qid int64 1 4.65M | metadata listlengths 3 3 | prompt stringlengths 31 25.8k | chosen stringlengths 17 28.2k | rejected stringlengths 19 40.5k | domain stringclasses 28
values |
|---|---|---|---|---|---|
92,506 | [
"https://mathoverflow.net/questions/92506",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4069/"
] | Let $V$ be a Zariski-closed subset of $\mathbb{A}^n_k$, where $k$ is an algebraically closed field. Assume that $V$ may be defined by polynomials of degree at most $d$ (or to put it otherwise $V$ is an intersection of hypersurfaces of degree at most $d$). My question is the following: is it also possible to define the ... | Here's a counterexample with $n=d=3$.<br>
Let $C$ be the rational curve $\lbrace (x,y,z) = (t,t^4,t^6) \rbrace$.
Then the space $S$ of cubics that vanish on $C$ is the span of
$\lbrace x^2 y - z, x^2 z - y^2, y^3 - z^2 \rbrace$.
But all such cubics vanish also on the line $y=z=0$. Therefore
we can take $V$ to be the z... | Edit: the first version of this was completely wrong, I hope this one works.
Take a line $L$ in $\mathbb P^3$, and two general surfaces of degree $d > 3$ passing through $L$; it is not hard to see that the only line contained in $S_1$ is $L$. Their intersection is the union of $L$ with an irreducible curve $C$ of d... | https://mathoverflow.net |
62,729 | [
"https://mathoverflow.net/questions/62729",
"https://mathoverflow.net",
"https://mathoverflow.net/users/14621/"
] | Let $A$ be a C*-algebra and let $\alpha$ be an action of the circle group $S_1$ on $A$ (Gauge action).
We define the following map:
$$E:A\rightarrow A;\quad E(a):=\int\alpha_t(a)\textrm{d} t.$$
My question is why $E$ is a conditional expectation into the fixed point algebra for the gauge action?
For example if we take... | For your first question, you can check that $E(a)$ is invariant with respect to the action, by using that the Haar measure is left-invariant. The other properties are also straightforward to check.
With respect to your second question, note that $\alpha_t(S_j^*) = e^{-it} S_j^*$. Consider an element $S_{i_1} \cdots S_... | Let $G$ be a compact group with Haar measure $\mu$ normalised such that $\mu (G)=1$. Let $A$ be a C*-algebra and $\alpha : G\rightarrow \text{Aut} (A)$ a continuous action of $G$ on $A$. Define
$$
E(x)=\int_{G}\alpha_{t}(x)d\mu(t).
$$
We claim that this is a faithful conditional expectation of $A$ onto the fixed-point ... | https://mathoverflow.net |
338,337 | [
"https://softwareengineering.stackexchange.com/questions/338337",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/37922/"
] | I'm building a RESTful API that uses JWT tokens for user authentication (issued by a <code>login</code> endpoint and sent in all headers afterwards), and the tokens need to be refreshed after a fixed amount of time (invoking a <code>renew</code> endpoint, which returns a renewed token).
It's possible that an user's AP... | This sounds like a case of <em>authentication</em> versus <em>authorization</em>.
JWTs are cryptographically signed claims about the originator of a request. A JWT might contain claims like "This request is for user X" and "User X has an administrator roles". Obtaining and providing this proof through passwords, signa... | Your API session is a thing which should not exist in a RESTful world at all. RESTful operations are supposed to be stateless, session contains state and thus has no place in a RESTful world.
The JWT should be your only way to determine whether a user is still eligible to access an endpoint or not. A session should pl... | https://softwareengineering.stackexchange.com |
1,968,416 | [
"https://math.stackexchange.com/questions/1968416",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/378610/"
] | What is the correspondence between the number of significant figures in base 10 and base 2?
Or in other words, if two decimal numbers are equal to $N_{10}$ significant figures, what is the number of significant figures, $N_{2}$, to which I need to compare them in binary, so that I get the same result?
Example: Let $... | For fixed $N_{10}$, no such $N_2$ exists.
<strong>Proof:</strong><br>
Given a tentative $N_2$, set $n = 2^{N_2} + 1$ (two $1$s with $(N_2 - 1)$ $0$s between). Then in base $2$, $n\cdot 2^k$ always rounds up to $(n+1)\cdot 2^k$. However $n\cdot 2^k - 1$ rounds down to $(n-1)\cdot 2^k$. Since $n\cdot 2^k$ and $n\cdot 2^... | Significant figures are a rough way of assessing the relative accuracy of a number. $98$ and $11$ each have two significant figures, but if you assign a maximum absolute error of $\pm 0.5$ to each, the relative error is about $0.5\%$ in the first case and $4.5\%$ in the other. A factor of $3$ is in line with the roug... | https://math.stackexchange.com |
249,501 | [
"https://mathoverflow.net/questions/249501",
"https://mathoverflow.net",
"https://mathoverflow.net/users/34538/"
] | Let $B_n$ be the boolean lattice of rank $n$. Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum, respectively.<br>
We identify the notion of edge with the notion of interval $[a,b]$ of cardinal $2$.<br>
We propose to label every edge with the symbols $\alpha$ or $\beta$, such that:
<ol>
<li>For every maxima... | Yes. For $n=1$, the result is clear as we must choose $\alpha$ for the one and only edge. For $n>1$, by the second condition there must be an atom $a \in B_n$ such that $[\hat 0, a]$ is not labeled with $\alpha$. Now for the first condition to hold every maximal chain in $[a, \hat 1] \cong B_{n-1}$ must have exactly... | John Machacek's elegant answer solves the question, but we can also answer the question by characterizing all the labelings. For each labeling $\lambda$, consider a labeling $\pi$ of elements in $B_n$ such that $\pi(a)$ is the number of edges with label $\alpha$ on each chain from $\hat 0$ to $a$, for every $a\in B_n$.... | https://mathoverflow.net |
199,565 | [
"https://softwareengineering.stackexchange.com/questions/199565",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/66745/"
] | I want to learn C but I tend to learn best when I have a project to work on. I've never done iPhone development, so I'm hoping to kill two birds with one stone. Will learning objective c also teach me to program c reasonably well, or are they too dissimilar?
EDIT: I'm mostly wondering if objective-c/iphone-development... | Based on your edit, I would say that Obj-C is not the best way to go.
Obj-C has a lot of nice things as part of the framework that make it a higher level language, which will mask the very gotchas that you want to experience in C. In particular, memory management in Obj-C is a big topic in its own right, and it is ver... | <code>90%</code> of what you'll learn doing a project in object-c will be related to Apple. As <code>90%</code> would be to Android if you learn Google's SDK using Java.
That makes it difficult to gain experience from the remaining <code>10%</code>.
The reverse way to look at is a <code>C</code> programmer who's neve... | https://softwareengineering.stackexchange.com |
673,559 | [
"https://physics.stackexchange.com/questions/673559",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/317469/"
] | I cannot understand that how can the frictional force applied a box on a conveyor belt be in the same direction of motion? How is frictional force responsible for the forward movement of a box on conveyor belt?
| Revised answer. Sorry I misread your question earlier.
You wish to estimate the mean and standard deviation of the mean for the time of an oscillation. You chose to take a random sample of 5 oscillation times, and you have 50 random samples, each of size 5. From each sample of 5 you should evaluate the oscillation t... | The standard deviation is only defined for multiple measurements.
If we have only done 1 measurement, we cannot define a standard deviation for this single measurement. Using the formula will give us standard deviaion = 0.
So we have to use our multiple measurements (5 in this case) in order to find the error estimate ... | https://physics.stackexchange.com |
82,438 | [
"https://mathoverflow.net/questions/82438",
"https://mathoverflow.net",
"https://mathoverflow.net/users/-1/"
] | Let $(M,hyp)$ be a closed hyperbolic surface. fix a point $m$ in $M$ and denote by $G=\pi_1(M,m) $.
now let $\alpha$ and $\beta$ in $G$ such that $\alpha$ and $\beta$ does not commute . my first question is why $<\alpha,\beta>$ is of inifinite index in $G$ and my second question: Let $n$ be the least positive i... | The answer to the first question: because it is free, and every subgroup of finite index is the fundamental group of a finite cover of the surface (which is again a hyperbolic surface). The answer to the second question is "infinity". Take any free non-abelian subgroup $F$ of the surface group. The group $F$, being fre... | These questions are the subject of a very nice paper of W. Jaco:
William Jaco. On certain subgroups of the fundamental group of a closed surface. Proc. Cam- bridge Philos. Soc., 67:17–18, 1970.
| https://mathoverflow.net |
2,484,716 | [
"https://math.stackexchange.com/questions/2484716",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/492147/"
] | <blockquote>
Prove that for every odd number $n$, it holds that $n^2+3$ is not not divisible
by $8$.
</blockquote>
<h3>My idea:</h3>
Let $n=2k+1$ for $k \in \mathbb{N}$, which implies
$$n^2+3=(2k+1)^2+3=4k^2+4k+1+3=4k^2+4k+4$$
How can I conclude that I cannot divide $4k^2+4k+4$ by $8$?
| Because $k(k+1)$ is divisible by $2$ and $4$ is not divisible by $8$.
| There's a much simpler proof using $n^2 \equiv 1 \mod 8$ for odd $n$, which is easy to prove exhaustively.
| https://math.stackexchange.com |
64,949 | [
"https://physics.stackexchange.com/questions/64949",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/24610/"
] | How can I determine whether the mass of an object is evenly distributed without doing any permanent damage? Suppose I got all the typical lab equipment. I guess I can calculate its center of mass and compare with experiment result or measure its moment of inertia among other things, but is there a way to be 99.9% sure?... | <h2>Malicious counter example</h2>
The desired object is a sphere of radius $R$ and mass $M$ with uniform density $\rho = \frac{M}{V} = \frac{3}{4} \frac{M}{\pi R^3}$ and moment of inertia $I = \frac{2}{5} M R^2 = \frac{8}{15} \rho \pi R^5$.
Now, we design a false object, also spherically symmetric but consisting of ... | If you have a rigid mass distribution sealed inside a black box, then the only things you can observe about its motion are its velocity vector and its angular velocity vector as functions of time. These can be predicted if you know the total force and total torque that act, plus the mass, center of mass, and moment of ... | https://physics.stackexchange.com |
257,071 | [
"https://softwareengineering.stackexchange.com/questions/257071",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/29020/"
] | In eager languages like Scheme and Python, you can use a lambda expression without parameters to delay evaluation, e.g. in Scheme (Chicken Scheme):
<pre><code>#;1> (define (make-thunk x) (lambda () (+ x 1)))
#;2> (define t (make-thunk 1))
#;3> (t)
2
</code></pre>
In line 2, <code>t</code> is bound to the une... | Well, the other answers cover what <code>\() -> "something"</code> means in Haskell: an unary function that takes <code>()</code> as argument.
<ul>
<li>What is a function without arguments? – A value. Actually, it can occasionally be useful to think of variables as nullary functions that evaluate to their value. Th... | You're misinterpreting what <code>()</code> means in Haskell. It isn't the lack of a value, it is rather the <em>only</em> value of the <strong>Unit</strong> type (the type itself being referred to by an empty set of parentheses <code>()</code>).
Since lambdas can be constructed to use pattern matching, the lambda exp... | https://softwareengineering.stackexchange.com |
180,952 | [
"https://electronics.stackexchange.com/questions/180952",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/41126/"
] | Here is a picture of a 1970's arc welding station.
There is a 9 inch bar of metal surrounded by 88 windings of 4mm copper, that used to be on the end of the secondary winding, which is made up of about 40 turns of 5mm copper.
What's it for?
<img src="https://i.stack.imgur.com/X44N4.jpg" alt="enter image description... | There's normally an inductor in series with the secondary (and a rectifier). That is probably the inductor.
There should also be a bar that slides through the slot in the transformer to adjust the welding current.
| Seems to be a choke (acts as flywheel to smooth rotation).
I think the (giant) rectifier diodes have been cut out?
