qid int64 1 4.65M | metadata listlengths 3 3 | prompt stringlengths 31 25.8k | chosen stringlengths 17 28.2k | rejected stringlengths 19 40.5k | domain stringclasses 28
values |
|---|---|---|---|---|---|
194,852 | [
"https://mathoverflow.net/questions/194852",
"https://mathoverflow.net",
"https://mathoverflow.net/users/1056/"
] | The following question appeared in my research:
Let $G_1,G_2,G_3$ all be subgroups of $S_n$, and consider the sum
$$
\sum_{g_i \in G_i, g_1g_2g_3 = id} \epsilon(g_1)
$$
that is, we only consider triplets whose product is the identity permutation,
and we sum all the signs of $g_1$.
The question is: <em>is this sum al... | There is an example in $S_5$ where the sum is negative. Take $G_1 = \langle (12345), (1325) \rangle$, $G_2 = \langle (135) \rangle$, $G_3 = \langle (35), (235) \rangle$. Then
$ \bigl\{(g_2,g_3) : g_2 \in G_2, g_3 \in G_3, g_2g_3 \in G_1 \bigr\} = \bigl\{ (\mathrm{id}, \mathrm{id}), ((135), (25)), ((153), (23))\bigr\}... | If $G_1 = S_n$ as in your edit, then every pair $(g_2,g_3) \in G_2 \times G_3$ occurs exactly once in the sum, and for each such summand we have $\epsilon(g_1) = \epsilon(g_2g_3)$. So the sum simplifies to
$$\sum_{g_2 \in G_2}\sum_{g_3 \in G_3} \epsilon(g_2g_3) = \left( \sum_{g_2 \in G_2} \epsilon(g_2)\right)\left(\sum... | https://mathoverflow.net |
4,156,683 | [
"https://math.stackexchange.com/questions/4156683",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/636226/"
] | <blockquote>
Find the Poles of <span class="math-container">$f(z)=\frac{z}{1-(e^z)^2}$</span>
</blockquote>
The possible poles of <span class="math-container">$f$</span> are all the points <span class="math-container">$1-(e^z)^2=0 \Rightarrow z=k\pi i , k \in \mathbb{Z}$</span>
My question is: how I can prove that ind... | To see if a suspected pole in <span class="math-container">$z_0$</span> is indeed a pole (and supposing you don't want to distinguish essential singularities and poles) it suffices to find a sequence <span class="math-container">$(z_n)_n$</span> with <span class="math-container">$z_n\to z_0$</span> such that <span clas... | Assume that <span class="math-container">$k$</span> is a non-zero integer. Note that
<span class="math-container">\begin{align*}
1 - (e^z )^2& = 1 - e^{2z} = 1 - e^{2\pi ik} e^{2(z - \pi ik)} = 1 - e^{2(z - \pi ik)} \\ & = 1 - \left( {1 + 2(z - \pi ik) + \mathcal{O}((z - \pi ik)^2 )} \right) =- 2(z - \pi ik)... | https://math.stackexchange.com |
15,444 | [
"https://electronics.stackexchange.com/questions/15444",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/2486/"
] | Given that the voltage regulator on my motorbike is set to 14.3 volts (so that it charges the 12 volt battery effectively) and given that the alternator is what provides power for the electrical system while the battery is charging, does this mean that the components in the system rated at 12 volts are being run above ... | As Kevin said, anything intended for automotive use will have this taken into account. These systems may be commonly described as "12V" to the public, but it's well known the voltage will be higher.
Designing electronics for automotive "12V" power can be a challenge. The fact that the battery charging voltage, and t... | This is safe, and will not shorten the lifetime of those components. This is not because a component rated for 12 V will not be damaged by 14.3 V, but because it's actually designed for that rating.
I'm not sure why your bike charges at 14.3 V, but the automotive standard is 13.8 V. Neither of th... | https://electronics.stackexchange.com |
626,756 | [
"https://physics.stackexchange.com/questions/626756",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/280091/"
] | Suppose on a summer afternoon, you have two chairs one made of <strong>steel</strong> and other of <strong>wood</strong>. Which one is better to sit on if you prefer colder one, provided they are at the same temperature?<br>Would your answer vary if both the chairs are now replaced by a <strong>solid wooden box</strong... | <strong>Potential of one is this and the other is this</strong> is a misnomer at best. The potential energy is stored in the fields and is simply given by-
<span class="math-container">$$U=-\dfrac{Gm_1m_2}{r}$$</span>
where <span class="math-container">$r$</span> is the distance between them.
There is no meaning to the... | If you have two equal masses, each at a distance of , r, from the center of mass, and you move them one at a time out to infinity, then the work to move the first one will be W = Gmm/(2r), but the work to move the second one will be zero, since there is no longer a force on it from the other mass. The initial total pot... | https://physics.stackexchange.com |
43,930 | [
"https://stats.stackexchange.com/questions/43930",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/10675/"
] | I'm trying to understand the philosophy behind using a Generalized Linear Model (GLM) vs a Linear Model (LM). I've created an example data set below where:
$$\log(y) = x + \varepsilon $$
The example does not have the error $\varepsilon$ as a function of the magnitude of $y$, so I would assume that a linear model of ... | Good effort for thinking through this issue. Here's an incomplete answer, but some starters for the next steps.
First, the AIC scores - based on likelihoods - are on different scales because of the different distributions and link functions, so aren't comparable. Your sum of squares and mean sum of squares have been... | In a more general way, $E[\ln(Y|x)]$ and $\ln([E(Y|X])$ are not the same. Also the variance assumptions made by GLM are more flexible than in OLS, and for certain modeling situation as counts variance can be different taking distinct distribution families.
About the distribution family in my opinion is a question abou... | https://stats.stackexchange.com |
114,618 | [
"https://electronics.stackexchange.com/questions/114618",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/15289/"
] | I was trying to measure the current draw of a USB-powered device (a Raspberry Pi) to determine the suitability of powering from an external battery pack. So I took a USB cable, split the +5v (red) line, placed a DMM on the line, plugged it in, and got.... nothing. The pi wouldn't boot. Replacing the DMM with a jumpe... | Do this:
<ol>
<li>Unplug the cable from the host and the device.</li>
<li>Make S1.</li>
<li>Adjust the power supply for 5V out as read on V.</li>
<li>Open S1</li>
<li>Connect the cable to the host.</li>
<li>Connect the cable to the device.</li>
<li>Make S1.</li>
<li>If required, readjust the power supply for 5V across... | This is why you need two multimeters. Then you could measure the voltage to see if it has sagged.
What device are you using to supply power? A computer or a USB charger? Computers can be funny about this sort of thing because they limit current to 100mA by default and rely on there being a current spike to detect when... | https://electronics.stackexchange.com |
1,384,646 | [
"https://math.stackexchange.com/questions/1384646",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/118494/"
] | There exists a map $f: \Bbb Z\rightarrow \Bbb Q $ such that $f$ is
A. Bijective and increasing
B. Onto and decreasing
C. Bijective and satisfies $f(n)\ge 0$ if $n\le 0$
D. Has uncountable images
Now option D. is absurd . Option C. is given to be the correct answer.I was thinking sin... | The problem is that, while the cardinality of $\mathbb{Z}$ and $\mathbb{Q}$ is the same, the topology is different. Consider the following: Given two integers $a$ and $b$ with $a < b$, can we say how many integers $c$ satisfy $a < c < b$? Now ask the same question for the rational numbers.
This leads to pro... | For A, some <span class="math-container">$f(n)$</span> lies between <span class="math-container">$f(0)$</span> and <span class="math-container">$f(1)$</span>, which requires the integer <span class="math-container">$n$</span> to lie between <span class="math-container">$0$</span> and <span class="math-container">$1$</s... | https://math.stackexchange.com |
196,207 | [
"https://softwareengineering.stackexchange.com/questions/196207",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/81895/"
] | So I am working on a small Project at my company, I am a mere trainee engineer (halfway through a Software Engineering degree), where we are developing an intranet - based project management and people management system - it incorporates Employee appraisals, personal skill management and development and so on.
Ive bee... | Make sure you get input from the more senior member regarding security. Employee information is not public record.
<strong>Here are some time savers:</strong>
<strong>Management Brains</strong> - Are they aware this started as the 3 person job and is now down to one? Someone has to do the math and adjust.
<strong>Ex... | If you mean that the distinct types of tasks were first divided among three, I suggest you request a weekly session (several hours) with the senior engineer to discuss progress, issues, ideas. It would be very unwise from the company standpoint to let you do it all alone, regardless of your level of expertise.
| https://softwareengineering.stackexchange.com |
2,079,726 | [
"https://math.stackexchange.com/questions/2079726",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/403395/"
] | Is it possible to find the sum of the series:
$$\frac{1}{p} + \frac{2}{p^2} +\frac{3}{p^3} +\dots+\frac{n}{p^n}\dots$$
Does this series converge? ($p$ is finite number greater than $1$)
| Hint:
$$\frac{1}{1-x}=1+x+x^2+\dots$$
Differentiate this formula (it is uniformly convergent for $|x|<1$) and multiply with x. Finally set $x=1/p$.
| The exact value is $\frac{\frac{1}{p}}{(1-\frac{1}{p})^{2}}$ when $-1<\frac{1}{p}<1$. Since $p>1$ in this case, the series converges.
Proof:
$$
\frac{1}{p}+\frac{2}{p^{2}}+\frac{3}{p^{3}}...
$$
$$
=(\frac{1}{p}+\frac{1}{p^{2}}+\frac{1}{p^{3}}...)+(\frac{1}{p^{2}}+\frac{1}{p^{3}}+\frac{1}{p^{4}}...)+(\frac{1}{... | https://math.stackexchange.com |
470,401 | [
"https://math.stackexchange.com/questions/470401",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/24690/"
] | Could any one tell me number of element in a principal ideal domain can be $25/36/35/15$ ?
I just know a principal ideal domain is generated by a single element. what the knowledge I need to find this result?
Thank you
| First, you need to know that any finite integral domain is a field. Then you should know that there is a finite field of cardinality $n$ if and only if $n$ is a prime power.
Note that what you said is probably not what you meant:
<blockquote>
a principal ideal domain is generated by a single element.
</blockquote>
... | 25 elements can have a principal ideal domain because for if n is p^2 then there exist a field of order P^2 moreover since a field is pid then every ideal is principal.
| https://math.stackexchange.com |
32,886 | [
"https://scicomp.stackexchange.com/questions/32886",
"https://scicomp.stackexchange.com",
"https://scicomp.stackexchange.com/users/32039/"
] | <h1>Goal</h1>
I'm trying to write code to compute the normalized Gaussian in the following,
<span class="math-container">$$
\begin{equation}
\int_{-\infty}^{\infty} \frac{1}{ \sigma \sqrt{2 \pi}} \exp\bigg( - \frac{(x - \mu)^{2}}{2 \sigma^{2}}\bigg)dx
\label{1}
\tag{1}
\end{equation}
$$</span>
where <span class="ma... | In the example output from your code, <span class="math-container">$\sigma$</span> is huge, i.e. the Gaussian is extremely broad. The variable <code>s</code> you define as the pre-factor for the argument of the corresponding exponential is then only <span class="math-container">$\approx -1\cdot{}10^{-15}$</span>, which... | No need for quadrature; your integral can be computed analytically. For this, you'll have to do the variable transformation <span class="math-container">$y = (x-\mu) / \sqrt{2} \sigma$</span>:
<span class="math-container">$$
\int\limits_{-\infty}^{\infty} k\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) {\rm d}x = k\sqr... | https://scicomp.stackexchange.com |
309,487 | [
"https://electronics.stackexchange.com/questions/309487",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/139284/"
] | How does one use <code>exit(EXIT_SUCCESS)</code> or <code>exit(EXIT_FAILURE)</code> in microcontroller C programming. I know it works on application level C programs when you include <code>#include<stdlib.h></code>.
The idea is to exit the main loop altogether when some unexpected value is returned, for example... | A good way to search in eagle is to use the * character. For example <code>"LM*"</code> will bring up all parts staring with LM and <code>"*LM*"</code> will bring up all parts containg LM.
