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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542,576 | [
"https://physics.stackexchange.com/questions/542576",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/196421/"
] | let's consider a particle in a 2D-Box of length <span class="math-container">$L_x, L_y$</span>.
Hamiltonian: <span class="math-container">$\hat{H}=-\frac{\hbar^2}{2m}(\partial_x^2 + \partial_y^2)$</span>
Potential: <span class="math-container">$V(x,y)=\begin{cases} 0, & 0\leq x \leq L_x, \ 0\leq y \leq L_y\\ \inf... | Let me try it this way.
<span class="math-container">$$f(x)+g(y)=E$$</span>
Take the partial derivative with respect to <span class="math-container">$x$</span> on both sides. Since <span class="math-container">$g$</span> depends only on <span class="math-container">$y$</span>, and <span class="math-container">$E$</spa... | I think we can have a proof by contradiction.
If we assume <span class="math-container">$f(x)$</span> is not constant while <span class="math-container">$g(y)\equiv Y$</span>, then we would have <span class="math-container">$max\{f(x)\}\neq min\{f(x)\}$</span>, which means <span class="math-container">$$max\{f(x)-g(y... | https://physics.stackexchange.com |
540,251 | [
"https://math.stackexchange.com/questions/540251",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/66639/"
] | Find numbers $a$ , $b$ , $p$ and $q$ so that:
$$x^3+15x^2+3x+5=p(x-a)^3+q(x-b)^3$$
| $$3pa^2+3qb^2=3\Rightarrow pa^2+qb^2=1$$
You have also $p+q=1$. So $pa^2+qb^2=p+q=1$ from here we conclude that $a=\pm1,b=\pm1$.
$$-3pa-3qb=15\Rightarrow pa+qb=-5$$
Using $pa+qb=-5$ and $p+q=1$ with $a=1,b=-1$ you find $p=-2$ and $q=+3$
Using $pa+qb=-5$ and $p+q=1$ with $a=-1,b=1$ you find $p=3$ and $q=-2$.
Therefore... | $(p,q)$ and $(pa^2,qb^2)$ both obey the same pair of linear equations.<br>
So either the pair of equations is degenerate, or the two vectors are equal.
| https://math.stackexchange.com |
241,656 | [
"https://stats.stackexchange.com/questions/241656",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/135602/"
] | I am trying to assess a model's prediction performance, and one metric I look at is the percentage of new observations that fall within the 95% tolerance interval, and see whether or not it is actually close to 95%. If the percentage of new observations that fall in the tolerance interval is much lower than 95%, could ... | Are you talking about a frequentist prediction interval or a Bayesian one? The frequentist prediction interval absolutely doesn't have to contain 95% of the new observations. The frequentist prediction <em>procedure</em> is such that, if you collect a vast amount of samples of size $N$ and for each of these samples you... | There is no relation between a 'correct' coverage probability and predictive performance. Take e.g. as prediction interval the range from the 2.5% to the 97.5% quantile of the responses in the training set. Its coverage probability will probably not be too far away from 95% in an independent test set but the predictive... | https://stats.stackexchange.com |
3,411,085 | [
"https://math.stackexchange.com/questions/3411085",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/687973/"
] | I've got confused about the question. There are several same questions, still I got confused.
<blockquote>
A deck of cards is shuffled and then divided into two halves of 26 cards each. A card is drawn from one of the halves; it turns out to be an ace. The ace is then placed in the second half-deck. The half is then... | Let <span class="math-container">$f(n)$</span> be the number you're looking for for a given <span class="math-container">$n$</span>. Let's work by strong induction.
Consider <span class="math-container">$f(n+1)$</span>. It could either have one term (which happens in one way) or more than one term. In this second c... | Let the number be <span class="math-container">$z + 1$</span>. Jot down <span class="math-container">$1$</span> on your paper.
Now decide whether you want to add <span class="math-container">$1$</span> to the <em>last</em> number you already wrote down (i.e. make it <span class="math-container">$2$</span>) or proceed ... | https://math.stackexchange.com |
179,068 | [
"https://physics.stackexchange.com/questions/179068",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/9604/"
] | I just studied atomic orbitals in a theoretical QM class, and I'm left with several questions, that are probably more questions in quantum chemistry:
<ul>
<li>Many orbitals seem to have a preferred axis - how is that axis
"chosen"? Does it move constantly, or is it actually a fixed
direction in space? Why is there an ... | Regarding your first two points:
The symmetry axis of an orbital is free for a free atom. If it's bound to some other atom through one of these one-dimensionally elongated orbitals, the orientation of one orbital is fixed.
If you take e.g. carbon, silicon or germanium, you have one s orbital and three p orbitals, whic... | $$
\newcommand{\ket}[1]{| #1 \rangle}
$$
I'll try to answer the last two.
<blockquote>
with an arbitrary superposition, the probability density for the electron could be anything - can we actually find the coefficients of the superposition an electron actually is in?
</blockquote>
I'm a little confused about what ... | https://physics.stackexchange.com |
20,759 | [
"https://quantumcomputing.stackexchange.com/questions/20759",
"https://quantumcomputing.stackexchange.com",
"https://quantumcomputing.stackexchange.com/users/14239/"
] | For some arbitrary state <span class="math-container">$|\psi\rangle = c_0|x_0\rangle + c_1|x_1\rangle + c_2|x_2\rangle ... + c_{2^n}|x_{2^n}\rangle$</span>, where each of <span class="math-container">$|x_i\rangle$</span> is a basis state, and each of <span class="math-container">$c_i$</span> is the corresponding comple... | I would call it a "term", same as a term in different kinds of mathematical expressions like sums or products.
I don't think I've seen a special math notation for terms. People usually spell out the complex amplitudes separately from the basis states, since the states will usually be part of some transformati... | I typically refer to it loosely as the "contribution" of the state <span class="math-container">$|x_i\rangle$</span>, or verbosely the "projection" of <span class="math-container">$|\psi\rangle$</span> onto <span class="math-container">$|x_i\rangle$</span>. As for denoting it, I'll often write out t... | https://quantumcomputing.stackexchange.com |
38,032 | [
"https://datascience.stackexchange.com/questions/38032",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/58849/"
] | I have installed Anaconda, but every time I open Terminal I have to go give the command:
<pre><code>export PATH=~/anaconda3/bin:$PATH
</code></pre>
How can I fix this issue?
| You can make sure that command is executed for every terminal (meaning Anaconda will be found) by adding it to your user's <strong>bash profile</strong>.
Open a terminal and follow these steps:
<ol>
<li>open the terminal profile: <code>gedit ~/.bashrc</code></li>
<li>at the end of the file, add: <code>export PATH=~/a... | Write your command in your <code>.bashrc </code> (access at <code> ~/.bashrc </code>)
It will be executed each time you call a terminal.
| https://datascience.stackexchange.com |
4,792 | [
"https://cardano.stackexchange.com/questions/4792",
"https://cardano.stackexchange.com",
"https://cardano.stackexchange.com/users/4986/"
] | I see the Cardano epoch nonce being used as a source of randomness. How can a non-technical person verify if the nonce used is actually legitimate? Is there a source where I can find the nonce for a specific epoch without running a node myself?
| In order to get transaction id using cardano-serialization-lib you should convert transaction id to bytes and convert them to hex string.
<pre><code>Buffer.from(utxo.input().transaction_id().to_bytes()).toString('hex')
</code></pre>
| Try this one: utxo.input().transaction_id()
| https://cardano.stackexchange.com |
2,658,238 | [
"https://math.stackexchange.com/questions/2658238",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/533239/"
] | Gentlemen,
I have small confusion in finding donditional probability in the "Cookies Problem" describe below:
Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random.
If ... | When we say something is “given,” we can say that we already observed it. In this case, we already observed you choosing bowl 1, so we don’t need to consider the probability of you choosing it. So if I asked you, what’s the probability that you pull the plain cookie GIVEN that you absolutely must pull from bowl 1, then... | Intuitively: It is the probability of picking the cookie <em>given</em> that it was bowl one. You do not include the probability for picking that bowl because it was <em>given</em> that it was picked.
| https://math.stackexchange.com |
29,888 | [
"https://scicomp.stackexchange.com/questions/29888",
"https://scicomp.stackexchange.com",
"https://scicomp.stackexchange.com/users/28245/"
] | I have a matrix, whose size scales as $2^N$ (assume even $N$). In each row of the matrix, only about $2^{N/2}$ of the entries are filled ($N$ can be somewhere between 10 and 40, depending on what's feasible computationally). I'm not sure whether using sparse matrices would be suitable for storage and computation - ther... | This kind of scaling is fairly common and sparse direct factorization methods are commonly used on matrices of up to a few hundred thousand rows and columns. By the time you get to $N=20$, you’ll have a million rows with a thousand nonzero entries per column. This is getting into the range where sparse direct factori... | This really comes down to what methods you are intending to use. If you store the matrix dense then you will use dense factorization routines, the cost of which will scale cubically with the matrix size, and so $2^{3N}$ for your case. That limits the $N$ dramatically, even on today's best supercomputers.
You might be ... | https://scicomp.stackexchange.com |
163,167 | [
"https://security.stackexchange.com/questions/163167",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/152281/"
] | A website of my client was compromised by a person claiming he/she has now access to the database. From the fact that this person sent an email saying "contact me and I will tell you where the hole is" I can only hope the access is limited to read-only.
I inspected access logs and found a script & unescaped GET re... | The tool I required to scan that website for more issues & to try to exploit them && fix them was sqlmap. I hope it helps somebody in the future.
| It doesn't matter which tool was used.
Even if you knew which tool was used, it doesn't mean that the tool would necessarily be able to exploit all that can be exploited.
If your app is vulnerable to SQL injection, then in all likelihood no tool at all is needed by even a moderately-skilled attacker to do anything t... | https://security.stackexchange.com |
33,135 | [
"https://chemistry.stackexchange.com/questions/33135",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/13221/"
] | I have seen two different notations for <strong>sodium acetate</strong>. The first one is:
$$\text{NaCH}_3\text{COO}$$
The second one is:
$$\text{CH}_3\text{COONa}$$
Now I'm confused, which one is the best to use?
| There is nothing wrong with either formula. And you can use even more:
<ul>
<li>$\ce{NaC2H3O2}$</li>
<li>$\ce{C2H3NaO2}$</li>
</ul>
It really depends on which point you want to bring across.
$$\ce{NaCH3COO}$$
This formula, being analogous to formulae like $\ce{NaCl}$ stresses the inorganic salt view more. It shows ... | Ok , both of the formulas are absolutely correct.It depends on what you want to signify , basically if you want to look from the point of view of organic chemistry then you take the second formula. And If you any to look from just the point of view of just ions then you can render the positive ion($\ce{Na+} $) to shift... | https://chemistry.stackexchange.com |
2,377 | [
"https://stats.stackexchange.com/questions/2377",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/795/"
] | I am curious if there is a transform which alters the skew of a random variable without affecting the kurtosis. This would be analogous to how an affine transform of a RV affects the mean and variance, but not the skew and kurtosis (partly because the skew and kurtosis are defined to be invariant to changes in scale). ... | My answer is the beginnings of a total hack, but I am not aware of any established way to do what you ask.
My first step would be to rank order your dataset you can find the proportional position in your dataset and then transform it to a normal distribution, this method was used in Reynolds & Hewitt, 1996. See... | Another possible interesting technique has come to mind, though this doesn't quite answer the question, is to transform a sample to have a fixed sample L-skew and sample L-kurtosis (as well as a fixed mean and L-scale). These four constraints are linear in the order statistics. To keep the transform monotonic on a samp... | https://stats.stackexchange.com |
611,445 | [
"https://physics.stackexchange.com/questions/611445",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/172936/"
] | <strong>Do identical, equally spaced lenses always have the same magnification, or does it depend on the sample's distance to the first lens?</strong>
This question comes from trying to solve:
<em>"There are 4 identical and equally spaced lenses between the sample and the camera. The distance from the first to the... | "Does the magnification solely depend on its shape/material (in which case they would all be the same)?"
