qid int64 1 4.65M | metadata listlengths 3 3 | prompt stringlengths 31 25.8k | chosen stringlengths 17 28.2k | rejected stringlengths 19 40.5k | domain stringclasses 28
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43,033 | [
"https://security.stackexchange.com/questions/43033",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/31302/"
] | I am trying to develop a PHP function to calculate NTLMv2 hash by providing server & client challenge, domain, account and password. The resulting hash appears to be not correct. Can someone guide me about the correct way:
Here is the algorithm in PHP language:
<pre><code>function GenerateNTMLv2($password, $acco... | I sorted out the correct algorithm for PHP. Thanks for the contribution. Following is the code:
<pre><code>function GenerateNTMLv2($password, $account, $domain, $client_challenge, $server_challenge) {
$unicode_password= iconv ( 'UTF-8', 'UTF-16LE', $password );
$NTLM_Key = mhash ( MHASH_MD4, $unicode_passwor... | You probably have encoding issues:
<ul>
<li>Beware that hash function process <em>bytes</em> but your strings may contain <em>characters</em> or even <em>hexadecimal digits</em>. For instance, your <code>$client_challenge</code> is a string which begins with <code>0101</code>, probably meaning that it <em>should</em> ... | https://security.stackexchange.com |
178,354 | [
"https://mathoverflow.net/questions/178354",
"https://mathoverflow.net",
"https://mathoverflow.net/users/6989/"
] | The most general way I can formulate my question is the following:
Question 1: <em>Given a Gorenstein quotient ring $S$ of a polynomial ring over a field $K$, can one construct a (topological) space $X$ such that the (even degree part of the) singular cohomology ring of $X$ with coefficients in $K$ is isomorphic to $... | I did not understand exactly your question. What is important here, is the ring $k$. If $k=\mathbf{Z}$ then the problem you are asking for is hard, if $k$ is any commutative ring then the problem is very hard. If $k$ is a field of characteristic 0 or $\mathbf{F}_{p}$ then there is always a solution. When the characteri... | As Matthias Wendt says, the general problem is hard. However, if $K$ has characteristic zero, the problem is completely solved by Sullivan's approach to rational homotopy theory: such a topological space always exists. See his paper "Infinitesimal computations in topology". Briefly you should take your algebra $S$, con... | https://mathoverflow.net |
57,073 | [
"https://electronics.stackexchange.com/questions/57073",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/11836/"
] | I have a system that was designed for testing digital components. I have a device in the system which is used for generation and acquisition of digital signals. Now, I will happily admit to not fully understanding everything I'm about to say, but:
<ul>
<li>The outputs of the device have a 50Ω nominal impedance,</li>... | First: Yes, your understanding is essentially correct, other than the issue being voltage and not charge.
<strong>Here is my analogy:</strong>
Consider a <strong>door</strong> to a house, with really smooth hinges, and no bolt or latch. The door is so light and so well-hinged that the slightest breeze would cause it ... | A pull up/down does three things.
1, it stabilizes the line, with a fixed reference (V+ for a pull up, or Gnd for a pull down, in most cases). The line will not float around. This could also be done without the pull-up, by directly connecting it to V+ or Gnd. This is a problem, which part 2 fixes.
2, <strong>it prote... | https://electronics.stackexchange.com |
60,342 | [
"https://chemistry.stackexchange.com/questions/60342",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/1358/"
] | Just reading through organic chemistry book (Modern Projects in Organic Chemistry by Mohrig, et al.), and it seems like the choice of solvent is usually arbitrary.
One of the experiments designed uses hypochlorite ion as the oxidizer and acetone as a solvent to oxidize 4-<em>tert</em>-butylcyclohexanol, but it isn't ... | There is no list per se, but there is certainly a process to determine appropriate solvents for a given reaction. For the specific reaction you mention, here are some criteria you would consider to select a solvent:
<ol>
<li>Because this is an oxidation reaction, choose a solvent that cannot be oxidized, known as an ... | I did organic chemistry in the lab for 9 years, and never once was there such a list. There are general rules for protic vs. aprotic and polar vs. non-polar, but a lot of times, when a reaction is published, the authors did a study of different kind of solvents and selected the "best" based on yield, reaction rate, eas... | https://chemistry.stackexchange.com |
285,540 | [
"https://softwareengineering.stackexchange.com/questions/285540",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/182092/"
] | Often times at work we opt to create views in the database to expose the data that we want to work with instead of building some monster query in our code. Being somewhat new to this field my assumptions are that the view is just another query that is added onto whatever query I pass at run-time.
Sometimes we are ret... | <blockquote>
is it appropriate to hard-code strings that are very unlikely to change during the lifetime of an application? Of course I can't guarantee 100% that they won't change, but the risk vs cost is almost trivial to weigh in my eyes - <strong>hardcoding is the better idea here</strong>
</blockquote>
It looks ... | You are reasoning at the wrong scope.
You haven't hardcoded only individual verbs. <strong>You have hardcoded the <em>language</em> and its <em>rules</em>.</strong> This, in turn, means that your application cannot be used for any other language, and cannot be extended with other rules.
If this is your intent (i.e. u... | https://softwareengineering.stackexchange.com |
635,704 | [
"https://electronics.stackexchange.com/questions/635704",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/153707/"
] | I want to be able to accurately measure the static 3D orientation of a drone I am building.
I understand one way to do this is to use a gyro sensor with continuous integration.
I want to combine this data with magnetic/compass data for improved accuracy and possibly automatic calibration.
I have also heard of "ori... | This can be done (transistors are very flexible and can by used in a myriad of ways). It can be useful (mostly in ICs) as part of a 'current mirror' where you want 2 or more devices to run the same current.
In addition, a diode like this can be 'better' than a standard (e.g. 1N4148) diode with lower leakage (but worse ... | Yes. This is commonly called a "diode connected transistor".
| https://electronics.stackexchange.com |
109,448 | [
"https://mathoverflow.net/questions/109448",
"https://mathoverflow.net",
"https://mathoverflow.net/users/9091/"
] | Let $H$ be a Hilbert space, $A$ be a normal bounded operator on $H$ with spectrum $\sigma(A)=\{\lambda\in \mathbb{C}\;|\;A-\lambda Id \text{ is not invertible }\}$. Is $\sigma\left(\dfrac{A-A^*}{2i}\right)$ the set of imaginary part of the elements of $\sigma(A)$ ?
Thanks.
| There is probably an elementary proof, but that's an immediate consequence of the continuous functional calculus :
$ \frac{A-A^*}{2i} = f (A) $
where $f$ is the imaginary part function on $\mathbb{C}$. And when you apply a continuous function $f$ to a normal operator $A$ you have : $Spec(f(A)) = f(Spec(A))$ (you ca... | The spectral theorme assert that there exist a unitary operator $U$ sucht that
$U^{*}AU$ is the multiplication operator by $\lambda$, $\lambda \in \mathbb{C}$. With the same unitary opeartor $U$, we have the real part (resp. the immaginary part) of $A$ mapped to $real(\lambda)$ resp. $imag(\lambda)$, with $real(A)=(1... | https://mathoverflow.net |
322,140 | [
"https://mathoverflow.net/questions/322140",
"https://mathoverflow.net",
"https://mathoverflow.net/users/78539/"
] | Let <span class="math-container">$\Theta$</span> be a subset of a metric space. Suppose <span class="math-container">$(X_\theta)_{\theta \in \Theta}$</span> is a random process on <span class="math-container">$\Theta$</span> which is <span class="math-container">$L$</span>-Lipschitz and with the property that there exi... | If for all <span class="math-container">$\theta \in\Theta$</span> we have <span class="math-container">$P(X_\theta\ge\epsilon)\le
A\exp(-B\epsilon^2)$</span> and <span class="math-container">$\Theta$</span> has <span class="math-container">$\epsilon$</span>-packing number <span class="math-container">$M(\epsilon)$</spa... | Disclaimer: This is just a detailed version of Aryeh's answer.
So, let <span class="math-container">$\mathcal C_\epsilon$</span> be <span class="math-container">$\epsilon/2$</span>-cover for <span class="math-container">$\Theta$</span> of minimal cardinality <span class="math-container">$N(\Theta;\epsilon/2)$</span>. ... | https://mathoverflow.net |
438,636 | [
"https://physics.stackexchange.com/questions/438636",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/209814/"
] | If gravity disappeared, would Newton's third law make everything that was pressed to the ground by gravity get pushed upwards?
| As other answers explain, Newton's third law wouldn't push you upwards, because reaction disappears as soon of action (gravity) vanishes.
However, we need to keep in mind that we are siting on several thousand kilometres of rock heavily compressed by its own weight. If weight suddenly disappears, that rock will react... | Yes, but in almost all cases the push would be imperceptible.
Reaction forces from surfaces occur when the molecules in the wall are displaced from their equilibrium position. The harder they are pushed the more they are displaced, and the more they are displaced the harder they pushed back. When you stand on a surfac... | https://physics.stackexchange.com |
14,865 | [
"https://physics.stackexchange.com/questions/14865",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/4459/"
] | Does it make sense to say, "The speed of light varies?" Some may say right off the bat "Yes, it changes as a wave passes through a different medium." However, I'd like to say no, because when I hear someone say the speed of light, I always think of the constant $c$ (unless the medium is specified to not be a vacuum, ... | I would generally say that most physicists mean "speed of light in a vacuum" when they say "speed of light," and therefore would say that the "speed of light is constant." If it is in a field that often deals with light propagation in materials (optics, condensed matter), people are usually pretty careful to say "speed... | The speed of light is constant. The reason the observed speed of light seems to slow down as it passes through a medium is because the photons interact with the particles of the medium. The photons never actually slow down, we just perceive their interaction with the medium.
| https://physics.stackexchange.com |
1,347,130 | [
"https://math.stackexchange.com/questions/1347130",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/124537/"
] | An island people does not use odd numbers. instead of counting 1,2,3,4,5,6 they count as 2,4,6,8,20,22....what number they use instead of 111?
<img src="https://i.stack.imgur.com/V4vbW.png" alt="enter image description here">
for 50, they use 400, so for 100 they use 800, so for 811 they should use 822. Is there any... | I guess it is sufficient to compute the numbers to base 5 and then replace the digits 0,1,2,3,4 by 0,2,4,6,8.
So $111= 4*5^2 + 2*5^1 +1*5^0$ yields the representation 421 and then you get 842 as a result.
<hr>
More generally, if they only use $k$ symbols as digits, than the way they count is exactly the same as coun... | Your table revealed the key there is a pattern for 4 numbers on a sequence such as n+n+2+n+4 +n+6 and then it stops and we + 12 and then we go back to our 4 sequence (look horizontal line , second line ) and if you look to the 4th line , horizontally you will see that the same pattern is repeated , but as you may have... | https://math.stackexchange.com |
733,040 | [
"https://physics.stackexchange.com/questions/733040",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/337240/"
] | I've always found using the righthand rule to remember how forces, B-fields, and particle velocities to be intellectually cheating myself a bit. It feels like being able to multiply numbers by using your fingers without knowing what multiplication really is. Yet, when I try the "just think about it approach" ... | <blockquote>
How can I think about the directions of the electromagnetic right hand rule vectors to intuitively understand what's going on, rather than using the equivalent of a cheat sheet?
</blockquote>
You can't intuitively understand it because it is a convention. We chose the "right-hand" rule and it is ... | To expand on hft's answer: in my opinion, the most "physically mature" way to think about this is to internalize the (rather Zen) understanding that pseudovectors (like torque or magnetic field), which result from cross products, <em>don't have any physically meaningful oriented direction at all</em>.
I used ... | https://physics.stackexchange.com |
14,719 | [
"https://quantumcomputing.stackexchange.com/questions/14719",
"https://quantumcomputing.stackexchange.com",
"https://quantumcomputing.stackexchange.com/users/13859/"
] | I have successfully installed the Qiskit in Anaconda by using <code>pip install qiskit</code>.
But I can't import Qiskit and get the following error.
