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19,326 | [
"https://softwareengineering.stackexchange.com/questions/19326",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/6894/"
] | I've had experience as a developer or team lead on several projects that have been outsourced and have seen less than stellar results in all cases. Most of these projects have employed a waterfall philosophy, with large kickoff meetings, months-long requirements gathering phases, plenty of conference calls, and innumer... | I have done it and I prefer it. The contract will have to be on a time and materials basis, rather than a fixed bid. The fixed bid theoretically moves some risk to the vendor and may be more competitively priced but in reality it generates a need for so much more work like you describe, not to mention fights over chang... | Agile development works best when it is fully implemented. Trying to combine just one or two agile concepts with waterfall development is a recipe for failure. Both clients and development teams must full embrace agile methodology. This means that both sides need to completely understand the advantages and restrictions... | https://softwareengineering.stackexchange.com |
27,469 | [
"https://scicomp.stackexchange.com/questions/27469",
"https://scicomp.stackexchange.com",
"https://scicomp.stackexchange.com/users/24942/"
] | I have two matrices $A, B$ coming from a finite element discretization of a system of partial differential equations. $A$ represents the system matrix and is symmetric and indefinite. $B$ is symmetric positive definite and is used as a preconditioner. Direct solver is used for computation of action of $B^{-1}$. The pre... | Look at the operators $$D_+^2 u = \frac{u_{n+2} - 2 u_{n+1} + u_n}{\Delta x^2}.$$
If you taylor expand this for small $\Delta x$ you arrive at
$$D_+^2 u = u_{xx} - \Delta x u_{xxx} + O(\Delta x^2)$$.
Thus $D_+^2 = D^2$ in the limit as $\Delta x \to 0$ (as it should), but note the error is first order in $\Delta x$ and ... | Your matrices are not well defined. The second derivative operator for example, takes three consecutive data values and outputs one second derivative value. That is not expressed by writing a square matrix.
There is a well-recognised Calculus of Finite Differences. You can find books with that phrase in the title, an... | https://scicomp.stackexchange.com |
198,164 | [
"https://softwareengineering.stackexchange.com/questions/198164",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/43804/"
] | I have a View Controller or a View that implements <code>delegate</code> and <code>dataSource</code> methods of <code>UITableView</code>.
These methods are not going to be used outside this class.
So is it a good idea to make the declaration of the <code>UITableViewDataSource</code> and <code>UITableViewDelegate</cod... | <blockquote>
So is it a good idea to make the declaration of the
UITableViewDataSource and UITableViewDelegate inside the .m file with
a private category?
</blockquote>
That depends. Is the fact that <code>MyViewController</code> these protocols a private implementation detail, or something that you want to comm... | Technically, it's of no consequence. Interface Builder will recognize that your class implements the protocols and offer a connection.
Apple code puts the protocols implemented in the header, and it's common practice. So anyone browsing your code will expect to see them there.
I think you are over thinking it hiding... | https://softwareengineering.stackexchange.com |
440,364 | [
"https://physics.stackexchange.com/questions/440364",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/210564/"
] | According to my understanding:
<ul>
<li>two orthogonal forces aren't related and two orthogonal vectors don't affect each other</li>
<li>the force of static friction <span class="math-container">$F_s$</span> depends on the normal force <span class="math-container">$F_n$</span>, so <span class="math-container">$$F_s = ... | Re. "two orthogonal forces aren't related and two orthogonal vectors don't affect each other"
Your equation <span class="math-container">$F_s = \mu_s F_n$</span> is not a vector equation. <span class="math-container">$F_s$</span> and <span class="math-container">$F_n$</span> are both scalars. If they were vectors, a... | There are two different things here that need to be firmed up:
<ol>
<li>The first is the notion that "two orthogonal vectors don't affect each other". I am not sure what is meant by this statement, but I suppose you are thinking how that <em>adding</em> one of a pair of orthogonal vectors to a given vector does not af... | https://physics.stackexchange.com |
23,668 | [
"https://chemistry.stackexchange.com/questions/23668",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/9298/"
] | What is the $\ce{OH}$ functional group in alcohol (ethanol)? My teacher told me it isn't the same as the $\ce{OH-}$ ion or hydroxide.<br/>
But it does donate one electron to the respective carbon chain and it's very tempting to say it's hydroxide as it fits perfectly into the role. What exactly is it?
| You could call ethyl alcohol "ethyl hydroxide" analogously to "sodium hydroxide". But it is a poor analogy, since (unlike NaOH) ethyl alcohol is not an ionic compound, does not dissociate in water to form ethyl cation and hydroxide anion, and does not raise the pH when dissolved in water.
| Well, I don’t understand what you are trying to ask.
But the $\ce{OH}$ you are bothered about can be thought exactly as that of $\ce{OH}$ of water ($\ce{H2O}$)
$\ce{C2H5-OH}$ in the same way as $\ce{H-OH}$
| https://chemistry.stackexchange.com |
1,347,425 | [
"https://math.stackexchange.com/questions/1347425",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/218389/"
] | To be specific, say we are given that $(a_n)$ is a sequence of real numbers such that \begin{equation} \sum_{n=1}^{\infty} n^3 a_n^2 < \infty. \end{equation}
Is it then true that
$$ \sum_{n=1}^{\infty} a_n < \infty ?$$
I believe the answer is yes, because it seems that $$\sum_{n=1}^{\infty} n^3 a_n^2 < \infty... | The sum $\sum n^3 a_n^2$ converges, and therefore $\lim_{n\to\infty} n^3a_n^2=0$. So after a while the terms $n^3a_n^2$ are $\lt 1$. So after a while $|a_n|\lt \frac{1}{n^{3/2}}$. By the Comparison Test we have absolute convergence, and therefore convergence.
| By the Cauchy-Schwarz inequality,
$$ \left(\sum_{n\geq 1}n^3 a_n^2\right)\cdot\left( \sum_{n\geq 1}\frac{1}{n^3}\right) \geq \left(\sum_{n\geq 1}|a_n|\right)^2 \tag{1}$$
hence:
$$ \sum_{n\geq 1}|a_n| \leq \sqrt{\zeta(3)\cdot\sum_{n\geq 1}n^3 a_n^2}.\tag{2}$$
| https://math.stackexchange.com |
99,766 | [
"https://electronics.stackexchange.com/questions/99766",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/23330/"
] | Ok This might sound dumb but I am trying to get an equation for the radiation intensity of a pyramidal horn antenna in spherical coordinates. I got a whole bunch of books and they all have the same equation:
$$
\mathbf{E}_\theta = \frac{j\, e^{-j k r}\, E_0\, A\, B\, (1 + \cos(\theta))\, \sin(\phi) ...}{8\, \lambda\, r... | For a compound box antenna the formula is similar and this might be useful: -
<img src="https://i.stack.imgur.com/pM66w.jpg" alt="enter image description here">
Obviously they are not using A and B to denote the aperture dimensions but, I suspect that "r" is going to be exactly the same i.e. distance between centre o... | Spherical coordinates are normally denoted by r, Θ, and φ. Your equation is in terms of those coordinates. r is the distance from the source origin (horn antenna) to the point given by the values of r, Θ, and φ. Thus you already have the answer to your question.
| https://electronics.stackexchange.com |
128,734 | [
"https://softwareengineering.stackexchange.com/questions/128734",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/26215/"
] | In Javascript there are a few clearly prominent techniques for create and manage classes/namespaces in javascript.
I am curious what situations warrant using one technique vs. the other. I want to pick one and stick with it moving forward.
I write enterprise code that is maintained and shared across multiple teams,... | <strong>Update</strong>
I now have a much better understanding of javascript and feel I can properly address the question. I think it was a poorly worded, but very important javascript topic to address.
The Self Executing Anonymous Function pattern is not one which requires use of the new keyword if you avoid the use... | Self executing anonymous functions are used to automate script execution without hooking into external events (i.e. window.onload).
In this example it is used to form the classic Module pattern, the primary purpose of which is to introduce a namespace into the global environment, and provide <strong>encapsulation</st... | https://softwareengineering.stackexchange.com |
575,853 | [
"https://physics.stackexchange.com/questions/575853",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/212124/"
] | I live near equator region of Earth. During winters specifically, I see water vapors forming over the surface of the water body <strong>in the morning</strong>. Why do they form?
| Air has less thermal mass than water. Without solar radiation, on a still night the air temperature will drop a lot, while a large body of water may cool very little. This can leave the surface of the water significantly above the dewpoint of the surrounding air.
Evaporation from the warmer water condenses as it is c... | In modern climates <em>dew</em> and <em>fog</em> are often formed in the morning as a result of condensation, since night temperatures are colder than the temperatures in the daytime. <em>Dew point</em> is a term that specifically means the temperature when the vapor becomes saturated and its excess must appear as cond... | https://physics.stackexchange.com |
19,250 | [
"https://electronics.stackexchange.com/questions/19250",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/5617/"
] | How is it possible to construct a "light intensity varier" (increse/decrease lighting) of a 220V bulb using pwm? The main requirement is to make use of pwm
I was thinking of this connection:
<pre><code>220V mains --> rectifier --> smoothing --> regular (to convert to dc) --> relay --> 220V bulb
where o... | If the relay is a mechanical one it's an Awfully Bad Idea™. Relays need several ms to activate and release, so the switching frequency will be limited to about 100Hz. Your actual control cycle will be much slower than this so you'll end up with noticeable <strong>flicker</strong> to start with. The relay will also make... | I take it by "220V bulb" you mean a normal standard household 220V AC bulb?
Rctifying (and smoothing) 220VAC would give you something more in the order of 310VDC which would not be good for the bulb.
So switching to using straight AC might be an option, yes?
Alas, no - not with PWM.
You'd be wanting to switch the A... | https://electronics.stackexchange.com |
9,642 | [
"https://chemistry.stackexchange.com/questions/9642",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/4356/"
] | <blockquote>
<strong>Question</strong><br>
Use activities to calculate the pH of each of the following solutions, being sure to use $\mathrm{\alpha}$ values and the extended Debye-Huckel equation.
The first one the concentration is $0.020 \mathrm{M}$ $\ce{NaCl}$
with the answer being : $\mathrm{pH}$ = $6.99$... | Let's first look at ionic strength:
Ionic strength is a measure of the total ion concentration in solution, however, species with higher charge can also have elctrostatic interactions with other species.
$$\mu = \frac{1}{2} \sum_i C_iZ_i^2 $$
What is the strength of 0.02 M NaCl solution?
$$\mu = \frac{1}{2}([\ce{Na... | Using the formula as:
$$\log \gamma_\ce{H+} = \frac{- 0.51(z_i^2)\sqrt{\mu}}{1 + \sqrt{\mu}}$$
instead of your formula , this formula gives more accurate answer:
$$\frac{-0.51 \times \sqrt{0.02}}{1+\sqrt{0.02}}=-0.063188664981$$
then
$$10^{-(-0.063188664981)}
=1.063188664981$$
then
$$\mathrm{pH}=-\log(1.063188664981 ... | https://chemistry.stackexchange.com |
283,250 | [
"https://softwareengineering.stackexchange.com/questions/283250",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/179235/"
] | Ive started coding for Wordpress and the coding standards recommend using Yoda conditions.
<pre><code>if ( true == $the_force ) {
$victorious = you_will( $be );
}
</code></pre>
But I also do some Android development, Javascript and have plans for robotics programming and some languages don't like Yoda style.
Im ... | Coding standards are there to make it easy to visually parse the code. They are often different for each language, and rightly so.
If someone else is going to read this code, then keep to the Wordpress conventions when programming for Wordpress. If you are the only person who will read the code, then do what makes it ... | Unless you are an extremely error-prone programmer with no code reviewers or automated linters to back you up, this particular choice is unlikely to be anything other than a stylistic preference with potentially minor benefits, so go right ahead.
