qid int64 1 4.65M | metadata listlengths 3 3 | prompt stringlengths 31 25.8k | chosen stringlengths 17 28.2k | rejected stringlengths 19 40.5k | domain stringclasses 28
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2,585 | [
"https://physics.stackexchange.com/questions/2585",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/304/"
] | It is true for fermions in the same potential that the total wavefunction of two particles must be antisymmetric with respect to exchange of electrons. Which means the spin wavefunction is given by
$\chi=\frac{1}{\sqrt{2}}[\chi_+ (1)\chi_- (2)-\chi_+ (2)\chi_- (1)] $
which looks very much like the bell state,
$\be... | if you replace "0" and "1" by "up" and "down", you get a similar state for two spins - which is referred to as the singlet. All these states are mathematically analogous except that the states "0" or "1", or "up" and "down", or "plus" and "minus" (as indices of your $\chi$) may mean physically different things - i.e. t... | Every state can be written in the way you mention, in terms of two states 1 and 2, for an appropriate choice of states 1 and 2. By itself it does not indicate any entanglement. What makes a state entangled is a specific property of the two states 1 and 2, namely that they are physical states belonging to two subsystems... | https://physics.stackexchange.com |
1,617,289 | [
"https://math.stackexchange.com/questions/1617289",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/300044/"
] | Let $S=\{(x_1,x_2,x_3,x_4,x_5)\in\mathbb{R^5}|x_1=x_3=x_5,x_2-x_4=2x_1-x_3\}$ is a subspace of $\mathbb{R^5}$. Find a basis of $S$. Expand a basis to a basis of $\mathbb{R^5}$.
Question: How to find a basis and dimension of $(x_1,x_2,x_3,x_4,x_5)\in\mathbb{R^5}$ without first expanding a basis?
One of the vectors tha... | Recall that the lexicographic order on $A\times B$ is essentially to take $A$ and replace each point with a copy of $B$.
So only in one of these lexicographic orders every element has an immediate successor. And having an immediate successor is preserved under isomorphisms.
(Generally, to show two orders are not isom... | To prove two structures aren't isomorphic, one good approach is to find a property that one structure satisfies that the other doesn't; and then show that property is preserved by isomorphisms.
For the specific example: have you tried drawing the two linear orders in question? What's a property $\mathbb{Q}$ has, that ... | https://math.stackexchange.com |
1,856,656 | [
"https://math.stackexchange.com/questions/1856656",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/353330/"
] | Why is it that in the form $y=a(x-h)^2+k$ of a parabola, that $(h,k)$ is the coordinate of the vertex? I am reviewing Algebra and cannot find a reasonable explanation anywhere.The highest level of math I am familiar is with Pre-Calculus. Thanks!
| Consider the graph of the parabola $y=ax^2$. Its vertex is clearly at $(0,0)$. Now, if you replace $x$ with $x-h$ in any equation, its graph gets shifted to the right by a distance of $h$. Similarly, replacing $y$ by $y-k$ shifts the graph up by $k$. If we make both of these substitutions in the above equation of the p... | Since $(x-h)^2$ is always nonnegative, for any value of $x$, it's at its smallest when the thing in parens is zero, i.e., when $x = h$. If $a$ is positive, this will be where the parabola has its minimum, i.e., $x = h$ is the $x$-coord of the vertex. (If $a$ is negative, this is the location of the max, but the same re... | https://math.stackexchange.com |
496,211 | [
"https://electronics.stackexchange.com/questions/496211",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/177402/"
] | I have to decide on a way to measure the efficiency of an algorithm running on a microcontroller/microprocessor.
My thought is that I can use the runtime of the algorithm to complete a certain task as an indirect measure of how the power consumption of that algorithm compares to another algorithm's power consumption. ... | There are many things that alter the power consumption of the device, but assuming your just computing an algorithm with all unused peripherals switching off, e.g. ADC (pulsing current each sample), GPIO (changing state consumes a small amount of current), watchdog circuit disabled (runs a clock and triggers an interru... | <blockquote>
Is microcontroller power consumption directly related to its operation time?
</blockquote>
Kind of. What draws most current is the CPU clock and any active hardware peripherals such as GPIO. Hardware peripherals being a story of their own since each one has unique power consumption characteristics.
The... | https://electronics.stackexchange.com |
55,850 | [
"https://stats.stackexchange.com/questions/55850",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/20735/"
] | I have a 1D normal distribution with mean $\mu$ and standard deviation $\sigma$. Given a new number $x$, I want to assign a confidence value to how likely $x$ is to have been drawn from the distribution $\mathcal{N}(\mu,\sigma)$
1 - the cdf gives me $P(X > x)$ but I'm not sure this is what I'm looking for.
Any hel... | Let's say you get $x = \mu$. While you could say that's the <em>most</em> likely possible value for $x$ if it did come from $N(\mu,\sigma^2)$, you're not in a position to say that the probability that it <em>did</em> come from that normal is anything. To make such a statement would require something like a Bayesian poi... | For a measure based on distance from the mean, try this:
Use 2P[X > x] for x > mean,
2P[X < x] for x < mean, and
1 for x = mean
| https://stats.stackexchange.com |
42,132 | [
"https://astronomy.stackexchange.com/questions/42132",
"https://astronomy.stackexchange.com",
"https://astronomy.stackexchange.com/users/38195/"
] | Astronomers or Scientists say that Jupiter's Mass would have to be multiplied 80 times to become a star (M Mass). What if Jupiter was denser, would there be thermonuclear energy because of the density?
| To achieve thermonuclear fusion you need high temperatures and high density. If you were able (by magic) to compress and heat the core of Jupiter then fusion would occur.
But there is no mechanism that can cause Jupiter to increase it's temperature and density like this. In the core of a proto-star, gravity will provi... | The minimum mass for a brown dwarf is about 12-13 jupiter masses, and averagely they're 75, so theres no chance that Jupiter would be a star at its current weight.
| https://astronomy.stackexchange.com |
246,172 | [
"https://electronics.stackexchange.com/questions/246172",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/98226/"
] | I am currently designing a PCB layout for my team's high current, high voltage project and am having trouble with choosing my trace width.
The circuit is composed of 12 inductors placed in series with short traces connecting them together (about 5mm max length).
We are feeding 10μs pulses at ~300A/1000V at a maximum r... | It's not the <em>current</em> that matters, it is heat.
The current will heat up your copper on top of the FR4.
The heat generated is current squared time resistance of (trace + solder joints + component leads).
The square in there is what makes the high currents so scary, even ath the 1:10k duty cycle you have on the... | 10us pulses at 10 Hz, so a duty cycle of 0.0001?
300A, so I^2T is 90000 * 0.0001 = 90.
Square root of 90 = ~9.5, so you need at a minimum a track that can handle 10A continuous (I would use something good for a few times that, maybe base on the size of the inductor pads), job done.
Now finding inductors to take th... | https://electronics.stackexchange.com |
1,564,736 | [
"https://math.stackexchange.com/questions/1564736",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/296752/"
] | Let $U = P_3(\Bbb R)$, the vector space of polynomials of degree at most $3$ in a formal variable $t$.
Let $f : U → \Bbb R$ be the linear map defined by $f (u) = u′(1)$, where $u′(1)$ is the derivative of $u$ (with respect to $t$) evaluated at $t = 1$
(i) Determine the matrix of f with respect to the bases {$1$,$t$,... | Looks like you’ve found a basis for the kernel, but there’s a more systematic way to go about this once you have the rref matrix. Find the columns that don’t have pivots (leading entries). Those will be the free coordinates in the kernel basis. Set each one to one and the rest to zero in turn, and solve for the remaini... | Here is another approach which uses the polynomial structure. First, let s=t-1. Then every polynomial can be written as $u=a_0+a_1 s + a_2 s^2 + a_3 s^3$ and now the map is $f(u) = a_1$. Thus, the kernel is spanned by $1,s^2,s^3$ since these polynomials and their combinations have $a_1=0$. Thus, $1,(t-1)^2,(t-1)^3$ is ... | https://math.stackexchange.com |
19,388 | [
"https://mathoverflow.net/questions/19388",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3238/"
] | I've always thought of the degree of a subvariety of projective space as the degree of the divisor that defines the (given) embedding into projective space. It's been pointed out to me that this works only for curves. Now I'm confused: is there a similar characterization of the degree of a general subvariety of some pr... | If $X\subset \mathbb P^n$ is a subvariety of dimension $m$ embedded by a linear system $V \subset H^0(X,\mathcal O_X(D))$ then the degree of $X$ is equal to $D^m$.
| See the section about intersection of chapter I of hartshorne. Intuitively degree is the number of intersection of the dimension r subvariety and the GENERIC dimension n-r linear subspace if they are both in P^n. A rigorous definition was given in that section by the leading coefficient of the Hilbert polynomial.
| https://mathoverflow.net |
176,089 | [
"https://softwareengineering.stackexchange.com/questions/176089",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/68815/"
] | We are starting an Agile project, and the product owner has made a start adding user stories to the product backlog (mostly epics at this stage).
Who would be responsible for breaking down epics into smaller user stories - is it left to the product owner, or would you expect the rest of the team to be involved (bearin... | As a general rule, the whole team is responsible for breaking down the epics prior to implementing them. It's fine to put the epic in the backlog and prioritize it, and then break it down just before the iteration in which it is planned for implementation. When breaking it down, the product owner (or other customer and... | IMO, the team would be responsible for breaking down the epics into smaller user stories which while in the beginning will be a challenge is likely something that will improve over time.
The team should be taking on an understanding of the domain and accepting their part in ramping up in getting the product built. Wh... | https://softwareengineering.stackexchange.com |
503,488 | [
"https://physics.stackexchange.com/questions/503488",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/196529/"
] | When an object moves in a circle, there's an acceleration towards the center of the circle, the centripetal acceleration, which also gives us the centrifugal force (since <span class="math-container">$F = ma$</span> is the equation for a force and the acceleration of an object, therefore, is caused by a force). But acc... | This is a common misinterpretation of Newton's third law, often stated as "to every action, there's an equal and opposite reaction." As you surmise, "action" and "reaction" refer to forces. However, they refer to forces acting on <em>different things</em>. Otherwise, nothing could accelerate, ever: if every force were... | Lets look at the Earth-moon system for an example. The centripetal force is Earth's gravity, keeping the Moon from flying away. But this works both ways, the Earth is pulled towards the Moon just as hard as the moon is pulled towards the Earth.
In your car example, the angle of the front tires means some percentage ... | https://physics.stackexchange.com |
74,840 | [
"https://softwareengineering.stackexchange.com/questions/74840",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/4112/"
] | My company is considering consolidating all their tier-1 (i.e. top end production) applications and sites into one all-encompassing code base.
The theory is that their permissions, design and overall functionality can be homogenised and centrally managed.
I have no end of concerns about this approach since the data s... | <h2>Bad idea</h2>
This reaction is based on the following assumptions:
<ol>
<li>There are a lot of fairly disparate applications being homogenised</li>
<li>There are many teams working on the different applications</li>
<li>There is no well-respected and authoritative software architect actively managing the applicat... | We have been working on this at my company. I think it is possible and there are obvious benefits (you mentioned them), however, this will probably be a 5 to 7 year project at an absolute minimum, and requires basically everything to be rewritten. If you can get sign off on something like that, then I would say go for ... | https://softwareengineering.stackexchange.com |
279,201 | [
"https://dba.stackexchange.com/questions/279201",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/218181/"
] | i have a table with the below columns
<pre><code>name varchar(250)
type varchar(25)
</code></pre>
is there a way for me to structure a query to return all records if the type parameter is null?
Im aware that i can do something like
<pre><code>select * from tbl where type = Coalesce(@param, 'a');
</code></pre>
which wil... | Use the placeholder for strings:
<pre><code>select * from tbl where type LIKE Coalesce(@param, '%');
</code></pre>
| Personally, I'd construct the query in the application, adding only those parameters that the user has specified.
