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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409,784 | [
"https://electronics.stackexchange.com/questions/409784",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/131989/"
] | What are some practical limitations to the speed of a 1-Wire interface?
| Generally speaking, there are two primary factors that affect the design of a one-wire bus.
The first is the edge speed that you can get with the pullup device, which is usually a resistor, but sometimes enhanced with a transistor.
The second is the accuracy of the timing that each device on the bus can maintain. Mos... | The bit-times are fixed in hardware. There's a 16kBaud standard mode and a 140kBaud overdrive mode.
What you can do with these bits depends on the slave chip. The DS2405 for example lets you address an individual chip on the bus, then pass bit slots and use them for continous PIO data – standard mode only.
Other chip... | https://electronics.stackexchange.com |
49,283 | [
"https://electronics.stackexchange.com/questions/49283",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/771/"
] | What is the (or is there a) practical difference between designating a layer of a printed circuit board as a power plane as compared making a polygon cover the layer and naming it the same as the power net? I'm thinking of EAGLE when I ask this question, but I think there are analogies in other ECAD tools. I know Sunst... | I don't know Eagle, but in other tools the difference has to do with how the gerbers are generated.
For a power plane, the gerbers will be generated as a negative image. The gerber file will indicate features for regions or shapes where copper should be <strong>removed</strong>.
For signal layers, the gerbers will i... | I don't see how you set the difference in Eagle, unless you mean the net class. The only difference for net classes is for rules in the DRC (and exclusion from the auto-router), for instance you can define specific spacing that you want to have between power traces (or planes/polygons) and other signals or a different ... | https://electronics.stackexchange.com |
74,899 | [
"https://dba.stackexchange.com/questions/74899",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/23722/"
] | I wonder if there any case that a join can not be converted to a sub-query.
BTW: I am not discussing advantages and disadvantage of using sub-query or using join.
| As far as I know, a <code>JOIN [INNER JOIN]</code>, from the theoretical point of view, is a projection of a <code>CROSS JOIN</code> (every combination of two tables). If you can get a cross join using a subquery and apply any function on it, then you have a perfect substitute. I think you can always transform:
<pre><... | I would say the statement is <strong>false</strong>, though it's difficult to prove a negative.
I don't believe a <code>CROSS JOIN</code> could be done with sub queries
<pre><code>SELECT *
FROM T1 CROSS JOIN T2
</code></pre>
or for that matter any join that delivers a many to many result
| https://dba.stackexchange.com |
35,431 | [
"https://astronomy.stackexchange.com/questions/35431",
"https://astronomy.stackexchange.com",
"https://astronomy.stackexchange.com/users/28496/"
] | This might sound like a strange question, but something got me thinking about it recently.
The opacity of plasma in stellar interiors can get quite high, making for shorter free-paths for photons. In these conditions I guess that the light you could theoretically gather, supposing you have a pair of indestructuble eyes... | No, it's not. The radiation field in the interior of the Sun is very close to a blackbody spectrum.
If you look in any particular direction the brightness (power per unit area) you see is <span class="math-container">$\sigma T^4$</span>, where <span class="math-container">$\sigma$</span> is Stefan's constant. Even at a... | Coming from a different direction as @Rob's, <em>Opacity</em> and <em>Thermal Radiation</em> are orthogonal properties of a material. The photon flux at the interior of the sun is very high, so it is definitely not <em>dark</em>. However, it is <em>opaque</em> to virtually all light outside the sun.
To provide an anal... | https://astronomy.stackexchange.com |
56,432 | [
"https://physics.stackexchange.com/questions/56432",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/17650/"
] | <h2>Introduction</h2>
I am trying to determine the viscosity of a fluid. Therefore, I let a sphere of known mass <code>m</code> and radius <code>r</code> fall (sink) into the fluid. I then measure the time it takes to sink/travel a given height.
Then there are the following forces having a effect on the sphere:
$F_g... | All the following is simply a simple application of math. If I'm wrong anywhere (and I'm bound to be), please feel free to point it out to me or just edit the post, if you like.
The largest planetoid is Ceres, at almost 1000 km in diameter. If we assume that your planetoid was a bit larger than this, and that Ceres wa... | Your problem is that if the projectile is moving faster than the spaceships, which are in orbit round the planet, then the projectile velocity will be too great to be affected by the gravity of an asteroid. A large moon might do it, but then it wouldn't happen quickly.
How about having a small (e.g. asteroid mass) bla... | https://physics.stackexchange.com |
8,472 | [
"https://physics.stackexchange.com/questions/8472",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/2578/"
] | Say we start with an entangled electron--positron pair and we separate them. I want to take the entangled electron and bind it to a proton or the outer shell of an atom.
Is it possible to do this while having the entangled electron still remain entangled with the positron? Would the electromagnetic interaction with t... | You say:
<blockquote>
I want to take the entangled electron and bound it to a proton or the outer shell of an atom.
</blockquote>
The act of binding the electron with a proton, or an outer shell of an atom, means that some energy in the form of electromagnetic radiation will be given up by the electron in order to... | I'm not going to answer your electron<->atom+photon interaction question because my atomic physics is veeery rusty..
But regardless, suppose it works like you write, what you are proposing is simply a spin-measurement device. By measuring the positron in the "other end", you collapse the wavefunction consisting of ... | https://physics.stackexchange.com |
271,415 | [
"https://stats.stackexchange.com/questions/271415",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/155638/"
] | I'm working on a stats project, but in the assignment there is some notation I haven't been able to find in the book or googling. in the below, I don't know which equation N(u,o^2) refers to;
<img src="https://i24.photobucket.com/albums/c18/alanek2002/stat%20help%202_zpspl1wy9su.png?t=1491068704" alt="part1">
with mea... | let $X_i\sim Binom(N,p)$ and n be number of simulations
we know
$E(\bar{X_i})=E(\frac{\sum X_i}{n})=\frac{\sum E(X_i)}{n}=\frac{n*E(X_i)}{n}(because~X_i's~are~identical~for~each~simulation)=Np$
and
$E(\bar{X_i^2})=E(\frac{\sum X_i^2}{n})=\frac{\sum E(X_i^2)}{n}=\frac{n*E(X_i^2)}{n}(because~X_i's~are~identical~for~e... | First - n∗p∗(1−p) is not .00188 - it's much larger (n is a million!). Second - of course the numbers you get are large - a million is a large number, and the variance is the square of the standard deviation and therefore is very large.י
| https://stats.stackexchange.com |
706,790 | [
"https://physics.stackexchange.com/questions/706790",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/12884/"
] | The introduction of the angular momentum as <span class="math-container">$\vec l = \vec r \times \vec p$</span> is also true for point particles. So <span class="math-container">$\vec l$</span> must refer to the orbital angular momentum (and not the "spin") in such a case.
If I have a particle moving straight... | Imagine you had a particle traveling at constant <span class="math-container">$\mathbf{v_0} = \dot{z_0}\mathbf{\hat{k}}$</span> upwards. At time <span class="math-container">$t = 0$</span>, your particle is at <span class="math-container">$\mathbf{r} = z_0 \mathbf{\hat{k}}$</span> and you apply an horizontal force <spa... | Angular momentum(and torque) does not necessarily mean that there are rotations in place. you can define angular momentum for straight lines, however, that isn't very useful to solve any problem.
If you have a particle moving in a straight line and apply a force perpendicular to it, it won't rotate unless there is a c... | https://physics.stackexchange.com |
281,167 | [
"https://mathoverflow.net/questions/281167",
"https://mathoverflow.net",
"https://mathoverflow.net/users/114537/"
] | Let $(X, \leq)$ be a partial order and $\Sigma_X$ a $\sigma$-algebra on $X$. Is the set $\{(x, y) \in X\times X \mid x \leq y\}$ measurable with respect to the product $\sigma$-algebra?
| This can fail, even if you assume that the lower cones of the order are measurable with respect to the algebra $\Sigma$. For example, consider the real numbers $X=\mathbb{R}$ and let $\Sigma$ be the usual Borel algebra. Let's assume the continuum hypothesis, and let $\leq$ be a well-order of the reals in order type $\o... | Here is another counterexample.
I claim that the set $\{(x,y)\in X\times X|x\leq y\}$ is never measurable whenever $X$ has cardinality greater than the continuum. Suppose that $\{(x,y)\in X\times X|x\leq y\}$ is measurable. Then the set
$\{(x,y)\in X\times X|x\geq y\}$ is also measurable. Therefore, the diagonal
$\{(x... | https://mathoverflow.net |
22,340 | [
"https://ai.stackexchange.com/questions/22340",
"https://ai.stackexchange.com",
"https://ai.stackexchange.com/users/-1/"
] | Let's say we have a captcha system that consists of a greyscale picture (of a part of a street or something akin to re-captcha), divided into 9 blocks, with 2 missing pieces.
You need to choose the appropriate missing pieces from over 15 possibilities to complete the picture.
The puzzle pieces have their edges processe... | Scaling only makes sense when there is something that reacts to that scale. Decision Trees though, just make a cut at a certain number.
Imagine: For a feature that goes from 0 to 100 a cut at 50 may be improving performance. Scaling this down to 0 to 1 making the cut a 0.5 doesn't change a thing.
Now on the other hand ... | Feature scaling happens to be a problem when a model is characterized by having a distance metric (or another kind of numerical evaluation for that matter). Therefore models such as support vector machines, neural networks, distance based clustering methods (e.g. k means) and linear/logistic regression are prone to cha... | https://ai.stackexchange.com |
91,466 | [
"https://dba.stackexchange.com/questions/91466",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/58238/"
] | I have a problem on my cloud server with 2GB of RAM and 40GB Hard Disk that's hosting a Wordpress website that receives around 1000-2000 hits a day. My problem is that MySQL is always crashing and my only option is to reboot the server. I noticed that logs were not enabled so I enabled them and here is what i found:
<... | You have a memory problem, something very typical in a memory-bound system, and that can be confirmed on the line:
<pre><code>150207 17:31:42 InnoDB: Fatal error: cannot allocate memory for the buffer pool
</code></pre>
As InnoDB cannot allocate memory for its buffer pool, it fails, and MySQL cannot start if the Inno... | Is WP using MyISAM? Or InnoDB?
If MyISAM, then set <code>innodb_buffer_pool_size=0</code>. If InnoDB, decrease innodb_buffer_pool_size to, say, 300M and key_buffer = 10M.
<code>table_open_cache</code> and <code>table_cache</code> (they are the same thing) could be lowered to 300.
Turn off the query_cache -- <code>... | https://dba.stackexchange.com |
481,042 | [
"https://physics.stackexchange.com/questions/481042",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/232244/"
] | If an electromagnetic wave is constituted by a continually accelerating charge, then how can a photon be neutral?
| All gases, including ideal gases, are compressible. That's because the molecules of gases are far apart and can readily be brought together by pressure. The specific heats of gases depend on the process. For an ideal gas, <span class="math-container">$C_p$</span> and <span class="math-container">$C_v$</span> are the sp... | Who said ideal gas is incompressible? Its (isothermal) compressibility is given by
<span class="math-container">$$\beta_{T}=-\dfrac{1}{V}\left(\dfrac{\partial V}{\partial P}\right)_{T}=-\dfrac{1}{V}\left(\dfrac{\partial}{\partial P}\dfrac{NK_{\rm B}T}{P}\right)_{T}=\dfrac{1}{V}\dfrac{NK_{\rm B}T}{P^{2}}=\dfrac{1}{P}$$... | https://physics.stackexchange.com |
597,512 | [
"https://physics.stackexchange.com/questions/597512",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/260203/"
] | I believe most particle detectors are based on either the photoelectric effect, or simply on excitation of atoms by light. Then, the energy resulting from this process is converted into something we associate with the detection of a particle/field excitation. If that is the case, every field/particle we measure has to ... | Electromagnetic interactions are usually involved, but this not necessarily require that the detection becomes one of detecting an electromagnetic field. The example that sprang immediately to mind for me was the bubble chamber and the cloud chamber (see below).