It's worth noting that coding styles shouldn't necessarily be similar ac... | https://softwareengineering.stackexchange.com |
299,525 | [
"https://physics.stackexchange.com/questions/299525",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/117991/"
] | A satellite revolves around the earth. Air pressure inside the satellite is maintained at 76 cm of Hg. What will be the height of mercury column in a barometer tube 1m long placed in the satellite.
I think it should be the same 76 cm. Can the rotation of the satellite affect the air pressure inside the room?
| It would not be the same 76 cm, because a satellite in orbit is in free fall. So the mercury in the barometer will behave as if it is weightless. All the weight of the mercury is essentially "used up" in maintaining centripetal acceleration in the circular orbit. If, as usual, the tubing above the mercury is initial... | If the pressure inside is maintained to standard 1 atm by means of suppose a heavy gas, then it would mean that when the satellite revolves the entire satellite system is under free fall, all the gas atoms ,the barometer tube ,the mercury in it ,all are under free fall . Thus in such a situation the gas atoms won't pro... | https://physics.stackexchange.com |
356,336 | [
"https://electronics.stackexchange.com/questions/356336",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/90278/"
] | I want to know if there is an approximate or exact way to know the consumption in watts of household appliances in a house from the monthly consumption that reaches the bill.
Taking into account that it arrives in kWh and I want to convert any value, which would be the best option.
<strong>305 kWh</strong> to W
| If something uses N kWh in a 30-day month, has an average consumption of
$$N \frac{1000\text{ W/kW}}{720\text{ h/month}}\text{ watts}$$
In other words, to convert "energy" to "power", divide by "time".
<hr>
Just as an example, suppose you use a 1500 W toaster for 3 minutes to make toast for breakfast. That would be... | If <strong>energy</strong> consumption in one month is 305kWh the mean (i.e. average) <strong>power</strong> consumption during that month is \$\frac{305kWh}{30 \times 24h}=\frac{305000}{720}W\approx 423W\$ (assuming a 30 day month).
| https://electronics.stackexchange.com |
530,905 | [
"https://math.stackexchange.com/questions/530905",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/100712/"
] | Given a fair dice, what is the expected number of sixes that get tossed before throwing 2 sixes in a row? Would I start by finding all possible sets that occur without two sixes in a row and multiply them by the corresponding probability? ie
$$\sum_{k=0}^\infty (A+B)^k $$ where A is all sequences that don't contain a... | The set of sequences prior to the final two sixes that end the game is actually given by
$$\sum_{k=0}^\infty (A+BA)^k $$
The generating function that counts the number of sixes and non-sixes prior to the final two sixes is<br>
$$C(a,b)=\sum_{k=0}^\infty (a+ba)^k={1\over 1-(a+ba)}.$$
You turn this into a joint probabi... | Denote by $E_\ne$ the expected number of additional sixes until the game ends when there was no six immediately before, and by $E_=$ the expected number of additional sixes when there was a six immediately before. Then
$$E_\ne={5\over6}E_\ne+{1\over6}(1+E_=),\quad E_= ={5\over6}E_\ne+{1\over6}\ 1\ ,$$
from which we obt... | https://math.stackexchange.com |
729,839 | [
"https://physics.stackexchange.com/questions/729839",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/346966/"
] | I've been studying physics recently particularly the uniform circular motion topic. I have read about an example that an object attached to a string moving in a vertical circular path is not a uniform circular motion due to the weight of the object always acting downward so the tension in the string varies at each poin... | An object <em>can</em> move in a vertical circular path at constant speed as long as there is a mechanism for exerting a torque on the object so as to keep the kinetic energy of the object constant (if there is no torque exerted then the kinetic energy of the object varies because its potential energy varies as it move... | We simplify the problem so it is a uniform circular motion.
In the real world it's not just like free falling body (think about air resistance).
| https://physics.stackexchange.com |
376,775 | [
"https://physics.stackexchange.com/questions/376775",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/117467/"
] | Homogeneity means that something is the same in every point in space, whereas isotropic means that there is no directionality. Uniformity implies there is "no variation".
But what is actually the difference between being homogeneous and uniform, e.g. for a field?
| In my experience, 'homogeneous' is related to mixture, a property of a multi constituent body, and 'uniform' a property of a a single entity. Example: Eletric field can be uniform and a distribution of matter can be homogeneous.
In fact, the general meaning of 'homogeneous' is: of the same kind; alike; consisting of p... | Similar to the answer of Nogueira and others comments.
It seems to me that homogeneity is used when matter is considered (tough, it can be in the presence of other components, as in cosmology) while uniformity has a more geometrical connotation. Therefore we use homogeneous for a mixture (meaning fine enough, if not ... | https://physics.stackexchange.com |
217,864 | [
"https://electronics.stackexchange.com/questions/217864",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/52544/"
] | Folks,
What is the difference between "Amp pk or Apk" vs "Amp pk-pk (or Amp peak to peak?)"
Are they the same thing?
| Amp pk is the peak value of the current while Amp pk-pk is the peak to peak value of the current. Thus they are not the same. For example, current in the form of a sine wave that has a peak value of 1 ampere has a peak to peak value of 2 amperes.
| Voltage has a "peak to peak" ration, which is usually written as Vp-p. However, unlike voltage, which is actually a measure of <em>difference</em> betwee2 potentials (rather than an absolute value), current is usally expressed as an absolute value (in number of columbs passing a given point per second [amps]).
Thus, w... | https://electronics.stackexchange.com |
255,113 | [
"https://security.stackexchange.com/questions/255113",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/266990/"
] | I have a simple Angular application and I deployed it on an Nginx server in a POD on a Kubernetes environment. I realised that I am now able to access the /etc/passwd file on the POD, without even having to login into the POD. Since this is the /etc/passwd file for a POD, should I be concerned?
A sample command that I ... | First, your calculations are way off. 62<sup>17</sup> is about 10<sup>30</sup>.
<blockquote>
at a rate of 10 million attempts per second
</blockquote>
Here is another problem. You cannot throw 10 million requests per second at Google without being blocked even before the first second ends. Google will surely find the b... | Let me flip your entire approach (and logic) around.
Because you know the search space, 62<sup>17</sup>, you already know all the URLs.
What you want to know is which of those URLs <em>return content</em>.
So, sure, you could ping each one and look for image files being returned, and you could create a list of all URLs... | https://security.stackexchange.com |
182,723 | [
"https://dba.stackexchange.com/questions/182723",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/10567/"
] | I'm having a brain-not-working moment and could really use some help. This is SQL Server 2014 Standard ed.
I have 3 Tables - for simplicity, we'll call them "name", "address", and "international". Sample data below:
<pre><code>Name
-----
Id Sequence Name
--- -------- ------
101 1 Pluto
102 1 ... | There is almost certainly a better way to do this but this one works and will probably get you started on a better solution.
<pre><code>SELECT nameid.id,
STUFF((SELECT '|' + RTRIM(res.name)
FROM name res
WHERE nameid.id = res.id
FOR XML PATH('') , TYPE).v... | That works! Thank you! :-)
I ended up making a small adjustment/change but it works! Thank you! I need to keep the names in the right order so I adjusted the order numbers. I need to retrieve only the first 3 name sequences even if there are more.
<pre><code> ;WITH myCTE AS (
SELECT id, seq as ord, ... | https://dba.stackexchange.com |
220,887 | [
"https://mathoverflow.net/questions/220887",
"https://mathoverflow.net",
"https://mathoverflow.net/users/56819/"
] | The simplest form of Finsler metric is:
$ds (X, dx)=\sqrt[4]{g_{\alpha, \beta, \gamma, \delta}(X).dx^{\alpha}dx^{\beta}dx^{\gamma}dx^{\delta}}$, where $g_{\alpha, \beta, \gamma, \delta}$ is a fourth degree polynomial that smoothly depends on $X$.
My question is: is it always true that $s^4(X)$, the fourth power of the... | It turns out that the answer is 'No, the fourth power of the geodesic distance from a point $p$ in a Finsler space $M$ whose norm, raised to the fourth power, is a smooth convex quartic form is <em>not</em> generally a smooth function on a neighborhood of $p$'.
While the desired smoothness does hold in the Minkowski... | Looking on the Minkowski Finlser metric (when the Finsler function does not depend on the point of the manifold) we see that the 4th power of the distance function can not be very smooth. Indeed, for the Minkowski metric generated by the norm $F:R^n \to R^n$ the function $s$ for $O=\vec 0$ is simply $s(y)=F(y)$. So ... | https://mathoverflow.net |
50,442 | [
"https://astronomy.stackexchange.com/questions/50442",
"https://astronomy.stackexchange.com",
"https://astronomy.stackexchange.com/users/46022/"
] | If we had a very powerful telescope with say a 150-meter diameter, placed in orbit around earth, would it be possible to get a side view rather than a top view of a person standing on the moon?
| Yes.
The person just has to be standing somewhere near the <em>limb</em> of the moon. That is near the edge of the apparent disc of the moon.
We see everything near the limb of the moon from the side, at all times. This is because the moon is (roughly) a sphere. It's not flat.
| Yes, but only just. The best-case (called "diffraction-limited") angular resolution of a telescope (i.e. the smallest detail it can make out) is given by
<span class="math-container">$$ \theta = 1.22 \frac{\lambda}{D} $$</span>
Where <span class="math-container">$\lambda$</span> is the wavelength of light you... | https://astronomy.stackexchange.com |
84,052 | [
"https://electronics.stackexchange.com/questions/84052",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/7548/"
] | So I know thermo cameras are sold. Police/Fire use them all the time. You point your little hand held black box at a bush and you can see if there is a creature living in there or at a burn zone and see where the hot spots are.
Is there a device that will do the same for RF signals? VHF up to 2.4GHz
I was thinking ... | So I find it very interesting that all answers until now seem to think in terms of pre-1900s radio technology. In order to productively think about portable or reasonably sized radio imaging techniques, you have to think a bit differently.
The way to receive electromagnetic waves is to produce a material that is opaqu... | The longer the wavelength of the radiation, the larger a sensor you need to detect it. Radio waves, with a wavelength starting in the millimeters, require far too large a sensor to detect in the same manner.
| https://electronics.stackexchange.com |
146,579 | [
"https://electronics.stackexchange.com/questions/146579",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/50937/"
] | <img src="https://i.stack.imgur.com/CmDO9.png" alt="circuit schematic" />
I've designed this circuit myself aiming to generate a sine wave at the collector of the transistor, but it doesn't work. The current at the collector is DC.
<ul>
<li>Is it a simulator error or the is design wrong?</li>
<li>What modifications sho... | The basic problem is that there is no feedback thru the resonating components. Note that they are tied to the base directly from the power supply. There is nothing that will make them oscillate.
There are many L-C oscillator circuits out there. No matter what topology you use, the frequency-setting components eithe... | There is no reason for the transistor to change its operatin point in steady state conditions (after the input transient).
An oscillator needs its perturbation at its output to feed into its input. This is not happening here.
| https://electronics.stackexchange.com |
284,202 | [
"https://mathoverflow.net/questions/284202",
"https://mathoverflow.net",
"https://mathoverflow.net/users/114580/"
] | A pair $(X,\tau )$ is called a generalized topological space if $\tau$ is collection of subsets of $X$ so that $\emptyset \in \tau$ and $\tau$ is closed under arbitrary unions. A subset $A$ of GTS $(X,\tau )$ is strongly nowhere dense if for any nonempty open set $U\in \tau$, there exists nonempty open set $V \subset ... | I add another answer in light of your comment to the answer above. Here I construct a generalized topological space (which is not a topological space) along with a non-empty strongly nowhere dense subset.
Let $X = \{0,1,2,3\}$ and let $$\tau = \big\{\{0,1\}, \{1,2\}, \{0,1,2\}, X\big\}.$$
Note that $X$ is the only me... | Let $(X,\tau)$ be any generalized topological space. If $A\subseteq X$ is open and $A\neq \emptyset$, it cannot be be strongly nowhere dense: Let $U=A$, then for every nonempty open set $V\subseteq U$ we have $V \cap A = V\cap U = V \neq \emptyset$.