<pre><code> ---------------------------------------------------------------------------
OSError Traceback (most recent call last)
<... | Reinstalling numpy didn't work for me.
I delete the <code>C:\Users\USERNAME\AppData\Roaming\Python\Python38\*</code> folder because that was left behind by a previous Python which I had uninstalled.
And this work for me.
Thanks to everyone who all answered my questions and I appreciate it.
| Try the following command and get back if errors persist
<code>from qiskit import *</code>
| https://quantumcomputing.stackexchange.com |
3,483,901 | [
"https://math.stackexchange.com/questions/3483901",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/623665/"
] | I understand that the harmonic series diverges because of the comparison test: <span class="math-container">$$1+\frac12+\frac13+\frac14+\frac15+\frac16+\frac17+\frac18+\cdots\\\ge1+\frac12+\frac14+\frac14+\frac18+\frac18+\frac18+\frac18+\cdots$$</span>However, I cannot intuitively understand why this is the case if <sp... | We have <span class="math-container">$(4+r)^2=(4-r)^2+4^2$</span> by drawing a right triangle. Thus <span class="math-container">$r=1$</span>. Thus the area is <span class="math-container">$\pi$</span>.
| Draw a right-angled triangle with vertices at the centre of the big square, the small circle and on one of the centres of the big semi-circles. Let r be the radius of the small circle. What do you see?
| https://math.stackexchange.com |
239,202 | [
"https://mathoverflow.net/questions/239202",
"https://mathoverflow.net",
"https://mathoverflow.net/users/13915/"
] | Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$ on $\{ 0<\delta < \varepsilon\}$, and that it satisfies the eikonal equation
$$ \| \nabla \delta \| = 1, \qquad... | Since this question seems to have attracted some interest, I will post my own answer (which is a proof of the statement in the comments by Anton).
<strong>If $u : M \to \mathbb{R}$ is a $C^2$ solution of the Eikonal equation
$$ \| \nabla u\| = 1, $$
then $\mathrm{Hess}(u)$, which a priori is only continuous, is smooth... | If $u$ is smooth, the left-hand-side in $(\star\star)$ could be rewritten as $(\nabla u)(\Delta u)$. Assuming that $u$ is $C^2$ this expression $(\nabla u)(\Delta u)$ is always defined, while your original expression $\nabla u \cdot \nabla \Delta u$ might be undefined.
Indeed assume $\gamma$ is a unit-speed geodesic s... | https://mathoverflow.net |
276,045 | [
"https://physics.stackexchange.com/questions/276045",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/116137/"
] | I haven't actually been able to find any information about this, I know LIGO insulates against certain types of EM radiation but I was wondering about radio waves in particular. Can anyone help?
| The final velocity of the dropped ball is the same as the initial velocity of the thrown ball because
<ul>
<li>they experience the same acceleration</li>
<li>they travel the same vertical distance</li>
<li>they take the same time to travel the same distance. </li>
</ul>
without all of these three conditions the velo... | This is not true. Let's see this in full details:
If one body is being droped and the other is being thrown up, forgetting about air resistance, the only important force is the weight of the bodies.
So, for starters, lets consider one body of mass $m$. The weight is $\mathbf{F} = -mg\mathbf{\hat{y}}$. Now, applying N... | https://physics.stackexchange.com |
79,674 | [
"https://mathoverflow.net/questions/79674",
"https://mathoverflow.net",
"https://mathoverflow.net/users/9915/"
] | Dear all,
I am thinking about a problem as follows:
Suppose a simply-connected 2-dimensional manifold has an $S^1$ boundary, is it homeomorphic to the open disk $D^2$? In fact, I would like to understand the general higher-dimensional case, i.e., how to decide the homeomorphism type of a (connected) manifold $M$ via ... | Yes, assuming the manifold compact. One way to see this is this: if we glue a disk along the boundary, we get a manifold which is simply-connected manifold by Seifert-van Kampen, and closed, hence a sphere by the classification theorem.
The same argument plus the Whitehead theorem and the topological Poincare conjectu... | I'm wondering whether there isn't a much easier argument for 2-dimensional case, without using Seifert-van Kampen or the classification of surfaces (which is a middling hard theorem).<br><br>
Fix the boundary of a collar $f\colon\, S^1\to M$ of the $S^1$-boundary of your simply-connected 2-manifold M, and contract it ... | https://mathoverflow.net |
108,009 | [
"https://physics.stackexchange.com/questions/108009",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/21146/"
] | When we deal with Special Relativity and we start considering spacetime instead of space and time each at once, we usually see books saying that we consider a space with four coordinate $x^\alpha$ with $x^0 = ct$. We also consider this manifold to be $\mathbb{R}^4$ and give to it the metric tensor $g = \operatorname{di... | But that's exactly the deeper meaning! Setting things up so that all coordinates are in the same units (besides being a reasonable requirement for $x^\mu$ to be considered a four-vector) is a constant reminder that time is really not that different from space. In fact, if it weren't for that sign in the metric, spaceti... | <blockquote>
Why do we set x0=ct instead of x0=t?
</blockquote>
Nobody holds us to set x0=t and measure the time in centimeters of the travel of the clock's pointer. Or measure it by the change in the circumference of the earth during the day night cycle, or... It is a matter of calibrating time to centimeters in te... | https://physics.stackexchange.com |
672,297 | [
"https://math.stackexchange.com/questions/672297",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/113640/"
] | I want to prove the interval $(1,5)$ is a convex set.
A convex set is a set having all the convex linear combinations of its point in it, where a convex linear combination is a linear combination of the form $X=(1-\alpha)X_1+ \alpha X_2$ where $\alpha$ lies between $0$ and $1$ and the sum of coefficients of above lin... | You just have to prove that for every $\alpha \in (0,1)$ and every pair $x_1$, $x_2$ $\in (1,5)$, $\alpha x_1 + (1-\alpha)x_2$ also lies in $(1,5)$.
Suppose without loss of generality that $x_1 \leq x_2$. Then for every $\alpha \in (0,1)$ (which implies that both $\alpha$ and $1-\alpha$ are positive)
$\alpha x_1 + (1... | If $x_1\leq x_2$ then $$x=(1-\alpha)x_1+\alpha x_2=x_1+\alpha(x_2-x_1)$$ If $\alpha \in [0,1]$ then $$0\leq \alpha(x_2-x_1)\leq x_2-x_1$$ so: $$x_1\leq x\leq x_1+(x_2-x_1)=x_2$$
Then $x_1,x_2\in (1,5)$ gives: $$1< x_1\leq x\leq x_2<5$$
| https://math.stackexchange.com |
251,361 | [
"https://stats.stackexchange.com/questions/251361",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/141981/"
] | I sometimes see job descriptions where R is posted as analytical tool while others mention R as programming language.
What is the difference between programming in R and performing analysis in R?
UPDATE as of 10/17/2020:
lately, new elements popping in job descriptions made me realize that my old question was too vag... | There really is not much of a difference. Or it is a difference that does not matter much.
One could say that programming is writing a set of steps to do some process, while analysis is actually <strong>doing</strong> the process. So strictly, you could say you are programming R when you design a function or procedure... | R is both, its a statistical tool but there is no drop and click interface you have to use the R programming language to perform the analysis you want to do. I would assume performing analysis in R and R programming are synonymous unless the context states otherwise: for example, if someone wants a R package built they... | https://stats.stackexchange.com |
21,321 | [
"https://electronics.stackexchange.com/questions/21321",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/6272/"
] | Lets say an ideal AA battery I have provides 2.0 Watt hours. It could be different, just a quick search for this example.
My very simple load is the battery connected to a 5mm LED. The LED has a forward voltage of 2 or so volts and ideal forward current is 20 mA for good lighting.
Here are my calculations:
<pre><cod... | Your AA battery won't provide the 2V for the LED, so you'll need two of them in series. I guess the 1.2V is the cell voltage, so you have 2.4V for two cells. Your LED would then be dissipating 20mA x 2V = 40mW, but the total load on the battery is 20mA x 2.4V = 48mW. The extra 8mW will be dissipated in the LED's series... | <ul>
<li>See "glossary" at end re mAh, Watts = power , Watt.second = energy..</li>
<li>Batteries should be rated in Ah capacity for calculation purposes when a linear regulator or constant current load is used. </li>
<li>Use of Wh capacity may be more appropriate when a switching regulator is used, provided that correc... | https://electronics.stackexchange.com |
10,990 | [
"https://cstheory.stackexchange.com/questions/10990",
"https://cstheory.stackexchange.com",
"https://cstheory.stackexchange.com/users/186/"
] | As you know there are many anomolies for the single tape Turing machines when the time is $o(n^2)$: multi-tape TM simulation, simulation of larger tape alphabet with just $\{0,1,b\}$, time constructability, non-tightness of time hierarchy theorem, ...
Also results like $\mathsf{DTime}(o(n\lg n)=\mathsf{Reg}$, and very... | The other answers look very nice. I'd like to share a comment Russell Impagliazzo made years ago in a lecture, which has stuck with me ever since.
<blockquote>
I think Turing may have preferred a single tape TM due to physical plausibility.
</blockquote>
I pointed Russell to this thread days ago but, seeing as he'... | I've seen texts define TIME( $f(n)$ ) using multi-tape Turing machines, but Sipser uses a single tape machine. You've almost surely first encountered this material through Sipser because it's so fabulously well written.
There is a crystal clear pedagogical reason why Sipser does this, namely the course just naturally... | https://cstheory.stackexchange.com |
12,262 | [
"https://physics.stackexchange.com/questions/12262",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/4034/"
] | If we were to build a high speed rail up the side of a mountain like in some Science Fiction movies, what is the velocity needed at the point of leaving the mountain excluding angular momentum from Earth’s rotation to achieve escape velocity?
| The total escape velocity is about 11,200 m/s (approximately 7 miles/sec.) in a direction roughly tangential to the earth's surface. At 30 degrees north latitude (e.g., somewhere in southern Texas) the eath's spin would contribute about 400 m/s to the tangential velocity, so actual speed, relative to the earth's surfa... | Escape velocity has got nothing to do with mass of the rocket.
$v_e = \sqrt{\frac{2GM}{(r_0+h)}}$
$r_0 -$ Radius of the earth<br>
$h -$ Height of the mountain
| https://physics.stackexchange.com |
223,140 | [
"https://dba.stackexchange.com/questions/223140",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/153693/"
] | Suppose I have the following table<br>
<strong>data(partitioned_key_index, some_dummy_measure)</strong>
Assume that partitions are of an equal large size. with Oracle 11g.
The end result should be like this
<pre><code>select partitioned_key_index, sum(some_dummy_measure)
from data group by partitioned_ke... | The database does this by default, no hints/tuning needed.
<pre><code>create table data(partitioned_key_index number, some_dummy_measure number)
partition by list (partitioned_key_index)
(
partition p1 values (1),
partition p2 values (2),
partition p3 values (3),
partition p4 values (4),
par... | Your intuition is correct. So far Oracle can not iterate over partitions and have a different exec plan for each of them. The only exception is new 12c feature, where a local index can be missing on some partitions.
General recommendation is that each partition should have similar statistical characteristic - or you wi... | https://dba.stackexchange.com |
130,356 | [
"https://cs.stackexchange.com/questions/130356",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/126621/"
] | Suppose the following is true for some f(n) and g(n):
<span class="math-container">$f(n) = \Theta(g(n))$</span>
Does that mean <span class="math-container">$f(n/k) = \Theta(g(n))$</span> for any value of <span class="math-container">$k>0$</span>?
