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http://tex.stackexchange.com/questions/35356/strange-looking-table | # Strange looking table
I am trying to print this rather odd looking table, below
I know there is an package called exam, though I sort of want to do this "my own style"
I did try to make the table below, but I was not even close in doing so.
Most importantly is that the table looks good, not that it is an exact replica of the table. Could anyone be so kind as to help me? I have started learning latex recently, and stumbling down the path of learning =)
EDIT: One could remove the vertical lines furthest to the left and right aswell, then the table would be more inline witht the standard of tables.
Edit2:
\documentclass[10pt,a4paper]{article}
\usepackage{mathtools}
\usepackage{booktabs}
\usepackage{multirow}
\usepackage{multicol}
\begin{document}
\noindent \begin{tabular*}{\textwidth}{@{}clccccccccccccccc@{}}
\toprule
& & & & & & Sum \\
\cmidrule{1-1}\cmidrule(l){2-16}\cmidrule(l){17-17}
\multirow{2}{*}{Part 1} & Problem & 1a1) & 1a2) & 1b1) & 1b2) & 1c & 1d1) & 1d2) & 1e & 1f & 2a & 2b & 2c \\
& Score & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 24 \\
\midrule
\multirow{2}{*}{Part 1} & Problem & 1a1) & 1a2) & 1b1) & 1b2) & 1c & 1d1) & 1d2) & 1e & 1f & 2a & 2b & 2c \\
& Score & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 24 \\
\cmidrule{1-1}\cmidrule(l){2-17}\cmidrule(l){17-17}
& \multicolumn{5}{r}{Total number of points} & 14 \\
\bottomrule
\end{tabular*}
\end{document}
-
Can you post what you tried so we have something to start with. This should be compilable in that it should include the \documentclass and all the appropriate packages needed fir your version. – Peter Grill Nov 19 '11 at 0:19
This table is not at all strange looking. :) – Werner Nov 19 '11 at 0:19
I tried looking into the multirow and multicolumn packages as it seems this would work best. Will update with an minimal example. Reason why I did not do this, is as all my attempts to create the table would not even compile. Also using booktabs for extra nice tables. – N3buchadnezzar Nov 19 '11 at 0:31
Updated again. Atleast it works for me now, but itss still broken. – N3buchadnezzar Nov 19 '11 at 3:59
@N3buchadnezzar: what do you mean with "broken"? – Gonzalo Medina Nov 19 '11 at 4:25
I would try changing the table layout, first of all, suppressing the vertical rules. Using the features provided by the booktabs package you can improve your tables layout. Here are two possibilities:
\documentclass{article}
\usepackage{multirow}
\usepackage{booktabs}
\begin{document}
\noindent\begin{tabular}{@{}clccccc@{}}
\toprule
& & & & & & Sum \\
\cmidrule{1-1}\cmidrule(l){2-6}\cmidrule(l){7-7}
\multirow{2}{*}{Part 1} & Problem & 1a & 2b & 3a & 5b & \\
& Score & 2 & 2 & 1 & 2 & 7 \\
\midrule
\multirow{2}{*}{Part 1} & Problem & 6a & 6b & 7a & 8b & \\
& Score & 2 & 2 & 1 & 2 & 7 \\
\cmidrule{1-1}\cmidrule(l){2-6}\cmidrule(l){7-7}
& \multicolumn{5}{r}{Total number of points} & 14 \\
\bottomrule
\end{tabular}
\vspace{2cm}
\noindent\begin{tabular}{@{}lccccccccc@{}}
\toprule
& \multicolumn{8}{c}{Part} & Total \\
\cmidrule{2-9}
& \multicolumn{4}{c}{1} & \multicolumn{4}{c}{2} \\
\cmidrule(r){1-1}\cmidrule(lr){2-5}\cmidrule(lr){6-9}\cmidrule(l){10-10}
Problem & 1a & 2b & 3a & 5b & 6a & 6b & 7a & 8b \\
Score & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 2 \\
\cmidrule(r){1-1}\cmidrule(lr){2-5}\cmidrule(lr){6-9}\cmidrule(l){10-10}
Sum & \multicolumn{4}{c}{7} & \multicolumn{4}{c}{7} & 14 \\
\bottomrule
\end{tabular}
\end{document}
Here's a third variant:
\documentclass{article}
\usepackage{multirow}
\usepackage{booktabs}
\begin{document}
\noindent\begin{tabular}{@{}cccc@{}}
\toprule
Part & Problem & Score & Sum \\
\cmidrule(r){1-1}\cmidrule(lr){2-2}\cmidrule(lr){3-3}\cmidrule(l){4-4}
\multirow{4}{*}{1} & 1a & 2 & \multirow{4}{*}{7} \\
& 2b & 2 & \\
& 3a & 1 & \\
& 5b & 2 & \\
\midrule
\multirow{4}{*}{2} & 6a & 2 & \multirow{4}{*}{7} \\
& 6b & 2 & \\
& 7a & 1 & \\
& 8b & 2 & \\
\midrule
& \multicolumn{2}{c}{Total} & 14 \\
\bottomrule
\end{tabular}
\end{document}
-
That looks really good =) I think I will use the bottom one. I have tried for almost 30 minutes now to produce something good. Could you perhaps take a look at my code and tell me why it will not compile? If i remove the multicol arguments in the second row, it compiles. I guess I need to learn this stuff too and not just copy =) Great job once again! – N3buchadnezzar Nov 19 '11 at 1:24
@N3buchadnezzar: the second argument of \multicolumn can only take one value: l, r, c, p{<length>} (or any other column type, but only one value). – Gonzalo Medina Nov 19 '11 at 1:32
@N3buchadnezzar: also, the third argument of \multicolumn cannot contain alignment characters &. The third argument is used for the material (text, in most cases) that will span the selected number of columns. – Gonzalo Medina Nov 19 '11 at 2:33
Gah! Your code breaks down, with the number of elements I need in that table... I updated my first post. Perhaps your last solution is the best? – N3buchadnezzar Nov 19 '11 at 3:37
@N3buchadnezzar: there are some strategies to reduce a table width: 1) Change \tabcolsep (default value = 6pt); you can say, for example, \setlength\tabcolsep{3pt} just before \begin{tabular}. 2) Reduce the font size; for this, you can use one of th font switches \small, \footnotesize; again this goes right before \begin{tabular}. To keep those changes local, you can use a pair of braces: {\small\begin{tabular}{...}...\end{tabular}}`. Of course you can try one or both methods. – Gonzalo Medina Nov 19 '11 at 3:59 | {"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": 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": 0.8874163627624512, "perplexity": 787.2728226798628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375095677.90/warc/CC-MAIN-20150627031815-00309-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://alanarnholt.github.io/BSDA/reference/Blood.html | Data for Exercise 7.84
Blood
## Format
A data frame/tibble with 15 observations on the following two variables
machine
blood pressure recorded from an automated blood pressure machine
expert
blood pressure recorded by an expert using an at-home device
## References
Kitchens, L. J. (2003) Basic Statistics and Data Analysis. Pacific Grove, CA: Brooks/Cole, a division of Thomson Learning.
## Examples
DIFF <- Blood$machine - Blood$expert
shapiro.test(DIFF)
#>
#> Shapiro-Wilk normality test
#>
#> data: DIFF
#> W = 0.92609, p-value = 0.2383
#>
qqnorm(DIFF)
qqline(DIFF)
rm(DIFF)
t.test(Blood$machine, Blood$expert, paired = TRUE)
#>
#> Paired t-test
#>
#> data: Blood$machine and Blood$expert
#> t = 0.68162, df = 14, p-value = 0.5066
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#> -2.146615 4.146615
#> sample estimates:
#> mean of the differences
#> 1
#> | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.22561924159526825, "perplexity": 8985.245440790015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662556725.76/warc/CC-MAIN-20220523071517-20220523101517-00124.warc.gz"} |
https://www.gamedev.net/forums/topic/184330-scale-and-wireframe/ | #### Archived
This topic is now archived and is closed to further replies.
# Scale and Wireframe
## Recommended Posts
I have two questions. I''m currently working on a Pong game in 3D. The first question I have is what scale should the meshes be made? Is a paddle with a width of 500 too big or does it not matter as long as everything ''matches''? Second question is is there a way to display only the wireframes of my meshes? Thanks!
##### Share on other sites
Scale wise, you can make your own scale. A value of 1 can represent 1 meter, 1 km, 1 cm or anything. You just have to make sure your coordinates are relative. (Also check your caps for the maximum coordinate size, but they are so huge that you''re pretty safe no matter what.
Haven''t used meshes in DX, so I don''t know about the second question.
##### Share on other sites
I don''t even know what meshes are yet but i''m using this for wireframe :
m_pD3DDevice->SetRenderState(D3DRS_FILLMODE, 3DFILL_WIREFRAME);
hope it works for you too
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• 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": 0.1737823188304901, "perplexity": 2581.4350951238}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806660.82/warc/CC-MAIN-20171122194844-20171122214844-00177.warc.gz"} |
https://tex.stackexchange.com/questions/496088/what-is-the-correct-way-to-replace-csxdef-foreach-with-pure-expl3-code | # What is the correct way to replace “csxdef/foreach” with pure expl3 code?
I'm trying to remove the dependencies of the packages pgffor and etoolbox from a code I own which uses expl3, but I don't know which is the correct equivalent of \csxdef in expl3. At the moment I have the following code, which works, but I don't know if it's the right way to replace it.
\documentclass{article}
\usepackage{xparse}
\usepackage{etoolbox}
\usepackage{pgffor}
\ExplSyntaxOn
\NewDocumentCommand{\mycmdone}{ m m }
{
\_csxdef_pgfetoolbox:nn { #1 } { #2 }
}
\cs_new_protected_nopar:Npn \_csxdef_pgfetoolbox:nn #1 #2
{
\foreach \x [count=\n] in { #2 } { \csxdef{#1\n}{\x} } % save in \#1<n>
}
\NewDocumentCommand{\mycmdtwo}{ m m }
{
\_csxdef_expl:nn { #1 } { #2 }
}
\cs_new_protected_nopar:Npn \_csxdef_expl:nn #1 #2
{
\clist_set:Nn \l_tmpa_clist {#2}
\int_step_inline:nn { \clist_count:N \l_tmpa_clist }
{
\cs_set:cpx {l_#1##1:} { \clist_item:Nn \l_tmpa_clist { ##1 } }
}
}
\ExplSyntaxOff
\begin{document}
Test mycmdone
\mycmdone{A}{X,Y,Z,W}
\csuse{A1} \csuse{A2} \csuse{A3} \csuse{A4}
% repeat
\mycmdone{A}{X,Y,Z,W}
\csuse{A1} \csuse{A2} \csuse{A3} \csuse{A4}
\par
Test mycmdtwo
\mycmdtwo{B}{P,Q,R}
\ExplSyntaxOn
\use:c{l_B1:} ~ \use:c{l_B2:} ~ \use:c{l_B3:}
\ExplSyntaxOff
% repeat
Test mycmdtwo
\mycmdtwo{B}{P,Q,R}
\ExplSyntaxOn
\use:c{l_B1:} ~ \use:c{l_B2:} ~ \use:c{l_B3:}
\ExplSyntaxOff
\end{document}
Commands take two arguments, where the second is separated by commas and is not used directly in the document, \csuse is passed as an argument to another code I have defined. I don't know if what I should use is \cs_new: or \l_#1#2_tl or if expl3 has a function that already does this (I don't want to reinvent the wheel).
Greetings.
This is more of a comment that is too long for a comment....
Since it seems that you want to be able to redefine these commands I think that you need to use \cs_set:cpx (or \cs_gset:cpx or \cs_set:cx or \cs_set:cx), but you don't need two functions here and, instead, the following is sufficient:
\documentclass{article}
\usepackage{xparse}
\usepackage{etoolbox}
\usepackage{pgffor}
\ExplSyntaxOn
\NewDocumentCommand{\mycmdtwo}{ m m }
{
\clist_set:Nn \l_tmpa_clist {#2}
\int_step_inline:nn {\clist_count:N \l_tmpa_clist}
{
\cs_set:cpx {l_#1##1:} { \clist_item:Nn \l_tmpa_clist {##1} }
}
}
\NewDocumentCommand\mycmd{m}{\use:c{l_#1:}}
\ExplSyntaxOff
\begin{document}
Test mycmdtwo
\mycmdtwo{B}{P,Q,R}
\mycmd{B1} \mycmd{B2} \mycmd{B3}
% repeat
Test mycmdtwo
\mycmdtwo{B}{P,Q,R}
\mycmd{B1} \mycmd{B2} \mycmd{B3}
\end{document}
I have also given a "helper" command \mycmd.
You can also use \cs_set:cx or \cs_gset:cx but, as Joseph points out in the comments, \cs_set:cpx and \cs_gset:cpx are much faster.
• Thank you very much for the explanation, in fact the saying using pgffor|csxdef I took it from (tex.stackexchange.com/a/378258/7832) which is an answer from yourself. – Pablo González L Jun 17 '19 at 1:25
• The p versions are much faster ... – Joseph Wright Jun 20 '19 at 12:02
• @JosephWright Thanks for pointing this out. – Andrew Jun 20 '19 at 22:06
I'd do a little different. The code as you put it iterates through the comma list 2+<num> times, with <num> being the number of items in the list. The first iteration is in \clist_set:Nn, which loops through it trimming spaces. The second iteration is in \clist_count:N, which loops again to count the number of items. And as many iterations as there are items in the list with \clist_item:Nn, which doesn't just skip to the item you want, but scans each item until it finds the one you want. It's rather suboptimal. In fact, using l3benchmark to measure the performance, 10 thousand iterations of your code runs in 1.32 s, while the code below runs in 0.39 s, more than 3x speedup.
I suggest using \clist_map_inline:nn (or something similar), which will iterate only once through the list, and use an auxiliary counter to keep track of the item number.
I'd say that you should use a <tl var> here with \tl_new:c and \tl_set:cx rather than \cs_set:cx (which, by the way, is much slower than \cs_set:cpx). And, since the generated items aren't functions (in expl3 terminology), but variables, this alternative seems better. Then the accessor function can just be changed to be \tl_use:c, which will check if the requested variable exists and issue an error if appropriate. Using \use:c here is not desired because if the requested variable does not exist it will expand to \relax and will not raise an error.
Finally, if you plan to reuse the created token lists, then instead of \tl_new:c (which raises an error if the variable exists), you can use \tl_clear_new:c which will clear the variable, or create it, depending if it existed or not.
\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\int_new:N \l_pablo_tmp_int
\NewDocumentCommand{\mycmdtwo}{ m m }
{
\int_zero:N \l_pablo_tmp_int
\clist_map_inline:nn {#2}
{
\int_incr:N \l_pablo_tmp_int
\tl_clear_new:c { l_pablo_#1 \int_use:N \l_pablo_tmp_int _tl } % Redundant here, but good practice in general
\tl_set:cx { l_pablo_#1 \int_use:N \l_pablo_tmp_int _tl } {##1}
}
}
\NewDocumentCommand\mycmd {m} { \tl_use:c { l_pablo_#1_tl } }
\ExplSyntaxOff
\begin{document}
Test mycmdtwo
\mycmdtwo{B}{P,Q,R}
\mycmd{B1} \mycmd{B2} \mycmd{B3}
% repeat
Test mycmdtwo
\mycmdtwo{B}{P,Q,R}
\mycmd{B1} \mycmd{B2} \mycmd{B3}
\end{document}
TeXhackers note: \tl_set:cx does \cs_set_nopar:cpx (i.e., \expandafter\xdef\csname...\endcsname, so the \tl_new:c is redundant. But it is in the best practices to declare the variable before using it.
• Thanks for the answer, I had my doubts about the correct use of cs_new vs \tl_new, putting \tl_new:c is almost obligatory in my case, otherwise it doesn't happen when using debug_on:n {...} (which I think is good practice :). Query ... Is there a difference between use:c and \tl_use:c? – Pablo González L Jun 17 '19 at 3:33
• @PabloGonzálezL \use:c{name} is exactly \csname name\endcsname. If \name exists it will be used and if it is not TeX will implicitly \let\name\relax and use it with no error message. \tl_use:c uses \tl_use:N, which checks if the tl var exists before using it and prints and error message if it does not. So again, in the realm of best practices, \tl_use:c is a better option because it will not silently do something you don't want it to. (Sorry for the delay, I posted the answer right before going to bed and just woke up :-) – Phelype Oleinik Jun 17 '19 at 8:56
• ...Okay, I understand, one last query (if you can add to your answer). If I need to rewrite \mycmdtwo (repeat) should I occupy \use:c or should I change \tl_new:c to \tl_clear_new:c ?...Saludos – Pablo González L Jun 17 '19 at 11:26
• @PabloGonzálezL If you want to do \mycmdtwo{B}{P,Q,R} again to overwrite the token lists, then \tl_clear_new:c is the way to go. I'll add to the answer. Only in very specific situations \use:c will be the correct choice, mainly due to its implicit behaviour. – Phelype Oleinik Jun 17 '19 at 11:33
• Thank you very much for the explanation and analysis. – Pablo González L Jun 17 '19 at 20:18
Token list variables are different from functions. Your code is meant to store token lists, rather than define actions.
\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand{\mycmd}{mm}
{
\seq_set_from_clist:Nn \l__pablo_mycmd_seq { #2 }
\seq_indexed_map_inline:Nn \l__pablo_mycmd_seq
{
\tl_clear_new:c { l_pablo_#1##1_tl }
\tl_set:cn { l_pablo_#1##1_tl } { ##2 }
}
}
\NewExpandableDocumentCommand{\pablouse}{m}
{
\tl_use:c { l_pablo_#1_tl }
}
\ExplSyntaxOff
\begin{document}
Test mycmd
\mycmd{A}{X,Y,Z,W}
\pablouse{A1} \pablouse{A2} \pablouse{A3} \pablouse{A4}
\end{document}
A benchmark of my solution compared with Phelype Oleynik's yields
8.22e-5 seconds (252 ops)
1.08e-4 seconds (321 ops)
Top is my solution, bottom is Phelype's.
• Thank you, I missed \seq_indexed_map_inline:Nn when reading the documentation. – Pablo González L Jun 20 '19 at 11:06
• Query, compared to the answer given by @Phelype Oleinik, which is more efficient? – Pablo González L Jun 20 '19 at 16:42
• @PabloGonzálezL I did the benchmark. – egreg Jun 22 '19 at 17:24
• Thank you very much...again. Saludos – Pablo González L Jun 22 '19 at 20:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.702775239944458, "perplexity": 3648.7149125040787}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579250607407.48/warc/CC-MAIN-20200122191620-20200122220620-00521.warc.gz"} |
http://www.snowland.se/tag/function/ | Change a folder LastWriteTime based on files within it
A few days ago I wrote a script that copies some files. Did notice that everything except the date on the folders were ok. So I added a few more lines of powershell code.
Did find a few suggestions on the web, but I like this one…. Since I wrote it. 😛
```Function Set-FolderDate {
Param (
[string] \$Path
)
Trap [Exception] {
Write-Debug \$("TRAPPED: " + \$_.Exception.Message);
Write-Verbose "Could not change date on folder (Folder open in explorer?)"
Continue
}
# Get latest filedate in folder
\$LatestFile = Get-ChildItem \$Path | Sort-Object LastWriteTime -Descending | Select-Object -First 1
# Change the date, if needed
\$Folder = Get-Item \$path
if (\$LatestFile.LastWriteTime -ne \$Folder.LastWriteTime) {
Write-Verbose "Changing date on folder '\$(\$Path)' to '\$(\$LatestFile.LastWriteTime)' taken from '\$(\$LatestFile)'"
\$Folder.LastWriteTime = \$LatestFile.LastWriteTime
}
Return \$Folder
}
Set-FolderDate -Path "D:\temp"
``` | {"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": 0.84536212682724, "perplexity": 1100.8538981455115}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647681.81/warc/CC-MAIN-20180321180325-20180321200325-00541.warc.gz"} |
https://motls.blogspot.com/2015/03/dwarf-galaxy-37-sigma-evidence-for-dark.html | ## Tuesday, March 10, 2015 ... //
### Dwarf galaxy 3.7-sigma evidence for dark matter claimed
A new astro-ph paper cross-linked to hep-ph,
Evidence for Gamma-ray Emission from the Newly Discovered Dwarf Galaxy Reticulum 2
by Geringer-Sameth and 6 co-authors (Carnegie-Mellon/Brown/Cambridge), boasts that they have looked into the Fermi-LAT telescope data more carefully than the Fermi folks and found something that looks like a 3.7-sigma (or, with model-independent background estimates, 2.3-sigma) deviation from the expected gamma-ray backgrounds at frequencies $2$-$10\GeV$.
The approximate shape of Reticulum 2, a dwarf galaxy, and the light it emits.
If that deviation is a sign of new physics, and it is a huge "If" indeed, the interpretation is that the signal is due to a pairwise self-annihilating particle whose mass is a few times $100\GeV$.
The signal-resembling observations have been made in Reticulum 2, a dwarf galaxy. One must realize that Reticulum 2 was "cherry-picked" as the "most signal-like dwarf galaxy" among a few dozens of dwarf galaxies which should reduce your faith (or certainty) that the signal is real.
An excess: in the middle of the picture, the red crosses seem to be above the solid black line.
Reticulum means a "small reticle" in Latin, and a "reticle" is a web, typically a system of thin lines (with a cross) in the circular visual field of a telescope. More importantly, Reticulum 2 (Ret2) is a dwarf galaxy which is really close – it is about 30 kpc away from Pilsen. ;-) It's further than Prague but it is very close – almost exactly equal to the diameter of the Milky Way (according to Wolfram Alpha). Ret2 is the third closest dwarf galaxy after Segue 1 (Seg1, 23 kpc) and Sagittarius (24 kpc).
Ret2 is one of the 9 new low-luminosity objects. It (or they) may be viewed as "satellites of the Milky Way". Among the 9 guys, it's probably the most obvious one. Along with Eridanus 2 and no one else, it shows a statistically significant elongation (others are round) which may perhaps increase the density at some points and improve the odds that self-annihilation of dark matter takes place there.
The signal-like optimistic interpretation is that a new particle, such as a superpartner, pairwise annihilates first into a pair of Standard Model $X\bar X$ particles before they create some gamma-rays. We don't know what the $X\bar X$ pair is. Depending on whether $X\bar X$ is $b\bar b$ or $\tau^+\tau^-$ or $hh$ or $\mu^+\mu^-$, the mass $m$ and $J\langle\sigma v \rangle$ of the dark matter particle may have different values – the mass is distributed in rather wide intervals roughly between $50$ and $1,000\GeV$.
It's obviously incorrect to say that the dark matter has been discovered. After all, the significance level wouldn't be sufficient for a hard discovery even if one forgot about all the other possible reasons to question the bold claim suggested in this paper (Ret2 was cherry-picked; Fermi folks haven't noticed that; even according to these 7 authors, it may be due to astrophysical sources, etc.). Nevertheless, the evidence has some beef and it's plausible that the signal is real and in the future, with the hindsight, these folks will be quoted as those who already saw the actual proof of the dark matter.
Thanks to Matt Buckley
#### snail feedback (12) :
I'm not a physicist, so please forgive my ignorant comment: I suspect dark matter is just as likely to exist as Gravity is likely to be in inisotropic. Maybe the fundamental solution needs to be revisited.
"between 50 and 1000GeV" Between a chromium atom and 5.77 top quarks. The particle is stable for the current life of the universe and has no interaction except gravitation. It felicitously self-annihilates without providing universal background radiation or a beacon near galactic central black holes.
Does it also fart rainbows?
I've never quite understood what thermal equilibrium of the universe means physically if dark matter essentially only interacts gravitationally, or the consequences on cosmic expansion. Or how dark matter gets cold. So in trying to picture dark matter decay, it seems like dark matter should be fairly hot, implying many heavy unstable particles still around to decay. From that it seems like the difficulty in spotting the radiation is already fairly strong evidence against the predominance of primordial dark matter. Is there some reason why decay of unstable dark matter would be faster?
Refresher course for the simple minded. If dark matter does not interact except gravitationally how can they produce photons? Also, is it possible that the dark matter world is as complex -- with atoms, molecules, etc., or something comparable -- as the world we know?
no, skittles.
Why do you suspect that these possibilities are equal? I'd say that's because you don't understand them enough to feel the difference.
You can't just state that the fundamental principles should be changed. You should actually construct a viable model. And the fundamental principles are fundamental because they are very hard to dismiss. That's the theoretical problem with MOND (putting aside the observations) it's not so trivial to make consistent fundamental theory for it as you may think. And be sure, you will have to introduce new fields and particles.
On the other hand some dark matter candidates like SUSY particles would be the truly revolutionary discovery
Interesting article. But I do not understand the reason for cherry picking low luminosity objects. I would think that emission of lot of visible light would not obscure detection of gamma rays!
I like your depiction of the dwarf galaxy, Reticulum 2 ;). I think this is what we call a Leprechaun in Ireland. There was another preprint yesterday from the Fermi-LAT and DES collaborations where they reported no excess of gamma ray radiation from 8 dwarf galaxies found by the Dark Energy Survey (http://arxiv.org/pdf/1503.02632v1.pdf). In this preprint Reticulum 2 is DES J0335.65403.
Low luminosity objects will have fewer astrophysical backgrounds like neutron stars, et cetera. The concentration of baryons to dark matter seems to drop in these objects, so the signal to noise of any dark matter signal could go up, all other things being equal.
Thanks for the reply. I understand their hypothesis now. Then, if a convincing signal of dark matter is not found in these objects, their hypothesis of a smaller baryon to dark matter in these objects would be proven wrong. | {"extraction_info": {"found_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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8344977498054504, "perplexity": 1040.8641954814111}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574532.44/warc/CC-MAIN-20190921145904-20190921171904-00314.warc.gz"} |
https://ask.sagemath.org/question/47774/determinants-of-matrices-with-symmetric-functions/?sort=votes | # Determinants of Matrices with Symmetric Functions edit
I'm interested in calculating the determinant of the matrix $A = (a_{i+j})$ for $0\leq i,j\leq n$ in which $a_{k} = \frac{h_k}{e_k}$ where $h_k, e_k$ are the homogeneous (elementary, resp.) symmetric functions of degree $k$. I would like to express the numerator of $\det(A)$ in terms of monomial symmetric functions.
I have written some code that I think does the job, but it seems an incredibly hacky way of doing it in my opinion. Since I am quite new to sage I was wondering whether someone might be able to take a look at what I've done and suggest a more sage-like way to tackle this problem? My code is below.
I define a set of functions:
# set up two matrices containing symbolic functions hh(i+j), ee(i+j), which are later replaced with symmetric functions h and e
def heMatrices(N):
hh = function('hh')
ee = function('ee')
H = matrix(SR,N,N,lambda i,j:hh(i+j))
E = matrix(SR,N,N,lambda i,j:ee(i+j))
return H,E
# Pointwise divides two matrices (a bit like Hadamard division??)
def elementwiseDivision( M, N ):
assert( M.parent() == N.parent() )
nc, nr = M.ncols(), M.nrows()
A = copy( M.parent().zero() )
for r in range(nr):
for c in range(nc):
A[r,c] = M[r,c] / N[r,c]
return A
# extracts numerator of determinant
def extractNumerator(M):
return det(M).numerator()
# extracts denominator of determinant
def extractDenominator(M):
return det(M).denominator()
# This is dodgy - treates expression as a python string and replaces e(x) with e[x], h(x) with h[x]
def convertToSymmetricString(expr):
st = str(expr)
st = st.replace('(','[')
st = st.replace(')',']')
st = st.replace('ee','e')
st = st.replace('hh','h')
return st
# Puts together the above functions.
def evaluateDeterminant(N):
h,e = heMatrices(N)
A = elementwiseDivision(h,e)
srExpression = extractNumerator(A)
sageExpression = convertToSymmetricString(srExpression)
return sageExpression
Then I open sage in the command line and type the following:
sage: Sym = SymmetricFunctions(QQ)
sage: Sym.inject_shorthands(verbose=False)
sage: ee = evaluateDeterminant(2)
sage: ans = sage_eval(ee, locals=vars())
sage: ans = m(ans)
which returns
2*m[2, 1, 1] + 2*m[2, 2] + 2*m[3, 1] + m[4]
which I believe is what I'm after.
edit retag close merge delete
Sort by » oldest newest most voted
Yeah that's quite hacky. Try this instead:
N = 2
R = PolynomialRing(QQ, names=['h_{}'.format(k) for k in range(2*N)] + ['e_{}'.format(k) for k in range(2*N)])
H = R.gens()[0:2*N]
E = R.gens()[2*N:]
M = Matrix(R.fraction_field(), N, lambda i, j: H[i+j]/E[i+j])
Sym = SymmetricFunctions(QQ)
e = Sym.elementary()
h = Sym.homogeneous()
m = Sym.monomial()
he_subs = dict(zip(R.gens(), [h[k] for k in range(2*N)] + [e[k] for k in range(2*N)]))
print(m(M.det().numerator().subs(he_subs)))
A symbolic alternative (basically, a cleaner version of what you did):
N = 4
Sym = SymmetricFunctions(QQ)
e = Sym.elementary()
h = Sym.homogeneous()
m = Sym.monomial()
H = function('H', nargs=1, print_func=lambda self, *args: 'h[{}]'.format(args[0]))
E = function('E', nargs=1, print_func=lambda self, *args: 'e[{}]'.format(args[0]))
M = Matrix(SR, N, lambda i, j: H(i+j)/E(i+j))
print(m(sage_eval(str(M.det().numerator()), locals={'h': h, 'e': e})))
more
( 2019-09-08 13:35:04 -0600 )edit
Many thanks! Interestingly it seems as though for n = 4 your solution takes an age to finish computing whereas my original solution returns an answer within around 90 minutes (on my not-very-good computer). I'm not sure how much I can really trust my answer though.
( 2019-09-09 05:13:23 -0600 )edit | {"extraction_info": {"found_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": 0.3782762587070465, "perplexity": 7530.145612337022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579250589861.0/warc/CC-MAIN-20200117152059-20200117180059-00525.warc.gz"} |
https://archive.softwareheritage.org/browse/content/sha1_git:2f8d77e142aa4e0fe8b75b03de27804d9a58c972/?path=a6f55739957004d562490e2b8fda2d38d136a287/rope_range.Rd | rope_range.Rd
% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/rope_range.R
\name{rope_range}
\alias{rope_range}
\title{Find Default Equivalence (ROPE) Region Bounds}
\usage{
rope_range(x, ...)
}
\arguments{
\item{x}{A \code{stanreg}, \code{brmsfit} or \code{BFBayesFactor} object.}
\item{...}{Currently not used.}
}
\description{
This function attempts at automatically finding suitable "default"
values for the Region Of Practical Equivalence (ROPE).
}
\details{
\cite{Kruschke (2018)} suggests that the region of practical
equivalence could be set, by default, to a range from \code{-0.1} to
\code{0.1} of a standardized parameter (negligible effect size
according to Cohen, 1988).
\itemize{
\item For \strong{linear models (lm)}, this can be generalised to \ifelse{html}{\out{-0.1 * SD<sub>y</sub>, 0.1 * SD<sub>y</sub>}}{\eqn{[-0.1*SD_{y}, 0.1*SD_{y}]}}.
\item For \strong{logistic models}, the parameters expressed in log odds ratio can be converted to standardized difference through the formula \ifelse{html}{\out{√(3)/π}}{\eqn{\sqrt{3}/\pi}}, resulting in a range of \code{-0.055} to \code{-0.055}.
\item For other models with \strong{binary outcome}, it is strongly recommended to manually specify the rope argument. Currently, the same default is applied that for logistic models.
\item For \strong{t-tests}, the standard deviation of the response is used, similarly to linear models (see above).
\item For \strong{correlations}, \code{-0.05, 0.05} is used, i.e., half the value of a negligible correlation as suggested by Cohen's (1988) rules of thumb.
\item For all other models, \code{-0.1, 0.1} is used to determine the ROPE limits, but it is strongly advised to specify it manually.
}
}
\examples{
library(rstanarm)
model <- stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200)
rope_range(model)
\dontrun{
library(rstanarm)
model <- stan_glm(vs ~ mpg, data = mtcars, family = "binomial")
rope_range(model)
library(brms)
model <- brm(mpg ~ wt + cyl, data = mtcars)
rope_range(model)
library(BayesFactor)
bf <- ttestBF(x = rnorm(100, 1, 1))
rope_range(bf)
}
}
\references{
Kruschke, J. K. (2018). Rejecting or accepting parameter values in Bayesian estimation. Advances in Methods and Practices in Psychological Science, 1(2), 270-280. \doi{10.1177/2515245918771304}.
} | {"extraction_info": {"found_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": 0.568780243396759, "perplexity": 9241.802961734975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780054023.35/warc/CC-MAIN-20210917024943-20210917054943-00150.warc.gz"} |
http://tex.stackexchange.com/questions/67552/in-tikz-how-to-make-the-text-on-an-edge-have-the-same-direction-as-the-edges | # In Tikz, how to make the text on an edge have the same direction as the edge's?
I have created a simple node-edge graph in which I would like to have the texts on each edge aligned to the direction of the edge.
\documentclass[class=minimal,border=0pt]{standalone}
\usepackage{pgf}
\usepackage{tikz}
\usetikzlibrary{arrows,automata}
\usepackage[latin1]{inputenc}
\begin{document}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.8cm,
semithick]
\tikzstyle{every state}=[fill=red,draw=none,text=white]
\node[state] (D) {$D$};
\node[state] (C) [below right of=D] {$C$};
\node[state] (B) [above right of=D] {$B$};
\node[state] (A) [below right of=B] {$A$};
\path (D) edge node {tdc} (C)
(B) edge node {tbc} (C)
(C) edge node {tca} (A);
\end{tikzpicture}
\end{document}
For example, I would like the text tbc rotates -90 degrees automatically to be aligned to the direction of the edge connects the node B to the node C. Is that possible?
-
Try adding the key sloped to the relevant nodes (PGF Manual Section 16.8). Does that do what you want? – Loop Space Aug 17 '12 at 12:24
Thanks! That's the solution! I was trying to figure out how to use text along path... however, the text is not aligned to the center of edge. any solution? – Rasoul Aug 17 '12 at 12:30
The answer to the question of how to get the text to go along the path is to use the sloped key. This rotates the node to match the tangent of the path at the point at which the node would be placed (see Section 16.8 of the PGF Manual).
However, as noted in the comments, this doesn't quite work as desired. This is because of the auto key. This shifts the node off the path. It does so by placing one of the node anchors at the placement point, the anchor chosen appropriately for the tangent of the path (I haven't looked at the code so I'm guessing as to how it chooses it). The problem comes with the fact that the anchor is chosen first and then the node is rotated. What ought to happen is that the node is rotated and then the anchor is chosen. But actually, this doesn't need any complicated code since the anchor will always be one of north or south (if allow upside down is set then it will always be south). So simply setting anchor=south instead of auto will do.
Here's a simple example demonstrating the above analysis. The first three nodes are not sloped, the second three are. The first of the triples has no (other) options, the second is auto and the third is anchor=south. It's clear from the 2nd and 5th that the node has effectively been rotated around its anchor point.
Here's your code with this change (plus a couple of minor stylistic changes: tikz loads pgf automatically, and \tikzstyle is depreciated).
\documentclass{standalone}
%\url{http://tex.stackexchange.com/q/67552/86}
\usepackage{tikz}
\usetikzlibrary{arrows,automata}
\usepackage[latin1]{inputenc}
\begin{document}
\begin{tikzpicture}[
->,
>=stealth',
shorten >=1pt,
auto,
node distance=2.8cm,
semithick,
every state/.style={fill=red,draw=none,text=white},
]
\node[state] (D) {$D$};
\node[state] (C) [below right of=D] {$C$};
\node[state] (B) [above right of=D] {$B$};
\node[state] (A) [below right of=B] {$A$};
\path[every node/.style={sloped,anchor=south,auto=false}]
(D) edge node {tdc} (C)
(B) edge node {tbc} (C)
(C) edge node {tca} (A);
\end{tikzpicture}
\end{document}
(NB the auto=false isn't necessary as the anchor=south overrides the auto key)
And here's the result:
-
Thanks for your detailed answer. It helped me to get it to work properly. – Rasoul Aug 18 '12 at 16:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8515960574150085, "perplexity": 973.5528059383994}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988924.75/warc/CC-MAIN-20150728002308-00214-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/yo-yo-and-torque.146284/ | # Yo-yo and torque
1. Dec 1, 2006
### igor123d
1. A yo-yo lies on a frictionless table. If you apply a horizontal force F to the string to the right, how would the yo-yo move (linearly and rotationally)
2. EF=ma ET=I(alpha)
3. Initially I thought that the yo-yo would move to the right and rotate counter-clockwise (because the string lies below the com). On the test in class that was the right answer apparently. But I looked online and one study guide said that there would be no rotation and the yo-yo would be moving to the right. (which is how I had ultimately answered the question). The idea is that the sum of all forces is ma and that the only external force is the pulling force, F. So if it would rotate, some kinetic energy would go to rotational thus linear would be smaller and a would not be F/m. But how could there be no torque if the force is applied at a point that is not the center of mass and has a component perpendicular to the distance?
Last edited: Dec 1, 2006
2. Dec 1, 2006
### OlderDan
I assume this is a qualitative question. Is the string pulling right or left?
3. Dec 1, 2006
### turdferguson
The answer is all of the above, it depends on the angle from horizontal. It has to do with where the torque is applied. If you pick the correct angle theta, the force is on the instantaneous axis of rotation and produces no torque, only translation. If its below (in your case it was horizontal), then your answer is correct. If the angle is greater than the axis, the yoyo will move in the opposite direction from which you pulled it.
4. Dec 1, 2006
### igor123d
Oh sorry forgot to mention, to the right.
5. Dec 1, 2006
### igor123d
Well but in this specific example, the force actos horizontally on a point below the center of mass, so there would have to be torque, right?
6. Dec 1, 2006
### igor123d
By the way here is an image of a similar situation (look at b):
7. Dec 1, 2006
### OlderDan
Then you are correct. The only horizontal force acting is F, so there will be an accleration F/m to the right. If the string is below the CM there will be a torqe CCW of F*r where r is the radius of the inner spool, so there will be a CCW rotation with angular acceleration.
If there were sufficient friction to prevent slipping instead of no friction, the yo-yo would wind itself up on the string as it rolled to the right. Friction would oppose F and produce a greater CW torque than the CCW torque of the string.
8. Dec 1, 2006
### igor123d
But would this work with conservation of energy? I mean if the only force is F then the total work would have to be Fx. But in this case, you would also have rotational energy, so you would have Fx+K rotational. How could this work?
9. Dec 1, 2006
### OlderDan
It is a bit tricky to calculate the work. Think of you doing work on the string, and the string being an energy transmitter to the yo-yo. You apply a force over a distance. How does that distance compare to the distance the yo-yo moves?
10. Dec 2, 2006
### igor123d
Well I guess since the string moves around the yo-yo, its distance would be R*theta. Then would it also have a translational part x, so that d=R*theta+x
But since the rotation is not what's casuing the linear motion, how would x be related to r*theta?
11. Dec 2, 2006
### Hootenanny
Staff Emeritus
Some really useful comments you've been posting recently....
12. Dec 2, 2006
### OlderDan
The relationship between x and r*θ will depend on the moment of inertia of the yo-yo. What is important is the relationship between W = F*(x + r*θ) and the total kinetic energy. Does F*x equal the translational KE and F*r*θ equal the rotational KE?
13. Dec 2, 2006
### igor123d
Yes, I guess they will. Ok thanks for your help Dan, this makes sense now. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8900046348571777, "perplexity": 719.775559825865}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719033.33/warc/CC-MAIN-20161020183839-00288-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/finding-the-electric-field-magnitude-in-each-region-of-a-sphere.771855/ | # Finding the electric field magnitude in each region of a sphere
1. Sep 19, 2014
### Joa Boaz
1. The problem statement, all variables and given/known data
A charged insulating spherical shell has winner radius of a/3 an an outer radius of a. The cross section is as shown on the picture. The outer shell has a non-constant volume charge density of ρ = 6*α*r^3. Find the electric field magnitude in each region (outside the object, in the shell itself, and in the cavity contained in the shell). Find the electric potential in each region, and sketch a graph of the electric potential as a function of radius for r = 0 to r = 2a and label the potential at r = a/3, a, and 2a. The reference point for the potential is that V = 0 to r = ∞
2. Relevant equations
Not sure but,
E = (1/4∏*ε) * Q/r^2
E = (1/4∏*ε) * (Q/r^3) * r
3. The attempt at a solution
q(inside) = -Q(sphere) = - 4/3 * ∏ * (a^3) ρ = (-4/3)*∏*((a^6)/729)*6α = (-8/729)*(a^6)*∏*α
q(outise) = - q(inside) = (8/729)*∏*α
σ(outer) = q(outside)/4∏*a^2 = (2/729)*(a^4)*∏*α
Not sure if the above is correct, but I am not sure what I have to do after this. Please can anyone help, thank you.
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2. Sep 20, 2014
### vanhees71
I'm not so sure about your notation. In such cases it's much easier to integrate the electrostatic Poisson equation,
$$\Delta \Phi=-\frac{1}{\epsilon_0} \rho.$$
In this case you have spherical symmetry and thus $\Phi=\Phi(r)$, where $r=|\vec{r}|$ is the distance from the origin (the center of your spheres). Just look up the Laplace operator in spherical coordinates and solve the differential equation in the different regions. The constants are determined by the appropriate boundary conditions.
3. Sep 20, 2014
### Joa Boaz
Thank you for your help. But I am not sure if I am allowed to use the Laplace operator since the professor has not gone over the Laplace transform. Is there any other way to tackle the above question without using Laplace transform?
Thank you.
4. Sep 20, 2014
### vanhees71
It's not about the Laplace transform but the Laplace operator. In Cartesian coordinates it reads
$$\Delta \Phi=\partial_x^2 \Phi + \partial_y^2 \Phi + \partial_z^2 \Phi.$$
5. Sep 20, 2014
### vela
Staff Emeritus
It's not very clear at all what you're doing. As far as I can tell, you found a similar example in the book or your notes and just plugged in the values from this problem into the equations from the example. That approach doesn't require any real understanding, and it won't help you in the long run. In this case, you seemed to have picked an example that doesn't apply to the described situation.
Try again, this time justifying why you're taking each step. To find the electric field, use Gauss's law. You have spherical symmetry here, so use it. Because the charge density is non-uniform, you'll need to integrate to find the enclosed charge.
6. Sep 20, 2014
### Joa Boaz
I do really appreciate your help. It seems that my level of physics is very low because I just don't see how I can use partial derivatives on this problem. But I really do appreciate your help.
7. Sep 20, 2014
### Dr Transport
This problem is solvable by Gauss's Law since it is spherically symmetric.
Last edited: Sep 20, 2014
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# Unsteady forces on an accelerating plate and application to hovering insect flight
Pullin, D. I. and Wang, Z. Jane (2004) Unsteady forces on an accelerating plate and application to hovering insect flight. Journal of Fluid Mechanics, 509 . pp. 1-21. ISSN 0021-1120. http://resolver.caltech.edu/CaltechAUTHORS:PULjfm04
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## Abstract
The aerodynamic forces on a flat plate accelerating from rest at fixed incidence in two-dimensional power-law flow are studied analytically and numerically. An inviscid approximation is made in which separation at the two plate edges is modelled by growing spiral vortex sheets, whose evolution is determined by the Birkhoff–Rott equation. A solution based on a similarity expansion is developed, valid when the scale of the separated vortex is much smaller than the plate dimension. The leading order is given by the well-known similarity growth of a vortex sheet from a semi-infinite flat plate, while equations at the second order describe the asymmetric sweeping effect of that component of the free-stream parallel to the plate. Owing to subtle cancellation, the unsteady vortex force exerted on the plate during the starting motion is independent of the sweeping effect and is determined by the similarity solution, to the order calculated. This gives a mechanism for dynamic stall based on a combination of unsteady vortex lift and pure added mass; the incidence angle for maximum vortex lift is $\arccos \sqrt{3/8}\,{\approx}\,52.2^\circ$ independent of the acceleration profile. Circulation on the flat plate makes no direct contribution. Both lift and drag force predictions from the unsteady inviscid theory are compared with those obtained from numerical solutions of the two-dimensional unsteady Navier–Stokes equations for an ellipse of high aspect ratio, and with predictions of Wagner's classical theory. There is good agreement with numerical results at high incidence and moderate Reynolds number. The force per unit span predicted by the vortex theory is evaluated for parameters typical of insect wings and is found to be in reasonable agreement with numerical simulations. Estimates for the shed circulation and the size of the start-up vortices are also obtained. The significance of this flow as a mechanism for insect hovering flight is discussed.
Item Type: Article "Reprinted with the permission of Cambridge University Press." Received August 7 2003, Revised November 10 2003, Published Online 7 June 2004 D. I. P. wishes to thank P. E. Dimotakis for helpful discussions. Z. J.W. wishes to acknowledge the support by NSF Early Career grant, ONR YIP, AFOSR and the Packard Foundation. CaltechAUTHORS:PULjfm04 http://resolver.caltech.edu/CaltechAUTHORS:PULjfm04 http://dx.doi.org/10.1017/S0022112004008821 No commercial reproduction, distribution, display or performance rights in this work are provided. 1313 CaltechAUTHORS Archive Administrator 10 Jan 2006 26 Dec 2012 08:43
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https://kar.kent.ac.uk/53812/ | # A preconditioner for the Ohta-Kawasaki equation
Farrell, Patrick E, Pearson, John W (2017) A preconditioner for the Ohta-Kawasaki equation. SIAM Journal on Matrix Analysis and Applications, 38 (1). pp. 217-225. ISSN 0895-4798. E-ISSN 1095-7162. (doi:10.1137/16M1065483)
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## Abstract
We propose a new preconditioner for the Ohta-Kawasaki equation, a nonlocal Cahn-Hilliard equation that describes the evolution of diblock copolymer melts. We devise a computable approximation to the inverse of the Schur complement of the coupled second-order formulation via a matching strategy. The preconditioner achieves mesh independence: as the mesh is refined, the number of Krylov iterations required for its solution remains approximately constant. In addition, the preconditioner is robust with respect to the interfacial thickness parameter if a timestep criterion is satisfied. This enables the highly resolved finite element simulation of three-dimensional diblock copolymer melts with over one billion degrees of freedom.
Item Type: Article 10.1137/16M1065483 Q Science > QA Mathematics (inc Computing science) > QA297 Numerical analysisQ Science > QA Mathematics (inc Computing science) > QA377 Partial differential equationsQ Science > QD Chemistry > QD473 Physical properties in relation to structure Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics John Pearson 20 Jan 2016 17:21 UTC 29 May 2019 16:54 UTC https://kar.kent.ac.uk/id/eprint/53812 (The current URI for this page, for reference purposes) | {"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": 0.9321808218955994, "perplexity": 3388.6288311531716}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1568514574039.24/warc/CC-MAIN-20190920134548-20190920160548-00445.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcdsb.2016003 | # American Institute of Mathematical Sciences
July 2016, 21(5): 1421-1434. doi: 10.3934/dcdsb.2016003
## Free boundary problem of Barenblatt equation in stochastic control
1 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China 2 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China
Received January 2014 Revised March 2014 Published April 2016
The following type of parabolic Barenblatt equations
min {$\partial_t V - \mathcal{L}_1 V, \partial_t V-\mathcal{L}_2 V$} = 0
is studied, where $\mathcal{L}_1$ and $\mathcal{L}_2$ are different elliptic operators of second order. The (unknown) free boundary of the problem is a divisional curve, which is the optimal insured boundary in our stochastic control problem. It will be proved that the free boundary is a differentiable curve.
To the best of our knowledge, this is the first result on free boundary for Barenblatt Equation. We will establish the model and verification theorem by the use of stochastic analysis. The existence of classical solution to the HJB equation and the differentiability of free boundary are obtained by PDE techniques.
Citation: Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
##### References:
show all references
##### References:
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2018 Impact Factor: 1.008 | {"extraction_info": {"found_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": 0.5521870255470276, "perplexity": 5052.422695838861}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986688826.38/warc/CC-MAIN-20191019040458-20191019063958-00167.warc.gz"} |
http://engineersphere.com/sine-functions | # Sine Functions
### Why do we need Sine functions?
Much of what you will work with in electronics requires alternating current, meaning you will encounter sine waves, square waves, and triangle waves. It might have been a few years since trigonometry, but you will quickly realize that mastering the basics of trigonometry is essential for your success in electrical engineering. Sine functions are one thing you will be seeing a lot of in electrical engineering. There is no avoiding them. Later in the curriculum, you will even learn how to break down square waves and triangle waves into series of sine waves.
The general equation for sinusoidal voltage signal is given by
$v(t) = A \sin(\omega t + \Phi) + V_{DC} = A \sin( 2 \pi f t + \Phi) + V_{DC}$
Where A is the amplitude, $\omega$ is the angular frequency, t is time, $\Phi$ is phase shift, and $V_{DC}$ is a constant voltage.
Likewise, current follows the same general equation.
$i(t) = A \sin(\omega t + \Phi) + I_{DC} = A \sin( 2 \pi f t + \Phi) + I_{DC}$
The equation for the sine wave in Figure 1 is $v(t) = 4\cdot\sin(2 \pi \cdot 1000t) + 2V$. The following sections will explain how to derive the equation for this sine wave as well as any other sine function that you may encounter.
### Amplitude
The amplitude (A) is found by taking the difference between the highest point and the lowest point of a waveform and dividing by two. In Figure 2, the highest point (VH) is 6 V while the lowest point (VL) is -2 V. The amplitude, therefore, is 4 V.
$A = \frac{V_{H} - V_{L}}{2} = \frac{6-(-2)}{2} = 4V$
### Period, Frequency, and Angular Frequency
The period (T) of a waveform is defined as the time it takes for that waveform to complete one cycle. Figure 2 shows two periods, or cycles, of a sine wave. For this sine wave, the period is equal to 1 ms. Identifying the period length allows you to derive the frequency (f) and angular frequency (ω) of the signal.
$\omega = 2 \pi \cdot f$ $\omega = \frac{2\pi}{T} = \frac{2\pi}{.001} = 6283 rad/s$ $f = \frac{1}{T} = \frac{1}{.001} = 1000Hz = 1kHz$
Frequency is defined as the number of cycles a waveform completes in one second. One cycle per second is known as a Hertz (Hz). The waveform above has a frequency of 1000 cycles per second, or 1 kHz.
The units for angular frequency are radians per second (rad/s). Angular frequency is similar to the frequency measured in Hertz, except now we are measuring radians rather than cycles. Because one cycle is equal toradians (360°), the angular frequency is derived by multiplying the frequency by. A common error in homework problems occurs when students neglect to convert one measure of frequency to another. Pay attention to which form you should be using and always remember the equation $\omega = 2\pi \cdot f$.
### Offset Voltage
When a DC voltage is applied to an AC voltage signal, the AC signal is said to have a DC offset voltage. A signal’s offset voltage can also be seen as the signal’s average voltage. You can use integration to find the average value of a time-varying function, but for sine functions, it is easier to use observation.
Consider the sine wave in Figure 3. The equation for this sine wave is $v(t) = 4 \sin(6283t)$. The waveform is oscillating around zero and has no offset. In contrast, the sine wave in Figure 4 has an offset of 2 V. This time, the waveform is oscillating around 2 and its equation is $v(t) = 4 \sin(6283t) + 2V$.
When determining the offset of a sine function, remember that an offset is a constant value. Adding a constant to a sine function only changes the level about which the sine wave oscillates.
### Phase Shift
A sine function has a phase shift when it does not begin its cycle at t = 0. A phase shift is illustrated in the figure to the right. To calculate phase shift, use the equations,
$\Phi_{degrees} = \frac{t_{1} - t_{2}}{T} \cdot 360 ^ {\circ}$ $\Phi_{radians} = \frac{t_{1} - t_{2}}{T} 2\pi rad$
where t1 is the waveform’s original starting point, t2 is where the wave’s starting point has been shifted to, and T is the period.
Example: Find the phase shift of the sine function in Figure 5.
Solution: The sine wave has been shifted from its original starting position (t1) of 0 ms to its new starting point (t2) at -0.4 ms. Therefore,
t1 = 0 ms, t2 = -0.4 ms, and T = 1 ms. Solving for degrees, the phase shift is equal to 144°.
$\phi = \frac{0-(-0.4 ms)}{1 ms} \cdot 360^ {\circ} = 144^ {\circ}$
Defining phase shift with respect to t = 0 works fine in mathematics, but real life won’t tell you when time is equal to zero. In labs this semester you will be observing signals on an oscilloscope. For a single signal, you can assume that there is no phase shift. Phase shift becomes important when more than one signal is involved. There are times when knowing the phase shift between two signals is crucial (the design of an audio system is one instance).
In instances with multiple signals, the phase shift of one signal is measured relative to another signal. One wave is your reference point from which you measure all other waves. Figure 6 shows V1 as the reference and V2 as the wave that has been shifted. Figure 7 shows the opposite. Once you decide on a reference, the same formula given above applies.
$\phi = \frac{t_{1} - t_{2}}{T} \cdot 360^{\circ}$
The sign of the phase shift will vary depending on which signal you use as a reference. The phase shift from V1 to V2 in Figure 6 is positive because V2 is shifted to an earlier time. This is a confusing concept for many people. Though it appears that V1 is ahead of V2, this is not the case. V1 starts its cycle a full 3 ms before V2 does. V1, therefore, is leading V2.
Another way of explaining this is by using the equation. In Figure 6, t1 – t2 must be positive because t1 is larger. In Figure 7, t1 – t2 must be negative.
$\phi_{figure6} = \frac{3 ms}{15 ms} \cdot 360^ {\circ} = 72^ {\circ}$
$\phi_{figure7} = \frac{-3 ms}{15 ms} \cdot 360^ {\circ} = -72^ {\circ}$
Written by Ryan Eatinger ([email protected]). Thank you! | {"extraction_info": {"found_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": 19, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8754701018333435, "perplexity": 673.8797890430773}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824293.62/warc/CC-MAIN-20171020173404-20171020193404-00118.warc.gz"} |
https://brilliant.org/problems/reducible-1000-degree-polynomials/ | # Reducible 1000 degree Polynomials
For how many positive integers $n<1000$ can the polynomial $f_n(x)=x^{1000}+x^n+2$ be written as a product of two non-constant polynomials with integer coefficients?
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "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": 0.7017106413841248, "perplexity": 229.3952420052745}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1590348509264.96/warc/CC-MAIN-20200606000537-20200606030537-00304.warc.gz"} |
https://www.springer.com/us/book/9783642204869?wt_mc=ThirdParty.RD.3.EPR653.About_eBook | CYBER DEAL: 50% off all Springer eBooks | Get this offer!
# Inorganic Chemistry in Tables
Authors: Turova, Nataliya
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• Valuable supplement to Inorganic Chemistry textbooks
• Easy to use
• Each reference scheme presents information on one element, its derivatives and their transformation
• Information on each double-sided sheet usually covered within one whole big chapter of a textbook
• Clearly structured which simplifies considerably the search for necessary information.
•
see more benefits
eBook $54.99 price for USA in USD • ISBN 978-3-642-20487-6 • Digitally watermarked, DRM-free • Included format: PDF • ebooks can be used on all reading devices • Immediate eBook download after purchase Softcover$69.99
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The present supplement to Inorganic Chemistry courses is developed in the form of reference schemes, presenting the information on one or several related element derivatives and their mutual transformations within one double-sided sheet. The compounds are placed from left to right corresponding to the increase in the formal oxidation number of the element considered. For each distinct oxidation state the upper position in the column is occupied by an oxide, its hydrated forms, followed then by basic (and oxo-) and normal salts. The position of each compound in this scheme is unambiguously determined in this approach by the central atom oxidation number (in the horizontal direction) and the nature of ligand (in the vertical one), which simplifies considerably the search for necessary information. The mutual transformations are displayed by arrows accompanied by the reagents or other factors responsible for the reaction (red arrows mean oxidation, green arrows mean reduction, black arrows – if the oxidation number is not changed). Modern training programs require the mastering of a tremendous amount of data. The present tables should serve as a useful addition to textbooks and lectures.
Dr. Nataliya Turova is Senior Research of Inorganic Chemistry at Lomonosov Moscow State University.
Her field of research comprises chemistry of coordinated compounds of metals with organic ligands. Her main work concerns investigating synthesis, physical and chemical properties and structure of metal alkoxides as well as theoretical principles of their use for the preparation of oxides by sol-gel method. She is the author of more than 400 articles published in Russian and international academic journals among them Polyhedron, J. Chem. Soc., Chem. Commun., Dalton Trans., Inorg. Chem. Commun., Inorg. Chim. Acta, Eur. J. Inorg. Chem., Z. Anorg. Chem., J. Sol-Gel Sci. Techn., J. Mater. Sci.
Based on Dr. Turova’s longstanding teaching experience in inorganic chemistry to students and postgraduates she wrote the reference- and schoolbook “Inorganic Chemistry in Tables” by which she aimed to simplify for the reader the searching of information scattered among handbooks, monographs or articles and making logical connections between separate binding classes, methods of their synthesis and further transformations. In the last 30 years the “Tables” were regularly reissued in Russia, were published in Japan (Kodansha Ltd., 1981) and are successfully used by students, postgraduates, lecturers and researchers of various specialties. In 2009 she prepared an abridged version of the “Tables” provided for schools of profound learning chemistry.
• Introduction
Pages 2-4
Turova, Dr. Nataliya
• Abbreviation
Pages 5-5
Turova, Dr. Nataliya
• Halogens, Astatine
Pages 6-11
Turova, Dr. Nataliya
• Manganese
Pages 12-13
Turova, Dr. Nataliya
• Technetium, Rhenium
Pages 14-15
Turova, Dr. Nataliya
eBook $54.99 price for USA in USD • ISBN 978-3-642-20487-6 • Digitally watermarked, DRM-free • Included format: PDF • ebooks can be used on all reading devices • Immediate eBook download after purchase Softcover$69.99
price for USA in USD
## Bibliographic Information
Bibliographic Information
Book Title
Inorganic Chemistry in Tables
Authors
2011
Publisher
Springer-Verlag Berlin Heidelberg
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-20487-6
DOI
10.1007/978-3-642-20487-6
Softcover ISBN
978-3-642-20486-9
Edition Number
1
Number of Pages
IV, 157
Number of Illustrations
44 b/w illustrations, 110 illustrations in colour | {"extraction_info": {"found_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": 0.17708630859851837, "perplexity": 8605.195702319224}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141195656.78/warc/CC-MAIN-20201128125557-20201128155557-00327.warc.gz"} |
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# Epson PriorityFAX 3000 Users manual Download
Transcript
ActFax Server
User’s Manual
8th Extended Edition
ActFax Communication-Software GmbH
http://www.actfax.com
[email protected]
Remark
This product or parts of it are only allowed to be used or reproduced according to the terms of the license agreement. Every other use of the software is not permitted. No parts of this manual are allowed to be reproduced,
transferred, translated or stored on other media without the written declaration of consent of ActFax Communication. Changes of the content of this manual are reserved. The contents of this manual do not give a claim of
any kind.
According to the law, it is allowed to create a backup copy of the software for archiving purposes. Every other
copy of the software – regardless of the purpose – is not permitted and will be prosecuted by law.
Microsoft® and Windows® are registered trademarks of the Microsoft Corporation.
UNIX® is a registered trademark of the American Telephone and Telegraph Company.
Pentium® is a registered trademark of the Intel Corporation.
Hewlett-Packard® is a registered trademark of the Hewlett-Packard Company.
ZyXEL™ is a registered trademark of the ZyXEL Communication Corporation.
The use of any trademark that is not explicitly named here, does not claim the right for free use of it.
Part. No. 2511-S02-E
ActFax User’s Manual
2
INSTALLATION. YOU ARE ONLY AUTHORIZED TO USE THE SOFTWARE WHEN YOU AGREE WITH THE TERMS OF
THE FOLLOWING LICENSE AGREEMENT. BY USING THE SOFTWARE YOU PROFESS TO THE TERMS OF THAT AGREEMENT.
IF YOU DO NOT AGREE WITH THAT LICENSE AGREEMENT YOU EITHER HAVE TO REMOVE THE SOFTWARE FROM
YOUR HARD DISK OR YOU HAVE TO RETURN THE COMPLETE SOFTWARE AND GET BACK THE MONEY.
By paying the license fee the licensee is granted the right to use the software under the following conditions:
USE: It is only allowed to use the software on the number of computers that is named in the license agreement.
It is not allowed to reassemble or recompile the software, except it is explicit allowed by law.
COPIES AND MODIFICATIONS: It is allowed to make copies of the software for archive purposes or when it is
necessary to transfer the software to another computer. It is not allowed to make copies of the software for any
other purposes.
OWNERSHIP: You agree that you have no rights on the software except the right for using the programs as defined in that agreement and the physical media. You know and accept that the software is protected by copyright
laws. When you violate this law you can be made liable by the manufacturer.
ASSIGNMENT OF RIGHTS: You are allowed to transfer the right of using the software to others when the terms
of this agreement are accepted. When you transfer the right for using that software you are no longer allowed
to use the software.
SUBLICENSES: You are not allowed to borrow the software to others or distribute copies of the software. If you
plan to redistribute or modify the software you need the assent of the manufacturer.
EXPIRATION OF THIS AGREEMENT: When you violate this software license agreement the manufacturer can request the customer to undo that defamation of the agreement. When the customer is not following that request
within 30 days the manufacturer can revoke the right for using the software.
UPDATES AND ENHANCEMENTS: The customer accepts that there is no right for updates and enhancements that
the manufacturer offers at a later time.
SOFTWARE WARRANTY
WARRANTY PERIOD: The manufacturer guaranties for the time of 6 months that the software will work in a
manner that the main use of the software is not impaired. This warranty is only valid if you have installed all of
the program files correctly. From the current position it is not possible to guaranty that the software is absolutely
free of errors. If the software does not work in the desired manner within that warranty period you can ask for
replacement or reparation. If it is not possible for the manufacturer to repair the error within a appropriate time
the customer can cancel the contract with full return of the money.
MEDIA: The manufacturer guaranties for the time of 6 months that the enclosed media are free of material faults.
If you detect errors on the media within the warranty period you can order a replacement media from the manufacturer. If it is not possible for the manufacturer to replace the media within a appropriate time the customer
can cancel the contract with full return of the money.
ActFax User’s Manual
3
CLAIM OF WARRANTY: You have to inform the manufacturer at least 30 days after the end of the warranty period
to report your claim of warranty.
LIMITATIONS IN WARRANTY: There is no warranty that exceeds the above listed terms of warranty. There are no
other agreements of any kind. The warranty period is limited to 6 months except otherwise defined by law.
INDEMNIFICATION AND EXCLUSION OF LIABILITY: THERE IS NO RIGHT OF INDEMNIFICATION OF ANY KIND
AGAINST THE MANUFACURER OR ANY OF ITS DISTRIBUTION PARTNERS REGARDLESS OF THE REASONS.
This license agreement and the warranty terms only define the conditions between the manufacturer and the
customer. Other agreements or warranty terms that are part of the package of other distributors are not practicable.
ActFax User’s Manual
4
1. Introduction
10
1.1. ActFax vs. ActiveFax .................................................................................. 10
1.2. Using the Manual ...................................................................................... 10
1.2.1. Help System ........................................................................................................... 10
1.2.2. Information Symbols .............................................................................................. 11
1.3. What is ActiveFax? .................................................................................... 11
1.4. System Requirements................................................................................ 11
1.4.1. System Requirements Fax Server .......................................................................... 12
1.4.2. System Requirements Fax Client ........................................................................... 12
1.5. Software Concept ...................................................................................... 12
1.5.1. Outgoing Faxes (Sending) ...................................................................................... 13
1.5.2. Outgoing E-Mails .................................................................................................... 14
1.5.3. Incoming Faxes (Receiving) .................................................................................... 14
1.5.4. Fax-On-Demand ..................................................................................................... 14
1.6. Screen Description .................................................................................... 15
2. ActiveFax Installation
17
2.1. Program Overview ..................................................................................... 17
2.2. Running the Setup ..................................................................................... 18
2.3. Customizing Program Settings ................................................................... 21
2.4. Establishing a Client Connection ................................................................ 21
3. How to ...
23
3.1. Create a New Fax Message ........................................................................ 23
3.1.1. Faxing from Windows Applications ....................................................................... 23
3.1.2. Instant Fax Messages ............................................................................................. 24
3.1.3. File System ............................................................................................................. 25
3.1.4. Named Pipes .......................................................................................................... 25
3.1.4.1. Example in Programming Language C ................................................................................ 26
3.2. Access Fax Messages on other Computers ................................................ 28
3.2.1. Fax Client Installation ............................................................................................. 28
3.2.2. Establishing a Client Connection ............................................................................ 28
3.3. Send Fax Messages from UNIX / Linux ....................................................... 30
ActFax User’s Manual
5
3.3.1. Configuring LPD/LPR Printers in UNIX.................................................................... 30
3.3.1.1. IBM RS/6000 (AIX) .............................................................................................................. 30
3.3.1.2. HP-9000 (HP/UX) ................................................................................................................ 30
3.3.1.3. Other UNIX Systems ........................................................................................................... 30
3.3.2. Sending Fax Messages with LPD/LPR ..................................................................... 31
3.3.3. Alternatives to LPD/LPR ......................................................................................... 31
3.3.3.1. FTP, TFTP and RAW Sockets ............................................................................................... 31
3.3.3.2. Serial Connection (RS-232) ................................................................................................. 31
3.3.3.3. File System (NFS, Samba) ................................................................................................... 31
3.3.3.4. Printer Commands (HP-LaserJet PCL, Epson-LQ, Postscript, PDF) ...................................... 32
3.3.3.5. Data Fields .......................................................................................................................... 32
3.4. Enter valid Fax Numbers ............................................................................ 33
3.4.1. Examples for valid fax numbers ............................................................................. 33
3.5. Adjust the Screen View.............................................................................. 34
3.6. Change the Faxlist Columns ....................................................................... 35
3.6.1. Columns ................................................................................................................. 35
3.6.2. Colors ..................................................................................................................... 35
3.7. Select an Entry in the Faxlist ...................................................................... 37
3.7.1. Selecting Entries ..................................................................................................... 37
3.7.2. Sorting Entries ........................................................................................................ 37
3.7.3. Searching for Entries .............................................................................................. 37
3.8. Automatically Redial failed Fax Transmissions........................................... 39
3.8.1. Automatic Redialing ............................................................................................... 39
3.8.2. Send a Fax Message again ..................................................................................... 39
3.9. Automatically Print Fax Messages ............................................................. 41
3.9.1. Sending Report ....................................................................................................... 41
3.9.2. Compressed Printing .............................................................................................. 41
3.9.3. General Settings ..................................................................................................... 42
3.9.4. User-dependent Settings ....................................................................................... 42
3.10. Protect Faxes against Unauthorized Access ............................................. 44
3.10.2. Security Settings................................................................................................... 44
3.11. Block Fax Numbers (Blacklist) .................................................................. 46
3.11.1. Block Outgoing Fax Calls ...................................................................................... 46
3.11.2. Block Incoming Fax Calls ...................................................................................... 46
3.12. Use the Fax-On-Demand Server .............................................................. 48
3.12.1. Fax Polling ............................................................................................................ 48
3.12.2. Creating Fax-On-Demand Documents ................................................................. 48
3.13. Move Fax Messages to the Archive ......................................................... 50
3.13.1. Automatic Archive................................................................................................ 50
ActFax User’s Manual
6
3.13.2. Manual Archive .................................................................................................... 51
3.13.3. Individual Archive Folders .................................................................................... 51
3.14. Use Multiple Fax Servers ......................................................................... 52
3.14.1. Configure Fax Clients ........................................................................................... 52
3.14.2. Add ActiveFax Printers ......................................................................................... 53
3.14.3. Configure ActiveFax Printers................................................................................ 53
3.15. Create a Cover Page or Overlay ............................................................... 55
3.15.1. What is a Cover Page / Overlay? .......................................................................... 55
3.15.2. Creating Cover Pages / Overlays .......................................................................... 55
3.15.3. Using Cover Pages / Overlays............................................................................... 56
3.15.4. Using Color Images............................................................................................... 56
4. Configuration
58
4.1.1. User Permissions .................................................................................................... 59
4.1.2. Alias Names ............................................................................................................ 60
4.1.3. Predefined Settings ................................................................................................ 60
4.1.4. Automatic Printing ................................................................................................. 61
4.1.6. Fax Forwarding ....................................................................................................... 62
4.1.7. Archive (Export) ..................................................................................................... 63
4.1.8. Group of Users ....................................................................................................... 64
4.1.8.1. Passive Group Members ..................................................................................................... 64
4.1.10. Routing of Inbound Fax Messages ....................................................................... 65
4.1.10.1. Routing using Direct Dial Numbers (MSN, DDI, DID, DTMF) ............................................ 65
4.1.10.2. Routing using CSID (Sender Identification) ....................................................................... 66
4.1.10.3. Routing using Modem ...................................................................................................... 67
4.1.10.4. Manual Routing ................................................................................................................ 67
4.1.10.5. Order of Routing Methods ............................................................................................... 68
4.2. Phone Book ............................................................................................... 69
4.2.1. Importing the Phone Book ..................................................................................... 70
4.2.1.1. Import from ASCII File ........................................................................................................ 70
4.2.1.2. Import from ODBC Database .............................................................................................. 71
4.2.2. Exporting the Phone Book ..................................................................................... 71
4.3. Transmission Protocol ............................................................................... 72
4.3.1. Printing the Transmission Protocol ........................................................................ 72
4.3.1.1. Layout ................................................................................................................................. 73
4.3.1.2. Automatic Printing.............................................................................................................. 73
4.3.1.3. Sending Report ................................................................................................................... 73
4.3.2. Exporting the Transmission Protocol ..................................................................... 73
4.3.3. Archiving the Transmission Protocol ..................................................................... 74
ActFax User’s Manual
7
4.4. Charge Optimization.................................................................................. 75
4.4.1. Delayed Transmissions ........................................................................................... 75
4.4.2. Time Limitation ...................................................................................................... 76
4.4.3. Least Cost Routing.................................................................................................. 77
4.5. Modem & ISDN.......................................................................................... 78
4.5.1. Voice over IP (VoIP / T.38) ..................................................................................... 78
4.5.3. Modem Configuration ............................................................................................ 79
4.5.4. Modem Status ........................................................................................................ 81
4.6. E-Mail ........................................................................................................ 82
4.6.1. SMTP Server (Mail Server) Configuration .............................................................. 82
4.6.2. Mail Server Authentication .................................................................................... 83
4.6.3. E-Mail Options........................................................................................................ 83
4.7. ODBC Database ......................................................................................... 85
4.7.1. Selecting a Data Source.......................................................................................... 85
4.7.2. Importing the Phone Book ..................................................................................... 86
4.7.3. Exporting the Transmission Protocol ..................................................................... 87
4.8. Cost Account Manager .............................................................................. 89
4.8.1. Create Cost Account Codes .................................................................................... 89
4.8.2. Select Cost Account Codes ..................................................................................... 89
4.9. Network Scanners ..................................................................................... 91
4.9.1. Installation ............................................................................................................. 91
4.10. External Configuration File options.cfg .................................................... 92
5. Data Fields
93
5.1. Why do I need Data Fields? ....................................................................... 93
5.2. Syntax of Data Fields ................................................................................. 93
5.2.1. Masking Data Fields ............................................................................................... 93
5.3. Overview of the Data Fields....................................................................... 93
5.3.1. Sender Fields .......................................................................................................... 94
5.3.2. Recipient Fields ...................................................................................................... 94
5.3.3. Common Fields....................................................................................................... 95
5.3.4. Special Data Fields ................................................................................................. 98
5.4. Examples for Data Fields ........................................................................... 98
5.5. Embedding Bitmaps into Fax Messages ..................................................... 99
5.6. Embedding Data Fields into Applications................................................. 100
5.6.1. Windows Applications.......................................................................................... 100
ActFax User’s Manual
8
5.6.1.1. Reference File for Data Fields ........................................................................................... 101
5.6.1.2. Split Print Jobs (Mail Merge Documents) ......................................................................... 102
5.6.1.3. Example in Programming Language C .............................................................................. 102
5.6.2. UNIX, Linux and other Operating Systems ........................................................... 102
5.6.2.1. Example in Programming Language C .............................................................................. 103
5.6.2.2. Example in Programming Language INFORMIX 4GL ......................................................... 103
104
6.3. Fax Client Updates ................................................................................... 105
6.4. Update Period ......................................................................................... 105
7. Appendix
106
7.1. Glossary ................................................................................................... 106
7.2. Keyboard Hotkeys ................................................................................... 110
7.3. Frequently Asked Questions - FAQ .......................................................... 111
7.4. Problem Solutions ................................................................................... 113
7.5. Sample Applications ................................................................................ 115
7.5.1. Windows Application (WinApp.exe) .................................................................... 115
7.5.1.1. Program Summary ............................................................................................................ 115
7.5.1.2. Source Code ...................................................................................................................... 116
7.5.2. Socket Application (Socket.exe) ........................................................................... 118
7.5.2.1. Program Summary ............................................................................................................ 119
7.5.2.2. Source Code ...................................................................................................................... 119
7.6. Index ....................................................................................................... 121
ActFax User’s Manual
9
1. Introduction
1.1. ActFax vs. ActiveFax
The terms “ActFax” and “ActiveFax” are used synonymously in this manual and also in the
software itself. Both terms refer exactly to the same software. The reason why the software
is known under ActFax and under ActiveFax is due to historical reasons. To preserve highest
backward compatibility with existing installations, no adjustments to the names have been
made in the software. Whenever this manual or the software refers to ActFax or ActiveFax,
the same software product is addressed.
1.2. Using the Manual
This manual describes how to install, configure and use ActiveFax. The topics of the manual
are subdivided into different chapters. The first two chapters give an overview of the program
features and the installation of the software. The other chapters contain information about
the configuration of the software and a description of routine tasks. The addendum at the end
of the manual includes the glossary, an overview of useful keyboard shortcuts, the frequently
asked questions (FAQ), a troubleshooting guide, sample applications and the index.
1.2.1. Help System
In addition to the contents of this manual, further help and information can also be found in
ActiveFax’ help system. Using the context sensitive online help of ActiveFax, detailed information is available for all functions and program options. The help system can be activated
either with the menu option Help / Contents and Index or with the Help button.
ActFax User’s Manual
10
1.2.2. Information Symbols
This manual uses graphical symbols to direct your attention to important text sections. These
symbols are used to highlight cross references to additional information about specific topics,
critical settings or other notes referring to a previously discussed topic.
The “Information” symbol is used to direct your attention to additional information
of the same topic. This symbol is also used to highlight useful tips simplifying daily
work.
The “Attention” symbol is used to direct your attention to common misunderstandings, error sources and critical settings.
The “Stop” symbol is used to direct your attention to very critical settings that might
cause loss of data.
1.3. What is ActiveFax?
ActiveFax is one of the most powerful network fax solutions currently on the market. The ActiveFax software enables you to manage all your fax documents with just a few mouse clicks,
without complicated and extensive administration work. Messages can be transmitted either
by fax or as an email. The sophisticated user and security concept guarantees straightforward
all fax messages is available from any workplace through the fax client program.
Of course ActiveFax also supports all features and functions you can expected from a state-ofthe-art network fax solution, like detailed transmission protocols, global and private phone
books, cover pages and overlays, as well as automatic transmission time optimization, least
cost routing and inbound fax routing. Furthermore, ActiveFax can also be used as a fax-ondemand server.
In addition to creating fax messages with any Windows application, ActiveFax can also be used
to generate faxes from non-Windows operating systems like UNIX or Linux. The support of
the network printer protocol LPD/LPR and printer commands of HP-LaserJet (PCL), Epson-LQ
and Postscript as well as PDF guarantees simple and quick integration of fax and email services
in UNIX and in Linux environments.
1.4. System Requirements
When running ActiveFax, it is recommended to use a system with at least the following minimum requirements: The operating system can be Windows XP / 2003 / Vista / 2008 / 7 / 2012
/ 8 / 10 (32-bit or 64-bit). You also need a fax modem, an ISDN adapter or a fax board from
Brooktrout or Intel/Dialogic. To use the software on multiple computers, there also needs to
be a TCP/IP network connection (LAN) available. The server computer should be at least an
ActFax User’s Manual
11
Intel® Pentium model with 300 MHz and 256 MB RAM. The display adapter should be configured to a resolution of at least 1024x768 and 64k colors.
Keep in mind, that these values are just approximate values. ActiveFax can also be installed
on hardware equipment that does not meet these requirements; however performance of the
software decreases significantly then.
1.4.1. System Requirements Fax Server
Resource
Minimum
Recommended
Requirement
Requirement
Processor (CPU)
Working Memory (RAM)
Available Disk Space
Operating System
Intel® Pentium/ 300 MHz
128 MB
100 MB
800x600, 256 colors
Windows XP / SP3
Network
TCP/IP or NetBeui
Intel® i3 / 1 GHz
512 MB
1000 MB
1024x768, 16M colors
Windows 2003 / 2008 / 7 / 2012 / 8 /
10 (32-bit or 64-bit)
TCP/IP
1.4.2. System Requirements Fax Client
Resource
Minimum
Recommended
Requirement
Requirement
Processor (CPU)
Working Memory (RAM)
Available Disk Space
Operating System
Intel® Pentium/ 300 MHz
64 MB
20 MB
800x600, 256 colors
Windows XP / SP3
Network
TCP/IP or NetBeui
Intel® i3 / 1 GHz
256 MB
100 MB
1024x768, 16M colors
Windows XP / 2003 / Vista / 2008 / 7 /
2012 / 8 / 10 (32-bit or 64-bit)
TCP/IP
1.5. Software Concept
The primary idea behind ActiveFax is making all tasks as automated as possible. This has been
consequently implemented in all parts of the software. The automation already starts at the
installation of the software. According to the slogan “unboxing and starting”, all hardware
connected to the system is fully automatically detected and configured. Thus there is almost
no need for administrative work. To ensure high flexibility, all parts of the software can be
individually configured to meet your requirements and preferences.
The core part of ActiveFax is the fax server. The fax server stores and manages all fax and email
messages and controls all communication tasks (communication to the outside world through
modems, ISDN adapters and fax boards and internal communication through the LAN). The
fax client component of ActiveFax is used to access the fax documents from any computer in
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12
the network. The fax services provided by ActiveFax can be subdivided into three main categories: Outgoing fax messages (sending), incoming fax messages (receiving) and fax-on-demand (fax polling).
1.5.1. Outgoing Faxes (Sending)
Outgoing fax documents can be created in various ways with ActiveFax. The easiest way to
create a new fax messages is printing a document from a Windows application (i.e. Microsoft®
Word). ActiveFax also supports a large number of communication services, so fax messages
can also be created from UNIX, Linux and other non-Windows operating systems. Especially
when using UNIX or Linux, the LPD/LPR printer service can be used to create new fax messages; with LPD/LPR, ActiveFax is accessed exactly the same way as every other physical
printer in the network.
Especially when sending fax messages from own (self-programmed) applications (i.e. ERP programs) the fax parameters (recipient, subject, priority, ...) can already be defined inside the
application. That way, the fax transmission can be done fully automatically without user interaction (the user does not need to enter the recipient of the fax message twice in that case).
When using that way of server connection, the fax parameters are transmitted to the fax
server through data fields. Since data fields are embedded directly into the fax document,
there is no need to deal with complex programming techniques like DDE (you just add the
parameters the same way as normal text directly to the document).
The transmission of pending fax documents is fully automatically processed by the fax server.
Depending on the preferred transmission time and the priority of the fax messages, documents will be transmitted according to their chronological order. Fax messages for unreacha-
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ble recipients (i.e. because of a busy line) will be automatically repeated periodically. The period between the redial attempts can be individually configured. If a fax message cannot be
delivered within a pre-configured number of attempts, the message will be marked as “undeliverable”. Such fax messages are listed separately by ActiveFax. Furthermore, the permission
for the individual users can also be configured to request sending clearance from an authorized user to start transmission of outgoing fax messages.
1.5.2. Outgoing E-Mails
Transmission of emails is done exactly the same way as sending fax messages. The only difference is that you have to enter the email address of the recipient instead of the fax number.
Depending on the configuration of the fax server, ActiveFax tries to convert documents to text
format whenever possible to send the message directly in the body of the email. If the conversion to text format cannot be done, the document is sent as an email attachment in PDF,
TIFF or GIF format. Delivery of the emails through the Internet is done fully automatically
through a direct connection to an SMTP server (mail server) or a dialup connection with RAS
(Remote Access Service).
1.5.3. Incoming Faxes (Receiving)
One of the main advantages of a fax server is the capability of routing inbound fax messages
to individual users. ActiveFax supports different methods of inbound fax routing. The most
reliable method is using a dedicated direct dial number for each user. That way fax messages
can be sent directly to individual users; the number of misrouted messages normally tends to
zero with that type of routing. Due to technical reasons, direct dial numbers are only available
when using an ISDN adapter or a DID (Direct Inward Dialing) capable modem or fax board.
Another way of inbound fax routing is CSID routing (sender identification). When using that
routing method, a phone book entry with the senders fax number (CSID) is mapped to a specific user. Inbound fax routing can also be done based on the modem (fax number) a fax is
received on. When using that method, specific user entries are mapped to the available modems (fax numbers).
Notification of new fax messages can be done in multiple ways. When using the fax client, the
receipt of new fax messages can be signaled with a popup window automatically displayed on
the client computer. As an alternative it is also possible to automatically print incoming fax
messages. The automatic printing function can be configured to use an individual printer for
each user, so it would be possible to print inbound fax messages directly on a user’s workplace
printer. To save paper and costs it is also possible to print fax messages in compressed format.
That way, multiple pages of a fax messages will be scaled down and printed on a single sheet
of paper. What would also be possible is to automatically forward fax messages by email or
to automatically export the faxes to a predefined directory. When forwarding faxes to multiple email addresses, separate the email addresses with a semicolon.
1.5.4. Fax-On-Demand
ActiveFax also includes a fax-on-demand server. “Fax-On-Demand” means, that documents
are made available to be requested from other fax machines. That way, different types of
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documents (i.e. price lists, data sheets, ....) can be made accessible to a large number of people. A fax-on-demand document is either mapped to a specific fax modem (fax number) or
when using an ISDN adapter to a direct dial number. The steps required to create fax-on-demand documents are exactly the same as for creating normal fax messages.
Beside the function of using ActiveFax as a fax-on-demand server, the software can also be
used to request documents from other fax-on-demand systems. This means that ActiveFax
can be used to receive a fax document by calling the number of a different fax-on-demand
system.
1.6. Screen Description
The Faxlist contains a list of all fax messages stored on the system. By selecting an entry
of the faxlist, the selected fax message is automatically loaded and displayed. By doubleclicking or using the Faxlist menu, the settings of a faxlist entry can be displayed and
modified.
The Fax Selection Window is used to select the type of fax message that should be displayed in the faxlist. The fax selection window also includes the recycle bin and the archive.
The User Selection Window is used to select the user or group that should be displayed
in the faxlist.
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The Communication Window displays the status of all communication services controlled by ActiveFax. With a mouse double-click or with the Communication menu, the
settings for the individual communication services can be configured.
The Fax Window displays the fax message currently selected in the faxlist. The zoom
factor of the fax message can be changed with the left and right mouse button or with
the zoom field of the toolbar. The number of fax pages displayed in the fax window can
be changed with the menu View / View Properties.
The Page Selection Window is used to change between the pages of a multi-page fax
message.
The Toolbar contains buttons for important and frequently used functions for fast and
The Status Line displays the date and time as well as other status information of the fax
software.
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2. ActiveFax Installation
2.1. Program Overview
ActiveFax mainly consists of two components, the fax server and the fax client program. The
installation of the fax server needs to be done only once on a single computer; all other computers in the network have the fax client installed to automatically load fax documents directly
from the fax server. The fax client needs to be installed on all computers in the network that
should be able to create or access fax messages.
It is recommended to install the fax server part of ActiveFax on a powerful system designed
for server operation. Especially when using delayed fax transmissions you should also take
care that the fax server computer is running 24 hours, 7 days a week and that the computer
is not turned off during night hours.
The above sample configuration shows a typical ActiveFax installation. This configuration contains the fax server as the core part of the faxing system, several fax clients and a UNIX server
that has been integrated using the network protocol LPD/LPR. Documents on paper are
scanned and submitted to the fax server with the HP Digital Sender network scanner.
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2.2. Running the Setup
Before you start with the installation of ActiveFax, you should define which computers in
the network should have the fax client software installed and which computer should be
used for the installation of the fax server software.
ActiveFax is available in a 32-bit and in a 64bit version. If the installation is done on a 64bit Windows system, it is recommended to
use the 64-bit version of ActiveFax for the installation. If it is required to communicate
with other 32-bit programs (i.e. Microsoft®
Exchange or Outlook) from within the fax
server or fax client, you need to use the 32-bit
ActiveFax version also for installations on 64bit Windows systems. You can mix any combination of 32-bit and 64-bit ActiveFax installations throughout the network.
It is recommended to install the fax server
first. After the installation of the fax server has
been successfully completed install the fax clients. To execute the installation of ActiveFax,
1) Insert the program CD into the CD-ROM
drive and start the Setup program or run
actfax_setup_en.exe (32-bit version) or
actfax_setup_x64_en.exe (64-bit version).
Please note that most PCs automatically
instructions of the Setup program that
guides you through the installation procedure.
2) Choose the directory for the installation of
the software. The default directory for the
installation is the default Windows program directory (C:\Program Files\ActiveFax). The “Search” button can be used to
change the target directory or drive for
the installation.
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3) Choose the installation procedure now.
You can select between a complete installation (server and client), a server-only installation or a client-only installation. Take
care that the fax server is normally only installed on a single computer in the network. If the fax server has already been installed, you typically choose a client installation here.
4) Select the program components that
should be installed. According to the previously selected installation procedure,
the correct settings are already preset
here. Make sure there is enough disk
space available on the installation drive.
5) Choose the program group that should be
used to create the program icons.
6) Enter the name of the licensee and the license key. If you do not yet have a valid
00000-00000-00000-00000 to register a
fully functioning demo version of ActiveFax. There are no restrictions or limitations in program features when you register a demo version of ActiveFax, but there
is a “Demoversion” watermark printed on
the faxes.
7) Specify if the fax server (or fax client)
should be automatically started during
system startup. It is strongly recommended to automatically start the fax
server; the fax server is started as a service in that case and is running right after
the computer has been booted, also when
no user is logged on to the system.
8) Turn on all modems connected to the system and start the fax server. ActiveFax
now scans the system for available fax modems, fax boards and ISDN adapters and
configures them.
9) Enter your name (company name) and fax
number and your area code. This information is displayed in the header of the
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fax messages. If you plan to also send
prefix needed for outgoing calls, specify
the dial prefix in the next step. Especially
when using phone systems (PBX), you normally have to specify a dial prefix to get an
outside line.
After the setup program has been finished, ActiveFax is completely installed and ready for a
first test fax message. To create a test fax, just start any Windows application (i.e. Microsoft®
Word or WordPad) and print a document to the ActiveFax printer. When the fax dialog window appears, enter the fax number of the recipient (light-blue input field) and confirm the fax
with “OK”. The fax message is now automatically transmitted by the fax server as soon as a
modem becomes available.
For an automatic installation of ActiveFax it is also possible to run the setup with command
line parameters. A complete list of all command line parameters can be displayed when you
running the setup with command line parameters can also be found in the ActiveFax
Knowledge Base at http://www.actfax.com/en/kbase.html?id=6822.
Please note that it is not required to restart the computer after the installation of ActiveFax.
All configuration settings are already active right after the setup program has been finished.
