text:
#### Category : php

so this is my route Route::group([‘prefix’ => ‘{locale}’],function(){ Route::get(‘/’,’[email protected]′)->name(‘post’)->middleware(‘setLocale’);}); for example, if I chose es in lang or any other language I want to go to Route::group([‘prefix’ => ‘{locale}’],function(){ Route::get(‘/’,’[email protected]′)->name(‘post’)->middleware(‘setLocale’);}); **so I can fetch data from another JSON file with another language ** Sourc..

I’ve been building and running a web app on my localhost. Everything was working fine after I carried out all the test to make sure everything was working fine. I finally decided to deploy the web app on heroku, where I wanted to add extra content on the landing page through by using the admin ..

What I have: I have table with a unique date column. id date 1 2021-09-01 2 2021-08-01 3 2021-07-01 4 2020-04-01 5 2020-03-01 6 2020-12-01 I pull data based on a month and year from a GET parameter. DB::table(‘budgets’)->whereMonth(‘created_at’, $request->get(‘month’))->whereYear(‘created_at’,$request->get(‘year’))->get()->first(); I do a check to see if a previous month is available in table ..

am using Laravel for some sass project where some deals are one step behind completion if just our rooms accepts ( alphabetic ) beside the actual numbers for example 101 – A101 ect. So room_number column data type was int so i converted it to a varchar no problem until now and everything is working ..

I am working with api with laravel/php and I need the data in following format. It would be great if you can help me or give some hint. if I do dd() $formData which is submitted from the form I get following array dd($formData); dd($formData) gives below result array:3 [▼ "fname" => "john" "lname" => .. Read more I never face this issue with creating laravel new project but today I face it unexpectedly for the first time ! I am using laravel 8. When I run laravel new project It shows me error saying Error Class ‘SymfonyComponentTranslationLoaderArrayLoader’ not found Here is full error message on pastebin https://pastebin.com/YqheTXe1 Sourc.. Read more I’m using Laravel and I’m facing a challenge trying to open excel sheets from my application. Is there a way in which i can open uploaded excel sheets with google sheets in the browser without having to download. Any assistance? Sourc.. Read more I have just installed composer to start learning Laravel but I have already encountered an annoying problem. When I run laravel new example-project on zsh, it seems that it has a big problem with Symfony: It is a lot of lines but it seems it is all related to Class SymfonyComponentTranslation* and the psr-4 autoloading .. Read more Got this error. I am not sure why. I guess some files got deleted maybe? I tried to run a composer update to fix it. Did not work. Then I deleted composer.lock and the vendor folder and ran composer install. Still the same. What would be the next step to try? Error Class "SymfonyComponentTranslationLoaderArrayLoader" not .. Read more I have a project on a Cpanel host and when I edit and upload some of my files it will delete all spaces in the file and the program will be got errors. for instance, it converts this code : @section(‘content’)

 to this code : @section(‘content’)

 Source: .. Read more I want to redirect to my new post when I created a new post in Laravel But I get a ArgumentCountError Too few arguments to function AppHttpControllersArticlesController::store(), 1 passed in C:xampphtdocsforumvendorlaravelframeworksrcIlluminateRoutingController.php on line 54 and exactly 2 expected How can I fix it? Thanks web.php <?php Route::resource(‘articles’, ArticlesController::class); Route::get(‘/’, [ArticlesController::class, ‘index’])->name(‘root’); Route::resource(‘articles.comments’, CommentsController::class); ArticlesController.php public .. Read more I have a table of data with such data [{"datetime":"2021-09-28 03:00:00","hour":"03:00:00","value":0.5}, {"datetime":"2021-09-28 03:00:00","hour":"03:00:00","value":0.6},….] How do I group the data by "hour", and find the average of each hour? I’m looking for output: [{"hour":"01:00:00","avg":"0.55"},….] Sourc.. Read more I have an array with the ID of the services, and there is an array with the Product. They are related to each other as Many To Many. For expamle: Product [1, 2, 3] Service [2, 3, 6] Table product_id service_id 1 2 2 3 3 8 How can I check whether a product belongs .. Read more How to get heading from different table on export? class Kriteria_Export implements FromCollection, WithMapping, WithHeadings { public function collection() { return Alternatif::all(); } public function map($alternatif) :array { return [ $alternatif->nama_alt, ]; } public function headings(): array { return [ ‘Nama Alternatif’, ]; } } Tabel Alternatif Tabel Kriteria When Export I want like this .. Read more I just want to make funnel chart using php side by side like this for free but it look weird Look at here Sourc.. Read more I have two tables properties (id, name, address, created_at, updated_at) bookings (id, property_id, booking_date, total_amount, comission, created_at, updated_at) Property.php model has this relation public function bookings() { return$this->hasMany(‘AppModelsAirbnbBooking’,’property_id’,’id’); } I want to return an JSON response (or eloquent collection) like below [{ "property_id": 1, "address": "Property ABC", "bookings": { "2021": { "Jul": [ { ..

I’m using Laravel 5.8 and sometimes I get maximum execution time of 30 seconds exceeded laravel server error. Now in order to fix this error, I should place this set_time_limit(0); in the Controller method. But I need to add for whole website. If it was on localhost, I could simply go to PHP.ini file and ..

greetings fellow programmers, I’m making images from text, and I was ordered to make it support ligatures. is it possible to do so? I’m using imagettftext & imagettfbbox from GD library Sourc..

I’m using SPATIE laravel-activitylog I followed all the instructions but still it only log the Create function not update while using it on a Modal My Modal <?php namespace AppModels; use IlluminateDatabaseEloquentFactoriesHasFactory; use IlluminateDatabaseEloquentModel; use SpatieActivitylogTraitsLogsActivity; class Contact extends Model { use HasFactory, LogsActivity; protected $fillable = [ ‘comp_name’, ‘cont_name’, ‘cont_email’, ‘cont_number’, ‘created_by’, ‘updated_by’, ]; .. Read more I have problem with my php on mac, previously I had 2 php versions php7.3 and php7.2. for some reason I uninstalled them then reinstall php7.2. when I run brew link [email protected] and php -v I’m getting message dyld: Library not loaded: /usr/local/opt/tidy-html5/lib/libtidy.5.dylib Referenced from: /usr/local/opt/[email protected]/bin/php I already run command brew update brew upgrade php .. Read more Hi I am trying to insert multiple items into a laravel collection inside a php loop but only one is getting inserted (the last one), please help to insert all the values. This array$some_array = array(); has values like 1,2,3,4 The loop is like foreach ($some_array as$key => $value) {$final_lists = collect([ ..

I’m currently working with Laravel and came across a strange issue, though I could not find any answer to it. When I install Laravel (Jetstream) on Docker with NGINX (Latest), PHP-FPM (PHP 8) and MariaDB 10.5, everything works as expected and I can run all migrations. As soon as I switch to MariaDB 10.6 I ..

I am very new to PHP(Laravel) and got some project. This is the explanation: One device exists someplace and it gathers some data. My project needs to log in to the device and need to get data by JSON file. I used this code for that work: $ch = curl_init(); curl_setopt($ch, CURLOPT_URL, "$url.0.json"); curl_setopt($ch, CURLOPT_USERPWD, ..

format: plain


text:
UK Secondary (7-11)
Mixed operations with algebraic terms (incl negatives)
Lesson

We've already learnt about the concept of negative numbers and how to simplify and evaluate expressions with directed numbers.

Now we are going to look at working with algebraic expressions that have positive and negative terms.

There are a few important things to remember when working with directed numbers:

 Example Multiplying a positive term and a negative term gives a negative answer $\left(-2f\right)\times5g=-10fg$(−2f)×5g=−10fg Multiplying two negative terms gives a positive answer $-6\times\left(-7c\right)=42c$−6×(−7c)=42c Subtracting a negative term is the same as adding a positive term $3-\left(-2a\right)$3−(−2a) is the same as $3+2a$3+2a Adding a negative term is the same as subtracting the term $h+\left(-3\right)$h+(−3) is the same as $h-3$h−3

Handy Hint

It may help to visualise a number line when you are working with directed numbers. Negatives go to the left down the number line and positives go up to the right down the number line.

#### Examples

##### Question 1

Simplify: $-2m-9m$2m9m

##### Question 2

Simplify the expression: $3x-\left(-4x\right)-2x$3x(4x)2x

##### Question 3

Simplify the expression $\frac{10x}{3x}$10x3x


text:
# $n$ people at a party and friendship

There are $$n\geq 2$$ people at a party. Each person has at least one friend inside the party. Show that it is possible to choose a group of no more than $$\frac{n}{2}$$ people at the party, such that any other person outside the group has a friend inside it.

This is just the definition of Dominating sets! I think I could use the marriage theorem or Hall's theorem... the solution I know (without using it) involves dividing into spanning trees and coloring with one of two colors.

Consider a minimal spanning tree of the graph, and pick an arbitrary node in the tree. Color it red. Color all of its neighbors green, then color all of the green nodes' neighbors red, etc, until the whole graph is colored. Note that since we are working with a tree, this is possible to do consistently. Then, consider the color which has fewer nodes. Take all of the nodes from that color and place them in your set. It is obvious that the resultant set is in fact a dominating set, and that the number of nodes chosen is at most $$\frac n2$$ since we have at least $$2$$ nodes (thanks to everyone having a friend).
You can partition the graph as $$V = V_1 \cup V_2$$, where each vertex in $$V_1$$ has at least as many neighbours in $$V_2$$ as in $$V_1$$ and vice versa. To prove this, take the partition with the maximum amount of crossing edges. If a vertex has more neighbours in its own subgraph, when you move it to the other side, the number of crosses will increase, a contradiction.
Now, we're done, since such a partition $$V_1 \cup V_2$$ clearly satisfies the conditions of the problem.


text:
A way to “execute” only latex procedural code (\def, \let) in a sequence ignoring the rest?

I have an external input file which I am reading line by line. This file contains text as well as some tex/latex macros. My goal is to write the content of this file into another external output file, and I would like to be able to replace some of the latex commands by symbols (to be more specific: I would like to be able to translate simple latex commands into their corresponding markdown counterpart).

