In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.
In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.
The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.
Now in my understanding from text ...just to clarify with you guys; the pde is of dimension 2 as ##t## and ##x## are the indepedent variables or it may also be considered to be of dimension 1, that is if there is a clear distinction between time and space variables.
Your insight on this is...
Can a vector subspace have the same dimension as the space it is part of?
If so, can such a subspace have a Cartesian equation?
if so, can you give an example.
Thanks in advance;
Hi, I Learned that we can add higher dimensional operator but they are non-renormalizable - but effect of higher dimensional operator is vanishes at low energy - my question is than how can we add higher dimensional operator at higher energy - like dimension 5 operator ( weinberg operator)...
hi, when we say in SM , we can add terms having dimension 4 or less than that- in this to what dimension we are refering ? kindly help how do you measure the dimension of terms in Lagrangian. thanks
vector<OP> negate (vector<OP> a) {
a.insert(a.begin(), neg);
return a;
}
vector<vector<OP>> negate (vector<vector<OP>> a) {
for (int i=0; i<a.size(); i++)
a[i] = negate(a[i]); // reference to 'negate' is ambiguous?
return a;
}
OP is an enum here. Why can't C++...
Consider a hypothetical five dimensional flat spacetime ##\mathbb{R}^5## with coordinates ##x, y, z, w, t##.
Now imagine that the hypersurface ##\Sigma =\mathbb{R}^3## of ##x, y, z## moves with constant rate ##r## along the coordinate ##w##, i.e. ##dw/dt=r##. Assuming that ##t \in (-\infty, +...
Any help to understand how the authors of this paper
Fine Tuning Problem of the Cosmological Constant in a Generalized Randall-Sundrum Model
calculted this size of the extra dimension Equ. (3.8) from the scale factor defined by Equ. (3.3) ? Specifically, this paragraph after Equ. (3.8)
-...
I have a question about operators in finite dimension Hilbert space.
I will describe the context before asking the question.
Assume we have two quantum states | \Psi_{1} \rangle and | \Psi_{2} \rangle .
Both of the quantum states are elements of the Hilbert space, thus | \Psi_{1} \rangle , |...
##f : [0,2] \to R##. ##f## is continuous and is defined as follows:
$$
f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$
$$
f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$
##V = \text{space of all such f}##
What would the basis for V? Well, for ##x \in [0,1]## the basis for ##V##...
Say that I want to reconstruct the Lorenz system only using one of the original three variables using Takens theorem. What methods can I use to estimate the embedding dimension else from the False nearest Neighbor method (FNN)? The reason I wonder is because I want to compare the accuracy of FNN...
I am studying Tunable couplers with Programmable Integrated photonics by Jose Capmany.
In this textbook, what is the physical meaning of signal s1 and s2? Is it an electric field intensity or the phase of electromagnetic wave?
And what is the physical dimension of transduction coefficient here?
It is given that a theory is invariant under the length scaling:\begin{align*}
x &\rightarrow \lambda x \\
\phi(x) &\rightarrow \lambda^{-D} \phi(\lambda^{-1} x)
\end{align*}for some ##D## to be determined. The action of a real scalar field is here:\begin{align*}
S = \int d^4 x...
According to e.g. Keith Conrad (https://kconrad.math.uconn.edu/blurbs/ choose Complexification) If W is a vector in the vector space R2, then the complexification of R2, labelled R2(c), is a vector space W⊕W, elements of which are pairs (W,W) that satisfy the multiplication rule for complex...
hi guys
our professor asked us to confirm the units of volume charge density ρ and also the surface charge density σ of a dielectric material given by
$$
\rho = \frac{-1}{4\pi k} \vec{E}\cdot\;grad(k)
$$
$$
\sigma= \frac{-(k-1)}{4\pi} \vec{E_{1}}\cdot\;\vec{n}
$$
I am somehow confused about the...
Hi all,
I am currently studying renormalization group and beta functions. Since I'm not in school there is no one to fix my mis-understandings if any, so I'd really appreciate some feedback.
PART I:
I wrote this short summary of what I understand of the beta function:
Is this reasoning...