I know that for the above to be true, there must exist positive consta... | Take <span class="math-container">$f(n) = n^4$</span>. For this function, if you double the argument, the result is multiplied by 16, that is by a constant factor. Now take <span class="math-container">$g(n) = 2^n$</span>. For this function, if you double the argument, the result is squared, and since <span class="math... | Suppose <span class="math-container">$f(n) = 2^n$</span>. Then <span class="math-container">$f(n / 2) = 2^{n / 2} = \sqrt{2^n}$</span> and setting e.g. <span class="math-container">$g(n) = 2^n$</span> as well shows <span class="math-container">$f(n) \in \Theta(g(n))$</span> but <span class="math-container">$f(n / 2) \n... | https://cs.stackexchange.com |
2,555,914 | [
"https://math.stackexchange.com/questions/2555914",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/436619/"
] | Let H be a Hilbert Space. Show that the dual space $H'$ of $H$ is a Hilbert Space with inner product $\langle \cdot, \cdot \rangle_1$ defined by $$ \langle f_z , f_v \rangle_1 = \overline{ \langle z,v\rangle}=\langle v,z\rangle,$$ where $f_z(x)=\langle x,z\rangle$, with $\langle \cdot, \cdot\rangle$ is the inner produc... | The map $\phi : H \to H'$ given by $\phi(v) = f_v$, where $f_v(x) = \langle x, v\rangle$, for $x \in H$ is an antilinear bijective isometry. Bijectivity and norm-preservation both follow from the Riesz Representation Theorem, and antilinearity can be easily verified:
$$\phi(\alpha v + \beta w)(x) = f_{\alpha v + \beta... | So $f_{z_{n}}-f_{z_{m}}=f_{z_{n}-z_{m}}$ and hence if $d(f_{z_{n}},f_{z_{m}})<\epsilon$, this means $\left<f_{z_{n}-z_{m}},f_{z_{n}-z_{m}}\right>\leq\epsilon^{2}$, or $\left<z_{n}-z_{m},z_{n}-z_{m}\right><\epsilon^{2}$, the latter is $\|z_{n}-z_{m}\|<\epsilon$, as $H$ is complete, there is some $z\... | https://math.stackexchange.com |
82,324 | [
"https://electronics.stackexchange.com/questions/82324",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/29036/"
] | I have a compact digital photo camera that will be on for very long periods(for a time-lapse video). It is powered by a <strong>Lithium-Ion 3,7V, 720mAh</strong> battery
I want to supply 4V to the battery terminals, while the battery is in the digital camera. This will power the camera and occasionally charge the batt... | Updated October 2022:
This applies to Lithium Ion cells (LiIon or LiPo) rated at a nominal 3.6V or 3.7V.
Floating at 4.0 volt is safe.<br />
Floating at 4.2V is fatal (for the battery and possibly for others as the battery may catch fire.)
If floating at 4.0V (or below) ENSURE that the voltage is well regulated and nev... | Yes, since the Li-Ion battery will only absorb as much current as required from the supply. Note that this assumes that the battery isn't "smart" in that it requires signaling from the device in order to begin charging.
Also note that this is not true of NiCad or NiMH unless the current is throttled to very low values... | https://electronics.stackexchange.com |
242,404 | [
"https://stats.stackexchange.com/questions/242404",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/136236/"
] | My current understanding is
1) if strata are not specified, then boot randomly selects rows with replacement from the entire dataset. If dataset is actually stratified then boot would often return uneven sample sizes. In contrast 2) if strata are specified, then boot randomly selects rows with replacement from within e... | Your understanding 1) and 2) are correct; the strata option makes bootstrap samples within each strata independently, whereas the non-strata option makes bootstrap samples of all the data meaning that bootstrap samples will not contain the same number of samples from within each strata.
As to why they are similar; it... | This is how I interpreted the correct answer from svendvn above in case anybody wishes to see the code:
<pre><code>require(boot)
# relative yield takes a matrix or dataframe and finds the ratio
# of the means: treatmentMean/controlMean.
# data structure:
# first column is strata, control = 1 and treatment = 2
# secon... | https://stats.stackexchange.com |
153,274 | [
"https://dba.stackexchange.com/questions/153274",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/24418/"
] | Scenario:
<ul>
<li>Table <code>MY_TABLE</code> has primary key constraint <code>PK_MY_TABLE</code></li>
<li>Table <code>MY_TABLE</code> has also an index called <code>IDX_PK_MY_TABLE</code> which is enforces the uniqueness for constraint <code>PK_MY_TABLE</code>.</li>
</ul>
If I disable constraint <code>PK_MY_TABLE</... | It's easily tested.
There are two possible scenarios.
<strong>Scenario 1:</strong>
Scenario 1 is creation of a PK using an existing index:
<pre><code>SQL> create table MY_TABLE
2 (
3 pk number
4 );
Table created.
SQL>
</code></pre>
Add the index to it:
<pre><code>SQL> create unique index IDX_... | Depends on how you disable the constraint.
<pre><code> alter table my_table modify primary key disable;
</code></pre>
The above drops the index, so when you enable the constraint, the index will be rebuilt. The below preserves the index:
<pre><code>alter table my_table modify primary key disable keep index;
</code><... | https://dba.stackexchange.com |
50,272 | [
"https://stats.stackexchange.com/questions/50272",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/9572/"
] | Given:
<ol>
<li>A coin with <strong>unknown</strong> bias $p$ (Head).</li>
<li>A strictly positive <strong>real</strong> $a > 0$.</li>
</ol>
Problem:
Generate a random Bernoulli variate with bias $p^{a}$.
Does anyone know how to do this? For instance, when $a$ is a positive integer, then one can flip the coin $a... | <strong>We can solve this via a couple of "tricks" and a little math.</strong>
Here is the basic algorithm:
<ol>
<li>Generate a Geometric random variable with probability of success $p$.</li>
<li>The outcome of this random variable determines a fixed known value $f_n \in [0,1]$.</li>
<li>Generate a $\mathrm{Ber}(f_n)... | Is the following answer silly?
If $X_1,\dots,X_n$ are independent $\mathrm{Ber}(p)$ and $Y_n$ has distribution $\mathrm{Ber}\left(\left(\sum_{i=1}^n X_i/n \right)^a\right)$, then $Y_n$ will be approximately distributed as $\mathrm{Ber}(p^a)$, when $n\to\infty$.
Hence, if you don't know $p$, but you can toss this coin... | https://stats.stackexchange.com |
43,457 | [
"https://mechanics.stackexchange.com/questions/43457",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/24522/"
] | I have a 2000 1.8L Mazda Protege and I just replaced the engine with a 2.0L JDM swap. The new engine is installed and appears to run <em>great</em>, it idles smoothly and revs fine, sounds great.
Oddly, after I replaced the engine, the automatic transmission will not go into gear. Actually it <em>did</em> go into gear... | <strong>UPDATE: We pulled the engine and trans today, and put the correct flexplate on, which definitely has a different geometry and sits closer to the trans (i.e. not pulling the torque converter out). There was no evidence of spline or seal damage. Put it all back together and ... it runs, drives, and shifts perfect... | My name is Robert I work at a transmission shop. I've been doing this kind of work for 20 some odd years. Start your car and note where the transmission fluid is at on the dipstick. Shut the car off wait for about 2 minutes recheck the fluid if there's no changein fluid level you broke the pump. The mechanism that tur... | https://mechanics.stackexchange.com |
274,732 | [
"https://softwareengineering.stackexchange.com/questions/274732",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/153817/"
] | If a coder writes code that nobody other than he can understand, and code reviews always end with the reviewer scratching his head or holding their head in their hands, is this a clear sign that the coder is simply not cut-out for professional programming? Would this be enough to warrant a career change? How importan... | The first rule of any professional software engineer is to write code that is comprehensible. The second example looks like an optimized example for an older, non-optimizing compiler or just someone who happens to want to express themselves with bitwise operators. It's pretty clear what's going on if we are familiar wi... | There are several questions that you raise.
1) Is this a clear sign that the coder is not cut out for professional programming?
<ul>
<li>No. Developers often go through stages where they learn about an idea and want to apply it. Do they always apply these ideas efficiently and/or effectively. No. Mistakes are mad... | https://softwareengineering.stackexchange.com |
3,368,557 | [
"https://math.stackexchange.com/questions/3368557",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/708257/"
] | I understand how to determine how many total possible 5-digit numbers with no repeating numbers you can find, I am just not sure how to figure out the last part
| By symmetry, having 4 before 2 and 2 before 4 are essentially the same case. Since there are a total of <span class="math-container">$5!=120$</span> combinations, there are <span class="math-container">$\frac{120}{2}=60$</span> numbers with 4 before 2.
| There are 5 non repeatable digits there are <span class="math-container">$ 5! $</span> possible sequences of such digits.
If there was a sixth digit in a <em>fixed</em> place (the end or beginning) there would still be <span class="math-container">$ 5! $</span> possible sequences of the digits.
The requirement is tha... | https://math.stackexchange.com |
2,109,452 | [
"https://math.stackexchange.com/questions/2109452",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/381431/"
] | If $A^2 = I$, then $A^n = I$ for all integers $n \geq 2$ (matrices)
Let $A=\pmatrix{1&0\\0&1}$
We see that $A^2$ = $\pmatrix{1&0\\0&1}\pmatrix{1&0\\0&1} = \pmatrix{1&0\\0&1}$
It seems that $A^n = I$ is true then, since I can just keep multiplying by the same matrix to get the identity... | Search for Involutory Matrices.
If
$$\begin{bmatrix}
a & b \\
c & -a
\end{bmatrix}$$
is such that $a^2+bc=1$, then $A^2=I$ but not neccesarily $A^3=I$. Take for example, $a=0$ and $b=c=1,$ then
$$A^2=\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}\cdot \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}... | Really? If $A^2=I$, we must have $A^3=A$. Is your $A$ equal to $I$?
| https://math.stackexchange.com |
410,866 | [
"https://softwareengineering.stackexchange.com/questions/410866",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/259571/"
] | I joined a company and the project I'm working on have an event handler that extends the FirstChanceException functionality to log the exceptions. The logs are stored in a file and not registered to the console.
All the controllers actions have try catch blocks and they ignore the exception in the catch (because it's b... | This is a bad practice for two major reasons:
<ol>
<li>Extra code has been written - and has to continue to be written - for what the framework already does quite well. Register a logging exception handler within the pipeline framework being used itself and remove the <code>try</code>..<code>catch</code> (empty) parad... | I don't see anything wrong with having a logging mechanism hook into the FirstChanceException handler. It might be the best way to guarantee that <em>all</em> exceptions get logged.
I <em>do</em> have a problem with controllers that consume exceptions without so much as telling the user that their request has failed. ... | https://softwareengineering.stackexchange.com |
133,435 | [
"https://dba.stackexchange.com/questions/133435",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/90577/"
] | My app runs a lot of queries like this
<pre><code>SELECT ... FROM Table
WHERE CategoryID IN (SELECT CategoryID FROM OtherTable WHERE ..)
</code></pre>
To save performance I decided to cache the second subquery in the application's memory (it rarely changes) and change the DB-query to
<pre><code>SELECT ... FROM T... | OK, after experimenting a bit further with queries and a <code>set statistics time on</code>, here's what I found:
<strong>On a small IN-list (4-12 items) - ALMOST identical</strong>
The performance is almost identical, the hardcoded variant uses a little bit more CPU, the overall "elapsed time" is either same or a l... | A lot of this depends on what else your query is doing. If that predicate is one of many, and isn't particularly useful for an index, then you may be seeing that predicate in a Predicate property (being a residual predicate) in your plan.
Alternatively, if you have an index on CategoryID, then are you wanting it to do... | https://dba.stackexchange.com |
39,872 | [
"https://security.stackexchange.com/questions/39872",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/11291/"
] | Recently a provider (of SIP trunking services) I subscribe to sent me a strange email. It claimed that someone in another country attempted to reset the password to my account and was unsuccessful in answering my security question.
The provider's response to this event was to reset my password.
<blockquote>
Dear Cu... | This is an absolute breach of security. Even if their policy was somehow sound, sending the password in <em>plaintext</em> to you in an email means that the reset is useless, and as you said, if the attacker had access to your email the security questions wouldn't do squat.