Is that for AC or DC welding operation? Looks like AC without the rectifier but not sure why you would have a choke on AC. If it's AC there need not be diodes or rectifiers, just conductors.
| https://electronics.stackexchange.com |
2,189 | [
"https://dsp.stackexchange.com/questions/2189",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/1141/"
] | I have a stochastic process completely uncorrelated. Why the autocorrelation function has a delta of dirac in the origin?
Which is the reason of that?
$R_{XX}({\tau})=A{\delta(\tau)}$
where $R_{XX}$ is the autocorrelation function of $X$ process and $A$ is a constant dependent by process spectrum
| It has a delta at the origin because you are multiplying your time series with itself (i.e. there is no time delay). Any non-zero time series correlated with itself will give a positive value.
| As you noted, the wide-sense stationary stochastic process $X(t)$ is uncorrelated with itself for all nonzero time lags $\tau$. For $\tau=0$, it's impossible for a process to be uncorrelated with itself (unless $X(t)$ is the trivial process $X(t)=0$). Look at the definition for $R_{XX}(\tau)$:
$$
R_{XX}(\tau) = E(X(t... | https://dsp.stackexchange.com |
38,269 | [
"https://security.stackexchange.com/questions/38269",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/27830/"
] | I ran <code>openssl speed</code> on my Ubuntu computer. Some results:
<pre><code>Doing md4 for 3s on 16 size blocks: 9063888 md4's in 3.00s
Doing md4 for 3s on 64 size blocks: 7105157 md4's in 3.00s
Doing md4 for 3s on 256 size blocks: 4507884 md4's in 3.00s
Doing md4 for 3s on 1024 size blocks: 1809773 md4's in 2.99s... | MD4 is a hash function; it processes input messages which are sequences of bits of arbitrary length. Processing time is <em>roughly</em> proportional to the input length, but with starting and finishing a hash function invocation imply some fixed overhead, partly due to the algorithm (initialization, padding...), partl... | <ol>
<li>It ran md4 as many times as it could for 3 seconds</li>
<li>This is how many times it managed to complete in 3 seconds</li>
<li>This is the size of blocks used. You can see how much longer a block of 8192 takes than a block of 16</li>
</ol>
| https://security.stackexchange.com |
101,793 | [
"https://cs.stackexchange.com/questions/101793",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/71228/"
] | Here is the Python code. The solution is fairly common and is seen in most textbooks like 'Cracking the Coding Interview' and 'Element of Programming Interviews'.
<pre><code>class TreeNode:
def __init__(self, x):
self.right = None
self.left = None
self.val = x
</code></pre>
Here is the sol... | <blockquote>
I understand I need to find constants <span class="math-container">$c_1$</span>, <span class="math-container">$c_2$</span>, <span class="math-container">$n_0 > 0$</span> where
<span class="math-container">$0 \leq c_1n^b \leq (n+a)^b \leq c_2n^b$</span> for all <span class="math-container">$n \geq n_0$</... | If n is large then we have n/2 ≤ n + a ≤ 2n.
We therefore have <span class="math-container">$(1/2)^b n^b ≤ n^b ≤ 2^b n^b$</span>.
There you have your two constants: <span class="math-container">$(1/2)^b$</span> and <span class="math-container">$2^b$</span>.
| https://cs.stackexchange.com |
467,877 | [
"https://physics.stackexchange.com/questions/467877",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/164488/"
] | The commutator of position and momentum, <span class="math-container">$[x,p_x]$</span>, has a unique value given by <span class="math-container">$i\hbar$</span>. Why doesn't the anticommutator <span class="math-container">$\{x,p_x\}$</span> also have a definite value?
| If both the commutator and anticommutator have unique values, then by linearity, that would imply that <span class="math-container">$\hat{x}\hat{p}$</span> has a unique value. This would be inconsistent with the fact that the matrix element <span class="math-container">$\langle x |\hat{x}\hat{p}| p\rangle= xp\langle x ... | A consistent (and physically relevant) quantum theory can either postulate the commutator of <span class="math-container">$\hat x$</span> and <span class="math-container">$\hat p$</span> or their anticommutator. Since <span class="math-container">$\{\hat x,\hat p\}=[\hat x,\hat p]+2\hat p\hat x$</span>, if a quantum th... | https://physics.stackexchange.com |
1,214,369 | [
"https://math.stackexchange.com/questions/1214369",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/177326/"
] | Let $k$ be a finite extension of $\mathbb{Q}$ and $|\cdot|$ an absolute value on it (either Archimedean or not). Let $L$ be the completion of $k$ with respect to this value, and take any irreducible polynomial $f(X)$ in $k[X]$.
<blockquote>
Is is true that $f(X)$ is still separable on $L[X]$, i.e. every irreducibile... | Here’s the way to look at this problem. Consider your polynomial $f(X)\in k[X]$. It’s separable if and only if $f$ and $f'$ are relatively prime. But from the equation $Af+Bf'=1$, you see that $f$ and $f'$ remain relatively prime in no matter what extension field $K\supset k$ you look at.
| In characteristic $0$, all irreducible polynomials are separable, meaning that their roots in a splitting field are all simple roots. From this description it is also clear that separable polynomials remain separable over extensions fields.
| https://math.stackexchange.com |
102,345 | [
"https://physics.stackexchange.com/questions/102345",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/28118/"
] | This is what I understand so far: in a conductor, the ions have a weak pull on the valence electrons. So when an electric field is applied, the free electrons are able to easily move about. Makes sense.
In a neutral conductor with no electric field, the free electrons aren't bound to any ions. Why? I understand that t... | An atom in isolation offers a potential well, and electrons form bound states in the well. The energy of those bound states can be calculated exactly in the case of a single-electron (hydrogen-like) atoms or by variational computational methods for more complicated cases.
Now when you put several atoms together in a t... | In a single free atom, electrons have well defined energy levels and are somewhat bound to atom. Consider the following quantum mechanical model of atom to get an idea about an isolated atom.<br>
<img src="https://i.stack.imgur.com/wMjhh.jpg" alt="enter image description here">
When all this isolated atoms come togeth... | https://physics.stackexchange.com |
23,744 | [
"https://mechanics.stackexchange.com/questions/23744",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/13825/"
] | I live in a climate that necessitates changing the wheels to ones with studded winter tires and back again to ones with summer times twice a year.
After several years of owning the car, I noticed that the aluminium wheels got stuck to the wheel hubs, necessitating the use of a rubber mallet to cause the wheels to move... | I've worked at a garage. We use a rubber mallet. Hit it like your doing lug nuts. Hit it, rotate 90*, hit it, rotate 45*, and so on so you are not always hitting on the same spot.
We would normally put some grease on the hubs. You can use anti-seize also. I wouldn't worry about water moving it around. It is sq... | I have successfully used a "Karate Kick" after watching it on an Eric the Car Guy video and yes it did work. You get your leg underneath the car and do a quick kick to the bottom of the tire on the rubber. Make sure the car is well supported and won't move. I never get under the car unless I have at least 2 or 3 sup... | https://mechanics.stackexchange.com |
254,576 | [
"https://security.stackexchange.com/questions/254576",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/124704/"
] | Why don't some services offer Google/Facebook/Apple/Twitter login? Namely Crypto exchanges.
I assume they want as many users as possible & this is a great way to get more. Is there some sort of security vulnerability associated with them?
Edit: For Google & Apple login since both offer email services (gmail &am... | There are a variety of reasons that a company may not want to offer a federated login option. Some of them include the following:
<ul>
<li>People don't necessarily protect their social media accounts very well. A company may want the ability to require a strong password or 2FA to log in, and that's harder to do when ... | In addition to the excellent reasons already mentioned in the other answer:
<ul>
<li>Single sign-on / federated authentication means the identity provider knows what site/service you're signing into, and when. Lots of people might not want Google or Apple knowing about every site they sign into, and in particular might... | https://security.stackexchange.com |
205,587 | [
"https://dba.stackexchange.com/questions/205587",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/150484/"
] | <h1>Question</h1>
How to improve this query to increase the speed of the query shown below from <strong>30+ seconds</strong> to milliseconds? I am using <strong>PostgreSQL: v 9.6.6</strong>.
<h1>Context</h1>
I have a time series table <code>buildings.hispoint</code> that stores historical data for data points in a t... | An index on <code>(point_id, datetime, value)</code> will likely speed up the query, as it will only have to do an index seek, read only the relevant part of the index and also have available (from the index) all the values of the <code>value</code> column to calculate the <code>MIN</code> and <code>MAX</code>.
An ind... | You're casting to float in runtime,
<pre><code>MAX(value::float) - MIN(value::float) AS aggregation_value
</code></pre>
Just change the <code>value</code> type to float if you want that effect.
<pre><code>ALTER TABLE buildings.hispoint
ALTER COLUMN value
SET DATA TYPE TO float;
</code></pre>
Also what's the poi... | https://dba.stackexchange.com |
86,342 | [
"https://electronics.stackexchange.com/questions/86342",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/29920/"
] | I'm struggling to understand the practical difference between carbon composition resistors and metal film resistors. From what I've read, metal film resistors tend to be available in more precise tolerances than carbon composition ones. Other than that, are there any differences between the two?
| Carbon composition resistors have higher noise and are prone to aging (change it's resistance with the time). Additionally they can change its resistance because of high voltage or high temperatures applied.
That is why the precision resistors with tolerance less than 1% are always made with metal film technology.
Th... | Metal film can be made to tighter tolerance, is more stable, and is less noisy. Nowadays there is very little reason to use a carbon composition resistor. Metal film resistors are made in such large numbers that there is no longer a cost advantage to using carbon composition.
| https://electronics.stackexchange.com |
1,458,890 | [
"https://math.stackexchange.com/questions/1458890",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/271427/"
] | The following question is copied word for word from my textbook, which is what causes me to be so confused about the contradiction that it implies.
<strong>The question:</strong>
For gases under certain conditions, there is a relationship between
the pressure of the gas, its volume, and its temperature as given by w... | Notice, in the given equation $$PV=mRT\implies R=\frac{PV}{mT}$$
$m\ (kg)$ is the mass of gas & $R$ is called specific gas constant which has unit $\frac{joule}{kg\cdot K}$ as you have calculated.
Now, the gas equation in term of Universal gas constant $\bar{R}$ is given as $$PV=n\bar{R}T\implies \bar{R}=\frac{... | There is a universal gas constant (also called the molar gas constant) that can be used for any ideal gas in the equation $P V = n R T$. There is also an individual gas constant (also called the engineering gas constant or the specific gas constant) for each gas or mixture (oxygen, air, etc) that can be used in the eq... | https://math.stackexchange.com |
44,914 | [
"https://electronics.stackexchange.com/questions/44914",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/4647/"
] | I am working on a project that has a micro-controller that needs to keep track of time over long periods, the project will be powered of mains power for the vast majority of the operation time, but I wanted to incorporate a backup battery for blackouts, or I'm clumsy and knock the plug out or something.
I have a few t... | EDIT: Button is connected like this:
<img src="https://i.stack.imgur.com/khFZk.png" alt="ButtonConnection">
If there was no pull-up, one would put the following in setup()
<pre><code>pinMode(button, INPUT);
digitalWrite(button, HIGH);
</code></pre>
An interrupt is an event that 'pauses' the main program, runs a fu... | The way your code is written, it is spending a <em>lot</em> of time in <code>delay(5000)</code> calls. This could easily explain missed button presses.
Every time the Arduino operating system calls your <code>loop()</code> function, as long as the state of the button is stable, you'll enter the <code>if (val == val2) ... | https://electronics.stackexchange.com |
563 | [
"https://chemistry.stackexchange.com/questions/563",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/74/"
] | This is my first time taking a $\ce{{}^{13}C-NMR}$ spectrum and while I'm aware the signal to noise ratio is much lower compared to $\ce{{}^{1}H-NMR}$, I didn't realize it was quite this bad. I took a spectrum and I only see the $\ce{CDCl3}$ triplet. What concentration range is generally appropriate given say 256 scans... | At 256 scans you should see something if your compound has reasonable concentration. The rule of thumb I was taught as an undergrad was ~10 mg of compound for a $^{13}$C scan in under a mL of solvent (400-500 uL if you are measuring precisely, modified by probe size and whatnot of course)
As for an exact concentratio... | There are several reasons why $\mathrm{^{13}C}$ experiments are far less sensitive than proton experiments.
