What is Fundamental: Definition and 946 Discussions
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.
There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells.
Fire, Gravity, Electromagnetism, Atoms, DNA, Steam power, Nuclear, Quarks. All of these things have one thing in common. They are fundamental aspects of the universe that humans have uncovered and given names. However, all of these great discoveries occured quite some time ago and as a science...
What is the way to compute ##\pi_1(PGL_2(R))##?
Is it related to defining an action of ##PGL_2(R)## on ##S^3##?
it would be helpful if you can provide me with relevant information regarding this
Let ## \mathbf{x''} = A\mathbf{x} ## be a homogenous second order system of linear differential equations where
##
A = \begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
## and ##
\mathbf{x} = \begin{bmatrix}
x(t)\\ y(t))
\end{bmatrix}
##
Now to solve this equation we transform it into a 4x4...
Hello,
I want to prove that for any lattice ##\Lambda = \Lambda(B)## (the ##B## is a basis) the orthogonalized parallelepiped ##P(B^*) = B^*\left[-0.5,0.5\right)^n## is a fundamental region of a lattice.
If I wanted to show this, I would try to establish a volume argument here. After all, the...
I'm referring to this result:
But I'm not sure what happens if I apply a linear differential operator to both sides (like a derivation ##D##) - more specifically I'm not sure at what point should each term be evaluated. Acting ##D## on both sides I'll get...
As far as I learned, the following statements should be correct (for closed systems, no chemical reactions), irrespective whether the process is reversible or irreversible (since S and V are state variables):
dU=TdS−pdV
dU=dQ+dW
Does this imply, that the statement:
dU≤TdS−pdV
is wrong?
This...
Hello all,
Textbooks define fundamental or base quantities as those quantities which are not expressed in terms of other quantities and they define derived quantities as those quantities which are expressed in terms of other quantities. I have the basic understanding that the choice of a set of...
Trying to understand something fundamental about how magnetic fields are generated by moving electrons in a conductor. I have read many forums, studied Emag and am left with more questions. Looking for some practical insight not Bio-Savart derivations, etc. These still do not explain why the...
Is it just me or are there some fundamental problems with this exam question ? What is driving me bananas is you have a moving truck (constant velocity). Then the truck "accelerates" but the block in the back of the truck "stays in the same place". Does this mean relative to the ground...
Why are Planck units considered fundamental ?
After all, there is a square root in their expressions.
And the value of the Planck mass is 20 orders of magnitude greater than the average values of elementary particles.
$$ m_{pl}=\sqrt{\frac{1}{2\pi}\cdot \frac{hc}{G}}$$
The order of the value of...
There are some theoretical processes (like vacuum decay in quantum field theory) that could change the physical constants of the universe. Similarly, in inflation theory, various models predict that multiple regions that would stop inflating would become "bubble universes" perhaps with different...
We think of length and time as the first fundamental quantities and velocity as the first derived quantity but any two determine the third so we would be completely justified in defining velocity as a fundamental quantity and one of length or time as the other, with the remaining being the first...
Fundamental units are defined as - each of a set of unrelated units of measurement, which are arbitrarily defined and from which other units are derived.
while Supplementary units are - Supplementary units are the dimensionless units that are used along with the base units to form derived units...
I am given this system of differential equations;
$$ x_1'=2t^2x_1+3t^2x_2+t^5 $$
$$ x_2' =-2t^2x_1-3t^2x_2 +t^2 $$
Now the first question states the following;
Find a fundamental matrix of the corresponding homogeneous system and
explain exactly how you arrive at independent solutions
And the...
I can't help but find the concept of atoms a bit weird. If everything is made of atoms, and atoms are mostly empty space, couldn't that be evidence towards simulation theory? If we can never actually touch anything, and everything is just "levitating" on a force field, couldn't that also be...
I would like to ask scientists or anybody: what do you think about qwarks, do you consider them to be the littiest parts of matter or that matter is cyclically or infinitaly smalling, like numbers, or that there are some smaller parts, which have not been observed, but they have an end in their...
As we know that their are four fundamental force in nature
Namely Electromagnetic force, strong and weak nuclear force and the gravity
But we study regarding their properties
And compare their properties we see a strange thing
That for the case of strong nuclear force enspite of being the...
Summary:: I am looking to take a engineering classes relevant to mechanical and or aerospace engineering. I am challenging my assumption that Statics & Dynamics is the most important and foundational/fundamental course to either mechanical or aerospace engineering (maybe fluid dynamics for...
What do you guys think of this soberly elegant proposal by Sean Carroll?
Reality as a Vector in Hilbert Space
Fundamental reality lives in Hilbert space and everything else (space, fields, particles...) is emergent. Seems to me a step in the right conceptual direction.
I am not sure if this is the correct place to post so I am very sorry if I am posting in the wrong place.
I am looking to have the four fundamental forces of physics explained as simply as possible, I have been doing some online research to try to understand it and I am having a hard time...
What property of a sinusoid makes it so special? I understand Fourier analysis, but really you could do Fourier using any periodic function as the building block.
Sinusoids really do seem to be fundamental though, if you narrow the pass band of a filter with any random signal you will get a...
The Fundamental Theorem of Arithmetic essentially states that any positive whole number n can be written as:
##n = p_1^{a_1} \cdot p_2^{a_2} \cdot p_3^{a_3} \cdot \dots##
where ##p_1##, ##p_2##, ##p_3##, etc. are all the primes, and ##a_1##, ##a_2##, ##a_3##, etc. are non-negative integers...
