Dimension Definition and 929 Threads

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.

Here I will read on the topic of the speed of light, and there I will read on the topic of the frequency of light, but never except in my topics, then only within my own entry within topic, do the two get overlayed dimensionally and light get unified into a single two-dimensional, possibly...

Poll Question:Do You Believe that 'Time' is a Dimension on its own??Choices: Yes No

This is my idea. It's not yet a theory or even a hypothesis; just an idea:What is time? Time is the fourth dimension. I've wondered if the fourth dimension could actually be visualized. I noticed on one of these threads, someone questioned why we haven't seen people from the future in our time...

is their a fifth dimention we know there is four dimentions (strait line,....time)may be there is a fifth one what about:1. (- time) that what we call traveling back by time2. (- subestance) !!3..............ISLAM IS ON THE WAY

quote:by the bookt=elapsed timeT=time for one roten=rotesf=frequencyt=nTfT=1f=1/T=n/tIf n is a pure number then why they say number of rotes?Do you think that rote is a dimension?quote:by the bookc=speed of lightf=frequencyw=wave lengthc=fw but since f=n/Tc=nw/T

Poll Question:All right, I'm sick of this contraversy between different views of what "time" is. It seems to me that time is a dimension (or, specifically, "coordinate"). Some people seem to think otherwise. As I can't quite understand their views, and I don't want to mis-quote someone, I will...

Hey people, here's a thought.....i think most people agree that there are more than just three dimentions to the universe. so, isnt it entirely possible that if you travelled at the speed of light (just so you can survive the trip) in one direction long enough you'd always end up back where you...

There IS NO SUCH THING AS TIME!!!!!!!!!!Time is just a way of discussing with other human beings parts of our daily lives. It was MADE UP. Therefore it is NOT another dimension...duh...

To those who disagree with the notion of time being a dimension, I previously asked in 'A ? On Time',(not in a vile way) "why do you disagree with some of the most brilliant minds of physics?"There weren't any replies, so I was wondering if it may have been overlooked.-Casey"We see the universe...

I stumbled across a e-book claiming to explain the 4 dimension simply. Instead it broke down like 800 math formulas on me. Can anyone give me a overview of it? Just trying to get an idea what there talking about..."Remember I got the Power to rip a driver from his Eddie Bauer at 90 mph" ---Eminem

Poll Question:Do you think that there is a possibility of a fifth dimension?--If you do believe in a fifth demension, explain what you think it is.--Paranoia: a healthy understanding of the way the universe worksChoices: Yes No Maybe, I'm not sure

Is it not at all possible that the fourth dimension can be spatial??We dance round in a ring and suppose,but the secret sits in the middle and knows.- Robert Frost

Thus is there a 5th dimension within atoms or external to atoms?will this be part of extended fields of limited Times or infinite Times? closed or open dimension?Life is a Song of SearchThe Uni-verse Sings on high.

Credit to Mentat.How can something infinite expand? Mentat asked this very question. I too am curious to know.

I'd like to know what time is... Ok a dimension, I can put my finger on that. But is time really just what defines "movement"? Where one point in 3 dimensional space moves to another. Or is it "how long it takes to do something". It's hard for me to explain this. If we didn't have time (as a...

I've just been playing around with an idea that is partly philosophical, but which definitely deserves to be discussed here.Consider the 4 dimensions. Of these 'time' is the most obvious to define... as 'change'.The other 3 spatial dimensions are tricky to define. What is 'length'? The distance...

It seems possible to me that the apparent randomness in Quantum Mechanics could be the result of another time like dimension opposing ours and canceling out the laws of entropy. Furthermore, this could also explain Relativistic effects and preserve the arrow of time. It could possibly be a...

Any ideas as to the possibility that dark matter is simply either a "compacted or uncompacted extra dimension?

Poll Question:How many of you think time is the 4th dimmensionAReidChoices: yes no

Reading old text about the limit of large N_c, there are some remarks telling that for finite N we have a colour tube and that it becomes a string in the limit. It sounds as the extra dimension of M-Theory, and I wonder if the math of both calculations are related somehow. Is the transition to...

As I understand it, dimension is a way of describing direction, with the first three spatial dimensions being straight lines which extend infinitely in one direction, perpendicular to each other. In string theories, several additional dimensions are required, sometimes up to nine or 10, I...