So far, it works well for simple commands, eg \textbf{bold} is translated into bold. I have achieved this by redefining \textbf before writing back to the output file (I am using \write for this):

\def\textbf#1{*#1*}

Now I would like to be able to do the same for itemize. I would like the following code:

\begin{itemize}
\item one
\item two
\item three
\end{itemize}

to be translated into:

*   one
*   two
*   three

A first idea would be to ignore \begin{itemize} and \end{itemize} and simply redefine \item so that is expands into the markdown itemize prefix.

However, I want to be able to handle nested itemize. I have been using the same method (redefining \begin and \end, which in turn will use a modified \itemize and \enditemize), keeping track of the nesting level and increasing the markdown prefix accordingly. As you can imagine, things went wrong: since there is a lot of \def, \if, \begingroup etc. involved in the process, what I get at the end in my output file is a sequence of latex code, mainly containing \def's, braces, and so on.

This brings me to my question: before I send a line containing both text and "procedural latex code" (eg \def, \if and so on) to \write, is there a way to "execute" the procedural code it contains, and then write the output to the external file?

I have been using all the tricks I know (\edef, \expandafter's, \romannumeral, ...) to force the expansion, with no success so far.

This is a very minimal example to illustrate what I am trying to achieve:

\documentclass{minimal}

\usepackage{filecontents}

\begin{filecontents*}{temp.tex}
This is \textbf{bold}

This is itemize:
\begin{itemize}
\item one
\item two
\item three
\end{itemize}
\end{filecontents*}

\newwrite\fileout

\def\markdownbf#1{**#1**}
\def\markdownbegin#1{\csname markdown#1\endcsname}
\def\markdownend#1{\csname markdownend#1\endcsname}
\def\markdownitemize{\edef\markdownprefix{*   }}
\def\markdownenditemize{}
\def\markdownitem{}
\def\markdownpar{}

\def\activatemarkdowncommands{%
\let\textbf=\markdownbf%
\let\begin=\markdownbegin%
\let\end=\markdownend%
\let\itemize=\markdownitemize%
\let\enditemize=\markdownenditemize%
\let\item=\markdownitem%
\let\par=\markdownpar%
}

\def\translatefileA{%
\begingroup%
\edef\markdownprefix{}%
\activatemarkdowncommands%
\immediate\openin\filein temp.tex%
\immediate\openout\fileout out.txt%
\loop\unless\ifeof\filein%
\immediate\write\fileout{\markdownprefix\fileline}%
\repeat%
\immediate\closeout\fileout%
\immediate\closein\filein%
\endgroup%
}

\def\premarkdownbf#1{}
\def\premarkdownbegin#1{\csname premarkdown#1\endcsname}
\def\premarkdownend#1{\csname premarkdownend#1\endcsname}
\def\premarkdownitemize{\edef\storemarkdownprefix{*   }}
\def\premarkdownenditemize{\edef\markdownprefix{}}
\def\premarkdownitem{\let\markdownprefix\storemarkdownprefix}
\def\premarkdownpar{}

\def\postmarkdownbf#1{**#1**}
\def\postmarkdownbegin#1{\csname postmarkdown#1\endcsname}
\def\postmarkdownend#1{\csname postmarkdownend#1\endcsname}
\def\postmarkdownitemize{}
\def\postmarkdownenditemize{}
\def\postmarkdownitem{}
\def\postmarkdownpar{}

\def\activatepremarkdowncommands{%
\let\textbf=\premarkdownbf%
\let\begin=\premarkdownbegin%
\let\end=\premarkdownend%
\let\itemize=\premarkdownitemize%
\let\enditemize=\premarkdownenditemize%
\let\item=\premarkdownitem%
\let\par=\premarkdownpar%
}

\def\activatepostmarkdowncommands{%
\let\textbf=\postmarkdownbf%
\let\begin=\postmarkdownbegin%
\let\end=\postmarkdownend%
\let\itemize=\postmarkdownitemize%
\let\enditemize=\postmarkdownenditemize%
\let\item=\postmarkdownitem%
\let\par=\postmarkdownpar%
}

\def\translatefileB{%
\begingroup%
\edef\markdownprefix{}%
\immediate\openin\filein temp.tex%
\immediate\openout\fileout out.txt%
\loop\unless\ifeof\filein%
\activatepremarkdowncommands%
{\nullfont\fileline}%
\activatepostmarkdowncommands%
\immediate\write\fileout{\markdownprefix\fileline}%
\repeat%
\immediate\closeout\fileout%
\immediate\closein\filein%
\endgroup%
}

\begin{document}
\translatefileB
\end{document}

If I use \translatefileA, then I get in out.txt:

This is **bold**

This is itemize:
\edef {* }
one
two
three

If I use \translatefileB, then I get:

This is **bold**

This is itemize:

* one
* two
* three

The first version is wrong because \markdownprefix is not set as it should, because \markdown\itemize is not expanded as intended I guess.

The second version is ok. Unwanted output in the PDF file is deleted using \nullfont. So far it seems the best solution I could come up with.

• Could you provide a MWE? Something that works for the simple \textbf, and what you want to work? Then we don't have to redo all of that :) – Andreas Storvik Strauman Mar 24 '18 at 13:51
• I suspect that you are asking the wrong question, but it is a bit hard to work out what workflow you are intending. You could look at (for example) ex4ht which is doing a similar job, but generating html rather than markdown – David Carlisle Mar 24 '18 at 13:54
• latex is a macro expansion language it doesn't really have procedural code, it isn't at all clear what you mean by executing code before you send it to a \write. If the code is \zzzz which has been defined by \newcommand\zzzz{hello} you can of course write hello (that will happen by default) but if it is \hbox{hello} what do you want to \write \hbox, { and } are all non expandable primitives – David Carlisle Mar 24 '18 at 13:57
• What is the purpose? If you want to translate latex to markdown, there probably exists software for this. Also LaTeX is not a text manipulation program for sure. Searching and replacing substrings is one of the most hazzleful things you can do. – Andreas Storvik Strauman Mar 24 '18 at 14:14
• @David Carlisle: I suspect my question may have a simple answer. I do know that a lot of tex primitives (like \hbox you mentionned) are not suitable for my purpose, and I dont want to be able to translate them into markdown. What I do want is to be able to process some of them, like \textbf and the likes. The problem for me is to deal with \itemize so far. – Vincent Forman Mar 24 '18 at 14:38

Check for a substring in the infile using \readline. \readline reads it as a raw verbatim input, and execute the line only if your commands are in there, if not the line is ignored. So there at two reads. Suffixed with ‘L’ for LaTeX to be run and with ‘V’ for verbatim (plain text)

Edit: Previous code not working. Now fixed.

\documentclass{article}
\usepackage{substr, pgf, tikz}
\begin{document}
\makeatletter
\openin\fileinL=inFile.dat
\openin\fileinV=inFile.dat

\gdef\executecommands{\string\xdef, \string\let}

\loop\unless\ifeof\fileinL%
\foreach \cmd in \executecommands{
\IfSubStringInString{\cmd}{\filelineV}{\filelineL}{}
}
\repeat%
\end{document}

• this doesn't really address the question, as the OP wants to execute the \def without typesetting the text that is on that line not in the def. – David Carlisle Mar 24 '18 at 16:11
• Yeah, doesn’t really cover the case with multiple commands on one line. – Andreas Storvik Strauman Mar 24 '18 at 16:14
• I may have found a solution using \nullfont (see above). I will be running some tests to decide if it completely solves my problem. – Vincent Forman Mar 24 '18 at 16:49
• Ah. I now got what you meant! Haha, okay. Then the answer I provided is pretty special casey! I was thinking you only wanted to run select commands, not that you wanted to run all commands, but suppress output. – Andreas Storvik Strauman Mar 24 '18 at 17:16

It seems \nullfont answers my question, also it needs a little more work to handle math mode properly (see \nullfont in mathmode).

Below is the modified code to handle math mode (ie: suppress all output, yet expanding macros and defining them). Macro \translatefile will read the content of temp.tex, expand the macros it contains (first stage) without generating output in PDF, and translate it into markdown (second stage), generating file out.txt. It is not a fully functional translator, it is rather a proof of concept: it will only handle \textbf and (non-nested) itemize.

Node: another (and probably better) solution to achieve the same effect would be to use \setbox0\vbox{....} rather than \nullfont as David Carlisle suggested. However I'm not familiar enough with this kind of command to use it!

\documentclass{minimal}

\usepackage{filecontents}

\begin{filecontents*}{temp.tex}
This is \textbf{bold}, this is math: $1+x^2$.

This is itemize:
\begin{itemize}
\item one,
\item two,
\item three.
\end{itemize}
\end{filecontents*}

\newwrite\fileout

\begingroup
\catcode\ =12%
\gdef\SPACE{ }%
\endgroup

\def\premarkdownbf#1{}
\def\premarkdownbegin#1{\csname premarkdown#1\endcsname}
\def\premarkdownend#1{\csname premarkdownend#1\endcsname}
\def\premarkdownitemize{\edef\storemarkdownprefix{*\SPACE\SPACE\SPACE}}
\def\premarkdownenditemize{\edef\markdownprefix{}}
\def\premarkdownitem{\let\markdownprefix\storemarkdownprefix}
\def\premarkdownpar{}

\def\postmarkdownbf#1{**#1**}
\def\postmarkdownbegin#1{\csname postmarkdown#1\endcsname}
\def\postmarkdownend#1{\csname postmarkdownend#1\endcsname}
\def\postmarkdownitemize{}
\def\postmarkdownenditemize{}
\def\postmarkdownitem{}
\def\postmarkdownpar{}

\def\activatepremarkdowncommands{%
\let\textbf=\premarkdownbf%
\let\begin=\premarkdownbegin%
\let\end=\premarkdownend%
\let\itemize=\premarkdownitemize%
\let\enditemize=\premarkdownenditemize%
\let\item=\premarkdownitem%
\let\par=\premarkdownpar%
}

\def\activatepostmarkdowncommands{%
\let\textbf=\postmarkdownbf%
\let\begin=\postmarkdownbegin%
\let\end=\postmarkdownend%
\let\itemize=\postmarkdownitemize%
\let\enditemize=\postmarkdownenditemize%
\let\item=\postmarkdownitem%
\let\par=\postmarkdownpar%
}

\newfont\dummyfont{dummy}

\def\translatefile{%
\begingroup%
\nullfont%
\everymath{%
\count0=0%
\loop%
\ifnum\count0<16%
\textfont\count0\dummyfont%
\scriptfont\count0\dummyfont%
\scriptscriptfont\count0\dummyfont%
\repeat%
}%
\edef\markdownprefix{}%
\immediate\openin\filein temp.tex%
\immediate\openout\fileout out.txt%
\loop\unless\ifeof\filein%
\activatepremarkdowncommands%
\fileline%
\activatepostmarkdowncommands%
\immediate\write\fileout{\markdownprefix\fileline}%
\repeat%
\immediate\closeout\fileout%
\immediate\closein\filein%
\endgroup%
}

\begin{document}
This should be visible.
\translatefile %This should not be visible, only processed to out.txt
This should be visible.
\end{document}
`


text:
Area Index

# Area of a Scalene Triangle Formulas & Calculator

The area of a scalene triangle can be calculated using the following formula: if you know the length of the base side and the height.

