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# What distinguishes time from space in Quantum Field Theory?

Consider the following expression for a general QFT action:

$$S ~=~ \int_0^t\mathrm dt~L ~=~\int_0^t\mathrm dt\int_\mathbb {R^3}\mathrm d^3x~\mathcal L ~=~\int\mathrm d^4x~\mathcal L.$$

Here we have clearly singled out time as the "worldline parameter" for calculating $S$, because we want to calculate how a system evolves in time (in some frame). But what's special about time (in particular this choice of time coordinate)? Of course time is manifestly distinguished from space by the metric $\eta_{\mu\nu}$, but a-priori is there anything special about time (it's not clear that we would have any entropy considerations, for example)? In particular we can consider Lorentz transformations that take this time $t$ to some other coordinate. It feels like in principle we should be able to chose any 3-D subspace of $\mathbb R^4$ and flow that along the spacetime manifold.

I'd really appreciate someone with a detailed understanding of the physics and/or geometry of the situation giving some clarifying remarks (I understand there may not be a clear-cut answer).

(Btw. I am aware of the aware of various arrows of time and have read some relevant discussion, e.g. What grounds the difference between space and time?)

• In QFT, there is no time operator. For example, in the $\mathbf{k}$ representation of QFT, the fields are promoted to operators. But there does not seem to exist any such operator $\hat{T}$ such that $$\hat{T}\psi=t\psi$$. – user28355 Jan 12 '14 at 23:34
• A priory -- everything is special about time. F.e. You can not go back in time, even though this is heavily used in physical arguments :) This formula and in particular FIELD theory has nothing to do with time. Same as mechanics has little to do with time. Nobody said time is the same as space. Somebody said time LOOKS like space. Apes also look like humans. – Asphir Dom Jan 26 '14 at 22:48

Indeed, you are free to choose any time-like direction as a direction in which the system evolves. For example the ADM formalism of General Relativity is based on this idea. In this formalism you take a 3-dimensional space-like surface as our world. Then the time will be related to the normal to this surface.

Very good detailed description of the ADM construction can be found in "A relativist's toolkit" by Poisson.

I guess that the demand that evolution should go in a time-like direction is related to some causality features. What is true is that you cannot do a Lorentz rotation from a space-like to a time-like direction.

• Bojowald's Canonical Gravity and Applications is also a great detail-description of this canonical "Space + time" decomposition... – Alex Nelson Jan 21 '14 at 21:34

Here two physically relevant facts distinguishing space form time in Minkowski spacetime (however everything follows can be generalized to every spacetime with suitable requirements).

1) This is general. Consider, in Minkowski spacetime, two events $p$ and $q$. If $p$ and $q$ stay in the $3$-space of a reference frame, then among the curves with spacelike tangent vector joining these points there is not the shortest curve, but you can always find a curve with length smaller than any fixed $\epsilon>0$. Obviously, if you consider only the curves lying to the rest space, there is the shortest one: the segment joining $p$ and $q$.

If $p$ and $q$ stay along a curve with timelike vector, for instance on the temporal axis of a reference frame, there is the longest curve joining them (instead of the shortest one).

2) This is properly referring to Quantum Field Theory. More precisely to fields solving $\delta \int_{\Omega} {\cal L}(\phi, \partial \phi) d^4x =0$ with $\phi$ vanishing on the boundary of $\Omega$, also before being quantized.

Assuming that the Lagrangian is quadratic in the first derivatives of $\phi$ with the metric $\eta$ as quadratic form, you obtain equations for $\phi$.

These equations, to be solved, need initial conditions. That is, you have to assign $\phi$ and $\partial_n \phi$ on a $3$-surface, whose $n$ is the normal unit vector.

Well, the problem is well-posed, i.e. there exists a unique solution and it continuously depends on initial data (with a suitable choice of topology and in a certain class or sufficiently regular initial data) if the surface can be described by spatial coordinates, for example the rest $3$-space of an inertial frame (but the result is general with some requirements I do not describe here). Otherwise, if just one coordinate on the $3$-surface is timelike the problem gives rise to some pathology (lack of existence, lack of uniqueness or discontinuous dependence from initial data). Without well-posedness of initial data problem to quantize fields (with some undefined meaning for "to quantize" since the standard procedure is not feasible) turns out to be very difficult.

In (classical or quantum) field theory, time is distinguished by a (classical) observer. The observer moves along a world line $x(s)$ parameterized by its eigentime $s$, with timelike derivative $\dot x(s)$. This distinguishes a direction of time, and the observer's space is orthogonal to it.

Different observers will generally have different directions of time, and any two of them can be turned into each other by a proper Loretz transformation.

Note that proper Loretz transformation always transform timelike vectors into timelike vectors, never into lightlike, spacelike, antilightlike, or antitimelike vectors. (For example, there is no Lorentz transformation that transforms the $t$-axis (i.e., $x_0$-axis) into the $x_1$-axis.) Thus one is not completely free to select a 3D subspace, but only those 3D affine subspaces that are orthogonal to a timelike vector (and hence consist of mutually spacelike points). These are indeed fully equivalent, for each one there is an observer for which it defines its space.

See also Chapter A7: Time and space in my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html

An important area of QFT where time is particularly special is the fact that the inner product of two fields is done over all space at a particular time (or over a spacelike hypersurface, if you're in curved spacetime).

In relativity itself, just the fact that the metric is (-+++), or (+---), depending, already has a lot of consequences separating time from space in many ways, especially the causal structure of spacetime.

It feels like in principle we should be able to chose any 3-D subspace of R4 and flow that along the spacetime manifold.

This in particular is a problem because while it has been proven that if you have a nicely behaved spacetime, you can take a spacelike hypersurface and "flow" it along a timelike curve, a generic spacetime with an arbitrary hypersurface may not work. This is related to the Cauchy problem of general relativity (the ability to extend the local solution of a PDE to the entire spacetime).

As the other answers have pointed out, there are geometric differences between space and time. However, if you want to know "how a field evolves" you have to specify what you mean. For example, we usually assume we can measure an entire field $\phi(x, y, z)$ at every point in space at a specific time. There are, however, applications where you might be measuring the field at a specific plane in space over time -- in this case you might make the longitudinal direction normal to the plane your independent variable and time as an independent variable. This is particularly useful in boundary value problems, which are not a big deal in QFT from what I understand but they could be elsewhere.

This appears in accelerator physics applications, where you might want to specify your coordinates in terms of a closed ideal trajectory and deviations from it, to which end your "time" is interpreted as the time of flight it took for a particle to get to that particular azimuth in a storage ring.

I think in practice, for most field theoretical applications, time is just the more convenient independent variable since you are not usually riding on a Galilean transformation of the coordinates and you are not dealing with boundary value problems.


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# Thread: Vector parallel to plane? Goes through origin?

1. ## Vector parallel to plane? Goes through origin?

Can someone please tell me how to answer this question?
My Algebra is not good so please in simple language.

http://i53.tinypic.com/6jj4hf.png
(didn't let me insert the image)

2. Why not just write it in? It's not that difficult. The given vector is <5, -24, 50> and the plane is [tex]x= \lambda<4, 5, 2>+ \mu<-5, 5, 3>+ <5, 2, -2>.

Is the vector parallel to the plane? Are there values for $\lambda$ and $\mu$ such that $\lambda<4, 5, 2>+ \mu<-5, 5, 3>= <5, -24, 50>$?

Is the vector in the plane? What does that mean? A vector is "movable" and does not actually lie in any plane. Are you interpreting <5,-24,50> as the vector from (0, 0, 0) to (5, -24, 50)? If so are there values for $\lambda$ and $\mu$ such that $\lambda<4, 5, 2>+ \mu<-5, 5, 3>= <5, -24, 50>$ and are there values for $\lambda$ and $\mu$ such that $\lambda<4, 5, 2>+ \mu<-5, 5, 3>= <0, 0, 0>$?

format: plain


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# How to prove the formula for nearest integer for a given real number x?

I'm reading Concrete Mathematics. The problem is from Exercise 3.2 of this book.

$$\lfloor x + 0.5\rfloor, \text{if \{x\} = 0.5 then ceil should be considered}\\ \lceil x - 0.5 \rceil, \text{if \{x\} = 0.5 then floor should be considered}$$ where {x} = fractional part of x

The problem is I'm not able to prove the above equations. How to prove it ?

