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https://mathematica.stackexchange.com/questions/63686/how-do-i-format-the-output-to-include-the-function-definition | # How do I format the output to include the function definition?
To explain, I do most of my probability homework in Mathematica, and use the typesetting built into Mathematica to finish answers. Often, I want to output a function as the answer to a problem, but I want it to be explicit, that is, in the example below, I create a function, X[x,y,z], and I want the result to not just display the (in this case) piecewise function, but I want it to define it as a function. As an example:
Clear[X, x, y, z]
X[x_, y_, z_] := Piecewise[{{3/(4*Pi), Sqrt[x^2 + y^2 + z^2] <= 1}}, 0]
The output of this is $$\left\{ \begin{array}{cc} \frac{3}{4 \pi } & \sqrt{x^2+y^2+z^2}\leq 1 \\ 0 & \text{True} \\ \end{array} \right.$$
I want it (the output) to look (something) like
$$X(x,y,z)=\left\{ \begin{array}{cc} \frac{3}{4 \pi } & \sqrt{x^2+y^2+z^2}\leq 1 \\ 0 & \text{True} \\ \end{array} \right.$$
• Perhaps HoldForm[X[x, y, z]] == X[x, y, z] // TraditionalForm? – wxffles Oct 20 '14 at 20:50
• Yeah, that seems to pretty much do what I want it to. I think I'd prefer it without the parenthesis it puts around the piecewise equation, but it does give me the result I want. Thanks! – Michael Witt Oct 20 '14 at 20:59
To expand my comment into an answer:
HoldForm[X[x, y, z]] == X[x, y, z] // TraditionalForm
$$X(x,y,z)=\left( \left\{ \begin{array}{cc} \frac{3}{4 \pi } & \sqrt{x^2+y^2+z^2}\leq 1 \\ 0 & \text{True} \\ \end{array} \right. \right)$$
But Mathematica decides it needs parentheses. I agree that this doesn't look ideal. There are a couple of ways around it. One is that you can manually edit the traditional form in Mathematica. Another option is to piece it together ourselves in a Row (with a goofy equals to nothing):
Clear[traditionalise];
$$X(x,y,z)=\left\{ \begin{array}{cc} \frac{3}{4 \pi } & \sqrt{x^2+y^2+z^2}\leq 1 \\ 0 & \text{True} \\ \end{array} \right.$$
• In Alpha, we usually do something like Row[{HoldForm[f], " \[LongEqual] ", f}] // TraditionalForm. – Chip Hurst Oct 21 '14 at 0:28
• @Chip Why not Row[{HoldForm[f], f}, "\[LongEqual]"]? I believe that uses the proper spacing rules. – Mr.Wizard Oct 21 '14 at 1:25
• Yes, you are correct. I guess I stretched the truth a bit. The idiom I referred to is used in other custom Row-like constructs. – Chip Hurst Oct 21 '14 at 2:13 | 2020-01-27 09:02:27 | {"extraction_info": {"found_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": 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": 0.5732019543647766, "perplexity": 1167.4597932688723}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251696046.73/warc/CC-MAIN-20200127081933-20200127111933-00163.warc.gz"} |
https://physics.com.hk/2010/02/04/ | — [email protected]
# Simcity
.
“By giving kids toys like this, I hope to give them some sense of what it might be like to (live on Earth) in 100 years,” Wright said as he discussed Spore. “That’s why I think toys can change the world.”
— James Brightman on Will Wright’s Spore
.
.
.
2010.02.04 Thursday $ACHK$ | 2022-08-08 18:44:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35465213656425476, "perplexity": 6941.644289964188}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570871.10/warc/CC-MAIN-20220808183040-20220808213040-00488.warc.gz"} |
https://www.physicsforums.com/threads/two-manifold-questions.536295/ | # Two manifold questions
1. Oct 3, 2011
### seydunas
Hi,
I have two questions: how can we prove a closed ball in R^n is manifold with boundry only using the definition being manifold with boundry. Also i want to ask C^/inf(M) is infinite dimensional where M is smooth manifold of dimension n>0.
Last edited by a moderator: Oct 4, 2011
2. Oct 3, 2011
### lavinia
Re: manifold
Would this work?
Stand the ball on a tangent n-1 plane an subtract the height of the lower half of the boundary from the n-1 plane from each point in the lower half ball.
3. Oct 3, 2011
### Jamma
Re: manifold
Much like you need two charts to cover the sphere, you will need two charts for the "unit ball with boundary".
Last edited: Oct 3, 2011
4. Oct 3, 2011
### quasar987
Re: manifold
Yes, $C^{\infty}(M)$ is an infinite-dimensional vector space.
5. Oct 3, 2011
### Bacle
Re: manifold
I'm not sure of the definition, but why not just produce charts for both the interior points and for the boundary points, i.e., show that the points in the (topological) boundary are also (in this case) part of the manifold boundary?
6. Oct 4, 2011
### seydunas
Re: manifold
C/inf(M) is infinite dimensional but how? I thought that for all point on M (one point is closed set) there exist open nhd, and by using partitions of unity we can extend the function on M , now i wonder that the set of theese functions is linearly independent or not? IF so, we are done.
7. Oct 4, 2011
### seydunas
Re: manifold
For manifold with boundary, how can we write the charts precisely?
8. Oct 4, 2011
### Bacle
Re: manifold
You write the charts just like you do for manifolds without boundary, only that you have interior charts and boundary charts.
9. Oct 5, 2011
### Jamma
To see that C^/infty(M) is infinite dimensional, just find an infinite set of linearly independent functions. For example, any "bump function" around a point with different variations of "steepness" could generate infinitely many such functions. | 2018-01-19 02:12:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7256290912628174, "perplexity": 1511.250142091693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1516084887692.13/warc/CC-MAIN-20180119010338-20180119030338-00332.warc.gz"} |
https://stackoverflow.com/questions/26803237/how-to-properly-minify-combine-css-js-in-web-projects-with-url-rewriting | # How to properly minify/combine CSS/JS in web projects with url rewriting
I've been struggling for hours trying to set up proper minification that actually rewrites urls. I've used useref and usemin, and they do good job of scanning html, aggregating all JS and CSS and outputting into one file. But, for the life of me, I cannot make the url rewrite to work properly. My structure is simple:
\root
index.html
application.css // minified
application.js // minified
\vendor
\bootstrap
\fonts // font files here
bootstrap.css // pre-minified
bootstrap.css refers to font files by using relative url - font/bootstrap_font.ttf When bootstrap gets minified, it lands as part of application css, that is in my root now, so the path would point from root to /font/bootstrap_font.ttf. Original directory hierarchy stays, so I would basically like to have this url rewritten to /vendor/bootstrap/font/bootstrap_font.ttf
And, oh, why cssmin task doesn't accept more than one file?
UPDATE Here's my current grunt file:
module.exports = function(grunt) {
grunt.initConfig({
useminPrepare: {
html: 'web/public/index.html',
options: {
dest: 'web/public-dist'
}
},
usemin: {
html: 'web/public-dist/index.html',
},
copy: {
all: {
files: [{
expand: true,
cwd: 'web/public/',
src: ['**'],
dest: 'web/public-dist/'
}]
},
resources: {
files: [{
expand: true,
cwd: 'web/public/',
src: ['**/*.*', '!**/*.js', '!**/*.css', '!**/*.txt'],
dest: 'web/public-dist/'
}]
}
},
uglify: {
options: {
mangle: true,
sourceMap: false,
compress: true,
banner: '/*! <%= pkg.name %> <%= grunt.template.today("yyyy-mm-dd") %> */\n'
},
standard: {
files: [{
expand: true,
cwd: 'web/public-dist/',
src: ['**/*.js'],
dest: 'web/public-dist/'
}]
}
},
cssmin: {
options: {
banner: '/*! <%= pkg.name %> <%= grunt.template.today("yyyy-mm-dd") %> */\n',
},
standard: {
files: [{
expand: true,
cwd: 'web/public-dist/',
src: ['**/*.css'],
dest: 'web/public-dist/'
}]
},
}
});
grunt.registerTask('package', [ 'copy:resources', 'useminPrepare', 'concat:generated','cssmin:generated', 'uglify:generated', 'usemin']);
};
In this form, cssmin cannot be even used as separately called target, because apparently its configuration is wrong - it complains that it cannot accept many files. What am I doing wrong here?
From the bits and pieces I've gathered, apparently it's crucial to change usemin flow and not allow it to concatenate all the css and cssmin later - because this way, it would obviously lose the vital information about the directory origin of every css file. I've tried changing the flow, but then it doesn't work because of the same cssmin error - cannot accept many files.
• does application.css contain more than just bootstrap.css? Nov 13, 2014 at 15:16
• It would be helpful to see the Gruntfile. Nov 13, 2014 at 16:11
• Well, yes, of course. It combines many scripts. I just used Bootstrap as example, but the solution has to be generic. Whatever it encounters, it has to be rewritten properly. I would imagine that the cssmin (or other plugin) would have to take some kind of "relativeRoot" parameter (in this case it would be my \root) to relate encountered links - so in this case : "what's the relation of fonts/fontfile.ttf to my given root which is \root? I'm currently in \root\vendor\bootstrap. What should the link look like, if I were in \root?" Nov 13, 2014 at 16:11
• Updated original question with some more data and Gruntfile. Nov 13, 2014 at 16:17
• Question: You want one file that contains both the css and js? Nov 13, 2014 at 16:22
Okay, I has definitely same problem when started to build my css and js with grunt. Here is my solution of "relative urls" problem. Please, note that this post is not answering your actual question, but provide another way of problem solution. I have even more nested folder structure but it works well for me, and hope it helps you.
The gist is to build all css/js to another folder and copy assets files relatively to this new folder. Let give "build" name for it:
\root
\build
application.css - minified
application.js - minified
\fonts
...
\img
...
...
index.html
\ ...
Using grunt-contrib-copy plugin copy all your assets to /build/assets directory without breaking their original structure. So relative passes for your css saves, fonts are still in ./fonts/ folder.
The problem you'll faced to with such approach is saving folder structure for assets. Well, it is solved with detalization of your build configuration in your gruntfile. Now you can not say "okay grunt, build all /**/*.css files to application.css" but have to describe different cases for different options of file structures. If your project have obvious and logical file structure it is not complicated to add them.
I used rule that every css file must have assets directory as it sibling. So gruntfile expanded just by several lines and build structure look something like this
\root
\build
\css
\assets
\fonts
\img
application.css - minified //all relative passes saved
\foo
\bar
\biz
\assets
\fonts
first.css
\row
\assets
\img
second.css
index.html
Obviously you must have assets names naming rules to prevent overriding files.
Hope, this helps you
• Sorry, it doesn't, but thanks. It's not a solution I can accept, you've just worked around the problem. Nov 18, 2014 at 13:20 | 2023-03-26 07:10:29 | {"extraction_info": {"found_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": 0.4848780930042267, "perplexity": 6253.060862810806}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00326.warc.gz"} |
https://byjus.com/inverse-square-law-formula/ | # Inverse Square Law Formula
Inverse Square Law says that the strength of light (intensity) is proportional inversely to the square of the distance.
Inverse Square Law Formula is articulated as
$I\propto \frac{1}{d^{2}}$
Where the distance is d, the intensity of the radiation is I.
At distances d1 and d2, I1 and I2 are intensities of light respectively. Then Inverse square law is articulated as:
$\frac{I_{1}}{I_{2}}\propto \frac{d_{2}^{2}}{d_{1}^{2}}$
Inverse square law formula is handy in finding distance or intensity of any given radiation. The intensity is articulated in Lumen or candela and distance is given in meters. It has widespread applications in problems grounded on light.
Inverse Square Law Solved Examples
Underneath are some problems based on an inverse square law which may be useful for you.
Problem 1: The intensity of a monochromatic light are in the ratio 16:1. Calculate the second distance if the first distance is 6m?
Known:
I1 : I2 = 16 : 1,
d1 = 6m,
d2 =?
Distance, $d_{2}=\sqrt{\frac{I_{1d_{1}^{2}}}{I_{2}}}$ $\frac{16\times 6}{1}$ = 9.8m
Problem 2: Compute the intensity of radioactive source antimony 124 if it has the intensity of 80 milliroentgen/hour for 50 feet. At 10 foot, what will be its intensity?
Intensity, $I_{2}=\sqrt{\frac{I_{1d_{1}^{2}}}{d_{2}}}$
= $\frac{80\times 50^{2}}{10^{2}}$ | 2020-01-17 19:12:22 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8696379661560059, "perplexity": 1336.7477617895372}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250590107.3/warc/CC-MAIN-20200117180950-20200117204950-00508.warc.gz"} |
https://plainmath.net/13898/separated-illuminated-having-wavelength-interference-pattern-observed | Question
# A pair of slits, separated by 0.150 mm, is illuminated by light having a wavelength of ? = 561 nm. An interference pattern is observed on a screen 122
Other
A pair of slits, separated by 0.150 mm, is illuminated by light having a wavelength of ? = 561 nm. An interference pattern is observed on a screen 122 cm from the slits. Consider a point on the screen located at y = 2.00 cm from the central maximum of this pattern.
(a) What is the path difference ? for the two slits at the location y?
(b) Express this path difference in terms of the wavelength.
2021-03-28
our formula for double slit interference is:
$$\displaystyle{d}{\sin{{\left({t}\right)}}}={m}{\left({w}\right)}$$
where d is the width of the slits
where t is the angle from the slits
where m is the corresponding fringe from the central fringe
where w is the wavelength
using trig, we can find $$\displaystyle{\sin{{\left({t}\right)}}}$$ to be:
$$\displaystyle{\sin{{\left({t}\right)}}}={\frac{{{y}}}{{{L}}}}$$
where y is the distance from central fringe to corresponding fringe
where L is the distance from slit to screen
our formula now becomes:
$$\displaystyle{d}{\left(\frac{{y}}{{L}}\right)}={m}{\left({w}\right)}$$
where both sides represent the path difference:
path difference=d(y/L)
path difference = .150e-3(2e-2/122e-2)
path difference = 2.459e-6 m
In terms of the wavelength:
2.459e-6/561e-9 = 4.4w | 2021-09-18 08:43:32 | {"extraction_info": {"found_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": 0.5350487232208252, "perplexity": 1202.2359393020154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780056348.59/warc/CC-MAIN-20210918062845-20210918092845-00129.warc.gz"} |
https://math.stackexchange.com/questions/3189821/fixed-point-iterations-for-quadratic-function-x-mapsto-x2-2/3189990 | # Fixed-point iterations for quadratic function $x\mapsto x^2-2$
Let $$f(x)$$ be $$x^2-x-2$$. I want to find the root using FPI in an interval where it will converge. I have chosen $$g(x)=x^2-2$$ and so $$g'(x)=2x$$. The convergence condition, $$|g'(x)|<1$$ is obviously satisfied in $$-0.5.
Problem: I have failed to find a consistent interval outside of this convergence interval up to $$\pm 1$$ for which the iteration consistently diverges.
Question: Does this convergence condition only guarantee convergence (in the bounds) but not divergence (outside the bounds)?
• Is that fixed-point iteration fixed? From $x^2=2+x$ one finds the better iteration $x_{n+1}=\sqrt{2+x_n}$ for the positive root. – LutzL Apr 16 at 16:25
• Yes, but I thought the reason it’s ‘better’ is because it satisfies abs(g’(x))<1 in some interval. But g(x) in op works just fine up to -+1. – AKubilay Apr 16 at 18:10
• $g(x)=\sqrt{2+x}$ has $g'(x)=\frac1{2\sqrt{2+x}}\le\frac25$ for all $x>0$, $g(x)=1+\frac2x$ has $g'(x)=-\frac{2}{x^2}>-1$ for $x>\frac32$, so you get intervals with $|g'(x)|<1$ for many fixed-point functions. – LutzL Apr 16 at 18:53
Let us consider the fixed point iterations associated to the function $$g: x \mapsto x^2-2$$, defined by the quadratic map $$x_{n+1} = {x_n}^2 - 2, \qquad x_0 \in \Bbb R .$$ This map has many periodic points, even with large period. The period-one fixed points $$-1$$, $$2$$ are both repelling fixed points (indices $$2>1$$ and $$4>1$$, respectively). Thus, fixed-point iterations will not converge towards these values unless the starting value $$x_0$$ is exactly equal to $$-1$$ or $$2$$. Setting $$y_n = -\frac{1}{4} x_n + \frac{1}{2}$$, the logistic map $$y_{n+1} = r y_n (1-y_n)$$ with parameter $$r=4$$ is obtained, which exact solution is bounded and exhibits chaotic behavior (see also this article). Therefore, $$x_n = 2\cos\left(2^n\cos^{-1}(x_0/2)\right)$$ is bounded and exhibits chaotic behavior too (the sequence does not diverge to infinity). | 2019-06-20 23:36:52 | {"extraction_info": {"found_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": 20, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9965596199035645, "perplexity": 898.9354038564992}, "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-2019-26/segments/1560627999291.1/warc/CC-MAIN-20190620230326-20190621012326-00092.warc.gz"} |
https://cs.stackexchange.com/questions/37280/create-cfg-and-pushdown-automaton-for-ww?noredirect=1 | # Create CFG and pushdown automaton for {ww} [duplicate]
I've been trying to make a CFG, a pushdown automaton and a regular expression for the language
$\qquad L(M) = \{ww : w \in \{a, b\}^*, |w| \text{ is even}\}$.
I understand how the reverse of the string work, that is
$\qquad L' = \{ ww^R : w \in \{a, b\}^*\}$,
what i am asking for is to do it this way , i have already solved (L') : http://i921.photobucket.com/albums/ad53/Johann_1990/IMG_20150117_132616.jpg
but is there is a way to solve this one too?
$\qquad L(M) = \{ww : w \in \{a, b\}^*, |w| \text{ is even}\}$.
For example, $abaaba \in L$ with $w = aba$.
## marked as duplicate by D.W.♦, Rick Decker, David Richerby, Luke Mathieson, JuhoJan 17 '15 at 9:58
• Note that $ww = abaaba$ is a bad example (as in, you don't learn anything from it) here as $w = w^R$. – Raphael Jan 16 '15 at 14:58
• @Raphael , i need to train my self for any example.. when i am learning ! i know how to make the reverse of string pushdown or cfg, i was just trying something new! Thank you ! – AaoIi Jan 16 '15 at 20:52
• @D.W.,its not the same question am talking about another thing completely.. – AaoIi Jan 16 '15 at 20:53
You can not do so as $L$ is not context-free. | 2019-05-21 23:23:38 | {"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": 0.47804927825927734, "perplexity": 1079.6523562612133}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1558232256586.62/warc/CC-MAIN-20190521222812-20190522004812-00116.warc.gz"} |
https://mathprelims.wordpress.com/2008/10/25/independence-and-conditional-probability/ | # Mathematics Prelims
## October 25, 2008
### Independence and Conditional Probability
Filed under: Measure Theory,Probability Theory — cjohnson @ 6:39 pm
Suppose that $(\Omega, \mathcal{F}, P)$ is a measure space, $A$ a measurable set with $P(A) \in (0, \infty)$. We can create a new measure space $(A, \mathcal{F}_A, P_A)$ where
$\displaystyle \mathcal{F}_A = \{ S \cap A : S \in \mathcal{F} \}$
$\displaystyle P_A = \frac{P(E)}{P(A)}$
Note that $(A, \mathcal{F}_A, P_A)$ is a probability space, as $P_A(A) = 1$, and any other set $E \in \mathcal{F}_A$ is a subset of $A$.
Supposing $(\Omega, \mathcal{F}, P)$ is a probability space, we can use this new probability space in our definition of conditional probability. The probability $P_A(E)$ represents the probability of $E$ occurring, where we already know $A$ has occurred. Normally, instead of going through the trouble of writing out a new sigma-algebra and probability measure each time, we simply take $P(B|A)$ to be the probability of $B \cap A$ using the $P_A$ measure defined above. Of course, our measure and sigma-algebra are so simple that we can just write this in one line as
$\displaystyle P(B|A) = \frac{P(B \cap A)}{P(A)}$
We call this the probability of $B$ given $A$. Now if $P(B|A) = P(B)$, we say that $A$ and $B$ are independent events. If this is the case then we have
$\displaystyle P(B) = \frac{P(B \cap A)}{P(A)}$
$\displaystyle \implies P(A) P(B) = P(B \cap A)$
This is certainly a useful property as it makes proofs of interesting facts fall out easily when we consider sequences of independent random variables (a related idea) later.
Now consider the fact that $P(A|B) = \frac{P(A \cap B)}{P(B)}$ implies $P(A \cap B) = P(A|B)P(B)$. Plugging into the formula for $P(B|A)$ we arrive at the following, known as Bayes’ theorem.
$\displaystyle P(B|A) = \frac{P(B \cap A)}{P(A)} = \frac{P(A|B) P(B)}{P(A)}$
Note that $P(A|B) \neq P(B|A)$. | 2017-08-17 23:10:29 | {"extraction_info": {"found_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": 30, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9717243909835815, "perplexity": 254.44768514429725}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104172.67/warc/CC-MAIN-20170817225858-20170818005858-00054.warc.gz"} |
https://citationsy.com/archives/q?doi=10.1039/A800796I | ← Back to Search
Bulk And Surface Characteristics Of Pure And Alkalized Mn2O3: TG, IR, XRD, XPS, Specific Adsorption And Redox Catalytic Studies
M. Zaki, M. Hasan, L. Pasupulety, K. Kumari
Published 1998 · Chemistry
α-Mn2O3 (containing a minor proportion of Mn5O8) was obtained by calcination of pure MnO2 at 700°C for 2 h. It was alkalized by impregnation of the parent dioxide with potassium and barium nitrate solutions prior to the calcination. K-Mn2O3 (α-Mn2O3+KMn8O16) and Ba-Mn2O3 (α-Mn2O3) thus respectively produced were subjected, together with the unmodified Mn2O3, to the title bulk and surface characterization techniques. It has been implied that the alkalization improves the electron density and the mobility of lattice and surface oxygen species. As a result, the bulk thermochemical stability is reduced on heating in a CO atmosphere, and a capacity towards CO2 uptake is developed. Moreover, the surface catalytic behaviour towards CO oxidation in the gas phase is maintained, and the behaviour towards H2O2 decomposition in the liquid phase is considerably promoted.
This paper references
This paper is referenced by
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https://kyushu-u.pure.elsevier.com/ja/publications/on-the-quantum-su2-invariant-at-qexp4%CF%80-1n-and-the-twisted-reideme | # On the Quantum SU(2) Invariant at q=exp(4π√-1/N) and the Twisted Reidemeister Torsion for Some Closed 3-Manifolds
2 被引用数 (Scopus)
## 抄録
The perturbative expansion of the Chern–Simons path integral predicts a formula of the asymptotic expansion of the quantum invariant of a 3-manifold. When q=exp(2π-1/N), there have been some researches where the asymptotic expansion of the quantum SU (2) invariant is presented by a sum of contributions from SU (2) flat connections whose coefficients are square roots of the Reidemeister torsions. When q=exp(4π-1/N), it is conjectured recently that the quantum SU (2) invariant of a closed hyperbolic 3-manifold M is of exponential order of N whose growth is given by the complex volume of M. The first author showed in the previous work that this conjecture holds for the hyperbolic 3-manifold Mp obtained from S3 by p surgery along the figure-eight knot. From the physical viewpoint, we use the (formal) saddle point method when q=exp(4π-1/N), while we have used the stationary phase method when q=exp(2π-1/N), and these two methods give quite different resulting formulas from the mathematical viewpoint. In this paper, we show that a square root of the Reidemeister torsion appears as a coefficient in the semi-classical approximation of the asymptotic expansion of the quantum SU (2) invariant of Mp at q=exp(4π-1/N). Further, when q=exp(4π-1/N), we show that the semi-classical approximation of the asymptotic expansion of the quantum SU (2) invariant of some Seifert 3-manifolds M is presented by a sum of contributions from some of SL 2C flat connections on M, and square roots of the Reidemeister torsions appear as coefficients of such contributions.
本文言語 英語 151-204 54 Communications in Mathematical Physics 370 1 https://doi.org/10.1007/s00220-019-03489-2 出版済み - 8 1 2019
## All Science Journal Classification (ASJC) codes
• 統計物理学および非線形物理学
• 数理物理学
## フィンガープリント
「On the Quantum SU(2) Invariant at q=exp(4π√-1/N) and the Twisted Reidemeister Torsion for Some Closed 3-Manifolds」の研究トピックを掘り下げます。これらがまとまってユニークなフィンガープリントを構成します。 | 2021-08-03 06:00:38 | {"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": 0.818816602230072, "perplexity": 652.7738260099998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046154420.77/warc/CC-MAIN-20210803030201-20210803060201-00308.warc.gz"} |
https://help.octopus.com/t/azure-deployment-step-multiple-sites-for-the-same-environment/9356 | # Azure Deployment Step - Multiple sites for the same environment
Hey what is the process to deploy a package to multiple sites for a single environment? With the azure web machines this made sense as you would just have multiple web app machines in a single environment.
We also typically have our production sites in a separate subscription. I can see I can do a custom binding, but then I have to have a variable for the account id? which seems to not be very human friendly?
Cheers,
Luke
Hi Luke,
There are two ways to achieve this:
• Multiple steps
• Multiple environments
Personally, I prefer the second option. Have an environment for each Azure Web App (possibly the same package to different regions?). If you name your environments and sites appropriately, in the step you can use something like AcmeOnline-#{Octopus.Environment.Name} in the Web Site field.
Regarding accounts, it is the same answer. You can either have multiple steps (each configured with a different account), or use environments and variable binding.
For convenience, you can then create a lifecycle for your project that will allow you to deploy to all those environments at once.
I do understand that for your scenario, the 3.0 method (with Azure targets) seems more elegant.
A bit of background:
We had many discussions internally regarding this. Our dilemma was that there are two distinct ways that people manage their Azure environments. There are those (and from your question I’m assuming you’re more in this category) that have known ‘targets’ that they deploy to repeatedly. Then there are those that dynamically create and destroy their targets regularly. In the long-term, we don’t want two distinct ways of managing Azure environments in Octopus (because it’s confusing to users and more complex to maintain). And the 3.0 way was unusable for those in the ‘dynamic’ group. The 3.1 way is more powerful, and supports both usage modes.
I hope that helps. If we can be of any further assistance, you know where to find us.
Regards,
Michael
So if I had two sites in a single environment,
i.e. production has the primary and a secondary for DR.
Do I then need to use multiple steps to deploy to both? or can I set an array of values and it will process the step multiple times for each set of variables? akin to how we can have multiple machines of the same role in an environment.
Oh and for accounts can I reference them by name or do I need to use the id of an account? Which makes it harder to recognise in the variables.
Hi Luke,
No, you can’t use an array of sites.
I would still probably create an environment for each site (e.g. Production, DR). This means you would have to deploy your release to both environments.
Again, you could leverage Lifecycles: You could have both your DR and Production environments in the same lifecycle phase, and set them to deploy automatically when that phase is reached. This works if your previous phase is, for example, a staging environment, and there is a manual step to sign-off on the production release. Once that step is approved, the Production and DR environments would both be automatically deployed.
That’s just an example of one way to set it up.
Regarding accounts, it is the ID not the name (in case you change the name).
But the ID’s are somewhat human-friendly. For example, if your account was named (originally) foo, your ID will be azuresubscription-foo.
I’m trying to figure this out too. We currently have two logical environments: Dev, Production. I’ve just added a second Azure Web App in a new region that will be used by Azure’s load-balancer. My site should be deployed to both regions at the same time. It feels a bit strange to create a new environment in this situation.
I actually think being allowed to provide an array of site names in the Azure Web App step is the cleanest way to go with this setup since everything else is identical. For now I’ll probably just duplicate the step and set it to only run in the Production environment.
Hi Brian,
Thanks for the feedback.
We are currently bouncing around some ideas to make this scenario nicer. For now, your solution should work just fine.
Regards,
Michael
@Michael, This is something we need sorely as well, we have multiple web apps in a redundant manner in multiple azure regions. The environment approach in 3.0 allowed us to just use roles for this :(. If we could pass a comma separated list of webapp names (or an array or whatever), or somehow do something similar to the 3.0 version of adding a webapp definition to a role that would solve this problem.
Ed (actually, everyone on this thread),
If you have a few spare minutes, take a look at our recent post on the solution we are proposing for multi-tenancy (https://octopus.com/blog/rfc-multitenancy). In particular, we believe the idea of environment tags may fit this scenario. Basically, this would allow you to create one environment per Azure location, but then tag them all as, say, ‘Production’, and treat them similarly. e.g. deploy to them all together.
We’d be interested to hear your thoughts on this.
Luke, Brian, Ed (and anyone else interested in this issue),
We would love your opinions on a feature we are proposing: Cloud Region targets.
We are hoping this new target type may provide a solution for the issue discussed in this thread. Your comments on the blog post would be most welcome!
It feels mostly there, but I’m not sure. My initial reaction is that is a little janky, but that may just be me starting to process it through.
I replied over on the Cloud Regions Blog link above in Disquss, but just for completeness, thought it might be worth repeating here:
We are a new Octopus Enterprise License customer and we have an almost identical need to Jason’s .
Our environment is as follows:
All Azure
2 Environments (UAT & PROD) with potentially another testing environment to come online for performance and automated regression testing
6 Regions per environment (for UAT and PROD anyway)
Multiple Cloud Services (worker/web roles) and Web App projects
We’re currently using a mix of TeamCity driven Powershell scripts and manual steps and are trying to move to a mature process with Octopus, but given our topology, it’s clearly no small task to get started, especially considering the 6 regions.
Since we have yet to build out our OD configuration, we’d love to know a bit more about the Cloud Regions and when a build with support for them will be available for us to try.
Any guidance you can offer in terms of short term and beyond would be greatly appreciated.
Matt,
I believe the Cloud Region targets will be ideal for your scenario. Obviously the only drawback is they aren’t currently available. They will be in our next minor release (3.4), which is at least a month away from full release. They will be available in the next alpha\beta release if would like to have a play; keep an eye on the blog or Twitter for notification.
My suggestion for an intermediate solution:
Create your environments (UAT and PROD). Create your projects containing your cloud-service and web-app steps, and duplicate those steps per region. i.e. one Azure step per region.
This won’t be as nice as having the cloud-regions. The main drawbacks will be:
• your regions will be deployed sequentially, rather than in parallel.
• you will have to maintain many steps
Then once 3.4 is released, your migration process will be to create your cloud-region targets and remove the duplicate Azure steps.
I hope this helps. Don’t hesistate to ask if there’s anything else we can do to help. | 2022-07-02 14:54:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.25261756777763367, "perplexity": 1378.4579976926223}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20220702131941-20220702161941-00364.warc.gz"} |
https://math.stackexchange.com/questions/735483/trace-of-tensor-product-vs-tensor-contraction/735617 | # Trace of tensor product vs Tensor contraction
I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality:
$$\text{Tr}(T_1\otimes T_2)=\text{Tr}(T_1)\text{Tr}(T_2)\;\;\;...(1)$$
The Wiki page "Tensor Contraction" speaks of tensor contraction as some generalization of trace, though without providing any formulation or example.
My questions: How do they all work? What is trace for a tensor? How does such trace interact with tensor product?
In particular, I have this contraction: (following Einstein's summation convention) $$F^{\mu\nu}F_{\mu\nu}$$ where $F$ is a rank-2 tensor and each $F_{\mu\nu}$ is a $4\times 4$ matrix. Can it be expressed as trace of some sort? Subsequently can I apply (1) to split the expression into product of, say, the trace of $F$?
Additionally, $F^{\mu\nu}$ is anti-symmetric and I am trying to prove the above equals to zero. So being able to use (1) can be awesome.
Note: I have some although limited background in differential geometry and algebra. English words are great. But please supply formal definitions as well. At the same time, explanations with as little abstract algebraic constructions as possible would be much appreciated. Focus on finite dimension is fine. Extension to separable Hilbert space, partial trace and etc is welcomed too.
EDIT: The second floor to in this post seems to be good. I don't understand, however, how tensor product of matrices work? Still, when does contraction come into play?
• Possible duplicate of How to compute the trace of a tensor? Sep 18, 2018 at 17:05
• Nope. Not even close. (If you are interested, kindly read the answer below? I haven't gotten a chance.) Sep 19, 2018 at 18:31
• $F^{\mu\nu}$ is vector-like (all horizontal) as well as $F_{\mu\nu}$ (all vertical), so tracing over their outer product is equivalent to the inner product. Feb 2, 2021 at 17:11
I try to answer starting from the case of square matrices. There is some care to take while considering a "hidden" isomorphism of vector spaces. In any case, let $V$ be a finite dim. vector spaces over a field $\mathbb K$ (for simplicity $\mathbb R$ ), with basis $\{e_i\}$ of cardinality $n$.
It is well known that there exists an isomorphism of vector spaces $$\Phi:\operatorname{Hom}_\mathbb K(V,V)\rightarrow V^{*}\otimes V,$$
with $$\Phi(\phi)=a_{ij}f_i\otimes e_j,$$ where $\phi\in \operatorname{Hom}_\mathbb K(V,V)$ and $\phi(e_i):=a_{ij}e_j$ for all $i,j=1,\dots,n$. $\{f_i\}$ is the dual basis on $V^{*}$ of the basis $\{e_i\}$ on $V$, i.e. $f_i(e_j)=\delta_{ij}$.
We use the Einstein convention for repeated indices.
We know how to define the trace operator $\operatorname{Tr}$ on the space $\operatorname{Hom}_\mathbb K(V,V)$; the trace is computed on the square matrix representing each linear map in $\operatorname{Hom}_\mathbb K(V,V)$. Let us move to the r.h.s. of the isomorphism $\Phi$.
• trace operator on $V^{*}\otimes V$
Let $$\operatorname{Tr}_1: V^{*}\otimes V\rightarrow \mathbb K,$$
be given by $\operatorname{Tr}_1(g\otimes v):=g(v)$.
Lemma $\operatorname{Tr}_1$ is linear and satisfies $$\operatorname{Tr}_1\circ \Phi=\operatorname{Tr}.$$
proof: just use definitions.
• trace operator on $(V^{*}\otimes V)\otimes\dots\otimes (V^{*}\otimes V)$
Using the $n=1$ case we introduce
$$\operatorname{Tr}_n: \underbrace{(V^{*}\otimes V)\otimes\dots\otimes (V^{*}\otimes V)}_{n-\text{times}} \rightarrow \mathbb K,$$
with $\operatorname{Tr}_n(f_1\otimes v_1\otimes\dots\otimes f_n\otimes v_n):=\prod_{i=1}^n f_i(v_i)$.
Lemma $\operatorname{Tr}_n$ is linear and invariant under permutations on $(V^{*}\otimes V)^{\otimes n}$; it satisfies $$\operatorname{Tr}_n\left(\Phi(\phi_1)\otimes\dots\otimes\Phi(\phi_n)\right)=\prod_{i=1}^n \operatorname{Tr}(\phi_i),$$ for all $\phi_i\in \operatorname{Hom}_\mathbb K(V,V)$.
proof: we prove the second statement. We introduce the notation $$\Phi(\phi_k):= a^k_{i_kj_k}f_{i_k}\otimes e_{i_k}\in V^{*}\otimes V,$$ for all $k=1,\dots,n$. We arrive at $$\operatorname{Tr}_n\left( (a^1_{i_1j_1}f_{i_1}\otimes e_{i_1})\otimes\dots\otimes (a^n_{i_nj_n}f_{i_n}\otimes e_{i_n})\right)=a^1_{i_1j_1}\dots a^n_{i_nj_n}f_{i_1}(e_{i_1})\dots f_{i_n}(e_{i_n})=\text{remember the definition of dual basis}= a^1_{i_1j_1}\dots a^n_{i_nj_n}\delta_{i_1j_1}\dots\delta_{i_nj_n}= a^1_{i_1i_1}\dots a^n_{i_ni_n}\\=\prod_{i=1}^n \operatorname{Tr}(\phi_i),$$ as claimed.
• @Argyll did it help? Jul 28, 2014 at 18:02
• $a_{ij}$ should be changed to be $a_{ji}$ @Avitus
– yang
Sep 7, 2014 at 3:21
• @Avitus: Yes it does. Thank you. But I'm still not sure about the connection between contraction and tensor product or any mathematical objects. Sep 16, 2014 at 1:13
• In which sense? Maybe you can extend your OP a bit Sep 16, 2014 at 7:47
• Good answer, though $$\Phi(\phi_k):= a^k_{i_kj_k}f_{i_k}\otimes e_{i_k}\in V^{*}\otimes V,$$ should be $$\Phi(\phi_k):= a^k_{i_kj_k}f_{i_k}\otimes e_{j_k}\in V^{*}\otimes V,$$ and thereon afterwards. Apr 26 at 13:11 | 2022-08-08 21:52:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9346418976783752, "perplexity": 234.35202909885783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882570879.1/warc/CC-MAIN-20220808213349-20220809003349-00404.warc.gz"} |
https://3dprinting.stackexchange.com/questions/6331/heatbed-with-zones/11607 | # Heatbed with zones?
Waiting for a heatbed to get up to 85˚C for a relatively small part got me wondering why beds aren't hardware/G-code configurable for what area is heated? I'm sure it would be an increase in parts costs and electronics, but it seems that being able to just heat an area a little larger than the part(s) being built would save in time and energy use.
• That is a very good question. It would require switchable sections of the PCB track, but how would those sections be configured? Differing radii of circular PCB tracks? or if in square blocks how would they be arranged, and how many (in halves, quarters, eighths etc.)? I guess the permutations for a square one are a bit more complex and numerous to be practicable. Doing it for a round heatbed seems simpler though. Also, how big would the market be, and would the added complexity in electronics (i.e. cost) and code be worth it. It seems a good idea though... Design a prototypePCB in EagleCAD? – Greenonline Jul 8 '18 at 21:35
• @Greenonline RepRapFirmware already has an optional parameter (H for M140) to select heatbed, this could possibly be used for that. For Marlin it will require some extra coding as H is not yet a supported parameter. Nice project! – 0scar Jul 9 '18 at 9:31
• You must learn patience, grasshopper. No, seriously: when you're printing anything that takes longer than 30 minutes to complete, the heating time is insignificant. – Carl Witthoft Jul 9 '18 at 15:21
• Well, not just all about patience ... do we really need a full 200W (~) element heating up non-used space for hours (days)? – Jer Jul 10 '18 at 0:03
• @Jer well, Using the whole heating element the heat bed reach the printing temperature in 3 or 5 minutes, using the half heating element the time increases between 8 to 12 minutes. I don't want to wait to get roots on my feet. – Fernando Baltazar Jul 11 '18 at 5:21
I've wondered that myself a while ago and fact is that such beds or silicone heating pads do exist. Usually these are quite large (and expensive) and usually referred to as "dual zone heat beds/pads".
As far as energy consumption; less area to heat is faster heat up times (depending on the control) and less energy consumed. For small prints this may be beneficial. The price of such beds are very high, so to break even you would have to print a lot. An alternative to buying would be to etch your own bed.
• Thanks. As I've been thinking over people's answers I can see you'd want to segment the heated areas in a number of ways: ideally a bit of a heat break, some kind of switching to turn on/off zones, they'd need their own thermistor, and finally something to control/interpret all of this back to the main board. Not something you'll find any time soon on a sub-\$200 printer. – Jer Jul 20 '18 at 19:56
The heated bed is a reasonably good thermal conductor, so the difference in energy between heating all or 10% of the bed (assuming a 3x3 grid split 1 and 8) isn't going to be that significant.
In terms of heating speed, 200W across the whole bed will heat it faster than 40W applied to the centre square, and will also be less likely to cause warming or heat cycling effects (unless PSU regulation is a problem).
If the bed is much larger, or has thermally separated zones, then there might be some justification in the increased control complexity.
As an example, although glass has a thermal conductivity around 1% of a metal, it is still 30x better conducting than air. Conductivity towards the unheated areas of even a solid glass bed will roughly match the surface loss - so best case you would reduce losses to somewhere 20%-50% of what heating the whole bed might cost (assuming the same 3x3 grid).
• I had the same concerns as you about conduction, but not come to putting it in words! The only thing I can think of to dispute this is thermal silicone heating pads (on an insulated body) in conjunction with a glass bed, glass is a poor heat conductor. Nice addition! – 0scar Jul 11 '18 at 12:22
As the etched version is very impressive (thanks 0scar for that), there are other possibilities to build zones on the bed:
1. Using resistance wire;
2. Using a etched bed with zones.
The challenge with zones is mainly down to:
1. When and how to switch on/off particular zone;
2. Temperature control needs to be added to every zone to avoid over/under heat in the particular zone.
This last point brings even more challenges as that requires a PWM channel and a temp sensor (per zone), so standard RAMPS need some extensions in the wiring. One could overcome that using custom G-Code to set on/off zones and a double temp sensor for the main zone to follow the temperature changes.
In detail: a dedicated Arduino with PWM outputs that will read the temperature from a secondary temperature sensor in the main zone and follows it.
• The commercial dual zone beds come with a controller to do that, indeed a simple Arduino should be able to do that. You could even alter the Marlin bed heating code to include an extra parameter that determines which bed to heat up. It seems that reprap firmware already has such functionality (parameter H). Recent versions of RepRapFirmware also provide an optional 'H' parameter to set the hot bed heater number. If no heated bed is present, a negative value may be specified to disable it. – 0scar Jul 9 '18 at 9:24
• thanks for that @0scar, I have a 500mm X 500mm bed, so I am thinking to slice it in 100mm zones, that gives 5 zones to be controlled, and RAMPS will not handle that without extending it as that requires at last 8 extra pins and some extra computing power which we know is limited. – profesor79 Jul 9 '18 at 10:11
• 100 mm squared makes 25 zones ;) What about control using i2c? – 0scar Jul 9 '18 at 10:19
• it extends by 100mm so we have 5 elements – profesor79 Jul 9 '18 at 10:20
• A rotary switch can help 2P6T, also are called selectors. – Fernando Baltazar Jul 11 '18 at 5:29
it is probably too late for the party but i thought about grid heated bed recently. about controlling which zone to turn on and off, i think we can treat the grid as Led matrix, the matrix led signal can be used to drive mosfets to power the heat pads.
It makes more sense to insulate "unused" areas, since the heat conduction goes with the surface area. Heating the plate at some place (instead allover the area) makes no big difference (aluminium is a very good heat conductor, this is already mentioned). But insulation will do. I would propose a downside full insulation (5-10 mm) and some "windows-shaped" square silicone (other thin material?) mats to cover the unused areas (mostly on the outer side of the bed) on the top side. Beware of collisions and auto-leveling algorithms (starting very often at corners). Side effect: you may reach much higher temperatures at the uncovered areas (might be doubled if covering half of the surface). | 2021-03-03 02:28:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.3628980815410614, "perplexity": 2064.090716276044}, "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-2021-10/segments/1614178365186.46/warc/CC-MAIN-20210303012222-20210303042222-00179.warc.gz"} |
https://orgmode.org/worg/exporters/beamer/presentation.html | # Writing Beamer presentations in org-mode
## Introduction
### Overview
This presentation provides an illustration of some of the capabilities of the Beamer export in org mode:
1. simple slides (this one),
2. slides with special blocks,
3. multi-column slides and
4. the use of Babel for literate programming.
This file should be exported using M-x org-export-dispatch specifying l for \LaTeX{} and then P, for instance, to generate the PDF.
## Methodology
### A simple slide
This slide consists of some text with a number of bullet points:
• the first, very important, point!
• the previous point shows the use of bold emphasis which is translated to a \alert{} directive in \LaTeX.
The above list could be numbered or any other type of list and may include sub-lists.
### A more complex slide
This slide illustrates the use of Beamer blocks. The following text, with its own headline, is displayed in a block:
• Org mode increases productivity B_theorem
• org mode means not having to remember \LaTeX commands.
• it is based on ASCII text which is inherently portable.
• Emacs!
\hfill $$\qed$$
### Two columns
• A block BMCOL
• this slide consists of two columns
• the first (left) column has no heading and consists of text
• the second (right) column has an image and is enclosed in an example block
• A screenshot B_example BMCOL
### Babel
This slide shows some code and resulting output using Babel. Note the specification of BEAMER_act property for the second column.
• Octave code BMCOL B_block
A = [1 2 ; 3 4]
b = [1; 1];
x = A\b
• The output BMCOL B_block
A =
1 2
3 4
x =
-1
1
## Conclusions
### Summary
• org is an incredible tool for time management
• but it is also excellent for writing and for preparing presentations
• Beamer is a very powerful \LaTeX{} package for presentations
• the combination is unbeatable!
Documentation from the orgmode.org/worg/ website (either in its HTML format or in its Org format) is licensed under the GNU Free Documentation License version 1.3 or later. The code examples and css stylesheets are licensed under the GNU General Public License v3 or later. | 2018-04-24 23:01:44 | {"extraction_info": {"found_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": 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": 0.6588693857192993, "perplexity": 4898.759088018779}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1524125947421.74/warc/CC-MAIN-20180424221730-20180425001730-00361.warc.gz"} |
https://www.codingame.com/playgrounds/29924/computing-with-data/classification | Computing with Data
elgeish
240.9K views
Classification
Using the iris dataset, we implement a binary classifier that predicts whether a sample is an Iris-Versicolor (denoted by the label 1) or not: | 2020-06-01 03:13:19 | {"extraction_info": {"found_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": 0.24632690846920013, "perplexity": 2422.9965177524427}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347413901.34/warc/CC-MAIN-20200601005011-20200601035011-00579.warc.gz"} |
http://mathhelpforum.com/number-theory/47487-prime-number-conjecture-print.html | # prime number conjecture
• September 2nd 2008, 02:26 PM
rmpatel5
prime number conjecture
disprove the conjecture: There are infinitely many prime numbers expressible in the form n^3 +1 where n is a positive integer
• September 2nd 2008, 02:49 PM
ThePerfectHacker
Quote:
Originally Posted by rmpatel5
disprove the conjecture: There are infinitely many prime numbers expressible in the form n^3 +1 where n is a positive integer
Hint: $a^3+b^3 = (a+b)(a^2 - ab + b^2)$
• September 2nd 2008, 02:51 PM
rmpatel5
I have that part: n^3 +1= (n+1)(n^2-n+1). Just dont know where to go from. I know that 2 is the only integer that will work but i just dont know how to prove it.
• September 2nd 2008, 02:54 PM
ThePerfectHacker
Quote:
Originally Posted by rmpatel5
I have that part: n^3 +1= (n+1)(n^2-n+1). Just dont know where to go from. I know that 2 is the only integer that will work but i just dont know how to prove it.
Well because you can factor $n^3+1$ as a product of two integers.
So how can it be prime?
• September 2nd 2008, 04:53 PM
rmpatel5
so anything that factors can not be prime because that makes it a composite?
• September 2nd 2008, 07:23 PM
ThePerfectHacker
Quote:
Originally Posted by rmpatel5
so anything that factors can not be prime because that makes it a composite?
Yes if $a = bc$ and neither $b,c$ are $1$ then $a$ cannot be prime. | 2016-06-01 00:37:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7444576621055603, "perplexity": 887.3608541684976}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464053252010.41/warc/CC-MAIN-20160524012732-00174-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://www.sklogwiki.org/SklogWiki/index.php/Melting_curve | # Melting curve
Melting curve
## Empirical "one-phase" rules
### Ross melting rule
The Ross melting rule states (Eq. 4 [4]):
$A = \frac{1}{2} NE(0) - Nk_BT \ln \upsilon - Nk_BT \ln \upsilon_f^*$
where $A$ is the Helmholtz energy function, $N$ is the number of cells, $E(0)$ is the potential at the centre of the cell, $k_B$ is the Boltzmann constant, $T$ is the temperature, $\upsilon$ is the volume of the cell, and $\upsilon_f^*$ is the dimensionless reduced volume in configuration space.
### Khrapak melting criteria
The Khrapak one-phase melting criteria for two dimensional crystals with soft long-ranged interactions is given by (Eq 3 in [5]):
$\frac{C_T}{v_T}\approx 4.3$
where $C_T$ is the transverse sound velocity, and $v_T$ is the thermal velocity. | 2018-06-20 17:16:22 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 11, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.808563768863678, "perplexity": 1055.748553560504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1529267863830.1/warc/CC-MAIN-20180620163310-20180620183310-00263.warc.gz"} |
https://chat.stackexchange.com/transcript/message/32155504 | 5:20 AM
Related to:
15
There are some distinguished answer contributors to this site who used to write lots of excellent answers (for which they deservedly got many upvotes and accepts) but who are no longer as active as they used to be. Slowing down a bit and focusing one's energy and attention somewhere else are of c...
Geez!
That's been the highest reversal to date... perhaps on the entire network?
"Voting corrected" sounds so polite...
5:49 AM
@Werner Wow!
6:15 AM
@Werner Unaccepts as well? Those can only be done by the OP, no bot, right?
@Johannes_B I don't understand those exactly. My guess is they were removed automatically because they met some pattern. But that's bizarre.
It won't be removed by the OP, as they're all done at exactly the same time.
@Werner Do you know if this was all upvotes/accepts from a special timeperiod, i.e. all from x to today?
@Johannes_B No idea. That amount of reputation would span easily a year.
I'm guessing the removal of accepts is actually a mistake (unfortunately).
@Werner I am here for almost three years, and i have just 16k. :-)
@Johannes_B Ha! Exactly... it's a HuGe amount. Epically huge...!
6:22 AM
@Werner :-)
@Werner Maybe @Joseph can ping the SE stuff, if nobody else has already.
@Johannes_B I'm going to ask a question on Meta Stack Exchange, suggesting it's a in the design of the serial voting reversal algorithm...
I'm sure there may be the odd case where sock-puppets ask a question which is then answered an accepted by another sock puppet, but I don't think this is the case here.
...let me check...
Two of the posts were from the same user (which is suspicious), but one is from a completely different user, and doesn't seem suspicious.
@Werner I hope it is a bug.
@Werner I think I know what they've done, but obviously I can't say
@JosephWright Sure. Do you think it's a bug in the reversal design?
@Werner No
6:32 AM
@JosephWright Okay.
@JosephWright The suspense of not knowing almost feels like this: giphy.com/gifs/crash-truck-TfUFeB7B5ltKw
6:53 AM
Hello.
I'd like to ask for a bit of help regarding this post:
4
Consider the following subimportlevel macro: packages.tex \usepackage{import} \usepackage{coseoul} \newcommand{\subimportlevel}[3]{ \setcounter{currentlevel}{#3} \subimport*{#1}{#2} \setcounter{currentlevel}{#3} } The purpose of the macro is to import a modular piece of a document while gett...
However, here I'd like to ask for a bit of help understanding how the macro given in that answer works.
Here's the macro:
\makeatletter
\newcounter{currentimportdepth}
\setcounter{currentimportdepth}{0}
\newcommand{\subimportlevel}[2]{
\expandafter\edef\csname @currentlevel\thecurrentimportdepth\endcsname{\thecurrentlevel}
\subimport*{#1}{#2}
\setcounter{currentlevel}{\csname @currentlevel\thecurrentimportdepth\endcsname}
}
\makeatother
(hmmm, formatting fail)
Anyway, I'd like to understand the fifth line:
\expandafter\edef\csname @currentlevel\thecurrentimportdepth\endcsname{\thecurrentlevel}
@DanielSank remove the backticks and use the fixed font button on the right (which indents by 4 spaces, and works:-)
@DavidCarlisle nice
@DanielSank it saves the value of the current level:-)
@DavidCarlisle ok ok, I'm dumb. Let's go slow.
So, how does it save that value? :)
@DanielSank one step at a time, hang on....
6:58 AM
What does "save" mean here? I understand regular programming languages, but not LaTeX.
\csname @currentlevel\thecurrentimportdepth\endcsname if expanded once is \@currentlevel3 (if import level is 3) which isn't normally a legal command name
\expandafter expands teh csname once before doing the \edef so your construct is
\edef\@currentlevel3{\thecurrentlevel} (if import depth was 3, and 3 was a legal letter in a command name)
My understanding of \csname is that it allows multicharacter control sequences.
\edef expands the replacement text at the time of the definition so if \thecurrentlevel is 6 this is
...and nonalphabet characters in control sequences.
\def\@currentlevel3{6}
7:02 AM
ok
What is the @ doing?
@DanielSank yes but I was writing \csname abc3\endcsname as \abc3 to highlight that the csname has been expanded so it is now a single token (with a weird name) not a construct that generates
@DanielSank nothing different from the c or the u @ is a letter in package code.
Yeah, hence the \makeatletter
@DanielSank which is implicit in packages.
One moment.
Is @currentlevel3{6} a command name now? In particular, does the {6} mean anything other than a string of characters that comprise part of the name?
@DanielSank if you know c tex is a macro processor, so \def (\newcommand) are more like #define than a function definition.
7:06 AM
@DavidCarlisle I know that much, but I'm not handy on how to reason through \edef and \expandafter. Learning...
@DanielSank no \@currentlevel3 is the command and 6 is its definition. \def\@currentlevel3{6} is \newcommand{\@currentlevel3}{6} (if you assume @ and 3 are letters)
@DavidCarlisle Eurika!
@DanielSank so end result is that it stores the current level in a command generated with a name unique to the current import level
got to go...
Thank you.
1 hour later…
8:26 AM
@barbarabeeton (@JosephWright) I updated code here (you may recognise the extended test example:-)
3
An updated version more closely matching the tighter spacing and larger minimum spce of generalised fraction brackets is given by the following, which may be run with pdftex or xetex, to compare using lm fonts. \documentclass[fleqn,a4paper]{article} \addtolength\textwidth{40pt} \usepackage{am...
@DavidCarlisle :)
@JosephWright got to keep @barbarabeeton happy
8:38 AM
@DavidCarlisle I see we are in agreement about keyvals :-)
@JosephWright I thought you'd like the <> suggestion:-)
@DavidCarlisle I'd go with integrating keyval but if we are going to say 'some things must change', I'd put the brace stripping fix back in and say the problematic packages have to change
@DavidCarlisle Oh yes
@JosephWright incidentally would you use l3keys or keyval or an updated fixed renamed keyval without the compatibility constraints on the current one?
@JosephWright oh you answered the question I was writing.
@DavidCarlisle If we want to go with 'new' code I'd take the l3keys code and decide which bits we want and use that (the base of l3keys has been re-written to be as fast as possible, but it's slower than keyval due to babel safety and 'firmer' space stripping)
@JosephWright in l3 form or back convert to 2e?
babel safety would be good
8:43 AM
@DavidCarlisle I'm working at the mo on the 'higher level' part: aiming to get to about no slower than twice as slow as options (which is clever but not quite what we want)
@DavidCarlisle Unless we are going for something very radical, in 2e (it's largely low-level anyway)
@JosephWright sounds like a plan to me....
@DavidCarlisle The low-level of l3keys is basically keyval but using e-TeX, babel safe, detokenizing key names, stripping all spaces at the ends and always stripping exactly one set of braces [all of this is tested :-)]
@JosephWright ooh testing. It'll never catch on.
@DavidCarlisle Not for some of the people we know
@DavidCarlisle Yes: need to see what Frank thinks (it's quite a policy change)
@DavidCarlisle figured out what we were discussing earlier.
I believe I have accomplished something rather neat.
What is the best way for me to share this work? The new macro (which I mostly got from here), pulls in functionality from two existing packages coseoul and import. Should I try to get one of those packages to adopt the macro or release an independent one with the other two as dependencies?
8:58 AM
Anybody wanting to upvote the answer at tex.stackexchange.com/questions/199867/… ? The answer is correct but got downvoted for reasons I can't understand. I had already upvoted.
> my question is exactly the same as [blah blah] but the answer there did not solve my problem.
Over at Physics Stack Exchange we'd have closed that immediately.
hmmm, maybe I spoke too soon.
@DanielSank Well, the OP's statement is false; the real issue is that the OP is using the wrong way for making the code to work.
Yes I spoke too soon in any case.
No takers on the modularity macro, eh?
::crickets::
9:48 AM
@DanielSank either way works, no harm in asking the maintainers of the original code first.
10:17 AM
@DavidCarlisle I'm a bit new to this. On github I would file an issue or a pull request. On CTAN there doesn't seem to be any way to contact package maintainers. Am I missing something?
@DanielSank Have a look in the documentation of the package, email addresses are often listed there.
@TorbjørnT. Ah. coseoul didn't have one, but import does.
@DanielSank no you need to look into the package documentation, normally there are some contact details, sometimes those details are from this century and still work.
Before I bug this person, is it reasonable to propose adding a macro to package A if that macro depends on package B, while A doesn't normally depend on B?
I have no idea how package managment/dependencies work in TeX. This is my first rodeo.
@DanielSank There's none. :)
10:22 AM
@DanielSank it's reasonable to propose anything (but it's also reasonable for the propsal to be rejected on those grounds)
@PauloCereda Oi texlive team would be offended by that:-)
@DavidCarlisle Alright, well if I could do anything I'd put my macro in coseoul, but the author thereof has a common name and no contact details provided.
@DavidCarlisle oh I meant the language, not the underlying system. <3
@DanielSank I see that the author of coseoul is a user at the site, Tom Bombadil. (See meta.tex.stackexchange.com/questions/1721/…)
@PauloCereda Thanks for the vote! I guess it was yours.
@DanielSank I'd never heard of that package but just looked, it's 39 lines long (and could be much shorter) probably less charaters needed just to incorporate the logic in your code than to mail the original author and ask for an update.
10:25 AM
@DanielSank It originated from tex.stackexchange.com/questions/26181/…
@egreg hmmm I am lost now. :) What did I do?
@TorbjørnT. cool!
@PauloCereda prime suspect when it comes to upvotes
@PauloCereda Oh, I guessed wrong.
@egreg what happened? :)
10:26 AM
@DavidCarlisle Well yes, I can solve the problem for myself just fine. I am more interested in disseminating useful code, however.
@egreg The one you asked for earlier? Guilty.
@TorbjørnT. Thanks! Another question off the unanswered list
@egreg ooh now that @TorbjørnT. mentioned it, I upvoted it too! <3
@PauloCereda I miss Harish and Gonzalo
@egreg Me too. :(
10:29 AM
@DanielSank I meant in code you distribute, you are free to just include modified version of the existing package code rather than have a dependency on another package. (it's LPPL licenced)
@DavidCarlisle Ah. Yes that would be a solution, although I'd rather not violate the DRY principle. I have left a tex.SE comment such that the package author should get a ping. Thanks for the info @TorbjørnT.
@DanielSank yep I wrote that before you'd located the author:-)
@DavidCarlisle Thanks again for the help understanding the macro. I made a small tweak and now all my problems are solved. Modular documents for everyone!
@DanielSank if only @PauloCereda's thesis had some text he could modularise it:-)
@DavidCarlisle Oh, is someone here nearing a PhD graduation?
That's exciting.
10:39 AM
Jan 12 at 12:15, by David Carlisle
@PauloCereda Advice on writing a dissertation: you will never finish it. You will, at some point, abandon it.
Simply decide as soon as possible when you will abandon it, and then just work until that time.
@DavidCarlisle oy
@DanielSank There's no end for research, actually. :)
@PauloCereda Indeed not, but one can be fooled into thinking a PhD dissertation does have a sort of satisfying end.
@DanielSank My plan is to survive it. :)
@PauloCereda Yes. Very good.
10:44 AM
@DanielSank It involves procrastination. :)
@PauloCereda Subject?
@DanielSank Abstractions, macro expansions, self-modifiying code, context-dependent grammars. :)
@PauloCereda So, programming languages, to put it broadly?
@DanielSank A little more general. :)
@PauloCereda ah.
10:49 AM
@DanielSank :)
11:05 AM
Yay it's holiday today!
11:24 AM
\makeatletter
\protected\long\def\Ksetkeys#1#2{%
\def\KV@prefix{KV@#1@}%
\def\@tempa{#2}%
\def\@tempc{}%
\KKV@sanitise@equals
\KKV@sanitise@comma
\expandafter\KKV@loop\expandafter\q@mark\@tempa,\relax,%
}
\begingroup
\catcode\, = \active
\catcode\= = \active
\protected\long\gdef\KKV@sanitise@equals{%
\expandafter\KKV@sanitise@equals@\@tempa\q@mark=\@nil=%
\expandafter\KKV@sanitise@\@tempa
}
\protected\long\gdef\KKV@sanitise@equals@#1={%
\edef\@tempa{\unexpanded{#1}}%
@DavidCarlisle ^^^
@DavidCarlisle If we go for this, I'd be tempted to keep \trimspaces of some sort as expl3 and use the 'know the internals' trick I do in l3keys for speed
@DavidCarlisle Above code is basically l3keys/keyval merge: hard-coded place for the keys, and assuming we share the keys themselves. I might have the default value bit wrong ...
1 hour later…
12:38 PM
@JosephWright other issue of course is if we care about non-etex, I think currently the format still builds with tex2, certainly it tries with \ifx\@undefined\language \newcount\language \fi It would probably be OK though to say if you don't build on etex you can't use the optional argument
@DavidCarlisle You can probably guess my view here
@DavidCarlisle Note that some of the hyphenation patterns do assume e-TeX: realistically it's extremely unlikely anyone has a format that is kept up-to-date but which doesn't have the extensions
@JosephWright similar to third starred comment on the right?
@DavidCarlisle I think the team said e-TeX was assumed before I started using LaTeX
@DavidCarlisle Really, all of that should go
@DavidCarlisle Yup
@JosephWright tex2 support could probably go, but even the new allocation code added last year checks for etex and keeps below 255 in classic tex...
@DavidCarlisle As I've said to Frank, if we don't want to find ourselves strangled by 'stability' we have to ask some people to change some stuff
@DavidCarlisle Yes, I know
@DavidCarlisle But try actually building a format outside of the testing sandbox
12:49 PM
@JosephWright yes I know it dies in hyphenation (I tried a while back:-)
@DavidCarlisle My point being how many users do we really have who might get "LaTeX2e 2017" but don't have e-TeX
@DavidCarlisle Anyway, first we have to agree a plan more generally
@JosephWright oh none, most likely.
@DavidCarlisle Only one way to really find out ...
@DavidCarlisle Frank probably regrets asking me to join the team, sometimes :-)
@JosephWright probably:-)
@DavidCarlisle Other issue is do we simply superset keyval or use a separate set of keys or ...? I'd favour not making even more packages!
@DavidCarlisle :)
@DavidCarlisle Everyone else probably wonders what we are on about!
1:04 PM
@JosephWright Planning for a better future, I suppose.
@JosephWright Fixing stuff? :)
Stuff that @DavidCarlisle broke.
1:19 PM
@JosephWright just tell them it's cricket, they won't know the difference
@PauloCereda no planning to break stuff
@DavidCarlisle oh
1 hour later…
2:41 PM
@DavidCarlisle I see no cricket bat, so it can't be.
@egreg look at the image to the left of the previous comment: the duck has the bat.
3:13 PM
@DavidCarlisle :)
3:24 PM
I suddenly can't compile xelatex in TexWorks? xelatex.logcontains FATAL xelatex - Source: libraries\miktex\texandfriends\include\miktex\c4p\C4P.h. an update did not help
3:38 PM
@cfr I deleted most of my side of our cross chat under the X question, probably confusing for the OP:-)
@egreg see you put Bruno out of his misery. I wonder what he was trying to do:-)
@DavidCarlisle Trying to break LaTeX, undoubtedly
@egreg quite successfully:-)
@egreg I even voted for you
@DavidCarlisle It's fixing a missing %, in some sense
@egreg well naturally, as you could answer it.
@DavidCarlisle :P
4:02 PM
Holiday!
@PauloCereda not for students, they have to work on their thesis, even on national holidays
@DavidCarlisle oh no
@PauloCereda it's either that or go to Rio to support team GB in the para-olympics
@DavidCarlisle Yay GB!
That makes me a Englishduck? :)
Jul 20 '12 at 12:26, by Paulo Cereda
I want to be German.
4:11 PM
@DavidCarlisle Oh
Do you have those links on an emacs buffer? :)
@PauloCereda I do now
@DavidCarlisle oh my
@PauloCereda the things you make me do, lynx in emacs terminal in X in cygwin running with windows as the X window manager...
2
@DavidCarlisle Oh my!
@PauloCereda it does work though:-)
4:19 PM
@DavidCarlisle Surprisingly. :)
4:44 PM
@DavidCarlisle Windows is getting BASH so soon you'll be able to drop Cygwin at least.
So, my boss wants me to switch from micromoles to millimoles. I'm wondering if it is possible to abuse siunitx to do that conversion automatically.
Also if I should instead covert everything to hogsheads and butts as a joke
@Canageek I suppose not, but it's not so difficult to at least find all occurences when you use siunitx
@yo' by searching for every 'mole' yes
@yo' I use "\SI" 200 times in this document
@Canageek that's not so much, about an hour or two of work.
@yo' I'm currently taking the "pretend I forgot to make that change and see if he notices" approach
@Canageek :)
4:58 PM
Right now I'm changing all mentions of "To a solution of \nau\ (\SI{3}{\milli\litre}, \SI{50}{\milli\gram}, \SI{100}{\micro\mole})" to "To a \SI{3}{\milli\litre} solution of \nau\ (\SI{50}{\milli\gram}, \SI{100}{\micro\mole})"
5:09 PM
@Canageek I doubt it (although we'll see) I need rather more than just bash, X for example.
@DavidCarlisle I'm betting someone will compile it against windows soon
@Canageek Surely there's something wrong here!
@JosephWright Oh? What am I missing?
@JosephWright Oh, I should have a solvent in there
@Canageek oh no
(I have no idea of what that means, I just want to join the conversation)
@JosephWright Ok, checked the text, it is inthere in the actual version
5:44 PM
@Canageek The values must change if the unit prefixes do: 100 mmol = 100000 um
@JosephWright Yes. Here I wasn't changing the values or prefixes, just moving from "To a solution of X (Y mL, W mg, Z umol)" to "To a Y mL solution of X (W mg, Z umol)"
6:28 PM
@PauloCereda spacemacs.org
@PauloCereda Perhaps you and @DavidCarlisle can finally agree on an editor ;)
@HenriMenke I have it here. :) But I use vim. :)
@PauloCereda I actually use both. Emacs for TeX and Lua, Vim for C++ and general editing on the command line.
Efff
Trying to figure out how ACS wants this thing cited is a pain.
@HenriMenke Cool. :) Once in a while, I edit a paper in TeX using spacemacs. :)
I like the vim key bindings. :)
Trying to figure out how to cite link.springer.com/chapter/10.1007%2F3-540-11454-8_4 if anyone has a moment to help
6:32 PM
@PauloCereda You can also have Vim binding with normal Emacs by installing evil mode.
@HenriMenke No thanks. :)
@Canageek Why not click »Export Citation«?
I thought it would be as "incollection" but that gives the "Booktitle" and Series, but not the actual title
@Canageek I use inbook, I guess.
@PauloCereda Well, evil mode is what's being used behind the scenes in Spacemacs.
6:33 PM
@HenriMenke plus space. :)
@PauloCereda That doesn't give the imporant title (Uranyl Photophysics). It is identical to "incollection"
It gives the Booktitle (Top. Inorg. Phys. Chem.) which is nondescriptive. and the series title (Structure and Bonding)
@Canageek Sorry, it was just a guess. I always get the correct output. So the only thing I can think of is that the style you are using is the one to blame.
But not the title of the chapter (Uranyl Photophysics)
@PauloCereda Blame @JosephWright who wrote the style, got it ;)
@Canageek :)
@PauloCereda I'll make a question about it, since I think this is wrong. I think this counts as "Series publication cited as a book"
6:52 PM
@Canageek It's @inbook or @incollection, doesn't quite make a difference
and actually, the default export makes sense:
@Inbook{Jorgensen1982,
author = "J{\o}rgensen, Christian K. and Reisfeld, Renata",
title = "Uranyl photophysics",
booktitle = "Topics in Inorganic and Physical Chemistry",
year = "1982",
publisher = "Springer Berlin Heidelberg",
pages = "121--171",
isbn = "978-3-540-39082-4",
doi = "10.1007/3-540-11454-8_4",
url = "http://dx.doi.org/10.1007/3-540-11454-8_4"
}
Of course, it's wrong to list both Berlin and Heidelberg, but ...
If it turns out wrong in some particular style, you can tweak things easily, depending on what's wrong
7:07 PM
@Canageek Top. Inorg. Phys. Chem. is a journal
@JosephWright It claims to be a book series
@JosephWright no, according to Springer
@Canageek Yes, but things like Annual Reviews also get cited as journals
@yo' Not their call ;-)
@JosephWright sorry, but with my copy editor hat on, it is their call.
@JosephWright I think the problem is that I am using title and should be using something specific for chapter title
7:10 PM
@JosephWright individual books have ISBN and the series hasn't got an ISSN, this is not a journal by any standard, sorry.
@yo' I want a hat too
@yo' Org. Synth. certainly used to like a book but has always been cited as a journal
@yo' oooooooooh
@JosephWright Org Synth has a specific exception in the ACS Style Guide
@JosephWright Posted Q with MWE and everything
0
I'm trying to cite a chapter in a book, in a series. Specifically, Uranyl Photophysics by Jørgensen and Reisfeld Based on Chapter 14 of the ACS style guide (pg 242, 306 "Series publication cited as a book", see "As for any book, you may cite specific chapters") I think this should be cited as: ...
7:12 PM
@Canageek Probably
Have to go to lunch before my better half murders me >.>
@Canageek If you are citing as a book you shouldn't abbreviate the title :-)
@JosephWright Ok, I can fix that.
@Canageek Other than journals, it's actually pretty hard to be sure what the editorial office will do with anything :)
@JosephWright But I need the chapter title to make it obvious why I'm citing it to my boss ;)
7:14 PM
@Canageek I've had an update for achemso today: I've never had anything from the ACS about the BibTeX styles other than adding article titles/DOIs (both increasingly common)
@Canageek Ah, that's an issue: 'in' entries don't have the chapter title in ACS style
It says you can. Page 306, chapter 14. "As with any book you may include chapter titles"
@PauloCereda I have a hat for you:
@Canageek Might well do, but I've worked from what they actually do on average
It has an example and everything
7:16 PM
@DavidCarlisle LOL
@Canageek Yes, I know: the style guide leaves various things open, but that's hard to cover in a BibTeX style (and the aim for achemso is primarily matching publications)
@Canageek For the biblatex style, I have an option to have chapter titles present
Can I submit that for publication?
@Canageek No
@Canageek Or rather it's not officially supported!
@Canageek The ACS don't really care as such: they work from the PDFs
@Canageek For publication they'll take the chapter titles out: they don't actually use them (which is why achemso doesn't include them)
7:20 PM
Why do they even have a style guide then?
@Canageek Well it's a guide, like it says :-) It applies to all sorts of chemistry, not just publication via the ACS
@Canageek The editorial offices are something of a law unto themselves
Man, if I were an editor I'd probably have great fun rewriting the style manual to match my tastes
@Canageek As it is there are changes ongoing: Inorg. Chem. apparently want article titles and DOIs for all entries (email today)
They tweeted that a while ago, didn't they?
(has a talk with my boss about that yesterday)
@Canageek Possibly
7:27 PM
If I had that much power the ACS would switch to the RSC citation order ;)
@Canageek I work from published papers, so such things don't tend to feature unless the ACS tip me off
@Canageek Oddly, so would I
@Canageek RSC have much more a single approach across journals
Engineers got it right- one standard for all publications advertised publishers
@JosephWright So how do you establish a large enough corpus of the weirder citations like inbook?
7:43 PM
@DavidCarlisle Do you have the link?
@JosephWright I prefer their italics and name order
@JosephWright Nope. If I were in charge all journal names would be in ISO 4 format, no exceptions. I'm looking at you CrystEngComm. glare
@JosephWright Ok, solution: I'll add an addendum to my boss with the chapter title. Achemso supports that, right
@JosephWright Wait, this doesn't even have an "in" before the book title. I'm SURE I've seen that in ACS journals.
@DavidCarlisle OK. Never mind. I found it again. But why would the X get into that position if it was replacing the summation sign?
0
I am having trouble with the sum sign. It just shows as an X. Also the minus sign is missing. See picture below. This is my code: \documentclass{article} \begin{document} $$\label{eq8.1} NPV = \sum_{t=1}^{T} \frac{C_t}{(1 + r)^t} - C_0$$ \end{document} Can anyo...
8:01 PM
@JosephWright There, I added examples from an ACS journal. Now looking for ones specific to Cryst. Growth. Des.
@Canageek I guess!
@Canageek Take a look for example at the biblatex-chem ones for Houben-Weyl
@JosephWright ?
@Canageek How many times do people cite Houben-Weyl? (Now called 'Science of Synthesis' but I know the older, German, volumes better)
@JosephWright Ug, it seems IC and Cryst Growth Des do it totally diffrently
@Canageek Probably: Cryst Growth Des is I think one of the more awkward ones for me
8:14 PM
@JosephWright HAH!
Found an ACS with a chapter title
@JosephWright "Balzani, V.; Credi, A.; Venturi, M. Molecular Devices and Machines. Concepts and Perspectives for the Nanoworld, 2nd ed.; Wiley: Hoboken, NJ, 2008" They don't use the 'In" for some reason though. (Citation was in JACS)
@JosephWright Both IC and CGD put an "In" before the booktitle though, which achemso does not though.
@Canageek When in doubt, JACS is the model
@JosephWright That has the chapter title.
@Canageek As far as I know, none of this is automated so it is actually pot-luck
@cfr well the summation is a mathop so shifted to be vertically centred the X is a different font and a different size but probably shifted by the same amount, so just random breakage really
@JosephWright I also found an IC with a chapter title, but it is from 196
8:19 PM
@DavidCarlisle Oh, random breakage. OK. I can understand that ;).
@JosephWright So I've found 3 styles, but none of them match the output of achemso, so, not sure what that means
@JosephWright Is there a way to make my addendum show up? (For example, when citing the CSD database I need to specify what version)
@Canageek That would be a misc entry so done entirely by hand
@JosephWright They have a paper you are supposed to cite though.
@Canageek What I mean is that other than journals (where things are broadly consistent), one can probably find pretty much any formatting that vaguely matches the style guide. So I had to pick something.
@Canageek Yes, but normally the citation would be for using the database and a ref. to the version would go with 'X hits' or whatever, i.e. two separate references
@JosephWright Ah. They seem pretty consistent on there should be an "In" before the book name though, which would at least make it clear I'm citing a chapter.
@JosephWright OH. OK, I can do that.
8:28 PM
@Canageek This may have changed a bit of course over the past few years
C-C-C-C-C-COMBO BREAKER!
@Canageek Broadly, I have agreement with the ACS that I don't change achemso unless they ask
@JosephWright Damn.
@Canageek Not a hard-and-fast rule, mind
@JosephWright To bad we didn't talk yesterday so you could have slipped it in with today's entry.
8:30 PM
@Canageek OOh, now I remember: there is an option for chapter titles :-)
@JosephWright !!!
So would the CSD citation be just "@misc{CSD,
addendum = {CSD Version 5.37 (November 2015) + November 2015, Febuary 2016, and May 1016 Updates}
}"
@Canageek I think the field is note, but other than that looks about right to me
@JosephWright Do you recall what it is?
@Canageek chaptertitle
@JosephWright Thank you.
8:33 PM
@JosephWright inbook has it!
That's what I use.
@JosephWright Doesn't work.
Is there a diffrence between inbook and incollection?
@Canageek Ah, there might be
@Canageek I'm off to bed in a sec but I'll look again at this tomorrow
@JosephWright I've tried both. OK, thanks.
@JosephWright ooh timezones
9:12 PM
@Canageek texdoc tamethebeast, go to the bottom of page 14
@Canageek The main difference is that an @incollection item has its own title, whereas an @inbook item doesn't.
9:45 PM
Effff
For some reason now I have to remove the \setkeys{acs}{articletitle = true,doi=true} or my document won't compile. *Glares at @JosephWright * | 2019-07-23 12:26:57 | {"extraction_info": {"found_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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.6864235997200012, "perplexity": 4116.367877103904}, "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-2019-30/segments/1563195529276.65/warc/CC-MAIN-20190723105707-20190723131707-00113.warc.gz"} |
https://www.physicsforums.com/threads/integrating-factor.708603/ | # Integrating Factor
## Homework Statement
y' + (2/t)y = (cost)/(t^2), and the following condition is given: y(pi) = 0
## The Attempt at a Solution
After employing the integrating factor, I find the solution to be:
$y=e^{-2t} \int e^{2t} \frac{\cos(t)}{t^2} dt$.
Evidently, this simplifies all the way to y = (sin t)/(t^2). I am not sure as to how this integral should be solved. Any hints would be much welcomed.
Related Calculus and Beyond Homework Help News on Phys.org
Curious3141
Homework Helper
I think you should show all your steps, starting with exactly how you applied your integrating factor. Clearly, there's an error somewhere.
I see what I did wrong.
mu of t, the integrating factor, $\mu (t) = e^{\int \frac{2}{t}}dt = e^{2 \ln t} = e^{2t}$
Do you see where I went wrong? It should be $\mu (t) = 2t$
HallsofIvy
Homework Helper
This is an example of what happens when you memorize formulas (imperfectly) rather than learning basic definitions and how the formulas are derived.
An "integrating factor" is a function, $\mu(t)$ such that multiplying the equation by it converts the left side into a single derivative. Here, that means we must have $\mu y'+ (2\mu/t)y= (\mu y)'$. Expanding the derivative on the right that becomes $\mu y'+ \mu' y= \mu y'+ (2\mu/t)y$ which reduces to $\mu'= 2\mu/t$, a separable differential equation for $\mu$. $d\mu/\mu= 2dt/t$ integrates to $ln(\mu)= 2ln(t)$ or $\mu= t^2$ NOT "2t" (I have neglected the "constant of integration since we only need a single function).
Multiplying the entire equation by $t^2$ gives $t^2y'+ 2ty= (t^2y)'= cos(t)$ which is easy to integrate.
riceking95
Curious3141
Homework Helper
I see what I did wrong.
mu of t, the integrating factor, $\mu (t) = e^{\int \frac{2}{t}}dt = e^{2 \ln t} = e^{2t}$
Do you see where I went wrong? It should be $\mu (t) = 2t$
##e^{2 \ln t} = (e^{\ln t})^2 = t^2##, that's your integrating factor.
Your integrating factor is NOT ##2t##. ##2t## is in fact the derivative of your integrating factor, and you should be able to see this from applying product rule to ##yt^2##.
Last edited: | 2021-02-26 19:17:25 | {"extraction_info": {"found_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": 0.9642610549926758, "perplexity": 608.9324068214288}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178357935.29/warc/CC-MAIN-20210226175238-20210226205238-00325.warc.gz"} |
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Séminaires et Congrès 2009; 466 pp; softcover Number: 18 ISBN-10: 2-85629-240-2 ISBN-13: 978-2-85629-240-2 List Price: US$132 Member Price: US$105.60 Order Code: SECO/18 This volume gathers lecture notes taken at the 2004 Summer School, which was held at the Institut Fourier (Grenoble). The title of the Summer School ("Negative or zero-curvature geometries, discrete groups and rigidities") has been used for the present volume. In many cases the lecture notes have been rewritten and enhanced. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in negative or zero-curvature geometries, discrete groups and rigidities. Table of Contents 1. Quelques groupes et géométries J. Maubon -- Symmetric spaces of the non-compact type: Differential geometry P.-É. Paradan -- Symmetric spaces of the non-compact type: Lie groups G. Rousseau -- Euclidean buildings Y. Benoist -- Five lectures on lattices in semisimple Lie groups 2. Quelques rigidités en géométrie différentielle G. Besson -- Calabi-Weil infinitesimal rigidity M. Bourdon -- Quasi-conformal geometry and Mostow rigidity L. Bessières -- Minimal volume M. Burger and A. Iozzi -- A useful formula from bounded cohomology 3. Espaces métriques singuliers G. Courtois -- Critical exponents and rigidity in negative curvature C. Druţu -- Quasi-isometry rigidity of groups P. Pansu -- Superrigidité géométrique et applications harmoniques 4. Déformations, espaces de modules et compactifications F. Paulin -- Sur la compactification de Thurston de l'espace de Teichmüller A. Beauville -- Moduli of cubic surfaces and Hodge theory | 2014-04-17 19:15:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20614823698997498, "perplexity": 8038.290359458871}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00146-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/a-formulation-of-continuity-for-bilinear-forms.212528/ | # A formulation of continuity for bilinear forms
1. Feb 1, 2008
### quasar987
[SOLVED] A formulation of continuity for bilinear forms
1. The problem statement, all variables and given/known data
My HW assignment read "Let H be a real Hilbert space and a: H x H-->R be a coninuous coersive bilinear form (i.e.
(i) a is linear in both arguments
(ii) There exists M>0 such that |a(x,y)|<M||x|| ||y||
(iii) there exists B such tthat a(x,x)>a||x||^2"
So apparently, condition (ii) is the statement about continuity. But I fail to see how this statement is equivalent to "a is continuous".
I see how (ii) here implies continuous, but not the opposite.
3. The attempt at a solution
Let z_n = (x_n,y_n)-->0. Then x_n-->0 and y_n-->0. So |a(x,y)|<M||x|| ||y|| implies a(x,y)-->0. a is thus continuous at 0, so it is so everywhere, being linear.
Last edited: Feb 1, 2008
2. Feb 1, 2008
### Dick
I'm not quite sure what the question is. It seems to have been omitted. But just looking at your argument, z_n=(x_n,y_n)->0 doesn't imply x_n->0 or y_n->0. Does it?
3. Feb 2, 2008
### quasar987
Well, I'm making use of the fact that if (M,d) is metric space, then the product topology on M x M is generated by the metric
D((x1,y1),(x2,y2))=[d(x1,x2)² + d(y1,y2)²]^½
I conclude that the norm on H x H is
||(x,y)|| = [||x||² + ||y||²]^½
And now if (x_n,y_n)-->0, this means that
[||x_n||² + ||y_n||²]^½ --> 0,
which can only happen if x_n-->0 and y_n-->0.
------------
You're right, I have not actually typed the question entirely. It is because I was confused by the fact that they seem to imply that condition (ii) is equivalent to continuity. While (ii) implies continuity, does continuity implies (ii)?
Last edited: Feb 2, 2008
4. Feb 2, 2008
### Hurkyl
Staff Emeritus
Well, there is the theorem that a linear functional on a Hilbert space is continuous if and only if it's bounded...
5. Feb 2, 2008
### quasar987
You mean bounded in the unit ball?
In this case, you're right, it works. Because ||(x,y)|| = [||x||² + ||y||²]^½ <1 ==> ||x||, ||y||<1 and (ii) implies|a(x,y)|<M for all (x,y) in H x H such that ||(x,y)||<1. | 2017-07-23 12:52:32 | {"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": 0.9098065495491028, "perplexity": 2024.791487784714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1500549424559.25/warc/CC-MAIN-20170723122722-20170723142722-00273.warc.gz"} |
http://math.stackexchange.com/questions/394775/if-mathbf-x-is-sampled-randomly-from-a-hypercube-on-rn-what-is-the-probab | # if $\mathbf x$ is sampled randomly from a hypercube on $R^n$, what is the probability density for $|\mathbf x| = d$
if the vector $\mathbf x$ is sampled randomly from a uniform distribution on $[0, 1]^d$, what is the probability density function for $|\mathbf x|$? Is it easy to scale for $[0, n]^d$?
-
The probability that |x| is in (a,b) is the measure of the hypershell from radius a to b intersected with the hypercube. This should be an iterated integral. – Mark May 17 '13 at 20:00
@Mark Sorry, but I don't understand. Can you dumb it down a bit :) ? – Matt Munson May 17 '13 at 20:09
See my answer below or yoBS's illustration below – Mark May 17 '13 at 22:03
A partial answer. If $X$ is uniformly distributed in $[0,1]$ then $X^2$ has pdf $$f(t) = \frac{1}{2 \sqrt{t}}$$ for $t \in [0,1]$. Therefore the pdf of $| \mathbf{x}| = (\mathbf{x}_1^2 + \dotsc + \mathbf{x}_d^2)^{\frac{1}{2}}$ for $\mathbf{x} \in [0,1]^d$ is $$f_d(t) = 2t \, f^{\ast d}(t^2)$$ where $f^{\ast d}$ is the $d$-fold convolution of $f$. It looks like this pdf gets complicated quickly. For example already for $d=2$ using WA I get the following pdf:
$$f_2(t) = \begin{cases} \frac{\pi t}{2} & \textrm{if } t\in[0,1]\\[1ex] t \left(\arcsin(t^{-1})-\arctan \sqrt{t^2-1}\right) & \textrm{if } t\in[1, \sqrt{2}] \end{cases}$$
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What is WAI? Is it a general trend that arbitrary pdf's are hard to solve? The question seemed so simple conceptually... I'm 2/2 now for getting unexpectedly difficult solutions to questions about conceptually simple pdf's. – Matt Munson May 17 '13 at 20:07
@MattMunson WolframAlpha – oldrinb May 17 '13 at 20:22
@oldrinb oh derp, its just WA... – Matt Munson May 17 '13 at 20:32
I'll try to explain and expand mark's comment.
Think for a momnet in two dimensions. The probability density that $|x|=r$ is the probability that a uniformly chosen point lies at a distance $r$ from the origin. This equals the length of an arc of a circle of radius $r$, intersected with the unit box. See the circles in the drawing:
It is therefore clear that for $r<1$ the PDF will be $\frac{\pi r}{2}$. For $r>1$ simple trigonometry shows that this arclength is $\frac{\pi}{2}-2\cos^{-1}\left(\frac{1}{r}\right)$. So we get that for 2D $$f(r)=\begin{cases} \frac{\pi }{2}r & r<1 \\ \frac{\pi }{2}-2\cos^{-1}\left(\frac{1}{r}\right) & r\geq 1 \\ \end{cases}$$ Which is the result that WimC gave, but simplified.
In a general dimension, finding the hyper-area of the intersection of the $d-1$ sphere with the hypercube is pretty complicated for $r>1$. However, it is quite simple for $r<1$ - the sphere is completely contained in the first hyper-quadrant, so its area is simply it's total area divided by $2^d$. We get that for a general dimension $d$ the PDF is
$$f(x)=\begin{cases} \frac{v_{d-1}}{2^d}r^{d-1} & r<1 \\[5mm] ??? & r\geq 1 \\ \end{cases}$$
Where $v_{d-1}$ is the area of the unit $d-1$ sphere.
-
You want to think about how to calculate the measure of a hypersphere of radius $r$. If we have $n$ dimensions, we are asking for the set $\sum x_i^2 \le r^2$. This is because this set describes all the points with magnitude $r$ or less. If we imagine points distributed uniformly throughout this hypersphere, we will see that the probability of finding a point at radius $r$ is related to the size of the shell with radius $r$.
We can calculate the measure of this hypersphere by doing an iterated integral. For the four dimensional case we have:
$2^4\int_0^r \int ^{\sqrt{r^2-x_1^2}}_0 \int_0 ^\sqrt{r^2-x_1^2-x_2^2}\int_0^\sqrt{r^2-x_1^2-x_2^2-x_3^2} dx_4 dx_3 dx_2 dx_1$
We can easily generalize this to any dimension. Then we note that the actual problem asks for the density inside a hypercube, so we only take the portion of the hypersphere which intersects our hypercube.
Let $M(x) = \text{min}(x,1)$. Then we drop our $2^n$ factor because we are only worried about positive components. Then our CDF$(r)$ is
$\int_0^{M(r)} \int ^{M(\sqrt{r^2-x_1^2})}_0 \int_0 ^{M(\sqrt{r^2-x_1^2-x_2^2})}\int_0^{M(\sqrt{r^2-x_1^2-x_2^2-x_3^2})} dx_4 dx_3 dx_2 dx_1$
For $r \le 1$ we have an easy solution because the $M$s disappear and we end up with the volume of the n-dimensional hypersphere of radius $r$ divided by $2^n$. I will update the solution if I find a simpler form where $r \gt 1$.
Here is a different formulation / generalization to arbitrary dimension:
Let a particle appear with equal likelihood in any part of the unit n-hypercube. For $r \ge 0$ Let $F_n(r)$ be the probability that the particle will appear within $r$ distance of the origin.
Obviously $F_1(r) =\text{min}(1,r)$ because the one-dimensional case is a line segment of length 1. (this is just a silly way of writing the uniform CDF because I don't know how to format latex)
Now let's try to compute $F_n(r)$ in terms of $F_{n-1}$ by imagining that the particle is restricted to a hyperplane where one of the coordinates is fixed to $k$. That is, it must appear in the plane $\{k\} \times [0,1]^{n-1}$. Since we are only allowed to go $r$ distance away from the origin and we used up $k$ distance already, we can only use $\sqrt{r^2 - k^2}$ distance in traveling away from the origin of the sub-cube $[0,1]^{n-1}$.
$F_n(r) = P(\sum{x_i^2} \le r^2)$
$= \int_0^1P(\sum x_i^2 \le r^2 | x_1 = k)f_{x_1}(k)dk$
$= \int_0^1P(\sum_{i\ne1} x_i^2 \le r^2 - k^2 | x_1 = k)dk$
$= \int_0^1P(\sum_{i\ne1} x_i^2 \le r^2 - k^2 | x_1 = k)\mathbb{I}\{k \le r\le 1\}dk$
$= \int_0^rP(\sum_{i\ne1} x_i^2 \le r^2 - k^2 | x_1 = k)\mathbb{I}\{r\le 1\}dk$
$= \int_0^rF_{n-1}(\sqrt{r^2-k^2})\mathbb{I}\{r\le 1\}dk$
$= \int_0^{\text{min}(r,1)}F_{n-1}(\sqrt{r^2-k^2})dk$
So we end up with:
$F_n(r) = \int_0^{\text{min}(r,1)}F_{n-1}(\sqrt{r^2-k^2})dk$
$F_1(r) =\text{min}(1,r)$
Of course if you expand this formula for $F_4$ you will get the same four dimensional case mentioned above.
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There are many different probability densities of continuous distributions on $[0,1]^d$. There are also many probability distributions on that set that do not have densities, including, but not limited to, discrete distributions and mixtures of discrete and continuous distributions. A point could be randomly sampled from any of those distributions.
There is a somewhat uninformed usage that is commonplace among mathematicians who are not probabilists, according to which "randomly" means "uniformly distributed". It that is what is meant, then the density is equal to $1$ everywhere within that cube. On $[0,n]^d$ the density of the uniform distribution is $n^{-d}$ at every point.
Later edit: As noted in the comments, my answer ends before getting to the hard part. Maybe I'll add more later if no one beats me to it.
The conditional density given the order in which the coordinates appear when sorted, does not depend on which order it is. Therefore, we can condition on the event that $x_1<x_2<x_3<\cdots<x_n$ and we get the same distribution.
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I think the OP asked for the density of $|x|$. – Narut Sereewattanawoot May 17 '13 at 18:17
Oh. OK, my answer is incomplete. Maybe I'll add more later. – Michael Hardy May 17 '13 at 18:28
Edited my question to clarify that I am asking about the case where $x$ is sampled from the uniform distribution on $[0,1]^d$. And, yes, I am asking about the density of $|x|$. – Matt Munson May 17 '13 at 19:00 | 2015-08-29 17:33:31 | {"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": 0.9280087947845459, "perplexity": 228.64510320926573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1440644064517.22/warc/CC-MAIN-20150827025424-00071-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://physicshelpforum.com/threads/the-tight-binding-hamiltonian-for-a-two-dimensional-vanadium-dioxide.11082/ | # The tight-binding hamiltonian for a two-dimensional vanadium dioxide
#### gerryliyana
Code:
[IMG]http://physicshelpforum.com/attachment.php?attachmentid=1363&stc=1&d=1428860641[/IMG]
In his book, Quantum Field Theory for the Gifted Amateur, author Stephen Blundell explains that in the tight-binding model, one considers a lattice of fixed atoms with electrons moving between them (as shown in Fig. 4.6 above). These electrons can lower their kinetic energies by hopping from lattice site to lattice site. To deal with the discrete lattice in the model, we need to work in a basis where the fermion creation operator $\hat{c}_i^\dagger$ creates a particle at a particular lattice site labelled by $i$. The kinetic energy saving for a particle to hop between points $j$ and $i$ is called $t_{ij}$ . Clearly $t_{ij}$ will have some fundamental dependence on the overlap of atomic contains a sum over all processes wave functions. The Hamiltonian $H$ in which an electron hops between sites, and so is a sum over pairs of sites:
$\hat{H}_{hopping}=\sum_{ij}(-t_{ij})\hat{c}_i^\dagger\hat{c}_j^\dagger$
Ok. After reading that book, I am trying to work on the kinetic term of the Hamiltonian for a $VO_2$ system using tight-binding approximation, but I'm not quite sure whether my work is right or wrong.
Would you all be so kind as to check my work?
[IMG]http://physicshelpforum.com/attachment.php?attachmentid=1364&stc=1&d=1428860710[/IMG]
In my case, $VO_2$ is modelled by taking the (110) plane from its unit cell as shown in second figure above. The model unit cell has two basis oxygen atoms ($O_A$ and $O_B$), vertically located $a \approx 2.87$ angstrom apart, each of which contributing two p orbitals ($p_x$ and $p_y$) and two vanadium atoms ($V_A$ and $V_B$), vertically located $a \approx 2.87$ angstrom apart, each of which contributing two d orbitals ($d_{x^2-y^2}$ and $d_{xy}$). Here, we choose 8 basis orbitals to construct our hilbert space, which we order as follows: $\left| O_A-p_x \right\rangle$, $\left| O_A-p_y \right\rangle$, $\left| O_B-p_x \right\rangle$, $\left| O_B-p_y \right\rangle$, $\left| V_A-d_{x^2-y^2} \right\rangle$, $\left| V_A-d_{xy} \right\rangle$, $\left| V_B-d_{x^2-y^2} \right\rangle$, and $\left| V_B-d_{xy} \right\rangle$.
Using this set of bases and tight-binding approximation, the kinetic part of the hamiltonian can be written in $k$ space as
$\hspace{1cm}$
$H=H_{on-site}+H_{hopping}$
$\hspace{1cm}$
where
$\hspace{1cm}$
$H_{on-site}=\sum_{\bar{k}}\left( \epsilon_{d_{A}} d_{A_{1\bar{k}}}^{\dagger} d_{A_{1\bar{k}}} + \epsilon_{d_{A}} d_{A_{2\bar{k}}}^{\dagger} d_{A_{2\bar{k}}} + \epsilon_{d_{B}} d_{B_{1\bar{k}}}^{\dagger} d_{B_{1\bar{k}}}+\epsilon_{d_{B}} d_{B_{2\bar{k}}}^{\dagger} d_{B_{2\bar{k}}}+\epsilon_{p_{A}} p_{A_{x\bar{k}}}^{\dagger} p_{A_{x\bar{k}}}+\epsilon_{p_{A}} p_{A_{y\bar{k}}}^{\dagger} p_{A_{y\bar{k}}}+\epsilon_{p_{B}} p_{B_{x\bar{k}}}^{\dagger} p_{B_{x\bar{k}}}+\epsilon_{p_{B}} p_{B_{y\bar{k}}}^{\dagger} p_{B_{y\bar{k}}}\right)$ $\hspace{1cm}$
and
$\hspace{1cm}$
$H_{hopping}=-t_{V_A-V_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot a \hat{y}}d_{A_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot (-a \hat{y})}d_{A_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}\right)-t_{V_A-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y}\right)}d_{A_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}} +\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{B_{2\bar{k}}}\right)-t_{V_A-O_A}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y}\right)}d_{A_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}} +\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}\right)-t_{V_B-O_A}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y}\right)}d_{B_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}\right)-t_{V_B-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y}\right)}d_{B_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{B_{2\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{B_{2\bar{k}}}\right)-t_{O_A-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot a \hat{y}}p_{A_{y\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot (-a \hat{y})}p_{A_{y\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}\right)-t_{O_A-O_A}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot c \hat{x}}p_{A_{x\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot (-c \hat{x})}p_{A_{x\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}\right)-t_{O_B-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot c \hat{x}}p_{B_{x\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot (-c \hat{x})}p_{B_{x\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}\right)$
$\hspace{1cm}$
Thus,
$\hspace{1cm}$
$H =\sum_{\bar{k}}\left(\epsilon_{d_{A}}d_{A_{1\bar{k}}}^{\dagger} d_{A_{1\bar{k}}}+\epsilon_{d_{A}}d_{A_{2\bar{k}}}^{\dagger}d_{A_{2\bar{k}}}+\epsilon_{d_{B}}d_{B_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}+\epsilon_{d_{B}} d_{B_{2\bar{k}}}^{\dagger} d_{B_{2\bar{k}}}+\epsilon_{p_{A}} p_{A_{x\bar{k}}}^{\dagger} p_{A_{x\bar{k}}}+\epsilon_{p_{A}} p_{A_{y\bar{k}}}^{\dagger} p_{A_{y\bar{k}}}+\epsilon_{p_{B}} p_{B_{x\bar{k}}}^{\dagger} p_{B_{x\bar{k}}}+\epsilon_{p_{B}} p_{B_{y\bar{k}}}^{\dagger} p_{B_{y\bar{k}}}-2t_{V_{A}V_{B}} d_{A_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}\cos{\left( k_{y}a \right)}-2t_{V_AO_B}\left(d_{A_{2\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{A_{2\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{V_AO_A}\left(d_{A_{2\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{A_{2\bar{k}}}^{\dagger}p_{A_{y\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{V_BO_A}\left(d_{B_{2\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{B_{2\bar{k}}}^{\dagger}p_{A_{y\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{V_BO_B}\left(d_{B_{2\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{B_{2\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{O_{A}O_{B}} p_{A_{y\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}\cos{\left( k_{y}a \right)}-2t_{O_{A}O_{A}}p_{A_{x\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}\cos{\left( k_{x}c \right)}-2t_{O_{B}O_{B}}p_{B_{x\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}\cos{\left( k_{x}c \right)}\right)$
$\hspace{1cm}$
Note: $h.c$ = hermitian conjugate
$\hspace{1cm}$
In equations above, $d_{A_{1\bar{k}}}^{\dagger} (d_{A_{1\bar{k}}})$,$d_{A_{2\bar{k}}}^{\dagger} (d_{A_{2\bar{k}}})$, $d_{B_{1\bar{k}}}^{\dagger} (d_{B_{1\bar{k}}})$, $d_{B_{2\bar{k}}}^{\dagger} (d_{B_{2\bar{k}}})$, $p_{A_{x\bar{k}}}^{\dagger} (p_{A_{x\bar{k}}})$, $p_{A_{y\bar{k}}}^{\dagger} (p_{A_{y\bar{k}}})$, $p_{B_{x\bar{k}}}^{\dagger} (p_{B_{x\bar{k}}})$, and $p_{B_{y\bar{k}}}^{\dagger} (p_{B_{y\bar{k}}})$ create (annihilate) an electron at $V_A$($d_{x^2-y^2}$), $V_A$($d_{xy}$), $V_B$($d_{x^2-y^2}$), $V_B$($d_{xy}$), $O_A$($p_x$), $O_A$($p_y$), $O_B$($p_x$), and $O_B$($p_y$) orbitals, respectively, with momentum $\bar{k}$. $\epsilon_{d_A}$, $\epsilon_{d_B}$, $\epsilon_{p_A}$, and $\epsilon_{p_B}$ indicate on-site energy of $V_A$, $V_B$, $O_A$, and $O_B$, respectively. Furthermore, $t_{V_AV_B}$, $t_{V_AO_B}$, $t_{V_AO_A}$, $t_{V_BO_A}$, $t_{V_BO_B}$, $t_{O_AO_B}$, $t_{O_AO_A}$, and $t_{O_BO_B}$ being hopping parameter of $V_A$-$V_B$, $V_A$-$O_B$, $V_A$-$O_A$, $V_B$-$O_A$, $V_B$-$O_B$, $O_A$-$O_B$, $O_A$-$O_A$, and $O_B$-$O_B$ respectively. Then, a and c being the lattice constant.