Although eagle, like every other circuit design tool does not have everything. Designing your own parts can be an essential tool.... | I have been using eagle for a while now and I used to face the same problems in the beginning. Searching for the exact package that you are using might be a tedious process and since the package has been made by someone else, you must verify whether the package is exactly same as what you are using or not by taking a p... | https://electronics.stackexchange.com |
99,605 | [
"https://softwareengineering.stackexchange.com/questions/99605",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/14754/"
] | I've heard the person at the keyboard named the "driver", and the other person named the "navigator". I've imagined rally car racers, where the person at the wheel just <em>cannot</em> keep up with <em>everything</em> that's happening, and they <em>must</em> have the help of the "navigator", who's shouting volumes and... | The driver (or less commonly pilot) has hands on with the keyboard and is right there, banging out the code.
The navigator (or observer, or less commonly co-driver or co-pilot) is sitting alongside with the reference documents making sure the code is going the right way.
The navigator has a better perspective of wha... | The way I think of it is nothing more than the driver holds the steering wheel (keyboard), the other guy needs a title and the rally metaphor works for most geeks.
I never thought of of extending the metaphor to the tree as suggested by @Bart
| https://softwareengineering.stackexchange.com |
607,821 | [
"https://electronics.stackexchange.com/questions/607821",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/2575/"
] | When I grew up these things were pretty commonplace:
<img src="https://i.stack.imgur.com/YCYML.png" alt="Water heater spiral" />
In case you've never seen one, it's a very primitive water heater that you simply place inside a pot of water (or a jar, or even a cup of tea-to-be).
Today they're almost extinct, for obvious... | Heating elements don't usually explode, they are made out of metal, metal usually melts at high temperatures. (unless there was something inside the coil which is unlikely)
What the warning was most likely attached to was: "don't plug in the element, then put it in the water". If you do that, then the water d... | You could make something like that to survive operation outside the water but it would require an expensive swaged heater, and would get dangerously (red) hot outside of liquid.
In fact those things are an aluminum tube with a fiberglass sleeve inside and inside the sleeve is some nichrome wire. That's it. They're depe... | https://electronics.stackexchange.com |
118,757 | [
"https://dba.stackexchange.com/questions/118757",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/80463/"
] | I want to check col1, and if there's 2 of the same in col1, copy the value in col2 that's != 0 to the col2 that is 0. The data comes in from an Excel file that I can't alter. See example below
Input
<pre><code>+----------+-------+
| prime | % |
+----------+-------+
| a1 | 3 |
| a1 | 0 |
| b... | This solution gives you the desired output.
<pre><code>DECLARE @TempTable TABLE
(
Prime CHAR(2) NOT NULL
, Percentage INT NOT NULL
);
INSERT INTO @TempTable
( Prime, Percentage )
VALUES ( 'a1', 3 )
, ( 'a1', 0 )
, ( 'b2', 25 )
, ( 'b3', 42 )
, ( 'c4', 23 )
, ... | This query gives you the correct output:
<pre><code>create table #data (prime char(2), pct int)
insert into #data(prime, pct) values
('a1', 3)
, ('a1', 0)
, ('b2', 25)
, ('b3', 42)
, ('c4', 23)
, ('c4', 0)
Update d1 set pct = d2.pct
From #data d1
Inner Join #data d2 on d2.prime = d1.prime
Wher... | https://dba.stackexchange.com |
59,455 | [
"https://dba.stackexchange.com/questions/59455",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/34663/"
] | I was in a meeting with my boss and technical support leader and they asked me for a database model to a very simple application. That application requires: employees information, some location data, and etc, etc, etc.
One of their database table was:
<ul>
<li>employee
<pre><code> id
name
lastname
username
... | I currently work in this field. This is a case where a surrogate key should be used.
Our software probably deals with a much wider range of scenarios than yours, but that extra flexibility may mean a lot for you in the future.
<ul>
<li>Student numbers (overall) may change quite frequently depending on how the numbers... | I would go with the surrogate key approach. I'm not familiar with matriculation number, but i expect it to be large enough so that having a surrogate integer might be the preferred alternative. Having a larger PK, means larger indexes, which means less performance for reads and writes, plus more space usage, and more s... | https://dba.stackexchange.com |
3,833,908 | [
"https://math.stackexchange.com/questions/3833908",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/745500/"
] | I have this task and I would like to hear ideas/solution for it.
There are 30 students on an exam.
The questions are 20.
Every student draws 2 question and the condition to pass the exam is true answer on the both questions.
There are 3 groups of students:
<ol>
<li>5 students knows all questions <span class="math-conta... | Take for example the ten students who know only five of the twenty questions. In order for a student in this group to pass the exam, both questions they are given must be among the five that they know how to answer. There are <span class="math-container">$\binom{5}{2} = 10$</span> ways to choose two questions from th... | For 1 <span class="math-container">$$P(\text{student passes}) = \frac{5}{30} \cdot 1 + \frac{15}{30}\frac{\binom{10}{2}}{\binom{20}{2}} + \frac{10}{30}\frac{\binom{5}{2}}{\binom{20}{2}} $$</span> splitting on the type of student and if we know e.g. 10 questions, we have to pick 2 from the good questions over 2 of all ... | https://math.stackexchange.com |
29,830 | [
"https://cs.stackexchange.com/questions/29830",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/21646/"
] | In some places in the world, people don't usually have access to (and hence little knowledge of) computers, and even if they have, hard- and software are outdated and usage plagued by power outages and such. Access to (good) books also tends to be lacking. How can I teach computer science under such circumstances?
I'... | Asking how you can study computer science without computers is a bit like asking how you can study cosmology without telescopes. Sure, it's nice to be able to look at the things you're studying and it's often very helpful to be able to play around with things. But there's a whole lot you can do without access to a comp... | Many computer science theories are purely theoretical and can be solved using pen(cil) and paper. Getting information about them and communicating them to the outside world requires a computer.
| https://cs.stackexchange.com |
8,112 | [
"https://electronics.stackexchange.com/questions/8112",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/2172/"
] | What's a good way to reduce the output from a 9V battery to the 1.8V to 5V required by an ATmega328 controller? The context is a small robotics platform with low power requirements (very slow movement).
| I would use a 7805 to get 5 volts simple circuit. Here is a image:
<img src="https://i.stack.imgur.com/MApBh.gif" alt="7805">
idea: please make sure that caps are ceramic/polymer caps. The ceramic caps only have
low ESR value. specially the right hand one.
| I might consider just providing the ATmega328 unregulated power unless you have something that actually requires it on the board. 2-3 AA cells should remain within 1.8 to 5.5 V over their life.
A schottky might be prudent if you insert the batteries the wrong way (probably will need 3 cells then), and you could add a... | https://electronics.stackexchange.com |
193,924 | [
"https://electronics.stackexchange.com/questions/193924",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/70923/"
] | I've noticed the term <em>telecom applications</em> or just <em>telecom</em> in a few datasheets I've come across, and in the book I'm working through, the phrase <em>"In high-speed dc-dc switchers for the telecom market ..."</em> appears.
What kind of requirements does that term imply? Does that mean a device that ap... | I know of at least one requirement for the telecom market and the DC-DC switching (or even linear) voltage regulators involved - they need to handle +-48V on the input. This is quite high for modern voltage regulators in most of the industries/fields I've dealt with, and this high voltage is a remnant of the old days ... | "Telecom" power supplies tend to have the following requirements or side-effects:
<ul>
<li><strong>Robustness</strong>, because you don't want to see your internet connection
fail, and because they are sometimes installed in remote locations,
making service very expensive.</li>
<li><strong>Hot-Swap</strong> is OK, bec... | https://electronics.stackexchange.com |
1,878,755 | [
"https://math.stackexchange.com/questions/1878755",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/-1/"
] | <blockquote>
Integrate $$\int \frac{x\ln (x)}{(x^2-1)^{3/2}} dx$$.
</blockquote>
<strong>My Try:</strong>
$$\int \frac{x\ln (x)}{x^3(1-1/x^2)^{3/2}} dx=\int \frac{\frac{1}{x^2}\ln (x)}{(1-\frac{1}{x^2})^{3/2}} dx.$$
Putting $x=\frac{1}{t}$,
I get
$$\int \frac{t^2\ln (\frac{1}{t})}{(1-t^2)^{3/2}(-t^2)} dt=\int \frac{... | One may first integrate by parts,
$$
\int \frac{x\ln (x)}{(x^2-1)^{3/2}} \:dx=-\frac{\ln (x)}{(x^2-1)^{1/2}} +\int \frac1{x(x^2-1)^{1/2}} \:dx
$$ then using the change of variable $u=\dfrac1{(x^2-1)^{1/2}}$, one gets
$$
\int \frac1{x(x^2-1)^{1/2}} \:dx=-\int \frac1{1+u^2} \:du=-\arctan \left(\dfrac1{(x^2-1)^{1/2}}\righ... | Check that
$$\int\frac1{t^{3/2}}dx=-2t^{-1/2}+C\implies \int\frac{f'(x)\,dx}{f(x)^{3/2}} =-2f(x)^{-1/2}+C$$
and now just observe that $\;2x=(x^2-1)'\;$ , so by parts:$${}$$
$$\begin{cases}&u=\log x,&u'=\cfrac1x\\{}\\&v'=\cfrac x{(x-1)^{3/2}},&v=-\cfrac1{\sqrt{x^2-1}}\end{cases}\;\;\implies$$$${}$$
$... | https://math.stackexchange.com |
188,420 | [
"https://physics.stackexchange.com/questions/188420",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/77256/"
] | In a Phet Lab simulation, which is all I have at the moment to do my learning on electric field forces, the point represented as colored on the diagram below is shown to have an electric field magnitude of 0. I can see why from the arrowhead diagrams, but how can I explain this? Is it simply that the repulsion of the e... | <blockquote>
Is it simply that the repulsion of the equidistant point charges creates a dead zone?
</blockquote>
It has nothing to do with repulsion or attraction. To find the electric field at some location due to a set of point charges, you have to add the electric field contribution due to each of the point char... | Electric field is zero in that point because the sum of electric field vectors have same intensity and direction, but are opposite. That point is halfway between two like charges.
| https://physics.stackexchange.com |
169,447 | [
"https://electronics.stackexchange.com/questions/169447",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/56106/"
] | <img src="https://i.stack.imgur.com/NkuFz.png" alt="Red outline">
What is the current source for in the schmitt trigger? I can't understand this. Can't see any resistor either.
| This appears to be a misuse of the symbol for a current source. Obviously, it should be a voltage source (which sets the comparison threshold), but whoever made the diagram (engineers, marketing, etc.) was too lazy or stupid to use the correct symbol.
| Maybe try this circuit instead: -
<img src="https://i.stack.imgur.com/YS6qm.jpg" alt="enter image description here">
| https://electronics.stackexchange.com |
366,812 | [
"https://mathoverflow.net/questions/366812",
"https://mathoverflow.net",
"https://mathoverflow.net/users/112207/"
] | Let <span class="math-container">$ \Bbb S^{d-1}=\{(x_1,\cdots ,x_d): x_1^2+ \cdots +x_d^2=1\}\subset \Bbb R^d$</span> be the unit
sphere. Let <span class="math-container">$\nabla u= (\partial_{x_1}u,\cdots, \partial_{x_d}u)$</span> be the gradient of a function <span class="math-container">$u\in C_c^\infty(\Omega)$</sp... | I do not believe there is a simple formula to express <span class="math-container">$I(u)$</span>, but for sure for most of the functions the inequality
<span class="math-container">$$
I(u)<\int_\Omega |\nabla u|^p\, dx
$$</span>
is sharp. For example if <span class="math-container">$\Omega=B$</span> is a ball and <s... | More of a comment than an answer, really, but too long for the comment box: For a fixed smooth function <span class="math-container">$u$</span> the map
<span class="math-container">$$
e\mapsto U(e):=\int_\Omega |\nabla u(x)\cdot e|^p dx
$$</span>
is differentiable as a function of <span class="math-container">$e\in\mat... | https://mathoverflow.net |
151,288 | [
"https://physics.stackexchange.com/questions/151288",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/23517/"
] | I am supposed to calculate the change in energy upon changing
both the temperature from $T_1$ to $T_2$ and the volume from $V_1$ to $V_2$.