In the realm of simple geometrical optics, the magnification of a lens is given by q/p, where q is the image distance and p is the object distance. So it's not really a property of the lens but rather of t... | I suspect the purpose of this problem is to see if you know how to use the lensmaker's formula, rather than to design a practical lens system. So IMHO the right approach would be to construct a formula for total magnification, with the given parameters as constraints, set magnification at <span class="math-container">... | https://physics.stackexchange.com |
159,614 | [
"https://mathoverflow.net/questions/159614",
"https://mathoverflow.net",
"https://mathoverflow.net/users/22051/"
] | Let $D$ be a $(v,k,\lambda)$-design. By the <strong>domination number of $D$</strong> I mean the domination number $\gamma(L(D))$ of the bipartite incidence graph of $D$.
<blockquote>
Is $\gamma(L(D))$ determined only by $v,k$, and $\lambda$,
irrespective of the actual structure of $D$?
</blockquote>
I can prove ... | Well, interestingly enough, all the 80 Steiner triple systems on 15 points have minimum dominating sets of size 10 - there is more tradeoff between the points/blocks than I first recognised.
But if we go a bit bigger then we can find some variation. Here is a $2$-$(25,4,1)$ design with 50 blocks.
<pre><code>0 1 2 3; ... | Gordon has done a proper search of $(15,3,1)$-designs. I guess my incorrect reasoning does lead to a computer-free proof for (15,3,13)-designs. This is kind of cheating though, because there are repeated blocks if one takes 13 copies of a STS. The idea may work for smaller $\lambda$; see my own comment below.
Follow... | https://mathoverflow.net |
554,621 | [
"https://physics.stackexchange.com/questions/554621",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/262648/"
] | To explain my question, I'll use an example. Let's say that I have a soda can and I apply <span class="math-container">$10$</span> N force from the right as well as from the left. Here, the net force is zero, yet the can will get deformed.
My question is: How can an object be deformed with no net force?
| “Net” force is <em>zero</em> only for the whole can as an object of both forces — it doesn't change position (or — more accurate — acceleration) of its mass center.
For individual parts (or particles) of the can the situation is different — the sum of forces are (in general) not zero.
| Net force equals zero is a schema that works for rigid bodies, that are by definition unbreakable or non elastic.
Real objects are elastic so a net force equal zero is obtained by means of a distribution of deformation inside the body that, due to elasticity (linear or not linear) creates an internal distribution of f... | https://physics.stackexchange.com |
403,695 | [
"https://softwareengineering.stackexchange.com/questions/403695",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/353366/"
] | My manager asked me how much it costs (in money, currency, $) to make a REST API request from our client application to one of our services. We do not use cloud, we have on-prem servers.
The payload size averages around 5kb, and the client app makes around 750 million requests a week.
I don't even know where to sta... | To make such a calculation, you start by making a list of cost factors and find out the weekly costs for each of them:
<ul>
<li>costs for your networking provider</li>
<li>the hardware (renting price, or purchasing price divided by deprecation period, broken down to a price/week), including backup systems, backup stor... | It is a relevant question when hosting on cloud which may be the reason behind the question. There it is normal for each service have a price table e.g $ 0.05 for 1000 requests or something like that.
If you are hosting on-prem the answer is amortized between the hardware. The people maintaining the hardware and the i... | https://softwareengineering.stackexchange.com |
37,580 | [
"https://softwareengineering.stackexchange.com/questions/37580",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/1973/"
] | I've never been very good at maintaining a coherent bunch of variable names for interfacing with XML files because I never name the variables in my interfaces the same way across my source.
There are Elements, Attributes, Documents, NodeLists, Nodes, DocumentFragments and other stuff. What's a good scheme for keeping... | It's unclear whether you're asking about the variables that you use in code to hold values retrieved from an XML file, or variables used to refer to parts of the file itself. In the former case, I don't see why you'd care that the data came from XML versus, say, a database.
In the latter case, I defined constants to r... | I personally use names like <pre>docNode, rootElem, customAttr</pre> and the like. There's definitely not a method for it in Hungarian Notation because they're all different objects and their implementations can be wildly different across various XML packages, unlike say a Boolean.
| https://softwareengineering.stackexchange.com |
104,738 | [
"https://security.stackexchange.com/questions/104738",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/79556/"
] | I have read some articles about IP spoofing. Briefly speaking, IP-spoofing means the attacker uses a fake IP address in the IP header, to pretend this IP packet is sent by another machine.
The attacker can use it to do something like Denial-of-Service attack (with other advance tech) ... and hide the real source IP ad... | Within a collision domain there is no filtering on the wire. So the reply will be delivered to the attackers NIC.
Normally the attackers NIC would drop such packets without passing them to the host but if the attacker puts thier nic in "promiscuous mode" they will get the response.
P.S. if you actually meant broadcas... | The IP still resolves to the MAC stored in your targets ARP table and that is the address if the local network. So yes, you don't see the replies until you poison the ARP cache of your target.
| https://security.stackexchange.com |
605,781 | [
"https://physics.stackexchange.com/questions/605781",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/139383/"
] | This might be a stupid question, but I actually think that it is not so obvious:
When solving Maxwell equations, depending on the problem usually a charge and current density are assumed.
However, how can there be any current without charge density?
I understand that in most problems where this question arises there is... | <blockquote>
From the example I understand that the net charge vanished but that there can be a current. But I was wondering whether this is just a good approximation and that actually one could take into account that there exist charges.
</blockquote>
Nobody is saying that charges don't exist. If <span class="math-co... | This is an interesting question and points to a possible misconception induced by the usual presentation of the subject.
The facts are that, at the macroscopic level, we have two quite independent concepts: charge and current. If one would have only the knowledge of Coulomb's law and its consequences and Ampere's law a... | https://physics.stackexchange.com |
65,037 | [
"https://softwareengineering.stackexchange.com/questions/65037",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/14954/"
] | How useful is UML if you are a developer working alone in a project?
| It can be very useful if the project is large enough that you have trouble keeping everything straight in your head. Getting something out onto paper/diagrams can also help the design and problem-solving process, at least for me.
...I also admit that for personal projects my diagrams aren't as formal as they would be ... | <strong>It is very useful and valuable</strong>.
As others have said it is best for communication and at a cursory glance you could say "only 1 developer so... no communication needed" but I don't think that's true.
So who is the UML and the communication for?
<ol>
<li><strong>You!</strong> - Yes you. When you go a... | https://softwareengineering.stackexchange.com |
1,640,179 | [
"https://math.stackexchange.com/questions/1640179",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/140396/"
] | $$3\sin x + 4\cos x = 2$$
To solve an equation like the one above, we were taught to use the double angle identity formula to get two equations in the form of $R\cos\alpha = y$ where $R$ is a coefficient and $\alpha$ is the second angle being added to $x$ when using the double angle identity.
<blockquote>
Why can'... | $$\begin{align}3\sin x + 4\cos x &= \sqrt{3^2 + 4^2 }\sin\left(x+\arctan\frac{4}{3}\right) \\
&= 5\sin(x+\arctan\frac{4}{3})\end{align}$$
<hr>
In general, if $$a\sin x + b\cos x = c$$
$$\frac{a\sin x}{\sqrt{a^2 + b^2}}+\frac{b \cos x}{\sqrt{a^2 + b^2}} = \frac{c}{\sqrt{a^2 + b^2}}$$
Let $\cos\phi = \dfrac{... | <strong>A different approach</strong>:
You can trade the trigonometric functions for a rational expression with
$$3\sin x+4\cos x=3\frac{2t}{1+t^2}+4\frac{1-t^2}{1+t^2}=2.$$
Thus, the quadratic equation
$$6t^2-6t-2=0,$$ which you can readily solve.
Then
$$\tan x=\frac{2t}{1-t^2}$$
and
$$x=\arctan\frac{-12\pm2\s... | https://math.stackexchange.com |
28,352 | [
"https://softwareengineering.stackexchange.com/questions/28352",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/6415/"
] | I have an abstract class that will have many implementors. There are, of course, many places to put abstract classes in a Java package structure. Should the abstract class be in the same package as the implementors, or should it be in an "abstract package" package, or somewhere else? Please explain why you would place ... | I would put it (like any other class) in the package it belongs most from it's content. That mostly - but not everytime - means, that you put it in the same package as implementing classes. Sometimes you put an abstract class in the same package as some other class using this abstract class - it's part of the API of th... | I was just wondering about the same thing (but for interface classes). So lets add a post to this discussion.
I have a package <code>com.cie.devices</code> where I put devices which I control using Java software. Now comes the time where there are enough devices to start building a hierarchy of devices. I was wonderin... | https://softwareengineering.stackexchange.com |
2,327,246 | [
"https://math.stackexchange.com/questions/2327246",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/257841/"
] | If $y=f(x)$ be the equation of the parabola which is touched by the line $y=x$ at the point where $x=1$.Prove that $f'(1)=1$ and $2f(0)=1-f'(0)$
<hr>
Let $f(x)=ax^2+bx+c$ be the parabola.Since the parabola is touched by the line $y=x$ at $x=1$.So $f'(1)=1$,because slope of $y=x$ is $1$.<br>
But i cannot prove the sec... | If $f''<0$, then $f'$ is decreasing. Suppose that $f(x_1)=f(x_2)=0$, with $0<x_0<x_1<1$. Then, by Rolle's theorem, $f'(x_2)=0$ for some $x_2\in[x_1,x_0]$. So, $f'(x)\leqslant 0$ if $x\geqslant x_2$ and so $f$ is decreasing on $[x_2,1]$. But $f(1)>0$ and $f(x_1)=0$.
| Hint. $f''(x)<0$ means that $f'$ is strictly decreasing. Therefore there is a unique maximum point $x_0$ in $(0,1)$ where $f'(x_0)=0$. Note that $f(0)<0$ and $f(1)>0$ imply that $f(x_0)>0$, $f$ is strictly increasing in $[0,x_0]$ and strictly decreasing in $[x_0,1]$ Can you take it from here?
| https://math.stackexchange.com |
157,927 | [
"https://dba.stackexchange.com/questions/157927",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/108666/"
] | If I have two tables;
<pre><code>Table A
id | etc
1 | etc
2 | etc
etc
</code></pre>
And
<pre><code>Table B
id | A_id | etc
1 | 2 | etc
2 | 2 | etc
3 | 1 | etc
etc
</code></pre>
The relationship between Table A and Table B is defined in the database as a foreign key with cascading delete. <strong>Do I ... | You always have to specify the columns you are joining on, regardless of any foreign key relationships.
Why? Because otherwise you couldn't do a full outer join if a foreign key existed between the tables, which would lead to identical queries producing different results for non-obvious reasons.
Also, what if you exp... | Yes, you still have to explicitly declare the join columns.
| https://dba.stackexchange.com |
380,684 | [
"https://stats.stackexchange.com/questions/380684",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/151862/"
] | We are minimizing either the <span class="math-container">$1$</span> norm of the residuals, least absolute value, or the <span class="math-container">$2$</span> norm, least squares.
Least absolute value:
<span class="math-container">$$\min_\beta||y - x \beta||_1$$</span>
Least squares:
<span class="math-container"... | They aren't different by a constant. If that were so, you would be right. But what the formula you post says is that they are different by <em>at least</em> some constant and <em>at most</em> some other constant. Further, the norms are the single numbers that are the total for all the values. But the individual contri... | Let <span class="math-container">$x$</span> be scalar and equal to one to consider the most simple example. Then consider minimizing
<span class="math-container">$$\min_\beta \mathbb E[(Y-\beta)^2]$$</span>
The first order condition:
<span class="math-container">$$0 = -2 \int_{-\infty}^\infty (Y - \beta)dF(y) = -2(\... | https://stats.stackexchange.com |
381,599 | [
"https://electronics.stackexchange.com/questions/381599",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/155758/"
] | I just learned transient circuits and we solved them both in the time domain and the Laplace domain and I can't understand why we would use the Laplace domain as it seems it is a lot more simple to use the time domain especially in series RLC circuits.
| It's easy to solve simple filter circuits in the time domain but there becomes a tipping-point where most engineers would prefer to solve problems in the frequency domain and apply (for example) a step-function. Finding the inverse Laplace is fairly straightforward because of Laplace tables.
