text:
# LaTeXTools: New Keybindings!

Heads-up: the keybindings for the LaTeXTools plugin have changed. Make sure to read the README file for details: go to https://github.com/SublimeText/LaTeXTools and scroll down (or click the “Read More” link near the project description).

Why change things? When I started working on this plugin, my objective was to emulate the functionality of the LaTeX bundle for TextMate. In particular, I tried to use the same keyboard shortcuts. However, this slowly led to an accumulation of problems. One was that, as I added features to the plugin, it became harder and harder to assign them to shortcuts that were not already used by Sublime Text 2 itself. The other problem is actually due to ST2’s greatest feature: it’s a cross-platform editor, and the plugin must also work equally well on Windows and Mac, with full Linux support also coming soon (yes!). However, different OS’s have slightly different conventions and expectations about what certain keys do. Bottom line: things were getting unmanageable.

Enter the new keyboard shortcut scheme. It’s actually very simple:

1. The build command is still bound to ctrl-b on Windows and Linux, and cmd-b on OS X. For the time being, the Goto Anything functionality is also still bound to ctrl-r / cmd-r.
2. All other plugin commands use ctrl-l or cmd-l followed by another key or key combination. That’s “ell” for LaTeX.

Example: to do a forward search (i.e. jump to the point int he PDF file corresponding to the current cursor position) on Windows, enter Ctrl-l followed by j. This is abbreviated as ctrl-l,j (the same convention used by Sublime Text). On OS X, this is cmd-l,j. In fact, from now on, I shall write C-l to mean either ctrl-l or cmd-l, depending on the platform.

I have tried to use mostly C-l plus a single keystroke, with reasonable mnemonics. For instance: “j” for jump to PDF, “c” for LaTeX command based on the current word, etc. Wrapping commands use the C- key twice: so, wrap in bold is C-l,C-b, etc. Reference and cite completion uses C-l,Ctrl-space (yes, even on OS X, that’s Command-ell, Control-space), but you can also use C-l,x (for cross-reference). Finally, to toggle the focus from/to the PDF upon compilation, there is a three-key sequence: C-l,t,f. I’ll try to reserve these for toggles and other less frequently used stuff.

Now, on ST2, the C-l shortcut is bound to “Extend selection to line”. That’s reassigned to C-l,C-l. The nice thing is that no other standard ST2 keyboard shortcut is affected.

As an additional benefit, these bindings are consistent with the “nice” (or “compact”) on-screen keyboard in Windows 8, which only has a Ctrl key (no Windows key, no Alt key).

Enjoy!

### 52 responses to “LaTeXTools: New Keybindings!”

1. Satyen

Thanks for continuing to make your wonderful plugin even better!!

So I was having trouble with completion in \cite{}, and on digging into the error messages I figured that the problem was some entries in my bib file with leading spaces before @article etc. Also some entries in my bib file were commented out; the plugin was still picking up the title and author list from those entries but missed picking up the keyword because the line didn’t start with @. Also, if the author list was split over multiple lines, sometimes the plugin wouldn’t pick it all up.

Anyway, I realized that an easy fix to this was to use the .bbl files that bibtex generates, instead of using the .bib files directly. The advantage is that .bbl files are much better structured than a typical .bib file (which is usually generated by copy-pasting bibtex entries from the web). But bibtex is already smart enough to get around formatting issues and produce a nice structured output in the form of a .bbl file. Also, the .bbl file aggregates entries from all the included .bib files, and *only* those that are actually referenced in the tex file, so you can avoid the recursive search for .bib files and superfluous bibliography entries to boot.

The only catch is that if you haven’t generated the .bbl file yet, or if you have a .bbl file but it doesn’t yet have your keyword, then the completion won’t work of course. A simple fix is to continue processing bib files *after* the bbl file, discarding all keywords previously seen.

With a quick and dirty change to the latex_cite_completions.py I was able to use a .bbl file in the completion, making for a prettier output. I wanted to bring this idea for your consideration, and I can mail you my code if you like (although I haven’t implemented the fix above).

2. robin

Hallo,
how can I view pdf (before ”ctrl+alt+v” ) on OSX? now is it ”super+L” or? I hit ”cmd+L”. nothing happens. and what is the ”super” key?

• cmd+L, v to view Cmd+L, j for forward search

• robin

i tried ”cmd+L, v” but , nothing happens

• If you are on Linux, can you try it now? There was a bug that I fixed in the last version. I also fixed things so file names with spaces are supported for forward/inverse search (again, in Linux). Things should be fine on other platforms.

• robin

I am using OS X .Still can not view the PDF with command ”cmd+L, v”. Now i just open the PDF with Skim at first, it will automatically update the pdf each time when i build again.

• Does “jump to pdf” (Cmd+L,J) work?

• BTW, sorry to say this, but Cmd+L,V works for me (OS X 10.7.5)

On Tue, Sep 25, 2012 at 7:41 AM, Marciano Siniscalchi wrote: > Does “jump to pdf” (Cmd+L,J) work? >

• robin

does not work ”Cmd+L, j ” on my 10.8.2

• Just to be clear–“cmd+L” is not “cmd-shift-L”. Can you try again?

3. Andrei

Hello! When I try to compile tex file (with latextools in sublime text 2 on windows 7 russian and miktex) with a cyrillic chars in path, nothing happends. In console i see “[u’texify’, u’-b’, u’-p’, u’–tex-option=”–synctex=1″‘, u’E:\\Documents\\sublime\\\u0436\u0431\u043a.sublime\\jbk.tex’]”, where “\\u0436\u0431\u043a” is corrupted cyrillic chars. I dont know where i should paste this issue and paste it here, excuse me.

• Thanks for the report. Unfortunately, unicode chars in the file path will cause trouble. I’m not sure there’s much I can do; I have another similar bug report on OS X. Question: can you run texify from the command line on that same path?

• Andrei

Yes, I can run texify from the command line on that same path. It is works correctly.

4. Anil

First off, thanks so much for taking the time to build this package for Sublime text. It’s essential for my workflow. My question is whether you have a cheatsheet that lists out the keyboard shortcuts for your package in one place.

• That’s a good idea. I’ll put it on my TODO list (sadly still pretty long!)

5. Anil

I just tried using the new keybindings for the first time (on my Mac). And I realized that for me, Cmd+# is a shortcut to jump to the file in the #th tab of the current window. Do you have to disable/override that manually?

I can send a screenshot of the appropriate menu item if that helps.

• IT’s Cmd-L (“ell”), not Cmd-1 (as in “one”).

• Anil

Talk about blind. Thanks!

6. Hello Marciano,

Thank you for all the wonderful work on this plugin. I use it on OS X, and I’ve run into an annoyance that you may or may not be able to help with.

The way I work with a .tex file is that I’ll have Skim open on one side of the screen and ST2 open on the other. I’ll usually have the PDF zoomed in so that the text width fills the window.

If I am working on an equation and hit the build command, the plugin will do a forward search and move the PDF in the Skim window so that the beginning of the equation is on the left side of the window. Then in order to get the PDF centered again the way I want it, I have to move it back.

Is there a way to disable the forward search when doing a build? I still want to have the capability to do a forward search, but I only want to do it when I actually want it. I believe this is how TeXShop works.

Thanks!

• You’re the second person to ask, so I guess I’ll have to implement it 🙂

BTW, I personally also *hate* that Skim behavior. Basically, when you do a forward search, it does not preserve the horizontal position (unless you have zoomed out so that you see the entire page, margin to margin). I think Yap on Windows used to not do that at all… unfortunately, it’s not something I can fix, except by not doing a forward search of course.

• It’s good to hear that I’m not the only one with this problem. A setting in LaTeXTools Preferences.sublime-settings to turn off forward search after building would be nice. 🙂

• Sorry for the double reply — it’s now in. See the README. Hope you like it 🙂

7. @ntmoe the latest commits added this. See the README. Hope you like it 🙂

8. I tried it out this afternoon. I really like the new option! The plug-in now works as I had wanted. Thanks so much!

9. Hi, I am a Mac user. Loved this. but my skim doesn’t launch when compiling (cmd+b). may I know how I can make it work?

10. Gabriel Droulers

Hi, I’m considering changing from texshop to Sublime text 2 with your tool for Latex. However, I can’t seem to find how to compile the bibtex data in my latex document. Does latexmk take care of this? I use a single .bib file placed in my texmf tree… is there a way to make sublime compiler see it?

Also, I can’t seem to do backward search with cmd+shift+click from skim, is this normal?

11. Xiaowei

Hi, first of all, thanks a lot for your efforts on LaTexTools!! I’m addicted to ST2 immediately for my TeX editing. However, after several rounds of googling, I still couldn’t figure out how to set up the building system correctly to enable bibtex. I used to do pdflatex + bibtex + pdflatex + pdflatex to get the bib file working. How should I modify the LaTeX.sublime-build file?

• You don’t need to. Latexmk should take care of it automatically. You do need to have a bibliography command in your source though.

• Xiaowei

Thanks! I’m not quite sure about what you mean by “have a bibliography command in the source”. But this is what I do. I add an argument “-bibtex” to the LaTeX.sublime-build file, i.e. change the corresponding session to

“cmd”: [“latexmk”, “-bibtex”,
“-e”, “\\\$pdflatex = ‘pdflatex %O -synctex=1 %S'”,
“-silent”,
“-f”, “-pdf”],

12. Gary Church

Hello.
I’m new to Sublime Text 2 and to your LaTeXTools plugin.
Since I’m a mathematics professor, most of my editing involves
mathematical expressions. One thing that I frustrates me is
entering nested fractions like “\frac{3 + \frac{2}{x}}{\frac{1}{x} – \frac{x}{3}}”
I can’t seem to get this to work using Sublime Text and your plugin. The outermost
\frac expands fine but I can’t get the nested \frac’s to expand. What I end up getting
is “\frac{3 + fra}{fra}”. Am I doing something wrong?
Thanks,
Gary

13. Shut the fuck up your plugin is awesome! Dear god, I had no idea forward and inverse search were even possible. And I also get \ref and \cite lookups. I love your plugin man. Thanks!!

