-off frequency for the guide mode. The vane-type guiding structure has been shown to limit the upper frequency of the passband of the plasma mode, compared with that of the plasma surface wave. In addition, the passband of the plasma mode increases with the plasma density while it becomes narrower for the guide mode. The influences of the parameters of the guiding structure and plasma density on the propagation of waves are also presented. 1. SAR observations of waves in ice De Carolis, Giacomo 2003-03-01 Ocean waves properties propagating in grease ice composed of frazil and pancakes as observed by SAR images are discussed. An ERS-2 SAR scene relevant to the Greenland Sea in an area where the Odden ice tongue developed in 1997 is considered as case study. The scene includes open sea and ice covered waters where a wave field is traveling from the open sea region. Wind induced features known as "wind rolls" can be distinguished, allowing the estimation of the wind vector. Hence the related wind generated ocean waves can be retrieved using a SAR spectral inversion procedure. The wave field is tracked while it propagates inside the ice field, thus allowing the estimation of the wave changes. Under the assumption of continuum medium, physical ice properties are then retrieved using a special SAR inversion procedure in conjunction with a recently developed wave propagation model in sea ice. The model assumes both the ice layer and the water beneath it as a system of viscous fluids. As a result, the changes suffered by the ocean wave spectrum in terms of wave dispersion and energy attenuation are related to sea ice properties such as concentration and thickness. Although the free parameters to be inverted are the ice thickness and viscosity and the water viscosity, the ice thickness is the only parameter of geophysical interest. Results are finally compared with external ice parameters information. 2. High latitude electromagnetic plasma wave emissions NASA Technical Reports Server (NTRS) Gurnett, D. A. 1983-01-01 The principal types of electromagnetic plasma wave emission produced in the high latitude auroral regions are reviewed. Three types of radiation are described: auroral kilometric radiation, auroral hiss, and Z mode radiation. Auroral kilometric radiation is a very intense radio emission generated in the free space R-X mode by electrons associated with the formation of discrete auroral arcs in the local evening. Theories suggest that this radiation is an electron cyclotron resonance instability driven by an enhanced loss cone in the auroral acceleration region at altitudes of about 1 to 2 R sub E. Auroral hiss is a somewhat weaker whistler mode emission generated by low energy (100 eV to 10 keV) auroral electrons. The auroral hiss usually has a V shaped frequency time spectrum caused by a freqency dependent beaming of the whistler mode into a conical beam directed upward or downward along the magnetic field. 3. Channeled particle acceleration by plasma waves in metals SciTech Connect Chen, P.; Noble, R.J. 1987-01-01 A solid state accelerator concept utilizing particle acceleration along crystal channels by longitudinal electron plasma waves in a metal is presented. Acceleration gradients of order 100 GV/cm are theoretically possible. Particle dechanneling due to electron multiple scattering can be eliminated with a sufficiently high acceleration gradient. Plasma wave dissipation and generation in metals are also discussed. 4. Electromagnetic-wave excitation in a large laboratory beam-plasma system NASA Technical Reports Server (NTRS) Whelan, D. A.; Stenzel, R. L. 1981-01-01 The mechanism by which unstable electrostatic waves of a beam-plasma system are converted into observed electromagnetic waves is of current interest in space physics and in tokamak fusion research. The process involved in the conversion of electrostatic to electromagnetic waves at the critical layer is well understood. However, the radiation from uniform plasmas cannot be explained on the basis of this process. In connection with certain difficulties, it has not yet been possible to establish the involved emission processes by means of experimental observations. In the considered investigation these difficulties are overcome by employing a large laboratory plasma in a parameter range suitable for detailed diagnostics. A finite-diameter electron beam is injected into a uniform quiescent afterglow plasma of dimensions large compared with electromagnetic wavelengths. The considered generation mechanism concerning the electromagnetic waves is conclusively confirmed by observing the temporal evolution of an instability 5. Ionospheric Plasma Disturbances and Effects on Radio Waves DTIC Science & Technology 2007-11-02 power HF waves. This study will be based on to propose future heating experiments in Alaska, using the newly constructed HAARP facility. 2. Summary...unlimited 13. ABSTRACT (Maximum 200 words) Ionospheric plasma heating experiments were conducted at Arecibo to investigate generation of ionospheric plasma...Plasma Research Group at MIT’s Plasma Science and Fusion Center has been conducting ionospheric plasma heating experiments at Arecibo, using the 6. Observation of gravity-capillary wave turbulence. PubMed Falcon, Eric; Laroche, Claude; Fauve, Stéphan 2007-03-02 We report the observation of the crossover between gravity and capillary wave turbulence on the surface of mercury. The probability density functions of the turbulent wave height are found to be asymmetric and thus non-Gaussian. The surface wave height displays power-law spectra in both regimes. In the capillary region, the exponent is in fair agreement with weak turbulence theory. In the gravity region, it depends on the forcing parameters. This can be related to the finite size of the container. In addition, the scaling of those spectra with the mean energy flux is found in disagreement with weak turbulence theory for both regimes. 7. Relationship between directions of wave and energy propagation for cold plasma waves NASA Technical Reports Server (NTRS) Musielak, Zdzislaw E. 1986-01-01 The dispersion relation for plasma waves is considered in the 'cold' plasma approximation. General formulas for the dependence of the phase and group velocities on the direction of wave propagation with respect to the local magnetic field are obtained for a cold magnetized plasma. The principal cold plasma resonances and cut-off frequencies are defined for an arbitrary angle and are used to establish basic regimes of frequency where the cold plasma waves can propagate or can be evanescent. The relationship between direction of wave and energy propagation, for cold plasma waves in hydrogen atmosphere, is presented in the form of angle diagrams (angle between group velocity and magnetic field versus angle between phase velocity and magnetic field) and polar diagrams (also referred to as 'Friedrich's diagrams') for different directions of wave propagation. Morphological features of the diagrams as well as some critical angles of propagation are discussed. 8. On the rogue waves propagation in non-Maxwellian complex space plasmas SciTech Connect El-Tantawy, S. A. El-Awady, E. I.; Tribeche, M. E-mail: [email protected] 2015-11-15 The implications of the non-Maxwellian electron distributions (nonthermal/or suprathermal/or nonextensive distributions) are examined on the dust-ion acoustic (DIA) rogue/freak waves in a dusty warm plasma. Using a reductive perturbation technique, the basic set of fluid equations is reduced to a nonlinear Schrödinger equation. The latter is used to study the nonlinear evolution of modulationally unstable DIA wavepackets and to describe the rogue waves (RWs) propagation. Rogue waves are large-amplitude short-lived wave groups, routinely observed in space plasmas. The possible region for the rogue waves to exist is defined precisely for typical parameters of space plasmas. It is shown that the RWs strengthen for decreasing plasma nonthermality and increasing superthermality. For nonextensive electrons, the RWs amplitude exhibits a bit more complex behavior, depending on the entropic index q. Moreover, our numerical results reveal that the RWs exist with all values of the ion-to-electron temperature ratio σ for nonthermal and superthermal distributions and there is no limitation for the freak waves to propagate in both two distributions in the present plasma system. But, for nonextensive electron distribution, the bright- and dark-type waves can propagate in this case, which means that there is a limitation for the existence of freak waves. Our systematic investigation should be useful in understanding the properties of DIA solitary waves that may occur in non-Maxwellian space plasmas. 9. On the rogue waves propagation in non-Maxwellian complex space plasmas El-Tantawy, S. A.; El-Awady, E. I.; Tribeche, M. 2015-11-01 The implications of the non-Maxwellian electron distributions (nonthermal/or suprathermal/or nonextensive distributions) are examined on the dust-ion acoustic (DIA) rogue/freak waves in a dusty warm plasma. Using a reductive perturbation technique, the basic set of fluid equations is reduced to a nonlinear Schrödinger equation. The latter is used to study the nonlinear evolution of modulationally unstable DIA wavepackets and to describe the rogue waves (RWs) propagation. Rogue waves are large-amplitude short-lived wave groups, routinely observed in space plasmas. The possible region for the rogue waves to exist is defined precisely for typical parameters of space plasmas. It is shown that the RWs strengthen for decreasing plasma nonthermality and increasing superthermality. For nonextensive electrons, the RWs amplitude exhibits a bit more complex behavior, depending on the entropic index q. Moreover, our numerical results reveal that the RWs exist with all values of the ion-to-electron temperature ratio σ for nonthermal and superthermal distributions and there is no limitation for the freak waves to propagate in both two distributions in the present plasma system. But, for nonextensive electron distribution, the bright- and dark-type waves can propagate in this case, which means that there is a limitation for the existence of freak waves. Our systematic investigation should be useful in understanding the properties of DIA solitary waves that may occur in non-Maxwellian space plasmas. 10. Nonlinear absorption of Alfven wave in dissipative plasma SciTech Connect Taiurskii, A. A. Gavrikov, M. B. 2015-10-28 We propose a method for studying absorption of Alfven wave propagation in a homogeneous non-isothermal plasma along a constant magnetic field, and relaxation of electron and ion temperatures in the A-wave. The absorption of a A-wave by the plasma arises due to dissipative effects - magnetic and hydrodynamic viscosities of electrons and ions and their elastic interaction. The method is based on the exact solution of two-fluid electromagnetic hydrodynamics of the plasma, which for A-wave, as shown in the work, are reduced to a nonlinear system of ordinary differential equations. 11. Alfven wave dispersion behavior in single- and multicomponent plasmas SciTech Connect Rahbarnia, K.; Grulke, O.; Klinger, T.; Ullrich, S.; Sauer, K. 2010-03-15 Dispersion relations of driven Alfven waves (AWs) are measured in single- and multicomponent plasmas consisting of mixtures of argon, helium, and oxygen in a magnetized linear cylindrical plasma device VINETA [C. Franck, O. Grulke, and T. Klinger, Phys. Plasmas 9, 3254 (2002)]. The decomposition of the measured three-dimensional magnetic field fluctuations and the corresponding parallel current pattern reveals that the wave field is a superposition of L- and R-wave components. The dispersion relation measurements agree well with calculations based on a multifluid Hall-magnetohydrodynamic model if the plasma resistivity is correctly taken into account. 12. Surface electromagnetic wave equations in a warm magnetized quantum plasma SciTech Connect Li, Chunhua; Yang, Weihong; Wu, Zhengwei; Chu, Paul K. 2014-07-15 Based on the single-fluid plasma model, a theoretical investigation of surface electromagnetic waves in a warm quantum magnetized inhomogeneous plasma is presented. The surface electromagnetic waves are assumed to propagate on the plane between a vacuum and a warm quantum magnetized plasma. The quantum magnetohydrodynamic model includes quantum diffraction effect (Bohm potential), and quantum statistical pressure is used to derive the new dispersion relation of surface electromagnetic waves. And the general dispersion relation is analyzed in some special cases of interest. It is shown that surface plasma oscillations can be propagated due to quantum effects, and the propagation velocity is enhanced. Furthermore, the external magnetic field has a significant effect on surface wave's dispersion equation. Our work should be of a useful tool for investigating the physical characteristic of surface waves and physical properties of the bounded quantum plasmas. 13. ICE/ISEE plasma wave data analysis NASA Technical Reports Server (NTRS) Greenstadt, E. W.; Moses, S. L. 1993-01-01 This report is one of the final processing of ICE plasma wave (pw) data and analysis of late ISEE 3, ICE cometary, and ICE cruise trajectory data, where coronal mass ejections (CME's) were the first locus of attention. Interest in CME's inspired an effort to represent our pw data in a condensed spectrogram format that facilitated rapid digestion of interplanetary phenomena on long (greater than 1 day) time scales. The format serendipitously allowed us to also examine earth-orbiting data from a new perspective, invigorating older areas of investigation in Earth's immediate environment. We, therefore, continued to examine with great interest the last year of ISEE 3's precomet phase, when it spent considerable time far downwind from Earth, recording for days on end conditions upstream, downstream, and across the very weak, distant flank bow shock. Among other motivations has been the apparent similarity of some shock and post shock structures to the signatures of the bow wave surrounding comet Giacobini-Zinner, whose ICE-phase data we revisited. 14. Wave rectification in plasma sheaths surrounding electric field antennas NASA Technical Reports Server (NTRS) Boehm, M. H.; Carlson, C. W.; Mcfadden, J. P.; Clemmons, J. H.; Ergun, R. E.; Mozer, F. S. 1994-01-01 Combined measurements of Langmuir or broadband whistler wave intensity and lower-frequency electric field waveforms, all at 10-microsecond time resolution, were made on several recent sounding rockets in the auroral ionosphere. It is found that Langmuir and whistler waves are partically rectified in the plasma sheaths surrounding the payload and the spheres used as antennas. This sheath rectification occurs whenever the high frequency (HF) potential across the sheath becomes of the same order as the electron temperature or higher, for wave frequencies near or above the ion plasma frequency. This rectification can introduce false low-frequency waves into measurements of electric field spectra when strong high-frequency waves are present. Second harmonic signals are also generated, although at much lower levels. The effect occurs in many different plasma conditions, primarily producing false waves at frequencies that are low enough for the antenna coupling to the plasma to be resistive. 15. Scattering of radio frequency waves by cylindrical density filaments in tokamak plasmas Ram, Abhay K.; Hizanidis, Kyriakos 2016-02-01 In tokamak fusion plasmas, coherent fluctuations in the form of blobs or filaments are routinely observed in the scrape-off layer. Radio frequency (RF) electromagnetic waves, excited by antenna structures placed near the wall of a tokamak, have to propagate through the scrape-off layer before reaching the core of the plasma. While the effect of fluctuations on the properties of RF waves has not been quantified experimentally, it is of interest to carry out a theoretical study to determine if fluctuations can affect the propagation characteristics of RF waves. Usually, the difference between the plasma density inside the filament and the background plasma density is sizable, the ratio of the density difference to the background density being of order one. Generally, this precludes the use of geometrical optics in determining the effect of fluctuations, since the relevant ratio has to be much less than one, typically, of the order of 10% or less. In this paper, a full-wave, analytical model is developed for the scattering of a RF plane wave by a cylindrical plasma filament. It is assumed that the plasma inside and outside the filament is cold and uniform and that the major axis of the filament is aligned along the toroidal magnetic field. The ratio of the density inside the filament to the density of the background plasma is not restricted. The theoretical framework applies to the scattering of any cold plasma wave. In order to satisfy the boundary conditions at the interface between the filament and the background plasma, the electromagnetic fields inside and outside the filament need to have the same k∥ , the wave vector parallel to the ambient magnetic field, as the incident plane wave. Consequently, in contrast to the scattering of a RF wave by a spherical blob [Ram et al., Phys. Plasmas 20, 056110-1-056110-10 (2013)], the scattering by a field-aligned filament does not broaden the k∥ spectrum. However, the filament induces side-scattering leading to surface 16. Plasma wave signatures in the magnetotail reconnection region - MHD simulation and ray tracing NASA Technical Reports Server (NTRS) Omura, Yoshiharu; Green, James L. 1993-01-01 An MHD simulation was performed to obtain a self-consistent model of magnetic field and plasma density near the X point reconnection region. The MHD model was used to perform extensive ray tracing calculations in order to clarify the propagation characteristics of the plasma waves near the X point reconnection region. The dynamic wave spectra possibly observed by the Geotail spacecraft during a typical cross-tail trajectory are reconstructed. By comparing the extensive ray tracing calculations with the plasma wave data from Geotail, it is possible to perform a kind of 'remote sensing' to identify the location and structure of potential X point reconnection regions. 17. Experimental quiescent drifting dusty plasmas and temporal dust acoustic wave growth SciTech Connect Heinrich, J. R.; Kim, S.-H.; Meyer, J. K.; Merlino, R. L. 2011-11-15 We report on dust acoustic wave growth rate measurements taken in a dc (anode glow) discharge plasma device. By introducing a mesh with a variable bias 12-17 cm from the anode, we developed a technique to produce a drifting dusty plasma. A secondary dust cloud, free of dust acoustic waves, was trapped adjacent to the anode side of the mesh. When the mesh was returned to its floating potential, the secondary cloud was released and streamed towards the anode and primary dust cloud, spontaneously exciting dust acoustic waves. The amplitude growth of the excited dust acoustic waves was measured directly along with the wavelength and Doppler shifted frequency. These measurements were compared to fluid and kinetic dust acoustic wave theories. As the wave growth saturated a transition from linear to nonlinear waves was observed. The merging of the secondary and primary dust clouds was also observed. 18. Dispersion relation of electrostatic ion cyclotron waves in multi-component magneto-plasma SciTech Connect Khaira, Vibhooti Ahirwar, G. 2015-07-31 Electrostatic ion cyclotron waves in multi component plasma composed of electrons (denoted by e{sup −}), hydrogen ions (denoted by H{sup +}), helium ions (denoted by He{sup +}) and positively charged oxygen ions (denoted by O{sup +})in magnetized cold plasma. The wave is assumed to propagate perpendicular to the static magnetic field. It is found that the addition of heavy ions in the plasma dispersion modified the lower hybrid mode and also allowed an ion-ion mode. The frequencies of the lower hybrid and ion- ion hybrid modes are derived using cold plasma theory. It is observed that the effect of multi-ionfor different plasma densities on electrostatic ion cyclotron waves is to enhance the wave frequencies. The results are interpreted for the magnetosphere has been applied parameters by auroral acceleration region. 19. Flow induced dust acoustic shock waves in a complex plasma Jaiswal, Surabhi; Bandyopadhyay, Pintu; Sen, Abhijit 2015-11-01 We report on experimental observations of particle flow induced large amplitude shock waves in a dusty plasma. These dust acoustic shocks (DAS) are observed for strongly supersonic flows and have been studied in a U-shaped Dusty Plasma Experimental (DPEx) device for charged kaolin dust in a background of Argon plasma. The strong flow of the dust fluid is induced by adjusting the pumping speed and neutral gas flow into the device. An isolated copper wire mounted on the cathode acts as a potential barrier to the flow of dust particles. A sudden change of the dust density near the potential hill is used to trigger the onset of high velocity dust acoustic shocks. The dynamics of the shocks are captured by fast video pictures of the structures that are illuminated by a laser sheet beam. The physical characteristics of the shock are delineated from a parametric scan of their dynamical properties over a range of plasma parameters and flow speeds. Details of these observations and a physical explanation based on model calculations will be presented. 20. Simulation of laser-driven plasma beat-wave propagation in collisional weakly relativistic plasmas Kaur, Maninder; Nandan Gupta, Devki 2016-11-01 The process of interaction of lasers beating in a plasma has been explored by virtue of particle-in-cell (PIC) simulations in the presence of electron-ion collisions. A plasma beat wave is resonantly excited by ponderomotive force by two relatively long laser pulses of different frequencies. The amplitude of the plasma wave become maximum, when the difference in the frequencies is equal to the plasma frequency. We propose to demonstrate the energy transfer between the laser beat wave and the plasma wave in the presence of electron-ion collision in nearly relativistic regime with 2D-PIC simulations. The relativistic effect and electron-ion collision both affect the energy transfer between the interacting waves. The finding of simulation results shows that there is a considerable decay in the plasma wave and the field energy over time in the presence of electron-ion collisions. 1. The ''phase velocity'' of nonlinear plasma waves in the laser beat-wave accelerator SciTech Connect Spence, W.L. 1985-04-01 A calculational scheme for beat-wave accelerators is introduced that includes all orders in velocity and in plasma density, and additionally accounts for the influence of plasma nonlinearities on the wave's phase velocity. The main assumption is that the laser frequencies are very large compared to the plasma frequency - under which it is possible to sum up all orders of forward Raman scattering. It is found that the nonlinear plasma wave does not have simply a single phase velocity, but that the beat-wave which drives it is usefully described by a non-local ''effective phase velocity'' function. A time-space domain approach is followed. (LEW) 2. Surface wave and linear operating mode of a plasma antenna SciTech Connect Bogachev, N. N. Bogdankevich, I. L.; Gusein-zade, N. G.; Rukhadze, A. A. 2015-10-15 The relation between the propagation conditions of a surface electromagnetic wave along a finiteradius plasma cylinder and the linear operating mode of a plasma antenna is investigated. The solution to the dispersion relation for a surface wave propagating along a finite-radius plasma cylinder is analyzed for weakly and strongly collisional plasmas. Computer simulations of an asymmetrical plasma dipole antenna are performed using the KARAT code, wherein the dielectric properties of plasma are described in terms of the Drude model. The plasma parameters corresponding to the linear operating mode of a plasma antenna are determined. It is demonstrated that the characteristics of the plasma antenna in this mode are close to those of an analogous metal antenna. 3. Waves: The Radio and Plasma Wave Investigation on the Wind Spacecraft Bougeret, J.-L.; Kaiser, M. L.; Kellogg, P. J.; Manning, R.; Goetz, K.; Monson, S. J.; Monge, N.; Friel, L.; Meetre, C. A.; Perche, C.; Sitruk, L.; Hoang, S. 1995-02-01 The WAVES investigation on the WIND spacecraft will provide comprehensive measurements of the radio and plasma wave phenomena which occur in Geospace. Analyses of these measurements, in coordination with the other onboard plasma, energetic particles, and field measurements will help us understand the kinetic processes that are important in the solar wind and in key boundary regions of the Geospace. These processes are then to be interpreted in conjunction with results from the other ISTP spacecraft in order to discern the measurements and parameters for mass, momentum, and energy flow throughout geospace. This investigation will also contribute to observations of radio waves emitted in regions where the solar wind is accelerated. The WAVES investigation comprises several innovations in this kind of instrumentation: among which the first use, to our knowledge, of neural networks in real-time on board a scientific spacecraft to analyze data and command observation modes, and the first use of a wavelet transform-like analysis in real time to perform a spectral analysis of a broad band signal. 4. The transmission of Alfven waves through the Io plasma torus Wright, A. N.; Schwartz, S. J. 1989-04-01 The nature of Alfven wave propagation through the Io plasma torus was investigated using a one-dimensional model with uniform magnetic field and an exponential density decrease to a constant value. The solution was interpreted in terms of a wave that is incident upon the torus, a reflected wave, and a wave that is transmitted through the torus. The results obtained indicate that Io's Alfven waves may not propagate completely through the plasma torus, and, thus, the WKB theory and ray tracing may not provide meaningful estimates of the energy transport. 5. Second harmonic plasma emission involving ion sound waves NASA Technical Reports Server (NTRS) Cairns, Iver H. 1987-01-01 The theory for second harmonic plasma emission by the weak turbulence (or random phase) processes L + L + or - S to T, proceeding in two three-wave steps, L + or - S to L prime and L + L prime to T, where L, S and T denote Langmuir, ion sound and electromagnetic waves, respectively, is developed. Kinematic constraints on the characteristics and growth lengths of waves participating in the wave processes, and constraints on the characteristics of the source plasma, are derived. Limits on the brightness temperature of the radiation and the levels of the L prime and S waves are determined. Expressions for the growth rates and path-integrated wave temperatures are derived for simple models of the wave spectra and source plasma. 6. Cyclotron waves in a non-neutral plasma column SciTech Connect Dubin, Daniel H. E. 2013-04-15 A kinetic theory of linear electrostatic plasma waves with frequencies near the cyclotron frequency {Omega}{sub c{sub s}} of a given plasma species s is developed for a multispecies non-neutral plasma column with general radial density and electric field profiles. Terms in the perturbed distribution function up to O(1/{Omega}{sub c{sub s}{sup 2}}) are kept, as are the effects of finite cyclotron radius r{sub c} up to O(r{sub c}{sup 2}). At this order, the equilibrium distribution is not Maxwellian if the plasma temperature or rotation frequency is not uniform. For r{sub c}{yields}0, the theory reproduces cold-fluid theory and predicts surface cyclotron waves propagating azimuthally. For finite r{sub c}, the wave equation predicts that the surface wave couples to radially and azimuthally propagating Bernstein waves, at locations where the wave frequency equals the local upper hybrid frequency. The equation also predicts a second set of Bernstein waves that do not couple to the surface wave, and therefore have no effect on the external potential. The wave equation is solved both numerically and analytically in the WKB approximation, and analytic dispersion relations for the waves are obtained. The theory predicts that both types of Bernstein wave are damped at resonances, which are locations where the Doppler-shifted wave frequency matches the local cyclotron frequency as seen in the rotating frame. 7. Whistler-mode Waves in a Hot Plasma Sazhin, Sergei 2005-10-01 The book provides an extensive theoretical treatment of whistler-mode propagation, instabilities and damping in a collisionless plasma. This book fills a gap between oversimplified analytical studies of these waves, based on the cold plasma approximation, and studies based on numerical methods. Although the book is primarily addressed to space plasma physicists and radio physicists, it will also prove useful to laboratory plasma physicists. Mathematical methods described in the book can be applied in a straightforward way to the analysis of other types of plasma waves. Problems included in this book, along with their solutions, allow it to be used as a textbook for postgraduate students. 8. ALFVEN WAVES IN A PARTIALLY IONIZED TWO-FLUID PLASMA SciTech Connect Soler, R.; Ballester, J. L.; Terradas, J.; Carbonell, M. E-mail: [email protected] E-mail: [email protected] 2013-04-20 Alfven waves are a particular class of magnetohydrodynamic waves relevant in many astrophysical and laboratory plasmas. In partially ionized plasmas the dynamics of Alfven waves is affected by the interaction between ionized and neutral species. Here we study Alfven waves in a partially ionized plasma from the theoretical point of view using the two-fluid description. We consider that the plasma is composed of an ion-electron fluid and a neutral fluid, which interact by means of particle collisions. To keep our investigation as general as possible, we take the neutral-ion collision frequency and the ionization degree as free parameters. First, we perform a normal mode analysis. We find the modification due to neutral-ion collisions of the wave frequencies and study the temporal and spatial attenuation of the waves. In addition, we discuss the presence of cutoff values of the wavelength that constrain the existence of oscillatory standing waves in weakly ionized plasmas. Later, we go beyond the normal mode approach and solve the initial-value problem in order to study the time-dependent evolution of the wave perturbations in the two fluids. An application to Alfven waves in the low solar atmospheric plasma is performed and the implication of partial ionization for the energy flux is discussed. 9. Plasma waves associated with the AMPTE artificial comet NASA Technical Reports Server (NTRS) Gurnett, D. A.; Anderson, R. R.; Haeusler, B.; Haerendel, G.; Bauer, O. H. 1985-01-01 Numerous plasma wave effects were detected by the AMPTE/IRM spacecraft during the artificial comet experiment on December 27, 1984. As the barium ion cloud produced by the explosion expanded over the spacecraft, emissions at the electron plasma frequency and ion plasma frequency provided a determination of the local electron density. The electron density in the diamagnetic cavity produced by the ion cloud reached a peak of more than 5 x 10 to the 5th per cu cm, then decayed smoothly as the cloud expanded, varying approximately as t exp-2. As the cloud began to move due to interactions with the solar wind, a region of compressed plasma was encountered on the upstream side of the diamagnetic cavity. The peak electron density in the compression region was about 1.5 x 10 to the 4th per cu cm. Later, a very intense (140 mVolt/m) broadband burst of electrostatic noise was encountered on the sunward side of the compression region. This noise has characteristics very similar to noise observed in the earth's bow shock, and is believed to be a shocklike interaction produced by an ion beam-plasma instability between the nearly stationary barium ions and the streaming solar wind protons. 10. Plasma wave system measurements of the Martian bow shock from the Phobos 2 spacecraft SciTech Connect Trotignon, J.G. ); Grard, R. ); Savin, S. ) 1991-07-01 The high-resolution data of the electric field observations performed by the plasma wave system (PWS) during some of the Martian bow shock intersections by Phobos 2 were analyzed. Plasma and wave detectors are very useful instruments for locating the shock transition region and studying structures in the upstream region, such as the foot or the electron foreshock. The electron plasma oscillations that develop in the latter give access to the plasma density of the solar wind. Shock surface models derived from the PWS data are compared to those obtained by other authors, and attention is paid to similarities and differences between the electric field measurements obtained for Mars, Venus, and Earth 11. Propagation of dust acoustic solitary waves in inhomogeneous plasma with dust charge fluctuations Gogoi, L. B.; Deka, P. N. 2017-03-01 Propagations of dust acoustic solitary waves are theoretically investigated in a collisionless, unmagnetized weakly inhomogeneous plasma. The plasma that is considered here consists of negatively charged dust grains and Boltzmann distributed electrons and ions in the presence of dust charge fluctuations. The fluid equations that we use for description of such plasmas are reduced to a modified Korteweg-de-Vries equation by employing a reductive perturbation method. In this investigation, we have used space-time stretched coordinates appropriate for the inhomogeneous plasmas. From the numerical results, we have observed a significant influence of inhomogeneity parameters on the propagation of dust acoustic solitary waves. 12. Electrotastic Solitary Waves (ESW) in the magnetotail: BEN wave forms observed by GEOTAIL NASA Technical Reports Server (NTRS) Matsumoto, H.; Kojima, H.; Miyatake, T.; Omura, Y.; Okada, M.; Nagano, I.; Tsutsui, M. 1994-01-01 Wave forms of BEN (Broadband Electrostatic Noise) in the geomagnetic tail were first detected by the Wave Form Capture reciever on the GEOTAIL spacecraft. The results show that most of the BEN in the plasma sheet boundary layer (PSBL) are not continuous broadband noise but are composed of a series of solitary pulses having a special form which we term 'Electrostatic Solitary Waves (ESW)'. A nonlinear BGK potential model is proposed as the generation mechanism for the ESW based upon a simple particle simulation which considers the highly nonlinear evolution of the electron beam instability. The wave forms produced by this simulation are very similar to those observed by GEOTAIL and suggest that the nonlinear dynamics of the electron beam play an essential role in the generation of ESW. 13. Variable dual-frequency electrostatic wave launcher for plasma applications. PubMed Jorns, Benjamin; Sorenson, Robert; Choueiri, Edgar 2011-12-01 A variable tuning system is presented for launching two electrostatic waves concurrently in a magnetized plasma. The purpose of this system is to satisfy the wave launching requirements for plasma applications where maximal power must be coupled into two carefully tuned electrostatic waves while minimizing erosion to the launching antenna. Two parallel LC traps with fixed inductors and variable capacitors are used to provide an impedance match between a two-wave source and a loop antenna placed outside the plasma. Equivalent circuit analysis is then employed to derive an analytical expression for the normalized, average magnetic flux density produced by the antenna in this system as a function of capacitance and frequency. It is found with this metric that the wave launcher can couple to electrostatic modes at two variable frequencies concurrently while attenuating noise from the source signal at undesired frequencies. An example based on an experiment for plasma heating with two electrostatic waves is used to demonstrate a procedure for tailoring the wave launcher to accommodate the frequency range and flux densities of a specific two-wave application. This example is also used to illustrate a method based on averaging over wave frequencies for evaluating the overall efficacy of the system. The wave launcher is shown to be particularly effective for the illustrative example--generating magnetic flux densities in excess of 50% of the ideal case at two variable frequencies concurrently--with a high adaptability to a number of plasma dynamics and heating applications. 