As an example of an RLC lo... | In this context, the Laplace transform is simply a tool that we use to deal with differential equations. I think it provides a slightly more intuitive feel for how responses change with frequency, but really, it's just a tool for dealing with differential equations.
The transform really starts to shine when you're ca... | https://electronics.stackexchange.com |
2,582 | [
"https://mechanics.stackexchange.com/questions/2582",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/970/"
] | Are there advantages go getting your car's oil changed at the dealership from which you purchased the car versus a "quick change" place like Jiffy Lube or Wal-Mart? I've used all three, and I feel the car is treated better at the dealer. Wal-Mart tends to leave a mess (oil in the seats, carpet, etc.)
| Honestly, I have worked at three different types of places.
<ul>
<li>First, it was aquick lube place in California. These can be good, if
they have good management. The bad comes in when you have bad
management, and they can "sell" things and not do them. These focus
mostly on services they sell, and really glance at... | Advantage being that if they spot a problem while doing the oil change that they are a LOT more likely to have appropriate parts on hand. A non-dealer shop would have to source the parts, adding more time to the job (and possibly even another trip to the shop).
| https://mechanics.stackexchange.com |
160,777 | [
"https://physics.stackexchange.com/questions/160777",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | <blockquote>
Consider a non-relativistic particle of mass $m$, moving along the $x$-axis in a potential $V(x) = m\omega^2x^2/2$. use path-integral methods to find the probability to find the particle between $x_1$ and $x_1 + dx_1$ if the particle is at $x_0$ at time $t = 0$.
</blockquote>
One finds the propagator to... | As $T = t_1 - t_0 \to 0$, we have$$\begin{align}\lim_{t_1 -~ t_0~ \to~ 0} \langle x_1, t_1\,|\,x_0, t_0\rangle &= \lim_{T~ \to~ 0}\left({{m\omega}\over{2\pi i\sin\omega T}}\right)^{1/2}\text{exp}\left[{{im\omega}\over{2\sin\omega T}}\left(\left(x_1^2 + x_0^2\right)\cos \omega T - 2x_0x_1\right)\right]\\ &=\lim_... | The probability density $$\lim_{x_1\to x_0} P(x_1, t_1; x_0, 0)$$ is basically the probability density for a particle at position $x_0$ to be back at that position after a time $t_1$ has passed.
| https://physics.stackexchange.com |
230,848 | [
"https://mathoverflow.net/questions/230848",
"https://mathoverflow.net",
"https://mathoverflow.net/users/86526/"
] | I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 :
<blockquote>
Let A be an unital C* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von
Neumann equivalence) in $M_∞(A)$ is cancellative.
</blockquote>
For the proof he proceeds as follow :
<ul>
<li>take s... | Theorem 7.1.2 in Murphy actually says that for $A$ a unital C$^*$-algebra the semigroup $K_0(A)^+$ is cancellative. But $K_0(A)^+$ is the semigroup of <em>stably equivalent</em> projections in $M_\infty(A)$ not just Murray von Neumann equivalent.
As a reminder $P,Q$ projections in $M_\infty(A)$ are stably equivalent i... | Let $R$ be a projection in $B(l^2)$ onto a subspace whose dimension and codimension are both infinite. Then $[P] \oplus [R] = [Q] \oplus [R]$ for any projections $P$ and $Q$ in $B(l^2)$, but not all such $P$ and $Q$ are equivalent.
| https://mathoverflow.net |
245,158 | [
"https://security.stackexchange.com/questions/245158",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/20141/"
] | I am looking for a login security measure where it is keylog and screen capture proof. Is there some type of login security like a 2FA without the need of a second device, but remembering a pattern or a formula which is used to solve a dynamic puzzle that is given to the user on login?
Say I am shown 100 words during l... | The difference between using some hardware backed key store (i.e. TPM, HSM, smartcard ...) and a "pure software" solution like <code>openssl genrsa</code> is not so much about the security of the key <strong>generation</strong> but about the security of the key <strong>storage</strong>.
HSM and similar are de... | It is a trade-off between
<ul>
<li>the security of storage (TPM is better, in theory it won't give off the key to anyone, it will just accept data to encrypt/sign for you with the key),</li>
<li>the security of implementation (openssl is opensource, TPM may have a nasty backdoor) and</li>
<li>the ease of use (a PEM fil... | https://security.stackexchange.com |
388,550 | [
"https://physics.stackexchange.com/questions/388550",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/177815/"
] | I'll start my master's in the next semester and I'll study QCD. I want a review for learn QCD and know what's going on in this field. I more interested in confinement.
| Short answer: <strong>no</strong>.
For example, gravity waves (i.e. ripples) on the surface of a liquid have both transverse and longitudinal motion, so they are not purely either.
| I was wondering if there is a difference between a transverse and longitudinal wave...
Imagine a rubber rod, it flexible to the sides so you can bend it and oscillate it like a string if you do it fast enough. That would be its transverse wave behavior.
Now the same rubber rod can be compressed or decompressed and th... | https://physics.stackexchange.com |
126,408 | [
"https://mathoverflow.net/questions/126408",
"https://mathoverflow.net",
"https://mathoverflow.net/users/16023/"
] | Let $\tilde{\mathbb N}$ be the Abelian semigroup (under addition) given by $\mathbb N\cup\{0,\infty\}$, and let $S_n$ be the Abelian monoid $\tilde{\mathbb N}^{2^n}$ under point-wise addition. Introduce the transition maps $\{\phi_n:S_{n+1}\to S_n\}_{n\in\mathbb N}$ defined by
$$\phi_{n-1}(n_1,n_2,n_3,n_4,\ldots,n_{2^... | You have $f(0,j) = 1 + f(0,j-1)$ so $f(0,j) = j$. I get
$$f(1,j) = j + 1 - \left(\frac{N-1}{N}\right)^j $$
$$ f(2,j) = j + 2 - 2 \frac{(N-1)^{j+1}}{N^{j+1}} - 2 \frac{(N-2)^j}{N^{j+1}}$$
$$ f(3,j) = j + 3 - 3 \frac{(N-1)^{j+2}}{N^{j+2}} - 6 \frac{(N-2)^{j+1}}{N^{j+2}} - 9 \frac{(N-3)^j}{N^{j+2}}$$
$$ f(4,j) = j + ... | This problem has a probabilistic interpretation. Namely, if one considers the Markov chain on the integer points in the rectangle $[0,N]\times [0,M]$ with the transition probabilities
$$
p\bigl((i,j),(i,j-1)\bigr) = \frac{N-i}{N} \;, \qquad p\bigl((i,j),(i-1,j)\bigr) = \frac{i}{N} \;,
$$
then $f(i,j)$ is the expected n... | https://mathoverflow.net |
43,532 | [
"https://mechanics.stackexchange.com/questions/43532",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/26966/"
] | I own a 1990 Suzuki GSXR 750, I, bought the bike about a month ago. The first time I took the bike out for a ride I noticed that when you accelerate hard and the engine is at high revs it feels as if the bike has got a dead spot and as if it's losing power. I didn't think much of it and rode the bike everyday to work t... | I'm driving a 2015 LEAF in S (base) trim. It has a CHAdeMO port and 6.6kW L2 charging.
The 2011 and 2012 LEAFs (like David B's) all had 3.3kW L2 charging, CHAdeMO or not. However, starting in 2013, CHAdeMO was an option on the S trim, standard on the SV and SL trim. And if you have CHAdeMO, you have 6.6kW L2.
| My 2011 Leaf has CHAdeMO rapid charging but has a 3.3kW internal charger for AC. So, having rapid charge is no guarantee of having the built-in 6.6kW charger.
| https://mechanics.stackexchange.com |
508,696 | [
"https://stats.stackexchange.com/questions/508696",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/310483/"
] | I am struggeling with a basic question and would be happy to get some pointers. I am trying to evaluate an algorithm <span class="math-container">$f$</span> which maps some sample <span class="math-container">$x$</span> onto a scalar <span class="math-container">$y\in\mathbb{R}$</span>, i.e. <span class="math-container... | In my final solution for my problem I closely followed the strategies detailed in [1]. As described in my question, my targets were two-fold:
<ol>
<li>Calculate a confidence-interval</li>
<li>Estimate the necessary sample size to determine a confidence interval of given width.</li>
</ol>
Among the strategies outlined i... | You say you have sufficient proof that <span class="math-container">$Z$</span> is normal <span class="math-container">$\mathcal{N}(\varepsilon,\sigma^2)$</span>.
<h2>Evaluate the goodness of fit</h2>
You can use the log-likelihood of the data <span class="math-container">$\mathcal{D}= \{z_1, z_2, \dots, z_n\}$</span> a... | https://stats.stackexchange.com |
18,720 | [
"https://softwareengineering.stackexchange.com/questions/18720",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/165/"
] | I've recently had a discussion with a coworker about versioning web applications.
I don't think you need it at all, and if you just want a sanity check to confirm your latest release is live, I think a date (YYMMDD) is probably good enough.
Am I off base? Am I missing the point? Should I use web application version... | If you resell the same web application to multiple customers, you absolutely should version it.
If it's a website that's only ever installed in one location, then the need is less dire, but it still likely couldn't hurt. You'd be less susceptible to timezone problems and such, too. Diagnosing an issue in a library ve... | If you can automate version number on your application's DLLs, it couldn't hurt. It will help you keep track of versions.
Generally, web apps are released only at one location where you are hosting it, so it's not as important as say a desktop application. It can help with roll-backs, bug tracking (i.e. which versio... | https://softwareengineering.stackexchange.com |
180,865 | [
"https://mathoverflow.net/questions/180865",
"https://mathoverflow.net",
"https://mathoverflow.net/users/41650/"
] | The other direction is well known.
I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove. I am also wondering whether there is a general nonsense style proof. It looks like the statement can be proved by playing adjunctions in several... | Well, as already explained in the comment by Vivek Shende, the answer is no: there are non-proper morphisms with push-forward that preserves coherence. I've thought a bit on this the last couple of years and on the "positive" side we have:
<strong>Theorem 1.</strong> If $f\colon X\to Y$ is a universally closed morphis... | If $f:X\to Y$ is separated of finite type between noetherian schemes and $f_*$ preserves coherence, then $f$ is proper. Here is a proof that follows the geometric idea (given in the comments of Piotr Achinger and Karl Schwede) to use a curve in $X$ that is not proper over $Y$ :
Since it is possible to extend a coheren... | https://mathoverflow.net |
381,057 | [
"https://mathoverflow.net/questions/381057",
"https://mathoverflow.net",
"https://mathoverflow.net/users/48162/"
] | Is there a triple of nonzero even integers <span class="math-container">$(a,b,c)$</span> that satisfies the following infinite system of congruences?