14. za

Hi !
I’m very impressed by your work and I love to use it.
Do you know if there is any option to toggle in order to not bring up the console if the build succeed without error? I’m pretty new to ST2 and your extesion.

Anyway, thanks for your work!

15. moses

First up thanks for sharing all this great work, an education. I am trying to figure out what I need to do to get this step working (!!!apologies if this is the wrong place to post this):

-compile using hotkey cmd+b.
-I want the Skim editor to open here and jump to the pdf spot of the corresponding cursor position in ST2.
-when I hit cmd+b, Marked opens, yes I had previously installed Marked prior to installing Skim. I am trying to figure out how to disable Marked such that I can use cmd+b to open skim.

When I check the installed packages in ST2, I have Markdown and Markdown (Pandoc). I have tried to Google my way to a solution but as yet I have not been successful. If you have any guidance, please let me know.
Further, when I check system preferences, keyboard shortcuts, see http://www.box.com/shared/aynp9w7qhyc7btb6lh6m, it appears I will have to configure the shortcuts. Any suggestions on how to import the configurations discussed above. Many Thanks.
Best,
Moses

• THis is really odd! The plugin invokes Skim directly, so whatever you have set as default PDF handler should not matter at all…

16. moses

Thanks for replying. Skim is not in the Build System under Tools (Marked is). Does installling your plugin put Skim there, or is there a step I have missed? Thanks

17. Gege

Hi and thanks you again for your great work ! I have question : is it possible to specify (for one time only) to compile with latex instead of pdftex … I have some pstrick figure in my document ….

• Gege

does anyone have any idea about that ? ty 🙂

18. C.J.

I love your plugin so far. However, I am having an issue with forward search on Mac OS 10.7 changing the focus to Skim. I thought focus should stay in the editor? Not sure if it matters, but I have the “keep_focus” preference set to true.
Thanks for any help you can provide.

• Hmmm… That should do it. There is a key combo that displays the current toggle settings in the status line (sorry I don’t remember what it is – please check the README). See if it says Focus-Editor. It may be that your settings are not being read in.

19. C.J.

Hmm. When I first open a .tex file, and check the toggle settings, it reports that “Keep focus: pref True toggle undefined”. The focus does stay in ST2 when I do a build though, just not with a forward search.

Could this be caused by Skim? Whenever I use the forward search, Skim is triggered and a popup with a status bar says Reloading Document… while it jumps to the desired spot in the pdf.

• Aaah… I see. I think that with a forward search the plugin always gives the focus to Skim. My thinking was, if the user only wants to redisplay, C-B is enough, so if s/he uses a forward search presumably s/he wants to look at the PDF. I’m not sure I want to change this behavior, because some users may find the change objectionable.

As for Skim, you need to disable the “Check for file changes” toggle in the Sync tab (Preferences).

20. C.J.

That’s understandable. The readme does say that focus should stay in ST2 when doing a forward search with c-l,j. So that’s why I was expecting that behavior.

I do have that setting in Skim disabled. Thanks again for the help.

21. Gege

I’m sorry to insist but I would really appreciate to know if there is a way from Latextools/ST2 for compiling with latex instead of pdflatex ? where can I specifify the compiler command ?
thanks in advance for your help and congrats again for your impressive work with latextools

• You can personalize the LaTeX.sublime-build file. First, copy it to ..Packages/User, otherwise it will get clobbered the next time you update LaTeXTools. Next, look at the relevant section (osx, windows or linux) and change the cmd setting to fit your needs.

22. Axel Holene

Hi! Thank you very much for this lovely plug-in! I have two questions:
1. Is there a way to have the ST2 widget/command window close when no error is found upon compilation?
2. Is it possible to have the compilation output in a Terminal.app window instead of in the ST2 widget/command window?
In advance, thanks! =)

• Hi,

1 is not currently possible, but could be added. I’ll consider this a feature request! 2 is quite a bit harder, especially because I’d really like to keep functionality the same across different platforms. So, it’s unlikely to happen.

Thanks for your feedback, M

23. Sep

Hello,

Thanks for the plugin. I’ve recently installed sublime 3 and everything works perfectly. However when I hit Ctrl-b to compile the tex file, I receive this error:

“Unable to find target command: make_pdf”

Googled it and figured someone else is also having the same problem. Can you help fixing it please?

Thanks,
Sep

• Hmmm… I”m surprised it works, because I haven’t ported it to st3 yet…

24. Tomasz

Hi Marciano, Have you ever used github as revision control for latex documents with multiple coauthors?

• Not really. I heard some people do. Have you?

• Tomasz

No, but I’m looking for a good solution. Dropbox is enough when it’s just you and another person. With 3 or more people this becomes a coordination problem and a lot of the discussion ends up in the email chain, which is a pain.

25. Miles

Hi,
On Windows 7 crtl+b works a treat for me but ctrl+l does not. ctrl+l performs “Expand Selection to Line” which is a standard Sublime 2 command that I use all the time when munging text. Is there a way to change the “primer” keystrokes to something else that isn’t already a standard sublime 2 shortcut?

Kind Regards,
Miles


text:
Free Version
Moderate

# Volume and Pressure Table

APPH12-G8E9H8

A group of students study the an ideal gas as it changes pressure through compression. The following data table shows the gradual values of pressure and volume at ten-second intervals. The pressure data have an uncertainty of $\pm 0.01\text{ atm}$. The volume data have uncertainty of $\pm 0.05\text{ ml}$.

Time (s) Volume (mL) Pressure (atm)
0 12.2 2.04
10 15.1 1.65
20 19.3 1.29
30 24.3 1.01
40 31.0 0.80

What type of thermodynamic process is shown by these data?

A

Isothermal.

B

Isobaric.

C

Isometric (isochoric).

D

format: plain


text:
Source term linearisation and Under-relaxation

as I learned from the awesome book1

If properly handled, source term improves solution stability. Consider a discretised conservation equation for a general variable \phi with source in element C written below:

$$a_{C}\bold \phi_{C} + \sum_{F} a_{F}\phi_{F} = Q_{C}^{\phi}V_{C}$$

If the RHS is large compared to the rest of the terms, the rate of convergence will be affected. In such cases, the rate of convergence can be improved by source term linearisation.

Source term linearisation

Here, Q_{C} is expanded using Taylor series up to first order.

$$Q_{C}^{\phi} = Q_{C}^{\phi^{*}}+\Big(\frac{\partial Q^{\phi} }{\partial \phi}\Big)_{C}^{*} (\phi_{C}-\phi_{C}^{*})$$

Hence the RHS of (1) can be written as

\begin{split}
Q_{C}^{\phi}V_{C} &=  Q_{C}^{\phi^{*}}V_{C}+\Big(\frac{\partial Q^{\phi} }{\partial \phi}\Big)_{C}^{*} (\phi_{C}-\phi_{C}^{*})V_{C}\\
&= (\frac{\partial Q^{\phi} }{\partial \phi}\Big)_{C}^{*} V_{C}\phi_{C} +\Big[ Q_{C}^{\phi^{*}} -  (\frac{\partial Q^{\phi} }{\partial \phi}\Big)_{C}^{*} \phi_{C}^{*} \Big]V_{C}\\
&= FluxC_{C} \phi_{C} + FluxV_{C}
\end{split}

First term in the RHS is treated as implicit and second as explicit. Substituting for RHS in (1)

$$[a_{C}-FluxC_{C}]\bold \phi_{C} + \sum_{F} a_{F}\phi_{F} = FluxV_{C}$$

FluxC_{C}should be -ve, else the system will loose diagonal dominance and Scarborough criterion shall not be fulfilled. This shall result in divergence. Note that, still a_{C} = -(a_{E}+a_{W}+a_{S}+a_{N}) pg # 215 , Moukalled et. al

Consider the case of radiation heat transfer. Q_{C} = A (T_{\inf}^{4} - T_{C}^{4}). If we take FluxC_{C}= 0 and FluxV_{C} = A (T_{\inf}^{4} - T_{C}^{4}) , for T_{C} > T_{\inf}, FluxV_{C} will be negative which may result in negative T_{C} value during iteration. If we employ source term linearisation, then FluxC_{C} = -4AT_{C}^{3} V_{C} and FluxV_{C} = A (T_{\inf}^{4} + 3T_{C}^{4}) V_{C} which is perfect because of negative FluxC_{C} and positive FluxV_{C}.