The natural abundance for $\mathrm{^{13}C}$ is only 1.1%, while $\mathrm{^{1}H}$ has a natural abundance of 99,9%. This difference alone means that if you don't label your sample with $\mathrm{^{13}C}$, the spec... | https://chemistry.stackexchange.com |
12,370 | [
"https://cs.stackexchange.com/questions/12370",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/7917/"
] | I'm exercising for an upcoming exam and I find this exercise:
<blockquote>
Say whether or not the language
$$L = \{a^jb^ia^{j-i}\mid i,j \ge 0\ , j > i\}$$
is a context-free language. Justify your answer.
</blockquote>
I have already tried (using the pumping lemma for CFL) with two different words:
$$w1 = \... | The following grammar shows the contextfreeness $L$:
<ul>
<li>$G = (N,T,P,S)$</li>
<li>$N = \{A, B, S\}$</li>
<li>$T = \{a, b\}$</li>
<li>$S = \{S\}$</li>
<li>$P = \{S \rightarrow aAa, A \rightarrow aAa, A
\rightarrow B, B \rightarrow aBb, B \rightarrow \lambda\}$</li>
</ul>
$L(G) = L$
| You are trying the wrong way. It is context-free.
(added.) The structure of the language (to guide the grammar construction) is better seen by setting $k = j-i$, so $j=k+i$. Then the language becomes $\{ a^ka^ib^ia^k \mid i\ge 0, k>0 \}$.
| https://cs.stackexchange.com |
165,070 | [
"https://physics.stackexchange.com/questions/165070",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/73160/"
] | Whenever we consider velocity, it is always from a frame of reference.
This means that if I am travelling .5c, I'm travelling at a speed between the speed of light and my current frame of references' speed.
Hence, is it theoretically possible for something to "not move"? I imagine that it would be completely incapab... | <blockquote>
This means that if I am travelling .5c, I'm travelling at a speed
between the speed of light and my current frame of references' speed.
</blockquote>
This isn't correct; you're at rest with respect to yourself. Uniform motion is relative. Two objects may have a <em>relative</em> speed of 0.5c but, a... | Quantum mechanically even at absolute zero objects have zero point energy. That is, they are "jiggling about". In a way this is a consequence of the uncertainty principle as they can't have both known location and momentum. I <em>think</em> this means that even in its own frame of reference no object is every truly "at... | https://physics.stackexchange.com |
635,952 | [
"https://physics.stackexchange.com/questions/635952",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/260493/"
] | Since we have to perform a relativistic transformation to transform the resonance peak energy from the lab frame to the resonances rest frame, I would find it surprising that we could use the total width of the resonance peak in the lab frame directly to calculate the resonance lifetime in its rest frame (i.e. just use... | The resonance energy width is <em>not</em> conserved between frames. However it is also not just as simple as the resonance width is always inversely proportional to the resonance particle lifetime in that frame.
A resonance particle is unstable and therefore exists as a superposition of many 'mass energy' eigenstates.... | The question is equivalent to 'can we use the lab frame lifetime to get the rest frame lifetime?'
The answer is yes, if you know the velocity you can just lorentz transform the time
| https://physics.stackexchange.com |
20,471 | [
"https://mathoverflow.net/questions/20471",
"https://mathoverflow.net",
"https://mathoverflow.net/users/1474/"
] | Why is it that every nontrivial word in a free group (it's easy to reduce to the case of, say, two generators) has a nontrivial image in some finite group? Equivalently, why is the natural map from a group to its profinite completion injective if the group is free?
Apparently, this follows from a result of Malcev's t... | Here is a direct proof for free groups.
Let $x_1,\dots,x_m$ be the generators of our group. Consider a word $x_{i_n}^{e_n}\dots x_{i_2}^{e_2}x_{i_1}^{e_1}$ where $e_i\in\{\pm 1\}$ and there are no cancellations (that is, $e_k=e_{k+1}$ if $i_k=i_{k+1}$).
I'm going to map this word to a nontrivial element of $S_{n+1}$,... | You can also use the fact that free groups are linear. For instance if $$a=\begin{pmatrix}
1 & 2 \\\\
0 & 1 \\ \end{pmatrix}$$ and $b$ is the transpose of $a$, a simple ping-pong argument shows that $a$ and $b$ generate a free subgroup of $SL(2,Z)$. Now any element of $SL(2,Z)$ will be non-trivial in reduction ... | https://mathoverflow.net |
193,682 | [
"https://softwareengineering.stackexchange.com/questions/193682",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/24110/"
] | I'm heading a new team of developers working on a software project that makes use of continuous integration (circleci) w/ a pretty fleshed out suite of busterjs unit/integration/acceptance tests. Our project is primarily written w/ coffeescript, and I try to make use of coffeescript-linter to ensure everyone working ou... | When you a) have a coding standard and b) have configured the linter to be consistent with the coding standard, that's when you start enforcing linting.
Put the linter on the CI server.
| The question is, how important is it to you that issues found by coffeescript-linter are addressed immediately.
If it is unacceptable to have code in the repository that gets flagged by the linter, then you should have the linter run as a commit hook so it can block the commit if there are issues.
If it is undesirabl... | https://softwareengineering.stackexchange.com |
86,332 | [
"https://stats.stackexchange.com/questions/86332",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/30201/"
] | What are some typical approaches to handling incomplete data in the Kalman Filter? I'm talking about the situation where some elements of the observed vector $y_t$ are missing, distinct from the case where an entire observed vector $y_t$ is missed. Another way of thinking about this would be that the dimension $p$ of t... | What is needed is simply to have a variable observation matrix, i.e. in the observation equation:
$$ \boldsymbol{Y_t} = \boldsymbol{A_t}\boldsymbol{\theta_t} + \boldsymbol{R_t}\boldsymbol{e_t} $$
matrix $\boldsymbol{A_t}$ (and $\boldsymbol{R_t}$) should omit at time $t$ the rows corresponding to <code>NA</code> entries... | Durbin and Koopman (2012) have a useful paragraph when it comes to missing observations.
They differentiate between two cases:
<ul>
<li>where all observations at time <span class="math-container">$t$</span> are missing and,</li>
<li>where a subset of observations at time <span class="math-container">$t$</span> are miss... | https://stats.stackexchange.com |
61,779 | [
"https://stats.stackexchange.com/questions/61779",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/26907/"
] | I have two test groups that conducted an online task measuring response times (avg, avg(congruent), avg(incongruent)). I expected one group to be faster than the other but it turned out exactly the opposite way (significant). So now I'm trying to find one or more factors that are responsible for this unexpected outcome... | You can probably look at your data in another way. Start with a t-test of the difference in average response time between the groups (that would be equivalent to a test that the Pearson – not Spearman – correlation between group membership and average time is 0). From there, you can easily add other variables in the mo... | By attempting several tests to produce a desired outcome you are greatly inflating your type I error rate. In fact, because the direction was <em>opposite</em> what you expected (or considered a significant result, but not significant in the right direction), in a way you're inflating type I and type II error rates sim... | https://stats.stackexchange.com |
74,538 | [
"https://stats.stackexchange.com/questions/74538",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/32222/"
] | I have wait-time data for a coffee shop for 4 weeks. Since the data is crowdsourced, it is sparse and non-uniform along the time. So my question is: how should I deal with this non-uniform data? What are the some methods in time series forecasting that can handle non-uniform data? Or is there any way to make the data u... | Transformations are like drugs ! Some are good for you and some aren't !. Haphazard selection of transformations should be studiously avoided.
a) One of the requirements in order to perform valid statistical tests of necessity is that the variance of the errors from the proposed model must not be proven to be non-con... | <strong>This question is answered beautifully by means of a <em>spread-versus-level</em> plot:</strong> a cube root transformation will stabilize the spreads of the data, providing a useful basis for further exploration and analysis.
<hr>
The data show a clear seasonality:
<pre><code>plot(y)
</code></pre>
<img src=... | https://stats.stackexchange.com |
415,259 | [
"https://electronics.stackexchange.com/questions/415259",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/208497/"
] | I have an SPST momentary tactile switch and want the pulse, whatever length, to be extended to 500ms (unless of course the button is pressed for longer than 500ms but we can ignore that if we hold the pulse form the rising edge). I don't know if there is a single IC to hold a pulse like this or if I can do it with pass... | I'm not sure how to give you a hint without giving you the complete answer.
But consider the difference between these two situations:
<ul>
<li>If <strong>any</strong> of the inputs to the upper encoder are asserted, then its outputs become the overall outputs, and Y2 needs to be asserted.</li>
<li>If <strong>none</st... | Connect Y0, Y1 to both 4:2 in parallel.
Use Y0 as the 3rd "address" line and a NOT gate to enable one or the other.
| https://electronics.stackexchange.com |
63,174 | [
"https://chemistry.stackexchange.com/questions/63174",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/22766/"
] | For instance, the ground-state configuration of N atom is a $p^3$ configuration of all parallel spins and one electron in each $2p$ orbital, which has:
<ul>
<li>Total spin angular momentum, $S = 3 * \frac{1}{2} = \frac{3}{2}$</li>
<li>Spin multiplicity, $2S + 1 = 2(\frac{3}{2}) + 1 = 4$</li>
<li>Total orbital angular ... | The full calculation is laid out below. Start by calculating the spin, orbital and total of all angular momentum. Each electron has spin quantum number $s = 1/2$ and magnetic quantum number $m_s = \pm 1/2$. Orbitals have angular momentum of $s=0, p=1, d=2, f=3$ etc.
The total spin angular momentum is the series of va... | *sigh *
Face-palm moment: I've already chosen an $S_{max}$, and when the subshell is exactly half-filled, $L_{max} = 0$, so $J = S_{max}$. And that would be why it's ${}^{4}S_{3/2}$, rather than ${}^{4}S_{1/2}$.
But if anyone has anything more to add, feel free to edit this answer.
<hr>
Here's what I would do if I... | https://chemistry.stackexchange.com |
1,241,358 | [
"https://math.stackexchange.com/questions/1241358",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/164148/"
] | Let G be a group, and $H \trianglelefteq G$, $K \trianglelefteq G$.
Prove that if $H \cap K = \{1\}$ and $ G / H $ and $ G/ K $ are abelian, then G is abelian.
I've tried to give a proof by contradiction but couldn't really get anywhere. If $ ab \ne ba $ then $\pi_{H}(ab) = \pi_{H}(ba)$ and I get a similar result wit... | The obvious homomorphism $\phi: G\to G/H\times G/K$ is injective, since the kernel is the set of elements in $H\cap K$. Since both quotients are abelian, the product is abelian, and $G$ is isomorphic to a subgroup of the product, and is thus abelian itself.
| Let $a, b \in G$. Consider $[a, b]:=aba^{-1}b^{-1}$ which is frequently called the commutator. (Up to choice of convention.)
Then the hypotheses tell us that $[a, b] \in H$ and $[a, b] \in K$ because $ab=ba$ mod $H$ or $K$.
| https://math.stackexchange.com |
265,942 | [
"https://math.stackexchange.com/questions/265942",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/54373/"
] | Which of the following subsets of <span class="math-container">$ \mathbb{R^2}$</span> are compact?