<span class="math-container">$$
a+b+c\equiv 0 \pmod{4} \\
a+3b+3c\equiv 0 \pmod{8} \\
3a+5b+9c\equiv 0 \pmod{16} \\
9a+15b+19c\equiv 0 \pmod{32} \\
\vdots \\
s_na + t_nb ... | Let <span class="math-container">$u_n = a s_n + b t_n + c s_{n+1}$</span>. The stronger claim is true: for large enough values of <span class="math-container">$n$</span>,<br />
the number <span class="math-container">$u_n$</span> will be exactly divisible
by a fixed power of <span class="math-container">$2$</span> that... | <span class="math-container">$u_n=s_na + t_nb + s_{n+1}c$</span> satisfies the same recurrence relation as <span class="math-container">$s_n$</span> and <span class="math-container">$t_n$</span>: <span class="math-container">$u_n = u_{n-1} +2u_{n-2} + 4u_{n-3}$</span>. The question is whether <span class="math-containe... | https://mathoverflow.net |
72,814 | [
"https://physics.stackexchange.com/questions/72814",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/26419/"
] | Charging by induction seems very unintuitive to me. Let me explain:
Consider a positively charged rod and a neutrally charged conductor. You can charge it by induction by earthing it and not lose charge on the rod. You can then use this potential difference in the spherical conductor as energy. This can be repeated ma... | <blockquote>
You can then use this potential difference in the spherical conductor as energy. This can be repeated many times. Does this not violate the law of conservation of energy?
</blockquote>
Conservation of energy applies to closed systems. If you connect your conductor to the ground, it is no longer a close... | As far as I am aware, charging by friction occurs between insulators.
In this situation, friction allows a charge to build up due to the energy provided by friction itself (kinetic energy). When energy is being applied, the charge builds up, and when energy ceases to be applied, the potential difference remains as th... | https://physics.stackexchange.com |
40,769 | [
"https://cstheory.stackexchange.com/questions/40769",
"https://cstheory.stackexchange.com",
"https://cstheory.stackexchange.com/users/43272/"
] | I was fooling around with some concept and was wondering if this viewpoint is explored at all. Let INJ-GRAPH be the subcategory of graphs (with morphisms as homomorphisms) whose morphisms consist only of the graph homomorphisms which are injective as sets. Then clearly for such a homomorphism $f : G \to H$ of graphs, t... | This question should (and will) probably be migrated to cs.se.
In the meantime, consider the computation tree of the depicted structure: in almost all paths, $p$ is seen only finitely often, making the premise of $GFp\to GFq$ false, so the formula <em>is</em> satisfied there.
However, there is one path, namely $s_0^\o... | <h2>The flaw in your reasoning</h2>
According to you GF p is false ( and it is not satisfied by model speaking more properly). Yet your fault in assuming, than akin to Boolean logic, it make the whole LTL formula true. It does not. The only way is to evaluate an LTL formula is on all the model paths (see page 17 of yo... | https://cstheory.stackexchange.com |
360,507 | [
"https://stats.stackexchange.com/questions/360507",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/37541/"
] | I am using the <code>forecast</code> package and the <code>auto.arima</code> function.
This function tries different arima model with different p and q parameters and selects the best one by AIC.
I tried increasing the default values, so <code>auto.arima</code> searches more possible models and indeed the AIC gets low... | It is not very surprising that you get lower AIC when increasing the range, overfitting is always an issue when you try multiple models. AIC or measuring MAPE on the test set helps reduce the risk but once you consider too many models they can mislead you as well.
| You have a supervised model, you always have the risk of overfitting or high model variance. You should test for instance, how perturbation affects the model. Does removing one point completely change the model?
If you add more models, there is a higher chance that one of does will do better in that specific training ... | https://stats.stackexchange.com |
16,660 | [
"https://astronomy.stackexchange.com/questions/16660",
"https://astronomy.stackexchange.com",
"https://astronomy.stackexchange.com/users/6610/"
] | The moon, for example, appears very bright in the night sky, but quite dark from photographs on/near its surface.
Similarly, Mars and Venus appear as bright dots in the sky but in closer photographs they're quite dark.
| The Moon, Mars, Venus are seen against a dark sky using an eye whose aperture is adjusted to the average brightness of the visual field. So bright small sources are over-exposed and so appear bright.
Photographs of planetary surfaces are taken with aperture and exposure set for proper exposure. So appear closer to how... | To add to Conrad Turner, if you look at the moon during the day against a bright sky, then i looks quite "washed out" and much more like the photographs taken from near its surface.
| https://astronomy.stackexchange.com |
531,024 | [
"https://physics.stackexchange.com/questions/531024",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/254088/"
] | If electricity is produced by flow of electron, that is current, then current should be measured in electron per second, why is it measured in coulombs per second?
| <blockquote>
If electricity is produced by flow of electron, that is current,
</blockquote>
Current and electricity is <em>not</em> only electrons flowing.
<ul>
<li>Sure, it is <strong>electrons</strong> flowing in metal wires, but</li>
<li>it is <strong>holes</strong> (positively charged) in semiconductors such as... | The word current is often used by itself instead of the longer, more formal "electric current". Electric current is defined as the rate at which charge flows through a surface. An ampere is the unit used to measure electric current.
Since charge is measured in coulombs and time is measured in seconds, an ampere is th... | https://physics.stackexchange.com |
2,335,632 | [
"https://math.stackexchange.com/questions/2335632",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/372198/"
] | I am trying to evaluate numerically :
\begin{equation}
G = \frac{-1}{4\pi}\sum_{l=0}^{\infty}\frac{2l+1}{\frac{l(l+1)}{R^2}+\frac{1}{L_d^2}}P_l(\cos(\gamma))
\end{equation}
Where $P_l$ is the $l_{th}$ Legendre Polynomial, $R = 6371000, Ld = 1000000$ and $\cos(\gamma)\in [-1,1) $. I know that the series converges for an... | <strong>Hint:</strong> You know that $0 \leq \arccos(x) \leq \pi$ for all $-1 \leq x \leq 1$ because of the way $\arccos$ is defined. So $0 \leq \frac{1}{2}\arccos(\frac19) \leq \frac12\pi$. This gives you the sign of $\sin(\frac{1}{2}\arccos(\frac19))$.
Now you can use the formula $\sin(\frac{x}{2})=\pm\sqrt{\frac{1... | $$\sin\left(\frac{1}{2}\arccos\frac{1}{9}\right)=\sqrt{\frac{1-\frac{1}{9}}{2}}=\frac{2}{3}$$
The right formula is
$$|\sin\frac{x}{2}|=\sqrt{\frac{1-\cos{x}}{2}}$$
and since it's obvious that $\sin\left(\frac{1}{2}\arccos\frac{1}{9}\right)>0$, we get, which got.
| https://math.stackexchange.com |
271,283 | [
"https://softwareengineering.stackexchange.com/questions/271283",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/126755/"
] | What i would like to do is the following.<br>
(Not sure if this question should go to StackOverflow, or here)
Consider a datastructure like this:
<pre><code>interface IAction {
IAction[] afterActions()
IAction[] beforeActions()
void execute()
}
</code></pre>
<strong>What I would like to do is <code>execu... | First of all, you should have a (let's call it) <code>ready</code> collection of IAction's whose dependencies were already met. This collection starts empty, of course. Then you should iterate over your IAction instances and check whether all of its dependencies were met. You do this by iterating over all the dependenc... | For question 3, you will need to recursively iterate over the graph of IAction instances entirely. Recursive iteration is a basic operation. If you don't know how to do that, read up on recursive tree traversal. You can detect cyclic dependencies in at least two ways:
<ul>
<li>Use a class instead of an interface, and ... | https://softwareengineering.stackexchange.com |
7,256 | [
"https://robotics.stackexchange.com/questions/7256",
"https://robotics.stackexchange.com",
"https://robotics.stackexchange.com/users/1831/"
] | I'm looking to build a sensor which will detect the level of liquid in a tube.
It does not have to be precisely accurate, just detect whether the level is approximately above a certain height.
<img src="https://i.stack.imgur.com/4USD9.jpg" alt="tube">
The liquid level can be seen in the red oval
<img src="https://i... | If you look in the industry, they usually use capacitive proximity sensors.
This sensors can be adjusted easily to sense behind the glass. In addition I can tell you, that this measurement is highly reliable. Usually they start around 70$, but you find also some cheaper DIY-solutions.
Another possibility is the use f... | Here's a few ideas:
<ul>
<li>Insert two bare wires as probes and measure the resistance between them. The resistance should drop dramatically once the liquid touches both sensors.</li>
<li>Insert a sonar into one end that points directly into the tube. This method should give you a simple measurement of how far the su... | https://robotics.stackexchange.com |
67,432 | [
"https://physics.stackexchange.com/questions/67432",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/25508/"
] | This is a question about a simple thing. The simplified expression for maximum height in vertical throw is $h=\frac{v^2}{2g}$ , could anyone explain <strong>intuitively</strong> (analogies are welcome) why there is a factor 2 on the denominator? And I mean intuitively, so not by transforming formulas etc, but by saying... | The best intuition is a calculation but in this simple case, the calculation is really intuitive so you shouldn't turn off when you hear the word "calculation".
The height reached by initial velocity $v$ is the height of the object after the initial velocity $v$ drops to $0$ (and then reverts the sign) because of the ... | The factor $2$ comes from the equation $v_f^2 = v_i^2 - \color{#C00}2gh$. That's how we derive the equation for max height.$$v_f^2 = v_i^2 - 2gh$$Now, we want $v_f$ to be $0$ (because at the maximum height, the speed is always zero). Calculating $h$, we have:$$ 0 = v^2 - 2gh \Rightarrow -v^2 = -2gh \Rightarrow v^2 = 2... | https://physics.stackexchange.com |
127,610 | [
"https://security.stackexchange.com/questions/127610",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/114146/"
] | A threat is defined as something that might do harm but once a threat happens do we still refer to it as a threat?
For instance, if malware infects our computer, do we still refer to the malware as a threat?
| You may be looking for 'breach'. As described below, the original threat remains, a breach has occurred, and additional new unmitigated threats may exist where the attacker may be able to do additional damage now that they're inside.
<strong>Threat</strong>: the possibility that something could try to happen
<strong... | A threat has nothing to with a malware. It is basically a certain probability that an vulnerability gets exposed.
A malware is an exploited vulnerability by some kind.
| https://security.stackexchange.com |
131,364 | [
"https://mathoverflow.net/questions/131364",
"https://mathoverflow.net",
"https://mathoverflow.net/users/766/"
] | This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question will still be clear.
Let $G$ be a finite group. Traditionally, a character $\chi: G \to {\bf C}$ on $G$ is defined as being the trace of a f... | I spent a long time writing an answer to this question, but MO did not believe I was a human being ( I did mis-spell one of the test words, but everyone deserves a second chance, I think ), so it seems to have disappeared. I am not sure I have the energy to do it again right now, but here (in precis, though not precise... | It might be useful here to think about the Morava $K$-theory rings $K(n)^*(BG)$ (for finite groups $G$). For the trivial group you get the graded ring $k=\mathbb{Z}/p[v_n,v_n^{-1}]$, where $|v_n|=2p^n-2$; for general groups you get a finitely generated graded module over $k$ (and all such modules are free). There is ... | https://mathoverflow.net |
198,636 | [
"https://softwareengineering.stackexchange.com/questions/198636",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/47330/"
] | Are there any languages that support foreach over multiple lists at once? Something like this:
<pre><code> foreach (string a in names, string b in places, string c in colors) {
// do stuff with a, b, c
}
</code></pre>
Sure, this can be done with a regular for loop:
<pre><code> for (int n = 0; n<max, n++) ... | Sure, Python can do that:
<pre><code>for a, b, c in zip(names, places, colors):
pass
</code></pre>
<code>zip</code> builds a list of tuples from the iterables passed to it. Python's <code>for</code> loop supports "unpacking", that is to say the tuples are split into their components. It's a weak form of pattern m... | <ul>
<li>In C#, you can transform multiple collections into one collection of tuples, then use a <code>foreach</code>. There is no specific language syntax, but .NET Framework still allows you to do it.</li>
<li>In SQL, you can use <code>join</code>. I'm not sure if the "languages" you're looking for extend to SQL.</li... | https://softwareengineering.stackexchange.com |
16,087 | [
"https://quant.stackexchange.com/questions/16087",
"https://quant.stackexchange.com",
"https://quant.stackexchange.com/users/8047/"
] | I load swap rates data from Bloomberg in excel using the command
<pre><code>=BDH(Ticker,"Px Last",Start_Date,End_Date,"Dts",FALSE)
</code></pre>
The tickers I use are "USSW1 Curncy", "EUSA1 Curncy", "BPSW1 Curncy", etc...