• What is the question? I don't have the book. Sep 10 '17 at 8:49
• how to prove the above two formulas ? Sep 10 '17 at 8:50
• I was puzzled for a moment. I guess that he means: "prove that these formula give you the nearest integer". Sep 10 '17 at 8:51
• yes correct @badjohn Sep 10 '17 at 8:52

Since $\lfloor y\rfloor \le y<\lfloor y\rfloor+1$ and there does not exist an integer $n$ with $\lfloor y\rfloor <n<\lfloor y\rfloor+1$, the only candidates for the nearest integer to $y$ are $\lfloor y\rfloor$ and $\lfloor y\rfloor+1$. The distances are $|y-\lfloor y\rfloor|=y-\lfloor y\rfloor =\{y\}$ and $|y-\lfloor y\rfloor-1|=1-y+\lfloor y\rfloor =1-\{y\}$, respectively. To find the nearest, we have to compare $\{y\}$ against $1-\{y\}$ or equivalently, $2\{y\}$ againts $1$. Thus

• If $0\le \{y\}<\frac12$, then the nearest integer is $\lfloor y\rfloor$
• If $\frac12 <\{y\}<1$, then the nearest integer is $\lfloor y\rfloor+1$
• If $\{y\}=\frac12$, then both $\lfloor y\rfloor$ and $\lfloor y\rfloor+1$ are nearest integers.

Remains to show that these piecewise definitions coincide with $\lfloor y+\frac12\rfloor$ (the treatment of $\lceil y-\frac12\rceil$ is similar).

• If $0\le\{y\}<\frac12$ then $y+\frac12<\lfloor y\rfloor +\frac12+\frac12=\lfloor y\rfloor +1$ and hence $\lfloor y+\frac12\rfloor <\lfloor y\rfloor +1$. Together with $\lfloor y+\frac12\rfloor \ge\lfloor y\rfloor$ we conclude that $\lfloor y+\frac12\rfloor = \lfloor y\rfloor$ as desired.
• If $\frac12<\{y\}<1$ then $y+\frac12>\lfloor y\rfloor +\frac12+\frac12=\lfloor y\rfloor+1\in\Bbb Z$ and hence $\lfloor y+\frac12\rfloor \ge\lfloor y\rfloor +1$. Together with $\lfloor y+\frac12\rfloor\le \lfloor y+1\rfloor=\lfloor y\rfloor +1$, we conclude $\lfloor y+\frac12\rfloor =\lfloor y\rfloor +1$.
• If $\{y\}=\frac12$, one readily verifies that $\lfloor y+\frac12\rfloor=\lfloor y\rfloor +1$

In summary, for all $y\in\Bbb R$, $\lfloor y+\frac12\rfloor$ is a nearest integer of $y$.

• Very nice proof. I have just one question. Considering the fact that you know solution, You use different conditions (in this case 3) and arrive at single common expression (in this case $\lfloor y + \frac{1}{2} \rfloor$). Suppose you don't know what is the solution, how will you know what is the common single expression that you need to stop at, from all different conditions ? Sep 10 '17 at 9:31


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# ordinary differential equations example

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# Error when creating table with more than 9 columns using {*} shortcut [duplicate]

I can't imagine that this hasn't been asked before, but googling doesn't help me find the answer.

Creating a table with 9 columns using the shortcut {*9{l}} works fine:

\documentclass{article}
\begin{document}
\begin{tabular}{*9{l}}
\end{tabular}
\end{document}

But when I get into double digits, I get an error:

\documentclass{article}
\begin{document}
\begin{tabular}{*10{l}}
\end{tabular}
\end{document}

! LaTeX Error: Illegal character in array arg.

How can I use the shortcut {*} with >9 columns?

## marked as duplicate by Sverre, Community♦Sep 4 '15 at 10:28

• Maybe \begin{tabular}{*{10}{l}}? Seemed to work for me. – moewe Sep 4 '15 at 10:06

You will have to wrap the relevant number into curly braces if it is not single digit.

So

\documentclass{article}
\begin{document}
\begin{tabular}{*{12}{l}}
1&2&3&4&5&6&7&8&9&10&11&12\\
\end{tabular}
\end{document}

should cut it.

• I usually wait 24+ hrs before accepting an answer, but I can't imagine anyone else wanting to provide an alternative answer (that's better) than this. – Sverre Sep 4 '15 at 10:10
• @Sverre An alternative answer (only to support you imagination}: \documentclass{article} \begin{document} \catcode`?=\active \def?{12} %\begin{tabular}{{12}{l}} \begin{tabular}{?{l}} 1&2&3&4&5&6&7&8&9&10&11&12\\ \end{tabular} \end{document} :-) – Przemysław Scherwentke Sep 4 '15 at 11:03


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# 2016 Moderator Election Q&A - Question Collection

In connection with the moderator elections, we will be holding a Q&A with the candidates. This will be an opportunity for members of the community to pose questions to the candidates on the topic of moderation. Participation is completely voluntary.

The purpose of this thread was to collect questions for the questionnaire. The questionnaire is now live, and you may find it here.

Here's how it'll work:

• During the nomination phase, (so, until Monday, July 18th at 20:00:00Z UTC, or 4:00 pm EDT on the same day, give or take time to arrive for closure), this question will be open to collect potential questions from the users of the site. Post answers to this question containing any questions you would like to ask the candidates. Please only post one question per answer.

• We, the Community Team, will be providing a small selection of generic questions. The first two will be guaranteed to be included, the latter ones are if the community doesn't supply enough questions. This will be done in a single post, unlike the prior instruction.

• This is a perfect opportunity to voice questions that are specific to your community and issues that you are running into at current.

• At the end of the phase, the Community Team will select up to 8 of the top voted questions submitted by the community provided in this thread, to use in addition to the aforementioned 2 guaranteed questions. We reserve some editorial control in the selection of the questions and may opt not to select a question that is tangential or irrelevant to moderation or the election. That said, if I have concerns about any questions in this fashion, I will be sure to point this out in comments before the decision making time.

• Once questions have been selected, a new question will be opened to host the actual questionnaire for the candidates, containing 10 questions in total.

• This is not the only option that users have for gathering information on candidates. As a community, you are still free to, for example, hold a live chat session with your candidates to ask further questions, or perhaps clarifications from what is provided in the Q&A.

If you have any questions or feedback about this process, feel free to post as a comment here.

• Am I missing something here? Have we begun nominating moderators? Where is the nomination post? – GentlePurpleRain Jul 11 '16 at 20:21
• @GentlePurpleRain I think the sidebar just hasn't updated yet. The election is active, though, and shows up at http://puzzling.stackexchange.com/election. – user20 Jul 11 '16 at 20:30
• Off-topic, but I thought it was funny that ~11.7% of the letters used in the body of the text went towards spelling "question(s)". – SpiritFryer Jul 15 '16 at 5:18

On multiple occasions, there have been calls to make the dynamics of PuzzlingSE deviate a bit from those of StackExchange in general. For example, suggestions to increase question upvote to 10 rep to reward questions with superb effort, option to award bounties to questions etc. How do you feel about such special rules for PuzzlingSE and what effort would you make to support/discourage these changes?

• This is a really good question. – ABcDexter Jul 12 '16 at 9:16
• This is particularly relevant because Puzzling is an outlier (well, mostly) on the Stack Exchange network in that most "questions" aren't actually "questions" at all. Moderators are the primary communication channel between a SE site and the SE dev/community team, which is especially important for a site like Puzzling. – Doorknob Jul 12 '16 at 11:02
• I think that because puzzling is currently a very small comminity, in comparison to others, the reputation ampunts it gives should be higher as an incentive. However, that would mean giving older users an extreme jump in reputation. That could throw off sitewide statistics. Therefore, I am against such a change. – user64742 Jul 17 '16 at 21:22
• @TheGreatDuck: I have several issues with your response. Firstly, I don't think Puzzling can be called a "very small community" anymore. The fact that the site has graduated completely shows that there is a critical mass of users. Additionally, the change in question bonus is not needed to provide greater incentive to new users. It is in fact a way that we can truly identify those users who put considerable effort into making high quality content. – CodeNewbie Jul 18 '16 at 5:00
• @TheGreatDuck: If it grants older users (who have 100+ upvotes on quite a few of their questions) a rep jump, I would not find that undeserved. Even if the decision to increase rep for question upvotes is passed and the changes are not applied retrospectively, it still ensures that the right kind of questions garner the right kind of reward. – CodeNewbie Jul 18 '16 at 5:09
• @CodeNewbie I am thinking of global SE rep counts. Also, puzzling is way smaller than the other big communities on SE. It may be a big community but for a SE community, it is one of the smaller ones. It would rank lower in site population statistics. I find this good, IMO. A smaller site is easier to manage. – user64742 Jul 18 '16 at 5:14

As a moderator, your close, reopen, delete, and undelete votes will become binding. How will this affect the way you currently vote, especially with close/reopen votes? In general, will your moderation style be more lasseiz-faire or proactive? How so?