When using Brooktrout fax boards (i.e. TR114, TR1034 or Trufax) you should install the drivers
for the fax board before doing the installation of the fax server. Because Brooktrout does not
directly provide drivers for their fax boards, drivers for Brooktrout fax boards are available at
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2.3. Customizing Program Settings
Most settings and parameters of the fax server are already initialized by the setup program
with default values. These default values reflect the most common settings that normally do
not need to be changed. However if you need to change some settings of the fax server, this
can be done with the relating menu options directly on the fax server. The table below shows
a listing of the most common settings:
Setting
Modem Configuration
Email Configuration
Users and Groups
Access Rights
Default Sender
Fax Resolution
Archive Settings
Automatic Printing
Fax Title
Next Fax ID Number
Automatic Startup
Communication / Modem
Communication / E-Mail
Extras / Security Settings
Extras / Predefined Settings
Extras / Options / Modem
Extras / Options / Archive
Extras / Options / Printing
Extras / Options / Fax Title
Extras / Options / General
File / Automatic Startup
Please note that the above table is only a short summary of the most common configuration
settings of ActiveFax. A detailed description of all settings can be found in the related chapters
2.4. Establishing a Client Connection
When the fax client (or ActiveFax printer driver) is started for the first time, it tries to find the
fax server in the network to establish a connection. ActiveFax supports three different ways
the client can connect to the fax server.
Normally the connection between the fax server and the fax client is done through one of the
network protocols TCP/IP or NetBeui (Windows Network). If the fax server and fax client are
installed on the same computer, it is also possible to use a direct connection (without network). Since the fax client automatically scans the network for all available fax servers, there
is no need to recall network or IP addresses. The first fax server found by the automatic detection will be displayed and used as the default server.
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The IP address of the fax server can also be set directly at the installation of the fax client when
you run the setup with the command line parameter -IP:IP address.
Example:
actfax_setup_en.exe -Client -IP:192.168.0.1
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3. How to ...
3.1. Create a New Fax Message
New fax messages can be created in various ways with ActiveFax. The easiest and most common way to create new fax messages is the printing function of a Windows application. As an
alternative it would also be possible to create “Instant Fax Messages” with ActiveFax. Instant
fax messages are used for short messages with just a few lines of text.
With Named Pipes or RAW Sockets it is possible to create fax messages without using the
Windows printer driver (no GDI interface). That way, fax messages can be created directly
from inside an application without using the Windows printing subsystem.
3.1.1. Faxing from Windows Applications
Start the Windows application that should be used to create the fax message (i.e. Microsoft® Word).
Select the printing function of the application and print the document to the ActiveFax
printer.
A dialog box appears. Enter the fax number of the recipient (light blue input field) there.
Confirm the fax with OK.
After the document has been “printed” that way, it is automatically transferred to the fax
server. Depending on the preferred transmission time and priority, the fax message is then
automatically transmitted as soon as a modem becomes available.
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3.1.2. Instant Fax Messages
Fax messages often only contain just a few lines of text. If you do not want to create a Word
document for such small messages, it is also possible to create the fax as an instant fax messages directly within the fax client program. An instant fax message is built by an optional
cover page and the text of the fax message.
It is not recommended to use instant fax messages for faxes larger than a single page or for
faxes that require complex formatting of the text. In such cases, it is recommended to write
the message with traditional text processing software and to print the document to the ActiveFax printer.
Formatting limitations of instant fax messages only allow the same font to be used for the
complete message. A preview window shows the appearance of the fax message in minimized
view.
To create a new instant fax message, follow these steps:
Choose the menu option File / New Instant Fax Message.
Optionally select a cover page.
Write the text of the fax message.
Specify the recipient of the fax message.
Confirm the fax with OK.
The main advantage of instant fax messages is that it is possible to subsequently modify the
text of the message through the fax dialog window. That way it is not required to create a
completely new document when you encounter spelling mistakes after the fax has been created.
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3.1.3. File System
Another powerful method of sending fax jobs to ActiveFax is the file system method. Using
that method, the files to be faxed just need to be stored in a predefined directory. ActiveFax
automatically scans this directory for new fax jobs and imports them for further processing.
When using the file system method, it is also possible to use data fields directly as part of the
file name, which makes it very simple to set the recipients fax number. The file system method
can be configured with the menu Communication / File System on the fax server.
Sample File Name:
C:\Import\Fax\[email protected] Duncan [email protected]@F211 [email protected]
When importing files from a network share, take care to always use the complete UNC path
for the network share (i.e. \\server\share). Do not use mapped drive letters, since mapped
drive letters are normally not visible to processes running as a service.
If you need to use special characters not allowed in a file name (i.e. * or ?) as part of the data
fields used in the file name, it would be possible to code such characters with a % sign, followed by the ASCII code of the character in hexadecimal format (i.e. * can be coded as %2A).
3.1.4. Named Pipes
When using “Named Pipes” it is possible to easily create fax messages directly from within
applications without using the ActiveFax printer driver. When not using the ActiveFax printer
driver, the main difference is, that the fax message is sent in plain text format in that case.
Quite similar to fax transmissions from UNIX or Linux systems, printer commands of HP-LaserJet (PCL), Epson-LQ and optionally Postscript and PDF can be used to format the document.
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Data fields can be added the same way as printer commands; they are written directly to the
document. Named Pipes can be used as an alternative when you do not want to work with
the Windows graphic subsystem (GDI).
Take care that Named Pipes are only available on fax servers running on Windows XP / 2003
/ Vista / 2008 / 7 / 2012 / 8 / 10 systems.
To use Named Pipes, just a few small modifications have to be done in your application. Follow
these steps to integrate the communication through Named Pipes with your application:
The Named Pipe needs to be configured on the fax server. To do so, choose the menu
option Communication / RAW Server and create a new entry for a NetBeui connection. The
name of the Named Pipe can be individually chosen (the default pipe name is “ActiveFax”).
Open the Named Pipe with writing permission in your application. This can be done with
normal API file functions. The file name is built in the format “\\servername\pipe\pipename” (i.e. \\faxserver\pipe\activefax). To open a Named Pipe, use the API function call
CreateHandle() or any corresponding function of your programming language.
Send the fax message directly to the previously opened file. You can use the API function
call WriteFile() or any corresponding function of your programming language to write the
data to the file. It is also possible to use printer commands and data fields in that case.
Close the Named Pipe with the API function call CloseHandle() or any corresponding function of your programming language. By closing the file (Named Pipe), the fax server automatically gets a signal that the fax message has been finished and can be transmitted.
The function calls for opening, writing and closing files (Named Pipes) may be known under
tool (i.e. Visual-C, Visual Basic, Delphi, Power Builder, ...).
3.1.4.1. Example in Programming Language C
#include <windows.h>
HANDLE
DWORD
BYTE
BYTE
DWORD
hFile;
dwError;
szText[1024];
szFax[128];
dwWritten;
int main(void)
{
hFile = CreateFile("\\\\faxserver\\pipe\\activefax",
NULL,
OPEN_EXISTING,
0,
NULL);
if (hFile == INVALID_HANDLE_VALUE) {
dwError = GetLastError();
return 1;
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}
lstrcpy(szFax, "555-123-4567");
wsprintf(szText, "This is a [email protected] %[email protected]", szFax);
WriteFile(hFile, szText, lstrlen(szText), &dwWritten, NULL);
CloseHandle(hFile);
return 0;
}
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3.2. Access Fax Messages on other Computers
The fax client can be used to display and access fax messages from other computers in the
network. Through a network connection to the fax server, access to all fax messages is available from any computer in the network with the fax client program installed.
3.2.1. Fax Client Installation
The fax client needs be installed on every computer that should be used to display, create or
control fax messages. The installation of the fax client is done with the same setup program
as the installation of the fax server. When running the setup, take care to choose the option
for a Client Installation, since a complete installation takes more disk space as required and
the fax server part of ActiveFax should not be installed on client systems.
3.2.2. Establishing a Client Connection
The first time the fax client is started the network is automatically scanned for all available fax
servers. After the scan completed you can choose the preferred fax server from the list (normally there is only a single fax server in a network).
When using a router for the network connection of the fax client, it could happen that the fax
server cannot be automatically detected by the fax client because of blocked broadcast packages used to identify the fax server in the network. In that case you have to manually enter
the IP address or hostname of the fax server.
The connection between the fax server and the fax client is established with one of the network protocols TCP/IP or NetBeui (Windows Network). Take care that connections with NetBeui are only possible when the fax server has been installed on Windows a XP / 2003 / Vista
/ 2008 / 7 / 2012 / 8 / 10 system. If you have both of these network protocols available, it is
recommended to always use the TCP/IP protocol, since TCP/IP requires less resources and is
a bit faster compared to NetBeui.
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Please note that the connection settings can be subsequently changed with the menu option
Communication / Network Settings.
The update of the data on the fax client is done automatically. Whenever the configuration of
the fax server changes or when a fax message is created or modified, the data on the fax client
is automatically refreshed on all client computers in the network. All clients have the same
data set at any time that way.
Take care that the total number of fax clients allowed to connect to the fax server is limited
by the license. If you exceed the number of users registered with the license, a message is
displayed on the fax client that exceeds the license limit; fax clients exceeding the license limit
cannot connect to the fax server. Fax clients that connected to the fax server before the license limit has been reached are not affected.
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3.3. Send Fax Messages from UNIX / Linux
In addition to the capability of sending fax messages from Windows applications using the
ActiveFax printer driver, it is also possible to create fax messages from other operating systems, like UNIX or Linux. Normally the connection between the UNIX / Linux server and ActiveFax is done through the LPD/LPR (Line Printer Daemon) protocol. Since this protocol is
based on the TCP/IP standard and is therefore included with all UNIX systems and Linux, it is
the preferred method for sending fax messages from UNIX.
When using LPD/LPR, ActiveFax is accessed exactly the same way as any other network
printer. The configuration of the printer in UNIX is done the same way as the configuration of
any other printer server (i.e. Extended Systems, AXIS, Emulex, etc.). The host name of the
remote printer is the address of the computer where ActiveFax has been installed. The queue
name can be individually chosen, since it is ignored by ActiveFax when using default settings.
By default the queue name “fax” is used.
3.3.1. Configuring LPD/LPR Printers in UNIX
3.3.1.1. IBM RS/6000 (AIX)
Start the system administrator program “smit”.
Choose the menu options Devices, Printer/Plotter, Manage Remote Printer, Client Services,
Remote Printer Queues and Add a Remote Queue.
Fill in the dialog box. In the Destination Host field enter the host name of the computer
where ActiveFax has been installed. The field Queue Name can be filled with any queue
name (i.e. “fax”).
3.3.1.2. HP-9000 (HP/UX)
Start the system administrator program “sam”.
Choose the menu option Printers and Plotters, Actions and Add Remote Printer.
Fill in the dialog box. In the Remote Printer Name field enter the host name of the computer where ActiveFax has been installed. The field Queue Name can be filled with any
queue name (i.e. “fax”). You should also enable the option Remote Printer is on a BSD
System.
3.3.1.3. Other UNIX Systems
To configure an LPD/LPR printer for other UNIX systems, it is recommended to consult your
system documentation. Usually an LPD/LPR printer is created with an entry in the
“/etc/printcap” file. Printer entries in the /etc/printcap file are created based to the following
scheme:
printername:\
:rm=hostname:\
:rp=queuename:
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fax:\
:rm=192.168.0.1:\
:rp=fax:
30
On some UNIX systems (i.e. SCO UNIX) it could also be required to activate the LPD/LPR protocol. For SCO UNIX this can be done with the command “mkdev rlp”.
3.3.2. Sending Fax Messages with LPD/LPR
After the network printer has been created on the UNIX system, the “lp” command can be
used to send print jobs to ActiveFax. To create a new print job with “lp”, use one of the following commands:
lp –dprintername filename
cat filename | lp –dprintername
Using the “lpstat –t” command or “lpstat –oprintername” the current printer status can be
displayed.
3.3.3. Alternatives to LPD/LPR
The LPD/LPR protocol is for sure the easiest and most powerful method to send fax messages
from UNIX systems. If it is not possible to use an LPD/LPR printer, ActiveFax also supports
numerous other protocols to be used instead of LPD/LPR. A short overview about these protocols can be found in the following sections.
3.3.3.1. FTP, TFTP and RAW Sockets
Using the TCP/IP network protocol it is also possible to send fax messages with FTP, TFTP and
RAW sockets. When using the FTP or TFTP protocol, the files to be faxed just need to be transferred to the virtual FTP server built-in with ActiveFax. The steps are exactly the same as for
RAW sockets, a direct TCP/IP connection to the fax server is established on a pre-set TCP/IP
port. The fax data is sent directly through that connection without any underlying protocol.
The fax message is finished by simply closing the TCP/IP connection.
3.3.3.2. Serial Connection (RS-232)
If there is no network connection available, data exchange with ActiveFax can also be done
through a serial interface. Using that method the UNIX system is connected through a serial
cable to the fax server PC. On the UNIX system it is just required to configure a printer that
sends the data to ActiveFax through the serial interface.
3.3.3.3. File System (NFS, Samba)
Another way to send fax data to ActiveFax is using the file system. When using that method
of data exchange, a predefined directory of the UNIX system is mounted (shared) on the fax
server computer. This can be done quite easily with software tools like NFS or Samba. The
mounted directory is automatically scanned for new fax files to be processed in that case.
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3.3.3.4. Printer Commands (HP-LaserJet PCL, Epson-LQ, Postscript, PDF)
Fax jobs sent from UNIX systems can also contain printer commands of HP-LaserJet (PCL and
HPGL), Epson-LQ and optionally Postscript and PDF. That way the printer output does not need
to be modified and you can send exactly the same output to the fax server that is normally
sent to physical printers. For direct processing of Postscript and PDF files it is required to have
the add-on software Ghostscript installed on the fax server (www.ghostscript.com).
3.3.3.5. Data Fields
To set the recipients fax number or other parameters of a fax messages (i.e. subject, priority,
etc.) already from within an application, it is possible to use data fields. Just like printer commands, data fields are embedded directly in the document to be faxed.
in this manual in the chapter Data Fields and in the online help of the fax server.
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3.4. Enter valid Fax Numbers
There is no special format required for fax numbers processed by ActiveFax. It is permitted to
use special characters like spaces, slashes, dashes or dots to format a fax number. Such characters are ignored by the fax server and are automatically removed for dialing. In general, fax
numbers can be entered in international format as well as in national format. Depending on
the local area code of your location the fax server automatically adjusts the number dialed by
the modem. It makes no difference if the fax number is entered as a local number or as a
complete number including country code and local area code. For a uniform appearance, it is
recommended to enter fax numbers always in the same format.
ActiveFax also supports internal fax numbers that do not use the dial prefix for dialing. By
default, such numbers are marked with an “X” character at the beginning of the number. The
character used to mark internal numbers can be changed in the modem configuration.
3.4.1. Examples for valid fax numbers
Own Country Code: 1 (USA)
Own Area Code:
712
Entered Fax Number
Dialed Fax Number
+1 934 431 7633
365-874-1297
712.887.3274
+49 89 102030-40
X125
19344317633
13658741297
8873274
011498910203040
125 (no dial prefix!)
Own Country Code: 49 (Germany)
Own Area Code:
089
Entered Fax Number
Dialed Fax Number
+49 40 102030-40
+43 1 98765
0043/1/987-65
0049 89 102030-40
089/102030-40
04010203040
0043198765
0043198765
10203040
10203040
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Through the menu View / View Properties you can change the number of pages to be displayed
on the screen. You can set the number of pages to be displayed horizontally and the number
of pages to be displayed vertically.
Especially when using wide screens (16:9 aspect ratio) it makes sense to configure the number
of pages to be displayed on the screen to 2 or 3 pages, because the space on the screen is
used more efficiently in that case and fax pages in portrait format fit better on the screen.
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3.6. Change the Faxlist Columns
3.6.1. Columns
The columns displayed in the faxlist can be individually changed through the menu Faxlist /
Faxlist Properties. You can configure the set of columns to be displayed in the faxlist as well as
the position of each column within the faxlist.
Please note that these settings can be individually configured for every fax client and for the
fax server.
3.6.2. Colors
In ActiveFax every status has a different color assigned to differentiate between the current
status of the fax messages without having to read the status text. The color mapping for the
fax status can be individually changed through the menu Faxlist / Faxlist Properties.
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Please note that these settings can be individually configured for every fax client and for the
fax server.
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3.7. Select an Entry in the Faxlist
3.7.1. Selecting Entries
Entries in the faxlist can be selected either with the left mouse button or with the arrow keys
on the keyboard. To execute a function (i.e. locking or deleting a fax) on more than one faxlist
entry, it would be possible to select multiple entries at the same time. The Ctrl key and the
Shift key are used to select more than one entry in that case. When using the Ctrl key, individual entries are selected, when using the Shift key, entries in a from/to range are selected.
To select all entries of the faxlist, the menu option Faxlist / Select all Entries can be used.
3.7.2. Sorting Entries
By default, the faxlist is sorted by creation date and time in descending order (latest entries
displayed first). The sort order can be individually changed to meet your requirements. The
field selection as well as the sort order (ascending or descending) can be separately configured
for each column. To change the sort order for the faxlist, follow these steps:
Use the left mouse button to click at the header of a column in the faxlist. The faxlist is
now sorted by this column.
If you click at the same column header again, the sort order changes from ascending to
descending and vice versa.
To add additional sort criteria (i.e. sort by date first, then by time), press and hold the Shift
key and click on the next column header to be added as a sort criteria.
The number of columns that can be used to sort the faxlist is not limited. Columns currently
used as sort criteria are marked with an arrow symbol in the column header. Sort criteria settings are automatically stored when the program is closed.
3.7.3. Searching for Entries
On systems with thousands of fax messages in the faxlist, finding a specific document can be
very difficult. ActiveFax offers various functions that help finding fax messages again very fast.
To search for a specific document, follow these steps:
Enter a known search term in the search field of the faxlist (i.e. subject, recipient name,
fax number, ...). It is also possible to enter multiple search terms, separated with a pipe
sign (|).
Select the user or group of the owner of the fax message in the user selection window.
Select the fax status of the fax message in the fax selection window (i.e. undelivered, sent,
Change the sort order of the faxlist by clicking on the column headers of the faxlist.
The most efficient way to search for a fax message is entering a search term. When entering
significant search terms the faxlist is normally reduced to just a few matching entries.
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Please note that you do not need to enter the exact phrase when searching for a specific entry.
It is normally sufficient to just enter a small part of the term you are searching for. For example
if you want to search for a fax message that was sent to “George Smith”, you can enter just
“Smi” or “george smi”. The search function is not case sensitive. It is also possible to search
for more than one search term. In that case you have to separate the different search terms
with a pipe sign (|). For example to search for the search terms “Smith” and “Los Angeles”,
enter “Smith|Angeles” in the search field.
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3.8. Automatically Redial failed Fax Transmissions
3.8.1. Automatic Redialing
Failed fax transmissions are automatically redialed by the fax server. You can individually configure the number of attempts and the delay after each failed attempt through the menu Extras / Options / Redialing. If the number of transmission attempts should be set to less than
the default number of 10 attempts, you need to configure a delay of 0 hours and 0 minutes
for one of the attempts.
The delay between the redial attempts should increase with each failed attempt, because it is
very likely that there is a long-lasting interruption on the receiving fax machine when several
consecutive transmission attempts fail. By choosing an increasing delay for each failed attempt, you can avoid that all redialing attempts are done within a short period of time.
3.8.2. Send a Fax Message again
If the transmission of a fax messages fails with all transmission attempts, the status of the fax
messages changes to Undeliverable. No further transmission attempts are done for that fax
message then. You should check if the fax number is correct in that case and change the fax
number as required. When changing the fax number of an undeliverable fax message, ActiveFax automatically asks the user if the message should be sent again. As an alternative it would
also be possible to manually send the fax message again through the menu Faxlist / Send
again.
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If a user should be informed about the status of a fax transmission by email, you can configure
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3.9. Automatically Print Fax Messages
Automatic printing of fax messages can be done in various ways with ActiveFax. Fax messages
can be printed either on a specific printer or in dependence of the owner of the fax to individual printers.
3.9.1. Sending Report
The automatic printing function can be configured to either print the complete fax message
in original format or as a sending report. The sending report is made up of status information
of the fax transmission and a preview of the first pages of the fax message printed on a single
sheet of paper. The number of fax pages printed on the sending report can be configured with
the menu Extras / Options in the Printing tab.
3.9.2. Compressed Printing
Compressed printing of fax messages helps to protect the environment and also reduces printing costs. When using compressed printing, multiple pages of a fax message are printed on a
single sheet of paper in compressed format. The number of fax pages to be printed on a single
page can be individually configured.
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3.9.3. General Settings
To activate automatic printing of inbound fax messages, follow these steps:
Choose the menu Extras / Options.
Choose the Printing tab.
Enable the option Automatically print incoming fax messages after receipt and select the
preferred printer.
Optionally choose the Compression option to print multiple pages on a single sheet of paper.
Optionally activate the Print Report parameter to print a sending report instead of the
complete fax message.
Complete the configuration with OK.
Please note, that automatic printing of the transmission protocol can also be configured
through the Printing tab in the menu Extras / Options.
3.9.4. User-dependent Settings
Automatic printing can also be individually configured for each user. Depending on the owner
of a fax message, printing can be done on different printers that way. To configure user-dependent printing, follow these steps:
Choose the menu option Extras / User Administrator or use the corresponding button in
the toolbar.
Select the user entry and press the Modify button.
Change to the Automatic Printing tab.
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Enable the option Use user-defined settings for automatic printing and select the preferred
printer.
Optionally choose the Compression option to print multiple pages on a single sheet of paper.
Optionally activate the Print Report parameter to print a sending report instead of the
complete fax message.
Complete the configuration with OK.
Routing of inbound fax messages to individual user entries can be done in various ways. More
information about inbound fax routing is available in the chapter User Administrator in this
manual.
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3.10. Protect Faxes against Unauthorized Access
Since fax messages often contain confidential information that should not be read by unauthorized persons, ActiveFax supports security features to protect privacy of the documents.
That way fax messages can be reliably protected against unauthorized access.
The base of every reliable security concept should be a well-considered user structure. It is
recommended to create a separate user account for every person working with the software
and to only grant those permissions necessarily needed for daily work. Each user account
should also be protected with a reliable password that is only known to the owner of the user
account.
ActiveFax also supports groups of users. Each user can be a member of multiple groups (i.e.
purchasing department, sales, ...). The main advantage of building groups of users is to grant
all members of a group access to each other’s fax messages (depending on permission settings).
3.10.2. Security Settings
To activate access verification, security settings need to be activated on the fax server. Follow
these steps to enable security settings:
Choose the menu option Extras / Security Settings or use the corresponding button in the
toolbar.
Selected the preferred security level for server access and for client access.
Complete the configuration with OK.
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Please note that security settings for the fax server and the fax client can be individually configured. To guarantee a high security level it is recommended to grant access to the fax server
This ensures that the fax server configuration cannot be modified without having a valid Administrator password and access to the fax client is only possible with a valid user account.
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3.11. Block Fax Numbers (Blacklist)
To avoid that ActiveFax answers fax calls from certain fax numbers or that outgoing fax messages are sent to specific fax numbers, it would be possible to create a blacklist through the
menu Communication / Modem / Blacklist. The blacklist contains all fax numbers that should
be blocked by the fax server.
3.11.1. Block Outgoing Fax Calls
If outgoing fax calls should be blocked by the fax server, take care to enable the option for
outgoing faxes when you create the entry for the blacklist. When a user creates a fax message
for a fax number stored in the blacklist, the fax message is automatically locked by the fax
server. Sending such fax messages is still possible, when the fax is manually unlocked.
Please note, that ActiveFax also supports wildcards (*) when you enter the fax number. When
using wildcards, it would be possible to block a complete range of fax numbers with just a
single entry. For example, to block all fax numbers starting with 555, enter 555* in the fax
number field.
3.11.2. Block Incoming Fax Calls
If incoming fax calls should be blocked by the fax server, take care to enable the option for
incoming faxes when you create the entry for the blacklist. When the fax server detects an
incoming call from a fax number stored in the blacklist, the call is automatically blocked by the
fax server.
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Keep in mind that incoming fax calls can only be blocked when the sending fax machine also
submits a fax number.
Please note, that ActiveFax also supports wildcards (*) when you enter the fax number. When
using wildcards, it would be possible to block a complete range of fax numbers with just a
single entry. For example, to block all fax numbers starting with 555, enter 555* in the fax
number field.
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3.12. Use the Fax-On-Demand Server
The fax-on-demand capabilities of ActiveFax can be used in two ways. It would be possible to
receive documents from other fax-on-demand servers and to use ActiveFax as a fax-on-demand server to make fax documents available to others.
3.12.1. Fax Polling
A fax polling request (this means to receive a fax document from a fax-on-demand server) can
be done with the following steps:
Choose the menu option Communication / Execute Polling (Fax-On-Demand).
Enter the fax number of the fax-on-demand server or use the phone book to import an
existing phone book entry.
Complete the request with OK.
The polling request is executed as soon as a modem becomes available. After the polling request has been successfully completed, the document is available in the faxlist as a received
fax message.
3.12.2. Creating Fax-On-Demand Documents
ActiveFax can also be used to make fax documents available for polling to other fax machines.
In that case, the document gets an individual fax modem or extension number (when using
ISDN) assigned. Especially when using ISDN adapters the number of documents that can be
made available for polling is not limited, since every document is identified by its own direct
dial number. To create a new fax-on-demand document, follow these steps:
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Create the fax-on-demand document with a Windows application of your choice (i.e. Microsoft® Word).
Choose the printing function of the application (i.e. the menu File / Print in Microsoft®
Word).
Select the ActiveFax printer to start the print job.
Change to the More Settings tab in the fax dialog window and activate the Create Fax-OnDemand Document option there.
Change to the Settings tab and select the modem to be used for the document. When
using ISDN it is also recommended to enter a direct dial number (MSN) to be used for the
document.
Complete the document with OK.
After the document has been “printed” that way, the fax-on-demand message is automatically
transferred to the fax server and is available for requests then. Take care that any existing faxon-demand document for the same modem or direct dial number (MSN) is automatically
moved to the archive when a new fax-on-demand document is created.
A counter with the total number of requests for the fax-on-demand document is available
through the fax dialog window of the document. A complete list of fax numbers that already
requested the document can also be found in the transmission protocol.
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3.13. Move Fax Messages to the Archive
To avoid that the faxlist with the fax messages becomes too big and confusing over the years,
old fax messages should be moved to the archive. ActiveFax supports an automatic archive
method as well as manual archiving. The fax server automatically creates an archive folder for
each month.
Please note, that ActiveFax only allows fax messages with the status Sent or Read to be moved
to the archive. This is for security reasons to avoid that fax messages that have not yet been
sent or read are moved to an archive folder by mistake. If it should still be possible to also
move fax messages with a different status to the archive, you can activate this with the parameter ArchiveIgnoreStatus in the options.cfg file.
3.13.1. Automatic Archive
By default the fax server automatically moves fax messages to the archive after 14 days. Only
fax messages with the status Sent and Read are automatically archived in that case. Fax messages that have been locked are also excluded from automatic archiving.
The settings for automatic archiving of fax messages and transmission protocol entries can be
changed with the menu Extras / Options / Archive on the fax server.
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3.13.2. Manual Archive
Using the menu Faxlist / Move to archive or with a right-click at the faxlist entry it is also possible to manually move fax messages to the archive. As an alternative it would also be possible
to manually move fax messages to an archive folder using drag-and-drop when moving the
faxlist entry over to an archive folder.
3.13.3. Individual Archive Folders
As an alternative to the archive folders automatically created by the system for each month,
it would also be possible to create your own individual folder structure for the archive. Individual archive folders can be created with the menu File / Folders / New Folder or with a rightclick at the Archive icon in the fax selection window. Individual archive folders are organized
in a tree structure; the level of subfolders is not limited.
Please note, that the permission to create or modify individual folders can be separately configured for each user.
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3.14. Use Multiple Fax Servers
When you have more than one fax server installed in the network (i.e. a separate fax server
for each branch office), it would be possible to create multiple ActiveFax printers and to add
multiple desktop icons for the fax client on the user’s workstations to connect directly to a
specific fax server. The address of the fax server does not need to be changed manually to
connect to a specific fax server in that case then.
3.14.1. Configure Fax Clients
When running multiple fax servers in the network, it would be possible to create a separate
icon for each fax server connection on the Windows desktop. The desktop icon should include
the name or the location of the fax server (i.e. ActiveFax Client New York). Through the Location field in the properties of the fax client icon, you can set the address of the fax server with
When the fax client is started with the parameter -Server:IP-Address, it is possible to run multiple instances of the fax client. This allows a fax client window to be open for each fax server
in the network.
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To create new fax jobs directly on a specific fax server, it would be possible to add a separate
ActiveFax printer for each fax server. The printer name should include the name or the location of the fax server (i.e. ActiveFax New York). The additional ActiveFax printers can be created directly through the ActiveFax Setup when you run the Setup with the command line
It is also possible to use the -AddPrinter parameter multiple times to add more than one
printer in a single Setup call.
3.14.3. Configure ActiveFax Printers
After an ActiveFax printer has been added for a connection with a specific fax server, you need
to configure the address of the fax server. You can configure the address of the fax server
through the Location field in the properties of the ActiveFax printer with the -Server:IP-Address parameter.
Take care, that none of the ActiveFax printers is in use when you install an update for ActiveFax
at a later time, because otherwise the printer driver cannot be updated.
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3.15. Create a Cover Page or Overlay
3.15.1. What is a Cover Page / Overlay?
In general, cover pages and overlays are created exactly the same way with ActiveFax. A cover
page is a separate page added as the first page of a fax message. Cover pages normally contain
information about the sender, recipient, subject and additional information like date, time
and the number of pages of the document.
Compared to cover pages, overlays are not added as a separate page at the beginning of a fax
message. Overlays are always displayed at the same page as the fax message (the image is
overlaying the fax page, just like printing on a form). That way it is possible to add bitmaps
(i.e. company logos, order forms, …) to fax messages in a very simple way.
3.15.2. Creating Cover Pages / Overlays
When creating a cover page or overlay, it is not differentiated between both. Cover pages and
overlays are both created exactly the same way through the Cover Page Designer of ActiveFax.
Just the way how a cover page or overlay is added to a fax messages determines whether it
will be used as a cover page or an overlay. To create a new cover page or overlay, follow these
steps:
Start the Cover Page Designer with the menu Extras / Cover Page Designer or use the corresponding button in the toolbar.
Select one of the design tools (i.e. text, graphics, data field, …) and design the cover page
or overlay.
Save the cover page and close the Cover Page Designer with the menu File / Close Cover
Page Designer.
It is recommended to save cover pages and overlays to the fax server main directory (C:\Program Files\ActiveFax\Server by default), because only .cov files stored in that directory are
automatically displayed in the dropdown list showing available cover pages and overlays.
Please note that the menu Extras / Page Format can be used to configure the pages on which
an overlay should be visible. That way it would for example be possible to specify that an
overlay should only be displayed on the first page or on the last page of a fax message.
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3.15.3. Using Cover Pages / Overlays
Cover pages and overlays can be added to fax messages in various ways. One method is to
choose the cover page or overlay in the fax dialog window of the ActiveFax printer. With the
menu Extras / Predefined Settings or Extras / User Administrator / Modify / Predefined Settings
it would be possible to configure default cover pages and overlays that are automatically
added for new fax messages. Another way to add cover pages and overlays to fax messages is
in the online help of the fax server and subsequent chapters of this manual.
3.15.4. Using Color Images
When cover pages or overlays are also sent by email, it is recommended that you also provide
a .jpg file of the image. The monochrome .bmp file is used whenever the message is sent by
fax; when the message is sent by email, the fax server uses the .jpg file for the PDF document
sent with the email. If the .bmp file and the .jpg file have the same file name (i.e. logo.bmp
and logo.jpg), the .jpg file is automatically selected when an image is added to the cover page
or overlay.
For best quality, you should only use .jpg files in the format RGB 24-bit (without alpha channel)
and a resolution of 200 dpi or better. The width and height of the .jpg file should be identical
with the width and height of the .bmp file.
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4. Configuration
The User Administrator of ActiveFax is used to manage the accounts for the users and to grant
individual permissions to different users. Users can also be part of a group; each user can also
be a member of multiple groups. There are two predefined users in the User Administrator,
the Administrator and the Unknown user; these user accounts cannot be deleted.
Each entry in the User Administrator is identified by a unique user name. It is recommended
to use the first name or some kind of nickname for the users to get short user names that are
easy to remember.
Each user entry can also have a unique direct dial number (MSN) assigned. With that direct
dial number it is possible to automatically route inbound fax messages directly to a specific
user. Direct dial numbers are only available with ISDN adapters or DID (Direct Inward Dialing)
capable modems and fax boards. When using normal fax modems, direct dial numbers are not
available due to technical reasons.
To avoid that fax messages are not processed during a long absence of a user, it is possible to
configure absence substitutions. The user defined as the absence substitution has full access
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to all fax messages of the absent user. Permission settings are however not transferred when
specifying absence substitutions.
To clearly define responsibilities and to increase security, it is recommended to only grant
those permissions to a user that are necessarily needed for daily work.
4.1.1. User Permissions
Permission
Description
Enables the user to modify important configuration settings of the fax server.
Enables the user to access all fax messages of users that are members of the same group (exception: private fax messages).
Enables the user to access all fax messages (exception: private fax messages).
Access permission for the central phone book.
Write Permission to Central Phone Book
Write permission for the central phone book.
Access and write permission for the transmission
protocol.
Automatic Sending Clearance
Enables the user to send fax messages without
confirmation of a supervisor.
Grant Sending Clearance to other Users
Enables the user to grant sending clearance to
other users.
Permission for International Calls
Enables the user to send fax messages to recipients with international fax numbers.
Execute Polling (Fax-On-Demand)
Enables the user to request fax-on-demand documents from a fax-on-demand server.
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Enables the user to create and manage fax-on-demand documents of the fax server.
Permission to create, rename and delete individual user folders.
Create Private Fax Messages
Permission to create a private fax message, only
visible to the owner of the fax message.
Permission to Change Owner of a Fax
Permission to change the owner of a fax message.
Delete Fax Messages
Permission to delete fax messages.
4.1.2. Alias Names
Many users are known under different names or have multiple logins or user accounts (i.e. on
UNIX systems). In such a case it is possible to configure all alias names for a user through the
Alias Names tab. ActiveFax automatically checks for alias names and replaces them with the
main user name, so it makes no difference which name is used to create a fax message.
4.1.3. Predefined Settings
With the Predefined Settings tab it is possible to individually configure different settings for
each user. You can choose between default sender settings and user-defined sender settings.
Through that configuration screen it is also possible to specify default cover pages and overlays to be used for new fax messages.
The default sender settings (without user reference) can be configured with the menu Extras
/ Predefined Setting.
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4.1.4. Automatic Printing
The settings for automatic printing can be individually configured for each user. That way it is
possible to automatically print fax messages or sending reports directly on the workplace
printer of a user. Select if you want to use the default settings or the user-dependent settings
for automatic printing.
The default settings for automatic printing can be configured through the Printing tab of the
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Email notifications are used to notify a user about the status of outbound and inbound faxes.
For notifications of outbound faxes, the notification for failed fax transmissions is sent after
the last failed transmission attempt (when the status of the fax message changes to “Undeliverable”). As an optional parameter it would also be possible to attach the original fax message
to the email as an attachment. For received faxes you can also automatically change the status
of the faxlist entry to “read” when the notification email is sent.
Email notifications that should be sent to more than one email address can be entered when
you separate the email addresses with a semicolon (i.e. [email protected]; [email protected]; [email protected]).
If the user’s email account requires separate credentials for the login at the mail server, you
can configure this through the “Authentication” button.
If email notifications for outgoing fax transmissions should only be sent for failed fax transmissions, it would be possible to configure this through the menu Extras / Options / General /
4.1.6. Fax Forwarding
Fax forwarding is used to automatically send inbound fax messages to other fax numbers or
email addresses. That way it is possible to forward fax messages received for a specific user to
other fax machines or email addresses.
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For a cost-saving forwarding of fax messages to other fax machines, it is possible to automatically set the priority of forwarded faxes to Low. In dependence of the charge optimization
settings, fax messages are forwarded during night hours that way.
4.1.7. Archive (Export)
The settings for the archive function (fax export) can be individually configured for each user.
That way it is possible to use a different export directory for each user. Select if you want to
use the default settings or the user-dependent settings for the archive.
The default settings for the archive function can be configured through the Archive tab of the
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4.1.8. Group of Users
For a better overview about the users stored in the User Administrator, users should be arranged in groups. Every user can be a members of an unlimited number of groups. The group
membership of a user also controls access to the fax messages of other group members. By
default, a user can only see fax messages of other users of the same group.
A user only has access to fax messages of other group members when the permission setting
To avoid that a user can see all faxes stored on the system, it is important to have the option
“Access to all users” disabled for the user. If this permission is enabled for a user, the user has
4.1.8.1. Passive Group Members
If a user should be a member of a group, but other group members should not be able to see
that user, you can define the user as a passive group member. To define a user as a passive
group member, right-click at the user or group entry in the group selection window and select
the option “Passive Group Member” from the menu.
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When a user is a passive member of a group, the fax messages of that user are not visible to
other group members. The user itself however has access to fax messages of all other (nonpassive) group members. A passive group membership is typically used for team leaders that
should have access to the fax messages of all team members, but the team members should
By default, the login of a user is done through the user account and password stored in the
User Administrator of ActiveFax. That way, any user can login with an individual user account
from any workstation in the network. As an alternative it would also be possible to use the
user name from the Windows login for the login of the user. You can activate this with the
Please note, that you still need to create the user accounts in the ActiveFax User Administrator, even when the Windows user name is used for the client login. This is required, because
otherwise it would not be possible to configure individual settings for the users (i.e. notifications for fax transmissions).
When using the Windows user name for the client login, it is not required to set a password
for the users in the User Administrator. Since user authentication has already been done with
the Windows login, no further authentication needs to be done by ActiveFax, so no passwords
are required for the user accounts.
4.1.10. Routing of Inbound Fax Messages
Routing of inbound fax messages can be done in various ways. Depending on the technical
capabilities of the modem or ISDN adapter and the phone line, not all routing methods are
always supported. ActiveFax uses the routing methods in the following order.
4.1.10.1. Routing using Direct Dial Numbers (MSN, DDI, DID, DTMF)
This method of inbound fax routing is the most reliable way to route faxes, since every user
has its own unique fax number in that case. Due to technical reasons, this routing method is
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only available with ISDN adapters or DID (Direct Inward Dialing) capable fax modems or fax
boards. It is not possible to use direct dial numbers with normal fax modems. To use direct
dial capabilities of an ISDN adapter, it is required that the ISDN line supports either the MSN
(Multiple Subscriber Number) or DDI (Direct Dial In) service. Activation of these services is
normally done directly by the phone network provider.
The direct dial numbers for the different users can be configured through the Direct Dial (MSN)
field when you modify the settings for a user in the User Administrator. You just need to enter
the direct dial number there; it is not required to enter the complete fax number.
When using a phone system (PBX) it could be required to configure the ISDN line (S0 bus) to
4.1.10.2. Routing using CSID (Sender Identification)
This kind of inbound fax routing is based on the sender identification (CSID) of a fax message.
Based on user names mapped to phone book entries, faxes received from known fax numbers
are automatically routed to individual users. To map a user to a phone book entry, follow
these steps:
Open the phone book with the menu Extras / Phone Book or use the corresponding button
in the toolbar.
Create a new phone book entry or select an existing entry.
Modify the entry with the Modify button or double-click on the entry.
Enter the sender identification in the CSID field. Please note that it is not required to fill in
the CSID field if the CSID and the fax number are identical. Special characters, like spaces,
slashes, dashes or dots are ignored when the fax number and CSID are compared.
Select the User that should be mapped to the phone book entry.
Complete the phone book entry with OK.
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4.1.10.3. Routing using Modem
When using this kind of inbound fax routing, a user is mapped directly to a modem (fax number). The limitation of that routing method is that it can only be used for a limited number of
users, since a dedicated modem (fax line) is required for each user that should be routed. To
configure modem based routing, follow these steps:
Choose the menu Communication / Modem or double-click on the corresponding icon in
the communication window.
Select the modem and press the Modify button or double-click on the selected entry.
Select the user name that should be mapped to the modem in the Default User for Incoming Faxes field.
Complete the configuration with OK.
4.1.10.4. Manual Routing
Manual routing is used when none of the above routing methods can be used. When using
manual routing, the fax document needs to be manually opened on the fax client to check
which user is the recipient of the fax message. The user has to be manually entered through
the fax dialog window in that case. To specify the user for a fax message, follow these steps:
Select the fax message with the left mouse button and find out to which user the fax messages belongs.
Double-click on the faxlist entry to display the fax dialog windows and select the user name
or move the faxlist entry over to the new user name in the user selection window left of
the faxlist window using drag-and-drop.
Complete the fax dialog window with OK.
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4.1.10.5. Order of Routing Methods
The user for inbound fax messages is identified according to the following order of routing
methods:
Check of direct dial numbers (if available).
Check of sender identification (CSID) using the phone book.
Check of default user mapped to the modem (default user is Administrator).
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4.2. Phone Book
Phone book entries in ActiveFax can be stored either in a global (central) or local phone book.
The central phone book is shared by all users and can be accessed from any client computer
in the network. The private phone book is stored directly with a user account and can only be
accessed by the owner of the phone book. Phone book entries that should be visible to all
users should be added to the central phone book; for confidential or private phone book entries, the user’s private phone book should be the preferred choice.
It is recommended to organize phone book entries in groups, for example grouped by branch
or department. Doing so makes it much easier to find phone book entries again. To search for
a specific phone book entry, the search function of the phone book can be used.
It is also possible to select more than one phone book entry at the same time by using the Ctrl
or Shift key. Especially for fax mailings, faxes often need to be sent to a complete group of
recipients. Using the right mouse button and the menu Select All it would be possible to select
all entries displayed in the phone book.
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4.2.1. Importing the Phone Book
Phone book entries can be imported in two ways with ActiveFax. One method is to import the
phone book entries from an external ASCII file (text file). The second method is to import the
phone book entries from an external database using the ODBC standard.
4.2.1.1. Import from ASCII File
To import phone book entries from an external ASCII file, follow these steps:
Select the Options / Import from File button.
Enter the File Name of the import file or use the Search File button.
Specify the Character Set and the Field Delimiter for the import file.
Specify the field order for the import file. Fields not included in the import file are automatically left blank.
Check the settings in the preview window and start the import of the phone book entries.
At the import process of the phone book entries the field ID-Number (i.e. customer or supplier
number) and the fields Name 1 and Fax Number are compared. If there is a matching entry
found in the phone book for these fields, the phone book entry is updated; otherwise a new
entry is created in the phone book.
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4.2.1.2. Import from ODBC Database
Another method to import phone book entries is to bind the phone book to an ODBC data
source. An external database is automatically checked for new and modified phone book entries in that case. More information about the import of phone book entries from an ODBC
data source can be found in the chapter ODBC Database of this manual.
4.2.2. Exporting the Phone Book
Phone book entries can be exported to an ASCII file (text file) with the Options / Export to File
button. The export file always includes all data fields of the selected phone book entries. Before the phone book is exported, make sure to configure the correct character set and field
delimiter.
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4.3. Transmission Protocol
The transmission protocol stores information about all outbound, inbound and fax-on-demand fax transmissions. This includes successful transmissions as well as status information
for incomplete or failed transmission attempts. When using ISDN adapters with the ISDN service AOC (Advice of Charge) activated, the transmission protocol also includes charging information. That way it would be possible to calculate the total charges for a user or cost account
code.
Detailed information for a transmission protocol entry can be displayed with the Details button or when you double-click at the selected protocol entry.
4.3.1. Printing the Transmission Protocol
The transmission protocol can be printed in three different ways. Use one of the buttons described below to print the protocol entries.
Print Option
Description
Print All Entries
All entries currently displayed in the transmission protocol list are
printed.
Only new entries that have not yet been printed are printed.
Print only New Entries
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Print only Selected Entries
Only selected entries of the transmission protocol list are printed.
4.3.1.1. Layout
Through the Options tab the sort order and layout for the printout of the transmission protocol can be changed. You can change between single-line and double-line layout there.
4.3.1.2. Automatic Printing
The transmission protocol can also be automatically printed. Use the Printing tab in the menu
Extras / Options to configure automatic printing of the transmission protocol.
4.3.1.3. Sending Report
As an alternative to the printout of the transmission protocols it is also possible to automatically print a sending report for each fax message. Sending reports are automatically printed
right after a fax message has been transmitted. The configuration for sending reports is done
through the Printing tab of the menu Extras / Options or with the User Administrator if you
need to configure individual printers for each user.
4.3.2. Exporting the Transmission Protocol
The entries of the transmission protocol can be exported in three different ways. Use one of
the methods below to do the export:
Use the Export button to export all selected protocol entries to an external ASCII file (text
file).
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Use ODBC data exchange to automatically export the transmission protocol entries to an
external database.
Use FTP to retrieve the transmission protocol from other computers in the network (i.e.
UNIX or Linux).
4.3.3. Archiving the Transmission Protocol
The entries of the transmission protocol can be automatically moved to an internal archive to
avoid that the protocol becomes too large over the years. Automatic archiving can be configured with the Archive tab in the menu Extras / Options. Old protocol entries are automatically
moved to the archive after 14 days by default.
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4.4. Charge Optimization
4.4.1. Delayed Transmissions
Depending on the fax volume and the location of the recipients it would be possible to reduce
phone charges by using delayed transmissions. When using automatic transmission delays,
the best (cheapest) transmission time is automatically calculated based on the priority of a fax
message and the phone rates of your phone network provider.
To activate automatic optimization of the transmission time, follow these steps:
Choose the menu Extras / Charge Optimization or use the corresponding button in the
toolbar.
Activate the option Enable Automatic Optimization of Charges.
Configure the maximum transmission delay admitted for the single priority levels.
Change to the Rates tab and configure the rates for the different days of the week and
times of the day.
Complete the configuration with OK.
Take care that automatic charge optimization is mainly affected by the priority of the fax messages. This makes it necessary to set the priority level of low-priority fax messages to “low”.
As an alternative it would also be possible to manually change the preferred transmission time
for a fax message. That way large fax mailings can be sent during night hours or at the weekend.
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4.4.2. Time Limitation
To avoid that fax messages are sent during specific times of the day or days of the week, it
would be possible to configure a time frame for each priority level. Depending on the priority
level of a fax, faxes are sent only during these times then. As an alternative it would also be
possible to configure time limitations to be used to suspend fax transmissions for a specific
time frame.
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4.4.3. Least Cost Routing
Another way to reduce phone charges is using least cost routing. Least cost routing can be
used if you have more than one phone network provider; based on the area code of the fax
number and the time of the day, ActiveFax automatically uses the best (cheapest) provider for
the transmission.
To activate least cost routing, follow these steps:
Choose the menu Extras / Modem or double-click on the corresponding icon in the communication window.
Change to the Least Cost Routing tab.
Activate the option Enable Least Cost Routing for outgoing calls.
Add an entry for each area code / time combination. Enter the area code first, followed by
the weekday, time and net access number.
Complete the configuration with OK.
Please note that it is not required to enter the complete area code when using least cost routing. “8” for example covers all fax numbers beginning with “8” (i.e. 89, 873, ...).
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4.5. Modem & ISDN
ActiveFax supports fax modems of all fax class standards, ISDN adapters compatible with the
CAPI 2.0 standard as well as dedicated fax boards from Brooktrout and Intel/Dialogic. See the
summary below for an overview of all fax standards supported by ActiveFax.
Modem Class
Standard
Description
Fax Class 1 / 1.0
TIA/EIA 578
This fax standard is supported by virtually all modems.
Fax Class 2
SP-2388, TR-29.2
This standard is normally only supported by old modems and has been replaced with Fax Class 2.0.
Fax Class 2.0
TIA/EIA 592
This standard is supported by most modern modem
types.
ISDN
CAPI 2.0
This standard is supported by virtually all ISDN adapters. Take care that the ISDN adapter also needs to support the Fax Group 3 (T.30) service.
BFAX
Brooktrout Fax and Voice API This standard supports Brooktrout TR1034, TR114 and
(BfvAPI)
Trufax fax boards.
GFAX
Intel/Dialogic Gammalink API This standard support Intel/Dialogic fax boards of the
CPi series.
The number of modems, ISDN adapters and fax boards supported by ActiveFax is not limited.
It is also possible to use different modems, ISDN adapters and fax boards on a single system.
Please note that the term “modem” is used for fax modems as well as ISDN adapters and fax
boards. Otherwise explicitly noted, there is no difference between fax modems, ISDN adapters
and fax boards when the term “modem” is used.
4.5.1. Voice over IP (VoIP / T.38)
Instead of using normal phone lines to transmit faxes, it would also be possible to use ActiveFax in combination with VoIP. When using VoIP, a special interface software is used instead
of the fax modem or fax board. ActiveFax supports the XCAPI software from TE-Systems
(www.xcapi.com) and the SoftIP software from Dialogic (www.dialogic.com) for VoIP integration.
When using VoIP for fax transmissions, it is recommended to make sure that the VoIP connection also supports the T.38 (Fax over IP) standard. VoIP connections without T.38 support are
often not reliable enough for solid fax transmissions.
Compared to normal analog fax modems, using ISDN adapters offers several advantages. ISDN
adapters might also be slightly cheaper compared to fax modems, especially when using ISDN
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Availability of direct dial numbers (individual fax number for each user). Required ISDN
service: MSN (Multiple Subscriber Number) or DDI (Direct Dial In).
Recording of transmission charges. Required ISDN service: AOC (Advice of Charge).
Simultaneous fax transmissions on multiple phone lines (channels). When using BRI (Basic
Rate Interface) 2 channels are supported by default; when using PRI (Primary Rate Interface) up to 30 channels are available with a single ISDN controller.
4.5.3. Modem Configuration
The setup program of ActiveFax automatically detects and configures all modems, ISDN adapters and fax boards connected to the system. To subsequently add or modify modems, use the
menu option Communication / Modem or double-click on the modem icon in the communication window.
Use the New or Modify button to create a new entry or to modify an existing entry in the
modem list.
Select the COM-Port (interface) the modem is connected to (i.e. COM01, COM02, ISDN,
etc.). Optionally use the Port Settings button to change the configuration of the COM port.
Set the Modem Type or use the Auto Detection button to automatically detect the modem
type.
When using an ISDN adapter, it is also recommended to enter at least one MSN (direct dial
number) in the MSN field. It is possible to configure up to three MSN here. Additional MSN
for the different users can be added through the User Administrator. Please note that ActiveFax answers to all incoming calls (also global calls), if you do not configure at least one
valid MSN either in the modem configuration or the User Administrator.
Depending on the settings of your phone system (PBX) it could also be required to configure additional parameters (i.e. dial prefix, dial method, etc.).
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Complete the configuration with OK.
When using an old phone system (PBX) or the modem is connected to an analog phone system, it could be required to use pulse dialing instead of tone dialing. Take care to choose the
correct dialing method in that case, since outgoing calls cannot be successfully dialed otherwise.
When using an ISDN adapter, you should take care to configure a valid MSN (direct dial number) either in the modem configuration or the User Administrator (at least for one user). If
there is no MSN configured for the fax server, ActiveFax answers to all incoming calls (also
global calls).
The Least Cost Routing tab can be used to automatically select the best (cheapest) phone network provider. Depending on the transmission time and area code of a fax message, net access
be found in the chapter Charge Optimization of this manual or in the online help of ActiveFax.
When using an ISDN adapter, multiple phone lines (B-channels) are available. To avoid that
ActiveFax uses all available channels for fax transmissions, the number of channels used for
faxing can be limited with the Extended button. Specify the total number of channels and the
number of channels used for outgoing calls here. That way it would be possible to keep some
channels in spare for other purposes (i.e. phone calls, Internet connections, ...).
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4.5.4. Modem Status
To display the current status of a modem, click on the modem icon in the communication
window (i.e. Modem on ISDN01). The status window shows the following information about
the fax transmission:
Current status (waiting, sending, receiving, error, etc.).
Other party (fax number and name).
Transmission parameters (transmission rate, resolution, compression).
Additional modem information (direct dial code, charge).
Transmission progress for the current page.
Start time of the transmission.
Duration of the transmission.
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4.6. E-Mail
As an alternative to fax transmissions, ActiveFax can also be used to send any document by
email. Depending on the configuration of the fax server, ActiveFax tries to convert the document to text format whenever possible. If the conversion to text format cannot be done, the
document is sent as an email attachment in PDF, TIFF or GIF format.
4.6.1. SMTP Server (Mail Server) Configuration
Delivery of emails is done through an SMTP server (Simple Mail Transfer Protocol). To configure the email service in ActiveFax, you just need to specify the address of your SMTP server
and the type of Internet connection for the fax server PC. If you do not know the address of
Enter the hostname or IP address of the SMTP server.
Specify whether you have direct access to the Internet or if you need to use dialup connections.
If you need to use dialup connections, configure the settings for the Remote Access Service
(RAS).
If you are using an external mail server (i.e. the mail server from your Internet Service Provider), it is strongly recommended to also enable one of the encryption protocols SSL/TLS or
STARTTLS for encrypted communication between ActiveFax and the mail server.
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4.6.2. Mail Server Authentication
Many mail servers require the user to login on the mail server to send emails to protect the
mail server from unauthorized access by spam senders. If your mail server requires authentication, you can configure this through the Authentication button.
If your mail server requires the email address used for authentication to be identical with the
email address of the sender, you need to configure individual authentication settings for each
user. The user-dependent authentication settings can be configured through the menu Extras
4.6.3. E-Mail Options
The settings of the email service can be individually configured to set the default attachment
format and other parameters.
The option Use individual user names for the “From” field of an email is used to define if
the complete user name should be used in the “From” field of the email. If this option is
disabled, the Name field (company name) is used instead.
The option Convert fax message to text format whenever possible controls if the fax server
should try to convert documents to text format (ASCII or HTML) whenever possible. If the
conversion cannot be done, the document is sent as an email attachment in PDF, TIFF or
GIF format.
The option Request receipt confirmation for emails is used to request a confirmation about
the successful receipt of the email from the email recipient.
The option Enable cover page and overlays for emails defines if cover pages and overlays
should be activated for messages sent by email.
The option Enable fax title for messages sent as an attachment controls if the header of
the fax message should be added to emails.
The File format for fax messages sets the file format for email attachments. Documents
are sent as an email attachment whenever it is not possible to convert a document to text
format. The default file format is PDF.
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The option Send emails always immediately is used to configure whether an email should
be delivered immediately (as soon as it is received by the fax server) or if the fax server
should wait for a predefined number of emails to be in the transmission queue. Especially
when using dialup connections for Internet access, this option can be very useful to reduce
phone charges.
The setting Share communication port for fax and email has to be activated if you share
the same modem for fax transmissions and dialup connections for Internet access. The fax
server automatically disconnects the modem connection in that case to give the Windows
RAS manager the chance to do a dialup connection to the Internet.
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4.7. ODBC Database
Using the ODBC data exchange standard, data can be exchanged between ActiveFax and external databases. Since Windows ODBC drivers are available for virtually all database products,
ActiveFax can be easily integrated with such databases. ActiveFax supports the automatic import of the phone book as well as the automatic export of the transmission protocol through
ODBC.
4.7.1. Selecting a Data Source
To use ODBC data exchange with ActiveFax, a data source has to be selected first. Follow these
steps to select an ODBC data source:
Choose the menu Communication / ODBC Database or double-click on the corresponding
icon in the communication window.
Selected a Data Source from the list of available data sources.
Enter the User Name and Password for the database connection. Please note that this
information is not needed for all database types.
Test the database connection with the Test Connection button.
Take care that the data source needs to be created through the Windows Control Panel first.
the manual of your database product. To make sure the data source can also be accessed
when the fax server is running as a service, it is recommended to only use System data sources
(no User data sources).
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Some ODBC drivers (i.e. some version of INFORMIX CLI) fail to release allocated system resources after a database connection has been closed. To save system resources, it is recommended to disable the option Automatic Disconnect when Idle in that case.
The ODBC drivers of some database products are not fully compatible with the ODBC data
exchange standard. When using such drivers, it could happen that data exchange with ActiveFax does not work as expected. In such cases you should try to get the latest ODBC driver
4.7.2. Importing the Phone Book
The import of the phone book is done fully automatically; the fax server checks the database
in predefined time intervals for new and modified phone book entries in that case. The import
function compares the fields ID-Number (i.e. customer or supplier number), Name 1 and Fax
Number to find existing phone book entries. If an existing entry is found, the entry is updated;
otherwise a new entry is created in the phone book.
Please note that the import of the phone book from an ODBC data source is always done for
the Central Phone Book.
To configure the automatic import of phone book entries from an ODBC data source, follow
these steps:
Change to the Phone Book tab.
Set the time interval to be used for the automatic update of the phone book data.
Select the Table that contains the phone book entries from the list of available database
tables. A default table can be created with the Use Default Table button. The default table
is created with all fields available in the phone book.
Configure the field mapping to set the relationship between database columns and phone
book fields. Fields not included in the database table are automatically imported with default or empty values. If you use the default table to import the phone book entries, the
field mapping is automatically configured by ActiveFax.
Complete the configuration with OK.
Please note that the conversion of different data types is automatically done by the fax server
whenever possible.
An alternative way to import phone book entries is to import the phone book data from an
Book of this manual.
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4.7.3. Exporting the Transmission Protocol
Transmission protocol entries are automatically exported right after a fax transmission completes. As soon as a new entry has been added to the transmission protocol, ActiveFax automatically connects to the database and exports the new entry.
If the database connection should be unavailable, the transmission protocol entries are automatically stored in the background and are exported as soon as the database becomes available again.
To configure the automatic export of the transmission protocol to an ODBC data source, follow
these steps:
Change to the Transmission Protocol tab.
Select the table to be used for the export of the transmission protocol entries. A default
table can be created with the Use Default Table button. The default table is created with
all fields available in the transmission protocol.
Configure the field mapping to set the relationship between the database columns and
the transmission protocol fields. Fields not included in the database table are automatically ignored. If you use the default table for the export of the transmission protocol entries, the field mapping is automatically configured by ActiveFax.
Complete the configuration with OK.
Please note that conversion of different data types is automatically done by the fax server
whenever possible.
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An alternative way to export the transmission protocol is the export to an ASCII file (text file).
this manual.
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4.8. Cost Account Manager
Use the Cost Account Manager to assign individual cost account codes to fax messages. Cost
account codes are mainly used by the accounting department.
4.8.1. Create Cost Account Codes
Through the menu Extras / Cost Account Manager the list of available cost account codes can
be created or changed. It would also be possible to enable an option to force users to always
enter a cost account code. With this option enabled, fax messages cannot be created by a user
without entering a cost account code. It would also be possible to enable an option to validate
cost account codes. When having this option enabled, only cost account codes defined in the
list of available cost account codes can be entered.
4.8.2. Select Cost Account Codes
The selection of cost account codes can be done either through a dropdown list or through a
separate selection window. The advantage of using the selection window is that you also have
a search function available. Especially when working with a large catalog of cost account
codes, the search function makes it very efficient to search for specific cost account codes.
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Please note that the entries displayed in the transmission protocol can also be selected based
on cost account codes.
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4.9. Network Scanners
Documents on paper can be automatically transmitted using special network scanners. You
can use any scanner with ActiveFax that is supported either by the HP Digital Sending Software
or by the Xerox Network Fax Server Enablement Kit (i.e. HP Digital Sender 9250C or Xerox
Workcentre). As an alternative it would also be possible to use any other type of scanner (i.e.
TWAIN compatible scanners) to send faxes on paper when you scan the document with the
software shipped with the scanner and print the scanned image to the ActiveFax printer.
4.9.1. Installation
ActiveFax is already pre-configured to connect to network scanners, so you just have to do the
installation and configuration of the network scanner in that case. Due to technical reasons,
the fax service of HP network scanners can only be used when the HP Digital Sending Software
has been installed on a Windows XP / 2003 / Vista / 2008 / 7 / 2012 / 8 / 10 system. More
information about the HP Digital Sending Software can be found in the administration manual
of the scanner in the chapter LAN Fax-Products.
For the installation and configuration of HP network scanners you should take special care of
the following settings:
The data exchange directory of the scanner (default directory is hpfscan) is automatically
created during the installation of ActiveFax in the ActiveFax base directory (i.e. C:\Program
Files\ActiveFax\hpfscan). Take care to also configure this directory in the scanner software
when you install the network scanner software.
The Data Exchange File Format for HP network scanners needs to be configured to PCL5
Packbits.
website at www.hp.com.
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4.10. External Configuration File options.cfg
Because ActiveFax offers a huge number of configuration parameters, rarely used parameters
are configured through an external configuration file to avoid that the user interface of the
fax server becomes overloaded with an unmanageable number of settings.
The configuration file options.cfg is located in the fax server main directory (C:\Program
Files\ActiveFax\Server\options.cfg by default). This file is a normal text file that can be modified with any text editor (i.e. Windows Notepad). It is also possible to modify the options.cfg
file directly through the fax server user interface with the menu Extras / Options / General /
Extended Options / External Parameters.
Take care to once stop and restart the fax server whenever the options.cfg file is modified,
because modifications on that file only become active when the fax server is restarted.
An overview of all parameters supported by the options.cfg file can be found in the fax server
help. When modifying the options.cfg file through the fax server user interface, the parameter
list can also be displayed through the Parameter Overview button.
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5. Data Fields
5.1. Why do I need Data Fields?
Each parameter of a fax message (i.e. fax number, priority, subject, ...) is stored by the fax
server in data fields with unique field numbers. That way it is possible to set the value for such
data fields (i.e. the recipients fax number) already from within an application. The user does
not need to enter this information again when the fax is created.
5.2. Syntax of Data Fields
Data fields are always formed using the same syntax and can always be written in plain text.
The following sections describe how data fields are used and include a reference for all data
@Fnnn [email protected]
@F307 Purchase Order [email protected]
@F211 [email protected]
Syntax:
Examples:
A data field always starts with the character sequence @F followed by the 3-digit field number
(nnn). The field number is followed by an optional space character and the content of the data
field (xxxxxx). The end of a data fields is always marked with the @ character. The character
set used for data fields depends on the character set used in the fax message and is automatically set by ActiveFax.
If the content of a data field contains the @ character, you have to “mask” the @ character
with a backslash (\). Otherwise the @ character would be treated as the end of the data field
and the content of the data field would be truncated. It is not necessarily required to mask
the @ sign for the email data fields @F111, @F212 and @F607, since ActiveFax automatically
detects the @ sign of the email address for such data fields.
Example:
@F212 michael.miller\@[email protected]
Please note that it is only required to mask the @ character. Other characters (including the
backslash itself) do not need to be masked.
5.3. Overview of the Data Fields
Since ActiveFax supports a total number of more than 50 different data fields, the fields have
been subdivided into three groups (sender fields, recipient fields and common fields). Depending on the group, the data fields start with different field numbers.
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Please note that only fields containing data need to be integrated with your application. Data
fields not specified by your application are automatically filled with default values by the fax
server. The most important data field is the data field for the recipients fax number (field
@F211), since this field is at least required to automatically deliver a fax message. It is recommended to also use other data fields with additional information, for example the recipient’s
name (field @F201) or the subject of the message (field @F307).
A detailed summary of all data fields can be found in the following sections. More information
5.3.1. Sender Fields
This group of data fields contains all parameters for the sender of a fax message. It is recommended to specify at least the fields Name 1 and Fax Number, since this information is printed
on the fax title of a message. Please note that default values for the sender parameters can
be specified through the menu Extras / Predefined Settings or with the User Administrator
Field Number
Field Name
Description
@F101
Name 1
Name 1 of the sender
@F102
Name 2
Name 2 of the sender
@F103
Name 3
Name 3 of the sender
@F104
Name 4
Name 4 of the sender
@F105
Name 5
Name 5 of the sender
@F106
Department
Department of the sender
@F107
CC
CC of the sender
@F108
Phone 1
Phone number 1 of the sender
@F109
Phone 2
Phone number 2 of the sender
@F110
Fax Number
Fax number of the sender
@F111
E-Mail
5.3.2. Recipient Fields
This group of data fields contains all parameters for the recipient of a fax message. It is recommended to specify at least the field Fax Number, since this field is always needed to automatically delivery a fax message.
Field Number
Field Name
Description
@F201
Name 1
Name 1 of the recipient
@F202
Name 2
Name 2 of the recipient
@F203
Name 3
Name 3 of the recipient
@F204
Name 4
Name 4 of the recipient
@F205
Name 5
Name 5 of the recipient
@F206
Department
Department of the recipient
@F207
Attention Of
Attention-of of the recipient
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@F208
CC
CC of the recipient
@F209
Phone 1
Phone number 1 of the recipient
@F210
Phone 2
Phone number 2 of the recipient
@F211
Fax Number
Fax number of the recipient
@F212
E-Mail
@F213
Communication Service
Communication service Fax or Email (F=Fax,
E=Email). Only needed if the fields @F211 and
@F212 are both set
@F299
Next Recipient
Delimiter for the next recipient
5.3.3. Common Fields
This group of data fields contains all parameters addressing neither the sender nor the recipient of a fax message.
Field Number
Field Name
Description
@F301
Priority
Priority of the fax message (1=very high, 25=high,
50=normal, 99=low)
@F302
Transmission Attempts
Number of transmission attempts so far (this field is
automatically filled)
@F303
Transmission Date
Preferred transmission date of the fax message
@F304
Transmission Time
Preferred transmission time of the fax message in the
format HH:MM
@F305
Cover Page
Cover page for the fax message
@F306
Overlay
Overlay for the fax message
@F307
Subject
Subject of the fax message
@F308
Free Text 1
@F309
Free Text 2
@F310
Free Text 3
@F311
User Name
User name of the fax message
@F312
Modem
Preferred modem (i.e. COM1, COM2, ISDN)
@F313
Resolution
Preferred resolution (0=standard, 1=normal, 2=fine)
@F314
Lock
Lock status of the fax message (0=not locked,
1=locked)
@F315
Cost Account
Cost account code for the fax message
@F316
Cover Page Text
Text on the cover page. Line breaks can be added
with \n
@F317
Cover Page Text (cont.)
This field is used in combination with @F316 to split
long text to multiple data fields. The text in the
@F317 field is always appended at the end of the
text. The data field @F317 can be used as often as required
@F320
ID-Number Phonebook
Complete recipient’s data from the phone book entry
with the specified ID number
@F350
From Page
Send from page n
@F351
To Page
Send to page n
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@F360
Private Fax Message
Mark fax message as private (0=public, 1=private)
@F370
Fax-On-Demand
Create a fax-on-demand document (0=normal Fax,
1=fax-on-demand document)
@F500
Reference File
This file will be evaluated for additional data fields
and is deleted then. If you do not use this field, ActiveFax tries to search for data fields in the file
Fields.dat in the installation directory (usually C:\Program Files\ActiveFax)
@F501
Automatic Printing
No dialog box to enter the recipient is displayed. Optional parameter (0=never display dialog window,
1=only display dialog window for messages without a
recipient, 2=always display dialog window)
@F502
Ignore Pages
This data field is used to ignore pages at the end of a
fax message. A positive parameter sets the total number of pages (without the cover page) that should be
displayed. A negative parameter is used to set the
number of pages that should be ignored at the end of
the message
@F503
Print Fax
This data field can be used to automatically print a fax
after it has been received by the server. As an optional parameter you can specify the printer name
and, separated with a comma, the options D to delete
the fax after it has been printed and the option H, to
ignore the fax header for printing. Example: @F503
Laserjet,[email protected]
@F504
Number of Copies
This data field sets the number of copies when a fax
message is automatically printed
@F505
Export Fax
This data field can be used to automatically export a
fax after it has been received by the server. As an optional parameter you can specify the file name for the
export file and, separated with a comma, the options
D to delete the fax after it has been exported and the
option H, to ignore the fax header for the export. Example: @F505 c:\export\test.pdf,[email protected]
@F506
Export Format
Specify export file format (tif, gif, bmp, pdf) and/or
resolution. Example: @F506 gif,[email protected]
@F507
Automatically Delete Fax
Automatically delete fax after transmission (1=move
fax to recycle bin, 2=delete fax). Example: @F507 [email protected]
@F555
Control Command
Send control command to ActiveFax.
@F555 DELETE [email protected]
@F555 RESEND [email protected]
@F555 LOCK [email protected]
@F555 UNLOCK [email protected]
@F555 PRINT Fax-ID [printername]@
@F555 EXPORT Fax-ID [filename]@
Example: @F555 DELETE [email protected]
@F599
New Fax Job
Start a new fax job within the document. This command is used for mail merge documents, when the individual pages of a document should be sent to different recipients. Each time you use that field on a page,
a new fax job is started at the beginning of that page
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@F600
E-Mail Bitmap Format
The email is always sent as an attachment. ActiveFax
does not try to convert the message into text format
@F601
E-Mail Line Break
Number of characters after which ActiveFax forces an
automatic line break when converting an email to text
format (20-999). Normally this value is automatically
calculated according to the width of a page
@F602
E-Mail Attachment
The email is always sent as an attachment, even when
the original message is in text format. Optionally it is
also possible to specify the attachment name with
this data field (i.e. @F602 [email protected])
@F603
E-Mail Body Text
Body text for emails that are sent as an attachment. If
this field is not used, the standard text is used instead. Line breaks can be added with \n
@F604
E-Mail Body Text (cont.)
This field is used in combination with @F603 to split
long text to multiple data fields. The text in the
@F604 field is always appended at the end of the
text. The data field @F604 can be used as often as required
@F605
E-Mail Format
Specify the email file format (tif, gif, bmp, pdf) and/or
the resolution. Example: @F605 gif,[email protected]
@F606
E-Mail Attachment
Append files in original format to an email. Multiple
files can be separated with a comma (,). The files
need to be located on the fax server. As optional parameters you can use D (Delete = delete file after
transmission) and R (Required = file is required, otherwise the message is not sent). Example: @F606
c:\dat\prices.pdf, c:\word\mailing.doc, [email protected]
@F607
This data field is used to specify a reply-to email adfdress that is different from the senders email address
@F608
E-Mail Confirmation
This data field is used to request a confirmation from
the recipient of the email that the email has been
@F700
Accumulated Document
This data field creates an accumulated document with
the given reference number. If an accumulated document with that reference number already exists, the
print job is added. As an optional parameter you can
also specify a timeout in seconds after that the accumulated document is automatically terminated. The
reference number needs to be a numeric value between 1 and 999,999,999. Examples: @F700 [email protected],
@F700 1234,[email protected]
@F701
Accumulated Doc End
This data field terminates an accumulated document
with the given reference number. Example: @F701
[email protected]
@F702
Attachment
Append files to the fax. The file format for the attachment needs to be either a text file or a file created
through the ActiveFax printer driver. Multiple files
can be separated with a comma (,). The files need to
be located on the fax server. Example: @F702
c:\data\[email protected]
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@F703
Overlay on/off
Turns an overlay on/off (0=off, 1=on). When using 0+
or 1+, the overlay is turned on/off on the following
page rather than on the current page
@F000
Import Image
Embed a bitmap into the document
5.3.4. Special Data Fields
Please note that not all of the above data fields are data fields in the literal sense. Special fields
are for example @F299, @F500, @F501, @F599 and @F000. The special meaning of these
fields is described on the following pages.
5.4. Examples for Data Fields
It is recommended to have a look at the sample file “sample.txt” to see how data fields are
used. This file is automatically copied to the installation directory of the fax server. The file
demonstrates how data fields, bitmap files and document formation with HP-LaserJet (PCL)
printer commands can be done. Especially when sending fax messages from UNIX or Linux
systems, this file would be a good point to start when integrating data fields.
5.4.1. Example 1
Recipient .......................... +43 1 1122 3344-12
Subject ............................. Purchase Order 123456
@F211 +43 1 1122 [email protected]
@F307 Purchase Order [email protected]
5.4.2. Example 2
Sender .............................. Duncan Inc., +1 555-123-4567
Recipient .......................... Brown Import & Export, 0043 1 9072544
@F101 Duncan [email protected]@F110 +1 [email protected]
@F201 Brown Import & [email protected]@F211 0043 1 [email protected]
5.4.3. Example 3
Sender .............................. Duncan Inc., 555-123-4567
Recipient 1 ....................... Miller Ltd., Fax: 444-110-2020
Recipient 2 ....................... Smith Inc., Fax: 333-220-3030
Recipient 3 ....................... Brown Corp., E-Mail: [email protected]
Subject ............................. Latest Pricelist
Xmit Date ......................... 31.01.2015
Xmit Time ......................... 20:15
Priority ............................. Low
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@F101
@F201
@F201
@F201
@F307
@F301
Duncan [email protected]@F110 [email protected]
Miller [email protected]@F211 [email protected]@[email protected]
Smith [email protected]@F211 [email protected]@[email protected]
Brown [email protected]@F212 [email protected]@
Latest [email protected]@F303 [email protected]@F304 20:[email protected]
[email protected]
The data field @F299 has been used as a delimiter between the single recipients in that
example. Since this data field is only used as a delimiter, no content is required for that
field.
5.4.4. Example 4
This example demonstrates how to embed bitmaps into a fax message using the special
in the following section of this chapter.
Yours sincerely
@F000 [email protected]
5.5. Embedding Bitmaps into Fax Messages
ActiveFax can be used to embed bitmaps at any position of a fax message. That way it would
be possible to easily integrate individual company logos or signatures. Embedded bitmaps are
normally used for fax messages that have been created on UNIX or Linux systems or other
non-Windows operating systems.
Quite similar to the syntax of data fields, bitmaps can be embedded into fax documents using
the following syntax:
Syntax: @F000 file[,X<pos>][,Y<pos>][,width-mm[,height-mm]]@
Example: @F000 sign.bmp,[email protected]
The parameters X<pos> and Y<pos> as well as width-mm and height-mm are optional. If you
do not specify these parameters, the size of the bitmap is automatically calculated. ActiveFax
uses a default resolution of 300 dpi when calculating the size of the bitmap in that case. The
parameters X<pos> and Y<pos> can be used to specify the exact position of the bitmap in mm
(e.g. X100,Y75) measured from the upper left corner.
It is recommended to only use bitmaps of the type Uncompressed Windows Bitmap (.bmp). It
is recommended to use .bmp files in monochrome format (1-bit color depth). If you do not
specify a path for the bitmap file, the file is automatically searched in the installation directory
of the fax server (normally C:\Program Files\ActiveFax\Server).
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5.5.1. Example 1
Bitmap with a default resolution of 300 dpi.
Yours sincerely
@F000 [email protected]
5.5.2. Example 2
Bitmap located in the directory C:\SCAN with a width of 50 mm (height calculated automatically).
Yours sincerely
@F000 c:\scan\mike.bmp,[email protected]
5.5.3. Example 3
Bitmap with a width of 50 mm and a height of 30 mm at position 100 mm x 75 mm.
Yours sincerely
@F000 mike.bmp,50,30,X100,[email protected]
When having a .jpg file with the same file name as the original .bmp file stored at the same
location as the .bmp file, the .jpg file is automatically used by the fax server whenever PDF
documents are generated (i.e. companylogo.bmp and companylogo.jpg). The .jpg file should
be of the same size as the.bmp file; the color format of the .jpg file should be RGB 24-bit. That
way it would be possible to automatically use color bitmaps and logos for PDF documents
generated by the fax server.
The number of bitmaps that can be embedded into a single fax document is not limited. For
an example of embedded bitmaps, have a look at the enclosed sample file “sample.txt”. This
file is automatically copied to the installation directory of the fax server and is a good point to
start when adding bitmaps to documents.
5.6. Embedding Data Fields into Applications
5.6.1. Windows Applications
The easiest way to specify data fields from within Windows applications is to add them to the
document name when printing. You just need to add the data fields at the end or at the beginning of the document name in that case.
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Example:
Document Name .............. Purchase Order 12345
Recipient .......................... Duncan Inc.
Fax Number ...................... 555-123-4567
Purchase Order [email protected] Duncan [email protected]@F211 [email protected]
As an alternative it would also be possible to embed data fields directly in the document.
When using this method, it is important to format the data fields with the “ActiveFax” font. If
you do not format data fields with that font, the data fields are ignored and printed as visible
text on the document.
Take care, that no additional text styles (bold, italic, underline) are allowed when formatting
data fields with the ActiveFax font.
Have a look at the enclosed sample file “Sample.rtf” for an example of data fields used directly
in a document. This file is automatically copied to the installation directory of the fax server
(C:\Program Files\ActiveFax\Server by default) and is a good point to start when adding data
fields to a document. You can open that file with Microsoft® Word or Windows WordPad.
5.6.1.1. Reference File for Data Fields
Another way to specify data fields is using an external reference file. The advantage of this
method is that you do not have to set all data fields directly in the document name or the
document itself. When using this method you just have to add a single data field, which is a
reference to the file with the remaining data fields. The link to the reference file is done with
the data field @F500 [email protected] in that case. Take care to use a unique file name for the
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reference file to avoid conflicts with other fax jobs. After the print job has been processed, the
reference file is automatically deleted by the fax server to ensure it is not re-used for other
print jobs.
Example:
Document Name .............. Purchase Order 12345
Reference File .................. c:\tmp\ref251173.dat
Purchase Order [email protected] c:\tmp\[email protected]
In that example the data fields are located in the file c:\tmp\ref251173.dat. There is no
special format required for the data fields in the reference file. It would be possible to write
each data field to a separate line or to write all data fields in a single line.
5.6.1.2. Split Print Jobs (Mail Merge Documents)
Sometimes it can be required to break up a single print job into multiple fax jobs (i.e. when
using the mail merge function of Microsoft® Word). In that case the data field @[email protected] can
be used to start a new fax job within a print job. When using the data field @[email protected] on a page,
a new fax job is started on that page including all following pages up to the next @[email protected] data
field. This data field is necessarily required when printing mail merge documents from Microsoft® Word, since Word generates a single huge print job for all fax pages of the mail merge
document.
5.6.1.3. Example in Programming Language C
DOCINFO
BYTE
BYTE
BYTE
DocInfo;
szText[1024];
szName[128];
szFax[128];
lstrcpy(szName, "Duncan Inc.");
lstrcpy(szFax, "555-123-4567");
wsprintf(szText, "Purchase [email protected] %[email protected]@F211 %[email protected]", szName, szFax);
DocInfo.cbSize = sizeof(DOCINFO);
DocInfo.lpszDocName = szText;
DocInfo.lpszDatatype = NULL;
.....
5.6.2. UNIX, Linux and other Operating Systems
To embed data fields in UNIX, Linux or other non-Windows operating systems, you can insert
the data fields directly into the document as normal text (no special font is needed). The fax
server automatically filters and evaluates the data fields, so they are not visible on the fax
message. Such data fields can be added at any position of the document.
Have a look at the enclosed sample file “sample.txt” for an example of data fields added to
documents in UNIX and Linux. This file is automatically copied to the installation directory of
the fax server and would be a good point to start for own projects. Please note that it is also
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possible to use printer commands of HP-LaserJet (PCL), Epson-LQ and optionally Postscript
and PDF to format fax messages.
5.6.2.1. Example in Programming Language C
char
char
int
char
char
szName[128];
szFax[128];
nPriority;
szSubject[128];
szText[1024];
lstrcpy(szName, "Duncan Inc.");
lstrcpy(szFax, "555-123-4567");
nPriority = 1;
lstrcpy(szSubject, "Purchase Order 12345");
wsprintf(szText, "@F201 %[email protected]@F211 %[email protected]@F301 %[email protected]@F307 %[email protected]",
szName, szFax, nPriority, szSubject);
.....
5.6.2.2. Example in Programming Language INFORMIX 4GL
DEFINE faxdata RECORD
name
fax
priority
subject
END RECORD
CHAR(128),
CHAR(128),
SMALLINT,
CHAR(128)
faxdata.name = "Duncan Inc."
faxdata.fax = "555-123-4567"
faxdata.priority = 1
faxdata.subject = "Purchase Order 12345"
PRINT "@F201
PRINT "@F211
PRINT "@F301
PRINT "@F307
.....
",
",
",
",
ActFax User’s Manual
faxdata.name CLIPPED, "@";
faxdata.fax CLIPPED, "@";
faxdata.priority USING "#&", "@";
faxdata.subject CLIPPED, "@"
103
ActiveFax can notify you whenever a new software update is released. The automatic notification about new software releases can be configured with the menu Help / Check for Updates. By default, the update notification is already enabled. As soon as a new software update
is released a notification message will be displayed on the fax server. If you have enabled email
Before an update is downloaded from the Internet and installed on the system, you always
need to confirm the update process. Under no circumstances, updates are automatically installed without asking the Administrator of the fax server first.
Please note, that the check for new updates can only be done when the fax server PC is connected to the Internet. The fax server uses HTTP port 80 to check for updates. If this port is
blocked by your firewall, it would be important to configure an exception for that port in the
firewall.
Please note, that you always only need to install the latest update. Any updates that have
been released between your existing ActiveFax version and the latest ActiveFax version do not
need to be installed.
If your ActiveFax installation is a mixed installation of 32-bit and 64-bit systems or when using
an installation with multiple languages, the fax server automatically downloads all files required for the update of the fax server and all fax clients.
new software release can therefore be displayed with a delay of a few days compared to the
release date of the update.
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A manual check for new software updates can be done at any time through the menu Help /
As soon as an update has been installed on the fax server, the update is also automatically
on the fax clients can be configured with the menu Help / Check for Updates. The notification
of fax clients is not enabled by default. The installation of an update on the fax client is always
only done when the user confirms the update. Under no circumstances an update is automatically installed on a fax client without asking the user first.
6.4. Update Period
New ActiveFax licenses automatically include free updates for 2 years. If you want to install
an update after the update period has been reached, an update package would be required
website www.actfax.com.
Using the menu File / License Settings you can check the update period included with your
The lifetime of a license is not affected by the update period. In general, licenses are always
valid without a time limit. After the update period of a license has been reached, the license
remains valid and the software can still be used without any limitations in the version that is
installed on the system.
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7. Appendix
7.1. Glossary
16-bit: The term “16-bit” is used for applications that are designed for the operating system Windows
3.x. 16-bit applications can also be executed on newer Windows version, but they do not fully use the
new features of these operating systems.
32-bit: The term “32-bit” is used for applications designed for Windows 95 and newer. A 32-bit application is using the functions of the operating system more efficient than a 16-bit application and is
therefore running faster and more stable. 32-bit applications cannot be executed on Windows 3.x.
64-bit: Starting with Windows XP, Microsoft® offers Windows as a 32-bit and as a 64-bit version. The
main advantage of 64-bit Windows is that it can address more physical memory compared to 32-bit
Windows. The overall performance of the fax server is almost identical on 32-bit and 64-bit Windows
systems.
Analog: The opposite of “digital”. Analog numbers can have any values, whereas digital numbers can
only have values of a defined scale. For fax transmissions the term “analog” is usually used for the
common phone network, whereas the term “digital” is used as a synonym for ISDN.
ANSI: Abbreviation for “American National Standard Institute”. The ANSI standard for example defines
the ANSI character set or the programming language ANSI C.
API: Abbreviation for “Application Programming Interface”. The API defines a group of functions that
is used for the processing of specific tasks.
ASCII: Abbreviation for “American Standard Code for Information Interchange”. This code defines
unique numbers for characters, numbers and other special and control characters. The values of the
ASCII code are between 0 and 255 (at 7-bit between 0 and 127).
Baudrate: Unit for the step rate of a modem. Baudot, who is giving the Baudrate its name, was a French
engineer, who developed the Baudot-Code, which was used prior to the ASCII-Code. Please note that
the terms Baudrate and Bitrate do not identify the same unit.
Bit. A bit is the smallest possible information unit on a computer. A bit can have two different conditions (0 or 1). By concatenating multiple bits, every kind of information can be stored. In computers, 8
single bits usually build one so called Byte. Therefore a byte can store 256 different values.
Bitrate: Unit for the data bits that are transferred in a given time period. This unit usually describes
the numbers of transferred bits within one second. Please note that the term Baudrate and Bitrate do
not identify the same unit.
Broadcast: The term “Broadcast” means a sending method that accesses all devices in a network.
Broadcast calls are mainly used to search for resources in a network.
CAPI: Abbreviation for “Common ISDN API”. This defacto standard of the German company AVM is
defining a programming interface for ISDN adapters of different manufacturers. ISDN adapters, which
support at least CAPI version 2.0, can be used for the transmission of fax messages.
CCITT: Abbreviation for “Comité Consultatif International Téléphonique et Télégraphique”. This committee is responsible for the standardization of telecommunication standards. See also ITU.
Class 1/2/2.0: Standard for fax modems. Depending on the modem type, usually at least one of these
standards is supported by a fax modem.
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Client: A Client is a software program that is used to retrieve and process data from a so called Server.
COM-Port: Other name for a serial interface (Communication Port) of a personal computer.
CSID: The term “CSID” is used for the sender identification of a fax message. The CSID is transmitted
with every fax messages and usually contains the fax numbers of the sender in international format.
DCE: Abbreviation for “Data Communication Equipment”. The term “DCE” is usually used for modem
devices.
DLL: Abbreviation for “Dynamic Link Library”. Other than EXE files, DLL files cannot be directly executed. A DLL usually contains program code that is dynamically loaded by other executable programs.
DTE: Abbreviation for “Data Terminal Equipment”. The term “DTE” is usually used for a computer.
Digital: The opposite of “analog”. Digital numbers can only have values of a specified scale, whereas
analog numbers can have any values. For fax transmissions the term “analog” is usually used for the
common phone network, whereas the term “digital” is used as a synonym for ISDN.
EIA: Abbreviation for “Electrical Industry Association”, the Association of the American Electronic Industry. The EIA for example, has standardized the serial interface (RS-232).
Fax: A fax or also called facsimile is used to exchange image data between two fax machines. The
images are usually compressed according to the compression standard “G3”.
Fax-On-Demand: The term “Fax-On-Demand” means, receipt of a document from a so called Fax-OnDemand Server.
FTP: Abbreviation for “File Transfer Protocol”. This protocol is mainly used for copying files. The FTP
protocol, which was initially only used on UNIX system, is nowadays also used on the Internet. In ActiveFax, the FTP protocol can also be used for the creation of fax messages.
G3 Fax Mode: The fax mode usually used for fax transmissions. The “G3” standard is specifying the
compression method, the transfer speed (maximum 14.400 bps) and other parameters.
GDI: Abbreviation for “Graphics Device Interface”. Under Microsoft® Windows, the GDI is used for
painting the contents of a window and for printing purposes.
Handshake: At the beginning of each fax transmission, the transmission speed has to be negotiated
between the two fax devices (modems). This synchronizing phase is also called handshaking phase.
HDLC: Abbreviation for “High Level Data Link Control”, the protocol used for synchrony data exchange.
ISDN: Abbreviation for “Integrated Services Digital Network”. Using ISDN, all data and speech information is transmitted in digital format. Since this way of data exchange is usually much more stable
and faster than analog data exchange, it is perfect for computer use. Other than the “normal” phone
network, ISDN offers a lot of extra services, like direct dial information or charging information.
ITU: The new name of CCITT.
Least Cost Routing: Least Cost Routing is used to automatically identify the cheapest possible phone
connection for a specific destination.
LPD: Abbreviation for “Line Printer Daemon”. This protocol is usually used on UNIX systems for print
jobs on network based printers.
Modem: Abbreviation of the two words MOdulator and DEModulator. A modem is transforming the
digital signals of a computer into analog signals (sounds). The other modem device is reconverting
these sounds back to digital signals
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MSN: Abbreviation for “Multiple Subscriber Number”. A MSN is used in ISDN and specifies different
unique phone numbers for a single phone line.
Named Pipe: Named Pipes are used in the NetBeui network protocol for data exchange. Named Pipes
can be used in ActiveFax to directly send fax messages to the Fax Server. Named Pipes have to be
specified using the format “\\server\pipe\pipename”.
NetBeui: The NetBeui protocol, also called Windows Network, is used for the data exchange between
two computers. The NetBeui protocol is only compatible with the Windows operating system and can
also be used with ActiveFax when the Fax Server has been installed on either Windows XP / 2003 /
Vista / 2008 / 7 / 2012 / 8 / 10.
ODBC: Abbreviation for “Open Database Connectivity”. This standard is used for the data exchange
between applications and database servers.
Offline: The condition of a modem or other device, which means, that the device do not have an active
connection.
Online: The condition of a modem or other device, which means, that the device is having an active
connection.
Other Party: The term “Other Party” identifies the other fax device or fax modem.
Overlay: An Overlay is used in ActiveFax to fade in some text or bitmaps (i.e. logos) into a fax message.
