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683,931 | [
"https://physics.stackexchange.com/questions/683931",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/323182/"
] | We know that voltage is nothing but the energy hold by <span class="math-container">$1\,\text C$</span> of charge and <span class="math-container">$1\,\text C=6.25\cdot10^{18}$</span> electrons, so does that mean whatever the value of potential is, that will always be holded by this much electrons only?
If this is tru... | The potential difference between two points is the work done per unit charge when we move a charge between those points i.e. it is the work divided by the charge.
So suppose we move a charge <span class="math-container">$Q$</span> and the work done is <span class="math-container">$W$</span>, then the potential differen... | A current of 4 amps indicates the 4 coulombs of charge are transferred each second. Each of those dissipates 20 joules of energy in the resistor. What's the problem?
| https://physics.stackexchange.com |
266,501 | [
"https://physics.stackexchange.com/questions/266501",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/85745/"
] | Can a black hole, in principle, fall into another black hole's innermost stable circular orbit?
| The innermost stable orbit is a theoretical concept that applies to a "test particle" orbiting a black hole. A test particle in this case would mean that it has negligible mass and so does not alter the metric (Schwarzschild, or more realistically Kerr) it is in.
If an orbiting black hole were <em>much</em> less massi... | On top of Rob Jeffries' answer, I'd like to add that gravitational wave radiation makes the black hole lose orbital energy $E$. But for really huge black holes this won't make much of a difference ( see the calculation).
<strong>Very rough estimates for the energy:</strong>
Let's take an order of magnitude estimate:
... | https://physics.stackexchange.com |
50,267 | [
"https://cs.stackexchange.com/questions/50267",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/22548/"
] | In class, we were given the following problem:
<blockquote>
We are given a directed graph G = (V, E) on which each edge (u, v) ∈ E has an associated
value r(u, v) which is a real number in the range 0 ≤ r(u, v) ≤ 1 that represents the reliability
of a communication channel from vertex u to vertex v. We interpret... | Your algorithm makes the wrong choice between the following two paths:
<ol>
<li>5 channels with a reliability of 50% (combined reliability 3.125%), weight $5 \cdot {1 \over 0.50} = 10$.</li>
<li>A single channel with a reliability of 8%, weight ${1 \over 0.08} = 12.5$.</li>
</ol>
| "Any function that maps smaller fractions to bigger numbers can be used" - This is correct but not what you are doing. What you are assuming is, for positive numbers <i>a, b, c</i> and <i>d</i>:
\begin{equation}
a + b > c + d \implies \frac{a + b}{a \times b} < \frac{c + d}{c \times d}
\end{equation}
Which is eqi... | https://cs.stackexchange.com |
4,452 | [
"https://mechanics.stackexchange.com/questions/4452",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/2274/"
] | It it worth using a motor oil conditioner? Do they actually help keep an engine running better, or are they just marketing ploys?
| Just use the manufacturers reccomended oil, oil additives for the most part are gimmicks. If you really want to protect your engine use a quality synthetic like Mobil 1 or the like. Modern engines have more consistent tolerances and can even run oil for longer periods of time. Stick to manufacturer recommendations, ... | I don't know about automotive oils specifically, but there are aviation oil additives that have proven benefits. So, it's certainly possible that an additive could have benefit. I'm always very suspicious of them though.
| https://mechanics.stackexchange.com |
218,096 | [
"https://softwareengineering.stackexchange.com/questions/218096",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/62387/"
] | I need to write some algorithms for a PIC micro controller. AFAIK, the official tools support either assembler or a subset of C.
My goal is to write the algorithms in a generic and reusable way without losing any runtime or memory performance. And if possible, I would like to do this without increasing the development... | Unless you have at least 3 use-cases where the algorithms <em>will</em> be used with different types and sizes, you are not going to do good job of the genericity anyway. So don't bother too much.
I'd recommend writing it in plain C using custom <code>typedefs</code> for all types (except things that are obviously <co... | The best way to handle this is to actually write your algorithms in a language independent pseudo code. Then, you can create a reference implementation in a language of choice, like C, carefully noting the edge cases that make this implementation tied to a particular memory model, data type model, byte sizes, bit hand... | https://softwareengineering.stackexchange.com |
498,300 | [
"https://electronics.stackexchange.com/questions/498300",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/251807/"
] | I've been attempting to recreate the effect of an old NTSC television broadcast lately with the use of DVDs. The DVD-Video spec allows for up to 720 x 480 pixels at a display rate of 29.97 frames per second, interlaced.
My DVD player is outputting composite video to an RF modulator which is then connected to my TV and... | That would depend on the specific modulator, but RF signal output is just basically the same composite video it gets but modulated to RF.
So the RF output will be NTSC-M if the baseband CVBS that is sent to modulator is also NTSC-M compliant, and that the modulator modulates it in NTSC-M compliant way.
Some sources c... | The DVD player composite encoder will low-pass filter the 720x480 YCbCr digital picture prior to encoding in NTSC or PAL. This is to comply with the spec for baseband composite, allowing the video signal to fit in the allotted RF bandwidth prior to modulation.
If the composite video exceeds that allotted 4.2MHz bandwi... | https://electronics.stackexchange.com |
24,266 | [
"https://biology.stackexchange.com/questions/24266",
"https://biology.stackexchange.com",
"https://biology.stackexchange.com/users/10318/"
] | T-cells are important for efficient vaccination since they have:
<ul>
<li>A. T-cell receptor (TCR)</li>
<li>B. CD4+</li>
<li>C. CD8+</li>
<li>D. Memory</li>
</ul>
My gut is telling me it's A since the TCR would recognize the antigens but I want to make sure. Thanks!
| vaccination is injecting the low or non-toxic substances that having a pathogen's identifying characteristics into human or animal body. When the animals exposed to the non-injurious pathogens, the immune system will produce a protective material, such as immune hormones, physiologically active substances and specific ... | I think T cell when presented to antigen (here vaccine) will produce memory T cell, which on next encounter with the same organism may act much faster and efficiently .so i think the correct answer is option D
| https://biology.stackexchange.com |
312,679 | [
"https://mathoverflow.net/questions/312679",
"https://mathoverflow.net",
"https://mathoverflow.net/users/130047/"
] | Find the number of ways of selecting k cells from a <span class="math-container">$(2\times n)$</span>-board such that no two selected cells share a side (non-adjacent).
For <span class="math-container">$n=3$</span> and <span class="math-container">$k=2$</span>, the answer is <span class="math-container">$8$</span>; f... | <span class="math-container">$\dots$</span> continued from above: to find the coefficient of <span class="math-container">$t^nq^k$</span>, rewrite the generating function as
<span class="math-container">\begin{align}G(t,q):=\frac{t+(2t+t^2)q}{1-t-(t+t^2)q}
&=\frac{t+(2t+t^2)q}{1-t}\cdot\frac1{1-\left(\frac{t+t^2}{1... | Let <span class="math-container">$P_n(q)$</span> be the polynomial such that the coefficient of <span class="math-container">$q^k$</span> is the number of <span class="math-container">$2\times n$</span>-strips with <span class="math-container">$k$</span> selected boxes.
Furthermore, let <span class="math-container">$A... | https://mathoverflow.net |
138,867 | [
"https://physics.stackexchange.com/questions/138867",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/60433/"
] | A person jumping from a higher building is likely to suffer less injury than the person jumping from a short building as the person from large building fall long enough to attain drag force which means the acceleration = 0 m/s square and body falls with constant speed?<br />
I got this doubt after reading drag force p... | No. Is it the final speed or final acceleration that determines a person's injuries when he/she hits the floor ?
A skydiver can reach terminal velocities of between 60-90 m/s. At terminal velocity, a = 0.
If you jump off a chair, you'll probably reach velocities of less than 1m/s. However, your acceleration will be $... | Actually the forces are proportional to the impact momentum. Before impact you have $J=m v$ momentum and it needs to be 0 after the impact (unless you bounce!). Newtons laws tells us that $J=\int F(t)\,{\rm d}t$ so the force needed depends on the total momentum $J$ and the time it takes for the impact event to happen.
| https://physics.stackexchange.com |
353,835 | [
"https://physics.stackexchange.com/questions/353835",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/167461/"
] | An object can float in saltwater while sinking in tap water since the density of tap water is less than that of saltwater. Then the weight of tap water displaced is larger than that of saltwater, but buoyant force is larger in saltwater, which contradicts Archimedes principle, which states buoyancy is equal to the weig... | <blockquote>
But what should I do when I want to obtain some of these quantities?
</blockquote>
In general, the wavefunction encodes the probability amplitudes for a particular value of an observable to be measured. Only in the special case that the wavefunction corresponds to an eigenstate of the observable can on... | $\hat{O}$ will have a set of eigenvectors $\phi_i$ and eigenvalues $o_i$.
You can expand your wave-function $\psi$ in terms of the basis set $\phi_i$, i.e. $\psi=\sum_i c_i\phi_i$.
If you make a single measurement then your wave-function $\psi$ will collapse to one of the eigenvectors $\phi_i$ with probability $|c_i|... | https://physics.stackexchange.com |
408,281 | [
"https://softwareengineering.stackexchange.com/questions/408281",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/104335/"
] | I mainly work on custom web applications that have just one production deployment.
While we are moving to continuous delivery, I was wondering if that approach reduces the need to make settings configurable.
For example, I was writing code that sends confirmation e-mails from a web server through Exchange.
Quite so... | You have several assumptions about the production environment that may be true today, but may not be tomorrow. Anything specific to that environment - even if you only expect a single installation - should be in a configuration file. Then changes, when needed, can be made immediately without having to wait for a buil... | One significant benefit in having a CI/CD pipeline is the ease at which it's possible to deploy into multiple different environments, where each environment has its own configuration. For example, it's frequently desirable to have a staging environment in which some users or stakeholders can review changes before anyt... | https://softwareengineering.stackexchange.com |
163,273 | [
"https://dba.stackexchange.com/questions/163273",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/23271/"
] | I'm having a really hard time diagnosing and working around a race condition that is causing problems on my slave and knocking it offline due to a foreign key relationship. We're using <code>MIXED</code> format replication in a basic <code>master -> slave</code> relationship.
The situation is thus, we have two ta... | So in the end, @jynus suggested in the OP comments that in their experience, replication issues like this often came down to the filters and unexpected behaviours resulting from them which seems to have led to a solution...
Because we needed the filters for a number of reasons, instead of eliminating them, I tried to ... | (Not likely to solve the question, but too big for a Comment)
<ul>
<li><code>CHAR(32) utf8</code> takes 96 bytes. This is especially bad if the string is hex or ascii anyway. If variable length, use <code>VARCHAR</code>. If always ascii, use <code>CHARACTER SET ascii</code> (or <code>latin1</code>). (The charset s... | https://dba.stackexchange.com |
390,863 | [
"https://mathoverflow.net/questions/390863",
"https://mathoverflow.net",
"https://mathoverflow.net/users/105900/"
] | Let <span class="math-container">$X$</span> be an incomplete Alexandrov space with sec <span class="math-container">$\ge -1$</span> in the sense that for any point in <span class="math-container">$X$</span> there exists a small neighborhood in which the four-points criterion is satisfied.
Suppose <span class="math-cont... | Yes, the conclusion is exactly the main result of Petrunin's paper "A globalization for non-complete but geodesic spaces", Mathematische Annalen volume 366, pages387–393(2016).
| I believe this is true by the work of Nan Li in Theorem A of "Globalization with probabilistic convexity", Journal of Topology and Analysis 07 (2015), no. 04, 719–735.
| https://mathoverflow.net |
60,971 | [
"https://datascience.stackexchange.com/questions/60971",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/75897/"
] | I am applying a ML model (LGBM binary classifier) to data and would now like to identify the part of data where I have a low ratio of false-negatives (false-postives are not such a problem) and as much as possible true-negatives.