So $A = \emptyset$ is the only strongly nowhere dense subset of $(X,\... | https://mathoverflow.net |
859,909 | [
"https://math.stackexchange.com/questions/859909",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/157095/"
] | How is $$log_42= \frac{1}{2}$$ ?
Any formula to how we calculate this?
I know i am confused when base is larger digit than log value term.
| In general $$\log_{g}(a)=\frac{\log(a)}{\log(g)}.$$ So $\log_4(2)=\frac{\log(2)}{\log(4)}=\frac{\log(2)}{2\log(2)}=\frac{1}{2}$.
| The question "what is $\log 2$ to base 4?" is equivalent to the question "what power of 4 is equal to 2?", by the definition of what a logarithm to a base means.
Thus, you just have to ask yourself what number we need to insert into this:
$$4^w = 2$$ to make it work.
| https://math.stackexchange.com |
35,643 | [
"https://mathoverflow.net/questions/35643",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4047/"
] | Suppose I have a symmetric $N \times N$ matrix A which has a one-dimensional Nullspace $N$. A is positive definite on $N^\bot$. In my case $N$ is the space of constant vectors (i.e. generated by the all-one vector).
I have to solve the problem $Ax = b$, with $b \in R(A)$ which has infinitely many solutions. I am l... | I'd suggest you to shift away the singularity: solve $(A+ee^T)y=b$ instead and then orthogonalize $y$ with respect to $e$ to get $x$. $A+ee^T$ is not sparse but you can compute matrix-vector products cheaply, and that's all you need for CG.
EDIT: forgot to define it, $e$ is the vector of all ones
| Plain CG, without the regularization, will converge to the pseudo-inverse solution provided you start the iteration with $y=0$. The regularized (preconditioned) solution will be different and the operation count will depend on the sparsity, so it's good to maintain this if you can. I guess partly it must depend on whe... | https://mathoverflow.net |
44,226 | [
"https://mathoverflow.net/questions/44226",
"https://mathoverflow.net",
"https://mathoverflow.net/users/-1/"
] | Let $D$ denote the unit complex 1-dimensional disc, together with the hyperbolic metric $h_D=\frac{4dz\wedge d\bar{z}}{(1-|z|^2)^2}$of curvature $-1$. By Nash's embedding theorem, we can always embed the disc $D$ real-analytically and isometrically into real Euclidean space ${\mathbb{R}}^n$ for some large $n$. (I think... | The answer is 'no', there is no holomorphic curve in $\mathbb{C}^n$ (for any $n$) such that the induced metric has constant negative curvature. To my knowledge, this was first proved by E. Calabi many many years ago, essentially using the structure equations for holomorphic curves in $\mathbb{C}^n$. The proof is easy... | There is a body of work on embedding Riemann surfaces into $\mathbb{C}^n$ for small $n$ (it's possible for large $n$). If I recall correctly, Riemann surfaces can be holomorphically embedded in $\mathbb{C}^3$ easily, but the sharp lower bound that is known for complex manifolds of complex dimension $n > 1$ (which i... | https://mathoverflow.net |
206,365 | [
"https://stats.stackexchange.com/questions/206365",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/11363/"
] | Let's say I have a system, where given a current state $s$ and my action $a$ I can sample successor states $s'$. I am given a policy $\pi$ such that $\pi(s,a)$ is the probability of action $a$ taken in state $s$.
Let's say, I would like to evaluate $V^\pi$ using this sampler. That's easy, at least in theory: I start w... | In statistics, bias is clearly a property of the estimator.
I share your observation that bias is often incorrectly applied to estimates. Your example seems rather innocent in that regard, because a well-meaning instructor could argue that your students assumed that the error of the estimates is so small that it's OK ... | Bias is the property of an estimator.
An <em>estimator</em> is itself a random variable and has a distribution (with a mean and variance). When an estimator has an expected value that is equal to the true, unknown value that it's trying to estimate we say the estimator is unbiased.
Now, when we calculate an <em>est... | https://stats.stackexchange.com |
57,064 | [
"https://stats.stackexchange.com/questions/57064",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/24770/"
] | I have a set of data points with an uncertainty on each point. From these data points I can fit a line, who's which the slope is a significant value. How do I use the information I have to get an estimate on the $ \sigma $ of my final value? All I can find online is information on $ R^2$ and $ \chi ^2 $. Bonus points i... | <pre><code>What is the expected distribution of residuals?
</code></pre>
It varies with the model in ways that make this impossible to answer generally.
<pre><code>For example, should the residuals be distributed normally?
</code></pre>
Not generally, no.
| There is a whole cottage industry centered around designing residuals for GLMs that are more symmetric or even approximately "normal" (i.e. Gaussian), e.g. Pearson residuals, Anscombe residuals, (adjusted) deviance residuals, etc. See for example Chapter 6 of James W. Hardin and Joseph M. Hilbe (2007) "Generalized Line... | https://stats.stackexchange.com |
46,958 | [
"https://dba.stackexchange.com/questions/46958",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/26051/"
] | I have been asked for recommendation on the optimal RDBMS for a large amount of data. The largest table would contain 2TB of data. My own largest table is only 400GB (mysql, performs very well).
The individual rows are going to be short, no blobs/etc, just a lookup table.
Is PostgreSQL nowadays performing better th... | I would go with MySQL (InnoDB engine of course) if your table is partitioned. PostgreSQL partitioning implementation with master-child tables and trigger based partition selection looks somewhat complicated for me. On the other hand in Oracle to use partitions one have to use Enterprise Edition ($47500 for 2 cores!) wi... | The size of the tables really doesn't matter all that much. It's all about how well indexed the tables are and how many people are using the database at once.
In my case I've got clients with billions of rows in a single table where the table size is about 1 TB and another 2 billion row table which has blobs. These t... | https://dba.stackexchange.com |
4,319,557 | [
"https://math.stackexchange.com/questions/4319557",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/884388/"
] | Are there any nowhere continuous functions <span class="math-container">$f: \mathbb{R} \to \mathbb{R}$</span> satisfying the identity <span class="math-container">$f(f(x)) = (f(x))^2$</span> <span class="math-container">$\forall x \in \mathbb{R}$</span>?
| Take some indicator-like function
<span class="math-container">$$
f\colon \Bbb R \to \Bbb R, x\mapsto \begin{cases}1 &\text{if}\ x\in\Bbb Q,\\ -1 &\text{if}\ x\in\Bbb R\setminus \Bbb Q.\end{cases}
$$</span>
Then <span class="math-container">$f(-1) = f(1) = 1 = (-1)^2=1^2$</span>.
| Perhaps useful to see where this example comes from. The first step is to note that if you have a function of the form
<span class="math-container">$$
g_{a, b}(x) = a\mathbf{1}_{\mathbb{Q}}(x) + b\mathbf{1}_{\mathbb{R} \setminus \mathbb{Q}}(x),
$$</span>
then this is everywhere discontinuous provided that <span class=... | https://math.stackexchange.com |
678,194 | [
"https://physics.stackexchange.com/questions/678194",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/312432/"
] | In the real world, it seems that traveling backwards in time is impossible, but do we have a theorem in physics that would imply this fact?
Some people (including Feynman) describe antiparticles as moving in the opposite direction of time coordinate axis. For example, the Dirac field involves an integral of the term
<s... | There are two aspects to this. First we can clarify the quantum field theory a bit. Secondly we need to distinguish the use of the concept "time" in discussions of quantum unitary evolution (Schrodinger's equation) from the concept of "time" in complicated evolution leading to thermodynamic irrevers... | "It seems that nothing is able to forbid the existence of a system with time reversed, but also we're unable to detect it (or we did but we didn't know that). I'm looking for someone who has an explanation for this."
First, let's define what we mean by "time." Take all the particles in the universe... | https://physics.stackexchange.com |
126,784 | [
"https://softwareengineering.stackexchange.com/questions/126784",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/20487/"
] | I've just started my first IT job in a team with two developers (junior and senior), a project leader and an architect. My role is not yet completely defined and I said I'd be ok with both programming or testing. For now they put me on use case analysis since the PL says that it would leverage my analytical skills (I'm... | <blockquote>
who's usually responsible for use case analysis?
</blockquote>
Everyone.
<blockquote>
How usual is it to put a newbie on it?
</blockquote>
Not too unusual. You may have appropriate problem domain knowledge that makes you more valuable working with the users.
<blockquote>
The hidden fear I have i... | Were you hired as a software developer? It doesn't sound like it to me.
You said your role is not completely defined. This means they can ask you to do almost anything that is realistically related to IT. This includes analysis work. And, if they want to leverage your analytical skills, it sounds like you might be doi... | https://softwareengineering.stackexchange.com |
40,703 | [
"https://dba.stackexchange.com/questions/40703",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/6173/"
] | I have a <code>#tempTable</code> that was created using
<pre><code>SELECT *
INTO #tempTable
FROM OPENROWSET('Microsoft.Ace.OLEDB.12.0', 'Excel 8.0;Database=MyFileName.xls',
'SELECT * FROM [Sheet1$]')
</code></pre>
Is there an easy way to copy the table structure of the <code>#tempTable</code> to a new physical table... | Why not use the same approach you used to create the #temp table?
<pre><code>SELECT * INTO dbo.NewTable FROM #tempTable;
</code></pre>
| Yes, just do :
<pre><code>select *
into dbo.NewTable
FROM #tempTable
</code></pre>
| https://dba.stackexchange.com |
374,138 | [
"https://physics.stackexchange.com/questions/374138",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/166180/"
] | I've recently read about the Cobalt-60 experiment and Parity violation on weak interactions, what says the parity is not a symmetry on weak interactions.
But if I did not understand wrong EM and strong interactions parity isn't violated, so is it a symmetry? Could we apply the noether's theorem to reflections just lik... | Noether's theorem only applies to continuous transformations. Parity is discrete.
| The conserved quantity is parity. Even states remain even, odd states remain odd. Discrete symmetries lead to discrete conserved quantities.
| https://physics.stackexchange.com |
289,690 | [
"https://physics.stackexchange.com/questions/289690",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/21146/"
] | Let us consider a system with a certain volume $V$ and at a particular temperature $T_0$. Suppose further that the pressure is increased adiabatically from $P_0$ to $P_1$. I want to be able to determine the change in temperature $\Delta T$.
My initial idea was: the change in entropy can be written (using the Gibbs rep... | Given any small volume in velocity space, the patch near the origin will have the highest probability of having particles. But there aren't many patches near 0.
Not far from the origin, the probability is still pretty high, and there is more volume to integrate over.
Farther away, there is lots of volume to integra... | This question bothered me for a long time also, and I do not find statisfying answers elsewhere. I just leave here an explanation that I think is correct.
The reason for that the Maxwell-Boltzmann distribution <span class="math-container">$g(v)=C_1 e^{(-C_2 v^2)} 4\pi v^2$</span> seems nonintuitive comes from two aspec... | https://physics.stackexchange.com |
50,007 | [
"https://mechanics.stackexchange.com/questions/50007",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/12816/"
] | I'm seeing auto-darkening welding helmets selling for barely more than the price of a conventional mask. But I also rarely see professional welders using auto-darkening hoods. Is there a reason for that?
As an amateur welder I find it tricky to setup on a weld, drop my mask, and then strike a clean arc before I can ... | There are a few reasons.
Firstly the first few generations of auto lenses had several drawbacks: they were expensive, had a restricted field of vision, and in some cases didn't give such a clear view of the weld as fixed shade lenses – especially metal coated ones. This is particularly important for MMA and TIG weldi... | Just because the cost of auto darkening is the same now, that hasn't always been the case. When they were new, the tech was much more expensive. When you're starting out and buying tools, you don't have much money, so you get the basics you can afford and build from that.