PC: Abbreviation for “Personal Computer”. This term is usually used for all computers that are compatible to IBM Personal Computers.
Polling: The term “Polling” is used for the receipt of a document from a so called Fax-On-Demand
Server.
Processor: The processor is often called the brain of a computer. All important tasks are controlled by
the processor of a computer.
Queue: The term “Queue” is usually used in the LPD protocol. A unique queue name is used to identify
a single printer on LPD.
RAM: Abbreviation for “Random Access Memory”. The RAM is the so called working memory of the
computer. The content of the RAM is lost as soon as the computer is turned off, so the RAM can only
be used to temporary store information.
Serial Interface: Communication interface, where the data is sent one bit after the other. With serial
data exchange, you only need to have one data line for sending and one line for receiving.
Server: A Server is a software program that stores data which is made available to so called Client
applications.
Service: A service is used by the Windows operating system and means an application that is already
executed during the boot time of the computer. One of the big advantages of a service is that it is
always active, even when there is no user logged on.
SMTP: Abbreviation for “Simple Mail Transfer Protocol”. This protocol is used in the Internet for E-Mail
transmissions.
Stream Socket: A data connection that is based on the TCP/IP network standard.
T.4: Standard for the compression of fax data.
T.30: Standard for the transmission of fax data.
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T.38: Standard for the transmission of fax data over IP networks.
TCP/IP: The TCP/IP protocol is a network protocol which is mainly used on the UNIX operating system
and on the Internet. Nowadays TCP/IP is also widely used on Windows systems.
TFTP: Abbreviation for “Trivial File Transfer Protocol”. This protocol is mainly used on the UNIX operating system and its primary use is the transfer of files. Nowadays the TFTP protocol is only used for
some special reasons. In ActiveFax, TFTP can also be used for the creation of fax messages.
Thread: A thread is a part of an application which is usually executing independently from the rest of
the application.
Timeout: A timeout is a specified amount of time, a given task may last. If a task is not completing
within the timeout period, the specific task usually will be aborted.
UNC: The default format of a path on the Windows network. The format of an UNC is “\\server\export\path\filename”.
Unicode: A 16-bit character set, which can be used to code all currently existing characters. The operating system Windows NT is using Unicode only on system level.
UNIX: An operating system, which has been developed by AT&T. UNIX is mainly used for multi user
systems. Nowadays there exist a lot of UNIX derivatives; the operating system LINUX is one of the most
famous UNIX derivatives. Other important UNIX derivatives are HP/UX, AIX and SCO-UNIX.
VoIP: Abbreviation for “Voice over IP”. This standard is used for phone calls done through IP networks.
Web Browser: A Web Browser is a software, that is used to display so called web pages on the local
computer (i.e. Netscape, Internet Explorer, ...).
Windows: The term “Windows” usually means the operating system of Microsoft®.
World Wide Web: Also known as “the Web”. The World Wide Web is the global part of the Internet.
Web Pages and other resources are linked together by so called Hypertext-Links. Therefore it is possible to access every web page from any location. Web Browsers are used to display the pages of the
World Wide Web.
WWW: Abbreviation for “World Wide Web”.
WYSIWYG: Abbreviation for “What you see is what you get”. This means, that the document displayed
on the screen is exactly of the same shape than the printout.
XON/XOFF: Software method that is controlling the flow of the data transmission over a serial interface. The XON/XOFF protocol is using the ASCII codes 17 and 19 to control the flow of the transmitted
data.
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7.2. Keyboard Hotkeys
Hotkey
Function
F3
F4
F5
F6
F7
F10
Phone Book
Transmission Protocol
Charge Optimization
Execute Polling (Fax-On-Demand)
Cover Page Designer
Ctrl 1
Ctrl 2
Ctrl 3
Ctrl A
Ctrl C
Ctrl F
Ctrl N
Ctrl O
Ctrl P
Ctrl S
Ctrl Z
Rotate incoming fax message by 90°
Rotate incoming fax message by 180°
Rotate incoming fax message by 270°
Select all entries of the faxlist
Show or hide the communication window
Show or hide the faxlist
Create a new Instant Fax Message
Create a new fax message from a file
Print
Save
Suspend outgoing fax messages
Ctrl Page-down
Ctrl Page-up
Display next page
Display previous page
Alt Enter
Alt F4
Display fax dialog
Close a window or the entire application
Cursor down
Cursor up
Cursor left
Cursor right
Page-down
Page-up
Scroll the fax view down one line
Scroll the fax view up one line
Scroll the fax view to the left
Scroll the fax view to the right
Scroll the fax view down one page
Scroll the fax view up one page
Ctrl & Left mouse button
Shift & Left mouse button
Select multiple entries of a list view
Select a from/to range of a list view
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7.3. Frequently Asked Questions - FAQ
to the questions are kept short, since detailed information about specific topics is also available in the corresponding chapters of this manual and in the online help of ActiveFax.
Please note that additional FAQ are also available through the official ActiveFax website at
www.actfax.com/en/faqs.html.
Question:
Is it possible to automatically start the fax server when the computer is booting?
Using the menu option File / Automatic Startup, ActiveFax can be configured to
automatically start during boot time.
Question:
Is it possible to select multiple entries of the faxlist at the same time?
Using the Ctrl key and Shift key it is possible to select more than one entry of
any list (not only the faxlist).
Question:
Is it possible to change the sort order of the faxlist?
Clicking with the left mouse button on a column header changes the sort order
of the faxlist. The Shift key can be used to specify multiple columns for sorting.
Question:
How can I quickly find a fax message again?
The fastest way to find a specific fax message is to specify a search term in the
search field of the faxlist.
Question:
How can I send a fax message from UNIX or Linux systems?
Usually fax messages from UNIX or Linux systems are sent using the LPD/LPR
protocol. When using LPD/LPR, ActiveFax is accessed exactly the same way as
any other network printer.
Question:
Is it possible to change the default resolution used for outgoing fax messages?
Using the Modem tab of menu option Extras / Options it is possible to change
the preferred resolution (fine or normal) for outgoing as well as for incoming
fax messages.
Question:
Is it possible to change the time interval for redialing?
Using the Redialing tab of the menu option Extras / Options it is possible to
specify an individual delay for each transmission attempt.
Question:
How can I archive a fax message?
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Fax messages are normally automatically archived by ActiveFax. Through the
Archive tab of menu option Extras / Options, the settings for the archive function can be individually configured. Please note that it is also possible to manually move fax messages to the archive.
Question:
Is it possible to automatically print fax messages or sending reports?
Through the Printing tab of the menu option Extras / Options the settings for
the automatic printing function and the sending report can be configured.
Please note that the settings for automatic printing can also be individually configured for each user through the User Administrator.
Question:
Is it possible to automatically print the transmission protocol?
Using the Printing tab of the menu option Extras / Options the settings for automatic printing of the transmission protocol can be configured.
Question:
Is it possible to change the information printed in the fax title?
Using the Fax Title tab of the menu option Extras / Options the appearance of
the fax title can be individually changed.
Question:
Is it possible to display a notification message upon receipt of a new fax message?
Using the General tab of the menu option Extras / Options it is possible to activate this feature with the option Display notification message upon receipt of
new fax messages.
Question:
Is it possible to automatically use the best (cheapest) transmission time for a
fax message?
Using the menu option Extras / Charge Optimization it is possible to configure
the settings for the automatic optimization of transmission times.
Question:
Where can I specify the default parameters for the sender of a fax message?
Using the menu option Extras / Predefined Settings the default parameters for
the sender of fax messages can be configured. Please note that it is also possible
to individually configure different settings for each user with the User Administrator.
Question:
Is it possible to protect fax message against unauthorized access?
Using the menu option Extras / Security Settings it is possible to activate the
security settings of ActiveFax.
Question:
Is it possible to create private fax messages only visible to the owner of the fax?
Using the fax dialog window and the More Settings tab you can activate the
option Private Fax Message to mark a fax message as private.
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7.4. Problem Solutions
Please note that additional problem solutions are also available through the Knowledge Base
on the official ActiveFax website at www.actfax.com/en/kbase.html.
Problem:
Solution:
The modem is dialing, but the connection cannot be established.
Check whether tone or pulse dialing has to be used with the phone line. You
should also make sure that you do not have to use a dial prefix for outgoing
calls. When using a phone system (PBX) you should take care, that the option
Wait for Dial Tone before Dialing is turned off. You should also try to dial the
number with a different communication program (i.e. HyperTerminal) to check
if the modem is ok.
Problem:
Solution:
During the transmission of fax messages I often get transmission errors.
Go to the modem configuration (menu Communication / Modem) and press the
Extended button to change the settings for the modem. Disable the options Enable MR-Compression, Error Correction Mode and Allow Transfer Rates faster
than 9.600 bps there.
Problem:
The connection between the fax server and the fax client cannot be established
using the TCP/IP protocol.
Use the PING command to make sure that the connection between the server
and the client computer is ok. Make sure that the TCP/IP protocol is properly
installed on the client and server computer. It could also help to reboot the
system.
Solution:
Problem:
Solution:
The connection between the fax server and the fax client cannot be established
using the NetBeui protocol (Windows Network).
Check if the computer name of the server PC is listed in the Windows network
list or use the Windows search function to check if the computer can be found
in the network. Make sure that you have permission to connect to the server
computer. It could also help to reboot the system.
Problem:
Solution:
The TCP/IP port for the LPD server is already in use by another application.
Close the other application or stop the service of that application. Please note
that ActiveFax can also be used to redirect LPD print jobs to local printers.
Problem:
Solution:
The printout of fax messages takes very long on laser printers.
Use a maximum resolution of 300 dpi for printing in that case. Especially when
using HP laser printers with insufficient memory installed you should disable
the option Page Protection in the printer properties to save resources.
Problem:
Solution:
Not all serial interfaces (COM ports) are listed in the modem configuration.
The automatic port detection of Windows is not working correctly in that case.
It is recommended to disable automatic COM port detection using the General
tab of the menu option Extras / Options.
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Problem:
Solution:
Received fax messages and fax messages from Windows applications are displayed incompletely.
Choose the menu option Extras / Options and disable the option Enable Processing of Bitmaps above the 64K-Limit in the General tab.
Problem:
Solution:
The system is very slow during transmission of fax messages.
Choose the menu option Extras / Options and disable the option Enable
Realtime Communication with the Modem Devices (Realtime-Priority) in the
General tab.
Problem:
Data exchange with an ODBC database is not working and the program is terminating with a “General Protection Fault” message.
This is usually caused by a bug in the ODBC driver. In such case, you should try
to get the latest version of the ODBC driver from the manufacturer of the database.
Solution:
Problem:
Solution:
Problem:
Solution:
When sending fax messages using an ISDN adapter, it sometimes happens, that
fax messages are transmitted twice as long as the original document.
The ISDN adapter ignores the fax parameter for the resolution of the remote
fax machine in that case and it is recommended to install the latest version of
the ISDN driver (CAPI driver).
I have forgotten the Administrator password and cannot access the fax server
anymore.
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7.5. Sample Applications
Integration of ActiveFax in other applications is very simple. When using data fields, it is possible to specify the fax parameters (i.e. recipients fax number, subject, priority, etc.) already
from within an application. Especially applications not running on Windows based operating
systems (i.e. UNIX, Linux, DOS, etc.) can take advantage of the support of HP-LaserJet (PCL)
and Epson-LQ printer commands for fax formatting.
This chapter includes two examples that demonstrate how data fields can be embedded with
applications. The examples have been designed as simple as possible to point out how data
fields are added to the program code.
Please note that the source codes for the sample applications as well as the executable files
are available in the ActiveFax installation directory on the fax server. The sample files are located in the Server\Samples directory on the fax server.
7.5.1. Windows Application (WinApp.exe)
This example in programming language C is used to demonstrate how data fields can be used
with Windows applications. This sample application uses normal Windows API calls for printing.
7.5.1.1. Program Summary
Program starts at WinMain()
Open the dialog box IDD_MAIN with DialogBox()
Initialize the dialog box with WM_INITDIALOG
Call the function PrintDocument() in WM_COMMAND
Evaluate the dialog box fields with SendDlgItemMessage()
Generate the document name for printing (add data fields)
Check if we want to print without displaying the fax dialog
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Open the “ActiveFax” printer with CreateDC()
Create a new document with StartDoc() and StartPage()
Write the message text with DrawText()
Close the document with EndPage() and EndDoc()
Close the printer with DeleteDC()
7.5.1.2. Source Code
#include <windows.h>
#include "resource.h"
// This font is used for the message text
LOGFONT LogFontText = {20, 0, 0, 0, FW_NORMAL, 0, 0, 0, DEFAULT_CHARSET,
OUT_DEFAULT_PRECIS, CLIP_DEFAULT_PRECIS, DEFAULT_QUALITY,
DEFAULT_PITCH | FF_DONTCARE, "Times New Roman"};
// Function prototypes
LRESULT WINAPI DialogProcMain(HWND, UINT, WPARAM, LPARAM);
BOOL PrintDocument(HWND);
//-----------------------------------------------------------------------------------------// This is the main Windows function
//-----------------------------------------------------------------------------------------int WINAPI WinMain(HINSTANCE
hInstance,
HINSTANCE
hPrevInstance,
LPSTR
lpCmdLine,
int
nCmdShow)
{
// Create the main dialog window
DialogBox(hInstance, MAKEINTRESOURCE(IDD_MAIN), NULL, DialogProcMain);
return 0;
}
//-----------------------------------------------------------------------------------------// This function handles the messages of the dialog box
//-----------------------------------------------------------------------------------------LRESULT WINAPI DialogProcMain(HWND
hWnd,
UINT
uMsg,
WPARAM
wParam,
LPARAM
lParam)
{
switch (uMsg) {
case WM_INITDIALOG:
// Initialize the dialog box items
SendDlgItemMessage(hWnd, ID_PRIORITY_NORMAL, BM_SETCHECK, BST_CHECKED, 0);
SendDlgItemMessage(hWnd, ID_MESSAGE_TEXT, WM_SETTEXT, 0,
(LPARAM) "Enter the text of the fax message here!");
return TRUE;
break;
case WM_COMMAND:
switch (LOWORD(wParam)) {
case ID_PRINT:
// Execute the printing routine
PrintDocument(hWnd);
break;
case ID_CANCEL:
// Terminate the application
EndDialog(hWnd, FALSE);
break;
}
break;
}
return FALSE;
}
//-----------------------------------------------------------------------------------------// This function is used to print (fax) the message
//-----------------------------------------------------------------------------------------BOOL PrintDocument(HWND hWnd)
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{
BYTE
BYTE
BYTE
BYTE
int
BOOL
RECT
DOCINFO
HDC
HANDLE
HANDLE
szMessageText[1024];
szDocumentName[512];
szFaxNumber[128];
szSubject[128];
nPriority;
bAuto;
rRectText;
DocInfo;
hDC;
hFontText;
hFontOrig;
// Retrieve the message text
SendDlgItemMessage(hWnd, ID_MESSAGE_TEXT, WM_GETTEXT, sizeof(szMessageText),
(LPARAM) szMessageText);
// Retrieve the fax number
SendDlgItemMessage(hWnd, ID_FAX_NUMBER, WM_GETTEXT, sizeof(szFaxNumber),
(LPARAM) szFaxNumber);
// Retrieve the subject
SendDlgItemMessage(hWnd, ID_SUBJECT, WM_GETTEXT, sizeof(szSubject), (LPARAM) szSubject);
// Retrieve the priority and set the correspond priority value (1, 25, 50 or 99)
if (SendDlgItemMessage(hWnd, ID_PRIORITY_LOW, BM_GETCHECK, 0, 0) == BST_CHECKED) {
nPriority = 99;
}
if (SendDlgItemMessage(hWnd, ID_PRIORITY_NORMAL, BM_GETCHECK, 0, 0) == BST_CHECKED) {
nPriority = 50;
}
if (SendDlgItemMessage(hWnd, ID_PRIORITY_HIGH, BM_GETCHECK, 0, 0) == BST_CHECKED) {
nPriority = 25;
}
if (SendDlgItemMessage(hWnd, ID_PRIORITY_VERY_HIGH, BM_GETCHECK, 0, 0) == BST_CHECKED) {
nPriority = 1;
}
// Retrieve the options
bAuto = (BOOL) (SendDlgItemMessage(hWnd, ID_AUTO, BM_GETCHECK, 0, 0) == BST_CHECKED);
// Create the document name. The document name itself is "Testfax",
// the rest are data fields
wsprintf(szDocumentName, "[email protected] %[email protected]@F307 %[email protected]@F301 %[email protected]",
szFaxNumber, szSubject, nPriority);
// Check whether we want to see the fax dialog
if (bAuto) {
lstrcat(szDokumentname, "@[email protected]");
}
else {
lstrcat(szDokumentname, "@F501 [email protected]");
}
// Open the printer
hDC = CreateDC(NULL, "ActiveFax", NULL, NULL); // The printer name is always ActiveFax
if (hDC == NULL) {
MessageBox(hWnd, "The printer 'ActiveFax' cannot be found!", "Error",
MB_OK | MB_ICONSTOP);
return FALSE;
}
// Set the document information structure
ZeroMemory(&DocInfo, sizeof(DocInfo));
DocInfo.cbSize = sizeof(DOCINFO);
DocInfo.lpszDocName = szDocumentName;
// Start a new document
StartDoc(hDC, &DocInfo);
// Create a new page
StartPage(hDC);
// Create and select an object for the font
hFontText = CreateFontIndirect(&LogFontText);
hFontOrig = SelectObject(hDC, hFontText);
// Draw the message text
SetRect(&rRectText, 50, 100, 2300, 3000);
DrawText(hDC, szMessageText, lstrlen(szMessageText), &rRectText,
DT_NOPREFIX | DT_WORDBREAK);
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// Unselect and delete the font
SelectObject(hDC, hFontOrig);
DeleteObject(hFontText);
// Finish the page
EndPage(hDC);
// Finish the document
EndDoc(hDC);
// Close the printer
DeleteDC(hDC);
return TRUE;
}
7.5.2. Socket Application (Socket.exe)
This sample application in programming language C demonstrates how data fields can be used
in a program that uses TCP/IP Sockets to connect to the fax server. Since sockets are available
in virtually all programming languages and operating systems (Windows, DOS, OS/2, UNIX,
Linux, etc.) this is a simply way to send faxes if no other connection to the fax server is available.
The connection to the fax server is done through stream sockets in that example. Please note
that you first have to configure a RAW socket with the TCP/IP port number 3000 on the fax
server; otherwise the connection to the fax server cannot be established. Follow these steps
to configure a RAW socket on the fax server:
Choose the menu option Communication / RAW Server or double-click on the corresponding icon in the communication window.
Press the New button.
Set the option TCP/IP Connection with Stream Socket and enter the Port Number 3000.
Complete the configuration with OK.
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7.5.2.1. Program Summary
Program starts at main()
Data input with gets()
Creating the fax message text with sprintf()
Initialization of the socket library WinSock with WSAStartup()
Create a new socket with socket()
Establish a connection to the fax server (IP address 89.1.0.1, port 3000) with connect()
Send the fax message with send()
Close the socket with shutdown() and closesocket()
Release the socket library with WSACleanup()
7.5.2.2. Source Code
#include <stdio.h>
#include <winsock.h>
// Function prototypes
int SendFax(char *, char *, char *);
//-----------------------------------------------------------------------------------------// Main function of the program
//-----------------------------------------------------------------------------------------int main(void)
{
char
szFaxNumber[128];
char
szSubject[128];
char
szMessageText[512];
char
szIP[128];
char
szPort[128];
char
szData[1024];
// Enter the data
printf("Enter the fax number
gets_s(szFaxNumber, 128);
printf("Enter the subject
gets_s(szSubject, 128);
: ");
: ");
printf("Text of the fax message: ");
gets_s(szMessageText, 512);
printf("\n");
gets_s(szIP, 128);
: ");
printf("Port number
: ");
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gets_s(szPort, 128);
printf("\n");
// Create a string for the fax message
wsprintf(szData, "\033(s5H"
"\033&d0D"
"%s\n\n"
"\033(s10H"
"\033&[email protected]"
"%s"
"@F211 %[email protected]@F307 %[email protected]",
szSubject, szMessageText, szFaxNumber, szSubject);
// Send the fax message
SendFax(szData, szIP, szPort);
getchar();
return 0;
}
//-----------------------------------------------------------------------------------------// Send the fax message
//-----------------------------------------------------------------------------------------int SendFax(char *MessageData, char *IP, char *Port)
{
SOCKET
Socket;
int
nResult;
// Initialize Windows Sockets
if (nResult != 0) {
printf("Error %d at WSAStartup()\n", nResult);
return 0;
}
// Create a new socket
Socket = socket(PF_INET, SOCK_STREAM, IPPROTO_TCP);
if (Socket == INVALID_SOCKET) {
printf("Error %d at socket()\n", WSAGetLastError());
return 0;
}
// Establish a new connection
if (nResult == SOCKET_ERROR) {
printf("Error %d at connect()\n", WSAGetLastError());
return 0;
}
// Send fax data
nResult = send(Socket, MessageData, lstrlen(MessageData), 0);
if (nResult == SOCKET_ERROR) {
printf("Error %d at send()\n", WSAGetLastError());
return 0;
}
// Close the socket
shutdown(Socket, 1);
closesocket(Socket);
WSACleanup();
printf("The fax message has been created successfully\n");
return 1;
}
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7.6. Index
A
Access Verification ........................................ 44
Active Directory ............................................. 65
ActiveFax Installation .................................... 17
Alias Names ................................................... 60
Alternatives to LPD/LPR ................................ 31
Appendix ..................................................... 106
Archive........................................................... 50
Automatic Archive ................................... 50
Individual Archive Folders........................ 51
Manual Archive ........................................ 51
Authentication .............................................. 83
Automatic Printing ........................................ 61
General Settings....................................... 42
User Depending Settings ......................... 42
Automatic Redialing ...................................... 39
B
Bitness
32-bit........................................................ 18
64-bit........................................................ 18
Blacklist ......................................................... 46
Block Incoming Fax Calls .......................... 46
Block Outgoing Fax Calls .......................... 46
Brooktrout ..................................................... 78
C
Charge Optimization ..................................... 75
Compressed Printing ..................................... 41
Concept
Fax-On-Demand ....................................... 14
Incoming Fax Message (Receiving) .......... 14
Configuration................................................. 58
Configuration Example .................................. 17
Configuring LPD Printers in UNIX .................. 30
HP-9000 (HP/UX) ..................................... 30
IBM RS/6000 (AIX) ................................... 30
Contents .......................................................... 5
Cost Account Codes....................................... 89
Create....................................................... 89
Select........................................................ 89
ActFax User’s Manual
Cover Page Designer ..................................... 55
Creating Cover Pages / Overlays ............. 55
Using Color Images.................................. 56
Using Cover Pages / Overlays .................. 56
What is a Cover Page / Overlay? ............. 55
CSID....................................................14, 66, 68
Customizing Program Settings ...................... 21
D
Data Fields .................................................... 32
Examples for Data Fields ......................... 98
Overview of the Data Fields .................... 93
Syntax of Data Fields ............................... 93
Why do I need Data Fields?..................... 93
Database ....................................................... 85
DDI ................................................................ 65
Delayed Transmissions ................................. 75
Demoversion ................................................ 18
Direct Dial Number ....................................... 14
DTMF ............................................................ 65
E
E-Mail ............................................................ 82
Embedding Bitmaps into Fax Messages ....... 99
Embedding Data Fields into Applications ... 100
Epson-LQ....................................................... 32
Establishing a Client Connection ............ 21, 28
Examples for valid Fax Numbers .................. 33
External Configuration File options.cfg ........ 92
F
FAQ ............................................................. 111
Fax Forwarding ............................................. 62
Fax Numbers ................................................. 33
Examples ................................................. 33
Format ..................................................... 33
Faxing from Windows Applications .............. 23
Faxlist ...................................................... 35, 37
Colors ...................................................... 35
Columns................................................... 35
Searching for Entries ............................... 37
Selecting Entries ...................................... 37
121
Sorting Entries.......................................... 37
Fax-On-Demand ...................................... 14, 48
Creating a Fax-On-Demand Document .... 48
Execute Polling ......................................... 48
File System .................................................... 25
File System (NFS, Samba) .............................. 31
Formatting Fax Messages.............................. 32
FTP ........................................................... 31, 74
FTP, TFTP and RAW Sockets .......................... 31
G
Glossary ....................................................... 106
H
Hardware Requirements ............................... 11
Help System................................................... 10
Hotkeys........................................................ 110
How to ... ....................................................... 23
access fax messages on other computers?
............................................................ 28
adjust the screen view? ........................... 34
automatically print a fax message? ......... 41
automatically redial failed fax
transmissions? .................................... 39
block fax numbers? .................................. 46
change the faxlist columns? .................... 35
create a cover page or overlay? .............. 55
create a new fax message? ...................... 23
enter a fax number? ................................ 33
move fax messages to the archive? ......... 50
protect the data from unauthorized
access?................................................ 44
select an entry in the faxlist? ................... 37
send a fax messages from UNIX? ............. 30
use multiple fax servers? ......................... 52
use the Fax-On-Demand Server? ............. 48
HP Digital Sending Software.......................... 91
HP LaserJet .................................................... 32
HP-9000 (HP/UX) ........................................... 30
HPGL .............................................................. 32
I
IBM RS/6000 (AIX) ......................................... 30
Incoming Fax Messages................................. 14
Installation of the Fax Client ......................... 28
Installation procedure ................................... 18
Installing ActiveFax ........................................ 17
Customizing Program Settings ................. 21
ActFax User’s Manual
Establishing a Client Connection ............. 21
Program Overview .................................. 17
Running the Setup ................................... 18
Instant Fax Messages.................................... 24
Introduction .................................................. 10
J
JPG Files ................................................ 56, 100
K
Keyboard Hotkeys....................................... 110
Knowledge Base.......................................... 113
L
Least Cost Routing .................................. 77, 80
LPD/LPR .......................................11, 13, 17, 30
M
Mail Merge Documents .............................. 102
Mail Server.................................................... 82
MSN .............................................................. 65
Multiple Fax Servers ..................................... 52
Configure ActiveFax Printers ................... 53
Configure Fax Clients............................... 52
N
Named Pipes ................................................. 25
NetBeui ..............................................12, 21, 28
Network Scanners......................................... 91
NFS ................................................................ 31
O
ODBC Database............................................. 85
options.cfg file .............................................. 92
Outgoing E-Mails .......................................... 14
Outgoing Fax Messages ................................ 13
P
Passive Group Members............................... 64
Permissions ............................................. 44, 59
Phone Book................................................... 69
Postscript ...................................................... 32
Predefined Settings ...................................... 60
Problem Solutions ...................................... 113
Program Overview ........................................ 17
122
Fax Server ................................................ 12
R
RAS ................................................................ 82
RAW Sockets ................................................. 31
Recycle Bin .................................................... 15
Redialing ........................................................ 39
Remote Access Service (RAS) ........................ 82
Report............................................................ 41
Routing of Incoming Fax Messages ............... 65
RS/232 Connection ........................................ 31
Running the Setup ......................................... 18
S
Samba ............................................................ 31
Sample Applications .................................... 115
Sample Configuration .................................... 17
SCO UNIX ....................................................... 31
Screen Description ........................................ 15
Security Settings ............................................ 44
Sending Fax Messages with LPD/LPR ............ 31
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` | {"extraction_info": {"found_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": 0.24309402704238892, "perplexity": 7902.590298261987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590901.10/warc/CC-MAIN-20180719125339-20180719145339-00213.warc.gz"} |
https://www.arxiv-vanity.com/papers/hep-ph/9406360/ | Saclay, T94/078 Orsay, LPTHE 94/58 (June 1994)
INTERMITTENCY AND EXOTIC CHANNELS
A. Bialas On leave from: Institute of Physics, Jagellonian University, Reymonta 4, PL-30 059 Cracow, POLAND
LPTHE, Bâtiment 211, Université Paris-Sud
F-91405 Orsay, FRANCE
R. Peschanski
CEA, Service de Physique Théorique, CE de Saclay
F-91191 Gif-sur-Yvette Cedex, FRANCE
ABSTRACT
It is pointed out that accurate measurements of short-range two-particle correlations in like-charge and in channels should be very helpful in determining the origin of the “intermittency” phenomenon observed recently for the like-charge pion pairs. Submitted to Phys. Rev. D (Brief Reports)
It is now well established that the correlations between two pions at small relative momenta observed in different hadron-induced processes of multiparticle production are drastically different for the like and unlike pairs. The like pairs show the so-called “intermittency” i.e. a strong (probably power-law) rise of the correlation function at small where
Q2=(p1−p2)2=(p1+p2)2−4m2π
is the square of the difference of particle momenta. This phenomenon finds a natural explanation in terms of quantum interference between identical particles (HBT correlations which is now commonly accepted
This interpretation implies that the observed correlations have “geometrical” origin, i.e. they are related to the space-time structure of the volume from which the pions are emitted Before this picture is accepted, however, it is important to discuss if there perhaps exists other dynamical explanations for this phenomenon. In fact, it was already a long time ago pointed out and recalled recently that these effects can be related to another remarkable difference between and channels, namely their resonance structure. While the system contains many resonances, they are absent in the “exotic” channels.
It is of course an entirely open question if this difference implies or not a different behaviour of correlations at very low invariant mass. However, there exist an argument suggesting that this may actually be the case It can be summarized as follows.
In the intermediate energy region, the amplitudes describing the cross-section are known to be dominated by exchange of Regge singularities in the -channel. The principle of duality implies that in the exotic channel the contribution of leading Regge poles exactly cancel each other We do not consider the Pomeron exchange contribution, as it is irrelevant for short-range correlations we are interested here.. Consequently, an exotic channel is dominated by exchange of singularities with low intercept It follows that the energy-dependent part of the cross-section is expected to fall with increasing invariant mass of the system following the power law
σ∼(M2)α−1=(p1+p2)2(α−1)
with
Since there are no resonances in the exotic channel, this smooth behaviour is expected to hold even at relatively low energies. It is therefore perhaps not unreasonable to speculate that it continues even down to the threshold Of course the phase-space factors at the threshold must be corrected for. This, however, is to a large extent taken care of, when one considers normalized correlation functions .. This possibility is particularly suggestive, when compared to the data of Refs. which show precisely the same power law A closer look at the data shows that they follow a power law in rather than This is not a serious difficulty: the asymptotic formula (2), justified at large cannot distinguish these two possibilities. at small and medium . In contrast, for non-exotic channel the leading Regge poles do contribute, the intercept is much higher and, consequently, the expected dependence much weaker than in (2). Even more important, the presence of prominent resonances in the channel disturbs significantly the simple power-law behaviour and thus precludes its trivial continuation to the threshold.
This completes the argument. Admittedly, it is not very convincing as it requires a rather bold extrapolation. We feel, however, that it should not be entirely dismissed, particularly in view of the experimental evidence shown in and Indeed, if one can (at least partly) explain the difference between like-change and unlike-change correlations without invoking quantum (HBT) interference, the interpretation of the existing data may be profoundly modified. Therefore it seems important to check carrefully this possibility.
The main point of the present note is to indicate that there exists a feasible way to test these ideas and perhaps even to give access to a relative strength of the effects of quantum (HBT) interference and those of the exotic nature of the like-charge interactions. The idea is there exist particle combinations which are either
(a) exotic but not identical
(b) identical but non-exotic
In the category (a) one can list and pairs. Since the particles are not identical, quantum interference cannot be responsible for short-range correlations. Therefore if the effect is present in channels it is probably due to their “exoticity”.
In the category (b) one has channel. It is formed by identical particles, so if quantum interference is at work we expect a pattern similar to channel. On the other hand system forms with probability and with probability Only state is exotic and thus if the resonance structure is responsible we expect a weaker short-range correlation effect than in system.
The early data of NA22 coll. and the more recent data of Aleph coll. indicate that the very short range correlations in the exotic channels, if present at all, are substantially weaker than in the channel. However, we feel that a precise comparison of the behavior at very small and at medium is necessary to assess correctly the role of exoticity in the channel.
To summarize, we suggest that accurate measurements of short-range correlations in and channels, should help to determine the origin of the phenomenon of “intermittency”. It seems to us important to clarify this point before one can safely conclude that the present data can be attributed solely to the effect of quantum (HBT) interference.
ACKNOWLEDGEMENTS
We would like to thank Wolfram Kittel for the correspondence about data and Andrzej Krzywicki for interesting comments. This research was supported in part by the KBN grant
REFERENCES
1 F. Mandl and B. Buschbeck, Vienna-preprint HEPHY-PUB-590-93 (May 1993); and Proc. XXII Int. Symp. on Multiparticle Dynamics, Santiago de Compostela Spain, 1992, p.561. A. Pajares (ed.) World Scientific, Singapore.
2 N.A. Agababyan et al., NA 22 coll., Z. Phys. C59 (1993) 405.
3 A. Bialas and R. Peschanski, Nucl. Phys. B273 (1986) 703. For a recent review, see E.A. DeWolf, I.M. Dremin and W. Kittel, to be published in Phys. Rep. C.
4 R. Hanbury-Brown and R. Q. Twiss : Phil Mag. 45 (1954) 663; For a review see D.H. Boal et al., Rev. Mod. Phys. 62 (1990) 553.
5 H. Bächler et al., NA 35 Coll., Zeit. Phys. C61 (1994) 551.
6 P. Grassberger, Nucl. Phys. B120 (1977) 231.
7 A. Bialas, Acta Phys. Pol. B23 (1992) 561.
8 E.L. Berger, R. Singer, G.M. Thomas and T. Kafka, Phys. Rev. D15 (1977) 206.
9 I.V. Ajinenko et al., NA22 Coll., Zeit. Phys. C61 (1994) 567.
10 M. Adamus et al., NA22 Coll., Zeit. Phys. C37 (1988) 347, fig.3 e,f.
11 D. Decamp et al., NA22 Coll., Zeit. Phys. C54 (1992) 75, fig.7. | {"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": 0.8952919244766235, "perplexity": 1388.6212039334548}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038056325.1/warc/CC-MAIN-20210416100222-20210416130222-00105.warc.gz"} |
http://mathhelpforum.com/advanced-math-topics/38431-advanced-transposition-problem.html | # Math Help - advanced transposition problem
I'm having massive difficulty in rearranging the following to find d
c/h=(d^2)*[1-exp-(2r^2/d^2)]
taking the natural log of both sides normally helps when there is an exp function present but here it just gets in the way - no matter how hard I try I can't get d on its own (d in terms of c and r).
can anyone help?
showing the steps you took would help me tackle similar problems in the future so I'd be extremely grateful for this.
Dan
2. There is no Algebra I method to solve such things.
Have a look at Lambert's W-Function. This will suggest to you various methods that have been contemplated to deal with such structures.
Lambert W-Function -- from Wolfram MathWorld
Lambert W function - Wikipedia, the free encyclopedia
Lambert's W-Function
On the other hand, if you do not already know of the difficulty of solving what you attempted to solve, these discussions may not suggest much that you will find useful.
On still another hand, have you considered what happens as 'd' increases without bound? $\frac{c}{h} = -2r^{2}$ Does that lead to anything useful?
On yet another hand, after substituting $q = \frac{1}{d^{2}}$ and rearranging, we have $\frac{cq}{h} = 1-e^{2qr^{2}}$
...and we can contemplate what happens for q = 0.
3. ## Thanks
Many thanks for such a comprehensive reply. It's beyond the scope of my current level of understanding but i'll endevour to fill in the blanks by doing some background reading.
Thanks once again
Dan | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6575744152069092, "perplexity": 680.0315850016055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1430454576828.73/warc/CC-MAIN-20150501042936-00069-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://gitlab.math.unistra.fr/patapon/patapon/-/commit/48bf31eea8ac7aea526c285566689667442f9a73 | Commit 48bf31ee by ph
### up
parent 5c6b6a87
... ... @@ -851,7 +851,7 @@ \subsection{Smooth vortex (performance test)} The smooth vortex test is a classical test for MHD codes. It is described for instance in \cite{dumbserJCP2016}. Because this is a exact solution, for instance in \cite{dumbserJCP2016}. Because this is an exact solution, it allows to assess the accuracy of the solver. Here we also used this test to evaluate the efficiency of the parallel implementation. The test case is built upon a single vortex, which is a stationary ... ... @@ -938,7 +938,7 @@ \end{tabular}\caption{Convergence and performance study\label{tab:Convergence-and-performance}. Some tests are done in single precision (float32) and others in double precision (float64). The efficiency'' is a comparison for N=1024 with the slowest device. "CU" means "Compute Units": it is the number of activated cores in a CPU computation or of OpenCL processing elements for a GPU computation. } with the slowest device. "CU" means "Compute Units": it is the number of activated cores in a CPU computation or of OpenCL compute units for a GPU computation. } \end{table} ... ...
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http://mathhelpforum.com/advanced-algebra/177458-degree-basis.html | 1. ## Degree/Basis
1.Find the degree and basis for Q(3^1/2,7^1/2) over Q.
2.For any positive integers a, b, show that Q(a^1/2+b^1/2)=Q(a^1/2,b^1/2)
Ideas:
1. Well I know if I looked at (3)^1/2 over Q
Then (3)^1/2 has minimal polynomial x^2-3, so degree 2 over Q
(7)^1/2 has minimal polynomial x^2-7 so degree 2 over Q
so entire thing has degree 2.
2.I started by computing the minimal polynomial of a^1/2+b^1/2 over Q
x=a^1/2+b^1/2
x-a^1/2=b^1/2
x^2+2(a)^(1/2)x+a=3
x^2+a-3=2(a)^1/2
x^4+x^2a-3x^2+x^2a+a^2-3a-3x^2-3a+9=4a
x^4+2x^2a-6x^2+a^2-6a+9-4a=0
I don't know how that would hekp me, though
2. For 1.
Ideas:
1. Well I know if I looked at (3)^1/2 over Q
Then (3)^1/2 has minimal polynomial x^2-3, so degree 2 over Q
a basis is: $1, \sqrt{3}$
(7)^1/2 has minimal polynomial x^2-7 so degree 2 over Q
a basis is $1, \sqrt{7}$
so entire thing has degree 2.
this is wrong.
a basis for the extension $\mathbb Q (\sqrt{3}, \sqrt{7})/ \mathbb Q$ is: $1, \sqrt{3}, \sqrt{7}, \sqrt{21}$ so the degree is 4.
3. 1. are you sure about this? remember Q(√3) is a subfield of R. the factorization of x^2 - 7 in R is (x + √7)(x - √7). if Q(√3,√7) has degree 2 over Q, wouldn't that force Q(√3,√7) to have degree 1 over Q(√3)? isn't this the same as saying √7 = a+b√3 for some rational a,b?
i think you should look for a polynomial of degree 4 as the minimal polynomial. the basis elements should be obvious to write down.
it is my contention that Q(√3,√7) = [Q(√3)](√7).
2. think about the basis elements you (hopefully) found for part (1). can you write √a + √b in terms of those basis elements, adapted to a,b instead of 3,7?
if so, then Q(√a+√b) is contained in Q(√a,√b). look at your minimal polynomial. what can you say about the dimension over Q of Q(√a+√b)? can you use a theorem from linear algebra about subspaces here?
4. for 2, I guess I should show (a)^1/2 + (b)^1/2 is in Q((a)^1/2, (b)^1/2) and a^1/2, b^1/2 in Q((a)^1/2 + (b)^1/2)
would this involve degrees and basises?
a basis for Q((a)^1/2 , (b)^1/2) is 1, a^1/2, b^1/2, (ab)^1/2
a^1/2+ b^1/2 in Q((a)^1/2 , (b)^1/2)
Q((a)^1/2 + (b)^1/2) has basis 1, (a)^1/2 + (b)^1/2?
Not quite sure about finding this basis
5. 1.basis elements are 1, root 3, root 7, and root(21)
2.root a + root b= root a+ root b since root a and root b would be basis elements
looking at the minimal polynomial, has dimension 4
Would I use this theorem: Let V ba an n dimensional vector space. If W is any subspace of V, then W=V iff W has dimension n?
6. that would be the one. you could show that you could express √a and √b in terms of powers of (√a+√b), but that seems like more work to me.
7. √a and √b in terms of powers of (√a+√b), but that seems like more work to me.
I want to look at both ways, so if I were to do it this way:
x=(√a+√b)
x^2=a+2√ab+b
???
8. well, the elements 1, √a+√b, (√a+√b)^2 = (a+b) + 2√a√b, (√a+√b)^3 = (2a+b)√a + (a+2b)√b all have to be linearly independent, since [Q(√a+√b):Q] = 4.
so if we set √a+√b = u, √a = (1/(a-b))(u^3 - (a+2b)u), and √b = (1/(b-a))(u^3 - (2a+b)u) (you might want to double-check my algebra).
this shows that Q(√a,√b) is contained in Q(√a+√b) (note as well, that this gives a different basis for Q(√a,√b):
(it is an interesting exercise to show that if we set √a = x, √b = y, that the set {1,x+y,a+b+2xy,(2a+b)x+(a+2b)y} is linearly independent if {1,x,y,xy} is).
9. ok that makes sense
10. Ok this question is related to 1.
What about degree and basis for Q(2^1/2,2^1/3) over Q
2^1/2 has minimal polynomial x^2-2
a basis would be 1,2^1/2
2^1/3 has minimal polynomial x^3-2
basis would be 1, 2^1/3,2^2/3
basis for Q(2^1/2,2^1/3) has degree 5 and basis 1, 2^1/3,2^2/3, 2^1/2,2^1/6?
11. no, 2*3 is not 5.
x^3 - 2 still doesn't have any roots in Q(√2). any root still lies in a cubic extension of Q(√2).
look at your set of "powers" of two: {0, 1/6, 1/3, 1/2, 2/3}. can you see what is missing from that list?
hint: 1/3 = 2/6, 1/2 = 3/6, 2/3 = 4/6....
12. degree 6
we need 5/6
13. Originally Posted by Deveno
well, the elements 1, √a+√b, (√a+√b)^2 = (a+b) + 2√a√b, (√a+√b)^3 = (2a+b)√a + (a+2b)√b all have to be linearly independent, since [Q(√a+√b):Q] = 4.
so if we set √a+√b = u, √a = (1/(a-b))(u^3 - (a+2b)u), and √b = (1/(b-a))(u^3 - (2a+b)u) (you might want to double-check my algebra).
this shows that Q(√a,√b) is contained in Q(√a+√b) (note as well, that this gives a different basis for Q(√a,√b):
(it is an interesting exercise to show that if we set √a = x, √b = y, that the set {1,x+y,a+b+xy,(2a+b)x+(a+2b)y} is linearly independent if {1,x,y,xy} is).
I thing I understand the first part
Now for showing Q(√a+√b) is contained in Q(√a,√b)
Let √a = x, √b = y
I have x+y=√a+√b
(x+y)^2=a+b+2xy
14. my bad, that was a typo, it should be a+b+2xy | {"extraction_info": {"found_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": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8998640179634094, "perplexity": 2593.462330852159}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164954485/warc/CC-MAIN-20131204134914-00094-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-much-is-70-degrees-fahrenheit-in-celsius | Chemistry
Topics
# How much is 70 degrees Fahrenheit in Celsius?
Mar 12, 2017
${70}^{\circ} F = 21. {\overline{1}}^{\circ} C$
#### Explanation:
To convert Fahrenheit to Celsius, the normal method is to subtract $32$ then divide by $1.8$ (i.e. $\frac{9}{5}$).
So in our example:
$\frac{5}{9} \left(70 - 32\right) = \frac{5}{9} \left(38\right) = \frac{190}{9} = 21. \overline{1}$
Alternatively, since $- {40}^{\circ} F = - {40}^{\circ} C$, we can convert by adding $40$, multiplying by $\frac{5}{9}$, then subtracting $40$:
$\frac{5}{9} \left(70 + 40\right) - 40 = \frac{5}{9} \left(110\right) - 40 = \frac{550}{9} - 40 = 61. \overline{1} - 40 = 21. \overline{1}$
The reverse conversion from Celsius to Fahrenheit can be done by either of the following:
• Multiply by $\frac{9}{5}$ then add $32$.
• Add $40$, multiply by $\frac{9}{5}$, then subtract $40$.
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http://quant.stackexchange.com/tags/numerical-methods/hot?filter=year | # Tag Info
## Hot answers tagged numerical-methods
4
The method described in Hallerbach (2004) always worked well for me. We derive an estimator for Black-Scholes-Merton implied volatility that, when compared to the familiar Corrado & Miller [JBaF, 1996] estimator, has substantially higher approximation accuracy and extends over a wider region of moneyness.
3
I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector $z \sim N(\mathbb{0},I_n)$ (where $z,0 \in \mathbb{R}^{n\times1}$ and $I_n \in \mathbb{R}^{n\times n}$ is the identity matrix) and you want to transform it into a multivariate normal $x \sim N(\mu,\Sigma)$ you do it the ...
2
There is a qualitative shift in the shape of the density. When V is small it is monotone decaying. When V is large it looks more like a Gaussian. Another reason he uses two schemes is that he wants match two moments of the density. When V is small, the moment matching equations for the quadratic Gaussian are unsolvable. When V is large they are unsolvable ...
2
Peter Jaeckel wrote a paper just on how to solve this problem: By Implication (July 2006; Wilmott, pages 60-66, November 2006). Probably the most complicated trivial issue in financial mathematics: how to compute Black's implied volatility robustly, simply, efficiently, and fast downloadable from jaeckel.org In my experience the most important thing is to ...
2
To keep things simple let's assume you have a perfect random number generator (i.e. I will discuss only the statistics not the numerics of the problem). I will also focus on the practical matter and gloss over some mathematical details. From a practical perspective "convergence" means that you will never get an exact answer from Monte-Carlo but ...
2
the output of an MC simulation depends on the random numbers used and if the distribution used is not too weird, after 10,000 runs you will get an answer that is distributed $$\mu + \frac{\sigma}{\sqrt{n}} Z,$$ with $Z$ a standard normal. Here $n=10,000.$ With $\mu$ the quantity you want and $\sigma$ the standard deviation. So you won't get precisely the ...
1
It appears that you are plotting your analytical delta as a % of the delta of the underlying. This is why the delta converges to 100% As for the numerical delta, it could be that you are not adjusting for the DV01 of the underlying. This would explain why the numerical delta still increases as the option gets more in the money and why the distortion is ...
1
Presumably you are trying to use a finite difference method to solve a differential equation. The non-uniformity of the grid has an impact on accuracy. Hence, it is useful to include a parameter in the grid-generation algorithm that controls the rate at which the spacing increases away from the boundary. There are many approaches for generating non-uniform ...
1
The only special function needed for computing Black-Scholes option prices is the cumulative normal function ("N" or "Phi") or equivalently the error function ("erf"). These are very widely available with good standard library implementations. The erf function in single and double precision is part of the c99 and c++11 math standard libraries. For your ...
1
Below is the root search algorithm code I wrote in college. This is written in octave. It's simple to understand and re-write in C++. Develop numerical methods algos as a separate module and integrate with your pricing and other code I want to WARN you to re-check for bugs. It always converges for my objective functions First function is Dekker method ...
1
Bracketing methods such as Bisection and Regula Falsi are always known to converge but they are very slow. Newton Raphson and secant methods are fast (quadratic convergence) but has convergence problems. Google for Newton Raphson convergence pitfalls. Classical ones such as"Trapped in local minima", "Diverge instead of converge" etc Algorithms such as ...
1
Use your total wealth allocated to the trades as denominator. Total wealth allocated would include all collateral. In this way you (or your broker) make sure that the denominator is always positive. Presumably this would also reflect what you really want to track. The only problem that remains is what amount of your wealth needs to be allocated. But this is ...
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http://physics.stackexchange.com/questions/9296/how-does-carbon-dioxide-or-water-vapour-absorb-thermal-infra-red-radiation-from | # How does carbon dioxide or water vapour absorb thermal infra red radiation from the sun?
We are all told at school water vapour and carbon dioxide are the top two greenhouse gases, and that they absorb thermal infra red radiation, trap heat and warm up the Earth. My question is how do they do that? Why can't Oxygen or Nitrogen or any other gas not absorb infra red radiation as well as water vapour or CO2?
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As you can see on these absorption spectra for $\textrm{H}_2\textrm{O}$ and $\textrm{C}\textrm{O}_2$, both molecules have moderate to strong absorbtion in the mid-IR wavelengths, with the absorption of $\textrm{C}\textrm{O}_2$ extending out into the longer wavelengths.
Other molecules common in the atmosphere don't have such strong absorption at the wavelengths given off by thermal radiation. If you are asking why that is, I'm afraid I can't give you a very detailed answer, except to say that the absorption spectra of molecules (and atoms) is governed by quantum mechanics. Maybe somebody else can explain how they would be calculated from principles, but that is beyond my education.
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The absorption of IR radiation is due to vibrations of molecules. When a vibration causes change in charge distribution (or dipole moment to be more specific) the IR radiation is absorbed.
Generally, hetero-polar molecules, like $H_2O \; and \; CO_2$, have permanent dipole moment. The external oscillating electric field in this case perturbs the Hamiltonian and causes IR absorption. Hence, they contribute to "Green House effect" by absorbtion of heat.
$H_2O$, which is non-linear molecule, has three fundamental modes of vibrations. Symmetrical Stretching, asymmetrical Stretching and scissoring (bending). $CO_2$, which is linear molecule, has four fundamental modes of vibrations. Symmetrical stretching, asymmetrical stretching and two degenerate scissoring modes, in planes perpendicular. The symmetric stretching mode in $CO_2$ does not produce or absorb any IR, as it does not cause change in dipole moment, but other modes do change charge distribution causing absorption of IR.
Well, the homo-polar molecules, like $N_2 \; and \; O_2$, does not have any permanent dipole moment. The external oscillating electric field does not perturb the Hamiltonian for nuclear motion, and does not absorb IR, though it perturbs Hamiltonian for electronic motion. Hence, do not contribute for "Green House effect".
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Chutsu was seen here last time 1st of may. And homo-polar is a nonsenseword. This words are used in chemistry for different types of bonds, but for polarity of molecules this words are nonsense. – Georg Oct 27 '11 at 16:50
In some quantity, everything in the air including nitrogen and oxygen absorbs and emits black body radiation at frequencies which overlap the frequencies absorbed by CO2. In fact, the only reason why there is IR in the air is because the surface of the earth emits black body radiation in proportion to its temperature. The air then does the same thing at some level. The question then is, in what quantity is the atmosphere absorbing and emitting black body radiation. The emissivity of nitrogen and oxygen gasses should be closed to 100%, since they do not reflect IR significantly. But the larger question is how does the quantity of black body absorption compare to the fingerprint absorption of CO2. Actual measurements and numbers do not seem to exist. So promoters use computer models to divide up the heat of the atmosphere between pollutants such as CO2 and everything else. They then pull such numbers out of the hat which say increases in CO2 levels will create a global temperature increase of about 6 deg. This is about 20% of the 33 deg. which the atmosphere is said to contribute to the temperature of the globe.There are about 30 times as many water vapor molecules in the air as CO2 molecules, and water vapor has a more effective fingerprint spectrum. It is also much more variable. This means water vapor will swamp whatever CO2 does. It is obviously not being honest to say CO2 does twenty percent of the heating, when there is a hundred times as much water vapor doing the same thing.
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The real fact is that all of the atmosphere is principally heated by thermal conduction and convection - heating the atmosphere by absorbing infrared radiation is negligible.
Water vapour is accepted to be about 1% of a normal atmosphere. At the Earth's surface the mass of a cubic metre of air is about 1.205 kg. As we go up in the atmosphere pressure decreases and thus so does the mass of air per cubic metre.
At 1% water vapour concentration by volume or about 1.5% by weight there are about 19 grams per cubic metre at sea level - significantly less at higher altitudes. Kiehl & Trenberth claim greenhouse gases produce "backradiation" of 324 Watts / square metre.
A watt is equivalent to a Newton metre per second. A newton is a mass of 1 kilogram accelerated by gravity - 9.8 metres per second.
Hence 324 Watts per square metre is equivalent to 324 Newton metres per second.
324 Newtons is 33 kg subjected to gravity's acceleration.
If anyone can explain to me how a mere 12 grams of water vapour in every cubic metre of normal air at sea level can produce that sort of energy I would feel much enlightened - I simply say it is impossible without furnace like temperatures.
As for Carbon Dioxide at 0.04% by volume which is 0.06% by weight it constitutes approximately 0.732 grams per cubic metre - again significantly less at high altitudes as it is heavier than air.
By similar logic if any one can explain how such an insignificant mass of Carbon Dioxide can produce any detectable quantity of radiation I would feel much enlightened !!
As data from the Moon orbiting satellites a heated surface radiating to space cools at a very slow rate.
True the Moon can become very cold on the night side but that lunar night lasts 354 Earth hours and the surface cools about 290 degrees during that 354 hours - less than a degree per hour.
At that rate the dark side of the Earth would "cool" less than 12 degrees before the Sun rose again.
It is the period of rotation of the Earth that is significant into determining how much the Earth cools.
It is the power of the Solar radiation that determines the maximum daytime temperature.
The atmosphere and oceans reduce the temperature the Solar radiation is capable of causing and there is simple proof in that the Moon's surfaces are heated to over the boiling point of water whilst the Earth never approaches those extremes except when conduction/convection is suppressed or prevented entirely.
Finally do you really believe that although Nitrogen and Oxygen do not significantly absorb Infrared radiation they do not radiate Infrared radiation ?
A well known fact of physics is that everything that has a temperature above absolute zero radiates electromagnetic radiation !
At ambient air temperatures well established physics, Wien's law, predicts that the intensity of the emitted electromagnetic radiation is at a peak well within the Infrared radiation wavelength band.
Almost all of the electromagnetic radiation in the atmosphere is coming from Nitrogen and Oxygen which constitute about 99% of the atmosphere.
The only way Nitrogen and Oxygen would not constitute the vast bulk of this atmospheric radiation is if they were at absolute zero and all the warmth you feel came from 19 grams per cubic metre water vapour and 0.732 grams per cubic metre CO2.
I think that proposition is absurd !
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https://tex.stackexchange.com/questions/295317/beamer-custom-sidebar-with-navigation-within-subsubsections | # Beamer custom sidebar with navigation within subsubsections
I would like to add a left handed sidebar to my beamer presentation slides, however there are some specifics as to how I would like to do this:
• The sidebar would only be present on specified pages (the majority of slides will not have the sidebar)
• The side bar would not show navigation to any of the sections or subsections for the rest of the presentation, instead it would have links to a series of other slides that would all also have this sidebar.
• This sidebar would be able to be called multiple times in the presentation where it would follow the same format, but each time called it would be in a different place, referencing slides in that location.
• I don't want this to effect the top navigation bar at all
• Ideally, I would be putting everything that the side bar would be referring to in its own subsubsection, but not all subsubsections would have this sidebar
It might help to know what this is for; I want to create a sidebar that will show steps of how to solve a math problem, where the content in the main part of the slide is the specifics for the particular problem.
This is some code to provide something to work with:
\documentclass[14pt]{beamer}
\usetheme{Frankfurt}
\begin{document}
\section{Section 1}
\subsection{Section 1.1}
\begin{frame}
\frametitle{1.1.1}
\end{frame}
\begin{frame}
\frametitle{1.1.2}
\end{frame}
\subsection{Section 1.2}
\begin{frame}
\frametitle{1.2.1}
\end{frame}
\begin{frame}
\frametitle{1.2.2}
\end{frame}
\subsection{Section 1.3}
\begin{frame}
\frametitle{1.3.1}
\end{frame}
\begin{frame}
\frametitle{1.3.2}
\end{frame}
\section{Section 2}
\subsection{Section 2.1}
\subsubsection{Section 2.1.1}
\begin{frame}
\frametitle{2.1.1 part A}
This would be the start to the problem
\end{frame}
\begin{frame}
\frametitle{2.1.1 part B}
This would start the solution
\end{frame}
\begin{frame}
\frametitle{2.1.1 part C}
This would have the next step
\end{frame}
\begin{frame}
\frametitle{2.1.1 part D}
This next step process would continue here (and to further slides if needed)
\end{frame}
\subsection{Section 2.2}
\begin{frame}
\frametitle{2.2.1}
\end{frame}
\begin{frame}
\frametitle{2.2.2}
\end{frame}
\end{document}
Where in this document I would like to have the sidebar be in the subsubsection part of this document. This would be a customized sidebar, but first would just like to focus on getting it there.
Ideally, the sidebar in that subsubsection would have text that would link to the various slides in the subsubsection (but not necessarily all slides).
• Please read the following SE question on how to write MWE for your questions: meta.tex.stackexchange.com/questions/228/… Most of the code you posted is completely unnecessary. Feb 22 '16 at 19:02
• I wanted the complete header that I'm using in case it caused some potential problems with the solution, but your point is well taken. I will edit my post down to not include all of that code. Feb 22 '16 at 19:13
Your problem sounds too specific to me, that a automated solution would be worth it. So why not just add the menu you need manually. You just have to add labels to the frames you want to link to. Then you can link to them from wherever you want – like a custom sidebar menu for example.
\documentclass{beamer}
\begin{document}
\section{Section 2}
\subsection{Section 2.1}
\subsubsection{Section 2.1.1}
\begin{frame}[label=problem]{2.1.1 part A}
\begin{minipage}{0.25\linewidth}
\footnotesize
\begin{enumerate}
\end{enumerate}
\end{minipage}%
\begin{minipage}{0.75\linewidth}
This would be the start to the problem
\end{minipage}
\end{frame}
\begin{frame}[label=solution]{2.1.1 part B}
\begin{minipage}{0.25\linewidth}
\footnotesize
\begin{enumerate}
\end{enumerate}
\end{minipage}%
\begin{minipage}{0.75\linewidth}
This would start the solution
\end{minipage}
\end{frame}
\begin{frame}[label=next]{2.1.1 part C}
\begin{minipage}{0.25\linewidth}
\footnotesize
\begin{enumerate}
\end{enumerate}
\end{minipage}%
\begin{minipage}{0.75\linewidth}
This would have the next step
\end{minipage}%
\end{frame}
\begin{frame}{2.1.1 part D}
This next step process would continue here (and to further slides if needed)
\end{frame}
\subsection{Section 2.2}
\begin{frame}{2.2.1}
\end{frame}
\begin{frame}{2.2.2}
\end{frame}
\end{document}
• Thanks for taking a look at it. I'm wanting to have some of the customization possibilities that come with the sidebar rather than putting it as a minipage, but didn't see how to do that myself. Also, don't know if this happens for you, but when I click on the links to the slides as is coded above, it doesn't take me to that actual slide, but rather half way down that slide (where the text is). Also, I will likely have 20+ of these, rather than a few isolated instances, so automation would be helpful. Feb 22 '16 at 19:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6439240574836731, "perplexity": 1021.7146716931352}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585045.2/warc/CC-MAIN-20211016231019-20211017021019-00488.warc.gz"} |
https://brilliant.org/discussions/thread/new-profiles/ | # New profiles!
Hey everyone, One of the feature requests that we've seen frequently is that when you come across an amazing user-submitted solution or discussion post, you want to see other solutions or posts by that same user. We've made this and much more possible in our recent update to everyone's profile pages. Check it out and let us know what you think!
Cheers, The team at Brilliant
Note by Suyeon Khim
6 years, 11 months ago
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph 1
paragraph 2
[example link](https://brilliant.org)example link
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$
Sort by:
Furthermore,
Note that you can now attempt to solve any problem solved by a person who's profile you click on, regardless of what level it is in. So if you have ever been curious what people in other levels are solving you can now find out by clicking on their profiles.
Staff - 6 years, 11 months ago
Thank you so much for this update!!! I love being able to solve so many more problems :)
- 6 years, 11 months ago
Do you think you can develop some sort of search bar. So you can search for people you know? That would be awesome!
- 6 years, 11 months ago
Great work! I love the new profiles.
Now that you are tweaking, can you do a little addition to "Past Solutions" section? I have been a member of this site from March 2013. Each week I left some of the problems unsolved. Recently, I have thought of returning back to those problems and try them again but it takes a lot of time to reach the previous problem sets. I mean I have to scroll down a lot and click "Load Previous Week" a numerous times to reach the first problem set I attempted. If possible, can you make this job a little easier for us, for example by adding a drop-down menu which lists all the problem sets?
EDIT: Moreover, I wanted to report that the scroll bar which appears in the Past Solutions section, does not function at all in Google Chrome. I often need to switch to Mozilla Firefox for that.
Thanks!
- 6 years, 11 months ago
Yes, the scroll bar doesn't work for me too. I opened a discussion thread about that but got no response.
- 6 years, 11 months ago
Just what I've been waiting for, thank you so much! I also like the details about every little thing, such as number of problems submitted. The only thing I'm still waiting for is the section which shows how many solutions have been featured. This came on Brilliant and was removed for some reason.
Also, instead of featured solutions, voting goes on forever, which I think is less of a good idea than the featured solution idea.
- 6 years, 11 months ago
This is awesome. It's also great how we can even submit solutions for problems not in our problem set. Here are some minor suggestions:
• Just like the solution vote count is shown in green, it would be cool if that would be done too for the topic vote count and comment vote count.
• Maybe you can divide all activitiy of a user in tabs, e.g.: "Solves", "Solutions", "Comments", "Topics", and "Votes" for a better overview.
• When visiting a problem from somebody else's problem set, it would be rather handy to have a badge or something similar indicate from which level it is.
- 6 years, 11 months ago
I think your first point is a good idea, and we will probably do it soonish.
With your second point, are you asking for a way to easily filter your feed, or another person's feed? For example If I went to your profile and only wanted to see your discussions comments, I could click a button and it would only display your discussions comments chronologically.
The last point is a little more complicated because there is so much overlap between levels, but we can definitely find someway to communicate what the difficulty of the problem is in the scheme of Brilliant.
Staff - 6 years, 11 months ago
I was talking about just any feed, actually, so I could see a list of just the problems someone has solved recently, or the comment's he's made, etc.
Oh yes, I forgot about the overlap. Maybe the number of point that could be earned could be displayed, although that might be confusing, because you can't actually earn points on a problem not for your problem set. Maybe you could show the problem sets it appears in.
- 6 years, 11 months ago
excellent update!!!
- 6 years, 11 months ago
One thing, though, I don't know if I like the lifetime points disappearing under quick hover-profiles. I used that a lot. Unless Brilliant is trying to get us away from profiling and stereotyping by lifetime points, which makes sense since "lifetime" is relative.
- 6 years, 11 months ago
I see the lifetime points section.
- 6 years, 11 months ago
"quick hover-over profiles"... You must have special vision enabling you to see what doesn't exist.
- 6 years, 11 months ago
These unlabeled colorless bars are different...
- 6 years, 11 months ago
So many changes everywhere all in 24 hours... My brain... aaahhhh... This is not correct discussion but I don't want to start another one- My friends are empty because I don't have Facebook. I just have anonymous blurred imaginary friends. That is kind of sad and unnerving, knowing that blur will always be there.
- 6 years, 11 months ago
Hahaha, Brilliant is always growing and changing... :)
We'll make the "blurry friends" smarter soon—you are probably going to have to tolerate it through the weekend though...
Staff - 6 years, 11 months ago
Now there are lots of colorful bars and problems are now called "a problem". I liked seeing the title of the problem.
- 6 years, 11 months ago
It says "a problem" in an event stream only when there isn't a name for the problem. This usually happens when the problem is in the Practice section...
Staff - 6 years, 11 months ago
Oooohhhhhhh... I'm not very smart. Thanks for revealing the obvious.
- 6 years, 11 months ago
Great work done by brilliant. I really appreciate that. Can you also add a section for Logical Reasoning. That would be great.
- 6 years, 11 months ago
I like the possibilities for many prompts instead of "intellectually exciting discoveries" and such. It leaves it open for more information.
- 6 years, 11 months ago
Keep in mind, you can also write your own prompt. You don't have to be limited by the random prompts we supply.
Staff - 6 years, 11 months ago
We can update our status too :) Thanks, Brilliant!!
- 6 years, 11 months ago
Your status displays in the little box when people hover over your profile with a cursor, so everybody can get a quick sense of what's going on with you.
Staff - 6 years, 11 months ago
Nice, all updates on Brilliant, Amazing!!
- 6 years, 11 months ago
from its too?but we can't to mention people but this is amazing
- 6 years, 11 months ago
It is possible that the ability to mention people in conversations you think are relevant to them will come out in the future. Like all things on Brilliant, your profiles will evolve.
Staff - 6 years, 11 months ago
This is the Amaziing of Brilliant Future sir.Two Thumbs For Brilliant (y)
- 6 years, 11 months ago
In my homepage, i am shown that i am in level 4 in number theory but on my profile it shows me that i am at level 3.. Please help..
- 6 years, 11 months ago
Its fixed now...Thanks.
- 6 years, 11 months ago
This is beyond amazing. Good job guys :)
- 6 years, 11 months ago
In my profile, i am shown that i am in level 5 in number theory but on problem page, shows me that i am at level 4.. Please help..Thank's
- 6 years, 11 months ago
You probably was well in ur last answers, so now you've been promoted. In the next update u will answer the questions of level 5. :)
- 6 years, 11 months ago
I love it!
- 6 years, 11 months ago
The new profiles are brilliant
- 6 years, 11 months ago
ohh its wonderful....thanx...
- 6 years, 11 months ago
Feature Request: We could add a messageboard to everyone's profile that would be much like a talk page
- 5 years, 9 months ago
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8078915476799011, "perplexity": 2160.904231198634}, "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-2020-34/segments/1596439736962.52/warc/CC-MAIN-20200806121241-20200806151241-00081.warc.gz"} |
http://superuser.com/users/196505/edi9999?tab=activity | # edi9999
less info
reputation
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bio website javascript-ninja.fr location New York, United States age 22 member for 1 year, 6 months seen Aug 5 at 15:46 profile views 7
Creator of DocxgenJS, UI Interface Developper, JS, Coffeescript and Backbone Lover (sometimes Angular too :p)
# 22 Actions
Jun12 comment How do I modify a style so that it will appear as a section heading in a Table of Contents? I don't think they is a way to do this. The right way to do that is to `add 1` to all of your Headings Jun2 awarded Commentator Jun2 comment Is there a way to embed hidden information in a paragraph in Microsoft Word Can't you use the commenting functionnality ? Feb6 awarded Teacher Feb6 answered Copying all files from one folder to an other with command line in windows Feb6 comment Copying all files from one folder to an other with command line in windows I needed it to be done in command line because it's a task I wanted to do often (it's part off the creation of an installer) Feb6 comment Copying all files from one folder to an other with command line in windows Without the /s it works, except that the folders aren't copied Feb6 comment Copying all files from one folder to an other with command line in windows This command : `copy /s C:\wamp\www\poker-tell\server\*.* C:\wamp\www\poker-tell\build\window` raises the same error Feb6 comment Copying all files from one folder to an other with command line in windows I would like to copy subdirectories too... Feb6 comment Copying all files from one folder to an other with command line in windows I get "The syntax of the command is incorrect. " My command : `copy C:\\wamp\\www\\poker-tell\\server\\*.* C:\\wamp\\www\\poker-tell\\build\\window /s` Feb6 comment Copying all files from one folder to an other with command line in windows I get "insufficient memory" (seems that the content of the paths of some of the files exceed 254) Feb6 comment Copying all files from one folder to an other with command line in windows Actually this doesn't work either because I would like to copy the folders that are in src too Feb6 awarded Student Feb6 asked Copying all files from one folder to an other with command line in windows Sep26 awarded Autobiographer Sep10 comment converting html into word doc or pdf preserving styles (or how to use git for word documents) So you mean automatically using one command line ? Sep10 answered converting html into word doc or pdf preserving styles (or how to use git for word documents) Jul31 awarded Editor Jul31 revised how to use codefolding feature of notepad++ for normal text documents Image Inline, corrected typos Jul31 suggested suggested edit on how to use codefolding feature of notepad++ for normal text documents | {"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": 0.8833510279655457, "perplexity": 4593.745164563457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1409535917663.12/warc/CC-MAIN-20140901014517-00001-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.bartleby.com/solution-answer/chapter-111-problem-4e-precalculus-mathematics-for-calculus-6th-edition-6th-edition/9780840068071/11e50de8-c2b1-11e8-9bb5-0ece094302b6 | # The equation relating x , y to z for the given conditions.
BuyFind
### Precalculus: Mathematics for Calcu...
6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind
### Precalculus: Mathematics for Calcu...
6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
#### Solutions
Chapter 1.11, Problem 4E
To determine
Expert Solution
## Answer to Problem 4E
The quantities x,y and z are related by equation z=12xy .
### Explanation of Solution
Given:
z is proportional to the product of x and y and z is 10 when x is 4 and y is 5.
Definition:
If the quantities x,y and z are related by an equation,
z=kxy (1)
For some constant k0 , we say that z is proportional to the product of x and y. The constant k is called the constant of proportionality.
Calculation:
It is given that z is proportional to the product of x and y.
Compare the above equation with the definition of proportionality in equation(1).
z=kxy (2)
Substitute 10 for z, 4 for x and 5 for y in equation (2).
10=k×4×510=20k
Simplify the above equation to find the value of k.
10=20kk=1020k=12
Substitute the value of k in equation (2)
z=12xy
Therefore, the quantities x,y and z are related by equation z=12xy .
### Have a homework question?
Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!
© 2021 bartleby. All Rights Reserved. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8204250335693359, "perplexity": 2051.9526924159527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057775.50/warc/CC-MAIN-20210925202717-20210925232717-00395.warc.gz"} |
https://earthscience.stackexchange.com/questions/9911/what-are-the-formulae-for-calculating-the-thermal-heating-effect-of-back-radiati | # What are the formulae for calculating the thermal heating effect of back radiation in greenhouse gases?
The ability of greenhouse gases to trap heat and re-radiate some of it back to the original heat source looks like a useful thermal property which could possibly be incorporated in things like double glazing, wall insulation, etc. What are the formulae which will enable us to calculate this additional thermal effect on top of the normal conductive heat loss parameters, heat capacity, U values etc ? We would also like to find details of the original experiments which were used to confirm the accuracy of said formulae as all I can find with internet searches are the well know school type experiments with plastic bottles and heat lamps which demonstrate the principle but are not sufficiently quantitatively accurate. We asked a similar question in Physics Stack Exchange as we considered it a question relating to fundamental physical (thermal) properties of gases but it got little response.
• I think this question is appropriate for this site, given the generality, but you might also have better luck with a more specific question about double glazing over at physics.stackexchange.com – naught101 Mar 17 '17 at 2:10
As you have mentioned already, there are several physical phenomena which are responsible for heat and energy transfer. I'll just list them again for completeness, although you already addressed them in your question.
1. Conduction
2. Convection
3. Latent heat of vaporization
To get to your question: Radiation is the transport of energy by means of electromagnetic waves emitted from a body of a absoulte temperature uneqal zero Kelvin.
What does that mean? Radiation does not rely on contact or a transporting fluid as do conduction and convection. Instead it is equivalent to the fourth power of its absolute Temperature. This is what we know as "Black Body Radiation".
The equation for this is:
$$E = \sigma * T^4$$
where σ is the Stefan-Boltzman Constant and T is the absolute Temperature in Kelvin. This law is obtained by integrating Planck's law over the solid angle and wavelength, in order to determinate the energy which is transmitted not only by one particular wavelength, but the whole spectrum and in all directions. Combined with Kirchhoffs Law, which says, that the absorbtivity of an object equals its emissivity, you can calculate the radiated Energy of any object or atmospheric layer. The emissivity (ε) of a body is a value between zero and one with one being the perfect black body. For most things you can find values in the web. It also explains why aluminum foil is such a good insulator.
$$E = ε * \sigma * T^4$$
To put all of this in an example: Everything around you radiates heat. Beginning with the most obvious, the sun. The sun's surface has an approximate temperature of 6000K. This means it emits radiation of the following spectrum:
As you can see, the maximum of the energy of the sun is radiated in the part which we know as visible light. Also a big part of the radiated energy is absorbed in the atmosphere (the difference between yellow and red graphs). Responsible for these are particularly Ozone (O3), Water (H2O) and (CO2) which are also our most important greenhouse gases.
What happens, is that the radiated energy of a particular wavelength "hits" the molecules and excites them into a state of higher kinetic energy. The average random kinetic energy of a substance is what we know as the temperature of this substance. Hence, also the atmosphere acts like a black body and radiates energy back to space but also back to earth.
Unfortunately I can't really say much about the second part of your question. I don't know about the original experiments. But I suggest a read about Gustav Kirchhoff, who introduced the idea in 1860 which then has been proceeded by Josef Stefan in 1879 and Ludwig Boltzman (1884). Also Planck's works from 1900 could be helpful.
I hope I was able to point you in the right direction with your studies.
It is important to remember a number of things about the basic Stefan-Boltzmann equation. It is supposed to apply to a spherical black body at uniform temperature. This is a hypothetical thing ! It is spherical because if you have an irregular shaped object parts of it can radiate back onto itself. It is also supposed to be a body i.e. have a surface from which the radiation can be emitted (and also absorbed). This makes it trickier to use with something that is partially transparent to radiation i.e. some radiation can pass right through it (like gases). Of course we can adjust the basic S-B equation with an emissivity factor where a real world object is concerned. Finding an emissivity value for a gas is a bit tricky too. Especially when mixed with other gases. Even if pure, there can be a quite wide range of values, pure CO2 could be as low as 0.002 .
When you have done all that however you need to understand something basic. The S-B equation gives you a figure for the radiative flux. It is NOT a figure for heat transfer. To obtain a figure for the actual heat transfer you need the temperature of the cold object (double glazing gas filling or earth atmosphere) and the temperature of the hot object (interior room or earth surface). S-B will give you the radiative HEAT transfer from the hot source to the cold sink. There is no heat transfer the other way even though there is radiation transfer both ways. Consider it as a NET radiative flux transfer if you like. The radiative flux which flows from the cold object to the hot one changes the net outflow of actual heat from the hot object, i.e. its rate of cooling will be slowed. This is effectively the same type of thing as insulation in a conductive or convective sense. The more insulation there is the slower the rate of cooling of the hot object.
So for the Earth's atmosphere we can see that the so called "back radiation" (same as the measurable down welling long wave IR) are not transfering heat from anything but are in fact changing the net radiative flux at the earth's surface and thus slowing down it's rate of heat loss. The back radiation is not making the surface of the earth hotter per se but making it less cool when it loses heat. If you can get an equivalent effect to the atmospheric greenhouse gas effect by putting pure CO2 in a double glazing unit I wouldn't like to guess. I suspect the effect may be small but some lab experiments carefully designed to just measure radiation effects and eliminate conduction/convection would be a good idea. I am not aware of any past experiments having been done on this. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7625482678413391, "perplexity": 409.98680836878657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038064520.8/warc/CC-MAIN-20210411144457-20210411174457-00441.warc.gz"} |
http://tex.stackexchange.com/questions/65668/how-to-rotate-the-label-such-that-its-base-line-is-parallel-to-the-bisector?answertab=active | # How to rotate the label such that its base line is parallel to the bisector?
How to rotate the label counter clockwise such that its base line is parallel to the bisector?
\documentclass[border=0pt,pstricks]{standalone}
\usepackage{pst-eucl}
\psset{PointName=none,PointSymbol=none}
\begin{document}
\begin{pspicture}[showgrid=false](6,6)
\pstGeonode[CurveType=polyline](1,1){A}(5,5){B}(4,1){C}
\pstBissectBAC[linestyle=dotted,linecolor=red]{A}{B}{C}{C'}
\pstMarkAngle[MarkAngleRadius=1.5,LabelSep=0.75]{A}{B}{C}{\tiny$180^\circ-\theta$}
\end{pspicture}
\end{document}
-
\documentclass[border=0pt,pstricks]{standalone}
\usepackage{pst-eucl}
\psset{PointName=none,PointSymbol=none}
\begin{document}
\begin{pspicture}[showgrid=false](6,6)
\pstGeonode[CurveType=polyline](1,1){A}(5,5){B}(4,1){C}
\pstBissectBAC[linestyle=dotted,linecolor=red]{A}{B}{C}{B'}
\pcline[linestyle=none](B')(B)\ncput[nrot=:U,npos=0.84]{\tiny$180^\circ-\theta$}
\end{pspicture}
\end{document}
a solution without knowing B'
\documentclass{article}
\usepackage{pst-eucl}
\begin{document}
\begin{pspicture}[showgrid=false](6,6)
\pstGeonode[CurveType=polyline](1,1){A}(5,5){B}(4,1){C}
\pcline[linestyle=none](!
\psGetNodeCenter{A} \psGetNodeCenter{B} \psGetNodeCenter{C}
/LengthBA B.x A.x sub dup mul B.y A.y sub dup mul add sqrt def
/LengthBC C.x A.x sub dup mul C.y A.y sub dup mul add sqrt def
/Factor LengthBC LengthBA div def
A.x B.x sub Factor mul B.x add /A.x ED
A.y B.y sub Factor mul B.y add /A.y ED
A.x C.x add 2 div A.y C.y add 2 div )(B)\ncput[nrot=:U,npos=0.84]{\tiny$180^\circ-\theta$}
\end{pspicture}
\end{document}
-
It becomes more complicated :-) – kiss my armpit Aug 10 '12 at 14:27
no, because you do not need any coodinate – Herbert Aug 10 '12 at 14:30
I meant that your solution depends on B' while my solution does not. However, I cannot rotate the label clockwise. – kiss my armpit Aug 10 '12 at 14:35
no I do not need the B'. I used it only while it was defined. See edit for another solution – Herbert Aug 10 '12 at 15:15
The calculation in the second example will put the label on the median rather than on the angle bisector. – kiss my armpit Aug 10 '12 at 17:49
I got the solution even though it is not elegant enough.
\documentclass[border=0pt,pstricks]{standalone}
\usepackage{pst-eucl}
\psset{PointName=none,PointSymbol=none}
\begin{document}
\begin{pspicture}[showgrid=false](6,6)
\pstGeonode[CurveType=polyline](1,1){A}(5,5){B}(4,1){C}
\pstMarkAngle[MarkAngleRadius=1.6,LabelSep=1]{A}{B}{C}{\rput{(B)}(0,0){\tiny$180^\circ-\theta$}}
\end{pspicture}
\end{document}
Clockwise rotation:
\documentclass[border=0pt,pstricks]{standalone}
\usepackage{pst-eucl}
\psset{PointName=none,PointSymbol=none}
\begin{document}
\begin{pspicture}[showgrid=false](6,6)
\pstGeonode[CurveType=polyline](1,1){A}(5,5){B}(4,1){C}
\pstMarkAngle[MarkAngleRadius=1.6,LabelSep=1]{A}{B}{C}{\rput{!\psGetNodeCenter{B} B.y B.x atan 180 sub}(0,0){\tiny$180^\circ-\theta$}}
\end{pspicture}
\end{document}
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http://mathhelpforum.com/geometry/217122-calculating-radius-circle-simple-trigonometry-geometry.html | # Math Help - Calculating the radius of a circle - simple trigonometry/geometry
1. ## Calculating the radius of a circle - simple trigonometry/geometry
From the circle detailed in this video:
To find the radius of the inner circle, why cannot we take the line down from OC to form a right angled triangle, and then do
tan(pi/6)=r/6?
Since tan is opposite over adjacent, why would this not work?
Apparently the answer is r=2, that's now what you would get doing the above. Why is it wrong?
2. ## Re: Calculating the radius of a circle - simple trigonometry/geometry
Hey Mukilab.
I don't know what triangle you are referring to, but it isn't in any way reflecting the radius of the circle. What the video did is the best way to figure out the problem IMO.
3. ## Re: Calculating the radius of a circle - simple trigonometry/geometry
Ah nevermind, I started drawing the circle/triangle on paint and I realised my mistake.
Thank you for taking the time to post anyway | {"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": 0.8830304145812988, "perplexity": 562.6610181098512}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736682773.27/warc/CC-MAIN-20151001215802-00114-ip-10-137-6-227.ec2.internal.warc.gz"} |
http://openvsp.org/wiki/doku.php?id=install | install
# Windows
OpenVSP should run on Windows without installation. Simply unzip the archive to the location of your choosing and double click vsp.exe.
# MacOS
Unzip the archive to the location of your choosing and run the vsp executable.
# Linux
For Ubuntu 18.04, a deb package is available at the OpenVSP downloads page.
For other distributions, there are no pre-compiled Linux binaries produced at this time. In order to run OpenVSP on Linux, you must compile it yourself.
For Debian based Linux distributions, see the following: Ubuntu Instructions
For ArchLinux and its derivatives, OpenVSP is available in the AUR as openvsp. It is recommended to install the package using an AUR helper such as yay.
To install OpenVSP on ArchLinux, simply run the following:
$yay -S openvsp This will pull all the dependencies needed to compile OpenVSP. Then it will automatically fetch, compile and install OpenVSP in your system. # FreeBSD OpenVSP is available from the FreeBSD's port collection. Installation instructions: Method 1: $ cd /usr/ports/cad/openvsp $make install (as root) Method 2: $ portinstall openvsp (as root)
This will pull all the dependencies needed to compile OpenVSP. Then it will automatically fetch, compile and install OpenVSP in your system. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.16493850946426392, "perplexity": 8876.222911497316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579250597458.22/warc/CC-MAIN-20200120052454-20200120080454-00251.warc.gz"} |
https://de.zxc.wiki/wiki/Einsteinium | # Einsteinium
properties
[ Rn ] 5 f 11 7 s 2
99 it
General
Name , symbol , atomic number Einsteinium, Es, 99
Element category Actinoids
Group , period , block Ac , 7 , f
CAS number 7429-92-7
Atomic
Atomic mass 252 and
Electron configuration [ Rn ] 5 f 11 7 s 2
1. Ionization energy 6th.36758 (25) eV614.38 kJ / mol
2. Ionization energy 12.2 (4) eV1 180 kJ / mol
3. Ionization energy 22nd.7 (4) eV2 190 kJ / mol
4. Ionization energy 38.8 (4) eV3 740 kJ / mol
5. Ionization energy 54.1 (1.9) eV5 220 kJ / mol
Physically
Physical state firmly
Modifications 1
Crystal structure cubic
density (calculated) 8.84 g / cm 3
Melting point 1133 K (860 ° C)
boiling point 1269 K (996 ° C)
Chemically
Oxidation states +2, +3 , (+4)
Normal potential (Es 3+ + 3 e - → Es)
Isotopes
isotope NH t 1/2 ZA ZE (M eV ) ZP
248 it {syn.} 27 min ε (≈ 100%) 248 Cf
α (≈ 0.25%) 244 Bk
SF (3 · 10 −5 %) ? ?
249 it {syn.} 102.2 min ε (≈ 100%) 249 Cf
α (≈ 0.57%) 245 Bk
250 it {syn.} 8.6 h ε (> 97%) 2.100 250 cf
α (?) 6.880 246 Bk
251 it {syn.} 33 h ε (?) 0.367 251 Cf
α (0.5%) 6,597 247 Bk
252 it {syn.} 471.7 d α (78%) 6.760 248 Bk
ε (22%) 1,260 252 Cf
253 it {syn.} 20.47 d α (100%) 6.739 249 Bk
SF (8.7 10 −6 %) ? ?
254 it {syn.} 275.7 d α (≈ 100%) 6.618 250 Bk
ε (0.03%) 254 Cf
254 m es {syn.} 39.3 h β - (98%) 254 m
SF (<3%) ? ?
α (0.32%) 250 Bk
255 It {syn.} 39.8 d β - (92.0%) 0.288 255 ft
α (8.0%) 251 Bk
For other isotopes see list of isotopes
Hazard and safety information
GHS hazard labeling
no classification available
As far as possible and customary, SI units are used.
Unless otherwise noted, the data given apply to standard conditions .
Einsteinium is an exclusively artificially produced chemical element with the element symbol Es and the atomic number 99. In the periodic table it is in the group of actinides ( 7th period , f-block ) and is one of the transuranic elements . Einsteinium is a radioactive metal, which can only be produced with great effort in just weighable quantities. It was discovered in 1952 after the test of the first American hydrogen bomb and named in honor of Albert Einstein , who, however, had nothing to do with the discovery or research of Einsteinium. It is produced in very small quantities in nuclear reactors . The metal and its compounds are obtained in small quantities primarily for study purposes.
## history
Einsteinium was discovered by Ivy Mike after the explosion .
Elution curves :
chromatographic separation of Fm (100), Es (99), Cf, Bk, Cm, Am.
Albert Einstein, 1921, photograph by Ferdinand Schmutzer
Einsteinium was found along with fermium after testing the first American hydrogen bomb, Ivy Mike , on November 1, 1952 on the Eniwetok Atoll . The first samples were obtained on filter papers that were carried when flying through the explosion cloud. Larger amounts were later isolated from corals. For reasons of military secrecy, the results were initially not published.
An initial investigation of the remains of the explosion had shown the formation of a new plutonium isotope 244 Pu, which could only have resulted from the uptake of six neutrons by a uranium- 238 core and two subsequent β-decays .
${\ displaystyle \ mathrm {^ {238} _ {\ 92} U \ {\ xrightarrow [{- 2 \ \ beta ^ {-}}] {+ \ 6 \ (n, \ gamma)}} \ _ {\ 94} ^ {244} Pu}}$
At the time, it was believed that the absorption of neutrons by a heavy nucleus was a rare occurrence. However, the identification of 244 Pu led to the conclusion that uranium nuclei can capture many neutrons, resulting in new elements.
The dissolved actinide ions were separated in the presence of a citric acid / ammonium citrate buffer in a weakly acidic medium ( pH ≈ 3.5) with ion exchangers at an elevated temperature. Element 99 (Einsteinium) was quickly detected; the isotope 253 Es, a high-energy α-emitter (6.6 MeV), was first found. It is created by capturing 15 neutrons from 238 U, followed by seven β decays.
This formation through continued neutron capture was possible because at the moment of detonation the neutron flux density was so high that most of the - radioactive - atomic nuclei that had formed in the meantime had not decayed by the next neutron capture. With a very high neutron flux, the mass number increases sharply without the atomic number changing. Only then do the resulting unstable nuclides decay over many β-decays to stable or unstable nuclides with a high atomic number:
${\ displaystyle \ mathrm {^ {238} _ {\ 92} U \ {\ xrightarrow [{- 7 \ \ beta ^ {-}}] {+ \ 15, \ 16, \ 17 \ (n, \ gamma) }} \ _ {\ \ \ \ \ \ \ \ \ \ \ \ 99} ^ {253, \ 254, \ 255} Es}}$
In September 1953, there was no telling when the results of the teams in Berkeley , Argonne and Los Alamos would be published. It was decided to produce the new elements through bombardment experiments; At the same time, it was assured that these results would not fall under secrecy and could therefore be published. Einsteinium isotopes were produced shortly afterwards at the University of California Radiation Laboratory by bombarding uranium ( 238 U) with nitrogen ( 14 N). It was noted that there is research on this element that has so far been kept secret. Isotopes of the two newly discovered elements were generated by irradiating the plutonium isotope 239 Pu, and the results were published in five publications in quick succession. The final reactions from Californium are:
${\ displaystyle \ mathrm {^ {252} _ {\ 98} Cf \ {\ xrightarrow {(n, \ gamma)}} \ _ {\ 98} ^ {253} Cf \ {\ xrightarrow [{17.81 \ d}] {\ beta ^ {-}}} \ _ {\ 99} ^ {253} Es \ {\ xrightarrow {(n, \ gamma)}} \ _ {\ 99} ^ {254} Es \ {\ xrightarrow [{}] {\ beta ^ {-}}} \ _ {100} ^ {254} Fm}}$
The Berkeley team was also concerned that another group of researchers might discover and publish the lighter isotopes of element 100 by ion bombardment before they could publish their classified research. Because at the end of 1953 and at the beginning of 1954 a working group from the Nobel Institute for Physics in Stockholm fired at uranium nuclei with oxygen nuclei; the isotope with the mass number 250 of the element 100 ( 250 μm) was formed.
The Berkeley team has already published some results on the chemical properties of both elements. Finally, the results of the thermonuclear explosion were released in 1955 and then published.