14. Variable dual-frequency electrostatic wave launcher for plasma applications Jorns, Benjamin; Sorenson, Robert; Choueiri, Edgar 2011-12-01 A variable tuning system is presented for launching two electrostatic waves concurrently in a magnetized plasma. The purpose of this system is to satisfy the wave launching requirements for plasma applications where maximal power must be coupled into two carefully tuned electrostatic waves while minimizing erosion to the launching antenna. Two parallel LC traps with fixed inductors and variable capacitors are used to provide an impedance match between a two-wave source and a loop antenna placed outside the plasma. Equivalent circuit analysis is then employed to derive an analytical expression for the normalized, average magnetic flux density produced by the antenna in this system as a function of capacitance and frequency. It is found with this metric that the wave launcher can couple to electrostatic modes at two variable frequencies concurrently while attenuating noise from the source signal at undesired frequencies. An example based on an experiment for plasma heating with two electrostatic waves is used to demonstrate a procedure for tailoring the wave launcher to accommodate the frequency range and flux densities of a specific two-wave application. This example is also used to illustrate a method based on averaging over wave frequencies for evaluating the overall efficacy of the system. The wave launcher is shown to be particularly effective for the illustrative example—generating magnetic flux densities in excess of 50% of the ideal case at two variable frequencies concurrently—with a high adaptability to a number of plasma dynamics and heating applications. 15. Trapped electron acceleration by a laser-driven relativistic plasma wave Everett, M.; Lal, A.; Gordon, D.; Clayton, C. E.; Marsh, K. A.; Joshi, C. 1994-04-01 THE aim of new approaches for high-energy particle acceleration1 is to push the acceleration rate beyond the limit (~100 MeV m-1) imposed by radio-frequency breakdown in conventional accelerators. Relativistic plasma waves, having phase velocities very close to the speed of light, have been proposed2-6 as a means of accelerating charged particles, and this has recently been demonstrated7,8. Here we show that the charged particles can be trapped by relativistic plasma waves-a necessary condition for obtaining the maximum amount of energy theoretically possible for such schemes. In our experiments, plasma waves are excited in a hydrogen plasma by beats induced by two collinear laser beams, the difference in whose frequencies matches the plasma frequency. Electrons with an energy of 2 MeV are injected into the excited plasma, and the energy spectrum of the exiting electrons is analysed. We detect electrons with velocities exceeding that of the plasma wave, demonstrating that some electrons are 'trapped' by the wave potential and therefore move synchronously with the plasma wave. We observe a maximum energy gain of 28 MeV, corresponding to an acceleration rate of about 2.8 GeV m-1. 16. Nonextensivity effect on radio-wave transmission in plasma sheath Mousavi, A.; Esfandiari-Kalejahi, A.; Akbari-Moghanjoughi, M. 2016-04-01 In this paper, new theoretical findings on the application of magnetic field in effective transmission of electromagnetic (EM) waves through a plasma sheath around a hypersonic vehicle are reported. The results are obtained by assuming the plasma sheath to consist of nonextensive electrons and thermal ions. The expressions for the electric field and effective collision frequency are derived analytically in the framework of nonextensive statistics. Examination of the reflection, transmission, and absorption coefficients regarding the strength of the ambient magnetic field shows the significance of q-nonextensive parameter effect on these entities. For small values of the magnetic field, the transmission coefficient increases to unity only in the range of - 1 < q < 1 . It is also found that the EM wave transmission through the nonextensive plasma sheath can take place using lower magnetic field strengths in the presence of superthermal electrons compared with that of Maxwellian ones. It is observed that superthermal electrons, with nonextensive parameter, q < 1, play a dominant role in overcoming the radio blackout for hypersonic flights. 17. Radio and Plasma Waves Synergistic Science Opportunities with EJSM Cecconi, Baptiste; André, Nicolas; Bougeret, Jean-Louis 2010-05-01 The radio and plasma wave (RPW) diagnostics provide a unique access to critical parameters of space plasma, in particular in planetary and satellite environments. Concerning giant planets, this has been demonstrated by major results obtained by the radio investigation on the Galileo and Cassini spacecraft, but also during the Ulysses, Voyager, and Pioneer flybys of Jupiter. Several other missions, past or in flight, demonstrate the uniqueness and relevance of RPW diagnostics to basic problems of astrophysics. The EJSM mission consists of two platforms operating in the Jupiter environment: the NASA-led Jupiter Europa Orbiter (JEO), and the ESA-led Jupiter Ganymede Orbiter (JGO). JEO and JGO will execute a choreographed exploration of the Jupiter System before settling into orbit around Europa and Ganymede, respectively. The EJSM mission architecture hence offers unique opportunities for synergistic and complementary observations that significantly enhance the overall science return of the mission. In this paper, we will first review new and unique science aspects of the Jupiter system that may benefit from different capabilities of RPW investigations onboard JGO and/or JEO: spectral and polarization information, mapping of radio sources, measurements of in situ plasma waves, currents, thermal noise, dust and nano-particle detection and characterization. We will then illustrate unique synergistic and complementary science opportunities offered by RPW investigations onboard JGO and/or JEO, both in terms of Satellite science and in terms of Magnetospheric Science. 18. Exchange interaction effects on waves in magnetized quantum plasmas SciTech Connect Trukhanova, Mariya Iv. Andreev, Pavel A. 2015-02-15 We have applied the many-particle quantum hydrodynamics that includes the Coulomb exchange interaction to magnetized quantum plasmas. We considered a number of wave phenomena that are affected by the Coulomb exchange interaction. Since the Coulomb exchange interaction affects the longitudinal and transverse-longitudinal waves, we focused our attention on the Langmuir waves, the Trivelpiece-Gould waves, the ion-acoustic waves in non-isothermal magnetized plasmas, the dispersion of the longitudinal low-frequency ion-acoustic waves, and low-frequency electromagnetic waves at T{sub e} ≫ T{sub i}. We have studied the dispersion of these waves and present the numeric simulation of their dispersion properties. 19. Electromagnetic rogue waves in beam-plasma interactions Veldes, G. P.; Borhanian, J.; McKerr, M.; Saxena, V.; Frantzeskakis, D. J.; Kourakis, I. 2013-06-01 The occurrence of rogue waves (freak waves) associated with electromagnetic pulse propagation interacting with a plasma is investigated, from first principles. A multiscale technique is employed to solve the fluid Maxwell equations describing weakly nonlinear circularly polarized electromagnetic pulses in magnetized plasmas. A nonlinear Schrödinger (NLS) type equation is shown to govern the amplitude of the vector potential. A set of non-stationary envelope solutions of the NLS equation are considered as potential candidates for the modeling of rogue waves (freak waves) in beam-plasma interactions, namely in the form of the Peregrine soliton, the Akhmediev breather and the Kuznetsov-Ma breather. The variation of the structural properties of the latter structures with relevant plasma parameters is investigated, in particular focusing on the ratio between the (magnetic field dependent) cyclotron (gyro-)frequency and the plasma frequency. 20. Cross-Frequency Coupling of Plasma Waves in the Magnetosphere Khazanov, G. V. 2014-12-01 Wave-particle and wave-wave interactions are crucial elements of magnetosphere and ionosphere plasma dynamics. Such interactions provide a channel of energy redistribution between different plasma populations, and lead to connections between physical processes developing on different spatial and temporal scales. The lower hybrid waves (LHWs) are particularly interesting for plasma dynamics, because they couple well with both electrons and ions. The excitation of LHWs is a widely discussed mechanism of interaction between plasma species in space and is one of the unresolved questions of magnetospheric multi-ion plasmas. It is demonstrated that large-amplitude Alfven and/or EMIC waves, in particular those associated with lower frequency (LF) turbulence, may generate LHWs in the auroral zone and ring current region and in some cases this serves as the Alfven and/or EMIC waves saturation mechanism. We believe that this described scenario, as well as some other cross-frequency coupling of plasma waves processes that will be discussed in this presentation, can play a vital role in various parts of the magnetospheric plasma, especially in the places under investigation by the NASA THEMIS and Van Allen Probes (formerly known as the Radiation Belt Storm Probes (RBSP)) missions. 1. A region of intense plasma wave turbulence on auroral field lines NASA Technical Reports Server (NTRS) Gurnett, D. A.; Frank, L. A. 1976-01-01 This report presents a detailed study of the plasma wave turbulence observed by HAWKEYE-1 and IMP-6 on high latitude auroral field lines and investigates the relationship of this turbulence to magnetic field and plasma measurements obtained in the same region. 2. Nonlocal theory of electromagnetic wave decay into two electromagnetic waves in a rippled density plasma channel SciTech Connect Sati, Priti; Tripathi, V. K. 2012-12-15 Parametric decay of a large amplitude electromagnetic wave into two electromagnetic modes in a rippled density plasma channel is investigated. The channel is taken to possess step density profile besides a density ripple of axial wave vector. The density ripple accounts for the momentum mismatch between the interacting waves and facilitates nonlinear coupling. For a given pump wave frequency, the requisite ripple wave number varies only a little w.r.t. the frequency of the low frequency decay wave. The radial localization of electromagnetic wave reduces the growth rate of the parametric instability. The growth rate decreases with the frequency of low frequency electromagnetic wave. 3. Wave enhancement of electron runaway rate in a collisional plasma SciTech Connect An, Z.; Liu, C.; Lee, Y.; Boyd, D. 1982-06-01 The effects of plasma waves on the electron runaway production rate is studied. For a wave packet with a one-dimensional spectrum directed along the electric field and with a phase velocity range containing the critical velocity v/sub c/ for runaway, the runaway production rate is found to be enhanced by many orders of magnitude. For an isotropic wave spectrum, however, the runaway production rate is reduced because of the wave-enhanced pitch angle scattering. 4. Transition of electromagnetic wave by suddenly created magneto plasma Kuo, Spencer P. 2017-02-01 The theory of the interaction of electromagnetic waves with a suddenly created magneto plasma is presented. It is shown that a linearly polarized wave propagating along the magnetic field is converted into a frequency upshifted two forward and two backward propagating waves; in each propagation direction, one is right hand circular polarization and the other one is left hand circular polarization. A static wiggler magnetic field is also produced. The combined forward and backward waves are amplitude modulated with rotating polarizations. The extent of the frequency upshift increases with the increases of the plasma density and the background magnetic field intensity. By increasing the background magnetic field, the required plasma density for the frequency upshift is reduced; consequently, the drop rate of the conversion efficiency with the increase in the frequency upshift of the combined forward wave can be reduced considerably; the conversion efficiency of the combined backward wave also increases. 5. Theory of ground surface plasma wave associated with pre-earthquake electrical charges Fujii, Masafumi 2013-03-01 is shown theoretically that if mobile electrical charge exists on the surface of the ground, a ground surface plasma wave is induced by radio waves. If the electrical charges are generated by tectonic stresses acting on crustal rocks prior to major earthquakes, the detection of a ground surface plasma wave could be used as a pre-earthquake electromagnetic phenomenon. The ground surface plasma wave has a dispersion relation, i.e., the relation between frequency and wavelength, similar to that of the free-space plane wave in the atmosphere over the radio broadcast frequency range. It allows for a strong coupling between these two types of waves. This is a mode of electromagnetic wave propagation that has not been previously reported. Numerical analysis demonstrates (1) the propagation of the ground surface plasma wave along a curved surface beyond the line of sight, (2) anomalous scattering by ground surface roughness, and (3) the generation of cross-polarized waves due to the scattering. These results all agree well with radio wave anomalies observed before large earthquakes. 6. Plasma wave turbulence in the strong coupling region at comet Giacobini-Zinner NASA Technical Reports Server (NTRS) Coroniti, F. V.; Kennel, C. F.; Scarf, F. L.; Smith, E. J.; Tsurutani, B. T.; Bame, S. J.; Thomsen, M. F.; Hynds, R.; Wenzel, K. P. 1986-01-01 Within 100,000 km of comet Giacobini-Zinner's nucleus, strong plasma wave turbulence was detected by the ICE electric and magnetic field wave instruments. The spatial profiles of the wave amplitudes are compared with measurements of the heavy ion fluxes of cometary origin, the plasma electron density, and the magnetic field strength. The general similarity of the wave and heavy ion profiles suggest that the waves might be generated by free energy in the pick-up ion distribution function. However, the expected parallel streaming instability of electrostatic modes generates waves with frequencies that are too low to explain the observations. The observed low frequency magnetic turbulence is plausibly explained by the lower hybrid loss-cone instability of heavy ions. 7. Wave mode identification of electrostatic noise observed with ISEE 3 in the deep tail boundary layer NASA Technical Reports Server (NTRS) Tsutsui, M.; Matsumoto, H.; Strangeway, R. J.; Tsurutani, B. T.; Phillips, J. L. 1991-01-01 The characteristics of the VLF electrostatic noise observed with ISEE 3 in the low-latitude boundary layer of distant geomagnetic tail are examined using a display format for the wave dynamic spectra different from that used by Scarf et al. (1984). It is shown that the observed noise is composed of impulsive bursts. The results of the detailed analysis of the noise parameters are used to develop a model of plasma wave behavior in the plasma rest frame. A hypothesis is proposed that the wide frequency extent of the noise spectra is composed of Doppler effects of waves propagating nearly omnidirectionally within the plasma rest frame, which is moving with the electron bulk speed. On the basis of this hypothesis, the wavelength of the observed waves were determined from the width of the frequency extent and the measured electron bulk speed. It is shown that the wavelength ranges from 2 to 8 times the plasma Debye length. 8. Geotail MCA plasma wave data analysis NASA Technical Reports Server (NTRS) Anderson, Roger R. 1994-01-01 NASA Grant NAG 5-2346 supports the data analysis effort at The University of Iowa for the GEOTAIL Multi-Channel Analyzer (MCA) which is a part of the GEOTAIL Plasma Wave Instrument (PWI). At the beginning of this reporting period we had just begun to receive our GEOTAIL Sirius data on CD-ROMs. Much programming effort went into adapting and refining the data analysis programs to include the CD-ROM inputs. Programs were also developed to display the high-frequency-resolution PWI Sweep Frequency Analyzer (SFA) data and to include in all the various plot products the electron cyclotron frequency derived from the magnitude of the magnetic field extracted from the GEOTAIL Magnetic Field (MGF) data included in the GEOTAIL Sirius data. We also developed programs to use the MGF data residing in the Institute of Space and Astronautical Science (ISAS) GEOTAIL Scientific Data Base (SDB). Our programmers also developed programs and provided technical support for the GEOTAIL data analysis efforts of Co-lnvestigator William W. L. Taylor at Nichols Research Corporation (NRC). At the end of this report we have included brief summaries of the NRC effort and the progress being made. 9. Effect of wave localization on plasma instabilities. Ph.D. Thesis NASA Technical Reports Server (NTRS) Levedahl, William Kirk 1987-01-01 The Anderson model of wave localization in random media is involved to study the effect of solar wind density turbulence on plasma processes associated with the solar type III radio burst. ISEE-3 satellite data indicate that a possible model for the type III process is the parametric decay of Langmuir waves excited by solar flare electron streams into daughter electromagnetic and ion acoustic waves. The threshold for this instability, however, is much higher than observed Langmuir wave levels because of rapid wave convection of the transverse electromagnetic daughter wave in the case where the solar wind is assumed homogeneous. Langmuir and transverse waves near critical density satisfy the Ioffe-Reigel criteria for wave localization in the solar wind with observed density fluctuations -1 percent. Numerical simulations of wave propagation in random media confirm the localization length predictions of Escande and Souillard for stationary density fluctations. For mobile density fluctuations localized wave packets spread at the propagation velocity of the density fluctuations rather than the group velocity of the waves. Computer simulations using a linearized hybrid code show that an electron beam will excite localized Langmuir waves in a plasma with density turbulence. An action principle approach is used to develop a theory of non-linear wave processes when waves are localized. A theory of resonant particles diffusion by localized waves is developed to explain the saturation of the beam-plasma instability. It is argued that localization of electromagnetic waves will allow the instability threshold to be exceeded for the parametric decay discussed above. 10. The ISPM unified radio and plasma wave experiment NASA Technical Reports Server (NTRS) Stone, R. G.; Caldwell, J.; Deconchy, Y.; Deschanciaux, C.; Ebbett, R.; Epstein, G.; Groetz, K.; Harvey, C. C.; Hoang, S.; Howard, R. 1983-01-01 Hardware for the International Solar Polar Mission (ISPM) Unified Radio and Plasma (URAP) wave experiment is presented. The URAP determines direction and polarization of distant radio sources for remote sensing of the heliosphere, and studies local wave phenomena which determine the transport coefficients of the ambient plasma. Electric and magnetic field antennas and preamplifiers; the electromagnetic compatibility plan and grounding; radio astronomy and plasma frequency receivers; a fast Fourier transformation data processing unit waveform analyzer; dc voltage measurements; a fast envelope sampler for the solar wind, and plasmas near Jupiter; a sounder; and a power converter are described. 11. Freak waves in negative-ion plasmas: an experiment revisited Kourakis, Ioannis; Elkamash, Ibrahem; Reville, Brian 2016-10-01 Extreme events in the form of rogue waves (freak waves) occur widely in the open sea. These are space- and time-localised excitations, which appear unexpectedly and are characterised by a significant amplitude. Beyond ocean dynamics, the mechanisms underlying rogue wave formation are now being investigated in various physical contexts, including materials science, nonlinear optics and plasma physics, to mention but a few. We have undertaken an investigation, from first principles, of the occurrence of rogue waves associated with the propagation of electrostatic wavepackets in plasmas. Motivated by recent experimental considerations involving freak waves in negative-ion plasmas (NIP), we have addresed the occurrence of freak waves in NIP from first principles. An extended range of plasma parameter values was identified, where freak wave formation is possible, in terms of relevant plasma parameters. Our results extend -and partly contradict- the underlying assumptions in the interpretation of the aforementioned experiment, where a critical plasma configuration was considered and a Gardner equation approach was adopted. This work was supported from CPP/QUB funding. One of us (I. Elkamash) acknowledges financial support by an Egyptian Government fellowship. 12. On the generation of plasma waves in Saturn's inner magnetosphere Barbosa, D. D.; Kurth, W. S. 1993-06-01 Voyager 1 plasma wave measurements of Saturn's inner magnetosphere are reviewed with regard to interpretative aspects of the wave spectrum. A comparison of the wave emission profile with the electron plasma frequency obtained from in situ measurements of the thermal ion density shows good agreement with various features in the wave data identified as electrostatic modes and electromagnetic radio waves. Theoretical calculations of the critical flux of superthermal electrons able to generate whistler-mode waves and electrostatic electron cyclotron harmonic waves through a loss-cone instability are presented. The comparison of model results with electron measurements shows excellent agreement, thereby lending support to the conclusion that a moderate perpendicular anisotropy in the hot electron distribution is present in the equatorial region of L = 5-8. 13. Cyclotron maser and plasma wave growth in magnetic loops NASA Technical Reports Server (NTRS) Hamilton, Russell J.; Petrosian, Vahe 1990-01-01 Cyclotron maser and plasma wave growth which results from electrons accelerated in magnetic loops are studied. The evolution of the accelerated electron distribution is determined by solving the kinetic equation including Coulomb collisions and magnetic convergence. It is found that for modest values of the column depth of the loop the growth rates of instabilities are significantly reduced and that the reduction is much larger for the cyclotron modes than for the plasma wave modes. The large decrease in the growth rate with column depth suggests that solar coronal densities must be much lower than commonly accepted in order for the cyclotron maser to operate. The density depletion has to be similar to that which occurs during auroral kilometric radiation events in the magnetosphere. The resulting distributions are much more complicated than the idealized distributions used in many theoretical studies, but the fastest growing mode can still simply be determined by the ratio of electron plasma to gyrofrequency, U=omega(sub p)/Omega(sub e). However, the dominant modes are different than for the idealized situations with growth of the z-mode largest for U approximately less than 0.5, and second harmonic x-mode (s=2) or fundamental o-mode (s=1) the dominant modes for 0.5 approximately less than U approximately less than 1. The electron distributions typically contain more than one inverted feature which could give rise to wave growth. It is shown that this can result in simultaneous amplification of more than one mode with each mode driven by a different feature and can be observed, for example, by differences in the rise times of the right and left circularly polarized components of the associated spike bursts. 14. Internal Wave Observations in Drake Passage Firing, Y. L.; Chereskin, T. K. 2012-12-01 Internal wave energy in Drake Passage is investigated using an ongoing time series (>8 years) of shipboard acoustic Doppler current profiler (SADCP) data collected on transits of the U.S. Antarctic supply vessel as well as 287 full-depth lowered ADCP and CTD profiles made on five process cruises that were part of the cDrake experiment (cdrake.org). The lateral and vertical distributions of upward- and downward-propgating internal wave energy are examined in the context of local bathymetry and background currents. Downward-propagating energy predominates in the surface layer, but over steep topography in some parts of Drake Passage upward-propagating energy is elevated even 1000 m above topography. The generation of internal wave energy by geostrophic flow over topography in the area is estimated and compared to the total observed internal wave energy during the cruises, while the time variability of this contribution to the internal wave energy is investigated using a 4-year time series of bottom currents from the cDrake project. Shear spectra and the shear-strain relationship are compared with the Garrett and Munk model (Garrett and Munk, 1975) and with the spectral shapes and variance ratios found in other regions by other authors. 15. MAGNETOACOUSTIC WAVES IN A PARTIALLY IONIZED TWO-FLUID PLASMA SciTech Connect Soler, Roberto; Ballester, Jose Luis; Carbonell, Marc E-mail: [email protected] 2013-11-01 Compressible disturbances propagate in a plasma in the form of magnetoacoustic waves driven by both gas pressure and magnetic forces. In partially ionized plasmas the dynamics of ionized and neutral species are coupled due to ion-neutral collisions. As a consequence, magnetoacoustic waves propagating through a partially ionized medium are affected by ion-neutral coupling. The degree to which the behavior of the classic waves is modified depends on the physical properties of the various species and on the relative value of the wave frequency compared to the ion-neutral collision frequency. Here, we perform a comprehensive theoretical investigation of magnetoacoustic wave propagation in a partially ionized plasma using the two-fluid formalism. We consider an extensive range of values for the collision frequency, ionization ratio, and plasma β, so that the results are applicable to a wide variety of astrophysical plasmas. We determine the modification of the wave frequencies and study the frictional damping due to ion-neutral collisions. Approximate analytic expressions for the frequencies are given in the limit case of strongly coupled ions and neutrals, while numerically obtained dispersion diagrams are provided for arbitrary collision frequencies. In addition, we discuss the presence of cutoffs in the dispersion diagrams that constrain wave propagation for certain combinations of parameters. A specific application to propagation of compressible waves in the solar chromosphere is given. 16. Surface Waves and Landau Resonant Heating in Unmagnetized Bounded Plasmas Bowers, Kevin 2001-10-01 Owing to the large areas and high plasma densities found in some recently developed devices [1], electrostatic theories of plasma resonances and surface wave [2-3] propagation in such devices are suspect as the size of the device is much larger than the free space wavelength associated with the peak plasma frequency. Accordingly, an electromagnetic model of surface wave propagation has been developed appropriate for large area plasmas. The predicted wave dispersion of the two models differs for extremely long wavelengths but is degenerate in devices small compared with wavelength. First principles particle-in-cell simulations using new techniques developed for the demanding simulation regime have been conducted which support these results. Given the slow wave character and boundary localized fields of surface waves, a periodic electrode may be used to resonantly excite a strong wave-particle interaction between surface waves and electrons. At saturation, the electron velocity distribution is enhanced above the phase velocity of the applied wave and suppressed below. The use of this technique (Landau resonant heating'') to selectively heat the electron high energy tail to enhance electron-impact ionization is demonstrated using particle-in-cell simulation. [1] Matsumoto (Sumitomo Metal Industries). Private Communication. July 1999. [2] Nickel, Parker, Gould. Phys. Fluids. 7:1489. 1964. [3] Cooperberg. Phys. Plasmas. Vol. 5, No. 4, April 1998. 17. Observation of mutual neutralization in detached plasma Akira, Tonegawa; Isao, Shirota; Ken'ichi, Yoshida; Masataka, Ono; Kazutaka, Kawamura; Tuguhiro, Watanabe; Nobuyoshi, Ohyabu; Hajime, Suzuki; Kazuo, Takayama 2001-10-01 Mutual neutralization in collisions between negative ions and positive ions in molecular activated recombination (MAR) has been observed in a high density magnetized sheet plasma source TPDSHEET-IV(Test Plasma produced by Directed current for SHEET plasma) device. Measurements of the negative ion density of hydrogen atom, the electron density, electron temperature, and the heat load to the target plate were carried out in hydrogen high density plasma with hydrogen gas puff. A cylindrical probe made of tungsten ( 0.4 x 2 cm) was used to measure the spatial profiles of H- by a probe-assisted laser photodetachment The Balmer spectra of visible light emission from hydrogen or helium atoms were detected in front of the target plate. A small amount of secondary hydrogen gas puffing into a hydrogen plasma reduced strongly the heat flux to the target and increased rapidly the density of negative ions of hydrogen atom in the circumference of the plasma, while the conventional radiative and three-body recombination processes were disappeared. These results can be well explained by taking the charge exchange recombination of MAR in the detached plasma into account. 18. Linear and Nonlinear MHD Wave Processes in Plasmas. Final Report SciTech Connect Tataronis, J. A. 2004-06-01 This program treats theoretically low frequency linear and nonlinear wave processes in magnetized plasmas. A primary objective has been to evaluate the effectiveness of MHD waves to heat plasma and drive current in toroidal configurations. The research covers the following topics: (1) the existence and properties of the MHD continua in plasma equilibria without spatial symmetry; (2) low frequency nonresonant current drive and nonlinear Alfven wave effects; and (3) nonlinear electron acceleration by rf and random plasma waves. Results have contributed to the fundamental knowledge base of MHD activity in symmetric and asymmetric toroidal plasmas. Among the accomplishments of this research effort, the following are highlighted: Identification of the MHD continuum mode singularities in toroidal geometry. Derivation of a third order ordinary differential equation that governs nonlinear current drive in the singular layers of the Alfvkn continuum modes in axisymmetric toroidal geometry. Bounded solutions of this ODE implies a net average current parallel to the toroidal equilibrium magnetic field. Discovery of a new unstable continuum of the linearized MHD equation in axially periodic circular plasma cylinders with shear and incompressibility. This continuum, which we named “accumulation continuum” and which is related to ballooning modes, arises as discrete unstable eigenfrequency accumulate on the imaginary frequency axis in the limit of large mode numbers. Development of techniques to control nonlinear electron acceleration through the action of multiple coherent and random plasmas waves. Two important elements of this program aye student participation and student training in plasma theory. 19. Enhanced generation of a second-harmonic wave in a composite of metamaterial and microwave plasma with various permittivities. PubMed Iwai, Akinori; Nakamura, Yoshihiro; Sakai, Osamu 2015-09-01 The generation of a second-harmonic wave, which is one typical nonlinear feature, is enhanced in a composite of plasma and metamaterial. When we generate plasma by an injection of microwaves, whose frequencies are fundamental, we observe intensified second-harmonic waves in the cases of negative-refractive-index states in which both metamaterial permeability and plasma permittivity are negative for the fundamental waves. We performed the measurements at multiple levels of microwave input power up to 300 W to regulate permittivity in the negative polarity for the fundamental wave and in the transient region, including the positive-zero-negative values, for the second-harmonic wave. We clarified that the observed enhancement results from high electron density in negative-permittivity plasma, the propagating fundamental frequency wave not being attenuated in the negative-refractive-index state, and partial phase matching between the fundamental and second-harmonic waves. 20. Coronal Loops: Observations and Modeling of Confined Plasma. PubMed Reale, Fabio Coronal loops are the building blocks of the X-ray bright solar corona. They owe their brightness to the dense confined plasma, and this review focuses on loops mostly as structures confining plasma. After a brief historical overview, the review is divided into two separate but not independent parts: the first illustrates the observational framework, the second reviews the theoretical knowledge. Quiescent loops and their confined plasma are considered and, therefore, topics such as loop oscillations and flaring loops (except for non-solar ones, which provide information on stellar loops) are not specifically addressed here. The observational section discusses the classification, populations, and the morphology of coronal loops, its relationship with the magnetic field, and the loop stranded structure. The section continues with the thermal properties and diagnostics of the loop plasma, according to the classification into hot, warm, and cool loops. Then, temporal analyses of loops and the observations of plasma dynamics, hot and cool flows, and waves are illustrated. In the modeling section, some basics of loop physics are provided, supplying fundamental scaling laws and timescales, a useful tool for consultation. The concept of loop modeling is introduced and models are divided into those treating loops as monolithic and static, and those resolving loops into thin and dynamic strands. More specific discussions address modeling the loop fine structure and the plasma flowing along the loops. Special attention is devoted to the question of loop heating, with separate discussion of wave (AC) and impulsive (DC) heating. Large-scale models including atmosphere boxes and the magnetic field are also discussed. Finally, a brief discussion about stellar coronal loops is followed by highlights and open questions. 