• I will be less likely to vote to close or delete things. In general, I am against closing except where it aids the community. For instance, a spam post will get deleted, but a porrly written riddle or one that is hard to solve will merely recieve an edit or an upvote from me as I think harder riddles are actually better than easy riddles. – user64742 Jul 17 '16 at 21:24

What is your personal opinion the "purpose" of PuzzlingSE should be?

( Yes, there is no overall consensus on this question, yet. This makes it even more imporant to know what would-be-moderators think. )

• A fun place where one can stop-by and relax solving puzzles or posting ideas
• A competition site where users compete to solve tricky puzzles
• A comunity-curated repository of creative, new and good puzzles
• A playground for both puzzlers ans puzzle-creators to test their ideas
• A knowledge database for all facts and questions concerning puzzles and puzzling
• ...
• This should be asked as it is a really important question. Good work :) – ABcDexter Jul 17 '16 at 11:13
• I think puzzling is a site where users create puzzles for other users to solve. This comprises the majority of the site. There are also puzzle related questions as well. – user64742 Jul 17 '16 at 21:38
• @BmyGuest I wish to answer them, unless that is a problem? – user64742 Jul 17 '16 at 21:41

How much time per week (minimum) do you think that you can invest on moderation over the long term and continuously, and what would you do if you realize that life made it impossible to fulfill this commitment?

• I think that this question is the most important for this community coz we have already witnessed the unwillingness of moderation of this community of two moderators out of three. – manshu Jul 12 '16 at 10:59
• While one moderator has been very active on site, I wouldn't go as far as calling the others unwilling. Less active yes, but they did chip in during serious disputes – BmyGuest Jul 12 '16 at 22:18
• Sorry, I must have gone blind/inactive during those serious disputes. – manshu Jul 14 '16 at 6:08
• @manshu Way before your time on site (according to your 7months membership). There were rather heated discussion of the way the site should progress, as originally it was a site much like all other SE sites with questions about puzzles and puzzling, but not puzzles themselves. Not everybody agreed with the way the site was moving, and while it is hugely popular (I think), there is still need to steer & moderate its direction. – BmyGuest Jul 14 '16 at 6:41
• Per week? Probably around 50 hours. I already spend that much time on SE, and I see no reason why my activity will not increase. Once my college semester starts in the fall, it will be even easier for me to find access to a computer and whatnot. So if anything, I only see it increasing from here. As for what I would do if I found it to be impossible...? The only way that is happening is if the computers break, and in that situation... i will drop off never to be seen again. – user64742 Jul 17 '16 at 21:24

As is witnessed in every human interaction, and has been seen in the past on this site, no one gets along with everyone. As a moderator, how would you remain neutral towards a user/users who you find yourself disliking?

• The simple fact of the matter is that I really do not care what others think of me, and while posts are annoying, I rarely remember names let alone a list of people I do not like. The only people I dislike are those who are flat out rude or disrepectful. Besides, on a community based in Q&A I rarely get to know other users in any way other than them being well known. – user64742 Jul 17 '16 at 21:41

Scenario: An user posts a strangely-written challenge. Some people flag, post comments doubting the OP's intention, downvote or vote to close it because it seems to be unclear, off-topic, overly-broad or something like that. After some debate in comments and/or chat, the OP insists that it is a very clever puzzle and that it is perfectly valid, possibly even triggering a close-reopen war.

Questions:

1. How do you react to that sort of puzzles?
2. What if the user is very experienced in this site and already posted lots of extremely well-written questions or answers?
3. What if it is a newbie posting his first question instead?
4. What if after being severely downvoted, ranted and attacked, when the solution is revealed, it is shown to be a brillant puzzle afterall?
5. What if after being severely upvoted and praised, even if no one could understand anything initially, when the solution is revealed, it was in fact shown to be a very stupid and lame puzzle?

Pardon me if this is a lot of questions smashed together, but all those questions in fact are very closely related and expected to be answered together within a single answer.

The purpose of this question is to evaluate how a moderator candidate would handle such situation, something that is somehat common in puzzling, but probably never happens anywhere else in the Stack Exchange network.

• I would not look at the opinions of others or how it was closed or reopened. Instead, I would attempt to determine a solution on my own if only to see if the question does in fact make sense and to prove those who are trigger happy wrong. Then, if the answer is revealed and seems logical I would proceed to leave a comment/answer explaining said logic. If this devolves into an vote war, I will probably post that we choose a common ground to have one final vote so as not to annoy other users. If it is still an unclear decision, then I will insist it stay open so as not to punish the user for it. – user64742 Jul 17 '16 at 21:55
• TLDR: better to leave a post in the middle of a controversial decision open and risk mild confusion then to punish a user for something we cannot even make clear of ourselves. We should have our affairs in order with a majority aggreement before we decide to randomly close a post. Of course, this only applies to a close vote war. Otherwise I am likely to ignore it other than an idle pole at the post out of curiosity. – user64742 Jul 17 '16 at 21:58
• @TheGreatDuck Good to know. However, when the nomination phase ends, the 8 highest voted questions from this questionnaire will be joined together and posted as a new meta-question in which each moderator candidate will be able to answer (and debate to some degree). So, instead of answering here in comments, there would be a proper place for answering them altogether, just hold on your impulse for a few more hours! If some question that you wanted to answer don't get into the list, I see no problem if you answer it anyway. – Victor Stafusa Jul 17 '16 at 22:38
• @TheGreatDuck Also, it is good that you are formulating your answers in advance. Hope to see you in the soon-to-be-posted meta-question! – Victor Stafusa Jul 17 '16 at 22:39
• I was not aware of this. I thought it was just a general forum for the mods to answer questions. – user64742 Jul 18 '16 at 2:26
• @TheGreatDuck Post here your answers: meta.puzzling.stackexchange.com/q/5114/5044 – Victor Stafusa Jul 19 '16 at 6:44

Do you consider your role as a moderator to be...

• Passive/reactive - taking action when prompted by other users. Having a close ear to the comunity, but keeping back with own opinions and ideas if not prompted for them. Mainly acting to solve/disarm disputes and keeping this a friendly place.

• Proactive - trying to push the site forward. Initiating new discussions/developments, i.e. starting meta-post questions to get the comunity developing itself over time.

• Editorial - Constantly monitoring and editing content according to established guidelines and enforcing these guidelines, but not influencing what the guidelines are...

• ...

How will being a moderator change your participation as user on this site?

For example, will you still post your own puzzle-questions and solutions, or will the diamond at your name make you more reluctant to do either?

• I will still post the same. I will be more active in reviewing and scrutinizing posts but little will change. – user64742 Jul 17 '16 at 22:02

### What makes you emotionally charged when dealing with moderator tasks or user-interaction on site?

Yes, you're calm and level-headed and stuff, but is there really nothing that rises your blood-pressure and makes you potentially judge hastily/emotionally?

And if you notice that your temper goes up, how do you usually handle it?

How would you handle a situation in which a user is upset at a moderator action that you have taken? For example, if someone posted on meta "This mod deleted my question/answer/comment and is abusing their power," how would you react?