<ol>
<li><span class="math-container">$\displaystyle \{(x, y) : xy = 1\}$</span>
</li>
<li><span class="math-container">$\displaystyle \{(x, y) : x^{\large\frac{2}{3}} + y^{\large\frac{2}{3}} = 1\}$</span>
</li>
<li><span... | Since the prime graph of $G$ is disconnected, and since every group of order $24$ is solvable by Burnside's theorem, $G$ is either Frobenius or $2$-Frobenius. If $G$ were Frobenius, then $G=K\rtimes C$ where $|C|$ and $|K|$ are coprime and $|C|$ divides $|K|-1$, but this is impossible since $|G|=2^3\times 3$. Thus $G... | Suppose that $H=\{e,a\}$ is a normal subgroup of order $2$. Let $x\in G$ have order $3$. Since $H\lhd G$, therefore $x\{e,a\}=\{e,a\}x$. Hence, $xa=ax$. Thus, $|xa|=\frac{|x||a|}{\gcd(|x|,|a|)}=6$. (contradiction)
| https://math.stackexchange.com |
150,470 | [
"https://electronics.stackexchange.com/questions/150470",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/64977/"
] | I'm using an optoisolator for high side switching a Nixie tube. For some reason, the optoisolator is not fully turning off, allowing the a very minimal amount of current to flow through the Nixie tube.
How can I make sure the optoisolator is fully off, so no current will flow through the Nixie?
I'm switching the opt... | A lot depends on exactly what parts you're trying to characterize. At a minimum, you need a DMM and a variable power supply. The supply should be able to provide, let's say, 12 volts at 1 or 2 amps. It should have both variable voltage and variable current limit settings. Let's take your examples.
LEDs - Set your powe... | For simple home-brew circuits component specs don't need to be that precise. Just component experience, looking at it, a multimeter, and maybe a function generator and dc power supply go a long way.
If you want the complete specifications you'd need a huge amount of very expensive T&M equipment, not to mention T&... | https://electronics.stackexchange.com |
2,387,781 | [
"https://math.stackexchange.com/questions/2387781",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/444436/"
] | I have a problem, where the equation of my ellipse is:
<br>$$1=\frac{x^2}{75}+\frac{y^2}{56.25}$$
<br>and I have been able to translate the equation to a complex form of
<br>$$|z+2.5\sqrt{3}|+|z-2.5\sqrt{3}|=10\sqrt{3}.$$
<br> Now the problem is being able to solve and show the method of substituting and rearranging f... | The derivative and integral are <em>almost</em> inverse functions, so in turn, you are <em>almost</em> correct. For simple polynomials, one multiplies by the power and then removes 1 from the power, and the other adds 1 to the power and divide by the new power.
For more complex functions, you can consider it visually,... | Since$$\int_a^xf'(t)\,\mathrm dt=f(x)-f(a),\tag{1}$$the short answer is that the integral of the derivative is the original function, up to a constant.
Of course, $(1)$ isn't true without restrictions. But if $f'$ is continuous, then, yes, $(1)$ holds.
| https://math.stackexchange.com |
588,329 | [
"https://physics.stackexchange.com/questions/588329",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/261461/"
] | So, I have been studying thermodynamics for a while and some mathematical steps involving partial derivatives have now started to hurt my head.
First of all, I understand that whenever we take a <span class="math-container">$PVT$</span> system, there are only two independent variables for any process to occur. Using th... | I think @GiorgioP answer quite nicely explains how to understand partial derivatives role in the total derivative. I will show you, how is the formula
<span class="math-container">$$\Big(\dfrac{\partial S}{\partial T} \Big)_P=\Big(\dfrac{\partial S}{\partial T} \Big)_V +\Big(\dfrac{\partial S}{\partial V} \Big)_T \Big ... | Clarifying such an issue goes in parallel between Math and Physics.
An explicit rewording of the key relation helps a lot.
<span class="math-container">$$dS=\Big(\dfrac{\partial S}{\partial T} \Big)_V dT +\Big(\dfrac{\partial S}{\partial V} \Big)_TdV ~~~~~~~~~~~~~~~~~~~[1]$$</span>
is the general expression of the diff... | https://physics.stackexchange.com |
9,519 | [
"https://cs.stackexchange.com/questions/9519",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/6697/"
] | I am having problems resolving the following question:
<blockquote>
Given some problem $X$. If there exists a polynomial time reduction from (for example) $\mbox{SAT}$ to $X$, $(\mbox{SAT} \leq_{p} X)$ and since we know that $\mbox{SAT}$ is $\mbox{NP-complete}$, to show that $X$ is $\mbox{NP-complete}$ is it necessa... | the reduction only shows that X is at least as hard as SAT (NP-HARD). This could mean that X is in a class suspected to be harder than NP such as PSPACE or EXPTIME. To show that X is NP-complete you must show that it is also in NP. If you try it out you will see that you can easily reduce SAT to problems in EXPTIME tha... | Here is a very general negative answer. Consider the following language:
$$ L = \{\langle M,x,o \rangle : M \text{ outputs } o \text{ on input } x\}, $$
where $o \in \Sigma^* \cup \{ \bot \}$ and $\langle \cdot,\cdot,\cdot \rangle$ is any polytime pairing function (we can even demand it to be logspace, or even weaker).... | https://cs.stackexchange.com |
47,833 | [
"https://physics.stackexchange.com/questions/47833",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/7527/"
] | Consider a Photon from Sun and travels with a velocity $c$. Now think we are that photon. For us, it looks like Sun is moving away from us with a velocity $c$. So, why don't we get attracted back towards Sun, because the mass of Sun would be infinite for us since it moves away from us with a velocity $c$.
| You have completely mixed the modern and classical concepts of relativity. If you're talking about mass increment, you shouldn't calculate speed of Sun based on absolute time & space notion.
For you as a photon, space will be contracted to zero and time will be dilated to infinity. So, you can't calculate a speed ... | When anything moves at the speed of light, all of our physical models break down. If you were to watch a spaceship speed up to the speed of light, you would see a clock on the ship slow down and come to a complete stop when it hit the speed of light (assuming you could even see it at this point). The ship would also co... | https://physics.stackexchange.com |
300,492 | [
"https://stats.stackexchange.com/questions/300492",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/26084/"
] | I call the function estimation based classifiers as deterministic, the ones which estimates the $f(x) = a'x+b$ directly, rather than estimating the conditional or joint probabilities directly. For example, SVM outputs a score which corresponds to $f(x) = a'x+b$ and the classification rule is $sign(f(x))$. Now, my ques... | Your approach is about as sensible as doing the same with another statistical distribution. There are some problems that need to be addressed.
Regarding p-values, if you allocate half of your $\alpha$ level to each one sided test you should be on the safe side. That's a Bonferroni correction for multiple testing. Othe... | Well, the problem is, you are trying to use hypothesis testing for something it can't really do, which is claiming that a certain (even approximate) value GAVE rise to something you observed.
What in my opinion would work as a heuristic is that you take your lower ($0.5 - \delta$) and upper bound($0.5 + \delta$) and ... | https://stats.stackexchange.com |
5,952 | [
"https://quantumcomputing.stackexchange.com/questions/5952",
"https://quantumcomputing.stackexchange.com",
"https://quantumcomputing.stackexchange.com/users/6254/"
] | Say we have a single qubit with some density matrix, for example lets say we have the density matrix <span class="math-container">$\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}$</span>. I would like to know what is the procedure for checking whether this state is pure or mixed.
I know that a pure state is... | First, the example that you give is not a density matrix (they must have trace 1).
Second, you’re asking how to go from the matrix into an operator representation that is not unique. So, there are many ways of doing this. However, a particularly natural way of decomposing it is using the spectral decomposition. The we... | By spectral theorem density matrices are diagonizable, since they are hermitian (also they are positive semi-definite and have trace 1). That means that there is a set of <span class="math-container">$n$</span> non-negative eigenvalues <span class="math-container">$\lambda_i$</span> with <span class="math-container">$n... | https://quantumcomputing.stackexchange.com |
178,761 | [
"https://security.stackexchange.com/questions/178761",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/169675/"
] | I am currently building an multiplayer application game on Android, and the server-side logic is deployed using Cloud Functions, which is a server-less environment. The application logic is as follows: two users are given a question (i.e. the same one) and they both try to answer it. When one user answers the question,... | By not using TLS you are giving up the integrity and confidentiality. Your connection can be monitored or altered and there is nothing there to guarentee the endpoint is who they say they are.
<ul>
<li>An internal attacker can man-in-the-middle your http connection reading and altering data. </li>
<li>An employee cou... | HTTPS is used to protect the connection against sniffing, manipulation and against impersonating an endpoint. How much this protection is needed in your local network depends on how much you can trust your network. As always, there is no absolute security but one has to find a balance between the achievable security an... | https://security.stackexchange.com |
364,104 | [
"https://physics.stackexchange.com/questions/364104",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/159281/"
] | In scalar QED, under a gauge transformation $$A_{\mu}\to A_{\mu}+\partial_{\mu}\alpha(x)$$
$\phi$ "can transform as"
$$\phi \to e^{-i\alpha(x)}\phi$$
(source: Schwartz's QFT 8.49). Why is this true? Why does a gauge transformation in $A_{\mu}$ do anything at all to $\phi$, which is a different field?
Or are we jus... | <blockquote>
Are we just saying we can construct a theory such that this holds, so let's just declare this to be true and see what happens?
</blockquote>
Yes, this is basically the case. But the assumption is not as arbitrary as it might seem at first. It's not like people went around trying various transformations ... | "Or are we just saying we can construct a theory such that this holds, so let's just declare this to be true and see what happens?"
This is precisely what happens. The essence of scalar QED is that it's a theory of a photon $A_\mu$ and a scalar $\phi$ which both transform nontrivially under the same local $U(1)$ symme... | https://physics.stackexchange.com |
469,234 | [
"https://physics.stackexchange.com/questions/469234",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/222575/"
] | <blockquote>
Two space shuttles <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are moving in opposite directions. A person on Earth measures <span class="math-container">$v_A = 0,75 c$</span> and <span class="math-container">$v_B = -0,85 c$</span>. What is the velocity of <span cl... | I replace "W" with "H", for Houston. They're space shuttles, after all. The velocity addition formula is for stacking up reference frames, not starting in the middle:
<span class="math-container">$$ v_{AB} = \frac{v_{AH}+v_{HB}}{1+\frac{v_{AH}v_{HB}}{c^2}}$$</span>
where <span class="math-container">$v_{xy}$</span> r... | Ignoring relativitiy for a minute,
<span class="math-container">$v_{BA}=v_{BW}+v_{WA}$</span> is the composition ["addition"] of velocities
in Galilean physics.
Since <span class="math-container">$v_{QP}=-v_{PQ}$</span>,
then
<span class="math-container">$$v_{BA}=v_{BW}+(-v_{AW})=v_{BW}-v_{AW}$$</span> is the formula ... | https://physics.stackexchange.com |
558,503 | [
"https://stats.stackexchange.com/questions/558503",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/331502/"
] | I'm trying to calculate a p-value for two samples and am having some issues understanding the role bootstrapping might play.
The data I have comes from an MRMC study: Five readers took a five-hundred question test creating five accuracy scores to form sample 1. Some time later, the same five readers took the same test,... | Bootstrapping done properly will not inflate your metrics. Bootstrapping allows you obtain the estimate of the probability distribution of the test statistic. Getting the p-values from this distribution involves using the actual sample size.
Here's an example of incorrect application of bootstrapping. A scientist was a... | First note that a bootstrap sample (draws with replacement) must contain exactly the same number of observations (independent draws) like the original data. I wonder however how bootstrap could be useful in this case.
As an alternative to the parametric paired t-test, there is a Monte Carlo test, but it is not based on... | https://stats.stackexchange.com |
372,030 | [
"https://electronics.stackexchange.com/questions/372030",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/187651/"
] | I have been searching for this answer for a long time and I haven't found any info on the way severed lines are repaired. There must be tens of thousands of tiny wires that must reconnected, and in the proper order, before the line can be put back into service. What tool and technique is required to achieve such a daun... | That's not completely unreasonable. Your distortion seems to last for about 3 usec.
First, your bias voltage (3 diode drops) is about 2 volts. Note that the data sheet give 2 volts as the minimum gate threshold, and it could be as much as 4. At high load impedances the FETs are close enough to turn-on that you don't ... | It is best to address your open loop performance before the opamp goes in .Increased idle current is good .I run about 600mA on my car radio circuits .This means that operation is mainly class A .Place a 1 microfarad cap between the gates for reduced distortion at reduced idle currents .
| https://electronics.stackexchange.com |
509,165 | [
"https://physics.stackexchange.com/questions/509165",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/206462/"
] | Suppose there is a beam of atoms moving with constant velocity in empty space. There are no collisions between the atoms. The temperature is essentially zero since velocities of atoms are all equal to the velocity of the center of mass.