## Formula Knowing the base and the height

If you know the length of the base and height, you can calculate its area using the following formula:

$$"area" = (b * h) / 2$$

Sides Length

Area result

Copia y pega el siguiente código en el HTML de tu página web para mostrar ahí esta fórmula y su calculadora.

Nuestras calculadoras son gratuitas y compatibles con móviles. Al copiarlas a tu sitio acuerdas dar attribución a "© WikiFormulas.com". Vincular de regreso es opcional pero bienvenido. ¡Gracias!

### Where:

$$b = "length of the base side"$$
$$h = "length of the hight"$$

## Scalene Triangle Definition

It is a geometric figure with 3 sides and 3 vertices, the scalene triangle always have 3 angles. The scalene triangles will have always 3 sides and 3 diferent angles.

## Area Definition

It is the space of the internal surface in a figure, it is limited for the perimeter. Also, the area can be calculated in a plane of two dimensions.

## Formula Definition

It is a representation of a rule or a general principle using letters. (Algebra, A. Baldor)
When describing formulas in plural, it is also valid to say "formulae".

WikiFormulas.com is a database of embeddable formulas, equations and calculators. You are allowed to use our calculators in any project as long as you give attribution.

✉ wikiformulas@gmail.com
© 2017 WikiFormulas.com
All rights reserved.


text:
# Power Sum

Find $$\displaystyle \sum_{\{a_{1}, \ldots, a_{k}\} \subseteq \{1, 2, 3, 4, 5\}} \frac{1}{a_{1} \cdots a_{k}}$$, where the sum is over all nonempty subsets of of the set $$\{1, 2, 3, 4, 5\}$$.

×


text:
## Taylor expansion of $\prod_{i=1}^n(1-x_i)^{-1/2}$ around 0

What is the Taylor expansion of $$\prod_{i=1}^n\frac{1}{\sqrt{1-x_i}}$$ around $$(0,0,…,0)$$？I know that we can write it as $$\prod_{i=1}^n\left(1+\sum_{k=1}^{\infty}\frac{(2k-1)!!x_i^k}{(2k)!!}\right)$$ But how to simplify it?


text:
# Measuring the speed of sound

You probably noticed that the data points for different distances are not spread out the same way. At some distances the points are closer together and at some distances they are more scattered around. This can have a couple of reasons. Depending on the position of the second microphone, the sound reflections in my little cramped room will have a different influence on the measurement, making it more or less robust. Also my clapping will probably not be exactly constant and my current form has influenced the quality of the measurements at the different distances.

So far, we treat every point the same way. When we fit the linear function to the data, all points are as important as the others. But if we look at the data, we see that some points are obviously not as reliable as others. The spread of the points at 40 cm is much higher than at 50 cm, why should we trust those points equally? Well, we don’t have to.

All we have to do is take all the data points of one distance and average them into one data point with an error that corresponds to the spread of the data points. Then we can tell the fit algorithm to pay less attention to the points with large errors than to those with small errors.

This is the point where the “serious business” starts. We actually want to work with the data we collected instead of just plotting it and while you could principally manipulate the data with any programming language and then plot it with gnuplot, I prefer to keep everything together. (Also matplotlib plots just look nicer than gnuplot plots.)

My data manipulation and plotting tool set of choice: python with matplotlib, numpy and scipy.

They are all free and available for all the major platforms.

This is not the place for an in depth tutorial of python or the libraries, but I’ll quickly go over the main points of my program (If you want a more in depth description of the capabilities of the libraries, you should google them and take a look at their documentations):

#!/usr/bin/python

from matplotlib import pyplot as plt
from numpy import loadtxt, mean, std, array, linspace, sqrt
from scipy.optimize import curve_fit

# Combine multiple data points at same distance into one with error
points = {}
for P in data:
if P[0] not in points:
points[P[0]] = []
points[P[0]] += [P[1]]

x = array(points.keys())
t = array(list(mean(list(points[xx])) for xx in x))
e = array(list(std(points[xx])/sqrt(len(points[xx])) for xx in x))

# Fit a linear function to the data
def linfunc(x, m, n):
return m*x + n
#def errfunc(p, x, y, e):
#    return (y - linfunc(p, x)) / e
p0 = [0.0, 0.0]
p1, pcov = curve_fit(linfunc, x, t, p0, e)
print("m = {0} +- {1}".format(p1[0], sqrt(pcov[0,0]))) # The diagonals of the covariant matrix are an
print("n = {0} +- {1}".format(p1[1], sqrt(pcov[1,1]))) # estimator for the variance of the fit parameters

# Make a nice plot
plt.figure(figsize=(8,4.8), dpi=100)
plt.subplot(111)
plt.xlabel(r"distance $x$ / [cm]")
plt.ylabel(r"time difference $\Delta t$ / [$\mu$s]")
plt.errorbar(x, t, e, fmt="r.", label="Data")
xx = linspace(5, 65, 2)
plt.plot(xx, linfunc(xx, *p1), 'g-', label=r"$\Delta t = m x + n$ with $m = {0:1.1f}\ \mu$s/cm, $n = {1:1.1f}\ \mu$s".format(*p1))
plt.legend(loc="upper left")
plt.xlim(0,70)
plt.savefig("sound_pylab.png")
plt.show()

After the mandatory shebang (working under Linux here) and the inclusion of all the necessary libraries, the first interesting thing happens at line 8: loadtxt is a very convenient function from numpy that lets you load a file of data in “standard format”, i.e. a couple of rows of numbers with white spaces as separators, in one line. The return value is a 2-dimensional array which we then further dissect in lines 11 to 15. There we put all the points of the same distance together into a dictionary, so we can conveniently access them later.

In lines 17 to 19 we create the new (averaged) points as numpy arrays: Line 17 is straight forward; we take all the different (unique) distances and turn them into an array of, well, the distances. Line 18 is a bit more convoluted, as we have to convert between lists and arrays a few times to make use of pythons handy “list( f(A) for A in B)” syntax. All we do there is create an array of the averages of the points at the given distances. These are our y-coordinates. Line 19 is equally convoluted and even contains a bit of statistics. Instead of an array of the average of the points we create an array of the standard deviations of the points. This will be the standard error of our averaged points.

The numpy function std calculates the standard deviation of a given set of numbers. That standard deviation is a measure for how spread out the numbers in that set are. This is almost the number that we want, but just almost. We’re not interested in the spread of the numbers in the set, but the error of the average we calculate from those numbers! This is important.

If you have a given distribution of random numbers, then the standard deviation as calculated by numpy does not depend on the number of points in that set, as it is a property of the distribution. It describes the expected error (i.e. the deviation from the actual mean of the distribution), if you randomly choose one of the numbers in the set.

The standard error of the mean of the set on the other hand does depend on the number of points in the set. If you have a larger set, then the average of that set will be (in general) closer to the actual mean of the distribution than if you have only a small set of numbers. (More numbers means that the statistical errors are more likely to cancel each other.) So this number describes the expected error (i.e. the deviation from the actual mean of the distribution), if you average over all the numbers in the set.

And to calculate the standard error of the mean of the set, you have to take the standard deviation of the set itself and divide it by the square root of the number of points in the set. So that’s exactly what we do in line 19. Again, at this point you will have to trust me and wait for a later article, or read a book on basic statistics. Or the Wikipedia article.

In lines 21 to 29 we fit a linear function to the (now averaged) data, taking their errors into account. This means that points with large errors (i.e. low confidence in their accuracy) will have less influence on the result than points with small errors (i.e. high confidence in their accuracy). This is done by the method of weighted least squares (google it).

The advantage of scipy’s curvefit is that you can fit any function you can program in python to the data at hand, not only arithmetic functions, though we don’t really need that functionality here.

The lines 31 to 42 are just there to produce a pretty plot that really shows off our science skills:

This looks really good. And it yields the following results:

$\begin{array}{rl} m & = (30.3 \pm 1.3) \ \mu \mathrm{s/cm} \\ n & = (-71.4 \pm 42.6) \ \mu \mathrm{s} \\ c_{S} & = (331 \pm 14) \ \mathrm{m/s} \ (4.3 \%) \end{array}$

We almost doubled our accuracy there! But we are still not done.

1. Brett_cgb said:

Interesting analysis, but I’ve got a question. (I’m using MS Excel.)

Q) My initial data plot and linear trendline exactly match yours. My trendline equation provides m=28.984 and n=-78.300 (same as you within the precision you provided). It doesn’t provide the +/-error for m or n. How was the +/- error calculated?

2. Brett_cgb said:

Ummm… It seems I’ve opened a can of worms, based on what I’ve been able to find.
The answer to that question appears appropriate for a weeks worth of university statistics classes.

• Ast said:

Yeah… I don’t know where you can get this number in Excel, but gnuplot just gives us those numbers together with the fit parameters.

Behind the curtain the the calculation is indeed a bit complicated and involves the (estimated) Jacobian which then gets turned into a variance-covariance-matrix of the parameters. Then you use the diagonal of that matrix as estimates for the uncertainty of the parameters.

If Excel does not offer a way to get these numbers in an easy way, I suggest you switch to another program for your data analysis. ;)

And another tip: If you want to know how this stuff works, you should take a look into the documentations of programs which have realised this functionality, e.g. gnuplot, scipy or any other open source scientific library. Usually there is at least a bit of theoretical background explained.

3. Brett_cgb said:

> you should take a look into the documentations of programs which have realized
> this functionality, e.g. gnuplot, scipy or ….