• Just delete the meta post + permanent ban. Anyone who provoke a diamond should burn in SE hell ! – Fabich Jul 12 '16 at 9:20
• I'll write my view point as a detailed answer. Then wait for the community's take, it's up to them to decide ( by up/down voting on question and answer). – ABcDexter Jul 12 '16 at 17:50
• @Lordofdark That's the spirit! No chance for complaints. Mods should be unreachable. – Avigrail Jul 13 '16 at 14:00
• If I was the one who is the moderator in question, I will likely just apologize and explain my behavior. I won't care what the upvotes/downvotes say. Until I am told otherwise, I will stand by my decision. If the post was spam, then it should stay closed or deleted. I cannot conceive any other instance of me deleting a post. – user64742 Jul 17 '16 at 22:01

Here is a set of general questions, gathered as very common questions asked every election. As mentioned in the instructions, the first two questions are guaranteed to show up in the Q&A, while the others are if there aren't enough questions (or, if you like one enough, you may split it off as a separate answer for review within the community's 8).

• How would you deal with a user who produced a steady stream of valuable answers, but tends to generate a large number of arguments/flags from comments?
• How would you handle a situation where another mod closed/deleted/etc a question that you feel shouldn't have been?

• In your opinion, what do moderators do?
• A diamond will be attached to everything you say and have said in the past, including questions, answers and comments. Everything you will do will be seen under a different light. How do you feel about that?
• In what way do you feel that being a moderator will make you more effective as opposed to simply reaching 10k or 20k rep?

When you evaluate the opinion of other users from their posts (comments, answers, discussions), what influences their 'opinion weight' for you? (And why?)

• Just the pure facts which were posted
• Style and spelling of the posts
• Complexity and detail of the posts
• User reputation on puzzling
• Overall user-reputation
• History and track-record of the user
• The comunities reaction to the post
• Whether or not the opinion matches yours
• ...
• Well, everything except for reputation has a little bit of a play in it. Ordinarily I would say a user's track record isn't relevant however, a notorious troll saying "let's add a new tag called "graphical"" is going to be met with skepticism. However, that's where the communities opinion comes into play. If the community is vividly claiming such a post to be spam and the user had a history of bad behavior, it's a no brainer. Obviously, I would still use my own opinion. After all, my purpose would be to NOT jump on the bandwagon. Otherwise, there would be no point for elected mods. – user64742 Jul 19 '16 at 1:19
• One could just arrange for everything to be voted on by high rep user in different queues. Then it would be the popular opinion most of the time. – user64742 Jul 19 '16 at 1:19

How would you handle a new user, the 7th you've seen today, and the 352nd this month, on this website that posts one or two frowned upon (don't want to say low-quality) questions like the type of those listed below?

• What's the next number: 1, 19, $\pi$, $a$, $IV$ ?
• I have 10 sheep and my neighbor has -2. How many do we have in total?

[Thanks @BmyGuest for the improvement]

• Good one. I would maybe extend to "How would you handle the 1000th new user...." as it will be a questons of patience and effort. – BmyGuest Jul 12 '16 at 11:10
• @BmyGuest Ha. good idea. I will edit it. – Marius Jul 12 '16 at 11:11
• Dude the first puzzle is greater than most of the number sequences :p – Fabich Jul 12 '16 at 11:43
• @Lordofdark. Do you have a solution for it? Because I don't. I will give you a bounty even. – Marius Jul 12 '16 at 11:45
• @Marius I have. It is 42. – dryairship Jul 12 '16 at 11:49
• @Hackerdarshi. nope. I just solved it and the answer is 'bugs bunny'. – Marius Jul 13 '16 at 10:57
• @Marius Ok. But I know that my solution of the second question is correct. It is ∞, because if the neighbor has -2 sheep, then he is a ghost, who has, of course, also (along with -2 sheep), ∞ sheep, and the total is again ∞. – dryairship Jul 13 '16 at 12:03
• Nope. The answer is 0, because the first one said the number in binary. – Marius Jul 13 '16 at 12:09
• If you are lost and lonely in a remote mid-ocean island, and your dog's boat which was flying got all the five tires flat, how many timber will Sandy need in order to build a stone car, so a half-chicken would lay an half-egg that takes your aunt out of the island in a nuclear rocket landing on Mars? – Victor Stafusa Jul 13 '16 at 17:58
• @VictorStafusa: I vote to close as off-topic. – CodeNewbie Jul 14 '16 at 4:53
• @VictorStafusa The answer is (i^i)*V*dx/dt where i is (-1)^1/2, V is the volume of the stone car and dx/dt is the differentiation of the distance of the island from Mars with respect to time. – dryairship Jul 17 '16 at 12:06
• I will tell them as I have always said, please ensure that your post has a unique solution. If it does and it is clear, then there is no need to do a thing. If it has many other (more than 15) possible solutions that fit the conditions you gave without error, then please delete the post. We do not want to induce high traffic questions due to wild mass guessing. However, high traffic questions due to amazing quality are welcome. And then I might link to the help center. – user64742 Jul 17 '16 at 22:05

A diamond will be attached to everything you say and have said in the past, including questions, answers and comments. Everything you will do will be seen under a different light. How do you feel about that?

I'm shamelessly copying this filler question from Grace Note's answer, because I think it's important enough to put on the full list.

What do you think this site's biggest challenge is? (E.g. post quality or quantity, too many/few closures, bad tools/guidance, etc.) What do you think should be done about it (by anybody: moderators, users in general, Stack Exchange staff, ...)?

• This is by far the most important question! – Avigrail Jul 14 '16 at 7:48
• This is an example of a post that I would delete. It is something that has the makings of a spam bot. There is a weird meaningless statement, a link, and an avatar that looks like a company logo. Spambot indeed, lol! (Just kidding. I know this is a fake spam post.) – user64742 Jul 17 '16 at 22:07
• @TheGreatDuck Not spam! British in-joke. In this country every aspiring politician has to run the gauntlet of politically influential online forum Mumsnet, and they always get asked 'the biscuit question '. I plan to base my voting in this mod-election entirely on biscuit preference (not really - but maybe 75%)... – A E Jul 18 '16 at 17:41
• Oh, I thought it was a fake spambot post as a joke. Y'know cause a moderator questionairre containing spam would be ironic. – user64742 Jul 18 '16 at 20:15

What questions (pick at least 10) from this site do you think are representative high-quality questions this site would need more of?

Note, the focus here is on what you think are examples of high quality not on what type of puzzle is welcome here.

Similarly, what answers (pick at least 3) from this site do you think are representative high-quality answers to questions?

This question shows the 'content knowledge' a candidate has as well as his opinion on 'quality standards' by example (as there are no strict comunity rules on quality yet.)

As an side-effect, we might get a few "curated" lists to be kept on meta which we can link to when explaining newcomers what good content is like.

• I don't think the answers to this question are relevant. This will not make you a good moderator or a bad one. It will just highlight the preferences for different puzzle type. For example I like alphametic puzzles but I know they are not really the most intriguing puzzles on this website. But I still like them because ... I have no reason. And it's not (only) a moderators job to keep a high quality on a website. It takes a team effort. If 100 people say a question has high quality and the moderator does not like it this still means it's a high quality question. – Marius Jul 12 '16 at 10:54
• @Marius: I was thinking the same first, but as we do not have real "guidelines" for quality content on this site - and have been strugglign to define one as a comunity ever since early alpha - I for once would like to give my moderator-vote to a person whos 'overall ideas' I share. As "quality" is sooo hard to define for this site, "defenition by example" is the next-best proxy I can think of. – BmyGuest Jul 12 '16 at 10:59
• I understand your idea. I've read your post above 3 times. I agree that some guidelines need to be established and is not an easy job to do. But I'm almost sure this is not the way to do it. I can predict what's going to happen. Some of the people will list the top questions in their favorite tag, others will just pick 10 from the top 20 most voted questions. – Marius Jul 12 '16 at 11:02
• @Marius Fair point, and UV/DV on this post will help sort the questionaire. I agree that "some" candidates will only do this the "quick and dirty" way, but that by itself is informative. The questions is also a way for the candiates to "show-off" their knowledge of existing content, which would be helpful. – BmyGuest Jul 12 '16 at 11:05
• oh...and by the way. I didn't want to start a DV trend on this post. It was just my opinion and I added a comment to why I think so. It seams the other downvoters didn't want to give a reason. Come on people. you can write 10 words about why you don't like this. – Marius Jul 12 '16 at 11:07
• Personal opinions on what constitutes a good question aren't really that relevant - it's the SE standards that matter. – Mithical Jul 12 '16 at 11:23
• No downvote, but I think this would just be some high-voted questions. – palsch Jul 12 '16 at 11:33

What moderating experience have you had before this?