Here are some particle detection methods in which we don'... | In order to see particles in the detector, we need them to <em>interact</em> somehow with the detector material itself in order to get a signal. Since there are 4 known kinds of <em>interactions</em> in nature (electromagnetic, strong, weak and gravitational) of which the electromagnetic field interacts the easiest, we... | https://physics.stackexchange.com |
382,850 | [
"https://softwareengineering.stackexchange.com/questions/382850",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/88371/"
] | Consider these 2 types of implementation:
<pre><code>public int add(int x, int y) {
return mysteriousAdd(x, y);
}
public int add(int x, int y) {
int output = mysteriousAdd(x, y);
return output;
}
</code></pre>
A colleague of mine says that second implementation is better because during debugging we get ... | If you compare the assembly/bytecode/IL/etc output of two functions like that, you should find them both to be the same. All but the most bush-league of compilers will optimize out the extra value.
So any performance/memory considerations should be thrown out.
<strong>So, make your choice regarding readability.</str... | Demonstrate in your programming environment that you can read the result of the function call without a variable <em>exactly as easy</em> without the temporary variable. I'm using what I think is quite a good coding environment, and I can't.
So demonstrate it. If you can, your colleague has learnt something. If you ca... | https://softwareengineering.stackexchange.com |
431,751 | [
"https://physics.stackexchange.com/questions/431751",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/208207/"
] | My chemistry teacher used to tell us that if you had a soup bowl with only electrons in it, the explosion could make you fly to Pluto. Was he right? Could this happen?
| The answer would depend how densely the electrons are packed. Let's say we have 1 kg of electrons, meaning we would have about <span class="math-container">$N = 10^{30}$</span> of them. For simplicity, let's approximate by arranging all of these electrons arranged in a spherical shell of radius <span class="math-contai... | A uniform sphere of radius <span class="math-container">$r$</span> and total charge <span class="math-container">$Q$</span> has an electric potential energy of <span class="math-container">$3Q^2/20\pi \epsilon_0 r$</span>.
Let's say your bowl is like a sphere of radius 5 cm. If it's all water ( molecular weight 18<spa... | https://physics.stackexchange.com |
470,830 | [
"https://physics.stackexchange.com/questions/470830",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | <span class="math-container">$\newcommand{\force}{\mathbf{F}}$</span>
<span class="math-container">$\newcommand{\velocity}{\mathbf{v}}$</span>
<span class="math-container">$\newcommand{\position}{\mathbf{r}}$</span>
Here's an incomplete proof of the work-energy theorem:
<span class="math-container">$$W = \int_1^2\for... | Based on our discussion in the comments and chat, it looks like you are considering a situation where the mass is lost by the system in such a way that the forces on the system caused by the loss of mass cancels out, so all that matters is the net force acting on the object. Then Newton's second law has the form <span ... | You are right. The correct version of work energy theorem states that the work done equals change in total relativistic energy and not just kinetic energy:
<span class="math-container">$W = \Delta(mc^2)$</span>
Here m is not rest mass <span class="math-container">$m_0$</span> but the total relativistic mass <span cla... | https://physics.stackexchange.com |
258,336 | [
"https://stats.stackexchange.com/questions/258336",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/135544/"
] | I'm comparing two test methods, A and B, for asphalt density against the reference "true" density. I have paired measurements for each method and the reference. From a paired t-test of Method A vs. Reference, the p-value is 0.40. For Method B vs Reference, the p-value is 0.0007.
So, I reject the null hypothesis that ... | Technically, it is not possible to draw any conclusion about the "sameness" of the two groups from the p value. The p-value is the probability of observing the data giving that the null hypothesis is right. An high p-value means that assumes H0 is right simply means that given that hypothesis, it is very likely that yo... | I don't have enough points to comment right now, though I'm sure someone can come up with a more complete answer.
The simple answer is no you can't claim that reference and Method A are the same. There is no p-value that will allow you to make that claim. Your alternative hypothesis was that the $\mu_A \neq \mu_{Ref}$... | https://stats.stackexchange.com |
178,381 | [
"https://mathoverflow.net/questions/178381",
"https://mathoverflow.net",
"https://mathoverflow.net/users/18943/"
] | Let $X$ be a compact Hausdorff topological set, and $Y$ be its closed subset. Is the ideal of functions vanishing on $Y$
$$
I=\{f\in C(X):\ \forall y\in Y\ f(y)=0\}
$$
complementable (as a closed subspace) in $C(X)$ (as a Banach space)?
This is true in the case when $X\subseteq {\mathbb R}^n$ (this follows from: Stein... | Not in general.
It's well-known in Banach space theory that the ideal $c_0$ in $\ell^\infty$ is not complemented (see e.g. Albiac & Kalton).
By the Gelfand representation, $\ell^\infty \simeq C(\beta \mathbb{N})$ as a $C^\ast$-algebra. This maps $c_0$ to the ideal of functions on $\beta \mathbb{N}$ that vanish on... | Alexander Shamov answered your question with a classical example, but you might be interested in a modern example constructed by Piotr Koszmider. There is an infinite connected compact Hausdorff space $K$ s.t. $C(K)$ has no complemented subspaces that are both infinite dimensional and infinite codimensional. In par... | https://mathoverflow.net |
26,209 | [
"https://datascience.stackexchange.com/questions/26209",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/44057/"
] | <h1>Details:</h1>
<strong>GPU</strong>: GTX 1080
<strong>Training</strong>: ~1.1 Million images belonging to 10 classes
<strong>Validation</strong>: ~150 Thousand images belonging to 10 classes
<strong>Time per Epoch</strong>: ~10 hours
I've setup CUDA, cuDNN and Tensorflow( Tensorflow GPU as well).
I don't think my mo... | That's about expected. If you divide the number of seconds by the number of images you processed, you get 33 milliseconds per image, which seems about right for such a small network. Larger networks usually take in the ballpark of 50 to 200 milliseconds per image.
Yes, a large dense layer is likely to hurt your perfo... | As shimao said, that's about what you'd expect. Despite not having many layers, an input size of 512x512 is a large image to be convolving over. The large computation time is likely more due to convolving 64 filters over the large image, instead of the fully connected layers.
The network you've put together has a funn... | https://datascience.stackexchange.com |
108,298 | [
"https://chemistry.stackexchange.com/questions/108298",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/24079/"
] | <blockquote>
An ionic compound has a solubility of <span class="math-container">$1\ \mathrm M$</span> in water at <span class="math-container">$25\ ^\circ \mathrm C$</span> and its solubility increases as the temperature is raised. What are the signs of <span class="math-container">$\Delta H^\circ$</span> and <span c... | @Raphaël did the hard part. ΔH is larger than zero.
<blockquote>
[OP] Since the the solubility increases as the temperature is raised (ΔG∘ becomes more negative), I know that ΔS∘ is positive
</blockquote>
This argument linking solubility and Gibbs energy is incorrect. Le Chatelier or the Van't Hoff equation connect... | I would solve this question rather simply.
Let's consider the general equation <span class="math-container">$\ce{ X-Y_{(s)} \rightarrow X^+_{(aq)} + Y^-_{(aq)}}$</span> with a <span class="math-container">$\Delta H = C$</span>. Intuitively, if the reaction is endothermic (<span class="math-container">$\Delta H > 0$... | https://chemistry.stackexchange.com |
2,884,788 | [
"https://math.stackexchange.com/questions/2884788",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/584749/"
] | I am beginning a differential equation textbook, and in the first problem section I encountered a question regarding the change in a population of mice. I am given that $\frac{dP}{dt} = kP^2$, $P(0) = 2$, and that $\frac{dP}{dt} = 1$ when $P = 10$. It is simple to find the value of $k$, but I am struggling to figure ou... | Hint:
$$x^2(y^2-1)-2x(1+y)-(y+1)^2=0$$
What if $y+1=0?$
Else $$x^2(y-1)-2x-(y+1)=0$$
Method$\#1:$
$$x=\dfrac{2\pm\sqrt{4+4(y^2-1)}}{2(y-1)}=\dfrac{1\pm y}{1+y}$$
Now $\dfrac{1-y}{1+y}=\dfrac2{1+y}-1$
$\implies1+y$ must divide $2$
Method $\#2:$
By symmetry, $x+1$ must be a factor of $$x^2(y-1)-2x-(y+1)$$
What... | Or $$x^2y^2 = (x+y)^2+2(x+y)+1 \implies x^2y^2 = (x+y+1)^2$$
<strong>1. case:</strong> $$xy =x+y+1\implies (x-1)(y-1)=2\implies ....$$
<strong>2. case:</strong> $$xy =-x-y-1\implies (x+1)(y+1)=0\implies ....$$
| https://math.stackexchange.com |
59,770 | [
"https://mathoverflow.net/questions/59770",
"https://mathoverflow.net",
"https://mathoverflow.net/users/13625/"
] | A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all simple using the concept of Riemann surfaces. But unfortunately, I just can't find it back. Does someone know if such a r... | This is widely open. Moreover, I think we will prove the Riemann Hypothesis much earlier than the simplicity of the zeros (if true). The latter is somehow much more accidental, the only reasonable argument I know in favor of it is "why would two zeros ever coincide"? Note, however, that some automorphic $L$-functions d... | To the best of my knowledge, it is still an open question as to whether all the zeros are simple. If you could find that article....
For what it's worth, any number of "proofs" of the Riemann Hypothesis have appeared on the ArXiv. Here are a few (I've not included three more that were withdrawn by the authors).
1006... | https://mathoverflow.net |
3,922 | [
"https://physics.stackexchange.com/questions/3922",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/1602/"
] | I have read in a couple of places that $\psi(p)$ and $\psi(q)$ are Fourier transforms of one another (e.g. Penrose). But isn't a Fourier transform simply a decomposition of a function into a sum or integral of other functions? Whereas the position and momentum wavefunctions are essentially different but related. The... | Dear user1602, yes, $\psi(x)$ and $\tilde\psi(p)$ are Fourier transforms of one another. This answers the only real question you have asked. So if one knows the exact wave function as a function of position, one also knows the wave function as a function of momentum, and vice versa.
In particular, there is no "wave fu... | If you have a state $|\psi\rangle $, the position wave function is:
\begin{equation}
\langle \vec{x}\mid \psi\rangle
\end{equation}
and the momentum wave function is:
\begin{equation}
\langle \vec{p}\mid \psi\rangle = \int\;d\vec{x}\; \langle\vec{p}\mid\vec{x}\rangle\langle \vec{x}\mid\psi\rangle
\end{equation}
The... | https://physics.stackexchange.com |
2,306,015 | [
"https://math.stackexchange.com/questions/2306015",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/437278/"
] | I could really use some help here:
Prove that $17|2x+3y\iff 17|9x+5y$.
I don't even know how to start. Just pointing me at a similar problem that has a full solution would be helpful enough, but of course I will be extremely thankful if someone could explain how to approach solving this! :)
| Let's see the problem in terms of linear algebra.
Let $u=(2,3)$, $v=(9,5)$, and $w=(x,y)$. These are vectors in $\mathbb F_{17}^2$.
Then, $17\mid 2x+3y\iff 17\mid 9x+5y$ iff $\langle u,w \rangle = 0 \iff \langle v,w \rangle = 0 $, and this happens iff $u^{\perp} = v^{\perp}$, which happens iff $u$ and $v$ generate th... | Note
<blockquote>
$$17x+17y = 9x+5y + 4\cdot (2x+3y).$$
</blockquote>
The rest follows immediately since $\gcd(17,4)=1$ since $at - au$ is always divisible by $a$. Here $a=17$ $t=x+y$ and $u$ is either ${1\over 17}(9x+7y)$ or ${4\over 17}(2x+3y)$.
| https://math.stackexchange.com |
660,413 | [
"https://physics.stackexchange.com/questions/660413",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/311740/"
] | In 17th century, Sir Isaac Newton gave us the universal law of gravitation which stated that gravity is an inverse square force. In 1915, Albert Einstein recognised gravity as a curvature of space-time caused by the presence of mass and energy in it. Does that mean Newton was wrong? Is Newton's law not true? It did pre... | The job of physics is to construct models that are able to explain and predict empirical observations. You can never be completely sure that a given model is the "true" description, only that it, at the very least, captures facets of the truth by successfully accounting for certain observations.