1. Inverse mirror plasma experimental device (IMPED) - a magnetized linear plasma device for wave studies Bose, Sayak; Chattopadhyay, P. K.; Ghosh, J.; Sengupta, S.; Saxena, Y. C.; Pal, R. 2015-04-01 In a quasineutral plasma, electrons undergo collective oscillations, known as plasma oscillations, when perturbed locally. The oscillations propagate due to finite temperature effects. However, the wave can lose the phase coherence between constituting oscillators in an inhomogeneous plasma (phase mixing) because of the dependence of plasma oscillation frequency on plasma density. The longitudinal electric field associated with the wave may be used to accelerate electrons to high energies by exciting large amplitude wave. However when the maximum amplitude of the wave is reached that plasma can sustain, the wave breaks. The phenomena of wave breaking and phase mixing have applications in plasma heating and particle acceleration. For detailed experimental investigation of these phenomena a new device, inverse mirror plasma experimental device (IMPED), has been designed and fabricated. The detailed considerations taken before designing the device, so that different aspects of these phenomena can be studied in a controlled manner, are described. Specifications of different components of the IMPED machine and their flexibility aspects in upgrading, if necessary, are discussed. Initial results meeting the prerequisite condition of the plasma for such study, such as a quiescent, collisionless and uniform plasma, are presented. The machine produces δnnoise/n <= 1%, Luniform ~ 120 cm at argon filling pressure of ~10-4 mbar and axial magnetic field of B = 1090 G. 2. Influence of Plasma Pressure Fluctuation on RF Wave Propagation Liu, Zhiwei; Bao, Weimin; Li, Xiaoping; Liu, Donglin; Zhou, Hui 2016-02-01 Pressure fluctuations in the plasma sheath from spacecraft reentry affect radio-frequency (RF) wave propagation. The influence of these fluctuations on wave propagation and wave properties is studied using methods derived by synthesizing the compressible turbulent flow theory, plasma theory, and electromagnetic wave theory. We study these influences on wave propagation at GPS and Ka frequencies during typical reentry by adopting stratified modeling. We analyzed the variations in reflection and transmission properties induced by pressure fluctuations. Our results show that, at the GPS frequency, if the waves are not totally reflected then the pressure fluctuations can remarkably affect reflection, transmission, and absorption properties. In extreme situations, the fluctuations can even cause blackout. At the Ka frequency, the influences are obvious when the waves are not totally transmitted. The influences are more pronounced at the GPS frequency than at the Ka frequency. This suggests that the latter can mitigate blackout by reducing both the reflection and the absorption of waves, as well as the influences of plasma fluctuations on wave propagation. Given that communication links with the reentry vehicles are susceptible to plasma pressure fluctuations, the influences on link budgets should be taken into consideration. supported by the National Basic Research Program of China (No. 2014CB340205) and National Natural Science Foundation of China (No. 61301173) 3. Plasma wave experiment for the ISEE-3 mission NASA Technical Reports Server (NTRS) Scarf, F. L. 1982-01-01 Analysis of data from a scientific instrument designed to study solar wind and plasma wave phenomena on the ISEE-3 mission is presented. The performance of work on the data analysis phase is summarized. 4. Langmuir rogue waves in electron-positron plasmas SciTech Connect Moslem, W. M. 2011-03-15 Progress in understanding the nonlinear Langmuir rogue waves which accompany collisionless electron-positron (e-p) plasmas is presented. The nonlinearity of the system results from the nonlinear coupling between small, but finite, amplitude Langmuir waves and quasistationary density perturbations in an e-p plasma. The nonlinear Schroedinger equation is derived for the Langmuir waves' electric field envelope, accounting for small, but finite, amplitude quasistationary plasma slow motion describing the Langmuir waves' ponderomotive force. Numerical calculations reveal that the rogue structures strongly depend on the electron/positron density and temperature, as well as the group velocity of the envelope wave. The present study might be helpful to understand the excitation of nonlinear rogue pulses in astrophysical environments, such as in active galactic nuclei, in pulsar magnetospheres, in neutron stars, etc. 5. DEMETER Observations of Equatorial Plasma Depletions and Related Ionospheric Phenomena Berthelier, J.; Malingre, M.; Pfaff, R.; Jasperse, J.; Parrot, M. 2008-12-01 DEMETER, the first micro-satellite of the CNES MYRIAD program, was launched from Baikonour on June 29, 2004 on a nearly circular, quasi helio-synchronous polar orbit at ~ 715 km altitude. The DEMETER mission focuses primarily on the search for a possible coupling between seismic activity and ionospheric disturbances as well as on the effects of natural phenomena such as tropospheric thunderstorms and man-made activities on the ionosphere. The scientific payload provides fairly complete measurements of the ionospheric plasma, energetic particles above ~ 70 keV, and plasma waves, up to 20 kHz for the magnetic and 3.3 MHz for the electric components. Several studies related to space weather and ionospheric physics have been conducted over the past years. Following a brief description of the payload and the satellite modes of operation, this presentation will focus on a set of results that provide a new insight into the physics of instabilities in the night-time equatorial ionosphere. The observations were performed during the major magnetic storm of November 2004. Deep plasma depletions were observed on several night-time passes at low latitudes characterized by the decrease of the plasma density by nearly 3 orders of magnitude relative to the undisturbed plasma, and a significant abundance of molecular ions. These features can be best interpreted as resulting from the rise of the F-layer above the satellite altitude over an extended region of the ionosphere. In one of the passes, DEMETER was operated in the Burst mode and the corresponding high resolution data allowed for the discovery of two unexpected phenomena. The first one is the existence of high intensity monochromatic wave packets at the LH frequency that develop during the decay phase of intense bursts of broadband LH turbulence. The broadband LH turbulence is triggered by whistlers emitted by lightning from atmospheric thunderstorms beneath the satellite. The second unexpected feature is the detection of a 6. EMIC wave scale size in the inner magnetosphere: Observations from the dual Van Allen Probes Blum, L. W.; Bonnell, J. W.; Agapitov, O.; Paulson, K.; Kletzing, C. 2017-02-01 Estimating the spatial scales of electromagnetic ion cyclotron (EMIC) waves is critical for quantifying their overall scattering efficiency and effects on thermal plasma, ring current, and radiation belt particles. Using measurements from the dual Van Allen Probes in 2013-2014, we characterize the spatial and temporal extents of regions of EMIC wave activity and how these depend on local time and radial distance within the inner magnetosphere. Observations are categorized into three types—waves observed by only one spacecraft, waves measured by both spacecraft simultaneously, and waves observed by both spacecraft with some time lag. Analysis reveals that dayside (and H+ band) EMIC waves more frequently span larger spatial areas, while nightside (and He+ band) waves are more often localized but can persist many hours. These investigations give insight into the nature of EMIC wave generation and support more accurate quantification of their effects on the ring current and outer radiation belt. 7. Filamentation of magnetosonic wave and generation of magnetic turbulence in laser plasma interaction SciTech Connect Modi, K. V.; Tiwary, Prem Pyari; Singh, Ram Kishor Sharma, R. P.; Satsangi, V. R. 2014-10-15 This paper presents a theoretical model for the magnetic turbulence in laser plasma interaction due to the nonlinear coupling of magnetosonic wave with ion acoustic wave in overdense plasma. For this study, dynamical equations of magnetosonic waves and the ion acoustic waves have been developed in the presence of ponderomotive force due to the pump magnetosonic wave. Slowly converging and diverging behavior has been studied semi-analytically, this results in the formation of filaments of the magnetosonic wave. Numerical simulation has also been carried out to study nonlinear stage. From the results, it has been found that the localized structures become quite complex in nature. Further, power spectrum has been studied. Results show that the spectral index follows (∼k{sup −2.0}) scaling at smaller scale. Relevance of the present investigation has been shown with the experimental observation. 8. Evolution of rogue waves in dusty plasmas SciTech Connect Tolba, R. E. El-Bedwehy, N. A.; Moslem, W. M.; El-Labany, S. K. 2015-04-15 The evolution of rogue waves associated with the dynamics of positively charged dust grains that interact with streaming electrons and ions is investigated. Using a perturbation method, the basic set of fluid equations is reduced to a nonlinear Schrödinger equation (NLSE). The rational solution of the NLSE is presented, which proposed as an effective tool for studying the rogue waves in Jupiter. It is found that the existence region of rogue waves depends on the dust-acoustic speed and the streaming densities of the ions and electrons. Furthermore, the supersonic rogue waves are much taller than the subsonic rogue waves by ∼25 times. 9. Solitary surface waves on a magnetized plasma cylinder Gradov, O. M.; Stenflo, L.; Sünder, D. 1985-02-01 We analyse high-frequency electrostatic solitary surface waves that propagate along a plasma cylinder in the presence of a constant axial magnetic field. The width of such a solitary wave, which is found to be inversely proportional to its amplitude, is expressed as a function of the magnitude of the external magnetic field. 10. Polar Plasma Wave Investigation Data Analysis in the Extended Mission NASA Technical Reports Server (NTRS) Gurnett, Donald A.; Menietti, J. D. 2003-01-01 The low latitude boundary layer (LLBL) is a region where solar wind momentum and energy is transferred to the magnetosphere. Enhanced "broadband" electric plasma waves from less than 5 Hz to l0(exp 5) Hz and magnetic waves from less than 5 Hz to the electron cyclotron frequency are characteristic of the LLBL. Analyses of Polar plasma waves show that these "broadband" waves are actually discrete electrostatic and electromagnetic modes as well as solitary bipolar pulses (electron holes). It is noted that all wave modes can be generated by approx. 100 eV to approx. 10 keV auroral electrons and protons. We will review wave-particle interactions, with focus on cross- diffusion rates and the contributions of such interactions toward the formation of the boundary layer. In summary, we will present a scenario where the global solar wind-magnetosphere interaction is responsible for the auroral zone particle beams, and hence for the generation of plasma waves and the formation of the boundary layer. It is speculated that all planetary magnetospheres will have boundary layers and they will be characterized by similar currents and plasma wave modes. 11. Polar Plasma Wave Investigation Data Analysis in the Extended Mission NASA Technical Reports Server (NTRS) Gurnett, Donald A. 2004-01-01 The low latitude boundary layer (LLBL) is a region where solar wind momentum and energy is transferred to the magnetosphere. Enhanced "broadband" electric plasma waves from less than 5 Hz to 10(exp 5) Hz and magnetic waves from less than 5 Hz to the electron cyclotron frequency are characteristic of the LLBL. Analyses of Polar plasma waves show that these "broadband" waves are actually discrete electrostatic and electromagnetic modes as well as solitary bipolar pulses (electron holes). It is noted that all wave modes can be generated by approx. 100 eV to approx. 10 keV auroral electrons and protons. We will review wave-particle interactions, with focus on cross-diffusion rates and the contributions of such interactions toward the formation of the boundary layer. In summary, we will present a scenario where the global solar wind-magnetosphere interaction is responsible for the auroral zone particle beams, and hence for the generation of plasma waves and the formation of the boundary layer. It is speculated that all planetary magnetospheres will have boundary layers and they will be characterized by similar currents and plasma wave modes. 12. Wave Properties of Equatorial Magnetosonic Waves as Observed by Cluster Balikhin, M. A.; Walker, S. N.; Shprits, Y. 2014-12-01 A survey of the Cluster STAFF data set shows a number of periods in which Equatorial Magnetosonic Waves display a discrete spectrum. In some of these instances, the frequency of emissions varies in the same fashion as the background magnetic field, indicating that the wars are observed within their source region. This paper analyses the propagation characteristics of these emissions and investigates the appropriateness of the quasi-linear assumption of a gaussian spectrum used in the numerical modelling of their role in the electron dynamics within the radiation belts based in the Chirikov resonance overlap criterion. 13. Modulation instability and rogue wave structures of positron-acoustic waves in q-nonextensive plasmas Bains, A. S.; Tribeche, Mouloud; Saini, N. S.; Gill, T. S. 2017-01-01 A theoretical investigation is made to study envelope excitations and rogue wave structures of the newly predicted positron-acoustic waves (PAWs) in a plasma with nonextensive electrons and nonextensive hot positrons. The reductive perturbation technique (RPT) is used to derive a nonlinear Schrödinger equation-like (NLSE) which governs the modulational instability (MI) of the PAWs. The NLSE admits localized envelope solitary wave solutions of bright and dark type. These envelope solutions depend upon the intrinsic plasma parameters. It is found that the MI of the PAWs is significantly affected by nonextensivity and other plasma parameters. Further, the analysis is extended for the rogue wave structures of the PAWs. The findings of the present investigation should be useful in understanding the acceleration mechanism of stable electrostatic wave packets in four components nonextensive plasmas. 14. Ambipolar potential effect on a drift-wave mode in a tandem-mirror plasma SciTech Connect Mase, A.; Jeong, J.H.; Itakura, A.; Ishii, K.; Inutake, M.; Miyoshi, S. ) 1990-05-07 The {bold k}-{omega} spectra of low-frequency waves which exist in a tandem-mirror plasma are observed by using the Fraunhofer-diffraction method. The observed dispersion relations are in good agreement with those of drift waves including a Doppler shift due to {bold E}{times}{bold B} rotation velocity. The fluctuation level is observed to depend sensitively on the radial profile of a plasma potential. It has a maximum value when a slightly negative electric field is formed, and decreases with increase in an electric field regardless of its sign. 15. Excitation, propagation and damping of helicon waves in a high density, low temperature plasma Caneses, J. F.; Blackwell, B. D. 2015-11-01 The MAGnetized Plasma Interaction Experiment (MAGPIE) is a helicon linear plasma device built to study fusion relevant plasma-surface interactions. In this work, we investigate helicon wave propagation in high density (1018-1019 m-3) low temperature (2-4 eV) magnetized (50-200 G) hydrogen plasma produced by a half-helical antenna operated at 7 MHz and 20 kW. Using the cold dielectric tensor with collisional terms (electron-neutral and Coulomb), helicon wave damping is calculated along the length of MAGPIE using a WKB approximation. Comparison with experiment indicates that wave damping, under these conditions, is entirely collisional. Numerical results from a fully electromagnetic wave code and 2D wavefield measurements indicate that helicon waves are excited at the plasma edge by the antenna's transverse current straps while the helical straps play a secondary role. These waves propagate towards the center of the discharge along the whistler wave ray direction (19 degrees to the background magnetic field), interfere on-axis and form the axial interference pattern commonly observed in helicon devices. 16. Method of accelerating photons by a relativistic plasma wave DOEpatents Dawson, John M.; Wilks, Scott C. 1990-01-01 Photons of a laser pulse have their group velocity accelerated in a plasma as they are placed on a downward density gradient of a plasma wave of which the phase velocity nearly matches the group velocity of the photons. This acceleration results in a frequency upshift. If the unperturbed plasma has a slight density gradient in the direction of propagation, the photon frequencies can be continuously upshifted to significantly greater values. 17. Gradient instabilities of electromagnetic waves in Hall thruster plasma SciTech Connect Tomilin, Dmitry 2013-04-15 This paper presents a linear analysis of gradient plasma instabilities in Hall thrusters. The study obtains and analyzes the dispersion equation of high-frequency electromagnetic waves based on the two-fluid model of a cold plasma. The regions of parameters corresponding to unstable high frequency modes are determined and the dependence of the increments and intrinsic frequencies on plasma parameters is obtained. The obtained results agree with those of previously published studies. 18. Kinetic full wave analyses of O-X-B mode conversion of EC waves in tokamak plasmas Fukuyama, Atsushi; Khan, Shabbir Ahmad; Igami, Hiroe; Idei, Hiroshi 2016-10-01 For heating and current drive in a high-density plasma of tokamak, especially spherical tokamak, the use of electron Bernstein waves and the O-X-B mode conversion were proposed and experimental observations have been reported. In order to evaluate the power deposition profile and the current drive efficiency, kinetic full wave analysis using an integral form of dielectric tensor has been developed. The incident angle dependence of wave structure and O-X-B mode conversion efficiency is examined using one-dimensional analysis in the major radius direction. Two-dimensional analyses on the horizontal plane and the poloidal plane are also conducted, and the wave structure and the power deposition profile are compared with those of previous analyses using ray tracing method and cold plasma approximation. This work is supported by JSPS KAKENHI Grant Number JP26630471. 19. Interference patterns in the Spacelab 2 plasma wave data - oblique electrostatic waves generated by the electron beam SciTech Connect Feng, Wei; Gurnett, D.A.; Cairns, I.H. ) 1992-11-01 During the Spacelab 2 mission the University of Iowa's Plasma Diagnostics Package (PDP) explored the plasma environment around the shuttle. Wideband spectrograms of plasma waves were obtained from the PDP at frequencies of 0-30 kHz and at distances up to 400 m from the shuttle. Strong low-frequency (below 10 kHz) electric field noise was observed in the wideband data during two periods in which an electron beam was ejected from the shuttle. This noise shows clear evidence of interference patterns caused by the finite (3.89 m) antenna length. The low-frequency noise was the most dominant type of noise produced by the ejected electron beam. Analysis of antenna interference patterns generated by these waves permits a determination of the wavelength, the direction of propagation, and the location of the source region. The observed waves have a linear dispersion relation very similar to that of ion acoustic waves. The waves are believed to be oblique ion acoustic or high-order ion cyclotron waves generated by a current of ambient electrons returning to the shuttle in response to the ejected electron beam. 31 refs. 20. Interaction of electromagnetic wave with quantum over dense plasma layer Rajaei, Leila 2016-10-01 The anomalous transmission of electromagnetic wave in the cold over dense plasma is investigated using the quantum hydrodynamic approach. The quantum effect on the dispersion relation of the surface wave excited by the electromagnetic radiation is evaluated and compared with the classical regimes. It is shown that the quantum dispersion curve, in comparison with its classical behavior, has an asymptotic approach at larger wave numbers. Investigating the transmission conditions, the effects of the main different parameters of the model such as the plasma density and Fermi velocity on the rate of transmission are scrutinized. 1. Ion temperature in plasmas with intrinsic Alfven waves SciTech Connect Wu, C. S.; Yoon, P. H.; Wang, C. B. 2014-10-15 This Brief Communication clarifies the physics of non-resonant heating of protons by low-frequency Alfvenic turbulence. On the basis of general definition for wave energy density in plasmas, it is shown that the wave magnetic field energy is equivalent to the kinetic energy density of the ions, whose motion is induced by the wave magnetic field, thus providing a self-consistent description of the non-resonant heating by Alfvenic turbulence. Although the study is motivated by the research on the solar corona, the present discussion is only concerned with the plasma physics of the heating process. 2. Ion temperature in plasmas with intrinsic Alfven waves Wu, C. S.; Yoon, P. H.; Wang, C. B. 2014-10-01 This Brief Communication clarifies the physics of non-resonant heating of protons by low-frequency Alfvenic turbulence. On the basis of general definition for wave energy density in plasmas, it is shown that the wave magnetic field energy is equivalent to the kinetic energy density of the ions, whose motion is induced by the wave magnetic field, thus providing a self-consistent description of the non-resonant heating by Alfvenic turbulence. Although the study is motivated by the research on the solar corona, the present discussion is only concerned with the plasma physics of the heating process. 3. Wave propagation in a quasi-chemical equilibrium plasma NASA Technical Reports Server (NTRS) Fang, T.-M.; Baum, H. R. 1975-01-01 Wave propagation in a quasi-chemical equilibrium plasma is studied. The plasma is infinite and without external fields. The chemical reactions are assumed to result from the ionization and recombination processes. When the gas is near equilibrium, the dominant role describing the evolution of a reacting plasma is played by the global conservation equations. These equations are first derived and then used to study the small amplitude wave motion for a near-equilibrium situation. Nontrivial damping effects have been obtained by including the conduction current terms. 4. Hydrodynamic Waves and Correlation Functions in Dusty Plasmas Bhattacharjee, A.; Wang, Xiaogang 1997-11-01 A hydrodynamic description of strongly coupled dusty plasmas is given when physical quantities vary slowly in space and time and the system can be assumed to be in local thermodynamic equilibrium. The linear waves in such a system are analyzed. In particular, a dispersion equation is derived for low-frequency dust acoustic waves, including collisional damping effects, and compared with experimental results. The linear response of the system is calculated from the fluctuation-dissipation theorem and the hydrodynamic equations. The requirement that these two calculations coincide constrains the particle correlation function for slowly varying perturbations [L. P. Kadanoff and P. C. Martin, Ann. Phys. 24, 419 (1963)]. It is shown that in the presence of the slow dust-acoustic waves, the dust auto-correlation function is of the Debye-Hekel form and the shielding distance is the dust Debye length. In the short-wavelength regime, an integral equation is derived from kinetic theory and solved numerically to yield particle correlation functions that display liquid-like'' behavior and have been observed experimentally [R. A.. Quinn, C. Cui, J. Goree, J. B. Pieper, H. Thomas and G. E. Morfill, Phys. Rev. E 53, R2049 (1996)]. 5. Helicon wave coupling in KSTAR plasmas for off-axis current drive in high electron pressure plasmas Wang, S. J.; Wi, H. H.; Kim, H. J.; Kim, J.; Jeong, J. H.; Kwak, J. G. 2017-04-01 6. High amplitude waves in the expanding solar wind plasma SciTech Connect Schmidt, J. M.; Velli, M.; Grappin, R. 1996-07-20 We simulated the 1 D nonlinear time-evolution of high-amplitude Alfven, slow and fast magnetoacustic waves in the solar wind propagating outward at different angles to the mean magnetic (spiral) field, using the expanding box model. The simulation results for Alfven waves and fast magnetoacustic waves fit the observational constraints in the solar wind best, showing decreasing trends for energies and other rms-quantities due to expansion and the appearance of inward propagating waves as minor species in the wind. Inward propagating waves are generated by reflection of Alfven waves propagating at large angles to the magnetic field or they coincide with the occurrence of compressible fluctuations. It is the generation of sound due to ponderomotive forces of the Alfven wave which we can detect in the latter case. For slow magnetoacustic waves we find a kind of oscillation of the character of the wave between a sound wave and an Alfven wave. This is the more, the slow magnetoacustic wave is close to a sound wave in the beginning. On the other hand, fast magnetoacustic waves are much more dissipated than the other wave-types and their general behaviour is close to the Alfven. The normalized cross-helicity {sigma}{sub c} is close to one for Alfven-waves and this quantity is decreasing slightly when density-fluctuations are generated. {sigma}{sub c} decreases significantly when the waves are close to perpendicular propagation. Then, the waves are close to quasi-static structures. 7. The effect of lower hybrid waves on JET plasma rotation Nave, M. F. F.; Kirov, K.; Bernardo, J.; Brix, M.; Ferreira, J.; Giroud, C.; Hawkes, N.; Hellsten, T.; Jonsson, T.; Mailloux, J.; Ongena, J.; Parra, F.; Contributors, JET 2017-03-01 This paper reports on observations of rotation in JET plasmas with lower hybrid current drive. Lower hybrid (LH) has a clear impact on rotation. The changes in core rotation can be either in the co- or counter-current directions. Experimental features that could determine the direction of rotation were investigated. Changes from co- to counter-rotation as the q-profile evolves from above unity to below unity suggests that magnetic shear could be important. However, LH can drive either co- or counter-rotation in discharges with similar magnetic shear and at the same plasma current. It is not clear if a slightly lower density is significant. A power scan at fixed density, shows a lower hybrid power threshold around 3 MW. For smaller LH powers, counter rotation increases with power, while for larger powers a trend towards co-rotation is found. The estimated counter-torque from the LH waves, would not explain the observed angular frequencies, neither would it explain the observation of co-rotation. 8. Low-Frequency Waves in Cold Three-Component Plasmas Fu, Qiang; Tang, Ying; Zhao, Jinsong; Lu, Jianyong 2016-09-01 The dispersion relation and electromagnetic polarization of the plasma waves are comprehensively studied in cold electron, proton, and heavy charged particle plasmas. Three modes are classified as the fast, intermediate, and slow mode waves according to different phase velocities. When plasmas contain positively-charged particles, the fast and intermediate modes can interact at the small propagating angles, whereas the two modes are separate at the large propagating angles. The near-parallel intermediate and slow waves experience the linear polarization, circular polarization, and linear polarization again, with the increasing wave number. The wave number regime corresponding to the above circular polarization shrinks as the propagating angle increases. Moreover, the fast and intermediate modes cause the reverse change of the electromagnetic polarization at the special wave number. While the heavy particles carry the negative charges, the dispersion relations of the fast and intermediate modes are always separate, being independent of the propagating angles. Furthermore, this study gives new expressions of the three resonance frequencies corresponding to the highly-oblique propagation waves in the general three-component plasmas, and shows the dependence of the resonance frequencies on the propagating angle, the concentration of the heavy particle, and the mass ratio among different kinds of particles. supported by National Natural Science Foundation of China (Nos. 11303099, 41531071 and 41574158), and the Youth Innovation Promotion Association CAS 9. Stimulation of plasma waves by electron guns on the ISEE-1 satellite NASA Technical Reports Server (NTRS) Lebreton, J.-P.; Torbert, R.; Anderson, R.; Harvey, C. 1982-01-01 The results of the ISEE-1 satellite experiment relating to observations of the waves stimulated during electron injections, when the spacecraft is passing through the magnetosphere, the magnetosheath, and the solar wind, are discussed. It is shown that the injection of an electron beam current of the order of 10 to 60 microamperes with energies ranging from 0 to 40 eV produces enhancements in the electric wave spectrum. An attempt has been made to identify the low-frequency electrostatic wave observed below the ion plasma frequency as an ion acoustic mode, although the excitation mechanism is not clear. A coupling mechanism between the electron plasma mode and streaming electrons with energies higher than the thermal speed of the cold electron population has been proposed to explain the observations above the electron plasma frequency. 10. Self-excited dust-acoustic waves in an electron-depleted nanodusty plasma SciTech Connect Tadsen, Benjamin Greiner, Franko; Groth, Sebastian; Piel, Alexander 2015-11-15 A dust density wave field is observed in a cloud of nanodust particles confined in a radio frequency plasma. Simultaneous measurements of the dust properties, grain size and density, as well as the wave parameters, frequency and wave number, allow for an estimate of the ion density, ion drift velocity, and the dust charge using a hybrid model for the wave dispersion. It appears that the charge on the dust grains in the cloud is drastically reduced to tens of elementary charges compared with isolated dust particles in a plasma. The charge is much higher at the cloud's periphery, i.e., towards the void in the plasma center and also towards the outer edge of the cloud. 11. Terahertz generation by beating two Langmuir waves in a warm and collisional plasma SciTech Connect Zhang, Xiao-Bo; Qiao, Xin; Cheng, Li-Hong; Tang, Rong-An; Zhang, Ai-Xia; Xue, Ju-Kui 2015-09-15 Terahertz (THz) radiation generated by beating of two Langmuir waves in a warm and collisional plasma is discussed theoretically. The critical angle between the two Langmuir waves and the critical wave-length (wave vector) of Langmuir waves for generating THz radiation are obtained analytically. Furthermore, the maximum radiation energy is obtained. We find that the critical angle, the critical wave-length, and the generated radiation energy strongly depend on plasma temperature and wave-length of the Langmuir waves. That is, the THz radiation generated by beating of two Langmuir waves in a warm and collisional plasma can be controlled by adjusting the plasma temperature and the Langmuir wave-length. 12. Ion-acoustic solitary waves in a positron beam plasma with electron trapping and nonextensivity effects Ali Shan, S.; -Ur-Rehman, Aman; Mushtaq, A. 2017-03-01 Ion-acoustic solitary waves (IASWs) are investigated in a plasma having a cold positron beam fluid, electrons following a vortex-like distribution with entropic index q, and dynamic ions. Using a standard procedure, a pseudo-potential energy equation is derived. The presence of nonextensive q - distributed trapped electrons and cold positron beam has been shown to influence the small amplitude soliton structure quite significantly. From the analysis of our results, it is shown that compressive IASWs are supported in this plasma model. As the real plasma situations are observed with plasma species having a relative flow, our present analysis should be beneficial for comprehending the electrostatic solitary structures observed in fusion plasma devices and positron winds observed in astrophysical plasmas. 13. Generation of Alfven waves by high power pulse at the electron plasma frequency van Compernolle, Bart Gilbert in excellent agreement with the observed Alfven waves. The field aligned suprathermal electrons in this work are a by-product of the plasma-microwave interaction. In space and laboratory plasmas, there are many instances in which pulses of field aligned electrons are observed, generated by various processes (e.g. in laser-produced-plasma experiments [VGV01, VGV03]). Cherenkov radiation of Alfven waves is of importance in all these cases, as long as the speed of the electrons is on the order of the Alfven speed. 14. Wave-particle and wave-wave interactions in hot plasmas: a French historical point of view Laval, Guy; Pesme, Denis; Adam, Jean-Claude 2016-11-01 The first researches on nuclear fusion for energy applications marked the entrance of hot plasmas into the laboratory. It became necessary to understand the behavior of such plasmas and to learn how to manipulate them. Theoreticians and experimentalists, building on the foundations of empirical laws, had to construct this new plasma physics from first principles and to explain the results of more and more complicated experiments. Along this line, two important topics emerged: wave-particle and wave-wave interactions. Here, their history is recalled as it has been lived by a French team from the end of the sixties to the beginning of the twenty-first century. 15. Comparison of deep space and near-earth observations of plasma turbulence at solar wind discontinuities NASA Technical Reports Server (NTRS) Scarf, F. L.; Fredricks, R. W.; Green, I. M. 1972-01-01 Simultaneous observations of plasma waves from the electric field instruments on Pioneer 9 and OGO 5 are used to illustrate the difference between near-earth and deep space conditions. It is shown that the experimental study of true interplanetary wave-particle interactions is difficult to carry out from an earth orbiter because the earth provides significant fluxes of nonthermal particles that generate intense plasma turbulence in the upstream region. 16. Wave-particle interactions induced by SEPAC on Spacelab 1 Wave observations NASA Technical Reports Server (NTRS) Taylor, W. W. L.; Obayashi, T.; Kawashima, N.; Sasaki, S.; Yanagisawa, M.; Burch, J. L.; Reasoner, D. L.; Roberts, W. T. 1985-01-01 Space experiments with particle accelerators (SEPAC) flew on Spacelab 1 in November and December 1983. SEPAC included an accelerator which emitted electrons into the ionospheric plasma with energies up to 5 keV and currents up to 300 mA. The SEPAC equipment also included an energetic plasma generator, a neutral gas generator, and an extensive array of diagnostics. The diagnostics included plasma wave detectors, and energetic electron analyzer, a photometer, a high sensitivity television camera, a Langmuir probe and a pressure gage. Twenty-eight experiments were performed during the mission to investigate beam-plasma interactions, electron beam dynamics, plasma beam propagation, and vehicle charging. The wave-particle interactions were monitored by the plasma wave instrumentation, by the energetic electron detector and by the optical detectors. All show evidence of wave-particle interactions, which are described in this paper. 17. Arbitrary electron acoustic waves in degenerate dense plasmas Rahman, Ata-ur; Mushtaq, A.; Qamar, A.; Neelam, S. 2016-12-01 A theoretical investigation is carried out of the nonlinear dynamics of electron-acoustic waves in a collisionless and unmagnetized plasma whose constituents are non-degenerate cold electrons, ultra-relativistic degenerate electrons, and stationary ions. A dispersion relation is derived for linear EAWs. An energy integral equation involving the Sagdeev potential is derived, and basic properties of the large amplitude solitary structures are investigated in such a degenerate dense plasma. It is shown that only negative large amplitude EA solitary waves can exist in such a plasma system. The present analysis may be important to understand the collective interactions in degenerate dense plasmas, occurring in dense astrophysical environments as well as in laser-solid density plasma interaction experiments. 18. Parametric decay of an extraordinary electromagnetic wave in relativistic plasma SciTech Connect Dorofeenko, V. G.; Krasovitskiy, V. B.; Turikov, V. A. 2015-03-15 Parametric instability of an extraordinary electromagnetic wave in plasma preheated to a relativistic temperature is considered. A set of self-similar nonlinear differential equations taking into account the electron “thermal” mass is derived and investigated. Small perturbations of the parameters of the heated plasma are analyzed in the linear approximation by using the dispersion relation determining the phase velocities of the fast and slow extraordinary waves. In contrast to cold plasma, the evanescence zone in the frequency range above the electron upper hybrid frequency vanishes and the asymptotes of both branches converge. Theoretical analysis of the set of nonlinear equations shows that the growth rate of decay instability increases with increasing initial temperature of plasma electrons. This result is qualitatively confirmed by numerical simulations of plasma heating by a laser pulse injected from vacuum. 