• I don't particularly like this question, because this is like expecting a young vibrant 25 year old with 30+ years of industry experience. A new moderator could be just as good as someone with experience if they have spent enough time on the site and understand how SE works and what PuzzlingSE needs to grow within that framework. – CodeNewbie Jul 14 '16 at 4:56


text:
# Integral t = tan(x/2) substitution

 HW Helper P: 2,685 $$\int \frac{dx}{5 - 3\sin x + 4 cos x}$$ I know I have to use t = tan(x/2) substitution, and after i do that and symplify I get: $$\int \frac{dt}{2t^2 - 3t + 1}$$ I dont know where to go from here. If anyone can see where to go, please help. Also, if there is an easier way to do this, please tell me!! Thanks.
 HW Helper P: 2,685 Integral t = tan(x/2) substitution ok i got an answer. $$| \ln \frac{\tan(x/2) - 1}{2\tan(x/2) - 1)^2} + C$$ This seems reasonable. If anyone can find a mistake, please tell me. Thanks for the help.
 P: 167 Hmm, I think there is something wrong with your expression $$\int \frac{dt}{2t^2 - 3t + 1}$$ I tried doing the substitution myself and the expression in the denominator was a "perfect square". Maybe you can show us how you did the substitution, or you can try checking your work again...


text:
# Is it a known phenomenon for the variance of a component (GMM) to increase without stopping?

I know it can happen for it to decrease dramatically as it overfits on a single datapoint. But I've never read about a component "taking everything over". See the following images (circles are stddevs). Could this be (relatively) normal behaviour?

edit: For some reason changing my gaussian function made the behaviour sane, note both return the same output for given inputs:

iter 300!

• How are you fitting the mixture model.. is it EM? What does the circles stand for... a contour plot of sort? – Suren Jan 23 '18 at 0:33
• Yes EM. Contour is stddev of the component (root of variance, covar matrix is diagonal). – Nimitz14 Jan 23 '18 at 9:11
• It is very much possible to end up with smaller size cluster, if you are fitting an incorrect model, say larger number of clusters. By looking at this data, even though it is 2D, it is difficult to say how many clusters would be appropriate or what type of mixture model you should fit. I'd recommend trying something like mclust software to see what it does, and then compare that with outcomes of your algorithm. – Suren Jan 24 '18 at 0:27


text:
# Mechanical–electrical analogies

Mechanical–electrical analogies are the representation of mechanical systems as electrical networks. At first, such analogies were used in reverse to help explain electrical phenomena in familiar mechanical terms. James Clerk Maxwell introduced analogies of this sort in the 19th century. However, as electrical network analysis matured it was found that certain mechanical problems could more easily be solved through an electrical analogy. Theoretical developments in the electrical domain[note 1] that were particularly useful were the representation of an electrical network as an abstract topological diagram (the circuit diagram) using the lumped element model and the ability of network analysis to synthesise a network to meet a prescribed frequency function.

This approach is especially useful in the design of mechanical filters—these use mechanical devices to implement an electrical function. However, the technique can be used to solve purely mechanical problems, and can also be extended into other, unrelated, energy domains. Nowadays, analysis by analogy is a standard design tool wherever more than one energy domain is involved. It has the major advantage that the entire system can be represented in a unified, coherent way. Electrical analogies are particularly used by transducer designers, by their nature they cross energy domains, and in control systems, whose sensors and actuators will typically be domain-crossing transducers. A given system being represented by an electrical analogy may conceivably have no electrical parts at all. For this reason domain-neutral terminology is preferred when developing network diagrams for control systems.

Mechanical–electrical analogies are developed by finding relationships between variables in one domain that have a mathematical form identical to variables in the other domain. There is no one, unique way of doing this; numerous analogies are theoretically possible, but there are two analogies that are widely used: the impedance analogy and the mobility analogy. The impedance analogy makes force and voltage analogous while the mobility analogy makes force and current analogous. By itself, that is not enough to fully define the analogy, a second variable must be chosen. A common choice is to make pairs of power conjugate variables analogous. These are variables which when multiplied together have units of power. In the impedance analogy, for instance, this results in force and velocity being analogous to voltage and current respectively.

Variations of these analogies are used for rotating mechanical systems, such as in electric motors. In the impedance analogy, instead of force, torque is made analogous to voltage. It is perfectly possible that both versions of the analogy are needed in, say, a system that includes rotating and reciprocating parts, in which case a force-torque analogy is required within the mechanical domain and a force-torque-voltage analogy to the electrical domain. Another variation is required for acoustical systems; here pressure and voltage are made analogous (impedance analogy). In the impedance analogy, the ratio of the power conjugate variables is always a quantity analogous to electrical impedance. For instance force/velocity is mechanical impedance. The mobility analogy does not preserve this analogy between impedances across domains, but it does have another advantage over the impedance analogy. In the mobility analogy the topology of networks is preserved, a mechanical network diagram has the same topology as its analogous electrical network diagram.

## Applications

Mechanical–electrical analogies are used to represent the function of a mechanical system as an equivalent electrical system by drawing analogies between mechanical and electrical parameters. A mechanical system by itself can be so represented, but analogies are of greatest use in electromechanical systems where there is a connection between mechanical and electrical parts. Analogies are especially useful in analysing mechanical filters. These are filters constructed of mechanical parts but designed to work in an electrical circuit through transducers. Circuit theory is well developed in the electrical domain in general and in particular there is a wealth of filter theory available. Mechanical systems can make use of this electrical theory in mechanical designs through a mechanical–electrical analogy.[1]

Mechanical–electrical analogies are useful in general where the system includes transducers between different energy domains.[note 1] Another area of application is the mechanical parts of acoustic systems such as the pickup and tonearm of record players. This was of some importance in early phonographs where the audio is transmitted from the pickup needle to the horn through various mechanical components entirely without electrical amplification. Early phonographs suffered badly from unwanted resonances in the mechanical parts. It was found that these could be eliminated by treating the mechanical parts as components of a low-pass filter which has the effect of flattening out the passband.[2]

Electrical analogies of mechanical systems can be used just as a teaching aid, to help understand the behaviour of the mechanical system. In former times, up to about the early 20th century, it was more likely that the reverse analogy would be used; mechanical analogies were formed of the then little understood electrical phenomena.[3]

## Forming an analogy

Electrical systems are commonly described by means of a circuit diagram. These are network diagrams that describe the topology of the electrical system using a specialised graph notation. The circuit diagram does not try and represent the true physical dimensions of the electrical components or their actual spatial relationship to each other. This is possible because the electrical components are represented as ideal lumped elements, that is, the element is treated as if it is occupying a single point (lumped at that point). Non-ideal components can be accommodated in this model by using more than one element to represent the component. For instance, a coil intended for use as an inductor has resistance as well as inductance. This can be represented on the circuit diagram as a resistor in series with an inductor.[4] Thus, the first step in forming an analogy of a mechanical system is to describe it as a mechanical network in a similar way, that is, as a topological graph of ideal elements.[5] Alternative, more abstract, representations to the circuit diagram are possible, for instance the bond graph.[6]

A mechanical network diagram of a simple resonator (top) and one possible electrical analogy for it (bottom)

In an electrical network diagram, limited to linear systems, there are three passive elements: resistance, inductance, and capacitance; and two active elements: the voltage generator, and the current generator.[note 2] The mechanical analogs of these elements can be used to construct a mechanical network diagram. What the mechanical analogs of these elements are depends on what variables are chosen to be the fundamental variables. There is a wide choice of variables that can be used, but most commonly used are a power conjugate pair of variables (described below) and the pair of Hamiltonian variables derived from these.[7]

There is a limit to the applicability of this lumped element model. The model works well if the components are small enough that the time taken for a wave to cross them is insignificant, or equivalently, if there is no significant phase difference in the wave either side of the component. What amounts to significant depends on how accurate the model is required to be, but a common rule of thumb is to require components to be smaller than one sixteenth of a wavelength.[8] Since wavelength decreases with frequency, this puts an upper limit on the frequency that can be covered in this kind of design. This limit is much lower in the mechanical domain than the equivalent limit in the electrical domain. This is because the much higher propagation speeds in the electrical domain lead to longer wavelengths (mechanical vibrations in steel propagate at about 6,000 m/s,[9] electromagnetic waves in common cable types propagate at about 2 x 108 m/s[10]). For instance, traditional mechanical filters are only made up to around 600 kHz[11] (although MEMS devices can operate at much higher frequencies due to their very small size). In the electrical domain, on the other hand, the transition from the lumped element model to the distributed element model occurs in the hundreds of megahertz region.[12]

