Can the roots of spacetime interval be linearly added up?

Question: It seems ridiculous to add up roots of two intervals to get the total interval. Considering 3 events, $\Delta s_{13}$ shouldn't be equal to $\Delta s_{12}$+$\Delta s_{23}$. But if we consider the events in a moving particle's worldline, it seems reasonable to add up the proper times. What's wrong with the logic?
Answer: The square root form for $\Delta s$ in terms of $v$ is correct for infinitesimal intervals, and really you should integrate it to find a finite spacetime interval:
$$\Delta s_{12}=\int_1^2 \sqrt{ds^2}.$$
So indeed if you consider three events,
$$\Delta s_{13}=\int_1^3 \sqrt{ds^2}=\int_1^2 \sqrt{ds^2}+\int_2^3 \sqrt{ds^2}=\Delta s_{12}+\Delta s_{23}.$$
Of course if your trajectory has constant velocity the integral simplifies and it looks like you're summing the two square roots.
Comment: To put it in a slightly more wordy but common-sense way: usually in 3D space when we add two vectors we have $|\vec v_1+\vec v_2|<|\vec v_1|+|\vec v_2|$ but when the vectors point in the same direction the $<$ turns into $=$, and in full generality we say $\le$. Spacetime has an analogous sense in which two 4-vectors can point in the "same direction," in which case their addition formula is direct. As long as events happen at the same place in a coordinate system, those spacetime intervals are precisely parallel and can be added just like $|\vec v_1+\vec v_2|=|\vec v_1|+|\vec v_2|$ can in 3D.
Comment: @CRDrost That's a good exposition - put it as a separate answer - I'd upvote it - even if it does seem trivial to you. There are loads of highschool and freshmen physicists on this site and one shouldn't underestimate the worth of good technical writing even for very simple things.
Comment: I think it's important to translate the two separate comments: what I was saying is that $\Delta s$ is not the norm of any vector, so you need to calculate its length in spacetime through the integral above. What you're saying is that if $\Delta s$ can be identified with the norm of a vector, the trajectory is a straight line, and then the sum of the norms is the norm of the sum if you break the trajectory into two separate parts.
Comment: @user121664 Thanks for your answer first. But shouldn't $\Delta s_{12}=((\Delta x)^2-(c\Delta t)^2)^{1/2}$? Why is it stated as the integral of infinitesimal roots of intervals?
Comment: That formula is valid to calculate the spacetime distance between two events *along a straight line*. If the events 1,2,3 are along a straight line, then you can "sum the square roots", as you say, which is what CR was getting at. If the particle is not moving in a straight line, that formula is not valid, and you need to use the integral form in my answer. Since that is the general rule, proper times do sum. See also the answer by @Electrodynamist.
Comment: @JasonTao suppose that you have a general path through Minkowski space, some parameter $q:~0\to 1$ with some path $(w(q),~\vec x(q))$ where $w = c t,$ and you want to know the time difference as measured by a clock which travels along that path. Then you have to add together a lot of tiny proper times and the general expression is going to be $\Delta\tau = \int d\tau = \int \sqrt{(dw)^2 - (d\vec x)^2} = \int_0^1 dq~\sqrt{(w'(q))^2 - (\vec x'(q))^2}.$ Does that help?
Answer: For timelike intervals (the intervals that can be travelled by a particle),
$d\tau^2 = g_{\alpha \beta} dx^{\alpha}dx^{\beta}$
Hence, $d\tau = \sqrt{g_{\alpha \beta} dx^{\alpha}dx^{\beta}}$
Thus, the total interval $\tau_{13} = \int_1^3 \sqrt{g_{\alpha \beta} dx^{\alpha}dx^{\beta}}$
Therefore,
$\tau_{13} = \int_1^2 \sqrt{g_{\alpha \beta} dx^{\alpha}dx^{\beta}} + \int_2^3 \sqrt{g_{\alpha \beta} dx^{\alpha}dx^{\beta}} = \tau_{12} + \tau_{23}$ . Always.
The point is $\tau_{13}^2 \neq \tau_{12}^2+\tau_{23}^2$.
Because, $d\tau^2 =  {g_{\alpha \beta} dx^{\alpha}dx^{\beta}} = g_{\alpha\beta}U^{\alpha}U^{\beta} d\lambda^2$
($\lambda$ is the affine parameter used to paramatrize the curve (path) along which the particle travels and $U^{\mu} := \dfrac{dx^{\mu}}{d\lambda}$)
Thus, in order to integrate, we first take the square root and then only perform the integration. There is no meaning to directly saying $\int_1^3 d\tau^2$.
So, $\tau_{13}^2 = \big(\int_1^3 \sqrt{g_{\alpha \beta} dx^{\alpha}dx^{\beta}}\big)^2$ NOT that $\tau_{13} = \sqrt{\int_1^3 {g_{\alpha \beta} dx^{\alpha}dx^{\beta}}}$.
'Reduced Mathematics' Version:
Your intuition is right. $\tau_{13} = \tau_{12} + \tau_{23}$. Reason is simply that we first take square root of $d\tau^2$ and then integrate - not the other way around. Thus, $\tau_{13}^2 \neq \tau_{12}^2+\tau_{23}^2$ but  $\tau_{13} = \tau_{12} + \tau_{23}$.


What is the meaning of $\sum_{i,j = 1}^n$?

Question: In the multi-dimensional Ito formula, this notation pops up:$$\sum_{i,j = 1}^n$$
What does that even mean? Do we take $i = 1$, then $j = 1, ....,n$, then $i =2$ and $j = 1,...,n$, and so on? Or do we take both $i$ and $j$ equal to 1, and then both $i$ and $j$ equal to 2, and so on? If it is the latter, then why don't they just write $\sum_{k=1}^n$, and then everywhere inside the summation, exchange both $j$ and $i$ for $k$? If it is the former, then surely the double summation notation makes the point clearer and is also better suited for calculation (i.e, if you want to take a constant in the inner sum out into the outer).
Comment: https://en.wikipedia.org/wiki/Summation#Capital-sigma_notation
Comment: There is always a sliding scale between clarity and conciseness. For two variables a double sum might be fine, but how would you handle the case of, say, five or more variables? (Not as uncommon as one would think.)
Answer: It has the following meaning
$$\sum_{i,j=1}^n=\sum_{i=1}^n\sum_{j=1}^n=\sum_{j=1}^n\sum_{i=1}^n$$
Since the sums are finite, the order of summation does not matter. This notation is convenient because it takes up less space than writing it out.
Answer: It's the summation of a square array. Here's a concrete example: $$\sum_{i, j = 1}^5 \gcd(i, j).$$ You could start with $\gcd(1, 1)$, then $\gcd(1, 2)$, $\gcd(1, 3)$, and so on and so forth until reaching $\gcd(5, 5)$. But like Moop said, addition is commutative, so you could fix $j$ while you iterate $i$, or you could proceed in diagonals, or you could even go through it randomly (of course keeping track of what you've already calculated).
I personally would prefer $$\sum_{i = 1}^5 \sum_{j = 1}^5 \gcd(i, j)$$ because you can more easily change that to a nonsquare rectangular array if you need to. But it could be a parsing speed bump if someone gets temporarily confused and thinks there is multiplication involved.

Or do we take both $i$ and $j$ equal to 1, and then both $i$ and $j$ equal to 2, and so on?

As you've already noted, that would be inefficient, since $i$ and $j$ could be collapsed to a single variable, like you said. it's the kind of mistake I would not expect in a book or a peer-reviewed journal article. But on a post on this website? It could happen, I presume we're all human here.


