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
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|---|---|---|---|---|---|
417,827 | [
"https://physics.stackexchange.com/questions/417827",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/201563/"
] | <strong>Baseline facts:</strong>
<pre><code>- speed of spin of planet Earth: <b>1670 km/h</b> (counterclockwise & tilted 23,4°)
- speed of Earth's rotation around Sun: <b>108000 km/h</b> (counterclockwise)
- speed of our Solar System: <b>700000 km/h</b> (towards star Vega / constal. Lyra)
- spee... | First you have to find a frame of reference. For the sake of simplicity, let us assume that there exist two star systems, and all the velocities you have mentioned in your question are identical to both the systems <strong>with respect to</strong> the assumed frame of reference. So, now we have a frame of reference, an... | <blockquote>
speed of Earth's rotation around Sun<br>
would case to zero<br>
and we would survive
</blockquote>
How so? If Earth gets velocity of zero, instead of spinning around the Sun it will be attracted to it and get melted :) Changing the speeds arbitrarily will change relative positions of planets and oth... | https://physics.stackexchange.com |
443,904 | [
"https://softwareengineering.stackexchange.com/questions/443904",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/412921/"
] | I am thinking how to develop an application in a DDD way, and now I am thinking about the concurrency part.
In some examples I have seen that in the domain classes are injected with dependency injection an implementation of the repository, and also in the root aggregate is added a field version, to control the concurre... | There is nothing inherently wrong in adding purely technical attributes to entities (or other domain objects) which are exclusive required for implementing things like persistence or concurrent access. Depending on
<ul>
<li>your organizational environment</li>
<li>your tooling</li>
<li>the way you use to talk with your... | One possible way to keep your domain model "pure" from persistence-related implementation details is not to persist your domain classes directly. Instead, you define a persistence model, and map between both models accordingly. Then, instead of defining repositories for your domain classes, you would define t... | https://softwareengineering.stackexchange.com |
32,325 | [
"https://cs.stackexchange.com/questions/32325",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/1342/"
] | In mathematics, there are many existence proofs that are non-constructive, so we know that a certain object exists although we don't know how to find it.
I am looking for similar results in computer science. In particular: is there a problem that we can prove it is decidable without showing an algorithm for it? I.e. w... | The simplest case I know of an algorithm that exists, though it is not
known which algorithm, concerns finite state automata.
The quotient $L_1/L_2$ of a language $L_1$ by a language $L_2$ is
defined as $L_1/L_2=\{x \mid \exists y\in L_2 \text{ such that } xy\in L_1\}$.
It is easily proved that regular set are closed... | To expand on Hendrick's original comment, consider this problem
<blockquote>
Given an integer $n\ge 0$ is there a run of $n$ or more consecutive 7s in the decimal expansion of $\pi$?
</blockquote>
This problem is decidable, since one of two cases may obtain:
<ol>
<li>There is an integer $N$ for which the decimal e... | https://cs.stackexchange.com |
12,734 | [
"https://ai.stackexchange.com/questions/12734",
"https://ai.stackexchange.com",
"https://ai.stackexchange.com/users/23993/"
] | The question is little bit broad, but I could not find any concrete explanation anywhere, hence decided to ask the experts here.
I have trained a classifier model for binary classification task. Now I am trying to fine tune the model. With different sets of hyperparameters I am getting different sets of accuracy on my... | You should only look for the cross-validation score. If this set is large enough, it will give you an accurate prediction of how your model will act for unseen data.
Your case is exceptional. The fitted model which is obviously overfitted actually performs better on the cross-validation set. This means in turn that y... | Assuming that your cross-validation scores(both on train set and test set) indicate model's prediction performance correctly, you should definitely decide which trained model to use based on your <strong>validation accuracy only,</strong> regardless your model is overfitted or not.
| https://ai.stackexchange.com |
210,758 | [
"https://mathoverflow.net/questions/210758",
"https://mathoverflow.net",
"https://mathoverflow.net/users/45564/"
] | Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that can give bounds or estimates for the probability that the kernel of $M$ contains at least one short non-zero vector $v\i... | For $m$ around $n/log n$ the probability goes to $0$, with $(0,1)$ or $(-1,1)$, it doesn't matter.
Combinatorial Lemma: Any set of subsets of $\{1,\dots, n\}$ which contains no pair of subsets $A, B$ with $A \subset B$ has size at most $\begin{pmatrix} n \\\lfloor n/2 \rfloor \end{pmatrix}$ elements.
Proof: Consider ... | Well, the probability that the kernel of $M$ is nontrivial is bounded above by around $(3/4)^n,$ by Tao-Vu (J. Amer. Math. Soc. 20 (2007), 603-628), so there is that.
| https://mathoverflow.net |
155,894 | [
"https://security.stackexchange.com/questions/155894",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/144174/"
] | I enabled the "Guest Wifi Network" from the router configuration menu.
Now I have two different wifi networks on my router (with different IP address).
The first one is for "trusted" devices - my professional computer and my smartphone. Only thoses devices are allowed (MAC address filter).
The other one is for "untr... | In principle a DDOS attack can send completely legitimate requests and thus overload the server. For example, just use a botnet to visit some persons blog and the server will potentially collapse under the load. There is no real way to protect against this because every request looks like a real request.
However, the ... | A answer has been accepted here - and it does make some valid points, but did not mention reflected amplification attacks; with tcp only small amounts of data are exchanged (and must be exchanged between both the attacker and victim) before processing, memory and io resources can be consumed. A slowloris type attack as... | https://security.stackexchange.com |
48,683 | [
"https://mathoverflow.net/questions/48683",
"https://mathoverflow.net",
"https://mathoverflow.net/users/2625/"
] | The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra
$$
k[X_1,...,X_n]/I_\Delta
$$
where $I_\Delta$ is the ideal generated by the $X_{i_1}...X_{i_r}$ with ${i_1,...,i_r}\notin \Delta$.
Somebody told me that this construction helps to study varieties $k[... | First when it comes to comparison with the simplicial complex it should be
realised that the Stanley-Reisner ring corresponds to the cone over the complex.
There is a non-homogeneous version of it where one replaces the linear subspace
through the origin by the affine space parallel to the linear space but passing
thro... | I think you're asking if there's a direct geometric relationship between an algebraic variety $X=V(I)$ (i.e., the zero set of an ideal $I$) and the Stanley-Reisner complex $\Delta_I$ or $\Delta_{in(I)}$ --- in other words, does the variety look like the simplicial complex? In general, I think the answer can be quite s... | https://mathoverflow.net |
4,085,170 | [
"https://math.stackexchange.com/questions/4085170",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/424260/"
] | Show that if natural <span class="math-container">$m$</span> is even, then
<span class="math-container">$2^{m} -1$</span> is not prime.
As <span class="math-container">$m$</span> is even, so <span class="math-container">$m= 2n, n \in \mathbb{N}$</span>. Hence,
<span class="math-container">$2^{2n} -1 \implies (2^n)^2 -... | First of all, the claim is false; if <span class="math-container">$ m $</span> is <span class="math-container">$ 2 $</span>, then <span class="math-container">$ 2 ^ m - 1 $</span> <em>is</em> prime. However, for all even <span class="math-container">$ m > 2 $</span>, the claim is correct; in fact, in that case, <sp... | If you take <span class="math-container">$2^{2n}-1$</span> mod <span class="math-container">$3$</span>:
<span class="math-container">$$\begin{align}
2^{2n}-1&=4^{n}-1\\
&\equiv1^{n}-1\\
&\equiv0
\end{align}$$</span>
So <span class="math-container">$3$</span> divides <span class="math-container">$2^{2n}-1$</... | https://math.stackexchange.com |
423,233 | [
"https://physics.stackexchange.com/questions/423233",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/204233/"
] | Today I was presented with the following scenario: A syringe with one end sealed and with a frictionless piston is dipped into cold water. What is the change in frequency of collision(ie, more or less)?
In my opinion, it should be lesser. When temperature decreases, average velocity of the gas particles decreases a... | Here is a simple answer: the collision rate can be said to be proportional to the ratio between velocity and the total surface area of the piston.
$PV=nRT$. Hence, $V\propto T$. The total volume is $\pi r^2h$, so we can say $V\propto h$, where $h$ is the height of the column of gas.
Next, $KE=\frac{3}{2}kT$, wher... | We need to know what pressure is. Pressure in this case is the force applied by the gases particule per surface unit. It is in fact a mean value like many macroscopic value in thermodynamic
<blockquote>
All the computations are done without the numerical constants
</blockquote>
To compute it let's say that the var... | https://physics.stackexchange.com |
418,069 | [
"https://mathoverflow.net/questions/418069",
"https://mathoverflow.net",
"https://mathoverflow.net/users/241460/"
] | <blockquote>
Let <span class="math-container">$ 0<a<b $</span>, <span class="math-container">$ f\in C^1\left([0,b]\right)$</span>. Assume that <span class="math-container">$ f $</span> is concave on <span class="math-container">$ [0,a] $</span> and convex on <span class="math-container">$ [a,b] $</span> with <spa... | Denote <span class="math-container">$b/n=s$</span>. We want a pointwise bound <span class="math-container">$$f(x)\leqslant f(s)+(x-s)f'(s),\label{1}\tag{$\heartsuit$}$$</span> then summing \eqref{1} up for <span class="math-container">$x=a_1,\ldots,a_n$</span> we get the desired inequality. Note that if <span class="ma... | Without loss of generality (wlog),
<span class="math-container">\begin{equation*}
0\le a_1\le\cdots\le a_p\le a\le a_{p+1}\le\cdots\le a_n\le b \tag{1}\label{1}
\end{equation*}</span>
for some <span class="math-container">$p\in\{0,\dots,n\}$</span>. Also, by approximation, wlog <span class="math-container">$f$</spa... | https://mathoverflow.net |
57,586 | [
"https://mathoverflow.net/questions/57586",
"https://mathoverflow.net",
"https://mathoverflow.net/users/11765/"
] | So let $X$ be a finite CW complex which is connected.
Q1: Is $\pi_1(X)$ necssarily a finitely presented group?
If the answer is yes, then how does prove it. I've tried to prove it using
an induction argument but I'm stuck... So every time one glues a cell then one needs
to show that this only throws in finitely many... | Let $X$ be a CW-complex, and write $X_k$ for the $k$-skeleton. The cellular approximation theorem says that any based map $S^1\to X$ is homotopic to a cellular map, and that any two cellular maps that are homotopic are homotopic via a cellular homotopy. This means that the map $\pi_1(X_1)\to\pi_1(X_2)$ is surjective,... | What Richard Kent said, although it probably deserves to be expanded on.
Suppose for simplicity that $X'$ is obtained from a CW complex $X$ by attaching an $n$-disk $D^n = \{x: |x| \leq 1\}$ along a map $S^{n-1} \to X$, where $n > 2$. To see that the inclusion $X \to X'$ induces an isomorphism on $\pi_1$, use the ... | https://mathoverflow.net |
403,441 | [
"https://softwareengineering.stackexchange.com/questions/403441",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/354435/"
] | I am trying to create a season schedule for our Knowledge Bowl league (middle school trivia). I am NOT very mathematical nor a "techie". Here are my constraints:
<ol>
<li>There are seven schools participating; each school has three teams. Example: Tenino Black, Tenino Red, and Tenino White This means there will b... | You're not missing anything.
To get the most up to date state, you need to query it (and even that will be delayed by the latency of your request). Caching it, or waiting for some event/message means you're necessarily working with potentially stale state.
Technically, you get to decide what trade-off to make. Quer... | <ol>
<li>If your database supports rolling back changes, you could check at the beginning that the user is permitted to make the request, allow the request to go through, and then check again that they are still allowed to make the request. If they are no longer allowed, roll back the changes, otherwise complete the r... | https://softwareengineering.stackexchange.com |
581,131 | [
"https://electronics.stackexchange.com/questions/581131",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/199042/"
] | Many have a hate relationship with ST's HAL. I have a like/hate one and was wondering if my approach will bite me later on.
I like the peripheral and clock setup of current CubeMx. The code generated is so much more readable than my own bare metal stuff and it is complete all the time. And I believe it will help me wit... | You usually get the bad with the good. In a recent project I had to read pins statuses even though they were set as outputs. This is where the bare metal stuff comes in handy.