Now I was wondering whether this solution is correct:
We can treat both transition independent from each other, as the energy is not path-dependent and therefore we could have $... | Your idea is correct but the calculation is not. For $N$ moles of an ideal gas you have $U = cNRT$, look that if $T$ doesn't change, $U$ also doesn't change. So you start at the state $(T_1,V_1)$ and want to go to the state $(T_2,V_2)$. You then use two process corresponding to the following sequence of states:
$$(T_1... | No, your solution is not correct. The energy difference is by definition the first term in your formula, you should drop the second term. The work done during the process is path dependent, so you don't have enough information to calculate the work done, neither the transferred heat (which is the sum of the energy diff... | https://physics.stackexchange.com |
380,102 | [
"https://softwareengineering.stackexchange.com/questions/380102",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/269824/"
] | I am making a shopping cart. A shopping cart will have a total amount of money you've got to pay for the products you've selected.
I'd like to approach the problem using both OOP (encapsulating it) and more anaemically or functionally (without encapsulation).
I'm using the simplest Product model:
<pre><code>public c... | Unless we're talking about tens of thousands of products, recalculating the total price every time is not worth considering in terms of performance. My guess is in a typical shopping list, you're rarely going to see more than 10 items. However, if that's not your case, feel free to write me in the comments and I'll r... | You are re-implementing <code>ICollection</code> too much here.
<h2>d) Get-only Properties</h2>
<pre><code>public class ShoppingCart
{
public ICollection<Product> Products { get; } = new List<Product>();
public decimal TotalPrice => Products.Sum(p => p.Price);
}
</code></pre>
Your ORM shoul... | https://softwareengineering.stackexchange.com |
233,740 | [
"https://security.stackexchange.com/questions/233740",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/237241/"
] | I've been unable to find any research or information on this.
Google periodically signs me out and forces me to sign back in. I have multiple devices and multiple google accounts so it's a bit frustrating but that's just how it is. However I was thinking about whether this practice is actually secure.
<ol>
<li>It see... | I agree with your thought that forcing users to re-authenticate frequently weakens security in the ways you describe. However there are also other security and performance factors at play here that may not be as obvious.
When you log in, what happens in the background is that the server issues some kind of "authen... | Many users use password managers. That's why entering password more often or less often doesn't make much difference.
<blockquote>
<ol>
<li>It seems to encourage easy-to-remember / easy-to-type passwords over longer stronger passwords</li>
</ol>
</blockquote>
It depends. For users that care about security there may be ... | https://security.stackexchange.com |
33,918 | [
"https://cs.stackexchange.com/questions/33918",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/23833/"
] | I'm still a bit confused with the terms "input length" and "input size" when used to analyze and describe the asymptomatic upper bound for an algorithm
Seems that input length for the algorithm depends a lot of the kind of data and the algorithm you are talking about.
Some authors refer to input length to the size of... | In the most formal sense, the size of the input is measured in reference to a Turing Machine implementation of the algorithm, and it is the number of alphabet symbols needed to encode the input.
This is of course rather abstract, and is very difficult to work with in practice, or at least very annoying - we would need... | It depends on your model of computation and also on the unfortunately sometimes on the algorithm itself.
<ul>
<li>If your model of computation is a <strong>Turing machine</strong>, then the size of the input is the number of cells occupied by the input. So if your input is $ababcd$ then the input has length 6. </li>
<... | https://cs.stackexchange.com |
4,839 | [
"https://earthscience.stackexchange.com/questions/4839",
"https://earthscience.stackexchange.com",
"https://earthscience.stackexchange.com/users/1371/"
] | If there were lots of geothermal plants—even mobile ones—near a volcano, how much power could this provide? Could the sapping of some of the heat energy make the volcano less likely to erupt?
| The question is a little off the mark, as volcanoes are just the surface expression of a much larger system. A more general idea is to target shallow magma chambers, and the hydrothermal fluids they act upon. This is being done all over the world, in tectonically active areas where there is a lot of molten rock at reas... | Volcanoes erupt due to increase in pressure, within the magma chamber (often 3-10 km deep), due to exolusion of volatiles. The magma chamber is usually heated from below. Geothermal plants in volcanic regions exploit heat from hot water in the "shallow subsurface", few hundreds of meters deep (and rarely exceeding 1-2 ... | https://earthscience.stackexchange.com |
120,421 | [
"https://mathoverflow.net/questions/120421",
"https://mathoverflow.net",
"https://mathoverflow.net/users/13356/"
] | Is there some regularizing version of Schwartz kernel theorem for topological spaces, i.e., in the form of
<blockquote>
Every continuous linear map $A\colon C\prime(X_2) \to C(X_1)$ is given by a kernel $k \in C(X_1 \times X_2)$?
</blockquote>
Here $C(\cdot)$ is the space of continuous functions, equipped with a su... | As has been pointed out by Peter Michor above, this is false, even for compact spaces. However, it is perhaps of interest that one can characterise in a natural way those operators from $C(K_1)'$ to $C(K_2)$ which are represented by kernels of the above type. They are the linear operators whose restrictions to the un... | If $X_i$ are both compact, then $C(X_1\times X_2)= C(X_1)\hat{\hat\otimes} C(X_2)$ for the completed inductive tensor product. So you only get the closure of the space of continuous finite rank operators in the space of all continuous operators in this way.
If $X_i$ are locally compact, then the same works for $C_0(X_i... | https://mathoverflow.net |
67,271 | [
"https://mathoverflow.net/questions/67271",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3410/"
] | This came up in a question on the xkcd forums. Is it possible to have a nonconstructive metaproof, i.e. a proof that there exists a proof in some formal system which does not construct said proof? Are there any known examples, preferably with some well-known formal system like PA?
Conversely, is it possible to prove a... | In theory, David’s answer is correct. Nevertheless, in practice it is perfectly possible to prove the existence of a proof non-constructively (such as by manipulating models and then appealing to the completeness theorem) where no one has a clue how to actually find the proof.
One example which springs to mind is Jaco... | If the proof system is recursively axiomatizable, this situation cannot occur.
If there exists a proof of $\Theta$, there exists an algorithm to find that proof. Namely, search the recursively enumerable set of deductions until you find a proof of $\Theta$. This must terminate, as we have proved that $\Theta$ is prova... | https://mathoverflow.net |
238,253 | [
"https://physics.stackexchange.com/questions/238253",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/2260/"
] | Is there any connection between the graviton, which is listed as a hypothetical particle, and the newly detected gravitational waves?
| They are both postulated by two different theories. Gravitons by standard model, and gravitational waves by general relativity. There is no theoretical link between the two at this time.
| If wave-particle duality holds for gravity then the graviton would be to gravitational waves what the photon is to electromagnetic waves.
| https://physics.stackexchange.com |
1,764,983 | [
"https://math.stackexchange.com/questions/1764983",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/278230/"
] | Which option(s) below have the values of $a$ and $L$ that satisfy the following equation?
$$\frac{\int_0^{4π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt}{\int_0^{π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt}=L$$
<blockquote>
Options:
<ol>
<li>$a=2$, $L=\frac{e^{4\pi}-1}{e^{\pi}-1}$</li>
<li>$a=4$, $L=\frac{e^{4\pi}-1}{... | $\begin{array}\\
\int_0^{4π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt
&=\sum_{k=0}^3 \int_{k\pi}^{(k+1)π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt\\
&=\sum_{k=0}^3 \int_{0}^{π}e^{t+k\pi}(\sin^{6}(a(t+k\pi))+\cos^{4}(a(t+k\pi)))\,dt\\
&=\sum_{k=0}^3 e^{k\pi}\int_{0}^{π}e^{t}(\sin^{6}(at))+\cos^{4}(at)))\,dt
\qquad\tex... | Hint use $\sin(x)=R(e^{ix}),\cos(x)=I(e^{ix})$ or $\sin(x)=\frac{e^{ix}-e^{-ix}}{2i},\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$
| https://math.stackexchange.com |
312,219 | [
"https://softwareengineering.stackexchange.com/questions/312219",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/24822/"
] | What is easier to understand, a big boolean statement (quite complex), or the same statement broken down into predicate methods (lots of extra code to read)?
Option 1, the big boolean expression:
<pre><code> private static bool ContextMatchesProp(CurrentSearchContext context, TValToMatch propVal)
{
re... | <blockquote>
What is easier to understand
</blockquote>
The latter approach. It's not only easier to understand but it is easier to write, test, refactor and extend as well.
Each required condition can be safely decoupled and handled in it's own way.
<blockquote>
it's problematic because you have to read all the... | In general, the latter is preferred.
It makes the call site more reusable. It supports DRY (meaning you have less places to change when the criteria change, and can do it more reliably). And very often those sub-criteria are things that will be reused independently elsewhere, allowing you to do that.
Oh, and it makes... | https://softwareengineering.stackexchange.com |
356,366 | [
"https://physics.stackexchange.com/questions/356366",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/149370/"
] | In free body diagrams, such as a beam attached horizontally to a wall, $F_g$ is always shown acting on the center of gravity of an object.
My question - is this the case in real life, where gravity only acts on this point of the object? Or is gravity acting on all parts of the object, but that point is at the exact ... | Gravity (treated as homogenous) is acting the same on all parts of the object, but if the object is rigid, internal forces allow the simplification that the centre of mass is where all the force acts.
Torque:
There is equal mass on both sides of the centre of mass so there is no net torque about it. If the pivot is at... | Gravity pulls in every single particle in the object.
This means that gravity causes a <em>torque</em> at every single particle around some other point.
<ul>
<li>If you look at a point to the left, then all those torques sum up to a <em>net</em> torque which pulls <em>clockwise</em>.</li>
<li>If you look at a point t... | https://physics.stackexchange.com |
429,349 | [
"https://physics.stackexchange.com/questions/429349",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/151236/"
] | From my professor's notes on statistical mechanics. <span class="math-container">$\left|\bf{k}\right\rangle$</span> is eigenstate of the hamiltonian of the free particle with periodic boundary conditions:
<span class="math-container">$$
\left\langle{\bf r}|{\bf k}\right\rangle = \frac{1}{\sqrt{V}} e^{i\bf{k}\cdot\bf{r}... | I've relabeled <span class="math-container">$\bf r,\, r'$</span> from the initial integral to:
<span class="math-container">$$ \delta_{\bf k,k'} \frac{\hbar^2k^2}{2m} = \frac{1}{V}\int\mathrm{d}{\bf r'_1}\int\mathrm{d}{\bf r_1}e^{-i\bf k' \cdot r'_1}\langle{\bf r'_1}|T_1|{\bf r_1}\rangle e^{i\bf k \cdot r_1} $$</span>... | In order to derive the last equation from the one before it, multiply both sides by $e^{-i \boldsymbol{k}\cdot \boldsymbol{r}''}e^{i \boldsymbol{k}'\cdot \boldsymbol{r}'''}$, and sum over both $\boldsymbol{k}$ and $\boldsymbol{k}'$, i.e. perform inverse Fourier transforms. Using the identity
$$
\sum_{\boldsymbol{k}} e^... | https://physics.stackexchange.com |
295,486 | [
"https://electronics.stackexchange.com/questions/295486",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/118072/"
] | When there is I2C lock caused by slave device driving the SDA line low you need to manually generate a few clock signals by doing a simple bit banging on the SCL pin. I thought that this need to be done with the same frequency as the I2C communication was running before the lock up but it also works for me if I do a bi... | <blockquote>
Does clock speed matter when recovering from I2C bus lock up?
</blockquote>
<strong>Simple answer:</strong> No.
<strong>Detailed answer:</strong> As you know, slaves are not programmed to run at a specified frequency because SCLK is generated and "served" by the master. To ensure proper resetting, jus... | It seems your problem is that some IIC bus slave is holding SDA low when it shouldn't.
Your solution is to give the slave clock pulses until it finally lets go of SDA. These are ordinary clocks from the slave's point of view. You can use whatever clock speed the bus and that slave can support.
As soon as the slave ... | https://electronics.stackexchange.com |
175,802 | [
"https://security.stackexchange.com/questions/175802",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/166511/"
] | I have looked all over online as well as this site to try to find out more information regarding the security of this, but haven't found anything. In my particular case, the product is a website, but I think this question applies for any software that hosts a large number of users.