Patankar’s implicit relaxation

Consider a conservation equation of the following form

$$a_{C}\bold \phi_{C} + \sum_{F} a_{F}\phi_{F} = b_{C}$$

Adding an subtracting previous value of variable,

\begin{split}
\phi_{C} &=  \frac{b_{C} - \sum a_{F}\phi_{F}}{a_{C}}\\
&=\phi_{C}^{*} + \frac{b_{C} - \sum a_{F}\phi_{F}}{a_{C}} - \phi_{C}^{*} \\
\end{split}

In order to relax, we apply \lambda to the second term in RHS , (which is the difference with previous value)

\phi_{C} =\phi_{C}^{*} + \lambda  \Big(\frac{b_{C} - \sum a_{F}\phi_{F}}{a_{C}} - \phi_{C}^{*} \Big)\\

rearranging,

\frac{a_{C}}{\lambda}\phi_{C}+ \sum a_{F}\phi_{F} = b_{C}+ a_{C}( \frac{1}{\lambda}-1)\phi_{C}^{*}

which can be written in the original form as (4) by reconsidering,

\begin{split}
\frac{a_{C}}{\lambda} &\rightarrow a_{C}\\ b_{C}+ a_{C}( \frac{1}{\lambda}-1)\phi_{C}^{*}  &\rightarrow b_{C}
\end{split}

It is clear that this method improves the diagonal dominance if \lambda <1

E-Factor method

In this method (4) is written in the following way:

\begin{split}
a_{C}\phi_{C} &=  b_{C} - \sum a_{F}\phi_{F}\\
&=\lambda( b_{C} - \sum a_{F}\phi_{F})+ (1-\lambda)a_{C}\phi_{C}^{*}\\
&=\frac{E}{1+E}( b_{C} - \sum a_{F}\phi_{F})+ \frac{1}{1+E}a_{C}\phi_{C}^{*} \text{, replacing }\lambda \text{ by} \frac{E}{1+E} \\

\end{split}
a_{C}\phi_{C}(1+\frac{1}{E}) +\sum a_{F}\phi_{F} =b_{C} +\frac{1}{E} a_{C}\phi_{C}^{*}

This under-relaxation effect can be interpreted in terms of some artificial transient time scale that advances \phi_{C} at each iteration. The time step \Delta t is proportional to \Delta t^{*} which is the characteristic time scale defined as

\Delta t^{*} = \frac{\rho_{C}V_{C}}{a_{C}}

Characteristic time scale is related to the time needed to diffuse and convect a change of \phi_{C} across the element. Hence E is like element CFL number. The following points rae to be noted:

• The \Delta t^{*} is proportional to element volume.
• Solution in smaller element advance slower compared to a bigger element.
• This is also a characteristic of Patankar’s implicit relaxation.
• E and \lambda are related by E = \frac{1}{1-\lambda}
• General values E: 4 – 10, \lambda : 0.75 – 0.9.

False transient relaxation

This can be viewed as a modification of Euler’s first order implicit transient method. Here, diagonal dominance id enhanced by adding pseudo transient term to diagonal coefficient and a pseudo previous time term to RHS for balance.Equation (4) is return as following:

$$(a_{C}+a_{C_{0}})\bold \phi_{C} + \sum_{F} a_{F}\phi_{F} = b_{C}+a_{C_{0}}\phi_{C} ^{*}$$

where,

a_{C_{0}} = \frac{\rho_{C}V_{C}}{\Delta t}

where, \Delta t is a user supplied time step.

The following points shall be noted for under relaxation in general

• Different equations have Different URF
• Different parts of the domain may employ Different URF
• Different iteration may have Different URF value.

Source

1. Moukalled F, Mangani L, Darwish M. The finite volume method in computational fluid dynamics. Berlin, Germany:: Springer; 2016.


text:
# Tangent to a circle

1. Mar 5, 2014

### sooyong94

1. The problem statement, all variables and given/known data
Given that the line $y=mx+c$ is a tangent to the circle $(x-a)^{2} +(y-b)^{2} =r^{2}$, show that $(1+m^{2}) r^{2}=(c-b+ma)^{2}$

2. Relevant equations
Quadratic discriminant, sum and product of roots

3. The attempt at a solution
I substituted y=mx+c into the equation of the circle, and this is what I have:
$(x-a)^{2} +(mx+c-b)^{2} =r^{2}$
Then I expanded and simplified into this:
$(1+m^{2})x^{2}+(2mc-2mb-2a)x+a^{2}+b^{2}-2bc+c^{2}-r^{2}=0$

Now if I have to use the quadratic discriminant it would be tedious to work with, unfortunately. :(

Last edited by a moderator: Mar 20, 2014
2. Mar 5, 2014

### scurty

There's a lot of Algebraic manipulation in this problem.

First hint: Find m, which is the same as $\frac{dy}{dx}$
Second hint: solve for c in terms of only a, b, x, y
Third hint: Show both sides of the equations are equal to each other (again, only in terms of a, b, x, y).

Edit: This is in the Precalculus section. Have you learned how to find $\frac{dy}{dx}$ yet?

Last edited: Mar 5, 2014
3. Mar 20, 2014

### utkarshakash

Use the concept that the radius of circle is equal to the distance of the given line from the centre of circle. You already know the formula for finding the distance of any point from a line and in this case it is simply the centre of circle (a,b). Equate both and you're done.

4. Mar 20, 2014

### pasmith

You should have expanded only so far as
$$(1 + m^2)x^2 + 2(m(c-b) - a)x + a^2 + (c-b)^2 - r^2 = 0$$
Keeping (c-b) as (c-b) throughout will make your calculations much less tedious.


text:
Jordan Santell open web engineer

# L-systems

Biologist Aristid Lindenmayer created Lindenmayer systems, or L-systems, in 1968 as a way of formalizing patterns of bacteria growth. L-systems are a recursive, string-rewriting framework, commonly used today in computer graphics to visualize and simulate organic growth, with applications in plant development, procedural content generation, and fractal-like art.

The following describes L-system fundamentals, how they can be visually represented, and several classes of L-systems, like context-sensitive L-systems and stochastic L-systems. Much of the following has been derived from Przemyslaw Prusinkiewicz and Lindenmayer's seminal work, The Algorithmic Beauty of Plants.

## String Rewriting

Fundamentally, an L-system is a set of rules that describe how to iteratively transform a string of symbols. A string, in this context, is a series of symbols, like "$a$" or "$ababaabaaaa$", and can be thought of as a word comprised of characters. Each rule, known as a production, describes the transformation of one symbol to another symbol, series of symbols, or no symbol at all. On each iteration, the productions are applied to each character simultaneously, resulting in a new series of symbols.

Productions in this rewriting system can be described with "before" and "after" states, often described as the predecessor and successor; for example, the production $a \longrightarrow ab$ represents that the symbol $a$ transforms into the symbols $ab$ on every iteration. The length of derivation, or the number of iterations, is represented by $n$. Given a word $a$ and productions $a \longrightarrow ab$ and $b \longrightarrow a$, the following illustrates how the word transforms over several iterations, from $n=0$ to $n=4$.

### Formalizing L-systems

L-systems are formalized as a tuple with the following definition:

$G = \langle V, w, P \rangle$

Where the components are:

• $V$, the alphabet, or all potential symbols in the string.
• $w$, the starting word, also known as the axiom, comprised of symbols from $V$.
• $P$, a series of productions describing the transformations or rules.

The L-system in Figure 1 can be formalized by defining its axiom ($w$) and a series of productions ($p_{1}, ..., p_{n}$) as:

\begin{aligned} w &: a \\ p_{1} &: b \longrightarrow a \\ p_{2} &: a \longrightarrow ab \\ \end{aligned}

The alphabet of all valid symbols can be inferred. It is implied that a symbol without a matching production has an identity production, e.g. $a \longrightarrow a$.

### D0L-systems

This fundamental form of an L-system is described as a deterministic, context-free L-system, or D0L-system (or sometimes DOL-system). 0L-systems are context-free, meaning that each predecessor is transformed regardless of its position in the string and its neighbors. Deterministic L-systems always produce the same result given the same configurations, as there is only one matching production for each predecessor.

## Graphical Representation of L-systems

L-systems can be represented visually via turtle graphics, of Logo fame. While L-systems are string rewriting systems, these strings are comprised of symbols, each which can represent some command. A turtle in computer graphics is similar to a pen plotter drawing lines in a 2D space. Imagine giving instructions to a pen plotter to draw a square: "draw 1cm. turn right. draw 1cm. turn right. draw 1cm. turn right. draw 1cm". Though plotters don't really have an orientation, an L-system's turtle can be represented by Cartesian coordinates $x$ and $y$, and an angle $\alpha$ that describes its forward direction. From there, symbols in a string can represent commands to change the state of the turtle.

To move a turtle around in 2D, symbols must be chosen to represent movement and rotation. The symbols $F$, $+$ and $-$ will be used here, as they are commonly selected for these commands in L-system interpreters. After deriving the result of an L-system using its production rules, the string can then be parsed from left to right, with the following symbols modifying the turtle state:

• $F$ move forward by $d$ units while drawing a line.
• $+$ rotate left by angle $\delta$.
• $-$ rotate right by angle $\delta$.

The variables $\delta$ and $d$ are global values indicating the magnitude of each symbol's rotation or movement. In non-parametric L-systems, each symbol's rotation and movement magnitude is a constant in the system.

Following the line in Figure 2 from the bottom left corner, the string can be read as "forward, forward, forward, right, forward, forward, right..." and so on.

A turtle may be decoupled from an L-system. The L-system has a starting string and a set of productions and outputs the resulting string. A turtle may take that final string as an input, and output some visual representation. For example, many of the illustrations shown here use the same L-system solvers, while using different turtles where appropriate, like one turtle built using CanvasRenderingContext2D and another using WebGL.

### Space-Filling Curves

Space-filling curves can be formalized via L-systems, resulting in a recursive, fractal-like pattern. More specifically, FASS curves, defined as space-filling, self-avoiding, simple, and self-similar. That is, a single, non-overlapping, recursive, continuous curve.

\begin{aligned} \delta &: 90 \\ w &: X \\ p_{1} &: X \longrightarrow + Y F - X F X - F Y + \\ p_{2} &: Y \longrightarrow - X F + Y F Y + F X - \\ \end{aligned}

The Hilbert Curve (Figure 3) is an example of a FASS curve that can be represented as an L-system. Considered a Node-rewriting technique, this L-system's productions declare that on each iteration, $X$ and $Y$ symbols are replaced with more $F$, $X$ and $Y$ symbols. With the angle $\delta$ defined as 90, this results in recursively generated square wave shape along a curve. While $F$, $+$ and $-$ are interpreted by the turtle, other symbols can be used for productions. In this case, $X$ and $Y$ are ignored when rendering, and only relevant when rewriting the string and matching productions.

### 3D Interpretation

In addition to a turtle traversing on a 2D plane, symbols may be introduced that instruct the turtle to draw in 3D. The Algorithmic Beauty of Plants uses the following symbols to control rendering in three dimensions:

• $+$ turn left by angle $\delta$.
• $-$ turn right by angle $\delta$.
• $\&$ pitch down by angle $\delta$.
• $\wedge$ pitch up by angle $\delta$.
• $\backslash$ roll left by angle $\delta$.
• $/$ roll right by angle $\delta$.
• $|$ turn around by 180°.