In some cases it is possible to continue using a topological network diagram even when components needing a distributed element analysis are present. In the electrical domain, a transmission line, a basic distributed element component, can be included in the model with the introduction of the additional element of electrical length.[13] The transmission line is a special case because it is invariant along its length and hence the full geometry need not be modelled.[14] Another way of dealing with distributed elements is to use a finite element analysis whereby the distributed element is approximated by a large number of small lumped elements. Just such an approach was used in one paper to model the cochlea of the human ear.[15] Another condition required of electrical systems for the application of the lumped element model is that no significant fields exist outside the component since these can couple to other unrelated components.[16] However, these effects can often be modelled by introducing some virtual lumped elements called strays or parasitics.[17] An analog of this in mechanical systems is vibration in one component being coupled to an unrelated component.[18]

### Power conjugate variables

The power conjugate variables are a pair of variables whose product is power. In the electrical domain the power conjugate variables chosen are invariably voltage (v) and current (i). Thus, the power conjugate variables in the mechanical domain are analogs. However, this is not enough to make the choice of mechanical fundamental variables unique. The usual choice for a translational mechanical system is force (F) and velocity (u) but it is not the only choice. A different pair may be more appropriate for a system with a different geometry, such as a rotational system.[19]

Even after the mechanical fundamental variables have been chosen, there is still not a unique set of analogs. There are two ways that the two pairs of power conjugate variables can be associated with each other in the analogy. For instance the associations F with v and u with i can be made. However, the alternative associations u with v and F with i are also possible. This leads to two classes of analogies, the impedance analogies and the mobility analogies.[20] These analogies are the dual of each other. The same mechanical network has analogs in two different electrical networks. These two electrical networks are the dual circuits of each other.[21]

### Hamiltonian variables

The Hamiltonian variables, also called the energy variables, are those variables r = (q, p), which are conjugate according to Hamilton's equations:[22]

${\displaystyle {\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}\quad ,\quad {\frac {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}=+{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}}}$

Further, the time derivatives of the Hamiltonian variables are the power conjugate variables.

The Hamiltonian variables in the electrical domain are charge (q) and flux linkage (λ) because,

${\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q}}=-{\frac {d\lambda }{dt}}=v}$ (Faraday's law of induction) and, ${\displaystyle {\frac {\partial {\mathcal {H}}}{\partial \lambda }}={\frac {dq}{dt}}=i}$

In the translational mechanical domain the Hamiltonian variables are distance displacement (x) and momentum (p) because,

${\displaystyle {\frac {\partial {\mathcal {H}}}{\partial x}}=-{\frac {dp}{dt}}=F}$ (Newton's second law of motion) and, ${\displaystyle {\frac {\partial {\mathcal {H}}}{\partial p}}={\frac {dx}{dt}}=u}$

There is a corresponding relationship for other analogies and sets of variables.[23] The Hamiltonian variables are also called the energy variables. The integrand of a power conjugate variable with respect to a Hamiltonian variable is a measure of energy. For instance,

${\displaystyle \int Fdx}$ and, ${\displaystyle \int udp}$

are both expressions of energy. They can also be called generalised momentum and generalised displacement after their analogs in the mechanical domain. Some authors discourage this terminology because it is not domain neutral. Likewise, the use of the terms I-type and V-type (after current and voltage) is also discouraged.[24]

## Classes of analogy

There are two principle classes of analogy in use. The impedance analogy (also called the Maxwell analogy) preserves the analogy between mechanical, acoustical and electrical impedance but does not preserve the topology of networks. The mechanical network is arranged differently to its analogous electrical network. The mobility analogy (also called the Firestone analogy) preserves network topologies at the expense of losing the analogy between impedances across energy domains. There is also the through and across analogy, also called the Trent analogy. The through and across analogy between the electrical and mechanical domain is the same as in the mobility analogy. However, the analogy between the electrical and acoustical domains is like the impedance analogy. Analogies between the mechanical and acoustical domain in the through and across analogy have a dual relationship with both the impedance analogy and mobility analogy.[25]

Different fundamental variables are chosen for mechanical translation and rotational systems leading to two variants for each of the analogies. For instance, linear distance is the displacement variable in a translational system, but this is not so appropriate for rotating systems where angle is used instead. Acoustical analogies have also been included in the descriptions as a third variant. While acoustical energy is ultimately mechanical in nature, it is treated in the literature as an instance of a different energy domain, the fluid domain, and has different fundamental variables. Analogies between all three domains − electrical, mechanical and acoustical − are required to fully represent electromechanical audio systems.[26]

### Impedance analogies

Impedance analogies, also called the Maxwell analogy, classify the two variables making up the power conjugate pair as an effort variable and a flow variable. The effort variable in an energy domain is the variable analogous to force in the mechanical domain. The flow variable in an energy domain is the variable analogous to velocity in the mechanical domain. Power conjugate variables in the analog domain are chosen that bear some resemblance to force and velocity.[27]

In the electrical domain, the effort variable is voltage and the flow variable is electrical current. The ratio of voltage to current is electrical resistance (Ohm's law). The ratio of the effort variable to the flow variable in other domains is also described as resistance. Oscillating voltages and currents give rise to the concept of electrical impedance when there is a phase difference between them. Impedance can be thought of as an extension to the concept of resistance. Resistance is associated with energy dissipation. Impedance encompasses energy storage as well as energy dissipation.

The impedance analogy gives rise to the concept of impedance in other energy domains (but measured in different units).[28] The translational impedance analogy describes mechanical systems moving in a single linear dimension and gives rise to the idea of mechanical impedance. The unit of mechanical impedance is the mechanical ohm; in SI units this is N-s/m, or Kg/s.[29] The rotational impedance analogy describes rotating mechanical systems and gives rise to the idea of rotational impedance. The unit of rotational impedance in the SI system is N-m-s/rad.[30] The acoustical impedance analogy gives rise to the idea of acoustic impedance. The unit of acoustic impedance is the acoustic ohm; in SI units this is N-s/m5.[31]

Variables[32]
Type Mechanical translation

variable

Mechanical rotation

variable

Acoustical variable Analogous

electrical variable

Power conjugate pair Effort variable Force Torque Pressure Voltage
Flow variable Velocity Angular velocity Volume flow rate Current
Hamiltonian variables Effort Hamiltonian Momentum Angular momentum Pressure-momentum Flux linkage
Flow Hamiltonian Displacement Angle Volume Charge
Elements Damping Rotational resistance Acoustic resistance Resistance
Mass Moment of inertia Acoustic mass[note 3] Inductance
Compliance Rotational compliance Acoustic compliance Capacitance
Mechanical impedance Mechanical impedance Acoustic impedance Electrical impedance

### Mobility analogies

Mobility analogies, also called the Firestone analogy, are the electrical duals of impedance analogies. That is, the effort variable in the mechanical domain is analogous to current (the flow variable) in the electrical domain, and the flow variable in the mechanical domain is analogous to voltage (the effort variable) in the electrical domain. The electrical network representing the mechanical system is the dual network of that in the impedance analogy.[33]

The mobility analogy is characterised by admittance in the same way that the impedance analogy is characterised by impedance. Admittance is the algebraic inverse of impedance. In the mechanical domain, mechanical admittance is more usually called mobility.[34]

Variables[35]
Type Mechanical translation

variable

Mechanical rotation

variable

Acoustical variable Analogous

electrical variable

Power conjugate pair Effort variable Force Torque Pressure Current
Flow variable Velocity Angular velocity Volume flow rate Voltage
Hamiltonian variables Effort Hamiltonian Momentum Angular momentum Pressure-momentum Charge
Flow Hamiltonian Displacement Angle Volume Flux linkage
Elements Responsiveness[note 4] Rotational responsiveness Acoustic conductance Resistance
Mass Moment of inertia Acoustic mass Capacitance
Compliance Rotational compliance Acoustic compliance Inductance
Mobility Rotational mobility Acoustic admittance Electrical impedance

### Through and across analogies

Through and across analogies, also called the Trent analogy, classify the two variables making up the power conjugate pair as an across variable and a through variable. The across variable is a variable that appears across the two terminals of an element. The across variable is measured relative to the element terminals. The through variable is a variable that passes through, or acts through an element, that is, it has the same value at both terminals of the element. The benefit of the through and across analogy is that when the through Hamiltonian variable is chosen to be a conserved quantity, Kirchhoff's node rule can be used, and the model will have the same topology as the real system.