I know there are numerous websites o... | Many people have looked at the reasons not to allow name changes from both a security and a community standpoint. However, there are plenty of legitimate reasons to allow username changes, <em>even if the username is separate from the display name</em>, for example:
<ul>
<li>Someone has changed their real life name or... | I would say as long as they are not able to change their unique identifier. I.e. they can change the name they show up as, but that name is tied back to an unchanging user ID number (this will make your DBAs happier too). I'd also make sure user's couldn't change their name to an old name of another user (to help mitig... | https://security.stackexchange.com |
1,177,218 | [
"https://math.stackexchange.com/questions/1177218",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/221284/"
] | Let $A$ be the event "only $2$ different faces in $4$ rolls of a die."
At each roll there's $6$ possibilities, so:<br>
$$\omega = 6\cdot 6\cdot 6\cdot 6$$<br>
Considering that it can be only two kinds of faces, for the first one there's $6$ possibilities, for the second one $5$ possibilities, for the third and forth t... | Your method undercounts the number of ways of getting two kinds of faces. You've assumed the second die roll is different from the first, when there is no reason to believe this.
There are $\binom{6}{2}$ pairs of faces. You can either (1) roll $3$ or one face and one of the other, or (2) roll $2$ of each face. There a... | I would say there are $6\choose2$ ways to pick the two faces and $\frac{4!}{2!2!}$ ways to arrange those two pairs of numbers.
| https://math.stackexchange.com |
639,144 | [
"https://physics.stackexchange.com/questions/639144",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/300594/"
] | I'm trying to solve problem 14.31 from University Physics with Modern Physics (13th E.) by Young & Freedman.
<blockquote>
Q: "You are watching an object moving in SHM. When the object is displaced <span class="math-container">$0.6\ m$</span> to the right of its equilibrium position, it has a velocity of <span ... | Presumably you know that experiments demonstrated the falsehood of the Classical model.
The model was based on the simple idea that the more energy you hit the electrode with, the more energy it would give each electron. Light was treated as a wave of such energy.
In that model, it follows that greater wave amplitude (... | Beach ball A and beach ball B are in a depression on sand. Ropes are connected to the balls. A low frequency is sent through the rope to ball A. A high frequency is sent through the rope to ball B.
Ball A sways back and forth, then leaves the depression. The kinetic energy of ball A is random value between zero and the... | https://physics.stackexchange.com |
81,976 | [
"https://physics.stackexchange.com/questions/81976",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | Do two resistors in parallel dissipate more power for a fixed applied voltage compared to the same two resistors in series?
| The most straightforward way to reason about this doesn't require much math.
The power delivered by the voltage source to either pair of resistors is <em>inversely proportion</em> to their combined resistance, i.e, if the combined resistance is greater, the power delivered is smaller.
$$p_R = \dfrac{V^2}{R}$$
Now, r... | In general, if the power consumed would depend on the circuit structure. But for a simple case, such as two resistors connected in series versus the same resistors connected in parallel (with identical voltage sources in both), the power dissipated in the parallel combination will be greater.
| https://physics.stackexchange.com |
3,817,589 | [
"https://math.stackexchange.com/questions/3817589",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/686945/"
] | Evaluate the limit
<span class="math-container">$$\lim_{x\to\infty}\frac{x^{\log_2 x}}{(\log_2 x)^x}$$</span>
I tried to evaluate this limit by taking natural log and using L'Hospital rule. However, it seems that both methods make the expression even harder to evaluate. I would be thankful for any help.
| We have that
<span class="math-container">$$\frac{x^{\log_2 x}}{(\log_2 x)^x}=2^{(\log_2 x)^2-x\log_2(\log_2 x)}$$</span>
and
<span class="math-container">$$(\log_2 x)^2-x\log_2(\log_2 x)=x\left(\frac{(\log_2 x)^2}x-\log_2(\log_2 x)\right)\to -\infty$$</span>
| Introduce <span class="math-container">$u=\log_2x$</span> i.e. <span class="math-container">$x=2^u$</span>. Your limit becomes:
<span class="math-container">$$\lim_{u\to\infty}\frac{(2^u)^u}{u^{(2^u)}}=\lim_{u\to\infty}\frac{2^{(u^2)}}{u^{(2^u)}}=\lim_{u\to\infty}\frac{2^{(u^2)}}{u^{(u^2)}}\cdot\frac{1}{u^{(2^u-u^2)}}=... | https://math.stackexchange.com |
53,088 | [
"https://softwareengineering.stackexchange.com/questions/53088",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/-1/"
] | Is it advisable to create multiple classes within a .cs file or should each .cs file have an individual class?
For example:
<pre><code>public class Items
{
public class Animal
{
}
public class Person
{
}
public class Object
{
}
}
</code></pre>
Dodging the fact for a minute that ... | The example you gave is actually fine in my opinion. You are declaring <strong>inner classes</strong>, so it is perfectly sensible to keep them in the <strong>same file</strong>. The only way around this would be to make your <code>Items</code> class a partial class and split it over multiple files. I'd consider this ... | To be brutally honest, please don't add more than one root class to a file. At my last job, there were files with not only multiple classes, but multiple namespaces and this stretched into the multiple thousands of lines of code. <strong>Very</strong> difficult to try to follow.
If there are classes that are closely... | https://softwareengineering.stackexchange.com |
142,067 | [
"https://security.stackexchange.com/questions/142067",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/129913/"
] | I recently graduated and now work as a software developer but at some point in my career, I would like to transition into Penetration Testing.
I know that pen testers sometimes write their own tools but I am curious to know if they follow and use Design Patterns, SOLID principles and aim to write clean code, or is it ... | There's a big distinction between pen testers who write a bit of code, and developers of security tools.
During a pen test, you may need to write some custom code. For example, if you find server-side template injection, there's likely to be no off-the-shelf exploit that works in this exact scenario. But with a little... | For pentesting work, you will likely use plenty of various libraries and I think it's the main aspect of the job.
Knowing this, it's also a bit of work to organize these projects, have all libraries on your system and so on. One project / folder per each utility you create is pretty a lot of job.
For pentesting utili... | https://security.stackexchange.com |
2,684 | [
"https://physics.stackexchange.com/questions/2684",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/724/"
] | Assume a static metric with (known) components $g_{\mu\nu}$. I'd like to know what is the gravitational pull $g$ of a test particle placed on an arbitrary point $X$.
The gravitational pull being defined as the acceleration the particle suffers as measured by an observer sitting in a reference frame fixed at the origin... | This problem is somewhat ill-formed. It's difficult to even define what 'static at the origin' means--static relative to what? What I will do is show the steps one needs to take to derive the motion of a particle, however. The following will be more constructive than explanatory, as explaining all of this thoroughly... | Firstly, calculate the 4-velocity $\mathbf u$ of your observer, stationary or not. The 4-acceleration $\mathbf a$ is given by the covariant derivative $\nabla_{\mathbf u}\mathbf u$. To compute this, look up the formula for covariant derivative in a given coordinate system. The magnitude of acceleration felt by the obse... | https://physics.stackexchange.com |
1,766,613 | [
"https://math.stackexchange.com/questions/1766613",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/328763/"
] | <blockquote>
Let $V$ be a finite dimensional real vector space with inner product $\langle \, , \rangle$ and let $W$ be a subspace of $V$. The <em>orthogonal complement</em> of $W$ is defined as
$$
W^\perp= \left\{ v \in V \,:\, \langle v,w \rangle = 0 \text{ for all } w \in W \right\}.
$$
Prove the following: $\... | Let $\beta=\{w_1,w_2,\ldots,w_k\}$ and $\gamma=\{x_1,x_2,\ldots,x_m\}$ be the bases for $W$ and $W^\perp$, respectively. It suffices to show that
$$\beta\cup\gamma=\{w_1,w_2,\ldots,w_k,x_1,x_2,\ldots,x_m\}$$
is a basis for $V$.
Given $v\in V$, then it is well-known that $v=v_1+v_2$ for some $v_1\in W$ and $v_2\in W^\pe... | <strong>Hint:</strong>
Take a basis $w_1,\dots,w_r$ of $W$, and consider the linear forms on $V$ defined by $w_i^*:v\mapsto\langle w_i,v\rangle$.
These linear forms are linearly independent, hence the solutions of the system of equations $w_i^*(v)=0,\ i=1,\dots r$ has codimension $r$ by the <em>rank-nullity theorem<... | https://math.stackexchange.com |
272,166 | [
"https://mathoverflow.net/questions/272166",
"https://mathoverflow.net",
"https://mathoverflow.net/users/109569/"
] | Assume that $\Re(z)>1$ and $x\in \mathbb R$ with $ x > -3$.
Is it possible to compute $$\sum_{n=0}^{\infty} \sum_{k=0}^{3^n-1} \frac1 { \left(3\times3^n+kx\right)^z }\ ?$$
I believe that this sum is related in some way to the Riemann zeta function, but I can not prove it.
| This is just a <strong>formal computation</strong> to relate your function with Hurwitz zeta-function
$$
\zeta(s,x)=\sum_{k\ge0}\frac1{(x+k)^s}.
$$
I <strong>assume</strong> convergence and a correct domain of definition:
\begin{align}
\notag\sum_{n\ge0}\sum_{k=0}^{3^n-1}\frac1{\left(3^{n+1}+kx\right)^z}&=
\sum_{... | I don't think there is much hope for an exact expression; for small $x$ a series expansion gives closed-form results, the first three terms are
$$\sum_{n=0}^{\infty} \sum_{k=0}^{3^n-1} \frac1 { \left(3^{n+1}+kx\right)^z }=\frac{3^{2z-1} \left(4-3^{z}-3^{1-z}\right)}{\left(3^{z-1}-1\right) \left(3^z-1\right) \left(3^{z... | https://mathoverflow.net |
285,876 | [
"https://softwareengineering.stackexchange.com/questions/285876",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/55516/"
] | Are events which are intended to tell listeners to carry out explicit actions an anti-pattern?
Imagine a worker service which raises the event <code>WorkStarted</code> at its beginning and the event <code>WorkFinised</code> at its end.
These events convey information about what that worker service <strong>did</strong... | The way to decide what an event should represent is the good old Single Responsibility Principle.
If, for instance, you have a long-running tax computation process and an incremental GUI visualization that displays its progress, then the tax processor should raise "started", "progress" or "finish" events. A GUI class ... | It is none of your business what I intend to do when your API raises an event. Maybe I will show an icon, maybe I will order coffee for the lead developer, start a music video, whatever I do, it's my business not yours. The point of the event is to notify those that are interested that something has happened, after t... | https://softwareengineering.stackexchange.com |
311,116 | [
"https://dba.stackexchange.com/questions/311116",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/251302/"
] | I want to change the path where the <code>ibtmp1</code> is stored in mysql 8.0.25 on debian 11. How to do this? I see:
<pre><code>mysql> SELECT * FROM INFORMATION_SCHEMA.FILES WHERE TABLESPACE_NAME='innodb_temporary'\G
*************************** 1. row ***************************
FILE_ID: 4294967293
... | Rick James' answer made me aware that (whyever) the conf file was not read. So I copied <code>/etc/mysql/mysql.conf.d/mysqld.cnf</code> to <code>/etc/mysql/my.cnf</code>. That one <em>is</em> read fine.
I then had to define (in <code>my.cnf</code>):
<pre><code>[mysqld]
innodb_tmpdir=$yourNewTmpPath
innodb_temp_data_fil... | <pre><code>mysqld --help --verbose
</code></pre>
At about line 13, it will say where the config file is. It is probably not spelled the way you spelled it. Mine says
<pre><code>...
Default options are read from the following files in the given order:
/etc/my.cnf /etc/mysql/my.cnf ~/.my.cnf
...
</code></pre>
| https://dba.stackexchange.com |
109,063 | [
"https://math.stackexchange.com/questions/109063",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/24940/"
] | Probably dumb question, but I would like to ask it anyway.