Use of trunks, branches and tags in a project

Question: I know the theory but how is svn used from a practical point of view for a project? Say, I have a web site with features that are going to be modifed or added. In what cases would new trunks, branches and tags be used?
Comment: see this answer http://stackoverflow.com/questions/5151760/release-from-svn-using-export-and-then-how-to-upgrade/5151821#5151821
Comment: trunks is one. Usually represents Main development line. Branches are created for bug fixes or for some alternate development line. Tags are like mile-stone, frozen branches, represents a bug free, tested release point in dev timeline.
Answer: Different organizations do very different things.  The solution that Alex gives above is a common one, however it encounters the issue that you'll only discover what other branches have conflicts with yours after you land your branch.  This causes people to have to debug conflicts in stuff that they have not looked at in some time.
The other common approach that I have encountered is to do all development in trunk, and make developers make all commits small, standalone, commits.  There are a variety of ways to add a feature and make it invisible by default, but turned on in your development copy.  Use them.  This approach requires care from the developers, but avoids pain from managing conflicts between longer lived branches.
A lot of the people who use Alex' solution will jump up and down and say, "That will never work for anything beyond a tiny team!"  To that I respond that there is nothing wrong with small teams, their productivity typically far exceeds anything a large team can do.  And that strategy can scale if developers have discipline, for instance Google uses it.  If you want to see an actual project that uses that strategy, take a look at http://llvm.org/.
And some advice.  If you want to follow Alex' strategy, I strongly recommend using git instead of svn.  It handles branches much better than svn does.  If you want to follow the strategy that I suggest and your team is not a small team with people you trust, you need to use a code review tool like http://code.google.com/appengine/articles/rietveld.html to reduce the obvious problems that can arise.
Comment: Good points @btilly, I'm experimenting with Git for my current project - so far I like it very much - but I won't go to of topic here :-)
Answer: Have a look at these two articles:
http://www.snuffybear.com/ucm_branch.htm
http://www.vance.com/steve/perforce/Branching_Strategies.html
The the link from SnuffyBear speaks more about the strategy that you seem to want to employ whereas the other article talks about other strategies such as branch by release. Each article outlines the advantages and disadvantages of each strategy.
Answer: I'd say a single trunk where "stable" very minor changes (perhaps small bug fixes) are made, that will not break your build.
Branches should be made for new features / large changes. The branch should be kept up to date with changes to your trunk, whilst the branch is in operation.
Once your new feature has been completed the branch should be merged back into the trunk and then the branch can be deleted.
Tags are for releases. e.g. v1.2
Comment: Thanks @James, btw. I must add a note that I myself have not been using SVN for long. My post is more a combination of the vast number of things I have read on the net and what I'm experimenting with myself.
Comment: Thanks Alex. Very clear explanation. To reformulate, it sounds like additional trunks are rarely used in a typical project and branches are used for new features. I'd already assumed that tags were used for releases.
Comment: That would only be true if you have only a single release in preparation otherwise you need multiple branches which represent the particular releases which are in preparation (like 1.0, 1.1 and 2.0)...
Answer: My humble observations: Merging branches is scary. In many cases, changes made by different developers are completely separate and so merge smoothly. But in any busy project, there are going to be times when changes conflict. If you're lucky, SVN recognizes the conflict and gives you errors at merge times. If you're less lucky, SVN doesn't catch it but it fails to compile. Either way, somebody now has to figure out how to put the changes together. Sometimes this is obvious and easy. But I've had many times where I've had to get the developers who made the changes together so we could figure out what to do.
If you're very unlucky, neither SVN nor the compiler see a problem, conflicting changes go to production, and the program behaves incorrectly.
From this observation I come to two conclusions: (a) Make as few branches as possible. Or more accurately, develop your strategy to do as few merges as possible. And (b) Do merges on code while it is still in testing.
For a while my company had a branching strategy that said that each project got its own branch, tested off of that branch, and when we were ready to deploy to production, we merged all the branches to be included in that release, compiled and deployed. This turned out to be a really really bad idea. Somebody would be trying to resolve merge conflicts the day before deploying, and the results of the merge were never tested. A lot of subtle bugs crept into production.
For a whlie we used this strategy: Most development is done in trunk. When a release is ready to be turned over to the test group, we peel off a branch for it. Then work on the next release proceeds in trunk. Any bug fixes to the current release are done in the branch, and the changes in this branch are periodically merged back into trunk. Occassionally it was necessary to have work on 3 releases going on simultaneously, i.e. one release was in testing, another was getting close to being ready for testing, and now we need to start on the next release. In that case we'd have the test branch, trunk for the current release, and a "pre" branch for the next release. When the current release went to the test group, we would then merge the "pre" branch into trunk and it became the current release.
We have recently begun experimenting with a slightly different strategy. When a release is in testing, we still peel off a separate branch for it. But any fixes coming out of testing are made in trunk, and then those fixes are merged from trunk into the test branch. This means that all development happens in trunk, and merges are always from trunk to somewhere else. This has two big advantages: One, developers are always testing with trunk, so we have a higher confidence that the code in trunk is good. The testing group will test against the release branch, so we have confidence that the release branch is good. That is, we have some confidence that both branches will be tested. When you make changes in a branch and then merge back to trunk. there's little assurance that anyone will test the results of that merge. Two, trunk always has the full "blame" history of every module. When you merge from branches back to trunk. the history in trunk attributes all changes from the branch to the person who did the merge, rather than the person who really made the change, and the comment tends to be an uninformative "merged from brach xyz". When you merge from trunk to a branch, sure the branch now shows the "wrong" person and an uninformative comment, but trunk has a good history. There's one place to go for history instead of trying to chase down branches.
Comment: You say: "If you're very unlucky, neither SVN nor the compiler see a problem, conflicting changes go to production, and the program behaves incorrectly." from that i follow that you have no unit- and/or integration tests which verify the results of a merge. My statement: No Unit/Integration Tests than no Merging! And that is true for any version control tool.
Comment: You wrote: "This turned out to be a really really bad idea. Somebody would be trying to resolve merge conflicts the day before deploying, and the results of the merge were never tested." this is called the "Big Bang Integration" which fails all the time. In particular the larger the team become the more complicated the merges become. The simple solution is synchronize the branches with the trunk (or the stable line) as often as possible. The result is having only small merges which can be handled simple.
Comment: Interesting stuff, especially I didn't realise about blame not behaving properly after a merge.
Comment: @khmarbaise: That was my point: We were doing merges just before deploying to production, with no testing of the result of the merge. Someone got the ... interesting ... idea that if the individual changes are tested and work, that the results of the merge should work, so there's no need to "retest". As I said further on in the post, this was a really really bad idea.


Which background processes can I disable to speed up my Mac

Question: If I'm on an old Mac and I just want the basics (web browsing, text editing, multitasking) to run a lot faster and I don't mind losing out on bells and whistles or temporarily disabling unneeded background processes, what processes would this be and how could I enable/disable them with a simple shell script?
Answer: I suspect there is not much you can do here. This is because most background processes only run when they are needed (see "On Demand")
Best to look at Activity Monitor to see if there is anything taking up CPU.
My guess is for text editing and multitasking you will have no speed issues.
For web browsing the browser and especially add ins will take up CPU and memory thus requiring swapping.


Does Odo ever return to being a changeling after he had become human?

Question: At the end of season 4 of Star Trek: Deep Space Nine, in the episode, "Broken Link," Odo is made human.  Does he ever return to being a changeling?  He is still human in the first episode of season 5. DS9 Season 5
Comment: Doesn't knowing this remove some of the tension and anticipation of watching the rest of the series?
Answer: Star Trek Deep Space Nine Season 5 Episode 12 has an infant changeling basically absorbing into him turning him back into a changeling.
Comment: Watched that episode tonight.  I had originally recorded all episodes on VHS and watched them, but forgot most of the shows.  They are all new to me again now.


use of "a" before the uncountable word "consensus"

Question: All major English dictionaries say that consesnsus is uncountable, but right under this warning they provide example sentences like the following:"We reached a consesnsus." How come?
Comment: Some nouns are both countable and uncountable, eg "love" as in "a love that dare not speak its name".
Comment: ***consensus***:  noun, ***plural consensuses***. [http://www.dictionary.com/browse/consensus]
Comment: The more rigorous test (see Huddleston & Pullum) for count-noun usage is not whether an indefinite article may be used with the usage, but whether numerals may. As 'two ...', 'one ...', 'many consensuses' are not usually considered acceptable, this is a non-count usage.
Comment: "Consensus" is a non-count noun. The presence of the article "a" does not mean the noun is countable. A noun is only a count noun if it can combine with the cardinal numbers "one, two, three" etc. A few non-count nouns (like "regard", "knowledge" and "consensus") can combine with "a", but that doesn't mean they are count nouns; they are non-count nouns.
Comment: Consensus can be countable. The consensus amongst doctors is that medicine should be the highest paid profession, but the  consensus amongst hospital managers is that management should be. How do we reconcile these two consensuses?  Or a third consensus amongst teachers about pay in education. .
Answer: Consensus can either be singular or noncountable.
As in the following examples from Learner's Dictionary:
Singular Examples:

The (general) consensus (of the group) was to go ahead with the plan.
Scientists have not reached a consensus on the cause of the disease.
  [=scientists do not yet agree about the cause of the disease]
There is a growing consensus [=more and more people agree] about/on
  the need for further investigation.
What is the consensus of opinion among the experts? [=what do the
  experts all say?]

Noncount examples

Everyone on the council seems to understand the need for consensus.
There is a lack of consensus among the citizens.
The decision was made by consensus

When you use the singular the emphasis is placed on the individual consensus. Perhaps the consensus took much time to hammer out, perhaps there was much compromise on the way, regardless the singular form places the reader's attention on the individual consensus, and the specifics associated with that unique consensus.
The uncounted form instead places the emphasis more abstractly on the fact that consensus was reached, needs to be reached, etc.
Comment: Ask yourself this: can "consensus" combine with the cardinal numbers "one, two, three" etc. If it can, it's a count noun; if not it's a non-count noun. Can you say "three consensuses"? Of course not, at least not in the example given. "Consensus" can only be a non-count noun.


Replacing wall sconce (that has on/off swtich) with ones that do not. Need options for control

Question: I have several very old wall sconces that have their own individual on/off toggle switches that I would like to switch out.  Unfortunately there is no wall switch that will allow for control of these.  I have found some wireless receiver controllers that might work - X10, EnOcean.  I wanted to see what people's opinions were/prior experiences with these devices.  I'm also open to any suggestions short of rewiring everything.  That will be done but we're not ready for a major remodel.  Thank you for your help.
Elmo
Comment: Perhaps the new sconce can have a switch added, similar to the existing.
Answer: You have options, they actually make smart light bulbs you can power on and off with an app, but beyond that you can always find a wall scone made of aluminum and drill out a canopy switch, connect it to the positive on the fixture and problem solved. More invasive would be snaking a 14/2 to a future switch box location, install both black and white to the switch (mark the white wire with black tape to prevent  accidents in the future. Then you need to connect the feed wire (the positive wire that carrys the power to the light) to one side of the new wire, and the positive going to the light fixture to the other side of the new wire. Hope this helps
Comment: I hadn't thought of smart bulbs. That might be the way to go. I'll have to check prices though,  they used to be very expensive. Any experience with the rf switches I mentioned.
Answer: I have used the x10 in the past and it worked well as long as the circuit for the device was not over loaded. I found that using total wattage of less than 75% of the rated value they held up for many years. The other issue I found with them the range can be quite a bit shorter if installed in a home with steel path & plaster. From memory I thought the signal traveled on the wires but did find 1 case where the home base module was about 100' from the garage the garage light controls were intermittent until the base was closer possibly the different outlet the base was now on the same leg that the lights were on I don't remember but those were the only 2 problems I ever found with them. Note this was prior to LED lighting they powered fluorescent and incandescent lights.


Can the price of a Dex pair be tracked using the data in cexplorer / db-sync?