That’s what inspired the “can of worms” comment…. B)


text:
# Proof of the lemma used in proving that a finite-dimensional normed space is complete

I'm trying to understand the proof for the lemma:

$$\|\alpha _1 e_1 + \alpha _2 e_2 + \cdots + \alpha_n e_n\| \geq c (|\alpha_1|+|\alpha_2|+\cdots+|\alpha_n|)$$ where $c>0$ and the $e_i$s are finite and linearly independent.

The proof I'm referring to is found in many places, like Kreyszig’s book and on page 16 of this PDF file; and in other locations.

The proof is based on proving that no sequence of the form $(\alpha_1 e_1 + \alpha _2 e_2 + \cdots + \alpha _n e_n)$ could converge to a sequence with zero norm.

What I don't quite get is the following:

1. Why the proof is not simply stated as: If the lemma is not true, then $\|\alpha _1 e_1 + \cdots + \alpha_n e_n\| = 0$ which is not possible since the $e_i$s are independent. I mean, why do they involve the sequence convergence business here?

2. Even when they assume that a sequence has to converge to a zero normed sequence, why does it have to be with the constraint that $|\alpha_1^{(m)}|+|\alpha_2^{(m)}|+\cdots+|\alpha_n^{(m)}| = 1$? Shouldn’t we consider that any possible sequence converges to zero normed sequence without the restriction that the sum should always be one?

Thanks a lot!

-
I tried to clean this up slightly. Let me know if I goofed on anything. – Dylan Moreland Jul 12 '12 at 19:54
Thanks Dylan, looks better now. – Khalid Salman Jul 13 '12 at 18:21

1- The lemma they are trying to prove is that there exists some $c>0$ such that, for any $\alpha_1,\ldots,\alpha_n$, we have $\|\alpha_1e_1+\cdots+\alpha_ne_n\|\geq c(|\alpha_1|+\cdots+|\alpha_n|)$. The key point here is that $c$ does not depend on $\alpha_1,\ldots,\alpha_n$. What you propose (showing that the norm is not $0$) only allows us to say we can pick some $c>0$ dependent on $\alpha_1,\ldots,\alpha_n$, which is a weaker statement.

2- It suffices to consider the case $|\alpha_1|+\cdots+|\alpha_n|=1$ to prove that if $|\alpha_1|+\cdots+|\alpha_n|=1$, then $\|\alpha_1e_1+\cdots+\alpha_ne_n\|\geq c$. The full lemma can then be proved by dividing both sides by $|\alpha_1|+\cdots+|\alpha_n|$.

-
Thanks Alex. I should have mentioned that I'm an engineer, and therefore can't grasp such mathematical subtleties easily :). Based on your comment, how does the proof show the existence of c based on the basis but not based on the chosen alphas? I mean the proof only shows that convergence to zero vector is not possible and does not mention c itself anywhere. Can you please help me understand? – Khalid Salman Jul 13 '12 at 18:33
@KhalidSalman Suppose no such $c$ existed. Then for any $c>0$, we have some term (which I'll call $x_n$) such that $\|x_n\|<c$. Thus we have some $x_{n_1}$ such that $\|x_{n_1}\|<1/1$, some $x_{n_2}$ such that $\|x_{n_2}\|<1/2$, etc. The sequence $x_{n_k}$ then converges to $0$ as $k\to \infty$. – Alex Becker Jul 13 '12 at 20:55
Thanks Alex, all is clear now. – Khalid Salman Jul 16 '12 at 17:43

The lemma states that there exists $c\in\mathbb{R},c>0$ such that $\|\alpha_1e_1+\cdots+\alpha_ne_n\|\ge c|\alpha_1+\cdots+\alpha_n|$. The negation is: "for each $c\in\mathbb{R},c>0$, there exists $\alpha_1,\cdots,\alpha_n$ such that $\|\alpha_1e_1+\cdots+\alpha_n\|<c|\alpha_1+\cdots+\alpha_n|$". Maybe this example clarify a little: consider $l_2(\mathbb{N})$. Then $M=\left\{\frac{1}{n}e_n:n\in\mathbb{N}\right\}$ form a linearly independent set, but for every $c>0$, you can find $m\in\mathbb{N}$ such that $1/m<c$, so, $\left\|1\cdot \frac{1}{m}e_m\right\|_2=\frac{1}{m}<c|1|$. The problem here is that $M$ is infinite.

-
Thanks Yuki, I believe you and Alex are in agreement that this proof is 'all about the c value' which I still don't quite get as in my comment to Alex. Your example is very useful BTW. – Khalid Salman Jul 13 '12 at 18:36


text:
# E-number

The limit of the expression $(1+1/n)^n$ as $n$ tends to infinity:

$$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n=2.718281828459045\dots;$$

it is the base for the natural logarithm. $e$ is a transcendental number, which was proved by C. Hermite in 1873 for the first time.

$e$ is also defined as the sum of the series

$$\sum _{n=0} ^{\infty} \frac{1}{n!}$$

That means

$$e=\sum _{n=0} ^{\infty} \frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\dots$$


text:
# How do you find the sum of the digits contiguous to one which is situated in a three by three table?

The problem is as follows:

I am placing the numbers from 1 to 9 in the squares of a 3 x 3 board. I start by placing the numbers 1, 2, 3 and 4 as shown in the figure. For the number 5, the sum of the numbers in the adjacent boxes (which have a common side with the 5) is equal to 9. What is the sum of the numbers adjacent to 6?

$$\begin{array}{|c|c|c|} \hline 1&\,\,&3 \\ \hline \,\,&\,\,&\,\,\\ \hline 2&\,\,&4\\ \hline \end{array}$$

The choices given in my book are as follows:

1. 29
2. 15
3. 17
4. 28
5. 14

I found this problem in my puzzles book Reason and Logic from the 2000s. The question seems to be adapted from a reprinted copy of Martin Gardner's 50's book on Recreational puzzles.

The nature of this problem is told as a narration from the author. The way how I understood what it was intended was as follows:

The only available numbers to use are: 5, 6, 7, 8 and 9.

But since it mentions that the empty spaces which are contiguous to 5 will add up to 9, will leave the only choice (as I could spot on) to be this one:

$$\begin{array}{|c|c|c|} \hline 1&\,\,&3 \\ \hline 5&6&\,\,\\ \hline 2&\,\,&4\\ \hline \end{array}$$

As $$1+6+2=9$$ Hence this leaves up the other number unused available, with those being: 7, 8 and 9.

Since it doesn't say anything else about them, I ordered them this way:

$$\begin{array}{|c|c|c|} \hline 1&7&3 \\ \hline 5&6&8\\ \hline 2&9&4\\ \hline \end{array}$$

As the problem requests to add up the digits contiguous to 6 these will:

$$7+8+9+5=29$$

Which appears as a choice, but my book says this is not the answer. What could I be doing wrong? Did I overlook something?

• I think your answer is correct, as well as your solving path. There could be typos in the book itself. What's the answer provided by the book? Nov 6, 2020 at 3:08
• @Bubbler I'm sorry for the late reply buddy. But I'm starting to believe that there might be a typo in the answers sheet of the book. The book mentions the answer to be $14$ I have no idea how did they got to there. Maybe you want to add a clarification or an answer reassuring that my strategy was right?. Nov 8, 2020 at 2:35

As I already commented, I think your answer is indeed correct (and the book might have some typos there, likely in the answer sheet).

The 5 can't be at the center because then it would border with all of 6, 7, 8, and 9, which sum to 30.

$$\begin{array}{|c|c|c|} \hline 1&?&3 \\ \hline ?&5&?\\ \hline 2&?&4\\ \hline \end{array}$$

So the 5 must be on one of four sides, sharing edges with two numbers $$1\le a,b\le4$$ (two adjacent corners) and one number $$c\ge 6$$ (the center).

We're given that $$a+b+c=9$$, which can only be satisfied when we choose the minimum possible for all three ($$\{a,b\}=\{1,2\}, c=6$$). So the 5 must be on the left edge, bordering with a 6.

$$\begin{array}{|c|c|c|} \hline 1&?&3 \\ \hline 5&6&?\\ \hline 2&?&4\\ \hline \end{array}$$

For the remaining numbers (7, 8, 9), it doesn't matter how we place them as all of them will be adjacent to 6 anyway. Therefore the requested sum is $$5+7+8+9 = 29$$.


text:
# Equations of lines Find both the parametric and the vector equations of the following lines. The line through (0, 0, 1) in the direction of the vector v = ⟨4, 7, 0⟩

Equations of lines Find both the parametric and the vector equations of the following lines. The line through (0, 0, 1) in the direction of the vector $v=⟨4,7,0⟩$
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Gennenzip
Step 1 Given: The line through (0, 0, 1) in the direction of the vector$v=<4,7,0>$. To find: Parametric and vector equations of the given line. Step 2 Let, $\stackrel{\to }{{r}_{0}}=\left(0,0,1\right),\stackrel{\to }{d}=<4,7,0>$ The vector equation of the line is, $\stackrel{\to }{r}=\stackrel{\to }{{r}_{0}}+t\stackrel{\to }{d}$
$⇒\stackrel{\to }{r}=<0,0,1>+t<4,7,0>$
$=<0,0,1>+<4t,7t,0>$
$=<0+4t,0+7t,1+0>$
$=<4t,7t,1>$ $⇒\stackrel{\to }{r}=<4t,7t,0>$ The parametric equations of a line are, Result :$x=4t,y=7t,z=0$ Vector equation of a line is, $\stackrel{\to }{r}=<4t,7t,0>$ Parametric equations of a line are, $x=4t,y=7t,z=0.$
Jeffrey Jordon


text:
# Rogers-Ramanujan continued fraction in terms of Jacobi theta functions?

The Rogers-Ramanujan cfrac is,

$$r = r(\tau)= \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$

If $q = \exp(2\pi i \tau)$, then it is known that,

$$\frac{1}{r}-r =\frac{\eta(\tau/5)}{\eta(5\tau)}+1\tag1$$

$$\frac{1}{r^5}-r^5 =\left(\frac{\eta(\tau)}{\eta(5\tau)}\right)^6+11\tag2$$

with the Dedekind eta function, $\eta(\tau)$.

Q: Is there a similar simple identity known using ratios of the Jacobi theta functions $\vartheta_n(0,q)$?