Ultimately, the Berkeley team's priority was universally recognized, as their five publications preceded the Swedish publication and they could rely on the previously secret results of the 1952 thermonuclear explosion. This was associated with the privilege of naming the new elements. They decided to name them after famous scientists who had already died. It was quickly agreed to give the names in honor of Albert Einstein and Enrico Fermi , both of whom had recently died: “We suggest for the name for the element with the atomic number 99, einsteinium (symbol E) after Albert Einstein and for the name for the element with atomic number 100, fermium (symbol Fm), after Enrico Fermi. "The announcement for the two newly discovered elements Einsteinium and Fermium was made by Albert Ghiorso at the 1st Geneva Atomic Conference , which took place from 8 to Took place on August 20, 1955. The element symbol for Einsteinium was later changed from E to Es .
## Isotopes
All 17 nuclides and 3 nuclear isomers known to date are radioactive and unstable. The known mass numbers ranging from 241 to 258. The longest half-life has the isotope 252 may making it on earth no natural deposits give more with 471.7 days. 254 It has a half-life of 275.7 days, 255 Es of 39.8 days and 253 Es of 20.47 days. All other radioactive isotopes have half-lives below 40 hours, the majority of them are below 30 minutes. Of the 3 core isomers, 254 m Es is the most stable with t ½ = 39.3 hours.
## Extraction and presentation
Einsteinium is created by bombarding lighter actinides with neutrons in a nuclear reactor. The main source is the 85 MW high-flux isotope reactor at Oak Ridge National Laboratory in Tennessee, USA, which is set up for the production of transcurium elements (Z> 96).
In 1961 enough Einsteinium was synthesized to produce a weighable amount of the isotope 253 Es. This sample weighed about 0.01 mg and was used to make Mendelevium . Another Einsteinium was made at Oak Ridge National Laboratory by bombarding 239 Pu with neutrons . Approximately 3 milligrams were obtained in four years of continuous irradiation from one kilogram of plutonium and subsequent separation.
### Extraction of Einsteinium isotopes
Small amounts of einsteinium and fermium were isolated and separated from plutonium , which was irradiated with neutrons. Four Einsteinium isotopes were found (with details of the half-lives measured at the time):
• 253 Es: α-emitters with t ½ = 20.03 days and with a spontaneous fission half-life of 7 × 10 5 years
• 254 m Es: β-emitter with t ½ = 38.5 hours
• 254 Es: α emitters with t ½ ≈ 320 days
• 255 Es: β-emitters with t ½ = 24 days
Two fermium isotopes were found:
• 254 Fm: α-emitters with t ½ = 3.24 hours and with a spontaneous fission half-life of 246 days
• 255 Fm: α-emitter with t ½ = 21.5 hours
Bombarding uranium with five-fold ionized nitrogen and six-fold ionized oxygen atoms also produced Einsteinium and Fermium isotopes.
The isotope 248 Es was identified when 249 Cf was bombarded with deuterium. It decays mainly through electron capture (ε) with a half-life of 25 ± 5 minutes but also through the emission of α-particles (6.87 ± 0.02 MeV). The ratio (ε / α) of ∼ 400 could be identified by the amount of 248 Cf generated by electron capture .
${\ displaystyle \ mathrm {^ {249} _ {\ 98} Cf \ + \ _ {1} ^ {2} D \ \ longrightarrow \ _ {\ 99} ^ {248} Es \ + \ 3 \ _ {0 } ^ {1} n \ quad \ left (^ {248} _ {\ 99} Es \ {\ xrightarrow [{27 \ min}] {\ epsilon}} \ _ {\ 98} ^ {248} Cf \ right )}}$
The isotopes 249 Es, 250 Es, 251 Es and 252 Es were generated by bombarding 249 Bk with α-particles. Thereby 4 to 1 neutrons can leave the nucleus, so that the formation of four different isotopes is possible.
${\ displaystyle \ mathrm {^ {249} _ {\ 97} Bk \ {\ xrightarrow [{- \ 4 \ to \ 1 \ n}] {+ \ alpha}} \ _ {\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 99} ^ {249, \ 250, \ 251, \ 252} Es}}$
Although the isotope 252 Es has the longest half-life, the isotope 253 Es is more easily accessible and is mainly used to determine the chemical properties. It was obtained by irradiating 100 to 200 μg 252 Cf with thermal neutrons (flux density: 2 to 5 × 10 14 neutrons × cm −2 s −1 , time period: 500 to 900 h). Ammonium α-hydroxyisobutyrate was used for the separation .
${\ displaystyle \ mathrm {^ {252} _ {\ 98} Cf \ {\ xrightarrow {(n, \ gamma)}} \ _ {\ 98} ^ {253} Cf \ {\ xrightarrow [{17.81 \ d}] {\ beta ^ {-}}} \ _ {\ 99} ^ {253} Es}}$
### Depiction of elementary Einsteinium
Einsteinium is obtained by reducing Einsteinium (III) fluoride with lithium or Einsteinium (III) oxide with lanthanum .
${\ displaystyle {\ ce {EsF3 + 3 Li -> Es + 3 LiF}}}$
${\ displaystyle {\ ce {Es2O3 + 2 La -> 2 Es + La2O3}}}$
## properties
Quartz ampoule (9 mm diameter) with approx. 300 micrograms Es-253. The glow is created by the intense radiation. (Black and white photography)
In the periodic table , the einsteinium with atomic number 99 is in the actinoid series, its predecessor is the californium, the following element is the fermium. Its analogue in the series of lanthanides is the holmium .
### Physical Properties
Einsteinium is a radioactive metal with a melting point of 860 ° C, a boiling point of 996 ° C and a density of 8.84 g / cm 3 . It crystallizes in the cubic crystal system in the space group Fm 3 m (space group no. 225) with the lattice parameter a = 575 pm, which corresponds to a face-centered cubic lattice (fcc) or a cubic closest packing of spheres with the stacking sequence ABC. The radioactivity is so strong that it destroys the metal grid. The metal is divalent and has a noticeably high volatility.
### Chemical properties
Like all actinides, Einsteinium is very reactive. The trivalent oxidation state is most stable in aqueous solution , but bivalent and tetravalent compounds are also known. Bivalent compounds could already be represented as solids; the tetravalent state could already be postulated during chemical transport in tracer quantities, but a final confirmation is still pending. Aqueous solutions with Es 3+ ions are pale pink in color.
## safety instructions
Classifications according to the CLP regulation are not available because they only include chemical hazard and play a completely subordinate role compared to the hazards based on radioactivity . The latter also only applies if the amount of substance involved is relevant.
## use
Einsteinium is mainly used in the production of higher transuranic elements and transactinides . Otherwise, the metal and its compounds are primarily obtained in small quantities for study purposes.
→ Category: Einsteinium connection
The study of einsteinium compounds is limited by several factors:
• The most readily available isotope 253 It is only available once or twice a year on a microgram scale.
• Long- range orders in solids are very quickly destroyed by the intense alpha radiation generated in around 20 days' half-life .
• The radioactive decay produces the isotopes 249 Bk and 249 Cf, which quickly contaminate the sample. The rate is around 3% per day.
${\ displaystyle \ mathrm {^ {253} _ {\ 99} Es \ {\ xrightarrow [{20 \ d}] {\ alpha}} \ _ {\ 97} ^ {249} Bk \ {\ xrightarrow [{314 \ d}] {\ beta ^ {-}}} \ _ {\ 98} ^ {249} Cf}}$
### Oxides
Einsteinium (III) oxide (Es 2 O 3 ) was obtained by calcining the corresponding nitrate in submicrogram quantities. The lattice parameter of the body-centered cubic crystal is 1076.6 (6) pm . A monoclinic and a hexagonal lanthanum (III) oxide structure are also known.
### Oxyhalides
The oxyhalides Einsteinium (III) oxychloride (EsOCl), Einsteinium (III) oxibromide (EsOBr) and Einsteinium (III) oxyiodide (EsOI) are known. Einsteinium (III) oxychloride has a tetragonal structure of the PbFCl type.
### Halides
Einsteinium (III) iodide (EsI 3 ), photographed in transmitted light
Halides are known to have the +2 and +3 oxidation states. The most stable level +3 is known for all compounds from fluorine to iodine and is also stable in aqueous solution.
Oxidation number F. Cl Br I. +3 Einsteinium (III) fluoride EsF 3 Einsteinium (III) chloride EsCl 3 orange Einsteinium (III) bromide EsBr 3 white-yellow Einsteinium (III) iodide EsI 3 amber colored +2 Einsteinium (II) chloride EsCl 2 Einsteinium (II) bromide EsBr 2 Einsteinium (II) iodide EsI 2
Einsteinium (III) fluoride (EsF 3 ) can be prepared by precipitation from Einsteinium (III) chloride solutions with fluoride, as well as from Einsteinium (III) oxide by reaction with ClF 3 or F 2 at 1-2 atmospheres pressure and 300-400 ° C. The crystal structure could not be determined, but it is assumed that it corresponds to the LaF 3 type , as with Berkelium (III) fluoride (BkF 3 ) and Californium (III) fluoride (CfF 3 ) .
Einsteinium (III) chloride (EsCl 3 ) is an orange-colored solid and forms a hexagonal structure of the UCl 3 type , with the Es atom being coordinated 9 times.
Einsteinium (III) bromide (EsBr 3 ) is a white-yellow solid and forms a monoclinic structure of the AlCl 3 type .
Einsteinium (III) iodide (EsI 3 ) is an amber-colored solid and forms a hexagonal structure of the BiI 3 type .
The divalent compounds of Einsteinium are produced by reducing the trivalent halides with hydrogen .
${\ displaystyle {\ ce {2 EsX3 + H2 -> 2 EsX2 + 2 HX}}}$
For einsteinium (II) chloride (EsCl 2 ), einsteinium (II) bromide (EsBr 2 ) and einsteinium (II) iodide (EsI 2 ), no detailed crystallographic data are known, but measured values of absorption bands are known.
## literature
Commons : Einsteinium - collection of images, videos and audio files
Wiktionary: Einsteinium - explanations of meanings, word origins, synonyms, translations
## Individual evidence
1. The values of the atomic and physical properties (Infobox) are, unless otherwise stated, taken from: Richard G. Haire: Einsteinium , in: Lester R. Morss, Norman M. Edelstein, Jean Fuger (Ed.): The Chemistry of the Actinide and Transactinide Elements , Springer, Dordrecht 2006; ISBN 1-4020-3555-1 , pp. 1577-1620.
2. Entry on einsteinium in Kramida, A., Ralchenko, Yu., Reader, J. and NIST ASD Team (2019): NIST Atomic Spectra Database (ver. 5.7.1) . Ed .: NIST , Gaithersburg, MD. doi : 10.18434 / T4W30F ( https://physics.nist.gov/asd ). Retrieved June 13, 2020.
3. entry on einsteinium at WebElements, https://www.webelements.com , accessed on June 13, 2020.
4. The hazards emanating from radioactivity do not belong to the properties to be classified according to the GHS labeling. With regard to other hazards, this element has either not yet been classified or a reliable and citable source has not yet been found.
5. Albert Ghiorso: Einsteinium and Fermium , Chemical & Engineering News, 2003.
6. Albert Ghiorso , G. Bernard Rossi, Bernard G. Harvey, Stanley G. Thompson: Reactions of U 238 with Cyclotron-Produced Nitrogen Ions , in: Physical Review , 1954 , 93 (1), pp. 257-257 ( doi: 10.1103 / PhysRev.93.257 ).
7. SG Thompson, A. Ghiorso, BG Harvey, GR Choppin: Transcurium Isotopes Produced in the Neutron Irradiation of Plutonium , in: Physical Review , 1954 , 93 (4), pp. 908-908 ( doi: 10.1103 / PhysRev.93.908 ) .
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10. PR Fields, MH Studier, JF Mech, H. Diamond, AM Friedman, LB Magnusson, JR Huizenga: Additional Properties of Isotopes of Elements 99 and 100 , in: Physical Review , 1954 , 94 (1), pp. 209-210 ( doi: 10.1103 / PhysRev.94.209 ).
11. GR Choppin, SG Thompson, A. Ghiorso, BG Harvey: Nuclear Properties of Some Isotopes of Californium, Elements 99 and 100 , in: Physical Review , 1954 , 94 (4), pp. 1080-1081 ( doi: 10.1103 / PhysRev .94.1080 ).
12. Hugo Atterling, Wilhelm Forsling, Lennart W. Holm, Lars Melander, Björn Åström: Element 100 Produced by Means of Cyclotron-Accelerated Oxygen Ions , in: Physical Review , 1954 , 95 (2), pp. 585-586 ( doi: 10.1103 / PhysRev.95.585.2 ).
13. ^ GT Seaborg , SG Thompson, BG Harvey, GR Choppin: Chemical Properties of Elements 99 and 100 ( abstract ; machine script (July 23, 1954), Radiation Laboratory, University of California, Berkeley, UCRL-2591 (Rev.) ).
14. ^ SG Thompson, BG Harvey, GR Choppin, GT Seaborg: Chemical Properties of Elements 99 and 100 , in: J. Am. Chem. Soc. , 1954 , 76 (24), pp. 6229-6236 ( doi: 10.1021 / ja01653a004 ).
15. ^ A b A. Ghiorso, SG Thompson, GH Higgins, GT Seaborg ( Radiation Laboratory and Department of Chemistry, University of California, Berkeley, California ), MH Studier, PR Fields, SM Fried, H. Diamond, JF Mech, GL Pyle , JR Huizenga, A. Hirsch, WM Manning ( Argonne National Laboratory, Lemont, Illinois ), CI Browne, HL Smith, RW Spence ( Los Alamos Scientific Laboratory, Los Alamos, New Mexico ): New Elements Einsteinium and Fermium, Atomic Numbers 99 and 100 , in: Physical Review , 1955 , 99 (3), pp. 1048-1049 ( doi: 10.1103 / PhysRev.99.1048 ; Maschinoscript (June 9, 1955), Lawrence Berkeley National Laboratory. Paper UCRL-3036 ).
16. ^ PR Fields, MH Studier, H. Diamond, JF Mech, MG Inghram, GL Pyle, CM Stevens, S. Fried, WM Manning ( Argonne National Laboratory, Lemont, Illinois ); A. Ghiorso, SG Thompson, GH Higgins, GT Seaborg ( University of California, Berkeley, California ): Transplutonium Elements in Thermonuclear Test Debris , in: Physical Review , 1956 , 102 (1), pp. 180-182 ( doi: 10.1103 /PhysRev.102.180 ).
17. a b Richard G. Haire: Einsteinium , in: Lester R. Morss, Norman M. Edelstein, Jean Fuger (eds.): The Chemistry of the Actinide and Transactinide Elements , Springer, Dordrecht 2006; ISBN 1-4020-3555-1 , pp. 1577-1620.
18. G. Pfennig, H. Klewe-Nebenius, W. Seelmann-Eggebert (eds.): Karlsruher Nuklidkarte , 7th edition, 2006.
19. Irshad Ahmad, Frank Wagner, Jr .: Half-life of the longest-lived Einsteinium Isotope- 252 Es , in: J. Inorg. Nucl. Chem. , 1977 , 39 (9), pp. 1509-1511 ( doi: 10.1016 / 0022-1902 (77) 80089-4 ).
20. William McHarris, FS Stephens, F. Asaro, I. Perlman: Decay scheme of einsteinium-254 , in: Physical Review , 1966 , 144 (3), pp 1031-1045 ( doi: 10.1103 / PhysRev.144.1031 ).
21. G. Audi, O. Bersillon, J. Blachot, AH Wapstra: The NUBASE evaluation of nuclear and decay properties , in: Nuclear Physics A , 729, 2003, pp. 3–128. doi : 10.1016 / j.nuclphysa.2003.11.001 . ( PDF ; 1.0 MB).
22. ^ High Flux Isotope Reactor , Oak Ridge National Laboratory; Retrieved September 23, 2010.
23. Darleane C. Hoffman, Albert Ghiorso, Glenn Theodore Seaborg: The Transuranium People: The Inside Story , Imperial College Press, 2000, ISBN 978-1-86094-087-3 , pp. 190–191 ( limited preview in Google Book search).
24. M. Jones, RP Schuman, JP Butler, G. Cowper, TA Eastwood, HG Jackson: Isotopes of Einsteinium and Fermium Produced by Neutron Irradiation of Plutonium , in: Physical Review , 1956 , 102 (1), pp. 203-207 ( doi: 10.1103 / PhysRev.102.203 ).
25. LI Guseva, KV Filippova, Yu. B. Gerlit, VA Druin, BF Myasoedov, NI Tarantin: Experiments on the Production of Einsteinium and Fermium with a Cyclotron , in: Journal of Nuclear Energy , 1954 , 3 (4), pp. 341–346 (translated in November 1956) ( doi: 10.1016 / 0891-3919 (56) 90064-X ).
26. A. Chetham-Strode, LW Holm: New Isotope Einsteinium-248 , in: Physical Review , 1956 , 104 (5), pp. 1314-1314 ( doi: 10.1103 / PhysRev.104.1314 ).
27. ^ Bernard G. Harvey, Alfred Chetham-Strode, Albert Ghiorso, Gregory R. Choppin, Stanley G. Thompson: New Isotopes of Einsteinium , in: Physical Review , 1956 , 104 (5), pp. 1315-1319 ( doi: 10.1103 /PhysRev.104.1315 ).
28. SA Kulyukhin, LN Auerman, VL Novichenko, NB Mikheev, IA Rumer, AN Kamenskaya, LA Goncharov, AI Smirnov: Production of Microgram Quantities of Einsteinium-253 by the Reactor Irradiation of Californium , in: Inorganica Chimica Acta , 1985 , 110 ( 1), pp. 25-26 ( doi: 10.1016 / S0020-1693 (00) 81347-X ).
29. BB Cunningham, TC Parsons, USAEC Doc. UCRL-20426 (1971), p. 239.
30. ^ A b R. G. Haire, RD Baybarz: Studies of einsteinium metal , in: Journal de Physique Colloques , 1979 , 40 (C4), pp. 101-102 ( doi: 10.1051 / jphyscol: 1979431 ; PDF ; PDF ( machine script ) ).
31. ^ RG Haire: Properties of the Transplutonium Metals (Am-Fm) , in: Metals Handbook, Vol. 2, 10th Edition, (ASM International, Materials Park, Ohio, 1990), pp. 1198-1201.
32. Phillip D. Kleinschmidt, John W. Ward, George M. Matlack, Richard G. Haire: Henry's Law vaporization studies and thermodynamics of einsteinium ‐ 253 metal dissolved in ytterbium , in: J. Chem. Phys. , 1984 , 81 , pp. 473-477 ( doi: 10.1063 / 1.447328 ).
33. ^ AF Holleman , E. Wiberg , N. Wiberg : Textbook of Inorganic Chemistry . 102nd edition. Walter de Gruyter, Berlin 2007, ISBN 978-3-11-017770-1 , p. 1956.
34. ^ A b D. D. Ensor, JR Peterson, RG Haire, JP Young: Absorption spectrophotometric study of 253 EsF 3 and its decay products in the bulk-phase solid state , in: J. Inorg. Nucl. Chem. , 1981 , 43 (10), pp. 2425-2427 ( doi: 10.1016 / 0022-1902 (81) 80274-6 ).
35. a b R. G. Haire, RD Baybarz: Identification and Analysis of Einsteinium Sesquioxide by Electron Diffraction , in: J. Inorg. Nucl. Chem. , 1973 , 35 (2), pp. 489-496 ( doi: 10.1016 / 0022-1902 (73) 80561-5 ).
36. RG Haire, L. Eyring, in: Handbook on the Physics and Chemistry of Rare Earths , vol. 18 Lanthanoids and Actinides Chemistry (ed. By KA Gscheidner, Jr., L. Eyring, GR Choppin, GH Lander), North-Holland, New York 1994, pp. 414-505.
37. Jump up ↑ a b c d e J. P. Young, RG Haire, JR Peterson, DD Ensor, RL Fellow: Chemical Consequences of Radioactive Decay. 2. Spectrophotometric Study of the Ingrowth of Berkelium-249 and Californium-249 into Halides of Einsteinium-253 , in: Inorg. Chem. , 1981 , 20 (11), pp. 3979-3983 ( doi: 10.1021 / ic50225a076 ).
38. ^ A b J. R. Peterson, DD Ensor, RL Fellows, RG Haire, JP Young: Preparation, characterization, and decay of einsteinium (II) in the solid state , in: Journal de Physique Colloques , 1979 , 40 (C4), p. 111-113 ( doi: 10.1051 / jphyscol: 1979435 ; PDF ).
39. ^ AF Holleman , E. Wiberg , N. Wiberg : Textbook of Inorganic Chemistry . 102nd edition. Walter de Gruyter, Berlin 2007, ISBN 978-3-11-017770-1 , p. 1969.
40. JP Young, RG Haire, RL Fellows, JR Peterson: Spectrophotometric studies of transcurium element halides and oxyhalides in the solid state , in: Journal of Radioanalytical Chemistry , 1978 , 43 (2), pp. 479-488 ( doi: 10.1007 / BF02519508 ).
41. DK Fujita, BB Cunningham, TC Parsons, JR Peterson: The solution absorption spectrum of Es 3+ , in: Inorg. Nucl. Chem. Lett. , 1969 , 5 (4), pp. 245-250 ( doi: 10.1016 / 0020-1650 (69) 80192-3 ).
42. DK Fujita, BB Cunningham, TC Parsons: Crystal structures and lattice parameters of einsteinium trichloride and einsteinium oxychloride , in: Inorg. Nucl. Chem. Lett. , 1969 , 5 (4), pp. 307-313 ( doi: 10.1016 / 0020-1650 (69) 80203-5 ).
43. RL Fellows, JR Peterson, M. Noé, JP Young, RG Haire: X-ray diffraction and spectroscopic studies of crystalline einsteinium (III) bromide, 253 EsBr 3 , in: Inorg. Nucl. Chem. Lett. , 1975 , 11 (11), pp. 737-742 ( doi: 10.1016 / 0020-1650 (75) 80090-0 ).
44. ^ RG Haire, ORNL Report 5485, 1978 .
45. ^ JR Peterson: Chemical Properties of Einsteinium: Part II , in: GT Seaborg (Ed.): Proceedings of the 'Symposium Commemorating the 25th Anniversary of Elements 99 and 100' , January 23, 1978; Report LBL-7701, April 1979, pp. 55-64.
46. RL Fellows, JP Young, RG Haire, JR Peterson, in: The Rare Earths in Modern Science and Technology (edited by GJ McCarthy and JJ Rhyne), Plenum Press, New York 1977, pp. 493-499.
47. JP Young, RG Haire, RL Fellows, M. Noé, JR Peterson: Spectroscopic and X-ray Diffraction Studies of the Bromides of Cf-249 and Es-253 , in: Transplutonium 1975 (edited by W. Müller and R. Lindner), North Holland, Amsterdam 1976, pp. 227-234.
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http://www.maa.org/press/periodicals/convergence/sums-of-powers-of-positive-integers-pythagoras-c-570-500-bce-turkey-greece-italy | Sums of Powers of Positive Integers - Pythagoras (c. 570-500 BCE), Turkey-Greece-Italy
Author(s):
Janet Beery (University of Redlands)
The Greek mathematician, philosopher, and mystic Pythagoras is said to have lived with his followers, the Pythagoreans, at Croton in what is now southern Italy. The Pythagoreans took as their motto “All is Number” and were believed to have experimented with number properties by arranging pebbles on a flat surface. As a result, they saw what we would describe as a sum of successive positive integers as a triangle or triangular number (Fig. 1).
Figure 1. The first four triangular numbers
Their pebble experiments led them to see that two copies of the same triangular number could be fitted together to form an oblong number; hence, for example, twice the triangular number 15 = 1 + 2 + 3 + 4 + 5 could be viewed as the oblong number 5 x 6 = 30 (Fig. 2).
Figure 2. Twice a triangular number is an oblong number, or, in modern notation, $2(1 + 2 + 3 + \cdots + n) = n(n + 1).$ Here we see that $2(1 + 2 + 3 + 4 + 5) = 5\cdot 6.$
Equivalently, any triangular number was half an oblong number; for example,
$1 + 2 + 3 + 4 + 5 = {{5 \cdot 6} \over 2},$
and, in general,
$1 + 2 + 3 + \cdots + n = {{n(n + 1)} \over 2}$
for any positive integer n.
Exercise 1: Use pebbles to illustrate that the sum of every two consecutive triangular numbers is a square number. Hint: Examples of pairs of consecutive triangular numbers include 6 and 10, 10 and 15, and 15 and 21.
The solution to this exercise is available here.
Janet Beery (University of Redlands), "Sums of Powers of Positive Integers - Pythagoras (c. 570-500 BCE), Turkey-Greece-Italy," Convergence (July 2010), DOI:10.4169/loci003284 | {"extraction_info": {"found_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": 0.5827512145042419, "perplexity": 921.1356759046046}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661905.96/warc/CC-MAIN-20160924173741-00275-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://en.m.wikiversity.org/wiki/Logical_disjunction | # Logical disjunction
Logical disjunction, also called logical alternation, is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false.
The truth table of ${\displaystyle p~\operatorname {OR} ~q,}$ also written ${\displaystyle p\lor q,\!}$ appears below:
${\displaystyle p\!}$ ${\displaystyle q\!}$ ${\displaystyle p\lor q}$ ${\displaystyle \operatorname {F} }$ ${\displaystyle \operatorname {F} }$ ${\displaystyle \operatorname {F} }$ ${\displaystyle \operatorname {F} }$ ${\displaystyle \operatorname {T} }$ ${\displaystyle \operatorname {T} }$ ${\displaystyle \operatorname {T} }$ ${\displaystyle \operatorname {F} }$ ${\displaystyle \operatorname {T} }$ ${\displaystyle \operatorname {T} }$ ${\displaystyle \operatorname {T} }$ ${\displaystyle \operatorname {T} }$
## Document history
Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 18, "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": 0.6967623233795166, "perplexity": 618.4752526256788}, "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-2022-49/segments/1669446710801.42/warc/CC-MAIN-20221201053355-20221201083355-00174.warc.gz"} |
https://www.discretization.de/movies/ | Movies
Some of our movies can be found on our Youtube channel.
The Discrete Charm of Geometry
A team of mathematicians is working together on a big project. Excitement of discovery, hope and disappointment, competition and recognition are shown from an infinitely close distance. Scientists united by the idea of discretization, which, in short, means: constructing continuous objects from basic building blocks. Akin to the scientists’ search for the right discretization of continuum, this film itself is composed of fragments – individual characters of different ages, temperaments and scientific approaches – which form a single continuous melody. The question of where the boundaries lie between mathematics and the lives of those who are involved in it and how much they are willing to sacrifice is as important as the search for precise scientific answers. A unique and unprecedented dive into the unknown world of mathematicians.
conform!
How can you make good flat maps of the round earth?' Our story begins with Mercator's world map of 1569, the first angle-preserving (or 'conformal') world map. His idea fell on fruitful soil, from which a new branch of mathematics has developed. The movie shows some of the highlights of this development, yielding a series of elegant visual forms which arise as 'conformal maps' on a variety of surfaces in 2- and 3-D. Featuring non-technical language, a simple aesthetic, compelling animation, and an original score, the movie builds an accessible bridge from everyday experience to a beautiful but little-known mathematical theory that continues to bear technological fruit today in fields such as computer graphics and architecture.
Koebe Polyhedra and Minimal Surfaces
This short experimental animation film explores the abstract mathematical world of Koebe Polyhedra and minimal surfaces. A 3D voyage thought the amazing world of discrete geometry, based on modern mathematical research.
Georg Cantor - Der Entdecker der Unendlichkeiten
Infinity has fascinated mankind for centuries. We wish for eternal life - looking up, space marks the final frontier. The idea of infinity for mathematics was invented by Georg Cantor; the 6th January 2018 marked the hundredth anniversary of his death. This documentary presents Georg Cantor’s life and his exploration of infinity for mathematics.
Saxonia-Entertainment produced this documentary in collaboration with the SFB/TRR 109 and the TU Berlin for the MDR which will present it on Sunday, 4th March 2018 at 22:20 h.
For further information please visit Saxonia-Entertainment or MDR.
Six university professors of math and physics wanted to share the beauty of their research.They organize a competition with the hope of inspiring artists to create ground breaking pieces, based on modern science.
A jury of six experts in art, fashion and architecture were invited to judge the results. One may think, that in contrast to mathematical beauty, artistic beauty can be appreciated without a complete understanding.
Unexpectedly, mathematical rigor is replaced by artistic rigor; the overconfidence of scientists in the beauty of their research clashes with the modern artists world view.
Math Circles Around the World
Every week, hundreds of children in different cities of the world meet to solve complex problems. Who they are, why they do it and how, in the movie “Mathematical Circles Around the World”.
Lobachevsky Space
Like in Lobachevsky geometry where the space is in excess and the lines behave unusually, in the film "Lobachevsky space" unusual mathematics is spiced up by a good portion of background history and current politics. Past and present, science and politics, Russia and Germany, Lobachevsky and Gauss are confronted in a polyphonic dialogue. A modern portrait of a lonely genius, rector of the Kazan University, whose pioneering ideas forestalled the development of mathematics and science in general. Lobachevsky story reemerges through the observation of the daily life of scientists in Kazan, Berlin and Göttingen.
Schrödinger's Smoke (SIGGRAPH 2016)
We describe a new approach for the purely Eulerian simulation of incompressible fluids. In it, the fluid state is represented by a ℂ²-valued wave function evolving under the Schrödinger equation subject to incompressibility constraints. The underlying dynamical system is Hamiltonian and governed by the kinetic energy of the fluid together with an energy of Landau-Lifshitz type. The latter ensures that dynamics due to thin vortical structures, all important for visual simulation, are faithfully reproduced. This enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments, even on modestly sized grids. Our implementation uses a simple splitting method for time integration, employing the FFT for Schrödinger evolution as well as constraint projection. Using a standard penalty method we also allow arbitrary obstacles. The resulting algorithm is simple, unconditionally stable, and efficient. In particular it does not require any Lagrangian techniques for advection or to counteract the loss of vorticity. We demonstrate its use in a variety of scenarios, compare it with experiments, and evaluate it against benchmark tests. A full implementation is included in the ancillary materials.
Inside Fluids: Clebsch Maps for Visualization and Processing (SIGGRAPH 2017)
© 2017 California Institute of Technology, Technische Universität Berlin (Photo: Business Wire)
Clebsch maps encode velocity fields through functions. These functions contain valuable information about the velocity field. For example, closed integral curves of the associated vorticity field are level lines of the vorticity Clebsch map. This makes Clebsch maps useful for visualization and fluid dynamics analysis. Additionally they can be used in the context of simulations to enhance flows through the introduction of subgrid vorticity. In this paper we study spherical Clebsch maps, which are particularly attractive. Elucidating their geometric structure, we show that such maps can be found as minimizers of a non-linear Dirichlet energy. To illustrate our approach we use a number of benchmark problems and apply it to numerically given flow fields.
Shape from Metric (SIGGRAPH 2018)
We study the isometric immersion problem for orientable surface triangle meshes endowed with only a metric: given the combinatorics of the mesh together with edge lengths, approximate an isometric immersion into $\mathbb{R}^3$. To address this challenge we develop a discrete theory for surface immersions into $\mathbb{R}^3$. It precisely characterizes a discrete immersion, up to subdivision and small perturbations. In particular our discrete theory correctly represents the topology of the space of immersions, i.e., the regular homotopy classes which represent its connected components. Our approach relies on unit quaternions to represent triangle orientations and to encode, in their parallel transport, the topology of the immersion. In unison with this theory we develop a computational apparatus based on a variational principle. Minimizing a non-linear Dirichlet energy optimally finds extrinsic geometry for the given intrinsic geometry and ensures low metric approximation error.
We demonstrate our algorithm with a number of applications from mathematical visualization and art directed isometric shape deformation, which mimics the behavior of thin materials with high membrane stiffness.
Watch the movie on https://youtu.be/FIVqa794w3U | {"extraction_info": {"found_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": 0.16926756501197815, "perplexity": 1599.1464804939806}, "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-2023-14/segments/1679296943704.21/warc/CC-MAIN-20230321162614-20230321192614-00493.warc.gz"} |
https://www.physicsforums.com/threads/kinematics-driving-question.260961/ | # Kinematics, driving question
1. Oct 1, 2008
### Jennifer001
1. The problem statement, all variables and given/known data
you're driving down the highway late one night at 20ms when a deep steps onto the road 35m infront of you. your reaction time before steeping on the brakes is 0.5s and the max deceleration of ur car is 10m/s^2
what is the max speed you could have and still not hit the deer?
deltaX=35m V=?
a=-10m/s^2 reaction time = 0.5s
2. Relevant equations
Xf= Xi+ViT+1/2aT^2
Vf-Vi=aT
Vf^2=Vi^2+1/2a(deltaX)
3. The attempt at a solution
so this is what i tried
i pluged the numbers into the first equation to get
35=0+ViT+1/2(-10)T^2
and i have 2 missing variables so i tried to get it from the other equation
Vf-Vi=aT
0-Vi=(-10)T then that didnt work so i tried the other equation
Vf^2=Vi^2+1/2a(deltaX)
0=Vi^2+1/2(-10)35
Vi= 13.23m/s
and i know that Vi can't be right because i did the question before it and it had Vi of 20m/s and it was 5m from hitting the deer...i dont know what i did wrong can someone help please?
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. Oct 1, 2008
### HallsofIvy
Staff Emeritus
In order to JUST miss hitting the deer, it must take all 35 meters to slow to 0 m/s. So you have TWO of the equations you just gave: 35=0+ViT+1/2(-10)T^2 for the 35 m and 0-Vi=(-10)T for the time to slow to 0. Now you have two equations in the two "unknown" numbers Vi and T. 0-Vi= -10T tells you that Vi= 10T. Replace Vi by that in the first equation to get a simple equation for T alone. Once you have found T, put that into Vi= 10T to find Vi.
Similar Discussions: Kinematics, driving question | {"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": 0.8263382315635681, "perplexity": 1483.949907741207}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187828189.71/warc/CC-MAIN-20171024071819-20171024091819-00188.warc.gz"} |
http://cdsweb.cern.ch/collection/L3%20Theses?ln=pt | 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.
# L3 Theses
2017-09-23
06:42
Messung der Myonpaarproduktion im Prozess e+ e- --> mu+ mu- (gamma) bei Schwerpunktsenergien von 89 GeV bis 183 GeV / Siedenburg, Thorsten Presented are the total cross-sections and forward-backward-asymmetries of the reaction at center of mass energies between 89 GeV and 183 GeV at the LEP-accelerator measured with the L3-Detector from 1995 to 1997 [...] PITHA-00-22 ; CERN-THESIS-2000-060. - 2000. - 125 p.
Fulltext - Fulltext - Full text
2017-09-23
06:42
Kosmische Myonen im L3-Detektor / Saidi, Rachid Durch die Untersuchung des Mondschattens in der primaren kosmischen Strahlung konnen Informationen uber die Winkelau osung des L3-Detektors gewonnen werden, sowie mit ausreichender Statistik das Verhaltnis von Antiprotonen zu Protonen fur Protonenergien um 1 TeV abgeschatzt werden [...] CERN-THESIS-2000-066. - 2000. - 66 p.
SERVER - Fulltext - Full text - Full text
2017-09-23
06:42
Analysis of Multi Muon Events in the L3 Detector / Schmitt, Volker The muon density distribution in air showers initiated by osmi parti les is sensitive to the hemi al omposition of osmi rays [...] CERN-THESIS-2000-067. - 2000. - 77 p.
Full text - Full text
2017-09-22
06:23
Inclusive D*(+/-) production in two photon collisions at LEP / Prokofiev, Denis Olegovich In this thesis I present my results on the measurement of the open charm production in two-photon collision events done with the L3 detector at Large Electron Positron machine (LEP) [...] UMI-30-78734 ; CERN-THESIS-2001-073. - 2001. - 113 p.
UMI - Purdue University
2017-09-22
06:23
Measurement of bottom quark production in two photon collisions / Saremi, Sepehr The cross section for bottom quark production in two-photon collisions, sigma( e+e- → e+e- bb¯X), is measured for the first time [...] UMI-30-42649 ; CERN-THESIS-2001-076. - 2001. - 160 p.
UMI - Full text - Full text
2017-09-21
14:04
Search for scalar leptons at LEP with the L3 detector / Xia, Lei In this thesis, I present a search for scalar leptons in e+e- annihilation using the L3 detector at LEP [...] UMI-30-69427 ; CERN-THESIS-2002-097. - 2002. - 180 p.
CT-THESIS - Fulltext - Full text
2017-09-21
14:04
Searches for the Standard Model Higgs boson in the leptonic decay channels / Yamamoto, Jin UMI-30-68997 ; CERN-THESIS-2002-098. - 2002. - 125 p.
UMI
2017-09-21
14:04
Measurements of four fermion cross-sections at LEP / Kopal, Miroslav The production of four fermions in e+e− collisions at LEP allows the verification of the Standard Model of the Electroweak Interactions in the Charged and Neutral Current Sectors [...] UMI-30-99810 ; CERN-THESIS-2002-099. - 2002. - 124 p.
2017-09-19
11:15
Precision measurement of the W-boson properties with the L3 detector at LEP / Pal, Imre UMI-31-66688 ; CERN-THESIS-2004-095. - 2004. - 174 p.
UMI
2017-09-19
11:15
Search for Antimatter in Cosmic Rays by using the Shadow of the Moon with the Muon Detector L3 / Parriaud, Jean-Francois CERN-THESIS-2003-062. - 2003. - 147 p.
SERVER | {"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": 0.9414825439453125, "perplexity": 14039.563605052917}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039560245.87/warc/CC-MAIN-20210422013104-20210422043104-00016.warc.gz"} |
https://mooseframework.inl.gov/source/markers/ErrorFractionMarker.html | # Errorfractionmarker
Marks elements for refinement or coarsening based on the fraction of the min/max error from the supplied indicator.
## Description
The ErrorFractionMarker utilizes the value from an Indicator as a measure of "error" on each element. Using this error approximation the following algorithm is applied:
ErrorFractionMarker example calculation.
1. The elements are sorted by increasing error.
2. The elements comprising the "refine" fraction, from highest error to lowest, of the total error are marked for refinement.
3. The elements comprising the "coarsen" fraction, from lowest error to highest, of the total error are marked for refinement.
## Example Input Syntax
[Adaptivity]
[./Indicators]
[./error]
type = AnalyticalIndicator
variable = u
function = solution
[../]
[../]
[./Markers]
[./marker]
type = ErrorFractionMarker
coarsen = 0.1
indicator = error
refine = 0.3
[../]
[../]
[]
(test/tests/markers/error_fraction_marker/error_fraction_marker_test.i)
## Input Parameters
• indicatorThe name of the Indicator that this Marker uses.
C++ Type:IndicatorName
Options:
Description:The name of the Indicator that this Marker uses.
### Required Parameters
• coarsen0Elements within this percentage of the min error will be coarsened. Must be between 0 and 1!
Default:0
C++ Type:double
Options:
Description:Elements within this percentage of the min error will be coarsened. Must be between 0 and 1!
• clear_extremesTrueWhether or not to clear the extremes during each error calculation. Changing this to false will result in the global extremes ever encountered during the run to be used as the min and max error.
Default:True
C++ Type:bool
Options:
Description:Whether or not to clear the extremes during each error calculation. Changing this to false will result in the global extremes ever encountered during the run to be used as the min and max error.
• refine0Elements within this percentage of the max error will be refined. Must be between 0 and 1!
Default:0
C++ Type:double
Options:
Description:Elements within this percentage of the max error will be refined. Must be between 0 and 1!
• blockThe list of block ids (SubdomainID) that this object will be applied
C++ Type:std::vector
Options:
Description:The list of block ids (SubdomainID) that this object will be applied
### Optional Parameters
• outputsVector of output names were you would like to restrict the output of variables(s) associated with this object
C++ Type:std::vector
Options:
Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object
• enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.
• control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector
Options:
Description:Adds user-defined labels for accessing object parameters via control logic. | {"extraction_info": {"found_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": 0.16011415421962738, "perplexity": 3441.197734102124}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376829140.81/warc/CC-MAIN-20181218102019-20181218124019-00088.warc.gz"} |
http://sachinashanbhag.blogspot.com/2012/03/nonsingular-and-nondefective-random.html | ## Thursday, March 1, 2012
### Nonsingular and Nondefective Random Matrices
In a previous post, I posed the folllowing question:
Consider again, a random square n by n matrix A, whose entries are restricted to the set of integers {-p, -p+1, ..., 0, ... p-1, p}. Each of the 2p+1 values are equally probable.
• What is the probability that this matrix is nonsingular?
• What is the probability that this matrix is nondefective?
For n = 3 and 4, using a simple Monte Carlo method, this is what I get. As p increases, we approach the continuous distribution of entries asymptotically.
Blue lines are for n=3, and green for n=4. Triangles and squares denote the probability that a random matrix is nonsingular, and nondefective, respectively
As we can see, both singular and defective matrices are extremely rare for large p. Between the two, defectiveness is a rarer feature. | {"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": 0.9356228113174438, "perplexity": 858.5306296866261}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102891.30/warc/CC-MAIN-20170817032523-20170817052523-00430.warc.gz"} |
https://cracku.in/blog/number-series-questions-for-ssc-gd-pdf/ | 0
429
# Number Series Questions For SSC GD PDF
Question 1: In the following question, select the missing number from the given series:
6, 19, 54, 167, 494, ?
a) 1491
b) 1553
c) 1361
d) 1642
Question 2: In the following question, select the missing number from the given series:
67, 70, 74, 77, 81, 84 ?
a) 87
b) 88
c) 86
d) 89
Question 3: In the following question, a series is given with one or more number (s) missing. Choose the correct alternative from the given options.
22, 22, 23, 20, 16, 17, 17, ?, ?, 8
a) 18, 9
b) 12, 13
c) 10, 9
d) 13, 10
Question 4: In the following question, a series is given with one or more number (s) missing. Choose the correct alternative from the given options.
?, 5, 30, 186, 1309, 10480
a) 0.25
b) 0.75
c) 1.00
d) 0
Question 5: In the following question, a series is given with one or more number (s) missing. Choose the correct alternative from the given options.
7, 51, 8, 65, 9, ?
a) 79
b) 80
c) 81
d) 82
Question 6: In the following question, a series is given with one or more number (s) missing. Choose the correct alternative from the given options.
0.2, 0.16, 0.072, 0.0256, ?
a) 0.0016
b) 0.004
c) 0.00512
d) 0.008
Question 7: Which of the following replaces ‘?’ in the following series?
10,21,45,84,500,?
a) 1050
b) 1045
c) 985
d) 1025
Instructions
In the following question, select the missing number from the given series.
Question 8: 84, 42, 44, 22, 24, 12, ?
a) 20
b) 14
c) 24
d) 28
Question 9: 1, 4, 13, 40, 121, ?
a) 284
b) 286
c) 364
d) 396
Instructions
In the following question, select the missing number from the given series.
Question 10: 2, 7, 22, 67, ?
a) 197
b) 198
c) 200
d) 202
Question 11: 1, 8, 29, 92, 281, ?
a) 567
b) 628
c) 776
d) 848
Instructions
In the following question, select the missing number from the given series.
Question 12: 1, 7, 3, 9, 6, 12, 10, 16, 15, ?
a) 18
b) 15
c) 20
d) 21
Question 13: 7, 10, 14, 19, 25, ?
a) 32
b) 36
c) 38
d) 40
Question 14: In the following question, select the missing number from the given series.
?, 5, 15, 45, 113
a) 1
b) 2
c) 3
d) 4
Question 15: In the following question, select the missing number from the given series.
1, 1, 3, 4, 5, 9, 7, 16, 9, 25, 11, ?
a) 17
b) 36
c) 49
d) 37
Instructions
In the following question, select the missing number from the given series.
Question 16: 2.2, 14.8, 40, 90.4, ?
a) 191.2
b) 194.4
c) 196.2
d) 208.4
Question 17: 84, 42, 28, 21, ?
a) 10.5
b) 16.8
c) 18.4
d) 19.6
Question 18: In the following series find 20th number
9, 5, 1, -3 -7, -11,……..
a) -64
b) -75
c) -70
d) -67
Instructions
A series is given, with one from missing, Choose the correct alternative from the given ones that will complete the series.
Question 19: 16, 61, 25, 52, 36, 63, 49, ?
a) 36
b) 94
c) 72
d) 46
Question 20: Find the wrong number in the given series ?
15, 28, 30, 39, 48
a) 28
b) 15
c) 30
d) 39
Difference between the numbers is in the form of $[x][x^{2}]$
$10+[1][1^{2}]=10+11=21$
$21+[2][2^{2}]=21+24=45$
$45+[3][3^{2}]=45+39=84$
$84+[4][4^{2}]=84+416=500$
Next number:
$500+[5][5^{2}]=500+525=1025$
The given series is an arithmetic progression with first term $a=9$ and common difference $d=-4$
$n^{th}$ term in an A.P. = $A_n=a+(n-1)d$
=> $A_{20}=9+(20-1)\times(-4)$
= $9+(19)(-4)$
= $9-76=-67$
=> Ans – (D) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8662180304527283, "perplexity": 388.16329215612336}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610703644033.96/warc/CC-MAIN-20210125185643-20210125215643-00167.warc.gz"} |
https://www.physicsforums.com/threads/classical-equations-and-light.935851/ | # B Classical equations and light
1. Jan 1, 2018
### JulianM
But momentum = mass x velocity so that part of the equation is E2 = c.(mv)
If the mass is zero then that formulation also yields zero energy (which we know is not true)
Last edited by a moderator: Jan 1, 2018
2. Jan 1, 2018
### DrStupid
This definition is based on another concept of mass. Today "mass" means invariant mass and that results in another equation for momentum.
3. Jan 1, 2018
### PeroK
The momentum of a photon is given by $p = hf/c$. Not by $p = mv$.
In fact, $p = mv$ is a non-relativistic approximation of the momentum of a massive particle. The correct equation is:
$p = \gamma mv$, where $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$
4. Jan 1, 2018
### JulianM
So lightarrow was wrong?
5. Jan 1, 2018
### DrStupid
No, he was and is still right.
6. Jan 1, 2018
### JulianM
Regardless of whether you include gamma this equation still contains mass.
Anything multiplied by zero is still zero.
7. Jan 1, 2018
### PeroK
For a photon $E = cp$. So, he is right and you are not.
$p = mv$, as I have already explained, is a classical equation for the momentum of a particle that is not valid in relativity - for any particle, massless or otherwise.
Last edited by a moderator: Jan 1, 2018
8. Jan 1, 2018
### DrStupid
Again, this "m" is an outdated concept of mass. It dates back to classical mechanics and is not used anymore because it is frame dependent in relativity. With the modern concept of mass (invariant mass) the equations for momentum and energy are
$p = \frac{{m \cdot v}}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}$
$E = \frac{{m \cdot c^2 }}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}$
Of course they cannot be used for photons (because v=c), but they result in equations like
$p = \frac{{E \cdot v}}{{c^2 }}$
and
$E^2 = m^2 \cdot c^4 + p^2 c^2$
which are valid for v=c and m=0.
Last edited by a moderator: Jan 1, 2018
9. Jan 1, 2018
### PeroK
That equation is for a massive particle. It doesn't apply for a photon. In fact, it is not $0$ for a photon, it is undefined. Since the photon has zero mass and speed $c$ you would get:
$p = \frac{mc}{0} = \frac00$
Which is undefined. So, you have to look elsewhere for the equation that gives the momentum of a photon.
10. Jan 1, 2018
### JulianM
So what is the equation in relativity? It can't be just multiplying momentum by gamma as PeroK did (above). That still gives zero energy.
11. Jan 1, 2018
### PeroK
The general equation for momentum is:
$p^2c^2 = E^2 - m^2c^4$
That works for both massive and massless particles.
For a massive particle the energy is $E = \gamma mc^2$
And, for a photon the energy is $E = hf$.
12. Jan 1, 2018
### DrStupid
But the limit for $v \to c$ (and therefore $m \to 0$) is defined.
13. Jan 1, 2018
### JulianM
How does
$p^2c^2 = E^2 - m^2c^4$
work for a massless particle if p= m.v
That would give (for v = c)
( mc^2).c^2 = E^2 - m^2.c^4
is equivalent to
E^2 = m^2.c^4 - m^2.c^4
There's something wrong, surely it cannot apply to a massless particle which posses energy
14. Jan 1, 2018
### PeroK
What will it take for you to accept that $p = mv$ is not a valid equation?
15. Jan 1, 2018
### Orodruin
Staff Emeritus
Again, as you have been told repeatedly, p = mv is not a valid equation relativistically. Repeating the same mistake does not make it true.
16. Jan 1, 2018
### JulianM
Well now you have me very confused. How do we know when to apply relativistic equations or not?
17. Jan 1, 2018
### phinds
For a massless particle, you have no choice. For massive particles, the relativistic equation is always true to but the slower the particle is, the less difference there is between the results of the classical equation and the relativistic equation. At human-scale speeds, the classical equation always gives results that are useful.
18. Jan 1, 2018
### PeroK
In principle, the relativistic equations are always valid and the classical equations are an approximation. If that approximation is good enough, then you can apply the classical equations.
If the velocities involved are a significant proportion of the speed of light, then you definitely need the relativistic equations.
In practice, it depends how accurate you need your answer. Time calculations for GPS satellites must be so accurate that relativistic equations are needed despite the modest speeds. But, normal engineering calculations are usually more than accurate enough with classical equations.
19. Jan 1, 2018
### ZapperZ
Staff Emeritus
No, in THIS case, momentum is defined via the wavenumber, i.e.
p=ħk
So there is no requirement for a "mass" here.
Zz.
20. Jan 1, 2018
### Staff: Mentor
You can always apply the relativistic equations, and you will always get the right answer. They're more exact than the non-relativistic ones.
However, the relativistic equations are also more complicated and using them is more work, so we don't use them when the relativistic effects are small enough to ignore. For example: what is the kinetic energy and the momentum of a one-kilogram mass moving at ten meters per second? Try calculating that using the classical formulas and using the relativistic ones. Which was more work? And what difference did it make?
The classical formulas can be used any time the speeds involved are small compared with the speed of light (equivalently, $\gamma$ is so close to 1 that you're OK with the approximation $\gamma=1$). Clearly this is not the case for particles travelling at the speed of light, so you know that the classical formulas can't be used here. | {"extraction_info": {"found_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": 0.9619095921516418, "perplexity": 964.8592539168455}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1524125945037.60/warc/CC-MAIN-20180421051736-20180421071736-00354.warc.gz"} |
http://math.stackexchange.com/questions/241439/how-do-i-determine-coefficients-of-a-square-function-in-this-example | # How do I determine coefficients of a square function in this example?
This point goes into plot of the function: P (-8, 6.25)
I also know that: C= 11/12
The unknowns are A and B.
Do I have enough data to solve this problem?
-
$$\;\;\;\;\;a(-8)^2+b(-8)+\frac{11}{12}=\frac{25}{4}=6.25\Longleftrightarrow 64a-8b+\frac{11}{12}=\frac{25}{6}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8639755845069885, "perplexity": 1869.3689323055914}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049272823.52/warc/CC-MAIN-20160524002112-00150-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://farside.ph.utexas.edu/teaching/329/lectures/node83.html | Next: 2-d problem with Neumann Up: The diffusion equation Previous: An improved 1-d solution
## 2-d problem with Dirichlet boundary conditions
Let us consider the solution of the diffusion equation in two dimensions. Suppose that
(214)
for , and . Suppose that satisfies mixed boundary conditions in the -direction:
(215)
at , and
(216)
at . Here, , , etc., are known functions of , whereas , are known functions of and . Furthermore, suppose that satisfies the following simple Dirichlet boundary conditions in the -direction:
(217)
As before, we discretize in time on the uniform grid , for . Furthermore, in the -direction, we discretize on the uniform grid , for , where . Finally, in the -direction, we discretize on the uniform grid , for , where . Adopting the Crank-Nicholson temporal differencing scheme discussed in Sect. 6.6, and the second-order spatial differencing scheme outlined in Sect. 5.2, Eq. (214) yields
(218)
where . The discretized boundary conditions take the form
(219) (220)
plus
(221)
Here, , etc., and , etc.
Adopting the Fourier method introduced in Sect. 5.7, we write the in terms of their Fourier-sine harmonics:
(222)
which automatically satisfies the boundary conditions (221). The above expression can be inverted to give (see Sect. 5.9)
(223)
When Eq. (218) is written in terms of the , it reduces to
(224)
for , and . Here, , and . Moreover, the boundary conditions (219) and (220) yield
(225) (226)
where
(227)
etc. Equations (224)--(226) constitute a set of uncoupled tridiagonal matrix equations for the , with one equation for each separate value of .
In order to advance our solution by one time-step, we first Fourier transform the and the boundary conditions, according to Eqs. (223) and (227). Next, we invert the tridiagonal equations (224)--(226) to obtain the . Finally, we reconstruct the via Eq. (222).
Next: 2-d problem with Neumann Up: The diffusion equation Previous: An improved 1-d solution
Richard Fitzpatrick 2006-03-29 | {"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": 0.9880627393722534, "perplexity": 1363.7504573924564}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038078021.18/warc/CC-MAIN-20210414185709-20210414215709-00315.warc.gz"} |
http://www.koreascience.or.kr/article/JAKO201511639883964.page | # 방열 기능형 고성능 스마트 전파흡수체 제조 방법 개발 및 전망
Kim, Dong Il;Jeon, Yong Bok
김동일;전용복
• Accepted : 2015.10.07
• Published : 2015.10.30
• 21 102
#### Abstract
With the rapid progress of electronics and radio communication technology, human enjoys greater freedom in information communication. However, EMW(Electro-Magnetic Wave) environments have become more complicate and difficult to control. Thus, international organizations, such as the American National Standard Institution(ANSI), Federal Communications Commission(FCC), the Comite Internationale Special des Perturbations Radio Electrique(CISPR), etc, have provided standard for controlling the EM wave environments and for the countermeasure of the electromagnetic compatibility(EMC). In this paper, fabrication of the smart EMW absorber which has heat radiating function and high performance absorption abilities were suggested. Furthermore, we prospected future smart EMW absorbers. The designed smart EMW absorber is fabricated following process. Firstly, we applied high temperature heat treated to a mixture of Iron-oxide($Fe_2O_3$) and ceramics. Secondly, we applied low temperature heat treated to the mixture of heat treated material and a carbon material. Lastly, we made apertures on the absorber. The designed smart EM wave absorber has the absorption ability of more than 20 dB from 2 GHz to 2.45 GHz band, respectively. Thus, it is respected that these results can be applied as various EMC devices in electronic, communication, and controlling systems.
#### Keywords
Electro-Magnetic Wave Environment;Heat Radiating Function;Iron-Oxide;Heat Treatment;Smart EMW Absorber
#### References
1. D. I. Kim, EMW Absorber Engineering, Dae-Young Pub. Co., 2006.
2. O. Hashimoto, Introduction to Wave Absorber, Morikita Publishing Co., 1997.
3. J. M. Song, H. J. Yoon, and Dong Il Kim, et al., "Dependence of electromagnetic wave absorption on ferrite particle size in sheet-type absorbers", J. Korea Phys. Soc., vol. 42, no. 5, pp. 671-675, 2003.
4. C. B. Mo, et al., "Fabrcating process and characristics of CNT/metal nano composite material", Korea Institute of Fine Engineering, vol. 24, no. 8, pp. 7-14, 2007.
5. J. H. Seong, et al., "Effects of mechanical characteristics on crystalization by variation of CNT and carbon- black reinforced rubber composite material", J. of Korea Institute of Mechanics, vol. 35, no. 9, pp. 999-1005, 2011.
6. A. Mehdipour, I. D. Rosca, C. W. Trueman, A. Sebak, and Suong Van Hoa, "Multi-wall carbon nanotube-epoxy composites with high shielding effectiveness for aeronautic applications", IEEE Transactions on Electromagnetic Compatibility, vol. 54, Issue. 1, pp. 28-36, 2012. https://doi.org/10.1109/TEMC.2011.2174241
7. E. Decrossas, M. A. El Sabbagh, S. M. El-Ghazaly, and V. F. Hanna, "Engineered carbon-nanotubes-based composite material for RF applications", IEEE Transactions on Electromagnetic Compatibility, vol. 54, Issue 1, pp. 52-59, Feb. 2003.
8. D. I. Kim, "Design of super wide-band ferrite EMW absorber in multi-layer type", J. of KIEES, vol. 7, no. 4, pp. 346-352, Oct. 1996.
9. Dong Soo Choi, Dong Il Kim, Do Yeol Kim, and Dong Han Choi, "A study on smart heat radiating sheet for protecting electronic equipments on the ship", International Journal of Navigation and Port Research, vol. 35, no. 7, pp. 569-573, Jul. 2011. https://doi.org/10.5394/KINPR.2011.35.7.569
10. 김동일, 최동한, 김기만, "옻의 특징과 옻을 지지재로 사용한 전자파 흡수체의 두께에 따른 전파흡수 특성 분석", 한국항해항만학회지, 28, pp. 861-867, 2004년 12월. | {"extraction_info": {"found_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": 0.6079639792442322, "perplexity": 19552.283916051863}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578527135.18/warc/CC-MAIN-20190419041415-20190419063415-00412.warc.gz"} |
http://math.stackexchange.com/questions/58914/a2-i-det-a0-implies-ai-is-non-singular | # $A^2=I,\det A>0$ implies $A+I$ is non-singular
Question:
If a square matrix $A$ satisfies $A^2=I$ and $\det A>0$, show that $A+I$ is non-singular.
I have tried to suppose a non-zero vector $x$ s.t. $Ax=x$ but fail to make a contradiction.
And I tried to find the inverse matrix of $A+I$ directly, suppose $(A+I)^{-1}=\alpha I +\beta A$, but it still doesn't work.
(Update: According to the two answers, this question itself is incorrect.)
-
This seems to be false: consider the $3 \times 3$ diagonal matrix with diagonal entries $1,-1,-1$. Similarly, taking an $n \times n$ diagonal matrix with two entries $-1$ and all the rest equal to $1$ gives a counterexample for any $n \geq 2$.
(A comment on how I came up with this: the matrix $A+I$ is singular iff $-I-A$ is singular iff $-1$ is an eigenvalue of $A$. The condition $A^2 = I$ forces each of the eigenvalues to be $\pm 1$ and the condition $\operatorname{det} A > 0$ forces the number of instances of $-1$ to be even, but this is not enough to give the result.)
Of course the above reasoning would also lead to Ricky Demer's counterexample, and probably should have. For some reason I thought of the above first.
-
$-I$ with even size is a counterexample.
-
$(I+A)(I-A)=0$, so $I+A$ is invertible (for $A$ satisfying $A^2=I$) if and only if $A=I$.
This works for $A$ in any ring with unit [in which $2$ is invertible], not only a ring of square matrices.
-
Why $I+A$ is invertible implies $A=I$? There may be $A$ s.t. $A^2=I$ and $A\not=I,-I$. – NGY Aug 22 '11 at 11:03
If I+A has inverse B, (I+A)(I-A)=0 implies B(I+A)(I-A)=B.0 hence I-A=0. – Did Aug 22 '11 at 11:58
@Didier: thanks. Also, to go in the other direction, from A=I to (I+A) being invertible, I should have specified that there be no 2-torsion. – zyx Aug 23 '11 at 3:11
A counter example is take matrix $A$ with diagonal entries 1 and -1. – srijan May 12 '12 at 6:27 | {"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": 0.97335284948349, "perplexity": 276.19685402671604}, "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-2014-52/segments/1418802770324.129/warc/CC-MAIN-20141217075250-00029-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://physicshelpforum.com/kinematics-dynamics/4791-position-time.html | Physics Help Forum position time
Kinematics and Dynamics Kinematics and Dynamics Physics Help Forum
Aug 16th 2010, 09:31 AM #1 Senior Member Join Date: Mar 2009 Location: Yadupatti Bazar,Sitamarhi Posts: 137 position time can anyone give me two examples having velocity and displcement just in opposite direction( if vel is +ve then disp is -ve and vice versa)? Thanks __________________ If science solves one problem, it creates ten other problems.
Aug 16th 2010, 10:10 AM #2 Physics Team Join Date: Jun 2010 Location: Morristown, NJ USA Posts: 2,352 I'll give you one example, then perhaps you can think of another on your own. When you throw a ball into the air, after some period of time it reaches its maximum height and then starts to fall back to the ground. While it is falling, and before it hits the ground, the displacement of the ball is positive (it's above the ground) but its velocity is negatrive (it is falling). I'm sure you can think of other example involving things like bouncing basketballs, or yo-yo's.
Aug 16th 2010, 11:25 AM #3 Senior Member Join Date: Mar 2009 Location: Yadupatti Bazar,Sitamarhi Posts: 137 suppose I take upward direction as +ve . Now, i throw a ball verticaly upward with a vel v. It reaches at maximum height +s(disp +ve, velocity also +ve) then falls down by covering -s(disp -ve,vel also -ve). Thus total disp: +s-s=0. I think in ur case it should be 2s( so how is it possible?) __________________ If science solves one problem, it creates ten other problems.
Aug 17th 2010, 10:17 PM #4 Senior Member Join Date: Mar 2009 Location: Yadupatti Bazar,Sitamarhi Posts: 137 can anyone comment? __________________ If science solves one problem, it creates ten other problems.
Aug 18th 2010, 06:31 AM #5 Senior Member Join Date: Mar 2009 Location: Yadupatti Bazar,Sitamarhi Posts: 137 i have got my answer. __________________ If science solves one problem, it creates ten other problems.
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Similar Physics Forum Discussions Thread Thread Starter Forum Replies Last Post kerrymaid Kinematics and Dynamics 0 Jan 26th 2011 01:11 PM rodjav305 Kinematics and Dynamics 10 Oct 8th 2010 12:13 PM alessandro Kinematics and Dynamics 1 Jan 31st 2010 08:38 AM christina Kinematics and Dynamics 4 Dec 8th 2009 08:59 AM s3a Kinematics and Dynamics 4 Mar 12th 2009 07:40 AM | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8351492881774902, "perplexity": 5682.421140286371}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1568514576355.92/warc/CC-MAIN-20190923105314-20190923131314-00348.warc.gz"} |
https://mathematica.stackexchange.com/questions/212286/cannot-remove-ragged-edges-of-transformedregion-plot | # Cannot remove ragged edges of TransformedRegion plot
I am working with two mappings of a region in the complex plane and I can't eliminate the ragged edges of the region in the second map and do not see anyway to add points to make the perimeter smooth and was wondering if there is an option I'm not aware of?. The maps are:
$$r1=\{z\in\mathbb{C} : |\log(z)\big|\leq 1\}$$
$$r2=\{z^{1/z}: z\in \text{r1}\}$$
I'm using the following code:
r1 = ImplicitRegion[
Abs[Log[r Exp[I t]]] <= 1, {{r, 0, 3}, {t, -Pi, Pi}}]
r1Plot = Region[r1, Axes -> True, PlotRange -> 3]
g[{r_, t_}] := ReIm@((r Exp[I t])^(1/(r Exp[I t])));
r2 = TransformedRegion[r1, g];
r2Plot = Region[r2, Axes -> True, PlotRange -> 3]
This produces (the perimeter edges are more noticeable in the notebook):
• r1Plot = Style[Region[r1, Axes -> True, PlotRange -> 3, PerformanceGoal -> "Quality"], Antialiasing -> True]. I don't know how to avoid the mesh lines artifacts which shows up when using Antialiasing, however. See: mathematica.stackexchange.com/questions/381/…, mathematica.stackexchange.com/q/2629/55405 – WeavingBird1917 Jan 2 at 13:23
• Thanks. However, it's the second plot that has the polygon-like perimeter on part of the region I'm trying to smooth-out. And when I use those options on this plot, I still obtain the rough edges. Also, I know the region should be smooth by other methods. – Dominic Jan 2 at 13:36
If you use BoundaryDiscretizeRegion, you can control the resolution of the boundary:
BoundaryDiscretizeRegion[r2, MaxCellMeasure -> "Length" -> 0.005] | {"extraction_info": {"found_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": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.15145476162433624, "perplexity": 2413.6134236675957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735964.82/warc/CC-MAIN-20200805183003-20200805213003-00499.warc.gz"} |
https://arxiv.org/abs/1406.3700 | cs.CC
# Title:The Parameterized Complexity of k-Biclique
Authors:Bingkai Lin
Abstract: Given a graph $G$ and a parameter $k$, the $k$-biclique problem asks whether $G$ contains a complete bipartite subgraph $K_{k,k}$. This is the most easily stated problem on graphs whose parameterized complexity is still unknown. We provide an fpt-reduction from $k$-clique to $k$-biclique, thus solving this longstanding open problem.
Our reduction use a class of bipartite graphs with a threshold property of independent interest. More specifically, for positive integers $n$, $s$ and $t$, we consider a bipartite graph $G=(A\;\dot\cup\;B, E)$ such that $A$ can be partitioned into $A=V_1\;\dot\cup \;V_2\;\dot\cup\cdots\dot\cup\; V_n$ and for every $s$ distinct indices $i_1\cdots i_s$, there exist $v_{i_1}\in V_{i_1}\cdots v_{i_s}\in V_{i_s}$ such that $v_{i_1}\cdots v_{i_s}$ have at least $t+1$ common neighbors in $B$; on the other hand, every $s+1$ distinct vertices in $A$ have at most $t$ common neighbors in $B$.
Using the Paley-type graphs and Weil's character sum theorem, we show that for $t=(s+1)!$ and $n$ large enough, such threshold bipartite graphs can be computed in $n^{O(1)}$. One corollary of our reduction is that there is no $f(k)\cdot n^{o(k)}$ time algorithm to decide whether a graph contains a subgraph isomorphic to $K_{k!,k!}$ unless the ETH(Exponential Time Hypothesis) fails. We also provide a probabilistic construction with better parameters $t=\Theta(s^2)$, which indicates that $k$-biclique has no $f(k)\cdot n^{o(\sqrt{k})}$-time algorithm unless 3-SAT with $m$ clauses can be solved in $2^{o(m)}$-time with high probability. Our result also implies the dichotomy classification of the parameterized complexity of cardinality constrain satisfaction problem and the inapproximability of maximum $k$-intersection problem.
Subjects: Computational Complexity (cs.CC) ACM classes: F.2.2 Cite as: arXiv:1406.3700 [cs.CC] (or arXiv:1406.3700v3 [cs.CC] for this version)
## Submission history
From: Bingkai Lin [view email]
[v1] Sat, 14 Jun 2014 08:20:39 UTC (21 KB)
[v2] Sat, 1 Nov 2014 05:46:37 UTC (25 KB)
[v3] Sat, 8 Jun 2019 02:09:26 UTC (27 KB) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7938564419746399, "perplexity": 568.388140462046}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1560627998509.15/warc/CC-MAIN-20190617143050-20190617165050-00359.warc.gz"} |
https://pub.uni-bielefeld.de/record/2331672 | ### Probing Backreaction Effects with Supernova Data
Seikel M, Schwarz D (2009)
In: PoS Cosmology. 029.
Konferenzbeitrag | Veröffentlicht | Englisch
Es wurden keine Dateien hochgeladen. Nur Publikationsnachweis!
Autor*in
Seikel, Marina; Schwarz, DominikUniBi
Abstract / Bemerkung
As the Einstein equations are non-linear, spatial averaging and temporal evolution do not commute. Therefore, the evolution of the averaged universe is affected by inhomogeneities. It is, however, highly controversial how large these cosmological backreaction effects are. We use the supernova data of the Constitution set up to a redshift of 0.1 in order to analyse to what extent the measurement of the Hubble constant is affected. The size of the effect depends on the size of the volume that is averaged over. The observational results are then compared to the theory of the backreaction mechanism.
Erscheinungsjahr
2009
Titel des Konferenzbandes
PoS Cosmology
Seite(n)
029
Konferenz
International Workshop on Cosmic Structure and Evolution
Konferenzort
Bielefeld , Germany
Page URI
https://pub.uni-bielefeld.de/record/2331672
### Zitieren
Seikel M, Schwarz D. Probing Backreaction Effects with Supernova Data. In: PoS Cosmology. 2009: 029.
Seikel, M., & Schwarz, D. (2009). Probing Backreaction Effects with Supernova Data. PoS Cosmology, 029
Seikel, M., and Schwarz, D. (2009). “Probing Backreaction Effects with Supernova Data” in PoS Cosmology 029.
Seikel, M., & Schwarz, D., 2009. Probing Backreaction Effects with Supernova Data. In PoS Cosmology. pp. 029.
M. Seikel and D. Schwarz, “Probing Backreaction Effects with Supernova Data”, PoS Cosmology, 2009, pp.029.
Seikel, M., Schwarz, D.: Probing Backreaction Effects with Supernova Data. PoS Cosmology. p. 029. (2009).
Seikel, Marina, and Schwarz, Dominik. “Probing Backreaction Effects with Supernova Data”. PoS Cosmology. 2009. 029.
Open Data PUB
arXiv: 0912.2308
Inspire: 839834 | {"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": 0.8895872831344604, "perplexity": 19873.31120514238}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1603107912807.78/warc/CC-MAIN-20201031032847-20201031062847-00244.warc.gz"} |
https://icsecbsemath.com/2019/02/10/class-9-theorems-on-area-lecture-notes/ | Area Axioms
Area Axiom: Every polygonal region has an area and is measured in square units.
Congruent Area Axiom: If $\triangle ABC$ and $\triangle DEF$ are two congruent triangles, then $ar(\triangle ABC) = ar(\triangle DEF)$. ie. two congruent regions have equal area.
Rectangle Area Axiom: If $ABCD$ is a rectangular region such that $AB = a$ units and $BC = b$ units, then $ar(ABCD) = ab$ square units.
Parallelograms on the same base and between the same parallels
Theorem 1: A diagonal of a parallelogram divides it into two triangles of same area
In this case $ar (\triangle ABC) = ar (\triangle ADC)$
Also $ar (\triangle ABD) = ar (\triangle BCD)$
Theorem 2: Parallelograms on the same base and between the same parallels are equal in area.
$ar (ABCD) = AB \times h$
$ar (ABFE) = AB \times h$
$\therefore ar (ABCD) = ar (ABFE)$
Theorem 3: Area of a parallelogram is the product of its base and the corresponding altitude.
Let the two adjacent sides of the parallelogram be $a$ and $b$. Area $= Base \times Height$
Triangle Area Axiom
Theorem 4: The area of a triangle is half the product of any of its sides and the corresponding altitude.
$a, b, c$ denotes the sides of the Triangle. Then:
$\displaystyle \text{Area } = \frac{1}{2} \times Base \times Height = \frac{1}{2} bh$
Area $= \sqrt{s(s-a)(s-b)(s-c)}$. This is known as Heron’s Theorem.
Theorem 5: If a triangle and parallelogram are on the same base and between the same parallels, the area of the triangle is equal to the half of the parallelogram.
$ar (ABCD) = AB \times h$
$\displaystyle ar ( \triangle ABE) = \frac{1}{2} \times AB \times h$
$ar ( \triangle ABE) = ar (ABCD)$
Trapezium Area Axiom
Theorem 6: The area of a trapezium is half the product of its heights and the sum of parallel sides.
A trapezium is a quadrilateral two of whose sides are parallel. A trapezium whose non-parallel sides are equal is known as an isosceles trapezium.
Let $a$ and $b$ be the parallel sides and $h$ be the distance between the parallel sides. $\displaystyle \text{Then Area } = \frac{1}{2} (a+b) \times h$
Triangles on the same base and between the same parallels
Theorem 7: Triangles on the same base and the same parallels have the same area.
$\displaystyle ar ( \triangle ABD) = \frac{1}{2} \times AB \times h$
$\displaystyle ar ( \triangle ABC) = \frac{1}{2} \times AB \times h$
$\displaystyle \therefore ar ( \triangle ABD) = ar ( \triangle ABC)$
Theorem 8: Triangles having equal areas and having one side of one of the triangles equal to one side of the other triangle, have their corresponding altitudes the same.
Theorem 9: Two triangles having the same bases (or equal bases) and equal area lie between the same parallels. | {"extraction_info": {"found_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": 56, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8732918500900269, "perplexity": 342.26660405096226}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585504.90/warc/CC-MAIN-20211022084005-20211022114005-00279.warc.gz"} |
https://mathhelpboards.com/threads/testing-latex-powered-by-mathjax.26/ | ### Welcome to our community
• Moderator
• #1
#### Chris L T521
##### Well-known member
Staff member
Jan 26, 2012
995
$\int_0^{\infty}e^{-x^2}\,dx=\frac{\sqrt{\pi}}{2}$
This is inline: $$\frac{\sqrt{b^2-4ac}}{2a}$$
Other centered equation
$$\Gamma(x)=\int_0^{\infty}e^{-t}t^{x-1}\,dt$$
Last edited:
#### The Chaz
##### Member
Jan 26, 2012
24
So LaTex in brackets is centered? Not sure I get the syntax, though I'm trying to scroll through it on the mobile...
#### Ackbach
##### Indicium Physicus
Staff member
Jan 26, 2012
4,203
So \ ( and \ ), with no spaces, gives you inline. \ [ and \ ], with no spaces, gives you displayed. $$x^{2}+4x-4=0$$, but $x^{2}+4x-4=0.$
The first is inline, the second displayed.
#### Prove It
##### Well-known member
MHB Math Helper
Jan 26, 2012
1,469
How odd. I was able to do centred fine using the dollar signs, but the others wouldn't work...
#### Ackbach
##### Indicium Physicus
Staff member
Jan 26, 2012
4,203
How odd. I was able to do centred fine using the dollar signs, but the others wouldn't work...
Yeah, it looks like the double-dollar signs are enabled for displayed equations, but the single are not enabled for inline, although MathJax is capable of doing those as well. Chris mentioned to me in an email that the reason for that is there are quite a few algebra problems concerning money, and newbies would have a hard time escaping those characters.
#### ThePerfectHacker
##### Well-known member
Jan 26, 2012
236
$$\Gamma(x)=\int_0^{\infty}e^{xt}t^{x-1}\,dt$$
Apparently LaTeX does not check for math mistakes.
#### Prove It
##### Well-known member
MHB Math Helper
Jan 26, 2012
1,469
Yeah, it looks like the double-dollar signs are enabled for displayed equations, but the single are not enabled for inline, although MathJax is capable of doing those as well. Chris mentioned to me in an email that the reason for that is there are quite a few algebra problems concerning money, and newbies would have a hard time escaping those characters.
I actually meant starting with the square and round brackets didn't work. Maybe I wasn't doing it right...
Test: $$(a + b)^n = \sum_{r = 0}^n {n\choose{r}}a^{n-r}b^r$$
$\lim_{h \to 0}\frac{\sin{h}}{h} = 1$
Hmm, now it works ><
#### CaptainBlack
##### Well-known member
Jan 26, 2012
890
Test:
Some text then inline $$f(x)=\sin(x^2/23)$$ OK
$f(x)=\frac{1}{\sin(x)}$ OK
$$\displaystyle f(x)=\frac{1}{\sin(x)}$$.....$$f(x)=\frac{1}{\sin(x)}$$ - just messing with the inline form
more text.
The thing that I don't like about the MathJax \ [ ... \ ] delimiters is the amount of white space inserted before and after the LaTeX
Note the [ noparse] [ /noparse] taggs are not working
CB
• Moderator
• #9
#### Chris L T521
##### Well-known member
Staff member
Jan 26, 2012
995
Test:
Some text then inline $$f(x)=\sin(x^2/23)$$ OK
$f(x)=\frac{1}{\sin(x)}$ OK
$$\displaystyle f(x)=\frac{1}{\sin(x)}$$.....$$f(x)=\frac{1}{\sin(x)}$$ - just messing with the inline form
more text.
The thing that I don't like about the MathJax \ [ ... \ ] delimiters is the amount of white space inserted before and after the LaTeX
Note the [ noparse] [ /noparse] taggs are not working
CB
Since this being called from the MathJax js file, there was no need to create a BBcode for it. Hence, noparse will have no effect on it.
#### Ackbach
##### Indicium Physicus
Staff member
Jan 26, 2012
4,203
Since this being called from the MathJax js file, there was no need to create a BBcode for it. Hence, noparse will have no effect on it.
So question: how do we output raw LaTeX code? Verbatim?
• Moderator
• #11
#### Chris L T521
##### Well-known member
Staff member
Jan 26, 2012
995
So question: how do we output raw LaTeX code? Verbatim?
It turns out you can do that using the code tags. For example,
Code:
$\displaystyle\int_{\partial M} \omega$
would compile to $$\large{\displaystyle{\int_{\partial M} \omega}}$$.
Last edited by a moderator:
#### sbhatnagar
##### Active member
Jan 27, 2012
95
Writing Inline LaTeX
LaTeX without "\displaystyle"
Code:\int_{0}^{\infty}\frac{\ln(x)}{(x-a)(x-b)}dx
$$\int_{0}^{\infty}\frac{\ln(x)}{(x-a)(x-b)}dx$$
LaTeX with "\displaystyle"
Code:\displaystyle \int_{0}^{\infty}\frac{\ln(x)}{(x-a)(x-b)}dx
$$\displaystyle \int_{0}^{\infty}\frac{\ln(x)}{(x-a)(x-b)}dx$$
#### Markov
##### Member
Feb 1, 2012
149
Re: Writing Inline LaTeX
I just realized the LaTeX is showing itself nicely! That's good.
Perhaps a little suggestion: I love to render LaTeX by using the dollar sign, but, is there a way to render fractions, sums, integrals, etc by not using \displaystyle? It's annoying to write it everytime!
#### Ackbach
##### Indicium Physicus
Staff member
Jan 26, 2012
4,203
Re: Writing Inline LaTeX
Try double-dollar signs (two to open, two to close).
#### Markov
##### Member
Feb 1, 2012
149
Re: Writing Inline LaTeX
Well it didn't work that well, because by using doble dollar sign, it will center the content, and I don't always need to center the content.
#### afwings
##### New member
Mar 7, 2012
9
Just thought I'd test out some things too. Here's an inline equation $$u = \phi \cdot \exp \left\{ {{\textstyle{1 \over 2}}\sigma \left( {x + y} \right)} \right\}$$ and here's a display (centered) equation ${\rm{Bond Value}} = {\rm{C}}\left[ {\frac{{1 - \frac{1}{{{{\left( {1 + {\rm{r}}} \right)}^{\rm{t}}}}}}}{{\rm{r}}}} \right] + \frac{{{\rm{FV}}}}{{{{\left( {1 + {\rm{r}}} \right)}^{\rm{t}}}}}$
So, how'd that work out?
#### afwings
##### New member
Mar 7, 2012
9
Just thought I'd test out some things too. Here's an inline equation $$u = \phi \cdot \exp \left\{ {{\textstyle{1 \over 2}}\sigma \left( {x + y} \right)} \right\}$$ and...
Ok, that's something I just noticed -- following the MathJax close delimiter -- \) -- even if I add a space, the space doesn't show up in the post. Wonder what happens if I add 2 spaces? Here's a shot at it: $${b^2} - 4ac % MathType!MTEF!2!1!+- % feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj % MCPbqedmvETj2BSbqefm0B1jxALjhiov2DGCKCLv2AGW0B3bqefqvA % Tv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYhf9irFfeu0dXdh9 % vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqai-hGuQ8kuc9pgc9 % q8qqaq-dir-f0-yqaiVgFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaGaamOyamaaCaaaleqabaGaaGOmaaaakiabgkHi % TiaaisdacaWGHbGaam4yaaaa!4201!$$ ok, I added 2 spaces, and still one showed up. Nice to know. (If it's already been pointed out, sorry I missed it.)
Trying something else. Here's a simple matrix: $\left[ {\begin{array}{*{20}{c}}{10}&{20}\\{30}&4\end{array}} \right] % MathType!MTEF!2!1!+- % feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj % MCPbqedmvETj2BSbqefm0B1jxALjhiov2DGCKCLv2AGW0B3bqefqvA % Tv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYhf9irFfeu0dXdh9 % vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqai-hGuQ8kuc9pgc9 % q8qqaq-dir-f0-yqaiVgFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWaamWaaeaafaqabeGacaaabaGaaGymaiaaicda % aeaacaaIYaGaaGimaaqaaiaaiodacaaIWaaabaGaaGinaaaaaiaawU % facaGLDbaaaaa!43CF!$
#### Jameson
Staff member
Jan 26, 2012
4,093
I'm not following what's going on in the matrix you made. Is that the intended output or did something go wrong?
#### afwings
##### New member
Mar 7, 2012
9
I'm not following what's going on in the matrix you made. Is that the intended output or did something go wrong?
Jameson, thanks for following up on that. There are really 2 issues shown in my post, neither of which I believe to be MHB's error.
1. When viewing the forum in FF16 for Windows (and only in that configuration), the character immediately following an inline equation rendered by MathJax overlaps the right edge of the equation. If a period follows the equation, the period ends up inside the equation, like this one (again, this effect only happens on Windows, only in Firefox, and only with Math Renderer > MathML): $$\left( {x,y} \right) % MathType!MTEF!2!1!+- % feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj % MCPbqedmvETj2BSbqefm0B1jxALjhiov2DGCKCLv2AGW0B3bqefqvA % Tv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYhf9irFfeu0dXdh9 % vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqai-hGuQ8kuc9pgc9 % q8qqaq-dir-f0-yqaiVgFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWaaeWaaeaacaWG4bGaaiilaiaadMhaaiaawIca % caGLPaaaaaa!40E2!$$. I've filed this as a MathJax bug, but it may be a Firefox bug. There are other things going on in Firefox 16. (And it may already be corrected; hard to tell.)
2. The second issue is related to how MathType codes matrices with the Math Help Boards translator, which will be added to Cut & Copy Preferences when MathType 6.9 is released (no projected release date yet). We've already corrected this.
Bob
#### Jameson
Staff member
Jan 26, 2012
4,093
Hi Bob,
Glad it's all working then. I notice in your post how a long string of code renders a very short $$\displaystyle \left( {x,y} \right)$$. I'm guessing this is part of the Math Type plugin?
Anyway, it sounds like you're more than on top of things so I won't ask you to explain it all to me. Just let me know if there's anything I can do and if any of the issues are on our end
Best,
Jameson
#### Jay
##### New member
Aug 19, 2014
14
So question: how do we output raw LaTeX code? Verbatim?
You insert LaTeX code verbatim like this using CODE and /CODE tags enclosed within square brackets. Everything between the tags will be listed verbatim.
Code:
$$X = \sum_{n=0}^5\left(\left(\sum_{j=1}^k A_{X,n,j} \cdot {cos(B_{X,n,j} + t \cdot C_{X,n,j})}\right) \cdot t^n\right)$$
which means:
$$X = \sum_{n=0}^5\left(\left(\sum_{j=1}^k A_{X,n,j} \cdot {cos(B_{X,n,j} + t \cdot C_{X,n,j})}\right) \cdot t^n\right)$$
Last edited: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8584708571434021, "perplexity": 4662.702133482664}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320304471.99/warc/CC-MAIN-20220124023407-20220124053407-00493.warc.gz"} |
http://mathhelpforum.com/number-theory/202559-investigate-equality.html | # Math Help - Investigate Equality
1. ## Investigate Equality
Problem: Investigate following equality:
$\frac{2}{\pi }=\frac{\sqrt{2}}{2} \frac{\sqrt{2+\sqrt{2}}}{2} \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} \text{...}$
Thanks.
2. ## Re: Investigate Equality
What is there to investigate? Do you want to prove the equality? Do you want to formulate a partial product and see if it converges?
4. ## Re: Investigate Equality
My thorough investigation revealed that this is Viète's formula. Now what?
5. ## Re: Investigate Equality
Thanks a lot Vlasev, you are truly a good n efficient investigator...good to know the origin of the problem.
6. ## Re: Investigate Equality
You can define a circle to be a regular polygon with an infinite number of sides. We can define \displaystyle \begin{align*} \pi \end{align*} as the circumference of a circle of diameter equal to 1 unit. Therefore, we can get an approximation for \displaystyle \begin{align*} \pi \end{align*} by evaluating the perimeter of said regular polygon.
Some algebraic manipulation of this will give the result you are trying to prove.
7. ## Re: Investigate Equality
This explanation is truly enlightening. Thanks a lot ProveIt.
8. ## Re: Investigate Equality
That's indeed a magnificent proof! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.994012176990509, "perplexity": 2370.008897634011}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802777454.142/warc/CC-MAIN-20141217075257-00068-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://thebeardsage.com/short-primer-on-probability/ | # Short Primer on Probability
## Fundamentals
A random variable is denoted in capital, and the values it can take is denoted in small .
Consider a collection of random variables . Random variables can be thought of as features of a particular domain of interest.
For example, the result of a coin toss can be represented using a single random variable, . This variable can take either of the categorical values or . If the same coin is tossed times, this can be represented using variables . Each of these values can be either or .
## Joint Probability
An expression of the form is called a joint probability function over the variables . The joint probability is defined when the values of are respectively. This is denoted by the expression
This is sometimes abbreviated as .
For a fair coin toss, . If a fair coin is tossed five times, .
The joint probability function satisfies
## Marginal Probability
The marginal probability of one of the random variables can be computed if the values of all of the joint probabilities for a set of random variables are known.
For example, the marginal probability is defined to be the sum of all those joint probabilities for which
When dealing with propositional variables (True/False) TrueFalse is denoted as .
## Conditional Probabilities
The conditional probability of given is denoted by .
where is the joint probability of and and is the marginal probability of . Thus
Joint conditional probabilities of several variables conditioned on several other variables is expressed as
A joint probability can be expressed in terms of a chain of conditional probabilities.
The general form of this chain rule is
## Bayes Rule
Different possible orders give different expressions but they all have the same value for the same set of variable values. Since the order of variables is not important
Which gives Bayes’ Rule
## Probabilistic Inference
In set notation , , The variables having the values respectively is denoted by , where and are ordered lists.
For a set , the variables in a subset of are given as evidence.
For example, consider . The evidence is being false. In other words equates to .
Thus need not be computed.
## Conditional Independence
A variable is conditionally independent of a set of variables given a set if
(1)
(2)
Saying that is conditionally independent of given also means that is conditionally independent of given . The same result also applies to sets and .
As a generalization of pairwise independence, the variables are mutually conditionally independent, given a set if each of the variables is conditionally independent of all of the others given .
(3)
When is empty
This implies that the variables are unconditionally independent.
### Thank You
• Mark – for pointing out typos and errors
## 2 Comments →Short Primer on Probability
1. Mark
Another great article! Nice, clear explanations.
I do have some suggestions for improvement though:
It seems there is an error in the Bayes Rule section. It should be P(V_i|V_j) = P(V_j|V_i)P(V_i)/P(V_j)
I would also recommend explaining how you introduced the P and not P random variables in the Probabilistic Inference section as it is not quite clear how that works out.
Finally, it seems your Latex implementation had trouble rendering for the fourth paragraph in the Conditional Independence section (\textbf{mutually conditionally independent}), right above equation (3).
1. TheBeard
I have expanded the Probabilistic Inference section and fixed the errors you have pointed out. Thank you. | {"extraction_info": {"found_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": 0.9611799716949463, "perplexity": 577.5965722055848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030334644.42/warc/CC-MAIN-20220926020051-20220926050051-00671.warc.gz"} |
http://openstudy.com/updates/503e0c29e4b0c29115be4cfe | Here's the question you clicked on:
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## anonymous 3 years ago y varies inversely as twice x. When x = 4, y = 1. Find y when x = 5. Delete Cancel Submit
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1. anonymous
• 3 years ago
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y varies inversely as twice x will be something like $y = \frac{k}{2x}$ does that help? or do you need more hints?
2. anonymous
• 3 years ago
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i need more hints
3. anonymous
• 3 years ago
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good day vegas14, this is an example of a proportionality problem. what you need to do is read the problem word for word, each word will be giving you hint as to how to approach the problem. I will list the possible cases in this type of problem, I hope that it will help you. caseI: if the problem states that "y varies directly to x" it means that y is directly proportional to x "but", it will be proportional only if you multiply a proportionality constant "k" $y \alpha x$ this equation (eq1) symbolizes the direct proportionality of y to x, to be able to replace the proportionality symbol alpha to an equal (=) sign you must multiply x to a proportionality constant which is "k". therefore transforming the equation to, $y=k(x)$
4. anonymous
• 3 years ago
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caseII: (this is the case of your problem) when the problem states that "y varies inversely as x" it means that your equation would be like this, $y \alpha \frac{ 1 }{ x }$ from this equation you should again have to replace the proportionality symbol "alpha" to an equal (=) sign. transforming the equation to be,$y=k(\frac{ 1 }{ x })$ which if simplified will become $y=\frac{ k }{ x }$
5. anonymous
• 3 years ago
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of course problems may vary in this cases, for example "it may state twice as x therefore you should use "2x" or thrice as x use "3x" or may be as the square of x use $x^{2}$ it will now depend on your own analysis of the problem.