Another aspect is speed and handling interrupts. Handling multiple external interrupts that may be spaced very closely apart is best done with b... | Yes you can mix HAL code with whatever custom code you like. The initialization of clocks and peripheral by using the mouse is quite nice and easy so in practice you have a prototype working within minutes instead of spending hours of reading the reference manual. And it still provides an example how to use the periphe... | https://electronics.stackexchange.com |
1,183,943 | [
"https://math.stackexchange.com/questions/1183943",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/84097/"
] | What would be the easiest way to solve the following set of equations:$$
x + y^2 = 7 $$$$ x^2 + y = 11$$
I've been trying substitution method but end up in a $4$th degree bi-quadratic equation. Thanks for your time!
EDIT: Final equation: $y^4 - 14y^2 + y + 38 = 0$. I'm only concerned about the real solutions. Is it ... | It is obvious that in calculating n!, you are multiplying at least n/2 numbers which are greater than n/2.
$(n!)^{1/n} > ((n/2)^{n/2})^{1/n} = (n/2)^{1/2}$ which is clearly divergent.
| According to Stirling's approximation
$$n!\sim\sqrt{2n\pi}(\frac ne)^n$$
so $$\sqrt[n]{n!}\sim(2n\pi)^{\frac{1}{2n}}(\frac ne)$$
since $$\lim_{n\to \infty}(2n\pi)^{\frac{1}{2n}}=1$$
$viz.$ $$\lim_{n\to\infty}(2n\pi)^{\frac2n}(\frac ne)=\infty$$
| https://math.stackexchange.com |
317,944 | [
"https://softwareengineering.stackexchange.com/questions/317944",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/30714/"
] | I would like to know if it is possible to include source code licensed under:
<ul>
<li>Mozilla Public License (MPL) version 2.0 </li>
</ul>
In a project licensed with:
<ul>
<li>The MIT License (MIT)</li>
</ul>
If yes I would like to ask:
<ul>
<li>Can I modify the code from MPL, but keeping their credits?</li>
</ul... | Yes, you can combine code that is licensed under MPL 2.0 into a project that contains source code from one or more other licenses, including the MIT License. However, the individual source files licensed under the MPL will need to remain licensed under the MPL, including any modifications that you make to the MPL-licen... | You can mix MPL and MIT licensed code in the same project.
What you can't do (without the permission of it's copyright holders) is remove the MPL license from the code. This means that if you or anyone else wants to distribute the combined source tree or binaries built from both the MIT and MPL parts of it it they mu... | https://softwareengineering.stackexchange.com |
195,825 | [
"https://softwareengineering.stackexchange.com/questions/195825",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/85632/"
] | I am reading the book <code>programming in Lua</code>. It said that
<blockquote>
Closures provide a valuable tool in many contexts. As we have seen, they are
useful as arguments to higher-order functions such as sort. Closures are valuable for functions that build other functions too, like our newCounter example;... | Guy 1 to Guy 2: hey dude I wanna do something when a user clicks in there, call me back when that happens alright?
Guy 2 calls back Guy 1 when a user clicks here.
| No, it will never <em>return</em> the action. The button will <em>execute</em> it when the user clicks on it. And that is the answer. You need callback functions when you want to define actions that should happen in another component in reaction to an event you don't control (the event loop, which is part of the system... | https://softwareengineering.stackexchange.com |
3,895,530 | [
"https://math.stackexchange.com/questions/3895530",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/810667/"
] | Determine all prime numbers p such that p^2+2 is also a prime number.
I have only found 3, and I think that's the only number but I have trouble writing proof for this. But I know that if p is a prime number other than 3 then p^2+2 will be divisible with 3.
| Suppose <span class="math-container">$p$</span> is prime and different from <span class="math-container">$3$</span>. Write <span class="math-container">$p^2+2$</span> as <span class="math-container">$$p^2+2=(p^2-1)+3=(p-1)(p+1)+3.$$</span> Since <span class="math-container">$3\nmid p$</span>, <span class="math-containe... | Primes larger than <span class="math-container">$3$</span> cannot be divisible by either <span class="math-container">$2$</span> or <span class="math-container">$3$</span>. Numbers with this property, which include all primes other than <span class="math-container">$2$</span> or <span class="math-container">$3$</span>,... | https://math.stackexchange.com |
323,495 | [
"https://dba.stackexchange.com/questions/323495",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/268993/"
] | I have 3 MySQL tables: products, orders and users.
I want to show all ordered products where not all products are "pre-orders" or "discounts". Note: one user can have multiple orders.
I use a subquery to count products with "discount" or "preorder" in the title and compare that w... | Use a subquery to get all <code>order_id</code>'s of orders with at least an eligible product, then join it with the rest.
<pre><code>SELECT *
FROM products p1 JOIN orders o1 ON o1.barcode = p1.barcode
JOIN users ON users.id = o1.user_id
JOIN (SELECT order_id FROM orders
JOIN products ON orders.barcode=produ... | First, I'll give you a hint of an answer:
Since the <code>LIKE</code> will be the slowest part, let's start with "count items with "discount" or "preorder" in the title and compare that with a count of all ordered items. If the result is not the same, I show all items."
That should be equi... | https://dba.stackexchange.com |
371,570 | [
"https://softwareengineering.stackexchange.com/questions/371570",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/234196/"
] | Currently, we've been working on the architecture team on defining the database models. What has been troubling me is my superior's advice when it comes to working with <em>audit fields</em>.
He advocates for the <code>updated_at</code> and <code>updated_by</code> fields to be Nullable, reasoning that these should be ... | <blockquote>
What has been troubling me is my superior's advice ...
</blockquote>
Have you asked your superior these questions with an attitude of being teachable? Honestly this is going to be your best approach.
<blockquote>
Which approach is better?
</blockquote>
<strong>Neither.</strong> They represent the ... | Either approach is workable and there's nothing inherently wrong with either. I would tend to prefer the approach you are advocating mainly because nulls are a pain to work with in SQL due to three-value logic. In general, it's better to avoid them. As you say, updated = created tells you the same thing as having th... | https://softwareengineering.stackexchange.com |
704,435 | [
"https://physics.stackexchange.com/questions/704435",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/299419/"
] | I am studying Electromagnetism from Griffiths and in the book it is stated that diagonal elements of Maxwell's tensor represent pressure. I want to calculate pressure on the wirings of an infinitely long solenoid with constant current.
I found <span class="math-container">$$T_{xx} = T_{yy} = -\frac{\mu_0n^2I^2}{2} $$<... | <blockquote>
My question is how do we get the answer from those three components of stress tensor ?
</blockquote>
What you are asking is not clear. Are you asking how to get the off-diagonal components? Or are you asking at how you arrived at the three expression for the diagonal components that you wrote in your post?... | It seems like you're curious as to what the three different components of the stress tensor mean. Roughly speaking, <span class="math-container">$T_{xx}$</span>, <span class="math-container">$T_{yy}$</span>, and <span class="math-container">$T_{zz}$</span> tell you how much force per area is being exerted across a pla... | https://physics.stackexchange.com |
126,296 | [
"https://security.stackexchange.com/questions/126296",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/92885/"
] | Nowadays, when someone is making an online payment, the associated bank often requires an authentication step.
As far as I know, there are mainly two ways to do this:
<ol>
<li><strong>SMS Verification</strong>.
The Bank sends an SMS to the client with a specific code that should be stated on the authentication web pa... | I've been doing professional pentesting for quite a while covering many of the banks having a permit here in Switzerland. Traditionally they were using <strong>username/password</strong> only.
Then they switched to <strong>TAN</strong> (transaction authentication number) handed out on paper. I just know 2 remaining ba... | I've never heard of just using the date of birth for authentication. It's a bad idea as a date of birth is non-revocable. Meaning if it is "compromised" you can't change it.
If it is the only means of authentication for consulting account or transactions it's a <em>REALLY BAD IDEA</em>.
But that brings me to my sec... | https://security.stackexchange.com |
324,341 | [
"https://mathoverflow.net/questions/324341",
"https://mathoverflow.net",
"https://mathoverflow.net/users/116167/"
] | I have a following sum:
<span class="math-container">$S_g=\sum_{k=0}^g k\binom{4g+2}{2k}$</span>
I can transform it into a different sum
<span class="math-container">$S_g=(2g+1)\sum_{k=1}^g\binom{4g+1}{2k-1}$</span>
What is the closed form or what is the method to deal with any of above sums?
| According to Maple,
<span class="math-container">$$ S_g = \left( g + \frac12\right) \left(16^g - {4 g \choose 2g}\right) $$</span>
| The Mathematica command
<pre><code>Sum[k*Binomial[4 g + 2, 2 k], {k, 0, g}]//FullSimplify
</code></pre>
performs
<span class="math-container">$$16^g g-\frac{\Gamma (4 g+3) \, _3F_2\left(2,1-g,\frac{3}{2}-g;g+\frac{5}{2},g+3;1\right)}{\Gamma (2 g-1) \Gamma (2 g+5)}.$$</span>
| https://mathoverflow.net |
182,860 | [
"https://dba.stackexchange.com/questions/182860",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/33583/"
] | Apologies if I haven't used the correct terminology, I am not a DBA, but right now, I am all that is here!
Situation is last week, a database server was misbehaving (it was about 6 years old). The hosting company built a new server, and moved all the current data across to the new server, which is running well.
Howev... | <h2>Not directly, but here's a work-around</h2>
The way you do partial restores of individual tables or rows in SQL Server is to restore the backups to a different database name and then manually move your specific missing data from the restored database back to your production DB.
This involves a couple of steps depen... | No, you cannot do what you mentioned in your question.
I am sure there are many ways to recover the data you need. One way is to:
<ul>
<li>Restore the backup as a different(name) database. </li>
<li>Reconcile data using this source and your current live database. How you do that? Will entirely depend upon with wha... | https://dba.stackexchange.com |
293,663 | [
"https://softwareengineering.stackexchange.com/questions/293663",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/191858/"
] | I have a hard time understanding the use of MSB (Most significant bit) vs. LSB (Least significant bit) and why they're used at all. It is my understanding that it has to do with Endianness, but I cannot get my head around it.
Basically: If I read a specification on X and they specify that the data needs to be read LSB... | Endianness for binary (base 2) is no different than endianness for any other base.
Without a specification telling you which side the most significant and least significant decimals were, you wouldn't know whether
<pre><code>1234
</code></pre>
is one-thousand-two-hundred-and-thirty-four or four-thousand-three-hundre... | When a byte is serialized into a stream of bits and transmitted serially, it becomes important to know whether it's transmitted LSbit-first or MSbit-first. The transmitter can send the bits in either order, and the contract whether it's LSbit-first or MSbit-first is established in the spec (or datasheet). For example... | https://softwareengineering.stackexchange.com |
372,818 | [
"https://softwareengineering.stackexchange.com/questions/372818",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/-1/"
] | Suppose if I have an abstract <code>Weapon</code> class, and the subclass <code>ReloadableWeapon</code> which implements the interface Reloadable.
<pre><code>interface Reloadable {
void Reload();
}
public abstract class Weapon{
@override
public abstract void attack(Player enemy){}
}
public final class Rel... | There is no "right" solution, this is fully context dependend. In an ideal design, you start with
<pre><code>ReloadableWeapon rw = new ReloadableWeapon()
</code></pre>
in a context where <code>ReloadableWeapon</code> is known, and now you can pass <code>rw</code> to functions which expect a <code>Weapon</code>, and a... | What I would like to point, is that by respect of the Liskov substitution principle : "objects in a program should be replaceable with instances of their subtypes without altering the correctness of that program.", the creation of the ReloadableWeapon should not create new instructions in higher level
In your example... | https://softwareengineering.stackexchange.com |
231,404 | [
"https://security.stackexchange.com/questions/231404",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/29770/"
] | Suppose we're using secure boot and remote attestation to prove to a server what client software is talking to it.