Like the 2D Hilbert Curve (Figure 3), a three-dimensional version can also be created (Figure 4) using these additional symbols, resulting in a 3D FASS curve.

\begin{aligned} \delta &: 90 \\ w &: X \\ p_{1} &: X \longrightarrow & \wedge \backslash X F \wedge \backslash X F \\ & & X - F \wedge / / X F \\ & & X \& F + / / X F \\ & & X - F / X - / \\ \end{aligned}

## Bracketed L-Systems

The space-filling Hilbert curve can be represented as a single, continuous line. For organic, tree-like structures, branching is used to represent a diverging fork. Two new symbols, square brackets, are introduced to represent a tree in an L-system's string, with an opening bracket indicating the start of a new branch, with the remaining symbols between the brackets being members of that branch. Symbols after the end bracket indicate returning to the point of the branch's origin. A stack is used to implement branching, storing the state of the turtle on the stack.

• $[$ push the current turtle state onto the stack.
• $]$ pop the top state from the stack and this becomes the current turtle state.

Symbols in a branch are transformed and replaced just as they were outside of a branch. This allows recursive, fractal-like behavior, with each branch forking into more branches, and so on.

## Context-sensitive L-systems

Rather than productions evaluating symbols in isolation (context-free), rules may be defined that only matches a symbol when it proceeds or succeeds another specific symbol. Context-sensitive L-systems contain production rules that specify symbols that must come before or after the predecessor in order to match, as opposed to context-free systems that evaluate predecessors in isolation.

These context rules are defined using $<$ and $>$ in the production rule, adjacent to the predecessor. In Figure 5, the first production rule only matches when the $a$ symbol is immediately after $b$, thus replacing the $a$ predecessor with its successor, $b$. This results in the $b$ symbol moving towards the right:

\begin{aligned} &baaaa \\ &abaaa \\ &aabaa \\ &aaaba \\ &aaaab \\ \end{aligned}

A similar system could be defined that propagates $b$ from right to left, defined via production $a > b \longrightarrow b$, replacing $a$ when there is a $b$ after it in the string.

L-systems with these one-sided context productions may be considered 1L-systems. Productions may also have both a before-context and an after-context in systems considered 2L-systems. They can be represented as:

$A < X > B \longrightarrow Y$

This production rule indicates that the predecessor $X$ will be replaced by successor $Y$ when it is between an $A$ and a $B$.

## Stochastic L-systems

The previously described systems are all deterministic; the same system with the same input will always generate the same result. Stochastic L-systems are non-deterministic, defined by several productions that match the same predecessor, chosen randomly given their weight on each iteration. The following production rules define that on each iteration, $X$ has a 50% chance to be rewritten as $A$, and a 50% chance to be rewritten as $B$.

\begin{aligned} X & \xrightarrow{.5} A \\ X & \xrightarrow{.5} B \\ \end{aligned}

This non-determinism is useful for procedurally creating variety and the seemingly random results of nature.

## Resources

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# Posts Tagged ‘ ANOVA ’

## Study design – test power, test rejection rate, permutation test

August 16, 2013
By

In an experiment involving social policy, a question has arisen as to what statistical method to use. The outcome is 0/1 -- let's say, whether a criminal defendant shows up for trial or not. Then there are a control group and 4 treatment groups (thus, ...

## Some Common Approaches for Analyzing Likert Scales and Other Categorical Data

July 2, 2013
By
$Some Common Approaches for Analyzing Likert Scales and Other Categorical Data$

Analyzing Likert scale responses really comes down to what you want to accomplish (e.g. Are you trying to provide a formal report with probabilities or are you trying to simply understand the data better). Sometimes a couple of graphs are sufficient and a formalize statistical test isn’t even necessary. However, with how easy it is […]

June 13, 2013
By

## Neuroscience, statistical power and how to increase it

April 21, 2013
By

There has been quite a bit of buzz recently about the Button et al. Nature Reviews Neuroscience paper on statistical power. Several similar reviews have been published in psychology and other disciplines and come to broadly the same conclusion - that m...

## Using multilevel models to get accurate inferences for repeated measures ANOVA designs

April 18, 2013
By

It is now increasingly common for experimental psychologists (among others) to use multilevel models (also known as linear mixed models) to analyze data that used to be shoe-horned into a repeated measures ANOVA design. Chapter 18 of Serious Stats introduces multilevel models by considering them as an extension of repeated measures ANOVA models that can […]

## Analyzing Pre-Post Data with Repeated Measures or ANCOVA

January 22, 2013
By

This kind of situation happens all the time, in which a colleague, a reviewer, or a statistical consultant insists that you need to do the analysis differently. Sometimes they're right, but sometimes, as was true here, the two analyses answer different research questions.

## Confusing Statistical Term #7: GLM

August 9, 2012
By

Like some of the other terms in our list--level and beta--GLM has two different meanings. It's a little different than the others, though, because it's an abbreviation for two different terms: General Linear Model and Generalized Linear Model. It's extra confusing because their names are so similar on top of having the same abbreviation.

## Serious stats book officially published

July 5, 2012
By

My serious stats book is officially published (in the UK at least). The US release date is next month (August 7th). I'm not sure why the release is later (possibly extra shipping time for the books). The earlier European release date is I suppose compe...

## Confidence intervals with tiers: functions for between-subjects (independent measures) ANOVA

June 21, 2012
By

In a previous post I showed how to plot difference-adjusted CIs for between-subjects (independent measures) ANOVA designs (see here). The rationale behind this kind of graphical display is introduced in Chapter 3 of Serious stats (and summarized in my earlier blog post). In a between-subjects – or in indeed in a within-subjects (repeated measures) – design […]

## The stimuli-as-fixed-effect fallacy

June 21, 2012
By

Neuroskeptic has just blogged on a new paper by Judd, Westfall and Kenny on Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely ignored problem. I can't access the original pap...


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rivet is hosted by Hepforge, IPPP Durham

## Rivet analyses reference

### TASSO_1982_I179022

$\rho^0$ spectrum in $e^+e^-$ collisions at $E_{\text{CMS}}=34$ GeV
Experiment: TASSO (Petra)
Inspire ID: 179022
Status: VALIDATED
Authors:
• Peter Richardson
References:
• Phys.Lett. B117 (1982) 135-140, 1982
Beams: e- e+
Beam energies: ANY
Run details:
Spectrum for $\rho^0$ production measured by the TASSO experiment in $e^+e^-$ collisions at $E_{\text{CMS}}=34$ GeV at PETRA
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 // -*- C++ -*- #include "Rivet/Analysis.hh" #include "Rivet/Projections/Beam.hh" #include "Rivet/Projections/UnstableParticles.hh" namespace Rivet { /// @brief Add a short analysis description here class TASSO_1982_I179022 : public Analysis { public: /// Constructor DEFAULT_RIVET_ANALYSIS_CTOR(TASSO_1982_I179022); /// @name Analysis methods //@{ /// Book histograms and initialise projections before the run void init() { declare(Beam(), "Beams"); declare(UnstableParticles(), "UFS"); // Book histograms book(_h_rho, 1, 1, 1); } /// Perform the per-event analysis void analyze(const Event& event) { // Get beams and average beam momentum const ParticlePair& beams = apply(event, "Beams").beams(); const double meanBeamMom = ( beams.first.p3().mod() + beams.second.p3().mod() ) / 2.0; MSG_DEBUG("Avg beam momentum = " << meanBeamMom); for (const Particle& p : apply(event, "UFS").particles(Cuts::pid==113)) { double xE = p.E()/meanBeamMom; double beta = p.p3().mod()/p.E(); _h_rho->fill(xE,1./beta); } } /// Normalise histograms etc., after the run void finalize() { scale(_h_rho, sqr(sqrtS())*crossSection()/microbarn/sumOfWeights()); // norm to cross section } //@} /// @name Histograms //@{ Histo1DPtr _h_rho; //@} }; // The hook for the plugin system DECLARE_RIVET_PLUGIN(TASSO_1982_I179022); }


text:
# OpenGraph DrawingFramework

v. 2022.02 (Dogwood)

ogdf::StaticSPQRTree Class Reference

Linear-time implementation of static SPQR-trees. More...

#include <ogdf/decomposition/StaticSPQRTree.h>

Inheritance diagram for ogdf::StaticSPQRTree:

## Public Member Functions

StaticSPQRTree (const Graph &G)
Creates an SPQR tree T for graph G rooted at the first edge of G. More...

StaticSPQRTree (const Graph &G, edge e)
Creates an SPQR tree T for graph G rooted at the edge e. More...

StaticSPQRTree (const Graph &G, Triconnectivity &tricComp)
Creates an SPQR tree T for graph G rooted at the first edge of G. More...

~StaticSPQRTree ()
Destructor. More...

Access operations
const GraphoriginalGraph () const override
Returns a reference to the original graph G. More...

const Graphtree () const override
Returns a reference to the tree T. More...

edge rootEdge () const override
Returns the edge of G at which T is rooted. More...

node rootNode () const override
Returns the root node of T. More...

int numberOfSNodes () const override
Returns the number of S-nodes in T. More...

int numberOfPNodes () const override
Returns the number of P-nodes in T. More...

int numberOfRNodes () const override
Returns the number of R-nodes in T. More...

NodeType typeOf (node v) const override
Returns the type of node v. More...

List< nodenodesOfType (NodeType t) const override
Returns the list of all nodes with type t. More...

Skeletonskeleton (node v) const override
Returns the skeleton of node v. More...

edge skeletonEdgeSrc (edge e) const
Returns the edge in skeleton of source(e) that corresponds to tree edge e. More...

edge skeletonEdgeTgt (edge e) const
Returns the edge in skeleton of target(e) that corresponds to tree edge e. More...

const SkeletonskeletonOfReal (edge e) const override
Returns the skeleton that contains the real edge e. More...

edge copyOfReal (edge e) const override
Returns the skeleton edge that corresponds to the real edge e. More...