Failing that, try something like this:
<pre><code>select field1, field2, field3, ...
from table1
where ( type = @param or @param is null )
</code></pre>
| https://dba.stackexchange.com |
138,364 | [
"https://physics.stackexchange.com/questions/138364",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/52858/"
] | Consider a pendulum with a bob and a massless, rigid, hinged rod attached to the bob. The bob is at rest at the bottom most position. Neglecting friction, is it possible to impart such a velocity (parallel to the horizontal) to the bob so as to make it stay upright in an unstable equilibrium. I know that if I impart $\... | If you initially give to the bob a velocity $\sqrt{4rg}$, it will actually take an <strong>infinite time</strong> for the bob to reach the top! A little lesser velocity will cause the bob to stop earlier and come back toward the initial point, while a little greater one will take the bob over the top (the motion will c... | Since it is a rigid rod, you are probably right. If the rigid rod is replaced with a string, then your teacher would be right, as the velocity at the top must be non zero in order for the string to remain tight and not collapse before reaching the top.
In real life scenarios, however, it is nearly impossible to maint... | https://physics.stackexchange.com |
540,973 | [
"https://physics.stackexchange.com/questions/540973",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/245502/"
] | The condition for normalization for a ket vector is <span class="math-container">$$\langle A \mid A\rangle = 1.$$</span> However, to test if ket <span class="math-container">$\mid A \rangle$</span> is normalized, should I form the inner product with its complex conjugate <span class="math-container">$\langle A^* \mid$<... | Here <span class="math-container">$A$</span> is just a quantum number (or a set of quantum numbers) labeling a vector, so it need not be conjugated. Products such as <span class="math-container">$\langle A^*|A\rangle$</span> do happen, e.g., when dealing with coherent states, but in this case <span class="math-containe... | The easy way to manipulate kets is to use the resolution of unity as much as possible
<span class="math-container">$$\int d^3x |x\rangle \langle x| = 1 $$</span>
Similarly in momentum space
<span class="math-container">$$\int d^3p |p\rangle \langle p| = 1 $$</span>
Then you have
<span class="math-container">$$ \lan... | https://physics.stackexchange.com |
42,118 | [
"https://security.stackexchange.com/questions/42118",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/30577/"
] | Say a small company just earned an ISO 27001 accreditation. What happens if during one of the audits it is found that the company didn't follow the ISMS? Is there a certain warning process that company has to go through? I basically want to know if there are instances when an ISO 27001 accreditation gets retracted and ... | I could not find anything about this particular ISO 97001 on the ISO website or any other related sites. But I saw you placed a tag ISO 27000 and that you mentioned ISMS, so I suppose you are talking about ISO 27001 which is Information Security Management System.
Normally, with all ISO certification, once you are ac... | Well, not all non conformities should be treated alike and this is why I think that the information security standard is subjective.
Speaking of the scenarios that may (not necessarily limited) apply here:
<ol>
<li>a need improvement is raised rather than a non conformity and a
follow-up surveillance audit (differen... | https://security.stackexchange.com |
38,763 | [
"https://mathoverflow.net/questions/38763",
"https://mathoverflow.net",
"https://mathoverflow.net/users/9189/"
] | Is $L^p(\mathbb{R}) \setminus 0$ contractible? My intuition says that the answer is yes, but I'm afraid that this is based on thinking of this as somehow similar to a limit of $\mathbb{R}^n \setminus 0$ as n approaches $\infty$, which is of course nonsense. In any case, every contraction I've tried ends up making som... | Here is something really cheap and dirty. Let $p<+\infty$. Take $g=\frac{1}{1+x^2}$. Then $f(x,t)=e^{-(1+|x|)t/(1-t)}f(x)$ ($0\le t\le 1$) is a continuous contraction of $L^p\setminus\{g\}$ to $0$. (the reason is that your only chance to hit $g$ is to start with it because $g(x)e^{s(1+|x|)}$ is not in $L^p$ for $s&g... | An infinite dimensional Banach space is homeomorphic to itself minus a point. Maybe R.D. Anderson or V. Klee proved this.
| https://mathoverflow.net |
76,638 | [
"https://mathoverflow.net/questions/76638",
"https://mathoverflow.net",
"https://mathoverflow.net/users/18122/"
] | Given a class function $f: G \to \mathbb Q$, where $G$ is a finite abelian group, is there an easy way to decide whether $f$ is an element of the rational representation ring $R_{\mathbb Q}(G)$, i.e. whether $f$ is a virtual character of some representations of $G$?
If it makes things easier you could also assume that... | Further to Ben's answer, it might be useful to picture the situation in $\mathbb{C}$. (Of course in $\mathbb{C}$ every domain is a domain of holomorphy, but we can still exhibit the same phenomenon that causes us to need the more complicated definition.)
The principal branch of the logarithm $f := \operatorname{Log}$ ... | Quoting Steven Krantz, Function Theory of Several Complex Variables, p. 6: the definition of domain of holomorphy is complicated because we must allow for the possibility (when dealing with an arbitrary open set $U$ rather than a smooth domain $\Omega$) that $\partial U$ may intersect itself.
Picture a cigar, and then... | https://mathoverflow.net |
510,405 | [
"https://physics.stackexchange.com/questions/510405",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/245835/"
] | Peoples ! Thanks you for your time.
I have a question about acceleration equation.
<span class="math-container">$$\begin{bmatrix}f_{x}\\ f_{y}\\ f_{z}\end{bmatrix}=\begin{bmatrix}\dot{u}\\\dot{v}\\ \dot{w}\end{bmatrix}+\begin{bmatrix}0 & w & -v \\ -w & 0 & u \\ v & -u & 0\end{bmatrix}\begin{... | There are two points to be clarified here.
<ul>
<li>The normal reaction force from the surface is a self-adjusting force. In particular, it can take any value so as to prevent the object in contact from penetrating. So, if an object resting on a surface has a weight <span class="math-container">$w$</span> then the no... | You are correct that there must initially be a net upward force, no matter how small and how brief, to get the object going. But in addition to balancing the forces immediately after to achieve constant velocity there must be a net downward force just before reaching the height such that the object will come to rest, i... | https://physics.stackexchange.com |
1,787,609 | [
"https://math.stackexchange.com/questions/1787609",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/203149/"
] | The purpose is to find the number of ways to color $2n$ following integers in red and blue, such that if $i$ is red, so is $i-1$.
I tried to use Inclusion-Exclusion principle, but I got stuck in the calculations.
It is a little tricky.
I am trying to count the number of cases where there are $k$ bad tuples $\colo... | $$\frac{\sqrt{2}}{\sqrt{x-3}}=\frac{\sqrt{2}\cdot\sqrt{x-3}}{\sqrt{x-3}\cdot\sqrt{x-3}}=\frac{\sqrt{2\cdot(x-3)}}{\left(\sqrt{x-3}\right)^2}=\frac{\sqrt{2x-6}}{x-3}$$
| The answer key is right (it's you): You must have multiplied numerator by <span class="math-container">$\sqrt{x-3}$</span>, not <span class="math-container">$\sqrt{x+3}$</span>.
| https://math.stackexchange.com |
3,611,723 | [
"https://math.stackexchange.com/questions/3611723",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/322814/"
] | Let <span class="math-container">$V$</span> be a finite-dimensional vector space and <span class="math-container">$T:V\rightarrow V$</span> be linear.
(a) Suppose that <span class="math-container">$V = R(T) + N(T)$</span>. Prove that <span class="math-container">$V = R(T)\oplus N(T)$</span>.
(b) Suppose that <span cl... | <strong>IMO 1997, Problem B2</strong>
Find all pairs <span class="math-container">$(a, b)$</span> of positive integers that satisfy <span class="math-container">$a^{b^2} = b^a$</span>.
<strong>Answer</strong>
<span class="math-container">$(1,1)$</span>, <span class="math-container">$(16,2)$</span>, <span class="math... | Render <span class="math-container">$a=tb^2$</span>. Then
<span class="math-container">$(tb^2)^{b^2}=b^{tb^2}$</span>
<span class="math-container">$tb^2=b^t$</span> (<span class="math-container">$b$</span> is assumed nonzero)
<span class="math-container">$t=b^{t-2}$</span> (<span class="math-container">$t$</span> i... | https://math.stackexchange.com |
16,583 | [
"https://mathoverflow.net/questions/16583",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4279/"
] | Let $x = \pi/(2k+1)$, for $k>0$.
Prove that<br>
$$
\cos(x)\cos(2x)\cos(3x)\dots\cos(kx) = \frac{1}{2^k}
$$
I've confirmed this numerically for $n$ from $1$ to $30$.
I'm finding it surprisingly difficult using standard trigonometric formula manipulation.
Even for the case $k = 2$, I needed to actually work out $\cos... | Let
$S(x)=\prod_{j=1}^k \text{sin}(jx)$
and
$C(x)=\prod_{j=1}^k \text{cos}(jx)$.
Let x = $\frac{\pi}{2k+1}$.
Then $S(2x) = S(x)$ (from $\text{sin}(\pi-x)=\text{sin}(x)$), and $S(2x)=2^kS(x)C(x)$ (from $\text{sin}(2x)=2\text{sin}(x)\text{cos}(x)$), from which the result follows.
Steve
| Hint: multiply by sin(x)
| https://mathoverflow.net |
196,718 | [
"https://physics.stackexchange.com/questions/196718",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/86496/"
] | Am I missing something?
<img src="https://i.stack.imgur.com/Zkxb4.jpg" alt="enter image description here">
It seems obvious to me that at $+A$ and $-A$, the spring has restorative forces equal in magnitude but opposite in direction.
But since gravity is always pulling it down, the spring at position $-A$ must have le... | The equilibrium position in this case is not where the spring is not stretched, it is actually stretched by a $\Delta x$ amount with $F_{spring}(0) = k\Delta x$.
So the spring force on point A is a little smaller than in point -A, since $ F_{spring}(A) = -k(A-\Delta x)$ and $ F_{spring}(-A) = k(A+\Delta x)$ so it co... | In accordance with Hooke's law the force is linear with distance. Incorporating gravity only means that the equillibirum position of the spring has changed, the "zero" around which it oscillates. The gravitational pull is already compensated by the spring. Thus the magnitude of the force is euqal at $-A$ and $+A$.
Edi... | https://physics.stackexchange.com |
581,253 | [
"https://physics.stackexchange.com/questions/581253",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | Why is it recommended to lower the pressure of tire when going through sand?
please help me I have tried very long thinking
this question is from my textbook but no answer
| Our objective is to not get our car stuck
for that we want increase the surface area of the tire which is in contact with the sand
we can do that by letting some air to increase surface area of contact and decreasing pressure exerted on the sand by the tire
<strong>I hope it is known that pressure is inverly proportio... | Normally, when driving on a firm surface, we want tires to be as firm as possible because (a) a soft tire is inefficient since energy is lost because the tire is continuously being deformed and (b) driving on soft tires can damage the tires. But in sand or on soft ground not getting stuck it is a higher priority. Partl... | https://physics.stackexchange.com |
726,810 | [
"https://physics.stackexchange.com/questions/726810",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/344534/"
] | The book starts with the equation
<span class="math-container">\begin{equation}
I(D) = \lambda^2 \int \frac{d^Dp_1}{(2\pi)^D}\frac{d^Dp_2}{(2\pi)^D} \frac{1}{\mathbf{p}_1^2 + m^2} \frac{1}{\mathbf{p}_2^2+m^2} \frac{1}{(\mathbf{p}_1+\mathbf{p}_2 + \mathbf{q})^2+m^2}\vphantom{\dfrac{a}{b}}
\tag{8.74}\label{8.74}
\end{equ... | Starting with the integral,
<span class="math-container">\begin{equation}
I(d) = \int dq_1dq_2 \frac{1}{(q_1^2+m^2)(q_2^2+m^2)[(q_1+q_2+P)^2+m^2]},
\tag{1}\label{1}
\end{equation}</span>
and inserting the partial q in (1), we have
<span class="math-container">\begin{equation}
I(d) = \frac{1}{2d}\int dq_1dq_2 \left(\fra... | <span class="math-container">$\newcommand{\bl}[1]{\boldsymbol{#1}}
\newcommand{\e}{\bl=}
\newcommand{\p}{\bl+}
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\newcommand{\les}{\bl&... | https://physics.stackexchange.com |
65,963 | [
"https://mathoverflow.net/questions/65963",
"https://mathoverflow.net",
"https://mathoverflow.net/users/1223/"
] | The following theorem, due to Regev, is one of the cornerstones of the theory of PI algebras (i.e., associative algebras satisfying a nontrivial polynomial identity):
Let $A$, $B$ be two PI algebras over a field $K$. Then their tensor product $A \otimes_K B$ is PI.