6. anonymous
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i really wanted to solve the problem for you, but it might be best if you do it on your own, just to test yourself if you really understand the logic behind =) but just to give you more tips: 1) Apply CaseII 2) Find the proportionality constant "k" ---- just this I will not give you which on the given you will use to solve for "k" =) 3) Solve for y(new) I hope this posts will help you. will check your answers soon. . . =)
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy. | {"extraction_info": {"found_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": 0.973931610584259, "perplexity": 1390.408053582421}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1464049270555.40/warc/CC-MAIN-20160524002110-00096-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://gorchilin.com/articles/energy/RLC_3?lang=en | 2017-07-27
Research website of Vyacheslav Gorchilin
All articles
Parametric change of inductance in a RL-circuit. Back EMF
In this note first we prove the theoretical possibility of obtaining additional energy from the back EMF in the inductor core. The proof is based on the classic formulas of electrical engineering. According to the classification here will be considered the generator of the first kind of first order partial cycle PCCIE.
In our time, divorced many myths and legends about the inexhaustible possibilities of back EMF (AEDS) in the inductor. According to some researchers, OADS can give more energy than it expended, and their experiences, in some cases, confirm this. Theorists explain such powers by the ether theory or the unused energy of the atoms of ferromagnetic materials. We will try to draw conclusions based on mathematics and the theory of electrical circuits [1], which worked well and can fully display the necessary processes. Thus, the author reiterates the idea that issledovatelyam it is not necessary to go into the jungle of other theories and hypotheses, enough of a different angle to look at the classics.
Next, we will show that the school or even high school — a nonparametric OATS — can increase the efficiency of the second kind $$(K_{\eta2})$$, so the outset that we consider only parametric coil which changes its inductance depending on the magnetic field $$(H)$$ in it. And since this tension is directly proportional to the current, as parameter we will have to address it. To convert the current back to $$H$$ is not extremely difficult, you need to know the parameters of the coil and core, i.e. the parameters of a particular device.
In the previous section we showed that a parametric change in capacitance in the full cycle: charge-discharge, does not increase $$K_{\eta2}$$. The same can be said of inductance: it is enough to replace the voltage on the currents and to change some of the ratios in the diff. equation. But with the coil inductance, which will contain a ferromagnetic core, we can do a little differently.
A typical graph of the magnetic permeability of the core is $$(\mu)$$ on the magnetic field $$H$$ to the left [2]. But the tension is proportional to the current, and the permeability of the inductance, this means that we can build a graph $$L(I)$$, which is proportional to $$\mu(H)$$. Thus, we obtain the parametric dependence of the inductance $$L$$ from the current $$I$$ flowing in it, and with this dependence, we will continue to work.
As an example, compare the two charts to the actual measured characteristics of the ferrite from the flyback transformer: graph 1, graph 2. On the second chart the y coordinate $$M(I)$$ shows how changing the inductance of the coil compared to the original, and $$I$$ is the current in the coil.
Recall that in this note we investigate AEDS, therefore, assume that the initial current $$I_0$$ in the coil already, and it will pass through the electric circuit consisting of inductance and resistance. How there came a shock we will discuss later, but for now we will use the classical formula for the calculation of the energy in the coil at the time of closure of the SW key: $W_L = \frac{L_0 I_0^2}{2} \qquad (3.1)$ where $$L_0$$ is the inductance of the coil at the time of closure of key SW. Now our task is this: to properly use this energy. It is necessary to find a mode in which the current in the circuit will produce the most work on active loading. The classic formula for this energy is: $W_R = R \int_0^T I(t)^2 dt \qquad (3.2)$ Where $$I(t)$$ is the current in the circuit depending on time, which is unknown to us. In order to find it, it is necessary to compile the differential equation for the transition process in our scheme [1]: ${L \over R} I(t)' + I(t) = 0 \qquad (3.3)$ where the inductance $$L$$ is a parameter from the current, and the current in turn — on time. For the solution of the equation it remains to determine the dependence of the inductance from the current for which we can take the polynomial $$M(I)$$ with coefficients $$..k_1 k_4$$. Its convenience — flexibility for various dependencies: $L = L_S\, M(I), \quad M(I) = {1 + k_1 I(t) + k_2 I(t)^2 \over 1 + k_3 I(t) + k_4 I(t)^3} \qquad (3.4)$ where $$L_S$$ is the initial inductance of the coil (no current). $$M(I)$$ must be proportional to $$\mu$$ from $$H$$ to the real core.
We introduce the time constant of the circuit $$\tau = L_S/R$$, which will record the final shape of the diff. equation: $\tau\, M(I)\, I' + I = 0 , \quad I=I(t) \qquad (3.5)$ the Analytical solution of this equation in this form is complicated, so we use mathematical editor MathCAD and get it in a numerical form. In the editor we appreciate the real time operation of the circuit after closing of the key, we denote it $$T$$ and make this time a finite number.
Why do we need time?
Indeed, if we want to obtain the dependence of the increase in efficiency of $$\mu$$ (see top graph), why should we use one additional variable — the time? Will try to fix it. We rewrite (3.6) in another form: $-\tau\, M(I)\, I' = I, \quad I=I(t) \qquad (3.6)$ and substitute into the formula (3.2): $W_R = R \int_0^T \left[\tau\, M(I)\, I'\right]^2 dt \qquad (3.7)$ After some mathematical operations, the author obtained a rather unusual integral in which energy dissipation on resistance does not depend on $$t$$, and even from the resistance: $W_R = \frac{L_S}{2} \int_0^{I_0^2} M(\sqrt{J})\ dJ , \quad J = I^2 \qquad (3.8)$ Here you need to pay attention to the fact that in paginegialle functions are all $$I$$ change for $$J$$ by the rule: $$I=\sqrt{J}$$. To find the final formula calculates the energy gain, divide the energy dispersed on the active resistance on the starting energy in the coil is: $K_{\eta2} = {W_R \over W_L} \qquad (3.9)$ Given that $W_L = {L_0\, I_0^2 \over 2} = {L_S\, M(I_0)\, I_0^2 \over 2}, \quad M(I_0)= {1 + k_1 I_0 + I_0 k_2^2 \over 1 + I_0 k_3 + k_4 I_0^3} \qquad (3.10)$ substitute the previously obtained results and we get an incredibly interesting pattern that does not depend on the time coordinate: $K_{\eta2} = \frac{1}{M(I_0)\, I_0^2} \int_0^{I_0^2} M(\sqrt{J})\, dJ , \quad J = I^2 \qquad (3.11)$ Or this same formula in another form: $K_{\eta2} = \frac{2}{M(I_0)\, I_0^2} \int_0^{I_0} M(I) \, I \, dI \qquad (3.12)$ Well, if we all want the same thing to Express through the magnetic permeability of the core and the magnetic field $$\mu(H)$$, then the formula (3.12), simply replace the appropriate letters: $K_{\eta2} = \frac{2}{\mu (H_0)\, H_0^2} \int_0^{H_0} \mu (H) \, H \, dH \qquad (3.13)$
Equations (3.11-3.13) and are the mathematical expression of free energy for back EMF in the coil core! A more General approach for the derivation of this formula see here
From it immediately clear that if a parametric dependency is missing, i.e. $$M(I)=M(J)=1$$, then a raise is not: $$K_{\eta2}=1$$. We have to learn to use this formula and find out whether there can be $$K_{\eta2}$$ is greater than one.
What is the result?
For example, take $$M(I)$$ from formulas (3.4) and substitute it into (3.11) replacing $$I$$ to $$J$$ by the rule: $$I=\sqrt{J}$$. The time dependence of ex and obtain: $K_{\eta2} = \frac{1}{M(I_0)\, I_0^2} \int_0^{I_0^2} {1 + k_1 J^{0.5} + k_2 J \over 1 + k_3 ^{0.5} + k_4 ^{1.5}}\, dJ \qquad (3.14)$ Again for example, enter the following current, and the coefficients are $$I_0=1.4, k_1=10, k_2=0, k_3=0, k_4=5$$. The left side shows a graph of the inductance of the coil from the current in accordance with these coefficients. As you can see, we work in growing, and in the drop-down field magnetization curve of the core and primary inductance is less than the maximum about 4 times, which is close to the real values of permeability of ferromagnets. Solving the integral (3.14) we get $$K_{\eta2}=2.07$$ is greater than one!
To substitute other factors and to get your results you can in the editor MathCAD downloading there this program or to test it in the calculator. It a visible pattern, which confirms the assumption that the drop down part of the curve $$\mu$$ from $$H$$ gives the largest contribution to the energy increase. In addition to choosing the work area on the chart, to achieve a positive result, it is necessary that the device provide access to the coil for the desired mode of current and magnetic field strength.
The rationale for the occurrence of additional energy in such devices, see here
What in practice?
You need to understand that we get according to equations (3.11-3.13) is the marginal value. Depending on the material of the core, when working on the right drop-down plot of $$M(I)$$, you may experience a variety of losses, mostly heat. But they will not always be equal to this increase, and therefore is described here the way to super devices still remains open. Izvestia of materials, good performance $$K_{\eta2}$$ should give Permalloy, on which chart, you can easily find the working site, even better — Metglas [8]. A little worse, in this sense, the situation with ferrites. Promising look some modern steels and certain core construction [5], however, we must not forget that part of the energy can be spent on Foucault.
Permalloy is working in a relatively small frequency and therefore the device running on it cannot develop large capacities. Ferrite can withstand orders of magnitude for large frequencies, but it is much more fragile. However, as a result, it can operate at high capacity.
The question remains open: how do you receive the initial current $$I_0$$ in the coil and how long it takes for the energy? For answer there are different approaches. One of them suggests that the core need to podmanivaya perpendicular to the primary field [3], and it can be done with short pulse whose energy is several times lower than required due to the inertia of a ferromagnet. The second approach [4-6] involves the ramp-up without additional perpendicular field. The third offers mechanical excitation of the core is a permanent magnet, and thus the initial current appears in the coil [7].
Given all the above it seems to us a very real seredinny receiving devices based on coils with ferromagnetic core. As you know, if math gave the green light, then the practical implementation will not take long.
The results of this work was done by a specialized calculatorthat allows you to find the dependence of magnetic permeability of any core on the magnetic field and to calculate it is potentially achievable increment of efficiency of the second kind.
The materials used | {"extraction_info": {"found_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": 0.8721423149108887, "perplexity": 298.5374576890251}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221211935.42/warc/CC-MAIN-20180817084620-20180817104620-00475.warc.gz"} |
https://www.gamedev.net/forums/topic/645924-sfml-crash-when-debugging-but-ok-on-release/ | # SFML crash when debugging, but ok on release.
## Recommended Posts
Meerul264 130
I tried to search regarding this problem on google but couldn't find a similar case. The problems I found is vice versa of mine - crash when release but ok when debugging.
I already checked my project's configuration a dozen times but still can't find whats wrong.
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alh420 5995
In what way is it crashing?
My first guess is that it's because of uninitialized data. The value of the unintialized data will be different in release and debug, and might make one of the cases crash.
My second guess is misuse of some pointer that either writes outside a buffer, or in already freed memory. That could also lead to one case crashing and the other not.
In both cases it can go either way, release working and debug crashing, or debug working but release crashing.
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SiCrane 11839
For SFML another possibility is that you've linked to the release version of the library in debug mode.
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Meerul264 130
In what way is it crashing?
Thanks for replying. The crash is that, it just produce white blank window. And when I try to click the window, it wouldn't respond (like when trying to click 'x' button to close the window). Also, this doesn't happen before I changed the code a bit. After I received this crash, I undo the changes I had made, but still get the crash. Why did this happen?
For SFML another possibility is that you've linked to the release version of the library in debug mode.
I don't think so.
Under my Configuration -> Debug -> Linker -> Input -> Additional dependencies:
sfml-graphics-d.lib
sfml-window-d.lib
sfml-system-d.lib
If I'm not mistaken, "-d" for debug, without the "-d" for release.
Also, I just realised I got this output when debugging:
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\dinput.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\hid.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\wintrust.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\crypt32.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\msasn1.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\ole32.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Unloaded 'C:\Windows\SysWOW64\ole32.dll'
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\apphelp.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\cmd.exe'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Unloaded 'C:\Windows\SysWOW64\cmd.exe'
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\cmd.exe'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Unloaded 'C:\Windows\SysWOW64\cmd.exe'
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\cmd.exe'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Unloaded 'C:\Windows\SysWOW64\cmd.exe'
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\cmd.exe'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Unloaded 'C:\Windows\SysWOW64\cmd.exe'
[...and the lists of similar kind of error goes on...]
I hope this can give you the clue to find what causes this crash. Thanks for the reply really appreciate it.
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Meerul264 130
I also got this, is this important?
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Users\Amirul\Documents\Visual Studio 2012\Projects\Dragon Villa SFML\Dragon Villa SFML\sfml-graphics-d-2.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Users\Amirul\Documents\Visual Studio 2012\Projects\Dragon Villa SFML\Dragon Villa SFML\sfml-window-d-2.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Users\Amirul\Documents\Visual Studio 2012\Projects\Dragon Villa SFML\Dragon Villa SFML\sfml-system-d-2.dll'. Cannot find or open the PDB file.
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Here's my general process for solving these kinds of problems:
Step 1: "Rebuild all". This might fix your problem on its own.
Step 2: Scatter a few breakpoints around areas where you think the crash is occurring.
Step 3: Run your debugger, and when you hit a debugger, start stepping through the code (using your debugger's 'step' feature) bit by bit until it reaches the point where it crashes. This is how you know what line is causing the crash.
Step 4: Find out why that line is crashing. Is a pointer uninitialized? Are you dividing by zero? Was the object already destroyed?
Step 5: Fix the problem. If you need help with that part, after you've already found the cause of the problem, then show us the part that's crashing, and we might have a few ideas if, by that point, you haven't already solved it yourself - which is usually the case.
It's also not a bad idea to occasionally run your code through a static analyzer like cppcheck. Static analyzers read your code, without compiling it, and look for common mistakes that compile fine but don't work fine.
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Dragonsoulj 3212
In what way is it crashing?
Thanks for replying. The crash is that, it just produce white blank window. And when I try to click the window, it wouldn't respond (like when trying to click 'x' button to close the window). Also, this doesn't happen before I changed the code a bit. After I received this crash, I undo the changes I had made, but still get the crash. Why did this happen?
For SFML another possibility is that you've linked to the release version of the library in debug mode.
I don't think so.
Under my Configuration -> Debug -> Linker -> Input -> Additional dependencies:
sfml-graphics-d.lib
sfml-window-d.lib
sfml-system-d.lib
If I'm not mistaken, "-d" for debug, without the "-d" for release.
Also, I just realised I got this output when debugging:
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\dinput.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\hid.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\wintrust.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\crypt32.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\msasn1.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\ole32.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Unloaded 'C:\Windows\SysWOW64\ole32.dll'
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\apphelp.dll'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\cmd.exe'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Unloaded 'C:\Windows\SysWOW64\cmd.exe'
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\cmd.exe'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Unloaded 'C:\Windows\SysWOW64\cmd.exe'
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\cmd.exe'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Unloaded 'C:\Windows\SysWOW64\cmd.exe'
'Dragon Villa SFML.exe' (Win32): Loaded 'C:\Windows\SysWOW64\cmd.exe'. Cannot find or open the PDB file.
'Dragon Villa SFML.exe' (Win32): Unloaded 'C:\Windows\SysWOW64\cmd.exe'
[...and the lists of similar kind of error goes on...]
I hope this can give you the clue to find what causes this crash. Thanks for the reply really appreciate it.
Anytime I get a white screen it is because I am missing a .dll in the folder.
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Meerul264 130
start stepping through the code (using your debugger's 'step' feature)
Lol thanks for introducing me to this interesting feature. i wouldn't know this exist.
Anyway i tried it out and found out that in main:
TEXT.Set_Fonts(); // problem avoided when I commented this out and came back when I uncomment this
TEXT.Set_Texts();
So I guess that is the root of the problem. But I couldn't figure out why is it so.
//in Texts.cpp
void Texts::Set_Fonts()
{
{
for(int i = 0; i < 10; i++)
{
text[i].setFont(font);
promptedText[i].setFont(font);
}
FPS.setFont(font);
}
else
{
std::cout << "Failed to load fonts " << std::endl;
system("pause");
}
}
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Did you step through ("step into") that function also?
My guess is that this part of the documentation might be causing some difficulty.
The sf::Text only keeps a pointer to the sf::Font. You have to make sure the sf::Font stays around for longer than the sf::Texts are in use.
Edited by Servant of the Lord
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Meerul264 130
I figured out the problem. It was really a silly mistake. I made the array promptedText[] to have a size of 2 elements ( I should make it at least 10 ) and I forgot to change it to 10 . Really silly mistake.. But this make me wonder why it gives a green light on release.
Oh well I got it now. Thanks man
Edited by Meerul264
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Yeah, that kind of mistake occurs alot. Part of programming is learning good practices that reduces or removes those kind of mistakes.
Two solutions:
1) Don't use "magic numbers" in your code. '10' in your code doesn't mean anything, and neither does '2'. Changing them in one place requires you to remember changing them in other places (sometimes dozens). Instead, it's much better to replace magic numbers with well-named constant variables. Changing the constant's value automaticly makes all the code that uses the constant still work properly.
Example:
//The value is only set once, in a specific location.
const int PROMPTED_TEXT_COUNT = 10;
//In one location:
sf::Text promptedText[PROMPTED_TEXT_COUNT];
//In another location:
for(size_t i = 0; i < PROMPTED_TEXT_COUNT; i++) { ... }
2) You could make it a std::vector. Or, if you're using C++11, you could make it a std::array. std::vectors keep track of their size as part of their class, and can still be accessed through the [] subscript operator during for() loops, and can also be used with vector.at(n) which double-checks that that location is valid before accessing.
Your for-loop would then look like:
//If you're using C++11's new range-for() syntax:
for(sf::Text &text : promptedText)
{
text.setFont(font);
}
//If you're not using C++11's range-for():
for(size_t i = 0; i < promptedText.size(); i++)
{
promptedText[i].setFont(font);
}
I'd also make sure "text[i].setFont(font);" and "promptedText[i].setFont(font);" are in different for()-loops, unless you're 100% confident that they will always forever have the same number of elements as each other.
Edited by Servant of the Lord
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Meerul264 130
//If you're not using C++11's range-for():
for(int i = 0; i < promptedText.size(); i++)
{
promptedText[i].setFont(font);
}
You are using promptedText.size(). Doesn't that require promptedText.push_back(sf::Text &) first? Is this possible if I were to initialise the font of this text before i begin the game loop (i.e before window is open) ?
Or you are saying that I have to implement the std::vector's reserve() or resize() first?
Edited by Meerul264
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Oh, yes, most certainly resize to the correct size first. I was (incorrectly) assuming your code was already doing that in your existing array elsewhere.
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Meerul264 130
I see. Thanks again :)
## Create an account
Register a new account | {"extraction_info": {"found_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": 0.22994227707386017, "perplexity": 9143.404117040633}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818696182.97/warc/CC-MAIN-20170926141625-20170926161625-00002.warc.gz"} |
http://mathhelpforum.com/algebra/25824-first-digit-after-comma.html | # Math Help - First digit after comma
1. ## First digit after comma
Find first digit after comma of expression $(\sqrt2+\sqrt3)^{2008}$
2. Originally Posted by terafull
Find first digit after comma of expression $(\sqrt2+\sqrt3)^{2008}$
What comma?
3. Originally Posted by Plato
What comma?
I think he means "decimal point" for us US people.
-Dan
4. Yes, I meant decimal point.
5. $2008log(\sqrt{2}+\sqrt{3})=999.5728$
Now, take $10^{.5728}=3.7394$
Those are the first few digits of this big number with about 1000 digits.
6. I dont understand why like this. And 3 is the first digit after decimal point or 7?
$(\sqrt2+\sqrt3)^{2008}=10^{2008log(\sqrt2+\sqrt3)} =10^{999.5728}=10^{999}*10^{0.5728}$
And what now?
7. Originally Posted by galactus
$2008log(\sqrt{2}+\sqrt{3})=999.5728$
Now, take $10^{.5728}=3.7394$
Those are the first few digits of this big number with about 1000 digits.
But I must know first digit after decimal point.
8. I pretty much gave you the answer. Look at it. There are 1000 digits in the number. Therefore, counting off blocks of 3 from the far right of this huge number gives us 3,739,..................
So, the next number after the comma is 7.
You could even run it through a calculator. My TI-92 gives me
3.7392622716E999
So, there are 999 digits after the 3 and the first one is 7.
BTW, do you use decimal points instead of commas in Europe or wherever?.
In the US, it's commas to separate the 1000's.
For instance, 1,000,000,000
Would you write 1.000.000.000?
If so, that's wacky and wrong. Do it the right way.
9. Originally Posted by galactus
I pretty much gave you the answer. Look at it. There are 1000 digits in the number. Therefore, counting off blocks of 3 from the far right of this huge number gives us 3,739,..................
So, the next number after the comma is 7.
You could even run it through a calculator. My TI-92 gives me
3.7392622716E999
So, there are 999 digits after the 3 and the first one is 7.
BTW, do you use decimal points instead of commas in Europe or wherever?.
In the US, it's commas to separate the 1000's.
For instance, 1,000,000,000
Would you write 1.000.000.000?
If so, that's wacky and wrong. Do it the right way.
OK, thanks. I mean decimal point, not comma.. So, I must find first digit after decimal point of this expression.
In Europe are 1/4=0,25 and 10^5=10.000 or =10 000. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46589213609695435, "perplexity": 1437.5332383761338}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657131545.81/warc/CC-MAIN-20140914011211-00223-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://socratic.org/questions/what-intermolecular-forces-are-present-in-c-2h-8o-1-propanol | Chemistry
Topics
# What intermolecular forces are present in C_2H_8O (1-Propanol)?
## ${C}_{2} {H}_{8} O$
Dec 6, 2016
The most obvious one in $\text{hydrogen bonding}$.
#### Explanation:
One of the best indicators of intermolecular interaction is the normal boiling point. Molecules with significant intermolecular interaction tend to have higher boiling points. Water, a small molecule, has an exceptionally high boiling point because of intermolecular hydrogen bonding, which persists BETWEEN molecules:
H-O^(delta-)-H^(delta+)cdotsO^(delta-)-H^(delta+)""_2
Now $\text{1-propanol}$ has a normal boiling point of $97 - 98$ ""^@C. And we compare this to that of $\text{isopropanol}$, $82.6$ ""^@C, and $\text{ethanol}$, $78.0$ ""^@C.
Given these data, there is another contributor to intermolecular interaction, and here it is the non-polar interaction between hydrocarbyl chains. The longer the chain, the greater the chain-chain interaction, and in long chain alcohols (and alkanes), boiling points will be elevated. Compare the boiling point of $\text{n-butanol,}$ $117.7$ ""^@C, and $\text{n-pentanol,}$ $137 - 139$ ""^@C.
##### Impact of this question
4801 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 18, "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": 0.46977499127388, "perplexity": 2633.939849512711}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986670928.29/warc/CC-MAIN-20191016213112-20191017000612-00365.warc.gz"} |
http://mathhelpforum.com/calculus/11275-calculus-vector-valued-func.html | # Thread: Calculus of Vector-Valued Func.
1. ## Calculus of Vector-Valued Func.
1.) Given: r(t)=tcos(t)i+e(t^2)j+ln(t)k,
what is r'(t)?
2.) Suppose r(t) is a vector-valued func.,
What geometric/graphical info does r'(a) tell us?
3.) Let r(t)=cos(t)i+sin(t)j and s(t)=sin(5t)i+cos(5t)j.
(a) What does the graphs of r(t) and s(t) look like?
(b) Suppose the graphs of two vector-valued func. r(t) and s(t) are the same, then must r'(0)=s'(0)? (Explain whether this is a new result, or was it also true for functions f(x) and g(x)?)
2. Hello, fifthrapiers!
Here's some help . . .
1) Given: . $\vec{r}(t)\;=\;t\cos(t)\vec{i} + e^{t^2}\vec{j} + \ln(t)\vec{k}$. . Find $\vec{r'}(t)$
$\vec{r'}(t)\;=\;(\cos t - t\sin t)\vec{i} + \left(2te^{t^2}\right)\vec{j} + \left(\frac{1}{t}\right)\vec{k}$
3) Let $r(t)\:=\:\cos(t)i + \sin(t)j$ and $s(t)\:=\:\sin(5t)i + \cos(5t)j$
(a) What does the graphs of $r(t)$ and $s(t)$ look like?
Both are unit circles, centered at the origin.
(b) Suppose the graphs of two vector-valued func. $r(t)$ and $s(t)$ are the same,
. . .then must $r'(0) = s'(0)$? . no
Consider: . $\begin{array}{cc}\vec{r}(t) \;= & \cos(t)\vec{i} + \sin(t)\vec{j} \\ \vec{s}(t) \:= & \cos(t + \pi)\vec{i} + \sin(t + \pi)\vec{j} \end{array}$
When $t = 0:$
. . $\begin{array}{ccc}r(0) = \langle 1,\,0\rangle & \text{ and } & r'(0) = \langle 0,\,1\rangle\!:\;\uparrow \\ \\
s(0) = \langle \text{-}1,0\rangle & \text{ and } &s'(0) = \langle 0,\,\text{-}1\rangle\!: \;\downarrow\end{array}$
. . . different derivatives
Explain whether this is a new result,
or was it also true for functions $f(x)$ and $g(x)$ ?
This is a new result.
With rectangular functions, $y = f(x)$ and $y = g(x)$,
. . if their graphs are identical, then the functions are identical.
. . Hence, their derivatives are equal.
With parametric functions, a graph can be generated in a number of ways.
. . As seen in part (b), their derivatives need not be equal. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 17, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9071699976921082, "perplexity": 1937.1617174127648}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988725451.13/warc/CC-MAIN-20161020183845-00090-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/361856/why-is-quantum-mechanics-called-01-dimensional-qft/361858 | # Why is quantum mechanics called 0+1 dimensional QFT?
I know that quantum mechanics is sometimes called 0+1 dimensional quantum field theory. What is the meaning? How should we understand it?
In field theory, a field can be thought of as a map from the spacetime $M$, usually a Lorentzian manifold---a particularly popular choice is $\Bbb R^{1,n-1}$ (Minkowski space)---to some other space. For instance, a scalar field $\phi$ can be viewed as a map $\phi:M\to \Bbb R$, or equivalently as a global section of the trivial (real) line bundle over $M$. The spacetime $M^n$ has one timelike direction and $n-1$ spacelike directions, and one can say that one studies $(n-1)+1$-dimensional field theory.
When doing mechanics, what do we use for a "field"? The position of the particle(s)! The position depends only on time, and hence we have maps $x_i:\Bbb R\to \Bbb R^n$ (in case the space in which the particles move is not simply $\Bbb R^n$, the target may be some other Riemannian manifold of dimension $n$) and fitting this into the general QFT picture we can identify $\Bbb R$ as our "spacetime", where we now have no spatial directions and only a timeline direction. Thus, we may call the theory a $0+1$-dimensional field theory. After quantization, one obtains a $0+1$-dimensional quantum field theory. | {"extraction_info": {"found_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": 0.8996157646179199, "perplexity": 252.17069335200685}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1573496669755.17/warc/CC-MAIN-20191118104047-20191118132047-00089.warc.gz"} |
https://socratic.org/questions/what-is-the-value-of-cos-75 | Trigonometry
Questions Topics
# What is the value of cos 75°?
Then teach the underlying concepts
Don't copy without citing sources
preview
?
#### Explanation
Explain in detail...
#### Explanation:
I want someone to double check my answer
35
Nov 21, 2015
$\frac{\sqrt{6} - \sqrt{2}}{4}$
#### Explanation:
Apply the trig identity: cos (a + b) = cos a.cos b - sin a.sin b
$\cos \left(75\right) = \cos \left(30 + 45\right) = \cos 30 \cdot \cos 45 - \sin 30 \cdot \sin 45 =$
= $\left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{6} - \sqrt{2}}{4}$
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https://forum.math2market.de/index.php?PHPSESSID=456c8d3b3f1f950d342df2f337eabb4b&action=profile;area=showposts;sa=messages;u=2 | ### Show Posts
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### Messages - Aaron Widera
Pages: [1] 2
1
##### WeaveGeo / Re: how do i create a larger weave
« on: October 12, 2021, 09:38:26 AM »
Hi Sarah,
you can change the size of the weaves in WeaveGeo by heading to the Domain tab and can add periodic cells of the weave in X-/Weft-Direction or Y-/Warp-Direction to the weave by chaning the Unit Cells option here.
2
##### Forum FAQs / Re: Frequently asked questions:
« on: September 29, 2021, 02:14:38 PM »
How do I use the Latex Editor?
• You can use the Latex Editor by clicking on the fMath formatting option above the text area.
• There are two different options:
• One that creates formulas inline $$A = \pi r^{2}$$
• And one that creates formulas in a new paragraph $A = \pi r^{2}$
• For latex code in a new paragraph, enter your latex code between this operators:
Code: [Select]
$A = \pi r^{2}$
• For latex code in line, enter your latex code between this two operators:
Code: [Select]
[latex=inline] A = \pi r^{2}[/latex]
• You can create code blocks as seen above by using the code block tags:
Code: [Select]
[code] Your code here [/code ]
3
##### Forum FAQs / Re: Frequently asked questions:
« on: September 29, 2021, 02:08:26 PM »
• How do I start a new board at the front page?
• It is not intended to change the boards on the forums starting page. The starting page is read only.
• How do I embedd an image into my posts?
• You can attach your image to the post and use the [ attachimg=1] to insert the first attachment as inline attachment. To insert the second attachment use [ attachimg=2].
• You can upload your image to a web hosting service and use the tag:
Code: [Select]
[img]IMAGEPATH[/img]
4
##### Forum FAQs / Re: Frequently asked questions:
« on: September 29, 2021, 02:04:29 PM »
5
##### Forum FAQs / Re: Frequently asked questions:
« on: September 29, 2021, 01:58:04 PM »
6
##### Property prediction (-Dict modules) / MOVED: Regarding how to create a macro for calculations with varying temperatures
« on: September 27, 2021, 09:39:16 AM »
7
##### Modelling of composites / Re: Modelling Rovings on the Basis of Support Points
« on: August 18, 2021, 02:33:18 PM »
Hi Marvin,
I prepared you an example, I assume, that youzr python script to create the roving support points is named PolyNomial.py.
It should look like that:
Code: [Select]
def test_func(parameter1, parameter2): x = [[2e-05, 2e-05, 2e-05],[4.5e-05, 2e-05, 2e-05],[7e-05, 3e-05, 2e-05],[9.5e-05, 4e-05, 3e-05]] y = [[2e-05, 2e-05, 2e-05],[2e-05, 4.5e-05, 2e-05],[3e-05, 7e-05, 2e-05]] return [x, y]So it is just a function "test_func" that gets two parameters. Then it calculates and get two list (x and y) with the support points. In this case the list is set, but you just need to enter you polynomial function.
Then the script returns the two lists combined as [x, y]
Now for the important part, you can copy and paste this code block into a blank python script and use it, I will use comments marked by # to comment in the script what is happening:
Code: [Select]
Header = { 'Release' : '2021', }Description = ''''''Variables = { 'NumberOfVariables' : 0, }# until here it is just header and initialization which is not importantimport PolyNomial #here we import the python scriptRoving_list = PolyNomial.test_func(123, 456) # by using .test_func(para1,para2) we are using the function from the script PolyNomial with the parameters para1 and para2. And save the returned values in the Roving_listdiam_1 = 2e-05 #Here you can set the Diameter of the Rovings. Here both Rovings in x and y directio have the same size. You could adapt it like below.diam_2 = 1e-05number_of_rovings = len(Roving_list)#now we define a dictionary that is filled with all information GeoDict needs to create the GAD-Objects#You might need to adjust the voxellenght and Domainsize (NX, NY, NZ) below by calculating the domainsizeGadAdd_args = { 'ResultFileName' : 'GadAdd.gdr', 'KeepStructure' : 0, 'NumberOfObjects' : number_of_rovings, 'Domain' : { 'PeriodicX' : False, 'PeriodicY' : False, 'PeriodicZ' : False, 'OriginX' : (0, 'm'), 'OriginY' : (0, 'm'), 'OriginZ' : (0, 'm'), 'VoxelLength' : (1e-06, 'm'), 'DomainMode' : 'VoxelNumber', # Possible values: VoxelNumber, Length, VoxelNumberAndLength 'NX' : 100, 'NY' : 100, 'NZ' : 100, 'Material' : { 'Type' : 'Fluid', # Possible values: Fluid, Solid, Porous 'Name' : 'Undefined', 'Information' : '', }, 'OverlapMode' : 'GivenMaterial', # Possible values: OverlapMaterial, NewMaterial, OldMaterial, GivenMaterial 'OverlapMaterialID' : 3, 'NumOverlapRules' : 0, 'HollowMaterialID' : 0, 'PostProcessing' : { 'ResolveOverlap' : False, 'MarkContactVoxels' : False, 'ContactMaterialID' : 15, }, }, }#This is the basic dictionary and now we fill it with the two Rovings. I Have it variable so when the polynomial script later returns more support lists, it should also work.for i in range(number_of_rovings): GadAdd_args[f'Object{i+1}:MaterialID'] = 1 GadAdd_args[f'Object{i+1}:Type'] = 'CurvedEllipticalFiber' GadAdd_args[f'Object{i+1}:NumberOfSegments'] = len(Roving_list[i])-1 GadAdd_args[f'Object{i+1}:FiberEndType1'] = 'Flat' GadAdd_args[f'Object{i+1}:FiberEndType2'] = 'Flat' GadAdd_args[f'Object{i+1}:Diameter1'] = diam_1 GadAdd_args[f'Object{i+1}:Diameter2'] = diam_2 GadAdd_args[f'Object{i+1}:Perpendicular'] = [-1,0,0] #we iterate over the Roving number i and fill the Dictionary with Object1, Object2. #The command GadAdd_args[f'Object{i+1}:Diameter1'] = diam_1 takes the dictionary GaddAdd_args at fills it at the entry Object i :Diameter1 with the value diam1 for j in range(len(Roving_list[i])): GadAdd_args[f'Object{i+1}:Point{j+1}'] = Roving_list[i][j] #each support point generates a new entry in the dictionary, here we fill the dictionary with the entry from the Roving_list. Where i means whether it is Roving x or y and j means the current support point. gd.runCmd("GadGeo:GadAdd", GadAdd_args, Header['Release'])
If you copy code block 1 into a scirpt called PolyNomial.py and code block 2 into another python script "Rovingcreation.py" and then exectue the Rovingcreation.py it should generate a small example.
Then you need to replace the test_func by your real polynomial function and it should still work. It is important that PolyNomial.py and Rovingcration.py are placed in the same folder.
Please let me know if you have any questions, because coding can always become a bit tricky.
8
##### Modelling of foams / Re: How to convert the channel structure to aerogel structure by GrainGeo?
« on: August 16, 2021, 05:54:51 PM »
You are welcome :)
By the way this is very intersting application and combination of the FoamGeo and GrainGeo modules.
Hopefully we are hearing about your work with GeoDict in one way or the other someday in the future.
9
##### Modelling of foams / Re: How to convert the channel structure to aerogel structure by GrainGeo?
« on: August 16, 2021, 10:20:09 AM »
Hi Evren,
sure we help you how to make such a Aerogel structure. You already are on the right track, you just need some more extra steps.
So when you have the channel structure you have to:
• Go to Model:ProcessGeo:Invert and invert the structure.
• Go to GrainGeo:CreateGrains and select
• Create in Current Domain and Keep Current Objects / Structure
• Select the option Without (Remove) Object Overlap and click Edit Here you set the Overlap SVP / (%) to 0.1
• Set the Add Object solid Volume Percentage to a bit less than the amount of Pore in the inverted structure. So for example the inverted structure has 12% porosity, than you choose 10%
• Finally you can set the diameter distributioin of your spheres as you need them and click on OK and Generate
• When it is finished you are left with the Aerogel and the remains of the inverted structure. Just Reassign this material ID to Pore (ID 00) and you have only the Aerogel left
That should give you the Aerogel. Please let me briefly know if it helped or if you have any questions left?
Have a nice day,
Aaron
10
##### Modelling of composites / Re: Modelling Rovings on the Basis of Support Points
« on: August 10, 2021, 05:04:00 PM »
Hi Marvin,
yes it is possible to create elliptical rovings with the support points automated by using the "Add GAD Object" option. You can use a python macro for that, the tricky question would be how to access the data form the excel sheet.
Can you tell me a bit more how the support points in the sheet are formatted?
Then I can explain you how you on that expample how to create a macro for that. So can you show a table here to show me how the data looks like? For example:
Rovings Roving 1 Coordinates Roving 1 Diameters Roving 2 Coordinates Roving 2 Diameters Support Point 1 X, Y, Z Diameter1, Diameter2, Diameter3 X, Y, Z Diameter1, Diameter2, Diameter3 Support Point 2 X, Y, Z Diameter1, Diameter2, Diameter3 X, Y, Z Diameter1, Diameter2, Diameter3
further support points
When I understand the table structure I can explain how to make the macro.
11
##### Modelling of composites / Re: Voids in a composite
« on: June 25, 2021, 11:11:33 AM »
This is some wht depending. How did you acquire that model?
Is the model a CT-Scan you imported to GeoDict or is it a digital model you generated inside GeoDict?
12
##### Modelling of foams / Re: Anisotropic foam
« on: June 25, 2021, 11:05:02 AM »
In the FoamGeo module in the "Basis Geometry" tab you can set the "Basis Pore-Geometry" option to "Use Current Analytic Geometry (GAD)". This option generates the foam on the basis of the curretnly loaded structure in GeoDict.
The loaded structure needs to be:
• A periodic structure
• An analytic strucutre, consisting only of GAD objects. Indicated by the green dot besides the "Objects"number in the top left corner
• Consists only of objects of sphere or ellipsoid type
So you need to generate an anisotropic ellipsoid structure on which you use FoamGeo on.
13
##### FiberGeo / Re: When do I need to use Isolation Distance in FiberGeo?
« on: June 25, 2021, 10:24:38 AM »
Hello Tabea,
This is a option where fibers are created such that a structure with non-touching fibers is made. In that option the fibers are placed in the structure and when they violate the given isolation distance to another fiber they are removed and a new fiber is placed and checked for violation. This happens until the stopping criterion is reached (for example a given Solid Volume Percentage or a given numbre of fibers).
But in general this approach can take much time for high solid volume fractions so in general we recommen to use the "Remove Overlap" option, here you can also set an Isolation distance.
14
##### DiffuDict / Re: Off-diagonal entries in matrix of Relative Diffusivity
« on: June 25, 2021, 10:06:15 AM »
Hello,
of course we can:
If you calculate the Diffusivity in x-direction, the entry in the main diagonal x_11 gives you the diffusivity in x-direction. The off-diagonal entries x_12, x_13, x_21 and x_31 give you the amount of diffusion going into y- and z-direction although the driving force, the concentration gradient, is only given in x -direction.
The same goes for Diffusivity in y- and z- direction.
Was that understandable?
15
##### for CT and µCT images / Re: Coordinate Transformation after Cropping
« on: June 16, 2021, 05:52:05 PM »
Hi Marvin,
yes such a transformation is possible with python and know how ProcessGeo Crop saves the settings to the gps file.
As an example say you have a 200x300x400 (X x Y x Z) structure with a voxel length of 2 µm. And you are looking at the X-Y Plane for the Coordination investigation.
Now you crop the structure in X direction for 20 voxel from the lower sider and 50 voxel from the higher side. That means the X direction is 70 voxels shorter (so it has 130 voxels in total).
Similar you crop the structure in Y direction for 30 voxel from the lower sider and 60 voxel from the higher side. That means the X direction is 90 voxels shorter (so it has 210 voxels in total).
This would mean that the X-Y Coordinates of your new Origin are at 40 µm x 60 µm of your old Origin.
The corresponding ProcessGeo Crop gps looks like:
Code: [Select]
<ProcessGeo> <Crop> XMinus 20 XPlus 50 YMinus 30 YPlus 60 ZMinus 0 ZPlus 0 </Crop> </ProcessGeo>So when you obtain coordiantes from the new system X' and Y' you can transform them back to the old system by:
$$X_{old} = X' - 20 * 2 µm$$ and $$Y_{old} = Y' - 30 * 2 µm$$
This you can automate by using GeoPython to access the Crop Seetings, here is a small example code:
Code: [Select]
import stringmapx_new = 1.04*1e-4y_new = 1.56*1e-4voxellength = 2*1e-6gpsMap = stringmap.parseGDR('PathToGPSFile/File.gps')x_crop = gpsMap.get('ProcessGeo:Crop:XMinus')y_crop = gpsMap.get('ProcessGeo:Crop:YMinus')x_trans = x_new + x_crop*voxellengthy_trans = y_new + y_crop*voxellengthIn this example your x and y coordiante values in the new system are 104 µm which is 1.04e-4 m and 156 µm which is 1.56e-4 m.
And this code extracts the cropped value in x and y direction and uses it to transform the coordinates into the original coordiante system. Those coordinates are saved in the variable x_trans and y_trans | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6709709167480469, "perplexity": 7487.931162145185}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585196.73/warc/CC-MAIN-20211018031901-20211018061901-00167.warc.gz"} |
https://mathematica.stackexchange.com/questions/146294/possible-bug-in-export-with-characterencoding-printableascii | # Possible bug in Export with CharacterEncoding -> “PrintableASCII”
In the Documentation we read (emphasis is mine):
By default, the Wolfram System uses the character encoding "PrintableASCII" when saving notebooks and packages. This means that when special characters are written out to files or external programs, they are represented purely as sequences of ordinary characters. This uniform representation is crucial in allowing special characters in the Wolfram Language to be used in a way that does not depend on the details of particular computer systems.
When creating packages and notebooks, special characters are always written out using full names.
ExportString["Lamé \[LongRightArrow] αβ+", "Package"]
"(* Created by Wolfram Mathematica 10.0 : www.wolfram.com *)
\"Lam\\[EAcute] \\[LongRightArrow] \\[Alpha]\\[Beta]+\"
"
In InputForm, all special characters are written out fully when using "PrintableASCII".
ToString["Lamé \[LongRightArrow] αβ+", InputForm,
CharacterEncoding -> "PrintableASCII"]
"\"Lam\\[EAcute] \\[LongRightArrow] \\[Alpha]\\[Beta]+\""
In the above-cited examples the character é is converted into the corresponding Mathematica's platform-independent representation \[EAcute]. This is expected since this character isn't PrintableASCIIQ:
PrintableASCIIQ@"é"
ToCharacterCode@"é"
ToCharacterCode["é", "ASCII"]
False
{233}
None
Hence I expect that Exporting this character with CharacterEncoding -> "PrintableASCII" will also give me \[EAcute]:
Export["test1.txt", "é", "Text",
CharacterEncoding -> "PrintableASCII"] // SystemOpen
Export["test2.txt", "é", "String",
CharacterEncoding -> "PrintableASCII"] // SystemOpen
The first input produces a file containing e', the second – a file with é.
Is this behavior correct? How can I export strings using only "PrintableASCII" and writing all the special characters in their FullForm as it happens when I Export as "Package"?
### Further considerations
Wrapping the string by InputForm doesn't work:
Export["test3.txt", InputForm@"é", "String",
CharacterEncoding -> "PrintableASCII"] // SystemOpen
produces a file containing "é" literally. It is worth to note that for ExportString wrapping by InputForm works:
ExportString[InputForm@"é", "String", CharacterEncoding -> "PrintableASCII"]
"\"\\[EAcute]\""
Wrapping by FullForm works but is unacceptable due to the quotation marks added:
Export["test4.txt", FullForm@"é", "String",
CharacterEncoding -> "PrintableASCII"] // SystemOpen
produces a file containing "\[EAcute]" literally.
Exporting as "Package" doesn't serve as a workaround because of the header added and also because strings are exported with the quotation marks.
The same happens with CharacterEncoding -> None.
I'm afraid support didn't answer your question correctly. However, "String" is one of the most confusing formats we have, so I am not totally suprised.
If you look at the "Background and Context" on ref/format/String, you'll see
• Arbitrary binary data represented as a Wolfram Language string.
• Used for importing or exporting entire raw binary data.
Note in particular that it is used for binary data--binary data in general does not use character encodings. The intended use case is something along the lines of Import["foo.png", "String"], when you wish to operate directly on the bytes of the PNG rather than the image it contains. In the import case, each byte gets mapped to the corresponding code point in the range 0-255. In the export case, we have to do something if a special character > 255 appears, so it is converted to its long name. You can think of this as akin to ToString[expr, CharacterEncoding->"ISO8859-1"]. In short, "String" is very different from "Text", which attempts to interpret textual data.
In a world where we have ByteArray, we problably wouldn't have had the "String" format, as the ByteArray would be the natural data structure. We don't yet have this format, but it is certainly on our roadmap. For now, you can Export lists a ByteArray using the "Byte" format, and import "Byte" format and then pass it to ByteArray to convert it from a lists of bytes to a proper ByteArray object.
I received a response from the official tech support on this issue ([CASE:3897096]):
As I understand, you wish to Export characters using the ASCII or PrintableASCII character sets, but Export is not giving the correct output for the character "é".
There seems to be a disregard for certain characters when Exporting with ASCII as the CharacterEncoding, regardless of their character codes. However, as a workaround, I have modified an example found in the CharacterEncoding documentation to get the desired output. Please see attached notebook.
The code shows some characters that do not have the desired output and uses the self-defined function fullname[] in order to achieve the requested Export. Please note that this workaround will work for those characters that do have a full name.
The excerpt from the attached notebook:
Helper function in use:
fullname[c_String] :=
StringReplace[ToString[c, InputForm, CharacterEncoding -> "ASCII"],
"\"\\[" ~~ a__ ~~ "]\"" :> "\\[" ~~ a ~~ "]"]
Comparison:
Export["test2.txt", {"\[CirclePlus]", "\[Wolf]", "\[Del]", "\[RuleDelayed]",
"\[CloverLeaf]", "\[Mu]", "\[PartialD]"}, "String",
CharacterEncoding -> "PrintableASCII"] // SystemOpen
Export["test2.txt",
fullname /@ {"\[CirclePlus]", "\[Wolf]", "\[Del]", "\[RuleDelayed]",
"\[CloverLeaf]", "\[Mu]", "\[PartialD]"}, "String",
CharacterEncoding -> "PrintableASCII"] // SystemOpen
The proposed workaround is (by the essence) the same as the one proposed in this answer with only difference that it won't handle correctly the characters without a full name. So that solution is better :).
Also the support guy points out that the set of "disregarded" characters is wider than I originally supposed: apart from extended ASCII characters with codes 128-255 it contains also some named Unicode characters like \[CirclePlus]. Although in the response it isn't stated clearly that this behavior is a bug, I assume that usage of words "workaround" and "disregard" in this context implies that current behavior is wrong. So I'll tag this appropriately.
• About the "extra" disregarded characters. That's because "String" is using OutputForm, and therefore some characters are using ascii-approximations, as explained in the tutorial you linked to in your question. So \[CirclePlus] gets converted into (+). This may not be the most useful behavior, but again, this is not the intended use case, so I won't lose any sleep over it. – Itai Seggev Jul 9 '17 at 23:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.2937783896923065, "perplexity": 3643.731659934919}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232262369.94/warc/CC-MAIN-20190527105804-20190527131804-00484.warc.gz"} |
https://enacademic.com/dic.nsf/enwiki/26564/Thermodynamic_temperature | # Thermodynamic temperature
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an “absolute” scale because it is the measure of the fundamental property underlying temperature: its "null" or zero point, absolute zero, is the temperature at which the particle constituents of matter have minimal motion and can be no colder.
## Overview
[
Fig. 1 The "translational motion" of fundamental particles of nature such as atoms and molecules gives a substance its temperature. Here, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. These room-temperature atoms have a certain, average speed (slowed down here two trillion fold). At any given instant however, a particular helium atom may be moving much faster than average while another may be nearly motionless. Five atoms are colored red to facilitate following their motions.] Temperature arises from the random submicroscopic vibrations of the particle constituents of matter. These motions comprise the kinetic energy in a substance. More specifically, the thermodynamic temperature of any bulk quantity of matter is the measure of the average kinetic energy of a certain kind of vibrational motion of its constituent particles called "translational motions." Translational motions are ordinary, whole-body movements in three-dimensional space whereby particles move about and exchange energy in collisions. "Fig. 1 "at right shows translational motion in gases; " "below shows translational motion in solids. Thermodynamic temperature’s null point, absolute zero, is the temperature at which the particle constituents of matter are as close as possible to complete rest; that is, they have motion, retaining only quantum mechanical motion. While scientists are achieving temperatures ever closer to absolute zero, they can not fully achieve a state of "“zero”" temperature. However, even if scientists could remove "all" kinetic heat energy from matter, quantum mechanical "zero-point energy" (ZPE) causes particle motion that can never be eliminated. Encyclopedia Britannica Online [http://britannica.com/eb/article-9078341 defines zero-point] energy as the “"vibrational energy that molecules retain even at the absolute zero of temperature."” ZPE is the result of all-pervasive energy fields in the vacuum between the fundamental particles of nature; it is responsible for the Casimir effect and other phenomena. See " [http://calphysics.org/zpe.html Zero Point Energy and Zero Point Field] ", which is an excellent explanation of ZPE by Calphysics Institute. See also " [http://www.phys.ualberta.ca/~therman/lowtemp/projects1.htm Solid Helium] " by the University of Alberta’s Department of Physics to learn more about ZPE’s effect on Bose–Einstein condensates of helium.
Although absolute zero ("T"=0) is not a state of zero molecular motion, it "is "the point of zero temperature and, in accordance with the Boltzmann constant, is also the point of zero particle kinetic energy and zero kinetic velocity. To understand how atoms can have zero kinetic velocity and simultaneously be vibrating due to ZPE, consider the following thought experiment: two "T"=0 helium atoms in zero gravity are carefully positioned and observed to have an average separation of 620 pm between them (a gap of ten atomic diameters). It’s an “average” separation because ZPE causes them to jostle about their fixed positions. Then one atom is given a kinetic kick of precisely 83 yoctokelvin (1 yK = 1 × 10–24 K). This is done in a way that directs this atom’s velocity vector at the other atom. With 83 yK of kinetic energy between them, the 620-pm gap through their common barycenter would close at a rate of 719 pm/s and they would collide after 0.862 second. This is the same speed as shown in the "Fig. 1 "animation above. Before being given the kinetic kick, both "T"=0 atoms had zero kinetic energy and zero kinetic velocity because they could persist indefinitely in that state and relative orientation even though both were being jostled by ZPE. At "T"=0, no kinetic energy is available for transfer to other systems. The Boltzmann constant and its related formulas describe the realm of particle kinetics and velocity vectors whereas ZPE is an energy field that jostles particles in ways described by the mathematics of quantum mechanics. In atomic and molecular collisions in gases, ZPE introduces a degree of "chaos", i.e., unpredictability, to rebound kinetics; it is as likely that there will be "less" ZPE-induced particle motion after a given collision as "more." This random nature of ZPE is why it has no net effect upon either the pressure or volume of any "bulk quantity" (a statistically significant quantity of particles) of "T">0 K gases. However, in "T"=0 condensed matter; e.g., solids and liquids, ZPE causes inter-atomic jostling where atoms would otherwise be perfectly stationary. Inasmuch as the real-world effects that ZPE has on substances can vary as one alters a thermodynamic system (for example, due to ZPE, helium won’t freeze unless under a pressure of at least 25 bar), ZPE is very much a form of heat energy and may properly be included when tallying a substance’s internal energy.
Note too that absolute zero serves as the baseline atop which thermodynamics and its equations are founded because they deal with the exchange of heat energy between "“systems”" (a plurality of particles and fields modeled as an average). Accordingly, one may examine ZPE-induced particle motion "within" a system that is at absolute zero but there can never be a net outflow of heat energy from such a system. Also, the peak emittance wavelength of black-body radiation shifts to infinity at absolute zero; indeed, a peak no longer exists and black-body photons can no longer escape. Due to the influence of ZPE however, "virtual" photons are still emitted at "T"=0. Such photons are called “virtual” because they can’t be intercepted and observed. Furthermore, this "zero-point radiation" has a unique "zero-point spectrum." However, even though a "T"=0 system emits zero-point radiation, no net heat flow "Q" out of such a system can occur because if the surrounding environment is at a temperature greater than "T"=0, heat will flow inward, and if the surrounding environment is at "T"=0, there will be an equal flux of ZP radiation both inward and outward. A similar "Q "equilibrium exists at "T"=0 with the ZPE-induced “spontaneous” emission of photons (which is more properly called a "stimulated" emission in this context). The graph at upper right illustrates the relationship of absolute zero to zero-point energy. The graph also helps in the understanding of how zero-point energy got its name: it is the vibrational energy matter retains at the "“zero kelvin point.”" Citation: "Derivation of the classical electromagnetic zero-point radiation spectrum via a classical thermodynamic operation involving van der Waals forces", Daniel C. Cole, Physical Review A, Third Series 42, Number 4, 15 August 1990, Pg. 1847–1862.] Zero kinetic energy remains in a substance at absolute zero (see "Heat energy at absolute zero", below).
Throughout the scientific world where measurements are made in SI units, thermodynamic temperature is measured in kelvins (symbol: K). Many engineering fields in the U.S. however, measure thermodynamic temperature using the Rankine scale.
By [http://www1.bipm.org/en/si/si_brochure/chapter2/2-1/2-1-1/kelvin.html international agreement,] the unit “kelvin” and its scale are defined by two points: absolute zero, and the triple point of Vienna Standard Mean Ocean Water (water with a specified blend of hydrogen and oxygen isotopes). Absolute zero—the coldest possible temperature—is defined as being precisely 0 K "and" −273.15 °C. The triple point of water is defined as being precisely 273.16 K "and" 0.01 °C. This definition does three things:
#It fixes the magnitude of the kelvin unit as being precisely 1 part in 273.16 parts the difference between absolute zero and the triple point of water;
#It establishes that one kelvin has precisely the same magnitude as a one-degree increment on the Celsius scale; and
#It establishes the difference between the two scales’ null points as being precisely 273.15 kelvins (0 K = −273.15 °C and 273.16 K = 0.01 °C).
Temperatures expressed in kelvins are converted to degrees Rankine simply by multiplying by 1.8 as follows: "T"K × 1.8 = "T"°R, where "T"K and "T"°R are temperatures in kelvins and degrees Rankine respectively. Temperatures expressed in Rankine are converted to kelvins by "dividing" by 1.8 as follows: "T"°R ÷ 1.8 = "T"K.
## Table of thermodynamic temperatures
The full range of the thermodynamic temperature scale and some notable points along it are shown in the table below.
A For Vienna Standard Mean Ocean Water at one standard atmosphere (101.325 kPa) when calibrated strictly per the two-point definition of thermodynamic temperature.
B The 2500 K value is approximate. The 273.15 K difference between K and °C is rounded to 300 K to avoid false precision in the Celsius value.
C For a true blackbody (which tungsten filaments are not). Tungsten filaments’ emissivity is greater at shorter wavelengths, which makes them appear whiter.
D Effective photosphere temperature. The 273.15 K difference between K and °C is rounded to 273 K to avoid false precision in the Celsius value.
E The 273.15 K difference between K and °C is ignored to avoid false precision in the Celsius value.
F For a true blackbody (which the plasma was not). The Z machine’s dominant emission originated from 40 MK electrons (soft x–ray emissions) within the plasma.
The relationship of temperature, motions, conduction, and heat energy
The nature of kinetic energy, translational motion, and temperature
At its simplest, “temperature” arises from the kinetic energy of the vibrational motions of matter’s particle constituents (molecules, atoms, and subatomic particles). The full variety of these kinetic motions contribute to the total heat energy in a substance. The relationship of kinetic energy, mass, and velocity is given by the formula "Ek" = frac|2"m" • "v" 2. [At non-relativistic temperatures of less than about 30 GK, classical mechanics are sufficient to calculate the velocity of particles. At 30 GK, individual neutrons (the constituent of neutron stars and one of the few materials in the universe with temperatures in this range) have a 1.0042 γ (gamma or Lorentz factor). Thus, the classic Newtonian formula for kinetic energy is in error less than half a percent for temperatures less than 30 GK.] Accordingly, particles with one unit of mass moving at one unit of velocity have precisely the same kinetic energy—and precisely the same temperature—as those with four times the mass but half the velocity.
.
The extent to which the kinetic energy of translational motion of an individual atom or molecule (particle) in a gas contributes to the pressure and volume of that gas is a proportional function of thermodynamic temperature as established by the Boltzmann constant (symbol: "kB"). The Boltzmann constant also relates the thermodynamic temperature of a gas to the mean kinetic energy of an individual particle’s translational motion as follows:
:"Emean" = frac|3|2"kBT" ::where…::"Emean" is the mean kinetic energy in joules (symbol: J)::"kB" = val|1.3806504|(24)|e=-23|u=J/K::"T" is the thermodynamic temperature in kelvins
While the Boltzmann constant is useful for finding the mean kinetic energy of a particle, it’s important to note that even when a substance is isolated and in thermodynamic equilibrium (all parts are at a uniform temperature and no heat is going into or out of it), the translational motions of individual atoms and molecules occurs across a wide range of speeds (see animation in "Fig. 1 "above). At any one instant, the proportion of particles moving at a given speed within this range is determined by probability as described by the Maxwell–Boltzmann distribution. The graph shown here in "Fig. 2 " shows the speed distribution of 5500 K helium atoms. They have a "most probable" speed of 4.780 km/s (0.2092 s/km). However, a certain proportion of atoms at any given instant are moving faster while others are moving relatively slowly; some are momentarily at a virtual standstill (off the "x"–axis to the right). This graph uses "inverse speed" for its "x"–axis so the shape of the curve can easily be compared to the curves in "" below. In both graphs, zero on the "x"–axis represents infinite temperature. Additionally, the "x" and "y"–axis on both graphs are scaled proportionally.
The high speeds of translational motion
Although very specialized laboratory equipment is required to directly detect translational motions, the resultant collisions by atoms or molecules with small particles suspended in a fluid produces Brownian motion that can be seen with an ordinary microscope. The translational motions of elementary particles are "very" fast [Even room–temperature air has an average molecular translational "speed" (not vector-isolated velocity) of 1822 km/hour. This is relatively fast for something the size of a molecule considering there are roughly 2.42 × 1016 of them crowded into a single cubic millimeter. Assumptions: Average molecular weight of wet air = 28.838 g/mol and "T" = 296.15 K. Assumption’s primary variables: An altitude of 194 meters above mean sea level (the world–wide median altitude of human habitation), an indoor temperature of 23 °C, a dewpoint of 9 °C (40.85% relative humidity), and 760 mmHg (101.325 kPa) sea level–corrected barometric pressure.] and temperatures close to absolute zero are required to directly observe them. For instance, when scientists at the NIST achieved a record-setting cold temperature of 700 nK (billionths of a kelvin) in 1994, they used optical lattice laser equipment to adiabatically cool caesium atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7 mm per second to in order to calculate their temperature. [Citation: "Adiabatic Cooling of Cesium to 700 nK in an Optical Lattice", A. Kastberg "et al"., Physical Review Letters 74, No. 9, 27 Feb. 1995, Pg. 1542. It’s noteworthy that a record cold temperature of 450 pK in a Bose–Einstein condensate of sodium atoms (achieved by A. E. Leanhardt "et al". of MIT) equates to an average vector-isolated atom velocity of 0.4 mm/s and an average atom speed of 0.7 mm/s.] Formulas for calculating the velocity and speed of translational motion are given in the following footnote.The rate of translational motion of atoms and molecules is calculated based on thermodynamic temperature as follows:
:$ilde\left\{v\right\} = sqrt\left\{frac$k_Bover 2} cdot T}mover 2}::where…::$ilde\left\{v\right\}$ is the vector-isolated mean velocity of translational particle motion in m/s::"kB" (Boltzmann constant) = 1.380 6504(24) × 10−23 J/K::"T" is the thermodynamic temperature in kelvins::"m" is the molecular mass of substance in kg/particle
In the above formula, molecular mass, "m", in kg/particle is the quotient of a substance’s molar mass (also known as "atomic weight", "atomic mass", "relative atomic mass", and "unified atomic mass units") in g/mol or daltons divided by 6.022 141 79(30) × 1026 (which is the Avogadro constant times one thousand). For diatomic molecules such as H2, N2, and O2, multiply atomic weight by two before plugging it into the above formula.
The mean "speed" (not vector-isolated velocity) of an atom or molecule along any arbitrary path is calculated as follows:
:$ilde\left\{s\right\} = ilde\left\{v\right\} cdot sqrt\left\{3\right\}$::where…::$ilde\left\{s\right\}$ is the mean speed of translational particle motion in m/s
Note that the mean energy of the translational motions of a substance’s constituent particles correlates to their mean "speed", not velocity. Thus, substituting $ilde\left\{s\right\}$ for "v" in the classic formula for kinetic energy, "Ek" = frac|2"m" • "v" 2 produces precisely the same value as does "Emean" = 3/2"kBT" (as shown in the section titled "The nature of kinetic energy, translational motion, and temperature)".
Note too that the Boltzmann constant and its related formulas establish that absolute zero is the point of both zero kinetic energy of particle motion and zero kinetic velocity (see also "Note 1" above).]
The internal motions of molecules and specific heat
[
Fig. 3 Molecules have internal structure because they are composed of atoms that have different ways of moving within molecules. Being able to store kinetic energy in these "internal degrees of freedom" contributes to a substance’s "specific heat capacity", allowing it to contain more heat energy at the same temperature.] There are other forms of heat energy besides the kinetic energy of translational motion. As can be seen in the animation at right, molecules are complex objects; they are a population of atoms and thermal agitation can strain their internal chemical bonds in three different ways: via rotation, bond length, and bond angle movements. These are all types of "internal degrees of freedom". This makes molecules distinct from "monatomic" substances (consisting of individual atoms) like the noble gases helium and argon, which have only the three translational degrees of freedom. Kinetic energy is stored in molecules’ internal degrees of freedom, which gives them an "internal temperature." Even though these motions are called “internal,” the external portions of molecules still move—rather like the jiggling of a stationary water balloon. This permits the two-way exchange of kinetic energy between internal motions and translational motions with each molecular collision. Accordingly, as heat is removed from molecules, both their kinetic temperature (the kinetic energy of translational motion) and their internal temperature simultaneously diminish in equal proportions. This phenomenon is described by the equipartition theorem, which states that for any bulk quantity of a substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active degrees of freedom available to the particles. Since the internal temperature of molecules are usually equal to their kinetic temperature, the distinction is usually of interest only in the detailed study of non-local thermodynamic equilibrium (LTE) phenomena such as combustion, the sublimation of solids, and the diffusion of hot gases in a partial vacuum.
The kinetic energy stored internally in molecules allows a substance to contain more heat energy at a given temperature (and in the case of gases, at a given pressure and volume), and to absorb more of it for a given temperature increase. This is because any kinetic energy that is, at a given instant, bound in internal motions is not at that same instant contributing to the molecules’ translational motions. [The internal degrees of freedom of molecules cause their external surfaces to vibrate and can also produce overall spinning motions (what can be likened to the jiggling and spinning of an otherwise stationary water balloon). If one examines a "single" molecule as it impacts a containers’ wall, some of the kinetic energy borne in the molecule’s internal degrees of freedom can constructively add to its translational motion during the instant of the collision and extra kinetic energy will be transferred into the container’s wall. This would induce an extra, localized, impulse-like contribution to the average pressure on the container. However, since the internal motions of molecules are random, they have an equal probability of "destructively" interfering with translational motion during a collision with a container’s walls or another molecule. Averaged across any bulk quantity of a gas, the internal thermal motions of molecules have zero net effect upon the temperature, pressure, or volume of a gas. Molecules’ internal degrees of freedom simply provide additional locations where kinetic energy is stored. This is precisely why molecular-based gases have greater specific heat capacity than monatomic gases (where additional heat energy must be added to achieve a given temperature rise).] This extra kinetic energy simply increases the amount of heat energy a substance absorbs for a given temperature rise. This property is known as a substance’s specific heat capacity.
Different molecules absorb different amounts of heat energy for each incremental increase in temperature; that is, they have different specific heat capacities. High specific heat capacity arises, in part, because certain substances’ molecules possess more internal degrees of freedom than others do. For instance, room-temperature nitrogen, which is a diatomic molecule, has "five" active degrees of freedom: the three comprising translational motion plus two rotational degrees of freedom internally. Not surprisingly, in accordance with the equipartition theorem, nitrogen has five-thirds the specific heat capacity per mole (a specific number of molecules) as do the monatomic gases. [When measured at constant-volume since different amounts of work must be performed if measured at constant-pressure. Nitrogen’s "CvH" (100 kPa, 20 °C) equals 20.8 J mol–1 K–1 vs. the monatomic gases, which equal 12.4717 J mol–1 K–1. Citations: [http://www.whfreeman.com/ W.H. Freeman’s] "Physical Chemistry", Part 3: Change ( [http://www.whfreeman.com/college/pdfs/pchem8e/PC8eC21.pdf 422 kB PDF, here] ), Exercise 21.20b, Pg. 787. Also [http://www.gsu.edu/ Georgia State University’s] " [http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/shegas.html Molar Specific Heats of Gases] ".] Another example is gasoline (see table showing its specific heat capacity). Gasoline can absorb a large amount of heat energy per mole with only a modest temperature change because each molecule comprises an average of 21 atoms and therefore has many internal degrees of freedom. Even larger, more complex molecules can have dozens of internal degrees of freedom.
The diffusion of heat energy: Entropy, phonons, and mobile conduction electrons
[
Fig. 4 The temperature-induced translational motion of particles in solids takes the form of "phonons. "Shown here are phonons with identical amplitudes but with wavelengths ranging from 2 to 12 molecules.] "Heat conduction "is the diffusion of heat energy from hot parts of a system to cold. A “system” can be either a single bulk entity or a plurality of discrete bulk entities. The term “bulk” in this context means a statistically significant quantity of particles (which can be a microscopic amount). Whenever heat energy diffuses within an isolated system, temperature differences within the system decrease (and entropy increases).
One particular heat conduction mechanism occurs when translational motion—the particle motion underlying temperature—transfers momentum from particle to particle in collisions. In gases, these translational motions are of the nature shown above in "Fig. 1. "As can be seen in that animation, not only does momentum (heat) diffuse throughout the volume of the gas through serial collisions, but entire molecules or atoms can advance forward into new territory, bringing their kinetic energy with them. Consequently, temperature differences equalize throughout gases very quickly—especially for light atoms or molecules; convection speeds this process even more. [The "speed" at which thermal energy equalizes throughout the volume of a gas is very rapid. However, since gases have extremely low density relative to solids, the "heat flux"—the thermal power conducting through a unit area—through gases is comparatively low. This is why the dead-air spaces in multi-pane windows have insulating qualities.]
Translational motion in "solids "however, takes the form of "phonons "(see "Fig. 4" at right). Phonons are constrained, quantized wave packets traveling at the speed of sound for a given substance. The manner in which phonons interact within a solid determines a variety of its properties, including its thermal conductivity. In electrically insulating solids, phonon-based heat conduction is "usually" inefficient [Diamond is a notable exception. Due to the highly quantized modes of phonon vibration occurring in its rigid crystal lattice, not only does diamond have exceptionally "poor" specific heat capacity, it also has exceptionally "high" thermal conductivity.] and such solids are considered "thermal insulators" (such as glass, plastic, rubber, ceramic, and rock). This is because in solids, atoms and molecules are locked into place relative to their neighbors and are not free to roam.
Metals however, are not restricted to only phonon-based heat conduction. Heat energy conducts through metals extraordinarily quickly because instead of direct molecule-to-molecule collisions, the vast majority of heat energy is mediated via very light, mobile "conduction electrons." This is why there is a near-perfect correlation between metals’ thermal conductivity and their electrical conductivity. [Correlation is 752 (W m−1 K−1) / (MS•cm), σ = 81, through a 7:1 range in conductivity. Value and standard deviation based on data for Ag, Cu, Au, Al, Ca, Be, Mg, Rh, Ir, Zn, Co, Ni, Os, Fe, Pa, Pt, and Sn. Citation: Data from "CRC Handbook of Chemistry and Physics", 1st Student Edition and [http://www.webelements.com/ this link] to Web Elements’ home page.] Conduction electrons imbue metals with their extraordinary conductivity because they are "delocalized," i.e. not tied to a specific atom, and behave rather like a sort of “quantum gas” due to the effects of "zero-point energy" (for more on ZPE, see "Note 1" below). Furthermore, electrons are relatively light with a rest mass only frac|1836th that of a proton. This is about the same ratio as a .22 Short bullet (29 grains or 1.88 g) compared to the rifle that shoots it. As Isaac Newton wrote with his ,
:"“Law #3: All forces occur in pairs, and these two forces":" are equal in magnitude and opposite in direction.”"
However, a bullet accelerates faster than a rifle given an equal force. Since kinetic energy increases as the square of velocity, nearly all the kinetic energy goes into the bullet, not the rifle, even though both experience the same force from the expanding propellant gases. In the same manner—because they are much less massive—heat energy is readily borne by mobile conduction electrons. Additionally, because they’re delocalized and "very" fast, kinetic heat energy conducts extremely quickly through metals with abundant conduction electrons.
The diffusion of heat energy: Black-body radiation
", above).
Black-body radiation diffuses heat energy throughout a substance as the photons are absorbed by neighboring atoms, transferring momentum in the process. Black-body photons also easily escape from a substance and can be absorbed by the ambient environment; kinetic energy is lost in the process.
As established by the Stefan–Boltzmann law, the intensity of black-body radiation increases as the fourth power of absolute temperature. Thus, a black body at 824 K (just short of glowing dull red) emits "60 times" the radiant power as it does at 296 K (room temperature). This is why one can so easily feel the radiant heat from hot objects at a distance. At higher temperatures, such as those found in an incandescent lamp, black-body radiation can be the principal mechanism by which heat energy escapes a system.
The heat of phase changes
The kinetic energy of particle motion is just one contributor to the total heat energy in a substance; another is "phase transitions", which are the potential energy of molecular bonds that can form in a substance as it cools (such as during condensing and freezing). The heat energy required for a phase transition is called "latent heat." This phenomenon may more easily be grasped by considering it in the reverse direction: latent heat is the energy required to "break" chemical bonds (such as during evaporation and melting). Most everyone is familiar with the effects of phase transitions; for instance, steam at 100 °C can cause severe burns much faster than the 100 °C air from a hair dryer. This occurs because a large amount of latent heat is liberated as steam condenses into liquid water on the skin.
Even though heat energy is liberated or absorbed during phase transitions, pure chemical elements, compounds, and eutectic
At one specific thermodynamic point, the melting point (which is 0 °C across a wide pressure range in the case of water), all the atoms or molecules are—on average—at the maximum energy threshold their chemical bonds can withstand without breaking away from the lattice. Chemical bonds are quantized forces: they either hold fast, or break; there is no in-between state. Consequently, when a substance is at its melting point, every joule of added heat energy only breaks the bonds of a specific quantity of its atoms or molecules, [Water’s enthalpy of fusion (0 °C, 101.325 kPa) equates to 0.062284 eV per molecule so adding one joule of heat energy to 0 °C water ice causes 1.0021 × 1020 water molecules to break away from the crystal lattice and become liquid.] converting them into a liquid of precisely the same temperature; no kinetic energy is added to translational motion (which is what gives substances their temperature). The effect is rather like popcorn: at a certain temperature, additional heat energy can’t make the kernels any hotter until the transition (popping) is complete. If the process is reversed (as in the freezing of a liquid), heat energy must be removed from a substance.
As stated above, the heat energy required for a phase transition is called "latent heat." In the specific cases of melting and freezing, it’s called "enthalpy of fusion" or "heat of fusion." If the molecular bonds in a crystal lattice are strong, the heat of fusion can be relatively great, typically in the range of 6 to 30 kJ per mole for water and most of the metallic elements. [Water’s enthalpy of fusion is 6.0095 kJ mol−1 K−1 (0 °C, 101.325 kPa). Citation: "Water Structure and Science, Water Properties, Enthalpy of fusion, (0 °C, 101.325 kPa)" (by London South Bank University). [http://www.lsbu.ac.uk/water/data.html Link to Web site.] The only metals with enthalpies of fusion "not" in the range of 6–30 J mol−1 K−1 are (on the high side): Ta, W, and Re; and (on the low side) most of the group 1 (alkaline) metals plus Ga, In, Hg, Tl, Pb, and Np. Citation: [http://www.webelements.com/ This link] to Web Elements’ home page.] If the substance is one of the monatomic gases, (which have little tendency to form molecular bonds) the heat of fusion is more modest, ranging from 0.021 to 2.3 kJ per mole. [Xenon value citation: [http://www.webelements.com/webelements/elements/text/Xe/heat.html This link] to WebElements’ xenon data (available values range from 2.3 to 3.1 kJ mol−1). It is also noteworthy that helium’s heat of fusion of only 0.021 kJ mol−1 is so weak of a bonding force that zero-point energy prevents helium from freezing unless it is under a pressure of at least 25 atmospheres.] Relatively speaking, phase transitions can be truly energetic events. To completely melt ice at 0 °C into water at 0 °C, one must add roughly 80 times the heat energy as is required to increase the temperature of the same mass of liquid water by one degree Celsius. The metals’ ratios are even greater, typically in the range of 400 to 1200 times. [Citation: Data from "CRC Handbook of Chemistry and Physics", 1st Student Edition and [http://www.webelements.com/ this link] to Web Elements’ home page.] And the phase transition of boiling is much more energetic than freezing. For instance, the energy required to completely boil or vaporize water (what is known as "enthalpy of vaporization") is roughly "540 times" that required for a one-degree increase. [H2O specific heat capacity, "Cp" = 0.075327 kJ mol−1 K−1 (25 °C); Enthalpy of fusion = 6.0095 kJ mol−1 (0 °C, 101.325 kPa); Enthalpy of vaporization (liquid) = 40.657 kJ mol−1 (100 °C). Citation: "Water Structure and Science, Water Properties" (by London South Bank University). [http://www.lsbu.ac.uk/water/data.html Link to Web site.] ]
Water’s sizable enthalpy of vaporization is why one’s skin can be burned so quickly as steam condenses on it (heading from red to green in "Fig. 7 "above). In the opposite direction, this is why one’s skin feels cool as liquid water on it evaporates (a process that occurs at a sub-ambient wet-bulb temperature that is dependent on relative humidity). Water’s highly energetic enthalpy of vaporization is also an important factor underlying why “solar pool covers” (floating, insulated blankets that cover swimming pools when not in use) are so effective at reducing heating costs: they prevent evaporation. For instance, the evaporation of just 20 mm of water from a 1.29-meter-deep pool chills its water 8.4 degrees Celsius.
Internal energy
The total kinetic energy of all particle motion—including that of conduction electrons—plus the potential energy of phase changes, plus zero-point energy comprise the "internal energy" of a substance, which is its total heat energy. The term "internal energy" mustn’t be confused with "internal degrees of freedom." Whereas the "internal degrees of freedom of molecules" refers to one particular place where kinetic energy is bound, the "internal energy of a substance" comprises all forms of heat energy.
Heat energy at absolute zero
As a substance cools, different forms of heat energy and their related effects simultaneously decrease in magnitude: the latent heat of available phase transitions are liberated as a substance changes from a less ordered state to a more ordered state; the translational motions of atoms and molecules diminish (their kinetic temperature decreases); the internal motions of molecules diminish (their internal temperature decreases); conduction electrons (if the substance is an electrical conductor) travel "somewhat" slower; [ Mobile conduction electrons are "delocalized," i.e. not tied to a specific atom, and behave rather like a sort of “quantum gas” due to the effects of zero-point energy. Consequently, even at absolute zero, conduction electrons still move between atoms at the "Fermi velocity" of about 1.6 × 106 m/s. Kinetic heat energy adds to this speed and also causes delocalized electrons to travel farther away from the nuclei.] and black-body radiation’s peak emittance wavelength increases (the photons’ energy decreases). When the particles of a substance are as close as possible to complete rest and retain only ZPE-induced quantum mechanical motion, the substance is at the temperature of absolute zero ("T"=0).
Note that whereas absolute zero is the point of zero thermodynamic temperature and is also the point at which the particle constituents of matter have minimal motion, absolute zero is not necessarily the point at which a substance contains zero heat energy; one must be very precise with what one means by “heat energy.” Often, all the phase changes that "can" occur in a substance, "will" have occurred by the time it reaches absolute zero. However, this is not always the case. Notably, "T"=0 helium remains liquid at room pressure and must be under a pressure of at least 25 bar to crystallize. This is because helium’s heat of fusion—the energy required to melt helium ice—is so low (only 21 J mol−1) that the motion-inducing effect of zero-point energy is sufficient to prevent it from freezing at lower pressures. Only if under at least 25 bar of pressure will this latent heat energy be liberated as helium freezes while approaching absolute zero. A further complication is that many solids change their crystal structure to more compact arrangements at extremely high pressures (up to millions of bars). These are known as "solid-solid phase transitions" wherein latent heat is liberated as a crystal lattice changes to a more thermodynamically favorable, compact one.
The above complexities make for rather cumbersome blanket statements regarding the internal energy in "T"=0 substances. Regardless of pressure though, what "can" be said is that at absolute zero, all solids with a lowest-energy crystal lattice such those with a "closest-packed arrangement" (see "Fig. 8," above left) contain minimal internal energy, retaining only that due to the ever-present background of zero-point energy. [No other crystal structure can exceed the 74.048% packing density of a "closest-packed arrangement." The two regular crystal lattices found in nature that have this density are "hexagonal close packed" (HCP) and "face-centered cubic" (FCC). These regular lattices are at the lowest possible energy state. Diamond is a closest-packed structure with an FCC crystal lattice. Note too that suitable crystalline chemical "compounds", although usually composed of atoms of different sizes, can be considered as “closest-packed structures” when considered at the molecular level. One such compound is the common mineral known as "magnesium aluminum spinel" (MgAl2O4). It has a face-centered cubic crystal lattice and no change in pressure can produce a lattice with a lower energy state.] One can also say that for a given substance at constant pressure, absolute zero is the point of lowest "enthalpy" (a measure of work potential that takes internal energy, pressure, and volume into consideration). [Nearly half of the 92 naturally occurring chemical elements that can freeze under a vacuum also have a closest-packed crystal lattice. This set includes beryllium, osmium, neon, and iridium (but excludes helium), and therefore have zero latent heat of phase transitions to contribute to internal energy (symbol: "U)". In the calculation of enthalpy (formula: "H" = "U" + "pV)", internal energy may exclude different sources of heat energy—particularly ZPE—depending on the nature of the analysis. Accordingly, all "T"=0 closest-packed matter under a perfect vacuum has either minimal or zero enthalpy, depending on the nature of the analysis. Citation: "Use Of Legendre Transforms In Chemical Thermodynamics", Robert A. Alberty, Pure Appl.Chem., 73, No.8, 2001, 1349–1380 ( [http://iupac.org/publications/pac/2001/pdf/7308x1349.pdf 400 kB PDF, here] ).] Lastly, it is always true to say that all "T"=0 substances contain zero kinetic heat energy.
## Practical applications for thermodynamic temperature
Thermodynamic temperature is useful not only for scientists, it can also be useful for lay-people in many disciplines involving gases. By expressing variables in absolute terms and applying Gay–Lussac’s law of temperature/pressure proportionality, the solutions to familiar problems are straightforward. For instance, how is the pressure in an automobile tire affected by temperature? If the tire has a “cold” pressure of 200 kPa-gage , then in absolute terms—relative to a vacuum—its pressure is 300 kPa-absolute. [Pressure also must be in absolute terms. The air still in a tire at 0 kPa-gage expands too as it gets hotter. It’s not uncommon for engineers to overlook that one must work in terms of absolute pressure when compensating for temperature. For instance, a dominant manufacturer of aircraft tires published a document on temperature-compensating tire pressure, which used gage pressure in the formula. However, the high gage pressures involved (180 psi ≈ 12.4 bar) means the error would be quite small. With low-pressure automobile tires, where gage pressures are typically around 2 bar, failing to adjust to absolute pressure results in a significant error. Referenced document: "Aircraft Tire Ratings" ( [http://airmichelin.com/pdfs/05%20-%20Aircraft%20Tire%20Ratings.pdf 155 kB PDF, here] ).] [Regarding the spelling “gage” vs. “gauge” in the context of pressures measured relative to atmospheric pressure, the preferred spelling varies by country and even by industry. Further, both spellings are often used "within" a particular industry or country. Industries in British English-speaking countries typically use the spelling “gauge pressure” to distinguish it from the pressure-measuring instrument, which in the U.K., is spelled “pressure gage.” For the same reason, many of the largest American manufacturers of pressure transducers and instrumentation use the spelling “gage pressure”—the convention used here—in their formal documentation to distinguish it from the instrument, which is spelled “pressure gauge.” (see "Honeywell-Sensotec’s" [http://sensotec.com/pressurefaq.shtml FAQ page] and Fluke Corporation’s [http://us.fluke.com/usen/Home/Search.asp?txtSearchBox=%22gage+pressure%22&x=0&y=0 product search page] ).] [A difference of 100 kPa is used here instead of the 101.325 kPa value of one standard atmosphere. In 1982, the International Union of Pure and Applied Chemistry (IUPAC) recommended that for the purposes of specifying the physical properties of substances, “"the standard pressure"” (atmospheric pressure) should be defined as precisely 100kPa (≈750.062Torr). Besides being a round number, this had a very practical effect: relatively few people live and work at precisely sea level; 100kPa equates to the mean pressure at an altitude of about 112 meters, which is closer to the 194–meter, worldwide median altitude of human habitation. For especially low-pressure or high-accuracy work, true atmospheric pressure must be measured. Citation: IUPAC.org, Gold Book, " [http://goldbook.iupac.org/S05921.html Standard Pressure] "] Room temperature (“cold” in tire terms) is 296 K. What would the tire pressure be if was 20 °C hotter? The answer is frac|316 K|296 K = 6.8% greater thermodynamic temperature "and" absolute pressure; that is, a pressure of 320 kPa-absolute and 220 kPa-gage.
## The origin of heat energy on Earth
Earth’s proximity to the Sun is why most everything near Earth’s surface is warm with a temperature substantially above absolute zero. [The deepest ocean depths (3 to 10 km) are no colder than about 274.7 – 275.7 K (1.5 – 2.5 °C). Even the world-record cold surface temperature established on July 21, 1983 at Vostok Station, Antarctica is 184 K (a reported value of −89.2 °C). The residual heat of gravitational contraction left over from earth’s formation, tidal friction, and the decay of radioisotopes in earth’s core provide insufficient heat to maintain earth’s surface, oceans, and atmosphere “substantially above” absolute zero in this context. Also, the qualification of “most-everything” provides for the exclusion of lava flows, which derive their temperature from these deep-earth sources of heat.] Solar radiation constantly replenishes heat energy that Earth loses into space and a relatively stable state of equilibrium is achieved. Because of the wide variety of heat diffusion mechanisms (one of which is black-body radiation which occurs at the speed of light), objects on Earth rarely vary too far from the global mean surface and air temperature of 287 to 288 K (14 to 15 °C). The more an object’s or system’s temperature varies from this average, the more rapidly it tends to come back into equilibrium with the ambient environment.
## History of thermodynamic temperature
* 1702–1703: Guillaume Amontons (1663 – 1705) published two papers that may be used to credit him as being the first researcher to deduce the existence of a fundamental (thermodynamic) temperature scale featuring an absolute zero. He made the discovery while endeavoring to improve upon the air thermometers in use at the time. His J-tube thermometers comprised a mercury column that was supported by a fixed mass of air entrapped within the sensing portion of the thermometer. In thermodynamic terms, his thermometers relied upon the volume / temperature relationship of gas under constant pressure. His measurements of the boiling point of water and the melting point of ice showed that regardless of the mass of air trapped inside his thermometers or the weight of mercury the air was supporting, the reduction in air volume at the ice point was always the same ratio. This observation led him to posit that a sufficient reduction in temperature would reduce the air volume to zero. In fact, his calculations projected that absolute zero was equivalent to −240 degrees on today’s Celsius scale—only 33.15 degrees short of the true value of −273.15 °C.
* 1742: (1701 – 1744) created a “backwards” version of the modern Celsius temperature scale whereby zero represented the boiling point of water and 100 represented the melting point of ice. In his paper "Observations of two persistent degrees on a thermometer," he recounted his experiments showing that ice’s melting point was effectively unaffected by pressure. He also determined with remarkable precision how water’s boiling point varied as a function of atmospheric pressure. He proposed that zero on his temperature scale (water’s boiling point) would be calibrated at the mean barometric pressure at mean sea level.
* 1744: ) [http://www.bipm.org/en/committees/cipm/cipm-1948.html formally adopted] “degree Celsius” (symbol: °C) in 1948.According to "The Oxford English Dictionary" (OED), the term “Celsius’s thermometer” had been used at least as early as 1797. Further, the term “The Celsius or Centigrade thermometer” was again used in reference to a particular type of thermometer at least as early as 1850. The OED also cites this 1928 reporting of a temperature: “My altitude was about 5,800 metres, the temperature was 28° Celsius.” However, dictionaries seek to find the earliest use of a word or term and are not a useful resource as regards the terminology used throughout the history of science. According to several writings of Dr. Terry Quinn CBE FRS, Director of the BIPM (1988 – 2004), including "Temperature Scales from the early days of thermometry to the 21st century" ( [http://www.imeko.org/publications/tc12-2004/PTC12-2004-PL-001.pdf 148 kB PDF, here] ) as well as "Temperature" (2nd Edition / 1990 / Academic Press / 0125696817), the term "Celsius" in connection with the centigrade scale was not used whatsoever by the scientific or thermometry communities until after the CIPM and CGPM adopted the term in 1948. The BIPM wasn’t even aware that “degree Celsius” was in sporadic, non-scientific use before that time. It’s also noteworthy that the twelve-volume, 1933 edition of OED didn’t even have a listing for the word "Celsius" (but did have listings for both "centigrade" and "centesimal" in the context of temperature measurement). The 1948 adoption of "Celsius" accomplished three objectives::1) All common temperature scales would have their units named after someone closely associated with them; namely, Kelvin, Celsius, Fahrenheit, Réaumur and Rankine.
2) Notwithstanding the important contribution of Linnaeus who gave the Celsius scale its modern form, Celsius’s name was the obvious choice because it began with the letter C. Thus, the symbol °C that for centuries had been used in association with the name "centigrade" could continue to be used and would simultaneously inherit an intuitive association with the new name.
3) The new name eliminated the ambiguity of the term “centigrade,” freeing it to refer exclusively to the French-language name for the unit of angular measurement.]
* 1777: In his book "Pyrometrie" (Berlin: [http://www.spiess-verlage.de/html/haude___spener.html Haude & Spener,] 1779) completed four months before his death, Johann Heinrich Lambert (1728 – 1777)—sometimes incorrectly referred to as Joseph Lambert—proposed an absolute temperature scale based on the pressure / temperature relationship of a fixed volume of gas. This is distinct from the volume / temperature relationship of gas under constant pressure that Guillaume Amontons discovered 75 years earlier. Lambert stated that absolute zero was the point where a simple straight-line extrapolation reached zero gas pressure and was equal to −270 °C.
* Circa 1787: Notwithstanding the work of Guillaume Amontons 85 years earlier, Jacques Alexandre César Charles (1746 – 1823) is often credited with “discovering”, but not publishing, that the volume of a gas under constant pressure is proportional to its absolute temperature. The formula he created was "V"1/"T"1 = "V"2/"T"2.
* 1802: Joseph Louis Gay-Lussac (1778 – 1850) published work (acknowledging the unpublished lab notes of Jacques Charles fifteen years earlier) describing how the volume of gas under constant pressure changes linearly with its absolute (thermodynamic) temperature. This behavior is called Charles’s Law and is one of the gas laws. His are the first known formulas to use the number “273” for the expansion coefficient of gas relative to the melting point of ice (indicating that absolute zero was equivalent to −273 °C).