<h1>Background</h1>
The data I am classifying is from a system, that includes many compl... | In the end, I have simply solved this problem by maintaining a constant number of training iterations independently of the size of the training dataset used. This way, I am able to test how the performance varies as the size of the training dataset increases, without also increasing the number of training iterations an... | In RL this is known as the exploration-exploitation trade-off, so i don't think you can avoid this regularizing the neural network, but rather the action policy it learns. The agent you are training is learning to solve a specific task and its policy is "over-fitting". You could tackle this in several ways, one of whic... | https://datascience.stackexchange.com |
127,939 | [
"https://mathoverflow.net/questions/127939",
"https://mathoverflow.net",
"https://mathoverflow.net/users/-1/"
] | This is probably standard for group-theorists:
Let $G$ be a finite group. Is it true that the intersection of all normalizers of subgroups equals the center?
If so, where do I find a proof? What about the same question for infinite groups?
The original question can be reformulated as follows: Let $G$ be a finite group... | No. There are non-abelian groups $G$ for which all subgroups are normal, such as the quaternion group of order 8. So the intersection of all normalizers is just $G$.
| The intersection of all normalizers of subgroups of a group $G$ is called the norm of $G$. By a result of Schenkman [E. Schenkman, On the norm of a group, Illinois J. Math., 7 (1960) 150-152] the norm of a group always lies in $Z_2(G)$ of the upper central series.
So the question has positive answer if for example th... | https://mathoverflow.net |
177,894 | [
"https://stats.stackexchange.com/questions/177894",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/10598/"
] | This is a question about statistical language. Do you regress the IV on the DV, or do you regress the DV on the IV? Which is the correct way of saying this?
| Traditionally speaking, one regresses the <em>dependent</em> variable (the Y, the outcome) on the <em>independent</em> variable (the X, the input). However, this is such an egregious abuse of statistical language, many disciplines have abandoned such verbiage altogether. The mistake is that "dependence" (in the proper ... | The question prejudges another question, good terminology for the variables concerned. Let's take that first.
<strong>DV</strong> is common, but not universal, shorthand for <strong>dependent variable</strong>. It's probably old-fashioned to remind that DV has often been used to mean <em>Deo volente</em>, God willing... | https://stats.stackexchange.com |
38,004 | [
"https://mathoverflow.net/questions/38004",
"https://mathoverflow.net",
"https://mathoverflow.net/users/-1/"
] | Is there a classification theorem for "finitely generated abelian groups with a distinguished element"? If it helps, you can restrict this to the cases where the order of each element divides the order of the distinguished element.
My idea would be to try to use this for a classification theorem for rings with a fini... | You say that we may assume that the order of every element divides the order of the distinguished element. That's the easy case. If the distinguished element generates a finite cyclic group $C$ of order $n$ and every element has order dividing $n$, then in fact $C$ is a direct summand. ("For the ring $\mathbb Z/n$ the ... | This is only a partial answer assuming the "if this help" (interpreted in a certain way, hoping it is the intended one).
Let G be a finite abelian group and g in G such that the order of each element of G divides the order of g. (I assume this means that the order of g is finite.)
There exist n_1,...,n_r such that n_... | https://mathoverflow.net |
473 | [
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"https://datascience.stackexchange.com",
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] | The usual definition of regression (as far as I am aware) is <em>predicting a continuous output variable from a given set of input variables</em>.
Logistic regression is a binary classification algorithm, so it produces a categorical output.
Is it really a regression algorithm? If so, why?
| Logistic regression is regression, first and foremost. It becomes a classifier by adding a decision rule. I will give an example that goes backwards. That is, instead of taking data and fitting a model, I'm going to start with the model in order to show how this is truly a regression problem.
In logistic regression, w... | <strong>Short Answer</strong>
Yes, logistic regression is a regression algorithm and it does predict a continuous outcome: the probability of an event. That we use it as a binary classifier is due to the interpretation of the outcome.
<strong>Detail</strong>
Logistic regression is a type of generalize linear regres... | https://datascience.stackexchange.com |
57,590 | [
"https://electronics.stackexchange.com/questions/57590",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/16367/"
] | I am doing a project in which I am getting data from some switches and push buttons on the pins of 8051 uC. Now I have to transmit this data on the port pins serially to the PC.
At the PC end, I am receiving the data in C# .Net application using the serial port class.
Now the issue is that I want that the data receiv... | I agree with the OP in that the critical part of the function of a D-latch lies in it's SR-latch, so I will focus on that part only for the moment.
Since we're dealing with NANDs here, I would derive the SR-latch function or truth table from that of a NAND with the inputs A and B and the output Z:
$$
\array{A&B&a... | In your first picture, consider the top NAND gate. We know that a NAND gate is such that, if any input is 0, then the output must be 1 (the only way the output of a NAND gate is 0, is if both inputs are 1). Thus, since ~S = 0, then Q = 1, regardless of the second input to the top NAND gate.
Since you now know both inp... | https://electronics.stackexchange.com |
46,147 | [
"https://datascience.stackexchange.com/questions/46147",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/68208/"
] | <strong>Information available</strong>
Consider that there are <em>N users</em> on a platform. Every user adds items that they like on their profile. These items have static attributes that describe the product.
<pre><code>User A:
Row | Attribute a | Attribute b | Attribute c
Item 1| 0.593 | 0.7852 | 0... | You can perform clustering of your customers based on a distance function.
Definition might look like this:
<ol>
<li>First, calculate euclidean distances between the first item of the first customer's basket and all of the items in the second customer's basket.</li>
<li>Then find out, what is the closest item from sec... | Well you could try unsupervised clustering. You may want to leave out the user and item label to start. Depending on how much data you have and guesses at how many "categories" you might end up with you can use K-means or Mean sift clustering. The idea would be you let the similarities be worked out so that you group t... | https://datascience.stackexchange.com |
375,600 | [
"https://softwareengineering.stackexchange.com/questions/375600",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/28921/"
] | On one hand, the invariants should be protected (To make invalid states impossible early and preferably at compile time rather than run time), and on the other hand, passing domain knowledge out of the domain is wrong.
On one hand, the value objects protect our domain invariants at first place, on the other hand using... | Not everything has to be perfectly object-oriented, even if you have to call nearly everything a <code>class</code> in C#.
Your first design is a reasonable approximation of an union in C#. You could make it slightly safer by preventing other implementations of the validation result interface, e.g. by using a base cla... | In your concrete example pattern matching can be implemented with polymorphic features of OO.
<pre><code>public interface IValidationResult
{
void Print();
}
public class FailedValidation : IValidationResult
{
private readonly string _errorMessage;
public FailedValidation(string errorMessage) => _erro... | https://softwareengineering.stackexchange.com |
145,272 | [
"https://stats.stackexchange.com/questions/145272",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/73001/"
] | I'm trying to understand why the softmax function is defined as such:
$\frac{e^{z_{j}}} {\Sigma^{K}_{k=1}{e^{z_{k}}}} = \sigma(z)$
I understand how this normalizes the data and properly maps to some range (0, 1) but the different between weight probabilities varies exponentially rather than linearly. Is there a reaso... | The categorical distribution is the minimum assumptive distribution over the support of "a finite set of mutually exclusive outcomes" given the sufficient statistic of "which outcome happened". In other words, using any other distribution would be an additional assumption. Without any prior knowledge, you must assume... | I know this is a late post, but I do feel like there would still be some value in providing some justification for those who happen to land here.
You're not entirely wrong. It is arbitrary to a certain extent, but perhaps arbitrary is the wrong word. It is more like a design choice. Let me explain.
It turns out that... | https://stats.stackexchange.com |
445,707 | [
"https://stats.stackexchange.com/questions/445707",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/222605/"
] | Note that <span class="math-container">$a$</span> has a mean of 0.
My approach:
<span class="math-container">$$X_t=X_{t-1}+a_t$$</span>
<span class="math-container">$$E[X_{t+1}\mid X_1 + \dots+X_{t-1}]$$</span>
<span class="math-container">$$=E[X_{t-1}+2a\mid X_1 + \dots+X_{t-1}]$$</span>
<span class="math-container"... | <span class="math-container">\begin{align}
E[X_{t+1} \mid X_1, \ldots, X_t]
&= E[X_t + a_{t+1} \mid X_1, \ldots, X_t]
\\
&= X_t + E[a_{t+1} \mid X_1, \ldots, X_t]
\\
&= X_t
\end{align}</span>
| Let <span class="math-container">$\{X_t\}_{t\geq 1}$</span> be a sequence of independent random variables such that <span class="math-container">$\Pr\{X_t=1\}=\Pr\{X_t=-1\}=1/2$</span>. Define <span class="math-container">$\mathscr{F_t}=\sigma(X_1,\dots,X_t)$</span> and <span class="math-container">$M_t=X_1+\dots+X_t$<... | https://stats.stackexchange.com |
108,690 | [
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"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/104629/"
] | I'm trying to prove a language is not regular through using pumping lemma, but can't seem to come up with any way of doing it.
The alphabet is:
<span class="math-container">$$ \Sigma = \{c, d\} $$</span>
The language is:
<span class="math-container">$$ A = \{z \in \Sigma^* \mid c(z) > d(z)\} , \text{where $c(w)... | You need to show that with that arbitrary pumping length <span class="math-container">$P$</span>, for <strong>any</strong> partition <span class="math-container">$c^{(P+1)}d^P = xyz$</span> such that <span class="math-container">$|y|>0$</span> and <span class="math-container">$|xy|\leq P$</span>, there is some <span... | You need to pick a string that's in the language (i.e., has fewer <span class="math-container">$d$</span>s than <span class="math-container">$c$</span>s) but, when pumped, creates strings that are not in the language (more <span class="math-container">$d$</span>s than <span class="math-container">$c$</span>s). Think ab... | https://cs.stackexchange.com |
76,411 | [
"https://security.stackexchange.com/questions/76411",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/60935/"
] | This question is with regards to mobile apps and apps stores.
Is it possible to program a mobile app that looks like a legitimate app, say for a bank, with the aim that people will download and use it, and key in their user id and password.
This 'fake' mobile app will then capture their password and send this info ... | Well, the app is certainly possible. If the app was created for good by a bank, it can be recreated for evil. Especially if the evil app doesn't actually need to handle banking transactions. It could just scoop the credentials and crash.
I think the other half of this question is "Are people dumb enough to fall for t... | To expand on Ohnana 's answer, yes this is possible.
If you were to create an application that essentially looked exactly the same as the bank's login screen, it could scrape the credentials and crash like they suggest. However, when you do that, and people start reviewing your app, it will be heavily rated on the bad... | https://security.stackexchange.com |
12,200 | [
"https://physics.stackexchange.com/questions/12200",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/652/"
] | Why is electricity not transmitted wirelessly such that we don't need to span cables on the earth's surface? As in: electricity is transmitted wirelessly from the power plant to the household.
| Electricity is the flow of electrical charge - generally electrically charged particles called electrons in a wire. It can't flow through air, except in the form of electrically charged particles of air - as in a spark or lightning stroke.
Magnetic fields can travel in air, so you can send electricity by using it to m... | Power is transmitted wirelessly in many applications, just not where transmission efficiency is high importance. And that is the reason it is not done more: efficiency. At low frequencies, galvanic (that means metal) conduction of electricity is many, many, many orders of magnitude more efficient than, say, air.
At... | https://physics.stackexchange.com |
26,284 | [
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"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/13651/"
] | I am curious as to how useful the engine temperature sensor actually is. It tells you the coolants temperature obviously, but I have read that the engine actually operates hotter than the coolant. For example, oil temperature can vary from 90 degrees Celsius to around 120 depending on engine load.
Hypothetically spea... | Orthodox antifreeze is based on ethelyne Glycol. It does depress the freezing point and raise the boiling point of the cooling system. This is desirable and the suggested concentration is roughly proportional to how cold you think it's going to get. As a coolant, water just can't be beaten so that's why it's still used... | It is common to measure engine temperature at the coolant, however newer systems that are more accurate are taking their place.
While the coolant temperature can vary, the engine is maintained at a constant 93 degrees celsius. This is because, combustion and lubrication is at the most optimum at this point.