Question: Is there a way to query the price of a pair that's on a dex independent of the dex? For example: MIN/ADA or others? I am trying to see how the data in the cexplorer database could be used to do this.
Comment: `cardano-db-sync` does not track native asset prices.
Comment: I do understand cardano-db-sync does not track prices of anything per-se, but I was asking if there is a method for extracting that information given the data available in cexplorer - i.e. a way of doing it yourself, not a way cardano-db-sync does it for you :|
Comment: I would be surprised if all of the data you need is even in `db-sync`.
Answer: This can be done. I used to do it via the client. One can index the current price by looking at the script address of the dex and filtering the native asset to get the utxo that represents the liquidity pool. From that you can deduce the price by just dividing, price = native asset amount in pool / ada amount in pool. I used a bash script to watch this price together with awk, something like
watch -n 1 'cardano cli query address $(cat addresOfDex) | awk \'*filter for line that contains native asset or pool id*\' | awk \'*devide the correct things*\'

Note that I ommitted the filters as I do not remember them. Db-sync can do something similar of course.
Comment: Thank you for the guidance to a possible solution!
Comment: This is a useful answer except the equation should be: price in ada = ada amount / native asset amount


BH1750 Sensor ICs - Block Diagram - Why it has capacitor connect output to input of OpAmp?

Question:
The image from BH1750FVI datasheet. http://rohmfs.rohm.com/en/products/databook/datasheet/ic/sensor/light/bh1750fvi-e.pdf (page 4)
I don't know why it has the capacitor connect output and input of OpAmp? Can anyone help me about this?
Comment: Hint: active low pass filter.
Answer: The block diagram is only an approximation of the whole circuit.  But essentially a capacitor in the feedback turns the amplifier into an integrator.
The photodiode sensor on the front end of the diagram probably only outputs micro amps.  Using an integrator probably allows them to achieve more accurate measurements because it reduces the bandwidth, and thus the noise power.  Integrating a weak noisy signal for a long time can give you a clean precise signal.


Youngs double slit experiment

Question: When the Young double slit experiment is performed with 2 separate slits..and interference is observed does Diffraction not take place in the 2 separate slits and thus affect the interference of 2 waves from the slits?
Comment: Possible Duplicate of : https://physics.stackexchange.com/questions/219594/does-diffraction-take-place-in-youngs-double-slit-experiment#comment472925_219595   (Couldn't flag the question)
Comment: whenever an EM wave, any wave,  encounters *any* finite obstacle, say a screen with a slit, then there is always diffraction.
Comment: So how does that diffraction affect the double slit interference pattern?
Comment: There would be no interference on the screen unless there is diffraction at the slits ... right?
Comment: Perhaps this ishelpful to draw the full picture: [What is the explanation for the phenomenon that single photons produce fringes even behind single edges?](http://physics.stackexchange.com/questions/325936/what-is-the-explanation-for-the-phenomenon-that-single-photons-produce-fringes-e?noredirect=1#comment731059_325936)) and [If I put one of the slits from the Young's double slit experiment apparatus into a different medium what happens to the fringe width?](http://physics.stackexchange.com/questions/322398/if-i-put-one-of-the-slits-from-the-youngs-double-slit-experiment-apparatus-into)
Answer: Diffraction
Wikipedia states that diffraction

is defined as the bending of light around the corners of an obstacle or aperture into the region of geometrical shadow of the obstacle.

Due to the first meaning coming from Francesco Maria Grimaldi diffraction refers also to the phenomenon of fringes behind obstacles:

light break up; that is, that parts of the compound [i.e., the beam of light], separated by division

Deflection
Experiments with point-like sources and the observation of fringes (intensity distributions) behind obstacles are done with electrons too (Bi-prism experiment by Möllenstedt and Jönnson, see Zeitschrift für Naturforschung 10 a S. 256 a). In contrast to the definition of diffraction as bending radiation into the region of geometrical shadow the results of this experiments have shown that the electrons get deflected only away from the geometrical shadow:

Some years before this biprizm experiment Hans Boersch published Fresnel'sche Elektronenbeugung (an article from 1940 behind a paywall :-0). It was shown that electrons behind an edge get deflected (Beugung = deflection):

Interference
To your question

does diffraction not take place in the 2 separate slits and thus affect the interference of 2 waves from the slits?

and according to the above mentioned experiments is given the following answer:

Not only behind a separate slit diffraction occurs but deflection occurs behind every (sharp or thin) edge.
The intensity distribution (fringes) starts for EM radiation inside the geometrical shadow and for electrons at potential-free edges outside the geometrical shadow. So for photons and for electrons the diffraction happens in different ways.
Yes you are right with the implication in your question in the sense that somehow the intensity distribution of the four edges of a double slit is the sum of the distribution behind every of the edges. You could call it interference or simply intensity distribution.
Answer: Which one is primary, diffraction or interference, is an ancient and rather sterile question because one does not exist without the other. Yes, one can imagine non-physical but purely mathematical notions of two idealized plane or spherical waves that interfere with each other without anything diffracting but there are no idealized plane or spherical waves. Those are idealized. Real waves that are generated from real antennas do diffract at their sources, in fact that diffraction combined with interference will produce the radiation pattern, never one without the other. (One can even say that an ideal hemi-spherical wave source is really just the ideal diffraction of a pure plane wave source from a point - see Huygens' principle, so even there is diffraction). When an idealized plane wave hits one or two or many slits, the waves passing through the slits will suffer diffraction (see comment above by  @garyp ) and the sum of those diffracted waves interfere with each other the result of which when collimated produces the interference pattern on the screen. This is all formulated mathematically in the Huygens-Fresnel principle and in the corresponding Kirchhoff integral, see https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle
Comment: Ok.. So is it like the interfering waves in a slit again interfere to produce a wave pattern on the screen? As, interference should take place individually in those slits as in the single slit expt?
Comment: It's hard to understand what you're asking..., but I think the answer is yes. In the real world, the double slit experiment requires you to consider _both_ single-slit interference effects _and_ double-slit effects at the same time. Most treatments oversimplify the situation by assuming the slits are like "point sources", but that's only an approximation.

Youngs double slit experiment


Email reply should be captured to lead object record from where email is sent

Question: I have a scenario Like.
From lead object through activity email is sent to customer if customer replies to that email i have to show that reply in activity of that particular record in lead object.can u please help me in solving the issue with out third party apps.I have tried with einstein capture but when same email id is used for different records both the data in mails is coming in both records which have same email id.
Comment: Hello , Does this discussion answer your question [Tie Email Responses back to Lead Object](https://salesforce.stackexchange.com/questions/287368/tie-email-responses-back-to-lead-object) ?
Answer: To achieve this functionality of displaying customer replies in the activity of the corresponding lead record in Salesforce, please follow below

Enable Email-to-Case or Email-to-Lead: Salesforce provides a standard feature called Email-to-Case or Email-to-Lead, which allows you to automatically create cases or leads from incoming emails. You need to enable this feature in your Salesforce org and configure it accordingly.

Set up Email Services: Email Services in Salesforce allow you to define an email address that can receive incoming emails and process them using Apex classes. You can set up an Email Service that will handle the incoming customer replies.

Create an Apex Class: Write an Apex class that will be associated with the Email Service. This class should extract the necessary information from the incoming email and create an activity record on the corresponding lead object.

sample code
<code>global class LeadEmailHandler implements Messaging.InboundEmailHandler {
    global Messaging.InboundEmailResult handleInboundEmail(Messaging.InboundEmail email, Messaging.InboundEnvelope envelope) {
        Messaging.InboundEmailResult result = new Messaging.InboundEmailResult();

        // Extract necessary information from the email, such as sender, subject, body, etc.
        String senderEmail = email.fromAddress;
        String subject = email.subject;
        String body = email.plainTextBody;

        // Use the extracted information to find the corresponding lead record
        Lead leadRecord = [SELECT Id FROM Lead WHERE Email = :senderEmail LIMIT 1];

        // Create an activity record and associate it with the lead record
        Task activity = new Task();
        activity.Subject = subject;
        activity.Description = body;
        activity.WhoId = leadRecord.Id;
        activity.WhatId = leadRecord.Id;
        activity.OwnerId = UserInfo.getUserId();

        // Insert the activity record
        insert activity;

        return result;
    }
}
</code>
Please let me know if you have any doubt
Comment: Hiee gopal I will try this scenario what you have explained can we connect through zoom or any means so that u can guide me thanks in advance
Comment: @AdityaSD it's very easy please try the same as I wrote in the example it will work, if it doesn't work you can share the issue here I will be happy to help!!


