Dimension Definition and 881 Threads

This fact blew my mind when I realized it: in 2-dimensional space one can make a bounded set into its strict subset by means of rotation. I still cannot comprehend how coud this be possible and how much of math gets blown by it. I think, it is used to prove the Tarski paradox. While most...

A map of a four-dimensional planet is three dimensional, so such can exist in our Universe. I made one and posted a video to the Internet. This is all based on William Kingdon Clifford's math from the 19th century. It works like this. A 4D planet has two perpendicular planes of rotation...

I am a retired lawyer who has been pondering tiny 5th dimensions since I first heard about them in 1986. I think I sort of understand the Kaluza-Klein theory in a geometrical sense. I'm trying to connect the dots of what I think I understand to what the standard model predicts. Does anybody here...

If so what does the reflection look like?

I think cooking time ##t## is proportional to mass ##m## of the turkey .

I've been wrestling with something for a bit. I have one pound of copper (99%) in cube-form. Roughly 1.5 inches on a side. It's essentially a paperweight. Are those dimensions in the ballpark for relatively pure copper?

I think 1+1,1+2 are known to exist. 1+3 is basically the Millenium prize, are there more options? Is string theory basically QFT in disguise? just with more than 3+1 dimensions. I guess the next Millenium (3000) prize would be to fill the gaps of Witten's M theory conjecture in his seminal...

Suppose you have space with time and four perpendicular Euclidian dimensions. Assume that atoms exist and the laws of physics are the same. This doesn't seem likely, it's just intellectual exercise. Then assume things on Earth have the same number of atoms they have in 4D. The main result...

I have found an interesting rabbit hole, because I thought the question of why we live in 3+1 was mainly a matter of footnotes and off-press debates. But it seems if was touched early by Weyl, Ehrenfest and Whitrow https://einsteinpapers.press.princeton.edu/vol13-doc/764 And then elaborated...

Is there something beyond Holors? :cool:

I know I'm completely wrong about this, but it's been a really fun thought experiment for me. I clearly have no physics training, so apologies for basic mistakes, incorrect terminology, etc.. I would love a deeper explanation of what I'm getting wrong! So in a 1D universe, you see in 0D (a...

If you and your frame are following a square path in space, and the Universe is the other frame, each leg of the square path instantly changes every photon in the Universe to suit your dimensional perspective of speed, length, frequency, and time. Is this new light situation actually different...

Hello, I would like to calculate the "weight" of a piece of metal. It seems a little confusing that weight is measured in Newtons or force. So I'm looking for a little help in understanding. if I consider a piece of average aluminum. Dimensions: Length: 60.96 mm Height: 60.96 mm Thickness...

Given a complex matrix ##A\in M_n(\mathbb{C})##, let ##X_A## be the subspace of ##M_n(\mathbb{C})## consisting of all the complex matrices ##M## commuting with ##A## (i.e., ##MA = AM##). Suppose ##A## has ##n## distinct eigenvalues. Find the dimension of ##X_A##.

Color charge is not scalar. Still, do their components have dimensions (in metrological terms)?

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, +...

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##...

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 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]...

They're dimensions, so they DO have a metric equation, right? Does energy flow cyclically between pairs of dimensions? To me, that's what rotation is.

program testmatek implicit none integer :: Nc=1000 ,k integer,parameter :: N=2 REAL :: kx ,a0=1.0 ,t0=0.25 DOUBLE PRECISION :: pi=4*ATAN(1.) COMPLEX , PARAMETER :: i=(0,1) complex :: ek1 ,ek2 complex :: MATRIX_ek(N,N) open(1,file='matek.txt') MATRIX_ek(N,N)=(0,0) DO k=-Nc,+Nc...

##| 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...

I know that the rank of A and A^T are equal, and that the statement follows from there, but I have no idea how to prove it.

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)##...