Newton's law ... | What we know is that gravity is an effect that causes the motion of particles to deviate from straight (Euclidian) lines, that it is an attractive rather than a repulsive effect, that it weakens with distance, that the effect propagates at the speed of light, that the strength of the effect is related to the degree to ... | https://physics.stackexchange.com |
164,668 | [
"https://softwareengineering.stackexchange.com/questions/164668",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/25476/"
] | I need some expert answers to help me determine the most efficient algorithm in this scenario.
Consider the following data structures:
<pre><code>type B { A parent; }
type A {
set<B> children;
integer minimumChildrenAllowed;
integer maximumChildrenAllowed;
}
</code></pre>
I have a situation where I n... | I would:
<ol>
<li>Calculate max*countOfA and min*CountOfA</li>
<li>If you don't have enough B to fill all the minimum A values you can't actually meet the criteria you have listed.</li>
<li>If you don't have enough A then add the minimum number you need (ceiling of (SumOfAMaxes-B)/DefaultMaximum). Update the sum of m... | Assuming you have existing A's with various numbers of B's already assigned to them, my first pass at this would probably be:
Sort the A's first on their number of B's (will call that the ChildCount), and then on a random number as the second sort factor. So you have a list of A's grouped by their ChildCount, lowest ... | https://softwareengineering.stackexchange.com |
494,428 | [
"https://physics.stackexchange.com/questions/494428",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/236411/"
] | Say I move from point A to point B. I carry an apple in my hand. What causes the apple to move from point A to B? In other words, what force is responsible for keeping the apple stuck to my hand and accelerate with the same acceleration as me? Is it friction or normal reaction?
What is the work done by each? Please me... | Puk is correct, but I'd like to elaborate on his answer a little bit. If you just place the apple on your hand (and don't grip it), the friction from your hand is responsible for moving the apple. Let's assume that you, your hand, and the apple are moving to the right.
Now think of it this way: when the hand begins mo... | The normal force is always at least partially responsible. Without the normal force, there is no friction. To what extent friction and normal force separately play a role depends on how you hold the apple. If you grip it tightly, it is likely mostly the normal force keeping it moving with you. If you place it on the pa... | https://physics.stackexchange.com |
16,406 | [
"https://cs.stackexchange.com/questions/16406",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/140/"
] | Is there a name for this infinite language?
$$L = \{ 0(,1)^*(,2)^*(,3)^*....(,n)^* \;\mid n \geq 1\}$$
A string $w \in L$ is simply an ordered list of increasing integers in which every integer $\geq 1$ can be repeated 0 or more times, for example "0", "0,1", "0,1,2,2", "0,1,2,3", "0,1,1,4,4,4,21,21".
Or perhaps a ... | Non-decreasing sequences of natural numbers.
| I don't know of any standard name, but I think calling it the language of sorted lists (or: sorted lists of integers) would be reasonable.
| https://cs.stackexchange.com |
307,657 | [
"https://dba.stackexchange.com/questions/307657",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/162483/"
] | I have a multi-join SQL query.
<pre><code>SELECT A.COMPANYEMAIL, A.CONTRACTNAME,
B.CLAIMNAME, C.PARTNAME, C.PARTPRICE,
C.DAMAGEAMOUNT, C.DAMAGEDATE
FROM CONTRACT A
JOIN CLAIM B ON A.ID=B.CONTRACTID
JOIN PART C ON B.ID=C.CLAIMID
WHERE A.COMPANYEMAIL='XYZ@ABC.COM'
ORDER BY DAMAGEDATE DESC
</code></pre>
The ID column in... | <blockquote>
<ol>
<li>Create non clustered index on B.CONTRACTID and C.CLAIMID</li>
</ol>
</blockquote>
No, by themselves, these are non-covering and would leave you with Key Lookup operations on a large tables. A Key Lookup is a row based operation, meaning if you feed it 10 million rows, it'll turn around and execute... | <ol>
<li>Yes, it is good practice to index your foreign key columns.
</li>
<li>Yes, try to cover where you can. Test by adding non covering elements as included columns, or key columns.
</li>
<li>I'd focus on what is being filtered in the <code>where</code>.
<pre><code>create nonclustered index [ix_Contract_CompanyEmai... | https://dba.stackexchange.com |
530,101 | [
"https://electronics.stackexchange.com/questions/530101",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/267588/"
] | Couldn’t appliances like fans also use AC motors? What is the advantage of converting to DC?
| When they do it's because they are controlled by electronic devices. It's easier to control DC than AC. The conversion also produce a lower DC voltage used by the electronic and safer for the user.
| Most household fans do not use DC motors but use AC motors to avoid conversion.
| https://electronics.stackexchange.com |
68,533 | [
"https://mathoverflow.net/questions/68533",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3692/"
] | Kurepa Hypothesis says there is a Kurepa tree, which is a $\omega_1$-tree has at least $\omega_2$ many branches.
It is known that beginning from a model with an inaccessible cardinal $\kappa$, after collapes $\kappa$ to $\omega_2$ using the Levy collape, then in the generic extension, Kurepa Hypothesis fails.
In above ... | The answer is yes, and merely forcing over your model to add additional subsets of $\omega_1$ will pump up the value of $2^{\omega_1}$, while not creating Kurepa trees.
Specifically, let us start in $V$, where $\kappa$ is an inaccessible cardinal, and suppose also that the GCH holds. You mentioned the result of Silve... | This paper contains several results of the kind: Keith Devlin:
$\aleph _{1}$-trees, Ann. Math. Logic 13(1978), 267–330.
| https://mathoverflow.net |
2,347,807 | [
"https://math.stackexchange.com/questions/2347807",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/442267/"
] | I have a function that I want to find the derivative of using the difference quotient definition of a derivative. The function is:
$$f(x)=\frac{\sqrt{x}}{x+1}$$
therefore, using the difference quotient definition, we have:
$$f'(x)=\lim_{h\to0}\frac{1}{h}\left(\frac{\sqrt{x+h}}{x+h+1}-\frac{\sqrt{x}}{x+1}\right)$$
t... | You properly wrote$$\lim_{h\to0}\frac{1}{h}\left(\frac{(x+1)\sqrt{x+h}-(x+h+1)\sqrt{x}}{(x+1)(x+h+1)}\right)$$ Now, it is clear that the problem is within $\sqrt{x+h}$. But, you could use the generalized binomial theorem to write $$\sqrt{x+h}=\sqrt{x}+\frac{h}{2 \sqrt{x}}+\cdots$$ So, for the numerator $$A=(x+1)\sqrt{x... | Alternatively, you can use the definition
$$f'(a)=\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}.$$
This gives
$$\frac{f(x)-f(a)}{x-a}=\frac{\frac{\sqrt x}{x+1}-\frac{\sqrt a}{a+1}}{x-a}=\frac{(a+1)\sqrt x-(x+1)\sqrt a}{(a+1)(x+1)(x-a)}$$
$$=\frac{(a\sqrt x-x\sqrt a)+(\sqrt x-\sqrt a)}{(a+1)(x+1)(x-a)}=\frac{-\sqrt {a... | https://math.stackexchange.com |
152,565 | [
"https://electronics.stackexchange.com/questions/152565",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/65988/"
] | <strong>Background:</strong> I am familiar with interfacing <em>a few</em> incremental/quadrature encoders to a single MCU chip. My go-to chip <code>STM32F10x</code> can decode 4-5 encoders even in a LQFP64 package (e.g. <code>STM32F100R8T6</code> has 4), which had always been more than enough for me... <em>until now</... | In the end the decision was to group them by 4 as @spehro-pefhany suggested in comment, and connect them to a bunch of MCUs, which can then communicate on SPI bus.
The main advantage is less cable mess, as I can place the MCUs close to the encoders.
| A small FPGA is the way to go here. You would need a very large CPLD to manage this many encoders - they are not useful for much more than glue logic. Generally a CPLD gives you one flip-flop per pin. An FPGA has logic resources that are specifically designed for building things like accumulators and shift registers... | https://electronics.stackexchange.com |
32,820 | [
"https://physics.stackexchange.com/questions/32820",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/7951/"
] | I have a slight difficulty understanding the solution to the following problem:
A light inextensible string with a mass $M$ at one end passes over a pulley at a distance $a$ from a vertically fixed rod. At the other end of the string is a ring of mass $m(M>m)$ which slides smoothly on the vertical rod as shown in t... | For your first question: obtaining equation (2) from (1) is basically just a matter of taking the derivative. Consider this expression that relates an infinitesimal change in the value of a function $f(x,b)$ to the infinitesimal changes in the values of its arguments:
$$\delta f(x,b) = \frac{\partial f(x,b)}{\partial ... | What I recommend to you is, please come up with your solution and put it here. Then we can let you know where you had done the mistake. However, since you also said that - you had a problem in understanding the solution of the above problem...I will try my best to make you understand this solution.
When you look into ... | https://physics.stackexchange.com |
190,452 | [
"https://dba.stackexchange.com/questions/190452",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/123345/"
] | I have to run a query where I need to find all records that were created OR updated within a date range.
The problem with this is that the data set is large and so a query in the past couple years can return a few million rows.
I think there are a couple problems here.
MySQL isn't going to be able to choose a key/... | <code>OR</code> is a difficult to optimize. 'Often' the fix is to rewrite using <code>UNION</code>.
<pre><code>( SELECT ... WHERE created_at > X )
UNION DISTINCT
( SELECT ... WHERE updated_at > X )
</code></pre>
and have 2 separate indexes:
<pre><code>INDEX(created_at)
INDEX(updated_at)
</code></pre>
Each <c... | I usually implement updated to be set to the creation date on creation. It makes queries like this trivial as you only need to query on the update date.
Using triggers for both ensures they are correctly maintained.
On insert:
created, updated = NOW
On update:
created = old.created (prevents modificatio... | https://dba.stackexchange.com |
61,749 | [
"https://chemistry.stackexchange.com/questions/61749",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/20633/"
] | During chemical reactions, the bonds between atoms break or form to either absorb or release energy. The result is a change to the potential energy of the system. The heat absorbed or released from a system under constant pressure is known as enthalpy.
Mathematically, we can think of the enthalpy of reaction as the di... | No: bond strengths as such do not change with temperature. The potential energy surface holding the atoms together, i.e. all the bonds, is constant for a given molecule.
However, all molecules have vibrational and rotational energy levels and molecules also have bulk kinetic energy due to thermal motion in a gas or l... | I think the answer is already in the question. You're talking about <em>enthalpy</em>, so we're measuring change in energy under <em>constant pressure</em>. This means that the system is able to do work as a result of volume changes. Volumes do change as a function of temperature, and different substances have differen... | https://chemistry.stackexchange.com |
861,103 | [
"https://math.stackexchange.com/questions/861103",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/153695/"
] | How many ways can a Doctor go to the Hospital on $5$ days of January (which has $31$ days) such that no two visits are on consecutive days?
I think the solution is: $\displaystyle\binom{27}{5}$<br>
But I have no idea if I'm right.
| The argument goes: after the first four hospital days ($H$), there must be a non-hospital day ($NH$). So we find a permutation of $$\{H,H,H,H,H,\overbrace{NH,NH,\ldots,NH}^{27}\}$$ of which there are $\binom{27}{5}$ then add $NH$ after the first four $H$s.
<hr>
For example, the permutation might come out as:
$$(NH,... | You can represent the 26 days when the doctor does not go to the hospital by 26 sticks, which create 27 gaps. If you select 5 of these gaps, this determines the 5 days when the doctor does go in; so this shows that your answer of $\binom{27}{5}$ is correct.