Mann-Whitney (U) Test at N>20

Question: I am a high school Psychology student. I just performed a study where two separate groups of sizes 26 and 28 took the same test under different conditions. For the data analysis my professor has instructed me to use the Mann-Whitney test, which is unfortunate because both have sample sizes greater than 20. I saw something on the Wikipedia article about how the sample distributions are approximately normal at that point, but I have no idea how to use the z variable it gives me because I hardly know anything about statistics. Can somebody give a noob a simple process for performing the Mann-Whitney test correcting for the sample size?
Comment: There is no requirement that sample size for Mann-Whitney be smaller than 20.
Comment: Yeah, seemed arbitrary to me, especially when we were instructed to throw out data to reach that sample size!
Comment: Oh my.  If that was a public school, I would ask for my taxes back.
Comment: Glad to hear it worked out.
Answer: The unpaired t-test is efectively "how to use the z variable" provided by the normal approximation, as you asked. In many circumstances the t-test (based on underlying normal distributions) may be as good as or better than the Mann-Whitney test, and as @AdamO notes the two tests tend to give similar results in practice.
But you have been asked to use the Mann-Whitney test, and there's no reason not to give it a try. There's no theoretical problem in using the Mann-Whitney test with samples of this (or any) size, and there's no need to "correct" for sample size. For example, the accepted answer on this page argues "you can't shoot yourself in the foot by using the Mann-Whitney test instead of the t-test, but the converse is not true." The Mann-Whitney test is based on comparisons of ranks among cases, rather than the actual values. This means it doesn't make strong assumptions about the underlying distributions, but it does raise some interesting issues having to do with ties, as discussed on this page. Modern statistical packages have ways of taking ties into account.
As you are just starting to learn how to apply statistics to real problems, you might find it informative to use both the Mann-Whitney test and an unpaired t-test on your data. Also, you should know that most analysts wouldn't consider this a "large" data set; that usually means more like thousands of cases (some might say millions) and up. But yes, this is the size of samples for which normal approximations are often good enough.
Answer: There are potentially two problems you are assessing: 1. The T-distribution's approximation to the normal curve in large N and 2. the use/misuse of the Mann Whitney test when distributional assumptions are not met (based on qualitative inspection).
For 1:
The Mann-Whitney test is just a T-test that replaces the actual observed values with their relative rank in a pooled sample. So, for instance, the smallest (or largest, it doesn't matter) value in the combined 54 observation sample is assigned value 1,the next in order the value 2, and so on. Then a T-test is applied to compare a difference in ranks.
We use a T-test instead of a Z-test when the variance of a sample is to be estimated from the same data whence we estimate the mean. This is to account for the bias that comes from "using the data twice". This bias correction is expressed as a number of degrees of freedom, for instance, a DF of 5 will give you a T-distribution that is very "stout" (large SD relative to a normal curve). As the DF of a T-distribution become numerous (say, 50 or more), the distribution becomes very regularly normal.
Arguably, when the T-statistic is appropriate in small samples, it should never be relaxed based on any condition of the sample size. If the degrees of freedom are large enough to give sampling distributions to test statistics which are approximately normal, we can be confident the T-distribution will reflect that as a consequence of its numerous degrees of freedom.
For 2:
Often in science, the T-test is taught that it is a test of mean differences in normally distributed data. It is true that this test will be exact when this is the case. However, if the data are non-normal, the t-test is "robust" to departures from normality. So even non-normal data, like proportions, test scores, ranks, etc. can be tested for mean differences and the t-test does "pretty good" (this is actually a mathematically rigorous definition).
The Mann-Whitney test was introduced as one of many statistical alternative tests based on the "ranks" of the data, as I described earlier. Therefore, many people touted such tests as being robust to outliers (which are NOT rigorously defined and scientifically dubious). This was further taken to mean that these tests can be applied when distributional assumptions about normality are violated.
It turns out that the Mann Whitney and the T-test usually give very similar inference. In fact, the t-test is more efficient (meaning that, if one were to sample more and more data, the t-test gets you the right answer quicker). The Mann-Whitney test, in fact, does rely on some assumptions (such as symmetric distributions) in order to make sure the inference is correct (about differences in medians). For this reason, I usually prefer running plain vanilla t tests for nearly all inference.
Mann-Whitney (for reasons unknown to me) became popular for small sample inference. I suspect it's because it's difficult to inspect histograms for approximate normality in small samples. However, due to the efficiency consideration I mentioned before, I think this only serves to give one a more conservative test (which is equivalent to throwing data away).
Comment: `"The [MW] test... [relies] on some assumptions (such as symmetric distributions) in order to make sure the inference is correct (about differences in medians)."` The ***about differences in medians*** here is critical to the claim.  The usual null hypothesis of the Mann-Whitney test --- that of stochastic equality --- doesn't make assumptions about the shape of the distributions.  It is only if we want to test a different hypothesis, about the median, that we need to make these assumptions. I don't know why people want to constrain the perfectly useful and interesting MW test in this manner.

Mann-Whitney (U) Test at N>20


ID of images from gallery

Question: Does anyone know how to get ID's of images from wordpress gallery in post? I have search entire internet and there is nothing to this subject.
Answer: You can find out the IDs of which images are included in a gallery by switching to the Text editor tab and look for a piece of code like this:
[gallery columns="3" link="file" ids="615,619,618,617,616"]

The comma-separated list of image IDs are enclosed in the ids attribute

If you're looking to find out IDs of existing images in order to manually create a gallery, you can visit the Media > Library admin menu and click on the Edit link under an image. In your browser's address bar, you should see something similar to the following:
http://example.com/wp-admin/post.php?post=622&action=edit

The number after post= is the image ID. In this case, the image ID is 622.
Comment: I don't want to search ids manually... I need some loop for it.
Comment: So you want to retrieve the IDs of images attached to a gallery using PHP code?
Comment: I want to get images "src" so yes I want theyr id
Answer: Assuming that bungeshea is right and you want to retrieve the IDs of images attached to a gallery using PHP, you can filter 'shortcode_atts_gallery', like this:
add_filter ('shortcode_atts_gallery', 'wsec_filter_gallery_atts', 10, 3);

function wpsec_filter_gallery_atts ($out, $pairs, $atts) {
    $ids = $out['include'];
    // Do what you want with the ids
    return $out;
}

ID of images from gallery


How can the upper limit for a star's mass be calculated?

Question: I know there is a theoretical mass limit for stars around $300 M_\odot$. But I cannot find a way to calculate that number. I tried to use the "Eddington Limit", but I don't know if I'm on the right track. Any suggestions to calculate that limit?
Comment: I don't think there is an absolute limit to the mass of a star. It also depends on metallicity. PopIII could have had even 1000 solar masses (e.g. see Hirano et al. 2014 10.1088/0004-637X/781/2/60). But sure, at solar metallicity, stars more massive than 200 or 300 solar masses would not be able to retain their envelope for long, due to the Eddington limit, as you suggest. I don't think this is easy to prove with pen and paper, though, but I'm ready to be proved wrong with amazing answers
Answer: A basic argument would go something like this:
If you assume that the opacity in the star's envelope is dominated by Thomson scattering from free electrons, then it is simple enough to show (e.g. see here), that for the gravitational force on the envelope to exceed the outwardly directed radiation pressure, then the stellar luminosity must obey
$$ L < 1.2\times 10^{31} \left( \frac{M}{M_\odot}\right)\ {\rm Watts}\ , $$
the so-called Eddington luminosity limit.
From there, if you have a prescription for the luminosity of a high mass star as a function of its mass; for example
$$L \sim 3.8\times 10^{26} \left(\frac{M}{M_\odot}\right)^{3}\ {\rm Watts}\, $$
is a commonly used approximation.
Just substitute this in for the left hand side of the first equation and obtain
$$ \left(\frac{M}{M_\odot}\right)^{2} < 3.2\times 10^4 $$
$$ M < 178 M_\odot$$
The exact figure is much more uncertain than this because (i) the opacity in the envelope isn't just due to Thomson scattering and in particular is highly dependent on metallicity, (ii) the luminosity isn't just some simple power-law scaling with the stellar mass, (iii) it doesn't consider rotation or mass-loss and (iv) the above argument crudely assumes you can calculate an Eddington limit for the whole star, whereas it might be exceeded in some parts of the envelope and not others.
The only way you are going to calculate a more accurate number is by doing a full simulation of the evolution of a massive star - e.g. Sanyal et al. (2015).


What is the Godox Alt / Mask feature?

Question: I've read about an "alt/mask" feature on some of Godox's studio lights (AD200, AD600, QT600II). It looks like it's some method for creating a firing sequence among multiple lights. What exactly is this feature, and what is it good for?
Answer: The MASK / ALT Feature
The MASK or ALT custom function on some of Godox's higher-end strobes defines groups and a firing sequence for those groups for multiple lights. Successive presses of the shutter will fire a different group of lights each time. All lights must be in M mode. It's called MASK with only two groups because it's often used to create silhouette shots/masks for product photography with one group for the backdrop and one for the image. It's called ALT by Godox when 2-4 groups can be assigned, not just two.
On the AD200, AD300, and the QT II/QT III series of AC monolights, one of the custom functions is MASK, and you can set it either to group n1 or n2 for alternate firing.
On the AD400, AD600, and AD1200, there are two custom function settings: UNIT and ALT.
The UNIT setting defines how many groups of lights you want the firing sequence to cycle through (2-4).  The ALT value sets which group the light is in (1-4).
Once everything is set, firing rotates through the defined groups of lights in sequence.  So, group 1's lights fire first, then on the next shot, the second group of lights, and so on.
Uses
There are two main uses this function is good for.
The first is for high-volume product photography, where you often want to mask out the background. If you define your key lights as one group, and your background lights as the other, you'll get a shot of the subject, and then get a white-background/black silhouette shot, perfect for defining a mask, as well.  This is why the two-group setting is also called MASK on the AD600.
The second usage is to cut down the amount of time required to recycle a light. You can alternate firing among multiple units, effectively halving/thirding/quartering the recycle time between bursts, depending on how many flashes you have set up in this manner.  Burst shooting with sports can be possible.
See also:

Godox video on Youtube about the feature
agv8or's post about the feature on POTN
Hensel freemask video on Youtube
Karl Taylor video on youtube about using alt with Siros lights for basketball
Answer: What is this mask?
(I will prepare a couple of diagrams for this)
A mask is an image of the silhouette of what you need to be masked to be used later in post to cut the object and make a transparent background.
If you take a shot of something with a white background and no illumination in the front, you can use that as a mask.
So the idea is taking two shots, one illuminating your subject and other illuminating only the background.
If you want to use this for a person you need to take those two images in a rapid succession.
How can I do this?
This is not an exclusive feature, but something you can create with some strobes that can, in optical slave mode, sync to the second strobe. (S2)
You fire a shot with some flashes sync to the master shot, either optically, by cable or radio.
And immediately you shoot the second photo hoping that only the master flash is firing, and this second shoot triggers one flash that is only lighting the background. This flash thinks that the first shoot was only the "pre shoot" and did not fired, now it does fire when receives the second burst of light.
This means a couple of things.

This master flash needs not to be fired at full power si it can shoot immediately the second burst.

All the other flashes should be at full power so they do not have time to shoot again immediately.

This could potentially work too playing, for example, using a different channel and firing one group of flashes first and then the second group.

If the object is not moving at all, you have time to shoot this flashes in different ways, simply taking a shot first to illuminate the object turning on and off different flashes.

What is the Godox Alt / Mask feature?


Microsoft Exchange CDOEX in a clustered environment

Question: We have a VS2005 native C++ application which connects to Exchange
server using CDOEX and CDOEXM to create and modify calendar entries
for multiple user folders. It's a command line utility which must be installed on the same server where the exchange server is installed. What is the correct procedure to use this
utility in a clustered environment? Any help would be appreciated.
Answer: Might I suggest you take a look at.
http://msdn.microsoft.com/en-us/library/aa373130%28VS.85%29.aspx
for general information on how failover clustering works.
But I would also like to point out that going forward ( ie exchange 2007 , 2010) you don't get CDOEXM.  And Instead you will have to find some way of integrating with powershell instead.  There are classes to explicitly run powershell scripting from inside a program, but they are C# so you either migrate all your code to C# or use some bastardized blend of IJW.