What stops an attacker from doing this:
<ul>
<li>Start a legitimate copy of the client software on machine A.</li>
<li>Get a remote attestation challenge from the server using a modified copy on machine ... | You don't. A TPM only protects against “mild” physical attacks. For example, it does protect against plugging in an alternate hard disk and booting from that. If the disk is encrypted with a key in the TPM, it protects against taking the hard disk out and plugging it into another machine. But it doesn't protect against... | The scenario you had described and the question that arises from it are both legitimate. In 2020, there exist on-chip implementations of TPM (aka fTPM), which alleviate the issue of attacks directed at the <em>bus</em>. But, you should be asking a different question, namely how does the TPM implementation know that t... | https://security.stackexchange.com |
260,842 | [
"https://math.stackexchange.com/questions/260842",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/33814/"
] | <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are holomorphic functions in <span class="math-container">$G \subset \mathbb C$</span> and continuous on the boundary of <span class="math-container">$G$</span>. Prove that <span class="math-container">$|f| + |g|$</span> gets its maximu... | I think <span class="math-container">$G$</span> is supposed to be bounded and hence <span class="math-container">$\overline{G}$</span> is compact. By continuity of <span class="math-container">$|f|+|g|$</span> and compactness of <span class="math-container">$\overline{G}$</span>, there exists <span class="math-containe... | If $\omega \in \mathbb{C}$ and $|\omega| = 1$ then $f + \omega g$ is holomorphic on $G$ and $|f + \omega g| \leq |f| + |g|$. Given some fixed $z \in G$ there is such an omega (depending on $z$) such that $|f(z) + \omega g(z)| = |f(z)| + |g(z)|$. Now apply the maximum modulus principle to $f + \omega g$.
| https://math.stackexchange.com |
110,685 | [
"https://security.stackexchange.com/questions/110685",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/70407/"
] | Someone has definitely gained access to two of my alternate emails. I know this by looking at the login activity. By looking at the email logs I discovered that the IP that is always gaining entry to my email is not my own.
I haven't changed my passwords yet because I'd like to somehow figure out who is doing this. Wh... | TL;DR:
<ul>
<li>Change your passwords</li>
<li>Enable two-factor authentication to prevent attackers from changing your password</li>
<li>Warn your sysadmin</li>
</ul>
<hr>
You should change your passwords ASAP. From a machine that you trust.
What good is it going to do that the attacker can still log in too? What ... | First of all, I would change my password.
Secondly, if at all possible, turn on two-factor authentication of some kind, so that you will be able to reset your password <em>if</em> someone should get access again, and actually change your password (is this possible with Hotmail? I know GMail supports it). You <strong>... | https://security.stackexchange.com |
746,249 | [
"https://math.stackexchange.com/questions/746249",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/5775/"
] | <blockquote>
Determine whether this matrix' columns are linearly dependent or not.
$$\begin{bmatrix} 1 & 0 & 2 \\ 0 & -1 & -2 \\ 2 & -2 & 0 \end{bmatrix}$$
</blockquote>
The determinant is $0$ - therefore they are linearly dependent!
<blockquote>
Without making any calculations.
</bl... | Hint:
Write a linear combination of column 1 and 2, that is $2C_1 + 2 C_2$.
| Note that the dimension of the column space is equal to that of the row space. Therefore, simply perform Gaussian elimination to determine the number of linearly independent rows (and hence the number of linearly independent columns).
In other words, the number of linearly independent columns will equal the number of... | https://math.stackexchange.com |
235,530 | [
"https://physics.stackexchange.com/questions/235530",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/107006/"
] | I am thinking on a object, e.g. ball or planet that starts rotating with increasing speed. Let's assume that his speed get's closer to the speed of light, what happens to this object? There are several forces acting. But I always get caught thinking that it will get heavier and heavier because of the additional energy ... | A phenomenon which has been observed in stars and planets is that when the body rotates faster and faster, the object becomes more and more elongated, thus some points in the object are farther away from the rotation axis, increasing the relative moment of inertia and requiring more torque to accelerate the body by the... | Note that "c" means linear velocity, not angular. Then, you would refer to tangential velocity.
If an object rotated at relativistic (tangential) velocity, then each "shell" will have a different space and time compression. By contraction of the tangent lengthes, the perceived length of large circles would be smaller ... | https://physics.stackexchange.com |
23,737 | [
"https://scicomp.stackexchange.com/questions/23737",
"https://scicomp.stackexchange.com",
"https://scicomp.stackexchange.com/users/8213/"
] | My problem is
$$\text{minimize}: \phantom{2} f(x) \\
\text{subject to }: \phantom{2} x_4 \ge 0$$
where $x=(x_1,x_2,x_3,x_4)$.
I know that the fourth component $x_4$ of the desired local minimizer is greater than zero, so the inequality constraint at the solution is inactive. However, there are other local minimizers... | There is really no particularly good answer that is generically true for any $f(x)$. My recommendation is to read the book by Nocedal & Wright on Numerical Optimization (one of my all-time favorite books) which covers the log-barrier method you use and many others. For example, you could consider active set methods... | An important special case of constrained optimization is constituted by optimization problems with simple bounds:
$$\min_{x \in \mathbb R^n} \quad f(x) \quad \text{s.t.} \quad x \geq 0$$
Assume that $f$ is continuously differentiable. Show that the first order necessary conditions for this problem are equivalent to ... | https://scicomp.stackexchange.com |
4,062,930 | [
"https://math.stackexchange.com/questions/4062930",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/854466/"
] | Given the equation <span class="math-container">$x^2-14x+m=0$</span>, where <span class="math-container">$m$</span> is a positive integer. If roots <span class="math-container">$p_1, p_2$</span> are distinct, positive, prime integers, then find the value of <span class="math-container">$K=(p_1+p_2)^2+2p_1p_2$</span>
I ... | We are given column vectors
<span class="math-container">$$ x = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \quad
l = \begin{pmatrix} l_1 \\ l_2 \\ l_3 \end{pmatrix}, \quad
m = \begin{pmatrix} m_1 \\ m_2 \\ m_3 \end{pmatrix}.\tag{1} $$</span>
Note that
<span class="math-container">$$ l\,m^T := \begin{pmatrix} l_1m... | Just as a sanity check...
<span class="math-container">$C := lm^T + ml^T = \begin{pmatrix}
2l_1m_1 & l_1m_2+l_2m_1 & l_1m_3+l_3m_1 \\
l_1m_2+l_2m_1 & 2l_2m_2 & l_2m_3+l_3m_2 \\
l_1m_3+l_3m_1 & l_2m_3+l_3m_2 & 2l_3m_3
\end{pmatrix}.$</span>
Then <span class="math-container">$C x = \begin{pmatrix}... | https://math.stackexchange.com |
4,005 | [
"https://cogsci.stackexchange.com/questions/4005",
"https://cogsci.stackexchange.com",
"https://cogsci.stackexchange.com/users/3249/"
] | I heard this comment on a radio program about Child parenting by psychologists: "A child will not comprehend an explanation that, a parent loves each sibling equally, as they do not have abstract thinking". This lead me to ponder and formulate the question.
| As often in this area, the problem seems to be entirely created by confusions about definitions and terminology and the difficulties resulting from the use of common-sense concepts like “surprise” in scientific theories. One answer would be “Why not?” Basically you seem to object to the definition of surprise used by s... | Also.
Appraisal theory argues that a cognitive mechanism assesses the meaning of a situation before initiating an emotion. It does not argue that this mechanism operates consciously, so demonstrating that it doesn't doesn't impinge on the validity of appraisal theory at all.
Mismatch negativity may be part of the pro... | https://cogsci.stackexchange.com |
690,942 | [
"https://math.stackexchange.com/questions/690942",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/51434/"
] | Given two real sequences $\{ a_n \}$ and $\{ b_n \}$, where $n \ge 0$, the convolution operation (denoted $\ast$) is defined as $\{ c_n \} = \{ a_n \} \ast \{ b_n \}$, where $c_n = \sum_{k=0}^{n} a_k b_{n-k}$. I want to prove that convolution is associative, i.e., $(\{ a_n \} \ast \{ b_n \}) \ast \{ c_n \} = \{ a_n \} ... | Define $\{d_n\}_n$ to be the result of the first of your sum. Let $\{ e_n \}_n$ the second one. It is enough to prove that for each $n \in \mathbb{N}, \ d_n = e_n$. Start by recalling that finite addition over $\mathbb{N}$ is associative, distributive and commutative (related to the multiplication obviously). Since th... | I'd like to present another solution in form of a straight forward computation with one substitution.
First, define $q := p + r$. Then
\begin{align}
(a*(b*c))[n] &= \sum_{p=0}^{N-1}a[p](b*c)[n-p] = \sum_{p=0}^{N-1}a[p] \sum_{r=0}^{N-1}b[r]c[(n-p)-r] \\
&= \sum_{p=0}^{N-1} \sum_{r=0}^{N-1} a[p]b[r]c[(n-p)-r] \... | https://math.stackexchange.com |
41,247 | [
"https://stats.stackexchange.com/questions/41247",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/16280/"
] | I am interested in how uncertainty can be accounted for when considering the risk of extinction of a species. Forgive me for extending a rather tired thought experiment, but at least it's familiar territory and I hope it illustrates what I am trying to understand.
Let's say that Schrödinger was not satisfied with kill... | Let $p_k, k=1, \ldots, K=15$ be the survival probabilities for individual snow cats.
<ol>
<li>Initialize the number of cats accounted for so far $k \leftarrow 0$, the vector of the binomial survival probabilities $(\pi_0^{(0)}, \pi_1^{(0)}, \ldots, \pi_K^{(0)}) \leftarrow (1, 0, \ldots, 0, 0)$</li>
<li>While there sti... | It's unclear what "risk of extinction category" means, but it appears the question asks to compute the distribution of the sum of 15 independent binomial variates having the given expectations. This is a convolution and it's most efficiently done with the Fast Fourier Transform.
Here is an example in <code>R</code>, ... | https://stats.stackexchange.com |
3,772,045 | [
"https://math.stackexchange.com/questions/3772045",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/325417/"
] | Given <span class="math-container">$f$</span> is a Lebesgue measurable function and <span class="math-container">$\int_0^1 x^{2n}f\,d\mu = 0 \quad \forall n$</span>, then show that <span class="math-container">$f = 0$</span> a.e.
Of course, if it was given that <span class="math-container">$f \geq 0$</span> then this w... | Here is a sketch of the proof. I believe you can fill in the details to complete the proof:
<blockquote>
<strong>Assumption.</strong> <span class="math-container">$f$</span> is integrable on <span class="math-container">$[0,1]$</span> and satisfies <span class="math-container">$\int_{0}^{1}x^{2n}f(x)\,\mathrm{d}x=0$</s... | Make the substitution <span class="math-container">$x=\sqrt y$</span> to see that <span class="math-container">$\int_0^{1}y^{n}g(y)dy=0$</span> for all <span class="math-container">$n$</span> where <span class="math-container">$g(y)=f(y)y^{-1/2}$</span>. (Integrability of <span class="math-container">$g$</span> follows... | https://math.stackexchange.com |
2,963,995 | [
"https://math.stackexchange.com/questions/2963995",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/64460/"
] | Today I was trying to prove that
<span class="math-container">$$\lim_{(x, y) \to (0, 0)}\dfrac {x^2y^2}{x^2+y^2} = 0 $$</span>
I got really lucky because the AM-GM inequality directly applies here to give us
<span class="math-container">$$\dfrac {x^2y^2}{x^2+y^2} \le \dfrac {x^4 + y^4}{x^2+y^2} \le \dfrac {(x^2+y^2)... | For the 3<span class="math-container">$\implies$</span>1 part, I would proceed like follows:
Since <span class="math-container">$[a]\cap [b] \neq \varnothing$</span>, there is some <span class="math-container">$c \in [a] \cap [b]$</span>, i.e. <span class="math-container">$c \sim a$</span> and <span class="math-contai... | Complete Proof (Credit goes to xbh for providing key step):
If a=b, then a and b must have the same equivalence class (by definition). Then if a and b have the same equivalence class, it follows their intersection cannot be empty (as two elements that have the same equivalence class cannot be disjoint). Lastly, since... | https://math.stackexchange.com |
27,943 | [
"https://math.stackexchange.com/questions/27943",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/8373/"
] | I'm a uni student doing a real analysis course and am finding it very interesting, but at the same time very confusing.
One question that has me stumped is how to get a countable set of functions from a set $S$, where $S$ is a set of all functions $u: \mathbb{N} → \{ 0,1,2 \}$. They also tell us that a function $v(n)$... | I think you may be confusing a few things here.