Consider the following "proof" of this theorem. Sin... | I spoke about this with Louis Rowen (who, among other, wrote a few books on the subject) and here is what I got from this conversation:
<ol>
<li>There is no circular dependency in this argument.</li>
<li>The Razmyslov-Kemer-Brown theorem is more difficult and complicated result than Regev's theorem, so there is no muc... | Your proof depends on $A$ and $B$ being finitely generated: $C[[x]]$ is a PI-algebra whose Jacobson radical is not nilpotent. Regev's theorem works, in general, if I remember correctly...
| https://mathoverflow.net |
373,873 | [
"https://softwareengineering.stackexchange.com/questions/373873",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/92672/"
] | After having studied GoF Prototype pattern and taken a look into Java implementation using the <code>Object#clone()</code> method I am still confused about when to use this pattern in a real application?
Please, provide some use-cases from Java applications, frameworks, etc. if there are any.
| Posting comment as answer (elaborated slightly):
A possible real world application might be say, when you need to create a spreadsheet containing many cells. Rather than set the style for each newly created cell to override the default stylings, you'd use a Prototype pattern to create a template cell, and clone that ... | Here's an example from the GoF book itself. Suppose you are developing a GUI/application framework that, among other things, provides a button control that is designed to be a way to create <em>application-specific</em> objects. This means that the framework can't know what these objects are; instead it has to provide ... | https://softwareengineering.stackexchange.com |
633,855 | [
"https://math.stackexchange.com/questions/633855",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/120421/"
] | As a strict beginner, I have been warned about getting mired in the natural language terms "inclusion" and "belonging," and this makes perfect sense to me. However, I have a problem about the characterization of "elements" and "subsets" which is related but harder for me to resolve.
For the following set:
<em>X</em> ... | You're completely correct. You ask "is it worthwhile to note this difference?" and my answer would be "yes, that's a good thing to have noted for yourself." But in practice, most sets we encounter in math tend to contain just one kind of thing (either numbers, or sets of numbers, or functions, or ...), so this kind of ... | This situation comes up very often in set theory (considered as a subject in its own right), since there the only objects we have are sets. That is, whenever we have a set $x$, all of its elements are sets as well.
So, we might have sets $x$ and $y$ such that $x\in y$ <em>and</em> $x\subseteq y$. But $x$ is still the ... | https://math.stackexchange.com |
65,045 | [
"https://dba.stackexchange.com/questions/65045",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/6096/"
] | I have a SQL Server 2005 instance that has worked fine for years. There have been no changes that I am aware of. Recently, I found that if I stop and then start the instance in SQL Server Configuration Manager, I get the error:
<code>The request failed or the service did not respond in a timely fashion. Consult the ev... | Well, my teammate found the answer to the problem.
He changed some details in the file 'tnsnames.ora' at ORACLE_HOME\NETWORK\ADMIN. He just replaced “hostname” by “localhost”.
<code>WP5 =</code><br>
<code>(DESCRIPTION =</code><br>
<code>(ADDRESS = (PROTOCOL = TCP)(HOST = <b>localhost</b>)(PORT = 1521... | Did you set the ORACLE_SID?
I think the problem is there
| https://dba.stackexchange.com |
678,574 | [
"https://physics.stackexchange.com/questions/678574",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/319971/"
] | I'm sorry if this is a naïve question, but it's been bugging me for a bit. If a spacecraft were traveling in a perfect space with unlimited fuel, could it not, assuming it is not influenced by other objects in space, continuously accelerate until it reaches or exceeds the speed of light?
| No it couldn't. One way to see this is to use the equation for kinetic energy, <span class="math-container">$E_k$</span>, of a body of (invariant) mass <span class="math-container">$m$</span>, as derived from the Special Theory of Relativity...
<span class="math-container">$$E_k=(\gamma - 1)mc^2\ \ \ \ \ \ \ \text{in w... | Let me concentrate on the part of the question about the accelerating spacecraft with the idealized engine (not running out of fuel). So imagine that you are sitting on this spaceship. The engine is running constantly at the same rate, nothing changes at all.
So imagine that the spacecraft is standing still. Then the e... | https://physics.stackexchange.com |
63,805 | [
"https://electronics.stackexchange.com/questions/63805",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/9459/"
] | Is it safe, or maybe I should I say "logical", to take an AC adaptor and splice the wires, reconnect them and solder together with a USB cable to power a device?
I had an old gameboy advance sp laying around and had an idea (sometimes my ideas are good, but most of the time they are not so good), well I cut the ac ada... | It <em>can</em> be safe, and it <em>can</em> be logical, but based on your description, it is neither. By "AC adapter" I am assuming you are talking about a regulated wall pack (wall wart, as some people say)...
Every device has different power requirements: voltage level and current input. It is becoming easier to mi... | With respect to @Kurt E Clothier's answer above, there are some clarifications I feel that need to be made.
You can't "overcurrent" a device. It just won't use the excess. If your AC adapter has higher current rating it won't destroy your device.
You can certainly exceed the device voltage which is why his AC adapte... | https://electronics.stackexchange.com |
194,866 | [
"https://physics.stackexchange.com/questions/194866",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/82015/"
] | Imagine that there's a force that is given to you in $psi$, say $1450\ psi$. And there's some body of weight say, $100\ kg$.
What I want to know is that, how much $psi$ will I need to lift the "body" above the ground? Is there any formula to calculate the same?
Moreover, I'd like to know that how much $kg$ can $1450... | To know the force, you'd know the area. Since Psi is a unit of pressure, you could have almost have no pressure and lift thousands of tons. On the contrary, you could have just a tiny fraction of a millionth of a gram but a very high pressure.
Since pressure is defined as force per unit area, $p = \frac{F}{A} \Leftrig... | To lift 100 kg (220 lbs) using a pressure of 1450 psi (about 100 atm), the minimum area needed to apply this force is:
$$area=\frac {220\ lbs}{1450\ psi}$$
$$area=0.152\ in^2$$
| https://physics.stackexchange.com |
450,229 | [
"https://physics.stackexchange.com/questions/450229",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/99567/"
] | If I travel to a 3d location where Einstein was known to have existed at some other time, I do not expect to see him, because his 4d location is different! Why then can I send a clock around a black hole and bring it back, verify that it's different from my clock???? How can I look at two clocks at the same time, if th... | The two clocks traveled different paths. Even in Newtonian physcics, the two paths can have different lengths in space. If two cars clear their trip odometers, then travel two different roads to the same location, they may have different values on arrival. In general relativity, the paths can also have diffferent l... | Your 2 clocks travelled on 2 different worldlines through 4D spacetime since they were synchronised, but now they are united at almost the same spacetime location (event), so you can easily compare them.
| https://physics.stackexchange.com |
403,062 | [
"https://electronics.stackexchange.com/questions/403062",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/202270/"
] | I am designing medical data acquisition equipment that requires moderately fast data transfer. As part of the design we are using a custom 100BASE-T PoE architecture (and cabling) to the data acquisition headstage, together with an intermediate "dumb" device to (1) provide medical-level isolation, (2) inject PoE power,... | You should be able to get away with two sets of magnetics, that is effectively what happens in POE midspan injectors (one transformer at each end plus one for the midspan injector).
| You do not need to invent bicycle - I am sure there're hub or ethernet switch devices available on the market certified for medical applications able to provide PoE to its connected devices. If you have custom connector in your design, then just provide appropriate cable with RJ-45 at one end and custom connector at an... | https://electronics.stackexchange.com |
172,283 | [
"https://physics.stackexchange.com/questions/172283",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/76125/"
] | What does it mean to have a negative electric potential? not talking about potential difference or voltage.
| Potentials are defined up to an arbitrary constant, so there is no particular meaning associated to a negative potential. It is the difference of potential that really matters, since the arbitrary constant is then washed away.
| Yes you can have negative electrostatic potentials. Consider a system of 4 charges $q$ and four $-q$ organized in a cube of lengths $d$ such that no charges of the same sign are adjacent to each other. This system has potential energy $$U=-\frac{q^2}{\pi\epsilon_0 d}\left(3+\frac{1}{\sqrt{3}}-\frac{3}{\sqrt{2}}\right)\... | https://physics.stackexchange.com |
62,980 | [
"https://physics.stackexchange.com/questions/62980",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/11625/"
] | One would like to motivate the classical partition function in the following way: in the limit that the spacing between the energies (generally on the order of $h$) becomes small relative to the energies themselves, one might write
$$Z_{quantum}=\sum_i e^{-\beta E_i} \stackrel{?}{\rightarrow} \int e^{-\beta E}dE$$
(w... | I prefer to see it in the following way:
\begin{equation}
Z_{quantum} \equiv \sum_{m} e^{-\beta E_m}
\end{equation}
Where $m$ is a quantum microstate eigenstate of the Hamiltonian. Now, you can split the sum into two parts; a sum over quantum microstates that yields the same energy eigenvalue $E_n$ and a sum over all... | I assume the system to be $N$ particles with potential energy $U(\vec{R})$ and kinetic energy $\frac{1}{2}(\dot{\vec{R}},M\dot{\vec{R}})$ where $R$ is a $3N$-dimensional vector in the configuration space and $M$ is the mass matrix. In particular, I assume that the particle numbers are fixed -- there is no interactions ... | https://physics.stackexchange.com |
467,360 | [
"https://stats.stackexchange.com/questions/467360",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/285655/"
] | So, consider a function <span class="math-container">$F(x,\theta)$</span> that needs to be linear in relation to the parameters <span class="math-container">$\theta$</span>. If
<span class="math-container">$y_i = \alpha\beta + \beta^2x + \epsilon$</span>
Then, it is possible to linearize it by defining <span class="m... | The following argument indicates how to address such questions generally.
Let's suppose there is a vector parameter <span class="math-container">$\theta\in\Theta\subset\mathbb{R}^p$</span> and a one-to-one differentiable reparameterization <span class="math-container">$\alpha = h(\theta)$</span> (where <span class="ma... | Your first example is a model with two effective parameters:<span class="math-container">$$y=\beta_0+\beta_x x+\varepsilon$$</span>
You have two degrees of freedom <span class="math-container">$\alpha,\beta$</span> so you were able to linearize the model. Having the same degrees of freedom is not a sufficient condition... | https://stats.stackexchange.com |
167,503 | [
"https://softwareengineering.stackexchange.com/questions/167503",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/12893/"
] | Will performance be better (quicker) if I manually compile the source for a software component for the actual machine that it will be used on, compared to if the source was compiled on another platform perhaps for many different architectures? I got some good results compiling source that I downloaded and I wonder whet... | In many (if not most) cases, yes. This because the compiler can produce native code optimized for that particular CPU and environment. The code is more "targeted".
| Compiling from sources gives you an advantage of setting the compiler flags the way you want for your specific platform. Downloading a package that has been compiled with the same exact settings as yours would offer no difference, but changing the settings from the defaults can get you big improvements.
For example, i... | https://softwareengineering.stackexchange.com |
17,156 | [
"https://mechanics.stackexchange.com/questions/17156",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/10585/"
] | I'm currently trying to replace the starter on a 2006 Chevy Cobalt.