If you start with a manual hood, you're goin... | https://mechanics.stackexchange.com |
74,359 | [
"https://security.stackexchange.com/questions/74359",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/55406/"
] | I am making a web site which can be considered similar to Reddit in some ways. I need to implement a sign up form so I will store informations about the users. My idea is to store the username, password, email address, and if they want also the gender, birthday and country.
There are many SSL certificates, from free to... | When you do the DDoS, you will be sending a flood of information. If you fully own the server, and you are sending it from your own server, then the 'send' and 'receive' points will be fine. But you still have to account for all the other machines inbetween.
If this is fully in an internal network you own, then there ... | If the server's on your own network, feel free. But if it's hosted somewhere, then your DDOS could impact other people using the same network path or other infrastructure. You could ask permission, but they'll almost certainly say no - SPs are generally comfortable with pentesting but nobody fools around with DDOS te... | https://security.stackexchange.com |
296,073 | [
"https://math.stackexchange.com/questions/296073",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/44402/"
] | So this is my question:<br>
Compute $7^{818} \pmod {1637}$ using no more than 14 multiplications mod 1637. (You should of course verify that 1637 is prime if you plan to use Fermat's Theorem.)
I would know how to do this problem if the power that 7 is raised to is higher than 1637. So if I had this:<br>
$$ 2^{200}\pmo... | <ol>
<li>$7^2=7\times 7$</li>
<li>$7^3=7^2\times7$</li>
<li>$7^6=7^3\times7^3$</li>
<li>$7^{12}=7^6\times7^6$</li>
<li>$7^{24}=7^{12}\times7^{12}$</li>
<li>$7^{48}=7^{24}\times7^{24}$</li>
<li>$7^{51}=7^{48}\times7^3$</li>
<li>$7^{102}=7^{51}\times7^{51}$</li>
<li>$7^{204}=7^{102}\times7^{102}$</li>
<li>$7^{408}=7^{204... | Using the binary representation of 818, yields:
$$818_{10} = 1100110010_2$$
There are ones in the $2^{1}, 2^{4}, 2^{5}, 2^{8}, 2^{9}$ positions.
So we can write:
$$7^{818} \pmod {1637} = 7^{(2+16+32+256+512)} \pmod {1637}$$
Using repeated squaring (this hugely reduces your workload), we have:
$$7^{2} \equiv 49 \p... | https://math.stackexchange.com |
4,052 | [
"https://dba.stackexchange.com/questions/4052",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/-1/"
] | Using the psql command line tool, how do I list all postgres tables in one particular schema
| <code>\dt schemaname.*</code> will do what you want.
| In addition to the <code>\dt</code> match, you can also look into the database catalog:
<pre><code>SELECT nspname||'.'||relname AS full_rel_name
FROM pg_class, pg_namespace
WHERE relnamespace = pg_namespace.oid
AND nspname = 'yourschemaname'
AND relkind = 'r';
</code></pre>
You can also do it with the more s... | https://dba.stackexchange.com |
267,226 | [
"https://mathoverflow.net/questions/267226",
"https://mathoverflow.net",
"https://mathoverflow.net/users/108456/"
] | In my physics research I came across a mathematical proposition (translated into the mathematical language from the physical problem) that I feel to be true, and would like to prove it:
<strong>Proposition: Consider the function $f(m)=\cos(\frac{2m+1}{N}\pi)$ where the integer $m$ satisfies $0\leq m<N$ and the inte... | This proof uses the reference suggested by Felipe Voloch in a comment below the answer provided by Anthony Quas:
In Conway and Jones, Acta Arith. XXX (1976) 229-240, Theorem 7 (abbreviated to CJ7 in the following discussions) states that:
Suppose we have at most four distinct rational multiples of $\pi$ lying strictl... | Let $\zeta=e^{2\pi i/N}$ and $\alpha=e^{2\pi i/(2N)}$. Your assumption was $\Re(\alpha(\zeta^{m_3}-2\zeta^{m_2}+\zeta^{m_1}))=0$. This is equivalent to $\alpha(\zeta^{m_3}-2\zeta^{m_2}+\zeta^{m_1}))+\bar\alpha(\zeta^{-m_3}-2\zeta^{-m_2}+\zeta^{-m_1})=0$. Multiplying by $\bar\alpha$ and using the fact that $\alpha^2=\ze... | https://mathoverflow.net |
14,499 | [
"https://mathoverflow.net/questions/14499",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3873/"
] | Forgive the elementary nature of the question. I understand that the second order Peano Axioms are categorical in the sense that all their models are isomorphic. This equivalence class of models is taken to be the definition of natural numbers.
My question is that: Is the theory defined by those second order axioms c... | Note that there are two different types of models of second-order logic: standard models, where second-order quantified variables range over all subsets of the domain; and Henkin models, where second-order quantified variables are allowed to range over a proper subset of the full power-set.
Henkin proved the completen... | If a theory has only one model, then of course it is complete, in the sense that any
statement or its negation is a consequence of the theory, because either it holds in the unique model or it doesn't.
As you mention, there is only one model of second-order PA, and this is the model of the natural numbers. Thus, sec... | https://mathoverflow.net |
29,466 | [
"https://stats.stackexchange.com/questions/29466",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/9490/"
] | Given a set $A$ of $n=kl$ positive real numbers, let $a_{ij}$ be a random number removed from the set $A$ (without replacement) for $i=1 \dots k$ and $j=1 \dots l$. What is the distribution and expected value of the following expression?
$$
\prod_{i=1}^{k} \sum_{j=1}^{l} a_{ij}
$$
Note: Using a set of 30 numbers for... | Provided $(a_{ij})$ is not "too bad," <strong>for large $k$ and large $l$ the distribution is approximately lognormal.</strong>
This follows because the sums will be approximately normal (by virtue of the Central Limit Theorem) and (for large $k$) almost iid. The product of a large number of iid (or almost iid) rando... | If my assumptions are right then contrary to the smooth curve shown above the distribution is actually discrete. The probability is P[X=Lrs] =1/(kl) for each r=1,2,...,k and s=1,2,...,l Lrs = ∏ ∑ aij summed over all i not equal to r and j not equal to s. The expectation is ∑∑Lij/(kl) for i=1,2,...k and j=1,2,...,l. ... | https://stats.stackexchange.com |
60,474 | [
"https://electronics.stackexchange.com/questions/60474",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/3337/"
] | Inside a defunct garage door opener I found this IC:
<img src="https://farm9.staticflickr.com/8377/8547054800_f2f46109d2_z.jpg" alt="33192C">
I tried searching for <em>33192C</em> with variations of 'datasheet' and such keywords, but I got routed to the usual datasheet farms who are happy to let me know that they don... | Looking through the Microchip KeeLoq "code hopping encoder designed for secure Remote Keyless Entry (RKE) systems" datasheets, I'm 99% sure that this is one of them, either rebranded for a customer, or a close custom version.
<img src="https://i.stack.imgur.com/jjKrR.png" alt="enter image description here">
Look for ... | That is a microchip logo. They bought a Israeli rolling code generator company for garage openers in the mid -90's. So that might help you. Called KeeLoq (I think)
| https://electronics.stackexchange.com |
121,400 | [
"https://physics.stackexchange.com/questions/121400",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/51858/"
] | I have a question about the following problem:
<blockquote>
<img src="https://i.stack.imgur.com/nOtw4.png" alt="enter image description here">
</blockquote>
I got the solution $v=\frac{M+m}{m} \sqrt{2gh}$.
But my real question is in the following picture:
<blockquote>
<img src="https://i.stack.imgur.com/gVtk4.p... | The equation in the green box has the correct dimensions, but you are correct, the two sides of the equation in the blue box must have the same dimensions too (and should probably be cm/s in the example shown, but as we can't see what M/m is or what units anything else has been given in, the whole thing is baffling.
U... | Units of measurements and conversion factors can be included in a numeric constant. When doing this it is necessary to specify the units of each quantity in the expression and it is mandatory to use that units.
This may look confusing, but if you master the technique you can write very simple formulae fixing a number ... | https://physics.stackexchange.com |
160,057 | [
"https://security.stackexchange.com/questions/160057",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/148954/"
] | I am learning about file formats. I have an application that will do analytics and will have some personal information. I need to make it such that only some people are able to access data and even those people get very limited access according to an api that I will provide. I am thinking about encryption as well as us... | <blockquote>
I am thinking about encryption as well as using a proprietary format to make it difficult to reverse engineer anything.....
I want to create a file format that will be <strong>difficult to reverse engineer so that application is safe</strong>.
</blockquote>
A file format which is difficult to reverse... | Creating a file format that is difficult to reverse-engineer is something you shouldn't aim for because it's likely to be unnecessarily time-consuming and in the end ineffective.
If a user has access to the application, it's often trivial to reverse-engineer the format by altering small amounts of data in the applicat... | https://security.stackexchange.com |
77,504 | [
"https://softwareengineering.stackexchange.com/questions/77504",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/22390/"
] | I name my variables using the .Net conventions:
<ul>
<li>camelCase for variables and fields (I tend to use _camelCase for private fields in a class)</li>
<li>PascalCase for methods, properties and classes</li>
</ul>
The only place I deviate is on constants and Enums where I actually prefer the Java SCREAMING_CAPS sty... | The only thing you should care about is that you're working in a team where people don't care about cleaning up things a bit. That is very sad.
Do what you do, continue to use the modern style and invite people (but not force them) to adopt it as well. It will take time of course. After some time you will see whether ... | I think you might need to ask yourself whether or not the hungarian notation is affecting your personal output/quality, or if it is more just hurting your ego. By ego, I mean that things generally work fine, but you would never want someone you respect from outside to see the shamefully outdated code. While I think th... | https://softwareengineering.stackexchange.com |
4,619 | [
"https://electronics.stackexchange.com/questions/4619",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/1319/"
] | I need a number of digital outputs to connect my computer to the real world, however it seems that this job is not nearly as easy as I had hoped.
I've looked into a number of different methods, ranging from dedicated digital I/O cards, micro controllers with USB interfaces, serial ports, parallel ports, ect. However a... | Sounds like you intend to flash some christmas lights. :-)
Anything wrong with using an arduino or similar?
It would be fairly easy to expand the number of IO ports if the number of ports say on the mega 1280/2560 aren't enough. You can drive it via serial/USB terminal. You could use shift registers on the output pins... | The serial to parallel shift register will work. Using the SPI port you will
have no problem with the 1KHz update rate. IIRC on an ATmega328 with an 8MHz crystal (or higher) you should be able to get 1Mbits per second. A lot of other microcontrollers
will work as well.
Another option is to use multiple microcontroller... | https://electronics.stackexchange.com |
23,102 | [
"https://earthscience.stackexchange.com/questions/23102",
"https://earthscience.stackexchange.com",
"https://earthscience.stackexchange.com/users/23287/"
] | How does manure & other organic matter improve soil structure? Based on my understanding, soil structure refers to the arrangement of the aggregates within the soil; a <em>good</em> soil structure prevents waterlogging, "nutrient-lockup" (and ultimately the death of the crops). However, I don't understand... | <h2>Surface area</h2>
Organic material in soil is finely divided and provides a large surface area to mass ratio. Clay particles have a similarly large surface area, but most clays stick to each other because of their micro-structure and chemistry, making soils less permeable and prone to over-compaction.
Organic matte... | Manure is natural fertilizer.
Organic material has Carbon in it, also contains proteins, carbs, etc. All the stuff that break down and feed plants over time. Plants don't feed directly but micros and fungi do and break it down to water soluable nutrients.
As a structure organic matter retains water, clay and sand are l... | https://earthscience.stackexchange.com |
49,357 | [
"https://mathoverflow.net/questions/49357",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3901/"
] | In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to the exceptional Lie Groups. I am not precisely sure what I am looking for, but ostensibly there should be answers forth coming from other who ... | I promised Sean a detailed answer, so here it is.