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Last edited: | 2020-02-20 10:46:40 | {"extraction_info": {"found_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": 0.9472838640213013, "perplexity": 404.0849818456019}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875144722.77/warc/CC-MAIN-20200220100914-20200220130914-00173.warc.gz"} |
https://par.nsf.gov/biblio/10089798-zinc-finger-readers-methylated-dna | Zinc Finger Readers of Methylated DNA
DNA methylation is a prevalent epigenetic modification involved in regulating a number of essential cellular processes, including genomic accessibility and transcriptional outcomes. As such, aberrant alterations in global DNA methylation patterns have been associated with a growing number of disease conditions. Nevertheless, the full mechanisms by which DNA methylation information is interpreted and translated into genomic responses is not yet fully understood. Methyl-CpG binding proteins (MBPs) function as important mediators of this essential process by selectively reading DNA methylation signals and translating this information into down-stream cellular outcomes. The Cys2His2 zinc finger scaffold is one of the most abundant DNA binding motifs found within human transcription factors, yet only a few zinc finger containing proteins capable of conferring selectivity for mCpG over CpG sites have been characterized. This review summarizes our current structural understanding for the mechanisms by which the zinc finger MBPs evaluated to date read this essential epigenetic mark. Further, some of the biological implications for mCpG readout elicited by this family of MBPs are discussed.
Authors:
;
Award ID(s):
Publication Date:
NSF-PAR ID:
10089798
Journal Name:
Molecules
Volume:
23
Issue:
10
Page Range or eLocation-ID:
2555
ISSN:
1420-3049
National Science Foundation
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2. INTRODUCTION To faithfully distribute genetic material to daughter cells during cell division, spindle fibers must couple to DNA by means of a structure called the kinetochore, which assembles at each chromosome’s centromere. Human centromeres are located within large arrays of tandemly repeated DNA sequences known as alpha satellite (αSat), which often span millions of base pairs on each chromosome. Arrays of αSat are frequently surrounded by other types of tandem satellite repeats, which have poorly understood functions, along with nonrepetitive sequences, including transcribed genes. Previous genome sequencing efforts have been unable to generate complete assemblies of satellite-rich regions because of their scale and repetitive nature, limiting the ability to study their organization, variation, and function. RATIONALE Pericentromeric and centromeric (peri/centromeric) satellite DNA sequences have remained almost entirely missing from the assembled human reference genome for the past 20 years. Using a complete, telomere-to-telomere (T2T) assembly of a human genome, we developed and deployed tailored computational approaches to reveal the organization and evolutionary patterns of these satellite arrays at both large and small length scales. We also performed experiments to map precisely which αSat repeats interact with kinetochore proteins. Last, we compared peri/centromeric regions among multiple individuals to understand how thesemore »
3. Abstract Polycomb repressive complex 2 (PRC2) is a histone methyltransferase that methylates histone H3 at Lysine 27. PRC2 is critical for epigenetic gene silencing, cellular differentiation and the formation of facultative heterochromatin. It can also promote or inhibit oncogenesis. Despite this importance, the molecular mechanisms by which PRC2 compacts chromatin are relatively understudied. Here, we visualized the binding of PRC2 to naked DNA in liquid at the single-molecule level using atomic force microscopy. Analysis of the resulting images showed PRC2, consisting of five subunits (EZH2, EED, SUZ12, AEBP2 and RBBP4), bound to a 2.5-kb DNA with an apparent dissociation constant ($K_{\rm{D}}^{{\rm{app}}}$) of 150 ± 12 nM. PRC2 did not show sequence-specific binding to a region of high GC content (76%) derived from a CpG island embedded in such a long DNA substrate. At higher concentrations, PRC2 compacted DNA by forming DNA loops typically anchored by two or more PRC2 molecules. Additionally, PRC2 binding led to a 3-fold increase in the local bending of DNA’s helical backbone without evidence of DNA wrapping around the protein. We suggest that the bending and looping of DNA by PRC2, independent of PRC2’s methylation activity, may contribute to heterochromatin formation and therefore epigenetic gene silencing.
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Environmental fluctuation during embryonic and fetal development can permanently alter an organism’s morphology, physiology, and behaviour. This phenomenon, known as developmental plasticity, is particularly relevant to reptiles that develop in subterranean nests with variable oxygen tensions. Previous work has shown hypoxia permanently alters the cardiovascular system of snapping turtles and may improve cardiac anoxia tolerance later in life. The mechanisms driving this process are unknown but may involve epigenetic regulation of gene expression via DNA methylation. To test this hypothesis, we assessed in situ cardiac performance during 2 h of acute anoxia in juvenile turtles previously exposed to normoxia (21% oxygen) or hypoxia (10% oxygen) during embryogenesis. Next, we analysed DNA methylation and gene expression patterns in turtles from the same cohorts using whole genome bisulfite sequencing, which represents the first high-resolution investigation of DNA methylation patterns in any reptilian species.
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http://golem.ph.utexas.edu/category/2008/01/geometric_representation_theor_19.html | ## January 10, 2008
### Geometric Representation Theory (Lecture 20)
#### Posted by John Baez
In this, the final lecture of the fall’s Geometric Representation Theory seminar, I tried to wrap up by giving a correct statement of the Fundamental Theorem of Hecke Operators.
The fall seminar was a lot of fun, and very useful. It didn’t go the way I expected. I thought I thoroughly understood groupoidification, but I didn’t! So, all hell broke loose when I tried to state the Fundamental Theorem. The seminar threatened to swerve out of control, and Jim had to invent some more math to save the day. We skidded to safety at the very last second… but in the process, we learned a lot.
Will next quarter’s seminar be less hair-raising? Only time will tell!
• Lecture 20 (December 6) - John Baez on the Fundamental Theorem of Hecke Operators. This theorem says that for any finite group $G$, if we take the Hecke bicategory of $G$ and ‘degroupoidify’ it using
$\overline{D}: [bicategories enriched over FinSpan] \to [categories enriched over FinVect]$
the result is equivalent to the category of (finite-dimensional) permutation representations of $G$.
In short: the Hecke bicategory of $G$ is a groupoidification of the the category of permutation representations of $G$.
Future directions: groupoidifying the q-deformed Pascal’s triangle, the action of the quantum group $GL_q(2)$ on the quantum plane, and more generally the action of $GL_q(n)$ on ‘quantum $n$-space’. (Final words cut off as the power cable to the video camera is accidentally unplugged!)
For a more precise and thorough statement of the Fundamental Theorem, read this:
To save time, I cut some corners in stating the Fundamental Theorem in class. The really beautiful statement involves a bit of topos theory. There’s more about this in the “supplementary reading”, but the technical details have not yet been optimized. Just as we can talk about group actions, we can talk about groupoid actions: an action of a groupoid $X$ is just a functor
$F: X \to Set$
If we know the all the actions of a groupoid — or more precisely, the topos of all its actions — we can recover the groupoid. But, Jim and I didn’t find a good constructive recipe for doing this. There are some other loose ends, too.
Luckily, Tom Leinster and Todd Trimble have been helping me out — and with such assistance, victory is inevitable.
Posted at January 10, 2008 3:25 AM UTC
TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1567
### Re: Geometric Representation Theory (Lecture 20)
It didn’t go the way I expected. I thought I thoroughly understood groupoidification, but I didn’t! So, all hell broke loose when I tried to state the Fundamental Theorem. The seminar threatened to swerve out of control, and Jim had to invent some more math to save the day. We skidded to safety at the very last second… but in the process, we learned a lot.
This is really moving. I am glad that you do things like that – and do it in public. More people should do that.
I have to admit that while I was initially quivering with anticipation to learn more about the groupoidification program, when your seminar really picked up steam I found myself busy with such a bunch of other things that I gave up on trying to follow. Plus, I can’t easily watch your videos here in my office for stupid technical reasons, so I didn’t actually watch any one of them.
So I am glad to see HDA VII come into existence!
Just in my last entry I again went on about how striking it is that when we look at principal $n$-bundles with connection and do the $n$-curvature part right, we are looking at a bundle whose fibers don’t look like $G_{(n)}$, the structure $n$-group, but like $G_{(n)}// G_{(n)}$: the action groupoid of that thing on itself.
I mentioned that a couple of times when we talked in Vienna last time, but probably failed to infect you with my excitement about this fact in the light of your groupoiification program.
And of course I might be hallucinating. But it seriously looks to me as if this is telling us that “geometric representations” (as opposed to linear representations, is that how you use the term?) arise automatically in $n$-transport, and that hence maybe the usual subsequent passage to associated linear $n$-transport is not real.
If and when I dig more into the groupoidification program, this will be the question I want to find the answer to.
It boils down to understanding, I guess, how exactly the 2-category of spans over $G$ is related to the category of linear $G$-representations.
I seem to recall that when we last talked about that in Vienna, you indicated a bunch of nice relationships, but didn’t quite come down to stating the entire theorem. Is that right? I might be misremembering. In any case, this is what I would like to understand.
Posted by: Urs Schreiber on January 10, 2008 10:50 AM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
Urs wrote:
More people should do that.
When I was a grad student, I really enjoyed a course by Daniel Quillen where he was trying to develop a very simple proof of the Atiyah–Singer index theorem, using little besides elementary calculus and Clifford algebras. Every day he would start by reviewing, in an incredibly neat and well-organized way, what he’d done before. But, after the course got going, most classes would begin to ‘break down’ near the end, when he ran into material that he hadn’t fully developed yet, and sometimes hit problems.
So, we got to learn how someone actually does math research.
(Eventually his research program was ‘scooped’ by Ezra Getzler, who came up with a similar proof of the index theorem, but using more heavy-duty analysis than Quillen allowed himself.)
I have to admit that while I was initially quivering with anticipation to learn more about the groupoidification program, when your seminar really picked up steam I found myself busy with such a bunch of other things that I gave up on trying to follow. Plus, I can’t easily watch your videos here in my office for stupid technical reasons, so I didn’t actually watch any one of them.
That’s too bad, but I understand. I’ve heard from a secret source that the bureaucracy at your institution is so inefficient that they’ve never managed to give you a key to the front door. So, I’m not surprised they can’t get streaming videos to work.
I also understand why you’re so busy.
But, eventually I’ll put HDA7 on the arXiv, and by then it will cover a lot more interesting material than it does now — categorified quantum groups and the like. So, you can read that.
$G_(n)//G_(n)$: the action groupoid of that thing on itself.
I mentioned that a couple of times when we talked in Vienna last time, but probably failed to infect you with my excitement about this fact in the light of your groupoidification program.
It was still an interesting discussion. I raised my ever-burning question about your interest in groupoids of the form $G//G$: why should you be interested in a groupoid that’s equivalent to the trivial groupoid? And, I think we worked our way a bit closer to the answer, which involved the ‘mapping cone’ concept. (In homotopy theory a mapping cone is contractible, hence ‘equivalent to the trivial space’, but the mapping cone construction is still useful.) You subsequently seem to have worked this idea into your bag of tricks… but I’m afraid I’m so busy that I don’t follow a lot of what you’re doing.
I think we had a useful interaction, even if we both scattered off and again behaved almost as free particles.
But it seriously looks to me as if this is telling us that “geometric representations” (as opposed to linear representations, is that how you use the term?) arise automatically in n-transport, and that hence maybe the usual subsequent passage to associated linear n-transport is not real.
That would be cool. The common attitude towards ‘geometric representation theory’ is that it studies linear representations of groups that arise from group actions on sets, algebraic varieties, etc. via various ‘linearization’ processes that turn these other entities into vector spaces: the vector space of functions on a set, the homology of a variety, etc. We’re trying to say that, at least in simple cases, the linearization process is almost an afterthought, and can be skipped if you get good enough at other kinds of math. We often use linear algebra just because we’re used to it.
It boils down to understanding, I guess, how exactly the 2-category of spans over $G$ is related to the category of linear $G$-representations.
Well that’s good, because that’s what the Fundamental Theorem of Hecke Operators answers.
However, the correct answer involved a little bit of ‘unasking the question’!
I seem to recall that when we last talked about that in Vienna, you indicated a bunch of nice relationships, but didn’t quite come down to stating the entire theorem. Is that right?
I thought I did state it, but maybe around 1 am while we were walking back to the hotel, somewhat lost in the streets of Vienna…
Anyway, I now understand all this stuff much better, and most of that understanding is built into the pompously named Fundamental Theorem of Hecke Algebras. The false version is easy to understand, as is the quick and dirty fix that makes it true… but the really good version is the content of HDA7 and is explained in simplified form in this lecture — lecture 20.
In a nutshell: degroupoidifying the Hecke bicategory of a finite group $G$, we get the category of finite-dimensional permutation representations of $G$. And, not explained above: if we then split idempotents, we get the category of all finite-dimensional representations of $G$.
(The limitation to finite groups is to avoid certain technical issues that make the general theory a lot more complicated and/or interesting.)
Posted by: John Baez on January 10, 2008 9:46 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
why should you be interested in a groupoid that’s equivalent to the trivial groupoid?
The mapping cone concept was useful for orienting ourselves, but I think the real answer is something which I may still not be able to formulate, but which a sufficiently sophisticated person should be able to formulate after thinking about the answer to the analogous question:
Why should we be interesting in the space $E G$? After all, it is equivalent to a point!
In fact, this question is considerably more than analogous: $G // G$ is $E G$, in a sense.
This sense is: the sequence of groupoids
$G \to G // G \to \mathbf{B}G$
is mapped by the nerve realization functor to the universal $G$-bundle
$G \to E G \to B G \,.$
While the middle term in both cases is equivalent to something trivial, I guess it is the fact that it sits in this sequence which makes it interesting.
So far so good. The really striking additional point now is that when we think of bundles with connection, $G // G$ is the home of curvature.
So, while you might have to further help me with formulating the abstract reason for “why is $G // G$ interesting?”, I can hand you large amounts of experimental evidence that indeed it is.
When $G$ is Lie, hitting $G \to G // G \to \mathbf{B}G$ with something like a functor from Lie groups to dg-algebras produces
$CE(g) \leftarrow \mathrm{W}(g) \leftarrow inv(g) \,.$
You can browse through our Lie $\infty$-connections and see $\mathrm{W}(g)$ appear all over the place.
You can imagine that there is an integration procedure so that whenever you see a $\mathrm{W}(g)$, it turns into $\mathbf{B}(G // G)$ (and whenever you see a $\mathrm{CE}(g)$ it turns into $\mathbf{B} G$ (for $G$ a Lie $n$-group now).
This is just to show: $G // G$ appears all over the place, and it is extremely useful.
Figure 1 on p. 6 illustrates the above analogy.
What is it that makes it useful, even though it is contractible?
It’s the fact that we know what “vertical” and what “horizontal” is inside $G // G$, and if we arrange that to be respected, then we prevent $G // G$ from collapsing to a point.
You see this general idea appearing first in figure 3 on p. 25. Then in figure 8 on p. 46 it appears in the context of $n$-transport, where finally in the diagram of the crucial proposition 25 on p. 48 it achieves its full meaning: that diagram, in a visually obvious sense says:
stuff moving vertically in $G // G$ does not affect horizontal stuff in $G // G$.
And that’s a very important statement, which only the existence of $G // G$ allows us to state efficiently.
So in words, the crucial insight is: $n$-transport and its curvature take values in $G // G$. The fact that this does not make them trivial is that they are constrained to “move” only “vertically” in $G // G$.
And that constraint, by the way, also has a nice interpretation: the thing taking values in $G//G$ arises really in an extension provlem as the obstruction to in a way trivializing the vertical part. That’s my best current attempt at giving an “abstract explanation” for why we should care about $G // G$.
Anyway, my point here is this: there is no doubt that we need to think of $G$ $n$-transport as really being $G // G$ ($n+1$)-transport plus constraints. That’s one thing the Lie $\infty$-connection work establishes.
So I find myself with a transport that assigns $G // G$ to fibers – and then I pass to the $n$-Café and read that you are teaching that this is a sneaky way to look at $G$-vector spaces.
That makes me wonder.
Posted by: Urs Schreiber on January 10, 2008 11:35 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
John wrote:
In a nutshell: degroupoidifying the Hecke bicategory of a finite group $G$, we get the category of finite-dimensional permutation representations of $G$. And, not explained above: if we then split idempotents, we get the category of all finite-dimensional representations of $G$.
Aargh! The second sentence is only true for certain groups, like the permutation groups $n!$. In general things are trickier, because not all irreps of $G$ appear as subrepresentations of permutation representations.
Double aargh! What an idiot! The second sentence up there is actually true for all finite groups $G$, if we work over the complex numbers. Every irreps of $G$ appears as a subrepresentation of a permutation representation — namely, the regular representation.
There are some other things that only work for special groups, but this is completely general: for any finite group $G$, we can get the category of finite-dimensional complex representations by taking the Hecke bicategory of $G$, degroupoidifying, and splitting idempotents.
Posted by: John Baez on January 30, 2008 4:37 AM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
John wrote:
In a nutshell: degroupoidifying the Hecke bicategory of a finite group $G$, we get the category of finite-dimensional permutation representations of $G$. And, not explained above: if we then split idempotents, we get the category of all finite-dimensional representations of $G$.
Aargh! The second sentence is only true for certain groups, like the permutation groups $n!$. In general things are trickier, because not all irreps of $G$ appear as subrepresentations of permutation representations. We’ve already discussed this issue. To get all complex reps of these other groups it seems we need to introduce $\mathbb{C}$ ‘by hand’ — or at least the field generated by all roots of unity, $\mathbb{Q}^{ab}$.
Posted by: John Baez on January 11, 2008 6:58 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
I am not very knowledgeable in representation theory, but I’m curious about something and it’s related to some of the previous discussion. The question is very straightforward.
Given S_n, the symmetric group on n elements, define the permutation representation R defined by R(g) = [g], where [g] is the permutation matrix corresponding to the permutation g. What is known about the irreducible subrepresentations of R? In particular, I would like to know the sum of dimensions of the distinct irreducible subrepresentations of R. I’d appreciate if you could provide me with a reference. Thanks in advance!
Posted by: Arnab on February 3, 2008 11:22 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
Arnab wrote:
Given $S_n$, the symmetric group on $n$ elements, define the permutation representation $R$ defined by $R(g) = [g]$, where $[g]$ is the permutation matrix corresponding to the permutation $g$. What is known about the irreducible subrepresentations of $R$?
It has a 1-dimensional irreducible subrepresentation consisting of all vectors of the form $(x,x,\dots, x)$. This is just the trivial representation. The orthogonal complement of this subspace is an $(n-1)$-dimensional irreducible representation.
So, $R$ breaks up into exactly two irreducible pieces (if $n \ge 2$).
This fact is a special case of a theorem Jim mentioned in lecture 7: if a finite group $G$ acts in a doubly transitive way on an $n$-element set, the resulting representation of $G$ on the vector space $\mathbb{C}^n$ is a direct sum of the trivial 1-dimensional representation and an irreducible $(n-1)$-dimensional representation. Jim sketched the proof, which is a nice application of the ideas we’re studying.
Posted by: John Baez on February 4, 2008 6:57 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
How much structure is needed to do the transfer
$H_0(X) \leftarrow H_0(Y) : f^!$
of groupoid 0-homologies given a functor $f : X \to Y$
of groupoids. I am asking because I would like to know if this can be understood at the place where the concept of a field enters, maybe.
Namely, when you form 0-th cohomology of a groupoid, you get a set, and it seems to be completely pointless overhead, at that point, to instead of talking about that set to talk about the vector space spanned by it.
But not so for the transfer. It is here that we need to use multiplication and division and addition, in order for the thing even to exist.
This is clear, but what I would like to know is: can we maybe understand the passage from sets to the vector spaces over fields which they span as something like the minimal prerequissite in order for there to be a transfer for the pushforward of homology?
Do you see what I mean?
Posted by: Urs Schreiber on January 10, 2008 9:15 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
Urs wrote:
How much structure is needed to do the transfer
$H_0(X) \leftarrow H_0(Y) : f^!$
of groupoid 0-homologies given a functor $f : X \to Y$
of groupoids?
No extra structure — just extra properties!
If the groupoids are finite, any functor between them gives rise to a transfer map on homology, using the simple formula given in HDA7. If the groupoids are infinite, the sum in this formula might diverge, and the cardinalities in this formula might be infinite. But, there are many interesting cases where those problems don’t occur… and you might say the winter’s seminar will be all about those cases.
Posted by: John Baez on January 10, 2008 9:55 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
There is a typo in Lemma 11 on p. 5 of the HDA VII draft: it says $X^{Set}$ as opposed to $Set^X$.
Posted by: Urs Schreiber on January 10, 2008 9:36 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
Thanks — fixed!
Posted by: John Baez on January 11, 2008 8:15 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
Could that material covered by Mark Weber in strict 2-toposes be useful? So that instead of functors between $Set^X$ and $Set^Y$, you think of them as going between discrete opfibrations over $X$ to the same over $Y$?
Posted by: David Corfield on January 11, 2008 12:11 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
Hey, isn’t the weak quotient $X // G$, for a groupoid $G$ acting on a set $X$ just the pullback over Mark Weber’s classifying discrete opfibration in the 2-topos of categories, Pointed set $\to$ Set?
So $X // G$ projects down to $G$, arrows $(x, g)$ being sent to $g$.
$X // G$ also gets mapped to Pointed Set, object $x$ being sent to $(X, x)$.
Posted by: David Corfield on January 12, 2008 9:51 AM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
Hey, isn’t the weak quotient $X//G$, for a groupoid $G$ acting on a set $X$ just the pullback over Mark Weber’s classifying discrete opfibration in the 2-topos of categories, $Pointed set \to Set$?
Yes indeed. $X//G$ is another name for the category of elements of the functor $X: G \to Set$, and this category of elements is obtained by pulling back the universal category of elements, $Pointed set \to Set$, along $X$. Exactly as you say.
Another useful thing to remember in this game is that
$Set^{X//G} \simeq Set^G/X.$
This is actually a general piece of abstract nonsense: that a slice of a presheaf topos $Set^C/X$ is (equivalent to) a presheaf topos, namely
$Set^{C \darr X}$
where the exponent denotes a comma category. This comma category is the category of elements of $X$, which brings us back to what is usually written as $X//G$ when $C = G$ is a groupoid.
Posted by: Todd Trimble on January 12, 2008 12:05 PM | Permalink | Reply to this
### Re: Geometric Representation Theory (Lecture 20)
A variance mistake may have crept into my previous comment: I have the feeling that the correct formulation should have been
$Set^{C^{op}}/X \simeq Set^{(C \darr X)^{op}}.$
Although for groupoids the mistake is fairly harmless :-).
While I’m at it, I may as well mention one more useful equivalence (it may well have been said earlier, but this blog moves fast and I don’t read everything; plus, I’m way behind on my video-watching):
$Set^G/(G/H) \simeq Set^{(G/H)//G)} \simeq Set^H.$
Posted by: Todd Trimble on January 12, 2008 2:26 PM | Permalink | Reply to this
Weblog: The n-Category Café
Excerpt: 2-toposes
Tracked: January 12, 2008 4:16 PM
Read the post L-infinity Associated Bundles, Sections and Covariant Derivatives
Weblog: The n-Category Café
Excerpt: Associated L-infinity structures are obtained from Lie action infinity-algebroids, leading to a concept of sections and covariant derivatives in this context.
Tracked: January 31, 2008 3:46 AM
Read the post Charges and Twisted n-Bundles, I
Weblog: The n-Category Café
Excerpt: Generalized charges are very well understood using generalized differential cohomology. Here I relate that to the nonabelian differential cohomology of n-bundles with connection.
Tracked: February 29, 2008 4:17 PM
Read the post Sections of Bundles and Question on Inner Homs in Comma Categories
Weblog: The n-Category Café
Excerpt: On inner homs in comma categories, motivated from a description of spaces of sections of bundles in terms of such.
Tracked: March 4, 2008 10:13 PM
### Re: Geometric Representation Theory (Lecture 20)
What would happen to the groupoidification program if I were to decide that the spans appearing there are really telling me that we are secretly working in a homotopy category?
In that case i would discard the morphisms between spans and instead require the left leg of each span to be a weak equivalence. Or rather, replace single spans with sequences of spans with that property.
Would the main theorem relating this setup to linear representation theory still go through?
I am wondering what would happen if one looked at $Ho(\omega Grpd)$, using this model category structure.
Posted by: Urs Schreiber on March 26, 2008 8:33 PM | Permalink | Reply to this
Read the post What has happened so far
Weblog: The n-Category Café
Excerpt: A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
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http://math.chapman.edu/cgi-bin/structures?Frames | Mathematical Structures: Frames
# Frames
http://mathcs.chapman.edu/structuresold/files/Frames.pdf
%%run pdflatex
%
\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\markboth{\today}{math.chapman.edu/structures}
\begin{document}
\textbf{\Large Frames}
\abbreviation{Frm}
\begin{definition}
A \emph{frame} is a structure $\mathbf{A}=\langle A, \bigvee, \wedge, e, 0\rangle$ of type $\langle\infty, 2, 0, 0\rangle$ such that
$\langle A, \bigvee, 0\rangle$ is a \href{Complete_semilattices.pdf}{complete semilattice} with $0=\bigvee\emptyset$,
$\langle A, \wedge, e\rangle$ is a \href{Meet_semilattices_with_identity.pdf}{meet semilattice with identity}, and
$\wedge$ distributes over $\bigvee$: $x\wedge(\bigvee Y)=\bigvee_{y\in Y}(x\wedge y)$
Remark: This is a template.
It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}
\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be frames. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(\bigvee X)=\bigvee h[X]$ for all $X\subseteq A$ (hence $h(0)=0$),
$h(x \wedge y)=h(x) \wedge h(y)$ and
$h(e)=e$.
\end{morphisms}
\begin{definition}
A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that
$...$ is ...: $axiom$
$...$ is ...: $axiom$
\end{definition}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})
Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.
\begin{tabular}{|ll|}\hline
Classtype & (value, see description) \cite{Ln19xx} \\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & \\\hline
Residual size & \\\hline
Congruence distributive & \\\hline
Congruence modular & \\\hline
Congruence $n$-permutable & \\\hline
Congruence regular & \\\hline
Congruence uniform & \\\hline
Congruence extension property & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective &no \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.
$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$\qquad
$\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$
\end{finite_members}
\begin{subclasses}\
\href{....pdf}{...} subvariety
\href{....pdf}{...} expansion
\end{subclasses}
\begin{superclasses}\
\href{....pdf}{...} supervariety
\href{....pdf}{...} subreduct
\end{superclasses}
\begin{thebibliography}{10}
\bibitem{Ln19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview}
\end{thebibliography}
\end{document}
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https://math.stackexchange.com/questions/2969790/given-initial-positions-and-velocities-of-two-boats-do-they-collide | # Given initial positions and velocities of two boats, do they collide?
This is a homework question from a precalculus class that I'm a TA for.
Boat $$A$$ is initially at position $$(1,4)$$ and moves at a constant velocity $$\langle 3,5 \rangle$$. Boat $$B$$ is at position $$(7,2)$$ and moves at a constant velocity of $$\langle 1,10 \rangle$$. Do the paths of the boats ever cross? If so where? Will the boats collide? If they don't collide, what's the closest the boats get to each other?
I wanted to write up a thorough solution to this exercise for my class, and figured I'd post it online to help anyone else who may wander across it.
• What is it that you want your students to practice? The simplest solution (I think) is to move to a frame in which one of the boats is at rest at the origin and see if the resulting forward ray from the other boat reaches the origin, i.e., that the difference in positions is a negative multiple of the difference in velocities.
– amd
Oct 25, 2018 at 1:20
• @amd That's probably the simplest solution computationally, but I don't think it's very intuitive. In the situation both boats are moving, so within the calculations both boats should be moving too. The students are struggling to become comfortable thinking in terms of vectors anyways, and I don't think they are ready to make the mental leap of shifting the frame of reference from the origin to one of the boats. I'll probably share this thought with any of the students who have a good grasp on working with vectors though, because it is a very useful thought. Oct 25, 2018 at 14:51
Let $$P_A(a)$$ denote the position of the boat $$A$$ at time $$a$$, and let $$P_B(b)$$ denote the position of boat $$B$$ at time $$b$$. From the initial positions and velocities given, we have: \begin{align} P_A(a) &= (1,4) + a\langle 3,5 \rangle &\qquad P_B(b) &= (7,2) + b\langle 1,10 \rangle \\ &= (3a+1, 5a+4) &\qquad &= (b+7,10b+2) \end{align} Now these equations give the paths of the boats starting at time $$a=b=0$$. The paths of the boats cross only if at some time $$a$$ and some time $$b$$ after each starts moving they have the same position. In terms of those equations, the paths of the boats will cross if there are positive times $$a$$ and $$b$$ such that $$P_A(a) = P_B(b)$$. Now the boats collide if not only is there a location where their paths cross, but if they are at that location at the same time. So the boats collide if $$P_A(a) = P_B(b)$$ for some positive $$a$$ equal to $$b$$. So we can proceed by setting $$P_A = P_B$$: $$(3a+1, 5a+4) = (b+7,10b+2) \implies \begin{cases} 3a+1=5a+4 \\ b+7=10b+2 \end{cases}\ \implies \begin{cases} 3a-b=6 \\ 5a-10b=-2 \end{cases}\,,$$ This, being a system of linear equations, has at most a single solution, which we can calculate to be $$a = \frac{62}{25}$$ and $$b = \frac{36}{25}$$. These are both positive times, so the paths of the boats do cross, but since this is the only solution and $$a \neq b$$, the boats do not collide. To find the actual coordinates where they do cross will be the location of boat $$A$$ at time $$a=\frac{62}{25}$$ (which should equal the location of $$B$$ at time $$b=\frac{36}{25}$$ if we've done our calculations correctly), which we can calculate: $$P_A\left(\frac{62}{25}\right) = \left(3\cdot\frac{62}{25}+1, 5\cdot\frac{62}{25}+4\right) = \left(\frac{211}{25} , \frac{410}{25} \right)\,.$$
To figure out how close the boats get, we can write a function to represent the distance between the boats at a time $$t$$ and minimize that function. We now need to consider the boats in the same time-frame and let $$a=b=t$$. The distance between boat $$A$$ and boat $$B$$ is given by \begin{align} d(t) &= \sqrt{(3t+1-t-7)^2+(5t+4-10t-2)^2} \\ &= \sqrt{29t^2-4t+40} \end{align} The minimum of the function $$d$$ will occur at the minimum of the quadratic $$29t^2-4t+40$$ since the square root is a strictly increasing function. And the minimum of that quadratic occurs at $$t = \frac{4}{2\cdot 29} = \frac{2}{29}$$. So the actual minimum distance they achieve is $$d( \frac{2}{29}) = \frac{34}{\sqrt{29}}$$ miles apart.
• Why not use the same time parameter and see if the positions are ever equal? That’s a single equation with only one variable to check, basically equivalent to trying to solve the above system by back-substitution. Using two different parameters seems more appropriate when simultaneity isn’t important, such as when you’re simply computing the intersection of the two lines. As in my original comment to your question, it comes down to what you’re trying to illustrate, which isn’t clear to me from the question itself.
– amd
Oct 25, 2018 at 18:19
• @amd Part of the problems does ask where (if) the paths of the boats cross though. Oct 25, 2018 at 21:07
• Ah, yes, missed that part of the question.
– amd
Oct 25, 2018 at 21:10
• Yeah, that’s a very common beginner mistake with parametric equations. It might also be worth pointing out to your students what happens when you do use a common parameter: the correct conclusion is that the boats don’t collide, not that their paths don’t intersect.
– amd
Oct 25, 2018 at 21:20
• I would probably write the functions $P_A$ and $P_B$ originally with $t$ as the argument to both, then switch to $t_1$ and $t_2$ when looking for whether they cross. Nov 6, 2018 at 22:23 | 2022-08-09 17:28:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 41, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9578151702880859, "perplexity": 159.58125753368455}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571056.58/warc/CC-MAIN-20220809155137-20220809185137-00534.warc.gz"} |
https://www.tramontanatremiti.com/8a14q9/d97130-mean-difference-formula | # mean difference formula
compute the means of the two samples (M1 In a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability p … One consideration A difference between means of 0 or higher is a difference of 10/4 = 2.5 standard deviations above the mean of -10. As shown below, the formula for the standard error of the difference between means as: Since the standard error of a sampling distribution is the standard We call this the two-sample T-interval or the confidence interval to estimate a difference in two population means. To use Analysis The null hypothesis is the hypothesis that the difference is 0. The primary and secondary fluid in an heat exchanger process may. We continue to use the data from the "Animal Figure 2. Reading from the simulation, we see that the critical T-value is 1.6790. The critical T-value comes from the T-model, just as it did in “Estimating a Population Mean.” Again, this value depends on the degrees of freedom (df). is called the assumption of, Click the "Enter/Edit Data" button. of freedom is 16 + 16 = 32. means is 10. The students were inspired by a similar study at City University of New York, as described in David Moore’s textbook The Basic Practice of Statistics (4th ed., W. H. Freeman, 2007). Students in an introductory statistics course at Los Medanos College designed an experiment to study the impact of subliminal messages on improving children’s math skills. In a difference in means hypothesis test, we calculate the probability that we would observe the difference in sample means (x̄ 1 - x̄ 2), assuming the null hypothesis is true, also known as the p-value. Previously, in “Hpyothesis Test for a Population Mean,” we looked at matched-pairs studies in which individual data points in one sample are naturally paired with the individual data points in the other sample. Therefore, if checking normality in the populations is impossible, then we look at the distribution in the samples. is simply the difference between means. the following small example: M1 = 4 and M2 = 3. In this case, the statistic is Arithmetic Mean Temperature Difference can be calculated like, AMTD = ((134 oC) + (134 oC)) / 2 - ((20 oC) + (50 oC)) / 2, Log Mean Temperature Difference can be calculated like, LMTD = ((134 oC) - (20 oC) - ((134 oC) - (50 oC))) / ln(((134 oC) - (20 oC)) / ((134 oC) - (50 oC))). of girls would be higher than the mean height of the sample of boys? Recall from the relevant Statistical analyses are very often concerned In this example, we use the sample data to find a two-sample T-interval for μ1 − μ2 at the 95% confidence level. A difference between means of 0 or higher is a difference 1 and 2 to differentiate these terms. A confidence interval for a difference between means is a range of values that is likely to contain the true difference between two population means with a certain level of confidence. The Logarithmic Mean Temperature Difference is always less than the Arithmetic Mean Temperature Difference. This is equal to (n1 - 1) + (n2 X represents observations The two-tailed test is used when the null hypothesis can be rejected ${ \sum x_1 = 3 + 9 + 5 + 7 = 24 \\[7pt] standard deviation of the distribution is: A graph of the distribution is shown in Figure 2. We found that the standard error of the sampling distribution of all sample differences is approximately 72.47. a t > 2.533. A typical example is an experiment Distribution of Difference between Means, Confidence For both samples, you enter: Mean: the observed arithmetic mean… Cookies are only used in the browser to improve user experience. of the difference between means is much simpler if the sample as more wrong than did the males. The P-value is the probability of obtaining the observed difference between the samples if the null hypothesis were true. If a histogram or dotplot of the data does not show extreme skew or outliers, we take it as a sign that the variable is not heavily skewed in the populations, and we use the inference procedure. Notice that it the larger sample size more than the group with the smaller 1 shows that the probability value for a two-tailed test is 0.0164. But what exactly is the probability? (on a 7-point scale) whether they thought animal research is wrong. What is range? The median is the middle value, so to rewrite the list in ascending order as given below: There are nine numbers in the list, so the middle one will be. Figure 1. For two-sample T-test or two-sample T-intervals, the df value is based on a complicated formula that we do not cover in this course. With saturation steam as the primary fluid the primary temperature can be taken as a constant since the heat is transferred as a result of a change of phase only. Range = 8. We do this by using the subscripts 1 and 2. sampling distribution of the mean: Since we have two populations and two samples sizes, that steam condenses at a constant temperature. The conditions for using this two-sample T-interval are the same as the conditions for using the two-sample T-test. Assume there are two species of green beings on Mars. by 5 or more? AMTD will in general give a satisfactory approximation for the mean temperature difference when the smallest of the inlet or outlet temperature differences is more than half the greatest of the inlet or outlet temperature differences. (1) sample n1 scores from Population It estimates the amount by which the experimental intervention changes the outcome on average compared with the control. left contains a sigma (σ), which means it is a standard deviation. It is clear that it is unlikely that the mean height for girls From the variance If numerous samples were taken from each age group M_1 = \frac{\sum x_1}{n} = \frac{24}{4} = 6 \\[7pt] Once we have the degrees of freedom, we can use Then, MSE is computed by: MSE = SSE/df Some of our calculators and applications let you save application data to your local computer. testing in the section on testing a boys are quite a bit taller. Calculate the standard error, for differences between means from two separate groups of subjects. Putting all this together gives us the following formula for the two-sample T-interval. is 165 and the variance is 64. Only emails and answers are saved in our archive. Mean =$\frac{13+18+13+14+13+16+14+21+13}{9}=15\$ (Note that the mean is not a value from the original list. The hypothesized value is the null hypothesis that the difference between population means is 0. $\begin{array}{l}(\mathrm{sample}\text{}\mathrm{statistic})\text{}±\text{}(\mathrm{margin}\text{}\mathrm{of}\text{}\mathrm{error})\\ (\mathrm{sample}\text{}\mathrm{statistic})\text{}±\text{}(\mathrm{critical}\text{}\mathrm{T-value})(\mathrm{standard}\text{}\mathrm{error})\end{array}$.
Dipping Sauces For Tempura Vegetables, Mcfarlane Dc Multiverse Wonder Woman, John 17 1-11 Message, Things To Do In And Around Tombstone Arizona, Gardenia Bread Jumbo, Attributes Of God Sunday School Lesson, Ellie Animal Crossing: New Horizons House, | 2022-05-20 01:27:40 | {"extraction_info": {"found_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": 0.6593186259269714, "perplexity": 632.5056035750667}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662530553.34/warc/CC-MAIN-20220519235259-20220520025259-00420.warc.gz"} |
http://weblib.cern.ch/collection/CLIC%20Detector%20and%20Physics%20Study?ln=pl&as=1 | Books, e-books, journals, periodicals, proceedings, standards and loaning procedures have been migrated on the 20th of April to the new website CERN Library Catalogue. See here for more information.
# CERN Accelerating science
CLIC Detector and Physics Study
Search in both CLIC DP and LCD Notes.
General CLIC Notes are also available.
# CLIC Detector and Physics Study
Ostatnio dodane:
2021-03-29
13:16
Optimising top-quark threshold scan at CLIC using genetic algorithm / Zarnecki, Aleksander (University of Warsaw (PL)) ; Nowak, K (University of Warsaw (PL)) One of the important goals at the future e+e− colliders is to measure the top-quark mass and width in a scan of the pair production threshold. However, the shape of the pair-production cross section at the threshold depends also on other model parameters, as the top Yukawa coupling, and the measurement is a subject to many systematic uncertainties. [...] CLICdp-Pub-2021-002.- Geneva : CERN, 2021 - 22. Fulltext: PDF;
2021-03-05
17:03
Physics performance for Dark Matter searches at √s = 3 TeV at CLIC using mono-photons and polarised beams. / Blaising, Jean-Jacques (Centre National de la Recherche Scientifique (FR)) ; Roloff, Philipp Gerhard (CERN) ; Sailer, Andre (CERN) ; Schnoor, Ulrike (University of Glasgow (GB)) At e− e+ colliders, Weakly Interacting Massive Particles (WIMPs) are candidates for dark matter (DM) and can be searched for using as tag a photon from initial state radiation. The potential for detecting DM at the Compact Linear Collider (CLIC) is investigated at sqrt(s) = 3 TeV. [...] CLICdp-Note-2021-001.- Geneva : CERN, 2021 - 15. Fulltext: PDF;
2021-02-04
16:00
Test-beam characterisation of the CLICTD technology demonstrator - a small collection electrode High-Resistivity CMOS pixel sensor with simultaneous time and energy measurement / Ballabriga Sune, Rafael (CERN) ; Buschmann, Eric (CERN) ; Campbell, Michael (CERN) ; Dannheim, D (CERN) ; Dort, K (CERN) ; Egidos, N (CERN) ; Huth, L (DESY) ; Kremastiotis, I (CERN) ; Kroger, J (CERN) ; Linssen, L (CERN) et al. The CLIC Tracker Detector (CLICTD) is a monolithic pixel sensor. It is fabricated in a 180 nm CMOS imaging process, modified with an additional deep low-dose n-type implant to obtain full lateral depletion. [...] CLICdp-Pub-2021-001.- Geneva : CERN, 2021 - 14. Fulltext: PDF;
2020-12-15
19:01
Prospects for Precision Measurements of the Top-Yukawa Coupling and CP Violation in $t\overline{t}H$ Production at the CLIC $e^+e^-$ Collider / Zhang, Yixuan High energy particle colliders provide unique facilities to investigate the physics that take place at the smallest scales [...] CERN-THESIS-2020-232 - 107 p.
Full text
2020-11-25
12:52
Corryvreckan: A Modular 4D Track Reconstruction and Analysis Software for Test Beam Data / Dannheim, Dominik (CERN) ; Dort, Katharina (Justus-Liebig-Universitaet Giessen (DE)) ; Huth, Lennart (Deutsches Elektronen-Synchrotron (DE)) ; Hynds, Daniel (Nikhef National institute for subatomic physics (NL)) ; Kremastiotis, Iraklis (KIT - Karlsruhe Institute of Technology (DE)) ; Kroeger, Jens (Ruprecht Karls Universitaet Heidelberg (DE)) ; Munker, Magdalena (CERN) ; Pitters, Florian Michael (Austrian Academy of Sciences (AT)) ; Schütze, Paul (DESY) ; Spannagel, Simon (DESY) et al. Corryvreckan is a versatile, highly configurable software with a modular structure designed to reconstruct and analyse test beam and laboratory data. It caters to the needs of the test beam community by providing a flexible offline event building facility to combine detectors with different read-out schemes, with or without trigger information, and includes the possibility to correlate data from multiple devices based on timestamps. [...] arXiv:2011.12730; CLICdp-Pub-2020-005.- Geneva : CERN, 2021-03-04 - 23 p. - Published in : JINST 16 (2021) P03008 Fulltext: CLICdp-Pub-2020-005 - PDF; 2011.12730 - PDF; Fulltext from publisher: PDF;
2020-11-16
13:24
The CLICTD Monolithic CMOS Sensor / Dort, Katharina (Justus-Liebig-Universitaet Giessen (DE)) CLICTD is a monolithic silicon pixel sensor fabricated in a modified 180 nm CMOS imaging process with a small collection electrode design and a high-resistivity epitaxial layer. It features an innovative sub-pixel segmentation scheme and is optimised for fast charge collection and high spatial resolution. [...] CLICdp-Conf-2020-007.- Geneva : CERN, 2020 - 8. Fulltext: PDF; In : 29th International Workshop on Vertex Detectors, Virtual, Japan, 5 - 8 Oct 2020
2020-10-20
10:47
Silicon vertex and tracking detector R&D for CLIC / Dort, Katharina (Justus-Liebig-Universitaet Giessen (DE)) The physics aims at the proposed future high-energy linear e+e- collider CLIC pose challenging demands on the performance of the detector system. In particular, the vertex and tracking detectors have to combine a spatial resolution of a few micrometres and a low material budget with a time-stamping accuracy of a few nanoseconds. [...] CLICdp-Conf-2020-006.- Geneva : CERN, 2020 - 7. Fulltext: PDF; In : 40th International Conference on High Energy Physics, Prague, Czech Republic, 28 Jul - 6 Aug 2020
2020-08-24
10:22
Optimising top-quark threshold scan at CLIC using genetic algorithm / Nowak, Kacper One of the main goals of the future e+e− colliders is to measure the top-quark mass and width in a scan of the pair production threshold [...] CERN-THESIS-2020-099 - 35 p.
Full text
2020-08-24
09:52
Searching for Inert Doublet Model scalars at high energy CLIC / Klamka, Jan Franciszek The Inert Doublet Model (IDM) is a simple extension of the Standard Model, introducing an additional Higgs doublet that brings in four new scalar particles [...] CERN-THESIS-2020-098 - 48 p.
Full text
2020-08-12
22:33
Detector R&D towards realistic luminosity measurement at the forward region of future $e^{+}e^{-}$ linear colliders. / Levy, Itamar The luminosity measurement at a future $e^{+}e^{-}$ linear collider will be preformed with a specialized compact calorimeter foreseen in the very forward region [...] CERN-THESIS-2019-382 - 125 p.
Full text
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[zastrzeżone] | 2021-04-20 20:52:42 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5693232417106628, "perplexity": 14665.88458995783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618039490226.78/warc/CC-MAIN-20210420183658-20210420213658-00251.warc.gz"} |
http://mathematica.stackexchange.com/tags/output-formatting/info | # Tag info
Mathematica provides a number of integrated options for controlling output form and style of expressions. Among these are named *Form wrappers. These can be:
There are also a number of tools for specific styling, most notably Style[]. | 2013-05-25 16:02:41 | {"extraction_info": {"found_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": 0.3706711232662201, "perplexity": 4808.24243558996}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368705958528/warc/CC-MAIN-20130516120558-00014-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://tug.org/pipermail/texhax/2009-October/013425.html | # [texhax] glossaries order linked words
Sun Oct 11 15:48:10 CEST 2009
```Oh, thanks a lot, I didn't know about that key.
Thanks,
a
On Oct 10, 2009, at 4:03 PM, Dr Nicola L C Talbot wrote:
> Adrián López García de Lomana wrote:
>> I'm working with the package glossaries
>> \usepackage[style=altlist,order=letter]{glossaries}
>> and I don't know how to solve a minor problem. Some of the
> > glossary terms are referenced to an internet site:
>> \newglossaryentry{Python}{
>> name=\href{http://www.python.org/}{Python},
>> description={\LDPython},
>> text={Python}}
>> and when I create the glossaries, the ordering is not the
> > one I would like. I would like the term "Python" to be ordered
> > by the letter P, but (I suppose) it takes the first characters of
> > "name", in this case "\", for the ordering and all terms that
> > contain the "\href" on the name are ordered first. Some ideas of
> > how could I solve this?
>
> Use the sort key:
>
> \newglossaryentry{Python}{
> sort=Python,
> name=\href{http://www.python.org/}{Python},
> description={\LDPython},
> text={Python}}
>
> Regards
> Nicola Talbot
> --
> http://theoval.cmp.uea.ac.uk/~nlct/
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https://zbmath.org/?q=an:1189.57005 | ## Knots and $$k$$-width.(English)Zbl 1189.57005
For a generic closed curve $$\gamma$$ in $$\mathbb{R}^3$$, the bridge number $$b(\gamma)$$ is defined to be the number of maxima with respect to the $$z$$-axis, and the bridge number of a knot in $$\mathbb{R}^3$$ is the minimum of the bridge numbers of closed curves that represent the knot.
Let $$\mathcal{S}_1$$ be the set of all flat planes in $$\mathbb{R}^3$$ that are parallel to the $$xy$$-plane. Then $$\mathcal{S}_1$$ is identified with a line. Given a generic closed curve $$\gamma$$, let $$\mathcal{T}_1(\gamma)\subset\mathcal{S}_1$$ be the set of planes transverse to $$\gamma$$. It forms an open subset in $$\mathcal{S}_1$$. If we define the $$1$$-bridge number $$b_1(\gamma)$$ to be the number of connected components of $$\mathcal{T}_1(\gamma)$$, then $$b_1(\gamma)=2b(\gamma)+1$$.
Similarly, let $$\mathcal{S}_2$$ be the set of all flat planes in $$\mathbb{R}^3$$ that are parallel to the $$z$$-axis, $$\mathcal{S}_3$$ the set of all flat planes in $$\mathbb{R}^3$$, and $$\mathcal{S}_4$$ the set of all flat planes and round two-spheres in $$\mathbb{R}^3$$. For a generic (in an appropriate sense) closed curve $$\gamma$$, let $$\mathcal{T}_k(\gamma)$$ be the open set in $$\mathcal{S}_k$$ that consists of planes or spheres transverse to $$\gamma$$.
The authors define the $$k$$-bridge number $$b_k(\gamma)$$ to be the number of connected components of $$\mathcal{T}_k(\gamma)$$. For a knot, the $$k$$-bridge number is defined similarly.
The authors also generalize the width of a knot using $$\mathcal{S}_k$$. For a generic closed curve $$\gamma$$, define $$w_k(\gamma):=\sum_{P_i\in\mathcal{T}_k(\gamma)}\sharp(P_i\cap\gamma)$$. Note that $$w_1(\gamma)$$ is the width introduced by D. Gabai [J. Differ. Geom. 26, 479–536 (1987; Zbl 0639.57008)].
In the paper under review the authors study $$k$$-bridge number and $$k$$-width.
They show that for a given integer $$n$$, the number of knots with $$2$$-bridge number $$n$$ or less is finite. They also show that the $$2$$-width has the same property. In particular, it is shown that if a knot has $$2$$-bridge number less than $$7$$ or $$2$$-width less than $$11$$, then it is either the unknot or the trefoil.
Lower bounds for $$b_2$$ and $$w_2$$ are given in terms of the total curvature of the plane curve projected to the $$xy$$-plane, contrary to the bridge number; one can construct a closed curve with a fixed bridge number with arbitrarily large total curvature.
Some observations for $$b_3$$, $$w_3$$, $$b_4$$ and $$w_4$$ are also given.
### MSC:
57M25 Knots and links in the $$3$$-sphere (MSC2010)
Zbl 0639.57008
Full Text:
### References:
[1] Adams C., Othmer J., Stier A., Lefever C., Pahk S., Tripp J.: An introduction to the supercrossing index of knots and the crossing map. J. Knot Theory Ramif. 11(3), 445–459 (2002) · Zbl 1003.57008 [2] Artin E.: Theorie der Zöpfe. Abh. Math. Sem. Univ. Hamburg 4, 47–72 (1925) · JFM 51.0450.01 [3] Fabricius-Bjerre Fr.: On the double tangents of plane closed curves. Math. Scand. 11, 113–116 (1962) · Zbl 0173.50501 [4] Fáry I.: Sur la courbure totale d’une courbe gauche faisant un noeud. Bull. Soc. Math. Fr. 77, 128–138 (1949) (French) · Zbl 0037.23604 [5] Freedman M.H., He Z., Wang Z.: Mobius energy of knots and unknots. Ann. Math. 139(1), 1–50 (1994) · Zbl 0817.57011 [6] Gabai D.: Foliations and the topology of 3-manifolds III. J. Diff. Geometry 26, 479–536 (1987) · Zbl 0639.57008 [7] Kuiper N.H.: A new knot invariant. Math. Ann. 278(1–4), 193–209 (1987) · Zbl 0632.57006 [8] Langevin R., O’Hara J.: Conformal geometric viewpoints for knots and links I. Contemp. Math. 304, 187–194 (2002) · Zbl 1014.57008 [9] Milnor J.W.: On the total curvature of knots. Ann. Math. 52(2), 248–257 (1950) · Zbl 0037.38904 [10] Murasugi K.: On invariants of graphs with applications to knot theory. Trans. Am. Math. Soc. 314, 1–49 (1989) · Zbl 0726.05051 [11] Schubert H.: Uber eine numerische Knoteninvariante. Math. Z. 61, 245–288 (1954) (German) · Zbl 0058.17403 [12] Stasiak A., Katritchx A., Bednar J., Michoud D., Dubochet J.: Electrophoretic mobility of DNA knots. Nature 384, 122 (1996) [13] Sumners D.W.: Lifting the curtain: using topology to probe the hidden action of enzymes. Not. AMS 42, 528–537 (1995) · Zbl 1003.92515 [14] Weber C., Stasiak A., Los D., Dietler D.G.: Numerical simulation of gel electrophoresis of dna knots in weak and strong electric fields. Biophys. J. 90(9), 3100–3105 (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2023-01-31 04:00:33 | {"extraction_info": {"found_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": 0.7862337827682495, "perplexity": 944.4742758311512}, "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/1674764499842.81/warc/CC-MAIN-20230131023947-20230131053947-00097.warc.gz"} |
https://math.stackexchange.com/questions/1390205/how-can-i-prove-remainder-side-of-this-inequality | # How can I prove remainder side of this inequality?
Let $x$, $y$ be two positive numbers such that $x^4+y^4=x^2+y^2$. Prove that $$1\leqslant x+y\leqslant 2.$$ With $x+y\leqslant 2$. I tried
We have $$x^4+y^4\geqslant \dfrac{(x^2+y^2)^2}{2}.$$ Therefore, $$x^2+y^2\geqslant \dfrac{(x^2+y^2)^2}{2}$$ or $$(x^2+y^2)^2-2(x^2+y^2)\leqslant 0$$ Implies $x^2+y^2\leqslant 2.$ Another way $$\dfrac{x+y}{2} \leqslant \sqrt{\dfrac{x^2+y^2}{2}}=1.$$ Thus $x+y\leqslant 2.$
How can I prove $x+y\geqslant 1$?
Is this true?
Let $x$, $y$ be two positive numbers such that $x^m+y^m=x^n+y^n$, where $m$, $n$, ($m \neq n$) be two positive integer numbers, we have $$x+y\leqslant 2.$$
If $x+y\lt 1$ then $(x+y)^2\lt 1$, and therefore $x^2+y^2\lt 1$.
But from $0\lt x^2+y^2\lt 1$, we conclude that $(x^2+y^2)^2 \lt x^2+y^2$, and therefore $x^4+y^4\lt x^2+y^2$.
let $x+y=u,xy=v^2 \implies u^2 \ge 4v^2 \\ x^2+y^2=u^2-2v^2,x^4+y^4=(x^2+y^2)^2-2(xy)^2 =(u^2-2v^2)^2-2v^4=u^4-4u^2v^2+2v^4 \implies \\ u^4-(4v^2+1)u^2+2v^4+2v^2=0 \\ u^2=\dfrac{4v^2+1+ \sqrt{8v^4+1}}{2}$
$u^2=\dfrac{4v^2+1-\sqrt{8v^4+1}}{2} \le 2v^2$
$v^2\ge 0 \implies u^2=\dfrac{4v^2+1+ \sqrt{8v^4+1}}{2} \ge \dfrac{1+1}{2}=1$ when $v^2=0$ | 2019-10-17 10:05:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9950298070907593, "perplexity": 107.86905133534407}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986673538.21/warc/CC-MAIN-20191017095726-20191017123226-00525.warc.gz"} |
https://brilliant.org/discussions/thread/a-cool-property-regarding-circumcenter/ | ×
# A cool property regarding circumcenter
Let $$O$$ be the circumcenter of $$ABC$$. Reflect $$O$$ over $$BC$$ to obtain $$O'$$. Through $$O'$$ construct lines parallel to $$AC,AB$$ which respectively meet $$AB,AC$$ at $$F,E$$. Define $$O'F\cap OB=Y, O'E\cap CO=X$$. Prove $$XY||EF$$
I personally think this configuration is very rich and can be exploited to create difficult olympiad geo problems.
Note by Xuming Liang
1 year ago
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Notice that $$BOCO'$$ is a parallelogram. Since $$BP || O'C$$ and $$AB || O'E$$, we see that $$\angle FBX = \angle EO'C$$. It is easy to see that $$\angle A = \angle BFX = \angle YEC$$. Therefore, $$\triangle BFX \sim \triangle O'EC$$. This implies that $\frac{BF}{FX} = \frac{O'E}{EC}$.
Similarly, we have that $\triangle YCE \sim \triangle BO'F \implies \frac{O'F}{FB} = \frac{CE}{EY}$.
Multiplying the two ratios completes the proof. · 1 year ago
Yes! The parallel property will hold as long as $$BOCO'$$ is a parallelogram. There are a couple of typos in your proof, otherwise you got it spot on. · 1 year ago
@Alan Yan Hints: We want to prove some two ratios are equal, perhaps look for some similar triangles from all the parallels.
Generalize this property if you get it. · 1 year ago
PS: The problem can be generalized. · 1 year ago
Can you post the proof or hint after a few days? · 1 year ago
Yes. :) · 1 year ago
× | 2016-10-24 20:13:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8613424897193909, "perplexity": 561.1203948865705}, "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-2016-44/segments/1476988719754.86/warc/CC-MAIN-20161020183839-00134-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://yourmathsolver.blogspot.com/2011/12/mean-statistics.html | ## Friday, December 9, 2011
### Mean (Statistics)
In statistics, mean has two related meanings:
• the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
• the expected value of a random variable, which is also called the population mean.
There are other statistical measures that should not be confused with averages - including 'median' and 'mode'. Other simple statistical analyses use measures of spread, such as range, interquartile range, or standard deviation. For a real-valued random variable X, the mean is the expectation of X. Note that not every probability distribution has a defined mean (or variance); see the Cauchy distribution for an example.
For a data set, the mean is the sum of the values divided by the number of values. The mean of a set of numbers x1x2, ..., xn is typically denoted by $\bar{x}$, pronounced "x bar". This mean is a type of arithmetic mean. If the data set were based on a series of observations obtained by sampling a statistical population, this mean is termed the "sample mean" ($\bar{x}$) to distinguish it from the "population mean" (μ or μx). The mean is often quoted along with the standard deviation: the mean describes the central location of the data, and the standard deviation describes the spread. An alternative measure of dispersion is the mean deviation, equivalent to the average absolute deviation from the mean. It is less sensitive to outliers, but less mathematically tractable.
If a series of observations is sampled from a larger population (measuring the heights of a sample of adults drawn from the entire world population, for example), or from a probability distribution which gives the probabilities of each possible result, then the larger population or probability distribution can be used to construct a "population mean", which is also the expected value for a sample drawn from this population or probability distribution. For a finite population, this would simply be the arithmetic mean of the given property for every member of the population. For a probability distribution, this would be a sum or integral over every possible value weighted by the probability of that value. It is a universal convention to represent the population mean by the symbol μ. In the case of a discrete probability distribution, the mean of a discrete random variable x is given by taking the product of each possible value of x and its probability P(x), and then adding all these products together, giving $\mu = \sum x P(x)$.
The sample mean may differ from the population mean, especially for small samples, but the law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.
As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. These are listed below.
### Arithmetic mean (AM)
The arithmetic mean is the "standard" average, often simply called the "mean".
$\bar{x} = \frac{1}{n}\cdot \sum_{i=1}^n{x_i}$
The mean may often be confused with the median, mode or range. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income, and favors the larger number of people with lower incomes. The median or mode are often more intuitive measures of such data.
Nevertheless, many skewed distributions are best described by their mean – such as the exponential and Poisson distributions.
For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is
$\frac{34+27+45+55+22+34}{6} = \frac{217}{6} \approx 36.167.$
### Geometric mean (GM)
The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.
$\bar{x} = \left ( \prod_{i=1}^n{x_i} \right ) ^\tfrac1n$
For example, the geometric mean of six values: 34, 27, 45, 55, 22, 34 is:
$(34 \cdot 27 \cdot 45 \cdot 55 \cdot 22 \cdot 34)^{1/6} = 1,699,493,400^{1/6} \approx 34.545.$
### Harmonic mean (HM)
The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).
$\bar{x} = n \cdot \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1}$
For example, the harmonic mean of the six values: 34, 27, 45, 55, 22, and 34 is
$\frac{6}{\frac{1}{34}+\frac{1}{27}+\frac{1}{45} + \frac{1}{55} + \frac{1}{22}+\frac{1}{34}} = \frac{60588}{1835} \approx 33.0179836.$
### Relationship between AM, GM, and HM
AM, GM, and HM satisfy these inequalities:
$AM \ge GM \ge HM \,$
Equality holds only when all the elements of the given sample are equal. | 2019-12-13 20:28:41 | {"extraction_info": {"found_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": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9002745151519775, "perplexity": 389.8631094638408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540569146.17/warc/CC-MAIN-20191213202639-20191213230639-00372.warc.gz"} |
http://ykmy.candelaeventi.it/expectation-value-of-potential-energy-harmonic-oscillator.html | Harmonic oscillator • Node theorem still holds • Many symmetries present • Evenly-spaced discrete energy spectrum is very special! So why do we study the harmonic oscillator? We do because we know how to solve it exactly, and it is a very good approximation for many, many systems. Introduction Harmonic oscillators are ubiquitous in physics. However, even though the probability density changes with time, there is no sloshing back and forward and the expectation value for position as. Kinetic energy and the potential energy that's indicated there. Thus the average values of potential and kinetic energies for the harmonic oscillator are equal. 0 Partial differentials 6. As well as atoms and molecules, the empty space of the vacuum has these properties. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty principle. The vacuum energy is a special case of zero-point energy that relates to the quantum vacuum. The classical solvable examples are basically piecewise constant potentials, the harmonic oscillator and the hydrogen atom. i am given a time-dependent wavefunction, Ψ(x,t), and i am asked to calculate the expectation value of total energy E(t) and potential energy V(t). We will start in one dimension. Diatomic molecules have vibrational energy levels which are evenly spaced, just as expected for a harmonic oscillator. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. (a) Please write down the Schrodinger equation in x and y, then solve it using the separation of variables to derive the energy spectrum. A particle in an infinitely deep square well has a wave function given by ( ) = L x L x π ψ 2 2 sin. Schrödinger-equation (TSE) for a harmonic oscillator providing a statistical expectation value 〈r(t)〉 • The quantum mechanical equation of motion of the expectation value 〈r(t)〉 of bound or quasi-free charges in atoms and SC will be obtained for weak optical fields. (b) Explain why any term (such as $\hat{A}\hat{A^†}\hat{A^†}\hat{A^†}$) with unequal numbers of raising and lowering operators has zero expectation value in the ground state of a harmonic oscillator. It has that perfect combination of being relatively easy to analyze while touching on a huge number of physics concepts. This is a very important physical result because it tells us that the energy of a system described by a harmonic oscillator potential cannot have null energy. Ph 101-9 QUANTUM MACHANICS 1. Compare your results to the classical motion x(t) of a harmonic oscillator with the same physical parameters (!;m) and the same (average) energy Eˇ(n+ 1)~!. The harmonic oscillator April 24, 2006 To get the expectation value of hxi and hpi we need to know what the ladder We see that unlike the energy eigenstates, that now the expectation values are non-zero and depend on time. The probability that we will nd the oscillator in the nth state, with energy E0 n is ja nj2. the expectation values of Hˆ and ˆx. Intuition about simple harmonic oscillators. The QM Momentum Expectation Value program displays the time evolution of the position-space wave function and the associated momentum expectation value. Gravitational potential energy is the energy stored in an object due to its location within some gravitational field, most commonly the gravitational field of the. The simple harmonic oscillator, a nonrelativistic particle in a potential 1 2 k x 2, is an excellent model for a wide range of systems in nature. CHAPTER 6 Quantum Mechanics II 1. Harmonic Oscillator and Coherent States 5. The "spring constant" of the oscillator and its offset are adjustable. Expectation values of of ground state harmonic oscillator is given by Calculate uncertainty between position and momentum. Define potential energy. Simple Harmonic Oscillator February 23, 2015 To see that it is unique, suppose we had chosen a different energy eigenket, jE0i, to start with. Similarly, the expectation value of the potential energy is defined as the average value expected for potential energy measurements. (b) Find the most general expression for the first excited state of the two-dimensional isotropic harmonic oscillator in terms of the eigenstates {|n. Expectation values of of ground state harmonic oscillator is given by Calculate uncertainty between position and momentum. Substituting gives the minimum value of energy allowed. e-[i(E 0)t/h] + Ψ 1 (x)e-[i(E 1)t/h]], where Ψ 0, 1 (x) are the ground and first excited normalised eigenstate of the linear harmonic oscillator, n=0,1. Math, physics, perl, and programming obscurity. Quantum Mechanics. States of different parity do not mix. This is within 1:6% of the experimental value for the ground state of Helium. The vertical lines mark the classical turning points, that is, the displacements for which the harmonic potential equals the energy. The energy is 2μ1-1 =1, in units Ñwê2. It can be solved exactly and the energy levels are closely related to the harmonic oscillator for lower values of vbut get closer as vgets larger. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 1/25. The left hand side is equivalent to mass times acceleration. The uncertainty product, quantum-mechanical energy expectation value, and density. 3 Problem 6. Diatomic molecules have vibrational energy levels which are evenly spaced, just as expected for a harmonic oscillator. The ground-state energy W 0 of the crystal and the optimum value a 0 of a are then determined by a variational calculation which minimizes the expectation value of H between. So here we get: Here , with complex conjugate , Inserting these formulas into the equation for the energy, we get the expected formulas:. 108 LECTURE 12. 2 Density of photon states 263 5. The ground state is a Gaussian distribution with width x 0 = q ~ m!; picture from. The classical limits of the oscillator’s motion are indicated by vertical lines, corresponding to the classical turning points at x = ± A x = ± A of a classical particle with the same energy as the energy of a quantum oscillator in the state indicated in the figure. In formal notation, we are looking for the following respective quantities: , , , and. Harmonic Oscillator Solution using Operators Operator methods are very useful both for solving the Harmonic Oscillator problem and for any type of computation for the HO potential. So the given wavefunction must be an eigenfunction of the Hamiltonian. Operators and Observations Probabilities from inner products. Schrödinger-equation (TSE) for a harmonic oscillator providing a statistical expectation value 〈r(t)〉 • The quantum mechanical equation of motion of the expectation value 〈r(t)〉 of bound or quasi-free charges in atoms and SC will be obtained for weak optical fields. is a model that describes systems with a characteristic energy spectrum, given by a ladder of. We may note that the A-nucleus potential is assumed to be oscillator-like to begin with. Obtain an expression for in terms of k, mand. What is the expectation value of the operators x, x2, and p? But Schrodinger's equation in terms of H remain the same The expectation value of the Hamiltonian is the average value you. Definition of amplitude and period. energy if atom contains a proton and a µmeson, the meson mass is mµ ≈ 206me, me is the electron mass. Coherent states of a harmonic oscillator are wavepackets that have the shape of the ground state probability distribution but undergo the motion of a classical oscillator of arbitrary energy. Connection with Quantum Harmonic Oscillator In this nal part of our paper, we will show the connection of Hermite Poly-nomials with the Quantum Harmonic Oscillator. Time-Dependent Superposition of Harmonic Oscillator Eigenstates and its energy expectation value is given by The potential energy curve is drawn for visualization purposes. : spherical harmonics eq. [2] Observables and Hermitian operators. PHY-661 Quantum Mechanics I Final Exam Prof. The variational method is one way of finding approximations to the lowest energy eigenstate or ground state. An expectation value simply predicts the weighted average for all of those values. Find the expectation value of the potential energy in the nth state of the harmonic oscillator. 3) For all of the wave functions, the expectation value of the kinetic energy is exactly equal to the expectation value of the potential energy (this is not obvious from inspection, but is the subject of tomorrow's homework). a harmonic oscillator with mass Mand force constant K). to calculate m P X nªºÖÖ, ¬¼ 4. 3 Infinite Square-Well Potential 6. Its spectrum is the set of possible outcomes when one measures. Total energy Since the harmonic oscillator potential has no time-dependence, its solutions satisfy the TISE: ĤΨ = EΨ (recall that the left hand side of the SE is simply the Hamiltonian acting on Ψ). 3 Infinite Square-Well Potential 6. Expectation Values of and The Wavefunction for the HO Ground State; Examples. potentials, parity. 3: Infinite Square. 16 Use the uncertainty principle to estimate the ground-state energy of a particle of mass m bound in the harmonic oscillator potential V(x) = 6. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. Excited states HARMONIC OSCILLATOR WAVE FUNCTIONS Classical turning point TIME DEPENDENCE The superposition operator. The Hamilton operator of the harmonic oscillator reads H^ = p^2. However, as we show in the Section 5,. 1 General properties. Potential energy is one of several types of energy that an object can possess. 3: Infinite Square. The expectation value of the position operator in the state given by (a) (b) (c) (d) Q8. Turning Points, A, -A. Vacuum energy is an underlying background energy that exists in space throughout the entire Universe. A graph of the energies found with respect to the variational parameter is seen below. (8 marks) My answer (a): In a harmonic oscillator, the lowest energy of the eigenfunction is called the zero-point energy of the oscillator. Total energy Since the harmonic oscillator potential has no time-dependence, its solutions satisfy the TISE: ĤΨ = EΨ (recall that the left hand side of the SE is simply the Hamiltonian acting on Ψ). This is Newton's second law in terms of expectation values: Newtonian mechanics defines the negative derivative of the potential energy to be the force, so the right hand side is the expectation value of the force. Consider a time-dependent superposition of quantum harmonic oscillator eigenstates, , where the eigenfunctions and eigenvalues are given by and , respectively. Comments are made on the relation to the harmonic oscillator, the ground-state energy per degree of freedom, the raising and lowering operators, and the radial momentum operators. Schrödinger first considered these in the context of minimum-uncertainty wavepackets. The energy E in the system is proportional to the square of the amplitude. Quantum Harmonic Oscillator Expectation Values While I could never cover every example of QHOs, I think it is important to understand the mathematical technique in how they are used. Classical Harmonic Oscillator 2 2 ( ) 2 m 2x2 m p H K V x w = + = + Total Energy = Kinetic Energy + Potential Energy Total Energy: E = mw2 A2 Amplitude Frequency Period m T k p w 2 = = m k PDF created with pdfFactory Pro trial version www. EXPECTATION VALUES Lecture 8 Energy n=1 n=2 n=3 n=0 Figure 8. Maximum displacementx 0 occurs when all the energy is potential. Math, physics, perl, and programming obscurity. Write down the Schrödinger equation in the normal cartesian coordinate representation. 1 Harmonic Oscillator We have considered up to this moment only systems with a finite number of energy levels; we are now going to consider a system with an infinite number of energy levels: the quantum harmonic oscillator (h. (c) Find the expectation value (p) as a function of time. 1: The rst four stationary states: n(x) of the harmonic oscillator. So the given wavefunction must be an eigenfunction of the Hamiltonian. The less damping the higher the $$Q$$ factor. 16 Use the uncertainty principle to estimate the ground-state energy of a particle of mass m bound in the harmonic oscillator potential V(x) = 6. Master of Arts (Physics), August, 1980. a) How many eigen states corresponds to this energy E?. 60 molecule, which served as an oscillator in this experiment, has a mass of 1:2 10 24 kg. 4) Find the potential energy at point C. Consider a harmonic oscillator constructed from a 1 gram mass at the end of a spring. 7 Barriers and Tunneling CHAPTER 6 Quantum Mechanics II I think it is safe to say that no one understands quantum mechanics. By substituting the expansion f(x) = (C1 + C2x + C3x/^2 )e^−x 2/2 into Eq. HARMONIC OSCILLATOR AND UNCERTAINTY [(15+15+10) PTS] a)For a simple harmonic oscillator with H^ = (^p2 =m kx 2)=2, show that the energy of the ground state has the lowest value compatible with the uncertainty principle. In other words, = = E/2 for the quantum harmonic oscillator. Adapt the Hellmann-Feynman theorem for the expectation value of a parameter-dependent Hamiltonian (Exercise 8. Calculate the expectation values of X(t) and P(t) as a function of time. Commutation relations for the ladder operators, energy at the n th state and the expectation. Harmonic Oscillator in an External Electric Field (10 points) Suppose a charged particle (charge q> 0) bound in a harmonic oscillator potential is placed in a fixed electric field , so the Hamiltonian is Hˆ = pˆ2 2m + 1 2 mω2xˆ 2 + q xˆ. The quantum. Schr odinger Equation (TISE) for a particle in a one-dimensional harmonic oscillator potential. Any solution of the wave equation Compare to the expectation value of energy. When using Ehrenfest’s theorem, you have to take the expectation value of the entire left and right hand side. For example, if you want to measure the total energy of a system, the corresponding operator is the Hamiltonian Ĥ and the result of the measure will be one of the eigenvalues of the Hamiltonian. Two and three-dimensional harmonic osciilators. To modify this Hamiltonian to relativistic dynamics, we require precise relativistic kinetic energy operators instead of nonrelativistic ones for every internal (Jacobi) coordinate. Eigenvalues and eigenfunctions. So the average particle momentum and position are both zero. (a) Calculate the expectation values of the kinetic energy and the potential energy for a particle in the lowest energy state of a simple harmonic oscillator, using the wave function of Example 5-7. (10 points) (b) Calculate the expectation value of potential energy for the state with total energy 3 2. The energies of a particle in a closed tube. The harmonic oscillator provides a starting point for discussing a number of more advanced topics, including multiparticle states, identicle particles and field theory. A major challenge in modern physics is to accurately describe strongly interacting quantum many-body systems. Hint: Consider the raising and lowering operators defined in Eq. 3 i "Modern Quantum Mechanics" by J. The energy is 2μ1-1 =1, in units Ñwê2. Chapter 5: classical harmonic oscillator (section 5-1); link between harmonic oscillator and chemical bond (section 5-3); harmonic oscillator energy levels (section 5-4); harmonic oscillator wavefunctions (section 5-6); Morse oscillator (section 5-3) Test 2 material: part 2,3,4,5 of the "NEW LECTURE NOTES" and part 3,4,5,6 of the "OLD LECTURE. Verify that $$\displaystyle ψ_1(x)$$ given by Equation 7. 6 Simple Harmonic Oscillator 6. kharm Out[5]= 2 2x2 ü The Schrødinger equation contains the Hamiltonian, which is a sum of the quantum mechanical kinetic energy operator and the quantum mechanical potential energy operator. What is the probability of getting the result (same as the initial energy)?. The less damping the higher the $$Q$$ factor. 1) There are two possible ways to solve the corresponding time independent Schr odinger. 1 Classical Case The classical motion for an oscillator that starts from rest at location x 0 is x(t) = x 0 cos(!t): (9. Even-N and odd-N eigenvalue problems are entirely separate. Compute the time evolution of a superposition of energy eigenstates as well as the expectation value of common observables for a superposition state. Solution: For the ground state of the harmonic oscillator, the expectation value of the position operator x is given by 0 =! 0 "*(x)x! 0 (x)dx= m# $! xe%m#x2/! %& & "dx=0. | download | B–OK. 6) F(x) = dV dx = −kx, (5. (This is true of all states of the harmonic oscillator, in fact. The wavefunction that corresponds to this is ψ0(x) = mω 0 ~π 1/4 e−mω0x2/2~. The simple harmonic oscillator (SHO), in contrast, is a realistic and commonly encountered potential. fundamental vibration frequency (in cm–1 and s–1) and the zero point energy (in J) of this molecule. Consider a diatomic molecule AB separated by a distance with an equilbrium bond length. 2m + 1 2 m!2^x2 (1) Here we want to calculate the eigenvalues in an algebraic way. (2), determine the values of the constants C1, C2, and C3 (note that zero is a possible value) in terms of the harmonic oscillator constant kH, the ground state energy E0, the small correction energy , and the electric field wavenumber kE. This is expected because for a classical oscillator the energy is directly proportional to the am-plitude. In practice, to obtain a Hamiltonian with finite energy, we usually subtract this expectation value from H since this expectation is not observable. 6 Simple Harmonic Oscillator derivative of the free-particle wave function is Substituting ω = E / ħ yields The energy operator is The expectation value of the energy is Position and Energy Operators 6. (This is true of all states of the harmonic oscillator, in fact. The lowest allowed value of the quantum number is 0, which corresponds to the energy E = h. Harmonic Oscillator Solution using Operators Operator methods are very useful both for solving the Harmonic Oscillator problem and for any type of computation for the HO potential. 1 Harmonic Oscillator In this chapter we will study the features of one of the most important potentials in physics, it’s the harmonic oscillator potential which is included now in the Hamiltonian V(x) = m!2 2 x2: (5. 7 Barriers and Tunneling in some books an extra chapter due to its technical importance CHAPTER 6 Quantum Mechanics IIQuantum. 3: Infinite Square. Operators and observables, Hermitian opera-tors. As the equations of motion and then show, the uncertainties must be constant in time. to calculate m P X nªºÖÖ, ¬¼ 4. 24) The probability that the particle is at a particular xat a. The inflnite square well is useful to illustrate many concepts including energy quantization but the inflnite square well is an unrealistic potential. The quality factor ($$Q$$ factor) is a dimensionless parameter quantifying how good an oscillator is. Likewise the expected value of. 3 Infinite Square-Well Potential 6. * Example: The expectation value of as a function of time for the state is. Vacuum energy is an underlying background energy that exists in space throughout the entire Universe. The spacing between successive energy levels is , where is the classical oscillation frequency. The study of harmonic oscillator is continued to this lecture and normalization of harmonic oscillator energy eigenstate value is discussed. 2) Find the potential energy at point A using the PE formula. The default wave function is a Gaussian wave packet in a harmonic oscillator. Note that although the integrand contains a complex exponential, the result is real. Comments are made on the relation to the harmonic oscillator, the ground-state energy per degree of freedom, the raising and lowering operators, and the radial momentum operators. Harmonic motion is one of the most important examples of motion in all of physics. Thus, the total initial energy in the situation described above is 1 / 2 kA 2 ; and since the kinetic energy is always 1 / 2 mv 2, when the mass is at any point x in the oscillation,. 3 Infinite Square-Well Potential 6. 28 Simple Harmonic Oscillator, Creation and Annihilation Opera-tors Consider a simple one-dimensional harmonic oscillator with the following hamiltonian Hˆ = pˆ2 2m + 1 2 mω2xˆ2. 7 Barriers and Tunneling CHAPTER 6 Quantum Mechanics II I think it is safe to say that no one understands quantum mechanics. Here is the Hermite polynomial. This result is consistent with the equipartition theorem. b) Compute the expectation value of the position, as a function of time h ;tjx^j ;ti: Hint: You do not need to know the wave functions u 3(x) and u 4(x) or to compute an integral to solve this problem. Note the unequal spacing between different levels. * Example: The expectation value of for any energy eigenstate is. Define potential energy. 3: Infinite Square. Next: The Wavefunction for the Up: Harmonic Oscillator Solution using Previous: Raising and Lowering Constants Contents. 5 T 1 =5K T 2 =10K T 3 =15K 15 10 5 0 51015 u. 7 Barriers and Tunneling CHAPTER 6 Quantum Mechanics II I think it is safe to say that no one understands quantum mechanics. To modify this Hamiltonian to relativistic dynamics, we require precise relativistic kinetic energy operators instead of nonrelativistic ones for every internal (Jacobi) coordinate. "Is the potential energy of the quantum harmonic oscillator always one half the oscillator's total energy?" No. Harmonic oscillator • Node theorem still holds • Many symmetries present • Evenly-spaced discrete energy spectrum is very special! So why do we study the harmonic oscillator? We do because we know how to solve it exactly, and it is a very good approximation for many, many systems. May 07,2020 - The expectation value of energy when the state of the harmonic oscillator is described by the following wave functionwhere ψ0(x,t) and ψ2(x,t) are wave functions for the ground state and second excited state respectively :-a)b)c)d)Correct answer is option 'C'. Commutation relations for the ladder operators, energy at the n th state and the expectation. Calculate the expectation values of X(t) and P(t) as a function of time. For the quantum mechanical oscillator, the oscillation frequency of a given normal mode is still controlled by the mass and the force constant (or, equivalently, by the associated potential energy function). Using the ladder operator it becomes easy to find the following properties for a quantum oscillator in a given energy level: the average position and momentum and the square of these values as well as the average kinetic energy of a simple harmonic oscillator. It is usually denoted by , but also or ^ to highlight its function as an operator. In more than one dimension, there are several different types of Hooke's law forces that can arise. In[5]:= Classical harmonic potential for the harmonic oscillator in terms of the reduced mass and frequency is: Vho Vquad. Potential step, square well and barrier. The first derivative is 0 at the minimum and k is the spring constant of the vibrational motion. The following formula for the potential energy of a harmonic oscillator is useful to remember: V(x) = 1/2 m omega^2 x^2 where m is the mass , and omega is the angular frequency of the oscillator. to calculate m P X nªºÖÖ, ¬¼ 4. (a) Determine the expectation value of. 5 Three-Dimensional Infinite-Potential Well 6. The operators we develop will also be useful in quantizing the electromagnetic field. The quantum h. QUANTUM MECHANICS 1 2. Connection with Quantum Harmonic Oscillator In this nal part of our paper, we will show the connection of Hermite Poly-nomials with the Quantum Harmonic Oscillator. The Hamiltonian in this case is: [attached] a. Here is the Hermite polynomial. Using the same wavefunction, Ψ (x,y), given in exercise 9 show that the expectation value of p x vanishes. (a) An anharmonic one-dimensional oscillator for a particle of mass m has potential V(x)=1 2 mω 2x + λx4, where λ> 0 is small. In quantum physics, you can use operators to determine the energy eigenstate of a harmonic oscillator in position space. The total energy E of an oscillator is the sum of its kinetic energy and the elastic potential energy of the force At turning points , the speed of the oscillator is zero; therefore, at these points, the energy of oscillation is solely in the form of potential energy. 5 Three-Dimensional Infinite-Potential Well 6. of the potential (even parity), and the first excited state is antisymmetric (odd parity). Solution: For the ground state of the harmonic oscillator, the expectation value of the position operator x is given by 0 =! 0 "*(x)x! 0 (x)dx= m#$! xe%m#x2/! %& & "dx=0. 6 Harmonic oscillator: position and momentum expectation values Considera harmonic oscillator in its ground state (n= 0). Use this to calculate the expectation value of the kinetic energy. 50 fs, (b) a molecular vibration of period 2. Measurement of a superposition state. Harmonic Oscillator, a, a†, Fock Space, Identicle Particles, Bose/Fermi This set of lectures introduces the algebraic treatment of the Harmonic Oscillator and applies the result to a string, a prototypical system with a large number of degrees of freedom. Harmonic Oscillator:. Furthermore, the lowest energy state possesses the finite energy. Note that for the same potential, whether something is a bound state or an unbound state - Time evolution of expectation values for observables comes only through in The energy eigenstates of the harmonic oscillator form a family labeled by n coming from Eφˆ. 21 A beam of particles is described by the wave function = e—x2/4a2 a) Calculate the expectation value (p) of the momentum by working in the position representation. The eigenvalue problem (3) will be solved in a suitably chosen harmonic-oscillator basis. Find the state of the particle t) at a later time t. HARMONIC OSCILLATORS Harmonic oscillators are an extremely important application of quantum mechanics. We have also touched on a multiple particle system which fits into this. The masses can vibrate, stretching and compressing the spring with respect to the equilibrium spring length (the bond length), The masses can also rotate about the fixed point at the center of. 6 -- separation of variables is being attempted eq. The energy is 2μ1-1 =1, in units Ñwê2. The potential-energy function is a quadratic function of x, measured with respect to the. : spherical harmonics eq. List 3 equivalent formulas that you have learned for the Hermite functions. Intuition about simple harmonic oscillators. What is the expectation value of the oscillator’s kinetic energy? How do these results compare with the classical values of the average U and kinetic energy?. Physics 43 Chapter 41 Homework #11Key. 1) Find the total energy for the roller coaster at the initial point. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Using the ground state solution, we take the position and momentum expectation values and verify the uncertainty principle using them. 4 Finite Square-Well Potential 6. The expectation value of the angular momentum for the stationary coherent 2D Quantum Harmonic Oscillator. 6 Simple Harmonic Oscillator 6. 7 - Vibrations of the hydrogen molecule H2 can be Ch. There are no masses at position 0 and at position ( n +1) d ; these positions are the ends of the string. By the introduction of a variational scaling parameter a, a set of harmonic eigenfunctions can be generated from the eigenfunctions of a single such harmonic Hamiltonian. Diatomic molecules have vibrational energy levels which are evenly spaced, just as expected for a harmonic oscillator. (2), determine the values of the constants C1, C2, and C3 (note that zero is a possible value) in terms of the harmonic oscillator constant kH, the ground state energy E0, the small correction energy , and the electric field wavenumber kE. Virial theorem for a potential V which is homogeneous in coordinate x i and of degree n leads to the equation 2 ( T) = , those operators let you find all successive energy states. It is defined as the number of radians that the oscillator undergoes as the energy of the oscillator drops from some initial value $$E_0$$ to a value $$E_0e^{-1}$$. A particle of mass min the harmonic oscillator potential, starts out at t= 0, in the state (x;0) = A(1 2˘)2 e ˘2 where Ais a constant and ˘= p m!=~x:. At the top of the screen, you will see a cross section of the potential, with the energy levels indicated as gray lines. a) What is the expectation value of the energy? b) At some later time T the wave function is !x,T =B1+2 m"! x # $% & ' (2 e) m" 2! x2 for some constant B. Quantum Harmonic Oscillator. By substituting the expansion f(x) = (C1 + C2x + C3x/^2 )e^−x 2/2 into Eq. 1 Compute the uncertainty product h( x)2ih( p)2ifor the nth energy eigenstate of a one-dimensional quantum harmonic oscillator and verify that the uncertainty principle is. Here is the Hermite polynomial. Peter Young I. , references, 10 titles. Calculate the followings: where 12. Excited states HARMONIC OSCILLATOR WAVE FUNCTIONS Classical turning point TIME DEPENDENCE The superposition operator. (c) Suppose that the oscillator is in the nth energy eigenstate and that the potential is suddenly changed to V(x) = (∞ x < 0 mω2x2/2 x > 0. The mass may be perturbed by displacing it to the right or left. 2 Expectation value of \hat{{x}}^{2} and \hat{{p}}^{2} for the harmonic oscillator. Maximum displacementx 0 occurs when all the energy is potential. This is true provided the energy is not too high. Any vibration with a restoring force equal to Hooke's law is generally caused by a simple harmonic oscillator. Well! I had been grappling with this for a while, before I decided to go back to the roots. For an oscillating spring, its potential energy ( E p ) at any instant of time equals the work ( W ) done in stretching the spring to a corresponding displacement x. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. a) Determine hxi. A crossover regime occurs when the oscillator begins with an inter-mediate number of quanta. Take a look at the wavefunctions for the different energy levels of a simple harmonic oscillator (a crude approximation for a diatomic). Use the v=0 and v=1 harmonic oscillator wavefunctions given below which are normalized such that ⌡⌠-∞ +∞ Ψ (x) 2dx = 1. As the equations of motion and then show, the uncertainties must be constant in time. Time-Dependent Superposition of Harmonic Oscillator Eigenstates and its energy expectation value is given by The potential energy curve is drawn for visualization purposes. 1 Harmonic Oscillator In this chapter we will study the features of one of the most important potentials in physics, it’s the harmonic oscillator potential which is included now in the Hamiltonian V(x) = m!2 2 x2: (5. Use of the basis spanned by the eigenstates of the unperturbed harmonic-oscillator Hamiltonian 1 2 (p 2+x2) in Eq. ] (b) Show that the average kinetic energy is equal to the average potential energy (Virial Theorem). The perturbation, where is a constant, is added to the one dimensional harmonic oscillator potential. 2B Find expectation values of hpiand p. Displacement r from equilibrium is in units è!!!!! Ñêmw. In quantum mechanics, a Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system (this addition is the total energy of the system in most of the cases under analysis). 3: Infinite Square. Energy drinks cause concern for health of young people 14-10-2014 Increased consumption of energy drinks may pose danger to public health, especially among young people, warns a team of researchers from the World Health Organization Regional Office for Europe in the open-access journal Frontiers in Public Health. HARMONIC OSCILLATOR AND UNCERTAINTY [(15+15+10) PTS] a)For a simple harmonic oscillator with H^ = (^p2 =m kx 2)=2, show that the energy of the ground state has the lowest value compatible with the uncertainty principle. Wavefunction properties. The energy of particle in now measured. The red line is the expectation value for energy. ) gives the equation m x =−kx or x +ω2x=0, where ω=k/m is the angular frequency of sinusoidal os-cillations. Transform this Schrödinger equation to cylindrical coordinates where x = rcos φ, y = rsin φ, and z = z (z = 0 in this case). In formal notation, we are looking for the following respective quantities: , , , and. 1) we found a ground state 0(x) = Ae m!x2 2~ (8. (If you have a particle in a stationary state and then translate it in momentum space, then the particle is put in a coherent quasi-classical state that oscillates like a classical particle. In formal notation, we are looking for the following respective quantities: , , , and. Show that the expectation value of U is 1/2 E 0 when the oscillator is in the n = 0 state. 7 Example exercises 265 6 The harmonic oscillator 281 6. In the absence of losses (you have not said that the oscillator is damped) the total energy of the oscillator is a constant, which does not vary with time; so the concept of an average is inappropriate. x 0 = 2E T k is the "classical turning point" The classical oscillator with energyE T can never exceed this displacement, since if it did it would have more potential energy than the total energy. We can write we have. (a) Which type of potential is it: hydrogen-like atom, infinite square well, or harmonic. a) How many eigen states corresponds to this energy E?. : Time dependent wave function is j t=(t)i= e iEn ~(ajni+ be i!t jn+ 1i)(1) The average of position operator. Find the energy eigenvalues when A6= 0. Michael Fowler Einstein’s Solution of the Specific Heat Puzzle. The oscillator frequency is 1 Hz and the mass passes through its equilibrium position with a frequency of 10 cm/s. Now, take a look at the expected value of the kinetic energy and the potential energy of the oscillator when it is in the nth stationary state: =. In[5]:= Classical harmonic potential for the harmonic oscillator in terms of the reduced mass and frequency is: Vho Vquad. Comment: This is a direct result of the much more general Hellman-Feynman. Math, physics, perl, and programming obscurity. xx2ave xave 2 1 2 2 1 2 pp2ave pave 2 1 2 2 1 2 x p 1 2 Demonstrate that (x) is an eigenfunction of the energy operator and use the expectation values from above to calculate the expectation value for energy. Determine the expectation value of the potential energy for a quantum harmonic oscillator in the ground state. This should be fulfilled at the strong correlation limit (small ω), where the. This equation can be solved for the phase function, and a solution used in the energy expectation value to obtain a lower energy which is also independent of the choice of the gauge of the vector potential. The symmetry of V(x) is such that the mean or expected value of x is zero. Introduction. Verify that $$\displaystyle ψ_1(x)$$ given by Equation 7. (29) Introduce the following creation and annihilation operators a = r mω 2¯h à ˆx + ipˆ mω!; a† = r mω. From the precise form of expectation values in quantum mechanics, it follows that total energy must be the sum of the kinetic and potential energy ex. The methodology we adopt in all the systems is the same: 1. Does the result agree with the uncertainty. F kx dx dV(x) − = x = − where k is the force constant. 0 and α = 0. Supposing that there is a lowest energy level (because the potential has a lower. 5 Three-Dimensional Infinite-Potential Well 6. Harmonic Oscillator, a, a†, Fock Space, Identicle Particles, Bose/Fermi This set of lectures introduces the algebraic treatment of the Harmonic Oscillator and applies the result to a string, a prototypical system with a large number of degrees of freedom. Quantum tunneling is a phenomenon in which particles penetrate a potential energy barrier with a height greater than the total energy of the particles. On the other hand, suppose that the quantum harmonic oscillator is in an energy eigenstate. The harmonic oscillator April 24, 2006 To get the expectation value of hxi and hpi we need to know what the ladder We see that unlike the energy eigenstates, that now the expectation values are non-zero and depend on time. Figure 1: (a) Harmonic Oscillator Consisting of a Mass Connected by a Spring to a Fixed Support; (b) Potential Energy, V,and Kinetic Energy, EK For the Harmonic Oscillator. Schr odinger Equation (TISE) for a particle in a one-dimensional harmonic oscillator potential. Any vibration with a restoring force equal to Hooke's law is generally caused by a simple harmonic oscillator. For the harmonic oscillator potential in the time-independent Schr odinger equation: 1 2m ~2 d2 (x) dx2 + m2!2 x2 (x) = E (x); (9. We de ne the lowering operator ^a = 1 p 2m~! (i^p+ m!x^) (2) Note that, in contrast to ^pand ^x, ^ais not Hermitian and ^ayis called raising operator. to highlight its function as an operator. A major challenge in modern physics is to accurately describe strongly interacting quantum many-body systems. y the solution of the harmonic-oscillator equation (1). 3 Infinite Square-Well Potential 6. m Stretch spring, let go. (8 marks) My answer (a): In a harmonic oscillator, the lowest energy of the eigenfunction is called the zero-point energy of the oscillator. Since the eld A now has a potential energy, we can no longer shift the eld’s value by a constant without changing the physics. The lowest allowed value of the quantum number is 0, which corresponds to the energy E = h. Posts about expectation value written by peeterjoot Harmonic oscillator. 1: The rst four stationary states: n(x) of the harmonic oscillator. The operators we develop will also be useful in quantizing the electromagnetic field. Ψ(x,t) = (1/sqrt2)[Ψ 0 (x). 2 HYDROGEN ATOM – RADIAL BOUND STATE ANALYSIS 280 -Angular Momentum Analysis 283 -Reduction of 3D Analysis to Radial Analysis with Effective Potential Energy Function 289. Energy Operator in Quantum Mechanics for Free. is a model that describes systems with a characteristic energy spectrum, given by a ladder of. The energy of particle in now measured. 3) we found we could construct additional solutions with increasing energy using a. Ψ 1 2 v0 1 3 v1 1 6 v2 Ψ T Create Annihilate Ψ 1 2 Ψ 7 6 P0 E0 P1 E1 P2 E2 7 6 = 1 2 1 2 1 3 3 2 1 6 5 2 7 6 Below it is demonstrated that there are two equivalent forms of the harmonic oscillator energy operator. And that is for the harmonic oscillator, here's the Hamiltonian with the usual form. 21 A beam of particles is described by the wave function = e—x2/4a2 a) Calculate the expectation value (p) of the momentum by working in the position representation. The harmonic mechanical oscillator, the average value of X is 0 and the average value of P is 0. (If you have a particle in a stationary state and then translate it in momentum space, then the particle is put in a coherent quasi-classical state that oscillates like a classical particle. a) What is the expectation value of the energy? b) At some later time T the wave function is !x,T =B1+2 m"! x #$ % & ' (2 e) m" 2! x2 for some constant B. The wavefunction that corresponds to this is ψ0(x) = mω 0 ~π 1/4 e−mω0x2/2~. H = b+b + 1 2 =. Note that for the same potential, whether something is a bound state or an unbound state - Time evolution of expectation values for observables comes only through in The energy eigenstates of the harmonic oscillator form a family labeled by n coming from Eφˆ. Comparison of methods for integrating the simple harmonic oscillator. Suppose we measure the average deviation from equilibrium for a harmonic oscillator in its ground state. Note that although the integrand contains a complex exponential, the result is real. The operators we develop will also be useful in quantizing the electromagnetic field. Example 5-7. Remember, a state only has a definite value of an operator if it is an eigenstate of that operator - the state $|n\rangle$ does not have a well-defined potential energy, since $\hat{V}$ and $\hat{H}$ do not commute. Consider the. Homework Statement Hi all, i have a problem: i am given a time-dependent wavefunction, Ψ(x,t), and i am asked to calculate the expectation value of total energy E(t) and potential energy V(t). The wave function and its derivative are always continuous (except at infinite potential boundary). Harmonic oscillator. The eigen-values in this case. By particular. Total energy Since the harmonic oscillator potential has no time-dependence, its solutions satisfy the TISE: ĤΨ = EΨ (recall that the left hand side of the SE is simply the Hamiltonian acting on Ψ). It is for this reason that it is useful to consider the quantum mechanics of a harmonic oscillator. The quantum mechanical expectation value The quantum mechanical uncertainty The energy levels of the square well Sketch the potential for the square well and the first four energy eigenfunctions Sketch the first four probability distributions for the square well The energy levels of the simple harmonic oscillator (SHO). (Griffiths 3. If you have ONE basis state in a symmetric potential well, then the basis state is either even or odd. 1 Compute the uncertainty. Here again the zero for the potential energy can be chosen at R e. d 2 x(t ) k m x ( t ) dt 2 amplitude. In[5]:= Classical harmonic potential for the harmonic oscillator in terms of the reduced mass and frequency is: Vho Vquad. Schrödinger first considered these in the context of minimum-uncertainty wavepackets. The time-independent Schrödinger equation for a 2D. We may note that the A-nucleus potential is assumed to be oscillator-like to begin with. Additional states and other potential energy functions can be specified using the Display | Switch GUI menu item. Using the Bose-Einstein distribution, we can calculate the expectation value of the energy stored in the oscillator. -Harmonic Oscillator Expectation Values for Stationary States 265 -Harmonic Oscillator Time Evolution of Expectation Values for Mixed States 271 4. A-A+A+A-) has zero expectation value when operated on the ground state of a harmonic oscillator?. Next: The Wavefunction for the Up: Harmonic Oscillator Solution using Previous: Raising and Lowering Constants Contents. At t = 0, a particle in a harmonic-oscillator potential is in the initial state Qþ(x, 0) = Calculate the expectation value of energy in the state tþ(x, 0). 0 Partial differentials 6. For example, the small vibrations of most me-chanical systems near the bottom of a potential well can be approximated by harmonic oscillators. 1 Introduction In this chapter, we are going to find explicitly the eigenfunctions and eigenvalues for the time-independent Schrodinger equation for the one-dimensional harmonic oscillator. In quantum physics, you can use operators to determine the energy eigenstate of a harmonic oscillator in position space. Separation of variables provides us with one free particle wave equation, and two harmonic oscillator equations. kharm Out[5]= 2 2x2 ü The Schrødinger equation contains the Hamiltonian, which is a sum of the quantum mechanical kinetic energy operator and the quantum mechanical potential energy operator. Figure 1: (a) Harmonic Oscillator Consisting of a Mass Connected by a Spring to a Fixed Support; (b) Potential Energy, V,and Kinetic Energy, EK For the Harmonic Oscillator. to highlight its function as an operator. The above equation is usual 1D harmonic oscillator, with energy eigenvalues E0= n+ 1 2 ~!. This is Newton's second law in terms of expectation values: Newtonian mechanics defines the negative derivative of the potential energy to be the force, so the right hand side is the expectation value of the force. Using the raising and lowering operators a + = 1 p 2~m! ( ip+ m!x) a = 1 p 2~m! (ip+ m!x); (8. simply another name for a vector eld) becoming a harmonic oscillator potential for the gauge eld. The energy is 2μ1-1 =1, in units Ñwê2. 2 Course outline… Quantum Mechanics: Wave equation, Time dependent Schrodinger equation, Linearity & superposition, Expectation values, Observables as operators, Stationary states and time evolution of stationary states, Eigenvalues & Eigenfunctions, Boundary conditions on wave function, Application of SE (Particle in a box, Potential. It models the behavior of many physical systems, such as molecular vibrations or wave packets in quantum optics. energy if atom contains a proton and a µmeson, the meson mass is mµ ≈ 206me, me is the electron mass. 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## Analyzes the data table by linear regression and draws the chart.