When I uninstalled the starter, I remove two screws holding the starter and another two, one that holds the ground and one that holds the current but after I removed the whole thing I realized that there are three wires total, the ground and other tw... | Actually, the wires that you have are thus:
Big fat gnarly wire is the main power wire, as you surmised. It comes directly from the battery. It provides power to the solenoid, not to the starter though (well, ultimately it gets to the starter ... I'll explain). There are two parts to the starter (in most vehicles). Th... | The purple wire is actually for power for fog lights the lt model has the sensor on it then it clicks into the fog light harness from there
| https://mechanics.stackexchange.com |
129,777 | [
"https://physics.stackexchange.com/questions/129777",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/56756/"
] | Imagine a book that we lift it with a force that is exactly equal to the force of gravity so the forces cancel out and the book moves with a constant velocity.
Consider the situation after the book has been lifted, and it has come to rest once again. According to the work and kinetic energy laws
$\Delta W = \Delta K... | You say:
<blockquote>
Imagine a book that we lift it with a force that is exactly equal to the force of gravity so the forces cancel out and the book moves with a constant velocity.
</blockquote>
so I'm guessing your reasoning is that the net force on the book is zero so the amount of work done on the book is zero.... | Let's say the book starts and stops from rest, as I believe you are assuming. The motion within this interval is unimportant, as you'll see.
The increase in gravitational potential energy of the Earth-book system came from your body. You did positive work on the system since your hand force and displacement are in the... | https://physics.stackexchange.com |
1,381,941 | [
"https://math.stackexchange.com/questions/1381941",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/248257/"
] | <blockquote>
Show by differentiating that $\ln x$ is a concave function of $x$. Deduce that if $p,q,x,y$ are positive real numbers with ${1\over p}+{1\over q}=1$, then $$xy \lt {x^p\over p}+{y^q\over q}$$
</blockquote>
I can see why $f(x)=\ln x$ is concave by differentiating it twice giving $-{1\over x^2} \lt 0$, bu... | Note that
$$\ln(x y)= \frac{1}{p} \ln x^p + \frac{1}{q} \ln y^q ,$$
Since $q^{-1}+p^{-1}=1$, we can use Jensen's inequality to show that
$$\ln(x y)=\frac{1}{p} \ln x^p + \frac{1}{q} \ln y^q \leq \ln \left(\frac{x^{p}}{p} +\frac{y^q}{q} \right)$$
Now, just take the exponential to get the answer. not that the inequalit... | As $\log$ is concave, you have, for all $u,v>0$ and for all $\lambda \in (0,1)$,
$\log(\lambda u+(1-\lambda)v)>\lambda\log(u)+(1-\lambda)\log(v)$
Now try to apply this inequality to your problem.
| https://math.stackexchange.com |
306,637 | [
"https://physics.stackexchange.com/questions/306637",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/141913/"
] | If a neutron has a mass ($1.674929\times 10^{-27} kg$) and all mass is defined as energy in the formula $E=mc^2$, does that mean that a neutron has an electromagnetic charge but is overall neutral, or do we understand this as having absolutely no charge at all and only "nuclear" energy?
Is there any other evidence of... | It is an experimental fact that neutrons have no net charge. They have a magnetic dipole moment, which points to having charged constituents (quarks), the charges of which algebraically sum to $0$. (It is not the only evidence for constituents in the neutron: scattering experiments also indicate the presence of consti... | The fact that the electromagnetic force doesn't affect the neutron is precisely <strong>the</strong> evidence that the neutron has no charge, by definition. Electric charge is a measure of the interaction of an object with the electromagnetic field; if something has no charge then electromagnetism doesn't affect it, an... | https://physics.stackexchange.com |
41,652 | [
"https://mechanics.stackexchange.com/questions/41652",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/13160/"
] | after installing the new headers on my car I am having an issue with the bolts not going in all the way in both from the cat end and the block end and I am afraid of snapping them from using too much force. The thing is they came out very easily without even having to use penetrating fluid but now they are giving me tr... | There are three potential reasons why a battery could be under the trunk.
One is improved weight distribution. This is the deciding factor in high-performance cars that often are rear wheel drive or all wheel drive. Because the engine is typically under the hood, cars naturally have more weight on the front wheels tha... | I can speak for the W12 Touareg (and V10 TDI), which employs a two-battery design. Neither battery under the engine bonnet/hood, primarily because space is at a premium (so much so that the engine and transmission have to be disconnected from the drivetrain in order to replace front arm bushings).
<ul>
<li>The larger ... | https://mechanics.stackexchange.com |
24,656 | [
"https://chemistry.stackexchange.com/questions/24656",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/13213/"
] | What mass of precipitate forms when a solution containing 6.24 g of potassium sulfide is reacted with a solution containing 19.2 g of barium nitrate?
I have already identified the limiting reagent $\left(\text{K}_2 \text{S}\right)$ as well as the mass of the precipitate.
My question, however, is: why is the Barium Su... | So basically this problem is bad and incomplete (they tend to be at this level) in that it doesn't give you the concentrations/volumes of the reagents/solutions, so we end up making several assumptions. First of all, <strong>precipitates cannot be aqueous</strong>, by definition they are solids. So the way you end up s... | Generally speaking, if the problem doesn't give concentration or enough information to find concentration, you can't know whether all or only a portion of the product will dissolve. You need to know the solubility product constant (Ksp), the final volume, and the amount of each ion to calculate the amount that will di... | https://chemistry.stackexchange.com |
618,753 | [
"https://physics.stackexchange.com/questions/618753",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/285636/"
] | I am studying Maxwell Boltzmann distribution law, In the derivation I came across a statement and I am not sure if my interpretation is right or wrong.
Statement is "since velocity space has been assumed to be isotropic, the density of velocity points can be taken as independent of inclination of <span class="math... | Because of Newton’s third law, for every action there is an equal but opposite reaction. When your feet push on the ground, the ground pushes back. The force that your foot exerts on a flat ground is mainly determined by friction. The higher the friction, the more efficiently you can walk. If the frictional force is de... | Soft sand deforms when you step on it. The force applied by your foot to the sand moves through the deformation distance, performing work on the sand which your muscles provide. That work does not assist your propulsion; it is lost to the sand.
The sleek flat floor does not deform when you walk on it and therefore does... | https://physics.stackexchange.com |
503,159 | [
"https://physics.stackexchange.com/questions/503159",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/217760/"
] | Due to rotation and low friction, can the Earth be considered a gyroscope? If so, any interesting implications to this? Thanks
| A spinning celestial body can indeed be considered to be a gyroscope.
Part of the motion pattern of the Earth is that it is subject to a gyroscopic precession with a period of about 26.000 years. The common name for this gyroscopic precession is 'precession of the equinoxes'. The precession of the equinoxes was notice... | The gyroscopic motion of the Earth is the reason we have seasons. For half the year the northern hemisphere is closer to the sun than the southern hemisphere, and for the other half of the year the opposite is true. This is because the axial direction of the Earth stays fixed as it rotates around the sun, a direct resu... | https://physics.stackexchange.com |
213,944 | [
"https://security.stackexchange.com/questions/213944",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/201889/"
] | I need to send verification emails for things like
<ul>
<li>email verification</li>
<li>password reset</li>
<li>email change</li>
<li>password change</li>
</ul>
In the past most webapps would send an email with a clickable <strong>verification link</strong> that I'd click to go back to the site and complete the proc... | The overall goal is to verify that a user can be reached through the email address.
This is done by creating a token, and any way of delivering that token back to your server will do, whether it is having the user click a link or copy/paste (or retype) a code.
Since your list includes password resets and email resets... | For me it was a matter of how efficient the verification process was and how well it actually worked in practice.
<strong>[ verification code ]</strong>
<ul>
<li>can be generated with the specific complexity you need/want</li>
<li>will most likely reach target</li>
<li>can be protected itself in various ways from int... | https://security.stackexchange.com |
398,128 | [
"https://mathoverflow.net/questions/398128",
"https://mathoverflow.net",
"https://mathoverflow.net/users/134512/"
] | Atiyah duality is the equivalence <span class="math-container">$M/\partial M \simeq (M^{-T(M)})^\vee$</span>, i.e. the Spanier-Whitehead dual of the space <span class="math-container">$M/\partial M$</span> is the Thom complex of the stable normal bundle of <span class="math-container">$M$</span>. The theorem is proven ... | Here is another short construction which is much simpler and just takes a few lines.
<ol>
<li>Let <span class="math-container">$M$</span> be a closed <span class="math-container">$n$</span>-manifold. Consider the diagonal <span class="math-container">$M \to M \times M$</span>. It is an embedding. Take its Pontryagin-Th... | <ol>
<li>Assume <span class="math-container">$M$</span> is a closed, smooth manifold of dimension <span class="math-container">$n$</span>. Let <span class="math-container">$\tau^+$</span> be the fiberwise one point compactification of its tangent bundle. This is a fiberwise <span class="math-container">$n$</span>-spher... | https://mathoverflow.net |
35,042 | [
"https://stats.stackexchange.com/questions/35042",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/13568/"
] | If I have random variables $X_1,X_2,\ldots,X_n$ that are Poisson distributed with parameters $\lambda_1, \lambda_2,\ldots, \lambda_n$, what is the distribution of $Y=\left\lfloor\frac{\sum_{i=1}^n X_i}{n}\right\rfloor$ (i.e. the integer floor of the average)?
A sum of Poissons is also Poisson, but I am not confident e... | A generalization of the question asks for the distribution of $Y = \lfloor X/m \rfloor$ when the distribution of $X$ is known and supported on the natural numbers. (In the question, $X$ has a Poisson distribution of parameter $\lambda = \lambda_1 + \lambda_2 + \cdots + \lambda_n$ and $m=n$.)
The distribution of $Y$ i... | As Michael Chernick says, if the individual random variables are independent then the the sum is Poisson with parameter (mean and variance) $\sum_{i=1}^{n} \lambda_i$ which you might call $\lambda$.
Dividing by $n$ reduces the mean to $\lambda / n$ and variance $\lambda / n^2$ so the variance will be less than the equ... | https://stats.stackexchange.com |
21,021 | [
"https://stats.stackexchange.com/questions/21021",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/8504/"
] | Disclaimer: I know absolutely nothing about statistics. I've had trouble searching for answers to my question, as I don't have much knowledge about the terminology of statistics.
I'm currently trying to plot a graph with two sets of values that are widely different. This doesn't really matter, but I'm doing this in P... | If you are interested in the changes as a fraction, then simply plot the logarithm of the values. A fixed distance in log space is a fixed fractional change, so if one line is steeper than the other it is changing more rapidly.
The log scale may also allow you to conveniently get both sets of values onto one graph wit... | You ask "is this correct?" and "is there a better way to do it?" but the answers to these questions depend on what exactly you are trying to do. A statistical graph is "wrong" only if it does things like distort the data; it is "bad" if it is hard to read, etc.
Are you interested in the <em>difference</em> between ... | https://stats.stackexchange.com |
128,025 | [
"https://electronics.stackexchange.com/questions/128025",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/52166/"
] | If we apply a voltage at the primary terminals of a transformer then for this we can get a voltage at the secondary. But they have 180 degree phase difference. What is the reason behind this??
I want to know why the phase difference between primary voltage and secondary voltage is 180 degree.
| Imagine you wound an inductor but instead of one thicker wire you used two thinner wires (this is commonplace by the way). An AC voltage is applied on both wires - lets call the AC voltage AC live and AC neutral and lets say neutral is connected to 0V for reference.
No problem so far?