Update operations
node rootTreeAt (edge e) override
Roots T at edge e and returns the new root node of T. More...

node rootTreeAt (node v) override
Roots T at node v and returns v. More...

Public Member Functions inherited from ogdf::SPQRTree
virtual ~SPQRTree ()

virtual List< nodenodesOfType (NodeType t) const =0
Returns the list of all nodes with type t. More...

void pertinentGraph (node v, PertinentGraph &Gp) const
Returns the pertinent graph of tree node v in Gp. More...

void directSkEdge (node vT, edge e, node src)

void replaceSkEdgeByPeak (node vT, edge e)

## Protected Member Functions

void cpRec (node v, PertinentGraph &Gp) const override
Recursively performs the task of adding edges (and nodes) to the pertinent graph Gp for each involved skeleton graph. More...

void init (edge e)
Initialization (called by constructor). More...

void init (edge eRef, Triconnectivity &tricComp)
Initialization (called by constructor). More...

void rootRec (node v, edge ef)
Recursively performs rooting of tree. More...

Protected Member Functions inherited from ogdf::SPQRTree
edge cpAddEdge (edge eOrig, PertinentGraph &Gp) const
Add an edge to Gp corresponding to eOrig. More...

node cpAddNode (node vOrig, PertinentGraph &Gp) const
Add a node to Gp corresponding to vOrig if required. More...

## Protected Attributes

EdgeArray< edgem_copyOf
skeleton edge corresponding to real edge e More...

int m_numP
number of P-nodes More...

int m_numR
number of R-nodes More...

int m_numS
number of S-nodes More...

const Graphm_pGraph
pointer to original graph More...

edge m_rootEdge
edge of G at which T is rooted More...

node m_rootNode
root node of T More...

NodeArray< StaticSkeleton * > m_sk
pointer to skeleton of a node in T More...

EdgeArray< edgem_skEdgeSrc
corresponding edge in skeleton(source(e)) More...

EdgeArray< edgem_skEdgeTgt
corresponding edge in skeleton(target(e)) More...

EdgeArray< StaticSkeleton * > m_skOf
skeleton containing real edge e More...

Graph m_tree
underlying tree graph More...

NodeArray< NodeTypem_type
type of nodes in T More...

Protected Attributes inherited from ogdf::SPQRTree
NodeArray< node > * m_cpV
node in pertinent graph corresponding to an original node (auxiliary member) More...

list of added nodes (auxiliary member) More...

## Friends

class StaticSkeleton

Public Types inherited from ogdf::SPQRTree
enum  NodeType { NodeType::SNode, NodeType::PNode, NodeType::RNode }
The type of a tree node in T. More...

## Detailed Description

Linear-time implementation of static SPQR-trees.

The class StaticSPQRTree maintains the arrangement of the triconnected components of a biconnected multi-graph G [Hopcroft, Tarjan 1973] as a so-called SPQR tree T [Fi Battista, Tamassia, 1996]. We call G the original graph of T. The class StaticSPQRTree supports only the statical construction of an SPQR-tree for a given graph G, dynamic updates are not supported.

Each node of the tree has an associated type (represented by SPQRTree::NodeType), which is either SNode, PNode, or RNode, and a skeleton (represented by the class StaticSkeleton). The skeletons of the nodes of T are in one-to-one correspondence to the triconnected components of G, i.e., S-nodes correspond to polygons, P-nodes to bonds, and R-nodes to triconnected graphs.

In our representation of SPQR-trees, Q-nodes are omitted. Instead, the skeleton S of a node v in T contains two types of edges: real edges, which correspond to edges in G, and virtual edges, which correspond to edges in T having v as an endpoint. There is a special edge er in G at which T is rooted, i.e., the root node of T is the node whose skeleton contains the real edge corresponding to er.

The reference edge of the skeleton of the root node is er, the reference edge of the skeleton S of a non-root node v is the virtual edge in S that corresponds to the tree edge (parent(v),v).

Definition at line 73 of file StaticSPQRTree.h.

## ◆ StaticSPQRTree() [1/3]

 ogdf::StaticSPQRTree::StaticSPQRTree ( const Graph & G )
inlineexplicit

Creates an SPQR tree T for graph G rooted at the first edge of G.

Precondition
G is biconnected and contains at least 3 nodes, or G has exactly 2 nodes and at least 3 edges.

Definition at line 85 of file StaticSPQRTree.h.

## ◆ StaticSPQRTree() [2/3]

 ogdf::StaticSPQRTree::StaticSPQRTree ( const Graph & G, edge e )
inline

Creates an SPQR tree T for graph G rooted at the edge e.

Precondition
e is in G, G is biconnected and contains at least 3 nodes, or G has exactly 2 nodes and at least 3 edges.

Definition at line 92 of file StaticSPQRTree.h.

## ◆ StaticSPQRTree() [3/3]

 ogdf::StaticSPQRTree::StaticSPQRTree ( const Graph & G, Triconnectivity & tricComp )
inline

Creates an SPQR tree T for graph G rooted at the first edge of G.

Precondition
G is biconnected and contains at least 3 nodes, or G has exactly 2 nodes and at least 3 edges.

Definition at line 99 of file StaticSPQRTree.h.

## ◆ ~StaticSPQRTree()

 ogdf::StaticSPQRTree::~StaticSPQRTree ( )

Destructor.

## ◆ copyOfReal()

 edge ogdf::StaticSPQRTree::copyOfReal ( edge e ) const
inlineoverridevirtual

Returns the skeleton edge that corresponds to the real edge e.

Precondition
e is an edge in G

Implements ogdf::SPQRTree.

Definition at line 165 of file StaticSPQRTree.h.

## ◆ cpRec()

 void ogdf::StaticSPQRTree::cpRec ( node v, PertinentGraph & Gp ) const
inlineoverrideprotectedvirtual

Recursively performs the task of adding edges (and nodes) to the pertinent graph Gp for each involved skeleton graph.

Implements ogdf::SPQRTree.

Definition at line 200 of file StaticSPQRTree.h.

## ◆ init() [1/2]

 void ogdf::StaticSPQRTree::init ( edge e )
protected

Initialization (called by constructor).

## ◆ init() [2/2]

 void ogdf::StaticSPQRTree::init ( edge eRef, Triconnectivity & tricComp )
protected

Initialization (called by constructor).

## ◆ nodesOfType()

 List ogdf::StaticSPQRTree::nodesOfType ( NodeType t ) const
override

Returns the list of all nodes with type t.

## ◆ numberOfPNodes()

 int ogdf::StaticSPQRTree::numberOfPNodes ( ) const
inlineoverridevirtual

Returns the number of P-nodes in T.

Implements ogdf::SPQRTree.

Definition at line 123 of file StaticSPQRTree.h.

## ◆ numberOfRNodes()

 int ogdf::StaticSPQRTree::numberOfRNodes ( ) const
inlineoverridevirtual

Returns the number of R-nodes in T.

Implements ogdf::SPQRTree.

Definition at line 126 of file StaticSPQRTree.h.

## ◆ numberOfSNodes()

 int ogdf::StaticSPQRTree::numberOfSNodes ( ) const
inlineoverridevirtual

Returns the number of S-nodes in T.

Implements ogdf::SPQRTree.

Definition at line 120 of file StaticSPQRTree.h.

## ◆ originalGraph()

 const Graph& ogdf::StaticSPQRTree::originalGraph ( ) const
inlineoverridevirtual

Returns a reference to the original graph G.

Implements ogdf::SPQRTree.

Definition at line 108 of file StaticSPQRTree.h.

## ◆ rootEdge()

 edge ogdf::StaticSPQRTree::rootEdge ( ) const
inlineoverridevirtual

Returns the edge of G at which T is rooted.

Implements ogdf::SPQRTree.

Definition at line 114 of file StaticSPQRTree.h.

## ◆ rootNode()

 node ogdf::StaticSPQRTree::rootNode ( ) const
inlineoverridevirtual

Returns the root node of T.

Implements ogdf::SPQRTree.

Definition at line 117 of file StaticSPQRTree.h.

## ◆ rootRec()

 void ogdf::StaticSPQRTree::rootRec ( node v, edge ef )
protected

Recursively performs rooting of tree.

## ◆ rootTreeAt() [1/2]

 node ogdf::StaticSPQRTree::rootTreeAt ( edge e )
overridevirtual

Roots T at edge e and returns the new root node of T.

Precondition
e is an edge in G

Implements ogdf::SPQRTree.

## ◆ rootTreeAt() [2/2]

 node ogdf::StaticSPQRTree::rootTreeAt ( node v )
overridevirtual

Roots T at node v and returns v.

Precondition
v is a node in T

Implements ogdf::SPQRTree.

## ◆ skeleton()

 Skeleton& ogdf::StaticSPQRTree::skeleton ( node v ) const
inlineoverridevirtual

Returns the skeleton of node v.

Precondition
v is a node in T

Implements ogdf::SPQRTree.

Definition at line 141 of file StaticSPQRTree.h.

## ◆ skeletonEdgeSrc()

 edge ogdf::StaticSPQRTree::skeletonEdgeSrc ( edge e ) const
inline

Returns the edge in skeleton of source(e) that corresponds to tree edge e.

Precondition
e is an edge in T

Definition at line 147 of file StaticSPQRTree.h.

## ◆ skeletonEdgeTgt()

 edge ogdf::StaticSPQRTree::skeletonEdgeTgt ( edge e ) const
inline

Returns the edge in skeleton of target(e) that corresponds to tree edge e.

Precondition
e is an edge in T

Definition at line 153 of file StaticSPQRTree.h.

## ◆ skeletonOfReal()

 const Skeleton& ogdf::StaticSPQRTree::skeletonOfReal ( edge e ) const
inlineoverridevirtual

Returns the skeleton that contains the real edge e.

Precondition
e is an edge in G

Implements ogdf::SPQRTree.