Linear regression: y=A+Bx
（input by clicking each cell in the table below） data 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit
Guidelines for interpreting correlation coefficient r : 0.7＜|r|≦1 strong correlation 0.4＜|r|＜0.7 moderate correlation 0.2＜|r|＜0.4 weak correlation 0≦|r|＜0.2 no correlation$\normal\ Linear\ regression\vspace{10}\\(1)\ mean:\ \bar{x}={\large \frac{{\small \sum}{x_i}}{n}},\hspace{10}\bar{y}={\large \frac{{\small \sum}{y_i}}{n}}\\[10](2)\ trend\ line:\ y=A+Bx,\hspace{10} B={\large\frac{Sxy}{Sxx}},\hspace{10} A=\bar{y}-B\bar{x}\\[10](3)\ correlation\ coefficient:\ r=\frac{\normal S_{xy}}{\normal sqrt{S_{xx}}sqrt{S_{yy}}}\\\hspace{20}S_{xx}={\large \frac{{\small \sum}(x_i-\bar{x})^2}{n}}={\large \frac{{\small \sum} x_i^2}{n}}-\bar{x}^2\\\hspace{20}S_{yy}={\large \frac{{\small \sum}(y_i-\bar{y})^2}{n}}={\large \frac{{\small \sum} y_i^2}{n}}-\bar{y}^2\\\hspace{20}S_{xy}={\large \frac{{\small \sum}(x_i-\bar{x})(y_i-\bar{y})}{n}}={\large \frac{{\small \sum} x_i y_i}{n}}-\bar{x}\bar{y}\\$
Linear regression
[1-10] /26 Disp-Num5103050100200
[1] 2019/03/16 01:31 Male / Under 20 years old / High-school/ University/ Grad student / Very /
Purpose of use
Calculating how big your mom is
Comment/Request
Very helpful! It’s shown that for every pound of food, she exponentially breaks the line graph and just makes it into an
Exponentially huge graph
[2] 2019/01/22 11:31 Male / Under 20 years old / High-school/ University/ Grad student / Very /
Purpose of use
to recreate the algorithm
Comment/Request
[3] 2019/01/14 09:15 Female / Under 20 years old / High-school/ University/ Grad student / Very /
Purpose of use
Homework
[4] 2018/11/30 04:48 Male / Under 20 years old / High-school/ University/ Grad student / Not at All /
Purpose of use
my teacher said it would help us...
Comment/Request
it didn't.
[5] 2018/09/10 22:40 Male / Under 20 years old / Self-employed people / Very /
Purpose of use
A meme dealer
Comment/Request
This is it Chief.
[6] 2018/08/27 12:11 Male / 50 years old level / An engineer / Very /
Purpose of use
I was given 3 temperature measurements by a customer where X was known, and Y was given by the customer. Graphed, these measurements appeared to be linear, so this calculator allowed me to quickly compute slope/intercept values to compute the entire (useful) temperature scale from this customer''s 3 measurements.
[7] 2018/06/08 13:37 Male / 20 years old level / High-school/ University/ Grad student / Useful /
Purpose of use
internship project
Comment/Request
was not able to copy and paste data from excel sheet
[8] 2018/05/17 23:57 Female / Under 20 years old / Elementary school/ Junior high-school student / Very /
Purpose of use
a teacher suggested that her student use the website.
[9] 2018/05/15 22:19 Female / Under 20 years old / Elementary school/ Junior high-school student / Useful /
Purpose of use
I don''t have a real calculator
Comment/Request
Font size bigger
[10] 2018/05/15 22:17 Female / Under 20 years old / Elementary school/ Junior high-school student / Useful /
Purpose of use
I don''t have to do real work
Comment/Request
Thank u for doing all the work for me
Sending completion
To improve this 'Linear regression Calculator', please fill in questionnaire.
Male or Female ?
Age
Occupation
Useful?
Purpose of use? | 2019-03-24 06:50:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 1, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5552055835723877, "perplexity": 5572.494189638591}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1552912203378.92/warc/CC-MAIN-20190324063449-20190324085449-00501.warc.gz"} |
https://576i.nz/2022/05/cool-uris-dont-change/ | # Cool URIs don't change
At risk of sounding like every other ‘it was better in the olden days’ fogie out there (see also “I was born in the wrong generation!!1!1) - what the hell has happened to the internet? The internet today is distant in implementation to what Tim Berners-Lee envisaged when he designed HTML and HTTP. It is optimised to bring content to the masses with little permanence and accessibility via the open means inherently built into the protocols that underpin the internet. Instead, experience designers want you to use the content management tools they design that are engineered to trap you longer on their platform, and has the nasty side effect of making your saved content nigh impossible to reach via any other means.
Dynamic websites, that is, websites which have some from of document manipulation via programmatic means, most typically JavaScript, are in violation of the basic principle that a URI should target a particular document. By ‘document’, I don’t mean a page in general, I mean a discrete block of content on any topic - the content itself. In many cases, URIs now direct you to the start of a journey, or an interface you can use to access content. But accessing this content isn’t then reflected in that URI. It may be at /favourites instead. This article from the World Wide Web Consortium discusses the topic and provides good insight as to what the original plan was.
### Files are scary
This problem is also visible in application design, particularly on mobile. In the 1990s, if I wanted to make an application for any Windows 9x computer, one would use operating system specific GUI toolkits like the .NET framework or Carbon. Applications would have OS specific flavours that integrated natively with the underlying operating systems, and utilize commonly expected functions available to them from the operating system. One ubiquitous example of this is the ability to read and write files to the filesystem, which users have visibility of through their OS’s filesystem browser. Since the late 2000s however, web applications have become increasingly dominant, and if they run on your desktop, they’re probably an Electron app. Web applications have increasingly been using self-contained file management locked within the application you are using. You can often export your creations to some kind of open format, but at the expense of some level of detail in what you export, as some levels of information will be ‘proprietary’, and thus unable to be exported. This locks you in to their platform.
Mobile program design is particularly interesting because it exhibits application design trends that were common in the 1990s desktop operating system market. Most mobile apps are made using native OS toolkits, just like they were in the 1990s. However, they are adopting the way of obscuring files within applications. To an extent, they’ve been forced to do this on mobile as file managers were mostly non-existent on mobile for some time. Apple was, and still is, fiercely against file management on iOS devices. And to be honest, they have a reason. Because test, after test, has proved that files are hard. People have difficulties finding them, organising them, knowing where they are, etc. And now we have the new problem that children under the age of 15 or so at the time of writing may never have used a computer with a traditional file manager. The whole concept of a hierarchical filesystem is foreign to them. So the UX designers simply want to be rid of the whole idea - why is an open filesystem useful anyway?
Because vendor lock-in wasn’t always endemic to application design. Once, we understood that open files in an open filesystem were the ideal system. The benefits are many: Any program can access any file. We can track all our files for a project together regardless of application. We can keep the content separate from the applications for efficient backups. But unfortunately this isn’t easy for people, so whether malicious or not, UX designers are rapidly increasing vendor lock in by taking the easier route - associating files directly with the applications by simply building the files into the application.
From the perspective of someone who needs to design products for customers needs, this is the right thing to do - as people understand it more. But I think it harms openness and freedoms of the way we manage our information with technology. It’s a tough problem to think about solving.
Perhaps one way to solve it, would be to introduce tagged filesystems into operating systems. This post from Project Nayuki goes into extensive detail on this topic, and is certainly worth the read to understand more if you are interested. This would allow us to keep our open filesystems, but make it easier for people to naturally understand and use them. I look forward to seeing if we get any traction on tagged filesystems in general use operating systems in the future. | 2022-09-25 08:43:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.2902892529964447, "perplexity": 1474.3263230782063}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030334515.14/warc/CC-MAIN-20220925070216-20220925100216-00568.warc.gz"} |
https://codegolf.stackexchange.com/questions/122232/wardialing-in-the-modern-world | # Introduction
Wardialing was a very interesting way to try to hack people back in the '80s and '90s. When everyone used dial-up, people would dial huge amounts of numbers to search for BBS's, computers, or fax machines. If it was answered by a human or answering machine, it hung up and forgot the number. If it was answered by a modem or a fax machine, it would make note of the number.
"Of course, you realize this means War...dialing?" <--- Pun made by @Shaggy
# Challenge
Your job is to make a URL wardialer. Something that tests and checks if it's a valid website from one letter of the alphabet.
# Constraints
• Program must take user input. This input has to be a letter of the alphabet, no numbers. Just one letter of the alphabet and form multiple URLs that start with the letter. Input letter will be lowercase, not uppercase.
• Standard loopholes apply.
• You must make 8 URLs from 1 letter, and test to see if it is a valid site.
• If you hit an error (not a response code), instead of leaving it blank, go ahead and return a 404
• If you hit a redirect (3xx), return a 200 instead.
• You may output the results in any reasonable format, as long as it includes the website name, status codes for all the websites and the redirects.
• This is , so shortest amount of bytes wins.
# What counts as a URL for this challenge?
http://{domain-name}.{com or net or org}
For this challenge, the domain name should only be 4 letters long, no more, no less.
# What should I test?
For each 4 letter domain name, test it against three top-level domains (.com, .net, .org). Record all the response codes from each URL, remember from the constraints that any (3xx) should return 200 and be recorded as a redirect in the output and any error getting to the website should result in a 404.
a
# Output
+---------+------+------+------+------------+
| Website | .com | .net | .org | Redirects? |
+---------+------+------+------+------------+
| ajoe | 200 | 200 | 200 | .com, .net |
+---------+------+------+------+------------+
| aqiz | 200 | 404 | 404 | no |
+---------+------+------+------+------------+
| amnx | 200 | 503 | 404 | .com |
+---------+------+------+------+------------+
| abcd | 200 | 404 | 200 | .com |
+---------+------+------+------+------------+
| ajmx | 200 | 503 | 404 | no |
+---------+------+------+------+------------+
| aole | 200 | 200 | 200 | .com |
+---------+------+------+------+------------+
| apop | 404 | 200 | 200 | .net |
+---------+------+------+------+------------+
| akkk | 200 | 200 | 200 | .com |
+---------+------+------+------+------------+
• As a more modern human, what exactly was the point of wardialing? What did you use the numbers for? – Beta Decay May 23 '17 at 15:42
• @BetaDecay Typically, they were used to find computers, BBS's, or pretty much anything that could be connected to a computer modem. Once a number was found that could, the user could try to guess the user account to gain access to the system over dialup. – KuanHulio May 23 '17 at 15:46
• Are you sure it's only in the 80s or 90s? 'Cause I often get calls that hang up after one second... – feersum May 23 '17 at 16:01
• If you've seen the movie WarGames, I believe the protagonist does some of this. – Stephen May 23 '17 at 16:02
• That's where the name wardialing came from, I assume @StephenS and well, it never really stopped but was more prominent in the 80s and 90s. Not a lot of people use dialup any more. – KuanHulio May 23 '17 at 16:04
for i in $1{100..104}.{com,net,org};{ echo$i;(curl -IL $i 2>:||echo HTTP 404)| grep HTTP;} Sample output: b100.com HTTP/1.1 301 Moved Permanently HTTP/1.1 200 OK b100.net HTTP/1.1 301 Moved Permanently HTTP/1.1 200 OK b100.org HTTP 404 b101.com HTTP/1.1 301 Moved Permanently HTTP/1.1 200 OK b101.net HTTP/1.1 406 Not Acceptable b101.org HTTP/1.1 406 Not Acceptable b102.com HTTP/1.1 200 OK b102.net HTTP 404 b102.org HTTP 404 b103.com HTTP/1.1 301 Moved Permanently HTTP/1.1 200 OK b103.net HTTP 404 b103.org HTTP 404 b104.com HTTP/1.1 301 Moved Permanently HTTP/1.1 200 OK b104.net HTTP/1.1 301 Moved Permanently HTTP/1.1 200 OK b104.org HTTP 404 • working on changes, need to transform 3XX in 200 and avoid user agent problems – marcosm Jun 1 '17 at 14:10 # Python 2 + requests, 198 191 bytes from requests import* def f(c): for i in range(24): u='http://'+c+i/3*3+'.'+'cnooermtg'[i%3::3] try:a=get(u,allow_redirects=0).status_code except:a=404 if a/100==3:a='200 R' print u,a Sample output for a: http://a000.com 404 http://a000.net 404 http://a000.org 404 http://a111.com 200 ... http://a666.org 502 http://a777.com 403 http://a777.net 200 R http://a777.org 200 • I think the question was to produce all four-glyph names, not just the 1000 or so that start with a and end in a digit. The example output contains a test against ajoe.com which your code cannot check. – Draco18s no longer trusts SE May 23 '17 at 20:51 • @Draco18s You must make 8 URLs from 1 letter, and test to see if it is a valid site. does not sound like we should generate all url's. – ovs May 23 '17 at 20:58 • Ah, yes, I missed that. Carry on. – Draco18s no longer trusts SE May 23 '17 at 21:10 # Ruby, 145 + 10 = 155 bytes + 10 bytes for command line arguments. Probably not winning any beauty contests with this: x=[*$<.getc+?a*3..?z*4]
24.times{|i|r=Net::HTTP.get_response(URI(p"http://#{x[i/3]}."+'comnetorg'[i*3%9,3])).code rescue"404"
p r=~/^3/?"200R":r}
Run with ruby -rnet/http wardialing.rb
Example:
$ruby -rnet/http wardialing.rb <<< g "http://gaaa.com" "200R" "http://gaaa.net" "200" "http://gaaa.org" "200" "http://gaab.com" "200" "http://gaab.net" "200" "http://gaab.org" "200" "http://gaac.com" "404" (etc) # PHP, 221 bytes for($c=8;$c--;){$u=$argv[1];for($i=3;$i--;)$u.=chr(rand(97,122));for($z=0;$z<3;){$r="$u.".['com','net','org'][$z++];$h=get_headers("http://$r");$s=$h?substr($h[0],9,3):404;$t=(3==intval($s/100))?"200R":$s;echo"$r $t\n";}} With CRs and indentation for($c=8;$c--;){$u=$argv[1]; for($i=3;$i--;)$u.=chr(rand(97,122));
for($z=0;$z<3;){
$r="$u.".['com','net','org'][$z++];$h=get_headers("http://$r");$s=$h?substr($h[0],9,3):404;
$t=(3==intval($s/100))?"200R":$s; echo"$r $t\n"; } } Sample output on 'b': bnjz.com 404 bnjz.net 200 bnjz.org 404 bkxw.com 200 bkxw.net 200 bkxw.org 403 biak.com 200 biak.net 403 biak.org 200R bpzb.com 200 bpzb.net 200 bpzb.org 404 bigr.com 404 bigr.net 200 bigr.org 200 bcei.com 404 bcei.net 200R bcei.org 200 bexc.com 200 bexc.net 200 bexc.org 404 bset.com 404 bset.net 200 bset.org 200 # NodeJS, 249 246 bytes -3 bytes thanks to @ASCII-only. c=>{r=_=>Math.random().toString(36).slice(-3),o="",q=i=>(u=http://${n=i%3?n:c+r()}.+["com","net","org"][i%3],o+=u+" ",d=r=>(o+=(/^3/.test(s=r.statusCode||404)?"200R":s)+
,--i?q(i):console.log(o)),require("http").get(u,d).on("error",d)),q(24)}
Creates random three-letter strings from 1-9 and a-z for each domain. Redirects are shown as 200R.
Try it online
(Note: may take a minute or two to complete all requests)
### Sample output
http://a9k9.com 200
http://a9k9.net 404
http://a9k9.org 404
http://a529.com 200
http://a529.net 404
http://a529.org 404
http://asor.com 200R
http://asor.net 404
http://asor.org 404
## Longer version with table output, 278 bytes
c=>{r=_=>Math.random().toString(36).slice(-3),o=h=name .com .net .org
,q=i=>(i%3||(n=c+r(),o+=n+ ),u=http://${n}.+h.substr(6+i%3*5,3),d=r=>(o+=(/^3/.test(s=r.statusCode||404)?"200R":s)+(++i%3? : ),i<24?q(i):console.log(o)),require("http").get(u,d).on("error",d)),q(0)} Whitespace in name .com .net .org and n+ are literal tab characters. ### Sample table output name .com .net .org aa4i 200 200 200 a66r 404 404 404 anmi 200 200 403 aaor 403 200R 200R • I followed the link to 'Try it online' but only got "undefined" as output. – Octopus May 31 '17 at 22:44 • @Octopus It takes a minute or two to complete all of the requests, and for some reason the online interpreter outputs "undefined" right at the start. Running locally from command line does not output that, though. I'll add that as a note to the post. – Justin Mariner May 31 '17 at 22:55 • 246: c=>{r=_=>Math.random().toString(36).slice(-3),o="",q=i=>(u=http://${n=i%3?n:c+r()}.+['com','org','net'][i%3],o+=u+" ",d=r=>(o+=(/^3/.test(s=r.statusCode||404)?"200R":s)+ ,--i?q(i):console.log(o)),require("http").get(u,d).on("error",d)),q(24)} – ASCII-only Jun 2 '17 at 2:24
• @ASCII-only thanks, updated. – Justin Mariner Jun 2 '17 at 18:02
# q/kdb+, 392378300 276 bytes
Solution:
t:{
w:y,($:)x; r:@[{"J"$-3#12#($":http://",x)y}[w;];"GET / HTTP/1.1\r\nHost:",w,4#"\r\n";404]; (x,wr)!(r;$y;$[(r>299)&r<400;[r:200;x];]) } f:{ R:(w,x,r)xcols(,/)each(x)t\:/:{x,/:3 cut 24?.Q.a}y; (Website,x,$"Redirects?")xcol update r:No from R where r=
}[.com.org.net;]
Example:
q)f"a"
Website .com .org .net Redirects?
---------------------------------
amik 200 200 200 .net
abjl 200 200 503 No
afuw 403 200 200 No
agby 200 200 200 No
afen 200 200 200 .net
ajch 200 404 200 No
aaro 200 200 200 No
ajsr 200 404 503 No
Notes:
Current solution is fairly hefty... Half the code is trying to fetch a URL, the other half is presenting the results nicely, unsure how much further I can golf this.
Explanation:
Function t
w:y,($:)x // convert suffix (.com) to string, prepend host and save in w @[x;y;404] // try function x with parameter y, on error return 404 "GET /..." // HTTP GET request in shortest form :http://x y // open web connection to x, send request y -3#12# // truncate output to 12 chars, take last 3 chars "J"$ // cast result to long
\$[x;y;z] // if x then y, else z
(x)!(y) // create a dictionary with x as keys, y as values
Function f
24?.Q.a // take 24 random characters from a-z
x,/:3 cut // cut into 3-character lists, prepend input 'x'
t\:/: // call function t with each combination of left and right
(,/) each // raze (reduce) each result
xcols // re-order columns into output format
xcol // rename columns | 2020-01-24 00:34:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.19601906836032867, "perplexity": 4203.433415281382}, "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-2020-05/segments/1579250614086.44/warc/CC-MAIN-20200123221108-20200124010108-00018.warc.gz"} |
https://tex.stackexchange.com/questions/54683/how-to-produce-structure-as-in-attached-image | # How to produce structure as in attached image
I remember seeing similar question (image wise) here on this forum but did not bookmark it :(
If anyone knows and can point me to the example or give me some ideas to get started would be great help. I am working on reproducing an old book which has around 4000 linesas shown in the attached image.
Text on each line is called "Sutra". Number on the left is a unique serial number. Some sutras have a number on the right, which means part or whole text that sutra is to be added to the sutra number shown on the right.
The part which gets transferred to below sutra should be shown as bold or colored. Sometimes its part and sometimes its the whole sutra that gets transferred as can be see in "6"
I also would like to show the text thats getting transferred to the sutra below on across the left line in scriptsize text.
In the shown example the span is limited to 3 sutras however in the later part it can span across ~100 sutras - in such cases the left line should spread across pages. the flow of words is always downwards, there will not be any case where the text flows upwards.
PS: numbers in front of Sutra 11 - 12, 19, please do change the question title that make more sense than what I have given.
Perhpas the tkz-linknodes package can be useful for your purpose; here's a little schematic example:
\documentclass{article}
\usepackage{graphicx}
\tikzset{ArrowStyle/.style={text=black,shorten >= 15pt,shorten <= 15pt}}
\tikzset{LabelStyle/.style={pos=0.25,right,font=\scriptsize}}
\tikzset{NodeStyle/.style={inner sep=0pt}}
\begin{document}
\begin{enumerate}
\begin{NodesList}[margin=15cm]
\item Fourth item.
\item Fifth item.
\end{NodesList}
\end{enumerate}
\end{document}
\documentclass{article}
\usepackage[explicit]{titlesec}
\usepackage{hyperref}
\usepackage{lipsum}
\titleformat{\section}
{}{}{1em}{}
\begin{document}
\tableofcontents
\section{Test Section One}
\lipsum[1-10]
\section{Test Section Two}
\lipsum[1-10]
\end{document} | 2019-11-17 02:53:51 | {"extraction_info": {"found_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": 0.5556372404098511, "perplexity": 1275.6014000992109}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20191117014405-20191117042405-00484.warc.gz"} |
https://solvedlib.com/n/4-imagine-that-a-sperm-with-six-chromosomes-fertilizes-an,17351361 | # 4. Imagine that a sperm with six chromosomes fertilizes an ovumwith seven chromosomes. How might this affect the zygote anddevelopment
###### Question:
4. Imagine that a sperm with six chromosomes fertilizes an ovum with seven chromosomes. How might this affect the zygote and development of the embryo?
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Zbl 0877.35042
Marcus, M.; Véron, L.
Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations.
(English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, No.2, 237-274 (1997). ISSN 0294-1449
The authors study properties of positive solutions of $$\Delta u+ hu-ku^p= f\tag1$$ in a (possibly) nonsmooth $N$-dimensional domain $\Omega$, $N\ge 2$, subject to the condition $$u(x)\to\infty\quad\text{if}\quad \delta(x):= \text{dist}(x,\partial\Omega)\to 0.\tag2$$ Here $p>1$ and $h$, $k$, $f$ are continuous in $\overline\Omega$ with $k>0$ and $f\ge 0$. Positive solutions of (1) satisfying (2) are called large solutions. A central point of this paper is the following localization principle: let $\Omega$ be a (not necessarily bounded) domain having the graph property and suppose $u$ is a positive solution of (1) satisfying $u(x)\to\infty$ locally uniformly as $x\to\Gamma$, where $\Gamma\subset\partial\Omega$ is relatively open. If $v$ is a large solution, then $v(x)/u(x)\to 1$ locally uniformly as $x\to\Gamma$. Closely related to this is a uniqueness result for large solutions in bounded domains having the graph property. For bounded Lipschitz domains the authors prove the existence of positive constants $c_1\le c_2$ such that the (unique) large solution $u$ of (1) satisfies $c_1\delta(x)^{-{2\over p-1}}\le u(x)\le c_2\delta(x)^{-{2\over p-1}}$ for all $x\in\Omega$. This is also a consequence of the localization principle and an existence theorem, obtained for large solutions in bounded domains satisfying the exterior cone condition.\par If the domain is not Lipschitz, the rate of blow-up at the boundary may be lower. This is proved for domains having a re-entrant cusp in the case $p\ge(N-1)/(N-3)$. Finally, the authors discuss the dependence of large solutions on the function $k$ and the domain $\Omega$.
[R.Beyerstedt (Aachen)]
MSC 2000:
*35J60 Nonlinear elliptic equations
35J67 Boundary values of solutions of elliptic equations
35B40 Asymptotic behavior of solutions of PDE
Keywords: uniqueness for large solutions; large solutions; localization principle; rate of blow-up at the boundary
Highlights
Master Server | 2013-05-25 16:44:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8921032547950745, "perplexity": 425.1897853015742}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368705976722/warc/CC-MAIN-20130516120616-00063-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://socratic.org/questions/what-is-the-derivative-of-x-x-2-1 | # What is the derivative of (x/(x^2+1))?
Jun 4, 2015
Using quotient rule, which states that for $y = f \frac{x}{g} \left(x\right)$,
$y ' = \frac{f ' \left(x\right) g \left(x\right) - f \left(x\right) g ' \left(x\right)}{g \left(x\right)} ^ 2$,
We can proceed to derivate this with no big problems.
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1 \cdot \left({x}^{2} + 1\right) - x \cdot 2 x}{{x}^{2} + 1} ^ 2$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{x}^{2} + 1 - 2 {x}^{2}}{{x}^{2} + 1} ^ 2$
$\textcolor{g r e e n}{\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- {x}^{2} + 1}{{x}^{2} + 1} ^ 2}$ | 2021-01-27 06:22:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "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": 0.4208396375179291, "perplexity": 2149.673172214562}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610704821253.82/warc/CC-MAIN-20210127055122-20210127085122-00627.warc.gz"} |
http://math.stackexchange.com/questions/158861/queries-regarding-newtons-method | # Queries regarding Newton's method
I am currently trying to study the Newton's method of optimization through this wiki article http://en.wikipedia.org/wiki/Newton%27s_method_in_optimization. However, I didn't get this concept about constructing the sequence xn and approximating the objective function by a quadratic function around xn.
Can anyone provide me some good references. I mean why are are constructing that sequence xn? Any geometric visualization will be helpful I guess.
- | 2014-03-12 08:58:07 | {"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": 0.9246083498001099, "perplexity": 837.8470016206751}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394021547621/warc/CC-MAIN-20140305121227-00003-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.asknumbers.com/InchesToFeetConversion.aspx | # Inches to Feet Conversion
Inches to feet (in to ft) conversion factor is 0.083333. To find out how many feet in inches, multiply by the conversion factor or use the converter.
1 Inch = 0.0833333333 (1/12) Foot
Inch is a comonly used unit of length in imperial and customary measurement systems. The word inch comes from latin and means one-twelfth of something, in this case 1/12 of a foot. The abbreviation is "in" and also the symbol is a double prime (").
Foot is one of the most common length units since ancient times, since they were using human body parts for length measurements. In 1959, international foot is defined to be exactly as 0.3048 metre whereas 1 survey foot is 3.280833333 metre. The abbreviation is "ft" and the symbol is (') prime.
If scroll down, you may also create and print your own custom conversion table create your table.
Converter
Enter an inch value that you want to convert into feet and click on the "convert" button.
For feet to inches converter, go to feet to inches
For all length units, go to length conversion
Create Custom Conversion Table
## Conversion Table
InchFootInchFootInchFootInchFoot
1 in 0.08 ft 26 in 2.17 ft 51 in 4.25 ft 76 in 6.33 ft
2 in 0.17 ft 27 in 2.25 ft 52 in 4.33 ft 77 in 6.42 ft
3 in 0.25 ft 28 in 2.33 ft 53 in 4.42 ft 78 in 6.50 ft
4 in 0.33 ft 29 in 2.42 ft 54 in 4.50 ft 79 in 6.58 ft
5 in 0.42 ft 30 in 2.50 ft 55 in 4.58 ft 80 in 6.67 ft
6 in 0.50 ft 31 in 2.58 ft 56 in 4.67 ft 81 in 6.75 ft
7 in 0.58 ft 32 in 2.67 ft 57 in 4.75 ft 82 in 6.83 ft
8 in 0.67 ft 33 in 2.75 ft 58 in 4.83 ft 83 in 6.92 ft
9 in 0.75 ft 34 in 2.83 ft 59 in 4.92 ft 84 in 7.00 ft
10 in 0.83 ft 35 in 2.92 ft 60 in 5.00 ft 85 in 7.08 ft
11 in 0.92 ft 36 in 3.00 ft 61 in 5.08 ft 86 in 7.17 ft
12 in 1.00 ft 37 in 3.08 ft 62 in 5.17 ft 87 in 7.25 ft
13 in 1.08 ft 38 in 3.17 ft 63 in 5.25 ft 88 in 7.33 ft
14 in 1.17 ft 39 in 3.25 ft 64 in 5.33 ft 89 in 7.42 ft
15 in 1.25 ft 40 in 3.33 ft 65 in 5.42 ft 90 in 7.50 ft
16 in 1.33 ft 41 in 3.42 ft 66 in 5.50 ft 100 in 8.33 ft
17 in 1.42 ft 42 in 3.50 ft 67 in 5.58 ft 125 in 10.42 ft
18 in 1.50 ft 43 in 3.58 ft 68 in 5.67 ft 150 in 12.50 ft
19 in 1.58 ft 44 in 3.67 ft 69 in 5.75 ft 175 in 14.58 ft
20 in 1.67 ft 45 in 3.75 ft 70 in 5.83 ft 200 in 16.67 ft
21 in 1.75 ft 46 in 3.83 ft 71 in 5.92 ft 250 in 20.83 ft
22 in 1.83 ft 47 in 3.92 ft 72 in 6.00 ft 300 in 25.00 ft
23 in 1.92 ft 48 in 4.00 ft 73 in 6.08 ft 500 in 41.67 ft
24 in 2.00 ft 49 in 4.08 ft 74 in 6.17 ft 750 in 62.50 ft
25 in 2.08 ft 50 in 4.17 ft 75 in 6.25 ft 1000 in 83.33 ft | 2017-10-19 23:39:56 | {"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": 0.8476780652999878, "perplexity": 3850.031771745026}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187823482.25/warc/CC-MAIN-20171019231858-20171020011858-00509.warc.gz"} |
https://byjus.com/question-answer/if-t-is-the-surface-tension-of-a-fluid-then-the-energy-needed-to-break/ | Question
# If $$T$$ is the surface tension of a fluid, then the energy needed to break a liquid drop of radius $$R$$ into $$64$$ equal drops is :
A
6πR2T
B
πR2T
C
12πR2T
D
8πR2T
Solution
## The correct option is B $$12 \pi R^2 T$$Let the radius of the small drop is $$r$$. Since, the volume of the bigger drop will be equal to the $$64$$ smaller drops. therefore, $$\frac{4}{3}\pi {R^3} = 64 \times \frac{4}{3}\pi {r^3}$$ $$R = 4r$$ The surface energy of the larger drop is given as, $${E_1} = \left( {4\pi {R^2}} \right)T$$ The surface energy of the small drops is given as, $${E_2} = 64 \times \left( {4\pi {r^2}} \right)T$$ $$= 64 \times \left( {4\pi {{\left( {\frac{R}{4}} \right)}^2}} \right)T$$ $$= \left( {16\pi {R^2}} \right)T$$ The required energy is given as, $$\Delta E = {E_2} - {E_1}$$ $$= \left( {16\pi {R^2}} \right)T - \left( {4\pi {R^2}} \right)T$$ $$= \left( {12\pi {R^2}} \right)T$$ Thus, the required energy to create the $$64$$ small drops is $$\left( {12\pi {R^2}} \right)T$$.Physics
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How do you solve this system of equations: 6x + 5y = 300 and 3x + 7y = 285#?... | 2022-07-06 07:18:27 | {"extraction_info": {"found_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": 0.2554364800453186, "perplexity": 1852.1985508443308}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656104668059.88/warc/CC-MAIN-20220706060502-20220706090502-00474.warc.gz"} |
https://www.effortlessmath.com/math-topics/top-10-grade-4-georgia-milestones-assessment-system-math-practice-questions/ | # Top 10 4th Grade Georgia Milestones Assessment System Math Practice Questions
The best way to prepare for your 4th Grade Georgia Milestones Assessment System Math test is to work through as many 4th Grade Georgia Milestones Assessment System Math practice questions as possible. Here are the top 10 4th Grade Georgia Milestones Assessment System Math practice questions to help you review the most important 4th Grade Georgia Milestones Assessment System Math concepts. These 4th Grade Georgia Milestones Assessment System Math practice questions are designed to cover mathematics concepts and topics that are found on the actual test. The questions have been fully updated to reflect the latest 2022 4th Grade Georgia Milestones Assessment System guidelines. Answers and full explanations are provided at the end of the post.
Start your 4th Grade Georgia Milestones Assessment System Math test prep journey right now with these sample 4th Grade Georgia Milestones Assessment System Math questions.
## 4th Grade Georgia Milestones Assessment System Math Practice Questions
1- Joe has 855 crayons. What is this number rounded to the nearest ten? _________
2- Peter’s pencil is $$\frac{12}{100}$$ of a meter long. What is the length, in meters, of Peter’s pencil written as a decimal?
A. 0.12
B. 1.02
C. 1.2
D. 12.100
3- There are 7 days in a week. There are 28 days in the month of February. How many times as many days are there in February than are in one week?
A. 4
B. 7
C. 21
D. 35
4- A football team is buying new uniforms. Each uniform costs $24. The team wants to buy 14 uniforms. Which equation represents a way to find the total cost of the uniforms? A. $$(20 × 10) + (4 × 4) = 200 + 16$$ B. $$(20 × 4) + (10 × 4) = 80 + 40$$ C. $$(24 × 10) + (24 × 4) = 240 + 96$$ D. $$(24 × 4) + (4 × 14) = 96 + 56$$ 5- A number sentence such as $$31 + Z = 98$$ can be called an equation. If this equation is true, then which of the following equations is not true? A. $$98 – 31 = Z$$ B. $$98 – Z = 31$$ C. $$Z – 31 = 98$$ D. $$Z + 31 = 98$$ 6- Circle a reasonable measurement for the angle: A. $$35^\circ$$ B. $$90^\circ$$ C. $$180^\circ$$ D. $$240^\circ$$ 7- Ella described a number using these clues: Three-digit odd numbers that have a 6 in the hundreds place and a 3 in the tens place Which number could fit Ella’s description? A. 627 B. 637 C. 632 D. 636 8- Tam has 390 cards. He wants to put them in boxes of 30 cards. How many boxes does he need? A. 7 B. 9 C. 11 D. 13 9- If this clock shows a time in the morning, what time was it 6 hours and 30 minutes ago? A. 07:45 AM B. 05:45 AM C. 07:45 PM D. 05:45 PM 10- Use the table below to answer the question. The students in the fourth-grade class voted for their favorite sport. Which bar graph shows the results of the student’s vote? A. B. C. D. ## Best 4th Grade Georgia Milestones Assessment System Math Exercise Resource for 2022 ## Answers: 1- 860 We round the number up to the nearest ten if the last digit in the number is 5, 6, 7, 8, or 9. We round the number down to the nearest ten if the last digit in the number is 1, 2, 3, or 4. If the last digit is 0, then we do not have to do any rounding, because it is already to the ten. Therefore, a rounded number of 855 to the nearest ten is 860. 2- A $$\frac{12}{100}$$ is equal to 0.12 3- A 7 days = 1 week 28 days $$= (28 ÷ 7)= 4$$ weeks 4- C Football team should buy 14 uniforms that each uniform cost$24 so they should pay (14 $$×$$ $24)$336.
Therefore, choice C is correct answer:
$$(24 × 10) + (24 × 4) =24(10+14) =24×14 = 336$$
5- C
These: $$98 – 31 = Z$$
$$98 – Z = 31$$
And $$Z + 31 = 98$$ are equal.
6- A
This angle is less than $$90^\circ$$. just choice A shows an angle less than $$90^\circ$$.
7- B
Three-digit odd numbers that have a 6 in the hundreds place and a 3 in the tens place are 631, 633, 635, 637, 639. 637 is one of the alternatives.
8- D
Tam wants to divide his 390 cards into boxes of 30 cards. So he needs $$390÷30=13$$ boxes.
9- C
Subtract hours: $$2 – 6 = -4$$
Subtract the minutes: $$15 – 30 = – 15$$
The minutes are less than 0, so:
• Add 60 to minutes ( $$-15 +60 =45$$ minutes)
• Subtract 1 from hours $$(-4 – 1 = -5)$$ the hours are less than 0, add 24: $$(24 – 5 =19)$$
The answer is 19:45 that is equal to 7:45
10- A
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SAVE $5 It was$18.99 now it is \$13.99 | 2022-08-18 07:20:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28640133142471313, "perplexity": 2884.4712441138204}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573172.64/warc/CC-MAIN-20220818063910-20220818093910-00262.warc.gz"} |
https://s99917.gridserver.com/9rphq5/actinide-series-properties-ef736c | These elements, usually considered ranging from atomic number 89 to atomic number 103 on the periodic table, have interesting properties, and play a key role in nuclear chemistry. The informal chemical symbol An is used in general discussions of actinide chemistry to refer to any actinide. The Pa metal is malleable, ductile, silvery, and has a melting point of about 1565. The nuclear energy creates a longer lifetime use of the device. It is soft enough to be scratched with a knife and melts at 1750. has purple and blue-white fluorides, has green chlorides, has green and red-orange and dark red bromides and has brown iodides. The 14 electrons are being added into 5f except in thorium (Z = 90) but filling a 5f continues again after thorium till 5f orbitals are complete at Z = 103 (Lawrencium). Lanthanide and Actinide Series are both referred to as Rare Earth Metals. The oxyhalides of Actinides are not binary but some are formed by earlier Actinides, for example, an aqueous Protactinium fluoride reacts with air to make PaO2F, Thorium can form ThOX2, Neptunium can make NpOF3, Plutonium forms PuOF4 and UO2F2 is made by Uranium (III) oxide and Hydrofluoric acid. Properties of Actinides As mentioned earlier, elements in the actinide family are heavy due to their large atomic mass. Actinium is used in scientific and medical research as a neutron source, indicator, and gamma source. Actinide Series + properties give you a broad overview of these metals from multiple angels. The actinides are the elements in the bottom of these two rows, while the top row is the lanthanide series. Third, septins are more stable than actin and MTs and, thereby, septins may provide a spatial memory for the re-growth of dynamic actin filaments and MTs at specific intracellular regions. The most common and known element is Uranium, which is used as nuclear fuel when its converted into plutonium, through a nuclear reaction. It has wide coordination chemistry with oxygen donor ligands. Thorium can be precipitated from ammonia with aluminum or from aluminum as a fluoride or oxalate. Two ways of placing the lanthanides and actinides within a periodic table are including them in their corresponding rows with the transition metals, which makes the table wider, or ballooning them out, making a three-dimensional table. Thorium metal is bright and silvery-white and tarnishes to a dull black color when exposed to air. The reason for this arrangement unlike other conventional electron configurations such as Na with configuration of [Ne]3s2. The energies of the 5f orbital drop rapidly with increasing atomic number. Actinides share the following properties: For the most part, we don't often encounter these radioactive elements in daily life. It is dense, silvery and a reactive metal; more reactive than uranium or neptunium. Was a power source for orbiting satellites and missions to take pictures of Jupiter. All the actinoid elements are heavy metals and, as such, are toxic, just as lead is toxic; relatively large amounts ingested over a long period cause serious illness.But, with the exception of the long-lived thorium and uranium isotopes, the real danger with the actinoid elements lies in the radioactive properties of these elements. Which Actinides were discovered in nature? Although they may have radioactivity that causes problems, they are useful to study to understand the trends in the Actinide series. All the actinide series elements have common 7s 2 configuration and variable occupancy of 5f and 6d subshells. She has taught science courses at the high school, college, and graduate levels. These extra neutrons help create a chain reaction as more neutrons come into contact with Uranium. These examples illustrate their importance in understanding key concepts in nuclear chemistry and related topics. Lanthanide and Actinide Chemistry. There are 15 actinide elements. These elements have no stable isotopes. Where Is Uranium Found on the Periodic Table? Microscopic amounts of Plutonium are made by neutron capture by Uranium, and yet occur naturally. Physiological properties of the actinoids. General Properties of Actinide Series (5f block elements) In 1923 Neils Bohr postulated the existence of an actinide series analogous to the lanthanide series. Some can be cut with a knife. The solubility of Actinide hydroxides or hydrous oxides in strong ammonium carbonate solutions allow the separation or Uranium and Thorium from other members of the Ammonium hydroxide group, such as Fe, Ti, Al and other rare earth metals. The actinides (An) may be prepared by reduction of AnF3 or AnF4 with vapors of Li, Mg, Ca, or Ba at 1100-1400 C. beta particle-ejected electron from nucleus of U, beta particle-ejected electron from nucleus of Np, beta particle-ejected electron from nucleus of Pa, beta particle-ejected electron from nucleus of Th, information contact us at [email protected], status page at https://status.libretexts.org. They react with boiling water or dilute acid to release hydrogen gas. The halides are very important binary compounds, sometimes the most important. Why Lanthanides and Actinides Are Separate on the Periodic Table, Transition Metals and the Properties of the Element Group, Periodic Table Study Guide - Introduction & History, Possible Production of Elements of Atomic Number Higher than 92, The Elements: A Visual Exploration of Every Known Atom in the Universe, Ph.D., Biomedical Sciences, University of Tennessee at Knoxville, B.A., Physics and Mathematics, Hastings College. Uranium hydride reacts with hydrogen gas at 250 degrees and swells up into a fine black powder, which is pyrophoric in air. The primary use of the actinide elements is as nuclear reactor fuel and in the production of nuclear weapons. Americium is found in smoke detectors. Uranium (V): rare oxidation states, unstable, six and seven coordination. The 5f orbitals are not shielded by the filled 6s and 6p subshells. All members of the series have similar chemical properties. The metal is a slivery, ductile and very malleable. Chichester, West Sussex, England; Hoboken, NJ: Wiley, 2006. forms in the 2, 3, and 4 oxidation states. Which Actinides were the first to be discovered? The nucleus of Uranium emits radioactivity in the form of alpha particles. The actinide series includes elements with the atomic numbers 89 to 103. The energy in the 6d orbitals is lower in energy than in the 5f orbitals. That is , Cf ( OH ) 3 is more covalent than Ac ( OH ) 3. Properties and Reactions of the Actinide Series of Elements At the bottom of the periodic table is a special group of metallic radioactive elements called actinides or actinoids. When attacked by air, it forms a green-gray oxide coating. The actinides (sometimes called actinoids) occupy the "bottom line" of the periodic table — a row of elements normally separated from the others, placed at the foot of the chart along with the lanthanides. If the conditions are right, the nuclear reactions can become chain reactions. General Properties and Reactions of The Actinides, [ "article:topic", "fundamental", "Solubility", "Precipitation", "Rare Earth Metals", "Actinide", "transition metal", "Actinide series", "showtoc:no", "Gamma Rays", "halides", "Uranium", "Nuclear Fission", "Alpha Particles", "Beta Particles", "Klaproth", "Berezelius" ]. The earliest actinides have a closer relation to the transition metals, where the oxidation state is equal to the number of electrons on the outer shell. has an eight coordination number (dodecahedron, sq antiprism). Actinide elements are all radioactive. The extra proton causes the atomic number to increase by one unit, but the mass number is unchanged. © 2023 by the Smith Family. These elements all have a high diversity in oxidation numbers and all are radioactive. New York: The Macmillian Company, 1963. are the most important, but compounds of the ions are well defined. The Actinide series contains elements with atomic numbers 89 to 103 and is the sixth group in the periodic table. *This is a picture of U3O8, a uranium, pitchblende ore, by Geomartin. Beta particles are deflected by the electric and magnetic fields in the opposite direction from alpha particles. Since the f-shell is nominally full in the ground state electron configuration for both these metals, they behave most like d-block metals out of all the lanthanides and actinides, and thus exhibit the most similarities in properties with Sc and Y. Although the chemical properties of thorium and uranium had been studied for over a century, and those of actinium and protactinium for over ?fty years, all of the chemical properties of neptunium and heavier elements as well as a great deal of uranium chemistry had been discovered since 1940. The Actinide series contains elements with atomic numbers 89 to 103 and is the third group in the periodic table. The actinides display less similarity in their chemical properties than the lanthanide series, exhibiting a wider range of oxidation states, which initially led to confusion as to whether actinium, thorium, and uranium should be considered d-block elements. Some of the properties of Actinides are: Binding energies of 5 f electrons are lower The usual list of elements in the actinide series is: For more information contact us at [email protected] or check out our status page at https://status.libretexts.org. The extra proton causes the atomic number to increase by one unit, but the mass number is unchanged. Chem. These elements are the backbone of nuclear fission technologies for electricity supply, with important applications in other strategic fields, from water management to space exploration and human health. Another property of actinides is their radioactivity. Because they are not as big or massive as alpha particles, they are deflected more strongly than alpha particles are. Actinides have been crucial in understanding nuclear chemistry and have provided valuable usage today, such as nuclear power. forms halides in the oxidation state of +3 only. $\ce{ ^{238}U \rightarrow ^{234}Th + ^4_2He}$. The +3 and +4 O.S. An example of pitchblende is located in the picture below. The sum of mass numbers must be the same on both sides. The identity and function of the causal genes in these susceptibility loci remain, however, elusive. The main Uranium ore is U3O8 and is known as pitchblende, because it occurs in black, pitch-like masses. What makes the Actinides you answered in the first questions different than the rest? It is 150 times more radioactive than radium. )-Neptunium is unknown in this oxide state, but scientists assume it does exist. Missed the LibreFest? $^{239}Np \rightarrow ^{239}Pu+ \beta^-$, $^{237}U \rightarrow ^{237}Np+ \beta^-$, $^{234}Pa \rightarrow ^{234}U + \beta^-$, $^{234}Th \rightarrow ^{234}Pa + \beta^-$. Relativistic effects influence the shielding characteristics of inner electrons. As a group, they are significant largely because of their radioactivity.Although several members of the group, including uranium (the most familiar), occur naturally, most are man-made. forms halides in oxidation states from +3 to +6. A majority of actinides form halides with halogens at specific temperatures and trihalides are the most well known halides. All are unstable and reactive due to atomic number above 83 (nuclear stability). The usual list of elements in the actinide series is: *actin* แปลว่าอะไร ความหมาย คำแปล หมายความว่า ตัวอย่างประโยค It does not have absorption in the UV visible region between 400-1000nm. Depending on your interpretation of the periodicity of the elements, the series begins with actinium or thorium, continuing to lawrencium. Gamma Rays are a form of electromagnetic energy such as visible light and hence are undeflected by electric and magnetic fields. Watch the recordings here on Youtube! Lanthanide and Actinide Series are both referred to as Rare Earth Metals. Im Sinne des Begriffs gehört Actinium nicht zu den Actiniumähnlichen, jedoch folgt di… The dihalides become insoluble for the first time. The atomic mass of hydrogen is 1 therefore; one can have a clear idea of comparatively how heavier these elements are. Lanthanides share the following common properties: Silvery-white metals that tarnish when exposed to air, forming their oxides. Zugerechnet werden ihr das Actinium und die 14 im Periodensystem folgenden Elemente: Thorium, Protactinium, Uran und die Transurane Neptunium, Plutonium, Americium, Curium, Berkelium, Californium, Einsteinium, Fermium, Mendelevium, Nobelium und Lawrencium. The series is the row below the Lanthanide series, which is located underneath the main body of the periodic table. The Actinide series contains elements with atomic numbers 89 to 103 and is the third group in the periodic table. The lanthanide and actinide series derive properties from the f-block electrons. General Properties of Actinide Series . The 14 electrons are being added into 5f except in thorium (Z = 90) but filling a 5f continues again after thorium till 5f orbitals are complete at Z = 103 (Lawrencium). Actinides require special handling, because many of them are radioactive and/or unstable. Depending on your interpretation of the periodicity of the elements, the series begins with actinium or thorium, continuing to lawrencium. of +3 to +7 in compounds. Depending on your interpretation of the periodicity of the elements, the series begins with actinium or thorium, continuing to lawrencium. Those specific isotopes have various but extremely high half-lives, meaning they decay slowly, which makes it easy for scientists to study and experiment with each of them over long periods of time. It generated under 1.5 kW of heat converted to electricity by thermoelectric elements. is restricted to iodides like ThI. This uncontrolled energy eventually leads to an explosion, which is the basis of the atomic bomb. forms many complexes with neutral donors. Actinium and Protactinium are found in small portions in nature, as decay products of 253-Uranium and 238-Uranium. Its only oxidation state is +3 and it only forms with chloride, fluorine, and even oxygen. They fill 5f orbital, 6d orbital, then 7s orbital. has six coordinate (dimeric), eight coordinate (dodecahedral and square antiprismatic), seven coordinate, and nine coodrdinate. Moses, Alfred J.. Analytical Chemistry of the Elements. 235-Uranium is bombarded with neutrons and turns into 236-Uranium. In the actinide seriesit has been used extensively for the determination of uranyl (UO22+) species and for the Cm3+complex speciation. Small quantities of plutonium and neptunium are present in uranium orders. Most are prone to hybridization. These elements are pyrophoric (spontaneously ignite in the air), particularly as finely divided powders. The elements belonging to this series has an atomic mass ranging from 227g/mol to 262g/mol. The +4 state is more stable in the Actinide series than in the Lanthanides. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The evidence that shows Actinides are very reactive are they have low solubilities as halide compounds; and the lighter, trivalent Actinide compounds are unstable in aqueous solutions and can be easily precipitated out of acidic and basic environments. By using ThoughtCo, you accept our, Identifying Element Blocks on the Periodic Table. The uranium in uranium hydride has an oxidation state of +3, which is strongly reducing, so it forms a stable hydride. Not all Actinides have the same properties when it comes solubility and precipitation, but most of the Actinides have similar traits and characteristics. The series is the row below the Lanthanide series, which is located underneath the main body of the periodic table. Houndsmill, Basingstoke, Hampshire and London: Macmillan Education Company Ltd, 1991. Actinoide [-noˈiːdə] (Actiniumähnliche; griech. The metals tarnish readily in air. of +3 to +6. It's shown in the bottom row of the periodic table. [#]-mass number of the longest living isotope is given when the element has no stable nucleotides. Reacts with hydrogen on warming to form hydrides, Yields compounds on warming with group 5 and group 6 non-metals. Biochemical Properties of l-Afadin It was examined by competition experiments whether the Mono Q sample of l-afadin (including s-afadin) bound along the sides of F-actin or at its ends. During a beta particle equation,a neutron within the nucleus of an atom is converted to a proton or electron spontaneously. The following are examples of Californium halides: CfCl. All 36 known isotopes of actinium are radioactive. The general electronic configuration of actinides is [Rn] 5f 0,1-14 6d 0,1-2 7s 2 where Rn stands for radon core. 1. However, throughout the series, the changes of the chemical and physical properties is very less. At the bottom of the periodic table is a special group of metallic radioactive elements called actinides or actinoids. There are 15 actinide elements. Gamma rays ($$\gamma$$) are radiation that is emitted when the nucleus is in an excited state due to excess energy. The proton remains in the nucleus and the electron is produced as a beta particle. Alpha particles produce ions but have weak penetrating power that can easily be stopped by sheets of paper. The lanthanide series is a unique class of 15 elements with relatively similar chemical properties. Uranium is used in nuclear reactors in the form of fuel in rods suspended in water under a pressure of 70 to 150 atmospheres. When we think of beta decay processes, we can look at the following equations and imagine a neutron within the nucleus of an atom converting to a proton or electron spontaneously. The first Actinides to be discovered were Thorium and Uranium. Comparative data are presented on the electronic configurations, oxidation states, oxidation–reduction (redox) potentials, thermochemical data, crystal structures, and ionic radii of the actinide elements, together with 15.1 Introduction 1753 Morss, Lester R., Edelstein Norman M., and Fuger Jean. Difference Between Actinides and Lanthanides Definition. Legal. In actinide series elements , ... As a result, the covalent properties of actinide metal hydroxide compounds regularly increases [according to Fajan’s rule,]from actinium to lawrencium. The trihalides are important and have pastel colors: fluoride and chloride are pink, bromide is white-pale yellow, and iodide is pale yellow. ThoughtCo uses cookies to provide you with a great user experience. Their positive charge allows them to be deflected by electric and magnetic fields. It tarnishes in air slowly and dissolves in dilute hydrochloric acid quickly. Dr. Helmenstine holds a Ph.D. in biomedical sciences and is a science writer, educator, and consultant. Actinide Series Properties. In 1923 Neils Bohr postulated the existence of an actinide series analogous to the lanthanide series. Dordrecht: Springer, 2006. The 241-Americium isotope is used as ionizing sources for smoke detectors. The following are the different oxides of the Actinide elements: (? Actinides: Actinides are chemical elements that can be found in the actinide series of the f block in the periodic table of elements. They all form binary compounds, such as trihalides. forms many different types of halides and several oxyhalides. The Actinide are a series of fifteen metal elements with atomic number 89 through 103. The actinoid (according to IUPAC terminology) (previously actinide) series encompasses the 15 chemical elements that lie between actinium and lawrencium included on … However, these two rows of elements are metals, sometimes considered a subset of the transition metals group. The actinide or actinoid series is the lower series of the f-block, which starts from actinium (atomic number 89) to lawrencium (atomic number 103). The Actinides that were discovered in small portions in nature were Actinium and Protactinium. Chem. They have atomic numbers ranging from 57 to 71, which corresponds to the filling of the 4 f orbitals with 14 electrons. The lighter trivalent, having a valence of three electrons, are unstable in aqueous solutions, such as transplutonium elements. All these elements are silver-colored metals that are solid at room temperature and pressure. Beta particles are electrons that came from the nuclei of atoms in nuclear decay processes. Pu+4, Th+4, Np+4, UO+2 and all trivalent actinides are precipitated by carbonate-free ammonium hydroxide. All are radioactive. In nuclear chemistry, the actinide concept proposed that the actinides form a second inner transition series homologous to the lanthanides.Its origins stem from observation of lanthanide-like properties in transuranic elements in contrast to the distinct complex chemistry of previously known actinides. Reacts readily with hot water to prevent substances from coming into contact in nuclear reactors. Magnetic Properties of Actinides: All actinides are paramagnetic in nature, which depends on the presence of unpaired electrons. Cotton, Simon. When the last member of the actinide series, element 103 or Lawrencium, was discovered, I was at school doing my A levels. The interaction of Actinides when radioactive with different types of phosphors will produce pulses of light. Lanthanide and Actinide Series are both referred to as Rare Earth Metals. Proudly created with Wix.com Wix.com Now , with increasing covalent character, their basic property decreases . Actinides are very dense metals with distinctive structures. 25 Nuclear Chemistry. Have questions or comments? Moreover, enthusiasts are also catered with the detailed breakdown of the atomic, optical and chemical behaviour of the metals. This was not unexpected as half lives had been getting shorter right along the actinide series. The isotope found had a mass of 258 and it didn't hang about for long - it had a half-life of just 3.8 seconds. Actinides display several valence states, typically more than the lanthanides. Ch. It has surface oxidation when exposed to air. The most common oxides are of the form M2O3, where M would be one of the elements in the Actinide series. Many times a beta particle is small enough that its charge could be ignored during calculations. The following are some examples of gamma ray reactions: $\ce{^{230}Th + ^1_0n \rightarrow ^{231}Th +} \gamma$, $\ce{^{238}U + ^1_0n \rightarrow ^{239}U +} \gamma$, $\ce{^{230}Th \rightarrow ^{230}Th +} \gamma$. The only oxidation state s are +3 and +2 and it only forms chloride and iodide halides. of Actinium different than the other Actinides. It has 6 allotropic metal forms, which makes it unusual. New Jersey 2007. Actinides are in the f-block of the periodic table. If they occur naturally, it is part of a decay scheme of a heavier element. The bulk of actinide use goes to energy production and defense operations. Members of the actinide series can lose multiple electrons to form a variety of different ions. The informal chemical symbol An is used in general discussions of actinoid chemistry to refer to any actinoid. They can form at normal pressure between room temperature and its melting point, 640. Trace Determination and Chemical Properties of Actinides by TRLIF Time-resolved laser-induced fluorescence (TRLIF) is a very sensitive and selective technique. nide and actinide transition series provides valuable insights into the properties of both. All actinides are radioactive, paramagnetic, and, with the exception of actinium, have several crystalline phases. The 236-Uranium then converts into smaller pieces, 2-3 neutrons are released along with energy. It is a silvery metal, with a melting point of 637. Trivalent Actinides can be separated from slightly acidic solutions as phosphates, mildly acidic solutions as oxalates, strongly acidic solutions as fluorides, and from basic solutions as hydroxides or hydrous oxides. ACTINIDES CONCEPT. This property appears to be driven by specific septin paralogs and isoforms with actin- and MT-binding domains, and may be further influenced by the subunit identity of septin complexes. The proton remains in the nucleus and the electron is produced as a beta particle. but only fluorides are in above +4. The actinides of light weight have a higher valence; for example, oxidation states of 4, 5 and 6 where they are more stable; and they form compounds with low solubility. The actinide series is included in some definitions of the rare earth elements. It forms insoluble fluoride and oxalate (Ac, The chemistry in the +2 and +3 O.S. It was the first transuranium element to be discovered in 1940. The actinoid /ˈæktɪnɔɪd/ series encompasses the 15 metallic chemical elements with atomic numbers from 89 to 103, actinium through lawrencium. Numerous allotropes can be formed—plutonium has at least six allotropes. It can also react with many other compounds at certain temperatures to make halides, oxides, and other compounds, including the following: 235U + 1n → 236U → fission fragments + neutrons + 3.20 x10-11 J. Are actinide series properties to study to understand the trends in the periodic table of elements below lanthanide! Have the same properties when it comes solubility and precipitation, but.. Ammonium hydroxide smoke detectors an atomic mass of 4 and an atomic number the. First questions different than the rest an atomic number of the actinides occur in nature as sea water minerals! Their oxides 6d1 7s2 chemical Physics 2020, 22 ( 4 ), eight coordinate dimeric...: Es2O these different from the f-block of the actinide series of fifteen metal elements with atomic number through. Rules for writing a nuclear equation can apply to other radioactive decay processes orbitals are not by... Loci linked to impaired β-cell function band structure and magnetic properties of actinides are examined, and Jean... Or massive as alpha particles produce large numbers of ions because of the atomic, optical chemical... Science courses at the high school, college, and the electron configuration of [ Ne ] 3s2 dilute! Tarnish when exposed to air, forming their oxides discussions of actinide use goes energy. Of their collisions and near collisions with atoms, while the top row is the row below the main ore! T2D locus is associated with decreased insulin secretion in response to oral glucose in... Actinide seriesit has been in existence longer than any other actinide a slivery, ductile and very malleable 5f so! 6D orbitals is lower in energy than in the opposite direction from alpha particles are deflected more strongly alpha! These two rows, while the top row is the first questions different than rest! The 15 metallic chemical elements that can easily be stopped by sheets of paper influence the shielding 5f... Sometimes referred to as rare Earth metals radioactive decay processes, such as nuclear fuels and in weaponry with and. F electrons are lower actinides CONCEPT group 6 non-metals common 7s 2 Rn... Also catered with the exception of lawrencium, a d-block element included in some definitions the! Studies have identified several type 2 diabetes ( T2D ) risk loci linked to impaired function... Check out our status page at https: //status.libretexts.org has at least allotropes! Of their collisions and near collisions with atoms, while the top row is the lanthanide series is in. Series includes elements with the detailed breakdown of the elements, ’ leading to the atomic mass of 4 an! And crystals luminescent decomposes at higher temperatures to hydrogen and uranium powders glucose stimulation in humans often these! Of lanthanides in it 2 configuration and variable occupancy of 5f and 6d.. For more information contact us at info @ libretexts.org or check out status... And 4 oxidation states from +3 to +6 eventually leads to an explosion, is... Only oxidation state of +3, which is the sixth group in +2... As nuclear fuels and actinide series properties weaponry pieces, 2-3 neutrons are released along energy! And square antiprismatic ), particularly as finely divided powders substances from coming contact! Glass and crystals luminescent ignite in the bottom row of the periodic table and even oxygen ( 4,. | 2022-01-19 01:21:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6343437433242798, "perplexity": 2723.8785555575123}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320301217.83/warc/CC-MAIN-20220119003144-20220119033144-00591.warc.gz"} |
https://cs.stackexchange.com/questions/33062/logic-and-functional-programming/39548 | # Logic and Functional programming [closed]
I have a subject Introduction to logic and functional programming but the course is not provided in detail. This is the provided course.
Introduction to declarative programming paradigms. The functional style of programming, paradigms of development of functional programs, use of higher order functional and pattern matching. Introduction to lambda calculus. Interpreters for functional languages and abstract machines for lazy and eager lambda calculi, Types, type- checking and their relationship to logic. Logic as a system for declarative programming. The use of pattern-matching and programming of higher order functions within a logic programming framework. Introduction to symbolic processing. The use of resolution and theorem-proving techniques in logic programming. The relationship between logic programming and functional programming.
I want to ask that, I have the SICP book. Will it be enough for the preparation of course?
For example: I don't know what topic can be under The functional style of programming, paradigms of development of functional programs.
Please guide me if anyone can in a proper direction. I cannot ask teachers as there are none for this subject.
## closed as unclear what you're asking by D.W.♦, Luke Mathieson, David Richerby, András Salamon, JuhoFeb 19 '15 at 7:34
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
• Unfortunately, we can not know. Only the people who have designed the course will. I don't know how a course without any teachers and/or literature recommendations can exist, but there you are. Maybe we can help you Computer Science Chat; otherwise you might try asking former students. – Raphael Nov 13 '14 at 16:03
• @Raphael These are recommended literature 1. S. Arun-Kumar: Introduction to Logic for Computer Science 2. Structure and Interpretation of Computer Programs, H Abelson and G Sussmann, MIT Press. 3. John Kelly: The Essence of Logic, Prentice-Hall of India. 4. The Functional Approach to Programming, G. Cousineau and M. Mauny, Cambridge University Press – Totoro Nov 14 '14 at 9:11
• Well then, best start with these! – Raphael Nov 14 '14 at 9:15
• Yes, this is an excellent advice. But I disagree that translating imperative programs into functional programs is a good way of learning functional programming. Instead, think of problems you'd like to solve and write functional programs that solve them. May I suggest "calculate the list of the first $n$ primes". – Andrej Bauer Feb 18 '15 at 20:20 | 2019-11-12 11:50:19 | {"extraction_info": {"found_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": 0.43581366539001465, "perplexity": 1287.2887567418697}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1573496665521.72/warc/CC-MAIN-20191112101343-20191112125343-00060.warc.gz"} |
http://mathhelpforum.com/discrete-math/205520-combinatorics-tim-tams.html | # Thread: Combinatorics - Tim Tams!
1. ## Combinatorics - Tim Tams!
I have been attempting to figure out the below question for over two hours. Could someone please help me towards a solution. Combination formula is not working for me.
"In how many ways can a packet of 24 TimTams be distributed amongst 6 chocoholics, so that nobody gets
more than 8 Tim Tams."
2. ## Re: Combinatorics - Tim Tams!
Originally Posted by floorplay
Combination formula is not working for me.
"In how many ways can a packet of 24 TimTams be distributed amongst 6 chocoholics, so that nobody gets more than 8 Tim Tams."
You wrote "Combination formula is not working for me." I have no doubt that that is absolutely the case. This would be a nightmare of a problem if one tries to use inclusion/exclusion or some such.
3. ## Re: Combinatorics - Tim Tams!
lol the 'combination formula'. We started binomial theorem only the other day so I probably would have never got that answer. Thanks for the explanation and link.
4. ## Re: Combinatorics - Tim Tams!
I think we can do this with generating functions. Take the coeff of x^24 in f(x)=(1+x+x^2+x^3+...+x^8)^6 | 2017-08-21 22:29:31 | {"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": 0.8170243501663208, "perplexity": 1672.2273567777595}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109670.98/warc/CC-MAIN-20170821211752-20170821231752-00296.warc.gz"} |
https://zbmath.org/?q=an%3A0938.65002 | The operation of a sequence transformation may often be referred to numbers placed in columns, each of which is associated with its index $$\tau(r)$$, the $$\tau(r)$$ $$(r \geq 0)$$ forming a strictly increasing sequence of nonnegative integers. The first column with index $$\tau(0) = 0$$ contains the successive members $$s(m)$$ $$(m \geq 0)$$ of the sequence $$\mathcal S$$ under transformation. The column having index $$\tau(r)$$ contains transformed values $$t(r,m|s(m),s(m+1),\ldots,s(m+\tau(r))$$ for $$m \geq 0$$ obtained, in particular, from $$\tau(r) + 1$$ successive members of $$\mathcal S$$. The further columns contain auxiliary numbers used in the construction of the transformed numbers. When $$\tau(r) = r$$ $$(r \geq 0)$$ there are no further columns; when $$\tau(r) = 2r$$ $$(r \geq 0)$$ the further columns are interlaced with their counterparts. It often occurs that when $$\mathcal S$$ has a special structure, the columns of numbers exhibit a corresponding pattern; for example, for some fixed $$n \geq 0$$ all entries in the column index $$\tau(n)$$ are the same, as is so, with $$\tau(n) = 2n$$, for the $$\varepsilon$$- and $$\rho$$-algorithms; certain linear processes and the $$q$$-$$d$$ algorithm produce other patterns. This property serves as the basis of a sequence extrapolation method: in the case of the $$\varepsilon$$-algorithm, the number $$t(r,0|s(0),\ldots,s(2r))$$ is determined from $$s(0),\ldots,s(2r)$$, the column with index $$2r$$ is filled with copies of it, the transformation is put into reverse to construct $$s'(m)$$ $$(m > 2r)$$. The extrapolated sequence $$\mathcal S'$$ of terms $$s'(m)$$ $$(m \geq 0)$$ is that sequence of special structure whose first $$2r + 1$$ members $$s'(m)$$ $$(0 \leq m \leq 2r)$$ agree with those of $$\mathcal S$$.
The progressive use of the $$\varepsilon$$-algorithm in this way is considered. It is remarked that if the algorithm is applied to the sequence $$\mathcal S'$$ to produce numbers in the column $$2r$$, they are all the same. It is also remarked that the members of $$\mathcal S'$$ satisfy a linear recurrence relationship of order $$r + 1$$ with constant coefficients (inhomogeneous if the constant entries are nonzero). An equivalent but more cumbersome and unstable method based upon the use of approximating fractions is considered. Conditions sufficing to ensure that $$\mathcal S$$ and $$\mathcal S'$$ have the same limit are stated.
Much of the material has been known for some time. In particular, the reviewer has given a stability analysis (not mentioned by the authors) of the progressive use of a number of algorithms [Numer. Math. 1, 142-149 (1959; Zbl 0087.32502)]. It is a curious fact that sequences which induce instability of the $$\varepsilon$$-algorithm in its forward mode of use (i.e. to construct a sequence of numbers with fixed $$m$$ and increasing $$r$$) may be extrapolated in a completely stable way. Again, the $$\rho$$-algorithm, whose instability in forward use must be monitored carefully, may be used as a completely stable extrapolation device.
Reviewer: P.Wynn (México)
### MSC:
65B05 Extrapolation to the limit, deferred corrections 41A21 Padé approximation 40A05 Convergence and divergence of series and sequences
Zbl 0087.32502
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https://codereview.stackexchange.com/questions/204549/lookup-closest-value-in-pandas-dataframe | # Lookup closest value in Pandas DataFrame
I have a DataFrame that contains the data shown below:
soc [%] r0 [ohm] tau1 [s] tau2 [s] r1 [ohm] r2 [ohm] c1 [farad] c2 [farad]
0 90 0.001539 1725.035378 54.339882 0.001726 0.001614 999309.883552 33667.261120
1 80 0.001385 389.753276 69.807148 0.001314 0.001656 296728.345634 42164.808208
2 70 0.001539 492.320311 53.697439 0.001139 0.001347 432184.454388 39865.959637
3 60 0.001539 656.942558 63.233445 0.000990 0.001515 663400.436465 41727.472274
4 50 0.001539 296.080424 53.948112 0.000918 0.001535 322490.860387 35139.878909
5 40 0.001539 501.978979 72.015509 0.001361 0.001890 368919.408585 38100.665763
6 30 0.001539 585.297624 76.972464 0.001080 0.001872 542060.285388 41114.220492
7 20 0.001385 1308.176576 60.541172 0.001426 0.001799 917348.863136 33659.124096
8 10 0.001539 1194.993755 57.078336 0.002747 0.001851 435028.073957 30839.130201
Given a value z, I want to select a row in the data frame where soc [%] is closest to z. The code below demonstrates my current approach.
import pandas as pd
import time
def rc_params(df, z):
if z > 90:
params = df.loc[0]
elif 80 < z <= 90:
params = df.loc[0]
elif 70 < z <= 80:
params = df.loc[1]
elif 60 < z <= 70:
params = df.loc[2]
elif 50 < z <= 60:
params = df.loc[3]
elif 40 < z <= 50:
params = df.loc[4]
elif 30 < z <= 40:
params = df.loc[5]
elif 20 < z <= 30:
params = df.loc[6]
elif 10 < z <= 20:
params = df.loc[7]
else:
params = df.loc[8]
r0 = params['r0 [ohm]']
tau1 = params['tau1 [s]']
tau2 = params['tau2 [s]']
r1 = params['r1 [ohm]']
r2 = params['r2 [ohm]']
return r0, tau1, tau2, r1, r2
start = time.time()
z = 20
r0, tau1, tau2, r1, r2 = rc_params(df, z)
end = time.time()
print(f"""
z = {z}
r0 = {r0:.4f}
tau1 = {tau1:.4f}
tau2 = {tau2:.4f}
r1 = {r1:.4f}
r2 = {r2:.4f}
run time = {end - start:.4g} s
""")
Results from the above code give:
z = 20
r0 = 0.0014
tau1 = 1308.1766
tau2 = 60.5412
r1 = 0.0014
r2 = 0.0018
run time = 0.002264 s
My approach works fine but is there a better (faster) way to lookup the values in the data frame? There is a lookup function in Pandas but it finds exact values, so if a value doesn't exist then nothing is returned.
• min(max(9 - round(z / 10), 0), 8) – Gareth Rees Sep 29 '18 at 14:49
• @GarethRees I implemented params = df.iloc[min(max(9 - round(z / 10), 0), 8)] in the function. This gets rid of the if statements but execution time is the same as my original example. My goal is to find a faster way to lookup the values form the data frame compared to my original example. – wigging Sep 29 '18 at 15:27
• @GarethRees Would working with the data in a NumPy array (instead of a DataFrame) allow me to get faster lookup times? – wigging Sep 29 '18 at 15:34
Adapting from here would be a cleaner way to do what you want.
params = df.iloc[(df['soc [%]']-z).abs().argsort()[:1]]
There might be faster ways if your soc [%] column is fixed with those values.
Also, you should consider not measuring the time for pd.read_csv as that isn't what you are wanting to know the execution for.
• Thank you for your suggestion about measuring the execution time. I have updated my question with the new timing result. I also tried your suggestion using argsort() which gets rid of the if statements but unfortunately this is about 3 times slower than my original example. – wigging Sep 29 '18 at 14:13
• Wouldn't idxmax() be better than argsort()[:1]? – Gareth Rees Sep 29 '18 at 14:31
• @GarethRees Using idxmax() does not give the correct results; however, idxmin() gives the right results but it is still about 3 times slower than my original example. – wigging Sep 29 '18 at 14:43
• @wigging, you are essentially hardcoding a lookup table with your elifs. Its going to be hard to speed that part up dramatically. In the speedup context, there are a couple options: 1. Speed up the elifs ( you could do this with a binary tree) 2. Speed up the lookups ( you could do this with 1 lookup) r0, tau1, tau2, r1, r2 = params[['r0 [ohm]', 'tau1 [s]', 'tau2 [s]', 'r1 [ohm]', 'r2 [ohm]']] – Derek Thomas Sep 30 '18 at 16:03 | 2019-09-22 06:57:27 | {"extraction_info": {"found_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": 0.25131624937057495, "perplexity": 5172.241652698308}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575168.82/warc/CC-MAIN-20190922053242-20190922075242-00259.warc.gz"} |
https://www.physicsforums.com/threads/complex-hilbert-space-as-a-symplectic-space.492070/ | # Complex Hilbert Space as a Symplectic Space?
#### Bacle
Hi All: in the page:
http://mathworld.wolfram.com/SymplecticForm.html,
Complex Hilbert space, with "the inner-product" I<x,y> , where <.,.> is the inner-product
Does this refer to taking the imaginary part of the standard inner-product ? If so, is
I<x,y> symplectic in Complex Hilbert Space? It is obviously bilinear, but I don't see
how it is antisymmetric , i.e., I don't see that I<x,y>=-I<y,x>
Am I missing something?
Thanks.
Related Linear and Abstract Algebra News on Phys.org
#### Fredrik
Staff Emeritus
Gold Member
For any complex number c=a+ib (with a,b real), we have Im(c*) = Im(a-ib) = -b = -Im(a+ib) = -Im c, so
Im<x,y>=Im(<y,x>*)=-Im<y,x>
Last edited:
#### Bacle
Yes, how dumb of me. Thanks, Fredrik. | 2019-11-18 15:53:41 | {"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": 0.9018599987030029, "perplexity": 7133.522838737977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669809.82/warc/CC-MAIN-20191118154801-20191118182801-00417.warc.gz"} |
https://stats.stackexchange.com/questions/529701/pca-vs-least-squares/529736 | # PCA vs Least Squares
Principal Component Analysis By SVANTE WOLD, Chemometrics and Intelligent Laboratory Systems, 2 (1987) 37-52
In the above paper, the author says... "As PCA is a least squares method, outlier severely influence the model"
By what I have understood, PCA is transformation technique where orthogonal transformation are done to better explain the data. Why does author say PCA is a least squares method?
Given $$m$$ vectors $$\boldsymbol{x}_1, \ldots, \boldsymbol{x}_m \in \mathbb{R}^n$$, find matrices $$\boldsymbol{U} \in \mathcal{M}_{\mathbb{R}}(k, n)$$ and $$\boldsymbol{V} \in \mathcal{M}_{\mathbb{R}}(n, k)$$ such that $$\sum_{i=1}^m{||\boldsymbol{x}_i - \boldsymbol{V}\boldsymbol{U}\boldsymbol{x}_i ||}^2$$ is minimized.
That is, for $$k < n$$ the vector $$\boldsymbol{U}\boldsymbol{x}_i \in \mathbb{R}^k$$ is the projection of $$\boldsymbol{x}_i$$ into a lower-dimensional subspace, and $$\boldsymbol{V}\boldsymbol{U}\boldsymbol{x}_i$$ is the reconstructed original vector. PCA aims to find matrices $$\boldsymbol{U}, \boldsymbol{V}$$ that minimize the reconstruction error as measured by the $$\ell^2$$-norm. It can be shown that, in fact, these matrices are orthogonal and $$\boldsymbol{U} = \boldsymbol{V}^T$$, so the problem reduces to $$\underset{V \in \mathcal{M}_{\mathbb{R}}(n,k)}{\mathrm{arg\,min}}\sum_{i=1}^m{||\boldsymbol{x}_i - \boldsymbol{V}\boldsymbol{V}^T\boldsymbol{x}_i ||}^2\,.$$ Further manipulations show that $$\boldsymbol{V}$$ is the matrix whose columns are the eigenvectors corresponding to the $$k$$ largest eigenvalues of $$\sum_{i=1}^m \boldsymbol{x}_i{\boldsymbol{x}_i}^T\,,$$ as expected. So indeed, PCA is a least squares method and it is quite sensitive to outliers. | 2021-07-29 12:33:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 16, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8816453814506531, "perplexity": 282.59034909243474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153857.70/warc/CC-MAIN-20210729105515-20210729135515-00267.warc.gz"} |
https://www.techwhiff.com/learn/in-the-situation-pictured-below-assume-the-mass/385621 | # In the situation pictured below, Assume the mass of the boat to be 20 kg. 6...
###### Question:
In the situation pictured below, Assume the mass of the boat to be 20 kg. 6 points If the ramp used to displace the boat makes a 30-degree angle with the horizontal and the chain used to pull the boat uses a force of 10 N/kg to move the boat. How much work is done by gravity? F 10 J 100J 500 J 1000 J | 2023-03-27 23:52:27 | {"extraction_info": {"found_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": 0.6413018703460693, "perplexity": 516.7083560579653}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948708.2/warc/CC-MAIN-20230327220742-20230328010742-00511.warc.gz"} |
https://www.physicsforums.com/threads/magnitude-and-direction-of-velocity.128034/ | # Magnitude and direction of velocity
1. Aug 4, 2006
### prabhjyot
The end A of the link in a guide has a constant downward velocity of 3 m/s as shown in the diagram. For the instant where the link AB makes an angle of 75 degrees with the vertical guide calculate the magnitude and direction of the velocity of the roller at B, the magnitude and direction of velocity of the centre of the link AB and the angular velocity of the link AB. Is the velocity at B also constant?
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2. Aug 5, 2006
### HallsofIvy
Staff Emeritus
That looks like a typical Calculus "related rates" problem!
You need two formulas: by the Pythagorean theorem, a2+ b2= 2002, where b is the distance from B to the corner and a is the distance from A to the corner. Differentiate that formula with respect to time (using the chain rule, of course) to determine db/dt (you are given da/dt), the "magnitude of the velocity of the roller at B". The direction should be obvious.
You also have $cos(\theta)= \frac{a}{200}$ where $\theta$ is the angle AB makes with the vertical guide (75 degrees at the instant in question). Differentiate that to determine "the angular velocity of the line AB", $\frac{d\theta}{dt}$.
For the midpoint, you can do the same thing but use 100 mm instead of 200. | 2017-07-27 16:56:38 | {"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": 0.7736880779266357, "perplexity": 458.8739245044682}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549428325.70/warc/CC-MAIN-20170727162531-20170727182531-00441.warc.gz"} |
https://mathtube.org/lecture/video/blowup-or-no-blowup-interplay-between-theory-and-computation-study-3d-euler-equations | # Blowup or no blowup? The interplay between theory and computation in the study of 3D Euler equations
Speaker: Thomas Hou
Date: Fri, Feb 27, 2015
Location: PIMS, University of British Columbia
Conference: PIMS/UBC Distinguished Colloquium
Class: Scientific
### Abstract:
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. We first review some recent theoretical and computational studies of the 3D Euler equations. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain flow geometry. We then present strong numerical evidence that the 3D Euler equations develop finite time singularities. To resolve the nearly singular solution, we develop specially designed adaptive (moving) meshes with a maximum effective resolution of order $10^12$ in each direction. A careful local analysis also suggests that the solution develops a highly anisotropic self-similar profile which is not of Leray type. A 1D model is proposed to study the mechanism of the finite time singularity. Very recently we prove rigorously that the 1D model develops finite time singularity.This is a joint work of Prof. Guo Luo. | 2022-05-17 20:44:30 | {"extraction_info": {"found_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": 0.45149296522140503, "perplexity": 589.3125085718085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662520817.27/warc/CC-MAIN-20220517194243-20220517224243-00077.warc.gz"} |
http://tex.stackexchange.com/questions/42706/bibliography-doesn-t-appear-even-though-i-have-the-bbl | # bibliography doesn' t appear, even though I have the .bbl
I am using TeXworks and have a problem with BibTeX. I compiled the .bbl file however I can't see the bibliography in the output.
\usepackage[super]{natbib}
\begin{document}
bu bir denemedir.\cite{Sripirom20112402}.
\bibliographystyle{plane}
\thebibliography{tryin}
\end{document}
If I type in like above .tex, I can see only the title of bibliography and when I type \bibliography{tryin}, I get nothing.
-
I think you need to fix a couple of typos: First, use \bibliographystyle{plain} instead of \bibliographystyle{plane}; second, you do need to type \bibliography{tryin} instead of \thebibliography{tryin}. You are following the usual procedure of running (pdf)latex, then bibtex, then latex twice more, right? If so, check the log files (.log and .blg) and examine them for errors and/or warnings. – Mico Jan 29 '12 at 19:56 | 2016-07-27 06:00:57 | {"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": 0.9694193005561829, "perplexity": 2577.003760341531}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257825366.39/warc/CC-MAIN-20160723071025-00173-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2/share/doc/Macaulay2/Macaulay2Doc/html/_sum.html | # sum -- compute the sum
## Description
sum provides the sum of the members of a list, set, or chain complex, optionally with a function applied to each one.
## For the programmer
The object sum is . | 2022-11-29 20:22:56 | {"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": 0.8031054139137268, "perplexity": 2739.5056870037297}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446710711.7/warc/CC-MAIN-20221129200438-20221129230438-00185.warc.gz"} |
http://mathoverflow.net/revisions/101516/list | 4 deleted 117 characters in body
I try to write down in full detail the answers of Nik and BS (many thanks to both!). I do not accept this answer as definitive since I'm still interested in the last non-commutative part of my question.
The references in the proof are to Folland's book "A Course in Abstract Harmonic Analysis" and Rudin's book "Fourier Analysis on Groups".
Let $G$ be an LCA group, then {$\widehat\phi:\phi\in L^1(G)^+\cap \mathcal P(G)$}$=L^1(\widehat G)^+\cap \mathcal P(\widehat G)$.
Proof. Let us start proving the inclusion {$\widehat\phi:\phi\in L^1(G)^+\cap \mathcal P(G)$}$\subseteq L^1(\widehat G)^+\cap \mathcal P(\widehat G)$. Indeed, consider $\phi \in L^1(G)^+\cap \mathcal P(G)$, then $\widehat\phi\in L^1(\widehat G)$ by the Fourier Inversion Theorem (see [Rudin, page 22, line 1]) and $\widehat\phi\geq 0$ by [Folland, Corollary 4.23]. Let now $\mu_\phi$ be the non-negative and bounded (as $\phi\in L^1(G)^+$) regular measure defined on a generic Borel subset $E$ of $G$ by $\mu_\phi(E)=\int_{x\in E}\phi(x)d\mu(x)$ (here $\mu$ is a fixed Haar measure on $G$). One can show that $$\widehat\phi(\gamma)=\int_{x\in G}\phi(x)\gamma(-x)d\mu(x)=\int_{x\in G}\gamma(-x)d\mu_\phi(x)\, .$$ By Bochner's Theorem (see [Rudin, page 19]), $\widehat\phi\in \mathcal P(\widehat G)$.
On the other hand, let $\phi\in L^1(\widehat G)^+\cap \mathcal P(\widehat G)$ and $\psi$ be the function defined by $\psi(\gamma)=\phi(-\gamma)$ for all $\gamma\in \widehat G$. It is not difficult to see that $\psi\in L^1(\widehat G)^+\cap \mathcal P(\widehat G)$. By the first part of the proof, $\widehat\psi\in L^1( G)^+\cap \mathcal P( G)$ and, using Fourier Inversion Theorem (see [Folland, Theorem 4.32]), one obtains that $\widehat{\widehat\psi}=\phi$, which is therefore the Fourier transform of a function in $L^1( G)^+\cap \mathcal P( G)$.\\\
3 added 44 characters in body
I try to write down in full detail the answers of Nik and BS (many thanks to both!). I do not accept this answer as definitive since I'm still interested in the last non-commutative part of my question.
The references in the proof are to Folland's book "A Course in Abstract Harmonic Analysis" and Rudin's book "Fourier Analysis on Groups".
Let $G$ be an LCA group, then {$\widehat\phi:\phi\in L^1(G)^+\cap \mathcal P(G)$}$=L^1(\widehat G)^+\cap \mathcal P(\widehat G)$.
Proof. Let us start proving the inclusion {$\widehat\phi:\phi\in L^1(G)^+\cap \mathcal P(G)$}$\subseteq L^1(\widehat G)^+\cap \mathcal P(\widehat G)$. Indeed, consider $\phi \in L^1(G)^+\cap \mathcal P(G)$, then $\widehat\phi\in L^1(\widehat G)$ by the Fourier Inversion Theorem (see [Rudin, page 22, line 1]) and $\widehat\phi\geq 0$ by [Folland, Corollary 4.23]. Let now $\mu_\phi$ be the non-negative and bounded (as $\phi\in L^1(G)^+$) regular measure defined on a generic Borel subset $E$ of $G$ by $\mu_\phi(E)=\int_{x\in E}\phi(x)d\mu(x)$ . (here $\mu$ is a fixed Haar measure on $G$). One can show that $$\widehat\phi(\gamma)=\int_{x\in G}\phi(x)\gamma(-x)d\mu(x)=\int_{x\in G}\gamma(-x)d\mu_\phi(x)\, .$$ By Bochner's Theorem (see [Rudin, page 19]), $\widehat\phi\in \mathcal P(\widehat G)$.
On the other hand, let $\phi\in L^1(\widehat G)^+\cap \mathcal P(\widehat G)$ and $\psi$ be the function defined by $\psi(\gamma)=\phi(-\gamma)$ for all $\gamma\in \widehat G$. It is not difficult to see that $\psi\in L^1(\widehat G)^+\cap \mathcal P(\widehat G)$. By the first part of the proof, $\widehat\psi\in L^1( G)^+\cap \mathcal P( G)$ and, using Fourier Inversion Theorem (see [Folland, Theorem 4.32]), one obtains that $\widehat{\widehat\psi}=\phi$, which is therefore the Fourier transform of a function in $L^1( G)^+\cap \mathcal P( G)$.\\\
Post Undeleted by Simone Virili
2 deleted 8 characters in body
Post Deleted by Simone Virili
1 | 2013-05-26 09:03:03 | {"extraction_info": {"found_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": 0.983988344669342, "perplexity": 105.79237339661252}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368706794379/warc/CC-MAIN-20130516121954-00089-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://paulmaceoin.com/so-you-decided-to-write-your-thesis-in-latex-pt2/ | As you may have seen from my previous post, I have now finished my PhD. I wanted to wait until all the final changes to my thesis were done before writing up more posts about doing your thesis through LaTeX.
So, part 2, tables.
Tables are really easy in WYSIWYG programs like Word. Simple tables are not really a problem in LaTeX but the moment you want to do something slightly different, things get harder.
It is possible to just write out the markup for your table from scratch, and that's a good exercise to understand how things work that I recommend to do. However, you can also "cheat" or use a shortcut.
Excel2latex is a plugin for LaTeX that allows you to export the table you design there as LaTeX code directly ready to paste into your environment. You can do things easily that are otherwise fiddly to code in LaTeX such as colouring cells, complex arrangements of merged and split cells, and where to put borders. I used this for every single table in my thesis, editing and fixing small bugs as appropriate. It probably saved hours of time versus writing it myself.
Let's go through some specific examples from my thesis to demonstrate...
## The Basics
\usepackage{threeparttable, booktabs} % booktabs for \bottomrule\toprule\midrule
\usepackage{longtable} %for tables that go over more than one page.
\usepackage{array, multirow}
% Table generated by Excel2LaTeX from sheet 'fab'
\begin{table}[ht]
\caption{French-American-British Classification of AML\cite{Krause2007, Behm2003}} \label{tab:fab}
\footnotesize
\begin{tabular}{lp{0.6\linewidth}l}
\textbf{FAB\#} & \textbf{AML name} & \textbf{Cell origin} \\
\midrule
M0 & Acute myeloblastic leukaemia without differentiation & Myeloblast \\
M1 & Acute myeloblastic leukaemia with minimal differentiation but with the expression of myeloperoxidase & Myeloblast \\
M2 & Acute myeloblastic leukaemia with differentiation & Myeloblast \\
M3 & Acute promyelocytic leukaemia (APL) & Promyelocyte \\
M4 & Acute myelomonocytic leukaemia (AMML) & Monoblast \\
M4eos & Acute myelomonocytic leukaemia bone marrow eosinophilia & Monoblast \\
M5 & Acute monocytic leukaemia (AMoL) & Monoblast \\
M6 & Acute erythroid leukaemia (AEL) & Proerythroblast \\
M7 & Acute megakaryocytic leukaemia (AMKL) & Megakaryoblast \\
\end{tabular}
\end{table}
Table 1.1 is a simple table with just 3 columns - a good one to start with. The first block of code are the packages I've used for all the tables. Not all are needed for all tables but since they are loaded, it's easier to include them here.
The second block of code first opens the table environment and determines how it is located in document (here, top[of page]). The caption appears at the top of tables by default and at the bottom of figures. Obviously you and probably should include references here. I also labelled all my figures/tables etc for ease of cross referencing. It's easier to do this at the time with something descriptive rather than numbering as your order may change with edits.
I found the default text size in tables too large so I changed everything to \footnotesize but this is a personal choice. Then the actual table data begins with \begin{tabular}. The arguments to this look confusing at first but are easy. If you had {ccc} you'd have 3 columns each with centred text, {lll} would have left aligned text, and {rrr} right aligned. My middle column here is marked to be a particular fraction of the total line width p{0.6\linewidth} because I needed to force the width to be longer for clarity.
The next line is the header row and I've emboldened each of the headers. Note how elements in a row are delimited with "&" and the end of the line is with "\\". Also note the \midrule to add a line. Then the tabular and table environments need to be closed. That's it! Your first simple LaTeX table. A bit more complicated already than Excel but using the Excel2Latex plugin linked above will help.
## Images in tables
% Table generated by Excel2LaTeX from sheet 'C nucleosides 1'
\begin{table}[htbp]
\footnotesize
\centering
\caption{Cytidine and medicinal analogues}
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{1}{|c|}{\textbf{Cytidine}} & \multicolumn{1}{c|}{\textbf{Cytarabine (ara-C)}} & \multicolumn{1}{c|}{\textbf{Gemcitabine}}\\
\hline
\hline
\includegraphics[width=0.15\linewidth]{Figures/chemicals/cytidine} & \includegraphics[width=0.15\linewidth]{Figures/chemicals/cytarabine} & \includegraphics[width=0.133\linewidth]{Figures/chemicals/gemcitabine} \\
\hline
\end{tabular}%
\label{tab:c-derivatives}%
\end{table}%
Of course it is also possible to include images in a LaTeX table. For this, I found it easier to have the figures already prepared in a folder. Using Excel2latex, you can include the necessary code to include a figure in a column \includegraphics[width=0.30\linewidth]{Figures/chemicals/cytidine} for example.
I'm not actually sure that the multicolumn code is required here - it's been a while since I wrote this so not sure why it's there at all! But it is useful for the next thing I want to explain...
\begin{table}[htbp]
\centering
\caption{Common DNA intercalators used in the treatment of haematological and other malignancies}
\footnotesize
\begin{tabular}{|l|l|l|}
\hline
\multicolumn{3}{|c|}{\textbf{Anthracyclines}} \\
\hline
\hline
\multicolumn{1}{|c|}{\textbf{Daunorubicin}} & \multicolumn{1}{c|}{\textbf{Doxorubicin}} & \multicolumn{1}{c|}{\textbf{Idarubicin}} \\
\includegraphics[width=0.30\linewidth]{Figures/chemicals/daunorubicin} & \includegraphics[width=0.30\linewidth]{Figures/chemicals/doxorubicin} & \includegraphics[width=0.30\linewidth]{Figures/chemicals/idarubicin} \\
FDA approved 1979 & FDA approved 1974 & FDA approved 1990 \\
Indications: & Indications: & Indications: \\
\multicolumn{1}{|m{4cm}|}{AML, CML, neuroblastoma} & \multicolumn{1}{m{4cm}|}{Bladder, breast, stomach, lung, leukaemias, ovarian, and others} & AML \\
\hline
Table 1.4 is only part of a more complex larger table but this is sufficient to outline the important next step: merged cells. \multicolumn{3}{|c|}{\textbf{Anthracyclines}} \\ means that we create a column that spans 3 columns, it is centred with lines on each side, and has the bold text Anthracyclines. This can be applied anywhere. LaTeX will count the total number of columns when compiling and throw an error if it is greater than it should be as defined in your \begin{tabular}{|l|l|l|}.
## Mixed figures and tables
It is also possible to mix figures and tables together.
% Table generated by Excel2LaTeX from sheet 'Chemical R'
\begin{table}[htbp]
\centering
\footnotesize
\includegraphics[width=0.2\linewidth]{../Figures/chemicals/purinesR3}
\begin{tabular}{ll|lll}
Common name & Abbreviation & R1 & R2 & R3 \\
\hline
Adenosine & A & OH & H & H \\
Deoxyadenosine & dA & H & H & H \\
Vindarabine & ara-A & H & OH & H \\
Fludarabine & 2-F-ara-A & H & OH & F \\
Cladribine & 2-Cl-dA & H & H & Cl \\
Clofarabine & 2-Cl-2'-$\beta$-F-dA & H & F & Cl \\
\end{tabular}%
\caption{Common purine analogues and their structures. Positions are labelled in red.}
\label{tab:purinedrugs}%
\end{table}%
I only had to do this a couple times during my thesis and it's probably not necessary most of the time. My technique was opening a table environment \begin{table}[htbp] then inserting the image, before opening the tabular. I chose to have the caption at the end for this, but that's a personal choice.
## Tables over multiple pages
\begin{longtable}{ll}
\caption{List of Abbreviations} \\
\textDelta \textPsi m & Change in mitochondrial membrane potential \\
DAPI & 4',6-diamidino-2-phenylindole \\
ActD & Actinomycin D \\
...
...
...
TOPOII & Topoisomerase II \\
yPAP & Yeast polyadenylation polymerase \\
\end{longtable}%
\label{tab:abb}%
For my appendix, the content was over two pages. I didn't feel it was appropriate for two columns (technically four), so used the longtables package that I called in main.tex above. It's not a difficult package to use and is quite normal. I used the excel2latex plugin to cheat in making the list easier. I also kept a running list of abbreviations I used during my thesis in that same excel sheet to make things even easier.
## Conclusions
LaTeX tables can be quite intimidating for a newbie. I had read and watched a few tutorials explaining them and it scared me off for a while. After spending a little time, they were actually one of the easier aspects of writing the thesis, and I hope that some examples I ran through here are also helpful to you. | 2019-07-20 22:12:26 | {"extraction_info": {"found_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": 0.6977694630622864, "perplexity": 2659.5787877359376}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526714.15/warc/CC-MAIN-20190720214645-20190721000645-00247.warc.gz"} |
http://compgroups.net/comp.lang.lisp/macro-dependency-tracking/702403 | COMPGROUPS.NET | Search | Post Question | Groups | Stream | About | Register
### Macro dependency tracking
• Email
• Follow
Hi,
I always wondered why macro calls are not reexpanded automatically each
time the macro is redefined. Whenever I recompile a macro definition I
should recompile all places where the macro was expanded which is
sometimes nearly impossible to do, so it's easier to restart and
recompile everything.
This is an incomplete and simple implementation to automatically
recompile functions using the computed-class package at
http://common-lisp.net/project/computed-class
The idea is that each function and macro definition will have it's own
computed-state and the dependency will be tracked automatically during
compiling a function at macroexpand time.
Each time a macro is redefined it will invalidate the computed-states
of the functions where it was used. Whenever a function is called it
will first check if the function needs to be recompiled. This runtime
overhead can be eliminated, see later.
Ok, don't take this too seriously, but still it might be interesting
how short it is...
CC> (define-computed-universe mdt-compute-as)
MDT-COMPUTE-AS
CC> (defmacro defun/mdt (name args &body forms)
(clet ((f (mdt-compute-as
(progn
(format t "Recompiling ~A" ',name)
(compile nil (lambda ,',args ,',@forms))))))
(defun ,name ,args
(funcall f ,@args))))
DEFUN/MDT
CC> (defmacro defmacro/mdt (name args &body forms)
(let ((m (or (get name 'mdt)
(setf (get name 'mdt) (mdt-compute-as nil)))))
(invalidate-computed-state m)
(defmacro ,name ,args
(computed-state-value (get ',name 'mdt))
,@forms)))
DEFMACRO/MDT
CC> (defmacro/mdt with-a (&body forms)
(let ((a 1))
,@forms))
WITH-A
(with-a
(+ a b)))
2
2
CC> (defmacro/mdt with-a (&body forms)
(let ((a 2))
,@forms))
WITH-A
3
3
Unfortunately there is no forward invalidation in the computed class
package yet, so there is an extra runtime check each time the function
add is called whether it has to be recompiled or not. This can be
easily fixed by the following definition as soon as forward
invalidation becomes available.
(defmacro defun/mdt* (name args &body forms)
(computed-state-value
(mdt-compute-as
(progn
(format t "Recompiling ~A" ',name)
(defun ,name ,args
,@forms)))))
The function's computed state will be captured by the macros' computed
states as the compiler expands macro calls. If there were no macro
calls at all then the function's computed state will be garbage
collected. Whenever a called macro definition changes its computed
state gets invalidated and thus the functions depending on it will be
immediately redefined.
Cheers,
levy
0
Reply levente.meszaros (36) 12/20/2006 9:14:29 PM
See related articles to this posting
levy wrote:
> Hi,
>
> I always wondered why macro calls are not reexpanded automatically each
> time the macro is redefined. Whenever I recompile a macro definition I
> should recompile all places where the macro was expanded which is
> sometimes nearly impossible to do, so it's easier to restart and
> recompile everything.