I was to solve this equation:
$z^4 = -16$
At first glance, the way to solve it would be (as any other equation):
$z^2 = \sqrt{-1} * 4 \lor z^2 = -\sqrt{-1} * 4$
$z^2 = 4i \lor z^2 = -4i$
$z = 2\sqrt{i} \lor z = -2\sqrt{i} \lor z = 2i\sqrt{i} \lor z = -2i\... | Say we want the two square roots of $4i$. The angle between the positive real axis and the line on which $4i$ lies is $90^\circ$ or $-270^\circ$. For the square root, you want half that angle, since angles are added when complex numbers are multiplied. It's enough to find one of the two square roots, since after tha... | Short way: polar coordinates.
$$
z^4=-16=(-1)2^4=2^4e^{i\pi}=2^4e^{i\pi+2k\pi i}=2^4e^{i\pi(2k+1)}\;,\;\;k\in\mathbb Z\;\;
$$
from which you get
$$
z=2e^{i\frac\pi 4(2k+1)}\;,\;\;k\in\mathbb Z
$$
which assumes only $4$ different values as $k\in\mathbb Z$, so we can write
$$
z_k=2e^{i\frac\pi 4(2k+1)}\;,\;\;k=0,1,2,3
$$... | https://math.stackexchange.com |
362,485 | [
"https://softwareengineering.stackexchange.com/questions/362485",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/291110/"
] | I'm designing a programming language for a game, to be used as the main way of interacting with the game, as opposed to using other methods of control, such as the arrow keys or mouse. My plan is to have the user control game objects through objects in the language.
<pre><code>if unit1.health > 20 {
unit1.att... | Make it possible to create a new object as a clone of an existing object with new properties.
So for example:
<pre><code>function Base() {
z = 5
return {
x: 2,
y: 4,
print_z: function() {
print(z)
}
}
}
function Child() {
return {
...Base(), // start w... | There are several alternatives to class-based inheritance that have already been explored
<ul>
<li>Prototype Delegation: objects can have one or more <em>special</em> references to other objects. When an object does not understand a message, the message is forwarded to the objects referenced by those <em>special</em> ... | https://softwareengineering.stackexchange.com |
13,984 | [
"https://stats.stackexchange.com/questions/13984",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/5346/"
] | I would like to combine the forecasted and backcasted (viz. the predicted past values) of a time-series data set into one time-series by minimizing the Mean Squared Prediction Error.
Say I have time series from 2001-2010 with a gap for the year 2007. I have been able to forecast 2007 using the 2001-2007 data (red lin... | Assuming you have the Squared Prediction Errors for the forecast and backcast individually I would recommend this:
Let w be a vector of length 12, let m be the month that you are interested in.
<pre><code>w=rep(NA,12);
for(w in 1:12){
w[m]=SPE_Backcast[m]/(SPE_Backcast[m]+SPE_Forecast[m]);
}
</code></pre>
Now w is th... | I find your suggested approach, of taking the means of the fore- and back-casts, interesting.
One thing that might be worth pointing out is that in any system exhibiting chaotic structure the forecasts are likely to be more accurate over shorter periods. That isn't the case for all systems, for example a damped pendul... | https://stats.stackexchange.com |
1,839,536 | [
"https://math.stackexchange.com/questions/1839536",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/42339/"
] | Here's an intentionally weird question for ye all.
Start off with the expression $x$.
<strong>Rule 0.</strong> We're allowed get a new expression from an old expression by replacing a subexpression with an $\mathbb{R}$-affine combination of itself. For instance, from $x$ we can get the expression $$\frac{1}{2}x+\frac... | Do you know that union of connected sets with a point in common is connected? If so, then the sphere is the union of two disks along their boundary, and disks are connected as you may well know.
Another simple solution is parametrizing the sphere, which takes the closed square surjectively in the sphere. Since continu... | Note that any two points on the sphere lie on some great circle. Once you figure out the equation of the great circle, then choosing any one of the arcs gives you a path between those two points.
Another method would be showing that $S^2$ is the continuous image of a connected set in $\mathbb{R^2}$. But the first met... | https://math.stackexchange.com |
17,877 | [
"https://stats.stackexchange.com/questions/17877",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/7194/"
] | I have a problem, which I think should be very common, but I mostly have been taught stochastic at university and most of my statistics knowledge is self-taught. Hence I have no clue on how to look for this on google or other search engines.
The basic question is, if I am testing a hypothesis against the null-Hypothes... | p*, the threshold below which we would reject the null hypothesis, is normally written as $\alpha$, and is known as the false-positive rate. A "false-positive" error is the error of rejecting the null hypothesis when it is actually true, and is also called a "Type-I" error. However, the interpretation of $\alpha$ mus... | "By accident" is not well defined here. The p-value is the chance of getting results as extreme or more extreme than those you got in a sample if the null hypothesis is true in the population.
| https://stats.stackexchange.com |
303,119 | [
"https://mathoverflow.net/questions/303119",
"https://mathoverflow.net",
"https://mathoverflow.net/users/125775/"
] | Consider, as a motivating example, the multiset $\left(\mathbb{P}_n,2\setminus\mathbf{0}\right)$ consisting of the underlying set of polynomials of order $n$ and lower, all with multiplicity $2$—expect for the polynomial identically equal to $0$, $\mathbf{0}$. (Heuristically, this is intended to be what occurs when you... | It is hard to posit more than one additive identity by the usual proof:
$e+f=f$ if $e$ is an additive identity and $e+f=e$ if $f$ is an additive identity hence $e=f$ if both are.
If the underlying field is $F=\{0,1\}$ then scalar multiplication is no issue and the vector space $W=F^{n+1}$ is essentially the double o... | The way I see it, a "multiset" is essentially a set $\tilde X$ together with a surjective map $\tilde X \buildrel\varphi\over\to X$, where $X$ is the "underlying set" of the multiset, such that the map $\varphi$ has finite fibers (the cardinal of each fiber $\varphi^{-1}(x)$ being the "multiplicity" of the element $x$ ... | https://mathoverflow.net |
90,642 | [
"https://security.stackexchange.com/questions/90642",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/77764/"
] | Do techniques exists to know if a machine is a honeypot (or suspicious indicators)?
What techniques can be used for a software inside the honeypot or outside it?
What techniques uses malware to prevent infecting honeypot?
| Yes.
The honeypot is not just there to attract an attacker, but also to gain information about them. So, a honeypot wants to log all actions of the attacker.
An attacker may now cause a log overflow (a technique seen in the wild is writing and deleting some file very often) and change their bahaviour on the informati... | Ideally, no.
A honeypot is a word used to define the purpose of a machine, but has no bearing on the machine itself, how it's configured or any particular software running on it.
That said, someone creating a honeypot typically has the goal of luring targets into expending resources or breaking OpSec. To do this, the... | https://security.stackexchange.com |
85,976 | [
"https://softwareengineering.stackexchange.com/questions/85976",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/3080/"
] | Today, Firefox 5 was released. If all goes according to plan, Firefox 7 will be out by the end of the year. Firefox has adopted the Google Chrome development model wherein version numbers are largely unimportant and so just supporting "the latest (publicly available) one" is probably the best strategy.
But how do you ... | Let's reflect a bit here for a minute - the only browser in the last decade to cause major problems for existing sites when it changed version number was Internet Explorer.
I don't recall a single instance of a client calling me and telling me their site is now broken with the latest version of Chrome or Firefox. The... | I think the continuous browsers versioning will signify the end of the approach known as "supporting all of the major browsers".
From now on it will be "supporting the two latest releases of the major browsers". Testing everything back too the origin has become to much to handle. The Stack Exchange team has adopted mo... | https://softwareengineering.stackexchange.com |
321,313 | [
"https://physics.stackexchange.com/questions/321313",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/149880/"
] | A charge $q$ in an electric potential $\phi$ acquires potential energy $q\phi$.
Likewise, a charge density $\rho(x)$ aquires potential energy density $\rho(x)\phi(x)$.
A moving charge $q$ in an electromagnetic 4-potential $A$ ``acquires'' an additional momentum $qA$ so that the canonical momentum equals $qA$ plus t... | Let me try to summarize what we learned: You asked whether we can write the standard equation of energy conservation in EM
$$
\partial_\mu T^{\mu\nu}= F^{\nu\alpha}j_\alpha ,
$$
which includes the work done by the currents as a source term, in such a way that total energy conservation is manifest, $\partial_\mu (T^{\mu... | <blockquote>
Q2. Is there a physically meaningful notion of the energy-momentum tensor T(x) which a 4-current density j(x) ``acquires'' in a 4-potential A(x)?
</blockquote>
Since you have in mind a non-dynamical, external current, to me it would make more sense to speak of the e-m tensor acquired <strong>by the fiel... | https://physics.stackexchange.com |
12,416 | [
"https://mechanics.stackexchange.com/questions/12416",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/7030/"
] | I'm going to be doing some camping out of my Kia, and want to make sure I will not run the battery down if I power various gadgets over night via the power adapter ports.
Specifically, I need to run a CPAP machine for sleep apnea, and I have the DC converter for the device. It draws 6 A @ 12 VDC, or 72 Watts.
Just ne... | One way round your problem would be too have a second battery, together with diode pack and wiring to charge it independantly. If your intention is too camp out in remote areas, you really dont want to saddle the vehicles system with any loads. Alternatively a generator for your machine. A continuous 6amp load overnigh... | It really depends on the battery, not the car. The car will have a very small drain for the radio presets and fob receiver, but not much. A typical Lead-Acid battery has around a 100 minute reserve capacity. Reserve capacity being how long it takes a 25 amp discharge to lower the voltage to unusable levels. That's roug... | https://mechanics.stackexchange.com |
200,878 | [
"https://physics.stackexchange.com/questions/200878",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/89449/"
] | Before I ask my question, let me just say, I know very little about particle physics and general relativity, so I may ask a obvious question or a question that makes little or no sense.
Now, what would stop this object accelerating to the speed of light:
If an object was in a perfect vacuum (no matter at all, except ... | A simple answer is that there is no point at which you are close to the speed of light, you might be close to the speed of light relative to other objects however you can keep trying to accelerate, and your slow relative to other objects. Your both fast and slow.
From the point of view of an object looking at you, the... | Many things stop you.
Firstly, it turns out forces do not cause accelerations. Forces increase momentum, and for slow speeds when you double the momentum you almost double your speed, and it is so close to doubling that for hundreds of years we thought it was doubling.
But now that we've learned how to make things go... | https://physics.stackexchange.com |
389,453 | [
"https://stats.stackexchange.com/questions/389453",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/221094/"
] | Assume we are given a joint distribution <span class="math-container">$P(X,Y)$</span> where <span class="math-container">$P(0,0)=0.1$</span>, <span class="math-container">$P(0,1) = 0.4$</span>, <span class="math-container">$P(1,0)=0.3$</span>, and <span class="math-container">$P(1,1)=0.2$</span>. The goal is to compute... | You have gone about this correctly, but the final answers are typically written as a function of <span class="math-container">$x$</span>. It's helpful, I think, to remember that <span class="math-container">$P(X|Y=1)$</span> is just shorthand for <span class="math-container">$P(X=x|Y=1)$</span> where <span class="math... | You are correct. You are supposed to be getting "multiple" probabilities.
You said that your goal is to figure out <span class="math-container">$P(X|Y = 1)$</span>. Well, in order to achieve your goal, you need to know what the probability that <span class="math-container">$X = 0$</span> given that <span class="math-... | https://stats.stackexchange.com |
39,344 | [
"https://electronics.stackexchange.com/questions/39344",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/5542/"
] | Year is 2012 and I can only find %5-tol resistors in the local market. They can make transistors at molecular scale, they can manufacture 14.318182MHz crystals, they can place trillions of flip-flops inside a memory chip.
Then why don't they start manufacturing %0.01-tol resistors? Is resistor manufacturing a more dif... | One more point worth considering: Maybe there's a problem with the local market?
In my local market, I have no problems getting 1% resistors and sometimes there's a larger choice of 1% resistors compared to 5% resistors. It's not always the question of can it be made but will people buy it too. Maybe your merchants fo... | The basic answer to your question is that for 99.9999% of the applications of resistors, the improved tolerance would have no value. Circuits are generally designed to work just fine with 5% and 10% resistors.
In the specific example you show, it really isn't the absolute tolerance of the resistors that's important, i... | https://electronics.stackexchange.com |
3,754,249 | [
"https://math.stackexchange.com/questions/3754249",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/737961/"
] | Given <span class="math-container">$\cos(a) + \cos(b) = 1$</span>, prove that <span class="math-container">$1 - s^2 - t^2 - 3s^2t^2 = 0$</span>, where <span class="math-container">$s = \tan(a/2)$</span> and <span class="math-container">$t = \tan(b/2)$</span>.