They should have done nothing as the securit... | No, it is not an appropriate response from the ISP. The attacker tried to reset the password, which shows that the attacker <strong>does not know</strong> the current password, and actually does not even try to guess it. Forcing a reset of that password cannot bring any good: it tries to fix exactly the part of the aut... | https://security.stackexchange.com |
198,025 | [
"https://softwareengineering.stackexchange.com/questions/198025",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/-1/"
] | Why not use a RTOS with microkernel architecture for web servers? The scheduler is deterministic and all requests will be handled quickly leading to faster response time? It's easy to extend an OS based on microkernel architecture, since everything is like a client-server communication via message passing. Also, the se... | <blockquote>
The scheduler is deterministic and all requests will be handled quickly leading to faster response time?
</blockquote>
Deterministic does not imply faster. An RTOS schedules guarantees that all requests will be served in some given time <em>T</em>, but a best effort scheduler may provide better average ... | A direct answer to your question: it doesn't matter. Most RTOS come with libraries that implement web servers.
But I guess a misconception in your question:
Realtime does not mean faster.
Realtime means that the time of the outcome is as much important as its value.
For example, a video decoder must deliver video f... | https://softwareengineering.stackexchange.com |
633,350 | [
"https://physics.stackexchange.com/questions/633350",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/285149/"
] | I just learned about sector velocity and how you can manipulate the equation to show that angular momentum <span class="math-container">$L$</span> is conserved by <span class="math-container">$L/2M$</span>. I know that sector velocity is constant and therefore angular momentum is constant. I have made it that far but I... | The angular momentum is given by a vector product <span class="math-container">$\vec{L}=m[\vec{r}\times\vec{v}]$</span>. One o the properties of the vector product is that the result is orthogonal to both factors. I.e. both <span class="math-container">$\vec{r}$</span> and <span class="math-container">$\vec{v}$</span> ... | It's because dust, gases, proto-planets, and other objects in solar systems have existed for sufficient time at a high enough density to interact strongly.
Many orbits exist initially, but any non-coplanar or highly elliptical orbits will eventually intersect other existing orbits. These intersections result in a coll... | https://physics.stackexchange.com |
142,912 | [
"https://physics.stackexchange.com/questions/142912",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/59161/"
] | Using positional index notation with tensors is common. For example, the following simple equation from Carroll's Spacetime and Geometry text (eq. 3.146):
$$
R = R^\mu_{\,\,\mu} = g^{\mu\nu}R_{\mu\nu}
$$
Where the Ricci scalar summation, $R^\mu_{\,\,\mu}$ uses positional offsets on the upper and lower indices.
So, I ... | On a two-index tensor, swapping the two indices is equivalent to transposing a matrix.
You may not see many authors spending a lot of effort on this issue simply because an awful lot of the tensors we deal with are symmetric. This includes the metric, Ricci tensor, Einstein tensor, and stress-energy tensor. Therefore ... | Comments to the question (v1):
<ul>
<li>Indices are raised and lowered vertically by the pertinent metric tensor of the theory.</li>
<li>The horizontal position of indices is important for a tensor that is not totally symmetric, e.g., the EM field strength $F_{\mu\nu}$ or the Riemann curvature tensor $R_{\mu\nu\lambda... | https://physics.stackexchange.com |
9,053 | [
"https://mechanics.stackexchange.com/questions/9053",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/4696/"
] | Have been driving my stang 2013 V6 for almost 2 years now. Right from the day 1 I have noticed that outside temperature somehow made my cars acceleration/ pickup however you call it become sluggish. I feel my car is more monster at night than at day I can easily get to 80 in no time and there have been times i felt un... | Yes, cold, humid air will make a difference. You can run richer mixture on a cold morning than on a hot afternoon, for example. On a small two-stroke you'll swap jets, while a modern (generous usage here) car adjust the mixture itself according to sensor data.
| Humid air will for sure make a difference, depending on exactly how hot it is where you live.
As for being able to cool the air down, there are a few different aftermarket options, but nothing that I'd reccomend. Cold-Air intakes don't make a difference. I knew of a few airboxes you can get that cool the air, but I d... | https://mechanics.stackexchange.com |
3,202,687 | [
"https://math.stackexchange.com/questions/3202687",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/668126/"
] | Let <span class="math-container">$A$</span> be the set of real numbers such that <span class="math-container">$0 < x \leq 1$</span> . For every <span class="math-container">$x \in A$</span>, let <span class="math-container">$E_{x}$</span> be the set of real numbers <span class="math-container">$y$</span> such that <... | Note that <span class="math-container">$\bigcap_{x\in A}E_x\subset(0,\infty)$</span>. If <span class="math-container">$y\in \bigcap_{x\in A}E_x\subset(0,\infty)$</span>, this means that <span class="math-container">$y<x$</span> for all <span class="math-container">$x>0$</span>. As Rudin says, this is a contradict... | Note that <span class="math-container">$x\notin E_x$</span> for every <span class="math-container">$x$</span>
| https://math.stackexchange.com |
3,961,981 | [
"https://math.stackexchange.com/questions/3961981",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/601685/"
] | What am I missing?
<span class="math-container">$$ x^2 + x + 1 = 0$$</span>
Then
<span class="math-container">$$ x^2 + x = -1$$</span>
<span class="math-container">$$ x(x+1) = -1$$</span>
<span class="math-container">$$ x(-x^2) = -1$$</span>
<span class="math-container">$$x^3 = 1 $$</span>
<span class="math-container">... | You start with a solution to the equation <span class="math-container">$x^2 + x + 1 = 0$</span> and correctly show that it is also a solution to the equation <span class="math-container">$x^3 = 1$</span>. You then observe that the only real solution to the second equation is <span class="math-container">$1$</span> (thi... | If you are assuming <span class="math-container">$x$</span> is a real number, then all of these steps are valid, and prove that the assumption <span class="math-container">$x^2+x+1=0$</span> gives a contradiction. In other words, this is just a proof that the equation <span class="math-container">$x^2+x+1=0$</span> ha... | https://math.stackexchange.com |
28,950 | [
"https://softwareengineering.stackexchange.com/questions/28950",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/2329/"
] | I'm studying Haskell for the purpose of understanding functional programming, with the expectation that I'll apply the insight that I gain in other languages (Groovy, Python, JavaScript mainly.)
I choose Haskell because I had the impression that it is very purely functional, and wouldn't allow for any reliance on stat... | I believe that understanding Haskell's type system is an amplifier to understanding functional programming.
The thing about purely functional programming is that in the absence of side-effects, which allow you to do all sorts of things implicitly, purely functional programming makes the structure of your programs much... | The most dynamically typed functional language is arguably Scheme. That said, Haskell's type system is an indicator of its purity. It's a question of "how does one measure purity?". Haskell's type system lets you easily cordon off impure actions in <code>IO</code>. To do that, you need a static type system.
But let's ... | https://softwareengineering.stackexchange.com |
150,430 | [
"https://electronics.stackexchange.com/questions/150430",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/64955/"
] | I'm planning to make a plant watering system . There is a beeper so it will alert us when the soil is dry.I have got a 12V DC motor . So i'll be removing this beeper and i need to add this DC motor.What all change are needed to made in the circuit so as to make this automatic.I can use my Laptop Adapter as the power so... | (EDITED) - Notice that the last Nand gate ahead of the beeper looks to be a slow oscillator, (the beeper likely rings as ON-OFF-ON-OFF....). The input to the last Nand changes from being either a low signal or a fast square wave signal, (which corresponds to wet soil then to dry soil conditions).
The diode in the fee... | Instead of the beeper, find a relay that has a coil compatible with the circuit's power supply and use the coil contacts for activating a motor running from 12V. If the circuit you have shown operates from 12V then it can be easier - use a P channel power mosfet instead of the BJT (Q1).
All this assumes that the circu... | https://electronics.stackexchange.com |
53,541 | [
"https://mathoverflow.net/questions/53541",
"https://mathoverflow.net",
"https://mathoverflow.net/users/12554/"
] | A relatively simple question, but I cannot seem to find anything relevant to this. In fact, I'm not even sure of "unbalanced design" is even the right terminology here, but it was suggested to me.
The problem may be described as follows. Let's say you have a partition of your population into some groups - A, B and C. ... | The group $S_3\wr ({\mathbb Z}_2 \times {\mathbb Z}_2 \times {\mathbb Z}_2)$ where $S_3$ is the symmetric group with 6 elements, ${\mathbb Z}_2$ is the group with 2 elements, $\wr$ is the wreath product.
The fact that it does not have a center is proved by inspection. The fact that it needs at least 3 generators fol... | The smallest example has order 18. It has a normal elementary abelian subgroup $N$ of order 9, and an element $t$ of order 2 such that $txt=x^{-1}$ for all $x \in N$.
| https://mathoverflow.net |
148,867 | [
"https://softwareengineering.stackexchange.com/questions/148867",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/37526/"
] | During all my professional lifetime that I have been programming in Windows enviroments, I advised my friends and coworkers to not use a DSN database connection in their applications, based on my own personal conclusions, but I really haven't seen any evidence of them.
I know that using a DSN connection via ODBC data ... | I'll go out on a limb and say no it is not "bad practice". It is a tool intended for flexible implementations. I might never use it myself as I'd prefer to use the native api for database connections but it clearly has a purpose. If you are writting client-ware to connect to arbitrary databases in different work env... | As KenK stated, an absolute "good" or "bad" is a tough call. One potential negative of using a DSN connection is that if changes need to be made it can be difficult to update all clients. Another is if developers need to switch servers for testing, it can be difficult to do so, particularly in corporate environments wh... | https://softwareengineering.stackexchange.com |
74,234 | [
"https://softwareengineering.stackexchange.com/questions/74234",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/23046/"
] | I am looking at how to justify changes to those who control budgets.
For example if we have Enterprise Library 4 on some systems and the latest version is 5, is it worth upgrading? How do you justify it?
Is it best to continually try to keep frameworks that systems use up to date. Or is it better to do a major lift e... | <strong>Cost-Benefit Analysis</strong>
Weigh up the cost of the upgrade against the benefits; estimate, include the non-specific benefits such as developer-satisfaction (why would I work on .NET 1.0 nowadays when I can get plenty of jobs working in 4.0?), but make it reasonable.
If the benefits outweigh the costs sho... | Using your example, I start with the question, what is there in EntLib 5 that I can't live without? If the list is very short or non-existant, well, that makes the desicion very easy. Usually the answer isn't that easy. Most of the time there are a handfull of "boy, that would be nice to have" or "they've made X a l... | https://softwareengineering.stackexchange.com |
1,715,573 | [
"https://math.stackexchange.com/questions/1715573",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/251829/"
] | <blockquote>
If $z^n+z^{-n}=2\cos(n\theta)$ show that $5z^4-z^3-6z^2-z+5=0 =>
> 10\cos^2(\theta)-\cos(\theta)-8=0$.
</blockquote>
I can prove $z^n+z^{-n}=2\cos(n\theta)$ with De Moivre's theorem (the sines cancel out and hence the result). But I don't know how I should apply this to this exercise. I don't know... | Assume $a>0$. One may write
$$
\begin{align}
\int_{0}^{3a} \frac{x\sqrt{3a-x}}{\sqrt{a+x}} dx &=a^2\int_{0}^{3} \frac{t\sqrt{3-t}}{\sqrt{1+t}} dt \qquad (x=at)
\\\\&=2a^2\int_{1}^{2} (u^2-1)\sqrt{4-u^2}\: du \qquad \left(u=\sqrt{1+t}\:\right)
\\\\&=8a^2\int_{\pi/6}^{\pi/2} (3-4\cos^2 v)\cos^2 v\: dv \qqu... | <strong>Hint:</strong>
This is classic. Use substitution:
$$t=\sqrt{\dfrac{3a-x}{a+x}}\iff x=a\frac{3-t^2}{t^2+1},\enspace t\ge 0$$
| https://math.stackexchange.com |
57,120 | [
"https://datascience.stackexchange.com/questions/57120",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/32243/"
] | I know that BPTT is the method to apply Back Propagation on RNN.