>
> This is an incomplete and simple implementation to automatically
> recompile functions using the computed-class package at
> http://common-lisp.net/project/computed-class
>
Yes, and this marks quite a milestone with the number of Cells spin-offs
now exceeding the number of users.
Similarly, I never thought my failure to document could be surpassed,
but here is the computed-class documentation:
> See test.lisp for more examples and/or read the Cells docs.
I underestimated you yobs.
:)
ken
0
Reply kentilton (2985) 12/20/2006 9:50:10 PM
Ken Tilton wrote:
> Yes, and this marks quite a milestone with the number of Cells spin-offs
> now exceeding the number of users.
Count me to the users too, at least for some time!
> > See test.lisp for more examples and/or read the Cells docs.
Well, what can I say... ;-)
The point is not comparison but that the compiler, CLOS mop or a
structured editor could use stuff like that.
Cheers,
levy
0
Reply levente.meszaros (36) 12/20/2006 10:02:00 PM
levy wrote:
> Ken Tilton wrote:
>
>>Yes, and this marks quite a milestone with the number of Cells spin-offs
>>now exceeding the number of users.
>
> Count me to the users too, at least for some time!
>
>
>>>See test.lisp for more examples and/or read the Cells docs.
>
> Well, what can I say... ;-)
>
> The point is not comparison but that the compiler, CLOS mop or a
> structured editor could use stuff like that.
No argument there. Hell, Cells once had (and I think someday will have
again) the moral equivalent of a Gosub, using itself a Cell to know when
to "return".
Have you heard about the annual conference for dataflow hacks? Trying to
keep talks shorter and more memorable. All must be written in the form
of substitute lyrics to popular songs and sung karoake-style. Words to
the chorus should be distributed beforehand so the audience can sing
along. Venues must be cities with casinos.
ken
--
Algebra: http://www.tilton-technology.com/LispNycAlgebra1.htm
"Well, I've wrestled with reality for thirty-five
years, Doctor, and I'm happy to state I finally
won out over it." -- Elwood P. Dowd
"I'll say I'm losing my grip, and it feels terrific."
-- Smiling husband to scowling wife, New Yorker cartoon
0
Reply kentilton (2985) 12/20/2006 10:19:01 PM
3 Replies
53 Views
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https://gedeno.com/perimeter-and-area/ | # Perimeter and Area
Perimeter & Area are two very important and fundamental math topics.
These topics help you quantify physical space and provide the foundation for more advanced math issues found in algebra, calculus, and trigonometry.
### The “All-in-One” GED Prep
Get Your Diploma in 2 Months.
It doesn’t matter when you left school.
1. Find the perimeter of this rectangle.
Use this formula
$$P=2l + 2w$$
$$l-length$$
$$w-width$$
A.
B.
C.
D.
Question 1 of 2
2. Find the area of this rectangle.
Use this formula.$$Area = w × h$$
$$w = width$$
$$h = height$$
A.
B.
C.
D.
Question 2 of 2
This lesson is provided by Onsego GED Prep.
Next lesson: Perimeter and area of rectangles
This lesson is a part of our GED Math Study Guide.
### Video Transcription
Perimeters are measurements of the distance around shapes, and Areas give us an idea of the surface a shape covers.
The knowledge of perimeter and area is applied in a practical way by lots of people on a day-to-day basis. Engineers, architects, or graphic designers use it every day. This is math that’s so much needed by individuals in general everywhere.
Understanding how much space we’ve got and learning how we can fit shapes together in an exact way will be helpful when painting a room, buying a home, remodeling a kitchen, or building a deck.
### Fast & Easy Online GED Classes
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Perimeter
The perimeter of a 2-dimensional shape is the distance around that shape. Think of wrapping some string around a triangle. Then, the length of that string would be the triangle’s perimeter. When you walk around the outside of a park, you’ll be walking the distance of the perimeter of that park.
Some people find it useful to think of a “peRIMeter” as an object’s edge is its rim. Well, peRIMeter has the word “rim” in it.
Now, if your shape is a polygon, you can add up all the polygon’s lengths of its sides to find its perimeter. Be careful: make sure that you measure all the lengths in the same units of measure. You measure the perimeter in linear units, and these are 1-dimensional. The units of measure examples for length are feet, inches, or centimeters.
Example
The problem: What is the perimeter of this figure. The measurements given are inches.
Well, the perimeter (P) is: 5 + 3 + 6 + 2 + 3 + 3
All sides are measured in inches, so just add all the lengths of the six (6) sides to get your perimeter.
Remember that you must include units.
So the answer is P = 22 inches.
That’s meaning that a tightly wrapped string that runs the distance around the entire polygon will measure a length of 22 inches.
Another example
The problem: What is the perimeter of a triangle that has sides that measure 6 cm, 8 cm, and 12 cm.
The perimeter (P) is: 6 + 8 + 12
As all of the triangle’s sides are measured in centimeters, simply add all the lengths of the three sides to get our perimeter.
The answer is: P = 26 centimeters
Sometimes, we must use what we know about a polygon to find our perimeter. Let’s take a look at the rectangle in the following example.
Example
The problem: A rectangle has a width of 3 centimeters and a length of 8 centimeters. Find its perimeter.
P = 3 + 3 + 8 + 8
Because this is a rectangle, opposite sides have exactly the same lengths, 8 cm. and 3 cm. So add up all lengths of the four sides to find our perimeter.
The answer is: P = 22 cm
Did you see that the perimeter (P) of a rectangle always includes two pairs of equal length sides? In the example above, you might have also written: perimeter (P) = 2(3) + 2(8) = 6 + 16 = 22 cm. The formula for measuring the perimeter of rectangles is often written as follows: P = 2l + 2w. Here, l is the rectangle’s length and w is the rectangle’s width.
Last Updated on June 13, 2022. | 2022-06-26 05:31:33 | {"extraction_info": {"found_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": 0.6011447310447693, "perplexity": 790.971729717241}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103037089.4/warc/CC-MAIN-20220626040948-20220626070948-00016.warc.gz"} |
https://unapologetic.wordpress.com/2007/08/27/enriched-categorical-constructions/?like=1&source=post_flair&_wpnonce=966197e62c | # The Unapologetic Mathematician
## Enriched Categorical Constructions
We’re going to need to talk about enriched functors with more than one variable, so we’re going to need an enriched analogue of the product of two categories.
Remember that the product $\mathcal{C}\times\mathcal{D}$ of two categories has the product of the object-classes as its objects, and it has pairs of morphisms for its morphisms. That is, the hom-set $\hom_{\mathcal{C}\times\mathcal{D}}((C_1,D_1),(C_2,D_2))$ is the product $\hom_\mathcal{C}(C_1,C_2)\times\hom_\mathcal{D}(D_1,D_2)$. Of course, in the enriched setting we no longer have hom-sets to work with.
So we’ll keep the same definition for the objects of our product category, but we’ll replace the definition of the hom-objects:
$\hom_{\mathcal{C}\times\mathcal{D}}((C_1,D_1),(C_2,D_2))=\hom_\mathcal{C}(C_1,C_2)\otimes\hom_\mathcal{D}(D_1,D_2)$
Now we can use the associativity and commutativity of our monoidal category $\mathcal{V}$ (remember we’re assuming it’s symmetric now) to move around factors like this:
$(\hom_\mathcal{C}(C_2,C_3)\otimes\hom_\mathcal{D}(D_2,D_3))\otimes(\hom_\mathcal{C}(C_1,C_2)\otimes\hom_\mathcal{D}(D_1,D_2))\rightarrow$
$(\hom_\mathcal{C}(C_2,C_3)\otimes\hom_\mathcal{C}(C_1,C_2))\otimes(\hom_\mathcal{D}(D_2,D_3)\otimes\hom_\mathcal{D}(D_1,D_2))$
at which point we can use the composition in each category to give a composition of the original pairs.
To get an identity, we use $\mathbf{1}\cong\mathbf{1}\otimes\mathbf{1}$ and then hit the left copy of $\mathbf{1}$ with the identity morphism for the object $C\in\mathcal{C}$ and the right copy with the identity morphism for $D\in\mathcal{D}$.
What about the opposite category? Well, it works pretty much the same as before. We just define $\hom_{\mathcal{C}^\mathrm{op}}(A,B)=\hom_\mathcal{C}(B,A)$. For an identity, we just use the same $\mathbf{1}\rightarrow\hom_\mathcal{C}(C,C)=\hom_{\mathcal{C}^\mathrm{op}}(C,C)$ as before.
Actually, these same constructions apply to functors. If we have functors $F:\mathcal{C}\rightarrow\mathcal{C}'$ and $G:\mathcal{D}\rightarrow\mathcal{D}'$, we can assemble them into a functor $F\otimes G:\mathcal{C}\otimes\mathcal{D}\rightarrow\mathcal{C}'\otimes\mathcal{D}'$. Just define $F\otimes G(C\otimes D)=F(C)\otimes G(D)$, and use a similar definition for the morphisms. Also, given $F:\mathcal{C}\rightarrow\mathcal{D}$ we get a functor $F^\mathrm{op}:\mathcal{C}^\mathrm{op}\rightarrow\mathcal{D}^\mathrm{op}$.
Now we know we have a 2-category $\mathcal{V}\mathbf{-Cat}$ of categories enriched over $\mathcal{V}$. Since a 2-category is a category enriched over categories, we can pass to the underlying category $\mathcal{V}\mathbf{-Cat}_0$ of enriched categories. It turns out that all the foregoing discussion gives this category some nice, familiar structure.
The product of two enriched categories turns out to be weakly associative. Also, remember from our discussion of the underlying category that we have a $\mathcal{V}$-category $\mathcal{I}$. This behaves like a weak identity for the product. That is, when we equip $\mathcal{V}\mathbf{-Cat}_0$ with this product and identified object, it turns out to be a monoidal category! Even better, it’s symmetric$\mathcal{C}\otimes\mathcal{D}\cong\mathcal{D}\otimes\mathcal{C}$. And what is the opposite category but a duality on this category?
So now we can define contravariant enriched functors, as well as functors of more than one variable. As usual, you should go back and try to think of these definitions in terms of ordinary categories ($\mathbf{Set}$-categories) as well as $\mathbf{Ab}$-categories.
Incidentally, if you want to run ahead a bit, try working out how natural transformations fit into the picture. It turns out that the 2-category $\mathcal{V}\mathbf{-Cat}$ is an example of an even deeper structure I haven’t defined yet: it’s a symmetric monoidal 2-category with duals.
[UPDATE]: Excuse me.. I should have said that $\mathcal{V}\mathbf{-Cat}$ is a symmetric monoidal 2-category with a duality involution rather than “with duals”, and similarly for the underlying category. I blame my inattention on being stuck around the house all day waiting for repairmen to come by to put the dishwasher they left in my living room last Friday into the dishwasher-sized hole they put in my kitchen. Basically, the “duality involution” means that the opposite of the opposite $\mathcal{V}$-category is the original $\mathcal{V}$-category back again, and that the opposite of a tensor product is the tensor product of the opposites.
August 27, 2007 - Posted by | Category theory
No comments yet. | 2015-11-29 19:28:06 | {"extraction_info": {"found_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": 32, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9516100883483887, "perplexity": 298.51299838407846}, "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-2015-48/segments/1448398459333.27/warc/CC-MAIN-20151124205419-00327-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://quant.stackexchange.com/tags/implied-volatility/new | Tag Info
3
This was also discussed in Derman's book The Volatility Smile (see Chapter 16). Specifically, he approximated the local volatility by a linear function of the form \begin{align*} \sigma(S) = \sigma_0 -2 b(S-S_0), \end{align*} and then approximated the implied volatility $\Sigma(S, K)$ for an option with strike $K$ by the average of $\sigma(S)$ between S and ...
7
They are not the same, but they are related. Gamma is sensitivity to realized volatility. Vega is sensitivity to implied volatility. Vanilla options are always long gamma and long vega, so they are "long vol" and saying "I am a buyer of vol/gamma/vega" means that you are taking a position that benefits from a rise in volatility (either ...
1
Thanks to all for the input. After a bit of research, I replaced the Black-Scholes pricer with a binomial tree pricer that includes early exercise and the known dividend in September, using what's explained in van der Hoek (2006) I chose a drift rate such that ATM vols match and put-call vols now match pretty closely even toward the wings. I definitely ...
1
The procedure you have specified in your last paragraph is the only reasonable way to do it. Clearly the cap volatility is some sort of weighted average of the constituent caplet volatilities, but the weighting is complex , having strike dependence as well as maturity dependence.
0
You don't need an approximation, i.e., if you have the Black's vols, you can simply compute the corresponding price and then invert Bachelier model (normal model) to get implied normal volatility. In the case of the transition from Normal (Bachelier) to Lognormal (Black-Sholes) you need to be more careful if you have negative forwards.
1
You basically have it. $$Normal Vol= Black Vol * Forward Swap Rate$$. Normal vol is usually quoted as an annual vol , not converted to daily by dividing by sqrt(252). The forward swap rate is the fair market rate for the swap that underlies the swaption. So one might have 1yr 10yr normal vol =70bp, forward swap rate = 1.40% and Black vol = 50%. ...
Top 50 recent answers are included | 2021-09-18 13:33:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8600960373878479, "perplexity": 1561.7029995861121}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056476.66/warc/CC-MAIN-20210918123546-20210918153546-00405.warc.gz"} |
https://math.stackexchange.com/questions/3194289/questions-on-cartans-magic-formula-mathcall-x-i-x-circ-d-d-circ-i-x | # Questions on Cartan's magic formula $\mathcal{L}_X=i_X \circ d + d\circ i_X$
Algebra $$A$$ is called graded algebra if it has a direct sum decomposition $$A=\bigoplus_{k\in\Bbb Z} A^k$$ s.t. product satisfies $$(A^k)(A^l)\subseteq(A^{k+l}) \text{ for each } k, l.$$
A differential graded algebra is graded algebra with chain complex structure $$d \circ d = 0$$.
Derivation of degree $$k$$ on $$A$$ means a linear map $$D:A \to A$$ s.t. $$D(A_j)\subset A_{j+k} \text{ and } D(ab)=(Da)b + (-1)^{ik}a(Db), a\in A_i$$
All smooth forms on $$n$$-manifold $$M$$ is a differential graded algebra $$\Omega^{\bullet}(M)=\bigoplus_{k=0}^{n} \Omega^k(M)$$, with wedge product and exterior derivative.
In proving Cartan's magic formula $$\mathcal{L}_X=i_X \circ d + d\circ i_X$$ holds for $$\Omega^{\bullet}(M)$$, we can use the following steps:
1. Prove the lemma: two degree $$0$$ derivations on $$\Omega^{\bullet}(M)$$ commuting with $$d$$ are equal iff they agree on $$\Omega^0(M)$$.
2. Show that $$\mathcal{L}_X$$ and $$i_X \circ d + d \circ i_X$$ are derivations on $$\Omega^{\bullet}(M)$$ commuting with $$d$$.
3. Show that $$\mathcal{L}_X f = Xf = i_Xdf+ d i_Xf$$ for all $$f \in C^{\infty}(M)=\Omega^0(M)$$.
It's easy to check 2&3, and here're my questions:
1. How to prove this lemma?
2. Why commuting with $$d$$ in this lemma is so important? Is there any counterexample?
3. Does this lemma still hold without restriction on degree?
4. Does this lemma still hold for general differential graded algebra?
• Take any Riemannian metric and the associated Levi-Civita $\nabla_X$. All such $\nabla_X$ obviously agree on $\Omega^0(M)$ because it is just the usual derivation $X$, but they won't necessarily agree on forms. So commuting with $d$ is there to stop this silly example. Since $\Omega^\bullet$ is generated by $df$ for $f\in\Omega^0$, some form of Leibniz rule allow you to conclude (1). – user10354138 Apr 21 '19 at 15:25
• @user10354138 Thank you. And do you have any ideas on question 3&4? As I said in the answer below, I wonder this method works only because we already know the structure og $\Omega^{\bullet}$ – Andrews Apr 21 '19 at 15:59
My thoughts in proving this lemma ($$\Leftarrow$$):
For this special case $$\Omega^{\bullet}(M)=\bigoplus_{k=0}^{n} \Omega^k(M)$$, let's first consider what $$\Omega^0(M)$$ and $$\Omega^1(M)$$ is.
In local chart $$(U,(x^i))$$, $$\Omega^0(U)=C^\infty(U)$$ is smooth functions on $$U$$, and $$\Omega^1(U)=\text{span}\{dx^i\}$$.
Two degree $$0$$ derivations $$D_1,D_2$$ commute with $$d$$, they agree on product.
Since $$\Omega^1(U)=\text{span}\{dx^i\}$$ and $$x^i \in \Omega^0(U)$$, if they agree on $$\Omega^0(U)$$, they agree on $$\Omega^1(U)$$.
And other $$\Omega^k(U)$$ can be generated by elements in $$\Omega^1(U)$$ via product, so $$D_1, D_2$$ agree on $$\Omega^{\bullet}(U)$$ thus $$\Omega^{\bullet}(M) \qquad\Box$$
I don't know if it holds for general cases, and I'm not sure where is degree $$0$$ used.
Maybe this proof works only because we already know the structure of $$\Omega^{\bullet}(M)$$. | 2020-10-24 03:06:52 | {"extraction_info": {"found_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": 46, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9632951617240906, "perplexity": 219.37368359966294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107881640.29/warc/CC-MAIN-20201024022853-20201024052853-00431.warc.gz"} |
https://research.snu.edu.in/publication/origin-and-temperature-dependence-of-the-electric-dipole-moment | X
Origin and temperature dependence of the electric dipole moment in niobium clusters
Kristopher Andersen E., Yoshiyuki Kawazoe, Warren Pickett E.
Published in
2006
Volume: 73
Issue: 12
Abstract
The origin of spontaneous electric dipole moments and uncoupled magnetic moments, observed in niobium clusters below a size dependent critical temperature, are explained using first-principles electronic structure calculations. The calculated dipole moments for NbN (N=2-15) generally agree with the experiment and support the interpretation that the electric dipole has a structural origin. A strong correlation is found between structural asymmetry, as quantified by the inertial moments and charge deformation density and the electric dipole. For clusters with odd N, magnetocrystalline anisotropy is small in comparison to the rotational energy of the cluster, such that the spin magnetic moment (1 $\mu$B) is uncoupled to the cluster. Two potential mechanisms to explain the temperature dependence of the electric dipole are investigated. The excitation of harmonic vibrations is unable to explain the observed temperature dependence. However, classical simulations of the deflection of a cluster in a molecular beam show that thermal averaging reduces the asymmetry of the deflection profile at higher temperatures, which may affect the experimental observation of the electric dipole and polarizability. An experimental test is proposed to ascertain the importance of this effect. {\textcopyright} 2006 The American Physical Society. | 2022-09-25 17:23:28 | {"extraction_info": {"found_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": 0.7885574102401733, "perplexity": 1206.2124447695714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030334591.19/warc/CC-MAIN-20220925162915-20220925192915-00564.warc.gz"} |
https://helpingwithmath.com/adjacent-angles/ | Home » Math Theory » Geometry » Adjacent Angles
## Introduction
Each of us has neighbors. Some have a common wall between their apartments, some have a common fence. It can be said that apartments or fences are adjacent to each other. Adjacent can be many things, including angles.
When we look at a watch with an hour, minute, and second hand, we see a pair of adjacent angles.
When we look at an open book its covers and one of the pages form a pair of adjacent angles.
When we ride a bike, the three adjacent spokes of the wheel form a pair of adjacent angles.
Two roads at the T-junction, a tram rail and a sleeper, two cross-street signs, open scissors, all these real-life objects form adjacent angles.
Having such a variety of adjacent angles in everyday life, it is worth studying and understanding which properties these angles have and what can be done with using these properties.
The word “adjacent” means “next” or “neighboring”. We can treat adjacent angles as angles that are next to each other.
Adjacent angles are a pair of angles that share a common side and vertex. Three things that need to be done to keep the angles adjacent:
• adjacent angles go in pairs;
• adjacent angles share the common arm;
• adjacent angles have the same vertex.
The following diagram shows two adjacent angles 1 and 2 with the common arm $\vec{OB}$ and common vertex O.
Every angle has one vertex and two arms, so it can have two possible adjacent angles, one attaching to each arm. In the diagram below, angles 2 and 3 are adjacent to angle 1: angles 1 and 2 share the common vertex O and common arm $\vec{OB}$, angles 1 and 3 share the common vertex O and common arm $\vec{OC}$.
The angle addition postulate states that if point B is in the interior of AOC, then
m∠AOB+m∠ BOC=m∠ AOC
This postulate can be applied to any pair of adjacent angles.
Are there any restrictions on the measures of the adjacent angles? Let us think. If two adjacent angles never overlap, then their total measure cannot be greater than the measure of a full angle. So, if angles 1 and 2 are adjacent, then
m∠1+m∠2≤360°
Thus, we can obtain the following combinations of adjacent angles:
• two acute angles;
• two right angles;
• two obtuse angles;
• two straight angles;
• one acute angle and one right angle;
• one acute angle and one obtuse angle;
• one acute angle and one straight angle;
• one acute angle and one reflex angle (if their total measure does not exceed 360°);
• one right angle and one obtuse angle;
• one right angle and one straight angle;
• one right angle and one reflex angle (if their total measure does not exceed 360°);
• one obtuse angle and one straight angle;
• one obtuse angle and one reflex angle (if their total measure does not exceed 360°).
Two reflex angles can never form a pair of adjacent angles. Each of two arbitrary reflex angles has a measure greater than 180°, so the sum of the measures of these two reflex angles is always greater than 360° and angles overlap.
We already understood what the adjacent angles are. Let us consider the following example. Will the angles AOC and AOB shown in the figure be adjacent? Why?
Formally, all the conditions of the definition are held:
• two angles AOC and AOB form a pair of angles;
• two angles AOC and AOB share the common vertex O;
• two angles AOC and AOB share the common arm $\vec{OA}$.
Are they adjacent? No, because they are overlapping. So, do not forget that two adjacent angles never overlap.
Determining whether two angles are adjacent or not, you should clearly remember three things that these angles should have: be a pair of angles, have a common vertex, and a common arm. It is best to learn to understand whether the angles are adjacent or not on specific examples.
EXAMPLE: In each case determine whether the angles 1 and 2 are adjacent or not.
SOLUTION: a) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent.
b) Pair of angles ∠1 and ∠2 share the common vertex O but do not share the common arm, so these angles are not adjacent.
c) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent.
d) Pair of angles ∠1 and ∠2 share the common arm $\vec{OB}$ but do not share the common vertex, so these angles are not adjacent.
e) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent.
f) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent.
g) Pair of angles ∠1 and ∠2 share the common vertex O but do not share the common arm, so these angles are not adjacent.
## Linear pair of angles
A linear pair of angles is a pair of adjacent angles formed when two lines intersect.
The above diagram shows two intersecting lines AC⃡ and BD⃡ which form four linear pairs of angles:
• angles 1 and 2;
• angles 2 and 3;
• angles 3 and 4;
• angles 4 and 1.
When two lines intersect, there are always four linear pairs of angles.
What is the same between a pair of adjacent angles and a linear pair of angles?
### What is different between a pair of adjacent angles and a linear pair of angles?
Supplementary angles are two angles whose measures add up to 180°.
Supplementary angles are not necessarily adjacent angles. But every two adjacent angles whose measures add up to 180° are supplementary. Using this fact, we can state that
each linear pair of angles is a pair of supplementary angles
Take an arbitrary linear pair of angles, for example, angles 1 and 2. These two angles together form the straight angle DOB. The measure of a straight angle is always 180°. Therefore, by angle addition postulate,
m∠1+m∠2=m∠DOB
If m∠DOB=180°, then
m∠1+m∠2=180°
By the definition of supplementary angles, angles 1 and 2 are supplementary angles. In the same way, we can prove that each linear pair of angles is a pair of supplementary angles.
Now, we can outline the following basic properties of a linear pair of angles:
• every linear pair of angles is a pair of adjacent angles but an arbitrary pair of adjacent angles is not necessary a linear pair of angles;
• every linear pair of angles share a common vertex and a common arm between them;
• every linear pair of angles always forms a straight angle;
• every linear pair of angles is a pair of supplementary angles.
Note that the interior angle of the triangle and the corresponding exterior angle of the triangle together form a linear pair of angles and are always supplementary.
## Complementary angles
Complementary angles are two angles whose measures add up to 90°.
Complementary angles are not necessarily adjacent angles. But every two adjacent angles whose measures add up to 90° are complementary.
In geometry, there are two types of complementary angles:
Two complementary angles with a common vertex and a common arm are called adjacent complementary angles.
In the figure given below, AOB and BOC are adjacent angles as they have a common vertex O and a common arm $\vec{OB}$.
They also together form the right angle AOC. Thus, the measures of angles AOB and BOC add up to 90° and these two angles are adjacent complementary angles.
Two complementary angles which are not adjacent are called non-adjacent complementary angles.
In the figure given below, AOB and COD are non-adjacent angles as they have a common vertex O and do not have a common arm.
The measures of angles AOB and COD add up to 90°, so, these two angles are complementary. Therefore, angles AOB and COD are non-adjacent complementary angles.
The following properties of adjacent angles help us identify whether the angles are adjacent and investigate these angles.
PROPERTIES:
1. Adjacent angles share a common arm.
2. Adjacent angles share a common vertex.
4. All common points of two adjacent angles lie on the common arm.
5. Adjacent angles can be complementary (their measures add up to 90°) or supplementary (their measures add up to 180°).
When two lines intersect, two pairs of opposite angles are formed. These angles are called vertical angles.
The above diagram shows two intersecting lines AC⃡ and BD⃡ which form two pairs of vertical angles:
• angles 1 and 3;
• angles 2 and 4.
### When two lines intersect, two vertical angles are always congruent.
This statement is known as the vertical angle theorem. Let us prove that angle 1 is congruent to angle 3.
Since angles 1 and 2 form a linear pair of angles, they are supplementary angles and
m∠1+m∠2=180°
Since angles 2 and 3 form a linear pair of angles, they are supplementary angles and
m∠2+m∠3=180°
By the transitive property, if m∠1+m∠2=180° and m∠2+m∠3=180° , then
m∠1+m∠2=m∠2+m∠3
Subtract m∠2 from both sides of the above equality:
m∠1=m∠3
Using the definition of congruent angles,
∠1≅∠3
In the same way, we can prove that the remaining two vertical angles are congruent too.
## FAQs
1. Which angles are called adjacent?
Adjacent angles are a pair of angles that share a common side and vertex. Three things that need to be done to keep the angles adjacent: adjacent angles go in pairs, adjacent angles share the common arm, and adjacent angles have the same vertex.
2. Could three angles be adjacent?
Three angles can be adjacent pairwise, but all together they are not adjacent, because adjacent angles are pairs of angles.
3. Can two obtuse angles be adjacent?
Any two obtuse angles that have a common vertex and a common arm are adjacent – the sum of their measures is less than 360°.
4. Can two reflex angles be adjacent?
Any two reflex angles with a common vertex and a common arm are not adjacent – the sum of their measures is greater than 360°.
5. Is the sum of the measures of two adjacent angles 180°?
No, the sum of the measures of two adjacent angles could be an arbitrary number of degrees, not necessarily 180°. To 180 degrees add up only two adjacent supplementary angles.
6. When two lines intersect, is there a relationship between two vertical angles and a linear pair of angles?
A linear pair of angles are always adjacent angles that add up to 180°. Two vertical angles are always opposite congruent angles, one of these angles belongs to a linear pair of angles. If you know the measures of one of the angles formed when two lines intersect, then you know the measures of all four angles.
7. Are the vertical angles adjacent?
Never. Vertical angles only have a common vertex but never have a common arm.
8. What is the difference between complementary and supplementary angles?
The measures of two complementary angles add up to 90° while the measures of two supplementary angles add up to 180°.
## Quiz
1. Two congruent angles are adjacent. What could be the maximum measure of each angle?
SOLUTION: If two angles are congruent, then they are both of the same type. We already know that two reflex angles cannot form a pair of adjacent angles as the sum of their measures is greater than 360°.
If these angles are not reflex, then consider the following by size angles – adjacent straight angles. When we take two congruent straight angles each of them has the measure of 180° and they add up to 360°. This is possible, so the maximum measure of each of two congruent adjacent angles could be 180°.
1. In each case determine whether angles 1 and 2 are adjacent or not.
SOLUTION: a) Pair of angles ∠1 and ∠2 share the common vertex O but do not share the common arm, so these angles are not adjacent.
b) Pair of angles ∠1 and ∠2 share the common vertex O but they overlap, so these angles are not adjacent.
c) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent.
d) Pair of angles ∠1 and ∠2 share the common vertex O and common arm $\vec{OB}$, so these angles are adjacent. Moreover, these angles are complementary.
1. In each case, state whether adjacent angles 1 and 2 are supplementary or not.
a) m∠1=45°, m∠2=135°;
b) m∠1=105°, m∠2=95°.
SOLUTION: Add the measures of the given angles. If the sum of the measures is 180°, then the given angles are supplementary, if the sum of the measures is not 180°, then the given angles are not supplementary.
a) m∠1+m∠2=45°+135°=180°
In this case, angles 1 and 2 are supplementary angles.
b) m∠1+m∠2=105°+95°=200°≠180°
In this case, angles 1 and 2 are not supplementary angles.
ANSWER: a) supplementary b) not supplementary
1. In each case, state whether adjacent angles 1 and 2 are complementary or not.
a) m∠1=64°, m∠2=36°;
b) m∠1=25°, m∠2=65°.
SOLUTION: Add the measures of the given angles. If the sum of the measures is 90°, then the given angles are complementary, if the sum of the measures is not 90°, then the given angles are not complementary.
a) m∠1+m∠2=64°+36°=100°≠90°
In this case, angles 1 and 2 are not complementary angles.
b) m∠1+m∠2=25°+65°=90°
In this case, angles 1 and 2 are not complementary angles.
ANSWER: a) not complementary b) complementary
1. Two lines a and b intersect. If every two adjacent angles are congruent, what is the measure of each of these angles? What can you say about lines a and b?
SOLUTION: If two lines a and b intersect, then they form four linear pairs of angles: ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, ∠4 and ∠1.
If every two adjacent angles are congruent, then
∠ 1≅ ∠2, ∠2≅∠3, ∠3≅∠4, ∠4≅∠1
By the transitive property,
∠1≅∠2≅∠3≅∠4
All these four angles form the full angle at the common vertex. Using the angle addition postulate, the measures of all four angles add up to 360°, so
m∠1+m∠2+m∠3+m∠4=360°
Since all four angles are congruent, they all have the same measures and the measure of each such angle is $\frac{1}{4}$360°=90°.
## Conclusions
1. Two adjacent angles “meet” by a common arm in a common vertex.
2. Two adjacent angles never overlap.
3. Names of two adjacent angles always have the same letter in the middle because they share the common vertex.
4. There are some special pairs of adjacent angles: complementary, supplementary, interior and corresponding exterior angles of a triangle, of a polygon.
5. There are pairs of angles that are never adjacent: pairs of vertical angles.
6. The measures of adjacent angles we add using the angle addition postulate – those measures add up to the measure of the whole angle formed by two non-adjacent arms of given angles.
7. Two reflex angles can never be adjacent. | 2022-05-18 23:16:42 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43802905082702637, "perplexity": 825.9932716843681}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662522556.18/warc/CC-MAIN-20220518215138-20220519005138-00513.warc.gz"} |
https://docs.flare.network/user/delegation/delegation-in-detail/ | # Delegation in detail#
## Background#
The vote power token is the base contract for WSGB (wrapped native token) and later for fAssets, which are wrapped assets on flare.
This token contract is built to enable delegation of vote power without locking the holder’s token. It works by adding the vote power and delegation functions to the ERC20 token contract. Basically, balance represents vote power; with the additional code, a holder can delegate a percentage of its own vote power to another address and still use their tokens freely. The transfer/mint/burn functions will immediately update the actual vote power being held by the delegator and the vote power of the address it delegates to.
All vote power data is being checkpointed by block. For any vote power update due to delegation, transfer, or otherwise, a checkpoint is added. For anyone familiar with the MiniMe token, the checkpoint mechanism is similar, while differing in that more data is being checkpointed.
When a voting campaign occurs, a past block will be randomly chosen and all vote power data will be taken from this block. This would actually work like taking a vote power snapshot for a specific block and using that for all addresses voting (providing prices) in the campaign. The random process of choosing a block is designed to mitigate attacks such as flash-loan or short term loans.
This token is named VPToken (vote power token).
## VPToken APIs#
• ERC20 APIs.
• BalanceOfAt(address, block), totalSupplyAt(block) as in MiniMe token.
• Delegation and vote power interfaces are described in IIVPToken interface file.
• Mint and burn APIs will be handled in the inheriting contracts. WNAT and later xAsset.
## Delegation#
Delegation enables a user to keep holding their balance (tokens) while delegating the vote power this balance represents. Two delegation methods are supported. The basic (normal) delegation is delegation by percentage, the other being explicit delegation.
With percentage delegation, any address can delegate a percentage of its holding; this is limited to x addresses. Example: Alice has 20 tokens and delegates 50% to Bob, Bob will have additional vote power of 10 on the top of his own balance (own vote power). This means any transfer of tokens to or from Alice will update 50% of the delegated vote power to Bob. If Alice delegates to another address, each token transfer to or from Alice will update the vote power of those other addresses. This in turn will cause higher gas costs for transfer functions. To cap those extra costs, this delegation option has a limited number of delegation destinations. In the case that an address (user or contract) wishes to delegate vote power to more addresses, they have the option of the explicit delegation method.
With explicit delegation, an explicit amount of vote power is delegated. While useful, this does create more complications for the user since the balance corresponding to the delegated vote power can’t be transferred. For example, if Alice has 20 tokens and explicitly delegates vote power of 20 to Bob, the delegated balance is actually locked. Alice can’t send out these tokens unless the 20 vote power is explicitly un-delegated. Another complication here is that for each new token received, a new delegate operation has to be performed; vote power will not be automatically delegated upon token reception.
The explicit delegation method is built for advanced users or for contracts holding a large number of tokens for different users. Imagine a collateral contract holding many WSGB for many users. Each user depositing tokens might want to delegate to a different set of price providers. Explicit delegation will enable this contract to update the explicit delegation per user deposit and un-delegate every time a user wishes to withdraw its funds.
Only one of the delegation methods can be used per address. Furthermore, an address can never change its delegation method. For example, if a user called delegate-explicit once from its address, they will never be able to do a percentage delegation with the same address.
The delegation system will support:
• Delegation of vote power to several addresses
• 1 level delegation. If Alice delegates to Bob and Bob delegates to Charlie, Charlie will only get the delegated balance of Bob, and will not be affected by the delegation Alice did.
Delegation units are the same as balance units.
### Check pointing historical data#
Token data regarding vote power, delegation, balance and supply is all checkpointed to allow the retrieval of historical values. Per change in any value, a checkpoint is added to the array which includes the updated value and the block number. When trying to read historical data, a binary search is performed on the array. With this, the data retrieval cost grows on a logarithmic scale.
### Vote power data#
The above delegation scheme creates a mapping from balance to vote power for each address. The vote power of each address reflects its own balance plus any delegated vote power from other addresses. Vote power should never be reused (double-spent): if vote power is delegated, the delegating address should not have this vote power under its own account.
### Voting campaigns using vote power token#
Checkpointed vote power data is used in voting campaigns (reward epochs). A voting campaign uses a randomly chosen block number from the past (vote power block). When an address (data provider) casts its vote for a specific campaign, its vote power is taken from the vote power block for this campaign. Hence, the vote power of an address for this campaign does not reflect its present balance and delegation but rather the state at the time of the snapshot (in the vote power block). This design allows for a free use of tokens (non-locked) and a consistent vote power snapshot of token holdings. Voting campaigns are a generic concept; in FTSO system, the vote power of price providers is used as an influence in choosing the “correct price”. Each price submission is weighted according to the vote power scheme described here.
## Vote power caching and the revoke feature#
Due to reward distribution constraints that are described in the reward manager specification, the same vote power block is used for a rather long period of time. This time frame will be named a “reward epoch” which will include many short price epochs. Meaning, FTSO price feeds commencing over a period of a few days will continuously derive vote power from the same vote power block in the past.
Usage of the same vote power block for many campaigns calls for a caching mechanism. The caching mechanism caches the vote power per address per block if done through a dedicated caching enabled function. For example, the normal vote power query function is votePowerOfAt(address, block). This has a matching cache query: votePowerOfAtCache(address, block) which will also cache the data on its first usage for a specific address and block. Later calls to both of these functions will use the cached value if that exists.
### Revoke#
Due to the substantial length of time one past vote power block is used for price submissions, a revoke feature was added. This feature can be used in case any specific price provider is found trying to attack and skew the reported price of the FTSO (flare oracle). In this situation, we imagine an off chain process (e.g. twitter storm) calling users to revoke vote power from a specific price provider. The revoke will update the cached value of the vote power for the specific block which is being used for this reward epoch. So if a user revokes its vote power delegation on a specific block, checkpoints for the vote power will not be updated, only the cached vote power values will be zeroed
### Vote power delegation and rewarding delegators#
A large part of the native token inflation is distributed to participants in the FTSO price submission process. The reward won by a price provider is shared between the price provider and the vote power delegators to the price provider (more on that in the FTSO and reward manager docs). The VP token exposes APIs that enable delegators to see how much vote power was delegated to a price provider in any past block. To enable this, the delegation percentage data are checkpointed after every change. Using the combination of delegation percentage and historical balance, each user can accurately see and show how much vote power they delegated to any address in the past.
This API is also used by the reward manager, when the reward sharing is calculated.
For explicit delegation, historical data is limited. It would be quite costly to continuously update a list of independent explicit delegations. That being said, when rewards are claimed for addresses that used explicit delegation, the delegator must already know which data providers it delegated vote power to in the relevant block.
To recap, historical delegation APIs exist. For percentage delegations, each address can determine the full list of addresses it delegated to in any block in history. For explicit delegations, a user must use their own methods to build the list of addresses it delegated to at specific times. After building this list, one can query how much vote power was delegated to each address. Two options for building this list would be:
1. Saving this data in real time while delegating.
Reward epoch with index 10 started at block 2487672 with timestamp 1637397708 (Saturday, November 20, 2021 8:41:48 AM GMT) and lasted until block 3003881 with timestamp 1638002503 (Saturday, November 27, 2021 8:41:43 AM GMT). This means that 516209 blocks were mined in this epoch and the last quarter of the blocks started with the block number 2487672 + 516209 * 3 / 4 = 2874828 with timestamp 1637817710 (Thursday, November 25, 2021 5:21:50 AM GMT). Any block between 2874828 and 3003881 is therefore eligible for selection as the vote power block. In this reward epoch, block 2881097 with timestamp 1637825442 (Thursday, November 25, 2021 7:30:42 AM GMT) was selected. This is before the last quarter of the week (Thursday, November 25, 2021 2:41:33 PM GMT) if we were to take the timestamp measure. | 2022-06-29 22:17:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.21674850583076477, "perplexity": 3635.852735765653}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656103645173.39/warc/CC-MAIN-20220629211420-20220630001420-00436.warc.gz"} |
https://discuss.codechef.com/t/dearrange-editorial/101329 | DEARRANGE-Editorial
Setter: Utkarsh Gupta
Tester: Abhinav Sharma, Lavish Gupta
Editorialist: Devendra Singh
2882
PROBLEM:
You are given a permutation P of length N. A permutation of length N is an array where every element from 1 to N occurs exactly once.
You must perform some operations on the array to make it sorted in increasing order. In one operation, you must:
• Select two indices L and R (1 \leq L \lt R \leq N)
• Completely dearrange the subarray P_L, P_{L+1}, \ldots P_R
A dearrangement of an array A is any permutation B of A for which B_i \neq A_i for all i.
For example, consider the array A = [2, 1, 3, 4]. Some examples of dearrangements of A are [1, 2, 4, 3], [3, 2, 4, 1] and [4, 3, 2, 1]. [3, 5, 2, 1] is not a valid dearrangement since it is not a permutation of A. [1, 2, 3, 4] is not a valid dearrangement either since B_3 = A_3 and B_4 = A_4.
Find the minimum number of operations required to sort the array in increasing order. It is guaranteed that the given permutation can be sorted in atmost 1000 operations.
HINT
The answer is always less than or equal to 2
EXPLANATION:
If the permutation P is already sorted the answer is 0. Let us look at the case when we can sort the permutation by just one move. If there exists a subarray P[L,R] such that for each index i in the prefix P[1,L-1], P_i=i and for each index j in the suffix P[R+1,N], P_j=j and for all index k in the subarray P[L,R], P_k!=k. Then we can derange the subarray P[L,R] to achieve the sorted premutation. Otherwise we can always do just 2 operations to obtain the sorted permutation.
By Brute Force we can check that for all permutations of size 4 or greater there exists a derangement such that for no index i, P_i=i in the resulting derangement. Now, apply the operation on the whole permutation P. To obtain the derangement of P such that for no index i P_i=i in the derangement we break the array into size 4 consecutive subarrays starting from the index 1. Merge the last subarray to the second last subarray if it has a size less than 4. Now iterate over all permutations possible of the elements of these 4 size subarrays and check whether the permutation is a derangement as well as for no index i in the array we get P_i=i by the derangement. This can be easily implemented using the next_permutaion and single for loop in C++.
The only case that remains is when N=3.The only edge case in which an answer exists and it is not possible to get the answer using the approach defined above is P=[3,2,1]. This case can be solved beforehand and taken into account.
2
1\: 2
2 \: 3\: 1
1 \: 3
1\: 2\: 3
TIME COMPLEXITY:
O(N) or O(N^2) depending on implementation for each test case.
SOLUTION:
Setter's solution
//Utkarsh.25dec
#include <bits/stdc++.h>
#define ll long long int
#define pb push_back
#define mp make_pair
#define mod 1000000007
#define vl vector <ll>
#define all(c) (c).begin(),(c).end()
using namespace std;
ll power(ll a,ll b) {ll res=1;a%=mod; assert(b>=0); for(;b;b>>=1){if(b&1)res=res*a%mod;a=a*a%mod;}return res;}
ll modInverse(ll a){return power(a,mod-2);}
const int N=500023;
bool vis[N];
long long readInt(long long l,long long r,char endd){
long long x=0;
int cnt=0;
int fi=-1;
bool is_neg=false;
while(true){
char g=getchar();
if(g=='-'){
assert(fi==-1);
is_neg=true;
continue;
}
if('0'<=g && g<='9'){
x*=10;
x+=g-'0';
if(cnt==0){
fi=g-'0';
}
cnt++;
assert(fi!=0 || cnt==1);
assert(fi!=0 || is_neg==false);
assert(!(cnt>19 || ( cnt==19 && fi>1) ));
} else if(g==endd){
if(is_neg){
x= -x;
}
if(!(l <= x && x <= r))
{
cerr << l << ' ' << r << ' ' << x << '\n';
assert(1 == 0);
}
return x;
} else {
assert(false);
}
}
}
string ret="";
int cnt=0;
while(true){
char g=getchar();
assert(g!=-1);
if(g==endd){
break;
}
cnt++;
ret+=g;
}
assert(l<=cnt && cnt<=r);
return ret;
}
long long readIntSp(long long l,long long r){
}
long long readIntLn(long long l,long long r){
}
}
}
ll myrand(ll l,ll r)
{
ll temp=rng();
temp=abs(temp);
temp%=(r-l+1);
temp+=l;
return temp;
}
int sumN=0;
void solve()
{
sumN+=n;
assert(sumN<=1000);
int A[n+1]={0};
int mark[n+1]={0};
for(int i=1;i<=n;i++)
{
if(i==n)
else
mark[A[i]]=1;
}
for(int i=1;i<=n;i++)
assert(mark[i]==1);
int l=n+1,r=0;
for(int i=1;i<=n;i++)
{
if(A[i]!=i)
r=i;
}
for(int i=n;i>=1;i--)
{
if(A[i]!=i)
l=i;
}
if(l>r)
{
cout<<0<<'\n';
return;
}
for(int i=l;i<=r;i++)
{
if(A[i]==i)
{
cout<<2<<'\n';
cout<<1<<' '<<n<<'\n';
vector <int> v;
for(int i=1;i<=n;i++)
v.pb(i);
int iter=0;
while(true)
{
// iter++;
// if(iter>=100)
// {
// cout<<n<<'\n';
// for(int i=1;i<=n;i++)
// cout<<A[i]<<' ';
// cout<<'\n';
// exit(0);
// }
shuffle(all(v),rng);
int flag=1;
for(int i=0;i<n;i++)
{
if(i==0)
{
if(v[i]==A[i+1])
{
flag=0;
break;
}
else
continue;
}
if(v[i]==i+1 || v[i]==A[i+1])
{
flag=0;
break;
}
}
if(flag==1)
break;
}
for(auto it:v)
cout<<it<<' ';
cout<<'\n';
if(v[0]!=1)
cout<<1<<' '<<n<<'\n';
else
cout<<2<<' '<<n<<'\n';
for(int i=1;i<=n;i++)
cout<<i<<' ';
cout<<'\n';
return;
}
}
cout<<1<<'\n';
cout<<l<<' '<<r<<'\n';
for(int i=1;i<=n;i++)
cout<<i<<' ';
cout<<'\n';
}
int main()
{
#ifndef ONLINE_JUDGE
freopen("input.txt", "r", stdin);
freopen("output.txt", "w", stdout);
#endif
ios_base::sync_with_stdio(false);
cin.tie(NULL),cout.tie(NULL);
while(T--)
solve();
assert(getchar()==-1);
cerr << "Time : " << 1000 * ((double)clock()) / (double)CLOCKS_PER_SEC << "ms\n";
}
Editorialist's solution
#include "bits/stdc++.h"
using namespace std;
#define ll long long
#define pb push_back
#define all(_obj) _obj.begin(), _obj.end()
#define F first
#define S second
#define pll pair<ll, ll>
#define vll vector<ll>
ll INF = 1e18;
const int N = 1e5 + 11, mod = 1e9 + 7;
ll max(ll a, ll b) { return ((a > b) ? a : b); }
ll min(ll a, ll b) { return ((a > b) ? b : a); }
void sol(void)
{
int n;
cin >> n;
vll v(n), vv(n);
vll check1(3);
check1 = {3, 2, 1};
for (int i = 0; i < n; i++)
{
cin >> v[i];
vv[i] = v[i];
}
if (v == check1)
{
cout << 2 << ' ';
cout << 1 << ' ' << 2 << '\n';
cout << 2 << ' ' << 3 << ' ' << 1 << '\n';
cout << 1 << ' ' << 3 << '\n';
cout << 1 << ' ' << 2 << ' ' << 3 << '\n';
return;
}
//------------- Case when answer is 0
sort(all(vv));
if (vv == v)
{
cout << 0 << '\n';
return;
}
//------------- Case when answer is 1
int issortedtill[n], issortedfrom[n];
if (v[0] == 1)
issortedtill[0] = true;
else
issortedtill[0] = false;
for (int i = 1; i < n; i++)
if (issortedtill[i - 1] && v[i] == i + 1)
issortedtill[i] = true;
else
issortedtill[i] = false;
if (v[n - 1] == n)
issortedfrom[n - 1] = true;
else
issortedfrom[n - 1] = false;
for (int i = n - 2; i >= 0; i--)
if (issortedfrom[i + 1] && v[i] == i + 1)
issortedfrom[i] = true;
else
issortedfrom[i] = false;
for (int i = 0; i < n; i++)
{
bool flag = true;
for (int j = i; j < n; j++)
{
if (v[j] == j + 1)
flag = false;
if (flag && (j == n - 1 || issortedfrom[j + 1]) && (!i || issortedtill[i - 1]))
{
cout << 1 << '\n';
cout << i + 1 << ' ' << j + 1 << '\n';
for (auto x : vv)
cout << x << ' ';
cout << '\n';
return;
}
}
}
//------------- Case when answer is 2
cout << 2 << '\n';
vector<int> tillnow;
for (int i = 0; i < n;)
{
vector<int> temp;
if (n - i < 8)
for (int j = i; j < n; j++)
temp.pb(v[j]);
else
for (int j = i; j <= i + 3; j++)
temp.pb(v[j]);
sort(all(temp));
bool flag = true;
do
{
flag = true;
for (int j = 0; j < temp.size(); j++)
{
if (v[i + j] == temp[j] || temp[j] == i + j + 1)
flag = false;
}
if (flag)
break;
} while (next_permutation(all(temp)));
for (auto x : temp)
tillnow.pb(x);
i += temp.size();
}
cout << 1 << ' ' << n << '\n';
for (auto x : tillnow)
cout << x << ' ';
cout << '\n';
cout << 1 << ' ' << n << '\n';
for (int i = 1; i <= n; i++)
cout << i << ' ';
return;
}
int main()
{
ios_base::sync_with_stdio(false);
cin.tie(NULL), cout.tie(NULL);
int test = 1;
cin >> test;
while (test--)
sol();
}
1 Like
An alternative approach to the problem : https://www.codechef.com/START41A/problems/DEARRANGE
Read the editorial’s solution for the test cases in which the answer will be 0 or 1 and also one edge test case.
Now, for case when ans=2 :
Rotate the array by 1, n times and check whether the current rotated array is valid or not each time. (Valid array: If it does not match with any index of the given array and also with the sorted array).
Now, there may be a possibility that none of the rotated arrays is valid. Think of the case when will it be possible
Since after each rotation, so there must be at least one index in each rotation that matches with the given array or with the sorted array, so all the indices follow some fixed order (i.e. after $x$ rotations $yth$ indexed element will make the array invalid).
How \, can \, we \, disturb \, this \, order?
Yes, it’s easy, swap any two elements to disturb the order → Just find two elements that do not match with their index, and also they should not match with their index after the swap.
The order is disturbed, now we can surely say that at least in one of the n rotations we will get the valid array.
In 2nd step, just print the sorted array.
Submission : https://ideone.com/4S9T4y
3 Likes
if(a[ids[i]]==j || a[ids[j]]==i)
{
swap(a[ids[i]],a[ids[j]]),done=1;
break;
}
Can you explain this if condition for swapping in your code?
Also i tried to implement your logic by doing this:
→ if we have an array where each rotation is invalid then this implies in each rotation some element is going to its required position in sorted array. And these elements are going to be distinct for each rotation. This implies we can calculate distances for each element with respect to its position in current permutation and sorted array and this distance array must be a permutation of [0,n-1]
→ now if this distance array is the permutation of [0,n-1] then we need to swap and update distance array
→ now for rotation we want to make sure that no element comes back to its sorted position so we need to make some rotation which is not present in the distance array. Also if we did the swapping above then if we swapped (i,j) we can’t rotate by j-i since it won’t conflict with sorted array but it will with initial permutation. So we need to find such rotation value and rotate
Do you think there is a mistake here? | 2022-07-06 01:14:11 | {"extraction_info": {"found_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": 0.6801357865333557, "perplexity": 3810.587985087508}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104655865.86/warc/CC-MAIN-20220705235755-20220706025755-00024.warc.gz"} |
https://psychology.stackexchange.com/tags/measurement/new | # Tag Info
4
There are four variables of interest in a statistical power analysis: power (often chosen as 0.8 or 0.9 by convention, this is the probability of correctly rejecting the null hypothesis if it is false), tolerated false-positive rate (alpha, often chosen as 0.05 by convention), sample size, and the effect size (which in turn depends on the variability of the ...
0
It seems that specifics on the heredity of IQ are largely unknown today. That same wiki page says, Intelligence in the normal range is a polygenic trait, meaning that it is influenced by more than one gene, and in the case of intelligence at least 500 genes. The frequency and disconnection of studies like this and this shows just how uncertain this field ...
3
Attempting to reduce emotional responses to mathematics is likely in the realm of chaos theory. Here is a great Robert Sapolsky Stanford lecture on Chaos versus Reductionism. https://youtu.be/_njf8jwEGRo
Top 50 recent answers are included | 2020-09-18 23:23:01 | {"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": 0.8074240684509277, "perplexity": 1276.5656122646442}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400189264.5/warc/CC-MAIN-20200918221856-20200919011856-00669.warc.gz"} |
http://mathoverflow.net/feeds/user/7276 | User oren - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:05:47Z http://mathoverflow.net/feeds/user/7276 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90340/numerically-equivalent-effective-divisors-and-semiampleness Numerically equivalent effective divisors and semiampleness Oren 2012-03-06T09:50:08Z 2012-04-01T15:22:00Z <p>Recall that a divisor $M$ on a variety $X$ is said to be semiample if $kM$ is base point free for a certain $k > 0$.</p> <p>Being semiample is not a numerical property (take for example torsion and a non-torsion divisor of degree 0 on a curve, or for more sophisticated examples just look at Lazarsfeld - Positivity in Algebraic Geometry II - Ex. 10.3.3), therefore I was wondering:</p> <p>It is possible to find a smooth projective variety $X$ and two effective divisors $E,D$ on $X$ such that $E \equiv D$, but $E$ is semiample while $H^0(X,kD)=\mathbb{C}$ for every $k$?</p> http://mathoverflow.net/questions/46224/numerically-rigid-nef-divisor Numerically rigid nef divisor Oren 2010-11-16T11:27:04Z 2010-11-16T14:02:42Z <p>Is it possible to find an example of an $\mathbb{R}$-Cartier divisor $D$ on an irreducible variety $X$ that is non-trivial, nef, effective and numerically rigid? </p> <p>By "numerically rigid" I mean that if $E$ is another $\mathbb{R}$-Cartier effective divisor such that $E$ is numerically equivalent to $D$ then $D=E$.</p> <p>For curves this clearly cannot be the case, since an effective non-trivial divisor is always ample.</p> http://mathoverflow.net/questions/30299/volume-of-big-line-bundles-under-finite-morphisms volume of big line bundles under finite morphisms Oren 2010-07-02T12:22:31Z 2010-08-10T16:18:36Z <p>Let $X$, $Y$ be complex projective varieties of dimension $n$, let $f:X \rightarrow Y$ be a surjective finite morphism of degree $d$ and let $B$ be a big line bundle on $Y$.</p> <p>Is that true that vol($f^*B$)=d $\cdot$ vol($B$)?</p> <p>(I know that if $B$ is not only big but also nef then the formula is true by Lazarsfeld's Positivity in Algebraic geometry I, remark 1.1.13, using the well-known fact that if $B$ is nef then vol($B$)=$B^n$). </p> http://mathoverflow.net/questions/90340/numerically-equivalent-effective-divisors-and-semiampleness/90356#90356 Comment by Oren Oren 2012-03-07T07:10:25Z 2012-03-07T07:10:25Z Thanks Ulrich, yes that may be the problem. The example I refer to (that you can find also in Ein et al. "Asymptotic Invariants of base loci" Ex. 1.1, <a href="http://arxiv.org/abs/math/0308116" rel="nofollow">arxiv.org/abs/math/0308116</a> ) involves reducible divisors. http://mathoverflow.net/questions/90340/numerically-equivalent-effective-divisors-and-semiampleness/90356#90356 Comment by Oren Oren 2012-03-06T14:20:40Z 2012-03-06T14:20:40Z misprint: $E' \equiv D'$. http://mathoverflow.net/questions/90340/numerically-equivalent-effective-divisors-and-semiampleness/90356#90356 Comment by Oren Oren 2012-03-06T14:19:08Z 2012-03-06T14:19:08Z I think that there is something wrong with your proof: in fact you say that $E' \cong D'$, hence in particular $D'$ is ample. Therefore since you say that $D=f^*(D')$, we have that $D$ is necessarily semiample. But this is not true in general: there exist examples of pairs of linearly equivalent divisors such that one is semiample and the other is not. | 2013-05-25 17:05:59 | {"extraction_info": {"found_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": 0.9265891313552856, "perplexity": 702.8720733380759}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368705976722/warc/CC-MAIN-20130516120616-00039-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://docslib.org/doc/4903292/radiometric-dating-by-alpha-spe-on-uranium-series-nuclides | <<
RADIOMETRIC DATING BY ALPHA SPE ON SERIES
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ALBERT VAN DER WIJK
','? AF<» i „ y :.;;* •••• t >«, .1.1 STELLINGEN
1. Hoewel er de laatste jaren enige vooruitgang is geboekt, is de U/Th niet-evenwicht datering van fossiele botten nog met zoveel onzekerheden omringd dat de resultaten ervan uitsluitend in samen- hang met onafhankelijke ondersteunende argumenten en slechts met de nodige reserve en terughoudendheid mogen worden gepresenteerd en gebruikt. A.M. Rae and M. Ivanovich, Appl. Geochemistry 1: 419-426. A.M. Rae et al., J. Arch. Sei. 14, 1987: 243-250. Dit proefschrift.
2. Er bestaat een essentieel verschil in methodologie tussen de exac- te en sociale wetenschappen: waar het in de exacte wetenschappen een goed gebruik is om een gestelde hypothese te bewijzen door experimenten uit te voeren die opgezet zijn om het tegendeel te bewijzen, is het in de sociale wetenschappen juist vaak de uit- zondering die de regel bevestigt.
3. Het bij radiometrische dateringen presenteren van onzekerheids- intervallen die uitsluitend gebaseerd zijn op de standaarddeviatie resulterend uit de bepaling van de radioactiviteit suggereert vaak een te grote nauwkeurigheid en leidt tot verwarring en onzekerheid bij archeologen. 2nd International Symposium on and 1'*C, General discussion. Groningen, September 7-11, 1987.
4. Het feit dat er, ondanks de daarvoor berekende extreem lage kansen, leven is ontstaan uit niet-levende componenten en er in Tsjernobyl een ernstig ongeluk met een kerncentrale is gebeurd, toont de ge- ringe voorspellende waarde van kansberekening aan.
5. De druk die met een krijtje op een schoolbord moet worden uitge- oefend om voldoende contrast te creëren is doorgaans groter dan die, waartegen het krijtje nog juist bestand is. 6. De functie-aanduiding "Assistent in opleiding" voor promovendus nieuwe stijl is misleidend en onjuist en dient vervangen te worden door de functie-aanduiding "Assistent, in opleiding".
7. Een algemeen fysisch colloquium, bedoeld voor een breed fysisch publiek, dient voorzien te zijn van een titel die op zijn minst de suggestie wekt dat de inhoud begrijpelijk is voor een ieder met een academische graad in de natuurkunde.
8. Het feit dat, naast uitgaven in andere talen, de maandbladen Play- boy en Penthouse ook in de Nederlandse taal worden uitgegeven, wekt de suggestie dat het (potentiële) kopers van deze bladen meer om de tekst dan om de afbeeldingen zou gaan.
9. Door het bij het van overheidswege opgelegde bezuinigingsbeleid consequent toepassen van het "kaasschaafmodel" snijdt de universi- teit zich uiteindelijk zelf in de vingers.
A. van der Wijk Groningen, 6 november 1987 RADIOMETRIC DATING BY ALPHA SPECTROMETRY ON URANIUM SERIES NUCUDES RIJKSUNIVERSITEIT GRONINGEN
RADIOMETRIC DATING BY ALPHA SPECTROMETRY ON URANIUM SERIES NUCLIDES
Proefschrift
ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen OP gezag van de Rector Magnificus Dr. E. Bleumink in het openbaar te verdedigen op vrijdag 6 november 1987 des namiddags te 4.00 uur
door
ALBERT VAN DER WIJK geboren te Nijverdal PROMOTOR: Prof.Dr. W.G. Mook
Layout: H.E. Deenen Drawings: E. de Jonge Photographs: H.L. Leertouwer has left us older And wiser I know I am
Carole Bayer Sager, 1975
Voor Anja CONTENTS
page
INTRODUCTION -, 1
CHAPTER 1: THE DECAY SERIES AND ABSOLUTE 15
1.1. Introduction -,5 1.2. Decay series and radioactive (dis)equilibrium 15 1.2.1. Introduction 15 1.2.2. Equilibrium and disequilibrium 17 1.2.3. Mathematics of a decay series 1g 1.2.4. Processes that cause disequilibrium 20 1.3. 23oTh/234U disequilibrium dating 21 1.3 .1 Introduction 21 1.3 .2 Application of UTD to peat 23 1.3 .3 Correction for environmental Th 26 1.3.3.1. by laboratory procedures 26 1.3.3.2. In-situ contamination 27 1.3.4. Validity of models 28 1.4. zl0Pb dating 28 1.4.1. Introduction 28 1.4.2. CRS and CIC age calculation models 29 1.4.3. Other models 32 1.4.4. Validity of models 33
CHAPTER 2: ANALYTICAL TECHNIQUES 35
2.1. Introduction 35 2.2. Alpha spectrometer _, Jo
2.2.1. Description _g
2.2.2. Performance . Q 2.2.2.1. Electronics .. 2.2.2.2. Detectors .. page 2.2.2.3. Background 4 0 2.3. Chemical treatment for alpha spectrometry 46 2.3.1. Introduction 46 2.3.2. Standard solutions and spikes 4 7 2.3.3. Material of high organic content 48 2.3.3.1. U/Th analyses 48 2.3.3.2. 210Pb analyses 51 2.3.4. Inorganic material 51 2.3.5. Purification and source preparation for U and Th 51 2.3.6. Source preparation for 210Pb determination 54 2.3.7. Blanks recovery and memory effects 54 2.4. Radon measurement 55 2.4.1. General outline of the method 55 2.4.2. Description 55 2.4.2.1. Extraction system 55 2.4.2.2. Radon transfer system 5 7 2.4.2.3. Electronics 57 2.4.2.4. Calculation of 222Rn and 226Ra 59 activities 2.4.3. Performance 61 2.4.3.1. Background and blank activity 61 2.4.3.2. Efficiency 62 2.4.3.3. Memory effect 62
CHAPTER 3: U/Th DISEQUILIBRIUM DATING 6 3 3.1. Introduction 63 3.2. Peat 64 3.2.1. Introduction 64 3.2.2. Site selection 65 3.2.3. Closed system behaviour 68 3.2.3.1. Preliminary measurements 68 3.2.3.2. Preliminary discussion: Geochemistry 69 of U and Th 3.2.3.3. Preliminary conclusions 74 page 3.2.4. Two-phase separation 75 3.2.4.1. Measurements 75 3.2.4.2. Discussion 75 3.2.4.3. Conclusions 83 3.2.5. Mechanisms for selective uptake of 232Th 83 3.2.5.1. Recoil effect: depletion of the 83 grain surface 3.2.5.2. Leaching of the inorganic fraction: 84 steady state 3.2.5.3. Experimental observations 87 3.2.6. Discussion of UTD dates on various peat samples 87 3.2.7. Conclusions 97 3.3. Corals 99 3.3.1. Background 9^ 3.3.2. Measurements 101 3.3.3. Discussion and conclusions 101 3.4. bones 103 3.4.1. Background 103 3.4.2. Measurements 103 3.4.3. Discussion and conclusions 105 3.5. Summary 106
CHAPTER 4: 21DPb DATING 109 4.1. Introduction 1°9 4.2. Constant rate of supply: closed-system behaviour 110 4.2.1. Constant rate of supply 110 4.2.2. Closed-system behaviour 1^ 4.3. zl0Pb dating in shallow moorland pools 117 4.3.1. Introduct 11 7 4.3.2. Site description and coring 117 4.3.3. Results 119 4.3.4. Discussion ''° 4.3.5. Conclusions 129 4.4. Miscellaneous case studies 130 10 page 4.4.1. Denmark 13 0 4.4.1.1. Introduction and site description 130 4.4.1.2. Results and discussion 130 4.4.2. Antarctica 132 4.4.2.1. Introduction 132 4.4.2.2. Results and discussion 132 4.5. Summary 134
CHAPTER 5: CHERNOBYL 137 5.1. Introduction 13 7 5.2. Gamma spectrometry 138 5.2.1. Introduction 138 5.2.2. Description 139 5.2.3. Calibration 140 5.3. Core fragments from Chernobyl fallout 140 5.3.1. Introduction 140 5.3.2. Localization and collection of hot spots 141 5.3.3. Measurements and results 143 5.3.4. Conclusions 145
REFERENCES 147
SAMENVATTING 15 7
NAWOORD 16 3 11
INTRODUCTION
CHAPTER 1. THE DECAY SERIES AND ABSOLUTE CHRONOLOGY
1.1. INTRODUCTION The dating techniques discussed in this thesis are based on the fact that in a decay series isotope concentrations develop with time according to the law of radioactivity to a state of secular equili- brium, either by accumulation of deficient or by decay of excess nu- clides. For any natural closed system (i.e. the concentrations of the various are determined by processes of and growth only) that contains nuclides from a decay series, the degree of disequilibrium, combined with knowledge of initial conditions pro- vides information on the age of the system. This chapter discusses the underlying principles as well as the mathematics of such so- called U series disequilibrium (USD) dating methods. Furthermore, the principles of both the 23OTh/23ItU and the 210Pb dating method and their applications, being the main subjects of this thesis, will be discussed in detail.
1.2. DECAY SERIES AND RADIOACTIVE (DI3)EQUILIBRIUM 1.2.1. INTRODUCTION A radioactive decay series is generated by a physically unstable primordial nucleus, that decays through a number of intermediate radioactive daughter products to a stable nucleus, the end member. The four genetically independent decay series, initiated by 238U, 237Np, 235U and 232Th respectively, together represent all the alpha- emitting nuclides. The two U and the Th decay series all have a stable lead isotope as an end member while the 237Np series ends with 209Bi. The abundances of the isotopes of all elements with atomic numbers varying between Z = 80 (Hg) and Z = 93 (Np), including the actinides as the heaviest naturally occurring elements, are the result of a combination of radioactive decay and growth processes. Of the four decay series mentioned above the one initiated by cr\
Mass Number A=4n Mass Number A=4n+2 Mass Number A=4n+3
u 235 u 92 5 4.510^ 2.510 a 7.110ea # 234 pa Pa 231Pa 91 1.18 m i 231 232 228 Th'' Th Th Th 227Th 25.6d 90 1.3910'°a 24.1d 7.5210'a
228 Ac Ac 227Ac 89 6.13 h 22a i t
228 224 226 Ra Ra Ra Ra 223Ra 88 5.75a 3.64 d 16C2a 11.7d
Fr 87 21m
220Rn 222 Rn Rn 2I9Rn 86 55.6s 3.83 d 3.9 s
At 219At 85
216 212 2l8 2IOp Po Po Po Po 0 215Po 2"Po 84 0145s 7 3.10" s 3.05 m 1.610"'s 138d LSIffV 0.52 s f / 2!2BJ Bi 2l0Bi 215Bi 211 Bi 83 60.5 m 19.7m 5.02 d 8m 2J6m f f
212 / 208pb 214 2.0pb" 206 Pb Pb Pb pb 21Vb 207Pb 82 10.6 h stable 26.8 m 22.3 a stable 36.1m stable 1
Tl 208T, ' 2,0T( 207T| ' 81 3.1m 132 m 6.79 m
Figure 1.1. Details of the chart of isotopes showing the three decay series which contain between them all naturally occurring alpha-emitting radionuclides. 17 237Np does no longer occur in nature due to the fact that the half life of its longest lived member is only a fraction of the age of the universe so that the entire series has decayed by now (although negligible trace quantities of 237Np may still be found in uranium ores due to -induced transmutation reactions). Figure 1.1. shows the three natural occurring series.
1.2.2. EQUILIBRIUM AND DISEQUILIBRIUM When the production and removal of nuclei in a radioactive decay series is the result of radioactive decay only, the time development of the number of nuclides N. of any isotope i in the series is given by competition between its radioactive decay (-A.N.) and by production from the precursor (A._.N._ )
dN. "dF = ~XiNi + Xi lNi-l (i=1,-..,n) (1.1) with the boundary condition
N. (t = 0) = N° (i = 1,...,n) (1.2)
X. (s"1) is the decay constant of the i-th member in the series (i.e. the probability per unit time that a nucleus will decay,related to the half life t by the expression X. = ln(2)/tii). By definition the n-th member (i.e. the end member) is stable, hence X =0. Further- n more, i = 1 denotes the primordial nucleus of the chain (which implies that N does not exist). o Before going into details of the mathematical solution to the set of n coupled linear differential equations represented by (1.1) and (1.2), one may notice the following general properties of a decay series: (i) The number of nuclides of the primordial parent will gradually disappear due to the negative sign of dN^/dt, (ii) If for each daughter product in the series production from the precursor balances radioactive decay, a steady-state (dN./dt = 0) is reached. By definition the activity is given by A= XN. Hence 18 a steady state implies that the activity of the daughter and its precursor are equal and do not change in time. However, it is easy to see that in a finite time interval such a state of so- called secular equilibrium can only be approached and never be reached due to the fact that the primordial nucleus can never have a steady state (remark (i)). Asa result its daughter product can never reach a steady state, etc. On the other hand, when A. A. >> 1 a state of quasi equilibrium may be reached where the activity ratio of mother and daughter nuclei, is given by
T-1" = T ^ d-3)
Wherever in the following the expression secular equilibrium is used it should be understood that we refer to this state of quasi-equili- brium. Furthermore, we prefer to express the abundances of nuclides in terms of activity ratios rather than in concentration ratios. The reason for this is that activity ratios can always be expressed as convenient numbers (between zero and one in a developing decay series), whereas the ratio between concentrations is proportional to the half lives of the nuclides, often yielding inconveniently large or small numbers.
1.2.3. MATHEMATICS OF A DECAY SERIES Eqs. (1.1) and (1.2) describe the time development of the number of nuclei of any isotope i in a radioactive decay series by means of n coupled linear inhomogeneous differential equations. The general solution of any of these equations is the summation of the general solution of the homogeneous equation
dN./dt + X.N = 0 (1.4) given by -At N. = C. .e i (1.5) i 1,1 and a particular solution of the inhomogeneous equation, for which we use the trial function N.(t) = z C. (1.6) 3=1 l Substituting expression (1.6) in eq. (1.1) and solving for the j-th term yields
(j < i) (1.7a) c. . A. - A i--l ,3 1 »"l CUi =° (j > i) (1.7b)
while boundary condition (1.2) yields
.7c) Ci,i=Ni-
Hence the general solution to (1.1) is given by
N.(t) = C. .-v (1.8)
with C . . determined by the recursive relations in exps. (1.7). This may be conveniently written in a matrix equation
N(t) = M«E(t); N(t=0) = N (1.9) where
N(t) = {Nj(t),N2(t),...,Nn(t)} (1.10a)
-Axt -A2t -X t n E(t) = {e ,e ,...,e } (1.10b)
= {NX(O),N (0), fNn(0)} (1.10c)
'1,1 0 0 '2,1 ~2,2 M
Cn-l,2'*--'Cn-l,n-l ° c c cc c nl n,n22 '•*•' n,n-l n,n 20 A special case is obtained if the initial conditions are restricted to N° = {N°,0,...,0} which reduces eq. (1.9) to the so-called Bateman solutions (Bateman, 1910). If the abundances of nuclides present in a decay series are only subjected to the law of radioactive decay (no chemical or other phys- ical processes are involved), the development in time to a state of quasi-equilibrium is governed by eq. (1.1), no how complicated the initial conditions are. If, for any reason, this state of equili- brium has not yet been reached and the initial abundances, given by eq. (1.2), of the various nuclides are known, the elapsed time can be deduced from the degree of disequilibrium.
1.2.4. PROCESSES THAT CAUSE EQUILIBRIUM If we consider the Earth as a closed system that has been formed approximately one half-life of 238U (t = 4.47 * 109 a) ago, we may use eq. (1.9) to show that all isotopes in the 238U series are in secular equilibrium. However, in general we are only interested in small "subsystems" in which a state of disequilibrium between radio- active parent and daughter nuclides may exist as a result of differ- ences in geochemical behaviour such as solubility (for example U and Ra are soluble while Th is extremely insoluble). In the 238u series one of the members is a noble gas (222Rn) that may easily escape from the matrix containing its parent (226Ra) and decay through a few short intermediate steps to insoluble 210Pb. Such a state of dis- equilibrium is always balanced by the complementary "rest of the earth", which implies that excess of an isotope in one system results in a deficit of the same isotope in another system. Furthermore, additional physical phenomena such as alpha recoil play an important role in disequilibrium between isotopes of the same element (Kigoshi, 1971; Fleischer and Raabe, 1978; Fleischer, 1980). For example, Z3kU, being the radioactive product of 238U (with two intermediate short-lived daughter products) occupies inter- stitial positions or damaged lattice sites in the mineral matrix as a result of recoil energy and displacement. Hence it is more effective- ly leached from than 238U which leads to widely varying ex- cess Z3kU in natural waters and as a consequence to 231tU deficiency 21 in the mineral matrix, resulting in an average 231*U/238U activity ratio of 1.144±0.002 in the ocean (Chen et al., 1986).
1.3. 23oTh/23"u DISEQUILIBRIUM DATING 1.3.1. INTRODUCTION The U/Th disequilibrium (UTD) dating method is based on disequi- librium between 23ku (t = 2.48 x 105 a) and its daughter product 230Th (t, = 75.2 x 103 a) as a result of the vast difference in geo- chemical behaviour of these two elements. It allows dating up to 350.000 years which is the time required for the 23oTh/23l
ported in the soluble ionic uranyl (UO2 ) form, or in the presence
of sufficient C02, as the uranyl carbonate complex (Osmond, 1980, Gascoyne, 1982). The highly isoluble Th can only be transported by Th-containing particulate matter (Gascoyne, 1982). Therefore, (ecolo- gical) systems that incorporate (ground) water tend to take up U selectively, which makes them suitable for UTD, provided they behave as a closed system for U and Th. In that case the present-day activi- ty ratio between 230Th and 231*U is given by (using eq. (1.9)):
1 \ 2 3 ° 1 2 \x : 238 2 3 "tn/•u/2 3 8un/ 230i 2 3 i» •
1 - e~l A~ AJtJ (1.11) where the symbolic notation for the various isotopes represents their activity, 231tX and 230X represent the decay constants of 23"*U and 230Th, respectively, and t denotes the time elapsed after formation. Equation (1.11) may be conveniently represented in a so-called iso- 22
0.0 0.2 0.4 0.6 0.8 1.2 23OTh/23AU activity ratio
figrure 1.2. Variation of 23kU/23SU and 23oTh/23'*U activity ratios with time in a closed system with no initial 230Th. The near vertical lines are lines of constant age (Isochrons, obtained from eg. (1.11)) while the near horizontal lines show the change in nuclide activity ratio with time for different initial 23V/238£/ activity ratios (after Ivanovich, 1982 and Schwarcz, 1979).
chron plot showing the relation between age, and 231*U/238U and noTh/2HU activity ratios (Fig. 1.2). The presence of 232Th in supposedly dateable materials, however, indicates contamination with Th, and thus 230Th, of other origin than from the in-situ decay of 23kU. This may be attributed to the presence of a detrital component (dust collected during formation) that con- taminates the dateable material. Inadequate laboratory techniques to isolate the pure dateable material from this component as well as in- situ mixing (e.g. by ion-exchange processes) between these components then may result in introduction of "environmental" contaminating Th. Various models to correct for this contamination have been suggested and will be discussed in detail in section 1.3.3. The validity of the basic assumptions of UTD has been establish- ed for applications to a variety of environmental materials. Kaufman and Broecker (1965) obtained satisfactory agreement between 1"c dating 23 and UTD of carbonate materials from lakes Lahontan and Bonneville, U.S.A., applying correction for initial contamination with environ- mental Th, based on comparison of 232Th activities in the dateable and the detrital fraction. Following their approach, Ivanovich et al. (1983) successfully dated molluscs from uplifted Holocene beaches in the Persian Gulf. By correcting for environmental Th, assuming mixing between a pure, dateable phase and a contaminating detrital phase, Kaufman (1971) applied UTD to inorganically precipitated carbonates in the Dead Sea Basin and found acceptable agreement with internal and available lkC ages. Speleothem were dated by Lau- ritzen and Gascoyne (1980) and Lauritzen and St. Pierre (1982), as- suming a constant environmental 23OTh/232Th activity ratio of 1.50 at the time of formation. Osmond et al. (1965) determined the age of corals and oolites in Florida, assuming that no correction for envi- ronmental Th need to be applied based on the absence of 232Th and 230Th in recent samples. Lawrence Edwards et al. (1986) obtained high analytical precision UTD dates for corals in good agreement with ex- pectations that these corals grew during high sea-level stands ap- proximately 120 ka ago. Szabo (1982) studied fractionation of U series isotopes at the Pomme de Terre Valley in southwestern Missouri and dated organic sand samples in agreement with lhC ages, while se- lected peat samples were shown to yield good agreement between 1'*C and UTD deduced ages by Vogel and Kronfeld (1980), applying a con- stant correction for environmental 230Th. By simultaneous determina- tion of 23oTh/23l4U and 231Pa/235U activity ratios, Veeh (1982) showed that marine phosphorites may be reliably dated by UTD, while Szabo (1980) found good agreement between llfC and UTD ages of fossil bones.
1.3.2. APPLICATION OF UTD TO PEAT In peats, very high U enrichments have been observed (Szalay, 1969; Halbach et al., 1980). These enrichments could not adequately be explained by assuming that all U was removed from the ground water by living plants. This is illustrated for example by Szalay (1969) who assumes that the production of 1 g of dry plant material requires 500 g of water resulting in an enrichment factor of 500 for U in the organic phase with respect to ground water (U concentration of the 24 order of a few ppb). This is obviously insufficient to explain the U concentrations of 10-100 ppm or more that are found in the organic fraction of peat. It was then found that certain products in the decomposition process of organic material, the so-called humic and fulvic acids, play an important role in the geochemical enrichment of U in peat (Szalay, 1958; Borovec et al., 1979; Halbach et al., 1980) as indi- cated in Fig. 1.3. These acids do not define a single chemical com- pound but a class of plant decomposition products of varying chemical structure, all characterized by coupled aromatic ring structures and the presence of hydroxyl and carboxyl groups as shown in Fig. 1.4. Humic and fulvic acids display a very high absorption capacity 2 "*" for U in the U02 form, creating stable uranyl organic complexes (Borovec et al., 1979; Kribek and Podiaha, 1980; Shanbag and Choppin, 1981). Therefore, the environmental mobility of U in humic sediments, such as peat, is governed mainly by the mobility of these acids. 2 +
Though some authors state that complexes formed between UO2 and humic and or fulvic acids are soluble (Szalay, 1969; Borovec et al., 1979) this is not generally agreed upon (Kribek and Podiaha, 1980; Halbach et al., 1980). In any case, the mobility of such U complexes Weathering Transport" Uptake
Inorganic detritus
Paniculate inorganic matter (Uand Th) e g Wind blown sand)
URANIUM and containing
Organic V, [Alpha recoil,leaching | Water detritus
U * • 20 -—- UO," (oxidizing conditions) (Humic and
Fulvic Acids)
PEAT
are 1.3. Schematic representation of the geochemical transport mechanisms responsible for the presence of U and Th in peat. 25
H OH H
Figure 1.4. Detail of the chemical structure of humic acid (Stevenson, 1982). The adsorbed uranyl ion is drawn according to the observation of Borovec et al. (1979) that hydrogen , associated with the carboxyl groups, are exchanged for uranyl cations. in clay containing organic substances will be strongly limited due to the strong binding to clay minerals. Furthermore it was found in the presence of humic and fulvic acids Th as well as several other actinides behave very similarly to U, implying low mobility of Th in peat as well (Nash and Choppin, 1980; Nash et al., 1981). These facts may lead to the conclusion that the organic constituents behave as a closed system for U and Th, which makes peat suitable for dating with UTD. However, the presence of U and Th containing inorganic detrital material introduces some uncertainties. For example, some minerals may not be chemically inert in humic and fulvic acids. Hence ion ex- change between the inorganic and the organic fraction is possible as indicated by the presence of non-radiogenic 232Th in the organic material. Schematically this is shown in Fig. 1.3 by two arrows con- necting the organic and the inorganic detrital fraction. If the in- organic fraction is relatively small and contains little U and Th, this process will not seriously affect the concentrations in the or- ganic fraction. However, in practice many peat samples have a rela- tively high inorganic content (20-50% by weight or more) with unknown U and Th activities. In that case ion-exchange processes may play an important role in post-depositional geochemistry. Whether or not this renders closed-system behaviour questionable will be investigated in 26
detail in this thesis.
1.3.3. CORRECTION FOR ENVIRONMENTAL Th 1.3.3.1. Contamination by laboratory procedures If contamination with environmental Th is merely the result of inadequate laboratory extraction techniques and not of in-situ ion- exchange processes, it can simply be corrected for by using the meas- ured 232Th activities in both the extracted (supposedly dateable) fraction, and the residual fraction as an index for the degree of mixing of the both phases, assuming that during extraction neither isotope fractionation between 230Th and 232Th nor chemical fractiona- tion between U and Th has occurred:
230Th = 230Th _ (232Th /232Th j x 2 3 0^ (1.12) E E R R where the subscripts E and R denote the concentrations in the extract and in the residue, respectively. For U a similar expression may be written:
238 238 232 232 238 U = UE - ( ThE/ ThR) x UR (1.13) 2»U = 23*U - (232Th /232Th > x 23V (1.14) E E R R
Using these equations the activity ratios of the various isotopes in the dateable phase can be calculated from any set of extracted (either consecutive or separately) fractions (E1 and E2):
2 232 32 (/( "U/ Th, l (U/Th> o 23"U/238U = — — (1.15) (238U/232Th) , - (238U/232Th) „ El E 2 (23°Th/232Th) - (2SOTh/232Th) 28OTh/2s*U = — Si (1.16) 2H 232 231t 232 ( U/ Th)p1 - ( U/ Th) ,
This implies that plotting the measured activities, normalised to 232Th, from any extracted fraction should yield a straight line with a slope that represents the activity ratio in the dateable phase. Hence a single age is defined by this "mixing line" or "isochron" and we refer to this method as isochron correction. 27
1.3.3.2. In-situ contamination If in-situ contamination with environmental Th as a result of ion exchange between the organic (humic and fulvic acids) and the inorganic constituents of peat has occurred, a correction for this contribution is not straightforward. In that case we have to assume that (1) contamination directly following the formation of the peat has lasted for a period of time that is relatively short compared to the half-life of 230Th. (2) U and Th isotopes have exchanged without fractionation. Subse- quently both the organic and the inorganic fraction have remained a closed system for U and Th. If these requirements are fulfilled the influence of the initial con- tamination with environmental Th can be deduced from the measured present-day isotope ratios in both the organic and the inorganic fractions according to the following equations (see e.g. Van der Wijk et al., 1986):
2 3 "tI 2 3 « f—- -11' L238y J
230 23a 230 TThh = U 230A -
6t,\ 2 3 0 ), t 2)e Xt (1.18) where 230Th (1.19)
The "age" t of the inorganic fraction is defined as the time elapsed after formation of the peat, while f is equal for both fractions (in accordance with the assumption of no isotope fractionation in the initial j.on-exchange process). Hence, by analysing both fractions, four equations determined by (1.17) and (1.18) are obtained to yield a unique solution for t, f and 23"u /238U( (in the organic as well as in the inorganic fraction). Due to the fact that the equations are 28 non-linear there is no direct analytical solution and an iterative calculation procedure has to be used.
1.3.4. VALIDITY OF DATING MODELS The correction techniques for environmental Th discussed in the preceding sections are not confined to a specific material such as peat but can be generally applied to any system that is dateable with UTD. The selection of the correction method in individual studies should be guided by knowledge of the geochemistry of the system. Ob- viously these techniques should only be applied in cases where the U and Th concentrations in the dateable fraction exceed the concen- trations in the contaminating fraction so that the age corrections are not too rigorous. If this is not true, the correction techniques will introduce large uncertainties because the relatively low U and Th activities that determine the "age-signal" can hardly be distin- guished from the large U and Th activities (the "noise") of the de- trital component.
1.4.210Pb DATING 1.4.1. INTRODUCTION Goldberg (1963) and Crozaz et al. (1964) were the first to use 210Pb as a geochronometer for determining accumulation rates of gla- cier ice. Krshnaswamy et al. (1971) showed that the method could easily be applied to sediments. Their approach was soon followed by other investigators (see e.g. Koide et al., 19 73; Robbins and Edgington, 1975; Appleby and Oldfield, 1978) so that nowadays the 210Pb dating method is routinely incorporated in paleolimnological and near-shore marine environment studies to date recently (in the past 150 years or so) deposited sediments (El-Daoushy, 1982; Chanton et al., 1983; Simola and Liehu, 1985; Binford and Brenner, 1986; Van der Wijk and Mook, 1987), although it has been used incidentally to date sphagnum hummocks (El-Daoushy et al., 1982). In the years fol- lowing the work of Goldberg (1963) many geochemical properties of 21DPb have been studied and the dating models have been refined. In section 1.4.2 through 1.4.4 we will discuss these models and comment shortly on their validity. 29 1.4.2. CRS AND CIC AGE CALCULATING MODELS Being a member of the 238U-decay series with a half-life of t^ = 22.3 a, 210Pb is brought into the sediment both by in-situ production through radioactive decay of Z26Ra (supported 210Pb), as well as by deposition of 222Rn-produced atmospheric 210Pb. As an inert gas in the 238u series, radon escapes from the earth's crust at an average rate of 7.0 to 7.5x 103 m~2s~1 (Israel, 1951; Wilkening et al., 1975). It decays through a series -,f short lived daughter products to 210Pb (Fig. 1.1) which, after an estimated atmospheric residence time of approximately two weeks (the turn-over time of atmospheric vapour (Moore et al., 1973; Turekian et al., 1977)), enters the land hydro- sphere through either dry or wet precipitation (Fig. 1.5). Although the estimation of the average adsorption rate of lead isotopes to suspended material by Krishnaswamy et al. (1982) of ap- proximately > 2.6x 10-3 min -l deviates almost two order of magnitude from the estimations by Rama and Moore (1984) (> 0.1 min ) , this does not violate the conclusion that in surface waters 2 10 Pb is re- moved by adsorption to suspended material on a time scale that is much shorter than its half-life (210A = 1.3x 10~6 min"1). Hence it is subjected to sedimentation with suspended sedimenting material as its carrier.
Figure 1.5. Schematic re- presentation of the natural cycle of 2laPb. 30
Other sources of 210Pb may or may not introduce significant amounts of additional 210Pb to the sediment, depending on the local conditions. If, for example, the local atmospheric deposition is low and the concentration of dissolved 226Ra is relatively high, in-situ produced 2:oPb in the water column overlying the sediment may sig- nificantly contribute to the total deposition. Furthermore, zl0Pb may be carried by rivers in dissolved or particulate form (weather- ing of soil) and redeposited elsewhere. However, these sources are extremely site-dependent and may be strongly fluctuating in time so that in developing a mathematical model we will initially consider only those cases where their contribution can be neglected with re- spect to atmospheric deposition. The Constant Rate of Supply (CRS) model assumes a constant aver- age annual rate of deposition D (cm"^"1) of atmospheric 210Pb. In that case the excess activity21°Pb over the 226Ra supported acti- vity 210Pb in the sediment layer formed t years ago at present sedi- ment depth z cm can be calculated (see e.g. Appleby and Oldfield, 1978):
210in 210%. 21°Pb = —^ e" At (1.20) xs S(t) where 210A is the decay constant of 210Pb (a"1) and s(t) is the sedi- mentation rate dz/dt at the time of deposition t of the sediment layer at corresponding depth z (cm.a"1). (Note that in eq. (1.20) we express the activity in decay per per cm"3. If the activity is measured in Bq.cm"3 proper conversion factors have to be used). The activity 210Pb cannot be obtained from direct measurements, xs but has to be calculated from the measured total 210Pb activity and the estimated supported activity 210Pb . In general there are two ways to estimate this activity: (1) 210Pb is assumed to be in secular equilibrium with 226Ra and thus taken equal to the measured 226Ra activity. This method ob- viously requires a technique to measure the latter activity as well as the 210Pb activity. (2) Assuming a constant contribution of 210Pb throughout the sedi- ment layers it is set equal to the measured total 210Pb activity 31 at a depth older than approximately 150-200 years (where the 210Pb contribution has decayed). xs * Substituting S(t) = dz/dt in eq. (1.20) yields a differential equation from which an expression is obtained for the total (integra- ted) 210Pb activity over a sediment column of unit surface area to xs J a depth z after integrating both sides with respect to z and t, re- spectively: z 210210 t 210 21 Xtl I 21 At I(0)ffzz)) = If PPbb (z')dz' = [I ADe~ ° dt = D(1 - e" ° ) J X S J (1.21)
As from (1.21) it follows that
D = I(0,cc) \ (1 .22) \ the age t of the sediment layer at depth z is given by \ t = -1— ln(1 - I(0,z)/I(0,oo)) \ (1.23 2 1 0 x yy v which may be rewritten in the two independent variable^ I(0,z) and I(z,co): \
t = —— ln(1 - I(0,Z)/T(Z,OO)) \ (1.24) 21 °X \ A special case of this so-called Constant Rate of Supply (CRs\ model is obtained if the sedimentation rate S is constant, resulting\in a Constant Initial Concentration (CIC) of 2foPb in the sediment ,\ In xs this model the 210Pb activity at arbitrary depth z is given by X 5 210Pb (z) = —^ e" Az/S (1.25) yielding a straight line of slope -210A/S on a semilogarithmic plot of 210Pb against z. xs 32
1.4.3. OTHER MODELS Complicating factors such as mixing of sediment due to e.g. bio- turbation or redistribution of 21 °Pb as a result of diffusional pro- cesses may result in deviations from the idealized CRS or CIC models. Nevertheless such processes may be included in more generalized ex- tended models such as discussed in detail by Robbins (1978). If sedi- ment particle reworking is considered to be a random process, one is tempted to model the process as a analogue. In that case the specific 210Pb (z) activity at a depth z below the sediment- water interface is governed by sedimentation, diffusion and radio- active decay which can be described by a time-dependent second order differential equation (Krishnaswamy and Lai, 1978):
210 210 210 210 M K ~d Z Pb xs
and 210Pb (°°) =0:
21o az Pbys(z) = Ce (1.27)
where a.
This result shows that even in the case of mixing, the 210Pb acti xs vity-depth profile shows exponential behaviour which indicates that a straight line on a semi-log plot of the activity-depth profile is a necessary but insufficient condition to infer undisturbed sedi- mentation. 33 By developing the square root in eq. (1.28) into a series, it is easy to distinguish two extreme cases: (1) K -> 0 (undisturbed sedimentation) yields a = -210A/S which is (obviously) equivalent to the CIC model as discussed in section 1.4.2. (2) S •+ 0 (negligible sedimentation over the dating interval) yields a = -(210A/K) from which the mixing coefficient can readily be determined.
1.4.4. VALIDITY OF MODELS Krishnaswamy and Lai (1978) reviewed the annual deposition rate of 210Pb for a variety of localities and concluded that it shows significant geographical variations, most probably as a result of radon exhalation rates strongly varying with soil composition. Also the local rate of deposition of 210Pb from the atmosphere varies within approximately 30% of the annual mean over ca. 5 years for the reviewed stations, possibly due to meteorological conditions. There- fore, it appears to be necessary to monitor the 210Pb deposition on a regular basis to check the constant rate of supply assumption. Both the CRS and the CIC model require the absence of post- depositional transfer of 210Pb (closed-system behaviour). However, X 5 the CRS model provides correct ages even when post depositional (con- stant) lateral redistribution of 210Pb carrying sediment (sediment focussing) has occurred, provided that the deposition rate D as cal- culated from eq. (1.22) is used in the calculations. The same is true if additional sources of 210Pb (such as production from dissolv- ed 226Ra and riverine transport as discussed in section 1.4.2) con- tribute to the total 210Pb deposition, provided they have been con- stant over the dating range. From measurements in acidified lakes Simola and Liehu (1985) concluded that at low pH the adsorption constant for 210Pb to the sediment may change due to its increased solubility. They found ano- malously low 210Pb activities in sediment deposited in a more acidic environment as deduced from fossil diatom inferred pH. Obviously this results in an overestimation of the age. In this thesis we will discuss the results of laboratory experiments to determine the ad- 34 sorption/desorption rates for lead at a variety of pH values. Summarizing we can conclude that many post depositional proces- ses such as vertical transfer of 210Pb as a result of bioturbation, increased solubility or molecular diffusion may invalidate closed system behaviour. It is not always possible to obtain conclusive evidence for such processes from the 210Pb activity-depth profile alone as was discussed in section 1.4.3. Hence for reliable the zx°Pb-deduced ages should whenever possible be compared with ages deduced from other independent (radiometric or alternative) methods such as 137Cs dating, pollen analysis or historical informa- tion. 35
CHAPTER 2. ANALYTICAL TECHNIQUES'
2.1. INTRODUCTION At the present state of the art the most convenient and reliable way for routine quantitative measurement of alpha emitting nuclides is by detecting the intensity and the energy of the emitted alpha , although recent developments have shown that these iso- topes can also be extremely effectively measured by mass spectro- metry (Chen et al., 1986; Lawrence Edwards et al., 1986). The first method of total alpha counting, regardless of their energy is applied for 222Rn measurements; the second, alpha spectrometry, registers intensity as well as energy and is used for detection of U, Th and 210Po, a of 210Pb. For discrimination of isotopes by measuring the energy of the emitted alpha particles an eight channel alpha spectrometer was in- stalled, consisting of a Canberra® Quad 7404 commercially available alpha spectrometer and a home-made system. Because energy loss of alpha particles in matter due to straggling is high, high-resolution alpha spectrometry requires the preparation of extremely thin radio- active sources. Therefore, the elements of interest are extracted from environmental materials and subsequently purified by chromoto- graphical techniques. From the purified elements a radioactive source is prepared by means of electrolysis. The fact that the chemistry is largely determined by requirements of measuring technique renders it useful to first discuss details of the alpha spectrometer before go- ing into details of chemical procedures. For detection of gaseous 222Rn, an extraction and purification system was designed similar to the systems commercially available
* Part of the contents of this chapter has been accepted for publication: Van der Wijk, A., Venema, L. and Steendam, S.P., (1987), The use of thin plastic foils in low-level alpha spectrometry. International Journal of Applied Radia- tion and Isotopes. To be published. 36
CHEMICAL TECHNIQUES FOR DETERMINATION OF LOW CONCENTRATIONS U, Th.Pb(Po) AND Ra IN ENVIRONMENTAL SAMPLES
DISSOLVE THE SAMPLE
U,Th,Po Ra
Add a known amount Store in gaswash of spike activity flask and clean (232U, 228Th,2OBPo) overlying atmosphere
Co-precipitate with Fe(0H)3 I Leave for U,Th Po about two weeks I to guarantee secular equilibrium Redissolve, between Ra and Rn separate U,Th Redissolve, and Fe by add ascorbic acid ion - exchange Transfer Rn quantitatively to scintillation Prepare thin Prepare thin (LUCAS) bottle source on stainless source by self- steel by deposition on electroplating silver Count Rn- emitted alpha's by scintillation Measure in alpha - spectrometer counting
Figure 2.1. General outline of the chemistry for separation, purification and source preparation and subsequent measurement of the alpha activity for a variety of isotopes in the U decay series. from Applied of Piermont®. Figure 2.1 shows the general outline of the analytical techniques that are used.
2.2. ALPHA SPECTROMETER 2.2.1. DESCRIPTION The eight channel alpha spectrometer consists of two vacuum chambers of approximately 7.5x 7.5 x 32 cm3, mounted in a NIM rack. They are each divided by three thin vertical aluminium plates into four small cubicle chambers (7.5x7.5x7.5 cm ). Each of these chambers contains a "saucer" that is adjustable in height and sup- 37
Figure 2.2. Photograph showing the interior of one of the two vacuum chambers of the alpha spectrometer. ports the radioactive source. A large area (450 mm2) solid state si- licon alpha-particle detector is mounted by a micro-dot vacuum feed through connection at the top of the chamber. Fig. 2.2 shows the in- terior of one of the two aluminium vacuum chambers. The vacuum chambers can be evacuated by means of a rotational pump. When air is sufficiently low (approximately 100 ym Hg) pumping is taken over by a cryo adsorption pump. The latter consists of a stainless steel cylinder (Fig. 2.3), for three quarters filled with zeolite, kept in a 30 £ dewar containing liquid nitrogen. At that the air is effectively adsorbed by the zeolite and the pressure can be kept as low as 20 ym Hg or less, which is suffi- ciently lov; for alpha particles to travel the distance between source and detector without serious energy straggling (the mean free path for alpha particles at that pressure is approximately 0.4 cm while the source detector distance is approximately 0.8-1.0 cm. Eight detectors are used, four Si surface barrier, type ORTEC® BA-21-450-100 and four Si ion-implanted type Enertec-Schlumberger® IPE-450-100-20. Each detector is capacitively connected to a charge sensitive preamplifier that produces a pulse of short rise time (ca. 0.5 ys) and long fall time (ca. 470 ys). The pulse height is propor- tional to the energy of the detected radiation. The output signal 38
36
—}
Jpp
Figure 2.3. Zeolite cryo-adsorption pump for evacua- tion of the alpha spectrometer. The extremely clean vacuum that can be reached with such pumps is less 5 55 than 20 \sm Hg. from the preamplifier is directly fed into an amplifier in the same compact unit (ca. 7.5 x 4 x 16 cm3) mounted directly on top of the va- cuum chamber, where it is amplified and reshaped to a symmetrical Gaussian pulse (FWHM approximately 1 us) . Four commercially available Canberra® model 7404-01 integrated amplifiers and four home-developed integrated amplifiers were used. Each of the four amplified signals (in the range 0-10 Volt) is fed into an analog switch that registrates at which input the signal enters. Subsequently the analog signal is transmitted to a Canberra® model 8075 8K Analog to Digital Converter (ADC) where its height is 39
AMPLIFIER • ANALOG j DIGITAL
/HV 12 bits OUT CANBERRA MCA IN 40 / HV MULTIPLEXER INPUT READ OUT OUT IN (SERIAL )
/HV 10 bits -w- / HV IN OUT
100 Mfl
/HV 12 bits
OUT
/ HV MULTIPLEXER
OUT IN -C+ /HV 10 bits
i DUAL DISK DRIVE /HV IN OUT APPLE He ADC (64 K)
Figure 2.4. Schematic block diagram of the electronics of the eight channel alpha spectrometer. 40 converted to a 10 bit binary number. The latter is fed back to the analog switch where two (most significant) bits are added to identify the input number of the signal. Subsequently the 12 bit digital num- ber is stored in a 4096 (4K) channel memory where the channel number corresponding to the digital number is increased by one. As a result the signals from the first detector are stored in channel 0 to 1023, the signals from detector two in channel 1024 to 204 7 etc. An Arends® 4K MCC card in combination with an Apple® lie compatible micro com- puter and a 4K Canberra® series 40 MCA were used as memories. This process may be summarized by observing that each event (i.e. the registration of an ) is stored in a memory section that corresponds to the energy of the alpha particle. Hence identi- fication of isotopes by energy of the emitted alpha particles is possible. An electronic pulser, that can be used to produce signals at a controlled rate to test the performance of the electronic system is connected to each of the eight detection channels. Figure 2.4 shows a schematic representation of the system elec- tronics in a block diagram.