First you speak of the set of all functions $\mathbb N \to \{0,1,2\}$, in other words sequences with values in $\mathbb N$, since such a function can be seen as the sequence
$$ f(0) , f(1) , f(2), f(3),\dots $$
(I'm assuming $0\in\mathbb N$, unaware of the convention you... | In general it is impossible to get a countable subset of an uncountable set without using the (countable) axiom of choice.
But in the case of functions $f:\mathbb{N}\to\{0,1,2\}$ you can find a countable subset of these functions without any use of the axiom of choice, using for example the fact that the natural numb... | https://math.stackexchange.com |
239,498 | [
"https://dba.stackexchange.com/questions/239498",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/124241/"
] | I am querying a sales order table and need to add a variable so that the query always returns a customer list for all customers that have not purchased this year. I don't want to hard code this years' date into WHERE or HAVING, but instead am interested to somehow have this dynamically pass the correct variable so tha... | There are many ways to get the beginning of this year, I think the most intuitive is <code>DATEFROMPARTS()</code>:
<pre><code>DECLARE @BeginningOfThisYear date = DATEFROMPARTS(YEAR(SYSUTCDATETIME()),1,1);
SELECT ... FROM dbo.Customers AS c
WHERE NOT EXISTS
(
SELECT 1 FROM dbo.Orders
WHERE CustomerID = c.Custo... | There could be many different ways to do this, this is one:
<pre><code>Declare @SO Table (SOID int,
date datetime)
insert into @so
(SOID, DATE)
VALUES
(1, dateadd(yy,-1,getdate())),
(2, dateadd(mm,-1,getdate()));
SELECT SOID, Date
from @SO
WHERE Year(date) = year(getdate())
</code></pre>
Same idea with a ... | https://dba.stackexchange.com |
156,902 | [
"https://cs.stackexchange.com/questions/156902",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/112217/"
] | I have sequence of integers <span class="math-container">$a_1, a_2, .., a_n$</span>,<br />
let <span class="math-container">$S_a = \sum_{i=1}^{N}{a_i}$</span>,<br />
for any <span class="math-container">$k \in (0; 1)$</span> I need an algorthim to that maps every <span class="math-container">$a_i$</span> into another i... | Compute the prefix sum of the <span class="math-container">$a_i$</span>, multiply all elements by <span class="math-container">$k$</span> and round. Now the desired integers are the pairwise differences of that sequence. The first condition is always honored.
E.g. <span class="math-container">$a=(3,6,2,7,9), k=0.8\to(3... | If <span class="math-container">$\sum_{i=0}^{n-1}b_i>k\sum_{i=0}^{n-1}a_i$</span> then <span class="math-container">$b_n = \lfloor ka_n\rfloor$</span>, else <span class="math-container">$b_n = \lceil ka_n \rceil$</span>. This requires only one pass through <span class="math-container">$\{a_n\}$</span>.
| https://cs.stackexchange.com |
310,882 | [
"https://softwareengineering.stackexchange.com/questions/310882",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/97834/"
] | An array of <em>T</em>, a pointer to <em>T</em>, and a reference to <em>T</em> (represented as <code>T[]</code>, <code>T*</code>, and <code>T&</code> respectively by <code>Type.ToString</code>) all share the similarity that they are constructed from an "element" type (which can be returned by <code>Type.GetElementT... | Generally, functions that take a type and return another type are called type constructors.
That term should be well known amongst functional languages, but less so in mainstream languages - often because the array notation is the only one used.
| In C#, T[] can be referred to as arrays or "covariant arrays", which both C# and Java have. Some language critiques think that the covariant arrays were a mistake, and would have been different had generics been in the language from the start.
C# does not have "reference to T" (e.g. <code>T&</code> as we find in ... | https://softwareengineering.stackexchange.com |
85,112 | [
"https://electronics.stackexchange.com/questions/85112",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/30175/"
] | <em>vectored interrupts are not possible if a cpu has single interrupt request line and single interrupt grant line while multiple interrupting devices are possible</em> .
Is the above statement is correct if yes then how.I am not very good in microprocessor but understand basic concepts.So if it is silly then please... | In a configuration where multiple devices are tied to a single interrupt line to the CPU, and the interrupts are simply combined together without identifying the source of the interrupt (via some priority encoding, or perhaps a big logical OR), then vectoring isn't possible. It looks essentially as if there is only a s... | It really depends on what kind of CPU you're talking about.
For example, the Z80 has a single interrupt request line, but it has a "vectored interrupt mode" that relies on a separate chip to prioritize multiple external interrupts and present a vector during the interrupt acknowledge cycle.
Other systems do it by dai... | https://electronics.stackexchange.com |
446,175 | [
"https://stats.stackexchange.com/questions/446175",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/265298/"
] | I confuse that the difference of Residuals deviance between binary and count data of the same data, by logistic regression in R. I'd like to know the way to calculate the both Residual deviance. Please give me some advice.
binary data
<pre><code>x<-c(2,2,2,2,2,3,3,3,3,3,5,5,5,5,5,6,6,6,6,6)
yesno<-c(1,1,0,0,0,1... | The summaries report twice the negative log likelihood (evaluated at the parameter estimates). <strong>They look inconsistent:</strong> one reports <span class="math-container">$24.7444$</span> while the other reports <span class="math-container">$9.096343.$</span> How can that be, when the parameter estimates and st... | I tried the case of proportion(=yes/yes+no), using above best answer. Yes, I got it.
But, I couldn’t understand the case without “weight=n”. A little bit more for complete understanding.
<pre><code>#-----with “weight=n”
modelcp<- glm(yp~x,family=binomial,weight=n)
(result<-summary(modelcp))
# Estimate... | https://stats.stackexchange.com |
13,878 | [
"https://datascience.stackexchange.com/questions/13878",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/24109/"
] | I have a series of datasets that are composed of 100 or so variables and a corresponding response variable. I am often faced with the question of trying to attribute differences in the response variable to the 100 or so variables.
My problem is that it is often difficult to compare two or more datasets and attribute ... | The statistics here are obviously very good, in fact too good for any practical data set. Your model is almost perfect... Unfortunately, it's practically useless and I'll explain.
In machine learning, if you see something like this you know you are in <strong>trouble</strong>. That can happen if there are problems wit... | In the dark I would say that it is suspicious. However this really depends on the problem you are solving. Certain datasets will have a lot of clear examples, a very easy task to solve etcetera.
What could have happened is that during the preprocessing somehow your targets leaked into your features. Some more advanced... | https://datascience.stackexchange.com |
394,445 | [
"https://softwareengineering.stackexchange.com/questions/394445",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/238251/"
] | I have several areas in a program where I am doing the following check on the same two booleans, but each spot has different text being written to a file via Stream Writer based on the value of the booleans.
Example
<pre><code>if (bool1 && bool2)
{
StreamWriter.WriteLine("Stuff to the file");
StreamWrit... | Another way to solve it using the Design Patterns <strong>Strategy</strong> and <strong>Factory Method</strong>:
An interface defines the strategy, in this case, it say that the strategy is write something, without details about what.
<pre class="lang-cs prettyprint-override"><code>interface IStrategy
{
void Writ... | Use a procedure to do the branching, either:
<pre><code>WriteLines(IEnumerable<string> linesBoth, IEnumerable<string> linesOnlyA, IEnumerable<string> linesOnlyB)
{
// ...
}
</code></pre>
or:
<pre><code>WithStream(Action<StreamWriter> onBoth, Action<StreamWriter> onA, Action<Strea... | https://softwareengineering.stackexchange.com |
8,734 | [
"https://stats.stackexchange.com/questions/8734",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/1381/"
] | Is there a well founded rule for the number of significant figures to publish?
Here are some specific examples / questions:
<ul>
<li>Is there any way to relate the number of significant figures to the coefficient of variation? For example, if the estimate is 12.3 and the CV is 50%, does that mean that the informatio... | I doubt there's a universal rule so I'm not going to make any up. I can share these thoughts and the reasons behind them:
<ul>
<li>When summaries reflect the data themselves--max, min, order statistics, etc.--<strong>use the same number of significant figures</strong> used to record the data in the first place. This... | I'd suggest 12 (1.2, 123.4). Omit the .3 since it's nearly meaningless, but many people when they see (1.2, 120) will assume that the last '0' in 120 is significant.
| https://stats.stackexchange.com |
192,800 | [
"https://security.stackexchange.com/questions/192800",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/165658/"
] | I am getting a XSS vulnerability through burp suite but when I inject script manually I don't get the XSS vulnerability.
Is this a reportable vulnerability?
| If you can't verify the result of an automated tool, it is not a good idea to report it's findings.
Automated tools are great, but they have a problem with false positives.
| Is this reflected XSS? One of the common reasons this can be difficult to reproduce is because of browser XSS filters. I recommend you use Firefox which does not have a filter.
To properly determine if this is exploitable you need to gain an understanding of what is happening - what characters get through the filter, ... | https://security.stackexchange.com |
48,660 | [
"https://cs.stackexchange.com/questions/48660",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/31126/"
] | <blockquote>
A system has 6 identical resources and <span class="math-container">$N$</span> processes competing for
them. Each process can request at most two requests. Which one of the
following values of <span class="math-container">$N$</span> could lead to a deadlock?
<ol>
<li>1</li>
<li>2</li>
<li>3</li>
<li>4</li>... | I agree that no deadlock is possible here. If there are three or fewer processes, there clearly cannot be a deadlock because there are enough resources for every process to just hold two resources the whole time.
So any deadlock must have at least four participating processes. To participate in a deadlock, a process ... | I would think that even with 5 processes, deadlock isn't possible. If 5 processes each need up to 2 of six identical resources, we could assume that each acquires one resources, and then we have 5 processes with a single resource left. Sure, four processes may have to wait when one process acquired the last resource, b... | https://cs.stackexchange.com |
372,283 | [
"https://mathoverflow.net/questions/372283",
"https://mathoverflow.net",
"https://mathoverflow.net/users/163893/"
] | Let <span class="math-container">$G$</span> be a 1-dimensional, commutative formal group over a ring <span class="math-container">$R$</span>. Give <span class="math-container">$G$</span> a coordinate <span class="math-container">$x$</span> and let <span class="math-container">$A\subset R$</span> be the subring generate... | This is not a reference, but a short proof.
We use the following (probably known, but see later) lemma on representing a symmetric tensor as a linear combination of rank-1 symmetric tensors.
<strong>Lemma.</strong> Let <span class="math-container">$A$</span> be a finite set, <span class="math-container">$K$</span> an i... | Concerning the reference request:
Several text books [1,2] give the theorem and proof for <A HREF="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial" rel="nofollow noreferrer">elementary symmetric polynomials</A> <span class="math-container">$s_k=$</span> sum of all <span class="math-container">$k\times k$<... | https://mathoverflow.net |
1,556,649 | [
"https://math.stackexchange.com/questions/1556649",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/-1/"
] | Let $\alpha^3-\alpha-1=0$, $K=\mathbb Q(\sqrt{-23})$, $K'=\mathbb Q(\alpha)$, and $L=\mathbb Q(\sqrt{-23},\alpha)$.
Then I am asked to show that the field extension $L/K$ is unramified.
I know that if $\mathfrak p\in\operatorname{Max}(\mathcal O_K)$ ramifies in $L$ then $\mathfrak p\mid\mathfrak d$ where $\mathfrak ... | Here’s an approach quite different from what you had in mind, purely local and not ideal-theoretic.
The field $K$ is ramified only at $23$, and since the $\Bbb Q$-discriminant of $k=\Bbb Q(\alpha)$ is of absolute value $23$ as well, the only possibility for ramification of $K'=Kk$ over $K$ is above the prime $23$, in ... | Observe that $$\begin{align}23 &= \alpha^3-\alpha-24=(\alpha-3)(\alpha^2+3\alpha+8) \\&=(\alpha-3)(\alpha-(-\frac{3}2+\sqrt {-23}))(\alpha-(-\frac{3}2-\sqrt {-23}))\end{align}$$
So $$(23,\alpha-10)\subset(\alpha-(-\frac 32\pm \sqrt{-23}),\alpha-10)\subset\mathcal O_L$$
| https://math.stackexchange.com |
13,587 | [
"https://astronomy.stackexchange.com/questions/13587",
"https://astronomy.stackexchange.com",
"https://astronomy.stackexchange.com/users/10697/"
] | I am looking for an answer as to why the geocentric gravitational constant, μ, defined as the product of the gravitational constant, G, and the mass of a body (earth in this case), M, can be calculated to a higher degree of precision than either G or M alone.