As José has already mentioned, it is only $G_2$ (of the five exceptional Lie groups) which can arise as the holonomy group of a Riemannian manifold. Berger's classification in the 1950's could not rule it out, and neither could he rule out the Lie group $\mathrm{Spin}(... | If you are willing to move away from Riemannian geometry, but still within
differential geometry, then
$G_2$ arises in a number of surprising and simple geometric circumstances,
but it is not the compact $G_2$ of the answers above, but rather the `split' $G_2$.
(Each simple Lie algebra has a compact real form and ... | https://mathoverflow.net |
282,270 | [
"https://physics.stackexchange.com/questions/282270",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | We know that Schwarzchild metric describes an asymptotically flat spacetime. This means that far away from the event horizon we can safely interpret the $r$ coordinate as distance from the center.
But when close enough to the event horizon the curvature becomes significant and our common sense of $r$ breaks.
The qu... | The Schwarzschild $r$ coordinate at a point is defined as the circumference of the circle passing through that point and centred on the mass divided by $2\pi$. This definition applies at all distances even inside the event horizon.
To see this take the Schwarzschild metric:
$$ ds^2 = -\left(1-\frac{r_s}{r}\right)dt^2... | You seem to have some serious conceptual misunderstandings about the subject. Co-ordinates are not "measured" they are defined on a space time manifold. Proper time on the other hand can be measured. So please think about the concepts correctly. I will recommend reading Einsteins works on the subject.
| https://physics.stackexchange.com |
175,940 | [
"https://dba.stackexchange.com/questions/175940",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/102975/"
] | I have two tables with following structure:
First Table:
<pre><code>ID Name Total_courses_allowed
1 Ahmad 4
2 Khan 3
3 Mahmood 2
</code></pre>
Second Table:
<pre><code>ID Course_attended emp_id
1 f 1
2 t 1
3 f 1
4 ... | I'm sorry if it is not well formatted but I'm on mobile.
You can get it by counting attended courses and getting those records where total is lower.
<pre><code>select id, name, total_couses_allowed
from table1
inner join (select emp_id, count(*) as courses
from table2
where course_attended = '... | <pre>
select id, name, Total_courses_allowed ,t2.real_Course_attended
from FirstTable t1
inner join (select emp_id, count(emp_id) as courses ,count(Course_attended) real_Course_attended
from SecondTable
where course_attended = 't'
group by emp_id,Course_attended) t2
on t1.id = t2.emp... | https://dba.stackexchange.com |
25,413 | [
"https://quant.stackexchange.com/questions/25413",
"https://quant.stackexchange.com",
"https://quant.stackexchange.com/users/20256/"
] | I have read that volatility smiles didn't show up for equity options until 1987. Can some one give me an example(s) of what markets now still don't have volatility smiles and what an explanation for them not having equity smiles would be?
thanks
| There are tons of market where vol smile doesn't exist - either because no one makes a market on the call/put options (private equity, physical real estate comes to mind) or only the ATM option gets traded infrequently. You can't have volatility smile without a vol market.
On the other hand (and maybe more relevant t... | The VIX, has a concave shape for its option's Implied volatility.
| https://quant.stackexchange.com |
164,119 | [
"https://stats.stackexchange.com/questions/164119",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/83069/"
] | If I have 2 data points like
<pre><code>16.70 (+/-4.33)
54.70 (+/-16.30)
</code></pre>
where 16.7 = mean and +/- 4.33 is the standard deviation, is it possible to calculate a range out of that? I would think that roughly, the min range here would be 16.7-4.33 and the max 54.7+16.30, so the range would be:
<pre><code... | Assume the population from which these data points are taken is normal (mean $\mu$, variance $\sigma^2$). I think your question is ill-posed because there <em>is</em> no population parameter that you can call "the" range here, and there is no use in trying to estimate it. Think of it this way: if the number of data poi... | If all you know is the population (i.e., true, not sample) mean and standard deviation, and you neither know nor assume anything else about the distribution of the random variable, then here's what you can conclude about the range of the random variable.
<ol>
<li>If standard deviation = 0, you can conclude that the ra... | https://stats.stackexchange.com |
78,507 | [
"https://electronics.stackexchange.com/questions/78507",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/27312/"
] | My system requires a power which is supplied by a Li-ion battery. However, I need to keep this battery charging at all time so it won't die. Is it possible to connect the battery to my system and also have it connected to the MAX1555 which is charging it? (think of it as using a cell phone while it's always plugged in... | What you're describing is an "always-plugged-in" device with a "backup" battery. In addition, the "backup" battery may be able to provide significantly more current than the built-in power source, for short cycles, and when the load is removed, the battery is charged back up.
LiPo batteries don't like staying at top v... | <strong>Think about what you are asking</strong>
You say: -
<blockquote>
I need to keep this Li-Po charging at all time so it won't die
</blockquote>
"Charging at all time" <em>MUST</em> mean you have power <strong>all the time</strong> so why not get rid of the LiPo and feed the circuit (load) from a regular pow... | https://electronics.stackexchange.com |
534,898 | [
"https://physics.stackexchange.com/questions/534898",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/247001/"
] | I have an <span class="math-container">$x-t$</span> graph (position-time) which its equation is <span class="math-container">$x=e^t$</span>.
If we get derivative of it, the velocity's equation will be <span class="math-container">$e^t$</span> too.
How can a particle's position and velocity be the same? what does it m... | Remember that these quantities have units. What does <span class="math-container">$\exp\left(1\,\mathrm{s}\right)$</span> mean? How does it compare to <span class="math-container">$\exp\left(1\,\mathrm{min}\right)$</span>?
Generally you can only take non-polynomial functions* of dimensionless quantities. So for your e... | There is no physical significance of the equations velocity and position w.r.t time being same simultaneously but what we can understand from this is that both phenomenon change simultaneously<br>
It is possible for position and velocity to be same but both of them depend on the frame of reference at any point of time ... | https://physics.stackexchange.com |
67,796 | [
"https://cs.stackexchange.com/questions/67796",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/29119/"
] | I'm not exactly sure whether this question belongs to this SE or not, but it doesn't seem to fit elsewhere.
Say a process behaves in a deterministic fashion (that is, it gives the same output when an input is entered over and over again), then <strong>can we build it as an algorithm?</strong>; if not, can you think of... | An algorithm must have a finite description, whereas a deterministic process according to your definition can use an infinite lookup table. For example there is a deterministic process that solves the halting problem for Turing machines, but there is no such algorithm.
| No. Deterministic, to me, means that the state of a system at any given point in time is a function of its state at any previous point in time. So all one needs to show a difference is an example of a system that evolves according to a function that isn't computable. For instance encode programs as Godel numbers. The s... | https://cs.stackexchange.com |
9,800 | [
"https://physics.stackexchange.com/questions/9800",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/904/"
] | If I see the steam coming out of cooling towers at an altitude of 200 meters, I cannot help but think that this energy is wasted. My question is: Why isn't this steam cooled enough to become water and used to create hydroelectric power due to the drop from 200 meters?
| There are a number of issues with recovering the energy (mechanical or thermal) from the stack exhaust of a cooling tower.
First, the temperature and energy (enthalpy) of the exhaust is too low to effectively recover. The exhaust is full saturated or over saturated air typically at a temperature of 110 F, not above 12... | Let us see -- how much water per second is getting at the height?
Let's estimate it as 1 ton of water per second.
Then the energy you can extract form the water is:<br>
$$ 1 [\mbox{ton/s}] * 200 [meters] * g \simeq 20000000 J/s = 2 \mbox{megaWatts}$$
Compared to hundreds or even thousands of megawatts produced and th... | https://physics.stackexchange.com |
2,480,085 | [
"https://math.stackexchange.com/questions/2480085",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/290738/"
] | <blockquote>
Show that the map $$\psi : \{0,1\}^\mathbb{N} \to \mathbb{R}, \;\;\;\; a=(a_n)_{n \in \mathbb{N}} \mapsto \sum^\infty_{n=1} \frac{a_n}{10^n} $$ is injective.
</blockquote>
In my head, and after a bit of thought, I could see that $\psi$ is indeed injective, but I'm unsure of how to explicitly show this f... | We have that
\begin{align}
f(a)=f(b)\iff f(a)-f(b)=0
&\iff \sum_{n=1}^{\infty}\frac{a_n-b_n}{10^n}=0\\
&\iff \sum_{n=1}^{\infty}\frac{a_n-b_n+1}{10^n}=\sum_{n=1}^{\infty}\frac{1}{10^n}
\end{align}
With $d_n=a_n-b_n+1$, we have that $d_n\in \{0,1,2\}$, so the LHS is the number with decimal representation $0.d_... | <strong>Hint:</strong> Decimal expansions of real numbers are (almost) unique, and the caveat leading to my inclusion of the word "almost" doesn't arise here.
| https://math.stackexchange.com |
7,512 | [
"https://chemistry.stackexchange.com/questions/7512",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/4042/"
] | Is it an ion or a neutral atom that is created through beta decay?
For example: $_6^{14}\mathrm C \to ~ _7^{14}\mathrm N + \mathrm e^- + \mathrm{v{_e}}^{-}$
Isn't $_7^{14}N$ an ion since neutron gives us only a proton and an electron, not a proton and 2 electrons? If so, why isn't it stated in the example above (it's... | An ion is created. Usually this is not noted down in the nuclear reaction because while talking of nuclear reactions we only concern ourselves about <em>nuclei</em>, not the entire atom. The reaction is to be read as "A 14-$\ce{C}$ nucleus decays into a 14-$\ce{N}$, an electron, and an antineutrino."
Also, note that t... | Further to @ManishEarth's answer, note that the distinction is important in doing mass-energy calculations. In the above example, you can find the energy released by subtracting the <strong><em>tabulated</em></strong> mass of a $N^{14}$ atom from the <strong><em>tabulated</em></strong> mass of the $C^{14}$ atom, and i... | https://chemistry.stackexchange.com |
1,824 | [
"https://ai.stackexchange.com/questions/1824",
"https://ai.stackexchange.com",
"https://ai.stackexchange.com/users/1982/"
] | I'm reading such nonsense about how an AI would turn the world into a supercomputer to solve a problem that it thought it needed to solve. That wouldn't be AI. That's procedural programming stuck in some loop nonsense. An AI would need to evolve and re-organise its neurons. It wouldn't be stuck to hardcode if it become... | <h2>It's a possible side effect</h2>
Any goal-oriented agent might, well, simply do things that achieve its goals while disregarding side effects that don't matter for these goals.
If my goals include a tidy living space, I may transform my yard to a nice, flat lawn or pavement while wiping out the complex ecosystem ... | It's not necessarily a nonsense. It all depends on the imposed criteria. Imagine the following. Say an advanced AI system is designed to control the stability of the local fauna and flora (area enclosed in some kind of a dome). It can control the pressure under the dome, the amount of light that goes through the dome e... | https://ai.stackexchange.com |
335,541 | [
"https://electronics.stackexchange.com/questions/335541",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/166383/"
] | I am doing a project and need to detect if a specific signal in the form of 2 + Asin(2πft) where A is between 1 and 4 and f is 20Hz - 20kHz. Once this id detected I will light an LED.
I was thinking of using op-amp comparators but they can only detect a range of voltages right?
| So, in the case that this is a Circuits 101 assignment, and accuracy isn't that important, and you have to use basic components: maybe do this in three stages, inaccurately, but enough for your assignment. This is kind of a draft, I'm not going to sit down and do the math, sorry:
<strong>First:</strong>
A bandpass f... | You are trying to detect a signal by both amplitude and frequency. You don't give any parameters of accuracy, response time, harmonic distortion, what signals can look like that you need to rejectd, etc, so anything will do.
This is a job for a microcontroller. The maximum range of your signal is -2 V to + 6 V. Use... | https://electronics.stackexchange.com |
483,464 | [
"https://physics.stackexchange.com/questions/483464",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | When I apply the Force to an area , say at an angle <span class="math-container">$\theta$</span> with the vertical, is the pressure defined as
<span class="math-container">$$ P =\frac{ \Delta F}{\Delta A_{Perpendicular} } =\frac{ \Delta F}{\Delta A \cos\theta} $$</span>
Or
<span class="math-container">$$ P =\frac{ \D... | <blockquote>
An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction.