If a globe is representative of the third dimension, what does a spinning globe represent? As we move through space and time, are we not interacting within the fourth dimension?
I've heard that there is some link between these two values (they're so close!) but I can't seem to find it anywhere. Can someone point me in the right direction? (there's also the J-invariant 196884, well you get the idea)
My intuition tells me this is a true statements so let's try to prove it.
The dimension is defined as the number of elements of a basis. Hence, we can work in terms of basis to prove the statement.
Given that ##U_3## appears on both sides of the inequality, let's get a basis for it. How...
I tried as first step to find Z_q the renormalization parameter, to do so I did the same procedure to find the renormalization parameter of the gauge field of the gluon A^a_\mu when a is representation index a \in {1,2,...,N^2-1} such that A^{a{(R)}}_{\mu}=\frac{1}{\sqrt{Z_A}}A^{a}_{\mu}...
I asked that question on another forum here. The 2 answers I got before the question was closed by an angry mod said:
You wouldn't understand the answer.
Don't ask that question. Ask about a Riemann sphere instead.
You're too lazy to look up the answer in [a GR textbook that I don't own]...
##| V_1 \rangle \in \mathbb{V}^{n_1}_1## and there is an orthonormal basis in ##\mathbb{V}^{n_1}_1##: ##|u_1\rangle, |u_2\rangle ... |u_{n_1}\rangle##
##| V_2 \rangle \in \mathbb{V}^{n_2}_2## and there is an orthonormal basis in ##\mathbb{V}^{n_2}_2##: ##|w_1\rangle, |w_2\rangle ...
hi guys
I was trying to find the matrix of the following linear transformation with respect to the standard basis, which is defined as
##\phi\;M_{2}(R) \;to\;M_{2}(R)\;; \phi(A)=\mu_{2*2}*A_{2*2}## ,
where ##\mu = (1 -1;-2 2)##
and i found the matrix that corresponds to this linear...
Hello everyone. I want to calculate the covariance matrix of a stochastic process using Matlab as
cov(listOfUVValues)
being the dimensions of listOfUVValues 211302*50. I get the following error:
Requested 211302x211302 (332.7GB) array exceeds maximum array size preference. Creation of...
I tried to do a Euler Lagrange equation to our Lagrangian:
$$\frac{S_\text{eff}}{T}=\int d^6x\left[(\nabla \phi)^2+(\nabla \sigma)^2+\lambda\sigma (\nabla \phi)^2\right]+\frac{S_{p.p}}{T}$$
and then I would like to solve the equation using perturbation theory when ##Q## or somehow...
In the following, I set the velocity of light unity.
I refer to theories of gravities in higher-dimensional spacetimes.
Newton` s constant converts the curvature scalar with dimension ##lenght^{-2}## into the matter Lagrangian with dimension ##energy/length^3##. So its dimension is...
I have a question from the youtube lecture
That part starts after 42 minutes and 47 seconds.
Balakrishnan said that if delta barriers are very distant (largely separated) then we have degeneracy. I do not understand how this is possible when in 1d problems there is no degeneracy for bond states.
Time independent Schroedinger equation in ##\delta## potential ##V(x)=-\lambda \delta(x)##, where ##\lambda >0## is given by
-\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi(x)-\lambda \delta(x)\psi(x)=E\psi(x).
How to find dimension of ##\lambda##? Dimension of ##V(x)## is
[V(x)]=ML^2T^{-2}.
Because...
Quickly estimate the value of this expression: sin(1.57). If your answer was about 1, then you assumed that 1.57 was a radian value. If your answer was about 0, you assumed that 1.57 was degrees. If you said that you can’t determine an answer, then you were expecting to see an angle...
total time: t = 36 mins = 0.6h = t1 + t2
=> t2 = t - t1 = 0.6h - 0.1h = 0.5h
Vmax = a1 x t1
Vat C = Vmax + a2t2
substitute Vmax in Vat C we have : 0 = a1 x 0.1h + (-600 km/h²) x 0.5h => a1 = 3000km/h²
Vmax = a1 x t1 = 3000 x 0.1 = 300km/h
I check the result by:
x1 = ½ a1 t1² = ½ . (3000)...