Which is works fine with RNN as it stops at certain point as changes approach to zero
but isn't it the exact Vanishing Gradient Problem?
if it is the same then Why does it have 2 names, one is a problem and one is a method.
if not then what am I missi... | I think the intuition here is that you want to push the negative loglikelihood (NLL) * the reward to be as negative as possible. Since our reward is often not differentiable because it is obtained through sampling, we can only change the NLL. For actions with high reward, we have greater "pressure" / gradient... | The actual loss is supposed to be <span class="math-container">$m.log\_prob(action) * reward$</span> without the negative sign. The default optimizer in Pytorch uses gradient descent methods, while the REINFORCE assumes gradient ascent update rule. To account for this the loss is made negative. This is clearly mentione... | https://datascience.stackexchange.com |
666,325 | [
"https://physics.stackexchange.com/questions/666325",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/311245/"
] | Say a ball is thrown from a height and it drops to the ground. Would the <span class="math-container">$v_f$</span> or final velocity be <span class="math-container">$0$</span> or would it be the velocity of the ball just before it hits the ground? If I assume the latter, I get the wrong answer according to my textbook.... | Any task or problem that asks for initial or final values, or any other values for that matter, <strong>must clearly define those values</strong>.
If you are asked to find final velocity, then you need the definition of "final" in that particular scenario.
<ul>
<li>Sure, the final velocity could have been def... | The final velocity refers to the velocity of the ball just before hitting the ground.Please check if you have computed the velocity vector correctly.
| https://physics.stackexchange.com |
204,355 | [
"https://softwareengineering.stackexchange.com/questions/204355",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/96092/"
] | If you have to pay the price of performance to make code more readable would you prefer that or you would always prefer performance over it?
| <strong>In layman's words:</strong>
<ul>
<li>Making the code ugly does not make it fast.</li>
<li>Making the code clean does not make it slow (or fast).</li>
<li>Making the code ugly does make the <strong>maintainer's</strong> work slower.</li>
<li>Clean code usually means the code was made by someone who cares.</li>
... | The general idea is to make the code as readable as possible first, and then only look at optimizing where there's a proven performance problem. In most cases a response time of, e.g. 5 ms instead of 1 ms isn't going to make a difference. 50 seconds instead of 10 seconds is a different matter.
| https://softwareengineering.stackexchange.com |
87,229 | [
"https://softwareengineering.stackexchange.com/questions/87229",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/3867/"
] | I built a lot of websites and applications during my life, I can put them under the following categories:
<ul>
<li>Did all the project as a freelancer -> For other companies </li>
<li>Did part of the project as a freelancer -> For other companies </li>
<li>Did all the project my self while was working for another ... | If you did the work and were paid by someone else to do that work, seek permission first. Depending on the various contracts and legal documents that you signed, you might or might not be able to share that work publicly for any number of reasons.
If you performed the work yourself for yourself when you were employed ... | [I am Not a Lawyer]
You're going to have copyright issues with your former employers if you don't first get permission to use those assets. Do you really want to start a new business by incurring potential liability?
The moral (probably legal, too) high ground would be to only use work you've done for yourself and o... | https://softwareengineering.stackexchange.com |
702,236 | [
"https://physics.stackexchange.com/questions/702236",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/332210/"
] | i was confused as to what will happen when two electrons which are entangled and then if one is exposed to light. will the absorption and emission theory still work and if so how will it work in this case?
| To have quantum entanglement one has to have a definite solution of the quantum mechanical equations describing the entangled system, i.e. a single wavefunction.
At the level of electrons it is photons that interact electromagnetically. But every new interaction requires a new quantum mechanical solution. If your two ... | If the two electrons only have their spins entangled (which is what we usually mean by entangled particles), then no. The interaction will be localized at the position of <span class="math-container">$a$</span>. To see that the photon's target was part of an entangled pair, you would probably need to repeat the experim... | https://physics.stackexchange.com |
31,157 | [
"https://chemistry.stackexchange.com/questions/31157",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/16114/"
] | I read my book and learned everything about enantiomers, racemization, $\mathrm{S_N1}$ and $\mathrm{S_N2}$ reactions, but when I'm trying to identify molecules as chiral or achiral, I get confused.
Consider the two molecules below:
<pre><code>1.CH3-CH-OH
|
Br
2.CH3-CH-Br
|
Br
</code></pre>
H... | Chirality can be determined by visualising the molecule in space. Once you do that, you can check to see if there exists a non-superimposable mirror image. If this is indeed the case, then your molecule is chiral.
You can look for an asymmetric carbon atom--because this often results in chirality.
An asymmetric carb... | Look for carbons with four <strong>different</strong> groups attached to identify potential chiral centers.
Draw your molecule with wedges and dashes and then draw a mirror image of the molecule. If the molecule in the mirror image is the same molecule, it is achiral. If they are different molecules, then it is chiral... | https://chemistry.stackexchange.com |
154,419 | [
"https://mathoverflow.net/questions/154419",
"https://mathoverflow.net",
"https://mathoverflow.net/users/45469/"
] | We have $n$ variables $a_0,a_1,\ldots,a_n$ such that $a_i\geq a_{i+1}$.
There are $k$ sets of linear inequality constraints on the $a_i$.
I need to check that any choice of $a_i$ satisfies at least one of the k sets of constraints.
If I see it correctly, essentially I would need to check that the union of the feasib... | The bad news is that many closed convex cones in infinite dimensions have empty interior, e.g., the natural cone of positive functions in $L^p(\Omega)$ for all $\Omega \subset \mathbb{R}^n$; or in $W^{1,p}(\Omega)$ for $\Omega \subset \mathbb{R}^n$ and $p < n$.
The positive cone in $L^2(\Omega)$ is self-dual (if yo... | Have a look on
E. Casini, E. Miglierina, cones with bounded nd unbounded bases and reflexivity, nonliniear Analysis
| https://mathoverflow.net |
611,315 | [
"https://electronics.stackexchange.com/questions/611315",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/211240/"
] | I understand that data needs to be valid during the setup and hold windows to prevent an unknown output or metastable condition. But what I don't understand is, <em>why</em> does this happen? <em>Why</em> is it that when data toggles during the setup and hold window, output is unknown?
I tried searching for this on the... | The fundamental problem is that the input is continuous, and the output is discrete. In synchronous systems, this is further compounded by the fact that the decision has to be made in a finite time.
The classic analog is balancing a pencil on its point. It's likely to fall one way or the other, but the better you balan... | Logic signals are just analog signals that are interpreted with a threshold. Digital circuits are also implemented with analog components.
The logic level is stable when voltages are well within the range of being below or above the threshold of changing between logic levels.
Whenever a signal is changing, the change i... | https://electronics.stackexchange.com |
18,568 | [
"https://quantumcomputing.stackexchange.com/questions/18568",
"https://quantumcomputing.stackexchange.com",
"https://quantumcomputing.stackexchange.com/users/16713/"
] | I want to build a quantum application, qApp if you will; how does one go about figuring out the right use-case and best framework to turn to, to start building these things?
Quantum REPL? How do I learn how to program qBits? Polarity? Spin? Quantum entanglement? How do you even program that? What are we even talking ab... | The simple answer is that you can't get the statevector information out of a real quantum computer; or, more generally, a quantum system. For the real computer to extract information from the qubits, they have to collapse to some basis state <span class="math-container">$\left(|0\rangle \; \text{or} \; |1\rangle \right... | Note that you do not measure anything before sending to QC. What you actually send to QC is a sequence of instructions telling QC what to do. The measurement is the last instruction necessary for obtaining results as pointed out in the other answer.
| https://quantumcomputing.stackexchange.com |
334,229 | [
"https://electronics.stackexchange.com/questions/334229",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/8784/"
] | I have an SPI module which has an initializer function:
<pre><code>static volatile uint8_t *registerOut, *registerIn, *registerDir;
static uint8_t clock, dataIn, dataOut;
void SpiInit(volatile uint8_t *pRegisterOut, volatile uint8_t *pRegisterIn,
volatile uint8_t *pRegisterDir, uint8_t pClock, uint8_t pD... | It's seems, like a small value filtering capacitor. It's purpose is to filter the high-frequency noise. It won't affect functionality of MOV.
| MOV is a voltage controlled R with some small C (pf). For best transient immunity both clamping and C voltage divider methods are needed for <strong>ESD with sub nanosecond rise times are faster than the diode clamps can respond or if lightning ionization via line for same reasons</strong>, so additional bandwidth red... | https://electronics.stackexchange.com |
88,627 | [
"https://math.stackexchange.com/questions/88627",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/16859/"
] | Let $F\subset\mathbb{R}$ be a measurable set of finite Lebesgue measure.
Find the limit
$$ \lim_{n \to \infty} \int_{F} \frac{dx}{2-\sin nx}.$$
Let $f_{n}(x)=\frac{1}{2-\sin nx}$. This function is bounded and
continuous. Thus, it is integrable. I even use Mathematica to compute the Riemann integral.
However, since w... | First note that
$$
\left|\frac{1}{2-\sin(nx)}\right|\le1\tag{1}
$$
Next note that for $z=\tan{\frac{x}{2}}$
$$
\begin{align}
\int_0^{2\pi}\frac{\mathrm{d}x}{2-\sin(x)}
&=\int_{-\infty}^\infty\frac{\frac{2\;\mathrm{d}z}{1+z^2}}{2-\frac{2z}{1+z^2}}\\
&=\int_{-\infty}^\infty\frac{\mathrm{d}z}{1-z+z^2}\\
&am... | Srivatsan's guess is correct. One way to see this is to write
$${1 \over 2 - \sin(nt)} = {1 \over 2} \bigg({1 \over 1 - {\sin(nt) \over 2}}\bigg)$$
$$= \sum_{k=0}^{\infty} {\sin^k(nt) \over 2^{k+1}}$$
If this sum is truncated at some $k = k_0$, the remainder is bounded by ${1 \over 2^{k_0}}$ in absolute value, and thu... | https://math.stackexchange.com |
429,310 | [
"https://softwareengineering.stackexchange.com/questions/429310",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/185535/"
] | I'm building a client for an API.
The API takes a query param that looks something like this <code>2-10</code>, that's a range and notice that it uses a hyphen (-).
Usually, data comes from another place using <code>en dash</code> (–) instead of <code>hyphen</code>.
If the api is called using an <code>en dash</code> in... | The user of an API doesn’t care where the data comes from; there is a contract between the API and the user. If the API uses a different service that makes trouble, the API must either report an error, or a correct result. A silent incorrect result is unacceptable, it means your API is broken.
Now from your description... | Your question is missing important details, some of which are mentioned in the comments, but the requirements are still muddy.
For example, when you use passive voice in "range is retrieved from another service" you leave unclear who does it, what the other service's API definition is, who could possibly be r... | https://softwareengineering.stackexchange.com |
175,916 | [
"https://softwareengineering.stackexchange.com/questions/175916",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/71338/"
] | I have at first sight a simple issue but can't wrap my head around on how to solve. I have an abstract class <code>Compound</code>. A <code>Compound</code> is made up of <code>Structures</code>. Then there is also a <code>Container</code> which holds 1 <code>Compound</code>.
A "special" implementation of <code>Compoun... | The solution I used is this:
<img src="https://i.stack.imgur.com/7ycaR.png" alt="Class Diagram">
And to reference the image in the question, a <code>Batch</code> extends Containable.
| <blockquote>
... to search for Containers which contain a Compound made up of a specific Structure requires that the "Path" from Container to Structure is well defined (Number of relationships or joins).
</blockquote>
Not if you are asking the container if it has such a structure yes or no, then only the container ... | https://softwareengineering.stackexchange.com |
111,112 | [
"https://chemistry.stackexchange.com/questions/111112",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/75844/"
] | <span class="math-container">$K$</span> represents the ratio of concentrations of molecules in a solution at equilibrium, which means that <span class="math-container">$Q_\mathrm{r}$</span> (that ratio at any given point) looks to be identical to <span class="math-container">$K$</span>. In other words, the molecules in... | <blockquote>
If K is large enough (bigger than <span class="math-container">$10^4$</span> in my curriculum), this means that the the concentration of the reactants is almost zero.