_source: wilkipedia
</blockquote>
In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a h... | <em><strong>When a ball hits a wall (ground), some quantity of its momentum must be transferred to the wall at rest.</strong></em>
That is correct.
<em><strong>If else the ball would bounce back with the exact same speed as it did when it collided with the wall. This would result in the ball traveling up to the exact h... | https://physics.stackexchange.com |
92,036 | [
"https://security.stackexchange.com/questions/92036",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/79001/"
] | I have searched and viewed most of the other questions and answers about implementing forgot password functionality on a website.
However, we have a unique situation. When a user registers an account and uses their company-based email as the main email account, how can we implement a forgot password scheme if they no ... | Ultimately, you need to establish some kind of secret with the client that is:
<ol>
<li>easily remembered</li>
<li>unrelated to their email/password combination</li>
<li>non-trivial to guess</li>
</ol>
Since you've ruled out security questions (which are iffy at best due to reason #3), you're left with a few options.... | You can offer entering a phone number and send password recovery message there (like Microsoft does) or using some things that only the user knows (e.g. ID of last ivoice)
| https://security.stackexchange.com |
165,525 | [
"https://dba.stackexchange.com/questions/165525",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/104370/"
] | I'm inserting data into a new table in a new database, based on a select to an existing table in an other database. The existing table has some 1500 million rows, and I am inner joining a table with about 10% of that. Both existing tables, are indexed on C5, while the the new table is indexed on C2, which is an unixtim... | You are totally trashing the index on NewTable.C2<br>
Just drop / disable the index and then build / rebuild it after the insert is complete
And you need to break this up as you are putting a massive load on the transaction log
If you want to optimize the select then just run the select alone
<pre><code>INSERT ... | Yes an index on the table you are selecting on will improve query performance (for the select).
However as the table you are creating is huge the query is going to take a long time anyway. Especially as a single insert.
Personally i would be looking for a way to break the insert into blocks or batches.
As pararazz... | https://dba.stackexchange.com |
259,359 | [
"https://dba.stackexchange.com/questions/259359",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/26130/"
] | I'm currently running PostgreSQL 9.4 in my production environment, but we're in the process of doing an OS upgrade on the server so I thought it might be time to upgrade our PostgreSQL version to the latest (12) so that I could potentially make use of some of the new features available. However, several of the tables i... | If the <code>oid</code>s of the tables are referenced in the code, you cannot simply get rid of them. Probably the old code relies on <code>oid</code> as an auto-generated identifier. Are there any primary keys or other indexes on these columns?
A way to preserve the <code>oid</code>s over an upgrade would be:
<ul>
<... | Here is a query to help find all tables that have the <code>WITH OIDS</code> set. It also generates the DDL to remove.
<pre><code> SELECT 'ALTER TABLE "' || n.nspname || '"."' || c.relname || '" SET WITHOUT OIDS;'
FROM pg_catalog.pg_class c
LEFT JOIN pg_catalog.pg_namespace n ON n.oid = c.relname... | https://dba.stackexchange.com |
451,116 | [
"https://physics.stackexchange.com/questions/451116",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/218397/"
] | A bit confused here - if a satellite approaches Earth, its displacement is in the same direction as the gravitational force (towards the centre of rotation). Therefore, gravity is doing positive work?
I read in my textbook that "negative work is done against the force of gravity" when a satellite approaches Earth. Is ... | The work done by a force on some object is defined as <span class="math-container">$\int\mathbf F\cdot\text d\mathbf x$</span>, where the line integral is over the path the object takes.
Therefore, if the object moves towards the Earth in the direction of the force of gravity, it must be that the force of gravity does... | <em>In this anwsner i will assume that there's only the gravitational force acting on the Satellite</em>.
The satellite is moving towards Earth i.e. the particle moves from <span class="math-container">$r+\delta$</span> to <span class="math-container">$r$</span>, then <span class="math-container">$W=\int_{r+\delta}^{r... | https://physics.stackexchange.com |
1,151 | [
"https://electronics.stackexchange.com/questions/1151",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/3/"
] | I just spent about 20 minutes trying to cut through a PCB with a hacksaw. These things are really tough.
What is the correct tool to cut PCB material, and get a nice straight line?
I saw something about a guillotine, but can't find a vendor.
| The first thing I should mention is safety. You need to wear breathing
protection since the dust created by cutting is very hazardous to breathe.
Safety glasses should always be worn. As Norm would say -- read and understand
all the manufacturers' instructions for your tools.
I use either a hacksaw, bandsaw or tin-snip... | I don't know if this is the best way, but I usually score it multiple times with a heavy-duty knife (on both sides) and snap it by hand. Obviously this won't work for any non-straight cuts. The knife I use is similar to one of these <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Cutter_Stanley.jpeg... | https://electronics.stackexchange.com |
46,022 | [
"https://mathoverflow.net/questions/46022",
"https://mathoverflow.net",
"https://mathoverflow.net/users/9534/"
] | I just found an English translation of Serre's FAC at Richard Borcherds' Algebraic Geometry course web page. I really want to read it sometime. I am beginner in Algebraic Geometry, just started learning Scheme theory (Hartshorne Ch II). My question is :
Shall I read coherent sheaves and the cohomology from the transla... | As always, the source you use may be related to what your goals are. To give some perspective, recall there are several ways to define sheaf cohomology, and Serre and Hartshorne feature different methods. Serre used Cech cohomology, and there the important long exact sequence property does not always hold. He was ab... | Dear J, here is a little technical warning which might be relevant to your question.
If you open Hartshorne and read the definition of "coherent" (Chapter II, §5, page 111) you might get the impression that the structural sheaf $ \mathcal O $ of a scheme is coherent: after all the scheme can be covered by affine ope... | https://mathoverflow.net |
92,978 | [
"https://physics.stackexchange.com/questions/92978",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/37039/"
] | Does a closed cage with an helium balloon weigh the same as an empty closed cage?
What if the balloon isn't touching the ceiling ? (It's a tall cage and we're weighing it in the process of the balloon elevating)
| It makes no difference whether or not the balloon touches the ceiling. The weight of the cage can always be determined from the contained mass, which is less with a helium ballon inside.
| A closed cage with a helium balloon doesn't weight the same as an empty cage because the mas of a helium balloon adds to the the mass of the empty cage. A cage with a helium balloon doesn't weight less it floats (according to the Archimedes Law) in a more dense medium.
| https://physics.stackexchange.com |
298,849 | [
"https://stats.stackexchange.com/questions/298849",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/133284/"
] | In the book Deep Learning, it says the softmax function is de facto a soft argmax function, and the corresponding soft version of the maximum function is <span class="math-container">$$\text{softmax} (z)^T z$$</span>
How to understand the latter?
| Consider the function
<span class="math-container">$\text{hardmax}(z)^Tz$</span>
for <span class="math-container">$z = [1, 2, 3, 4, 5]$</span>
where hardmax is a hard version of softmax, which returns 1 for the maximum component and 0 for all the other components.
Then we will have
<span class="math-container">$[0, 0, ... | <code>softmax</code> is a smooth approximation of the <code>argmax</code> function,* taking a vector and returning a vector:
<span class="math-container">$$\text{softmax}(x) = \frac{e^{\beta x}}{\sum{e^{\beta x}}} \to \text{argmax}(x)$$</span>
This takes a vector as input and returns a vector as output (a one-hot enco... | https://stats.stackexchange.com |
52,977 | [
"https://biology.stackexchange.com/questions/52977",
"https://biology.stackexchange.com",
"https://biology.stackexchange.com/users/23553/"
] | If life formed on earth by natural laws, why can't we observe the formation of life from matter today? Is it because this is a rare phenomenon? It seems just after formation of earth life formed on earth and around 2-3 billion years has passed. It must be enough time for life to form several times from scratch.
| The first life was probably so fragile and simple we would likely not even recognise it if it did appear. We don't know exactly how, or where, life first appeared but nearly all theories suggest simple collections of enzymes in protected environments that slowly gather the mechanisms needed for free-living life over ti... | Well.. consider this... once life already exist... and a new life form spontaneous came into came into existence, how likely is the newcomer going to be able to compete with the current contender, who now has time to develop all sorts of abilities from being in an arms race between members of its own group. (A bit like... | https://biology.stackexchange.com |
211,137 | [
"https://electronics.stackexchange.com/questions/211137",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/97002/"
] | I want to design a circuit to trigger a relay depending on the house current load:
When current load is below 12A the relay must turn on and when current load goes above 17A relay must turn off.
How can this be done knowing that the relay will switch a 4A 220Volt heater.
thank you
| No, the input current is <strong>not</strong> similar to the output current for any reasonable power supply circuit (things like dropping resistors don't count- you'd be throwing away 45W to get 0.7W). Things like transformers and switching power supplies (lossless to a first order) have output <strong>power</strong> t... | Your useage of 3.3 * 0.2 = 0.66 watts
You should have multiplied like this : 3.3v * .24 = 0.792 watts
For a rough estimate, you can use this method to estimate consumption at 220 VAC as : 0.792/220 = 3.6 mA.
However, this does not take into account the current drawn by the primary of the transformer alone. ... | https://electronics.stackexchange.com |
71,024 | [
"https://mathoverflow.net/questions/71024",
"https://mathoverflow.net",
"https://mathoverflow.net/users/10204/"
] | I am working on a problem were I encounter matrices of the form
<span class="math-container">$X = \begin{bmatrix}\frac{1}{1 - a_ib_j}\end{bmatrix}_{ij}$</span>
I am aware of Cauchy matrices, which have the form
<span class="math-container">$X = \begin{bmatrix}\frac{1}{a_i - b_j}\end{bmatrix}_{ij}$</span>
(sometimes wri... | Given two diagonal matrices $D_1,D_2$, matrices such that $\nabla(X):=D_1X-XD_2$ is low-rank are known in literature as <em>Cauchy-like</em> matrices. This includes your case, as $\operatorname{diag}(a_i^{-1})X-X\operatorname{diag}(b_j)$ is rank 1, assuming $a\neq 0$ as you did.
Cauchy-like matrices with <em>displacem... | Not an answer, but some comments and a suggestion.
The place I have previously seen $(1-a_ib_j)^{-1}$ is in Theorem~7.12.1 in <em>Enumerative Combinatorics</em> volume 2, by R. Stanley, where he mentions Cauchy's identity that describes $\prod_{ij}(1-a_ib_j)^{-1}$ as a sum of Schur functions.
In the notes to Chapter ... | https://mathoverflow.net |
487,872 | [
"https://math.stackexchange.com/questions/487872",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/59276/"
] | assume that $\frac xy \in \mathbb N$ , is it correct that $\frac xy mod (n)=\frac{x \ mod \ n}{y\ mod \ n} mod \ n $ ? if not then how to compute it ?
explination :
I am dealing with large numbers and I want to compute $\frac xy mod \ n$ , assume that x is very big integer ( can reach $100^{100}$) while $y$ is very ... | I'm assuming that by "$a \mod b$" you mean "the remainder when $a$ is divided by $b$" rather than the (more mathematical) equivalence class definition. To avoid confusion, I'll instead write "% n" for the remainder modulo $n$, and $\mod n$ for mathematical equivalence modulo $n$.