From wikipedia
<blockquote>
...for celestial bodies such... | Ignoring details such as the oblateness of the Earth, atmospheric drag, third body influences such as the Moon and the Sun, relativity, ..., the period of a satellite of negligible mass (even the International Space Station qualifies as a "satellite of negligible mass") is $T=2\pi\sqrt{\frac {a^3}{\mu_\mathrm{Earth}}}$... | $\mu$ is a quantity that can be easily observed (semi-) directly. It can be easily derived from orbital period of orbiting bodies or acceleration of falling bodies, even if their mass is significantly smaller than the mass of the body you're measuring $\mu$ for.
Now the only way to really obtain G is to divide $\mu$ b... | https://astronomy.stackexchange.com |
66,384 | [
"https://quant.stackexchange.com/questions/66384",
"https://quant.stackexchange.com",
"https://quant.stackexchange.com/users/58689/"
] | I'm currently trying to understand risk-neutral valuation and transforming real-world stochastic processes to their risk-neutral version. If I understood it correctly, the main point of risk-neutral valuation is to not have to deal with real-world drifts which are very difficult to estimate, but instead it requires to ... | Management fees are commonly charged on the asset base, in this case, the 10MM. There might be exceptions to this rule, but they are rare. FOr mutual funds, management fees are typically charged on a daily basis, say 2%/260 (260 being the proxy for the number of trading days in a year) of the prevailing net asset value... | From a managed fund operation perspective, the 2% charge is on how much funds one investor controls not the total FUM of the fund.
| https://quant.stackexchange.com |
502,515 | [
"https://stats.stackexchange.com/questions/502515",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/216170/"
] | 4 couples, 8 people total, participate in a Secret Santa gift exchange. Call the people A, B, C, D, E, F, G, H. Assume A+B are a married couple. Likewise, assume C+D, E+F, and G+H are all couples. All 8 people put their names on a piece of paper and then the people randomly draw names from the bowl. If a person draws... | If the order is random, then there are 8! different permutations that are equal likely. However, we can partition those 8! permutations by considering what position the first member of each couple is in the order of choosing. There are only 14 choices for the locations of the first member of each couple:<br />
1, 2, 3... | I am no expert at probability, but I think I can answer part a)
The probability for A to draw Bs name is <span class="math-container">$\frac17$</span> as choosing his own name would result in a redraw. This leaves 7 remaining names in the hat, so Bs chance to draw A is <span class="math-container">$\frac17$</span>.
Sim... | https://stats.stackexchange.com |
215,593 | [
"https://electronics.stackexchange.com/questions/215593",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/1074/"
] | I have a project where I'll need to generate some sound waves. In case it matters, the microcontroller I'll be using is an STM32L476RG which have an internal DAC.
The question I have is related with the output power/sound I'll get. I'm not sure if driving the speaker directly through the DAC will give me proper... | If you're going to be using just the STM to drive a speaker, the main limiting factor is the current handling capability of the pins of the microcontroller.
<ol>
<li>You didn't mention what power you want coming out of the speaker, but let's take a look at what the microcontroller can provide. Its working voltage ran... | The DAC is a low voltage very low current device .If all you are after is room filling volume then a complimentry Emitter Follower would be simple easy and cheap.Distortion will be very low .If you want extremely low distortion then you could do a cascode follower .Whether you run class A AB or B depends on the relat... | https://electronics.stackexchange.com |
468,841 | [
"https://stats.stackexchange.com/questions/468841",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/23732/"
] | Someone made this statement:
<pre><code>1 out of 1000 corresponds to a proportion of 0,1%. In this case you
have a chance of 37% of having 0 outcome, significance level 63%.
3 out of 1000 translating to a proportion of 0,3% has a 5% chance to hit 0.
</code></pre>
Are the above statement correct and how one arrives... | Ah. Well, this seems to be a <em>very</em> badly formulated way of saying:
<blockquote>
We perform 1000 independent trials, each with a success probability of 0.001. Then we have a chance of 37% to observe no successes at all.
</blockquote>
This is a true statement, <code>pbinom(0,1000,1/1000)</code>. Note how the ... | Another potential way of solving this problem is with a Poisson Distribution.
In this case <span class="math-container">$\lambda$</span> is 1/1000. So the probability of of observing 0 events with a single trial is:
<pre><code>ppois(0, 1/1000)
#[1] 0.9990005
</code></pre>
Thus it is 99.9% chance that no events will... | https://stats.stackexchange.com |
692,132 | [
"https://physics.stackexchange.com/questions/692132",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/326988/"
] | In polar coordinates, I know <span class="math-container">$$\mathrm{d}x = \cos\theta\, \mathrm{d}r - r \sin\theta \,\mathrm{d}\theta,$$</span> and <span class="math-container">$$\mathrm{d}y = \sin\theta \,\mathrm{d}r + r \cos\theta\, \mathrm{d}\theta,$$</span> thus I managed to successfully derive <span class="math-con... | Start with:
Position vector in polar coordinates
<span class="math-container">$$S(polar)=r\hat{r}$$</span>
With unit vectors <span class="math-container">$\hat{r}$</span> and <span class="math-container">$\hat{\theta}$</span>
<span class="math-container">$$\hat{r}=cos\theta \hat{x} + sin\theta \hat{y}$$</span>
<span c... | Notice that, in Cartesian coordinates, the arc length is given by
<span class="math-container">$$\mathrm{d}\vec{S} = \mathrm{d}x \, \hat{x} + \mathrm{d}y \, \hat{y}. (1)$$</span>
One can then notice that, in spherical coordinates, one has
<span class="math-container">\begin{gather}
\hat{x} = \cos\theta \, \hat{r} - \si... | https://physics.stackexchange.com |
2,373 | [
"https://electronics.stackexchange.com/questions/2373",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/-1/"
] | I have a switch marked as having a max current of 2A at 240Vac, and 6A at 125Vac.
What would its max current rating be at 12Vdc?
| Technically, it is impossible to tell: switch manufacturers have to deal with the subtleties of arcing when designing circuits, and AC behavior is different than DC behavior. Arcs will be extinguished much more easily with AC (since voltage passes through zero) than with DC.
It's odd that the rating drops so much at 2... | The general rule of thumb for switches with 125VAC or 240VAC ratings is that the switch
will operate satisfactory up to 30VDC at the highest rating listed on the
switch. IIRC I read this rule at the carlingtech.com site.
| https://electronics.stackexchange.com |
18,166 | [
"https://robotics.stackexchange.com/questions/18166",
"https://robotics.stackexchange.com",
"https://robotics.stackexchange.com/users/22280/"
] | I would like to control a system which orientation I measure with as quaternion based EKF in one degree of freedom. Obviously the simplest thing would be to convert the quaternion estimate to euler angles and take one of these. Unfortunately these euler angles are not stable. The other option I was thinking about was t... | You will have to specify the full quaternion in any case and that is because each w,x,y and z are a function of all the Euler angles.
Some ways to get by could be:
<ol>
<li>If you are concerned about an EKF generating noisy output in the other two angles: in the update step, you can hard code these known angles</li>... | If your system has a fixed rotation about some axis, then you should be able to design your KF about a single angle of rotation. You should be able to tailor your equations of motion and measurements to be a function of that angle.
| https://robotics.stackexchange.com |
99,135 | [
"https://stats.stackexchange.com/questions/99135",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/45739/"
] | I have two independent Poisson processes $A$ and $B$ with arrival rates $\lambda_A$ and $\lambda_B$, respectively. Now, the expected time for the arrival of the next item for the merged process should be $\frac {1}{\lambda_A+\lambda_B}$.
Assuming $T_{A+B}$ to be the arrival time for the next item of the combined proce... | heropup is right. The problem is that once you know that $X=A$, $X$ is not merely drawn from the exponential with rate $\lambda_A$ since you also know that the sampled value had to be small enough to win the comparison with the hypothetical sampled value from $B$.
So, the density given that $X=A$ is the renormalized ... | <span class="math-container">\begin{align}
& \Pr( T_{A+B} > t \mid X=A) \\[10pt]
= {} & \frac{\Pr(T_{A+B} > t\ \&\ X=A)}{\Pr(X=A)} \\[10pt]
= {} & \frac{\Pr(t < T_A < T_B)}{\Pr(X=A)}, \tag 1
\end{align}</span>
<span class="math-container">${}$</span>
<span class="math-container">\begin{alig... | https://stats.stackexchange.com |
52,666 | [
"https://math.stackexchange.com/questions/52666",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/4997/"
] | I really hate integration by parts, so when faced with $\int_{-\pi}^\pi x^2 \cos n x \, dx$ I tried writing it as
$$\int_{-\pi}^\pi x^2 \cos n x \, dx =<br>
\frac{d}{dn} \int_{-\pi}^\pi x \sin n x \, dx =
\frac{d^2}{dn^2} \int_{-\pi}^\pi \cos n x \, dx $$1
I have done something wrong. The integrand is continuously ... | It did work (except you are missing a negative sign). Remember $n$ is not always an integer, so that $$-\int_{-\pi}^\pi \cos(nx)dx=-\frac{2\sin(\pi n)}{n}.$$
Then $$\frac{d^2}{dn^2} \left(-\int_{-\pi}^\pi \cos(nx)dx\right) =\frac{d}{dn} \left( -\frac{2\pi\cos (\pi n)}{n}+\frac{2\sin(\pi n)}{n^2}\right)$$
$$=\frac{2... | Well, if there are cases in which you can't avoid integration by parts, then please use the below formula:
<ul>
<li>If $u$ and $v$ are functions of $x$ and dashes denote differentiation and suffixes integration with respect to $x$, then $$\small\int uv \ dx = uv_{1} -u'v_{2}+ u''v_{3} - u'''v_{... | https://math.stackexchange.com |
1,551,789 | [
"https://math.stackexchange.com/questions/1551789",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/286048/"
] | Exercise from Artin's 2nd edition of Algebra.
How many elements of order $5$ might be contained in a group of order $20$?
My attempt:
By the third Sylow Theorem, $|Syl_{5} (G)| = 1, 2, 4$ and it is also $\equiv 1 \mod 5$, so it is 1. Thus theree is a unique subgroup of order $5$, which contains four elements of orde... | Let $n_{5}$ denote the number of Sylow $5$-subgroups. By Sylow Theorems, $n_{5} \equiv 1 \mod 5$ and $n_{5}|4$ and both conditions imply that $n_{5} = 1$. Hence the number of non-identity elements of order $5$ is $n_{5}(5-1) = 4$.
| assume you have two distinct subgroup H and K of order 5 ( otherwise the are the same). Now think about the set HK.
| https://math.stackexchange.com |
226,871 | [
"https://mathoverflow.net/questions/226871",
"https://mathoverflow.net",
"https://mathoverflow.net/users/63877/"
] | Let $X$ be a proper smooth connected curve over an algebraically closed field $k$ of characteristic $0$, and suppose that $X$ is equipped with a $k$-linear action of a finite group $G$. It makes sense to form the quotient curve $Y := X/G$, and $Y$ is $k$-smooth because it is normal. Is it true that the pullback of diff... | Yes, the formula holds, even when the action is not free. Here is the principle of a proof for the case of a tame action (which covers the characteristic $0$ case). Denote by $\pi:X\to Y=X/G$ the quotient morphism.
First consider the exact sequence of $G$-sheaves on $X$
$$ 0 \to \pi^* \Omega^1_{Y/k} \to \Omega^1_{X/k... | <B>Counterexample in the non-tame case</B>. This is <I>not</I> what was asked by the OP. However, since it comes up in answers and comments above, I thought I would write down one example to illustrate the problem in the non-tame case.
Let $k$ be a field of characteristic $p$. Let $U$ be $\text{Spec}\ k[x][1/(x^{... | https://mathoverflow.net |
25,837 | [
"https://dsp.stackexchange.com/questions/25837",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/16089/"
] | I was reading from Internet about FIR filter design using MATLAB, and I saw parameters like passband frequency, stopband frequency, passband attuenation, stopband attuenation, etc. What made me confused was the parameter sampling frequency.