The effe... | https://mechanics.stackexchange.com |
2,087,809 | [
"https://math.stackexchange.com/questions/2087809",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/291078/"
] | I need to show that the inclusion map from the complex plane to the extended complex plane is continuous, where the metric on $\mathbb{C}$ is the usual one and on $\mathbb{C}_\infty$, it's:
$$d(z,w) = \frac{2|z-w|}{\sqrt{1+|z|^2}\sqrt{1+|w|^2}}$$
Since $\infty$ isn't really involved here, this basically boils down to... | The integrand is not analytic, by brute force:
\begin{align*}
\oint_{C} \bar{z}|z| \, dz &=
\int_{-1}^{1} x|x| dx+\int_{0}^{\pi} e^{-it} \times 1 \, d(e^{it}) \\
&= 0+\int_{0}^{\pi} i \, dt \\
&= i\pi
\end{align*}
Your result is right.
| Since <span class="math-container">$|z|=1$</span> on the integration path, you also have <span class="math-container">$|z|^2=z\overline z=1$</span> and thus <span class="math-container">$|z|\overline z=(1)(1/z)=1/z$</span>. So your differential is just <span class="math-container">$dz/z$</span> and the definite integr... | https://math.stackexchange.com |
1,700,069 | [
"https://math.stackexchange.com/questions/1700069",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/170847/"
] | Determine if the series is convergent.
$$\sum_{n=2}^\infty(\sqrt{n^2+4}-\sqrt{n^2-4})$$
I'm considering using the Limit Comparison test but I can't decide on a p-series to use. I conjugated the function and i got to.
$$\sum_{n=2}^\infty\frac{8}{(\sqrt{n^2+4}+\sqrt{n^2-4})}$$
The denominator is confusing me. I'm going ... | Keep going. The inequalities you derived imply
$${8\over\sqrt{n^2+4}+\sqrt{n^2-4}}\ge {8\over2n+2}={4\over n+1}$$
| Compare it to series $\sum_{n=1}^\infty \frac{1}{n}$.
$\frac{\sqrt{n^2+4}-\sqrt{n^2-4}}{\frac{1}{n}} = \frac{8n}{\sqrt{n^2+4}+\sqrt{n^2-4}} =\\
= \frac{8}{\sqrt{1+\frac{4}{n^2}}+\sqrt{1-\frac{4}{n^2}}} $
And it's equal to $4$, if $n\to\infty$.
(Of course, it works only if you have studied so far. In not, see other an... | https://math.stackexchange.com |
218,121 | [
"https://security.stackexchange.com/questions/218121",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/-1/"
] | It seems to be common knowledge that a VPN has the potential to protect data that is being transferred through an insecure network. Does a VPN also protect a secure network/router from malicious website data being sent to a device using that network with a VPN?
I am new to this site so please forgive any mistakes or i... | No.
A VPN provides an encrypted tunnel between an initial device A and a server B where the data is decrypted and processed, for example by forwarding onto a requested server C. Responses from C are sent back to B, encrypted, and sent to A. This means a man in the middle, M, can't determine what A is sending to C in... | what you are talking about is called "reverse tunnelling". It does work (sort of) and used to be used to transfer traffic between low security networks across a higher security bearer. The basic concept being that the payload traffic is encapsulated (by a device to the same security standard as the higher security bear... | https://security.stackexchange.com |
6,174 | [
"https://physics.stackexchange.com/questions/6174",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/386/"
] | That pretty much says it.
Suppose I have some powder of $NaCl$. It is kept in contact with itself in vacuum. You are free to remove all the disturbances that bother you.
Is that true that, well, there exist a ($\mbox{very}^\mbox{very}$ large) amount of time $T$ that for every moment $t>T$ you will have a single b... | If you wait long enough, it'll become a bunch of iron (assuming it's confined so that the atoms can't evaporate off as Marek pointed out). After all, an iron nucleus is more stable than any other nucleus. The probability of the other nuclei tunnelling together to form a big hunk of iron is absurdly low, but it's not st... | Technically, if you have infinite time to wait, then yes. This is based on the standard "if it isn't forbidden, it is compulsory" idea of QM. However, as other answers have correctly pointed out, there are other potential quantum fluctuations that are also available and may be more probable. Combined with the unlikelih... | https://physics.stackexchange.com |
52,661 | [
"https://electronics.stackexchange.com/questions/52661",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/17417/"
] | It seems this is the chicken and egg problem. To be able to determine the frequency response of a microphone one needs a speaker that has a known response. To be able to know the response of a speaker one needs a mic with a flat response.
Is there some trick used or some device(essentially a microphone) that is "prova... | Reciprocity calibration is the technique used to calibrate a microphone without the need for any calibrated sources. It relies on the basic physics of transducers that show that the ratio of the receiving response to transmitting response of most transducers (designated as reciprocal transducers) is independent of the... | One way is to use an event with a mathematically predictable frequency response. A spark of known short duration (i.e. much shorter than the shortest time of interest - 1 to 5 us for audio) in an acoustic free field, would do.
The support structure for microphone and spark gap, and the "free field" get in the way of p... | https://electronics.stackexchange.com |
108,769 | [
"https://dba.stackexchange.com/questions/108769",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/71903/"
] | I have a table similar to this (simplified):
<pre><code>CREATE TABLE books (
id INT AUTO_INCREMENT,
category INT NOT NULL,
PRIMARY KEY (id),
KEY (category)
);
</code></pre>
This table has over <code>10,000,000 rows</code>, at around <code>12 categories</code>. So each category has an average of <code>... | The query will be slow because cardinality of <code>category</code> index is low. There are 12 categories, so in average the query will read 1/12 part of the index. You can't improve this query.
Your original approach can improve overall performance. Just instead of manually updating <code>book_count</code> create a t... | my trick is, make temporary table with field category and units then u make schedule at night to counting it n save to your temporary table, then ur just read that table with just 12 row in it, very faster, but ur count H-1, u need liitle trick againt to add current day ...
| https://dba.stackexchange.com |
276,698 | [
"https://softwareengineering.stackexchange.com/questions/276698",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/171328/"
] | I asked a question about extending TreeMaps into "sort by value" TreeMaps on the "code review" site on stackoverflow. Based on the response, I re-wrote the code. I like it. But, it feels like I might be just using tricks. Or, maybe this is a legit way to write software? Can someone please take a look:
<pre><code>publi... | <code>MyMap</code> only overrides a single method. The interface implementation does nothing besides delegating to the <code>MyMap</code> instance. You could easily just write this in the interface implementation.
<pre><code>class TreeMapByValue implements SortedMapByValue, Comparator {
private Map<Object, Inte... | Is a and Has a are reasonable guidelines, very useful when you are learning, but in practice you should never use inheritence unless it's so blatently obvious that it's needed that there is just no question.
Inheritene often seems like a much neater solution, but people who have used it extensively have learned that i... | https://softwareengineering.stackexchange.com |
140,225 | [
"https://cs.stackexchange.com/questions/140225",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/136522/"
] | I'm not very sure how exactly a fractal number is stored nor how random function works in mordern programming languages.
But I am curious, will random function ever hits a hard coded decimal number?
I know mathematically it is impossible, but how about in computer science?
Such as the following javascript code, will it... | Your random number generator might not be able to produce all numbers. Without looking at Math.Random, if it returned an integer then it would never match 0.123456. If it returned a double precision floating point number 0 <= rnd < 1, it might produce a 53 bit integer from 0 to 2^53-1, and multiply by 2^-53. In t... | Well, it technically <em>is</em> possible (with a small caveat). This is due to the fact a computer has finite memory, and hence cannot represent all possible numbers, but rather only a very very small fraction of them (and thus the probability to "hit" some particular number it can represent is not 0). Howev... | https://cs.stackexchange.com |
3,890,141 | [
"https://math.stackexchange.com/questions/3890141",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/831690/"
] | The word FACETIOUS is the most common word in the English language which has each of the five vowels in order from left to right. How many ways can the letters in the word FACETIOUS be arranged such that all five vowels are in order from left to right?
I am wondering if there is an easier way than bashing and counting ... | We start with <span class="math-container">$9!$</span> and divide by <span class="math-container">$5!$</span> to account for the vowels. This is because there are <span class="math-container">$9!$</span> permutations without the extra restriction, and then one in every <span class="math-container">$5!$</span> of them h... | There are nine spaces, <span class="math-container">$5$</span> vowels that must be in order, and <span class="math-container">$4$</span> are consonants. You can pick any letter first and choose where to place it so lets choose to place the <span class="math-container">$4$</span> consonants first.
We can put the <span ... | https://math.stackexchange.com |
16,402 | [
"https://security.stackexchange.com/questions/16402",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/10815/"
] | Recently, on live chat, a representative of a web hosting company (will not disclose the company name for liability issues), asked me for my password to the server in plain text in order to verify the identity of the account holder (me)
He then proceeded to tell me that "your password does match it" -- He's reffering... | Being slightly in the hosting business I realize what you mean. To put it simply, it is <strong>extremely bad practice to ask clients to give their passwords</strong> in plain text but all companies do it. They either ask the full password or simply a few characters from the end.
My point is that, at some places like ... | Usually the hosting providers shouldn't ask for the customer personal details( like passwords for control panel , admin panel or FTP) via live chat. This information should be provided via tickets to and from the customer contact email address which he previously used to register with the hosting company.
| https://security.stackexchange.com |
134,286 | [
"https://dba.stackexchange.com/questions/134286",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/89816/"
] | I am currently setting up percona-online-schema-change to do a ALTER on a table with 300m+ rows.
I've verified that percona-online-schema-change can make the required ALTER.
The source table receives updates/inserts in large batches at erratic intervals.
During testing, with defaults values for percona-online-schema... | We ran an update with extra parameters and it completed.
In the end, we didn't alter the chunk-size but instead altered the default settings for the copy-rows process, to increase the number of retries and the interval between retries.
We also increased the innodb_lock_wait_timeout from the default 1 second to 5 sec... | You seem to be asking the right questions and that seems like a reasonable course of action.
Don't be surprised if this runs for a few weeks.
| https://dba.stackexchange.com |
478,029 | [
"https://stats.stackexchange.com/questions/478029",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/182657/"
] | There is one concept in Statistics that I don't feel clear, and I could not find it in textbooks. Why sometimes do people compare coefficient estimates with corresponding standard errors? Here is the context:
I am reading the book An Introduction to Categorical Data Analysis by Alan Agresti (2nd edition, the thin versi... | By saying "the estimates for weight and width are only slightly larger than their SE values" I believe the authors are underlining the (lack of) precision.
For example, let's imagine the coefficient is 100 and the SE is 4. Then, the 95% confidence interval will be 92.16 - 107.84.
On the other hand, let's imag... | We have point estimates and measures of dispersion. In this particular example, the coefficients are point estimates and the standard errors convey the dispersion. What is more, both are in the same unit (example, meters, pounds, etc.), and in the same scale. In other words, if we have a given coefficient (say, 1.35) ... | https://stats.stackexchange.com |
66,433 | [
"https://dba.stackexchange.com/questions/66433",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/39417/"
] | I managed to get MongoDB running as a three member replica on Windows 8, but I had problems with configuration files. If I used the new (MongoDB 2.6) file format (YAML) and used the replica set name option as:
<pre><code>replication:
replSetName: rs1
</code></pre>
(or any of the other variants I had tried in YAML... | I was not able to replicate, so here's my successful YAML config file (Windows 7 64 bit, MongoDB 2.6.1):
<pre><code>replication:
replSetName: "rs1"
oplogSizeMB: 512
storage:
dbPath: "data"
smallFiles: true
systemLog:
destination: file
path: "data/mongod.log"
</code></pre>
How I started it:
<p... | Ensure the configuration file uses ASCII encoding. mongod does not support configuration files with non-ASCII encoding, including UTF-8.
| https://dba.stackexchange.com |
385,388 | [
"https://physics.stackexchange.com/questions/385388",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/184327/"
] | They keep saying that spinning molten iron generates Earth's magnetic field.