Now let’s trap the atomic beam into a box. When atoms collide with walls, kinetic... | When you have a distribution of particles in a single energy state and compare that to a distribution with particles arranged in multiple energy states the entropy values will be significantly different. You can work that out for a even if you do not view it from the perspective of microstates and instead view the syst... | The first frame of reference is moving with the gas particles. The gas particles have ZERO velocity in their frame.
The next step is to "put them in a box" and have them "collide with the walls". The step does work on the particles. It causes them to move from one reference frame to another. The work that is done is t... | https://physics.stackexchange.com |
65,869 | [
"https://security.stackexchange.com/questions/65869",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/35502/"
] | So I am running pentest on a device running Win7 and admin user set group policy to disable all mass storage devices and all usb's.
I have access to local user without privs.
Although my usb stick wont work and group policy rules wont let me my USB flash, USB modem or anything but my iPhone works as a USB storage de... | As @guki117 pointed out in his answer, the iPhone is connecting as a camera.
You should be able to exploit that with a basic digital camera with a removable memory card, put whatever files you want on the card, put it back in the camera and connect the camera with USB. The files should be right there.
| It's most likely connecting as a Media Device or Camera. There's a seperate GPO for each type of removable media. Perhaps this particular policy only enforces against flash drives.
This is quite common on androids.. I can switch my device to be a Camera and bypass our GPO for blocking removable devices.
| https://security.stackexchange.com |
317,859 | [
"https://electronics.stackexchange.com/questions/317859",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/132863/"
] | For removing rosin flux from my board, should I use Acetone or Isopropyl Alcohol? I need an easily available solvent.
| Acetone being used to clean circuit boards can be problematic. The biggest issue that I found was that it leaves a residue in the board that still requires several other cleaning steps to get rid of including IPA and hot water.
Even IPA can leave a residue as it dries.
There is also the problem mentioned in the comm... | If you want real results to clean PCBs after rosin-based fluxes, you should use a specially-formulated solvent. Typically the solvents are based on IPA, but contain Toluene, Heptane, and Difluoroethane. Note - no acetone there.
| https://electronics.stackexchange.com |
3,243,170 | [
"https://math.stackexchange.com/questions/3243170",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/20019/"
] | If a population flips a coin, say, 10 times, what percentage of them will flip ten consecutive heads? Is it just as simple as <span class="math-container">$0.5^{10} \approx 0.1\% $</span> of the population will get this result? Or would be something along the lines of the portion of the normal distribution with area <s... | The <em>expected</em> percentage is <span class="math-container">$0.5^{10}$</span> ... but obviously the <em>actual</em> percentage can be different. In that sense you get a normal distribution of possible outcomes for that percentage ... but the area will still be 1: the sum of probabilities for each of those possible... | This is a straightforward application of the binomial probability distribution. If <span class="math-container">$Z$</span> is the number of heads in ten tosses, we have
<span class="math-container">$$P(Z = k) = {10\choose k}\cdot{1\over 2^{10}}$$</span>
for <span class="math-container">$0 \le k \le 10$</span>.
| https://math.stackexchange.com |
77,081 | [
"https://mechanics.stackexchange.com/questions/77081",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/29082/"
] | We have a 2004 van, it had never been in an accident, never needed more than regular maintenance service. All bells & whistles, excellent leather interior (no kids). It was bumped in the back, broken tail light, broken plastic outer bumper (not the steel under bumper) A 1/2" deep 5" long dent in the fende... | Keep in mind that an insurance policy is a contract between you and the insurance company. In return for your premiums, they agree to do certain things.
One of those things is to compensate you for damages which appears to be applicable in this case. So you have sustained a loss and the company is obligated to make ... | The owner's insurance policy may have an exclusion for vehicles which bear a "salvage" title.
If such an exclusion appears in the policy terms and conditions, then despite the OP's statement that the car is "fully insured"...it isn't, and the insurance company can decline payment for damages or inj... | https://mechanics.stackexchange.com |
322,788 | [
"https://physics.stackexchange.com/questions/322788",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/115962/"
] | I have 2 surfaces rubbing against one another (sliding past ) with some relative speed $v$ w.r.t. one another. Suppose, they slide for time $t$, like so. Is there then some formula for finding out, what would be the temperature rise in each of the surfaces in contact ?
My best guess is : $$\Delta T = \frac{K \cdot f \... | Olin Lathrop captured something important, which is that the word "burn" does not always align to the chemistry definition of the word. But I did want to answer your question.
Chocolate will burn in a "naked" pan because the heat is not evenly spread. In cooking, it's desirable to have the top surface of the pan hav... | When you use a bare pan it conducts heat from the burner and the metal will reach a very high temperature; enough to carmelize, and possibly burn the chocolate.
But when you put water in the pan, bring it to a boil the water basically <em>regulates</em> its temperature at phase transition (liquid to gas) to the boilin... | https://physics.stackexchange.com |
44,853 | [
"https://stats.stackexchange.com/questions/44853",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/14836/"
] | I'm not sure where to post this, so I thought I'd post it here. If there is another better place to ask this question then please could you let me know.
Let
$\beta$ = rate of infection, and $\gamma$ = rate of recovery.
I don't understand how the basic reproduction number (BRN) is $\frac{\beta}{\gamma}$?
I know the... | I'm just guessing here but...
The basic reproduction number is the expected number of secondary infections over the lifetime of the initial infection. Let $S$ be the number of secondary infections over the lifetime of the initial infection and $L$ be the lifetime of the initial infection.
$S|L$ can be modeled as a Po... | The answer to this doesn't necessarily rely on the distribution, it can be thought of as a simple problem of incoming infections vs. outgoing infections.
If you are trying to fill a bathtub, the water level will only rise if the rate of incoming water outpaces the rate of outgoing water. The same principle is true of ... | https://stats.stackexchange.com |
445,879 | [
"https://physics.stackexchange.com/questions/445879",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/215397/"
] | So today in my physics class we derived the center of mass of a uniform cone, and it all made sense, but near the end of class a student asked,
<blockquote>
"If you were to split an object into two parts with a plane through it's center of mass, would the two objects' masses be equal?"
</blockquote>
And it got me... | You have a conjecture:
<blockquote>
Cutting an object into two pieces using a plane through its center of mass yields two pieces with equal mass.
</blockquote>
You have found a counter-example to this, so it must be false in general.
The reason this does not work in your case is because if each each piece had the ... | Because mass that's further from the centre has more rotational inertia than mass closer. If you kick a can in its CM it moves without flipping end over end, if the can had more mass at the bottom its CM is lower. So it's not just the amount but the distance that's important.
| https://physics.stackexchange.com |
726,164 | [
"https://physics.stackexchange.com/questions/726164",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/165965/"
] | If you quickly pass your finger or hand through a candle or fire, it doesn't feel very hot (great party trick). Is this because your hand spends such little time in the fire that there isn't enough time for conduction to occur? Or, does the motion of your hand prevent the heat-transfer process from beginning (to a sign... | <blockquote>
Is this because your hand spends such little time in the fire that
there isn't enough time for conduction to occur?
</blockquote>
Pain occurs when the skin temperature reaches the threshold of pain. The temperature of the skin due to exposure to the candle flame is the result of the combination of the skin... | I am answering because I can't comment Bob D (my reputation is below 50)
As Bob D said for burn to occur the minimum intensity is 2.5 to 5 watt/cm² (in SI units 25000 to 50000 watt/m²)
So reason one is that skin exposure time to 80W candle is less than 0.125 to 0.0625 seconds (125 to 62.5 milliseconds)
And second reaso... | https://physics.stackexchange.com |
29,008 | [
"https://cs.stackexchange.com/questions/29008",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/9583/"
] | I'm trying to implement a feedforward neural network using a graph.
The thing is: I haven't found any example in which is used a graph data structure.
So far the examples I've found used arrays.
Can anyone please point me in the direction of some literature on the topic or some tutorial?
| Many implementations you can find out in the web are done on matrices (MATLAB for instance) since it provides a compact notation. Haykin's textbook on neural networks takes this approach. Matrices also provide a simple translation to hardware design (FPGA, ASIC, etc.). They are also more often implemented on the FPU.
... | You should use a directed graph since information passes in one way when running the network and in the other when training. Your data structure should support enumerating all incoming and all outgoing edges for each node, as well as enumerating nodes by their depth. Any data structure supporting these operations can b... | https://cs.stackexchange.com |
257,567 | [
"https://electronics.stackexchange.com/questions/257567",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/123543/"
] | i want to understand voltage.
voltage across a battery is the energy per charge that the battery can provide.
if we give more charge inside the battery we can get more overall energy in the battery output. meaning if the current entering the circuit is more(more charge per time),we get more overall energy/time and henc... | Datasheet 1st page says
<pre><code>3A Continuous Output Current
4A Peak Output Current
</code></pre>
<blockquote>
The peak current is usually limited by choice of L and IC heat loss and is exceeded by upper switch max current capacity.
</blockquote>
| In theory you could get 4-A output current, assuming very large heatsink pad and inductance enough large to minimize current ripple. But maximum duty cycle is 90 %, so there will be also switching at 340 kHz and total losses will probably overheat the poor IC. Nobody uses components at their maximum ratings, even if it... | https://electronics.stackexchange.com |
463,950 | [
"https://electronics.stackexchange.com/questions/463950",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/234607/"
] | I'm currently using MPLABX IDE to develop for a PIC16F877A microcontroller. I want to divide two numbers and find out the quotient (let the two numbers be 10 and 4) and store in a register.
I was thinking of a loop which increments the value of register by 1 every time 4 goes in 10 until it becomes zero or indivisible ... | Your approach ignores the fact that the minimum voltage that you can get at the output is not zero, whereas the approach you found takes that into account.
In other words, your simplified approach would result in a circuit that clips the negative peaks before the positive peaks reach their limit, reducing the symmetri... | If there is no load resistor, it makes sense to bias the collector half way between Vcc (18V) and Ve (1v in your case) for maximum collector swing in either direction. This is because, with no load resistor, the limits of the collector swing are 18V and about Ve+0.2V. (assuming the signal wiggle on the base is quite sm... | https://electronics.stackexchange.com |
83,458 | [
"https://dba.stackexchange.com/questions/83458",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/25073/"
] | I got issues with some geo IP query. So basically here is create code of table:
<pre><code>CREATE TABLE `geo_ip_city` (
`id` INT(5) UNSIGNED NOT NULL AUTO_INCREMENT,
`begin` BIGINT(20) NOT NULL DEFAULT '0',
`end` BIGINT(20) NOT NULL DEFAULT '0',
`code` VARCHAR(2) NOT NULL DEFAULT '',
`city` TINYINT... | If you are 100% sure that the intervals <code>(begin, end)</code> are never going to be overlapping, you can use this query, which only needs an index on <code>(begin)</code> or <code>(begin, end)</code> and will be much more efficient than what you have:
<pre><code>SELECT t.*
FROM
( SELECT g.*
FROM geo_ip_cit... | Your conditions are searching for ranges. Now imagine you're looking in a dictionary and looking up all words where the first letter is "greater" than <code>A</code>. Of how much use is the index? You want to narrow the search range down as much as possible. MySQL most of the time can only use one index per table. Comb... | https://dba.stackexchange.com |
1,920,093 | [
"https://math.stackexchange.com/questions/1920093",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/123276/"
] | Suppose I have a vector space $V$ of two dimensions, i.e., $V \subseteq \mathbb{R}^2$. Then, I define the column space to be $\mathcal{C}(V) = \{w: w = \sum_{i=1}^{2}c_i v_i\}$ where $c_i \in \mathbb{R}$.
Now, suppose that my orthogonal complement space is defined to be:
$$
\mathcal{C}^{\perp}(V) = \{w: <w,y> ... | First, note that vector spaces don't have a corresponding column space. Typically, only matrices have column spaces (defined to be all possible linear combinations of the matrix's column vectors).