Definition at line 159 of file StaticSPQRTree.h.

## ◆ tree()

 const Graph& ogdf::StaticSPQRTree::tree ( ) const
inlineoverridevirtual

Returns a reference to the tree T.

Implements ogdf::SPQRTree.

Definition at line 111 of file StaticSPQRTree.h.

## ◆ typeOf()

 NodeType ogdf::StaticSPQRTree::typeOf ( node v ) const
inlineoverridevirtual

Returns the type of node v.

Precondition
v is a node in T

Implements ogdf::SPQRTree.

Definition at line 132 of file StaticSPQRTree.h.

## ◆ StaticSkeleton

 friend class StaticSkeleton
friend

Definition at line 76 of file StaticSPQRTree.h.

## ◆ m_copyOf

 EdgeArray ogdf::StaticSPQRTree::m_copyOf
protected

skeleton edge corresponding to real edge e

Definition at line 233 of file StaticSPQRTree.h.

## ◆ m_numP

 int ogdf::StaticSPQRTree::m_numP
protected

number of P-nodes

Definition at line 223 of file StaticSPQRTree.h.

## ◆ m_numR

 int ogdf::StaticSPQRTree::m_numR
protected

number of R-nodes

Definition at line 224 of file StaticSPQRTree.h.

## ◆ m_numS

 int ogdf::StaticSPQRTree::m_numS
protected

number of S-nodes

Definition at line 222 of file StaticSPQRTree.h.

## ◆ m_pGraph

 const Graph* ogdf::StaticSPQRTree::m_pGraph
protected

pointer to original graph

Definition at line 216 of file StaticSPQRTree.h.

## ◆ m_rootEdge

 edge ogdf::StaticSPQRTree::m_rootEdge
protected

edge of G at which T is rooted

Definition at line 219 of file StaticSPQRTree.h.

## ◆ m_rootNode

 node ogdf::StaticSPQRTree::m_rootNode
protected

root node of T

Definition at line 220 of file StaticSPQRTree.h.

## ◆ m_sk

 NodeArray ogdf::StaticSPQRTree::m_sk
protected

pointer to skeleton of a node in T

Definition at line 228 of file StaticSPQRTree.h.

## ◆ m_skEdgeSrc

 EdgeArray ogdf::StaticSPQRTree::m_skEdgeSrc
protected

corresponding edge in skeleton(source(e))

Definition at line 229 of file StaticSPQRTree.h.

## ◆ m_skEdgeTgt

 EdgeArray ogdf::StaticSPQRTree::m_skEdgeTgt
protected

corresponding edge in skeleton(target(e))

Definition at line 230 of file StaticSPQRTree.h.

## ◆ m_skOf

 EdgeArray ogdf::StaticSPQRTree::m_skOf
protected

skeleton containing real edge e

Definition at line 232 of file StaticSPQRTree.h.

## ◆ m_tree

 Graph ogdf::StaticSPQRTree::m_tree
protected

underlying tree graph

Definition at line 217 of file StaticSPQRTree.h.

## ◆ m_type

 NodeArray ogdf::StaticSPQRTree::m_type
protected

type of nodes in T

Definition at line 226 of file StaticSPQRTree.h.

The documentation for this class was generated from the following file:


text:
## Statistical Science

### Could Fisher, Jeffreys and Neyman Have Agreed on Testing?

James O. Berger

#### Abstract

Ronald Fisher advocated testing using p-values, Harold Jeffreys proposed use of objective posterior probabilities of hypotheses and Jerzy Neyman recommended testing with fixed error probabilities. Each was quite critical of the other approaches. Most troubling for statistics and science is that the three approaches can lead to quite different practical conclusions.

This article focuses on discussion of the conditional frequentist approach to testing, which is argued to provide the basis for a methodological unification of the approaches of Fisher, Jeffreys and Neyman. The idea is to follow Fisher in using p-values to define the "strength of evidence" in data and to follow his approach of conditioning on strength of evidence; then follow Neyman by computing Type I and Type II error probabilities, but do so conditional on the strength of evidence in the data. The resulting conditional frequentist error probabilities equal the objective posterior probabilities of the hypotheses advocated by Jeffreys.

#### Article information

Source
Statist. Sci., Volume 18, Issue 1 (2003), 1-32.

Dates
First available in Project Euclid: 23 June 2003

https://projecteuclid.org/euclid.ss/1056397485

Digital Object Identifier
doi:10.1214/ss/1056397485

Mathematical Reviews number (MathSciNet)
MR1997064

Zentralblatt MATH identifier
1048.62006

#### Citation

Berger, James O. Could Fisher, Jeffreys and Neyman Have Agreed on Testing?. Statist. Sci. 18 (2003), no. 1, 1--32. doi:10.1214/ss/1056397485. https://projecteuclid.org/euclid.ss/1056397485

#### References

• Barnett, V. (1999). Comparative Statistical Inference, 3rd ed. Wiley, New York.
• Basu, D. (1975). Statistical information and likelihood (with discussion). Sankhyā Ser. A 37 1--71.
• Basu, D. (1977). On the elimination of nuisance parameters. J. Amer. Statist. Assoc. 72 355--366.
• Bayarri, M. J. and Berger, J. (2000). $P$-values for composite null models (with discussion). J. Amer. Statist. Assoc. 95 1127--1142, 1157--1170.
• Berger, J. (1985a). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, New York.
• Berger, J. (1985b). The frequentist viewpoint and conditioning. In Proc. Berkeley Conference in Honor of Jack Kiefer and Jerzy Neyman (L. Le Cam and R. Olshen, eds.) 1 15--44. Wadsworth, Belmont, CA.
• Berger, J. and Berry, D. (1988). Statistical analysis and the illusion of objectivity. American Scientist 76 159--165.
• Berger, J., Boukai, B. and Wang, Y. (1997). Unified frequentist and Bayesian testing of a precise hypothesis (with discussion). Statist. Sci. 12 133--160.
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text:
MathSciNet bibliographic data MR1175897 14J45 (14E30) Kawamata, Yujiro Boundedness of \$\bold Q\$$\bold Q$-Fano threefolds. Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), 439–445, Contemp. Math., 131, Part 3, Amer. Math. Soc., Providence, RI, 1992. Links to the journal or article are not yet available

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
# Trig Function

1. Nov 2, 2005

### Chop235

I am stuck on this problem:

It tells me to use the Half-Angle Formulas to find the exact value of the trig function.

sec((15*pi)/8)

I know I should convert that into the cos function and when I do I get:

cos(8/(15*pi))

But now I need to find cos(alpha/2) to be able to use the Half-Angle formula. I think what I can do is multiply it by 2 and then that would be my alpha, then I could just plug it in right? So then I would have:

cos((16/(15*pi))/2)

I am not sure if this is right so far or if I am even on the right track.

2. Nov 2, 2005

### verty

As far as I can see:

sec(15*pi/8)
= sec(2pi - pi/8)
= sec(pi/8)

3. Nov 2, 2005

### HallsofIvy

Staff Emeritus
NO!!! sec x= 1/(cos(x)) not cos(1/x)!
Use the Half-Angle Formula alright but to go the other way!
sec(15pi/8)= 1/cos(15pi/8) so you just need to find cos(15pi/8)
That is $$cos(15pi/8)= \sqrt{\frac{1}{2}(1+ cos(15pi/4))}$$.
cos(15pi/4) is easy.


text:
# What is the solution to the proportion x/(12-x)=5/15?

Apr 29, 2015

Solving this would require cross multiplication

we are given
$\frac{x}{12 - x}$ $=$ $\frac{5}{15}$

on cross multiplying:
15 is taken to the L.H.S numerator and (12 - x) is taken to R.H.S numerator

we get:
15$x$ $=$ 5(12 - $x$)
15$x$ $=$ 60 - 5$x$
20$x$ $=$ 60

$x$ $=$ 3

so our proportion is:

$\frac{3}{9}$ $=$$\frac{5}{15}$

note: on solving both fractions we get equal value of 0.33 on both sides thus they are in proportion as L.H.S $=$ R.H.S


text:
# zbMATH — the first resource for mathematics

## Chen, Xinxing

Compute Distance To:
 Author ID: chen.xinxing Published as: Chen, X.; Chen, X. X.; Chen, Xin Xing; Chen, XinXing; Chen, Xinxing
 Documents Indexed: 61 Publications since 1982, including 1 Book
all top 5

#### Co-Authors

 2 single-authored 5 Ying, Jiangang 2 Chen, Dayue 2 Yao, Qiang 1 Barlow, Martin T. 1 Derrida, Bernard 1 Hu, Yueyun 1 Lifshits, Mikhail A. 1 Shi, Zhan 1 Tan, Yongji 1 Xie, Jiansheng 1 Zhao, Minzhi
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#### Serials

 2 Stochastic Processes and their Applications 2 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 2 Science China. Mathematics 1 Journal of Statistical Physics 1 Mathematische Annalen 1 Chinese Annals of Mathematics. Series A 1 Science in China. Series A 1 Electronic Journal of Probability 1 Acta Mathematica Sinica. English Series 1 Acta Mathematica Sinica. Chinese Series
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#### Fields

 13 Probability theory and stochastic processes (60-XX) 6 Combinatorics (05-XX) 2 Partial differential equations (35-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Numerical analysis (65-XX)