</blockquote>
This statement is not always true - it depends on the stoichiometry of the reaction.
For the reaction <span class="math-co... | See my comment above.
As a numerical example, take acetic acid (<span class="math-container">$\ce{AcOH}$</span>), which has <span class="math-container">$K_\mathrm{a} = 1.8 \cdot 10^{-5}$</span>.
This means that:
<span class="math-container">$$K_\mathrm{a} = \frac{[\ce{AcO-}][\ce{H+}]}{[\ce{AcOH}]}$$</span>
And the... | https://chemistry.stackexchange.com |
343,549 | [
"https://mathoverflow.net/questions/343549",
"https://mathoverflow.net",
"https://mathoverflow.net/users/27767/"
] | For <span class="math-container">$\Omega$</span> a bounded open set of <span class="math-container">$\mathbf{R}^d$</span> and <span class="math-container">$f\in L^p(\Omega)$</span> the infimum
<span class="math-container">\begin{align*}
\inf_{C\in\mathbf{R}} \|f-C\|_p
\end{align*}</span>
is reached (by compactness). F... | Consider the case <span class="math-container">$p=6$</span> where <span class="math-container">$\Omega$</span> consists of three sets of equal measure on which <span class="math-container">$f$</span> takes values <span class="math-container">$0, 1, 3$</span> respectively. Then <span class="math-container">$C_6(f)$</sp... | The constant <span class="math-container">$C$</span> that you've constructed is called the <em>barycenter</em> in the metric geometry literature. Specifically, you're asking about the <span class="math-container">$p$</span>-barycenter of the measure <span class="math-container">$f_*m$</span> on <span class="math-conta... | https://mathoverflow.net |
618,816 | [
"https://physics.stackexchange.com/questions/618816",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | I have the following Lagrangian:
<span class="math-container">$$L=\frac{\mu}{2}\left(\dot{r}^2+r^2\dot\phi^2\right)-U(r),$$</span>
The Euler-Lagrange equations are thus:
<span class="math-container">$$\frac{d}{dt}\left(\mu r^2\dot\phi\right)=0$$</span>
<span class="math-container">$$\frac{d}{dt}(\mu \dot r)=\mu r\dot{\... | Yes, the term <span class="math-container">$\mu r^2\dot{\phi}$</span> represents the angular momentum, and the first Euler equation telling you that the angular momentum is the constant of motion for the given problem. The second Euler equation is just the radial force equation. If you can try to write newton's equati... | It is the generalized momentum conjugate to <span class="math-container">$\phi$</span>, defined as
<span class="math-container">$$p_\phi=\frac{\partial L}{\partial \dot{\phi}}.$$</span>
With your choice of generalized coordinates, <span class="math-container">$r$</span> and <span class="math-container">$\phi$</span>, i... | https://physics.stackexchange.com |
297,838 | [
"https://softwareengineering.stackexchange.com/questions/297838",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/150071/"
] | A colleague and I are working together on a Meteor app.
One of us thinks that the following code in two places should be wrapped in a function to avoid duplication -- the other thinks that it leads to needless indirection/complexity.
I won't tell you which is me, because I would like the answers to be unbiased :).
W... | It depends.
Assuming that the <code>subject</code> (and not the <em>sender</em>, for instance) is the only "repeated" parameter here, I'd turn the question back on you: <em>If the subject text changes in one place, does it necessarily change in both?</em>
DRY isn't so much about eliminating the amount you copy+paste.... | I would at least like to have the string constant <code>"Message from our company"</code> consolidated in one place. If this is standard boilerplate, then some UI guy/copy editor is going to want it changed at some point and would want it to be consistent across the app.
| https://softwareengineering.stackexchange.com |
42,405 | [
"https://electronics.stackexchange.com/questions/42405",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/7217/"
] | I need to build a VGA cable (assemble connectors to wire).<br>
I need a male-male cable.<br>
Is its wiring straight or cross? (should i connect 1-1,2-2,... or something else)
Thanks
| Yes, all VGA cables are wired 1-to-1, whether they are regular male-male cables or the relatively rarer male-female extension cables. The equipment at either end always has a female connector, and they're all wired the same way.
| In fact if you don't need pin 9 (option 5v) you could use an old male to male serial cable. Whic I have recently done, myself.
| https://electronics.stackexchange.com |
2,071,154 | [
"https://math.stackexchange.com/questions/2071154",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/291500/"
] | So far I've only seen examples of high-order (ie: second, third) partial derivatives of functions $f:\mathbb{R^n} \to \mathbb{R}$, but not any with $f:\mathbb{R^n} \to \mathbb{R^m}$.
Is there a reason for this? Is this because the definition of a partial derivative is a function $f: \mathbb{R^n} \to \mathbb{R}$?
Merr... | $$\frac {2^{4n-n^2-1}}{2^{2n}}=2^{4n-n^2-1-2n}=2^{2n-n^2-1}=2^{-(n-1)^2}$$
where I have used $\frac{a^b}{a^c}=a^{b-c}$
| Try taking out $2^n$ and $2^{n-1}$ from the denominator and the numerator. I got $$\frac{1}{2^{(n-1)^2}}.$$
If you can't get it let me know. I'll show the work.
| https://math.stackexchange.com |
6,758 | [
"https://stats.stackexchange.com/questions/6758",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/2817/"
] | I've fit a model with several independent variables, one of which is the lag of the dependent variable, using the dynlm package.
Assuming I have 1-step-ahead forecasts for my independent variables, how do I get 1-step-ahead forecasts for my dependent variables?
Here is an example:
<pre><code>library(dynlm)
y<-ar... | Congratulations, you have found a bug. Prediction for <code>dynlm</code> with new data is broken if lagged variables are used. To see why look at the output of
<pre><code>predict(model)
predict(model,newdata=data)
</code></pre>
The results should be the same, but they are not. Without <code>newdata</code> argument, ... | Following @md-azimul-haque 's request, I dug through my 4 years old source code, and found the following appropriately named function. Not sure if this is what @md-azimul-haque is looking for?
<pre><code># pass in training data, test data,
# it will step through one by one
# need to give dependent var name, so that i... | https://stats.stackexchange.com |
75,084 | [
"https://electronics.stackexchange.com/questions/75084",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/8322/"
] | Why might a multimeter ask for the "wrong" sizes of fuses?
On the front of my Fluke 87 multimeter, the two probe ports for amperage are labeled as fused at 10A (max) and 400mA (max). When opening the multimeter to check the fuses (both are burned out), however, the fuses are different -- 15A and 1A, respectively. Thei... | The actual text is:
<pre>
10A MAX
FUSED
</pre>
Maybe this should be read as "10A max" and "fused" — two different statements, not one sentence.
<img src="https://i.stack.imgur.com/s10ro.jpg" alt="Fluke Multimeter Port Labels">
What I am trying to say is that you should not read the two lines on the multimeter... | Something I am missing from the answers here but may still be of help to future readers is that:
Fuses are meant to avoid dangerous situations when a circuit fails, not to limit current!
The only thing of interest to the selection of a fuse is whether it does not impact the circuit in a negative way (i.e. in a multim... | https://electronics.stackexchange.com |
675,527 | [
"https://physics.stackexchange.com/questions/675527",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/318361/"
] | Angle of incidence = Angle of reflection is true experimentally.<br />
I just started highschool. Don't mind if this is a low level question. But I was just curious.
| A proof requires axioms, basic statements that can't be proven. In the case of physical principles, some of these will be just reliable observations.
Fermat's Least Time Principle tells us that light take the least time path between possible trajectories. If the speed of light remains constant in the material, the prin... | There are no mathematical proofs of physical phenomena. The universe is a real thing that is what it is; mathematics is a logic construct with no physical reality.
<em>Theories</em> explain the universe, often in terms of <em>models</em>. A model is a mathematical approximation of a system that we can manipulate with l... | https://physics.stackexchange.com |
1,489 | [
"https://mathoverflow.net/questions/1489",
"https://mathoverflow.net",
"https://mathoverflow.net/users/828/"
] | Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup generated?
| Rene Thom answered this in section II of "Quelques propriétés globales des variétés différentiables." Every class $x$ in $H_r(X; \mathbb Z)$ has some integral multiple $nx$ which is the fundamental class of a submanifold, so the homology is at least rationally generated by these fundamental classes.
Section II.11 wor... | This is a reply to Alon's comment, but it's too long to be a comment and is probably interesting enough to be an answer.
Here's an example Thom gives of a homology class that is not realized by a submanifold: let <span class="math-container">$X=S^7/\mathbb Z_3$</span>, with <span class="math-container">$\mathbb Z_3$</s... | https://mathoverflow.net |
349,238 | [
"https://physics.stackexchange.com/questions/349238",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/87169/"
] | I've come across a wavefunction $$\psi(x)=xe^{-\left(x\right)^2}$$ which is an odd function.
The expectation value is said to be zero . I don't know any good physical interpretation for why the expectation of momentum is zero by just looking at a function (not sure if there is one). But if there is, could I say that... | The title of your question is a little more general than the example you provide.
Consider for instance wavefunctions of the form $\psi_n(x)\sim e^{-(x-a)^2/2}H_n(x-a)$. These functions are even or odd about $x=a$. The probability densities $\vert\psi_n(x)\vert^2$ are symmetrical about $x=a$ and $\langle x\rangle=... | The mean momentum is $-i\hbar\int dx\psi^\ast\psi'$. If $\psi$ is real with a definite parity, the integrand is odd so the mean momentum is zero.
| https://physics.stackexchange.com |
35,697 | [
"https://quant.stackexchange.com/questions/35697",
"https://quant.stackexchange.com",
"https://quant.stackexchange.com/users/26887/"
] | Taken from a mid-July Wall Street Journal news story:
<code>Surging optimism in financial markets hasn’t translated into a big pickup in economic growth. Stocks hit records Friday and big U.S. banks reported stronger-than-expected earnings. But new government data showed consumers pulled back spending at mid-year eve... | You can price an asset paying $X_{t+1}$ in two ways:
$$P_t=\frac{1}{R_f}\sum_{\omega} Q(\omega)X_{t+1}(\omega)$$
$$P_t=\sum_{\omega} P(\omega)M_{t+1}(\omega)X_{t+1}(\omega)$$
As you can see, the price is making a joint statement (i.e. you can recover $Q(\omega)$) regarding both the probability of an event $P(\omega)$ a... | First of all, you need to understand risk-neutral measures are not meant to make <em>predictions</em> of future prices, but they are meant to allow <em>hedging</em> (ie <em>risk replication</em>). Historical measures on their side, are meant to <em>predict future prices</em>.
I used the plural because <strong>they are... | https://quant.stackexchange.com |
1,591,425 | [
"https://math.stackexchange.com/questions/1591425",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/249503/"
] | I just want to know what mistake am I doing in this question
<blockquote>
Show that the roots of the equation $(1+x)^{2n}+(1-x)^{2n}=0$ are given by $x=\pm i tan\frac{(2k+1)\pi}{4n}$ where $k=0,1,2,3,...,n-1$
</blockquote>
So I did the following steps:$$ (1+x)=[(-1)^{\frac{1}{2}}]^{\frac{1}{n}}(1-x) $$
$$(1+x)=(i)^... | Each is equal to its complex conjugate.
| Take a simple example of a 1-dimensional complex matrix: $[a+ib]$.
To be hermitian, you need to show that $[a+ib]=\overline{[a+ib]}=[a-ib]$, hence $b=0$.
| https://math.stackexchange.com |
196,383 | [
"https://mathoverflow.net/questions/196383",
"https://mathoverflow.net",
"https://mathoverflow.net/users/-1/"
] | Let $(X,L,\omega)$ be a projective variety with polarization $L$. then we can write
$$\dim H^0(X,L^k)=a_0k^n+a_1k^{n-1}+...$$
If $X$ is smooth then $a_0=Vol(X)$ and we can compute $a_i$.