But, even if that iron is somewhat ionized, those two lost electrons are still nearby in the fluid. So the spin of the ++ and the spin of the -- should cancel out any magnetic field they'd generate.
| The algebraic formulation is more general and takes into account many subtleties that arise in QFT and that are hidden in quantum mechanics.
In fact, in quantum mechanics the Stone-von Neumann theorem tells us that the irreducible representation of the algebra of quantum observables (more precisely, of the algebra of ... | The essential mistake in your reasoning is that you contrast the wrong notions of "state" - the algebraic states are not only supposed to be <em>pure</em> vector states, i.e. represented by vectors in the "natural" Hilbert space of the system, but they also include all <em>mixed</em> states, i.e. density matrices. Of c... | https://physics.stackexchange.com |
30,365 | [
"https://stats.stackexchange.com/questions/30365",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/11941/"
] | Today I came across a new topic called the Mathematical Expectation. The book I am following says, expectation is the arithmetic mean of random variable coming from any probability distribution. But, it defines expectation as the sum of product of some data and the probability of it. How can these two (average and expe... | Informally, a probability distribution defines the relative frequency of outcomes of a random variable - the expected value can be thought of as a weighted average of those outcomes (weighted by the relative frequency). Similarly, the expected value can be thought of as the arithmetic mean of a set of numbers generated... | The expectation is the average value or mean of a random variable not a probability distribution. As such it is for discrete random variables the weighted average of the values the random variable takes on where the weighting is according to the relative frequency of occurrence of those individual values. For an absol... | https://stats.stackexchange.com |
110,277 | [
"https://stats.stackexchange.com/questions/110277",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/52219/"
] | In Gaussian Process (GP), the kernel (co-variance function) is used to measure the similarity between one point and a given point. There are so many kernel functions for GP, and I wonder how to select a suitable kernel. For instance, if my time-series data are not periodic, should I choose the Squared Exponential (SE) ... | One possibility you might try is simulating Gaussian Processes with different kernels. In that way, you can get a feel for what the different kernels will produce. This can most easily be done by selecting a grid of values and simulating from the multivariate normal implied by that grid. To make things easier, just use... | Set aside a second set of training data, and "train" your model architecture using that.
i.e.
1) select an arbitrary kernel
2) train it using training set 1
3) evaluate it on training set 2 (using accuracy, precision, recall, whatever)
4) if !tired: goto 1)
5) else: return kernel with highest evaluation scor... | https://stats.stackexchange.com |
1,219,641 | [
"https://math.stackexchange.com/questions/1219641",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/89368/"
] | Stel $D$ a metric space. Let $K_1 \subset K_2 \subset K_3 \subset ...$ a serie of compact sets in $D$. I was wondering if $K = \bigcup_{n=1}^\infty K_n$ is compact too. If we take an open cover of $K$ then we can find for every $n$ a finite cover of $K_n$ but is there a way to extend to the case it is infinite?
| Ascending chains like that will not preserve compactness, since the finite subcover for each level might get bigger and bigger.
As an example, look at $[-1,1] \subset [-2,2] \subset [-3,3] \dots$ in $\mathbb{R}$. Each set is compact, but the union is not.
| If you want a finite interval example:
$$\;\left[\frac12,\,1\right]\subset\left[\frac13,\,1\right]\subset\ldots\subset\left[\frac1n,\,1\right]\subset\ldots$$
Each interval is closed and bounded and thus compact, yet their union is not:
$$\bigcup_{n=2}^\infty\left[\frac1n,\,1\right]=(0,1]$$
| https://math.stackexchange.com |
550,364 | [
"https://electronics.stackexchange.com/questions/550364",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/201217/"
] | I am having issues with reading 2 ADC channels in my code running in a PIC18F24K22.
One channel is used to read a potentiometer to control the speed of a motor ( PWM) and the other is reading a temperature sensor. They work fine individually but when I try to read both it does not seem to be working properly. As I inc... | This is another way to implement your ADC to temperature function:
<pre><code>/*
* File: AppLoop.c
* Author: dan1138
* Target: PIC18F24K22
* Compiler: XC8 v2.31
* IDE: MPLABX v5.45
*
* Created on March 7, 2021, 11:37 AM
*
* PIC18F24K22
* +---------:_:---------+
* ... | The loop is very fast, but when you do both channels it is still very fast but the motor rate is being updated only half as fast, but even so could be something like every 40 microseconds (12kHz). It is possible that updating the pwm (CCPR5L= store) is interfering with the pwm itself.
You said you put in a delay, but d... | https://electronics.stackexchange.com |
222,383 | [
"https://softwareengineering.stackexchange.com/questions/222383",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/11950/"
] | I have a method to be tested which depends on the OS, what's the best approach for testing the method depending on the OS? Is it to test on every OS that I have I require? Is there a better approach for running the tests on only one system?
Some more specific information is that I have a method which is dependent on ... | If you target multiple OSs, you most probably have to test your program on each of them. These tests should typically include running your test suite on the desired OS. So the straightforward solution for unit testing the above method is to adapt the expected result on <code>platform.system()</code> exactly the same wa... | Using the mock framework, you can patch <code>platform.system()</code>.
<pre class="lang-py prettyprint-override"><code>from unittest.mock import patch
…
def testUpdateStringWithDarwin():
with patch('platform.system', MagicMock(return_value="Darwin")):
result = updateString()
assert res... | https://softwareengineering.stackexchange.com |
4,842 | [
"https://chemistry.stackexchange.com/questions/4842",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/1554/"
] | A binary compound of lead and oxygen contains $90.66 \% \ce{Pb}$. What is the empirical formula for the compound? How do I use the percentage to get to the normal formula, then how do I change the normal formula to empirical?
| You will get the empirical formula straight from this data. In order to convert to a molecular formula, you need more information.
By way of another example:
<blockquote>
A binary compound of tin and chlorine contains 62.89% tin by mass. Determine the empirical formula.
</blockquote>
Our first goal is to figure ou... | I'm not going to this question for you. The point of this forum is to educate and not to do your homework.
Imagine you 100 g of compound XY where X and Y are elements.
You know the percentage of X is 70 and of Y is 30.
This means there must be 70 g and 30 g of X and Y respectively.
To convert this to moles you need... | https://chemistry.stackexchange.com |
625,446 | [
"https://physics.stackexchange.com/questions/625446",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/294533/"
] | Many posts here turn around the question <em><strong>how</strong></em> exactly spacetime symmetries are represented on (projective) Hilbert spaces in quantum mechanics. The question here is <em><strong>why</strong></em> quantum states should live in (projective) representation spaces of these symmetry groups in the fir... | Note that <span class="math-container">$c^x = \exp[x\ln(c)]$</span>, so for an operator <span class="math-container">$\hat A$</span> you would have
<span class="math-container">$$c^\hat A = \exp[\ln(c) \hat A] = \sum_{n=0}^\infty \frac{[\ln(c)]^n}{n!} \hat A^n$$</span>
| Write the operator in its eigenbasis. In the eigenbasis, treat each eigenvalue as a number. Compute the function written. Then, the eigenbasis of the resulting operator is the same, and the eigenvalues are the corresponding functions of the eigenvalues.
In formulas: For a hermitian operator <span class="math-container"... | https://physics.stackexchange.com |
559,293 | [
"https://electronics.stackexchange.com/questions/559293",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/139034/"
] | What is the memory map of w65c02s?
The datasheet is not clear at all compared to the atmega328p.
Can someone put up a chart and explain the memory space of this microprocessor?
| The 6502 was designed for a minimal system with an external 2316 ROM or 2716 ERPOM and the 6532 companion chip. That one had RAM, Timers and GPIO. You had to invent some glue logic from 74xx and/or 40xx chips yourself. That glue logic decided on the address map.
You could in fact have the reset vector hardwired to some... | <blockquote>
What is the memory map of 65c02w?
</blockquote>
Well, "vestigial" would be a misnomer, because that implies that it had one to begin with. "Mostly up to you" would be a better way to put it.
<blockquote>
The datasheet is not clear at all compared to the atmega328p.
</blockquote>
From t... | https://electronics.stackexchange.com |
384,098 | [
"https://physics.stackexchange.com/questions/384098",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/182938/"
] | Lets say a woman is attached to the center of a spinning disc with a rope ,and she also has friction between her shoes and the surface of the disc.
A problem I am doing says that the force pointing to the center (the centripetal force) includes friction and tension.
Tension i understand that it pulls her inwards. But... | First and foremost, take into account the misinterpretations you are making that @knzhou discusses in their answer.
As a secondary issue, let me mention an example of an operator similar to what you mention. One trivial example is $p^2$, which in the position representation is proportional to the laplacian, which yo... | No, you can only take the divergence of a vector field, and a wavefunction is a scalar. The angular momentum is not the curl either; curl is $\nabla \times$ while you have $\mathbf{r} \times \nabla$.
| https://physics.stackexchange.com |
18,920 | [
"https://earthscience.stackexchange.com/questions/18920",
"https://earthscience.stackexchange.com",
"https://earthscience.stackexchange.com/users/18652/"
] | Why doesnt mankind "collect" thermal energy (there has to be some way: thermal couplings, detour over chemical energy, whatever) and after it has been concentrated at one point, turn it to electrical energy (like steam turbines do) or at least radiate it to space?
| Collecting thermal energy is really hard. As others have said, things like heat pumps exist for moving heat around, but the laws of thermodynamics (which are fairly fundemantal in physics) require that moving heat around will always generate <em>more</em> heat.
Now, the amount of extra heat generated can be less than... | This is due to thermodynamics, the three laws of which can be summarized as 1) You can't win; 2) You can't even break even; 3) You can't leave the game.
The crucial point here is that heat engines don't actually work on heat, they work on temperature differences. So you can't really "collect" heat and turn i... | https://earthscience.stackexchange.com |
211,433 | [
"https://security.stackexchange.com/questions/211433",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/45914/"
] | Using DNSSEC you can be sure that you have the right IP for a domain and using a certificate for the IP signed by someone you trust you know you have the right IP.
Shouldn't this be enough to know the connection is correct? Why would the domain name be needed in the certificate used by the server?
| You pretty much hit the nail on the head when you said that you need physical access to the machine.
If you have physical access, you don't <em>need</em> to go through the official steps to reset the root password, as you can flips bits on the hard drive directly, if you know what you're doing. I.e., you can boot up ... | <blockquote>
How is this not a glaring security vulnerability?
</blockquote>
It is. Physical access to your system <em>is</em> the ultimate vulnerability.
<blockquote>
Is there a way to disable this 'feature' so that it cannot be changed from GRUB like this?
Can you do this in all other Linux distros as well... | https://security.stackexchange.com |
1,998 | [
"https://astronomy.stackexchange.com/questions/1998",
"https://astronomy.stackexchange.com",
"https://astronomy.stackexchange.com/users/1241/"
] | When we move our vehicle, we "feel" the speed of it. However, when stationary, we still are moving,but moving by quiet a speed. We know that earth moves with <code>X</code> speed, our own solar system moves at <code>Y</code> speed. etc.<br>
Why do we not feel this speed? Even sitting at my home, my mind must be gyratin... | In a car, you have a perception of speed because of (a) the "wind" passing by as you rush through the air which is not moving at the same speed as the vehicle, and (b) you perceive the stationary objects nearby as "moving" off into the distance behind.
As the earth moves in its orbit, you don't notice any "wind" from ... | From a (more) physics standpoint our acceleration on earth is basically zero from what we can feel. Just like the others that posted about cars traveling at a certain <em>constant</em> velocity, you won't feel a change. If a car is traveling at a <em>constant</em> velocity there is essentially no feelable force acting ... | https://astronomy.stackexchange.com |
2,116,730 | [
"https://math.stackexchange.com/questions/2116730",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/328576/"
] | I want to determine whether the following function is differentiable at $x=0$:
$f: \mathbb{R} \rightarrow \mathbb{R}$, $x \mapsto \frac{{\sin}^2(x)}{x^2}$ for $x\neq 0$ and $1$ for $x=0$.