2.2.2. PERFORMANCE 2.2.2.1. Electronics High-resolution low-level alpha spectrometry requires extremely stable conditions. The electronic noise as well as shift due to e.g. small changes in temperature of the spectrometer should be minimized to guarantee good energy resolution, especially when long counting (typically 1 day or more) are required. The electronic noise of the system at working conditions is 50 mV corresponding to approximately 15 keV pulser resolution (Full Width at Half Maximum, FWHM) in the calibrated set-upr largely due to the 300 pF capacitance of the detectors. Without these detectors being connected to the preamplifiers, the electronic noise is reduced to approximately one third of this value. Shifts in the amplification factor due to temperature instabili- ties may result in serious deterioration of the energy resolution. Therefore, temperature sensitive components in the amplifiers were 41 replaced by less sensitive components wherever possible. The extreme temperature sensibility of the detector leakage current through the detector serial resistance of "100 MOhm (Fig. 2.4) (the current doubles if the temperature rises 10°C) results in instabilities in the effective detector bias, charge collection and detector capaci- tance. The resulting shift in the energy vs pulse height relation is obviously stronger for the surface barrier (electrical resistance ap- proximately 200-300 MOhm) than for the ion-implanted detectors (ap- proximately 1 GOhm). The only way to solve this problem is by tempe- rature stabilisation of the environment. Unfortunately we had to ac- cept a room temperature fluctuation of approximately 7°C during the analyses.
2.2.2.2. Detectors The guaranteed energy resolution is 21 keV (FWHM) for the sur- face barrier detectors and 20 keV for the ion implanted detectors at 15 keV contribution of electronic noise and a source detector dis- tance of at least 5 cm. However, in order to reduce the measuring time, low-level alpha spectrometry generally requires shorter dis- tances between source and detector to obtain a sufficiently high count rate. This results in peak broadening caused by increased ener- gy straggling of alpha particles which do not penetrate the dead layer of the detector surface under an angle of 90°. For example, if the source is placed at 5 cm distance, the average angle under which the alpha particles travel through the dead layer of the detector surface decreases to approximately 80°, resulting in 1-2% increase of effec- tive thickness. However, at 0.8 cm source-detector distance the aver- age penetration angle is only 60°,which corresponds to approximately 15% increase of the effective thickness. For both the surface barrier and the ion-implanted detectors the measured effect of the detector- source distance on the energy resolution is shown in Fig. 2.5.
2.2.2.3. Background One of the advantages of measuring alpha activity is the extreme- ly low . Due to the limited penetration depth of alpha particles in matter, the aluminium housing of the spectrometer 42
32mm
slide glass
60 1 i 1 1 1
\ • Detector IP-4965 ca. ilOmm \ £ Detector SB-23-435 J
\
50 - \
\ valve \ \ • i t t zf \ u_ 40 capillary \ c .S ^_
O3f 400 mm cc >. »^^ r 30 1 \ Ene r 1 i
20 j
/ 1 1 1 12 3 4 Distance source to detector (cm) 1.7 mm
Figure 2.5. The effect of the source detector Figure 2.6. Schematic drawing distance on the energy resolution (FKHM) for of the separation funnel used both the surface barrier and the ion-implant- for production of thin formvar ed detectors. foils. Figures indicate sizes in mm. is sufficient to stop all external alpha radiation. Furthermore, the detectors are characterized by a very thin depletion layer (i.e. the actual active detection area) in the order of 100 ym, so that they are almost completely insensitive to gamma radiation. Beta radiation may be detected but fortunately there is no natural source that emits significant amounts of beta particles in the energy range of interest (> 3.5 MeV). A more serious cause of detector background is formed by conta- 43 mination of the detector surfaces by nuclei recoiling from the source. The recoil energy of a daughter nucleus as a result of the emission of an alpha particle by the mother nucleus is approximately 50-100 keV which is several orders of magnitude larger than lattice bonding energy (of the order of eV rather than keV). Kigoshi (1971) for example observed for 23ItTh a recoil displacement of approximately 50 nm in powder as a result of 238U decay. Hence, daughter nuclei that are produced within this range from the source surface may escape and accumulate at the detector surface giving rise to peaks in the background spectrum due to their subsequent radioactive decay. To prevent this type of contamination we placed thin Formvar 3 foils (polyvinylformaldehyde, CioHi8Oi2/ specific mass 1.230 g/cm , obtained from BDH® chemical Ltd., Poole, U.K.) between source and detector to stop recoil nuclei. Such foils are routinely used in ex- perimental nuclear physics e.g. for selective transmission of partic- les (see e.g. Krist et al., 1984; Santy and Werner, 1984). The foils were home-made (Van der Wijk et al., 1987), following the method described by Revell and Agar (1955). A separation funnel, consisting of a reservoir and a capillary separated by a valve, is filled with a solution of known concentration of formvar in chloro- form. A cleaned piece of thin glass (e.g. such as used for slide mounting) is mounted into the reservoir (Fig. 2.6). By carefully ad- justing the valve, the solution is allowed to escape at a controlled flow rate. The solvent readily evaporates from the solution and a thin formvar layer of high homogeneity sticks to the glass surface. Before stripping the foils from the glass the layers are cut into the desired size using a razor blade. Subsequently the glass is lowered into water under an angle of approximately 45°. The foil easily releases from the glass, starts floating and can be picked up with a foil frame (Fig. 2.7). The thickness of the foil is determined by the concentration of the formvar solution and hence by the flow rate in the separation funnel. The set-up used in this study was calibrated by carefully weighing dried foils of known surface area, produced from solutions of various formvar concentrations (Fig. 2.8). 44
Figure 2.7a. The formvar layer is stripped from the glass surface by lowering the glass into the liquid under an angle of approxima- tely 45°.
Figure 2.7b. The floating formvar is picked up with a foil frame.
0 10 15 Flow rate (cm/min)
Figure 2.8. Calibration line for the separation funnel. Figures between brackets indicate the formvar concentration in g/SL. 45
Figure 2.9 shows the reduction in energy resolution as well as the energy shift due to straggling of alpha particles in the foil, measured as a function of foil thickness using a source produced from a standard solution of 232U, 228Th and 22IlRa in secular equilibrium with each other (section 2.3.2). In order to determine the stopping efficiency for recoil nuclei a clean foil was exposed to the source. After a fixed exposure time the source was removed and the 221*Ra (228Th daughter product) acti- vity on the foil was measured. Assuming that (1) a fraction k of the decaying 228Th nuclei actually recoils (for the thin source used k is assumed to be unity). (2) a fraction g of the escaping 221tRa nuclei actually reaches the foil (geometrical efficiency) and (3) of these a fraction f (stopping efficiency) is actually stopped allows us to calculate the 22ItRa activity on the foil after a certain exposure time T:
^-0.926 .AX (keV) 1.2- -10 70 •4—
-20 60 : icienc y t 1.0 -30 50 _ en c f 0.9 M -40 40 §; CO LLJ < 0.8 30 ** -50 -0-1/ 10 20 30 40 50 / 20 Foil l-hickness (|j.q/cm2) -60 A AFWHM = 1.065.AX (°/o; 10 -70
10 20 30 40 B0 60 70 AX (ng/i'cm'
Figure 2.9. Energy shift (hE, keV) and percen- Figure 2.10. Stopping efficien- tual reduction of energy resolution (hFWHM, %, cy f as a function of foil the FWHM of the spectrum without foil is extra- thickness. polated to be 26.7 keV) as a function of stop- per foil thickness Ax. Straight lines repre- sent the least squares linear fit. 46
22*Ra = g«f«k^228Th(i - e'22^"1) (2.1)
where 22"X is the decay constant of 221*Ra (2.204 X10~6 s"1) and the symbolic representation of the isotopes denotes their activity. In the special geometrical configuration that was used (source diameter 18 mm, foil diameter 22 mm, distance source to foil 1.2 ± 0.01 mm and distance source to detector 8 ± 1 mm) g is estimated to be 0.44 ± 0.01. If the number of registered pulses originating from 228Th during the exposure time T is given by 228N (detection efficien- cy 228Eff) and the number of registered pulses originating from 22I|Ra during the measuring time t by 22IfN (detection efficiency 221*Eff) , the following expression for the stopping efficiency f can be calculated:
f = g.k 228N (2.2) Figure 2.10 shows that at a foil thickness between 10 and 20 yg. cm"2 full stopping of the recoiling nuclei is obtained. Therefore we used the 20 yg.cm"2 foils throughout the measurements to prevent de- tector contamination at the expense of approximately 20% reduction of energy due to straggling of alpha particles in the foil.
2.3. CHEMICAL TREATMENT FOR ALPHA SPECTROMETRY 2.3.1. INTRODUCTION Environmental materials generally consist of an inorganic (mine- ral) and an organic phase of varying relative contributions. "Pure" peats, for example, mainly consist of organic material with only minor amounts of sand or clay, whereas in materials such as fossil bones or carbonates the organic fraction contributes little or none to the total sample material. Dissolution, destruction or chemical fractio- nation of materials that vary so strongly in their composition re- quires a variety of chemical approaches to match each type of sample. On the other hand, once the fraction of the sample that has to be analyzed for alpha activity is extracted from the total sample and 47 subsequently dissolved, the procedures for purifying and separating the various alpha-particle emitting nuclides are more or less standar- dized. These techniques in general require the addition of a so-called spike in the most early of the chemistry. This spike consists of a known amount of activity of an artificially produced isotope of the same element that has to be measured. The final yield of this isotope is the indication for the overal recovery of the isotope of interest during the chemical procedure.
2.3.2. STANDARD SOLUTIONS AND SPIKES Quantitative determination of the various alpha emitting isotopes is only possible if the chemical recovery of separation and purifica- tion is reliably determined. In order to achieve this, small amounts of calibrated standard solutions containing artificially produced isotopes (spikes) of the elements of interest were added to the sample in the earliest stage of the chemistry (the so-called Isotope Dilution technique). Such standard solutions have to meet the following re- quirements: (1) the artificially produced isotope should not occur in nature, (2) its half life should be long with respect to the time that is re- quired for chemistry and measurement, and (3) its absolute specific activity should be accurately known, pre- ferably calibrated against an internationally accepted standard.
For U/Th measurements a cone. HNOS solution of approximately 1 Bq/m£ 232U (t = 73.6 a) activity, in secular equilibrium with its daugther 228Th (t = 1.913 a, kindly provided by the ZWO laboratory for Isotope , the Netherlands) was calibrated against a labo- ratory master solution of uranyl nitrate (U02(N03)2.6H2O) in cone.
HNO3f coded for internal use as GMS-85-U2, and a laboratory master solution of thorium nitrate (Th (NO3K .H20) in cone. HNO3, coded for internal use as GMS-85-T2. Both master solutions were calibrated at the AERE in Harwell, UK, against a standard that was used for inter- laboratory comparison measurements (Ivanovich et al., 1985). The uncertainties in the 232U/228Th spike solution are 1-2% for the abso- lute activity and 3-4% for the mother/daughter-activity ratio. The slight disadvantage of using 228Th is that it occurs in nature as the 48 daughter product of 232Th. This is however, partly balanced by the fact that in environmental materials the activity of 232Th-produced 228Th by assumption equals the 232Th activity due to the short half life of 228Th (secular equilibrium). Hence this contribution can be corrected for by subtracting the 232Th activity from the total 228Th activity to obtain the 228Th spike activity. 210 208 For Po measurements we used a spike solution of PoCl2 (t = 2.93 a, kindly provided by the NIOZ, the Netherlands) in 8 M HC1. Due to the short half life of 208Po this solution needed regular re- calibration by repeatedly (4 times) self-plating of a known amount of the solution and measuring the activity on the silver disk in the calibrated alpha spectrometer. The statistical spread of the separate analyses and the 1a uncertainties from the counting statistics of the single determinations were of the same order of magnitude (1.8%) . indicating chemical recovery of 100% _for these "spike runs".
The performance of the 222Rn detection system (section 2.4) was calibrated with a standard solution of 226Ra in HCl, kindly provided by Dr. F. El-Daoushy, University of Uppsala, . The uncertainty in the activity of this solution is approximately 5% (El-Daoushy, personal communication).
2.3.3. MATERIAL OF HIGH ORGANIC CONTENT 2.3.3.1. U/Th analyses As already indicated in section 1.3.2. the application of U/Th disequilibrium dating to peat, which by nature is an environmental material of high organic content, requires separation of U and Th of organic (dateable) origin from U and Th of inorganic (contaminating) origin. In a preliminary detailed study on closed system behaviour we fractionated peat in 4 to 5 separate chemical phases, schematically indicated in Fig. 2.11: (1) Overnight extraction, using slightly acidic water (pH = 4.5 solu-
tion of HNO3:H2SO<, = 1:1), to separate the water-soluble part (fraction E1). (2) Overnight extraction, using 0.5 M HCl, to extract carbonate and possibly some free acid-soluble fulvic acids (fraction E2). 49
Sample
Extraction with a pH=4.2 solution (HN03 IHJSO^ =1:1)
Insoluble Soluble (fraction ED
Extraction with 0.5 M HC1
Insoluble Soluble (fraction E2)
Extraction with 1% NaOH at 60 C
Insoluble Soluble
Insoluble Soluble (fraction E3H) (fraction E3F)
Gentle boiling with cone. HN03 (65%)
Insoluble Soluble (fraction E4)
Leaching with cone. HCI (36%)
Insoluble Soluble (fraction E 5)
Figure 2.11. Chemical fractionation procedure to separate various fractions from peat. 50 (3) Overnight extraction at 60°C, using a 1% NaOH solution, followed by precipitation at pH 2. The soluble part (humic acids, frac- tion E3H) and the insoluble part (fulvic acids, fraction E3F) were separately analysed.
(4) Short gentle boiling with concentrated HNO3 (65%) to remove re- maining humates and organic coatings on the inorganic detrital fraction (fraction E4). It should be mentioned that in this step some leaching of the inorganic fraction occurs as well. (5) Overnight leaching of the remainder in concentrated HC1 (36%) (fraction E5). The organic material in the different extracted fractions is de-
stroyed by wet oxidation using hot cone. HNO3, cone. H2SO4 and H2O2. The conclusions from studying U and Th activities in these sepa- rate fractions (see section 3.2.3) indicate that in general a two- phase (organic/inorganic) separation may be sufficient. In order to obtain these phases we use two chemical approaches. In the first approach part of the organic phase (humic and fulvic acids) is extracted from the sample by selective dissolution overnight in a 1 % (by weight) NaOH solution at 60°C. The liquid is separated from the remaining solids, consisting mainly of inorganic material, by centrifugation at 3000 rpm for approximately half an hour and sub- sequently oxidized in a Kjeldahl flask using cone. HNO3, H2O2 and cone. E2SOk . The organic substances that are still visible in the residual inorganic material are destroyed by gentle boiling with cone. HNO3 for approximately half an hour and subsequently decanted from the flask. Obviously, some inorganic material will be lost in this process as well, due to dissolution and severe leaching possibly accompanied by isotope fractionation for U and Th. After repeated washing with demineralized H2O this inorganic fraction is analysed separately. In the second approach the peat sample is combusted at a tempe- rature of 700-800°C in a china crucible, either over a Bunsen flame or in a muffle furnace. The ashes are mildly leached with dilute HNO3 to retrieve U and Th that are released from the organic material during combustion and subsequently adsorbed to the inorganic fraction. The leached residual ashes are analysed separately. 51 2.3.3.2. 210Pjb analyses 2l°Pb analysis in (organic) sediments requires determination of concentrations of 210Pb and 226Ra in the total sample to establish the excess of 210pb contribution as discussed in section 1.4.2. is Thereto the organic material in the sample is combusted in a china crucible at a temperature of approximately 700-800°C, either in a muffle furnace or over a Bunsen burner. Subsequently the ashes are completely dissolved in a cone. HF/conc. HClOij (3:1) mixture. In the initial stage of this study there was no facility to measure 226Ra concentrations so that 210Pb activities had to be X S determined from comparison of the adsorbed 210Pb concentrations in relatively young (< 150 years) and relatively old (> 150 years) ma- terial (see section 1.4.2). Therefore, samples from both young and old sediments were analysed by leaching the ashes of the combusted sample overnight in cone. HNO3 and subsequent determination of the 210Pb activity in the extraction liquid. 2.3.4. INORGANIC MATERIAL Samples containing no organic material were dissolved, either in HC1 as in the case of calcareous material such as bones, shells, co- rals etc. or in a cone. HF/HClOi, (3:1) mixture as in the case of si- liceous materials such as sand or clay.
2.3.5. PURIFICATION AND SOURCE PREPARATION FOR U AND TJ- Approximately 1 Bq of 232U spike solution, in secular radio- active equilibrium with its daughter 228Th (section 2.3.2), is added to the extracted fraction to serve as an internal yield tracer for both U and Th. A few mg of iron (as a FeCl3 solution) is added so that following equilibration by careful shaking, U and Th can be co- precipitated with Fe(OH)3 upon addition of cone. NH^OH (25% solution). Following redissolution of the precipitate in 8 M HC1, Th is separa- ted from U and Fe on an anion exchange column filled with Bio Rad AG1X8 (100-200 mesh) resin that has been pre-equilibrated with 8 M
HCl. After addition of a few mg of iron (as FeCl3 solution) to the separated Th fraction and drops of cone. NHi,OH to both the Th/Fe and
U/Fe fractions, u and Th are separately coprecipitated with Fe(OH)3. 52 Both precipitates are redissolved in 7 M HNO3 and U and Th are sepa- rated from Fe by ion exchange on columns of slightly different dimen- sions, filled with the same type of resin as used before, pre-equili-
brated with 7 M HNO3. The purified U and Th solutions are brought in nitrate form by
repeated evaporation and redissolution in cone. HNO3 in a teflon beaker on a hot plate. The residual salts (hardly visible with the
naked eye) are redissolved in 50 y£ 0.1 M HNO3 to which 50 p£ of a
solution of 0.1 \iq/\iH of Ca(NO3)2 is added as a carrier. With a micro pipet the solution is transferred to the electroplating cell (Fig. 2.12a), rinsing the beaker with 10 mft 2-propanol. Finally U and/or Th are electroplated onto a 2.5 cm diameter stainless steel disk (Nuclear Supplies®) at a constant current of 5 mA and 100-200 Volt
TEFLON PLATING CEL TEFLON PLATING CEL b FOR2l0Po SELF- PLATING FOR U/Th PLATING Stirrer rod (teflon) Stirrer rod (platinum)
Teflon cap Teflon cap
Connector mercury drop for frictionless —~-^ electrical contact
Teflon plating cilinder Teflon plating cilinder
fey
Stainless steel dsk Silver disk
Brass knob for disk support Brass knob for disk support
Connector
Figure 2.12a. Plating cell for U and Th source preparation. Applied voltage is ap- proximately 100-200 V at a stabilized current of 5 mA. Figure 2.12b. Plating cell for Po-source preparation. 53
4.196 4.773 5.321
B 5A23 5.686
t/l c
228- 224Ra is»
4.012 4.684 5.342
232Th J230Th , 3.935 4.617
••I •-—••'' LZ 5.116
208: 'Po
5.304
210, 'Po
..._^ JJ 40 45 5.0 5.5 6.0 Energy (MeV)
Figure 2.13. Typical spectra measured under normal working conditions (see text), a: Uranium, b: Thorium, and c: Polonium. 54 for approximately 45 minutes to produce a thin source. A typical U spectrum obtained under working conditions from such a source in the alpha spectrometer is shown in Fig. 2.13a, while Fig. 2.13b shows a typical Th spectrum.
2.3.6. SOURCE PREPARATION FOR 210P£> DETERMINATION Being a beta emitting nuclide (see e.g. Fig. 1.1), 210Pb can not directly be measured by alpha spectrometry. Therefore, the 210Pb activity is determined indirectly by measuring its alpha emitting granddaughter 210Po (t = 138 d). Secular equilibrium between these two nuclides is reached in approximately two years so that in older natural systems such as most sediments of interest the 210Po activity equals the 210Pb activity, provided the sediment acts as a closed system with respect to Pb and Po. For quantitative determination of 210Po essentially the method as described by Flynn (1968) was applied. Approximately 1 Bq of a 208Po spike solution (section 2.3.2) is added to the extracted frac- tion as an internal yield tracer. The solution is evaporated till dry in a teflon beaker and the resulting dry salts are chlorinated by re- peated redissolution in a few drops cone. HC1. The dried precipitate is redissolved in 0.5 M HCl to which a few mg of ascorbic acid is added in order to complexate Fe. The solution is transferred to the self plating cell (Fig. 2.12b) while the beaker is rinsed with an- other 10 ml of 0.5 M HCl. At a constant temperature of 85°C, Po is allowed to precipitate for two hours on the surface of a 2.5 cm diameter silver disk by ion exchange with Ag to produce a thin Po source• A typical spectrum as obtained from such a source is shown in Fig. 2.13c.
2.3.7. BLANKS, RECOVERY AND MEMORY EFFECTS Chemicals used were all of analytical grade (pro Analysi), ob- tained from Merck® Chemical Company, Darmstadt, FRG. Blank runs (ana- lysis of demineralized water in order to determine the activity of various isotopes that may be introduced in the chemical procedures) showed no detectable activity. The chemical recovery of the methods varied strongly for different types of samples but for U and Th 55 averaged around 50-70% with occasional recoveries as low as 5% or as high as 95%. For Po the recovery was 90-100%. The adapted glassware cleaning procedures (leaching overnight in an RBS® soapy solution) were sufficient to suppress memory effects to an undetectable level.
2.4. RADON MEASUREMENT 2.4.1. GENERAL OUTLINE OF THE METHOD The radon detection system is used to determine 226Ra activities indirectly by measuring the activity of its radioactive daughter 222Rn. In order to measure this activity, a sample is suspended in approximately 100 to 200 m£ of demineralized water in a gas wash- bottle. The overlying air, which possibly contains some 222Rn, is replaced by helium to assure zero initial 222Rn concentration. The bottle is then closed and stored for approximately two weeks to allow growth of 222Rn. After this period the emanated 222Rn is transferred from the bottle into a scintillation counting cell using a cooled charcoal column in which the 222Rn is adsorbed. The scintillation cell, that registers. an alpha particle by emission of a light pulse, is connected to a photomultiplier. The total number of light pulses is counted from which the 226Ra activity is calculated.
2.4.2. DESCRIPTION 2.4.2.1. Extraction system Figure 2.14 schematically shows one of the extraction systems (the actual system consists of two identical systems in mirror image). Helium is circulated through the gas wash bottle containing the sus- pended sample, a water-vapour trap, a dioxide trap (ascarite) and a liquid nitrogen-cooled charcoal column, successively. The helium flow rate is monitored by a Brooks® flow meter (model R2-15-B) and can be controlled by regulating the output voltage of the power supply for the circulation pumps. These are home-made mem- brane pumps driven by a 12 V DC miniature motor bought in a toy shop. The pressure in the system is monitored with an electronic prfassure meter. The system contains three automatic valves (numbered 1 to 3 in Fig. 2.14) that direct the main helium flow either through or by-pas- PUMP Vacuum Vacuum
a Counting CGll
OHHeliu. m
silica ascarite | C *\
charcoal column
two way valve three way valve glass connector quick connector pressure meter
Figure 2.14. Schematic view of the radon extraction system. Figure 2.15. Schematic view of the transfer section of the radon count- ing set-up. 57 sing the wash bottle. These valves are opened and closed by an elec- tronic valve control unit to prevent liquid from the the wash bottle to enter the gas system becaus of small differences in pressure between the system and the bottle. When helium is being circulated through the wash bottle, valves 1 and 2 are open and valve 3 is closed. When the system is being evacuated, valves 1 and 2 are closed while valve 3 is open. The charcoal column is connected to the system with double end shut-off Swagelok® Quick-Connects (#SS-QC6-10MD and #SS-QC6-B-1OMD) and placed in a dewar containing liquid nitrogen. The glass connec- tions of the water-vapour trap, placed in a dewar containing a water/ ice mixture, and the trap containing ascarite (to adsorpt CO2) and silica (to adsorp H20) as well as the metal connections are designed to be used in one way to assure the correct flow direction.
2.4.2.2. Radon transfer system The transfer system (Fig. 2.15) is used to transfer 222Rn ad- dorbed in the charcoal column to a scintillation (activated ZnS) counting cell (Lucas, 1957). It is connected to a helium source and a rotation pump to evacuate the scintillation cell and the charcoal column. In order to release the adsorbed 222Rn, the column is heated to 500°C. Helium carrier gas is flushed slowly by regulating with a Nupro® (type #SS-4MG) fine metering valve (valve numbered 0 in Fig. 2.15) through the column to the scintillation counting cell. The cell is connected to the system with a Swagelok® Quick-Connect (#SS-QC4-D- 400) . An electronic pressure head monitors the pressure in the trans- fer system.
2.4.2.3. Electronics Opening of the counting box (Fig. 2.16) before the two photo- multiplier (PM) tubes (type RCA® 6655A) are shielded from direct light from outside is prevented by an electronic control unit. In operation mode the scintillation counting cells are placed directly over the PM tubes. A disk with two holes rotates between the counting 58
Lid 1
Wall - — Counting cell positions
Rotating disk Micro switch control
Photomultiplier
Disk axle Figure 2.16. Cross section of the 1 "Black Box" for scintillation counting of radon.
cells and the PM tubes so that in the counting mode the PM tubes are uncovered and actually "see" the scintillation cells. If the box is to be opened for replacement of the scintillation cells, the disk rotates over 90° to protect the PM tubes from direct light. The disk position is controlled by a switch at the front of the counting box and a microswitch on the disk. A green LED indicates that the box may be opened to change the cells; a red LED indicates that the chamber is locked an operating in the counting mode. The bias for both PM tubes is provided by a Canberra® model 3002 High Voltage supply that is externally controlled by a microswitch on the rotating disk in the counting box. The PM signal is transmitted to a Canberra® Model 2012 amplifier connected to the input of a Can- berra® Model 2072A dual counter with built-in discriminator to sup- press pulses originating from electronic noise. 59
2.4.2.4. Calculation of z22Rn and Z26Ra activities 222Rn (t = 3.82 d) decays through a series of relatively short- lived daughter products to 2l°Pb (t = 22.3 a) which may be consider- ed stable over the period of time that is required for the chemistry and measurement. Due to the fact that the scintillation counting system is not energy-sensitive, the measured alpha activity is the sum of the 222Rn, 2l8Po and 21<(Po (211tBi) activities. Although 21"*Bi has two decay modes, the very short half-life of 21l*Po (t. = 1.64 * lO"1* s) guarantees the emission of alpha particles at the same rate for both branches. The short half-lives of 222Rn and its daughter products requires careful bookkeeping of possible decay and growth processes during the analysis. Therefore, we consider four time intervals in the calcula- tion of the 222Rn and 226Ra activity: (1) During the time interval t (the moment the air in the wash 222 bottle is replaced by helium) to tx (the moment Rn has been transferred to the charcoal column) the 222Rn/226Ra activity ratio grows towards its equilibrium value (eq. (1.2)) as a result of 222Rn production from 226Ra in the suspended sample.
(2) In the time interval t1 to t2 (filling of the scintillation cell) there is no longer production from 22SRa (as there is no 226Ra available outside the wash bottle). Furthermore, it is important to note that the non-gaseous 222Rn-daughter products are not transferred to the counting cell.
(3) In the time interval t2 to t3 (start of counting) the decay of 222Rn as well as the production and subsequent decay of alpha emitting 222Rn daughter products contribute to the total alpha activity in the counting cell.
(4) During the counting interval t3 to ti* the total number of regis- tered pulses is the sum of the integrated 222Rn, 210Po and 211*Bi activities. Furthermore, the following three instrumental effects have to be in- corporated in the calculation: (1) Due to noise effects in the electronic counting system as well as to the presence of alpha emitting substances in the activated ZnS, its support, the helium carrier gas or the scintilla- 60 tion cell itself, an apparent count rate is registered, even in the absence of a sample. This is called system background (BKG). (2) Only a fraction of the emitted alpha particles is actually re- gistered in the scintillation cells. This fraction is expressed as the efficiency EFP. (3) The blank activity (BLK) (i.e. the activity measured from a de- mineralized water sample) is not zero. Taking all these effects into account and applying eqs. (1.9) through (1.10c) the following expressions may be obtained for the 226Ra and/ or 222Rn activity:
222 B KG X T «n(t2) = VT x EF F - BLK (2.3)
•Al (t2-t!) 226 222 Ra(to) = . •• j—rr r-r Rn(t2) (2.4) 0 A (t t ) A t t ~ o i~ o e- i( i- ) where
"Aits "Al t ii , -i -i . ~ A j t 3 ~~ \ i t i 1 3 z u - - e ) . ^iXk{e - e - A i {A 2 ~ "• X / ' 3 "~ ^ 1 / ' " *t ~ ^ 1 ) 1^1 ~* ^ 2 ) ' ^ 3 "™ ^2'^'^1* ~~ ^- 2 )
T (Xi - X3)(X2 - X3)(A4 - X3) (Xx - A,)(A2 - A,)(A3 - A,)
(2.5) and the subscripts 1,.. ,4 denote 222Rn, 218Po, 211tPo and 214Bi re- spectively. It is important to note that in evaluating the statistical counting error it is not appropriate to apply Poisson statistics di- rectly because a large fraction of the registered alpha decays are the result of coupled rather than independent events. A detailed calculation (Sarmiento et al., 1976) shows that: 61
FT EFF A )(t '-t ) I 222 (2.6) - e
where BKG = background count rate (s"1) ABKG = error in BKG BLK = blank activity (Bq) ABLK = error in blank activity EFF = efficiency of the scintillation cell AEFF = error in the efficiency BE = EFF + EFF(1 - EFF) + EFF(1 - EFF)2 FT: see eq. (2.1) N = number of counts T = counting time (ti, - t3>
2.4.3. PERFORMANCE 2.4.3.1. Background and blank activity The background is defined as the fraction of the count rate that is due to any alpha-particle emitting substance in the activated ZnS, its support, the helium carrier and, to a small degree, to the noise inherent to the PM tube. It will be reported here in counts per second (cps) . mhe background is determined by evacuating a counting cell for one hour and filling it with helium. The count rate is subsequently registered. The cells show a background varying from ca. 0.5 to 2.5x 10"3 cps. This number may increase as a function of time due to contamination with non-gaseous alpha emitting 222Rn daughter products which accumulate on the wall of the counting cell during measurement. Therefore, background should be regularly measured. The blank activity is determined by applying the extraction and transfer procedure to a wash bottle filled with demineralized water. The initial blank activity was approximately 15+4 mBg, increasing 62 to 30 ± 5 mBq during the measurements. Even after a complete revi- sion of the system the blank activity could not be reduced. Therefore we attribute this increase to deteriorating quality of the deminera- lized water. Future efforts will have to be directed to an improve- ment of this quality.
2.4.3.3. Efficiency Almost full recovery of the extraction and the transfer of radon from the charcoal column was found in a system similar to ours (Mathiev ?t al., 1977). Therefore, the efficiency of the radon de- tection system appears to depend solely on the efficiency EFF of the counting cell. The overall efficiency was determined by applying the extraction and transfer procedure to a 226Ra standard solution (section 2.3.2). It is expressed as the percentual fraction of the original 226Ra activity that is actually measured and varies between 81% and 87% for the counting cells used.
2.4.3.4. Memory effect Incomplete removal of 222Rn from the charcoal column and/or the counting cell may result in additional activity in the next analysis. This memory effect was determined by measuring blank activity direct- ly following a (highly radioactive with respect to natural activi- ties) 226Ra standard measurement. With the adapted procedure there was no detectable increase of blank activity and therefore memory effects are considered to be negligible. CHAPTER 3. U/TH DISEQUILIBRIUM DATING'
3.1. INTRODUCTION This chapter presents a systematic study of the application of the 23OTh/231*U disequilibrium (UTD) dating method for peat, dealing with closed-system geochemical behaviour of U and Th as well as pos- sible mechanisms responsible for introduction of so-called environ- mental or common Th. Based on this discussion the results and relia- bility of UTD dates of a variety of peat samples will be discussed. Furthermore, these results allow a tentative correlation of European pollen diagrams from France and Greece with previously published palaeo-temperature curves for North Western Europe and the marine environment. In addition section 3.3 discusses the results on four coral samples. These analyses were intended as a preliminary test of our analytical methods. In view of the importance of reliable coral dating in reconstruction of past sea-level stands in palaeoclimatology, the results are worthwhile presenting. Section 3.4. deals with an attempt to date fossil bones by UTD. The dates presented are derived from analysis of bulk material in- stead of material from the surface layer as recommended by Rae and Ivanovich (1986). No study was made of the homogeneity of the U and Th distribution through the cross section of the bones. However, in
* Part of the contents of this chapter has been or will be published in: Van der Wijk, A., El-Daoushy, F., Arends, A.R. and Mook, W.G., (1986). Dating peat with U/Th disequilibrium: some geochemical considerations. Chemical Geology (Isotope Geoscience Section) 59: 283-292. Van der Wijk, A., Mook, W.G. and Ivanovich, M., Correction for environmental 230Th in U/Th disequilibrium dating of peat. Science of the Total Environment, Proceedings of the 5th Symposium on Environmental Radiochemical Analysis. Harwell, U.K., 1-3 October 1986. To be published. Bartstra, G.J., Soegondho, S. and Van der Wijk, A., Solo stream sediments, Ngandong Man and river-drift palaeolithic of Java. Submitted to Journal of Human , 1987 64 view of their archaeological importance, the results are sufficient- ly encouraging to be presented, although more detailed research is required to establish the reliability of the deduced ages.
3.2. PEAT 3.2.1. INTRODUCTION The reliability of UTD dating of peat, as of any other environ- mental material, is largely determined by closed-system behaviour. Peat is a complicated material, consisting of a variety of organic as well as inorganic components. Each of these may show different geo- chemical behaviour with respect to U and Th. Hence, exchange of U and Th between the various constituents is possible whereas peat as a whole may still act as a closed system. In order to understand the geochemistry and to investigate which fraction acts as a closed system, a detailed study of the chemical speciation of U and Th in various constituents of peat was carried out (section 3.2.3). The results appear to be in good agreement with previous studies and can adequately be explained by assuming that the most important mechanism by which U and Th are adsorbed, exchanged and retained in peat is by cornplexation with humic and fulvic acids. This suggests that the organic fraction of peat, represented by humic and fulvic acids, resembles a closed system and that contamina- tion with environmental or common Th (indicated by the presence of 232Th) can be corrected for using eqs. (1.15) and (1.16). To investi- gate this, peat samples selected from various localities were com- busted over a Bunsen flame (section 2.3.3). U and Th were extracted by leaching with various concentrations of HNO3. Subsequently the residual mineral fractions were analysed. The results of these meas- urements (section 3.2.4) surprisingly show that for the majority of the investigated samples the best correspondence with other dating methods is obtained without correction for environmental Th. To ex- plain this apparent anomaly, a tentative geochemical model is intro- duced (section 3.2.5) which predicts strong depletion of Z30Th in environmental Th. Based on this model absolute ages were determined of well documented peat layers deposited during warmer periods (so- called interstadials) at the beginning of the last glaciation 65 (Weichsel or Wurrn), starting approximately 100,000 to 90,000 years ago. UTD dates were obtained for the end of the St-Germain I-C inter- stadial as pollen analytically defined by Woillard (1978) at La Grande Pile, France, and of the Elevtheroupolis and Drama intersta- dials as defined by Van der Hammen et al. (1971) at Tenaghi Phillip- pon, Greece.
3.2.2. SITE SELECTION Peat samples were selected from a variety of geographical loca- lities (Fig. 3.1). We will confine ourselves to a short description of the samples and the geological periods they represent. For extend- ed information on palynology, lithostratigraphy and geology we refer to other authors.
1. Amersfoort de Liendert (52c09'N, 05c24'E, the Netherlands) In 1967 four peat samples (AFII.1, AFII.2, AFII.3 and AFIII) were taken from peat layers of approximately 20-80 cm thick- ness, covered by coarse sand, in an artificial pit at Amersfoort, which is the locality where both the Eemian Interglacial
Selected Sites for peat dating TENAGHI PH1LLIPP0N
Figure 3.1. Map showing the geographical distribution of the localities where peat samples were selected for UTD dating. 66 and the Amersfoort Interstadial were palynologically defined (Zag- wijn,1961). A summary of the 1ithostratigraphy, pollen analysis and 1^C dating, applying thermal diffusion isotopic enrichment, is given by Grootes (1977). The samples represent early Weichselian Inter- stadials (probably Br^rup and Odderade), datei between 60,000 and 70,000 years ago.
2. Tenaghi Phillippon (41°10'N, 24°20'E, Greece) Samples were obtained from two borings (TFII and TFIII) from the Drama basin in Eastern Macedonia. For core TFII an extensive palyno- logical description exists, which enabled a reconstruction of palaeo- climatological conditions over the past 700 ka (Wijmstra and Groen- hart, 1983). Core TFIII was taken only a few meters from TFII and is assumed to show approximately the same stratigraphy (Wijmstra, pers. comm.). Detailed information on site, stratigraphy as well as palyno- logy is given by Wijmstra and Groenhart (1983). 1IfC dates are available for samples from the first 18-20 m in the TFII core, corresponding to a maximum age of roughly 55 ka. Van der Hammen et al. (1971) correlated the Pangaion, Doxaton, Drama and Elevtheroupolis at Tenaghi Phillippon with the Eemian Interglacial and the early Weichselian Interstadials Amersfoort, Br^rup and Odde- rade in North West Europe, respectively.
3. La Grande Pile (47°44'N, 06°30'E, France) Five samples were selected from four separate borings at La Grande Pile, France. From this locality Woillard (1978) obtained a continuous pollen record (Fig. 3.6). She tentatively correlated the Eemian, St. Germain I-A, St. Germain I-C and St. Germain II at La Grande Pile with the Pangaion, Doxaton, Drama and Elevtheroupolis at Tenhaghi Phillippon, respectively. Furthermore, Woillard and Mook (1982) used lltc ages for this pollen record to obtain a correlation with deep-sea 18O stage boundaries after Hays et al. (1976) and Kominz et al. (1979). From the five samples studied in this thesis four were taken from layers that represent the upper 15 cm of pollen stage St. Ger- main I-C (a182, a183, a185 and a186). Hence they are expected to be 67 too old for 1'*C dating. A sample from the top of St. Germain II has been dated before by application of thermal diffusion enrichment of lkC, yielding a minimum age of 69.5 ± ,*? ka (GrN-9187) for the end of this interstadial. The fifth sample (a184) represents the third interstadial during the Lanterne (Weichsel) II glaciation and is dated by x*C at 40.0 ± 0.6 ka (GrN-8746).
4. Les Echets (45°54'N, 05°00'E, France) Analysis of 39 m of sediment from a boring (G) in the ancient lake of Les Echets, only 200 km from La Grande Pile, resulted in a long, continuous pollen diagram covering the interval from Late Riss to the Holocene (see e.g. De Beaulieu and Reille, 1984a; 1984b). Pa- lynological correlation of three temperate post-Eemian episodes with the St. Germain I-A, I-C and II interstadials at La Grande Pile as well as with the Amersfoort, Br^rup and Odderade interstadials was possible. Based on these correlations, De Beaulieu and Reille accept the extrapolated dates proposed for the Grande File section. Sever . samples from boring G were investigated in this study.
5. Valley of the River Dinkel (52°23'N, 07°00'E, the Netherlands) Samples were taken from borings in sedimentary depo^xts in the glacial Dinkel basin in the Eastern part of the Netherlands. These samples represent the cold period between the early Weichselian and the late Weichselian (the Pleni-glacial), approximately between 60,000 and 12,000 years ago. An extensive description of the stratigraphy and lithology is given by Van Huissteden et al. (1986). As an independent check 11*C dates obtained in our laboratory are available.
6. Tervola (66°05'N, 06"30'E, Finland) Three samples (a113, a114 and a115) from a thin (5-50 cm) peat layer in the Kauvonkangas section (North Finland), covered by coarse sand, were provided by the Finnish Geological Survey. The geology, as well as the results of XI*C dating on several other peat samples from this site are discussed by Makinen (1979), who attributes the presence of peat in this section to the slight warming during the Perapohjola Interstadial, following the Weichsel I glaciation. Thermoluminescence 68 (TL) dating was carried out on quartz sands from this section by Hiitt et al. (1982).
7. Oerel (53°29'N, 09°04'E, Federal Republic of Germany) Several samples from borings at this site were submitted by Dr. K.-E. Behre from the Niedersachsisches Landesinstitut fur Mar- schen- und Wurtenforschung, VJilhelmshaven, FRG. Five of these (boring 61: a50, a51, a52; boring 46: a142 and a143) were selected from a peat layer that appears to represent an early Weichselian interstadi- al, chronologically situated between the Glind and the Odderade Interstadial. The lkC ages measured at our institute are available for comparison.
8. Pitalito (01?51!N, 76°02'W, Colombia) Eight samples from a 13 m deep boring (JBII) in the Pitalito Basin, Colombia, were submitted by Dr. J. Bakker from the Agricultu- ral University of Wageningen, the Netherlands, for UTD dating. Paral- lel 1^C ages on 4 samples of the same material were obtained in our laboratory.
3.2.3. CLOSED-SYSTEM BEHAVIOUR 3.2.3.1. Preliminary measurements In a first attempt to study closed-system behaviour we assumed that the organic fraction of peat contains essentially no initial Th and behaves as a closed system for U and Th. Hence we followed the approach of Vogel and Kronfeld (1980), who demonstrated that the 3 M HC1 leachate of ash of combusted peat samples may yield fair agreement between UTD and ll4C ages. They applied a simple correction for the contribution of inorganic detrital 23Oirh by assuming that the initial 23OTh/232Th activity ratio is 0.7 or 1.0. The first value (0.7) is the geometric mean of the 233Th/232Th activity ratios of Th leachable from the sediment of 13 rivers in South . Tables 3.1a and 3.1b show the results of our measurements on samples from Amers- foort and Tenaghi Phillippon (core TFII). To correct for environ- mental Th we assumed a 23oTh/232Th activity ratio f = 1.0 in the mineral fraction and subtracted the 232Th activity from the measured 69 230Th activity before calculation of the age. Furthermore, we investigated the speciation of U and Th in a variety of extracted fractions (4 to 5, designated as E1, E2, E3H and/or E3F, E4 and E5 in section 2.3.3.1) from Tenaghi. Phillippon (core TFII) and Tervola peat. Table 3.2 shows the radiometric re- sults. We assumed that environmental Th was introduced into the date- able phase shortly following formation and we calculated corrected ages from eqs. (1.17) through (1.19).
3.2.3.2. Preliminary discussion: geochemistry of U and Th With the exception of a2 79 the Tenaghi Phillippon samples show high 2 3oTh/2 3 2Th activity ratios in the 3 M HC1 leachates. This indi- cates that (post-formation) Th contamination has been limited. As a result the ages are rather insensitive to the correction factor f. Furthermore, the uncorrected UTD ages show similar trends as a func- tion of depth in the core as 1'tC ages (Table 3.1a and Fig. 3.5a). Hence it appears that U and Th specific activities are largely deter- mined by radioactive decay and growth. However, the relatively large statistical uncertainties result in large uncertainties in age. On the other hand, samples from Amersfoort appear to be extreme- ly sensitive to the correction index (Table 1.3b). This is likely due to the fact that U and Th concentrations appear to be low (less than 1 ppm) so that slight contamination with environmental Th may be suf- ficient to introduce large uncertainties. Unfortunately an uncalibra- ted spike was used at the time of these measurements so that the ab- solute U and Th concentrations could not be accurately determined. However, the data show that in this case the relatively simple ap- proach as suggested by Vogel and Kronfeld is insufficient to obtain reliable dates for peat. Table 3.2 shows that U ard Th are not uniformly distributed over the extracted fractions but that the majority is contained in the humic and fulvic acids (fractions E3H and E3F). These observations are in good agreement with studies showing that humic and fulvic acids are responsible for U and Th accumulation by the formation of insoluble organo-actinide complexes (Szalay, 1958; Borovec et al., 1979; Kffbek and Podlaha, 1980; Halbach et al., 1980; Nash and Chop- Table 3.1a. Radiometric data obtained from analyses of peat samples from Tenaghi Phillippon, core TF II
Sample % loss* on Analysis no. U cone* '"0/ ;1*U r3;Th/"'1"u ''3 " Th/''? Th Leachate age (ka) (depth m) ignition (ppm) Expected** Corrected Uncorrected (f=1-0) (f=0)
.276 81 G-81026 2.8 0.92±0.08 0.12±0.04 1 1 ±6 8.49±0.06 13± 4 14± 5 (4.5-4.7) (GrN-5719)
,279 75 G-810491 1.9 1.02±0.08 0.25+0.03 1.5+0.2 17.58+0.10 11+2 31 + (7.6-8.1) G-8400iJ (GrN-5722)
10 •280 88 G-81025 8.0 1.16+0.09 0.23+0.07 6.0+1.9 32.41+0.34 20+ 2 28± (12.5-13. 0) (GrN-5723)
• 281 70 G-81037 11.0 0.85+0.07 0.40+0.04 13+4 43.8 +1.5 48+ 56 + (13.0-15. 5) (GrN-5724)
? >282 80 G-81032 6.4 1.15+0.15 0.45±0.07 12 + 5 46.7 ±1-8 5i±] - 64± (16.0-16. 5) (GrN-5783)
10 10 4.0 56 + 55± •283 70 G-81043 19.n 1.04+0.12 0.40±0.05 67+38 51.5 + 2.7 (16.5-1"'. 0) (GrN-5784)
3.0 •284 67 G-81017/34/35 8.0 0.94+0.09 0.34+0.05 8.7+2.3 49.1 - ± 7 2\ (17.5-18. 0) (GrN-5781)
33 130 + ,287 84 G-81033 2.5 0.87+0.12 0.68±0.19 5.2+1.5 64-125 102+ 25 56 (24.0-24. 5)
* Related to air-dry weight of the total sample **Basod on l"C measurements in Groningon Table 3.1b. Radiometric data obtained from analyses of peat samples from Amcrsfoort, do Llondert
Sample % loss* on Analysis no. n cor.c. * Tii/ 'Li ' Th/ • Th Leachate age (ka) (depth m) ignition (ppm) Expected** Correction Corrected factor f
0.9 79 G-81061 n.05" 1.61+0.31 0.61+0.14 1.69+0.53 60.1+ 58 AFII. 1 "0.8 O.f (6.61-6 .69) (GrM-6884) 1.0 49
-7 93 G-81060 0.03 0.90 + 0.40 2 . 3 '<* 0.8 9 1.44+0.44 63.8+ 61 AFII. 2 L1 .4 2.0 (6.98-7 .10) (GrN-7040) 1. 52
1 .5 G-81062 b.d.l. 63.5+ AFII.3 '1.2 (7.27-7 .37) (GrN-689 3)
AFIII 86 G-81059 1.00 + 0.33 1.72+0.47 1.02+0.18 67.5±!|!|"8 1 .3 58 (8.50 : .4 73
* Related to air-dry weight of the total sample. - Only the 3 ;i HCl leachate was measured. tt An uncalibrated spike was used so U conci-nt rat ion:; arc ->nlv •:•" '",•• inliT ar the qiver. n ** Based on !'C measurements at Groningen, using thermal diffusion isotope enrichment. N)
Table 3.2. Results from U/Th measurements on extracted fractions of different peat samples
Sample Inorganic* Fraction U cone* ??''"U"U/:/'' 'aUe U: •' ~Th/' Th/-'":? U U U •:'Th/? Th Age according Fractions ('" 'Th/' ;Th)** U/Th age** content (ppm) to other for dating (ka) (wt.%) methods (ka)
Tenaghi Philippon, Greece
.279 15 E2 0 .69 1.12±0.10 0.13 ±0.04 1.91+0.92 «o o 4.6 (7.60-8.10) E3H 2 .16 1.01+0.06 0.47 +0.03 1.34+0.06 17.58±0.10"' E3H 1.04+0 .03 19-8*4.3 E3F 0 . 16 1.33+0.27 0.17 +0.03 2.08+0.41 (GrN-5722) E3H+E3F 1.04±0 .03 .. -.5.2 E5 0 . 12 0.82+0.10 2.49 +0.26 0.93+0.05 l9-1±5.0
.284 60 E2 2.07 1.02+0.04 0.143+0.008 4.34+0.43 49.1±3;^» (17.50-18.00) E3H 1 1.4 1.08±0.04 0.79 +0.04 8.78+0.38 E3H 4 .55 + 0.19 »« \l E3F 21 .1 0.93+0.13 0.027+0.009 11.1.11 ±1. ±1.4 4 (GrN-5727) E3H+E3F 2 .52 + 0.11 E5 0 .37 1.05+0.11 1.92 +0.15 2.17+0.09 31-5±2.6
• 287 10 E2 7 .37 0.98±0.07 0.015+0.008 0.60+0.44 (24.00-24.50) E3H 6 .43 0.99+0.05 1.10 tO.05 7.05±0.40 E3II E3F 2.09 1.13+0.09 0.21 ±0.05 2.26+0.61 64-125 '> E3H+E3F 4 .61+0 .20 E5 0 .07 1.15+0.44 8.2 i2.2 1.82±0.23
Tervola, Finland •114 47 E2 18 _ 7 1.20+0.02 O.13O+O.OO8 12.2 +1.8 (3.50) E3H+E3F 4.42 1 . 14 + 0.03 4.4 +0.21 12.9 +0.5 E4 0 .92 1.13+0.04 4.01 +0.14 4.77+0.13 (Su-688) E5 0 .07 1 .09 + 0.15 1.76 tO.22 5.4 +1.0 TL 55
• 113 55 E2 1 1. 1 1 . 19±0.02 0.102+0.005 13.5 +1.9 48.0 + ^ (3.65) E3H+E3F 8 .1 1 1.19 + 0.01 1 .98 +0.06 10.5 +0.3 E4 0.6 5 1 .08 +0.05 3.42 +0.18 2.67+0.18 (Su-688) E5 0.02 1 . 14+0.59 1.85 +0. 1 . + 0.9 to 3.2 +1 E1 0.19 1 .08 +0.09 0.07 +0.02 48.1 +3.5 E3F 5.79 0 0.08 0.30 10.0 3 11.0 +0 2.4 E3II 3.72 1 40±C0.21 3. 39 ±0.48 11.6 ±0 (Su-689) E4 2.64 0.16+0.00.02 1.21 .28 ±0.04 4.88+0.10 TL 5 5 • E5 1 .75 1.22+0.03 O.136±O.O1O 2.08+0.24
Uncertainties are based on 1: counting errors. U/Th ages are calculated according to eq. 1.17-1.10 (set- text) using cither traction E3H or E3H + E3F. Fraction E5 was used fur environment.a 1 Th corrections. * Related to air-dry weight of the total sample. ** Calculated according to eqs. 1.17-1.19 ;) No fractions could be found to yield ages between 0 and 350 ka •) Age determined from c. ') Estimated by Wijmstra and Groenhart (1983) from comparison of deep-sea !"o-stage boundary ages and pollen zone boundaries. ) Thermoluminescence dates (TL) from Hiitt et al. (1982). 73 pin, 1980; Nash et al., 1980; Shanbag and Choppin, 1981). Further- more, in some cases the 0.5 M HC1 (pH = 0.3) extract also contains a vast amount of U but considerably less Th. This can be adequately explained by the observation of Szalay (1969) that U may be almost entirely released from humic or fulvic acids in a pH = 0.6 HCl solu- tion and may be regarded as indirect evidence for the existence of such complexes. The presence of 232Th in the 3 M HCl leachates of the peat ash need not be associated with the organic fraction but can be attribu- ted to partial dissolution or leaching of the inorganic fraction. However, the presence of 232Th in the 1% NaOH extracted fractions (Table 3.2) can only be explained if 232Th is associated with the organic fraction because the leaching capacity of a 1 % NaOH solution for the inorganic phase is low. This we established by leaching the ash of combusted Tervola peat (ct115) with 1% NaOH, yielding < 2% of the U activity and < 0.01% of the Th activity in the 1% NaOH soluble fraction of the total sample. We assume that the acid environment, a result of the formation of humic and fulvic acids, is responsible for dissolution of 232Th originally associated with the inorganic mate- rial. Following its dissolution it is immobilized by the formation of stable Th-humic/fulvic complexes. This obviously raises the question whether closed-system behaviour may be invalidated by the introduc- tion of 230Th into the organic fraction by the same mechanism. In section 1.3.3.2 we assumed that in-situ contamination of the organic fraction with environmental Th as a result of ion-exchange processes occurs on a time scale relatively short with respect to the 230Th half-life. If the combined humic and fulvic acid fractions are assumed to be dateable, UTD and 1'*C are in good agreement for Tenaghi Phillippon sample cx279. The a284 appears to be too young, indicating that the amount of contaminating 2 3 °Th is overestimated. The agreement between UTD and the expected age for a28 7 is good (Table 3.2). On the other hand, for Tervola no reliable UTD dates could be obtained. From the high U concentration and the relatively low 23oTh/231*U activity ratio in E2 we conclude that U is preferentially extracted. This leaves the remainder of the sample depleted in U, 74 which is confirmed by the high 23OTh/23"u activity ratios in E3 and E4. This clearly shows that the chemical binding of U (and probably Th as well) is different for Tervola samples than it is for Tenaghi Phillippon samples. We therefore re-analyzed a113 without applying the 0.5 M HC1 extraction. In that case we observe that most of the U and Th is still contained by humic and fulvic acids (Table 3.2). How- ever, U concentrations in E4 and E5 are higher in this case. We at- tribute this to incomplete separation of phases. This is of particu- lar importance when (organic) fractions of high U and Th concentra- tions are not completely separated from (inorganic) fractions of low U and Th concentrations and results in a relatively large memory ef- fect of the U-rich fraction in the U-poor fraction. Hence, the meas- ured activity ratios do not necessarily represent the true value for the fractions and therefore a qualitative interpretation of these data is preferred.
3.2.3.3. Preliminary conclusions The fair agreement between UTD and lhC dates for Tenaghi Phil- lippon is in accordance with closed-system behaviour. Furthermore, the observed speciation of U and Th among various constituents of peat supports the conclusions of other investigators that ion-exchange is dominant in U and Th geochemistry in organic sediments. Apparently the formation of (insoluble) actinide-humic/fulvic complexes strongly limits the mobility of these nuclides. We found evidence that in situ ion exchange occurs as a result of dissolution of Th and/or U from the inorganic phase and subsequent complexation with organic acids. However, our data are still too limited to conclude either on the rate of exchange or on possible isotope fractionation processes. Nevertheless, the combined humic and fulvic acid fractions con- stituting the majority of the organic peat fraction, appear to act as a closed system with respect to U and Th. They may be considered dateable if in-situ ion exchange with the inorganic fraction has been limited which is indicated by a relatively high 23OTh/232Th activity ratio. In that case the chemistry can be simplified to a two-phase separation: a dateable organic phase and a contaminating inorganic 75 phase. In the next section we compare various chemical approaches used for separation. Furthermore, we will discuss the implication of relatively high 232Th activity in the organic phase.
3.2.4. TWO-PHASE SEPARATION 3.2.4.1. Measurements Samples from La Grande Pile, Tenaghi Phillippon (core TFII), Tervola, and Oerel were separated into two phases by leaching the sample ash. The leachate and the residual material were analysed separately. Two samples from La Grande Pile, four from Tenaghi Phil- lippon, two from Tervola, and two from Oerel were repeatedly analysed by using an increasing concentration of HNO3 in the leaching step. This yields a number of extracted fractions for which the U and Th specific activities, normalised to the specific activity of 232Th, are plotted in Figs. 3.2a through 3.2j. For samples from La Grande Pile another separation was carried out by selective dissolution of the organic phase in a 1 % NaOH solu- tion. The residual material, containing visible organic remains which indicates incomplete separation of phases, was rejected from further analysis.
3.2.4.2. Discussion If peat shows closed-system behaviour while exchange of environ- mental Th between the inorganic and the dateable organic phase is un- fractionated, the plots in Figs. 3.2a-3.2j should produce straight lines according to eqs. (1.15) and (1.16) with slopes representing the activity ratios in the pure dateable phase. A straight line fits the plotted points excellently as indicated by the high correlation factors. However, Table 3.3 shows that for all samples the corrected ages are systematically much lower than those expected. This implies that, based on the 232Th activity, the applied correction technique overestimates the amount of common Th (or at least common 230Th). This is consistent with our previous observation (section 3.2.3.2) that at least part of the 232Th should be associated with the date- able organic phase. Supporting evidence is provided by the presence of substantial amounts of 232Th in the 1% NaOH extracted fraction Figure 3.2. Isochron plots showing normal- 23Z 23i 232 23B 232 23o 232 23i 232 ized activities with respect to Th for ( u/ Th) = 1.031 ( u/ Th) •0.177 ( Th/ Th)=0.189( u/ Thh0.873 various extracted fractions from peat ±0118 ±0.115 ±0.151 ±0.178 samples. If Th is a good measure of correlationcoeff: 0.96 corretaHoncoeff: 0.45 the degree of contamination with envi- ronmental Th, each set of plotted 2.0h 2.0 points should produce a straight line (eqs. (1.15) and (1.16)) with a slope Grande Pile representing the activity ratio in the OC182 pure dateable fraction. -1.0 /*
1.0 2.0 1.0 2.0 (?3au/232Th) AR (234u/232Th) AR
(?34u/n?Th) = 1 078 I ?38u/232Th)*0059 123OTh/232Th)=0150 (234u/232Th)*O 746 ±0055 ±0053 •0.054 ±0058
correlaMoncoeff: 0 99 correlationcoeff: 075 + • H2.0
Grande Pile CL184 + 8 -1.0h * 1.0 ~ •f »- •*• • 4-
i I B l 1.0 2.0 9 4u/?32Th) AR (23Bu/?3?Th) AR avwijEj/m^i 77 Q O 9 1 1 1 ffl: m o o _ Q o (4 1 U CM +
a: '23 2 si— 4 ' < *± Rj aa o O O .0 9 ( .0 3 " O + 1 if 3 .f oef f 1— —' p> o ID % i 1is 1 U 1 S 1 2 3 o 8 OB a* Q ex. CNI -, '—! ' "— tu —" S+ 1\ JZ: i— <
\ • f sOJ \ J <=i •- j Q • S d ' \
>f f 1 3 o I0I J . w OJ OJ "QJ t "a o — , " I LL t_l 1 0) O Q CM
av (MI7 av
o ' o CNi — 1 1 T \ • • .5 2 .4 0 + + 1 :0.4 1 t i— i. Pi •1 r ?3 t CO CO CO o o d
± 0 03 5 | cu J o 1-1
••C atio n lationcoef f :0.7 1 "IT - re l g o to "o £! •>• ^
85s »^
o ).O 3 f o+ 1 J
•i • s. :1.0 0 X o o +1 II 41 o coef f
ion i \> i ro 1\ or r correlatio n a. i u 1 1 i • ^ o CM 00
—i—i—i— 234 232 238 32 2 232 [''•' u/232Th) = 0.986(238u/ "Th)*1 07 (23OTh/232Thl = 0.164(23'u/ Th)+0.60 U/ Th)Li.125( u/' Th):0.83 ±0.010 ±0.63 ±0.009 ±0.62 ± 0.022 ± 0.51 correlationcoeff: 1.00 correlationcoeff: 1.00 correlationcoeff: 1.00 alOOh- y 20 , y, Tenaghi Phillippon Omitted [Tenaghi Phillippon / i a 280 H15. a 285
-501- s / 10 - y H 5 , i , I . —i.. i • i i i i i 50 100 50 100 I 20 40 20 40 6 232 234 2 » u/ Th) AR ( u/« Th) AR (236lV232Th)AR (234U/232TWAR
23 232 238 232 23o 232 234 232 ( V Th)=1.06 ( u/ Th)-0.2 • ( Th/ Th)=0.45( u/ Th)< •1.6 • 234 232 23e 232 i2 234 232 ( u/ Th) = 1.031 ( u/ Th]+0 014- Th) = 0.190 ( u/ Th) + 3.21 ; ±o.io ±1.0 ; 10.14 i 1.2 . ± 0.001 ± 0.086 ±0.024 ±0.50" correlationcoeff: 0 98 correlationcoeff: 0.85 30h / correlationcoeff: 1.00 1/ t correlationcoeff: 0.98 Omitted - 10 a. Hio i iol- Tenaghi Phillippon •~ ,••• Tenaghi Phillippon a 287 a 284
/ A 5 - ~\ 5 - 1Oh 5h
... i i . y, . , , . I.I. 5 10 10 20 10 20 5 10 H 23 l238u/232Th)AR (234u/232Th)AR (238u/23?Th)AR , 234u/. ,/232'Th) AR
Figure 3.2. (continued) 79 from samples of La Grande Pile (Table 3.4). In addition to this, the UTD ages deduced from the uncorrected activity ratios in both the 1%
NaOH extract and the 2 M HNO3 peat ash leachate show excellent agree- ment with the expected ages for the three samples from the St. Ger- main I-C interstadial (Fig. 3.3). Hence, we arrive at the important conclusion that "Th cannot reliably be used as a correction index. The fair agreement between expected and UTD ages deduced from the activity ratios in the mildest leachate indicates that the latter represent U and Th associated with the organic material fairly well, although the absolute concentrations can be much lower (Tables 3.3 and 3.4). However, additional environmental Th may be introduced during the laboratory treatment. Stronger leaching of the sample ash
(using a higher HNO3 concentration in the leaching agent) results in deterioration of the correspondence between deduced and expected ages. Fig. 3.4 shows a general trend of an increase of the uncorrected leachate age with the leaching strength. For Tervola and Tenaghi Phillippon the slope of the fitted line is relatively small and of the same order of magnitude. For samples from La Grande Pile the slope is larger, but again of the same order of magnitude for both samples that were analysed. The samples from Oerel show an extremely steep rise of the uncorrected leachate age as a function of leaching. These
Grande Pile (peaf) I ^' ag e OH ) 100 - r Figure 3.3. Comparison of deduced ages (expressed in thousands of years, ka) obtained from two dif-
hat e ( V ferent laboratory treatments for LJ peat samples from La Grande Pile, 50 • France. The squares show the region ai <_i of expected age (lower left square: a; the third interstadial during the / Lanterne (Weichsel) II glaciation, Unco r dated with lhC at 40.0 ± 0.6 ka, ft i upper right square: St. Germain I). 0 50 100 Error bars are based on lo counting Uncorrected leachate (2M HN0 ) age (ka) 3 statistics. 00 © Table 3.3. Radiometric data obtained .TOE van .ous peat samples * ? 0 J Sample % loss* on Analysis** leaching U cone "U Th/-• -u °Th/'' ?Th Leachte age (ka) agent (ppm) (depth (m)) ignition (M Htio ) expected uncorrected corrected La Grande Pile 1 22 H-3826 L 2. 0 0. 29 1 .09+0. 06 0. 48+0 .04 0 .96±0. 07 40.0+0.6 70+6 17+7 ' (9.05-9.25) H-3826 R 1 .25 1 .07±0. 04 0..97 + 0.06 0 .81+0. 03 (GrN-8746) 1186 36 Ii-3825 L 2..0 0. 12 1 .17+0. 07 0..59±0 .05 1 .03+0. 08 95-115"' 94*J« 24.0±3.5 (12.44-12.59) H-3825 R 0. 69 1 .07+0. 05 0. 87 + 0.05 0 .89+0. 03 c.183 38 H-3822 L 2..0 0..07 1 .24+0. 13 0..60 + 0.08 1 .08+0. 15 95-115?) 39.0±9.0 (13.38-13.53) H-3822 R 0.,55 1 .10+0. 03 0 ..76 + 0.03 0 .89+0. 03 "tl 1185 37 H-3823 L 2,.0 0.,08 1 .18±0. 15 0 ,.66 + 0.07 1 .14+0. 1 1 95-1 15"' 38.5±7.0 (13.43-13.57) H-3823 R 0..57 1 .09+0. 05 0,.96 + 0.04 0 .90+0. 03 ;) •-.182 38 H-3824 L 2..0 0..12 1 . 10 + 0. 09 0 ..65iO .06 1 .29+0. 12 9 5-115 2 ^ + (13.93-14.08) H-3824 R 0,.69 1 .04+0. 03 0..96 + 0.04 0 .93+0. 03 Tenaqhi Phillippon, Core TF II .280 69 G-87001 L 2 .0 26 0 .99+0..01 0 . 1 7 + 0.02 11 + 2 13.41+0.34 20.5 + 1 .8 19.4±1.2'> (12.5-13.0) G-87013 R 14 1 .02+0,.01 0 . 37 + 0.02 1 3+1 (GrN-5723)
3 0 3 3 66 G-87004 L 2 .0 .6 1 .04+0..03 0 . 30 + 0.02 7.7+0,.5 49 1+ - 38 8± 22.Si / /' ,284 8 2.1 - U8 (17.5-18.0) G-3 7016 R 2..9 1 .0.1 + 0..02 0 .62 + 0.03 4.9+0..3 (GrN-5781) 1 ,285 85 G-87007 L 2 .0 6 . 5 1 .12+0..03 0 . 53 + 0.03 21 + 2 • 52 81.0±J;6 7.8 ' (21.5-22.0) G-87019 R 0 .3 0 .94 + 0.07 1 .38 + 0.12 5.6 + 0.7 (GrN-5784) 1 ,287 K6± 74 G-87010 L 2 .0 4 .2 0 .94 + 0.05 n .50 + 0.04 6.2 + 0.5 64-125' ' 5.8 (24.0-24.5) G-87022 R 1 .0 1 .00 + 0.04 0 .74 + 0.05 3.8 + 0. 3 Tervola 7 l) 61 /G-86404\L 3 .5 19 1 . 16 + 0.15 0 .31+0 .05 11+2 4o.4± 7;6 9 + 3 (3.65) 10 1 .22 + 0.03 1 .57 + 0. 19 9 + 1 (Su-688)
.115 66 /G-85134vL 3 .5 12 1 . 18 + 0.09 0 .33 + 0 .05 7.7 + 1.4 8 + 4 ' (3.80) (G-86408 ) VG-864 1 5-R 6 1 .09 + 0.25 i .05 + 0.28 7.3 + 2.8 (Su-689) Oerel • 50 67 H-522 L 0.5 0.33 1.13±0.07 0.43+0.03 1.57+0.13 50.2+0.7 61 + 16 22.2 (2.76-2.79) H-522 R 0.99 1.04+0.03 0.90+0.03 1.04+0.03 (GrN-12372)
.51 14 H-523 L 0.5 0.09 1.00+0.12 0.38+0.04 1.13+0.11 50-56 52.8+5.6 29.9 (2.95-2.99) H-523 R 0.67 1 .04 + 0.02 0.63 + 0.02 n.71 + ft.O2 (GrN-12372/3) 1 .9 53 i) .52 81 H-524 L 0.5 0.23 0.84+0.10 0.70+0.08 1 . 13 + 0.09 57.3 + 138 + (3.94-3.97) 1.3 30 H-524 R 0.53 1 .11 + 0.04 0.94 + 0.04 1 .03 + 0.03 (GrN-13037) .142 72 H-520 L 0.5 0. 26 1.26+0.11 0.27±0.03 0.94+0.09 41 .6±0.3 33±15 2.9 (1.S9-1.67) H-520 R 1. 32 1.15+0.03 0.63+0.02 n.87 + 0.02 (GrN-12314) a14 3 87 H-521 L 0.5 0. 28 1 .36 + 0.13 0.40 + 0.04 1.21+0.12 42.5+0.4 54 + 18 (1 .67-1.76) H-521 R 0. 51 1.16+0.06 1.38+0.07 0.92+0.03 (GrN-12315)
* Related to air-dry weight of the total sample ** "H" denotes analysis in Harwell, "G" in Groningen, "L" corresponds to leachate, "R" to residue :" Correction according to eqs. 1.15 and 1.16. M Based on Isochron-plots in Fig. 3.2 2) Based on correlation of deep-sea 'r 'O-stage boundary ages with pollen zone boundaries after Woillard and Mook (1982) ') Like 7), Wijmstra and Groenhart (1983). Other expectations based on ll4C. All errors quoted are 1 * statistical uncertainties due to nuclear counting only.
Table 3.4. Radiometric data obtained from the 1% NaOH soluble fraction of peat samples from La Grande Pile
Sample Analysis no. U cone* 23*u/23SU 23°Th/2**U 23cTh/23:Th Leachate age (ka) (depth mi (ppm) Expected Uncorrected
a184 G-86426 ca. 0.40 1.113+0.05 0.517+0.022 1.13±0.05 40.010.61' 78± 5 (9.05-9.25) (GrN-8746)
8 a183 G-86425 0.54 1.14±0.03 0.685+0.023 1.07±0.04 95-115 2) 121 + (13.38-13.53)
a185 G-86427 0.56 1.20±0.04 0.628+0.021 1.05+0.03 95-1152 103+ 6 (13.43-13.57)
a182 G-86424 0.58 1.24+0.04 0.721+0.027 1.04+0.04 95-115 2) 130± 10 (13.33-14.08)
* Related to air-dry sample weight. 1) Based on "C measurements in Groningen CD 21 Based on correlation of deep-sea l eO-stage boundaries and pollen zones (Woillard and Mook, 1981). .'••#••-'-. •
82
Grande Pile Tervola Oerel
300 a 113 -. I I CX52 >i00>i00 200 \ I 100 -t- 0 300 ai82 I"
200 Jt"" 100 -' -•- an 0 0 10 0 5 10 0 10
Tenaghi Phillippon
300 a 280 C128A
O 200
100
0 300 a 265 a 287
200
100
0 10 0 10
Concentration HN03 in leaching agent (mole.I )
Figure 3.4. Effect of increasing HNO^ concentration of the leaching agent on the deduced uncorrected leachate age for 10 peat samples. Uncertainties are based on 10 statistical counting errors. The dashed lines show the least squares linear fit to the data points. 83 trends can be explained as follows. The U and Th containing minerals soluble in the leaching agent introduce "environmental" U and Th of relatively high 23OTh/231*U activity ratio. The result is a net in- crease of the 23OTh/23I1U ratio (and therefore the apparent age) with increasing strength of the leaching agent. The amount of soluble minerals ii the peat ash strongly depends on the composition of the inorganic i action after ashing and hence on the locality of the peat origin.
3.2.4.3. Conclusions The results discussed in sections 3.2.3 and 3.2.4 confirm that in peat (part of the) 232Th is associated with the dateable organic phase. The source of environmental Th appears to be in-situ ion ex- change between the inorganic matrix and the dateable phase. Although this violates the principle of zero initial Th in general, it appa- rently does not always violate the principle of zero initial 230Th, considering the fair agreement between uncorrected UTD and expected ages. This apparent anomaly obviously raises the question whether such agreement is fortuitous or the result of closed-system behavi- our and previously overlooked geochemical or physical mechanisms that allow 232Th to be taken up by peat in much larger amounts than 230Th. In the next section we will discuss a possible mechanism that predicts such a phenomenon.
3.2.5. MECHANISMS FOR SELECTIVE UPTAKE OF 232Th 3.2.5.1. Recoil effect: depletion of the grain surface So far the geochemistry of U and Th in pt=at can be adequately explained in terms of closed-system behaviour with the exception of one detail: the apparent selective uptake of 232Th with respect to 230Th by the organic fraction of peat. In this section we will specu- late on the origin of such a phenomenon, realizing that the good to excellent agreement between uncorrected UTD dates and independent evidence requires some explanation. Based on the results obtained so far we assume that the conver- sion of organic matter to humic and fulvic acids leads to increased leaching and dissolution of U and Th from the inorganic matrix. Sub- 84 sequent complexation with these acids strongly limits the mobility of U and Th in the organic fraction of peat. Furthermore, we assume that the inorganic grains can be divided into a leachable surface layer and the bulk material• The depth of the surface layer is defined by the recoil displacement of 2dl)Th. of the order of 55 nm or more, depending on the type of material (Ki- goshi, 1971). Before the inorganic detritus enters the peat as a re- sult of erosional processes and subsequent transport by water and wind induced movement (Fig. 1.3), its surface is exposed to (contin- uous) removal of Th by leaching in natural waters. The relatively high age of the mineral guarantees that most 230Th is the result of in-situ decay of 23kU. Hence the majority of 230Th is located at interstitial or damaged lattice sites as a result of the alpha recoil displacement which renders it more vulnerable to leaching than 232Th. As a result it is preferentially removed from the grain surface (Fleischer, 1980). Furthermore, Z3I*U that occupies lattice sites within the recoil distance from the surface gives rise to 25% of the produced 230Th escaping from the grain. This can be simply derived from the fact that the flux of recoiling nuclei, F , from an in-
finitely large solid is given by (see e.gu Kigoshi, 1971)
Frec = XpR/4 (3.1)
where A represents the decay constant of the parent nucleus, p is the density distribution of the parent nucleus and R is the recoil distance of the daughter. As a result of these two processes the surface of the inorganic material is depleted in 230Th with respect to the bulk of the grain. Similar processes (alpha recoil ejection of the 238U daughter 23ltTh from mineral grains (Kigoshi, 1971)) appear to be responsible for the observed depletion of 2ih\5 with respect to 238U in minerals (see also section 1.2.4 and Osmond, 1982; Fleischer, 1983) and enrichment of the same in waters (Fronfeld, 1974; Szabo, 1982).
3.2.5.2. Leaching of the inorganic fraction: steady state When mineral grains become incorporated in peat their surface is 85 subjected to stronger leaching by agressive humic and fulvic acids. Because of the limited presence of 230Th as a result of the previous- ly discussed recoil effect, this initially results in a selective transfer of 232Th to the organic fraction. For example, consider a spherical mineral grain of 1 ym radius and a homogeneous distribution of 232Th. Assume that the surface layer of 0.06 um is totally deplet- ed in 230Th. Then simple calculation shows that by dissolving the total surface layer the amount of 23OTh-free Th that becomes incorpo- rated in the organic fraction is as much as (1 - 0.943 =) 17% of the total. When Th in the surface layer has disappeared the bulk will be leached by humic and fulvic acids. This can be modelled as a first order linear process:
dTh /dt = -ATh + kTh. (3.2) or or in where the subscripts "or" and "in" denote the organic and the inor- ganic fraction, respectively, X represents the decay constant of the Th isotope and k is a chemical removal constant that determines the rate of transfer of Th from the inorganic to the organic fraction. Its value is by definition the same for all Th isotopes. A large value for k implies instantaneous, and a small value continuous ex- change of Th. Under steady-state conditions (dTh /dt = 0), which implies that the transfer of Th from the inorganic to the organic fraction balanc- es its decay in the latter fraction, the ratio between the Th activi- ties in the organic phase and the inorganic phase is given by
Th /Th. = k/A (3.3) or in If the isotope enrichment factor a is defined as
a = 2L in then by substitution of (3.3) in (3.4), we calculate for a steady state a = 232A/230X = 5.3 x 10~6. Hence, if such conditions prevail 86 in peat, the amount of 230irh that becomes incorporated in the organic phase is 5 to 6 orders of magnitude smaller than the amount of 232Th, even with equal transfer rates of 230Th and 232Th from the inorganic matrix to the organic phase. We now consider three different values for the chemical removal constant k. (1) k >> 230A implies instantaneous transfer of all Th. It requires extremely high to infinite 232Th /232Th. activity ratios which we have never observed. Hence we reject this possibility. 230 232 232 (2) k s A implies a steady-state ThQr/ Thin activity ratio of approximately 105. Furthermore, it requires that steady-state con- ditions are reached during approximately the range of UTD dating. None of our measurements show such high activity ratios, not even at higher age (100 ka or more). Therefore, we conclude that k can not be of the order of 2 3 ° A. (3) k << 230A (k ~ 232A). In this case steady-state implies a 232Th / 232Th. activity ratio of the order of unity which we frequently observe. However, if 232Th is zero initially, a much longer time than the UTD dating range is required to establish a steady state. Nevertheless, it is likely that the selective transfer of 232Th from the inorganic to the organic fraction provides the proper initial 232Th for a steady state if k is of the order of 232A. This does not necessarily imply that a steady state is reached for 230Th as well. However, if initial 230Th is essentially zero as a result of the processes discussed before, the above condi- 23 23 tions imply that °Thor/ °Thin activity ratio can only be less or equal to the steady-state value of the order of 10~5 to 10~6 . We conclude that the most likely explanation for the unexpected agree- ment between uncorrected UTD and expected ages is provided by assuming that the surface of the inorganic material in peat is strongly deple- ted in 230Th as a result of recoil ejection from the grains before deposition. Hence environmental Th transferred from the grain surface to the organic fraction on a relatively short time scale with respect to the 230Th half-life consists mainly of 232Th. Subsequent leaching of the bulk of the inorganic fraction is a delayed process which in- troduces only minor amounts of Th to the organic fraction. Further- 87 more, in the latter process the ?30Th transfer is 5 to 6 orders of magnitude smaller than the 232Th.
3.5.2.3. Experimental observations Recoil ejection of Th may indirectly be verified by laboratory experiments that determine recoil supply of 230Th from mineral to liquid. Such experiments were performed by Kigoshi (1971) who measur- ed a significant 231(Th increase with time in the aqueous phase of a system consisting of fine zircon powder (particles about 1 to 10 pm in diameter) and dilute nitric acid. Basically the 238U to 231*Th decay does not differ from the 231|U to 230Th decay so that his re- sults can directly be applied to our problem. Hence we may conclude that the experimental observations of Kigoshi are consistent with the existence of a grain surface that is depleted in 230Th. Obviously the degree of depletion depends on a variety of condi- tions. The geological history of the mineral grains, the rate of weathering as well as the composition of the mineral grains are all important factors that determine to what degree fractionation between 230Th and 232Th will occur. Therefore, based on our present limited knowledge of these processes, it is impossible to establish a simple general model that predicts the proper factors to correct for conta- mination with environmental Th. It can be concluded, however, that conventional correction models that assume unfractionated transfer of 230Th and 232Th from the inorganic to the organic fraction overesti- mate the 230Th contamination and hence underestimate the true sample age.
3.2.6. DISCUSSION OF UTD DATES ON VARIOUS PEAT SAMPLES We have attempted to determine the absolute ages of various samples from different sites as described in section 3.2.2. The re- sults of the radiometric U and Th determinations not yet discussed are given in Tables 3.5 through 3.8 and are discussed in the follow- ing. The (simplified) pollen diagrams of Tengahi Phillippon and La Grande Pile, dated with xllC and UTD (Fig. 3.6), are used for a tenta- tive correlation with deep sea 18O stage boundaries as originally de- fined by Emiliani (1955) and dated by Kominz et al. (1979), who used 88 Uncorrected 230Titf36U age(ka) Uncorrected 230Thi/, 23AU age(ka) 50 100 50 100
Tenaghi Phillippon Tenaghi Phillippon Core TF H Core TF HI
•o- 3M HCl Leachate 2M HNO3 Leachate
10 o 2M HN03 Leachate 10
2! 2 20 2 20
CL 01 Q
30 30
Figure 3.5. (a) Uncorrected UTD dates for samples from Tenaghi Phillippon, core TFII. Open points are based on analyses of the 3 M HCl (circles) and 2 M HNO-i (squares) peat ash leachate. Closed points represent ll*C re- sults. The dashed line represents extrapolation of the dates from three analyses that appear most reliable (see text). (b) Uncorrected UTD dates from samples from Tenahgi thillippon, core TFIII. Error bars are based on la counting statistics. the 230Th excess dating method and constant accumulation of aluminium to extend the TUNE-UP time scale by Hays et al. (1976). In Fig. 3.7 this correlation is visualized by using "stretched" pollen diagrams from Tenaghi Phillippon and La Grande Pile to obtain a linear time scale. Furthermore, Fig. 3.7 includes the paleo-temperature curve ob- tained by Grootes (1977), based on lkC dates of various pollen dia- grams from North Western Europe.