Will tennis ball produce same interference pattern in double slit experiment if everything was scaled up?

Question: Assume the same experimental setup as double slit experiment with electron but everything is exactly scaled up in size so that electron is replaced with tennis ball. Now we throw tennis balls at double slits, instead of electrons. Will tennis ball produce interference pattern same as electron when looked from similarly scaled distance?
Comment: Related: [De Broglie wavelength](https://en.wikipedia.org/wiki/Matter_wave#De_Broglie_hypothesis)
Comment: You may find this interesting.https://physics.stackexchange.com/q/280696/129433
Answer: You see the interference when the De Broglie wavelength of the object is comparable or larger than the distance between the slits. The De Broglie wavelength is $\lambda= {h \over m v}$. Since the electron has a very low mass $m$, it can have a "large" wavelength (say some nanometers), and therefore it is possible to show its interference pattern using a crystal lattice, in the same way it is done for X-rays.
A tennis ball, on the other hand, has a very large mass compared to the electron. Its wavelength will be incomparably smaller. So the problem is not scaling up the experiment, but scaling it down. There is of course some difficulty when trying to squeeze a tennis ball into a slit smaller then the Planck length.
Ok but, imagine that somebody could hypothetically arrange such experiment. Do we believe that we would see the interference pattern? The answer is that we don't know. The largest things we have measured the interference pattern of are some macro molecules. We don't know whether quantum mechanics holds up to the tennis ball scale or if some modification of the theory (quantum gravity?) is needed to describe large objects.
Comment: About the last statement, we do understand how quantum mechanics can be approximated to usual Newtonian mechanics at length scale comparable to tennis ball scale. Do you think quantum gravity is relevant for describing large objects like tennis ball?
Comment: @KP99 it may be, or it may be not. The fact is that we don't know because we have no experimental verification. What we do know is that naively extending quantum mechanics to large objects leads to [weird paradoxes](https://en.m.wikipedia.org/w/index.php?title=Wigner%27s_friend&action=edit&section=9). On the other hand, we have many modified theories of QM, that predict a different behaviour for macroscopic objects, such as [objective collapse theories](https://en.m.wikipedia.org/wiki/Objective-collapse_theory)
Answer: Macroscopic object vs. elementary particle
The main difference between a tennis ball and an electron is that the former is a complex object consisting of many atoms (and hence of many electrons, protons and neutrons), whereas the latter is an elementary particle.
The collision between a complex object and slits will likely excite low energy modes in the object (i.e., the ball) - that is the collision will be inelastic, and this will result in partial or complete destruction of the interference picture. To bypass this obstacle the ball has to be cooled to very low temperature and have a small velocity. Note that the same could happen, if we consider screen as a macroscopic object, which is plausible even in the case when the experiment is done with electrons - see this answer on a related question.
Buckyballs interference
Another thing to point out is that a double-slit experiment has been done with C60 molecules (buckyballs): Wave–particle duality of C60 molecules (see also here). This is somewhat similar to the tennis balle xperiment, although still on a much smaller scale.
Wavelength of a tennis ball
The other answers (correctly) pointed out the difficulties associated with a very short wave length of a macroscopic object. However, it is necessary to point out that such a composite object is not characterized by a single wavelength - what is meant here is the wave length of its center-of-mass. As long as the object does not interact with anything, the momentum of the center-of-mass is conserved and the object can be treated as a de Broglie wave, but this is manifestly not the case when it interacts with the screen/slits.
Launching the ball
Another difficulty is putting a ball in a state with a well-defined momentum. For electrons this is easily achieved by electrical means, for they are charged particles. E.g., in semiconductor realizations of the double-slit experiment one simply uses the lack of balance between right- and left-moving electrons when an electric potential is applied.
Launching a tennis ball by usual mechanical means is a very incoherent process. One could envisage something like cooling the ball to micro-Kelvin temeratures, giving it an excess electric charge and accelerating it with electric field.
Answer: The answer is an emphatic no.
For noticeable diffraction to occur, the width of the slits needs to be of the same order of magnitude as the wavelength associated with the object passing through them. In the case of a tennis ball, the wavelength would be extremely small compared to the ball itself, so a slit narrow enough to cause diffraction would be too small to allow the ball to pass.
Another reason why you would not see anything in your proposed experiment is that a tennis ball is vastly larger than an electron, so if you scaled up the viewing distance by the same factor, you would be standing around 1,000,000,000,000 kilometres from your apparatus, or about a light year.
Answer: YES, but the interference pattern will be of different shape.
https://www.desmos.com/calculator/hxjhr3eiyd
Look at this graph and change the l value. It is the wavelength of the wave.
Since the mass of a tennis ball is large compared that of an electron, its wavelength is very small according to the de-broglie equation.
So the interference pattern is, as you can see from the graph, the pattern we observe.


To put something in service of, or in lower priority to

Question: I'm looking for a slick way of saying a person is able to override their ego, or place their ego in the service of a higher goal. For example "He is able to subvert his ego for the needs of the team", but that seems a bit awkward to me.  Any better ideas?
Comment: I think the idea you have in mind is _subordinate_ (to make subject or subservient ," according to Merriam-Webster), not _subvert_ (to overturn or overthrow from the foundations: RUIN"). But _subordinate_ normally takes the preposition _to_, not _for_. So the sentence would read, "He is able to subordinate his ego to the needs of the team." But this suggestion is simply a word choice recommendation.
Answer: You could try rearranging the order:

He is able to put the needs of the team ahead of his ego.
Answer: You could say that "He is able to sacrifice his needs for those of his team."

the surrender or destruction of something prized or desirable for the sake of something considered as having a higher or more pressing claim.


Magnetic Declination-Why isn't it always 11.3?

Question: Why does magnetic declination change with area?
Comment: -1 Why should it have one value 11.3 everywhere?
Answer: Imagine I put two dots on a map and ask you what the angle between them is.
After some thought you should accuse me of asking a poorly formed question, because the angle between them depends on the location from which you measure it and can be anywhere between 0 and 180 degrees.
So, if we treat the rotational pole and the magnetic pole as two dots on the globe does it make any more sense to ask what is the angle between your shortest routes to them without first specifying where you start?
Answer: The magnetic declination is the horizontal deviation of the direction of a magnetic needle's north pole from the geographic direction of the north pole. There is a wide variation of the magnetic declination over the earth's surface. See Wikipedia https://en.wikipedia.org/wiki/Magnetic_declination .


RCCB tripping while checking the current between neutral and ground

Question: Normally, when the panel is working, it means Residual Current Circuit Breaker (a type of GFCI) can be in the on stage, but while checking the current between neutral and ground the RCCB is getting tripped. Can anybody please explain why this happens?
Comment: Are you sure you're checking *current* between neutral and ground?  Exactly how are you testing?
Answer: This is normal. There can be a slight potential difference between the neutral and ground at a receptacle. (This would be due to a load on the circuit.) If you provide a current path to ground, then current will flow in the neutral path and there would be no corresponding current in the hot. A difference in the current between the neutral and the hot will cause the RCB (GFCI in the US) to trip.
EDIT Another way to look at it is that connecting the neutral and the ground at a receptacle divides any current flowing in the neutral path thereby reducing its value below that flowing in the hot path. This causes a difference in the current flowing in the hot and neutral paths. This will trip a RCB if it is in the receptacle or in the circuit breaker for the circuit or the master breaker.
EDIT2 There has to be enough current flowing through the RCB for a connection between neutral and ground to trip the RCB, that is, the circuit must have a load somewhere causing current to flow through the RCB.
If a circuit is completely unloaded and a low resistance path is established between the neutral and the ground (such as a connection with an ammeter), then the RCB would not trip.
In the case of RCB receptacles they can be in a loaded circuit, but on spur (as they would be if they are connected by pigtailing) or they can be "distal to" (beyond) the location where the neutral and ground are connected. I think the RCB receptacles would trip in those cases, but I doubt this is relevant to the OP's question here. He is probably asking about an RCB breaker.
Comment: @EdBeal OP is measuring the current, not the voltage. Measuring the current happens with low resistance, measuring the voltage with high resistance.
Comment: I would be surprised that a meter could create enough of a load to trip a GFCI in the U.S. and that should happen at 6ma, from my reading an RCCB is set to 30ma so although your answer on the imbalance is correct having a meter trip would concern me.
Comment: Missed that an inline current test would trip unless the current was limited, I do have an adjustable tester that verifies the GFCI / RCCB trips at the proper value. Putting an amp meter from neutral to ground should trip the device as long as the meter fuse is large enough+
Answer: Working as intended. A neutral-ground fault is still a ground fault.
In fact, some US GFCIs actively check for this during their power-up self test.
GFCI/RCD isn't about voltage, it's about leakage.  Leakage on neutral can be just as deadly when combined with another problem.
Also, I'm not quite sure why you're checking for current between neutral and ground, since the RCD already does that.  Literally, the RCD compares current on the hot to current on the neutral, and they should be equal.  If they are not equal, it trips. The RCD does not examine or interact with ground at all, but I cannot imagine a way for there to be any neutral-ground current  and yet have the hot current and neutral current still be equal.


How do you handle dazing and stunning, with regard to solo monsters?

Question: In 4th edition, one of the biggest problems I've found with solos is that they get locked down way too easily--between stuns, dazes, immobilization, and other effects...they are far too effective when used on a single enemy threat.  On the other hand, I don't want to stop players from using them altogether, by removing any utility of the powers.  What tricks have worked to keep these poor, defenseless dragons (for instance) dangerous in your game?
Answer: Monster Manual 3 solos tend to use one of a few tricks to deal with this. Any one of them is pretty good; using multiples might be a bit much.

Give the solos saves at the start of their turn. If you do this, that saving throw probably shouldn't get the +5 bonus to saves. You can also tune this down a bit by only letting them make one save.
Allow the solos to save against "end of my next turn" effects. I like this one a lot, because right now missing can be more effective than hitting when a miss makes the effect have that duration.
Give the solos multiple initiative actions -- e.g., Tiamat acts several times per turn. This means they'll get more chances to save. Definitely useful.