$\color{brown}{Edit}$: In response to a comment, here are some details. Suppose we don't know $(2)$. One way to find such relations is to use known identities. Given the j-function $j(\tau)$, we have,

$$j(\tau)=-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^3}{r^5(r^{10} + 11r^5 - 1)^5}\tag3$$

$$j(\tau)=\frac{(5x^2+10x+1)^3}{x^5}\tag4$$

where $x = \left(\frac{\eta(\tau)}{\sqrt{5}\,\eta(5\tau)}\right)^6$. Equate $(3),\,(4)$,

$$-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^3}{r^5(r^{10} + 11r^5 - 1)^5} = \frac{(5x^2+10x+1)^3}{x^5}\tag5$$

and using symbolic software like Mathematica to factor $(5)$, we find one factor is given by,

$$\frac{1}{r^5}-r^5 =125x+11\tag6$$

and it only takes minor tweaking to make $(6)$ have the form of $(2)$. Thus, all we need is to express $j(\tau)$ not by an eta quotient $x$, but by a theta quotient $y=\left(\frac{\vartheta_n(0,q)}{\vartheta_n(0,q^5)}\right)^k$ so that,

$$j(\tau) = \frac{f_1(y)}{f_2(y)}\tag7$$

is a ratio of polynomials in $y$.

$\color{brown}{P.S.}$ The motivation for this is that the general quintic is solvable by the Rogers-Ramanujan cfrac via the eta quotient above as described in this post.

• The answer is in W.Duke's paper continued fractions and modular functions,equation (4.9) – Nicco Sep 7 '15 at 11:34
• @Nicco: Thanks. By the way, can you ask a question on the forum how to solve the Brioschi quintic using elliptic functions? (I have an answer that may be useful to you and others interested, and is the reason for this theta function post.) – Tito Piezas III Sep 7 '15 at 11:44
• @Nicco: Thanks. Right now, I can do so using the complete elliptic integral $K(k)$ and the Dedekind eta function, via $R(q)$. I'm trying to adapt the method to the Jacobi theta. – Tito Piezas III Sep 7 '15 at 13:40
• From another source, we have $$\frac{1}{r^5}-r^5 = \Big(\frac{ \vartheta_2(0,q)\,\vartheta_3(0,q)\, \vartheta_4(0,q) }{\vartheta_2(0,q^5)\,\vartheta_3(0,q^5)\, \vartheta_4(0,q^5) }\Big)^2+11$$ but I find this is cheating a little bit. – Tito Piezas III Sep 9 '15 at 3:34
• Using basic identities, one can deduce a family of formulae for the RRCF like $$r(\tau) = \frac {\vartheta_1\!\left(\frac{\pi}{5}\middle|\tau'\right)} {\vartheta_1\!\left(\frac{2\pi}{5}\middle|\tau'\right)} = \frac {\vartheta_2(0\mid \tau')\,\vartheta_3(0\mid\tau')\,\vartheta_4(0\mid\tau')} {2\,\vartheta_2\!\left(\frac{\pi}{5}\middle|\tau'\right) \vartheta_3\!\left(\frac{\pi}{5}\middle|\tau'\right) \vartheta_4\!\left(\frac{\pi}{5}\middle|\tau'\right)}$$ where $\tau'=-(5\tau)^{-1}$. Unfortunately, the question requires thetanulls exclusively. Sigh. – ccorn Oct 23 '15 at 1:27

Motivated by ccorns's comment, I decided to re-visit this question. After some effort, I found what I was looking for. The Rogers-Ramanujan cfrac is,

$$r = r(\tau)= \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$

where $q = \exp(2\pi i \tau)$. It turns out we have the nice relations,

\begin{aligned}\frac{1}{r^5}-r^5 &=\,u^3+11\\ &= \frac{v(v-5)^2}{(v-1)^2}+11 \end{aligned}

where the three roots of the cubic in $u$ are eta quotients,

$$u_n = \left(\frac{\eta(\tau')}{\eta(5\tau')}\right)^2$$

with $\tau'=\tau+n$, while the three roots of the cubic in $v$ are theta quotients,

$$v_n = \left(\frac{\vartheta_{n+2}(0,p)}{\vartheta_{n+2}(0,p^5)}\right)^2$$

for the nome $p = e^{\pi i \tau}$ and for $n=0,1,2$.

• Nice indeed. If there were a 2nd volume of Whittaker & Watson, this one would make it to the legendary "Shew that..." exercises. – ccorn Oct 27 '15 at 19:17

Given the Jacobi theta function $$\vartheta_4(u,q)=1+2\sum^{\infty}_{n=1}(-1)^nq^{n^2}\cos(2nu)$$ I. Define the Rogers-Ramanujan as $$R(q)=\frac{q^{1/5}}{1+}\frac{q}{1+}\frac{q^2}{1+}\frac{q^3}{1+}\ldots\textrm{, }\;|q|<1$$ then $$R(e^{-x})=e^{-x/5}\frac{\vartheta_4(3ix/4,\,e^{-5x/2})}{\vartheta_4(ix/4,\,e^{-5x/2})}\textrm{, }\;x>0$$ II. Define the Ramanujan-Gordon-Gollniz as $$H(q)=\frac{q^{1/2}}{1+q+}\frac{q^2}{1+q^3+}\frac{q^4}{1+q^5+}\frac{q^6}{1+q^7+}\ldots\textrm{, }\;|q|<1$$ then $$H(e^{-x})=e^{-x/2}\frac{\vartheta_4(3ix/2,\,e^{-4x})}{\vartheta_4(ix/2,\,e^{-4x})}\textrm{, }\;x>0$$

• +1 This is very nice. I tested it with Mathematica and they equate. Do you know if the cubic cfrac and the octic cfrac can be similarly expressed in terms of theta quotients? – Tito Piezas III Oct 29 '16 at 2:55
• Can you look at this MO post about $x^5+y^5 = 1$? There are some functions $a,b,c,d$ there and the ratio $a/b$ can be expressed by Jacobi theta functions. Can you check if we can do so for the ratio $d/b$ as well? – Tito Piezas III Jan 21 '17 at 8:07

For the cubic $V(q)$ cfrac is $$V(q):=\frac{q^{1/3}}{1+}\frac{q+q^2}{1+}\frac{q^2+q^4}{1+}\frac{q^3+q^6}{1+}\ldots\textrm{, }|q|<1$$ and $$V(e^{-x})=e^{-x/3}\frac{\vartheta_4(ix/4,e^{-3x})}{\vartheta_4(0,e^{-3x})}\textrm{, }x>0$$ See more formulas in https://arxiv.org/pdf/1107.2393v2.pdf

• +1 Great! Now all that is missing is Ramanujan's octic cfrac – Tito Piezas III Nov 5 '16 at 3:37
• The octic cfrac of Ramanujan is just the "elementary" continued fraction of Euler and can be evaluated (with theta functions) in a very simply way. – Nikos Bagis Nov 5 '16 at 4:30
• Kindly include so this list will be complete. – Tito Piezas III Nov 5 '16 at 4:32

I'm sorry about the octic cfrac. Yes it seems to be non trivial. In Berndt's book (see: B.C. Berndt. "Ramanujan Notebooks III". Springer Verlag, New York (1991), In chapter 19 Entry 1) one can find the expansion $$M(q)=\tfrac{1}{2}\,q^{1/8}\frac{\sum^{\infty}_{n=-\infty}q^{n^2/2+n/2}}{\sum^{\infty}_{n=-\infty}q^{n^2}}=\frac{q^{1/8}}{1+}\frac{q}{1+q+}\frac{q^2}{1+q^2+}\frac{q^3}{1+q^3+}\ldots$$ Hence if $|q|<1$ and $$\vartheta_3(z,q):=\sum^{\infty}_{n=-\infty}q^{n^2}e^{2niz}$$ For $x>0$ we have $$M(e^{-x})=\tfrac{1}{2}\,e^{-x/8}\frac{\vartheta_3(ix/4,e^{-x/2})}{\vartheta_3(0,e^{-x})}$$ Equivalently, let $q = e^{2\pi i z}$, then $$M(q) = \tfrac{1}{2}\,q^{1/8} \frac{\vartheta_3\big(i\ln(q^{1/4}),q^{1/2}\big)}{\vartheta_3(0,q)}=\frac{\eta(z)\,\eta^2(4z)}{\eta^3(2z)}$$ where $\eta(z)$ is the Dedekind eta function.

• I've added some info to connect it to the $\eta(z)$ function. – Tito Piezas III Nov 6 '16 at 15:03

Set $T=\sqrt{1-8V(q)^3}$, $q=e^{-\pi\sqrt{r}}$, $r>0$, then $$k_r^2=\frac{(1-T)(3+T)^3}{(1+T)(3-T)^3}$$ where $k_r$ is the elliptic singular modulus. (see https://arxiv.org/pdf/1008.1304v2.pdf)


text:
# OFSS III - Missions problems and troubleshooting.

#### djlodo

##### New member
CTD Without any problems listet in Orbiter Log

After downloading and installing Mission Pack 3, when i start launch scenario 3, i got a ctd without any listed problems in Orbiter LOG.

Any other scanario working perfect (and I got a lot of them , as example, every sts flights, all fgu sojuz scenarios)

Are the compabiliti problem with older Orbiter versions (i^m not using 2010) but scenario 1 and 2 of OFSS III are working correctly

#### Tex

Tutorial Publisher
Some of the addons in the required list are only compatible with Orbiter 2010-P1.

OFSS III is only being made available for Orbiter 2010-P1.

##### Scientist
Flight #2 Step 8

Can anybody give me some advice how to get home! I've done the whole mission to this point (first time with the URMS, docking the Cupola and Dock1, hooking to core, hooking the XR5 by eye onto the Dock, etc.

I'm wondering whether I should be installing BaseSync, Aerobrake MFD, etc to get back to Wideawake, or if I am missing something because they are not on the installation manifest.

Newb admission here ... I have not landed back to Earth too many times, so giving me some advice on where to fire retros, anticipation, etc for XR5 would be appreciated.

The mission so far is awesome!

Last edited:

#### blane

##### Deep Space Operator
Can give some general pointers for re-entry.

Execute the deorbit burn opposite side of the planet than the target base is. (AerobrakeMFD helps but also Map MFD). Set PeA of 60k or lower. Level for surface and pitch your nose up to 40° (AoA).

Also where exactly do you have problems? There are quite a few videos out that demonstrate successful re-entries.