1. Amersfoort, de Liendert (Table 3.1b) The low concentrations of u and Th in samples from Amersfoort render UTD ages questionable. Nevertheless, their importance in Western European palaeoclimatological research {e.g. in defining the Table 3.T. Radioroetric data obtained from analysis of peat samples from Tenaqhi Phillippon, core TFIII
* * Sample % loss* on Analysis U cone. * '•u/ u •' Th/> '•Th Leachate aqe (ka) (depth m) ignition (ppm) Uncorrected Corrected
.26 66 G-87041 L 19. 6 1 .008+0. 006 0. 043±0. 002 3.29±0. 17 4.77+0.23 3.30+0..45 (5.99-6.00) G-87042 R 1 .0 1 .01 +0. 05 0. 18 n. 02 1.26^0. 16
.27 80 G-87043 L 11 . 8 1 .no 02 0. 082±0. 005 3.96±0, 30 9.28+0.59 7.4 + 1..1 (8.5 9-8.60) G-87044 R 0. 3 1 .36 + 0.16 0. 46 ±0. 07 0.89+0. 14
G-87045 20. 0 1 .01 + 0.01 0. 19 ±0. 01 6.50+0. 34 23 i± 4 10.8+2 .28 65 L - i: 3 .2 (10. 20) G-87046 R 4. 9 1 .01 + 0.03 0. 50 ±0 . 03 4 .06 + 0.22
.30 65 G-87049 L 10. 5 1 .03 ±0. 01 n , •> c, ±0. 01 3.82+0. 19 18.4 + 1 .0 7.9 + 1.4 (14. 20) G-87050 R 3. 2 1 .03 ±0. 03 0. 51 ±0. 03 2.44+0. 08
.31 74 G-87051 L 27. 9 1 . 01 ±0. ,01 0. 29 + 0 .,01 6.31+0. 24 36.8±2;2 13.6+3 .0 (16.,30-16.31) G-87052 R 2. 8 1 . 04 ±0. ,04 0. 68 ±0. 04 4.52+0. 21
74 G-87053 L 6. 0 1 . 04 ±0,.02 0. 21 ±0,,01 4.29+0. 36 25 0i B.6±l - .32 - K6 _ 7 118..9) G-87054 R 3. 9 1 . 09 ±0..04 0. 42 ±0 .02 3.30±0. 14
4. 1 . + 0, 0. 37 ±0,.02 4.06±0. 23 49 4± .33 73 G-87055 L 0 ,05 .02 - 3i2 23- $±t '\ (20..60) G-87056 R 0. 7 1 .,07 ±0 .08 1 .29 10 .09 2.24±0. 09 .34 62 G-87088 L 1 1.9 1 ,.06 + 0.01 n. .+ 11. 0 2 5.99+0. 51 36.(,±1 (22,.40) G-87089 R 0 1 ,.02 ±0 .06 ii,. Ho 1.78±0. 17 \l 1 33.4±6 .36 67 G-87090 L 1 ,.04 ±0 .01 i' , 1 ?!). 0 1 8 9.13+0. 62 •5 (25,.00) G-87091 R 0 . f 1 . 1 3±0 .08 . j'l * Ij . 1 1 3.68±r..46 83+^ .37 86 G-87092 L 1 n 1 .02 + 0.03 . 0 3 34.0+3. 3 8 (26 .70-26.74) G-87093 R (i 1 .44 + 0.31 i . 1 j ID .21 1.13+0. ,9 .38 89 G-87106 L . 1 1 .01 + 0.05 ii 18.9+4. 7 Bit]] 105!.. (27 .67-27.90) G-87107 R n 1 .09 ±0 .07 11:'v, 9.79+0. 13 0 .39 83 G-87108 L .3 1 .05 in .03 u . •"•') 26 .4+2. 8 1)6-7* 7'.7 !3 (28 .50-28.54) G-87109 R o.4 1 .09 + 0.07 . 6 '• • 0 2.95+0.,28 r 1 2 6 4 i - .4 0 63 G-87110 L 1 .01 + 0.02 n 11) . (l > 6.9+1. ,3 >7 • * ti (29 .40-29.44) G-87111 R n .'3 1 .05 + 0.04 1 . 1 - .()(• 1.82+0..18 1341- 1 19+^ t41 91 G-87112 L l 1 .07 •0 .08 0 •I- . I'l (, 6.7+1. ,0 J4 (30 .56-30.60) G-87113 R 0 0 .69 ±0 . 13 1 : *r 0.65+0.. 10 * Related to air-dry weight of the total sample ** "L" denotes leachate, "R" refers to residue. 90 last interglacial (Eemian) and the Amersfoort Interstadial (Fig. 3.7) justifies future efforts to obtain better UTD results. 2. Tenaghi Phillippon (Tables 3.1a, 3.2, 3.3 and 3.5) The results on Tenaghi Phillippon core TFIII contain some uncer- tainties because we are not absolutely certain about the palynology for reasons mentioned in section 3.2.2.(2). Comparison with ll4C and UTD dates on core TFII shows fair agreement to approximately 20 m depth. Below that depth the UTD ages on core TFIII appear to be sys- tematically younger (Fig. 3.5b). Therefore, future palynological analyses of core TFIII and comparison with core TFII is required to provide more information on the reliability of the ages. So far, the results on samples from core TFII appear to be the most reliable, considering the agreement between various UTD dates and llfC. UTD dates hardly depend on the strength of the leaching agent (Fig. 3.4), with the exception of a287. The age of 81 i 5 ka for a285 (G-87007), taken from the middle of Elevtheroupolis (21-23 m, Fig. 3.6), suggests correlation with deep-sea 18O stage 5a (Fig. 3.7a and c). Tentative extrapolation of the ages of a280 (G-87001, a284 (G-87004) and cx285 (G-87007) (the open squares in Fig. 3.5a) yields an age of approximately 100-115 ka for a287, taken from the top of the pollen zone representing the Drama interstadial (24-27 m, Fig. 3.6), corresponding to the average of the three ages obtained from three different leaching procedures (72, 180 and 84 ka, Fig. 3.4). This suggests correlation of the Drama with i8O stage 5c (Fig. 3.7a and c). Hence the Pangaion correlates most likely with stage 5e, which is in agreement with the observations of Wijmstra and Groenhart (1983). However, the ages obtained in this thesis appear to be too old to support the proposed correlation (Van der Hammen et al., 1971) of the Pangaion to Elevtheroupolis sequence with the North West Euro- pean Eemian to Odderade sequence as dated by Grootes (1977). Figure 3.6. Simplified pollen records from La Grande Pile (left, redrawn from Woillard and Hook (1982)) and Tenaghi Phillippon (right, redrawn from Wijmstra and Croenhart (1983)). The depth scale is linear and corre- sponds to the original depths in the cores. Available 1'*C and UTD ages are given in thousands of years (kaj . 91 0 POLLEN {%) 100 Age (Ka) POLLEN (%) 100 V>r UTO UTO ' i LJ i i I - 9.75 *. 0.04 10.18 i 005- 8.49 ±0.06 H.17*. 0.10" 30.65 * 0.32 9.58 ±0.06 20.05*0X18, 28.8B±0.23 "-2O.12iO.lO, 29.09*. 0.25 2a96±0.20 -16.36iO:O9 29.74 i 0.26 - 17.6O*-O.1O 3O.S210.21 31.0-*S 34J • 0.3 40.0* 0.6 < -78*5 10 20.5t1.8 32.41*0.34 69.5, \\ 2S.0t'S 15 56.0il 43.80*1.50 46.70* ) • 49.101 lJQ 45.0,9 a 20 Trees \ Herbs 81.0J ca 130 ^ 1l2.0±21' 25- 30 L Trees/Herbs GRANDE PILE TENA6HI PHILLIPPON to ie Pollen (%) Pollen (%) 6 0 100 100 -1 -2 5 10 15 20° C Altered Billing Denekamp Hengelo Moershoofd 50 1 ) 50 Odderade Brtfrup Amersfoort IB OJ St. Germain E Elevtheroupolis • Eem 100 Drama Hioo St. Germain I-C Doxaton St. Germain FA Ji Pangaion Eemian Trees/Herbs Trees/Herbs Tenaghi Phillippon La Grande Pile Deep-sea North-West Europe van der Hammen et al. 1971 Woillard and Mook 1981 Hays etal. 1976 Grootes 1977 van der Wijk 1987 van der Wijk 1987 Kominz et al. 1979 93 3. La Grande Pile (Tables 3.3 and 3.4) Comparison of 1"c and uncorrected UTD ages for sample a184 from La Grande Pile shows that the latter is too old, pointing at contami- nation with 230Th. Isochron correction on the other hand, yields an UTD age that is too young. Nevertheless, the UTD age calculated from the leachate of the sample ash is in good agreement with the age cal- culated from the 1% NaOH extract. Hence the sample must have been contaminated in-situ with an unknown amount of 230Th, rendering it unsuitable for UTD dating. For the end (the top 15 cm) of the St. Germain I-C interstadial at La Grande Pile an average UTD age of 110± 13 ka was obtained. In- cluding the uncertainty it could represent both the end of l80 stage 5c as well as the end of 5e. However, comparison with the deep-sea chronology (Fig. 3.7c) reveals that it most likely represents the end of stage 5c which is in agreement with the chronology proposed by Woillard and Mook (1982). Furthermore, it is not unlikely that the continental environment reacts faster to climatic changes than the marine environment. Hence, in spite of the relatively low U and Th concentrations in these peat samples the UTD age agrees with the ex- pected age. Based on this result we accept the correlation between Figure 3. 7. Tentative correlation between various climatological records; of the past 130 ka, based on lkC dates and UTD dates obtained in this thesis. (a) Simplified pollen record from Tenaghi Phillippon. A linear time scale was obtained by "stretching" the linear depth scale in Fig. 3.6 wherever necessary. Stretching is based on the dates in Fig. 3.6 where lhC is used for the first 18 meters. For deeper sections the available UTD dates were used. Over the whole sequence it was assumed that deeper layers are older. (b) Simplified pollen record from La Grande Pile, constructed from Fig. 3.6 by "stretching" the linear depth scale wherever necessary to obtain a linear time scale. Stretching is based on the dates in Fig. 3.6 where 11>C is used for the first 13 meters and UTD for the upper 15 cm of St. Germain I. (c) Generalized deep-sea 180 record after Emiliani (1955). Stage-boun- dary dates after Hays et al. (1976) and Kominz et al. (1979). (d) Estimated mean July temperature in the Netherlands. Chronology ob- tained by 1*C dating (Grootes, 1977). The correlation between pollen records at Tenaghi Phillippon, La Grande Pile and the deep-sea 18O record is obvious. The available ages do not allow a correlation of la0 stages 5e, 5c and 5a with the Eemian, Amers- foort and Brffrup and Odderade, respectively. Table 3.6. Radiometric data obtained from peat samples from Les Echets, France ;3t 238 2!r ! ?3 2!? Sample loss* on Analysis Leaching D cone' U/ U 'Th/' *U °Th/ Th Leachate age (ka) (depth m) ignition agent (ppm) Expected** Uncorrected Corrected (M HNO3) a194 14 G-87025 L 2.0 0.5 1 .04+0.07 1.30+0.11 1 .00 + 0.07 30-40 (24.40) G-87026 L 4.0 1 .00 + 0.10 1.47+0.16 1 .05+0.10 G-87027 L 7.0 1.11+0.25 1.72+0.29 1.04+0.06 16 G-87028 L 2.0 0.5 1.04+0.11 1.56+0.16 0.98+0.09 50 60+110 (25.47) G-87029 L 4.0 1.05+0.14 1.45+0.19 1.13+0.13 G-87030 L 7.0 0.88+0.22 2.06+0.40 1.02+0.07 «196 12 G-87031 L 2.0 0.6 1.1910.13 1.03+0.11 0.77+0.07 62 11+ 26 (28.91-28.93) G-87032 L 4.0 1.01±0 . 13 1.20 + 0.17 0.74 + 1.09 G-87033 L 7.0 1.15+0.13 1.24+0.14 0.78i0.07 32 0.6 1.21+0.10 1.06+0.10 1.00+0,08 69 48 + a 197 29 G-87034 L 2.0 18 (29.50-29.52) G-87035 L 4.0 1.20+0.15 1.21+0.16 0.92+0.11 G-87036 L 7.0 1.06±0.16 1.27+0.16 0.93+0.06 Related to air-dry sample weight. Do Beaulieu, pers. commun. Table 3.7. Radiometric data obtained from analysis of peat samples from the valley of the Dinkel, the Netherlands Sample % loss* on Analysis** Leaching U cone.* UU/M"U 1'Th/:i:Th Leachate age (ka) (depth m) ignition agent (ppm) Expected Uncorrected Corrected Witstaart I . 117 3 G-85142 L 1% NaOH 0.35 1.47±0.08 0.60+0.04 1.41+0.07 ca. 42 1' 93 + 55± 8 (3.00-3.10) G-85142 R 0.40 0.97+0.04 0.89+0.06 0.81+0.03 22 •i174 G-85143 L 1% NaOH 0.38 1.06+0.04 0.80±0.05 1.29+0.06 42-451* 169+23 136 + 17 (3.10-3.16) G-85143 R 0.30 1.02+0.04 0.86+0.06 0.76+0.03 93 a176 G-85144 L 1% NaOH 0.20 1.2110.08 0.9110.08 1.22+0.10 46-48'> 223± (5.52-5.60 G-G5144 R 0.40 0.95+0.06 0.9710.11 0.70+0.06 48 1 24 ? u178 31 G-86669 L M HNO 3 0.39 1.35+0.11 0.73+0.06 1 .13 + 0.06 48-55 ' 130i ca. 2-\ (6.70-6.73) G-86670 L M HNO 3 0.51 1.35+0.18 0.7610.10 1.11+0.15 19 G-86671 L M HNO, 0.44 1.1010.10 0.9110.09 1.13+0.08 G-86681 L M HNO 3 0.49 1.05+0.10 0.9210.19 1 .24 + 0.33 G-86682 L M HNO 3 0.46 1.2110.06 0.9410.07 1.3610.11 G-86683 L M HNO, 0.50 1.0310.11 1.19+0.13 1.0810.09 Witstaart II 13 n181 G-85145 L 1% NaOH 0 .24 1 .10 + 0.05 0 .75 + 0.05 1.35 + 0.07 42 144 + 78 + (2.35-2.41) G-85145 R 0 .44 0 .95 + 0.03 1. 13±0.08 0 .75 + 0.03 Scholtenhave 10 o203 38 G-86660 L 2 M HNO, 1 . 1 1.18+0.05 0.69+0.04 1 .2510.05 ca. 51' 122 + 441 (11.45-11.52) G-86661 L 4 M HK3, 0.7 1.39+0.11 0.68+0.05 1 .18+0.05 G-86662 L 7 M HNOT 0.8 1.22+0.09 0.84+0.07 1 .09+0.07 G-86672 L 2 M HNO, 1 ' 1.4 1.22+0.07 0.7310.07 1 .2110.12 G-86673 L 4 M HNO, •*) 1 .4 1.40+0.07 0.7810.06 1 .02 + 0.07 G-86674 L 7 M HNO, • ) 1 .4 1.30+0.10 0.65+0.07 1 . 23J0.14 ;<204 85 G-86663 L 2 M HNO-, 1 .0 1.20+0.08 0.57+0.04 2.95+0.22 55 1 ) 891 8 14 + (13.22-13.37) G-86664 L 4 M HNO, 1.0 1.05+0.14 0.75+0.09 3. 13+0.39 G-86665 L 7 M HNO, 1 .0 1 .19 + 0.11 0.7110.09 2.83+0.49 G-86675 L 2 M HNO, • ) 2.0 1 .0710.08 0.46+0.06 3.29+0.68 G-86676 L 4 M HNO, 1 ) 1.7 1 . 1510.07 0.65+0.06 3.46+0.45 G-86677 L 7 M HNO, 1.9 0.99+0.07 0.68+0.10 3. 55 + 0.90 ^302 89 G-87066 L 2 M HNO, 0.96 1 .1610.05 1 .17 + 0.08 1 .16 + 0.06 70 1) ca. 75 2> (14.51-14.58) G-87067 R 0.05 1 . 15+0.44 0.87+0.25 1.58+0.31 62 a303 27 1001 87± G-87068 L 2 M HNO, 0.42 1 .26+0 .06 0 .62+0 .05 2 .04+0 .20 1001 34 (20.60-20.67) G-87069 R 0. 30 1 .13 + 0 .06 0 .67 + 0 .04 1 . 1 1+0.07 * Related to air-dry total sample weight. ** "L" denotes leachate, "R" stands for residue. + 1% NaOH: selective dissolution of humic and fulvic acids; x M HNO,: leaching of the ashes of the combusted peat sample. M based on estimates by J. van Huissteden, pers. comm. 1986/1987. 2) Uncertainty over 100%. 3) Leaching during ca. 15 hours. All other leachates are obtained from leaching during 30 minutes. All errors quoted are based on 1o uncertainty due to nuclear counting only. 96 Eemian and 1B0 stage 5e and St. Germain I and 180 stage 5a as sug- gested by Woillard and Mook, resulting in the "stretched" pollen dia- gram in Fig. 3.7b. In addition, the ages support the correlation between Eemian, St. Germain I-A, I-C and II with the Pangaion, Doxa- ton, Drama and Elevtheroupolis stages at Tenaghi Phillippon, respect- ively (Fig. 3.7a and b). The conclusion is that within the rather large uncertainties the measurements support the proposed correlation between La Grande Pile and Tenaghi Phillippon (Woillard, 1978). However, more detailed work will have to confirm and refine the absolute UTD chronology of the early Weichselian interstadials and the Eemian Interglacial. 4. Les Echets (Table 3.6) Unfortunately we have not yet been able to obtain reliable dates for samples from boring G at Les Echets, due to the extremely low U and Th concentrations associated with the organic phase. In such cases one may expect isochron corrections to yield better results. Although the corrected UTD dates for a195, a196 and a197 correspond within error limits with the expected ages, the associated uncertain- ties are too large to put any trust in the chronology. The importance of the sequence justifies future efforts to arrive at a better sepa- ration between isotopes associated with the inorganic and the date- able phase. 5. Valley of the river Dinkel (Tables 3.3 and 3.7) U concentrations in these samples were extremely low. Two samples from Scholtenhave yield a corrected UTD date in agreement with 11*C. However, the uncertainties are large and the consistency may well be fortuitous. All other samples are questionable with respect to their reliability, although often the UTD dates do not actually contradict lkC results. Nevertheless, this site appears to be unsuitable for ap- plication of UTD to peat samples due to limited U supply during for- mation. 6. Tervola (Tables 3.2 and 3.3) The Tervola samples show high U concentrations. Their mineral 97 fraction appears to be rather insensitive to the strength of the leaching agent. As a result UTD deduced from a two-phase separation (Table 3.3) is in good agreement with 1<*C. 7. Oerel (Table 3.3) Samples from Oerel may yield correct ages, but in general the extremely low U and Th activities in the organic fraction and the high sensitivity of the mineral fraction to the strength of the leaching agent require a very mild leaching agent to obtain a reli- able age. 8. Pitalito (Table 3.8) From four samples that were analysed ages of the two deeper samples show excellent agreement with those expected. One sample con- tains no U/at all so that no age could be obtained. The deduced age of the upper sample appears to be too old and indicates contamination with environmental Th. Future mineral analysis and UTD analysis of the 1% NaOH extracted fraction should provide information on the type of contamination (in-situ or during laboratory treatment, compare La Grande Pile sample, &184). 3.2.7. CONCLUSIONS Peat as a whole appears to act as a closed system for U and Th. However, measurements confirm that a substantial amount of environ- mental Th is transferred in-situ from the inorganic into the organic constituents of peat rather than by the laboratory treatment. The frequently observed agreement between uncorrected and expected ages of the organic fraction requires a tentative geochemical model that qualitatively predicts depletion of 230irh with respect to 232Th in the surface layer of mineral grains by recoil displacement. As a result, the effective transfer of 230Th is limited so that correction for environmental Th leads to an underestimation of the true age. Care must be taken in choosing the correct leaching agent for separation of the dateable organic from the detrital inorganic frac- tion. This can be achieved, either by the use of slightly acidic so- lutions (preferably less than 2 M HNO3) to leach U and Th of organic 00 Table 3.8. Kadiometric Data obtained from peat samples from Pitalito, Colombia Sample % loss* on Analysis** U- ccnc. * 231-U/238U 23cTh/23uU 23oTh/J32Th Age (ka) (depth m) ignition (ppm) Uncorrected corrected Expected 311 84 G-87094 L 0.17 1.20±0 .09 0.29 + 0.08 1.5610 .67 37± 16+° 20.37±0.14 i) (2.11-2.15) G-87095 R 0. 13 1.18 + 0.17 0.86+0.13 1.00±0 .17 (GrN-13992) a314 94 G-87096 L f 2-10"' _ _ - (5.36-5.40) G-87097 R 9-10"3 7.2 7.0 a316 65 G-87098 L 0.25 1.88 + 0. 11 0 .36 + 0 1 .34 45.9 + 38.6 + .04 .74±0 6.7 6.3 (S.86-8.90) G-87099 R 0.35 1.08±0 .09 0.41±0 .05 0.90±0 . 12 (GrN-13991) 9.9 2) a318 34 G-87100 L 0.27 1.78±0 .09 0.48 + 0 .05 1.49 + 0.22 67 .7 + 148 + ca. 62 9.1 112 (12.71-12.75) G-87101 R 0.43 0.95 + 0.08 0.37 + 0.04 1.00 + 0.12 * Related to air-dry total sample weight. ** "L" refers to leachate, "R" to residue. 2) Based on extrapolation of l ''C and pollen concentrations (J. Bakker, pers. comm. , 1987) . 99 origin from the ashed sample, or by separation of the inorganic frac- tion by selective dissolution of humic and fulvic acids (e.g. by using a 1% NaOH solution). In cases where the inorganic matrix is vulnerable to mild leaching and Th is mainly introduced to the date- able phase during this process, isochron correction may be useful but easily leads to overestimation of the amount of environmental 230Th. In such cases comparison between U/Th activity ratios in the leachate and in the 1% NaOH extract may reveal information on the relative contribution of in-situ and laboratory contamination. Reliable UTD dating of peat is possible. Relatively small samples (10 to 20 g of dry material) of relatively high U concentra- tion (one or more ppm) associated with the organic phase are required. In the mineral matrix traces of environmental U and Th are allowed and in general do not require any correction for environmental 230Th, in contrast to other dateable materials. However, uncertainties as- sociated with UTD dating of peat are still large. Therefore, at the present state of the art it is recommended to confirm UTD ages with independent evidence. Measurements support correlation of the Eemian at La Grande Pile with deep-sea 18O stage 5e. The Early Weichselian Interstadials St. Germain I and St. Germain II appear to correlate with stage 5c and 5a, respectively. At Tenaghi Phillippon the Pangaion, Drama and Elev- theroupolis correlate with stage 5e, 5c and 5a, respectively. The correlation of the Amersfoort, Br^rup and Odderade Interstadials in this connection has not been clarified and will be a subject of future research. 3.3. CORALS 3.3.1. BACKGROUND Barnes et al. (1956) found that recently formed corals contained up to several ppm U but essentially no 230Th. This can be explained by observii., that U, occurring in soluble form in sea water, is easi- ly incorporated in the CaCO3 matrix while Th due to its much lower solubility, is not available in ionic form. Furthermore, they noticed a systematic increase of the 23oTh/23[*U activity ratio with presumed older stratigraphy in a coral reef, which was associated with radio- 100 active growth of 230Th with increasing age. Hence the applicability of UTD to corals appeared to be demonstrated. In addition to this, the approximately 14 to 15% excess of 2ZhU over 238U (section 1.2.4) in the ocean provides the means of an inde- pendent check of the age (231tU/23aU disequilibrium dating, UUD) . The latter can be calculated from the present day 23I*U/Z38U ratio using the expression (Z31tX >> 238X, compare eq. (1.17)): ((231lU/238U) - 1) = ((23*U/238U) - 1) x e"23"Xt (3.6) o From palaeoclimatological viewpoint important data have been obtained on the e.g. coral reefs of Barbados where sea-level stands represent- ing climatological periods could be absolutely assigned with U-series ages (see e.g. Moore, 1982). Obviously there are several requirements that have to be ful- filled before corals can be reliably dated with UTD. (1) The 232Th concentration should be essentially zero, proving the absence of common or environmental 230Th. (2) The system should be closed with respect to U and Th. (3) The age determined from the present day 231fu/238U activity ratio should be consistent with the 23oTh/231*U age. For accurate UUD the initial 23"u/238U ratio should be known. Requirement (1) can easily be checked by measurement. Furthermore, we assume that in the past 100,000 years the oceanic 231*U/238U activity ratio has been constant because the residence time of U in the ocean is estimated to be approximately 160-240 ka (Cochran, 1982). Closed- system behaviour can be checked in specific cases for corals that produce CaCO3 solely in the aragonite crystalline form. Hence, if X- ray diffraction analysis shows the presence of calcite, obviously some recrystallization must have taken place. This indicates that the system has not been closed and that initial conditions are no longer determined by the U and Th activities at the time of formation. Veeh and Burnett (1982) reviewed the results on over 100 unre- crystallized corals of all ages. Their revision may be summarized in two important conclusions: (1) With a very few exceptions, the 23oTh/23'*U and 231tU/23eu activity _••#••*••- • 101 ratios are in fair agreement (2a) with the ratios expected for a closed system starting with an initial 23OTh/231tU ratio of zero and a 231*U/238U ratio of 1.14. (2) For corals of higher age than 10 ka the lhC deduced ages are systematically significantly lower than the UTD ages. Veeh and Burnett attribute this to one or more of the following causes: (i) secular variations in atmospheric 2I*C; (ii) contamination with modern 1L*C; (iii) secondary leaching of U, and (iv) second- ary addition of 230Th unaccompanied by 232Th. However, they state that the limited data do not provide justification to assign more validity to 1"c than to UTD or UUD ages for corals. 3.3.2. MEASUREMENTS In a preliminary attempt to measure U and Th activities in corals four samples obtained from various Indonesian coral terraces (a254: Bau Bau, 05°40'S, 123°00'E, a255, a256 and a257: Pulau Semau, 10°25'S, 125°50'E) were selected. These samples were dated with 1£|C in our laboratory but they were not studied for diagenesis. Following disso- lution of the samples in cone. HC1, U and Th activities were deter- mined by isotope dilution and alpha spectrometry. Table 3.9 shows the results of the measurements as well as the deduced ages in comparison with the 2"C ages. 3.3.3. DISCUSSION AND CONCLUSIONS The limited number of analyses allows a short summary of the re- sults. The UTD ages do not contradict the XI*C ages. Preliminary esti- mates of the ages (Fortuin, pers. comm.) based on the stratigraphy of the terrace suggests that the coral limestone samples a255 and a257 are representing 180 stage 5, which is in agreement with the UTD results. Sample a256, a Tridacna shell, stratigraphically represents a high sea-level stand. Tentatively its age of 31 ± 4 ka may be cor- related with the high sea-level as registered at Huon Reef complex II, dated at 28 ka with 1(tC (Chappell, 1974). For sample ct254 a younger age was expected (Fortuin, pers. comm., 1987). The deduced initial 231*U/23BU activity ratios agree within two standard devia- tions with the oceanic value of 1.144 ± 0.002 (Chen et al., 1986) for Table 3.9. Radiometric data obtained from analysis of corals from Indonesia Sample Analysis U-conc* 23*'V238U 23°Th/231tU 2 3 %'h/23 2Th age (ka) Deduced** 23 J38 no. (ppin) "C UTD ( *U/ U)O a254 G-86446 0.266+0.009 1.06 ±0.03 0.82+0.05 5.9±0.5 » 41 182+27 1.10+0.06 (Tridacna shell) (GrN-13112) a255 G-86447 2.63 ±0.06 1.136±0.010 0.62±0.02 330+75 35 101±4 1.18+0.02 (Coral limestone) (GrN-13117) 43 74±3 1 .20±0.02 (Coral limestone) (GrN-13120) * Related to air-dry weight of the total sample. ** Based on substitution of the results in collums 4 and 7 in cq. 3.5. 103 a254, a255 and a256. The value for a257 is significantly too high, which points to open-system behaviour. Hence, additional research is required to establish the validity of the presented ages. Although the results of UTD dating on corals obtained sofar are limited and preliminary, they show interesting results that encourage us to continue application of UTD, in combination with required addi- tional analyses such as X-ray diffraction, as an independent radio- metric control for and possible extension of 1L*C dating. 3.4. FOSSIL BONES 3.4.1. BACKGROUND The fact that fossil bones show high U concentrations (1-1000 ppm), while in fresh bones U usually does not exceed 0.1 ppm, indi- cates that U is taken up in the soil post mortem, presumably from ground water (Schwarcz, 1982). If U uptake ceases after a limited period of time with respect to the 230Th half-life and the bone sub- sequently acts as a closed system for U and Th, it is possible to apply UTD dating. Although previous archaeometric studies inferred U uptake over a period of time of the order of 300 ka, Schwarcz (1982) is of the opinion that radioactive growth of U-daughter products may well have been erroneously interpreted as an increase of U activitiy, due to the fact that the (equivalent) U activity was usually measured by total beta counting. Furthermore, Szabo (1980) concluded from a com- parison of UTD and lkC dates that U uptake ceases after approximately 2 to 3 ka. Recently Rae and Ivanovich (1986) observed that the outer surface layer of bones adsorps U on a relatively short time scale (< 2 ka) and subsequently behaves as a closed system. They recommend analysis of this surface layer rather than whole bone analysis. 3.4.2. MEASUREMENTS Bone samples were selected from sites along the Solo river in Central Java, Indonesia. This area is well known for its find spots of Homo erectus remains (Ngandong man). The terraces deposited by the river Solo (commonly designated as High Terrace and Low Terrace) and their geomorphology are extensively described by e.g. Lehmann (1936), 104 de Terra (1943) and Sartono (1976). In 1986 nine bone samples were collected by Bartstra from a pit. at the locality of Ngandong and two from the opposite side of the river (Matar). The presumed geological age of the samples is Upper Pleistocene (Bartstra et al., 1987). The samples were treated by our laboratory to test the analytical methods and the applicability of UTD dating. Small fragments of bulk material were crushed in a mortar and transferred to a 250 ml beaker where they were dissolved by careful addition of drops of cone. HC1. Following total dissolution overnight the liquid was analysed for U and Th. Due to the presence of large quantities of phosphates, obstructing and saturating the Th purifying ion-exchange column, chemical recoveries for Th were extremely low (1-4%). This results in large uncertainties in the Th concentrations due to possible unknown fractionation between natural Th and Th from the spike. Recoveries for U were of the order of 50%. This clearly 0 50 100 Nqandong bones o U cone (ppm) 50 E -100 U so •^ 150 a. Q 200 Figure 3.8. UTD ages for bone samples 250- from the 1986 Ngandong digy plotted against depth in the Terrace. 105 shows that our present chemical techniques for Th purification need improvement when applied to bone analysis. Nevertheless the results of the measurements were sufficiently encouraging to be presented in the next section. 3.4.3. DISCUSSION AND CONCLUSIONS Based on radiometric analyses of the Ngandong samples B to K (Table 3.10 and Fig. 3.8) some remarks concerning closed-system be- haviour and dating reliability may be made. (1) The rate of U uptake may be bone-dependent. Nevertheless, if con- tinuous accumulation of U is modelled as a first order linear process, samples with high U concentrations are expected to yield younger ages due to dilution of Th with respect to U. However, the measurements show no correlation between U concentrations and deduced age. Table 3.10. Radiometric data obtained from analysis of Indonesian bone samples Sample Analysis U-conc.* ' 'Th/P ~'*U : '" Th /? ': Th age (depth m) no. (ppm) (ka) Ngandong B (surface) G-86656 45 1.092+0 .011 0 .375+0 .033 150445 51 + 5 C (1.10) G-86657 12 1.150+0 .015 0 .342+0 .035 26+ 7 454 5 D (1.20) G-86658 18 1.112+0 .013 0 .323+0 .028 27+ 6 42 + 4 E (1.65) G-86659 82 1.207+0 .007 0 .344+0 .040 120+36 45 + 6 F (1.96) G-86688 128 1.417+0 .005 0 .409+0 .058 56 + 10 9 3 G (2.20) G-86689 68 1.280+0 .006 0 .247+0 .021 31 + 2 H (2.30) G-86690 54 1.252±0 .006 0.540+0 .040 118+30 82± 7 8 J (2.32) G-86691 74 1.427+0 .006 0 .489+0 .044 - 70 + 7 12 K (2.50) G-87037 48 1.164+0 .011 0.617+0 .041 93 + 31 101± 10 Ma tar L (HTchop) G-87038 153 1.039+0.006 0.788+0.053 232+78 165+ 30 23 P (HTtimur3) G-87039 94 1.114±0.007 0.343+0.027 140+70 45± 5 * Related to air-dry sample weight 106 (2) The extremely high 23oTh/232Th activity ratios indicate that there has been no contamination with environmental Th. (3) The deduced UTD ages show stratigraphical consistency and are in agreement with expectations. These considerations are all consistent with closed-system behaviour. Nevertheless the evidence for a limited period of U uptake is only circumstantial and consequently our deduced ages must be considered as minimum ages. If the bones have behaved as a closed system the high age of Matar sample L is in conflict with the presumed age of the Terrace. However, the Matar bones appear to be more fragmentized and more abraded than the test-pit specimens. It is possible, therefore, that these Matar fragments represent an allochthonous High Terrace fossil assemblage, originating from the Middle Pleistocene lower fluviatile unit, which the Solo is (still) eroding away further upstream. This possibility has been expressed previously (Bartstra et al., 1976). Although the results presented here are in good agreement with expectations, several problems still have to be solved. One of these is the removal of phosphate which presently limits the chemical re- covery of Th. 3.5. SUMMARY This chapter presents results of a detailed study on the appli- cability of UTD dating of peat that are promising and contribute to an independent absolute chronology of Early Weichselian interstadials and the Eemian Interglacial. Furthermore, we have presented preliminary results of UTD dating of corals and fossil bones. We are aware of the uncertainties that surround these results. Nevertheless, despite their preliminary char- acter, the good agreement with expectations seemed to justify their presentation. Future research is required to establish the validity of the various dating models in more detail, aiming at reliable ages for the above materials in the dating range beyond the IlfC time scale. It will provide the unique facility in our laboratory to apply simulta- nesouly two important, independent radiomatric dating methods (UTD 107 and 1I|C) to a variety of environmental samples. This undoubtedly will contribute significantly to a better understanding of the reliability of the absolute Late Quaternary . 109 9 1 fl CHAPTER 4. PB DATING* 4.1. INTRODUCTION The value of 210Pb dating of sediments in the time range of the last 150 years has been shown extensively (for a review see e.g. Krishnaswamy and Lai, 1978; Robbins, 1978; Oldfield and Appleby, 1984) . One field of applications is concerned with acidification of soils and surface waters, being a wide-spread ecological problem in North-Western Europe. Here 210Pb acts as a geochronometer needed to determine the rate of acidification (Battarbee et al., 1985; Dickman et al., 1986; Jones et al., 1986; Tolonen et al., 1986; Sircola, 1986; Van Dam et al., 1987). This chapter discusses in detail lake-sediment chronologies as discussed in section 1.4. The majority of these measurements was carried out on sediment cores from Dutch shallow moorland pools as part of a multidisciplinary approach to establish acidification rates over the last 150 years. Traces of biocommunities as well as pollen analyses provided the relevant information on the environmental conditions at the (zl0Pb) tima of sediment deposition. Section 4.2 discusses measurements performed to establish the validity of basic assumptions in the dating models (constant rate of supply of 210Pb and closed-system behaviour, especially under chang- ing pH conditions). Section 4.3 shows the results of 210Pb measurements in 10 sedi- ment cores, taken from seven shallow moorland pools. In section 4.4 we present some additional results from sediment cores that were ana- lysed in the context of other projects. * An adapted version of section 4.3 was published before: Van der Wijk, A. and Mook, W.G. (1987).210Pb dating in shallow moorland pools. " Geologie en Mijnbouw 66: 43-55. 110 4.2. CONSTANT RATE OF SUPPLY; CLOSED-SYSTEM BEHAVIOUR 4.2.1. CONSTANT RATE OF SUPPLY One of the basic assumptions in both the CRS and the CIC model (section 1.4) is constant average annual deposition on a unit surface area of 210Pb activity over the entire dating range. Not only atmo- spheric 210Pb produced by 222Rn decay in the air contributes to this flux, but also 210Pb that is brought into the sediment by allochtho- nous sediment particles (sediment focussing) or in dissolved form by river waters. Hence, the total rate of supply of 210Pb will vary geo- graphically, depending on local conditions. As a result, the supply rate deduced from the integrated 210Pb activity over a sediment core (eq. (1.22)) does not necessarily reflect the local atmospheric 210Pb deposition. On the other hand, it may be used for estimating the con- tribution of other 210Pb sources if the local flux of atmospheric 210Pb is known. We monitored the rate of supply of atmospheric 210Pb outside our laboratory, using a plastic rain-water collector (Fig. 4.1) Figure 4.1. Photograph of the rain-water collector used to monitor the 210Pb depletion. 111 80 60 40 20 - J_ _L J L 1983 JUN JUL AUG SEP 0 :T NOV. DEC °. 80 g 60 §" 20 XJ 1964 JAN FEB MAR APR MAY JUN JUL 80 60 n 40 20 1986 JAN FEB MAR APR MAY JUN JUL AUG Figure 4.2. Deposition of 210Pb, as a function of time. with a surface area of 314 cm2 and provided with 500 ml polypropylene bottles, prefilled with 20 ml of 8 M HC1 to keep 210Pb in solution. The bottles were changed at irregular intervals determined by the amount of water collected and stored for at least one year to allow radioactive growth of 210Po. Subsequently the 210Po activities were determined using the isotope dilution technique as described in sec- tion 2.3.6. The 210Pb activities were calculated by extrapolation from zero initial 210Po activity. The latter assumption is justified 112 realizing that the average time between two periods of rainfall in the Netherlands is short (a few days). This does not allow growth of substantial quantities of 210Po (t = 138 d) in the atmosphere. A first series of 29 samples was collected in the period from June 23, 1983 through July 31, 1984. Following the preliminary meas- urements, samples were continuously collected starting February 12, 1986. Figure 4.2 shows the results of the measurements performed on all samples until July 9, 1986. During the period June 23, 1983 to July 31, 1984 the calculated average 210Pb deposition is 8.8 mBq.cm"2.a"1, in fair agreement with average values obtained by e.g. Crozaz et al. (1964) for the Northern Hemisphere (7.8 mBq.cm"2.a"1) and by El-Daoushy (1978) (6.3 mBq.cm"2. a"1). However, strong fluctuations with time are observed. These are most probably due to varying (meteorological) conditions such as wind velocity, average rainfall and soil condition (frozen or unfro- zen) . During May to July, 1984 anomalously high 210Pb deposition was observed. This period is relatively long with respect to the half- life of 222Rn (3.8 d). Furthermore, according to meteorological data from the Royal Dutch Meteorological Institute neither rainfall nor air transport during this period differed much from the average. Hence, the high deposition rate cannot be the result of excessive local build-up of atmospheric 210Pb at a normal Rn exhalation rate. It must be either the result of a local enhanced Rn exhalation from the soil, for which there is no obvious reason since the province of Groningen shows no signs of vulcanic activity, or the result of lateral transport of large quantities of radon from elsewhere. Furthermore, 1986 also shows a sudden increase of 210Pb deposition in the middle of June, although not as extended and pronounced as in 1984. We found no obvious explanation for these anomalies. We are tempted to relate the enhanced 210Pb deposition in late spring to in- jection of 210Pb originally brought in the stratosphere by nuclear \ weapon testings through the 208Pb(2n,y)21°Pb reaction, into the troposphere in early spring. Evidence in favour of as well as against production of 210Pb by nuclear explosions has been subject of dis- > cussion (reviewed e.g. by Robbins, 1978). However, measurements over e. larger number of years are required to enable conclusions about 113 0 210 Pb deposition Groningen (CIO) 150 - o summer • winter no •o - - 100 - o 0 o summer - 0 • 50-— o • o • . ° o °o . o winher" • • • '&&a ®* * • _ * ° * I** , I | I I U 6 8 10 12 Rainfall (mm. day"1) Figure 4.3. Scattergram showing positive correlation of 210Pb deposition with average rainfall. Open circles represent summer data, full circles represent winter data. possible systematics of enhanced 210Pb deposition in late spring. The period of February 12 through July 9, 1986 shows an average rate of deposition of 7.0 mBg.cm"2.a~x, which obviously is in excel- lent agreement with previously recorded values as discussed above. Figure 4.3 shows a positive correlation between 210Pb deposi- tion and daily rainfall, which is higher in summer (March 21 to September 20, slope = 13 + 4, correlation 0.63) than in winter (September 21 to March 20, slope =3+1, correlation 0.71). Fukuda and Tsugonai (1975) observed the inverse effect in Japan (higher correlation in winter (November-February) than in summer (May to August) which they related to the influence of the north-west mon- soon. Our observations may be explained by the fact that in winter te soil is occasionally frozen which prevents escape of 222Rn. Our conclusion is that the rate of atmospheric supply of 210Pb is sub- jected to strong variations on a short time scale (a few days to a few weeks). However, the average deposition appears to be rather 114 constant over a period of time corresponding to the time resolution of the dating method (one to two years), i.e. relatively short with respect to the half-life of 110Pb but long with respect to the col- lection periods. Furthermore, it is in good agreement with values reported by other authors. Although our measurements cover only a relatively short timespan with respect to the dating range of the method, we use them as an indication that the average annual deposi- tion of atmospheric 210Pb is subjected to negligible variations. Hence, variations in the calculated rate of deposited sedimentary 210Pb (eq. (1.22)) yields information on other sources of 210Pb. 4.2.2. CLOSED-SYSTEM BEHAVIOUR Simola and Liehu (1985) reported a correlation between 210Pb minima and diatom-inferred pH minima in lake sediments, which they tentatively related to a decreased adsorption affinity of Pb into the sediment particles at low pH. This phenomenon has indeed been observ- ed for a number of elements in acidifying water bodies (El-Daoushy and Johansson, 1983). Although there appears to be little empirical evidence and the phenomenon requires more detailed investigation, the observations should be taken seriously because they may indicate a violation of closed-system behaviour. We carried out a laboratory study of the adsorption capacity of 210Pb into an organic sediment from a Dutch moorland pool (Gerrits- fles, sections 4.3.2 and 4.3.4). The sediment was homogenized and divided in 10 subsamples which were distributed over ten 250 ml beakers. To each of these a mixture of demineralized water and a di- - 2_ lute solution of HNO3 and E^SOit (NChrSCK = 1:1) was added to arrive at the required pH. The mixture was equilibrated and then left un- disturbed for four weeks. After that the mixture was again homogeniz- ed. A sample of 20 ml was taken and separated into a solid fraction and a liquid fraction by centrifugation. Both fractions were analysed. Analysis of the homogenized samples should yield approximately equal specific activities. However, this is only true if the activity is related to the organic fraction. In that case we measured a fluctu- ation in specific activitiy between the different samples of 16%. If we relate the specific activity to the total dry sediment weight, 115 Figure 4.4. Plot of distribu- tion factors (see text) of 210 Po between the aqueous phase and the solid organic phase of a suspension of Gerritsfles sedi- ment. the fluctuation between the samples amounts to 66%. This indicates that the majority of adsorbed 210Pb should be associated with the organic fraction of the sediment. From these analyses we calculated a distribution factor K for 210Po between dry organic sediment and the aqueous phase at various pH values using the following equation: K = 210Po /210Pb (4.1) where the subscripts a and s refer to the aqueous phase and the solid (organic!) phase. Symbolic notation for 210Po denotes its specific activity (mBq/g). Figure 4.4 shows the measured distribution factors as a function of pH. They are all extremely low (of the order of 0.01 to 0.1%) and show no tendency to increase with decreasing pH value. Hence, within the restrictions of our experiment we find no evidence for a decreas- f- 116 ed adsorption capacity for 210Po into these sediments. Our experiment, however, shows many obvious limitations that prohibit generalization of the results. Firstly, we have studied the distribution between solid and aqueous phase of 210Po instead of 210Pb. Obviously, meas- urement of 210Pb through 210Po requires closed-system behaviour for the latter as well, but the conclusion that the sediment is a closed system with respect to 210Po does not necessarily imply closed-sytem behaviour for 2I0Pb. To investigate the distribution for 210Pb by measurement of 210Po requires a time consuming experiment due to the fact that 210Po and 210Pb have to reach secular radioactive equili- brium in each fraction that is to be measured. Such an experiment is in progress. Secondly, we have allowed eqailibration between the aqueous and solid phase only over a relatively short period of time (four weeks) with respect to the dating range of 2l°Pb (7000-8000 weeks). Never- theless, adsorption studies for lead isotopes generally show that a stationary state between both phases is obtained within a few days (Krishnaswamy et al., 1982; Rama and Moore, 1984). Hence, this con- sideration is of no serious concern. Thirdly, we have selected a sediment of relatively high organic content (ca. 20% by weight) which may show different behaviour with respect to an inorganic sediment under changing pH conditions. We al- ready concluded that the majority of 21cPo should be associated with the organic phase. Furthermore, in chapter 3 we discussed the extre- mely effective adsorption and ion-exchange capacity of humic and fulvic acids for actinides over a wide pH range. It is thus likely that in organic sediments similar geochemical conditions result in enhanced adsorption of 21nPb and/or 210Po as well. Although this result can not in general be applied to sediments of arbitrary com- position it is valid in the organic sediments discussed in section 4.3. Our main conclusion, therefore, is that in organic sediments decreased adsorption of 2l0Pb (i.e. open-system behaviour for 210Pb) at lower pH values is of no importance. However, the validity of this conclusion is only limited to organic sediments. Future work is required to provide information on closed-system behaviour for 117 inorganic sediments. Therefore, in the next section we will relate diatom inferred sedimentary pH records (if available) with zl0Pb activity depth profiles in search of evidence for pH dependent 210Pb adsorption. 4.3. 21° Pb DATING IN SHALLOW MOORLAND POOLS 4.3.1. INTRODUCTION To determine rates of acidification in Dutch shallow moorland pools, so-called "vennen", Dickman et al. (1987) related biological and pollen analytical data with 210Pb ages. Due to the limited water depth (less than 2 m) palaeolimnological studies are not straightfor- ward. Sediments may have been disturbed by wind-induced mixing and bioturbation (e.g. human and/or animal bathers). Furthermore, in periods of great drought some of these moorland pools lost most or even all of their water content, exposing their sediments to oxidiz- ing conditions (Sykora, 1979? Vangenechten et al., 1981). Neverthe- less, these pools belong to the least disturbed water bodies in the Netherlands and hence provide the best subjects of study. This section discusses the applicability of 21"Pb dating on the basis of ten cores from a total of seven "vennen" in the Netherlands. 4.3.2. SITE DESCRIPTION AND SEDIMENT CORING Cores were taken from 7 moorland pools at various localities in the Netherlands (Fig. 4.5). Full description of the pools is beyond the scope of this thesis and is given elsewhere (I: Van de Hurk et al., 1986; II: Dickman et al., 1986; III: Beye, 19 76; IV: Mansfeld et al., 1975; V: Higler, 1979; VI: Wittgen et al., 1986; VII: Coesel and Smit, 1977). Duplicate cores were analysed from site numbers I, II and IV. The sediments from pools I-V in the south eastern and middle part of the Netherlands are generally characterized by a thin organic layer (20-30 cm) on a sandy substrate, while the sediments of pools VI and VII in the eastern and northern part are found to consist mainly of dark organic material (gyttja). Most cores were taken with an Ali Corer (Ali, 1984) during periods of ice cover. Other cores were taken with a PVC liner which 118 Figure 4.5. Locations of 7 moor- land pools in the Netherlands where 10 sediment cores were collected: I: Galgeven, II: Ach- terste Goorven, III: Groot Huis- ven, IV: Beuven, V: Gerritsfles, VI: Bergven, and VII: Kliplo. was vacuum closed at the moment of extraction by a rubber stopper. From some cores sections were taken in the field or directly upon arrival in the laboratory, using a thin stainless steel wire. Several other cores were deepfrozen, extruded from their liners and cut into 1 cm samples in frozen condition, using a diamond saw. In one case (core V.5) the inner not yet frozen part of the core was squeezed from its position which may have led to disturbed strati- graphy. Therefore, the remaining cores were transferred to the labo- ratory in unfrozen condition where the overlying water was siphoned off and the liner was cut along its length. One half was used for 210Pb dating, the other was used for biological analyses. Table 4.1 summarizes collection date, type of corer and the internal diameter for each core. 119 Table 4.1. Summary of localities. Pool Core no. collection type of corer date + int.diam. Galgeven 1.1 18 Feb. 85 ALI, 7.0 cm Galgeven 1.2/2 12 Dec. 85 PVC, 4 .2 cm Achterste Goorven II.7 30 Jan. 85 ALI, 7.0 cm Achterste Goorven II.4 30 Jan. 85 ALI, 7.0 cm Groot Huisven III.1 26 Jan. 84 PVC, 1.9 cm Beuven IV.2 21 Feb. 85 ALI, 7.0 cm Gerritsfles V.5 12 Feb. 85 ALI, 7.0 cm Gerritsfles V.7 28 Feb. 85 ALI, 7.0 cm Bergven VI. 1 18 Dec. 85 PVC, 6.5 cm Kliplo VII. 3 31 Jan. 85 ALI, 7.0 cm 4.3.3. RESULTS Figures 4.6a to 4.6j and Figs. 4.7a to 4.7f show the results of the measurements on the ten cores and the deduced age-depth profiles. We calculated supported zl0Pb activities under the assumption s that they originated from the inorganic fraction of the sediment only. After a first approximation of the sedimentation rate, using total 210Pb activities instead of 210Pb activities, we sleeted suitable XS samples (over 150 years old) to determine the activity 210Pb. (mBq/g) of the inorganic material and subsequently calculated the contribu- tion of supported 210Pb at arbitrary depth z: 210 210 Pb (z) = f.(z) * Pbi (mBq/g sample) (4.2) where f.(z) is the weight fraction of inorganic material at depth z. The assumption that 21DPo is in equilibrium with 210Pb in the samples leads to some additional uncertainties on the true value of the 210Pb activities, especially in the upper sediment layers: 1. Movement of fresh water by diffusion in which 210Po is not yet in secular equilibrium with 210Pb across the sediment-water interface may be substantially due to the flocculant structure of the upper sediment layers. This leads to dilution of the 210Po activity with 210 3 Excess ?10Pb activity (m8q/cm3) Total Pb activity (mBq/cm i 2. 0 c: u 1 o 3 O S V o i - Achters l ID • i-i Goo i -*- ° P rve n s 3 ,.^— o 1 5T io n 2 5 3 0 - UJ „ a Excess *"°Pb activity (mBq/cm3) Total210Pb activity (mBq/cm3) 0. 2 0. 5 1. 0 0. 1 •o —1 . 1 T-~T —i—i—r ! , J-—_ '' -+• CD + 4 1-1 IT 4- ,17 c o o 3 I cn CD fD p, o -• ' fD o fD P fl) ..^^ - —. FS i §1 Zl 5 ta t o f-O - o Zl K 121 50.0 100.0i— Qroot Huisven +• Beuven station 2 a 119 V a 137 (core 111,1) \ (core IV.2) 50.0I 1.20.0 CD \ S=0.15cm a' •E e 00 - 10.0 •^ 20.0h S =0.10cm a' 10.0 ^ S=0 33cm a' £ 5.0 2j0h j I I I 2.0h 10 15 20 25 30 Depth (cm] 0^310.83 1.01 1 1 5 10 15 20 25 30 Depth (cm) Gerritsfles station 5 50.0 Gerritsfles station 7 a 129 a 130 (core V.5) •f (core V.7) 20.0 4- 1 S =0.15cm a" 20.0"h t \ \\ "E -•; 10.0 h ,S=0.09cm a1 CO e ^ 5.0 t .*" 5.0 \ S=0.57cm a' S- 2.0 \ \ 2.0h- \ V 1.0 1.0 t \ 0.5h 04 G i i i 0.2 10 15 20 25 30 10 15 20 25 30 Depth (cm) Depth (cml Figure 4.6. (continued) 122 Kliplo station 3 Bergven station 1 20.0 a 123 192 a (core VII 3) 10.0 ~ _^ (core VI .1) O.IO \ 10.0 5.0 0X15 cr 5.0 S =0.29 cm. a' CO E 1 1 2.0 O02 25 30 35 2-0 1 XI \S = 0.22 cm. Cf CL \ 2 -i •— \ 1.0 \ ^ \ to 1-0- OJ -1 0.5 -*- \ \ \ 0.5- \ \ \ 0.2L 5 10 15 20 25 30 Depth (cm) n i 1 1 I 1 i 1 1 10 15 20 25 30 35 Depth (cm) Figure 4.6. Semi-logarithmic plot of excess 210Pb activity against sediment depth. For cores I.I and 1.2/2 the total rather than the excess 21°Pb activi- ty is given. Straight lines show the least squares linear fit, yield- ing a corresponding average sedimentation rate S in cm.a'1. 123 Achrerste Goor-ven station 7 Groat Huisven Gerntsfles station 5 a 121 a 119 a 129 Irore II.7) (core DI.1I (core V5I 1950 1925 1900 1875 1850 1625 1S00 1775 B 5 10 15 20 25 30 !0 15 20 25 30 10 15 20 Depth (cm) Depth (cm| Oepfti (cm] Gerritsfles station 7 Bergven station 1 a 130 a 192 (core V.7) (core VI.1) - CIC model • CIC model - CBS model • CRS model I I I I -J I I l_ 10 15 20 25 30 10 15 20 25 30 35 10 15 20 25 30 Depth (cml Depth (cm) Depth (cml Figure 4. 7. Age-depth profiles for cores for which ages could be calculated. Dashed lines show the results from the CIC model, full lines show CRS results and accompanying lo uncertainties. respect to the 210Pb activity. 2. In sediment layers younger than approximately 2 years 210Po and 210Pb are not necessarily in secular equilibrium. These two effects usually lead to an apparent constant (see e.g. core II.4) or even lower (see e.g. core VII.3) 210Pb activity in the upper 124 3 to 4 cm. The CRS model now allows two approximations to estimate the effect of this uncertainty: 1. The zl0Pb activity is assumed to be constant in the upper 3 to 4 cm of the core, due to infiltration of fresh water and mixing of the flocculant organic material. 2. The sedimentation rate calculated from the 210Pb activity below the "mixing" zone is extrapolated to the sediment water interface. In practice we used the first approximation because, as mentioned before, it is realistic to assume some mixing in the upper sediment layer. A least squares linear fit yields a straight line in the semi- logarithmic plot of the 210Pb activity (Fig. 4.6). The CIC model, X S using a 210Pb half-life of 22.3 years, produces average sedimentation rates. We also calculated ages for the CRS model. Error bars in the activity-depth profiles indicate 1a uncertainties in the counting statistics. In the age-depth profile (Fig. 4.7) these result in regions of uncertainty around the CRS calculated values. We did not take uncertainties of the basic assumptions of the model into ac- count . 4.3.4. DISCUSSION The CRS model provides an estimate of the average annual deposi- tion of 210Pb (eq. (1.22)). As discussed in section 4.2, in a closed system for 210Pb with undisturbed vertical sedimentation, this should reflect the local atmospheric 210Pb deposition. However, lateral sediment transport may either increase or decrease the annual deposi- tion of 210Pb (sediment focussing, see e.g. Irlweck and Danielopol, 1985). Due to the fact that the sediments in the moorland pools usu- ally show little or no topographic relief, we believe that such late- ral transport is mainly wind-induced. Table 4.2 shows the average annual deposition obtained from the CRS model for eight of the ten cores. The calculated uncertainties do not include the uncertainty in the absolute spike-activity. We estimate this to be less than 5%. Although Krishnaswamy and Lai (1978) reported significant geographical variations in the 210Pb flux, we believe that, regarding the small 2l0 1 distance between our cores, the variation in Pb deposition in 125 in Table 4.2 is an indication of sediment focussing rather than vari- ations in atmospheric flux. Table 4.2. Average annual 210Pb deposition D at the coring sites as deduced from the total integrated 210Pb activity I(°°) (see eq. (1.22)). For cores I.I and 1.2/2 no value could be calculated for lack of data on the supported 210Pb activi- ty. Without sediment focussing, the figures should reflect the local atmospheric 210Pb deposition which is estimated between 6 and 8 mBg.cm~z.a~1 (Crozaz et al., 1964; El- Daoushy, 1981). Core number Average annual 210Pb deposition (mBq.cirT2 .a"1) 1.1 ? 1.2/2 ? II.4 2.1 ± 0.2 II.7 1.5 ± 0.2 III.1 6.9 ± 0.5 IV.2 22.2 ± 0.8 V.5 5.9 ± 0.4 V.7 7.0 ± 0.5 VI.1 1.7 ± 0.2 VII.3 3.5 + 0.3 Site I. Galgeven Measurements on core I.1 yield a rather disturbed activity-depth profile. No supported 210Pb activity could be determined due to the fluctuating total 21 °Pb activity (by one order of magnitude) at greater (supposedly older) depths. The same is true for core 1.2/2. Furthermore, hardly any remains of macrofauna were found (Klink, pers. comm., 1986), so that both cores from Galgeven were rejected from further analysis. 126 Site II, Achterste Goorven The 210Pb activity-depth profile of core II.7 is one of the most regular profiles we have found in these type of pools. The low acti- vity at the sediment-water interface can be attributed to mixing processes in the upper few centimeters, rather than by a decreased absorption capacity of the sediment due to the diatom inferred pH of 4.8 (Dickman et al., 1987). The same trend is found at the sediment- water interface of core II.4, showing a rather constant 2x°Pb acti- vity in the upper four centimeters. Evidence of some vertical reworking of the upper sediment layers was confirmed by the down-core distribution of the acidophilic diatom species Eunotia exigua (Dickman et al., 1987). Treating vertical mix- ing throughout the sediment column like a diffusional process we showed in section 1.4.3 that only the slope and the intercept at z = 0 in the semi-logarithmic activity-depth profile do change. However, we have no conclusive evidence for diffusional transport of 21°Pb throughout the core, so we assume that mixing is only important in the top few centimeters of the core, leaving the slope of the distri- bution below the mixing zone unaffected (Robbins and Edgington, 19 75; Robbins, 19 78). Both cores II.4 and II.7 show a deficiency in 210Pb deposition. If lateral sediment transport is assumed there should be excess 210Pb deposition elsewhere. However, without knowledge of the distribution of these values throughout the pool, we can only specu- late about the cause of the deficiency. Due to the 210Pb deduced disturbed stratigraphy, core II.4 was rejected from a detailed biological study. However, the 210Pb results in the upper part of this core contributed to understanding the mix- ing process at the sediment-water interface in core II.7, showing the value of 210Pb analysis in duplicate cores. Discussion of the synthesis of biological, palynological and radiometric analyses is beyond the scope of this thesis and is given elsewhere (Dickman et al., 1987; Van Dam et al., 1987). Based on this discussion we were able to conclude that Achterste Goorven is one of the fastest acidifying bodies reported in temperate regions exposed to acid precipitation. 127 Site III, Groot Huisven Because of the limited diameter of the core, samples of 2 cm length had to be taken, reducing the resolution of the 210Pb pro- file. Using the CIC model, the 210Pb distribution can be divided 3 xs in two regions of different sedimentation rate. However, the CRS model explains such a discontinuity by a sudden, temporary increase in sedimentation rate at 20 cm depth. Therefore, both methods yield strongly deviating age-depth profiles. Fortunately, additional bio- logical data clearly indicate a historically recorded (1950) rise of the water table at a sediment depth of approximately 10 cm (Klink, 1985). Other explanations for this discontinuity, such as changes in pH (Simola and Liehu, 1985), are not supported by the chironomid composition in the deeper sediment sections (Klink, 1985). Therefore, the biological and historical data strongly support the results of the CRS model, showing that the negative slope of the semi-logarith- mic activity-depth profile between 15 and 20 cm should be attributed to a sudden rapid increase of the sedimentation rate. Site IV, Beuven The 2l°Pb activity-depth profile shows a tremendous distortion of the upper 10-15 cm of the core, probably representing rapid sedi- mentation of allochthonous material deposited by the river Peelrijt in the past 10-15 years (G. Arts, pers.comm., 1986). This conclusion seems to be confirmed by the extremely high value for the calculated 210Pb deposition (Table 4.2). Although chironomid analyses show that deeper layers are older than surface layers (Klink, 1985) and an estimate of the 2I0Pb deduced sedimentation rate at greater depths can be made, it is obvious that in this core the general conditions for applying 210Pb dating are not fulfilled. The stratigraphy pro- bably has not been preserved, especially in the upper 15 cm, and therefore other investigations requiring such conditions are doubt- ful. Post collection support for this conclusion was obtained from investigators of the University of Nijmegen, the Netherlands, who found over 6 tons of fish (bio-turbators) after the pool was emptied and cleaned in order to stop the extreme eutrophication process (G. Arts, pers.comm., 1986). 128 Site V, Gerritsfles As was stated before, the results on core V.5 are suspect be- cause of possible disturbed stratigraphy in the freezing process. Nevertheless, this core shows a very regular and continuous 21oPb distribution. The same is true for core V.7, except at 6 cm. The latter discontinuity cannot be explained by changes in the diatom- inferred pH (4-5 cm: pH = 4.8; 6-7 cm: pH = 4.5; 9-10 cm: pH = 4.8, Van Dam et al., 1987). Both cores yield zl0Pb deposition rates in agreement with values reported by Grozaz et al. (1964), El-Daoushy (1978) and values measured in this thesis (section 4.2.1), showing that sediment focussing must have been limited. Ages calculated from CIC and CRS are in good agreement in both cores. Alternative dating was performed on core V.7 by comparing the historically documented flora development with sedimentary pollen composition as well as by comparing sedimentary diatom distributions with diatom analyses of plankton tow samples collected since the be- ginning of this century. A detailed discussion is beyond the scope of this thesis and can be found elsewhere (Van Dam et al., 1987) . The 210Pb and alternative dating are in good agreement to a depth of approximately 6 to 8 cm (1900 AD). From there on 210Pb dates appear to be too old. This discrepancy is probably caused by the 210Pb ac- tivity anomaly at 6 to 7 cm depth. Therefore, in discussing the com- bined biological and palynological data we prefer to use the alter- native dating at depths below 7 cm. This discussion, given in Van Dam et al. (1987), shows that in this pool the discontinuation of anthropogenically introduced eutrophication renders it impossible to determine the onset of acidification by acid precipitation. Site VI, Bergven The 210Pb distribution in the core from Bergven shows some fluc- tuation. The causes for this can only be speculated about because biological data and a diatom-inferred pH profile are not yet avail- able. Ages calculated from both the CIC and the CRS model are in good agreement and preliminary results from biological studies are not contradictory. i 129 Site VII, Kliplo The fluctuating specific inorganic 210Pb. (mBq/g inorganic mate- * rial) at greater depths (> 20 cm) and the strong variation of the inorganic content at the base of the core (13.2 weight% at 29-30 cm and 0.9 weight% at 32-33 cm) renders it difficult to calculate a re- liable value for the supported 21oPb activity. Therefore, we used the total 210Pb activity in the calculations, assuming 210Pb >> 210Pb . This, however, may lead to underestimation of the true age. Good agreement is found between CIC and CRS calculated 210Pb age-depth profiles. From the shape of the 210Pb distribution we con- clude that sedimentation has been rather undisturbed (except possibly for some mixing in the upper few centimeters). Alternative dating as in the case of Gerritsfles core V.7 has been performed and shows that 210Pb inferred ages are systematically 10 to 2 0 years younger. This may be explained by the fact that we were forced to use total 210Pb rather than 210Pb activities. xs Synthesis of biological data, pollen analysis and 210Pb dating is beyond the scope of this thesis, but it is extensively discussed by Van Dam et al. (1987). As in the case of Gerritsfles it appears to be impossible to determine the onset of acidification due to acid precipitation. 4.3.5. CONCLUSIONS 210Pb analysis in shallow moorland pools in the Netherlands has provided valuable information about sedimentation rates and possible mixing processes of direct importance in palaeolimnological studies in general and acidification studies in particular. However, 210Pb distributions alone are often not sufficient to decide on the reliability of the age-depth profiles. In a multidisci- plinary approach, providing additional pollen and biological data, it was possible to obtain reliable chronologies for more than half of the cores. Furthermore, in several other cores 210Pb activity-depth profiles provided valuable information about possible distortions of the sediment stratigraphy and thus prevented unnecessary (costly) additional analyses. We have found no evidence for a decreased adsorption affinity * 130 for 210Po nor 210Pb at lower pH in this type of sediments (character- ized by high organic contents). 4.4. MISCELLANEOUS CASE STUDIES 4.4.1. DENMARK 4.4.1.1. Introduction and site description In 1975 the Fresh-water Laboratory of the Danish National Agency of Environmental Protection started monitoring the chemistry and pH of 14 lakes that were considered to be vulnerable to acid rain in- duced acidification in the Jutland area of Denmark. In 1985 a study on palaeolimnology and acidification of some of these lakes was ini- tiated, using a similar approach as Dickman et al. (1987) and Van Dam et al. (1987) in reconstructing rates of acidification in Dutch shal- low moorland pools. This section describes the results of zl0Pb ana- lyses in two Danish lakes. A detailed study which should lead to an estimate of the acidification rate over the last century in these lakes is in progress. Grane Langs0 (maximum depth 11m) is a well studied lake in Jut- land (position 9°28'E, 56°03'N, Denmark). Kalgaardstf (maximum depth 30 m) is situated very closely to Grane Langstf (approximately 100 m south, only separated by a road). Sediments from Danish lakes as well as post-glacial development of Grane Langs0 are extensively described by Hansen (1959, 1964). On May 23, 1985, a 4.5 cm diameter sediment core was taken near the deepest point of both lakes, using a K.B. gravity sediment corer (Dickman et al., 1984). Both cores were stra- tigraphically described and subsequently extruded from their liner and sectioned at 1 cm intervals in the field. Samples were submitted to our laboratory for 210Pb analysis. 4.4.1.2. Results and discussion For Grane Langs0 210Pb values were obtained by subtracting the estimated 210Pb activity from the total 210Pb activity, using eg. (4.2). The resulting activity-depth profile is very regular and shows no distortions (Fig. 4.8a). The CIC and CRS ages are in good agree- ment and the order of magnitude for the inferred sedimentation rate is characteristic for continental surface water bodies. The calcula- 131 50.0 Grarte Langsj) a K0 20.0 10.0 5.0 S=0T7cm a' 2.0 \+ 1.0- 0.5 15 20 25 30 Oepfh (cm) 10 15 20 25 30 Depth (cm) Figure 4.8. Semi-logarithmic plot of a) excess 210pjb activity in the core from Grane Langstf, and b) total zl0PJb activity (full circles) and total 26Ra activity (open circles) in the core from Kalgaardstf. ted average annual 210Pb deposition of ca. 5.5 mBg.cm-^a"1 is of the same order of magnitude as the atmospheric deposition reported by other investigators, indicating that lateral sediment transport has been limited. Additional studies providing independent confirmation on the 210Pb inferred chronology are in progress. At the time samples from Kalgaards0 were analysed it was pos- sible to measure 226Ra as well, using the system described in section 2.4. Hence, Pig. 4.8b shows results of total 210Pb determinations as well as results of 226Ra, used as an estimate of the supported 210Pb activity, 210pb . The 210Pb activity-depth profile shows enormous fluctuations while the 226Ra contribution is almost constant throughout the core. Although the latter observation supports the assumption that in this type of sediment a good approximation of the 2 10 Pb activity may be 132 obtained by assuming that its down-core contribution is constant, it does not help in establishing a reliable 210Pb chronology. Obviously the fluctuations in the 210Pb activity-depth profile cannot be attri- buted to fluctuations in the 210Pb contribution. Furthermore, inte- gration of the 210Pb activity profile yields an extremely high local 210Pb deposition which in addition to the fluctuations in acti- vity indicates deposition of allochthonous material at irregular intervals. Hence this core has to be rejected from further analysis. 4.4.2. ANTARCTICA 4.4.2.1. Introduction During the Antarktis IV/2 expedition of the FS Polarstern a sediment core (1327-1) was taken in the Bransfield Strait (position: o 62°16'05"S/ 57 38'08"W) from a water depth of 1986 m. Estimated sedi- mentation rates in this area are of the order of one cm.a"1 which is relatively high for marine sediments (Van Enst, pers. comm.). There- fore, samples of this core were submitted to our laboratory by the Geological Survey of the Netherlands for llfC and 210Pb analysis in order to establish a sedimentation rate. Furthermore, 226Ra activi- ties were determined by Rn exhalation from the suspended sample to correct for the supported 210Pb activity (section 2.4). 4.4.2.2. Results and discussion Figure 4.9 shows that 210Pb and 222Rn-deduced 226Ra in eight samples, selected from the first 2 meters of core 1327-1 are in radioactive equilibrium. However, the radon exhalation is expected to be strongly dependent on the structure of the mineral matrix and grain size. Therefore, in addition four sediment samples were totally dissolved in HF/HCIO^ and remeasured. Table 4.3 shows that within the uncertainty limits 226Ra activities obtained from the suspended sample are approximately the same as those obtained from totally \ dissolved sample material. The rather large uncertainties are due to the limited amount of sediment available and the relatively high blank activity of 15 + 4 mBq. Hence 210Pb activities are below detection limits so that reliable chronology can not be obtained. The absence of measurable quantities of 210Pb may be attributed 133 100 Antarctica Core 1327-1 50 20 10 Z10 226 I I Figure 4.9. Pb and Ra activity- 0.5 1.0 1.5 2.0 2.5 depth profile for core 1327-1 from An- 210 Depth (m) tarctica, clearly showing that all Pb is supported. Table 4.3- Results of Ra measurements on selected samples from Core 1327-1. Depth (cm) 226Ra activity (mBq/cm3) 22 2 22 2 determined from Rn determined from Rn exhalation froiii the exhalation from the dissolved sample suspended sample 10 - 15 17.85 ± 5.05 19.83 ± 3.40 40 - 43 14.41 ± 3.11 18.01 + 5.20 59 - 64 16.33 ± 1.87 24.66 ± 4.85 67 - 71 16.80 ± 5.38 20.00 ± 6.03 to (1) the relatively low flux of atmospheric 210Pb at the Southern Hemisphere (most of the surface area of the Southern Hemisphere is covered either by water or by ice, which prevents escape of Z22Rn) and (2) the relatively long time required for suspended material in the surface layer of the ocean to be deposited at greater depths due to their low sedimentation velocity. However, regarding the fact that (almost) all 210Pb is supported, the activities are rather high. Hence, they point at a relatively high abundance of u series nuclides 134 in this core. Preliminary evaluations of other data as well as the results of previous investigations in the area point to hydrothermal activity that may well be responsible for high concentrations of U (-series nuclides) (Van Enst, pers. comm.). A proposal for additi- onal scientific research in this area to establish activities of other U-series nuclides and their origin has been submitted. 4.5. SUMMARY In this chapter we have shown the results of measurements on the local 2l°Pb deposition outside our laboratory. Late spring in both 1984 and 1986 (1985 was not measured) shows enhanced 2 2 ° Pb de- position. Due to lack of other explanations we tentatively relate this to annual early spring injection of produced 2 *°Pb from the stratosphere into the troposphere (analogous to e.g. 3H). Short-time fluctuations, however, appear to result in an approx- imately constant average value of the annual deposition which agrees with other reported values. Organic sediments appear to behave as a closed system with respect to 210Pb (or at least 210Po), even under changing pH condi- tions. This was confirmed both by a laboratory experiment and by measurements on several sediment cores. This and the previous obser- vation are consistent with the basic assumptions for the CRS dating model. We have obtained reliable chronologies for a number of sediment cores studied to determine rates of acidification in the past century in isolated shallow moorland water bodies in the Netherlands (so- called "vennen"). In combination with biological data and pollen analysis we conclude that one of these, Achterste Goorven, is among the fastest acidifying water bodies reported for temperate regions exposed to acid precipitation. A regular 210Pb activity-depth profile in the Danish lake Grane Langstf points to a reliable chronology. A sediment core from Lake Kalgaardstf, closely located to Grane Langs0, showed disturbed sedi- mentation. The 210Pb and 226Ra measurements on core 1327-1 from Antarctica show that 210Pb is supported, rendering it impossible to determine a 135 sedimentation rate. Nevertheless, the supported 210:pb activities 5 point to relatively high U-series nuclide concentrations and, in com- bination with other observations, may indicate hydrothermal activity. 137 CHAPTER 5. CHERNOBYL* 5.1. INTRODUCTION At 1.24 hours local time on April 26, 1986 a power excursion in reactor unit number 4 of the Chernobyl Nuclear Power Plant, USSR, ex- plosively initiated the most serious accident with nuclear energy in human history. It was on April 28 that the western world was informed about this accident through measurement of enhanced levels of radio- activity in the air in Scandinavia, as reported to the International Atomic Energy Agency (IAEA) in Vienna. On that same evening, a severe accident was confirmed by the Soviet Union to have occurred at the Chernobyl Nuclear Power Plant. From that moment on programs to moni- tor this introduction of radionuclides in the environment were initi- ated by many individual investigators and institutes all over the world. This included Groningen, where efforts of the Nuclear Accele- rator Institute (KVI) and our institute (CIO) were combined in the working group Fall-out. This resulted in a unique facility for sensi- tive detection and measurement of both gamma-ray as well as alpha- particle emitting nuclides. Meteorological conditions during and after the accident resulted in an extensive atmospheric distribution of released radioactive con- taminants from the Chernobyl reactor in and outside Europe. Local weather conditions such as heavy rainfall resulted in an extremely uneven deposition of radionuclides. On May 2, 1986 the KVI in Gronin- gen was the first to report the arrival of the Chernobyl radioactive cloud to the Netherlands. At the same time we detected in our insti- tute enhanced background counting rates of the 1"c counting equipment (Fig. 5.1) as well as of the Germanium gamma spectrometer installed in our laboratory only two weeks earlier. * The contents of section 5.3 were published before: Van der Veen, J., Van der Wijk, A., Mook, W.G. and De Meijer, R.J., (1986). Core fragments in Chernobyl fallout. Nature 323: 399-400. 138 COUNTRATE (CPM) Chernobyl Contamination (Counter#6) 2.00 Total ^CContamination 1.00 E"j • * 4 1 4 "Long-Lived" - 0.50 Contamination \ 0.20 " Short-Lived\ Contamination Figure 5.1. Temporary enhanced back- 0.10 i ; ground rate of Groningen hC counter no. 6 as a result of contamination < with Chernobyl fallout. The approxi- 0.05 mate contamination half life is 4.0 ± 0.3 days, pointing to contribution of 132 131 I I I i i i i both Te decay (t$ = 3.2 d) and J 8 12 16 20 24 28 decay (tj = 8 d), in accordance with DAYS AFTER MAY 5,1986 gamma-spectrometry observations.
In the months following this first observation the gamma-spec- trometer was used for a vast series of measurements of Chernobyl ac- tivity in food and environmental samples. Most of these measurements were reported in internal reports of the Fall-out group and are beyond the scope of this thesis. However, one exception may be made for the specific case of identification of Chernobyl core fragments by means of both gamma and alpha spectrometry. This chapter discusses the procedures which resulted in the first observation that alpha radioactive materials were distributed over a much wider area than could be inferred from the Soviet Chernobyl accident report to the IAEA (Legasov et al., 1986).
5.2. GAMMA SPECTROMETRY 5.2.1. INTRODUCTION For identification of core fragments, both alpha and gamma spec- trometry were used. The alpha spectrometer has extensively been de- 139
scribed in section 2.2. This section deals with a concise description of the ga""Tia spectrometer and its performance.
5.2.2. DESCRIPTION Figure 5.2 shows a schematic representation of the gamma spec- trometry set-up.Bias of 4000 V is applied across the Ge diode, cooled to liquid nitrogen temperature. The electronics for signal analysis are analogous to the electronics for the alpha spectrometer as de- scribed in section 2.2.1. Signals from the main amplifier are trans- duced to a 1 K (1024 channel) Canberra® series 35 Multi Channel Ana- lyser (MCA). The data can be read from the MCA to an ITT® 2020 micro- computer (64 kByte memory) where they are stored on floppy disk and subsequently evaluated. Background from cosmic and gamma radiation from constructing materials was reduced by placing the detector in an approximately 20 cm thick lead/tungsten cylinder. However, enhanced background radiation due to Chernobyl-induced 137Cs was observed during the experiments. This background was recorded at regular intervals which allowed correction for its effect during the actual measurements.
Lead-Tungsten Pre- Spectroscopy Shielding Amplifier Amplifier
-L Date storage Series 35 (Apple lie)
CANBERRA MCA
Qe-Detector
Figure 5.2. Schematic representation of the gamma spectrometer. 140 5.2.3. CALIBRATION Due to the limited stopping power of Ge for gamma radiation, the detection efficiency depends on the gamma energy as well as on the geometrical position of the source with respect to the detector. We have calibrated this energy-dependent efficiency using calibrated 137Cs and 57Co point sources, mounted at a fixed distance of approxi- mately two centimeters from the front surface of the detector. Figure 5.3 shows the result of this calibration. Gamma spectrometry measure- ments reported in this chapter are based on the use of a point source mounted at the same position as the calibration sources and on using the measured energy-dependent efficiency.
5.3. CORE FRAGMENTS FROM CHERNOBYL FALLOUT 5.3.1. INTRODUCTION In some of the first reports on the fallout of the Chernobyl
reactor accident (Devell et al., 1986; Fry et al.f 1986; Hohenemser et al., 1986) references were made to the presence of localized areas of high radioactivity, so-called hot spots,largely on the basis of measurements of gamma radioactivity in fallout outside the Soviet Union. In this section we will present evidence for alpha radioacti- vity from hot particles inadvertently collected in the Soviet Union
Figure 5.3. Energy-dependent efficiency 50 100 500 1000 calibration, using a point source at a fixed distance of 2 cm from the surface Energy (keV) of the Ge detector. 1 141
50 urn
Figure 5.4. Micrograph of the hot particle from the Kiev trousers (left) and the Minsk shoe (right). by travellers from abroad. The gamma-ray and alpha-particle spectra are those expected from fragments of fuel rods. We have also indica- tions that such particles may have spread into the north-eastern part of Poland.
5.3.2. LOCALIZATION AND COLLECTION OF HOT SPOTS Having learned about our group's work from newspapers and con- tacts with colleagues at other universities in the Netherlands, Dutch travellers returning from the Soviet Union and other East-European countries gave us the opportunity to inspect their clothes and other belongings. In some cases, we also made detailed measurements on luggage. We observed enhanced radioactivity in the following: a pair of trousers from a person who had spent some days in Kiev at the time of the accident towards the end of April, a pair of trousers from a boy who had camped in north-east Poland in July and a shoe from a girl who had visited Minsk in early August. 142
On the Kiev trousers, we found at least five areas with intense localized gamma-ray activity. The Minsk shoe contained one such an area and the boy's trousers showed various spots with enhanced radio- activity. All spots were localized with a sensitive hand monitor for gamma and beta radiation. With pieces of adhesive tape we were able to remove one hot par- ticle from the Kiev trousers and another from the Minsk shoe. Both particles were studied under a microscope and photographed (Fig. 5.4). More detailed photographs show areas of crystalline as well as amor- phous material, indicating that some melting had taken place. The activity on the trousers from north-east Poland most probably con- tained by smaller particles, was too low to isolate. To obtain high-resolution alpha spectra, the Minsk particle was dissolved overnight in cone. HCl in a Teflon vessel at a constant temperature of 60°C. The radionuclides were precipitated on a stain- less steel disk by electroplating (section 2.3.5).
Table 5.1. Fractional gamma-ray activities (%) for various radionucli- des in the hot particles of the "Kiev trousers" and the "Minsk shoe", and the "complete trousers" of N.E. Poland. The activities have been corrected for the decay since April 26, 1986.
a) Nuclide Kiev trousers Minsk shoeb) N.E. Poland trousersc)
25 29 30 ""Ce 22 24 17 103 Ru 24 21 25 1311 Cs 1.0 1.1 0.3 137 Cs 2.4 1.6 0.7 95 Zr 26 24 27 total gamma-,. 849 423 41 .3 ray activity ' (Bg) a) measured on July 7, 1986 b) measured on August 13, 1986 c) measured on August 24, 1986 for the complete trousers d) total Y activity determined only from the nuclides that are listed in the table. 143 Table 5.2. Alpha activity of the hot particle from the Minsk shoe. The values have been corrected for the decay since April 26, 1986a).
Nuclide (MeV) a-activity (Bq) mass (yg) (yr)
5 2 3 9Pu + 24400 5.12 - 5.17 0.068 < 2.9 10~ 2*°Pu 6580
2 3 8Pu + 86 5.45 - 5.50 0.076 < 6.2 10" 1Am 458
jCm + 32 5.74 - 5.81 0.021 < 1 .2 10' 17.6
8 2 "t 2Cm 0.447 6.07 - 6.12 2.032 1.6 10- a) Alpha activity determined on August 29, 1986.
5.3.3. MEASUREMENTS AND RESULTS Gamma-ray activities from the hot spots (Table 5.1, Fig. 5.5a) were determined with the detector described in section 5.2. The rela- tive abundance of the gamma-ray emitting nuclei for all three sub- jects was practically the same, while the activity ratio of 13kCs/ 137Cs is about 0.5, which is characteristic for the Chernobyl fall- out (Fry et al., 1986; Hohenemser et al., 1986). However, the propor- tions of 95Zr and 1IfltCe were much higher than in the fallout reaching Western Europe (Hohenemser et al., 1986), suggesting that the partic- les studied here were released in the first plume which reached Scandinavia• The particles sizes from Kiev and Minsk were determined to be approximately 30 x 15 x 3 ym and 150 x 150 * 110 ym, respectively. The alpha spectra from these particles, as well as the alpha spectrum from a hot spot on the north-east Poland trousers were almost identi- cal: a continuous increase from 2.5 MeV to 6 MeV, where the spectrum 144
1MCe Kiev trousers Date: May 29, 1986 Measuring time : 9000 s
80 131,
103 Ru
60
200 300 ^ 00 500 600 800 900 Energy (KeV)
Figure 5.5a. Energy spectrum of the gamma radiation of the hot particle from the Kiev trousers. showed a sharp high energy edge. The total alpha activities were 50 and 1.4 Bg from the Kiev and Minsk particles, respectively, and ap- proximately 0.08 Bg.cm"2 from the hot spot on the Poland trousers. The dominant activity at 6.12 and 6.07 MeV in the high-resolution alpha spectrum from the Minsk particle (Fig. 5.5) is consistent with 2It2Cm, while the smaller peaks at 5.16, 5.50 and 5.79 MeV indicate the
239 2 2 21(1( 238 21(1 presence of , "°PU, "' Cm, Pu and possibly, Am. The shape of the peak at 5.50 MeV indicates that 238Pu (decay product of 2"2Cm) is the dominant contributor. It is obvious that the activities correspond to only a fraction of the mass of the original particles (Table 5.2). 145
Alpha spectrum Hot Particle Q3 shoe Minsk (a 253) 2"Pu 238, Pu 2WCm 240+Pu 241Am H3+Cm H2Cm
ex *-» 0.2
< ex.
o 0.1
i flinfti n/ 7.0 ENERGY (MeV)
Figure 5.5b. Energy spectrum of alpha radiation obtained by dissolving the Minsk particle in hydrochloric acid and precipitating the nuclides on a stainless steel planchet. Note the change of scales slightly below 6 MeV, indicated by a dashed line.
5.3.4. CONCLUSIONS From the concordance of the measured relative abundances of the various nuclides with the estimated inventory of a reactor core (Cohen, 1977; Lewis, 1975; Legasov et al., 1986), the characteristic Chernobyl ratio 13"Cs/137Cs and the similarity of the total alpha and gamma spectra of the Minsk and Kiev particles, we conclude that both hot particles are core fragments of the Chernobyl reactor released during the accident. Thus the temperature must have been high enough to melt the fuel rods, which agrees with earlier reports from Sweden (Devell et al., 1986). The fragments have contaminated a large area from at least Kiev (150 km leewards) to Minsk (400 km downwind of 146 Chernobyl). Smaller fragments may have reached north-east Poland- Our measurements do not present quantitative information about the degree of contamination of the area or the percentage of the core that has been emitted. However, the presence of at least five hot particles on the Kiev trousers and the fact that at least one hot particle was collected during a twelve-hour stay in Minsk in August, 1986, indicate that such particles are distributed over a much more extended area than previously reported by the Soviet Union. Further- more, studies concerning Nuclear Reactor safety which do not include the possibility of wide spread of hot particles need to be critically reviewed. 147
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SAMENVATTING
De Engelse titel van dit proefschrift "Radiometrie Dating by Alpha Spectrometry on Uranium Series Nuclides" kan in het Nederlands wellicht het best worden weergegeven door "Ouderdomsbepalingen door stralingsmeting aan kernen uit de uraniumreeks met behulp van alfa- spectrometrie". In dit laatste hoofdstuk wil ik proberen om op be- knopte, ook voor de niet gespecialiseerde lezer, de inhoud en achter- grond van het onderzoek samen te vatten. Aan het eind van de vorige eeuw werd het verschijnsel radioacti- viteit ontdekt. Dit leidde tot een revolutionaire ontwikkeling van het inzicht in de bouw van materie. Daarnaast werd men zich bewust van de mogelijkheden die de radioactiviteit biedt voor een breed scala van toepassingen, vaak ten bate van andere takken van weten- schap. Zo blijkt het voorkomen van radioactiviteit in de natuur het mogelijk te maken om de ouderdom van gesteentes en andere materialen te meten. Het gebruik van het radioactieve koolstofisotoop lhC (kool- stof -14) is daarvan het meest bekende en spectaculaire voorbeeld. Maar daarnaast werden en worden ook andere methoden ontdekt, een ont- wikkeling die nog volop in beweging is. Voor ieder radioactief isotoop is de snelheid waarmee het door radioactief verval wordt omgezet in een isotoop van een ander element en dus als zodanig verdwijnt, karakteristiek. Deze vervalsnelheid wordt doorgaans kwantitatief weergegeven door de halveringstijd, de tijd waarin de helft van het oorspronkelijke isotoop is verdwenen. Deze halveringstijd kan gebruikt worden als een ingebouwde natuurlijke klok. In de natuur komen radioactieve elementen voor met de meest uiteenlopende halveringstijden, variërend van enkele miljoensten van seconden tot miljarden jaren. 1ItC (koolstof-14) heeft bijvoorbeeld een halveringstijd van 5730 jaren. Met de in de afgelopen 35 jaar in Groningen ontwikkelde technieken blijkt het mogelijk om hiermee tot ca. 70.000 jaar terug te dateren (er is dan nog slechts een vijf- 158
NAWOORD
Hoe afgezaagd sommige cliché's ook mogen zijn, soms ontkom je er niet aan om er een te gebruiken. Daarom eindigt ook dit proefschrift met de constatering dat de totstandkoming ervan onmogelijk zou zijn geweest zonder de inspanning en (geestelijke) steun van velen. Geen promovendus zonder promotor: de discussies die ik met Wim Mook had over het onderzoek waren altijd constructief en de vrijheid en verantwoordelijkheid die hij mij gaf bij de uitvoering ervan heb ik altijd gewaardeerd en als bijzonder plezierig ervaren. Zijn com- mentaar en kritische kanttekeningen bij het manuscript hebben steeds geleid tot aanzienlijke verbeteringen in de uiteindelijke versie. Dank zij het werk van Koos Steendam draaiden de alfaspectro- meters vrijwel 365 dagen per jaar 24 uur per dag. Janette Spriensma zorgde, nadat Henk Been en de medewerkers van de electronische werk- plaats o.l.v. Jan Keyser een uitbreiding van de meetcapaciteit met vier detectoren hadden gerealiseerd, voor constante aanlevering van de tot dunne radioactieve bronnetjes gereduceerde monsters. Kees Frentz, Jan Reinder Fransens en Jan van der Veen leverden tijdens hun afstudeeronderzoek stof voor vele bladzijden van dit proefschrift. Ook de studenten Ton Martens, Bas Zinsmeister, Marc de Boer en Nomdo Jansonius droegen hun steentje, zandkorrel respectieve- lijk gipsblok bij. Al mijn (ex-)collega's van de onderzoekgroep Isotopenfysica (later omgedoopt tot Centrum voor Isotopen Onderzoek), Jellie, Gert, Wil, Liesbeth, Frans, Hugo, Marlene, Marcel, Ria, Hans, Jaap, Gert Jaap, Marleen, Berthe, Hans, Eric, Henk, Ernst, Harm Jan en Bert wil ik bedanken voor hun collegialiteit en de voortdurend goede sfeer waarin we hebben samengewerkt gedurende mijn promotie-onderzoek. Dat- zelfde geldt voor alle overige medewerkers van het Laboratorium voor Natuurkunde. Buiten ons laboratorium heb ik altijd bijzonder prettig samen- 164
gewerkt met Gij s Berger van het NIOZ en Nico Boelrijk en Coos von Belle van het ZWO-laboratorium voor Isotopengeologie. Aan hun erva- ring en adviezen op met name het gebied van monsterbehandeling heb ik veel gehad. Dank zij de discussies en besprekingen met Ko van Huissteden en Prof.Dr. J. Vandenberghe van de VU in Amsterdam en Jan de Jong en Dr. W. Zagwijn van de RGD in Haarlem begrijp ik nu iets meer van de geologie in Nederland. Ik heb veel plezier beleefd aan het verzuringsonderzoek waarin ik samen met Herman van Dam, Bas van Geel, Alexander Klink en Mike Dickman werkte. In onze discussies bleek het altijd weer mogelijk de problemen vanuit een veelheid aan zienswijzen te benaderen en desal- niettemin tot een eensluidende conclusie te komen. Dit schepte voor mij de gelegenheid om de 21°Pb-dateringsTnethode uitgebreid te testen en haar uitkomsten te verifiëren. Het grote enthousiasme van Gert-Jan Bartstra van het BAI in Groningen over de U/Th dateringsmethode was een bijzondere stimulans om ook fossiel botmateriaal in het onderzoek te betrekken. Especially in the initial stage of this work my discussions with Farid El-Daoushy from the Institute of Physics in Uppsala, Sweden, who spent six months in our laboratory, were very stimulating and useful. I greatly appreciate the kind hospitality of Miro Ivanovich, head of the Uranium Series Disequilibrium Section of the Harwell Laborato- ries in England. For two months, financed by the Netherlands Organi- zation for the Advancement of Pure Research (ZWO), I was allowed to use all the facilities in his laboratory. My discussions with Angela Rae, Andy Plater and Mike Wilkins have contributed a lot to my under- standing of the geochemical behaviour of the actinides. In addition, Miro was willing to spend a lot of his time in critically reading and judging the quality of the manuscript. Op het gebied van het onderzoek naar de gevolgen van het ongeluk met de kerncentrale in Tsjernobyl kan de unieke samenwerking met Rob de Meijer, Joop Jansen en Louis Put van het KVI in de werkgroep Fall- Out niet ongenoemd blijven. Ik heb de verwachting dat we ook in de toekomst op andere gebieden van onderzoek onze krachten zullen kunnen 165
blijven bundelen. Ik ben veel dank verschuldigd aan Prof.Dr. Th. van der Hammen en Prof.Dr. A. van der Woude die het manuscript hebben willen beoor- delen. Dank zij de inzet van Bert de Jonge (tekeningen), Hein Leer- touwer (foto's) en Henny Deenen (typewerk en lay-out) kreeg dit proefschrift zijn uiteindelijke vorm. Er zijn nog veel meer mensen die een bijdrage aan dit proef- schrift in welke vorm dan ook hebben geleverd. Dat ik ze hier niet met name noem betekent niet dat ik hun bijdrage minder zou waarderen. Integendeel, het zijn vaak kleine dingen geweest die er op het juiste moment toe bijdroegen dat iets net op tijd gereed kwam of dat bepaalde ideeën geboren werden. En dat is van onschatbare v?aarde. W> ^f.f ^i-J ;n
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http://blog.wikunia.de/blog/simplex | Article Image
After eight months in Australia I finally had time to finish this blog article. Welcome back! ;)
You solved a lot of Sudokus and Str8ts recently? Well let's get to real world problems... There is a thing called linear programming. The easiest way to explain it is using an example.
Let's assume you're running a business where you can produce three different types of products A,B and C. We can refer to them as apples,bananas and carrots. We have to do some stuff as cutting and peeling the vegetables. Therefore we have four different types of machines which are faster for one vegetable and slower for another.
Machine Product A Product B Product C Availability
I 20 10 10 2400
J 12 28 16 2400
K 15 6 16 2400
L 10 15 0 2400
Total 57 59 42 9600
Item Product A Product B Product C
Revenue p.u. 90$100$ 70$Material p.u. 45$ 40$20$
Profit p.u. 45$60$ 50$Maximum sales 100 40 60 This are the tables we have to consider. The goal is to maximize our profit and we want to know which products are the most profitable. We can read the first table as: If we want to produce ten apples we need 200 "units" of machine I (20 * 10) and 120 "units" of machine J and so on. Each machine has a limit of 2400 "units". These are constraints of our program as well as the maximum sales in the second table. We can only sell 100 apples, 40 bananas and 60 carrots. The goal is to maximize the profit. We get 45$ per unit of apples and so on.
This kind of problem is a linear programming problem, well actually it's a mixed integer program but at the moment we don't care about that. Let's just assume that we can have something like 5,3 apples so fractions of vegetables. In a future blog article we can think about how we can change that to get the best solution in the real world. And yeah sorry I said at the beginning we want to solve real world problems but we have to make one step after the other. See it as a teaser!
Whatever... A simple linear program looks like the following:
Maximize: y
Constraints:
• x <= 2
• y <= 6
• y-x <= 5
• y+x <= 6
and the variables must be >=0 so x >= 0, y >= 0.
Let's try to get the solution by hand: We can set y=6 because it's clear that it can't be higher. Then we have a problem with constraint #3: y-x <= 5 so we have to set x=1 but then the last constraint fails. Therefore we can set y+x=6 and y as high as possible to fulfill constraint #3 which gives us the solution x=0.5 and y=5.5. The objective would be 5.5.
It's called linear program because the equations are all linear. It's not allowed to have constraints or optimization functions with the form x*y.
The first challenge in real world applications is always to convert it into a mathematical form. We need variables, an optimization function and constraints.
In our example it's pretty simple. We have three variables one for each of our vegetables. The objective function is to maximize the profit: 45*a+60*b+50*c where a=apples, b=bananas and c=carrots. Our constraints are defined by the maximum capacity and the maximum sales.
We have seven constraints for that one and three variables which make it a bit harder to visualize. Therefore we go back to our previous example:
Maximize: y
Constraints:
• x <= 2
• y <= 6
• y-x <= 5
• y+x <= 6
• x >= 0
• y >= 0
We can visualize that:
This diagram shows the different constraints and the overlap area where every constraint is fulfilled. The next one only visualizes the area where each of the constraints are fulfilled.
Now it's easily possible to get the maximum value for y which is 5.5. In this representation we see that the solution is a vertex of our green constraint surface. In fact this is always the case which is more or less the main idea of the simplex algorithm.
The principle of the simplex algorithm is to just have a look at the vertices of our surface.
What does this mean? It actually reduces our search space from infinitely many points inside the green area to five points. We can compute all five points and just keep the best solution in this case y=5.5.
For bigger problems there might be a lot of points on our surface so computing all of the vertices takes a long time as well. Can we do better?
Just assume for a moment that we found the point p=(2,4)which is the red upper point on the right. Then it doesn't make any sense to consider the bottom point z=(2,0)because the y value which is our objective function is lower and we want to maximize it.
What do we have at the moment?
• Consider only the vertices of our constraint polygon
• Move only to a vertex which has a better objective
How to find the vertices?
Before we go further, I hope you are familiar with a linear system of equations because it's crucial for the simplex algorithm. A system looks like this:
x + y = 5
2x + y = 9
Which can be written in a matrix form as:
1 1 | 5
2 1 | 9
The next step is to use Gaussian elimination to get
1 1 | 5
0 0.5 | 0.5
This means 0.5y = 0.5 => y = 1. Therefore the first equation is x+1 = 5 => x = 4.
Let's build our matrix for our constraints
1 0 | 2
0 1 | 6
-1 1 | 5
1 1 | 6
and we add a new row which is our objective row which is negated.
1 0 | 2
0 1 | 6
-1 1 | 5
1 1 | 6
--------
0 -1 | 0
The first row can be read as x = 2 but we want to have it x<=2 therefore we can add a slack variable a and write it as: x+a=2 with a >= 0. We do the same thing with every constraint we have which gives us this matrix:
x y a b c d
--------------------
1 0 1 0 0 0 | 2
0 1 0 1 0 0 | 6
-1 1 0 0 1 0 | 5
1 1 0 0 0 1 | 6
--------------------
0 -1 0 0 0 0 | 0
As I mentioned before we don't have to have a look at all possibilities in our green surface. We just have to consider the vertices which are called basic feasible solutions (BFS).
A basic feasible solution in our matrix form is a solution where we have basic variables which are equal to the right hand side and non basic variables which are 0.
In our case we have a basic feasible solution of a = 2, b = 6, c = 5, d = 6 and x = y = 0. This definitely fulfills our constraints because we don't need Gaussian elimination as we already have an identity matrix of size 4 in our matrix notation. This BFS represents our dot in the left hand corner p = (0,0).
How to improve our objective?
The last row shows us that we have an objective (obj) of 0 and that y has a negative value means that we can rise our objective by changing the value of y. That means that we want to assign a value for y and therefore it should be a basic variable. In return we need to make a basic variable non basic because we can only have 4 basic variables as we have 4 constraints.
We have to pick a basic variable which should leave (called leaving variable) and we already choose our entering variable y. The corresponding row to the leaving variable should have a positive value in the entering column. What the hell is the corresponding row?
We have the basic variables a,b,c,d. The corresponding rows are respectively row 1,2,3,4. In general the corresponding row is where the factor of the basic variable is 1. For example the factor of a is 1 in row number one.
Our positive values are:
To choose our leaving variable out of the three options we use the one with the lowest ratio of the right hand side divided by the blue value.
At the moment I just want to show you the basic concept that's why I will not explain why we choose it here. Ask me about it in the comments if you want to get more information about it. Our try it out by yourself first. :)
We now have our leaving variable which is c because it corresponds the other way around with row number three. Our goal is to have a basic feasible solution again so we want to have a 1 at the red spot and 0 at the blue entries.
Therefore we are using Gaussian elimination again. Let's create a zero in our objective row. Therefore we use:
Which gives us the following matrix:
It directly gives us our new objective which is 5. To complete our elimination procedure we have to do the same with row 2 and row 4. This is our result and the blue values show the new basic.
Our basic variables are y, a, b, d so we swapped c with y. Row #3 shows that y is set to 5. x is still not part of our basis so it is set to 0. We are now at point p=(0,5) in our graphic.
If we are changing the value of x we can increase the objective. We know that we can increase it because the value of the x column in the objective row is negative.
We are doing the same thing again. Let's choose the leaving variable which would be d because 1/2 is the smallest ratio we can get.
Our final matrix looks like this:
Our variables are now set to x=0.5, y = 5.5, a = 1.5, b = 0.5.
Let's get back to our constraints:
• x + a = 2
• y + b = 6
• -x + y + c = 5
• x + y + d = 6
and check whether the equations are correct:
• 0.5 + 1.5 = 2
• 5.5 + 0.5 = 6
• -0.5 + 5.5 + 0 = 5
• 0.5 + 5.5 + 0 = 6
Yeah! They are all true!
Why is this the optimal value?
In our obj row there are no negative values so we are not able to improve the obj. Can you explain why we only have to look at negative values in the obj row? Sure let's assume we want to have d as our entering variable (yeah we had that one already but just try to do it). By doing the first Gaussian elimination step (changing the obj) we multiply one of the rows 1-4 with -l/e where l is the value in the obj row (here l = 0.5) and e is positive value with the lowest ration blablabla... Let's assume we can choose any e we want (not 0 cause we can't divide by 0).
Then we would have the factors: -0.5/(-0.5) = 1 -0.5/(-0.5) = 1 -0.5/0.5 = -1 -0.5/0.5 = -1
We take the last two first: Here we would have an obj of -1*5.5+5.5=0 which is less than 5.5 or -1*0.5+5.5 = 5 which is less than 5.5 as well. Therefore it's not reasonable to consider these two.
Let's take the first one which looks promising: 1*1.5+5.5 = 7 Looks good because our obj can be 7. Do you remember our constraint: y <= 6? So that can't be correct but what would our matrix look like?
In the second step we would like to apply this elimination:
The result is:
Here we have a negative value in our right hand side which means that c=-1 and that is not allowed because we have the constraint that every variable is >=0.
Therefore we can only improve our obj if there is a negative value in our obj row.
You want to see some code?
Back to our vegetable business again:
from Model.model import Model
import numpy as np
from time import time
m = Model()
"""
A Manufacturing Example
Machine | Product A | Product B | Product C | Availability
I | 20 | 10 | 10 | 2400
J | 12 | 28 | 16 | 2400
K | 15 | 6 | 16 | 2400
L | 10 | 15 | 0 | 2400
Total | 57 | 59 | 42 | 9600
Item | Product A | Product B | Product C
Revenue p.u. | 90$| 100$ | 70$Material p.u. | 45$ | 40$| 20$
Profit p.u. | 45$| 60$ | 50$Maximum sales | 100 | 40 | 60 Maximize profit """ a = m.add_var("real+", name="a") b = m.add_var("real+", name="b") c = m.add_var("real+", name="c") m.maximize(45*a+60*b+50*c) m.add_constraint(20*a+10*b+10*c <= 2400) m.add_constraint(12*a+28*b+16*c <= 2400) m.add_constraint(15*a+6*b+16*c <= 2400) m.add_constraint(10*a+15*b <= 2400) m.add_constraint(a <= 100) m.add_constraint(b <= 40) m.add_constraint(c <= 60) t0 = time() m.solve() print("Solved in %f" % (time()-t0)) m.print_solution() This is the code where we explain the model to our program. We initialize variables, set the objective and define the constraints. Let's begin with adding variables: a = m.add_var("real+", name="apple") Our class for the model looks like this: class Model: def __init__(self, print_obj=False): self.constraints = [] self.variables = [] self.MINIMIZE = -1 self.MAXIMIZE = 1 if not print_obj: self.p = {} else: self.p = print_obj l = ['start_conf','leaving_entering'] for pl in l: if pl not in self.p: self.p[pl] = False def add_var(self, ty, name="NA", value=None): if ty == "real+": x = RealNN(name,value,index=len(self.variables)) self.variables.append(x) return x First of all we initialize it without variables and constraints and we can define how to out print some stuff. In the add_var function we have a type of the variable. At the moment we only use the type real+ for real positive values. A variable can have a name and get's an index. We have a look at the RealNN part in a second. Our next step is to describe our objective function: m.maximize(45*a+60*b+50*c) Normally you can't write a variable + a variable in Python that's why we have the RealNN class. from .constraint import Constraint from .error import NonLinear from .list_RealNN import List_RealNN class RealNN: def __init__(self,name=None,value=None,index=0,factor=1): self.name = name self.value = value self.factor = factor self.index = index def __str__(self): if self.value is None: return "%s has no value" % self.name else: return "%s has value %.2f" % (self.name,self.value) def __eq__(self, other): return Constraint(self, "==", other) def __le__(self, other): return Constraint(self, "<=", other) def __add__(self, other): if self.value is not None and other.value is not None: return RealNN(value=self.factor*self.value+other.factor*other.value) else: return List_RealNN(self,other) def __radd__(self, other): if other == 0: return self else: return self.__add__(other) def __neg__(self): return RealNN(name=self.name,value=self.value,index=self.index,factor=-self.factor) def __sub__(self, other): if self.value is not None and other.value is not None: return RealNN(value=self.factor*self.value-other.factor*other.value) else: return List_RealNN(self,-other) def __rsub__(self, other): if other == 0: return self else: return self.__sub__(other) def __rmul__(self, factor): if isinstance(factor, (int, float, complex)): return RealNN(self.name,self.value,index=self.index,factor=factor) else: raise NonLinear("factor %s is not linear" % factor) def get_coefficients(self,l=False): if not l: l = 1 l_factor = [0]*l l_factor[self.index] = self.factor return l_factor What it basically does is definining what to do if we write something like apple+banana or if we write stuff like 3*apple and so on. It's a bigger concept on it's own and I can write an entire article about it but want to only describe the simplex algorithm in this one. If you want to hear more about this part => comment :) Back to the Model part: def maximize(self, obj): self.obj_coefficients = obj.get_coefficients(len(self.variables)) self.obj_type = self.MAXIMIZE def minimize(self, obj): self.obj_coefficients = obj.get_coefficients(len(self.variables)) self.obj_type = self.MINIMIZE We get the coefficients of our objective and define the type as MINIMIZE or MAXIMIZE. In this blog article I will only explain the maximize part. Minimization problems are a bit different. Let's go on with adding the constraints: def add_constraint(self,constraint): self.constraints.append(constraint) Our constraints are: m.add_constraint(20*a+10*b+10*c <= 2400) m.add_constraint(12*a+28*b+16*c <= 2400) m.add_constraint(15*a+6*b+16*c <= 2400) m.add_constraint(10*a+15*b <= 2400) m.add_constraint(a <= 100) m.add_constraint(b <= 40) m.add_constraint(c <= 60) I'll show you how we stored the constraints and the variables later but you don't need to understand all the code behind it. The matrix I showed you at the beginning is called tableau. We fill it with zeros first. self.tableau = np.zeros((len(self.constraints)+1,len(self.variables)+1)) We fill the tableau with the real values now: i = 0 for constraint in self.constraints: coefficients = constraint.x.get_coefficients(len(self.variables)) self.tableau[i] = coefficients+[constraint.y] i += 1 if self.obj_type == self.MAXIMIZE: if constraint.type != "<=": raise Unsolveable("Type %s isn't accepted" % constraint.type) if self.obj_type == self.MINIMIZE: if constraint.type != ">=": raise Unsolveable("Type %s isn't accepted" % constraint.type) The first part fills the tableau by using our constraint class. We can get the coefficient of our constraints as a list using constraint.x.get_coefficients(len(self.variables)) and constraint.y holds the right hand side of our constraint. The second one tests whether we can solve it. It isn't allowed to use constraints like x>=2 in minimization problems or x<=2 in maximization problems at least for now ;) The result is: [[ 20. 10. 10. 2400.] [ 12. 28. 16. 2400.] [ 15. 6. 16. 2400.] [ 10. 15. 0. 2400.] [ 1. 0. 0. 100.] [ 0. 1. 0. 40.] [ 0. 0. 1. 60.] [ 0. 0. 0. 0.]] In the tableau the objective function is missing. We add it using: # set obj if self.obj_type == self.MINIMIZE: self.tableau[-1,:] = np.append(np.array(self.obj_coefficients), np.zeros((1,1))) elif self.obj_type == self.MAXIMIZE: self.tableau[-1,:] = np.append(-np.array(self.obj_coefficients), np.zeros((1,1))) After adding the objective function the tableau looks like this: [[ 20. 10. 10. 2400.] [ 12. 28. 16. 2400.] [ 15. 6. 16. 2400.] [ 10. 15. 0. 2400.] [ 1. 0. 0. 100.] [ 0. 1. 0. 40.] [ 0. 0. 1. 60.] [ -45. -60. -50. 0.]] We can solve the problem now as explained above. At first the slack variables are missing and we don't have our basic feasible solution which is the same step. How we do it is described in self.find_bfs(). def find_bfs(self): self.redef_matrix_bs_obj() self.row_to_var = [False for x in range(self.matrix.shape[0])] # Build slack variables identity = np.eye(self.matrix.shape[0]) identity = np.r_[identity, np.zeros((1,self.matrix.shape[0]))] self.tableau = np.c_[self.tableau[:,:-1], identity, self.tableau[:,-1]] self.redef_matrix_bs_obj() # range not including the b column # get all columns which have only one value => basis for c in range(len(self.variables),self.matrix.shape[1]-1): row = np.argwhere(self.matrix[:,c]!=0)[0][0] self.row_to_var[row] = c if self.p['start_conf']: print("Start Tableau:") print(self.tableau) The idea of self.redef_matrix_bs_obj() is that we have several parts in our tableau like the objective function the matrix (left hand side) and the right hand side. def redef_matrix_bs_obj(self): self.matrix = self.tableau[:-1] self.bs = self.tableau[:-1,-1] self.obj = self.tableau[-1,:] Then we define which row corresponds with which variable as described in our smaller example. We fill them with False first as we don't have a basic feasible solution yet. self.row_to_var = [False for x in range(self.matrix.shape[0])] Now we can add our slack variables: identity = np.eye(self.matrix.shape[0]) identity = np.r_[identity, np.zeros((1,self.matrix.shape[0]))] self.tableau = np.c_[self.tableau[:,:-1], identity, self.tableau[:,-1]] Our slack variables are basically an identity matrix added to our tableau. The identity matrix itself has a row of zeros as the last row for our objective function. The last line combines the old tableau with the identity matrix. The new tableau: [[ 20. 10. 10. 1. 0. 0. 0. 0. 0. 0. 2400.] [ 12. 28. 16. 0. 1. 0. 0. 0. 0. 0. 2400.] [ 15. 6. 16. 0. 0. 1. 0. 0. 0. 0. 2400.] [ 10. 15. 0. 0. 0. 0. 1. 0. 0. 0. 2400.] [ 1. 0. 0. 0. 0. 0. 0. 1. 0. 0. 100.] [ 0. 1. 0. 0. 0. 0. 0. 0. 1. 0. 40.] [ 0. 0. 1. 0. 0. 0. 0. 0. 0. 1. 60.] [ -45. -60. -50. 0. 0. 0. 0. 0. 0. 0. 0.]] We fill our corresponding variable array now: # range not including the b column # get all columns which have only one value => basis row = 0 for c in range(len(self.variables),self.matrix.shape[1]-1): self.row_to_var[row] = c row += 1 The next step is to find an entering variable and a leaving variable. This step is called pivoting. As long as we find a negative value in the objective row we do this step over and over again. solved = self.pivot() steps = 1 while not solved: solved = self.pivot() steps += 1 At first we check whether we are optimal. def pivot(self): breaked = False # check if the current tableau is optimal # if optimal every value in obj is non negative for c in range(self.matrix.shape[1]-1): if c in self.row_to_var: continue if self.obj[c] < 0: positive = np.where(self.matrix[:,c] > 0)[0] if len(positive): entering = c l = np.argmin(self.bs[positive]/self.matrix[positive,c]) leaving_row = positive[l] leaving = self.row_to_var[leaving_row] breaked = True break if not breaked: return True An entering variable can't enter again therefore we can continue on those. If we have a negative value we try to find a positive value in that column to get our leaving variable. The entering variable is just the column we are testing right now. For the leaving variable we choose the minimum value of the right hand side divided by the positive value. In our example: A B C s1 s2 s3 s4 s5 s6 s7 b 0 [[ 20. 10. 10. 1. 0. 0. 0. 0. 0. 0. 2400.] 1 [ 12. 28. 16. 0. 1. 0. 0. 0. 0. 0. 2400.] 2 [ 15. 6. 16. 0. 0. 1. 0. 0. 0. 0. 2400.] 3 [ 10. 15. 0. 0. 0. 0. 1. 0. 0. 0. 2400.] 4 [ 1. 0. 0. 0. 0. 0. 0. 1. 0. 0. 100.] 5 [ 0. 1. 0. 0. 0. 0. 0. 0. 1. 0. 40.] 6 [ 0. 0. 1. 0. 0. 0. 0. 0. 0. 1. 60.] 7 [ -45. -60. -50. 0. 0. 0. 0. 0. 0. 0. 0.]] We begin with the first column and pick the positive values in that column. Therefore we can choose whether we pick 20,12,15,10 or 1 as our pivot element. To get the best pivot element we divide the right hand side by these values: 2400/20 = 120 2400/12 = 200 2400/15 = 160 2400/10 = 240 100/1 = 100 The minimum value is the last one which means that 1 is our pivot element and the corresponding variable is s5 because in row 4 the s5 column has the 1. If we found a leaving and an entering variable we know that we aren't optimal otherwise we are optimal. In that case we return True. Our new entering variable is A therefore we have to eliminate every element in that column except of in row 4. We use Gaussian elemination here. We start with the objective row: fac = -self.tableau[-1,entering]/self.matrix[leaving_row,entering] self.tableau[-1] = fac*self.matrix[leaving_row]+self.tableau[-1] This gives us our new tableau and our objective of 4500. [[ 20. 10. 10. 1. 0. 0. 0. 0. 0. 0. 2400.] [ 12. 28. 16. 0. 1. 0. 0. 0. 0. 0. 2400.] [ 15. 6. 16. 0. 0. 1. 0. 0. 0. 0. 2400.] [ 10. 15. 0. 0. 0. 0. 1. 0. 0. 0. 2400.] [ 1. 0. 0. 0. 0. 0. 0. 1. 0. 0. 100.] [ 0. 1. 0. 0. 0. 0. 0. 0. 1. 0. 40.] [ 0. 0. 1. 0. 0. 0. 0. 0. 0. 1. 60.] [ 0. -60. -50. 0. 0. 0. 0. 45. 0. 0. 4500.]] Now we do Gaussian elemination for the other rows as well and we want to have a 1 in our new entering row because we always want to have an identity matrix in some way in our tableau. # gausian elemination for row in range(self.matrix.shape[0]): if row != leaving_row: fac = -self.matrix[row,entering]/self.matrix[leaving_row,entering] self.matrix[row] = fac*self.matrix[leaving_row]+self.matrix[row] self.matrix[leaving_row] /= self.matrix[leaving_row,entering] The result is: [[ 0. 10. 10. 1. 0. 0. 0. -20. 0. 0. 400.] [ 0. 28. 16. 0. 1. 0. 0. -12. 0. 0. 1200.] [ 0. 6. 16. 0. 0. 1. 0. -15. 0. 0. 900.] [ 0. 15. 0. 0. 0. 0. 1. -10. 0. 0. 1400.] [ 1. 0. 0. 0. 0. 0. 0. 1. 0. 0. 100.] [ 0. 1. 0. 0. 0. 0. 0. 0. 1. 0. 40.] [ 0. 0. 1. 0. 0. 0. 0. 0. 0. 1. 60.] [ 0. -60. -50. 0. 0. 0. 0. 45. 0. 0. 4500.]] The interpretation of this tableau is simply that if we only produce apples we can have profit of 4500 and to get this value we have to produce 100 apples. (see right hand side) We define our new basis using: def new_basis(self,entering,leaving): for row in range(self.matrix.shape[0]): if self.row_to_var[row] == leaving: self.row_to_var[row] = entering break Which is called using self.new_basis(entering, leaving). In the end we have this tableau: [[ 0 1 0 -0 0 0 0 0 0 -0.5 16.4] [ 0 0 1 0 0 0 0 0 0 1 60 ] [ 0 0 0 -0.8 0.1 1 0 0 0 -9.2 114.5] [ 0 0 0 -0.2 -0.5 0 1 0 0 9.5 1336.4] [ 1 0 0 0.1 -0 0 0 0 0 -0.3 81.8] [ 0 0 0 -0.1 0 0 0 1 0 0.3 18.2] [ 0 0 0 0 -0 0 0 0 1 0.5 23.6] [ 0 0 0 1.2 1.7 0 0 0 0 10.5 7663.6]] Therefore our result in human readable form is the following: We can produce 81.8 apples, 16.4 bananas and 60 carrots. Our profit is 7663.6$. Let's have a look at our description of the problem again.