I also recommend not having solos fight by themselves, despite the name. I think most fights are more interesting if there are some monsters alongside the solos. Even if a stunlock does happen, the other monsters remain dangerous. See also Greg Bilsland's recent post on that subject.
Finally, if you want to try something really unexpected, read the Angry DM's posts on D&D Boss Fight design. He explains exactly how to build solos that split each fight into three acts, which keeps the fight fresh throughout and reduces the dazing and stunning problem. His stuff is superb.
Comment: I also seem to remember seeing somewhere that Essentials will have some solutions to that.
Comment: Quite probably. It's definitely something that needs some help.
Comment: The blog there on D&D Boss Fights is excellent, and some of those ideas sound just like what I'm looking for.  Thanks a lot.
Answer: Sly Flourish recently wrote a blog entry with new monster powers, one of which, Brutal Shakeoff, is designed to let solos shake off status effects rather than get locked down.  In brief, the solo can sacrifice HP to save.  That blog is a great source of other monster fixes and optimizations, check it out.
Answer: My technique for handling this situation is this: If the players have a method for locking down a solo monster (or any monster, really) and I don't have any means for countering it, I just let it happen and don't sweat it. Some players have obviously invested a certain amount of time and effort developing their characters into being able to do this exact thing, so if that's their plan, I think its ok to let it happen and play out by the rules. Sometimes players come up with a great plan that easily beats a monster. There's nothing wrong with that. D&D is more than just the battles, anyhow.
Answer: Depending on the abilities that cause the status, you might be able to throw some minions at the party as well simply as fodder.  Sure, the party can still affect the main enemy, but they still have a lot of little minions to deal with while hopefully the main enemy successfully breaks free from the daze/stun/etc.
Another situation is to have back-to-back encounters where the party doesn't get the time to rest and thus earn encounter abilities back.  Maybe the dragon had a deal with a group of local kobolds that the dragon would offer protection and safety right outside of the dragon's lair in return for treasure and crunchy PCs with ketchup food.  If a kobold flees the encounter to alert the dragon and the PCs pursue it, the encounter suddenly becomes that much more difficult for them if they've spent their encounter (or daily) abilities.
Along those same lines, have the party encounter the solo enemy after a few other encounters for the day.  Hopefully they've already burned through an ability or two that they really should have held in reserve.
Comment: Heh. If I wanted to be really cute, using the dragon example, I'd have a bunch of minions who had the power Rouse the Master. Standard action, grant their master a save with no bonuses. Alternatively, if the solo cares about the minions, their deaths could trigger a save.
Answer: You can also just plain remove the ability for stuns to function continously. Something like every other round stunned, the solo monster can act normally. It still rewards stunning but limits it.
Alternatively, I heard some people give the save bonus a commulative +1. A solo's save the first time is +5, the second time +6 and so on.
Making Solo Fights have more than just the solo has already been mentioned
Hope that helps.


Can an operational amplifier circuit be made entirely out of diode Nand and Nor gates?

Question: I'm not sure whether this question has been asked before, but in researching my question, I came across this user's post (at If-else decision structure using op amps or any other non-programmable electronic device), which appears to ask my question in reverse. This user seems to be asking (effectively) whether OP Amp circuitry can be constructed in such a way as to mimic or "emulate" logical operations (or by extension, logic gates). My question, which is precisely the reverse of that, is can a circuit, consisting entirely of particular combinations of NAND and NOR diode logic gates, be constructed to function like an Operational Amplifier? (I recently asked an electrical engineer that question, and his brief answer was "no". But he never explained why he believed that it could not be done. If it cannot be done, there should be a rational reason why that is the case.) I suppose that an equivalent alternative way of phrasing my question might be, can there exist an algorithm for simulating all of the operations of an Operational Amplifier? --John
Answer:
Can An Operational Amplifier Circuit Be Made Entirely Out Of Diode Nand And Nor Gates?

This apparently simple-enough question is somewhat ambiguous and can be answered several ways.

Spehro has assumed that you convert the input to a digital value and perform digital arithmetic on it. So he says the answer is yes.
ScottMcP takes your question at face value, notes that you say "diode gates", which have no gain, assumes that you want to use the digital gate as an analog amplifier, and as diode gates have analog gains of < unity, says no.
WhatRoughBeast also takes the "digital gate as analog amplifier" and notes that the digital gate lacks features needed to make an opamp. But he also notes that you can make a single input amplifier using a digital gate as an analog gain stage.

Placeholder adopts "digital gate as analog stage" approach BUT goes the second mile and shows how a "true" [tm] differential input amplifier could be constructed.

There another option, not covered so far. That is, inputting analog signal into a digital gate which is acting as a "digital" oscillator (albeit using an analog timing delay) so that the gate outputs a digital signal, but the gate's digital operation (in this case the oscillator mark-space ratio) is affected by the analog signal, allowing amplification in the digital domain and then conversion back to audio.

Below I cover the "digital gate converted to an analog amplifier" aspects and then introduce the rarely seen [draws deep breath] "digital gate acting as an analog amplifier while producing digital output which can be easily be converted back to analog" concept. Usually more fun than useful, but potentially of value 'in an extreme situation'.

So:
Gate is digital, signals are analog"
No, not if the gates are used as digital logic gates and the amplifier deals only with analog signals.
Diode resistor logic gates with no gain.
And, also no, if the gates have no "gain".

Diode & resistor only gates have no gain - but they are also never used in anything practical because of loading by subsequent stages.

"DTL" gates operating in analog mode.
Only "sort of".
DTL = Diode-Transistor-Logic gates can have gain and can be made into (not very good) amplifiers by using them "strangely". The series diode connected inputs place constraints on what can be achieved with analog inputs. You'd have to be very keen.
More modern digital gates in analog mode:
If you are prepared to use digital gates in analog mode, as is done in some applications, then yes, it can be done. This is relatively unusual in everyday use and is usually a compromise or "done for fun" or extreme cost cutting.
General background:
An "Op-Amp"is an analog signal amplifier with high gain (Vout/Vin = "high", a high input impedance and a low output impedance. I could explain what "analog" means in this context but, if that's unclear, it's an essential part to understanding the question, let alone the answer, and you can very easily research it yourself.
If SOME digital gates have connections made from output to input via a resistor or resistor network, they can be biased into a linear state and then used as an analog amplifier.
Some Diode AND and OR gates MAY be able to be biased in this manner and so used as an amplifier. This is not using them in their intended mode and the amplifier quality would be poor. The diodes would add very significant distortion. Even older RTL logic may work better than this but it's still not a true digital gate when used this way.
Example of a modern CMOS gate being used in an analog biased mode to create a lineara amplifier.
Circuit is from here
They say:

CMOS inverters can be used as linear amplifiers where negative feedback is applied. Best linearity is achieved with feedback applied around three inverters which gives almost perfect linearity up to a dynamic output of 5 V peak to peak with a 10 V supply rail The gain is set by the ratio of Rl and R2 and the values are typical for a gain of 100. The high frequency response with the values given is almost flat to 20 kHz. The frequency response is determined by Cl and C2..
This circuit is not suitable for low level signals because the signal to noise ratio is only approx 50 dB to 5 V output with the typical values shown.

Analog amplifier with purely digital gate:
You will only rarely find this concept described. It works.
BUT (and many people are not aware of this) you can use some sorts of digital gates as analog amplifiers while remaining in a digital mode. This is a rather (or very) special case.
eg if a 74C14 Schmitt triggered input inverter has a resistor connected from output to input it will oscillate (for a certain range of resistors), using stray capacitance as part of the RC timing circuit. If you now AC couple an AC signal to the input the output will be pulse width modulated. If you low pass filter the output signal you can get amplified analog. Ask me how I know :-). The secret to "keeping it digital" is the Schmitt trigger input. The gate cannot be biased into a linear mode due to hysteresis and oscillates instead.

fosc should be >> finput. As the gate input is driven above half way by the incoming AC the negative to positive transition times of the oscillation are skewed, and similarly as Von falls below half voltage, but with opposite effect on mark space. The resultant output is variable mark-space PWM driven by Vin. Filtering this AC signal recovers the amplified signal.
Diagram based on this original
Comment: "digital gate converted to an analog amplifier" I believe they call them Class-D amplifiers. They are the cheapest and most efficient amplifier designs available. It is likely your portable electronics uses one.
Comment: The easy answer is no, I like that Sphero found a yes.  (oops Spehro, couldn't you just change your name Sphero, it's got such a nice round feel :^) @Spehro Pefhany
Comment: @DietrichEpp In my example with the Schmitt trigger input gate then, no, the gate's output transistors are running as pure switches. The output (which I show as "PWM") is a rail to rail 'square wave' with variable mark-space. It's Class D, with the relative magnitude of mark and space periods affected by how much the analog input assists or opposes the feedback via Rosc. The gain of the amplifier is affected by the ratio of Rosc and input impedance. If the input can perturb the mean current substantially the mark:space is affected more and the filtered output has a greater signal component.
Comment: @Aron -Indeed. Class D it is - but you'll seldom see it achieved this informally. It's very easy to look at the circuit and think it must be running in linear mode - and, without Schmitt trigger input it would be.
Comment: Ah... I'm used to seeing the little squiggly hysteresis mark, I didn't bother reading the notes.
Comment: @DietrichEpp Sorry. My bad. Circuit lab did not have a Schmitt symbol. If I'd been rigorous I could have copied the diagram to an editor and added it. Trying to draw inside components with lines fails. There's probably some way to do it.Next time :-) ...
Comment: @DietrichEpp Hysteresis symbol added :-)
Answer: The heart of the operational amplifier is the difference amplifier, where a current is divided between a pair of highly matched input transistors. This is what allows high gain combined with low offsets. Logic gates simply don't have the geometry to do this.
So the answer is no.
That said, logic inverters (and by extension almost any inverting gate such as a NAND or NOR), has the potential to be used as a linear amplifier - provided you can provide proper biasing and don't expect great performance. This approach is commonly used in producing low-performance crystal oscillators, which don't require purity of output waveform or exceptional stability. Back many years ago I seem to recall the occasional circuit idea using CD4000-series inverters as AC-coupled amplifiers, with such use limited to the non-buffered gates. But these applications are not operational amplifiers, merely more-or-less adequate single-ended amps. I also seem to recall that they had very limited linearity.
Comment: See my 'digital gate in digital mode as analog / PWM /  analog amplifier' example :-). More fun than value.
Answer: Sure, if you have digital numbers for the input voltages an op amp merely does the calculation Vout = Av(Vin+ - Vin-).  Easily done with gates (and either NAND or NOR gates would be sufficient).
NAND and NOR gates require more than just diodes, they require transistors, of course. OR and AND gates are not sufficient.
If you do the calculation periodically (meaning some kind of clock) you can simulate an analog op-amp arbitrarily closely, assuming the calculations are fast enough and the digitization is below the noise floor of the analog op-amp.
Answer: One of the simplest op-amps looks like this:

simulate this circuit – Schematic created using CircuitLab
And you can build it using NAND gates:

simulate this circuit

simulate this circuit
So the answer is YES.
Keep in mind that this will be crap, and will operate only over  avery narrow range.  The primary tool for operation that we have in the design of op-amps from transistors is playing with the widths and length of the transistors.  We could do this here a little bit, especially in the diff pair as I could parallel several NAND gates.
Of course I haven't given you the necessary background to understand how this works, but the circuits (topology) are identical. To clarify things, if something is not hooked up, then ignore it and if you trace the connections you can see that I have duplicated the op-amp using 5 NANDs.
Answer: An operational amplifier must have gain. There is no way to get gain out of diodes.
Comment: There's also no way to get an inversion out of diodes alone. Since the question is about NAND and NOR gates, clearly we're talking about circuits that contain active devices.


Is This a Bad Infodump?

Question: I'm kind of in a dilemma. If I was to write a prologue to a fantasy story about the history of the kingdom and the political scope of what is happening there, is it necessarily a bad infodump? All of this knowledge is incredibly important to understand the context of the story, even at the beginning, and I make the narrator explain it in an interesting way (as though he was telling a story around a campfire or something). It's meant to have a sort of "movie-montage" feel to it. It's a backstory/history, but it's meant to have an interesting, almost action-packed feel even though it's only brief snippets of scenes. (If any of you have watched The Hobbit movies, you understand the beginning prologues are kind of interesting. It's a bit like that). So, is this a bad infodump, or can it be good if I make it interesting?
Comment: What exactly is "this" infodump?
Comment: @Alexander A prologue to the story that discusses all the necessary world lore. OP is asking the question in a general sense, rather than pointing to a specific example.
Comment: @F1Krazy there is a wide variety in how this can be done, ranging from very good to very bad.
Comment: Do your main characters know the kingdom's history from the start, or would it be possible to have them learn this over the course of the story?
Comment: Ok. I understand what you guys are saying. I think I can mix most of the infodump into the story. However, I will make the first page a letter from the narrator, because that part is necessary for the plot. But thank you for the advice.
Answer: It is probably still a bad infodump, at least if you intend to sell the story to a traditional publisher.
Because it is not immediate, we don't know who the main character is, and all the "action" is in the past and has nothing to do with the present.
I have been told (by a real publisher) they he wants his MC up front, page 1, and they want them interacting with other people by page 2. At least a conversation. He doesn't want a "contemplative" opening, or an "in transit" opening (somebody thinking about life, the universe and everything while on the bus to work).
My advice is to dump the reader into the middle of your MC's everyday life, give the MC some normal-life problem to solve that requires human interaction, to help reveal their character. Let the reader figure out what is going on. If the background is truly important, it should affect the actions and decisions of the MC, and the MC can "remember" those pieces as they go. It is okay to leave the reader in the dark and reveal the back story slowly. Not just okay, but better, it builds mystery, and interest.
The problem with infodumps, no matter how interesting you try to make them, is they are just a lot of information you want the reader to memorize, and readers cannot memorize much. You are saying "Remember this guy [you never heard of] wants this girl [you never heard of] but she loves this other guy [you never heard of] because ...
The reader is out. Readers remember things by following the hero they are interested in through scenes and vicariously experiencing, through their imagination, what the hero is experiencing. They remember the scenes, and the people they interacted with, and what was said. They remember what is important, so they only remember the bits of the background story that actually has an impact on the MC and what they do and say.
One thing you can do to aid that is to make some character fairly uninformed about the backstory, so your MC or other characters must explain it, or alternatively, make your MC uninformed about aspects of the backstory, so he needs to be informed about it by other characters. Or some mix like that; nobody knows everything, but due to the mission elements are discovered.
Do not make the mistake of thinking that because some best-selling or famous author got away with something, you can too. JK Rowling is the richest author in the world, but on the craft of writing she is pedestrian, at least for adults. She appeals to the middle grade, non-adult teens especially. On the imaginative front she is stellar, off the charts, very good at combining fantasy and mystery, and that is what has carried her books to #1. After she was rejected by nearly every publisher in England, by her account. (Probably because her writing is pedestrian, and modern publishers don't read very far before pitching stories into their reject pile: Which her book was rescued from, by a publisher's child.)
JRR Tolkien is an excellent writer, also with stellar imagination. I wouldn't count on being a one-in-a-hundred-million writer. If you are, that's great, but you will still have a better chance if you avoid the obvious modern prohibitions that every publisher knows: Don't start with an infodump. Don't try to "make it work". Don't start with a prologue, or wall of text, or wall of dialogue, or a "contemplative" opening.
Open on the MC's normal world, solving their normal world issues, interacting with their culture and other people. Figure out what is actually necessary from the backstory to justify the MC's actions and those of others, and incorporate that as you go. Thousands of best-sellers do it, thousands of good movies do it, you can do it too.
Answer: There's a much better way to handle backstory you want your reader to learn. Handled appropriately, you can make learning your backstory not merely tolerable - you can make it something your readers are hungry to find out! The key is, as other answers have mentioned, to drip feed the backstory and context throughout your story.
At the beginning of your story, you don't want your readers to know what's going on. You want to give them just enough that they know you know what's going on. This makes the backstory a compelling mystery, and every little drip you give your readers from then on sheds light on why the main character's story matters.
It's a bit self-indulgent, but anime shows pull this off all the time. In Spirited Away, the main character Chihiro suddenly finds herself trapped working in a magical bathhouse. For the entire first half of the movie, she understand absolutely nothing about how the bathhouse or the magical world it exists in works, and neither does the audience. She eventually learns just enough to break a curse on her family and escape, but even that information is never infodumped. She, and the audience, learn all of it by reading in-between the lines of what happens and inferring how things work. As another example, the show Full Metal Alchemist has an incredibly rich magic system and elaborate world, but the only thing we know after the first episode is how wrong alchemy can go when used carelessly. The story ultimately includes elements from grudges from old wars that have never been forgiven, shadowy cabals that have been using dark magic for generations, desperate wizards willing to throw away their humanity for a taste of more power, and the horrifying implications of what powers alchemy - but we know none of this at the start of the show, and all of it is revealed very gradually.
This approach is also used by master writers. Terry Pratchett uses this all the time in the Discworld novels. The Discworld is a fully developed world that's reused across the entire franchsie, and Pratchett doesn't shy away from explaining how the overall world works at the beginning of his novels. But most of the individual novels are focused on one specific mystery. He consistently introduces the question of the mystery at the beginning of the novel, then very gradually chips away at it until one specific and compelling aspect of the Discworld has been explored in detail. In Going Postal, an all-powerful despot finds himself forced to strong-arm a petty criminal into running a post office that's been abandoned for decades. The question at the beginning is, why in the world does the despot need to pull so many strings just to get the mail to go out? By the end of the novel, we learn about a power struggle between the most powerful men in the city, with the post office at the heart of it - but there are only the vaugest hints of this by the time the protagonist shows up at the post office. And in Soul Music, Death becomes caught up in the surreal case of a musician who is simultaneously dead and very much alive, playing the best music anyone has heard in generations. It turns out that the poor bard is being possessed by an ancient entity so powerful that even Death is pushed to his limits struggling against it - but we don't learn exactly what this entity is until the climax! The entire book gradually gives us more and more information about this mystery, so that the reveal feels like meeting an old friend. But the full context isn't given to the reader until the very end.
In Life, the Universe and Everything, Douglass Adams writes about the galaxy being threatened by a violent race of aliens who aim to revive an ancient war using a superweapon that will wipe out all life. We don't find out about the evil aliens, let alone their history or aims, for about half the book. The Lion, the Witch, and the Wardrobe takes us into Narnia and carries the children through the beginning of their adventure before we meet Aslan or learn the full extent of the White Witch's despotic rule. Harry Potter and the Sorcerer's Stone doesn't begin any worldbuilding infodumping at all until Harry escapes the Durselys, and the mystery of the titular Sorcerer's Stone isn't fully solved until Quirrel and Voldermort explain the final missing pieces to Harry in the climax.
In all of these cases, by withholding context and backstory at the beginning and gradually explaining it, the writers keep the audience and readers curious about what's going on. Every time a new piece of information is given out, the readers need to reconsider everything they've learned in its light, and readers are invited to guess at what's truly happening.
Answer: I know the question is months old but you probably didn't yet finish writing the book and you certainly haven't selected an answer, so here's mine.
Prologues are stories, and stories aren't bad. What matters is execution.
Let's see what makes prologues bad so we can think about overcoming those aspects.

prologues are boring if/when, instead of immersing readers into the world, they work as a filter--like reading from a history textbook. The obvious solution is to use POV characters, dramatize the key points (throw away the non-key points or weave them later into the story), and use all-senses immersion to draw readers into the world as opposed to keeping them at a distance.
prologues are infodumps if they're written as exposition about a world no one cares about yet. The solution to this is to write the same information as action (i.e. dramatized scenes).
but if you write the backstory as dramatized scenes, the problem becomes that readers may bond with someone else than the MC, usually someone who'll die or be irrelevant to the story. This is more upsetting when there's a switcheroo where the writing makes them think the POV is the MC, and then backtracks on that promise. The solution to that is twofold: a) making the correct promise so readers know  the POV is indeed a throwaway one, so they won't bond and be disappointed and b) immersive writing that makes reading entertaining enough to overcompensate (visceral all-senses details, characterization through action, interesting bits of inner monologue suggesting the worldbuilding etc). I'll return to this point.