I have tried using the identity <span class="math-containe... | <span class="math-container">$\frac{1-t^2}{1+t^2} +\frac{1-s^2}{1+s^2}=1$</span>
<span class="math-container">$1+s^2-t^2-s^2t^2+1+t^2-s^2-s^2t^2=1+s^2+t^2+s^2t^2$</span>
<span class="math-container">$\Rightarrow 1-s^2-t^2-3s^2t^2=0$</span>
| Hint:
<span class="math-container">$$\dfrac{1-s^2}{1+s^2}+\dfrac{1-t^2}{1+t^2}=1$$</span>
<span class="math-container">$$\iff\dfrac{1-s^2}{1+s^2}=1-\dfrac{1-t^2}{1+t^2}$$</span>
<span class="math-container">$$\iff(1-s^2)(1+t^2)=2t^2(1+s^2)$$</span>
| https://math.stackexchange.com |
110,954 | [
"https://security.stackexchange.com/questions/110954",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/95803/"
] | <blockquote>
Would adding two factor authentication to an application increase the
attack surface of this application?
</blockquote>
I think it would, since by adding complexity the likelihood of a vulnerability in my application is increased and so it opens another path for the attacker to exploit the system.
W... | To directly answer the question; it depends on the implementation and/or use-case.
It's not as simple as "more functionality equals greater risk". 2-Factor authentication only ensures that there are at least two channels used as intended and thereby prove that the authentication claim is <em>more than likely</em> cor... | Two factor authentication is a great way to help prevent unauthorized access especially if you use a product that's already been in development and had time to mature. Should you choose to roll your own 2-factor from scratch you expose yourself to many risks thus widening you attack surface. When looking for a provider... | https://security.stackexchange.com |
88,109 | [
"https://physics.stackexchange.com/questions/88109",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/34849/"
] | <blockquote>
<strong>Question:</strong>
A police boat is chasing a boat with criminals along a straight river
by moving against the stream. The speed of the river stream is 3 miles
per hour, the speed of the boat with criminals relative to the river
is 30 miles per hour, and the police boat is 4 miles per hou... | All correct. Your values for $ \dot{x}_P$ and $ \dot{x}_C$ are negative <strong>because you defined $ \dot{x}_R$ to be positive</strong> (both boats are moving against the river). No problem there, this is completely up to you, but to satisfy the requirement of the question you are required to state that motion of the ... | Note that the speed of the criminals relative to the river ($27\,\mathrm{mph}$) is larger than the speed of the stone relative to them ($11\,\mathrm{mph}$). Then, the resulting velocity would be expected to be in the negative direction
| https://physics.stackexchange.com |
2,599,235 | [
"https://math.stackexchange.com/questions/2599235",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/230803/"
] | <blockquote>
Let <span class="math-container">$f$</span> be a real-valued function on <span class="math-container">$\Bbb{R}$</span>. Consider the functions
<span class="math-container">$$w_j(x) = \sup \left\{ \left|f(u)-f(v)\right| : u,v \in \left[x-\frac{1}{j},x+\frac{1}{j}\right] \right\}$$</span>
where <... | First lets write down a definition of continuity. A function $f$ is continuous at $x$ iff for every $n \in \mathbb{N}$ there exists a $j \in \mathbb{N}$ such that $\sup \{ |f(u) - f(v)| : u,v \in [x- \frac{1}{j}, x + \frac{1}{j}] \} < \frac{1}{n}$. (This is worded a little differently to the usual $\varepsilon$-$\de... | Note that for a fixed $x$ $w_j$ is decreasing as a function of $j$ and non-negative and hence $A_{j, n} \subseteq A_{j+1, n}$. If $x\in C$ then the oscillation of $f$ at $x$ is $0$ or in other words $\lim_{j\to\infty} w_{j} (x) =0$ and therefore for every $n\in\mathbb {N} $ there is some $j\in\mathbb{N} $ such that $w_... | https://math.stackexchange.com |
139,616 | [
"https://physics.stackexchange.com/questions/139616",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/41092/"
] | Both $\hat{a}^\dagger\hat{a}$ and $\hat{a}\hat{a}^\dagger$ are Hermitian, how do we know which one represents the particle number?
| We require that the number operator have the following property:
$$\hat n |0\rangle = 0.$$
We know that
$$\hat a |0\rangle = 0$$
and we know that
$$\hat a |1\rangle = |0\rangle$$
and we know that
$$\hat a^{\dagger} |0\rangle = |1\rangle. $$
Thus, it follows that
$$\hat a \hat a^{\dagger} \ne \hat n$$
since
... | Since you define e.g. in the bosonic case
$c_j^\dagger: H_N^S \rightarrow H_{N+1}^S,\quad c_j^\dagger | \ldots n_j \ldots \rangle := \sqrt{n_j+1} |\ldots n_j+1 \ldots \rangle$
$c_j: H_N^S \rightarrow H_{N-1}^S,\quad c_j | \ldots n_j \ldots \rangle := \sqrt{n_j} |\ldots n_j-1 \ldots \rangle$
it makes more sence to ... | https://physics.stackexchange.com |
40,459 | [
"https://stats.stackexchange.com/questions/40459",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/1390/"
] | I conducted a computer-based assessment of different methods of fitting a particular type of model used in the palaeo sciences. I had a large-ish training set and so I randomly (stratified random sampling) set aside a test set. I fitted $m$ different methods to the training set samples and using the $m$ resulting model... | You are essentially doing some form of cross-validation here for each of your <em>m</em> methods and would then like to see which method performed better. The results between runs will definitely be dependent, since they are based on the same data and you have overlap between your train/test sets. The question is wheth... | May not really understand what you have done but
<blockquote>
for Run I am assuming that the RMSEP values for that run are correlated to some degree
</blockquote>
Yes, that reflects how challenging the test set was in that run
<blockquote>
but are uncorrelated between runs
</blockquote>
No, given the way you ... | https://stats.stackexchange.com |
184,579 | [
"https://dba.stackexchange.com/questions/184579",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/61777/"
] | Setting up SSISDB with the new AlwaysOn support in SQL 2016 Enterprise. This creates two jobs, one of which is the SSIS Failover Monitor Job. The secondary is not readable. The code for this job is:
<pre><code>DECLARE @role int
DECLARE @status tinyint
SET @role = (SELECT [role] FROM [sys].[dm_hadr_availability_replica... | Consider wrapping the statement in the <code>BEGIN END</code> block inside of <code>sp_executeSQL</code> so that it bypasses the syntax checking until runtime.
<pre><code>DECLARE @role int
SET @role = (SELECT [role] FROM [sys].[dm_hadr_availability_replica_states] hars INNER JOIN [sys].[availability_databases_cluster]... | Here is an improved version of the above answer which uses the @@SERVERNAME global variable to populate the @server_name parameter.
This meas that the script will work without modification on any SQL Instance
<pre><code>DECLARE @role INT
DECLARE @sqlCmd NVARCHAR(MAX)
SET @role = (SELECT [role] FROM [sys].[dm_hadr_ava... | https://dba.stackexchange.com |
975,098 | [
"https://math.stackexchange.com/questions/975098",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/144272/"
] | On page thirty of Matsumura, it says that $\Bbb{Z}$ has krull dimension 1 because every prime ideal is maximal. I understand this because for any prime p you have $0 \subset p$.
However, for artinian rings, it says that the dimesion is zero because every prime ideal is maximal. This is where I'm confused. As above, we... | Hint: Is $0$ a prime ideal in all rings? If $0$ is a prime ideal, and it is maximal, is there any other prime ideal?
| Every prime ideal $\mathfrak p$ in an Artinian ring $R$ is maximal. To see this, let $x \in R / \mathfrak p$, $x \ne 0$. Consider the descending chain $(x) \supset (x^2) \supset \cdots$ to show that $x$ is invertible.
On the other hand, $\mathbb Z$ is a principal ideal domain. Every <em>nonzero</em> prime ideal in a P... | https://math.stackexchange.com |
353,184 | [
"https://physics.stackexchange.com/questions/353184",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/84158/"
] | If you had two objects simplified as points. Then you calculated the combined gravitational field for both objects. This gravitational field would determine how a beam of light would pass between the objects.
But now how does each object react to this gravitational field. Does it feel the combined field. Or only the f... | I will answer the question in the domain of classical as opposed to quantum physics.
I will use electromagnetism first, in order to present the concepts, and then comment briefly on how they apply to gravity. Gravity is more subtle when we allow for General Relativity.
So here is how it goes for electromagnetism and ... | Remember that an object can't exert a force on itself, because of Newton's third law. I don't think this is directly related to General Relativity.
Let's say the two heavy objects are the sun and the earth, and the thing between them is the moon. (We recently had an eclipse, so this seems appropriate!) Then you're rig... | https://physics.stackexchange.com |
5,710 | [
"https://chemistry.stackexchange.com/questions/5710",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/1888/"
] | The reaction of sodium with water is quite infamous.
$\ce{2Na + 2H_2O \rightarrow 2NaOH + H_2}$
But exactly why does $\ce{Na}$ displace $\ce H$ in $\ce{H_2O}$?
| Alkali metals in general are extremely reactive metals. They are highly reducing in nature. Hence, they react with water forming their corresponding hydroxides evolving dihydrogen gas.
In case of sodium - as cited in your question - its reduction potential is $\pu{-2.7109 V}$ hence it easily reduces hydrogen.
Also, a... | Sodium is more "electropositive" than hydrogen as it is down on the periodic table. This makes it more reactive to say elements like fluorine or oxygen. In this same way it is reactive to ions like hydroxide. Since water can turn into hydroxide and hydrogen ion the sodium ion is able to displace the hydrogen ion. This ... | https://chemistry.stackexchange.com |
2,686 | [
"https://stats.stackexchange.com/questions/2686",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/104/"
] | Help me here, please. Perhaps before even giving me an answer you may need to help me ask the question. I have never learned about time series analysis and do not know if that is indeed what I need. I have never learned about time smoothed averages and do not know if that is indeed what I need. My statistics background... | Higher correlation within subject gets you more power when the test being done is a differencing, equivalent to a paired t-test. The standard deviation used in calculating effect size is multiplied by $\sqrt{1-\rho}$. The standard deviation for difference scores (for a one-sample test) is $SD\sqrt{2-2\rho}$. This al... | <ul>
<li>All else being equal, bigger correlations in a repeated measures design means more power</li>
<li>A larger correlation means that more of the variance in the dependent variable is explained by systematic effects of cases/individuals</li>
<li>Prior research or previous datasets are useful for estimating the cor... | https://stats.stackexchange.com |
19,892 | [
"https://cs.stackexchange.com/questions/19892",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/5291/"
] | Almost all type-theoretical treatments of references that I've studied introduce references as accompanied with at least three operations (sometimes including the fourth):
<ol>
<li>Construction (allocation): $ \text{ref } e $</li>
<li>Elimination (dereferencing): $ !e $</li>
<li>Updating: $ e_1 := e_2 $</li>
<li>(not ... | References are useful even if you don't update the object, since copying objects takes memory and time. When passing a reference, only a pointer is passed. But suppose that you were passing a string - it could take far longer to copy the entire string. This is why language like C++ support <em>constant references</em>,... | Another thought, it's very easy to implement parametric polymorphism if all objects are boxed. Indeed that's how data in Haskell, *ML, and most other languages are stored.
Sure the operations are not explicit, but
<pre><code>case Just 1 {- Allocation -} of
Just x {- dereferencing/fetching -} -> x + 1
</code></pr... | https://cs.stackexchange.com |
71,683 | [
"https://electronics.stackexchange.com/questions/71683",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/22844/"
] | A book about electrical machines states as below
<pre><code>what happens in a generator of this sort when the load is increased? When
the load supplied by the generator is increased, IL (and therefore IA) increases. As
the armature current increases, the lARA drop increases, so the terminal voltage of
the generator fa... | "Load" is a term for how hard the device has to work to power whatever is attached to it. "High load" means the output power is high, "low load" means the output power is low. You can increase the load by, say, connecting a second device to the output of the generator. Assuming the voltage is constant, the current will... | The 'load' is the power drawn from the generator. Once you get that definition, the rest should be clear.
| https://electronics.stackexchange.com |
107,871 | [
"https://electronics.stackexchange.com/questions/107871",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/39903/"
] | I'm working on repairing a swimming pool hot tub heater. Feed is 240v 1Ph A/C power and it uses 18 gauge wire throughout. Apparently something got wet and I can see signs of a thermal event within the an electrical connector joint. I'm replacing the connector and the terminals with all new ones. (And yes, I will che... | From Tyco Electronics Soft Shell Pin and Socket Connectors, page 173.