To answer your question, it depends on the dimension of the column space.
<ul>
<li>If $\dim \mathcal C(A) = 0$ so that $\... | We speak about orthogonal <strong>complement</strong> of a subspace of a vector space.
Since $V$ is already of dimension $2$, then $V^\perp$ has dimension $0$ <strong>in</strong> $\mathbb{R}^2$.
More generally, in $\mathbb{R}^n$, for every subspace $W$ of $\mathbb{R}^n$, $\dim (W)+\dim (W^\perp)=n$ by rank nullity th... | https://math.stackexchange.com |
67,813 | [
"https://softwareengineering.stackexchange.com/questions/67813",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/20955/"
] | How many of you actually work out the exercises when learning from a book (any programming related book), I'm currently working my way through a C++ book and find that some of the exercises I feel I can complete rather easily I skip. Do most people do this? Or do they read the whole book and come back to exercises that... | I find it to be helpful to actually type in the solutions to the exercises and run them. Sometimes you'll get the answer on the first try, and sometimes it's a little bit trickier than it first looked. You'll never know what you're missing until you have working code.
One huge benefit to typing in the exercises your... | I find that I tend to skip trivial exercises unless I'm still at the stage where I'm having some difficulty even wrapping my head around the syntax of a new language. They are there to give you something to practice as much as to provide a thought-provoking challenge.
| https://softwareengineering.stackexchange.com |
57,411 | [
"https://cs.stackexchange.com/questions/57411",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/6915/"
] | <strong>Question:</strong> Find out next increasing value of each element in this below array.
<pre><code>int[] array = { 5, 2, 7, 10, 4, 12}
</code></pre>
e.g) <BR>
5's nextIncreasingValue: 7 <BR>
2's nextIncreasingValue: 7 <BR>
7's nextIncreasingValue: 10 <BR>
... <BR>
12's nextIncreasingVa... | It <strong>is</strong> $O(n^2)$. Consider <code>array = [n, n-1, n-2, ..., 1]</code>.
<hr>
BTW, you could implement this in $O(n)$ by scanning from the end of the array:
<ul>
<li>Suppose the next-increasing-value (NIV) of <code>a[i+1]</code> is <code>x</code>.</li>
<li>We want to find the NIV of <code>a[i]</code>, w... | This should be a comment, but I don't have that rep yet on this stackexchange.
kennytm's answer is right, I just want to add my rule of thumb:
whenever you are iterating over an array/list/other linear data structure, regardless of how many elements you skip or how early you break out, the big-O notation is always O(... | https://cs.stackexchange.com |
306,912 | [
"https://physics.stackexchange.com/questions/306912",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/142419/"
] | The problem is:
<blockquote>
A lunch pail is accidentally kicked off as steel beam on a building under construction. Suppose that the original horizontal speed is $1.5$ m/s. How far does the lunch pail fall after it travels $3.50$m horizontally?
</blockquote>
So, a fairly easy problem:
<ol>
<li>$t=\frac{d}{v} = \f... | <ul>
<li>When it is 3.5 m out horizontally, it is the $y$ meters down vertically. </li>
<li>So, if it is the 3.5 m out horizontally <strong>after time $t$,</strong> where is it then vertically at that moment? </li>
</ul>
It is the $y$ meters down, of course. The horizontal and vertical distances share the time $t$.
| The time in the equation represents some time during the projectile motion. The equations all relate to what you choose as the coordinates and initial conditions.
In this case you want to find how far it falls after it travels a horizontal distance. So the time in this case should represent time in the air until the... | https://physics.stackexchange.com |
65,139 | [
"https://softwareengineering.stackexchange.com/questions/65139",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/12750/"
] | In Python, and most likely many other programming languages, common data structures can be found as an integrated part of the <em>core language</em> with their own dedicated syntax. If we put LISP's integrated list syntax aside, I can't think of any other languages that I know which provides some kind of data structure... | It depends what the language is for.
Some examples (somewhat stolen from other answers):
<ul>
<li>Perl has special syntax for hashtables, arrays, strings. Perl is often used for scripting, these are useful for scripting.</li>
<li>Matlab has special syntax for lists, matrices, structures. Matlab is for doing matrix ... | Perl has hashmaps and PL/SQL supports records, and I have very foggy memories of matlab having syntax to support vectors and matrices of all different dimensions (though I could be wrong about this one and it could be argues that these are data <em>types</em> not data <em>structures</em>)... I would say that having som... | https://softwareengineering.stackexchange.com |
111,476 | [
"https://mathoverflow.net/questions/111476",
"https://mathoverflow.net",
"https://mathoverflow.net/users/27221/"
] | I want to derive a Gronwall-type inequality from the inequality below. Here all the functions are nonnegative, continuous and if you need some assumptions you may use that.
$$ f^2(t) \leqslant g^2(t) + \int_0^t (f(s) +c) f(s) ds \;\;\;\; (t \in [0,T]) $$
So please help!
| Consider the function
$$h(t):=\int_0^t f(s)e^{t-s}ds\,,$$
which solves the ODE $h'=h+f$ with $h(0)=0$, so $$h(t):=\int_0^t \Big(h(s)+f(s)\Big)ds\, .$$
Adding the term $-c\, h(t)$ to both sides, your inequality takes a more familiar form of a Gronwall inequality:
$$f(t)^2-c\;h(t)\le g(t)^2 + \int_0^t \Big(f(s)^2 -c\;h... | Consider g=0, c=2, f(t)=t.
By the way, what did you mean by "if you need some assumptions?" Are there folks out there who prove things without them?
| https://mathoverflow.net |
281,166 | [
"https://physics.stackexchange.com/questions/281166",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/130637/"
] | I've witnessed a cable rupture some time ago where it took 1-2 seconds for the energy of the apartment complex (100 metres away) to fall down.
Given the fact that theoretically (as far as I know) energy/electricity speed should be close to the velocity of <strong><em>c</em></strong>, why does it took that time for the... | I suspect that what you are really after is that $\epsilon$ is invariant under Lorentz transformations between inertial observes. In fact, we have a well known formula for the determinat of an operator:
$$ \epsilon_{i_1...i_n}A^{i_1}_{\ \ j_1}...A^{i_n}_{\ \ j_n} = \mathrm{det} A \epsilon_{j_1...j_n}. $$
All Lorentz tr... | If we are referring to the same Levi-Civita symbol, i.e. the completely antisymmetric symbol, then this is not a tensor. It is a tensor density of weight $-1$, which means that under a general coordinate transformation it gets multiplied by the inverse Jacobian of the transformation. So it is normal that you can't prov... | https://physics.stackexchange.com |
55,961 | [
"https://cs.stackexchange.com/questions/55961",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/8767/"
] | In a tree, I want to refer to a particular child of a node, the child of this child, the child of this child of this child, and then the child of this child of this child of this child. For instance, in a binary tree, I would like to refer to the right sibling of the node, the right child of this one, etc..
Can I call... | No, the phrase "the first four immediate descendants" doesn't say what you want. In particular, "first four" sounds like you're talking about four vertices, not four <em>levels</em> of vertices. If I read "immediate descendants" I'd be confused because it sounds like it should mean the same as "children" but surely yo... | There is no such name. The best way to describe what you want is, well, to specify it using language or mathematics. Don't expect an existing name for every concept you ever dream up; this is why we have language, so we can express novel concepts.
Personally, I don't find "the first four immediate descendants of a n... | https://cs.stackexchange.com |
11,117 | [
"https://mathoverflow.net/questions/11117",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3067/"
] | Any Eilenberg-MacLane space $K(A,n)$ for abelian $A$ can be given the structure of an $H$-space by lifting the addition on $A$ to a continuous map $K(A\times A,n)=K(A,n)\times K(A,n)\to K(A,n)$.
Does somebody know an explicit way to describe this structure in the cases $K({\mathbb Z}/2{\mathbb Z},1)={\mathbb R}P^\inft... | Look at $\mathbb R^\infty\setminus 0$ as the space of non-zero polynomials, which you can multiply. Pass to the quotient to construct the projective space and, from the multiplication, its $H$-space product.
The complex case is quite the same.
<strong>NB:</strong> Jason asks in a comment below if this is the same $H$... | There's also a different way of writing down the $H$-space structure, that I like for its algebro-geometric flavor. (I'll talk about $\mathbb{C}P^\infty$ here, and $\mathbb{R}P^\infty$ should be analogous.)
Regarding $\mathbb{C}P^\infty$ as a classifying space for complex line bundles, we know that this $H$-space str... | https://mathoverflow.net |
38,661 | [
"https://softwareengineering.stackexchange.com/questions/38661",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/22523/"
] | I have done or was part of some mobile projects that are uploaded in their respective markets and currently live, and I am already mentioning status as LIVE for respective projects.
One of my friend -who is actually a web developer- has recently suggested me to put the urls of the application if the project is live. A... | How many of these sites do you have? <strong>I can guarantee that a prospective employer is not going to click on more than 2 or 3</strong>.
Rather than listing them all on your CV, I would set up a "portfolio" website, where you can list all of your projects. Include a screenshot, a description of the project, your r... | I sometimes feel a little nervous clicking a shortened link. Perhaps you could do both, especially in a printed resume where someone will need to type the URL:
<blockquote>
Developed and deployed a missile tracking system (http://www.missiletrackingsystem.foo/entrypoint/starthere.abd - bit.ly/1234) which saved my em... | https://softwareengineering.stackexchange.com |
236,351 | [
"https://softwareengineering.stackexchange.com/questions/236351",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/126589/"
] | When I compile a C program, it is compiled first to assembly code, then assembled into machine code. I'm curious why it doesn't just convert straight to machine code in the first place.
| It depends on the compiler and the options you provide to the compiler. These days the most widely used compilers will write out machine language by default, but will generate an assembly listing if you request it. It can be helpful to have the assembly listing because a) sometimes compilers have bugs and you want to c... | It's possible for compilers to generate code which is suitable for use as input to a processor (for direct execution), a linker, an assembler, or some other kind of building tool. Which format is useful often depends upon what other code it will need to be combined with. If a compiler for language X which generates o... | https://softwareengineering.stackexchange.com |
146,279 | [
"https://security.stackexchange.com/questions/146279",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/128429/"
] | How does full-disk encryption work when the disk or system unexpectedly loses power? I assume that it doesn't just quickly re-encrypt the data in a split-second, so how does it protect data on the disk after power loss?
Does it keep most data encrypted and decrypt files when they are accessed? If so, wouldn't this als... | Full disk encryption ensures that data is never present on the drive in an unencrypted state, so sudden power loss does not result in any unencrypted data remaining on the drive. When your computer has to access data, it will read the encrypted data, then use the key you provided (also in RAM) so the data that gets sto... | Full disk encryption sits between the filesystem and the storage drive and seamlessly translates between the drive's encrypted sectors and the decrypted ones which it presents to the filesystem (which in turn translates those to files and folders). When the filesystem writes a sector the FDE layer encrypts that sector ... | https://security.stackexchange.com |
3,043,321 | [
"https://math.stackexchange.com/questions/3043321",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/612168/"
] | I fully apologize if this is a duplicate, but I really can’t find the answer to this online here. But a proof of the rule
<span class="math-container">$$\int a^x dx= \frac{a^x}{\ln(a)}+C$$</span>
would be really helpful. I need it for something I’m doing. This is all I need. I have no sources on the problem as I want... | I assume it's kosher to use the exponential integral with base <span class="math-container">$e$</span>, i.e. <span class="math-container">$\int e^x dx = e^x +C$</span>? Or, more generally, for a constant <span class="math-container">$k$</span>,
<span class="math-container">$$\int e^{kx}dx = \frac{1}{k}e^{kx}+C$$</span... | Just differentiate the right hand side and see what you get. Note that
<span class="math-container">$$
\frac{d}{dx}\left(\frac{a^x}{\ln a}\right)=\frac{1}{\ln a}\frac{d}{dx}(e^{x\ln a})=\frac{1}{\ln a}\times e^{x\ln a}\times \ln a=a^x
$$</span>
where we used the chain rule in the second equality.
| https://math.stackexchange.com |
293,856 | [
"https://mathoverflow.net/questions/293856",
"https://mathoverflow.net",
"https://mathoverflow.net/users/9833/"
] | I need the following result (which I believe to be true though I was too lazy to write down a complete proof).