<hr>
Another way to see this is to let $x_1, \cdots, x_6$ b... | https://math.stackexchange.com |
424,505 | [
"https://electronics.stackexchange.com/questions/424505",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/184100/"
] | I have an active high reset (RST) pin that is connected to a push-pull output. I am trying to connect another input to this RST pin, but this input is unable to drive the RST line high. Is this not possible because of the other output that is connected to the RST line? Can multiple push-pull outputs not drive a single ... | No, multiple push-pull outputs cannot drive the same pin.
If you want the "default" state of this pin to be low then you need to add a pulldown resistor to it, perhaps 10k<span class="math-container">\$\Omega\$</span> to ground. Then use drivers that can only drive high, such as PMOS/PNP transistors or tristate driver... | Run each push-pull output thru a schottkey diode. Then they can only drive high, and not low. 10K pulldown as noted above.
| https://electronics.stackexchange.com |
250,874 | [
"https://physics.stackexchange.com/questions/250874",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/114939/"
] | <blockquote>
A small hand grenade is thrown with an initial speed V0 forming an
angle ɵ with the horizontal ground.
Assume that at its highest point the grenade explodes and is split
into two identical parts. Knowing that one of the parts travels down
along the same path followed by the grenade from launch... | You can approach this in a couple of different ways.
1) The motion of the center of mass of the particles will follow the same path the grenade would have followed if it had not exploded.
2) The total momentum of the two pieces <em>immediately after</em> the explosion will be equal to the momentum of the grenade <em>im... | For the second part to travel down the path of the full grenade, it must have a velocity $-V_x$. By taking momentum into account, it can then be shown that the other part must have a horizontal velocity $3V_x$. Momentum is conserved, even though kinetic energy is not. You can then use these new velocities to calculate ... | https://physics.stackexchange.com |
300,920 | [
"https://electronics.stackexchange.com/questions/300920",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/146991/"
] | I'm still student so excuse me if the question is not professional.
Basically ADC can convert any voltage within decided range to the digital form.
Isn't that what exactly the digital voltmeter does?
Why do we need the complicated circuit of digital voltmeter if only ADC is enough to do the job for us?
| If you connect a 5V ADC to 120VAC, you will likely have a rather violent puff of smoke. The circuitry of a digital voltmeter is for scaling and protection: it scales a wide variety of voltages in ranges to fit a single range of voltages that is most optimal for the single ADC internal to the voltmeter. This circuitry i... | Bluntly, User Interface. The DMM has an ADC in it, and it has plenty of interface hardware that correctly scales the input for the ADC, provides an excitation voltage to facilitate resistance and capacitance measurements, reference resistors to allow current measurements, glue to drive the LCD display, etc.
Even if y... | https://electronics.stackexchange.com |
590,890 | [
"https://physics.stackexchange.com/questions/590890",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/278615/"
] | Consider a Lagrangian <span class="math-container">$\mathcal{L}$</span> which is function of, for example, some vector fields <span class="math-container">$A^\mu$</span> and tensor fields <span class="math-container">$B^{\mu\nu}$</span>. That is,
<span class="math-container">\begin{align}
\mathcal{L}=\mathcal{L}(A^\mu,... | <strong>Classical mechanics answer</strong>
A simple example of this would be to impose constraints that can't be solved simultaneously, using Lagrange multipliers. For example, let's take a particle in 2 spatial dimensions and require that it is simultaneously on a circle of radius <span class="math-container">$R$</sp... | It is probably easy to find a Lagrangian producing inconsistent equations of motion. To this end, one can, e.g., start with a complex Lagrangian. It is not obvious that such Lagrangians are necessarily unsatisfactory, because, while they do not seem to make sense in classical mechanics, the relevant path integral in qu... | https://physics.stackexchange.com |
551,727 | [
"https://electronics.stackexchange.com/questions/551727",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/278544/"
] | If the frequency of the load exceeds the SMPS switching frequency, what are the results?
| <blockquote>
If the frequency of the load exceeds the SMPS switching frequency, what are the results?
</blockquote>
If what you mean is the frequency of current drawn by the load...
Like any regulator, your SMPS controls its output impedance via feedback up to a certain frequency that is lower than the switching freque... | For a HF load greater than HF SMPS source, the voltage regulator depends on the capacitance ratio of load to source. Otherwise a resistive load will draw all it's current from the low ESR output caps as the SMPS keeps charging it up. In some cases, you might use an RC filter to reduce load regulation ripple.
| https://electronics.stackexchange.com |
3,467,592 | [
"https://math.stackexchange.com/questions/3467592",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/371990/"
] | Can someone please explain to me how I can show the following operator is compact?
<span class="math-container">$$ Af(x)=\int_0^xf(t)dt$$</span>
where <span class="math-container">$A:L_2[0,1]\rightarrow L_2[0,1]$</span>.
There are two definitions of compactness that I've encountered:
Def1. An operator <span class="ma... | To show that <span class="math-container">$A$</span> is bounded, Cauchy-Schwarz works well
<span class="math-container">$$
\|Af\|^2 = \int_0^1\left|\int_0^x f(t)dt\right|^2dx \\
\le \int_0^1 \int_0^x|f(t)|^2dt\int_0^xdt\,dx \\
= \int_0^1 \int_0^x|f(t)|^2dt xdx \\
\le \int_0^1 \int_0^1 |f(t)|^2dt \, 1 ... | <span class="math-container">$A$</span> is an integral operator with kernel <span class="math-container">$K(x,t) = \mathsf1_{(0,x)}(t)$</span> and
<span class="math-container">$$
\iint_{[0,1]^2} |K(x,t)|^2\ \mathsf d(x\times t) = \int_0^1\int_0^x\ \mathsf dt \ \mathsf dx = \frac 12<\infty,
$$</span>
so <span class="... | https://math.stackexchange.com |
238,674 | [
"https://mathoverflow.net/questions/238674",
"https://mathoverflow.net",
"https://mathoverflow.net/users/27767/"
] | Consider a vector field $F:\mathbb{R}^2\rightarrow \mathbb{R}^2$ of the following form $F(y_1,y_2)=(y_2,\mu(y_1))$, where $\mu\in\mathscr{C}^1(\mathbb{R})$ has appropriate growth so that the solutions of $Y'=F(Y)$ exists globally for any initial condition. Assume for some $a\in\mathbb{R}$ that $\mu(a)=0>\mu'(a)$.
T... | The period of the compact, non-singular orbit $\gamma$ is given by $$T(\gamma)=\oint_\gamma \tau$$ where $\tau$ is any differential $1$-form such that $\tau(F)=1$, <em>e.g.</em> $\tau:=\frac{\mathrm{d}y_2}{\mu(y_1)}$. From this expression you may be able to study the boundedness of $T$ as $\gamma$ closes on another sta... | For convenience, let's take $a=0$ and $\mu'(a) = -1$, so the linearized system has solutions $y_1 = r \sin(t)$, $y_2 = r \cos(t)$. Write a periodic solution of the
nonlinear system as $y_1 = r(t) \sin(\theta(t))$, $y_2 = r(t) \cos(\theta(t))$. We then get
$$ \dot{\theta} = \frac{\dot{y_1} \cos(\theta) - \dot{y_2} \s... | https://mathoverflow.net |
6,655 | [
"https://cs.stackexchange.com/questions/6655",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/4193/"
] | Suppose a <strong>build</strong> max-heap operation runs bubble down over a heap. How does its amortized cost equal $O(n)$?
| I assume that the operation <strong>build</strong> just turns an array into a heap by repairing the heap-property for every subtree bottom-up (let the operation for a single repair step called <strong>heapify</strong>).
It is not so hard to see, that <strong>heapify</strong> takes $O(h)$ steps, where $h$ is the height... | Note that here we don't have an arbitrary list of operations, we are talking about a single operation. So referring to it as amortized analysis can be confusing. Amortized analysis is usually used for an arbitrary list of operations, not a particular list of operations. We can use the methods for amortized analysis but... | https://cs.stackexchange.com |
147,215 | [
"https://chemistry.stackexchange.com/questions/147215",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/97925/"
] | I was trying to understand the derivation of the formula <span class="math-container">$G=G^\circ+RT\ln (p/p^\circ)$</span> starting with the following relationships:
<span class="math-container">$$
\begin{align}
\mathrm dU&=T\,\mathrm dS-p\,\mathrm dV\tag{1}\\
\mathrm dH&=\mathrm dU+p\,\mathrm dV+V\,\mathrm dp=... | For a continuous flow system operating at steady state in contact with a constant temperature reservoir at the same temperature as that of the entering fluid, and with negligible change in kinetic energy and gravitational potential energy, the reversible shaft work per mole that the system does on the surrounding is eq... | The sign convention you use for work 'done by' and work 'done on' the system is not correct when you attempt to interpret the <span class="math-container">$\Delta G$</span> equation.
The first law is:<br />
<span class="math-container">$$U = W + q$$</span>
<span class="math-container">$$dU = dW + dq$$</span>
<span clas... | https://chemistry.stackexchange.com |
191,309 | [
"https://softwareengineering.stackexchange.com/questions/191309",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/59642/"
] | I've just taken on a new job at a college as (the sole) Web application developer.
The college has a number of disparate but all pretty badly coded legacy systems. Mostly built in PHP they deal with things like attendance, exam results, marking etc.
My first job is to build a system that incorporates a lot of this da... | I would not expect any "agile development process" here as a solution to your current problem. First thing for you should be: <strong>clear your mission</strong>. That means:
<ul>
<li>clarify what your own responsibilities are</li>
<li>clarify what the responsibilities of the other stakeholders are</li>
<li>identify w... | <strong>Separate the ones who really want this project to work from the herd.</strong>
Due to a lot of politics, someone put together this meeting with a list of attendees where the membership was determined by who would get the most upset if I don't invite them. It happens. This goal was fullfilled but as the develop... | https://softwareengineering.stackexchange.com |
529,599 | [
"https://physics.stackexchange.com/questions/529599",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/201807/"
] | Conceptually, I've always understood entropy to be a statistical idea. For example, if you have a vacuum inside of a box and you place a handful of gas atoms on one side, the molecules have a higher statistical probability to spread out, versus remain in one concentrated spot. Therefore, on average, they will spread ar... | Here is a mathematically precise real-world example: Consider a monoatomic ideal gas of <span class="math-container">$N$</span> atoms in a box of constant volume <span class="math-container">$V$</span>, in contact with an environment of temperature <span class="math-container">$T$</span>.
Now imagine increasing the te... | Say you have a box. And you have <span class="math-container">$n$</span> balls numbered from <span class="math-container">$1$</span> to <span class="math-container">$n$</span>. Your goal is to put the balls inside the box. However, more the number of balls inside the box, the harder it is to add more. That’s just how t... | https://physics.stackexchange.com |
337,189 | [
"https://math.stackexchange.com/questions/337189",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/67737/"
] | Find the volume of the solid below the graph of the function $z = 81-x^2-3y^2$
above the region D in the xy-plane where D is the region between the parabola
$y^2 = 2x+4$ and the line $y = x-1$.
| First of all find the area $D$ by intersect to functions lying on the $xy$ plane: $$y^2=2x+4\\\ y=x-1$$ You get $$y_1=1-\sqrt{7},~~y_2=1+\sqrt{7},~~x_1=2-\sqrt{7},~~x_2=2+\sqrt{7}$$ so you should think about the following triple integrals: $$\int_{y_1}^{y_2}\int_{\frac{y^2}2-2}^{y+1}\int_0^{81-x^2-3y^2}dzdxdy$$
<img s... | (Note: as I believe this to be homework I am only suppling a strong hint. Please let me know if you need more or if it doesn't work) Solve first for the intersections of the two functions only involving x and y. Say that they intersect when $y = a$ and $y = b$. You then need to solve for: $$\int_{a}^{b}\int_{y^2/2-2... | https://math.stackexchange.com |
25,148 | [
"https://stats.stackexchange.com/questions/25148",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/10077/"
] | I'm having following problem while doing some analysis with R.