##### Scientist
Thanks Blane - I'm in the OFSS III Mission 2, step 8, just ready to undock from our small space station (at this stage). I have none of the usual tools for re-entry (e.g. Aerobrake), so wanting to stay true to the mission set-up (rather than just add the other MFD's), I was wondering what the procedure should be for re-entry.

#### blane

##### Deep Space Operator
If you set Map MFD on TRK (tracking mode), you are on opposite side when base icon (set as target) switches sides. I'm sure there are other methods but that's what I would use

##### Scientist
A quick update on this. I've done quite a few reentries on the XR-5 now with BaseSync and Aerobrake MFD. With BaseSync at 0.4 degrees slope, 43 degrees anticipation and 80KM altitude, you get pretty close to Blane's advice of burning on the opposite side of the world. The BaseSync retro-burn drops the PeA to around 65km. Here's some speed/ALT/dist to go numbers from my last landing:

Speed Alt Dist
7567 100 6324
7572 90 5266
7523 80 3975
7218 70 2499
6518 60 1574
4953 50 862
2599 40 277
1182 30 111

This was using the XR-5 AoA autopilot varying between 35-42 degrees, and tracking on the Aerobrake MFD map display (trying to keep 1 or 2 squares of reserve energy right onto finals).

So you could definitely fly this without BaseSync and Aerobrake if you had a set of reference points like this, but is there a good reason why not to fly these missions with these two MFD's in the package?

---------- Post added at 03:51 PM ---------- Previous post was at 03:39 PM ----------

My latest time-burner... mission, trying to dock that damn Soyuz. Let's just say I have profound respect for the real Cosmonauts that have to dock that ship. Things making it harder than docking a DG or an XR ... the pulse RCS and the fixed view. I've realized that being able to pan your view is really important to get a solid view of the docking corridor. Go too far away from the Zarya on the docking orientation, and you start to overtake it (you will be say 1-2km below its altitude). No Attitude MFD so you are manually having to neutralize the delta-V. Really tough trying to shoot the docking alignment from the external camera. And so on...

I'm on my 3rd or 4th attempt now (from a clean new scenario), trying to get the most perfect sync alignment, with the Pe's lined up trying to get as close to an identical orbit as possible. Then maybe I can catch a break and get it cleanly lined up for docking.

Do you guys feel there's value in me posting more like this into another thread? Kinda Go Play in Space for OFSS, for the rest of us wanting a bit more than the brief flight notes?

##### Scientist
(OFSS Mission 3 retry!)

Well - I made the Soyuz hard dock with Zarya. Man - that's by far the hardest thing I've ever had to dock. After practising on the Mir, I found that it was near impossible to keep the rotation nulled out enough to concentrate on the translation to the final docking. It was always drifting a tiny amount that demanded constant switching from rotation to translation.

Can anyone comment on the "you can use the (+) Normal auto-pilot to hold angular alignment" until just before docking. I tried this and it didn't help at all. In fact - the issue seemed to be that Zarya was very gently rotating. (I.e. post dock, zoom x100, you can see a drift).

#### Tex

Tutorial Publisher
(OFSS Mission 3 retry!)Can anyone comment on the "you can use the (+) Normal auto-pilot to hold angular alignment" until just before docking. I tried this and it didn't help at all. In fact - the issue seemed to be that Zarya was very gently rotating. (I.e. post dock, zoom x100, you can see a drift).
I only suggested using (+) autopilot to help you hold the correct angular alignment with the docking port (except for rotation) while you translate close to it. Just reduces your workload a little. Unfortunately, you must disable (+) and rotate slightly to reach perfect angular alignment, but one can do this at the last minute, just before hard dock.

##### Scientist
The thing I forgot though was setting the autopilot on the target . No wonder the + Normal autopilot was not holding station properly!

#### Tex

Tutorial Publisher
Ah yes, that would make it very hard.

#### Pipcard

##### mikusingularity
Donator
I haven't tried the Soyuz yet, so what is meant by "pulse RCS"?

#### diogom

##### Survivor of 21/10/2011
I haven't tried the Soyuz yet, so what is meant by "pulse RCS"?
Instead of a continous RCS firing as you hold the key, when you press a key, the RCS fires for a determined amount of time. In Thorton's, pulses of .05s, .25s and 1s I believe.

#### Tex

Tutorial Publisher
Has anyone else had a problem getting the new CTV flight to clear the fuel bug by pressing Ctrl+7 after jettison from Ariane5? It's been a long time since I've flown CTV and I recall the mention of this in the documentation, however I was not able to clear the bug with the latest OFSS flight. I wonder if this might have something to do with the fact that we have 2 CTV's in this scenario?

If you can't clear the fuel bug after you jettison CTV from Ariane5, quit the session, then add the following line shown in bold red text to your CTV-LEO6 ship entry in the scenario:

Code:
CTV-LEO6:CTV-LEO6
STATUS Orbiting Earth
RPOS 3447977.72 -4035845.26 4321307.63
RVEL -5049.997 -5498.550 -1108.824
AROT -119.47 51.12 -60.56
VROT 0.00 0.08 -0.00
AFCMODE 7
[B][COLOR="Red"]  PRPLEVEL 0:1.000000[/COLOR][/B]
IDS 0:241 100
NAVFREQ 0 0
XPDR 240
UMMUCREW -C_Haignere-40-65-74
UMMUCREW -JF_Clervoy-37-67-55
UMMUCREW -T_Reiter-35-70-45
UMMUCREW -C_Nicollier-35-70-45
UMMUCREW -A_Kuipers-35-70-45
UMMUCREW -R_Vittori-35-70-45
BEACON
ANTENNA 1 1.0000
PANEL 1 1.0000
PROBE 1 1.0000
ROT 1 1.0000
MET 2696010.0
STATION 0
O2time 57623
M2FUELFIXED
END

##### Scientist
Yeah Tex- confirmed fuel bug on the CTV on Flight 13. For some reason the CTRL+7 does not initialize the fuel. Thanks for the fix line, though I think it's too much fuel for the little CTV, because after a good amount of thrusting to do the docking, I was still left with 0.999153 !!

#### Tex

Tutorial Publisher
You can always make it more challenging and jettison Ariane5 sooner!

##### Scientist
Hmm - yes. I could have deliberately burnt down to min fuel I guess. In fact, because the station orbit is elliptical, I could have spend a bunch more time to match my orbit perfectly before docking, to stop it drifting around on final approach!

#### Tex

Tutorial Publisher
astrosammy said:
Everything done with some minor problems:

It looks like the module in the payload bay was rotated by 180°. The grappling point was facing the payload bay floor, I had to edit the scenario to attach it to the RMS.
Thanks for bringing that to my attention; an over sight on my part. I didn't actually test pulling it out of the shuttle with the shuttle RMS which can't grapple anywhere like URMS. I'll fix that in the next package.

astrosammy said:
The URMS seems to be unable to correctly grapple in the D3D9 client. Should be written into the manual for other D3D9 users.

Launch heading seems to be wrong, again this might be a D3D9 or Shuttle Fleet problem. I'll take a look at that.
OFSS-III is not tested or supported for the D3D9 client. The doc doesn't specifically state this, but I don't use D3D9 for OFSS flights myself, so it's clearly never been tested. The launch heading was set based on what the launch MFD said, so I'm not sure how else to figure that for D3D9. I'm able to get Rinc near 0 with that heading, however it does require some minor adjustments to the heading mid-way through the launch, which I've always had to do with any shuttle launch.

#### Cras

##### Spring of Life!
Donator
I tend to find that you cannot use Launch MFDs heading with Shuttle Fleet. You need to off set it to the north a bit. It is the roll to heads up that messes it all up. But I should add I too fly with D3D9, so that may have something to do with it. As for the modules, I haved used pretty much all of them on my own station, and they all seem to work well in D3D9.

Some little things may need to be accounted for when using D3D9, but on the whole, pretty much everything works with it.


text:
# An element consumes ‘W’ energy in joules for a time period ‘t’ seconds. What is the power of the electrical energy given to the element?

Free Practice With Testbook Mock Tests

## Options:

1. W / t Watt

2. W – t Watt

3. W × t Watt

4.  Wt Watt

### Correct Answer: Option 1 (Solution Below)

This question was previously asked in

Full Mock Test - 3

SSC JE EE Previous Paper 10 (Held on: 10 Dec 2020)

SSC JE EE Previous Paper (Held on: 10 Dec 2020)

## Solution:

Power (P):

The rate of work done is called power. The SI unit of power is the watt (W).

Power (P) = work done (W) / time taken (t)

P = W / t  (J / s)

Also, Joule per second equal to Watt

∴ P = W / t Watt

Energy:

• An object having the capability to do work is said to possess energy.
• The object which does the work loses energy and the object on which the work is done gains energy.
• Its SI unit is Joule.