Let $f$ be a function of two complex variables analytic at the origin and $a\not\in\mathbb{N}$. Then the differential equation
$$z'=\frac{az}{x}+f(x,z)$$
has an unique solution $z(x)$ which is analytic at... | These questions were studied for the first time by Briot and Bouquet in 19 century. For a modern reference see for example the book of E. Hille, Ordinary differential equations in the complex domain. The chapter is called Some equations of Briot and Bouguet.
Your case is actually simple: plug a formal power series for ... | Your equation falls in the category of regular singular differential equation. Writing your equation as
$$
x z'=a z+g(x,z),
\tag{$\ast$}$$
the singularity is called regular because the exponent of the singularity at zero is the same as the order of derivatives. Examples are Bessel equations
$
x^2z''+xz'+(x^2-\nu^2) z=... | https://mathoverflow.net |
45,341 | [
"https://electronics.stackexchange.com/questions/45341",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/1786/"
] | Okay imagine you have a voltage supply of 10V and one Resistor of 5 Ohms.
Now find the current : I=5/10 = 0.5A
So potential difference across that resistor : V=IR=10V .. which proves this statement I found in my lecture note :
<blockquote>
If no internal resistance is present in voltage supply, the potential diff... | <blockquote>
V=IR=10V .. which proves this statement I found in my lecture note
<blockquote>
If no internal resistance is present in voltage supply, the potential difference across the resistor is equal to supply voltage.
</blockquote>
</blockquote>
You don't need to measure anything to prove this statem... | <blockquote>
Okay imagine you have a voltage supply of 10V and one Resistor of 5 Ohms.
Now find the current : I=5/10 = 0.5A
</blockquote>
Wrong: if V = 10 V and R = 5 Ohms, then the current is given by Ohm's law:
$$
I = \frac{V}{R} = \frac{10}{5} = 2 \mathrm{\, A}
$$
That, given that the voltage supply is ide... | https://electronics.stackexchange.com |
3,622 | [
"https://softwareengineering.stackexchange.com/questions/3622",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/1785/"
] | I often develop an application entirely myself. But did I really?
I feel strange about that claim and never know when that is true.
I mean I designed it, coded it, but I used XYZ plug-in. Can I still claim that I did it all myself even though I didn't create the plugin I used?
Consider this conversation?
<blockquote>
<... | That depends on what "XYZ part" does. Most code libraries are pretty modular things that take care of one simple task. Even if someone else designed that, you're the one who integrated it into the overall vision of what your program is supposed to do.
Unless the third-party code takes care of a major part of the app... | Fundamentally be honest - at the end of the day you put the site or the app together so you can take credit for the end result as ultimately you did build it.
If you used a plugin you deserve credit for being smart enough to find and to know to use the plugin <em>but</em> in my case I do acknowledge (in some cases mak... | https://softwareengineering.stackexchange.com |
1,144,558 | [
"https://math.stackexchange.com/questions/1144558",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/181141/"
] | <strong>Problem:</strong>
<pre><code>When a company receives an order, there is a probability of .42 that
its value is over $1000. If an order is valued at over $1000, then there is a
probability of .63 that the customer will pay with a credit card.
</code></pre>
<strong>A.) What is the probability that the nex... | It is a lot simpler than you are making it to be. If $O$ is the event that the order is over 1000, and $C$ is the event that the order is paid with a credit card (regardless of the order amount), then you are told that $$\Pr[C \mid O] = 0.63,$$ hence $$\Pr[\bar C \mid O] = 1 - \Pr[C \mid O] = 1 - 0.63 = 0.37.$$ There... | Your first answer is correct. The probability of the conjunction of three independent events is the product of their individual probabilities.
Your second answer should be of the same form. However, here the probability of two dependent events must use conditional probability.
$$\mathsf P(\mathbf O \cap \math... | https://math.stackexchange.com |
263,073 | [
"https://security.stackexchange.com/questions/263073",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/259980/"
] | I'm pretty sure a lot of you are familiar with the "USB Rubber Ducky", the USB stick that emulates a keyboard. Is there software for Windows that will prevent this type of attack, while still letting me use legitimate keyboards? For example, is there a program that will prompt "Please enter the words you... | I'm not aware of any built into windows. However there are third party device control products that lock the screen when a new keyboard is added. This thwarts some of the "rubber ducky" attacks by preventing a simple set of keystrokes from running immediately on insertion of the usb device doing anything us... | I'm not aware of any such software, and in general, such software would likely be ineffective and potentially harmful. Consider the following scenarios.
First, one could create a real USB keyboard that contains a special chip that, upon activation with a special key sequence or upon remote activation, sends malicious ... | https://security.stackexchange.com |
220 | [
"https://cardano.stackexchange.com/questions/220",
"https://cardano.stackexchange.com",
"https://cardano.stackexchange.com/users/180/"
] | As the Cardano mainnet, will the Cardano testnet be maintained indefinitely and immutably or it will perish at some point in time?
| No, there are plans to reset the testnet in the future, probably sometime after Goguen.
But it is immutable forever, if you keep the blockchain data.
| Well, we can assume that it will be maintained indefinetely because Cardano will also will be under development.
In the same way as in system administration you have Dev, Staging and Production environments in Cardano we also need to have testnet before pushing that in mainnet.
| https://cardano.stackexchange.com |
342,489 | [
"https://stats.stackexchange.com/questions/342489",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/139047/"
] | I am trying to figure out how to make a correct statement with two values:
<code>X = 59.8%
Y = 26.7%</code>
Is this correct?
<code>X has 44.6% more apples than Y</code>
If I wanted to use the word "margin" what does that imply?
Can I say
<code>X has more apples than Y by a margin of 44.6%</code>
Also, is there... | In your specific case, as mentioned by whuber, both statements are incorrect.
First let me explain how they are incorrect using your example with simple figures.
<ul>
<li>The basket contained 100 apples to be shared between 5 people</li>
<li>X took 60% of the basket (= 60 apples)</li>
<li>Y took 27% of the basket (= ... | (1) X is 33.1 percentage points larger than Y.
(2) If you take this difference and divide it by X, you get 33.1/26.7 = 1.2397, so X is 124% more than Y.
(3) If you divide X by Y, you get 2.2397, so X is 224% of Y.
(4) Note that the number I got in (3) is 100% more than (2). This is because X is equal to Y plus the ... | https://stats.stackexchange.com |
9,144 | [
"https://dsp.stackexchange.com/questions/9144",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/1024/"
] | Take the simple frequency-domain band-pass filtering operation below . . .
<pre><code>NFFT = 128;
x = randn(NFFT,1);
H = zeros(NFFT,1);
H(10:20) = 1;
y = ifft(H.*fft(x), 'symmetric');
</code></pre>
This gives a real output because I'm using the conjugate symmetric flag to the ifft operation.
I want a function that r... | Conjugate symmetric means
$$f(-x) = f^{\ast}(x)$$
i.e. the sign of the imaginary part is opposite when $x<0$
The FFT of a real signal is conjugate symmetric. One half of the spectrum is the positive frequencies, and the other half is the negative. The negative coefficients are conjugate of the positive.
So if yo... | You just have to make sure that
$$H_k = H^*_{N-k},\quad k=1,2,\ldots N-1,\quad (N\ldots\text{FFT length})$$
and that $H_0$ is real-valued.
| https://dsp.stackexchange.com |
32,607 | [
"https://quant.stackexchange.com/questions/32607",
"https://quant.stackexchange.com",
"https://quant.stackexchange.com/users/19401/"
] | I need to prove that the butterfly condition is always positive under no arbitrage theorem. We are constructing a long butterfly using European call options
<pre><code>C(T,K+∆K) - 2C(K) + C(T,K-∆K) > 0 where ∆K < K
</code></pre>
I have managed to prove for greater than or equal to zero using the following steps... | You generally can't simply subtract two inequalities as you did in your attempt. Here are two approaches to solve your problem:
<strong>No-Arbitrage Argument</strong>
Assume that the initial value of the Butterfly spread was strictly negative $V_0 < 0$. Buying the butterfly spread would thus yield a strictly posit... | it's a model-free result. The pay-off is non-negative everywhere and positive somewhere.
Since it's non-negative everywhere, if its price was negative there would be a clear arbitrage.
We have to show positive. We assume that there is a positive probability that the stock lands in the region where the pay-off is po... | https://quant.stackexchange.com |
1,787,072 | [
"https://math.stackexchange.com/questions/1787072",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/327486/"
] | Boolean algebras aren't algebras (to the best of my understanding).
So why are they called algebras?
Wouldn't it make more sense to call them a "Boolean system" or a "Boology" or something else like that?
| Because Boole himself introduced the word "algebra" into the subject.
The term "algebra of logic" appears in Boole's 1854 book on Laws of Thought:
<blockquote>
Let us conceive, then, of an Algebra in which the symbols x, y, z, etc. admit indifferently of the values 0 and 1, and of these values alone. The laws, the ... | They are algebras, in the sense of <em>universal algebra</em> (where "algebra" is basically synonymous with "first-order structure", except that it requires the language to have no relation symbols).
In fact, I think this notion of algebra = algebraic structure long preceded the definition of algebra as (roughly) a mo... | https://math.stackexchange.com |
329,466 | [
"https://physics.stackexchange.com/questions/329466",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/148764/"
] | Suppose I push myself against the Earth:
<ul>
<li>Am I pushing the Earth away OR</li>
<li>Is it the Earth pushing me away?</li>
</ul>
| Suppose I push myself against the Earth:
<ul>
<li>I push the Earth away </li>
</ul>
<strong>AND</strong>
<ul>
<li>The Earth pushes me away with a force of the same magnitude and opposite in direction. </li>
</ul>
This is Newton's third law in action.
| Both. Force is defined as a rate of change of momentum, and net momentum change must be zero as momentum is conserved. If you and the earth can be seen to have momentum $0$ at $t=0$, at $t=t$ where you have jumped, and have upwards momentum, the earth must have 'downwards' momentum, such that $p_{net} = 0$ [and thus si... | https://physics.stackexchange.com |
411,418 | [
"https://softwareengineering.stackexchange.com/questions/411418",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/301421/"
] | I basically read across the entire internet that exceptions are bad, slow and should be avoided whenever possible. This confuses me because I thought I have a "valid" case to use exceptions.
Consider an input parser which transforms a string into an expression tree, that a calculator can evaluate. For me this seems pe... | There are no shortage of complaints about throwing and catching both within the same method. These are considered a form of goto, considered harmful.
Throwing an exception when we don't want to proceed — e.g. with parsing — is different and is reasonable usage of exceptions.