19. Modeling shear wave splitting observations from Iceland Fu, Y. V.; Li, A.; Ito, G.; Hung, S. 2010-12-01 The goal of this research is to investigate the sources of shear-wave splitting in Iceland using synthetic waveforms generated from a variety of models. We employ a pseudospectral method in waveform modeling that allows 3-D heterogeneity and anisotropy. Several 1-D and 2-D models have been tested for a vertically propagating plane shear wave. For the two-layer models with horizontal symmetry axes, our results show that the apparent fast direction is towards the fast orientation in the upper layer. This experiment may explain why shear wave splitting measurements tend to be correlated with surface geology. We have also tested models with lateral anisotropic variations including a dike and a plume. The anisotropic boundary can be well resolved based on the change of fast directions and delay times. The splitting parameters near the boundary are affected by the laterally varied structure and the affected distance depends on wavelength, which is about 40 km for periods of 4-6 s and 50 km for periods of 8-10 s. We are currently performing experiments on a radial flow model from a plume stem. Synthetic shear-wave splitting measurements will be conducted from two more realistic geodynamic models. The first one is the “radial flow” model with low Rayleigh number. The pounding plume material is much thicker than the lithosphere and therefore does not strongly “feel” the lithosphere thickening away from the axis. Thus the plume spreads as fast away from the axis as it does along it. The other one is the “channel flow” model with high Rayleigh number. In this model the plume stem is much narrower and the thickness of the pounding plume material beneath the lithosphere much thinner. Thus the very low viscosity plume material is channeled more along axis by the thickening lithosphere. Combing the synthetic with the observed splitting results, we expect to determine the best geodynamic models for Iceland that fit seismic constraints. 20. MESSENGER Orbital Observations of Large-Amplitude Kelvin-Helmholtz Waves at Mercury's Magnetopause NASA Technical Reports Server (NTRS) Sundberg, Torbjorn; Boardsen, Scott A.; Slavin, James A.; Anderson, Brian J.; Korth, Haje; Zurbuchen, Thomas H.; Raines, Jim M.; Solomon, Sean C. 2012-01-01 We present a survey of Kelvi\\ n-Helmholtz (KH) waves at Mercury's magnetopause during MESSENGER's first Mercury year in orb it. The waves were identified on the basis of the well-established sawtooth wave signatures that are associated with non-linear KH vortices at the magnetopause. MESSENGER frequently observed such KH waves in the dayside region of the magnetosphere where the magnetosheath flow velocity is still sub -sonic, which implies that instability growth rates at Mercury's magnetopau are much larger than at Earth. We attribute these greater rates to the limited wave energy dissipation in Mercury's highly resistive regolith. The wave amplitude was often on the order of ' 00 nT or more, and the wave periods were - 10- 20 s. A clear dawn-dusk asymmetry is present in the data, in that all of the observed wave events occurred in the post-noon and dusk-side sectors of the magnetopause. This asymmetry is like ly related to finite Larmor-radius effects and is in agreement with results from particle-in-cell simulations of the instability. The waves were observed almost exclusively during periods when the north-south component of the magnetosheath magnetic field was northward, a pattern similar to that for most terrestrial KH wave events. Accompanying plasma measurements show that the waves were associated with the transport of magnetosheath plasma into the magnetosphere. 1. Wave propagation in strongly dispersive superthermal dusty plasma El-Labany, S. K.; El-Shewy, E. K.; Abd El-Razek, H. N.; El-Rahman, A. A. 2017-04-01 The attributes of acoustic envelope waves in a collisionless dust ion unmagnetized plasmas model composed of cold ions, superthermal electrons and positive-negative dust grains have been studied. Using the derivative expansion technique in a strong dispersive medium, the system model is reduced to a nonlinearly form of Schrodinger equation (NLSE). Rational solution of NLSE in unstable region is responsible for the creation of large shape waves; namely rogue waves. The subjection of instability regions upon electron superthermality (via κ), carrier wave number and dusty grains charge is discussed. 2. Linear coupling of acoustic and cyclotron waves in plasma flows SciTech Connect Rogava, Andria; Gogoberidze, Grigol 2005-05-15 It is found that in magnetized electrostatic plasma flows the velocity shear couples ion-acoustic waves with ion-cyclotron waves and leads, under favorable conditions, to their efficient reciprocal transformations. It is shown that in a two-dimensional setup this coupling has a remarkable feature: it is governed by equations that are mathematically equal to the ones describing coupling of sound waves with internal gravity waves [Rogava and Mahajan, Phys. Rev. E 55, 1185 (1997)] in neutral fluids. For flows with low shearing rates a fully analytic, quantitative description of the coupling efficiency, based on a noteworthy quantum-mechanical analogy, is given and transformation coefficients are calculated. 3. MAVEN Observation of an Obliquely Propagating Low-Frequency Wave Upstream of Mars NASA Technical Reports Server (NTRS) Ruhunusiri, Suranga; Halekas, J. S.; Connerney, J. E. P.; Espley, J. R.; McFadden, J. P.; Mazelle, C.; Brain, D.; Collinson, G.; Harada, Y.; Larson, D. E.; Mitchell, D. L.; Livi, R.; Jakosky, B. M. 2016-01-01 We report Mars Atmosphere and Volatile EvolutioN (MAVEN) mission observations of a large amplitude low-frequency plasma wave that propagated oblique to the ambient magnetic field upstream of Mars along with a non-solar-wind plasma component that had a flow velocity perpendicular to the magnetic field. We consider nine possibilities for this wave that include various combinations of its propagation direction, polarization in the solar wind frame, and ion source responsible for its generation. Using the observed wave parameters and the measured plasma parameters as constraints, we uniquely identify the wave by systematically discarding these possibilities. We determine that the wave is a right-hand polarized wave that propagated upstream in the solar wind frame. We find two possibilities for the ion source that can be responsible for this wave generation. They are either newly born pickup protons or reflected solar wind protons from the bow shock.We determine that the observed non-solar-wind component is not responsible for the wave generation, and it is likely that the non-solar-wind component was merely perturbed by the passage of the wave. 4. Effects of nonlinear plasma wake field on the dust-lattice wave in complex plasmas Lee, Myoung-Jae; Jung, Young-Dae 2017-02-01 The influence of a nonlinear ion wake field on the dust-lattice wave is investigated in complex dusty plasmas. The dispersion relation for the dust-lattice wave is derived from the equation of motion including the contribution due to the nearest-neighbour dust grain interaction. The results show that the nonlinear wake-field effect increases the wave frequency, especially at the maximum peak positions. It is found that the oscillatory behaviour of the dust-lattice wave enhances with an increase of the spacing of the dust grains. It is also found that the amplitude of the dust-lattice wave significantly decreases with an increase of the inter-dust grain distance. In addition, it is found that the amplitude of the dust-lattice wave increases with increasing Debye length. The variation of the dust-lattice wave due to the Mach number and plasma parameters is also discussed. 5. Dispersion relations for electromagnetic wave propagation in chiral plasmas SciTech Connect Gao, M. X.; Guo, B. Peng, L.; Cai, X. 2014-11-15 The dispersion relations for electromagnetic wave propagation in chiral plasmas are derived using a simplified method and investigated in detail. With the help of the dispersion relations for each eignwave, we explore how the chiral plasmas exhibit negative refraction and investigate the frequency region for negative refraction. The results show that chirality can induce negative refraction in plasmas. Moreover, both the degree of chirality and the external magnetic field have a significant effect on the critical frequency and the bandwidth of the frequency for negative refraction in chiral plasmas. The parameter dependence of the effects is calculated and discussed. 6. Ion acoustic shock wave in collisional equal mass plasma SciTech Connect Adak, Ashish; Ghosh, Samiran; Chakrabarti, Nikhil 2015-10-15 The effect of ion-ion collision on the dynamics of nonlinear ion acoustic wave in an unmagnetized pair-ion plasma has been investigated. The two-fluid model has been used to describe the dynamics of both positive and negative ions with equal masses. It is well known that in the dynamics of the weakly nonlinear wave, the viscosity mediates wave dissipation in presence of weak nonlinearity and dispersion. This dissipation is responsible for the shock structures in pair-ion plasma. Here, it has been shown that the ion-ion collision in presence of collective phenomena mediated by the plasma current is the source of dissipation that causes the Burgers' term which is responsible for the shock structures in equal mass pair-ion plasma. The dynamics of the weakly nonlinear wave is governed by the Korteweg-de Vries Burgers equation. The analytical and numerical investigations revealed that the ion acoustic wave exhibits both oscillatory and monotonic shock structures depending on the frequency of ion-ion collision parameter. The results have been discussed in the context of the fullerene pair-ion plasma experiments. 7. Lower Hybrid Oscillations in Multicomponent Space Plasmas Subjected to Ion Cyclotron Waves NASA Technical Reports Server (NTRS) Khazanov, G. V.; Krivorutsky, E. N.; Moore, T. E.; Liemohn, M. W.; Horwitz, J. L. 1997-01-01 It is found that in multicomponent plasmas subjected to Alfven or fast magnetosonic waves, such as are observed in regions of the outer plasmasphere and ring current-plasmapause overlap, lower hybrid oscillations are generated. The addition of a minor heavy ion component to a proton-electron plasma significantly lowers the low-frequency electric wave amplitude needed for lower hybrid wave excitation. It is found that the lower hybrid wave energy density level is determined by the nonlinear process of induced scattering by ions and electrons; hydrogen ions in the region of resonant velocities are accelerated; and nonresonant particles are weakly heated due to the induced scattering. For a given example, the light resonant ions have an energy gain factor of 20, leading to the development of a high-energy tail in the H(+) distribution function due to low-frequency waves. 8. Ion-acoustic waves in a nonstationary ultra-cold neutral plasma SciTech Connect Mendonca, J. T.; Shukla, P. K. 2011-04-15 We consider the excitation and dispersion of electrostatic ion-acoustic (IA) waves in a nonstationary ultra-cold neutral plasma (UCNP). This can be seen as an extension of time-refraction models of photons and plasmons to the case of low-frequency IA waves in the UCNP. It is shown that temporal changes in the medium lead to a frequency-shift of the IA wave, and to the emission of the IA waves propagating in a direction opposite to each other. We consider an arbitrary temporal variation of the background plasma density, and determine the transmission and reflection coefficients. We also consider the influence of a fast ionization process, assumed inhomogeneous in volume and show that it excites a well-defined spectrum of ion-acoustic waves, which agree very well with a recent experimental observation. 9. Waves in a bounded quantum plasma with electron exchange-correlation effects SciTech Connect Ma Yutng; Mao Shenghng; Xue Juji 2011-10-15 Within a quantum hydrodynamic model, the collective excitations of the quantum plasma with electron exchange-correlation effects in a nano-cylindrical wave guide are studied both analytically and numerically. The influences of the electron exchange-correlation potential, the radius of the wave guide, and the quantum effect on the dispersion properties of the bounded quantum plasma are discussed. Significant frequency-shift induced by the electron exchange-correlation effect, the radius of the wave guide and the quantum correction are observed. It is found that the influence of the electron exchange-correlation, the radius of the wave guide and the quantum correction on the wave modes in a bounded nano-waveguide are strongly coupled. 10. Two dimensional PIC simulations of plasma heating by the dissipation of Alfven waves NASA Technical Reports Server (NTRS) Liewer, P. C.; Kruecken, T. J.; Ferraro, R. D.; Decyk, V. K.; Goldstein, B. E. 1992-01-01 Two dimensional plasma particle simulations of the evolution of large amplitude circularly polarized Alfven waves propagating parallel to the magnetic field show that the waves decay via both one- and two- dimensional parametric decay instabilities. For parameters studied, one-dimensional processes dominate the simulations, but two-dimensional decay processes, including the recently predicted filamentation instability are also observed. The daughter waves generated by the parametric decay are primarily damped by the ions, leading to ion heating. The parametric decay processes efficiently convert the ordered fluid ion motion in the Alfven wave into ion thermal energy. These processes may be important for the dissipation of Alfven waves in the solar wind, the corona and other space plasma environments. The computations were performed on the Intel Touchstone parallel supercomputer. 11. Hybrid Model of Inhomogeneous Solar Wind Plasma Heating by Alfven Wave Spectrum: Parametric Studies NASA Technical Reports Server (NTRS) Ofman, L. 2010-01-01 Observations of the solar wind plasma at 0.3 AU and beyond show that a turbulent spectrum of magnetic fluctuations is present. Remote sensing observations of the corona indicate that heavy ions are hotter than protons and their temperature is anisotropic (T(sub perpindicular / T(sub parallel) >> 1). We study the heating and the acceleration of multi-ion plasma in the solar wind by a turbulent spectrum of Alfvenic fluctuations using a 2-D hybrid numerical model. In the hybrid model the protons and heavy ions are treated kinetically as particles, while the electrons are included as neutralizing background fluid. This is the first two-dimensional hybrid parametric study of the solar wind plasma that includes an input turbulent wave spectrum guided by observation with inhomogeneous background density. We also investigate the effects of He++ ion beams in the inhomogeneous background plasma density on the heating of the solar wind plasma. The 2-D hybrid model treats parallel and oblique waves, together with cross-field inhomogeneity, self-consistently. We investigate the parametric dependence of the perpendicular heating, and the temperature anisotropy in the H+-He++ solar wind plasma. It was found that the scaling of the magnetic fluctuations power spectrum steepens in the higher-density regions, and the heating is channeled to these regions from the surrounding lower-density plasma due to wave refraction. The model parameters are applicable to the expected solar wind conditions at about 10 solar radii. 12. 'EXTREME ULTRAVIOLET WAVES' ARE WAVES: FIRST QUADRATURE OBSERVATIONS OF AN EXTREME ULTRAVIOLET WAVE FROM STEREO SciTech Connect Patsourakos, Spiros; Vourlidas, Angelos E-mail: [email protected] 2009-08-01 The nature of coronal mass ejection (CME)-associated low corona propagating disturbances, 'extreme ultraviolet (EUV) waves', has been controversial since their discovery by EIT on SOHO. The low-cadence, single-viewpoint EUV images and the lack of simultaneous inner corona white-light observations have hindered the resolution of the debate on whether they are true waves or just projections of the expanding CME. The operation of the twin EUV imagers and inner corona coronagraphs aboard STEREO has improved the situation dramatically. During early 2009, the STEREO Ahead (STA) and Behind (STB) spacecrafts observed the Sun in quadrature having a {approx}90 deg. angular separation. An EUV wave and CME erupted from active region 11012, on February 13, when the region was exactly at the limb for STA and hence at disk center for STB. The STEREO observations capture the development of a CME and its accompanying EUV wave not only with high cadence but also in quadrature. The resulting unprecedented data set allowed us to separate the CME structures from the EUV wave signatures and to determine without doubt the true nature of the wave. It is a fast-mode MHD wave after all. 13. Observation of Ion Cyclotron Heating in a Fast-flowing Plasma for an Advanced Plasma Thruster Ando, Akira; Hatanaka, Motoi; Shibata, Masaki; Tobari, Hiroyuki; Hattori, Kunihiko; Inutake, Masaaki 2004-11-01 In the Variable Specific Impulse Magnetoplasma Rocket (VASIMR) project in NASA, the combined system of the ion cyclotron heating and the magnetic nozzle is proposed to control a ratio of specific impulse to thrust at constant power. In order to establish the advanced plasma thruster, experiments of an ion heating and plasma acceleration by a magnetic nozzle are performed in a fast-flowing plasma in the HITOP device. A fast-flowing He plasma is produced by Magneto-Plasma-Dynamic Arcjet (MPDA) operated with an externally-applied magnetic field up to 1kG. RF waves with an ion cyclotron range of frequency (f=20-300kHz) is excited by a helically-wound antenna located downstream of the MPDA. Increases of an ion temperature and plasma stored energy measured by a diamagnetic coil clearly observed during the RF pulse. The heating efficiency is compared for various magnetic field configurations and strengths. There appears no indication of cyclotron resonance in a high density plasma where the ratio of ion cyclotron frequency to ion-ion collision one is below unity, because an ion-ion collisional effect is dominant. When the density becomes low and the ratio of ion cyclotron frequency to ion-ion collision one becomes high, features of ion cyclotron resonance are clearly appeared. The optimum magnetic field strength for the ion heating is slightly lower than that of the cyclotron resonance, which is caused by the Doppler effect due to the fast-flowing plasma. An ion energy distribution function is measured at a magnetic nozzle region by an electrostatic analyzer and increase of the parallel velocity is also observed. 14. A morphological study of waves in the thermosphere using DE-2 observations NASA Technical Reports Server (NTRS) Gross, S. H.; Kuo, S. P.; Shmoys, J. 1986-01-01 Theoretical model and data analysis of DE-2 observations for determining the correlation between the neutral wave activity and plasma irregularities have been presented. The relationships between the observed structure of the sources, precipitation and joule heating, and the fluctuations in neutral and plasma parameters are obtained by analyzing two measurements of neutral atmospheric wave activity and plasma irregularities by DE-2 during perigee passes at an altitude on the order of 300 to 350 km over the polar cap. A theoretical model based on thermal nonlinearity (joule heating) to give mode-mode coupling is developed to explore the role of neutral disturbance (winds and gravity waves) on the generation of plasma irregularities. 15. Filamentation of laser beam and suppression of stimulated Raman scattering due to localization of electron plasma wave Purohit, Gunjan; Sharma, Prerana; Sharma, R. P. 2012-02-01 This paper presents the effect of laser beam filamentation on the localization of electron plasma wave (EPW) and stimulated Raman scattering (SRS) in unmagnetized plasma when relativistic and ponderomotive nonlinearities are operative. The splitted profile of the laser beam is obtained due to uneven focusing of the off-axial rays. The semi-analytical solution of the nonlinearly coupled EPW equation in the presence of laser beam filaments has been found. It is observed that due to this nonlinear coupling between these two waves, localization of EPW takes place. Stimulated Raman scattering of this EPW is studied and back reflectivity has been calculated. Further, the localization of EPW affects the eigenfrequency and damping of plasma wave. The new enhanced damping of the plasma wave has been calculated and it is found that the SRS process gets suppressed due to the localization of plasma wave in laser beam filamentary structures. 16. Observation and Control of Hamiltonian Chaos in Wave-particle Interaction SciTech Connect Doveil, F.; Ruzzon, A.; Elskens, Y. 2010-11-23 Wave-particle interactions are central in plasma physics. The paradigm beam-plasma system can be advantageously replaced by a traveling wave tube (TWT) to allow their study in a much less noisy environment. This led to detailed analysis of the self-consistent interaction between unstable waves and an either cold or warm electron beam. More recently a test cold beam has been used to observe its interaction with externally excited wave(s). This allowed observing the main features of Hamiltonian chaos and testing a new method to efficiently channel chaotic transport in phase space. To simulate accurately and efficiently the particle dynamics in the TWT and other 1D particle-wave systems, a new symplectic, symmetric, second order numerical algorithm is developed, using particle position as the independent variable, with a fixed spatial step.This contribution reviews: presentation of the TWT and its connection to plasma physics, resonant interaction of a charged particle in electrostatic waves, observation of particle trapping and transition to chaos, test of control of chaos, and description of the simulation algorithm.The velocity distribution function of the electron beam is recorded with a trochoidal energy analyzer at the output of the TWT. An arbitrary waveform generator is used to launch a prescribed spectrum of waves along the 4m long helix of the TWT. The nonlinear synchronization of particles by a single wave, responsible for Landau damping, is observed. We explore the resonant velocity domain associated with a single wave as well as the transition to large scale chaos when the resonant domains of two waves and their secondary resonances overlap. This transition exhibits a devil's staircase behavior when increasing the excitation level in agreement with numerical simulation.A new strategy for control of chaos by building barriers of transport in phase space as well as its robustness is successfully tested. The underlying concepts extend far beyond the field of 17. Constraining the Braneworld with Gravitational Wave Observations NASA Technical Reports Server (NTRS) McWilliams, Sean T. 2011-01-01 Some braneworld models may have observable consequences that, if detected, would validate a requisite element of string theory. In the infinite Randall-Sundrum model (RS2), the AdS radius of curvature, L, of the extra dimension supports a single bound state of the massless graviton on the brane, thereby reproducing Newtonian gravity in the weak-field limit. However, using the AdS/CFT correspondence, it has been suggested that one possible consequence of RS2 is an enormous increase in Hawking radiation emitted by black holes. We utilize this possibility to derive two novel methods for constraining L via gravitational wave measurements. We show that the EMRI event rate detected by LISA can constrain L at the approximately 1 micron level for optimal cases, while the observation of a single galactic black hole binary with LISA results in an optimal constraint of L less than or equal to 5 microns. 18. On plasma rotation induced by waves in tokamaks SciTech Connect Guan, Xiaoyin; Dodin, I. Y.; Fisch, N. J.; Qin, Hong; Liu, Jian 2013-10-15 The momentum conservation for resonant wave-particle interactions, now proven rigorously and for general settings, is applied to explain in simple terms how tokamak plasma is spun up by the wave momentum perpendicular to the dc magnetic field. The perpendicular momentum is passed through resonant particles to the dc field and, giving rise to the radial electric field, is accumulated as a Poynting flux; the bulk plasma is then accelerated up to the electric drift velocity proportional to that flux, independently of collisions. The presence of this collisionless acceleration mechanism permits varying the ratio of the average kinetic momentum absorbed by the resonant-particle and bulk distributions depending on the orientation of the wave vector. Both toroidal and poloidal forces are calculated, and a fluid model is presented that yields the plasma velocity at equilibrium. 19. Asymmetric drift instability of magnetosonic waves in anisotropic plasmas Bashir, M. F.; Chen, Lunjin 2016-10-01 The general dispersion relation of obliquely propagating magneto-sonic (MS) waves for the inhomogeneous and anisotropic plasmas is analyzed including the effect of wave-particle interaction. The numerical analysis is performed without expanding both the plasma dispersion and the modified Bessel functions to highlight the effects of density inhomogeneity and the temperature anisotropy. The obtained results are compared with the recent work [Naim et al., Phys. Plasmas 22, 062117 (2015)], where only drift mode near the magnetosonic frequency is investigated. In our paper, we additionally analyzed two related modes depicting that the drift effect leads to an asymmetric behavior in the dispersion properties of drift MS waves. The possible application to the solar coronal heating problem has also been discussed. 20. Ion-acoustic cnoidal waves in a quantum plasma SciTech Connect Mahmood, S.; Haas, F. 2014-10-15 Nonlinear ion-acoustic cnoidal wave structures are studied in an unmagnetized quantum plasma. Using the reductive perturbation method, a Korteweg-de Vries equation is derived for appropriate boundary conditions and nonlinear periodic wave solutions are obtained. The corresponding analytical solution and numerical plots of the ion-acoustic cnoidal waves and solitons in the phase plane are presented using the Sagdeev pseudo-potential approach. The variations in the nonlinear potential of the ion-acoustic cnoidal waves are studied at different values of quantum parameter H{sub e} which is the ratio of electron plasmon energy to electron Fermi energy defined for degenerate electrons. It is found that both compressive and rarefactive ion-acoustic cnoidal wave structures are formed depending on the value of the quantum parameter. The dependence of the wavelength and frequency on nonlinear wave amplitude is also presented. 1. Strongly driven ion acoustic waves in laser produced plasmas SciTech Connect Baldis, H.A.; Labaune, C.; Renard, N. 1994-09-20 This paper present an experimental study of ion acoustic waves with wavenumbers corresponding to stimulated Brillouin scattering. Time resolved Thomson scattering in frequency and wavenumber space, has permitted to observe the dispersion relation of the waves as a function of the laser intensity. Apart from observing ion acoustic waves associated with a strong second component is observed at laser intensities above 10{sup 13}Wcm{sup {minus}2}. 2. High amplitude waves in the expanding solar wind plasma NASA Technical Reports Server (NTRS) Schmidt, J. M.; Velli, M.; Grappin, R. 1995-01-01 We simulated the 1-D nonlinear time-evolution of high-amplitude Alfven, slow and fast magnetoacoustic waves in the solar wind propagating outward at different angles to the mean magnetic (spiral) field, using the expanding box model. The simulation results for Alfven waves and fast magnetoacustic waves fit the observational constraints in the solar wind best, showing decreasing trends for energies and other rms-quantities due to expansion and the appearance of inward propagating waves as minor species in the wind. Inward propagating waves are generated by reflection of Alfven waves propagating at large angles to the magnetic field or they coincide with the occurrence of compressible fluctuations. In our simulations, fast and slow magnetoacoustic waves seem to have a level in the density-fluctuations which is too high when we compare with the observations. Furthermore, the evolution of energies for slow magnetoacoustic waves differs strongly from the evolution of fluctuation energies in situ. 3. SOLAR WIND STRAHL BROADENING BY SELF-GENERATED PLASMA WAVES SciTech Connect Pavan, J.; Gaelzer, R.; Vinas, A. F.; Yoon, P. H.; Ziebell, L. F. E-mail: [email protected] E-mail: [email protected] 2013-06-01 This Letter reports on the results of numerical simulations which may provide a possible explanation for the strahl broadening during quiet solar conditions. The relevant processes involved in the broadening are due to kinetic quasi-linear wave-particle interaction. Making use of static analytical electron distribution in an inhomogeneous field, it is found that self-generated electrostatic waves at the plasma frequency, i.e., Langmuir waves, are capable of scattering the strahl component, resulting in energy and pitch-angle diffusion that broadens its velocity distribution significantly. The present theoretical results provide an alternative or complementary explanation to the usual whistler diffusion scenario, suggesting that self-induced electrostatic waves at the plasma frequency might play a key role in broadening the solar wind strahl during quiet solar conditions. 4. Solar Wind Strahl Broadening by Self-Generated Plasma Waves NASA Technical Reports Server (NTRS) Pavan, J.; Vinas, A. F.; Yoon, P. H.; Ziebell, L. F.; Gaelzer, R. 2013-01-01 This Letter reports on the results of numerical simulations which may provide a possible explanation for the strahl broadening during quiet solar conditions. The relevant processes involved in the broadening are due to kinetic quasi-linear wave-particle interaction. Making use of static analytical electron distribution in an inhomogeneous field, it is found that self-generated electrostatic waves at the plasma frequency, i.e., Langmuir waves, are capable of scattering the strahl component, resulting in energy and pitch-angle diffusion that broadens its velocity distribution significantly. The present theoretical results provide an alternative or complementary explanation to the usual whistler diffusion scenario, suggesting that self-induced electrostatic waves at the plasma frequency might play a key role in broadening the solar wind strahl during quiet solar conditions. 5. Nonlinear Plasma Waves Excitation by Intense Ion Beams in Background Plasma SciTech Connect Igor D. Kaganovich; Edward A. Startsev; Ronald C. Davidson 2004-04-15 Plasma neutralization of an intense ion pulse is of interest for many applications, including plasma lenses, heavy ion fusion, cosmic ray propagation, etc. An analytical electron fluid model has been developed to describe the plasma response to a propagating ion beam. The model predicts very good charge neutralization during quasi-steady-state propagation, provided the beam pulse duration {tau}{sub b} is much longer than the electron plasma period 2{pi}/{omega}{sub p}, where {omega}{sub p} = (4{pi}e{sup 2}n{sub p}/m){sup 1/2} is the electron plasma frequency and n{sub p} is the background plasma density. In the opposite limit, the beam pulse excites large-amplitude plasma waves. If the beam density is larger than the background plasma density, the plasma waves break. Theoretical predictions are compared with the results of calculations utilizing a particle-in-cell (PIC) code. The cold electron fluid results agree well with the PIC simulations for ion beam propagation through a background plasma. The reduced fluid description derived in this paper can provide an important benchmark for numerical codes and yield scaling relations for different beam and plasma parameters. The visualization of numerical simulation data shows complex collective phenomena during beam entry and exit from the plasma. 6. Magnetospheric electron-velocity-distribution function information from wave observations Benson, Robert F.; ViñAs, Adolfo F.; Osherovich, Vladimir A.; Fainberg, Joseph; Purser, Carola M.; Adrian, Mark L.; Galkin, Ivan A.; Reinisch, Bodo W. 2013-08-01 The electron-velocity-distribution function was determined to be highly non-Maxwellian and more appropriate to a kappa distribution, with κ ≈ 2.0, near magnetic midnight in the low-latitude magnetosphere just outside a stable plasmasphere during extremely quiet geomagnetic conditions. The kappa results were based on sounder-stimulated Qn plasma resonances using the Radio Plasma Imager (RPI) on the IMAGE satellite; the state of the plasmasphere was determined from IMAGE/EUV observations. The Qn resonances correspond to the maximum frequencies of Bernstein-mode waves that are observed between the harmonics of the electron cyclotron frequency in the frequency domain above the upper-hybrid frequency. Here we present the results of a parametric investigation that included suprathermal electrons in the electron-velocity-distribution function used in the plasma-wave dispersion equation to calculate the Qn frequencies for a range of kappa and fpe/fce values for Qn resonances from Q1 to Q9. The Qn frequencies were also calculated using a Maxwellian distribution, and they were found to be greater than those calculated using a kappa distribution with the frequency differences increasing with increasing n for a fixed κ and with decreasing κ for a fixed n. The calculated fQn values have been incorporated into the RPI BinBrowser software providing a powerful tool for rapidly obtaining information on the nature of the magnetospheric electron-velocity-distribution function and the electron number density Ne. This capability enabled accurate (within a few percent) in situ Ne determinations to be made along the outbound orbital track as IMAGE moved away from the plasmapause. The extremely quiet geomagnetic conditions allowed IMAGE/EUV-extracted counts to be compared with the RPI-determined orbital-track Ne profile. The comparisons revealed remarkably similar Ne structures. 7. Observational Confirmations of Spiral Density Wave Theory Kennefick, Julia D.; Kennefick, Daniel; Shameer Abdeen, Mohamed; Berrier, Joel; Davis, Benjamin; Fusco, Michael; Pour Imani, Hamed; Shields, Doug; DMS, SINGS 2017-01-01 Using two techniques to reliably and accurately measure the pitch angles of spiral arms in late-type galaxies, we have compared pitch angles to directly measured black hole masses in local galaxies and demonstrated a strong correlation between them. Using the relation thus established we have developed a pitch angle distribution function of a statistically complete volume limited sample of nearby galaxies and developed a central black hole mass function for nearby spiral galaxies.We have further shown that density wave theory leads us to a three-way correlation between bulge mass, pitch angle, and disk gas density, and have used data from the Galaxy Disk Mass Survey to confirm this possible fundamental plane. Density wave theory also predicts that the pitch angle of spiral arms should change with observed waveband as each waveband is sampling a different stage in stellar population formation and evolution. We present evidence that this is indeed the case using a sample of galaxies from the Spitzer Infrared Nearby Galaxy Survey. Furthermore, the evolved spiral arms cross at the galaxy co-rotation radius. This gives a new method for determining the co-rotation radius of spiral galaxies that is found to agree with those found using previous methods. 