<blockquote>
"Notes:
1. Split pins recommended for use in housings having 6, 9, 12 and 15 circuits to <strong>reduce mating force</strong>."
</blockquote>
If you look at the TE document 110-213, page 3 under Mating force, you can see that the s... | It is called a <strong>bifurcated</strong> contact and the reliability of redundant contacts improves significantly, so if one fails at least another is available. But it gets more complicated with varying insertion and extraction forces, so the non-split "may be " better for secure retention forces. Then there is pl... | https://electronics.stackexchange.com |
3,354,523 | [
"https://math.stackexchange.com/questions/3354523",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/-1/"
] | I need to solve the integral
<span class="math-container">$\displaystyle \int _{-1}^{3} \left[\arctan \left(\dfrac{x}{x^2+1}\right) + \arctan \left(\dfrac{x^2 + 1}{x}\right)\right]dx$</span>.
I thought I could use the identity <span class="math-container">$\arctan(x) + \operatorname{arccot}(x) = \dfrac{\pi}{2}$</sp... | <span class="math-container">$\forall x\in\mathbb{R}^*,\,\arctan(x)+\arctan\left(\frac{1}{x}\right)=\mathrm{sgn}(x)\frac{\pi}{2}$</span>, split the integral into two integrals :
<span class="math-container">$$ \int _{-1}^{3} \left[\arctan \left(\dfrac{x}{x^2+1}\right) + \arctan \left(\dfrac{x^2 + 1}{x}\right)\right]dx=... | Remember that
<span class="math-container">$$\arctan x+\arctan\frac1x=\begin{cases}\cfrac\pi2,&x>0\\{}\\-\cfrac\pi2,&x<0\end{cases}$$</span>
so we can simply write your integral as
<span class="math-container">$$\int_{-1}^0-\frac\pi2\,dx+\int_0^3\frac\pi2\,dx=-\frac\pi2+\frac{3\pi}2=\pi$$</span>
| https://math.stackexchange.com |
58,950 | [
"https://softwareengineering.stackexchange.com/questions/58950",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/20368/"
] | I've been involved in several projects either as tester or developer. In many of projects there were following statuses for defects:
<ol>
<li>WON'T FIX</li>
<li>Cancelled</li>
</ol>
Do you use such statuses and how do you differ them? I ask, because most people can't explain the difference. My understanding is:
<st... | As others have noted, these status names aren't very clear. I would prefer more precise and detailed status names:
<ul>
<li><strong>Won't Fix</strong> (the cost of fixing this is not justified)</li>
<li><strong>Workaround Provided</strong> (and it is enough to make users happy)</li>
<li><strong>Not a Bug</strong> (but... | Taking your 2 descriptions:
<blockquote>
<strong>WON'T FIX</strong> - developer will not fix the
defect, due to it's not a defect;
<strong>Cancelled</strong> - defect should not be
fixed, because of lowest priority
</blockquote>
It is obvious that the intended difference is:
<blockquote>
<strong>WON'T F... | https://softwareengineering.stackexchange.com |
117,480 | [
"https://softwareengineering.stackexchange.com/questions/117480",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/37552/"
] | I am working on a library which had several headers that are meant to only be used by the library itself. I also have a few classes and functions in headers that I do not want the client to use. For instance, I have a header that contains debugging functions, but they should not be used outside of the library as it cou... | A common way to handle this is public and private headers.
Given that some things must appear in a header file, either for physical implementation reasons, or for performance/inlining reasons, you need to split somehow.
Have your public header include the private header; and add comment(s) nearby that this is an impl... | <blockquote>
Should this code be hidden from the client? If so, how?
</blockquote>
Off course. You do it by not providing the headers. That is, you just provide the shared libraries, and the headers declaring classes your client should use.
| https://softwareengineering.stackexchange.com |
2,355,905 | [
"https://math.stackexchange.com/questions/2355905",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/335742/"
] | <strong>Prove that every continuous function $f:[0,1] \rightarrow [0,1]$ has a fixed point.</strong>
What does the statement $f:[0,1] \rightarrow [0,1]$ mean? I have never studied function rigorously so I am not well-versed with the notation.
Can somebody help me out or provide some resources that may prove beneficia... | It means that the <strong>domain</strong> of the function is $[0,1]$ and that the <strong>codomain</strong> of the function is $[0,1]$. An example of such a function would be $x^2$, for example, or maybe $\sin(x)$ or an infinitely many other functions.
To solve your problem, here's a couple guidelines:
Take a look at... | The notation $f:X\to Y$, where $X$ and $Y$ are sets means that $f$ is a function that to each element in $X$ assigns a unique element in $Y$. $X$ and $Y$ could be intervals, like $[0,1]$, they could be thw whole number line $\Bbb R$, or any other set, like $\Bbb N$ or $\{a, b, c, d\}$.
$X$ and $Y$ may be the same set... | https://math.stackexchange.com |
241,864 | [
"https://electronics.stackexchange.com/questions/241864",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/22677/"
] | I was just wondering that since resistance is required for a potential difference to build up, what happens when we have a perfectly conducting wire, 0 resistance? Will current still flow through the medium? Or am I getting something wrong?
| Yep, basically you're thinking about this wrong.
Ohm's law basically says: "for a resistor in isolation, you can relate Voltage, Current and Resistance." in other words if you know any two of those quantities you can determine the third one.
There is nothing more to it. In this case, you know the <em>ideal</em> resi... | In a superconductor, there is no resistance and you can actually send current through it. The trick is here is that you can't use Ohm's law. Ohm's law apply at room temperature, quasi static and conservative system.
In a superconductor, the ohm's law doesn't hold and doesn't make any sense, infinite current doesn't m... | https://electronics.stackexchange.com |
38,862 | [
"https://softwareengineering.stackexchange.com/questions/38862",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/12041/"
] | I learned C++ in college (now we are moving to Java.) On the side I learned basic scheme, php and javascript. However my knowledge is basic enough that I wouldn't even know how to start building a website such as this.
The reason I ask is I want to further develop my web-development skills/knowledge by doing a pro... | <h2>My Advice:</h2>
Start off small, <strong>and</strong> from the very beginning:
<ul>
<li>First open Notepad or any text editor, get a HTML reference book or google it.</li>
<li>Learn to write good clean valid HTML. </li>
<li>Then learn CSS</li>
<li>Then learn Javascript.</li>
<li><strong>Stage 1</strong> Create a ... | I am also an student and from my own experience I've found that it is best way to proceed with a project is to put the general idea on paper.
The specific knowledge you need should come from what you are trying to do.
Worry about what you want to do first.
Then worry about the specifics later and learn as you go.
... | https://softwareengineering.stackexchange.com |
48,705 | [
"https://security.stackexchange.com/questions/48705",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/37190/"
] | Recently I noticed a lot of traffic generated by automated bots that were checking for vulnerabilities in my web server. They were mainly looking for ways to hack into plesk/wordpress which I don't even have running on the server.
Now my question is, what are hackers trying to accomplish by hacking the server. I mean,... | The ultimate goal? It heavily depends on the initiator of the attack, but here are some ideas:
Attacker interested in financial gains:
<ul>
<li>It may be worth huge amounts of effort to target high traffic webservers/domains that enjoy a good reputation (The domain name is associated with a trustful service) and then... | The traffic you are experiencing is merely the result of automatic scans by bots that all servers connected to the Internet will experience.
The target of such attacks are the lowest common denominator of servers out there -- badly configured with default passwords or unpatched with announced exploitable vulnerabiliti... | https://security.stackexchange.com |
678,675 | [
"https://math.stackexchange.com/questions/678675",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/129201/"
] | The problem is the following:
The symmetric coin is tossed 1600 times. What is the probability that the head will be shown up more than 1200 times?
<hr>
Attempt.
Using the formula $\mathbb{P}(|X-MX|)>e)≤ DX/e^2$ I put the numbers in it
$$\mathbb{P}(|X-800|>1200)\le 400/1200^2$$
But do not get the answer whi... | You did everything right, except that to find when it is greater than $1200$ you need $|X-800|>400$ not $|X-800|>1200$. Also note, that this will include cases $X>1200$ as well as $X<400$, and since the distribution is symmetric around $800$, all you need to do is divide $400/400^2$ by $2$.
| Let $X$ be the number of heads. Then $X$ has mean $\mu=800$ and variance $\sigma^2=(1600)(1/2)(1/2)=400$. The Chebyshev Inequality says that
$$\Pr(|X-\mu|\ge k\sigma)\le \frac{1}{k^2}.$$
In our case, we are interested in something related to the probability that $X\ge 1200$. Now $1200=\mu +k\sigma$, where $k=20$. So b... | https://math.stackexchange.com |
605,171 | [
"https://stats.stackexchange.com/questions/605171",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/357522/"
] | I have the following question. At first this seemed very silly but after thinking about it, I found my self struggling.
Given <span class="math-container">$X$</span>, a random variable, I should decide if the following sentences are right or wrong and explain why:
1 - <span class="math-container">$X$</span> and <span c... | <span class="math-container">$\DeclareMathOperator{\Var}{Var}$</span>
You can perform the similar calculation for variance by noting <span class="math-container">$\Var(Y) = E[Y^2] - (E[Y])^2$</span> for any random variable <span class="math-container">$Y$</span>:
<span class="math-container">\begin{align}
& E\left[... | <span class="math-container">$$\mathbb{E}[\hat{\theta}]=\mathbb{E}\left[\frac{1}{n\sigma}\sum_{i=1}^{n} \exp \left[-Y_i^2\left(\frac{1}{2\sigma^2}-\frac{1}{2}\right)\right]\phi(Y_i)\right]$$</span>
<span class="math-container">$$ = \frac{1}{n}\sum_{i=1}^{n} \mathbb{E}\left[\frac{1}{\sigma}\exp \left[-Y_i^2\left(\frac{1... | https://stats.stackexchange.com |
88,813 | [
"https://mathoverflow.net/questions/88813",
"https://mathoverflow.net",
"https://mathoverflow.net/users/7333/"
] | Hello Everybody,
Is there a proof of Sobolev embedding theorem without using the GNS or Morrey inequalities? If so, can you provide me with some references?
<strong>Background:</strong> I happened to attened a talk on Computational PDEs and Sobolev spaces. The speaker made a reference to a proof by S. L. Sobolev us... | There is an English translation of Sobolev's book containing his original proof:
Sobolev, S. L. Some applications of functional analysis in mathematical physics. Translated from the third Russian edition by Harold H. McFaden. With comments by V. P. Palamodov. Translations of Mathematical Monographs, 90. American Mathe... | I have no idea what Sobolev's proof using polynomials is, but I can suggest the following two proofs:
a) If all you want are the embedding theorms of a Sobolev space into an $L^p$ or $L^\infty$ space and do not need the sharp constant, then by far the easiest proof is using the original Gagliardo-Nirenberg inequality
... | https://mathoverflow.net |
16,824 | [
"https://dba.stackexchange.com/questions/16824",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/1945/"
] | I'm using BIDS to create a cube in SSAS 2008 r2.
I used the wizard to create a time dimension. Part of that configuration is selecting a date range.
After creation, I realized that I needed a wider date range than I originally specified.
I did find that I could open the dimension, go to properties, and redefine the... | Bug 11720698 with the title "SYS.WARNING_SETTINGS$ NOT CLEANED UP WHEN OBJECT IS DROPPED." may be causing this.
| Oracle inserts into <code>warning_settings$</code> when you compile an object that produces a compilation warning.
I traced an object creation with (<code>ALTER SESSION SET PLSQL_WARNINGS='ENABLE:ALL'</code>) & without (<code>ALTER SESSION SET PLSQL_WARNINGS='DISABLE:ALL'</code>) warnings enabled to show this.
Wi... | https://dba.stackexchange.com |
289,718 | [
"https://softwareengineering.stackexchange.com/questions/289718",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/91341/"
] | When I'm in charge, I typically model services separate from endpoints. For example: <code>Company.Project.Domain.dll</code> has all of the logic required to complete the services for that domain. In order to access it you either reference that library or, <strong>if I'm exposing the services through an endpoint</str... | Separate your concerns. Complicated business logic should <em>never</em> be intermingled with scheduling logic that is also complex.