If $X$ is non-smooth then how can we compute $a_i$?
| Sometimes one can at least compute $\chi(\mathcal{O}_X(X, \, L^k))$.
The formula that one expects will be of the form "ordinary Riemann-Roch formula plus correction terms depending on the singularities of $X$".
This is of course a bit vague, nevertheless the correction terms can be explicitly computed
in some particu... | I guess you are mostly interested in $a_0,a_1$ for applications to K-stability.
If $X$ is normal, then still $a_0 = \frac{L^n}{n!}$ and $a_1 = \frac{-K_X.L^{n-1}}{2(n-1)!}$. This follows from asymptotic Riemann-Roch for normal varieties, see for example Odaka's "A generalization of Ross-Thomas' slope theory" Lemma 3.5... | https://mathoverflow.net |
2,550,854 | [
"https://math.stackexchange.com/questions/2550854",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/-1/"
] | <blockquote>
Choose the transitive closure of the relation R={(a,b),(b,a),(b,c),(c,d),(c,e)}
</blockquote>
My answer was
<pre><code>(a,c),(b,d),(a,d),(b,e),(a,e)
</code></pre>
But it was wrong. The correct answer was:
<pre><code>R={(a,a),(a,b),(a,c),(a,d),(a,e),(b,a),(b,b),(b,c),(b,d),(b,e),(c,d),(c,e)}
</code><... | This is because $(a,b)$ and $(b,a)$ is in your relation that is $(a,b)$ gives us that $a$ is related to $b$ and $(b,a)$ gives us that $b$ is related to $a$. So for transitivity $a$ should be related to $a$ that is $(a,a)$ should be there.
| The closure og R always contains R. You need to iteratively add pairs to R by taking two existing ones with the form <code>(r,s)</code> and <code>(s,t)</code> and then add <code>(r,t)</code> if it isn't there already. I.e. since you have <code>(a,b)</code> and <code>(b,a)</code> you must add <code>(a,a)</code>.
| https://math.stackexchange.com |
208,792 | [
"https://electronics.stackexchange.com/questions/208792",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/95792/"
] | For example, is there any material I can place over a capacitive sensor, or the finger or conductor approaching it, so that the sensor will no longer activate?
| A capacitive touch sensor sort of forms a parallel plate capacitor with your appendages. As capacitance increases above a threshold, the sensor triggers. So, let us examine the capacitance of a parallel plate capacitor and see what we can do to reduce it:
$$
C = \frac{k \epsilon_0 A}{d}
$$
where \$ \epsilon_0\$ is th... | Interpose a conductor - a sheet of copper, or tinfoil - between finger and capacitive touch sensor - and ground the conductor. (It must be insulated from the capacitive sensor, of course).
Your finger will form capacitance to the ground plane but not the sensor behind it.
This assumes the sensor reacts to changes in... | https://electronics.stackexchange.com |
247,541 | [
"https://physics.stackexchange.com/questions/247541",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/101677/"
] | I've seen multiple questions about a pendulum on a train and most say to use $T = 2 \pi (L/F)^{1/2}$ and I have done this to compare the pendulum's periods before being on a train and then once its on the train I am aware the period on the train should be shorter however I am trying to prove this.
My problem is resolvi... | I infer from the formula and your question, that you are talking about a train in a turn, that experiences centripetal force.
In this case, the "effective $g$" is greater than just the earth attraction.
In the simplest case you assume, that the new force is perpendicular to gravitation and that you know the paramete... | Your question is incomplete. Are we talking about a train that is in a steady state with a constant speed and no turns, a train that is undergoing constant acceleration, or a train that is undergoing changing acceleration?
If the train is in a steady state then the pendulum is adequately modelled by the equation you p... | https://physics.stackexchange.com |
20,019 | [
"https://mathoverflow.net/questions/20019",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4923/"
] | Say I have $X_{ij}$, $j \le i$ with the property that $X_{ij}$ are centered and identically distributed and $E(X_{ij} X_{ij'}) = o(\exp(-i)))$. Then does $\sum_j X_{ij}$ have Gaussian domain of attraction?
As a related question, if $X_1, X_2, X_3$ are identically distributed and centered and $E(X_i X_j) = c$, what bo... | Not necessarily. One has to impose more restrictive mixing and moment conditions. A classical book is:
Ibragimov I.A., Linnik Yu.V. <em>Independent and stationary sequences of random variables</em>
There is a long-standing question asked by Ibragimov: is $\phi$-mixing and finiteness of second moment sufficient for CL... | Your double subscripts are extraneous. Let's consider a simpler situation, where we have a single family of random variables $\{X_i\}$.
As Yuri Bakhtin says above, your condition is not sufficient for a CLT to hold. Here is a simpler situation, however: suppose that $X_i$ and $X_j$ satisfy finite-range dependence. ... | https://mathoverflow.net |
254,832 | [
"https://security.stackexchange.com/questions/254832",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/265562/"
] | <strong>Standard Password</strong>
When I create an account somewhere, like let's say a crypto exchange, I'm forced to choose a username and a strong password. Let's say I choose:
<pre><code>username: thirdratecoder
password: someLongPasswordWithMixedCaseAndTheNumber555_*&
</code></pre>
Now, that's a pretty obscure... | <blockquote>
Why is the use of Authy TFA more secure than just having a long password? Presumably the user will record his Authy code right along side his normal password.
</blockquote>
No, the user <em>should</em> not do this.
The user should record the code in a different system, or a system where it's not easily ret... | TFA as in <strong>TWO</strong> Factor Authentification is an <strong>additional</strong> protection.
Ideally your <em>Password</em> is "<em>What You Know</em>".
Authy or similar is "<em>What You Have</em>".
<pre><code>Well sort-of in the case of Authy and its ilk.
Because it's software it could be i... | https://security.stackexchange.com |
14,596 | [
"https://cs.stackexchange.com/questions/14596",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/9978/"
] | Why is the reduction $\textbf{SAT} \leq_P \textbf{3SAT}$ possible, but $\textbf{SAT} \leq_P \textbf{2SAT}$ not possible, given, that $\textbf{SAT}$ is $\textbf{NP}$-complete, $\textbf{2SAT} \in \textbf{NP}$ and $\textbf{3SAT} \in \textbf{NP}$?
| I believe the asker is wanting to know specifically why the standard approach to reducing $\text{SAT}$ to $\text{3SAT}$ does not continue to extend to the $\text{2SAT}$ case. Here is a walkthrough as to why.
<strong>Conversion of a SAT Instance to a 3-SAT Instance:</strong>
For convenience, let us assume everything i... | First of all, "not possible" is only under the assumption that $P\neq NP$.
While it is true that $2SAT\in NP$, we also know that $2SAT\in P$. Thus, if indeed $SAT\leq_p 2SAT$, then $P=NP$.
The reduction $SAT\le_p 3SAT$ is possible since we know that $3SAT$ is $NP$-complete, and therefore there is a reduction from eve... | https://cs.stackexchange.com |
572,027 | [
"https://physics.stackexchange.com/questions/572027",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/31748/"
] | Imagine two <span class="math-container">$ideal$</span> black bodies, one at temperature <span class="math-container">$T_1$</span> and the other at <span class="math-container">$T_2$</span>, <span class="math-container">$T_1 \ne T_2$</span>, both are in thermal equilibrium with their respective heat baths and now we se... | <blockquote>
After a while I expect that the two cavities (black bodies) thus connected will thermally equilibrate, they will assume the same temperature so that the total flux from one will equal the total flux from the other.
</blockquote>
I don't think this is consistent with the description of the setup. If all the... | Examine the power spectrum from both objects. For any fraction of the spectrum you want to look at, the hotter body has greater power than the cooler body. There is no portion of the spectrum where this fails to hold. At any frequency, the power difference exists and is always in the same direction.
Therefore if any... | https://physics.stackexchange.com |
718,956 | [
"https://math.stackexchange.com/questions/718956",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/113241/"
] | A popular proof of the irrationality of $\sqrt2$ is to first assume that the number is rational. This means that $\sqrt2=a/b$ where $a$ and $b$ are integers. Another assumption is that $a$ and $b$ are coprime. It turns out that this leads to a contradiction. And the conclusion is that $\sqrt2$ is not rational (because ... | You don't <em>need</em> to assume that $a$ and $b$ are coprime.
If $\sqrt 2= a/b$ with $a,b\in \mathbb Z$, then $2b^2=a^2$. Now count the number of factors of $2$ on each side: on the left, you get an odd number of factors of $2$, while on the right you get an even number of factors.
| That $a$ and $b$ are coprime is not an assumption, it is simply the fact that any rational number can be written in a way so that the numerator and denominator are coprime. The only assumption being made is that $\sqrt 2$ is rational.
| https://math.stackexchange.com |
83,176 | [
"https://mathoverflow.net/questions/83176",
"https://mathoverflow.net",
"https://mathoverflow.net/users/19874/"
] | Dear all,
giving a support class for PDE lecture i am wondering is there an easy argument for :
Why the boundary regularity of the domain important for the regularity of the solution of the weak form of the Poisson equation with Dirichlet boundary conditions?
Thank you,
Sebastian
| You might start by looking at the book by Grisvard (Elliptic problems in nonsmooth domains). For instance, in Theorem 3.1.1.1 he proves a very precise identity which shows basically the following: if you want to estimate ANY second derivative of a function $u$ defined on a domain $\Omega$ in terms of the laplacian $\De... | I am not sure if this helps when teaching a basic PDE class, but this is certainly a useful understanding:
Elliptic problems can be interpreted via diffusion processes. The solution at a point $x$ can be written as expectation of the boundary condition at the (random) exit point for the diffusion emitted from $x$ and... | https://mathoverflow.net |
2,166 | [
"https://physics.stackexchange.com/questions/2166",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/847/"
] | I read somewhere that according to relativity, it is possible - involving black holes and other stuff - to jump into the past. Is it possible for anything to go back in time either continuously or by jumping?
| It is mathematically possible to create some instances in which an object goes back in time <em>relative to some observer</em>. For example, simply going faster than light causes such an effect, but of course, speed of light is the limit for any massive object.
While it is mathematically possible, there are many parad... | There are solutions to Einstein's field equations, which have closed timelike curves. For example Godel's solution. Would that constitute time travel, if you can reach the same point on your world line in finite time? One objection may be that such solution do not describe the universe, but examples as the Tipler's cyl... | https://physics.stackexchange.com |
286,918 | [
"https://physics.stackexchange.com/questions/286918",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/98940/"
] | I have read the pages suggested as similar and not found the answer to my question, or if it's there I didn't understand it. I have seen radial probability density described on various different webpages as:
$$r^2 R_{nl}^2(r)$$
$$4 \pi r^2 R_{nl}^2(r)$$
$$R_{nl}^2(r)$$
Where $R_{nl}(r)$ is the radial wavefunction. I've... | Those factors usually come out when separating the wave-function for a 3-dimensional problem (like the Hydrogen atom) in its radial and angular parts:
$$ \psi(r, \theta, \phi) = R(r) Y(\theta, \phi). $$
If you are then interested in the probability of finding the particle (or whatever you are studying) at a distance f... | The radial part of the Schroedinger equation is solved using $u(r) \equiv R(r) r$. Here, the $R(r)$ is the actual radial part of the wavefunction. Hence the $r^2R_{nl}^2$ comes from that. The substitution is made so that it looks analogous to the classical energy equation with a central potential(effective). The $4\pi$... | https://physics.stackexchange.com |
860,283 | [
"https://math.stackexchange.com/questions/860283",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/156019/"
] | This is what I did for:
$$\lim \limits_{x \to 0} \space \cot{x}-{1 \over x}$$
<ol>
<li>Check form: $\infty - \infty$.</li>
<li>Rearrange it to be a quotient:
$$\begin{align} \\
& =\lim \limits_{x \to 0} \space {\cos{x} \over \sin{x}}-{1 \over x} \\
&\\
& =\lim \limits_{x \to 0} \space {x\cos{x} - \sin{x}... | You did indeed miss a simplification:
$$\lim \limits_{x \to 0} \space {(-x\sin{x} + \cos{x}) - \cos{x} \over (x\cos{x} + \sin{x})}=\lim \limits_{x \to 0} \space {-x\sin{x}\over (x\cos{x} + \sin{x})}$$
It's not a lot simpler, but it is simpler.
| Speaking of simplicity, often by L'Hospital rule can be applied in combination with so-called remarkable limits. If this exercise can do so:
$$\lim \limits_{x \to 0}( \cot{x}-{1 \over x})=\lim \limits_{x \to 0} \frac{x\cos x-\sin x}{x\sin x} = \lim \limits_{x \to 0} \frac{x\cos x-\sin x}{x^2}\cdot\lim \limits_{x \to 0}... | https://math.stackexchange.com |
1,579,778 | [
"https://math.stackexchange.com/questions/1579778",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/238738/"
] | I know that if $A$ is any algebra of subsets of a set $X$ and if $\mu$ is a measure defined on A then this measure $\mu$ can be extended to a measure $\mu^*$ on a $\sigma$ algebra $A^*$ of subsets of X which contain $A$
I understand the above but not the stuff below.