Hi, if I have a equation like (just a random eq.) p_dot = S(omega)*p. where p = [x, y, z] is the original states, omega = [p, q, r] and S - skew symmetric. How does the equation appear if i only want a system to have the state z? do I get z_dot = -q*x + p*y. Or is the symmetric not valid so I...

Imagine we draw a two dimensional finite plane with coordinate axes; for simplicity, let's make it a square. Now, suppose we add a third dimension that represents the possible distances between any two points on the square. Now we have a three dimensional space. What shape will that space have...

The farthest distance of two places in an area is 200 km. If someone wants to make a map of that area on a 1 m × 1 m paper, the possible scale to make it is ... a. 1 : 210 b. 1 : 2.100 c. 1 : 21.000 d. 1 : 210.000 Can you help? The 200 and 210 makes me think that the distance on map won't be an...

Hello, It has been a long time since I first looked at this, so thought I might ask for some help in clarifying this problem: Is an equation of the form --> Velocity = (Distance) * (Trigonometric function) a valid one in physics? If so, what is the relationship of trigonometric functions...

If ##\textbf{u}_1,...,\textbf{u}_n## form a basis in a linear space, how does one determine the dimension of the span ##\textbf{u}_1-\textbf{u}_2, \textbf{u}_2-\textbf{u}_3,...,\textbf{u}_n-\textbf{u}_1##? Since ##\textbf{u}_1,...,\textbf{u}_n## form a basis, they're linearly independent. If one...

Summary: Integrating the 1 dimensional MB Distribution in terms of translational kinetic energy up to infinity, does not yield ##\frac{1}{2}k_BT## as it should be. The format for the 3 dimensional Maxwell-Boltzmann Distribution is ##A\cdot e^{-\frac{E}{k_BT}} \cdot g(E)## in which ##A## can be...

Problem Statement: Trying to understand the principles for the equation e=mc2. Relevant Equations: E=mc2 So just started at a physics A level and so far loving it. I understand this question has indeed been asked before, however for different reasons. I understand the purpose of e=mc2 and I...

as calculations are technically difficult in curved spaces, I wonder if we would obtain the same results by adding one additional (virtual) dimension in order to embed the space in a higher order Euclidean volume, just to facilitate the treatments? (for example embed a 3D hypersphere in a 4D...

If we set a dimension for the unobservable, we may stumble on a unifying theory for the large and small. 3D + Time + Waves When I say Waves, I'm talking about the waves a particle becomes when it is unobserved and going through the double slit. If waves only exist as math, observation pulls...

Motivated by some apparently intractible unsolved problems, I see cosmology as a beautiful mathematical description of a strongly flawed paradigm. This forum wisely does not allow laying out alternate paradigms, so I try to ask questions to guide my immature understanding , in the spirit of...

Suppose we use fractional derivatives (https://en.m.wikipedia.org/wiki/Fractional_calculus) in GR, hence we have a local group symmetry ##SO(3-\epsilon,1+\epsilon)## does any reference exist about an equation for ##\epsilon## ?, since it could depend on coordinates too.

Homework Statement Problem given to me for an assignment in a math course. Haven't learned about roots of unity at all though. Finding this problem super tricky any help would be appreciated. Screenshot of problem below. [/B] Homework Equations Unsure of relevant equations The Attempt at...

<Moderator's note: Moved from a technical forum and thus no template.> I am at the beginners level of linear algebra and having problem of the intersection of matrices. Your kind help is much appreciated for the following question Let\quad M1=\begin{Bmatrix} x & -x \\ y & z \end{Bmatrix},\quad...

In several places, for example https://xxx.lanl.gov/pdf/chao-dyn/9406003v1, it is claimed that the Riemann zeta function is a fractal under the assumption of a positive result for the Riemann Hypothesis, because (1) the Voronin Universality Theorem, and (2) if the RH is true, then the zeta...

Let $$\left[\begin{array}{rrrrrrr} 1 & 0 & -1 & 0 & 1 & 0 & 3\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 & 4 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 1 & 3 \end{array}\right]$$ Find a basis for the null space of A, the dimension of the null space of A, and...

Let ##(x_1,x_2,x_3)=\vec{r}(\theta,\phi)## the parametrization of a usual sphere. If we consider a projection in two dimension ##(a,b)=\vec{f}(x_1,x_2,x_3)## Then I don't understand how to use the metric, since it is ##g_{ij}=\langle \frac{\partial\vec{f}}{\partial...