But what is the actual use of dimension? We can do dimension analysis but it can be simply changed into unit analysis and the result will be the same. So why introduce dimension for physical quantities? Why unit is not enough?
Thanks
Hello
This is not a homework, this is my own experiment to understand how the motion works. Please, follow my question here below:
I have a hot wheels race with a slope with 10 degrees where I use a small car which departs from the top to the bottom. I have taken 5 times the time to get an...
I am stuck on finding the dimension of the subspace. Here's what I have so far.
Proof: Let ##W = \lbrace x \in V : [x, y] = 0\rbrace##. We see ##[0, y] = 0##, so ##W## is non empty. Let ##u, v \in W## and ##\alpha, \beta## be scalars. Then ##[\alpha u + \beta v, y] = \alpha [u, y] + \beta [v...
I am working on related rates problems involving figuring out how area of a square increases per second based on how much one side increases per second (or how the area of a circle increases based on increase of the radius, etc.). I was wondering about the practical significance of problems like...
This proof was in my book.
Tensor product definition according to my book: $$V⊗W=\{f: V^*\times W^*\rightarrow k | \textrm {f is bilinear}\}$$ wher ##V^*## and ##W^*## are the dual spaces for V and W respectively.
I don't understand the step where they say ##(e_i⊗f_j)(φ,ψ) = φ(e_i)ψ(f_j)##...
Below is the whole code. I can't change the whole code, I can only change the "Kalman class".
The Kalman class in the code below is my attempt to solve the problem.
But the code doesn't work well.
I have written these 5 equations in the Kalman-filter algorithm:
State Extrapolation Equation...
Is the intersection of a 4D line segment and a 3D polyhedron in 4D a point in 4D, if they at all intersect? Intuitively, it looks like so. But I am not sure about it and how to prove it.
I always thought that one independent equation cuts down the dimension by 1, so if we had two planes, say x - y - z = 1 and x + y + z = 1, then because these are two independent equations, the dimension of the intersection should be 1 because each plane is cutting down the dimension by 1.
Using...
At least according to Tim Anderson Ph.D who wrote the paper in Physics Review.
https://news.knowledia.com/US/en/articles/a-5th-dimension-may-explain-quantum-theory-the-infinite-universe-medium-6f1d6fd371e068a07f357b9babe9ab2eec06d034
What do you make of this?
"The paper simply presents...
It's just think that if we measure 1 second with a clock we should be able to "see" a 300'000km long piece of something in space or not ?
Or does the time extension only has to be understood as a set of numbers indicating timelaps, so that there is no "geometry" of time ?
Suppose the universe were described by internal geometry by a ball, i.e. the metric where :
$$diag(1,r^2,r^2 sin(\theta)^2)$$
Now if we go to exterior geometry and suppose there existed a 4th timelike dimension the manifold were for example modelized by :
$$\left(\begin{array}{c}...
The perimeter of a circle increases by radius, the surface area of a ball increase by radius(which is height which is the third dimension if the ball is a planet like the Earth), and the universe is expanding by time, can we say that the fourth dimension is time by this ?
To form a 2-torus, a narrow tube can be bent into a loop and joined end to end:
But instead of forming this loop in our three-dimensional space, the loop can also be formed in a direction perpendicular to three-dimensional space, moving it into the fourth dimension of space.
What's the name of...
My thoughts are:
a) it should just be N^2
b) just N since they're identical
c) due to Pauli exclusion would it be N^2 - N since they have to be different states?
adding all the torques around the red circle position (taking clockside direction as positive ):
-M*g*L*sin(theta)-k*x*y=I *w (considering that the suspension bar is of negligible mass as the problem indicates )here "x" is the normal "x" of hooke's law (I don't know exactly what it is for a...
A topless square box is made by cutting little squares out of the four corners of a square sheet of metal 12 inches on a side, and then folding up the resulting flaps. What is the largest side area which can be made in this way?
What information I have so far is that since the side of the...
This is wild.
I was always fascinated with the Mandelbrot set, as well as the bifurcation diagram. I had no idea the Mandelbrot diagram was a different visualization of the bifurcation diagram.
Question: is this video accurate? I always question the veracity of YouTube science videos.