#### Citations contained in zbMATH Open

31 Publications have been cited 239 times in 231 Documents Cited by Year
Existence and uniqueness theorem for uncertain differential equations. Zbl 1196.34005
Chen, X.; Liu, B.
2010
Herringbone buckling patterns of compressed thin films on compliant substrates. Zbl 1111.74362
Chen, X.; Hutchinson, John W.
2004
A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations. Zbl 1114.74056
Chen, X.; Toh, K. C.; Phoon, K. K.
2006
Effect of surface/interface stress on the plastic deformation of nanoporous materials and nanocomposites. Zbl 1454.74008
Zhang, W. X.; Wang, T. J.; Chen, X.
2010
The complete convergence theorem holds for contact processes on open clusters of $$\mathbb Z^{d }\times \mathbb Z^{+}$$. Zbl 1177.82039
Chen, Xinxing; Yao, Qiang
2009
Wave propagation under curvature effects in a heterogeneous medium. Zbl 0878.35009
Chen, X.; Namah, G.
1997
Determining plastic properties of a material with residual stress by using conical indentation. Zbl 1178.74072
Yan, Jin; Karlsson, A. M.; Chen, X.
2007
Invariants of the velocity-gradient tensor in a spatially developing inhomogeneous turbulent flow. Zbl 1383.76179
Buxton, O. R. H.; Breda, M.; Chen, X.
2017
Sparse envelope model: efficient estimation and response variable selection in multivariate linear regression. Zbl 07072139
Su, Z.; Zhu, G.; Chen, X.; Yang, Y.
2016
The complete convergence theorem holds for contact processes in a random environment on $${\mathbb Z}^d \times {\mathbb Z}^{+}$$. Zbl 1247.60137
Yao, Qiang; Chen, Xinxing
2012
Two random walks on the open cluster of $$\mathbb Z^{2}$$ meet infinitely often. Zbl 1216.05147
Chen, Xinxing; Chen, Dayue
2010
Chen, X.; Kim, K. S.
2003
Gaussian bounds and parabolic Harnack inequality on locally irregular graphs. Zbl 1352.60126
Barlow, Martin T.; Chen, Xinxing
2016
Some sufficient conditions for infinite collisions of simple random walks on a wedge comb. Zbl 1244.60069
Chen, Xinxing; Chen, Dayue
2011
Partitioned versus global Krylov subspace iterative methods for FE solution of 3-D Biot’s problem. Zbl 1173.74405
Chen, X.; Phoon, K. K.; Toh, K. C.
2007
Blank development and the prediction of earing in cup drawing. Zbl 0843.73035
Chen, X.; Sowerby, R.
1996
Structure evolution at early stage of boundary-layer transition: simulation and experiment. Zbl 07193492
Jiang, X. Y.; Lee, C. B.; Chen, X.; Smith, C. R.; Linden, P. F.
2020
Pointwise upper estimates for transition probabilities of continuous time random walks on graphs. Zbl 1361.60030
Chen, Xinxing
2017
Snap-through buckling of initially curved microbeam subject to an electrostatic force. Zbl 1371.74106
Chen, X.; Meguid, S. A.
2015
Coalescence of drops with mobile interfaces in a quiescent fluid. Zbl 1291.76048
Nemer, M. B.; Santoro, P.; Chen, X.; Bławzdziewicz, J.; Loewenberg, M.
2013
Multistage scenario-based interval-stochastic programming for planning water resources allocation. Zbl 1418.90183
Li, Y. P.; Huang, G. H.; Chen, X.
2009
A novel DCT-based algorithm for computing the modulated complex lapped transform. Zbl 1374.94695
Chen, X.; Dai, Q.
2006
On multiscale significance of Rice’s normality structure. Zbl 1192.74009
Yang, Q.; Chen, X.; Zhou, W. Y.
2006
The effects of thin and ultrathin liquid films on dynamic wetting. Zbl 1186.76101
Chen, X.; Ramé, E.; Garoff, S.
2003
Random coloring evolution on graphs. Zbl 1206.60068
Chen, Xin Xing; Ying, Jian Gang
2010
Effects of superimposed hydrostatic pressure on fracture in round bars under tension. Zbl 1183.74246
Peng, J.; Wu, P. D.; Huang, Y.; Chen, X. X.; Lloyd, D. J.; Embury, J. D.; Neale, K. W.
2009
The Markov chain asymptotics of random mapping graphs. Zbl 1117.05098
Chen, Xinxing; Ying, Jiangang
2007
Solutions for the deformations and stability of elastoplastic hollow cylinders subjected to boundary pressures. Zbl 0943.74016
Chen, X.; Tan, C. P.; Haberfield, C. M.
1999
Analysis of arbitrary Mindlin plates or bridge decks by spline finite strip method. Zbl 0877.73078
Ng, S. F.; Chen, X.
1995
Effect of frictional force and nose shape on axisymmetric deformations of a thick thermoviscoplastic target. Zbl 0848.73017
Batra, R. C.; Chen, X.
1994
Effect of frictional force on the steady state axisymmetric deformations of a viscoplastic target. Zbl 0825.73636
Chen, X.; Batra, R. C.
1993
Structure evolution at early stage of boundary-layer transition: simulation and experiment. Zbl 07193492
Jiang, X. Y.; Lee, C. B.; Chen, X.; Smith, C. R.; Linden, P. F.
2020
Invariants of the velocity-gradient tensor in a spatially developing inhomogeneous turbulent flow. Zbl 1383.76179
Buxton, O. R. H.; Breda, M.; Chen, X.
2017
Pointwise upper estimates for transition probabilities of continuous time random walks on graphs. Zbl 1361.60030
Chen, Xinxing
2017
Sparse envelope model: efficient estimation and response variable selection in multivariate linear regression. Zbl 07072139
Su, Z.; Zhu, G.; Chen, X.; Yang, Y.
2016
Gaussian bounds and parabolic Harnack inequality on locally irregular graphs. Zbl 1352.60126
Barlow, Martin T.; Chen, Xinxing
2016
Snap-through buckling of initially curved microbeam subject to an electrostatic force. Zbl 1371.74106
Chen, X.; Meguid, S. A.
2015
Coalescence of drops with mobile interfaces in a quiescent fluid. Zbl 1291.76048
Nemer, M. B.; Santoro, P.; Chen, X.; Bławzdziewicz, J.; Loewenberg, M.
2013
The complete convergence theorem holds for contact processes in a random environment on $${\mathbb Z}^d \times {\mathbb Z}^{+}$$. Zbl 1247.60137
Yao, Qiang; Chen, Xinxing
2012
Some sufficient conditions for infinite collisions of simple random walks on a wedge comb. Zbl 1244.60069
Chen, Xinxing; Chen, Dayue
2011
Existence and uniqueness theorem for uncertain differential equations. Zbl 1196.34005
Chen, X.; Liu, B.
2010
Effect of surface/interface stress on the plastic deformation of nanoporous materials and nanocomposites. Zbl 1454.74008
Zhang, W. X.; Wang, T. J.; Chen, X.
2010
Two random walks on the open cluster of $$\mathbb Z^{2}$$ meet infinitely often. Zbl 1216.05147
Chen, Xinxing; Chen, Dayue
2010
Random coloring evolution on graphs. Zbl 1206.60068
Chen, Xin Xing; Ying, Jian Gang
2010
The complete convergence theorem holds for contact processes on open clusters of $$\mathbb Z^{d }\times \mathbb Z^{+}$$. Zbl 1177.82039
Chen, Xinxing; Yao, Qiang
2009
Multistage scenario-based interval-stochastic programming for planning water resources allocation. Zbl 1418.90183
Li, Y. P.; Huang, G. H.; Chen, X.
2009
Effects of superimposed hydrostatic pressure on fracture in round bars under tension. Zbl 1183.74246
Peng, J.; Wu, P. D.; Huang, Y.; Chen, X. X.; Lloyd, D. J.; Embury, J. D.; Neale, K. W.
2009
Determining plastic properties of a material with residual stress by using conical indentation. Zbl 1178.74072
Yan, Jin; Karlsson, A. M.; Chen, X.
2007
Partitioned versus global Krylov subspace iterative methods for FE solution of 3-D Biot’s problem. Zbl 1173.74405
Chen, X.; Phoon, K. K.; Toh, K. C.
2007
The Markov chain asymptotics of random mapping graphs. Zbl 1117.05098
Chen, Xinxing; Ying, Jiangang
2007
A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations. Zbl 1114.74056
Chen, X.; Toh, K. C.; Phoon, K. K.
2006
A novel DCT-based algorithm for computing the modulated complex lapped transform. Zbl 1374.94695
Chen, X.; Dai, Q.
2006
On multiscale significance of Rice’s normality structure. Zbl 1192.74009
Yang, Q.; Chen, X.; Zhou, W. Y.
2006
Herringbone buckling patterns of compressed thin films on compliant substrates. Zbl 1111.74362
Chen, X.; Hutchinson, John W.
2004
Chen, X.; Kim, K. S.
2003
The effects of thin and ultrathin liquid films on dynamic wetting. Zbl 1186.76101
Chen, X.; Ramé, E.; Garoff, S.
2003
Solutions for the deformations and stability of elastoplastic hollow cylinders subjected to boundary pressures. Zbl 0943.74016
Chen, X.; Tan, C. P.; Haberfield, C. M.
1999
Wave propagation under curvature effects in a heterogeneous medium. Zbl 0878.35009
Chen, X.; Namah, G.
1997
Blank development and the prediction of earing in cup drawing. Zbl 0843.73035
Chen, X.; Sowerby, R.
1996
Analysis of arbitrary Mindlin plates or bridge decks by spline finite strip method. Zbl 0877.73078
Ng, S. F.; Chen, X.
1995
Effect of frictional force and nose shape on axisymmetric deformations of a thick thermoviscoplastic target. Zbl 0848.73017
Batra, R. C.; Chen, X.
1994
Effect of frictional force on the steady state axisymmetric deformations of a viscoplastic target. Zbl 0825.73636
Chen, X.; Batra, R. C.
1993
all top 5