First of all, I would like to know what this "Curncy" keyword stand for. Then as I am also inte... | This means that a linear regression is not the best model for your data. You may want to try a regularized regression (LASSO/Ridge) to see if penalizing the coefficients will help.
| Regression analysis, as a minimization of the sum of squared errors, does not require normality of the error term.
The requirements are that errors are homoscedastic and uncorrelated. And these are the fundamental assumptions (together with exogeneity). Then estimators are unbiased, optimal (exhibit the minimum varia... | https://quant.stackexchange.com |
85,048 | [
"https://electronics.stackexchange.com/questions/85048",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/4475/"
] | I am uploading software via serial ISP in to an LPC81x series microcontroller. This is a Cortex-M0+ chip. After software upload, I would like to start the execution of the uploaded software without having to reset the CPU externally.
There is a "Go " command in the serial ISP and the "lpc21isp" software uses it as su... | This question is flawed because at the point when the switch closes \$V_O\$ is undefined by any of the circuit components. Nothing in the circuit prior to t=0 defines \$V_O\$.
<blockquote>
At t<0 vs is unplugged and the rest of circuit is dead. va=v 0 =0V
</blockquote>
No this is not true.
Just think about it. ... | <strong>Rule number 1</strong>: If the opamp works with negative feedback and is not saturated, the voltages on the two inputs are always equal.
<strong>Rule number 2</strong>: The input current is always 0.
So, the current through the resistor is equal to the current through the capacitor and is:
$$
I_C = I_R = \fr... | https://electronics.stackexchange.com |
356,880 | [
"https://electronics.stackexchange.com/questions/356880",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/178486/"
] | I once have a tablet (Samsung Galaxy Tab 3 GT-P5210) that has an IR blaster built to it.
I sometimes use it to control the TV on our house if the remote is lost.
Now, I don't understand that if I use it to control the TV, they are saying that the TV will be damaged.
Is this true, or is this some kind of a myth, so t... | Since IR is just light that is invisible to the human eye, it can't really damage the TV. They switch the IR LED in the remote on and off at a given frequency to represent a set digital value, that the TV then decodes.
Blocking other IR signals is a different topic, but it's highly unlikely. Btw, you can see IR throu... | The Samsung tool could accidentally issue RC codes meant for service technicians only. The last CRT I had died slowly, and the technician came and took out his service remote and could adjust the TV through it. It was even possible to select the high voltage level - that was meant to have the same electronics for diffe... | https://electronics.stackexchange.com |
307,871 | [
"https://physics.stackexchange.com/questions/307871",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/142275/"
] | (Edit, Question was previously: "Can a changing electric field produce a magnetic field?")
I've just began studying Maxwell's equations today and what really had my attention is Ampere's law, the second term in particular.
$$\int\vec B \cdot d\vec l=μ_0I_{encl}+μ_0ε_0\frac{dΦ_E}{dt}$$
Does this mean that a changing e... | <blockquote>
But some websites state that as long as there is no current - charge
movement , there is no magnetic field being created.
</blockquote>
This is incorrect. Light propagation happens because a changing electric field can produce a magnetic field just like a changing magnetic field produces an electric f... | I will try to put this very simply. As we know, a straight current carrying wire produces a magnetic field encircling the conducting wire. Also, as theoreticaly suggested, a displacement current is set up between the plates of the capacitor when there is a change in electric field (generally due to change in charge tha... | https://physics.stackexchange.com |
241,650 | [
"https://math.stackexchange.com/questions/241650",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/47988/"
] | The question was:
<blockquote>
Guess the limit of the series, and prove your guess from the definition of limits (without any of the theorems thought in class).
</blockquote>
I guessed that the limit is $0$, and then looking at the definition of a limit I have:
$\forall\epsilon>0\:\exists n_{0}>0\:\forall n&g... | $$2^n = e^{n\log2 } = 1 + n \log 2 + \dfrac{n^2 \log^2(2)}2 + \cdots \geq 1 + \dfrac{n^2 \log^2(2)}2$$
Hence, we get that $$2^n - 1 \geq \dfrac{n^2 \log^2(2)}2$$
Hence, $$\dfrac{n}{2^n-1} \leq \dfrac{2n}{n^2 (\log 2)^2} = \dfrac{2}{n (\log 2)^2}$$
Now choose $N = \left \lceil \dfrac{2}{\epsilon (\log 2)^2} \right \rcei... | <strong>Hint:</strong><br>
Use induction to prove that $2^n-1>n^2 \ \forall n \geq 5 $.
Conclude that $\dfrac{n}{2^n-1}<\dfrac{1}{n} \ \forall n \geq5$.
Now it should be easy. For $\epsilon>0$ if $n_0=\max{\{5,\lceil\frac{1}{\epsilon}\rceil+1\}}$ then $\dfrac{n}{2^n-1}<\epsilon \ \ \forall n \geq n_0$.
| https://math.stackexchange.com |
2,901,783 | [
"https://math.stackexchange.com/questions/2901783",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/-1/"
] | I am having trouble solving a multi part question.
Express $ \frac x{x^2-3x + 2} $ in the partial fraction form.
The answer I got was $\frac2{x-2}-\frac1{x-1}$ .
The problem comes when they asked:
Show that, if $x$ is so small that $x^3$ and higher powers of $x$ can be neglected, then:
$$\frac x{x^2-3x +2}\approx\... | I believe you are already done and just have linguistic issue; I think that "if $x$ is so small that $x^3$ and higher powers of $x$ can be neglected, then [...]" should be interpreted as "if $x$ has property $P(x)$, then [...]" where $P(x)\equiv$ "$x^3$ and higher powers can be neglected". That is, you are not supposed... | Just cross out terms with $x^3$, $x^4$, ... in the approximation.
The question assumes these terms are too small, and we don't care.
For example, if I know x is between 0 and 0.1. Then $x^3$ is at most 0.001. $x^4$ is even smaller.
I think the sum of the terms $x^3$, $x^4$, ... are at least smaller than $x^2$, whic... | https://math.stackexchange.com |
15,992 | [
"https://physics.stackexchange.com/questions/15992",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/5793/"
] | I've been trying to figure this one out for a while on my own, so I'd like to ask for your help if you could offer some.
The task states:
<blockquote>
A heater made out of a wire with a diameter <span class="math-container">$R = 0.2\text{ mm}$</span>, length <span class="math-container">$4\pi\text{ m}$</span> and elect... | You may consider this question from perspective of energy view: the electric energy is consumed by the resistor and convert this energy to the thermal energy (the source of heat that heat up the water). So from this point of view, if you can assume the 100% electric energy converting to thermal energy, and usually, thi... | Find the resistance of the particular wire. Then calculate the power it uses. Assume this power is dissipated as heat. Find how much energy is converted to thermal energy by heat in 10 minutes. Use the equation Q = mc ΔT to find the mass of the water.
| https://physics.stackexchange.com |
25,625 | [
"https://astronomy.stackexchange.com/questions/25625",
"https://astronomy.stackexchange.com",
"https://astronomy.stackexchange.com/users/22406/"
] | If I take a shot longer than 20 seconds, the stars get a tail. I have a Celestron Advanced GT mount with starshoot autoguide scope. I use a cannon can on a 9" Celestron scope. I can't get a picture without having tails on my stars. It only looks good if I keep the pictures under 20 seconds.
EDIT :
I did the polar ali... | If you can't get any longer than 20 seconds, chances are it's your polar alignment. The mount you have is a motorised GoTo EQ mount (I believe), but if these aren't aligned properly, then they can still have trouble.
Also, depending on your software, try a 2 or 3 star align to get a better accuracy.
Also check the ... | you need a motorized equatorial mount, which tracks the stars in such a manner as to cancel the rotation of the earth and eliminate the tails on your photos. These are commercially available for many common types of telescope mounts.
| https://astronomy.stackexchange.com |
80,400 | [
"https://physics.stackexchange.com/questions/80400",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/30457/"
] | I'm having trouble conceptualizing why the voltage drop between two points of an ideal wire (i.e. no resistance) is $0~V$. Using Ohm's Law, the equation is such:
$$
V = IR \\
V = I(0~\Omega) \\
V = 0$$
However, conceptually I can't see how there is no change in energy between these two points.
It is my understanding... | The key thing is that there is NO electric field within the perfect wire. So, there is no force acting on the electron, and thus no work done on it (while it's in the perfect wire).
This goes back to the definition of a perfect conductor (which the perfect wire is). Within a perfect conductor, there is no electric fie... | $F$ is not greater than 0 in an ideal wire, think 'frictionless surface' if it helps.
In this idealisation, the electrons are considered to move from 4 to 3 without effort... Therefore there is no need to invoke any energetic loss.
If this doesn't appeal then you need to drop the idealisation and consider resistivity... | https://physics.stackexchange.com |
249,950 | [
"https://softwareengineering.stackexchange.com/questions/249950",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/57792/"
] | Suppose I have distinct classes that have the same behavior, which can be represented like this:
<pre><code>public interface Behavior {
void operationA();
}
public class ImplementerA implements Behavior {
public void operationA() {
}
}
public class ImplementerB implements Behavior {
public void oper... | Violating a best practice is not an exceptional condition. If you want to declare that 01/01/0001 is an invalid value, then throw an exception for that, specifically.
Looking forward to the day when my IDE throws an <code>InvalidBraceStyleException</code>.
| A date of <code>01/01/0001</code> would usually indicate a bug. Someone probably has used a default-initialized variable.
Your intention basically is to have a runtime assert to guard against a specific class of bugs. That's a valid thing to do. In fact it is done all the time.
Consider having this validation as a la... | https://softwareengineering.stackexchange.com |
271,163 | [
"https://softwareengineering.stackexchange.com/questions/271163",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/109112/"
] | I'm not sure that this is entirely on-topic, but I'll try to make it so.
I have an online service (API) and an Android application that uses it for <em>all actions in the app</em>. At the current moment, nothing is stored locally, so I have to depend on the API being online, and would like to warn the user when it's no... | I would Check Availability on <strong>Failed</strong> Action. Just issue your calls normally and if you get a failure, then call the Service Availability function to verify.
Assume service is up, if one gets and error, then issue the additional call to verify availability.
| Just because a http-ping at time <em>t</em> gets a response, you've still got a large uncertainty that a request at time <em>t'</em> will complete.
I suggest you not ping the server if it can be avoided because it is a poor proxy measure of actual availability in the future. If a request is made and no response is rec... | https://softwareengineering.stackexchange.com |
309,308 | [
"https://physics.stackexchange.com/questions/309308",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/139371/"
] | I can state my question with an example. We say that the hydrogen atom is in the ground or excited state. But should it be the electron which is in the ground or the excited state.
Like a question states that an electron collides & excites a ground state hydrogen atom to the 3rd excited state. What does that mean?... | No it is the whole atom that is in a certain state i.e. has a certain energy.
The proton is about 2000 times heavier than the electron, so at least when first teaching students about the hydrogen atom we tend to assume the proton is fixed. That is the electron moves in a fixed Coulomb potential. If we make this assump... | I'm not sure but I think that in the nucleus itself there are energy levels so maybe the fact that the electron moves up an energy level, that affects the nucleus and raises the protons for example by an energy level, because remember if the electrons need more em force to keep them from escaping the atom because they ... | https://physics.stackexchange.com |
236,853 | [
"https://softwareengineering.stackexchange.com/questions/236853",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/52871/"
] | Is there a performance difference between using <code>++x</code> over <code>x++</code> increment operators?
I've see developers use <code>++x</code> in loops where I would normally write <code>x++</code> out of habit.