I have a dataframe like this:
<pre><code>Name | Group | Count
Person 1 | A | 3
Person 2 | A | 1
Person 3 | A | 0
Person 1 | B | 5
Person 2 | B | 0
Person 3 | B | 1
Person 1 | C | 1
</code></pre>
and I'd need to "expand" it (not sure if the right term) t... | While it is a very useful package, I think <em>reshape</em> is overkill in this case, <em>rep</em> can do the job.
Here are some example data:
<pre><code>df <- data.frame(
name=c("Person 1", "Person 2", "Person 3", "Person 1", "Person 2", "Person 3"),
group=c("A", "A", "A", "B", "B", "B"),
count=c(3... | You can use the untable function from the reshape package.
Given the above df (by @Gaël Laurans)
<pre><code>library(reshape)
expanded<-untable(df[,c(1,2)], num=df[,3])
name group
1 Person 1 A
1.1 Person 1 A
1.2 Person 1 A
2 Person 2 A
4 Person 1 B
4.1 Person 1 B
4.2 Person 1 B
4... | https://stats.stackexchange.com |
53,956 | [
"https://datascience.stackexchange.com/questions/53956",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/38315/"
] | I want to compute the "distance" between users in order to return the top n similar users, for any given user.
For each user a have a bunch of features.
This is close to a recommendation system, however I don't have ratings and I can't assign each user to different products, hence I'll need to compare "how similar" Us... | As others also pointed out as long as you have numeric data (or data that can be converted to numeric) you can use some sort of distance measure between the users.
Simple solution is the <strong>euclidean distance</strong> (or some other like minkowski or manhattan). The gotcha with them that they are sensitive for ha... | No need for algorithms, or recommendation systems. You have:
<blockquote>
For each user a have a bunch of features.
</blockquote>
As long as they're numeric, or can be made numeric (e.g. aggregating the values or one-hot-encoding them), you already have distances. What you may not have is the proper variance acro... | https://datascience.stackexchange.com |
7,349 | [
"https://softwareengineering.stackexchange.com/questions/7349",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/-1/"
] | As a <strong>general purpose</strong> programmer, what should you learn first and what should you learn later on?
Here are some skills I wonder about...
<ul>
<li>SQL </li>
<li>Regular Expressions </li>
<li>Multi-threading / Concurrency </li>
<li>Functional Programming </li>
<li>Graphics </li>
<li>The mastery of ... | SQL -- learn immediately. Don't worry about DBA stuff, but master queries, table and view creation, some optimization, joins, etc.
Regular Expressions -- Handy but almost never critical. Any time I need a RegEx I just google around for it and test until I find a good one.
Multi-threading / Concurrency -- Useful to ... | In my experience, programmers who are "trying to be good programmers" by learning things like they would learn basic math are never as good as those who program with a purpose.
Just learn what you need to do to accomplish an idea you have; learning any of the points you listed is useless if you're not going to use it.... | https://softwareengineering.stackexchange.com |
8,657 | [
"https://mechanics.stackexchange.com/questions/8657",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/4418/"
] | I have a '97 civic sedan.<br/>
I want to lower the front of my car and lift the rear, or keep it standard. <br/>
now the body of the car is suspended a little higher than 100mm over the tyre of the car, but it seems low already.
I want the car lower, but not ridiculously low, a space should still exist between the whe... | Checking several on line vendors specs the 40mm or 60mm refers to the amount of lowering from stock. If your stock spring leaves 100mm of tire clearance a 40mm spring would leave 60 mm of clearance. Several things must be taken into account prior to selecting your springs. Your car may already be lower than stock due t... | There's really not enough information to accurately answer this question, but I'll try.
If you're talking about 40 mm or 60 mm as the spring diameter, the spring diameter affects the spring rate. Assuming all other attributes of the spring are equal, such as length, number of coils, etc, then a 40 mm spring will have... | https://mechanics.stackexchange.com |
8,412 | [
"https://chemistry.stackexchange.com/questions/8412",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/4432/"
] | What is hyperconjugation and how do you know if there is hyperconjugation in a compound? Which orbitals are involved in hyperconjugation?
| In the ethyl cation, one can write the following resonance structures:
<img src="https://i.stack.imgur.com/KRgPz.jpg" alt="Resonance by hyperconjugation" />
The structure on the right can be actually written 3 times since there are 3 hydrogens available. Note also that the "ethylenic-like" structure on the ri... | Hyperconjugation is the interaction between adjacent orbital (empty <span class="math-container">$\mathrm{p}$</span> for cation, <span class="math-container">$\pi$</span> for alkene) with <span class="math-container">$\sigma$</span> bond, while common conjugation is between adjacent orbital and <span class="math-contai... | https://chemistry.stackexchange.com |
53,346 | [
"https://mathoverflow.net/questions/53346",
"https://mathoverflow.net",
"https://mathoverflow.net/users/12518/"
] | Let $x,y$ be vectors in $\mathbb{R}^n$ and let's use the notation $\hat x$ for the vector $x$ with its components sorted in increasing order.
The Hardy-Littlewood-Polya inequality states that
$$ x\cdot y \leq \hat x\cdot \hat y.$$
Let us also use the notation $xy\in\mathbb{R}^n$ to denote the coordinate-wise product of... | Here is a Matlab script that will generate a quick counterexample for you:
<pre><code>function [x,y]=testIneq(n, p, q, r)
% x and y are length n vectors
% Try: [x,y]=testIneq(2,1,2,3) to get a counterexample!
flag = 1;
iter = 0;
while (flag)
iter = iter + 1;
x = randn(n,1);
y = randn(n,1);
xh = so... | Rather for a counterexample. Let's say all coordinates are positive.
The inequality is equivalent to the claim that $f(t)=\frac{\|xy\|_t}{\|\hat x\hat y\|_t}$ satisfies $f(s)f(t)\le f(1)$ for $s\le 1\le t$ ($1\le p$ is not really a restriction due to the possibility to raise to positive powers inside and outside, so o... | https://mathoverflow.net |
38,760 | [
"https://datascience.stackexchange.com/questions/38760",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/59608/"
] | I'm using keras. I have one classification problem. The output should be either 0 or 1. I trained my model and I'm getting 86.59 accuracy. But when i check the predicted output what I'm seeing is all ones. I tried creating a categorical classifier with two nodes and tried the same. The test accuracy is 86.59% but when ... | As with most problems like this, it is always best to see the dataset upfront to gain a full understanding.
That said, if your categorical dependent variable is between 0 and 1, have you ensured that the independent variables in your dataset have also been scaled in this way? From looking at your code, it doesn't look... | Did you check how many samples are there for each class. I suspect imbalance class problem here. If you have majority of images with class=1. Classifier will be biased towards predicting everything as class 1. Try to balance your classes by either duplicating minor class images or deleting major class images.
| https://datascience.stackexchange.com |
188,066 | [
"https://security.stackexchange.com/questions/188066",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/180566/"
] | I know ISP see the hostnames of devices connected to them
Does ISP see the hostname of VirtualBox Machines ?
Also do they see MAC address of Virtual Machines ?
| So here's the TL;DR:
<h1>It's easier for you to maintain a watchful eye on the laptop than it is an empty desk other people can access in a large area to either infect it, or walk away with it</h1>
It's basically the same reason you don't leave your laptop in a public place:
<blockquote>
You don't know who has acc... | Physical access to any computer is typically a large security risk. Unless your system is locked down with defenses such as FDE, secure boot enablement, an admin password set on the BIOS, etc it is typically a quick process to gain a system-level shell on a computer running either Mac or Windows when physical access is... | https://security.stackexchange.com |
285,538 | [
"https://physics.stackexchange.com/questions/285538",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/132687/"
] | I am a third-year high schooler and I remember Michio Kaku once said (in some documentary which I can't sadly find now) That only pushing forces exist. Pulling a door is actually your fingers pushing it from the other side. Gravity is the curvature of space-time so it is pushing it towards earth. The charges probably p... | Not all forces push, some can pull, like gravity or the electrostatic force between oppositely charged particles. What Michio Kaku probably meant is that only <em>contact</em> forces exist: in other words, objects can only affect each other if the are exactly at the same point - they can't exert a force on something f... | In electromagnetism, the force between two electric charges $q_1, q_2$ is
$$
\vec F = \frac1{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat r.
$$
There are two types of charge which add, in equal quantities, to zero, so we call them "positive" and "negative" charges. And what you can see is that same-sign charges feel a re... | https://physics.stackexchange.com |
223,148 | [
"https://softwareengineering.stackexchange.com/questions/223148",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/113677/"
] | I'm a self-taught programmer. Up until now I haven't had too much problems building things I have wanted to. Historically, given an issue, especially one I have caused I haven't had much of a problem figuring it out. This all changed about two months ago, when I was offered my first position at a company whom I became ... | I have had good success to learn new code bases with the following approach. Strive to understand:
<ol>
<li>Purpose: understand what the system or framework does, conceptually speaking</li>
<li>Architecture: create a view on the overall architecture of the system or framework</li>
<li>Components: learn how major parts... | You said you're a self-taught programmer and by this question I assume you don't have prior experience with such big projects.
Personally, I have less experience than you so do not take my words as gift of god.
Honestly, I think it's perfectly normal to be overwhelmed from such big projects when you don't have prior e... | https://softwareengineering.stackexchange.com |
284,495 | [
"https://physics.stackexchange.com/questions/284495",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/129383/"
] | I came across this problem which mentioned that a coin which would otherwise sink in water, would float if it was flattened by a hammer.
What i dont understand is how that is possible, since the volume would not change and that implies that the density of the coin doesn't change either.
If we use archimed principle or... | If you take a pin and place it into water very carefully with the pin pointed down, no matter how careful you are, the pin will sink. However, you can easily do the experiment at home where you carefully lay a pin on its side on the surface of the water and presto, it floats. The volume doesn't change, but the surface ... | If a coin is flattened it doesn't lose any volume, which is why it is counter intuitive. If however you would flatten a coin, and then place it on still water (carefully on top), then the surface area is larger, and the surface tension of the water could (if the coin is sufficiently thin) hold the coin up.
The coin wo... | https://physics.stackexchange.com |
117,213 | [
"https://mathoverflow.net/questions/117213",
"https://mathoverflow.net",
"https://mathoverflow.net/users/30196/"
] | Can somebody tell me of other applications of Floer homology besides the proof of the Arnold conjecture.
Every answer would be appreciated.
| Floer homology has, in one form or another, become ubiquitous in symplectic geometry and to give a complete list of its applications would be a mammoth task. Here are a few.
1) One early incarnation of Floer homology was in instanton gauge theory. Floer's instanton invariant is a homology group associated to a homolog... | Lagrangian intersection Floer homology can be used to define topological invariants for knots and 3-manifold. A very successful example is Heegaard Floer homology: although it is now an almost completely combinatorial theory, its first definition used Floer homology.
In short the construction works like this: by more ... | https://mathoverflow.net |
335,921 | [
"https://softwareengineering.stackexchange.com/questions/335921",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/237893/"
] | I'm reading the book Clean Code by Uncle Bob. I'm also enrolled in a data structures & performance course and reading several algorithms and data structures books.
One immediately apparent difference is that in Clean Code, Uncle Bob criticizes non-semantic variable names such as "k", "x", "j", etc for general use ... | <blockquote>
why the drastic difference in engineering vs. data structure code presentation?
</blockquote>
It is an unfortunate intersection between mathematics and computer science.
In mathematics, each variable is often denoted with a single symbol. There are several reasons for that; among those is to avoid the ... | Different people have different styles, that's why you get differences in style. People also have different learning styles.
One way of learning algorithms that works for some people is reproducing the algorithm yourself, bit by bit, in your own words. That way you can replace less meaningful variable names with mean... | https://softwareengineering.stackexchange.com |
475,670 | [
"https://electronics.stackexchange.com/questions/475670",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/136142/"
] | <strong>Updated/Corrected question:</strong>
Realizing what confuses me the most, I will make this more straight-forward. Say that a transistor works in the forward active region and it receives an ac input signal at its base.