8. On Variational Methods in the Physics of Plasma Waves SciTech Connect I.Y. Dodin 2013-03-08 A fi rst-principle variational approach to adiabatic collisionless plasma waves is described. The focus is made on one-dimensional electrostatic oscillations, including phase-mixed electron plasma waves (EPW) with trapped particles, such as Bernstein-Greene-Kruskal modes. The well known Whitham's theory is extended by an explicit calculation of the EPW Lagrangian, which is related to the oscillation-center energies of individual particles in a periodic fi eld, and those are found by a quadrature. Some paradigmatic physics of EPW is discussed for illustration purposes. __________________________________________________ 9. Is dust acoustic wave a new plasma acoustic mode? Dwivedi, C. B. 1997-09-01 In this Brief Communication, the claim of the novelty of the dust acoustic wave in a dusty plasma within the constant dust charge model is questioned. Conceptual lacunas behind the claim have been highlighted and appropriate physical arguments have been forwarded against the claim. It is demonstrated that the so-called dust acoustic wave could better be termed as a general acoustic fluctuation response with a dominant characteristic feature of the acoustic-like mode (ALM) fluctuation response reported by Dwivedi et al. [J. Plasma Phys. 41, 219 (1989)]. It is suggested that both correct and more usable nomenclature of the ALM should be the so-called acoustic mode. 10. Relativistic (covariant) kinetic theory of linear plasma waves and instabilities SciTech Connect Lazar, M.; Schlickeiser, R. 2006-06-19 The fundamental kinetic description is of vital importance in high-energy astrophysics and fusion plasmas where wave phenomena evolve on scales small comparing with binary collision scales. A rigorous relativistic analysis is required even for nonrelativistic plasma temperatures for which the classical theory yielded unphysical results: e.g. collisonless damping of superluminal waves (phase velocity exceeds the speed of light). The existing nonrelativistic approaches are now improved by covariantly correct dispersion theory. As an important application, the Weibel instability has been recently investigated and confirmed as the source of primordial magnetic field in the intergalactic medium. 11. On the rogue wave propagation in ion pair superthermal plasma SciTech Connect Abdelwahed, H. G. E-mail: [email protected]; Zahran, M. A.; El-Shewy, E. K. Elwakil, S. A. 2016-02-15 Effects of superthermal electron on the features of nonlinear acoustic waves in unmagnetized collisionless ion pair plasma with superthermal electrons have been examined. The system equations are reduced in the form of the nonlinear Schrodinger equation. The rogue wave characteristics dependences on the ionic density ratio (ν = n{sub –0}/n{sub +0}), ionic mass ratio (Q = m{sub +}/m{sub −}), and superthermality index (κ) are investigated. It is worth mentioning that the results present in this work could be applicable in the Earth's ionosphere plasmas. 12. Predicting EMIC wave properties from ring current plasma conditions Cowee, M.; Fu, X.; Jordanova, V. 2015-12-01 Recently, sophisticated computer models have shown that accurate, dynamic modelling of the energetic electrons in the radiation belt requires global and real-time plasma and wave conditions. Data provided by in-situ spacecraft measurement are too sparse to supply enough inputs for continuous global modeling of the radiation belt. Here we present a model to predict amplitude, peak frequency and spectral width of the electromagnetic ion cyclotron (EMIC) wave from the anisotropic ring current ion distributions, which are the source of the wave. The model is derived from hybrid simulations in a large initial parameter space for plasmas consisting of electrons, protons, and helium ions. Key parameters include the ratio of plasma frequency to ion gyrofrequency, the density, temperature and anisotropy of hot ions, and the cold-ion composition. The results show that amplitude, peak frequency and spectral width of EMIC waves can be related to linear properties of the anisotropy-driven instability, e.g. growth rate and plasma beta, through simple analytic formulas. Combined with a dynamic ring current model, this model can provide global EMIC wave information needed for radiation-belt modeling. 13. Surface wave propagation in non-ideal plasmas Pandey, B. P.; Dwivedi, C. B. 2015-03-01 The properties of surface waves in a partially ionized, compressible magnetized plasma slab are investigated in this work. The waves are affected by the non-ideal magnetohydrodynamic (MHD) effects which causes finite drift of the magnetic field in the medium. When the magnetic field drift is ignored, the characteristics of the wave propagation in a partially ionized plasma fluid is similar to the fully ionized ideal MHD except now the propagation properties depend on the fractional ionization as well as on the compressibility of the medium. The phase velocity of the sausage and kink waves increases marginally (by a few per cent) due to the compressibility of the medium in both ideal as well as Hall-diffusion-dominated regimes. However, unlike ideal regime, only waves below certain cut-off frequency can propagate in the medium in Hall dominated regime. This cut-off for a thin slab has a weak dependence on the plasma beta whereas for thick slab no such dependence exists. More importantly, since the cut-off is introduced by the Hall diffusion, the fractional ionization of the medium is more important than the plasma compressibility in determining such a cut-off. Therefore, for both compressible as well incompressible medium, the surface modes of shorter wavelength are permitted with increasing ionization in the medium. We discuss the relevance of these results in the context of solar photosphere-chromosphere. 14. Nonlinear Electron Acoustic Waves in Dissipative Plasma with Superthermal Electrons El-Hanbaly, A. M.; El-Shewy, E. K.; Kassem, A. I.; Darweesh, H. F. 2016-01-01 The nonlinear properties of small amplitude electron-acoustic ( EA) solitary and shock waves in a homogeneous system of unmagnetized collisionless plasma consisted of a cold electron fluid and superthermal hot electrons obeying superthermal distribution, and stationary ions have been investigated. A reductive perturbation method was employed to obtain the Kadomstev-Petviashvili-Burgers (KP-Brugers) equation. Some solutions of physical interest are obtained. These solutions are related to soliton, monotonic and oscillatory shock waves and their behaviour are shown graphically. The formation of these solutions depends crucially on the value of the Burgers term and the plasma parameters as well. By using the tangent hyperbolic (tanh) method, another interesting type of solution which is a combination between shock and soliton waves is obtained. The topology of phase portrait and potential diagram of the KP-Brugers equation is investigated.The advantage of using this method is that one can predict different classes of the travelling wave solutions according to different phase orbits. The obtained results may be helpful in better understanding of waves propagation in various space plasma environments as well as in inertial confinement fusion laboratory plasmas. 15. Method for generating a plasma wave to accelerate electrons DOEpatents Umstadter, D.; Esarey, E.; Kim, J.K. 1997-06-10 The invention provides a method and apparatus for generating large amplitude nonlinear plasma waves, driven by an optimized train of independently adjustable, intense laser pulses. In the method, optimal pulse widths, interpulse spacing, and intensity profiles of each pulse are determined for each pulse in a series of pulses. A resonant region of the plasma wave phase space is found where the plasma wave is driven most efficiently by the laser pulses. The accelerator system of the invention comprises several parts: the laser system, with its pulse-shaping subsystem; the electron gun system, also called beam source, which preferably comprises photo cathode electron source and RF-LINAC accelerator; electron photo-cathode triggering system; the electron diagnostics; and the feedback system between the electron diagnostics and the laser system. The system also includes plasma source including vacuum chamber, magnetic lens, and magnetic field means. The laser system produces a train of pulses that has been optimized to maximize the axial electric field amplitude of the plasma wave, and thus the electron acceleration, using the method of the invention. 21 figs. 16. Method for generating a plasma wave to accelerate electrons DOEpatents Umstadter, Donald; Esarey, Eric; Kim, Joon K. 1997-01-01 The invention provides a method and apparatus for generating large amplitude nonlinear plasma waves, driven by an optimized train of independently adjustable, intense laser pulses. In the method, optimal pulse widths, interpulse spacing, and intensity profiles of each pulse are determined for each pulse in a series of pulses. A resonant region of the plasma wave phase space is found where the plasma wave is driven most efficiently by the laser pulses. The accelerator system of the invention comprises several parts: the laser system, with its pulse-shaping subsystem; the electron gun system, also called beam source, which preferably comprises photo cathode electron source and RF-LINAC accelerator; electron photo-cathode triggering system; the electron diagnostics; and the feedback system between the electron diagnostics and the laser system. The system also includes plasma source including vacuum chamber, magnetic lens, and magnetic field means. The laser system produces a train of pulses that has been optimized to maximize the axial electric field amplitude of the plasma wave, and thus the electron acceleration, using the method of the invention. 17. Stability of shock waves in high temperature plasmas SciTech Connect Das, Madhusmita; Bhattacharya, Chandrani; Menon, S. V. G. 2011-10-15 The Dyakov-Kontorovich criteria for spontaneous emission of acoustic waves behind shock fronts are investigated for high temperature aluminum and beryllium plasmas. To this end, the Dyakov and critical stability parameters are calculated from Rankine-Hugoniot curves using a more realistic equation of state (EOS). The cold and ionic contributions to the EOS are obtained via scaled binding energy and mean field theory, respectively. A screened hydrogenic model, including l-splitting, is used to calculate the bound electron contribution to the electronic EOS. The free electron EOS is obtained from Fermi-Dirac statistics. Predictions of the model for ionization curves and shock Hugoniot are found to be in excellent agreement with available experimental and theoretical data. It is observed that the electronic EOS has significant effect on the stability of the planar shock front. While the shock is stable for low temperatures and pressures, instability sets in as temperature rises. The basic reason is ionization of electronic shells and consequent increase in electronic specific heat. The temperatures and densities of the unstable region correspond to those where electronic shells get ionized. With the correct modeling of bound electrons, we find that shock instability for Al occurs at a compression ratio {approx}5.4, contrary to the value {approx}3 reported in the literature. Free electrons generated in the ionization process carry energy from the shock front, thereby giving rise to spontaneously emitted waves, which decay the shock front. 18. Nonlinear Generalized Hydrodynamic Wave Equations in Strongly Coupled Dusty Plasmas SciTech Connect Veeresha, B. M.; Sen, A.; Kaw, P. K. 2008-09-07 A set of nonlinear equations for the study of low frequency waves in a strongly coupled dusty plasma medium is derived using the phenomenological generalized hydrodynamic (GH) model and is used to study the modulational stability of dust acoustic waves to parallel perturbations. Dust compressibility contributions arising from strong Coulomb coupling effects are found to introduce significant modifications in the threshold and range of the instability domain. 19. Ion-Acoustic Waves in Self-Gravitaing Dusty Plasma SciTech Connect Kumar, Nagendra; Kumar, Vinod; Kumar, Anil 2008-09-07 The propagation and damping of low frequency ion-acoustic waves in steady state, unmagnetised, self-gravitating dusty plasma are studied taking into account two important damping mechanisms creation damping and Tromso damping. It is found that imaginary part of wave number is independent of frequency in case of creation damping. But when we consider the case of creation and Tromso damping together, an additional contribution to damping appears with the increase in frequency attributed to Tromso effect. 20. Obliquely Propagating Electromagnetic Waves in Magnetized Kappa Plasmas Gaelzer, R. 2015-12-01 The effects of velocity distribution functions (VDFs) that exhibit a power-law dependence on the high-energy tail have been the subjectof intense research by the space plasma community. Such functions, known as kappa or superthermal distributions, have beenfound to provide a better fitting to the VDF measured by spacecraft in the solar wind. One of the problems that is being addressed on this new light is the temperature anisotropy of solar wind protons and electrons. An anisotropic kappa VDF contains a large amount of free energy that can excite waves in the solar wind. Conversely, the wave-particle interaction is important to determine the shape of theobserved particle distributions.In the literature, the general treatment for waves excited by (bi-)Maxwellian plasmas is well-established. However, for kappa distributions, either isotropic or anisotropic, the wave characteristics have been studied mostly for the limiting cases of purely parallel or perpendicular propagation. Contributions for the general case of obliquely-propagating electromagnetic waves have been scarcely reported so far. The absence of a general treatment prevents a complete analysis of the wave-particle interaction in kappa plasmas, since some instabilities, such as the firehose, can operate simultaneously both in the parallel and oblique directions.In a recent work [1], we have obtained expressions for the dielectric tensor and dispersion relations for the low-frequency, quasi-perpendicular dispersive Alfvén waves resulting from a kappa VDF. In the present work, we generalize the formalism introduced by [1] for the general case of electrostatic and/or electromagnetic waves propagating in a kappa plasma in any frequency range and for arbitrary angles.We employ an isotropic distribution, but the methods used here can be easily applied to more general anisotropic distributions,such as the bi-kappa or product-bi-kappa. [1] R. Gaelzer and L. F. Ziebell, Journal of Geophysical Research 119, 9334 1. Plasma characterization using ultraviolet Thomson scattering from ion-acoustic and electron plasma waves (invited) Follett, R. K.; Delettrez, J. A.; Edgell, D. H.; Henchen, R. J.; Katz, J.; Myatt, J. F.; Froula, D. H. 2016-11-01 Collective Thomson scattering is a technique for measuring the plasma conditions in laser-plasma experiments. Simultaneous measurements of ion-acoustic and electron plasma-wave spectra were obtained using a 263.25-nm Thomson-scattering probe beam. A fully reflective collection system was used to record light scattered from electron plasma waves at electron densities greater than 1021 cm-3, which produced scattering peaks near 200 nm. An accurate analysis of the experimental Thomson-scattering spectra required accounting for plasma gradients, instrument sensitivity, optical effects, and background radiation. Practical techniques for including these effects when fitting Thomson-scattering spectra are presented and applied to the measured spectra to show the improvements in plasma characterization. 2. Magnetospheric plasma - Sources, wave-particle interactions and acceleration mechanisms. NASA Technical Reports Server (NTRS) Speiser, T. W. 1971-01-01 Some of the basic problems associated with magnetospheric physics are reviewed. The sources of magnetospheric plasma, with auroral particles included as a subset, are discussed. The possible ways in which the solar wind plasma can gain access to the magnetosphere are outlined. Some important consequences of wave-particle interactions are examined. Finally, the basic mechanisms which energize or accelerate particles by reconnection and convection are explained. 3. Linear waves in a resistive plasma with Hall current SciTech Connect Almaguer, J.A. ) 1992-10-01 Dispersion relations for the case of a magnetized plasma are determined taking into account the Hall current and a constant resistivity, {eta}, in Ohm's law. It is found that the Hall effect is relevant only for parallel (to the equilibrium magnetic field) wave numbers in the case of uniform plasmas, giving place to a dispersive behavior. In particular, the cases of {eta}{r arrow}0 and small (nonzero) resistivity are discussed. 4. Ion acoustic waves at comet 67P/Churyumov-Gerasimenko. Observations and computations Gunell, H.; Nilsson, H.; Hamrin, M.; Eriksson, A.; Odelstad, E.; Maggiolo, R.; Henri, P.; Vallieres, X.; Altwegg, K.; Tzou, C.-Y.; Rubin, M.; Glassmeier, K.-H.; Stenberg Wieser, G.; Simon Wedlund, C.; De Keyser, J.; Dhooghe, F.; Cessateur, G.; Gibbons, A. 2017-03-01 Context. On 20 January 2015 the Rosetta spacecraft was at a heliocentric distance of 2.5 AU, accompanying comet 67P/Churyumov-Gerasimenko on its journey toward the Sun. The Ion Composition Analyser (RPC-ICA), other instruments of the Rosetta Plasma Consortium, and the ROSINA instrument made observations relevant to the generation of plasma waves in the cometary environment. Aims: Observations of plasma waves by the Rosetta Plasma Consortium Langmuir probe (RPC-LAP) can be explained by dispersion relations calculated based on measurements of ions by the Rosetta Plasma Consortium Ion Composition Analyser (RPC-ICA), and this gives insight into the relationship between plasma phenomena and the neutral coma, which is observed by the Comet Pressure Sensor of the Rosetta Orbiter Spectrometer for Ion and Neutral Analysis instrument (ROSINA-COPS). Methods: We use the simple pole expansion technique to compute dispersion relations for waves on ion timescales based on the observed ion distribution functions. These dispersion relations are then compared to the waves that are observed. Data from the instruments RPC-LAP, RPC-ICA and the mutual impedance probe (RPC-MIP) are compared to find the best estimate of the plasma density. Results: We find that ion acoustic waves are present in the plasma at comet 67P/Churyumov-Gerasimenko, where the major ion species is H2O+. The bulk of the ion distribution is cold, kBTi = 0.01 eV when the ion acoustic waves are observed. At times when the neutral density is high, ions are heated through acceleration by the solar wind electric field and scattered in collisions with the neutrals. This process heats the ions to about 1 eV, which leads to significant damping of the ion acoustic waves. Conclusions: In conclusion, we show that ion acoustic waves appear in the H2O+ plasmas at comet 67P/Churyumov-Gerasimenko and how the interaction between the neutral and ion populations affects the wave properties. Computer code for the dispersion analysis is 5. Excitation of Plasma Waves in Aurora by Electron Beams NASA Technical Reports Server (NTRS) daSilva, C. E.; Vinas, A. F.; deAssis, A. S.; deAzevedo, C. A. 1996-01-01 In this paper, we study numerically the excitation of plasma waves by electron beams, in the auroral region above 2000 km of altitude. We have solved the fully kinetic dispersion relation, using numerical method and found the real frequency and the growth rate of the plasma wave modes. We have examined the instability properties of low-frequency waves such as the Electromagnetic Ion Cyclotron (EMIC) wave as well as Lower-Hybrid (LH) wave in the range of high-frequency. In all cases, the source of free energy are electron beams propagating parallel to the geomagnetic field. We present some features of the growth rate modes, when the cold plasma parameters are changed, such as background electrons and ions species (H(+) and O(+)) temperature, density or the electron beam density and/or drift velocity. These results can be used in a test-particle simulation code, to investigate the ion acceleration and their implication in the auroral acceleration processes, by wave-particle interaction. 6. Breathing rogue wave observed in numerical experiment. PubMed Ruban, V P 2006-09-01 Numerical simulations of the recently derived fully nonlinear equations of motion for long-crested water waves [V. P. Ruban, Phys. Rev. E 71, 055303(R) (2005)] with quasirandom initial conditions are reported, which show the spontaneous formation of a single extreme wave on deep water. This rogue wave behaves in an oscillating manner and exists for a relatively long time (many wave periods) without significant change of its maximal amplitude. 7. Laboratory observation of elastic waves in solids Rossing, Thomas D.; Russell, Daniel A. 1990-12-01 Compressional, torsional, and bending waves in bars and plates can be studied with simple apparatus in the laboratory. Although compressional and torsional waves show little or no dispersion, bending waves propagate at a speed proportional to (f)1/2. Reflections at boundaries lead to standing waves that determine the vibrational mode shapes and mode frequencies. Boundary conditions include free edges, simply supported edges, and clamped edges. Typical mode shapes and mode frequencies for rectangular bars, circular plates, and square plates are described. 8. Computational study of nonlinear plasma waves: 1: Simulation model and monochromatic wave propagation NASA Technical Reports Server (NTRS) Matda, Y.; Crawford, F. W. 1974-01-01 An economical low noise plasma simulation model is applied to a series of problems associated with electrostatic wave propagation in a one-dimensional, collisionless, Maxwellian plasma, in the absence of magnetic field. The model is described and tested, first in the absence of an applied signal, and then with a small amplitude perturbation, to establish the low noise features and to verify the theoretical linear dispersion relation at wave energy levels as low as 0.000,001 of the plasma thermal energy. The method is then used to study propagation of an essentially monochromatic plane wave. Results on amplitude oscillation and nonlinear frequency shift are compared with available theories. The additional phenomena of sideband instability and satellite growth, stimulated by large amplitude wave propagation and the resulting particle trapping, are described. 9. Electromagnetic waves near the proton cyclotron frequency: Stereo observations SciTech Connect Jian, L. K.; Wei, H. Y.; Russell, C. T.; Luhmann, J. G.; Klecker, B.; Omidi, N.; Isenberg, P. A.; Goldstein, M. L.; Figueroa-Viñas, A.; Blanco-Cano, X. 2014-05-10 Transverse, near-circularly polarized, parallel-propagating electromagnetic waves around the proton cyclotron frequency were found sporadically in the solar wind throughout the inner heliosphere. They could play an important role in heating and accelerating the solar wind. These low-frequency waves (LFWs) are intermittent but often occur in prolonged bursts lasting over 10 minutes, named 'LFW storms'. Through a comprehensive survey of them from Solar Terrestrial Relations Observatory A using dynamic spectral wave analysis, we have identified 241 LFW storms in 2008, present 0.9% of the time. They are left-hand (LH) or right-hand (RH) polarized in the spacecraft frame with similar characteristics, probably due to Doppler shift of the same type of waves or waves of intrinsically different polarities. In rare cases, the opposite polarities are observed closely in time or even simultaneously. Having ruled out interplanetary coronal mass ejections, shocks, energetic particles, comets, planets, and interstellar ions as LFW sources, we discuss the remaining generation scenarios: LH ion cyclotron instability driven by greater perpendicular temperature than parallel temperature or by ring-beam distribution, and RH ion fire hose instability driven by inverse temperature anisotropy or by cool ion beams. The investigation of solar wind conditions is compromised by the bias of the one-dimensional Maxwellian fit used for plasma data calibration. However, the LFW storms are preferentially detected in rarefaction regions following fast winds and when the magnetic field is radial. This preference may be related to the ion cyclotron anisotropy instability in fast wind and the minimum in damping along the radial field. 10. Linear and nonlinear heavy ion-acoustic waves in a strongly coupled plasma SciTech Connect Ema, S. A. Mamun, A. A.; Hossen, M. R. 2015-09-15 A theoretical study on the propagation of linear and nonlinear heavy ion-acoustic (HIA) waves in an unmagnetized, collisionless, strongly coupled plasma system has been carried out. The plasma system is assumed to contain adiabatic positively charged inertial heavy ion fluids, nonextensive distributed electrons, and Maxwellian light ions. The normal mode analysis is used to study the linear behaviour. On the other hand, the well-known reductive perturbation technique is used to derive the nonlinear dynamical equations, namely, Burgers equation and Korteweg-de Vries (K-dV) equation. They are also numerically analyzed in order to investigate the basic features of shock and solitary waves. The adiabatic effects on the HIA shock and solitary waves propagating in such a strongly coupled plasma are taken into account. It has been observed that the roles of the adiabatic positively charged heavy ions, nonextensivity of electrons, and other plasma parameters arised in this investigation have significantly modified the basic features (viz., polarity, amplitude, width, etc.) of the HIA solitary/shock waves. The findings of our results obtained from this theoretical investigation may be useful in understanding the linear as well as nonlinear phenomena associated with the HIA waves both in space and laboratory plasmas. 11. High Latitude Electromagnetic Plasma Wave Emissions. DTIC Science & Technology 1982-05-01 1980), and possibly Uranus (Brown, 1976). Auroral hiss emissions have also been detected near the Io plasma torus at Jupi- ter (Gurnett et al., 1979...between 130 kHz and 2,600 kHz, Astrophys. J., 180:359. Brown, L. W., 1976, Possible radio emission from Uranus at 0.5 MHz, Astrophys. J., 207:L202. 28 12. Nonlinear dust-acoustic waves in a strongly coupled dusty plasma with vortexlike ion distribution SciTech Connect Anowar, M. G. M.; Rahman, M. S.; Mamun, A. A. 2009-05-15 The nonlinear features of dust-acoustic (DA) waves in a strongly coupled unmagnetized dusty plasma (containing electrons following Boltzmann distribution, ions obeying vortexlike distribution, and negatively charged mobile dust) are investigated by using reductive perturbation method. It is observed that the nonlinear propagation of the DA waves gives rise to solitary structures when the strong correlation is absent and gives rise to shock structures when the strong correlation among the dust grains is present. The condition for the formation of oscillatory and monotonic shock structures is also found. The implications of our result in space and laboratory dusty plasmas are discussed. 13. Pioneer Venus observations of plasma and field structure in the near wake of Venus NASA Technical Reports Server (NTRS) Luhmann, J. G.; Russell, C. T.; Brace, L. H.; Knudsen, W. C.; Taylor, H. A.; Scarf, F. L.; Colburn, D. S.; Barnes, A. 1982-01-01 Ionospheric plasma density depletions or 'holes' are observed by the Pioneer Venus orbiter in association with radial magnetic fields in the near wake of Venus. This report presents examples of the collected observations of these unexpected features of the Venus nightside ionosphere obtained by the Langmuir probe, magnetometer, ion mass spectrometer, retarding potential analyzer, plasma analyzer, and electric field experiments. The connection between plasma density depletions and temperature changes, changes in ion composition, plasma wave emissions, and magnetic fields with a substantial radial component is illustrated. Mechanisms that may be responsible for the formation and maintenance of holes are suggested. 14. Nonlinear waves in dense dusty plasmas with high fugacity Rao, N. N.; Shukla, P. K. 2001-01-01 Nonlinear propagation of small, but finite, amplitude electrostatic dust waves has been investigated in the low as well as high fugacity regimes by deriving the corresponding Boussinesq equation which, for unidirectional propagation, reduces to the Korteweg-de Vries equation. The dust-acoustic wave (DAW) solitons are shown to correspond to the tenuous (low fugacity) dusty plasmas, while in the dense (high fugacity) regime the solitons are associated with the dust-Coulomb waves (DCWs). Unlike the DAW solitons which are (dust) density compressional and supersonic, the DCW solitons are (dust) density rarefactive and propagate with super-Coulombic speeds. 15. Observation of Phillips's spectrum in Faraday waves Castillo, Gustavo; Falcon, Claudio 2016-11-01 We consider the problem of wave turbulence generated by singularities from an experimental point of view. We study a system of Faraday waves interacting with waves generated by a wave-maker driven with a random forcing. We measure the temporal fluctuations of the surface wave amplitude at a given location and we show that for a wide range of forcing parameters the surface height displays a power-law spectra that greatly differs from the one predicted by the WT theory. In the capillary region the power spectrum turns out to be proportional to f-5, which we believe is due to singularities moving across the system. Proyecto Postdoctorado Fondecyt Nro 3160032. 16. Nonlinear Coherent Structures of Alfvén Wave in a Collisional Plasma Jana, Sayanee; Ghosh, Samiran; Chakrabarti, Nikhil 2016-10-01 The Alfvén wave dynamics is investigated in the framework of Lagrangian two-fluid model in a cold magnetized collisional plasma in presence of finite electron inertia. In the quasi-linear limit, the dynamics of the nonlinear Alfvén wave is shown to be governed by a modified Korteweg-de Vries Burgers (mKdVB) equation. In this mKdVB equation, the electron inertia is found to act as a source of dispersion and the electro-ion collision serves as a dissipation. In the long wavelength limit, we have also investigated wave modulation characteristics of the nonlinear Alfvén wave. The dynamics of this modulated wave is shown to be governed by a damped nonlinear Schrödinger equation (NLSE). These nonlinear equations are analysed by means of analytical and numerical simulation to elucidate the various aspects of the phase-space dynamics of the nonlinear wave. Results reveal that nonlinear Alfvén wave exhibits shock, envelope and breather like structures. Numerical simulations also predict the formation of Alfvénic rogue waves, rogue wave holes and giant breathers. These results could be useful for understanding the salient features of the Alfvénic magnetic field structures from observational data in very low- βmagnetized collisional plasmas in space and laboratory. 17. Theory of high-frequency waves in a coaxial plasma wave guide Maraghechi, B.; Farrokhi, B.; Willett, J. E. 1999-10-01 An analysis of the high-frequency eigenmodes of a coaxial wave guide containing a magnetized annular plasma column is presented. A transcendental equation is derived from the boundary conditions in the form of an eighth-order determinant equated to zero. Simultaneous solution of this determinantal equation and a polynomial equation derived from the wave equation yields the dispersion relations for the eigenmodes. By reduction of the order of the determinant the appropriate transcendental equation is easily obtained for some special cases, e.g., partially filled coaxial wave guide. The electrostatic treatment of a coaxial cylindrical wave guide is also presented. The corresponding transcendental equation is reduced to some special cases, e.g., conventional wave guide containing an annular plasma column under electrostatic approximation. Numerical solutions are obtained for some azimuthally symmetric EH (perturbed TM) and HE (perturbed TE) wave guide modes, cyclotron modes, and space-charge modes. A strong dependence of the frequencies of these electromagnetic-electrostatic waves on the radii of the coaxial wave guide and the plasma column is revealed. 18. RF wave propagation and scattering in turbulent tokamak plasmas SciTech Connect Horton, W. Michoski, C.; Peysson, Y.; Decker, J. 2015-12-10 Drift wave turbulence driven by the steep electron and ion temperature gradients in H-mode divertor tokamaks produce scattering of the RF waves used for heating and current drive. The X-ray emission spectra produced by the fast electrons require the turbulence broaden RF wave spectrum. Both the 5 GHz Lower Hybrid waves and the 170 GHz electron cyclotron [EC] RF waves experience scattering and diffraction by the electron density fluctuations. With strong LHCD there are bifurcations in the coupled turbulent transport dynamics giving improved steady-state confinement states. The stochastic scattering of the RF rays makes the prediction of the distribution of the rays and the associated particle heating a statistical problem. Thus, we introduce a Fokker-Planck equation for the probably density of the RF rays. The general frame work of the coupled system of coupled high frequency current driving rays with the low-frequency turbulent transport determines the profiles of the plasma density and temperatures. 19. Linear Electrostatic Waves in Unmagnetized Arbitrarily Degenerate Quantum Plasmas Rightley, Shane; Uzdensky, Dmitri 2012-10-01 Plasmas in which the inter-particle spacing approaches the thermal de Broglie wavelength are subject to quantum statistical effects due to Pauli exclusion, and many familiar plasma phenomena are modified on such length scales because of the Heisenberg uncertainty principle. The question of how to model these quantum plasmas is a naturally interesting one, as it pushes the envelope of our knowledge of plasma physics and applies the well-established principles of quantum mechanics in a novel context. Such models are important for microelectronic systems, dense laser-produced plasmas, and some extreme astrophysical environments. For completely degenerate plasmas, both kinetic and fluid theories have already been developed. In this presentation, unmagnetized Fermi-Dirac equilibrium plasmas with finite temperature and arbitrary degree of degeneracy are considered. Linear dispersion relations for electrostatic waves and oscillations, including Landau damping, are derived and analyzed. The analysis is carried out using a self-consistent mean-field quantum kinetic model (the Wigner-Poisson system). Growth of waves due to kinetic instabilities, such as the Buneman and bump-on-tail instabilities, is also considered. 20. Dust Detection Using Radio and Plasma Wave Instruments in the Solar System Ye, S.; Gurnett, D. A.; Kurth, W. S.; Averkamp, T. F.; Kempf, S.; Hsu, S.; Srama, R.; Grün, E.; Morooka, M. W.; Sakai, S.; Wahlund, J. E. 2014-12-01 Nanometer to micrometer sized dust particles pervade our solar system. The origins of these dust particles include asteroid collisions, cometary activity, and geothermal activity of the planetary moons, for example, the water dust cloud ejected from Saturn's moon Enceladus. Radio and plasma wave instruments have been used to detect such dust particles via voltage pulses induced by impacts on the spacecraft body and antennas. The first detection of such dust impacts occurred when Voyager 1 passed through Saturn's ring plane. Since then, dust impacts have been detected by radio and plasma wave instruments on many spacecraft, including ISEE-3, Cassini, and STEREO. In this presentation, we review the detection of dust particles in the solar system using radio and plasma wave instruments aboard various spacecraft since the Voyager era. We also show characteristics of the dust particles derived from recent observations by Cassini RPWS in Saturn's magnetosphere. The dust size distribution and density are consistent with those measured by the conventional dust detectors. A new method of measuring the electron density inside the Enceladus plume based on plasma oscillations observed after dust impacts will also be discussed. The dust measurement by radio and plasma wave instruments complements that by conventional dust detectors and provide important information about the spatial distribution of dust particles due to less pointing constraints and the larger detection area. 1. Non-linear Langmuir waves in a warm quantum plasma SciTech Connect Dubinov, Alexander E. Kitaev, Ilya N. 2014-10-15 A non-linear differential equation describing the Langmuir waves in a warm quantum electron-ion plasma has been derived. Its numerical solutions of the equation show that ordinary electronic oscillations, similar to the classical oscillations, occur along with small-scale quantum Langmuir oscillations induced by the Bohm quantum force. 