For example;
<pre><code> for(int x=0; x < 10; ++x)
</code></pre>
There would be no difference i... | For simple integers; <strong>NO</strong>, any decent peephole optimizer will see the value isn't used and use the best option available.
For more complex (overloaded) types in C++ (like iterators); possibly, the pre and post increment expand to different methods which may be completely different. Typically the post in... | No, there is no consistent performance difference between these operators that holds true across a range of languages.
In most languages I would expect them to behave identically, down to the point of generating identical code. The reason is that each produces the identical outcome, and provided the compiler writer is... | https://softwareengineering.stackexchange.com |
41,679 | [
"https://mathoverflow.net/questions/41679",
"https://mathoverflow.net",
"https://mathoverflow.net/users/9926/"
] | 3 players are playing a game where they get to pick independently without knowing the other players picks one of 2 prizes (A,B) and the payout is (a,b) for the two prizes, divided by how many people picked the specific prize.
For example, if the prizes are (3,1) and 2 people picks A and 1 person picks B, the 2 people ... | This looks like a homework exercise on mixed strategy Nash equilibria.
My ASCII art skills are a little rusty, but let's write up a "nice" litte 2x2x2 cube:
<pre>
+--------------+-----------------+
C plays 3 / / / |
+--------------+----------------+ |
C plays... | The accepted answer correctly finds one equilibrium but overlooks three others: Any two players can choose $3$ while the other chooses $2$.
| https://mathoverflow.net |
43,066 | [
"https://cs.stackexchange.com/questions/43066",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/34120/"
] | I have a problem with this exercise:
<blockquote>
Let G be the following ambiguous grammar for the λ-calculus:
<code>E → v | λv.E | EE | (E)</code>
where E is the single non-terminal symbol, λv.E represents abstraction w.r.t. the variable v in E, and EE represents application.
<ol>
<li>Define an <s... | <h2>Compatibility of left associativity and LL(1) parsing</h2>
You just hit one of the major inconsistencies in the use of
context-free (CF) syntax. People want to choose grammars so that the
parse-tree will reflect the intended structure of the sentence, close to its semantics, especially in the case
of non associat... | My attempt:
<pre><code>E -> A | λv.E
A -> FA'
A' -> A | ɛ
F -> (E) | v
</code></pre>
This grammar is LL(1) and should respect the required properties.
| https://cs.stackexchange.com |
72,636 | [
"https://physics.stackexchange.com/questions/72636",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/12980/"
] | I have a simple question: Does gravity slow down a horizontally thrown baseball?
Assuming when a baseball is thrown it has a vertical velocity as well, does it slow does the ball?
Any help is much appreciated.
| (I can't quite comment on the previous post, so I'll have to write a new answer).
If we set the curvature of the Earth to be non-negligible in our problem, yes, gravity would slow the baseball down by an extremely tiny amount, but, if we exclude this case (which, again, I stress to be many orders of magnitude below an... | No, gravitational acceleration is vertically downwards. It does not affect the horizontal velocity of any projectile.
| https://physics.stackexchange.com |
51,965 | [
"https://physics.stackexchange.com/questions/51965",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/20142/"
] | I'm puzzled about the potential energy of a spring. A spring is a conservative system. So the potential energy should be defined only up to a constant -- can be defined to be 0 anywhere. However, one cannot escape needing the un-stretched length of the spring.
For example, the difference in gravitational potential ... | The potential energy only being defined up to a constant does not imply that potential energy differences only depend on differences in position.
To see this mathematically, assume that a function $U$ has the property that
$U(x_2)-U(x_1) = f(x_2-x_1)$
for some function $f$. Then if we take $x_2 = x+\Delta x$ and ... | Choosing a different $x_0$ is not choosing a different $U_0$ - the force changes. Your $\Delta U$ is a work done by the elastic force. It, of course, depends on the force initial value.
| https://physics.stackexchange.com |
1,110,101 | [
"https://math.stackexchange.com/questions/1110101",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/294365/"
] | Assume that $\sum a_n$ is convergent and $\sum b_n$ is absolutely convergent.To show that $\sum a_nb_n$ is absolutely convergent
My try::Consider the sequence of partial sums of $\sum |a_nb_n|$
$S_n=|a_1b_1|+|a_2b_2|+......+|a_nb_n|$
Now $(a_1b_1+a_2b_2+......+a_nb_n)\leq ( (a_1^2+a_2^2+..+a_n^2))^\frac{1}{2}((b_1^... | Hint: for large enough $n$, $|a_n|\leq 1$, say for $n\geq N$... So $$\sum_{n=0}^{N+p}|a_nb_n|\leq \underbrace{\sum_{n=0}^{N-1}|a_nb_n|}_{C}+\sum_{n=N}^{N+p}|b_n|\leq C+\sum_{n=0}^{+\infty}|b_n|\,.$$
| Hint: The sequence $(a_n)$ converges to $0$ and is bounded. Show that partial sums of $\sum a_nb_n$ form a Cauchy sequence.
| https://math.stackexchange.com |
2,336,602 | [
"https://math.stackexchange.com/questions/2336602",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/403569/"
] | Let $\psi: V \rightarrow V$ be a linear transformation and let $\psi^n$ be n- <em>th</em> power of $\psi$.<br>
Let's consider a polynomial $P = a_nx^n + ...+a_0$ and let's define $P(\psi) = a_n\psi^n+...+a_1\psi+a_0$<br>
The task is to decide whether $P(\psi)$ is a linear transformation.<br>
What I know is that $\psi^n... | First prove that
<blockquote>
If $A$ and $B$ are linear transformations(with same domain and range), then $A+B$ is also a linear transformation.
</blockquote>
Then prove that
<blockquote>
For a constant $c$, $cA$ is also a linear transformation.
</blockquote>
Once you know these two, can you answer your questi... | The set of linear maps $V\to W$ is always a vector space (for the obvious notions of addition and scalar multiplication), so it is closed under forming linear combinations. If moreover $W=V$, then the space is closed under composition as well, so when it contains$~\psi$ it also contains all its powers$~\psi^k$ for $k\i... | https://math.stackexchange.com |
738,744 | [
"https://physics.stackexchange.com/questions/738744",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/209383/"
] | The dualities described the theories under certain transformations to be equivalent, i.e. <span class="math-container">$T$</span>-duality described the equivalence between theories with <span class="math-container">$R$</span> and <span class="math-container">$\frac{1}{R}$</span>. However, this looked very much like a s... | I don't think you've gotten a full answer yet. John is right that dualities, trialities, etc would not give rise to conserved charges <em>even if they were symmetries</em> because <span class="math-container">$\mathbb{Z}_2$</span>, <span class="math-container">$\mathbb{Z}_3$</span>, etc are discrete. But there are cont... | If I remember it right, Noether's theorem is about continuous symmetries. Duality is in general an integer one, so no conserved charges are expected, at least at the level of most literal reading of Noether's statement.
| https://physics.stackexchange.com |
21,197 | [
"https://dba.stackexchange.com/questions/21197",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/6390/"
] | The BOOLEAN datatype in Access (ie, Jet/ACE) returns a -1 for True and 0 for False; the field is always required (i.e., it cannot be set to NULL). The SQL Server BIT type returns 1 for True and 0 for False and also allows NULLs.
I'm looking for a T-SQL column definition using a combination of smallint, check constrai... | To address your concerns about BIT:
<ol>
<li>You can set your BIT column to NOT NULL.</li>
<li>You can use -1 when <em>setting</em> a BIT column to "true".</li>
<li>You can create a view that translates to -1, but +1 should be fine anyway unless your application explicitly checks for the numeric -1 (anything but zero ... | This will do it:
<pre><code>MyField smallint NOT NULL CONSTRAINT MyField_YesNo CHECK (MyField = 0 OR MyField = -1)
</code></pre>
| https://dba.stackexchange.com |
82,099 | [
"https://softwareengineering.stackexchange.com/questions/82099",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/27181/"
] | My team leader has this terrible habit of mucking with the database schema, and making changes that would cause severe breakage on the code base (without really consulting me on how the changes would affect the code base).
Normally, I would just live with it, but we have a deadline in 2 weeks and this has been happen... | "Deadline is in two weeks; we need to freeze the schema if we're going to hit it."
| You are going to have to be more forceful and make sure that soon (as in yesterday, the day before yesterday, or last month) you settle on a schema and move forward. There's no sensible way that you can continue to develop an app with a database that is a moving target.
| https://softwareengineering.stackexchange.com |
247,034 | [
"https://mathoverflow.net/questions/247034",
"https://mathoverflow.net",
"https://mathoverflow.net/users/10898/"
] | Let $d$ be a positive, non-square integer, and define $c_d$ to be the smallest positive number with the following property: for all pairs of co-prime integers $(p,q)$ with $q > 0$, the inequality
$$\displaystyle \left \lvert \frac{p}{q} - \sqrt{d} \right \rvert > \frac{c_d}{q^2}$$
holds. The existence of such a... | Edit: I answered a different question than the one asked. The question asks for the least $c_d = q^2|\sqrt{d}-p/q|$. I evaluated the liminf.
If $d = a^2+b$ with $1\le b \le 2a$ then the simple continued fraction for $\sqrt{d}$ is preperiodic, and the period ends with $2a$, which is the largest coefficient, with $a$ be... | This feels like the kind of thing that's easier to figure out than find in the literature. Seems that the optimal $c_d$ comes from the minimal positive solution of $p^2 - d \, q^2 = +1$, and is thus less than $\frac1{2\sqrt d}$ but very close to $\frac1{2\sqrt d}$ if the fundamental solution is at all large.
| https://mathoverflow.net |
370,464 | [
"https://mathoverflow.net/questions/370464",
"https://mathoverflow.net",
"https://mathoverflow.net/users/10366/"
] | Let <span class="math-container">$p$</span> be a prime, and let <span class="math-container">$\mathcal{U}$</span> be a family of finite <span class="math-container">$p$</span>-groups such that
<ol>
<li>Any group isomorphic to a group in <span class="math-container">$\mathcal{U}$</span> is also in <span class="math-cont... | No. Fix an odd prime <span class="math-container">$p$</span>. Let <span class="math-container">$H_p$</span> be a non-abelian group of order <span class="math-container">$p^3$</span> and exponent <span class="math-container">$p$</span> (this is unique to isomorphism).
Let <span class="math-container">$\mathcal{C}_p$</sp... | This is more or less the same as YCor's answer but perhaps more elementary. Let <span class="math-container">$p$</span> be an odd prime, and let <span class="math-container">$\mathcal{U}$</span> be the class of <span class="math-container">$p$</span>-groups in which all elements of order <span class="math-container">$... | https://mathoverflow.net |
665,839 | [
"https://physics.stackexchange.com/questions/665839",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/313803/"
] | So I just started studying physics recently, and I came up with a question that I'm not sure how to solve, and I think it may be because of some fundamental misunderstanding. I don't want anyone to answer the problem, I'm just providing it for context.
The problem: Let's say a man is a distance <strong>h</strong> from ... | The reason why you are confused is that you are muddling a number of concepts.
Yes, gravity is the only force acting on the man while he is falling, so a force equal and opposite to gravity is sufficient to counteract the force of gravity. That happens when you are standing on the ground, for example, when the reaction... | The conceptual problem is balancing forces, when it requires balancing momenta. Of course:
<span class="math-container">$$ F = \frac{dp}{dt} $$</span>
The gravitational force is indeed constant, so after falling for a time <span class="math-container">$t$</span>:
<span class="math-container">$$ p = \int_0^t F(t')dt'=F\... | https://physics.stackexchange.com |
62,642 | [
"https://mathoverflow.net/questions/62642",
"https://mathoverflow.net",
"https://mathoverflow.net/users/14354/"
] | Does any one know why $d_3: H^* (X, K^0(point))\rightarrow H^{*+3}(X,K^0(point))$ is actually extended $Sq^3$ to $\mathbb{Z} $ coefficient.
| This follows from the following considerations:
<ol>
<li>This differential in the Atiyah-Hirzebruch spectral sequence must be a stable cohomology operation for general nonsense reasons (the first nonvanishing differential always is, no matter what the generalized cohomology theory is).</li>
<li>There are exactly two s... | Okay, I can't pass up the chance to try and be more industrious than Tyler (this is really a comment on Tyler's answer).