I don't understand the significance of sampling frequency with respect to a fi... | All digital filter frequency parameters (passband begin frequency, passband end frequency, and stopband begin frequency) are stated in terms of an input signal sequence's Fs sampling frequency. For a lowpass filter example, if I said a lowpass filter's passband width (it's "cutoff" frequency) is 0.2, I'm saying that th... | Signals that you can handle in the computer have to be sampled into discrete values. You do this by measuring the voltage on the analog input many thousand times per second. Exactly how often you make that measurement is the sampling rate.
Typical rates for PC audio systems are 44100Hz, 22050 Hz, and 11025 Hz.
When... | https://dsp.stackexchange.com |
484,972 | [
"https://electronics.stackexchange.com/questions/484972",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/210429/"
] | Reflections in a transmission line is causing ringing in square wave because certain frequencies are getting enhanced, but when frequency of the square wave changes won't the amplitude of those enhanced frequency decrease or increase because the amplitude of higher harmonics decrease or increase when we change the freq... | That is because the edges of the square wave is identical regardless of the rate of when the edges happen. So what happens at the edge does not depend on the frequency of the square wave.
| Here is the logic of solving math problems using the frequency domain:
<ul>
<li>This differential equation stuff is <em>hard</em>. In fact, it's practically impossible, and there's all this <em>housekeeping</em> mixed in. I wish there were a better way!</li>
<li>Oh look, if I'm analyzing a linear, time-invariant sys... | https://electronics.stackexchange.com |
594,978 | [
"https://physics.stackexchange.com/questions/594978",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/164488/"
] | If an observer moves in an accelerated frame in flat spacetime, the vacuum looks like a thermal distribution of particles to that observer. This is the Unruh effect.
<ul>
<li>Is it a special relativistic (SR) effect or a general relativistic (GR) effect? On one hand, it's flat spacetime which seems that SR is enough. B... | It is a SR phenomenon <strong>since it would be true even if GR were false</strong>: no equivalence principle, no Einstein equations, etc. (and thus it is independent of it). The said effect is a consequence of properties of the unique Poincare invariant vacuum state which satisfies a peculiar property (KMS condition) ... | This is not an <em>either - or</em> situation. One should make a distinction between the <em>physics</em> of the effect, in this case the accelerating detector coupled to a quantum field registering the thermal spectrum of excitations of that field, and the <em>conceptual framework</em> used to explain the effect. Both... | https://physics.stackexchange.com |
173,582 | [
"https://mathoverflow.net/questions/173582",
"https://mathoverflow.net",
"https://mathoverflow.net/users/36688/"
] | Let $\Delta ABC$ be a triangle in the plane. Let $P_{1}, P_{2}, P_{3}$ be the intersection points of bisectors, medians and altitudes, respectively. We define the quantity:
\begin{equation}
Q(\Delta ABC)=\frac{\mathcal{A}(\Delta P{1}P_{2}P_{3})}{\mathcal{A}(\Delta ABC)}
\end{equation}<br>
where $\mathcal{A}$ is the ar... | I can say something for the triangles with acute angles. But the obtuse angles may be a bit of a problem, since the intersection of the altitudes will be outside of the triangle.
For the triangle with acute angles, the supremum is definitely less than $1$. In fact, it would be true for any $P_1$ and $P_3$, so long as ... | There is a systematic method to attack you problem in the general case of three arbitrary given triangle centres (I am using the concepts and notations of the "Encyclopedia of Triangle Centers" which can be consulted online). We consider three centre functions <span class="math-container">$f$</span>, <span c... | https://mathoverflow.net |
131,148 | [
"https://electronics.stackexchange.com/questions/131148",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/43172/"
] | I've just had a PCB manufactured for me, but it seems like the solder mask is only applied to where the copper is.
I'm trying to figure out if it's my fault or not.
According to my understanding there supposed to be solder mask <strong>everywhere</strong> on the board <strong>except</strong> the places in red (pads/vi... | That looks correct.
Whether it's your fault or not depends on what you told the manufacturer the file names represent- the names in your title are standard for Altium, but other EDA software uses different names. Even so, I would expect the people at most suppliers to flag it and ask questions, but some folks are doi... | Everything looks okay to me.
Gerber files can be either "positive" or "negative". Solder mask is usually given as a negative layer, meaning the gerber shows where material should be removed rather than where it should be retained.
The reason is to reduce the size of the file. You can see it's a lot less work to say... | https://electronics.stackexchange.com |
97,394 | [
"https://mathoverflow.net/questions/97394",
"https://mathoverflow.net",
"https://mathoverflow.net/users/22907/"
] | Let $x_1,x_2,...,x_k$ be irrational number,is it always true that:
$\liminf_{n\rightarrow\infty} \sum_{i=1}^k (nx_i)=0$ (where $(x)$ denotes the fractional part of $x$)
If not,what are the necessary and sufficient conditions that {$x_i$} must satisfy so that $\liminf_{n\rightarrow\infty} \sum_{i=1}^k (nx_i)=0$
| Here is a counterexample: $x_1 = \sqrt2, x_2 = 1-\sqrt2$.
For $n>0$, $(nx_1) + (n x_2) =1$ so the limit infimum is $1$, not $0$.
I think the other answers assumed that you meant $x-[x]$, where $[x]$ is the nearest integer to $x$, instead of $(x)$.
If there is no rational dependency, then the multiples are dense i... | If you take any vector in Euclidean space, and look at its integer multiples modulo the integer lattice, we know what happens qualitatively. They are dense in a certain subtorus of the obvious torus. That's a version of Kronecker's theorem. Your question is about being near the origin infinitely often. So I say it's tr... | https://mathoverflow.net |
3,798,475 | [
"https://math.stackexchange.com/questions/3798475",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/726524/"
] | In my textbook for discrete mathematics the following is stated:
<blockquote>
<strong>Theorem</strong>: <span class="math-container">$\mathbb Z_p$</span> is a field if and only if <span class="math-container">$p$</span> is prime.
In the following we denote the field with <span class="math-container">$p$</span> elements... | Yes, there are finite fields other than <span class="math-container">$\Bbb Z_p$</span>. The cardinal of such a field is always the power of a prime number. And, yes, there is a field with <span class="math-container">$4$</span> elements. It can be defined as <span class="math-container">$\Bbb Z_2[x]/\langle x^2+x+1\ran... | Any finite field must be order of <span class="math-container">$p^{n}$</span> , where <span class="math-container">$p$</span> prime and <span class="math-container">$n$</span> is some non negative integer.
For order <span class="math-container">$p^{n}$</span> , just take Algebraic extension of the field <span class="ma... | https://math.stackexchange.com |
99,019 | [
"https://cs.stackexchange.com/questions/99019",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/95452/"
] | My question is that can we say that runtime of the binary search is <span class="math-container">$\Omega(\log n)$</span>?
I know it is both <span class="math-container">$\Omega(1)$</span> and <span class="math-container">$O(1)$</span> for the best case, and <span class="math-container">$\Omega(\log n)$</span> and <sp... | To prove the lowerbound of <span class="math-container">$\Omega(\mathrm{log}n)$</span>, it is best to use an adversary argument.
For each <span class="math-container">$n$</span>, you can fix some pattern of access when running binary search, i.e. a sequence of <span class="math-container">$Left-Half$</span>, <span cla... | First, you can analyze the time complexity of binary search in whatever case you wish, say "best case" and "worst case". In the best case, you use <span class="math-container">$f(n)$</span> time, while in the worst case you use <span class="math-container">$g(n)$</span> time. These are two functions of <span class="mat... | https://cs.stackexchange.com |
57,584 | [
"https://physics.stackexchange.com/questions/57584",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/20263/"
] | Is this a correct definition of electrostatic induction, The <strong>production</strong> of opposite charges on a neutral body when a charged body is brought near to it? I think it's not the production but redistribution.
| Charge is conserved, so it is never "produced". You are correct that it is redistributed in the scenario you describe, which is sometimes called electrostatic induction, not to be confused with electromagnetic induction.
| Induction is the process of appearance of opposite charge on the surface of a neutral conductor,when a charged body is brought near it.
| https://physics.stackexchange.com |
313,021 | [
"https://softwareengineering.stackexchange.com/questions/313021",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/220846/"
] | I am working as SAP developer, where in many cases you have traditional requirements to applications (reports):
<ul>
<li>Read in some data from the database or a file</li>
<li>Do some magic with that data, e.g. do various calculations</li>
<li>Write out the results into the database or a file again</li>
</ul>
This is... | I think your question goes beyond some magical explanation as to why OO is better. In fact, that question has already been asked on this site.
You made a management mistake. You can't take a complex subject, do some minimal training with not follow-up or management buy-in and think you're going to get people to change... | I'm sure this isn't what you want to hear, but <em>you can't</em>. The best you can do is to write your code in an OO fashion (when it's appropriate to) and refactor their code into something OO (again, when appropriate). Some of them will see you moving more quickly and become interested; most won't. Write the best co... | https://softwareengineering.stackexchange.com |
502,024 | [
"https://math.stackexchange.com/questions/502024",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/96301/"
] | <blockquote>
$$\lim_{x \to 0} \frac{2}{x^2}.$$
</blockquote>
If we apply L'Hopital rule, Then the procedure would go: $0/2x$, and then $0/2$ which is zero. What is wrong with this application L'Hopital rule, as it clearly seems wrong..
| In order to use the $0/0$ case of L'Hospital's rule, we require that both the numerator <em>and</em> the denominator tend to $0$ at the appropriate point. The numerator does not tend to $0$.
| L'Hopital's rule states that
if $$\lim_{x \to C} f(x) = 0$$
and $$\lim_{x \to C} g(x) = 0$$
then
$$\lim_{x \to C} \dfrac{f(x)}{g(x)} = \lim_{x \to C} \dfrac{f'(x)}{g'(x)}$$
Another form of the rule is:
if $$\lim_{x \to C} f(x) = \infty$$
and $$\lim_{x \to C} g(x) = \infty$$
then
$$\lim_{x \to C} \dfrac{f(x)}{g(x)} =... | https://math.stackexchange.com |
181,525 | [
"https://security.stackexchange.com/questions/181525",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/172905/"
] | I have implemented PBKDF2 authentication for some web services.
The client is given the following information, so they can duplicate the PBKDF2 function:
<ul>
<li>the hashing algorithm (SHA256)</li>
<li>a secret password that is 30 bytes long</li>
<li>the number of iterations (4096)</li>
<li>the hash length (32)</li>... | The point of using salt is to prevent brute force attacks against multiple passwords at once. If they all have different salts your dictionary or rainbow table is only valid for one specific salt.
So, indeed using different salts for the same password is not helpful.
N.B.: If somebody can break TLS connections they w... | Honestly? There is no need, its just extra processing overhead. If you create a unique salt, you can use it for all of your communications and the only way to break it would be the attacker got your salt, then used the salt to attack your password. If you used a complex password, little to 0 chance anything will happen... | https://security.stackexchange.com |
595,434 | [
"https://math.stackexchange.com/questions/595434",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/86863/"
] | I am trying to derive the formula for the variance of a geometric distribution and am stuck at the following problem:
I need to find the sum to infinity for the following series:
$1+3(1-p)+5(1-p)^2+7(1-p)^3+\dots$ where $p$ where $p$ is a constant $0<p<1$
The series looks like the sum of the odd numbers and th... | Write $q = \sqrt{1-p}$. Then your series becomes
$$1\cdot q^0 + 3\cdot q^2 + 5\cdot q^4 + 7\cdot q^6 +\dotsb = \sum_{k=0}^\infty (2k+1)\cdot q^{2k}.$$
The latter is easily recognised as
$$\frac{d}{dq} \sum_{k=0}^\infty q^{2k+1} = \frac{d}{dq} q\sum_{k=0}^\infty (q^2)^k = \frac{d}{dq} \frac{q}{1-q^2} = \frac{(1-q^2) ... | <strong>Sketch:</strong>
Let $T=\sum_{k=0}^\infty (2k+1)q^k$ be the series you want to evaluate ($q=1-p$) and $S=\sum_{k=0}^\infty 2kq^k$. We have
$$
T=S+\sum_{k=0}^\infty q^k,
$$
where we can calculate the second series with the formula for geometric series. So you can figure out $T$ by calculating $S$. Now
$$
S=2\s... | https://math.stackexchange.com |
45,287 | [
"https://cs.stackexchange.com/questions/45287",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/15063/"
] | I just read in <em>Effective Java</em> about the <code>hashCode</code> method:
<blockquote>
<ol>
<li>Store some constant nonzero value, say, 17, in an int variable called result.</li>
<li>For each significant field f in your object (each field taken into account by the equals method, that is), do the following:
... | If the hashed codes are $x_1,\ldots,x_n$ (in that order), then the resulting hash value is
$$
x_n + 31 x_{n-1} + 31^2 x_{n-2} + \cdots + 31^{n-1} x_1 + 31^n \cdot 17 \pmod{2^{32}},
$$
assuming <code>int</code> is 32 bits, and ignoring signed/unsigned distinctions. In view of this, the integers 17 and 31 should satisfy ... | In abstract terms, we are looking at a mapping from triples of numbers to integers. If you unfold what they do there, you get
$\qquad\displaystyle f(a,b,c) = 17 \cdot 31^3 + a \cdot 31^2 + b \cdot 31 + c$.