* 1848: , (1824 – 1907) also known as Lord Kelvin, wrote in his paper, " [http://zapatopi.net/kelvin/papers/on_an_absolute_thermometric_scale.html On an Absolute Thermometric Scale] ," of the need for a scale whereby “infinite cold” (absolute zero) was the scale’s null point, and which used the degree Celsius for its unit increment. Like Gay-Lussac, Thomson calculated that absolute zero was equivalent to −273 °C on the air thermometers of the time. This absolute scale is known today as the Kelvin thermodynamic temperature scale. It’s noteworthy that Thomson’s value of “−273” was actually derived from 0.00366, which was the accepted expansion coefficient of gas per degree Celsius relative to the ice point. The inverse of −0.00366 expressed to five significant digits is −273.22 °C which is remarkably close to the true value of −273.15 °C.
* 1859: William John Macquorn Rankine (1820 – 1872) proposed a thermodynamic temperature scale similar to William Thomson’s but which used the degree Fahrenheit for its unit increment. This absolute scale is known today as the Rankine thermodynamic temperature scale.
* 1877 - 1884: (1844 – 1906) made major contributions to thermodynamics through an understanding of the role that particle kinetics and black-body radiation played. His name is now attached to several of the formulas used today in thermodynamics.
* Circa 1930s: Gas thermometry experiments carefully calibrated to the melting point of ice and boiling point of water showed that absolute zero was equivalent to −273.15 °C.
* 1948: [http://www.bipm.fr/en/CGPM/db/9/3/ Resolution 3] of the 9th CGPM (Conférence Générale des Poids et Mesures, also known as the General Conference on Weights and Measures) fixed the triple point of water at precisely 0.01 °C. At this time, the triple point still had no formal definition for its equivalent kelvin value, which the resolution declared “will be fixed at a later date.” The implication is that "if" the value of absolute zero measured in the 1930s was truly −273.15 °C, then the triple point of water (0.01 °C) was equivalent to 273.16 K. Additionally, both the CIPM (Comité international des poids et mesures, also known as the International Committee for Weights and Measures) and the CGPM [http://www.bipm.org/en/committees/cipm/cipm-1948.html formally adopted] the name “Celsius” for the “degree Celsius” and the “Celsius temperature scale.”
* 1954: [http://www.bipm.fr/en/CGPM/db/10/3/ Resolution 3] of the 10th CGPM gave the Kelvin scale its modern definition by choosing the triple point of water as its second defining point and assigned it a temperature of precisely 273.16 kelvin (what was actually written 273.16 “degrees Kelvin” at the time). This, in combination with Resolution 3 of the 9th CGPM, had the effect of defining absolute zero as being precisely zero kelvin and −273.15 °C.
* 1967/1968: [http://www.bipm.fr/en/CGPM/db/13/3/ Resolution 3] of the 13th CGPM renamed the unit increment of thermodynamic temperature “kelvin”, symbol K, replacing “degree absolute”, symbol °K. Further, feeling it useful to more explicitly define the magnitude of the unit increment, the 13th CGPM also decided in [http://www.bipm.fr/en/CGPM/db/13/4/ Resolution 4] that “The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.”
* 2005: The CIPM (Comité International des Poids et Mesures, also known as the International Committee for Weights and Measures) [http://www.bipm.fr/en/si/si_brochure/chapter2/2-1/kelvin.html affirmed] that for the purposes of delineating the temperature of the triple point of water, the definition of the Kelvin thermodynamic temperature scale would refer to water having an isotopic composition defined as being precisely equal to the nominal specification of Vienna Standard Mean Ocean Water.
## Derivations of thermodynamic temperature
Strictly speaking, the temperature of a system is well-defined only if its particles (atoms, molecules, electrons, photons) are at equilibrium, so that their energies obey a Boltzmann distribution (or its quantum mechanical counterpart). There are many possible scales of temperature, derived from a variety of observations of physical phenomena. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the efficiency of idealized heat engines. Thus the "ratio" "T"2/"T"1 of two temperatures "T"1 and"T"2 is the same in all absolute scales.
Loosely stated, temperature controls the flow of heat between two systems, and the universe as a whole, as with any natural system, tends to progress so as to maximize entropy. This suggests that there should be a relationship between temperature and entropy. To elucidate this, consider first the relationship between heat, work and temperature. One way to study this is to analyse a heat engine, which is a device for converting heat into mechanical work, such as the Carnot heat engine. Such a heat engine functions by using a temperature gradient between a high temperature "T"H and a low temperature "T"C to generate work, and the work done (per cycle, say) by the heat engine is equal to the difference between the heat energy "q"H put into the system at the high temperature the heat "q"C ejected at the low temperature (in that cycle). The efficiency of the engine is the work divided by the heat put into the system or
:$extrm\left\{efficiency\right\} = frac \left\{w_\left\{cy${q_H} = frac{q_H-q_C}{q_H} = 1 - frac{q_C}{q_H} qquad (1)
where wcy is the work done per cycle. Thus the efficiency depends only on qC/qH. Because "q"C and "q"H correspond to heat transfer at the temperatures "T"C and "T"H, respectively, the ratio "q"C/"q"H should be a function "f" of these temperatures:
:$frac\left\{q_C\right\}\left\{q_H\right\} = f\left(T_H,T_C\right)qquad \left(2\right).$
Carnot’s theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, a heat engine operating between temperatures "T"1 and "T"3 must have the same efficiency as one consisting of two cycles, one between "T"1 and another (intermediate) temperature "T"2, and the second between "T"2 and "T"3. This can only be the case if
:$f\left(T_1,T_3\right) = frac\left\{q_3\right\}\left\{q_1\right\} = frac\left\{q_2 q_3\right\} \left\{q_1 q_2\right\} = f\left(T_1,T_2\right)f\left(T_2,T_3\right).$
Now specialize to the case that $T_1$ is a fixed reference temperature: the temperature of the triple point of water. Then for any "T"2 and "T"3,:$f\left(T_2,T_3\right) = frac\left\{f\left(T_1,T_3\right)\right\}\left\{f\left(T_1,T_2\right)\right\} = frac\left\{273.16 cdot f\left(T_1,T_3\right)\right\}\left\{273.16 cdot f\left(T_1,T_2\right)\right\}.$Therefore if thermodynamic temperature is defined by:$T = 273.16 cdot f\left(T_1,T\right) ,$then the function "f", viewed as a function of thermodynamic temperature, is simply:$f\left(T_2,T_3\right) = frac\left\{T_3\right\}\left\{T_2\right\},$and the reference temperature "T"1 will have the value 273.16. (Of course any reference temperature and any positive numerical value could be used — the choice here corresponds to the Kelvin scale.)
It follows immediately that:$frac\left\{q_C\right\}\left\{q_H\right\} = f\left(T_H,T_C\right) = frac\left\{T_C\right\}\left\{T_H\right\}.qquad \left(3\right).$
Substituting Equation 3 back into Equation 1 gives a relationship for the efficiency in terms of temperature:
:$extrm\left\{efficiency\right\} = 1 - frac\left\{q_C\right\}\left\{q_H\right\} = 1 - frac\left\{T_C\right\}\left\{T_H\right\}qquad \left(4\right).$
Notice that for "T"C=0 the efficiency is 100% and that efficiency becomes greater than 100% for "T"C<0. Since an efficiency greater than 100% violates the first law of thermodynamics, this requires that zero must be the minimum possible temperature. This has an intuitive interpretation: temperature is the motion of particles, so no system can, on average, have less motion than the minimum permitted by quantum physics. In fact, as of June 2006, the coldest man-made temperature was 450 pK.
Subtracting the right hand side of Equation 4 from the middle portion and rearranging gives
:$frac \left\{q_H\right\}\left\{T_H\right\} - frac\left\{q_C\right\}\left\{T_C\right\} = 0,$
where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function "S" (i.e., a function which depends only on the state of the system, not on how it reached that state) defined (up to an additive constant) by
:$dS = frac \left\{dq_mathrm\left\{rev\left\{T\right\}qquad \left(5\right),$
where the subscript indicates heat transfer in a reversible process. The function "S" corresponds to the entropy of the system, mentioned previously, and the change of "S" around any cycle is zero (as is necessary for any state function). Equation 5 can be rearranged to get an alternative definition for temperature in terms of entropy and heat:
:$T = frac\left\{dq_mathrm\left\{rev\left\{dS\right\}.$
For a system in which the entropy "S" is a function "S"("E") of its energy "E", the thermodynamic temperature "T" is therefore given by
:$frac\left\{1\right\}\left\{T\right\} = frac\left\{dS\right\}\left\{dE\right\},$ so that the reciprocal of the thermodynamic temperature is the rate of increase of entropy with energy.
* Absolute zero
* Black body
* Boiling
* Boltzmann constant
* Brownian motion
* Carnot heat engine
* Celsius
* Chemical bond
* Condensation
* Convection
* Degrees of freedom
* Delocalized electron
* Diffusion
* Elastic collision
* Electron
* Energy
* Energy conversion efficiency
* Enthalpy
* Entropy
* Evaporation
* Fahrenheit
* First law of thermodynamics
* Freezing
* Gas laws
* Heat
* Heat conduction
* Heat engine
* Internal energy
* ITS-90
* Ideal gas law
* Joule
* Kelvin
* Kinetic energy
* Latent heat
* Laws of thermodynamics
* Maxwell–Boltzmann distribution
* Melting
* Mole
* Molecule
* Orders of magnitude (temperature)
* Phase transition
* Phonon
* Planck’s law of black body radiation
* Potential energy
* Quantum mechanics:
** Introduction to quantum mechanics
** Quantum mechanics (main article)
* Rankine scale
* Specific heat capacity
* Standard enthalpy change of fusion
* Standard enthalpy change of vaporization
* Stefan–Boltzmann law
* Sublimation
* Temperature
* Temperature conversion formulas
* Thermal conductivity
* Thermodynamic equations
* Thermodynamic equilibrium
* Thermodynamics
*
* Timeline of temperature and pressure
measurement technology
* Triple point
* Universal gas constant
* Vienna Standard Mean Ocean Water (VSMOW)
* Wien’s displacement law
* Work (Mechanical)
* Work (thermodynamics)
* Zero-point energy
## Notes
"In the following notes, wherever numeric equalities are shown in ‘concise form’—such as" 1.85487(14) × 1043"—the two digits between the parentheses denotes the uncertainty at "1σ" standard deviation "(68%" confidence level) in the two least significant digits of the significand."
* " [http://www.chm.davidson.edu/ChemistryApplets/KineticMolecularTheory/index.html Kinetic Molecular Theory of Gases.] " An excellent explanation (with interactive animations) of the kinetic motion of molecules and how it affects matter. By David N. Blauch, [http://www.chm.davidson.edu/ Department of Chemistry] , [http://www2.davidson.edu/index.asp Davidson College] .
* " [http://www.calphysics.org/zpe.html Zero Point Energy and Zero Point Field.] " A Web site with in-depth explanations of a variety of quantum effects. By Bernard Haisch, of [http://www.calphysics.org/index.html Calphysics Institute] .
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### Look at other dictionaries:
• thermodynamic temperature — termodinaminė temperatūra statusas T sritis Standartizacija ir metrologija apibrėžtis Vienas iš pagrindinių SI dydžių, kurio matavimo vienetas kelvinas (K) yra vandens trigubojo taško termodinaminės temperatūros 1/273,16 dalis. atitikmenys: angl … Penkiakalbis aiškinamasis metrologijos terminų žodynas
• thermodynamic temperature — termodinaminė temperatūra statusas T sritis fizika atitikmenys: angl. thermodynamic temperature vok. thermodynamische Temperatur, f rus. термодинамическая температура, f pranc. température thermodynamique, f … Fizikos terminų žodynas
• thermodynamic temperature — termodinaminė temperatūra statusas T sritis Energetika apibrėžtis Temperatūra, apibūdinanti medžiagos dalelių šiluminio judėjimo intensyvumą ir nepriklausanti nuo pasirinktos medžiagos ir matavimo būdo. Žymima T. Termodinaminės temperatūros… … Aiškinamasis šiluminės ir branduolinės technikos terminų žodynas
• thermodynamic temperature — noun Temperature defined in terms of the laws of thermodynamics rather than the properties of a real material: expressed in kelvins … Wiktionary
• thermodynamic temperature scale — termodinaminė temperatūros skalė statusas T sritis Energetika apibrėžtis Nepriklauso nuo termometrinės medžiagos ir turi vieną atskaitos tašką – vandens trigubąjį tašką, kuriam suteikta T = 273,16 K vertė. Termodinaminė temperatūros skalė… … Aiškinamasis šiluminės ir branduolinės technikos terminų žodynas
• thermodynamic temperature scale — noun a temperature scale of which the unit is the kelvin (K) which is ¹⁄₂₇₃.₁₆ of the thermodynamic temperature of the triple point of water … Australian English dictionary
• thermodynamic temperature scale — termodinaminė temperatūros skalė statusas T sritis Standartizacija ir metrologija apibrėžtis Temperatūros skalė, pagrįsta absoliučiuoju nuliu, t. y. žemiausia temperatūra, kurią teoriškai galima būtų pasiekti ir kuri yra 273,16 °C žemiau ledo… … Penkiakalbis aiškinamasis metrologijos terminų žodynas
• thermodynamic temperature scale — termodinaminės temperatūros skalė statusas T sritis fizika atitikmenys: angl. thermodynamic temperature scale vok. thermodynamische Temperaturskala, f rus. термодинамическая шкала температур, f pranc. échelle thermodynamique des températures, f … Fizikos terminų žodynas | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 18, "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": 0.8314574360847473, "perplexity": 2009.9445610117443}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738950.61/warc/CC-MAIN-20200813014639-20200813044639-00029.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/138104-mean-variance-two-stochastic-variables-combined-print.html | # Mean and variance of two stochastic variables combined
• April 9th 2010, 05:30 AM
Eva BSc
Mean and variance of two stochastic variables combined
Hi,
I'm stuck with a little problem that I need to use for my research.
Let X and N be stochastic random variables. I need to show that:
Mean:
E(sum(X)) = E(N) * E(X)
Variance:
var(sum(X)) = E(N) * var(X) + var(N) * (E(X))^2
Anyone knowing to do this?
Cheers, Eva
• April 9th 2010, 08:16 AM
Anonymous1
Quote:
Originally Posted by Eva BSc
Mean:
E(sum(X)) = E(N) * E(X)
When $N=n, \ \ S = \sum_i X_i$ has $E(S)=nE(X_i)$
Breaking things down according to the value of $N,$ we have...
$E[S] = \sum_{n=0}^{\infty} E(S|N=n)P(N=n) = \sum_{n=0}^{\infty} nE(X_i)P(N=n) = E(N)E(X_i)$
• April 9th 2010, 08:27 AM
Anonymous1
Quote:
Originally Posted by Eva BSc
Variance:
var(sum(X)) = E(N) * var(X) + var(N) * (E(X))^2
When $N=n, \ \ S = \sum_i X_i$ has $Var(S) = nVar(X_i)$ and hence,
$E(S^2|N=n)= nVar(X_i) + (nE(X_i))^2$
Then,
$E[S^2] = \sum_{n=0}^{\infty} E(S^2|N=n)P(N=n) =$ $\sum_{n=0}^{\infty} \{nVar(X_i) + n^2 (E(X_i))^2 \}P(N=n) = E(N)Var(X_i) + E(N^2) (E(X_i))^2$
So,
$Var(S) = E(S^2) - (E(S))^2 =$ $E(N)Var(X_i) + E(N^2)(E(X_i))^2 - (E(N)E(X_i))^2 = E(N)Var(X_i) + Var(N)(E(X_i))^2$
• April 9th 2010, 08:30 AM
Eva BSc
Thnx!
• April 9th 2010, 08:30 AM
Anonymous1
I had to prove this in my homework a while back. (Nerd)
• April 9th 2010, 08:38 AM
cgiulz
I've used this fact before, but never seen its proof. Thanks!
• April 10th 2010, 03:30 AM
Moo
• April 10th 2010, 12:26 PM
Anonymous1
Just to confirm, my instructor verified that the proof above is valid and correct... | {"extraction_info": {"found_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": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5973220467567444, "perplexity": 5743.226906906484}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1419447555354.63/warc/CC-MAIN-20141224185915-00007-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://stats.libretexts.org/Bookshelves/Applied_Statistics/Book%3A_Biological_Statistics_(McDonald)/1.0 | # 1: Basics
• 1.1: Data analysis steps
• 1.2: Types of Biological Variables
One of the first steps in deciding which statistical test to use is determining what kinds of variables you have. When you know what the relevant variables are, what kind of variables they are, and what your null and alternative hypotheses are, it's usually pretty easy to figure out which test you should use. I classify variables into three types: measurement variables, nominal variables, and ranked variables.
• 1.3: Probability
When dealing with probabilities in biology, you are often working with theoretical expectations, not population samples. For example, in a genetic cross of two individual Drosophila melanogaster that are heterozygous at the vestigial locus, Mendel's theory predicts that the probability of an offspring individual being a recessive homozygote (having teeny-tiny wings) is one-fourth, or 0.25. This is equivalent to saying that one-fourth of a population of offspring will have tiny wings.
• 1.4: Basic Concepts of Hypothesis Testing
The technique used by the vast majority of biologists, and the technique that most of this handbook describes, is sometimes called "frequentist" or "classical" statistics. It involves testing a null hypothesis by comparing the data you observe in your experiment with the predictions of a null hypothesis. You estimate what the probability would be of obtaining the observed results, or something more extreme, if the null hypothesis were true.
• 1.5: Confounding Variables
A confounding variable is a variable that may affect the dependent variable. This can lead to erroneous conclusions about the relationship between the independent and dependent variables. You deal with confounding variables by controlling them; by matching; by randomizing; or by statistical control.
## Contributor
• John H. McDonald (University of Delaware) | {"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": 0.8294841647148132, "perplexity": 1270.4354430074698}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986666959.47/warc/CC-MAIN-20191016090425-20191016113925-00410.warc.gz"} |
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# If f(x) = 4x^2 + px, what is the minimum value of f(x)?
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[GMAT math practice question]
If $$f(x) = 4x^2 + px$$, what is the minimum value of $$f(x)$$?
$$1) f(1) = -4$$
$$2) f(2) = 0$$
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Expert Joined: 02 Aug 2009 Posts: 7099 Re: If f(x) = 4x^2 + px, what is the minimum value of f(x)? [#permalink] ### Show Tags 04 Jul 2018, 01:41 MathRevolution wrote: [GMAT math practice question] If $$f(x) = 4x^2 + px$$, what is the minimum value of $$f(x)$$? $$1) f(1) = -4$$ $$2) f(2) = 0$$ Minimum value of $$4x^2+px$$ is at 8x+p=0 or when $$x=\frac{-p}{8}$$ So min value can be found if value of p is known... 1) f(1)=-4 f(x)=4x^2+px so f(1)=4*1^2+p*1=4+p=-4........ p=-8 So min value be at x=-8/-8=1 f(x)=4x^2-8x=4*(1)^2-8*1=-4 Sufficient 2)f(2)=0 f(x)=4x^2+px so f(2)=4*2^2+p*2=16+2p=0....... p=16/(-2)=-8 So min value be at x=-8/-8=1 f(x)=4x^2-8x=4*(1)^2-8*1=-4 Sufficient D _________________ 1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html GMAT online Tutor Senior Manager Joined: 04 Aug 2010 Posts: 310 Schools: Dartmouth College Re: If f(x) = 4x^2 + px, what is the minimum value of f(x)? [#permalink] ### Show Tags 04 Jul 2018, 01:52 3 MathRevolution wrote: [GMAT math practice question] If $$f(x) = 4x^2 + px$$, what is the minimum value of $$f(x)$$? $$1) f(1) = -4$$ $$2) f(2) = 0$$ To determine the minimum value of the graph, we need to know the value of p. Question stem, rephrased: What is the value of p? Statement 1: Substituting $$x=1$$ and $$f(x)=-4$$ into $$f(x) = 4x^2 + px$$, we get: $$-4 = 4(1^2) + p(1)$$ Since we can solve for $$p$$, SUFFICIENT. Statement 2: Substituting $$x=2$$ and $$f(x)=0$$ into $$f(x) = 4x^2 + px$$, we get: $$0 = 4(2^2) + p(2)$$ Since we can solve for $$p$$, SUFFICIENT. _________________ GMAT and GRE Tutor Over 1800 followers Click here to learn more [email protected] New York, NY If you find one of my posts helpful, please take a moment to click on the "Kudos" icon. Available for tutoring in NYC and long-distance. For more information, please email me at [email protected]. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6619 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If f(x) = 4x^2 + px, what is the minimum value of f(x)? [#permalink] ### Show Tags 06 Jul 2018, 00:41 => Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. Since we have 1 variable (p) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first. Condition 1) Since $$f(1) = -4,$$ we have $$4 + p = -4$$ and p = $$-8.$$ $$f(x) = 4x^2 -8x = 4( x^2 - 2x + 1 - 1 ) = 4(x-1)^2 – 4.$$ The minimum of $$f(x)$$ is $$-4$$. Condition 1) is sufficient. Condition 2) Since $$f(2) = 0$$, we have $$16 + 2p = 0$$ and $$p = -8.$$ $$f(x) = 4x^2 -8x = 4( x^2 - 2x + 1 - 1 ) = 4(x-1)^2 – 4.$$ The minimum of $$f(x)$$ is $$-4.$$ Condition 2) is sufficient. Therefore, D is the answer. Answer: D Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2). If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: If f(x) = 4x^2 + px, what is the minimum value of f(x)? &nbs [#permalink] 06 Jul 2018, 00:41
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https://jerseycityremodelers.com/32yb4k/uniform-priors-in-stan-cbb12e | Now, let’s load the data: If for some reason the column names are not read in properly, you can change column names using: What research question can we ask with these data? In this case this uniform prior is improper, because these intervals are unbounded. Let’s compare to our previous estimate with “lm”: Figure 3. It’s a great resource for understanding and diagnosing problems with Stan, and by posting problems you encounter you are helping yourself, and giving back to the community. The Stan documentation discourages uniform priors. We’re going to use normal priors with small standard deviations. } parameter can have every one-dimensional prior implemented in Stan, for instance uniform, Cauchy or even Gamma priors. We can also extract some of the key summary statistics from our simple model, so that we can compare them with the outputs of the Stan models later. Now let’s turn that into a dataframe for inputting into a Stan model. Figure 7. The examples are related to Bayesian data analysis course . However, uniform priors have very little place outside of textbooks. We can also look at the full posterior of our parameters by extracting them from the model object. Created by Max Farrell & Isla Myers-Smith. We can investigate mean posterior prediction per datapoint vs the observed value for each datapoint (default line is 1:1). While inferences based on the posterior mode are convenient, often other summaries of the posterior distribution are also of interest. y ~ normal(alpha + x * beta , sigma); The log of the uniform complementary cumulative distribution function (Double-check in the Stan manual to see which sampling statements have corresponding rng functions already coded up.). You just need to think carefully about each modelling decision you make! For traceplots, we can view them directly from the posterior: Figure 6. Use improper uniform priors from −∞−∞ to ∞∞ for all model parameters. Each row is an iteration (single posterior estimate) from the model. If $$\alpha \in \mathbb{R}$$ and $$\beta \in (\alpha,\infty)$$, then for R uniform_rng(reals alpha, reals beta) vectorized PRNG functions. A goal of the Stan development team is to make Bayesian modelling more accessible with clear syntax, a better sampler (sampling here refers to drawing samples out of the Bayesian posterior distribution), and integration with many platforms and including R, RStudio, ggplot2, and Shiny. Below, we explain its usage and list some common prior dist… And in a future tutorial, we will introduce the concept of a mixture model where two different distributions are modelled at the same time - a great way to deal with zero inflation in your proportion or count data! If you think diffuse inverse gamma priors are the answer, that was the second anti-pattern I alluded to earlier. Figure 2. If we want to use a previously written .stan file, we use the file argument in the stan_model() function. Comparing estimates across random posterior draws. As a negative side e ect of this exibility, correlations between them cannot be modeled as parameters. See our Terms of Use and our Data Privacy policy. All the files you need to complete this tutorial can be downloaded from this Github repository. model { Stan statements are processed sequentially and allow … Check out some Stan models in the ecological literature to see how those Bayesian models are reported. This year’s NCAA shooting contest was a thriller that saw Cassandra Brown of the Portland Pilots win the grand prize. You usually only need to worry is this number is less than 1/100th or 1/1000th of We can get summary statistics for parameter estimates, and sampler diagnostics by executing the name of the object: What does the model output show you? Typically, the data generating functions will be the distributions you used in the model block but with an _rng suffix. Suppose that instead of a uniform prior, we use the prior ⇠ Beta(↵,). real uniform_cdf(reals y, reals alpha, reals beta) The uniform cumulative distribution function of y given lower bound alpha and upper bound beta. First, let’s find a dataset where we can fit a simple linear model. set_prior is used to define prior distributions for parameters in brms models. If you are new to Stan, you can join the mailing list. This got me thinking, just how good is Cassandra Brown? | {"extraction_info": {"found_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": 0.603280246257782, "perplexity": 1281.1930718122946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964363216.90/warc/CC-MAIN-20211205191620-20211205221620-00212.warc.gz"} |
https://www.albert.io/ie/ap-physics-c-e-and-m/direction-of-emf-1 | Free Version
Moderate
# Direction of EMF 1
APPHEM-0U81L4
A solenoid is hooked up to a battery and switch as shown in the figure below:
In the moments just after the switch is closed, what direction is the emf of the solenoid coil?
A
Location $A$ is positive and Location $B$ is negative.
B
Location $A$ is positive and Location $B$ is positive.
C
Location$A$ is negative and Location $B$ is negative.
D
Location $A$ is negative and Location $B$ is positive.
E
There would be no emf on the solenoid coil. | {"extraction_info": {"found_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": 0.44025006890296936, "perplexity": 813.160111247991}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170286.6/warc/CC-MAIN-20170219104610-00391-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/a-n-th-lg-n.35904/ | # A[n] = Θ(lg n)?
1. Jul 20, 2004
### e(ho0n3
a[n] = Θ(lg n)!?
Suppose {a[n]} is an increasing sequence and whenever m divides n, then
a[n] = a[n/m] + d
where m is a positive integer and d is a real number. Show that a[n] = Θ(lg n).
a[n/m] + d ≤ a[n/2] + d, and with this I can show that a[n] = O(lg n). All I need to do is show that a[n] = Ω(lg n). This is where I'm stuck. Any hints?
2. Jul 20, 2004
### Hurkyl
Staff Emeritus
To show that something is in omega(log x), it suffices to find some sequence {x_i} such that a[x_i] grows like log x.
By the way, I'm disconcerted with your reasoning that it is in O(log x); what if n is odd? I suppose you're just abbreviating the complete proof you have.
3. Jul 21, 2004
### e(ho0n3
Sorry. What I meant was
$$a_n = a_{\lfloor n/2 \rfloor} + d$$
My head is a total blank. I can't think of anything. | {"extraction_info": {"found_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": 0.8493074774742126, "perplexity": 2402.271343606918}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988718278.43/warc/CC-MAIN-20161020183838-00036-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://webwork.maa.org/moodle/mod/forum/discuss.php?d=6581&parent=17639 | WeBWorK Problems
Re: Why comments and latex \( do not work?
by Alex Jordan -
Number of replies: 0
This problem uses PGML, with the statement wrapped in BEGIN_PGML...END_PGML. The math delimiters for PGML are different than for regular PG. They are not slash-parens. Here is some documentation:
https://webwork.maa.org/wiki/Mathematical_notation_-_PGML
For inline math, the delimiters are [...].
For a comment in PGML, see: | {"extraction_info": {"found_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": 0.3604127764701843, "perplexity": 20434.945306327994}, "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/1656103645173.39/warc/CC-MAIN-20220629211420-20220630001420-00462.warc.gz"} |
https://codegolf.stackexchange.com/questions/47417/evaluate-a-chain-of-inequalities/47442 | # Evaluate a chain of inequalities
Write code to evaluate whether a chain of inequalities is true or false. An example input is the string
3<=4!=9>3==3
This is true because each of its components is true:
(3<=4) and (4!=9) and (9>3) and (3==3)
Input:
A string that represents a chain of one or more inequalities. The allowed comparison operators are
== equals
!= does not equal
> is greater than
>= is greater than or equal to
< is less than
<= is less than or equal to
The allowed numbers are single-digit numbers 0 through 9. There won't be any spaces, parentheses, or other symbols.
Output:
The correctness of the inequality as a consistent Truthy or Falsey value. Consistent means every Truthy output is the same and every Falsey output is the same.
Restriction:
The intent of this challenge is for you to write code that processes the inequalities, rather than have them be evaluating them as code, even for a single inequality in the chain. As such, methods like Python's eval and exec that evaluate or execute code are banned. So are functions which look up a method or operator given its name as a string. Nor is it allowed to launching processes or programs to do the evaluation for you.
Test cases:
3<=4!=9>3==3
True
3<=4!=4
False
5>5
False
8==8<9>0!=2>=1
True
• Can we accept input with Unicode inequality signs like ≤ and ≥ instead of <= and >=? – FUZxxl Mar 5 '15 at 12:11
• @FUZxxl You can't. – xnor Mar 5 '15 at 21:58
# Ruby, 71 + 1 = 72
With command-line flag -n, run
p (0..99).none?{|i|~/#{a=i%10}(#{%w/!=|. <?=* >?=*/[a<=>b=i/10]})#{b}/}
Generates all possible failing regular expressions, and checks whether the input string matches any of them. Outputs true if none do, else false. Takes input via STDIN, separated by newlines.
Tricks:
• We get all possible pairs of digits by looping from 0 to 99 and extracting the 10s and 1s digits.
• The only actual comparison we do is a<=>b, which returns -1,0,or 1 for less than, equal to, or greater. These all slice to different elements of a three-string array, finding the regular expression for comparisons that don't match.
• What a clever strategy! – xnor Mar 5 '15 at 23:05
# Perl, 82
$_=<>;($'<=>$&)-61+ord$1&&($2&&$&==$')^('$'lt$1)&&die"\n"while/\d(.)(=?)/g;print 1 Prints 1 when true and a blank line when false, since the empty string is Perl's main falsey value. The while loop goes over the string matching the regex \d(.)(=?). Then the variables $1 and $2 correspond to the characters of the operator, and the special variables $& and \$' will behave as the two operands in a numerical context. The operands are compared with <=> and the result matched against the first character of the operator. Then equality and inequality is dealt with specially.
# CJam, 60 bytes
This code seems a bit ugly and potentially not fully optimized, but it's the best I've got so far.
Try it online.
q_A,sSer_S%@@-(])1\@z{~:X\:^i['=")<"P"(>"P'<'^'>PPP]=~e&X}/;
### Explanation
q "Read the input";
_A,sSer "Copy the input and replace each digit with a space";
_S% "Split around spaces to obtain the operation list";
@@- "Remove operations from the input to obtain the operand list";
(])1\@z "Remove the first operand from the list to be the initial left
operand, initialize the result to 1 (true), and pair up the
operations and remaining operands";
{ "For each operation-operand pair:";
~:X "Let the operand be the right operand of this operation";
\:^i "Hash the operation (bitwise XOR of all characters)";
[ "Begin cases:";
'= " 0: Equals";
")<" " 1: Less than or equal to";
P " 2: (Invalid)";
"(>" " 3: Greater than or equal to";
P " 4: (Invalid)";
'< " 5: Less than";
'^ " 6: Bitwise XOR (stand-in for not equal to)";
'> " 7: Greater than";
P " 8: (Invalid)";
P " 9: (Invalid)";
P "10: (Invalid)";
]=~ "Execute the case selected by the operation hash modulo 11";
e& "Compute the logical AND of the result and the value produced
by this operation to be the new result";
X "Let the right operand be the new left operand";
}/ "End for each";
; "Clean up and implicitly print result";
# JavaScript (ES6) 110 116
Straightforward:scan string, c is current digit, l is last digit, o is operator.
F=x=>(l='',[for(c of x)10-c?(v=!l||v&&(o<'<'?l!=c:(o[1]&&c==l)||(o<'='?l<c:o<'>'?c==l:l>c)),l=c,o=''):o+=c],v)
Test In Firefox / FireBug console
;['3<=4!=9>3==3','3<=4!=4','5>5','8==8<9>0!=2>=1']
.forEach(s=>console.log(s,F(s)))
3<=4!=9>3==3 true
3<=4!=4 false
5>5 false
8==8<9>0!=2>=1 true
# Haskell, 156 bytes
r a=read[a]::Int
l"!"=(/=)
l"="=(==)
l">"=(>=)
l"<"=(<=)
k">"=(>)
k"<"=(<)
[]#_=1<2
(a:'=':b:c)#i=l[a]i(r b)&&c#r b
(a:b:c)#i=k[a]i(r b)&&c#r b
f(h:t)=t#r h
Usage example:
f "3<=4!=9>3==3" -> True
f "3<=4!=4" -> False
f "5>5" -> False
f "8==8<9>0!=2>=1" -> True
Ungolfed version:
digitToInt d = read [d] :: Int
lookup2 "!" = (/=)
lookup2 "=" = (==)
lookup2 ">" = (>=)
lookup2 "<" = (<=)
lookup1 ">" = (>)
lookup1 "<" = (<)
eval [] _ = True
eval (op:'=':d:rest) i = lookup2 [op] i (digitToInt d) && eval rest (digitToInt d)
eval (op:d:rest) i = lookup1 [op] i (digitToInt d) && eval rest (digitToInt d)
evalChain (hd:rest) = eval rest (digitToInt hd)
eval takes two arguments: the string to parse (starting always with a comparison operator) and a number i which is the left argument for comparison (and was the right argument in the previous round). The operator is returned by lookup2 if it's a two character operator (check only 1st char, because 2nd is always =) and by lookup1 if it's just a single character. eval calls itself recursively and combines all return values with logical and &&.
# Common Lisp - 300185 169 165
(lambda(s)(loop for(a o b)on(mapcar'read-from-string(cdr(ppcre:split"([0-9]+)"s :with-registers-p t)))by #'cddr always(if o(funcall(case o(=='=)(!='/=)(t o))a b)t)))
# Example
(mapcar (lambda(s) ...)
'("2<=3<=6>2<10!=3"
"3<=4!=9>3==3"
"3<=4!=4"
"5>5"
"8==8<9>0!=2>=1"))
=> (T T NIL NIL T)
# Explanation
(lambda (s)
(loop for (a o b) on (mapcar
'read-from-string
(cdr
(cl-ppcre:split "([0-9]+)" s
:with-registers-p t))) by #'cddr
always (if o
(funcall (case o
(== '=)
(!= '/=)
(t o))
a b)
t)))
• ppcre:split splits on digits; for example:
(ppcre:split "([0-9]+)" "2<=3<=6>2<10!=3" :with-registers-p t)
=> ("" "2" "<=" "3" "<=" "6" ">" "2" "<" "10" "!=" "3")
Notice the first empty string, which is discarded using cdr
• Mapping read-from-string to this list calls the read function for each strings, which returns symbols and numbers.
• loop for (a op b) on '(3 < 5 > 2) by #'cddr iterates over the list by a step of 2 and thus binds a, op and b as follows, for each successive pass.
a op b
----------
3 < 5
5 > 2
2 nil nil
• always checks whether the next expression is always true: either the operator is nil (see above) or the result of the comparison holds (see below).
• the case selects a Common-Lisp comparison function, according to the symbol previously read; since some operators are identical in Lisp and the given language we can simply return o in the default case.
# Python 2, 95 102
t=1
n=o=3
for c in map(ord,raw_input()):
o+=c
if 47<c<58:t&=627>>(o-c+3*cmp(n,c))%13;n=c;o=0
print t
The loop is a straightforward pass through the string one character at a time. The t&=... part is where the magic happens. Basically, I hash the operator together with the value of cmp(lhs,rhs) (-1, 0, or 1 depending on if lhs is less, equal, or greater than rhs). The result is a key into a lookup table that gives 0 or 1 depending on whether the numbers compare properly given that operator. What lookup table, you ask? It's the number 627 = 0001001110011 in binary. Bitwise operators do the rest.
This works on the four given test cases; let me know if you find an error for another case. I haven't tested it very rigorously.
• You need to take in a as input. – xnor Mar 6 '15 at 4:36
• @xnor Whoops. Corrected. – DLosc Mar 6 '15 at 4:38
# Javascript 101 bytes
a different approach from the js solution posted here
F=(s,i=0,l=o="")=>[...s].every(c=>c>=0?[l^c,l==c,,l<c,l>c,l<=c,,l>=c]["!==<><=>=".search(o,l=c,o="")]:o+=c,l=o="")
console.log(F("3<=4!=9>3==3")==true)
console.log(F("3<=4!=4")==false)
console.log(F("5>5")==false)
console.log(F("8==8<9>0!=2>=1")==true)
# Java 8, 283 bytes
s->{String[]a=s.split("\\d"),b=s.split("\\D+");int i=0,r=1,x,y;for(;i<a.length-1;)if((x=new Byte(b[i]))!=(y=new Byte(b[++i]))&(a[i].equals("=="))|(a[i].equals("!=")&x==y)|(a[i].equals(">")&x<=y)|(a[i].equals(">=")&x<y)|(a[i].equals("<")&x>=y)|(a[i].equals("<=")&x>y))r--;return r>0;}
Explanation:
Try it here.
s->{ // Method with String parameter and boolean return-type
String[]a=s.split("\\d"), // All the inequalities
b=s.split("\\D+"); // All the digits
int i=0, // Index-integer (starting at 0)
r=1, // Flag integer for the result, starting at 1
x,y; // Temp integer x and y
for(;i<a.length-1;) // Loop from 0 to the length - 1
if((x=new Byte(b[i]))!=(y=new Byte(b[++i]))&(a[i].equals("=="))
// If "==" and x and y as int are not equal:
|(a[i].equals("!=")&x==y) // Or "!=" and x and y are equal
|(a[i].equals(">")&x<=y) // Or ">" and x is smaller or equal to y
|(a[i].equals(">=")&x<y) // Or ">=" and x is smaller than y
|(a[i].equals("<")&x>=y) // Or "<" and x is larger or equal to y
|(a[i].equals("<=")&x>y)) // Or "<=" and x is larger than y
r--; // Decrease r by 1
// End of loop (implicit / single-line body)
return r>0; // Return if r is still 1
} // End of method | {"extraction_info": {"found_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": 0.3089540898799896, "perplexity": 6250.11474992339}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540527010.70/warc/CC-MAIN-20191210070602-20191210094602-00129.warc.gz"} |
http://hiking-calculator.deadunicornz.org/ | # Hiking calculator
## Build the path by clicking on the map to get range and time estimates
Metric unit system
Imperial unit system
$V = V_0 * K_{slope} * K_{elevation} * K_{backpack}$
Where $V_0$ is a maximum safe and stable speed. Hiker achieves it descending -2.86° slope. Default $V_0 = 6 {km \over h}$ translates into 5 $km \over h$ on a flat terrain.
$K_{slope} = e^{-3.5 \left|{h \over l} + 0.05\right|}$ is defined by Tobler's hiking function. $h$ is elevation gain/loss and $l$ is a distance.
$K_{elevation} = e^{-9.80665 * 0.0289644 * (h - h_{acc}) \over {8.3144598 * 288.15}} = e^{-0.00011855(h - h_{acc})}$ and derived from the Barometric formula for atmosphere density. The idea here is that speed is decreased with atmosphere density decreasing, $h$ is elevation and $h_{acc}$ the elevation you are acclimatized for.
$K_{backpack} = (1 - {m_b \over 100})$ coefficient determines how backpack mass slows you down. $m_b$ is backpack mass to body mass ratio, percents.
Disclaimer: The calculator is intended for ballpark estimates. If you're not hiking in a ballpark use your discretion. | {"extraction_info": {"found_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": 0.8059558272361755, "perplexity": 3531.812269070359}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948517845.16/warc/CC-MAIN-20171212173259-20171212193259-00043.warc.gz"} |
https://crypto.stackexchange.com/questions/29713/inverting-rsa-function/29729 | # Inverting RSA function
I am in high school and I am writing a paper on RSA. I want to show that low values of the public key exponent can make it easy to 'invert' the function so that the encrypted message can be recovered. How is this done? I have tried to read the Handbook of Applied Cryptography but it's not making sense...
Would this be inversion: Taking cipher text, and continuously adding the modulus, then taking the eth root of the sum. If the result is an integer, then it is the plaintext message.
• PS. My paper is just covering plain old RSA, no padding or anything :) – EVELYN Oct 9 '15 at 2:32
• I suppose it should be the private key and not the public one. – user27950 Oct 9 '15 at 4:54
• RSA without padding is vulnerable, and should not be presented in introductory material without mention that it must not be used for serious purposes (except for random messages almost as wide as the modulus). Combined with low value of the public exponent, RSA without padding is even more vulnerable, but usually not because it is easier to invert the public function (the exception to that being in the end of poncho's answer) – fgrieu Oct 9 '15 at 13:09
I want to show that low values of the public key exponent can make it easy to 'invert' the function so that the encrypted message can be recovered.
That is not known to be true; as long as the modulus is large enough to make factorization infeasible, there is no known way to compute e-th roots in general.
Now, if the plaintext $p$ is small enough that $p^e < N$, then it is easy to recover $p$ (just take the e-th root over the integers, which is an easy problem). That's as close as we know to the result you're trying to get.
Would this be inversion: Taking cipher text, and continuously adding the modulus, then taking the eth root of the sum. If the result is an integer, then it is the plaintext message.
If $N$ is small enough to make work in a practical amount of time, then $N$ is small enough to factor.
• I'm reading the last paragraph of the question as a simple extension of the e-th root attack, working when $p^e<k\cdot N$ for moderate $k$. I'm sure $k\approx2^{20}$ is feasible. I wish I knew how large $k$ needs to be in order to avoid a more sophisticated attack. This answer to a significantly different question does not tell. – fgrieu Oct 9 '15 at 16:26
• Hello, could you please explain k? I have not encountered a constant or variable k in my readings. – EVELYN Oct 10 '15 at 9:57
• @EVELYN : $\:$ k is a positive integer, and in this case that's all it is. $\;\;\;\;$ – user991 Oct 11 '15 at 8:39
While Poncho's answer is fully valid I'll hereby extend the part on the proposed idea being inferior to factoring.
I want to show that low values of the public key exponent can make it easy to 'invert' the function so that the encrypted message can be recovered.
As poncho said this is only feasible if the message's length is smaller than the modulus length divided by e, i.e. when no modular reduction takes place.
Would this be inversion: Taking cipher text, and continuously adding the modulus, then taking the eth root of the sum. If the result is an integer, then it is the plaintext message.
The problem of this approach is the run-time. For this approach to be viable you need to beat the General Number Field Sieve (GNFS) in terms of performance at least for some messages. So let's assume the best case scenario, where your algorithm starts operating: if $||m||\geq ||N||/e$, so for example for a 1025-bit message for 3072-bit RSA with $e=3$. To be beat the GNFS, you need to be able to carry out the attack on less than $2^{256}$ operations for this scenario. Clearly, the ciphertext will have length $||c||=3072$ bits with high probability and the term you're looking for $m^e$ has $1025*3=3075$ bits. So you have to test every single (or at least half) of all the multiples of $N$ in the 3075 bit range, which are $2^3=8$, which is indeed feasible. But which area of messages can you break using this technique? You can break any message for which the above equation holds and for which $(||m||-||N||/e)*e<2^{60}$, meaning for $e=3$ you "can break 20 more bits" (f.ex. 1025-1045). If you consider your technique "good" if it stays below factoring, you can even extend that to 80 bits. But in practice this kind of attack is already countered by randomized padding.
• The approach in the question remains feasible when the message's length is slightly higher than the modulus length divided by $e$, at least up to $20/e$ extra bits, with modest means. I do not know what the upper limit is; this answer to a significantly different question does not tell. – fgrieu Oct 9 '15 at 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": 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": 0.8214862942695618, "perplexity": 398.86665974370806}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038075074.29/warc/CC-MAIN-20210413213655-20210414003655-00283.warc.gz"} |
https://wiki.freepascal.org/index.php?title=NumLib_Documentation&oldid=128664 | # NumLib Documentation
## Introduction
NumLib is a collection of routines for numerical mathematics. It was written by C.J.J.M. van Ginneken, W.J.P.M. Kortsmit, and L. Reij at the Technical University of Einhoven (Netherlands) and donated to the Free Pascal project. Its beginnings date back to 1965. Originally written in Algol 60, ported to poorly formatted Pascal with strange procedure names, it is hard stuff, in particular because an official documentation is not readily available (well... -- Dutch documentation files in TeX can be found at http://www.stack.nl/~marcov/numlib/).
But after all, this library contains a series of very useful routines grouped into several units:
• Calculation of special functions (unit spe)
• Solution of systems of linear equations (unit sle)
• Finding roots of equations (unit roo)
• Fitting routines (unit ipf)
• Basic operations with matrices and vectors (unit omv)
• Matrix inversion (unit inv)
• Calculation of determinants (unit det)
• Calculation of eigenvalues (unit eig)
Therefore, it is the intention of this wiki article to create some useful documentation. Each routine described here was tested and its description will be accompanied by a working code example.
Note: This page heavily makes use of mathematical formulas written in LaTeX. In order to see these formulas in the browser JavaScript must be enabled.
### Naming convention of routines
Would you imagine that a procedure named spears calculates the arcsin? The original NumLib routines limit the procedure/function names to 6 characters (maybe a limitation of Algol?). The first three characters always are identical with the name of the unit in which they are implemented. This leaves only three characters to differentiate between the routines...
Routines later added to the original library are given more conventional names, such as normaldist for calculation of the normal distribution.
## Data types and declarations (unit typ)
### Basic floating point type, ArbFloat
In NumLib, a new type ArbFloat is defined which is used as floating point type throughout the entire library. This allows to switch between IEEE Double (64bit) and Extended(80bit) (though big endian state is unknown at present).
At the time of this writing the DEFINE ArbExtended is enabled, and therefore ArbFloat is identical to the standard type extended. In order to change the floating point type define or undefine ArbExtended and adapt the machineconstants to the type selected. ### Basic integer type, ArbInt Because in plain FPC mode the type Integer is 16-bits (for TP compatibility), and in Delphi 32-bits, all integers were changed to ArbInt. The basic idea is the same as with ArbFloat. ### Vectors NumLib often passes vectors to procedures as one ArbFloat plus some integer value. This ArbFloat value is the first value of an array containing the vector, the integer is the count of elements used from this array. Internally an array type is mapped over the first value in order to access the other array elements. This way both 0- and 1-based arrays as well as static and dynamic arrays can be passed to the functions and procedures of NumLib. procedure DoSomething(var AVector: ArbFloat; n: Integer); type TLocalVector = array[0..MaxElements] of ArbFloat; var i: Integer; x: TLocalVector absolute AVector; begin for i:=0 to n-1 do // do something with x[i] end; var x: array[1..100]; // Note: This array begins with index 1! ... DoSomething(x[1], 100); ### Matrices A matrix is a two-dimensional arrangement of numbers in m rows and n columns. ${A} = \left[ \begin{array}{cccc} a_{11} & a_{12} & a_{13} & \ldots & a_{1n}\\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & a_{m3} & \ldots & a_{mn} \end{array} \right]$ The matrix elements aij are addressed by row and column indexes, i and j, respectively. In mathematics, the index values usually start with 1 (i = 1..m, j = 1..n) while in computing the first index is preferred to be 0 (i = 0...m-1, j = 0...n-1). #### Generic matrices In this type of matrix, no assumption is made on symmetry and shape of the data arrangement. A constant (real) matrix can be declared in Pascal as a two-dimensional array of floating point numbers: A: array[1..m, 1..n] of ArbFloat (or A: array[0..m-1, 0..n-1] of ArbFloat in 0-based index notation). Similarly, a constant complex matrix can be declared as an array[1..m, 1..n] of complex. When a NumLib procedure needs a matrix the first (top left) matrix element must be passed as a var parameter, along with the number of rows and columns. It is even possible to allocate larger arrays as needed, and in this case the allocated row length (or number of columns) must be specified. Alternatively a matrix also can be specified as a one-dimensional array. This works because all data are found within the same memory block. The array index, k must be converted to the 2D matrix indexes, i, j: k := i * n + j <==> i = k div n, j := k mod n {for 0-based array and matrix indexes} k := (i-1) * n + j <==> i := (k-1) div n + 1, j := (k-1) mod n + 1 {for 1-based array and matrix indexes} Using a one-dimensional array is also the correct way to dynamically allocate memory for a matrix. The first idea of using an array of array of ArbFloat will not work because this does not allocate a single memory block. Instead, individual memory blocks will be allocated for the inner array (rows), and these pointers will be stored in the outer array - this way of storing 2D data is not compatible with the requirements of NumLib. Here is a simple example how to create, populate and read a matrix dynamically $A = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right]$ var A: array of Arbfloat; // Note: the index in a dynamic array is 0-based. ij: Integer; i, j: Integer; p: ^ArbFloat; m, n: Integer; begin m := 2; n := 3; SetLength(A, m, n); A[0] := 1; A[1] := 2; A[2] := 3; A[3] := 4; A[4] := 5; A[5] := 6; for i := 0 to m-1 do begin for j := 0 to n-1 do begin ij := i * n + j; Write(A[ij]); end; WriteLn; end; // or p := @A[0]; // pointer to first array element for i:=0 to m - 1 do begin for j := 0 to n - 1 do begin WriteLn(p^); inc(p); // advance pointer to next array element end; WriteLn; end; #### Symmetric matrices The elements of a symmetric matrix are mirrored at the main diagonal, i.e. aij = aji. In other words: a symmetric matrix is equal to its transpose: A = AT. NumLib contains some routine optimized for such matrices. They require the matrix being stored like a generic matrix in a rectangular m x n array or a linear m*n array. However, only the half of the matrix below the diagonal must be populated (but it can be, of course). #### Symmetric positive definite matrices A matrix A is positive definite if the product dT A d is a positive number for any non-zero vector d. A matrix is symmetric positive definite if, additionally, AT = A. This type of matrices has some interesting properties (determinant is never 0, eigenvalues are real and positive). Therefore NumLib offers specialized routines for many matrix problems with symmetric positive matrices which are superior to those for general matrices. The matrix is stored in the conventional way like a rectangular 2D array of m rows and n columns or a linear 1D array with m*n elements. #### Band matrices A n x n band matrix has zero elements everywhere except for a band along the main diagonal characterized by a left and right bandwidth (l and r, respectively). The NumLib procedures for band matrices assume that the matrix stores only the elements of the band. It is expected the the matrix is passed as a 1D array. Starting at the top/left corner point of the matrix (element A11) the diagonal elements are collected by running along the rows of the matrix, from left to right and from top to bottom. In total, there will be n (l+1+r) - (r(r+1) + l(l+1)) div 2 elements. The calling procedure must allocate a 1D array of at least this size for the diagonal elements. Note that a band matrix cannot be passed as a 2D array. As an example, suppose this 7 x 7 matrix with left bandwidth 2 and right bandwidth 1: $A = \begin{bmatrix} A_{11} & A_{12} & 0 & 0 & 0 & 0 & 0 \\ A_{21} & A_{22} & A_{23} & 0 & 0 & 0 & 0 \\ A_{31} & A_{32} & A_{33} & A_{34} & 0 & 0 & 0 \\ 0 & A_{42} & A_{43} & A_{44} & A_{45} & 0 & 0 \\ 0 & 0 & A_{53} & A_{54} & A_{55} & A_{56} & 0 \\ 0 & 0 & 0 & A_{64} & A_{65} & A_{66} & A_{67} \\ 0 & 0 & 0 & 0 & A_{75} & A_{76} & A_{77} \end{bmatrix}$ The 1D data array to be passed to the NumLib routines consists of these elements const arr: array[1..24] of ArbFloat = ( A11, A22, A21, A22, A23, A31, A32, A33, A34, A42, A43, A44, A45, A53, A54, A55, A56, A64, A65, A66, A67, A75, A76, A77 ); #### Symmetric band matrices In a symmetric matrix, the matrix elements can be mirrored at the main diagonal of the matrix, i.e. Aij = Aji. All elements out side a band of width w to the left and right of the main diagonal are zero. Like with a generic band matrix, NumLib assumes that the only the elements of the diagonals are passed as a 1D array to the procedures dedicated to this matrix type. Since in a symmetric band matrix the left and right band widths are equal it is assumed that only the left band and the main diagonal are stored. Again the relevant diagonal elements are collected by running across the rows of the matrix from left to right and from top to bottom, stopping in each row at the main diagonal. In total, there will be n (w+1) - w(w+1) div 2 elements in the data array. As an example consider this 7 x 7 symmetric band matrix with band width 2: $A = \begin{bmatrix} A_{11} & A_{21} & A_{31} & 0 & 0 & 0 & 0 \\ A_{21} & A_{22} & A_{32} & A_{42} & 0 & 0 & 0 \\ A_{31} & A_{32} & A_{33} & A_{43} & A_{53} & 0 & 0 \\ 0 & A_{42} & A_{43} & A_{44} & A_{54} & A_{64} & 0 \\ 0 & 0 & A_{53} & A_{54} & A_{55} & A_{65} & A_{75} \\ 0 & 0 & 0 & A_{64} & A_{65} & A_{66} & A_{76} \\ 0 & 0 & 0 & 0 & A_{75} & A_{76} & A_{77} \end{bmatrix}$ The 1D array for the NumLib procedures is const arr: array[1..18] of ArbFloat = ( A11, A21, A22, A31, A32, A33, A42, A43, A44, A53, A54, A55, A64, A65, A66, A75, A76, A77 ); #### Tridiagonal matrices A tridiagonal matrix has zero elements everywhere except for the main diagonal and the diagonal immediately above and below it. This way the matrix can be stored more efficiently than a generic matrix because all the empty elements can be dropped. The elements of an n x n square tridiagonal matrix can be stored as three vectors • an array l with n-1 elements for the subdiagonal • an array d with n elements for the main diagonal, and • an array u with n-1 elements for the superdiagonal $A= \begin{bmatrix} d[0] & u[0] & & \dots & 0 \\ l[1] & d[1] & u[1] & & \\ & l[2] & d[2] & u[2] & \\ \vdots & & \ddots & \ddots & \ddots \\ & & l[n-1] & d[n-1] & u[n-1] \\ 0 & & & l[n] & d[n] \end{bmatrix}$ #### Symmetric tridiagonal matrices In these tridiagonal matrices, the subdiagonal and superdiagonal elements are equal. Consequently, NumLib requests only two, instead of three, one-dimensional arrays: • an array d with n elements for the main diagonal, and • an array c with n-1 elements for the superdiagonal and subdiagonal $A= \begin{bmatrix} d[0] & c[1] & & \dots & 0 \\ c[1] & d[1] & c[2] & & \\ & c[2] & d[2] & c[3] & \\ \vdots & & \ddots & \ddots & \ddots \\ & & c[n-1] & d[n-1] & c[n] \\ 0 & & & c[n] & d[n] \end{bmatrix}$ ### Complex numbers NumLib defines the type complex as a Turbo Pascal object. Real and imaginary parts are denoted as xreal and imag, respectively. Methods for basic operations are provided: type complex = object xreal, imag : ArbFloat; procedure Init(r, i: ArbFloat); // Initializes the number with real and imaginary parts r and i, resp procedure Add(c: complex); // Adds another complex number procedure Sub(c: complex); // Subtracts a complex number function Inp(z:complex):ArbFloat; // Calculates the inner product (a.Re * b.Re + a.Im * b.Im) procedure Conjugate; // Conjugates the value (i.e. negates the imaginary part) procedure Scale(s: ArbFloat); // Multiplies real and imaginary part by the real value s function Norm: ArbFloat; // Calculates the "norm", i.e. sqr(Re) + sqr(Im) function Size: ArbFloat; // Size = sqrt(Norm) function Re: ArbFloat; // Returns the real part of the complex number procedure Unary; // Negates real and imaginary parts function Im: ArbFloat; // Returns the imaginary part of the complex number function Arg: ArbFloat; // Calculates the phase phi of the complex number z = x + y i = r exp(i phi) procedure MinC(c: complex); // Converts itself to the minimum of real and imaginary parts of itself and another value, c procedure MaxC(c: complex); // Converts itself to the maximum of real and imaginary ports of itself and another value, c procedure TransF(var t: complex); end; The real and imaginary parts of a complex number can also be retrieved by these stand-alone functions: function Re(c: complex): ArbFloat; function Im(c: complex): ArbFloat; ## Basic operations with matrices and vectors (unit omv) ### Inner product of two vectors The result of the inner product of two vectors a = [a1, a2, ... an] and b = [b1, b2, ... bn] (also called "dot product" or "scalar product") is a scalar, i.e. a single number. It is calculated as the sum of the element-by-element products of the vectors. Both vectors must have the same lengths. $\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^n a_i b_i = a_1 b_1 + a_2 b_2 + \dots + a_n b_n$ NumLib provides the function omvinp for calculation of the inner product: function omvinp(var a, b: ArbFloat; n: ArbInt): ArbFloat; • a and b are the first elements of 1-dimensional arrays representing the vectors a and b, respectively. • n defines the dimension of the vectors (count of array elements). Both vectors must have the same number of elements. Example Calculate the dot product of the vectors a = [0, 1, 2, 2, -1] and b = [3, -1, -2, 2, -1]. program dot_product; uses typ, omv; const n = 5; a: array[0..n-1] of ArbFloat = (0, 1, 2, 2, -1); b: array[0..n-1] of Arbfloat = (3, -1, -2, 2, -1); var ab: ArbFloat; i: Integer; begin ab := omvinp(a[0], b[0], n); Write('Vector a = ['); for i:= Low(a) to High(a) do Write(a[i]:5:0); WriteLn(' ]'); Write('Vector b = ['); for i:= Low(b) to High(b) do Write(b[i]:5:0); WriteLn(' ]'); Write('Dot product a b = '); WriteLn(ab:0:0); end. ### Product of two matrices The product of a m x n matrix A and a n x p matrix B is a m x p matrix C in which the elements are calculated as the inner product of the row vectors of A and the column vectors of B (see here for more details): $C_{ij} = \sum_{k=0}^n A_{ik} B_{kj}$ The NumLib function to perform this multipication is omvmmm ("mmm" = multiply matrix by matrix): procedure omvmmm(var a: ArbFloat; m, n, rwa: ArbInt; var b: ArbFloat; p, rwb: ArbInt; var c: ArbFloat; rwc: ArbInt); • a is the first element of the first input matrix A. The array will not be changed. • m specifies the number of rows (or: column length) of matrix A. • n specifies the number or columns (or: row length) of matrix A. • rwa is the allocated row length of matrix A. This means that the array can be allocated larger than actually needed. Of course, rwa cannot be smaller than n. • b is the first element of the second input matrix B. Again, this array will not be changed. • p specifies the number or columns (or: row length) of matrix B. The number of rows is already defined by the number of columns of matrix A, n. • rwb is the allocated row length of matrix B. rwb cannot be smaller than p. • c is the first element of the output matrix C. After the calculation, this and the following array elements will contain the elements of the product matrix. • The size of the output matrix is m rows x p columns as set up by the input matrices A and B. • rwc is the allocated row length of matrix C. rwc cannot be smaller than p. Example Calculate the product of the matrices $A = \begin{bmatrix} 3 & \ 5 & 4 & 1 & -4 \\ -2 & \ 3 & 4 & 1 & 0 \\ 0 & \ 1 & -1 & -2 & 5 \end{bmatrix} \ \ \ \ \ B = \begin{bmatrix} 3 & 4 & -2 & 0 \\ 0 & -2 & 2 & 1 \\ -1 & 0 & 3 & 4 \\ -2 & -3 & 0 & 1 \\ 1 & 3 & 0 & -1 \end{bmatrix}$ program matrix_multiplication; uses typ, omv; const m = 3; n = 5; p = 4; A: array[1..m, 1..n] of ArbFloat = ( ( 3, 5, 4, 1, -4), (-2, 3, 4, 1, 0), ( 0, 1, -1, -2, 5) ); B: array[1..n, 1..p] of ArbFloat = ( ( 3, 4, -2, 0), ( 0, -2, 2, 1), (-1, 0, 3, 4), (-2, -3, 0, 1), ( 1, 3, 0, -1) ); var C: array[1..m, 1..p] of ArbFloat; i, j: Integer; begin // Calculate product omvmmm( A[1,1], m, n, n, B[1,1], p, p, C[1,1], p ); // Print A WriteLn('A = '); for i:= 1 to m do begin for j := 1 to n do Write(A[i, j]:10:0); WriteLn; end; WriteLn; // Print B WriteLn('B = '); for i:= 1 to n do begin for j := 1 to p do Write(B[i, j]:10:0); WriteLn; end; WriteLn; // Print C WriteLn('C = A B = '); for i:= 1 to m do begin for j := 1 to p do Write(C[i, j]:10:0); WriteLn; end; end. ### Product of a matrix with a vector A matrix A of size m x n multiplied with a vector b of length n yields another vector of length m. The elements of the result vector c are the dot products of the matrix rows with b. Note that the length of vector b must be equal to the number of columns of matrix A. \begin{align*} \mathbf{c} = A\ \mathbf{b}= \left[ \begin{array}{cccc} a_{11} & a_{12} & a_{13} & \ldots & a_{1n}\\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & a_{m3} & \ldots & a_{mn} \end{array} \right] \ \ \left[ \begin{array}{c} b_1\\ b_2\\ b_3\\ \vdots\\ b_n \end{array} \right] = \left[ \begin{array}{c} a_{11}b_1+a_{12}b_2 + a_{13}b_3 + \cdots + a_{1n} b_n\\ a_{21}b_1+a_{22}b_2 + a_{23}b_3 + \cdots + a_{2n} b_n\\ \vdots\\ a_{m1}b_1+a_{m2}b_2 + a_{m3}b_3 + \cdots + a_{mn} b_n\\ \end{array} \right]. \end{align*} NumLib's procedure for this task is called omvmmv ("mmv" = multiply a matrix with a vector): procedure omvmmv(var a: ArbFloat; m, n, rwidth: ArbInt; var b, c: ArbFloat); Example Calculate the product \begin{align*} \mathbf{c} = A\ \mathbf{b}= \left[ \begin{array}{cccc} 3 & 5 & 4 & 1 & -4\\ -2 & 3 & 4 & 1 & 0\\ 0 & 1 & -1 & -2 & 5 \end{array} \right] \ \ \left[ \begin{array}{c} 3\\ 0\\ -1\\ -2\\ 1 \end{array} \right] \end{align*} program matrix_vector_multiplication; uses typ, omv; const m = 3; n = 5; p = 4; A: array[1..m, 1..n] of ArbFloat = ( ( 3, 5, 4, 1, -4), (-2, 3, 4, 1, 0), ( 0, 1, -1, -2, 5) ); b: array[1..n] of ArbFloat = ( 3, 0, -1, -2, 1 ); var c: array[1..m] of ArbFloat; i, j: Integer; begin // Calculate product c = A b omvmmv( A[1,1], m, n, n, b[1], c[1] ); // Print A WriteLn('A = '); for i:= 1 to m do begin for j := 1 to n do Write(A[i, j]:10:0); WriteLn; end; WriteLn; // Print vector b WriteLn('b = '); for i:= 1 to n do Write(b[i]:10:0); WriteLn; WriteLn; // Print result vector c WriteLn('c = A b = '); for i:= 1 to m do Write(c[i]:10:0); WriteLn; end. ### Transpose matrix The transpose matrix AT of matrix A is obtained by flipping rows and columns: \begin{align*} {A} = \left[ \begin{array}{cccc} a_{11} & a_{12} & a_{13} & \ldots & a_{1n}\\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & a_{m3} & \ldots & a_{mn} \end{array} \right] \ \ \ {A^T} = \left[ \begin{array}{cccc} a_{11} & a_{21} & \ldots & a_{m1}\\ a_{12} & a_{22} & \ldots & a_{m2}\\ a_{13} & a_{23} & \ldots & a_{m3} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1n} & a_{2n} & \ldots & a{mn} \end{array} \right] \end{align*} Use the procedure omvtrm to perform this operation with NumLib: procedure omvtrm( var a: ArbFloat; m, n, rwa: ArbInt; var c: ArbFloat; rwc: ArbInt ); • a denotes the first element of the input matrix A. The elements of this array are not changed by the procedure. • m is the number of rows of matrix A. • n is the number of columns of matrix A. • rwa is the number of allocated columns of A. This way the array of A can be larger than acutally needed. • c is the first element of the transposed matrix AT. It has n rows and m columns. • rwc is the number of allocated columns for the transposed matrix. Example Obtain the transpose of the 2x4 matrix ${A} = \left[ \begin{array}{cccc} 1 & 2 & 3 & 4 \\ 11 & 22 & 33 & 44 \end{array} \right]$ program transpose; uses typ, omv; const m = 2; n = 4; A: array[1..m, 1..n] of ArbFloat = ( ( 1, 2, 3, 4), (11, 22, 33, 44) ); var C: array[1..n, 1..m] of ArbFloat; i, j: Integer; begin // Transpose A omvtrm( A[1,1], m, n, n, C[1,1], m ); // Print A WriteLn('A = '); for i:= 1 to m do begin for j := 1 to n do Write(A[i, j]:10:0); WriteLn; end; WriteLn; // Print C WriteLn('AT = '); for i:= 1 to n do begin for j := 1 to m do Write(C[i, j]:10:0); WriteLn; end; end. ### Norm of vectors and matrices A norm is a function which assigns a positive "length" to a vector a or a matrix M. In case of vectors, NumLib supports these norms: • The 1-norm of a vector is defined as the sum of the absolute values of the vector components. It is also called "Taxicab" or "Manhattan" norm because it corresponds to the distance that a taxis has to drive in a rectangular grid of streets. $\|a\|_1 = \sum_{i=1}^n |{a_i}|$ • The 2-norm (also: Euclidian norm) corresponds to the distance of point a from the origin. $\|a\|_2 = \sqrt{\sum_{i=1}^n {a_i}^2}$ • The maximum infinite norm of a vector is the maximum absolute value of its components. $\|a\|_\infty = \max({a_1}, {a_2}, ... {a_n})$ The norms of matrices, as implemented by NumLib, are usually defined by row or column vectors: • The 1-norm of a matrix is the maximum of the absolute column sums: $\|M\|_1 = \max_{1 \le j \le {n}} \sum_{i=1}^m|M_{ij}|$ • The maximum infinite norm of a matrix is the maximum of the absolute row sum. $\|M\|_\infty = \max_{1 \le i \le\ m} \sum_{j=1}^n |M_{ij}|$ • Only the Frobenius norm is calculated by individual matrix elements. It corresponds to the 2-norm of vectors, only the sum runs over all matrix elements: $\|M\|_F = \sqrt{\sum_{i=1}^m \sum_{j=1}^n {M_{ij}}^2}$ These are the NumLib procedure for calculation of norms: function omvn1v(var a: ArbFloat; n: ArbInt): ArbFloat; // 1-norm of a vector function omvn1m(var a: ArbFloat; m, n, rwidth: ArbInt): ArbFloat; // 1-norm of a matrix function omvn2v(var a: ArbFloat; n: ArbInt): ArbFloat; // 2-norm of a vector function omvnfm(Var a: ArbFloat; m, n, rwidth: ArbInt): ArbFloat; // Frobenius norm of a matrix function omvnmv(var a: ArbFloat; n: ArbInt): ArbFloat; // Maximum infinite norm of a vector function omvnmm(var a: ArbFloat; m, n, rwidth: ArbInt): ArbFloat; // Maximum infinite norm of a matrix • a is the first element of the vector or matrix of which the norm is to be calculated • m, in case of a matrix norm, is the number of matrix rows. • n is the number of vector elements in case of a vector norm, or the number of columns in case of a matrix norm. • rwidth, in case of a matrix norm, is the allocated count of columns. This way a larger matrix can be allocated than actually needed. Example program norm; uses typ, omv; const n = 2; m = 4; a: array[0..m-1] of ArbFloat = (0, 1, 2, -3); b: array[0..m-1, 0..n-1] of Arbfloat = ((3, -1), (-2, 2), (0, -1), (2, 1)); var i, j: Integer; begin Write('Vector a = ['); for i:= Low(a) to High(a) do Write(a[i]:5:0); WriteLn(' ]'); WriteLn(' 1-norm of vector a: ', omvn1v(a[0], m):0:3); WriteLn(' 2-norm of vector a: ', omvn2v(a[0], m):0:3); WriteLn(' maximum norm of vector a: ', omvnmv(a[0], m):0:3); WriteLn; WriteLn('Matrix b = '); for i:= 0 to m-1 do begin for j := 0 to n-1 do Write(b[i,j]:5:0); WriteLn; end; WriteLn(' 1-norm of matrix b: ', omvn1m(b[0, 0], m, n, n):0:3); WriteLn('Forbenius norm of matrix b: ', omvnfm(b[0, 0], m, n, n):0:3); WriteLn(' maximum norm of matrix b: ', omvnmm(b[0, 0], m, n, n):0:3); ReadLn; end. ## Determinant of a matrix (unit det) Unit det provides a series of routines for calculation of the determinant of a square matrix. Besides a procedure for general matrices there are also routines optimized for special matrix types (tridiagonal, symmetric). In order to avoid possible overflows and underflows a factor may be split off during calculation, and the determinant is returned as a floating point number f times a multiplier 8k. In many cases, however, k = 0, and the determinant simply is equal to f. ### Determinant of a standard matrix The term "standard matrix" means here a matrix with is stored in the standard NumLib storage as described here, i.e. the matrix is stored as a 2D or 1D array of real numbers. The 2D array has at least n rows and n columns, the 1D array is dimensioned to contain at least '2 elements. Without making any assumptions on the properties of the matrix the determinant can be calculated by the procedure detgen. More efficient algorithms can be used, however, if the matrix has these particular properties: • symmetric matrix --> use procedure detgsy. • symmetric positive definite matrix --> use procedure getgpd. procedure detgen(n, rwidth: ArbInt; var a, f: ArbFloat; var k, term: ArbInt); // --> generic matrix procedure detgsy(n, rwidth: ArbInt; var a, f: ArbFloat; var k, term: ArbInt); // --> symmetric matrix procedure detgpd(n, rwidth: ArbInt; var a, f: ArbFloat; var k, term: ArbInt); // --> symmetric positive definite matrix • n determines the number of rows and columns of the input matrix (it must be square matrix). • rwidth specifies the allocated row length of the input matrix. This takes care of the fact that matrices can be allocated larger than actually needed. • a is the first (i.e. top/left) element of the input matrix. The entire matrix must be contained within the same allocated memory block. • f and k return the determinant of the matrix. The determinant is given by f * 8k. Usually k = 0, and the determinant simply is equal to f. Example Calculate the determinant of the 4x4 matrix $A= \left[ \begin{array}{rrrr} 4 & 2 & 4 & 1 \\ 30 & 20 & 45 & 12 \\ 20 & 15 & 36 & 10 \\ 35 & 28 & 70 & 20 \end{array} \right]$ The matrix is positive definite because all elements are positive and the product bT A b cannot become zero or negative for any non-zero vector b. While the routine detgpd is the best for this matrix type the code below contains also commented-out calls to the other routines. program determinant_spd_matrix; // "spd" = symmetric positive definite uses math, typ, det; const n = 4; const a: array[1..n, 1..n] of ArbFloat = ( ( 5, 7, 6, 5), ( 7, 10, 8, 7), ( 6, 8, 10, 9), ( 5, 7, 9, 10) ); var f: ArbFloat; k: Integer; term: Integer = 0; i, j: Integer; begin // write input matrix WriteLn('A = '); for i := 1 to n do begin for j := 1 to n do Write(a[i, j]:10:0); WriteLn; end; WriteLn; // Calculate determinant of symmetric positive definite matrix detgpd(n, n, a[1, 1], f, k, term); // or uncomment to use one of the other methods // detgen(n, n, a[1, 1], f, k, term); // detgsy(n, n, a[1, 1], f, k, term); if term = 1 then WriteLn('Determinant of A = ', f * intpower(8, k):0:9, ' (k = ', k, ')') else WriteLn('Incorrect input'); end. ### Determinant of a band matrix If a matrix has the shape of a band matrix then its determinant is best calculated by using the procedure detgba procedure detgba(n, l, r: ArbInt; var a, f: ArbFloat; var k, term:ArbInt); • n denotes the number of columns and rows of the matrix (it must be square). • l identifies the left bandwidth, i.e. how many diagonals the band extends to the left, or below, the main diagonal. • r identifies the right bandwidth, i.e. how many diagons the band extends to the right, or above, the main diagonsl. • a specifies the first element of a 1D array which contains the elements of the diagonal band, see this. This array contains only the band elements and is obtained by running across the rows of the band matrix from left to right and top to bottom, starting at element A11. It must be dimensioned to contain at least n*(l+1+r) - (r*(r+1)+l*(l+1)) div 2 elements. • f and k return the determinant of the matrix. The determinant is given by f * 8k. Usually k = 0, and the determinant simply is equal to f. Example Calculate the determinant of the 5x5 band matrix with left bandwidth l = 2 and right bandwidth r = 1 $A = \begin{bmatrix} 1 & 2 & 0 & 0 & 0 \\ 2 & 1 & 1 & 0 & 0 \\ 1 & 3 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \end{bmatrix}$ program determinant_band_matrix; uses math, typ, det; const n = 5; L = 2; R = 1; nelements = n*(L+1+R) - (R*(R+1) + L*(L+1)) div 2; const // Diagonal elements arranged in rows, elements outside of the band omitted! a: array[1..nelements] of Arbfloat = ( 1, 2, 2, 1, 1, 1, 3, 0, 1, 1, 1, 1, 1, 1, 1, 1 ); var f: ArbFloat; term: Integer = 0; i, i1, i2, j, k: Integer; begin // Write diagonal elements WriteLn('n = ', n); Writeln('Left band width L = ', L); WriteLn('Right band width R = ', R); Write('Diagonal elements of A = ', a[1]:0:0); for k := 2 to nelements do Write(a[k]:5:0); WriteLn; WriteLn; // write reconstructed band input matrix (not needed for calculation) WriteLn('Reconstructed A = '); k := 1; for j := 1 to n do begin i1 := j - L; i2 := i1 + L + R; for i:=1 to i1-1 do Write(0.0:5:0); for i := i1 to i2 do if (i >= 1) and (i <= n) then begin Write(a[k]:5:0); inc(k); end; for i:= i2 + 1 to n do if i <= n then Write(0.0:5:0); WriteLn; end; WriteLn; // Calculate determinant detgba(n, l, r, a[1], f, k, term); if term = 1 then WriteLn('Determinant of A = ', f * intpower(8, k):0:9, ' (k = ', k, ')') else WriteLn('Incorrect input'); end. ### Determinant of a symmetric positive definite band matrix If a matrix has the shape of a band matrix and is symmetric positive definite then detgpb is the dedicated routine for calculation of its determinant. procedure detgpb(n, w: ArbInt; var a, f: ArbFloat; var k, term:ArbInt); • n denotes the number of columns and rows of the matrix (it must be square). • w identifies the bandwidth, i.e. how many diagonals the band extends to the left or to the right of the main diagonal. Left and right bandwidth must be equal. • a specifies the first element of a 1D array which contains the elements of the diagonal band, see this. This array contains only the left band elements and the elements of the main diagonal. It is obtained by running across the rows of the band matrix from left to right and top to bottom, but stopping at the main diagonal - the right band is ignored since it is equal to the left band. The array must be dimensioned to contain at least n*(w+1) - (w*(w+1)) div 2 elements. • f and k return the determinant of the matrix. The determinant is given by f * 8k. Usually k = 0, and the determinant simply is equal to f. Example Calculate the determinant of the 7x7 symmetric band matrix with bandwidth w = 2 $A= \begin{bmatrix} 5 & -4 & 1 & & & & \\ -4 & 6 & -4 & 1 & & & \\ 1 & -4 & 6 & -4 & 1 & & \\ & 1 & -4 & 6 & -4 & 1 & \\ & & 1 & -4 & 6 & -4 & 1 \\ & & & 1 & -4 & 6 & -4 \\ & & & & 1 & -4 & 5 \end{bmatrix}$ program determinant_spd_band_matrix; // "spd" = symmetric positive definite uses math, typ, det; const n = 7; w = 2; nelements = n * (w + 1) - (w * (w + 1)) div 2; const // Band elements arragned in rows, elements outside of the band omitted! a: array[1..nelements] of Arbfloat = ( 5, -4, 6, 1, -4, 6, 1, -4, 6, 1, -4, 6, 1, -4, 6, 1, -4, 5 ); function MatrixIndexToArrayIndex(i, j, n, w: Integer): Integer; function DiagElementIndex(i: Integer): Integer; var k: Integer; begin Result := 1; if i = 1 then exit; // Rows truncated at left for k := 2 to w do begin Result := Result + k; if k = i then exit; end; // full rows and those truncated at right for k := w+1 to n do begin Result := Result + w + 1; if k = i then exit; end; Result := n; end; var d: Integer; begin if j > i then begin Result := MatrixIndexToArrayIndex(j, i, n, w); exit; end; Result := DiagElementIndex(i); if (i = j) then exit; d := i - j; if d > w then Result := -1 else begin dec(Result, d); if (Result < 1) then Result := -1 end; end; var i, j, k: Integer; f: ArbFloat; term: Integer = 0; begin // Write diagonal elements WriteLn('n = ', n); Writeln('(One-sided) band width W = ', w); Write('Diagonal elements of A = ', a[1]:0:0); for k := 2 to nelements do Write(a[k]:3:0); WriteLn; WriteLn; // write reconstructed band input matrix (not needed for calculation) WriteLn('Reconstructed A = '); k := 1; for i := 1 to n do begin for j := 1 to n do begin k := MatrixIndexToArrayIndex(i, j, n, w); if k = -1 then Write(0.0:3:0) else Write(a[k]:3:0); end; WriteLn; end; WriteLn; // Calculate determinant detgpb(n, w, a[1], f, k, term); if term = 1 then WriteLn('Determinant of A = ', f * intpower(8, k):0:9, ' (k = ', k, ')') else WriteLn('Incorrect input'); end. ### Determinant of a tridiagonal matrix Calculation of the determinant of a tridiagonal matrix is done by the procedure detgtr: procedure detgtr(n: ArbInt; var l, d, u, f: ArbFloat; var k, term:ArbInt); • n denotes the number of columns and rows of the matrix (it must be square). • l specifies the first element in the subdiagonal of the matrix. This 1D array must be dimensioned to at least n-1 elements. • d specifies the first element along the main diagonal of the matrix. This 1D array must be dimensioned to contain at least n elements. • u specifies the first element along the superdiagonal of the matrix. The array must be dimensioned to have at least n-1 elements. • f and k return the determinant of the matrix. The determinant is given by f * 8k. Usually k = 0, and the determinant simply is equal to f. Example Calculate the determinant of the matrix $A= \left( \begin{array}{rrrr} 1 & 1 & & \\ 2 & 1 & 1 & \\ & 1 & 1 & 0 \\ & & 1 & 2 \end{array} \right)$ program determinant_tridiagonal_matrix; {mode objfpc}{\$H+}
uses
math, typ, det;
const
n = 4;
const
u: array[0..n-2] of ArbFloat = (-1, -1, -1 ); // superdiagonal
d: array[0..n-1] of ArbFloat = ( 2, 2, 2, 2); // main diagonal
l: array[1..n-1] of ArbFloat = ( -1, -1, -1); // subdiagonal
var
f: ArbFloat;
k: Integer;
term: Integer = 0;
i, j: Integer;
begin
// write input matrix
WriteLn('A = ');
for i := 0 to n-1 do begin
for j := 0 to i-2 do
Write(0.0:5:0);
if i > 0 then
Write(l[i]:5:0);
Write(d[i]:5:0);
if i < n-1 then
Write(u[i]:5:0);
for j:=i+2 to n-1 do
Write(0.0:5:0);
WriteLn;
end;
WriteLn;
// Calculate determinant
detgtr(n, l[1], d[0], u[0], f, k, term);
if term = 1 then
WriteLn('Determinant of A = ', f * intpower(8, k):0:9, ' (k = ', k, ')')
else
WriteLn('Incorrect input');
end.
## Inverse of a matrix (unit inv)
Suppose a square matrix A. Then the matrix A-1 is the inverse of A if the product A-1 A is the identity matrix I (all elements zero, except for the diagonal elements which are 1).
The inverse matrix exists if the determinant of A is not zero.
$A^{-1} A = I = \begin{bmatrix} 1 & & 0 \\ & \ddots & \\ 0 & & 1 \end{bmatrix}$
NumLib provides various routines for calculation of the inverse matrix. Which algorithm is used depends on the properties of the matrix A. All routines discussed here have in common that the matrix is stored in the standard NumLib way, i.e. as a 2D array of real numbers with n rows and m columns, or as a 1D array with n m elements.
No special requirements are made for the matrix passed to invgen, except that it is square. The calculation in this routine is based on LU decomposition with partial pivoting. If the matrix is symmetric, it can be processed by invgsy which uses a reduction to tridiagonal form internally. A symmetric positive definite matrix can be passed to invgpd where the inverse is calculated by Cholesky decomposition.
procedure invgen(n, rwidth: ArbInt; var ai: ArbFloat; var term: ArbInt); // --> generic matrix
procedure invgsy(n, rwidth: ArbInt; var ai: ArbFloat; var term: ArbInt); // --> symmetric matrix
procedure invgpd(n, rwidth: ArbInt; var ai: ArbFloat; var term: ArbInt); // --> symmetric positive definite matrix
• n indicates the size of the matrix.
• rwidth is length of the rows of the two-dimensional array which holds the matrix. Normally n = rwidth, but the matrix could have been declared larger than actually needed; in this case rwidth refers to the declared size of the data array.
• ai is the first (top-left) element of the matrix in the data array. After the calculation the input data are overwritten by the elements of the inverse matrix in case of successful completion (term = 1); in other cases, the matrix elements are undefined.
• term provides an error code:
• 1 - successful completion, the elements of the inverse matrix can be found at the same place as the input matrix.
• 2 - the inverse could not be calculated because the input matrix is (almost) singular.
• 3 - incorrect input data, n < 1.
Example
Calculate the inverse of the symmetric matrix $A = \left[ \begin{array}{rrrr} 5 & 7 & 6 & 5 \\ 7 & 10 & 8 & 7 \\ 6 & 8 & 10 & 9 \\ 5 & 7 & 9 & 10 \end{array} \right]$ This matrix is symmetric and positive definite (the product bT A b with any vector b = [b1, b2, b3]T cannot become zero or negative since all elements of this matrix are positive). Therefore, invgpd is best-suited for this task although the other routines can be used as well (uncomment their calls below).
program inverse_matrix;
uses
typ, inv, omv;
const
n = 4;
D = 5;
var
// a is the input matrix to be inverted
// Note that this is matrix must be symmetric positive definite.
a: array[1..n, 1..n] of ArbFloat = (
(5, 7, 6, 5),
(7, 10, 8, 7),
(6, 8, 10, 9),
(5, 7, 9, 10)
);
b: array[1..n, 1..n] of Arbfloat;
c: array[1..n, 1..n] of ArbFloat;
term: Integer = 0;
i, j: Integer;
begin
// write input matrix
WriteLn('a = ');
for i := 1 to n do begin
for j := 1 to n do
Write(a[i, j]:10:D);
WriteLn;
end;
WriteLn;
// Store input matrix because it will be overwritten by the inverse of a
for i := 1 to n do
for j := 1 to n do
b[i, j] := a[i, j];
// Calculate inverse -- uncomment the procedure to be used.
//invgen(n, n, a[1, 1], term);
//invgsy(n, n, a[1, 1], term);
invgpd(n, n, a[1, 1], term);
// write inverse
WriteLn('a^(-1) = ');
for i := 1 to n do begin
for j := 1 to n do
Write(a[i, j]:10:D);
WriteLn;
end;
WriteLn;
// Check validity of result by multiplying inverse with saved input matrix
omvmmm(a[1, 1], n, n, n,
b[1, 1], n, n,
c[1, 1], n);
// write inverse
WriteLn('a^(-1) x a = ');
for i := 1 to n do begin
for j := 1 to n do
Write(c[i, j]:10:D);
WriteLn;
end;
end.
## Solving systems of linear equations (unit sle)
A system of linear equations (or linear system) is a collection of two or more linear equations involving the same set of variables, x1 ... xn.
$\begin{cases} a_{11} x_1 + a_{12} x_2 + \dots + a_{1n} x_n = b_1 \\ a_{21} x_1 + a_{22} x_2 + \dots + a_{2n} x_n = b_n \\ \ \ \ \vdots \\ a_{m1} x_1 + a_{m2} x_2 + \dots + a_{mn} x_n = b_m \end{cases}$
In order to find a solution which satisfies all equations simultaneously the system of equations is usually converted to a matrix equation A x = b, where
$A = \left [ \begin{array}{ccc} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{array} \right ], \ \mathbf{x} = \left [ \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \right ], \ \mathbf{b} = \left [ \begin{array}{c} b_1 \\ b_2 \\ \vdots \\ b_m \end{array} \right ]$
NumLib offers several procedures to solve the matrix equation depending on the properties of the matrix.
### Square matrices with standard storage
Here we describe several routines in which the matrix of the equation system is square, i.e. there are as many equations as unknowns. This matrix is assumed to be passed to the procedures in the standard NumLib way as the first element of a 2D or 1D array. The 2D array must be dimensioned to contain at least n rows and n columns, the 1D array must contain at least n2 elements.
The most general procedure is slegen and applied to any square matrix A. The system is solved using the Gauss elimination procedure with partial pivoting.
If the matrix A is symmetric then the reduction to tridiagonal form generally is more stable. This is done in routine slegsy.
If A, furthermore, is a symmetric positive definite matrix, then the Cholesky method is best-suited. This is applied in routine slegpd.
procedure slegen(n, rwidth: ArbInt; var a, b, x, ca: ArbFloat; var term: ArbInt); // generic matrix
procedure slegsy(n, rwidth: ArbInt; var a, b, x, ca: ArbFloat; var term: ArbInt); // symmetric matrix
procedure slegpd(n, rwidth: ArbInt; var a, b, x, ca: ArbFloat; var term: ArbInt); // symmetric positive definite matrix
• n is the number of unknown variables. It must be the same as the number of columns and rows of the coefficent matrix.
• rwidth identifies the allocated number of columns of the coefficient matrix. This is useful if the matrix is allocated larger than needed. It is required that n <= rwidth.
• a is the first (i.e. top left) element of the coefficient matrix A; the other elements must follow within the same allocated memory block. The matrix will not be changed during the calculation.
• b is the first element of the array containing the constant vector b. The array length at least must be equal to n. The vector will not be changed during the calculation.
• x is the first element of the array to receive the solution vector x. It must be allocated to contain at least n values.
• ca is a parameter to describe the accuracy of the solution.
• term returns an error code:
• 1 - successful completion, the solution vector x is valid
• 2 - the solution could not have been determined because the matrix is (almost) singular.
• 3 - error in input values: n < 1.