Clearly, this function is continuous $\forall x \in \mathbb{R}$ and the derivative exists $\forall x \in \mathbb{R}\backslash\{... | Using l'Hôpital, the limit can be computed
$$
\lim_{h\to 0} \frac{\sin^2h-h^2}{h^3} = \lim_{h\to 0} \frac{2\sin h\cos h-2h}{3h^2} = \lim_{h\to 0} \frac{2\cos 2h-2}{6h} = \lim_{h\to 0} \frac{-4\sin 2h}{6}=0
$$.
Another alternative perhaps more in line with your thinking, is to consider the function
$$
g(x) = \operato... | Use $a^2- b^2= (a- b)(a+ b)$ to factor $sin^2(h)- h^2$ as $(sin(h)- h)(sin(h)+ h)$ so that $\frac{sin^2(h)- h^2}{h^2}= \left(\frac{sin(h)- h}{h}\right)\left(\frac{sin(h)+ h)}{h}\right)$$= \left(\frac{sin(h)}{h}- 1\right)\left(\frac{sin(h)}{h}+ 1\right)$. Since the limit of $\frac{sin(h)}{h}$ is 1, that becomes $(1- 1)... | https://math.stackexchange.com |
126,659 | [
"https://physics.stackexchange.com/questions/126659",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/21270/"
] | The $10$ generators of the Poincare group $P(1;3)$ are $M^{\mu\nu}$ and $P^\mu$. These generators can be determined explicitly in the matrix form. However, I have found that $M^{\mu\nu}$ and $P^\mu$ are often written in terms of position $x^\mu$ and momentum $p^\mu$ as
$$ M^{\mu\nu} = x^\mu p^\nu - x^\nu p^\mu $$
and
... | One obtains those expressions by considering a particular action of the Poincare group on fields.
Consider, for example, a single real scalar field $\phi:\mathbb R^{3,1}\to\mathbb R$. Let $\mathcal F$ denote the space of such fields. Define an action $\rho_\mathcal F$ of $P(3,1)$ acting on $\mathcal F$ as follows
\b... | The generators of isometry, also generators of the Poincare group, are the Killing vectors, hence we need the Killing vectors of Minkowski spacetime, $ds^2 = -dt^2 + dx^2 + dy^2+ dz^2$. The defining equation of the Killing vectors in terms of their components ($\xi = \xi^\mu \partial_\mu$ is a Killing vector with compo... | https://physics.stackexchange.com |
33,842 | [
"https://mathoverflow.net/questions/33842",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3545/"
] | The Suzuki and Ree groups are usually treated at the level of points. For example, if $F$ is a perfect field of characteristic $3$, then the Chevalley group $G_2(F)$ has an unusual automorphism of order $2$, which switches long root subgroups with short root subgroups. The fixed points of this automorphism, form a su... | It is not really a question of inner forms. What happens is that the
<em>algebraic group</em> $G_2$ has an extra endomorphism $\varphi$ whose square
is the Frobenius map (over the appropriate finite field). Just as for any
algebraic group over a finite field $F$ its rational points over $F$ are the
fixed points of the ... | To supplement Torsten's account, the original Suzuki groups of type $C_2$ in characteristic 2 resulted from a purely group-theoretic investigation but were then recovered in the algebraic group setting. The Ree groups of types $F_4, G_2$ in respective characteristics 2, 3 were constructed inside the Chevalley groups ... | https://mathoverflow.net |
373,326 | [
"https://electronics.stackexchange.com/questions/373326",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/167399/"
] | I am asked to calculate the phase difference between the two signals below.
$$i_1 = -4\sin(377t + 55^\circ) \hspace{0.2cm}\quad \mathrm{and}\quad \hspace{0.2cm} i_2 = 5\cos(377t - 65^\circ)$$
What I did was to convert $$i_1 = -4\sin(377t + 55^\circ)$$
to
$$i_1 = 4\cos(377t + 145^\circ)$$
Then find the difference by ... | Both phase differences are correct and, in fact, indistinguishable. There is an infinite number of ways writing the same sine wave, just by adding any integer multiple of 2*pi (or 360 degrees) to the phase.
The problem is with trying to apply the concept of "leading" and "lagging" to sine waves. Strictly speaking a ... | Both of the results you got are right. You feel it confusing, because you imagine it as a running race, where one has to be first and the other the second, but in fact these are <strong>periodic</strong> signals. Which means that the leading is always relative.
Try to sketch the signals and select a local maximum of ... | https://electronics.stackexchange.com |
436,134 | [
"https://mathoverflow.net/questions/436134",
"https://mathoverflow.net",
"https://mathoverflow.net/users/57975/"
] | For a positive integer <span class="math-container">$n$</span>, the prime omega function value <span class="math-container">$\Omega(n):=\sum_{p\mid n}{\nu_p(n)}$</span> counts the number of prime divisors of <span class="math-container">$n$</span> with multiplicities. A result of Hardy and Wright, [1, Theorem 430 on p.... | Yes, this is true.
First, let us observe that replacing prime powers with primes cannot make a difference, and the same goes to replacing <span class="math-container">$\Omega$</span> with <span class="math-container">$\omega$</span>.
H. Halberstam proved that
<span class="math-container">$$\frac{\sum_{p \le x} \omega(p... | One can prove that in fact the function mpe has bounded average which clearly improves <span class="math-container">$\log \log$</span>. If a positive integer n has <span class="math-container">$mpe(n)=m$</span> then there exists a prime power p^m that divides n hence the cardinality of such integers below x will be at ... | https://mathoverflow.net |
414,877 | [
"https://mathoverflow.net/questions/414877",
"https://mathoverflow.net",
"https://mathoverflow.net/users/89064/"
] | Find all integer solutions to the equation
<span class="math-container">$$
y(x^2+1)=z^2+1.
$$</span>
There is, for example, an infinite family of solutions <span class="math-container">$x=u$</span>, <span class="math-container">$y=(uv\pm1)^2+v^2$</span>, <span class="math-container">$z=(u^2+1)v \pm u$</span>, <span cla... | The equation says that <span class="math-container">$z^2 + 1 \equiv 0 \mod (x^2+1)$</span>. For each positive integer <span class="math-container">$x$</span>, you can enumerate the square roots of <span class="math-container">$-1$</span> in the integers mod <span class="math-container">$x^2+1$</span> as you remark in... | This answer (which I made Community Wiki) attempts an explicit summary of how to ``list" explicitly the solutions of the given equation, given the answer of Robert Israel and the comment of David Speyer which followed it.
It's convenient to deal with the prime <span class="math-container">$2$</span> first, but thi... | https://mathoverflow.net |
84,414 | [
"https://mathoverflow.net/questions/84414",
"https://mathoverflow.net",
"https://mathoverflow.net/users/100/"
] | Here is a basic question. When does $H^1_{et}(X,\mathbb{Z})$ vanish? Using the exact sequence of constant etale sheaves $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\rightarrow 0$, it is enough to show that $H^1_{et}(X,\mathbb{Q})$ vanishes. It is known, for instance by 2.1 of Deninger's ... | The standard example is a copy of $\mathbb A^1_k$, where $k$ is an algebraically closed field, with two points glued. In algebraic terms, $X = \mathop{\rm Spec}R$, where $R := k[x,y]/(y^2 - x^3 + x^2)$. Consider the finite morphism $\pi\colon \mathbb A^1 \to X$, which yields an exact sequence
$$
0 \to \mathbb Z_X \to \... | More generally, if $X$ is proper over an algebraically closed field, then $H^1(X,\mathbb Z)$
is isomorphic to the cocharacter module of the maximal torus of the Picard variety
$Hom(\mathbb G_m,Pic^0)$.
| https://mathoverflow.net |
101,725 | [
"https://mathoverflow.net/questions/101725",
"https://mathoverflow.net",
"https://mathoverflow.net/users/1355/"
] | I have never seen any algebraic number theory book discuss the origin of the term "ray class group." Does anyone know where the word "ray" comes from in this context? I always thought it might be a person, but I never see it capitalized.
For quick background: the ray class group for a modulus $\mathfrak{m}$ of a numbe... | There are many introductions to number theory, but few are as original (my term for what others would call weird) as Fueter's "Synthetische Zahlentheorie" published in 1925.
It starts with elementary number theory, and discusses the arithmetic of cyclotomic
fields up to the Dedekind zeta function and applications to qu... | The word <em>ray</em> comes from the German <em>Strahl</em>, as in <em>Strahlklassengruppe</em> which Hasse uses but which goes back to Fueter, as Franz points out in a comment below.
| https://mathoverflow.net |
17,070 | [
"https://chemistry.stackexchange.com/questions/17070",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/5084/"
] | Does pyridine or furan have the more shielded proton NMR signals?
I know that both are aromatic, so both have rather deshielded signals due to ring current combining constructively with the applied magnetic field.
However, which would have the more shielded signal? I know that shielded protons have more electron d... | Here is the proton nmr spectrum for pyridine.
<img src="https://i.stack.imgur.com/Kj9cw.jpg" alt="enter image description here">
Here is the proton nmr spectrum for furan
<img src="https://i.stack.imgur.com/Wd0NX.jpg" alt="enter image description here">
For comparison, benzene has a single proton nmr signal at 7.27... | Ron's answer addresses the empircal data, but I feel compelled to comment that your assertion that pyridine has a nitrogen who's "lone pair participating in resonance" is not correct.
In pyridine, nitrogen is sp2 hybridized and forming a pi-bond necessarily puts the lone pair perpendicular to the ring, and so it may <... | https://chemistry.stackexchange.com |
246,211 | [
"https://softwareengineering.stackexchange.com/questions/246211",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/138876/"
] | Suppose there are <code>n</code> lines for a hotline.
Whenever a customer calls the hotline, the call is forwarded to one of the <code>n</code> lines. And I want to assign percentage of calling to each of the n lines. Suppose there are two lines and one line is assigned 60% and other is 40%, the total number of calls... | Do some bookkeeping about the already taken calls and calculate their distribution over the n lines. This gives you n percentage values (your already achieved distribution), which can be compared to the n percentages you want to achieve. Whenever a new call comes in, assign that call to the line with the highest deviat... | <ul>
<li>Let's suppose the number of workers is less than 100</li>
<li>Create an array of workers with a capacity of 100</li>
<li>Put in that array a worker a number of times equal to the percentage of calls he should get, for example if worker1 should get 30% of all calls, then put him in positions 0 to 29 of the arra... | https://softwareengineering.stackexchange.com |
146,654 | [
"https://dba.stackexchange.com/questions/146654",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/1782/"
] | Using SQL Server Management Studio, open a query window and execute
<pre><code>BEGIN TRAN
</code></pre>
In another window execute
<pre><code>select [text],
from sys.sysprocesses
cross apply sys.dm_exec_sql_text(sql_handle)
where status = 'sleeping'
and open_tran = 1
</code></pre>
I see <code>select @@trancount</cod... | This is a SQL Server Management Studio thing. At least SSMS version 13.0.15500.91. This version of SSMS will count the number of open transactions on your behalf using the same connection as your query window.
Here's how I found out,
<ul>
<li>I connected to another server running 2012 and also saw <code>SELECT @@TRA... | I don't think it's a Management Studio thing--I've seen this with other application before, but haven't personally seen it with SSMS. That being said, I wouldn't be surprised if SSMS did something like this in some scenario.
If you grab a few more columns from sys.sysprocesses, you might be able to track back to who ... | https://dba.stackexchange.com |
59,221 | [
"https://softwareengineering.stackexchange.com/questions/59221",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/20395/"
] | I am a non-programmer that has used the services of : freelancer, odesk, etc
I've tried asking for what i need but, I can't find anyone who can show me any type of example similar to what I request in the specs for the web-programming.