The idea is to encode everything into one number in a <em>surjective</em> way, that is no two triples map to the... | https://cs.stackexchange.com |
2,555,182 | [
"https://math.stackexchange.com/questions/2555182",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/458544/"
] | I am new to complex number system and I want the value of $ (1+\omega-\omega^2)^7 $ . Where $ \omega $ is cube root of unity.
(Cube roots of unity: $ 1, e^{±i\frac{2\pi}{3}} $ )
I want to simply get the value of the expression in easy way.
| Note that $1+\omega+\omega^2=0$ when $\omega$ is a cube root of unity $\neq 1$. Using this fact, we have, $$(1+\omega-\omega^2)^7=((1+\omega+\omega^2)-2\omega^2)^7=(-2\omega^2)^7=-128\omega^{14}=-128(\omega^3)^4\omega^2=-128\omega^2$$
| 1+$\omega$ +$\omega^2$ = 0 (this holds since $\omega^3$=1 => $\omega^3$-1=0 => ($\omega$ -1)(1+$\omega$ +$\omega^2$)=0
and here $\omega$ $\neq$ 1 )
This gives:
1+$\omega$ =-$\omega^2$
Substitute it in $(1+ω−\omega^2)^7$,
you get
$-2\times(\omega^2)^7$= $(-2)^7$*$\omega^14$=$-128\times\omega^2$
where ω is cube roo... | https://math.stackexchange.com |
314,803 | [
"https://physics.stackexchange.com/questions/314803",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/121312/"
] | <blockquote>
A particle is projected up an inclined plane of base angle $\beta$ with the horizontal with an initial velocity $V$. The particle collides elastically with the incline and rebounds vertically. If the particle reaches back the point of projection after time<br>
$\large T = \dfrac {aV}{g \sqrt{1+b\sin^2\... | when we drive , the frictional force between the tires and the road increases the temperature of the air inside the tire.
According to Gay-Lussac's Law,
p ∝ T ; [When volume is constant]
so the increase in temperature also increases tire pressure.
| The tyre does heat up, <strong>but not because of friction:</strong> the tyre does not slide along the road and the friction between tyre and road therefore does no work.
What does happen is that the tyre is not perfectly circular. It is squashed by the weight of the vehicle. As the vehicle moves, the part of the tyre... | https://physics.stackexchange.com |
2,525 | [
"https://mathoverflow.net/questions/2525",
"https://mathoverflow.net",
"https://mathoverflow.net/users/290/"
] | I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:
<ul>
<li>The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.</li>
<li>The Krull dimension, based on chains of prime ideals.</li>
<li>The transcendence degree of the fraction field of... | In a non-commutative ring, you need to be careful with what you even mean by a prime ideal, and usually there are very few two-sided ideals you might call prime. Oh, and even in the cases when there is a nice ring of fractions, it won't be a field, and so transcedence degree is still bad.
My personal favorite notion ... | In non-commutative algebra Krull dimension has been generalised by Gabriel & Rentschler. A decent account of it can be found in Chapter 6 of McConnell and Robson's book on non-commutative Noetherian rings.
The basic idea is as follows: An artinian module has Krull dimension 0.
A module that does not have Krull ... | https://mathoverflow.net |
2,011 | [
"https://cstheory.stackexchange.com/questions/2011",
"https://cstheory.stackexchange.com",
"https://cstheory.stackexchange.com/users/434/"
] | Suppose $Z_G(J,h)$ is a partition function of Ising model with coupling $J$ and magnetic field $h$ on graph $G$. What is the complexity of finding the gradient of Z at $\mathbf{0}$?
Specifically, if $n$ is the number of vertices, $E$ is a set of edges, $\mathcal{X}=\{1,-1\}^n$, the gradient is of the following functio... | The key difference is that you have a field tracking the size of the list. This field can be accessed in constant time. The field must be updated for every addition or removal. If you compute the size of the list by counting the elements, then it will be linear.
| It is common to implement lists using arrays for obvious reasons (random access). If you have an implementation that extends the array by a constant number of fields $k$ every time it is full, you can achieve constant time for figuring out the number of elements without having additional information stored.
Execute a ... | https://cstheory.stackexchange.com |
274,652 | [
"https://electronics.stackexchange.com/questions/274652",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/120256/"
] | I am learning about headphones and microphones and I have 2 questions.
I bought a few pairs of headphones from the dollar store and took one pair apart. There is a weak little magnet in the center with very thin insulated wire coiled around it and there is a very delicate plastic sheet covering this little assembly. N... | In most cases, headphone transducers will also work as dynamic microphone transducers. However, they put out a VERY SMALL amount of signal which you will never see on a regular meter. No it cannot be "amplified" by simply applying a DC voltage (from a battery or whatever). You need a high-gain AC amplifier. Specificall... | Plug the headphones into the 'line in' socket on a PC. Use the sound-in application to select that input, and to use as high a gain as possible for that input.
Use an application like Audacity (free, cross platform), to record, view, and further amplify the sounds.
| https://electronics.stackexchange.com |
157,326 | [
"https://stats.stackexchange.com/questions/157326",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/79971/"
] | I have a rating system which can take 5 possible outcomes, which are coded to:
<pre><code>0 : No problem
1 : Minor problem
2 : Mild problem
3 : Moderate problem
4 : Severe problem
</code></pre>
This scale relates to severity of individual's medical conditions and I have 12 separate items (each with the same 5 possibl... | As you point out, these weights are too close from each others. More than a change of analysis, I would suggest to keep the sum (or mean) but change the weights.
I would suggest three ways depending to the degree of time you dispose and rigour you want :
<ul>
<li>Simplest one : use weights that makes more sense to ... | A simple match solution is to apply exponentiation to your values (with two, or three, or more - even not integer - as exponent), and then go for sum or mean. This leads to an 'auto' higher weight for mild to severe problems.
$$ v = \frac{\sum_1^n x^a}{ n} , \quad a> 1$$
With your example values ($n=12$), $a=3$ ex... | https://stats.stackexchange.com |
4,104 | [
"https://mathoverflow.net/questions/4104",
"https://mathoverflow.net",
"https://mathoverflow.net/users/828/"
] | By the GAGA principle we know that a holomorphic vector bundle E->X is analitically isomorphic to an algebraic one, say F->X, and by definition F is locally trivial in the Zariski topology. But since the isomorphism between E and F is analytic, I fail to see if this implies that E is Zariski locally trivial too.
I hop... | You get (analytic) trivializations of E over Zariski-open sets just by composing a trivialization of E with the isomorphism between E and F. Of course, you do not get algebraic trivializations, but for this you would need an algebraic structure on E in the first place.
| It's probably worth noting that etale-locally trivial principal GL(n)-bundles are automatically Zariski-locally trivial. This isn't necessarily true for general G.
| https://mathoverflow.net |
392,238 | [
"https://softwareengineering.stackexchange.com/questions/392238",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/327693/"
] | I have a <code>User</code> entity, which <em>may</em> have a <code>RefreshToken</code> (for authentication).
Notes:
<ul>
<li>A refresh token doesn't have "identity", but is related to a single user - it is only valid for that user. In the db that means a foreign key to users table. In Entity Framework I can however m... | The refresh token is solely defined by its values. And there is no continuity of the object when the values change, because it would be replaced by a new token. This is why it is a value object.
This does not prevent that one of the value refers to a given user-id, which gives the impression of continuity between succ... | I don't consider a RefreshToken being a part of the domain model in the first place: It is purely application specific (authentication / authorization of the application's user).
Having said that, if it <em>were</em> part of the domain model, its name "token" implies, that it does have an identity (which is a... | https://softwareengineering.stackexchange.com |
172,930 | [
"https://mathoverflow.net/questions/172930",
"https://mathoverflow.net",
"https://mathoverflow.net/users/38075/"
] | Suppose I have a dg algebra $(A,d)$ and a chain complex $M^\bullet$ of semi-free $(A,d)$ modules. I am hoping it is true that $ Tot^\coprod (M^\bullet)$ is again a semi-free $(A,d)$ module. Is this so? And if yes, what filtration should one choose?
The definition of semi-free I am using is: $X$ is semi-free if $X$ is ... | If $M_\bullet$ is not bounded above, then the totalization may not be semi-free.
For example, let $k$ be a field, and $A=k[x]/(x^2)$, considered as a DG algebra concentrated in degree zero, with trivial differential. Then the chain complex
$$\dots\to A\to A\to A\to A\to\dots,$$
where all the differentials are multipli... | So here is a partial answer, If $M_\bullet$ chain complex of dg $A$ modules then $\text{Tot}^\coprod M$ can be realized as a dg $A$ module whose underlying graded module is given by
$ \coprod\limits_{i \in \mathbb{Z}} [-i] M_i $
and whose differential is
$d = d_{[-i]M_i} + d^i_{M_\bullet}$
Now suppose that $M_\bu... | https://mathoverflow.net |
224,822 | [
"https://physics.stackexchange.com/questions/224822",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/40266/"
] | When we solve the Schrodinger equation for the time-evolution operator:
\begin{equation}
i\hbar\frac{\partial}{\partial t}U(t,t_{0})=HU(t,t_{0}),
\end{equation}
We have three cases to be treated separately:
Case 1. The Hamiltonian operator $H$ is independent of time:
\begin{equation}
U(t,t_{0})=\exp\left[\frac{-iH... | You can prove this without looking at any of the specific cases by doing a first-order perturbation of the differential equation that defines the time-evolution operator.
We start with
$$
i \hbar \frac{\partial}{\partial t} U(t, t_0) = H U(t, t_0).
$$
Or, rearranging some terms,
$$
\frac{\partial}{\partial t} U(t, ... | Hermiticity of $H$ for all three cases combined together can be shown directly from the Schrödinger equation. To do this, first take the derivative of the unitarity relation $U(t,t_{0}) U^{\dagger}(t,t_{0}) = I$ with respect to $t$, which gives
\begin{equation}
U(t,t_{0}) \left[\frac{\partial}{\partial t}U^{\dagger}(t,... | https://physics.stackexchange.com |
176,366 | [
"https://security.stackexchange.com/questions/176366",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/18521/"
] | I'm trying to get a better understanding of authorized_keys in the context of an SSH server.
So if I have a linux server and I've placed a clients public SSH key into my servers authorized_keys file, that client may now login to the server. Presumably, this is the public key from a specific user on that client (new k... | The authorized_keys file is not a per-server thing - it's a per-user-on-server thing.
If you want someone to access your server as "user1", you place their public key in <code>~user1/.ssh/authorized_keys</code>. Now they can login as user1, and only user1.
If you want them to be able to login as other users, you need... | The authorized_keys file of an user account includes the public key of the client <strong>user</strong> which is allowed to login on this account and not the host key of the client host. This means a user could use the same key pair from multiple systems to login into a specific account. Also, multiple users on the sam... | https://security.stackexchange.com |
204,234 | [
"https://electronics.stackexchange.com/questions/204234",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/93329/"
] | My question arises from interest mainly in Outrunner motors which are used on RC planes/drones.
With that being said, I understand that these motors are controlled by an Electronic Speed Controller (ESC), which is used to switch DC power into two of the three different phases of the motor at a specific time by monito... | Brushless DC means exactly that - a DC motor with brushless commutation. The controller's job is to switch the phases in and out at the correct rotor <em>angles</em> (not frequency) just like the commutator in a brushed motor. The motor spins at whatever speed it wants to, depending on supply voltage and load.