Example
Solve this system of linear equations:
$\begin{cases} 5 x_1 + 7 x_2 + 6 x_3 + 5 x_4 = 57 \\ 7 x_1 + 10 x_2 + 8 x_3 + 7 x_4 = 79 \\ 6 x_1 + 8 x_2 + 10 x_3 + 9 x_4 = 88 \\ 5 x_1 + 7 x_2 + 9 x_3 + 10 x_4 = 86 \end{cases} \qquad \Rightarrow \qquad A= \left[ \begin{array}{rrrr} 5 & 7 & 6 & 5 \\ 7 & 10 & 8 & 7 \\ 6 & 8 & 10 & 9 \\ 5 & 7 & 9 & 10 \end{array} \right] , \; \mathbf{b} = \left[ \begin{array}{r} 57 \\ 79 \\ 88 \\ 86 \end{array} \right]$
This matrix is symmetric positive definite. Therefore, all routines presented here can be applied. Uncomment the requested line in the code below. As a test the result vector x is multiplied to matrix A, and the result must equal the constant vector b.
program solve_sys_lin_eq;
uses
typ, sle, omv;
const
n = 4;
A: array[1..n, 1..n] of ArbFloat = (
( 5, 7, 6, 5),
( 7, 10, 8, 7),
( 6, 8, 10, 9),
( 5, 7, 9, 10)
);
b: array[1..n] of ArbFloat = (
57, 79, 88, 86
);
var
x: array[1..n] of ArbFloat;
b_test: array[1..n] of ArbFloat;
ca: ArbFloat;
i, j: Integer;
term: Integer;
begin
WriteLn('Solve matrix system A x = b');
WriteLn;
// Print A
WriteLn('Matrix A = ');
for i:= 1 to n do begin
for j := 1 to n do
Write(A[i, j]:10:0);
WriteLn;
end;
WriteLn;
// Print b
WriteLn('Vector b = ');
for i:= 1 to n do
Write(b[i]:10:0);
WriteLn;
WriteLn;
// Solve - select one of these methods
slegen(n, n, A[1, 1], b[1], x[1], ca, term);
// slegsy(n, n, A[1, 1], b[1], x[1], ca, term);
// slegpd(n, n, A[1, 1], b[1], x[1], ca, term);
if term = 1 then begin
WriteLn('Solution vector x = ');
for i:= 1 to n do
Write(x[i]:10:0);
WriteLn;
WriteLn;
omvmmv(A[1,1], n, n, n, x[1], b_test[1]);
WriteLn('Check result: A x = (must be equal to b)');
for i:= 1 to n do
Write(b_test[i]:10:0);
WriteLn;
end
else
WriteLn('Error');
end.
### Band matrix
A special solution, slegba, is implemented for a band matrix, i.e. a matrix in which all elements are zero outside a band of width l below and of width r above the main diagonal.
procedure slegba(n, l, r: ArbInt; var a, b, x, ca: ArbFloat; var term:ArbInt);
• n denotes the number of columns and rows of the matrix (it must be square).
• l identifies the left bandwidth, i.e. the number of diagonals the band extends below (or to the left of) the main diagonal.
• r identifies the right bandwidth, i.e. the number of diagonals the band extends abofe (or to the right of) the main diagonsl.
• a specifies the first element of a 1D array which contains the elements of the diagonal band, see this. This array contains only the band elements and is obtained by running across the rows of the band matrix from left to right and top to bottom, starting at element A11. It must be dimensioned to contain at least n*(l+1+r) - (r*(r+1)+l*(l+1)) div 2 elements. Note that a 2D array cannot be used for this routine.
• b is the first element of the array containing the constant vector b. The array length at least must be equal to n. The vector will not be changed during the calculation.
• x is the first element of the array to receive the solution vector x. It must be allocated to contain at least n values.
• ca is a parameter to describe the accuracy of the solution.
• term returns an error code:
• 1 - successful completion, the solution vector x is valid
• 2 - the solution could not have been determined because the matrix is (almost) singular.
• 3 - error in input values: n < 1, l < 0, l >= n, r < 0, or r >= n
Example
Solve this system of linear equations:
$\begin{array}{ccccccc} 5 x_1 & - & 4 x_2 & + & x_3 & & & & & & & & & = 0 \\ -4 x_1 & + & 6 x_2 & - & 4 x_3 & + & x_4 & & & & & & & = 0 \\ x_1 & - & 4 x_2 & + & 6 x_3 & - & 4 x_4 & & & & & & & = 0 \\ & & x_2 & - & 4 x_3 & + & 6 x_4 & - & 4 x_5 & & & & & = 1 \\ & & & & x_3 & - & 4 x_4 & + & 6 x_5 & - & 4 x_6 & + & x_7 & = 0 \\ & & & & & & x_4 & - & 4 x_5 & + & 6 x_6 & - & 4 x_7 & = 0 \\ & & & & & & & & x_5 & - & 4 x_6 & + & 5 x_7 & = 0 \\ \end{array} \qquad \Rightarrow \qquad A= \left[ \begin{array}{rrrr} 5 & -4 & 1 & 0 & 0 & 0 & 0 \\ -4 & 6 & -4 & 1 & 0 & 0 & 0 \\ 1 & -4 & 6 & -4 & 1 & 0 & 0 \\ 0 & 1 & -4 & 6 & -4 & 1 & 0 \\ 0 & 0 & 1 & -4 & 6 & -4 & 1 \\ 0 & 0 & 0 & 1 & -4 & 6 & -4 \\ 0 & 0 & 0 & 0 & 1 & -4 & 5 \end{array} \right] , \; \mathbf{b} = \left[ \begin{array}{r} 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \right]$
program solve_band;
uses
typ, sle, omv;
const
n = 7;
L = 2;
R = 2;
nelements = n * (L + 1 + R) - (L * (L + 1) + R * (R + 1)) div 2;
// Band elements arranged in rows, elements outside of the band omitted!
a: array[1..nelements] of Arbfloat = (
5, -4, 1,
-4, 6, -4, 1,
1, -4, 6, -4, 1,
1, -4, 6, -4, 1,
1, -4, 6, -4, 1,
1, -4, 6, -4,
1, -4, 5
);
b: array[1..n] of ArbFloat = (
0, 0, 0, 1, 0, 0, 0
);
var
x: array[1..n] of ArbFloat;
a_test: array[1..n, 1..n] of ArbFloat;
b_test: array[1..n] of ArbFloat;
ca: ArbFloat;
i, j, k: Integer;
term: Integer;
begin
WriteLn('Solve matrix system A x = b where A is a band matrix');
WriteLn;
// Write diagonal elements
WriteLn('Matrix A:');
WriteLn(' n = ', n);
Writeln(' Left band width L = ', L);
WriteLn(' Right band width R = ', R);
WriteLn(' Diagonal elements of A = ');
for k := 1 to nelements do
Write(a[k]:5:0);
WriteLn;
WriteLn;
// Solve
slegba(n, l, r, A[1], b[1], x[1], ca, term);
if term = 1 then begin
// To test the result we multiply A with the solution vector x and check
// whether the result is equal to b.
// Since mutliplication of a matrix stored like a band matrix is quite
// cumbersome we copy the band matrix into a standard matrix.
FillChar(a_test, SizeOf(a_test), 0);
k := 1;
for i:=1 to L do
for j := i-L to i+R do
if (j >= 1) and (j <= n) then begin
a_test[i,j] := A[k];
inc(k);
end;
for i:= L+1 to n-R do
for j := i-L to i+R do begin
a_test[i,j] := A[k];
inc(k);
end;
for i := n-R+1 to n do
for j := i-L to i+R do
if j <= n then begin
a_test[i,j] := A[k];
inc(k);
end;
// Print A
WriteLn('Matrix A =');
for i:=1 to n do begin
for j:=1 to n do
Write(a_test[i,j]:5:0);
WriteLn;
end;
WriteLn;
// Print b
WriteLn('Vector b = ');
for i:= 1 to n do
Write(b[i]:10:0);
WriteLn;
WriteLn;
// Print solution
WriteLn('Solution vector x = ');
for i:= 1 to n do
Write(x[i]:10:3);
WriteLn;
WriteLn;
omvmmv(a_test[1,1], n, n, n, x[1], b_test[1]);
WriteLn('Check result: A x = (must be equal to b)');
for i:= 1 to n do
Write(b_test[i]:10:0);
WriteLn;
end
else
WriteLn('Error');
end.
### Symmetric positive definite band matrix
The optimized solution method for a symmetric positive band matrix is called slegpb.
procedure slegpb(n, w: ArbInt; var a, b, x, ca: ArbFloat; var term: ArbInt);
• n denotes the number of columns and rows of the matrix (it must be square).
• w identifies the (one-sided) bandwidth, i.e. the number of diagonals the band extends below (or to the left of) the main diagonal. This is the same value as the number of diagonals the band extends above (or to the right of) the main diagonsl.
• a specifies the first element of a 1D array which contains the elements of the diagonal band, see this. This array contains only the band elements and is obtained by running across the rows of the band matrix from left to right and top to bottom, starting at element A11, but stopping at the main diagonal. This means that the array contains only the left band and the main diagonal elements. It must be dimensioned to contain at least n*(w+1) - (w*(w+1)) div 2 elements. Note that a 2D array cannot be used for this routine.
• b is the first element of the array containing the constant vector b. The array length at least must be equal to n. The vector will not be changed during the calculation.
• x is the first element of the array to receive the solution vector x. It must be allocated to contain at least n values. If term is not 1 then the solution array contains undefined data.
• ca is a parameter to describe the accuracy of the solution.
• term returns an error code:
• 1 - successful completion, the solution vector x is valid
• 2 - the solution could not have been determined because the matrix is (almost) singular.
• 3 - error in input values: n<1, w<0, or w>=n
Example
The matrix of the band matrix example is symmetrical positive definite. Therefore, procedure slegpb can be applied as well:
program solve_spd; // "spd" = symmetric positive definite
uses
typ, sle, omv;
const
n = 7;
w = 2;
nelements = n * (w + 1) - (w * (w + 1)) div 2;
// Band elements arranged in rows, elements outside of the band as well as right diagonals omitted!
a: array[1..nelements] of Arbfloat = (
5,
-4, 6,
1, -4, 6,
1, -4, 6,
1, -4, 6,
1, -4, 6,
1, -4, 5
);
b: array[1..n] of ArbFloat = (
0, 0, 0, 1, 0, 0, 0
);
var
x: array[1..n] of ArbFloat;
a_test: array[1..n, 1..n] of ArbFloat;
b_test: array[1..n] of ArbFloat;
ca: ArbFloat;
i, j, k: Integer;
term: Integer;
begin
WriteLn('Solve matrix system A x = b where A is a symmetric positive definite band matrix');
WriteLn;
// Write diagonal elements
WriteLn('Matrix A:');
WriteLn(' n = ', n);
Writeln(' (One-sided) band width w = ', w);
WriteLn(' Diagonal elements of A = ');
for k := 1 to nelements do
Write(a[k]:5:0);
WriteLn;
WriteLn;
// Solve
slegpb(n, w, A[1], b[1], x[1], ca, term);
if term = 1 then begin
// To test the result we multiply A with the solution vector x and check
// whether the result is equal to b.
// Since mutliplication of a matrix stored like a band matrix is quite
// cumbersome we copy the band matrix into a standard matrix.
FillChar(a_test, SizeOf(a_test), 0);
i := 1;
k := 1;
while (k <= nelements) and (i <= n) do begin
for j := i - w to i do
if (j >= 1) and (j <= n) then begin
a_test[i, j] := a[k];
a_test[j, i] := a[k];
inc(k);
end;
inc(i);
end;
// Print A
WriteLn('Matrix A =');
for i:=1 to n do begin
for j:=1 to n do
Write(a_test[i,j]:5:0);
WriteLn;
end;
WriteLn;
// Print b
WriteLn('Vector b = ');
for i:= 1 to n do
Write(b[i]:10:0);
WriteLn;
WriteLn;
// Print solution
WriteLn('Solution vector x = ');
for i:= 1 to n do
Write(x[i]:10:3);
WriteLn;
WriteLn;
omvmmv(a_test[1,1], n, n, n, x[1], b_test[1]);
WriteLn('Check result: A x = (must be equal to b)');
for i:= 1 to n do
Write(b_test[i]:10:3);
WriteLn;
end
else
WriteLn('Error');
end.
### Tridiagonal matrix
NumLib offers two specialized procedures for solving a system of linear equations based on a tridiagonal matrix A. Both rely on the memory-optimized storage of tridiagonal matrices as described here. They differ in the internal calculation algorithm and in the behavior with critical matrices.
procedure sledtr(n: ArbInt; var l, d, u, b, x: ArbFloat; var term: ArbInt);
procedure slegtr(n: ArbInt; var l, d, u, b, x, ca: ArbFloat; var term: ArbInt);
• n is the number of unknown variables. It must be the same as the number of columns and rows of the coefficent matrix.
• l specifies the first element in the subdiagonal of the matrix A. This 1D array must be dimensioned to at least n-1 elements.
• d specifies the first element along the main diagonal of the matrix A. This 1D array must be dimensioned to contain at least n elements.
• u specifies the first element along the superdiagonal of the matrix A. The array must be dimensioned to have at least n-1 elements.
• b is the first element of the array containing the constant vector b. The array length at least must be equal to n. The vector will not be changed during the calculation.
• x is the first element of the array to receive the solution vector x. It must be allocated to contain at least n values.
• ca is a parameter to describe the accuracy of the solution.
• term returns an error code:
• 1 - successful completion, the solution vector x is valid
• 2 - the solution could not have been determined because the matrix is (almost) singular.
• 3 - error in input values: n < 1.
sledtr is numerically stable if matrix A fulfills one of these conditions:
• A is regular (i.e. its inverse matrix exists), and A is columnar-diagonally dominant; this means:
• |d1| ≥ |l2|,
• |di| ≥ |ui-1| + |li+1|, i = 2, ..., n-1,
• |dn| ≥ |un-1|
• A is regular, and A is diagonally dominant; this means:
• |d1| ≥ |l2|,
• |di| ≥ |ui| + |li|, i = 2, ..., n-1,
• |dn| ≥ |un|
• A is symmetric and positive-definite.
However, this procedure does not provide the parameter ca from which the accuracy of the determined solution can be evaluated. If this is needed the (less stable) procedure slegtr must be used.
Example Solve this tridiagonal system of linear equations for n = 8:
$\begin{cases} 188 x_1 - 100 x_2 = 0 \\ -100 x_1 + 188 x_2 -100 x_3 = 0 \\ \vdots \\ -100 x_{n-2} + 188 x_{n-1} - 100 x_n = 0 \\ -100 x_{n-1} + 188 x_n = 0 \end{cases} \qquad \Rightarrow \qquad A= \left[ \begin{array}{rrrrr} 188 & -100 & & & 0 \\ -100 & 188 & -100 & & \\ & \ddots & \ddots & \ddots & \\ & & -100 & 188 & -100 \\ & & & -100 & 188 \\ \end{array} \right] , \; \mathbf{b} = \left[ \begin{array}{r} 88 \\ -12 \\ \vdots \\ -12 \\ 88 \end{array} \right]$
program solve_tridiag;
uses
typ, sle;
const
n = 8;
u: array[1..n-1] of ArbFloat = (-100, -100, -100, -100, -100, -100, -100 );
d: array[1..n] of ArbFloat = ( 188, 188, 188, 188, 188, 188, 188, 188);
l: array[2..n] of ArbFloat = ( -100, -100, -100, -100, -100, -100, -100);
b: array[1..n] of ArbFloat = ( 88, -12, -12, -12, -12, -12, -12, 88);
var
A: array[1..n, 1..n] of ArbFloat;
x: array[1..n] of ArbFloat;
b_test: array[1..n] of ArbFloat;
ca: ArbFloat;
i, j: Integer;
term: Integer;
begin
WriteLn('Solve tridiagonal matrix system A x = b');
WriteLn;
Write('Superdiagonal of A:':25);
for i := 1 to n-1 do
Write(u[i]:10:0);
WriteLn;
Write('Main diagonal of A:':25);
for i:= 1 to n do
Write(d[i]:10:0);
WriteLn;
Write('Subdiagonal of A:':25);
Write('':10);
for i:=2 to n do
Write(l[i]:10:0);
WriteLn;
Write('Vector b:':25);
for i:=1 to n do
Write(b[i]:10:0);
WriteLn;
// Solve
slegtr(n, l[2], d[1], u[1], b[1], x[1], ca, term);
{ or:
sledtr(n, l[2], d[1], u[1], b[1], x[1], term);
}
if term = 1 then begin
Write('Solution vector x:':25);
for i:= 1 to n do
Write(x[i]:10:0);
WriteLn;
// Multiply A with x to test the result
// NumLib does not have a routine to multiply a tridiagonal matrix with a
// vector... Let's do it manually.
b_test[1] := d[1]*x[1] + u[1]*x[2];
for i := 2 to n-1 do
b_test[i] := l[i]*x[i-1] + d[i]*x[i] + u[i]*x[i+1];
b_test[n] := l[n]*x[n-1] + d[n]*x[n];
Write('Check b = A x:':25);
for i:= 1 to n do
Write(b_test[i]:10:0);
WriteLn;
end
else
WriteLn('Error');
end.
### Overdetermined systems (least squares)
Unlike the other routines in unit sle which require a square matrix A, slegls can solve linear systems described by a rectangular matrix which has more rows than colums, or speaking in terms of equations, has more equations than unknowns. Such a system generally is not solvable exactly. But an approximate solution can be found such that the sum of squared residuals of each equation, i.e. the norm $\|\mathbf{b} - A \mathbf{x}\|_2$ is as small as possible. The most prominent application of this technique is fitting of equations to data (regression analysis).
procedure slegls(var a: ArbFloat; m, n, rwidtha: ArbInt; var b, x: ArbFloat; var term: ArbInt);
• a is the first (i.e. top left) element of an array with the coefficient matrix A; the other elements must follow within the same allocated memory block. The array will not be changed during the calculation.
• m is the number of rows of the matrix A (or: the number of equations).
• n is the number of columns of the matrix A (or: the number of unknown variables). n must not be larger than m.
• rwidth identifies the allocated number of columns of the coefficient matrix A. This is useful if the matrix is allocated larger than needed. It is required that n <= rwidth.
• b is the first element of the array containing the constant vector b. The array length must correspond to the number of matrix rows m, but the array can be allocated to be larger.
• x is the first element of the array to receive the solution vector x. The array length must correspond to the number of matrix columns n, but the array can be allocated to be larger.
• term returns an error code:
• 1 - successful completion, the solution vector x is valid
• 2 - there is no unambiguous solution because the columns of the matrix are linearly dependant..
• 3 - error in input values: n < 1, or n > m.
The method is based on reduction of the array A to upper triangle shape through Householder transformations.
Example
Find the least-squares solution for the system A x = b of 4 equations and 3 unknowns with
$A= \left( \begin{array}{rrr} 1 & 0 & 1 \\ 1 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{array} \right), \ \ b= \left( \begin{array}{r} 21 \\ 39 \\ 21 \\ 30 \end{array} \right).$
program solve_leastsquares;
uses
typ, sle, omv;
const
m = 4;
n = 3;
A: array[1..m, 1..n] of ArbFloat = (
(1, 0, 1),
(1, 1, 1),
(0, 1, 0),
(1, 1, 0)
);
b: array[1..m] of ArbFloat = (
21, 39, 21, 30);
var
x: array[1..n] of ArbFloat;
term: ArbInt;
i, j: Integer;
b_test: array[1..m] of ArbFloat;
sum: ArbFloat;
begin
WriteLn('Solve A x = b with the least-squares method');
WriteLn;
// Display input data
WriteLn('A = ');
for i := 1 to m do
begin
for j := 1 to n do
Write(A[i, j]:10:0);
WriteLn;
end;
WriteLn;
WriteLn('b = ');
for i := 1 to m do
Write(b[i]:10:0);
WriteLn;
// Calculate and show solution
slegls(a[1,1], m, n, n, b[1], x[1], term);
WriteLn;
WriteLn('Solution x = ');
for j:= 1 to n do
Write(x[j]:10:0);
WriteLn;
// Calculate and display residuals
WriteLn;
WriteLn('Residuals A x - b = ');
sum := 0;
omvmmv(a[1,1], m, n, n, x[1], b_test[1]);
for i:=1 to m do begin
Write((b_test[i] - b[i]):10:0);
sum := sum + sqr(b_test[i] - b[i]);
end;
WriteLn;
// Sum of squared residuals
WriteLn;
WriteLn('Sum of squared residuals');
WriteLn(sum:10:0);
WriteLn;
WriteLn('----------------------------------------------------------------------------');
WriteLn;
// Modify solution to show that the sum of squared residuals increases';
WriteLn('Modified solution x'' (to show that it has a larger sum of squared residuals)');
x[1] := x[1] + 1;
x[2] := x[2] - 1;
WriteLn;
for j:=1 to n do
Write(x[j]:10:0);
omvmmv(a[1,1], m, n, n, x[1], b_test[1]);
sum := 0;
for i:=1 to m do
sum := sum + sqr(b_test[i] - b[i]);
WriteLn;
WriteLn;
WriteLn('Sum of squared residuals');
WriteLn(sum:10:0);
end.
## Eigenvalues and eigenvectors (unit eig)
A vector is an eigenvector if it does not change its direction after a linear transformation has been applied to it. In matrix terms: if a non-zero vector x of dimension n is multiplied to a square nxn matrix A and the product is the same vector multiplied by a scalar λ then x is an eigenvector, and λ is called an eigenvalue:
$A \mathbf{x} = \lambda \mathbf{x} \qquad \Rightarrow \qquad A \mathbf{x} - \lambda \mathbf{x} = 0$
NumLib offers a variety of routines for calculation of eigenvectors and eigenvalues. They are optimized for the matrix type. Every matrix type will be supported by two or four procedures of various degree of complexity; they are identified by appended numbers 1 and 2, or 1 to 4:
• The procedures with appended 1 calculate all eigenvalues.
• The procedures with the appended 2 do the same but only for some eigenvalues which have an index in a specified range (λk1...λk2 out of λ1...λn).
• The procedures with the appended 3 calculate all eigenvalues and all eigenvectors.
• The procedures with the appended 4 do the same, but again return only some eigenvalues and eigenvectors having an index in the specified interval.
### Matrices with general storage
The routines discussed here assume that the n x n matrix for which the eigenvalues are to be calculated is stored in the conventional way as a 2D (or 1D) array of ArbFloat values.
The NumLib method for determining the eigenvalues and eigenvectors of a generic matrix are called eigge1 and eigge3 ("ge" = "generic"). Optimized calculation schemes exists for generic symmetric matrices and symmetric positive definite matrices; the corresponding routines are called eiggs1, eiggs2, eiggs3, and eiggs4 for generic symmetric matrices ("gs" = "generic symmetric")., and eiggg1, eiggg2, eiggg3, eiggg4 for symmetric positive definite matrices ("gg" = ??)
// Generic matrix (without any special symmetries)
procedure eigge1(var a: ArbFloat; n, rwidth: ArbInt; var lam: complex; var term: ArbInt); // all eigenvalues
procedure eigge3(var a: ArbFloat; n, rwidtha: ArbInt; var lam, x: complex; rwidthx: ArbInt; var term: ArbInt); // all eigenvalues and eigenvectors
// Generic symmetric matrix
procedure eiggs1(var a: ArbFloat; n, rwidth: ArbInt; var lam: ArbFloat; var term: ArbInt); // all eigenvalues
procedure eiggs2(var a: ArbFloat; n, rwidth, k1, k2: ArbInt; var lam: ArbFloat; var term: ArbInt); // some eigenvalues (index k1..k2)
procedure eiggs3(var a: ArbFloat; n, rwidtha: ArbInt; var lam, x: ArbFloat; var term: ArbInt); // all eigenvalues and eigenvectors
procedure eiggs4(var a: ArbFloat; n, rwidtha, k1, k2: ArbInt; var lam, x: ArbFloat; var term: ArbInt); // some eigenvalues and eigenvectors (index k1..k2)
// Symmetric positive definite matrix
procedure eiggg1(var a: ArbFloat; n, rwidth: ArbInt; var lam: ArbFloat; var term: ArbInt); // all eigenvalues
procedure eiggg2(var a: ArbFloat; n, rwidth, k1, k2: ArbInt; var lam: ArbFloat; var term: ArbInt); // some eigenvalues (index k1..k2)
procedure eiggg3(var a: ArbFloat; n, rwidtha: ArbInt; var lam, x: ArbFloat; var term: ArbInt); // all eigenvalues and eigenvectors
procedure eiggg4(var a: ArbFloat; n, rwidtha, k1, k2: ArbInt; var lam, x: ArbFloat; var term: ArbInt); // some eigenvalues and eigenvectors (index k1..k2)
• a is the first element of an array containing the matrix A for which the eigenvalue/eigenvector has to be calculated. The array must be dimensioned to provide space for at least n2 floating point values.
• n specifies the size of the matrix A, i.e. the number of rows or columns. Note that the input matrix must be square, i.e. the number of rows and columns is equal.
• rwidth is the allocated row length of the array a. It can be larger than n if the array is allocated larger than necessary, but normally rwidth = n.
• lam is the first element of an array receiving the calculated eigenvalues. In case of a generic matrix (eigge1, eigge2) the eigenvalues can be complex; therefore, the array must be dimensioned for values of type complex as declared in unit typ. In the other cases (eiggs1..4 or eiggg1..4) the eigenvalues are real, and the array must be dimensioned for datatype ArbFloat. Since a nxn matrix has n eigenvalues the array must be allocated for at least n values, in case of the procedures with appended 2 or 4 only k2-k1+1 values are sufficient (see below).
• term returns an error code:
• 1 -- successful calculation
• 2 -- calculation failed
• 3 -- error in input data: n<1, k1<1, k1>k2, or k2>n.
Additionally, in case of procedure eigge3, eigge4 , eiggg3 or eiggg4:
• x is the first element of a matrix to receive the calculated eigenvectors. Again, the eigenvectors of a generic matrix can have complex components. Therefore, the matrix must be declared for the datatype complex, and it must be large enough to hold at least n2 values, in case of eigge4 or eiggg4 n * (k2-k1+1). If the matrix is symmetric or positive definite, the eigenvectors are real, and the array must be declared for datatype ArbFloat. In any case, the eigenvectors are normalized to unit length and arranged in the columns of this matrix.
• rwidthx denotes the allocated row length of the matrix x. Thus it is possible to dimension the result matrix larger than actually needed.
Additionally, in case of procedures eiggs2, eiggs4, eiggg2 and eiggg4:
• k1 and k2 define the interval of indexes k for which the eigenvalues (λk) and eigenvalues (ck) are to be calculated. They are integers and must be ordered such that 1<=k1<=k2<=n.
Example
Calculate the eigenvalues and eigenvectors of the matrix
$A= \begin{bmatrix} 8 & -1 & -5 \\ -4 & 4 & -2 \\ 18 & -5 & -7 \end{bmatrix}$
program eig_general_matrix;
uses
SysUtils, math,
typ, eig;
const
n = 3;
D = 3;
var
// a is the input matrix
a: array[1..n, 1..n] of ArbFloat = (
( 8, -1, -5),
(-4, 4, -2),
(18, -5, -7)
);
lambda: array[1..n] of complex;
x: array[1..n, 1..n] of complex;
term: Integer = 0;
i, j: Integer;
function ComplexToStr(z: complex; Decimals: Integer): String;
var
sgn: array[boolean] of string = ('+', '-');
begin
Result := Format('%.*f %s %.*fi', [Decimals, z.Re, sgn[z.Im < 0], Decimals, abs(z.Im)]);
end;
begin
// write input matrix
WriteLn('a = ');
for i := 1 to n do begin
for j := 1 to n do
Write(a[i, j]:10:D);
WriteLn;
end;
WriteLn;
// Calculate eigenvalues/vectors
eigge3(a[1,1], n, n, lambda[1], x[1,1], n, term);
// write eigenvalues
WriteLn('Eigenvalues: lambda = ');
for i := 1 to n do
Write(ComplexToStr(lambda[i], D):25);
WriteLn;
WriteLn;
// Write eigenvectors
WriteLn('Eigenvectors (as columns): x = ');
for i := 1 to n do begin
for j := 1 to n do
Write(ComplexToStr(x[i, j], D):25);
WriteLn;
end;
end.
### Symmetric band matrices
NumLib provides four routines to calculate eigenvalues and eigenvectors of symmetric band matrices: eigbs1...eigbs4 ("bs" = band and symmetric):
// All eigenvalues
procedure eigbs1(var a: ArbFloat; n, w: ArbInt; var lam: ArbFloat; var term: ArbInt);
// Some eigenvalues (index k1..k2)
procedure eigbs2(var a: ArbFloat; n, w, k1, k2: ArbInt; var lam: ArbFloat; var term: ArbInt);
// All eigenvalues and all eigenvectors
procedure eigbs3(var a: ArbFloat; n, w: ArbInt; var lam, x: ArbFloat; rwidthx: ArbInt; var term: ArbInt);
// Some eigenvalues, some eigenvectors (index k1..k2)
procedure eigbs4(var a: ArbFloat; n, w, k1, k2: ArbInt; var lam, x: ArbFloat; rwidthx: ArbInt; var m2, term: ArbInt);
• a specifies the first element of a 1D array which contains the diagonal elements of the left band and the main diagonal. It is obtained by running across the rows of the band matrix from left to right and top to bottom, but stopping at the main diagonal - the right band is ignored since it is equal to the left band. The array must be dimensioned to contain at least n*(w+1) - (w*(w+1)) div 2 elements.
• n denotes the number of columns and rows of the matrix (it must be square).
• w identifies the bandwidth, i.e. how many diagonals the band extends to the left or to the right of the main diagonal. Left and right bandwidth must be equal because the matrix is symmetric.
• lam is the first element of a real-valued array receiving the calculated eigenvalues. The eigenvalues of a symmetric matric are real values. Since an n x n matrix has n eigenvalues the array must be allocated for at least n ArbFloat values.
• x, in case of routines eigbs3 and eigbs4, is the first element of a matrix to receive the calculated eigenvectors. The matrix must be declared for the datatype Arbfloat, and it must be large enough to hold at least n x n values. The eigenvectors are normalized to unit length and are arranged in the columns of this matrix.
• rwidthx, in case of routines eigbs3 and eigbs4, denotes the allocated row length of the matrix x. Thus it is possible to dimension the result matrix larger than actually needed.
• k1 and k2, in case of procedures eigbs2 and eigbs4, define the interval of indexes k for which the eigenvalues λk and eigenvalues xk are to be calculated. They are integers and must be ordered such that 1<=k1<=k2<=n.
• m2, in case of routine eigbs4, indicates the index of the largest eigenvalue for which an eigenvector has been calculated.
• term returns an error code:
• 1 -- successful calculation
• 2 -- calculation failed
• 3 -- error in input parameters: n<1, w<0, w>=n, k1<1, k1>k2, or k2>n.
If the bandwidth w > n/3 then it is better to calculate all eigenvalues and eigenvectors even if not all of them are needed.
Example
Calculate the eigenvalues and eigenvectors of the symmetric 7 x 7 matrix $A= \begin{bmatrix} 5 & -4 & 1 & & & & 0 \\ -4 & 6 & -4 & 1 & & & \\ 1 & -4 & 6 & -4 & 1 & & \\ & 1 & -4 & 6 & -4 & 1 & \\ & & 1 & -4 & 6 & -4 & 1 \\ & & & 1 & -4 & 6 & -4 \\ 0 & & & & 1 & -4 & 5 \end{bmatrix}$ The eigenvalues are $\lambda_k = 16\sin^{4}\frac{k\pi}{16}, \qquad k=1,2,\ldots,7$
program eig_symmband_matrix;
uses
SysUtils, math,
typ, eig;
const
n = 7;
w = 2;
nelements = n * (w + 1) - (w * (w + 1)) div 2;
D = 3;
// a contains the elements of the left and main diagonals of the input matrix
a: array[1..nelements] of ArbFloat = (
5,
-4, 6,
1, -4, 6,
1, -4, 6,
1, -4, 6,
1, -4, 6,
1, -4, 5 );
function MatrixIndexToArrayIndex(i, j, n, w: Integer): Integer;
function DiagElementIndex(i: Integer): Integer;
var
k: Integer;
begin
Result := 1;
if i = 1 then
exit;
// Rows truncated at left
for k := 2 to w do begin
Result := Result + k;
if k = i then
exit;
end;
// full rows and those truncated at right
for k := w+1 to n do begin
Result := Result + w + 1;
if k = i then
exit;
end;
Result := n;
end;
var
d: Integer;
begin
if j > i then begin
Result := MatrixIndexToArrayIndex(j, i, n, w);
exit;
end;
Result := DiagElementIndex(i);
if (i = j) then
exit;
d := i - j;
if d > w then
Result := -1
else begin
dec(Result, d);
if (Result < 1) then
Result := -1
end;
end;
var
lambda: array[1..n] of ArbFloat;
x: array[1..n, 1..n] of ArbFloat;
term: Integer = 0;
i, j, k: Integer;
begin
// Write diagonal elements
WriteLn('n = ', n);
Writeln('(One-sided) band width w = ', w);
Write('Diagonal elements of A = ', a[1]:0:0);
for k := 2 to nelements do
Write(a[k]:3:0);
WriteLn;
WriteLn;
// write reconstructed band input matrix (not needed for calculation)
WriteLn('Reconstructed A = ');
k := 1;
for i := 1 to n do begin
for j := 1 to n do begin
k := MatrixIndexToArrayIndex(i, j, n, w);
if k = -1 then
Write(0.0:3:0)
else
Write(a[k]:3:0);
end;
WriteLn;
end;
WriteLn;
// Calculate eigenvalues/vectors
eigbs3(a[1], n, w, lambda[1], x[1,1], n, term);
if term <> 1 then begin
WriteLn('term = ', term, ' --> ERROR');
halt;
end;
// Write expected results of eigenvalues
WriteLn('Expected eigenvalues:');
for i := 1 to n do
Write(16 * intpower(sin(i*pi/16), 4):15:D);
WriteLn;
WriteLn;
// write eigenvalues
WriteLn('Calculated eigenvalues: lambda = ');
for i := 1 to n do
Write(lambda[i]:15:D);
WriteLn;
WriteLn;
// Write eigenvectors
WriteLn('Eigenvectors (as columns): x = ');
for i := 1 to n do begin
for j := 1 to n do
Write(x[i, j]:15:D);
WriteLn;
end;
end.
### Symmetric tridiagonal matrices
This type of matrices is treated best by the routines eigts1..eigts4:
// All eigenvalues
procedure eigts1(var d, cd: ArbFloat; n: ArbInt; var lam: ArbFloat; var term: ArbInt);
// Some eigenvalues (with indices k1..k2)
procedure eigts2(var d, cd: ArbFloat; n, k1, k2: ArbInt; var lam: ArbFloat; var term: ArbInt);
// All eigenvalues and all eigenvectors
procedure eigts3(var d, cd: ArbFloat; n: ArbInt; var lam, x: ArbFloat; rwidth: ArbInt; var term: ArbInt);
// Some eigenvalues and eigenvectors (with indices k1..k2)
procedure eigts4(var d, cd: ArbFloat; n, k1, k2: ArbInt; var lam, x: ArbFloat; rwidth: ArbInt; var m2, term: ArbInt);
• d specifies the first element along the main diagonal of the matrix. This 1D array must be dimensioned to contain at least n elements.
• c specifies the first element in the subdiagonal of the matrix. This 1D array must be dimensioned to at least n-1 elements.
• n denotes the number of columns or rows of the matrix (it must be square).
• k1 and k2 define the interval of indexes k for which the eigenvalues (λk) and eigenvalues (ck) are to be calculated. They are integers and must be ordered such that 1<=k1<=k2<=n.
• lam is the first element of an array receiving the calculated eigenvalues. Since there are n real eigenvalues the array must be dimensioned to contain at least n values of type ArbFloat. In case of the routines with appended 2 and 4, it is sufficient to provide space only for k2 - k1 + 1 values.
• x is the first element of an array to receive the calculated eigenvectors. Since the eigenvectors are real the array must be prepared for data type ArbFloat. In case of routine eigts3, the array must be dimensioned for at least n2 elements, in case of routine eigts4 it must provide space for at least n * (k2-k1+1) values. The eigenvectors are normalized to unit length and arranged in the columns of this matrix.
• rwidthx denotes the allocated row length of the matrix x. Thus it is possible to dimension the result matrix larger than actually needed.
• term returns an error code:
• 1 -- successful calculation
• 2 -- calculation failed
• 3 -- error in input data: n<1, k1<1, k1>k2, or k2>n.
Example
Calculate eigenvalues and eigenvectors of the matrix $A= \begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 2 & 0 \\ 0 & 2 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix}$ The expected eigenvalues are $-\sqrt{2},\ 2-\sqrt{2},\ \sqrt{2},\ 2+\sqrt{2}$
program eig_symmtridiag_matrix;
uses
typ, eig;
const
n = 4;
// a contains the elements of the main diagonal of the input matrix
d: array[1..n] of ArbFloat = (1, 1, 1, 1);
// c contains the elements of the subdiagonal
c: array[2..n] of ArbFloat = (1, 2, 1);
var
lambda: array[1..n] of ArbFloat;
x: array[1..n, 1..n] of ArbFloat;
term: Integer = 0;
i, j, k: Integer;
begin
// Write diagonals elements
WriteLn('n = ', n);
Write('Elements of main diagonal = ', d[1]:0:0);
for k := 2 to n do
Write(d[k]:3:0);
WriteLn;
Write('Elements of subdiagonal = ', ' ':3, c[2]:0:0);
for k := 3 to n do
Write(c[k]:3:0);
WriteLn;
WriteLn;
// write reconstructed band input matrix (not needed for calculation)
WriteLn('Reconstructed A = ');
for i := 1 to n do begin
for j := 1 to n do begin
if j = i then
Write(d[i]:3:0)
else if (j = i-1) then
Write(c[i]:3:0)
else if (j = i+1) then
Write(c[i+1]:3:0)
else
Write(0.0:3:0);
end;
WriteLn;
end;
WriteLn;
// Calculate eigenvalues/vectors
eigts3(d[1], c[2], n, lambda[1], x[1,1], n, term);
if term <> 1 then begin
WriteLn('term = ', term, ' --> ERROR');
halt;
end;
// Write expected results of eigenvalues
WriteLn('Expected eigenvalues:');
Write(-sqrt(2):15:3, 2-sqrt(2):15:3, sqrt(2):15:3, 2+sqrt(2):15:3);
WriteLn;
WriteLn;
// write eigenvalues
WriteLn('Calculated eigenvalues: lambda = ');
for i := 1 to n do
Write(lambda[i]:15:3);
WriteLn;
WriteLn;
// Write eigenvectors
WriteLn('Eigenvectors (as columns): x = ');
for i := 1 to n do begin
for j := 1 to n do
Write(x[i, j]:15:3);
WriteLn;
end;
end.
## Finding the roots of a function (unit roo)
The roots are the x values at which a function f(x) is zero.
### Roots of a polynomial
A polynomial of degree n
$z^n + a_1 z^{n-1} + a_2 z^{n-2} + ... + a_{n-1} z + a_n = 0$
always has n, not necessarily distinct, complex solutions. The datatype complex has been described in section complex numbers. These roots can be calculated by the function roopol:
procedure roopol(var a: ArbFloat; n: ArbInt; var z: complex; var k, term: ArbInt);
• a is the first element of an array containing the polynomial coefficients. Note that it is assumed that the polynomial has been normalized such that the coefficient of the highest-degree term is 1; this coefficient is not contained in the array. Therefore, the array must be dimensioned for at least n values. Since only real polynomials are handled here the array elements must be of datatype ArbFloat. The polynomial coefficients must be ordered from highest to lowest degree. Note that the data organization of the array is different from other polynomial routines in this library.
• n is the degree of the polynomial. Must be a positive integer.
• z is the first element in an array of complex values which returns the roots of the polynomial. The array must be dimensioned to contain at least n values. The returned values are undefined if an error had occurred (term <> 1).
• k returns how many roots were found. Always must be equal to n, otherwise an error has occured.
• term returns an error code:
• 1 -- sucessful completion, the result array contains valid data.
• 2 -- Not all roots could be detected, i.e. k < n.
• 3 -- Error in input data: n < 1.
Example
Calculate the roots of the polynomial z5 + 3 z4 + 4 z3 - 8 z2. The expected zero points are:
$z_1=0,\ z_2=0,\ z_3=1,\ z_4=-2+2i,\ z_5=-2-2i$
program polynomial;
uses
SysUtils, typ, roo;
const
n = 5;
a: array[1..n] of ArbFloat = (3, 4, -8, 0, 0);
var
z: array[1..n] of complex;
k: ArbInt;
term: ArbInt;
i: Integer;
c: complex;
function ComplexToStr(z: complex; Decimals: Integer): string;
const
SIGN: array[boolean] of string = ('+', '-');
begin
Result := Format('%.*f %s %.*f i', [Decimals, z.re, SIGN[z.im <0], Decimals, abs(z.im)]);
end;
begin
// Solve equation
roopol(a[1], n, z[1], k, term);
if term = 1 then begin
// Display results
WriteLn('Results of procedure roopol:');
for i:=1 to n do
WriteLn(' Solution #', i, ': ', ComplexToStr(z[i], 6):20);
WriteLn;
// Display expected results
Writeln('Expected results:');
c.Init(0, 0); // z1 = 0
WriteLn(' Solution #1: ', complexToStr(c, 6):20);
c.Init(0, 0); // z2 = 0
WriteLn(' Solution #2: ', complexToStr(c, 6):20);
c.Init(1, 0); // z3 = 1
WriteLn(' Solution #3: ', complexToStr(c, 6):20);
c.Init(-2, +2); // z4 = -2 + 2 i
WriteLn(' Solution #4: ', complexToStr(c, 6):20);
c.Init(-2, -2); // z4 = -2 - 2 i
WriteLn(' Solution #5: ', complexToStr(c, 6):20);
end else
WriteLn('ERROR');
end.
### Roots of the quadratic equation
${z}^2 + {p} {z} + {q} = 0$
is a special polynomal of degree 2. It always has two, not necessarily different, complex roots. These solutions can be determined by the procedure rooqua.
procedure rooqua(p, q: ArbFloat; var z1, z2: complex);
• p and q are the (real) coefficients of the quadratic equation
• z1 and z2 return the two complex roots. See unit typ for the declaration of the type complex
Note that rooqua assumes that the coefficient of the quadratic term has been normalized to be 1.
Example
Determine the roots of the equation z2 + 2 z + 5 = 0.
program quadratic_equ;
uses
SysUtils, typ, roo;
var
z1, z2: complex;
const
SIGN: array[boolean] of string = ('+', '-');
begin
rooqua(2, 5, z1, z2);
WriteLn(Format('1st solution: %g %s %g i', [z1.re, SIGN[z1.im < 0], abs(z1.im)]));
WriteLn(Format('2nd solution: %g %s %g i', [z2.re, SIGN[z2.im < 0], abs(z2.im)]));
end.
### Roots of the binomial equation
The equation
$z^n = a$
is another special polynomial in which all terms except for the highest- and lowest-order terms have zero coefficients. It has n complex solutions which can be calculated by the procedure roobin:
procedure roobin(n: ArbInt; a: complex; var z: complex; var term: ArbInt);
• n is the exponent in the binomial equation used. n must be a positive integer.
• a is the constant term a at the right side of the binomial equation. It is expected to be a complex value (see section on complex numbers).
• z is the first element of an array of data type complex to receive the results of the procedure. It must be dimensioned for at least n complex values.
• term returns an error code:
• 1 -- successful termination
• 2 -- error in input data: n < 1
Example
Calculate the roots of the equation
$z^4 = -1$
The exact solutions are
$z_1 = \frac{1}{2} \sqrt{2} (1+i) ,\ z_2 = \frac{1}{2} \sqrt{2} (1-i) ,\ z_3= \frac{1}{2} \sqrt{2} (-1+i) ,\ z_4= \frac{1}{2} \sqrt{2} (-1-i)$
program binomial;
uses
SysUtils, typ, roo;
const
n = 4;
var
z: array[1..n] of complex;
term: ArbInt;
i: Integer;
a: complex;
c: complex;
function ComplexToStr(z: complex; Decimals: Integer): string;
const
SIGN: array[boolean] of string = ('+', '-');
begin
Result := Format('%.*g %s %.*g i', [Decimals, z.re, SIGN[z.im < 0], Decimals, abs(z.im)]);
end;
begin
// Prepare constant term as a complex value
a.Init(-1, 0);
// Solve equation
roobin(n, a, z[1], term);
if term = 1 then begin
// Display results
WriteLn('Results of procedure roobin:');
for i:=1 to n do
WriteLn(' Solution #', i, ': ', ComplexToStr(z[i], 6):20);
WriteLn;
// Display expected results
Writeln('Expected results:');
c.Init(1, 1);
c.Scale(0.5*sqrt(2));
WriteLn(' Solution #1: ', complexToStr(c, 6):20);
c.Init(1, -1);
c.Scale(0.5*sqrt(2));
WriteLn(' Solution #2: ', complexToStr(c, 6):20);
c.Init(-1, 1);
c.Scale(0.5*sqrt(2));
WriteLn(' Solution #3: ', complexToStr(c, 6):20);
c.Init(-1, -1);
c.Scale(0.5*sqrt(2));
WriteLn(' Solution #4: ', complexToStr(c, 6):20);
end else
WriteLn('ERROR');
end.
### Bisection
In the bisection method two x values a and b are estimated to be around the expected root such that the function values have opposite signs at a and b. The center point of the interval is determined, and the subinterval for which the function has opposite signs at its endpoints is selected for a new iteration. The process ends when a given precision, i.e. interval length, is achieved.
In NumLib, this approach is supported by the procedure roof1r:
procedure roof1r(f: rfunc1r; a, b, ae, re: ArbFloat; var x: ArbFloat; var term: ArbInt);
• f is the function for which the root is to be determined. It must be a function of one floating point argument (type ArbFloat). The type of the function, rfunc1r, is declared in unit typ.
• a and b are the endpoints of the test interval. The root must be located between these two values, i. e. the function values f(a) and f(b) must have different signs.
• ae and re determine the absolute and relative precision, respectively, with which the root will be determined. re is relative to the maximum of abs(a) and abs(b). Note that precision and speed are conflicting issues. Highest accuracy is achieved if ae is given as MachEps (see unit typ). Both parameters must not be negative.
• x returns the value of the found root.
• term returns whether the process has been successful:
• 1 - successful termination, a zero point has been found with absolute accuracy ae or relative accuracy re
• 2 - the required accuracy of the root could not be reached; However, the value of x is called the "best achievable" approach
• 3 - error in input parameters: ae < 0 or re < 0, or f(a)*f(b) > 0
Example
The following program determines the square root of 2. This is the x value at which the function f(x) = x^2 - 2 is zero. Since f(1) = 1^2 - 2 = -1 < 0 and f(2) = 2^2 - 2 = 2 > 0 we can assume a and b to be 1 and 2, respectively.
program bisection_demo;
uses
typ, roo;
function f(x: ArbFloat): ArbFloat;
begin
Result := x*x - 2;
end;
var
x: ArbFloat = 0.0;
term : ArbInt;
begin
roof1r(@f, 1.0, 2.0, 1e-9, 0, x, term);
WriteLn('Bisection result ', x);
WriteLn('sqrt(2) ', sqrt(2.0));
end.
### Roots of a system of nonlinear equations
Procedure roofnr finds the roots of a system of (nonlinear) equations:
$f_{i}(x_1,x_2,\ldots,x_n)=0, \; i=1,2,\ldots,n$
procedure roofnr(f: roofnrfunc; n: ArbInt; var x, residu: ArbFloat; ra: ArbFloat; var term: ArbInt);
• f contains the address of the procedure that calculates the function values fi(x1, x2, ..., xn). The type of the function, roofnrfunc, is declared in unit typ to be
type
roofnrfunc = procedure(var x, fx: ArbFloat; var deff: boolean);
• Here, x is the first element of an array of at least n ArbFloat values which provides the xj values at which the functions are to be calculated.
• fx is the first element of another array of n ArbFloat values to receive the result of each function fi. Suppose, we want to solve the following system of equations
$\begin{array}{l} f_1(x_1, x_2) = x_1^2 + x_2^2 - 2 = 0 \\ f_2(x_1, x_2) = -(x_1 - 1)^2 + x_2 = 0 \end{array}$
Then the function to be used as f can be written like this:
procedure func(var x1, f1x: real; var deff: boolean);
var
x: array[1..2] of real absolute x1;
f: array[1..2] of real absolute f1x;
begin
f[1] := sqr(x[1]) + sqr(x[2]) - 2;
f[2] := -sqr(x[1] - 1) + x[2];
end;
• The boolean parameter deff provides a way to stop the root finding process if some condition is met. The above example has two solutions, namely the intersection of a circle with a parabola. If only the intersection with x[1] >= 1 is requested then the process can be stopped as soon as a value x[1] < 1 is reached by setting deff to false:
procedure func(var x1, f1x: real; var deff: boolean); far;
var
x: array[1..2] of real absolute x1;
f: array[1..2] of real absolute f1x;
begin
deff := x[1] >= 1;
if deff then begin
f[1] := sqr(x[1]) + sqr(x[2]) - 2;
f[2] := -sqr(x[1] - 1) + x[2];
end;
end;
• n is the number of equations or independent x values (xi) in the system. Both must be the same.
• x is the first element of an array of at least n ArbFloat values which serves for both input and output. On input the array must contain estimates of the roots to be used as starting points of the iterative calculation. On output, the array is overwritten with the zero points found. Inspect the returned value of term to decide whether the output array contains a valid solution.
• residu is the 2-norm of the vector of residuals in the calculated solution
$\|f\|_2 = \sqrt{\Sigma_{i=1}^{n} f_i^2}$
• ra contains the relative accuracy with which the desired solution is to be calculated. Recommended values: 10-3, 10-5, 10-8 if ArbFloat is single, real, or double.
• term returns an error code:
• 1 -- successfull completion, a solution has been found with the desired accuracy
• 2 -- the desired accuracy could not be reached, the returned values of x are the best ones achievable.
• 3 -- Incorrect input data: n < 0, or re < 0
• 4 -- The calculation process has been stopped because an internal limit for the count of function calls has been reached. Retry with other starting values for x.
• 5 -- Not enough progess, there may be no solution at all. If the solution is close to 0 choose another starting value.
• 6 -- The procedure wants to calculate a value outside the range defined by deff.
Example
Calculate a solutions of the equation system
$\begin{array}{l} f_1(x_1, x_2) = x_1^2 + x_2^2 - 2 = 0 \\ f_2(x_1, x_2) = -(x_1 - 1)^2 + x_2 = 0 \end{array}$
program nonlinsystem;
uses
SysUtils, typ, roo;
const
n = 2;
ra = 1e-10;
var
x: array[1..n] of ArbFloat;
f_check: array[1..n] of ArbFloat;
residu: ArbFloat;
i: Integer;
term: Integer;
deff: Boolean;
procedure funcs(var x0, fx: ArbFloat; var deff: boolean);
var
xloc: array[1..n] of ArbFloat absolute x0;
f: array[1..n] of ArbFloat absolute fx;
x, y, z: ArbFloat;
begin
x := xloc[1];
y := xloc[2];
f[1] := sqr(x) + sqr(y) - 2;
f[2] := -sqr(x - 1) + y;
end;
begin
// Initial guessed values
x[1] := 0;
x[2] := 0;
// Solve equation
roofnr(@funcs, n, x[1], residu, ra, term);
WriteLn('term = ', term);
WriteLn;
if term in [1, 2] then begin
// Display results
WriteLn('Results found by procedure roofnr:');
for i:=1 to n do
WriteLn('Solution #' + IntToStr(i)+': ':20, x[i]:0:6);
WriteLn;
WriteLn('Norm of residuals: ', residu:0:15);
WriteLn;
// Test functions:
f(x[1], f_check[1], deff);
WriteLn('Check if the function values really are 0 at the found roots:');
for i := 1 to n do
WriteLn('Function #' + IntToStr(i) + ': ':20, f_check[i]:0:15);
end else
WriteLn('ERROR');
end.
## Numerical integration of a function (unit int)
The NumLib function int1fr calculates the integral of a given function between limits a and b with a specified absolute accuracy ae:
procedure int1fr(f: rfunc1r; a, b, ae: ArbFloat; var integral, err: ArbFloat; var term: ArbInt);
• f points to the function to be integrated. It must be a function of a single real variable (type ArbFloat) and return an Arbfloat. See the type rfunc1r declared in unit typ.
• a, b are the limits of integration. a and/or b can attain the values +/-Infinity for integrating over an infinite interval. The order of a and b is handled in the mathematically correct sense.
• ae determines the absolute accuracy requested.
• integreal returns the value of the integral. It is only valid if term = 1.
• err returns the achieved accuracy if the specified accuracy could not be reached. term has the value 2 in this case.
• term returns an error code:
• 1 - successful termination, the integral could be calculated with absolute accuracy ae.
• 2 - the requested accuracy could not be reached. But the integral is approximated within the accuracy err.
• 3 - incorrect input data: ae < 0, or a = b = infinity, or a = b = -infinity
• 4 - the integral could not be calculated: divergence, or too-slow convergence.
Example
Calculate the integral
$\int_a^b \frac 1 {x^2} \mathrm{d}x$
for several integration limits a and b. Since the function diverges at x = 0 the interval from a to b must not contain this point. The analytic result is $-1/{b} + 1/{a}$
program integrate;
uses
SysUtils, typ, int;
function recipx2(x: ArbFloat): ArbFloat;
begin
Result := 1.0 / sqr(x);
end;
function integral_recipx2(a, b: ArbFloat): Arbfloat;
begin
if a = 0 then
a := Infinity
else if a = Infinity then
a := 0.0
else
a := -1/a;
if b = 0 then
b := Infinity
else if b = Infinity then
b := 0.0
else
b := -1/b;
Result := b - a;
end;
procedure Execute(a, b: ArbFloat);
var
err: ArbFloat = 0.0;
term: ArbInt = 0;
integral: ArbFloat = 0.0;
begin
try
int1fr(@recipx2, a, b, 1e-9, integral, err, term);
except
term := 4;
end;
Write(' The integral from ' + FloatToStr(a) + ' to ' + FloatToStr(b));
case term of
1: WriteLn(' is ', integral:0:9, ', exected: ', integral_recipx2(a, b):0:9);
2: WriteLn(' is ', integral:0:9, ', error: ', err:0:9, ', exected: ', integral_recipx2(a, b):0:9);
3: WriteLn(' cannot be calculated: Error in input data');
4: WriteLn(' cannot be calculated: Divergent, or calculation converges too slowly.');
end;
end;
begin
WriteLn('Integral of f(x) = 1/x^2');
Execute(1.0, 2.0);
Execute(1.0, 1.0);
Execute(2.0, 1.0);
Execute(1.0, Infinity);
Execute(-Infinity, -1.0);
Execute(0.0, Infinity);
// NOTE: The next line will raise an exception if run in the IDE. This will not happen outside the IDE.
// Integrate(-1.0, Infinity);
end.
## Ordinary differential equations (unit ode)
### Solving a single first-order differential equation
The routine odeiv1 solves an initial value problem described by a first-order differential equation of the form $\begin{cases} y' = f(x, y), \qquad x \in [a, b]\\ y(a) = \alpha \end{cases}$ with given a, b, f and initial condition y(a) = α.
procedure odeiv1(f: rfunc2r; a, ya: ArbFloat; var b, yb: ArbFloat; ae: ArbFloat; var term: ArbInt);
• f is the funtion to be processed. It depends on two real variables and returns a real value, as declared in unit typ as type rfunc2r = function(x, y: ArbFloat): ArbFloat.
• a is the x value of the starting point of the interval in which the solution is determined.
• ya is the value α of the function at the starting point a (initial value).
• b is the x value of the end point of the interval in which the solution is determined. The case a > b is allowed. If, after the calculation, the error code term is 2 then the interval has been changed, and b contains the new endpoint to which the solution could be calculated with the desired accuracy, ae. In all other cases, b is unchanged.
• yb returns the result of the calculation if term < 3. It is the value of the function at the end point b of the interval. If term = 3 then the result is undefined.
• ae defines the absolute accuracy with which the value y(b) must be determined.
• term returns an error code:
• 1 - successful completion
• 2 - the solution could not reach the point b with the required accuracy. However, yb is a sufficiently accurate approximation of y at the delivered b.
• 3 - input error: ae <= 0
The algorithm is adaptive and is based on an explicit one-step Runge-Kutta method of order five.
The procedure is not suitable for a rigid differential comparison.
The accuracy can not be guaranteed in all cases. In so-called unstable problems, e.g. small variations in y(a) give large variations in y(b), the error may be significantly larger than what the result of the procedure suggests. Call the procedure with different values of ae to investigate this case.
If one wants to solve the initial value problem for a number of points, e.g. from x = 0 to x = 1 with step size 0.1, then it is advisable to "integrate" and thus not to restart at x = 0 with every new step.
Example
Solve the differential equation y" = -10 (y - x2) with the initial condition y(0) = 0 for x = 0, 0.5, ..., 5. Compare with the exact solution y(x) = -0.02 exp(-10 x) + x2 - 0.2 x + 0.02
program solve_ode;
uses
typ, ode;
function f(x, y: ArbFloat): Arbfloat;
begin
Result := -10 * (y - sqr(x));
end;
function exact(x: real): real; far;
begin
Result := -0.02*exp(-10*x) + sqr(x) - 0.2*x + 0.02;
end;
const
d = 0.5; // interval length
ae = 1e-5;
n = 10;
var
a, b: ArbFloat;
ya, yb: ArbFloat;
term: ArbInt;
i: Arbint;
begin
// Set initial conditions
a := 0.0;
b := a + d;
ya := 0.0;
WriteLn('x':12, 'y':12, 'exact':12, 'error code':17);
WriteLn;
WriteLn(a:12:5, ya:12:5, exact(a):12:5, '-':17);
for i := 1 to n do begin
odeiv1(@f, a, ya, b, yb, ae, term);
WriteLn(b:12:5, yb:12:5, exact(b):12:5, term:17);
a := b;
ya := yb;
b := b + d;
end;
end.
### Solving a system of first-order differential equations
An initial value problem described by system of first-order differential equations can be solved by the procedure odeiv2: $\begin{cases} \mathbf{y}' = \mathbf{f}(x, \mathbf{y}), \qquad x \in [a, b]\\ \mathbf{y}(a) = \mathbf{\alpha} \end{cases}$
where
• y is an unknown n vector
y: [a, b] --> Rn
or: y = [y1(x), y2(x), ..., yn(x)]
• f is a vector function
f: [a, b] --> Rn
or: f(x, y) = [f1(x,y), f2(x,y), ..., fn(x,y)]
• a, b, and the initial conditions α = y(a) = [y1(a), y2(a), ... yn(a)] are given.
The algorithm is based on an explicit one-step Runge-Kutta method of order five with variable step size.
procedure odeiv2(f: oderk1n; a: ArbFloat; var ya, b, yb: ArbFloat; n: ArbInt; ae: ArbFloat; var term: ArbInt);
• The parameter f contains the name of the procedure that calculates the function values fi(x, yi). The declaration of the procedure must match the type oderk1n = procedure(x: ArbFloat; var y, fxy: ArbFloat) declared in unit typ. Here x is the coordinate in the interval [a, b], and y is the first value of an array of y values to be used in each equation of the ODE system. The function result is returned in another array which is specified by its first value fxy. Both arrays must be dimensioned to provide at least n elements.
• a is the begin of the calculation interval.
• ya is the first element of an array containing the initial conditions for each ODE of the system, i.e. the function values yi(a) = αi. The array must be allocated to containt at least n elements.
• b is the end of the calculation interval. The case a > b is allowed. If after the calculation term is 2 then the interval has been changed, and b contains the new endpoint to which the solution could have been calculated with the desired accuracy, ae. In all other cases, b is unchanged.
• yb is the first element of an array which receives the results of the calculation. It must be dimensioned to contain at least n elements.
• n denotes the number of equations in the system.
• ae specifies the absolute accuracy with which the solution must be calculated.
• term is an error code:
• 1 - successful completion.
• 2 - the solution could not reach the point b with the required accuracy. However, the values in the yb array are sufficiently accurate approximations of the y values at the delivered b.
• 3 - input error: n < 1, or ae <= 0.
Example
Integrate the following initial value problem between x = 0 and x = 1 with step size 0.1: $\begin{cases} y'_{1}=2xy_{1}+y_{2} \\ y'_{2}=-y_{1}+2xy_{2}\\ y_1(0)=0,\ y_2(0)=1 \end{cases}$
The exact solutions are $\begin{cases} y_1(x)=\exp({x^2}) \sin(x) \\ y_2(x)=\exp({x^2}) \cos(x) \end{cases}$
program solve_ode_sys;
uses
typ, ode;
const
ae = 1e-5;
procedure f(x: ArbFloat; var y, fxy: ArbFloat);
var
_y: array[1..2] of ArbFloat absolute y;
_fxy: array[1..2] of ArbFloat absolute fxy;
begin
_fxy[1] := 2*x*_y[1] + _y[2];
_fxy[2] := -_y[1] + 2*x*_y[2]
end;
function exact1(x: ArbFloat): ArbFloat;
begin
Result := exp(x*x) * sin(x);
end;
function exact2(x: ArbFloat): ArbFloat;
begin
Result := exp(x*x) * cos(x);
end;
var
a, b, d: ArbFloat;
ya: array[1..2] of ArbFloat;
yb: array[1..2] of ArbFloat;
term: ArbInt;
i: Arbint;
n: ArbInt;
begin
// Set initial conditions
a := 0.0;
b := 0.1;
d := b - a;
ya[1] := 0.0;
ya[2] := 1.0;
n := 10;
WriteLn('x':12, 'y[1]':12, 'y[2]':12, 'exact[1]':12, 'exact[2]':12, 'error code':17);
WriteLn;
WriteLn(a:12:5, ya[1]:12:5, ya[2]:12:5, exact1(a):12:5, exact2(a):12:5, '-':17);
for i := 1 to n do begin
odeiv2(@f, a, ya[1], b, yb[1], 2, ae, term);
WriteLn(b:12:5, yb[1]:12:5, yb[2]:12:5, exact1(b):12:5, exact2(b):12:5, term:17);
a := b;
ya[1] := yb[1];
ya[2] := yb[2];
b := b + d;
end;
end.
## Interpolation and fitting (unit ipf)
Unit ipf contains routines for
• fitting a set of data points with a polynomial,
• interpolating or fitting a set of data points with a natural cubic spline. A spline is a piecewise defined function of polynomials which has a high degree of smoothness at the connection points of the polynomials ("knots"). In case of a natural cubic spline, the polynomials have degree 3 and their second derivative is zero at the first and last knot.
Fitting means: determine an approximating smooth function such that the deviations from the data points are at minimum.
Interpolation means: determine a function that runs exactly through the data points.
Use of polynomials or splines is recommended unless the data are known to belong to a known function in which there are still some unknown parameters, for example, the data are measurements of the function f(x) = a + b e-c x.
In the most common case, the data contain measurement errors and appear to follow the shape of a "simple" function. Therefore, it is recommended to first try to fit with a low-degree polynomial, e.g. not higher than degree 5. If this is not successful or if the shape of the function is more complicated try to fit with a spline.
If the data contain no (or very little) measurement noise, it is almost always wise to interpolate the data with a spline.
### Fitting with a polynomial
For this purpose, the routine ipfpol can be applied. It uses given data points (xi, yi), i = 1, ..., m, and n given coefficients b0, ..., bn to calculate a polynomial $\operatorname{p}(x) = b_0 + b_1 x + \dots + b_n x^n$
In the sense of the least squares fitting, the coefficients b0 ... bn are adjusted to minimize the expression
$\sum_{i=1}^m {(\operatorname{p}(x) - y_i)^2}$
procedure ipfpol(m, n: ArbInt; var x, y, b: ArbFloat; var term: ArbInt);
• m denotes the number of data points. It is required that m ≥ 1
• n is the degree of the polynomial to be fitted. It is required that n > 0.
• x is the first element of a one-dimensional array containing the x values of the data points. The array must be allocated to contain at least m values.
• y is the first element of a one-dimensional array containing the y values of the data points. The array must be dimensioned to contain at least m values.
• b is the first element of a one-dimenisonal array in which the determined polynomial coefficients are returned if term is 1. The array must have been dimensioned to provide space for at least n + 1 values. The returned coefficients are ordered from lowest to highest degree.
• term returns an error code:
• 1 - successful completion, the array b contains valid coefficients
• 3 - input error: n < 0, or m < 1.
After the coefficients b0, ..., bn once have been determined the value of p(x) can be calculated for any x using the procedure spepol from the unit spe.
Note: With this routine, of course, an interpolating polynomial can also be determined by equaling n to m - 1.
Example
Fit a polynomial of degree 2 to these data points: $\begin{array}{ccc} \hline i & x_i & y_i \\ \hline 1 & 0.00 & 1.26 \\ 2 & 0.08 & 1.37 \\ 3 & 0.22 & 1.72 \\ 4 & 0.33 & 2.08 \\ 5 & 0.46 & 2.31 \\ 6 & 0.52 & 2.64 \\ 7 & 0.67 & 3.12 \\ 8 & 0.81 & 3.48 \end{array}$
program polynomial_fit;
uses
SysUtils, StrUtils, typ, ipf, spe;
const
m = 8;
n = 2;
x: array[1..m] of Arbfloat = (
0.00, 0.08, 0.22, 0.33, 0.46, 0.52, 0.67, 0.81
);
y: array[1..m] of Arbfloat = (
1.26, 1.37, 1.72, 2.08, 2.31, 2.64, 3.12, 3.48
);
var
b: array[0..n] of ArbFloat;
i: Integer;
term: ArbInt;
xint, yint: ArbFloat;
begin
WriteLn('Fitting a polynomial of degree 2: p(x) = b[0] + b[1] x + b[2] x^2');
WriteLn;
WriteLn('Data points');
WriteLn('i':10, 'xi':10, 'yi':10);
for i:=1 to m do
WriteLn(i:10, x[i]:10:2, y[i]:10:2);
WriteLn;
// Execute fit
ipfpol(m, n, x[1], y[1], b[0], term);
// Display fitted polynomial coefficients
WriteLn('Fitted coefficients b = ');
WriteLn(b[0]:10:6, b[1]:10:6, b[2]:10:6);
WriteLn;
// Interpolate and display fitted polynomial for some x values
WriteLn('Some interpolated data');
WriteLn('':10, 'x':10, 'y':10);
for i:= 1 to 5 do begin
xint := 0.2*i;
yint := spepol(xint, b[0], n);
WriteLn('':10, xint:10:2, yint:10:2);
end;
end.
### Interpolation with a natural cubic spline
The routine ipfisn can be used for this purpose. It calculates the parameters of the spline. Once these parameters are known, the value of the spline can be calculated at any point using the ipfspn procedure.
Assume that (xi, yi), i = 0, ..., n' are the given data points with x0 < x1 < ... < xn. Then assume a function g(x) which is twice continuously differentiable (i.e., the first and second derivatives exist and are continuous) and which interpolates the data (i.e., g(xi = yi). Imagine g(x) as the shape of a thin flexible bar. Then the bending energy (in first approximation) is proportional to
$\int_{-\infty}^{\infty} g''(x)^2 dx$
The interpolating natural spline s can be understood as the function g which minimizes this integral and therefore the bending energy.
The spline s has the following properties:
• s is a cubic polynomial on the interval [xi, xi+1]
• it is twice continuously differentiable on (-∞, ∞)
• s"(x0) = s"(xn) = 0
• on (-∞, x0) and (xn, ∞) s is linear.
procedure ipfisn(n: ArbInt; var x, y, d2s: ArbFloat; var term: ArbInt);
• Calculates the second derivatives of the splines at the knots xi.
• n is the index of the last data point, i.e. there are n+1 data points.
• x is the first element of the array with the x coordinates of the data points. It must be dimensioned to contain at last n+1 values. The standard declaration would be var x: array[0..n] of ArbFloat. The data points must be in strictly ascending order, i.e. there must not be any data points for which xi >= xi+1.
• y is the first element of the array with the y coordinates of the data points. It must be dimensioned to contain at last n+1 values. The standard declaration would be var y: array[0..n] of ArbFloat.
• d2s is the first element of the array which receives the second derivatives of the splines at the knots if the returned error code term is 1. Note that the array does not contain values for the first and last data point because they are automatically set to 0 (natural splines). Therefore the array must be dimensioned to contain at least n-1 elements. The standard declaration would be: var d2s: array[1..n-1] of ArbFloat
• term returns an error code:
• 1 - successful completion, the array beginning at d2s contains valid data
• 2 - calculation of the second derivatives was not successful
• 3 - incorrect input parameters: n ≤ 1, or xixi+1
function ipfspn(n: ArbInt; var x, y, d2s: ArbFloat; t: ArbFloat; var term: ArbInt): ArbFloat;
• Calculates the value of the spline function at any coordinate t. The calculation uses the second derivatives determined by ipnisn.
• n, x, y, d2s have the same meaning as above
• term returns an error code:
• 1 - successful completion
• 3 - The calculation could not be executed because n ≤ 1.
Example
Interpolate a natural cubic spline through the following data points: $\begin{array}{ccc} \hline i & x_i & y_i \\ \hline 0 & 0.00 & 0.980 \\ 1 & 0.09 & 0.927 \\ 2 & 0.22 & 0.732 \\ 3 & 0.34 & 0.542 \\ 4 & 0.47 & 0.385 \\ 5 & 0.58 & 0.292 \\ 6 & 0.65 & 0.248 \\ 7 & 0.80 & 0.179 \\ 8 & 0.93 & 0.139 \end{array}$
program spline_interpolation;
uses
typ, ipf;
const
n = 8;
x: array[0..n] of Arbfloat = (
0.00, 0.09, 0.22, 0.34, 0.47, 0.58, 0.65, 0.80, 0.93
);
y: array[0..n] of Arbfloat = (
0.990, 0.927, 0.734, 0.542, 0.388, 0.292, 0.248, 0.179, 0.139
);
var
s: array[0..n] of ArbFloat;
d2s: array[1..n-1] of ArbFloat;
i: Integer;
term: ArbInt;
xint, yint: ArbFloat;
begin
WriteLn('Interpolation with a natural cubic spline');
WriteLn;
WriteLn('Data points');
WriteLn('i':10, 'xi':10, 'yi':10);
for i:=0 to n do
WriteLn(i:10, x[i]:10:2, y[i]:10:3);
WriteLn;
// Interpolation
ipfisn(n, x[0], y[0], d2s[1], term);
// Display 2nd derivatives of splines at xi
WriteLn('s"(xi)');
Write(0.0:10:6); // 2nd derivative of 1st point is 0
for i:=1 to n-1 do
Write(d2s[i]:10:6);
WriteLn(0.0:10:6); // 2nd derivative of last point is 0
WriteLn;
// Interpolate for some x values
WriteLn('Some interpolated data');
WriteLn('':10, 'x':10, 'y':10);
for i:= 1 to 5 do begin
xint := 0.2 * i;
yint := ipfspn(n, x[0], y[0], d2s[1], xint, term);
WriteLn('':10, xint:10:2, yint:10:2);
end;
end.
### Fitting with a natural cubic spline
The routine ipffsn can be used for this purpose. This routine calculates the spline-fit parameters. Once these parameters are known the value of the spline can be calculated at any point using the procedure ipfspn.
Following the model with the bending energy in the previous chapter the calculated fit minimizes
$\int_{-\infty}^{\infty} g''(x)^2 dx + \lambda \sum_{i=0}^n (g(x_i) - y_i)^2$
where the parameter λ must be determined. It is a measure of the compromise between the "smoothness" of the fit - which is measured by the integral (the bending energy) - and the deviation of the fit to the given data - which is measured by the sum of squares. The solution is again a natural cubic spline.
Warning: The procedure ipffsn is buggy and crashes. Therefore, it is not documented at the moment.
## Special functions (unit spe)
### Evaluation of a polynomial
Horner's scheme provides an efficient method for evaluating a polynomial at a specific x value:
$\operatorname{p}(x) = a_0 + a_1 x + a_2 x^2 + ... + a_n x^n \ \Rightarrow \ \operatorname{p}(x) = a_0 + x (a_1 + x (a_2 + \dots + x (a_{n-1} + x a_n)))$
This method is applied in NumLib's procedure spepol:
function spepol(x: ArbFloat; var a: ArbFloat; n: ArbInt): ArbFloat;
• x is the argument at which the polynomial is evaluated.
• a is the first element of an array containing the polynomial coefficients a0, a1, a2, ... an. The coefficient must be ordered from lowest to highest degree.
• n is the degree of the polynomial, i.e. there are n+1 elements in the array.
Example
Evaluate the polynomial
$\operatorname{p}(x) = 2 x^3 - 6 x^2 + 2 x - 1$
at x = 3. Note that the terms are not in the correct order for spepol and rearrange them to
$\operatorname{p}(x) = -1 + 2 x - 6 x^2 + 2 x^3$
program polynomial;
uses
typ, spe;
const
n = 3;
a: array[0..n] of ArbFloat = (-1, 2, -6, 2);
var
x, y: ArbFloat;
begin
x := 3.0;
y := spepol(x, a[0], n);
WriteLn(y:0:6);
end.
### Error function
The error function erf(x) and its complement, the complementary error function erfc(x), are the lower and upper integrals of the Gaussian function, normalized to unity:
$\operatorname{erf}(x) = \frac {2} {\sqrt{\pi}} \int_0^x e^{-{t}^2} dt$
$\operatorname{erfc}(x) = \frac {2} {\sqrt{\pi}} \int_x^\infty e^{-{t}^2} dt = {1} - \operatorname{erf}(x)$
Both functions can be calculated by the NumLib functions speerf and speefc, respectively
function speerf(x: ArbFloat): ArbFloat; // --> erf(x)
function speefc(x: ArbFloat): ArbFloat; // --> erfc(x)
Example
A table of some values of the error function and the complementary error function is calculated in the following demo project:
program erf_Table;
uses
SysUtils, StrUtils, typ, spe;
const
Wx = 7;
Wy = 20;
D = 6;
var
i: Integer;
x: ArbFloat;
fs: TFormatSettings;
begin
fs := DefaultFormatSettings;
fs.DecimalSeparator := '.';
WriteLn('x':Wx, 'erf(x)':Wy+1, 'erfc(x)':Wy+1);
WriteLn;
for i:= 0 to 20 do begin
x := 0.1*i;
WriteLn(Format('%*.1f %*.*f %*.*f', [
Wx, x,
Wy, D, speerf(x),
Wy, D, speefc(x)
], fs));
end;
end.
### Normal distribution
The (cumulative) normal distribution is the integral of the Gaussian function with zero mean and unit standard deviation, i.e.
$\operatorname{N}(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} \operatorname{exp}(\frac{t^2}{2}) dt = \frac{1}{2}[1 + \operatorname{erf} (\frac{x}{\sqrt{2}})]$
Suppose a set of random floating point numbers which are distributed according to this formula. Then N(x) measures the probability that a number from this set is smaller than x.
In NumLib, the normal distribution is calculated by the function normaldist:
function normaldist(x: ArbFloat): ArbFloat;
• x can take any value between -∞ and +∞.
• The function result is between 0 and 1.
The inverse normal distribution of y returns the x value for which the normal distribution N(x) has the value y.
function invnormaldist(y: ArbFloat): ArbFloat;
• y is allowed only between 0 and 1.
• The function can take any value between -∞ and +∞.
Note: The functions normaldist and invnormaldist are not available in fpc 3.0.x and older.
Example
Calculate the probabilities for x = -10, -8, ..., 12 for a normal distribution with mean μ = 2 and standard deviation σ = 5. In order to reduce this distribution to the standard normal distribution implemented by NumLib, we introduce a transformation xr = (x - μ)/σ.
program normdist;
uses
typ, spe;
const
mu = 2.0;
sigma = 5.0;
x: array[0..12] of ArbFloat = (-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, 12, 14);
var
xr, y: ArbFloat;
i: Integer;
begin
WriteLn('Normal distribution with mean ', mu:0:1, ' and standard deviation ', sigma:0:1);
WriteLn;
WriteLn('x':20, 'y':20);
for i:= 0 to High(x) do begin
xr := (x[i] - mu) / sigma;
y := normaldist(xr);
WriteLn(x[i]:20:1, y:20:6);
end;
end.
### Gamma function
The gamma function is needed by many probability functions. It is defined by the integral
$\Gamma({x}) = \int_0^{\infty}t^{x-1} e^{-t} dt$
NumLib provides two functions for its calculation:
function spegam(x: ArbFloat): ArbFloat;
function spelga(x: ArbFloat): ArbFloat;
The first one, spegam, calculates the function directly. But since the gamma function grows rapidly for even not-too large arguments this calculation very easily overflows.
The second function, spelga calculates the natural logarithm of the gamma function which is more suitable to combinatorial calculations where multiplying and dividing the large gamma values can be avoided by adding or subtracting their logarithms.
Example
The following demo project prints a table of some values of the gamma function and of its logarithm. Note that spegam(x) overflows above about x = 170 for data type extended.
program Gamma_Table;
uses
SysUtils, StrUtils, typ, spe;
const
VALUES: array[0..2] of ArbFloat = (1, 2, 5);
Wx = 7;
Wy = 30;
Wln = 20;
var
i: Integer;
x: ArbFloat;
magnitude: ArbFloat;
begin
WriteLn('x':Wx, 'Gamma(x)':Wy, 'ln(Gamma(x))':Wln);
WriteLn;
magnitude := 1E-3;
while magnitude <= 1000 do begin
for i := 0 to High(VALUES) do begin
x := VALUES[i] * magnitude;
if x <= 170 then // Extended overflow above 170
Write(FormatFloat('0.000', spegam(x)):Wy)
else
Write('overflow':Wy);
WriteLn(spelga(x):Wln:3);
if abs(x-1000) < 1E-6 then
break;
end;
magnitude := magnitude * 10;
end;
end.
### Incomplete gamma function
The lower and upper incomplete gamma functions are defined by
$\operatorname{P}({s},{x}) = \frac{1}{\Gamma({s})} \int_0^{x}t^{s-1} e^{-t} dt$
$\operatorname{Q}({s},{x}) = \frac{1}{\Gamma({s})} \int_{x}^{\infty}t^{s-1} e^{-t} dt = 1 - \operatorname{P}({s}, {x})$
function gammap(s, x: ArbFloat): ArbFloat;
function gammaq(s, x: ArbFloat): ArbFloat;
• The parameter s must be positive: s > 0
• The argument x must not be negative: x ≥ 0
• The function results are in the interval [0, 1].
Note: The functions gammap and gammaq are not available in fpc v3.0.x and older.
Example
The following project prints out a table of gammap and gammaq values:
program IncompleteGamma_Table;
uses
SysUtils, StrUtils, typ, spe;
const
s = 1.25;
Wx = 7;
Wy = 20;
D = 6;
var
i: Integer;
x: ArbFloat;
fs: TFormatSettings;
begin
fs := DefaultFormatSettings;
fs.DecimalSeparator := '.';
WriteLn('x':Wx, 'GammaP('+FloatToStr(s,fs)+', x)':Wy+1, 'GammaQ('+FloatToStr(s,fs)+', x)':Wy+1);
WriteLn;
for i := 0 to 20 do begin
x := 0.5*i;
WriteLn(Format('%*.1f %*.*f %*.*f', [
Wx, x,
Wy, D, gammap(s, x),
Wy, D, gammaq(s, x)
], fs));
end;
end.
### Beta function
The Beta function is another basis of important functions used in statistics. It is defined as
$\operatorname{B}(a, b) = \frac{{\Gamma(a)}{\Gamma(b)}}{\Gamma(a+b)} = \int_0^1{t^{x-1} (1-t)^{y-1} dt}$
The NumLib routine for its calculation is
function beta(a, b: ArbFloat): ArbFloat;
• The arguments a and b must be positive: a > 0 and b > 1.
• The calculation is based on the logarithms of the gamma function and therefore does not suffer from the overflow issues of the gamma function.
Note: The function beta is not available in fpc v3.0.x or older.
### Incomplete beta function
The (regularized) incomplete beta function is defined by
$\operatorname{I}_x(a,b) = \frac {1}{\operatorname{B}(ab)} \int_0^x{t^{x-1} (1-t)^{y-1} dt}$
function betai(a, b, x: ArbFloat): ArbFloat;
• The arguments a and b must be positive: a > 0 and b > 1.
• x is only allowed to be in the range 0 ≤ x ≤ 1.
• The function result is in the interval [0, 1].
The inverse incomplete beta function determines the argument x for which the passed parameter is equal to Ix(a, b):
function invbetai(a, b, y: ArbFloat): ArbFloat; The arguments a and b must be positive: a > 0 and b > 1. y is only allowed to be in the range 0 ≤ x ≤ 1. The function result is in the interval [0, 1]. Note: The functions betai and invbetai are not available in fpc v3.0.x and older. Example Calculate the incomplete beta function for some values of x, a, b, and apply the result to the inverse incomplete beta function to demonstrate the validity of the calculations. program incomplete_beta_func; uses SysUtils, typ, spe; procedure TestBetai(a, b, x: ArbFloat); var xc, y: ArbFloat; begin y := betai(a, b, x); xc := invbetai(a, b, y, macheps); WriteLn(' x = ', x:0:6, ' ---> y = ', y:0:9, ' ---> x = ', xc:0:6); end; begin WriteLn('Incomplete beta function and its inverse'); WriteLn; WriteLn(' y = betai(1, 3, x) x = invbetai(1, 3, y)'); TestBetaI(1, 3, 0.001); TestBetaI(1, 3, 0.01); TestBetaI(1, 3, 0.1); TestBetaI(1, 3, 0.2); TestBetaI(1, 3, 0.9); TestBetaI(1, 3, 0.99); TestBetaI(1, 3, 0.999); WriteLn; WriteLn(' y = betai(1, 1, x) x = invbetai(1, 1, y)'); TestBetaI(1, 1, 0.001); TestBetaI(1, 1, 0.01); TestBetaI(1, 1, 0.1); TestBetaI(1, 1, 0.2); TestBetaI(1, 1, 0.9); TestBetaI(1, 1, 0.99); TestBetaI(1, 1, 0.999); WriteLn; WriteLn(' y = betai(3, 1, x) x = invbetai(3, 1, y)'); TestBetaI(3, 1, 0.001); TestBetaI(3, 1, 0.01); TestBetaI(3, 1, 0.1); TestBetaI(3, 1, 0.2); TestBetaI(3, 1, 0.9); TestBetaI(3, 1, 0.99); TestBetaI(3, 1, 0.999); WriteLn; end. Bessel functionsThe Bessel functions are solutions of the Bessel differential equation: ${x}^2 y'' + x y' + ({x}^2 - \alpha^2) {y} = 0$The Bessel functions of the first kind, or cylindrical harmonics, Jα(x), are the solutions which are not singular at the origin, in contrast to the Bessel functions of the second kind, Yα(x). Yα(x) is only defined for x > 0. Example Calculate the probability that random normal-distributed numbers are within 1, 2, and 3 standard deviations around the mean. Calculate the maximum deviation of normal-distributed deviates (in units of standard deviations) from the mean at probabilities 0.90, 0.95, 0.98, 0.99 and 0.999. The solutions for a purely imaginary argument are called modified Bessel functions of the first kind, Iα(x), and modified Bessel function of the second kind, Kα(x). Kα(x) is only defined for x > 0. NumLib implements only the solutions for the parameters α = 0 and α = 1: function spebj0(x: ArbFloat): ArbFloat; // Bessel function of the first kind, J0 (alpha = 0) function spebj1(x: ArbFloat): ArbFloat; // Bessel function of the first kind, J1 (alpha = 1) function speby0(x: ArbFloat): ArbFloat; // Bessel function of the second kind, Y0 (alpha = 0) function speby1(x: ArbFloat): ArbFloat; // Bessel function of the second kind, Y1 (alpha = 1) function spebi0(x: ArbFloat): ArbFloat; // modified Bessel function of the first kind, I0 (alpha = 0) function spebi1(x: ArbFloat): ArbFloat; // modified Bessel function of the first kind, I1 (alpha = 1) function spebk0(x: ArbFloat): ArbFloat; // modified Bessel function of the second kind, K0 (alpha = 0) function spebk1(x: ArbFloat): ArbFloat; // modified Bessel function of the second kind, K1 (alpha = 1) Other functionsThe unit spe contains some other functions which will not be documented here because they are already available in th FPC's standard unit math. Function Equivalent function in math Description function speent(x: ArbFloat): LongInt; floor(x) Entier function, calculates first integer smaller than or equal to x function spemax(a, b: Arbfloat): ArbFloat; max(a, b) Maximum of two floating point values function spepow(a, b: ArbFloat): ArbFloat; power(a, b) Calculates ab function spesgn(x: ArbFloat): ArbInt; sign(x) Returns the sign of x (-1 for x < 0, 0 for x = 0, +1 for x > 0) function spears(x: ArbFloat): ArbFloat; arcsin(x) Inverse function of sin(x) function spearc(x: ArbFloat): ArbFloat; arccos(x) Inverse function of cos(x) function spesih(x: ArbFloat): ArbFloat; sinh(x) Hyperbolic sine function specoh(x: ArbFloat): ArbFloat; cosh(x) Hyperbolic cosine function spetah(x: ArbFloat): ArbFloat; tanh(x) Hyperbolic tangent function speash(x: ArbFloat): ArbFloat; arcsinh(x) Inverse of the hyperbolic sine function speach(x: ArbFloat): ArbFloat; arccosh(x) Inverse of the hyperbolic cosine function speath(x: ArbFloat): ArbFloat; arctanh(x) Inverse of the hyperbolic tangent | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6581600308418274, "perplexity": 2523.8424012721784}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400198942.13/warc/CC-MAIN-20200921050331-20200921080331-00073.warc.gz"} |
https://eprints.soton.ac.uk/436064/ | The University of Southampton
University of Southampton Institutional Repository
# Lattice-enhanced Fano resonances from bound states in the continuum metasurfaces
Tan, Thomas C.W., Plum, Eric and Singh, Ranjan (2020) Lattice-enhanced Fano resonances from bound states in the continuum metasurfaces. Advanced Optical Materials, 8 (6), 1-10, [1901572].
Record type: Article
## Abstract
Fano resonances in metamaterials are known for their high quality (Q) factor and high sensitivity to external perturbations, which makes them attractive for sensors, lasers, non-linear and slow light devices. However, Fano resonances with higher Q factors obtained through structural optimization of individual resonators are accompanied by lower resonance intensity, thereby limiting the overall figure of merit (FoM) of the resonance. This article reports a strategy for simultaneously enhancing the Q factor and FoM of Fano resonances in terahertz metamaterials. Coupling of the Fano resonance, which arises from a symmetry-protected bound state in continuum, to the first order lattice mode of the metamaterial array leads to stronger field confinement and substantial enhancement of both Q factor and FoM. As such enhancement occurs in planar metamaterials independently of the resonator geometry, the proposed approach can be utilized for a wide range of high-Q and high-sensitivity terahertz metadevices.
Text
FanoBIC-accepted version - Accepted Manuscript
Restricted to Repository staff only until 13 January 2021.
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Accepted/In Press date: 19 November 2019
e-pub ahead of print date: 13 January 2020
Published date: 19 March 2020
Keywords: Metamaterial, Fano resonance, Bound states in continuum, BIC, Lattice mode, Terahertz
## Identifiers
Local EPrints ID: 436064
URI: http://eprints.soton.ac.uk/id/eprint/436064
ISSN: 2195-1071
ORCID for Eric Plum: orcid.org/0000-0002-1552-1840
## Catalogue record
Date deposited: 27 Nov 2019 17:30
## Contributors
Author: Thomas C.W. Tan
Author: Eric Plum
Author: Ranjan Singh | {"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": 0.8342658281326294, "perplexity": 14376.135964961002}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737645.2/warc/CC-MAIN-20200808110257-20200808140257-00385.warc.gz"} |
https://space.stackexchange.com/questions/30501/how-could-the-2018-08-30-soyuz-ms-09-iss-leak-be-so-slow/30504 | # How could the 2018-08-30 Soyuz MS-09 / ISS leak be so slow?
The leak reportedly led the ISS to lose 0.8 millibars of air pressure per hour, which is both big considering what was at stake and low for a 2-mm hole when the outside is near to complete vacuum. Am I missing something? How could the German astronaut even block it with his finger without any damage if complete vacuum was on the other side?
Per this comment, the value of 0.8 mbar/hr seems to come from:
• Small holes and vacuum in general aren't as lethal as sci-fi would make you think. In fact, humans can produce quite a bit of suction by themselves. If you hold a drinking straw to your skin and suck in using your mouth, you can achieve down to around %10 air pressure and you won't be injured or probably even uncomfortable. See here someone measures human suction power: youtu.be/ANVI04QmthE – Dragongeek Sep 6 '18 at 21:51
• I was blocking a 10-bar hose of about 5mm diameter with my finger just fine. Blocking a hole against 1 bar suction is really nothing. – SF. Sep 6 '18 at 22:42
• There is a video on youtube by CodysLab who showed exactly that: blocking a tiny hole in vacuum against the pressure. – PlasmaHH Sep 7 '18 at 11:08
• I'm curious, where did the "0.8 millibars of air pressure per hour" figure come from exactly? Do you have a source? – uhoh Sep 7 '18 at 18:16
• @uhoh the only source is Stephen Clark from spaceflightnow.com/2018/08/30/…. There is no any mentions of the leak rate on other sources, including NASA site. But usually spaceflightnow.com is rather reliable source. – Heopps Sep 8 '18 at 6:28
The ISS is at 1 bar, i.e. 1 kgf/cm2, or 10 gramsf/mm2. So the pressure on that 2 mm hole is 31.4 gramsf, well within the range a human finger can handle.