Writing isn't an exact science so, rather than thinking in terms of 'bad' and 'good', I prefer to think in terms of 'what can I do to make this exact story work?' In light of that idea, here are execution tricks that should make your prologue work.
1). Dramatize it.
You already said you want your prologue to be somewhat of a montage.
Now, we're taught as writers that putting information on the page in the character's voice is already much more entertaining than writing plain exposition. But inner monologue still isn't the best way to write the backstory. For instance, I'm struggling to read Kingdom of Liars by Nick Martell (recommended by Brandon Sanderson). The problem is that the main character's voice is, at best, ponderous. It's unfunny, it's unfocused, it's long winded, and frankly the narrative doesn't seem to know where it's headed. This pile-up of negative qualities kills pacing (where pacing = the narrative seeming to progress in some direction). Kingdom of Liars opens to a long (or at least slow) internal narrative where the character outlines something about his father being a traitor, and him being judged at the time for some obscure act. The fact is the guy is in a courtroom but the narration is kept close in his head which makes it feel like nothing happens because the camera is angled inward rather than outward. So unless the voice is funny-as-hell Pratchettian and you have something interesting to say, inner monologue falls flat in comparison with action.
Returning to the idea of montage, what I'd do is write mini-scenes with the key points of the historical conflict I need to convey. No war to make it exciting? No problem. Equally well, I'd open with (I wrote this example on the spot--sadly I can't seem to format it any better):

Queen Galpharmina tapped her fingers on the cold golden ram head of the throne's armrest, keeping her face straight as she measured the Envoy. The sweaty itch under her klakhor wool wig made her scalp twitch--it turned out klakhor was as bad as sheep in this heat wave.
The Queen kept her unblinking gaze on the Envoy and gave the silence time to mount as the Envoy stewed. In the meantime, she made a mental note to try the peacock feathers wig tomorrow.
The richly clothed man standing before the throne's foot very slowly blanched under her stare. Worry lines deepened on his face and his hands shot up to grab onto the thick rings of his Baramettal collars, the insignia of his rank.
In the quiet, Galpharmina's words resounded like the new durasteel cannon she witnessed tested only yesterday. "Kill the Envoy."
The Envoy gasped, falling to his knees, and he opened his mouth in shock.
A deafening din erupted, filling the throne room as hundreds of voices whispered, shouted, and begged. At the back of the crowd, several courtiers shuffled hurriedly to the exits.
Galpharmina made another mental note with their names. Effortlessly, her voice covered the crowd as she breathed in and let her Power Octaves flow out her throat to amplify her sound. "Send his flayed carcass to Mad King Phear." She used that name on purpose, so he will hear of her contempt.
The Envoy's lips moved like those of a beached fish but no words came out. He placed his hands on the floor as if to stop himself from falling.
Galpharmina's trusted advisor burst out of the crowd and made for the throne, his white hair in disarray and his velvets disheveled as if he ran. He kneeled next to the Envoy, in an exaggeration of court manners.
"What is it, Treak?" Galpharmina asked as if she didn't know what he was about to say. As if they hadn't rehearsed this scene weeks in advance.
"My Queen," Treak gasped, "think of the consequences! King Phear will declare war."
On cue, Galpharmina said what she'd--quite literally--been itching to say for hours. "Then let him taste our new artillery."
The Envoy's eyes shot up at her, their whites bulging like those of frightened horses.
The crowd's din peaked and cheers rose from the throng of courtiers in the audience.
"She did it!"
"The Queen has durasteel!"

This scene can, of course, also be written as "The Queendom of Batakria developed new artillery in 1765 to face the threat from the north. Their breakthrough was based on discovering a new metal infused with magical properties".
Write these mini-scenes viscerally. Write them in super close POV to make the readers enjoy the throwaway characters and breathe the world. Use all-senses immersion to make the experience of reading these mini-scenes as much about being transported to that world as they're about the world information you want to convey.
Make the right promises. If you write mini-scenes dramatizing the key points of a long war, possibly one spanning centuries, all the POV characters will be dead at the end. Readers often hate prologues because they've been given the wrong promise. They dislike getting invested into a character only for them to die unexpectedly 2 pages later (unless it's a modern fantasy based on subverting tropes and hence is killing the Chosen One to show it). So make it obvious from the first lines that all these characters will die and this is for the story's benefit. In the example above, while readers are probably amused by Galpharmina and Treak's scheme to maximize the impact of the announcement, they won't be shocked to discover she's dead for 500 years when the main story starts.
Keep the mini-scenes short. The shorter the mini-scenes that dramatize the key points of the Ten Thousand Years War are, the less time readers have to become invested in characters, and they can have fun instead. Sure, you want readers be invested in your characters, but in the right ones, not the throwaways.
Keep it interesting. Give each mini-scene a payoff. A super cool artifact is found, or a key person is saved, or a disaster is stopped with the prices of the character's lives (so they die but they win), or a cool fact about the world is revealed. If you use worldbuilding information as payoffs rather than presenting it as exposition, its entertainment value increases tenfold.
Write interesting characters. This is a special point, because I just said that readers shouldn't get attached to these throwaway characters, and that's true because you don't want to put readers off before they start liking the book. But the POV characters should still be cool and interesting to read about. Opening with incompetent, unsympathetic characters give a tone to the book and that tone will feel like stale coffee that's been sitting in the mug for days. Sketching the characters lightly through action, and making them chase the key point of the scene will make them dynamic and interesting. You don't need fully developed characters, you just need to have them do little quirky things that make them stand out.
3). Give your prologue an arc.
Outline the key points of the Ten Thousand Year War of Heavens and make the scenes build on each other as if it's an episodic story with an opening, a plot, and a conclusion. Like Asimov did with the Foundation. This will give it the meaning and direction prologues often lack because the throughline of the conflict can be traced through the scenes.
3). Only write the interesting bits.
By which I mean the key highlights of the Ten Thousand Year War of Heavens that shaped your world. Start the scene at the latest possible moment and end it early.
I know, defining "interesting" is the hardest thing but here's my shot. Interesting means a) stuff that doesn't happen in our world, i.e. is unique to the secondary world you're writing and b) stuff that's outrageous enough that it makes readers think "wait, what? how does this work?" Anything that readers already read elsewhere is in the great trove of common knowledge. You can safely rely on that knowledge of readers and skip the parts they're familiar with from Game of Thrones or Lost or whatever. Skip to the good bits that make your world unique and stir curiosity.
4). Call it something else than prologue.
Why do you necessarily need to call it a prologue? How about "Before the Beginning"? Or "How the Helionisil Archipelago Got Destroyed". Or something quirky like "The Evolution of Battle Customs During the Ten Thousand Year War Before the Jade Emperor of Heaven Descended the Black Gate Carried by the Nine Winds of Perfection."
Whatever you name it, make that name to be a hook. Make it interesting such that anyone who picks up the book will read it and say "oh I gotta know what this is about". If you hook readers well, they'll cut you more slack and be willing to wait and see where you're leading them.
Writing is like stage magic. The essence of the story stays the same, whereas the entertainment comes from how you do it. You don't have to be honest and call it a prologue. Make like a magician and call it something enticing. Storytelling is like hiding your kid's spinach in their fruit juice because they find a green drink more exciting and you can trick them by blending spinach in. Hide the concepts in the storytelling in a way that makes them exciting.
If this is what you were already planning to do, you'll be fine.
Anyway, you can do it. Good luck!
Answer: You will likely receive answers along the lines of that as long as it is appropriate, does not spoil, and does not distract the reader, than it is perfectly Okay. But honestly, you have not provided enough information for a more detailed or in-depth answer IMO.
Comment: All in all the "infodump" is a page and a half, important to the story, and not super boring.
Answer: Yes, it's bad. Don't do it. Infodumps in the prologue are the sign of a true amateur. To the average reader 'prologue' means 'optional'. Most will skip the appetiser in search of the meat.
In this circumstance you cannot equate movies with books. People with cite the Star Wars prologue as a defence but Star Wars is a film to be VIEWED. The prologue is one of the few occasions where the customer is required to READ. If they'd wanted to read they'd have stayed at home with a good book and not ventured out to the theatre.
The only reason for an infodump is if the information cannot be revealed via a scene.
Sorry.
Answer: You can move the "infodump" out of the main narrative, and include it as reference material in the front-matter or appendices.
For example, instead of your MC reciting the list of important events leading to the start of the story, include these in a timeline at the front of the book.
The reader will not read it before they start, but they will refer back to it as events are mentioned in the text. Or you can use a footnote to suggest a reader refer to it.
Likewise with maps, geneologies, Dramatis Personae, etc.
NB: historical novelists have the same problem and handle it (I think) slightly better than fantasy writers. CJ Sansom's "Shardlake" series and Colleen McCullough's "Masters of Rome" are good examples to follow.
Answer: Infodumps are not a great way to start anything.
Ultimately, if you need an infodump to explain things at the start, then you are starting in the wrong place.
There is rarely information so important that it can't be weaved in later.

Is This a Bad Infodump?


Converting NAD27 geographic to grid

Question: I have a series of lat/long coordinates in NAD27 that i need to convert to NAD27 zone 17n eastings/northings. I have ARCGIS 9.3 but no direct and reliable means for the transformation. Any ideas?
Answer: your data probably needs to be in UTM NAD27 zone 17 N ( EPSG:26717)
this projection is available in ArcGIS 9.3, so you can use  the "project" tool to change the refence system. (no transformation needed because you are in the same datum)
for UTM projection based on a list of coordinates , you can use earthpoint (batch mode available)

Converting NAD27 geographic to grid