I'll try to explain why there's a non-zero $d_3$ differential in the AHSS for $\mathbb{R}P^2 \times \mathbb{R}P^4$.
The K-theory of $\mathbb{R}P^{2k}$ is $\mathbb{Z} \oplus \mathbb{Z}/2^k$ in dim... | https://mathoverflow.net |
330,996 | [
"https://softwareengineering.stackexchange.com/questions/330996",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/196423/"
] | When writing something that creates many (1000s) of small objects often, should you try to minimize it for performance? Especially if you don't know what system it will be run on, from low to high end desktops or even mobile. For mobile, I heard that creating a lot of objects hinders the performance a good bit, though ... | In general, <strong>no</strong>, you shouldn't avoid creating objects for fear of performance loss. There are several reasons for this.
<ol>
<li>Using objects is kind of the point of using Java. Avoiding them preemptively is a sign that the decision to use Java might not have been the right one to begin with.</li>
<li... | When contemplating impacts on performance before you've written code you should assume you don't know what you're doing.
There is a very small space overhead for objects in java versus primitives. The tinier your objects are, the larger the percentage of overhead will be. However, I learned long ago that things that... | https://softwareengineering.stackexchange.com |
26,629 | [
"https://dsp.stackexchange.com/questions/26629",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/17944/"
] | I need to design a filter able to take a measurement of alternate current and shift the phase of the input by 90°.
The filter should work between 40 and 60Hz and at this range should have unity gain. Outside this range I don`t care about the gain or phase.
I research about Hilbert filter and sounds that this kind ... | @Rodrigo PG: Your Matlab Hilbert transformer is causal. This means, in order to synchronize-in-time your $y$ sequence with your $yf$ sequence, you need to delay your $y$ sequence by (length of $b$ minus 1)/2 samples when the length of $b$ is odd. This will delay your input $y$ by the same amount as the Hilbert transfor... | Hilbert Transformers are non-causal, i.e. they need to be delayed to be implementable. So you get the 90 degree phase-shift plus a bulk delay of 20 samples (half the filter length).
You see the 90 degree phase shift if you delay the original signal by 20 samples as well.
EDIT for Part 2:
Your lower bandpass cutoff ... | https://dsp.stackexchange.com |
1,662,398 | [
"https://math.stackexchange.com/questions/1662398",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/160288/"
] | I am currently studying for my upcoming midterm and I am stumped on this example provided in the slides. Basically here is the question:
<blockquote>
Given 35 computers, what is the probability that more than 10 computers are in use(active)? We are told that each computer is only active 10% of the time. The answer g... | As you have correctly noted, the probability of exactly 10 being active at any one time is
$${35 \choose 10} \cdot (0.10)^{10} \cdot (0.90)^{25}.$$
More generally, the probability of exactly $k$ being active at any time is
$${35 \choose k} \cdot (0.10)^{k} \cdot (0.90)^{35-k}.$$
Since you want the probability of more t... | The expression you have subtracted from $1$ is the correct computation of the probability that exactly $10$ computers are active. The question asks for the probability that more than $10$ are active, so you should sum the chances that $11, 12, 13, \dots$ are active. As you only expect $3.5$ to be active, these will dec... | https://math.stackexchange.com |
596,666 | [
"https://physics.stackexchange.com/questions/596666",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/280881/"
] | Light is an electromagnetic wave composed of electric and magnetic components.
I recently read that the velocity of light is in the direction <span class="math-container">$\mathbf{E}\times\mathbf{B}$</span> where <span class="math-container">$\mathbf{E}$</span> and <span class="math-container">$\mathbf{B}$</span> are t... | In the inertial frame (not the accelerating frame):
<ul>
<li>When standing in a bus on roller skates, when the bus brakes you will move forwards. As you described yourself, it seems as though you are being pushed forward with some non-existing centrifugal "force". But in reality there is no such force - it is... | <blockquote>
The first point seems reasonable when describing the pseudoforce on
the upper part of our body in a bus when the bus suddenly stops or
starts.
However, when describing pseudoforces like centrifugal force, it makes
no sense to me. Can anyone explain how is it so?
</blockquote>
The first example, where the b... | https://physics.stackexchange.com |
1,710,523 | [
"https://math.stackexchange.com/questions/1710523",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/266798/"
] | Im asked to find the arc length of :
$$
\int_{-2}^{x}\sqrt{3w^4-1}dw
$$
where x is between -2 and -1.
Do I find the integral just as I would normally find it then find the arc length of that? Im a little confused on the notation I guess.
| The function $$f(x)=\int_{-2}^x\sqrt{3w^4-1}\,dw$$ is a curve in the plane. The formula for arc-length of a curve is $$L=\int_a^b\sqrt{(f'(x))^2+1}\,dx.$$ The fundamental theorem of calculus tells us that $$f'(x)=\sqrt{2x^4-1}.$$ Therefore, you need to solve the integral $$L=\int_{-2}^{-1}\sqrt{(\sqrt{3x^4-1})^2+1}\,dx... | Use the formula for the arc length:
$$\int_{-2}^{-1}\sqrt{1+\frac{\partial}{\partial x}\left[\int_{-2}^{x}\sqrt{3w^4-1}\space\text{d}w\right]^2}\space\text{d}x=$$
<hr>
Using the fundamental theorem of calculus:
$$\frac{\partial}{\partial x}\left[\int_{-2}^{x}\sqrt{3w^4-1}\space\text{d}w\right]=\frac{\sqrt{-\left(3x... | https://math.stackexchange.com |
86,437 | [
"https://stats.stackexchange.com/questions/86437",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/40183/"
] | I'm doing my dissertation in Accounting and I developed a questionnaire to test certain hypotheses on the relationship between independent variables and a dependent variable. All were measured on a 5-point Likert scale. I am using SPSS to analyse the data. The variables are all non-normal (I used Kolmogorov-Smirnov tes... | This can't be answered by us, only by you. But I can give you the information you need to make the decision.
In correlation (whether Spearman or Pearson) we are looking at the relationship between two variables. Neither is dependent or independent - they are treated equally.
In regression (of any sort) one variable ... | To report the relashionship between variables, <strong>you may prefer a scatterplot</strong>. As part of the Royal Society of Chemistry's Analytical Methods Committee, Prof Brian Ripley used to write short expository notes for ANALYST. The note entitle "Uses (Proper and Improper) of Correlation Coefficients", which app... | https://stats.stackexchange.com |
3,759,154 | [
"https://math.stackexchange.com/questions/3759154",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/709703/"
] | I start by trying to see the homomorphism between <span class="math-container">$\mathbb{Z}[x]$</span> and <span class="math-container">$\mathbb{Z_2}[x]$</span>.I define the homomorphism by <span class="math-container">$\phi:(ax^i)=a(mod 2)x^i$</span>.It is trivial to see that it a homomorphism. The kernel of the homomo... | You don't need completeness of <span class="math-container">$X$</span>. This is true in normed spaces. Your last idea is a good one:
We may assume <span class="math-container">$x \neq 0$</span>.
Define the functional
<span class="math-container">$$\varphi: \Bbb{C}x \to \Bbb{C}: \lambda x \mapsto \lambda \Vert x \Vert$$... | Yes, this is a perfect application. You can start by defining a functional on the one-dimensional space spanned by <span class="math-container">$x$</span> as <span class="math-container">$\phi(ax)=a\|x\|$</span> which has norm <span class="math-container">$=1$</span> and extend using HB.
| https://math.stackexchange.com |
359,660 | [
"https://stats.stackexchange.com/questions/359660",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/216093/"
] | So I have the players "fantasy score" stat. This stat is the sum of their ingame stats multiplied by preset weights. These weights are from the video game DOTA 2's official fantasy points weights. For example:
<pre><code>Points for kills are calculated by the number of kills *.3
Points for deaths are calculated by st... | Since you’ve clarified in comments that you’re trying to construct intuitive measures, I think it’s worth considering what would be required to construct a bounded scale for unbounded inputs using this data.
You state in the question that the scores are unbounded. So we need a scheme to map real numbers to (0,10). Th... | You should try to determine the distribution of the Dota 2 scores. Once you know the distribution you can use the quantile function for the distribution to see where the values lie on a 0 to 1 scale. From that 0 to 1 scale you should be able to convert them to 1 to 10. You can use the scipy package in python to compute... | https://stats.stackexchange.com |
888,021 | [
"https://math.stackexchange.com/questions/888021",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/35855/"
] | Suppose that $u=\lambda id-s$, as $s$ is a linear map as ($s\circ s = id)$ and $\lambda \in \mathbb{R} \setminus\{-1,1\}$.
How can prove that $u$ is invertible and find the expression of $u^{-1}$ (the inverse of u)?
| To approach this and related problems it can make sense to start with formal manipulations (the validity of which is established at the end, or possibly this can even be circumvented).
To simplify notation I set $c = \lambda^{-1}$ and $v= c u = id - cs$.
You like know that the inverse of $1 - q = \sum_{n=0}^{\inf... | $u$ is a linear map, and as such is invertible if and only if it has a trivial kernel. Take a look at $x\in \ker(u)$, so $u(x) = 0$. Since $u(x) = \lambda x - s(x)$, this means that $$s(x) = \lambda x$$
But, from $s(x) = \lambda x$, it follows that $s(s(x))= s(\lambda x)$, meaning that (since $s$ is linear) $$s(s(x)) ... | https://math.stackexchange.com |
83,324 | [
"https://mathoverflow.net/questions/83324",
"https://mathoverflow.net",
"https://mathoverflow.net/users/12597/"
] | Hi MathOverflow,
I'm not sure if it makes sense to ask this question in the general setting, but:
Are there any necessary conditions for a function, such that if $N$ is a not <em>Lebesgue</em> measurable, $f(N)$ is <em>Lebesgue</em> measurable?
I am working on a problem, which seems to suggest that there are no 'tri... | My guess is that the characterization is the following:
<blockquote>
A function $f$ maps every non-measurable set into a measurable set if and only if the domain or the image of $f$ has measure zero.
</blockquote>
One direction is trivial. For the other direction assume that the image of $f$ is positive. Take a no... | Suppose $A \subset I = [0,1]$ is Lebegue non-measurable, $B \subseteq I$ Lebesgue measurable, and $f: I \to I$ is a measurable function with $A = f^{-1}(B)$. By inner regularity, $B$ is the disjoint union of sets $C$ and $D$ where $C$ is an $F_\sigma$ and $D$ has measure 0. Then $A$ is the disjoint union of $f^{-1}(C... | https://mathoverflow.net |
109,031 | [
"https://physics.stackexchange.com/questions/109031",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/37677/"
] | What I am really asking is are there other functions that, like $\sin()$ and $\cos()$ are <strong>bounded</strong> from above and below, and <strong>periodic</strong>?
If there are, why are they never used to describe oscillations in Physics?
EDIT:
Actually I have just thought of a <em>cycloid</em>, which indeed is ... | Part of it is that Newtonian mechanics is described in terms of <em>calculus</em>.
When we consider vibrational motions, we're talking about some particle that tends to not be displaced from some equilibrium position. That is, the force on the particle, at displacement $x$, $F(x)$, is equal to some function of displac... | One of the big reasons not discussed above is Fourier theory -- any function $f(x)$ can be expressed in the form $f(x) = \int dk\, A(k)e^{ikx}$, which basically means that any function can be decomposed into an infinite sum of sines and cosines. Since this is the case and dealing with sine and cosine is mathematically... | https://physics.stackexchange.com |
219,543 | [
"https://softwareengineering.stackexchange.com/questions/219543",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/7020/"
] | Should a class know about its subclasses? Should a class do something that is specific for a given subclass for instance?