Now, imagine you only applied AC... | The phase difference exists or not depending on whether the primary and secondary turns are wound in the same direction or not, which will make which end of the secondary you compare with what end of the primary in phase or not.
| https://electronics.stackexchange.com |
415,583 | [
"https://physics.stackexchange.com/questions/415583",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/200020/"
] | I have spent an hour with all I had got to figure out the diffrence between them : So What I understood is please correct me if I'm wrong (I know I will be dead wrong) That electrostatic potential occurs in stationary charges and that this term means The potential or the ability of a charge to do work and for Electric ... | For small changes in the state of an ideal gas you can derive from $pV=nRT$ (by taking logs and then taking differentials) the relationship$$\text{d} \ln V=\frac{1}{T}\text{d}T-\frac{1}{p}\text{d}p.$$This has the same form as Chester Miller's relationship for a solid, so you can regard $\frac{1}{T}$ as his $\alpha$, an... | $$dlnV=\alpha dT-\beta dP$$ where $\alpha$ is the coefficient of volume thermal expansion and $\beta$ is the bulk compressibility (both functions of temperature and pressure).
| https://physics.stackexchange.com |
239 | [
"https://mathoverflow.net/questions/239",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3/"
] | I'd love to have a list of 'small' 2-knots, for some sense of small. It's not clear what one should filter by, but there are two obvious candidates
<ul>
<li>Write a movie presentation, and count the frames.</li>
<li>Project the 2-knot to R^3, and count the triple points.</li>
</ul>
Does anyone know if this has been a... | There is a table in Yoshikowa's paper based upon the representation as graphs with certain markings of vertices. The markings indicate which directions are the A and B smoothings. The table is quite small.
Kamada's book on Surface braids has a nice table of 2-knots that don't have triple points. There are tons of the... | Another complexity filter could be the number of 4-dimensional simplices in a triangulation of the complement.
2-knots complements have simple ideal triangulations. The proof is very similar to the proof for 1-knots, viewed through the lens of Morse theory on manifolds with boundary. ie: the Wirtinger presentation i... | https://mathoverflow.net |
88,380 | [
"https://mathoverflow.net/questions/88380",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4085/"
] | Consider Higher order predicate logic over dependent type theory (DPL) as defined in Chapter 11 of B. Jacobs's book "Categorical Logic and Type Theory" (though I think this question applies to first-order predicate logic too).
In the category $\mathbb{P}$ of propositions-in-dependent-type-contexts, an object is a well... | My understanding of the situation is that you have to correctly understand what the morphisms do. A geometric picture might be helpful.
Let us interpret a context $\Gamma = x_1 : A_1, \ldots, x_n : A_n$ as a cartesian product $|\Gamma|$ of topological spaces. A proposition in a context $\Gamma \vdash \phi$ is interpre... | I'm not sure what you mean by the individual propositional objects having models or not.
In any case, as you said, as morphism in $\mathbb P$ simply <em>is</em> a context morphism compatible with the preordering. So $A$ and $B$ are definitely not isomorphic in $\mathbb P$. In a model, $A$ is the initial subobject of t... | https://mathoverflow.net |
118,275 | [
"https://mathoverflow.net/questions/118275",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4231/"
] | In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with <em>split</em> Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A}$ and writes $\langle \mathcal{A} \rangle$ for its split Grothendieck group: the free abelian group on objects ... | These groups are mentioned in [Swan '68 - Algebraic K-Theory, p.69]. He constructs $K_0(\mathcal{A}, S)$ for a class $S$ of exact sequences in $\mathcal{A}$. Take the free abelian group mod the relations from sequences in $S$. For example the class of all exact sequences for the Grothendieck-group $K_0(\mathcal{A})$ or... | The split Grothendieck group for vector bundles on a complete variety appears in Nori's PhD thesis on the fundamental group scheme; this was published in the Proceedings of the Indian Academy of Science in 1981. It is used to define and study finite vector bundles. Nori does not give any references, so as far I know th... | https://mathoverflow.net |
2,549,651 | [
"https://math.stackexchange.com/questions/2549651",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/509051/"
] | I solved the limit
$$\lim_{x \to \infty} x(\ln(1+x) - \ln(x))$$
by writing it as $\lim_{x \to \infty} \frac{\ln(\frac{1+x}{x})}{\frac{1}{x}}$ and applying l'Hopital rule but is it possible to solve it without using l'Hopital rule?
| \begin{align*}
x(\ln(1+x)-\ln(x))=x\ln\left(1+\dfrac{1}{x}\right)=\ln\left(1+\dfrac{1}{x}\right)^{x}\rightarrow\ln e=1,
\end{align*}
if you accept that $\left(1+\dfrac{1}{x}\right)^{x}\rightarrow e$ as $x\rightarrow\infty$.
| By MVT there is $c \in [x,x+1]$ such as $\ln(1+x)-\ln(x)=1/c$ so $x(\ln(1+x)-\ln(x))=x/c$
$c \in [x, x+1]$ therefore $x/c \in [x/(x+1), 1]$.
So by the squeeze theorem lim is $1$.
| https://math.stackexchange.com |
15,071 | [
"https://quant.stackexchange.com/questions/15071",
"https://quant.stackexchange.com",
"https://quant.stackexchange.com/users/17133/"
] | We see in the market that a implied volatility surface is not flat. Based on this observation different models were developed to capture the structure, e.g. CEV / SABR.
A measure often used for the skew is a risk reversal, i.e.
$$\sigma_{25,c}-\sigma_{25,p}$$
and butterfly
$$\frac{\sigma_{25,c}+\sigma_{25,p}}{2}-\... | You are absolutely correct that they should be seen as approximations. While it would be nice to let h go to zero in a mathematical sense this is of course impossible in real life as the options are only traded in particular intervals. While the smallest interval may be less than 25, for historical reasons traders h... | A measure of vol skew which is used is dsigma/dk or dc/dk. You first need to build arbitrage free volatility curve for that. Rr and fly are just used to get the pillar points and only in fx. The IR market directly gives pillar points I.e sigma(k). I would suggest you read Gatheral book if you want to know in detail of ... | https://quant.stackexchange.com |
4,592,387 | [
"https://math.stackexchange.com/questions/4592387",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/19331/"
] | Let <span class="math-container">$U,V$</span> be two connected, bounded open sets in the plane, and assume that <span class="math-container">$\partial V \subset U$</span>, where <span class="math-container">$\partial V$</span> is the boundary of <span class="math-container">$V$</span>.
Is it true that <span class="math... | Try
<span class="math-container">$$
U = \{z : 1<|z|<3\}\\
V = \{z : |z|<2\}
$$</span>
Show <span class="math-container">$\partial V \subset U$</span> but <span class="math-container">$V \not\subset U$</span>.
<hr />
Now try with hypothesis "simply connected".
| No, this does not follow.
As a counterexample consider <span class="math-container">$V$</span> to be the open unit disk and <span class="math-container">$U$</span> an open disk of radius larger than one with its center removed.
Then <span class="math-container">$\partial V = S^1 \subset U$</span> but <span class="math-... | https://math.stackexchange.com |
139,780 | [
"https://security.stackexchange.com/questions/139780",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/543/"
] | What are the relative merits / deficiencies of a Privileged Access Management (PAM) service compared with a strict policy of named users (no account sharing)?
By PAM, I'm specifically referring to devices which act as proxies, identifying the end-user and substituting shared credentials to access a target account, e.g... | I don't think it's reasonable to limit the response to a single commercial
product. Looking at the question more generally:
<ul>
<li>There are a few basic strategies to control access to high-privileged
access (root, Administrator and the like) on systems:
<ul>
<li>Create a personal admin account for every human tha... | As OP mentioned CyberArk, this would answer part of query.
CyberArk can be set to
<ul>
<li>2FA logon to CyberArk before view the list of target device account</li>
<li>change password with password history on device like Windows or service like Facebook or CyberArk logon itself
<ul>
<li>automatically</li>
<li><stron... | https://security.stackexchange.com |
420,331 | [
"https://stats.stackexchange.com/questions/420331",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/255267/"
] | It seems like a basic questions, but I found so many answers on the internet...
I have an experiment in which I observe 3 plants per modality. I have 6 different treatments; All the conditions are okay to run an ANOVA ( normality, homogeneity and independance). Is it correct to run an ANOVA with only 3 replications p... | ANOVA doesn't have a minimum sample (or category) size requirement, as long as it can be estimated. The only difference between an ANOVA and Kruskal-Wallis is that KW is the non-parametric version of ANOVA, used when the assumptions of ANOVA do not hold.
The reason why someone could suggest using a KW instead of an AN... | Yes, it's fine to do an ANOVA. A Kruskal-Wallis test is basically an ANOVA on the ranks instead of the values themselves. If you do the ANOVA and it turns out that the residual distributions are a problem (and cannot be dealt with via a transformation or generalized linear model), you can do a Kruskal-Wallis instead.
... | https://stats.stackexchange.com |
16,897 | [
"https://mechanics.stackexchange.com/questions/16897",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/6919/"
] | My cousin told me when he enter his car, he put the key to ignition then count to 3 and start the engine.
The main reason of him doing that is to give enough time to fluids (I'm not sure which one) to fill to the engine (I'm skeptic, need more info on this).
He told me that in <em>"ze good ol'days"</em>, car would so... | The only type of vehicle you need wait on is a diesel powered one. The reason for that is you need to wait (in most of them) for the glow plugs to heat the combustion chamber. Modern fuel injected cars will keep their fuel pressure after shutdown. It will stay there (usually) for a good 24 hours (<strong>Note:</strong>... | I used to do this with my Ford Fiesta and Puma - it's all about fuel. When the ignition key is turned on, the fuel pump runs for a few seconds, then shuts off. This ensures there is enough fuel pressure in the rail to start (since a richer mixture is required to start when cold). The Puma did not like me turning the ke... | https://mechanics.stackexchange.com |
716,150 | [
"https://physics.stackexchange.com/questions/716150",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/312538/"
] | I am just confirming here how the concept of electrostatic shielding helps in preventing lightining to be not getting into the car inside .
<blockquote>
So we know car has a metal body outside , so first consider there is no lightning happening outside so the outer surface has no charge at all of car body . And the who... | The situation is nearly electrostatic/magnetostatic because the car is small compared to the lightning bolt. A pulse of lightning gathers its energy from a field that is kilometers in extent. At the speed of light, this takes microseconds. A representative model of lightning surges commonly used by electrical engineers... | lightning is an electrical current. this means it consists of an electrical carrier, carrying the electrical charge through a path. this path, starts from a highly charged cloud, and ends in the earth soil, for it has the least possible voltage. everything in nature, if not manipulated by an ulterior force, takes the ... | https://physics.stackexchange.com |
178,685 | [
"https://physics.stackexchange.com/questions/178685",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/78895/"
] | If the centre of mass of a system of particles is at the origin, does this mean that if there is a particle on the positive X-axis, there must be atleast one particle on the negative X-axis?
I feel that this is true, since to get $X_{COM}$ as zero there should be a particle on the negative X -axis.
Am I on the ri... | After doing some more research I found the answer to my question.
The method I proposed was actually one of the first methods for hydrogen-boron fusion that was tested. It's called "fixed/solid target proton-boron-11 fusion". Experimentation very quickly showed that the method could not work because of two big problem... | Usually when you find the term "shielding" as it refers to electrons and fusion, it refers to the electron's aiding fusion, not absorbing energy from an incoming proton.
The electrons screen the nucleus' positive charge until the proton gets much closer to the nucleus than it would if the nucleus were fully or partial... | https://physics.stackexchange.com |
2,863,407 | [
"https://math.stackexchange.com/questions/2863407",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/430142/"
] | Am I right that the rows of an $(n-1)\times n$ matrix, $\mathbf{A}_{(n-1)\times n}$, are linearly dependent iff
$$A_{n,i}=0,\forall i\in\{1,..,n\}\text{, where }A_{n,i}\text{ is the }(n,i)\text{-th cofactor of }\mathbf{A}_{n\times n},$$
where $\mathbf{A}_{n\times n}$ is $\mathbf{A}_{(n-1)\times n}$ with a blank row ad... | Since you are talking about dependent types, I suggest
$(a:A) \to B_a$
or
$(a \in A) \to B_a,$
where the former is more "typish" and the latter more mathematical.