<strong>Is there a chance that the change in the base voltage will decrease Vbe and force ... | <blockquote>
Is there a chance that the change in the base voltage will decrease Vbe and force the transistor into the cut-off region?
</blockquote>
Yes this is possible, and it would result in non-linear operation of your amplifier.
<blockquote>
Is there any chance this might happen before first entering the sa... | I guess this question is more about vocabulary than about BJTs.
"AC small signal analysis" or "small signal conditions" is a useful approximation to simplify the analysis of a circuit.
This approximation assumes the AC signal voltage wiggles around the DC operating point in a "small enough" interval that the operati... | https://electronics.stackexchange.com |
17,601 | [
"https://engineering.stackexchange.com/questions/17601",
"https://engineering.stackexchange.com",
"https://engineering.stackexchange.com/users/13340/"
] | I am Comp-sci student doing a software engineering paper about designing a program, planning the software like you were actually planning a real-life software project, short of writing actual pseudo-code or code. For my project, I picked doing an app that can detect a physical man in the middle attack. Like monitoring ... | many radio receivers contain a circuit called an IF oscillator. the output of this oscillator is added to the radio frequency signal that the radio is receiving to downconvert the incoming high frequency to a lower frequency that is easier to process. Old radios used IF oscillators that were powerful enough to actually... | Yes, kind of.
You could create a sensitive/specialized metal detector, which would induce/detect a current in any piece of metal (including an antenna)
There are antennas specifically designed to be detected (rfid technology); you broadcast a particular frequency, and they resonate and respond. However the ratio of ... | https://engineering.stackexchange.com |
20,940 | [
"https://softwareengineering.stackexchange.com/questions/20940",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/194/"
] | Rails, to me, seems like a perfect level of abstraction for most types of web development. However, having watched some seasoned Rails consultants build an application, I'm finding that they use Gems to an extent that it's often pretty difficult to figure out what's going on in the code. As an intermediate-level Rails ... | Staying current is very challenging in the Rails community because of the proliferation of frameworks/components and the speed with which they are adopted by the mainstream. It means it takes work to stay current enough to understand new code, and that old code gets old very fast...
I program Rails only as a hobby and... | Your question reminded me of a point Paul Graham makes in <em>On Lisp</em>:
<blockquote>If people complain that using utilities makes your code hard to read, they
probably don’t realize what the code would look like if you hadn’t used them.
Bottom-up programming makes what would otherwise be a large program look
like ... | https://softwareengineering.stackexchange.com |
46,384 | [
"https://chemistry.stackexchange.com/questions/46384",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/25666/"
] | As I know, chlorophyll is green as it absorbs all wavelengths but green. It absorbs red, yellow, blue, purple, etc. Thus, the plant does not utilize green light.
i] Why is chlorophyll green? Is there 'chlorophyll' of other colors?
ii] Is the green chlorophyll an 'evolutionary mistake'?
iii] If chlorophyll was black... | i] There is chlorophyll-b (yellow-green) and carotenoids (yellowish).
ii] The green chlorophyll isn't an evolutionary mistake (unless, of course, you take the viewpoint that evolution IS just mistakes). Plant's get plenty of sunlight. The sun is way stronger than indoor lights used to grow plants. They don't really ne... | To give you a quick start and hint to your questions,
(i) Chlorophyll is green, because it reflects off the green light. Yes, there are chlorophyll of other colours. If you look at the absorption and action spectra of photosynthesis, you can see that there are chlorophyll a, b, carotenoid that absorbs various wavelen... | https://chemistry.stackexchange.com |
223,318 | [
"https://electronics.stackexchange.com/questions/223318",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/84339/"
] | I am using <strong>4 TEC1-12706 thermoelectric cooler</strong> on a suitcase, by cutting 4 holes to fit these through and sealed them. All of them are connected to <strong>12V DC supply</strong> from a SMPS and each of them have attached <strong>heatsinks</strong> with 12V DC <strong>fans</strong> on their <strong>hot ... | My guess is your peltier are not enough to cool the whole suitcase (or not being applied current enough). Here is my reasoning:
1 - No heatsink on cold side + water: I guess this means when nothing is transfering heat from the suitcase to the peltier, the peltier gets cold enough to condense water.
2 - Heatsink on co... | Peltier junctions produce a temperature difference between one side and the other. This temperature difference is inversely proportional to how much heat (in watts) it is pumping out of the cold side into the hot side. You've already seen that this happens very readily. If you have no fans or heatsink on the cold si... | https://electronics.stackexchange.com |
183,131 | [
"https://stats.stackexchange.com/questions/183131",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/95988/"
] | Is it compulsory to have a single neuron in the output layer?
Or is it ok to have multiple neurons in output layer?
| One neuron in the output layer means regression, multiple neurons means classification. Simple as that. An artificial neuron has inherently a continuous value output, therefore regression can be regarded as the natural function of the network where a single output neuron is required. Classification is achieved by means... | Usually you have multiple output neurons in NN if you are solving multiclass classification problem. Otherwise it doesn't make sense, but anyway nothing stops you from having multiple outputs.
| https://stats.stackexchange.com |
15,071 | [
"https://electronics.stackexchange.com/questions/15071",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/4436/"
] | From the perspective of data capture, how is this achieved? If I wanted to implement a home-made digital device to capture high frequency analogue signals, what are my options? So far, I've only come up with some fairly useless ideas for designs!
Using a PIC microprocessor, the A/D sample rate on a 18f series I believ... | The entry level DSO Rigol 1052E (the one I own and 100 MHz capable with software change) uses an Analog Devices AD9288. This is a dual channel ADC with 8 bit parallel outputs and samples at either 40 or 100 million samples per second (depending on speed grad of chip). Although the Rigol is a 1 Gig samples per second,... | I presume they use <strong>Flash ADCs</strong>. These have the advantage that the conversion is immediate, while SA (Successive Approximation) ADCs like used in most microcontrollers perform an algorithm that requires a number of steps. A disadvantage of Flash ADCs is that they are rather heavy on hardware (an 8-bit AD... | https://electronics.stackexchange.com |
110,102 | [
"https://cs.stackexchange.com/questions/110102",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/106030/"
] | For demonstration purposes I was wondering about some very easy to grasp LC abstractions and I came to the idea of a function that simply eats its argument, and nothing more.
If you apply λx. (Yes no lambda term after the point) to an argument the abstraction reduces to nothing.
Although this isn't very useful for c... | No, <span class="math-container">$\lambda x.$</span> is not a syntactically valid expression. By the definition of the syntax of lamda terms,
<blockquote>
<span class="math-container">$\lambda x.M$</span> is a term if <span class="math-container">$x$</span> is a variable and <span class="math-container">$M$</span> i... | I am sorry but writing just λx. Is not usefull in any way. And it is not a function that eats his argument, it is a undefined function like writing <span class="math-container">$f(x)=$</span>
| https://cs.stackexchange.com |
67,343 | [
"https://mathoverflow.net/questions/67343",
"https://mathoverflow.net",
"https://mathoverflow.net/users/15488/"
] | Let $A$ be a commutative algebra (over the complex numbers, with a unit) and let $M$ be a finitely generated projective $A$-module, and let $m_1,\ldots,m_n$ be a set of generators of $M$. The Dual Basis Theorem states that there exists $m_1^\ast,\ldots,m_n^\ast\in M^\ast$ such that $x=\sum m_i^\ast(x)m_i$ for all $x\in... | More abstractly: Let $A$ be a commutative unital ring and let $M$ be a f.g. projective $A$-module. The rank $rk_P(M)$ of $M$ at a prime ideal may be defined as the vector space dimension of $K\otimes_A M$ where $K$ is the residue field. Because projective modules of local rings are free, this is a locally constant fun... | For any homomorphism $\varphi : A \to \mathbb C$, $\varphi({\rm tr}({\bf 1}))$ will be the corresponding trace for the finite-dimensional vector space $M \otimes_A \mathbb C$, and hence be equal to the rank of $M$ at $\varphi$.
So, if $A$ is some algebra of continuous functions on a connected space, then the trace ${... | https://mathoverflow.net |
509,467 | [
"https://physics.stackexchange.com/questions/509467",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/226137/"
] | Without invoking Hermitian conjugation (which, to my mind at least, would beg the question), what is it that makes the bra <span class="math-container">$\langle\psi\rvert$</span> the unique dual of the ket <span class="math-container">$\rvert\psi\rangle$</span>?
For instance I know that provided <span class="math-con... | Two complex topological vector spaces <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are said to be in duality if there is a sesquilinear map
<span class="math-container">$$b:X\times Y\to \mathbb{C}\; .$$</span>
The idea is that, given such map, a dual action of <span class="math-c... | Of course if you just define it as you said then you can have many such functionals which map this ket to real numbers. That is why we should define such functionals which when acting on a basis give ones and zeroes. That would be the basis for the space of functionals. Such space is called dual and the basis is dual b... | https://physics.stackexchange.com |
33,220 | [
"https://mechanics.stackexchange.com/questions/33220",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/19994/"
] | i am having hero passion pro new model in the morning when i start my bike with kick i have on the choke the bike started without no hesitation but when i pull of the choke after 15 seconds
and restarted it ,its not restarting with reserve or in an on state since the fuel is full its about 8 litres petrol guys please ... | Turn the choke ON to start a cold engine. Once it is running, set it to PARTIAL until the engine is warm. Once the engine has been running a few minutes, set the choke to OFF.
Choke helps the engine run rich. This is necessary when the engine is cold. The engine will start easiest with the choke on. The engine will ru... | I believe you have a carburetor engine, and you need to adjust your idle, make a richer air/fuel mix. Your Idle mix is too lean, so you can't start it. When you get it richer, you might need to adjust a throttle.
<hr>
If you have only 1 screw on a carburetor, it'll be a throttle adjustment for idle. Try to start it w... | https://mechanics.stackexchange.com |
3,814,167 | [
"https://math.stackexchange.com/questions/3814167",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/676335/"
] | I started to solve this problem as one which could be decomposed using partial fractions, and I think that you can do it, but the problem is that you end up with an imaginary number in the formulation and my solution is not matching up to the known solution.
Here is my attempt via partial fractions:
<span class="math-c... | Your error is that no constants <span class="math-container">$A,\,B$</span> exist satisfying your putative identity. Since fractional powers are involved, partial fractions won't work, except perhaps with a clever substitution. If you want to do this without trigonometry, I recommend <span class="math-container">$u=x^a... | Following is the most intuitive solution of this problem I can think of.
<span class="math-container">\begin{align}
\int\dfrac{1}{(x-1)^\frac32x^\frac12}dx&=\int\frac{1}{x^2\left(\dfrac{x-1}{x}\right)^\frac32}dx\\
&=\int\frac{du}{u^\frac32}&&\left(\text{Substituting }u=1-\frac1x\right)\\
&=-\frac{2}... | https://math.stackexchange.com |
724 | [
"https://mathoverflow.net/questions/724",
"https://mathoverflow.net",
"https://mathoverflow.net/users/382/"
] | Pretty much exactly what it says on the tin. Let G be a connected graph; then the Tutte polynomial T_G(x,y) carries a lot of information about G. However, it obviously doesn't encode everything about the graph, since there are examples of non-isomorphic graphs with the same Tutte polynomial.
My question is, what infor... | No-one so far has mentioned matroids. The Tutte polynomial encodes some of the information from the cycle matroid of the graph. Two graphs with the same cycle matroid (and number of vertices) have the same Tutte polynomials. So if a graph property is not determined by the cycle matroid (and the number of vertices) then... | The real thing to focus on is the rank polynomial R(x,y) = T(x+1,y+1).
This is a generating function that counts the number of subsets of edges according to their size and their rank (where "rank" = "matroid rank" which in the graph case is equivalent to #edges - #components).