Thus, in the electrical domain the across variable is voltage and the through variable is current. In the mechanical domain the analogous variables are velocity and force, as in the mobility analogy.[36] In the acoustic system, pressure is an across variable because pressure is measured relative to the two terminals of an element, not as an absolute pressure. It is thus not analogous to force which is a through variable, even though pressure is in units of force per area. Forces act through an element; a rod with a force applied to the top will transmit the same force to an element connected to its bottom. Thus, in the through and across analogy the mechanical domain is analogous to the electrical domain like the mobility analogy, but the acoustical domain is analogous to the electrical domain like the impedance analogy.[37]

Variables[38]
Type Mechanical translation

variable

Mechanical rotation

variable

Acoustical variable Analogous

electrical variable

Power conjugate pair Across variable Velocity Angular velocity Pressure Voltage
Through variable Force Torque Volume flow rate Current
Hamiltonian variables Across Hamiltonian Displacement Angle Pressure-momentum Flux linkage
Through Hamiltonian Linear momentum Angular momentum Volume Charge

### Other energy domains

The electrical analogy can be extended to many other energy domains. In the field of sensors and actuators, and for control systems using them, it is a common method of analysis to develop an electrical analogy of the entire system. Since sensors can be sensing a variable in any energy domain, and likewise outputs from the system can be in any energy domain, analogies for all energy domains are required. The following table gives a summary of the most common power conjugate variables used to form analogies.[39]

Energy domain analogies[40]
Energy domain Effort variable Flow variable
Electrical Voltage Current
Mechanical Force Velocity
Fluid Pressure Volume flow rate
Thermal Temperature difference Entropy flow rate
Magnetic Magnetomotive force (mmf) Magnetic flux rate of change
Chemical Chemical potential Molar flow rate

It is perhaps more common in the thermal domain to choose temperature and thermal power as the fundamental variables because, unlike entropy, they can be measured directly. The concept of thermal resistance is based on this analogy. However, these are not power conjugate variables and are not fully compatible with the other variables in the table. An integrated electrical analogy across multiple domains that includes this thermal analogy will not correctly model energy flows.[41]

Similarly, the commonly seen analogy using mmf and magnetic flux as the fundamental variables, which gives rise to the concept of magnetic reluctance, does not correctly model energy flow. The variable pair mmf and magnetic flux is not a power conjugate pair. This reluctance model is sometimes called the reluctance-resistance model since it makes these two quantities analogous. The analogy shown in the table, which does use a power conjugate pair, is sometimes called the gyrator–capacitor model.[42]

## Transducers

A transducer is a device that takes energy from one domain as input and converts it to another energy domain as output. They are often reversible, but are rarely used in that way. Transducers have many uses and there are many kinds, in electromechanical systems they can be used as actuators and sensors. In audio electronics they provide the conversion between the electrical and acoustical domains. The transducer provides the link between the mechanical and electrical domains and thus a network representation is required for it in order to develop a unified electrical analogy.[43] To do this the concept of port from the electrical domain is extended into other domains.[44]

Transducers have (at least[note 5]) two ports, one port in the mechanical domain and one in the electrical domain, and are analogous to electrical two-port networks. This is to be compared to the elements discussed so far which are all one-ports. Two-port networks can be represented as a 2×2 matrix, or equivalently, as a network of two dependent generators and two impedances or admittances. There are six canonical forms of these representations: impedance parameters, chain parameters, hybrid parameters and their inverses. Any of them can be used. However, the representation of a passive transducer converting between analogous variables (for instance an effort variable to another effort variable in the impedance analogy) can be simplified by replacing the dependent generators with a transformer.[45]

On the other hand, a transducer converting non-analogous power conjugate variables cannot be represented by a transformer. The two-port element in the electrical domain that does this is called a gyrator. This device converts voltages to currents and currents to voltages. By analogy, a transducer that converts non-analogous variables between energy domains is also called a gyrator. For instance, electromagnetic transducers convert current to force and velocity to voltage.[46] In the impedance analogy such a transducer is a gyrator.[47] Whether a transducer is a gyrator or a transformer is analogy related; the same electromagnetic transducer in the mobility analogy is a transformer because it is converting between analogous variables.[48]

## History

James Clerk Maxwell developed very detailed mechanical analogies of electrical phenomena. He was the first to associate force with voltage (1873) and consequently is usually credited with founding the impedance analogy.[49] This was the earliest mechanical–electrical analogy.[50] However, the term impedance was not coined until 1886, long after Maxwell's death, by Oliver Heaviside.[51] The idea of complex impedance was introduced by Arthur E. Kennelly in 1893, and the concept of impedance was not extended into the mechanical domain until 1920 by Kennelly and Arthur Gordon Webster.[52]

Maxwell's purpose in constructing this analogy was not to represent mechanical systems in terms of electrical networks. Rather, it was to explain electrical phenomena in more familiar mechanical terms.[53] When George Ashley Campbell first demonstrated the use of loading coils to improve telephone lines in 1899, he calculated the distance needed between coils by analogy with the work of Charles Godfrey on mechanical lines loaded with periodic weights.[54] As electrical phenomena became better understood the reverse of this analogy, using electrical analogies to explain mechanical systems, started to become more common. Indeed, the lumped element abstract topology of electrical analysis has much to offer problems in the mechanical domain, and other energy domains for that matter. By 1900 the electrical analogy of the mechanical domain was becoming commonplace. From about 1920 the electrical analogy became a standard analysis tool. Vannevar Bush was a pioneer of this kind of modelling in his development of analogue computers, and a coherent presentation of this method was presented in a 1925 paper by Clifford A. Nickle.[55]

The application of electrical network analysis, most especially the newly developed field of filter theory, to mechanical and acoustic systems led to huge improvements in performance. According to Warren P. Mason the efficiency of ship electric foghorns grew from less than one per cent to 50 per cent. The bandwidth of mechanical phonographs grew from three to five octaves when the mechanical parts of the sound transmission were designed as if they were the elements of an electric filter ( see also Mechanical filter § Sound reproduction). Remarkably, the conversion efficiency was improved at the same time (the usual situation with amplifying systems is that gain can be traded for bandwidth such that the gain-bandwidth product remains constant).[56]

In 1933 Floyd A. Firestone proposed a new analogy, the mobility analogy, in which force is analogous to current instead of voltage. Firestone introduced the concept of across and through variables in this paper and presented a structure for extending the analogy into other energy domains.[57] A variation of the force-current analogy was proposed by Horace M. Trent in 1955 and it is this version that is generally meant by the through and across analogy.[58] Trent used a linear graph method of representing networks which has resulted in the force-current analogy historically being associated with linear graphs. The force-voltage analogy is historically used with bond graph representations, introduced in 1960 by Henry Paynter, however, it is possible to use either analogy with either representation if desired.[59]

## Notes

1. ^ a b An energy domain pertains to a system or subsystem in which the energy and forces are all of a particular kind such as electrical, mechanical, acoustical, thermal, and so on.
2. ^ The five-element scheme can be extended to active devices such as transistors by the use of two-port networks containing dependent generators provided the transistor is operating in a substantially linear region.
3. ^ Acoustic mass does not have units of mass. In the SI system it has units of kg/m4 (Barron, p. 333)
4. ^ Responsiveness is the inverse of mechanical resistance (Seely et al., p. 200)
5. ^ Piezoelectric transducers are frequently modelled as three-port devices, one electrical and two mechanical, because mechanical vibrations are induced on both sides of the crystal (Cheeke, pp. 213-214).