Therefore, implement whatever smart solution you want, but wrap it into a well-named method and then schedule <em>that</em>.
| If your implementation of the heavy logic is an ordenary, thread-free method it is much easier to write testcode (unit/integration-Tests) for it and to debug it. Writing junit-tests for threads is a nightmare.
| https://softwareengineering.stackexchange.com |
501,986 | [
"https://stats.stackexchange.com/questions/501986",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/266347/"
] | I have independent 2 samples (both with <span class="math-container">$n = 200$</span>) and i want to show that the true mean of sample b is higher than the true mean of sample a i.e. <span class="math-container">$H_0 : \mu_B \le \mu_A$</span> and <span class="math-container">$H1: \mu_B \gt \mu_A$</span>.
I wanted to co... | Just to add to/expand on @abstrusiosity's answer:
<ol>
<li>As mentioned above, 0 falls between <code>-1062.14</code> and <code>Inf</code> - so if you were using the confidence interval to make null hypothesis testing decisions you <em>would</em> fail to reject the null in this case. (assuming the null is that the diffe... | The result from R says you don't reject the null.
The result gives you the confidence interval of the difference. It says <span class="math-container">$-1062.14 \lt \mu_B-\mu_A \lt \infty$</span>. That means <span class="math-container">$\mu_B$</span> could be less than <span class="math-container">$\mu_A$</span> and... | https://stats.stackexchange.com |
452,228 | [
"https://physics.stackexchange.com/questions/452228",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/218915/"
] | I know, that <span class="math-container">$$ \uparrow \equiv \left[ \begin{array} { l } { 1 } \\ { 0 } \end{array} \right] $$</span> and <span class="math-container">$$ \bigg| \frac { X - i Y } { \sqrt { 2 } } \bigg \rangle = \sqrt { \frac { 3 } { 8 \pi } } \frac { x - iy } { r } $$</span> but I don't know <span cla... | When there are kets from different spaces put together, it is assumed that there is a hidden tensor product. So
<span class="math-container">$|\psi\ \rangle\ |\uparrow\ \rangle\ $</span> actually means <span class="math-container">$|\psi\rangle \otimes |\uparrow\ \rangle$</span>.
In the same way, notation can put ev... | Well, your question is a little bit confusing, and being careful with notations is really important in quantum mechanics. However, I think I may help you. This resultant state
<span class="math-container">$$\bigg| \frac { X - i Y } { \sqrt { 2 } } \uparrow \bigg\rangle$$</span>
that you get, is probably the tensor pr... | https://physics.stackexchange.com |
3,254,950 | [
"https://math.stackexchange.com/questions/3254950",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/671365/"
] | Hello I am trying to prove the following problem. Let G be a group <span class="math-container">$$A\le G , B\le G, A\cdot B = \{\;ab\;|\;a\in A, b\in B\;\}$$</span> I have to prove that <span class="math-container">$$\vert {A}\cdot{B}|=\frac{\vert{A}\vert\cdot\vert{B}\vert}{A\cap B}$$</span> So I am trying this approac... | We need to prove that <span class="math-container">$$\sum_{cyc}(x^3+3x^2y+3x^2z+2xyz)+\sum_{cyc}3xyz\geq 4\sum_{cyc}(x^2y+x^2z+xyz)$$</span> or <span class="math-container">$$\sum_{cyc}(x^3-x^2y-x^2z+xyz)\geq0,$$</span> which is Schur.
Your way is wrong because you took too strong estimation, that got a wrong inequali... | If we write <span class="math-container">$x+y+z=1$</span> (we can do that) then we have to prove <span class="math-container">$$1+9xyz\geq 4(xy+yz+zx)$$</span> Clearly one of <span class="math-container">$x,y,z$</span> is smaller than <span class="math-container">$4/9$</span>, we can assume <span class="math-container"... | https://math.stackexchange.com |
66,381 | [
"https://math.stackexchange.com/questions/66381",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/11533/"
] | I have been trying to come up with a proof, but I don't think I am making leaps in logic and assuming things that I'm possibly trying to prove, if anyone could help correct/(or completely discard) what I have done so far and help me towards a rigerous proof I would greatly appreciate it. This is what I have come up wit... | The intersection of any number of closed subsets if <strong>always</strong> closed.
Let $X$ be a topological space. Suppose $C_i$ are closed sets ($i\in I$ any index set). Set $U_i=\{x\in X\mid x\notin C_i\}$ the complement of $C_i$ is an open set.
The union of any number of open sets is open, therefore $U=\bigcup_{... | Suppose the definition of a closed set is being taken as "a set which contains all its accumulation points". Then examine an accumulation point of the intersection. All of the points in the intersection are contained in each one of the original collection of closed sets. Looking at the closed sets one by one, because t... | https://math.stackexchange.com |
89,642 | [
"https://security.stackexchange.com/questions/89642",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/76855/"
] | If my website is targeted for a DDoS attack after I have been paid for completing the website, and I get an angry phone call from the client regarding outage of service, what do I do?
It hasn't actually happened yet, but the idea haunts me.
| The following is all hypothetical:
First off you should NEVER sign a SLA in this case, or guarantee any uptime whatsoever. (you are delivering a website, not the service to host that)
Secondly, a hosting company should be used who can defend against a DoS attack in some way. (be aware of SLA's and their limitations)
... | As Lawri points out, for the most part the site being DDoSed is <em>not your problem</em>. It's up to the hosting provider to take the steps necessary to mitigate (not stop; there aren't really any ways to completely <em>stop</em> one) a DDoS attack.
Note the qualifier: "<em>for the most part</em>".
There <em>is</em... | https://security.stackexchange.com |
41,967 | [
"https://security.stackexchange.com/questions/41967",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/29394/"
] | As a home internet user, why does my ISP block my communication with any other client on the same subnet as mine? It is done for security obviously, but what is the risk?
Update: as clarified by Tylerl, the issue is related to non broadcast nature of the end user connection to POP. Users are only able to communicate w... | As a rule, no; ISPs don't block communication between peers within their network. Though also as a rule, home Internet users don't allow any inbound connections, so the lack of connectivity is impossible to test without the cooperation of someone else on the same network.
Also, communication between subscriber peers i... | This is the first time I have ever heard of this. It makes no sense. Why do you think it could have a security reason? I know I have never had any block like this with any of my ISPs.
I would suggest it is the configurations on your system and the person you are trying to connect to that should be checked.
Dont expec... | https://security.stackexchange.com |
37,241 | [
"https://math.stackexchange.com/questions/37241",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/7571/"
] | I haven't done too many proofs, but I'd like to attempt more and thought I'd take a shot at this one, which is from Penrose's book "The Road To Reality."
The problem is to prove that the function given by $y = 0, x \leq 0$ and $y = e^{-1/x}, x > 0$ is $C^\infty$ smooth.
When x is less than 0, there's no problem a... | Good start! Here's a little hint: you don't have to compute the $n$th derivative of $e^{-1/x}$ explicitly (that would be a mess). It's enough to show what structure it has, namely "a polynomial in $1/x$, times $e^{-1/x}$". This will allow you to deduce that it tends to zero as $x\to 0^+$.
| The derivatives are on the positive side do not seem impossible to calculate. I think they are
$$f^{(n)}(x) = e^{-1/x} \sum_{i=n+1}^{2n} {i \choose 2n-i}{i-1 \choose 2n-i}(2n-i)! (-x)^{-i}$$
as can be shown by induction. They have the limit $0$ as $x \to 0$ from above (easiest to see by first letting $y=1/x$ and ... | https://math.stackexchange.com |
561,221 | [
"https://physics.stackexchange.com/questions/561221",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | <blockquote>
The Lagrangian for the motion of a particle with mass <span class="math-container">$m$</span> and charge <span class="math-container">$q$</span> in a constant magnetic field <span class="math-container">$B$</span> is given by <span class="math-container">$$\mathcal{L}(x,v)=\frac{m}{2}\left|v\right|^2-\frac... | you have to show that <span class="math-container">$\mathcal{L}'=\mathcal{L}$</span>
with:
<span class="math-container">$$\mathcal{L}=\frac{m}{2}\vec{v}^T\,\vec{v}-\frac{q}{2c}\vec{v}^T\,\vec{\omega}$$</span>
where <span class="math-container">$\vec{\omega}=\vec{x}\times\vec{B}$</span>
<span class="math-container">$$\m... | The conservation of the second term seems intuitive, as simply a preservation of volume, formed by vectors <span class="math-container">$v, x, B$</span>. Nevertheless, I also present a way to deduce this for infinitesimal transformation, which will imply for a finite rotation. Under a small rotation with <span class="m... | https://physics.stackexchange.com |
99,803 | [
"https://stats.stackexchange.com/questions/99803",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/46011/"
] | I have a simulated distribution with mean 12.53% and standard deviation 11.83%. The sample size is big enough (10,000) to assume it is a Normal distribution.
How do I properly test if the value "26.05%" is significantly larger than the mean 12.53%?
Can anyone please help me to write the null hypothesis, as well as t... | I think you're making life hard for yourself there.
You just need to use a few elementary properties of variances and covariances.
Here's one approach:
<ul>
<li>start with the algebraic definition of your random walk process.</li>
<li>derive $\text{Var}(y_t)$ in terms of $\text{Var}(y_{t-1})$ and the variance of the... | For each integer $t$, $Y_t = \sum_{i=1}^t X_i$ where the $X_i$ are iid random
variables. From the independence of the $X_i$, it follows that $\operatorname{var}(Y_t) = \sum_{i=1}^t \operatorname{var}(X_i) = t\sigma^2$. For integer $h$, let
$W = \sum_{i=t+1}^{t+h} X_i$ and note that $W$ and $Y_t$ are independent
rando... | https://stats.stackexchange.com |
46,867 | [
"https://stats.stackexchange.com/questions/46867",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/8336/"
] | Having a little trouble with this one:
Suppose $X_1, X_2, \ldots $ are iid standard normal random variables. Let $W_n = \sqrt{n} \frac{X_1 + \cdots + X_n}{X_1^2 + \cdots + X_n^2}$. Find the limiting distribution of $W_n$ as $n \to \infty$.
Too bad convergence in distribution isn't closed under division. Can't get slu... | Notice that $\frac{X_1 + X_2 + \ldots+ X_n}{\sqrt n}$ is a a standard gaussian variable. Then apply Slutsky's theorem.
| The numerator is normally distributed with mean $0$ and variance $n$, and the denominator is distributed $\chi^2$ with $n$ degrees of freedom. Start there.
| https://stats.stackexchange.com |
269,506 | [
"https://softwareengineering.stackexchange.com/questions/269506",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/111590/"
] | As a mid-level developer on my team, I participate in requirements gathering/scope planning meetings of projects that I will be part of. I have been finding it difficult to come up with the questions that add value to the discussion or to my knowledge. After some self analysis, I found that knowing that there is a seni... | Answering your questions in order:
<ol>
<li>The mindset going into this in terms of preparation, is to ask the right questions so that the user knows <strong>exactly what they want</strong>. This is much more difficult than it seems. I need to emphasize <strong>asking the right questions</strong>. Be specific, if ther... | Users most times know exactly their daily work and which problems they have to solve. What they typically don't know is
a) how the software exactly has to look like to support that process.
b) how they can explain their process to you.
So focus on questions to understand what the user <em>really</em> want (their go... | https://softwareengineering.stackexchange.com |
28,009 | [
"https://biology.stackexchange.com/questions/28009",
"https://biology.stackexchange.com",
"https://biology.stackexchange.com/users/4819/"
] | Does the heart no longer function? Does it go into ventricular fibrillation?
What happens?
| Damage to the brainstem - the part of the brain responsible for controlling breathing and the heart - has different effects on the heart and the lungs.
Every heartbeat is not directed by the brain. Instead, the brain has a "throttle" to the heart, by which it can regulate the heart rate up and down, but the actual hea... | The heart beats are initiated by the pacemaker cells which are located in the heart itself. So absence of neural input will not affect heart function by much. Though heart can function without neural input, it will not be able to respond to stimuli. The human heart rate is regulated beat by beat through the vagus nerve... | https://biology.stackexchange.com |
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