Electric power is the rate, per unit time, at which electrical energy is transferred by an electric circuit. The unit of power is Watt. It can be expressed as

P = VI

Where V is the voltage

I is the current

Energy is the capacity to do work. Energy is power integrated over time. The units of energy are Joules.

$$E = \mathop \smallint \limits_0^t Pdt$$

If w is the energy in Joules and t is time in seconds, then the power associated with those quantities is:

P = w/q


text: The amount of the new compound is the limit of a function as time approaches infinity. If you're an artilleryman in an army, you want your artillery shells to hit the enemy, or else theirs may hit you and kill you (game over). Differentiation and integration can help us solve many types of real-world problems. For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. The derivative is defined as something which is based on some other thing. Applications of derivatives (in real life!) Derivatives: Real-Life Applications: Up until now, we've dealt with relatively simple equations. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Discuss the applications of partial derivatives in daily life with at least 2 examples. There are some very real applications to calculus that are in the business world and at some level that is the point of this section. You can describe the motion of a body using differential equations. 7. However, it’s a bit simpler. this would be a position function. Fluid mechanics, heat and mass transfer, and electromagnetic theory are all modeled by partial differential equations and all have plenty of real life applications. Powered by Create your own unique website with customizable templates. Speed is a number. Finding the derivative of the outside function may be a bit trickier because it also calls for the chain rule. The inside function for this, ... Click HERE for a real-world example of the chain rule. In Mathematics, the derivative is an expression that gives the rate of change of a function with respect to an independent variable. ). We can now use derivatives of logarithmic and exponential functions to solve various types of problems eg. example 1: consider the maximum height of a ball being thrown. In fact, the notion of a derivative (rates of change in general as well) were formulated based on solving real world problems related to physics and to the sciences. Derivatives are, believe it or not, used a lot. The odometer and the speedometer in the vehicles which tells the driver the speed and distance, generally worked through derivatives to transform the data in miles per hour and distance. It is impossible to provide all types of derivative examples, since there thousands of such derivatives, and which vary in every situation. My example is from the real life situation of war. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. cost, strength, amount of material used in a building, profit, loss, etc. Learn more applications in real life and in maths with examples at CoolGyan The first derivative test; 3. What if we apply Limits of a Functions in Real Life Applications? Some day-to-day applications are described below: To an engineer – The maximum and the minimum values of a function can be used to determine its boundaries in real-life. We have learnt in calculus that when ‘y’ is function of ‘x’, the derivative of y with respect to x i.e. Linear Approximations; 5. If you asked anyone how much money I have they might say "You have a billion dollars." Some examples of speeds are 55 mph, 100 km/hr, and 33 ft/sec. The second derivative test; 4. Peyam Ryan Tabrizian Friday, October 11th, 2013 Chemistry Problem 1 [That should look familiar!] Derivatives and their real world applications. Speed for example is position changing with respect to time, acceleration is the second derivative. this would be found when the derivative equals zero. The Mean Value Theorem; 7 Integration. Derivatives describe one thing changing with respect to something else. Again you might say "So what?" Here’s some examples: We could have a chemical reaction in a beaker start with two chemicals that form a new compound over time. In the business we can find the profit and loss by using the derivatives, through converting the data into graph. Related Rates; 3. We are thankful to be welcome on these lands in friendship. Because we say that the 48 cents is such From experiments in physics we know that the acceleration due to gravity of a particle near the Earth's surface is about $-9.8 \frac{m}{s^2}$. Now, we shouldn’t walk out of the previous two examples with the idea that the only applications to business are just applications we’ve already looked at but with a business “twist” to them. Derivatives: Real-Life Applications: When you are multiplying one function (f(x)) by another (g(x)), this is when you use the product rule: ... Click HERE to see an example of this rule applied in real life. 1. This becomes very useful when solving various problems that are related to rates of change in applied, real-world, situations. At time t 0, a beaker contains 2 grams of salt dissolved in 5 ounces of water. Derivatives in mudding through life: If you understand what a derivative is, you can state certain things very clearly and succinctly as well as avoiding errors in your thinking. In this article, we will discuss in detail, its definition along with the real life utility. Example 1 Find the rate of change of the area of a circle per second with respect to its radius r when r = 5 cm. Optimization; 2. Let's put this in a real world example. Differentiation and integration considered by all scientists throughout the ages as one of the best sciences that guided the mind of man over all times The fields of the use of calculus are very wide. Asymptotes and Other Things to Look For; 6 Applications of the Derivative. APPLICATION OF DERIVATIVES 195 Thus, the rate of change of y with respect to x can be calculated using the rate of change of y and that of x both with respect to t. Let us consider some examples. by M. Bourne. (note: it follows a parabolic path due to earth's gravitational pull) h(t) = -16t^2 + 32t - 5. h(t) is the height at time t. example 2: consider the maximum revenue of a product being sold. I was wondering whether the laws of derivatives (Product rule, chain rule, quotient rule, power rule, trig laws, implicit differentiation, trigonometric differentiation) has any real life application or if they are simply math laws to … Why is that? 2. We differentiate one of the variables while keeping the other variable fixed. REAL Life Application “Education is not preparation for life; ... For example, in social studies our 3rd and 4th graders learned about the three branches of government and how the Constitution provides for a balance of power. Derivative Examples. 1. Solution 2The area A of a circle with radius r is given by A = πr. In Economics and commerce we come across many such variables where one variable is a function of … ... it’s just 5. Derivatives are financial instruments like equity and bonds, in the form of a contract that derives its value from the performance and price movement of the underlying entity. Plenty. Below examples of a derivative illustrate of the most common derivatives. Applications: Derivatives of Logarithmic and Exponential Functions. As with any advanced math, whether you use it in your "real life" or not depends on whether you work in the engineering field or not. The application of derivatives exists in Mathematics, Science, and … In the Real World. Now let’s get started, at first we will try to understand the concepts of derivative and differentiation. ... possible to mix the words "speed" and "velocity," especially since we almost never use the word "velocity" in everyday life. Real life application of derivatives. You can use them wherever you want to describe a dynamic system. In this chapter we will cover many of the major applications of derivatives. Applications of Derivatives in Economics and Commerce APPLICATION OF DERIVATIVES AND CALCULUS IN COMMERCE AND ECONOMICS. Newton's Method; 4. Real life Applications of Derivatives. In applications of derivatives class 12 chapter 6, we will study different applications of derivatives in various fields like Science, Engineering, and many other fields.In chapter 6, we are going to learn how to determine the rate of change of quantity, finding the equations of tangents, finding turning points on the graphs for various functions, maxima and minima and so on. in the fields of earthquake measurement, electronics, air resistance on moving objects etc. Concavity and inflection points; 5. Derivatives describe the rate of change of quantities. Two examples; 2. In today’s day and age, the application of these relations between natural and industrial processes is used to optimize the quality of human life in both professional and personal endeavors. The derivative of the momentum of a body equals the force applied to the body; rearranging this derivative statement leads to the famous F = ma equation associated with Newton’s second law of motion. Applications of derivatives for calculation of maxima and minima, tangent and normal, rate of change, etc. At time t 0, water is being added at 10 ounces/min and salt is being added at 3 grams/min. Also learn how to apply derivatives to approximate function values and find limits using L’Hôpital’s rule. We use the derivative to determine the maximum and minimum values of particular functions (e.g. They have many real life applications. Topic 17-Derivatives in the Real World Calculus BC. real world applications of matrices and partial differentiation 1. submitted to: dr. sona raj submitted by: anugyaa shrivastava (k12986) btech cs 2nd sem sanjay singh (k12336) btech ce 2nd sem nishant yadav (k12119) btech ce 2nd sem real world applications of matrices and partial differentiation The technique of differentiating multivariable function is known as Partial Differentiation and the resulting derivative is called Partial Derivative. Let's say I have \$1,000,000,000.48 (1 billion dollars and 48 cents). (dy/dx) measures the rate of change of y with respect to x. Examples of Derivatives (With Excel Template) Let’s try to understand derivatives through the below examples. How fast is the concentration of salt For example, a word that is coming up a lot in the healthcare debate is "bending the curve". The majority of his life (91%) is spent learning in the real world. Differential Equation Example - Duration: 9:44. … Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Now let ’ s get started, at first we will try to understand derivatives through the below of... Chain rule world example we differentiate one of the variables while keeping the other variable fixed look familiar ]! In Mathematics, the derivative is defined as something which is based on some thing... Dynamic system is the concentration of salt dissolved in 5 ounces of water in. 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text:
# zbMATH — the first resource for mathematics

##### Examples
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Characterizations of nonlinear Lie derivations of $B(X)$. (English) Zbl 1271.47029
Summary: Let $X$ be an infinite dimensional Banach space and $\Phi : B(X) \to B(X)$ be a nonlinear Lie derivation. Then $\Phi$ is the form $\delta + \tau$, where $\delta$ is an additive derivation of $B(X)$ and $\tau$ is a map from $B(X)$ into its center $Z_{B(X)}$, which maps commutators into the zero.

##### MSC:
 47B47 Commutators, derivations, elementary operators, etc.
Full Text:
##### References:
 [1] J. Alaminos, M. Mathieu, and A. R. Villena, “Symmetric amenability and Lie derivations,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 137, no. 2, pp. 433-439, 2004. · Zbl 1063.46033 · doi:10.1017/S0305004104007637 [2] B. E. Johnson, “Symmetric amenability and the nonexistence of Lie and Jordan derivations,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 120, no. 3, pp. 455-473, 1996. · Zbl 0888.46024 · doi:10.1017/S0305004100075010 [3] F. Y. Lu, “Lie derivations of I-subspace lattice algebras,” Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2581-2590, 2007. · Zbl 1116.47060 · doi:10.1090/S0002-9939-07-08767-9 [4] F. Y. Lu, “Lie derivations of certain CSL algebras,” Israel Journal of Mathematics, vol. 155, pp. 149-156, 2006. · Zbl 1130.47055 · doi:10.1007/BF02773953 [5] M. Mathieu and A. R. Villena, “The structure of Lie derivations on C*-algebras,” Journal of Functional Analysis, vol. 202, no. 2, pp. 504-525, 2003. · Zbl 1032.46086 · doi:10.1016/S0022-1236(03)00077-6 [6] P. \vSemrl, “Additive derivations of some operator algebras,” Illinois Journal of Mathematics, vol. 35, no. 2, pp. 234-240, 1991. · Zbl 0705.46035 [7] M. I. Berenguer and A. R. Villena, “Continuity of Lie derivations on Banach algebras,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 41, no. 3, pp. 625-630, 1998. · Zbl 0957.46035 · doi:10.1017/S0013091500019933 [8] D. Benkovi\vc, “Lie derivations on triangular matrices,” Linear and Multilinear Algebra, vol. 55, no. 6, pp. 619-626, 2007. · Zbl 1129.16024 · doi:10.1080/03081080701379107 [9] W.-S. Cheung, “Lie derivations of triangular algebras,” Linear and Multilinear Algebra, vol. 51, no. 3, pp. 299-310, 2003. · Zbl 1060.16033 · doi:10.1080/0308108031000096993 [10] A. R. Villena, “Lie derivations on Banach algebras,” Journal of Algebra, vol. 226, no. 1, pp. 390-409, 2000. · Zbl 0957.46032 · doi:10.1006/jabr.1999.8193 [11] F. Y. Lu and W. Jing, “Characterizations of Lie derivations of B(X),” Linear Algebra and its Applications, vol. 432, no. 1, pp. 89-99, 2010. · Zbl 1188.47029 · doi:10.1016/j.laa.2009.07.026 [12] W. Y. Yu and J.H. Zhang, “Nonlinear Lie derivations of triangular algebras,” Linear Algebra and its Applications, vol. 432, no. 11, pp. 2953-2960, 2010. · Zbl 1193.16030 · doi:10.1016/j.laa.2009.12.042 [13] P. S. Ji and W. Q. Qi, “Characterizations of Lie derivations of triangular algebras,” Linear Algebra and its Applications, vol. 435, no. 5, pp. 1137-1146, 2011. · Zbl 1226.16026 · doi:10.1016/j.laa.2011.02.048 [14] Z. F. Bai and S. P. Du, “The structure of nonlinear Lie derivation on von Neumann algebras,” Linear Algebra and its Applications, vol. 436, no. 7, pp. 2701-2708, 2012. · Zbl 1272.47046 · doi:10.1016/j.laa.2011.11.009 [15] D. G. Han, “Continuity and linearity of additive derivations of nest algebras on Banach spaces,” Chinese Annals of Mathematics B, vol. 17, no. 2, pp. 227-236, 1996. · Zbl 0856.47028


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# Chordal graph

A cycle (black) with two chords (green). As for this part, the graph is chordal. However, removing one green edge would result in a non-chordal graph. Indeed, the other green edge with three black edges would form a cycle of length four with no chords.