If you think "that's a function, not a pair", remember that in mathematics, a function $f : X \to Y$ is defined as a subset of $X \times Y$ such that for a... | Given an indexed family of sets $(B_a)_{a\in A}$, the set $\{\langle a,b\rangle\;|\;a\in A\wedge b\in B_a\}$ is just the disjoint union of the indexed family. Depending on context, you may see this as $$\biguplus_{a\in A}B_a$$ or $$\coprod_{a\in A}B_a$$ or $$\sum_{a\in A}B_a.$$
| https://math.stackexchange.com |
713,104 | [
"https://math.stackexchange.com/questions/713104",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/2538/"
] | Are there any combinatorial games whose order (in the usual addition of combinatorial games) is finite but neither $1$ nor $2$?
Finding examples of games of order $2$ is easy (for example any impartial game), but I have not been able to think up an example with finite order where the order did not come from some sort ... | $$f(n)=\sum_{i=1}^n \frac{i(i+1)}{2}$$
$$f(n)=\sum_{i=1}^n \frac{i^2}2 +\frac{i}2$$
Using two well known identitites,
$$f(n)=\frac12\left(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\right)$$
Simplifying:
$$f(n)=\frac{n(n+1)}{4}\left(\frac{2n+1}{3}+1\right)$$
$$f(n)=\frac{n(n+1)}{4}\left(\frac{2n+4}{3}\right)$$
$$f(n... | There is a general formula $$\sum_{k=1}^n\frac {k(k+1)\dots(k+r-1)}{r!}=\frac {n(n+1)\dots(n+r)}{(r+1)!}$$
Which can be proved by induction - base case $n=1$, both sides of the equation are equal to $1$.
Then $$\frac {n\left[(n+1)\dots(n+r)\right]}{(r+1)!}+\frac {\left[(n+1)\dots(n+r)\right]}{r!}=\frac {\left[(n+1)\d... | https://math.stackexchange.com |
2,314,009 | [
"https://math.stackexchange.com/questions/2314009",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/383926/"
] | Assume that $\mu$ is a probability measure on $\mathbb Z$ with some given properties.
I now want to show that $\mu$ is strictly positive.
I found a proof which starts like this:
Assume $\mu$ is not strictly positive.
Then there exists a $x_0$
such that $\mu(x_0)=0$ and $\mu(x_0+1)+\mu(x_0-1)>0$
M question: a meas... | Hint. Consider the substitution $u_n=n!b_n$ then recursion becomes
$$u_{n+1}=u_n+n.$$
| Further hint (with respect to Robert Z answer). Once you have $u_{n+1}=u_n+n$ you also have
$$ u_N-u_0=\sum_{n=0}^{N-1}\left(u_{n+1}-u_n\right) = \sum_{n=0}^{N-1} n = \frac{N(N-1)}{2} $$
from which $u_N = \frac{N(N-1)}{2}+u_0$ follows.
| https://math.stackexchange.com |
195,904 | [
"https://math.stackexchange.com/questions/195904",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/40337/"
] | I'm currently trying to solve a programming question that requires me to calculate all the integer solutions of the following equation:
$x^2-y^2 = 33$
I've been looking for a solution on the internet already but I couldn't find anything for this kind of equation. Is there any way to calculate and list the integer sol... | Suppose that $x=y+n$; then $x^2-y^2=y^2+2ny+n^2-y^2=2ny+n^2=n(2y+n)$. Thus, $n$ and $2y+n$ must be complementary factors of $33$: $1$ and $33$, or $3$ and $11$. The first pair gives you $2y+1=33$, so $y=16$ and $x=y+1=17$. The second gives you $2y+3=11$, so $y=4$ and $x=y+3=7$. As a check, $17^2-16^2=289-256=33=49-16=7... | We have $x^2-y^2=33$ iff $(x-y)(x+y)=33$. So to solve our equation we find all ordered pairs $(u,v)$ such that $uv=33$. Then we set $x-y=u$ and $x+y=v$, and solve.
We get
$x=\dfrac{v+u}{2}$ and $y=\dfrac{v-u}{2}$. Since $u$ and $v$ will be both odd, $u+v$ and $v-u$ will be even, so $x$ and $y$ will be integers.
The ... | https://math.stackexchange.com |
74,941 | [
"https://mathoverflow.net/questions/74941",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4721/"
] | Just a curiosity:
<blockquote>
Is there an assertion of which a proof (formalizable, say, in ZFC) is not known but a proof that it's <em>not</em> undecidable (in ZFC) <em>is</em> known?
</blockquote>
Edit: after the comments, I think the actual question was
<blockquote>
Is there an ("interesting") assertion of ... | If it's known that some statement $S$ is decidable in ZFC, then you can just run a computer program that enumerates all ZFC-proofs and stops when it finds a proof of $S$ or a proof of $\neg S$. By hypothesis, this algorithm is guaranteed to terminate. Therefore, the only possible obstacle separating decidable stateme... | It shouldn't be hard to find a large number $N$ such that no one knows a proof that $N$ is prime and no one knows a proof that $N$ is not prime. Yet the question of the primality of $N$ can't be undecidable - there is a simple (if impractical) algorithm for deciding it.
| https://mathoverflow.net |
63,722 | [
"https://datascience.stackexchange.com/questions/63722",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/85925/"
] | I have recently read about Fine Tuning, and what I want to know is, when we are fine-tuning our model is it necessary to Freeze the model and train only the top part of the model and then unfreeze some layers and again train the model or one can directly begin by unfreezing some layers? Till now, I have read that one d... | It can work either way. If you want to keep the exact feature extractors, then you should freeze everything except the "top" of the model. You can also unfreeze the whole model; the "top" of the model will be trained from scratch, and the feature extractors near the "bottom" of the model will be tweaked to work bette... | It is a good practice to freeze the early layers when you finetune the model. There are two reasons why you want to do this :
<ol>
<li>Your new layers is initialized randomly and will always start with very big loss. Big loss means big gradient and if the weights are not frozen this will be propagated . Your model wil... | https://datascience.stackexchange.com |
256,713 | [
"https://stats.stackexchange.com/questions/256713",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/93191/"
] | If I run one experiment, and obtain a p-value of $p=.03$ and I have <em>a priori</em> decided that my $\alpha=.05$. I have a significant a result, I conclude that my data reject $H0$.
If I run two identical experiments with the same study $\alpha$, both with a p-value of $p=.03$. Now as I have done two different tests... | You seem to be confusing the requirement of replication (which reduces Type I error rates) with the problem of multiple comparisons (which increases Type I error rates).
REQUIRING REPLICATION:
If you decide a priori to make significance require two independent tests to come out p < .05, you are actually REDUCING th... | I'm not sure that you actually ran two (or one hundred) experiments here.
Instead, it sounds like you ran a single experiment, testing a single hypothesis, twice. As such, your "family" of hypotheses hasn't gotten larger and there is no need to correct your $\alpha$ to maintain the same familywise error rate.
Reali... | https://stats.stackexchange.com |
50,209 | [
"https://quant.stackexchange.com/questions/50209",
"https://quant.stackexchange.com",
"https://quant.stackexchange.com/users/31372/"
] | I know some of base Python, but I have only briefly used numpy, pandas, etc...
I was wondering what's a good resource to learn Python specifically for quantitative finance. I know of plenty of books/resources to learn Python in general. I was wondering if there was a resource for learning Python specifically when it... | Try Hilpisch's books, especially <em>Python for Finance</em>. For derivatives pricing specifically, he wrote <em>Derivatives Analytics with Python</em>.
Hope that helps.
| I started with "Pandas for Data Analysis" by Wes Mckinney (the original developer of pandas) without ever using Python before.
After 3yrs I have the skills of a full stack developer, after some other projects using for example "Flask for Web Development" by Miguel Grinberg which is nice project for getting familiar wi... | https://quant.stackexchange.com |
53,274 | [
"https://softwareengineering.stackexchange.com/questions/53274",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/2210/"
] | I've often seen such comments be used:
<pre><code>function foo() {
...
} // foo
while (...) {
...
} // while
if (...) {
...
} // if
</code></pre>
and sometimes even as far as
<pre><code>if (condition) {
...
} // if (condition)
</code></pre>
I've never understood this practice and thus never applied it... | I would say if you code is so long that you can't easily follow your braces, your code needs refactoring, for most languages.
However, in templating languages (like PHP) it could be valid, because you might have a large block of HTML that separates the beginning and end of the condition or loop structure.
| It's a code-smell and usually a hangover from old-fashioned code style. Before decent IDEs refactoring was more difficult and not as common as it is now, hence methods were longer and these comments were there to help navigate them better.
| https://softwareengineering.stackexchange.com |
315,977 | [
"https://softwareengineering.stackexchange.com/questions/315977",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/163949/"
] | Given for example this code:
<pre><code>for(int i=1;i<n;i++)
for(int j=1; j < sqrt(i); j++)
foo(); //foo takes constant time
</code></pre>
can someone explain to me how to calculate the computational complexity ("Big O") of this kind of program?
| This is basically a math question, it might be better on "Computer Science" stackexchange or even SO. But I guess you can expect programmers to know it also, and Programmers is not only for "Soft" questions so maybe it is well placed here.
Anyways.
Based on the answers so far, I believe that Phillip Kendall has misre... | The complexity is quite obviously the sum of sqrt (i), for i = 1 to n. That kind of sum is usually quite hard to calculate in closed form. As an approximation you don't calculate the sum, but the integral of sqrt (i) over an interval of width n (since you were summing n values), and if you have no good reason to choose... | https://softwareengineering.stackexchange.com |
463,316 | [
"https://electronics.stackexchange.com/questions/463316",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/156012/"
] | I need to 8 channels of 100Vdc to 300vdc, load current 10mA to 80mA.
my idea is to use a step transformer to step up dc 5v 1A source to 300Vdc 50mA (typical application).
low voltage side of the transformer (5v side) will be controller with MOSFET using a microcontroller and the output side or high voltage side will ... | "Pulses" and large cone displacement suggests the PSU may be shutting down due to too much current demand, then restarting to see if the demand has gone away, shutting down again ad infinitum. 2A isn't very much, with a low impedance speaker.
The large displacement comes from C4 discharging and then re-charging each t... | Based on your description and measurements, when the PC volume is set to over 24% the audio signal voltage is too high for the amplifier. Suitable solutions to allow higher signal voltages at amplifier input connector include attenuating the PC output voltage with a resistor divider or volume control potentiometer befo... | https://electronics.stackexchange.com |
89,741 | [
"https://softwareengineering.stackexchange.com/questions/89741",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/25476/"
] | I inherited an existing code base for a product that is reprehensibly sloppy. The fundamental design is woefully inadequate which unfortunately I can do little about without a complete refactor (HIGH coupling, LOW cohesion, rampant duplication of code, no technical design documentation, integration tests instead of un... | Firstly I'd like to point out that unit tests are not a replacement for integration tests. The two need to exist side-by-side. Be grateful that you have integration tests, otherwise one of your small refactorings could well make one of the low-tolerance cash cows go ballistic on you.