Since you're assuming $\frac{x}{y} \i... | $$ x = ky \quad , \frac{x}{y} \pmod n = \frac{x}{y} - n\left \lfloor \frac{x}{ny} \right \rfloor $$
$$ \frac{x \pmod n}{y \pmod n} = \frac{x - n\left \lfloor \frac{x}{n} \right \rfloor}{y- n\left \lfloor \frac{y}{n} \right \rfloor } $$
| https://math.stackexchange.com |
117,609 | [
"https://mathoverflow.net/questions/117609",
"https://mathoverflow.net",
"https://mathoverflow.net/users/28037/"
] | Is there an injective and Riemann integrable map $f:\mathbb R^3\rightarrow\mathbb R$? (Of course such a map cannot be continuous.)
| Consider for example $f \colon [0,1]^3 \to \mathbb{R}$ defined as
$$f(x,y,z) = \sum_{i=0}^\infty 8^{-i}(x_i+2y_i+4z_i),$$
where $x_i = \left(\sum_{j=0}^{i-1}2^{i-j}x_j\right) - \lfloor x 2^i \rfloor$, and $y_i, z_i$ are defined the same way. In other words, $(x_i)_i$ is a specific dyadic representation of $x$.
<stro... | <strong>beginning</strong>
Start with the unit cube $E$ in $\mathbb R^3$ and the unit interval $[0,1]$ in $\mathbb R$. Choose an injective map $\phi : E \to [0,1]$. This is the remaining question: do this so that $\phi$ is Riemann integrable. That is, the set of discontinuities has measure zero.
Then cover $\mat... | https://mathoverflow.net |
1,105,001 | [
"https://math.stackexchange.com/questions/1105001",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/208024/"
] | I'm sure this question involves combinations, dividing the probabilities of each black card/ace over ${52}\choose{2}$ (for $2$ cards drawn) and adding them but I'm not getting the right answer. The given solution is: $\frac{183}{221}$.
Thanks!
| You are looking for the probability of drawing two cards that include a black card or an ace. When you are looking for the probability where multiple events can trigger the desired outcome (black card OR ace, AT LEAST one of the dice is a six, etc) sometimes it is easier to find the probability of the opposite happeni... | On same lines as turkeyhundt
We find the probability of opposite outcome
Drawing a red card which is not an ace has probability 24/52
Repeat this without replacement and the probability is (24-1)/(52-1)
So the required probability is 1-(24/52)(23/51)=175/221
| https://math.stackexchange.com |
38,007 | [
"https://mathoverflow.net/questions/38007",
"https://mathoverflow.net",
"https://mathoverflow.net/users/1132/"
] | I've been working through the heat-equation proof of the Atiyah-Singer index theorem. My question is what is the motivation for the definition of the index of an operator? I know there is the isomorphism between the homotopy classes of maps from a compact topological space to the space of Fredholm operators and the f... | Usually one really wants to know the dimension of the kernel of an operator rather than its index. The problem is that the dimension of the kernel is not a continuous function of the operator, so is very hard to compute in terms of topological data. The key point about the index is that it is a continuous function of t... | I would say that the basic reason why the index of an operator is an interesting number is that one may care about the dimension of the vector space of smooth solutions of the corresponding differential equation. The index theorem serves to describe this number in terms of the symbol of the operator (in one formulation... | https://mathoverflow.net |
30,173 | [
"https://math.stackexchange.com/questions/30173",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/8957/"
] | I am not sure how to solve these equations because the involve the ceiling function. I am trying to solve for $M$ and $L$ in terms of only $C$. Is there a way? If not, is there an algorithm that could calculate $M$ and $L$ values based on a given $C$ value? Here are the equations:
$$M = C + L - 1$$
$$L = \lceil\ ... | You can convert $L=\lceil 0.06 M \rceil$ into two inequalities,
$$0.06M\le L < 0.06M + 1\;,$$
together with the condition that $L$ is an integer. Then you can substitute $M$ from the first equation:
$$0.06(C+L-1)\le L < 0.06(C+L-1) + 1$$
and "solve" for $L$:
$$0.06(C-1)\le 0.94L < 0.06(C-1) + 1\;,$$
$$\f... | You can rewrite your problem as the intersection of two objects. Let $x=M$ and $y=L$ then the first equation is the parametric line $y=x+1-c$ and the second equation $y = \lceil 0.06x \rceil$ can be thought as a collection of horizontal lines (I let you figure out precisely what is that collection). Now all you want is... | https://math.stackexchange.com |
236,408 | [
"https://electronics.stackexchange.com/questions/236408",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/79236/"
] | Suppose we have a signed 12-bit output which is capable of representing both positive and negative numbers, how can this be converted to a +/- 10v signal using DACs, op-amps, etc.
For example, 1100 1011 0111 would result in a <em>negative voltage</em> and 0111 1101 1000 would result in a <em>positive voltage</em>.
Wh... | It depends a lot on the DAC (and also on the representation). You may be able to find a bipolar DAC that accepts a 2's complement input, for example. One common method is to use a unipolar 12-bit DAC and add an offset to the output so 0x000 would be about -10 and 0xFFF would be +10.
However, your example appears to i... | You have 12 bits to encode a range of 20V. So your resolution will be
<pre><code>20V / 2^12 = 0.0049V per LSB
</code></pre>
So in order to get <code>X</code> volts, you just divide <code>X</code> by <code>0.0049</code> and round it. It will give you the value which you convert into (signed, 2's complement for exam... | https://electronics.stackexchange.com |
726,875 | [
"https://physics.stackexchange.com/questions/726875",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/342755/"
] | I know that the invariant interval <span class="math-container">$I$</span> is the same in all reference frames. However, I don't know what is the physical meaning of <span class="math-container">$I$</span>. Is it just a quantity for us to check our answers?
| It actually suffices to know the expectation values of all projection operators of the form <span class="math-container">$P_\psi:=|\psi\rangle\langle \psi|$</span> for <span class="math-container">$\psi \in H$</span> (which are of course observables). Indeed, suppose we know
<span class="math-container">$$E_\psi:=\math... | Intuitively, if you know the expectation values of all possible observables, that should be enough to fix the state of the system. This almost sounds tautological, since the state is just a mathematical device to compute expectation values of observables. If you already have all these expectation values, then you have ... | https://physics.stackexchange.com |
28,406 | [
"https://dba.stackexchange.com/questions/28406",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/3787/"
] | I want to count two columns, but need to get the results of first column in one row.
<pre><code>SELECT country, COUNT(*)
FROM table1
GROUP BY country, type
</code></pre>
This query gives me
<pre><code>country type COUNT(*)
Canada first 22
Canada second 42
Canada third 15
Australia second ... | Looks like you want something like:
<pre><code>SELECT country
, sum(case when type = 'first' then 1 else 0 end) as type_first
, sum(case when type = 'second' then 1 else 0 end) as type_second
, sum(case when type = 'third' then 1 else 0 end) as type_third
FROM table1
GROUP BY country
</code></pre>
| <pre><code>SELECT
country,
SUM(IF(A.type='first' ,country_type_count,0)) type_first,
SUM(IF(A.type='second',country_type_count,0)) type_second,
SUM(IF(A.type='third' ,country_type_count,0)) type_third
FROM
(
SELECT country, type, COUNT(*) country_type_count
FROM table1 GROUP BY country, type
) A... | https://dba.stackexchange.com |
163,662 | [
"https://stats.stackexchange.com/questions/163662",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/83395/"
] | It's typical to initialize EM for Gaussian Mixture Models using the result of kmeans clustering. However, kmeans only gives you the means (centers) of the starting GMM, but EM initialization often requires a complete GMM description (that is, including the covariance matrices and weights).
Therefore what is a 'good' ... | k-means also tells you which data points belong to which cluster. A good starting estimate for the covariances should be the within-cluster covariances, and a good estimate for the weights should be the fractions of data points allocated to each cluster.
| Old question, but its worth saying anyway..
The EM algorithm iterates between finding the responsibilities for each data point (the probability they belong to each mixture) and calculating the modes of the distributions. Therefore you should only need to initialise the responsibilities OR the modes (but not both), thi... | https://stats.stackexchange.com |
916,049 | [
"https://math.stackexchange.com/questions/916049",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/148176/"
] | Under the transformation $\displaystyle w=\frac{z-1}{z+1}$, show that the map of the straight line x=y is a circle and find its centre and radius.
My attempt: Putting z=x+ix, $\displaystyle w=\frac{(x-1)+ix}{(x+1)+ix}$
$\displaystyle w=\frac{[(x-1)+ix][(x+1)-ix]}{[(x+1)+ix][(x+1)-ix]}$
$\displaystyle w=\frac{(2x^2-1... | $y = x^2 \implies y ' = 2x$: slope of tangent to curve at $x$.
$y = -x + k = (-1)x + k$, so slope of normal line is $-1$.
At what value of $x$ is $y' = 2x = -\frac 1{-1}=1\;?$
Solve for $x$, given $2x = 1$.
(Recall that the slope of the line normal to a curve at a point $(x, y)$ is the negative reciprocal of the s... | You have the curves $f(x)= k-x$ and $g(x)=x^2$. If the intersection is at $x^*$ the condition of normality is $g'(x^*)f'(x^*) = -1$
So $g'(x)f'(x) = -1\cdot 2x$. We can get the intersection by solving $x^*-k = f(x^*) = g(x^*) = x^{*2}$ for $x^*$.
| https://math.stackexchange.com |
186,811 | [
"https://physics.stackexchange.com/questions/186811",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/4962/"
] | The usual Lorentz force expression I am familiar with is this:
$$\vec F=q(\vec E+\vec v \times \vec B)$$
I have seen some other versions lately that include an extra factor $1/c$:
$$\vec F=q\left(\vec E+ \frac{1}{c} \vec v \times \vec B\right)$$
What is this $c$ and how is it included? I guess other parameters in t... | In the second formula, $c$ is the speed of light.
Both formulas use different system of units. The first one uses the SI: $q$ in coulombs, $\vec{E}$ in volts per meter and $\vec{B}$ in teslas. The second one uses gaussian units: $q$ in statcoulombs, $\vec{E}$ in statvolts per centimeter and $\vec{B}$ in gauss (being s... | it depends in the system (sI or CGS or...) that you work in. nothing special about it. and some times you get c=1 or h=1 )planck's constant just to have simpler forms which does not effect your wanted result.
| https://physics.stackexchange.com |
71,908 | [
"https://electronics.stackexchange.com/questions/71908",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/19987/"
] | In one of my classes, we skimmed over lithography, but mostly the optics side of things (the diffraction limit, liquid immersion to increase angle of incidence, etc).
One point that was never covered is how the light actually <em>dopes</em> the silicon, and creating a transistor. I've tried to stumble around on the n... | There are multiple steps but the basic process is that you use a photoresist.
At the beginning of a process step, a photoresist is "spun" on to the wafer. It is a very literal thing, they spin the wafer while dripping the polymer onto the surface which spreads out into a thin layer of precise thickness. This is cure... | To elaborate on the projection (imaging) step:
The original design of a microchip is "drawn" by some other means (e.g. electron microscopy) on a glass plate called <em>reticle</em>. The reticle is imaged on the photoresist with reduction (e.g. 4 times reduction in ASML machines), producing tiny structures. While all ... | https://electronics.stackexchange.com |
2,981,663 | [
"https://math.stackexchange.com/questions/2981663",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/611442/"
] | I am having a moment of weakness and lost the plot, why is this wrong
<img src="https://i.stack.imgur.com/T4b1C.png" alt="Image of sine plots">
I assumed adding <span class="math-container">$2\pi$</span> would give the same result, clearly not the case on the graph (using desmos graphing tool)
<span class="math-conta... | Note that desmos is plotting two different lines, the black vertical line
<span class="math-container">$$x = \frac16 \pi$$</span>
and the red vertical line
<span class="math-container">$$x = \frac16 \pi + 2\pi$$</span>
The purple graph is a graph of
<span class="math-container">$$\sin(\frac16 x)$$</span>
which wi... | You don't have <span class="math-container">$\sin x$</span>, but <span class="math-container">$\sin(x/6),$</span> so when you add <span class="math-container">$2\pi$</span> to <span class="math-container">$x,$</span> this is what happens:
<span class="math-container">$$\sin\left(\frac{1}{6}(x+2\pi)\right) = \sin\left(... | https://math.stackexchange.com |
441,792 | [
"https://physics.stackexchange.com/questions/441792",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/213376/"
] | I learned that Bohr explained line spectra by postulating that electrons can only be at certain discrete distances from the nucleus. Later, this theory was refuted/improved by de Broglie and Schrödinger. Since their theories, electrons were seen as standing waves and we can only know where they will <em>probably</em> b... | <blockquote>
Since their theories, electrons were seen as standing waves and we can only know where they will <em>probably</em> be. The regions with <span class="math-container">$90\%$</span> probability are called orbitals.