If the... | First, as somebody else pointed out, ESC stands for Electronic Speed Control, not Engine Speed Control. And BLDC motors do not experience any slip. They are synchronous motors.
Second, the quote you gave is confused. BLDC motors don't have a "DC" rotor. They have a permanent magnet rotor. "DC rotor" would only m... | https://electronics.stackexchange.com |
640,593 | [
"https://physics.stackexchange.com/questions/640593",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/209185/"
] | If I spin a magnet really fast, and put my finger near it, i get no electric shock.
However if i place a wire near that wire it will induce an emf across the wire, if i touch it, will i get a shock?
aka, does the introduction of the wire make it easier for me to get a shock? my intuition says yes, but when i actually t... | The answer is current actually does flow and charges do accumulate at ends. That's why arcs form in open circuits at high voltage. And as you mentioned the flow would be a momentary one and not continuous.
When we connect a battery to an open wire, the potential of the battery does push charges around. But as soon as t... | Current (or more properly, <em>conduction</em> current) can flow into an open wire and charge accumulate at the end IF there is <em>displacement</em> current to continue the circuit. Displacement current is not the flow of charge, but a change in the electric field. Two situations where this might happen is in an anten... | https://physics.stackexchange.com |
136,684 | [
"https://cs.stackexchange.com/questions/136684",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/92702/"
] | Solving <span class="math-container">$T(n) = 2T(\frac{n}{2}) + n\log(n)$</span> without master theorem, given <span class="math-container">$T(1) = 1$</span>
My approach with recurrence tree:
<span class="math-container">$n \sim n\log(n)$</span>
<span class="math-container">$\frac{n}{2} \sim 2 \frac{n}{2}\log(\frac{n}{2... | Let <span class="math-container">$a$</span> is the first element of <span class="math-container">$A$</span>. Let <span class="math-container">$A$</span> is composed of array <span class="math-container">$B = (b_{1},\dotsc,b_{n})$</span> and array <span class="math-container">$C = (c_{1},\dotsc,c_{n})$</span>.
Now, we w... | Here is the natural intuitive way to sort the array. The idea is to keep track of <em>two</em> inserting positions.
<ol start="0">
<li>(Data structure) A node contains a value, a pointer to its previous node, which can be null, and a pointer to its next node, which can be null.
</li>
<li>Initialize a two-way linked lis... | https://cs.stackexchange.com |
423,584 | [
"https://electronics.stackexchange.com/questions/423584",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/5778/"
] | In an extreme low power project (MCU+BLE SIP running on a CR2016 battery for year(s)) I would like to measure the power consumption under different environments/algorithms.
What would be the correct way or setup to measure micro (nano ??) watts? in a short duration and over a long period of time?
The input voltage ca... | Given R3, R4 & C1 that diode is doing mostly diddly squat.
Generally you add a diode in parallel with the driving resistor at a gate of a Mosfet to accelerate turn-on or turn-off by providing more drive to the Miller-multiplied Drain-Gate capacitance.
In this particular case the diode seems to be there to bypas... | That diode will make the FETs turn on faster and turn off more slowly. It's probably done to achieve some advantageous compromise between switching losses and reduced RF interference.
| https://electronics.stackexchange.com |
207,640 | [
"https://electronics.stackexchange.com/questions/207640",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/-1/"
] | I'm new in Verilog and I need to create a module that when Entry=1, it will increment 5'b000100 in the "Money" output.
This is the code that I made, but it's not working properly. The program it's not storing the total of the money(that's what I want my program to do).
For example,
Everytime a person enters a footba... | <pre><code> always@(Entry)
begin
Money = 6'b000000;
if (Entry)
Money = Money + 6'b000100;
end
</code></pre>
Whenever the Entry input changes, you set <code>Money</code> to 0, then maybe add 4.
So <code>Entry</code> will alw... | @ThePhoton's answer is correct as to one minor issue with your code that essentially makes the always block infer a multiplexer with 0 as one input and 4 as another input (with Entry being the select signal).
However, the think to remember is FPGA's like synchronous logic. Your code as it is written is completely asyn... | https://electronics.stackexchange.com |
40,890 | [
"https://cs.stackexchange.com/questions/40890",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/29555/"
] | You have two friends, call them A and B. They each are given two wine bottles: one bottle holds <code>k_1</code> litres and the other <code>k_2</code> litres. Both A and B can perform these operations:
- completely fill a wine bottle with wine
- completely empty a wine bottle
- pour the wine of one bottle into the oth... | Here's an efficient method that doesn't involve constructing a graph of the states and searching for a path to each target value. Let the capacities of the bottles be integers $K_1<K_2$, respectively, and denote by $(a_1, a_2)$ the current amounts in the bottles. You have six basic operations you can perform, for $i... | The idea is to construct a directed graph whose vertices are labelled by pairs of integers $(a_1,a_2)$, where $0 \leq a_i \leq k_i$ (I'm assuming that $k_1,k_2$ are integers). There is also an additional vertex $t$. The vertex $(a_1,a_2)$ represents the fact that bottle $i$ holds $a_i$ litres. Connect $(a_1,a_2)$ to $(... | https://cs.stackexchange.com |
212,929 | [
"https://mathoverflow.net/questions/212929",
"https://mathoverflow.net",
"https://mathoverflow.net/users/-1/"
] | For sake of simplicity, let's say that we live in $G = SL(2, \mathbb{C})$. Every conjugacy class of diagonalizable matrices $$[A] := \{gAg^{-1} \mid g \in G\}$$ can be identified with its set of eigenvalues $\{\lambda, \lambda^{-1}\}$. As such, the collection of diagonalizable matrices in $SL(2, \mathbb{C})$ can be ide... | You have a geometric invariant theory question here: your space is the space of the pairs of matrices, and the action is given by conjugation. One possible way to deal with this is to study the ring of invariant functions. In case of one matrix it really boils down to the ring generated by the coefficients of the chara... | It is fairly easy to do this directly. I consider the case that both $A$ and $B$ are individually diagonalizable, and not scalar. Note also that ${\rm SL}(2,\mathbb{C})$-conjugacy is equivalent to ${\rm GL}(2,\mathbb{C})$-conjugacy, since every matrix in ${\rm GL}(2,\mathbb{C})$ has a scalar multiple in ${\rm SL}(2,\ma... | https://mathoverflow.net |
422,559 | [
"https://physics.stackexchange.com/questions/422559",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/190972/"
] | Consider a massless, frictionless piston fit into an airtight container containing an ideal gas. And let us say that the gas undergoes quasi-static isothermal expansion by lowering the pressure applied by the piston slowly.
The piston moves a distance, say $dr$ in an instant in which the force applied on it by the gas ... | Work done is defined as $dW = \vec{F}.d\vec{r}$. Force on the piston is in direction of it's displacement, so according to definition of work done it will be positive. On the other hand, force on gas is opposite to the displacement of gas, so work done on gas comes out to be negative due to the $cos(\theta)$ term.
For... | <blockquote>
But for the corresponding work done to be negative of one another, the each gas particle must be displaced in the opposite direction with same direction dr, which is not necessarily true.
</blockquote>
I think I see where you're coming from. The piston is gaining energy (i.e. work done <em>on</em> the p... | https://physics.stackexchange.com |
366,078 | [
"https://physics.stackexchange.com/questions/366078",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/150539/"
] | It has been written in Goldstein's Classical Mechanics that in Lagrangian formalism, the independent dynamical variables are $q$ and $t$. That's why we represent state of a system in Lagrangian formalism using a point in configuration space. But throughout the calculations we treat $\dot{q}$ also as an independent vari... | We do not treat <span class="math-container">$\dot q$</span> as an independent variable in the derivation of the Euler-Lagrange equations. The rough answer is that <span class="math-container">$q$</span> and <span class="math-container">$\dot q$</span> are independent as inputs to the Lagrangian, but become linked onc... | The general Lagrangian formalism is developed in a manifold $j^1(E)$ with the structure of a <strong>jet bundle</strong> constructed out of a <strong>fiber bundle</strong> $E \to \mathbb R$.
In other words $E$ is locally the product of $Q$ and $\mathbb R$, where $Q$ is a manifold where configurations of the system ... | https://physics.stackexchange.com |
154,834 | [
"https://physics.stackexchange.com/questions/154834",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/37677/"
] | In classical mechanics, the usual formula to translate the evolution of a quantity as seen from an <em>inertial</em> frame of reference to a <em>rotational</em> frame is: $$\frac{d \textbf{A} }{dt} \vert_{Inertial} = \frac{d \textbf{A} }{dt} \vert_{Rotational} + \boldsymbol{\omega}\times\textbf{A};$$
How do we make th... | The best way to describe a rotating reference system is via a simple change of coordinates. If you have a description of a phenomenon in some set of inertial coordinates <span class="math-container">$\{t, x, y, z \}$</span>, then you can obtain a description of its motion in a rotating reference frame with coordinates... | The best start is to pull out a physics book and look at the various expressions for the wave equation. The simplest, and foundational for the invention of special relativity, is the form with no coefficients for the differentials. Now find the wave equation expressed in terms of spherical coordinates. We have coeff... | https://physics.stackexchange.com |
455,103 | [
"https://stats.stackexchange.com/questions/455103",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/276605/"
] | The assumption is that
<span class="math-container">$Y_1,...,Y_n$</span> are independent and <span class="math-container">$Y_i \sim N(\mu,\sigma^2)$</span>. The unknown parameters are <span class="math-container">$(\mu,\sigma^2) \in \mathbb R \times(0,\infty) $</span>.
Question: Why can't variance be 0? meaning: <spa... | You say in your question that the distributional form is an assumption (and therefore so is the allowable set of parameter values), so it is really up to you what assumption you wish to make. It is possible to extend the family of normal distributions to include the case where <span class="math-container">$\sigma=0$</... | In normal distribution
<span class="math-container">$$E(Y_i -\mu)^2=\sigma^2$$</span>
so if <span class="math-container">$\sigma^2\rightarrow 0^+$</span> so <span class="math-container">$$E(Y_i -\mu)^2\rightarrow 0^+$$</span> in hence
<span class="math-container">$$P(Y_i=\mu) \rightarrow 1$$</span> so <span class=... | https://stats.stackexchange.com |
48,968 | [
"https://mathoverflow.net/questions/48968",
"https://mathoverflow.net",
"https://mathoverflow.net/users/45/"
] | This is a simple terminology question: I want to know if the involution $z \mapsto z^{-1}$ on Laurent polynomials (over some ring, I happen to be working over $\mathbb{Z}$ but that's not important) has a special name.
My motivation is perhaps a little unusual for this site. I'm doing some computations that involve ma... | I would call it the antipode. If your base ring is commutative, then the Laurent polynomials are the coordinate ring of the multiplicative group, and the antipode gives you the inversion on the group scheme.
| I think the answer to the original question is that there is no special name for the involution (otherwise it would have turned up by now). My first encounter with it was in the 1979 Kazhdan-Lusztig paper on Hecke algebras, where they use a
bar notation and combine this involution on Laurent polynomials with the inve... | https://mathoverflow.net |
178,193 | [
"https://dba.stackexchange.com/questions/178193",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/128169/"
] | Consider the following table structure:
<pre><code>UserGroup(ie. 'Admin')
ID = 'Admin'
ApplicationFK = 'App1'
JoinTable
UserGroupFK = 'Admin'
PermissionFK = 'Approve'
PermissionValue = 'READ'
Permission(ie. 'Approve')
ID = 'Approve'
ApplicationFK = 'App2'
UserPermission
UserFK = 'Joh... | Making sure I understand/have correctly elaborated on @ypercubeᵀᴹ's comment/linked answer:
The correct structure should be:
<pre><code>UserGroup(ie. 'Admin')
ID = 'Admin'
ApplicationFK = 'App1'
[PK: ID, ApplicationFK]
JoinTable
UserGroupFK = 'Admin'
ApplicationFK = 'App1'
PermissionFK = 'Appr... | The question will be solved after you define primary keys correctly.
Please remember that FK must point to other table's primary key (or unique key).
It is legal (at least in SQL) to use multi-field FKs.
| https://dba.stackexchange.com |
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