2. Dust-Coulomb waves in dense dusty plasmas Rao, N. N. 1999-12-01 Dusty plasmas can be considered as tenuous, dilute or dense when the dust fugacity parameter f≡4πnd0λD2R˜NDR/λD satisfies f≪1, ˜1, or ≫1, where nd0, λD and R denote, respectively, the dust number density, the plasma Debye length and the dust grain size (radius), and ND=nd0λD3 is the dust plasma parameter. Dense dusty plasmas are shown to support a new kind of ultra low-frequency electrostatic dust mode which may be called the "Dust-Coulomb Wave" (DCW). In contrast to the dust-acoustic wave (DAW) and the dust-lattice wave (DLW) which exist even for constant grain charge, DCWs are accompanied by dust charge as well as number density perturbations which are proportional to each other. For frequencies much smaller than the grain charging frequency, DCWs propagate as normal modes with the phase speed CDC≡qd0/√mdR , where qd0 (md) is the charge (mass) of the dust grains. In the long wavelength limit, the DCW phase speed is much smaller than that of DAW (CDA), and scales as ˜CDA/√f . Thus, for a given wave number, the frequency regime for the existence of DCW is much lower than the DAW regime. A comparison between the three types of dust-modes (DCWs, DAWs, and DLWs) has been carried out. 3. Plasma wave experiment for the ISEE-3 mission NASA Technical Reports Server (NTRS) Scarf, F. L. 1983-01-01 An analysis of data from a scientific instrument designed to study solar wind and plasma wave phenomena on the ISEE-3 Mission is provided. Work on the data analysis phase of the contract from 1 October 1982 through 30 March 1983 is summarized. 4. New Observation of Wave Excitation and Inverse Cascade in the Foreshock Region He, Jiansen; Duan, Die; Yan, Limei; Huang, Shiyong; Tu, Chuanyi; Marsch, Eckart; Wang, Linghua; Tian, Hui 2016-04-01 Foreshock with nascent plasma turbulence is regarded as a fascinating region to understand the basic plasma physical processes, e.g., wave-particle interactions as well as wave-wave couplings. Although there have been a bunch of intensive studies on this topic, some key clues about the chain of the physical processes still lacks from observations, e.g., the co-existence of upstream energetic particles as the free energy source, excited pump waves as the wave seed, inverse cascaded daughter waves, and scattered energetic particles as the end of nonlinear processes. A relatively comprehensive case study with some new observations is presented in this work. In our case, upstream energetic protons drifting at tens of Alfvén speed with respect to the background plasma protons is observed from 3DP/PESA-High onboard the WIND spacecraft. When looking at the wave magnetic activities, we are surprised to find the co-existence of high-frequency (0.1-0.5 Hz) large-amplitude right-hand polarized (RHP) waves and low-frequency (0.02-0.1 Hz) small-amplitude left-hand polarized (LHP) waves in the spacecraft (SC) frame. The anti-correlation between magnetic and velocity fluctuations along with the sunward magnetic field direction indicates the low-frequency LHP waves in the SC frame is in fact the sunward upstream RHP waves in the solar wind frame. This new observation lays solid foundation for the applicability of plasma non-resonance instability theory and inverse cascade theory to the foreshock region, in which the downstream high-frequency RHP pump waves are excited by the upstream reflected energetic protons through non-resonance instability and low-frequency RHP daughter waves are generated by the pump waves due to nonlinear parametric decay. The weak signal of alpha particle flux in the foreshock region concerned is also favorable to the occurrence of nonlinear decay process. Furthermore, enhanced downstream energetic proton fluxes are found and inferred to be scattered by 5. Dust-acoustic solitary waves in a four-component adiabatic magnetized dusty plasma SciTech Connect Akhter, T. Mannan, A.; Mamun, A. A. 2013-07-15 Theoretical investigation has been made on obliquely propagating dust-acoustic (DA) solitary waves (SWs) in a magnetized dusty plasma which consists of non-inertial adiabatic electron and ion fluids, and inertial negatively as well as positively charged adiabatic dust fluids. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation which admits a solitary wave solution for small but finite amplitude limit. It has been shown that the basic features (speed, height, thickness, etc.) of such DA solitary structures are significantly modified by adiabaticity of plasma fluids, opposite polarity dust components, and the obliqueness of external magnetic field. The SWs have been changed from compressive to rarefactive depending on the value of {mu} (a parameter determining the number of positive dust present in this plasma model). The present investigation can be of relevance to the electrostatic solitary structures observed in various dusty plasma environments (viz. cometary tails, upper mesosphere, Jupiter's magnetosphere, etc.) 6. Laser-wakefield acceleration of monoenergetic electron beams in the first plasma-wave period. PubMed Mangles, S P D; Thomas, A G R; Kaluza, M C; Lundh, O; Lindau, F; Persson, A; Tsung, F S; Najmudin, Z; Mori, W B; Wahlström, C-G; Krushelnick, K 2006-06-02 Beam profile measurements of laser-wakefield accelerated electron bunches reveal that in the monoenergetic regime the electrons are injected and accelerated at the back of the first period of the plasma wave. With pulse durations ctau >or= lambda(p), we observe an elliptical beam profile with the axis of the ellipse parallel to the axis of the laser polarization. This increase in divergence in the laser polarization direction indicates that the electrons are accelerated within the laser pulse. Reducing the plasma density (decreasing ctau/lambda(p)) leads to a beam profile with less ellipticity, implying that the self-injection occurs at the rear of the first period of the plasma wave. This also demonstrates that the electron bunches are less than a plasma wavelength long, i.e., have a duration <25 fs. This interpretation is supported by 3D particle-in-cell simulations. 7. Generation of Quasi-monoenergetic High-energy Electron Beam by Plasma Wave SciTech Connect Koyama, K.; Saito, N.; Ogata, A.; Masuda, S.; Tanimoto, M.; Miura, E.; Kato, S.; Adachi, M 2004-12-07 We have demonstrated an acceleration of a quasi-monoenergetic electron beam by trapping electrons in a plasma wave. Experiments were performed by focusing 2-TW (50 fs) laser pulses on supersonic gas jet targets. An intensity was 5 x 1018W/cm2(a0 = 1.5). An electron density was estimated to be 1.3 x 1020cm-3. The quasi-monoenergetic electron beam at 7 MeV was observed with a peak to foot ratio of 10. An appearance of a Stokes Raman satellite in the forward scattering well correlated with the quasi-monoenergetic electron beam. A frequency shift of the satellite coincided with a plasma frequency at the measured plasma density. Appearance of the Raman satellite coincided with appearances of a fishbone structure in a side-scattering image. Supposing the fishbone structure originated from the plasma wave, an acceleration length was estimated to be 200 to 500 microns. 8. Dynamics of Rocky Mountain Lee Waves Observed During Success NASA Technical Reports Server (NTRS) Dean-Day, J.; Chan, K. R.; Bowen, S. W.; Bui, T. P.; Gary, B. L.; Chan, K. Roland (Technical Monitor) 1997-01-01 On two days during SUCCESS, the DC-8 sampled wave clouds which formed downstream of the ridges east of the Rocky Mountains. Wave morphology for both flights is deduced from temperature and 3-dimensional wind measurements from the MMS, isentrope profiles from the MTP, and linear perturbation theory. The waves observed on 960430 are smaller and found to be decaying with altitude, while the waves sampled on 960502 are vertically propagating and consist of larger, multiple wave scales. Wave orientations are consistent with the underlying topography and regions of high ice crystal concentration. Updraft velocities were estimated from the derived wave properties and are consistent with MMS vertical winds. 9. Revisiting linear plasma waves for finite value of the plasma parameter Grismayer, Thomas; Fahlen, Jay; Decyk, Viktor; Mori, Warren 2010-11-01 We investigate through theory and PIC simulations the Landau-damping of plasma waves with finite plasma parameter. We concentrate on the linear regime, γφB, where the waves are typically small and below the thermal noise. We simulate these condition using 1,2,3D electrostatic PIC codes (BEPS), noting that modern computers now allow us to simulate cases where (nλD^3 = [1e2;1e6]). We study these waves by using a subtraction technique in which two simulations are carried out. In the first, a small wave is initialized or driven, in the second no wave is excited. The results are subtracted to provide a clean signal that can be studied. As nλD^3 is decreased, the number of resonant electrons can be small for linear waves. We show how the damping changes as a result of having few resonant particles. We also find that for small nλD^3 fluctuations can cause the electrons to undergo collisions that eventually destroy the initial wave. A quantity of interest is the the life time of a particular mode which depends on the plasma parameter and the wave number. The life time is estimated and then compared with the numerical results. A surprising result is that even for large values of nλD^3 some non-Vlasov discreteness effects appear to be important. 10. Dust gravitational drift wave in complex plasma under gravity SciTech Connect Salahshoor, M. Niknam, A. R. 2014-12-15 The dispersion relation of electrostatic waves in a complex plasma under gravity is presented. It is assumed that the waves propagate parallel to the external fields. The effects of weak electric field, neutral drag force, and ion drag force are also taken into account. The dispersion relation is numerically examined in an appropriate parameter space in which the gravity plays the dominant role in the dynamics of microparticles. The numerical results show that, in the low pressure complex plasma under gravity, a low frequency drift wave can be developed in the long wavelength limit. The stability state of this wave is switched at a certain critical wavenumber in such a way that the damped mode is transformed into a growing one. Furthermore, the influence of the external fields on the dispersion properties is analyzed. It is shown that the wave instability is essentially due to the electrostatic streaming of plasma particles. It is also found that by increasing the electric field strength, the stability switching occurs at smaller wavenumbers. 11. Dust-acoustic rogue waves in a nonextensive plasma SciTech Connect Moslem, W. M.; Shukla, P. K.; Sabry, R.; El-Labany, S. K. 2011-12-15 We present an investigation for the generation of a dust-acoustic rogue wave in a dusty plasma composed of negatively charged dust grains, as well as nonextensive electrons and ions. For this purpose, the reductive perturbation technique is used to obtain a nonlinear Schroedinger equation. The critical wave-number threshold k{sub c}, which indicates where the modulational instability sets in, has been determined precisely for various regimes. Two different behaviors of k{sub c} against the nonextensive parameter q are found. For small k{sub c}, it is found that increasing q would lead to an increase of k{sub c} until q approaches a certain value q{sub c}, then further increase of q beyond q{sub c} decreases the value of k{sub c}. For large k{sub c}, the critical wave-number threshold k{sub c} is always increasing with q. Within the modulational instability region, a random perturbation of the amplitude grows and thus creates dust-acoustic rogue waves. In order to show that the characteristics of the rogue waves are influenced by the plasma parameters, the relevant numerical analysis of the appropriate nonlinear solution is presented. The nonlinear structure, as reported here, could be useful for controlling and maximizing highly energetic pulses in dusty plasmas. 12. Dust-acoustic rogue waves in a nonextensive plasma. PubMed Moslem, W M; Sabry, R; El-Labany, S K; Shukla, P K 2011-12-01 We present an investigation for the generation of a dust-acoustic rogue wave in a dusty plasma composed of negatively charged dust grains, as well as nonextensive electrons and ions. For this purpose, the reductive perturbation technique is used to obtain a nonlinear Schrödinger equation. The critical wave-number threshold k(c), which indicates where the modulational instability sets in, has been determined precisely for various regimes. Two different behaviors of k(c) against the nonextensive parameter q are found. For small k(c), it is found that increasing q would lead to an increase of k(c) until q approaches a certain value q(c), then further increase of q beyond q(c) decreases the value of k(c). For large k(c), the critical wave-number threshold k(c) is always increasing with q. Within the modulational instability region, a random perturbation of the amplitude grows and thus creates dust-acoustic rogue waves. In order to show that the characteristics of the rogue waves are influenced by the plasma parameters, the relevant numerical analysis of the appropriate nonlinear solution is presented. The nonlinear structure, as reported here, could be useful for controlling and maximizing highly energetic pulses in dusty plasmas. 13. Cluster observations of electrostatic solitary waves near the Earth's bow shock Hobara, Y.; Walker, S. N.; Balikhin, M.; Pokhotelov, O. A.; Gedalin, M.; Krasnoselskikh, V.; Hayakawa, M.; André, M.; Dunlop, M.; RèMe, H.; Fazakerley, A. 2008-05-01 consistent with the BGK (Bernstein-Greene-Kruskal) ion holes. The two classes of observed solitary waves may greatly influence the ambient plasma dynamics around the shock. The bipolar solitary waves do not exhibit a large net potential difference but may still play an important role in plasma thermalisation by particle scattering. Unipolar/tripolar solitary waves exhibit a remarkable net potential difference that may be responsible for the plasma energisation along the ambient magnetic field. 14. Observation of a hierarchy of up to fifth-order rogue waves in a water tank. PubMed Chabchoub, A; Hoffmann, N; Onorato, M; Slunyaev, A; Sergeeva, A; Pelinovsky, E; Akhmediev, N 2012-11-01 We present experimental observations of the hierarchy of rational breather solutions of the nonlinear Schrödinger equation (NLS) generated in a water wave tank. First, five breathers of the infinite hierarchy have been successfully generated, thus confirming the theoretical predictions of their existence. Breathers of orders higher than five appeared to be unstable relative to the wave-breaking effect of water waves. Due to the strong influence of the wave breaking and relatively small carrier steepness values of the experiment these results for the higher-order solutions do not directly explain the formation of giant oceanic rogue waves. However, our results are important in understanding the dynamics of rogue water waves and may initiate similar experiments in other nonlinear dispersive media such as fiber optics and plasma physics, where the wave propagation is governed by the NLS. 15. Dust kinetic Alfvén solitary and rogue waves in a superthermal dusty plasma SciTech Connect Saini, N. S. Singh, Manpreet; Bains, A. S. 2015-11-15 Dust kinetic Alfvén solitary waves (DKASWs) have been examined in a low-β dusty plasma comprising of negatively charged dust grains, superthermal electrons, and ions. A nonlinear Korteweg-de Vries (KdV) equation has been derived using the reductive perturbation method. The combined effects of superthermality of charged particles (via κ), plasma β, obliqueness of propagation (θ), and dust concentration (via f) on the shape and size of the DKASWs have been examined. Only negative potential (rarefactive) structures are observed. Further, characteristics of dust kinetic Alfvén rogue waves (DKARWs), by deriving the non-linear Schrödinger equation (NLSE) from the KdV equation, are studied. Rational solutions of NLSE show that rogue wave envelopes are supported by this plasma model. It is observed that the influence of various plasma parameters (superthermality, plasma β, obliqueness, and dust concentration) on the characteristics of the DKARWs is very significant. This fundamental study may be helpful in understanding the formation of coherent nonlinear structures in space and astrophysical plasma environments where superthermal particles are present. 16. Dust kinetic Alfvén solitary and rogue waves in a superthermal dusty plasma Saini, N. S.; Singh, Manpreet; Bains, A. S. 2015-11-01 Dust kinetic Alfvén solitary waves (DKASWs) have been examined in a low-β dusty plasma comprising of negatively charged dust grains, superthermal electrons, and ions. A nonlinear Korteweg-de Vries (KdV) equation has been derived using the reductive perturbation method. The combined effects of superthermality of charged particles (via κ), plasma β, obliqueness of propagation (θ), and dust concentration (via f) on the shape and size of the DKASWs have been examined. Only negative potential (rarefactive) structures are observed. Further, characteristics of dust kinetic Alfvén rogue waves (DKARWs), by deriving the non-linear Schrödinger equation (NLSE) from the KdV equation, are studied. Rational solutions of NLSE show that rogue wave envelopes are supported by this plasma model. It is observed that the influence of various plasma parameters (superthermality, plasma β, obliqueness, and dust concentration) on the characteristics of the DKARWs is very significant. This fundamental study may be helpful in understanding the formation of coherent nonlinear structures in space and astrophysical plasma environments where superthermal particles are present. 17. Gravity wave amplitudes changes observed in different airglow emissions: influence of wave breaking and observational selection Schmidt, Carsten; Wüst, Sabine; Hannawald, Patrick; Bittner, Michael 2016-04-01 The upper mesosphere lower thermosphere region is well known for enhanced gravity wave breaking. Airglow emissions originating in this height region provide a good possibility for detailed studies of gravity wave behavior in this altitude. Therefore, rotational temperatures and intensities of the OH(3-1), OH(4-2), OH(6-2) and O2b(0-1)-transitions recorded at the NDMC (Network for the Detection of Mesospheric Change) site Oberpfaffenhofen (48.1°N, 10.3°E), Germany are examined. First results indicate, that both significant amplitude growth from the lower (~87km) OH airglow emissions to the higher (~95km) O2 airglow emissions of more than 100% as well as strong damping can be observed. On several occasions OH- and O2-emissions show completely independent behavior - probably related to the complete breakup of a gravity wave. These amplitude changes are set into relation to emission layer height, vertical wavelength, absolute temperature and potential seasonal dependence. Observations from further NDMC sites in France, Germany and Austria are used to discuss the evolution of these waves on horizontal scales from 100km to 1000km. 18. Excitation of electrostatic waves in the electron cyclotron frequency range during magnetic reconnection in laboratory overdense plasmas SciTech Connect Kuwahata, A.; Igami, H.; Kawamori, E.; Kogi, Y.; Inomoto, M.; Ono, Y. 2014-10-15 We report the observation of electromagnetic radiation at high harmonics of the electron cyclotron frequency that was considered to be converted from electrostatic waves called electron Bernstein waves (EBWs) during magnetic reconnection in laboratory overdense plasmas. The excitation of EBWs was attributed to the thermalization of electrons accelerated by the reconnection electric field around the X-point. The radiative process discussed here is an acceptable explanation for observed radio waves pulsation associated with major flares. 19. Nonextensive dust acoustic waves in a charge varying dusty plasma Bacha, Mustapha; Tribeche, Mouloud 2012-01-01 Our recent analysis on nonlinear nonextensive dust-acoustic waves (DA) [Amour and Tribeche in Phys. Plasmas 17:063702, 2010] is extended to include self-consistent nonadiabatic grain charge fluctuation. The appropriate nonextensive electron charging current is rederived based on the orbit-limited motion theory. Our results reveal that the amplitude, strength and nature of the nonlinear DA waves (solitons and shocks) are extremely sensitive to the degree of ion nonextensivity. Stronger is the electron correlation, more important is the charge variation induced nonlinear wave damping. The anomalous dissipation effects may prevail over that dispersion as the electrons evolve far away from their Maxwellian equilibrium. Our investigation may be of wide relevance to astronomers and space scientists working on interstellar dusty plasmas where nonthermal distributions are turning out to be a very common and characteristic feature. 20. Solitary and freak waves in superthermal plasma with ion jet Abdelsalam, U. M.; Abdelsalam 2013-06-01 The nonlinear solitary and freak waves in a plasma composed of positive and negative ions, superthermal electrons, ion beam, and stationary dust particles have been investigated. The reductive perturbation method is used to obtain the Korteweg-de Vries (KdV) equation describing the system. The latter admits solitary wave solution, while the dynamics of the modulationally unstable wavepackets described by the KdV equation gives rise to the formation of freak/rogue excitation described by the nonlinear Schrödinger equation. In order to show that the characteristics of solitary and freak waves are influenced by plasma parameters, relevant numerical analysis of appropriate nonlinear solutions are presented. The results from this work predict nonlinear excitations that may associate with ion jet and superthermal electrons in Herbig-Haro objects. 1. Slow Mode Waves in the Heliospheric Plasma Sheet NASA Technical Reports Server (NTRS) Smith, Edward. J.; Zhou, Xiaoyan 2007-01-01 We report the results of a search for waves/turbulence in the Heliospheric Plasma Sheet (HPS) surrounding the Heliospheric Current Sheet (HCS). The HPS is treated as a distinctive heliospheric structure distinguished by relatively high Beta, slow speed plasma. The data used in the investigation are from a previously published study of the thicknesses of the HPS and HCS that were obtained in January to May 2004 when Ulysses was near aphelion at 5 AU. The advantage of using these data is that the HPS is thicker at large radial distances and the spacecraft spends longer intervals inside the plasma sheet. From the study of the magnetic field and solar wind velocity components, we conclude that, if Alfven waves are present, they are weak and are dominated by variations in the field magnitude, B, and solar wind density, NP, that are anti-correlated. 2. Wave-Wave Interactions in the Stratosphere: Observations during Quiet and Active Wintertime Periods. Smith, Anne K.; Gille, John C.; Lyjak, Lawrence V. 1984-02-01 Using satellite data from the Nimbus 7 LIMS instrument, a previous study by Smith showed that interactions among planetary waves 1, 2 and 3 in the stratosphere were significant during January 1979. That month was characterized by an exceptionally large wave 1 amplitude in the stratosphere. The present study extends the analysis to the period November 1978-March 1979 to determine the conditions under which wave-wave interactions have a significant effect on variations in wave activity and on wave-mean flow interactions. A quantitative measure of how wave-wave interactions affect the wave activity of zonal waves 1 and 2 is obtained from the potential enstrophy budget.The results demonstrate that the relative importance of wave-wave versus wave-mean flow interactions depends on the magnitude of the eddy mean wind and potential vorticity relative to the zonal means. When the zonal mean wind is weak, a relatively small amplitude wave tends to behave nonlinearly, whereas when the mean wind is strong, only large amplitude waves are significantly nonlinear. In the 1978-79 winter, the zonal mean wind was weaker and wave-wave interactions were more important in middle and late winter than during November-December.Further evidence is presented that the vacillation between waves 1 and 2, which has been observed in the winter stratosphere of both hemispheres, is as strongly influenced by wave-wave interactions in the stratosphere as by variations in the forcing from the troposphere. 3. Electrostatic surface waves on a plasma with non-uniform boundary 1990-10-01 A new analytical method is introduced to consider electrostatic surface waves propagating on a cold plasma. A very simple dispersion relation is derived for a plasma bounded by two dielectrics. Previous theory for solitary surface waves is also generalized. 4. Multisolitary plasma surface waves in the presence of an external pump field 1991-11-01 It is shown that an external electromagnetic pump wave, which interacts with the electrostatic surface oscillations in a semi-infinite plasma with a sharp boundary, can excite a sequence of solitary waves on the plasma surface. 5. Supersonic propagation of ionization waves in an under-dense, laser-produced plasma SciTech Connect Constantin, C; Back, C A; Fournier, K B; Gregori, G; Landen, O L; Glenzer, S H; Dewald, E L; Miller, M C 2004-10-22 We observe a laser-driven supersonic ionization wave heating a mm-scale plasma of sub-critical density up to 2-3 keV electron temperatures. Propagation velocities initially 10 times the sound speed were measured by means of time-resolved x-ray imaging diagnostics. The measured ionization wave trajectory is modeled analytically and by a 2D radiation-hydrodynamics code. The comparison to the modeling suggests that nonlocal heat transport effects may contribute to the attenuation of the heat wave propagation. 6. Ion-acoustic solitons, double layers and rogue waves in plasma having superthermal electrons Singh Saini, Nareshpal 2016-07-01 Most of the space and astrophysical plasmas contain different type of charged particles with non-Maxwellian velocity distributions (e.g., nonthermal, superthermal, Tsallis ). These distributions are commonly found in the auroral region of the Earth's magnetosphere, planetary magnetosphere, solar and stellar coronas, solar wind, etc. The observations from various satellite missions have confirmed the presence of superthermal particles in space and astrophysical environments. Over the last many years, there have been a much interest in studying the different kind of properties of the electrostatic nonlinear excitations (solitons, double layers, rogue waves etc.) in a multi-component plasmas in the presence of superthermal particles. It has been analyzed that superthermal distributions are more appropriate than Maxwellian distribution for the modeling of space data. It is interesting to study the dynamics of various kinds of solitary waves, Double layers, Shocks etc. in varieties of plasma systems containing different kind of species obeying Lorentzian (kappa-type)/Tsallis distribution. In this talk, I have focused on the study of large amplitude IA solitary structures (bipolar solitary structures, double layers etc.), modulational instability and rogue waves in multicomponent plasmas. The Sagdeev potential method has been employed to setup an energy balance equation, from which we have studied the characteristics of large amplitude solitary waves under the influence of superthermality of charged particles and other plasma parameters. The critical Mach number has been determined, above which solitary structures are observed and its variation with superthermality of electrons and other parameters has also been discussed. Double layers have also been discussed. Multiple scale reductive perturbation method has been employed to derive NLS equation. From the different kind of solutions of this equation, amplitude modulation of envelope solitons and rogue waves have been 7. {Interball-1 Plasma, Magnetic Field, and Energetic Particle Observations} NASA Technical Reports Server (NTRS) Sibeck, David G. 1998-01-01 Funding from NASA was received in two installments. The first installment supported research using Russian/Czech/Slovak/French Interball-1 plasma, magnetic field, and energetic particles observations in the vicinity of the magnetopause. The second installment provided salary support to review unsolicited proposals to NASA for data recovery and archiving, and also to survey ISTP data provision efforts. Two papers were published under the auspices of the grant. Sibeck et al. reported Interball-1 observations of a wave on the magnetopause with an amplitude in excess of 5 R(sub E), the largest ever reported to date. They attributed the wave to a hot flow anomaly striking the magnetopause and suggested that the hot flow anomaly itself formed during the interaction of an IMF discontinuity with the bow shock. Nemecek et al. used Interball-1's VDP Faraday cup to identify large transient increases in the magnetosheath density. They noted large variations in simultaneous Wind observations of the IMF cone angle, but were unable to establish any relationship between the cone angle variations at Wind and the density variations at Interball-1. Funds from the second installment were used to review over 20 proposals from various researchers in the scientific community who sought NASA support to restore or archive past observations. It also supported a survey of ISTP data provisions which was used as input to a Senior Review of ongoing NASA ISTP programs. 8. TG wave autoresonant control of plasma temperature SciTech Connect Kabantsev, A. A. Driscoll, C. F. 2015-06-29 The thermal correction term in the Trivelpiece-Gould (TG) wave’s frequency has been used to accurately control the temperature of electron plasma, by applying a swept-frequency continuous drive autoresonantly locked in balance with the cyclotron cooling. The electron temperature can be either “pegged” at a desired value (by constant drive frequency); or varied cyclically (following the tailored frequency course), with rates limited by the cooling time (on the way down) and by chosen drive amplitude (on the way up) 9. Statistical Features of EMIC Waves Observed on Van Allen Probes in the Inner Magnetosphere Lee, D. Y.; Roh, S. J.; Cho, J.; Shin, D. K.; Hwang, J.; Kim, K. C.; Kurth, W. S.; Kletzing, C.; Wygant, J. R.; Thaller, S. A. 2015-12-01 Electromagnetic ion cyclotron (EMIC) waves are one of the key plasma waves that can affect charged particle dynamics in the Earth's inner magnetosphere. Knowledge of global distribution of the EMIC waves is critical for accurately assessing the significance of its interaction with charged particles. With the Van Allen Probes EMFISIS observations, we have surveyed EMIC events for ~2.5 years period. We have identified well-defined, banded wave activities only, as distinguished from broad band wave activities. We have obtained global distribution of occurrence of the identified waves with distinction between H- and He-bands. We compare it with previous observations such as THEMIS and CRRES. For the identified events we have drawn all the basic wave properties including wave frequency, polarization, wave normal angle. In addition, we have distinguished the EMIC events that occur inside the plasmasphere and at the plasmapause from those outside the plasmasphere. Finally, we have tested solar wind and geomagnetic dependence of the wave events. We give discussions about implications of these observations on wave generation mechanism and interaction with radiation belt electrons. 10. Nonlinear Alfvén wave dynamics in plasmas Sarkar, Anwesa; Chakrabarti, Nikhil; Schamel, Hans 2015-07-01 Nonlinear Alfvén wave dynamics is presented using Lagrangian fluid approach in a compressible collisional magnetized plasma. In the framework of two fluid dynamics, finite electron inertia is shown to serve as a dispersive effect acting against the convective nonlinearity. In a moving frame, the Alfvén wave can, therefore, form an arbitrarily strong amplitude solitary wave structure due to the balance between nonlinearity and dispersion. Weak amplitude Alfvén waves are shown to be governed by a modified KdV equation, which extends for finite dissipation to a mKdV-Burgers equation. These equations have well known solutions. Next, we have analyzed the fourth order nonlinear Alfvén wave system of equations both numerically and by approximation method. The results indicate a collapse of the density and magnetic field irrespective of the presence of dispersion. The wave magnetic field, however, appears to be less singular showing collapse only when the dispersive effects are negligible. These results may contribute to our understanding of the generation of strongly localized magnetic fields (and currents) in plasmas and are expected to be of special importance in the astrophysical context of magnetic star formation. 11. Bernstein wave aided laser third harmonic generation in a plasma Tyagi, Yachna; Tripathi, Deepak; Kumar, Ashok 2016-09-01 The process of Bernstein wave aided resonant third harmonic generation of laser in a magnetized plasma is investigated. The extra-ordinary mode (X-mode) laser of frequency ω 0 and wave number k → 0 , travelling across the magnetic field in a plasma, exerts a second harmonic ponderomotive force on the electrons imparting them an oscillatory velocity v → 2 ω0 , 2 k → 0 . This velocity beats with the density perturbation due to the Bernstein wave to produce a density perturbation at cyclotron frequency shifted second harmonic. The density perturbation couples with the oscillatory velocity v → ω0 , k → 0 of X-mode of the laser to produce the cyclotron frequency shifted third harmonic current density leading to harmonic radiation. The phase matching condition for the up shifted frequency is satisfied when the Bernstein wave is nearly counter-propagating to the laser. As the transverse wave number of the Bernstein wave is large, it is effective in the phase matched third harmonic generation, when the laser frequency is not too far from the upper hybrid frequency. 12. Two-dimensional s-polarized solitary waves in relativistic plasmas. I. The fluid plasma model SciTech Connect Sanchez-Arriaga, G.; Lefebvre, E. 2011-09-15 The properties of two-dimensional linearly s-polarized solitary waves are investigated by fluid-Maxwell equations and particle-in-cell (PIC) simulations. These self-trapped electromagnetic waves appear during laser-plasma interactions, and they have a dominant electric field component E{sub z}, normal to the plane of the wave, that oscillates at a frequency below the electron plasma frequency {omega}{sub pe}. A set of equations that describe the waves are derived from the plasma fluid model in the case of cold or warm plasma and then solved numerically. The main features, including the maximum value of the vector potential amplitude, the total energy, the width, and the cavitation radius are presented as a function of the frequency. The amplitude of the vector potential increases monotonically as the frequency of the wave decreases, whereas the width reaches a minimum value at a frequency of the order of 0.82 {omega}{sub pe}. The results are compared with a set of PIC simulations where the solitary waves are excited by a high-intensity laser pulse. 13. Geotail MCA Plasma Wave Investigation Data Analysis NASA Technical Reports Server (NTRS) Anderson, Roger R. 1996-01-01 The goals of this program include identifying, studying, and understanding the source, movement, and dissipation of plasma mass, momentum, and energy between the Sun and Earth. The GEOTAIL spacecraft was built by the Japanese Institute of Space and Aeronautical Science and has provided extensive measurements of entry, storage, acceleration, and transport in the geomagnetic tail. Due to the GEOTAIL trajectory, which kept the spacecraft passing into the deep tail, GEOTAIL also made 'magnetopause skimming passes' which allowed measurements in the outer magnetosphere, magnetopause, bow shock, and upstream solar wind regions as well as in the lobe, magnetosheath, boundary layers, and central plasma sheet regions of the tail. In late 1994, after spending nearly 30 months primarily traversing the deep tail region, GEOTAIL began its near Earth phase where apogee was reduced first to about 50 Re and later to 30 Re and perigee was decreased to about 10 Re. The WIND spacecraft was launched on November 1, 1994, and the POLAR spacecraft was launched on February 24, 1996. These successful launches have dramatically increased the opportunities for GEOTAIL and the GGS spacecraft to conduct global research. 