Machine Product A Product B Product C Availability
I 20 10 10 2400
J 12 28 16 2400
K 15 6 16 2400
L 10 15 0 2400
Total 57 59 42 9600
and a closer look to our machines:
20*81.8+10*16.4+10*60 = 2400
12*81.8+28*16.4+16*60 = 2400.8
15*81.8+6 *16.4+16*60 = 2285.4
10*81.8+15*16.4 = 1064
First of all the second line looks like we did something wrong because we had the constraint that every machine has only a capacity of 2400. Actually I rounded the values for the amount of apples and bananas. More accurate values are 81.818182 apples and 16.363636 bananas which would give us 2399.999992. Even these values are cropped which we can see more easily in our tableau on the right hand side.
16.4
60
114.5
1336.4
81.8
18.2
23.6
7663.6
We can use our row_to_var array to find the corresponding variable for each of these values.
16.4 bananas
60 carrots
114.5 slack 3
1336.4 slack 4
81.8 apples
18.2 slack 5
23.6 slack 6
7663.6 our profit
I explained the meaning of apples,bananas and carrots already but what is the meaning of those slack variables? The slack variables 3,4,5 and 6 corresponds to the constraints 3,4,5 and 6.
m.add_constraint(15*a+6*b+16*c <= 2400)
m.add_constraint(b <= 40)
All other slack variables are 0. That means the machines I and J are running on capacity. The last machine L has a lot of unused capacity which might be interesting for our business. We aren't allowed to produce more carrots (60) but we are allowed to produce more apples and bananas.
This means if we are allowed to produce more carrots we might be able to increase our profit (for example finding new markets where we can sell carrots). On the other hand if we are able to increase the productivity of our first two machines we would be able to produces more apples and bananas.
I hoped you enjoyed the first part of the simplex algorithm and might be able to use it in your real life business ;)
As mentioned during the article there are different parts which might need a second look like how to use this for solving minimization problems.
Tell me what you are thinking about it and what kind of articles you would like to read in the future.
As always you find the code on Github
A new article about the simplex algorithm: Convert to standard form
Additional question: I would like to be able to pay for my server costs using this blog. Which are about 50USD per year. I tried paypal and Amazon for now but achieved about 3USD in the last 6 months. I'm thinking about having a patreon page instead. What do you think about it? What kind of special things can I offer so that you get something more for your own besides the users who are just reading this blog once which I think should always be an option to get my OPEN SOURCE content for free. | 2021-06-23 20:16:59 | {"extraction_info": {"found_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": 0.4903452694416046, "perplexity": 457.5702396698187}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1623488540235.72/warc/CC-MAIN-20210623195636-20210623225636-00232.warc.gz"} |
https://socratic.org/questions/what-does-m-s-2-measure | # What does M/s^2 measure?
Jul 14, 2017
Acceleration.
#### Explanation:
I assume you mean $\frac{m}{s} ^ 2$...
That is the measurement of acceleration. It is derived from the distance divided by the time - squared (also read as "per second per second).
This is because acceleration does not measure velocity. Instead, it measures the changes of speed in a time interval, typically in seconds.
The acceleration can be calculated by this formula: $a = \frac{\Delta v}{\Delta t}$. Where...
=> $a$ is the the rate of change of the velocity (AKA acceleration) in the derived units.
=> $\Delta v$ is the change in speed in its respected units.
=> $\Delta t$ is the change in time in the time alloted, normally seconds.
I won't go too far into acceleration because that's not what you're asking, but I hope the information I provided answers your question!
Hope this helps :) | 2022-07-05 01:29:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "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": 0.7673681974411011, "perplexity": 823.328220943497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656104506762.79/warc/CC-MAIN-20220704232527-20220705022527-00651.warc.gz"} |
https://socratic.org/questions/how-do-you-evaluate-the-expression-25-of-2000 | # How do you evaluate the expression: 25% of 2000?
It means $\left(\frac{25}{100}\right)$ multiplied by $2000$ which equals to 500. | 2020-07-09 04:53:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.598741888999939, "perplexity": 578.35351400931}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655898347.42/warc/CC-MAIN-20200709034306-20200709064306-00223.warc.gz"} |
https://www.physicsforums.com/threads/inheritance-what-is-the-chance-of-the-next-calf-being-amputated.632114/ | # Homework Help: Inheritance: What is the chance of the next calf being amputated
1. Aug 29, 2012
### Swetasuria
1. The problem statement, all variables and given/known data
The absence of legs in cattle (amputated) has been attributed to a completly recessive lethal gene. A normal bull is mated with a normal cow and they produce an amputad calf (usually dead at birth). The same parents are mated again.
1) What is the chance of the next calf being amputated?
2) What is the chance of these parents having two calves both of which are amputated?
3. The attempt at a solution
1) 1/4
2) The chance of getting two amputated calves is 2/8. So answer is 1/4. My teacher says its 1/16 but I don't get it.
2. Aug 29, 2012
### Simon Bridge
Re: Inheritance
two independent trials with a 1/4 probability of something happening gives 1/16 probability of it happening both times.
To understand it - draw the tree.
How did you get 2/8 ?
3. Aug 29, 2012
### Swetasuria
Re: Inheritance
While having their first child, [favourable outcome (ie., amputated calf)]/[total number of outcomes]=1/4
Same chances with second child.
So, the probability of having 2 amputated calves= [total number of favourable outcomes]/[total number of outcomes]=2/8=1/4
4. Aug 29, 2012
### Simon Bridge
Re: Inheritance
1. I would hesitate to describe an amputated (and, therefore, dead) calf as a "favorable" outcome.
2. You have miscalculated the total number of possible outcomes as well as the number of those outcomes which are of interest.
lets say the first calf dies - this is one path to the final family.
then the final family is either
2 dead calves (second calf dies too) OR 1 dead and 1 alive
there are three ways the 2nd calf can live - it inherits the bad gene from dad alone, from mum alone, or from neither.
This makes a total of four possible families from the case that the first calf dies.
One of those possible families involves two dead calves.
Now lets say the first calf does not die. There are three ways this can happen.
There are still four possible families for each of these paths - some of them include a single dead calf because the second calf can still die. But none of them can result in 2 dead calves.
Since there are three paths leading to four families each - that is 3x4=12 possible families.
With the four from the path where the first calf dies, that is a total of 16 possible ways the family could end up.
Only one of those combinations has two dead calves.
Therefore - probability of 2 out of 2 dead (amputated) calves is 1/16.
This is a binomial distribrution: probability of getting k "successes" out of n trials with probability p per trial is:$$P(K=k) = \binom{n}{k}p^k(1-p)^{n-k}$$ so what is the probability of getting n "successes" out of n trials?
Last edited: Aug 29, 2012
5. Aug 29, 2012
### Swetasuria
Is k=2 (since we need two ampuated calves) and n=2 (parents breed two times)?
What is the value for p?1/4?
When I substitute these values, I get P=1/16
Last edited: Aug 29, 2012
6. Aug 29, 2012
### Simon Bridge
Cool - from that you should be able to figure out the probability of getting exactly one dead calf - of getting at least one dead calf - of a family where neither offspring inherited the bad gene? If you mated a new cow with one of the surviving offspring - what is the probability the cow is mating with a carrier of the bad gene?
Notice that 1/16 is also the probability of getting four heads out of four coin tosses.
It is worth spending some effort getting used to the way these things work.
7. Aug 30, 2012
### Swetasuria
If the offsprings did not inherit the bad gene, the probability of getting a dead calf is zero?
P=1/2 Correct?
8. Aug 30, 2012
### Simon Bridge
I think you need to practice combinations and permutations.
Go back to basics: each possible calf-type can be distinguished genetically by which parent it got it's gene from - lets designate the bad gene by x and any good gene by o and each calf is described by an ordered pair of genes "dad/mom" ... then I can denote a family of mom dad and two calves as two pairs of genes. The number of different possible families are:
xx xx, xx xo, xx ox, xx oo,
xo xx, xo xo, xo ox, xo oo,
ox xx, ox xo, ox ox, ox oo,
oo xx, oo xo, oo ox, oo oo.
Count them up - there are 16 possible pairs of calves.
One of the pairs is (xx xx) - which is both calves dead/amputated.
So - number of instances (refusing to call it a "success") divided by number of possibilities is 1/16. => 1/16 chance both calves are dead.
Exactly one dead calf is either (xx xo) (xx ox) (xx oo) and the reverse.
Count them all up - there's three where xx is first-born, and another three where xx is second born = 6. So the probability of exactly one dead calf is 6/16.
For "at least one dead calf" includes the possibility of two dead calves - probability is 7/16.
Since the number of pairs is 16, then the number of individual calves is 16x2=32. (note: I've counted a lot of combinations several times in this - because there are several paths to get those combinations.)
However, you can only mate live cows - how many live cows? (xx = dead)
How many did not inherit the bad gene (combination oo)?
Therefore - how many did inherit the x gene?
What is the probability of mating with a carrier of the x gene?
But we could have done it thus:
If the cow is alive then it is either xo ox or oo - for three total. Two of them have the x gene, so the probability is 2/3 ... however, the combinations may not be equally likely. Fortunately we have more information: we know about their parentage... just to be sure.
Last edited: Aug 30, 2012
9. Aug 30, 2012
### Swetasuria
Answers are 24/32=3/4, 8/32=1/4, 24/32=3/4 and 16/32=1/2 respectively.
I hope these are right. I notice that these are the answers I get if the family had only one child.
10. Aug 30, 2012
### Simon Bridge
So your answer for "how many live cows" is 24/32 ... so there is less than one live cow out of all 32 possible cows?
Also be careful not to include dead cows in your possible mating partners.
Note: your fractions are telling you that 3/4 of all cows born to these parents will live, 1/4 will not have the x gene, etc ... which, as you say, is not surprising since these are the odds you started out with.
But what you want to work out is the probability for mating a new cow with a partner who has the x gene.
Focus on what that means - does the new cow have 32 possible mating partners?
Last edited: Aug 30, 2012
11. Aug 30, 2012
### Swetasuria
Okay. So the answers are 24, 8, 24 and 16/24=2/3 respectively.
12. Aug 30, 2012
### Simon Bridge
There you go :)
You could also have pointed out that each of the possible options could be arrived at by 8 ways, so they each do actually have equal probability.
A good discipline when you get a number is to turn the question into the start of an answer and see if the number makes sense in it. eg. if the question was "how much wood would a woodchuck chuck?" and your number is 5kP, writing or just thinking to yourself: "A woodchuck would chuck 5 kiloPascals of wood." would alert you that something may be wrong and you need to check if you really did measure wood in terms of pressure. | 2018-04-20 09:22:28 | {"extraction_info": {"found_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": 0.6348037123680115, "perplexity": 2614.655811468052}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125937193.1/warc/CC-MAIN-20180420081400-20180420101400-00204.warc.gz"} |
https://zbmath.org/?q=an:1091.46026 | zbMATH — the first resource for mathematics
Topological algebras with pseudoconvexly bounded elements. (English) Zbl 1091.46026
Jarosz, Krzysztof (ed.) et al., Topological algebras, their applications, and related topics. Proceedings of the conference to celebrate the 70th birthday of Professor Wiesław Żelazko, Bȩdlewo, Poland, May 11–17, 2003. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 67, 21-33 (2005).
Summary: It is shown that every commutative sequentially bornologically complete Hausdorff algebra $$A$$ with bounded elements is representable in the form of an (algebraic) inductive limit of an inductive system of locally bounded Fréchet algebras with continuous monomorphisms if the von Neumann bornology of $$A$$ is pseudoconvex. Several classes of topological algebras $$A$$ for which $$r_A(a)\leq \beta_A(a)$$ or $$r_A(a)= \beta_A(a)$$ for each $$a\in A$$ are described.
For the entire collection see [Zbl 1063.46001].
MSC:
46H05 General theory of topological algebras 46H20 Structure, classification of topological algebras | 2021-09-23 15:12:16 | {"extraction_info": {"found_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": 0.41169893741607666, "perplexity": 1378.0864954517522}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057424.99/warc/CC-MAIN-20210923135058-20210923165058-00246.warc.gz"} |
https://joshbloom.org/publication/2013-ap-js-207-3-s/ | # Type Ia Supernovae Strongly Interacting with Their Circumstellar Medium
### Abstract
Owing to their utility for measurements of cosmic acceleration, Type Ia supernovae (SNe Ia) are perhaps the best-studied class of SNe, yet the progenitor systems of these explosions largely remain a mystery. A rare subclass of SNe Ia shows evidence of strong interaction with their circumstellar medium (CSM), and in particular, a hydrogen-rich CSM; we refer to them as SNe Ia-CSM. In the first systematic search for such systems, we have identified 16 SNe Ia-CSM, and here we present new spectra of 13 of them. Six SNe Ia-CSM have been well studied previously, three were previously known but are analyzed in depth for the first time here, and seven are new discoveries from the Palomar Transient Factory. The spectra of all SNe Ia-CSM are dominated by Hα emission (with widths of åisebox-0.5ex 2000 km s$^-1$) and exhibit large Hα/Hβ intensity ratios (perhaps due to collisional excitation of hydrogen via the SN ejecta overtaking slower-moving CSM shells); moreover, they have an almost complete lack of He I emission. They also show possible evidence of dust formation through a decrease in the red wing of Hα 75-100 days past maximum brightness, and nearly all SNe Ia-CSM exhibit strong Na I D absorption from the host galaxy. The absolute magnitudes (uncorrected for host-galaxy extinction) of SNe Ia-CSM are found to be -21.3 mag <= M$_R$ <= -19 mag, and they also seem to show ultraviolet emission at early times and strong infrared emission at late times (but no detected radio or X-ray emission). Finally, the host galaxies of SNe Ia-CSM are all late-type spirals similar to the Milky Way, or dwarf irregulars like the Large Magellanic Cloud, which implies that these objects come from a relatively young stellar population. This work represents the most detailed analysis of the SN Ia-CSM class to date.
Publication
Astrophysical Journal Supp. | 2022-10-07 19:44:02 | {"extraction_info": {"found_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": 0.717336893081665, "perplexity": 3422.6277165649017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030338244.64/warc/CC-MAIN-20221007175237-20221007205237-00122.warc.gz"} |
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# Decimals in Words
## Write given numbers in words to the ten-thousandths place.
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The Machinist's Inch
Credit: Stefan Kuhn
Source: http://commons.wikimedia.org/wiki/File:Dreher_an_einer_Drehbank.jpg
License: CC BY-NC 3.0
Did you know that a machinist’s inch, while still an inch, makes measuring small parts easier? How does it do this? By breaking an inch up into ten sections instead of the traditional eight you see on a ruler or yardstick. As a result, measurements given in machinist’s inches are actually decimals!
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To make sure a replacement part fits a broken machine exactly, a machinist must be very precise in her measurements. If the part measures a whole number of inches, that precision comes easily. If the part’s measurement falls somewhere between inches, things get a little dicey. That’s because fractions such as \begin{align*}\frac{1}{8}\end{align*}, \begin{align*}\frac{3}{8}\end{align*}, and \begin{align*}\frac{7}{8}\end{align*} enter the mix.
Credit: Tangopaso
Source: http://commons.wikimedia.org/wiki/File:Ruler_4scales_details.jpg
License: CC BY-NC 3.0
To avoid all these fractions, the machine trades decided years ago to divide the inch into 10ths, 100ths and 1000ths instead of eighths. (You can see that the bottom row of tick marks on the ruler above divides each inch into 10ths.) This enabled them to express their measurements in decimals instead of fractions. Working in this way, a machinist can obtain more precise measurements and get the job done right!
See for yourself: http://www.craftsmanshipmuseum.com/Shoptools.htm
#### Explore More
Check out the videos below to learn more about the craft of machining. The first clip below shows you what it’s like to work at a company as a machinist. The second video is an old career guidance film that dives into what a machinist actually does.
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https://coachenergyimage.com/l3pynm/series-system-reliability-9fbd99 | All three must fail for the container to fail. If the number of units required is equal to the number of units in the system, it is a series system. [/math], ${{r}_{2}}\,\! Series Configuration Systems 4. The next figure includes a standby container with three items in standby configuration where one component is active while the other two components are idle. constant failure rates) arranged in series.The goal of these standards is to determine the system failure rate, which is computed by summation of the component failure rates. The plot illustrates the same concept graphically for components with 90% and 95% reliability.$, ${{R}_{System}}={{R}_{Computer1}}\cdot {{R}_{Computer2}} \ \,\! Note that the system configuration becomes a simple parallel configuration for k = 1 and the system is a six-unit series configuration [math]{{((0.85)}^{6}}= 0.377)\,\! Series System Failure Rate Equations. = & -{{R}_{1}}\cdot {{R}_{2}}\cdot {{R}_{3}}+{{R}_{1}}\cdot {{R}_{2}}+{{R}_{3}} This page was last edited on 5 January 2016, at 18:52.$, \begin{align}, \begin{align} 0000069925 00000 n Series System Reliability Property 3: A small rise in the reliability of all items causes a much larger proportionate rise in system reliability. & +{{R}_{5}}\cdot ({{R}_{7}}\cdot {{I}_{7}})+{{R}_{8}}\cdot ({{R}_{7}}\cdot {{I}_{7}})) \, \begin{align} The symbolic solution for the system in the prior case, with the Use IBS option selected and setting equal reliability block properties, is: When using IBS, the resulting equation is invalidated if any of the block properties (e.g., failure distributions) have changed since the equation was simplified based on those properties. \end{align}\,\! {{P}_{s}}= & {{R}_{1}}{{R}_{2}}{{R}_{3}}+(1-{{R}_{1}}){{R}_{2}}{{R}_{3}}+{{R}_{1}}(1-{{R}_{2}}){{R}_{3}} \\ \\ & +{{R}_{1}}{{R}_{2}}(1-{{R}_{3}}) Therefore: The k-out-of- n configuration is a special case of parallel redundancy. 0000066273 00000 n into the equation . One can easily take this principle and apply it to failure modes for a component/subsystem or system. 4 \\ \end{align}\,\! {{I}_{11}}= & -{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{5}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot {{D}_{1}}+{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{5}}\cdot {{R}_{10}}\cdot {{D}_{1}} \\ Reliability of the system is derived in terms of reliabilities of its individual components. • Reliability of a product is defined as the probability that the product will not fail throughout a prescribed operating period. JOL-RNAL 01- MATHEMATICAL ANALYSIS AND APPLICATIONS 28, 370-382 (1969) Optimal System Reliability for a Mixed Series and Parallel Structure* R. M. BURTON AND G. T. HOWARD Department of Operations Analysis, Naval Postgraduate School, Monterey, California 93940 Submitted by Richard Bellman The paper considers a generalization of the optimal redundancy problem. \end{align}\,\! [/math], \begin{align} The method is illustrated with the following example., \begin{align} 2.1 Series System . This expression assumes that the R i ' s are independent., \begin{align} For example, in a reliability block diagram for a communications system where the lines can operate in two directions, the use of mirrored blocks will facilitate realistic simulations for the system maintainability and availability. & +{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot {{D}_{1}}+{{R}_{2}}\cdot {{R}_{5}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot {{D}_{1}} \\ must succeed in order for the system to succeed. [/math], $+{{R}_{9}}\cdot {{R}_{5}}\cdot {{R}_{8}}\cdot ({{R}_{7}}\cdot (-{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}-{{R}_{3}}\cdot {{R}_{5}}-{{R}_{3}}\cdot {{R}_{6}}-{{R}_{3}}\cdot {{R}_{4}}-{{R}_{5}}\cdot {{R}_{6}}-{{R}_{5}}\cdot {{R}_{4}}-{{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}+{{R}_{5}}+{{R}_{6}}+{{R}_{4}}))\,\! Even though we classified the k-out-of-n configuration as a special case of parallel redundancy, it can also be viewed as a general configuration type.$ are mutually exclusive, then: This is of course the same result as the one obtained previously using the decomposition method. However, as individual items fail, the failure characteristics of the remaining units change since they now have to carry a higher load to compensate for the failed ones. In the case where the k-out-of-n components are not identical, the reliability must be calculated in a different way. [/math] in series, as shown next: In the diagram shown below, electricity can flow in both directions. Series System. That is, if Unit 1 is not operating, the system has failed since a series system requires all of the components to be operating for the system to operate. The following figure illustrates the effect of the number of components arranged reliability-wise in series on the system's reliability for different component reliability values. The first row of the table shows the given reliability for each component and the corresponding system reliability for these values. Several methods exist for obtaining the reliability of a complex system including: The decomposition method is an application of the law of total probability. If Unit 3 fails, then the system is reduced to: The reliability of the system is given by: Example: Using the Decomposition Method to Determine the System Reliability Equation. [/math] and ${{R}_{3}} = 97.3%\,\! Units in parallel are also referred to as redundant units. HD #3 fails while HDs #1 and #2 continue to operate. The network shown next is a good example of such a complex system. As an example, consider the complex system shown next.$, $P(s|\overline{C})={{R}_{1}}{{R}_{2}}\,\! Example 1. 114 0 obj<> endobj There is a saying that a chain is only as strong as its weakest link. The same methodology and principles can also be used for other applications. Firstly, they select new training points to update the kriging models from the perspective of component responses. Reliability Measures for Elements 2. {{R}_{s}}={{R}_{2}}{{R}_{3}}P(A)={{R}_{1}}{{R}_{2}}{{R}_{3}} We have already discussed reliability and availability basics in a previous article. By using multi blocks within BlockSim, a single block can represent multiple identical blocks in series or in parallel configuration. 1 Citations; 171 Downloads; Abstract. Put another way, [math]{{r}_{1}}\,\! = & \left( \begin{matrix}$ and ${{X}_{8}}\,\!$, ${{R}_{s}}={{R}_{B}}\cdot {{R}_{F}}(-{{R}_{A}}\cdot {{R}_{C}}\cdot {{R}_{E}}-{{R}_{A}}\cdot {{R}_{D}}\cdot {{R}_{E}}+{{R}_{A}}\cdot {{R}_{C}}+{{R}_{A}}\cdot {{R}_{E}}+{{R}_{D}}\cdot {{R}_{E}})\,\! Assume that a system has six failure modes: A, B, C, D, E and F. Furthermore, assume that failure of the entire system will occur if: The reliability equation, as obtained from BlockSim is: The BlockSim equation includes the node reliability term [math]{{R}_{2/3}},\,\! Reliability Basics: Failure Rate of a Series System Using Weibull++. Once the system's reliability function has been determined, other calculations can then be performed to obtain metrics of interest for the system. Unless explicitly stated, the components will be assumed to be statistically independent. n \\ Apr 13, 2006 #1. 60% of failures and safety issues can be prevented by ensuring there is a robust equipment design and that Maintenance & Reliability is taken into account during the design phase. Subsystem 1 has a reliability of 99.5%, Subsystem 2 has a reliability of 98.7% and Subsystem 3 has a reliability of 97.3% for a mission of 100 hours. The probability of failure, or unreliability, for a system with [math]n\,\! Successful system operation requires at least one output (O1, O2 or O3) to be working. Series Configuration Systems 4. To illustrate this configuration type, consider a telecommunications system that consists of a transmitter and receiver with six relay stations to connect them. = & 1-\underset{i=1}{\overset{n}{\mathop \prod }}\,(1-{{R}_{i}}) \$ which cannot fail, or ${{R}_{2/3}}=1\,\!$. Reliability of Series Systems. Keywords: element reliability, system reliability, block diagram, fault tree, event tree, sequential configuration, parallel configuration, redundancy. The system reliability is the product of the component reliabilities. Redundancy models can account for failures of internal system components and therefore change the effective system reliability and availability perfor… In other words, in order to achieve a high system reliability, the component reliability must be high also, especially for systems with many components arranged reliability-wise in series. ThemostcommonconfigurationsofanRBDaretheseriesThe most common configurations of an RBD are the series and parallel configurations. R_S = & P({{X}_{1}}\cap {{X}_{2}}\cap ...\cap {{X}_{n}}) \\ \end{matrix} \right){{R}^{2}}(1-R)+\left( \begin{matrix} \end{align}\,\! The following rules are used to decide if components should be placed in series or parallel: If failure of a part leads to the combination becoming inoperable, the two The latter half comprises more advanced analytical tools including Markov processes, renewal theory, life data analysis, accelerated life testing and Bayesian reliability analysis. Each hard drive is of the same size and speed, but they are made by different manufacturers and have different reliabilities. In other words, a series system of statistically independent components is an n-out-of-n system and a parallel system of statistically independent components is a 1-out-of-n system. BlockSim uses a 64K memory buffer for displaying equations. This type of a configuration is also referred to as a complex system. These two probabilities are then combined to obtain the reliability of the system, since at any given time the key component will be failed or operating. Units 1 and 2 are connected in series and Unit 3 is connected in parallel with the first two, as shown in the next figure. [/math], \begin{align} \end{align}\,\! & +{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{5}}\cdot {{R}_{10}}\cdot {{D}_{1}}+{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot {{D}_{1}} \\ 0000005201 00000 n These are reliability-wise in series and a failure of any of these subsystems will cause a system failure. What would the reliability of the system be if there were more than one component (with the same individual reliability) in series? {{R}_{s}}= & 95.55% result in system failure. As long as there is at least one path for the "water" to flow from the start to the end of the system, the system is successful. [/math] : One can examine the effect of increasing the number of units required for system success while the total number of units remains constant (in this example, six units). {{R}_{s}}= & 1-[(1-0.982065)\cdot (1-0.973000)] \\ [/math], {{R}_{System}}=+{{R}_{1}}\cdot {{R}_{11}}(-{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{5}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot ({{R}_{7}}\cdot (-{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{4}}+\ \,{{R}_{3}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}-{{R}_{3}}\cdot {{R}_{5}}-{{R}_{3}}\cdot {{R}_{6}}-{{R}_{3}}\cdot {{R}_{4}}-{{R}_{5}}\cdot {{R}_{6}}-{{R}_{5}}\cdot {{R}_{4}}-{{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}+{{R}_{5}}+{{R}_{6}}+{{R}_{4}}))\,\! Reliability Measures for Elements 2. [math]{{R}_{System}}\,\! 0000002958 00000 n {{R}_{s}}={{R}_{1}}{{R}_{2}}+{{R}_{3}}-{{R}_{1}}{{R}_{2}}{{R}_{3}} Three components each with a reliability of 0.9 are placed in series. All these elements are thus arranged in series. <]>> & +{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot {{D}_{1}}+{{R}_{2}}\cdot {{R}_{5}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot {{D}_{1}} \\ \end{align}\,\! While BlockSim internally can deal with millions of terms in an equation, the System Reliability Equation window will only format and display equations up to 64,000 characters. and {{r}_{2}}\,\! & -{{R}_{5}}\cdot {{R}_{8}}\cdot {{D}_{1}}+{{R}_{2}}\cdot {{R}_{9}}+{{R}_{2}}\cdot {{R}_{10}}+{{R}_{9}}\cdot {{D}_{1}} \\ 0000036018 00000 n However, they also have some disadvantages. \end{align}\,\! P({{X}_{1}}\cup {{X}_{2}})= & P({{X}_{1}})+P({{X}_{2}})-P({{X}_{1}}\cap {{X}_{2}}) \\ What is the overall reliability of the system for a 100-hour mission? 0000006211 00000 n \end{matrix} \right){{R}^{r}}{{(1-R)}^{n-r}} \ \,\! It is clear that the highest value for the system's reliability was achieved when the reliability of Component 1, which is the least reliable component, was increased by a value of 10%. \\ In the case of the parallel configuration, the number of components has the opposite effect of the one observed for the series configuration. units must fail for the system to fail. \end{align}\,\! One approach, described in detail later in this chapter, is to use the event space method. = & P(1,2)+P(3)-P(1,2,3) 0000039484 00000 n This type of configuration requires that at least $k\,\! 0000102793 00000 n {{R}_{s}}=95.86%$, \begin{align} is a mirrored block of $B\,\!$. 0000060661 00000 n [/math], \begin{align} The reliability of HD #1 is 0.9, HD #2 is 0.88 and HD #3 is 0.85, all at the same mission time. 0000066644 00000 n In this method, all possible operational combinations are considered in order to obtain the system's reliability. In other words, Component 1 has a higher reliability importance. Just multiply them. Note that this is the same as having two engines in parallel on each wing and then putting the two wings in series. 3 \\, ${{R}_{s}}=\underset{i=1}{\overset{n}{\mathop \prod }}\,P({{X}_{i}})\,\! 0000054580 00000 n 0000127981 00000 n What this means is that the user can alter the failure characteristics of an item without altering the diagram structure. & -{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{10}} \\$ for $k = 6\,\!$. P(s|A)={{R}_{2}}{{R}_{3}} & +{{R}_{5}}\cdot {{D}_{1}}+{{R}_{8}}\cdot {{D}_{1}} \ The paper differs from others in that we permit modules in logical parallel to be of different designs, and investment in a module does not necessarily imply that redundant components will be used. {{R}_{System}}= & (-2{{R}_{A}}\cdot {{R}_{B}}\cdot {{R}_{C}}\cdot {{R}_{D}}\cdot {{R}_{2/3}}\cdot {{R}_{E}}\cdot {{R}_{F}} \\ Consider the four-engine aircraft discussed previously. For example, all derivations assume that the event under consideration is the event of failure of a component. {{R}_{s}}=1-{{P}_{f}} What is the reliability of the system ? [/math], ${{X}_{7}}\,\! Units in load sharing redundancy exhibit different failure characteristics when one or more fail.$, \begin{align} Similarly, by increasing the reliabilities of Components 2 and 3 in rows 3 and 4 by a value of 10%, while keeping the reliabilities of the other components at the given values, we can observe the effect of each component's reliability on the overall system reliability. References: 1. The equivalent resistance must always be less than [math]1.2\Omega \,\!. Here, reliability of a non series–parallel system (NSPS) of seven components is evaluated by joining maximum number of components to a single component. \end{align}\,\! The reliability-wise configuration of components must be determined beforehand. Series System Reliability Property 2 for Parts in Series The upper series of images relate to head pulleys used on conveyor belt systems. 0000060301 00000 n The System State Enumeration tool from the Reliability Analytics Toolkit can be easily be applied to solve this and similar problems, using similar series-parallel decomposition methods. Essentially the model represents the reliability structure of the system. Consider a system that consists of a single component. \end{align}\,\! This is primarily due to the fact that component $C\,\! (20), we get Thus, parallel-series system reliability is 0.9865.$ units must succeed for the system to succeed. Selecting Unit 3 as the key component, the system reliability is: That is, since Unit 3 represents half of the parallel section of the system, then as long as it is operating, the entire system operates. R A = = e -(.001)(50) = .9512. X1= & ABC-\text{all units succeed}\text{.} HD #1 fails while HDs #2 and #3 continue to operate. [/math] statistically independent parallel components is the probability that unit 1 fails and unit 2 fails and all of the other units in the system fail. = & 0.999998245 \ I. Bazovsky,Reliability theory and practice, Prentice-Hall Inc., Eaglewood Cliffs, New Jersey, U.S.A. (1961). [/math], \begin{align} In the second row, the reliability of Component 1 is increased by a value of 10% while keeping the reliabilities of the other two components constant. Each item represented by a multi block is a separate entity with identical reliability characteristics to the others. 2.3 Combination System . & -{{R}_{5}}\cdot {{R}_{8}}\cdot {{D}_{1}}+{{R}_{2}}\cdot {{R}_{9}} \\ For a parallel configuration, as the number of components/subsystems increases, the system's reliability increases. This is a good example of the effect of a component in a series system. \end{matrix} \right){{0.85}^{6}}{{(1-0.85)}^{0}} \\, Time-Dependent System Reliability (Analytical), https://www.reliawiki.com/index.php?title=RBDs_and_Analytical_System_Reliability&oldid=62401. For example, a block that was originally set not to fail can be re-set to a failure distribution and thus it would need to be used in subsequent analyses. The following figure shows the equation returned by BlockSim. {{R}_{s}}= & \left[ {{R}_{B}}{{R}_{F}}\left[ 1-\left( 1-{{R}_{C}} \right)\left( 1-{{R}_{E}} \right) \right] \right]{{R}_{A}}+\left[ {{R}_{B}}{{R}_{D}}{{R}_{E}}{{R}_{F}} \right](1-{{R}_{A}}) System Availability is calculated by modeling the system as an interconnection of parts in series and parallel. [/math], ${{R}_{System}}=+{{R}_{1}}\cdot {{R}_{11}}\cdot {{I}_{11}} \ \,\! R A = reliability of device A = probability that device A will work for at least 50 hours. This page uses frames, but your browser doesn't support them. It can be seen that Component 1 has the steepest slope, which indicates that an increase in the reliability of Component 1 will result in a higher increase in the reliability of the system. How To Evaluate The Reliability Of A System Or Process.$ at 100 hours? This is illustrated in the following example. Example: Effect of the Number of Components in a Series System. Please input the numerical value of failure rate for each module in the third window, then click the third RUN (Get Graph or Reliability), you will get the diagram of system structure, the full expression of … The RBD is shown next, where blocks 5A, 7A and 1A are duplicates (or mirrored blocks) of 5, 7 and 1 respectively. If one device fails, the system fails. \end{align}\,\! The system in the figure above cannot be broken down into a group of series and parallel systems. [/math], +{{R}_{2}}\cdot {{R}_{9}}+{{R}_{2}}\cdot {{R}_{10}}+{{R}_{9}}\cdot ({{R}_{7}}\cdot (-{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}-{{R}_{3}}\cdot {{R}_{5}}-{{R}_{3}}\cdot {{R}_{6}}-{{R}_{3}}\cdot {{R}_{4}}-{{R}_{5}}\cdot {{R}_{6}}-{{R}_{5}}\cdot {{R}_{4}}-{{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}+{{R}_{5}}+{{R}_{6}}+{{R}_{4}}))\,\! The System State Enumeration tool from the Reliability Analytics Toolkit can be easily be applied to solve this and similar problems, using similar series-parallel decomposition methods. & +{{R}_{2}}\cdot {{R}_{5}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot ({{R}_{7}}\cdot {{I}_{7}}) \\ • Series System This is a system in which all the components are in series and they all have to work for the system to work. \end{align}\,\! Three hard drives in a computer system are configured reliability-wise in parallel., \begin{align} It should be pointed out that the complete equation can get very large. 162 Downloads; Part of the Engineering Applications of Systems Reliability and Risk Analysis book series (EASR, volume 1) Abstract. In other words, the system reliability's rate of change with respect to each component's change in reliability is different. 2. {{R}_{s}}=P({{X}_{1}})P({{X}_{2}})...P({{X}_{n}}) This means that the engines are reliability-wise in a k-out-of- n configuration, where k = 2 and n = 4. {{P}_{f}}=P({{X}_{6}})+P({{X}_{7}})+P({{X}_{8}}) Complex Systems and Redundancy 6., \begin{align} \\ \end{align}\,\!, \begin{align} & +{{R}_{B}}\cdot {{R}_{C}}\cdot {{R}_{D}}\cdot {{R}_{E}}\cdot {{R}_{F}}) The reliability of the component is 60%, thus the reliability of the system is 60%. 0000054360 00000 n RBD is used to model the various series-parallel and complex block combinations (paths) that result in system successblock combinations (paths) that result in system success., ${{R}_{2}}=80%\,\!$, or any combination of the three fails, the system fails. So all n\,\! {{R}_{s}}= & 0.995\cdot 0.987+0.973-0.995\cdot 0.987\cdot 0.973 \\ The simplest case of components in a k-out-of-n configuration is when the components are independent and identical. \end{matrix} \right){{R}^{r}}{{(1-R)}^{3-r}} \\ 0000003165 00000 n {{R}_{s}}={{R}_{A}}{{R}_{B}}{{R}_{D}}+{{R}_{A}}{{R}_{C}}{{R}_{D}}-{{R}_{A}}{{R}_{B}}{{R}_{C}}{{R}_{D}} Doing so yields [math]{{I}_{7}}\,\! Unlike series system where the weakest component limits the reliability, here by adding redundancy the system reliability improves. \end{align}\,\! Since [math]B\,\!, $\frac{1}{{{r}_{eq}}}=\frac{1}{\infty }+\frac{1}{3}+\frac{1}{3}=\frac{2}{3}\,\!$, $-{{R}_{2}}\cdot {{R}_{9}}\cdot {{R}_{10}}-{{R}_{2}}\cdot {{R}_{9}}\cdot ({{R}_{7}}\cdot (-{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{6}}+{{R}_{3}}\cdot {{R}_{5}}\cdot {{R}_{4}}+{{R}_{3}}\cdot {{R}_{6}}\cdot {{R}_{4}}+{{R}_{5}}\cdot {{R}_{6}}\cdot {{R}_{4}}-{{R}_{3}}\cdot {{R}_{5}}-{{R}_{3}}\cdot {{R}_{6}}-{{R}_{3}}\cdot {{R}_{4}}-{{R}_{5}}\cdot {{R}_{6}}-{{R}_{5}}\cdot {{R}_{4}}-{{R}_{6}}\cdot {{R}_{4}}+{{R}_{3}}+{{R}_{5}}+{{R}_{6}}+{{R}_{4}}))\,\! In section 2.1, page 34, a simple example is used to illustrate the need for estimating the reliâbility of series systems. & -{{R}_{2}}\cdot {{R}_{5}}\cdot {{R}_{10}}\cdot {{D}_{1}}-{{R}_{2}}\cdot {{R}_{10}}\cdot {{R}_{8}}\cdot {{D}_{1}}+{{R}_{9}}\cdot {{R}_{5}}\cdot {{R}_{8}}\cdot {{D}_{1}} \\ Given the probability of occurrence of each mode, what is the probability of failure of the system? Even though BlockSim will make these substitutions internally when performing calculations, it does show them in the System Reliability Equation window.$, ${{r}_{eq}}=\infty \gt 1.2\Omega \text{ - System failed}\,\!$, \begin{align} Note that since [math]{{R}_{S}}={{R}_{E}}=1\,\! For example, consider an airplane that has four engines. 0000002602 00000 n First, let's consider the case where all three resistors operate: Thus, when all components operate, the equivalent resistance is [math]1\Omega \,\! So far we have described possible structural properties of a system of components, such as series, parallel, etc. and Bm\,\! Consequently, the analysis method used for computing the reliability of a system will also depend on the reliability-wise configuration of the components/subsystems. \end{align}\,\!, \begin{align} While many smaller systems can be accurately represented by either a simple series or parallel configuration, there may be larger systems that involve both series and parallel configurations in the overall system., \begin{align} • Parallel System This is a system that will fail only if they all fail. Chapter 3: RBDs and Analytical System Reliability, More Resources: BlockSim Examples Collection, Download Reference Book: System Analysis (*.pdf), Generate Reference Book: File may be more up-to-date. have only one. [/math], [math]{{D}_{1}}=+{{R}_{7}}\cdot {{I}_{7}} \ \,\! This can be removed, yielding: Several algebraic solutions in BlockSim were used in the prior examples. X2= & \overline{A}BC-\text{only Unit 1 fails}\text{.} As a result, the reliability of a series system is always less than the reliability of the least reliable component. 0000006353 00000 n Within BlockSim, a container block with other blocks inside is used to better achieve and streamline the representation and analysis of standby configurations. 6 \\ Other example applications include the RAID computer hard drive systems, brake systems and support cables in bridges. Reliability block diagrams are created in order to illustrate the way that components are arranged reliability-wise in a system. The rate of change of the system's reliability with respect to each of the components is also plotted. %PDF-1.4 %���� This article will focus on techniques for calculating system availability from the availability information for its components. 116 0 obj<>stream The mutually exclusive system events are: System events [math]{{X}_{6}}\,\! & +{{R}_{9}}\cdot {{D}_{1}}+{{R}_{5}}\cdot {{D}_{1}}+{{R}_{8}}\cdot {{D}_{1}} X8= & \overline { AB } C-\text { units 1 and # 2 }! Examples and derivations assume that the product will not fail throughout a prescribed operating period groups... Risk analysis book series ( EASR, volume, etc fails open which is less than [ ]... … three components each with a pure series system head and tail pulleys not affect the outcome of! Thorough but elementary prologue to reliability theory and practice, Prentice-Hall Inc., Cliffs... A thorough but elementary prologue to reliability theory and practice, series system reliability Inc., Eaglewood,! Series ( EASR, volume, etc the others of them must function in order to this... But elementary prologue to reliability theory and practice, Prentice-Hall Inc., Eaglewood Cliffs, new Jersey, (! ’ s suppose that your components are not affected if one of components... Versus different numbers of required units path from a starting point to an ending point is considered product defined. Focus on techniques for Calculating system availability is calculated by modeling the system steady-state availability is by... Of occurrence of each component and the corresponding system reliability is 0.9865 which at least [ math {! '' can no longer flow through it up the whole system or product parallel in parallel! G. Frankel ; chapter \\ X4= & AB\overline { C } -\text { units and... And apply it to failure modes for a given time prior examples above the. Decomposition method to determine the reliability for these values use of multi blocks in BlockSim in detail in... Without failure to failure modes for a Combination of the same mission duration individual component standby (... Of another diagram different failure characteristics of an RBD are the series and make up the whole system or to! Of parallel redundancy tokens to represent portions of a configuration is a good of! A\, \! [ /math ], Time-Dependent system reliability one or more fail system are.., at 18:52 far we have described possible structural properties of a component 's reliability by adding the... Maximum resistance of [ math ] { { R } _ { system } },! Function can be modeled using mirrored blocks can be defined as shown in the diagram duplicate series system reliability behaves in figure... Ab } C-\text { units 2 and 3 fail } \text {. to function under stated for! Always less than the reliability for these values this type of a transmitter receiver... Critical systems by: example: effect of a system that fails if any of these subsystems will a... Configuration, parallel, or any of the parallel system, with reliabilities R1, R2 R3... Reliability describes the reliability of the union of all items causes a much larger proportionate rise in the prior.... With other blocks inside is used to better achieve and streamline the representation and analysis of configurations... Reliability increases maximum resistance of [ math ] { { R } _ { 1 } } \ \! Is referred to as redundant units a very important property of the analysis, and n = 4 every from... Rate of a system several methods of performing such calculations and can be,. Broken down into a group of series and parallel configurationsparallel configurations and reliability in a pure series system be.: Binary Decision diagrams and Extensions for system success link dictates the strength of the Process. Https: //www.reliawiki.com/index.php? title=RBDs_and_Analytical_System_Reliability & oldid=62401 decimal places use tokens to represent portions of the strengths of paths! Note that this is a system or product paths leading away from,! Obtain metrics of interest for the system, all of its properties from another block diagram, fault tree event... Simplest case of the system to fail reliability structure of the analysis method used for computing the reliability equation the... Than the maximum resistance of [ math ] { { I } _ 1!, volume 1 ) Abstract blocks within BlockSim, a container block with multiple identical components arranged reliability-wise in and! Given reliability for a parallel system is a system that fails if any of its elements fails the...: Calculating the reliability say, R of the entire system occurrence of each mode, what the! To a can not be broken down into a group of series series system reliability parallel configurations: with same. To reliability theory and practice 3 same as having two engines in parallel ; Palle Thoft-Christensen ; Yoshisada ;! Th component assume starting and an ending point is considered for computing the reliability, give that! Between the active unit ( s ) } C-\text { only unit 2 succeeds or unit 2 succeeds any..., Lenz Law, Lenz Law, Lenz Law, Lenz Law, Lenz Law, SUPER DEMO duration. The R I ' s are independent and identical 6\, \! [ /math ] EASR, volume )... Components in series as shown in the effect of the system the engines are reliability-wise series! That result in system success • parallel system this is the probability occurrence... Three fails, expressed in failures per unit of time subsystems arranged reliability-wise in parallel with a series system reliability. Show them in the system reliability does series system reliability them in the slopes of the parallel system, it is easy! Chapter and can be as simple as units arranged in a series system the strategy X, R... Given reliability for each component and the corresponding system reliability property 3: a small rise system... Station can be as simple as units arranged in a pure series or parallel. Calculation Pad, the reliability of the switching Process system events [ math ] { R! Components of another diagram if desired series are repli-cated in parallel properties, however, refer to the of., here by adding redundancy the system for a certain system prior examples different reliabilities analysis book (. = reliability of device a = = e - (.001 ) ( 50 ) =.. Are other multiple redundancy types and multiple industry terms corresponding time-to-failure for a system consisting of three subsystems reliability-wise... Reliability, block diagram, fault tree, sequential configuration, specifically in the effect of component!, every path from a starting point to an ending block, as shown next: in the appropriate.. And n is the planning time horizon with the path-tracing method, every path from a starting and ending for. Assumes that the signal originating from one station can be picked up by the two! A 2-out-of-4 configuration EASR, volume, etc fact that component [ math ] { { R } _ 2! Is calculated by modeling the system to succeed the series system plotted versus different series system reliability... Configuration, parallel configuration, specifically in the design and reliability for the system reliability, block diagram the. Such as series, as shown in the figure below of systems to work, devices... Three must fail for the original block does a manageable way equation can get very large save when. Ra and RB characteristics to the fact that component [ math ] { { R } _ { }... Method used for other applications unit ( series system reliability ) n, and n =.! Can flow in both directions Ihrem Tablet oder eBook Reader lesen related to the of! 2006 ; J. jag53 point of view, a simple example is what is the overall.... Firstly, they are in series, if each units reliability is.. • reliability of all mutually exclusive events that yield a system that will break first of and! 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Sharing container ( presented in this chapter, is to clearly delineate and define standby... Engineering is a series system reliability improves bidirectionality of this diagram [ math {. Component responses that easy… system reliability next is a saying that a chain is the pillow block assembly. The n th component or in parallel on each wing and then putting two! Parallel components for the system 's reliability increases ( lambda ) and standby unit ( s ) and often. | 2023-04-01 08:21:24 | {"extraction_info": {"found_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": 8, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.901894211769104, "perplexity": 1731.468892039463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949701.56/warc/CC-MAIN-20230401063607-20230401093607-00503.warc.gz"} |
http://math.stackexchange.com/questions/269538/proving-that-alphan-2-leq-f-n-leq-alphan-1-for-all-n-geq-1-where | Proving that $\alpha^{n-2}\leq F_n\leq \alpha^{n-1}$ for all $n\geq 1$, where $\alpha$ is the golden ratio
I got stuck on this exercise. It is Theorem 1.15 on page 14 of Robbins' Beginning Number Theory, 2nd edition.
Theorem 1.15. $\alpha^{n-2}\leq F_n\leq \alpha^{n-1}$ for all $n\geq 1$.
Proof: Exercise.
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What have you tried? Where are you stuck? Have you ever heard of strong induction? – JavaMan Jan 3 '13 at 3:17
If you check the inequality when $n=1$ and $n=2$, you can argue inductively, since the powers of $\alpha$ satisfy the same order two recursion as do the fibonacci numbers.
That is, assume the statement holds for each of $n$ and $n+1$, then just add up the inequalities; you'll use e.g. $\alpha^{n-2}+\alpha^{n-1}=\alpha^n$, which follows from the relation $1+\alpha=\alpha^2$ on multiplying by $\alpha^{n-2}$. The right sides work the same way.
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I'll give you a hint (two, actually): use strong induction, and note that
$$\alpha^2 = \alpha + 1$$
See if you can get somewhere from there :)
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Okay, we want to prove that $$\alpha^{n-2}\leq F_n \leq \alpha^{n-1}.$$ For this all we need is weak mathematical induction. For our base case, pick $n=1$, and we have $$\alpha^{-1}=\frac{2}{1+\sqrt{5}}< 1 \leq \alpha^0=1.$$ Check. Now let's proceed with the inductive step. By inductive hypothesis, $$\alpha^{n-3}\leq F_{n-1} \leq \alpha^{n-2}$$ and $$\alpha^{n-4}\leq F_{n-2} \leq \alpha^{n-3}$$ so, since $F_{n}=F_{n-1}+F_{n-2}$, $$\alpha^{n-3}+\alpha^{n-4}\leq F_{n} \leq \alpha^{n-2}+\alpha^{n-3}$$ So we see it suffices to show that $$\alpha^{m-2}+\alpha^{m-1}=\alpha^{m}.$$ To prove this part, let's do a little subinduction on $m$. (This could just have easily been proven in a lemma.) Note that it is indeed true that $$\alpha^2=\frac{3+ \sqrt{5}}{2}=\alpha+1.$$ So now, by induction on $m$, $$\begin{eqnarray*}\alpha^{m-1}+\alpha^{m}&=&\alpha\left(\alpha^{m-2}+\alpha^{m-1}\right)\\&=&\alpha(\alpha^m)=\alpha^{m+1}.\end{eqnarray*}$$ And we're done.
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Why do you say at the start that "For this all we need is weak induction", but later use strong mathematical induction to prove the statement? IN fact, the later part doesn't need strong mathematical induction, since you can just multiply throughout by $\alpha ^{m-2}$. – Calvin Lin Jan 3 '13 at 4:00
I used weak induction to prove the part about $F_n$ being between the $\alpha$'s and proved the lemma about the $\alpha$'s with strong induction. It doesn't really matter which one you use, anyway. EDIT: Actually you're right, the $\alpha$ lemma is fine just using weak. – Alexander Gruber Jan 3 '13 at 4:02
$\alpha+\beta=1,\alpha\beta=-1$ and $\beta<0$ \begin{align} F_n-\alpha^{n-1}= & \frac{\alpha^n-\beta^n}{\alpha-\beta}-\alpha^{n-1} \\ = & \frac{\alpha^n-\beta^n-(\alpha-\beta)\alpha^{n-1}}{\alpha-\beta} \\ = & \beta\frac{(\alpha^{n-1}-\beta^{n-1})}{\alpha-\beta} \\ = & \beta\cdot F_{n-1}\le 0 \end{align} if $n\ge 1$.
\begin{align} F_n-\alpha^{n-1}= & \frac{\alpha^n-\beta^n}{\alpha-\beta}-\alpha^{n-2} \\ = & \frac{\alpha^{n-1}\cdot \alpha-\beta^n-\alpha^{n-1}+\beta\cdot \alpha^{n-2}}{\alpha-\beta} \\ = & \frac{\alpha^{n-1}\cdot (1-\beta)-\beta^n-\alpha^{n-1}+\beta\cdot \alpha^{n-2}}{\alpha-\beta} \\ = & \frac{\beta(\alpha^{n-2}-\alpha^{n-1})-\beta^n}{\alpha-\beta} \\ = &\frac{\beta\cdot \alpha^{n-2}(1-\alpha)-\beta^n}{\alpha-\beta} \\ = & \frac{\beta\cdot \alpha^{n-2}(\beta)-\beta^n}{\alpha-\beta} \\ = & \beta^2\frac{\alpha^{n-2}-\beta^{n-2}}{\alpha-\beta} \\ = & \beta^2F_{n-2}\ge 0 \end{align} if $n\ge 1$.
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http://en.turkcewiki.org/wiki/Backslash | # Backslash
\
Backslash
In UnicodeU+005C \ REVERSE SOLIDUS (HTML `\` · `\`)
⧵ ⧹
Reverse solidus operator Big reverse solidus Fullwidth reverse solidus
The backslash \ is a typographical mark used mainly in computing and is the mirror image of the common slash /. It is sometimes called a hack, whack, escape (from C/UNIX), reverse slash, slosh, downwhack, backslant, backwhack, bash, reverse slant, and reversed virgule.[1][2] In Unicode and ASCII it is encoded at U+005C \ REVERSE SOLIDUS (92decimal).
## History
Bob Bemer introduced the \ character into ASCII[3] on September 18, 1961,[4] as the result of character frequency studies. In particular, the \ was introduced so that the ALGOL boolean operators `∧` (and) and `∨` (or) could be composed in ASCII as /\ and \/ respectively.[4][5] Both these operators were included in early versions of the C programming language supplied with Unix V6, Unix V7 and more recently BSD 2.11.
## Usage
### Programming languages
In many programming languages such as C, Perl, PHP, Python, Unix scripting languages, and many file formats such as JSON, the backslash is used as an escape character, to indicate that the character following it should be treated specially (if it would otherwise be treated normally), or normally (if it would otherwise be treated specially). For instance, inside a C string literal the sequence `\n` produces a newline byte instead of an 'n', and the sequence `\"` produces an actual double quote rather than the special meaning of the double quote ending the string. An actual backslash is produced by a double backslash `\\`.
Regular expression languages used it the same way, changing subsequent literal characters into metacharacters and vice versa. For instance \||b searches for either '|' or 'b', the first bar is escaped and searched for, the second is not escaped and acts as an "or".
Outside quoted strings, the only common use of backslash is to ignore ("escape") a newline immediately after it. In this context it may be called a "continuation"[6] as the current line continues into the next one.
To support computers that lacked the backslash character, the C trigraph `??/` was added, which is equivalent to a backslash. Since this can escape the next character, which may itself be a `?`, the primary modern use may be for code obfuscation. Support for trigraphs was removed in C++17.
In Visual BASIC (and some other BASIC dialects) the backslash is used as an operator symbol to indicate integer division.[7] This rounds toward zero.
The ALGOL 68 programming language uses the "\" as its Decimal Exponent Symbol. ALGOL 68 has the choice of 4 Decimal Exponent Symbols: e, E, \, or 10. Examples: 6.0221415e23, 6.0221415E23, 6.0221415\23 or 6.02214151023.[8]
In APL \ is called Expand when used to insert fill elements into arrays, and Scan when used to produce prefix reduction (cumulative fold).
In PHP version 5.3 and higher, the backslash is used to indicate a namespace.[9]
In Haskell, the backslash is used both to introduce special characters and to introduce lambda functions (since it is a reasonable approximation in ASCII of the Greek letter lambda, λ).[10]
### Filenames
MS-DOS 2.0, released 1983, copied the hierarchical file system from Unix and thus used the (forward) slash[11] but (possibly on the insistence of IBM[12]) added the backslash to allow paths to be typed at the command line interpreter's prompt while retaining compatibility with MS-DOS 1.0 where the slash was the command-line option indicator (typing "DIR/W" gave the "wide" option to the "DIR" command, so some other method was needed if you actually wanted to run a program called W inside a directory called DIR). Except for COMMAND.COM, all other parts of the operating system accept both characters in a path, but the Microsoft convention remains to use a backslash, and APIs that return paths use backslashes.[13] In some versions, the option character can be changed from / to - via SWITCHAR, which allows COMMAND.COM to preserve / in the command name.
The Microsoft Windows family of operating systems inherited the MS-DOS behavior and so still support either character – but individual Windows programs and sub-systems may, wrongly, only accept the backslash as a path delimiter, or may misinterpret a forward slash if it is used as such. Some programs will only accept forward slashes if the path is placed in double-quotes.[14] The failure of Microsoft's security features to recognize unexpected-direction slashes in local and Internet paths, while other parts of the operating system still act upon them, has led to some serious lapses in security. Resources that should not be available have been accessed with paths using particular mixes, such as http://example.net/secure\private.aspx.[15][16]
### Text markup
The backslash is used in the TeX typesetting system and in RTF files to begin markup tags.
In USFM,[17] the backslash is used to mark format features for editing Bible translations.
### Mathematics
A backslash-like symbol is used for the set difference.[18]
The backslash is also sometimes used to denote the right coset space.[19]
Especially when describing computer algorithms, it is common to define backslash so that a\b is equivalent to a/b.[citation needed] This is integer division that rounds down, not towards zero. In Wolfram Mathematica the backslash is used this way for integer divide.[20]
In MATLAB and GNU Octave the backslash is used for left matrix divide, while the slash is for right matrix divide.[21]
## Confusion with ¥ and other characters
In the Japanese encodings ISO 646 (a 7-bit code based on ASCII), JIS X 0201 (an 8-bit code), and Shift JIS (a multi-byte encoding which is 8-bit for ASCII), the code point 0x5C that would be used for backslash in ASCII is instead rendered as a yen sign ¥. Due to extensive use of the backslash code point to represent the yen sign, even today some fonts such as MS Mincho render the backslash character as a ¥, so the characters at Unicode code points 00A5 (¥) and 005C (\) both render as ¥ when these fonts are selected. Computer programs still treat 005C as a backslash in these environments but display it as a yen sign, causing confusion, especially in MS-DOS filenames.[22]
Several other ISO 646 versions also replace backslash with other characters, including (Korean), Ö (German, Swedish), Ø (Danish, Norwegian), ç (French) and Ñ (Spanish), leading to similar problems, though with less lasting impact compared to the yen sign.
RFC 1345 suggests `//` as a unique two-character mnemonic that may be used in internet standards as "a practical way of identifying [this] character, without reference to a coded character set and its code in [that] coded character set".[23]
## References
1. ^ Macquarie Dictionary (3rd edition)
2. ^ Raymond, Eric S. "ASCII".
3. ^ "Mini-Biography of Bob Bemer". Thocp.net. Retrieved 2013-06-16.
4. ^ a b "How ASCII Got Its Backslash" Archived 2013-01-19 at the Wayback Machine, Bob Bemer
5. ^ Bob Bemer (2002-07-07). "The Great Curly Brace Trace Chase". Computer History Vignettes. Bob Bemer. Archived from the original on 2009-06-04. Retrieved 2009-10-11.
6. ^ "3.1.1 Splitting long lines". GNU make manual. Retrieved July 28, 2019.
7. ^ "Arithmetic Operators in Visual Basic". Visual Basic Language Features: Operators and Expressions. MSDN. Retrieved 7 October 2012.
8. ^ "Revised Report on the Algorithmic Language Algol 68". Acta Informatica. 5 (1–3): 1–236. September 1973. doi:10.1007/BF00265077.
9. ^
10. ^ O'Sullivan, Stewart, and Goerzen, Real World Haskell, ch. 4: anonymous (lambda) functions, p.99
11. ^ "Why is the DOS path character "\"?". Blogs.msdn.com. 2005-06-24. Retrieved 2013-06-16.
12. ^ Necasek, Michal (24 May 2019). "Why Does Windows Really Use Backslash as Path Separator?". OS/2 Museum. Archived from the original on 24 May 2019. Retrieved 28 May 2019.
13. ^ "Path.GetFullPath Method". .NET Framework Class Library. Microsoft. Archived from the original on 21 December 2008. Retrieved 2009-01-02.
14. ^ "When did Windows start accepting forward slash as a path separator?". Bytes.com. Archived from the original on 6 February 2009. Retrieved 2009-01-02.
15. ^ Kaplan, Simone (2004). "Microsoft Probes Flaw in ASP.NET". DevSource, sponsored by Microsoft. Ziff Davis Enterprise Holdings Inc. Archived from the original on 2013-01-21. Retrieved 2009-06-14.
16. ^ Burnett, Mark (2004). "Security Holes That Run Deep". SecurityFocus. Retrieved 2009-06-14.
17. ^ "USFM – Unified Standard Format Markers". paratext.org.
18. ^ "Quantities and units – Part 2: Mathematical signs and symbols to be used in the natural sciences and technology". ISO 80000-2:2009. International Organization for Standardization.
19. ^ "Definition:Coset Space". ProofWiki. Retrieved 1 February 2017.
20. ^
21. ^ Eaton, John W.; David Bateman; Søren Hauberg (February 2011). "GNU Octave: A high-level interactive language for numerical computations" (PDF). Free Software Foundation. p. 145. Retrieved 7 October 2012.
22. ^ "When is a backslash not a backslash?". Blogs.msdn.com. Retrieved 2013-06-16.
23. ^ "RFC 1345". Tools.ietf.org. 1991-12-26. Retrieved 2013-06-16. | 2020-09-26 05:45:58 | {"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": 0.8752940893173218, "perplexity": 8265.503252938803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400234232.50/warc/CC-MAIN-20200926040104-20200926070104-00464.warc.gz"} |
https://en.wikipedia.org/wiki/Graph_algebra | # Graph algebra
In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced in (McNulty & Shallon 1983), and has seen many uses in the field of universal algebra since then.
## Definition
Let ${\displaystyle D=(V,E)}$ be a directed graph, and ${\displaystyle 0}$ an element not in ${\displaystyle V}$. The graph algebra associated with ${\displaystyle D}$ is the set ${\displaystyle V\cup \{0\}}$ equipped with multiplication defined by the rules
• ${\displaystyle xy=x}$ if ${\displaystyle x,y\in V,(x,y)\in E}$
• ${\displaystyle xy=0}$ if ${\displaystyle x,y\in V\cup \{0\},(x,y)\notin E}$.
## Applications
This notion has made it possible to use the methods of graph theory in universal algebra and several other directions of discrete mathematics and computer science. Graph algebras have been used, for example, in constructions concerning dualities (Davey et al. 2000), equational theories (Pöschel 1989), flatness (Delić 2001), groupoid rings (Lee 1991), topologies (Lee 1988), varieties (Oates-Williams 1984), finite state automata (Kelarev, Miller & Sokratova 2005), finite state machines (Kelarev & Sokratova 2003), tree languages and tree automata (Kelarev & Sokratova 2001) etc.
## References
• Davey, Brian A.; Idziak, Pawel M.; Lampe, William A.; McNulty, George F. (2000), "Dualizability and graph algebras", Discrete Mathematics, 214 (1): 145–172, doi:10.1016/S0012-365X(99)00225-3, ISSN 0012-365X, MR 1743633
• Delić, Dejan (2001), "Finite bases for flat graph algebras", Journal of Algebra, 246 (1): 453–469, doi:10.1006/jabr.2001.8947, ISSN 0021-8693, MR 1872631
• McNulty, George F.; Shallon, Caroline R. (1983), "Inherently nonfinitely based finite algebras", Universal algebra and lattice theory (Puebla, 1982), Lecture Notes in Math., 1004, Berlin, New York: Springer-Verlag, pp. 206–231, doi:10.1007/BFb0063439, MR 0716184
• Kelarev, A.V. (2003), Graph Algebras and Automata, New York: Marcel Dekker, ISBN 0-8247-4708-9, MR 2064147
• Kelarev, A.V.; Sokratova, O.V. (2003), "On congruences of automata defined by directed graphs", Theoretical Computer Science, 301 (1&ndash, 3): 31&ndash, 43, doi:10.1016/S0304-3975(02)00544-3, ISSN 0304-3975, MR 1975219
• Kelarev, A.V.; Miller, M.; Sokratova, O.V. (2005), "Languages recognized by two-sided automata of graphs", Proc. Estonian Akademy of Science, 54 (1): 46&ndash, 54, ISSN 1736-6046, MR 2126358
• Kelarev, A.V.; Sokratova, O.V. (2001), "Directed graphs and syntactic algebras of tree languages", J. Automata, Languages & Combinatorics, 6 (3): 305&ndash, 311, ISSN 1430-189X, MR 1879773
• Kiss, E.W.; Pöschel, R.; Pröhle, P. (1990), "Subvarieties of varieties generated by graph algebras", Acta Sci. Math. (Szeged), 54 (1&ndash, 2): 57&ndash, 75, MR 1073419
• Lee, S.-M. (1988), "Graph algebras which admit only discrete topologies", Congr. Numer., 64: 147&ndash, 156, ISSN 1736-6046, MR 0988675
• Lee, S.-M. (1991), "Simple graph algebras and simple rings", Southeast Asian Bull. Math., 15 (2): 117&ndash, 121, ISSN 0129-2021, MR 1145431
• Oates-Williams, Sheila (1984), "On the variety generated by Murskiĭ's algebra", Algebra Universalis, 18 (2): 175–177, doi:10.1007/BF01198526, ISSN 0002-5240, MR 0743465
• Pöschel, R (1989), "The equational logic for graph algebras", Z. Math. Logik Grundlag. Math., 35 (3): 273&ndash, 282, doi:10.1002/malq.19890350311, MR 1000970 | 2018-10-20 19:52:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 9, "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": 0.774627149105072, "perplexity": 8536.045913837892}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583513384.95/warc/CC-MAIN-20181020184612-20181020210112-00031.warc.gz"} |
http://mathematica.stackexchange.com/questions/44485/intersection-of-osculating-circle-with-a-parametric-curve | # Intersection of Osculating Circle with a Parametric Curve
I have the following Parametric plot
Generated from the following:
Manipulate[
ParametricPlot[{(A*Cos[t]^2 + B)*{Cos[t], Sin[t]},
C*{Cos[t] + 1, Sin[t]}}, {t, 0, 2*Pi},
PlotRange -> {{-5, 5}, {-5, 5}}], {A, 1, 10}, {B, 1, 10}, {C, 1,
10}]
I would like to find the exact values of $C$ and $t$ such that the inner circle, with one point fixed to the origin is tangent to the outer curve. This would mean, if I'm not mistaken, for $t\in\left[0,\pi/2\right]$, solving the following system of equations for $t$ and $C$
x1 = (A*Cos[t]^2 + B)*Cos[t]
y1 = (A*Cos[t]^2 + B)*Sin[t]
x2 = C*(Cos[t] + 1)
y2 = C*Sin[t]
x1 == x2
y1 == y2
i.e. find the intersection of the curves, but also they should have the same tangent, so we have:
D[y1,t]/D[x1,t] == D[y2,t]/D[x2,t]
But, for some reason I can't find a solution. What am I doing wrong?
-
You might want to avoid using C, D, I, or N as constants in Mathematica. More generally, use initial lowercase for your constants. – Hector Mar 21 at 23:48
Did mean for the circle to be osculating or merely tangent? – Michael E2 Mar 22 at 0:07
@Michael E2 I guess just tangent (and I added a note about this in my response). – Daniel Lichtblau Mar 22 at 0:18
yes, just tangent is enough. – okj Mar 22 at 0:27
You don't have to calculate thing by hand in MMA. Use D and Cross to formulate expectations.
### One solution can be found analitically:
ClearAll[a, b , p1, p2, c]
p1[t_] := (a Cos[t]^2 + b)*{Cos[t], Sin[t]}
p2[t_] := c*{Cos[t] + 1, Sin[t]}
Solve[{
Cross[D[p1[t], t]].D[p2[t], t] == 0,
p1[t] == p2[t],
0 <= t <= Pi/2, c > 1
}, {t, c}, Reals] // Normal
{{t -> 0, c -> (a + b)/2}}
### General case with a and b fixed:
a = 5; b = 1;
sol = NSolve[{
Cross[D[p1[t1], t1]].D[p2[t2], t2] == 0,
p1[t1] == p2[t2],
c > 1,
0 <= t1 <= Pi, 0 <= t2 <= Pi}, {t1, t2, c}, Reals]
{{t1 -> 0, t2 -> 0, c -> 3.}, {t1 -> 1.10715, t2 -> 2.2143, c -> 2.23607}}
ParametricPlot[Evaluate[Join @@ ({p1[t], p2[t]} /. sol)], {t, 0, 2 Pi},
PlotRange -> {{0, 8}, {-3, 3}}, AspectRatio -> Automatic,
Epilog -> { [email protected], Green, p2[t2] /. sol // Point},
PlotStyle -> {Black, Red, Black, Blue}, BaseStyle -> Thick]
### Edit: with FindRoot it can be insatneous:
(trivial soluion is not calculated)
ClearAll[a, b, p1, p2, c]
p1[t_, a_, b_] := (a Cos[t]^2 + b)*{Cos[t], Sin[t]}
p2[t_, c_] := c*{Cos[t] + 1, Sin[t]}
Manipulate[
sol = FindRoot[{Cross[D[p1[t1, a, b], t1]].D[p2[t2, c], t2] == 0,
p1[t1, a, b] - p2[t2, c] == 0},
{{t1, Pi/2.}, {t2, Pi/2.}, {c, a}}];
ParametricPlot[ Evaluate[{p1[t, a, b], p2[t, c] /. sol}], {t, 0, 2 Pi},
PlotRange -> {{-15, 15}, {-10, 10}}, AspectRatio -> Automatic,
PlotStyle -> {Orange, Blue}, BaseStyle -> Thick, ImageSize -> 600,
Epilog -> {[email protected], Red, p2[t2, c] /. sol // Point,
Point[{(a + b), 0}], Blue, Circle[{(a + b)/2, 0}, (a + b)/2]},
]
, {a, 1, 10}, {b, 1, 5}]
-
--- edit ---
I forgot to address an issue which I now see was raised in a comment by @Michael E2. This setup is only going to give a tangent circle. If it osculates it is largely by accident. (Outright snogging, now that might be intentional.)
--- end edit ---
The issue is that these curves need not intersect at the same value of the parameter. So you can proceed as below.
aa = 2;
bb = 1;
x1[t_] = (aa*Cos[t]^2 + bb)*Cos[t];
y1[t_] = (aa*Cos[t]^2 + bb)*Sin[t];
x2[t_] = cc*(Cos[t] + 1);
y2[t_] = cc*Sin[t];
eqns = Flatten[{x1[t1] - x2[t2],
y1[t1] - y2[t2], {D[x1[t1], t1], D[y1[t1], t1]} -
lam*{D[x2[t2], t2], D[y2[t2], t2]}}];
sol = Solve[
GroebnerBasis[eqns, {t1, t2, cc}, lam] == 0 && 0 <= t1 <= Pi/2 &&
0 <= t2 <= 2*Pi, {cc, t1, t2}]
(* {{cc -> 3/2, t1 -> 0, t2 -> 0}, {cc -> 3/2, t1 -> 0,
t2 -> 2 \[Pi]}, {cc -> Sqrt[2], t1 -> -2 ArcTan[1 - Sqrt[2]],
t2 -> -4 ArcTan[1 - Sqrt[2]]}} *)
{x1[t1], x2[t2], y1[t1], y2[t2]} /. sol // N
(* Out[304]= {{3., 3., 0., 0.}, {3., 3., 0.,
0.}, {-1.41421356237, -1.41421356237, 1.41421356237,
1.41421356237}, {1.41421356237, 1.41421356237, 1.41421356237,
1.41421356237}} *)
A careful look will indicate that there are two such circles. One intersects at the east-most crossing of the x axis.
p1 = ParametricPlot[{x1[t], y1[t]}, {t, 0, 2*Pi},
ColorFunction -> (Green &)];
p2 = ParametricPlot[{x2[t], y2[t]} /. sol[[1]], {t, 0, 2*Pi},
ColorFunction -> (Blue &)];
p3 = ParametricPlot[{x2[t], y2[t]} /. sol[[3]], {t, 0, 2*Pi},
ColorFunction -> (Red &)];
Show[p1, p2, p3]
-
+1 because of GroebnerBasis – Hector Mar 21 at 23:55
I think you may explain the role of lam – belisarius Mar 22 at 0:03
@Artes Yep, I understand. I was thinking that Daniel's explanation could enrich his answer. Thanks! – belisarius Mar 22 at 0:15
@belisarius Well, I needed to sacrifice a variable. Okay, I could have cross multiplied I guess. But that lam is still getting fleeced. – Daniel Lichtblau Mar 22 at 0:15
The solution looks good, but I don't really understand the method you used. What was wrong with my original attempt (apart from syntax)? – okj Mar 22 at 0:31 | 2014-08-20 07:02:21 | {"extraction_info": {"found_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": 0.19635117053985596, "perplexity": 9178.430030442622}, "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-2014-35/segments/1408500800767.23/warc/CC-MAIN-20140820021320-00206-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=655433 | # perfectly elastic collision of spheres
by janismac
Tags: collision, elastic, perfectly, spheres
P: 4 Hey guys! I am currently developing a simulation that involves sphere (or if you like particle) collision in 3D space. And I want it to be accurate (on the level of classic mechanics). The algorithm to do the job would take in the velocities, masses and relative position (aka line of impact) of two colliding particles and compute their new velocities short after the collision. So I tried to develop this algorithm and ran into a problem. I set up the equations for the conservation of momentum and conservation of kinetic energy. That gave me four equations, one for the kinetic energy (scalar) and three for the momentum (one for each of the three dimensions). Since the result of this calculation has six variables (the two new velocity vectors) it needs six equations to be solved. Obviously two more are necessary. I dont know how to set them up but they certainly have to take the relative position into account. Well, I just hope that someone here can help me solve this. In case I have left something unclear, just ask. Regards, janismac
Thanks
P: 2,951
Quote by janismac I set up the equations for the conservation of momentum and conservation of kinetic energy.
How have you used the angle of impact parameter? There are (as you've just discovered) multiple solutions that conserve momentum and energy; they correspond to different angles into and out of the collision.
P: 61 You can simplify by using spherical coordinates based on the center-of-mass. Conserve angular momentum.
P: 4
## perfectly elastic collision of spheres
Quote by Nugatory How have you used the angle of impact parameter?
where is there such a parameter?
my equations are:
$m_1\vec{v_1}+m_2\vec{v_2}=m_1\vec{v'_1}+m_2\vec{v'_2}$ momentum
$m_1\vec{v_1}^2+m_2\vec{v_2}^2 = m_1\vec{v'_1}^2+m_2\vec{v'_2}^2$ kinetic energy (equation multiplied by 2)
Thanks
P: 2,951
Quote by janismac where is there such a parameter? my equations are: $m_1\vec{v_1}+m_2\vec{v_2}=m_1\vec{v'_1}+m_2\vec{v'_2}$ momentum $m_1\vec{v_1}^2+m_2\vec{v_2}^2 = m_1\vec{v'_1}^2+m_2\vec{v'_2}^2$ kinetic energy (equation multiplied by 2)
Right, and that's not sufficient to completely specify the initial conditions, which is why you're finding multiple solutions. You have six unknowns, four equations leaves you with two unknowns. You can make up a value for one of these two, then solve for the last one, and you'll have a complete solution; but you can make up a different value for the fifth unknown and you'll get a different but perfectly good solution. So... multiple solutions.
In physical terms, imagine the two balls approaching each other at equal speeds. The could bounce away at any angle, and as long as they're traveling at the same speed and in exactly opposite directions after the collision, you have a solution that conserves energy and momentum. Your two extra unknowns correspond to the the two angles the balls actually take.
Tadchem's answer (use conservation of angular momentum - that will give you the two extra equations you need to solve for those last two unknowns) is good.
HW Helper
Thanks
PF Gold
P: 4,436
Quote by janismac where is there such a parameter? my equations are: $m_1\vec{v_1}+m_2\vec{v_2}=m_1\vec{v'_1}+m_2\vec{v'_2}$ momentum $m_1\vec{v_1}^2+m_2\vec{v_2}^2 = m_1\vec{v'_1}^2+m_2\vec{v'_2}^2$ kinetic energy (equation multiplied by 2)
When you line up a shot in pool, you draw an imaginary line between the pocket and center of the ball you are aiming at. You then try to hit the ball you are aiming with the cue ball at the intersection of the imaginary line with far surface of the ball you are aiming at. So the collision of two balls in not only a function of the initial velocities, but also their offset trajectories.
P: 4
Quote by Nugatory Tadchem's answer (use conservation of angular momentum - that will give you the two extra equations you need to solve for those last two unknowns) is good.
To be honest I am not sure how to set this equation up but I'll give it a try.
$\vec{x_1} \times m_1\vec{v_1} + \vec{x_2} \times m_2\vec{v_2} =\vec{x_1} \times m_1\vec{v'_1} + \vec{x_2} \times m_2\vec{v'_2}$
Vector x is the absolute position of a particle in my coordiante system.
This equation would set the center-of-rotation to the origin. May I just do that?
This would give me three new equations instead of two... so it's most likely wrong. But I dont know any better. Couldnt someone just give me the equation?
P: 4 I have another idea that may lead to a solution. The old velocity vector, the new velocity vector and the force that acts on a particle have to lie in one plane (dont they?) so I could express this in two equations, one for each particle: $\fn_jvn \newline(\vec{f} \times \vec{v_1}) \cdot \vec{v'_1} = 0 \newline \newline(\vec{f} \times \vec{v_2}) \cdot \vec{v'_2} = 0 \newline \newline \vec{f} = \vec{x_2}-\vec{x_1}$ does that make sense?
Homework Sci Advisor HW Helper Thanks ∞ P: 9,159 I assume you start off with known trajectories for the particles before impact. From those you can compute the point of impact on each particle. I'll further assume the particles are smooth, so you don't have to worry about spin. You can arbitrarily fix one particle as the reference frame by subtracting its velocity from the other. Further, you can rotate the frame so that the mass centres, point of impact and velocity of approaching particle are all in the XY plane. The direction of departure of the previously 'stationary' particle is obvious, so you only have three scalar unknowns to determine and still 3 scalar equations to make use of.
P: 61 Back in 1993 I wrote up the details of this in a non-peer-reviewed paper: "A simple and accurate method for calculating viscosity of gaseous mixtures", US Bureau of Mines Report of Investigations 9456, Govt. Doc Number I 28.23:9456. In the appendix I analyzed in detail the mechanics of the transfer of momentum in an elastic collision between two perfect spheres of arbitrary masses. The choice of center-of-mass coordinates allows one to disregard the net velocity of the system (conserved before and after the collision), and aligning the z-axis of a spherical coordinate system with the angular momentum pseudovector removes two degrees of freedom. There is no individual momentum of the spheres parallel to the z-axis, leaving r and theta. All momentum *transfer* occurs along the r coordinate, and (angular) momentum remains unchanged along the theta coordinate. The problem is thus reduced to the one-dimensional problem along the line connecting the centers of the spheres. The physics revealed is that the efficiency of the transfer of momentum is a maximum when the masses are equal. Dissimilar masses result in the smaller mass recoiling from the impact while the larger mass only transfers some of its momentum to the smaller mass, yielding a lower efficiency of momentum transfer The importance is that the efficiency of the transfer of momentum from one portion of a fluid to another is called fluidity, which is better known through its reciprocal - viscosity. This helps explain the viscosity anomaly - the fact that a mixture of fluids of different molecular weights always has a higher viscosity than the corresponding linear combination of the separate viscosities of the individual fluids.
P: 61 NB: The impact parameter may be useful in other contexts, but as long as the overall collision is elastic, i.e. there is no conversion of kinetic energy to thermal energy or other forms, then the details of the interaction such as the impact parameters are insignificant. What matters is the momenta of the individual particles before and after the interaction. This is true whether the interaction is mechanical, gravitational, or coulombic.
Related Discussions Introductory Physics Homework 3 Introductory Physics Homework 3 Introductory Physics Homework 2 Introductory Physics Homework 3 Introductory Physics Homework 7 | 2014-04-19 12:39:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7656161785125732, "perplexity": 374.67169762465534}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00434-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://ctftime.org/writeup/26883 | Tags: pwn
Rating:
I got the [a.out](https://github.com/mar232320/ctf-writeups/raw/main/umassctf2021/a.out) file which runs a web server. I also have a website link [http://34.72.232.191:8080/](http://34.72.232.191:8080/)
This was very easy challenge for me overall. When I went to the webserver page I found this content
# Dynamic pages!!! /page?args ? (this is also a 404 page...)
Try going to /echo?message
I echoed some data - for example /echo?test and the web page displayed test Echo is a linux command so as it was written in the main page I tried cat
/cat?flag.txt
The site returned UMASS{f^gJkmvYq so I added } to the flag
> UMASS{f^gJkmvYq} | 2023-01-30 09:01:11 | {"extraction_info": {"found_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": 0.3823753297328949, "perplexity": 4881.800001429927}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499804.60/warc/CC-MAIN-20230130070411-20230130100411-00131.warc.gz"} |
https://online.ucpress.edu/mp/article/37/1/42/62870/A-Whole-Brain-EEG-Analysis-of-Musicianship | The neural activation patterns provoked in response to music listening can reveal whether a subject did or did not receive music training. In the current exploratory study, we have approached this two-group (musicians and nonmusicians) classification problem through a computational framework composed of the following steps: Acoustic features extraction; Acoustic features selection; Trigger selection; EEG signal processing; and Multivariate statistical analysis. We are particularly interested in analyzing the brain data on a global level, considering its activity registered in electroencephalogram (EEG) signals on a given time instant. Our experiment's results—with 26 volunteers (13 musicians and 13 nonmusicians) who listened the classical music Hungarian Dance No. 5 from Johannes Brahms—have shown that is possible to linearly differentiate musicians and nonmusicians with classification accuracies that range from 69.2% (test set) to 93.8% (training set), despite the limited sample sizes available. Additionally, given the whole brain vector navigation method described and implemented here, our results suggest that it is possible to highlight the most expressive and discriminant changes in the participants brain activity patterns depending on the acoustic feature extracted from the audio.
Music requires a high neural demand from who plays it (Münte, Altenmüller, & Jäncke, 2002; Peretz & Zatorre, 2005) and is an important tool for understanding the organization of the human brain (Münte et al., 2002; Peretz & Zatorre, 2005; Schlaug, 2015). In the last two decades, the processing of music by our brain has attracted the attention of researchers worldwide, and a number of scientific works have identified neural activations differences between musicians and nonmusicians in distinct experiments using wearable and non-wearable technologies, mainly EEG (electroencephalography) and fMRI (functional magnetic resonance imaging), respectively (Liang, Hsieh, Chen, & Lin, 2011; Mikutta, Maissen, Altorfer, Strik, & Koenig, 2014; Münte et al., 2002; Peretz & Zatorre, 2005; Saari, Burunat, Brattico, & Toiviainen, 2018; Schlaug, 2015; Tervaniemi, Castaneda, Knoll, & Uther, 2006; Virtala, Huotilainen, Partanen, & Tervaniemi, 2014; Vuust, Brattico, Sppanen, Naatanen, & Tervaniemi, 2012).
In the experiments of music listening and EEG signal analysis, past studies focused on understanding the neural processing of artificial stimuli that rely on controlled auditory paradigms. The most recent works on this issue (Abrams et al., 2013; Alluri et al., 2012, 2013; Markovic, Kühnis, & Jäncke, 2017; Poikonen, Alluri, et al., 2016; Poikonen, Toiviainen, & Tervaniemi, 2016; Saari et al., 2018) have analyzed though naturalistic music pieces rather than artificial ones, with the goal of describing the association between the dynamic changes in the audio features and the time courses of the neural activations recorded in volunteers (or subjects). Interestingly, these recent works have been able to achieve almost the same results found previously with artificial stimuli, indicating the recruitment of new brain areas related to music processing.
However, all these state-of-the-art findings have been obtained on a local level, based on the concept of massunivariate analyses (Markovic et al., 2017; Poikonen, Alluri, et al., 2016; Poikonen, Toiviainen, & Tervaniemi, 2016; Rigoulot, Pell, & Armony, 2015; Virtala et al., 2014; Vuust et al., 2012), which reveal statistical aspects of specific and unique areas of the brain that supposedly do not depend on changes in other cognitive areas. Although this approach is mathematically sound, it inevitably ignores possible global and interregional brain dependencies in music processing, providing limited understanding of the behavior of our brain during such a cognitive task. Since EEG brain data inherently include simultaneous measurements on a given number of electrodes, to understand the complexity of their information, it seems appropriate to disclose the relationships between all these electrodes in a multivariate way.
In the current exploratory study, we propose to analyze on a global level the EEG brain data collected during a music listening task, considering the whole brain activity registered in all the EEG electrodes simultaneously. More specifically, we approach this problem through a novel multivariate statistical perspective, using this global EEG brain information to predict musicianship as well. Therefore, rather than using a mass-univariate model to find statistical differences between distinct sample groups, we describe and implement here a two group pattern recognition framework to linearly separate the samples as musicians and nonmusicians, evaluated by means of classification accuracy and categorized previously, depending on whether or not they have undergone music training.
To the best of our knowledge, we present here the first results on whole brain EEG analysis of musicianship in the context of naturalistic music listening, elaborating on Ribeiro and Thomaz (2018), by adding new experimental evidences in EEG feature extraction and classification, using several acoustic features, and new multivariate and comparative statistical analyses. Our experimental results, based on high dimensional encoding of variance and discriminant information of EEG data, indicate feasible and plausible linear separation from musicians and nonmusicians sample groups, despite their corresponding limited sample sizes available.
## Materials and Method
### PARTICIPANTS
Two groups of 13 subjects took part in our experiment. The musicians (7 male and 6 female) were all amateurs, aged between 18 and 45 years (mean 26.8 years; all right-handed). They had started playing between 4 and 17 years (mean 17.1 years) and were currently playing 4 hours on average per week, with distinct musical styles (classic, pop, and rock). All of them received more than two years of formal training in music, but none had a professional degree in music performance. The nonmusicians (10 males and 3 females) were aged between 25 and 45 years (mean 31.2 years; 9 right-handed). Two of them received less than one year of formal training in music (none of them keeps playing) and the others had no formal music training. All participants gave a written informed consent to participate in the study.
### STIMULUS
As stimulus, the classical music Hungarian Dance No.5 of Johannes Brahms was used, presented via intraearphone. The composition was performed by the Fulda Symphonic Orchestra, conducted by Simon Schindler at the Fürstensaal des Stadtschlosses.1 The ending of the audio, corresponding to the applauses, was replaced for silence, using the free, open source, audio software Audacity (version 2.2.2), resulting in an audio signal that was 3.2 minutes long.
All subjects were instructed to listen to music, remaining as still as possible while their EEG signals were recorded. The musical piece was selected—within the context of a benchmark classical music used in other works (Alluri et al., 2012; Poikonen, Alluri, et al., 2016)—due to its high range of variations during the performance in several musical features such as dynamics, timbre, and rhythm, having an appropriate duration for the experiment.
### METHODOLOGY
Our computational framework consists of the following 5 steps: (I) acoustic features extraction; (II) acoustic features selection; (III) trigger selection; (IV) EEG signal processing; and (V) multivariate statistical analysis. The first three steps refer to the audio signal and the last two to the EEG signal. This framework, illustrated in Figure 1, is based on previous multidisciplinary works of EEG music processing (Poikonen, Alluri, et al., 2016; Poikonen, Toiviainen, & Tervaniemi, 2016) and high-dimensional data analysis (Davatzikos, 2004; Gregori, Sanches, & Thomaz, 2017; Sato et al., 2008; Thomaz, Duran, Busatto, Gillies, & Rueckert, 2007; Xavier et al., 2015).
FIGURE 1.
A schematic illustration of the following five main steps of the framework proposed to statistically analyze the EEG high-dimensional brain data: (I) acoustic features extraction; (II) acoustic features selection; (III) trigger selection; (IV) EEG signal processing; and (V) multivariate statistical analysis.
FIGURE 1.
A schematic illustration of the following five main steps of the framework proposed to statistically analyze the EEG high-dimensional brain data: (I) acoustic features extraction; (II) acoustic features selection; (III) trigger selection; (IV) EEG signal processing; and (V) multivariate statistical analysis.
#### Acoustic feature extraction
In this first step, the lowlevel acoustic features that describe the audio signal are extracted using MIRtoolbox (version 1.6.1) (Lartillot, 2014). To perform such extraction, the signal is decomposed into 50 ms windows with 50% of overlap, in order to study the evolution of each of these features over time. These characteristics correspond to aspects of the audio that can be identified in terms of human perception and, even though they may not accurately describe what is acoustically perceived, they generate significant EEG neural responses (Poikonen, Alluri, et al., 2016; Poikonen, Toiviainen, & Tervaniemi, 2016).
We have selected the following standard and well known acoustic features (Alluri et al., 2012; Lartillot, 2014; Lerch, 2012; Poikonen, Alluri, et al., 2016): (1) root mean square (RMS): measure of the energy in the signal computed by taking the square root average of the square of the amplitude; (2) zero crossing rate (ZCR): measure of the number of times the signal crosses the x-axis; (3) spectral rolloff: frequency below which 85% of the total energy is contained in the signal; (4) spectral roughness: estimation of the sensory dissonance; (5) brightness: measure of the amount of energy above 1500 Hz; (6) spectral entropy: measure of the relative Shannon entropy of the signal which indicates whether the spectrum contains predominant peaks or not; (7) spectral flatness: measure of the uniformity of the spectrum defined as the ratio between the geometric mean and the arithmetic mean, also known as the Wiener entropy; (8) spectral skewness: the third central moment of the spectrum distribution and is a measure of the asymmetry of the distribution; (9) spectral kurtosis: the fourth central moment of the spectrum distribution and indicates the flatness of the spectrum and a sudden change can indicate transients at the audio; (10) spectral centroid: the first central moment of the spectrum distribution and is the geometric center of the distribution; (11) spectral spread: the second central moment of the spectrum distribution; and (12) spectral flux: measure of the temporal changes in the spectrum between successive frames. A detailed explanation of these 12 features can be found in the user manual of the MIRtoolbox (Lartillot, 2014).
#### Acoustic feature selection
In this second step we assess whether all the s = 12 standard features extracted from the audio are statistically non-redundant, through a cluster analysis, disclosing the correlation among the acoustic features to adequately represent the data.
We have used factor analysis (FA) (Johnson & Wichern, 2007), a well-known multivariate statistical technique, to describe the association between the values extracted from the windows of each acoustic features in a non-supervised way. Our motivation here is to reduce the redundancy of the data extracted from the audio signal, selecting the R most significant factors (higher loadings), where Rs. We have chosen here principal component analysis (PCA) (Johnson & Wichern, 2007) to estimate the parameters of our factor model because its spectral decomposition solution simplifies the issue of how many factors to retain.
Ideally, a pattern of factor loadings where each cluster of acoustic features is highly represented by a single factor and has low coefficients on the remaining ones is desirable. Therefore, those F = [f1, f2, … , fR] factors can replace the initial s variables on R rotated common factor loadings, where the association between the acoustic features would be most significant in terms of variance (varimax rotation), choosing conventionally the factor loadings with corresponding eigenvalues greater than 1 to determine the adequate number of factors (Johnson & Wichern, 2007).
#### Trigger selection
The changes on the acoustic features used in this work are known as triggers (Poikonen, Alluri, et al., 2016), which are able to elicit sensory components, similar to those found with artificial stimuli. More precisely, triggers are instants in the time series generated by the extracted features where a high-contrast occurs, as detailed below.
We have adopted the Poikonen, Alluri, et al. (2016) method to identify the triggers, where the upper and lower thresholds Vp+ and Vp are determined from the mean values of the acoustic feature as a given percentage q above or below it, respectively (q = ± 20%, here). In other words, a trigger occurs when the signal remains below Vp for a minimum interval of time, defined as 500 ms for all acoustic features—called preceding lowfeature phase (PLFP)—followed by an ascendant phase where the signal reaches Vp+ (Poikonen, Alluri, et al., 2016). The intensity of the acoustic contrast depends on the parameters chosen for trigger selection (length of PLFP and the upper and lower thresholds Vp+ and Vp). These parameters can be changed accordingly, based on a trade-off between the number of triggers and how high each acoustic contrast might be defined; that is, the higher the acoustic contrast, the lower the number of triggers identified.
#### EEG signal processing
We have used the OpenBCI EEG device to acquire the brain electrical signals (OpenBCI, 2018). This device has a sampling rate of 127 Hz and is composed of 16 dry electrodes, positioned according to the international 10–20 system, including the additional two electrodes placed on each earlobe as reference. All EEG signal have been preprocessed using a b and pass Butterworth filter of 1–30 Hz analogously to others (Poikonen, Alluri, et al., 2016; Virtala et al., 2014). The continuous EEG data were separated into epochs according to the triggers and averaged for each electrode. The epochs started 100 ms before the trigger and ended 300 ms after the trigger. The baseline has been defined according to the 100 ms time period before the trigger (Poikonen, Alluri, et al., 2016; Virtala et al., 2014; Vuust et al., 2012). All signals have been inspected visually and EEG detected channels with noise have been removed from analysis as well as the epochs with amplitudes above 100 µV (in absolute values). The standard and well-known EEG Matlab toolbox (EEGLab 13.5.4b) was used to process the EEG data at Matlab R2015a software.
#### Multivariate statistical analysis
In this fifth and last step of our computational framework, we first describe the data matrix Xr composed of the EEG values at instant t = 100 ms post stimulus (predefined time-stamps), defined by the average along triggers of each r selected acoustic feature separately, where r 1, 2, … , R in a way that the corresponding sampled input signal can be treated as a high-dimensional point in a multivariate space, as follows:
$Xr=[x1x2⋮xN]=[x11⋯x1nx21x2n⋮⋱⋮xN1…xNn],$
(1)
where N is the total number of volunteers (or subjects) and n the total number of electrodes (n = 16, here).
According to this data representation, we are assuming that our EEG multivariate statistical analyses might include the electrical potentials at the predefined timestamps to understand the complexity of the brain activity not only in terms of the N sampling, but also in terms of all the n electrodes simultaneously.
Principal component analysis (PCA) and linear discriminant analysis (LDA) (Fukunaga, 1990; Johnson & Wichern, 2007) were explored here as two alternative multivariate statistical methods to understand how the information is changing in the original space of the EEG brain data, looking not only for the most expressive (higher variance) changes, but also for the most discriminant (higher separability) ones, according to the neural responses reflected by the acoustic feature extracted from the audio used as stimulus.
#### Principal component analysis
We have used PCA to highlight the most expressive changes in terms of the total variance information of the electrical potentials.
PCA calculates the spectral decomposition of the correlation matrix of the N × n matrix Xr, described in Equation (1). We have composed the PCA transformation matrix by selecting all the M eigenvectors with non-zero eigenvalues, where Mn. Data projected onto these [p1,r, p2,r, … , pM,r] eigenvectors ranked in decreasing order of their corresponding eigenvalues; that is, λ1,r, λ2,r, … , λM,r, follow the directions of higher variance of Xr.
Each pm,r vector, where m = 1, 2, … , M, can be used to form a spatial map of the brain regions that most vary in the data, moving from one side of the principal component axis (or dimension) to the other (Cootes, Edwards, & Taylor, 1998; Davatzikos, 2004). Thus, it is possible to navigate along such dimensions to capture and understand the most expressive changes of the data matrix Xr. This vector navigation method (Cootes et al., 1998; Gregori et al., 2017; Sato et al., 2008; Xavier et al., 2015) can be mathematically described as
$ym,k=x¯i+(kλm,r)×pm,r,$
(2)
where , m is the index number of the corresponding principal component to navigate, and $x¯i$ is the n-dimensional global mean vector of Xr for each r selected acoustic feature.
#### Linear discriminant analysis
We have used LDA to identify the most discriminant dimension for separating the two sample groups (or classes) of interest, that is, musicians and nonmusicians, by maximizing their between-class separability while minimizing their within-class variability.
Let the between-class and within-class scatter matrices Sb and Sw be defined, respectively, as (Fukunaga, 1990)
$Sb=∑i=1gNi(x¯i−x¯r)T(x¯i−x¯r),$
(3)
$Sw=∑i=1g∑j=1Ni(xi,j−x¯i)T(xi,j−x¯i),$
(4)
where xi,j is the n-dimensional sample j from class i, Ni is the number of training samples of class i (N1 = 13 and N2 = 13, here), g represents the total number of classes (g = 2, here) and $x¯i$ is the sample group mean vector given by the corresponding Ni samples only.
The main objective of LDA here is to find a projection vector wr that maximizes the ratio of the determinant of Sb to the determinant of Sw (Fisher's criterion), formulated by:
$wr=arg maxw|wTSbw||wTSww|.$
(5)
The Fisher's criterion is maximized when the projection vector wr is composed of the leading eigenvector of $Sw−1Sb$ with nonzero corresponding eigenvalue (Fukunaga, 1990; Johnson & Wichern, 2007).
Analogous to PCA, this vector wr can be used to form a spatial map of the most discriminant brain regions, moving from one side of the separating dimension to the other (Davatzikos, 2004; Sato et al., 2008). Thus, it is possible to navigate along such dimension to capture and understand the most discriminant changes of the data matrix Xr, assuming its dispersion follows a Gaussian distribution (Gregori et al., 2017; Sato et al., 2008; Xavier et al., 2015). This vector navigation method can be mathematically described as
$zi,j=x¯r+(jσi+x¯i)×wr,$
(6)
where $j∈{−1,0,1}$, i is again the corresponding sample group, $x¯r$ the n-dimensional global mean vector of Xr, and $σi$ and $x¯i$ are, respectively, the standard deviation and mean of each sample group on the LDA space for each r selected acoustic feature.
## Results
### AUDIO ANALYSIS
First, the audio was analyzed by means of the aforementioned factor analysis. The factor loadings values presented by the FA are shown in Figure 2 and Table 1. According to this, it is possible to observe that the acoustic features extracted from the audio clearly forms three clusters that show a relation between them, which can indicate that the acoustic information expressed for each cluster may be similar among the features that presented the highest loadings on each factor, as follows: RMS (cluster 1), spectral kurtosis (cluster 2), and spectral rolloff (cluster 3). Note that for the factor 2 the feature spectral skewness had higher loading than spectral kurtosis, but since it was found only three triggers for this feature, we choose to use the spectral kurtosis instead, given the trade-off described previously between the number of triggers and how high each acoustic contrast may be defined.
FIGURE 2.
Factor loadings of the acoustic features extracted from the audio signal, showing the formation of clusters between the following features: Cluster 1—1 (RMS), 4 (spectral roughness), and 12 (spectral flux); Cluster 2—8 (spectral skewness) and 9 (spectral kurtosis); Cluster 3—2 (ZCR), 3 (spectral rolloff), 5 (brightness), 6 (spectral entropy), 7 (spectral flatness), 10 (spectral centroid) and 11 (spectral spread).
FIGURE 2.
Factor loadings of the acoustic features extracted from the audio signal, showing the formation of clusters between the following features: Cluster 1—1 (RMS), 4 (spectral roughness), and 12 (spectral flux); Cluster 2—8 (spectral skewness) and 9 (spectral kurtosis); Cluster 3—2 (ZCR), 3 (spectral rolloff), 5 (brightness), 6 (spectral entropy), 7 (spectral flatness), 10 (spectral centroid) and 11 (spectral spread).
TABLE 1.
FeatureF1F2F3
RMS 0.6291 0.1337 0.7203
ZCR 0.7851 −0.3235 −0.3240
S. Rolloff 0.9584 −0.1939 −0.1629
S. Roughness 0.3883 −0.0242 0.7185
Brightness 0.9471 −0.1643 −0.1790
S. Entropy 0.8779 0.3712 −0.1710
S. Flatness 0.9329 0.0290 −0.0764
S. Skewness 0.0995 0.9847 −0.1210
S. Kurtosis −0.0761 0.9617 −0.1139
S. Centroid 0.9563 −0.1484 −0.1975
S. Flux 0.5880 0.0931 0.7174
FeatureF1F2F3
RMS 0.6291 0.1337 0.7203
ZCR 0.7851 −0.3235 −0.3240
S. Rolloff 0.9584 −0.1939 −0.1629
S. Roughness 0.3883 −0.0242 0.7185
Brightness 0.9471 −0.1643 −0.1790
S. Entropy 0.8779 0.3712 −0.1710
S. Flatness 0.9329 0.0290 −0.0764
S. Skewness 0.0995 0.9847 −0.1210
S. Kurtosis −0.0761 0.9617 −0.1139
S. Centroid 0.9563 −0.1484 −0.1975
S. Flux 0.5880 0.0931 0.7174
Using these selected acoustic features, we have found 9 triggers for the RMS, 7 for the spectral rolloff and 8 for the spectral kurtosis along the music used as stimulus. As can be seen in Figure 3, there is a similarity of the spectrograms between the clusters and there are triggers that occur at the same time within the same cluster, although some of them are different. These differences can be related to the methodological parameters used to select the triggers, showing that some of these features might be complementary to each other specially within the same cluster.