(Equivalently, you can view it as count... | https://mathoverflow.net |
230,532 | [
"https://physics.stackexchange.com/questions/230532",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/87129/"
] | I was reading about energy, and I got to know that energy is the ability to do work. Then I read about work, and I found that almost everywhere, resources say that when an object applies force to move another object through a distance, it is said to do work, and the Math is that $W = Fs$(Even the tag on this site which... | To back up CuriousOne. A good example would be you sumo wrestling with your friend. Lets say your friend is far bigger/stronger than you. So you begin pushing one another. His force on you is greater and so you are obviously moving in what you would call your -x direction. Both of you move in the -x direction, but of c... | In another way, we can define work as "the change of energy".<br>
When you do work on a system, you change its energy. So positive means you're increasing the system's energy (like pushing a trolley forward), and negative means you're decreasing it.
In this context, it might be easier to see why some people are incli... | https://physics.stackexchange.com |
94,620 | [
"https://mathoverflow.net/questions/94620",
"https://mathoverflow.net",
"https://mathoverflow.net/users/-1/"
] | Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G_m$ and a non-singular scheme $X$ of finite type over $k.$
Let us define the following:
<strong>Condition 1:</strong> $X$ can be covered by $G_m$-invariant quasi-affine open subschemes.
In the paper "Some theorems on actions of algebraic gro... | It is a theorem of Sumihiro (Equivariant completion, Corollary 2) that a normal variety over an algebraically closed field with an action of a torus is covered by invariant affine open subsets.
Here, the normality hypothesis is necessary : the conclusion does not hold for the action of $\mathbb{G}_m$ on $\mathbb{P}^1$... | Every normal variety with an action of a torus is covered by invariant affine open subsets. This is proved in Hideyasu Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.
| https://mathoverflow.net |
211,073 | [
"https://softwareengineering.stackexchange.com/questions/211073",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/101909/"
] | In C and C++, the main method holds the filename in the first position of the array at argv[0]. In Java, however, the filename is not included in the args string array.
Is there a practical reason for this? I understand that this makes iterating through command line arguments 0-based instead of 1-based, but is there a... | In some cases a program can be run in different ways and exhibit different behavior on how it is called. If you call <code>vim</code> as <code>vi</code>, it runs in a compatibility mode. Sometimes it is to try to maintain one version of several related programs - for example <code>mailq</code> and <code>newaliases</c... | actually there's no benefit with it, it really depends on the syntax of the programming language you are using if it is 0-based or 1-based. the variable (you refer to as filename) also depends on the language, it can be different in other languages just follow the correct syntax of the language you are using.
| https://softwareengineering.stackexchange.com |
188,316 | [
"https://electronics.stackexchange.com/questions/188316",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/26394/"
] | I'm trying to analyze a boost converter circuit. I have all the specs for the inductor used, including inductance (6.8uH) and saturation current (1A). This runs from a 3V supply. I'm wondering if there is a formula for the maximum power than an inductor with a certain inductance and saturation current can transfer w... | With a fixed supply voltage and a fixed inductance and a known saturation current, you not only have the maximum energy that the coil can store:
$$E = 0.5 I_{sat}^2 L$$
but also how long it takes to charge the coil with that amount of energy:
$$t_{charge} = \frac{I_{sat} L}{V_{supply}}$$
The power transferred is eq... | Each switching cycle, energy is stored in the choke, then discharged into the load. Your limit is the total energy that can be stored in the choke, multiplied by the frequency the converter runs at. In your case, the choke can store \$\frac{1}{2}LI^2 = \frac{1}{2}(6.8x10^{-6})(1)^2=3.4\mu J\$. Multiply that by your swi... | https://electronics.stackexchange.com |
330,244 | [
"https://electronics.stackexchange.com/questions/330244",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/42950/"
] | I need to have a 5v supply with around an amp of current draw max, and a 6v supply with around a 4amps current draw max. I was thinking: instead of having two switching regulator circuits, just having one 5amp 9v switching regulator circuit feeding into an Lm7805 and an lm338t (linear regulators) to generate my 5v, and... | No, this sounds reasonable, if it is what you want to do. One problem is that you will drop 9-6=3 volt at 4 amperes, that's 12 watts. You need a hefty heat sink for that.
A better option is perhaps to have a 6V output from a switching regulator, and a 5V low-dropout linear regulator, converting 6V to 5V. The 5 volt re... | Using a switching regulator upfront and a linear regulator for final regulation can make sense if you want to reduce power wastage while reducing the amount of switching noise that reaches your final power rails.
I question why you would go as high as 9V though, nowadays you can get voltage regulators with dropout vol... | https://electronics.stackexchange.com |
1,141,964 | [
"https://math.stackexchange.com/questions/1141964",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/202435/"
] | I have here a linear transformation <span class="math-container">$T : P_3(\mathbb{R})\rightarrow P_3(\mathbb{R}) $</span> defined by:
<blockquote>
<span class="math-container">$ T(at^3 + bt^2 + ct + d) = (a-b)t^3 + (c-d)t $</span>
</blockquote>
I'm very very new in this subject and I'm not going well with polynomials. ... | The kernel is correct. Additionally, since the kernel depends on only two coefficients $a$ and $c$, it has dimension 2.
<br>
<strong>For the image:</strong>
Take any polynomial $p(t)=At^3+Bt^2+Ct+E$.
The question now is: How do $A,B,C,E$ have to look for there to exist some $a,b,c,d$ such that $T(at^3+bt^2+ct+d)=... | We can set up the matrix of the linear transformation $T:P_3(\mathbb{R})\rightarrow P_3(\mathbb{R})$, then find its null space and column space, respectively. First, if we agree to represent the third-order polynomial $P_3=at^3 + bt^2 + ct + d$ by the column vector $\begin{pmatrix}a &b& c& d\end{pmatrix}^T... | https://math.stackexchange.com |
2,276,114 | [
"https://math.stackexchange.com/questions/2276114",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/444440/"
] | If $$lim_{n\rightarrow \infty} \frac{1^a + 2^a +3^a +\cdots +n^a}{(n+1)^{a-1}\left( (na+1)+(na+2) + \cdots +(na+n)\right)}= \frac{1}{60}$$
Find the value of $a$.
Obviously, the denominator turns out to be $$(n+1)^{a-1}(n^2a+\frac{n(n+1)}{2})$$
But I am not able to simplify the numerator.Can the numerator be simplifie... | There is no closed formula for $1^a + 2^a + \dotsc + n^a$ with general $a$, but the asymptotic behaviour of that sum is easy enough, and the asymptotic behaviour is all that matters here. Show first that the sequence is unbounded for $a \leqslant -1$. For $a > -1$, write the term as
$$\frac{1^a + 2^a + \dotsc + n^a... | <em>This is not an answer but it is too long for a comment.</em>
Interested by the post and impressed by the solution proposed by Daniel Fischer, I have been wondering if, for finite values of $n$ and for any value of $a$ (integer, rational or not), we could obtain an approximation of<br>
$$S_n=\frac{\sum_{i=1}^n i^a... | https://math.stackexchange.com |
583,515 | [
"https://stats.stackexchange.com/questions/583515",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/363498/"
] | It requires the Gaussian process model output ranging between 0 and 1, is there any technique to constrain the Gaussian process model's output?
| You are not calculating the (conditional) variance of the square of a random variable, but the (conditional) <em>expectation</em> of the square of a random variable. The left hand side of <span class="math-container">$$\mathbb{V}(Y|X = x) = \mathbb{E}(Y^2|X = x)$$</span> is the conditional variance of <span class="mat... | Given that:
<span class="math-container">$$\Bbb E[g(Y)|X] = \int_{-\infty}^{\infty}{g(y)f_{Y|X}(x,y)\ dy} $$</span>
we need to find the <strong>conditional pdf</strong>, which is:
<span class="math-container">$$f_{Y|X}(x,y) = {f_{X,Y}(x,y)\over f_X(x)}$$</span>
Now for your specific case, <span class="math-container">$... | https://stats.stackexchange.com |
2,608,049 | [
"https://math.stackexchange.com/questions/2608049",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/484872/"
] | Let $\psi_{1} := (P \wedge Q \wedge R)\vee(\neg P\wedge Q\wedge\neg R)\vee(\neg P\wedge\neg Q\wedge R)\vee(\neg P \wedge\neg Q\wedge\neg R)$ and
Let $\psi_{2} := (\neg P \wedge \neg Q) \vee (\neg P \wedge \neg R) \vee (P \wedge Q \wedge R)$
I have proven so far, that the two formulas in DNF are equivalent to $P\Left... | HINT
Use:
<strong>Adjacency</strong>
$P \Leftrightarrow (P \lor Q) \land (P \lor \neg Q)$
$P \Leftrightarrow (P \land Q) \lor (P \land \neg Q)$
So, for example, you can combine the terms $\neg P \land \neg Q \land R$ and $\neg P \land \neg Q \land \neg R$ to $\neg P \land \neg Q$
In fact, that gets you almost the... | Hint: If you're familiar with distribution over two terms, break up $(X \vee Y \vee Z)$ into $((X \vee Y )\vee Z)$ and distribute twice.
| https://math.stackexchange.com |
296,793 | [
"https://softwareengineering.stackexchange.com/questions/296793",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/101669/"
] | I am designing a REST api backend that is meant to be consumed by a javascript front-end.
I am not sure how to communicate server-side business logic errors (e.g. a user trying to retrieve his password with an unknown email, etc.) to the front-end.
After setting a 40x status code on the response, I need to add furthe... | I would say that option 2 makes it easier to understand what happened. The client can know to look in the body for a message, which is the usual place information is found. In your option 1, not only would the client need to know to look at the headers (which is more work in many rest client libraries), it would also ... | The best way to do this, in my experience, is to describe a universal error structure for your API. Something like this might suffice:
<pre><code>{
"errorCategory": "Authentication/Authorization",
"errorMessage": "Unknown email provided as part of authentication",
"errorDetailURL": "http://api.company.com/docume... | https://softwareengineering.stackexchange.com |
202,956 | [
"https://mathoverflow.net/questions/202956",
"https://mathoverflow.net",
"https://mathoverflow.net/users/66688/"
] | Let $S^{d}$ denote the standard $d$-dimensional sphere. I heard from a physicist that from physical arguments they have been able to show that the vector bundle:
$E_{d} = TS^{d}\oplus \Lambda ^{d-2}T^{\ast}S^{d}$
is topologically trivial, meaning that it is parallelizable. The convention is that $\Lambda ^{0}T^{\ast... | It is true in general that $E_d$ is trivial. As remarked by Neil Strickland above, this boils down to showing that $TS^d\oplus\Lambda^2 TS^d$ is always a trivial bundle. To see this, represent $S^d$ as $SO(d+1)/SO(d)$. Then all bundles in question are homogeneous vector bundles (and the isomorphisms used in the comment... | Actually, this is simpler than I thought. The map
$$ (u,v) \mapsto x\wedge u + v $$
gives an isomorphism
$$ T_xS^n \oplus \Lambda^2 T_xS^n \to \Lambda^2(\mathbb{R}^{n+1}), $$
and this trivialises the bundle $T\oplus\Lambda^2T$.
| https://mathoverflow.net |
380,277 | [
"https://softwareengineering.stackexchange.com/questions/380277",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/304516/"
] | I have a couple of functions, each function verifies a set of rules and updates a common object. The common object is just a container that holds a list of rules that passed or failed. I would like to implement this as async. Is there any downside to this approach?
Are there any recommendations? The sample code I inte... | Having each async method update a common object risks running into threading/non-deterministic order issues with the results. So rather than have each async method update a common object, have each return a result, then update the results when all have completed:
<pre><code> public Results ValidateRule()
{
var c... | A big concern with asynchronous functions is state management. If your function requires data that might be changed by an asynchronous function, then its possible for your data to change or become stale before or mid execution, causing unexpected results that can be difficult to debug.
If you can set up your functions... | https://softwareengineering.stackexchange.com |
144,234 | [
"https://stats.stackexchange.com/questions/144234",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/72400/"
] | We're doing a simple exercise and we have the following situation:
if ( random(1-100) == 50 OR random( 1-100 ) == 50 ) THEN
[..]