Hello all,
I am currently working on the four fundamental spaces of a matrix. I have a question about the orthogonality of the
row space to the null space
column space to the left null space
------------------------------------------------
In the book of G. Strang there is this nice picture...
Hi all. We are computing the second fundamental form corresponding to a surface.
The correct answer should give the coefficients 2, 0, and -2 for IIuu, IIuv, IIvv
We have only been able to compute IIz, and not IIx, IIy where we have functions instead of 2, 0 and -2.
We are 99% certain that...
So I know the acceleration is 0, so the net force is 0.
QE=1.6E-3*9.9E3 k hat = 15.84 k hat (thats one force)
qv x B = q(v i hat + v j hat) x (0.51 i hat + 0.10 j hat)
=q(0.10 v k hat- 0.51 v k hat)
=q(-0.41 v k hat)
=-0.000656 v k hat
Now solving for velocity,
0.000656v=15.84...
So first I did the vector stuff.
r2-r1= 1.3 i hat-47.5 j hat-14.5 k hat
magnitude = 49.68
magnitude squared = 2468.19
Now plugging it all in...
F=9E9*6.3E-3*2.8E-3/2468.19=64.322
y vector, -47.5/49.68=-0.956119 j hat
Multiply this by force and I get -61.499 but answer should be -36.14
I know its something to do with my unit conversions because my answer is the right numbers but too small by a factor of 10000.
I convert the radius to meters by multiplying it by 1000. I convert the density to kg/m^3 by dividing it by 10.
I find the volume using the equation (4/3)*pi*radius^3...
So what I did was find each of the forces the masses had on m1 using the above equation.
From m2 I found 19.975 in the negative i hat and for m4 i found 29.96 in the positive k hat direction using the above equation.
For m3 I used pythagorean theorem to calculate r, which was 3.25 (so r^2 was...
Cosmological inflationary models are general models in the sense that they could be applied to a variety of fundamental theories. Most physicists working in inflation assume that there is only one (but yet unknown) fundamental theory which through inflation would produce multiple regions or...
Please forward to appropriate subforums as I wasn't sure where to post it.In the standard model there are the composite particles and elementary particles, for the elementary particles there is said to be an associated field and the particle is the excitation of that field.
Apart from asking how...
My first question is actually, what happens when any two objects get near each other? This question is often phrased as "Why can't you really touch anything?" or "Why can't you walk through walls?" I have heard two answers: 1. the repulsion between electrons 2. the Pauli exclusion principle...
When we have electrical resistance it is based on dissipation of energy, when we have reactance it is a consequence of charge accumulation in capacitance. Together they form impedance. But let's consider what exactly is responsible for dissipation in case of resistance? Is not it because of...
Does it not raise question to be able to build two parameters with the same dimension and order of magnitude of two fundamental constants from five parameters (other fundamental constants and measured properties)?
$$\sqrt{10\frac{\left(\varepsilon_0 e^{-2}\right)^3\left(k_B T\right)^4}{\rho_c}}...
According to the book I am using, one can decompose a finite abelian group uniquely as a direct sum of cyclic groups with prime power orders.
Uniquely meaning that the structures in the group somehow force you to one particular decomposition for any given group.
Unfortunately, the book gives no...
Physicist Joseph Polchinski wrote an article (https://arxiv.org/pdf/1412.5704.pdf) where he considered the possibility that all symmetries in nature may not be fundamental. He says at page 36:
"From more theoretical points of view, string theory appears to allow no exact global symmetries, and...
I recall my memory when I was a high school student. I first learned about energy from Newton Mechanics. The only forms of energy involved are kinetic energy and potential energy. At that time, energy is more like a mathematical shortcut derived from F = ma rather than a concrete physical...
Why is this wrong?:
That is, why is not f = 4fo?
Oh. And my figure 1 can be wrong, because maybe i got confused if the string was open or closed in the boundaries, anyway this change nothing the relations (1)
I have asked this question before but badly and just caused confusion. So I thought to ask it again but without muddled presentation.
If ##M## is a closed surface embedded in ##R^3## that has strictly positive Gauss curvature then its second fundamental form is positive definite and so is a...
This problem came from Problems, Section 3 Chapter 7 in ML Boas, Mathematical Methods in Physical Sciences. Boas suggested to make a computer plot. From my computer plot (I use online graphing calculator) and find that the period of the sum is 2.
Instead of using computer, I want to find the...
This is about possible problems with the equations of EM theory of Faraday and Maxwell.
EM theory states that when an electron moves a magnetic field is generated.
Does this field move with electron in the direction of the electron (if the electron is in motion)?
Or, does it stay still in the...
Apologies in advance if this is a naive question. From what I've understood, it's not particularly meaningful to talk about gravity as a force, since it induces the same acceleration (classically speaking) in everything.
Whatever device or accelerometer we use, every component of that too will...
I was originally going to start a thread on individual interpretations of quantum mechanics to try to understand them better, but after reading this article by @PeterDonis it might make more sense to use that as a starting point and try to understand how the different interpretations fit into...
If the Fine-structure constant was measured to deviate in the sky , and this deviation was directional, which fundamental invariance principle would be violated?
Quasar survey by VLT has observed deviations in the FSC that appear to be locked against directions of distant galaxies...
We know that the energy levels for electrons surrounding nucleus are quantized , only coming in discrete levels.
When I see the standard model of elementary particles table I notice specific masses for each of the particles whether they be quarks or leptons or bosons like the higgs.
I know that...