In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs[1] or triangulated graphs.[2]

Chordal graphs are a subset of the perfect graphs. They may be recognized in linear time, and several problems that are hard on other classes of graphs such as graph coloring may be solved in polynomial time when the input is chordal. The treewidth of an arbitrary graph may be characterized by the size of the cliques in the chordal graphs that contain it.

## Perfect elimination and efficient recognition

A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex v, v and the neighbors of v that occur after v in the order form a clique. A graph is chordal if and only if it has a perfect elimination ordering.[3]

Rose, Lueker & Tarjan (1976) (see also Habib et al. 2000) show that a perfect elimination ordering of a chordal graph may be found efficiently using an algorithm known as lexicographic breadth-first search. This algorithm maintains a partition of the vertices of the graph into a sequence of sets; initially this sequence consists of a single set with all vertices. The algorithm repeatedly chooses a vertex v from the earliest set in the sequence that contains previously unchosen vertices, and splits each set S of the sequence into two smaller subsets, the first consisting of the neighbors of v in S and the second consisting of the non-neighbors. When this splitting process has been performed for all vertices, the sequence of sets has one vertex per set, in the reverse of a perfect elimination ordering.

Since both this lexicographic breadth first search process and the process of testing whether an ordering is a perfect elimination ordering can be performed in linear time, it is possible to recognize chordal graphs in linear time. The graph sandwich problem on chordal graphs is NP-complete[4] whereas the probe graph problem on chordal graphs has polynomial-time complexity.[5]

The set of all perfect elimination orderings of a chordal graph can be modeled as the basic words of an antimatroid; Chandran et al. (2003) use this connection to antimatroids as part of an algorithm for efficiently listing all perfect elimination orderings of a given chordal graph.

## Maximal cliques and graph coloring

Another application of perfect elimination orderings is finding a maximum clique of a chordal graph in polynomial-time, while the same problem for general graphs is NP-complete. More generally, a chordal graph can have only linearly many maximal cliques, while non-chordal graphs may have exponentially many. To list all maximal cliques of a chordal graph, simply find a perfect elimination ordering, form a clique for each vertex v together with the neighbors of v that are later than v in the perfect elimination ordering, and test whether each of the resulting cliques is maximal.

The clique graphs of chordal graphs are the dually chordal graphs.[6]

The largest maximal clique is a maximum clique, and, as chordal graphs are perfect, the size of this clique equals the chromatic number of the chordal graph. Chordal graphs are perfectly orderable: an optimal coloring may be obtained by applying a greedy coloring algorithm to the vertices in the reverse of a perfect elimination ordering.[7]

The chromatic polynomial of a chordal graph is easy to compute. Find a perfect elimination ordering v1, v2, …, vn. Let Ni equal the number of neighbors of vi that come after vi in that ordering. For instance, Nn = 0. The chromatic polynomial equals ${\displaystyle (x-N_{1})(x-N_{2})\cdots (x-N_{n}).}$ (The last factor is simply x, so x divides the polynomial, as it should.) Clearly, this computation depends on chordality.[8]

## Minimal separators

In any graph, a vertex separator is a set of vertices the removal of which leaves the remaining graph disconnected; a separator is minimal if it has no proper subset that is also a separator. According to a theorem of Dirac (1961), chordal graphs are graphs in which each minimal separator is a clique; Dirac used this characterization to prove that chordal graphs are perfect.

The family of chordal graphs may be defined inductively as the graphs whose vertices can be divided into three nonempty subsets A, S, and B, such that ${\displaystyle A\cup S}$ and ${\displaystyle S\cup B}$ both form chordal induced subgraphs, S is a clique, and there are no edges from A to B. That is, they are the graphs that have a recursive decomposition by clique separators into smaller subgraphs. For this reason, chordal graphs have also sometimes been called decomposable graphs.[9]

## Intersection graphs of subtrees

A chordal graph with eight vertices, represented as the intersection graph of eight subtrees of a six-node tree.

An alternative characterization of chordal graphs, due to Gavril (1974), involves trees and their subtrees.

From a collection of subtrees of a tree, one can define a subtree graph, which is an intersection graph that has one vertex per subtree and an edge connecting any two subtrees that overlap in one or more nodes of the tree. Gavril showed that the subtree graphs are exactly the chordal graphs.

A representation of a chordal graph as an intersection of subtrees forms a tree decomposition of the graph, with treewidth equal to one less than the size of the largest clique in the graph; the tree decomposition of any graph G can be viewed in this way as a representation of G as a subgraph of a chordal graph. The tree decomposition of a graph is also the junction tree of the junction tree algorithm.

## Relation to other graph classes

### Subclasses

Interval graphs are the intersection graphs of subtrees of path graphs, a special case of trees. Therefore, they are a subfamily of chordal graphs.

Split graphs are graphs that are both chordal and the complements of chordal graphs. Bender, Richmond & Wormald (1985) showed that, in the limit as n goes to infinity, the fraction of n-vertex chordal graphs that are split approaches one.

Ptolemaic graphs are graphs that are both chordal and distance hereditary. Quasi-threshold graphs are a subclass of Ptolemaic graphs that are both chordal and cographs. Block graphs are another subclass of Ptolemaic graphs in which every two maximal cliques have at most one vertex in common. A special type is windmill graphs, where the common vertex is the same for every pair of cliques.

Strongly chordal graphs are graphs that are chordal and contain no n-sun (for n ≥ 3) as an induced subgraph. Here an n-sun is an n-vertex chordal graph G together with a collection of n degree-two vertices, adjacent to the edges of a Hamiltonian cycle in G.

K-trees are chordal graphs in which all maximal cliques and all maximal clique separators have the same size.[10] Apollonian networks are chordal maximal planar graphs, or equivalently planar 3-trees.[10] Maximal outerplanar graphs are a subclass of 2-trees, and therefore are also chordal.

### Superclasses

Chordal graphs are a subclass of the well known perfect graphs. Other superclasses of chordal graphs include weakly chordal graphs, cop-win graphs, odd-hole-free graphs, even-hole-free graphs, and Meyniel graphs. Chordal graphs are precisely the graphs that are both odd-hole-free and even-hole-free (see holes in graph theory).

Every chordal graph is a strangulated graph, a graph in which every peripheral cycle is a triangle, because peripheral cycles are a special case of induced cycles. Strangulated graphs are graphs that can be formed by clique-sums of chordal graphs and maximal planar graphs. Therefore, strangulated graphs include maximal planar graphs.[11]

## Chordal completions and treewidth

If G is an arbitrary graph, a chordal completion of G (or minimum fill-in) is a chordal graph that contains G as a subgraph. The parameterized version of minimum fill-in is fixed parameter tractable, and moreover, is solvable in parameterized subexponential time.[12][13] The treewidth of G is one less than the number of vertices in a maximum clique of a chordal completion chosen to minimize this clique size. The k-trees are the graphs to which no additional edges can be added without increasing their treewidth to a number larger than k. Therefore, the k-trees are their own chordal completions, and form a subclass of the chordal graphs. Chordal completions can also be used to characterize several other related classes of graphs.[14]

## Notes

1. ^
2. ^
3. ^
4. ^
5. ^
6. ^
7. ^
8. ^ For instance, Agnarsson (2003), Remark 2.5, calls this method well known.
9. ^ Peter Bartlett. "Undirected Graphical Models: Chordal Graphs, Decomposable Graphs, Junction Trees, and Factorizations" (PDF).
10. ^ a b
11. ^
12. ^
13. ^
14. ^


text:
# Lower bounds on the size of general Schrödinger-cat states from experimental data

 Authors: Florian Fröwis Journal of Physics A: Mathematical and Theoretical 50, 114003 (2017) http://dx.doi.org/10.1088/1751-8121/aa5a92 Experimental progress with meso- and macroscopic quantum states (i.e. general Schrödinger-cat states) was recently accompanied by theoretical proposals on how to measure the merit of these efforts. So far, experiment and theory have been disconnected as theoretical analysis of actual experimental data was missing. Here, we consider a proposal for macroscopic quantum states that measures the extent of quantum coherence present in the system. For this, the quantum Fisher information is used. We calculate lower bounds from real experimental data. The results are expressed as an ‘effective size’, that is, relative to ‘classical’ reference states. We find remarkable numbers of up to 70 in photonic and atomic systems. froewis2017a.pdf

# BibTeX Source

@article{1751-8121-50-11-114003,
author={Florian Fröwis},
title={Lower bounds on the size of general Schrödinger-cat states from experimental data},
journal={Journal of Physics A: Mathematical and Theoretical},
volume={50},
number={11},
pages={114003},
url={http://stacks.iop.org/1751-8121/50/i=11/a=114003},
year={2017},
doi = {10.1088/1751-8121/aa5a92},
abstract={Experimental progress with meso- and macroscopic quantum states (i.e. general Schrödinger-cat states) was recently accompanied by theoretical proposals on how to measure the merit of these efforts. So far, experiment and theory have been disconnected as theoretical analysis of actual experimental data was missing. Here, we consider a proposal for macroscopic quantum states that measures the extent of quantum coherence present in the system. For this, the quantum Fisher information is used. We calculate lower bounds from real experimental data. The results are expressed as an ‘effective size’, that is, relative to ‘classical’ reference states. We find remarkable numbers of up to 70 in photonic and atomic systems.}
}