</blockquote>
Not quite. The orbital is the standing wave itself.
<blockquote>
But how c... | <blockquote>
But how can line spectra be explained if electrons are not restricted to discrete distances but rather to orbital regions?
</blockquote>
It's because the electrons are restricted to discrete <em>energies</em>, not distances. This can be properly explained with quantum mechanics, but basically, electrons... | https://physics.stackexchange.com |
2,941,865 | [
"https://math.stackexchange.com/questions/2941865",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/468493/"
] | <span class="math-container">$\ L $</span> is the tangent line at <span class="math-container">$\ x = x_0 $</span> to the graph of <span class="math-container">$\ y= ax^3 +bx$</span>. I need to find the <span class="math-container">$\ x$</span> value of the second point where <span class="math-container">$\ L$</span> c... | <span class="math-container">\begin{eqnarray} x^3-3x_0^2x+2x_0^3 &=&x^3-x_0^2x-2x_0^2x+2x_0^3\\
&=& x(x^2-x_0^2)-2x_0^2(x-x_0)\\
& =& (x-x_0)\Big(x(x+x_0)-2x_0^2\Big)\\
&=&(x-x_0)\Big(x^2+xx_0-2x_0^2\Big)\\
&=&(x-x_0)(x+2x_0)(x-x_0)\\
&=&(x-x_0)^2(x+2x_0)
\end{eqnarray}<... | You have to solve the equation <span class="math-container">$4x^{3}-3x_0^{2}x-2x_0^{3}=0$</span> and you already know that <span class="math-container">$x=x_0$</span> Is one solution. The equation can be written as <span class="math-container">$(x-x_0)^{2}(x+2x_0)=0$</span>. Hence <span class="math-container">$x=-2x_0$... | https://math.stackexchange.com |
231,524 | [
"https://softwareengineering.stackexchange.com/questions/231524",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/122442/"
] | This is a front end project that is using socket.io for all communication with the server. This allows for push which is good, but recently we have been asked to provide a pause button. The pause button is intended on stopping the data from changing out on the ui if the user does not wish it to do so automatically.
W... | I have some cognitive dissonance regarding the pause button - as a user, I consider 'Pause' to mean - halt what the server is doing. From your requirement, though, it seems that the server is not affected by the button at all, which might be confusing, and even misleading, as the feedback to the user really stops. I be... | I presume that you need to update the UI with the most recent information when you un-pause the update.
This means that you want pausing to occur on the UI and not on the socket. Let the socket run freely, updating local state. For your UI, you use a copy of the local state for display and don't refresh the copy until... | https://softwareengineering.stackexchange.com |
595,265 | [
"https://physics.stackexchange.com/questions/595265",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/180589/"
] | We know that if there are two observers S and S', they will observe some events on the space-time and their relation is given as
<span class="math-container">$$\Delta x = \gamma(\Delta x' + \beta \Delta c t')$$</span>
<span class="math-container">$$\Delta ct = \gamma(\Delta ct' + \beta \Delta x')$$</span>
Now my proble... | If you are measuring the length of an object in <span class="math-container">$S$</span> or in <span class="math-container">$S'$</span>:
\begin{itemize}
\item $\Delta x$ is the distance between two events at the same time in frame $S$.
\item <span class="math-container">$\Delta x'$</span> is the distance between two eve... | I don't know if it helps, but one problem that I see with distances in relativity is how to measure it in typical examples:
A trip from earth to Alpha Centauri, that is 4 ly distance, at a speed of 0,5c, will take 8 years in earth time:
<span class="math-container">$t' = \gamma (t - vx) = 1,1547(8 - 0.5*4) = 6,9282$</s... | https://physics.stackexchange.com |
201,366 | [
"https://softwareengineering.stackexchange.com/questions/201366",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/36804/"
] | re-edited by author: no this is not 2 questions. This is one question about code review questions containing two separate points. Please do not edit my question.
For naming variables, the two sides are regarding using articles (a, an, the) in the variable name (e.g. theBook vs. book -or- parseTheString vs. parseStrin... | <h1>Articles in Variable Names</h1>
<strong>They add little to no value in most cases, but make the code longer to read.</strong>
Usually, I'd rarely have a the need for the distinction between a "a" or "the" in the flow of a function, so it doesn't make sense to use articles.
It's a formal programming language and ... | <blockquote>
For hard-coding strings, the three sides are whether to put all strings into constants (public static final) in a separate file (class or interface), put them all in the same file (private static final), or don't use constants for strings at all. (e.g. new Button("Click Me") vs. new Button(BUTTON_LABEL_C... | https://softwareengineering.stackexchange.com |
724,265 | [
"https://physics.stackexchange.com/questions/724265",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/240369/"
] | How work done is really understood?
I know that <span class="math-container">$W=F\cdot d$</span>. I am interested in the <strong>meaning of force here</strong> i.e.
<ol>
<li>Is it a continuous force applied till displacement? like the case of pulling trolley bag till displacement <span class="math-container">$d$</span>... | The formula <span class="math-container">$W=Fd$</span> gives you the work performed by a force <span class="math-container">$F$</span> which is acting constantly over a particle as it moves over a distance <span class="math-container">$d$</span>. The distance that the particle moves after the force stops does <strong>n... | In any formula, the symbols must be defined and the conditions under which the formula holds should be specified.
To me, <span class="math-container">$W=\vec F\cdot \vec d$</span> is shorthand for "the work done by a constant force <span class="math-container">$\vec F$</span> along a straight path represented by a... | https://physics.stackexchange.com |
1,597,891 | [
"https://math.stackexchange.com/questions/1597891",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/302178/"
] | Let $G$ be an abelian group of order $75=3\cdot 5^{2}$. Let $Aut(G)$ denote its group of automorphisms. Find all possible order of $Aut(G)$.
My approach is to first study its Sylow 5-subgroup. Since $n_{5}|3$ and $n_{5}\equiv 1\pmod{5}$, $n_{5}=1$. So $G$ has a unique Sylow 5-subgroup, denote $F$. By Sylow's Theorem, ... | Sylow theory is generally not useful for Abelian groups since we already know so much about their structure, and since every Abelian group has a unique $p$-Sylow subgroup when it exists.
Using the classification theorem, we have that $|{G}| = 75$ implies $G \cong \mathbb{Z}/3\mathbb{Z} \times (\mathbb{Z}/5\mathbb{Z})^... | $G$ is abelian, so Sylow subgroups are characteristic. Hence $G$ is product of characteristic Sylow subgroups, say $H_3$ and $H_5$. Then $Aut(G)\cong Aut(H_3)\times Aut(H_5)$.
$H_3$ is cyclic, whose automorphism group is well known. $H_5$ is either cyclic or $Z_5\times Z_5$. The automorphism groups in both cases in... | https://math.stackexchange.com |
281,331 | [
"https://dba.stackexchange.com/questions/281331",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/170931/"
] | In postgres I want to select rows where column1 matches column2 for corresponding row and returns result having same values as well as records consisting word from column1.
<pre><code>select * from table_name where "column1" ILIKE "column2";
</code></pre>
i am using above query but it returns exact... | Wouldn't a simple loop be enough?
<pre><code>create or replace function remove_vertices(par_geom geometry)
returns geometry as $$
DECLARE
declare invalid_pnts record;
begin
for invalid_pnts in select * from selfintersects(par_geom)
loop
-- do something with every record
end loop;
if not found then
... | This is really hard to answer, because you don't show the big picture. I would be inclined to do everything in a single statement that joins to the result of the function call, rather than storing the result somewhere. So that no intermediate storage is needed at all. But without seeing more of the code (the "call... | https://dba.stackexchange.com |
294,540 | [
"https://physics.stackexchange.com/questions/294540",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/37677/"
] | We have two particles that can be in either level $E_0 = 0$ or in level $E_1$.
If we treat them as Bose particles, then the partition function will be:
$$ Z = 1 + e^{-\beta E_1} + e^{-2\beta E_1}, $$
whereas if we treat them as classical indistinguishable particles we'd get:
$$ Z = \frac{(1+e^{-\beta E_1})^2}{2!} = ... | In quantum statistics there there is never any over-counting. In this system containing two particles (= a quantum many-body system) the only three possible many-body states are
$$\vert \psi_A \rangle = \vert E_0 \rangle \otimes \vert E_0 \rangle \qquad \text{with energy} \quad E = 0$$ (with which I denote particle on... | It comes from the first partition, your not considering all possible 2-states. Indeed, one possible state is both are in $0$, 2 other are one is in $0$ the other in $E_1$ and finally both in $E_1$. Because they are indistinguishable you must divide the whole thing by 2!.
| https://physics.stackexchange.com |
229,549 | [
"https://softwareengineering.stackexchange.com/questions/229549",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/60527/"
] | It's very common in .NET for an exception to be wrapped in several layers of "outer exceptions" which give marginally more contextual data. For example, in EF if your update fails, you get exceptions wrapped similar to this:
<ul>
<li><code>EntityException</code></li>
<li><code>DbUpdateException</code></li>
<li><code>... | to hide what the core cause was in places where it doesn't matter
the top level only needs to know that a storage exception occurred instead of an SQLException, which may not happen if you decide to migrate to a non-sql data store
not wrapping also leaks the abstraction and requires reimplementation of the top level ... | A compelling principle behind error handling is to handle the error at the layer best suited to deal with the error. Handling an error can involve retrying the operation, signaling the user, or just plain rolling over and dying.
Frequently, this means that error handling is taken care of fairly high up in the applica... | https://softwareengineering.stackexchange.com |
205,839 | [
"https://security.stackexchange.com/questions/205839",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/-1/"
] | I have successfully discovered a vulnerability in the New York Times website. Is there any known way I can report this? I have not attacked it. But I found a bypass. What should I do?
| They do not have a specific security contact listed, so you should email <code>help@nytimes.com</code> and ask for your message to be forwarded to their IT security department.
| I see that my answer is too late, but I'm going to give it anyway because I believe it will be useful to others, although likely Controversial.
<blockquote>
<strong>Do not report vulnerabilities by identifiable or traceable methods!</strong>
</blockquote>
In a reasonable and rational world, the recipient of your vu... | https://security.stackexchange.com |
4,181,347 | [
"https://math.stackexchange.com/questions/4181347",
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] | Let there be a decomposition <span class="math-container">$φ(x)=\displaystyle\sum_{n=1}^\infty c_n\sin{nx}$</span> on the segment <span class="math-container">$[0,π]$</span>, and on our segment <span class="math-container">$φ(x)≥0$</span>. How do I prove that the series <span class="math-container">$\displaystyle\sum_{... | That system has two and only two solutions:<span class="math-container">$$\left(\frac{\sqrt[4]8}{\sqrt3},-\frac1{\sqrt[4]2\sqrt3},\frac1{\sqrt[4]2\sqrt3}\right)\left(\longleftrightarrow\lambda=\frac{\sqrt{27}}{2\sqrt[4]2}\right)$$</span>and<span class="math-container">$$\left(-\frac{\sqrt[4]8}{\sqrt3},\frac1{\sqrt[4]2\... | Define <span class="math-container">$f$</span> by <span class="math-container">$f(x,y,z)=8x-y+z$</span> and <span class="math-container">$g$</span> by <span class="math-container">$g(x,y,z)=x^4+y^4+z^4-1$</span>. We seek the maximum and minimum of <span class="math-container">$f(x,y,z)$</span> subject to <span class="m... | https://math.stackexchange.com |
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