My instincts tells me that is a bad design, it seems like an anti-pattern of some sort.
| The answer implied by the concept of classes is "no".
Either whatever action, data or relation you're handling is part of all subclasses - then it should be handled in the superclass without checking the actual type. Or it applies only to some subclasses - then you'd have to perform run-time type checks to do the rig... | Not only <em>should</em> it not know, it simply <em>can't</em>! Usually, a class can be extended anytime, anywhere. It can be extended by classes that didn't even <em>exist</em> when it was written.
Some languages allow extending classes to be controlled by the superclass. In Scala, a class can be marked as <code>seal... | https://softwareengineering.stackexchange.com |
62,327 | [
"https://electronics.stackexchange.com/questions/62327",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/19817/"
] | <img src="https://i.stack.imgur.com/J4zbC.png" alt="this is the schematics">
I did this circuit in my protoboard, and it worked for a while, but the quality sound was terrible, and suddenly it stoped working. Can anyone explain me in which part of the circuit i did something wrong? any sugestions to improve the sound ... | One obvious problem is that you forgot to power the amp. Perhaps because of the double transistor gain stage, a little signal was making it onto the base of the first transistor. Connect the 9 V supply to pin 6, and put a 100 nF to 1 µF ceramic bypass cap immediately accross the chip.
Depending on the micropho... | Your circuit is DC-biased. Some current is always flowing through the speaker all the time. Let's figure out how much.
The datasheet claims "Self-centering output quiescent voltage" so let's assume the output DC voltage is 4.5 volts.
Now there will be two Si diode drops in the two output transistors. Therefore the... | https://electronics.stackexchange.com |
3,854,022 | [
"https://math.stackexchange.com/questions/3854022",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/833150/"
] | I am wondering how <span class="math-container">$\lim_{x \to 0} f(x) = 0$</span> but <span class="math-container">$\lim_{x \to 0} f(f(x)) = 1$</span> is possible. Since <span class="math-container">$\lim_{x \to 0} f(f(x))$</span> = <span class="math-container">$f(\lim_{x \to 0} f(x)) = \lim_{x \to 0} f(x) = 0$</span> b... | Take <span class="math-container">$$f(x) = \begin{cases}0 & x\neq 0 \\
1 & x=0\end{cases} $$</span>
Then <span class="math-container">$\lim_{x \to 0} f(x) = 0$</span> since in any neighbourhood of <span class="math-container">$0$</span> the function is <span class="math-container">$0$</span>. But note that <spa... | Let consider
<ul>
<li><span class="math-container">$f(x)=0$</span> for <span class="math-container">$x\neq 0$</span></li>
<li><span class="math-container">$f(0)=1$</span> for <span class="math-container">$x= 0$</span></li>
</ul>
then
<span class="math-container">$$f(x)\to 0$$</span>
<span class="math-container">$$f(f(x... | https://math.stackexchange.com |
175,881 | [
"https://electronics.stackexchange.com/questions/175881",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/46551/"
] | Currently I'm working on a C8051F120 MCU where external interrupts can be defined in two ways:
<ol>
<li>Edge sensitive (falling)</li>
<li>Level sensitive (low-level)</li>
</ol>
In level-sensitive interrupts as soon as the MCU detects a low level at the external pin it will execute the ISR which is the same as detecti... | Its exactlly what is says.
If edge interrupt is set, the ISR will only get fired on falling/rising edge of a pulse. While if level sensitive interrupt (as you say) is set the ISR will get fired everytime there is a low-level/high-level signal on the corresponding pin.
In short, edge interrupt gets fired only on chang... | A level sensitive interrupt and an edge sensitive interrupt are actually two quite different things. I'll try to give some general insights that might help you understand how other interrupts work too.
Let's assume that your CPU can execute code in two modes: normal mode, and interrupted mode. To go from normal mode t... | https://electronics.stackexchange.com |
3,169,668 | [
"https://math.stackexchange.com/questions/3169668",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/495992/"
] | If A and B don't commute are there counterexamples that AB is diagonalizable but BA not?
I read that if AB=BA then both AB and BA are diagonalizable.
| Try <span class="math-container">$A = \begin{bmatrix}0&0\\1&0\end{bmatrix}$</span> and <span class="math-container">$B = \begin{bmatrix}0&0\\0&1\end{bmatrix}$</span>
| This is true in the special case when both <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are invertible.
If we denote by <span class="math-container">$E_C(\lambda)$</span> the eigenspace of the matrix <span class="math-container">$C$</span> relative to the eigenvalue <span class="... | https://math.stackexchange.com |
2,063,865 | [
"https://math.stackexchange.com/questions/2063865",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/115030/"
] | Hi I know how to derive the following result below,
\begin{eqnarray}
\frac{\partial}{\partial X} Tr (X^\top A X B)= A^\top X B^\top +AXB\\
\frac{\partial}{\partial X^\top} Tr (X^\top A X B)=\left( A^\top X B^\top +AXB\right)^\top
\end{eqnarray}
where $X^\top$ is the real transpose of $X$.
However I am trying to d... | Consider the function
$$\eqalign{
f(Y) &= \operatorname{tr}(AYAY) = A^T:YAY \cr
}$$where colon denotes the Frobenius (aka double-dot) product.
Let's find the differential and gradient of this function
$$\eqalign{
df &= A^T:(dY\,AY+YA\,dY) \cr
&= 2\,A^TY^TA^T:dY \cr
\cr
\frac{\partial f}{\partial Y} &a... | Write it down in terms of components, you get
$$\frac{\partial}{\partial \bar{X}_{ij}} \mathop{\rm tr}(X X^H X X^H)
=\frac{\partial}{\partial \bar{X}_{ij}} ( X_{kl} \bar{X}_{ml} X_{mn} \bar{X}_{kn}), $$
where for some components you need to compute $\frac{d\bar{z}}{dz}$ which doesn't exist, since the function $z\mapsto... | https://math.stackexchange.com |
140,872 | [
"https://mathoverflow.net/questions/140872",
"https://mathoverflow.net",
"https://mathoverflow.net/users/32741/"
] | I have just started to read about operads, so this question might be silly.
So it seems to me that any "reasonable" class of algebras can actually be defined as a class of all algebras over a certain operad. For example, associative algebras are algebras over the associative operad $\mathcal{As}$, commutative algebras... | One example is a bialgebra. An operad only has operations taking several inputs and one output, so there is no room for comultiplication. On the other hand this is not a serious obstacle. A small modification of the defining axioms will accommodate also this example: bialgebras are algebras over a properad, rather than... | <strong>Self-distributive laws</strong>, defined by
$x(yz)=(xy)(xz)$
are an example of such a kind of algebras. They are useful in the study of braids and large cardinals.
Here the obstruction is that one need to duplicate the variable $x$ to write the axiom.
| https://mathoverflow.net |
54,323 | [
"https://physics.stackexchange.com/questions/54323",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/17374/"
] | Could someone explain to me what the collision of two photons would look like? Will they behave like,
<ol>
<li>Electromagnetic waves: they will interfere with each other and keep their wave nature</li>
<li>Particles: they will bounce like classical balls</li>
</ol>
I assume that energy of that system is too small to ... | Your assumption that pair production is ruled out, rules out* that two photons interact through higher-order processes. Quantum electrodynamics tells us that two photons cannot couple directly. That leaves us with classical electromagnetism, which tells us that electromagnetic waves pass through each other <em>without<... | A lowest order QED Feynman diagram for the process photon + photon $\rightarrow$ electron + positron looks like shown below (the time axis is the horizontal axis).
From the point of view of energy conservation, this process is only possible if sum of the energy of the photons is above twice the electron mass. In the c... | https://physics.stackexchange.com |
31,008 | [
"https://math.stackexchange.com/questions/31008",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/9122/"
] | This is a beginner question but bear with me, let's say i have thrown a dice <code>5 times</code>, and every time the result was <code>1</code>. The next time i throw it what are the chances/probabilities it will be 1 again? Is it still 1/6? If not how is this calculated?
What i am trying to understand is whether the ... | If you threw a fair dice $5$ times and got a $1$ every single time, then the event comprised of the $5$ throws together has a probability of $\frac{1}{6^5}$. The odds that the next throw results in a $1$ is $\frac{1}{6}$, because the events are independent.
In the case of independent events, it doesn't matter what has... | This is not a mathematical question. Mathematically speaking, independence is an assumption. That is, we can model the experiment you describe mathematically by several independent events, but nothing mathematically forces us to model the events as independent; the motivation for that comes from extra-mathematical sour... | https://math.stackexchange.com |
617,471 | [
"https://physics.stackexchange.com/questions/617471",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/99217/"
] | In the book of Edwards <em>The Theory of Polymer Dynamics</em>, at page 24 it is given that
<blockquote>
2.5 Excluded volume effect
In the models of polymers considered in the previous sections, the interaction among the polymer segments is
limited to within a few neighbours along the chain. In reality,
however, segmen... | The confusion stems from the fact that one commonly wants to consider what are called passive transformations as opposed to active transformations.
The idea can be seen in three dimensions and then generalized from there.
One should not write <span class="math-container">${\bf u} = u^a$</span>, it is an abuse of notati... | I agree that the wikipedia article is confusing. <span class="math-container">$A^\mu$</span> represents the components of the vector <span class="math-container">$\mathbf{A}$</span> with respect to the basis vectors.
| https://physics.stackexchange.com |
284,294 | [
"https://dba.stackexchange.com/questions/284294",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/103249/"
] | I had a query that was being blocked. Ran sp_whoisactive @ get_locks=1 and identified the blocking query and it's owner. The result indicated the table had been locked for over four hours. The developer stated it was a stored procedure that she executed and it completed in 21 seconds. She still had the SSMS window open... | <blockquote>
Is this standard behavior?
</blockquote>
No. There was either a transaction started before the stored procedure ran, or the session was running with SET IMPLICIT_TRANSACTIONS ON, which can be set to default for SSMS query windows.
In any case if the session was blocking with no query running, it had an op... | What David said, and also, when the window was closed the connection would have terminated and the open transaction would <strong>roll back</strong>. Any work done by that SP, all rows written by it, will be lost.
| https://dba.stackexchange.com |
532,351 | [
"https://physics.stackexchange.com/questions/532351",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/84729/"
] | A coordinate system is often thought of as labelling points in space as if a perpendicular were to be dropped from that point to each of the coordinate axes, and the values on each of the axes read of and recorded in a tuple <span class="math-container">$(x,y,z)$</span>. Consequently, the coordinates are interpreted as... | I liked the discussion in "A Geometric Approach to Differential Forms" by David Bachman, and "Tensors, Differential Forms And Variational Principles" Lovelock & Rund.
Firstly, vector spaces do not need coordinate systems. Vector spaces can be defined in abstract terms very well. A great book here is Halmos "Finite... | <blockquote>
Can coordinate systems exist without a vector space?
</blockquote>
Yes. For example, there is a coordinate system on the surface of the Earth called latitude and longitude. But the surface of a sphere is not a vector space.
<blockquote>
How can we express this mathematically?
</blockquote>
In the ca... | https://physics.stackexchange.com |
193,522 | [
"https://physics.stackexchange.com/questions/193522",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/59393/"
] | Why is ice more reflective (has higher albedo) than liquid water? They're both the same substance (water). Is something quantum mechanical involved?
| In fact ice is slightly less reflective than water. The reflectivity is related to the refractive index (in a rather complicated way) and the refractive index of ice is 1.31 while the refractive index of water is 1.33. The slightly lower refractive index of ice will cause a slightly lower reflectivity. In both cases th... | I think it is because of the fact that snow is white. We all know that black colour absorb most of the radiation, while white colour reflect most of the radiation.
I don't think there is something related yo quantum mechanics. In order to answer the question of subjects like physics and chemistry, we should first try t... | https://physics.stackexchange.com |
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