The random function may return any number between 1-100 1 and 100 included with a theoretic complete randomization.
I'd like to know what is the % of that IF statement to return TRUE.
an... | Let X = random(1-100) and Y = random( 1-100 ). That is uniformly distributed between the integers $1, 2, 3 \dots 100$
Then $P(X=50 \cup Y=50) = P(X=50)+P(Y=50)-P(X=50 \cap Y=50)$.
If the variables are independent then the probability should be: $0.01+0.01-0.01^2 = 0.0199 $. So, unless I misunderstand your statement,... | You can model this as a binomial probability.
$X \sim Binom(n=2, p = 1/100)$
$P(X\geq 1) = 0.0199$
| https://stats.stackexchange.com |
699,924 | [
"https://physics.stackexchange.com/questions/699924",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/301181/"
] | While solving a numerical, I learnt this data:
<blockquote>
take resistance of bulb filament as <span class="math-container">$1200 \Omega$</span> and resistance of electric heater coil as <span class="math-container">$100 \Omega$</span>.
</blockquote>
My question is if both bulb and heater are designed to produce heat ... | You're almost there. You recognise that for <span class="math-container">$H$</span> to be proportional to the resistance, <span class="math-container">$I$</span> must be held constant, and that for <span class="math-container">$H$</span> to be proportional to <span class="math-container">$\frac 1R$</span>, <span class=... | <blockquote>
if both bulb and heater are designed to produce heat
</blockquote>
The equation for electrical power on a resistor has many different forms
<span class="math-container">$$P = U I = I^2 R = U^2 / R$$</span>
All these equations are correct. In the context of the light bulb, the only difference is that you ha... | https://physics.stackexchange.com |
330,303 | [
"https://electronics.stackexchange.com/questions/330303",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/107412/"
] | Background:
I am using a shielded inductor in a switching power supply, and am investigating an EMI issue.
My hypothesis is that since the inductor is saturated, the magnetic field is no longer contained in the ferrite core, and the inductor becomes equivalent to an air core inductor. I know that air core inductors r... | Once the core is saturated, it doesn't look like a magnetic core anymore when you try to increate the magnetic field. The inductor becomes essentially air-core for any additional current.
While over-saturating a shielded inductor will cause the magnetic field to leak outside the shield more, this shouldn't cause much... | As inductor saturates, the rate of change of magnetic flux also decreases significantly, which in turns produces less EM radiation [you need changing flux for EM propagation]. Conversely, as the indcutor saturates, the current through it jumps high during saturation period in each cycle. That jump in current might prod... | https://electronics.stackexchange.com |
20,963 | [
"https://mathoverflow.net/questions/20963",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4706/"
] | It is known that Namba forcing is stationary-preserving and hence can be used in the setting of Martin's Maximum. Does this result in any striking consequences?
| I think that I may have found a suitable candidate; namely, the result of
Konig and Yoshinobu that $MM$ implies that there are no $\omega_{1}$-regressive $\omega_{2}$-Kurepa trees. The proof seems to have the same relatively direct flavor as those in Baumgartner's $PFA$ article.
| Magidor's proof that MM implies that there are no <em>good</em> scales of length lambda^+ for lambda>cf(lambda)=w, utilizes a Namba-style forcing.
| https://mathoverflow.net |
23,311 | [
"https://scicomp.stackexchange.com/questions/23311",
"https://scicomp.stackexchange.com",
"https://scicomp.stackexchange.com/users/18230/"
] | I am starting to dabble in MPI. I am fairly new to this area and I am currently reading the MPI standard.
I would like to write my first MPI program, a simple hello world program, on a cluster that I have. I am not sure if anyone is familiar with the IBM bladecenter. Supposedly, there is a midplane that connects all ... | The standard is purposefully silent on these issues preferring to leave such issue to library implementers to figure out. There's no need to standardize such mechanisms. Most network hardware has a unique (enough) identifier (MAC address or other GUID), and most operating systems associate a hostname with each node. Th... | <ol>
<li>Everything Bill Barth wrote is correct.</li>
<li>You are confused by MPI processes (or "ranks") and computers. If you do "mpirun -np 18 yourprogram" on a 4-core machine that is not connected to anything, then MPI_Comm_size will still report that there are 18 MPI processes. Not 1, or 4. To an extent you could s... | https://scicomp.stackexchange.com |
230,497 | [
"https://security.stackexchange.com/questions/230497",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/233120/"
] | Is it worth paying for a VPN to protect myself from Google tracking since, on an Android phone, you're logged into a Google account and Google knows who you are, anyway?
Google can still track your activities depending on what Android OS reports to them. A related question would be whether encrypted messaging and emai... | Specifically regarding VPNs, they are often marketed as a panacea to solve all security and privacy problems. While they can solve some problems, unfortunately much of it is just false hype and exaggerated for advertisement purposes.
VPNs protect network transit point-to-point from your device to the VPN server. They ... | Well, if you think of Google as a malicious adversary that is willing to go to any length to track you, then certainly the situation does seem rather hopeless. But just because Android <em>can</em> send decrypted messaging data to Google without your consent doesn't mean it will. While Android's default privacy setting... | https://security.stackexchange.com |
302,901 | [
"https://softwareengineering.stackexchange.com/questions/302901",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/192431/"
] | I need to represent an abstraction over various parts of the hardware for a game. I'm trying to decouple the code that does things like manage the logic of the game from the code that is API/platform specific or any ugly implementation details.
Like this:
<pre><code>std::unique_ptr<IDevice> device(CreateDevice(... | This is called the <em>Singleton Pattern</em> and is considered to be an anti-pattern by many people nowadays, since it hides dependencies and makes your code less maintainable.
The necessity for using the singleton pattern may arise from other bad design decisions, but I cannot judge since I know nothing about your a... | I would solve this by passing around the IDevice instance to the instances that need it, and only instantiate the class in the main function. This also makes the dependencies more explicit.
| https://softwareengineering.stackexchange.com |
594,311 | [
"https://physics.stackexchange.com/questions/594311",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/275819/"
] | I encountered the following integral in dimensional regularization
<span class="math-container">$$
I=\int d^d k \,e^{i\vec{k}\cdot \vec{x}}\frac{1}{\vec{k}^2}\frac{1}{(\vec{q}-\vec{k})^2},
$$</span>
say that we already Wick rotated the integral.
This looks like something which would be possible to evaluate but I'm not ... | The definition of particles in QFT is a bit technical than our usual notion of particles. A particle is an excitation of a field. For example, the Higgs boson is an excitation of the Higgs field. With this notion, we can say electrons are particles. However, the wave notion is also built-in in the excitation part of th... | I think most in quantum physics would say just the opposite, that there are no "grains of sand". Rather, Caltech theoretical physicist Sean Carroll put it this way: “To understand what is going on, you actually need to give up a little bit on the notion of particles..... The universe is full of fields, and ... | https://physics.stackexchange.com |
9,439 | [
"https://physics.stackexchange.com/questions/9439",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/3385/"
] | In a test I took recently in my beginning Physics class there was a question that really irked me:
<blockquote>
<em>A man is swimming upstream at 10 mph. The water is flowing downstream at 5 mph.</em>
<em>How fast is the man moving?</em>
</blockquote>
Well, you can actually take this question two ways:
<ol>
<li>Firstly... | David answer is ok.
A man is swimming upstream at 10 mph -- is undefined : <strong>in relation to</strong> the ground or in relation to the water?, <em>probably</em> in relation to the ground
water is flowing downstream at 5 mph. -- <em>probably</em> in relation to the ground
How fast is the man moving? -- ... | Nope. The question is ambiguous.
In any case, if there is a way to resolve this ambiguity, it's a question of language interpretation. The physics only really starts once you've figured out what the question means. (To be fair, it's often the case that most of the difficulty in doing a physics problem is just translat... | https://physics.stackexchange.com |
11,636 | [
"https://cstheory.stackexchange.com/questions/11636",
"https://cstheory.stackexchange.com",
"https://cstheory.stackexchange.com/users/608/"
] | In the preface of Friedman and Felleisen's book The Little Schemer it states:
<blockquote>
We could, for example, describe the entire technical content of this book in less than a page of mathematics, but a reader who understands that page has little need for that book.
</blockquote>
Has anyone got an online link ... | What Sam said.
Also, it's really <em>well</em> under a page. If you're familiar with evaluation contexts, you can specify the call-by-value lambda calculus like this:
Terms
$$M ::= x \mid (M \, M) \mid (\lambda x . M)$$
Values
$$V = (\lambda x . M)$$
Evaluation contexts
$$E ::= [\:] \mid ([\:] M) | (V [\:])$$
... | All you really need is the definition of the untyped $\lambda$-calculus, which you can find in numerous places. Everything else follows from that.
| https://cstheory.stackexchange.com |
70,762 | [
"https://quant.stackexchange.com/questions/70762",
"https://quant.stackexchange.com",
"https://quant.stackexchange.com/users/25156/"
] | Given historical asset prices at consistent time intervals, one can estimate annual volatility as:
SampleStDev(log(S<sub>i</sub>/S<sub>i-1</sub>)) / sqrt(interval)
What's the correct way to do this when the time intervals are inconsistent (e.g. S<sub>2</sub> was observed 1 day after S<sub>1</sub>, but S<sub>3</sub> was... | To make @nbbo2's answer more precise, let's assume that we observe various sums <span class="math-container">$z_{k,h}$</span> of independent and identically distributed random variables <span class="math-container">$x_i$</span> (i.e. returns),
<span class="math-container">$$
z_{k}\equiv \sum_{i=k-h_k+1}^k x_i
$$</span>... | Assume GBM. Suppose there are <span class="math-container">$n+1$</span> observations, indexed by <span class="math-container">$j=0,\cdots,n$</span>. The stock price is <span class="math-container">$S_j$</span> at time <span class="math-container">$T_j$</span> where time is measured in yearly units.
The annualized varia... | https://quant.stackexchange.com |
13,451 | [
"https://physics.stackexchange.com/questions/13451",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/4906/"
] | In Special Relativity, the spacetime interval between two events is $s^2 = -(c{\Delta}t)^2+({\Delta}x)^2+({\Delta}y)^2+({\Delta}z)^2$ giving the Minkowski metric $\eta_{\mu\nu}=\text{diag}(-1, 1, 1, 1)$. What is the justification for making time have a negative coefficient, and how closely is that related to the 2nd la... | I think this is a case of the mathematics being designed to model reality. As you say, making the time component of the metric positive would give a space that doesn't match what we observe. In particular, the negative component for time allows us to disconnect regions of space that aren't causally linked. In other wor... | There is a direct link between the minus sign in the metric and thermodynamics. Because the sign is negative, positive energies cannot be rotated to negative energies, and it makes sense to say that the energy of a physical system is always positive. This gives rise to thermodynamic partitioning.
Unlike energy, spatia... | https://physics.stackexchange.com |
3,992,770 | [
"https://math.stackexchange.com/questions/3992770",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/830301/"
] | I had difficulties figuring out the solutions to the following problem:
<blockquote>
Let <span class="math-container">$H$</span> be a separable infinite dimensional Hilbert space. Let
{<span class="math-container">${L_n}$</span>} be a sequence of linear finite rank operators from <span class="math-container">$H$</span>... | (i) <span class="math-container">$L_n$</span> are finite rank, so by definition are bounded. Since <span class="math-container">$L_nx$</span> converges, let <span class="math-container">$L(x):=\lim_{n\to\infty}L_nx$</span>. That <span class="math-container">$L$</span> is linear is easy to verify.
Also, <span class="mat... | Yes, Uniform Boundedness Principle give the first part.
Take <span class="math-container">$L_n$</span> independent of <span class="math-container">$n$</span> to get an example where the limit is compact.
Let <span class="math-container">$(e_n)$</span> be an orthonormal basis for <span class="math-container">$H$</span>.... | https://math.stackexchange.com |
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