## References

1. ^ Busch-Vishniac, p. 17
2. ^ Darlington, p. 7
3. ^ Care, pp. 74-77
4. ^ Chan, pp. 2-3
5. ^ Busch-Vishniac, p. 17
6. ^ Busch-Vishniac, pp. 17-18
• Borutzky
7. ^ Busch-Vishniac, pp. 18, 21
8. ^ Kleiner, p. 69
9. ^ Myers, p. 136
10. ^ White, p. 93
11. ^ Carr, pp. 170-172
12. ^ Froehlich & Kent, vol. 6, p. 434
13. ^ Joines et al., pp. 69-71
15. ^ Fukazawa & Tanaka, pp. 191-192
16. ^ Agarwal & Lang, pp. 9-11
17. ^ Semmlow, p. 405
18. ^ Sen, pp. 29, 41
19. ^ Busch-Vishniac, pp. 18-19
20. ^ Busch-Vishniac, p. 19
21. ^ Eargle, p. 5
22. ^ Hand, L. N.; Finch, J. D. (2008). Analytical Mechanics. Cambridge University Press. ISBN 978-0-521-57572-0.
23. ^ Busch-Vishniac, p. 21
24. ^ Borutzky, pp. 27-28
25. ^ Busch-Vishniac, pp. 18-20
26. ^ Kleiner, pp. 67-68
27. ^ Busch-Vishniac, p. 18
• Borutzsky, pp. 22-23
28. ^ Busch-Vishniac, p. 18
• de Silva, p. 132
29. ^ Kleiner, p. 15
30. ^ Beranek & Mellow, p. 94
31. ^ Kleiner, p. 84
32. ^ Busch-Vishniac, pp. 18, 21
33. ^ Eargle, pp.4-5
34. ^ Kleiner, p. 70
35. ^ Busch-Vishniac, pp. 18-19, 21
36. ^ Busch-Vishniac, pp. 19-20
• Jackson, p. 17
• Regtien, p. 20
37. ^ Busch-Vishniac, pp. 19-20
• de Silva, pp. 132-133
38. ^ Busch-Vishniac, pp. 18-21
39. ^ Busch-Vishniac, p. 17
40. ^ Busch-Vishniac, pp. 18-19
• Regtien, p. 21
• Borutzsky, p. 27
41. ^ Busch-Vishniac, p. 19
• Regtien, p. 21
42. ^ Hamill, p. 97
43. ^ Busch-Vishniac, pp. 11-12
44. ^ Janschek, p. 94
45. ^ Lenk et al., pp. 207-208
46. ^ Eargle, pp. 5-6
47. ^ Beranek & Mellow, pp. 70-71
• Lenk et al., p. 147
• Janschek. pp. 94-95
48. ^ Janschek, 95-96
49. ^ Bishop, p. 8.4
• Busch-Vishniac, p. 20
50. ^ Smith, p. 1648
51. ^ Martinsen & Grimnes, p. 287
52. ^ Hunt p. 66
53. ^ Care, p. 75
54. ^ Mason, p. 409
55. ^ Care, p. 76
56. ^ Mason, p. 405
57. ^ Bishop, p. 8.2
• Smith, p. 1648
58. ^ Busch-Vishniac, p. 19
59. ^ Bishop, p. 8.8

## Bibliography

• Agarwal, Anant; Lang, Jeffrey, Foundations of Analog and Digital Electronic Circuits, Morgan Kaufmann, 2005 ISBN 008050681X.
• Barron, Randall F., Industrial Noise Control and Acoustics, CRC Press, 2002 ISBN 0203910087.
• Beranek, Leo Leroy; Mellow, Tim J., Acoustics: Sound Fields and Transducers, Academic Press, 2012 ISBN 0123914213.
• Bishop, Robert H., Mechatronics: An Introduction, CRC Press, 2005 ISBN 1420037242.
• Borutzky, Wolfgang, Bond Graph Methodology, Springer, 2009 ISBN 1848828829.
• Busch-Vishniac, Ilene J., Electromechanical Sensors and Actuators, Springer Science & Business Media, 1999 ISBN 038798495X.
• Care, Charles, Technology for Modelling: Electrical Analogies, Engineering Practice, and the Development of Analogue Computing, Springer, 2010 ISBN 1848829485.
• Carr, Joseph J. RF Components and Circuits, Oxford: Newnes, 2002 ISBN 0750648449.
• Chan, Shu-Park, "Circuits: Introduction", pp. 2–4, in Dorf, Richard C. (ed), The Electrical Engineering Handbook, CRC Press, 1997 ISBN 1420049763.
• Cheeke, David N., Fundamentals and Applications of Ultrasonic Waves, CRC Press, 2012 ISBN 143985498X.
• Darlington, S, "A history of network synthesis and filter theory for circuits composed of resistors, inductors, and capacitors", IEEE Transactions on Circuits and Systems, vol. 31, pp. 3–13, 1984.
• de Silva, Clarence W., Vibration: Fundamentals and Practice, CRC Press, 2006 ISBN 0849319870.
• Eargle, John, Loudspeaker Handbook, Kluwer Academic Publishers, 2003 ISBN 1402075847.
• Firestone, Floyd A., "A new analogy between mechanical and electrical system elements", The Journal of the Acoustical Society of America, vol. 3, pp. 249–267, 1933.
• Froehlich, Fritz E.; Kent, Allen, The Froehlich/Kent Encyclopedia of Telecommunications, CRC Press, 1991 ISBN 0824729021.
• Fukazawa, Tatsuya; Tanaka, Yasuo, "Evoked otoacoustic emissions in a cochlear model", pp. 191–196 in Hohmann, D. (ed), ECoG, OAE and Intraoperative Monitoring: Proceedings of the First International Conference, Würzburg, Germany, September 20–24, 1992, Kugler Publications, 1993 ISBN 9062990975.
• Hamill, David C., "Lumped equivalent circuits of magnetic components: the gyrator-capacitor approach", IEEE Transactions on Power Electronics, vol. 8, iss. 2, pp. 97–103.
• Hunt, Frederick V., Electroacoustics: the Analysis of Transduction, and its Historical Background, Harvard University Press, 1954 OCLC 2042530.
• Jackson, Roger G., Novel Sensors and Sensing, CRC Press, 2004 ISBN 1420033808.
• Janschek, Klaus, Mechatronic Systems Design, Springer, 2011 ISBN 3642175317.
• Joines, William T.; Palmer, W. Devereux; Bernhard, Jennifer T., Microwave Transmission Line Circuits, Artech House, 2013 ISBN 1608075699.
• Kleiner, Mendel, Electroacoustics, CRC Press, 2013 ISBN 1439836183.
• Lenk, Arno; G. Ballas, Rüdiger; Werthschützky, Roland; Pfeifer, Günther, Electromechanical Systems in Microtechnology and Mechatronics, Springer, 2010 ISBN 3642108067.
• Lurie, Boris; Enright, Paul, Classical Feedback Control, CRC Press, 2011 ISBN 1439860173.
• Martinsen, Orjan G.; Grimnes, Sverre, Bioimpedance and Bioelectricity Basics, Academic Press, 2011 ISBN 0080568807.
• Mason, Warren P., "Electrical and mechanical analogies", Bell System Technical Journal, vol. 20, no. 4, pp. 405–414, October 1941.
• Myers, Rusty L., The Basics of Physics, Greenwood Publishing Group, 2006 ISBN 0313328579.
• Paynter, Henry M., Analysis and Design of Engineering Systems, MIT Press, 1961 OCLC 1670711.
• Radmanesh, Matthew M., Electronic Waves & Transmission Line Circuit Design, Author House, 2011 ISBN 1456752324.
• Regtien, Paul P. L., Sensors for Mechatronics, Elsevier, 2012 ISBN 0123944090.
• Seely, Samuel; Tarnoff, Norman H.; Holstein, David, Digital Computers in Engineering, Holt, Rinehart and Winston, 1970 OCLC 92614.
• Semmlow, John, Signals and Systems for Bioengineers, Academic Press, 2012 ISBN 0123849829.
• Sen, S. N., Acoustics, Waves and Oscillations, New Age International, 1990 ISBN 8122402666.
• Smith, Malcolm C., "Synthesis of mechanical networks: the inerter", IEEE Transactions on Automatic Control, vol. 47, iss. 10, pp. 1648–1662, October 2002.
• Trent, Horace M., "Isomorphisms between oriented linear graphs and lumped physical systems", The Journal of the Acoustical Society of America, vol. 27, pp. 500–526, 1955.
• White, Curt, Data Communications and Computer Networks, Cengage Learning, 2012 ISBN 1285225864.