#### Cited by 389 Authors

 12 Yao, Kai 12 Zhu, Yuanguo 9 Gao, Jinwu 7 Ferronato, Massimiliano 7 Gao, Yuan 7 Xue, Xiaofeng 7 Yang, Xiangfeng 6 Chen, Xiaowei 5 Gambolati, Giuseppe 5 Gao, Rong 5 Huang, Yonggang Young 5 Potier-Ferry, Michel 5 Rogers, John A. 5 Sheng, Yuhong 5 Zhang, Zhiqiang 4 Hu, Heng 4 Ji, Xiaoyu 4 Jiang, Hanqing 4 Ke, Hua 4 Peng, Jin 4 Ralescu, Dan A. 4 Wen, Meilin 4 Zhang, Bo 3 Audoly, Basile 3 Bergamaschi, Luca 3 Boudaoud, Arezki 3 Chen, Xinxing 3 Dormieux, Luc 3 Feng, Xiqiao 3 Huang, Xiaoxia 3 Kang, Rui 3 Kondo, Djimédo 3 Kumagai, Takashi 3 Lan, Yanfei 3 Liu, Baoding 3 Liu, Weiqi 3 Ning, Yufu 3 Su, Zhihua 3 Sun, Yiyao 3 Tang, Wansheng 3 Wang, Xiao 3 Yang, Yi 3 Zhang, Yi 2 Britaňák, Vladimír 2 Cailletaud, Georges 2 Chen, Dayue 2 Chen, Xinfu 2 Chen, Xiumei 2 Damil, Noureddine 2 Ding, Shanshan 2 Fang, Daining 2 Fei, Weiyin 2 Fu, Zongfei 2 Hutchinson, John Woodside 2 Jia, Lifen 2 Khang, Dahl-Young 2 Le, Minh-Quy 2 Lee, Chang-Boon 2 Li, Bo 2 Li, Bo 2 Linden, Paul F. 2 Liu, Yuhan 2 Lou, Bendong 2 Ou, Zhicheng 2 Pan, Yu 2 Phoon, Kok-Kwang 2 Smith, C. Ray 2 Su, Taoyong 2 Sun, Yugang 2 Suo, Zhiyong 2 Taleb, Lakhdar 2 Tao, Nana 2 Triantafyllidis, Nicolas 2 Wang, Jian 2 Wu, Huishan 2 Yao, Xiaohu 2 Zhang, Xiaoqing 2 Zhao, Ruiqing 2 Zhu, Guangyu 1 Abbas, Umber 1 Abe, Yoshihiro 1 An, Baiguo 1 An, Qi 1 Andersons, J. 1 Ansari, Reza 1 Arriëns, Huibert J. Lincklaen 1 Asghari, Mehran 1 Aubertin, Michel 1 Bai, Zhongzhi 1 Bakiler, A. Derya 1 Banabic, Dorel 1 Barber, Joel R. 1 Barlow, Martin T. 1 Basaran, Osman A. 1 Batra, Romesh C. 1 Bazhlekov, Ivan B. 1 Béda, Péter B. 1 Belouettar, Salim 1 Ben Bettaieb, Mohamed 1 Bigerelle, Maxence ...and 289 more Authors
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#### Cited in 83 Serials

 25 Fuzzy Optimization and Decision Making 23 Soft Computing 17 Applied Mathematics and Computation 14 Journal of the Mechanics and Physics of Solids 9 European Journal of Mechanics. A. Solids 8 International Journal of Solids and Structures 7 Journal of Fluid Mechanics 6 Acta Mechanica 6 International Journal of Engineering Science 5 Computer Methods in Applied Mechanics and Engineering 4 International Journal of Plasticity 4 Journal of Computational Physics 3 International Journal for Numerical and Analytical Methods in Geomechanics 3 International Journal for Numerical Methods in Engineering 3 Applied Mathematical Modelling 3 International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 3 Journal of Applied Mathematics 3 Journal of Intelligent and Fuzzy Systems 3 Frontiers of Mathematics in China 2 Archive for Rational Mechanics and Analysis 2 Computers & Mathematics with Applications 2 Journal of Statistical Physics 2 Physica A 2 Information Sciences 2 Journal of Computational and Applied Mathematics 2 Journal of Functional Analysis 2 Cybernetics and Systems 2 Signal Processing 2 Stochastic Processes and their Applications 2 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 2 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2 Physics of Fluids 2 Mathematical Problems in Engineering 2 European Journal of Control 2 Iranian Journal of Fuzzy Systems 2 International Journal of Systems Science. Principles and Applications of Systems and Integration 1 Biological Cybernetics 1 International Journal of General Systems 1 Journal of Mathematical Analysis and Applications 1 Mathematical Methods in the Applied Sciences 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 The Annals of Statistics 1 Journal of Differential Equations 1 Journal of the Mathematical Society of Japan 1 Meccanica 1 Tohoku Mathematical Journal. Second Series 1 Statistics & Probability Letters 1 Applied Numerical Mathematics 1 Acta Mathematicae Applicatae Sinica. English Series 1 Probability Theory and Related Fields 1 Numerical Methods for Partial Differential Equations 1 Mathematical and Computer Modelling 1 Science in China. Series A 1 European Journal of Operational Research 1 International Journal of Computer Mathematics 1 Archive of Applied Mechanics 1 Journal of Nonlinear Science 1 SIAM Journal on Scientific Computing 1 Applied Mathematics. Series B (English Edition) 1 Numerical Linear Algebra with Applications 1 Statistica Sinica 1 Complexity 1 ELA. The Electronic Journal of Linear Algebra 1 Mathematics and Mechanics of Solids 1 Abstract and Applied Analysis 1 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Discrete Dynamics in Nature and Society 1 Communications in Nonlinear Science and Numerical Simulation 1 Computational Geosciences 1 Engineering Computations 1 Stochastic Environmental Research and Risk Assessment 1 International Journal of Nonlinear Sciences and Numerical Simulation 1 Archives of Computational Methods in Engineering 1 Boundary Value Problems 1 Journal of Industrial and Management Optimization 1 Acta Mechanica Sinica 1 ALEA. Latin American Journal of Probability and Mathematical Statistics 1 Electronic Journal of Statistics 1 Mathematical Modelling of Natural Phenomena 1 ISRN Applied Mathematics 1 AMM. Applied Mathematics and Mechanics. (English Edition) 1 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences
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#### Cited in 33 Fields

 74 Mechanics of deformable solids (74-XX) 51 Probability theory and stochastic processes (60-XX) 36 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 34 Ordinary differential equations (34-XX) 26 Numerical analysis (65-XX) 18 Partial differential equations (35-XX) 18 Systems theory; control (93-XX) 16 Operations research, mathematical programming (90-XX) 14 Fluid mechanics (76-XX) 9 Measure and integration (28-XX) 6 Combinatorics (05-XX) 6 Calculus of variations and optimal control; optimization (49-XX) 6 Biology and other natural sciences (92-XX) 4 Statistics (62-XX) 4 Classical thermodynamics, heat transfer (80-XX) 4 Statistical mechanics, structure of matter (82-XX) 3 Real functions (26-XX) 3 Dynamical systems and ergodic theory (37-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Computer science (68-XX) 2 Mechanics of particles and systems (70-XX) 2 Geophysics (86-XX) 2 Information and communication theory, circuits (94-XX) 1 General and overarching topics; collections (00-XX) 1 History and biography (01-XX) 1 Mathematical logic and foundations (03-XX) 1 Functions of a complex variable (30-XX) 1 Potential theory (31-XX) 1 Difference and functional equations (39-XX) 1 Integral equations (45-XX) 1 Operator theory (47-XX) 1 Differential geometry (53-XX) 1 Optics, electromagnetic theory (78-XX)


text:
# 2sqrt (6a + 7) - 2a = 0? solve the equation

Apr 18, 2017

$a = 7$

#### Explanation:

Given: $2 \sqrt{6 a + 7} - 2 a = 0$

Restrict the domain so that the expression under the radical never becomes negative:

2sqrt (6a + 7) - 2a = 0; a>=-7/6

Divide the equation by 2:

sqrt (6a + 7) - a = 0; a>=-7/6

sqrt (6a + 7) = a; a>=-7/6

Square both sides:

6a + 7 = a^2; a>=-7/6

Subtract $6 a + 7$ from both sides:

0 = a^2-6a - 7; a>=-7/6

Factor:

$\left(a - 7\right) \left(a + 1\right) = 0$

$a = 7 \mathmr{and} a = - 1$

Please notice that I have can discarded the restriction because neither root violates it.

Check:

$2 \sqrt{6 \left(7\right) + 7} - 2 \left(7\right) = 0$
$2 \sqrt{6 \left(- 1\right) + 7} - 2 \left(- 1\right) = 0$

$0 = 0$
$2 = 0$

The negative root must be discarded; it is an erroneous root introduced by squaring.

$a = 7$ is the only correct answer.


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# VNote

## A Vim-inspired note-taking application with Markdown

A note-taking application with Markdown, which knows programmers and Markdown better.

Built with C++ and Qt, available on Linux, Windows, and macOS.

### Discussion

Hunter
Makers
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Hi @tamlok I'd love to know - what drove you to build VNote?
Le TanMaker@tamlok · Software Engineer
@abadesi Hi, I think most Markdown editors do not know Markdown well and they even treat side-by-side live preview as the advantage of their products. VNote also utilize lots of the pros of the code editors from a programmer's perspective.
@ozgrozer · Software developer
302.4 MB! Isn't that too much for a note taking app?
Le TanMaker@tamlok · Software Engineer
@ozgrozer Do you mean memory usage? VNote uses QtWebEngine, an embedded browser, to render the HTML content. And I admit that optimization is needed. :)
@ozgrozer · Software developer
@tamlok I mean size of the .app file. It's too much. What's inside of it? :) Apple's Notes.app is 13.3 MB. Evernote.app is 133.7 MB.
Le TanMaker@tamlok · Software Engineer
@ozgrozer I think it is the QWebEngineProcess inside it.
@yoaquim · Engineer, Manager
Good stuff, but... Why is this better than using plain-old vim to write up an MD file and then just using a Chrome plugin, or something similar, to preview that file? Interested if there are other features that make it stand out! Keep coding!
Le TanMaker@tamlok · Software Engineer
@yoaquim I think it is the INTEGRATION, as well as MathJax and PlantUML. Check out this ![image]() for the effect of VNote's In-Place Preview. :)