They have front ends and back ends, but they don't fulfill true "live" website re... | <blockquote>
I am a non-programmer that has used
the services of : freelancer, odesk,
etc
</blockquote>
There are very few really talented people on these services. Even the mediocre command salaries of $40k in the US, anyone competent earns $50k and up. The average bid on a site like Rent-A-Code [i've hired the... | <blockquote>
How do I figure how to evaluate a programmer ?
</blockquote>
How do you figure how to evaluate an automobile mechanic? A plumber? A doctor?
It's the same.
Trust. Experience. Skills.
| https://softwareengineering.stackexchange.com |
108,424 | [
"https://stats.stackexchange.com/questions/108424",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/16111/"
] | Say I am about to receive 5 cash prizes and I have the probability of receiving each cash prize. Let's denote a set of cash prizes with $k$. So, below is the set of cash prizes and the set of corresponding probabilities:
$$k=\{65,25,30,54,30\}$$
$$p(k)=\{0.8,0.5,0.25,0.2,0.4\}$$
Hence, the expected value of my cash p... | You don't state if the events that each of the prizes are won are independent of each other (and if not, what the form of dependence is), which you need if you want to work out the probability that the total value of prizes exceeds some amount. I'll assume independence for now.
There are also problems with the notatio... | Now the problem at hand is trivial as expected cash prize is not random if prize values and probabilities are fixed. Consequently, this probability equals zero or one with respect to correspondence between your calculated expected value and fixed sum.
If you want instead find $P(k > 85)$, I think that as it is a re... | https://stats.stackexchange.com |
43,714 | [
"https://physics.stackexchange.com/questions/43714",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/3397/"
] | Consider a theory of one complex scalar field with the following Lagrangian.
$$
\mathcal{L}=\partial _\mu \phi ^*\partial ^\mu \phi +\mu ^2\phi ^*\phi -\frac{\lambda}{2}(\phi ^*\phi )^2.
$$
The potential is
$$
V(\phi )=-\mu ^2|\phi |^2+\frac{\lambda}{2}|\phi|^4.
$$
The classic stable minimum of this potential is given... | As is easily checked, fields linear in creation and annihilation operators (and hence amenable to a particle interpretation) have zero vacuum expectation value. Thus the $\phi$ field with its nonvanishing vacuum expectation value cannot be given a particle interpretation. But the field $\psi=\phi-v$ has such an interpr... | Firstly, note that 'particles' are quantised small oscillations of a field.
Now, to the potential. You would find a 'mass' term expanding around any point, but you'll also generically find a linear term. If you tried to perturbatively quantise the 'small oscillations' around such a point, you would find non-zero Fey... | https://physics.stackexchange.com |
2,797 | [
"https://scicomp.stackexchange.com/questions/2797",
"https://scicomp.stackexchange.com",
"https://scicomp.stackexchange.com/users/59/"
] | I want to solve the following expression (used to obtain an analytic solution to a current distribution inside a workpiece):
$$a_{mn} = -\frac{\frac{4}{ab} \int_0^a \int_0^b f(x',y')\sin(px')\sin(qy')\mathrm{d}x'\mathrm{d}y'}{t\sinh(tc)}$$
Here. $a$,$b$ are scalar constants and $p = \frac{m\pi}{a}$, $q = \frac{n\pi}{... | It depends on how big $m$ and $n$ are (assuming that $x'$ ranges from $0$ to $a$, though the notation suggests the opposite if read literally). You should probably split the intervals into at least $m$ resp $n$ subintervals, integrate the pieces separately, and sum the results.
If $f$ is smooth, a Gaussian rule is pro... | If you solve this only for a single pair $m,n$, then high order Gaussian integration is likely the fastest way to do this. But if you want to build the elements of a matrix $a_{mn}$ for many different values $m,n$, then it is best to recognize that what you're computing here is a double Fourier transform, and it would ... | https://scicomp.stackexchange.com |
2,589,644 | [
"https://math.stackexchange.com/questions/2589644",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/429543/"
] | I'm attempting to solve this via induction.
The base case, $j=1$, is given by assumption. Then, we let our hypothesis be that nullity$(A^k)=$ nullity$(A^{k+s})$ for $j=s$, and show that nullity$(A^k)=$ nullity$(A^{k+s+1})$ for $j=s+1$. Thus, let $x$ be an element of the null space of $A^{k+s}$. That is, $A^{k+s}x=0$.... | Notice that $\ker T_2 \subset \ker T_1T_2$, so $\text{nullity}(T_2)\leq \text{nullity}(T_1T_2)$.
In particular, $\text{nullity}(T^k)\leq\text{nullity}(T^{k+1})$.
Assume that $k$ is such that $\text{nullity}(A^k)=\text{nullity}(A^{k+1})$.
Suppose there were $j\geq 1$ with $\text{nullity}(A^k)\neq \text{nullity}(A^{k+j}... | The condition $\text{nullity}(A^k)=\text{nullity}(A^{k+1})$ is the same as the condition $$A^{k+1}x=0\implies A^kx=0. \tag {1}$$ Prove this yourself.
Now we are to prove $\ker(A^{k+2})=\ker(A^{k+1})$ ($\ker A$ means nullspace of $A$), which would imply $\text{nullity}(A^{k+2})=\text{nullity}(A^{k+1})$. Observe $$A^{k+... | https://math.stackexchange.com |
213,680 | [
"https://electronics.stackexchange.com/questions/213680",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/56089/"
] | Okay, so I have what could be considered a silly question I should've probably learned when I started working electronics way back when, but I just now thought of it and haven't really needed to ask it until now.
I noticed on a AC-DC adapter yesterday that the ratings going in were 1.5A @ 120V, and the ratings coming ... | It depends on how the conversion is performed, and broadly there are two options:
<ul>
<li>linear converters have approximately the same input and output currents,</li>
<li>switching converters (and transformers) act like gearboxes, providing a different voltage/current ratio (which is like a torque/speed ratio) but s... | If you had that adapter at 12V @ 6A it can be converted to 5V.
If a linear regulator were used the current would be no more than 6A and the 5V output. The conversion drops 7V at up to 6A in heat in the linear regulator so 42W is cooking some regulator components.
If you use a suitably designed switching regulator to... | https://electronics.stackexchange.com |
371,845 | [
"https://electronics.stackexchange.com/questions/371845",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/176148/"
] | What would be the correct term for a clock input that isn't made to oscillate per se?
In an attempt to only allow input <strong>A</strong> to have any effect on a circuit at a chosen time, one could AND it with another input <strong>B</strong>.
Would input <strong>B</strong> still be referred to as a clock if it were... | The input B could be called "gate". Or "strobe", if the action happens on one of its edges.
It could be called "enable" or "chip select" if during it's on state it activates input A.
| Gated clocks are very common. If you're driving another chip with a SPI bus, it reduces EMC if you're not running that clock all the time. Also some SPI devices lack an enable input, so a gated clock is the only way to control transfers.
This would still be referred to as a "clock" for that SPI bus though.
| https://electronics.stackexchange.com |
2,425,671 | [
"https://math.stackexchange.com/questions/2425671",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/479343/"
] | Let $F$ be a function of $x$, $y$, and $f(x,y)$.
Does $\dfrac{\partial{F}}{\partial{x}}$ take into account the change in $f$ due to the change in $x$?
For instance let $F(x,y,f(x,y)) = x+y+f(x,y)$. Then is $\dfrac{\partial{F}}{\partial{x}} = 1$ or $1+\dfrac{\partial{f}}{\partial{x}}$?
| This is lousy notation, and completely ambigious as it is formulated.
To begin with, saying “F is a function of $x$, $y$ and $f(x,y)$” is bound to lead to confusion.
In order to even talk about partial derivatives unambiguously, you need to specify a coordinate system $(x,y,z)$, and define your function $F$ as a funct... | For a function $F(x,y,f(x,y))=x+y+f(x,y)$, the derivative with respect to $x$ is
$$\frac{\partial}{\partial x}(x+y+f(x,y))=\frac{\partial }{\partial x}x +\frac{\partial}{\partial x}y +\frac{\partial}{\partial x} f(x,y)$$
Assuming $y$ is not a function of $x$, this means
$$1+\frac{\partial}{\partial x} f(x,y)$$
| https://math.stackexchange.com |
382,280 | [
"https://softwareengineering.stackexchange.com/questions/382280",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/7783/"
] | I have a system that checks the status of a large number of entities on schedule every minute. For each entity, there would be a JSON file which has fields indicating the statuses for different attributes. The system dumps these JSON files on a network share.
Each run of the schedule that runs every minute generates a... | In my blunt opinion this is like a ridiculous way to use threads because what does user input involve? It involves checking to see if the user pressed/release some key on the keyboard or pushed/release some button on a mouse or joystick (and normally I wouldn't use a blocking function like <code>getch</code> for this).... | From a purely conceptual standpoint, <code>main()</code> is no different than any other thread in the same process. It just happens to be the one that was started first but still shares code and data with every other thread within the process.
Under the hood, there may be differences in how threads are implemented. ... | https://softwareengineering.stackexchange.com |
65,034 | [
"https://security.stackexchange.com/questions/65034",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/31356/"
] | Many of my customers refuse to sell or recycle their old hard drives or mobile phones because everybody now-days talks about how popular data recovery program's are. Isn't it enough to use a software to write a zero to every byte of the storage area in order to protect your deleted data?
What am I missing here? Are t... | Yes, it will be enough to fill the drive with zeros to defeat any data recovery program. However, a specialized laboratory breaking apart the drive and using special tools to inspect the platters may still be able to recover some data.
Tools for disk wiping overwrite the contents several times with some patterns (eg. ... | Hypothetically, even if the data is recoverable through an electron microscope, the costs are going to be astronomical (in the millions), and you would require a lot of proprietary manufacturer information to do so. The only entity that has the resources to pull it off would probably be the US government.
So unless yo... | https://security.stackexchange.com |
284,198 | [
"https://softwareengineering.stackexchange.com/questions/284198",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/165513/"
] | I'm developing a command-line tool that has many user options. Some of the options are lists of values. I'd like the program to be able to parse these listed options directly from the command line, and via a file (e.g. --opt=valA,valB and --opt=vals.txt).
Is it preferable to have two separate options for each case, or... | The test to apply is whether or not someone can definitively understand the intent of the arguments just by looking at them.
In your example, the interpretation of <code>--opt=foo</code> would vary depending on the program's surroundings. That could sow confusion or, worse, cause a security problem if there's a file ... | You've seen programs that interpret option arguments <em>both</em> as a filename <em>and</em> as literals? Depending on what, the state of the file system? That's an insidious defect or exploit waiting to happen. Never, never, ever do this.
| https://softwareengineering.stackexchange.com |
38,086 | [
"https://mechanics.stackexchange.com/questions/38086",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/13953/"
] | Will shifting my car to park while stopped at a red light cause any harm (engine, brakes, transmission, etc.) to my car? I do not apply the parking brakes. I have an automatic 2016 Toyota Corolla.
Some background: after my commute distance increased, I developed pain in my right knee. There are many traffic lights ... | In my experience, most manual cars can move along in 1st gear at idle – on a flat(ish) surface. I've only tried this in the process of teaching people to drive a manual. My hunch is that in 2nd and almost certainly in 3rd or higher you'd stall most engines.
But the question is "does the engine have enough torque (and ... | Obviously in a manual car with the clutch engaged there is a direct connection between the wheels and the engine. Also while switched on most engines have an idle setting designed to keep them running under no external load.
Whether a specific engine will drag itself forward on idle will depend on its torque characte... | https://mechanics.stackexchange.com |
25,749 | [
"https://dba.stackexchange.com/questions/25749",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/13340/"
] | I've got a stored procedure that refers to a linked server. In several places throughout the procedure I've got something like the following:
<pre><code>INSERT INTO [TableName]
(...Columns...)
SELECT ...Columns...
FROM [ServerName\InstanceName].[Catalogue].[dbo].[TableName]
WHERE TableNameID = @TableNameID
</code></pr... | The name of your linked server doesn't have to be the server's name. You can use a generic name.