14. Relativistic warm plasma theory of nonlinear laser-driven electron plasma waves. PubMed Schroeder, C B; Esarey, E 2010-05-01 A relativistic, warm fluid model of a nonequilibrium, collisionless plasma is developed and applied to examine nonlinear Langmuir waves excited by relativistically intense, short-pulse lasers. Closure of the covariant fluid theory is obtained via an asymptotic expansion assuming a nonrelativistic plasma temperature. The momentum spread is calculated in the presence of an intense laser field and shown to be intrinsically anisotropic. Coupling between the transverse and longitudinal momentum variances is enabled by the laser field. A generalized dispersion relation is derived for Langmuir waves in a thermal plasma in the presence of an intense laser field. Including thermal fluctuations in three-velocity-space dimensions, the properties of the nonlinear electron plasma wave, such as the plasma temperature evolution and nonlinear wavelength, are examined and the maximum amplitude of the nonlinear oscillation is derived. The presence of a relativistically intense laser pulse is shown to strongly influence the maximum plasma wave amplitude for nonrelativistic phase velocities owing to the coupling between the longitudinal and transverse momentum variances. 15. Relativistic warm plasma theory of nonlinear laser-driven electron plasma waves SciTech Connect Schroeder, Carl B.; Esarey, Eric 2010-06-30 A relativistic, warm fluid model of a nonequilibrium, collisionless plasma is developed and applied to examine nonlinear Langmuir waves excited by relativistically-intense, short-pulse lasers. Closure of the covariant fluid theory is obtained via an asymptotic expansion assuming a non-relativistic plasma temperature. The momentum spread is calculated in the presence of an intense laser field and shown to be intrinsically anisotropic. Coupling between the transverse and longitudinal momentum variances is enabled by the laser field. A generalized dispersion relation is derived for langmuir waves in a thermal plasma in the presence of an intense laser field. Including thermal fluctuations in three velocity-space dimensions, the properties of the nonlinear electron plasma wave, such as the plasma temperature evolution and nonlinear wavelength, are examined, and the maximum amplitude of the nonlinear oscillation is derived. The presence of a relativistically intense laser pulse is shown to strongly influence the maximum plasma wave amplitude for non-relativistic phase velocities owing to the coupling between the longitudinal and transverse momentum variances. 16. Reduction and analysis of data from the plasma wave instruments on the IMP-6 and IMP-8 spacecraft NASA Technical Reports Server (NTRS) Gurnett, D. A.; Anderson, R. R. 1983-01-01 The primary data reduction effort during the reporting period was to process summary plots of the IMP 8 plasma wave data and to submit these data to the National Space Science Data Center. Features of the electrostatic noise are compared with simultaneous observations of the magnetic field, plasma and energetic electrons. Spectral characteristics of the noise and the results of this comparison both suggest that in its high frequency part at least the noise does not belong to normal modes of plasma waves but represents either quasi-thermal noise in the non-Maxwellian plasma or artificial noise generated by spacecraft interaction with the medium. 17. Arbitrary amplitude kinetic Alfven solitary waves in two temperature electron superthermal plasma Singh, Manpreet; Singh Saini, Nareshpal; Ghai, Yashika 2016-07-01 Through various satellite missions it is observed that superthermal velocity distribution for particles is more appropriate for describing space and astrophysical plasmas. So it is appropriate to use superthermal distribution, which in the limiting case when spectral index κ is very large ( i.e. κ→∞), shifts to Maxwellian distribution. Two temperature electron plasmas have been observed in auroral regions by FAST satellite mission, and also by GEOTAIL and POLAR satellite in the magnetosphere. Kinetic Alfven waves arise when finite Larmor radius effect modifies the dispersion relation or characteristic perpendicular wavelength is comparable to electron inertial length. We have studied the kinetic Alfven waves (KAWs) in a plasma comprising of positively charged ions, superthermal hot electrons and Maxwellian distributed cold electrons. Sagdeev pseudo-potential has been employed to derive an energy balance equation. The critical Mach number has been determined from the expression of Sagdeev pseudo-potential to see the existence of solitary structures. It is observed that sub-Alfvenic compressive solitons and super-Alfvenic rarefactive solitons exist in this plasma model. It is also observed that various parameters such as superthermality of hot electrons, relative concentration of cold and hot electron species, Mach number, plasma beta, ion to cold electron temperature ratio and ion to hot electron temperature ratio have significant effect on the amplitude and width of the KAWs. Findings of this investigation may be useful to understand the dynamics of coherent non-linear structures (i.e. KAWs) in space and astrophysical plasmas. 18. Observation of cavitation during shock wave lithotripsy Bailey, Michael R.; Crum, Lawrence A.; Pishchalnikov, Yuri A.; McAteer, James A.; Pishchalnikova, Irina V.; Evan, Andrew P.; Sapozhnikov, Oleg A.; Cleveland, Robin O. 2005-04-01 A system was built to detect cavitation in pig kidney during shock wave lithotripsy (SWL) with a Dornier HM3 lithotripter. Active detection, using echo on B-mode ultrasound, and passive cavitation detection (PCD), using coincident signals on confocal, orthogonal receivers, were equally sensitive and were used to interrogate the renal collecting system (urine) and the kidney parenchyma (tissue). Cavitation was detected in urine immediately upon SW administration in urine or urine plus X-ray contrast agent, but in tissue, cavitation required hundreds of SWs to initiate. Localization of cavitation was confirmed by fluoroscopy, sonography, and by thermally marking the kidney using the PCD receivers as high intensity focused ultrasound sources. Cavitation collapse times in tissue and native urine were about the same but less than in urine after injection of X-ray contrast agent. Cavitation, especially in the urine space, was observed to evolve from a sparse field to a dense field with strong acoustic collapse emissions to a very dense field that no longer produced detectable collapse. The finding that cavitation occurs in kidney tissue is a critical step toward determining the mechanisms of tissue injury in SWL. [Work sup ported by NIH (DK43881, DK55674, FIRCA), ONRIFO, CRDF and NSBRI SMS00203. 19. Collisional damping of helicon waves in a high density hydrogen linear plasma device Caneses, Juan F.; Blackwell, Boyd D. 2016-10-01 In this paper, we investigate the propagation and damping of helicon waves along the length (50 cm) of a helicon-produced 20 kW hydrogen plasma ({{n}\\text{e}}˜ 1-2 × 1019 m-3, {{T}\\text{e}}˜ 1-6 eV, H2 8 mTorr) operated in a magnetic mirror configuration (antenna region: 50-200 G and mirror region: 800 G). Experimental results show the presence of traveling helicon waves (4-8 G and {λz}˜ 10-15 cm) propagating away from the antenna region which become collisionally absorbed within 40-50 cm. We describe the use of the WKB method to calculate wave damping and provide an expression to assess its validity based on experimental measurements. Theoretical calculations are consistent with experiment and indicate that for conditions where Coulomb collisions are dominant classical collisionality is sufficient to explain the observed wave damping along the length of the plasma column. Based on these results, we provide an expression for the scaling of helicon wave damping relevant to high density discharges and discuss the location of surfaces for plasma-material interaction studies in helicon based linear plasma devices. 20. Quenching Plasma Waves in Two Dimensional Electron Gas by a Femtosecond Laser Pulse Shur, Michael; Rudin, Sergey; Greg Rupper Collaboration; Andrey Muraviev Collaboration Plasmonic detectors of terahertz (THz) radiation using the plasma wave excitation in 2D electron gas are capable of detecting ultra short THz pulses. To study the plasma wave propagation and decay, we used femtosecond laser pulses to quench the plasma waves excited by a short THz pulse. The femtosecond laser pulse generates a large concentration of the electron-hole pairs effectively shorting the 2D electron gas channel and dramatically increasing the channel conductance. Immediately after the application of the femtosecond laser pulse, the equivalent circuit of the device reduces to the source and drain contact resistances connected by a short. The total response charge is equal to the integral of the current induced by the THz pulse from the moment of the THz pulse application to the moment of the femtosecond laser pulse application. This current is determined by the plasma wave rectification. Registering the charge as a function of the time delay between the THz and laser pulses allowed us to follow the plasmonic wave decay. We observed the decaying oscillations in a sample with a partially gated channel. The decay depends on the gate bias and reflects the interplay between the gated and ungated plasmons in the device channel. Army Research Office. 1. Extensions of 1d Bgk Electron Solitary Wave Solutions To 3d Magnetized and Unmagnetized Plasmas Chen, Li-Jen; Parks, George K. This paper will compare the key results for BGK electron solitary waves in 3D mag- netized and unmagnetized plasmas. For 3D magnetized plasmas with highly magnetic field-aligned electrons, our results predict that the parallel widths of the solitary waves can be smaller than one Debye length, the solitary waves can be large scale features of the magnetosphere, and the parallel width-amplitude relation has a dependence on the perpendicular size. We can thus obtain an estimate on the typical perpendicular size of the observed solitary waves assuming a series of consecutive solitary waves are in the same flux tude with a particular perpendicular span. In 3D unmagnetized plasma systems such as the neutral sheet and magnetic reconnection sites, our theory indi- cates that although mathematical solutions can be constructed as the time-stationary solutions for the nonlinear Vlasov-Poisson equations, there does not exist a param- eter range for the solutions to be physical. We conclude that single-humped solitary potential pulses cannot be self-consistently supported by charged particles in 3D un- magnetized plasmas. 2. Excitation of Electron Acoustic Waves in Plasmas of the SINP-MaPLE Device Chowdhury, Satyajit; Biswas, Subir; Chakrabrati, Nikhil; Pal, Rabindranath 2016-10-01 Electron acoustic wave (EAW) is the low frequency branch of the undamped electrostatic plasma wave and has low phase velocity. In order to overcome Landau damping the EAW needs a non-Maxwellian electron velocity distribution with a flat region near the phase velocity, or equivalently, a plasma with two temperature electron species with a relative velocity between them. The ECR produced plasmas of the MaPLE device at Saha Institute of Nuclear Physics provide such characteristics as observed by retarded field energy analyzer and single Langmuir probe. Experiments are carried out to exploit this feature by putting a negatively biased mesh launcher inside the plasma and energizing it with sinusoidal voltages from a function generator with frequencies varying near the ion plasma frequency. Circular mesh probes along the axis of the device serve as detectors for wave propagation. Experimental results show EAWs are indeed launched and propagate along the magnetic field direction. The dispersion curve experimentally obtained shows the phase velocity matching satisfactorily with the estimated theoretical values. Changing the bias on the launcher the electron distribution function is varied, which, in turn, controls the wave amplitude. Detailed experimental results will be presented. Department of Atomic Energy, Govt. of India. 3. Pulsed and continuous wave acrylic acid radio frequency plasma deposits: plasma and surface chemistry. PubMed Voronin, Sergey A; Zelzer, Mischa; Fotea, Catalin; Alexander, Morgan R; Bradley, James W 2007-04-05 Plasma polymers have been formed from acrylic acid using a pulsed power source. An on-pulse duration of 100 micros was used with a range of discharge off-times between 0 (continuous wave) and 20,000 micros. X-ray photoelectron spectroscopy (XPS) has been used in combination with trifluoroethanol (TFE) derivatization to quantify the surface concentration of the carboxylic acid functionality in the deposit. Retention of this functionality from the monomer varied from 2% to 65%. When input power was expressed as the time-averaged energy per monomer molecule, E(mean), the deposit chemistry achieved could be described using a single relationship for all deposition conditions. Deposition rates were monitored using a quartz crystal microbalance, which revealed a range from 20 to 200 microg m(-2) s(-1), and these fell as COOH functional retention increased. The flow rate was found to be the major determinant of the deposition rate, rather than being uniquely defined by E(mean), connected to the rate at which fresh monomer enters the system in the monomer deficient regime. The neutral species were collected in a time-averaged manner. As the energy delivered per molecule in the system (E(mean)) decreased, the amount of intact monomer increased, with the average neutral mass approaching 72 amu as E(mean) tends to zero. No neutral oligomeric species were detected. Langmuir probes have been used to determine the temporal evolution of the density and temperature of the electrons in the plasma and the plasma potential adjacent to the depositing film. It has been found that even 500 micros into the afterglow period that ionic densities are still significant, 5-10% of the on-time density, and that ion accelerating sheath potentials fall from 40 V in the on-time to a few volts in the off-time. We have made the first detailed, time- and energy-resolved mass spectrometry measurements in depositing acrylic acid plasma. These have allowed us to identify and quantify the positive ion 4. Fine Spectral Properties of Langmuir Waves Observed Upstream of the Saturn's Bowshock by the Cassini Wideband Receiver Hospodarsky, G. B.; Pisa, D.; Santolik, O.; Kurth, W. S.; Soucek, J.; Basovnik, M.; Gurnett, D. A.; Arridge, C. S. 2015-12-01 Langmuir waves are commonly observed in the upstream regions of planetary and interplanetary shock. Solar wind electrons accelerated at the shock front are reflected back into the solar wind and can form electron beams. In regions with beams, the electron distribution becomes unstable and electrostatic waves can be generated. The process of generation and the evolution of electrostatic waves strongly depends on the solar wind electron distribution and generally exhibits complex behavior. Langmuir waves can be identified as intense narrowband emission at a frequency very close to the local plasma frequency and weaker broadband waves below and above the plasma frequency deeper in the downstream region. We present a detailed study of Langmuir waves detected upstream of the Saturnian bowshock by the Cassini spacecraft. Using data from the Radio and Plasma Wave Science (RPWS), Magnetometer (MAG) and Cassini Plasma Spectrometer (CAPS) instruments we have analyzed several periods containing the extended waveform captures by the Wideband Receiver. Langmuir waves are a bursty emission highly controlled by variations in solar wind conditions. Unfortunately due to a combination of instrumental field of view and sampling period, it is often difficult to identify the electron distribution function that is unstable and able to generate Langmuir waves. We used an electrostatic version of particle-in-cell simulation of the Langmuir wave generation process to reproduce some of the more subtle observed spectral features and help understand the late stages of the instability and interactions in the solar wind plasma. 5. Magnetospheric Multiscale observations of large-amplitude, parallel, electrostatic waves associated with magnetic reconnection at the magnetopause Ergun, R. E.; Holmes, J. C.; Goodrich, K. A.; Wilder, F. D.; Stawarz, J. E.; Eriksson, S.; Newman, D. L.; Schwartz, S. J.; Goldman, M. V.; Sturner, A. P.; Malaspina, D. M.; Usanova, M. E.; Torbert, R. B.; Argall, M.; Lindqvist, P.-A.; Khotyaintsev, Y.; Burch, J. L.; Strangeway, R. J.; Russell, C. T.; Pollock, C. J.; Giles, B. L.; Dorelli, J. J. C.; Avanov, L.; Hesse, M.; Chen, L. J.; Lavraud, B.; Le Contel, O.; Retino, A.; Phan, T. D.; Eastwood, J. P.; Oieroset, M.; Drake, J.; Shay, M. A.; Cassak, P. A.; Nakamura, R.; Zhou, M.; Ashour-Abdalla, M.; André, M. 2016-06-01 We report observations from the Magnetospheric Multiscale satellites of large-amplitude, parallel, electrostatic waves associated with magnetic reconnection at the Earth's magnetopause. The observed waves have parallel electric fields (E\|\|) with amplitudes on the order of 100 mV/m and display nonlinear characteristics that suggest a possible net E\|\|. These waves are observed within the ion diffusion region and adjacent to (within several electron skin depths) the electron diffusion region. They are in or near the magnetosphere side current layer. Simulation results support that the strong electrostatic linear and nonlinear wave activities appear to be driven by a two stream instability, which is a consequence of mixing cold (<10 eV) plasma in the magnetosphere with warm (~100 eV) plasma from the magnetosheath on a freshly reconnected magnetic field line. The frequent observation of these waves suggests that cold plasma is often present near the magnetopause. 6. Exact Damping for Relativistic Plasma Waves Swanson, D. G. 2000-10-01 The damping coefficient for a relativistic plasma may be reduced to a single integral with no approximations through use of the Newberger sum rules when k_z=0. Expanding the integral in a series, the leading term agrees with the leading term of the weak relativistic function F_7/2(z), but the remaining terms are not alike. The single expansion parameter is proportional to λ z, indicating that the result may NOT be accurately expressed as a series involving products of Bessel functions of argument λ times functions F_q(z). Expressions for the imaginary parts of all dielectric tensor elements will be presented. The real parts of the tensor elements are not as simple, but because the elements are analytic, they must likewise be modified. 7. Initial thermal plasma observations from ISEE-1 NASA Technical Reports Server (NTRS) Baugher, C. R.; Chappell, C. R.; Horwitz, J. L.; Shelley, E. G.; Young, D. T. 1980-01-01 The initial measurements of magnetospheric thermal ions by the Plasma Composition Experiment on ISEE-1 are presented to demonstrate the surprising variety in this plasma population. The data provide evidence that the adiabatic mapping of the high latitude ionosphere to the equatorial plasma trough provides an insufficient description of the origin, transport, and accumulation processes which supply low energy ions to the outer plasmasphere and plasma trough. 8. Initial thermal plasma observations from ISEE-1 Baugher, C. R.; Chappell, C. R.; Horwitz, J. L.; Shelley, E. G.; Young, D. T. 1980-09-01 The initial measurements of magnetospheric thermal ions by the Plasma Composition Experiment on ISEE-1 are presented to demonstrate the surprising variety in this plasma population. The data provide evidence that the adiabatic mapping of the high latitude ionosphere to the equatorial plasma trough provides an insufficient description of the origin, transport, and accumulation processes which supply low energy ions to the outer plasmasphere and plasma trough. 9. Parametric Excitations of Fast Plasma Waves by Counter-propagating Laser Beams SciTech Connect G. Shvets; N.J. Fisch 2001-03-19 Short- and long-wavelength plasma waves can become strongly coupled in the presence of two counter-propagating laser pump pulses detuned by twice the cold plasma frequency. What makes this four-wave interaction important is that the growth rate of the plasma waves occurs much faster than in the more obvious co-propagating geometry. 10. Oblique solitary waves in a five component plasma Sijo, S.; Manesh, M.; Sreekala, G.; Neethu, T. W.; Renuka, G.; Venugopal, C. 2015-12-01 We investigate the influence of a second electron component on oblique dust ion acoustic solitary waves in a five component plasma consisting of positively and negatively charged dust, hydrogen ions, and hotter and colder electrons. Of these, the heavier dust and colder photo-electrons are of cometary origin while the other two are of solar origin; electron components are described by kappa distributions. The K-dV equation is derived, and different attributes of the soliton such as amplitude and width are plotted for parameters relevant to comet Halley. We find that the second electron component has a profound influence on the solitary wave, decreasing both its amplitude and width. The normalized hydrogen density strongly influences the solitary wave by decreasing its width; the amplitude of the solitary wave, however, increases with increasing solar electron temperatures. 11. Oblique solitary waves in a five component plasma SciTech Connect Sijo, S.; Manesh, M.; Sreekala, G.; Venugopal, C.; Neethu, T. W.; Renuka, G. 2015-12-15 We investigate the influence of a second electron component on oblique dust ion acoustic solitary waves in a five component plasma consisting of positively and negatively charged dust, hydrogen ions, and hotter and colder electrons. Of these, the heavier dust and colder photo-electrons are of cometary origin while the other two are of solar origin; electron components are described by kappa distributions. The K-dV equation is derived, and different attributes of the soliton such as amplitude and width are plotted for parameters relevant to comet Halley. We find that the second electron component has a profound influence on the solitary wave, decreasing both its amplitude and width. The normalized hydrogen density strongly influences the solitary wave by decreasing its width; the amplitude of the solitary wave, however, increases with increasing solar electron temperatures. 12. Survey of Coherent Approximately 1 Hz Waves in Mercury's Inner Magnetosphere from MESSENGER Observations NASA Technical Reports Server (NTRS) Boardsen, Scott A.; Slavin, James A.; Anderson, Brian J.; Korth, Haje; Schriver, David; Solomon, Sean C. 2012-01-01 We summarize observations by the MESSENGER spacecraft of highly coherent waves at frequencies between 0.4 and 5 Hz in Mercury's inner magnetosphere. This survey covers the time period from 24 March to 25 September 2011, or 2.1 Mercury years. These waves typically exhibit banded harmonic structure that drifts in frequency as the spacecraft traverses the magnetic equator. The waves are seen at all magnetic local times, but their observed rate of occurrence is much less on the dayside, at least in part the result of MESSENGER's orbit. On the nightside, on average, wave power is maximum near the equator and decreases with increasing magnetic latitude, consistent with an equatorial source. When the spacecraft traverses the plasma sheet during its equatorial crossings, wave power is a factor of 2 larger than for equatorial crossings that do not cross the plasma sheet. The waves are highly transverse at large magnetic latitudes but are more compressional near the equator. However, at the equator the transverse component of these waves increases relative to the compressional component as the degree of polarization decreases. Also, there is a substantial minority of events that are transverse at all magnetic latitudes, including the equator. A few of these latter events could be interpreted as ion cyclotron waves. In general, the waves tend to be strongly linear and characterized by values of the ellipticity less than 0.3 and wave-normal angles peaked near 90 deg. Their maxima in wave power at the equator coupled with their narrow-band character suggests that these waves might be generated locally in loss cone plasma characterized by high values of the ratio beta of plasma pressure to magnetic pressure. Presumably both electromagnetic ion cyclotron waves and electromagnetic ion Bernstein waves can be generated by ion loss cone distributions. If proton beta decreases with increasing magnetic latitude along a field line, then electromagnetic ion Bernstein waves are predicted 13. Fundamental emission via wave advection from a collapsing wave packet in electromagnetic strong plasma turbulence SciTech Connect Jenet, F. A.; Melatos, A.; Robinson, P. A. 2007-10-15 Zakharov simulations of nonlinear wave collapse in continuously driven two-dimensional, electromagnetic strong plasma turbulence with electron thermal speeds v{>=}0.01c show that for v < or approx. 0.1c, dipole radiation occurs near the plasma frequency, mainly near arrest, but for v > or approx. 0.1c, a new mechanism applies in which energy oscillates between trapped Langmuir and transverse modes until collapse is arrested, after which trapped transverse waves are advected into incoherent interpacket turbulence by an expanding annular density well, where they detrap. The multipole structure, Poynting flux, source current, and radiation angular momentum are computed. 14. Measurement of dispersion relation of waves in a tandem mirror plasma by the Fraunhofer-diffraction method SciTech Connect Mase, A.; Jeong, J.H.; Itakura, A.; Ishii, K.; Miyoshi, S. ) 1990-04-01 The Fraunhofer diffraction measurements from a tandem mirror plasma are reported. The successful use of a new multichannel detector array permits a detailed study of {bold k}{minus}{omega} spectra of long-wavelength waves with a few plasma shots. The observed dispersion relations are in good agreement with those of drift wave including a Doppler shift due to {bold E}{times}{bold B} rotation velocity. 15. Full wave simulation of waves in ECRIS plasmas based on the finite element method SciTech Connect Torrisi, G.; Mascali, D.; Neri, L.; Castro, G.; Patti, G.; Celona, L.; Gammino, S.; Ciavola, G.; Di Donato, L.; Sorbello, G.; Isernia, T. 2014-02-12 This paper describes the modeling and the full wave numerical simulation of electromagnetic waves propagation and absorption in an anisotropic magnetized plasma filling the resonant cavity of an electron cyclotron resonance ion source (ECRIS). The model assumes inhomogeneous, dispersive and tensorial constitutive relations. Maxwell's equations are solved by the finite element method (FEM), using the COMSOL Multiphysics{sup ®} suite. All the relevant details have been considered in the model, including the non uniform external magnetostatic field used for plasma confinement, the local electron density profile resulting in the full-3D non uniform magnetized plasma complex dielectric tensor. The more accurate plasma simulations clearly show the importance of cavity effect on wave propagation and the effects of a resonant surface. These studies are the pillars for an improved ECRIS plasma modeling, that is mandatory to optimize the ion source output (beam intensity distribution and charge state, especially). Any new project concerning the advanced ECRIS design will take benefit by an adequate modeling of self-consistent wave absorption simulations. 16. Pioneer 9 plasma wave and solar plasma measurements for the August 1972 storm period NASA Technical Reports Server (NTRS) Scarf, F. L.; Wolfe, J. H. 1974-01-01 The solar disturbances of August 1972 produced large-scale solar wind perturbations that were detected by the Pioneer 9 plasma probe, electric field detector, and magnetometer for an extended time period commencing early on August 3. During this ten-day interval the interplanetary plasma parameters at r approximately equal 0.8 AU varied over unusually wide ranges, so that the conditions for generation of high and low VLF wave levels could be identified fairly readily. It is demonstrated that no measurable signals were detected in the broadband electric field channel (sensitive to waves with f greater than or equal to 100 Hz in the spacecraft frame of reference) unless the proton density was high enough to yield a proton plasma frequency with f greater than or about equal to 100 Hz. The analysis suggests that waves related to ion acoustic oscillations were detected throughout the extended storm period. 17. Theory and observations of slow-mode solitons in space plasmas. PubMed Stasiewicz, K 2004-09-17 A generalized model for one-dimensional magnetosonic structures of large amplitude in space plasmas is presented. The model is verified with multipoint measurements on Cluster satellites in the magnetosheath and the boundary layer under conditions of plasma beta (plasma/magnetic pressure) between 0.1-10. We demonstrate good agreement between the model and observations of large amplitude structures and wave trains, which represent increases of magnetic field and plasma density 2-5 times the ambient values, or local decreases (holes) by approximately (50-80)%. Theoretically derived polarization and propagation properties of slow-mode nonlinear structures are also in agreement with in situ measurements in space. 18. Characterizing low frequency plasma waves at Mars with MAVEN Ruhunusiri, Suranga; Halekas, Jasper; Connerney, Jack; Espley, Jared; Larson, Davin; Mitchell, David L. 2015-04-01 We use the measurements from the Solar Wind Ion Analyzer (SWIA) and the magnetometer (MAG) instruments aboard the MAVEN spacecraft to characterize plasma waves in the Martian magnetosphere. SWIA is a toroidal energy analyzer that measures 3-d ion velocity distributions, and we use it for measuring ion moment fluctuations. MAG instrument, on the other hand, is a fluxgate magnetometer, and we use it for measuring magnetic field fluctuations. Mars is unique in the solar system because of two characteristics: it only has an induced magnetosphere with strong crustal fields at low altitudes, and it has an extended atmosphere due to its lower gravity. Due to these two characteristics, Mars presents a unique environment to study the interaction of a planetary magnetosphere and an exosphere with the solar wind. One consequence of this interaction is the excitation of low frequency plasma waves which have highest power near and below the proton gyrofrequency. Studying these waves is of interest because they can play a vital role in the mass and energy transport in the Martian magnetosphere. In this investigation, we use both ion moment fluctuations (density and velocity) and the magnetic field fluctuations to characterize these low frequency plasma waves. 19. Investigation of the Millimeter-Wave Plasma Assisted CVD Reactor SciTech Connect Vikharev, A; Gorbachev, A; Kozlov, A; Litvak, A; Bykov, Y; Caplan, M 2005-07-21 A polycrystalline diamond grown by the chemical vapor deposition (CVD) technique is recognized as a unique material for high power electronic devices owing to unrivaled combination of properties such as ultra-low microwave absorption, high thermal conductivity, high mechanical strength and chemical stability. Microwave vacuum windows for modern high power sources and transmission lines operating at the megawatt power level require high quality diamond disks with a diameter of several centimeters and a thickness of a few millimeters. The microwave plasma-assisted CVD technique exploited today to produce such disks has low deposition rate, which limits the availability of large size diamond disk windows. High-electron-density plasma generated by the millimeter-wave power was suggested for enhanced-growth-rate CVD. In this paper a general description of the 30 GHz gyrotron-based facility is presented. The output radiation of the gyrotron is converted into four wave-beams. Free localized plasma in the shape of a disk with diameter much larger than the wavelength of the radiation is formed in the intersection area of the wave-beams. The results of investigation of the plasma parameters, as well as the first results of diamond film deposition are presented. The prospects for commercially producing vacuum window diamond disks for high power microwave devices at much lower costs and processing times than currently available are outlined. 20. On the instability and energy flux of lower hybrid waves in the Venus plasma mantle NASA Technical Reports Server (NTRS) Strangeway, R. J.; Crawford, G. K. 1993-01-01 Waves generated near the lower hybrid resonance frequency by the modified two stream instability have been invoked as a possible source of energy flux into the topside ionosphere of Venus. These waves are observed above the ionopause in a region known as the plasma mantle. The plasma within the mantle appears to be a mixture of magnetosheath and ionospheric plasmas. Since the magnetosheath electrons and ions have temperatures of several tens of eV, any instability analysis of the modified two stream instability requires the inclusion of finite electron and ion temperatures. Finite temperature effects are likely to reduce the growth rate of the instability. Furthermore, the lower hybrid waves are only quasi-electrostatic, and the energy flux of the waves is mainly carried by parallel Poynting flux. The magnetic field in the mantle is draped over the ionopause. Lower hybrid waves therefore cannot transport any significant wave energy to lower altitudes, and so do not act as a source of additional heat to the topside ionosphere. 1. Statistical Analysis of EMIC Waves in Multiple Component Plasma Including Heavy Ions Matsuda, S.; Kasahara, Y.; Goto, Y. 2013-12-01 It is well known that Earth's radiation belts are located around geomagnetic equator, where wide ranges of energetic particles from several hundred keV to several tens MeV are contained. According to the recent study, it is suggested that ELF/VLF waves such as EMIC waves and chorus emissions deeply contribute to the generation and loss mechanism of relativistic electrons in the radiation belt. The ERG mission[1] is expected to provide important clues for solving plasma dynamics in the Earth's radiation belts by means of integrated observation of wide energy range of plasma particles and high resolution plasma waves. On the other hand, long-term observation data which covers over 2 cycles of solar activity obtained by the Akebono satellite is very valuable to work out the strategy of the ERG mission. The ELF receiver, which is a sub-system of the VLF instruments onboard Akebono, measures waveforms below 50 Hz for one component of electric field and three components of magnetic field, or waveforms below 100 Hz for one component of electric and magnetic field, respectively. It was reported that ion cyclotron waves were observed near magnetic equator by the receiver[2]. In our previous study[3], we introduced four events of characteristic EMIC waves observed by Akebono in April, 1989. These waves have sudden decrease of intensity just above half of proton cyclotron frequency changing along the trajectories of Akebono. Comparing the observed data with the dispersion relation in multiple species of ions under cold plasma approximation, we demonstrate that a few percent of 'M / Z = 2 ions (M = mass of ions, Z = charge of ions)' such as alpha particles (He++) or deuterons (D+) cause such characteristic attenuation of EMIC at lower hybrid frequency. In the present study, we performed polarization analysis and direction finding of the waves. It was found that these EMIC waves were left-handed polarized in the higher frequency part, while the polarization gradually changes to 2. Numerical Simulation of Waves Driven by Plasma Currents Generated by Low-Frequency Alfven Waves in a Multi-Ion Plasma NASA Technical Reports Server (NTRS) Singh, Nagendra; Khazanov, George 2003-01-01 When multi-ion plasma consisting of heavy and light ions is permeated by a lowfrequency Alfien (LFA) wave, the EXB and the polarization drifts of the different ion species and the electrons could be quite different. The relative drifts between the charged-particle species drive waves, which energize the plasma. Using 2.5-D particle-in-cell simulations, we study this process of wave generation and its nonlinear consequences in terms of acceleration and heating plasma. Specifically we study the situation for LFA wave frequency being lower than the heavyion cyclotron frequency in a multi-ion plasma. We impose such a wave to the plasma assuming that its wavelength is much larger than that of the waves generated by the relative drifts. For better understanding, the LFA-wave driven simulations are augmented by those driven by initialized ion beams. 