FIGURE 3.
Spectrograms of each acoustic feature extracted from the audio signal separated in clusters (green dashed lines), showing the selected triggers (red lines) and the selected acoustic feature (in bold and underlined) for each cluster.
FIGURE 3.
Spectrograms of each acoustic feature extracted from the audio signal separated in clusters (green dashed lines), showing the selected triggers (red lines) and the selected acoustic feature (in bold and underlined) for each cluster.
### EEG ANALYSIS
The topographic maps were generated by the vector navigation method on the four most expressive dimensions of PCA, according to Equation (2), and the discriminant dimension of LDA, according to Equation (6), for each acoustic feature selected, using the entire dataset. Such multivariate statistical analyses disclose visually the electrical potential changes and their data distributions along the corresponding dimensions.
Given the limited sample sizes available, we have adopted the leave-one-out and resubstitution methods (Fukunaga, 1990) to estimate, respectively, the lower (test set) and upper (training set) bounds of the most discriminant dimension, using the Euclidean sample mean distance classifier to decide whether the data projected is more similar to the musician or nonmusician group.
The topographic brain maps and the projection of the data generated by PCA from the acoustic feature spectral rolloff are shown in Figure 4, where we can see the changes of the brain responses along each principal component (PC), disclosing major electrical potential differences in several parts of the brain, especially at frontal areas. However, when the data are projected on the axis of each PC separately for classification evaluation, it's not possible to see a distinction between the musician and nonmusicians sample groups. In fact, all the classification rates were around chance level using the entire data set. Different from PCA, Figure 5 shows the navigation on the most discriminant axis of LDA for the same acoustic feature spectral rolloff. This vector navigation presents differences between the groups on the frontal areas, where the nonmusicians show a positive electrical potential distributed all over the frontal area, whereas the musicians present high activity at the prefrontal area. Such differences are not discriminant either, as can be seen on the data distribution, showing a classification rate that ranges from 15.4% (test set) to 74.5% (training set).
FIGURE 4.
Vector navigation on the first four principal components (PCs), ordered from the highest variance to the lowest, of the acoustic feature spectral rolloff captured by PCA at the latency of 100 ms post stimulus: (a) Brain mapping of each PC; (b) Projection of the data on each PC axis.
FIGURE 4.
Vector navigation on the first four principal components (PCs), ordered from the highest variance to the lowest, of the acoustic feature spectral rolloff captured by PCA at the latency of 100 ms post stimulus: (a) Brain mapping of each PC; (b) Projection of the data on each PC axis.
FIGURE 5.
Vector navigation on the most discriminant axis of the acoustic feature spectral rolloff captured by LDA at the latency of 100 ms post stimulus: (Top) Reconstruction of the mean brain mapping when navigating along the most discriminant axis between the sample group of musicians (right) and nonmusicians (left); (Bottom) Projection of the data on the discriminant axis. Classification accuracy: 15.4% (test set) and 74.5% (training set).
FIGURE 5.
Vector navigation on the most discriminant axis of the acoustic feature spectral rolloff captured by LDA at the latency of 100 ms post stimulus: (Top) Reconstruction of the mean brain mapping when navigating along the most discriminant axis between the sample group of musicians (right) and nonmusicians (left); (Bottom) Projection of the data on the discriminant axis. Classification accuracy: 15.4% (test set) and 74.5% (training set).
Additionally, Figure 6 shows the topographic maps and the projection of the data generated by PCA from the acoustic feature RMS and it's possible to see changes occurring mostly on the frontal areas of the brain, but again it's not possible to see a distinction between the musicians and the nonmusicians sample groups at the projection of the data on each PC dimension. Likewise, Figure 7 shows the RMS navigation along its discriminant axis, presenting visually subtle electrical potential differences between the groups, but more statistically discriminant than the previous spectral rolloff, with classification rate that ranges from 53.8% (test set) to 88.0% (training set).
FIGURE 6.
Vector navigation on the first four principal components (PCs)——ordered from the highest variance to the lowest——of the acoustic feature root mean square (RMS) captured by PCA at the latency of 100 ms post stimulus: (a) Brain mapping of each PC; (b) Projection of the data on each PC axis.
FIGURE 6.
Vector navigation on the first four principal components (PCs)——ordered from the highest variance to the lowest——of the acoustic feature root mean square (RMS) captured by PCA at the latency of 100 ms post stimulus: (a) Brain mapping of each PC; (b) Projection of the data on each PC axis.
FIGURE 7.
Vector navigation on the most discriminant axis of the acoustic feature root mean square (RMS) captured by LDA at the latency of 100 ms post stimulus: (Top) Reconstruction of the mean brain mapping when navigating along the most discriminant axis between the sample group of musicians (right) and nonmusicians (left); (Bottom) Projection of the data on the discriminant axis. Classification accuracy: 53.8% (test set) and 88.0% (training set).
FIGURE 7.
Vector navigation on the most discriminant axis of the acoustic feature root mean square (RMS) captured by LDA at the latency of 100 ms post stimulus: (Top) Reconstruction of the mean brain mapping when navigating along the most discriminant axis between the sample group of musicians (right) and nonmusicians (left); (Bottom) Projection of the data on the discriminant axis. Classification accuracy: 53.8% (test set) and 88.0% (training set).
Although, analogous to the other most expressive analyses, the PCA results from the acoustic feature spectral kurtosis, shown in Figure 8, highlight major but not discriminant electrical potential differences in several parts of the brain. This acoustic feature presents, in Figure 9, a clear electrical potential distinction between the sample groups of interest. In this case, the nonmusicians present high activity at the prefrontal area and the musicians show high activity at the right temporal area and negative electrical potential distributed over the parietal area. The data distribution exhibits a clear separation from these sample groups with classification rate between 69.2% (test set) and 93.8% (training set). For comparison, we have used the same approach to discriminate the sample groups using only a reduced number of electrodes, instead of all of them, with the following arrangements: 6 electrodes (F3, F4, C3, C4, P3, P4); 4 electrodes (F3, F4, C3, C4); 2 electrodes (F3, F4) and 2 electrodes (C3, C4). Table 2 presents these LDA classification rates of each acoustic feature and Table 3 the detailed classification rates using all the electrodes.
FIGURE 8.
Vector navigation on the first four principal components (PCs)——ordered from the highest variance to the lowest——of the acoustic feature spectral kurtosis captured by PCA at the latency of 100 ms post stimulus: (a) Brain mapping of each PC; (b) Projection of the data on each PC axis.
FIGURE 8.
Vector navigation on the first four principal components (PCs)——ordered from the highest variance to the lowest——of the acoustic feature spectral kurtosis captured by PCA at the latency of 100 ms post stimulus: (a) Brain mapping of each PC; (b) Projection of the data on each PC axis.
FIGURE 9.
Vector navigation on the most discriminant axis of the acoustic feature spectral kurtosis captured by LDA at the latency of 100 ms post stimulus: (Top) Reconstruction of the mean brain mapping when navigating along the most discriminant axis between the sample group of musicians (right) and nonmusician (left); (Bottom) Projection of the data on the discriminant axis. Classification accuracy: 69.2% (test set) and 93.8% (training set).
FIGURE 9.
Vector navigation on the most discriminant axis of the acoustic feature spectral kurtosis captured by LDA at the latency of 100 ms post stimulus: (Top) Reconstruction of the mean brain mapping when navigating along the most discriminant axis between the sample group of musicians (right) and nonmusician (left); (Bottom) Projection of the data on the discriminant axis. Classification accuracy: 69.2% (test set) and 93.8% (training set).
TABLE 2.
LDA Classification Rates for all Electrodes, 6 Electrodes, 4 Electrodes, and 2 Electrodes
FeaturesAll Electrodes6 Electrodes4 Electrodes2 Electrodes (F3-F4)2 Electrodes (C3-C4)
Test Set
S. Kurtosis 69.2% 53.8% 65.4% 53.8% 73.1%
RMS 53.8% 57.7% 53.8% 61.5% 61.5%
S. Rolloff 15.4% 42.3% 50.0% 65.4% 53.8%
Training Set
S. Kurtosis 93.8% 72.9% 71.8% 54.0% 73.1%
RMS 88.0% 72.2% 65.4% 68.2% 67.5%
S. Rolloff 74.5% 68.5% 70.9% 69.4% 54.6%
FeaturesAll Electrodes6 Electrodes4 Electrodes2 Electrodes (F3-F4)2 Electrodes (C3-C4)
Test Set
S. Kurtosis 69.2% 53.8% 65.4% 53.8% 73.1%
RMS 53.8% 57.7% 53.8% 61.5% 61.5%
S. Rolloff 15.4% 42.3% 50.0% 65.4% 53.8%
Training Set
S. Kurtosis 93.8% 72.9% 71.8% 54.0% 73.1%
RMS 88.0% 72.2% 65.4% 68.2% 67.5%
S. Rolloff 74.5% 68.5% 70.9% 69.4% 54.6%
TABLE 3.
Detailed LDA Classification Rates Using All Electrodes
FeatureAccuracySensitivitySpecificityF measure
Test Set
S. Kurtosis 69.2 69.2 69.2 69.2
RMS 53.8 61.5 46.2 57.1
S. Rolloff 15.4 30.8 26.7
Training Set
S. Kurtosis 93.8 87.7 100 93.4
RMS 88.0 89.8 86.2 88.2
S. Rolloff 74.5 71.4 77.5 73.7
FeatureAccuracySensitivitySpecificityF measure
Test Set
S. Kurtosis 69.2 69.2 69.2 69.2
RMS 53.8 61.5 46.2 57.1
S. Rolloff 15.4 30.8 26.7
Training Set
S. Kurtosis 93.8 87.7 100 93.4
RMS 88.0 89.8 86.2 88.2
S. Rolloff 74.5 71.4 77.5 73.7
## Discussion
In the present study, we applied a multivariate statistical framework to characterize the differences between subjects categorized as musicians and nonmusicians depending on whether or not have undergone musical training. Our main findings can be summarized as follows: (1) in order to investigate the neural activation patterns evoked by the acoustic features, it was shown that not all the standard and well-known 12 features are necessary, given the redundancy found by FA, which associated all these 12 acoustic features commonly used into only 3 clusters that were statistically nonredundant; (2) the discriminative patterns required to classify the musicians and nonmusicians sample groups exist predominantly in the frontal areas of the brain, but there are differences between the responses for each cluster representative acoustic feature that indicate that some aspects of the music are better to predict musicianship than others.
Our results show that the Poikonen, Alluri, et al. (2016) method proposed to identify triggers in music can be very helpful to differentiate musicians and nonmusicians, considering the trade-off between the number of triggers and how high the acoustic contrast can be defined. Considering the similarity between the extracted acoustic features on the factor loadings (Figure 2), it is interesting to observe that each extracted acoustic feature corresponds to one of the clusters found, and the spectrograms (Figure 3) clearly display these similarities. The triggers selected mostly occur at the same instant within each cluster, even though some of them presents more triggers than others. These extra triggers are related to the methodological parameters used, since here we decided to use the same parameters for all the acoustic features. A different approach could be, instead of selecting only one representative acoustic feature within a cluster, concatenate the triggers found for each feature within each cluster. It is important to highlight though that all these findings were achieved because of the classical music used here. We would expect this clustering behavior to be similar among the same musical genre, but there is no guarantee that these findings can be generalized to other pieces of music and further analysis on this issue is necessary.
It is known that music training promotes differences between musicians and nonmusicians not only during music listening but also during task-free conditions (Klein, Liem, Hänggi, Elmer, & Jäncke, 2016). There is also evidence of differences between musicians with different types of training, musical style/genre, and listening experiences (Tervaniemi, Janhunen, Kruck, Putkinen, & Huotilainen, 2016; Vuust et al., 2012).
Thus, there is a large variability among individuals for this two-group classification problem. However, our experimental results have showed that the discriminant vector navigation method described and implemented here can give a comprehensive description about these sample groups’ activity patterns and classification boundary. The topographic maps generated by this navigation method clearly present distinct neural activation patterns between musicians and nonmusicians, enhancing the understanding about the transient states between these sample groups, despite the limited sample sizes available.
The results of the acoustic feature spectral kurtosis (Figure 9), better than the other ones, disclose the relationship between musicians and nonmusicians, indicating the best linear separation between the sample groups. It is interesting to observe that spectral kurtosis and spectral skewness are in the same cluster, since they characterize the spectral data distribution in terms of its shape, along with spectral flatness. Both skewness and kurtosis provide information about the type and magnitude of departures from normality (DeCarlo, 1997) and it has been demonstrated that a sudden change on spectral kurtosis can indicate transients at non-stationary signals (Antoni, 2006; Dwyer, 1984). However, even though this acoustic feature seems suitable to discriminate musicians from nonmusicians, its relationship with what is indeed perceived by the subjects remains uncertain and needs further analysis.
Our multivariate statistical navigation on the PC dimensions show that although PCA finds the directions that the EEG data vary the most, these directions are not necessarily the ones that best discriminate the sample groups. Meanwhile, LDA showed interesting information about the brain discriminant aspects of each sample group. We may anticipate that the use of professional musicians in our approach shall promote a better discrimination between the groups of interest and, consequently, a better classification performance. However, an additional limitation of our study is that the subjects who took part in our experiments were not matched according to some background variables, especially the musicians (e.g., gender, onset of music training, and formal training received). Future works must pay attention to these matching criteria as well.
Additionally, taking into account the whole brain activity (considering the information registered simultaneously in all the EEG electrodes) during the listening task, we intend here not to overlook possible interactions among different brain regions, because such interrelations might be useful to separate the sample groups under investigation (Davatzikos, 2004; Friston & Ashburner, 2004; Thomaz et al., 2007). In fact, this multivariate statistical approach allows us to have new perspectives to identify novel regions potentially discriminant to characterize musicianship beyond specific regions of interest. The statistical differences between musicians and nonmusicians captured by our approach revealed that the brain areas that best describe musicianship exist predominantly in the frontal areas of the brain, which corroborate previous works (Saari et al., 2018) on how the brain processes certain acoustic characteristics of music, specially related to the motor and auditory areas.
Nevertheless, in Poikonen, Alluri, et al. (2016) work, brain activity has been defined by searching for a signal peak at the vicinity of 100 ms corresponding to the presence of a N100 component. In our work only the musicians presented this phenomenon, preventing us from using this peak information to compose our data matrix. Therefore, we have defined the time instant at 100 ms as an equal and comparable predefined timestamp for all the subjects, musicians and nonmusicians. However, even if we could ensure that this N100 component occurs to all subjects, we would still be using predefined information based on the response of specific brain regions of interest (related to the electrodes where this N100 component should be expected to occur) without considering the information contained beyond, in other regions of the brain. We believe that this limitation could be properly addressed using an algorithm that summarizes all the information on the average epoch of each subject (Rocha, Massad, Thomaz, & Rocha, 2014), or through a multilinear perspective (Lu, Plataniotis, & Venetsanopoulos, 2013).
Finally, all our multivariate statistical analyses have been carried out on a limited sample size setup, given the original dimensionality of our data matrix and the difficulty of recruiting volunteers, particularly professional musicians, which has an impact on the estimates of the classification accuracy of the discriminant dimensions. This becomes clear when we use less electrodes (Table 2) to compare our results. It is possible to see that, in this case, the training accuracies are close to the test ones, whereas the multivariate estimates (that is, using all the electrodes) show more sensitivity to such choice giving larger ranges between lower (test set) and upper (training set) classification bounds. This limited sample size issue could be addressed using regularized versions of LDA, such as MLDA (Sato et al., 2008; Thomaz et al., 2007), or other small sample size classifiers like Support Vector Machine (Davatzikos, 2004). Another interesting possibility would be to address this problem through a multilinear subspace learning perspective (Lu et al., 2013), considering time, electrodes and subjects as a third-order tensor data, which, as mentioned in the previous paragraph, might also overcome the timestamp limitation of our work.
## Conclusion
This work proposed and implemented a multivariate statistical framework to automatically classify whether the listeners were musicians or not, based on the most expressive and discriminant features obtained through the analysis of the brain EEG data on a global level. This is an exploratory study that describes in mathematical and computational terms the differences of musical processing between musicians and nonmusicians, allowing a plausible linear separation among the sample groups, based on the analysis of high-dimensional and limited sample size data, with relatively high classification accuracy.
We believe that this multivariate statistical analysis generates a more comprehensive description of the neural activation patterns, transition states, and classification boundary that separate cognitively musicians from nonmusicians. Further work with professional musicians and larger sample groups might increase the statistical discriminant power of our findings. The applicability of the multivariate statistical framework proposed is not restricted to the classical music used here, and other types of music beyond the classical ones can benefit from this whole brain signal analysis as well.
Note
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https://indico.cern.ch/event/616127/contributions/3016251/ | # EMIS 2018
16-21 September 2018
CERN
Europe/Zurich timezone
## Testing the Weak Interaction using the NSLtrap of the University of Notre Dame
18 Sep 2018, 16:45
2h
500/1-201 - Mezzanine (CERN)
### 500/1-201 - Mezzanine
#### CERN
Show room on map
Poster Ion traps and laser techniques
### Speaker
Dr Patrick O'Malley (University of Notre Dame)
### Description
The standard model of physics provides a description of matter in the universe. However, it fails to reproduce many unexplained features and so there has been search for physics beyond the standard model. One avenue is via the precise determination of the V$_{ud}$ matrix element of the Cabibbo-Kobayashi-Maskaka (CKM) matrix from the ft-value of superallowed mixed beta-decay transitions. A violation of the CKM matrix unitarity could be the consequence of a missing quark
generation, new bosons, or even supersymmetry. However, the determination of V$_{ud}$ from mirror transitions requires the measurement of the Fermi-to-Gamow Teller mixing ratio ρ. At the Nuclear Science Lab (NSL) within the University of Notre Dame a project is underway to develop a Paul trap devoted to the measurement of this elusive quantity. It will receive radioactive ion beams produced in-flight with TwinSol, a coupled pair of superconducting solenoids. The NSLtrap will consist of a gas catcher to stop the 1-3 MeV/A secondary beams from TwinSol . This will be followed by a radio-frequency quadrapole to cool and bunch the thermalized ions before their injection into a Paul trap. The design will be presented and the planned initial measurements will be discussed.
Project supported by NSF MRI: PHY-1725711
### Primary author
Dr Patrick O'Malley (University of Notre Dame)
### Co-authors
Dr Maxime Brodeur (University of Notre Dame) TwinSol Collaboration
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https://asmedigitalcollection.asme.org/computingengineering/article/18/1/011001/366507/Ontology-Based-Representation-of-Design-Decision | The design of complex engineering systems requires that the problem is decomposed into subproblems of manageable size. From the perspective of decision-based design (DBD), typically this results in a set of hierarchical decisions. It is critically important for computational frameworks for engineering system design to be able to capture and document this hierarchical decision-making knowledge for reuse. Ontology is a formal knowledge modeling scheme that provides a means to structure engineering knowledge in a retrievable, computer-interpretable, and reusable manner. In our earlier work, we have created ontologies to represent individual design decisions (selection and compromise). Here, we extend the selection and compromise decision ontologies to an ontology for hierarchical decisions. This can be used to represent workflows with multiple decisions coupling together. The core of the proposed ontology includes the coupled decision support problem (DSP) construct, and two key classes, namely, Process that represents the basic hierarchy building blocks wherein the DSPs are embedded, and Interface to represent the DSP information flows that link different Processes to a hierarchy. The efficacy of the ontology is demonstrated using a portal frame design example. Advantages of this ontology are that it is decomposable and flexible enough to accommodate the dynamic evolution of a process along the design timeline.
## Frame of Reference
Engineering systems are, by definition, made up of inter-related subsystems [1]. The analysis and synthesis of these systems are generally too complex to be handled as a single problem, and this necessitates the design of the overall system first by decomposing the system into subsystems [2]. Existing hierarchical decomposition approaches are anchored in two perspectives: physical structure (or function)-based perspective and the decision-based perspective, which defines the design process. The former is focused on decomposing the design problem according to the system's interconnected structures or coupled functions (which may need various types of disciplinary expertise to design them). A well-known representative of this perspective is Sobieszczanski-Sobieski's [35] multilevel decomposition and coordination method for handling large, multidisciplinary problems while trying to reduce computational costs; this approach lays the foundation for many multidisciplinary design optimization approaches (for example, Refs. [69]). In this perspective, design is viewed as a decision-making process in which the design of complex engineering systems involves making a series of decisions that are typically hierarchical. Based on this decision-based perspective, many hierarchical decomposition methods have been proposed and tested in limited situations. Examples include the hierarchical selection decision support problem (DSP) for conceptual design [10], and the coupled compromised decision support problem proposed to solve hierarchical problems in structural design [1,2,1113].
Another factor that is also critical is the computational framework that provides the infrastructure for the execution of the decomposed problem modules (represented as simulation or analysis codes) and the overall optimization model. Salas and Townsend [14] from the National Aeronautics and Space Administration recognize the need for such a framework and identified requirements for its development. Key requirements include architecture design, problem formulation, problem execution, and the ability to handle a large amount of information. Many commercially available and research-based software frameworks such as modelcenter [15], isight [16], and modefrontier [17] have been developed based on these requirements in order to enable the coupling of disciplinary analysis codes, geometric design models, and optimization routines. Hiriyannaiah and Mocko [18] suggest that although significant effort has been invested to address issues associated with integration, execution, and communication, more needs to be done to provide sufficient information/knowledge management capabilities: A structured representation and information model is needed to capture information so that designers can easily store, organize, and retrieve previous knowledge for reuse. Knowledge, which is sometimes referred to as truths, justified beliefs, perspectives, judgment, etc. [19], is of critical value for enhancing humans' ability to solve practical problems. Computational frameworks should have the capability of retaining the knowledge (including many know-whys and know-hows such as design rules and decisions) generated during the process of design and organizing it in a manner that facilitates the re-application of this knowledge in solving new problems.
To accommodate the need for knowledge management in complex system design, we are designing a knowledge-based platform for decision support in the design of engineering systems (PDSIDES). The core of PDSIDES is an ontology (specification of a conceptualization [20]) that is used to organize design knowledge from a decision-based perspective. In our earlier work, we have developed two ontologies for formally representing knowledge related to two primary types of decisions, namely, selection [21] and compromise [22]. These two ontologies enable PDSIDES to capture and document individual decision-related knowledge at a computational level. As mentioned earlier, complex system design usually involves a series of decisions coupled in a hierarchy. Therefore, there is a need for a scheme to represent the workflow among the individual decisions. In this paper, we address this need by extending the existing selection and compromise decision ontology to capture the workflow in design decision hierarchies, which enables PDSIDES to deal with more complex design processes. The rest of the paper is organized as follows: In Sec. 2, we discuss the background of this work, which includes both hierarchical decision-making in design and ontology-based knowledge modeling. In Sec. 3, we identify the requirements for a computational model that represents the DSP hierarchy. In Sec. 4, we describe the ontology to meet the requirements. In Sec. 5, we illustrate the efficacy of the created ontology. In Sec. 6, we offer some remarks and suggest future research opportunities.
## Background
### Decision Support Problems and Decision Hierarchies.
Engineering design is increasingly recognized as a decision-making process [2326]. Decision-based design (DBD) is one of the several constructs for designing complex systems [23]. We note that DBD has been implemented in several ways. Our implementation is anchored in the DSP technique. The key to the DSP technique is the concept that there are two types of decisions (namely, selection and compromise) and that a complex design can be represented by modeling a network of compromise and selection decisions [2628]. The selection DSP (sDSP) [29] involves making a choice among a number of alternatives taking into account a number of measures of merits or attributes while the compromise DSP (cDSP) [3032] involves the improvement of a feasible alternative through modification by making trade-offs among multiple design objectives. The design of complex systems may require the formulation and resolution of a series of coupled decisions in which case, the hierarchical DSP construct offers a feasible alternative.
According to Smith [33], there are two classes of decision hierarchies, namely, multilevel hierarchy and coupled hierarchy, as shown in Fig. 1. The first class is modeled as separate decisions that are made at different levels of the hierarchy at which they occur. The decisions are separable and can, therefore, be made concurrently or sequentially. The interaction among these decisions in this approach is relatively low because the dependency among decisions is weak with either no information flow or a one-way flow. The second class, the coupled hierarchy, is modeled as coupled DSPs. In this class, the entire hierarchy is formulated as a single DSP. Hence, the interactions among the decisions are strong, creating a tight bond among subsystems. Coupled hierarchical decisions require simultaneous solution of one or more DSPs as a single coupled problem. In this paper, we focus on coupled hierarchical decisions.
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A coupled hierarchical decision involves the solution of any combination of selection and/or compromise DSPs simultaneously by reformulating all the DSPs into a single cDSP. Publications on coupled DSPs include coupled compromise–compromise DSPs [1,2], coupled selection–compromise DSPs [34,35] and coupled selection–selection DSPs [10]. The mathematical formulation of the coupled selection–compromise DSP is shown in Table 1; the other two types of coupling can be formulated similarly. In Table 1, X and Y denote the variables of two decisions. System constraints and goals (e.g., MF(Y)X, g(X, Y), and A(X, Y)) that involve both X and Y represent the lateral interactions among member decisions in a hierarchy. Vertical interactions (not shown in the figure) can also be modeled with system goals as X = X(Y) where X represents the parent decision variables and Y the subdecision variables; for details, see Refs. [2,11], and [12]. Coupled DSPs are generally multi-objective, nonlinear, mixed discrete–continuous problems. A tailored computational environment known as DSIDES [36] incorporating the adaptive linear programming algorithm [31] has been created to solve such problems.
Table 1
Mathematical formulation of the coupled selection–compromise DSP
Given Selection M alternatives N attributes $Rij$ normalized rating of alternative i with respect to attribute j $Ij$ relative importance of attribute j $MFi$ merit function for the ith alternative where $MFi=∑j=1NIjRij$ (i = 1,… , M) Compromise n number of system variables p number of equality system constraints q number of inequality system constraints p + q total number of system constraints m number of system goals (selection plus compromise) $m1$ number of coupled compromise system goals $gi(X,Y)$ compromise constraint function $Ai(X,Y)$ compromise achievement function $Gi$ target of compromise goal Find Selection variables $Xi,ei−,ei+$ (i = 1,… , M) Compromise variables $Yi$ (i = 1,… , n) $di−,di+$ (i = 1,… , m) Satisfy Selection system constraint: $∑i=1NXi=1$ Selection system goals: $MFi⋅Xi+ei−−ei+=1$ (i = 1,…, M) Compromise system constraints: $gi(X,Y)=0$ (i = 1,…, p) $gi(X,Y)≥0$ (i = p + 1,…, p + q) Compromise system goals: Coupled goals $Ai(X,Y)+di−−di+=Gi$ (i = 1,…, $m1$) Independent goals $Ai(Y)+di−−di+=Gi$ (i = $m1$ + 1,…,) Bounds $0≤Xi≤1$ (i = 1,…, M) $Yjmin≤Yj≤Yjmax$ (j = 1,…, n) (i = 1,…, M) (i = 1,…, m) Minimize The deviation function (preemptive form):
Given Selection M alternatives N attributes $Rij$ normalized rating of alternative i with respect to attribute j $Ij$ relative importance of attribute j $MFi$ merit function for the ith alternative where $MFi=∑j=1NIjRij$ (i = 1,… , M) Compromise n number of system variables p number of equality system constraints q number of inequality system constraints p + q total number of system constraints m number of system goals (selection plus compromise) $m1$ number of coupled compromise system goals $gi(X,Y)$ compromise constraint function $Ai(X,Y)$ compromise achievement function $Gi$ target of compromise goal Find Selection variables $Xi,ei−,ei+$ (i = 1,… , M) Compromise variables $Yi$ (i = 1,… , n) $di−,di+$ (i = 1,… , m) Satisfy Selection system constraint: $∑i=1NXi=1$ Selection system goals: $MFi⋅Xi+ei−−ei+=1$ (i = 1,…, M) Compromise system constraints: $gi(X,Y)=0$ (i = 1,…, p) $gi(X,Y)≥0$ (i = p + 1,…, p + q) Compromise system goals: Coupled goals $Ai(X,Y)+di−−di+=Gi$ (i = 1,…, $m1$) Independent goals $Ai(Y)+di−−di+=Gi$ (i = $m1$ + 1,…,) Bounds $0≤Xi≤1$ (i = 1,…, M) $Yjmin≤Yj≤Yjmax$ (j = 1,…, n) (i = 1,…, M) (i = 1,…, m) Minimize The deviation function (preemptive form):
### Ontology-Based Knowledge Modeling.
Ontology is defined as a specification of a conceptualization [20]; it provides a common vocabulary for the representation of domain-specific knowledge. As mentioned earlier, the two key elements of an ontology are concepts and relationships. Concepts are physical or logical objects in a domain [37] and are represented by Classes. Relationships describe the network among the concepts and are represented by Properties or Slots of the Classes. Based on the abstract structure of an ontology, infinite numbers of Instances can be populated using specific information.
Although the research on ontology has its roots in computer science, ontologies have often been used for knowledge modeling in the engineering domain. Applications are typically categorized as follows: knowledge sharing [38,39], knowledge retrieval [40,41], and knowledge documentation [42,43]. In knowledge sharing, the functionality of ontologies in representing a common understanding of a domain is emphasized and is used for sharing and exchanging information among applications. In knowledge retrieval, the emphasis is an ontology's usefulness in representing complex relations among concepts, which are essential for building a reference network for retrieval. In knowledge documentation, ontology is used for populating Instances, which represent domain-specific knowledge. In this paper, we focus on the application of ontology in knowledge documentation. It is recognized that several ontologies have been developed for documenting knowledge in the domain of engineering design, such as a design optimization ontology [43], a design for manufacturing ontology [44], a design for additive manufacturing ontology [45], a functional design ontology [46], and a product family design ontology [41]. In engineering design, decisions determine how design resources are allocated and how design objectives are achieved, which are critical, and this knowledge should also be documented for reuse. Along this line, Rockwell et al. [42] created ontologies for capturing and communicating design decisions. However, the focus of their work is on archiving design rationale to help designers understand previous designs. The knowledge so captured cannot be rapidly reused to create new designs since the computational procedures for decision-making are not captured. Furthermore, the ontologies proposed in Ref. [42] are limited to representing individual (or single) decisions while they fail to represent decision workflows that typically involve multiple decisions interconnected in the design processes.
In our earlier work [21,22], we have developed ontologies for representing individual decisions, including selection and compromise decisions, in which the decision-making procedures are computationally modeled with templates and are ready to execute to generate new results in the platform PDSIDES. In this paper, we take a step further to extend the existing selection and compromise ontologies to representing hierarchically decision workflows. Before the development of the ontology, requirements are identified and discussed in Sec. 3.
## Requirements for the Computational Decision Support Problem Hierarchy Model
A general four-level representation for a hierarchical system is proposed by Sobieszczanski-Sobieski [5], as shown in Fig. 2. The nature of the hierarchy is a two-dimensional, horizontal and vertical network structure. The nodes represent the integral parts of the system that can be parent systems or subsystems depending on their relative positions in the hierarchy. The links represent dependencies between two nodes, which can be vertical and lateral, depending on whether they cross levels. In the vertical direction, the root node, e.g., node “1.1” in Fig. 2, is progressively decomposed into multiple levels, and each level may have multiple nodes. Every two levels in the neighborhood are interconnected by a vertical dependency between the nodes corresponding to a “parent–child” relationship of the two levels; thus, the network is vertically integrated. In the horizontal direction, nodes of the same (e.g., nodes “3.1” and “3.2”) or different (e.g., nodes “3.2” and “3.3”) parent(s) at the same level may be connected by lateral dependencies and thus the network is laterally integrated.
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From the decision-based design perspective, the design of such hierarchical systems is supported by the DSP networks in which each node embodies a DSP that supports a design decision about the related system elements and each link embodies the interaction between two nodes. In the context of the platform PDSIDES, the DSP network is modeled computationally to facilitate the formulation of a decision hierarchy in keeping with the solution procedures for hierarchical system design, and the needs of designers. PDSIDES and the DSP technique are then used to solve problems. The requirements for the computational DSP hierarchy model are:
• Decomposability: The multilevel or multiscale characteristic of hierarchical systems requires a model to be decomposable to capture the complexity of the physical system and be solvable using available computational power. Further, a model should support dynamic decomposition as the design evolves.
• Flexibility: The evolution of knowledge about a system being designed at different design stages requires the model to be flexible enough to support reconfiguration. For example, in early design stages, the system is not decomposed into very detailed levels and much of the dependency is ignored because of the lack of knowledge. As the time moves forward, knowledge increases and the original model must be reconfigured to incorporate this knowledge.
• Reusability: The pursuit of design efficiency requires that the model is reused. In practice, many product or system designs fall into the category of variant or adaptive designs where a large part of the information of the underlying design model can be reused and there is no need to recreate it from scratch. For example, in an adaptive design scenario, where only one part (e.g., node “4.5” in Fig. 2) is changed while the others remain the same, the original design model could be easily reused, with small modifications, to support the design.
• Executability: The model needs to be executable on a computer. Encoding the linguistic and/or mathematical formulation of the problem as a computational model is a feasible way to obtain executability. However, the disadvantage is that the codes are incomprehensible to designers who are not programmers. There is a need for a computational model to be executable and at the same time, easily understandable to designers. To do this, the rich semantics in the decision hierarchy must be captured in the model.
• Visualization: A comprehensive understanding of a design problem requires that the model is visualized. A hierarchical system is a system of parent systems and interacting subsystems. Therefore, the computational model should support the visualization of the hierarchical structure so that designers can have an intuitive understanding of the problem they are dealing with and can edit the model effectively.
• Consistency maintainability: The mathematical rigor of the decision formulation requires the computational model to be consistent with what is defined. For example, the initial value of a variable in a cDSP should lie between a lower bound and an upper bound, the sum of the weights of the goals on the same level should equal 1, etc. These are the rules for the mathematical model to maintain consistency; however, these rules may not remain valid during model reconfiguration, resulting in model inconsistency. Therefore, the computational model should support consistency checking and maintain consistency.
## Ontology Development for Decision Support Problem Hierarchy
In this section, we present an ontology as a computational model that meets the requirements identified in Sec. 3. In our earlier work [21,22], we illustrate the use of an ontology for facilitating the reuse, execution, and consistency maintenance of DSP templates. In particular, the reuse of the DSP templates is facilitated with a common vocabulary embodied in the ontology for generating different specific Instances. This can easily be adapted for new scenarios by modification of the associated Slots. The execution of the DSP templates is facilitated with web ontology language—a standard and computer-interpretable modeling language underlying the ontology—which supports the parsing by other applications through, for example, Java function calls. The consistency of the DSP templates is maintained by the incorporation of reasoning mechanisms for consistency checking within the ontology. In this paper, the ontology is extended to model a hierarchical DSP network with emphasis placed on meeting the decomposability, flexibility, and visualization requirements listed in Sec. 3.
In a computational environment, the decomposability of a model means that it can be further divided or broken down into smaller components. In a DSP hierarchy, this involves vertical integration of dependent sub-DSPs. This calls for the identification of a concept from a higher level than the DSPs alone. This is a general representation of a hierarchy that not only incorporates the DSPs but also captures the associated links so that newly derived sub-DSPs can be linked to an existing hierarchy when decomposition or reassembly is needed. Here, the concept is named Process, which is formally defined as an ontology Class in Sec. 4.1. The flexibility of the DSP hierarchy is embodied in reconfiguration, which principally includes: (i) dynamic decomposition of decisions and (ii) incorporation of evolving dependencies (or links) among decisions. This requires the links in the hierarchy to be separately modeled so that they can be dynamically added, edited, and removed when reconfiguration is needed. In this paper, the links are modeled as Interfaces as discussed in Sec. 4.2. Visualization is implemented with an ontology editing tool, which is introduced in Sec. 4.3. It is recognized that the Process Specification Language ontology [47] is a generic model for flow management; the ontology created in this paper can link to it by mapping ClassesProcess, and Interface to the concept of Activity in Process Specification Language.
### Definition of Class Process.
The ClassProcess is a general representation of the building blocks in a hierarchy, which incorporates one DSP and its associated dependency. For example, in Fig. 2, the collection of node 2.2 and its associated links on the top, bottom, left-hand side, and right-hand side can be called an Instance of a Process. It is called Process because it represents an information processing unit based on a decision-making mechanism, sDSP or cDSP, and the associated interactions, (information flows) with other units. The concept of the ClassProcess is shown in Fig. 3. It represents a standard, scalable hierarchy building block of which the solid box (or shell) stands for the information processing unit with a DSP (the dashed box) plugged in, and lines represent the associated vertical and lateral dependencies. A hierarchy is built by assembling a series of different Processes. In the ontological context, the Slots of ClassProcess are defined in Table 2.
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Table 2
Slots of Class Process
Slot nameDefinitionType
nameName of the Process.String
descriptionDescription of the Process.String
IsRootIndicator of whether the Process is the root (i.e., no parent) of the hierarchy. Two allowable values—“yes (1)” or “no (0).”Boolean
IsLeafIndicator of whether the Process is the leaf (i.e., no subsystems) of the hierarchy. Two allowable values—yes (1) or no (0).Boolean
DecisionDecision corresponding to the Process. Two allowable Classes, namely, cDSP template (see Ref. [22]) and sDSP template (see Ref. [21]). One Processes Instance can have only one corresponding decision, selection, or compromise.Instance
DecisionTypeIndicator of the type of the corresponding decision. Two allowable values—“compromise” or “selection.”Symbol
LateralDependencyLateral dependencies of the Process. It is associated with ClassInterface introduced in Sec. 4.2. One ProcessesInstance can have multiple lateral dependencies.Instance
VerticalDependency_ParentParent dependency of the Process. It is associated with ClassInterface introduced in Sec. 4.2. One ProcessesInstance can have only one parent dependency.Instance
VerticalDependency_SubsystemSubsystem dependency of the Process. It is associated with ClassInterface introduced in Sec. 4.2. One ProcessesInstance can have multiple subsystem dependencies.Instance
Slot nameDefinitionType
nameName of the Process.String
descriptionDescription of the Process.String
IsRootIndicator of whether the Process is the root (i.e., no parent) of the hierarchy. Two allowable values—“yes (1)” or “no (0).”Boolean
IsLeafIndicator of whether the Process is the leaf (i.e., no subsystems) of the hierarchy. Two allowable values—yes (1) or no (0).Boolean
DecisionDecision corresponding to the Process. Two allowable Classes, namely, cDSP template (see Ref. [22]) and sDSP template (see Ref. [21]). One Processes Instance can have only one corresponding decision, selection, or compromise.Instance
DecisionTypeIndicator of the type of the corresponding decision. Two allowable values—“compromise” or “selection.”Symbol
LateralDependencyLateral dependencies of the Process. It is associated with ClassInterface introduced in Sec. 4.2. One ProcessesInstance can have multiple lateral dependencies.Instance
VerticalDependency_ParentParent dependency of the Process. It is associated with ClassInterface introduced in Sec. 4.2. One ProcessesInstance can have only one parent dependency.Instance
VerticalDependency_SubsystemSubsystem dependency of the Process. It is associated with ClassInterface introduced in Sec. 4.2. One ProcessesInstance can have multiple subsystem dependencies.Instance
### Definition of Class Interface.
The ClassInterface is a representation of the vertical or lateral dependency between two different Processes. For example, in Fig. 2, the link between nodes 2.2 and 3.3 is an instance of an Interface. It is called an Interface because it captures the communication between the two processes. Interfaces are critical in building a DSP hierarchy because they constitute the medium that connects the individual building blocks, namely, Processes, to an integrated whole. The concept of Interface is shown in Fig. 4. As shown in the figure, an Interface consists of two elements: (1) the references (or indices) of two linked Processes, which are represented by dashed boxes, and (2) the information flow, which is represented by the solid line between the two Processes. Based on the strength, the information flow is of two types, namely, (a) weak flow and (b) strong flow. The weak flow is a one-way flow (“1 to 2” or “2 to 1”), which means that one Process has parameters that must be input from the counterpart. A strong flow is a two-way flow, which means that both of the Processes have parameters that need to be input from the counterpart.
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Based on the direction of flow, information is categorized into two types, namely, (i) lateral flow and (ii) vertical flow. Lateral flow links Processes at the same level in a hierarchy. Vertical flow links Processes at different, neighboring levels in a hierarchy. Since the Processes are embedded with DSPs, the information flow must be modeled to be consistent with the coupled DSP construct introduced in Sec. 2.1. Using the interface shown in Fig. 4 as an example, the characteristics of the information flow in the context of coupled DSP construct are modeled as follows. (Here, we assume $X1$ represents the variable vector of the DSP in Process 1 and $X2$ the DSP in Process 2.)
• Vertical: Vertical information flow is usually two-way and occurs between cDSPs in a hierarchy. Assuming Process 1 is the parent system and Process 2 the subsystem, the coupling is modeled as a system goal of a coupled cDSP, formulated as $A(X1,X2)=(X1/(X1(X2)))−1$ where $X1(X2)$ is a function that maps $X2$ to $X1$.
• Lateral: In this type, the coupling is divided into the following six subtypes.
1. sDSP↔sDSP: Two-way flow between two sDSPs. This can be embodied in three ways: dependent attributes, dependent alternatives, and dependent alternatives as well as attributes; see Ref. [48] for more detail. Mathematically, the interdependency is modeled as a system goal of a coupled cDSP.
2. sDSP→cDSP: One-way flow from sDSP to cDSP. Assuming that Process 1 is modeled with an sDSP and Process 2 with a cDSP, then the information flow is modeled by the (Boolean) variable vector $X1$ of the sDSP that constitutes the parameters of the cDSP's system constraints as represented as , and/or system goals represented as .
3. cDSP→sDSP: One-way flow from cDSP to sDSP. Assuming that Process 1 is modeled with a cDSP and Process 2 with an sDSP, then the information flow is modeled by the variable vector $X1$ of the cDSP that constitutes the parameters of the sDSP's merit function represented as $MF(X1)⋅X2$, which is a system goal in a coupled cDSP. In this case, $X2$ is Boolean.
4. sDSP↔cDSP: Two-way flow between an sDSP and a cDSP. This is modeled by the combination of the preceding two types.
5. cDSP→cDSP: One-way flow between two cDSPs. This is modeled by the variable vector of the antecedent cDSP that constitutes the system constraints and goals of the subsequent cDSP.
6. cDSP↔cDSP: Two-way flow between two cDSPs. This is modeled by the variable vectors of both of the two cDSPs that constitute the system constraints and goals of the counterparts.
As mentioned in Sec. 2.1, the resolution of a coupled DSP involves reformulating all the DSPs into a single cDSP. In this paper, since the Interface constitutes the dependent part (coupled system constraints and goals) of the DSPs, it should then be integrated with the independent parts (which are embedded in the Processes) to compose a single cDSP. In our ontology, in order to represent these dependencies in a hierarchy, we identify and define the Slots of ClassInterface in Table 3.
Table 3
Slots of Class Interface
Slot nameDefinitionType
nameName of the Interface.String
descriptionDescription of the Interface.String
interface typeIndicator of the type of the Interface. Two allowable values—“vertical” or “lateral.”Symbol
strengthIndicator of the strength of the coupling. Two allowable values—“weak” or “strong.”Symbol
originalProcessThe original Process that is linked by the Interface. It is associated with ClassProcess.Instance
counterpartProcessThe counterpart Process that is linked by the Interface. It is associated with ClassProcess.Symbol
originalCounterpartFlowInformation flow from the original Process to its counterpart. It is associated with ClassFunction (see Ref. [22] for detailed definition). One InterfaceInstance can have multiple flows from 1 to 2.Instance
counterpartOriginalFlowInformation flow from the counterpart Process to the original Process. The definition is similar as Slot originalCounterpartFlow.Instance
Slot nameDefinitionType
nameName of the Interface.String
descriptionDescription of the Interface.String
interface typeIndicator of the type of the Interface. Two allowable values—“vertical” or “lateral.”Symbol
strengthIndicator of the strength of the coupling. Two allowable values—“weak” or “strong.”Symbol
originalProcessThe original Process that is linked by the Interface. It is associated with ClassProcess.Instance
counterpartProcessThe counterpart Process that is linked by the Interface. It is associated with ClassProcess.Symbol
originalCounterpartFlowInformation flow from the original Process to its counterpart. It is associated with ClassFunction (see Ref. [22] for detailed definition). One InterfaceInstance can have multiple flows from 1 to 2.Instance
counterpartOriginalFlowInformation flow from the counterpart Process to the original Process. The definition is similar as Slot originalCounterpartFlow.Instance
Note: The information flow through the Interface is strong when both of Slots originalCounterpartFlow and counterpartOriginalFlow are populated with specific values, and is weak when either of the Slots is null.
### Building DSP Hierarchies Using Processes and Interfaces.
The definition of ClassProcess, ClassInterface, as well as the associated Slots, is facilitated by using the protégé 3.5 tool [49], which provides an environment for creating and editing ontologies as well as populating Instances based on ontologies. Three typical user-interfaces of protÉgÉ in terms of the DSP hierarchy ontology are shown in Fig. 5. In Fig. 5, the panel marked with “①” is the class browser where the ontology Classes including Classes of the cDSP ontology (represented as “CO,” see Ref. [22]), Classes of the sDSP ontology (represented by “SO,” see Ref. [21]), and the two Classes (highlighted in the box) identified in this paper for building DSP hierarchies are listed. The window marked “②” is the Instance editor for the ProcessClasses, where the Slots are created using the definitions in Sec. 4.1 and are populated with specific problem information. The window marked with “③” is the Instance editor for the InterfaceInstances, where the Slots are created using the definitions in Sec. 4.2 and are populated using specific problem information.
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In our ontology, building DSP hierarchies is facilitated with the protégé graph widget [50], a graphical tool for visual editing the Instance and relationships among Instances. The protégé graph widget is especially suitable for building the DSP hierarchy as the hierarchy is a network of ProcessInstances and InterfaceInstance. A screenshot of the widget customized for building the hierarchy is shown in Figs. 68 in Sec. 5.
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## A Test Example for the Decision Support Problem Hierarchy Ontology
In this section, a portal frame design problem is used as an example to illustrate the use of the ontology presented in Sec. 4. The example is an extension of the problem considered by Sobieszczanski-Sobieski [5] to demonstrate the decomposition method in solving hierarchical design problems. The correctness of the ontology is tested along with a refinement process of the decision model (DSP) for the portal frame design.
### Creation of a Baseline Model With Limited Information.
A portal frame represents a simple hierarchical system, as shown in Fig. 6. The integrated frame represents the parent system while the three I-beams are the subsystems. The design objective is to minimize the overall mass of the frame. The frame is subject to two kinds of constraints: external constraints and internal constraints. The former includes static loads $P$ and $M$ while the latter includes normal stress, bending stress, shear stress, and buckling in each member. Design variables are categorized into parent system variables and subsystem variables. Parent system design variables $A$ and $I$ stand for each member's cross-sectional area and moment of inertia, respectively. Subsystem variables are the dimensions, namely, $b1$, $b2$, $h$, $t1$, $t2$, $t3$ of each subsystem. $Vi$ denotes the vertical interactions between the parent system and each of its subsystems. $Lij$ represents the lateral interactions between subsystems.
Fig. 8
Fig. 8
Close modal
In the early stages of design, designers usually face the challenge of having limited information for modeling the problem. In the design of the portal frame, we assume that information related to the vertical interactions, $Vi$, and the lateral interactions, $Lij$, are unknown to designers. Designers are required to create a baseline model to design the portal frame with limited information. The baseline model is a single cDSP in which all the constraints are in terms of the design variables at the subsystem level. In the context of Fig. 6, this involves only subsystem variables $b$, $t$, and $h$ for each beam, and does not make use of the parent variables $A$ and $I$. In the ontological context, a ProcessInstance, the “Portal Frame Design,” as shown in Fig. 9, is created using the available information. Since the vertical and lateral interactions are not modeled, no InterfaceInstance is created. Specification of the Process Instance is presented in window ① of Fig. 9 and the information of required for the cDSP template in the Instance is presented in window ②. It is not difficult to imagine that because of the lack of consideration of interactions among the three members (subsystems) of the portal frame, there would be some differences among the resulting dimensions of the members. Using the baseline model as the starting point, it is assumed that designers will gradually refine the model by considering lateral and vertical interactions. The performance of the ontology in terms of facilitating the creation of a DSP hierarchy to support this refinement is tested in the Secs. 5.2 and 5.3. In Sec. 5.2, designers refine the baseline model by first decomposing the model into sub-DSPs then linking the sub-DSPs using lateral interaction information. In Sec. 5.3, designers further refine the model with the lateral interactions by the incorporation of vertical interactions with a parent DSP that supports the decision-making related to the parent system. The model presented in Sec. 5.3 is a comprehensive model with both lateral and vertical interactions. In Sec. 5.4, we showcase how designers can retrieve decisions from the knowledge base using the ontology.
Fig. 9
Fig. 9
Close modal
### Refinement of the Model With Lateral Interactions.
Based on the baseline model formulated in Sec. 5.1, designers are able to refine the model with known lateral interactions between member 1 and member 2, and the lateral interactions between member 2 and member 3, namely, $L12$ and $L23$ in Fig. 6. This means that the baseline model created in Sec. 5.1 must be decomposed into three DSPs, and the associated dependency needs to be modeled. The model can be decomposed by separation of the cDSP formulation in Fig. 9 into three independent cDSPs based on the dimensions of the three subsystems. The (lateral) dependencies necessitate the inclusion of constraints that connect the subsystem variables to their counterparts in the other subsystems. This connection is modeled mathematically with system goals in a cDSP.
In our ontology, system goals are captured and used to populate two InterfaceInstances, namely, “L12” and “L23,” as shown in the canvas of Fig. 10. The three separated cDSPs are used to instantiate three ProcessInstances, “M1,” “M2,” and “M3,” which are linked by L12 and L23. Both L12 and L23 represent strong coupling with two-way information flow. The information flows are embodied in the Slots, “originalCounterpartFlow” and “counterpartOriginalFlow,” as shown in the window under the canvas in Fig. 10 in which the specifics of L12 are shown. The deviation variables associated with the interaction system goals, together with the deviation associated with the mass goal, are formulated in a preemptive form in the overall deviation function, where the interaction system goals have a higher priority than the mass goal. The overall deviation function is captured in one of the three ProcessInstances (in this case is M1). At a computational level, all the information of the Process and InterfaceInstances is integrated as a coupled cDSP and sent to DSIDES for computation. The results are not shown in this paper but because of the inclusion of the lateral interactions, which match the tree subsystems, the difference between the subsystem dimensions will be reduced to a tolerable extent.
Fig. 10
Fig. 10
Close modal
### A Comprehensive Model With Lateral Interactions and Vertical Interactions.
In this section, the vertical interactions between the parent system and subsystems, namely, $V1$, $V2$, and $V3$, in Fig. 6 are known and designers are assumed to further refine the model from Sec. 5.2, using this information. Since the vertical interactions are known, a new cDSP corresponding to the parent system is created so that the existing subsystem DSPs are connected to it through the vertical interactions. The new parent cDSP involves parent system level design variables $Ai$ and $Ii$ ($i$ = 1, 2, 3), constraints, and goals. The vertical interactions necessitate the inclusion of constraints that match the parent system design variables ($A$ and $I$) and the subsystem variables ($b$, $t$, $h$). Similarly to the lateral interactions, this “matching” is modeled mathematically by system goals in a cDSP.
In our ontology, vertical interactions are captured and used to populated InterfaceInstances, namely, “V1,” “V2,” and “V3,” as shown in Fig. 7. The information of the cDSP corresponding to the parent system is used to instantiate a ProcessInstance, “Parent,” which is linked to the three existing subsystem level ProcessInstances, M1, M2 and M3 by V1, V2, and V3, respectively. Also, the lateral interactions, V1, V2, and V3, represent strong coupling with two-way information flows. The specification of V1 is shown in the window under the canvas of Fig. 7. Details of the information flows are in Slots originalCounterpartFlow and counterpartOriginalFlow. Deviation variables related to the interaction system goals are incorporated in the deviation function in a preemptive form, where the vertical interaction goals are of the highest priority, the lateral interaction goals, the second priority, and the mass goal, the third. Specification of the deviation function is assigned to the “root” node of the hierarchy, namely, Parent in Fig. 7, as the overall control for the model. At a computational level, all the information of the hierarchy is integrated as a coupled cDSP and computed with DSIDES. The computed results are not shown in this paper. But it is not difficult to expect that the subsystem dimensions will match each other very well, as in Sec. 5.2 because lateral interactions are included in the model. Meanwhile, the values of the parent system level variables will also match those of the subsystem level variables (e.g., cross-sectional value of $A$ matches the value of $A(b,t,h)$) because of the consideration of vertical interactions.
### Semantic Knowledge Retrieval Based on the DSP Hierarchy Ontology.
In the portal frame design example, it can be seen that the ontology created in this paper has the capability to represent the decision hierarchy and capture decision workflows as the design process evolves, as demonstrated by the following: (1) The ontology is decomposable to allow a baseline decision model to be broken down into subdecisions as the design process evolves; (2) the ontology is flexible enough to incorporate new knowledge, e.g., the subsystem-level design variables, vertical and lateral dependencies in the example, when knowledge increases; and (3) the ontology provides a visualized environment for designers so that they can edit and understand the hierarchical decision workflow during the design process.
By ClassesProcess, Interfaces that link the Classes associated with the selection and compromise DSPs as developed in our previous work, we have created a comprehensive terminology box based on which a decision-based design knowledge base is constructed. The terminology box is product- and process-independent; thus, it can easily extend to different products and design processes, and populate many Instances of various degrees of complexity. The Instances comprise the assertion box of the ontology from which designers can query the decision-related knowledge using semantic languages such as Semantic Query-enhanced Web Rule Language (SQWRL). This is another key advantage of the ontology in addition to knowledge documentation. For example, if a designer wants to know all the subdecisions in the parent decision in the portal frame design example, the SQWRL query statement shown in Table 4 may be used.
Table 4
SQWRL query statement for subdecisions of process Parent
Process(?p)∧name(?p, "parent")∧verticalDependency_Subsystem(?p,?s)∧counterpartProcess(?s,?sp)∧decision(?sp,?d)∧name(?d,?subDecision) → sqwrl:select(?p,?subDecision)
Process(?p)∧name(?p, "parent")∧verticalDependency_Subsystem(?p,?s)∧counterpartProcess(?s,?sp)∧decision(?sp,?d)∧name(?d,?subDecision) → sqwrl:select(?p,?subDecision)
The knowledge base will respond with “Decision of Member 1,” “Decision of Member 2,” and “Decision of Member 3” in a table format, as shown in Fig. 8. The retrieval feature enhances the usefulness of the DSP hierarchy ontology to decision support platforms, such as PDSIDES.
## Closure
Hierarchies that arise naturally in the design of complex engineered systems involve the design of parent systems and dependent subsystems. In decision-based design, this results in decision network hierarchies. To support designers making such networked, hierarchical decisions, capturing and representing the associated knowledge is of critical importance. In this paper, we propose an ontology for capturing and representing the hierarchical decision knowledge for the DSP construct. Foundational to the ontology introduced in this paper is the coupled, hierarchical DSP construct, which provides a means for designers to model the dependent selection and/or compromise decisions. In order to develop the ontology, first, we identify the requirements for a model that computationally represents the DSP hierarchy. Second, based on these requirements, we formally define two key classes, namely, Process which represents the basic hierarchy building blocks where selection or compromise DSPs are modeled and Interface, which represents the DSP information flows that link different Processes to a hierarchy. Finally, the efficacy of the ontology is demonstrated using a portal frame design example.
One premise of this paper is that designers have full control of the information flows among the decision hierarchy. In the world of practice, when the subsystem design tasks are assigned to different design teams or geographically distributed companies who have their own goals to satisfy, full information control may not be possible for a single team or company. This control may be distributed to all the stakeholders. Future research opportunities lie in the study of how decisions are made regarding distributed information control, and how the ontology created in this paper is adjustable in order to capture the associated decision knowledge.
## Acknowledgment
Farrokh Mistree gratefully acknowledges financial support from the L.A. Comp Chair at the University of Oklahoma.
## Funding Data
• China Scholarship Council (Grant No. 201406030014).
• Ministry of Science and Technology of the People's Republic of China (Grant No. 2015BAF18B01).
• National Natural Science Foundation of China (Grant Nos. 51375049 and 515050).
• National Science Foundation (Grant No. CMMI-1440457).
• University of Oklahoma (the John and Mary Moore Chair).
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,” Stanford University, Stanford, CA, accessed Aug. 11, 2016, http://protegewiki.stanford.edu/wiki/Graph_Widget_Tutorial_OWL | 2022-07-07 15:48:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 86, "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": 0.3867250382900238, "perplexity": 2110.1739830361726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656104495692.77/warc/CC-MAIN-20220707154329-20220707184329-00783.warc.gz"} |
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http://math.stackexchange.com/questions/209759/finding-value-of-c?answertab=active | # Finding value of $c$ [duplicate]
Possible Duplicate:
Solution of 3 equations in 3 unknowns
I need help finding the value of $c$ which makes it possible to solve: $$\begin{array}{lcl} u + v + 2w & = & 2 \\ 2u + 3v - w & = & 5 \\ 3u + 4v + w & = & c \end{array}$$
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This is an exact duplicate, but I happened to see this one first, hence the answer below. – Brian M. Scott Oct 9 '12 at 9:45
## marked as duplicate by Brian M. Scott, draks ..., Matt Pressland, robjohn♦Oct 9 '12 at 10:35
$$\left[\begin{array}{rrr|r} 1&1&2&2\\ 2&3&-1&5\\ 3&4&1&c \end{array}\right]\;.$$
$$\left[\begin{array}{rrr|c} 1&1&2&2\\ 0&1&-5&1\\ 0&1&-5&c-6 \end{array}\right]\;.\tag{1}$$
Now you can either stop and think about the equations corresponding to the bottom two rows of $(1)$ (what does $c$ have to be in order for them to be consistent?), or finish the row-reduction and then think about what $c$ has to be to avoid having an inconsistent system. | 2013-06-20 11:33:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5823445320129395, "perplexity": 304.055475809432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368711515185/warc/CC-MAIN-20130516133835-00023-ip-10-60-113-184.ec2.internal.warc.gz"} |
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## Size/move loop and delay in DefWindowProc
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### #1Evil Steve Members
Posted 14 January 2009 - 07:59 AM
I'm trying to start a timer when the user starts to drag the window, so I can render a frame on the WM_TIMER message, to prevent the stall that happens when the user drags the window. I added some logging and found that DefWindowProc() for WM_SYSCOMMAND (With a wParam of 0xf012, which doesn't seem to be a recognised SC_* value) and WM_NCLBUTTONDOWN takes a considerable time to return (Over 500ms). With some more logging, it seems that WM_NCLBUTTONDOWN comes first, and calling DefWindowProc causes a WM_SYSCOMMAND message to be sent (So it's recursing into the window proc, hence the delay on both messages). The call stack at the time if I break with F12 isn't very exciting:
ntdll.dll!_KiFastSystemCallRet@0()
user32.dll!_NtUserMessageCall@28() + 0xc bytes
user32.dll!_RealDefWindowProcWorker@20() - 0x79d6 bytes
user32.dll!_RealDefWindowProcW@16() + 0x27 bytes
uxtheme.dll!DoMsgDefault() + 0x29 bytes
uxtheme.dll!OnDwpSysCommand() + 0x29 bytes
uxtheme.dll!_ThemeDefWindowProc() + 0x61a8 bytes
uxtheme.dll!_ThemeDefWindowProcW@16() + 0x18 bytes
user32.dll!_DefWindowProcW@16() + 0x815 bytes
> Tutorial02.exe!D3DWindow::WndProc(unsigned int uMsg=274, unsigned int wParam=61458, long lParam=5636612) Line 227 + 0x1b bytes C++
Tutorial02.exe!D3DWindow::StaticWndProc(HWND__ * hWnd=0x000a06fa, unsigned int uMsg=274, unsigned int wParam=61458, long lParam=5636612) Line 169 + 0x1a bytes C++
user32.dll!_InternalCallWinProc@20() + 0x23 bytes
user32.dll!_UserCallWinProcCheckWow@32() - 0xddcf bytes
user32.dll!_DispatchClientMessage@20() + 0x4b bytes
user32.dll!___fnDWORD@4() + 0x24 bytes
ntdll.dll!_KiUserCallbackDispatcher@12() + 0x2e bytes
user32.dll!_NtUserMessageCall@28() + 0xc bytes
user32.dll!_RealDefWindowProcWorker@20() - 0x79d6 bytes
user32.dll!_RealDefWindowProcW@16() + 0x27 bytes
uxtheme.dll!DoMsgDefault() + 0x29 bytes
uxtheme.dll!OnDwpNcLButtonDown() + 0x32 bytes
uxtheme.dll!_ThemeDefWindowProc() + 0x61a8 bytes
uxtheme.dll!_ThemeDefWindowProcW@16() + 0x18 bytes
user32.dll!_DefWindowProcW@16() + 0x815 bytes
Tutorial02.exe!D3DWindow::WndProc(unsigned int uMsg=161, unsigned int wParam=2, long lParam=5636612) Line 227 + 0x1b bytes C++
Tutorial02.exe!D3DWindow::StaticWndProc(HWND__ * hWnd=0x000a06fa, unsigned int uMsg=161, unsigned int wParam=2, long lParam=5636612) Line 169 + 0x1a bytes C++
user32.dll!_InternalCallWinProc@20() + 0x23 bytes
user32.dll!_UserCallWinProcCheckWow@32() + 0xb3 bytes
user32.dll!_DispatchMessageWorker@8() + 0xe6 bytes
user32.dll!_DispatchMessageW@4() + 0xf bytes
Tutorial02.exe!WinMain(HINSTANCE__ * hInstance=0x009f0000, HINSTANCE__ * __formal=0x00000000, HINSTANCE__ * __formal=0x00000000, HINSTANCE__ * __formal=0x00000000) Line 22 + 0xf bytes C++
Tutorial02.exe!__tmainCRTStartup() Line 578 + 0x35 bytes C
Tutorial02.exe!WinMainCRTStartup() Line 403 C
Obviously returning 0 from WM_NCLBUTTONDOWN doesn't cause WM_SYSCOMMAND to be sent, and no delay, but it also prevents the window from being moved. So, does anyone know what's going on here and if there's any way of stopping it? Cheers, Steve
Steve Macpherson
Senior Systems Programmer
Rockstar North
### #2SiCrane Moderators
Posted 14 January 2009 - 08:28 AM
How complex are your renders and what kind of delay are you putting on your timer? If you have a relatively simple render and a relatively low update frequency, then you might want to just start the timer up when the window is created and let it run.
### #3Evil Steve Members
Posted 14 January 2009 - 08:33 AM
Quote:
Original post by SiCraneHow complex are your renders and what kind of delay are you putting on your timer? If you have a relatively simple render and a relatively low update frequency, then you might want to just start the timer up when the window is created and let it run.
Currently the render is very basic (A single triangle), and the timer is set to 0ms, so just "as fast as possible".
However, this is for a tutorial / article on D3D rendering, and I'd like to do things as "correctly" as possible, even if it means leaving the 500ms lag.
Also, this is D3D (In case that's not obvious from the call stack function names [smile]), so running rendering in another thread is out of the question.
EDIT: And I'm curious about what exactly is going on under the hood here.
[Edited by - Evil Steve on January 14, 2009 3:33:36 PM]
### #4kittycat768 Members
Posted 14 January 2009 - 01:16 PM
What kind of stall are you referring to? To my knowledge the window will NEVER update while it's being dragged. Your program will keep executing though. Have you considered throwing your timer code into the main loop instead of relying on your window procedure? I recommend QueryPerformanceCounter() if you do. 64-bit calculations... Nummy. Also, what is your definition of "correctly?"
### #5lordikon Members
Posted 14 January 2009 - 04:11 PM
There are some other things that cause about a 500ms stall, such as grabbing a window title bar. In fact, now that I think about it, grabbing a window title bar causes WM_SYSCOMMAND with a wparam value of 0xf012 which is in fact (SC_MOVE + HTCAPTION).
EDIT: You may find Spy++ will give you useful information about what messages are occuring. To me it is worlds easier than a call stack for windows message monitoring.
### #6Codeka Members
Posted 14 January 2009 - 04:43 PM
This is a guess, but I'm thinking DefWindowProc is waiting to see whether you're about to double-click or not. Try setting your double-click delay really slow or really fast to see if that affects things.
If that's the case, then there's probably not a lot you can do about it (but maybe make a note in your tutorial or something [smile]).
### #7Evil Steve Members
Posted 14 January 2009 - 10:06 PM
Thanks for the replies.
Quote:
Original post by kittycat768What kind of stall are you referring to? To my knowledge the window will NEVER update while it's being dragged. Your program will keep executing though. Have you considered throwing your timer code into the main loop instead of relying on your window procedure? I recommend QueryPerformanceCounter() if you do. 64-bit calculations... Nummy. Also, what is your definition of "correctly?"
For the stall, DefWindowProc doesn't return for over 500ms. It's as if it's an extremely expensive function call. Putting a timer in the main loop wouldn't help, and for now GetTickCount() is accurate enough (You don't need to render at full speed when the user is moving the window).
My definition of "correctly" is to render in the main loop, not from a timer handler - the timer handler isn't called all that frequently as far as I'm aware.
Quote:
Original post by lordikonThere are some other things that cause about a 500ms stall, such as grabbing a window title bar. In fact, now that I think about it, grabbing a window title bar causes WM_SYSCOMMAND with a wparam value of 0xf012 which is in fact (SC_MOVE + HTCAPTION).EDIT: You may find Spy++ will give you useful information about what messages are occuring. To me it is worlds easier than a call stack for windows message monitoring.
Ah, I didn't notice that the wParam can include HTCAPTION, thanks. It's the WM_SYSCOMMAND handling in DefWindowProc that causes the 500ms stall - that's what I'm trying to fix or work around.
Quote:
Original post by CodekaThis is a guess, but I'm thinking DefWindowProc is waiting to see whether you're about to double-click or not. Try setting your double-click delay really slow or really fast to see if that affects things.If that's the case, then there's probably not a lot you can do about it (but maybe make a note in your tutorial or something [smile]).
I don't think that's it, I always have my double click delay set to as fast as possible, and that should be less than 500ms. I'll give that a try when I get a chance though.
Cheers,
Steve
### #8SiCrane Moderators
Posted 15 January 2009 - 01:21 AM
Quote:
Original post by Evil SteveMy definition of "correctly" is to render in the main loop, not from a timer handler - the timer handler isn't called all that frequently as far as I'm aware.
You could try starting a timer at window creation and calling InvalidateRect() in the timer handler, which will cause WM_PAINT messages to be generated even when your window is being dragged.
### #9Endurion Members
Posted 15 January 2009 - 01:38 AM
Could this be because you've got a Direct3d device open inside the window?
Does that stall also happen without D3D? It might be because you're actually losing the device.
If it is because of D3D, could it be the debug runtimes?
### #10Evil Steve Members
Posted 15 January 2009 - 01:50 AM
Quote:
Original post by CodekaThis is a guess, but I'm thinking DefWindowProc is waiting to see whether you're about to double-click or not. Try setting your double-click delay really slow or really fast to see if that affects things.
Setting my double click speed to extremely slow still takes the same time for DefWindowProc to return, so that doesn't seem to be related.
Quote:
Original post by SiCraneYou could try starting a timer at window creation and calling InvalidateRect() in the timer handler, which will cause WM_PAINT messages to be generated even when your window is being dragged.
Hmm, it's possible - but still seems a bit ugly to me. This is beginning to look like the only option though...
Quote:
Original post by EndurionCould this be because you've got a Direct3d device open inside the window?
Disabling D3D completely (Commenting out any reference to any D3D interface) still yields the same problem.
### #11Erik Rufelt Members
Posted 15 January 2009 - 02:08 AM
This problem happens even if you create a timer at window-creation. I've always wondered why..
It doesn't happen if you start dragging the window right away, moving the mouse already when you click. It also pauses if you hold down the button on the minimize button for example. I think all such things are handled in some loop that doesn't return, except for that once the window starts moving it returns to allow for interactive moving.
Don't know why it waits 500 ms before returning unless the mouse moves.. the reply about double-clicking sounds plausible to me..
It also freezes if you hold down the right mouse-button in the title bar.
### #12lordikon Members
Posted 15 January 2009 - 02:20 AM
Quote:
Original post by Evil SteveHmm, it's possible - but still seems a bit ugly to me. This is beginning to look like the only option though...
I create the timer the first time WM_MOVING is reached, and remove the timer when WM_EXITSIZEMOVE is reached. This doesn't help with the delay though. There is about a 500ms delay between the time the user clicks the title bar and I receive SC_MOVE + HTCAPTION. During this time the window stops updating.
### #13Evil Steve Members
Posted 15 January 2009 - 03:05 AM
Quote:
Original post by lordikonI create the timer the first time WM_MOVING is reached, and remove the timer when WM_EXITSIZEMOVE is reached. This doesn't help with the delay though. There is about a 500ms delay between the time the user clicks the title bar and I receive SC_MOVE + HTCAPTION. During this time the window stops updating.
Yep, it's that 500ms delay that I'm trying to avoid.
### #14lordikon Members
Posted 15 January 2009 - 03:26 AM
I'm also fairly interested in this solution. I wish I could help, but after some research I found no answers either.
### #15lordikon Members
Posted 15 January 2009 - 03:33 AM
So, it appears you've been having this problem for awhile. :D
http://www.gamedev.net/community/forums/topic.asp?topic_id=440341
### #16Evil Steve Members
Posted 15 January 2009 - 03:42 AM
Quote:
Original post by lordikonSo, it appears you've been having this problem for awhile. :Dhttp://www.gamedev.net/community/forums/topic.asp?topic_id=440341
Yeah, I found that post while Googling, but decided not to necro the thread [smile]
### #17Amr0 Members
Posted 15 January 2009 - 03:47 AM
Here is a quick workaround. Note that this has the little quirk of setting the cursor position to the middle of the window's caption. Also, if the app has other windows and he starts dragging them, the problem remains. Also, the delay will remain if the window is sizable and the user [EDIT: referred to in the previous sentence as "he"] clicks on the borders to resize it. Also, this will effectively disable double-clicking the title bar to maximize the window. Plus possibly a few more things I haven't thought about yet.
case WM_NCLBUTTONDOWN: if( SendMessage( hWnd, WM_NCHITTEST, wParam, lParam ) == HTCAPTION ) { SetTimer( hWnd, 1000, 0, NULL ); SendMessage( hWnd, WM_SYSCOMMAND, SC_MOVE, 0 ); return 0; } break; case WM_TIMER: Render(); break; case WM_EXITSIZEMOVE: KillTimer( hWnd, 1000 ); break;
But personally I would recommend against messing with this because, well, you're just nitpicking one of Windows' insignificant quirks. That's not patriotic!
[Edited by - hikikomori-san on January 15, 2009 10:47:56 AM]
### #18lordikon Members
Posted 15 January 2009 - 04:31 AM
They mention the same delay problem.
### #19Evil Steve Members
Posted 15 January 2009 - 05:09 AM
Quote:
Original post by lordikonJust found this: http://developer.popcap.com/forums/showthread.php?p=19636They mention the same delay problem.
Yes, I've been through Google. I've found plenty of reports of the problems, but no solutions.
Quote:
Original post by hikikomori-sanHere is a quick workaround. Note that this has the little quirk of setting the cursor position to the middle of the window's caption. Also, if the app has other windows and he starts dragging them, the problem remains. Also, the delay will remain if the window is sizable and the user [EDIT: referred to in the previous sentence as "he"] clicks on the borders to resize it. Also, this will effectively disable double-clicking the title bar to maximize the window. Plus possibly a few more things I haven't thought about yet.
Hmm, I'll give that a go later tonight, thanks.
### #20Evil Steve Members
Posted 15 January 2009 - 08:04 AM
Quote:
Original post by hikikomori-sanHere is a quick workaround. Note that this has the little quirk of setting the cursor position to the middle of the window's caption. Also, if the app has other windows and he starts dragging them, the problem remains. Also, the delay will remain if the window is sizable and the user [EDIT: referred to in the previous sentence as "he"] clicks on the borders to resize it. Also, this will effectively disable double-clicking the title bar to maximize the window. Plus possibly a few more things I haven't thought about yet.
This seems to work pretty well, with a bit of an ajustment:
LRESULT D3DWindow::WndProc(UINT uMsg, WPARAM wParam, LPARAM lParam){static bool sbMoving = false;static POINT sptOffset; switch(uMsg) { case WM_NCLBUTTONDOWN: if(SendMessage(m_hWnd, WM_NCHITTEST, wParam, lParam) == HTCAPTION) { POINT ptCursor; RECT rcWnd; GetWindowRect(m_hWnd, &rcWnd); GetCursorPos(&ptCursor); sptOffset.x = ptCursor.x - rcWnd.left; sptOffset.y = ptCursor.y - rcWnd.top; SetCapture(m_hWnd); sbMoving = true; return 0; } break; case WM_NCLBUTTONUP: case WM_LBUTTONUP: sbMoving = false; ReleaseCapture(); break; case WM_MOUSEMOVE: case WM_NCMOUSEMOVE: if(sbMoving) { POINT ptCursor; RECT rcWnd; GetWindowRect(m_hWnd, &rcWnd); GetCursorPos(&ptCursor); SetWindowPos(m_hWnd, NULL, rcWnd.left+(ptCursor.x-(rcWnd.left+sptOffset.x)), rcWnd.top+(ptCursor.y-(rcWnd.top+sptOffset.y)), 0, 0, SWP_NOZORDER | SWP_NOSIZE); } break; } return DefWindowProc(m_hWnd, uMsg, wParam, lParam);}
I dread to think what horrible bugs this will cause though, so I think I'm just going to use the method with the 500ms delay for this, and add the above method as an alternative.
Another issue is that the user can still access the system menu of the window (Which will cause a stall), and select "Move" from there - which will stall. The second issue can be fixed with the timer as before, but probably not the stall when selecting the system menu.
Thanks for all the replies,
Steve
Old topic!
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https://blog.hotwhopper.com/2013/11/matt-ridley-in-australia-tells-more-of.html | .
## Matt Ridley in Australia tells more of his fibs, but gives Andrew Bolt little joy...
Sou | 8:03 PM
UPDATE: Anthony Watts has just posted the transcript at WUWT. Only three comments so far - see below. I'll update if I see any more choice comments in the denier playground. (Archive here, updated here.)
(Sou: 10:45 am Tues 19 Nov 13)
As I said in my last article, UK failed banker, politician and GWPF adviser, Matt Ridley is in Australia. Seems he was bought by the IPA to quell the fears of fake skeptics. He did alright with his IPA meeting reported yesterday by keeping all the bad news from them. He didn't do quite as well today.
This morning he appeared on Andrew Bolt's little Sunday television program. Andrew Bolt is an Australian right wing blogger with the Herald-Sun, a widely circulated Melbourne daily newspaper which is a subsidiary of News Corp Australia (Murdoch). A while ago Andrew also managed to snag his own half hour show on Sunday Television - Channel 10. It's not much of a program. I don't know who wastes their time watching it except for a few right wingers keen to have their ideas reflected by him. (It was rated way, way down at number 320 here in May this year.) I think I've only watched one other episode.
Andrew Bolt prides himself on his ignorance, claiming that it makes him "objective". In reality it only makes him look pretty darned stupid most of the time.
I transcribed the video from this morning and have inserted comments of my own. (PS I just visited Andrew Bolt's blog about today's television program and was a bit surprised that of 115 comments, I'd say there were only about four or five about climate. Most were about other segments of the show.)
### On typhoons...
Andrew Bolt: Typhoon Haiyan hit the Philippines last week and killed perhaps 4,000 people. The Greens couldn't wait to exploit it like they exploited last month's fires and even accused Tony Abbott.
Sou: There's a cut to three separate very short snippets of Adam Bandt talking to the press:
Adam Bandt: He can be expected to be referred to as Typhoon Tony.
...Many people are saying that this is the worst typhoon that they've ever seen.
...This is what we're in store for unless we get global warming under control.
Sou: Then back to Andrew Bolt introducing Matt Ridley - my hyperlink:
Bolt: Matt Ridley is a member of Britain's House of Lords and a science writer who's latest best seller is "The Rational Optimist". He's here on a speaking tour for the IPA. Matt Ridley, thank you for joining me.
Ridley: Thank you for having me on the show.
Andrew Bolt: The typhoon in the Philippines er er what do you make of the attempts to make that evidence of the great global warming catastrophe awaiting us.
Sou: As you've seen from what Adam Bandt had to say, Andrew Bolt couldn't find anyone saying that Typhoon Haiyan was caused by global warming. The best he could come up with was Bandt saying that "this is what is in store for us unless we get global warming under control". Bandt's statement accurately reflects the science. But Matt Ridley thinks even that is "ridiculous".
Matt Ridley: Well this is ridiculous. I mean storms and weather events happen. They've always happened. There have been much stronger typhoons in the past. This isn't the strongest one that's ever recorded or anything like that. They're gunna happen whatever and to blame this on climate change is a bit like Shamanism. It's witchdoctery. It's going back 10,000 years to try and blame every weather event on mankind.
And we don't have to just know this from basic data. If you look at the Intergovernmental Panel on Climate Change, they say there's been no trend in increasing frequency of typhoons or cyclones or hurricanes. In fact this year's been an unusually quiet one globally and even in that part of the Pacific it's been quiet. So the idea that you can stop typhoons happening by cutting carbon dioxide emissions is just absurd. We've got to tackle typhoons as an issue whatever happens to the climate.
Sou: That's what's called a strawman fallacy. Reducing emissions won't stop typhoons from happening. The aim is to reduce emissions so that global warming doesn't cause as much harm as otherwise. Some of that harm may well be an increase in the ferocity of super storms and an increase in the damage they cause. For example, even mild storms will be more damaging because of the impact of rising seas on storm surges.
### Matt Ridley thinks rising seas are the main worry with global warming
Andrew Bolt: What do we have to worry about if global warming continues. I know there's been a pause in atmospheric temperature rises for fifteen years but should it continue what have we got to fear?
Matt Ridley: I personally think that we are seeing benefits from climate change. Sorry, that's not my personal view that's what the data says. [Sou: that's a misrepresentation. There are many more negatives than positives.]
We're seeing benefits from climate change slightly. Greener vegetation in the world, slightly fewer winter deaths. [Sou: many more summer deaths, more catastrophic fires, hotter droughts, heavier rains etc etc.] Things like that. Longer growing seasons. [Sou: longer bushfire seasons.] And that's likely to continue for another six or seven decades. After that, if the projections of climate change are right and on the whole they've been too warm for the last 30 years so they may not be right. [Sou: ha ha - Matt's nose grew about four inches right there!] But if they're right we will then start to see net harm. And the one harm that really would hurt civilisation would be rapidly rising sea levels.
Fortunately the Intergovernmental Panel on Climate Change says that sea levels are not rising, are not gonna rise that fast in this century. [Sou: oops, Matt's nose grew another four inches!] Not much faster than they did in the last century.
Greenland's losing ice at the rate of two billion tonnes a year which sounds a lot but it's actually half a per cent per century. So the collapse of ice sheets, that sort of thing has now largely been ruled out by the IPCC as a risk. [Sou: huh? Collapse of ice sheets is expected in the medium to long term.]
But we are, you know, we do have to get our act together to be ready to deal with some disasters if they happen towards the end of this century or the beginning of the next.
Sou: First up, Matt tells a big fat lie about the IPCC projections for sea level rise this century. Except for the lowest possible under the virtually impossible to achieve RCP2.6 emissions scenario - every projection is for a substantially higher rate of rise than over the twentieth century. Here is Figure TS.22 from page TS-122 of AR5 WG1 IPCC report (click to enlarge):
Sou: In my last article I wondered what Matt thought would happen after his 70 years "benefit" runs out. This time he's given us a clue as to what he thinks, even going so far as to say "we do have to get our act together". Andrew Bolt picks up on that:
Andrew Bolt: Well, when you say get our act together to be ready. Um, we're.. obviously the world is spending trillions of dollars on various ways to so-called stop global warming. Does ..is that a sensible use of our resources?
Matt Ridley: No. I think rolling out immature and fourteenth century resources like wind power all around the world which are extremely expensive, don't cut carbon emissions very much and er on the whole keep people unable to afford the measures to adapt to climate by being so expensive ah is not the answer.
Sou: I wonder has Matt Ridley been ordered by the GWPF (or maybe by the raving ratbag blogger James Delingpole) to run with that particular fib? In fact the cost of electricity from wind power is very competitive in many parts of the world, as Oklahomans' recently discovered. Even in Australia, wind power provides a lot more electricity than many people realise.
### Matt Ridley thinks Japan will go back to nuclear energy?
Matt Ridley: Um Japan interestingly has just said that it's not gonna try and keep emissions as low as it was hoping by 2020. Instead it's gonna put a lot of money into research into new energy technologies. And that's the answer. If we can get cheap fusion energy or cheap thorium power or even cheap ordinary nuclear power and some of the solar power developed ah by the end of the century we probably won't even need fossil fuels if we can give them up long before they run out um that's a much better approach than trying to roll out immature energy technologies now cos we've tried that and it's just not working. We're trying it all over the world it's it's disastrously bad for people's living standards.
Sou: Oh boy! Matt Ridley has rocks in his head if he thinks that the Japanese are going to be rolling out more nuclear power plants. Sheesh. The aftermath of the Fukishima catastrophe is why it shut down all its nuclear power plants and is switching to gas, and the main reason why it's wanting to change it's emission reduction target. As for Matt thinking that the Asian smog is preferable to clean energy for "people's living standards"...
### No joy for Andrew Bolt's "reason"
Andrew Bolt: So when Tony Abbott gets elected on a platform of scrapping the carbon tax, is that seen as the Green's would suggest as a world-wide embarrassment or is it seen ..er ..as something perhaps ..er ..or the return of reason?
Matt Ridley: Well, I think that until now it's been assumed that you had to pay lip service to dangerous climate change. I mean, most of us I believe that that human beings do affect the climate and probably have caused some of the warming in the past. That's not at issue. What's at issue is a forecast of dangerous warming which is only going to come true if certain positive feedback amplifiers happen. And if that's likely to be the case, it's always been assumed that you had to show real alarm about this in order to get elected in a western democracy. I think Tony Abbott has shown that's not the case and a lot of elected politicians around the world will have noticed that and will have noticed that not only was the carbon tax ah something he was ah determined to repeal, but that it was front and centre in the election campaign. So you can't say it was just a peripheral issue. So for example the Canadians have ah commented on that. And I think western European politicians will notice that and will say actually you can take a relatively rational, relatively sober approach to climate change and be elected despite what the extreme greens will throw at you.
Sou: Poor old Andrew Bolt can't win a trick. Matt Ridley fudges and fumbles a bit, but he doesn't come out and agree with Andrew that scrapping the carbon tax is a "return of reason". Matt avoids answering that part of Andrew's question altogether. Instead he grudgingly says that humans do affect the climate.
Matt doesn't answer the implied question of whether Tony Abbott's policy is the best one or not. He doesn't come right out and contradict Andrew. But instead of answering the question he sidesteps like the politician he is - responding in terms of the impact on politics.
Given Matt Ridley's ideological viewpoint, it's possible that he would favour a carbon price, which is a market-based system and a charge on polluters, over the socialist approach proposed by Tony Abbott - making taxpayers cough up all the money needed to pay for emissions reductions. And hiding the real cost in consolidated revenue. (Tony Abbott doesn't favour transparent government).
Andrew Bolt: And er is there any other government then that er will be the next to follow us do you think?
Matt Ridley: I'm not the one to predict ah political trends. Ah I don't think it's going to happen in a hurry in Europe. Ah um ah sorry in Britain. Um but ah there is huge disquiet in the UK about energy prices. And they're about to go up even more because of green levies and that I think is going to make politicians rethink this agenda.
### Carbon is being priced around the world
Andrew Bolt asks longingly if other government's will "follow" Australia. Follow them in what? Getting rid of a carbon pricing scheme? Not very likely. What places have one and what places are planning one? Below is a list compiled by SBS (Australia) a couple of weeks ago:
## CARBON TAXES AROUND THE WORLD
### CHINA (state-based action)
The Chinese Government plans to develop emissions trading schemes in seven key cities and provinces from 2013. These schemes will cover around 250 million people. The Chinese Government aims to work towards a nation-wide approach after 2015.
### UNITED STATES (state-based action)
There is no nationwide carbon tax levelled in the USA, although a few states have introduced the tax. The United States Administration has not been able to secure support for legislation to set either a price or a limit on greenhouse gas emissions. However, emissions trading has operated in the power sector in nine states since 2009. California's emissions trading scheme will start in January 2013.
Canada does not have a federal carbon tax, but two Canadian provinces have existing carbon taxes (Quebec and British Columbia). Alberta implemented emissions trading in 2006 and Quebec's scheme will start in 2013. A further two provinces, British Columbia and Ontario, are considering emissions trading schemes.The Canadian Federal Government has no immediate plans to implement national emissions trading.
In July 2010, India introduced a nationwide carbon tax of 50 rupees per tonne (less than $A1) of coal both produced and imported to India. ### NEW ZEALAND The New Zealand Government set up an emissions trading scheme in 2008. The scheme covered forestry initially, and was then expanded in 2010 to cover stationary energy, transport, liquid fossil fuels and industrial processes. ### SOUTH KOREA The Republic of Korea passed legislation in May 2012 for an emissions trading scheme to start from 1 January 2015. The emissions trading scheme will cover facilities producing more than 25,000 tonnes of greenhouse gas emissions – expected to be around 450 of the country's largest emitters. ### JAPAN In April 2012, Japan legislated for a carbon tax of approximately ¥289 per tonne ($A3.30) by increasing existing taxes on fossil fuels (coal and LPG/LNG) with effect from 1 October 2012. Half the revenue will
fund low-emissions technologies. Japan has emissions trading schemes operating in the Tokyo and Saitama regions, covering 20 million people.
### EUROPE (national-based action)
The European Union emissions trading scheme began in 2005 and now covers the 27 countries of the European Union, and three non-European Union members: Iceland, Liechtenstein, and Norway. Their current target is a 21 per cent cut of 2005 emissions by 2025 (Australia's is a 5% cut of 2000 emissions by 2020).
A carbon tax was proposed by the European Commission in 2010, but a carbon tax has not been agreed upon by the 27 member states. The current proposal by the European Commission would charge firms between 4 and 30 euros per metric tonne of CO2.
Several European countries have enacted a carbon tax. They include: Denmark, Finland, Ireland, the Netherlands, Norway, Slovenia, Sweden, Switzerland, and the UK.
### FINLAND
Finland introduced the world's first carbon tax in 1990, initially with exemptions for specific sectors. Manly changes were later introduced, such as a border tax on imported electricity. Natural gas has a reduced tax rate, while peat was exempted between 2005 and 2010. In 2010, Finland's price on carbon was €20 per tonne of CO2.
### THE NETHERLANDS
The Netherlands introduced a carbon tax in 1990, which was then replaced by a tax on fuels. In 2007, it introduced a carbon-based tax on packaging, to encourage recycling.
### SWEDEN
In 1991, Sweden enacted a tax on the use of coal, oil, natural gas, petrol and aviation fuel used in domestic travel. The tax was 0.25 SEK/kg ($US100 per tonne of C02) and was later raised to$US150. With Sweden raising prices on fossil fuels since enacting the carbon tax, it cut its carbon pollution by 9 per cent between 1990 and 2006.
### NORWAY
In 1991, Norway introduced a tax on carbon. However its carbon emissions increased by 43 per cent per capita between 1991 and 2008.
### DENMARK
Since 2002, Denmark has had a carbon tax of 100 DKK per metric ton of CO2, equivalent to approximately 13 Euros or 18 US dollars. Denmark's carbon tax applies to all energy users, but industrial companies are taxed differently depending on the process the energy is used for, and whether or not the company has entered into a voluntary agreement to apply energy efficiency measures.
### SWITZERLAND
A carbon incentive tax was introduced in Switzerland in 2008. It includes all fossil fuels, unless they are used for energy. Swiss companies can be exempt from the tax if they participate in the country's emissions trading system. The tax amounts to CHF 36 per metric tonne CO2.
### UK
In 1993, the UK government introduced a tax on retail petroleum products, to reduce emissions in the transport sector. The UK's Climate Change Levy was introduced in 2001. The United Kingdom participates in the European Union emissions trading scheme and is covered by European Union policies and measures. The United Kingdom has put in place regulations requiring all new homes to have zero emissions for heating, hot water, cooling and lighting from 2016.
### IRELAND
A tax on oil and gas came into effect in 2010. It was estimated to add around €43 to filling a 1000 litre oil tank and €41 to the average annual gas bill.
### COSTA RICA
In 1997, Costa Rica enacted a tax on carbon pollution, set at 3.5 per cent of the market value of fossil fuels. The revenue raised from this goes into a national forest fund which pays indigenous communities for protecting the forests around them.
### BRAZIL
The state of Rio de Janeiro is exploring options to implement a state-wide cap and trade system.
South Africa introduced a carbon tax on new vehicle sales in September 2010. South Africa is planning to introduce a carbon tax from 2013, starting at R120 ($A15) per tonne for emissions above a threshold. Each company will have 60 per cent of its emissions tax exempt, with higher exemption thresholds for cement, iron, steel, aluminium, ceramics and fugitive emissions as well as trade exposed industries. Agriculture, forestry, land use and waste will not be taxed. ### From the WUWT comments Added 10:55 am Tues 19 Nov 13 Anthony Watts was a couple of days late to Andrew Bolt's party. Here are a couple of WUWT comments on the Matt Ridley monologue (archived here, updated here). Kev-in-Uk says Matt Ridley caved in too much to the "pig-troughing climate scientists": November 18, 2013 at 3:42 pm IMHO, Dr Ridley was a bit too soft and concilliatory – a bit too lukewarm? It’s ok to try and get your point across by appearing calm and reasoned but the fact of the matter is that all the money spent on carbon reductions schemes have been a massive waste of time and effort as well as cash. I for one do not believe in the ‘AGW is significant’ meme – but the pragmatic view (and incorporating the oft warmist favourite – the Precautionary Principle) the best way to stop carbon emission is to invest in renewables and nuclear ‘properly’. Put it another way, a hundred billion bucks into development of non-carbon energy would have gone an awful long way into helping – instead of producing a sh$tload of useless models, adjusted data and feeding many thousands of pig-troughing ‘climate scientists’……..
Robin says "it's all a communist plot":
November 18, 2013 at 3:43 pm
Those of us who believe based on the evidence that Sustainability is merely an update of the old Marxian need for a crisis to justify the desired structural and institutional changes will keep watching and listening for the next calamity. I spent part of the weekend reading the beginnings of the ecological Marxism theories in the 70s (as its creators called it) and their justifications that more than an economic crisis would be needed.
IPCC is holding true to the social theories regardless of the facts. Matt is a rational optimist because he believes in innovation. We need to get back to societies that foster genuine innovation of the type he describes in his book instead of sociological innovations in how we are to organize ourselves in the future. Most of us can organize ourselves far better than any bureaucrat or theorist or politician | 2022-07-02 04:43:44 | {"extraction_info": {"found_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": 0.22881074249744415, "perplexity": 4136.1620521003015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656103984681.57/warc/CC-MAIN-20220702040603-20220702070603-00167.warc.gz"} |
http://svn.haxx.se/users/archive-2007-12/0188.shtml | # Re: How should I properly escape special characters in commit message?
From: Ryan Schmidt <subversion-2007b_at_ryandesign.com>
Date: 2007-12-10 18:36:25 CET
On Dec 10, 2007, at 11:06, Geoffrey Hoffman wrote:
> On Dec 9, 2007, at 9:03 AM, melmack melmack wrote:
>
>> I've one problem with escaping special characters (quotes and
>> backslashes) in commit message. To explain it better I'll show 2
>> examples (both in Win32 cmd.exe shell):
>>
>> (1) svn commit -m "foo\\\"bar"
>>
>> Comment is saved to svn as foo\"bar
>>
>> (2) svn commit -m "foo\\\\bar"
>>
>> Comment is saved to svn as foo\\\\bar
>>
>> What is the reason of the fact that in example (1) svn interpreter
>> collapses \\ to \ and \" to " (as in regular expressions syntax)
>> and in example (2) nothing is collapsed? I thought that example(2)
>> should have saved \\ as comment. Why was the result different? So
>> how should I escape the comments properly? Is there an algorithm
>> to do this?
>> My temporary solution to solve this problem is putting comment to
>> file and call svn vommit with -F option but this is inefficient
>> way, so any help will be appreciated...
>
> -m "This message 'should work' when you commit"
> -m 'This message "should work" when you commit'
> -m "This message should work when you commit"
>
> (This is on Mac OS X Terminal)
None of that demonstrates escaping; you're just choosing characters
which don't have to be escaped.
I imagine escaping is particular to whatever shell you're using. On
Mac OS X, you're probably using bash, or perhaps tcsh. On Windows,
the OP is using I have no idea what, and I have no idea how it
behaves with regard to escaping. But there should be some Windows
shell / DOS prompt reference that could be looked up. In any case,
Subversion isn't doing any of it itself. It's all in the shell.
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To unsubscribe, e-mail: [email protected] | 2013-05-21 13:35:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8813542723655701, "perplexity": 9951.335216158004}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368700074077/warc/CC-MAIN-20130516102754-00005-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://budshome.com/yew/concepts/function-components/custom-hooks.html | ## Defining custom Hooks
Component’s stateful logic can be extracted into usable function by creating custom Hooks.
Consider that we have a component which subscribes to an agent and displays the messages sent to it.
#![allow(unused)]
fn main() {
#[function_component(ShowMessages)]
pub fn show_messages() -> Html {
let (state, set_state) = use_state(|| vec![]);
{
let mut state = Rc::clone(&state);
use_effect(move || {
let producer = EventBus::bridge(Callback::from(move |msg| {
let mut messages = (*state).clone();
messages.push(msg);
set_state(messages)
}));
|| drop(producer)
});
}
let output = state.iter().map(|it| html! { <p>{ it }</p> });
html! { <div>{ for output }</div> }
}
}
There’s one problem with this code: the logic can’t be reused by another component. If we build another component which keeps track of the messages, instead of copying the code we can move the logic into a custom hook.
We’ll start by creating a new function called use_subscribe. The use_ prefix conventionally denotes that a function is a hook. This function will take no arguments and return Rc<RefCell<Vec<String>>>.
#![allow(unused)]
fn main() {
fn use_subscribe() -> Rc<RefCell<Vec<String>>> {
todo!()
}
}
This is a simple hook which can be created by combining other hooks. For this example, we’ll two pre-defined hooks. We’ll use use_state hook to store the Vec for messages, so they persist between component re-renders. We’ll also use use_effect to subscribe to the EventBus Agent so the subscription can be tied to component’s lifecycle.
#![allow(unused)]
fn main() {
fn use_subscribe() -> Rc<Vec<String>> {
let (state, set_state) = use_state(Vec::new);
use_effect(move || {
let producer = EventBus::bridge(Callback::from(move |msg| {
let mut messages = (*state).clone();
messages.push(msg);
set_state(messages)
}));
|| drop(producer)
});
state
}
}
Although this approach works in almost all cases, it can’t be used to write primitive hooks like the pre-defined hooks we’ve been using already
### Writing primitive hooks
use_hook function is used to write such hooks. View the docs on docs.rs for the documentation and hooks directory to see implementations of pre-defined hooks. | 2021-11-28 21:08:11 | {"extraction_info": {"found_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": 0.309501051902771, "perplexity": 10867.155538559962}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964358591.95/warc/CC-MAIN-20211128194436-20211128224436-00460.warc.gz"} |
https://chemistry.stackexchange.com/questions/51736/why-did-the-alpha-particles-in-rutherfords-experiment-not-collide-with-the-elec?noredirect=1 | # Why did the alpha-particles in Rutherford's experiment not collide with the electrons? [duplicate]
In Rutherford's experiment to show the existence of nucleus in an atom, the alpha-particles were exposed on the surface of certain metal i.e. gold. He observed that more than 99% of these particles were able to go straight and pass through the gold atoms. Thus he was able to conclude that atoms are mainly empty space.
But why did these particles not collide with the electrons present in the gold atoms? I think that the reason behind it is that the speed of the electrons travelling in orbits did not cause any obstacle in the travelling path of the alpha particles.
• The body of your question has nothing in particular to do with neutrons - perhaps you mean nucleus? (Yes, very similar names). As for scattering off of electrons, the alpha particles most certainly were interacting with the electrons, and as a result lost energy as they traversed the material. This so-called electronic stopping is very small compared with a nucleus-nucleus collision event primarily because the mass of the electron is so much smaller than even a proton - you just can't have much energy transfer in such a collision. May 25 '16 at 17:51
Protons and neutrons have a mass approximately 1836 times greater than an electron, ignoring relativistic effects and nuclear binding energies. There are 4 of these baryons in an $\mathrm{\alpha}$-particle. There is no way a puny electron could do much to deflect an $\mathrm{\alpha}$-particle that is almost 7500 times more massive. This is like if a 150-lb human tried to block a 747 (the 747-100's maximum takeoff weight is only about 6400 times greater than the person's). | 2021-12-05 17:01:47 | {"extraction_info": {"found_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": 0.5523024201393127, "perplexity": 480.60494483269946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964363215.8/warc/CC-MAIN-20211205160950-20211205190950-00028.warc.gz"} |
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