If B is an arbitrary subset of $X$
Then $\mu^*(B... | Here the four lines are formal definition of outer measure, usually denoted by $\mu^*$. People often construct a measure from a outer measure. You can check this from the book by Royden, and many other standard real analysis books.
And I think you confuse the measure with the outer measure in your question. An outer m... | Every sequence of sets $(E_j)_j$ in $\mathcal A$ that covers $B$ will induce a value $\sum\mu E_j$.
The collection of these values has an infimum and $\mu^{*}B$ is defined to be that infimum.
| https://math.stackexchange.com |
313,736 | [
"https://softwareengineering.stackexchange.com/questions/313736",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/144365/"
] | I've been studying creational design patterns for the past week or so because I have a common use case that keeps coming up, and I can't figure out which pattern fits the bill.
Here is a simplified scenario: I have different types of notifications -- e.g. email, SMS (text message), etc. -- and I don't want the clien... | You've done a good job of abstracting the notification, but you haven't applied similar abstraction to the job of creating the notification target.
What you're doing is not a "Factory Method", because, as you are observing, it doesn't use a single method that decides, usually by the parameters supplied, which true cla... | It appears that abstract factory shouldn't accept parameters else that would mean the client does* know the details. I have been attempting the same for years but I'm finding its incorrect e.g.
<pre><code>// EmailMsgFactory implements MsgFactory
// SmtpMsgFactory implements MsgFactory
// but here you would still need ... | https://softwareengineering.stackexchange.com |
877,344 | [
"https://math.stackexchange.com/questions/877344",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/33907/"
] | I would like to prove $\lim_{n\to \infty}e^nn!-(n+1)^n=0$.
All I have really done is show $(n+1)^n=\sum_{i=0}^n\frac{n!}{(n+1)^i(i!)(n-i)!}$
| Stirling's formula seems to hold the answer.
$$ \frac{\alpha^n n!}{(n+1)^n} \sim \frac{\alpha^n\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}{n^n}\sim \left(\frac{\alpha}{e}\right)^n\sqrt{n} $$
Since the exponent grows/shrinks quicker than $\sqrt{n}$ then if $\alpha\geq e$ it will diverge to $+\infty$ and if $\alpha<e$ ... | (This is too long for a comment, but neither is it yet a full answer.)
Consider the Taylor series expansion of $e^x$.
\begin{align*}
e^x &= \sum_{k = 0}^\infty \frac{x^k}{k!}\\
&= \lim_{n \rightarrow \infty} \sum_{k = 0}^n \frac{x^k}{k!}
\end{align*}
It follows
\begin{align*}
\lim_{n \rightarrow \infty} n! ... | https://math.stackexchange.com |
1,652,765 | [
"https://math.stackexchange.com/questions/1652765",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/313072/"
] | I am having trouble proving this statement without using the limit rules. I know I start by assuming that the sequence $\{s_n-L\}$ converges to zero, therefore, for every number $ ϵ > 0 $, there is an integer $N$ such that
$\|s_n-L-0\|$ $< ϵ$, whenever <em>n</em> > N. How would I prove that $\lim_{n\to\infty}(s_... | <strong>HINT:</strong>
You stated the definition of the limit:
For all $\epsilon >0$ there is a number $N$ so that
$$-\epsilon <s_n-L<\epsilon \tag 1$$
whenever $n>N$.
Now, add $L$ from both sides of $(1)$. What can you conclude?
| It is a direct consequence of the definition of limit which I repeat for clarity:
<strong>Definition:</strong> <em>A sequence $\{a_{n}\}$ tends to a limit $A$ as $n \to \infty$ if for any arbitrary number $\epsilon > 0$ there is a positive integer $N$ such that $$|a_{n} - A| < \epsilon$$ whenever $n \geq N$.</em... | https://math.stackexchange.com |
182,396 | [
"https://softwareengineering.stackexchange.com/questions/182396",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/43808/"
] | I am designing a simple GUI that will allow a user to create a GPX route by clicking repeatedly on a map panel. I am faced with a design dilemma for how to represent the nodes and route. This construct can be abstracted as follows:
<ul>
<li>the user clicks <strong>n</strong> times on the screen in sequence to create... | Mutable circular references are nasty. Just use an ArrayList and don't worry about it.
<em>You can see all of the elements on screen</em>: it's not going to be so long that linear time takes any time.
re dude demanding an explanation why mutable circular references are nasty: like, as in bad code. It's an inefficient... | You really want to leverage Java-provided "list maintenance" functionality. Go with ease of use first and worry about performance later. "pre-optimization" is a dog chasing its tail.
<pre><code>Class Waypoint {
string name;
double lat; // I'm not saying double is necessarily the ideal type here.
double lo... | https://softwareengineering.stackexchange.com |
495,697 | [
"https://electronics.stackexchange.com/questions/495697",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/151108/"
] | One day, someone here, I forgot who, wrote: "On modern ic's, every input and output pin can be connected directly to either ground or Vcc." Talking about ologic ic's.
Is it true? And is true all the time?
I read also many comments recommending not to do that and always add a resistor of several KOhms.
Adding a resisto... | Logic inputs and outputs are different. Different families of logic IC (TTL, CMOS etc) behave differently. The manufacturers and commercial users anticipate things that hobby users don't usually. With almost every do/don't statement, there may be some exceptions that do not apply to hobby users.
1) Don't let CMOS inpu... | Logic IC outputs do have resistance, but that resistance is generally not intended to limit output current to a safe value - it is just a byproduct of the transistors used in the output circuit.
The input circuits of CMOS logic ICs have a very high impedance. If they are left unconnected, they will pick up random cha... | https://electronics.stackexchange.com |
531,368 | [
"https://electronics.stackexchange.com/questions/531368",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/245375/"
] | When we say, for example, an antenna has spec of 22 dBi;
Does it mean that its power is 22 dB stronger <em><strong>for any point at fixed direction</strong></em> than if it were to emit spherically?
| Say you have a source of 0 dBm power.
If you hook it up to an antenna with 22 dBi gain, then in the direction of the main lobe, your source appears to be 22 dB stronger than it would if it were connected to an ideal, isotropic radiator.
| For an antenna, a gain of 22 dBi means a gain of 22 dB <strong>with respect to a theoretical isotropic antenna</strong>.
If an isotrpoic antenna radiates 1 watt, it would radiate that power uniformly over the 4π steradians around it, so at a power density of about 80 mW per steradian.
For your antenna with 22 dBi gain,... | https://electronics.stackexchange.com |
121,519 | [
"https://softwareengineering.stackexchange.com/questions/121519",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/-1/"
] | We have unfinished tasks which are part of stories in sprint. They are not accepted by the PO.
They must be included for the next planning of sprint.
Obviously the stories must be moved, but what about the tasks? Shall they be reopened and moved for the next sprint within the stories or not?
| The approach I have used to address these issues is to address them at the next sprint planning meeting, with choices of:
<ul>
<li>Do we roll over all unfinished tasks automatically to the next iteration?</li>
<li>Do we review each one and re-evaluate whether we still want that in the next sprint?</li>
<li>Do we look ... | The question that comes to my mind is: Does it reall matter? If it only matters because of SCRUM-purenessness I would just go with what's most convenient. Remeber: Focusing on success is more important than following processes.
I think started tasks should be finished to what extent that is possible. If tasks are real... | https://softwareengineering.stackexchange.com |
102,830 | [
"https://dba.stackexchange.com/questions/102830",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/67524/"
] | I have a legacy PostgreSQL database source (ODBC) that I'm attempting to migrate to new SQL Server schema using SSIS. I'm getting a warning saying:
<blockquote>
'Row by Row' fetch method is enforced because the table has LOB column(s). Column content is LOB
</blockquote>
The thing is, none of the column really <em>... | Apparently this just boils down to SSIS treating any varchar larger than 128 as NTEXT. Not sure why. I can, however, go into the advanced properties of the ODBC source and change the types back to something like DT_WSTR. Which seems to work for the most part.
However, I did determine that a few of the tables I'm deali... | I used Data Conversion for the varchar larger than 128 as NTEXT but what removed the error for me eventually is the set Validate External Data to False.
| https://dba.stackexchange.com |
454,451 | [
"https://stats.stackexchange.com/questions/454451",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/276789/"
] | We conducted A/B/C test in our website and now i want to check the results. But the problem i faced is that there are 2 test groups and 1 control, and not 1 test and 1 control groups as usual. When there is 1 control group and 1 test group I use binomial test and it's ok. If I had quantitative data I'd use ANOVA but my... | <h4>This raises an under-appreciated point about the scope of "time-series analysis"</h4>
I think you have hit on an under-appreciated point in time-series analysis here, which is that mathematically these models can also legitimately model data indexed by some other variable that does not represent "tim... | I would say the simplest answer is that a time-series is any dataset where <strong>event time is a <em>feature</em></strong>. It doesn't <em>need</em> to be correlated with anything in any way,
but you also can't assume it's <em>not</em>.
Event time here is defined as a <em>unique</em> numeric index <span class="math... | https://stats.stackexchange.com |
139,397 | [
"https://stats.stackexchange.com/questions/139397",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/69869/"
] | So suppose you have one die. Person A rolls one die 3 times. Person B does the same after A is finished. What is the probability that person A's smallest number (of his/hers 3 rolls) is greater than person B's biggest number?
Extra addition: Now problem is the same but instead of 1 die we have 32 cards with numbers 1-... | <strong>Let's generalize the problems further.</strong>
<h3>The card game: drawing without replacement</h3>
Notice that according to the description of the card game, each set of three draws is <em>without</em> replacement. Generalize "three" by $k\ge 1$ and suppose there are $N$ distinct card values.
$A$ draws $k... | I didn't solve the problem analytically. However, I wrote a code in R to simulate it. For the cards the solution should be close to 4.5% and for the dice close to 3%. I hope this helps a bit in case to confirm an analytical solution.
<pre><code>n <- 10000 #number of trials
c <- rep(0,n)
d <- 32 #set to 32 fo... | https://stats.stackexchange.com |
457,208 | [
"https://physics.stackexchange.com/questions/457208",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | The Hamiltonian operator is given by <span class="math-container">$$\hat{H} = \frac{\hat{\textbf{p}}^2}{2m} + \hat{V}.$$</span> For this reason it's of interest to know how to find <span class="math-container">$\hat{V}$</span> in any representation. Is the potential operator---independent of a chosen basis---given by <... | The equation you write does not assume a basis. In other words, the equation:
<span class="math-container">$$\hat H=\frac{\hat p^2}{2m}+\hat V(\hat r,t)$$</span>
can be expressed in any basis we choose.
For example, in 1D we can choose to work in the position basis:
<span class="math-container">$$\hat H=-\frac{\hbar^... | Yes, usually the potential is written as a polynomial through a Taylor series which easily extends to the operator analog needed for quantum mechanics.
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