<pre><code>EXEC master.dbo.sp_addlinkedserver
@server = N'COMMONNAME',
@srvproduct=N'MSDASQL',
@provider=N'SQLNCLI',
@provstr=N'DRIVER={SQL Server};SERVER=ACTUALSERVERNAME;UID=user1;PWD=rosebud567;',
... | I like the idea of using a generic linked server name. However, in many environments this may not be possible, In this case, you can use dynamic SQl in your sp.
<pre><code>declare @linkedservername nvarchar(200)
declare @sql nvarchar(4000)
SET @linkedservername = CASE @@ServerName
WHEN '... | https://dba.stackexchange.com |
51,908 | [
"https://stats.stackexchange.com/questions/51908",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/21844/"
] | Is there an algorithm for fast nearest neighbor search of circular dimensions?
e.g.,
For a dimension based on "hour of day", a KD-tree would place 00:01 and 23:59 far apart. But the proper distance metric would yield the shortest distance (2min).
Angle is another such dimension, or seasons, or months, or ...
I was ... | How many circular dimensions are there? Two of the tricks described below trade off resources for representational convenience, unfortunately exponentially on the number of circular dimensions.
There are at least three tricks you can use:
<ol>
<li>Use coordinate patches: create many small kd-trees, overlapping so th... | Try a clustering algorithm like DBScan. R and Weka both have this algo. It's density based. So if the clusters are continuous in nature for the day, it should do a good job of making a cluster for a day.
| https://stats.stackexchange.com |
155,987 | [
"https://mathoverflow.net/questions/155987",
"https://mathoverflow.net",
"https://mathoverflow.net/users/45397/"
] | Let $X$ be a projective scheme of pure dimension $1$. Let $U$ be a open subscheme and $j:U \to X$ the open immersion. Let $\mathcal{F}$ be a coherent sheaf on $U$.
Denote by $j_!(\mathcal{F})$ the extension of $\mathcal{F}$ by zero (as mentioned in Hartshorne, Algebraic geometry Ex. II.$1.19$). Is it true that $H^1(U,\... | Actually, this is almost never true.
<strong>Claim</strong> Let $X$ be a separated one-dimensional scheme of finite type over an algebraically closed field. Let $U\subseteq X$ be a dense open set that does not contain any irreducible component of $X$. Further let $\mathscr F$ be a coherent sheaf on $X$. Then
<ol>
<... | No, this is not true. This works for extensions from <strong>closed</strong> subsets. Take for $X$ an irrational curve, so that $H^1(X;\mathcal{O}_X)\ne0$, and let $U=X\setminus\text{point}$ and $\mathcal{F}=\mathcal{O}_U$, which is the restriction of $\mathcal{O}_X$. You immediately get $H^1(X,j_!\mathcal{F})\twoheadr... | https://mathoverflow.net |
53,752 | [
"https://chemistry.stackexchange.com/questions/53752",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/13295/"
] | I am planning on making some homemade soap soon and have read online that breathing in the fume generated from the reaction between lye and water is not recommended because it is "hazardous."
I am a little bit confused because I thought the reaction between lye and water is simply a highly exothermic dissolution react... | The fumes will contain trace amounts of NaOH. The fumes are microscopic water droplets, and these droplets will still be caustic, as they are essentially a solution of sodium hydroxide.
There is no real chemical reaction going on here, but rather a dissociation of the sodium and hydroxide ions, which is an exothermic ... | Add the water to the lye - the slow reaction with slow diffusion means there are no fumes. Misinformation about this (reference Soap Queen's alarmist and inaccurate instructions about mixing water and lye) has been pervasive on soap making websites
| https://chemistry.stackexchange.com |
309,925 | [
"https://mathoverflow.net/questions/309925",
"https://mathoverflow.net",
"https://mathoverflow.net/users/110229/"
] | For a Weyl group $W$, I would like to know whether each $w\in W$ can be expressed as $w=s_{\alpha_1}s_{\alpha_2}\cdots s_{\alpha_k}$ for some distinct positive roots $\{\alpha_1, \alpha_2, \cdots, \alpha_k\}\subseteq \Phi^+$.
I know for type A, the above is true.
Since $W(A_n)\cong S_{n+1}$ with the map $s_{e_i-e_j}... | The answer is "yes" and there is a geometric explanation.
Let $\mathcal{H}$ denote the set of hyperplanes corresponding to the reflections $s_\alpha$ with $\alpha\in\Phi^+$ (note that $s_\alpha=s_{-\alpha}$), and let $\Sigma$ denote the connected components of $V\setminus\bigcup_{H\in\mathcal{H}}H$ (where $V$ is the v... | After reading your comments, I come up with the following proof. I would like to know whether my proof is correct or not.
For weyl group $W$, each $w\in W$ can be expressed as $w=s_{\beta_l}\cdots s_{\beta_{2}} s_{\beta_1}$ for some distinct positive roots $\{\beta_1, \beta_2, \cdots, \beta_l\}\subseteq \Phi^+$.
<str... | https://mathoverflow.net |
97,000 | [
"https://mathoverflow.net/questions/97000",
"https://mathoverflow.net",
"https://mathoverflow.net/users/23731/"
] | Hi guys just a quick questions
What are the real life application of catalan numbers?
Thanks a lot!
| In all seriousness, this type of enumerative combinatorics is very useful e.g. in figuring out expected running time for computer programs. An example involving Catalan numbers (which may seem a little contrived, but I think things like this probably do really come up) is: count how many times you go through the inner... | This is discussed at great length in:
"Catalan numbers with applications" by Thomas Koshy. The main application seems to be to make money for the publisher (the book is insanely expensive), but google books has extracts.
| https://mathoverflow.net |
163,445 | [
"https://mathoverflow.net/questions/163445",
"https://mathoverflow.net",
"https://mathoverflow.net/users/42006/"
] | I am looking for advice concerning a specific situation related to centre manifold theory (compare Perko 2001).
<h2>The part which is known</h2>
Let's consider a differential equation in higher-dimensional Euclidean space which admits equilibrium points. Centre manifold theory yields the following nice results for indi... | You will want to look for <em>Normally Hyperbolic Invariant Manifolds</em> or NHIMs. Briefly said, these are invariant submanifolds (i.e. a curve of equilibria is a special case) with hyperbolic dynamics in the normal directions: the spectrum (in this case: eigenvalues) in the normal directions must be bounded away fro... | I believe Jaap provided some great references for Normally Hyperbolic Invariant Manifolds (NHIM), but I want to address what happens in the special case when the NHIM is a manifold of equilibria, near points at which normal hyperbolicity of these manifolds is violated (i.e., you get a zero eigenvalue in a direction tha... | https://mathoverflow.net |
31,650 | [
"https://mechanics.stackexchange.com/questions/31650",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/2274/"
] | I have had a squeak when I turn my steering wheel for about 6 months now, and it's driving my insane. The squeak shows up after a day in the rain, and lasts about a week after the last time driving in wet conditions.
I checked all the standard places for a squeak such as the steering stabilizer and trackbar joints b... | You didn't mention checking the ball joints (which are usually greasable) which are at the top and bottom of the knuckle or the U-joints on the axle shafts. Those are the most likely candidates.
| My money is on the power steering pump belt being loose. In wet conditions this will squeal, it'll do it when the pump is placed under load (when turning the wheel) and do it worst when the steering rack is at the limit of it's travel.
It wouldn't surprise me if the pump is located near the passenger side wheel well.... | https://mechanics.stackexchange.com |
144,756 | [
"https://dba.stackexchange.com/questions/144756",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/100847/"
] | I'm searching about how to disable triggers, and seems like it's not possible under MariaDB.
My problem is, how to manage then, when you import data.
I have a database where almost all business logic is inside it with triggers and stored procs. Probably, there are records autogenerated by triggers and stored-procs tha... | Best practice...
A database is a place to store data. It is the 'source of truth'.
The application is the place for
* interpreting the data
* "Business logic"
* Formatting and pretty-printing of output
* Cleansing the data as it is stored into the database.
It is often wise (especially in more complex systems) to h... | All the DBMS vendors suggest to create all the bussines logic in the database because that will asures the dependency to specific DBMS.
To my students I recommend to analyze each situation, new capabilites like column store or in memory structures help you to improve your app performance moving some logic to the datab... | https://dba.stackexchange.com |
22,641 | [
"https://electronics.stackexchange.com/questions/22641",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/6696/"
] | I've purchased a large batch of items, varying from capacitors to potentiometers to ICs and would not like to ruin the investment in a careless mistake.
I have a metal filing cabinet from IKEA that sits between the carpet and my wooden desk, would there be any harm from to the components just being in there?
I have m... | Styrofoam is ESD death, alas.<br>
Some people wrap it in Al foil but blowing up a few photos of ICs with this done will show you that its very very easy to get a pin through a hole in the AL that does not touch metal but does touch foam. Murphy will have no problem t all blowing up your better ICs this way. Wrapping l... | Some additions to what Russel said:
<ol>
<li>Make your environment 'reasonably' ESD safe. that includes NO STYROFOAM. A somewhat conductive floor and desktop will help a lot (old-style linoleum, wood). Don't put on your wool clothing. Have a few plants and water them well: keep humidity > 50%. no problem in my country... | https://electronics.stackexchange.com |
336,903 | [
"https://stats.stackexchange.com/questions/336903",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/133220/"
] | I am working on a class project where I compare the performance of GAN and WGAN. Since the only difference between GAN and WGAN is the Wasserstein loss, I chose one neural network model architecture and trained both GAN and WGAN (so, only the loss functions differ).
However, WGAN performs much worse than GAN, and I'm ... | Usually, the same architecture and parameters would not be good for training both GAN and WGAN.
In a typical GAN, you want to avoid making the discriminator more powerful than the generator, and you want to avoid training the discriminator so much that it "overpowers" the generator and always finds the fakes.
In WG... | Try to substitute gradient clipping with gradient penalty in WGAN, if you haven't done so yet. The important thing is that you should NOT use batch normalization in WGAN discriminator, as it breaks the whole idea. The authors of the WGAN paper suggest to use layer norm.
| https://stats.stackexchange.com |
29,866 | [
"https://mathoverflow.net/questions/29866",
"https://mathoverflow.net",
"https://mathoverflow.net/users/7144/"
] | I was wondering what would be the best way to present your paper at a conference, if your paper is selected for "short communication", lasting for about 15 minutes?
Should you concentrate on the main results or the proofs?
And what should a first-time presenter be wary of?
Thanks in advance.
| The first priority is to state your main results and explain why they are interesting (e.g. how they fit in with related work). With only 15 minutes you do not have much time to discuss proofs, but it is nice to give a brief outline of the proof of your main result and what is involved.
As a first-time presenter, I w... | Two further remarks.
The time devoted to one aspect of your work has not to be correlated to the time you spent on it.
Do not speed up because you see someone nodding or sleeping.
| https://mathoverflow.net |
99,593 | [
"https://softwareengineering.stackexchange.com/questions/99593",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/6866/"
] | I don't see much of a difference between the concerns of exception handling and logging in that both are cross cutting concerns. What do you think? Shouldn't it be handled separately on its own rather than interleaved with the core logic a method is implementing?
<strong>EDIT</strong>: What I am trying to say, is that... | <strong>In some cases yes</strong>
In the cases where you have an exception that you want logged (which I would assume to be almost always) then yes the exception is tied to a cross cutting concern.
<strong>Most of the time no</strong>
However take the instance of a SocketListener, if a socketlistener throws an exce... | Logging is optional. Handling exceptions isn't.
Logging is pretty generic and while it sources from specific logic it feeds to a generic consumer. Exceptions are always specific to the logic and some code knowledgeable about that logic has to handle the mess it made.
At least in my limited use of the two that means... | https://softwareengineering.stackexchange.com |
128,513 | [
"https://mathoverflow.net/questions/128513",
"https://mathoverflow.net",
"https://mathoverflow.net/users/9248/"
] | It is well-known that there exist pseudo-Anosov automorphisms of surfaces that act trivially on the homology: they form the Torelli group. Similarly there exists pseudo-Anosov automorphisms that act periodically.