3. Generation and effects of EMIC waves observed by the Van Allen Probes on 18 March 2013 Zhang, J.; Saikin, A.; Gamayunov, K. V.; Spence, H. E.; Larsen, B.; Geoffrey, R.; Smith, C. W.; Torbert, R. B.; Kurth, W. S.; Kletzing, C. 2015-12-01 Electromagnetic ion cyclotron (EMIC) waves play a crucial role in particle dynamics in the Earth's magnetosphere. The free energy for EMIC wave generation is usually provided by the temperature anisotropy of the energetic ring current ions. EMIC waves can in turn cause particle energization and losses through resonant wave-particle interactions. Using measurements from the Van Allen Probes, we perform a case study of EMIC waves and associated plasma conditions observed on 18 March 2013. From 0204 to 0211 UT, the Van Allen Probe-B detected He+-band EMIC wave activity in the post-midnight sector (MLT=4.6-4.9) at very low L-shells (L=2.6-2.9). The event occurred right outside the inward-pushed plasmapause in the early recovery phase of an intense geomagnetic storm - min. Dst = -132 nT at 2100 UT on 17 March 2013. During this event, the fluxes of energetic (> 1 keV), anisotropic O+ dominate both the H+ and He+ fluxes in this energy range. Meanwhile, O+ fluxes at low energies (< 0.1 keV) are low compared to H+ and He+ fluxes in the same energy range. The fluxes of <0.1 keV He+ are clearly enhanced during the wave event, indicating a signature of wave heating. To further confirm the association of the observed plasma features with the EMIC waves, we calculate the electron minimum resonant energy (Emin) and pitch angle diffusion coefficient (Dαα) of the EMIC wave packets by using nominal ion composition, derived total ion density from the frequencies of upper hybrid resonance, and measured ambient and wave magnetic field. EMIC wave growth rates are also calculated to evaluate the role of loss-cone distributed ring current ions in the EMIC wave generation. 4. Waves in dusty plasmas and the concept of fugacity Rao, Nagesha N. 2000-10-01 The propagation of ultra low-frequency electrostatic modes in dusty plasmas has been reviewed in the light of the concept of dust fugacity (f ), which is defined by f≡4πnd0λD2R where nd0, λD and R are, respectively, the dust number density, the plasma Debye length and the grain size (radius). Dusty plasmas are defined to be tenuous, dilute or dense according as f<<1, ~1, or >>1, respectively. By using the fluid as well as the kinetic (Vlasov) theories, attention is focused on the Dust-Acoustic Waves'' (DAWs) and the Dust-Coulomb Waves'' (DCWs) which exist in the tenuous and the dense regimes, respectively. Unlike the DAWs which exist even for constant grain charge, the DCWs are the normal modes associated with grain charge fluctuations, and are driven by an effective pressure called Coulomb Pressure''. They can be considered as the electrostatic analogue of the hydromagnetic (Alfvén or magnetoacoustic) modes which are driven by the magnetic field pressure. In the dilute regime, the two modes merge into a single mode, which may be called the Dust Charge-Density Wave'' (DCDW). When the grains are closest, the DCW dispersion relation is identical with that of the Dust-Lattice Waves'' (DLWs). Dense dusty plasmas are shown to be governed by a new scale-length defined by λR≡1/4πnd0Rδ, where δ is a parameter related to the charging frequencies. The scale-length λR characterizes the effective shielding length due to the collective grain interactions, and plays a fundamental role in dense dusty plasmas, which is very similar to that of the Debye length (λD) of the tenuous regime. The frequency spectrum as well as the damping rates for the various dust modes have been analytically obtained, and compared with the numerical results. . 5. Observations of ubiquitous compressive waves in the Sun's chromosphere. PubMed Morton, Richard J; Verth, Gary; Jess, David B; Kuridze, David; Ruderman, Michael S; Mathioudakis, Mihalis; Erdélyi, Robertus 2012-01-01 The details of the mechanism(s) responsible for the observed heating and dynamics of the solar atmosphere still remain a mystery. Magnetohydrodynamic waves are thought to have a vital role in this process. Although it has been shown that incompressible waves are ubiquitous in off-limb solar atmospheric observations, their energy cannot be readily dissipated. Here we provide, for the first time, on-disk observation and identification of concurrent magnetohydrodynamic wave modes, both compressible and incompressible, in the solar chromosphere. The observed ubiquity and estimated energy flux associated with the detected magnetohydrodynamic waves suggest the chromosphere is a vast reservoir of wave energy with the potential to meet chromospheric and coronal heating requirements. We are also able to propose an upper bound on the flux of the observed wave energy that is able to reach the corona based on observational constraints, which has important implications for the suggested mechanism(s) for quiescent coronal heating. 6. Ion anisotropy driven waves in the earth's magnetosheath and plasma depletion layer SciTech Connect Denton, R.E.; Hudson, M.K. . Dept. of Physics and Astronomy); Anderson, B.J. . Applied Physics Lab.); Fuselier, S.A. ); Gary, S.P. ) 1993-01-01 Recent studies of low frequency waves ([omega][sub r] [le] [Omega][sub p], where [Omega][sub p] is the proton gyrofrequency) observed by AMPTE/CCE in the plasma depletion layer and magnetosheath proper arereviewed. These waves are shown to be well identified with ion cyclotron and mirror mode waves. By statistically analyzing the transitions between the magnetopause and time intervals with ion cyclotron and mirror mode waves, it is established that the regions in which ion cyclotron waves occur are between the magnetopause and the regions where the mirror mode is observed. This result is shown to follow from the fact that the wave spectral properties are ordered with respect to the proton parallel beta, [beta][sub [parallel]p]. The later result is predicted by linear Vlasov theory using a simple model for the magnetosheath and plasma depletion layer. Thus, the observed spectral type can be associated with relative distance from the magnetopause. The anisotropy-beta relation, A[sub p] [triple bond] (T[perpendicular]/T[sub [parallel 7. Conversion of ionospheric heater HF waves into electron acoustic waves in warm ionospheric plasma Lehtinen, N. G.; Inan, U. S.; Bunch, N. L. 2012-12-01 The Stanford full-wave method (StanfordFWM) was developed in order to calculate generation and propagation of electromagnetic waves in cold magnetized stratified plasmas. We generalize it by including the effects of electron temperature, by following a procedure analogous to that of [Budden and Jones, 1987, doi:10.1098/rspa.1987.0077]. The advantage of StanfordFWM is that it is intrinsically numerically stable against swamping'' by evanescent waves while in the method of Budden and Jones [1987] the problem of numerical swamping is severe ...'' The new method is used to calculate mode conversion between electron acoustic (Langmuir) and electromagnetic modes for propagation in a warm ionospheric plasma with a gradient of electron density and an arbitrary direction of the background geomagnetic field, in the vicinity of density corresponding to the plasma resonance. As a numerical check, we demonstrate good agreement with previous calculations of Budden and Jones [1987] obtained by a numerically-unstable full-wave method scheme; Mjolhus [1990, doi:10.1029/RS025i006p01321] obtained by the method of contour integration in the complex n-plane; and Kim et al [2008, doi:10.1063/1.2994719] using a numerical electron fluid simulation code. We demonstrate that under certain conditions the linear conversion of the ordinary HF electromagnetic waves radiated by an ionospheric heater into electron acoustic waves may be very efficient, with implications for the HF heating of the F-region of ionosphere. 8. Full wave simulation of lower hybrid waves in Maxwellian plasma based on the finite element method SciTech Connect Meneghini, O.; Shiraiwa, S.; Parker, R. 2009-09-15 A full wave simulation of the lower-hybrid (LH) wave based on the finite element method is presented. For the LH wave, the most important terms of the dielectric tensor are the cold plasma contribution and the electron Landau damping (ELD) term, which depends only on the component of the wave vector parallel to the background magnetic field. The nonlocal hot plasma ELD effect was expressed as a convolution integral along the magnetic field lines and the resultant integro-differential Helmholtz equation was solved iteratively. The LH wave propagation in a Maxwellian tokamak plasma based on the Alcator C experiment was simulated for electron temperatures in the range of 2.5-10 keV. Comparison with ray tracing simulations showed good agreement when the single pass damping is strong. The advantages of the new approach include a significant reduction of computational requirements compared to full wave spectral methods and seamless treatment of the core, the scrape off layer and the launcher regions. 9. The interaction between ULF waves and thermal plasma ions in the magnetosphere Zong, Qiugang 2016-07-01 During substorm activities, energetic particle injections associated with ULF waves have been detected when Cluster fleet was traveling inbound in the Southern Hemisphere. Substorm-injected energetic particles are strong and clearly modulated by these ULF waves. The ULF waves with the period of 1 min are probably the third harmonic mode. The periodic pitch angle dispersion signatures at 5.2-6.9 keV energy channel were detected by Cluster satellite. These thermal plasma have high coherence with the electric field of the third harmonic poloidal mode and satisfy the drift-bounce resonant condition of N = 2. In addition, ion outflows from the Earth's ionosphere (tens to hundreds of eV) are also observed to be modulated by these ULF waves. To the best of our knowledge, this is the first report to show that ULF waves can simultaneously interact with both substorm-injected "hot" particles from the magnetotail and cold outflow ions from the Earth's ionosphere. 10. Performance of an ion-cyclotron-wave plasma apparatus operated in the radiofrequency sustained mode NASA Technical Reports Server (NTRS) Swett, C. C.; Woollett, R. R. 1973-01-01 An experimental study has been made of an ion-cyclotron-wave apparatus operated in the RF-sustained mode, that is, a mode in which the Stix RF coil both propagates the waves and maintains the plasma. Problems associated with this method of operation are presented. Some factors that are important to the coupling of RF power are noted. In general, the wave propagation and wave damping data agree with theory. Some irregularities in wave fields are observed. Maximum ion temperature is 870 eV at a density of five times 10 to the 12th power cu cm and RF power of 90 kW. Coupling efficiency is 70 percent. 11. Nonplanar dust acoustic solitary waves in a strongly coupled dusty plasma with superthermal ions SciTech Connect El-Labany, S. K. Zedan, N. A.; El-Taibany, W. F. E-mail: [email protected]; El-Shamy, E. F. 2014-12-15 The nonplanar amplitude modulation of dust acoustic (DA) envelope solitary waves in a strongly coupled dusty plasma (SCDP) has been investigated. By using a reductive perturbation technique, a modified nonlinear Schrödinger equation (NLSE) including the effects of geometry, polarization, and ion superthermality is derived. The modulational instability (MI) of the nonlinear DA wave envelopes is investigated in both planar and nonplanar geometries. There are two stable regions for the DA wave propagation strongly affected by polarization and ion superthermality. Moreover, it is found that the nonlinear DA waves in spherical geometry are the more structurally stable. The larger growth rate of the nonlinear DA MI is observed in the cylindrical geometry. The salient characteristics of the MI in the nonplanar geometries cannot be found in the planar one. The DA wave propagation and the NLSE solutions are investigated both analytically and numerically. 12. Gravitational Waves: A New Observational Window NASA Technical Reports Server (NTRS) Camp, Jordan B. 2010-01-01 The era of gravitational wave astronomy is rapidly approaching, with a likely start date around the middle of this decade ' Gravitational waves, emitted by accelerated motions of very massive objects, provide detailed information about strong-field gravity and its sources, including black holes and neutron stars, that electromagnetic probes cannot access. In this talk I will discuss the anticipated sources and the status of the extremely sensitive detectors (both ground and space based) that will make gravitational wave detections possible. As ground based detectors are now taking data, I will show some initial science results related to measured upper limits on gravitational wave signals. Finally Z will describe new directions including advanced detectors and joint efforts with other fields of astronomy. 13. Waves generated in the vicinity of an argon plasma gun in the ionosphere NASA Technical Reports Server (NTRS) Cahill, L. J., Jr.; Arnoldy, R. L.; Lysak, R. L.; Peria, W.; Lynch, K. A. 1993-01-01 Wave and particle observations were made in the close vicinity of an argon plasma gun carned to over 600 km altitude on a sounding rocket. The gun was carned on a subpayload, separated from the main payload early in the flight. Twelve-second argon ion ejections were energized alternately with a peak energy of 100 or 200 eV. They produced waves, with multiple harmonics, in the range of ion cyclotron waves, 10 to 1000 Hz at rocket altitudes. Many of these waves could not be identified as corresponding to the cyclotron frequencies of any of the ions, argon or ambient, known to be present. In addition, the wave frequencies were observed to rise and fall and to change abruptly during a 12-s gun operation. The wave amplitudes, near a few hundred Hertz, were of the order of O. 1 V/m. Some of the waves may be ion-ion hybrid waves. Changes in ion populations were observed at the main payload and at the subpayload during gun operations. A gun-related, field-aligned, electron population also appeared. 14. Linear and Nonlinear Dust Acoustic Waves, Shocks and Stationary Structures in a dc-Glow-Discharge Dusty Plasma Merlino, Robert 2011-10-01 In 1990, Rao, Shukla, and Yu (Planet. Space Sci. 38, 543) predicted the existence of the dust acoustic (DA) wave, a low-frequency (~ few Hz), compressional dust density wave that propagates through a dusty plasma at a phase speed ~ several cm/s. The DA wave was first observed by Chu et. al., (J. Phys. D: Appl. Phys. 27, 296, 1994) in an rf-produced dusty plasma, and by Barkan et. al., (Barkan et. al. Phys. Plasmas 2, 2161, 1995) who obtained video images of the DA wave trains using light scattering from a dust suspension confined in an anodic glow discharge plasma formed within a Q machine plasma. The dispersion relation for DAWs was measured by Thompson et. al., (Phys. Plasmas 4, 2331, 1997) in a dc glow discharge dusty plasma by modulating the discharge current at a set frequency. DAWs have been investigated by many groups both in weakly-coupled and strongly-coupled dusty plasmas (E. Thomas, Jr., Contrib. Plasma Phys. 49, 316, 2009). In most experiments where DA waves are present, the wave amplitude is relatively high, indicating that they are nonlinear. In this talk, results of our recent experiments on DAWs will be presented. The following experiments, performed in a dc glow-discharge dusty plasma will be described: (1) Observations of spontaneously excited nonlinear, cylindrical DAWs, which exhibit confluence of waves propagating at different speeds. (2) Investigations of self-steepening DAWs that develop into DA shocks with thicknesses comparable to the interparticle separation (Heinrich et. al., Phys. Rev. Lett. 103, 115001, 2009). (3) Measurements of the linear growth rates of DAWs excited in merging dust clouds. (4) The formation of stationary, stable dust density structures appearing as non-propagating DAWs (Heinrich et. al., Phys. Rev. E, in press, 2011). This work was performed in collaboration with S. H. Kim, J. R. Heinrich, and J. K. Meyer. Work supported by DOE Grant No. DE-FG01-04ER54795 15. Electron beam driven lower hybrid waves in a dusty plasma SciTech Connect Prakash, Ved; Vijayshri; Sharma, Suresh C.; Gupta, Ruby 2013-05-15 An electron beam propagating through a magnetized dusty plasma drives electrostatic lower hybrid waves to instability via Cerenkov interaction. A dispersion relation and the growth rate of the instability for this process have been derived taking into account the dust charge fluctuations. The frequency and the growth rate of the unstable wave increase with the relative density of negatively charged dust grains. Moreover, the growth rate of the instability increases with beam density and scales as the one-third power of the beam density. In addition, the dependence of the growth rate on the beam velocity is also discussed. 16. Universal self-amplification channel for surface plasma waves Deng, Hai-Yao; Wakabayashi, Katsunori; Lam, Chi-Hang 2017-01-01 We present a theory of surface plasma waves (SPWs) in metals with arbitrary electronic collision rate τ-1. We show that there exists a universal intrinsic amplification channel for these waves, subsequent to the interplay between ballistic motions and the metal surface. We evaluate the corresponding amplification rate γ0, which turns out to be a sizable fraction of the SPW frequency ωs. We also find that the value of ωs depends on surface scattering properties, in contrast with the conventional theory. 17. Wave propagation in a moving cold magnetized plasma Hebenstreit, H. 1980-03-01 Polarization relations and dispersion equations are derived for media that are electrically anisotropic in the comoving frame. Three-dimensional calculations for media at rest recover the known dispersion equations, i.e., Astrom's dispersion equation for magnetized cold plasmas and Fresnel's wave normal equation for uniaxial crystals. An analogous four-dimensional calculation yields the generalization to moving media. The dispersion equations so obtained for moving gyrotropic media are then discussed qualitatively for various special media and special directions of wave propagation. Finally, the polarization relations are specialized to media gyrotropic in the comoving frame. 18. On the evolution of linear waves in cosmological plasmas SciTech Connect Dodin, I. Y.; Fisch, N. J. 2010-08-15 The scalings for basic plasma modes in the Friedmann-Robertson-Walker model of the expanding Universe are revised. Contrary to the existing literature, the wave collisionless evolution must comply with the action conservation theorem. The proper steps to deduce the action conservation from ab initio analytical calculations are presented, and discrepancies in the earlier papers are identified. In general, the cosmological wave evolution is more easily derived from the action conservation in the collisionless limit, whereas when collisions are essential, the statistical description must suffice, thereby ruling out the need for using dynamic equations in either case. 19. Theories of radio emissions and plasma waves. [in Jupiter magnetosphere NASA Technical Reports Server (NTRS) Goldstein, M. L.; Goertz, C. K. 1983-01-01 The complex region of Jupiter's radio emissions at decameter wavelengths, the so-called DAM, is considered, taking into account the basic theoretical ideas which underly both the older and newer theories and models. Linear theories are examined, giving attention to direct emission mechanisms, parallel propagation, perpendicular propagation, and indirect emission mechanisms. An investigation of nonlinear theories is also conducted. Three-wave interactions are discussed along with decay instabilities, and three-wave up-conversio. Aspects of the Io and plasma torus interaction are studied, and a mechanism by which Io can accelerate electrons is reviewed. 20. The Nonlinear Landau Damping Rate of a Driven Plasma Wave SciTech Connect Benisti, D; Strozzi, D J; Gremillet, L; Morice, O 2009-08-04 In this Letter, we discuss the concept of the nonlinear Landau damping rate, {nu}, of a driven electron plasma wave, and provide a very simple, practical, analytic formula for {nu} which agrees very well with results inferred from Vlasov simulations of stimulated Raman scattering. {nu} actually is more complicated an operator than a plain damping rate, and it may only be seen as such because it assumes almost constant values before abruptly dropping to 0. The decrease of {nu} to 0 is moreover shown to occur later when the wave amplitude varies in the direction transverse to its propagation.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9345300793647766, "perplexity": 2492.3886780434236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818690591.29/warc/CC-MAIN-20170925092813-20170925112813-00702.warc.gz"}
https://aas.org/archives/BAAS/v25n2/aas182/abshtml/S405.html	10 $\mu m$ Imaging of NGC 4151 and NGC 1068 Session 4 -- AGNs Display presentation, Monday, 9:20-6:30, Pauley Room ## [4.05] 10 $\mu m$ Imaging of NGC 4151 and NGC 1068 M. Meixner, J. R. Graham (UCB), G. W. Hawkins, C. J. Skinner (IGPP), E. Keto (LLNL), J. F. Arens, J. G. Jernigan (SSL) We have obtained mid infrared images of the nearby Seyfert nuclei NGC 1068 and NGC 4151 to study the spatial distribution of the emission. The data were obtained at UKIRT with the Berkeley mid-IR camera which is supported by IGPP and LEA at Lawrence Livermore National Laboratory. These observations will help distinguish between thermal and non-thermal emission mechanisms, and determine the relative contribution of emission from dust in a molecular torus and the narrow line region. The images at 8.5 and 12.5 $\mu m$ both have FWHM of $1.''3$ and very high signal to noise ($>$ 300). These data, when combined with an accurate measurement of the point spread function, permit us to constrain the sub--arcsecond structure of the emission. We confirm that the 10 $\mu m$ emission of NGC 1068 is spatially extended north--south (Becklin et al. 1976 ApJ 186 L69) but these new data show that the emission is very asymmetric. The emission consists of unresolved point source which accounts for $\simeq 40\%$ of the flux together with a jet like extended component centered 0.''3 further north. The extended component has a FWHM of 0.''88 in the north-south direction (60 $pc$ at a distance of 13 Mpc), and unresolved ($<0.''1$) east-west. The extended component is almost certainly thermal emission from dust. However, since the dust is located about 20 $pc$ from the nucleus, the emission is probably not due to dust that is in thermal equilibrium with the central engine. Rather, the emission is due to dust that is heated locally in circumnuclear star forming regions, stochastically heated small dust grains, or dust heated in the narrow line region by resonantly trapped Lyman $\alpha$.	{"extraction_info": {"found_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": 0.9101486802101135, "perplexity": 2260.327513559708}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257823670.44/warc/CC-MAIN-20160723071023-00268-ip-10-185-27-174.ec2.internal.warc.gz"}
https://brilliant.org/problems/ap-and-gp/	A.P. and G.P. Algebra Level pending If $$a,b,c$$ are in A.P. and $$x,y,z$$ are in G.P. , then what is the value of :$x^{(b-c)} \cdot y^{(c-a)} \cdot z^{(a-b)}$ ×	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8126583695411682, "perplexity": 1014.8854222689756}, "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/1508187823114.39/warc/CC-MAIN-20171018195607-20171018215607-00419.warc.gz"}
https://stats.stackexchange.com/questions/291016/acceptance-probability-for-metropolis-hastings-mcmc-on-multinomial-dirichlet-mod	Acceptance probability for Metropolis-Hastings MCMC on multinomial-Dirichlet model As an exercise to learn how to manually code MCMC, I've built a Metropolis-Hastings sampler on top of a multinomial-Dirichlet posterior distribution. Since a closed form solution exists, I can compare results from the MCMC with simulations from the actual posterior distribution. I'm using a Dirichlet proposal distribution with parameters equal to the latest probabilities in the chain times a scaling constant (~1000), which makes the expected value of the distribution equal to those probabilities, with the scaling constant controlling the variance. Since this distribution certainly isn't symmetric, I tried adding the ratio of the of values from the proposal distribution to the calculation of the acceptance probability. Doing this, however, seems to bias the results away from the results given by the closed form solution. The only way I've gotten results from the MCMC to match the results from the closed-form solution is to calculate the acceptance probability from the posterior distribution alone, as you would if the proposal distribution were symmetric. R code here: https://github.com/sivadivel/nps_stats/blob/master/manual_mcmc.R My question is: why is this the case? • A scaling constant of 1000 seems enormous and may have the consequence of turning the proposal into a Dirac mass. It has also the side effect of producing a normalising constant in the Dirichlet that may create an overflow: in R gamma(1000) returns Inf – Xi'an Jul 12 '17 at 7:07 • You're right Xi'an, using a scaling constant of 1000 produces an acceptance rate of 91% with a lot of autocorrelation in the chain, and lowering it to around 35 produces an acceptance rate of ~45% with much less autocorrelation. – Levi Davis Jul 12 '17 at 14:47 You either do a random walk (or not) with symmetric proposal distributions, or use a fixed asymmetric proposal distribution. According to the answer by David Marx under this question: I suppose you could perform a "pseudo" random walk with an asymmetric distribution which would cause the proposals to drift in the opposite direction of the skew (a left skewed distribution would favor proposals toward the right). This is probably why you observed: The only way I've gotten results from the MCMC to match the results from the closed-form solution is to calculate the acceptance probability from the posterior distribution alone, as you would if the proposal distribution were symmetric. And to calculate ppropos in your code, you should write: ppropos = log(ddirichlet(chain[i,1:3], a)) - log(ddirichlet(proposal, a)) with the assumption that the proposal distribution is $$Dir(a)$$ and fixed.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8836555480957031, "perplexity": 629.3695059687175}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038084765.46/warc/CC-MAIN-20210415095505-20210415125505-00227.warc.gz"}
http://pmelchior.net/blog/the-magic-of-proximal-operators.html	The magic of proximal operators This note is the first in a series, which summarizes of one of my deep dives and the basic elements of my python package [proxmin](https://github.com/pmelchior/proxmin). The problem at hand is minimizing a function $f$ under a non-differentiable constraint. The simplest common case is non-negativity, which one can express by a element-wise penalty function: $$g(x) = \begin{cases} 0 & \text{if}\ x \geq 0\\ \infty &\text{else}\end{cases}$$ so that the problem is to minimize $f+g$. If the objective function is the likelihood of a linear model $Ax=b$ with a known coefficient matrix $A$, the problem is known as Non-negative Least Squares, which can be solved with Quadratic Programming. However, we want to find ways to work with other, and potentially several simultaneous, non-differentiable constraints. Instead, let us find out what makes this problem hard. In short, it stems from two aspects, namely that $g$ is not differentiable at 0, otherwise we could simply follow the gradient of $f+g$, and, for good measure, that $g$ doesn't have a finite value for $x<0$. Here's where the proximal operators (alternatively called "proximity operators") come to the rescue. In what follows I will paraphrase key statements of the excellent review paper by [Parikh & Boyd (2013)](http://web.stanford.edu/~boyd/papers/prox_algs.html), which I highly recommend for a more thorough and still readable introduction to this topic. Instead of minimizing $g(x)$ directly, we can search for $$\underset{u}{\text{argmin}}\left\lbrace g(u) + \frac{1}{2\lambda}\lVert x-u\rVert_2^2\right\rbrace \equiv \text{prox}_{\lambda g}(x).$$ As long as a $g$ is a [closed proper convex function](https://en.wikipedia.org/wiki/Closed_convex_function), the argument of the minimization is strongly convex and not everywhere infinite, so it has a unique minimizer for every $x\in\mathbb{R}^N$. Even more so, one can show that the fixed points of the proximal operator $\text{prox}_{\lambda g}(x)$ are precisely the minimizers of $g$. In other words, $\text{prox}_{\lambda g}(x^\star)=x^\star$ if and only if $x^\star$ minimizes $g$. That's absolutely magical! Let me emphasize it again: > The proximal operator of $g$ yields finite results for all $x\in\mathbb{R}^N$, even if $g$ doesn't. Its fixed points are minimizers of $g$. Now you may ask: OK, that's all fine, but haven't we just moved the problem of minimizing $g$ into the definition of its proximal operator? Or in other words: how can we solve the problem? That's the next astonishing piece about proximal operators: many of them have analytic forms. Chief among them are the operators of indicator functions $$I_\mathcal{C}(x) = \begin{cases}0 & \text{if}\ x\in\mathcal{C}\\\infty& \text{else}\end{cases}$$ of a closed non-empty convex set $\mathcal{C}$. For them, it's quite obvious that the proximal operator is simply the Euclidean projection operator onto the set $$\text{prox}_{I_\mathcal{C}}(x) = \underset{u\in\mathcal{C}}{\text{argmin}} \lVert x-u\rVert_2^2,$$ and the scaling parameter $\lambda$ becomes meaningless. For non-negativity, the proximal operator is thus simply the element-wise projection onto the non-negative numbers, which we'll call $[.]_+$. Many other projections are known and analytic (think back to your geometry classes: straight lines, circles, cones; more complicated ones like norms, and ellipsoids … A quite comprehensive list is in section 6 of the Parikh paper and in the appendix of [Combettes & Pesquet (2010)](https://arxiv.org/abs/0912.3522v4)). Sometimes the subset projection becomes very tough, as my summer student this year has found out quickly. We were looking for a projection onto the set of monotonic functions, whose values $y_i = y(x_i)$ obey $\forall i: y_i \leq y_{i+1}\ \text{if}\ x_i \leq x_{i+1}$. This is a projection onto a polyhedron, which has no closed-form expression. Active-set or Quadratic Programming methods can solve it iteratively, but we decided to go with a different route (subject of a later post). Besides indicators, there are the penalty functions, which usually depend on the scaling parameter to set the strength of the penalty. Classic non-differentiable examples are the $\ell_1$ or $\ell_0$ sparsity penalty, whose proximal operators even have quite obvious names, namely the "soft thresholding" and "hard thresholding" operators: \begin{align} &\text{prox}_{\lambda \ell_0}(x) = \text{Hard}_\lambda(x) = \begin{cases}0 & \text{if}\ \|x\|<\lambda \\ x & \text{else}\end{cases}\\ &\text{prox}_{\lambda \ell_1}(x) = \text{Soft}_\lambda(x) = \text{sign}(x)\left[\|x\|-\lambda\right]_+ \end{align} More complicated ones can be constructed. To see how we need to introduce another concept that keeps popping up in the constrained optimization literature: the subderivative and its higher-dimensional version, the subgradient. It the generalization of the gradient to non-differentiable functions. Again: > The subderivative generalizes the derivative to functions that are not differentiable. It's quite straightforward, really: A subgradient of a function $g: \mathbb{R}^N\rightarrow(-\infty,\infty]$ at a point $x_0$ with $g(x_0) < \infty$ is a vector $\tau\in\mathbb{R}^N$ such that $g(x)\geq g(x_0) + \tau^\top (x-x_0)$. The set of all such subgradients, which may be empty, is called the subdifferential $\partial g(x_0)$. It handles infinite values, such as the ones that arise in the indicator functions, by yielding an empty set $\partial g(x_0)=\emptyset$ if $g(x_0) = \infty$. On the other hand, if $g$ is convex and differentiable at $x_0$, then $\partial g(x_0) = \lbrace\nabla g(x_0)\rbrace$. The concept of the subdifferential enables existence and convergence proofs. But, in addition, it can be used to validate and even derive the analytic form of a proximal operator. Because of the fixed-point property of the proximal operators, we have \begin{align} &p = \text{prox}_g(x) \Leftrightarrow x-p\in\partial g(p)\\ &p = \text{prox}_g(x) \Leftrightarrow x-p = \nabla g(p)\ \text{if}\ g\ \text{is differentiable at}\ p. \end{align} If we were interested to, say, derive the proximal-operator form of the Maximum Entropy regularization, that is we want to minimize $\text{S}(x) = -\sum_i^N x_i\ln(x_i)$, we can use the second equation above because the entropy is differentiable for $x_i>0$ and find that $$\text{prox}_{\lambda S}(x) = -\lambda \Re\left[W\left(-\exp\left(-\frac{x+\lambda}{\lambda}\right) \Big/ \lambda\right)\right],$$ with the Lambert-$W$ function. ## Proximal minimization We can now go to the most practically relevant aspects of proximal operators, namely the close relation to functional minimization. If $f$ is differentiable, one can write the limit $\lambda\rightarrow 0$ as $$\text{prox}_{\lambda f}(x) = (I + \lambda\nabla f)^{-1}(x).$$ If $f$ isn't differentiable, the subdifferential will formally replace the gradient and the following argument still holds in principle, but one has to be more careful (see Parikh & Boyd (2013), section 3.2, for what's called "resolvent of the subdifferential operator"). Back to the simpler case of a differentiable $f$. The first-order approximation $$\text{prox}_{\lambda f}(x) = x- \lambda\nabla f(x)$$ is equivalent to the usual gradient update with step size $\lambda$. The second-order form $$\text{prox}_{\lambda f}(x) = x- \left(\nabla^2f(x) + (1/\lambda)I\right)^{-1} \nabla f(x).$$ has a Tikhonov regularization with parameter $\lambda$, also known as the Levenberg-Marquard update. > Gradient and Levenberg-Marquard updates are proximal operators of first- and second-order approximations of $f$. Now also the fixed-point property makes intuitive sense: just run gradient updates until there's no change, and you'll end up at the minimum. This sets us up with the means to compute the minimum of $f+g$, both functions being closed proper convex, and $f$ being differentiable. Connecting the concepts introduced this note, we'll use the fixed-point optimization of $f$ with proximal operator of $g$: $$x^{k+1} \leftarrow \text{prox}_{\lambda^k g}\left(x^k- \lambda^k\nabla f(x^k)\right)$$ This update sequence (from step $k$ to $k+1$) is called Proximal Gradient Method (PGM). It constitutes a sequence of two proximal operators and is thus related to the method of [Alternating Projections](https://en.wikipedia.org/wiki/Projections_onto_convex_sets). It's simple and elegant, and works with any constraint $g$ (not just simple things like non-negativity). One critical aspect is the upper limit on $\lambda^k$: It is customarily set as $\lambda^k \in (0,1/L]$, where $L$ is the Lipschitz constant of $\nabla f(x^k)$, but it needs to be smaller than $2/L$. Otherwise, PGM will blow up rapidly without any warning! There a several tweaks to speed up the convergence, like over-relaxation and Nesterov acceleration, but I won't go into these here. This is just a ultra-short summary of the rich field of constrained optimization with proximal operators. I am absolutely stoked by what they allow me to do, which I will present in future notes.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9515779614448547, "perplexity": 425.19434723165193}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257649627.3/warc/CC-MAIN-20180324015136-20180324035136-00226.warc.gz"}

Subsets and Splits

No saved queries yet

Save your SQL queries to embed, download, and access them later. Queries will appear here once saved.