<hr>
However, in your c... | Exceptions should be used for exceptional cases. A parse error is not exceptional; an input that can be parsed correctly is probably more exceptional: -(
| https://softwareengineering.stackexchange.com |
61,671 | [
"https://dsp.stackexchange.com/questions/61671",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/43958/"
] | I'm studying some of the basic equalizing structres and I understand how Zero-forcing works, but it seems to me that a known channel impulse response is needed. Am I right? If so, what's the point? I mean, you're not likely going to know how the channel is, so how Zero-forcing is in any way useful?
| If the received signal can be written as
<span class="math-container">$$\mathbf{y} = \mathbf{H}\,\mathbf{x} + \mathbf{n}$$</span>
where <span class="math-container">$\mathbf{H}$</span> is the channel matrix, <span class="math-container">$\mathbf{x}$</span> is the transmitted vector, and <span class="math-container">$... | You need an estimate of the channel to receive the sequence but the zero-forcing equalizer does <strong>not</strong> need the channel response as an input. The zero forcing equalizer estimates the channel response. This can be done either with a training sequence, or can be decision directed when signal to noise ratios... | https://dsp.stackexchange.com |
138,652 | [
"https://physics.stackexchange.com/questions/138652",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/60400/"
] | I've read in many books and through many answers on this and other sites about this concept. Now I'm not sure what to "believe". In the subject at my university, we are taught that conducting wires are perfect. For example if in non-ideal wires electrons move randomly, but when you apply E-field they move in that direc... | If you have been given the explicit forms of the Hamiltonian $H$ and basis vectors $\vert j \rangle$, then you should compute the matrix elements directly as you suggested: $H_{ij} = \langle i \vert H \vert j \rangle$. However, based on the wording of the question, I suspect that this isn't the case. When the questions... | It seens that your question is about how to construct the $H$ matrix
,so since $<e_{i}|e_{j}> = \delta_{ij} $ and $H_{ij} = <e_{i}|H|e_{j}>$ you can use then to construct $H$ such ,
$H=\epsilon (|e_{1}><e_{1}|+|e_{1}><e_{2}|+|e_{2}><e_{1}|+|e_{2}><e_{2}|)$
The coefficient doesn'... | https://physics.stackexchange.com |
2,465,513 | [
"https://math.stackexchange.com/questions/2465513",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/362595/"
] | I'm new to 3D-coordinate systems and this is a question I faced in my textbook.
I first tried to imagine the entire thing in 2D: so the same question for the sphere $(x-1)^2+(y-1)^2=1$. It's much simpler to answer this. Keeping in mind that $\sin\left(\frac{\pi}{4}\right)=\cos\left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{... | As Ross says, the point of minimum and maximum distance from origin will lie on line joining centre of sphere with origin.
The centre of sphere is $\langle1,1,1\rangle$ and radius is $1$ unit. So a line passing through origin $O$ and centre $C$ will be $L$, given by:
$$\begin{align}
\vec{L}(r) &= \dfrac{r}{\sqrt{... | Draw a line from the origin through the center of the sphere. The point where it hits the sphere on the opposite side from the origin is the farthest point. Your answer is incorrect because the $z$ coordinate of the center is $-1$, not $+1$
| https://math.stackexchange.com |
81,534 | [
"https://mechanics.stackexchange.com/questions/81534",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/62027/"
] | I have a 2003 Ford Mustang with 225/55R16 tires and the original rims. All of the tires are very bald so I bought new 225/55R16 tires (with no rims). I'd like to install the new tires on the old rims. I'm being told that this can only be done at a shop; is there any way to do it from home?
| Mounting tires from home is not something that can be done safely, for 2 reasons:
<ol>
<li>Tires need to be stretched to get on and off the rim, there's a special tool to do this without damaging the tire.</li>
<li>Once the tire is on the rim, you need to balance the wheel and tire. This involves adding lead weights in... | It is possible for SOME . But a very bad idea for nearly everyone.
| https://mechanics.stackexchange.com |
7,679 | [
"https://chemistry.stackexchange.com/questions/7679",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/4119/"
] | I want to learn chemistry. I did not have the benefit of studying it when I was in highschool, and I was always intimidated by it (I feel it's more rote memorization rather than understanding - I'm more of a Physics and Math kinda guy).
Anyway, what are the concepts that I need to learn?
<strong>How, and what resourc... | You can get the valence electrons in an atom's electronic arrangement by consulting the periodic table:
<img src="https://i.stack.imgur.com/VrFqx.jpg" alt="enter image description here">
<ul>
<li>The Group 1 atoms have 1 valence electron.</li>
<li>The Group 2 atoms have 2 valence electrons.</li>
<li>The Group 3 atoms... | Yes there is! For all elements in the following groups;
Group 1A= +1
Group 2A= +2
Group 3A= +3
Group 4A= +4 or -4
Group 5A= -3
From group 5, the valency descends in this order;
-3, -2, -1, 0.
Group 8A elements have a valency of Zero.
I hate complicated stuff too lol.
| https://chemistry.stackexchange.com |
15,271 | [
"https://mathoverflow.net/questions/15271",
"https://mathoverflow.net",
"https://mathoverflow.net/users/717/"
] | (I removed my motivation because it may be misleading :) )
Let $A$ be a noetherian commutative ring and let $M \neq 0$ be a finitely generated zero-dimensional (i.e. $\mathrm{dim} \ \mathrm{Supp}(M) = 0$) $A$-module. Then the submodule $0 < M$ has primary decomposition $0 = \bigcap_{\mathfrak{p} \in \mathrm{Supp}(... | If $M$ is finitely generated and has $0$-dimensional support, then $M$ is in fact supported at finitely many maximal ideals (a $0$-dimensional closed subset of the Spec of a Noetherian ring is just a finite union of maximal ideals), and one has the isomorphism
$M = \oplus_{\mathfrak p} M_{\mathfrak p} = \oplus_{\math... | I could answer this now myself (so, sorry for asking!): From a general result (presented e.g. in Eisenbud's commutative algebra book) it follows that $M \lbrack \mathfrak{q}^\infty \rbrack = \bigcap_{\mathfrak{p} \in \mathrm{Supp}(M), \mathfrak{p} \neq \mathfrak{q}} M(\mathfrak{p})$ and then it follows that $\mathbf{j}... | https://mathoverflow.net |
203,936 | [
"https://softwareengineering.stackexchange.com/questions/203936",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/26494/"
] | I have VB.Net project, with a class with a couple of abstract (mustoverride) properties, that I am thinking about moving into an interface.
My question, is structurally how do I do this? Currently all of the classes that would be implementing this are derived from this base class, so it might make sense to put it in ... | Common guidance in .NET land, in most cases, is to have every class and interface in their own file. With a 1:1 correspondance between files and classes, navigation becomes easier. This argues against option #1.
Regarding the choice between #2 and #3, I would argue for the former, with some caveats. I don't think yo... | I would group all my classes and interfaces in a separate folder. Everything should be grouped by functionality and not by whether it is a base class, an interface or whatever.
What would happen if you had a complex design pattern, would you subdivide the interfaces in separate folders?
| https://softwareengineering.stackexchange.com |
206,967 | [
"https://softwareengineering.stackexchange.com/questions/206967",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/95501/"
] | <strong>Brief History</strong>
I am new to Automated Unit Testing and Mocking objects world; previously we used to do Unit Testing (including Integration Testing and we mistakenly referred that as Unit Testing) manually, but now <strike>we have planned to change all of that to</strike> I am pushing our organization to... | Unit tests are just one level of the hierarchy of automated tests. Unit tests exist to verify that the code you (the developer) have actually written behaves the way you thought it should when you wrote it.
There are two caveats inherent in unit testing. First, coverage is not exercise. You may execute every line of c... | You want functional tests that run through all components working together. A test where you give your application an input and have it work with all the actual objects and then check your output. These tests verify that the connection between the tests work (as well as mocked methods actually exist).
Even with writ... | https://softwareengineering.stackexchange.com |
579,856 | [
"https://math.stackexchange.com/questions/579856",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/111208/"
] | <blockquote>
$f: \mathbb N\to\mathbb N$, $f(m)+f(n)=f(m+n)$ for all $m,n\in\mathbb N$, and $f(5)=45$. Find $f(2013)$.
</blockquote>
I messed up my original posting, its fixed now.
I changed $m+ n$ to $f(m+n)$.
| The question now says that $f$ is linear (which is very different!).
Hint: $kf(1)=f(1)+\ldots +f(1)=f(1+\cdots +1)=f(k)$ and so if $f(5)=45$ then $5f(1)=f(5)=45$. Can you find $f(1)$ from this?
<hr>
<em>(This was the contradiction that appears using the previous, unedited question.)</em>
No such $f$ exists. If $\ma... | If $f(n+m)=f(n)+f(m)$, you notice that
$$
f(2)=f(1+1)=2f(1),
$$
$$
f(3)=f(2+1)=f(2)+f(1)=\cdots
$$
With an induction argument you should be able to show that $f(2013)=18117$.
| https://math.stackexchange.com |
2,166 | [
"https://computergraphics.stackexchange.com/questions/2166",
"https://computergraphics.stackexchange.com",
"https://computergraphics.stackexchange.com/users/56/"
] | I know that if you turn of vsync, it synchronizes rendering with the vertical redraw cycle to prevent tearing, and that doing so caps your rendering rate (FPS) at the monitor refresh rate, which is commonly 60hz / 60 fps, although other rates exist as well.
However, when you are not running at a full 60fps how does vs... | The short answer is It's Complicated™. :)
A lot of factors can affect frame timing (and the associated problem of animation juddering, due to the game's animation delta-times not matching actual frame delivery times). Some of the factors are: how CPU-limited versus GPU-limited the game is, how the game's code measures ... | It depends on how a missed frame is handled by the driver.
One option is to just wait until the next vsync, causing a hitch of 32 ms and if the application is just at the limit of 16 ms can cause fluctuations.
The next option is to queue the frame for display next frame but don't wait on it. This will still cause a v... | https://computergraphics.stackexchange.com |
2,481,884 | [
"https://math.stackexchange.com/questions/2481884",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/269497/"
] | <blockquote>
If $x,y,z$ are positive reals satisfying $\frac{1}{3} \le xy+yz+ zx \le 3$, then find the range of values of $x+y+z$
</blockquote>
This is from BMO 1993. I have applied AM GM and CS in every possible way my brain can think of. I do not think my working will be of any help to anyone, so I am not showing.... | For the lower bound use the fact that $(x+y+z)^2 \geq 3(xy+xz+yz) \geq 3$ so $x+y+z \geq 1$. The equality is satifised for $ x=y=z=\frac{1}{3}$
To prove there is no upper bound pick $x=n, y=\frac{1}{n}, z=\frac{1}{n} $ for some natural number $n$. Then the conditions are satisfied.
| We have
\begin{eqnarray*}
(x-y)^2+(y-z)^2+(z-x)^2 \geq 0. \\
\end{eqnarray*}
Now add $3(xy+yz+zx)$ and complete the square
\begin{eqnarray*}
(x+y+z)^2 \geq 3(xy+yz+zx) \geq 1. \\
\end{eqnarray*}
So $1$ is a lower bound.
| https://math.stackexchange.com |
540,657 | [
"https://physics.stackexchange.com/questions/540657",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/226851/"
] | Trying to understand quantum information. Need some help :(
What does this notation <span class="math-container">$$ \langle\alpha|\hat{n}|\alpha\rangle $$</span> mean? Here <span class="math-container">$$|\alpha\rangle$$</span> is a coherent state and <span class="math-container">$$\hat{n}$$</span> is photon-number op... | The previous answer gives the meaning, here is the calculation.
The coherent state <span class="math-container">$\mid \alpha\rangle$</span> is an eigenvectior of the annihilation operator
<span class="math-container">$\mathsf{a}$</span>, with
<span class="math-container">$$\mathsf{a}\mid \alpha\rangle= \alpha \mid \... | If you had many systems in the state <span class="math-container">$|\alpha\rangle$</span> and you were to make measurements of the photon number for each of those systems, then <span class="math-container">$\langle\alpha|\hat n|\alpha\rangle$</span> is the expectation value of those measurements.
This can be calculate... | https://physics.stackexchange.com |
103,287 | [
"https://mathoverflow.net/questions/103287",
"https://mathoverflow.net",
"https://mathoverflow.net/users/25054/"
] | My question is:
Can we judge a manifold that can admit a (p,q) metric?
I only know the case that the existen of a lorentz metric is equivalent to Euler Character is zero
| The criterion for existence of a $(p,q)$ metric is (assuming $p+q=dim X$) that the tangent bundle splits as a direct sum of two subbundles of dimensions $p$ and $q$.
EDIT : I doubt there is an easy algebraic topology criterion in general, as characteristic classes [EDIT after Lennart Meier's comment: other than Eule... | I believe that you're a bit mistaken about the final claim. The correct statement should be that:
<blockquote>
There exists a "time-orientable lorentz" (i.e. a lorentzian metric with a nowhere vanishing timelike vector field) metric if and only if there exists a nowhere vanishing vector field (which happens if and o... | https://mathoverflow.net |
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