On the other hand, given a fibered knot (in <span class="math-container">$S^3$</span>) with pseudo-Anosov m... | There exists an infinite family of quasipositive fibre surfaces of genus 3 with the same Alexander polynomial as the torus knot T(2,7). This answers Pierre's improved question, since quasipositive surfaces do not contain any essential unlinked annuli.
Let us first recall that the fibre surface S' of T(2,7) is obtained... | The simplest fibered knot with periodic homology monodromy is probably K = 11n74. Robert Riley dubbed this his "favorite knot" (see "Parabolic Representations of Knot Groups, I"). It is doubly slice, which means that K is the cross-section of an unknotted 2-sphere in 4-space.
The knot is a member of a family of knots... | https://mathoverflow.net |
197,199 | [
"https://physics.stackexchange.com/questions/197199",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/73136/"
] | Consider the case:
Assume molecules to sphere and their |<em>velocity</em>| as an indication of their temperature. Now for head on elastic collision (i.e transfer of heat), assume the momentum is conserved
Therefore doing math,
$m₁$ moving in $+x$ and $m₂$ in $−x$
$$(m₁×v₁)−(m₂×v₂) = (m₁−m₂)×v$$<br>
Where v is commo... | Zeroth law of thermodynamics? Zeroth law of thermodynamics is across systems. If they are colliding they are in the same system.
Temperate is not a common velocity. It is a measure of the average kinetic energy.
Common velocity? (m₁−m₂) × v is neither a proper application of zeroth law or temperature. You as... | Temperature is a statistical property of a large collection of particles moving and interacting at random. Talking about temperature for a pair particles colliding doesn't really make any sense. Your particles simply don't have a temperature and your result doesn't have anything to do with the zeroth law.
What you se... | https://physics.stackexchange.com |
104,933 | [
"https://mathoverflow.net/questions/104933",
"https://mathoverflow.net",
"https://mathoverflow.net/users/21892/"
] | In algebraic K-theory one defines $K_0(R)$ as the result of application of the Grothendieck construction to the semigroup of isomorphism classes of left f.g. projective $R$-modules.
But we can also consider the category of left f.g. $R$-modules and apply the same construction to obtain a group (let's call it $G(R)$).
W... | Typically the homomorphism here $K_0 \rightarrow G_0$ fails to be surjective. This shows up in a wide range of examples involving group algebras of finite groups (over fields of characteristic dividing the group order), restricted enveloping algebras of modular Lie algebras, etc. The homomorphism itself is often cal... | The map is an isomorphism iff every fg module has finite projective dimension iff (in the commutative case) R is regular.
| https://mathoverflow.net |
14,678 | [
"https://bioinformatics.stackexchange.com/questions/14678",
"https://bioinformatics.stackexchange.com",
"https://bioinformatics.stackexchange.com/users/9772/"
] | I have around 10,000 FASTA files of Influenza A virus.
These files contains sequences of each of the 8 segments of the viral genome and I want to separate each genome segment into a different folder and subdivide each directory into different files based on their geographic region of isolation.
In each FASTA file for e... | The general idea is:
<ol>
<li>iterate over all files
<ol>
<li>read the first line of each file, extract the <code>segment</code> bit</li>
<li>move the file based on that</li>
</ol>
</li>
</ol>
In Python, this can be done in many ways. Here’s one way:
<pre class="lang-py prettyprint-override"><code>import os
import re
... | One "out there" approach you could take is to create a fasta file containing your regions of interest (as separate sequences), and then map your fasta files to that reference fasta file, generating BAM/SAM output. After that, position-sort the BAM file, and you've now got a content-sorted list of sequences.
T... | https://bioinformatics.stackexchange.com |
166,848 | [
"https://physics.stackexchange.com/questions/166848",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/68625/"
] | She asked us if the body was accelerating or slowing down, and I immediately said that it was accelerating (because the $a=3>0$). Then she said that I was wrong because the direction of the acceleration vector was the opposite of the direction of initial speed($v_0=-3$). I do not understand why it slows down, becaus... | The initial velocity and acceleration here are in opposite directions. The magnitude of velocity (represented by $S=|\vec v|$) decreases upto a certain instant. (i.e. where $\vec v=0$).
Edit: Also, consider these graphs. ($t^.$ being the time where $v=0$) Note how the velocity increases but the magnitude of it (in the... | I understand what your teacher is saying, but I think she's wrong. In my physics classes we were told never to use the word "de"celeration, only acceleration. Why? For elegance reasons mostly.
<ul>
<li>Acceleration is a vector quantity, therefore its magnitude is always positive or zero with a direction. </li>
<li>Th... | https://physics.stackexchange.com |
40,235 | [
"https://dba.stackexchange.com/questions/40235",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/15787/"
] | We have a production box that has about 60 publications on it. Some of these are used, and some seem to not be... What I need to do is create, or find, a script that will get the count of subscribers/subscriptions per publication so I can easily identify unused publications.
We are using SQL Server 2005. :(
Thank you... | I was able to get the necessary info I needed by poking around a bit:
<pre><code>use Distribution;
select count(s.publisher_id) as 'Number of Subscribers', p.publication_id, p.Publisher_db, p.publication from MSpublications p
left outer join MSmerge_subscriptions s on p.publication_id = s.publication_id
group b... | Below script will give you information for all databases with replication info:
<pre><code>-- For Merge replication
-- Author: Kin Shah
DECLARE @Detail CHAR(1)
SET @Detail = 'Y'
CREATE TABLE #tmp_replcationInfo
(
articlename VARCHAR(128),
publisherserver VARCHAR(128),
publisherdb VARCHAR(128), ... | https://dba.stackexchange.com |
584,741 | [
"https://physics.stackexchange.com/questions/584741",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/260493/"
] | If we have two electrons in a state <span class="math-container">$|\psi\rangle=\frac{1}{\sqrt2}[|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle]$</span> and we measure the spin of the first electron to be up, does the wavefunction collapse into the state <span class="math-container">$|\psi\rangle=|\uparrow\downar... | I think the answer boils down to this - for electrons that can either exist in single particle spatial wavefunctions <span class="math-container">$\phi_A(r)$</span> and <span class="math-container">$\phi_B(r)$</span>, and single particle spin states <span class="math-container">$|\uparrow\rangle$</span> or <span class=... | I suppose <span class="math-container">$|A,B\rangle = |A\rangle |B\rangle$</span> is a "2 non-identical particle state" in your notation, so that your first spin state looks symmetric. This is impossible for identical electrons, unless there is a "spatial part" of the wave function that, in this cas... | https://physics.stackexchange.com |
408,237 | [
"https://mathoverflow.net/questions/408237",
"https://mathoverflow.net",
"https://mathoverflow.net/users/334560/"
] | Let <span class="math-container">$G$</span> be a reductive group and <span class="math-container">$X$</span> a smooth <span class="math-container">$G$</span>-variety. Then the fixed point subvariety <span class="math-container">$X^G$</span> is also smooth (this is theorem 13.1 of Milne's book on algebraic groups). Supp... | No. Let <span class="math-container">$C \subset \mathbb{P}^2$</span> be a smooth sextic curve, let <span class="math-container">$X$</span> be the double covering of <span class="math-container">$\mathbb{P}^2$</span> ramified over <span class="math-container">$C$</span>, and let <span class="math-container">$G = \mathbb... | We can generalize Sasha's counterexample in all dimensions as follows.
Let <span class="math-container">$f \colon X \to \mathbb{P}^n$</span> be a double cover branched over a smooth hypersurface <span class="math-container">$Y=Y_{2n+2}$</span> of degree <span class="math-container">$2n+2$</span>. Then <span class="math... | https://mathoverflow.net |
555,497 | [
"https://math.stackexchange.com/questions/555497",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/-1/"
] | let $x,y,z>0$ and such $xyz=1$ show that
$$\dfrac{1}{2(x+1)^2+1}+\dfrac{1}{2(y+1)^2+1}+\dfrac{1}{2(z+1)^2+1}\ge\dfrac{1}{3}$$
My try: I will find a value of the $k$ such
$$\dfrac{1}{2(x+1)^2+1}\ge\dfrac{1}{9}+k\ln{x}$$
note $\ln{x}+\ln{y}+\ln{z}=0$,so
$$\sum_{cyc}\dfrac{1}{2(x+1)^2+1}\ge\dfrac{1}{3}+k(\ln{x}+\ln{y}... | let $$x=\dfrac{bc}{a^2},y=\dfrac{ca}{b^2},z=\dfrac{ab}{c^2}$$
then we only prove follow inequality
$$\dfrac{a^4}{3a^4+2b^2c^2+4a^2bc}+\dfrac{b^4}{3b^4+2c^2a^2+4b^2ca}+\dfrac{c^4}{4c^4+2a^2b^2+4c^2ab}\ge\dfrac{1}{3}$$
By Cauchy-Schwarz inequality
\begin{align*}
\sum_{cyc}\dfrac{a^4}{3a^4+2b^2c^2+4a^2bc}&\ge\dfrac{... | $z=\dfrac{1}{xy}$, put in LHS and and clean the denominators, we have:
edit:
LHS-RHS=$ 9y^2x^4-8y^3x^3+2y^2x^3+9y^4x^2+2y^3x^2-9y^2x^2-8yx^2-8y^2x+2yx+9 \ge0 \iff $
$4y^2x^4-8yx^2+4\ge 0,\\4y^4x^2-8y^2x+4\ge0,\\5y^2x^4+5y^4x^2\ge 10x^3y^3,\\2y^2x^3+2y^3x^2\ge 4(xy)^{\frac{5}{2}} \iff\\ LHS \ge 2x^3y^3+4(xy)^{\frac{5}... | https://math.stackexchange.com |
213,484 | [
"https://electronics.stackexchange.com/questions/213484",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/84456/"
] | Some books talk about the natural response of the RLC circuit. This is when the voltage source is taken out from the circuit. Some other books talk about the transient response of an RLC circuit, which is the time it takes the circuit to reach to steady state. The equations seem similar. I googled and some people are s... | Natural response refers to the zero-input response, where only initial conditions generate the system response. Transient response refers to the system response to a time domain input signal, such as an impulse or a step. The exponential terms in both types of response will be closely related - having the same exponent... | I know this is a late answer, but I've just come across this question and wanted to add my point of view.
Suppose we have a generic dynamic (and linear) circuit with one or more external input sources and with some initial conditions at the dynamic elements. That implies that we have chosen an arbitrary time instant a... | https://electronics.stackexchange.com |
61,510 | [
"https://physics.stackexchange.com/questions/61510",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/23316/"
] | I would like to solve the angular part (the one for what is usually called the $\theta$ angle) of a time-independent 3D Schrodinger equation
$$
\frac{\mathrm{d}}{\mathrm{d}x}\left[ (1-x^2) \frac{\mathrm{d}P(x)}{\mathrm{d}x}
\right]+\left[ l(l+1) - \frac{m^2}{1-x^2} \right]P(x) = 0,
$$
where $l=0,1,2,\ldots$ and $m = -... | That's equivalent simply to $c\int dx/x$. Switch to the Euclidean spacetime, $k_0=ik_4$ where $(k_1,\dots k_4)$ is $k_E$; i.e. analytically continue in $k_0$ (Wick rotation). The integral is
$$\int \frac{i\cdot d^4 k_E}{(2\pi)^4} \frac{1}{(k_E^2)^2} \exp(ik\cdot \epsilon)$$
So it's proportional to the Fourier transform... | I will give another approach to this identity. First, we notice that
$$\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{(k^2-m^2)^2}e^{ik\cdot\epsilon}=-i\frac{\partial}{\partial m^2}D_F(x)\big|_{x=\epsilon}$$
For space-like vector $\epsilon^2=-r^2<0$, we have
$$D_F(x)=\frac{m}{4\pi^2r}K_1(mr)$$
whose derivation refers to We... | https://physics.stackexchange.com |
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