I would start to work on the compi... | Writing unit tests might be more valuable use of your time: it will give you some insights into the current workings of the codebase, and should you decide to start from what you have rather than rewriting everything from scratch, you'll have a solid base to make changes without taking too much risk. Chances are that i... | https://softwareengineering.stackexchange.com |
2,844,128 | [
"https://math.stackexchange.com/questions/2844128",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/574643/"
] | I need to evaluate the following limit without using L'Hopital's rule:
$$\lim _{x\to 0}\left(\frac{5-5\cos\left(2x\right)+\sin\left(4x\right)}{x}\right)$$
I thought the best way was to separate it in two limits:
$$\lim _{x\to 0}\left(\frac{5-5\cos\left(2x\right)}{x}\right)+\lim_{x\to \:0}\left(\frac{\sin\left(4x\rig... | Hint:Using $$\cos(2x) = \cos^2(x)-\sin^2(x) =1-\sin^2(x)-\sin^2(x)=1-2\sin^2(x),$$
we get $$\frac{5-5\cos(2x)}{x} = 10\sin(x)\cdot\frac{\sin(x)}{x}.$$
| <strong>Hint:</strong> Use these two facts and a little bit of algebra:
$$\lim_{x \to 0} \frac{1-\cos(x)}{x} = 0$$
$$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$$
Note that $$\lim_{x \to 0} 10 \times \frac{1-\cos(2x)}{2x} = 0$$
The most straightforward and rigorous way to see why the first relation holds is to use the Taylo... | https://math.stackexchange.com |
5,599 | [
"https://mechanics.stackexchange.com/questions/5599",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/2892/"
] | Why can some cars (mostly old Russian cars on YouTube) seemingly go through water while others can't? I'm talking about situations when the water level is at half the height of the car, so the engine is under the water. What are the differences between engines that break under water and those that don't?
| driving through water could potentially do a couple of different things:
First off, if the water level is over the tail pipe, the engine rpm must be kept high in order to keep the exhaust pressure up, so that water does not go up the tail pipe and stall the engine. If this was to happen, it wouldn't likely do damage, ... | No car is "waterproof" since it's need air at some point to run correctly, but you can make it more "water-friendly". My 1987 Toyota Land Cruiser have been modified to tolerate water crossing up to about mid-door. To do so you'll have to(non exhaustive list) :
<ul>
<li>Raise your air intake as much as you can, mostly... | https://mechanics.stackexchange.com |
166,021 | [
"https://physics.stackexchange.com/questions/166021",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/66165/"
] | I am trying to learn more about quantum mechanics. I am reading a book by Griffiths that I like. I'm trying to summarize what I've learned. So below I provided three assumptions. I'd like to know if they are correct.
Consider a particle in space and time.
<ol>
<li>We cannot know where the particle is with certainty... | <blockquote>
We cannot know where the particle is with certainty.
</blockquote>
The particle, in general, does not have a definite location to know.
<blockquote>
Under certain conditions, we can know and predict future probability
distributions.
</blockquote>
The evolution of the state is <em>determined</em> b... | <ol>
<li>Correct, expect just don't say "on average," just we will find the particle at different locations more often than others in accordance with the probability distribution described by a wave function.</li>
<li>Incorrect, you may always calculate the evolution of an initial state if you know the effective Hamilt... | https://physics.stackexchange.com |
1,553,464 | [
"https://math.stackexchange.com/questions/1553464",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/177463/"
] | Expanding the compressed title, the question is:
Is there a distance function $d$ on $\mathbb{R}^2$ such that a set is closed with respect to $d$ if and only if it is a finite union of affine subspaces of $\mathbb{R}^2$?
So I know that an affine subspace of $V$ over a field $k$ is a set of the form $v+U= \left \{ v... | Let $f_1(x)=-x^2$. Then $f_1'(x)=-2x<0$ and $g_1(x)=-x^2-(1-x))^2 = -2x^2+2x-1$ and $g_1'(x)=2-4x$ so $g(x)$ is decreasing when $x>\frac{1}{2}$.
On the other hand, if $f_2(x)=(1-x)^2$. Then $f_2'(x)=-2(1-x)<0$ and $g_2(x)=(1-x)^2+x^2=-g_1(x)$. So $g_2(x)$ is increasing precisely where $g_1(x)$ was not, and vi... | Can u conclude this inequality???
$$f(\frac{2}{8})+f(\frac{6}{8})<f(\frac{1}{8})+f(\frac{7}{8}) $$
So $D$ is the answer.
| https://math.stackexchange.com |
255,110 | [
"https://softwareengineering.stackexchange.com/questions/255110",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/123282/"
] | We started following the Agile Scrum methodology and we have completed about 10 sprints.
One observation I had is that not all in the team are taking up the responsibilities of completing the tasks and user stories by themselves and everytime they have to be instructed or allotted with some tasks. Also the estimate th... | If you only looked at agile because you were expecting an increased productivity be aware that agile (/scrum) is not a silver bullet. Yes, self empowered teams can become more productive, but they need help.
So get a coach. Agile is like playing chess. It takes 30 minutes to explain the rules and after that you can st... | At the risk of sounding a bit cheeky, one alternative is to actually use scrum. You're using something <em>like</em> scrum but isn't <em>actually</em> scrum.
If you have team members that aren't engaged, it sounds like you need an agile coach to get you past the learning phase.
| https://softwareengineering.stackexchange.com |
528,646 | [
"https://electronics.stackexchange.com/questions/528646",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/111377/"
] | I have an electronic part with a marking mbr10100, I know it is a Schottky diode for 10 A, 100 V. The thing is, it looks like it has two diodes inside. At least it shows so when I test it with a multimeter. My question is - does it have tolerance of 10 A for each in the pair or will it tolerate this voltage and current... | The "BLE rules" you refer to would be the "Core Specifications" of the Bluetooth Special Interest Group (Bluetooth SIG). As far as I know, they do not place a limit on how infrequently a device can advertise.
But you may run into problems connecting, if using a smartphone as your central device. Alt... | <blockquote>
Is it ok to increase the time between BLE advertisements to longer than 10.24 seconds?
</blockquote>
That would reduce the likelihood of being detected by something that intermittently scans, for example because it is itself battery powered.
<blockquote>
Our system is constantly scanning for our beacons so... | https://electronics.stackexchange.com |
190,189 | [
"https://dba.stackexchange.com/questions/190189",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/116213/"
] | My table:
<pre><code>Date Employee Status
-----------------------------
20171106 001 At work
20171107 001 Sick leave
20171108 001 At work
20171109 001 At work
20171111 001 Sick leave (A gap here)
20171112 001 Sick leave
20171115 001 ... | Yes Rick is right. Because all of nodes are down you have to bootstrap Cluster again.
You stopped all nodes gracefully.
If you know What is the last one (node) stopped, choice this node too bootstrap and then start another node.
But If you don't remember the last node which you have turned off, open grastate.dat (lo... | To know in which node was the last boot of the cluster,
assuming it was typed manually in the Linux command line,
you can use the linux command
$ history
at ALL the cluster nodes,
and see which node had the most recent command in history:
systemctl start mysql@bootstrap.service
| https://dba.stackexchange.com |
2,679,067 | [
"https://math.stackexchange.com/questions/2679067",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/348408/"
] | Say $n$ can be represented as a sum of three non-zero squares. (i.e. $n = a^2 +b^2+ c^2$, for some $n,a,b,c \in \mathbb{N}$)
Is it possible that every natural power of $n$ is also a sum of three non-zero
squares? (i.e. $n^k = x^2+y^2+z^2$ for $x,y,z,k \in \mathbb{N}$)
| Yes, more generally, if,
$$N = x_1^2+x_2^2+\dots+x_m^2$$
then one can always find integer $y_i$ such that,
$$N^k = y_1^2+y_2^2+\dots+y_m^2$$
<hr>
<strong>Proof</strong>: Use the expansion of
$$(a+i\sqrt{w})^k$$
To get,
$$a^2 + w = N,\\ (a^2 - w)^2 + (2 a)^2 w = (a^2 +
w)^2,\\ (a^3 - 3 a w)^2 + (3 a^2 - w)... | Yes, this is possible. Take $n=3$. Clearly $3=1^2+1^2+1^2$. Now any power of $3$ is also the sum of three squares by the Three-Squares Theorem, because $3^k$ is not of the form $4^n(8m+7)$. This may include zero summands, though. Otherwise one has to give a direct solution. If $n=3^k$ with $k=2m+1$, then $3^k=(3^m)^2+(... | https://math.stackexchange.com |
15,282 | [
"https://mathoverflow.net/questions/15282",
"https://mathoverflow.net",
"https://mathoverflow.net/users/78/"
] | This is probably a very elementary question in symplectic geometry, a subject I've picked up by osmosis rather than ever really learning.
Suppose I have a symplectic manifold $M$. I believe that a <em>Lagrangian fibration</em> of $M$ is a collection of immersed Lagrangian submanifolds so that as a fibered manifold loc... | Your Question 1 is called Darboux theorem for fibrations (see: Arnold, V., Givental, A., Symplectic geometry, Dynamical Systems IV, Symplectic Geometry and its Applications (Arnold, V., Novikov, S., eds.), Encyclopaedia of Math. Sciences 4, Springer-Verlag, Berlin-New York, 1990.)
Here is how to construct suitable Da... | It seems Question 2 is a special case of: What are the fiber-preserving symplectomorphisms of $T^*M$? This has a nice answer.
First, any diffeomorphism $f$ of $M$ defines a fiber-preserving symplectomorphism of $T^*M$, its cotangent lift, by
$(q,\xi) \mapsto (f(q),((df_q)^*)^{-1}\xi)$. These are exactly the fiber-pres... | https://mathoverflow.net |
4,268,476 | [
"https://math.stackexchange.com/questions/4268476",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/947491/"
] | Let <span class="math-container">$X_n$</span> be a bounded random process and let <span class="math-container">$T$</span> be a stopping time that is finite almost surely.
Suppose we have the inequality <span class="math-container">$E(X_n) \geq c$</span>, where <span class="math-container">$c$</span> is a constant.
Does... | This is not correct. Let <span class="math-container">$\mathbb{P}(X_1 = 2c) = \mathbb{P}(X_1 = 0) = \frac 12$</span> and <span class="math-container">$X_2 = 0$</span> if <span class="math-container">$X_1 = 2c$</span> and <span class="math-container">$X_2 = 2c$</span> if <span class="math-container">$X_1 = 0$</span>. ... | As long as <span class="math-container">$X_n$</span> is a supermartingale, meaning <span class="math-container">$E[X_{n+1}|X_n]\le X_n$</span> for all <span class="math-container">$n\in \mathbb N$</span>, then we can conclude that <span class="math-container">$E[X_n ^T]\ge E[X_n]$</span>. This follows from the general ... | https://math.stackexchange.com |
4,354,151 | [
"https://math.stackexchange.com/questions/4354151",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/924287/"
] | I have the system
<span class="math-container">$x'(t)=2x+8y$</span>
<span class="math-container">$y'(t)=-x-2y$</span>
Which has the I.C. <span class="math-container">$X(0)=(6,-2)$</span>. So I take that it means this:
<span class="math-container">$6=2x+8y$</span>
<span class="math-container">$-2=-x-2y$</span>
and then ... | If<span class="math-container">$$A=\begin{bmatrix}2&8\\-1&-2\end{bmatrix},$$</span>then<span class="math-container">$$\exp(tA)=\begin{bmatrix}\sin (2 t)+\cos (2 t) & 4 \sin (2 t) \\ -\frac{1}{2} \sin (2 t) & \cos (2 t)-\sin (2 t)\end{bmatrix}.$$</span>So, if <span class="math-container">$f(t)=\exp(tA).(... | In your case, the solution writes
<span class="math-container">$$
\mathbf{x}(t) =
a_1 e^{\lambda_1 t} \mathbf{v}_1 +
a_2 e^{\lambda_2 t} \mathbf{v}_2
$$</span>
using the eigenvector/eigenvalue of the matrix <span class="math-container">$\mathbf{A}$</span>
in the relation
<span class="math-container">$
\dot{\mathbf{x}}... | https://math.stackexchange.com |
261,056 | [
"https://stats.stackexchange.com/questions/261056",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/24406/"
] | I came across this in lecture but failed to understand why:
<blockquote>
Why does maximum likelihood estimation have issues with over fitting? Given data X and you want to estimate parameter theta.
</blockquote>
An example would be helpful.
| Maximum likelihood does not tell us much, besides that our estimate is the best one we can give based on the data. It does not tell us anything about the quality of the estimate, nor about how well we can actually predict anything from the estimates.