Also, the ISS is really big compared to the hole. It takes a long time for hundreds of m3 to evacuate through a 2 mm hole.
• thank you, intuitively I thought the pressure would be much higher on the finger and as you say 2 mm is not much compared to the volume of the ISS, so it makes sense! I guess I must have been influenced by SF movies where everything explodes as soon as there's a leak ^^ I should have made the calculation instead of relying on my physical intuition. – tsnobip Sep 6 '18 at 14:15
• @tsnobip check this video Use Your Finger to Stop a Leak on the Space Station? . You can see how easy it's to stop the leak with your finger without any issue (maybe just a small hickey on your finger :) ) – bitcell Sep 7 '18 at 6:17
• well nice finding @bitcell, that's exactly the kind of thing I was looking for! But still, I think I would have probably hesitated a bit before putting my finger on a leak of the ISS in real life! ^^ – tsnobip Sep 7 '18 at 9:05
• Pressure is force per unit area, not mass per unit area or mass. – David Richerby Sep 7 '18 at 18:11
• fixed the units. – Hobbes Sep 7 '18 at 19:34
This is the image of the hole (news source, although the image is from NASA)
The hole is 2mm in diameter. Even with a total vacuum on the other side, you're not talking a lot of volume getting through that hole. I used this calculator with a pressure gradient of 101kPa (ISS standard) and 0.1 kPa through a 2mm hole and got a water flow rate of ~0.1 cubic meters of water per hour (or 26.6 gallons per hour). For contrast, the US has a hard rule limit of 10 gallons per minute on gasoline pumps. As such, the pressure exerted through that 2mm hole is not going to be enough to pull much more than air or a nearby viscous liquid out.
• You don't need a flow rate calculator. Simply use $\dot p/p=\dot m/m = \dot V/V$ and convert that 0.8 mb/hr pressure loss to a mass loss of about 0.9 kg/hr or a volumetric flow rate of 0.8 m$^3$/hr. (I'm keeping all the numbers to one place of accuracy, since that's what was given originally.) – David Hammen Sep 6 '18 at 14:32
• Also, the hole wasn't all the way through from the interior of the hull to the exterior. It was a hole in the internal layer of the hull, with a bunch of kevlar, aluminum, and other materials in-between that would further insulate his skin and may have potentially lowered the leak rates even more. – Magic Octopus Urn Sep 6 '18 at 19:08
• Using water or gasoline flow rates for comparison is, at least without further discussion, pretty useless. One might also compute that only, like, two grams of honey would drip the hole per second or something, but this wouldn't tell us anything about how much air goes through it. – leftaroundabout Sep 7 '18 at 12:23
• @leftaroundabout My point (which I guess got lost here) is that if not much liquid can escape that hole, air (which is mostly empty space itself) would produce even less pressure. Hobbes' answer was better by calculating pressure itself. – Machavity Sep 7 '18 at 13:40
• You should have used the right one calculator for an air flow. The water flow rate calculator is the wrong one. – Uwe Dec 26 '19 at 21:00
It's not really that the leak was slow, more that it took some time to manifest:
Another source told the news agency the worker did not report the error and instead applied a sealant of some sort. After two months in orbit, the sealant apparently dried out, the source said, and was expelled by the cabin air pressure, opening up a leak.
(The article is slightly incorrect; MS-09 docked with the ISS on June 06 and the leak was detected on the 29/30 August, so it actually took almost three months to become apparent).
After the sealant failed, the leak was detected very quickly, and its small size meant that relatively little atmosphere escaped before the astronauts were able to patch it. Their temporary repair will not be a long-term issue for the ISS - the module with the leak is the orbital module of Soyuz, which is jettisoned to burn up into the atmosphere after the astronauts depart the ISS.
• According to the article I referenced, the worker used glue to fix the hole. Had they reported it, they could have fixed it by welding it.. – Machavity Sep 7 '18 at 13:37
• @tsnobip have you seen this answer yet? I think this addresses the timing nicely, but it may have been posted after you accepted the other answer. – uhoh Sep 7 '18 at 18:18
tl;dr:
How could the 2018-08-30 Soyuz MS-09 / ISS leak be so slow?
By being about 2 millimeters in diameter!
@DavidHammen's comment converts 0.8 mbar/hr to about 0.8 m^/hr air loss rate presumably at standard conditions. Let's see how that's done, how it checks against "a 2mm hole" and what it means if there were no response of any kind (human or make-up air).
He uses the first order relationships
$$\frac{\dot{p}}{p}=\frac{\dot{m}}{m}=\frac{\dot{V}}{V} = 0.8 \times 10^{-3}/hr$$
where I'm guessing $p$ is the pressure of the remaining air (assuming no make-up air and no change in temperature, which is reasonable considering the air is in intimate contact with so much solid surface area), $m$ is the mass of the remaining air, and $V$ is the equivalent volume of the remaining air if it were at standard conditions.
An ISS pressurized volume above about 938 m^3 (matches values on the internet) times $$0.8 \times 10^{-3}/hr$$ does indeed give about 0.8 m^3/hr!
Now let's see what a 2 mm hole in a thin plate is expected to do. I found two online calculators, although they may have somewhat different assumptions, and the hole has some depth (the wall thickness of the Soyuz at this location) and side roughness, but still we can try.
Number 1. gives 2.74E-04 m^3/sec or 1.0 m^3/hr, almost identical to the quoted 0.8 m^3/hr value, and number 2 gives something at least close; about 3 m^3/hr.
So baring any make-up air, that's about a 1% drop in pressure every ten hours. That's enough to be alarming, and of course would probably trigger an alarm in less than ten hours since strikes by meteorites and debris so likely over time that one would expect the ISS to be hyper-vigilant about leaks. | {"extraction_info": {"found_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": 0.4875454902648926, "perplexity": 1603.2896239261454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178374391.90/warc/CC-MAIN-20210306035529-20210306065529-00614.warc.gz"} |
http://clay6.com/qa/11146/the-linear-momentum-p-of-a-moving-body-varies-with-time-according-to-equati | Browse Questions
# The linear momentum p of a moving body varies with time according to equation $p=a+bt^2$ where a,b are constants. Net force acting on the body is
a) Constant
b) proportional to $t^2$
c) Inversely proportional to t
d) proportional to t
$F=\large\frac{dp}{dt}=\frac{d}{dt}$$(a+bt^2)$
$F=2bt$
$f \;\alpha\; t$
Net force acting on the body is proportional to t
Hence d is the correct answer.
edited Jan 26, 2014 by meena.p
+1 vote | {"extraction_info": {"found_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": 0.8251183032989502, "perplexity": 1410.9605479675272}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988720962.53/warc/CC-MAIN-20161020183840-00559-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://papers.nips.cc/paper/1989/hash/92c8c96e4c37100777c7190b76d28233-Abstract.html | #### Authors
Gintaras Reklaitis, Athanasios Tsirukis, Manoel Tenorio
#### Abstract
A nonlinear neural framework, called the Generalized Hopfield network, is proposed, which is able to solve in a parallel distributed manner systems of nonlinear equations. The method is applied to the general nonlinear optimization problem. We demonstrate GHNs implementing the three most important optimization algorithms, namely the Augmented Lagrangian, Generalized Reduced Gradient and Successive Quadratic Programming methods. The study results in a dynamic view of the optimization problem and offers a straightforward model for the parallelization of the optimization computations, thus significantly extending the practical limits of problems that can be formulated as an optimization problem and which can gain from the introduction of nonlinearities in their structure (eg. pattern recognition, supervised learning, design of content-addressable memories).
1 To whom correspondence should be addressed.
356
Reklaitis, Tsirukis and Tenorio
1 RELATED WORK The ability of networks of highly interconnected simple nonlinear analog processors (neurons) to solve complicated optimization problems was demonstrated in a series of papers by Hopfield and Tank (Hopfield, 1984), (Tank, 1986). The Hopfield computational model is almost exclusively applied to the solution of combinatorially complex linear decision problems (eg. Traveling Salesman Problem). Unfortunately such problems can not be solved with guaranteed quality, (Bruck, 1987), getting trapped in locally optimal solutions. Jeffrey and Rossner, (Jeffrey, 1986), extended Hopfield's technique to the nonlinear unconstrained optimization problem, using Cauchy dynamics. Kennedy and Chua, (Kennedy, 1988), presented an analog implementation of a network solving a nonlinear optimization problem. The underlying optimization algorithm is a simple transformation method, (Reklaitis, 1983), which is known to be relatively inefficient for large nonlinear optimization problems.
2 LINEAR HOPFIELD NETWORK (LHN) The computation in a Hopfield network is done by a collection of highly interconnected simple neurons. Each processing element, i, is characterized by the activation level, Ui, which is a function of the input received from the external environment, Ii, and the state of the other neurons. The activation level of i is transmitted to the other processors, after passing through a filter that converts Ui to a 0-1 binary value, Vi' The time behavior of the system is described by the following model:
~ T·V· - -' + I· ' ~ 'J J J | {"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": 0.8430560231208801, "perplexity": 1272.461616608657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030336921.76/warc/CC-MAIN-20221001195125-20221001225125-00255.warc.gz"} |
https://cstheory.stackexchange.com/questions/35991/np-hardness-on-cayley-graphs/35993 | # NP-hardness on Cayley graphs
What is known about complexity of NP-hard problems on Cayley graphs?
Suppose that the graph is given explicitly as the multiplication table of the group and the list of generators. So the input length is the size of the graph. Can we solve NP-complete problems on such graphs (maximum clique/max-cut) in polynomial time?
What about some special cases of groups? For example, $\mathbb{Z}_n$ (a.k.a. circulant graphs) or $\mathbb{Z}_2^{\log(n)}$. That is, the input to the problem is the set of generators (and $1^n$ to represent the size of the graph). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9370612502098083, "perplexity": 282.1104436839532}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316783.70/warc/CC-MAIN-20190822042502-20190822064502-00541.warc.gz"} |
http://www.math.u-szeged.hu/mathweb/index.php/hu/component/jevents/icalrepeat.detail/2021/09/23/1509/-/sadegh-marzban-university-of-szeged-a-hybrid-pde-abm-model-for-infection-dynamics-study-on-stochastic-variability-application-to-sars-cov-2-and-influe?Itemid=110 | Év szerint Hónap szerint Ugrás a hónaphoz
## Sadegh Marzban (University of Szeged): A hybrid PDE-ABM model for infection dynamics: study on stochastic variability, application to SARS-COV-2 and influenza, and exploring some treatment options
Csütörtök, 23. Szeptember 2021, 11:00 - 12:30
Abstract.
We propose a hybrid partial differential equation -- agent-based (PDE--ABM) model to describe the spatio-temporal viral dynamics in a cell population. The virus concentration is considered as a continuous variable and virus movement is modelled by diffusion, while changes in the states of cells (i.e. healthy, infected, dead) are represented by a stochastic agent-based model. The two subsystems are intertwined: the probability of an agent getting infected in the ABM depends on the local viral concentration, and the source term of viral production in the PDE is determined by the cells that are infected.
We develop a computational tool that allows us to study the hybrid system and the generated spatial patterns in detail. We systematically compare the outputs with a classical ODE system of viral dynamics, and find that the ODE model is a good approximation only if the diffusion coefficient is large.
We demonstrate that the model is able to predict SARS--CoV--2 infection dynamics, and replicate the output of in vitro experiments. Applying the model to influenza as well, we can gain insight into why the outcomes of these two infections are different.
Hely : Riesz Lecture Hall, 1st Floor, Bolyai Institute, Aradi Vértanúk tere 1., Szeged | {"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": 0.8074097037315369, "perplexity": 1814.6329929075528}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764499634.11/warc/CC-MAIN-20230128121809-20230128151809-00379.warc.gz"} |
https://www.assignmentexpert.com/homework-answers/economics/other/question-67442 | 68 719
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# Answer to Question #67442 in Other Economics for saad A. aljibreen
Question #67442
In a simultaneous move game, if both charge the low price (LP) they each earn $5K. If both charge the high price (HP) they each earn$10K. If they charge different prices, the one playing LP earns $15K and the other earns$1K. If the businesses play indefinitely, then a Nash Equilibrium is
a. for one firm to charge a HP forever
b. for your firm charge a LP when the other firm does
c. for each firm to charge HP until the rival does not, and then to charge LP
d. for each firm to charge LP until the rival does not, and then to charge a HP forever
A common solution to a coordination game is
a. to issue threats before the game starts
b. to change the payoffs
c. for one player to accommodate the other
d. for one player to go first
If the businesses play indefinitely, then a Nash Equilibrium is:
d. for each firm to charge LP until the rival does not, and then to charge a HP forever
A common solution to a coordination game is:
c. for one player to accommodate the other.
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for any assignment or question with DETAILED EXPLANATIONS! | {"extraction_info": {"found_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": 0.2934344708919525, "perplexity": 6094.733804283696}, "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-2019-04/segments/1547583700012.70/warc/CC-MAIN-20190120042010-20190120064010-00051.warc.gz"} |
http://mathhelpforum.com/calculus/17256-parametric-equations.html | Math Help - Parametric equations
1. Parametric equations
Hi,
I'm completely lost in this problem .... can someone help me pls?
Consider the parametric equations x= 4*cos^2(θ) and y = 2*sin(θ)
and fill this chart.
When t = -pi/2, x = ? and y = ?
When t = -pi/4, x = ? and y = ?
When t = pi/4, x = ? and y = ?
When t = pi/2, x = ? and y = ?
I also have to graph the equation in my pgrahing calculator. I tried to do it but I don't know if I did it right.
- Isabel
2. Originally Posted by cuteisa89
Hi,
I'm completely lost in this problem .... can someone help me pls?
Consider the parametric equations x= 4*cos^2(θ) and y = 2*sin(θ)
and fill this chart.
When t = -pi/2, x = ? and y = ?
When t = -pi/4, x = ? and y = ?
When t = pi/4, x = ? and y = ?
When t = pi/2, x = ? and y = ?
I also have to graph the equation in my pgrahing calculator. I tried to do it but I don't know if I did it right.
- Isabel
When you say "When t = -pi/2" do you mean "When $\theta$ = -pi/2?"
Just plug the angle into the given formulas for x and y.
For example:
$\theta = \pi/2 \implies x = 4cos^2(\pi/2) = 0 \text{ and }y = 2sin(\pi/2) = 2$
-Dan
3. Yeah.... t= θ.
Thanks I think I've got it now. I just wasn't sure how to do it properly.
Thanks Dan!
4. Originally Posted by cuteisa89
Hi,
I'm completely lost in this problem .... can someone help me pls?
Consider the parametric equations x= 4*cos^2(θ) and y = 2*sin(θ)
and fill this chart.
When t = -pi/2, x = ? and y = ?
When t = -pi/4, x = ? and y = ?
When t = pi/4, x = ? and y = ?
When t = pi/2, x = ? and y = ?
I also have to graph the equation in my pgrahing calculator. I tried to do it but I don't know if I did it right.
- Isabel
x=4(cost)^2=4(1-(sint)^2)=4-(2sint)^2=4-y^2
So it is a parabola like this:
Attached Thumbnails | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8724790215492249, "perplexity": 1207.8493070874908}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776438539.21/warc/CC-MAIN-20140707234038-00003-ip-10-180-212-248.ec2.internal.warc.gz"} |
https://web2.0calc.com/questions/nice_3 | +0
# Nice!
0
323
2
+816
Find the solutions to $$z^3 = -8.$$ Enter the solutions, separated by commas.
Dec 28, 2018
#1
+845
+1
if the question is find z because im not sure,
z^3 = -8
(-2)^3 = -8
therefore z = -2
Dec 28, 2018
edited by YEEEEEET Dec 28, 2018
edited by YEEEEEET Dec 28, 2018
#2
+109519
+3
That is correct Zeet.
The only real solution is z=-2
but I think mathtoo wants the complex solutions too. It is a cubic so there are 3 solutions altogether.
There will be equally spaced at $$\frac{2\pi}{3}$$ radians apart on the complex plane. So the others are at $$\frac{\pi}{3}\;\;and \;\;\frac{-\pi}{3}$$
The modulus or absolute value is 2
so
$$z=-2,\;\;2e^{(\pi/3)i},\;\;2e^{(-\pi/3)i}$$
$$2e^{(\frac{\pi}{3})i}\\=2[cos(\pi/3)+isin(\pi/3)]\\ =2[\frac{1}{2}+i\frac{\sqrt3}{2}]\\ =1+i\sqrt3\\~\\ 2e^{(\frac{\pi}{3})i}\\ =1-i\sqrt3$$
so
$$z=-2,\;\;1+i\sqrt3,\;\;1-i\sqrt3$$
Here is the diagramatic representation. Copied from Wolfram|Alpha.
Melody Dec 28, 2018
edited by Melody Dec 28, 2018 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9683364629745483, "perplexity": 12013.677723608334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1590347423915.42/warc/CC-MAIN-20200602064854-20200602094854-00506.warc.gz"} |
http://mathhelpforum.com/number-theory/227581-help-i-m-stuck-question.html | # Math Help - Help I'm stuck with this question:
1. ## Help I'm stuck with this question:
Prove that:
If n is any integer that is not divisible by 2 or 3 then n^2 mod 12 =1?
help i don't know what to do...
2. ## Re: Help I'm stuck with this question:
If $n\not| 2$ or $n\not| 3$, then $n^2(\text{mod} 12) \equiv 1$
Is this what you mean?
3. ## Re: Help I'm stuck with this question:
Originally Posted by mendax05
Prove that:
If n is any integer that is not divisible by 2 or 3 then n^2 mod 12 =1?
help i don't know what to do...
$\exists\ p \in \mathbb Z\ such\ that\ n = 3p + k,\ where\ k = 1\ or\ k = 2.$
$CASE\ A:\ k = 1.$
Assume p is odd. Then 3p is odd. So 3p + 1 is even. Thus n is even. But by hypothesis n is not evenly divisible by 2 and so is not even. Contradiction. So p is even.
$\exists\ q \in \mathbb Z\ such\ that\ 2q = p \implies n = 3p + 1 = 6q + 1 \implies$
$n^2 = 36q^2 + 12q + 1 \implies \dfrac{n^2}{12} = \dfrac{36q^2 + 12q + 1}{12} = 3q^2 + q + \dfrac{1}{12}.$ QED
$CASE\ B:\ k = 2.$
Assume p is even. Then 3p is even. So 3p + 2 is even. Thus n is even. But by hypothesis n is not evenly divisible by 2 and so is not even. Contradiction. So p is odd.
$\exists\ q \in \mathbb Z\ such\ that\ 2q + 1 = p \implies n = 3p + 2 = 3(2q + 1) + 2 = 6q + 5 \implies$
$n^2 = 36q^2 + 60q + 25 \implies \dfrac{n^2}{12} = \dfrac{36q^2 + 60q + 25}{12} = 3q^2 + 5q + 2 + \dfrac{1}{12}.$ QED
4. ## Re: Help I'm stuck with this question:
It finally dawned on me that I made this proof MUCH harder than it needed to be.
What we need is this lemma: for any three successive integers, exactly one is evenly divisible by 3.
$n\ not\ evenly\ divisible\ by\ 2 \implies n + 1\ and\ n - 1\ evenly\ divisible\ by\ 2.$
$n\ not\ evenly\ divisible\ by\ 3 \implies either\ n + 1\ or\ else\ n - 1\ evenly\ divisible\ by\ 3.$
$\therefore n \pm 1\ is\ evenly\ divisible\ by\ 2\ and\ 3 \implies n \pm 1\ is\ evenly\ divisible\ by\ 6 \implies$
$\exists\ p \in \mathbb Z\ such\ that\ 6p = n \pm 1 \implies n = 6p \pm 1 \implies n^2 = 36p^2 \pm 12p + 1 \implies$
$\dfrac{n^2}{12} = \dfrac{36p^2 \pm 12p + 1}{12} = 3p^2 \pm p + \dfrac{1}{12} = an\ integer + \dfrac{1}{12}. QED.$
5. ## Re: Help I'm stuck with this question:
please don't post multiple copies of the same question. I answered this question here as well. | {"extraction_info": {"found_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": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.937960684299469, "perplexity": 462.8406222471629}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1440644064538.31/warc/CC-MAIN-20150827025424-00204-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/linear-algebra-and-its-applications-5th-edition/chapter-1-linear-equations-in-linear-algebra-1-1-exercises-page-10/18 | # Chapter 1 - Linear Equations in Linear Algebra - 1.1 Exercises: 18
No. The planes do not have a point of intersection; i.e., the system of equations is not consistent.
#### Work Step by Step
Simplifying the augmented matrix to triangular form provides the relations: $x_1=\frac{-14}{6}$, $x_2=\frac{-5}{6}$, and $0=\frac{-5}{6}$. This final condition is not true, so there is no $(x_1, x_2)$ such that the equations for the three specified planes are satisfied.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | {"extraction_info": {"found_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": 0.8061339855194092, "perplexity": 414.07023493374857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589618.52/warc/CC-MAIN-20180717070721-20180717090721-00254.warc.gz"} |
https://proceedings.neurips.cc/paper/2020/hash/09ccf3183d9e90e5ae1f425d5f9b2c00-Abstract.html | #### Authors
Qian Huang, Horace He, Abhay Singh, Yan Zhang, Ser Nam Lim, Austin R. Benson
#### Abstract
<p>Incorporating relational reasoning into neural networks has greatly expanded their capabilities and scope. One defining trait of relational reasoning is that it operates on a set of entities, as opposed to standard vector representations. Existing end-to-end approaches for relational reasoning typically extract entities from inputs by directly interpreting the latent feature representations as a set. We show that these approaches do not respect set permutational invariance and thus have fundamental representational limitations. To resolve this limitation, we propose a simple and general network module called Set Refiner Network (SRN). We first use synthetic image experiments to demonstrate how our approach effectively decomposes objects without explicit supervision. Then, we insert our module into existing relational reasoning models and show that respecting set invariance leads to substantial gains in prediction performance and robustness on several relational reasoning tasks. Code can be found at github.com/CUAI/BetterSetRepresentations.</p> | {"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": 0.8799557685852051, "perplexity": 3366.089477514165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178359497.20/warc/CC-MAIN-20210227204637-20210227234637-00519.warc.gz"} |
https://bkamins.github.io/julialang/2022/06/17/missing.html | Introduction
Some time ago I have written a post about ABC of handling missing values in Julia. Its objective was to give an introduction to the topic for the newcomers. However, occasionally users complain that working with missing values in Julia is less convenient than in e.g. Python or R.
Such opinions are always debatable, so recently I decided to run a small pool on Julia Discourse about the skipmissing function. The question was if we want to shorten the skipmissing name into something that is more convenient to use in interactive work. To my surprise, a vast majority of voters preferred a verbose and explicit operation name. This preference regarding handling of missings, to my surprise, reminded me of several passages from The Zen of Python:
• Explicit is better than implicit.
• In the face of ambiguity, refuse the temptation to guess.
So given this preference, how should Julia users retain convenience. Let me share some of my thoughts on this topic that did not make into my previous post.
This post was written under Julia 1.7.2, Missings.jl 1.0.2, MissingsAsFalse.jl 0.1, and StatsBase.jl 0.33.16.
Verbosity
Indeed writing missing, skipmissing, and passmissing (the last one is defined in Missings.jl), in places where they are needed, might seem verbose. However, the good news is that most of the time you do not have to type them fully thanks to completions so in your editor/REPL:
• instead of missing write mis<tab>;
• instead of skipmissing write skipm<tab>;
• instead of passmissing write pas<tab>.
Let us compare. In R:
sum(c(1, NA), na.rm=T)
vs Julia:
sum(skipmissing([1, missing]))
seems longer. However, if you take into account the amount of typing you need to do (number of keystrokes) it is the same.
Missing values in logical conditions
As I have written in this post I personally strongly recommend using coalesce to handle missing values in logical conditions. This allows you, to follow the Explicit is better than implicit. principle by explicitly showing in the code if missing should be treated as true or as false.
Here is a short example:
julia> c = missing
missing
julia> c ? "true" : "false"
ERROR: TypeError: non-boolean (Missing) used in boolean context
julia> coalesce(c, false) ? "true" : "false"
"false"
However, some users find it more convenient to use @mfalse macro from the MissingsAsFalse.jl package:
julia> using MissingsAsFalse
julia> @mfalse c ? "true" : "false"
"false"
Correlation matrix with missing values
A common, and relatively complex case of handling missing values, is computing of correlation matrix of data that contains missings. Let us check what Julia offers here. We will use the pairwise function from StatsBase.jl.
julia> using Random
julia> using StatsBase
julia> using Statistics
julia> Random.seed!(1234);
julia> x = rand([1:10; missing], 16, 4)
16×4 Matrix{Union{Missing, Int64}}:
4 missing 2 10
7 8 missing 8
3 missing 7 5
10 9 8 10
4 2 7 6
5 6 1 missing
missing 7 8 9
9 3 2 6
6 7 10 missing
9 3 4 6
7 2 9 8
9 7 3 3
1 8 6 5
3 9 4 7
5 7 3 2
8 5 missing 4
julia> pairwise(cor, eachcol(x))
4×4 Matrix{Union{Missing, Float64}}:
1.0 missing missing missing
missing 1.0 missing missing
missing missing 1.0 missing
missing missing missing 1.0
julia> pairwise(cor, eachcol(x), skipmissing=:pairwise)
4×4 Matrix{Float64}:
1.0 -0.218849 -0.0177704 0.0922413
-0.218849 1.0 0.00973122 0.102969
-0.0177704 0.00973122 1.0 0.364821
0.0922413 0.102969 0.364821 1.0
julia> pairwise(cor, eachcol(x), skipmissing=:listwise)
4×4 Matrix{Float64}:
1.0 -0.229568 -0.100028 0.247283
-0.229568 1.0 -0.127153 -0.0420058
-0.100028 -0.127153 1.0 0.67068
0.247283 -0.0420058 0.67068 1.0
By default pairwise for cor returns missing when at least one of the columns contains missing values. Use :pairwise value of skipmissing keyword argument to skip entries with a missing value in either of the two vectors passed to cor and use :listwise to skip entries with a missing value in any of the vectors passed to pairwise.
In this example we see that since there are several ways how missing values should be handled by cor in pairwise computations, instead of using the skipmissing function, a keyword argument is used allowing to specify what the data scientist wants exactly.
A similar situation is in the subset and subset! functions from DataFrames.jl, that also take a skipmissing keyword argument, to simplify handling of logical conditions specifying which rows from a source data frame should be kept.
Conclusions
In Julia the approach is that handling of missing values is explicit. This choice is guided by the fact that then in the code it is explicitly visible what was the developer’s decision about how they should be treated. This choice makes code more verbose. Today I have tried to show that, especially with a good editor/REPL support, in practice it does not introduce a large overhead. | {"extraction_info": {"found_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": 0.5158239603042603, "perplexity": 2395.7970908860757}, "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/1656103915196.47/warc/CC-MAIN-20220630213820-20220701003820-00665.warc.gz"} |
https://lists.nongnu.org/archive/html/auctex-devel/2022-11/msg00019.html | auctex-devel
[Top][All Lists]
## Re: scope environment in TikZ
From: Arash Esbati Subject: Re: scope environment in TikZ Date: Thu, 10 Nov 2022 14:14:53 +0100 User-agent: Gnus/5.13 (Gnus v5.13) Emacs/29.0.50
Hi Keita,
Ikumi Keita <[email protected]> writes:
> The next thing I'd like to discuss is scope environment in TikZ.
You're drawing some TikZ pictures lately? ;-)
> The scope environment is practically always accompanied by an optional
> argument by its nature:
> \begin{tikzpicture}
> \begin{scope}[TikZ options...]
> ...
> \end{scope}
> \end{tikzpicture}
> In other words, its "optional" argument is virtually a kind of mandatory
> argument.
I'm not familiar with TikZ, but I had a brief look at the manual and
indeed, using the scope environment without options is pretty pointless.
> So I'm thinking to apply the attached patch. It modifies the AUCTeX
> behavior with respect to the scope environment to insert a bracket pair
> "[]" even when the user gave empty answer for query of optional
> argument. In addition, it moves the point inside in that empty "[]",
> assuming that the user actually expects to fill in that bracket
> immediately, possibly with the help of in-buffer completion facility.
Keep in mind that if you want to use in-buffer completion when you're
inside the brackets, you need (TeX-arg-key-val key=vals)' in the hook,
so I think the best would be if you could see if you achieve what you're
looking after with LaTeX-env-args' -- but I think that wouldn't work
because of the (save-excursion ...) it uses?
> I admit that it might be a bit too aggresive. What do you think about
> this idea?
See above. Patching ConTeXt-insert-environment' seems reasonable to
me.
Best, Arash
` | {"extraction_info": {"found_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": 0.9195736646652222, "perplexity": 6802.666797250063}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500619.96/warc/CC-MAIN-20230207134453-20230207164453-00466.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/search.php?author_id=16276&sr=posts&sid=6c724b216d8465756a23932c4c1ff857 | ## Search found 111 matches
Sat Mar 14, 2020 5:48 pm
Forum: Reaction Mechanisms, Reaction Profiles
Topic: Order of Reactions
Replies: 4
Views: 115
### Re: Order of Reactions
For an elementary step, the coefficients double as the order of the reactants.
The order for a non-elementary reaction cannot be determined from the formula, only experimentally.
Sat Mar 14, 2020 5:44 pm
Forum: General Rate Laws
Topic: Elementary steps
Replies: 3
Views: 140
### Re: Elementary steps
Elementary steps make up each baby-step of the reaction, and sum to the total reaction. The total reaction is usually a non-elementary step.
Sat Mar 14, 2020 5:43 pm
Forum: Ideal Gases
Topic: 5% Rule
Replies: 6
Views: 228
### Re: 5% Rule
You also know that you can use the approximation if your equilibrium constant, K, is small enough.
Sat Mar 14, 2020 5:41 pm
Forum: Ideal Gases
Topic: Kc
Replies: 7
Views: 169
### Re: Kc
You use any products and reactants so long as they are not liquids or solids.
Tue Mar 10, 2020 6:45 pm
Forum: Arrhenius Equation, Activation Energies, Catalysts
Topic: Activation Energy vs. Free Energy of Activation
Replies: 3
Views: 94
### Activation Energy vs. Free Energy of Activation
What is the difference between activation energy and free energy of activation?
Thu Mar 05, 2020 10:00 pm
Forum: Method of Initial Rates (To Determine n and k)
Topic: Determining n
Replies: 5
Views: 161
### Re: Determining n
Do you mean n in electrochemistry, for the number of moles of electrons being transferred? Or n in kinetics, for the order of the reaction?
Thu Mar 05, 2020 9:54 pm
Forum: Method of Initial Rates (To Determine n and k)
Topic: units of k
Replies: 9
Views: 212
### Re: units of k
If the reaction is zero-order, K has units of M/s
If the reaction is first-order, K has units of 1/s
If the reaction is second-order, K has units of 1/M.s
Thu Mar 05, 2020 4:48 pm
Forum: General Rate Laws
Topic: First vs Second Order Rate Laws
Replies: 3
Views: 78
### Re: First vs Second Order Rate Laws
Increasing the concentration of a reactant does NOT increase the rate of the reaction. But as stated above, an increase in concentration would have a greater effect on a second rate reaction than a first rate reaction due to the exponent.
Thu Mar 05, 2020 4:36 pm
Forum: Kinetics vs. Thermodynamics Controlling a Reaction
Topic: Comparing Experimental Data
Replies: 3
Views: 123
### Re: Comparing Experimental Data
We created a ratio of concentration 2/concentration 1, both of which were raised to the power of m. 2.0 was the value of that ratio (2.0^m). The ratio of the actual rates was also 2, so m must be 1.
Mon Mar 02, 2020 9:25 pm
Forum: Galvanic/Voltaic Cells, Calculating Standard Cell Potentials, Cell Diagrams
Topic: Inert Electrode
Replies: 1
Views: 73
### Inert Electrode
How do you know if a half reaction needs an inert electrode like Pt?
Sun Mar 01, 2020 10:34 pm
Forum: Balancing Redox Reactions
Topic: Self Test 6L.3
Replies: 1
Views: 84
### Self Test 6L.3
I don't understand how the textbook found this answer. The questions asks us to write the chemical equation corresponding to a cell, and I found the correct oxidation and reduction equations, but not the correct full equation. reduction : Hg2(NO3)2 + 2e- --> 2Hg + 2NO3- oxidation : 2Hg + 2HCl --> Hg...
Thu Feb 27, 2020 9:59 pm
Forum: Work, Gibbs Free Energy, Cell (Redox) Potentials
Topic: cell potential
Replies: 3
Views: 111
### Re: cell potential
Cell potential is greatest when the cell reaction first begins to occur. Following this, the concentrations change and the potential is no longer at its maximum value.
Thu Feb 27, 2020 9:57 pm
Forum: Interesting Applications: Rechargeable Batteries (Cell Phones, Notebooks, Cars), Fuel Cells (Space Shuttle), Photovoltaic Cells (Solar Panels), Electrolysis, Rust
Topic: Metal dissolution
Replies: 10
Views: 495
### Re: Metal dissolution
I understand how to determine this based on Ecell values, but how would we determine this based off electrochemical properties??
Thu Feb 27, 2020 9:54 pm
Forum: Balancing Redox Reactions
Topic: Acidic/Basic Solutions
Replies: 3
Views: 60
### Re: Acidic/Basic Solutions
We should be given that information because that affects how we solve the problem.
Thu Feb 27, 2020 9:50 pm
Forum: Balancing Redox Reactions
Topic: When to add H+ or H20
Replies: 19
Views: 444
### Re: When to add H+ or H20
You add H2O to balance the oxygen, and H+ to balance the hydrogen in an acidic solution.
You add OH- to balance the oxygen, and H2O to balance the hydrogen in a basic solution.
Thu Feb 27, 2020 9:48 pm
Forum: Work, Gibbs Free Energy, Cell (Redox) Potentials
Topic: Where to find Ecell values
Replies: 15
Views: 274
### Re: Where to find Ecell values
Does anyone know when you would flip the sign in a half relation? I know that if you flip the equation the sign flips, but then do you not do this when calculating E(cell)? like if they give you the E value in the problem and the equation has to be flipped do you flip the sign? The Ecell values giv...
Sat Feb 22, 2020 11:01 pm
Forum: Balancing Redox Reactions
Topic: Basic conditions
Replies: 6
Views: 124
### Re: Basic conditions
In an acidic solution, you add H+ to balance the hydrogen in the equation. In a basic solution, however, you add OH- to balance the hydrogens (and then add H2O to balance that change in oxygen).
Sat Feb 22, 2020 11:00 pm
Forum: Appications of the Nernst Equation (e.g., Concentration Cells, Non-Standard Cell Potentials, Calculating Equilibrium Constants and pH)
Topic: electrochemical series
Replies: 4
Views: 122
### Re: electrochemical series
This is just a table, like what can be found in Appendix 2A for enthalpy, entropy, etc.
Sat Feb 22, 2020 10:59 pm
Forum: Student Social/Study Group
Topic: Curve?
Replies: 50
Views: 2138
### Re: Curve?
There is no specific guideline for this, Lavelle just "curves" according to the final, total class averages. Not much hope though, considering the average on the midterm was fairly high.
Thu Feb 20, 2020 6:06 pm
Forum: Galvanic/Voltaic Cells, Calculating Standard Cell Potentials, Cell Diagrams
Topic: Reaction A
Replies: 1
Views: 61
### Reaction A
What is reaction A? The textbook keeps referring to a 'reaction A' in order to find the amount of moles in a reaction. Can someone please explain and/or provide me with the page in the textbook that introduces this?
Wed Feb 19, 2020 7:53 pm
Forum: Galvanic/Voltaic Cells, Calculating Standard Cell Potentials, Cell Diagrams
Topic: Hydrogen Galvanic Cell
Replies: 1
Views: 89
### Hydrogen Galvanic Cell
The textbook states "Not all electrode reactions include a conducting solid as a reactant or product. For example, to use the reduction 2H+ + 2e- --> H2 at an electrode, a chemically inert metallic conductor, such as an unreactive metal or graphite, must carry the electrons into or out of the e...
Sun Feb 16, 2020 2:02 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: Hess Law
Replies: 6
Views: 131
### Re: Hess Law
In order to use Hess's Law, you should be given multiple reactions so that you can add/subtract them as needed. Maybe if you aren't given multiple equations, you could use the reaction enthalpies for the formation of all the products and reactants, and create the equations yourself (ie creating an e...
Sun Feb 16, 2020 1:58 pm
Forum: Balancing Redox Reactions
Topic: half reaction
Replies: 8
Views: 147
### Re: half reaction
A half reaction is when a redox reaction is broken up into the reduction or oxidation components.
Tue Feb 11, 2020 10:43 pm
Forum: Entropy Changes Due to Changes in Volume and Temperature
Topic: Change in entropy of an irreversible process.
Replies: 3
Views: 62
### Re: Change in entropy of an irreversible process.
Jack Riley 4f wrote:delta S of the surroundings does equal 0 because it is free expansion, but delta S total must still increase so that only leaves delta S of the system to contribute to this. Thus, delta S total = delta S system =/= 0
Why does deltaS still have to increase?
Tue Feb 11, 2020 9:10 pm
Forum: Heat Capacities, Calorimeters & Calorimetry Calculations
Topic: deltaU for an ideal gas
Replies: 2
Views: 61
### deltaU for an ideal gas
Can someone please explain how to derive/where these equations come from?
deltaU = 3/2nRdeltaT = 0
and
Utot = 3/2nRT
both for an ideal gas.
Tue Feb 11, 2020 12:45 pm
Forum: Concepts & Calculations Using Second Law of Thermodynamics
Topic: Spontaneity - Enthalpy
Replies: 1
Views: 52
### Spontaneity - Enthalpy
Is a spontaneous process accompanied by an increase in entropy of the surroundings, or of the system?
Tue Feb 11, 2020 12:42 pm
Forum: Entropy Changes Due to Changes in Volume and Temperature
Topic: Change in entropy of an irreversible process.
Replies: 3
Views: 62
### Change in entropy of an irreversible process.
Why is deltaS of an irreversible process not equal to zero? For example, 4I.9 states that for the irreversible process, deltaS is the same as deltaS for the reversible process. This does not make sense to me, because the process in this example is also free expansion. Therefore, w= 0, deltaU= 0, and...
Tue Feb 11, 2020 12:37 pm
Forum: Entropy Changes Due to Changes in Volume and Temperature
Topic: How to think of entropy?
Replies: 2
Views: 47
### How to think of entropy?
Prof. Lavelle has said multiple times not to think of entropy as disorder. If we don't think of entropy as disorder, how should we think of it?
Sun Feb 09, 2020 2:16 pm
Forum: Gibbs Free Energy Concepts and Calculations
Topic: Entropy units
Replies: 7
Views: 180
### Re: Entropy units
I'm sure either are fine, but if you have a large number of joules (1000+), it might be more useful to provide the answer in kilojoules.
Sun Feb 09, 2020 2:15 pm
Forum: Non-Equilibrium Conditions & The Reaction Quotient
Topic: solids and liquids in the rxn quotient
Replies: 8
Views: 169
### Re: solids and liquids in the rxn quotient
Yes, liquids and solids are excluded from calculating K or Q.
Sun Feb 09, 2020 2:11 pm
Forum: Entropy Changes Due to Changes in Volume and Temperature
Topic: Cm
Replies: 4
Views: 69
### Re: Cm
Cm is the molar specific heat of the compound.
Sun Feb 09, 2020 2:10 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: ST 4A.1A [ENDORSED]
Replies: 3
Views: 337
### Re: ST 4A.1A[ENDORSED]
Chem_Mod wrote:What answer did you get? It might be an error because other students are getting different numbers as well. I got -.94kJ, not -.86kJ.
I got -.95kJ as well.
Tue Feb 04, 2020 2:21 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: C = K?
Replies: 5
Views: 143
### C = K?
Example 4A.4 in the textbook gives a change in temperature in celsius, then states that it is the same amount in kelvin. How is this possible?
Tue Feb 04, 2020 2:13 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: ST 4A.1A [ENDORSED]
Replies: 3
Views: 337
### ST 4A.1A[ENDORSED]
I've done this problem so many times and I can't get the right answer. Can someone please write out the solution? Water expands when it freezes. How much work does 100. g of water do when it freezes at 0 deg. C and pushes back the metal wall of a pipe that exerts an opposing pressure of 1070 atm? Th...
Wed Jan 29, 2020 7:25 pm
Forum: Heat Capacities, Calorimeters & Calorimetry Calculations
Topic: q vs deltaH
Replies: 6
Views: 71
### Re: q vs deltaH
q represents the transfer of heat to or from the system. Delta H represents the sum of the internal energy in the system, taking into account the system's pressure and volume when necessary.
Wed Jan 29, 2020 7:24 pm
Forum: Heat Capacities, Calorimeters & Calorimetry Calculations
Topic: memorize
Replies: 6
Views: 83
### Re: memorize
We should be given a table with a number of different values, including the heat capacity of various elements/compounds.
Wed Jan 29, 2020 7:23 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: Enthalpies
Replies: 4
Views: 68
### Re: Enthalpies
The standard enthalpy of formation is the change in enthalpy that accompanies the formation of one mole of a compound from its elements. The standard enthalpy of reaction is the change in enthalpy that occurs in a system when one mole of matter is transformed by a chemical reaction.
Wed Jan 29, 2020 7:19 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: Delta U
Replies: 6
Views: 64
### Re: Delta U
Delta U is the internal energy of the system/reaction, or the total store of energy in a system. One way delta U can be calculated is by subtracting the internal energy of the initial system from the internal energy of the final system.
Wed Jan 29, 2020 7:17 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: Exothermic and Endothermic
Replies: 11
Views: 439
### Re: Exothermic and Endothermic
The enthalpy for an exothermic reaction is negative because the system/reaction loses energy, and the surroundings gain energy. Similarly, an endothermic reaction requires energy, so the enthalpy value is positive.
Sun Jan 26, 2020 9:02 pm
Forum: Phase Changes & Related Calculations
Topic: Enthalpy is said to be additive
Replies: 10
Views: 122
### Re: Enthalpy is said to be additive
Enthalpy is additive because it is a state function, meaning it can be added or subtracted.
Sun Jan 26, 2020 9:01 pm
Forum: Phase Changes & Related Calculations
Topic: Heat v Enthalpy
Replies: 3
Views: 60
### Re: Heat v Enthalpy
Enthalpy is a state function that keeps tract of the losses of energy during heat transfer. Heat itself, however, like work, is not a state function because it depends on the path used to go from the initial to the final state.
Sun Jan 26, 2020 8:58 pm
Forum: Phase Changes & Related Calculations
Topic: Phase Changes
Replies: 7
Views: 99
### Re: Phase Changes
You should calculate the enthalpy, then add the enthalpy of the phase change.
Sun Jan 26, 2020 8:45 pm
Forum: Phase Changes & Related Calculations
Topic: Why does steam cause burns?
Replies: 29
Views: 372
### Re: Why does steam cause burns?
The steam has much higher enthalpy than the liquid, because more heat is required to achieve a gaseous state. Therefore, when it comes into contact with your skin and quickly condenses, it releases a large amount of heat in comparison to that that would be released by the liquid state.
Sat Jan 25, 2020 6:47 pm
Forum: Phase Changes & Related Calculations
Replies: 5
Views: 83
What order did Prof. Lavelle recommend we read the textbook in, since he said he was going out of order for the Thermochemistry outline?
Sat Jan 18, 2020 9:18 pm
Forum: Applying Le Chatelier's Principle to Changes in Chemical & Physical Conditions
Topic: 5.39
Replies: 1
Views: 29
### 5.39
5.39 part b asks us to find the new equilibrium concentrations of two gases when the volume of the flask they are contained in is reduced by half. The original concentration of one of the gases was 0.020 M. The solutions manual immediately finds that the new concentration of this gas is 0.040 M. How...
Fri Jan 17, 2020 3:23 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Kc & Kp
Replies: 12
Views: 116
### Re: Kc & Kp
Can we just use K and not specify Kc or Kp? Or would we be marked down?
Fri Jan 17, 2020 3:20 pm
Forum: Ideal Gases
Topic: PV=nRT equation manipulation
Replies: 13
Views: 259
### Re: PV=nRT equation manipulation
Manipulating the PV=nRT is helpful because P gives the partial pressure of gases, which we can use when the problem provides us only with molar concentrations.
Fri Jan 17, 2020 3:17 pm
Forum: Non-Equilibrium Conditions & The Reaction Quotient
Topic: Partial Pressure
Replies: 19
Views: 151
### Re: Partial Pressure
How can we convert between atm and bars? The examples in the book show that the conversion is not 1:1.
Fri Jan 17, 2020 3:11 pm
Forum: Applying Le Chatelier's Principle to Changes in Chemical & Physical Conditions
Topic: Inert Gas does not change pressure?
Replies: 7
Views: 103
### Re: Inert Gas does not change pressure?
Do you have to change the volume in order to change the pressure? Or would adding a non-inert gas affect pressure as well?
Wed Jan 15, 2020 9:10 pm
Forum: Non-Equilibrium Conditions & The Reaction Quotient
Topic: Self Test 5I.3b
Replies: 4
Views: 47
### Self Test 5I.3b
I'm not sure how to start the calculations for Self Test 5I.3b. Can anyone explain how to begin? Hydrogen chloride gas is added to a reaction vessel containing solid iodine until its partial pressure reaches 0.012 bar. At the temperature of the experiment, K = 3.5 x 10^-32 for 2HCl(g) + I2(s) -> 2HI...
Fri Jan 10, 2020 11:17 am
Forum: Ideal Gases
Topic: Understanding Q
Replies: 13
Views: 134
### Re: Understanding Q
I believe we would omit pure solids and liquids because Q is calculated in the same way as K.
Fri Jan 10, 2020 11:16 am
Forum: Ideal Gases
Topic: Solving for K (coefficients)
Replies: 11
Views: 156
### Re: Solving for K (coefficients)
The reactants/products don't have to be in a specific order in the calculation of K itself, but the stoichiometric coefficients should be assigned to their respective reactant/product.
Fri Jan 10, 2020 11:15 am
Forum: Ideal Gases
Topic: What is the Importance of homogeneous vs heterogeneous equilibria [ENDORSED]
Replies: 12
Views: 221
### Re: What is the Importance of homogeneous vs heterogeneous equilibria[ENDORSED]
I believe it is important so that we use the correct state when calculated K (using brackets for aqueous reactants/products, P for gaseous reactants/products, and knowing not to include pure solids and liquids in the calculation).
Fri Jan 10, 2020 11:13 am
Forum: Ideal Gases
Topic: K
Replies: 10
Views: 135
### Re: K
Kc and Kp are the same, they are just used for reactions in different states.
Fri Jan 10, 2020 11:12 am
Forum: Ideal Gases
Topic: Difference between K and Q
Replies: 9
Views: 84
### Re: Difference between K and Q
Q can be taken at any point in a reaction while K is taken when the reaction has reached equilibrium. If K = Q, then the reaction was in equilibrium at the time Q was taken. K and Q can be compared to each other in order to determine if the forward or reverse reaction is favored.
Thu Dec 05, 2019 5:10 pm
Forum: Naming
Topic: Ammine vs. Amine
Replies: 1
Views: 47
### Ammine vs. Amine
When are we supposed to use ammine, and when should we use amine?
For example, ethylenediamine (en) uses amine and has NH2 groups, but the ligand HN3 is called ammine. What is the reasoning? What about NH4+?
Wed Dec 04, 2019 11:29 pm
Forum: Shape, Structure, Coordination Number, Ligands
Topic: Ligand Names
Replies: 2
Views: 57
### Ligand Names
Are the ligands on the test (that we are expected to know) limited to the ligand table Prof. Lavelle created (named Ligand Names in Coordination Compounds)? Or are we expected to be able to recall more?
Wed Dec 04, 2019 11:11 pm
Forum: Naming
Topic: Naming Compounds
Replies: 2
Views: 55
### Re: Naming Compounds
I was hoping these wouldn't be on the final... but I'd like to know the answer as well.
Wed Dec 04, 2019 11:10 pm
Forum: Quantum Numbers and The H-Atom
Topic: 1E.5 part d
Replies: 1
Views: 102
### Re: 1E.5 part d
The more electron shielding an electron has, the lower the Zeff it experiences. Because the 1s orbital has more density near the nucleus than the 1p orbital, it is said to shield the 1p orbital from the full effective charge of the nucleus, Zeff. Key misunderstanding here is that the s and p orbital...
Wed Dec 04, 2019 10:58 pm
Forum: Properties & Structures of Inorganic & Organic Acids
Topic: Stronger acid?
Replies: 13
Views: 631
### Re: Stronger acid?
HCLO2 would be the stronger acid, because the greater the # of oxygen atoms attached to the central atom, the stronger the acid.
Wed Dec 04, 2019 6:14 pm
Forum: Calculating pH or pOH for Strong & Weak Acids & Bases
Topic: K constant and pK value
Replies: 3
Views: 50
### K constant and pK value
Do we need to know how to calculate the Ka, acid dissociation constant, and Kb, the base dissociation constant? Or are these the same as the K constant itself?
Also, is Kw within our scope?
Sun Dec 01, 2019 5:34 pm
Forum: Lewis Acids & Bases
Topic: acid v. base?
Replies: 16
Views: 283
### Re: acid v. base?
The Lewis structure of a Lewis acid typically has a positive charge, making them capable and likely to accept an electron pair. Lewis bases typically have a lone pair available to donate.
Sun Dec 01, 2019 5:28 pm
Forum: Conjugate Acids & Bases
Topic: NH3, acid or base?
Replies: 12
Views: 4052
### Re: NH3, acid or base?
Ammonia is considered a Bronsted base because it loses its lone pair and accepts a proton to become NH4+.
Sun Dec 01, 2019 5:20 pm
Forum: Calculating pH or pOH for Strong & Weak Acids & Bases
Topic: pH scale
Replies: 12
Views: 190
### Re: pH scale
pH cannot be negative, which is why the formula for calculating pH is -log(H3O+).
Sun Dec 01, 2019 5:15 pm
Forum: Industrial Examples
Topic: Biological examples of co-ordination compounds
Replies: 3
Views: 175
### Re: Biological examples of co-ordination compounds
Prof. Lavelle did spend a significant amount of time on cisplatin, so I think we should understand that and the distinction between cis and trans.
Sun Dec 01, 2019 5:10 pm
Forum: Naming
Topic: How to Know the Charge of Ions
Replies: 7
Views: 152
### How to Know the Charge of Ions
How are we expected to know the charge of ions quickly (SO4 2-, CO3 2-)? Are we going to be given the ligand table shared by Prof. Lavelle on the final?
If not, what is the method for accurately determining the charge of an ion?
Sat Nov 23, 2019 9:50 pm
Forum: Sigma & Pi Bonds
Topic: How to Find Sigma Bonds and Pi Bonds
Replies: 15
Views: 530
### Re: How to Find Sigma Bonds and Pi Bonds
The Lewis structure shows you how many single/double/triple bonds there are.
Single bond = 1 sigma bond
Double bond = 1 sigma bond + 1 pi bond
Triple bond = 1 sigma bond + 2 pi bonds
Sat Nov 23, 2019 9:47 pm
Forum: Hybridization
Topic: Hybridization Notation
Replies: 5
Views: 95
### Re: Hybridization Notation
It should be written sp^3d.
Sat Nov 23, 2019 9:46 pm
Forum: Bronsted Acids & Bases
Topic: Bronsted Acids vs Lewis acids
Replies: 4
Views: 55
### Re: Bronsted Acids vs Lewis acids
I think they're two different acid-base theories; the difference is that a Lewis theory states that an acid is an electron acceptor and a base is an electron donor, and the Bronsted theory states that an acid is a proton donor and a base is a proton acceptor.
Sat Nov 23, 2019 9:37 pm
Forum: Shape, Structure, Coordination Number, Ligands
Topic: Coordinate covalent bonds
Replies: 2
Views: 41
### Re: Coordinate covalent bonds
I think the difference is that in a covalent bond, both atoms contribute electrons to the bond, but in a coordinate covalent bond, both electrons of the bond are donated from one of the atoms.
Sat Nov 23, 2019 9:36 pm
Forum: Naming
Topic: Naming order
Replies: 2
Views: 62
### Re: Naming order
The ligands should be ordered alphabetically
Sun Nov 17, 2019 6:04 pm
Forum: Octet Exceptions
Topic: Octet expansion
Replies: 6
Views: 259
### Re: Octet expansion
Elements in the p-block of the 3rd period or lower can have expanded octets. I don't think we've learned the maximum amount of electrons they can hold.
Sun Nov 17, 2019 6:02 pm
Forum: Determining Molecular Shape (VSEPR)
Topic: single, double, and triple bonds
Replies: 6
Views: 142
### Re: single, double, and triple bonds
Regardless of how many electrons are held within the bond, they all act as a single unit because they all lie in the same region.
Sun Nov 17, 2019 6:00 pm
Forum: SI Units, Unit Conversions
Topic: Memorizing Conversions
Replies: 25
Views: 645
### Re: Memorizing Conversions
The constants and equations sheet is on the class website, and that's the same one given for the midterm and final. You should check it out, and if there's anything that is not on there, you should memorize it.
Sun Nov 17, 2019 5:56 pm
Forum: Determining Molecular Shape (VSEPR)
Replies: 7
Views: 137
Lavelle uses the class average at the end of the quarter to assign final grades.. I'd like to know an approximate of what an A would be though, percentage wise.
Sun Nov 17, 2019 5:54 pm
Forum: Formal Charge and Oxidation Numbers
Topic: Calculating formal charge
Replies: 8
Views: 259
### Re: Calculating formal charge
You should just calculate the formal charges of all the atoms, and use all of them to determine whether or not you need to change the structure (bonds/lone pairs).
Sun Nov 10, 2019 1:59 pm
Forum: Polarisability of Anions, The Polarizing Power of Cations
Topic: Can polar molecules be more polar than another?
Replies: 5
Views: 82
### Re: Can polar molecules be more polar than another?
Yes, some polar molecules can be more polar than another. This is related to the molecule's differences in electronegativity as well as their symmetry.
Sun Nov 10, 2019 1:55 pm
Forum: Dipole Moments
Topic: Boiling Point
Replies: 11
Views: 200
### Re: Boiling Point
Boiling points relate to dipole-dipole interactions because as the strength of these interactions increases, more energy is required to separate the molecules, leading to a proportional increase in boiling point.
Sun Nov 10, 2019 1:53 pm
Forum: Electronegativity
Topic: Ionization energy of O vs N
Replies: 6
Views: 338
### Re: Ionization energy of O vs N
It's easier understand this if you look at it visually, by drawing out the p-orbitals for each atom. N has one electron in each p-orbital, and O has one electron in 2 of the orbitals, and two electrons in one of the orbitals. N's electron configuration is more stable because its orbitals are half-fi...
Sun Nov 10, 2019 1:45 pm
Forum: Octet Exceptions
Topic: General principles of octet exception
Replies: 7
Views: 101
### Re: General principles of octet exception
When a central atom has empty d-orbitals close in energy to valence orbitals, it can accommodate an expanded octet. Basically, atoms in the p-block of period 3+ can accommodate an expanded octet, but the size of the central atom also matters. It has to be physically capable of forming bonds to more ...
Sun Nov 10, 2019 1:43 pm
Forum: SI Units, Unit Conversions
Topic: Homework from Outline 3 due?
Replies: 5
Views: 239
### Re: Homework from Outline 3 due?
I'd think the homework should be from Outline 3, considering the midterm only covered through 2D, and we haven't even really covered much of 2F or 2E yet.
Sat Nov 02, 2019 10:51 pm
Forum: Ionic & Covalent Bonds
Topic: ionic vs. covalent
Replies: 7
Views: 105
### Re: ionic vs. covalent
When drawing our Lewis structures, is it necessary to indicate if bonds are covalent or ionic?
Sat Nov 02, 2019 10:45 pm
Forum: Resonance Structures
Topic: Visual structure of resonance structure
Replies: 3
Views: 67
### Re: Visual structure of resonance structure
The electrons are not divided! Resonance just means that there are multiple possibilities for how the electrons are distributed around the structure.
Sat Nov 02, 2019 10:43 pm
Forum: Lewis Structures
Topic: Lewis Structures
Replies: 9
Views: 148
### Re: Lewis Structures
I think we're supposed to draw the one that is the most stable, i.e. lowest formal charges for each atom (I think that's what you're referring to by 'lowest energy').
Sat Nov 02, 2019 10:42 pm
Forum: Ionic & Covalent Bonds
Topic: Midterm
Replies: 28
Views: 490
### Re: Midterm
The midterm will include outlines 1 and 2, and most of the curriculum from outline 3. It stops where the Friday lecture ended.
Sat Nov 02, 2019 10:41 pm
Forum: Wave Functions and s-, p-, d-, f- Orbitals
Topic: Filling of Orbitals
Replies: 2
Views: 52
### Filling of Orbitals
Why is the d-orbital filled before the s-orbital is filled for some atoms? For example, the electron configuration for Silver is [KR] 4d^10 5s^1. Why is the electron configuration not [KR] 4d^9 5s^2?
Sun Oct 27, 2019 7:47 pm
Forum: Wave Functions and s-, p-, d-, f- Orbitals
Topic: p-orbitals
Replies: 11
Views: 138
### Re: p-orbitals
You should use the simplified form unless specified otherwise (i.e. if the question wants to know which orbitals are occupied).
Sun Oct 27, 2019 7:31 pm
Forum: Trends in The Periodic Table
Topic: Cations
Replies: 8
Views: 151
### Re: Cations
Cations are smaller than their parents atoms because they have less electrons. Similarly, anions are larger than their parent atoms, because they have more electrons.
Sun Oct 27, 2019 7:28 pm
Forum: Formal Charge and Oxidation Numbers
Topic: Formal Charge and Lone Pairs
Replies: 4
Views: 111
### Re: Formal Charge and Lone Pairs
You count the number of bonded electrons, not the number of pairs.
Sun Oct 27, 2019 7:16 pm
Forum: General Science Questions
Topic: Study Guides
Replies: 6
Views: 117
### Re: Study Guides
If you purchased the textbook bundle from UCLA, there is a resource called Sapling Learning that is similar to Prof. Lavelle's focus topics, and I believe they have practice problems as well.
Sun Oct 27, 2019 7:13 pm
Forum: Properties of Electrons
Topic: Electron removal
Replies: 11
Views: 184
### Re: Electron removal
Ionization energy decreases as the size of the orbital increases. This is because as the orbital size increases, electrons become farther from the nucleus, and therefore farther from its positive charge, making them easier to remove than core electrons.
Sun Oct 20, 2019 4:00 pm
Forum: Properties of Electrons
Topic: Shared Electrons
Replies: 4
Views: 93
### Re: Shared Electrons
This has to do with Hund's rule. Basically, if more than 1 orbital in a sub shell is available, you should add electrons with parallel spins to different orbitals before filling up a singular orbital. This configuration has slightly lower energy than that of a paired arrangement because it maximizes...
Sun Oct 20, 2019 3:51 pm
Forum: Properties of Light
Topic: Equations and Constants
Replies: 6
Views: 109
### Re: Equations and Constants
Also, I don't believe the conversions are provided on the tests (all the 10^x formulas).
Sun Oct 20, 2019 3:41 pm
Forum: Properties of Light
Topic: Unit for Wavelength
Replies: 34
Views: 533
### Re: Unit for Wavelength
The unit for wavelength is meters... but, we aren't provided the conversions to nanometers, picometers, etc. on tests, so you should memorize the conversions for these.
Sun Oct 20, 2019 3:37 pm
Forum: Properties of Light
Topic: Modules
Replies: 5
Views: 108
### Re: Modules
Sapling Learning is a similar kind of resource. You should have access to it if you purchased the textbook bundle through UCLA, and there are some learning objective courses on there.
Sun Oct 20, 2019 3:31 pm
Forum: Properties of Light
Topic: Balmer and Lyman series.
Replies: 2
Views: 82
### Balmer and Lyman series.
Can someone please explain the significant of the Balmer and Lyman series'? I understand they are both sequences of lines corresponding to atomic transitions, but what is their use in this level of chemistry? Are the series going to be incorporated into questions, or are they provided to increase th...
Sat Oct 12, 2019 11:13 pm
Forum: DeBroglie Equation
Topic: Knowing Which Equation to Use
Replies: 4
Views: 134
### Re: Knowing Which Equation to Use
I usually begin by circling the quantities and units given in the problem, and what type of unit I need to find. Then, I just find which equation allows me work between what is given and the unknown.
Sat Oct 12, 2019 11:10 pm
Forum: DeBroglie Equation
Topic: De Broglie's Equation
Replies: 2
Views: 53
### Re: De Broglie's Equation
De Broglie's equation is reserved for objects that mass, like electrons, cars, or baseballs. Since light is, by definition, mass-less, De Broglie's equation does not apply. The wavelength of light can be determined given the speed and frequency of the light wave, as wavelength = c/v.
Sat Oct 12, 2019 11:05 pm
Forum: Properties of Electrons
Topic: Wavelength and Electrons
Replies: 4
Views: 76
### Re: Wavelength and Electrons
Wavelength depends on the speed that the object is moving at. The wavelength of any given electron is not constant.
Wavelength can be determined using the De Broglie equation: h (Planck's constant) / (mass)(velocity) of the given object.
Sat Oct 12, 2019 10:55 pm
Forum: Properties of Light
Topic: Equations We’ve Learned So Far
Replies: 11
Views: 315
### Re: Equations We’ve Learned So Far
Anyone have any tips for distinguishing between formulas designated for electrons vs those related to light? I'm having trouble keeping all their meanings and uses straight. | {"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": 0.8250133395195007, "perplexity": 5388.553616925005}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038464146.56/warc/CC-MAIN-20210418013444-20210418043444-00612.warc.gz"} |
https://eulumdat.blogspot.com/2015/05/improved-checks-and-privacy-in.html | ## Wednesday, May 27, 2015
### IES syntax coloring, keyword content assist in EulumdatTools 0.14.6.v20160115
A new version of EulumdatTools 0.14.6.v20160115 was published on January 15 at http://www.fold1.com/eulumdattools/.
#### Improvements 0.14.6
• Improved IES syntax coloring, now also indicating periods as field separators
• Added a QuickFix for missing item code in standard version and duplicate item code in sandbox version for EULUMDAT files. You can fill in product code from filename or product name.
#### Improvements 0.14.5
• Code quality improvements
#### Improvements 0.14.4
• IES file keywords now have content assist, activated on CTRL+SPACE on Windows/Linux and CMD+SPACE on Mac
• IES files now have syntax coloring, with illegal characters immediately showing up in RED!
• Fixed issue with hard coded expiry date
#### Improvements 0.14.3
• All versions of EulumdatTools are now based on Eclipse 4.5.1 Mars. To upgrade reinstall from the download site listed below. The Mac version now looks like a regular Mac application.
• Make it possible to retain Testlab and Testdate/Issuedate on conversion of IES to EULUMDAT and vice versa. Based on a suggestion/request by Mikulas Parma.
• Allow setting of Test Lab name when there is a full license, to be used in conversions.
• Removed some annoying behavior where latest blog post opened in external browser
#### Improvements 0.14.2
• Fixed nasty conversion bug where IES to EULUMDAT conversion for IES files with absolute photometry had wrong DFF factors calculated. These can be caught and fixed by the Eulumdat validator, but now they are calculated correctly in the first place. If you have converted IES files to EULUMDAT with absolute photometry in the past and not validated the output, please reconvert your files!
• NOTE: the same bug affected the online IES to EULUMDAT converter at http://ies2eulumdat.appspot.com, so if you have used that, reconvert also!
#### Improvements 0.14.1
• A regression causing the IES validator to stop working was fixed (IES Validator Feature)
#### Improvements 0.14.0
• Checking your license will now use the https: protocol instead of http: increasing your privacy
• A problem where the cursor moved to start of line in the source editors when installed in Eclipse e4 was fixed
• You can read this blog right in EulumdatTools so you will always be up to date on latest developments
• EulumdatTools now contains a structured compare editor to make comparing 2 EULUMDAT files easier (see image) (Sandbox Feature)
• A check for overlapping C-planes, where there are C-planes for both 0° and 360° was added (Sandbox Feature)
#### Availability
Registered users should update using Help / Check for Updates... and install the needed features following the steps below, new users should install EulumdatTools first. New users can download an apply for 30 day free evaluation license at the installation site
#### Installation of EulumdatTools
For use in Eclipse you can use the standard Eclipse update site For stand alone RCP versions of EulumdatTools choose your platform and perform update after installation.:
#### Installation of Sandbox Feature
Select menu Help / Install New Software... and choose the Eulumdat Tools Sandbox / Sandbox Feature, press Next and follow through with the installation. Also see here | {"extraction_info": {"found_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": 0.3099767565727234, "perplexity": 13328.205312765616}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1568514570740.10/warc/CC-MAIN-20190915052433-20190915074433-00078.warc.gz"} |
http://www.exambeat.in/aptitude/races-and-games/1710/ | # Aptitude:: Races and Games
@ : Home > Aptitude > Races and Games > General Questions
### Exercise
"
#### In a 100 m race, A can beat B by 25 m and B can beat C by 4 m. In the same race, A can beat C by:
A. 21 m B. 26 m C. 28 m D. 29 m
#### A runs $\dpi{150}&space;1\tfrac{2}{3}$ times as fast as B. If A gives B a start of 80 m, how far must the winning post be so that A and B might reach it at the same time?
A. 200 m B. 300 m C. 270 m D. 160 m
#### In a 300 m race A beats B by 22.5 m or 6 seconds. B's time over the course is:
A. 86 sec B. 80 sec C. 76 sec D. None of these
#### A can run 22.5 m while B runs 25 m. In a kilometre race B beats A by:
A. 100 m B. C. 25 m D. 50 m
#### In a 200 metres race A beats B by 35 m or 7 seconds. A's time over the course is:
A. 40 sec B. 47 sec C. 33 sec D. None of these
Page 1 of 3
Submit* | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5496819019317627, "perplexity": 2582.129289879154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281574.78/warc/CC-MAIN-20170116095121-00103-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://blogs.mathworks.com/steve/2006/12/13/poly2mask-and-roipoly-part-2/ | # POLY2MASK and ROIPOLY – Part 2
I'm familiar with two basic ways to determine whether a point is inside a polygon. The first is to compute the winding number, which is the number of times that a closed curve encircles the point. This technique is used by the MATLAB function inpolygon.
When computing point-inside-polygon for all the points on a pixel grid, the crossing test is more commonly used. You follow a ray from one side of the image to the other, counting the number of times the ray crosses an edge of the polygon.
x1 = [30 80 60 30];
y1 = [30 50 90 30];
ray_x = [20 90];
ray_y = [70 70];
points_x = [30 60 80];
points_y = [70 70 70];
plot(x1,y1)
hold on
plot(ray_x,ray_y, 'r')
plot(points_x,points_y,'Marker','o','MarkerSize',4,'Color','k',...
'MarkerFaceColor','k','LineStyle','none')
text(30,70,'A','VerticalAlignment','bottom')
text(60,70,'B','VerticalAlignment','bottom')
text(80,70,'C','VerticalAlignment','bottom')
axis off
hold off
title('Polygon 1')
At point A, the ray has not crossed a polygon edge; A is outside the polygon. At point B, the ray has crossed a polygon edge one time; B is inside. At point C, the ray has crossed a polygon edge twice; C is outside.
Polygons can get much more complex than a triangle, of course. Here are a couple of convex, nonsimple polygons.
subplot(2,1,1)
plot([30 70 70 30 30], [30 70 30 70 30])
hold on
plot([20 80],[60 60],'r')
hold off
axis off
title('Polygon 2')
subplot(2,1,2)
plot([30 70 70 40 40 60 60 30 30], ...
[30 30 55 55 35 35 60 60 30])
hold on
plot([20 80],[45 45],'r')
plot(50,45,'Marker','o','MarkerSize',4,'MarkerFaceColor','k')
text(50,45,'D','VerticalAlignment','bottom')
ylim([25 75])
hold off
axis off
title('Polygon 3')
In polygon 2, the ray crosses into and then out of the polygon twice. Polygon 3 is even more interesting. Is point D inside or outside the polygon? The answer is a matter of convention. A commonly-used convention is the even-odd rule. A point is outside the polygon if there have been an even number of edge crossings. A point is inside there have been an odd number. By this rule, point D is outside the polygon.
The crossing test and the even-odd rule are easy to understand. The challenge is to avoid getting tripped up in the special cases. The troublesome cases include rays intersecting vertices, edge intersections, and coincident edges. Another case is when a ray is coincident with an edge. Here are some diagrams to illustrate. (Note that in polygon 7 there are two coincident edges on the left side of the polygon.)
subplot(2,2,1)
plot([20 30 40 30 20], [30 20 30 40 30]);
hold on
plot([10 50], [30 30], 'r')
hold off
axis off
title('Polygon 4'), ylim([10 50])
subplot(2,2,2)
plot([20 30 40 50 60 20], [30 10 20 10 30 30]);
hold on
plot([10 70], [10 10], 'r')
hold off
axis off
title('Polygon 5'), ylim([0 40])
subplot(2,2,3)
plot([20 50 20 50 20], [20 20 60 60 20]);
hold on
plot([10 60], [40 40], 'r')
hold off
axis off
title('Polygon 6'), ylim([10 70])
subplot(2,2,4)
plot([20 60 60 20 20 50 20 20], ...
[20 20 50 50 20 35 50 20]);
hold on
plot([10 70],[35 35],'r')
hold off
axis off
title('Polygon 7'), ylim([10 60])
In my next post on this topic, I'll get into the specific algorithm used by the Image Processing Toolbox functions polymask and roipoly.
Here's Part 1 of this topic.
Published with MATLAB® 7.3
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https://www.biostars.org/p/329832/ | 1
0
Entering edit mode
4.4 years ago
jonessara770 ▴ 240
Hi,
I am a newbie in Retrotransposable/repeat elements analysis.
I used UCSC table in order to get the sequence of RepeatMasker table in fasta format. When I download it through the webpage, the file is truncated at chr16 and does not have all chromosome.
Thanks Sara
repeat TE Retrotransposable/repeat • 2.8k views
0
Entering edit mode
fastq file of RepeatMasker UCSC table
Is that correct? I don't recollect there being an option to download fastq format files from table browser.
0
Entering edit mode
Tagging: genecats.ucsc
See the error below.
1
Entering edit mode
4.4 years ago
mbk0asis ▴ 660
I don't think you can download repetitive sequences directly from UCSC genome browser as genomax mentioned. Instead, get the bed file of RepeatMasker and whole genome sequence of your organism from UCSC genome browser, and use 'bedtools getfasta' to extract the sequences of retroelements.
0
Entering edit mode
You should be able to get the sequence using table browser (which is what OP was doing). It appears that their connection timed out and the entire file did not download.
0
Entering edit mode
Thanks! I tied this but the file is truncated.
0
Entering edit mode
Did you provide a file name to save the data to a file (rather than have to show up in the browser window)? If not try that.
0
Entering edit mode
Yes, I did and I save the file but it is truncated from chr18 upwards.
0
Entering edit mode
You are right in that the download from table browser is truncated after a certain length (I got a 1.6G file). You may want to get individual files for the remaining chromosomes and append them to the main.
It may be worth reporting this as a bug to UCSC support people. I will tag them here as well. | {"extraction_info": {"found_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": 0.32405585050582886, "perplexity": 5435.076923285139}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446711232.54/warc/CC-MAIN-20221208014204-20221208044204-00586.warc.gz"} |
https://cloud.originlab.com/doc/Tutorials/Fitting-NLFit-MultiVar | # 4.2.2.30 Nonlinear Multiple Variables Fitting
## Summary
Origin supports fitting functions with multiple dependent or independent variables. With the nonlinear fitting function, you can define multiple variables and separate them with semicolons. Since global fitting allows you to fit only one function at a time, this is a good way to defeat that limitation.
Origin ships with three built-in functions with multiple dependent and independent variables. These functions, available in the Multiple Variables category, are actually composites consisting of two ordinary functions. The GaussianLorentz function, for example, is a combination of the Gaussian and Lorentz functions, sharing y0 and xc:
$y_1 =y_0+\frac {A_1}{w_1\sqrt{\pi /2}}e^{-2\frac{(x-x_c)^2}{w_1^2}}$
$y_2=y_0+\frac{2A_2}\pi \frac {w_2} {4\left( x-x_c\right) ^2+w_2^2}$
This tutorial will demonstrate how to fit such multi-variable functions.
## What you will learn
• Use Nonlinear Multiple Variables Fitting to fit a curve with two different functions.
• Assign data to fitting variables.
## Steps
1. Start with a new project or create a new workbook and import the data file \samples\curve fitting\Gaussian.dat.
2. Highlight Column(A) and Column(B). In the main menu, click Analysis, then point to Fitting, and then click Nonlinear Curve Fit.
3. In the NLFit dialog’s left panel, select Function Selection. In the right panel, select Multiple Variables in the Category dropdown menu. In the Function dropdown menu, select GaussianLorentz.
As you can see on the Sample Curve tab, the equations in this fitting function share the same parameters, y0 and xc.
4. In the NLFit dialog’s left panel, select Data Selection. In the right panel, expand the Range node and assign data to the fitting variables. In this example, we have assigned column B to both y1 and y2, which means that both expressions will fit the same dataset.
5. Click Fit until converged to fit, then OK. In the results sheet, compare parameters A and w, with the Gaussian and Lorentz functions sharing the same offset and peak center. | {"extraction_info": {"found_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": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3278099298477173, "perplexity": 2424.583230392903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487623596.16/warc/CC-MAIN-20210616093937-20210616123937-00389.warc.gz"} |
https://andjournal.sgu.ru/en/rubrika/determinirovannyy-haos?page=4 | ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)
# Детерминированный хаос
## Experimental realization of lorenz model of liquid’s convective instability in vertical toroidal loop
Stable and unstable regimes of glycerine convection in vertical toroidal loop are investigated experimentally. The results of Fourier-analysis, DFA, wavelet-, and correlation analysis of liquid’s motion peculiarities are presented. Chaotic attractor with Lorenz-attractor signs is constructed.
## Verification of hyperbolicity conditions for a chaotic attractor in a system of coupled nonautonomous van der pol oscillators
We present a method and results of numerical computations on veri?cation of hyperbolic nature for the chaotic attractor in a system of two coupled nonautonomous van der Pol oscillators (Kuznetsov, Phys. Rev. Lett., 95, 2005, 144101). At selected parameter values, we indicate a toroidal domain in four-dimensional phase space of Poincar? e map (topologically, a direct product of a circle and a three-dimensional ball), which is mapped into itself and contains the attractor we analyze.
## Statistical properties of the intermittent transition to chaos in the quasi-periodically forced system
By the example of the quasi-periodically forced logistic map we investigate statistical properties of the transition from strange nonchaotic attractor to chaos in the system with intermittent dynamics. The probability characteristics of laminar and chaotic phase distributions, as well as scaling laws for distributions of local Lyapunov exponents are studied at parameter values near the transition point.
## Influence of noise on chaotic self-sustained oscillations in the regime of spiral attractor
In the present paper we analyze the in?uence of white and colored noise on chaotic selfsustained oscillations in the regime of spiral attractor. We study characteristics of instantaneous phase and spectra of noisy chaotic oscillations. The phenomenon of chaos synchronization by external narrow-band noise has been estimated. Synchronization phenomena under the in?uence of narrow-band noise signals with equal spectra and di?erent probability densities are compared.
## Critical behavior of asymmetrically coupled noisy driven nonidentical systems with period-doublings
We investigated the in?uence of external noise on the critical behavior typical to nonidentical coupled systems with period-doubling. We obtained the numerical value of the scaling factor for noise amplitude by means of the renormalization group analysis. Also we demonstrated the selfsimilar structure of the parameter plane near the critical point in the model system of two noisy driven coupled logistic maps.
## Peculiarities of complex dynamics and transitions to chaotic regimes in the model of two interacting systems with phase control
The work is devoted to investigation of complex dynamics in the model of two interacting systems with phase and delay control. Stability conditions of synchronous regime are determined. The processes of excitement of nonsynchronous regimes and transitions between them are considered. Scenarios of development of nonsynchronous regimes under variation of the systems parameters are determined. Routes to chaotic behavior of the model are discussed. Results are presented in the form of one-parameter bifurcation diagrams and phase portraits of the model attractors.
## About scaling properties of identical coupled logistic maps with two types of coupling without noise and under influence of external noise
In this paper the in?uence of noise in system of identical coupled logistic maps with two types of coupling – dissipative and inertial – is discussed. The corresponding renormalization group analysis is presented. Scaling property in the presence of noise is considered, and necessary illustrations in «numerical experiment style» are given.
## Analytical solution of spectral problem for the perron – frobenius operator of piece-wise linear chaotic maps
Spectral properties of the linear non-self-adjoint Perron – Frobenius operator of piece-wise linear chaotic maps having regular structure are investigated. Eigenfunctions of the operator are found in the form of Bernoulli and Euler polynomials. Corresponding eigenvalues are presented by negative powers of number of map brunches. The solution is obtained in general form by means of generating functions for eigenfunctions of the operator. Expressions for eigenfunctions and eigenvalues are di?erent for original and inverse maps having even and odd number of branches.
## Synchronization in systems with bimodal dynamics
Considering model with bimodal dynamics we investigate the synchronization of di?erent time scales. Transition between mode-locked and mode-unlocked chaotic attractors is investigated. It is shown that this transition involves a situation in which the synchronized chaotic attractor loses its band structure.
## Computation of Lyapunov exponents for spatially extended systems: advantages and limitations of various numerical methods
Problems emerging in computations of Lyapunov exponents for spatially extended systems are considered. We concentrate on the incorrect orthogonalization of high sized ill conditioned matrices appearing in course of the computation, and on large errors emerging under certain conditions if the finite difference numerical method is applied to solve equations. The practical guidelines helping to avoid the mentioned problems are represented. | {"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": 0.8818835020065308, "perplexity": 1714.8783367307035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046154459.22/warc/CC-MAIN-20210803124251-20210803154251-00322.warc.gz"} |
https://www.maplesoft.com/support/help/maple/view.aspx?path=Units/volume_flow&L=E | volume flow - Maple Help
Units of Volume Flow
Description
• Volume flow has the dimension length cubed over time. The SI composite unit of volume flow is the cubic meter per second.
• Maple knows the units of volume flow listed in the following table.
Name Symbols Context Alternate Spellings Prefixes gallon_per_minute gpm US_liquid * gallons_per_minute UK US_dry cubic_foot_per_minute cfm standard * cubic_feet_per_minute
An asterisk ( * ) indicates the default context, an at sign (@) indicates an abbreviation, and under the prefixes column, SI indicates that the unit takes all SI prefixes, IEC indicates that the unit takes IEC prefixes, and SI+ and SI- indicate that the unit takes only positive and negative SI prefixes, respectively. Refer to a unit in the Units package by indexing the name or symbol with the context, for example, gallon_per_minute[US_liquid] or cfm[standard]; or, if the context is indicated as the default, by using only the unit name or symbol, for example, gallon_per_minute or cfm.
The units of volume flow are defined as follows.
A US liquid, US dry, or UK gallon per minute is defined as a gallon of the same context per minute.
A standard cubic foot per minute is defined as a cubic standard foot per minute.
Examples
> $\mathrm{convert}\left('\mathrm{gpm}','\mathrm{dimensions}','\mathrm{base}'=\mathrm{true}\right)$
$\frac{{{\mathrm{length}}}^{{3}}}{{\mathrm{time}}}$ (1)
> $\mathrm{convert}\left(1,'\mathrm{units}','\mathrm{gpm}',\frac{'\mathrm{gal}'}{'\mathrm{min}'}\right)$
${1}$ (2)
> $\mathrm{convert}\left(1,'\mathrm{units}','\mathrm{gpm}\left[\mathrm{UK}\right]',\frac{'\mathrm{gal}\left[\mathrm{UK}\right]'}{'\mathrm{min}'}\right)$
${1}$ (3)
> $\mathrm{convert}\left(1,'\mathrm{units}','\mathrm{gpm}\left[\mathrm{US_dry}\right]',\frac{'\mathrm{gal}\left[\mathrm{US_dry}\right]'}{'\mathrm{min}'}\right)$
${1}$ (4)
> $\mathrm{convert}\left(1,'\mathrm{units}','\mathrm{cfm}',\frac{{'\mathrm{ft}'}^{3}}{'\mathrm{min}'}\right)$
${1}$ (5) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 10, "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": 0.930768609046936, "perplexity": 2979.0493041167238}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1637964363125.46/warc/CC-MAIN-20211204215252-20211205005252-00361.warc.gz"} |
http://nag.com/numeric/fl/nagdoc_fl24/html/D03/d03pyf.html | D03 Chapter Contents
D03 Chapter Introduction
NAG Library Manual
# NAG Library Routine DocumentD03PYF
Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
## 1 Purpose
D03PYF may be used in conjunction with either D03PDF/D03PDA or D03PJF/D03PJA. It computes the solution and its first derivative at user-specified points in the spatial coordinate.
## 2 Specification
SUBROUTINE D03PYF ( NPDE, U, NBKPTS, XBKPTS, NPOLY, NPTS, XP, INTPTS, ITYPE, UP, RSAVE, LRSAVE, IFAIL)
INTEGER NPDE, NBKPTS, NPOLY, NPTS, INTPTS, ITYPE, LRSAVE, IFAIL REAL (KIND=nag_wp) U(NPDE,NPTS), XBKPTS(NBKPTS), XP(INTPTS), UP(NPDE,INTPTS,ITYPE), RSAVE(LRSAVE)
## 3 Description
D03PYF is an interpolation routine for evaluating the solution of a system of partial differential equations (PDEs), or the PDE components of a system of PDEs with coupled ordinary differential equations (ODEs), at a set of user-specified points. The solution of a system of equations can be computed using D03PDF/D03PDA or D03PJF/D03PJA on a set of mesh points; D03PYF can then be employed to compute the solution at a set of points other than those originally used in D03PDF/D03PDA or D03PJF/D03PJA. It can also evaluate the first derivative of the solution. Polynomial interpolation is used between each of the break points ${\mathbf{XBKPTS}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NBKPTS}}$. When the derivative is needed (${\mathbf{ITYPE}}=2$), the array ${\mathbf{XP}}\left({\mathbf{INTPTS}}\right)$ must not contain any of the break points, as the method, and consequently the interpolation scheme, assumes that only the solution is continuous at these points.
None.
## 5 Parameters
Note: the parameters U, NPTS, NPDE, XBKPTS, NBKPTS, RSAVE and LRSAVE must be supplied unchanged from either D03PDF/D03PDA or D03PJF/D03PJA.
1: NPDE – INTEGERInput
On entry: the number of PDEs.
Constraint: ${\mathbf{NPDE}}\ge 1$.
2: U(NPDE,NPTS) – REAL (KIND=nag_wp) arrayInput
On entry: the PDE part of the original solution returned in the parameter U by the routine D03PDF/D03PDA or D03PJF/D03PJA.
3: NBKPTS – INTEGERInput
On entry: the number of break points.
Constraint: ${\mathbf{NBKPTS}}\ge 2$.
4: XBKPTS(NBKPTS) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{XBKPTS}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NBKPTS}}$, must contain the break points as used by D03PDF/D03PDA or D03PJF/D03PJA.
Constraint: ${\mathbf{XBKPTS}}\left(1\right)<{\mathbf{XBKPTS}}\left(2\right)<\cdots <{\mathbf{XBKPTS}}\left({\mathbf{NBKPTS}}\right)$.
5: NPOLY – INTEGERInput
On entry: the degree of the Chebyshev polynomial used for approximation as used by D03PDF/D03PDA or D03PJF/D03PJA.
Constraint: $1\le {\mathbf{NPOLY}}\le 49$.
6: NPTS – INTEGERInput
On entry: the number of mesh points as used by D03PDF/D03PDA or D03PJF/D03PJA.
Constraint: ${\mathbf{NPTS}}=\left({\mathbf{NBKPTS}}-1\right)×{\mathbf{NPOLY}}+1$.
7: XP(INTPTS) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{XP}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{INTPTS}}$, must contain the spatial interpolation points.
Constraints:
• ${\mathbf{XBKPTS}}\left(1\right)\le {\mathbf{XP}}\left(1\right)<{\mathbf{XP}}\left(2\right)<\cdots <{\mathbf{XP}}\left({\mathbf{INTPTS}}\right)\le {\mathbf{XBKPTS}}\left({\mathbf{NBKPTS}}\right)$;
• if ${\mathbf{ITYPE}}=2$, ${\mathbf{XP}}\left(\mathit{i}\right)\ne {\mathbf{XBKPTS}}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{INTPTS}}$ and $\mathit{j}=2,3,\dots ,{\mathbf{NBKPTS}}-1$.
8: INTPTS – INTEGERInput
On entry: the number of interpolation points.
Constraint: ${\mathbf{INTPTS}}\ge 1$.
9: ITYPE – INTEGERInput
On entry: specifies the interpolation to be performed.
${\mathbf{ITYPE}}=1$
The solution at the interpolation points are computed.
${\mathbf{ITYPE}}=2$
Both the solution and the first derivative at the interpolation points are computed.
Constraint: ${\mathbf{ITYPE}}=1$ or $2$.
10: UP(NPDE,INTPTS,ITYPE) – REAL (KIND=nag_wp) arrayOutput
On exit: if ${\mathbf{ITYPE}}=1$, ${\mathbf{UP}}\left(\mathit{i},\mathit{j},1\right)$, contains the value of the solution ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{\mathrm{out}}\right)$, at the interpolation points ${x}_{\mathit{j}}={\mathbf{XP}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{INTPTS}}$ and $\mathit{i}=1,2,\dots ,{\mathbf{NPDE}}$.
If ${\mathbf{ITYPE}}=2$, ${\mathbf{UP}}\left(\mathit{i},\mathit{j},1\right)$ contains ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{\mathrm{out}}\right)$ and ${\mathbf{UP}}\left(\mathit{i},\mathit{j},2\right)$ contains $\frac{\partial {U}_{\mathit{i}}}{\partial x}$ at these points.
11: RSAVE(LRSAVE) – REAL (KIND=nag_wp) arrayCommunication Array
The array RSAVE contains information required by D03PYF as returned by D03PDF/D03PDA or D03PJF/D03PJA. The contents of RSAVE must not be changed from the call to D03PDF/D03PDA or D03PJF/D03PJA. Some elements of this array are overwritten on exit.
12: LRSAVE – INTEGERInput
On entry: the size of the workspace RSAVE, as in D03PDF/D03PDA or D03PJF/D03PJA.
13: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
## 6 Error Indicators and Warnings
If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{ITYPE}}\ne 1$ or $2$, or ${\mathbf{NPOLY}}<1$, or ${\mathbf{NPDE}}<1$, or ${\mathbf{NBKPTS}}<2$, or ${\mathbf{INTPTS}}<1$, or ${\mathbf{NPTS}}\ne \left({\mathbf{NBKPTS}}-1\right)×{\mathbf{NPOLY}}+1$, or ${\mathbf{XBKPTS}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{NBKPTS}}$, are not ordered.
${\mathbf{IFAIL}}=2$
On entry, the interpolation points ${\mathbf{XP}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{INTPTS}}$, are not in strictly increasing order, or when ${\mathbf{ITYPE}}=2$, at least one of the interpolation points stored in XP is equal to one of the break points stored in XBKPTS.
${\mathbf{IFAIL}}=3$
You are attempting extrapolation, that is, one of the interpolation points ${\mathbf{XP}}\left(i\right)$, for some $i$, lies outside the interval [${\mathbf{XBKPTS}}\left(1\right),{\mathbf{XBKPTS}}\left({\mathbf{NBKPTS}}\right)$]. Extrapolation is not permitted.
## 7 Accuracy
See the documents for D03PDF/D03PDA or D03PJF/D03PJA. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 61, "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": 0.9940730929374695, "perplexity": 2969.950064281214}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931009004.88/warc/CC-MAIN-20141125155649-00133-ip-10-235-23-156.ec2.internal.warc.gz"} |
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