Overfitting means, we are estimating some parameters, which only hel... | Some models are just too flexible: In these cases, maximum likelihood estimators can effectively "memorize" the data---signal <em>and</em> noise. Such considerations motivate reducing the flexibility of some models, through, for instance, some kind of regularization.
| https://stats.stackexchange.com |
675,025 | [
"https://physics.stackexchange.com/questions/675025",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/290987/"
] | <em><strong>Specific heat is the amount of thermal energy required to increase/ decrease the temperature of a unit mass of a substance by 1K</strong></em>
This definition makes sense to me and is straightforward, however the sources I'm referring to say that specific heat is also a measure of a substance's storage abil... | Consider 1g of iron, which has a specific heat of 0.45 J/g<span class="math-container">$^\circ $</span>C. After heating it by 10 degrees, it stores an additional 4.5J of energy. You can compare this to other forms of potential energy, like compressing a spring, or raising a weight to some height, or charging a battery ... | If we heat an amount <span class="math-container">$m$</span> of a substance with specific heat capacity <span class="math-container">$c_p$</span>, from an initial temperaure <span class="math-container">$T_1$</span> to a final temperature <span class="math-container">$T_2$</span>, then the amount of heat energy <span c... | https://physics.stackexchange.com |
104,684 | [
"https://math.stackexchange.com/questions/104684",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/20944/"
] | <strong>A function $f(x)$ is defined and continuous on the interval $[0,2]$ and $f(0)=f(2)$.
Prove that the numbers $x,y$ on $[0,2]$ exist such that $y-x=1$ and $f(x) = f(y)$.</strong>
I can already guess this is going to involve the intermediate value theorem.
So far I've defined things as such:
I'm looking to sat... | if $g(0)$ is positive, $g(1)$ will be negative and vice versa, so the IVT provides a root. if both are zero, $g(0)=g(1)=0=f(1)-f(0)=f(2)-f(1)$ and you're done as well.
| Hint: You are told that $f(2)=f(0)$ (not $f(1)$ as you say). This gives you that $g(1)=-g(0)$
| https://math.stackexchange.com |
290,892 | [
"https://math.stackexchange.com/questions/290892",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/51744/"
] | I'm sorry in advance for how specific this problem is. I've been trying to make it as generic as possible but this is as far as I could get. I want to prove that a fixed point is attractive.
I didn't find much on Google about proving the attractiveness of a fixed point, but if there is a general methodology, that woul... | We will work on $[0,+\infty[$.
First observe that for $x\geq 0$:
$$
f(x)=x\quad\Leftrightarrow\quad ax+b=x^\alpha.
$$
Now draw the graphs of $ax+b$ and $x^\alpha$ to convince yourself that there could be $0$, $1$ or $2$ positive fixed points in general.
It turns out that the condition $b<b_0$ guarantees that ther... | The second derivative $f''$ is positive as is $f'$ and $f$ itself. In particular, $f(0) > 0.$ So, one possibility is no fixed points. Another is exactly one fixed point, as in $g(x) = e^{x-1}$ at $x=1,$ in which case the derivative is exactly one.
If, as you say, there are two fixed points, call them $0 < u <... | https://math.stackexchange.com |
5,916 | [
"https://softwareengineering.stackexchange.com/questions/5916",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/1130/"
] | Someone once said we should prefix all our methods with the <code>/// <summary></code> comment blocks (C#) but did not explain why.
I started to use them and found they annoyed me quite a bit, so stopped using them except for libraries and static methods. They're bulky and I'm always forgetting to update them. ... | <blockquote>
<strong>Use them as much as possible.</strong>
</blockquote>
Yes, those are special comments that become the documentation for the method. The contents of <code><summary></code>, the parameter tags, etc. that are generated show up in intellisense when you or someone else is getting ready to call y... | If your coding standard demands that you use such comments (and a coding standard for an API or a framework may demand that), then you have no choice, you have to use such comments.
Otherwise, consider seriously not using such comments. You can avoid them in most cases by changing your code like this:
<pre><code> ... | https://softwareengineering.stackexchange.com |
62,661 | [
"https://security.stackexchange.com/questions/62661",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/50562/"
] | It is really common (and I would say it is some kind of security basic) to not show on the login page if the username or the password was wrong when a user tries to log in.
One should show a generic message instead, like "Password or username are wrong".
The reason is not to show potential attackers which usernames ar... | No, you are correct that at some point during efforts to prevent attackers from determining valid user identities you will either have to lie to them or provide exceptionally vague error messages.
Your app could tell a user that "the requested username is unavailable" and not be specific as to whether it was already... | You're making the assumption that the system actually knows which field was entered incorrectly. There are several reasons this is not necessarily true.
One possibility is that it's a side effect of implementation. A simplistic method of looking up logins in a database might look something like (using <code>:n</code> ... | https://security.stackexchange.com |
174,685 | [
"https://physics.stackexchange.com/questions/174685",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/58482/"
] | If I have a Hermitian operator $H:V \to V$ on a finite-dimensional vector space $V$, and I write down its matrix representation in some basis $B$ with matrix representation being $[H]_B$, then in what cases is $[H]_B$ invertible? ( I assume being invertible is a property of the linear operator independent of the basis,... | It is invertible iff its determinate doesn't vanish
$$ \det([H]_B) \ne 0 $$
Note that this property of the determinate is invariant under a change of basis since:
\begin{align}
\det(S^{-1} \cdot [H]_B \cdot S) & = \det(S^{-1}) \cdot \det([H]_B) \cdot \det(S) = \frac{1}{\det(S)} \cdot \det([H]_B) \cdot \det(S) \\... | Well, this is a problem of linear algebra substantially. The basic idea is that a square matrix $A$ is invertible if and only if $detA\neq 0$. Then, if the matrix is diagonalizable, since the determinant is invariant under coordinate transformations $A'=C^{-1}AC^1$, when you compute the determinant you get that, you ge... | https://physics.stackexchange.com |
55,902 | [
"https://datascience.stackexchange.com/questions/55902",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/77900/"
] | I trained my data with XGBoost in python with GridSearchCV as follows:
<pre><code>parameters = {'nthread':[6],
'objective':['binary:logistic'],
'learning_rate': [0.01, 0.1],
'max_depth': [5,8,13],
'n_estimators': [200,500,1000,3000],
'seed': [1337]... | If you are after well calibrated scores (that is, the scores outputted by your model can be interpreted as probabilities in some sense, at least according to what you have observed empricially in your training set) then accuracy is not the right function to optimize (i would argue that it is never a good function to op... | Yeah trees aren't the best way to get probabilities, they're very good at hard predictions though.
<blockquote>
The decision tree probability estimates, which are a natural
calculation from the frequencies at the leaves, can be systematically skewed towards 0 and 1, as the leaves are essentially dominated by one c... | https://datascience.stackexchange.com |
117,185 | [
"https://electronics.stackexchange.com/questions/117185",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/29737/"
] | I want to create a simple sinusoidal oscillator (Wien-bridge with soft limiting), connected to a high gain amplifier.
I don't have much experience with audio, but I'm guessing a speaker needs a positive and negative voltage to function.
If I wanted to power this circuit with only a +5V rail, what's the easiest way to... | <blockquote>
I don't have much experience with audio, but I'm guessing a speaker
needs a positive and negative voltage to function.
</blockquote>
A loudspeaker is (typically) a two-terminal device so it simply requires <em>a</em> voltage across the terminals.
What you want to be sure to do is to eliminate any <em>const... | Yes, speakers work best when no DC component is included in the signal they are being driven with. There are several ways to drive a speaker from a single supply:<ol>
<li>For small currents, just do it anyway. You only have a 5 V supply, so putting the speaker between a transistor and the supply will work OK for a s... | https://electronics.stackexchange.com |
193,798 | [
"https://physics.stackexchange.com/questions/193798",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/77208/"
] | In <strong>The Feynman Lectures</strong>, in the chapter <strong>Characteristics of Force</strong>, In the section entitled <strong>Molecular forces</strong>, Feynman talks about the molecular forces, and then he states afterwards:
<img src="https://i.stack.imgur.com/O2BaC.jpg" alt="enter image description here">
<bl... | As the picture already shows this can only be considered an approximation for the force in a interval $x\in (d-\epsilon, d+\epsilon)$ for small $\epsilon$. Already the constant $k$ in your expression $F=-kx$ will be different from the constant $k$ in the picture in $F=\frac{k}{r^7}$ - basically, in Hooke's law you will... | Feynman may have been thinking that the same equations have the same solutions. That is, since the molecule obeys $F=-kx$ for small displacements, we can bring along all our previous solutions and intuitions we obtained in solving the problem of a mass on a spring. For example, the oscillation frequency is $\omega=k/m$... | https://physics.stackexchange.com |
722,068 | [
"https://physics.stackexchange.com/questions/722068",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/342531/"
] | When learning momentum in introductory physics, I was taught this formula: <span class="math-container">$m_1 v_{1i}+m_2 v_{2i}=m_1 v_{1f}+m_2 v_{2f}$</span>.
However, my new calculus-based textbook states advises to use the formula <span class="math-container">$v_{1f}=\frac{m_1-m_2}{m_1+m_2}v_{1i}+\frac{2m_2}{m_1+m_2}v... | The first formula is the total momentum of the system which does not change, On the left-hand side is the expression for the <em>before</em> impact momentum, and on the right-hand side the <em>after</em> impact momentum. The two are equal, regardless of the reference frame, as long as the observer is in an inertial fra... | You are clearly learning physics (good!). A very important lesson is that the physics is in the derivation of the formula, not in the result! So quoting the formula without knowing where it came from is a bad plan.
The formulae you quote express different physics. The first formula (from your 'introductory' text book) ... | https://physics.stackexchange.com |
413,846 | [
"https://physics.stackexchange.com/questions/413846",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/187139/"
] | In reading these discussions I often see these two different definitions assumed. Yet they are very different. Which is correct: Does gravity CAUSE the bending of spacetime, or IS gravity the bending of spacetime? Or do we not know? Or is it just semantics?
Would, in the absence of spacetime, my apple still fall ... | I think the correct answer should be that what we call gravity is a fictional force which we experience due to living in an accelerated reference frame (as opposed to an inertial one). Unlike other forces, the force of gravity disappears by a coordinate change. If a person is in a falling elevator, they experience free... | <blockquote>
Does gravity CAUSE the bending of spacetime?
</blockquote>
What causes the bending of spacetime is the presence of a massive object.
<blockquote>
IS gravity the bending of spacetime?
</blockquote>
Yes.
<blockquote>
Would, in the absence of spacetime, my apple still fall to the earth?
</blockquot... | https://physics.stackexchange.com |
93,551 | [
"https://physics.stackexchange.com/questions/93551",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/37334/"
] | Is the answer C?
Assume that you look at three objects that all float on water, but have different shapes and sizes. Which one has the greatest density?
A.The object that has the largest fraction of its volume submerged.
B.Impossible to determine in cases when the three objects have different volumes.
<strong>C.... | <blockquote>
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium.
</blockquote>
Yes, metals have infinite permit... | Unlike dielectrics the electrons in conductors are not bound to atoms, therefore, metals can be modeled as electrons in free space. Permittivity of conductors is, therefore, equal to the permittivity of free space.
| https://physics.stackexchange.com |
897,885 | [
"https://math.stackexchange.com/questions/897885",
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] | Ridiculously embarrassing question, but can $\frac{x^2-x}{x^2-25}$ be simplified to simply $\frac{1-x}{1-25}$?
Full thought process here is that this is essentially $\frac{x*x-x}{x*x-25}$ so the $x$s should cancel. The full problem is:$$\frac{x^2-x-30}{x^2-25}$$
<h2>sorry</h2>
I'm used to programming forums where a... | If you are unsure, then one way to check whether things like this <em>might</em> be true is to plug in a value for $x$. Let $x = 2$. We get:
$$\frac{x^2-x}{x^2 - 25} = \frac{2}{-21} \neq \frac{1-x}{1-25} = \frac{-1}{-24} = \frac{1}{24}$$
So in this case, you made a mistake somewhere.
Of course, if you plug in a ... | No. If you want to divide numerator and denominator by $x^2$, you will get $\dfrac{1-\frac1x}{1-\frac{25}{x^2}}$, which isn't really simpler. If you really want to do something to simplify it, you can rewrite it as
$$\frac{x^2-25-x+25}{x^2-25}=\frac{x^2-25}{x^2-25}+\frac{-x+25}{x^2-25}=1+\frac{25-x}{x^2-25}$$
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